Domain Wall Fermions Research Articles

Scale setting the Möbius domain wall fermion on gradient-flowed HISQ action using the omega baryon mass and the gradient-flow scales t0 and w0

We report on a subpercent scale determination using the omega baryon mass and gradient-flow methods. The calculations are performed on 22 ensembles of ${N}_{f}=2+1+1$ highly improved, rooted staggered sea-quark configurations generated by the MILC and CalLat Collaborations. The valence quark action used is M\"obius domain wall fermions solved on these configurations after a gradient-flow smearing is applied with a flowtime of ${t}_{\mathrm{gf}}=1$ in lattice units. The ensembles span four lattice spacings in the range $0.06\ensuremath{\lesssim}a\ensuremath{\lesssim}0.15\text{ }\text{ }\mathrm{fm}$, six pion masses in the range $130\ensuremath{\lesssim}{m}_{\ensuremath{\pi}}\ensuremath{\lesssim}400\text{ }\text{ }\mathrm{MeV}$ and multiple lattice volumes. On each ensemble, the gradient-flow scales ${t}_{0}/{a}^{2}$ and ${w}_{0}/a$ and the omega baryon mass $a{m}_{\mathrm{\ensuremath{\Omega}}}$ are computed. The dimensionless product of these quantities is then extrapolated to the continuum and infinite volume limits and interpolated to the physical light, strange and charm quark mass point in the isospin limit, resulting in the determination of $\sqrt{{t}_{0}}=0.1422(14)\text{ }\text{ }\mathrm{fm}$ and ${w}_{0}=0.1709(11)\text{ }\text{ }\mathrm{fm}$ with all sources of statistical and systematic uncertainty accounted for. The dominant uncertainty in both results is the stochastic uncertainty, though for $\sqrt{{t}_{0}}$ there are comparable continuum extrapolation uncertainties. For ${w}_{0}$, there is a clear path for a few-per-mille uncertainty just through improved stochastic precision, as recently obtained by the Budapest-Marseille-Wuppertal Collaboration.

Dislocations under gradient flow and their effect on the renormalized coupling

Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_\text{geom}=\langle F{\tilde F}\rangle /32\pi^2$ definition of the topological charge does not necessarily vanish even when the gauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations) not seen by the fermions may be promoted to instanton-like objects by the gradient flow. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/L\to 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $\beta$ function.

Near-conformal dynamics in a chirally broken system

Composite Higgs models must exhibit very different dynamics from quantum chromodynamics (QCD) regardless whether they describe the Higgs boson as a dilatonlike state or a pseudo-Nambu-Goldstone boson. Large separation of scales and large anomalous dimensions are frequently desired by phenomenological models. Mass-split systems are well-suited for composite Higgs models because they are governed by a conformal fixed point in the ultraviolet but are chirally broken in the infrared. In this work we use lattice field theory calculations with domain wall fermions to investigate a system with four light and six heavy flavors. We demonstrate how a nearby conformal fixed point affects the properties of the four light flavors that exhibit chiral symmetry breaking in the infrared. Specifically we describe hyperscaling of dimensionful physical quantities and determine the corresponding anomalous mass dimension. We obtain $y_m=1+\gamma^*= 1.47(5)$ suggesting that $N_f=10$ lies inside the conformal window. Comparing the low energy spectrum to predictions of dilaton chiral perturbation theory, we observe excellent agreement which supports the expectation that the 4+6 mass-split system exhibits near-conformal dynamics with a relatively light $0^{++}$ isosinglet scalar.

Charmed and ϕ meson decay constants from 2+1-flavor lattice QCD * *Supported by the National Key Research and Development Program of China (2017YFB0203200). This work was partially supported by the National Natural Science Foundation of China (NSFC) (11935017). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the

On a lattice with 2+1-flavor dynamical domain-wall fermions at the physical pion mass, we calculate the decay constants of , , and . The lattice size is , which corresponds to a spatial extension of fm, with a lattice spacing of fm. For the valence light, strange, and charm quarks, we use overlap fermions at several mass points close to their physical values. Our results at the physical point are MeV, MeV, MeV, MeV, and MeV. The couplings of and to the tensor current ( ) can be derived from ratios and , respectively, which are the first lattice quantum chromodynamics (QCD) results. We also obtain ratios and , which reflect the size of heavy quark symmetry breaking in charmed mesons. Ratios and can be taken as a measure of SU(3) flavor symmetry breaking.

Critical behavior in the single flavor Thirring model in 2+1D

Results of a lattice field theory simulation of the single-flavor Thirring model in 2+1 spacetime dimensions are presented. The lattice model is formulated using domain wall fermions as a means to recover the correct U(2) symmetries of the continuum model in the limit where wall separation $L_s\to\infty$. Simulations on $12^3, 16^3\times L_s$, varying self-interaction strength $g^2$ and bare mass $m$ are performed with $L_s = 8, \ldots 48$, and the results for the bilinear condensate $\langle\bar\psi\psi\rangle$ fitted to a model equation of state assuming a U(2)$\to$U(1)$\otimes$U(1) symmetry-breaking phase transition at a critical $g_c^2$. First estimates for $g^{-2}a$ and critical exponents are presented, showing small but significant departures from mean-field values. The results confirm that a symmetry-breaking transition does exist and therefore the critical number of flavors for the Thirring model $N_c > 1$. Results for both condensate and associated susceptibility are also obtained in the broken phase on $16^3\times48$, suggesting that here the $L_s\to\infty$ extrapolation is not yet under control. We also present results obtained with the associated 2+1$d$ truncated overlap operator DOL demonstrating exponential localisation, a necessary condition for the recovery of U(2) global symmetry, but that recovery of the Ginsparg-Wilson condition as $L_s\to\infty$ is extremely slow in the broken phase.

Ratio of strange to u/d momentum fraction in disconnected insertions

The ratio of the strange quark momentum fraction $\langle x\rangle_{s+\bar{s}}$ to that of light quark $u$ or $d$ in disconnected insertions (DI) is calculated on the lattice with overlap fermions on four domain wall fermion ensembles. These ensembles cover three lattice spacings, three volumes and several pion masses including the physical one, from which a global fitting is carried out. A complete nonperturbative renormalization and the mixing between the quark and glue operators are taken into account. We find the ratio to be $\langle x\rangle_{s+\bar{s}}/\langle x\rangle_{u+\bar{u}} ({\rm DI})=0.795(79)(77)$ at $\mu = 2$ GeV in the $\overline{{\rm MS}}$ scheme. This ratio can be used as a constraint to better determine the strange parton distribution especially in the small $x$ region in the global fittings of PDFs when the connected and disconnected sea are fitted and evolved separately, demonstrating a new way that connects lattice calculations with global analyses.

Beauty mesons in Nf=2+1+1+1 lattice QCD with exact chiral symmetry

We present the first study of $N_f=2+1+1+1$ lattice QCD with domain-wall quarks. The $(b, c, s)$ quarks are physical, while the $(u, d)$ quarks are heavier than their physical masses, with the pion mass $ \sim 700 $ MeV. The gauge ensemble is generated by hybrid Monte Carlo simulation with the Wilson gauge action for the gluons, and the optimal domain-wall fermion action for the quarks. Using point-to-point quark propagators, we measure the time-correlation functions of quark-antiquark meson interpolators with quark contents $\bar b b$, $\bar b c$, $\bar b s$, and $ \bar c c$, and obtain the masses of the low-lying mesons. They are in good agreement with the experimental values, plus some predictions which have not been observed in experiments. Moreover, we also determine the masses of $(b, c, s)$ quarks.

Casimir effect for lattice fermions

We propose a definition of the Casimir energy for free lattice fermions. From this definition, we study the Casimir effects for the massless or massive naive fermion, Wilson fermion, and (Möbius) domain-wall fermion in 1+1 dimensional spacetime with the spatial periodic or antiperiodic boundary condition. For the naive fermion, we find an oscillatory behavior of the Casimir energy, which is caused by the difference between odd and even lattice sizes. For the Wilson fermion, in the small lattice size of N≥3, the Casimir energy agrees very well with that of the continuum theory, which suggests that we can control the discretization artifacts for the Casimir effect measured in lattice simulations. We also investigate the dependence on the parameters tunable in Möbius domain-wall fermions. Our findings will be observed both in condensed matter systems and in lattice simulations with a small size.

FK/Fπ from Möbius domain-wall fermions solved on gradient-flowed HISQ ensembles

We report the results of a lattice quantum chromodynamics calculation of ${F}_{K}/{F}_{\ensuremath{\pi}}$ using M\"obius domain-wall fermions computed on gradient-flowed ${N}_{f}=2+1+1$ highly improved staggered quark (HISQ) ensembles. The calculation is performed with five values of the pion mass ranging from $130\ensuremath{\lesssim}{m}_{\ensuremath{\pi}}\ensuremath{\lesssim}400\text{ }\text{ }\mathrm{MeV}$, four lattice spacings of $a\ensuremath{\sim}0.15$, 0.12, 0.09 and 0.06 fm and multiple values of the lattice volume. The interpolation/extrapolation to the physical pion and kaon mass point, the continuum, and infinite volume limits are performed with a variety of different extrapolation functions utilizing both the relevant mixed-action effective field theory expressions as well as discretization-enhanced continuum chiral perturbation theory formulas. We find that the $a\ensuremath{\sim}0.06\text{ }\text{ }\mathrm{fm}$ ensemble is helpful, but not necessary to achieve a subpercent determination of ${F}_{K}/{F}_{\ensuremath{\pi}}$. We also include an estimate of the strong isospin breaking corrections and arrive at a final result of ${F}_{{\stackrel{^}{K}}^{+}}/{F}_{{\stackrel{^}{\ensuremath{\pi}}}^{+}}=1.1942(45)$ with all sources of statistical and systematic uncertainty included. This is consistent with the Flavour Lattice Averaging Group average value, providing an important benchmark for our lattice action. Combining our result with experimental measurements of the pion and kaon leptonic decays leads to a determination of $|{V}_{us}|/|{V}_{ud}|=0.2311(10)$.

The Atiyah–Patodi–Singer Index and Domain-Wall Fermion Dirac Operators

We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.

Towards a semiclassical description of QCD vacuum around Tc

We study the vacuum topology of 2+1 flavor QCD above the chiral crossover transition, at $T \lesssim 1.2~T_c$, on lattices of size $32^3\times 8$. Since overlap fermions have exact chiral symmetry and an index theorem even on a finite lattice, we use them to detect the topological content of gauge fields generated using domain wall fermion discretization for quarks. We further use different periodicity phases along the temporal direction for the valence overlap quarks, which allows us to probe different topological structures present in the gauge field ensembles, through its zero modes. This procedure provides strong evidences that fermion zero-modes can be quantitatively understood to arise due to different species of instanton-dyons. We estimate their relative abundances from the Dirac-eigenvalue density and resolve the so called "topological clusters" via multi-parameter fits to their density, providing therefore an understanding of the interactions between instanton-dyons. The typical separation between dyons we obtain, is $\sim 0.3$ fm. Surprisingly, it emerges out from this study that a semi-classical description of the fermionic zero modes in the QCD vacuum is quite accurate just above $T_c$.

Effective charge from lattice QCD * * Our calculations benefited from the following resources: CINES, GENCI, IDRIS (Project ID 52271); and the IN2P3 Computing Facility. Work supported by National Natural ScienceFoundation of China (11805097), Jiangsu Provincial Natural Science Foundation of China (BK20180323). Jiangsu Province Hundred Talents Plan for Professionals; Generalitat Valenciana, under grant Prometeo/2019/087; Spanish Ministry of Economy and Competitiveness

Using lattice configurations for quantum ​​​​​chromodynamics (QCD) generated with three domain-wall fermions at a physical pion mass, we obtain a parameter-free prediction of QCD’s renormalisation-group-invariant process-independent effective charge, . Owing to the dynamical breaking of scale invariance, evident in the emergence of a gluon mass-scale, GeV, this coupling saturates at infrared momenta: . Amongst other things: is almost identical to the process-dependent (PD) effective charge defined via the Bjorken sum rule; and also that PD charge which, employed in the one-loop evolution equations, delivers agreement between pion parton distribution functions computed at the hadronic scale and experiment. The diversity of unifying roles played by suggests that it is a strong candidate for that object which represents the interaction strength in QCD at any given momentum scale; and its properties support a conclusion that QCD is a mathematically well-defined quantum field theory in four dimensions.

Gradient flow step-scaling function for SU(3) with ten fundamental flavors

We calculate the step-scaling function, the lattice analog of the renormalization group $\ensuremath{\beta}$-function, for an SU(3) gauge theory with ten fundamental flavors. We present a detailed analysis including the study of systematic effects of our extensive data set generated with ten dynamical flavors using the Symanzik gauge action and three times stout smeared M\"obius domain wall fermions. Using up to ${32}^{4}$ volumes, we calculate renormalized couplings for different gradient flow schemes and determine the step-scaling $\ensuremath{\beta}$ function for a scale change $s=2$ on up to five different lattice volume pairs. In an accompanying paper we discuss that gradient flow can promote lattice dislocations to instantonlike objects, introducing nonperturbative lattice artifacts to the step-scaling function. Motivated by the observation that Wilson flow sufficiently suppresses these artifacts, we choose Wilson flow with the Symanzik operator as our preferred analysis. We study systematic effects by calculating the step-scaling function based on alternative flows (Zeuthen or Symanzik), alternative operators (Wilson plaquette, clover), and also explore the effects of the perturbative tree-level improvement. Further we investigate the effects due to the finite value of ${L}_{s}$.

First-Principles Calculation of Electroweak Box Diagrams from Lattice QCD.

We present the first realistic lattice QCD calculation of the γW-box diagrams relevant for beta decays. The nonperturbative low-momentum integral of the γW loop is calculated using a lattice QCD simulation, complemented by the perturbative QCD result at high momenta. Using the pion semileptonic decay as an example, we demonstrate the feasibility of the method. By using domain wall fermions at the physical pion mass with multiple lattice spacings and volumes, we obtain the axial γW-box correction to the semileptonic pion decay, □_{γW}^{VA}|_{π}=2.830(11)_{stat}(26)_{syst}×10^{-3}, with the total uncertainty controlled at the level of ∼1%. This study sheds light on the first-principles computation of the γW-box correction to the neutron decay, which plays a decisive role in the determination of |V_{ud}|.

Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD.

We report the first result for the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment with all errors systematically controlled. Several ensembles using 2+1 flavors of physical mass Möbius domain-wall fermions, generated by the RBC and UKQCD collaborations, are employed to take the continuum and infinite volume limits of finite volume lattice QED+QCD. We find a_{μ}^{HLbL}=7.87(3.06)_{stat}(1.77)_{sys}×10^{-10}. Our value is consistent with previous model results and leaves little room for this notoriously difficult hadronic contribution to explain the difference between the standard model and the BNL experiment.

The Atiyah–Patodi–Singer index on a lattice

Abstract We propose a nonperturbative formulation of the Atiyah–Patodi–Singer (APS) index in lattice gauge theory in four dimensions, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.

Roper state from overlap fermions

The Roper state is extracted with valence overlap fermions on a $2+1$-flavor domain-wall fermion lattice (spacing $a = 0.114$ fm and $m_{\pi} = 330$ MeV) using both the Sequential Empirical Bayes (SEB) method and the variational method. The results are consistent, provided that a large smearing-size interpolation operator is included in the variational calculation to have better overlap with the lowest radial excitation. Similar calculations carried out for an anisotropic clover lattice with similar parameters find the Roper $\approx 280$ MeV higher than that of the overlap fermion. The fact that the prediction of the Roper state by overlap fermions is consistently lower than those of clover fermions, chirally improved fermions, and twisted-mass fermions over a wide range of pion masses has been dubbed a "Roper puzzle." To understand the origin of this difference, we study the hairpin $Z$-diagram in the isovector scalar meson ($a_0$) correlator in the quenched approximation. Comparing the $a_0$ correlators for clover and overlap fermions, at a pion mass of 290 MeV, we find that the spectral weight of the ghost state with clover fermions is smaller than that of the overlap at $a = 0.12$ fm and $0.09$ fm, whereas the whole $a_0$ correlators of clover and overlap at $a = 0.06$ fm coincide within errors. This suggests that chiral symmetry is restored for clover at $a \le 0.06$ fm and that the Roper should come down at and below this $a$. We conclude that this work supports a resolution of the "Roper puzzle" due to $Z$-graph type chiral dynamics. This entails coupling to higher components in the Fock space (e.g. $N\pi$, $N\pi\pi$ states) to induce the effective flavor-spin interaction between quarks as prescribed in the chiral quark model, resulting in the parity-reversal pattern as observed in the experimental excited states of $N, \Delta$ and $\Lambda$.

Nucleon mass and isovector couplings in 2+1 -flavor dynamical domain-wall lattice QCD near physical mass

We report nucleon mass, isovector vector and axial-vector charges, and tensor and scalar couplings, calculated using two recent 2+1-flavor dynamical domain-wall fermions lattice-QCD ensembles generated jointly by the RIKEN-BNL-Columbia and UKQCD collaborations. These ensembles were generated with Iwasaki $\times$ dislocation-suppressing-determinant-ratio gauge action at inverse lattice spacing of 1.378(7) GeV and pion mass values of 249.4(3) and 172.3(3) MeV. The nucleon mass extrapolates to a value $m_N = 0.950(5)$ GeV at physical point. The isovector vector charge renormalizes to unity in the chiral limit, narrowly constraining excited-state contamination in the calculation. The ratio of the isovector axial-vector to vector charges shows a deficit of about ten percent. The tensor coupling no longer depends on mass and extrapolates to 1.04(5) in $\overline {\rm MS}$ 2-GeV renormalization at physical point, in a good agreement with the value obtained at the lightest mass in our previous calculations and other calculations that followed. The scalar charge, though noisier, does not show mass dependence and is in agreement with other calculations.

Gluon propagator and three-gluon vertex with dynamical quarks

We present a detailed analysis of the kinetic and mass terms associated with the Landau gauge gluon propagator in the presence of dynamical quarks, and a comprehensive dynamical study of certain special kinematic limits of the three-gluon vertex. Our approach capitalizes on results from recent lattice simulations with (2+1) domain wall fermions, a novel nonlinear treatment of the gluon mass equation, and the nonperturbative reconstruction of the longitudinal three-gluon vertex from its fundamental Slavnov–Taylor identities. Particular emphasis is placed on the persistence of the suppression displayed by certain combinations of the vertex form factors at intermediate and low momenta, already known from numerous pure Yang–Mills studies. One of our central findings is that the inclusion of dynamical quarks moderates the intensity of this phenomenon only mildly, leaving the asymptotic low-momentum behavior unaltered, but displaces the characteristic “zero crossing” deeper into the infrared region. In addition, the effect of the three-gluon vertex is explored at the level of the effective gauge coupling, whose size is considerably reduced with respect to its counterpart obtained from the ghost-gluon vertex. The main upshot of the above considerations is the further confirmation of the tightly interwoven dynamics between the two- and three-point sectors of QCD.

Gradient flow step-scaling function for SU(3) with twelve flavors

We calculate the step scaling function, the lattice analog of the renormalization group $\beta$-function, for an SU(3) gauge theory with twelve flavors. The gauge coupling of this system runs very slowly, which is reflected in a small step scaling function, making numerical simulations particularly challenging. We present a detailed analysis including the study of systematic effects of our extensive data set generated with twelve dynamical flavors using the Symanzik gauge action and three times stout smeared M\"obius domain wall fermions. Using up to $32^4$ volumes, we calculate renormalized couplings for different gradient flow schemes and determine the step-scaling $\beta$ function for a scale change $s=2$ on up to five different lattice volume pairs. Our preferred analysis is fully $O(a^2)$ Symanzik improved and uses Zeuthen flow combined with the Symanzik operator. We find an infrared fixed point within the range $5.2 \le g_c^2 \le 6.4$ in the $c=0.250$ finite volume gradient flow scheme. We account for systematic effects by calculating the step-scaling function based on alternative flows (Wilson or Symanzik) as well as operators (Wilson plaquette, clover) and also explore the effects of the perturbative tree-level improvement.