The International Symposium on Lattice Field Theory is an annual conference that attracts scientists from around the world. Originally started as a place for physicists to discuss their recent developments in lattice gauge theory, nowadays the conference is the largest of its type and has grown to include areas like algorithms and machine architectures, code development, chiral symmetry, physics beyond the standard model, and strongly interacting phenomena in low-dimensions.
The 39th Lattice conference will take place in Bonn, Germany, from August 8 to August 13 2022.
The scientific programme of this conference will include plenary talks and parallel sessions on the following topics:
The conference is supported by:
Recent results from lattice simulations of QCD at nonzero temperature and/or density and/or in presence of magnetic fields will be reviewed. Progress in our understanding of the phases and boundaries in the phase diagram, as well as on the calculation of thermodynamic quantities with relevant phenomenological consequences will be discussed.
As the precision test of the standard model has become accurate, the need for fine lattices has been increasing. However, as we approach the continuum limit, we get into the critical region of the theory and encounter critical slowing down. Among many studies tackling this problem, we develop the idea of trivializing map, whose use in lattice calculation was proposed by Luscher. With this field transformation, the theory of interest will be mapped to the strong coupling limit. Luscher gave an analytic formula to construct the trivializing map in the form of t-expansion, where t is the trivializing-flow time. In this work, we alternatively use the Schwinger-Dyson equation to obtain the trivializing map approximately. In this method, we choose a set of Wilson loops to include in the flow kernel by hand and determine their coefficients from the expectation values of the Wilson loops. The advantages of this method over the t-expansion are two-fold: (1) We can circumvent the rapid increase of necessary Wilson loops which we have in increasing the order of t-expansion because the basis can be chosen arbitrarily. (2) We can expect to obtain a reasonable approximation of the trivializing map also for large beta because the coefficients are determined from a non-perturbative evaluation of the expectation values. In this talk, we show preliminary results applying our method to pure Yang-Mills theory.
In this talk, we review recent advances in applying quantum computing to lattice field theory. Quantum technology offers the prospect to efficiently simulate sign-problem afflicted regimes in lattice field theory, such as the presence of topological terms, chemical potentials, and out-of-equilibrium dynamics. First proof-of-concept simulations of Abelian and non-Abelian gauge theories in (1+1)D and (2+1)D have been accomplished, and resource efficient formulations of gauge theories for quantum computations have been proposed. The path towards quantum simulations of (3+1)D particle physics requires many incremental steps, including algorithmic development, hardware improvement, methods for circuit design, as well as error mitigation and correction techniques. After reviewing these requirements and recent advances, we discuss the main challenges and future directions.
Reaching Exascale compute performance at an affordable budget requires increasingly heterogeneous HPC systems, which combine general purpose processing units (CPUs) with acceleration devices such as graphics processing units (GPUs) or many-core processors. The Modular Supercomputing Architecture (MSA) developed within the EU-funded DEEP project series breaks with traditional HPC system architectures by orchestrating these heterogeneous computing resources at system-level, organizing them in compute modules with diﬀerent hardware and performance characteristics. Modules with disruptive technologies, such as quantum devices, can also be included in a modular supercomputer to satisfy the needs of speciﬁc user communities. The goal is to provide cost-effective computing at extreme performance scales fitting the needs of a wide range of Computational Sciences.
This approach brings substantial beneﬁts for heterogeneous applications and workﬂows. In a modular supercomputer, each application can dynamically decide which kinds and how many nodes to use, mapping its intrinsic requirements and concurrency patterns onto the hardware. Codes that perform multi-physics or multi-scale simulations can run across compute modules due to a global system-software and programming environment. Application workflows that execute different actions after (or in parallel) to each other can also be distributed in order to run each workflow-component on the best suited hardware, and exchange data either directly (via message-passing communication) or via the filesystem. A modular supercomputing system can supply any combination or ratio of resources across modules and is not bound to fixed associations between, for instance, CPUs and accelerators as will be found in clusters of heterogeneous nodes. It is therefore ideal for supercomputer centers running a heterogeneous mix of applications (higher throughput and energy eﬃciency).
This talk will describe the Modular Supercomputing Architecture – which constitutes the central element in Europe’s roadmap to Exascale computing –, including its history, its role in Europe’s Exascale computing strategy, its hardware and software elements, and experiences from mapping applications and workflows to MSA systems.
Emerging sampling algorithms based on normalizing flows have the potential to solve ergodicity problems in lattice calculations. Furthermore, it has been noted that flows can be used to compute thermodynamic quantities which are difficult to access with traditional methods. This suggests that they are also applicable to the density-of-states approach to complex action problems. In particular, flow-based sampling may be used to compute the density directly, in contradistinction to the conventional strategy of reconstructing it via measuring and integrating the derivative of its logarithm. By circumventing this procedure, the accumulation of errors from the numerical integration is avoided completely and the overall normalization factor can be determined explicitly. In this proof-of-principle study, we demonstrate our method in the context of two-component scalar field theory where the O(2) symmetry is explicitly broken by an imaginary external field. First, we concentrate on the zero-dimensional case which can be solved exactly. We show that with our method, the Lee-Yang zeroes of the associated partition function can be successfully located. Subsequently, we confirm that the flow-based approach correctly reproduces the density computed with conventional methods in one- and two-dimensional models.
Normalizing flows (NFs) are a class of machine-learning algorithms that can be used to efficiently evaluate posterior approximations of statistical distributions. NFs work by constructing invertible and differentiable transformations that map sufficiently simple distributions to the target distribution, and provide a new, promising route to study quantum field theories regularized on a lattice. In this contribution, based on our recent work [arXiv:2201.08862], I explain how to combine NFs with stochastic updates, demonstrating that this theoretical framework is the same that underlies Monte Carlo simulations based on Jarzynski’s equality, and present examples of applications for the evaluation of free energies in lattice field theory.
This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to $2\beta$ on the real-time Schwinger-Keldysh contour with the $0+1$ dimensional anharmonic oscillator using $m=1, \; \lambda=24$, which was previously unattainable using the complex Langevin equation.
In this talk, we discuss gauge-equivariant architectures for flow-based sampling in fermionic lattice field theories with pseudofermions. We also discuss how flow-based sampling approaches can be improved by combination with standard techniques such as even/odd preconditioning and the Hasenbusch factorization. Numerical demonstrations in two-dimensional U(1) and SU(3) theories with $N_f=2$ flavors are provided.
Automatic Differentiation (AD) techniques allows to determine the Taylor expansion of any deterministic function. The generalization of these techniques to stochastic problems is not trivial. In this work we explore two approaches to extend the ideas of AD to stochastic processes, one based on reweighting and another one based on the ideas of numerical stochastic perturbation theory using the Hamiltonian formalism. We show that, when convergence can be guaranteed, the approach based on NSPT is able to converge to the Taylor expansion with a much smaller variance.
The numerical sign problem has been a major obstacle to first-principles calculations of many important systems, including QCD at finite density. The worldvolume tempered Lefschetz thimble method is a HMC algorithm which solves both the sign problem and the ergodicity problems simultaneously. In this algorithm, configurations explore the extended configuration space (worldvolume) that includes a region where the sign problem disappears and also a region where the ergodicity problem is mild. The computational cost of the algorithm is expected to be much lower than other related algorithms based on Lefschetz thimbles, because one no longer needs to calculate the Jacobian of the gradient flow of Picard and Lefschetz when generating configurations. In this talk, after reviewing the basics of the method, we apply the method to various lattice field theories suffering from the sign problem, and report on the numerical results together with the computational cost scaling with the lattice volume.
We present results for the energy levels for two pions and a kaon, and two kaons and a pion, all at maximal isospin, on CLS ensembles D200 and N203, with pion/kaon masses of
200/480 MeV and 340/440 MeV, respectively. We use multiple frames, and have determined many energy levels on each ensemble. We fit these levels, together with those for $2\pi^+$, $2K^+$ and $\pi^+ K^+$, to the predictions of the 2+1 three-particle quantization condition, and thus determine two- and three-particle K matrices. We compare our results to the expectations of chiral perturbation theory. Issues that arise in the implementation of the 2+1 quantization condition are discussed.
The quest of unraveling the nature of excited hadrons necessarily involves determination of universal (reaction independent) parameters of these states. Such determinations require input, either from experiment or theory.
Lattice gauge theory is the only tool available to us to tackle the non-perturbative dynamics of QCD encoded in the determined finite-volume interaction spectra. Many insights have been gained on resonant two-body systems in the past by studying such spectra. Now -- with the advent of the three-body finite-volume methods -- advances are being made towards more complex systems. This progress will be discussed in the talk, including theoretical developments and applications to phenomenologically interesting systems.
We study a three-particle resonance in Euclidean Lattice $\phi^4$ theory with two fields, having different masses and an interaction that makes the heavy field decay into three light. We observe the avoided-level crossings characteristic of a resonance and analyse the data with two formalisms FVU and RFT with an aim to determine the mass and the width of the resonance.
Recent years have witnessed a rapid growth of interest to the three-body
problem on the lattice. In this connection, the derivation of a relativistic-
invariant three-particle quantization condition, which relates the finite-volume
lattice spectrum to the infinite-volume observables in the three-particle sec-
tor, has become a major challenge. First and foremost, providing a manifestly
relativistic-invariant framework is important because the typical momenta of
light particles studied on the lattice are generally not small, as compared to
their mass. Moreover, Lorentz invariance puts stringent constraints on the
possible form of the two- and three-body interactions, reducing the number
of effective couplings needed for their parameterization. These constraints
are absent in the non-invariant formulations, leading to an inflation of the
number of independent parameters.
In the literature, there exist three different but conceptually equivalent
formulations of the three-particle quantization condition. In this talk, I shall
put the issue of the relativistic covariance of these formulations under a
renewed scrutiny. A novel formulation is suggested, which is devoid of some
shortcomings of the existing approaches related to the explicit non-covariance
of the three-particle propagator. The proposed approach is based on the
“covariant” NREFT framework. We reformulate this framework, choosing
the quantization axis along an arbitrary timelike unit vector vμ, demonstrate
the explicit relativistic invariance of the infinite-volume Faddeev equations
and derive the modified quantization condition. The relativistic invariance is
tested numerically, producing synthetic data for the energy levels in different
moving frames.
In this talk, I will present our recent results on two- and three-particle scattering in the O(3) non-linear sigma model in 1+1 dimensions. We focus on the isospin-1 and 2 channels for the two-particle case, and the isospin-2 and 3 channels for three particles. We perform numerical simulations at four values of the physical volume and three lattice spacings, using a three-cluster generalization of the cluster update algorithm. The lattice results for two particles are then compared against exact analytic predictions of the finite-volume energy levels obtained combining analytic results for the phase shifts and the (1+1)-dimensional two-particle scattering formalism. For the three-particle results, we use the relativistic field theory (RFT) approach to constrain the scheme-dependent three-body interaction.
We investigate the energy levels corresponding to the Roper resonance based on a two-flavor chiral effective Lagrangian at leading one-loop order. We show that the Roper mass can be extracted from these levels for not too large lattice volumes.
Further, to include three body dynamics, such as $N \pi \pi$, we introduce a non-relativistic effective field theory for the Roper resonance within a covariant particle-dimer picture. This particle-dimer approach is a suitable framework to investigate three particle scattering relevant for the Roper channel. We analyze the appearing dimer fields, calculate the energy levels of the Roper resonance in a finite volume and compare the obtained energy levels with the results from the fully relativistic chiral effective Lagrangian.
The Bielefeld Parma collaboration has recently put forward a method to investigate the QCD phase diagram based on the computation of Taylor series coefficients at both zero and imaginary values of the baryonic chemical potential. The method is based on the computation of multi-point Padè approximants. We review the methodological aspects of the computation and, in order to gain confidence in the approach, we report on the application of the method to the two-dimensional Ising model (probably the most popular arena for testing tools in the study of phase transitions). Besides showing the effectiveness of the multi-point Padè approach, we discuss what these results can suggest in view of further progress in the study of the QCD phase diagram.
We report updated results on the determination of Lee-Yang edge (LYE) singularities in $N_f=2+1$ QCD using highly improved staggered quarks (HISQ) with physical masses on $N_\tau=4$, $6$, $8$ lattices. The singularity structure of QCD in the complex $\mu_B$ plane is probed using conserved charges calculated at imaginary $\mu_B$. The location of the singularities is determined by studying the (uncancelled) poles of multi-point Padé approximants. We show that close to the Roberge-Weiss (RW) transition, the location of the LYE singularities scales according to the 3-$d$ $\mathbb{Z}_2$ universality class. By combining the new $N_\tau=6$ data with the $N_\tau=4$ data from our previous analysis we extract a rough estimate for the RW temperature in the continuum limit. We also discuss some preliminary results for the singularities close to the chiral phase transition obtained from simulations on $N_\tau=6$, $8$ lattices.
We calculate Fourier coefficients of the net-baryon number as a function
of a purely imaginary chemical potential. The asymptotic behavior of
these coefficients is governed by the singularity structure of the QCD
partition function and thus encodes information on phase
transitions. Although it is not easy to obtain a high number of Fourier
coefficients from lattice QCD data directly, models for these
coefficients have been constructed in the past. We investigate to what
extent our data is consistent with those models and estimate the
position of the nearest singularities in the complex chemical potential
plane. Our lattice data has been obtained from simulations with
(2+1)-flavors of highly improved staggered quarks (HISQ) at imaginary
chemical potential on $N_\tau=4$, $6$ and $8$ lattices at physical quark
masses. For the calculation of the Fourier coefficients we apply
asymptotic numerical quadrature designed for highly oscillatory integrals.
Knowledge of the screening masses at finite chemical potential can provide insight into the nature of the QCD phase diagram. However, lattice studies at finite chemical potential suffer from the well-known issue of the sign problem, which has made the calculation of observables such as screening correlators and screening masses at finite chemical potential quite challenging. One way to proceed is by expanding the observable in a Taylor series in the chemical potential and hence calculating the finite-density corrections to the observable. In this talk, we will use this approach to calculate the screening mass of the pseudoscalar meson at finite temperatures and chemical potential by expanding the screening correlator in a Taylor series in the chemical potential. We will present our results for the second derivative of the screening mass w.r.t. the chemical potential. Our calculation was done on $64^3 \times 8$ lattices generated using the (2+1) HISQ action.
In this talk we present our study of the electromagnetic conductivity in dense quark-gluon plasma obtained within lattice simulations with $𝑁_𝑓 = 2 + 1$ dynamical quarks. We employ stout improved rooted staggered quarks at the physical point and the tree-level Symanzik improved gauge action. The simulations are performed at imaginary chemical potential. To reconstruct electromagnetic conductivity from current-current correlators, we employ the Tikhonov regularisation method as well as the modified Backus-Gilbert method, computing the convolution of the spectral density with the target function. Our study indicates that electromagnetic conductivity of quark-gluon plasma rapidly grows with the real baryon density.
We study the (2+1)-dimensional Gross-Neveu model in an external magnetic field. The model, which serves as a toy model for QCD, has been predicted by mean-field studies to exhibit a very rich phase structure in the plane spanned by temperature and chemical potential as the external field is varied. We investigate what remains of this phase structure beyond the mean-field approximation. Our lattice results are consistent with the magnetic catalysis scenario, i.e. an increase of the chiral condensate with the magnetic field, both at finite temperature and chemical potential.
We show that a recently discovered non-perturbative field-theoretical mechanism giving mass to elementary fermions, is also capable of generating a mass for the electro-weak bosons and can thus be used as a viable alternative to the Higgs scenario. A detailed analysis of this remarkable feature shows that the non-perturbatively generated fermion and $W$ masses have the parametric form $m_{f}\sim C_f(\alpha)\Lambda_{RGI}$ and $M_W\sim g_w c_w(\alpha)\Lambda_{RGI}$, respectively, where the coefficients $C_f(\alpha)$ and $c_w(\alpha)$ are functions of the gauge couplings, $g_w$ is the weak coupling and $\Lambda_{RGI}$ is the RGI scale of the theory. In view of these expressions, we see that to match the experimental value of the top quark and $W$ masses, we need to conjecture the existence of a yet unobserved sector of massive fermions subjected, besides ordinary Standard Model interactions, to some kind of super-strong gauge interaction, so as to have the RGI scale of the whole theory in the TeV region. Though limited in its scope (in this talk we ignore hypercharge and leptons and discuss only the case of one family, neglecting weak isospin splitting), this approach opens the way to a solution of the mass naturalness problem and an understanding of the fermion mass hierarchy.
We present an update of our results for the ongoing work on the four-supercharge two-dimensional Yang–Mills theory discretized on a Euclidean torus using thermal boundary conditions. Although the theory under consideration does not have a gravity dual, we investigate whether it has features qualitatively similar to its sixteen-supercharge counterpart. Our investigation hints at a possible ‘spatial deconfinement’ transition in this theory similar to the maximal one with sixteen supercharges. We also analyse the behaviour of the scalars, Wilson lines, and the absence of supersymmetry breaking with a relatively large-N setup and various lattice sizes in different coupling (temperature) regimes and draw comparisons with the two-dimensional maximally supersymmetric Yang–Mills theory.
We present an update of our ongoing study of the SU(2) gauge theory with one flavor of Dirac fermion in the adjoint representation. Compared to our previous results we now have data at larger lattice volumes, smaller values of the fermion mass, and also larger values of $\beta$. We present data for the spectrum of mesons, baryons, glueballs, and the hybrid fermion-glue state, as well as new estimates of the mass anomalous dimension from both finite-size hyperscaling and the Dirac mode number, and discuss the implications of these data for the presence or otherwise of chiral symmetry breaking in this theory.
In this work we present perturbative results for the renormalization of the supercurrent operator, $S_\mu$, in ${\cal{N}}=1$ Supersymmetric Yang-Mills theory. At the quantum level, this operator mixes with both gauge invariant and noninvariant operators, which have the same global transformation properties. In total, there are 13 linearly independent mixing operators of the same and lower dimensionality. We determine, via lattice perturbation theory, the first two rows of the mixing matrix, which refer to the renormalizations of $S_\mu$, and of the gauge invariant mixing operator, $T_\mu$. To extract these mixing coefficients in the $\overline{MS}$ renormalization scheme and at one-loop order, we compute the relevant two-point and three-point Green’s functions of $S_\mu$ and $T_\mu$ in two regularizations: dimensional and lattice. On the lattice, we employ the plaquette gluonic action and for the gluinos we use the fermionic Wilson action with clover improvement.
Supersymmetry on the lattice is explicitly broken by the gluino mass and lattice artifacts. However, it can be restored in the continuum limit by fine tuning the parameters based on the renormalized Ward identities. On the renormalization step not only the mass but also the renormalization of the supercurrent needs to be addressed. Here we present a lattice investigation to obtain the renormalization factors of the supercurrent for $\cal{N} = 1$ SYM in a gauge invariant renormalization scheme (GIRS). We also provide the conversion factors which are necessary in order to translate our results to the continuum $\overline{MS}$ scheme.
Vector Boson scattering (VBS) is a central process in the search for physics beyond the SM at collider experiments. To correctly identify SM and BSM physics, such as composite Higgs scenarios, at these experiments, it is crucial to gain a clear picture of VBS-like processes.
In our study we therefore analyse this process in a reduced SM-setup for different physical scenarios. To this end we apply a Lüscher-type analysis to extract scattering properties and compare the results with (augmented) perturbative tree-level predictions.
We show that the nonperturbative approach suggests a composite structure for the scalar degree of freedom, being in line with previous investigations. Furthermore we present an alternative way of extracting resonance-like states from the spectrum by using the perturbative prediction as a tool.
COLA is a software library for lattice QCD written in modern Fortran and NVIDIA CUDA. Intel and NVIDIA have dominated the HPC domain for a long time, but the status quo has been changed with the recent advent of AMD-based systems in the supercomputing Top`500. Setonix is a next generation Cray AMD machine currently being installed at the Pawsey Supercomputing Centre in Perth, Australia. Setonix features both AMD CPUs and AMD Instinct GPUs. This talk will describe first experiences with porting COLA to the AMD platform.
In this talk I give an update on the status of the GPT software package. (https://github.com/lehner/gpt.)
Lyncs-API is a Python API for Lattice QCD. It aims to create a complete framework for easily running applications via Python. It implements low- and high-level tools, including interface to common LQCD libraries. Last year, at this conference, we presented the API to the community for the first time. In this talk we will give a status update on its development and show the potential of the API via some applications we have implemented this year.
We present progress in interfacing the Hybrid Monte Carlo implementation in the tmLQCD software suite with the QUDA library and compare its performance to our top of the line algorithms on CPU machines. We discuss the main challenges and overheads of our approach and scrutinize its fundamental architectural limitations before exploring ongoing improvements as well as current and future simulations.
In this talk we present work on extending the set of solvers for the inversion of the Dirac matrix for Wilson-Clover type fermions in Grid. Particular emphasis is put on the inexact deflation method put forward by Lüscher. Besides providing fast solves for configurations at the physical point one of the method’s central advantages is that it can be included into the HMC algorithm at relatively low computational cost. We assess the performance of our implementation of the algorithm on both CPU and GPU architectures and carry out comparisons with other solvers.
We report novel lattice QCD results for the three-gluon vertex from quenched lattice-QCD simulations. Using standard Wilson action, we have computed the three gluon vertex beyond the usual kinematic restriction to the symmetric (q² = r² = p²) and soft-gluon (p = 0) cases where it depends on a single momentum scale. We will present a detailed analysis of the asymmetric case (r² = q² ≠ p²) where the transversely projected vertex can be cast in terms of three independent tensors.
The lattice data show a clear dominance of the form-factor corresponding to the tree-level tensor.
For the general kinematical configuration (q² ≠ r² ≠ p²); we have computed the projection of the three-gluon vertex providing the relevant information on the ghost-gluon kernel-related function W(q²) that appears in the recently discussed smoking-gun signals of the Schwinger mechanism in QCD. This projection exhibits a striking scaling in terms of (q² + r² + p²)/2.
In this talk we present numerical simulations of N = 4 super Yang-Mills for 3 color gauge theory over a wide range of ’t Hooft coupling 5 ≤ λ ≤ 30 using a supersymmetric lattice action. By explicit computations of the fermion Pfaffian we present evidence that the theory possesses no sign problem and exists in a single phase out to arbitrarily strong coupling. Furthermore, preliminary work shows that Non-Abelian Coulomb potential extracted via Polyakov loop correlators shows the 1/R scaling and a dependence on square root of the 't Hooft coupling at large values of λ as expected from the holographic calculations.
Master-field simulations offer an approach to lattice QCD in which calculations are performed on a small number of large-volume gauge-field configurations. This is advantageous for simulations in which the global topological charge is frozen due to a very fine lattice spacing, as the effect of this on observables is suppressed by the spacetime volume. Here we make use of the recently developed Stabilised Wilson Fermions to investigate a variation of the master-field approach in which only the temporal direction (T) is taken larger than in traditional calculations. As compared to a hyper-cubic master-field geometry, this has the advantage that finite-L effects can be useful, e.g. for multi-hadron observables, while compared to open boundary conditions time-translation invariance is not lost.
In this proof-of-concept contribution, we study the idea of using very cold, i.e. long-T, lattices to topologically 'defrost' observables at fine lattice spacing. We identify the scalar-scalar meson two-point correlation function as useful probe and present first results from Nf=3 ensembles with time extents up to T=2304 and a lattice spacing of a=0.055fm.
We study two different SU(2) gauge-scalar theories in 3 and 4 spacetime
dimensions. Firstly, we focus on the 4 dimensional theory with 2 sets of
fundamental scalar (Higgs) fields, which is relevant to the 2 Higgs Doublet Model
(2HDM), a proposed extension to the Standard Model of particle physics.
The goal is to understand the particle spectrum of the theory at zero temperature
and the electroweak phase transition at finite temperature. We present exploratory
results on scale setting and the multi-parameter phase diagram of this theory.
On the other hand, we are interested in the 3 dimensional SU(2) theory with
multiple Higgs fields in the adjoint representation, that can be mapped to cuprate
systems in condensed matter physics which host a rich phase diagram including
high-Tc superconductivity. It has been proposed that the theory with 4 adjoint Higgs
fields can be used to explain the physics of hole-doped cuprates for a wide range
of parameters while the theory with 1 real adjoint Higgs field would describe the
physics of electron-doped cuprates. We show exploratory results on the phase
diagram of these theories.
Topological Data Analysis (TDA) is a field that leverages tools and ideas from algebraic topology to provide robust methods for analysing geometric and topological aspects of data. One of the principal tools of TDA, persistent homology, produces a quantitative description of how the connectivity and structure of data changes when viewed over a sequence of scales. We propose that this presents a means to directly probe topological objects in gauge theories. In this talk I will present recent work on using persistent homology to detect center vortices in SU(2) lattice gauge theory configurations in a gauge-invariant manner. I will introduce the basics of persistence, describe our construction, and demonstrate that the result is sensitive to vortices. Moreover, I will discuss how with simple machine learning, one can use the resulting persistence to quantitatively analyse the deconfinement transition via finite-size scaling, providing evidence on the role of vortices in relation to confinement in Yang-Mills theories.
The Hamiltonian formalism for lattice gauge theories has experienced a resurgence of interest in recent years due to its relevance for quantum simulation, a major goal of which is the solution of sign problems in QCD. The particular formulation of the Hamiltonian formalism is itself an important design decision, where factors to consider include (non)locality of the degrees of freedom, (non)Abelian constraints, and computational costs associated with simulating the Hamiltonian.
This work represents a key step toward understanding the costs and benefits associated with the loop-string-hadron (LSH) formulation of lattice gauge theories by generalizing the original SU(2) construction to SU(3) (in 1+1 dimensions). We show that the SU(3) LSH construction is indeed a straightforward generalization of its SU(2) counterpart with all salient theoretical features left intact---particularly the conversion of SU(3) Clebsch-Gordan coefficients into explicit functions of LSH number operators. The validity of the LSH approach is underscored by demonstrating numerical agreement with the better-known purely-fermionic formulation of the theory (with open boundary conditions).
The standard method for determining matrix elements in lattice QCD requires the computation of three-point correlation functions. This has the disadvantage of requiring two large time separations: one between the hadron source and operator and the other from the operator to the hadron sink. Here we consider an alternative formalism, based on the Dyson expansion leading to the Feynman-Hellmann theorem, which only requires the computation of two-point correlation functions. Both the cases of degenerate energy levels and quasi-degenerate energy levels which correspond to diagonal and transition matrix elements respectively are considered in this formalism. Numerical results for the Sigma to nucleon transition are presented in a further contribution by M. Batelaan.
Theoretical calculations of the transition form factors of the hyperons are an important component of the determination of the CKM matrix elements. These calculations historically have been performed using ratios of lattice three point functions and two-point functions to extract the form factors, this requires the careful balancing of control over excited states and the preservation of a strong signal. We present a novel method which uses the Feynman-Hellmann method to relate a shift in energy due to a perturbation to the required form factors, this method requires only the calculation of two-point functions. The formalism of this Method is expanded on in the presentation by R. Horsley, the details of the numerical computation and the results of the Sigma to nucleon transition will be presented here.
$V_{ub}$ is the smallest and least known of all CKM matrix elements; the community determines its current value primarily through the exclusive process $B\to\pi\ell\nu$. This talk will present our progress toward a lattice QCD determination of the $V_{ub}$ matrix element from a novel transition - $B \to \pi\pi\ell\nu$ process, where the $\pi\pi$ rescattering features the $\rho(770)$ resonance as an enhancement. We perform our calculation on $N_f=2+1$ clover fermions on a lattice of $L=3.6$ fm and a pion mass of $320$ MeV. After a brief overview of the theoretical framework, we will discuss some preliminary results.
Multi-particle states with additional pions are expected to result in a non-negligible excited-state contamination in lattice simulations at the physical point. We show that heavy meson chiral perturbation theory (HMChPT) can be employed to calculate the contamination due to two-particle $B\pi$ states in various $B$-meson observables like the decay constant $f_B$ and the $B^*B\pi$ coupling $g_{\pi}$. We work in the static limit and to next-to-leading order (NLO) in the chiral expansion. The $B\pi$ states are found to typically overestimate the observables at the few percent level depending on the size of two currently unknown NLO low-energy coefficients. A strategy to independently measure one of them with the 3-point function of the light axial vector current will be discussed.
Combining experimental input, perturbative calculations, and form factors computed in lattice QCD simulations, it is possible to deduce $|V_{ub}|$ from semileptonic decays of B mesons. But the results of the form factors are contaminated by excited-states, which may lead to noticeable systematic errors in the desired CKM matrix element.
This talk presents our recent computations of the dominant $B\pi$ excited-states contamination in $B \to \pi$ form factors in Heavy Meson Chiral Perturbation Theory. The results were obtained in the static limit and to NLO in the chiral expansion and include new, to date unknown, low energy constants. Depending on their value, the effects for lattice simulations can be considerable.
I will describe recent progress in the development of custom machine learning architectures based on flow models for the efficient sampling of gauge field configurations. I will present updates on the status of this program and outline the challenges and potential of the approach.
We present our attempts to control the sign problem by the path optimization method with emphasis on efficiency of the neural network. We found a gauge invariant neural network is successful in the 2-dimensional U(1) gauge theory with a complex coupling. We also investigate possibility of the improvement in the learning process.
We present a novel strategy to strongly reduce the severity of the sign problem, using line integrals along paths of changing imaginary action. Highly oscillating regions along these paths cancel out, decreasing their contributions. As a result, sampling with standard Monte-Carlo techniques becomes possible in cases which otherwise requires methods taking advantage of complex analysis, such as Lefschetz-thimbles or Complex Langevin. We lay out how to write down an ordinary differential equation for the line integrals. As an example of its usage, we apply the results to a 1d quantum mechanical anharmonic oscillator with a $x^4$ potential in real time, finite temperature.
At fine lattice spacings, lattice simulations are plagued by slow (topological) modes that give rise to large autocorrelation times. These in turn lead to statistical and systematic errors that are difficult to estimate. We study the problem and possible algorithmic solutions in 4-dimensional SU(3) gauge theory, with special focus on instanton updates and metadynamics.
A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which sampling is trivial. A series of seminal studies have demonstrated that approximations of trivializing maps can be 'machine-learned' by a class of invertible neural models called Normalizing Flows, constructed such that the Jacobian determinant of the transformation can be efficiently computed. Asymptotically exact sampling from the theory of interest can be performed by drawing samples from a simple distribution, passing them through the network, and reweighting the resulting configurations (e.g. using a Metropolis test). From a theoretical perspective, this approach has the potential to become more efficient than traditional Markov Chain Monte Carlo sampling techniques, where autocorrelations severely diminish the sampling efficiency on the approach to the continuum limit. A major caveat is that it is not yet well-understood how the size of models and the cost of training them is expected to scale. In previous work, we conducted an exploratory scaling study using two-dimensional $\phi^4$ theory with up to $20^2$ lattice sites, which suggested that training costs grow very quickly indeed. We present updated results using a more scalable architecture utilising convolutional neural networks, and discuss various factors contributing to the scalability of these methods.
The recent introduction of machine learning tecniques, especially normalizing flows, for the sampling of lattice gauge theories has shed some hope on improving the sampling efficiency of the traditional HMC algorithm. However, naive usage of normalizing flows has been shown to lead to bad scaling with the volume. In this talk we propose using local normalizing flows at a scale given by the correlation length. Even if naively these transformations have a very small acceptance, when combined with the HMC lead to algorithms with high acceptance and reduced autocorrelation times compared with HMC. Several scaling tests are performed in the $\phi^{4}$ theory in 2D.
Finite-volume pionless effective field theory is an efficient framework with which to perform the extrapolation of finite-volume lattice QCD calculations of multi-nucleon spectra and matrix elements to infinite volume and to nuclei with larger atomic number. In this contribution, a new implementation of this framework based on correlated Gaussian wavefunctions optimized using differentiable programming and using a solution of a generalised eigenvalue problem is discussed. This approach is found to be more efficient than previous stochastic implementations of the variational method, as it yields comparable representations of the wavefunctions of nuclei with atomic number $A \le 6$ with an order of magnitude fewer terms. Future applications to infinite-volume extrapolations of nuclear matrix elements will also be discussed.
This talk presents a new method for computing correlators for systems of many identical mesons. The method allows the computation of every meson correlator up to N mesons from propagators using only a single N by N eigendecomposition. This pushes the frontier of many-meson calculations from dozens to thousands, and as a demonstration I will present the computation of the maximal-isospin pion correlator for systems from 1 up to 6144 pions on an ensemble of Wilson fermions with slightly heavier than physical pions ($m_\pi \sim 170 \, {\rm MeV}$). In addition, I will cover some aspects of analyzing such correlators, a task complicated by the sheer scale of the correlators involved.
The formalism for relating finite-volume energies and matrix elements to scattering and decay amplitudes has been established for three-pion states with all possible isospins in the so called RFT (relativistic field theory) method. This necessarily leads to coupled-channel systems. The three-pion I=1 channel, for example, includes all two-pion isospins as sub-channels. In this talk I describe issues and strategies in implementing both the scattering and decay formalism in practice and show examples of the relations between finite- and infinite-volume quantities. I also describe an open source python library that supports the practical implementation.
The Lüscher scattering formalism, the standard approach for relating the discrete finite-volume energy spectrum to two-to-two scattering amplitudes, fails when analytically continued so far below the infinite-volume two-particle threshold that one encounters the t-channel cut. This is relevant, especially in baryon-baryon scattering applications, as finite-volume energies can be observed in this below-threshold regime, and it is not clear how to make use of them. In this talk we present a generalisation of the scattering formalism that resolves this issue, allowing one to also constrain scattering amplitudes on the t-channel cut.
The γ⋆γ⋆ → ππ scattering amplitude can help constrain hadronic contributions to the anomalous magnetic moment of the muon, as well as structural information of glueball and tetraquark candidates. To leading order in QED, this amplitude can be accessed from matrix elements from non-local products of electromagnetic currents evaluated in an infinitely large Minkowski spacetime. In this talk, we present a model-independent formalism to determine this amplitude from finite, Euclidean spacetime correlation functions.
Determining the internal structure of hadrons is a necessary step to advance our understanding of the dynamics of confined partons. Extracting form factors of resonances directly from lattice QCD requires a formal connection between the finite volume Euclidean correlation functions and the infinite volume Minkowski amplitudes. In this talk we describe a novel procedure to extract transitions that couple states with at most two nucleons by exploiting the finite volume of the lattice. Building on previous work pertaining to spinless systems, we describe how to achieve the description of the spin degrees-of-freedom given their non-trivial finite-volume interaction with an external local current of arbitrary Lorentz structure. We will present the main ingredients of our derivation, and an outlook for future calculations where we discuss a case study of the significance of the finite-volume corrections as a function of the binding energy of a deuteron-like state.
In QCD at large enough isospin chemical potential Bose-Einstein Condensation (BEC) takes place, separated from the normal phase by a phase transition. From previous studies the location of the BEC line at the physical point is known. In the chiral limit the condensation happens already at infinitesimally small isospin chemical potential for zero temperature. The zero-density chiral transition might then be affected, depending on the shape of the BEC boundary, by its proximity. As a first step towards the chiral limit, we perform simulations of 2+1 flavors QCD at half the physical quark masses. The position of the BEC transition is then extracted and compared with the results at physical masses.
At finite baryon chemical potential, the sign problem hinders Monte Carlo simulations which is remedied by a Dual Representation that makes the sign problem mild. At the strong coupling limit, the dual formulation with staggered quarks is well established. We have used this formulation to look at the quark mass dependence of the baryon mass and the nuclear transition which allows us to quantify the nuclear interaction. The results obtained are also compared with the mean field theory.
The Hamiltonian formulation of Lattice QCD with staggered fermions
in the strong coupling limit has no sign problem at non-zero baryon density
and allows for Quantum Monte Carlo simulations.
We have extended this formalism to two flavors,
and after a resummation, there is no sign problem
both for non-zero baryon and isospin chemical potential.
We report on recent progress on the implementation
of the Quantum Monte Carlo simulations and present results
on the baryon and isospin densities in the chiral limit.
These will be compared with Meanfield theory.
The thermodynamics of QCD with sufficiently heavy dynamical quarks can be described by a three-dimensional Polyakov loop effective theory, after a truncated character and hopping expansion. We investigate the resulting phase diagram for low temperatures by mean field methods. Taking into account chemical potentials both for baryon number and isospin, we obtain clear signals for a liquid-gas type transition to baryon matter at $\mu_I=0$ and a Bose-Einstein condensation transition at $\mu_B=0$ , as well as for their connection as a function of both chemical potentials.
At low temperature and large chemical potential QCD might exhibit a chiral inhomogeneous phase, as indicated by various simple low-energy models. One of these models is the 3+1-dimensional Nambu-Jona-Lasinio model, which is non-renormalizable -- rendering the results possibly dependent on the employed regularization scheme. While most previously published results regarding the inhomogeneous phase in this model were obtained with the Pauli-Villars or similar regularizations, this talk explores the dependence of this phase on different lattice regularizations. Furthermore, the lattice approach allows us to determine the energetically preferred shape of the condensate without a specific ansatz.
We studied the 2+1 dimensional XY model at nonzero chemical potential on deformed integration manifolds with the aim of alleviating its sign problem. We investigated several proposals for the deformations and managed to considerably improve on the severity of the sign problem with respect to standard reweighting approaches. In this talk I present numerical evidence that a significant reduction of the sign problem can be achieved which is exponential in both the squared chemical potential and the spatial volume. Furthermore, I discuss a new approach to the optimizaiton procedure, based on reweighting, that sensibly reduces its computational cost.
We use one-flavour QCD ($N_c=3$) as a proxy to understand $\mathcal{N}=1$ SYM. For our simulations, we use tree-level improved Wilson fermions and Symmanzik improved gauge action. The hadron spectrum is obtained using LapH for different masses and simulation volumes. We find that the ratio of pseudo-scalar over scalar is smaller than one, which is in line with expectations from theory. We also report on our efforts to increase the number of colours in our simulations.
Composite Higgs models are a popular solution to the Naturalness problem in the Higgs sector, where the mass of the Higgs bosons is explained in terms of Goldstone dynamics. We address a composite model described by a $SU(4)$ gauge group with fermions in the fundamental and two index anti-symmetric representations of the gauge group. We will show results from lattice simulations investigating the chiral limit of this theory with a focus on the multi-representation dynamics and the reconstruction of spectral densities from lattice correlators.
We present ongoing investigations of maximally supersymmetric Yang--Mills ($Q = 16$ SYM) theory in three space-time dimensions. This theory is conjectured to be holographically dual to higher-dimensional quantum gravity. Previous results focused on the homogeneous "D2" phase. The work in progress concerns phase transitions between this "D2" phase and a localized "D0" phase.
Maximally supersymmetric Yang--Mills theory ($\mathcal N = 4$ SYM) is conformal for any value of the coupling. Lattice regularization breaks conformality through the introduction of a non-zero lattice spacing and a finite lattice volume. I will present ongoing computations of conformal scaling dimensions in lattice $\mathcal N = 4$ SYM, based on a lattice formulation that exactly preserves a supersymmetry sub-algebra at non-zero lattice spacing. The main targets are the non-trivial anomalous dimension of the Konishi operator, as well as a mass anomalous dimension extracted from the eigenvalue mode number of the fermion operator. The latter is expected to vanish in the conformal continuum theory, providing insight into the interplay of lattice discretization and conformality.
On behalf of the Lattice Strong Dynamics (LSD) collaboration, we present first results of the SU(4) Gauge theory Stealth Dark Matter hadron spectrum using stochastic Laplacian Heaviside (sLapH) smearing. We compare our results to previous work in the context of our Stealth Dark Matter baryon scattering project.
In the context of Strongly Interacting Dark Matter theories dark isosinglet mesons might play an important role in the low-energy dynamics and might provide crucial signatures in collider and direct detection searches. We present first results in $Sp(4)$ gauge theory with $N_f=2$ fundamental Dirac flavours on the dark isosinglet pseudoscalar $\eta'$ and the isosinglet contribution to the dark $\pi^0$ meson in the case of strong isospin breaking. A preliminary investigation of the pion scattering lengths shows that the parameter space studied so far appears to be phenomenologically viable. Additionally, these results can be relevant to other BSM models such as composite Higgs scenarios.
Bandwidth and latencies are central performance limiters for Lattice QCD. To overcome bandwidth limiters one way is to reduce the number of bits need by e.g., mixed precision solvers. These provide great speedups but increase the relative importance of latency limiters. We discuss techniques that QUDA uses to reduce latencies from GPU-CPU and GPU-network transfers and their impact for strong-scaling HMC simulations, where these matter most.
MPI Job Manager (MPI_JM) is "scheduler" designed enable users to make maximum use of heterogenous architectures, particularly which require a "swarm" of independent MPI tasks is required for a complete calculation - such as lattice QCD calculations of correlation functions on pre-existing configurations. MPI_JM managers all these tasks through lightweight C++ code supported by Python3. MPI_JM allows users to describe the resource requirements of their tasks (GPU-intense, CPU-only, number of nodes, wall clock time, etc) as well as their dependencies. MPI_JM then schedules these tasks on an allocation on an HPC platform based upon user defined priority and dependencies. Jobs with GPU-intense and CPU-only requirements are placed upon the same nodes, maximizing the use of all node resources. This is all managed with a single mpirun
call, minimizing the requirements of the service nodes that manage an HPC system. Planned features include (among others):
Multiple job-configurations: as the wall clock of the allocation nears the end, the optimal run configuration may not have enough time to complete, but doubling the nodes at a performance loss would allow a job to complete in time. MPI_JM can try alternate configurations specified by the user, to use up the otherwise idle cycles towards the end of a job allocation
Try again: sometimes, the GPUs on a node will just fail to start up in time, causing a job to time out. MPI_JM can be instructed to try N-times before giving up and trying a new job, or removing those nodes from the allowed ones to be used in the allocation.
Use real wall-clock time rather than user specified estimate: Optinoally, MPI_JM will track performance of similar jobs in a database, and then use this information to provide more reliable estimates of wall-clock time requirements than what is specified by the user.
etc.
The 2D O(N) non-linear sigma models are exactly solvable theories and, on the lattice, they have many applications from statistical mechanics to QCD toy models. In this talk, I will consider a particular generalization of the O(N) model, i.e. the non-linear sigma model on the supersphere. The global symmetry group of this model – the OSp(N+2M|2M) supergroup – mixes bosonic and fermionic degrees of freedom, hence the sigma model can be thought of as a toy model for string worldsheet theories with target space supersymmetry. In this talk, I will describe the non-linear sigma model on the supersphere, its discretization on the lattice, its renormalization properties, and the relation between this model and its non-supersymmetric equivalent. I will also present our strategy for numerical simulations and some preliminary numerical results.
A generalization of Wilsonian lattice gauge theory may be obtained by considering the possible self-adjoint extensions of the electric field operator in the Hamiltonian formalism. In the special case of $3D$ $U(1)$ gauge theory these are parametrised by a phase $\theta$, and the ordinary Wilson theory is recovered for $\theta=0$. We consider the case $\theta=\pi$, which, upon dualization, turns into a theory of staggered integer and half-integer height variables. We investigate order parameters for the breaking of the relevant symmetries, and thus study the phase diagram of the theory, which could reveal a new universality class of $3D$ Abelian gauge theories with a broken $\mathbb{Z}_2$ symmetry absent in the ordinary theory.
Recent studies on the 't Hooft anomaly matching condition have suggested
a nontrivial phase structure in 4D SU($N$) gauge theory at $\theta=\pi$.
In the large-$N$ limit, it has been found that CP symmetry at $\theta=\pi$ is broken
in the confined phase, while it restores in the deconfined phase,
which is indeed one of the possible scenarios.
However, at small $N$, one may find other situations that are consistent
with the consequence of the anomaly matching condition.
Here we investigate this issue for $N=2$ by direct lattice calculations.
The crucial point to note is that the CP restoration can be probed
by the sudden change of the tail of the topological charge distribution at $\theta=0$,
which can be seen by simulating the theory at imaginary $\theta$ without the sign problem.
Our results suggest that the CP restoration at $\theta=\pi$ occurs at temperature
higher than the deconfining temperature unlike the situation in the large-$N$ limit.
The 3D Ising conformal field theory (CFT) describes different physical systems, such as uniaxial magnets or fluids, at their critical points. In absence of an analytical solution for the 3D Ising model, the scaling dimensions and operator product expansion (OPE) coefficients characterizing this CFT must be determined numerically. The currently most-cited values for these quantities have been obtained from the conformal bootstrap, while lattice calculations have so far only produced reliable results for the scaling dimensions involved in calculating the critical exponents. Using Quantum Finite Elements to investigate critical $\phi^4$-theory on $\mathbb{R}\times\mathbb{S}^2$, we have extracted scaling dimensions and OPE coefficients of the 3D Ising CFT by fitting the lattice four-point function with expectations from the operator product expansion for a radially quantized CFT and extrapolating to the continuum limit. This way, we have for the first time been able to use Monte Carlo simulations to compute the central charge of the theory, as well as scaling dimensions and OPE coefficients of high-spin operators.
For the 2d Ising model on a triangular lattice, we determine the exact values of the three critical coupling coefficients which restore conformal invariance in the continuum limit as a function of an affine transformation of the triangle geometry. On a torus with a non-trivial modular parameter, we present numerical results showing agreement with the exact CFT solution. Finally, we discuss how this method may be applied to simulate the critical Ising model on curved 2d simplicial manifolds.
We study the massless Schwinger model with an additional 4-fermi interaction and a topological term. For topological angle $\theta = \pi$ charge conjugation is implemented in a non-trivial way and we study its spontaneous breaking. We use staggered fermions and the Villain action for the gauge fields, where the topological term is an integer and charge conjugation at $\theta = \pi$ is an exact symmetry. The complex action problem is overcome by a suitable worldline/worldsheet representation. We find that as a function of the 4-fermi coupling the system has a critical point where charge conjugation is broken spontaneously and we present first results on the nature of the critical point.
CKM matrix elements can be obtained from lattice determinations of semileptonic decay form factors by combining them with experimental results for decay rates. We give a status update on our study using the Domain Wall Fermion action for up/down, strange and charm quarks to determine semileptonic form factors for $D \rightarrow \pi \ell \nu$, $D \rightarrow K \ell \nu$ and $D_s \rightarrow K \ell \nu$ decays. Data have been produced on three lattice spacings and pion masses in the range 250 Mev to 400 Mev, and preliminary form factor data are presented. These will subsequently be included in a global fit.
We present HPQCD's improved scalar, vector and tensor form factors for $B \to K$ semileptonic decays, using the heavy-HISQ formalism for more accurate normalisation of the weak currents. Working with masses close to the physical $b$ on the finest ensemble and including three ensembles with physical light quarks, we cover the full physical $q^2$ range with good precision. Our uncertainties at $q^2=0$ are a factor of three better than earlier work.
We compare Standard Model observables using our form factors to experimental measurements for the rare flavour changing neutral current processes $B \to K \ell^+\ell^-$ and $B \to K \nu\bar{\nu}$ and discuss the significance of the tensions that arise.
We study, with lattice QCD, the radiative leptonic decays $P\to \ell\,\nu_\ell\,\ell'^+\,\ell'^-$, where $P$ is a charged pseudoscalar meson and $\ell$ and $\ell^\prime$ are charged leptons. These processes are mediated by the emission of a virtual photon and, in addition to the ``point-like" contribution in which the virtual photon is emitted either from the lepton or the meson treated as a point-like particle, four structure-dependent (SD) form factors contribute to the amplitude.
We present a strategy for the extraction of the SD form factors and implement it in an exploratory lattice computation of the decay rates for the four channels of kaon decays ($\ell,\ell^\prime=e,\mu$). The lattice computation has been performed employing only one gauge ensemble, with simulated pion and kaon masses equal to $320$ and $530$ MeV, respectively.
It is the SD form factors which describe the interaction between the virtual photon and the internal hadronic structure of the decaying meson, and in our procedure we separate the SD and point-like contributions to the amplitudes. The form factors are extracted with good precision and used to reconstruct the branching ratio values, which are compared with the available experimental data.
These are very suppressed processes, which thus provide an excellent test of the Standard Model, and provide a useful avenue for the search for signatures of new physics.
In the region of hard photon energies, radiative leptonic decays represent important probes of the internal structure of hadrons.
Moreover, radiative decays can provide independent determinations of Cabibbo-Kobayashi-Maskawa matrix elements with respect to purely leptonic or semileptonic channels.
Prospects for a precise determination of leptonic decay rates with emission of a hard photon are particularly interesting, especially for the decays of heavy mesons for which currently only model-dependent predictions, based on QCD factorization and sum rules, are available to compare with existing experimental data.
We present a non-perturbative lattice calculation of the structure-dependent form factors which contribute to the amplitudes for the radiative decays $H \to \ell \nu_\ell \gamma$, where $H$ is a charged pseudoscalar meson, using the Domain Wall formulation of lattice fermions.
With moderate statistics, thanks to the use of a sine-cardinal-reconstruction technique and improved estimators, we are able to provide rather precise, first-principles results for the form factors in the full kinematical (photon-energy) range for both light and heavy mesons.
We developed a strategy to implement RI/MOM schemes on quark bilinear and four-quark operators. In these schemes, the momentum transfer is not restricted to the exceptional point or to the symmetric point. In particular, we study the convergence of the perturbative series and the potential to reduce some systematic errors (discretisation and chiral symmetry breaking effects). In particular, we observe a notable reduction of the pseudo-Goldstone pole contributions which could lead to a significant improvement for the renormalisation of some four-quark operators.
Title: Structure and geometry of 12C from a Wigner SU(4) symmetric interaction
The carbon-12 nucleus, one of the most crucial elements for life, is full of interesting structures and multifaceted complexity. One famous example is the first excited 0+ state, the so called Hoyle state. It can not be described by most of the ab initio calculations. Moreover, a lack of model-independent description for the shape also hinders an understanding of its geometric properties. Here we present calculations of 12C by nuclear lattice effective field theory using a simple nucleon–nucleon interaction that is independent of spin and isospin and therefore invariant under Wigner’s SU(4) symmetry. Despite the simplicity of the interaction, the agreement with experiment is impressive, not only for all the low-lying levels including the Hoyle state, but also properties such as the charge radius, density profiles, and BE2 transitions. Furthermore, we provide the first model-independent tomographic scan of the three-dimensional geometry for those nuclear states, which show many interesting shapes and features.
A recently re-discovered variant of the Backus-Gilbert algorithm for spectral reconstruction enables the controlled determination of smeared spectral densities from lattice field theory correlation functions. The particular advantage of this model-independent approach is the a priori specification of the kernel with which the underlying spectral density is smeared, allowing for variation of its peak position, smearing width, and functional form. If the unsmeared spectral density is sufficiently smooth in the neighborhood of a particular energy, it can be obtained from an extrapolation to zero smearing kernel width at fixed peak position.
The determination of scattering amplitudes is a natural application. As a proof-of-principle test, an inclusive rate is computed in the two-dimensional O(3) sigma model from a two-point correlation function of conserved currents. The results at finite and zero smearing radius are in good agreement with the known analytic form up to energies at which 40-particle states contribute, and are sensitive to the 4-particle contribution to the inclusive rate. The straight-forward adaptation to compute the R-ratio in lattice QCD from two-point functions of the electromagnetic current is briefly discussed.
A very rich place to look for phenomena to challenge our current understanding of physics is the flavor sector of the Standard Model (SM). In particular, the $|V_{cb}|$ matrix element of the CKM matrix is the subject of a long standing tension, depending on whether it is determined using inclusive or exclusive methods. On top of that, the SM theoretical calculations of some universality ratios $R(X)$ show large, unexplained tensions with experimental measurements.
Recently, there have been interesting efforts in Lattice QCD (LQCD) trying to cast some light onto the current situation. Calculations of the form factors of the gold-plated channels $B\to D^{(\ast)}\ell\nu$ at non-zero recoil are becoming the
norm, and when combined with the latest data coming from $B$ factories, they offer promising prospects of settling the matter.
In this talk, I will review the current status of the form factor LQCD calculations at non-zero recoil of the $B\to D^{(\ast)}\ell\nu$ channels, and their impact in the determination of $|V_{cb}|$ and $R(D^{(\ast)})$.
We review recent progress on heavy flavor physics from lattice QCD.
One of the most direct predictions of QCD is the existence of color-singlet states
called Glueballs, which emerge as a consequence of the gluon field self-interactions.
Despite the outstanding success of QCD as a theory of the strong interaction
and decades of experimental and theoretical efforts, all but the most basic properties of Glueballs are still being debated.
In this talk, I will review efforts aimed to understanding Glueballs and the current status of Glueball searches, including recent experimental results and lattice calculations.
The study of real-time evolution of quantum field theories is known to be an extremely challenging problem for classical computers. Due to a fundamentally different computational strategy, quantum computers hold the promise of allowing for detailed studies of these dynamics from first principles. However, much like with classical computations, it is important that quantum algorithms do not have a cost that scales exponentially with the volume. In this paper, we present an interesting test case: a formulation of a compact U(1) gauge theory in 2+1 dimensions. A naive implementation onto a quantum circuit has a gate count that scales exponentially with the volume. We discuss how to break this exponential scaling by performing an operator redefinition that reduces the non-locality of the Hamiltonian and also provide explicit implementations using the Walsh function formalism. While we study only one theory as a test case, we expect the exponential gate scaling to persist for formulations of other gauge theories, including non-Abelian theories in higher dimensions.
We propose a variational quantum eigensolver suitable for exploring the phase structure of the multi-flavor Schwinger model in the presence of a chemical potential. The parametric ansatz we design incorporates the symmetries of the model and can be implemented on both measurement-based and circuit-based quantum hardware. We numerically demonstrate that our ansatz is able to capture the phase structure of the model and allows for faithfully approximating the ground state. Our results show that our approach is suitable for current intermediate-scale quantum hardware and can be readily implemented on existing quantum devices.
With the long term perspective of using quantum computers for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of $\mathrm{SU}(2)$, such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.
Simulating SU$(N)$ gauge theories on a quantum computer requires some form of digitization of the gauge degrees of freedom. Recently, we have proposed discretisation schemes, which offer in contrast to finite subgroups the possibility to freely refine the discretisation. Here we present an approach to define the corresponding canonical momentum operators. We present results on the restoration of the fundamental commutation relations towards continuous gauge field degrees of freedom.
Sign problems in Monte Carlo simulations have long hindered studies of phase diagrams of lattice gauge theories (LGTs) at finite densities. Quantum computation of LGTs does not encounter sign problems, but preparing thermal states needed for a complete phase-diagram analysis on quantum devices is a difficult and resource-intensive process. Thermal Pure Quantum (TPQ) states have been proposed in recent years as an efficient method to reliably estimate thermal expectation values on a quantum computer. We propose a new form of TPQ states, called Physical Thermal Pure Quantum (PTPQ) states, to quantum compute thermal expectation values and non-equal time correlation functions of LGTs at finite temperature and density. We illustrate the approach by computing the chiral phase diagram of a toy theory accessible to near-term quantum hardware, 1+1 dimensional $\mathbb{Z}_2$ LGT coupled to staggered fermions, and analyze the resource requirement of the associated quantum algorithms. Our approach may open new paths forward in simulating the phase diagram of strong interactions in nature using the ever-improving quantum computers.
Quantum computing is a promising new computational paradigm which may allow one to address exponentially hard problems inaccessible in Euclidean lattice QCD. Those include real-time dynamics, matter at non-zero baryon density, field theories with non-trivial CP-violating terms and can often be traced to the sign problem that makes stochastic sampling methods inapplicable. As a prototypical example we consider a low-dimensional theory, Quantum Electrodynamics in 1+1 space-time dimensions with a theta term. Using staggered fermions, this model can be mapped to a quantum Ising-like model with nearest-neighbor interactions which is well-suited for digital gate-based quantum computers. We study and compare properties of three algorithms that can be employed for the initial state preparations: Quantum Adiabatic Evolution (QAE), Quantum Approximate Optimization Algorithm (QAOA) as well as recently proposed Rodeo Algorithm. Understanding their convergence properties may be helpful for designing optimal algorithms with minimal number of CNOT gates for near-term noisy intermediate scale quantum (NISQ) devices that are currently within technological reach.
Recently, a doubly charmed tetraquark $T_{cc}$ with flavor $cc\bar u\bar d$ just $0.36(4)~$MeV below $D^0D^{*+}$ threshold was discovered by the LHCb collaboration. We present the first lattice study of $DD^*$ scattering in this channel, involving rigorous determination of pole singularities in the related scattering amplitudes that point to the existence of $T_{cc}$. Working with a heavier than physical light quark mass, we find evidence for a shallow virtual bound state pole in the $DD^*$ scattering amplitude with $l=0$, which is likely related to $T_{cc}$.
The doubly charm tetraquark with exotic quark composition $cc\bar u \bar d$ is the longest-lived exotic hadron discovered in the experiment. Our lattice simulation establishes a virtual bound state pole in DD* scattering at $m_\pi\simeq 280$ MeV, which is likely related to this state. We discuss the expected dependence of this hadron on the light and heavy quark masses, and compare it to the lattice results.
We study a doubly-bottomed tetra-quark state $(bb\bar{u}\bar{d})$ with quantum number $I(J^P)=0(1^+)$, denoted by $T_{bb}$, in lattice QCD with the NRQCD quark action for $b$ quarks.
Employing $(2+1)$-flavor gauge configurations at $a \approx 0.09$ {fm} on $32^3\times 64$ lattices, we have extracted the coupled channel HAL QCD potential between $\bar{B}\bar{B}^*$ and $\bar{B}^* \bar{B}^*$, which predicts an existence of a bound $T_{bb}$ below the $\bar{B}\bar{B}^*$ threshold.
By extrapolating results at $m_\pi\approx 410,\, 570,\, 700$ {MeV} to the physical pion mass $m_\pi\approx140$ {MeV}, we obtain a biding energy with its statistical error as $E_{\rm binding}^{\rm (single)} = 155(17)$ MeV and $E_{\rm binding}^{\rm (coupled)} = 83(10)$ MeV, where "coupled" means that effects due to virtual $\bar{B}^* \bar{B}^*$ states are included through the coupled channel potential, while only a potential for a single $\bar{B}\bar{B}^*$ channel is used in the analysis for ``single".
A comparison shows that the effect from virtual $\bar{B}^* \bar{B}^*$ states is quite sizable to the binding energy of $T_{bb}$. We estimate systematic errors to be $\pm 20$ MeV at most, which are mainly caused by the NRQCD approximation for $b$ quarks.
We report progress on finite-volume determinations of heavylight-meson -- Goldstone boson scattering phase shifts using the Luescher method on CLS 2+1 flavor gauge field ensembles. In a first iteration we will focus on D-meson -- pion scattering in the elastic scattering region at various pion masses using ensembles with three lattice spacings. We employ ensembles on the CLS quark-mass trajectory with a fixed trace of the quark-mass matrix as well as ensembles with a strange-quark mass fixed close to its physical value, which will allow us to study both the light- and the strange quark-mass dependence of positive parity heavy-light hadrons close to threshold.
We present an investigation of the spectrum of exotic charmonium-like mesons using lattice QCD. The focus is on $\bar cc \bar qq$ $J^{PC}=1^{+\pm}$ states with isospin 1. Many mesons with properties incompatible with a $\bar cc$ structure have already been discovered, e. g. the $Z_c$ mesons with isospin 1. A lattice study of a four-quark system with two different total momenta and on two different lattice sizes is made. We extract the energy levels, use Lüscher's formalism, and determine the scattering length for the $D\bar D^*$ scattering close to the threshold. A comparison to results from the phenomenological approaches is made, and the constraints on the scattering length are given.
Optimized meson operators in the distillation framework are used to study the charmonium spectrum in two ensembles with two heavy dynamical quarks at half the physical charm quark mass but different lattice spacings. The use of optimal meson distillation profiles is shown to increase the overlap with the ground state significantly, as well as grant access to excited states, for multiple quantum numbers including hybrid states with very little additional cost. These same operators are also employed for the calculation of meson-glueball mixing.
We present results for the electromagnetic form factors of the proton and neutron computed on the Coordinated Lattice Simulations (CLS) ensembles with $N_\mathrm{f} = 2 + 1$ flavors of $\mathcal{O}(a)$-improved Wilson fermions and an $\mathcal{O}(a)$-improved conserved vector current. In order to estimate the excited-state contamination, we employ several source-sink separations and apply the summation method. The quark-disconnected diagrams entering the isoscalar quantities are computed explicitly. For this purpose, a stochastic estimation based on the one-end trick is performed, in combination with a frequency-splitting technique and the hopping parameter expansion. By these means, we obtain a clear signal for the form factors including the quark-disconnected contributions, which have a statistically significant effect on our results. From the $Q^2$-dependence of the form factors, we determine the electric and magnetic charge radii and the magnetic moment of the proton and neutron. The chiral interpolation is carried out by simultaneously fitting the pion mass and $Q^2$-dependence of our form factor data directly to the expressions resulting from covariant chiral perturbation theory including vector mesons. To assess the influence of systematic effects, we average over various cuts in the pion mass and the momentum transfer, as well as over different models for the lattice spacing dependence, using weights derived from the Akaike Information Criterion (AIC).
We present results for the nucleon electromagnetic form factors using $N_f$=2+1+1 twisted mass lattice QCD with clover improvement and with quarks with masses tuned to their physical values. Our preliminary analysis includes three ensembles at similar physical volume and lattice spacings $a\sim$0.08 fm, $\sim$0.07 fm, and $\sim$0.06 fm allowing us to take the continuum limit directly at the physical mass point. For each ensemble we assess excited state effects using several sink-source time separations in the range 0.8 fm - 1.6 fm, exponentially increasing statistics with the separation.
We present the results of a complete lattice calculation of the gravitational form factors (GFFs) of the proton and pion, including glue as well as connected and disconnected quark contributions, on an ensemble with 2+1 flavors of Wilson fermions with close-to-physical pion mass of 170 MeV. We use these results to predict full, physical densities of energy, pressure, and shear forces inside the proton and pion via the relation of GFFs with the energy-momentum tensor.
Nucleon isovector form factors calculated on a 2+1-flavor domain-wall-fermions ensemble with strange and degenerate up and down quarks at physical mass and lattice cut off, $a^{-1}$, of about 1.730(4) GeV, will be presented. The ensemble was generated jointly by RBC and UKQCD collaborations with a spatial extent of $48a$ or about 5.5 fm. The form factors are calculated in collaboration with LHP as well. The resulting shape parameters of the form factors such as vector-charge mean squared radius, $\langle r_1^2\rangle$, or anomalous magnetic moment, $F_2(0)$ appear less dependent on possible excited-state contaminations than the corresponding charges. This is likely because of larger statistical fluctuations at finite momentum transfer. For example, preliminary estimates are $\langle r_1^2\rangle \sim 0.142(13)\, \mbox{fm}^2$ and $F_2(0) \sim 3.22(8)$.
I will discuss progress on computing nucleon elastic form factors with the stochastic LapH method.
OR, if results these results are not yet ready,
I will discuss preliminary results on the nucleon-pion sigma term determined with O(30) HISQ ensembles with MDWF valence fermions. The nucleon spectrum results are determined at 7 pion masses in the range 130 < Mpi < 400 MeV, four lattice spacings in the range 0.06 < a < 0.15 fm, and several volumes. The nucleon-pion sigma term is determined through a derivative of the extrapolation of the nucleon mass to the physical point.
We present results for the isovector axial form factor of the nucleon computed on a set of $N_f = 2 + 1$ CLS ensembles with $\mathcal{O}(a)$-improved Wilson fermions and the Lüscher-Weisz gauge action. The set of ensembles covers a range of pion masses from 353 MeV down to the physical pion mass, and lattice spacings between 0.05 fm and 0.09 fm.
We use the summed operator insertion method (summation method) to suppress the contamination from excited states, and use the $z$-expansion to parametrise the $Q^2$-behaviour of the form factor. Systematic effects are taken into account by performing a number of fits with cuts for the pion mass and lattice spacing and with different ansätze for the chiral and infinite-volume extrapolations. Our final result for the $z$-expansion coefficients is provided by an Akaike-information-criterion based model average.
Recently an approximate SU(4) chiral spin-flavour symmetry was discovered in
multiplet patterns of QCD meson correlation functions, in a temperature range above
the chiral crossover. This symmetry is larger than the full chiral symmetry
of QCD with massless u,d-quarks. It can only arise effectively
when color-electric quark gluon interactions dominate the effective Dirac action
of QCD, which suggests that quarks remain bound in such a regime.
At temperatures about two to three times the crossover temperature, this
pattern disappears again, and the usual chiral symmetry is recovered.
We present additional evidence for this phenomenon based on meson screening
masses, and discuss how this chiral spin symmetric band continues into
the QCD phase diagram.
We investigate the phase structure of QCD with three degenerate quark flavors at finite temperature using Mobius domain wall fermions. To locate the critical endpoint and explore the order of phase transition on the diagonal line of the Columbia plot, we performed simulations at temperatures 131 and 196 MeV with lattice spacing $a\sim 0.12$ fm corresponding to temporal lattice extent $N_{\tau}=8,12$ with varying quark mass for two different volumes with aspect ratios $N_{\sigma}/N_{\tau}$ ranging from 2 to 3. By analyzing the volume and mass dependence of the chiral condensate, disconnected chiral susceptibility and Binder cumulant we find that there is a crossover at $m_q^{\mathrm{\overline {MS}}}(2\, \mathrm{GeV}) \sim 40\, \mathrm{MeV}$ for $\mathrm{T_{pc}}\sim$ 196 MeV and a transition point at $m_q^{\mathrm{\overline {MS}}}(2\, \mathrm{GeV}) \sim 4\, \mathrm{MeV}$ for $\mathrm{T}\sim$131 MeV on $24^3\times 12$ lattices. The $36^3\times12$ lattices are being investigated for the finite size scaling, we will present its result and discuss the nature of transition for $\mathrm{T}\sim$131 MeV.
The global center symmetry of quenched QCD at zero baryonic chemical potential is broken spontaneously at a critical temperature $T_c$ leading to a first-order phase transition. Including heavy dynamical quarks breaks the center symmetry explicitly and weakens the first-order phase transition for decreasing quark masses until it turns into a smooth crossover at a $Z_2$-critical point. We investigate the $Z_2$-critical quark mass value towards the continuum limit for $N_\text{f}=2$ flavors using lattice QCD in the staggered formulation. As part of a continued study, we present results from Monte-Carlo simulations on $N_\tau=8, 10$ lattices. Several aspect ratios and quark mass values were simulated in order to obtain the critical mass from a fit of the Polyakov loop to a kurtosis finite size scaling formula. Moreover, the possibility to develop a Ginzburg-Landau effective theory around the $Z_2$-critical point is explored. The coefficients of the Landau functional can be determined from fits of the Polyakov loop to the data as a function of the bare parameters.
The so called Columbia Plot summarises the order of the QCD thermal transition as a function of the number of quark flavours and their masses. Recently, it was demonstrated that the first-order chiral transition region, as seen for $N_f=3-6$ on coarse lattices, exhibits tricritical scaling while extrapolating to zero on sufficiently fine lattices. Here we extend these studies to imaginary baryon chemical potentials, using $N_f=4,5$. A similar shrinking of the first-order region is observed with decreasing lattice spacing, which is so far entirely similar to the situation at zero density.
QCD with infinite heavy quark masses exhibits a first-order thermal transition which is driven by the spontaneous breaking of the global $\mathcal{Z}_3$ center symmetry. We analyze the corresponding order parameter, namely the Polyakov loop and its moments, and show, with a rigorous finite size scaling, that in the continuum limit the transition is of first order. We show that the use of a parallel tempering algorithm can significantly reduce the large auto-correlation times, which are mainly caused by supercritical slowing down for first order phase transitions. As a result, we calculate the transition temperature $w_0 T_c$ with per-mill precision, and the latent heat, carrying out controlled continuum and infinite volume extrapolations.
Decreasing the quark masses weakens the transition until the latent heat vanishes at the critical mass. We give an update on our exploration of the heavy mass region with three flavors of staggered quarks.
The QCD crossover is marked by the rapid change in various observables such as the chiral condensate, the Polyakov loop or the topological susceptibility. We studied the topological properties in pure SU(3) gauge theory where the transition is first order.
Our study focused on the topological susceptibility and the $b_2$ coefficient of the expansion of the free energy density around $\theta=0$. There was already some evidence for the discontinuity in the topological susceptibility at the transition temperature in SU(N) Yang-Mills theories. We determined the continuum extrapolated value of this discontinuity for $N=3$ in the infinite volume limit. We also determined the temperature dependence of the $b_2$ coefficient directly at $\theta=0$ and by using information from imaginary $\theta$ simulations.
In our previous work, we showed that unresolved excited state contaminations provide a major source of systematic uncertainty in the calculation of the nucleon electric dipole moment due to the QCD topological term theta. Here we extend this result to the calculation of the nucleon electric dipole moment due to the quark chromo-electric dipole moment operator. We also show quantitatively the impact of mixing of the latter with lower-dimensional operators on the lattice. Finally, we present preliminary results from a unitary clover-on-clover calculation for the QCD topological term.
We report our calculation of the neutron electric dipole moment (EDM) induced by the theta term. We use overlap fermions on three 2+1-flavor RBC/UKQCD domain wall lattices with pion mass ranging from ~300 to ~500 MeV. The use of lattice chiral fermions guarantees a correct chiral limit even at finite lattice spacings and enables us to reliably extrapolate our result from heavy pion masses to the physical point. Furthermore, by utilizing the partially-quenched chiral extrapolation formula, several valence pion points are added to better constrain the chiral extrapolation. With the help of the cluster decomposition error reduction (CDER) technique and a large amount of statistics accumulated, the statistical uncertainty is effectively controlled. We also carefully check the systematic uncertainties from the two-state fits, the momentum extrapolation, the chiral extrapolation and the CDER technique.
At the TeV scale, low-energy precision observations of neutron characteristics provide unique probes of novel physics. Precision studies of neutron decay observables are susceptible to beyond the Standard Model (BSM) tensor and scalar interactions. The neutron electric dipole moment also has high sensitivity to new BSM CP-violating interactions. To fully utilise the potential of future experimental neutron physics programs, matrix elements of appropriate low-energy effective operators within neutron states must be precisely calculated. We present results from the QCDSF/UKQCD/CSSM collaboration for the isovector charges $g_T,~g_A$ and $g_S$ using lattice QCD methods and the Feynman-Hellmann theorem. We use a flavour symmetry breaking method to systematically approach the physical quark mass using ensembles that span five lattice spacings and three volumes. We extend this existing flavour breaking expansion to also account for lattice spacing and finite volume effects in order to quantify all systematic uncertainties.
By far the biggest contribution to hadronic vacuum polarization (HVP) arises from the two-pion channel. Its quark-mass dependence can be evaluated by combining dispersion relations with chiral perturbation theory, providing guidance on the functional form of chiral extrapolations, or even interpolations around the physical point. In addition, the approach allows one to estimate in a controlled way the isospin-breaking corrections that arise from the pion mass difference. As an application, I will present an updated estimate of phenomenological expectations for electromagnetic and strong isospin-breaking corrections to the HVP contribution in the anomalous magnetic moment of the muon.
In this contribution we report on progress in the determination of the isospin breaking corrections to the vector-vector correlator in QCD from the RBC/UKQCD collaborations. They are relevant to estimate the hadronic contributions to the muon anomalous magnetic moment directly from first-principles lattice QCD simulations, and indirectly from cross sections measured in tau decay experiments.
We present a calculation of the intermediate window quantity of the
hadronic vacuum polarization contribution to the muon g-2 using a
Lorentz-covariant coordinate-space method at a fixed pion mass of ~350
MeV. This method is more flexible in the choice of the integration
kernel than the time-momentum representation and gives a different
perspective on the systematic errors of the g-2 calculation. It
furthermore serves as a check for the recent results of the Mainz
group.
$3+1$ dimensional QED with massless electrons is chirally symmetric, at least in the perturbative regime. This is true, even though this symmetry is anomalous, because QED lacks instantons. However, in the presence of external magnetic fields, approximate calculations based on Schwinger-Dyson equations indicate that chiral symmetry is spontaneously broken. In a constant external magnetic field $B$, this produces a dynamical electron mass $\propto\sqrt{eB}$ and a chiral condensate $\propto(eB)^{3/2}$. The magnetic field catalyses chiral symmetry breaking with an effective dimensional reduction from $3+1$ dimensions to $1+1$ dimensions. We simulate lattice QED in a constant homogeneous magnetic field using the RHMC algorithm. We increase $\alpha$ from $1/137$ to $1/5$ to make the chiral symmetry breaking measurable. To access the chiral limit, we need to increase the lattice size. By using a large $eB$, the localization of the motion in the plane perpendicular to the magnetic field due to the dominance of low lying Landau levels means that we only need a large lattice size in the directions of $B$ and time. Our preliminary `data' show clear evidence for a non-zero condensate in the zero electron mass limit and hence chiral symmetry breaking.
Standard local updating algorithms experience a critical slowing down close to the continuum limit, which is particularly severe for topological observables. In practice, the Markov chain tends to remain trapped in a fixed topological sector. This problem further worsens at large $N$, and is known as $~\mathit{topological}$ $~\mathit{freezing}$.
To mitigate it, we adopt the parallel tempering on boundary conditions proposed by M. Hasenbusch. This algorithm allows to obtain a reduction of the auto-correlation time of the topological charge up to several orders of magnitude.
With this strategy we are able to provide the first computation of low-lying glueball masses at large $N$ free of any systematics related to topological freezing.
In order to understand the puzzle of the free energy of an individual quark in QCD, we explicitly construct ensembles with quark numbers $N_V\neq 0\!\mod 3$, corresponding to non-zero triality in a finite subvolume $V$ on the lattice. We first illustrate the basic idea in an effective Polyakov-loop theory for the heavy-dense limit of QCD, and then extend the construction to full Lattice QCD, where the electric center flux through the surface of $V$ has to be fixed at all times to account for Gauss's law. This requires introducing discrete Fourier transforms over closed center-vortex sheets around the spatial volume $V$ between all subsequent time slices, and generalizes the construction of 't Hooft's electric fluxes in the purge gauge theory. We derive this same result from a dualization of the Wilson fermion action, and from the transfer matrix formulation with a local $\mathbb Z_3$-Gauss law to restrict the dynamics to sectors with the required center charge in $V$.
Quark confinement is perhaps the most important emergent property of the theory of quantum chromodynamics. I review recent results studying centre vortices in SU(3) lattice gauge theory with dynamical quarks. Starting from the original Monte Carlo gauge fields, a vortex identification procedure yields vortex-removed and vortex-only backgrounds. The comparison between the original `untouched' Monte Carlo gauge fields and these so called vortex-modified ensembles supports the notion that centre vortices are fundamental to confinement in full QCD.
We compute the topological susceptibility of $N_f = 2+1$ QCD at physical point in a temperature range going from 200 to 600 MeV. We adopt a multicanonical approach to enhance topological fluctuations and a definition of the susceptibility based on the spectral projectors over the eigenmodes of the staggered Dirac operator. This method allows to reduce lattice artifacts affecting the standard gluonic definition, making the continuum limit extrapolation more reliable.
The Hamiltonian approach can be used successfully to study the real time evolution of a non-Abelian lattice gauge theory on the available noisy quantum computers. In this talk, results from the real time evolution of SU(2) pure gauge theory on IBM hardware are presented. The long real time evolution spanning dozens of Trotter steps with hundreds of CNOT gates and the observation of a traveling excitation on the lattice were made possible by using a comprehensive set of error mitigation techniques. Self-mitigation is our novel tool, which consists of using the same physics circuit as a noise-mitigation circuit.
Studies of the Schwinger model in the Hamiltonian formulation have hitherto used the Kogut-Susskind staggered approach. However, Wilson fermions offer an alternative approach and are often used in Monte Carlo simulations. Tensor networks allow the exploration of the Schwinger model even with a topological θ-term, where Monte Carlo methods would suffer from the sign problem. Here, we study the one-flavour Schwinger model with Wilson fermions and a topological θ-term using Matrix Product States (MPS) methods in the Hamiltonian formulation. The mass parameter in this model receives an additive renormalization shift from the Wilson term. In order to perform a continuum extrapolation, the knowledge of this shift is important. We present a method suitable for tensor networks that determines the mass renormalization using observables such as the electric field density, which vanish when the renormalized mass is zero. Using this shift, the continuum extrapolation is performed for various observables.
Quantum simulations of QCD require digitization of the infinite-dimensional gluon field. Schemes for doing this with the minimum amount of qubits are desirable. A practical digitization for SU(3) gauge theories via its discrete subgroup S(1080) has been shown to allow classical simulations down to a=0.08 fm and reproduce thermal and glueball spectrum using modified and improved actions. Together with primitive gates and improved Hamiltonians for non-abelian gauge theories, the time is approaching where more realistic quantum resource estimates will become possible.
Open quantum systems are good models of many interesting physical systems. Non-Hermitian Hamiltonians are known to describe, or at least approximate some of these open quantum systems well. Recently, there has been an increase in interest in quantum algorithms for simulating such Hamiltonians, such as the Quantum Imaginary Time Evolution algorithm, and other ones based on trace preserving quantum operations, using an enlarged Hilbert space. The focus of our work is on testing the near-term applicability of some of these NISQ-era algorithms on real, noisy quantum hardware. We will look at the 1D quantum Ising model in complex parameter space. Such models have a rich phase-structure in the complex plane and studying them would allow us to explore critical regions such as Lee-Yang edges and Fisher zeros. We will also discuss the applicability of these algorithms for ground-state preparation.
We determine the gradient flow scale $t_0$ at the physical point with an overall uncertainty of around $0.5\%$ using the $\Xi$ baryon mass as input. We utilise 47 CLS ensembles generated with $N_f=2+1$ non-perturbatively $O(a)$ improved Wilson dynamical fermions comprising six lattice spacings in the range $a=0.04-0.1$ fm, spatial volumes with $LM_\pi>4$ and pion masses ranging from around 420 MeV down to the physical point. Combined quark mass, continuum limit, and finite volume fits are performed to the baryon octet masses along three trajectories in the quark mass plane, which tightly constrains the mass of the $\Xi$ baryon at the physical point. The strange and light quark sigma terms are determined for all octet baryons from the dependence of the baryon masses on the renormalized quark masses.
The OpenLat initiative presents its results of lattice QCD simulations using Stabilized Wilson Fermions (SWF) using 2+1 quark flavors. Focusing on the $SU(3)$ flavor symmetric point $m_\pi=m_K=412$ MeV, four different lattice spacings ($a = 0.064,0.077,0.094,0.12$ fm) are used to perform the continuum limit to study the cutoff effects.
We present the results of basic gauge observables and of hadron masses, and their statistical properties like the autocorrelation. For the determination of the hadron masses we used a Bayesian analysis framework with constraints and model averaging to obtain the most unbiased results as possible.
We compute the static energy of a quark-antiquark pair in lattice QCD using a method which is not based on Wilson loops, but where the trial states are formed by eigenvector components of the covariant lattice Laplace operator. The computational effort of this method is significantly lower than the standard Wilson loop calculation, when computing the static potential not only for on-axis, but also for many off-axis quark-antiquark separations, i.e., when a fine spatial resolution is required, e.g., for string breaking calculations. We further improve the signal by using multiple eigenvector pairs, weighted with Gaussian profile functions of the eigenvalues, providing a basis for a generalized eigenvalue problem (GEVP), as it was recently introduced to improve distillation in meson spectroscopy. We show results from the new method for the static potential with dynamical fermions and demonstrate its efficiency compared to traditional Wilson loop calculations.
We present SU(3) lattice Yang-Mills data for hybrid static potentials from five ensembles with different small lattice spacings and the corresponding parametrizations for quark-antiquark separations $0.08 \text{ fm} \le r \le 1.12 \text{ fm}$. We remove lattice discretization errors at tree level of perturbation theory and at leading order in $a^2$ as well as the $a$-dependent self-energy. In particular the tree-level improvement of static potentials is discussed in detail and two approaches are compared. The resulting parametrizations are expected to represent continuum results for hybrid static potentials within statistical errors.
We study four-quark systems using lattice QCD, which consist of two heavy antiquarks (either $ \bar{b}\bar{b} $ or $\bar{b}\bar{c}$) and two light quarks (either $ ud $ or $ us $) and search for bound states in these channels. In addition to commonly used local interpolating operators we also employ scattering interpolating operators, which seem to be very important for an accurate extraction of possibly existing bound states as well as low-lying scattering states.
Moreover, we study the overlaps of trial states generated by our interpolating operators and low-lying energy eigenstates to obtain insights regarding the composition of the latter.
We present the leading order mixed-action effect $\Delta_{\rm mix}\equiv m_{\pi,{\rm vs}}^2-\frac{m_{\pi,{\rm vv}}^2+m_{\pi,{\rm ss}}^2}{2}$ using HISQ,
clover or overlap valence fermion actions on the gauge ensembles with kinds of
sea fermion actions among a widely used lattice spacing range $a\in [0.04,0.19]$~fm. The results suggest that $\Delta_{\rm mix}$ decreases on the forth order
of the lattice spacing on the gauge ensembles with the dynamical chiral sea
fermion, likes the Domain wall or HISQ fermion. When the clover sea fermion
action which has explicit chiral symmetry breaking is used in the ensemble, $\Delta_{\rm mix}$ can be much larger regardless of the valence fermion action
used.
We present the current status of our analyis of nucleon structure observables including isovector charges and twist-2 matrix elements as well as the nucleon mass. Results are computed on a large set of CLS $N_f=2+1$ gauge ensembles with $M_\pi\approx 0.130\mathrm{MeV} \ldots 350\mathrm{MeV}$, four values of the lattice spacing $a\approx0.05\mathrm{fm}\ldots0.09\mathrm{fm}$ and covering a large range of physical volumes. Compared to the results presented at last year's conference we have added data on a very fine and large box at small light quark mass ($T\times L^3 =192\times 96^3$, $M_\pi=172\mathrm{MeV}$, $a=0.05\mathrm{fm}$). Besides, additional (intermediate) source-sink separations have been computed on the coarser ensembles, further increasing effective statistics and allowing for a more fine-grained control in the treatment of the excited state contamination. Excited states in the nucleon matrix elements are tamed by a simultaneous, two-state fit ansatz using the summation method. The physical extrapolation for all observables including the nucleon mass can be carried out in a global fit.
The study of resonance form factors in lattice QCD is a challenging endeavor. Namely, the infinite-volume limit, $L \rightarrow \infty$, is not well defined in the matrix element (here, $L $ is the spacial extension of a rectangular lattice). This irregular behavior persists even after multiplying each external leg with the pertinent Lellouch-Lüscher factor and stems from the so-called triangle diagram.
In this talk, I shall discuss a novel method to tackle this problem in which the difficulty, related to the presence of the triangle diagram, never emerges. The approach is based on the study of two-particle scattering in a static, spatially periodic external field by using a generalization of the Lüscher method in the presence of such a field. In addition, I shall demonstrate that the resonance form factor in the Breit frame is given by the derivative of a resonance pole position in the complex plane with respect to the coupling constant of the external field. This result is a generalization of the well-known Feynman-Hellmann theorem for the form factor of a stable particle.
We present results of nucleon structure studies measured in 2+1 flavor QCD with the physical light quarks ($m_\pi$ = 135 MeV) in a large spatial extent of about 10 fm. Our calculations are carried out with the PACS10 gauge configurations generated by the PACS Collaboration with the stout-smeared $O(a)$ improved Wilson fermions and Iwasaki gauge action at $\beta$=1.82 and 2.00 corresponding to the lattice spacings of 0.085 fm (coarser) and 0.063 fm (finer) respectively. When we compute nucleon two-point and three-point functions, the all-mode-averaging technique is employed in order to reduce the statistical errors significantly without increasing computational costs. At both lattice spacings, we evaluate nucleon form factors associated with lepton-nucleon elastic scattering measurements.
We report on the recent progress of our analysis into nucleon sigma terms, as well as the singlet axial and tensor nucleon charges.
These are extracted from the CLS gauge configurations, which utilise the Lüscher-Weisz gluon action and the Sheikholeslami-Wohlert fermion action with $N_{f} = 2 + 1$ fermions, with pion masses ranging from the physical value up to 410 MeV, and lattice spacings covering a range between 0.09fm and 0.04fm.
We have employed a variety of methods to determine the necessary correlation functions, including the sequential source method for connected contributions, and the truncated solver method for disconnected contributions.
Extrapolation to the physical point involves leading order discretisation, chiral and finite-volume effects.
We present an analysis of the pion-nucleon sigma term on the CLS ensembles with $N_f = 2 + 1$ flavors of ${\cal O}(a)$-improved Wilson fermions. We perform a chiral interpolation based on ensembles with pion masses ranging from 130 MeV to roughly 350 MeV. The analysis covers four lattice spacings between $a\approx 0.05\mathrm{fm}\ldots0.09\mathrm{fm}$, allowing for an estimate of systematics associated with lattice artefacts.
A lot of progress has been made in the direct determination of nucleon sigma terms. Using similar methods we consider the sigma terms of the other octet baryons as well. These are determined on CLS gauge field ensembles employing the Lüscher-Weisz gluon action and the Sheikholeslami-Wohlert fermion action with $N_\mathrm{f} = 2 + 1$. The ensembles have pion masses ranging from ${410}\,\mathrm{MeV}$ down to the physical value and lattice spacings covering a range between ${0.039}\,\mathrm{fm}$ and ${0.098}\,\mathrm{fm}$. We present a preliminary chiral extrapolation for $a\approx 0.06$ fm along a trajectory where the sum of the sea quark masses is kept constant. We discuss multi-state fits to tackle the well-known problem of excited state contamination comparing ratio and summation method.
Using the high statistics datasets of the HotQCD Collaboration,
generated with the HISQ (2+1)-flavor action for light and strange quarks,
and treating the charm sector in the quenched approximation, we analyze
the second and fourth order cumulants of charm fluctuations and
the correlations of charm with lighter conserved flavor quantum numbers.
We can make use of a factor 100 larger statistics on $N_\tau =8$ lattices
and datasets on lattices with temporal extent $N_\tau=12$ and $16$, which
never have been used in studies of the charm fluctuations. This allows us to
perform the continuum limit for charm fluctuations in the quenched
approximation.
Analyzing correlations of charm fluctuations with baryon number and
electric charge fluctuations we can project onto charmed baryon and
meson correlations and compare results with quark model extended hadron
resonance gas model calculations. We aim at a precise determination of
the dissociation temperature of charmed hadrons and will probe the
sensitivity of the fluctuations observables to the presence of
multiple-charmed baryons.
We discuss results about inhomogeneous chiral phases, i.e. phases where in addition to chiral symmetry also translational symmetry is broken, in the $2+1$-dimensional Gross-Neveu model using the mean-field approximation. The phase diagram of the GN model is presented and the existence of inhomogeneous phases is ruled out. The non-existence of inhomogeneous phases will then be shown for a variety of Four-fermion and Yukawa models in 2+1 dimensions. Numerical minimizations of the effective action using two variants of naive fermions support our results.
The thermal photon emission rate is determined by the spatially transverse, in-medium spectral function of the electromagnetic current. Accessing the spectral function using Euclidean data is, however, a challenging problem due to the ill-posed nature of inverting the Laplace transform. In this contribution, we present the first results about testing the proposal of directly computing the analytic continuation of the retarded correlator at fixed, vanishing virtuality of the photon via the calculation of the appropriate Euclidean correlator at imaginary spatial momentum. We employ two-flavors of dynamical Wilson fermions at a temperature of 250 MeV.
We study the Wilson line correlation function in Coulomb gauge on $96^3 \times$ Nt lattices in 2+1 flavor QCD with physical strange quark and light quark masses corresponding to pion mass of 310 MeV, with the aim to determine the complex potential at non-zero temperature. In our calculation we use HISQ action in fixed scale approach with lattice spacing 1/a=7.1 GeV and Nt=56,36,32,28,24,20, corresponding to temperatures T=127, 197, 221, 253, 296, 354 MeV, respectively. To reduce the noise in the correlator calculations we apply gradient flow. From the analysis of the Wilson line correlation function we conclude that the corresponding spectral function is well described by a dominant peak. The peak position corresponds to the real part of the potential, while the effective width of the peak gives the imaginary part of the potential. We find that the real part of the potential is temperature independent and shows no sign of screening, while the imaginary part shows a strong temperature dependence.
We present a lattice determination of the disconnected contributions to the leading-order hadronic vacuum polarization (HVP) to the muon anomalous magnetic moment in the so-called short and intermediate time-distance windows. We employ gauge ensembles produced by the Extended Twisted Mass Collaboration (ETMC) with $N_f = 2+1+1$ flavors of Wilson twisted-mass clover-improved quarks with masses approximately tuned to their physical value. We take the continuum limit employing three lattice spacings at about 0.08, 0.07 and 0.06 fm.
We present new lattice results of the ETM Collaboration for the SM prediction of the so-called intermediate window (W) and short-distance (SD) contributions to the leading-order hadronic vacuum polarization (HVP) term of the muon anomalous magnetic moment, $a_\mu^{HVP}$.
Our results are obtained from extensive simulations of twisted mass lattice QCD with dynamical up, down, strange and charm quarks at physical mass values, different volumes, and lattice spacings down to $a\sim 0.057~{\rm fm}$. Our determinations of $a_\mu^{HVP}(W)$ and $a_\mu^{HVP}(SD)$ are compared with existing lattice results and with their dispersive counterparts based on experimental data for $e^+ e^−$ annihilation into hadrons. The comparison with dispersive data confirms the tension in $a_\mu^{HVP}(W)$ while showing no significant tension in $a_\mu^{HVP}(SD)$.
With the publication of the new measurement of the anomalous magnetic moment of the muon, the discrepancy between experiment and the data-driven theory prediction has increased to $4.2\sigma$. Recent lattice QCD calculations predict values for the hadronic vacuum polarization contribution that are larger than the data-driven estimates, bringing the Standard Model prediction closer to the experimental measurement. Euclidean time windows in the time-momentum representation of the hadronic vacuum polarization contribution to the muon $g-2$ can help clarify the discrepancy between the phenomenological and lattice predictions.
We present our calculation of the intermediate distance window contribution using $N_\mathrm{f}=2+1$ flavors of O$(a)$ improved Wilson quarks. We employ ensembles at six lattice spacings below $0.1\,$fm and pion masses down to the physical value. We present a detailed study of the continuum limit, using two discretizations of the vector current and two independent sets of improvement coefficients.
Our result at the physical point displays a tension of $3.8\sigma$ with a recent evaluation of the intermediate window based on the data-driven method.
We employ $N_\mathrm{f}=2+1$ flavours of $O(a)$ improved Wilson fermions to determine the hadronic vacuum polarisation functions $\bar{\Pi}^{\gamma\gamma}$ and $\bar{\Pi}^{\gamma Z}$ for Euclidean squared momenta $Q^2\leq 7$ $\mathrm{GeV}^2$. We extrapolate the results to the physical point using ensembles at four lattice spacings and several pion masses, including its physical value. We observe a tension of up to 3.5 standard deviations between our lattice results for $\Delta\alpha_\mathrm{had}^{(5)}(-Q^2)$ and estimates based on the $R$-ratio for space-like momenta in the range $3-7~\mathrm{GeV}^2$.
We discuss the conversion of our lattice result for the hadronic running of the electromagnetic coupling, $\Delta\alpha(-Q^2)$, computed for Euclidean momenta into an estimate for $\Delta\alpha^{(5)}_{\rm had}(M_Z^2)$ using the Euclidean split technique (Adler function approach). We focus specifically on the running in the spacelike regime from momentum scales below $7\,\rm GeV^2$ up to $M_Z^2$, which can be determined either in perturbative QCD or by using dispersion theory and experimentally determined hadronic cross sections. A detailed comparison with results from other lattice calculations and phenomenology is performed. We present an in-depth discussion of the relation to lattice estimates of the hadronic vacuum polarisation contribution to the muon $g-2$ and the implications for global electroweak fits.
We investigate the isospin symmetry breaking effects in the two-flavour Schwinger model. Specifically, we check a prediction by Howard Georgi about automatic fine-tuning effects, i.e. that the isospin breaking is suppressed exponentially in the fermion mass $m_f$.
We study non-invertible defects constructed from dualities in the Cardy-Rabinovici model. The Cardy-Rabinovici model is a four-dimensional $U(1)$ lattice gauge theory with both electrically and magnetically charged particles, which is used as a playground for investigating the dynamics of the Yang-Mills theory with $\theta$ angle. A notable feature of this model is that the conjectured phase diagram has the electromagnetic $SL(2, \mathbb{Z})$ invariance generated by $S$ and $T$ transformations. Although this model does not enjoy the $SL(2, \mathbb{Z})$ duality in a naive way, we notice that the $SL(2, \mathbb{Z})$ transformations can be understood as dualities between the Cardy-Rabinovici model and $\mathbb{Z}_N$ 1-form gauged one. Based on this observation, we construct non-invertible symmetries and determine their non-group-like fusion rules in a formal continuum description of the Cardy-Rabinovici model. Moreover, for some self-dual points, we find that this symmetry turns out to have a mixed gravitational anomaly, which rules out the trivially gapped phase. We also address how the conjectured phase diagram matches this anomaly. This talk is based on arXiv:2204.07440 [hep-th].
In this contribution, we report on our study of the properties of the Wilson flow and on the calculation of the topological susceptibility of $Sp(N_c=2N)$ gauge theories for $N=1,\cdots,4$.
The Wilson flow is shown to scale according to the quadratic Casimir operator of the gauge group, as was already observed for $SU(N_c)$, and the commonly used scales $t_0$ and $w_0$ are obtained
for a large interval of the inverse coupling for each probed value of $N_c$.
The continuum limit of the topological susceptibility is computed and it is conjectured that it scales with the dimension of the group. Our estimates of the topological susceptibility and the
measurements performed in the $SU(N_c)$ Yang-Mills theories by several independent collaborations allow us to test this conjecture and to obtain the universal large-$N$ limit of the rescaled topological susceptibility.
Charged particles in an Abelian Coulomb phase are non-local infra-particles that are surrounded by a cloud of soft photons which extends to infinity. Gauss' law prevents the existence of charged particles in a periodic volume. In a C-periodic volume, which is periodic up to charge conjugation, on the other hand, charged particles can exist. This includes vortices in the 3-d XY-model, magnetic monopoles in 4-d U(1) gauge theory, as well as protons and other charged particles in QCD coupled to QED. In four dimensions non-Abelian charges are confined. Hence, in an infinite volume non-Abelian infra-particles cost an infinite amount of energy. However, in a C-periodic volume non-Abelian infra-particles (whose energy increases linearly with the box size) can indeed exist. Investigating these states holds the promise of deepening our understanding of confinement.
We present our ongoing study of a set of solutions to the $SU(N)$ Yang-Mills equations of motion with fractional topological charge. The configurations are obtained numerically by minimising the action with gradient flow techniques on a torus of size $l^2\times\tilde{l}^2$ (with $\tilde{l}\equiv Nl$) and twisted boundary conditions. We pay special attention to the large $N$ limit, which is taken along a very peculiar sequence, with the number of colours $N$ and the magnetic flux $m$ selected respectively as the n and (n-2) terms of the Fibonacci sequence. We discuss the large $N$ scaling of the solutions and analyze several gauge invariant quantities as the Polyakov and Wilson loops. We also discuss the Hamiltonian limit, with one of the large directions sent to infinity, where these instantons represent tunnelling events between inequivalent pure gauge configurations.
We report recent progress in determining $\varepsilon_K$, the indirect CP violation parameter in the neutral kaon system, calculated using lattice QCD inputs such as $\hat{B}_K$, $\xi_0$, $\xi_2$, $|V_{us}|$, $|V_{cb}|$, and $m_c(m_c)$.
We present the first lattice study of dibaryons with highest bottom number. Utilizing a set of state-of-the-art lattice QCD ensembles and methodologies, we determine ground state of the dibaryon composed of two $\Omega_{bbb}$ baryons. We extract the related scattering amplitude in the $^1S_0$ channel and find a sub threshold pole, which is an unambiguous evidence for a deeply bound $\Omega_{bbb}$-$\Omega_{bbb}$ dibaryon. The binding energy of such a state as dictated by this pole singularity is $-89(_{-12}^{+16})$ MeV. We quantify various systematic uncertainties involved in this determination, including those related to the Coulomb repulsion between the bottom quarks.
The dominant contribution to the long distance region of any meson correlation function comes from the quark propagator's eigenmodes with the smallest eigenvalues. As precision demands for this region increase, methods that offer an exact determination of these low modes have become widely adopted as an effective tool for noise reduction. This work explores the effect of exact low modes on noise reduction for all-to-all as well as traditional wall-to-all propagator techniques. We focus on the connected light quark vector current two-point correlation function, a key observable for the hadronic vacuum polarization contribution to the muon's anomalous magnetic moment. For this analysis we use MILC's 2+1+1 Highly Improved Staggered Quark (HISQ) ensembles at lattice spacings as small as ~0.06 fm at physical mass.
We report recent progress in data analysis on two-point and three-point correlation functions. The data set of measurement is obtained using the Oktay-Kronfeld (OK) action for the heavy quarks (valence quarks) and the HISQ action for the light quarks on MILC HISQ a12m220 ensemble ($N_f = 2+1+1$ flavors).
Inclusive hadronic $\tau$ decays are particularly interesting from the phenomenological since they give access to $V_{ud}$ and $V_{us}$. A long-standing issue is the tension between the $V_{us}$ determinations coming from leptonic and semileptonic kaon decays and the ones obtained from inclusive hadronic $\tau$ decays. To date (as far as we know) the problem has been addressed indirectly by using sum-rule techniques and non-perturbative lattice inputs complemented with perturbative calculations. In this talk we discuss how a recent method for the extraction of smeared spectral densities from euclidean lattice correlators can profitably be used to obtain a direct lattice calculation of inclusive hadronic $\tau$ decay rates. No perturbative inputs are used in our approach as the decay rate is extracted directly from euclidean correlators with two insertions of the relevant hadronic weak current. We also present preliminary numerical results obtained by applying our method to the correlators produced by ETMC with $N_f=2+1+1$ dynamical flavours at $a=0.08\,\mathrm{fm}$ and physical pion masses.
In this study, we explore the distribution of energy-momentum tensor around the static quark and antiquark in SU(3) pure gauge theory at finite temperature. Double extrapolated transverse distributions on mid-plane of the flux tube have been presented for the first time at nonzero temperature. Also, we investigate the spatial distributions of the flux tube on the source plane obtaining from the stress tensor for several $q\bar{q}$ separations and temperatures above and below the critical temperature. The resultant distributions show the detailed structure of the flux tube. Finally, we show the dependence of $F_{stress}$ that is computed from the integral of the stress tensor on the distance between the quark and antiquark on a finer lattice.
We compute the spectra of flux tubes formed between a static quark antiquark pair up to a significant number of excitations and for eight symmetries of the flux tubes, up to $\Delta_u$, using pure gauge $SU(3)$ lattice QCD in 3+1 dimensions. To accomplish this goal, we employ a large set of appropriate operators, an anisotropic tadpole improved action, smearing techniques, and solve a generalized eigenvalue problem. Moreover, we compare our results with the Nambu-Goto string model to evaluate possible tensions which could be a signal for novel phenomena.
A study of heavy-light meson spectroscopy, specifically the excited and exotic spectra of $B$, $B_s$ and $B_c$ is presented. This work was done on an anisotropic lattice of volume $20^3 \times 128$, with (2+1) flavours of dynamical quarks. A large basis of suitable operators was used in a variational analysis to determine finite volume spectra grouped by lattice irrep for each meson. Spin-identified spectra were produced by assigning continuum quantum numbers $J^P$ to each energy level based on the distribution of dominant operator overlaps, up to spin $J=4$. By examining the operator-state overlaps for each energy level in the lattice irreps, candidate states for a lightest hybrid supermultiplet with $J^P = (0,1,2)^-$ were identified in $B$, $B_s$ and $B_c$.
In this work, we calculate the fine tuning of parameters in N = 1 Super- symmetric QCD, discretized on a Euclidean lattice. Specifically, we study the renormalization of the Yukawa (gluino-quark-squark interactions) and the quar- tic (four-squark interactions) couplings. At the quantum level, these interactions suffer from mixing with other operators which have the same transformation properties. We exploit the symmetries of the action, such as charge conjuga- tion and parity, in order to reduce the list of the mixing patterns. To deduce the renormalizations and the mixing coefficients we compute, perturbatively to one-loop and to the lowest order in the lattice spacing, the relevant three-point and four-point Green’s functions using both dimensional and lattice regulariza- tions. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use naive discretization. We obtain analytic expressions for the renormalization and mixing coefficients of the Yukawa couplings; they are func- tions of the number of colors Nc, the gauge parameter α, and the gauge coupling g. Furthermore, preliminary results on the quartic couplings are also presented.
Lattice scales defined using gradient flow are typically very precise, while also easy to calculate. However, different definitions of flows and operators can differ, suggesting possible systematical effects. Using the set of RBC-UKQCD 2+1 flavor domain wall fermion and Iwasaki gauge action ensembles, we explore differences between $\sqrt{t_0}$ and $w_0$ gradient flow scales, compare the impact of different operators to define the energy density, and study the effect of using tree-level improvement for the gradient flow. We find that tree level improvement, traditionally used in step-scaling studies, significantly reduces cut-off effects of the $t_0$ scale. Our findings should be generally applicable to any gauge action.
The Lambda parameter of three flavor QCD is obtained by computing the running of a
renormalized finite volume coupling from hadronic to very high energies where connection with perturbation theory can safely be made. The theory of decoupling allows us to perform the bulk of the computation in pure gauge theory. The missing piece is then an accurate matching of a massive three flavor coupling with the pure gauge one, in the continuum limit of both theories. A big challenge is to control the simultaneous continuum and decoupling limits, especially when chiral symmetry is broken by the discretization.
We refine our previous study of a $\bar{b}\bar{b}ud$ tetraquark resonance with quantum numbers $I(J^P) = 0(1^-)$ which is based on antistatic-antistatic-light-light lattice QCD potentials by including heavy quark spin effects via the mass difference of the $B$ and $B^\ast$ meson.
This leads to a coupled channel Schroedinger equation where the two channels correspond to $BB$ and $B^\ast B^ \ast$ respectively. We explore the existence of a tetraquark resonance by searching for $\mbox{T}$ matrix poles in the complex energy plane and find that the heavy quark spins have a significant impact. We also provide an outlook on a possible future finite volume scattering analysis of the same system carried out in full lattice QCD.
The calculation of disconnected diagram contributions to physical signals is a computationally expensive task in Lattice QCD. To extract the physical signal, the trace of the inverse Lattice Dirac operator, a large sparse matrix, must be stochastically estimated. Because the variance of the stochastic estimator is typically large, variance reduction techniques must be employed. Multilevel Monte Carlo (MLMC) methods reduce the variance of the trace estimator by utilizing a telescoping sequence of estimators. Frequency Splitting is one such method that uses a sequence of inverses of shifted operators to estimate the trace of the inverse lattice Dirac operator, however there is no a priori way to select the shifts that minimize the cost of the multilevel trace estimation. We present a sampling and interpolation scheme that is able to predict the variances associated with Frequency Splitting under displacements of the underlying space time lattice. The interpolation scheme is able to predict the variances to high accuracy and therefore choose shifts that correspond to an approximate minimum of the cost for the trace estimation. We show that Frequency Splitting with the chosen shifts displays significant speedups over multigrid deflation.
Staggered fermions, Karsten-Wilczek (KW) fermions and Borici-Creutz (BC) fermions all retain a remnant chiral symmetry. The price to be payed is that they are doubled, and the resulting taste symmetry is broken by cut-off effects. We measure the size of the taste symmetry violation by determining the low-lying eigenvalues of these fermion operators in the two-dimensional Schwinger model which admits, like QCD, a global topological charge of a given gauge configuration. A first result is that it matters whether the pertinent eigenmode is a would-be zero-mode or a non-topological mode. The intra-pair splittings of the fermion formulations mentioned are found to depend sensitively on the gauge coupling $\beta$. Still, it turns out that it is surprisingly difficult to verify standard Symanzik scaling for these taste-breaking effects.
We report recent progress in data analysis on two-point correlation functions with HYP-smeared staggered fermions using a sequential bayesian fitting method. We present details on data analysis and preliminary results for the meson spectrum.
We describe our implementation of a multigrid solver for Wilson clover fermions, which increases parallelism by solving for multiple right-hand sides (MRHS) simultaneously. The solver is based on Grid and thus runs on all computing architectures supported by the Grid framework. We present detailed benchmarks of the relevant kernels, such as hopping and clover term on the various multigrid levels, intergrid operators, and reductions. The benchmarks were performed on the JUWELS Booster system at FZ Jülich, which is based on Nvidia A100 GPUs. For example, solving a $48^4$ lattice on 16 GPUs, the overall speedup obtained solely from MRHS is about 7x.
QCD sum-rule mass predictions for tetraquark states provide insights on the interpretations and internal structure of experimentally-observed exotic mesons. However, the overwhelming majority of tetraquark QCD sum-rule analyses have been performed at leading order (LO), which raises questions about the underlying theoretical uncertainties from higher-loop corrections. The impact of next-to-leading order (NLO) perturbative effects are systematically examined in scalar ($J^{PC}=0^{++}$) isoscalar light-quark tetraquark systems where comprehensive LO sum-rule analyses have been performed and NLO perturbative corrections to the correlators have previously been calculated. Using the scalar-isoscalar state as a detailed case study to illustrate the differences between LO and NLO analyses, it is shown that NLO effects in individual Laplace sum-rules are numerically significant and have an important role in improving the reliability of the sum-rule analyses by widening the Borel window. However, ratios of sum-rules are found to be less sensitive to NLO effects with the additional advantage of cancelling the anomalous dimension that emerges from the NLO corrections. NLO mass predictions based on these sum-rule ratios are thus remarkably robust despite the slow perturbative convergence of the underlying correlator. The mass prediction $0.52\,\text{GeV}< m_{\sigma}<0.69\,\text{GeV}$ for the lightest scalar-isoscalar $\sigma$ state are in good agreement with the four-quark interpretation of the $f_0(500)$, and the relative coupling strengths of the $f_{0}(980)$ and $f_0(500)$ to the tetraquark current agree with the pattern found in chiral Lagrangian analyses. Effects of the $\sigma$ resonance width are studied for a different models, including resonance shapes inspired by chiral Lagranians.
When comparing the Lagrangian and Hamiltonian formulations of lattice gauge theories, a matching procedure is required to match the parameters and observables between these two formulations. For this, we take the continuum limit in time direction on the Lagrangian side, while keeping the spatial lattice spacing fixed. We study several observables for this nonperturbative matching and compare different ways to take the temporal continuum limit. We apply our approach to the pure U(1) lattice gauge theory in 2+1 dimensions.
The problem of having to reconstruct the decay rates and corresponding amplitudes of the single-exponential components of a noisy multi-exponential signal is common in many other areas of physics and engineering besides lattice field theory, and it can be helpful to study the methods devised and used for that purpose in those contexts in order to get a better handle on the problem of extracting masses and matrix elements from lattice correlators. Here we consider the use of Padé and Padé-Laplace methods, which have found wide use in laser fluorescence spectroscopy and beyond, emphasizing the importance of using robust Padé approximants to avoid spurious poles. To facilitate the accurate evaluation of the Laplace transform required for the Padé-Laplace method, we also present a novel approach to the numerical quadrature of multi-exponential functions.
Wilson-like Dirac operators can be written in the form $D=\gamma_\mu\nabla_\mu-\frac 12 \Delta$. For Wilson fermions the standard two-point derivative $\nabla_\mu^\mathrm{std}$ and 9-point Laplacian $\Delta^\mathrm{std}$ are used. For Brillouin fermions these are replaced by improved discretizations $\nabla_\mu^\mathrm{iso}$ and $\Delta^\mathrm{bri}$ which have 54- and 81-point stencils respectively. We derive the Feynman rules in lattice perturbation theory for the Brillouin action and apply them to the calculation of the improvement coefficient $c_\mathrm{SW}$, which, similar to the Wilson case, has a perturbative expansion of the form $ c_\mathrm{SW}=1+{c_\mathrm{SW}}^{(1)}g_0^2+\mathcal{O}(g_0^4)$.
We find ${c_\mathrm{SW}}^{(1)}_\mathrm{Brillouin}=0.16182118(1)$ compared to ${c_\mathrm{SW}}^{(1)}_\mathrm{Wilson}=0.26858825(1)$.
We report recent progress in data analysis on the correlation functions
of the semileptonic decays $B_{(s)} \to D_{(s)}\ell\nu$ form factors.
The data set of measurement is MILC HISQ ensemble for the light quarks
and Oktay-Kronfeld (OK) action for the heavy quarks: a12m310 ($N_f=2+1+1$ flavor)
We used sequential Bayesian method for the analysis and adopted Newton
method to find better initial guess.
We investigate $I = 0$ bottomonium bound states and resonances in S, P, D and F waves using lattice QCD static-static-light-light potentials. We consider five coupled channels, one confined quarkonium and four open $B^{(*)}\bar{B}^{(*)}$ and $B^{(*)}_s\bar{B}^{(*)}_s$ meson-meson channels and use the Born-Oppenheimer approximation and the emergent wave method to compute poles of the \box{T} matrix. We discuss results for masses and decay widths and compare them to existing experimental results. Moreover, we determine the quarkonium and meson-meson composition of these states to clarify, whether they are ordinary quarkonium or should rather be interpreted as tetraquarks.
In this study, we calculated the effect of self-interacting dark matter on neutron stars. Properties like the mass, radius and the tidal deformability are affected by the presence of dark matter in neutron stars. We show that the Love number can be used to probe the presence and the properties of dark matter inside of neutron stars in future gravitational wave measurements.
Low energy effective models are a useful tool to understand the mechanisms behind physical processes in QCD. They additionally provide ways to probe into regions of the QCD phase diagram that are harder to simulate on the lattice, e.g., small temperature, due to their lower UV cutoff, as well as more direct comparison with functional methods such as fRG. We present here lattice simulations of such an effective model: the quark-meson model. We simulate the theory via Stochastic Quantisation and report on the effects of employing coloured noise, a method that allows control over the momentum scale of the simulation.
We compute the pion and kaon matrix elements with non-local staple-shaped operators using an $N_f=2+1+1$ twisted mass fermion ensemble. The lattice has volume $24^3 * 48$, lattice spacing $a=0.093 \ fm$ and a pion mass of $350 \ MeV$. We employ momentum smearing to improve the signal as we increase the momentum. We explore momenta corresponding to $1.11 \ GeV$ and $2.78 \ GeV$. We also study the mixing pattern of the operators under renormalization and implement an RI/MOM scheme for the non-perturbative renormalization.
Bridge++ is a general-purpose code set for lattice QCD simulations aiming at a readable, extensible, and portable code while keeping practically high performance. The new version 2.0 employs machine-dependent optimization,extended from a fixed data layout in double precision only to a flexible data layout in float/double precision. In this talk, we report the performance on supercomputer Fugaku with Arm A64FX-SVE by Fujitsu.
Modern B-factory experiments, such as Belle II, are able to investigate physics anomalies with some
of the largest datasets ever produced. High luminosity datasets allow for precision measurements of
exclusive B-decays, such as in B → ℓν, which in turn reduce error in calculations of the correponding
CKM matrix element, Vub. This is especially important given the current tension between calculations
of Vub via exclusive decays and inclusive ones, the latter of which could hint towards the presence of
beyond Standard Model processes. While experimental error in Vub can be constrained with larger
datasets, controlling the error contributions from the relevant theory parameters, such as the B(s)
meson decay constant fB(s) , requires novel analysis.
This work will present the continuing efforts from the UKQCD/QCDSF/CSSM groups towards
improving calculations of fB(s) with lattice QCD techniques. This is performed on 2+1 flavour gauge
ensembles, where SU (3)f symmetry is broken in a controlled way. The heavy b-quark is treated with
an anisotropic clover-improved action and tuned to the physical properties of B and Bs mesons. Such
a tuning requires fitting approximately 1600 correlation functions, where individually optimising the
bounds of each fit is no longer feasible, and may lead to systematic fit uncertainties that are difficult to
quantify. A weighted-average across multiple fitting regions is implemented so as to improve practicality
and reduce the potential for bias in the final derivation of fB(s)
High statistics results for quantities like the gradient flow scale, the quark masses, the lower lying baryon spectrum and the baryon octet sigma terms determined on CLS ensembles with $N_f=2+1$ non-perturbatively $O(a)$ improved Wilson dynamical fermions are presented at this conference by the RQCD collaboration. In this contribution, we provide further details of the analysis focusing on systematics associated with the extraction of the lattice data including autocorrelations and the continuum, quark mass and finite volume extrapolations, including the fit forms employed.
Our exploratory study looks for direct access to the hadronic transition
amplitude at the resonance without resorting to the Lüscher formalism.
We study the decay $\Psi(3770)\to D\bar{D}$ by applying partially
twisted boundary conditions to the quenched charm quark,
circumventing possible problems with final state interactions.
If successful, we could compute the dependence of the transition amplitude
on the charm-quark mass, and test the predictions made by
phenomenological quark pair creation models.
Finally, we investigate if and to what extent an explicit extraction
of the excited state $\Psi(3770)$ with help of the GEVP is necessary for this analysis.
Fourier acceleration is a technique used in Hybrid Monte Carlo (HMC) simulations to decrease the autocorrelation length. In the weak interaction limit, Fourier acceleration eliminates the problem of critical slowing down. In this work, we show that by properly tuning the kinetic term in HMC simulations, Fourier acceleration can be applied effectively to a strongly interacting $\phi^4$ theory. We use this algorithm to study the linear sigma model with a large coupling constant and in the spontaneous symmetry breaking phase. We find that Fourier acceleration leads to a reduced autocorrelation length and faster thermalization. In addition, we find a method to make use of the tuned kinetic term in the Fourier-accelerated HMC to further reduce the statistical error of many observables. There are several on-going efforts that try to apply Fourier acceleration to non-Abelian gauge theories like QCD. We think some techniques developed in this work can help the application of Fourier acceleration to QCD.
Collins-Soper (CS) evolution kernel is critical to relate transverse-momentum-dependent parton distribution functions (TMDPDFs) at different scales. When the parton transverse momentum is small, $q_{T}\sim\Lambda_{\mathrm{QCD}}$, the CS kernel is non-perturbative; the determination of the CS kernel in the non-perturbative regime can only be done through experiment or first-principles calculations. Here, preliminary results are presented for a new calculation of the non-perturbative CS kernel using lattice QCD and Large-Momentum Effective Theory. This work improves the control over and reduces the systematic uncertainties compared to previous lattice QCD calculations, and is the first computation at close-to-physical valence and sea pion masses $m_{\pi} \approx 140\textrm{ MeV}$.
Isospin breaking corrections become relevant when aiming to quantify hadronic observables with uncertainties below the percent level. Discretising QED on the lattice is a non-trivial task and several suggested methodologies are available in the literature. Our work uses massive QED, which provides a fully local prescription of QED on the lattice. We present a status update of our ongoing computation of isospin breaking corrections to the spectrum and provide an outlook on future computations.
The automatic fine-tuning of isospin breaking effects by conformal coalescence found by Howard Georgi in the 2-flavor Schwinger model is studied. Numerical investigation of meson mass splitting confirms the exponential suppression of symmetry breaking effects.
The stochastic LapH method has proven to be successful in hadronic calculations. In this work, with charm light spectroscopy in mind, we set up and optimise the LapH procedure limiting ourselves to the evaluation of 2-point mesonic functions. The calculations are performed on CLS ensembles with $N_F = 2 +1$ Wilson-Clover fermions on a $32^3 \times 64$ lattice with open boundary conditions. The results showed are obtained with two software packages, namely Grid/Hadrons and Chroma-laph, and a brief comparison between the two is drawn.
We present first results of a recently started lattice QCD investigation of antiheavy-antiheavy-light-light tetraquark systems including scattering interpolating operators in correlation functions both at the source and at the sink. In particular, we discuss the importance of such scattering interpolating operators for a precise computation of the low-lying energy levels in $\bar b \bar b u d$ and $\bar b \bar b u s$ four-quark systems and corresponding scattering analyses.
Topological freezing is a well known problem in lattice simulations: with shrinking lattice spacing, a transition between topological sectors becomes increasingly improbable, leading to a problematic increase of the autocorrelation time. We present our investigation of metadynamics as a solution for topological freezing in the Schwinger model. Specifically, we take a closer look at the collective variable used in this process and its scaling behaviour. We visualize the effects of topological freezing and how metadynamics helps in that respect. Possible implications for and differences to four-dimensional SU(3) are briefly discussed.
We report on the non-perturbative determination of the improvement coefficient $c_A$ of the axial vector current $A^\mu(x)$ in three-flavour lattice QCD with stabilised Wilson-Clover fermions.
Our computational method exploits the PCAC relation for two different pseudo-scalar states within the Schrödinger functional, which are modelled by altering the spatial structures at the boundaries via properly chosen wavefunctions.
The lattice spacings considered span a range that matches the gauge field ensembles with the stabilised Wilson-Clover action being generated by the OPEN LATtice initiative.
In the same framework and using chiral Ward identities, we also present preliminary results on the renormalisation constants $Z_V$ and $Z_A$ of the vector and axial vector current, respectively.
Stabilized Wilson fermions are a reformulation of Wilson clover fermions that incorporates several numerical stabilizing techniques, but also a local change of the fermion action - the original clover term being replaced with an exponentiated version of it. We intend to apply the stabilized Wilson fermions toolbox to the thermodynamics of QCD, starting on the Nf=3 symmetric line on the Columbia plot, and to compare the results with those obtained with other fermion discretizations.
Computations within theories with complex actions are generally inaccessible by standard numerical techniques as they typically suffer from the numerical sign problem. The complex Langevin (CL) method aims to resolve this problem. In recent years CL has been successfully applied to various problems, e.g. the QCD equation of state for finite chemical potential, and therefore also may represent a promising method in other applications with similar numerical issues. However, CL in its original formulation is numerically unstable and therefore needs to be artificially stabilised to avoid wrong attractors of the distribution function as well as runaway instabilities.
In this work, we study the application of modern stabilisation techniques such as dynamical stabilisation and gauge cooling to CL simulations of real-time SU(2) Yang-Mills theory. We present preliminary numerical results demonstrating that stabilisation techniques may extend the applicability of CL in real-time gauge theories.
This poster reviews the recent HPQCD calcuation of $B_c^+ \to D^0 \ell^+ \nu$ and $B_c^+ \to D_s^+ \ell^+ \ell^- (\nu \bar\nu)$ form factors [Phys. Rev. D 105, 014503 (2022), arXiv:2108.11242]. We comment on prospects for experimental measurement of $B_c^+ \to D^0 \ell^+ \nu$ and implications for CKM matrix elements.
Increasing GPU power across a competitive market of various GPU manufacturers and GPU based supercomputers pushes lattice programmers to develop code usable for multiple APIs. In this poster we showcase SIMULATeQCD, a SImple MUlti-GPU LATtice code for QCD calculations, developed and used by the HotQCD collaboration for large-scale projects on both NVIDIA and AMD GPUs. Our code has been made publicly available on GitHub. We explain our design strategy, give a list of available features and modules, and provide our most recent benchmarks on state-of-the-art supercomputers.
We present preliminary results for the leading strange and charm connected contributions to the hadronic vacuum polarization contribution to the muon's g-2. Measurements are performed on the RC collaboration’s QCD ensembles, with $N_f =3+1$ $O(a)$ improved Wilson fermions and C boundary conditions. The HVP is computed on a single value of the lattice spacing and two lattice volumes. In addition, we compare the signal-to-noise ratio for different lattice discretizations of the vector current.
The I=1/2 and I=3/2 nucleon-pion scattering lengths are determined from a high-statistics computation on a single ensemble of gauge field configurations from the CLS consortium with dynamical up, down, and strange quarks and a pion mass $m_{\pi} = 200{\rm MeV}$. The stochastic-LapH approach to quark propagation enables the efficient computation of all required correlation functions, and a statistical precision is achieved which suggests that controlled computations at the physical point are possible. The I=3/2 p-wave scattering amplitude is also precisely determined, and is consistent with the presence of the narrow $\Delta(1232)$ resonance. Systematic errors due to excited states and the reduced symmetry of the finite volume are addressed, but the extrapolation to the continuum and physical quark masses is left to future work.
The quark-gluon vertex is an important object of QCD. Studies have shown that this quantity can be relevant for the dynamical chiral symmetry breaking pattern in the vacuum. The goal of our project is to obtain the quark-gluon vertex at finite temperature around the deconfinement/chiral transition using the tools provided by lattice QCD. It will be the first time that the quark-gluon vertex at finite temperature is determined using lattice QCD. The propagators, which are a by-product of this project, are also of interest in themselves. In this poster, we will describe our motivations and goals, some details of the determination and report on the status of the calculation.
We give an update on the ongoing effort of the RC$^\star$ collaboration to generate fully dynamical QCD+QED ensembles with C$^\star$ boundary conditions using the openQ$^\star$D code. The simulations were tuned to the U-symmetric point ($m_d=m_s$) with pions at $m_{\pi^{\pm}} \approx 400$ MeV. The splitting of the light mesons is used as one of three tuning observables and fixed to $m_{K^0}-m_{K^{\pm}} \approx 5$ MeV and $m_{K^0}-m_{K^{\pm}} \approx 25$ MeV on ensembles with renormalized electromagnetic coupling $\alpha_R \approx \alpha_{\text{phys.}}$ and $\alpha_R \approx 5.5 \alpha_{\text{phys.}}$ respectively. The tuning of the three independent quark masses to the desired line of constant physics is particularly challenging. In this poster we will define the chosen hadronic renormalization scheme, and we will present a tuning strategy based on a combination of mass reweighting and linear interpolation to explore the parameter space. We will comment on finite-volume effects comparing meson masses on two different volumes with $m_{\pi^\pm}L \approx 3.2$ and $m_{\pi^\pm}L \approx 5.1$. We will also provide some technical details on our updated strategy to calculate the sign of the fermionic Pfaffian, which arises in presence of C$^\star$ boundary conditions in place of the standard fermionic determinant. An overview of the QCD+QED configurations generated by the RC$^\star$ collaboration will be given in the companion talk presented by J. Lücke.
Gauge covariant smearing based on the 3D lattice Laplacian can be used to create extended operators that have better overlap with hadronic ground states. This is often done iteratively. For staggered quarks using two-link parallel transport preserves taste properties. We found that such iterative smearing was taking an inordinate amount of time when done on the CPU, so we have implemented the procedure in QUDA.
Instead of carrying out two consecutive parallel transports between nearest neighbor sites on each smearing iteration, we calculate the product of the two links joining next-to-nearest-neighbor sites once and reuse it for all iterations. This reduces both required floating point operations and communications.
We present the performance of this code on some recent GPUs.
We give an update on our ongoing studies of the light composite scalar in eight-flavor SU(3) gauge theory. The chiral limit of this theory can serve as the strong dynamics input to a number of composite Higgs models. Composite Higgs models of this type naturally produce $S$ and $T$ parameters of the size required to explain the new CDF $W$ mass measurement. We present our improved subtraction scheme for scalar correlators at zero spatial momentum. We compare this scheme with results from moving frames at non-zero spatial momentum where no subtraction is required but an assumption must be made regarding the dispersion relation. We also perform an infinite volume extrapolation. Our analysis includes full implementation of the Bayesian model averaging procedure of Jay-Neil which substantially reduces the systematic uncertainties of our previous published results. We will also present first results for the flavor-singlet scalar decay constant and the flavor non-singlet scalar meson mass and decay constant. Finally, we will present two competing EFT analyses: one assuming the light scalar is a pseudo-dilaton and another assuming the massless theory has a strongly-coupled IR fixed point.
Semileptonic heavy-to-heavy and heavy-to-light $B$ decays are very intriguing transitions since a long-standing tension affects the inclusive and the exclusive determinations of the CKM matrix elements $\vert V_{cb} \vert$ and $\vert V_{ub} \vert$. In the former case, another discrepancy exists between the SM expectations and the measurements of the ratios $R(D^{(*)})$, which are a test of Lepton Flavour Universality (LFU). In both cases, a central role is played by the hadronic Form Factors (FFs) describing these decays. Our goal is to re-examine the $b \to c$ and $b \to u$ quark transitions through the Dispersive Matrix (DM) approach, which is based on the non-perturbative determination of the dispersive bounds. It describes in a model-independent way the FFs in the full kinematical range, starting from existing Lattice QCD data at large momentum transfer. From the DM bands we obtain the new SM expectations $R(D)=0.296(8)$ and $R(D^*)=0.275(8)$, each of which is compatible with the corresponding average of measurements at the $1.3\sigma$ level. The value of $R(D^*)$ corresponds to the use of the recent FNAL LQCD results for the FFs as input for the DM approach. Then, by comparing the DM bands of the FFs with the experiments we obtain $\vert V_{ub}\vert = (3.85 \pm 0.27) \cdot 10^{-3}$ from $B \to \pi$ and $B_s \to K$ decays and $\vert V_{cb}\vert = (41.2 \pm 0.8) \cdot 10^{-3}$ from $B \to D^{(*)}$ and $B_s \to D_s^{(*)}$ decays. These values are compatible with the inclusive ones and with the indirect determinations from the Unitarity Triangle within the $1\sigma$ level.
The structure of hadrons relevant for deep-inelastic scattering are completely characterised by the Compton amplitude. The standard approach in structure function calculations is to utilise the operator product expansion where one computes the local matrix elements. However, it is well established that tackling anything beyond leading-twist presents additional challenges that are not easily overcome; complicating the investigations of hadron structure at a deeper level. Alternatively, it is possible to directly calculate the Compton amplitude by taking advantage of the Feynman-Hellmann approach. By working with the physical amplitude, the intricacies of operator mixing and renormalisation are circumvented. Additionally, higher-twist contributions become more accessible given precise enough data.
In this talk, we focus on the QCDSF/UKQCD Collaboration's advances in calculating the forward Compton amplitude via an implementation of the second-order Feynman-Hellmann theorem. We highlight our progress on investigating the low moments of unpolarised structure functions of the nucleon. We also have a glance at our progress on the polarised and off-forward cases.
Next generation high-precision neutrino scattering experiments have the goal of measuring the as-of-yet unknown parameters governing neutrino oscillation. This effort is hampered by the use of large nuclear targets: secondary interactions within a nucleus can confuse the interpretation of experimental data, leading to ambiguities about the initial neutrino interaction in scattering events. The distribution of energies for neutrino events must instead be inferred from the responses of a sum of dissimilar event topologies. For this reason, precise neutrino cross sections on nucleon targets are of vital importance to the neutrino oscillation experimental program. On the other hand, the necessary experimental data for neutrino scattering with elementary targets are scarce because of the weak interaction cross section, which leads to poorly-constrained nucleon and nuclear cross sections.
Lattice QCD is uniquely positioned to provide the requisite nucleon amplitudes needed to enable high-precision oscillation experiments. In particular, LQCD has the ability to probe axial matrix elements that are challenging to isolate or completely inaccessible to experiments. In this talk, I will discuss some of my work to quantify neutrino cross sections with realistic uncertainty estimates, primarily focusing on neutrino quasielastic scattering and the nucleon axial form factor. I will also outline how the needs of next-generation neutrino oscillation experimental programs can be met with modern dedicated LQCD computations.
We review progress on the lattice QCD calculation of parton structure in the nucleon, specifically that of the gluon. The structure of a hadron is typically described by $x$ dependent distributions, most notably the simplest case of the parton distribution function (PDF). Boosted hadronic matrix elements of operators, which are calculable in lattice QCD, can be related to the PDF indirectly. This relationship is not uniquely defined, both theoretically from how one organizes power corrections and practically due to limitations of lattice QCD calculations. I will present the latest lattice QCD calculations of these matrix elements and describe the practical approaches for relating that data to PDFs.
First-principles calculations of multi-hadron dynamics are a crucial goal for lattice QCD calculations. Significant progress has been achieved in developing, implementing and applying theoretical tools that connect finite-volume quantities to their infinite-volume counterparts. In this talk, I will review some recent theoretical developments and numerical results regarding multi-particle quantities in a finite volume. The focus will be laid on properties of resonances and observables involving nucleons.
I review the recent progresses on lattice calculations of hadron spectroscopy and interactions. The methods to precisely determine the energy eigenstates on lattice and subsequently extract the scattering information have been matured in the last years. After briefly introduce the methodology, I present the new results in the last couple of years, focus will be the results on the exotic hadrons beyond the conventional quark model, such as multi-quark states and glueballs. I will also discuss the existing challenges and future paths.
The International Lattice Data Grid (ILDG) started almost 20 years ago as a global community initiative to enable and coordinate sharing of gauge configurations within the lattice QCD community. We outline the basic ideas of ILDG and explain the urgent need to fully support the meanwhile established FAIR data management practices. We will report on recent activities within the ILDG and on ongoing efforts in migrating to modern technologies.
Using D-Wave's quantum annealer as a computing platform, we study lattice gauge theory with discrete gauge groups. As digitization of continuous gauge groups necessarily involves an approximation of the symmetry, we extend the formalism of previous studies on the annealer to finite, simply reducible gauge groups. As an example we use the dihedral group $D_n$ with $n=3,4$ on a two plaquette ladder for which we provide proof-of-principle calculations of the ground-state and employ the known time evolution formalism with Feynman clock states.
Future quantum computers will enable the study of real-time dynamics of non-perturbative quantum field theories without the introduction of the sign problem. We present ongoing progress on low-dimensional lattice systems which will serve as suitable testbeds for near-term quantum devices. The two systems studied to date are 0+1 dimensional supersymmetric quantum mechanics and the Wess-Zumino model in 1+1 dimensions. In both we comment on whether supersymmetry is dynamically broken for various superpotentials.
We present a tensor-network method for strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. We apply the method to the two-dimensional case and validate it by comparing results for the partition function, the chiral condensate and the baryon density with exact analytical expressions on small lattices up to volumes of $4\times4$. For larger two-dimensional volumes, we present first tensor results for the chiral condensate as a function of the mass and volume, and observe that the chiral symmetry is not broken dynamically in two dimensions. We also present tensor results for the number density as a function of the chemical potential, which hint at a first-order phase transition.
We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for the free energy agrees with the analytical results in weak and strong coupling regimes.
Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining methods that attempt to remove the CDL structure from tensor networks. Such approaches have been shown to be beneficial on two dimensional spin systems. We discuss how to adapt the removal of the CDL structure to tensor networks including fermions, and numerical results that contain some comparisons to the plain TRG, where significant differences are found, will be shown.
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a $q$-state clock model (infinite coupling). In the latter case, a $q$-state clock model can also be defined for non-integer values of $q$. Thus, such a limit can also be considered as an analytic continuation of an ordinary $q$-state clock model to non-integer $q$. The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer $q$, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods. The results may be relevant for configurable Rydberg-atom arrays.
Previous Lattice QCD calculations of nucleon transverse
momentum-dependent parton distributions (TMDs) focused
on the case of transversely polarized nucleons, and thus
did not encompass two leading-twist TMDs associated with
longitudinal polarization, namely, the helicity TMD and
the worm-gear TMD corresponding to transversely polarized
quarks in a longitudinally polarized nucleon. Based on a
definition of TMDs via hadronic matrix elements of quark
bilocal operators containing staple-shaped gauge connections,
TMD observables characterizing the aforementioned two TMDs
are evaluated, utilizing a RBC/UKQCD domain wall fermion
ensemble at the physical pion mass.
We report the first lattice QCD calculation of pion valence quark distribution with next-to-next-to-leading order perturbative matching correction, which is done using two fine lattices with spacings $a=0.04$ fm and $0.06$ fm and valence pion mass $m_\pi=300$ MeV, at boost momentum as large as $2.42$ GeV. As a crucial step to control the systematics, we renormalize the pion valence quasi distribution in the recently proposed hybrid scheme, which features a Wilson-line mass subtraction at large distances in coordinate space, and develop a procedure to match it to the $\overline{\rm MS}$ scheme. We demonstrate that the renormalization and the perturbative matching in Bjorken-$x$ space yield a reliable determination of the valence quark distribution for $x$ in range of 0.03 $\sim$ 0.80 with 5--20 % uncertainties.
We report a state-of-the-art lattice QCD calculation of the isovector quark transversity distribution of the proton in the continuum and physical limit using large-momentum effective theory. The calculation is done at three lattice spacings $a \approx$ {0.085, 0.064, 0.049} fm and various pion masses $m_{\pi} \approx$ {350, 280, 220} MeV, with the proton momenta up to 2.8 GeV. The result is non-perturbatively renormalized in the hybrid scheme with self renormalization which is the only infrared-free approach known so far, and extrapolated to the continuum, physical and infinite momentum limit. We also make a comparison with recent global analyses for the nucleon isovector quark transversity distribution.
We present an exploratory study of the quasi-beam function on a $N_f = 2 + 1 + 1$ twisted mass lattice of size $24^3 \times 48$, with a pion mass of $350$ MeV and of lattice spacing $0.093$ fm. We show preliminary results for longitudinal momentum of up to $1.7$ GeV and transverse separation of up to $0.28$ fm. We also discuss the possible renormalization of the bare matrix element using RI/MOM scheme and outline the next steps in extracting the continuum TMDPDF.
In this talk I will show our calculations of Collins-Soper kernel and soft function on a newly generated 2+1 flavor clover fermion CLS ensemble of size $48^4$ with $a=0.098$ fm. The light sea quark mass corresponds to a pion mass of 333 MeV for this ensemble and the valence quark mass to 662 MeV. We measure the large-momentum-transfer meson form factors and its transverse-momentum-depdent wave functions at momenta up to $P^z=12\frac{2\pi}{L}$. The Collins-Soper kernel and soft function are extracted from them using next-to-leading-order factorization based on large-momentum effective theory. Our results are in good agreement with literature.
We present results for the parton distribution functions (PDFs) of the nucleon at the physical point from lattice QCD utilizing a next-to-next-to-leading order (NNLO) matching. We consider two different strategies in our calculation. The first makes use of the short-distance factorization formalism to extract the first few Mellin moments in a model-independent way. In the second approach, we consider a matching in Bjorken-x space using the recently developed hybrid renormalization scheme.
We study phase structure and critical point of finite-temperature QCD with heavy quarks applying the hopping parameter expansion (HPE). We first study finite-size effects on the critical point on $N_t=4$ lattices with large spatial volumes taking the LO and NLO effects of the HPE, and find that the critical scaling of the Z(2) universality class expected around the critical point of two-flavor QCD is realized when the aspect ratio of the lattice is larger than about 9. This enables us to determine the critical point in the thermodynamic limit with high precisions. By a study of the convergence of the HPE, we confirm that the result of the critical point with the low orders of the HPE is reliable for $N_t=4$, while we need to incorporate higher order effects for $N_t \ge 6$. To extend the study to $N_t \ge 6$ lattices, we then develop a method to take the effects of higher-order terms of the HPE up to a sufficiently high order. We report on the status of our simulations on $N_t \ge 6$ lattices adopting the new method.
We present the latest results from the use of the Backus-Gilbert method for reconstructing the spectra of NRQCD bottomonium mesons using anisotropic FASTSUM ensembles at non-zero temperature. We focus in particular on results from the $\eta_b$, $\Upsilon$ and $\chi_{b1}$ generated from Tikhonov-regularized Backus-Gilbert coefficient sets. We extend previous work on the Laplace shifting theorem as a means of resolution improvement and present new results from its use. We conclude with a discussion of the limitations of the improvement routine and elucidate a connection with Parisi-Lepage statistical scaling.
We report preliminary progress in the calculation of the thermal interquark potential of bottomonium using the HAL QCD method with NRQCD quarks. We exploit the fast Fourier transform algorithm, using a momentum space representation, to efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus obtain the central potential for various temperatures. This work was performed on our anisotropic 2+1 flavour "Generation 2" FASTSUM ensembles.
The heavy quark diffusion coefficient is encoded in the spectral functions of the chromo-electric and the chromo-magnetic correlators, of which the latter describes the T/M contribution. We study these correlators at two different temperatures T=1.5Tc and T=10⁴Tc in the deconfined phase of SU(3) gauge theory. We use gradient flow for noise reduction. We perform both continuum and zero flow time limits to extract the heavy quark diffusion coefficient. Our results imply that the mass suppressed effects in the heavy quark diffusion coefficient are 20% for bottom quarks and 34% for charm quark at T=1.5Tc.
We present a novel approach to nonperturbatively estimate the heavy quark momentum diffusion coefficient, which is a key input for the theoretical description of heavy quarkonium production in heavy ion collisions, and is important for the understanding of the elliptic flow and nuclear suppression factor of heavy flavor hadrons. In the heavy quark limit, this coefficient is encoded in the spectral functions of color-electric and color-magnetic correlators that we calculate on the lattice to high precision by applying gradient flow. For the first time we apply the method to 2+1 flavor ensembles with temperatures between 200-350 MeV. Using our experience from quenched QCD, where we performed a detailed study of the lattice spacing and flow time dependence, we estimate the heavy quark diffusion coefficient using theoretically well-established model fits for the spectral reconstruction.
We present full QCD correlator data and corresponding reconstructed spectral functions in the pseudoscalar channel. Correlators are obtained using clover-improved Wilson fermions on $N_f=2+1$ HISQ lattices. We use gradient flow to check whether it reduces cut-off and mixed action effects. Valence quark masses are tuned to their physical values by comparing the mass spectrum obtained from the lattice QCD with experimental values at each flow time. For the spectral reconstruction, we use models based on perturbative spectral functions from different frequency regions like resummed thermal contributions around the threshold from pNRQCD and vacuum contributions well above the threshold. We show preliminary results of the reconstructed spectral function obtained for the first time in our study for full QCD. In addition, we compare the results with the previous continuum extrapolated results in the quenched approximation.
The renormalization group (RG) $\beta$ function describes the running of the renormalized coupling and connects the ultraviolet and infrared regimes of quantum field theories. Focusing at systems with SU(3) gauge group and fermions in the fundamental representation, we study how the RG $\beta$ function changes from a QCD-like system with $N_f=2$ flavors to a conformal system with $N_f=12$ flavors. Specifically we report on new results for simulations with $N_f=4$, 6, and 8 flavors and compare our findings to existing lattice determinations in the literature as well as perturbative predictions.
Our results are based on gradient flow measurements performed on dynamical gauge field configurations generated using Möbius domain wall fermions and Symanzik gauge action. In the case of $N_f=4$ and 6 flavor our investigations are limited by the emergence of confinement at stronger gauge couplings, whereas $N_f=8$ simulations run into an unphysical bulk phase transition.
We report on numerical results of masses and decay constants of the lightest pseudoscalar, vector and axial vector mesons in $Sp(4)$ lattice gauge theory with three Dirac flavours of fermions in the antisymmetric representation. In addition, we measure the masses of other flavoured mesons in the spin-0 and spin-1 channels, as well as the first excited state of the vector meson. Using the gradient flow method to set a common scale, we attempt to carry out the continuum extrapolation. In this setup, we also compute the masses of the chimera baryons composed of two fundamental and one antisymmetric fermion constituents.
Chimera baryons are an important feature of composite Higgs models, since they play role of top partner in partial top compositeness. In the realisation of the mechanism provided by $Sp(4)$ gauge theory, such exotic objects are composed of two fundamental and one antisymmetric fermion constituents. We perform lattice computations for the chimera baryon spectrum both in the quenched approximation and with three dynamical antisymmetric Dirac fermions. Masses of various chimera baryons with different quantum number will be presented, including the one known as the top partner.
Many models of composite dark matter feature a first-order confinement transition in the early universe, which would produce a stochastic background of gravitational waves that will be searched for by future gravitational-wave observatories. I will present work in progress using lattice field theory to predict the properties of such first-order transitions and the resulting spectrum of gravitational waves. Targeting both the thermal as well as the bulk phase transitions of SU(N) Yang-Mills theories, this work employs the Logarithmic Linear Relaxation (LLR) density of states algorithm to avoid long autocorrelations.
In the Holographic Model, the two-point function of Energy-Momentum Tensor (EMT) of the dual QFT can be mapped into the power spectrum of the Cosmic Microwave Background in the gravitational theory. However, the presence of divergent contact terms poses challenges in extracting a renormalized EMT two-point function on the lattice. Using a $\phi^4$ theory of adjoint scalars valued in the su(N) Lie Algebra as a proof-of-concept motivated by Holographic Cosmology, we apply a novel method for filtering out such contact terms by making use of infinitely differentiable "bump" functions which enforce a smooth window that excludes contributions at zero spatial separation. The process effectively removes the local contact terms and allows us to extract the continuum limit behaviour of the renormalized EMT two-point function.
The infrared effective theory of adjoint QCD with one Dirac
flavour is still under debate. The theory could be confining, conformal,
or fermionic fields could become the lightest fields in the IR. Chiral
symmetry seems to be important to answer this question. Previous
investigations have considered Wilson fermions breaking chiral symmetry.
We present here the first results for this theory based on overlap
fermions. These indicate a chiral symmetry breaking and formation of a
fermion condensate. We have also investigated the running coupling of
the theory, which indicates no IR conformality in the energy region we have explored.
We determine the strange quark mass and the isospin averaged up/down quark mass from QCD in the isospin limit. We utilize 46 CLS ensembles generated with $N_f=2+1$ non-perturbatively $O(a)$ improved Wilson fermions comprising six lattice spacings in the range $a=0.1$ fm down to $a=0.04$ fm, spatial volumes with $LM_\pi>4$ and pion masses ranging from around 420 MeV down to the physical point. The quark masses, obtained from axial Ward identities, are fitted simultaneously as functions of the squared pion and kaon masses with all correlations taken into account. The main source of uncertainty at present is from the renormalization of the quark masses and we check the universality of the continuum limit, employing different sets of renormalization constants, obtained from the step scaling function with Schrödinger boundary conditions as well as employing the RI'-SMOM scheme with a subsequent conversion to the $\overline{\rm MS}$ scheme at the three-loop level.
We present preliminary results for a scale setting procedure based on a mixed action strategy, consisting of Wilson twisted mass valence fermions at maximal twist on CLS ensembles with $N_f=2+1$ flavours of O(a)-improved Wilson sea quarks. Once the matching of valence and sea quark masses is performed, universality tests are carried out by comparing the continuum-limit results of the mixed action setup to those of the regularisation based solely on O(a)-improved Wilson fermions. The scale setting uses the pion and kaon decay constants, in units of flow scale $t_0$, obtained from combining computations with the unitary Wilson action and the mixed action. The proper isolation of ground states as well as the continuum-chiral extrapolations are evaluated through model variation techniques. An update on the determination of $t_0$ will be presented.
We report on the determination of light quark masses with three sea-quark flavours based on a mixed action with maximally twisted valence fermions on CLS ensembles with O($a$)-improved Wilson sea quarks. The renormalisation and the renormalisation group running over a wide range of scales are based on existing non-perturbative computations in the Schrodinger functional scheme. The determinations of the light ($u$,$d$) and strange quark masses are based on a combination of the results from the unitary Wilson action with those from the mixed action.
The decay constants of the kaon and pion provide important input into the determination of light CKM matrix elements. Here we present current progress in computing these quantities using the ensembles and analysis techniques employed by the Budapest-Marseille-Wuppertal collaboration in our recent determination of $a_{\mu}^{LO,HVP}$. This work will provide input on the current 2.6-sigma tension between lattice-based determinations and CKM unitarity. It also serves as a cross-check of the many aspects of the analysis shared with our $a_{\mu}^{LO,HVP}$ determination. While this analysis includes considerably more statistics than previous lattice calculations, the total errors on the ratio $f_K/f_\pi$ are similar to the best existing determinations. This is a result of our comprehensive and conservative approach to systematic error estimation, which was developed for $a_{\mu}^{LO,HVP}$.
We present our calculation of radiative correction to the pion and nucleon decay given by the $\gamma W$ box contribution and needed for the determination of $V_{ud}$.
The pion box contribution is computed on five 2+1+1-flavor HISQ ensembles using with Clover action.
The preliminary nucleon box contribution is being analyzed on one ensemble.
In both contributions, the loop momentum is integrated with discrete sums.
We report on our first set of results for charm physics, using a mixed-action setup with maximally twisted valence fermions on CLS $N_f = 2+1$ ensembles. This setup avoids the need of improvement coefficients to subtract ${\rm O}(am_c)$ effects. The charm quark mass, $D$ and $D_s$ decay constants, and some charmonium observables are computed on a subset of CLS ensembles, which allows to take the continuum limit and extrapolate to the physical pion mass, and assess the scaling properties. Special attention is paid to the implementation of techniques to deal with systematic uncertainties. Our results show excellent prospects for high-precision computations on the full set of ensembles.
We show how staggered fermions can be coupled to gravity by generalizing them to Kaehler-Dirac fermions. The latter experience a perturbative gravitational anomaly which breaks
a U(1) symmetry down to Z_4. This anomaly is captured exactly by the lattice theory. Furthermore we show that this theory exhibits a second
non-perturbative 't Hooft anomaly which can be seen
by considering propagation on non-orientable spaces. This anomaly can be
cancelled for multiples of two Kaehler-Dirac fields. This observation explains recent work that shows that multiples of two staggered fermions can be gapped without breaking symmetries.
This research aims to analyze the integrability condition of the chiral determinant of 4D overlap fermions and construct lattice chiral gauge theories.
$\quad$ We formulate the integrability condition with 5D and 6D lattice domain wall fermions. Our formulation parallels the recent cobordism classification of the global ‘t Hooft anomaly using the $\eta$-invariant based on the Dai-Freed theorem and the Atiya-Patodi-Singer index theorem in the continuum theory.
$\quad$ The necessary and sufficient condition for constructing a lattice chiral gauge theory comes down to the statement that "$\exp ( 2\pi i \eta ) = 1$ for any gauge configurations satisfying the admissibility condition in 5D lattice space.", where $\exp ( 2\pi i \eta ) $ is defined as the phase of the partition function of the 5D domain wall fermion.
We investigate the Casimir effect for relativistic lattice fermions, such as the naive fermion, Wilson fermion, and overlap fermion with the periodic or antiperiodic boundary condition. We also discuss anomalous behaviors for nonrelativistic particles. We apply our approaches to condensed matter systems described by low-energy effective Hamiltonian of Dirac semimetals such as Cd3As2 and Na3Bi.
Lattice simulations of Yang-Mills theories coupled with $N_f$ flavours of fermions in the adjoint representation provide a way to probe the non-perturbative regime of a plethora of different physical scenarios, such as Supersymmetric Yang-Mills theory to BSM models. Although the large-$N_c$ limit of these theories can give important insight into the strongly coupled regime of these models, the computational cost of standard lattice simulations involving dynamical adjoint fermions forces one to small-$N_c$ gauge groups. In this talk I am going to present how this large-$N_c$ limit is tackled on the lattice by exploiting volume reduction through twisted boundary conditions, which allows one to simulate these theories at high values of $N_c$ such as 289,361. I will emphasise our most recent results on Yang-Mills theory coupled with one Majorana adjoint fermions ($N_f=\frac{1}{2}$), which corresponds to $\mathcal{N}=1$ SUSY Yang-Mills.
Standard lattice formulations of non-relativistic Fermi gases with two spin components suffer from a sign problem in the cases of repulsive contact interactions and attractive contact interactions with spin imbalance. We discuss the nature of this sign problem and the applicability of the complex Langevin method in both cases. For repulsive interactions, we find the results to converge well using adaptive step size scaling and a Gaussian regulator to modify the lattice action. Finally, we present results on density profiles and correlations of a harmonically trapped, one dimensional system in both position and momentum space, which are also directly accessible via cold atoms experiments.
The non-local dependence of the fermion determinant on the gauge field limits our ability of simulating Quantum Chromodynamics on the lattice. Here we present a factorization of the gauge field dependence of the fermion determinant based on an overlapping four-dimensional domain decomposition of the lattice. The resulting action is block-local in the gauge and in the auxiliary bosonic fields. Possible applications are multi-level integration, master field simulations, and more efficient parallelizations of Monte Carlo algorithms and codes.
In this talk we present the first RBC-UKQCD lattice calculation of the leading isospin-breaking corrections to the ratio of leptonic decay rates of kaons and pions into muon and neutrino, $\Gamma(K_{\ell 2})/\Gamma(\pi_{\ell 2})$. This computation is performed using domain wall fermions with close-to-physical (light and strange) quark masses. The QED effects are implemented using a perturbative approach and infrared divergences are regulated according to the QED$_\mathrm{L}$ prescription. We describe the strategy to extract the relevant hadronic matrix elements from Euclidean correlation functions and we discuss the important role of finite volume effects in this calculation.
Analytical techniques to derive the finite-volume dependence of observables calculated in lattice simulations can be used to improve numerical determinations. With the need for (sub-)percent precision in lattice predictions, also isospin-breaking effects have to be considered. When including electromagnetism in the so-called QED$_{\textrm{L}}$ prescription, having good control of the associated finite-volume effects is particularly important, as the scaling with the spatial extent L typically is inverse polynomial. In this talk, we will discuss the finite-size effects in the RBC/UKQCD calculation of radiative corrections to leptonic decays. Particular emphasis will be put on the order $1/L^3$ contribution, where the non-locality of QED$_{\textrm{L}}$ plays an important role.
In lattice calculations including isospin-breaking effects, low-energy Standard Model predictions can be unambiguously obtained providing external inputs to define the quark masses, the QCD scale and the value of the electromagnetic coupling. However, there is phenomenological interest to define an isospin-symmetric value of a given observable, or to define the corrections coming from the strong or electromagnetic isospin-breaking effects separately. This separation is known to be prescription-dependent, and a diversity of such prescriptions is used across the lattice community. Since these quantities are actively used, for example in the context of the muon g-2 or radiative corrections to weak decays, the question of quantifying scheme dependency is relevant. In this talk we discuss a general framework to describe these ambiguities, and how to estimate them using lattice data or effective field theories.
We present the comparison of preliminary results of $D\to\pi$ semileptonic
decays from two related projects: the first one is based on unitary Wilson
fermions, and the second uses valence quarks rotated to maximal twist.
While these projects differ by their goals and strategies, both studies
are performed on CLS $N_f=2+1$ configurations, with similar analysis
techniques. The universality test can then be used as a non-trivial
validation of our calculations, in particular regarding the notoriously
difficult control of excited state contributions to form factors. Finally,
we will discuss the scaling of these two fermionic actions, compared to
their theoretical merits, with a focus on $O(am_c)$ and $O(ap)$ lattice
artefacts.
We present our results for the kaon semileptonic form factors using the two
sets of the PACS10 configuration, whose physical volumes are more than
(10 fm)$^4$ at the physical point. The lattice spacings are 0.063 and 0.085 fm.
The configurations were generated using the Iwasaki gauge action and $N_f=2+1$
stout-smeared nonperturbatively $O(a)$ improved Wilson quark action. From the
momentum transfer dependence of the form factors in the continuum limit, we
evaluate the slope and curvature for the form factors at the zero momentum
transfer. Furthermore, we calculate the phase space factor, which is used
to obtain $|V_{us}|$ through the kaon semileptonic decay. These results are
compared with previous lattice results and experimental values.
Worldline representations were established as a powerful tool for studying bosonic lattice field theories at finite density. For fermions, however, the worldlines still may carry signs that originate from the Dirac algebra and from the Grassmann nature of the fermion fields. We show that a density of states approach can be set up to deal with this remaining sign problem, where finite density is implemented by working with a fixed winding number of the fermion worldlines. We discuss the approach in detail and show first results of a numerical implementation in 2 dimensions.
The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "Rényi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering high-precision studies of Rényi entropies prohibitively expensive.
In this work, we propose a method to overcome the signal-to-noise ratio problem and show some first results for SU(N) in 3 and 4 dimensions.
We develop a method to improve on the statistical errors for higher moments using machine learning techniques. We present here results for the dual representation of the Ising model with an external field, derived via the high temperature expansion.
We compare two ways of measuring the same set of observables via machine learning: the first gives any higher moments but has larger statistical errors, the second provides only two point function but with small statistical errors. We use the decision tree method to train the correlations between the higher moments and the two point function and use the accurate data of these observable as a input data.
Supervised machine learning with a decoder-only CNN architecture is used to interpolate the chiral condensate in QCD simulations with five degenerate quark flavors in the HISQ action. From this a model for the probability distribution of the chiral condensate as function of lattice volume, light quark mass and gauge coupling is obtained. Using the model, first order and crossover regions can be classified, and the boundary between these regions can be marked by a critical mass. An extension of this model to studies of phase transitions in QCD with variable number of flavors is expected to be possible.
Deep generative models such as normalizing flows are suggested as alternatives to standard methods for generating lattice gauge field configurations. Previous studies on normalizing flows demonstrate proof of principle for simple models in two dimensions. However, further studies indicate that the training cost can be, in general, very high for large lattices. The poor scaling traits of current models indicate that moderate-size networks cannot efficiently handle the inherently multi-scale aspects of the problem, especially around critical points. In this talk, we explore current models that lead to poor acceptance rates for large lattices and explain how to use effective field theories as a guide to design models with improved scaling costs. Finally, we discuss alternative ways of handling poor acceptance rates for large lattices.
Many fascinating systems suffer from a severe (complex action) sign problem preventing us from simulating them with Markov Chain Monte Carlo. One promising method to alleviate the sign problem is the transformation towards Lefschetz Thimbles. Unfortunately, this suffers from poor scaling originating in numerically integrating of flow equations and evaluation of an induced Jacobian. In this talk we present a new preliminary Neural Network architecture based on complex-valued affine coupling layers. This network performs such a transformation efficiently, ultimately allowing simulation of systems with a severe sign problem. We test this method within the Hubbard Model at finite chemical potential, modelling strongly correlated electrons on a spatial lattice of ions.
We present results of the x-dependence of the unpolarized gluon PDF for the proton. We use an $N_f = 2+1+1$ ensemble of maximally twisted mass fermions with clover improvement and the Iwasaki improved gluon action. The quark masses are tuned so that the pion mass is $260$ MeV. We use a $32^3 \times 64$ lattice size with a lattice spacing $a=0.093$ fm giving a spatial extent of $3$ fm. We employ the pseudo-distribution approach and obtain the light-cone Ioffe time distribution (ITD) combining data for nucleon momentum boosts up to $1.67$ GeV and Wilson line length, $z$, up to $0.56$ fm. We explore systematic effects such as the dependence on the maximum value of $z$ entering the fits to obtain the ITD. We also study various options to reconstruct the x-dependence of the gluon PDF.
Precise exploration of the partonic structure of the nucleon is one of
the most important aims of high-energy physics. In recent years, it has
become possible to address this topic with first-principle Lattice QCD
investigations. In this talk, we focus on the so-called
pseudo-distribution approach to determine the isovector unpolarized
PDFs. In particular, we employ three lattice spacings to study
discretization effects and extract the distributions in the continuum
limit, at a pion mass of around 370 MeV. Also, for the first time with
pseudo-PDFs, we explore effects of the 2-loop matching from pseudo- to
light-cone distributions.
"We present results on the chiral-even twist-3 quark GPDs for the proton using one ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover term, corresponding to a pion mass of 260 MeV. We employ the quasi-distribution method which relates lattice matrix elements of non-local operators defined in coordinate space to the light-cone distributions in the momentum (x) space. The approach requires momentum-boosted proton states and a matching formalism computed in Large Momentum Effective Theory (LaMET). In our calculation, we use three values of the momentum boost, namely 0.83, 1.25, 1.67 GeV. The GPDs are defined in the symmetric (Breit) frame, which we implement here with 4-vector momentum transfer squared of 0, 0.69, and 1.39 GeV$^2$, all at zero skewness."
The Parton Distribution Functions (PDFs) encode the non-perturbative collinear dynamics of a hadron probed in inclusive and semi-inclusive scattering processes, and hence provide an avenue to address a number of key questions surrounding the structure of hadrons. This talk will summarize recent efforts of the HadStruc Collaboration to map out the leading-twist quark PDFs of the nucleon using Lattice QCD. This effort hinges on the computation of matrix elements of space-like parton bilinears, which factorize, akin to the QCD collinear factorization of hadronic cross sections, in a short-distance regime into the desired PDFs - ideas codified within the pseudo-distribution formalism. By exploiting the distillation spatial smearing paradigm, matrix elements of sufficient statistical quality are obtained such that the leading-twist PDFs and various systematic effects can be simultaneously quantified. Consistency of our obtained PDFs with phenomenological expectations is also explored.
We present a lattice QCD calculation towards determining gluon helicity distribution and how much of the proton’s spin budget is contributed by gluons. We consider matrix elements of bilocal operators composed of two gluon fields that can be used to determine the polarized gluon Ioffe-time distribution and the corresponding parton distribution function. We employ a high-statistics computation using a $32^3\times 64$ lattice ensemble with $358$ MeV pion mass and $0.094$ fm lattice spacing using a combination of numerical techniques previously proven successful for the case of unpolarized gluon distribution. An important outcome of this work is that we find a hint for a nonzero gluon spin contribution to the proton spin from the model-independent extraction of gluon helicity Ioffe-time distribution over a range of Ioffe-time, $\nu\leq 9$.
A major focus of the new Electron-Ion Collider will be the experimental determination of generalised parton distributions (GPDs). I will give an outline of the CSSM/QCDSF collaboration's determination of GPD properties from a lattice calculation of the off-forward Compton amplitude (OFCA). By determining the OFCA, we can access phenomenologically important properties such as scaling and non-leading-twist contributions, and the subtraction function. We calculated the OFCA for soft momentum transfer $t\in[0.3,1.2]$, and determine moments of the helicity-conserving and -flipping amplitudes, which reduce to their respective GPD moments at leading-twist.
The FASTSUM collaboration has developed a comprehensive research programme in thermal QCD using 2+1 flavour, anisotropic ensembles. In this talk, we summarise our recent results including hadron spectrum calculations using our “Generation 2L” ensembles. We will also report on our progress in obtaining anisotropic lattices with a temporal spacing of 17am, half that of our Generation 2L data, which we will use in future studies to reduce systematic effects.
Singly, doubly and triply charmed baryons are investigated at multiple temperatures using the anisotropic FASTSUM 'Generation 2L' ensemble. We discuss the temperature dependence of these baryons' spectrum in both parity channels with a focus on the confining phase. To further qualify the behaviour of these states around the pseudocritical temperature, the parity doubling due to the restoration of chiral symmetry is examined. The addition of heavier 'heavy' quarks and lighter 'light' quarks compared to our previous studies improves our understanding.
We present a strategy to study QCD non-perturbatively on the lattice at very high temperatures. This strategy exploits a non-perturbative, finite-volume, definition of the strong coupling constant to renormalize the theory. As a first application we compute the flavour non-singlet meson screening masses in a wide range of temperature, from $T\sim 1 $ GeV up to $\sim 160 $ GeV with three flavours in the chiral limit of QCD. Our results show very interesting features of the screening spectrum at very high temperatures. On the one hand the mass splitting between the vector and the pseudoscalar screening masses is clearly visible up to the electroweak scale and cannot be explained by the known NLO perturbative result. On the other hand the restoration of chiral symmetry manifests itself through the degeneracy of the pseudoscalar and the scalar channels and of the vector and the axial ones. This degeneracy pattern is the one expected by Ward identities associated with the presence of chiral symmetry.
It is known that contrary to expectations, the order parameter of chiral
symmetry breaking, the Dirac spectral density at zero virtuality, does not
vanish above the critical temperature of QCD. Instead, the spectral density
develops a pronounced peak at zero. We show that the spectral density in the
peak has large violations of the expected volume scaling. This anomalous
scaling and the statistics of these eigenmodes is consistent with them being
produced by mixing instanton and antiinstanton zero modes. Consequently, we
show that a nonvanishing topological susceptibility implies a finite density
of eigenvalues around zero, which can have implications on the restoration of
chiral symmetry above the critical temperature.
The interrelation between quantum anomalies and electromagnetic fields leads to a series of non-dissipative transport effects in QCD. In this work we study anomalous transport phenomena with lattice QCD simulations using improved staggered quarks in the presence of a background magnetic field. In particular, we calculate the conductivities both in the free case and in the interacting case, analysing the dependence of these coefficients with several parameters, such as the temperature and the quark mass.
Gradient flow can be used to describe a Wilsonian renormalization group transformation. In this talk, we use gradient flow to extract running mesonic and baryonic anomalous dimensions for an SU(3) gauge system with $N_f = 10$ fundamental flavors. Our results are important for constructing theories beyond the Standard Model describing fermion mass generation either by partial compositeness or 4-fermion interactions.
The IKKT matrix model in the large-$N$ limit is conjectured to be a non-perturbative definition of the ten-dimensional type IIB superstring theory. Due to the Pfaffian's inherently complex nature upon Euclideanization, the model has a severe sign problem. The phase of the Pfaffian plays a critical role in determining the correct vacuum of the model. In recent years, the complex Langevin method has been proved to tackle the sign problem successfully. In this talk, we discuss our results from the complex Langevin simulations of the Euclidean version of the IKKT model. We investigate the possibility of spontaneous breaking of $SO(10)$ rotational symmetry. The model must be deformed to evade the singular drift problem during complex Langevin simulations. We recover the original model in the vanishing deformation parameters limit. In addition to mass deformations that explicitly break supersymmetry, we introduce supersymmetry-preserving deformations with a Myers term. We conclude that the phase of the Pfaffian indeed induces the spontaneous breaking of the $SO(10)$ rotational symmetry in the Euclidean IKKT matrix model.
Past lattice simulations tentatively suggested that the spectrum of observable particles in BSM theories is qualitatively different than perturbatively expected. We expand on this using a GUT-like toy theory, SU(3) Yang-Mills coupled to a scalar `Higgs' in the fundamental representation. We show the most comprehensive spectroscopy to date, including all channels up to spin 2., and find it indeed in disagreement with perturbative expectations.
The discrepancy can be traced back to nontrivial field-theoretical effects arising from the requirement of gauge invariance. These results still appear to be consistent with a mechanism proposed by Fröhlich, Morchio and Strocchi, giving a possible analytical approach.
Beyond the standard model theories involving early universe first order phase transitions can lead to a gravitational wave background that may be measurable with improved detectors. Thermodynamic observables of the transition, such as the latent heat, determined through lattice simulations can be used to predict the expected signatures from a given theory and constrain physical models. Metastable dynamics around the phase transition make precise determination of these observables difficult and often lead to large uncontrolled numerical errors. In this talk, I will discuss a prototype lattice calculation in which the first order deconfinement transition in the strong Yang-Mills sector of the standard model is analysed using a novel lattice method, the logarithmic linear relaxation method. This method provides a determination of the density of states of the system with exponential error suppression. From this, thermodynamic observables can be reconstructed with a controlled error, providing a promising direction for accurate model predictions.
Computing CP-violating nucleonic matrix elements on the lattice allows one to place theoretical constraints on the couplings of effective interactions related to BSM sources of CP-violation. These interactions are related to local operators that mix under renormalization. Typically, this mixing is parametrized by the only scale available, the lattice spacing, and induces local divergences in the coefficients of lower-dimensional operators, obscuring the continuum limit. The gradient flow has become an attractive method to circumvent this problem. In adopting the flow to define renormalized operators, the renormalization and mixing scales are disentangled, allowing for a clean computation of the corresponding matching (Wilson) coefficients. Perturbative calculations within the gradient flow formalism can be used to fix the high-energy behavior of the matching coefficients, so that the matrix elements are renormalized across a wide range of energy scales. We present results on the renormalization and mixing of the gluon chromoelectric dipole moment (gCEDM) operator to one-loop order in perturbation theory. These include the power-divergent mixing of the gCEDM with the topological charge density and the logarithmic mixing with various dimension-six operators. We also discuss the construction of a basis compatible with the chiral anomaly.
The gradient flow has become a common tool for state-of-the-art lattice
calculations. I will present observations and selected results obtained with the gradient flow.
The gradient flow, which exponentially suppresses ultraviolet field fluctuations and thus removes ultraviolet divergences (up to a multiplicative fermionic wavefunction renormalization), can be used to describe real-space Wilsonian renormalization group transformations and determine the corresponding beta function. We recently proposed a new nonperturbative renormalization scheme for local composite fermionic operators that uses the gradient flow and is amenable to lattice QCD calculations. Here we present nonperturbative results for the beta function and the Lambda parameter in two flavour QCD, along with the nonperturbative running of quark bilinear operators, obtained using our gradient flow scheme.
We present new results for the pure gauge SU(3) static force computed in a novel way on the lattice. We use Wilson loops with a chromoelectric field insertion for measuring the force directly and compare it with the traditional way of performing a numerical derivative on the static potential. Extended Wilson loop calculations have a bad signal-to-noise ratio, and the use of discretized chromo field insertions causes finite extension effects. We extend our method to support the gradient flow algorithm to improve the signal-to-noise ratio and to challenge finite extension issues, which leads to a larger impact on the general usage of operators with chromo field insertions. Furthermore, we show that direct measurement of the static force can be used to extract the strong coupling constant $\alpha_s$ and to perform the scale setting.
Padé approximants are employed in order to study the analytic structure of the four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, are used as fitting functions to lattice data for the propagators, carefully propagating uncertainties due to the fit procedure taking into account all possible correlations. Applying this procedure systematically to the gluon propagator data, we observe the presence of a pair of complex poles at $p^2_{\mathrm{pole}} = (-0.37 \pm 0.05_{\mathrm{stat}} \pm 0.08_{\mathrm{sys}}) \pm (0.66 \pm 0.03_{\mathrm{stat}} \pm 0.02_{\mathrm{sys}}) i \,\, \mathrm{GeV}^2$, where the first error is statistical and the second systematic, and also a zero at the negative real axis of $p^2$, at $p^2_{\mathrm{zero}} = (-2.9 \pm 0.4_{\mathrm{stat}} \pm 0.9_{\mathrm{sys}}) \,\, \mathrm{GeV}^2$. For the ghost propagator, the Padés indicate the existence of the single pole at $p^2 = 0$, as expected. The presence of the pair of complex poles in the gluon propagator, already hinted upon in previous works, is now put into a more firm basis thanks to the model independence and careful error propagation of our procedure.
In this talk, I will revisit the emergence of de Sitter space in Euclidean dynamical triangulations (EDT). Working within the semi-classical approximation, it is possible to relate the lattice parameters entering the simulations to the partition function of Euclidean quantum gravity. We verify that the EDT geometries behave semi-classically, and by making contact with the Hawking-Moss instanton solution for the Euclidean partition function, we show how to extract a value of the renormalized Newton coupling from the simulations. I will discuss new ways to extract the necessary quantities from the lattice configurations and present an updated value for the renormalized Newton coupling.
In Non Destructive Testings (NDT), ultrasonic Time Revesal based Nonlinear Elastic Wave Spectroscopy (TR-NEWS) turned out to be an efficient method. In order to find out anomalies in the convolution of scattered phonetic waves one of which is time reversed (TR) phonon of the other, it is necessary to perform Fourier transforms of signals.
The energy flow of nonlinear waves detected in TR-NEWS has symmetry structure of quaternions, the path of phonetic waves are confined on a $2D$ plane spanned by $e_1, e_2$. The space can be regarded as projected one from the $(2+1)D$ space containing $e_1\wedge e_2$.
In one loop approximation, we consider 7 A type loops which sit on $2D$ plane spanned by $e_1, e_2$, and 13 B type loops which include a pair of path proportional to $e_1\wedge e_2$ and $e_2\wedge e_1$ that connect two $2D$ planes.
We adopt a model of bosonic phonons propagating in Fermi-sea of neutral Weyl spinors which follow the Clifford algebra. Configurations in momentum space is transformed to real position space via Clifford Fourier Transform (CFT).
We propose application of Machine Learning (ML) or Neural Network (NN) technique for the analysis of
optimal weight of 20 kind of topological loops.
Using different observables we test the approach to the continuum limit of several lattice gauge actions. We use lattice spacings in the range that are usually found in typical lattice QCD simulations. As observables we use different flow observables. This allows to check the scaling properties of the different discretizations with high statistical precision.
In this talk we present results on B-meson semileptonic decays using the highly improved staggered quark (HISQ) action for both valence and 2+1+1 sea quarks. The use of the highly improved action, combined with the MILC collaboration's gauge ensembles with lattice spacings down to ~0.03 fm, allows the b quark to be treated with the same discretization as the lighter quarks. The talk will focus on updated results for $B_{(s)} \to D_{(s)}$, $B_{(s)} \to K$, and $B \to \pi$ scalar and vector form factors.
We present new results on semileptonic decays of D-mesons using the highly improved staggered quark (HISQ) action for both valence and 2+1+1 sea quarks. Our calculation uses lattice spacings ranging from 0.12 fm down to 0.042 fm, including several ensembles with physical-mass pions. The focus on the talk will be on the vector and scalar form factors ($f_+$ and $f_0$) for the decays $D\to\pi$, $D\to K$ and $D_s \to K$. Phenomenological applications will be discussed.
We discuss progress towards the RBC & UKQCD collaborations' next generation of measurements of Standard Model direct CP-violation in kaon decays with G-parity boundary conditions, for which we aim to leverage the power of the upcoming exascale computers to perform the continuum limit and thus eliminate this dominant lattice systematic error.
Since our recent publication on direct CP violation and the Delta I = 1/2 rule in $K \to \pi\pi$ decay which was made with G-parity boundary conditions, we have revisited this problem with a conventional lattice setup employing periodic boundary conditions and two lattice spacings to check our previous result and to improve the precision. We show that the physical amplitude, which corresponds to an excited state in this case, can be obtained reliably with the Generalized Eigenvalue Problem (GEVP) method. Not only are periodic boundary conditions cheaper and allow the use of existing ensembles, but they provide a straightforward path to introduce electromagnetism and strong isospin symmetry breaking, which will be needed in the near future. In this talk, we show our preliminary results on $24^3$ and $32^3$ lattices with domain-wall fermions at physical masses and discuss the prospect of the high-precision calculation of $K \to \pi\pi$ decay with periodic boundary conditions.