In the SU(2) lattice gauge-Higgs theory with a fundamental-representation Higgs field, different remnant global subgroups of the local gauge symmetry — the Landau-gauge remnant underlying the Kugo-Ojima criterion, the Coulomb-gauge remnant underlying the confining color-Coulomb potential, and the dual U(1) probed by the monopole creation operator — are known to break spontaneously along different lines in the coupling plane, while the Fradkin-Shenker-Osterwalder-Seiler analyticity theorem shows that local gauge-invariant observables vary analytically through much of that plane. The open problem, posed explicitly by Caudy and Greensite (2008), is to decide which, if any, of these symmetry breakings signals a transition between physically distinct confining and non-confining phases, and more generally whether breaking of a global subgroup of a local gauge symmetry can ever be guaranteed to mark a change of physical state when no thermodynamic nonanalyticity accompanies it. Later work investigated the monopole operator (with negative evidence for its diagnostic value) and proposed a gauge-invariant custodial-symmetry (spin-glass) order parameter whose breaking is conjectured to coincide with a transition between two types of confinement, but no theorem or consensus resolution exists.

Background

The SU(2) lattice gauge-Higgs model is a lattice regularized non-Abelian gauge theory in which an SU(2) gauge field with Wilson action (coupling β=4/g2\beta = 4/g^2) interacts with a scalar Higgs field in the fundamental representation of the gauge group (gauge-Higgs coupling γ\gamma). It is the canonical laboratory for the interplay of confinement and the Higgs mechanism. Elitzur's theorem states that a local gauge symmetry cannot be spontaneously broken: without gauge fixing, the expectation value of any gauge-variant operator such as the Higgs field ϕ\phi is identically zero (S. Elitzur, Phys. Rev. D 12, 3978 (1975), DOI 10.1103/PhysRevD.12.3978). What can break is only a global subgroup of the local gauge symmetry, where 'global' means a subgroup specified by a number of parameters that does not grow with the spacetime volume. A gauge condition that removes only the local freedom, such as Landau gauge (∂μAμ=0\partial_\mu A_\mu = 0, imposed on all four spacetime components of the gauge field) or Coulomb gauge (∂iAi=0\partial_i A_i = 0, imposed only on the spatial components of the gauge field), leaves such a remnant global symmetry un-fixed, and gauge-variant operators can then serve as order parameters for its breaking. Different confinement criteria are tied to different remnants: the Kugo-Ojima criterion (covariant gauges) relates confinement to the unbroken realization of a global color symmetry surviving in Landau-type gauges; the Coulomb-confinement criterion ties confinement in Coulomb gauge to a nonvanishing asymptotic color-Coulomb potential, whose absence/presence is likewise tied to remnant symmetry breaking; and the dual-superconductor picture (monopole condensation) is probed by the Pisa monopole creation operator μ\mu, whose disorder transition is diagnosed by a volume-growing negative peak in ρ=dlog⁡⟨μ⟩/dβ\rho = d\log\langle\mu\rangle/d\beta (A. Di Giacomo, B. Lucini, L. Montesi, G. Paffuti, 'Color confinement and dual superconductivity of the vacuum. I', Phys. Rev. D 61, 034503 (2000), DOI 10.1103/PhysRevD.61.034503). Separately, the Fradkin-Shenker and Osterwalder-Seiler analyticity theorems establish that in gauge theories with Higgs fields in the fundamental representation there are regions of the coupling plane connecting the Higgs and strong-coupling confinement-like regimes in which the free energy and all local gauge-invariant observables vary analytically (E. Fradkin and S. H. Shenker, 'Phase diagrams of lattice gauge theories with Higgs fields', Phys. Rev. D 19, 3682 (1979), DOI 10.1103/PhysRevD.19.3682; K. Osterwalder and E. Seiler, 'Gauge field theories on a lattice', Ann. Phys. 110, 440 (1978), DOI 10.1016/0003-4916(78)90039-8). Caudy and Greensite mapped the remnant-symmetry breaking lines of the Landau and Coulomb remnants in the β\beta-γ\gamma plane: the two lines coincide with the thermodynamic transition line at β≥2\beta \ge 2 but diverge from each other at β<2\beta < 2, so the two order parameters cannot both mark a confinement-Higgs phase transition; the asymptotic string tension vanishes and the spectrum consists of color singlets everywhere at γ>0\gamma > 0, leaving unclear in what physical sense any of these transitions corresponds to a change of state. Related non-thermodynamic markers, such as the percolation (Kertész) line of center vortices — a geometric line defined by the onset of infinite vortex clusters, whose location depends on the type of object chosen (J. Greensite and Š. Olejník, 'Vortices, symmetry breaking and temporary confinement in SU(2) gauge-Higgs theory', Phys. Rev. D 74, 014502 (2006), arXiv:hep-lat/0603024) — show the same ambivalence: coincident with thermodynamic transitions in one regime, detached in another. A later research program introduced a different notion: custodial symmetry, a global symmetry acting on the Higgs field alone (for example a group of global Higgs-field transformations that commutes with every gauge transformation), detectable through a gauge-invariant but nonlocal Edwards-Anderson-type order parameter analogous to the spin-glass order parameter, together with a distinction between 'color confinement' (all asymptotic states are color singlets, which also holds in the Higgs regime) and the stronger 'separation-of-charge confinement' (isolated external charges cannot be separated with finite energy, the analogue of the Wilson area law).

Problem Statement

In the four-dimensional SU(2) lattice gauge-Higgs theory with a fundamental-representation Higgs field, decide which — if any — of the remnant global subgroups of the local gauge symmetry has a spontaneous breaking that is a physically meaningful diagnostic of a confining phase. The named candidates are: the Landau-gauge remnant global symmetry underlying the Kugo-Ojima confinement criterion, the Coulomb-gauge remnant global symmetry underlying the confining color-Coulomb potential, and the dual U(1) symmetry whose breaking is detected by the monopole creation operator of the dual-superconductor picture; the decision must not be limited to these candidates if another global subgroup of the local symmetry is the physically relevant one. The problem has three linked parts that jointly serve this single objective. (1) Determine whether the physical distinction that such a breaking would mark — a transition between physically distinct confining and non-confining (Higgs) phases — exists at all in this theory, given the Fradkin-Shenker-Osterwalder-Seiler analyticity of local gauge-invariant observables, and if it exists, characterize it in gauge-invariant terms (for example, as a transition between color confinement and separation-of-charge confinement, or as the onset or loss of magnetic disorder). (2) Decide, for each candidate subgroup — or for whatever subgroup part (1) singles out — whether its breaking line in the β\beta-γ\gamma coupling plane coincides with that physical distinction, and in particular whether its breaking away from the thermodynamic transition line (where Landau and Coulomb remnant breakings demonstrably diverge at β<2\beta < 2) carries any gauge-invariant phase information at all. (3) Settle the general question the source paper poses: whether spontaneous breaking of a global subgroup of a local gauge symmetry can ever be guaranteed to signal a change in the physical state when no thermodynamic nonanalyticity accompanies it. The question is resolved either by a criterion that identifies a symmetry whose breaking coincides with a gauge-invariantly detectable physical distinction in this theory class — the admissible class of symmetries being the global subgroups of the local gauge symmetry, including the three named candidates and any other global subgroup of the electric or dual magnetic symmetry left unfixed by a gauge choice, together with any gauge-invariant global symmetry of the Higgs sector, such as the custodial (spin-glass) symmetry — or by a demonstration covering all global subgroups of the local gauge symmetry and all gauge choices, not merely the three named candidates, that no remnant-global-subgroup breaking carries such information. The two branches differ in scope — the negative branch ranges over the remnant global subgroups of the local gauge symmetry, the positive branch over the wider admissible class — but each is a sufficient resolution on its own: answering either the remnant-subgroup question or the broader global-symmetry question completes the task.

References

  1. W. Caudy and J. Greensite, 'On the Ambiguity of Spontaneously Broken Gauge Symmetry', Phys. Rev. D 78, 025018 (2008), DOI 10.1103/PhysRevD.78.025018, arXiv:0712.0999, https://arxiv.org/abs/0712.0999
  2. S. Elitzur, 'Impossibility of Spontaneously Breaking Local Symmetries', Phys. Rev. D 12, 3978 (1975), DOI 10.1103/PhysRevD.12.3978, https://doi.org/10.1103/PhysRevD.12.3978
  3. E. Fradkin and S. H. Shenker, 'Phase Diagrams of Lattice Gauge Theories with Higgs Fields', Phys. Rev. D 19, 3682 (1979), DOI 10.1103/PhysRevD.19.3682, https://doi.org/10.1103/PhysRevD.19.3682
  4. K. Osterwalder and E. Seiler, 'Gauge field theories on a lattice', Ann. Phys. 110, 440 (1978), DOI 10.1016/0003-4916(78)90039-8, https://doi.org/10.1016/0003-4916(78)90039-8
  5. T. Kugo and I. Ojima, 'Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem', Prog. Theor. Phys. Suppl. 66, 1 (1979), DOI 10.1143/PTPS.66.1, https://doi.org/10.1143/PTPS.66.1
  6. A. Di Giacomo, B. Lucini, L. Montesi and G. Paffuti, 'Color confinement and dual superconductivity of the vacuum. I', Phys. Rev. D 61, 034503 (2000), DOI 10.1103/PhysRevD.61.034503, arXiv:hep-lat/9906024, https://arxiv.org/abs/hep-lat/9906024
  7. J. Greensite and Š. Olejník, 'Vortices, symmetry breaking and temporary confinement in SU(2) gauge-Higgs theory', Phys. Rev. D 74, 014502 (2006), DOI 10.1103/PhysRevD.74.014502, arXiv:hep-lat/0603024, https://arxiv.org/abs/hep-lat/0603024
  8. J. Greensite and B. Lucini, 'Is Confinement a Phase of Broken Dual Gauge Symmetry?', Phys. Rev. D 78, 085004 (2008), DOI 10.1103/PhysRevD.78.085004, arXiv:0806.2117, https://arxiv.org/abs/0806.2117
  9. C. Bonati, G. Cossu, A. D'Alessandro, M. D'Elia and A. Di Giacomo, 'On the phase diagram of the Higgs SU(2) model', PoS LAT2008, 252 (2008), DOI 10.22323/1.066.0252, arXiv:0901.4429, https://arxiv.org/abs/0901.4429
  10. C. Bonati, G. Cossu, M. D'Elia and A. Di Giacomo, 'Phase diagram of the lattice SU(2) Higgs model', Nucl. Phys. B 828, 313 (2010), DOI 10.1016/j.nuclphysb.2009.12.003, arXiv:0911.1721, https://arxiv.org/abs/0911.1721
  11. S. Friederich, 'Gauge Symmetry Breaking in Gauge Theories: In Search of Clarification', European Journal for Philosophy of Science 3, 203 (2013), DOI 10.1007/s13194-012-0061-y, arXiv:1107.4664, https://arxiv.org/abs/1107.4664
  12. J. Greensite and K. Matsuyama, 'Confinement criterion for gauge theories with matter fields', Phys. Rev. D 96, 094510 (2017), DOI 10.1103/PhysRevD.96.094510, arXiv:1708.08979, https://arxiv.org/abs/1708.08979
  13. J. Greensite and K. Matsuyama, 'What symmetry is actually broken in the Higgs phase of a gauge-Higgs theory?', Phys. Rev. D 98, 074504 (2018), DOI 10.1103/PhysRevD.98.074504, arXiv:1805.00985, https://arxiv.org/abs/1805.00985
  14. J. Greensite and K. Matsuyama, 'On the distinction between color confinement, and confinement', PoS Confinement2018, 046 (2018), DOI 10.22323/1.336.0046, arXiv:1811.01512, https://arxiv.org/abs/1811.01512
  15. J. Greensite and K. Matsuyama, 'Higgs phase as a spin glass and the transition between varieties of confinement', Phys. Rev. D 101, 054508 (2020), DOI 10.1103/PhysRevD.101.054508, arXiv:2001.03068, https://arxiv.org/abs/2001.03068
  16. J. Greensite and K. Matsuyama, 'Symmetry, Confinement, and the Higgs Phase', Symmetry 14, 177 (2022), DOI 10.3390/sym14010177, arXiv:2112.06421, https://arxiv.org/abs/2112.06421
  17. D. Ward, 'Higgs phase as a spin glass phase in D=5 dimensional SU(2) gauge-Higgs theory', Phys. Rev. D 105, 034516 (2022), DOI 10.1103/PhysRevD.105.034516, arXiv:2112.12220, https://arxiv.org/abs/2112.12220
  18. S. Gangwani, J. Greensite and A. Oualla, 'Aspects of the Higgs phase in SU(2) x U(1) lattice gauge Higgs theory', Phys. Rev. D 109, 054504 (2024), DOI 10.1103/PhysRevD.109.054504, arXiv:2312.02117, https://arxiv.org/abs/2312.02117
  19. C. Bonati, A. Pelissetto and E. Vicari, 'Three-dimensional Abelian and non-Abelian gauge Higgs theories', Phys. Rep. 1133, 2386 (2025), DOI 10.1016/j.physrep.2025.05.004, arXiv:2410.05823, https://arxiv.org/abs/2410.05823
  20. M. Grady, 'Solution to the gauge-Higgs analyticity paradox', arXiv:1502.04362 (2015), https://arxiv.org/abs/1502.04362