Modification of the operator-product expansion $T(\psi(x)\bar\psi(y))$ under general gauge transformations in the Thirring model

In the two-dimensional Thirring model, the short-distance operator-product expansion $T(\psi(x)\bar\psi(y))$ is known explicitly in Johnson's gauge, and Mueller, Trueman and Wilson (1972) determined how it transforms under the explicit exponential gauge $\psi\to e^{ia\chi}\psi$ with $\chi$ a free massive scalar: all singular coefficient functions acquire the same multiplicative short-distance phase, and a new operator insertion proportional to $(x-y)^\mu\partial_\mu\chi$ must be added. The Feynman-graph analysis of the model admits a strictly larger class of gauge transformations — current-kernel transformations $S_1[F]$ with an essentially arbitrary scalar kernel $F$ — which leave all observables invariant while shifting the fermion field's short-distance dimension by an arbitrary amount. Mueller–Trueman–Wilson showed that the anomalous dimension of the composite chiral densities is invariant under this general class, but left the corresponding modification of the operator-product expansion undetermined. The problem is to determine that modification: the transformation of the singular functions, the additional operator insertions and their short-distance coefficients, and the full operator content required to reproduce the gauge-transformed multi-point correlators, with the exponential-gauge result recovered as the special case. The modern bosonization/conformal-field-theory framework fixes the gauge-invariant operator algebra that gives the question its context, but the precise technical core remains open.

Background

The Thirring model is a quantum field theory in one space and one time dimension of a single Dirac fermion $\psi$ with a local current-current four-fermion interaction $\frac{g}{2}j_\mu j^\mu$, where $j_\mu=\bar\psi\gamma_\mu\psi$ is the conserved vector current; the 'massive' version adds a fermion mass term. The model is exactly solvable in the massless case: Johnson (1961) obtained the exact Green's functions, and the massive model's short-distance (ultraviolet) behavior is governed by the massless solution.

The operator-product expansion (OPE) expresses the product of two local fields at nearby points as a sum of local operators multiplied by singular c-number coefficient functions of the separation; it is the tool that defines composite operators and their scaling dimensions. An anomalous dimension is a scaling dimension that differs from the free-field (canonical) value because of the interaction. Wilson (1970) constructed the short-distance OPE for the Thirring model in the gauge of Johnson's exact solution ('Johnson's gauge'): with $\xi=x-y$,

$$T(\psi(x)\bar\psi(y)) = -iG(\xi)I + \sum_{j=1}^{4}C_j(\xi)\,\mathcal{O}_j(y) + \text{remainder},$$

where $G$ is the full fermion propagator, $I$ the identity, and the leading operators are the chiral scalar and pseudoscalar densities $\phi_\pm=:\bar\psi(1\pm\gamma_5)\psi:$ (here $\gamma_5$ is the chirality matrix and the colons denote normal ordering, the standard reordering of field-mode operators that subtracts vacuum expectation values) and the light-cone currents $j_\pm=:\bar\psi(\gamma^1\pm\gamma^0)\psi:$, the combinations of the vector current that measure right- and left-moving charge. The exact results are: the fermion dimension $d_\psi=\frac{1}{2}+\gamma$ with $\gamma=(\lambda^2/4\pi^2)(1-\lambda^2/4\pi^2)^{-1}$ in the coupling conventions of that paper; the currents keep their canonical dimension, fixed by the current algebra (the commutation relations of the conserved currents); and the chiral densities acquire the anomalous dimension $d_\phi=(1-\lambda/2\pi)/(1+\lambda/2\pi)$.

Mueller and Trueman (1971) analyzed the massive model in perturbation theory and found a QED-like gauge freedom at the level of Feynman graphs: the longitudinal part of the vacuum polarization is ambiguous, and, more generally, one may insert into time-ordered correlators the unitary operator

$$S_1[F]=\exp\Big\{\tfrac{i}{2}\int dz\,dz'\,j_\mu(z)j_\nu(z')\,\partial^\mu\partial^\nu F(z-z')\Big\}$$

with an essentially arbitrary scalar kernel $F$ (subject only to mild regularity). This multiplies the fermion two-point function by $e^{i[F(x-y)-F(0)]}$, leaves all $S$-matrix elements and conserved-current matrix elements invariant, and — choosing $F(x)\sim -ia\ln(x^2/x_0^2)$ near the origin — shifts the short-distance dimension of $\psi$ by an arbitrary amount $a$. Hence the anomalous dimension of the elementary fermion field is gauge-dependent and carries no intrinsic physical meaning; only composite, gauge-invariant operators have well-defined dimensions.

Mueller, Trueman and Wilson (1972) then asked what happens to the OPE itself under such gauge transformations. For the explicit exponential gauge $\psi\to e^{ia\chi}\psi$, with $\chi$ a free Hermitian scalar field of mass $M$, they found: the propagator and all singular OPE coefficients acquire the same multiplicative short-distance phase, $G(\xi)\to e^{-ia^2D^c(\xi)}G(\xi)$ and $C_j(\xi)\to e^{-ia^2D^c(\xi)}C_j(\xi)$, where $D^c(\xi)=\tfrac{i}{2\pi}K_0(\sqrt{-\xi^2M^2})$ is the causal function of the free massive scalar ($K_0$ is the modified Bessel function of the second kind; $D^c(\xi)\sim\tfrac{i}{4\pi}\ln(-\xi^2M^2)$ at short distance); and the expansion must be enlarged by a new operator insertion, $a\,e^{-ia^2D^c(\xi)}G(\xi)\,\xi^\mu\partial_\mu\chi(y)$, which arises from expanding the phase difference $e^{ia[\chi(x)-\chi(y)]}$ — without it, the OPE fails to reproduce the gauge-transformed four-point function. Johnson's gauge is the special gauge in which this extra term vanishes. They showed that the anomalous dimension $d_\phi$ of the chiral densities is invariant under the gauge transformation, argued that this invariance extends to the full graph-admissible class $S_1[F]$, and stated that for that general class the corresponding modification of the OPE was not determined in their work — the origin of the present problem.

For context, later developments fixed the surrounding landscape: the massive Thirring model is equivalent to the sine-Gordon model (Coleman 1975), the fermion can be constructed as a soliton (vertex) operator of a free boson (Mandelstam 1975), anomalous dimensions of fermion bilinears in four-fermion models follow from current algebra and scale invariance (Dashen–Frishman 1975), and the exact fermion correlation functions of the massless model form a multi-parameter family (the Klaiber parametrization of Johnson's solution) whose parameters realize precisely this fermion-field 'gauge' freedom. These results fix the gauge-invariant operator algebra, but they do not give the explicit OPE transformation law under the general gauge class that is asked for here.

Problem Statement

For the (massive) Thirring model, determine the modification of the short-distance operator-product expansion

$$T(\psi(x)\bar\psi(y)) = -iG(\xi)I + \sum_j C_j(\xi)\,\mathcal{O}_j(y) + \text{remainder},\qquad \xi = x-y,$$

under the general class of gauge transformations admitted by the Feynman-graph analysis — the current-kernel transformations $S_1[F]$ of Mueller–Trueman (1971) with a scalar kernel $F$ from the source's 'essentially arbitrary' mild-regularity class — beyond the explicit exponential phase gauge $\psi\to e^{ia\chi}\psi$ treated by Mueller–Trueman–Wilson (1972). The determination comprises: (i) the transformation law of the singular functions $G(\xi)$ and $C_j(\xi)$; (ii) the additional operator insertions that must appear — the general-gauge analogues of the insertion $\xi^\mu\partial_\mu\chi(y)$ of the free-scalar-phase case — including whether these insertions are local operators of the model's operator algebra (built from the currents $j_\pm$ and the chiral densities $\phi_\pm$) or necessarily new or non-local objects; (iii) their short-distance coefficient functions; and (iv) the full operator content required so that the expanded product reproduces the gauge-transformed multi-point correlators — all $n$-point functions, of which the four-point function is the first nontrivial case — order by order in the short-distance expansion. The exponential-phase-gauge results — the multiplicative phase $e^{-ia^2D^c(\xi)}$ on $G$ and on every $C_j$, together with the insertion $a\,e^{-ia^2D^c(\xi)}G(\xi)\xi^\mu\partial_\mu\chi(y)$ — must be recovered as the special case in which the kernel $F$ is proportional to the free massive scalar causal function $D^c$, and the gauge-invariance of the composite anomalous dimension $d_\phi$ must be preserved. The precise admissibility conditions on the kernel $F$ (the regularity and decay or support properties that bound the class) are part of what the determination must fix, not assumed input. Determine this modification, or else establish that no local-operator OPE of the stated form exists for general admissible kernels — beyond the question in (ii) of whether the insertions are local or non-local — in which case the class of admissible kernels that do admit such a local-operator OPE is also to be identified.

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