(that is the Luttinger model with ultraviolet cutoff and local interaction, which is equivalent to the Thirring model with fixed ultraviolet cutoff).

(2)The beta function for this special model (which is not solvable) is asymptotically vanishing, so that the effective coupling on large scales is essentially constant and of the same order of the coupling on small scales.

Up to now there are two different ways to prove property (2). The first one was developed in the last years and is based in an essential way on the exact Mattis-Lieb solution of the Luttinger model [1, 2]. More recently, we found a new proof, based on the Ward identities obtained by a chiral local gauge transformation, applied to the Tomonaga model with infrared cutoff [3, 4]. This is an old approach in the physical literature, but its implementation in an RG scheme is not trivial at all, because the ultraviolet and infrared cutoffs destroy local gauge invariance and produce not negligible correction terms with respect to the formal Ward identities.

The solution of the problem is in the use of a new set of identities, called “Correction Identities”, relating the corrections to the Schwinger functions and showing the phenomenon of chiral anomaly. By combining Ward and Correction identities with a Dyson equation, the vanishing of the Beta function follows, so that the infrared cutoff can be removed.

As a byproduct, even the ultraviolet cutoff can be removed, after a suitable ultraviolet renormalization, so that a Euclidean Quantum Field Theory corresponding to the Thirring model at imaginary time is constructed, for any value of the mass [5].

2 The Tomonaga Model with Infrared Cutoff

The model is not Hamiltonian and can be defined in terms of Grassmannian variables. It describes a system of two kinds of fermions with linear dispersion rela-

k D

where N is a normalization constant, D ψ is the Lebesgue Grassmannian measure, Z0 is a fixed constant, that we shall put equal to 1, and [Ch,0(k)]−1 is a smooth

to 1 in the interval {γ ≤ |k| ≤ 1}. The measure (1) is a Gaussian Grassmannian measure with propagator

The correlation functions of density and field operators for the Tomonaga model

The correlation functions will be graphically represented as in the examples of Fig. 1. They are of course well defined, if γ h is large enough; we want to discuss how to control the limit h → −∞.

Fig. 1 Graphical representation of a few correlation functions

3 The RG Analysis

We shall perform a multi-scale analysis of the functional (3), by using the identity

The decomposition (5) implies the following decomposition of the covariance (2) in single scale covariances:

as well as a corresponding factorization of the measure (1) as the product of the Gaussian Grassmannian measures with propagators gω(j )(k). Hence, we can perform iteratively the integration over the different scales, starting from j = 0. Moreover, after any integration step, we absorb in the remaining part of the free measure the terms linear in the momentum, so that at step j we get an expression of the type, see [4] for details:

where PZj(h),Ch,j has roughly the same form as (1), with the field renormalization constant Zj(h) in place of Z0, and the cutoff function

where · · · denotes the remainder, made of irrelevant terms, λ(h)j is the running cou-

pling and Zj(2,h) is the density renormalization constant.

Let us put Zj(1,h) = Zj(h); it is easy to see, by using the definitions and the support properties of the single scale propagators, that, if h < h,

Finally, if εh stays small for h → −∞, one can remove the infrared cutoff and show that [4]

λ−∞(λ) and ηi (λ−∞) being analytic functions. Moreover λ−∞(λ) is odd in λ and ηi (λ−∞) is even in λ−∞.

4 The Dyson Equation

To prove (14) is not an easy task, since the interaction is marginal, from the RG point of view, and indeed the bounds following from the RG analysis show a divergence linear in h. In order to clarify the origin of this apparent divergence and explain how to use Ward identities to solve the problem, it is convenient to consider the following Dyson equation:

whose graphical representation is given in Fig. 2.

Fig. 2 Graphical representation of the Dyson equation (16)

The RG analysis allows us to get rigorously dimensional bounds on the correlation functions. In particular, if we fix the external momenta in the Dyson equation so that