Coincidence of gauge-invariant custodial-symmetry breaking with the separation-of-charge-to-color confinement transition in gauge-Higgs theories

In lattice gauge-Higgs theories with a fundamental-representation Higgs field, Elitzur's theorem forbids spontaneous breaking of the local gauge symmetry, yet the Higgs and confining regions can be distinguished by gauge-invariant means: a non-local order parameter detects the spontaneous breaking of a global custodial symmetry, and the confining region obeys a stronger property called separation-of-charge ($S_c$) confinement. The source paper conjectured that the custodial-symmetry-breaking line coincides with the $S_c$-to-color ($C$) confinement transition throughout the coupling plane. The authors later argued analytically, via a reformulated order parameter, that the coincidence holds for theories without a massless phase, but the coincidence has not been independently verified, the analytic control of the $S_c$ region rests on strong-coupling bounds, and the phase structure in models where a massless phase or a second gauge group intervenes (compact U(1) with unit-charge matter, $SU(2)\times U(1)$) remains unmapped. The problem is to determine this Higgs-versus-confinement phase structure throughout the coupling plane and to establish, confirm, or refute the coincidence with confinement-type change at moderate and weak coupling.

Background

A gauge-Higgs theory is a lattice gauge theory, such as $SU(N)$ gauge theory coupled to a scalar (Higgs) field in the fundamental representation of the gauge group (often taken unimodular, i.e. of fixed unit modulus, so that only its gauge orientation is dynamical), with lattice couplings $\beta$ (gauge) and $\gamma$ (Higgs). Two classical results constrain its phase structure. Elitzur's theorem states that a local gauge symmetry cannot break spontaneously, so the gauge-fixed Higgs expectation value $\langle\phi\rangle$ is not a physical order parameter. The Osterwalder-Seiler and Fradkin-Shenker analyses showed that for fundamental matter the free energy is analytic in much of the coupling plane, suggesting that the Higgs and confinement regions are smoothly connected.

Frohlich, Morchio, and Strocchi nevertheless noted a residual distinction: besides the local gauge symmetry there is a global custodial symmetry acting only on the Higgs field (global $SU(2)$ for $N=2$, global $U(1)$ otherwise, for a single fundamental scalar). Greensite and Matsuyama (Phys. Rev. D 98, 074504 (2018)) constructed a non-local gauge-invariant order parameter $\Phi[U]$ for the breaking of this custodial symmetry, built from matter-like operators $V(\boldsymbol{x},\boldsymbol{y};U)$ that transform covariantly under gauge transformations constructed from the gauge field alone (pseudomatter operators, e.g. eigenmodes of the covariant Laplacian, or the Coulomb-gauge gauge transformation itself). Because the order parameter is non-local, its non-analyticity does not contradict the Osterwalder-Seiler theorem. Numerically they located the custodial-breaking transition line in the $(\beta,\gamma)$ plane for $SU(2)$ and $SU(3)$.

In parallel, Greensite and Matsuyama (Phys. Rev. D 96, 094510 (2017)) distinguished two types of confinement in theories with fundamental matter: color or $C$-confinement, meaning the asymptotic spectrum consists of color singlets, and the stronger separation-of-charge or $S_c$-confinement, meaning that for every bi-covariant dressing operator $V(\boldsymbol{x},\boldsymbol{y};U)$ built from the gauge field alone, the energy $E_V(R)$ of the corresponding static charge-anticharge state — a state with separated color charge unscreened by matter fields — obeys a common asymptotically linear lower bound, $E_V(R) \geq E_0(R)$ with $\lim_{R\to\infty} dE_0/dR = \sigma > 0$. $S_c$-confinement was proven to exist at strong coupling (via strong-coupling expansions and bounds) but not throughout the plane, so a boundary between the $S_c$ and $C$ regions must exist. The 2018 paper conjectured that the custodial-symmetry-breaking transition coincides with the $S_c$-to-$C$ transition, which would make confinement-versus-Higgs a sharp, gauge-invariant distinction.

A follow-up (Greensite and Matsuyama, Phys. Rev. D 101, 054508 (2020)) reformulated the order parameter in the spirit of the Edwards-Anderson spin-glass order parameter (computed from the Higgs field on a single time slice at fixed gauge field) and argued analytically, through physical gauges that leave a remnant global symmetry, that the spin-glass (broken custodial) phase is exactly the $C$-confining Higgs phase, while the unbroken phase is either $S_c$-confining or massless; in the absence of a massless phase the two transitions coincide. The authors state the original conjecture was 'almost correct' and holds in the new formulation. This argument, however, is a physics derivation rather than a rigorous theorem, has not been independently verified, and the analytic handle on $S_c$-confinement remains its strong-coupling bound. Independent frameworks in overlapping settings, such as the topological vortex order parameter and line-operator statistics of Cherman, Jacobson, Sen, and Yaffe, and the smooth matching of Aharonov-Bohm phases found by Hayashi in superfluid gauge theories, address phase boundaries with different criteria and do not settle the coincidence question. The relation to older confinement criteria is only partially clarified: the Fredenhagen-Marcu criterion (which tests confinement through the infinite-volume behavior of a matter-field two-point function) was argued to test $C$-confinement only, while a quantitative reconciliation with the Kugo-Ojima criterion (a confinement test in the BRST operator formalism of gauge quantization, where BRST symmetry — the residual global symmetry remaining after gauge fixing — determines whether physical states can carry global color charge) and with propagator-based signals is unaddressed.

Problem Statement

For lattice gauge-Higgs theories with a fundamental-representation Higgs field — namely $SU(2)$ and $SU(3)$ gauge theories with a unimodular (fixed unit modulus) Higgs field — determine the Higgs-versus-confinement phase structure throughout the $(\beta,\gamma)$ coupling plane by gauge-invariant means. Concretely: (i) locate the transition line for spontaneous breaking of the global custodial symmetry by non-local gauge-invariant order parameters; (ii) locate the boundary between separation-of-charge ($S_c$) confinement and color ($C$) confinement, i.e. the locus where the minimal energy $E_V(R)$ over bi-covariant dressing operators $V(\boldsymbol{x},\boldsymbol{y};U)$ — operators built from the gauge field alone that transform covariantly under gauge transformations at both endpoints — ceases to be bounded below by an asymptotically linear function $E_0(R)$ with $\lim_{R\to\infty} dE_0/dR = \sigma > 0$; because $S_c$-confinement requires such a lower bound for every bi-covariant $V$, a coupling point is placed in the $C$ region by exhibiting a single bi-covariant operator $V$ whose $E_V(R)$ grows sublinearly, whereas the $S_c$ side of the boundary can be asserted only where a common lower bound over the full class of such operators is established (currently the strong-coupling regime); and (iii) establish whether these two lines coincide (or provide a coupling point or operator $V$ where they do not), so that the broken-versus-unbroken realization of the global symmetry can be mapped against the confinement type. Parts (i) and (ii) remain required whichever way (iii) is settled: a lone coupling point or dressing operator $V$ at which the two lines fail to coincide settles (iii) by refutation but does not by itself substitute for locating the lines. Extending the analysis beyond this core — in particular to the compact U(1) unit-charge model, where a massless phase intervenes and the broken-versus-unbroken global U(1) mapping was explicitly left as a future investigation in the source paper, to the electroweak-like $SU(2)\times U(1)$ model, and to the reconciliation with the Kugo-Ojima or propagator-based confinement criteria — is an optional direction rather than part of the required core scope.

References

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