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arxiv: 2603.27608 · v3 · submitted 2026-03-29 · ✦ hep-lat

Recognition: 2 theorem links

· Lean Theorem

Domain wall fermions

Thomas Blum , Yigal Shamir
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:22 UTC · model grok-4.3

classification ✦ hep-lat
keywords domain wall fermionslattice QCDchiral symmetryGinsparg-Wilson relationWilson kernelMöbius fermionsresidual chiral breaking
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The pith

Domain wall fermions recover exact chiral symmetry when the fifth dimension becomes infinitely long.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Domain wall fermions are a lattice formulation for quarks that aims to preserve chiral symmetry. The paper proves that this symmetry is recovered exactly in the limit of infinite length for the auxiliary fifth direction. It derives the effective four-dimensional operator that results and shows it satisfies the Ginsparg-Wilson relation. For any practical finite fifth-direction size, some residual chiral symmetry breaking remains, and its size is governed by spectral properties of the Wilson kernel. This framework supports controlled lattice QCD calculations while approaching the ideal chiral limit.

Core claim

The domain wall fermion formulation recovers exact chiral symmetry in the limit of an infinite fifth direction. The effective four-dimensional operator obtained in this limit satisfies the Ginsparg-Wilson relation. For finite extent of the fifth direction, residual chiral symmetry breaking occurs and is controlled by the spectral features of the Wilson kernel. Various improvements, including Möbius fermions, are discussed to reduce this residual breaking.

What carries the argument

The domain wall fermion operator constructed with a finite fifth dimension, where the Wilson kernel in the extra dimension sets the rate at which chiral symmetry is approached.

If this is right

  • Exact chiral symmetry holds in the infinite fifth-direction limit.
  • The effective four-dimensional operator obeys the Ginsparg-Wilson relation.
  • Residual breaking for finite fifth-direction size is set by the Wilson kernel spectrum.
  • Möbius and other improvements reduce the residual effects for practical simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formulation enables accurate lattice computations of chiral-sensitive quantities such as meson masses and decay constants.
  • Tuning the kernel spectrum offers a practical route to smaller residual breaking without increasing computational cost.
  • Similar extra-dimension constructions may extend to other lattice fermion actions facing chiral symmetry issues.

Load-bearing premise

The spectral properties of the Wilson kernel determine the size of residual chiral symmetry breaking for any finite fifth-direction extent.

What would settle it

A lattice calculation in which the residual chiral symmetry breaking fails to decrease as the fifth dimension is lengthened, contrary to the spectral predictions of the Wilson kernel, would falsify the recovery mechanism.

Figures

Figures reproduced from arXiv: 2603.27608 by Thomas Blum, Yigal Shamir.

Figure 1
Figure 1. Figure 1: The triangle anomaly. The vertex on the left represents the di [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tadpole correction in one-loop lattice perturbation theory. [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Setting sun diagram, the dominant contribution to the quantum [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mixing of J a 5q with J a 5 , or J5q with J5. function. The quantum broadening of the wave function occurs because the virtual fermion propagating inside the diagram can have arbitrary momentum (with a match￾ing momentum carried by the gauge field). Hence, the least damped propagation in the fifth direction is always controlled by the maximum of e −α(p) over the Brillouin zone. More generally, in an arbitr… view at source ↗
Figure 5
Figure 5. Figure 5: Mixing of J5q with J5 only. Consider first the mixing of J a 5q with J a 5 for finite N5. As we will shortly see, this mixing arises at the one loop level from the diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic phase diagram of lattice QCD with two dynamical [PITH_FULL_IMAGE:figures/full_fig_p041_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: How the spectrum of HW changes as one adds to the gauge field (top to bottom) one dislocation, many dislocations, and a random ensemble of dislocations. 6.4 Low-lying eigenstates and their spectral density By now we have encountered several important features of domain wall fermions as well as overlap fermions which, one way or another, are controlled by the low￾lying spectrum of the Wilson kernel HW or of… view at source ↗
Figure 7
Figure 7. Figure 7: A dislocation is a small region of the lattice where many gauge links are [PITH_FULL_IMAGE:figures/full_fig_p055_7.png] view at source ↗
read the original abstract

We introduce the formulation of domain wall fermions in the context of lattice QCD. We prove the recovery of exact chiral symmetry in the limit of an infinite fifth direction, and derive the effective four-dimensional operator satisfying the Ginsparg-Wilson relation obtained in this limit. We discuss the residual breaking of chiral symmetry for finite extent of the fifth direction, and how it is affected by spectral features of the Wilson kernel. We also discuss various improvements of domain wall fermions including notably M\"obius fermions. These notes are a chapter contributed to the on-line book ``Lattice QCD at 50 years'' (LQCD@50).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces the formulation of domain wall fermions in lattice QCD. It proves the recovery of exact chiral symmetry in the limit of infinite fifth-direction extent, derives the effective four-dimensional operator satisfying the Ginsparg-Wilson relation in this limit, discusses residual chiral symmetry breaking for finite fifth-direction length as controlled by the spectrum of the Wilson kernel, and covers improvements including Möbius fermions. These notes constitute a contributed chapter to the online book 'Lattice QCD at 50 years'.

Significance. If the derivations hold, the paper supplies a clear pedagogical exposition of a standard lattice fermion formulation that exactly preserves chiral symmetry in the infinite-Ls limit and reduces to the overlap operator obeying the Ginsparg-Wilson relation. This is valuable for the historical and technical overview in the target book, particularly the spectral analysis of residual breaking and the treatment of Möbius improvements, which offer practical guidance for reducing discretization effects in modern lattice QCD simulations.

major comments (1)
  1. [Proof of chiral symmetry recovery (near abstract claim)] The central proof of exact chiral symmetry recovery in the infinite-Ls limit rests on the standard properties of the Wilson kernel and the Ginsparg-Wilson relation drawn from prior literature. The manuscript should explicitly state the spectral assumptions on the Wilson kernel (e.g., gap away from zero) that guarantee the limit exists uniformly for all gauge configurations, as this is the load-bearing step for the claim.
minor comments (3)
  1. [Residual breaking section] The discussion of residual breaking for finite Ls would benefit from a concrete numerical example or plot illustrating how the lowest eigenvalues of the Wilson kernel set the size of the breaking, to make the spectral control claim more tangible for readers.
  2. [Improvements section] Ensure the Möbius fermion improvement is given its own subsection with explicit equations for the modified kernel or domain-wall height parameter, as the abstract mentions it but the treatment appears brief.
  3. [Introduction and derivations] Add or verify citations to the foundational works (Kaplan 1992, Shamir 1993, Furman-Neuberger 1995) at the points where the Wilson kernel and Ginsparg-Wilson relation are first introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central proof of exact chiral symmetry recovery in the infinite-Ls limit rests on the standard properties of the Wilson kernel and the Ginsparg-Wilson relation drawn from prior literature. The manuscript should explicitly state the spectral assumptions on the Wilson kernel (e.g., gap away from zero) that guarantee the limit exists uniformly for all gauge configurations, as this is the load-bearing step for the claim.

    Authors: We agree that the derivation relies on spectral properties of the Wilson kernel. In the revised manuscript we will add an explicit statement of the assumptions: the normalized Wilson-Dirac operator is assumed to have no eigenvalues on the branch cut of the sign function (i.e., spectrum bounded away from zero in the interval [-1,1] after standard rescaling). This guarantees that the fifth-dimensional transfer matrix is well-defined and that the Ls → ∞ limit recovers the overlap operator satisfying the Ginsparg-Wilson relation for each fixed gauge configuration. We will also clarify that the convergence holds pointwise under this per-configuration assumption; a uniform gap over the entire gauge-field space is not required for the formal statement and is not generally present in QCD ensembles. This addition will be placed immediately after the statement of the infinite-Ls limit in the main text. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained from standard definitions

full rationale

The paper proves recovery of exact chiral symmetry as the fifth-direction extent Ls tends to infinity by direct construction of the effective 4D operator from the domain-wall formulation. This operator is shown to reduce to the overlap operator obeying the Ginsparg-Wilson relation. The proof proceeds from the definition of the Wilson kernel and the infinite-Ls limit without fitted parameters, self-referential predictions, or load-bearing self-citations. Residual chiral symmetry breaking for finite Ls is controlled by the spectrum of the Wilson kernel via standard spectral analysis. No step reduces the central claim to its own inputs by construction; the derivation is independent and matches established results in lattice QCD.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper reviews a standard construction that relies on the Wilson Dirac operator and the Ginsparg-Wilson relation already present in the prior literature; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Wilson kernel is a valid lattice Dirac operator whose spectrum controls residual chiral symmetry breaking.
    Invoked when discussing finite fifth-direction effects.
  • standard math The Ginsparg-Wilson relation is the appropriate lattice statement of chiral symmetry.
    Used to characterize the effective 4D operator in the infinite limit.

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Reference graph

Works this paper leans on

92 extracted references · 92 canonical work pages · 61 internal anchors

  1. [1]

    Karsten and J

    L.H. Karsten and J. Smit,Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly,Nucl. Phys. B183(1981) 103

    work page 1981

  2. [2]

    Nielsen and M

    H.B. Nielsen and M. Ninomiya,Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory,Nucl. Phys. B185(1981) 20

    work page 1981

  3. [3]

    Nielsen and M

    H.B. Nielsen and M. Ninomiya,Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,Nucl. Phys. B193(1981) 173

    work page 1981

  4. [4]

    Pelissetto,Lattice Nonlocal Chiral Fermions,Annals Phys.182(1988) 177

    A. Pelissetto,Lattice Nonlocal Chiral Fermions,Annals Phys.182(1988) 177

    work page 1988

  5. [5]

    Karsten,Lattice Fermions in Euclidean Space-time,Phys

    L.H. Karsten,Lattice Fermions in Euclidean Space-time,Phys. Lett. B104 (1981) 315

    work page 1981

  6. [6]

    Lattice Chiral Gauge Theories

    M. Golterman,Lattice chiral gauge theories,Nucl. Phys. B Proc. Suppl.94 (2001) 189 [hep-lat/0011027]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  7. [7]

    Golterman,Staggered fermions,2406.02906

    M. Golterman,Staggered fermions,2406.02906. [8]MILCcollaboration,Nonperturbative QCD Simulations with 2+1 Flavors of Improved Staggered Quarks,Rev. Mod. Phys.82(2010) 1349 [0903.3598]. 49See the companion chapter of theLQCD@50book by DeGrand [15]. 74

    work page arXiv 2010

  8. [8]

    A Method for Simulating Chiral Fermions on the Lattice

    D.B. Kaplan,A Method for simulating chiral fermions on the lattice,Phys. Lett. B288(1992) 342 [hep-lat/9206013]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  9. [9]

    DeGrand and C.E

    T. DeGrand and C.E. Detar,Lattice methods for quantum chromodynamics, World Scientific (2006)

    work page 2006

  10. [10]

    Smit,Introduction to Quantum Fields on a Lattice, vol

    J. Smit,Introduction to Quantum Fields on a Lattice, vol. 15, Cambridge University Press (2003), 10.1017/9781009402705

    work page doi:10.1017/9781009402705 2003

  11. [11]

    Gattringer and C.B

    C. Gattringer and C.B. Lang,Quantum chromodynamics on the lattice, vol. 788, Springer, Berlin (2010), 10.1007/978-3-642-01850-3

    work page doi:10.1007/978-3-642-01850-3 2010

  12. [12]

    Chiral Symmetry and Lattice Fermions

    D.B. Kaplan,Chiral Symmetry and Lattice Fermions, inLes Houches Summer School: Session 93: Modern perspectives in lattice QCD: Quantum field theory and high performance computing, pp. 223–272, 12, 2009 [0912.2560]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  13. [13]

    Exact chiral symmetry, topological charge and related topics

    F. Niedermayer,Exact chiral symmetry, topological charge and related topics, Nucl. Phys. B Proc. Suppl.73(1999) 105 [hep-lat/9810026]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  14. [14]

    DeGrand,The Ginsparg-Wilson relation and overlap fermions,2512.07743

    T. DeGrand,The Ginsparg-Wilson relation and overlap fermions,2512.07743

    work page arXiv

  15. [15]

    Ginsparg and K.G

    P.H. Ginsparg and K.G. Wilson,A Remnant of Chiral Symmetry on the Lattice,Phys. Rev. D25(1982) 2649

    work page 1982

  16. [16]

    Chiral Fermions from Lattice Boundaries

    Y. Shamir,Chiral fermions from lattice boundaries,Nucl. Phys. B406(1993) 90 [hep-lat/9303005]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  17. [17]

    Exactly massless quarks on the lattice

    H. Neuberger,Exactly massless quarks on the lattice,Phys. Lett. B417 (1998) 141 [hep-lat/9707022]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  18. [18]

    Critical Momenta of Lattice Chiral Fermions

    K. Jansen and M. Schmaltz,Critical momenta of lattice chiral fermions,Phys. Lett. B296(1992) 374 [hep-lat/9209002]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  19. [19]

    Axial symmetries in lattice QCD with Kaplan fermions

    V. Furman and Y. Shamir,Axial symmetries in lattice QCD with Kaplan fermions,Nucl. Phys. B439(1995) 54 [hep-lat/9405004]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  20. [20]

    Reducing Chiral Symmetry Violations in Lattice QCD with Domain-Wall Fermions

    Y. Shamir,Reducing chiral symmetry violations in lattice QCD with domain wall fermions,Phys. Rev. D59(1999) 054506 [hep-lat/9807012]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  21. [21]

    L¨ uscher,Construction of a Selfadjoint, Strictly Positive Transfer Matrix for Euclidean Lattice Gauge Theories,Commun

    M. L¨ uscher,Construction of a Selfadjoint, Strictly Positive Transfer Matrix for Euclidean Lattice Gauge Theories,Commun. Math. Phys.54(1977) 283

    work page 1977

  22. [22]

    Chiral Determinant as an Overlap of Two Vacua

    R. Narayanan and H. Neuberger,Chiral determinant as an overlap of two vacua,Nucl. Phys. B412(1994) 574 [hep-lat/9307006]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  23. [23]

    Chiral Fermions on the Lattice

    R. Narayanan and H. Neuberger,Chiral fermions on the lattice,Phys. Rev. Lett.71(1993) 3251 [hep-lat/9308011]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  24. [24]

    Locality properties of Neuberger's lattice Dirac operator

    P. Hernandez, K. Jansen and M. L¨ uscher,Locality properties of Neuberger’s lattice Dirac operator,Nucl. Phys. B552(1999) 363 [hep-lat/9808010]. 75

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  25. [25]

    Locality bound for effective four-dimensional action of domain-wall fermion

    Y. Kikukawa,Locality bound for effective four-dimensional action of domain wall fermion,Nucl. Phys. B584(2000) 511 [hep-lat/9912056]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  26. [26]

    Chern-Simons Currents and Chiral Fermions on the Lattice

    M.F.L. Golterman, K. Jansen and D.B. Kaplan,Chern-Simons currents and chiral fermions on the lattice,Phys. Lett. B301(1993) 219 [hep-lat/9209003]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  27. [27]

    Anomalies and chiral defect fermions

    Y. Shamir,Anomalies and chiral defects fermions,Nucl. Phys. B417(1994) 167 [hep-lat/9310006]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  28. [28]

    Callan, Jr

    C.G. Callan, Jr. and J.A. Harvey,Anomalies and Fermion Zero Modes on Strings and Domain Walls,Nucl. Phys. B250(1985) 427

    work page 1985

  29. [29]

    Vector like gauge theories with almost massless fermions on the lattice

    H. Neuberger,Vector-like gauge theories with almost massless fermions on the lattice,Phys. Rev. D57(1998) 5417 [hep-lat/9710089]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  30. [30]

    Golterman and Y

    M. Golterman and Y. Shamir,Propagator Zeros and Lattice Chiral Gauge Theories,Phys. Rev. Lett.132(2024) 081903 [2311.12790]

    work page arXiv 2024

  31. [31]

    Golterman and Y

    M. Golterman and Y. Shamir,Constraints on the symmetric mass generation paradigm for lattice chiral gauge theories,Phys. Rev. D113(2026) 014503 [2505.20436]

    work page arXiv 2026

  32. [32]

    Low energy effective action of domain-wall fermion and the Ginsparg-Wilson relation

    Y. Kikukawa and T. Noguchi,Low-energy effective action of domain wall fermion and the Ginsparg-Wilson relation,Nucl. Phys. B Proc. Suppl.83 (2000) 630 [hep-lat/9902022]. [34]RBC, UKQCDcollaboration,Domain wall QCD with physical quark masses, Phys. Rev. D93(2016) 074505 [1411.7017]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  33. [33]

    Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation

    M. L¨ uscher,Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation,Phys. Lett. B428(1998) 342 [hep-lat/9802011]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  34. [34]

    Effective theory for quenched lattice QCD and the Aoki phase

    M. Golterman, S.R. Sharpe and R.L. Singleton, Jr.,Effective theory for quenched lattice QCD and the Aaoki phase,Phys. Rev. D71(2005) 094503 [hep-lat/0501015]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  35. [35]

    Domain wall quarks and kaon weak matrix elements

    T. Blum and A. Soni,Domain wall quarks and kaon weak matrix elements, Phys. Rev. Lett.79(1997) 3595 [hep-lat/9706023]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  36. [36]

    Quenched Lattice QCD with Domain Wall Fermions and the Chiral Limit

    T. Blum et al.,Quenched lattice QCD with domain wall fermions and the chiral limit,Phys. Rev. D69(2004) 074502 [hep-lat/0007038]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  37. [37]

    One loop calculation in lattice QCD with domain-wall quarks

    S. Aoki and Y. Taniguchi,One loop calculation in lattice QCD with domain wall quarks,Phys. Rev. D59(1999) 054510 [hep-lat/9711004]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  38. [38]

    S. Aoki, T. Izubuchi, Y. Kuramashi and Y. Taniguchi,Perturbative renormalization factors of quark bilinear operators for domain wall QCD, Phys. Rev. D59(1999) 094505 [hep-lat/9810020]. 76

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  39. [39]

    New Domain-Wall Fermion Actions

    Y. Shamir,New domain wall fermion actions,Phys. Rev. D62(2000) 054513 [hep-lat/0003024]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  40. [40]

    On the Viability of Lattice Perturbation Theory

    G.P. Lepage and P.B. Mackenzie,On the viability of lattice perturbation theory,Phys. Rev. D48(1993) 2250 [hep-lat/9209022]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  41. [41]

    Chiral properties of domain-wall quarks in quenched QCD

    J.C. Collins,Renormalization : An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, vol. 26 of Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1984), 10.1017/9781009401807. [44]CP-PACScollaboration,Chiral properties of domain wall quarks in quenched QCD,Phys. Rev. D63(2001) ...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1017/9781009401807 1984

  42. [42]

    Future of Chiral Extrapolations with Domain Wall Fermions

    S.R. Sharpe,Future of Chiral Extrapolations with Domain Wall Fermions, in Workshop on Domain Wall Fermions at Ten Years, 6, 2007 [0706.0218]. [47]RBC, UKQCDcollaboration,Continuum Limit Physics from 2+1 Flavor Domain Wall QCD,Phys. Rev. D83(2011) 074508 [1011.0892]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  43. [43]

    Aoki,New Phase Structure for Lattice QCD with Wilson Fermions,Phys

    S. Aoki,New Phase Structure for Lattice QCD with Wilson Fermions,Phys. Rev. D30(1984) 2653

    work page 1984

  44. [44]

    Aoki,A Solution to the U(1) Problem on a Lattice,Phys

    S. Aoki,A Solution to the U(1) Problem on a Lattice,Phys. Rev. Lett.57 (1986) 3136

    work page 1986

  45. [45]

    Kawamoto and J

    N. Kawamoto and J. Smit,Effective Lagrangian and Dynamical Symmetry Breaking in Strongly Coupled Lattice QCD,Nucl. Phys. B192(1981) 100

    work page 1981

  46. [46]

    Spontaneous Flavor and Parity Breaking with Wilson Fermions

    S.R. Sharpe and R.L. Singleton, Jr,Spontaneous flavor and parity breaking with Wilson fermions,Phys. Rev. D58(1998) 074501 [hep-lat/9804028]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  47. [47]

    Localization in Lattice QCD

    M. Golterman and Y. Shamir,Localization in lattice QCD,Phys. Rev. D68 (2003) 074501 [hep-lat/0306002]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  48. [48]

    Spectral flow, condensate and topology in lattice QCD

    R.G. Edwards, U.M. Heller and R. Narayanan,Spectral flow, chiral condensate and topology in lattice QCD,Nucl. Phys. B535(1998) 403 [hep-lat/9802016]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  49. [49]

    Exact local fermionic zero modes

    F. Berruto, R. Narayanan and H. Neuberger,Exact local fermionic zero modes,Phys. Lett. B489(2000) 243 [hep-lat/0006030]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  50. [50]

    Chiral properties of domain-wall fermions from eigenvalues of 4 dimensional Wilson-Dirac operator

    S. Aoki and Y. Taniguchi,Chiral properties of domain wall fermions from eigenvalues of four-dimensional Wilson-Dirac operator,Phys. Rev. D65 (2002) 074502 [hep-lat/0109022]. 77

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  51. [51]

    Localization in lattice QCD (with emphasis on practical implications)

    M. Golterman and Y. Shamir,Localization in lattice QCD (with emphasis on practical implications),Nucl. Phys. B Proc. Suppl.129(2004) 149 [hep-lat/0309027]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  52. [52]

    McKane and M

    A.J. McKane and M. Stone,Localization as an alternative to Goldstone’s theorem,Annals Phys.131(1981) 36

    work page 1981

  53. [53]

    Mobility edge in lattice QCD

    M. Golterman, Y. Shamir and B. Svetitsky,Mobility edge in lattice QCD, Phys. Rev. D71(2005) 071502 [hep-lat/0407021]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  54. [54]

    Localization properties of lattice fermions with plaquette and improved gauge actions

    M. Golterman, Y. Shamir and B. Svetitsky,Localization properties of lattice fermions with plaquette and improved gauge actions,Phys. Rev. D72(2005) 034501 [hep-lat/0503037]. [60]RBC, UKQCDcollaboration,Localization and chiral symmetry in three flavor domain wall QCD,Phys. Rev. D77(2008) 014509 [0705.2340]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  55. [55]

    Ginsparg-Wilson Relation and Ultralocality

    I. Horvath,Ginsparg-Wilson relation and ultralocality,Phys. Rev. Lett.81 (1998) 4063 [hep-lat/9808002]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  56. [56]

    Domain Wall Fermions with Exact Chiral Symmetry

    R.G. Edwards and U.M. Heller,Domain wall fermions with exact chiral symmetry,Phys. Rev. D63(2001) 094505 [hep-lat/0005002]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  57. [57]

    Domain wall fermions with improved gauge actions

    Y. Aoki et al.,Domain wall fermions with improved gauge actions,Phys. Rev. D69(2004) 074504 [hep-lat/0211023]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  58. [58]

    Localization of lattice fermions: lessons for overlap

    B. Svetitsky, Y. Shamir and M. Golterman,Localization of lattice fermions: Lessons for overlap,PoSLAT2005(2006) 129 [hep-lat/0508015]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  59. [59]

    Renormalization group flow of SU(3) lattice gauge theory - Numerical studies in a two coupling space

    Y. Iwasaki and T. Yoshie,Renormalization Group Improved Action for SU(3) Lattice Gauge Theory and the String Tension,Phys. Lett. B143(1984) 449. [66]QCD-TAROcollaboration,Renormalization group flow of SU(3) lattice gauge theory: Numerical studies in a two coupling space,Nucl. Phys. B577 (2000) 263 [hep-lat/9911033]

    work page internal anchor Pith review Pith/arXiv arXiv 1984

  60. [60]

    Gap Domain Wall Fermions

    P.M. Vranas,Gap Domain Wall Fermions,Phys. Rev. D74(2006) 034512 [hep-lat/0606014]. [68]JLQCDcollaboration,Lattice gauge action suppressing near-zero modes of H(W),Phys. Rev. D74(2006) 094505 [hep-lat/0607020]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  61. [61]

    Controlling Residual Chiral Symmetry Breaking in Domain Wall Fermion Simulations

    D. Renfrew, T. Blum, N. Christ, R. Mawhinney and P. Vranas,Controlling Residual Chiral Symmetry Breaking in Domain Wall Fermion Simulations, PoSLATTICE2008(2008) 048 [0902.2587]. [70]RBC, UKQCDcollaboration,Direct CP violation and the∆I= 1/2rule in K→ππdecay from the standard model,Phys. Rev. D102(2020) 054509 [2004.09440]. 78 [71]RBC, UKQCDcollaboration,...

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  62. [62]

    Chao and N

    E.-H. Chao and N. Christ,Calculating the two-photon exchange contribution toK L→µ+µ− decay,Phys. Rev. D110(2024) 054514 [2406.07447]

    work page arXiv 2024

  63. [63]

    McGlynn,Advances in Lattice Quantum Chromodynamics, Ph.D

    G.E. McGlynn,Advances in Lattice Quantum Chromodynamics, Ph.D. thesis, Columbia U., 2016. 10.7916/D8T72HD7

    work page doi:10.7916/d8t72hd7 2016

  64. [64]

    The M\"obius Domain Wall Fermion Algorithm

    R.C. Brower, H. Neff and K. Orginos,The M¨ obius domain wall fermion algorithm,Comput. Phys. Commun.220(2017) 1 [1206.5214]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [65]

    Truncated Overlap Fermions: the link between Overlap and Domain Wall Fermions

    A. Bori¸ ci,Truncated overlap fermions: The Link between overlap and domain wall fermions,NATO Sci. Ser. C553(2000) 41 [hep-lat/9912040]. [76]UKQCDcollaboration,Conserved currents for M¨ obius Domain Wall Fermions,PoSLATTICE2014(2015) 087

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  66. [66]

    Optimal lattice domain-wall fermions

    T.-W. Chiu,Optimal domain wall fermions,Phys. Rev. Lett.90(2003) 071601 [hep-lat/0209153]. [78]TWQCDcollaboration,Chiral Symmetry and the Residual Mass in Lattice QCD with the Optimal Domain-Wall Fermion,Phys. Rev. D86(2012) 094508 [1205.6151]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  67. [67]

    Solutions of the Ginsparg-Wilson relation and improved domain wall fermions

    W. Bietenholz,Solutions of the Ginsparg-Wilson relation and improved domain wall fermions,Eur. Phys. J. C6(1999) 537 [hep-lat/9803023]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  68. [68]

    Optimised Dirac Operators on the Lattice: Construction, Properties and Applications

    W. Bietenholz,Optimised Dirac Operators on the Lattice: Construction, Properties and Applications,Fortsch. Phys.56(2008) 107 [hep-lat/0611030]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  69. [69]

    Reducing Residual-Mass Effects for Domain-Wall Fermions

    K. Jansen and K.-i. Nagai,Reducing residual mass effects for Domain-Wall fermions,JHEP12(2003) 038 [hep-lat/0305009]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  70. [70]

    A construction of lattice chiral gauge theories

    R. Narayanan and H. Neuberger,A Construction of lattice chiral gauge theories,Nucl. Phys. B443(1995) 305 [hep-th/9411108]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  71. [71]

    The relation between the waveguide and overlap implementations of Kaplan's domain wall fermions

    M.F.L. Golterman and Y. Shamir,The Relation between the waveguide and overlap implementations of Kaplan’s domain wall fermions,Phys. Lett. B353 (1995) 84 [hep-lat/9501035]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  72. [72]

    Abelian chiral gauge theories on the lattice with exact gauge invariance

    M. L¨ uscher,Abelian chiral gauge theories on the lattice with exact gauge invariance,Nucl. Phys. B549(1999) 295 [hep-lat/9811032]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  73. [73]

    Weyl fermions on the lattice and the non-abelian gauge anomaly

    M. L¨ uscher,Weyl fermions on the lattice and the nonAbelian gauge anomaly, Nucl. Phys. B568(2000) 162 [hep-lat/9904009]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  74. [74]

    Lattice regularization of chiral gauge theories to all orders of perturbation theory

    M. L¨ uscher,Lattice regularization of chiral gauge theories to all orders of perturbation theory,JHEP06(2000) 028 [hep-lat/0006014]. 79

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  75. [75]

    Global Anomalies in chiral gauge theories on the lattice

    O. B¨ ar and I. Campos,Global anomalies in chiral gauge theories on the lattice,Nucl. Phys. B581(2000) 499 [hep-lat/0001025]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  76. [76]

    Nonperturbative Regulator for Chiral Gauge Theories?

    D.M. Grabowska and D.B. Kaplan,Nonperturbative Regulator for Chiral Gauge Theories?,Phys. Rev. Lett.116(2016) 211602 [1511.03649]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  77. [77]

    Kaplan,Chiral Gauge Theory at the Boundary between Topological Phases,Phys

    D.B. Kaplan,Chiral Gauge Theory at the Boundary between Topological Phases,Phys. Rev. Lett.132(2024) 141603 [2312.01494]

    work page arXiv 2024

  78. [78]

    Kaplan and S

    D.B. Kaplan and S. Sen,Weyl Fermions on a Finite Lattice,Phys. Rev. Lett. 132(2024) 141604 [2312.04012]

    work page arXiv 2024

  79. [79]

    Golterman and Y

    M. Golterman and Y. Shamir,Conserved currents in five-dimensional proposals for lattice chiral gauge theories,Phys. Rev. D109(2024) 114519 [2404.16372]

    work page arXiv 2024

  80. [80]

    Kaplan and S

    D.B. Kaplan and S. Sen,Regulated chiral gauge theory and the strong CP problem,2412.02024. [93]TWQCDcollaboration,Exact Pseudofermion Action for Monte Carlo Simulation of Domain-Wall Fermion,Phys. Lett. B738(2014) 55 [1403.1683]

    work page arXiv 2014

Showing first 80 references.