On the finiteness of noncommutative supersymmetric QED$_3$ in the covariant superfield formulation

The three-dimensional noncommutative supersymmetric QED is investigated within the superfield approach. We prove the absence of UV/IR mixing in the theory at any loop order and demonstrate its one-loop finiteness.

One of the most remarkable properties of noncommutative theories consists of an unusual structure of divergences, the so called UV/IR mixing, that could lead to the appearance of infrared divergences [4,12]. It should be noticed that the cancellation of quadratic and linear ultraviolet divergences in commutative theories does not guarantee the absence of harmful infrared divergences in their noncommutative counterparts [13,14,15,16]. The elimination of such divergences is crucial since they may obstruct the development of a sound renormalization scheme, leading to the breakdown of the perturbative series.
Based on experience, it is natural to expect that supersymmetry could improve this situation [4,17]. In fact, the Wess-Zumino model [14] and the three-dimensional sigma-model [18] are renormalizable at all loop orders. This is furtherly supported by the results of [19] according to which the one-loop effective action in N = 1, 2 super-Yang-Mills theory contains only logarithmic divergences while for N = 4 the theory is one-loop finite [19,20].
In this paper we employ the covariant superfield formalism to study noncommutative supersymmetric QED 3 . We will prove that this theory is free of nonintegrable UV/IR divergences at any loop order. We shall also demonstrate that the model is one-loop finite.
The action of the three-dimensional N = 1 noncommutative supersymmetric QED is [21] S = 1 where is a superfield strength constructed from the spinor superpotential A α . Hereafter it is implicitly assumed that all commutators and anticommutators are Moyal ones. In this work we consider only space-space noncommutativity, to evade unitarity problems [22]. This action is invariant under the gauge transformations Then, we must add a gauge fixing term which we choose to be leading to the quadratic action The free gauge propagator is where C αβ = −C αβ is the second-rank antisymmetric symbol defined with the normalization C 12 = i. The most convenient choice for the gauge fixing parameter is ξ = 1, the Feynman gauge, in which the propagator collapses to The interaction part of the classical action in the pure gauge sector is The action of the associated Faddeev-Popov ghosts reads implying in the propagator We assume that the ghosts are in the adjoint representation. The total action is, then, given by To study the divergence structure of the model we shall start by determining the superficial degree of divergence ω associated to a generic supergraph. Explicitly, ω receives contributions from the propagators and, implicitly, from the supercovariant derivatives. This last dependence can be unveiled by the use of the conversion rule (12) and the identity (D 2 ) 2 = . Each loop contributes two power of momentum to ω. To see how this come about, notice that each integration over d 3 k is decreased by one power of momentum when contracting the corresponding loop into a point. It can be seen that, if V 3 , V 2 , V 1 , and V 0 are, respectively, the number of pure gauge vertices with three, two, one and none spinor derivatives, then, they altogether will contribute 3 2 V 3 + V 2 + 1 2 V 1 . Furthermore, V c gauge-ghost vertices will increase ω by 1 2 V c . Each gauge propagator (let their number be P A ) lowers ω by two, each ghost propagator (let their number be P c ) lowers ω by one. Moving a supercovariant derivative to an external field decreases ω by 1 2 (let N D be the number of spinor derivatives moved to the external fields). Putting everything together we may conclude that ω is given by The number of the ghost vertices is equal to the number of the ghost propagators, P c = V c , since the ghost propagators only form closed loops. Thus, after using the topological identity This power counting relationship characterizes noncommutative supersymmetric QED 3 as an UV super-renormalizable theory. It is easy to realize that linear divergences may come only from the graphs with V 3 = 2, or V 2 = 1, or V c = 2. These graphs are depicted in Fig. 1, they contribute to the two-point functions of A α field. In these graphs, a crossed line corresponds to a factor D α acting on the ghost propagator. A trigonometric factor e ik∧l − e il∧k = 2i sin(k ∧ l) originates from each commutator. By denoting the contributions of the graphs in Fig. 1 by I 1a , I 1b , and I 1c , respectively, we have Where not otherwise indicated it must be understood that the supercovariant derivatives act on the Grassmann variable θ 1 , also δ 12 = δ 2 (θ 1 − θ 2 ). In the expressions for the I 1 's the terms where covariant derivatives act on external fields were omitted because they do not produce linear divergences and UV/IR mixing (as we shall shortly verify, such terms give only finite contributions). In the formulae above they are indicated by the ellipsis. After some D-algebra transformations we arrive at Hence, the total one-loop two-point function of the gauge superfield, given by I 1 = I 1a +I 1b +I 1c , is free from both UV and UV/IR infrared singularities. The same situation takes place in the four-dimensional noncommutative supersymmetric QED [15,16]. It is also easy to show that the logarithmically divergent parts of I 1a , I 1b and I 1c , which involve derivatives of the gauge fields, turn out to be proportional to the integral and are therefore finite by symmetric integration. Thus, the logarithmic divergences in I 1a , I 1b , and I 1c are also absent, i.e., the two-point function of A α field is finite in the one-loop approximation. We already mentioned that linear divergences are possible only for V 2 = 1, or V 3 = 2, or V c = 2. Nevertheless, it is easy to see that two-and higher-loop graphs satisfying these conditions are just vacuum ones. Then, there are no linear UV and UV/IR infrared divergences beyond one-loop and, as consequence, the Green functions are free of nonintegrable infrared divergences at any loop order.
We examine next the structure of potentially logarithmic divergent diagrams. They corre- Notwithstanding, the contributions of these graphs turn out to be very similar among themselves so that the same mechanism of cancellation of divergences applies. As a prototype of this mechanism let us consider the supergraph with V 3 = 3 in Fig. 2. Its amplitude in the Feynman gauge reads By using the relationship (12) and the identity {D α , D 2 } = 0 we find that I 2 vanishes. The fact that this graph is finite is actually a gauge independent statement. Indeed, in an arbitrary gauge and after D-algebra transformations, I 2 is given by whose planar part is proportional to that of the integral in Eq. (17), which is finite. The nonplanar part of I (ξ) 2 is composed of two terms, one proportional to which is evidently finite, and the other proportional to a linear combination of integrals of the form Here,p αβ = Θ mn p n (σ m ) αβ , and Θ mn is the constant antisymmetric matrix characterizing the noncommutativity of the underlying space-time. As Θ 0i = 0, this last expression does not produce logarithmic divergences, which confirms the finiteness of the contribution I (ξ) 2 . The above mechanism also enforces the vanishing of UV logarithmic divergences and of UV/IR infrared logarithmic singularities from the graphs in Fig. 3. The UV finiteness of all these one-loop graphs may be proved in an analogous way. For example, in the Feynman gauge the one-loop graph with V 2 = 2 contains four spinor derivatives and its UV leading contribution is proportional to the finite integral in Eq. (17). A similar situation arises for the one-loop graph with V 2 = V 3 = 1. The one-loop graph with V 3 = 2 and V 2 = 1 contains 8 D-factors and, after using the identity (D 2 ) 2 = , either a finite contribution proportional to that in Eq. (17) or a finite term in which some derivatives are moved to the external fields could emerge. The others potentially divergent one-loop graphs correspond to V c = 4 or V 3 = 4 and for them the same mechanism applies and, hence, they are finite. As it can be checked, the same happens in an arbitrary covariant gauge. The vanishing of UV/IR infrared singularities for all these graphs has the same origin as that for the graph in Fig. 2.
Up to this point, the net result of our study is that the theory without matter turns out to be one-loop UV and IR finite. It is interesting to note that, in the framework of the background field method [21,23], all contributions to the effective action are superficially finite. From a formal viewpoint this is caused by the presence of two spinor derivatives in the expression for the strength W α in Eq. (2), which makes N D ≥ 4 in Eq. (14), since loop corrections must be at least of second order in the background strengths (compare with [19]). We also remark that Eq. (14) implies in the absence of divergences at three-and higher-loop orders, in agreement with the super-renormalizability of the theory. This concludes our analysis of the N = 1 supersymmetry.
We next study the interaction of the spinor gauge field with matter. To this end we add to (36) the matter action Here, φ a , a = 1, . . . , N, are scalar superfields andφ a their corresponding conjugate ones. We may also write The free propagator of the scalar fields is which, in momentum space, reads The superficial degree of divergence when matter is present is given by where, as before, V i is the number of pure gauge vertices with i spinor derivatives, E φ is the number of external scalar lines, N D is the number of spinor derivatives associated to external lines, V D φ is the number of triple vertices A α * φ a * ← → D α φ a , and V 0 φ is the number of quartic vertices φ a * φ a * A α * A α .
Graphs can now be split into those with E φ = 0 and those with E φ = 0. The leading UV divergence for those with E φ = 0 is ω = 3/2, corresponding to a tadpole graph which vanishes identically. What comes next are graphs with two external A α legs which are UV linearly divergent. They are depicted in Fig. 4. Graphs with three and four external A α legs are UV logarithmically divergent. The remaining ones are finite. As for the graphs with E φ = 0, only those with E φ = 2 are potentially UV logarithmically divergent, those with E φ > 2 are finite.
Graphs with E φ = 0 verify the conditions V 0 φ > 0 or V D φ > 0 which, unless for the tadpole graph already mentioned, imply that 1 , the corresponding supergraphs are superficially finite, according to (26). Since there are no external matter legs, each vertex of the one-loop graph must involve matter. Hence, we arrive at the following condition for ω being non negative The lower limit of the inequality corresponds to ω = 1, whereas the upper limit corresponds to ω = 0.
The UV linearly divergent case is only realized by the one-loop matter correction to the two-point function of the gauge field A α (Fig. 4). The graph a in Fig. 4 furnishes, where the indices 1 and 2 in the supercovariant derivatives designate the field to which the D operator is applied. Taking into account the explicit form of the propagators, we found which, after using D β2 δ 12 = −D β1 δ 12 , can be cast as where we have introduced the notation It is convenient to split I 4a into two parts, I 4a = I 4a +I (2) 4a , where I 4a and I (2) 4a are, respectively, associated to the first and second terms in the large brackets in the right hand side of Eq. (30). It is straightforward to verify that For the second term in the right hand side of Eq. (30) one analogously finds By adding Eqs. (32) and (33) we can cast the contribution from the graph a in Fig. 4 as The algebraic manipulations for the graph b in Fig. 4 are simpler and yield The complete correction to the two-point function is, therefore, We stress that the dangerous linear divergences have disappeared, i.e., the two-point function of A α field turns out to be free of UV/IR infrared singularities and, moreover, finite. This two-point function can be used for deriving the effective propagators in the 1 N expansion [24]. It remains to consider the graphs with ω = 0. It follows from (27), that the only remaining one-loop logarithmically divergent graphs involving matter are those ones depicted in Fig. 5. Nevertheless, a direct calculation shows that the planar contributions of the first three of these supergraphs are proportional to the integral in Eq. (17) whose divergent part is known to vanish. The divergent parts of their nonplanar contributions vanish in a way similar to that of the graphs in Figs. 2 and 3. The fourth graph is evidently finite. The last graph is finite by the same reason as the first three.
We shall next deal with the graphs with E φ > 0. Such graphs do not contain linear divergences, according to Eq. (26). Furthermore, the number of external scalar legs must be even since any vertex carries an even number of scalar fields, and only an even number of them can be contracted into propagators. As stated before, the logarithmic divergences in this case are possible only for E φ = 2, V D φ = 2 and for E φ = 2, V 0 φ = 1. These graphs are shown in Fig.  6. The graph a in 6 gives the contribution As before, the ellipsis stands for manifestly finite terms. After some simplifications, one obtains which is finite. The second graph in Fig. 6 yields the amplitude which vanishes identically because of k α α = 0. Therefore the two-point function of the scalar field is free from UV/IR mixing and, moreover, finite in any covariant gauge. It follows from Eq. (26) that the supergraphs with two or more external scalar legs and one or more gauge legs are also superficially finite.
To sum up we conclude that the three-dimensional noncommutative supersymmetric QED is one-loop UV and UV/IR infrared finite both without and with matter. A natural development of this work consists in the investigation of the possibility of appearance of divergences at two-loop order. Other possible developments are a detailed study of the 1/N expansion for the model involving many scalar fields and the analysis of spontaneous symmetry breaking and the Higgs mechanism.