# Superfield covariant analysis of the divergence structure of noncommutative supersymmetric QED

###### Abstract

Commutative supersymmetric Yang-Mills is known to be renormalizable for , while finite for . However, in the noncommutative version of the model (NCSQED) the UV/IR mechanism gives rise to infrared divergences which may spoil the perturbative expansion. In this work we pursue the study of the consistency of NCSQED by working systematically within the covariant superfield formulation. In the Landau gauge, it has already been shown for that the gauge field two-point function is free of harmful UV/IR infrared singularities, in the one-loop approximation. Here we show that this result holds without restrictions on the number of allowed supersymmetries and for any arbitrary covariant gauge. We also investigate the divergence structure of the gauge field three-point function in the one-loop approximation. It is first proved that the cancellation of the leading UV/IR infrared divergences is a gauge invariant statement. Surprisingly, we have also found that there exist subleading harmful UV/IR infrared singularities whose cancellation only takes place in a particular covariant gauge. Thus, we conclude that these last mentioned singularities are in the gauge sector and, therefore, do not jeopardize the perturbative expansion and/or the renormalization of the theory.

Also at] Department of Theoretical Physics, Tomsk State Pedagogical University Tomsk 634041, Russia (email: )

Formerly at] Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA (email: )

## I Introduction

During last years noncommutative (NC) field theories have been intensively studied. These theories emerged as the low energy limit of the open superstring in the presence of an external magnetic field (-field) SW although nowadays they are interesting in their own right (for a review see Nekr ; Szabo ; Girotti11 ).

The most striking property of noncommutative field theories is undoubtly the UV/IR mechanism, through which the ultraviolet divergences (UV) are partly converted into infrared (IR) ones Minw ; Mat ; RR1 . These infrared divergences footnote1 may be so severe that the perturbative expansion of the theory becomes meaningless. Hence, the key point about the consistency of a noncommutative field theory is whether these divergences cancel out.

So far, only one four-dimensional noncommutative theory is known to be renormalizable, the Wess-Zumino model Girotti1 ; Buchbinder1 . In this case supersymmetry plays an essential role because it improves the ultraviolet behavior and, therefore, the UV/IR mechanism only generates mild UV/IR infrared divergences which do not spoil the renormalization program. In three space-time dimensions we are aware of at least two noncommutative renormalizable models: the supersymmetric nonlinear sigma model sig and the supersymmetric linear sigma model in the limit linsm .

As for nonsupersymmetric gauge theories, the UV/IR mechanism breaks down the perturbative approach Mat ; RR1 ; Hay ; SJ ; Armoni ; Bonora ; FL ; Gur ; Nichol . Nevertheless, we can entertain the hope that noncommutative supersymmetric gauge theories are free from nonintegrable UV/IR infrared singularities and, furthermore, renormalizable. We are aware of the following results concerning noncommutative supersymmetric gauge field theories:

1) By working with the formalism of component fields Mat ; RR1 it has been shown that the dangerous UV/IR infrared divergences cancel in the one-loop contributions to the gauge field two and three-point functions. The two-point function turns out to contain quadratic and logarithmic UV divergences. Dimensional regularization takes care of the first ones while the last ones must be renormalized. As for the harmful infrared divergences originating through the UV/IR mechanism, they are only quadratic and cancel out within a supersymmetric multiplet. The three-point function is linearly UV divergent by power counting. However, this time, the leading UV divergences vanish by symmetric integration, while the IR poles originating from them cancel out among themselves.

2) By using the superfield formalism Bichl et al. bichl calculated, in the Landau gauge and in the absence of matter (), the one-loop contributions to the two-point function of the gauge superfield. Only quadratic and logarithmic UV divergences are present and one deals with them as indicated in the previous paragraph. The quadratic infrared poles in the nonplanar part of the amplitude again cancel while the linear ones do not arise. The superfield formulation represents an improvement with respect to the component field formulation because supersymmetry is explicitly preserved at all stages of the calculation.

3) Zanon and collaborators Za1 ; Zanon used the background field method to evaluate the one-loop contributions to the field strength two-point functions in supersymmetric Yang-Mills theories, where only logarithmic divergences were found. The three-point function was shown to vanish. For they demonstrated that, up to one loop, there are no divergences at all.

This paper is dedicated to pursue further the study, within the superfield formulation, of the consistency of NCSQED in an arbitrary covariant gauge. We analyze the divergence structure induced by the UV/IR mechanism in the two and three-point gauge field Green functions.

In Section 2 we establish our definitions and conventions and present the gauge invariant action describing the dynamics of NCSQED in superspace. Next, the gauge fixing and the Faddeev-Popov terms are found. Finally, we add chiral matter superfields and derive the Feynman rules of NCSQED with extended supersymmetry.

We start, in Section 3, by reviewing the cancellation of the leading UV/IR infrared divergences in the one-loop corrections to the two-point function of the gauge superfieldbichl . A straightforward generalization shows that these results also hold for extended supersymmetry and/or when the theory is formulated in an arbitrary covariant gauge.

In Section 4 we compute the one-loop corrections to the three-point functions of the gauge superfield in an arbitrary covariant gauge. This is done for . From power counting follows that the amplitude is at the most quadratically divergent. As far as the planar part is concerned, dimensional regularization takes care of the quadratic UV divergences, the linear ones vanish by symmetric integration, while the logarithmic divergences are to be eliminated through renormalization. As for the nonplanar part, the UV/IR mechanism will be seen not to give rise to quadratic IR divergences but only to linear and logarithmic ones. Interestingly enough, the linear IR divergences arise from two different sources: a) integrals which, by power counting, are quadratically UV divergent but whose Moyal phase factor not only regularizes them but also lowers the degree of the IR divergence, b) integrals which are linearly UV divergent by power counting but regularized by the noncommutativity. The softening mechanism mentioned in a) also contributes IR logarithmic divergences, which, nevertheless, do not jeopardize the perturbative expansion.

The conclusions are contained in Section 5.

## Ii The action and Feynman rules for NCSQED

### ii.1 The action

In superspace NCSQED is described by the nonpolynomial action SGRS ; footnoteSGRS

(1) |

where is the coupling constant, is a real vector gauge superfield,

(2) |

and denotes Moyal product of operators, i.e.,

(3) |

Here, is the antisymmetric real constant matrix characterizing the noncommutativity of the underlying space-time. The expression

(4) |

where

(5) |

will play a relevant role for determining the Feynman rules in the theory.

Under the group of gauge transformations

(6) |

with () a chiral (antichiral) superfield, transforms as follows

(7) |

thus leaving invariant.

In future, we shall be needing the expansion of in powers of , up to the order . To this end we first recall the identity Buchbinder2

(8) | |||||

Then, after by part integrations and by exploring the properties of the Moyal product footnote2 one obtains

(9) |

where

(10) |

(11) |

(12) |

(13) | |||||

As usual, gauge fixing is implemented by adding to the action the covariant term

(14) |

where is a real number labeling the gauge. Clearly,

(15) |

For the covariant gauge , the Faddeev-Popov determinant reads

(16) |

Here are the ghost fields while denotes the change in provoked by an infinitesimal gauge transformation. One readily obtains from (7) that

(17) |

where

(18) |

After recalling the Laurent expansion of , around , one arrives at

(19) | |||||

Therefore, by going back with Eq. (19) into Eq. (16) one finds for the ghost action the following expression

(20) |

where

(21) |

(22) |

(23) |

In addition to the real vector superfield we introduce now a chiral matter superfield in the adjoint representation. This enables us to construct a theory in which the supersymmetry is realized. The generalization to is straightforward and will be done afterwards. The corresponding action describing the free matter superfield as well as its interaction with the gauge superfield reads

(24) |

whose invariance under supergauge transformations follows from (7) together with

(25) |

The first four terms of the expansion of as a power series of ,

(26) |

are found to be

(27) |

(28) |

(29) |

(30) |

### ii.2 Feynman rules

From the quadratic part of the action one obtains, through standard manipulations, the free propagators

(31a) | |||

(31b) | |||

(31c) | |||

(31d) |

corresponding to the gauge, ghosts and matter superfields, respectively. They are depicted in Fig. 1.

On the other hand, the interacting part of the total action together with Eq. (II.1) enable us to find the elementary vertices in the theory. They are displayed in Fig. 2. In an obvious notation

(32) |

(33a) | |||

(33b) | |||

(33c) |

(34a) | |||

(34b) | |||

(34c) |

(35a) | |||

(35b) | |||

(35c) | |||

(35d) |

(36a) | |||

(36b) | |||

(36c) | |||

(36d) |

(37a) | |||

(37b) | |||

(37c) |

Here,

(38a) | |||

(38b) | |||

(38c) | |||

(38d) | |||

(38e) | |||

(38f) |

the momenta are taken positive when entering the vertex and momentum conservation holds in all vertices.

We close this Section by pointing out that the superficial degree of divergence of a generic Feynman graph is given by SGRS

(39) |

where is the number of external chiral lines. As known, in a noncommutative quantum field theory, a generic Feynman graph will decompose into planar and nonplanar parts. The superficial degree of UV divergence of the planar part is measured by . The nonplanar part is free of UV divergences but afflicted by IR singularities generated through the UV/IR mechanism Minw ; footnote1 , in this last connection also gives the highest possible degree of the IR divergences.

## Iii One loop-contributions to the vector gauge superfield two-point function .

The cancellation of the harmful UV/IR infrared divergences in was already proved in bichl for and by working in the Landau gauge. Here, the proof is generalized by showing that the just mentioned cancellation takes place for an arbitrary covariant gauge and extended supersymmetry.

Let us first concentrate on the graphs involving either a tadpole or a loop (see Fig. 3). Since there are no external chiral lines, . Now, only those graphs with all factors in the internal lines may exhibit quadratic UV divergences. Diagrams with a factor and/or a on the external lines can at the most be linearly divergent. Any other combination of ’s on the external lines corresponds to contributions which are logarithmically divergent or finite. These follows from the -algebra alone SGRS . However, one is to take into account also the noncommutativity, which gives origin to a trigonometric factor that modifies the Feynman integrands. The combination of these two ingredients rules out, for the diagrams under analysis, the UV and UV/IR infrared linearly divergent terms. Hence, in this case, only quadratic divergences may jeopardize the consistency of the theory. They are contained in graphs (a), (b) and (c) in Fig. 3.

From the Feynman rules derived in Section 2, we found that the contribution arising from the tadpole diagram is given by

(40) | |||||

Here, a factor coming from the permutation of the external legs has already been taken into account. Moreover, we note that the term proportional to in the right hand side of (31a) does not contribute.

From (38b) one finds that

(41) |

After -algebra manipulations, one ends up with

(42) |

where

(43) |

The planar part of only contains quadratic UV divergences, while the nonplanar one only develops quadratic IR infrared singularities.

The amplitudes associated with diagrams (b) and (c) of Fig. 3 are, respectively,

(44) | |||||

(45) | |||||

where the comes from the second order of the perturbative expansion. After standard rearrangements one gets

(46) | |||||

(47) | |||||

Here, is short for all terms which are at the most logarithmically divergent. Furthermore, from Eq. (38a)

(48) | |||||

As a result, the terms proportional to , in the second brackets in the right hand sides of Eqs. (46) and (47), drop out in the sum . On the other hand, the term proportional to in Eq. (46) survives. From power counting follows that such term might give rise to (dangerous) linear divergences. To see whether this really happens, we start by expanding

(49) |

around . It is then obvious that the would be linearly divergent integral

(50) |

vanishes by symmetric integration. As stated above, the even parity of the trigonometric factor in Eq. (43), eliminates the linear UV divergences and also the linear UV/IR infrared divergences. To summarize:

(51) |

We turn next into computing the ghost contributions to . A direct consequence of the -algebra is that graphs containing any of the vertices , , , or , depicted in Fig. 2, only contribute . We shall therefore concentrate on the diagrams which might provide quadratic and/or linear divergent contributions to . These are the graphs (d) and (e) of Fig. 3.

The calculation of the tadpole contributions (graphs (d) and (e) in Fig. 3) is straightforward and yields

(52) |

The same expression arises for . Then, after using (41), one obtains

(53) |

The evaluation of the ghost loop contributions (graphs (f) and (g) in Fig. 3) is a little bit more involved. By applying the Feynman rules we obtain

(54) | |||||