Fröb and Cadamuro rigorously constructed the simplest local composite operators of the massless sine-Gordon model in the finite regime β2<4π\beta^2<4\pi: they proved that ∂μφ ∂νφ\partial_\mu\varphi\,\partial_\nu\varphi and the stress tensor need renormalisation beyond free-field normal ordering at every perturbative order, that the renormalised perturbation series for their correlation functions with vertex operators converges (in Euclidean signature and in Lorentzian signature in arbitrary Hadamard states), and that a one-loop, ℏ\hbar-proportional quantum correction must be added to the stress tensor to obtain a conserved quantity. The model, being classically integrable, also possesses an infinite hierarchy of higher-spin conserved currents beyond the stress tensor, and the form-factor literature proposes that — in contrast to the stress tensor — these currents receive no quantum corrections. Extending the Fröb–Cadamuro results to this hierarchy is open: renormalised correlation functions involving the higher currents have not been constructed, convergence of their renormalised perturbation series at comparable generality is unproved, and the conservation status of the renormalised currents is unknown. Partial progress exists — Zanello proved super-renormalisability of all higher currents in perturbative algebraic QFT and, in his dissertation, summability of the expectation values of the renormalised current components for cutoffs of small support — but the analogues of the Fröb–Cadamuro theorems for the full hierarchy, which would settle whether the classical conservation laws survive quantisation without corrections, are missing.

Background

The massless sine-Gordon model is a cornerstone of exactly solvable quantum field theory in two dimensions. It describes a scalar field φ\varphi on the plane whose interaction density is g(x):cos⁡(βφ)(x):g(x):\cos(\beta\varphi)(x):, built on the free massless boson; here β\beta is the coupling and the 'adiabatic' cutoff g∈S(R2)g\in\mathcal{S}(\mathbb{R}^2) (a smooth function decaying at infinity) switches the interaction off at large distances and tames the infrared difficulties of the massless free field in two dimensions, which has no Fock vacuum. Normal ordering :⋅::\cdot: (subtracting contractions with the free two-point function) makes the vertex operators V±β=:e±iβφ:V_{\pm\beta}=:\mathrm{e}^{\pm i\beta\varphi}: well defined. The 'finite regime' β2<4π\beta^2<4\pi is the range in which these operators have finite scaling dimension; at β2=4π\beta^2=4\pi the model becomes a free massive Dirac fermion, the basis of Coleman's equivalence with the massive Thirring model, and the range 4π≤β2<8π4\pi\le\beta^2<8\pi is super-renormalisable but has additional ultraviolet subtleties.

Classically the model is integrable: it admits soliton solutions, a factorised two-particle scattering matrix conjectured by Zamolodchikov and Zamolodchikov to persist exactly in the quantum theory, and an infinite hierarchy of local conserved currents of increasing Lorentz spin. These currents can be constructed by combining Bäcklund transformations (a classical solution-generating symmetry of the sine-Gordon equation) with Noether's theorem; their components are obtained from an explicit recurrence relation, and the stress tensor is the first member of the hierarchy. In the form-factor programme, correlation functions of local fields are expanded in matrix elements between exact scattering states ('form factors'); Babujian, Karowski and collaborators computed the exact form factors of all higher currents of the sine-Gordon model, solved the eigenvalue problem for all higher conserved charges, and found agreement with perturbative Feynman-graph expansions. On this basis it has been proposed that — in contrast to the stress tensor — the higher currents do not receive quantum corrections, i.e. their exact interacting expressions coincide with the normal-ordered classical ones.

On the rigorous side, the model was constructed in perturbative algebraic quantum field theory (pAQFT) by Bahns and Rejzner. In pAQFT, interacting fields are algebraic objects defined by a formal perturbation series (the Gell-Mann–Low series with interaction cutoff gg), renormalisation is the extension of distributions to coinciding points in the Epstein–Glaser scheme, and physical states — in the Lorentzian case, Hadamard states, i.e. states whose two-point function has the universal short-distance singularity structure of the Minkowski vacuum — are chosen only after the algebra is fixed. Bahns and Rejzner proved that in the finite regime the perturbation series for the S-matrix, for the derivative ∂μφ\partial_\mu\varphi and for the vertex operators converge for cutoffs of sufficiently small support.

Fröb and Cadamuro extended this to the simplest composite local operators beyond vertex operators: Oμν=∂μφ ∂νφO_{\mu\nu}=\partial_\mu\varphi\,\partial_\nu\varphi and the stress tensor TμνT_{\mu\nu}. Working in the finite regime β2<4π\beta^2<4\pi, they proved (Theorems 1–6 of their paper) that: (i) after removal of infrared and ultraviolet cutoffs, the renormalised expectation values of OμνO_{\mu\nu} and TμνT_{\mu\nu} together with vertex operators are well-defined distributions, with additional (logarithmically cutoff-divergent) counterterms required at every order and the finite renormalisation freedom determined explicitly; (ii) the renormalised perturbation series for these expectation values converges — in Euclidean signature for non-negative Schwartz cutoffs gg, and in Lorentzian signature for arbitrary Schwartz gg with constants depending on a smallness condition on ∥g∥∞\|g\|_\infty and on standard positivity and growth hypotheses on the state-dependent part of the two-point function — in arbitrary Hadamard states; and (iii) the renormalised stress tensor is not conserved as it stands: a quantum correction proportional to ℏ\hbar, one-loop exact, must be added to obtain a conserved operator T^μν\hat T_{\mu\nu}.

The quantum status of the higher currents has a long history. Flume found that in conventional quantum perturbation theory all currents acquire anomalous terms; Kulish, Nissimov and collaborators argued from the exactly-solvable side that the infinite set of conservation laws survives quantisation; and Lowenstein and Speer rigorously proved the existence of infinitely many conserved currents in renormalised perturbation theory, order by order in the formal power series — a statement that proves neither convergence of the series nor the sharper property that the specific currents obtained from the classical recurrence are conserved without any renormalisation correction. In the modern constructive setting, the first step was taken by Zanello, who derived closed recursive formulas for the classical higher currents and proved in pAQFT that all of them are super-renormalisable by power counting, via a 'piecewise renormalisation' whose number of steps is bounded by the degree of the currents; in his dissertation he additionally proved summability of the expectation values of the renormalised interacting current components in Gaussian states, for cutoffs of sufficiently small support.

Problem Statement

Consider the massless sine-Gordon model in the finite regime β2<4π\beta^2<4\pi with adiabatic interaction cutoff g∈S(R2)g\in\mathcal{S}(\mathbb{R}^2), in the constructive setting of Fröb and Cadamuro (Euclidean functional-integral approach and Lorentzian pAQFT with Hadamard states). The hierarchy of classically conserved higher-spin currents obtained from the Bäcklund/Noether recurrence has the stress tensor as its lowest member; 'higher currents' means all members beyond the stress tensor. Prove for these currents the analogues of the Fröb–Cadamuro results:

  1. Renormalisation. After removal of the infrared and ultraviolet cutoffs, the renormalised correlation functions of the higher currents together with vertex operators are well-defined distributions — in Euclidean signature and in Lorentzian signature in arbitrary Hadamard states — with the required counterterms and the finite renormalisation freedom characterised explicitly (analogues of Theorems 1 and 4 of Fröb–Cadamuro).

  2. Convergence. The renormalised perturbation series for these correlation functions converges under the same hypotheses under which Fröb–Cadamuro established convergence for the stress tensor: non-negative Schwartz gg in the Euclidean case, and Schwartz gg with the stated smallness condition on ∥g∥∞\|g\|_\infty together with the standard positivity and growth hypotheses on the state-dependent part of the two-point function in the Lorentzian case (analogues of Theorems 2 and 5).

  3. Conservation and quantum corrections. Determine whether the renormalised higher currents are conserved in the quantum theory without any ℏ\hbar-dependent finite renormalisation, as proposed in the form-factor literature — that is, prove the corresponding conservation identities for the renormalised currents (the analogues of the expectation-level conservation theorem Fröb–Cadamuro proved for the corrected stress tensor T^μν\hat T_{\mu\nu}), or else exhibit the quantum correction that must be added to restore conservation, as Fröb–Cadamuro did for the stress tensor (where the correction is one-loop exact).

References

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