Exact spectral-function sum rules beyond asymptotic freedom: the domain of validity of the Bernard-Duncan-LoSecco-Weinberg relations

Bernard, Duncan, LoSecco and Weinberg (Phys. Rev. D 12, 792, 1975) used the operator-product expansion to determine the complete set of exact, quark-mass-dependent spectral-function sum rules for vector and axial-vector (and scalar/pseudoscalar) current propagators in a chiral SU(4) x SU(4) theory broken explicitly by quark masses and spontaneously; their derivation is rigorously valid only in asymptotically free theories, where all anomalous dimensions vanish at short distance and the Wilson coefficients scale canonically. Whether the same sum rules remain exact in theories that are not asymptotically free was explicitly left open and remains open: no general validity criterion beyond perturbative renormalization-group ones, and no proof or controlled refutation for a specified ultraviolet-complete non-asymptotically-free theory, exists in the literature. The 1975 phenomenological obstacles bundled with the question - inability to isolate vector from axial parts within the sum-rule set, saturation by low-lying mesons, and slowly convergent logarithmic sum rules - have since been overtaken: hadronic tau-decay data now separate the V and A spectral functions, Borel-type QCD sum rules brought saturation under control, renormalization-group-improved moment analyses incorporate the logarithmic corrections systematically, and lattice spectral-density reconstruction permits direct tests in strongly coupled gauge theories beyond QCD. The surviving open core is the domain-of-validity question, on which current strongly-coupled beyond-Standard-Model phenomenology still conditions ('as happens in asymptotically free gauge theories').

Background

Sum rules in quantum field theory are exact integral relations among the spectral functions of current two-point correlation functions. For a current $J$, the two-point function admits a Kallen-Lehmann representation: it is a superposition of free propagators weighted by spectral functions $\rho(s)$ of the invariant mass squared $s$, which are positive and physically measurable (e.g., through the cross section for producing the corresponding hadrons in $e^+e^-$ annihilation or in the hadronic decays of the $\tau$ lepton). A sum rule states that a moment $\int_0^\infty ds\, s^n\rho(s)$ of a spectral function, or of a linear combination of spectral functions, equals a low-energy constant; it is superconvergent when the integrand falls fast enough at $s\to\infty$ that the integral converges without subtractions.

The prototype is due to Weinberg (1967). Assuming the chiral current algebra (the framework of equal-time commutation relations among current components) of the vector current $J_V$ and the axial-vector current $J_A$ (its chiral partner; the combinations $J_V\pm J_A$ couple to right- and left-handed fields - the "$V\pm A$" sectors) together with sufficiently soft high-energy behavior of the difference $\rho_V-\rho_A$, two sum rules follow: $\int(\rho_V-\rho_A)ds=f_\pi^2$ and $\int s(\rho_V-\rho_A)ds=0$, where $f_\pi$ is the pion decay constant. Saturated by the $\rho$ and $a_1$ mesons (the lightest vector and axial-vector mesons, respectively), they correlate vector- and axial-vector-meson masses and couplings; together with the contemporaneous result of Das, Guralnik, Mathur, Low and Young, they predict the electromagnetic mass splitting of the pion. These derivations, however, rested on assumed - not derived - high-energy behavior.

Wilson's operator product expansion (OPE) - the short-distance expansion of a product of local operators into a series of local operators with computable coefficient functions ("Wilson coefficients") of known scaling - supplies the missing derivation. Bernard, Duncan, LoSecco and Weinberg (1975) (BDLW) gave a systematic procedure: the short-distance behavior of products of vector and axial-vector currents determines the convergence or superconvergence of the first and second spectral sum rules, together with a third sum rule involving only the spin-0 spectral function; conversely, the OPE identifies all - and only - those linear combinations of current propagators whose short-distance behavior is soft enough to yield superconvergent sum rules. Applied to a theory with chiral $\mathrm{SU}(4)\times\mathrm{SU}(4)$ symmetry broken explicitly by quark masses and spontaneously to any subgroup containing the symmetry of the mass matrix, this produced the complete set of exact, mass-dependent sum rules: the quark masses appear explicitly in the coefficients, and the relations reduce to the original first and second Weinberg sum rules in the chiral limit of the light quarks. An appendix proves that the Wilson coefficient functions exhibit the full symmetries of the Lagrangian, so spontaneous symmetry breaking does not spoil the argument.

The load-bearing assumption is asymptotic freedom: the effective coupling must vanish at short distances, as it does in QCD. Then all anomalous dimensions (the shifts of operator scaling dimensions away from their classical values) vanish asymptotically and the Wilson coefficients scale canonically, up to logarithms; BDLW's cancellation of the leading short-distance singularities then goes through exactly. BDLW state that the sum rules hold if the quark-mass matrix has symmetry $\mathrm{SU}(2)\times\mathrm{U}(1)\times\mathrm{U}(1)$ and the theory is asymptotically free; the derivation is "strictly true only for asymptotically free theories," with results "expected to apply for a range of other theories" but not demonstrated there. In a theory that is not asymptotically free - for example one flowing to a nontrivial (non-Gaussian) ultraviolet fixed point, a short-distance limit in which the coupling remains finite instead of vanishing - operators and coefficient functions acquire anomalous dimensions, the canonical cancellations behind superconvergence are unproven, and the generic expectation is that they fail. (Earlier attempts by Borchardt-Mathur and by Hagiwara-Mohapatra had to assume special values of the anomalous dimension of the quark-mass operator; the asymptotic-freedom assumption removes this.) The perturbative renormalization-group criteria of Weisberger (1976) formulate the superconvergence conditions in terms of the anomalous dimensions of the currents and of the symmetry-breaking operators at the short-distance fixed point; no comparably general criterion beyond that perturbative setting is known.

The question has present-day bite. In precision QCD, Weinberg-type sum rules and their logarithmic (renormalization-group) refinements underlie the moment analyses of hadronic $\tau$-decay data used to determine the strong coupling $\alpha_s$ and chiral low-energy constants. In theories of dynamical electroweak symmetry breaking - technicolor and composite-Higgs models, in which a new strongly coupled gauge sector plays the role that QCD plays for pions - the sum rules are routinely used as ultraviolet matching constraints, for example to bound the masses of heavy vector and axial-vector resonances through the oblique $S$ parameter (an electroweak precision observable sensitive to new strong dynamics). Whether those uses are theorems or assumptions depends precisely on the unresolved domain-of-validity question.

Problem Statement

Consider a quantum field theory with a global chiral symmetry $G$ (such as $\mathrm{SU}(N)\times\mathrm{SU}(N)$) broken (a) explicitly by a fermion mass matrix and (b) spontaneously to a subgroup containing the symmetry of the mass matrix, with conserved vector and axial-vector currents and their spectral functions. For asymptotically free theories, the operator-product-expansion analysis of Bernard, Duncan, LoSecco and Weinberg determines the complete set of exact, mass-dependent spectral-function sum rules: the linear combinations of $V\pm A$ (and scalar/pseudoscalar) spectral functions whose leading short-distance singularities cancel, so that their integrals converge to computable low-energy constants. The open problem is the domain of validity of these sum rules outside the asymptotically free class, in three linked parts:

1. Criteria. Establish necessary and/or sufficient conditions under which the superconvergent combinations remain superconvergent in theories that are not asymptotically free.

2. Determinations in concrete theory classes. For specified non-asymptotically-free theories - for example gauge theories above the boundary of asymptotic freedom, theories at a non-Gaussian ultraviolet fixed point, or strongly coupled sectors of beyond-Standard-Model models - determine whether the mass-dependent sum rules hold exactly, fail, or are modified (by non-canonical powers or logarithms). It suffices to give such a determination in any single specified theory of this kind; the listed classes are illustrative. Existing evidence is either perturbative, effective-model (the second Weinberg sum rule is violated by the condensate term in the Nambu-Jona-Lasinio model, a four-fermion effective theory of chiral symmetry breaking), or based on uncontrolled approximations (Schwinger-Dyson studies near conformality, i.e. integral equations for the full Green functions closed by truncation assumptions); no controlled nonperturbative determination for an ultraviolet-complete non-asymptotically-free theory is known.

3. Replacement structure. When exact equality fails, characterize what survives in its place: inequality-type constraints on spectral functions following from positivity, analyticity and ultraviolet data, or modified sum rules with anomalous-dimension-dependent powers.

Each of the three parts is a separately attackable sub-question that can be pursued on its own: a controlled result settling part 1 or part 2 answers the domain-of-validity question as posed, while part 3 is required only for the theories in which exact equality fails.

The original 1975 paper bundled this question with phenomenological obstacles - the absence of sum-rule combinations isolating $V$ from $A$, saturation by low-lying mesons with poorly known axial couplings and unknown mixing of the flavor-diagonal mesons (including 'paracharmonium', the spin-singlet charmonium), and slowly converging logarithmic sum rules. As shown by subsequent work, separated vector and axial spectral functions from hadronic tau decays, Borel-type sum-rule technology, renormalization-group-improved moment analyses, and lattice spectral-density reconstruction have overtaken those obstacles; the problem stated here is the surviving fundamental core, which recent work still leaves open.

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