Renormalization properties of the mass operator $A_{\mu }^a{A}_{\mu }^a$ in three-dimensional Yang-Mills theories in the Landau gauge

Abstract

Massive renormalizable Yang-Mills theories in three dimensions are analyzed within the algebraic renormalization in the Landau gauge. In analogy with the four-dimensional case, the renormalization of the mass operator $A_{\mu }^a{A}_{\mu }^a$ turns out to be expressed in terms of the fields and coupling constant renormalization factors. We verify the relation we obtain for the operator anomalous dimension by explicit calculations in the large N_f expansion. The generalization to other gauges such as the non-linear Curci–Ferrari gauge is briefly outlined.

Introduction

Recently, much work has been devoted to the study of the operator $A_{\mu }^a{A}_{\mu }^a$ in four-dimensional Yang-Mills theories in the Landau gauge, where a renormalizable effective potential for this operator can be consistently constructed [1], [2]. This has produced analytic evidence of a non-vanishing condensate $〈A_{\mu }^a{A}_{\mu }^a〉$, resulting in a dynamical mass generation for the gluons [1], [2]. A gluon mass in the Landau gauge has been reported in lattice simulations [3] as well as in a recent investigation of the Schwinger–Dyson equations [4]. Besides being multiplicatively renormalizable to all orders of perturbation theory in the Landau gauge, the operator $A_{\mu }^a{A}_{\mu }^a$ displays remarkable properties. In fact, it has been proven [5] by using BRST Ward identities that the anomalous dimension ${\gamma }_{A^2}(a)$ of the operator $A_{\mu }^a{A}_{\mu }^a$ in the Landau gauge is not an independent parameter, being expressed as a combination of the gauge β function, β(a), and of the anomalous dimension, γ_A (a), of the gauge field, according to the relation

$${\gamma }_{A^2}(a)=-\left(\frac{\beta (a)}{a}+{\gamma }_A(a)\right)\text{,}a=\frac{g^2}{16{\mathrm{\pi }}^2}\text{,}$$

which can be explicitly verified by means of the three loop computations available in [6]. The operator $A_{\mu }^a{A}_{\mu }^a$ turns out to be multiplicatively renormalizable also in the linear covariant gauges [7]. Its condensation and the ensuing dynamical gluon mass generation in this gauge have been discussed in [8].

Moreover, the operator $A_{\mu }^a{A}_{\mu }^a$ in the Landau gauge can be generalized to other gauges such as the Curci–Ferrari and maximal Abelian gauges. Indeed, as was shown in [9], [10], the mixed gluon–ghost operator²$(\frac{1}{2}{A}_{\mu }^a{A}_{\mu }^a+\alpha {\overline{c}}^a{c}^a)$ turns out to be BRST invariant on-shell, where α is the gauge parameter. In both gauges, the operator $(\frac{1}{2}{A}_{\mu }^a{A}_{\mu }^a+\alpha {\overline{c}}^a{c}^a)$ turns out to be multiplicatively renormalizable to all orders of perturbation theory and, as in the case of the Landau gauge, its anomalous dimension is not an independent parameter of the theory [11]. A detailed study of the analytic evaluation of the effective potential for the condensate $〈\frac{1}{2}{A}_{\mu }^a{A}_{\mu }^a+\alpha {\overline{c}}^a{c}^a〉$ in these gauges can be found in [12], [13]. In particular, it is worth emphasizing that in the case of the maximal Abelian gauge, the off-diagonal gluons become massive due to the gauge condensate $〈\frac{1}{2}{A}_{\mu }^a{A}_{\mu }^a+\alpha {\overline{c}}^a{c}^a〉$, a fact that can be interpreted as evidence for the Abelian dominance hypothesis underlying the dual superconductivity mechanism for color confinement.

The aim of this work is to analyze the renormalization properties of the operator $A_{\mu }^a{A}_{\mu }^a$ in three-dimensional Yang-Mills theories in the Landau gauge. This investigation might be useful in order to study by analytical methods the formation of the condensate $〈A_{\mu }^a{A}_{\mu }^a〉$ in three dimensions, whose relevance for the Yang-Mills theories at high temperatures has been pointed out long ago [14]. Furthermore, the possibility of a dynamical gluon mass generation related to the operator $A_{\mu }^a{A}_{\mu }^a$ could provide a suitable infrared cutoff which would prevent three-dimensional Yang-Mills theory from the well known infrared instabilities [15], due to its superrenormalizability.

The organization of the paper is as follows. In Section 2 we discuss the renormalizability of the three-dimensional Yang-Mills theory in the Landau gauge, when the operator $A_{\mu }^a{A}_{\mu }^a$ is added to the starting action in the form of a mass term, $m^2\int {\mathrm{d}}^3{\mathrm{xA}}_{\mu }^a{A}_{\mu }^a$. We shall be able to prove that the renormalization factor $Z_{m^2}$ of the mass parameter m² can be expressed in terms of the renormalization factors Z_A and Z_g of the gluon field and of the gauge coupling constant, according to

$$Z_{m^2}=Z_g{Z}_A^{-1/2}.$$

This relation represents the analogue in three dimensions of the Eq. (1). In Section 3 we give an explicit verification of the relation (2) by using the large N_f expansion method. In Section 4 we present the generalization to the non-linear Curci–Ferrari gauge.