Non-perturbative ultraviolet fate of quasi-stable (temporarily free) massive non-Abelian gauge theories

Politzer and Ross (1974) showed that non-Abelian gauge theories in which all vector bosons acquire masses through the Higgs mechanism are not asymptotically free once the scalar quartic couplings are included, but can be 'quasi-stable': over a large but finite momentum range their coupled renormalization-group flow mimics an ultraviolet-stable fixed point at the origin, before the scalar quartic combinations reach order unity and perturbative scaling breaks down totally. The open problem is the non-perturbative completion of that analysis: determine what happens beyond the perturbative breakdown — a Landau pole or triviality with no interacting continuum limit, an interacting ultraviolet fixed point (asymptotic safety), a cyclic (limit-cycle) flow, or other genuinely strong-coupling behavior — and determine non-perturbatively the basin of attraction and duration of the quasi-stable window. The original strong-interaction motivation is superseded by QCD and the timelike-continuation facet was settled by later two-loop timelike computations, but the non-perturbative ultraviolet fate of the fully-Higgsed class remains open; the electroweak Standard Model and massive-vector extensions such as coloron models (theories with massive colored gauge bosons), which retain an unbroken massless gauge sector, are closely related but are not themselves fully Higgsed.

Background

In the early 1970s, deep-inelastic electron–proton scattering exhibited approximate Bjorken scaling (the dimensionless structure functions depend only weakly on momentum transfer), while the discovery of asymptotic freedom in massless non-Abelian gauge theories explained why such scaling should hold up to logarithmic corrections [Gross and Wilczek 1973; Politzer 1973]. A non-Abelian gauge theory is a gauge theory whose symmetry group, such as SU(n), is generated by charges that do not commute; its renormalization-group (RG) flow, the dependence of the renormalized couplings on the momentum scale, can drive the gauge coupling $g$ toward zero at short distances with $\beta_g = -a(g/4\pi)^2 g + \dots$ (one-loop coefficient $a = \frac{11}{3}n - \frac{4}{3}c - \frac{m}{6}$ for $m$ complex fundamental scalar multiplets and fermions with trace normalization $c$), making the theory weakly coupled at high energies. Massless gauge theories, however, suffer severe infrared singularities, so it was natural to ask whether the strong interaction could instead be a gauge theory in which all vector bosons are massive through the Higgs mechanism — spontaneous symmetry breaking by elementary scalar fields with quartic self-couplings $h_i$, which yields massive gauge bosons while preserving renormalizability [Cheng, Eichten and Li 1974].

Politzer and Ross (1974) analyzed exactly this class: SU(n) gauge theory with $m$ complex scalar multiplets in the fundamental representation (the defining $n$-dimensional representation of the group; complete breaking of the gauge symmetry requires $m \ge n-1$) plus fermions. Their central result is that such theories are not asymptotically free as complete systems: the coupled one-loop evolution of the scalar quartics leaves the origin $g = h_i = 0$ not ultraviolet-attractive. Forming the ratios $\Gamma_i = h_i/g^2$, whose one-loop evolution is autonomous, they found that RG trajectories can pass through a narrow 'gap' where the effective gauge and quartic couplings all decrease together; on such trajectories the theory behaves for a large but finite range of RG time $t$ as if the origin were ultraviolet-stable. They called such theories quasi-stable (later authors said 'temporary freedom' or 'temporarily free'), and estimated, for SU(3) with $m=3$ and plausible strong-interaction initial conditions, weak-coupling windows of $\Delta t \approx 4$, large enough to mimic Bjorken scaling (up to powers of $\ln q^2$) over the experimentally accessible domain. Using positivity of one-loop beta functions in the spirit of Coleman and Gross [Coleman and Gross 1973], they further showed that temporary freedom essentially requires non-Abelian gauge fields: Abelian gauge couplings, pure Yukawa theories, and scalar theories with positive quartics all have couplings that grow monotonically. Finally, terms proportional to the products $\bar h_i \bar h_j$ of running quartic couplings enter the anomalous dimensions (the exponents governing the scale dependence of fields and operators) at the next order; once the $\bar h_i \bar h_j$ reach order unity the perturbative evolution, and with it the computed approximate scaling, breaks down completely, at a scale that depends sensitively on the initial conditions. The paper leaves open what happens after that breakdown, as well as two further issues: the relation between Lagrangian mass parameters and the physical spectrum (a strong-coupling question) and the ambiguity of analytically continuing spacelike (Euclidean) vacuum-polarization results to the timelike region needed for $\sigma(e^+e^- \to \mathrm{hadrons})$. The limit-cycle hypothesis for massive vector models of this class is documented in the later literature [Stamenov 1978].

The candidate concepts for the ultraviolet fate are standard. A Landau pole, as in QED, is a finite scale where the coupling diverges; in four-dimensional scalar $\phi^4$-type theories this is tied to triviality: as the ultraviolet cutoff is removed, the interaction must vanish, so no interacting continuum theory exists, a picture supported by non-perturbative lattice studies of the Higgs model [Dashen and Neuberger 1983; Lüscher and Weisz 1988]. An interacting ultraviolet fixed point of the full RG flow would instead define an asymptotically safe continuum theory; such fixed points have been constructed perturbatively in certain gauge-Yukawa theories (theories with both gauge and Yukawa scalar-fermion couplings) in the Veneziano limit (a large number of colors and fermion flavors with fixed ratio), precisely where asymptotic freedom is lost [Litim and Sannino 2014], and are advocated for beyond-asymptotic-freedom gauge theories at large flavor number [Antipin and Sannino 2018], but no construction covers the fully-Higgsed class. The flow could also close into a limit cycle, an RG flow returning periodically to weak coupling; no example is known in any relativistic four-dimensional gauge theory, the established examples being non-relativistic Hamiltonian systems [Glazek and Wilson 2002], though the hypothesis was explored phenomenologically for massive vector models [Stamenov 1978]. A further structural fact matters for any lattice determination: for Higgs fields in the fundamental representation the Higgs and confinement regimes of a gauge-Higgs theory are analytically connected [Fradkin and Shenker 1979], so no local order parameter separates them. What is missing is a non-perturbative determination of the ultraviolet fate of the quasi-stable class itself and of the basin of attraction of its weak-coupling window.

Problem Statement

For the class of four-dimensional SU(n) gauge theories in which the gauge symmetry is completely broken — $m \ge n-1$ complex scalar multiplets in the fundamental (defining) representation so that all vector bosons acquire masses through the Higgs mechanism — together with fermion matter, whose coupled renormalization-group flow exhibits quasi-stability (temporary freedom: all effective gauge and scalar quartic couplings decrease together over a large but finite range of momentum scales before the scalar quartic combinations $\bar h_i \bar h_j$ grow to order unity and invalidate perturbation theory):

(i) Determine the ultraviolet fate of this class beyond the perturbative breakdown scale: does the flow terminate in Landau-pole/trivial behavior in which no interacting continuum limit exists, in an interacting ultraviolet fixed point (asymptotic safety) with its critical behavior, in a cyclic (limit-cycle) flow returning periodically toward weak coupling, or in some other genuinely non-perturbative strong-coupling behavior? The determination must control the regime where the relevant couplings are of order unity and not rest solely on unresummed finite-order perturbation theory.

(ii) Determine the basin of attraction of the quasi-stable window: which non-perturbatively defined set of microscopic (low-scale) initial conditions of the couplings flows through the weak-coupling regime for a range of scales of the order of the perturbatively estimated quasi-stable window (RG time $\Delta t$ of order 4), and what fixes the extent of that window beyond leading order.

The class-level question of part (i) requires a uniform non-perturbative verdict holding for every member, or an explicit classification of the fate across the parameters $n$, $m$ and the fermion content. Part (ii) requires its own non-perturbative determination of the basin of attraction and of the extent of the quasi-stable window. The two parts share one objective — a quantitative, non-perturbative understanding of how initially strong couplings in massive non-Abelian gauge theories approach the weak regime and where and how that approach ends. The timelike-continuation facet has been resolved by later direct two-loop timelike computations, and the original strong-interaction motivation is superseded by QCD; those facets are not part of the open core.

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