Open completion problems for the Parisi-Zhang modified (quenched) Eguchi-Kawai reduction of large-$N$ gauge theory

In 1982 Parisi and Zhang proposed a 'modified Eguchi-Kawai model': a single-site (one-point) matrix model for large-$N$ gauge theory in which the role of space-time is played by fixed diagonal background phase matrices $\theta$, space-time momenta being replaced by differences of color indices. They showed that its planar Feynman rules and its $\theta$-averaged Wilson loops reproduce those of infinite-volume $U(N)$ gauge theory at leading order as $N\to\infty$, and they proposed a $U(N)$ one-site action, adding a gauge-fixing quadratic term and an infinitesimal gluon mass with the limits $N\to\infty$ taken before $\mu^2\to0$, to enforce the background, arguing only that this is 'very likely' to forbid the singular gauge transformations that would remove $\theta$. The in-principle validity of quenching-based reduction is actively contested: Bringoltz and Sharpe (2008) gave weak-coupling arguments and Monte-Carlo evidence for breakdown via dynamically generated correlations, while Neuberger (2020) argued that the 2008 analysis did not test the originally intended prescription and that the question remains open, requiring future extensive numerical work. The source paper's background-enforcing action was never directly revisited after 1984. The problem of settling and completing the modified Eguchi-Kawai reduction therefore remains open.

Background

In gauge theories with structure group $U(N)$ or $SU(N)$ (the color group, whose rank $N$ counts the number of color charges), 't Hooft's large-$N$ limit sends $N\to\infty$ while holding the 't Hooft coupling $\lambda=g^2N$ fixed. In this limit only Feynman diagrams that can be drawn on a plane without self-intersections (planar diagrams) survive at leading order, and expectation values of products of gauge-invariant traces factorize into products of the individual expectation values up to corrections of relative order $1/N$. Planar gauge theory is therefore a natural target for exact or nonperturbative methods, but it has resisted exact solution in three and four space-time dimensions.

On a lattice, a gauge theory is encoded in link variables $U_\mu(x)\in U(N)$ attached to the edges joining neighboring sites $x$ and $x+\hat\mu$ (with $\hat\mu$ the unit vectors of the $d$ space-time directions), with a Boltzmann weight built from traces of products of links around elementary square plaquettes (the Wilson action). The basic observables are Wilson loops, $\mathrm{tr}\,W(C)=\mathrm{tr}\prod_{\ell\in C}U_\ell$, traces of link products around closed contours $C$; their growth with loop area signals confinement. In the planar limit the loop expectation values obey a closed hierarchy of Dyson-Schwinger equations (also called loop or Makeenko-Migdal equations) relating the change of a loop expectation under variation of its contour to sums of expectations of loops with an inserted link; this hierarchy is independent of the lattice volume.

Eguchi and Kawai (1982) observed that, because the planar loop equations are volume-independent, the $d$-dimensional theory at $N=\infty$ should be reproduced by a single-site model containing just one link matrix per direction, provided two conditions hold: expectation values factorize as in infinite volume, and the symmetry under the group $(\mathbb{Z}_N)^d$ of transformations $U_\mu\mapsto z_\mu U_\mu$ with $z_\mu$ an $N$-th root of unity (the center symmetry, whose unbroken phase encodes the translation invariance that allows all lattice sites to be identified) is not spontaneously broken. Within months it was shown (Bhanot, Heller and Neuberger; Parisi, both 1982) that this center symmetry does break spontaneously at weak coupling, invalidating the original reduction in the continuum limit.

The cure explored in the source paper is quenching: one freezes the eigenvalue phases of the single-site link matrices. Writing the phases as fixed diagonal matrices $e^{i\theta_\mu}=\mathrm{diag}(e^{i\theta_\mu^1},\ldots,e^{i\theta_\mu^N})$ with real angles $\theta_\mu^a\in(-\pi,\pi]$, one studies the one-point theory at fixed $\theta$ and averages observables over the $\theta$. In the momentum-quenched prescription of Gross and Kitazawa (1982), planar equivalence with the ordinary theory holds at any fixed order in perturbation theory: differences of color (matrix) indices act as momenta, $p=\theta_a-\theta_b$, and sums over colors become momentum integrals, $\frac{1}{N}\sum_a f(\theta_a)\to\int\frac{d^Dp}{(2\pi)^D}f(p)$, provided the $\theta^a$ fill the Brillouin zone (the momentum-space unit cell of the lattice) uniformly. Whether this replacement, and the equivalence itself, hold nonperturbatively is a separate and much harder question.

The source paper, Parisi and Zhang's 'A modified Eguchi-Kawai model' (1982), develops this construction in both continuum and lattice form. In the continuum it imposes twisted boundary conditions $A_\mu(x+\hat\nu)=e^{i\theta^\nu}A_\mu(x)e^{-i\theta^\nu}$ and $\psi(x+\hat\nu)=e^{i\theta^\nu}\psi(x)$, with constant diagonal phase matrices $e^{i\theta^\nu}$; here 'twisted' refers to these diagonal phase shifts, not to the 't Hooft twist matrices of the separate twisted Eguchi-Kawai model described below. At a single point the resulting Feynman rules depend on the background only through phase differences, e.g. a gluon propagator $-\delta_{\mu\nu}/(\theta_a-\theta_b)^2$ for $a\ne b$, and one-loop self-energies and vertex corrections reproduce the planar continuum results under the identification $p=\theta_a$. On the lattice, with $U_\mu\in O(N)$ and $U_\mu(\theta)=U_\mu e^{i\theta_\mu}$, the one-site action is $-S(\theta)=\frac{\beta}{2}\sum_{\mu\ne\nu}\mathrm{tr}\,U_\mu(\theta)U_\nu(\theta)U_\mu^\dagger(\theta)U_\nu^\dagger(\theta)$ with $\beta$ proportional to $N/g^2$; observables are $\theta$-averages, $\langle\mathrm{tr}\,W(C)\rangle=\int\prod\frac{d\theta}{2\pi}\,\langle\mathrm{tr}\,W(C,\theta)\rangle_\theta$; open (non-closed) Wilson loops average to zero; and the $\theta$-averaged closed loops satisfy the same Dyson-Schwinger equations as the infinite-volume planar theory at leading order in $1/N$. The $O(N)$ and $U(N)$ versions agree at that order (the color-generator identity $\sum_j(T^j)_{ab}(T^j)_{cd}=\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}$ is common to both), although not exactly: under $O(N)$ rotations quarks and antiquarks transform in the same representation, so color lines carry no direction. For the $U(N)$ theory the authors propose enforcing the background by adding to the one-site action a gauge-fixing quadratic term $\frac{1}{2}g^{-2}\mathrm{tr}\big(\sum_\mu[U_\mu,e^{i\theta_\mu}]\big)^2$ and an infinitesimal gluon mass $\mu^2\sum_\mu(\mathrm{tr}\,U_\mu+\mathrm{tr}\,U_\mu^\dagger)$, with $N\to\infty$ taken before $\mu^2\to0$; they argue this is 'very likely' to forbid the singular gauge transformations that would otherwise remove $\theta$, and they leave open which of the two terms is needed. The claimed equivalence is explicitly leading-order and formal: the $O(1/N)$ differences, namely non-planar topologies, the singular regions where color indices coincide (the diagonal gluon components $A_{\mu,aa}$, which require the small mass), and the heuristic character of the color-sum-to-integral replacement, are acknowledged but not worked out.

A further motivation is the master field: because planar expectation values factorize, the large-$N$ functional integral is formally dominated by a single representative configuration, and an explicit construction of that configuration would amount to solving planar gauge theory. Constructive master fields are known only in special settings, such as two-dimensional Yang-Mills theory and the strong-coupling limit of lattice gauge theories whose plaquette weight is the heat kernel (the diffusion kernel on the group $U(N)$), not for four-dimensional Yang-Mills theory.

Later literature pursued three main routes. First, other quenching-type prescriptions: Klinkhamer (1983) showed that different quenching procedures may be inequivalent, and Rossi and Testa (1983) gave a nonperturbative proof of the quenching prescription, whose assumptions were later questioned. Second, the twisted Eguchi-Kawai (TEK) model of Gonzalez-Arroyo and Okawa (1983), which enforces the background by inserting fixed center elements (twist matrices) into the action rather than by quenching or gauge fixing; this became the mainstream route and works in modern simulations at very large $N$. Third, partial reduction (simulating a lattice of a few sites rather than a single site) and deformation methods, such as adjoint fermions (fermions coupled in the adjoint representation of the color group, whose feedback on the gauge fields stabilizes center symmetry) and trace deformation (augmenting the plaquette action with powers of the plaquette trace so as to disfavor center-symmetry breaking), which yield volume independence at finite accessible $N$. The validity of the quenching route itself was re-examined in 2008 and 2020 with conflicting conclusions; it has not been settled.

Problem Statement

Decide and complete the modified (quenched) Eguchi-Kawai reduction proposed by Parisi and Zhang ('A modified Eguchi-Kawai model', Physics Letters B 114 (1982) 319): determine whether the one-point (single-site) construction with an enforced diagonal phase background $\theta$, in the four-dimensional lattice formulation of the source, reproduces the planar sector of infinite-volume $U(N)$ gauge theory in the $N\to\infty$ limit (the documented 2008-2020 controversy over the validity of quenched reduction concerns $d=4$, and settling this four-dimensional case resolves the objective), and make the construction rigorous and minimal. The common objective uniting the components below is the validation and completion of that specific reduction; the components are the facets under which the source paper and the subsequent literature leave it open.

(i) Nonperturbative validity. Prove or refute, beyond finite-order perturbative matching and beyond the leading-order comparison of Dyson-Schwinger equations, that the $\theta$-averaged one-point model, with the background-enforcing modification of the action, coincides with infinite-volume planar $U(N)$ gauge theory in the $N\to\infty$ limit; in particular, determine whether dynamically generated correlations between the quenching phases and the Euclidean gauge-field components invalidate the equivalence at intermediate and weak coupling.

(ii) Minimal enforcement. Determine which modification of the $U(N)$ one-site action, the gauge-fixing quadratic term, the infinitesimal mass term, both, or an alternative, is necessary and sufficient to forbid the singular gauge transformations that remove $\theta$ (the source's gauge shifts subject to $[A,\theta]=0$, under which the diagonal background can be gauged away) while preserving the planar equivalence.

A refutation of (i) settles the task on its own, since (ii) presupposes an equivalence to preserve; only if (i) is established does the task additionally require settling (ii).

References

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