{'fraction_non_alphanumeric': 0.0352224253503961, 'fraction_numerical': 0.01230956733698964, 'mean_word_length': 4.726448011165387, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dynamical generation of mass in the noncommutative supersymmetric Schwinger model

Within the superfield formalism, we study the dynamical generation of mass to the gauge superfield in the noncommutative two-dimensional supersymmetric Schwinger model. We show that the radiatively generated mass does not depend on the noncommutative parameter up to one-loop order.


I. INTRODUCTION
In the last years, field theory models constructed in lower dimensions of space-time have been intensively discussed, because, through AdS/CF T correspondence [1], they could be related to more elaborated theories in higher dimensions. The supersymmetric gauge theories in lower dimensions considered as candidates to describe M 2-branes [2][3][4] attract main attention. Currently, a large number of papers is devoted to the study of several aspects of these theories, such as effective potential calculations [5][6][7], dualities [8][9][10][11] and generation of mass through the spontaneous symmetry breaking mechanism [12][13][14][15][16]. The Schwinger model, i.e. quantum electrodynamics in two dimensions of space-time, is of the special interest among the low-dimensional gauge theories since it possesses the interesting feature of dynamical generation of mass, and is known as an example of a confining model [17]. It is worth to mention that the two-dimensional noncommutative supersymmetric (SUSY) quantum electrodynamics is finite to all loop orders in perturbation theory [18], with the same conclusion is true for the three-dimensional commutative SUSY QED [19].
Throughout this paper, we are using the superfield formalism; it is the more convenient way to evaluate Feynman diagrams in SUSY models. It preserves a manifest supersymmetry in all stages of calculations, avoiding potential problems, such as, for example, the lacking of a supersymmetric renormalization presented in [20] is not a problem when supergraph techniques are used [21].
The present paper is organized as follows: in Sec. II we introduce the model and evaluate the quadratic part of the effective action for the gauge superfield, in order to observe the dynamical generation of mass to the noncommutative supersymmetric Schwinger model. We demonstrate that this effect is independent of the noncommutative parameter Θ, up to one-loop order, just as the non-supersymmetric version of present model [22] . In Sec. III we present our last comments and remarks.

A. Pure gauge theory
Our starting point is the classical action of the noncommutative SUSY Schwinger model (NSS), where [Γ β , {Γ β , Γ α } * ] * is the noncommutative gauge super-field strength which transforms covariantly, W α = e iK * W α * e iK , with K = K(x, θ) being a real scalar superfield. Essentially, as discussed in [18], there is no difference between conventions and notations for supersymmetric models defined in three and two dimensions of space-time. Therefore, we use the notations and conventions as adopted in [23]. The inclusion of a gauge fixing and the corresponding Faddeev-Popov ghosts terms is required to quantize this model.
For later purposes, let us write the quadratic part of the gauge superfield action, which is given by The propagators obtained from (1), for the pure gauge sector, can be cast as where For simplicity, but without loss of generality, we will work in the Feynman gauge, i.e., we choose ξ = 1.
The effective action receives one-loop contributions from the diagrams drawn in Fig. 1. Performing the D-algebra manipulations with the help of the Mathematica c packet SusyMath [24], we arrive at the following results. The supergraph 1(a) is vanishing, while the others contributions can be cast as Performing some algebraic manipulations and adding these two contributions, we have Using Feynman representation and trivial transformations we can rewrite the integral over k as where I 1 and I 2 are planar and nonplanar contributions.
It is easy to see that where . We note, however, that this integral diverges both at higher and lower limits, so, we implement the cutoff regularizations on both limits which yields where, at the end of the calculations, one must put 1 , 2 → 0.
Applying the results from [25] and regularizing the integral in the similar way, we find where p • p ≡ p m Θ mn Θ nl p l , that in two dimensions of space-time can be written as Θ 2 p 2 , once we assume Θ mn = Θ mn . Thus, we arrive at the final expression for the integral Unfortunately, this integral cannot be more simplified since the modified Bessel function cannot be expressed in terms of the elementary functions. However, for our aims, that is, or studying the mass generation, it is sufficient to use its asymptotic behavior for small arguments: K ±1 (s) = 1 s + cs + O(s 2 ), where c is a constant whose explicit value is not important. Therefore, in the limit of p 2 Θ 2 ∼ 0, we obtain that the integral I is of order of Θ 2 p 2 . Taking into account only two leading terms of the expansion of the modified Bessel function, we find that the terms singular in the limits 1 → 0 and 2 → 0 turn out to be completely cancelled, after which we can remove the regularization, and the integral over the Feynman parameter x is trivial. As a result, we find that the effective action is just given by We conclude that no mass is dynamically generated due to self-interacting gauge sector.

B. Matter superfields in fundamental representation
The form of the matter couplings depends on assumed noncommutative representation for matter superfields. Let us first consider matter superfields in the fundamental left-representation.
To the action (1) we add the following matter superfield action from which we obtain the matter superfield propagator given by The contributions due to matter coupling, Figures 1 (d) and (e), in the fundamental representation, can be cast as We can note that the logarithmic divergent terms cancel between each other, and the contribution to the quadratic part of effective action for the gauge superfield coming from matter sector is given by Summing up the classical and quantum parts of the effective action, we have To study dynamical generation of mass it is enough to evaluate the effective action in the limit p 2 → 0 [22]. Performing the integral d 2 k (2π) 2 1 k 2 (k+p) 2 with the help of an infrared regulator and taking p 2 → 0, we obtain where we find the presence of a massive pole for the perturbative full propagator arisen from the effective action (18), where M 2 γ = e 2 3 √ 3 . One should notice that the dynamically generated mass is independent of noncommutative parameter Θ.

C. Matter superfields in adjoint representation
When matter superfields are assumed to be in the noncommutative adjoint representation, the matter superfield action turns out to be The vertices of interaction written in terms of noncommutative Moyal phases are given by equations (A6) and (A7). This sector contributes to effective action with [18] S 1(d+e) =− e 2 2 Adding (20) with the contribution that come from the gauge sector (6), the quantum correction to the quadratic part of the gauge superfield effective action is given by One-loop quantum effects (21)  suggested in [26], we expect that the dynamical generation of mass as well as a Θ dependence in such mass can occur when three-loop Feynman supergraphs are taken into account. Actually, this work is in progress. Also, we expect that this approach can be useful for study of the non-Abelian extension of the Schwinger model and for studies of the three-dimensional noncommutative SUSY QED. We are going to discuss these problems in forthcoming papers.
Acknowledgements. This work was partially supported by Conselho Nacional de Desen-

Gauge superfield self-interactions and ghost couplings
The noncommutative vertices for the gauge superfield self-interaction and Fadeev-Popov ghost couplings are given by: where a ∧ b = a µ b ν Θ µν .

Matter superfield couplings: fundamental representation
When matter superfield is in the fundamental left representation, the noncommutative vertices can be cast as

Matter superfield couplings: adjoint representation
When matter superfield is in the adjoint representation, the noncommutative vertices look like