In his 1971 paper proving that spontaneously broken (Higgsed) Yang-Mills theories are renormalizable and unitary, 't Hooft explicitly acknowledged three unfinished issues: (1) the extension to arbitrary loop order of the combinatorial proof that anomalies are absent and the Ward identities hold; (2) the construction of a fully explicit gauge-covariant regularization scheme with a complete all-orders consistency proof of the renormalization procedure; (3) whether spontaneous hiding of gauge invariance, with the Yang-Mills bosons becoming massive vector particles, can occur in pure massless Yang-Mills theory without introducing elementary scalar fields. Items (1) and (2) were settled within a few years by dimensional regularization ('t Hooft-Veltman, 1972) and the all-orders renormalizability proofs of Lee-Zinn-Justin, Becchi-Rouet-Stora and Gross-Jackiw, and are now textbook material. Item (3) survives as a precise open problem in two coupled facets: the rigorous existence of a mass gap in four-dimensional pure Yang-Mills theory (the Clay Millennium 'Yang-Mills & the Mass Gap' problem, still unsolved), and the physical mechanism of dynamical gauge-boson mass generation (Schwinger-mechanism dynamical gluon mass and spontaneous BRST breaking versus a confinement-type gap without Higgs-like symmetry hiding), which remains actively contested without an accepted resolution.

Background

A Yang-Mills theory is a quantum field theory whose Lagrangian is invariant under local (spacetime-dependent) transformations of a non-Abelian gauge group such as SU(2), generalizing the gauge principle of electromagnetism. Quantizing such a theory requires choosing a gauge, which introduces unphysical Faddeev-Popov ghost fields (fictional scalar particles with Fermi statistics) whose contributions cancel the unphysical polarization states of the gauge fields. The residual symmetry of the gauge-fixed action is encoded in Ward identities (also called Slavnov-Taylor identities): relations among Green's functions that follow from the independence of physical amplitudes from the arbitrary gauge choice. Renormalizability - the property that ultraviolet divergences can all be absorbed into a finite number of parameters of the original Lagrangian - holds only if renormalization does not violate these identities. Quantum anomalies (such as the Adler-Bell-Jackiw anomaly, an unexpected symmetry violation caused by an illegal shift of integration variables in loop integrals, first found in the axial-current Ward identities) can spoil the identities and destroy renormalizability, so their absence must be proven.

A separate problem is mass. A naive Proca-type mass term for the gauge fields breaks local gauge invariance and yields a provably non-renormalizable theory. The Brout-Englert-Higgs mechanism evades this: a scalar field with a non-zero vacuum expectation value 'hides' the local gauge invariance (the vacuum is not invariant, although the Lagrangian is), some or all gauge bosons become massive vector particles, and the theory remains renormalizable and unitary. 't Hooft's 1971 paper, building on his companion proof for massless Yang-Mills fields (Nucl. Phys. B33 (1971) 173), constructed and controlled such models - an SU(2) model with an isospin-one scalar, an isospin-one-half doublet model, and an SU(2) times U(1) model with a photon and massive rho-like mesons - and thereby laid the perturbative foundation of the electroweak Standard Model. Within the paper, the proof of consistency rests on Ward identities derived from the independence of amplitudes from arbitrary source functions, and the paper states verbatim, in its introduction and section 4, that the all-orders extension of the combinatorial proof (including anomaly absence) is known to the author but 'rather involved and we shall not present it here', and that the one-loop gauge-covariant regulator (a fifth Minkowski dimension) could be replaced by 'more dimensions' for an all-orders consistency proof, likewise not presented.

Both of these technical gaps were closed almost immediately by the follow-up literature, which is now textbook. 't Hooft and Veltman (Nucl. Phys. B44 (1972) 189) introduced dimensional regularization, an explicit regularization and renormalization procedure in which, as their abstract states, 'in non-anomalous cases also Ward identities are satisfied at all stages'; Lee and Zinn-Justin (Phys. Rev. D5 (1972) 3121, 3137 and subsequent parts) gave the complete all-orders renormalizability proof for spontaneously broken gauge theories; Becchi, Rouet and Stora (Ann. Phys. 98 (1976) 287) proved the Slavnov identities to all orders within the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization framework (the standard order-by-order procedure for subtracting ultraviolet divergences) using what is now called BRST symmetry, a residual global fermionic symmetry of gauge-fixed actions; and Gross and Jackiw (Phys. Rev. D6 (1972) 477) derived the all-orders consistency conditions for anomalies, which are automatically satisfied in vector-like scalar-only models such as those of the source paper.

The third acknowledged issue has a different character and remains open. The paper's introduction states verbatim that 'it is very well possible that all Yang-Mills bosons become massive vector particles' when gauge invariance is hidden, and asks whether this can happen 'with massless Yang-Mills fields alone', whereas in the paper's models 'additional scalar fields are introduced'. Modern physics decomposes this question. Pure Yang-Mills theory is asymptotically free (its coupling weakens at short distances) and therefore strongly coupled in the infrared; lattice gauge theory (Wilson, 1974) showed quark confinement - the absence of isolated color charges from the physical spectrum, which contains only color-neutral bound states - in the strong-coupling regime and provided a manifestly gauge-invariant regulator. Elitzur (1975) proved that a local gauge symmetry cannot be spontaneously broken at all - gauge symmetry is a redundancy, not a physical symmetry - so 'hiding' must be assessed through gauge-invariant observables (the spectrum; Wilson loops, traces of the gauge field around closed contours whose area-law behavior signals confinement; and center symmetry, invariance under multiplication of the gauge field by elements of the center of the gauge group, which is the global remnant of local gauge invariance relevant in pure Yang-Mills theory) or through order parameters in gauge-fixed formulations (such as spontaneous breaking of BRST symmetry). 't Hooft (1978) and Fradkin and Shenker (1979) classified the possible phases of gauge theories. Numerical lattice studies support a positive mass gap (massive color-singlet, i.e. gauge-neutral, bound states called glueballs) in pure Yang-Mills theory, and in Landau gauge (a standard covariant gauge choice) the gluon propagator behaves as if the gluon had acquired a dynamical mass of order half a GeV. The mechanism side is studied through the Schwinger mechanism (Schwinger, 1962: a massless pole in the vacuum polarization makes the gauge boson massive without explicit symmetry breaking; exact in two dimensions), dynamical mass generation in continuum QCD (Cornwall, 1982, and the modern program based on Schwinger-Dyson equations - the exact infinite tower of integral equations relating the Green's functions of a quantum field theory - and the pinch technique, a reorganization of perturbation theory producing gauge-invariant effective propagators and vertices; reviewed by Aguilar, Binosi and Papavassiliou, 2016), Curci-Ferrari-type effective models (gauge-fixed Lagrangians augmented by gluon and ghost mass terms that softly break BRST symmetry) and spontaneous BRST breaking (Fazio and Smetana, 2026), the Gribov-Zwanziger scenario (a restriction of the gauge-fixed path integral designed to avoid Gribov copies, the redundant gauge-field configurations that a smooth gauge condition fails to remove, with characteristic modifications of the infrared gluon and ghost propagators), and 'Higgs-Schwinger mechanism' scenarios (Comitini, 2026). The rigorous existence side is precisely the Clay Mathematics Institute Millennium Prize problem 'Yang-Mills & the Mass Gap': construct four-dimensional quantum Yang-Mills theory and prove a mass gap - still officially unsolved.

Problem Statement

The source paper ('t Hooft, Nucl. Phys. B35 (1971) 167) acknowledged three unfinished issues; two of them - the all-orders combinatorial proof that anomalies are absent and the Ward identities hold, and the construction of an explicit gauge-covariant regularization scheme with a complete all-orders consistency proof of the renormalization procedure - are settled by dimensional regularization and the Lee-Zinn-Justin, Becchi-Rouet-Stora and Gross-Jackiw literature (references 3-7), and are not part of the open target. The open target is the source's third question.

Can spontaneous hiding of gauge invariance - in which some or all Yang-Mills bosons become massive vector particles - occur in a pure massless Yang-Mills theory, without the introduction of elementary scalar fields? For four-dimensional pure Yang-Mills theory (compact gauge group, no elementary scalar or fermion fields), determine, at the level of the continuum, infinite-volume, nonperturbative quantum theory: (a) existence - does the theory possess a dynamically generated mass scale, that is, a strictly positive mass gap; and (b) mechanism - by what mechanism is any such mass generated: Higgs/Schwinger-type dynamical mass generation for the gauge fields (as in dynamical-gluon-mass, Higgs-Schwinger and spontaneous-BRST-breaking scenarios), or a confinement-type gap without Higgs-like symmetry hiding?

The rigorous core of part (a) is the Clay Millennium Prize problem 'Yang-Mills & the Mass Gap' in its standard formulation (construct quantum Yang-Mills theory on four-dimensional Minkowski space and prove a mass gap), which is the authoritative formalization of the existence facet of this derived question. Solving (a) does not by itself decide (b): a strictly positive mass gap is a gauge-invariant spectral statement that does not determine whether the gauge fields acquire a dynamical (Higgs/Schwinger-type) mass or whether the gap is of confinement type, so the mechanism facet must be established on its own terms in the continuum theory. Lattice discretizations, gauge-fixed effective models (for example Curci-Ferrari or Gribov-Zwanziger type), and studies with added matter fields are admissible as supporting evidence but are not by themselves answers: a demonstration that holds only at finite lattice spacing, only in a fixed gauge, or only after adding elementary fields does not resolve the source question.

References

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  19. A.C. Aguilar, D. Binosi and J. Papavassiliou, "The gluon mass generation mechanism: a concise primer", Frontiers of Physics (Beijing) 11 (2016) 111203. arXiv:1511.08361, https://arxiv.org/abs/1511.08361; DOI: 10.1007/s11467-015-0517-6, https://doi.org/10.1007/s11467-015-0517-6.
  20. G. Comitini, "Nonperturbative Higgs-Schwinger mechanism at the origin of the gluon mass and color confinement", preprint (2026). arXiv:2603.22181, https://arxiv.org/abs/2603.22181. Unrefereed preprint; a contested proposal, not an accepted resolution of part (b).
  21. A.R. Fazio and A. Smetana, "Spontaneous BRST symmetry breaking in infrared QCD", preprint (2026). arXiv:2603.29401, https://arxiv.org/abs/2603.29401. Unrefereed preprint; a contested proposal, not an accepted resolution of part (b).
  22. Clay Mathematics Institute, "Yang-Mills & the Mass Gap", official Millennium Prize problem page, https://www.claymath.org/millennium/yang-mills-the-maths-gap/. Problem marked Unsolved; "no proof of this property is known".