Viability of the Parisi-Zhang (1983) alternative lattice fermion formulations: induced mass terms, decoupling of radiatively generated operators, and the continuum limit of the second-order and spontaneous-symmetry-breaking schemes

In 1983 Parisi and Zhang proposed three alternative formulations of lattice fermions intended to avoid the fermion-doubling problem while treating chiral symmetry better than Wilson or staggered fermions: (1) a second-order formulation in which the modulus of the Dirac determinant is represented by bosonic pseudofermions with kinetic operator $-\not D^2+m^2=-D^2+\tfrac{1}{2}\sigma_{\mu\nu}F_{\mu\nu}+m^2$, discretized by a lattice Laplacian whose momentum-space form $4\sin^2(p_\mu a/2)/a^2$ vanishes only at $p=0$ and hence has no doubler poles; (2) a left-handed version built from the non-Hermitian operator $-\not D_L^2$ targeting single-chirality (neutrino-like) fermions; and (3) a scheme in which spontaneous symmetry breaking by scalar fields gives doublers tree-level masses while the lattice action keeps a formal chiral symmetry that was argued to be gaugeable. The paper left the viability of all three schemes unestablished, and the specific questions raised by these schemes - the mechanism and magnitude of perturbatively generated mass terms, the decoupling of all radiatively induced operators (e.g. $\bar\phi\sigma_{\mu\nu}F_{\mu\nu}\phi$), the continuum fate of the non-Hermitian Lorentz-violating pieces of $-\not D_L^2$, the effect of lattice artifacts scaling as powers of $\ln a$, and the consistency of tuning a mass counterterm while gauging the formally chiral-symmetric action - were never answered in the subsequent literature, and the schemes themselves were never developed further. The underlying objectives were pursued by other routes: Ginsparg-Wilson/overlap and domain-wall fermions now solve the vector-like case, while genuinely chiral lattice gauge theories remain an open problem addressed by independent programs (mirror fermions and symmetric mass generation, bosonization in two dimensions). Determining whether the Parisi-Zhang formulations possess the intended continuum limits therefore remains open.

Background

Lattice regularization - placing a quantum field theory on a discrete space-time grid of spacing $a$ - is the standard nonperturbative definition of QCD-like gauge theories, but formulating fermions on a lattice meets a classic obstruction. The naive discretization of the Dirac operator, $\gamma_\mu\sin(p_\mu a)/a$ in momentum space, vanishes not only at $p=0$ but also at the corners of the Brillouin zone (the momentum-space unit cell of the lattice), so each continuous fermion field appears with an unwanted multiplicity of species, called doublers (16-fold in four dimensions). Nielsen and Ninomiya proved a no-go theorem: under assumptions of locality, translational invariance, and an exact chiral symmetry realized in the standard way, a lattice theory cannot describe a single doubler-free chiral fermion. The standard remedies each pay a price. The Wilson term (an explicit doubler-lifting mass term proportional to the lattice Laplacian) removes doublers but explicitly breaks chiral symmetry - the symmetry under independent phase rotations of left- and right-handed fermion components ($\gamma_5$ transformations), which is what forbids fermion mass terms - and forces tuning of a hopping parameter (the coefficient of nearest-neighbor fermion hopping on the lattice) to reach the chiral limit. Staggered fermions (Susskind) keep a remnant chiral symmetry at the cost of a complicated flavor structure. Banks and Casher had earlier introduced a lattice Dirac equation of second order in the covariant derivative precisely to avoid this 'spectral multiplication'.

Parisi and Zhang (1983) explored three alternatives in this tradition. First, using the identity $\det(i\not D+m)=[\det(-\not D^2+m^2)]^{1/2}\times(\mathrm{phase})$ with

$$-\not D^{2}=-D^{2}+\tfrac{1}{2}\sigma_{\mu\nu}F_{\mu\nu},$$

where $\sigma_{\mu\nu}=\tfrac{i}{2}[\gamma_\mu,\gamma_\nu]$ is the spin tensor and $F_{\mu\nu}$ the gauge field strength (so the field that represents the determinant carries a built-in anomalous-magnetic-moment coupling), the modulus of the fermion determinant is represented by bosonic spinor fields $\phi$ (pseudofermions) with a lattice-Laplacian discretization of $D^2$; since $4\sin^2(p_\mu a/2)/a^2$ vanishes only at $p=0$, the propagator $(i\not D+m)(-D^2+\tfrac{1}{2}\sigma_{\mu\nu}F_{\mu\nu}+m^2)^{-1}$ has no doubler poles. Second, the same trick applied to a left-handed (Weyl, i.e. single-chirality, neutrino-like) determinant, $\det(i\not D_L)=[\det(-\not D_L^2+\delta m^2)]^{1/2}$, where the second-order operator contains

$$\mathbf D_{L}^{2}=\mathbf D^{2}+\mathrm{i}D_{0}\,\boldsymbol{\tau}\cdot\mathbf D,$$

with $D_0$ and $\mathbf D$ the temporal and spatial covariant derivatives and $\boldsymbol\tau$ Pauli matrices mixing chirality; this operator is non-Hermitian in Euclidean space and contains terms that violate Lorentz (rotation) invariance. Third, a scheme in which scalar fields $(\sigma,\pi)$ with potential $\lambda(\sigma^2+\pi^2-\mu^2)^2$ spontaneously break a U(1) symmetry, while a Yukawa coupling mediated by the lattice Laplacian $\Delta$, $a^2g\bar\psi(\sigma+\mathrm{i}\gamma_5\pi)\Delta\psi$, gives doubler modes tree-level masses of order $g$ and physical fermions only one-loop masses of order $g^2$, leaving a formal chiral symmetry that the authors argued could be gauged without explicit symmetry-breaking counterterms. Their own one-loop analysis indicated that induced quartic pseudofermion interactions $(\bar\phi\Gamma\phi)^2$ scale to zero as $a\to0$, behaving like $(\ln a)^{-2+\gamma/b_0}$, where $\gamma$ is the anomalous dimension of the composite operator (the quantum shift of its scaling dimension) and $b_0$ the leading coefficient of the gauge beta function (the logarithmic running of the coupling).

Problem Statement

Determine whether the three lattice fermion formulations proposed by Parisi and Zhang (1983) possess continuum limits reproducing the intended continuum fermion theories, resolving the following five issues:

(i) Chiral symmetry and mass generation: the second-order/pseudofermion formulations obscure the global chiral ($\gamma_5$) symmetry and generate mass terms perturbatively even when the bare mass is set to zero; establish the mechanism and magnitude of this mass generation and its impact on chiral observables, in particular whether it can be tuned away or controlled.

(ii) Induced operators: beyond the one-loop analysis of quartic pseudofermion interactions $(\bar\phi\Gamma\phi)^2$, determine whether all other radiatively induced local operators - including $\bar\phi\sigma_{\mu\nu}F_{\mu\nu}\phi$ - are irrelevant and decouple as $a\to0$.

(iii) Left-handed operator: the left-handed second-order operator $-\not D_L^2$, containing $\mathbf D_L^2=\mathbf D^2+\mathrm{i}D_0\boldsymbol\tau\cdot\mathbf D$ and, like the second-order operator defined in the Background section, a magnetic-moment ($\sigma_{\mu\nu}F_{\mu\nu}$) coupling, is non-Hermitian and contains Lorentz-violating pieces (e.g. the $D_0\mathbf D$ term); determine whether and in what sense these pieces vanish or can be controlled in the continuum limit, and whether the construction yields the correct single-chirality continuum theory (correct propagator residues, no unphysical modes).

(iv) Logarithmic artifacts: lattice corrections to continuum dimensionless quantities in the second-order scheme scale as powers of $\ln a$ rather than as positive powers of $a$; quantify the effect of these logarithmic artifacts on continuum extrapolations.

(v) Spontaneous-symmetry-breaking scheme: the scheme still requires introducing and tuning a fermion-mass counterterm (the analogue of tuning the Wilson hopping parameter); determine whether this tuning is consistent with gauging the formally chiral-symmetric lattice action, i.e. whether the tuned, gauged theory realizes the intended chiral fermion content.

A determination establishes, for each scheme, either a controlled continuum limit with the intended particle content (including, for the left-handed and spontaneous-symmetry-breaking schemes, the single-chirality or chiral-gauge case) or a specific obstruction (for example a relevant induced operator that cannot be tuned away, a non-decoupling Lorentz-violating remnant, an unphysical pole structure, or an inconsistency with the Nielsen-Ninomiya theorem and anomaly matching, i.e. the requirement that the anomaly of the intended continuum theory be reproduced by the massless lattice spectrum), with the analysis expressed in terms of the operators and actions of the three 1983 formulations as defined in the Background section.

References

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