Completing the anomaly-matching proof of chiral symmetry breaking in QCD-like theories: the heavy-quark continuity gap for $N_f<p_{\min}$ and generalization beyond $SU(N_c)$ fundamentals

Whether confinement forces chiral symmetry breaking in four-dimensional QCD-like theories ($SU(N_c)$ Yang-Mills with $N_f$ massless Dirac flavors in the fundamental representation) has been studied through 't Hooft anomaly matching and persistent mass conditions since 1980. A recent proof program (Ciambriello, Contino, Luzio, Romano, Xu) reduced the question to algebra: by the downlifting strategy — an explicit construction that turns any integer solution of the anomaly and mass conditions for $N_f$ flavors into one for $N_f-1$ flavors at fixed $N_c$ — chiral symmetry breaking is now proven, assuming only a color-singlet infrared description, for every $N_f\ge p_{\min}(N_c)$, where $p_{\min}$ is the smallest prime factor of $N_c$ — including a full proof that the anomaly matching conditions have no integral solutions whenever $N_f$ is a multiple of a prime factor of $N_c$. Two gaps survive and form this problem. First, for $2\le N_f<p_{\min}(N_c)$ — most importantly two-flavor $SU(3)$ QCD — the proof still rests on an unproven continuity assumption: that no phase transition occurs when the remaining $p_{\min}-N_f$ quark masses are sent to infinity. Second, the algebraic proof has not been extended beyond $SU(N_c)$ with fundamentals, to other gauge groups or higher-dimensional fermion representations of vectorlike theories. Solving the first gap would elevate chiral symmetry breaking in real-QCD-like theories to an unconditional theorem.

Background

A QCD-like theory is four-dimensional $SU(N_c)$ Yang-Mills theory coupled to $N_f$ flavors (Dirac fermions) of massless quarks in the fundamental representation of the color group, with $N_c>2$ and $2\le N_f$ below the edge of the conformal window — the conformal window being the interval of flavor numbers in which the theory flows to an infrared conformal fixed point instead of confining — so that the theory is expected to confine. When the quarks are massless the global symmetry is the chiral group $SU(N_f)_L\times SU(N_f)_R\times U(1)_B$, where $L,R$ act on left- and right-handed quarks and $U(1)_B$ is baryon number. 't Hooft (1980) observed that if the vacuum preserved this chiral symmetry, the confined, color-singlet bound states would have to include massless fermions whose 't Hooft anomalies — the coefficients of the one-loop three-current and two-current-gravity triangle diagrams of the global symmetry — exactly reproduce those of the elementary quarks. These Anomaly Matching Conditions (AMC) form a system of linear Diophantine equations for the indices (number of left-handed minus right-handed massless states) of each irreducible representation. He also proposed decoupling conditions, reformulated by Preskill and Weinberg (1981) as the Persistent Mass Conditions (PMC): when some quarks are given large masses, any color-singlet bound state containing them must also become massive; Vafa and Witten (1984) derived the PMC from Euclidean positivity of vectorlike gauge theories. If the combined AMC+PMC system has no solution in nonnegative integers for some $N_c,N_f$, then in a confining theory chiral symmetry must spontaneously break. Work in the 1980s (Farrar; Cohen and Frishman; Schwimmer; and others) proved no-solution results only under assumptions on the hadron spectrum (e.g., purely baryonic spectra of massless fermions, or the validity of $N_f$-independence — the assumption that the solution set of AMC+PMC does not change with $N_f$). Ciambriello, Contino, Luzio, Romano, and Xu (arXiv:2212.02930, published as Phys. Rev. D 110, 114035) gave a proof of $N_f$-independence restricted to spectra interpolated by class-A tensor operators (tensors whose indices can all be contracted with flavor indices), and — in the companion paper arXiv:2404.02967 (Phys. Lett. B 862, 139367 (2025)) — introduced downlifting: an explicit construction that turns any integer solution of AMC+PMC for $N_f$ flavors into one for $N_f-1$ flavors at fixed $N_c$. Combined with the number-theoretic input that the $[SU(mp)_{L,R}]^2 U(1)_V$ AMC (where $U(1)_V$ denotes the vector baryon-number symmetry, i.e. the $U(1)_B$ defined above) have no integral solutions when $p$ is a prime factor of $N_c$ and $m$ any positive integer (a generalization, to arbitrary spectra, of the Preskill-Weinberg three-flavor argument), induction proves chiral symmetry breaking for all $N_f\ge p_{\min}(N_c)$, where $p_{\min}$ is the smallest prime factor of $N_c$, assuming only confinement in the sense of a color-singlet infrared description and making no assumption about which massless composites (baryons or exotics with antiquark constituents) appear. For $N_f<p_{\min}$, integer solutions of AMC+PMC do exist, and the argument instead gives masses to $p_{\min}-N_f$ flavors and takes them to infinity, invoking an unproven assumption that no phase transition occurs along this path. A pedagogical review by Xu (arXiv:2509.06190, published as Int. J. Mod. Phys. A 41, 2630001 (2026)) states the resulting no-go theorem and identifies exactly these gaps as the remaining challenges.

Problem Statement

The goal is to make the anomaly-matching proof of chiral symmetry breaking in confining QCD-like theories unconditional and general, closing the two gaps left by the downlifting proof of Ciambriello et al. (arXiv:2404.02967): (1) Continuity gap: for $SU(N_c)$ with $N_f$ fundamental massless flavors and $2\le N_f<p_{\min}(N_c)$ — in particular for two-flavor $SU(3)$ QCD — prove from first principles that chiral symmetry $SU(N_f)_L\times SU(N_f)_R\times U(1)_B$ is spontaneously broken in the confining regime without assuming the absence of phase transitions as the $p_{\min}-N_f$ remaining quark masses are sent to infinity. (2) Generalization: prove that confinement, in the sense of a color-singlet infrared description, implies spontaneous chiral symmetry breaking beyond $SU(N_c)$ with fundamental flavors — for other gauge groups and fermion representations of vectorlike theories — identifying the analogue of the persistent mass conditions where Vafa-Witten reasoning still applies. Both parts serve one objective: an assumption-free theorem stating that confinement, in the sense of a color-singlet infrared description, implies chiral symmetry breaking for all $N_f\ge2$ in QCD-like theories and their vectorlike generalizations.

References

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