Asymptotic freedom with radiatively induced Higgs self-couplings: the open problems left by Salam and Strathdee (1978)

The problem is the set of open questions left by Salam and Strathdee's 1978 argument that asymptotic freedom is not disturbed in a non-Abelian gauge theory whose bare Higgs self-couplings vanish - that is, that the effective Higgs couplings induced by gauge-particle exchange are finite and gauge invariant on the mass shell, so that radiative symmetry breaking is consistent with ultraviolet completeness and yields Higgs and vector masses of the same order. The 1978 argument rests on a one-loop, axial-gauge spectral analysis (the negativity of the spectral weight excluding spacelike CDD poles in the running coupling), an ansatz of replacing renormalized by running couplings whose self-consistent justification was not given, an approximated threshold treatment, and a gauge-dependent effective potential. The authors themselves listed what remains to be done: extension of the spectral analysis to arbitrary gauges, a self-consistent treatment of overlapping insertions, explicit computation of mass and multiparticle threshold effects, an off-shell gauge-invariant treatment of symmetry breaking, concrete model computations of the predicted mass hierarchy, and - speculatively - the same running-coupling methodology for quantum gravity. Later literature absorbed the perturbative parts (multi-loop beta functions for general gauge theories, Nielsen identities, triviality bounds), transformed the arbitrary-gauge question into the gauge-independent effective-charge program, refuted the perturbative gravity version (two-loop divergence of pure Einstein gravity) while spawning its nonperturbative descendant (asymptotic safety), and revived the core question as an active program of asymptotically free scaling solutions and complete asymptotic freedom in Higgs-Yukawa models, whose own practitioners state that no standard-model-like realization exists and that nonperturbative verification is pending. The problem remains open.

Background

Asymptotic freedom - the decrease of the effective coupling at short distances, which makes a theory more weakly coupled the higher the energy - is the property that makes non-Abelian gauge theories such as quantum chromodynamics ultraviolet-complete in perturbation theory. Renormalizable gauge theories that realize spontaneous symmetry breaking through the Higgs mechanism, however, require elementary scalar fields, and independent (noncomputable) quartic scalar self-couplings of the type commonly written as a term $\lambda\phi^4$ in the Lagrangian generically destabilize asymptotic freedom: the scalar contribution to the renormalization-group (RG) beta function has the wrong sign, as had been catalogued already by Cheng, Eichten and Li in their study of Higgs phenomena in asymptotically free gauge theories.

In 1978 Salam and Strathdee took a different route. Consider a non-Abelian gauge theory in which the bare scalar quartic couplings are simply absent from the Lagrangian. Scalar-scalar scattering amplitudes are then induced through gauge-meson exchange. If these induced amplitudes were truly logarithmically divergent, renormalized $\lambda$ counterterms would be needed and the scalar couplings would reappear to spoil asymptotic freedom. The paper argues that they are not: using renormalization-group-improved vertex and self-energy insertions in the unmodified graphs - well approximated, for the purpose of counting potential infinities, by replacing renormalized coupling constants with running coupling constants (the paper's 'Dyson insertions', in the spirit of Dyson's resummation of propagator and vertex corrections) - the induced effective Higgs couplings are convergent and gauge invariant on the mass shell (that is, for external particles carrying their physical, real momenta); consequently, at weak coupling, radiative symmetry breaking gives Higgs and vector masses of the same order of magnitude.

The technical machinery goes back to Gell-Mann and Low, who tied the effective coupling of quantum electrodynamics to the spectral function of the transverse photon propagator. In this spectral representation the running coupling is a dispersion integral over the spectral weight of the gauge propagator. If that spectral weight were positive - as it is in quantum electrodynamics - the dispersion integral would generate a spurious pole of the running coupling on the spacelike axis, a Castillejo-Dalitz-Dyson (CDD) pole, the spectral analogue of the Landau-pole obstruction (the growth of the effective charge with energy that signals the ultraviolet failure of QED-like theories). Frenkel and Taylor computed the one-loop spectral functions in the axial gauge (a gauge condition of the form $n\cdot A=0$ with fixed four-vector $n$, in which the coupling renormalization resides entirely in the gauge-particle propagator) and found the essential negativity of the spectral weight, here denoted $\tilde\rho_1$ - in the paper's words the 'hallmark of asymptotic freedom' - which guarantees the absence of spacelike CDD poles. The proof that vertex and scalar self-energy insertions cancel against each other (leaving only the gauge-propagator insertions that define the running coupling) was given in the paper's Appendix A via Ward identities, but only for a U(1) theory and for non-Abelian theories in the axial gauge.

The paper's conclusions rest on acknowledged approximations. The ansatz of replacing renormalized by running couplings is justified only for infinity counting and on-shell amplitudes; the self-consistent improvement that would treat overlapping insertions (which happen to be absent for scalar-scalar amplitudes but plague the improvement of the RG estimates themselves) is left open and expected to be 'severe'. The threshold and mass modifications of the spectral functions for gauge-boson mass $M^2\neq 0$ and realistic multiparticle thresholds are approximated by an ansatz rather than computed. The effective potential - the standard tool for radiative symmetry breaking since Coleman and Weinberg's work on radiative corrections as the origin of symmetry breaking - is gauge invariant only at vanishing external momentum, hence only for on-shell (massless) scalars, a weakness already understood through Nielsen's identity relating effective potentials in different gauges. The group-theoretic estimates of the scalar-to-vector mass hierarchy are order-of-magnitude only. Finally, the paper speculatively extends the same Dyson-insertion/running-coupling methodology to quantum gravity, suggesting that the running Newton constant decreases at short distances (an 'asymptotically free' gravity with an effective short-distance repulsion compensating Newtonian attraction), based on a coefficient it derives as positive in its Eq. (20), while itself hedging that gravity is only 'likely' asymptotically free.

The modern context frames these questions sharply. Pure $\lambda\phi^4$ theory is believed to be trivial (noninteracting) in four dimensions, and gauge theories with elementary scalars are believed nontrivial only for limited ranges of couplings, leading to upper bounds on Higgs masses (Callaway's 'triviality pursuit'). The two-loop RG for general gauge theories including scalar quartics is standard (Machacek and Vaughn), so the perturbative bookkeeping of the 1978 argument is now routine. The question of whether scalar self-interactions can be compatible with, or even part of, asymptotic freedom has been revived as an active program: 'complete asymptotic freedom', where all marginal couplings (gauge, Yukawa (fermion-scalar), and quartic) are simultaneously asymptotically free (Hansen et al.), and the 'asymptotically free scaling solutions' of non-Abelian Higgs models constructed with generalized boundary conditions and functional RG methods (the functional renormalization group: exact flow equations for the full effective action rather than for single couplings; Gies, Zambelli and collaborators), whose papers explicitly position themselves as descendants of the Salam-Strathdee program.

Problem Statement

The problem, as posed and explicitly left open by Salam and Strathdee (1978), is to complete the argument that asymptotic freedom is not disturbed in a non-Abelian gauge theory whose bare Higgs self-couplings vanish - that is, to establish, gauge-independently and beyond the one-loop, axial-gauge, threshold-approximation level of the 1978 analysis, that the effective Higgs couplings induced by gauge-particle exchange are finite and gauge invariant on the mass shell, with the radiatively induced symmetry breaking and its mass hierarchy under control. Concretely, as acknowledged in the source paper, the following must be done:

(1) Arbitrary-gauge spectral foundation: carry the Frenkel-Taylor one-loop axial-gauge spectral analysis through for arbitrary gauges, and trace the origin of the negativity of the spectral weight $\tilde\rho_1(x^2)$ - or at least of its dispersion integral - which is what excludes spacelike Castillejo-Dalitz-Dyson (CDD) poles in the running coupling; likewise extend the Ward-identity proof that vertex and scalar self-energy insertions cancel, given in the source only for a U(1) theory and for the axial gauge, to arbitrary gauges.

(2) Self-consistent higher-order improvement: construct the self-consistent methodology for improving the renormalization-group estimates of the running coupling and of the vertex and self-energy insertions themselves - the systematic replacement of renormalized by running couplings - treating the overlapping insertions that the source flags as 'severe' for this purpose.

(3) Explicit thresholds: compute explicitly, rather than approximate by an ansatz, the mass- and threshold-dependent modifications of the spectral functions for gauge-boson mass $M^2\neq 0$ and realistic multiparticle thresholds, including the finite-$k^2$ behavior and possible infrared subtleties.

(4) Off-shell gauge-invariant symmetry breaking: develop a fuller off-shell, gauge-invariant treatment of symmetry breaking with radiatively induced quartic terms, beyond the effective-potential method that is gauge invariant only for on-shell (massless) scalars.

(5) Concrete model computations: verify the group-theoretic estimates - the order-of-magnitude scalar-to-vector mass-squared ratio $\frac{g^{2}}{6\pi^{2}}\left[-\frac{11}{3}C(G)-\ldots S_3-\ldots F\right]^{1/2}$, further multiplied by a group-theoretic factor depending on the Higgs and fermion representation content (with $C(G)$ the quadratic Casimir of the gauge group and $S_3$, $F$ the scalar and fermion group-theoretic trace factors; the ellipses stand for the scalar- and fermion-sector coefficients as displayed in the source's Eq. (17a)), and the induced-coupling dependence on these factors - in concrete model computations including higher-loop corrections and explicit representation assignments, to assess the viability of the predicted Higgs-to-vector mass hierarchy at weak coupling.

(6) Quantum-gravity extension (the source's speculative component): determine whether the same Dyson-insertion/running-coupling methodology applies to quantum gravity - whether gravity Feynman rules and insertion prescriptions exist that avoid double counting and negative-norm states; whether a consistent renormalization group for the running gravitational coupling exists whose logarithmic factors do not conspire to produce anomalous power-law behavior of on-shell amplitudes; and whether the coefficient of $k^2$ in the source's Eq. (20) for the putative running Newton constant is confirmed to be positive in a controlled calculation - the inference on which the source's own hedged conclusion that gravity is 'likely' asymptotically free rests.

The central objective is the full, gauge-independent foundation for asymptotic freedom with radiatively induced Higgs couplings; the components above are parts of that objective.

References

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