Whether anomaly-free chiral gauge theories — fermions in complex representations of a gauge group, as in the SU(5) grand-unified example or the Standard Model — can be given a nonperturbative, gauge-invariant lattice definition by dynamically decoupling the unwanted mirror fermions is a problem posed in Eichten and Preskill's 1986 proposal, whose phase-structure questions remained explicitly undetermined there. Later work determined the phase structure of the original 1986 model negatively (no undoubled Weyl fermions; spontaneous symmetry breaking occurs), and transferred the decoupling strategy to Ginsparg–Wilson and symmetric-mass-generation formulations, where it remains unsolved: no mirror-decoupling phase producing a local, unitary chiral gauge theory has been demonstrated, and general constraints and no-go results have been derived against it. The surviving open core is the existence of a nonperturbative lattice construction of non-abelian chiral gauge theories via mirror-fermion decoupling.

Background

A chiral gauge theory is a gauge theory in which the gauge interaction couples differently to left- and right-handed fermion components, so that the fermions transform in complex representations of the gauge group and the gauge symmetry itself forbids mass terms for at least some of them — as in the Standard Model and in grand-unified theories such as SU(5). Giving such theories a nonperturbative definition by lattice regularization (discretizing Euclidean spacetime on a hypercubic grid of spacing aa) has been a central open problem of lattice field theory since the early 1980s.

The obstacle is fermion doubling: by the Nielsen–Ninomiya theorem, any free lattice fermion action that is local, translation invariant, and has an exactly conserved chiral symmetry necessarily propagates extra fermion species ("doublers") at the corners of the Brillouin zone. Wilson's remedy — a momentum-dependent term of order aa that gives the doublers masses of order the ultraviolet cutoff 1/a1/a — explicitly breaks chiral symmetry, which is acceptable in vector-like theories such as QCD but is fatal in a chiral gauge theory. Consequently, in a lattice regularization each desired chiral ("ordinary") mode is accompanied by unphysical opposite-chirality "mirror" modes, and the central difficulty is to remove the mirrors without breaking the gauge symmetry.

Eichten and Preskill (1986) proposed to remove the mirrors dynamically: introduce strong auxiliary interactions — local multifermion (four-fermion) couplings of strength λ\lambda — designed so that, in a strong-coupling expansion (an expansion in the hopping parameter, valid for large λ\lambda with the gauge coupling set to zero), mirror modes pair with composite fermion states carrying conjugate gauge quantum numbers to form gauge-invariant massive Dirac fermions, while the ordinary modes stay light. Derivative ("Wilson-like") four-fermion couplings of strength rr were added to lift the degeneracy between ordinary and mirror modes. The construction was validated only in the λ→∞\lambda\to\infty limit, where all fermions are massive. The paper's own open problem, stated in its Section 8 ("The Continuum Limit"), is the phase structure at intermediate λ\lambda: whether a region — a "wedge" in the λ\lambda–rr plane, or, if the critical coupling λc=0\lambda_c = 0, a critical line — exists in which mirror modes remain bound at the cutoff mass while the ordinary modes become massless, so that the continuum limit is a theory of undoubled chiral fermions. The paper enumerated the specific failure modes: spontaneous breaking of the exact lattice symmetries by the multifermion interactions, including the global groups enlarged for analytic convenience (SO(10), or SU(16)) and possibly gauge or Lorentz symmetry; the absence of the needed ordinary–mirror splitting; an unstable fixed point requiring the tuning of infinitely many parameters; and, for its SU(5) example (fermions in the 55 and 10∗10^{*} representations plus a singlet spectator field), an ambiguity in how the anomalous global U(1) is realized — unbroken with a massless composite fermion saturating the 't Hooft anomaly-matching condition, or spontaneously broken with a Goldstone boson. The 't Hooft anomaly-matching condition is the consistency requirement that the anomaly of an exact global symmetry, computed in the ultraviolet, must be reproduced in the infrared, e.g. by massless composite fermions or Goldstone bosons. The paper also identified the practical tools — auxiliary scalar variables that make the action quadratic in the fermions, integration of the fermions (yielding a nonlocal effective action that cannot be ignored), numerical searches for the critical point at vanishing gauge coupling, and high-order strong-coupling computations.

Later work transferred this "mirror-fermion decoupling" idea to a modern setting. The Ginsparg–Wilson relation — a modified lattice chiral symmetry realized by the overlap and domain-wall lattice Dirac operators (two standard lattice discretizations of the Dirac operator that obey this relation and thereby evade the doubling theorem) — permits an exact split of a lattice fermion field into light and mirror components, so that only the mirrors need participate in the strong interactions; Lüscher used such fermions to construct abelian chiral gauge theories with exact gauge invariance. Symmetric mass generation (SMG) denotes the current attempt to gap — i.e. give a mass to every state of — the mirror sector by strong interactions without any spontaneous symmetry breaking, so that the gauge field can then be switched on. These successors inherit precisely the Eichten–Preskill question: does a decoupling phase with the required properties exist, and does its continuum limit define a local, unitary chiral gauge theory?

Problem Statement

Can an anomaly-free chiral gauge theory — fermions transforming in complex representations of a gauge group, as in the SU(5) grand-unified example or the Standard Model — be realized as the continuum limit of a local, gauge-invariant lattice theory in which the unwanted mirror (doubler) fermion modes are decoupled dynamically? In the original Eichten–Preskill formulation and its modern descendants this requires determining, nonperturbatively:

  1. The phase structure in the plane of the strong auxiliary coupling λ\lambda (multifermion or Yukawa) and the Wilson-type splitting coupling rr: specifically, whether a decoupling region — a wedge, or a critical line if λc=0\lambda_c = 0 — exists in which mirror modes are bound at masses of order the cutoff 1/a1/a while the ordinary modes are exactly massless and interact weakly at distances much larger than the lattice spacing aa.
  2. Whether the exact lattice symmetries in that region — including the global groups enlarged for analytic convenience and, ultimately, the gauge symmetry and Lorentz invariance — remain unbroken, rather than being spontaneously broken by the strong mirror interactions.
  3. Whether the desired continuum limit is a stable fixed point, reachable by tuning only finitely many parameters rather than infinitely many.
  4. Whether the resulting continuum theory is a local, unitary chiral gauge theory that saturates the 't Hooft anomaly-matching conditions in the expected way — in the SU(5) example, unbroken global U(1) with a massless composite fermion versus spontaneous breaking with a Goldstone boson — as decided by spectrum and correlation-function calculations, including the behavior of mirror-sector two-point functions after the gauge field is turned on.

The specific 1986 model of Eichten and Preskill has since been analyzed and found not to realize the desired phase (see refs. 2–3). The problem that remains open is whether any lattice realization of the mirror-decoupling strategy — pre-Ginsparg–Wilson, Ginsparg–Wilson-based, or symmetric-mass-generation-based — can satisfy conditions 1–4.

References

  1. Estia Eichten and John Preskill, 'Chiral gauge theories on the lattice', Nuclear Physics B 268 (1986) 179–208, DOI 10.1016/0550-3213(86)90207-5, https://doi.org/10.1016/0550-3213(86)90207-5
  2. Maarten F. L. Golterman and Donald N. Petcher, 'On the Eichten-Preskill proposal for lattice chiral gauge theories', Nuclear Physics B (Proc. Suppl.) 26 (1992) 486–488, DOI 10.1016/0920-5632(92)90311-F, https://doi.org/10.1016/0920-5632(92)90311-F
  3. Maarten F. L. Golterman, Donald N. Petcher and Elena Rivas, 'Absence of chiral fermions in the Eichten-Preskill model', Nuclear Physics B 395 (1993) 596–622, DOI 10.1016/0550-3213(93)90049-U, arXiv:hep-lat/9206010, https://arxiv.org/abs/hep-lat/9206010
  4. Martin Lüscher, 'Abelian chiral gauge theories on the lattice with exact gauge invariance', Nuclear Physics B 549 (1999) 295–334, DOI 10.1016/S0550-3213(99)00115-7, arXiv:hep-lat/9811032, https://arxiv.org/abs/hep-lat/9811032
  5. Maarten Golterman, 'Lattice chiral gauge theories', Nuclear Physics B (Proc. Suppl.) 94 (2001) 189–203, DOI 10.1016/S0920-5632(01)00953-7, arXiv:hep-lat/0011027, https://arxiv.org/abs/hep-lat/0011027
  6. Erich Poppitz and Yanwen Shang, 'Lattice chirality, anomaly matching, and more on the (non)decoupling of mirror fermions', Journal of High Energy Physics 2009 (2009) 03:103, DOI 10.1088/1126-6708/2009/03/103, arXiv:0901.3402, https://arxiv.org/abs/0901.3402
  7. Erich Poppitz and Yanwen Shang, 'Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible?', International Journal of Modern Physics A 25 (2010) 2761–2813, DOI 10.1142/S0217751X10049852, arXiv:1003.5896, https://arxiv.org/abs/1003.5896
  8. Chen Chen, Joel Giedt and Erich Poppitz, 'On the decoupling of mirror fermions', Journal of High Energy Physics 2013 (2013) 04:131, DOI 10.1007/JHEP04(2013)131, arXiv:1211.6947, https://arxiv.org/abs/1211.6947
  9. Yoshio Kikukawa, 'Why is the mission impossible? — Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories', Progress of Theoretical and Experimental Physics 2019 (2019) 073B02, DOI 10.1093/ptep/ptz055, arXiv:1710.11101, https://arxiv.org/abs/1710.11101
  10. Dorota M. Grabowska and David B. Kaplan, 'Nonperturbative Regulator for Chiral Gauge Theories?', Physical Review Letters 116 (2016) 211602, DOI 10.1103/PhysRevLett.116.211602, arXiv:1511.03649, https://arxiv.org/abs/1511.03649
  11. Juven Wang and Xiao-Gang Wen, 'Nonperturbative definition of the standard models', Physical Review Research 2 (2020) 023356, DOI 10.1103/PhysRevResearch.2.023356, arXiv:1809.11171, https://arxiv.org/abs/1809.11171
  12. Maarten Golterman and Yigal Shamir, 'Propagator Zeros and Lattice Chiral Gauge Theories', Physical Review Letters 132 (2024) 081903, DOI 10.1103/PhysRevLett.132.081903, arXiv:2311.12790, https://arxiv.org/abs/2311.12790
  13. Maarten Golterman and Yigal Shamir, 'Constraints on the symmetric mass generation paradigm for lattice chiral gauge theories', Physical Review D 113 (2026) 014503, DOI 10.1103/qd1t-wzy1, arXiv:2505.20436, https://arxiv.org/abs/2505.20436