The Supersymmetric ()D Noncommutative Model in the Fundamental Representation
Abstract
In this paper we study the noncommutative supersymmetric model in dimensions, where the basic field is in the fundamental representation which, differently to the adjoint representation already studied in the literature, goes to the usual supersymmetric model in the commutative limit. We analyze the phase structure of the model and calculate the leading and subleading corrections in a expansion. We prove that the theory is free of nonintegrable UV/IR infrared singularities and is renormalizable in the leading order. The twopoint vertex function of the basic field is also calculated and renormalized in an explicitly supersymmetric way up to the subleading order.
pacs:
11.10.Nx, 11.10.Gh, 11.10.Lm, 11.15.qI Introduction
The model in dimensions was studied since the end of the 1970 decade, mainly because it is a reasonably simple scalar model which possesses gauge invariance, and it was found to reproduce several effects typical of more complicated gauge models in four spacetime dimensions, such as instantons solutions and confinement D'Adda:1978uc ; D'Adda:1978kp . The crucial simplifying aspect of the model is that the gauge field is nondynamical at the classical level, its dynamics being entirely generated by quantum corrections. The possibility of studying in a simpler setting some of the most crucial aspects of gauge theories has been one of the main sources of interest in the model.
The phase structure of the model, for example, was studied in Arefeva:1980ms , unveiling the existence of two phases. In one of them the symmetry is broken down to , whereas in the other the model remains symmetric and mass generation occurs for the fundamental bosonic fields. Afterwards, it was found Abdalla:1990qf that the coupling to fermions preserves the two phases structure but the long range force is hindered by the fermionic fields. More recently, the supersymmetric model was studied both in the WessZumino gauge Inami:2000eb as well as using a manifestly supersymmetric covariant formulation Cho:2003qn .
On the other hand, in the past few years there have been a great deal of interest in quantum field theories defined over a noncommutative spacetime Szabo:2001kg . Sources of this interest are, among others, their relations with string theory Seiberg:1999vs and quantum gravity Doplicher:1994tu . In particular, gauge theories defined in a noncommutative spacetime have been intensively studied, and several interesting effects were found, such as UV/IR mixing Minwalla:1999px ; Hayakawa:1999yt and strong restrictions of gauge groups and couplings to the matter Matsubara:2000gr ; Chaichian ; Ferrari2 . The gauge invariance of the quantum corrections to the effective action also becomes very nontrivial in the noncommutative setting Liu:2000ad ; Pernici:2000va .
Noncommutative gauge theories present a rich spectrum of phenomena, yet they can be also very complicated to deal with. So it was natural to investigate a simpler model with gauge invariance, such as the model. When extending this model to the noncommutative spacetime, one finds more than one possibility of coupling the Lagrange multiplier and the gauge fields to the basic bosonic fields. In the nonsupersymmetric case, the socalled fundamental and adjoint representations were considered in Asano:2003ix , using an expansion. In the fundamental representation, the model turned out to be renormalizable and free of dangerous infrared UV/IR singularities. However, in the adjoint representation, the appearance of nonintegrable UV/IR divergences presented itself as an obstacle to the consistency of the model in higher orders of the expansion.
It is now well known that supersymmetry helps in avoiding the UV/IR problem in noncommutative field theories Girotti:2000gc ; Chepelev:2000hm ; Matusis:2000jf . Indeed, a supersymmetric extension of the noncommutative model in the adjoint representation was studied in Asano:2004vq , showing that the aforementioned problem is not present. However, in that case, when the noncommutativity of the space is withdrawn the model does not return to the commutative supersymmetric , but to a free scalar theory, where the basic scalar field does not satisfy a constraint nor is coupled to an auxiliary gauge field.
In the present work we are going to analyze the supersymmetric noncommutative model in the fundamental representation, i.e., the case where the commutative limit really goes to the usual commutative supersymmetric model. We shall study the phase structure of the model and the issue of the UV/IR mixing in the Feynman integrals, aiming at establishing its renormalizability at the leading order in the expansion. We will also look at the subleading corrections to the twopoint vertex function of the scalar superfield, and show how the use of an explicitly supersymmetric quantization scheme avoids some difficulties with the usual (component) approach.
Other aspects of the noncommutative supersymmetric model in the fundamental representation were also focused in the recent literature. The structure of BPS and nonBPS solitons was first found to be quite similar to the commutative model Lee:2000ey ; Furuta:2002ty ; Furuta:2002nv ; Foda:2002nt , but then novel solutions with no commutative counterparts were found Otsu:2003fq . This raised questions about the equivalence between the and the nonlinear sigma model in the noncommutative case Otsu:2004fz . Alternatives to the above investigations, employing the SeibergWitten map, can be found in Ghosh:2003ka and Govindarajan:2004kk . More recently, the dynamics of the model in a nonanticommutative spacetime has also been investigated Inami:2004sq ; Araki:2005nn .
This paper is organized as follows. In Section II, we present the model in superfield formulation and study its phase structure. Leading order corrections to the effective action of the auxiliary and gauge superfields are calculated in Section III, and the renormalizability of the model at the leading order is discussed in Section IV. In Section V, the quadratic effective action of the scalar superfield is discussed at the subleading order. Finally, Section VI contains our conclusions and final remarks. In the Appendix, we explicitly give the component formulation of the superfield model studied by us.
Ii The Noncommutative Supersymmetric Model
The bosonic model, in commutative spacetime, when the matter field is in the fundamental representation, is defined by the action Arefeva:1980ms
(1) 
where is an Nuple of scalar fields and is a scalar Lagrange multiplier which enforces the constraint . The covariant derivative is , being an auxiliary vector gauge field, which classically is the composite field . The spacetime index runs over , and we use the metric .
The model defined by (1) can be generalized to a noncommutative spacetime where coordinates satisfy
(2) 
by substituting into (1) the usual product of functions by the Moyal product moyal . The divergence structure of the noncommutative model has been extensively discussed in Asano:2003ix ; Asano:2004vq . Here we shall focus on a noncommutative supersymmetric extension of the model (1) which, adopting the conventions of Gates:1983nr , is given by
(3) 
In this expression, , where with and with are, respectively, the bosonic and the grassmanian superspace coordinates, is an Nuple of scalar superfields, is a scalar Lagrange multiplier superfield, and is a twocomponent spinor auxiliary gauge superfield. The supercovariant spinorial derivative is given by , where . We remark that the Moyal product in (3) is defined by Chu:1999ij
(4) 
affecting only the bosonic coordinates , which are noncommutative, whereas the grassmanian coordinates satisfy the usual anticommutativity rule ^{1}^{1}1Notice, however, that one can consider the extension of the noncommutativity to the grassmanian superspace coordinates in four spacetime dimensions Seiberg . Recently, this idea has also been extended to threedimensional spacetime ournac . To avoid troubles with unitarity, we shall restrict the noncommutativity to the spatial bosonic coordinates, what amounts to consider Gomis:2000zz ^{2}^{2}2It is interesting to remember that unitarity violations are a peculiar aspect of the approach for noncommutative field theories we are considering. There are alternative proposals which do not suffer from this issue, see Doplicher:1994tu ; Bahns:2002vm ; Liao:2002xc ; Balachandran:2004rq .
The action above is U(N) globally invariant and U(1) gauge invariant. The infinitesimal gauge transformations are given by,
(5)  
where is a real scalar superfield. The ordering of and in the constraint term in Eq. (3) implies in the need of being transformed under the gauge transformation in order to maintain the invariance of the action (observe that the transformation of disappears when we take the commutative limit , as it should). Had we chosen the opposite order for and , then would not need to transform, but a mixing between the fields and would appear already in leading approximation (we will return to this point later).
By writting the original (unrenormalized) superfields in terms of renormalized ones through the definitions, , , and , the action gets written as,
(6)  
In this equation and in the remaining of this paper, we will not explicitly indicate the Moyal product, which should be understood to be present when multiplying fields in configuration space. Wave functions and coupling constant counterterms are defined through
(7) 
where is an arbitrary parameter with dimension of mass and is the adimensional renormalized coupling constant. Substituting these expressions in Eq. (6) and omitting the subindex R that indicates renormalized fields, we have
(8)  
From now on, we will not explicitly write the R subindex; all fields will be understood to be the renormalized ones.
To study the phase structure of the model let us suppose that the superfields and acquire constant nonvanishing vaccum expectation values (VEVs) and , this last one, for simplicity, supposed to be in the component ^{3}^{3}3We shall not consider possible quantum phases that could arise from classical noncommutative solitonic solutions, which can be singular at small Gopakumar:2000zd . . These VEVs will play the role of order parameters identifying the different phases we will find. By redefining the fields in term of new fields that have zero VEVs,
(9)  
the action of Eq. (8) is written as
where is a short for the counterterms action, and . The propagator for the first components of is given by
(11) 
where . The interaction vertices of the theory are
(12) 
where and denotes , and similarly for the other fields.
The condition that, in leading order of , the redefined fields and have zero vacuum expectation values imply in the equations,
(13) 
where the in the integral symbol and in represent an ultraviolet regulator. The first gap equation is represented graphically in Fig. 1. These equations are the same as the corresponding ones for the supersymmetric commutative model Inami:2000eb . The dependence on of the underlying noncommutativity of the spacetime, manifested through the phase factors appearing in the vertices, disappear due to the vanishing of the momentum entering through the external leg of or . In particular, this fact ensures that UV/IR infrared singularities do not appear in the gap equation. The scalar integral in (II) can be performed using dimensional reduction Siegel:1979wq , leading to
(14) 
hence the counterterm turns out to be finite, providing an arbitrary finite renormalization of the gap equation, which now reads,
(15) 
One convenient choice for the counterterm is , where is the same mass scale introduced in (II), so that the solution of the first of Eqs. (II) can be written as
(16) 
From the second of Eqs. (II) we see that the model presents two phases,

A broken U(N) phase in which has a vacuum expectation value and the fields remain massless (). This happens for
(17) 
A symmetric phase in which has an induced mass but . This happens for
(18)
From Eq. (18) one can immediately calculate the function in the symmetric phase,
(19) 
As can be read from this formula, goes to zero for , characterizing an ultraviolet fixed point at . This result is the same as the one for the corresponding commutative model Inami:2000eb . The same analysis for the behavior of the function in the broken phase leads to similar conclusions.
We stress that the choice is convenient, but not essential. Any value of provided that leads to the same phase structure, only the value of the critical changes. If , on the other hand, the symmetric phase does not exist, while for , the model does not have the broken phase. Finally, one could choose other regularization to calculate the divergent integral in Eq. (14), in which case it could be necessary the counterterm to contain an infinite renormalization to render the gap equation (II) finite. Even in this case, the above considerations would apply without any changes.
Iii Effective propagators at leading order
As we are mainly interested in studying the renormalization of the model and the two phases have the same ultraviolet behavior coleman we will work in the symmetric phase from now on, so that the action reads
(20)  
where is related to and by Eq. (18), and
(21)  
From these equations, we see that the propagator of the is also given by Eq. (11). This fact simplifies the perturbative calculations in the next subsections.
iii.1 The two point effective action of the Field
As we read from the Eq. (20), classically the field is purely a constraint field without dynamics. However, in leading order of it acquires a (nonlocal) kinetic term becoming a propagating field. In this approximation, the radiative corrections to its two point effective action (see Fig. 2) is given by
(22) 
The exponential factors in the two vertices cancel between themselves and the result is similar to the commutative one. The nonlocal character of this action is explicit in the factor ,
(23)  
The finiteness of is consistent with the absence of a counterterm in the original action. From Eq. (22), we arrive at the following propagator,
(24) 
which is regular in the infrared () while decreasing as in the ultraviolet limit ().
Since, by definition, the effective propagator is minus the inverse of the kernel in Eq. (22), in the supergraph formalism we still have the identity represented graphically in Fig. 3, which is known to hold in the usual commutative model Arefeva:1980ms . This powerfull identity is very important in the study of subleading quantum corrections to the vertex functions of the model, as we will comment in Section V.
iii.2 The two point effective action of the spinorial gauge potential
Another field whose dynamics is generated only at the quantum level is the spinorial superfield . The leading radiative correction to its two point effective action is represented in Fig. 4. The contribution of the graph 4a gives
(25) 
while the graph 4b yields
(26)  
Adding the two contributions above we get
(27)  
Individually, the graphs in Fig. 4 are linearly ultraviolet divergent, but their leading divergences cancel between themselves, so that contains at most logarithmic divergences. In fact, it turns out to be finite, since due to the symmetric integration in the loop momentum, and after using the identity
(28) 
Eq. (27) can be written as
(29) 
where
(30) 
corresponds to the linear part of the (noncommutative) Maxwell superfield strength. Using the relation Gates:1983nr , we can write Eq. (29) as
(31) 
where we omitted the explicit arguments of the external fields since they follow the same pattern as in the previous equations. In Eq. (31) the first and second terms are induced nonlocal Maxwell and ChernSimons terms. The local limit obtained by the approximation gives for the coefficient of the induced ChernSimons term the value .
To calculate the propagator of we need to fix a gauge. One frequent choice in the literature is the Wess Zumino gauge, which amounts to choosing the components and of the superpotential (see Eq. (A) in the Appendix) as vanishing; this choice greatly simplifies the calculation in terms of component fields, but it breaks manifest supersymmetry and can lead into difficulties. So, we will fix the gauge by adding to the action given by Eq. (31) a covariant nonlocal gauge fixing term,
(32) 
and the corresponding FaddeevPopov action,
(33) 
With this gauge choice, the part of the action quadratic in turns out to be
(34) 
from which follows the propagator
(35) 
or in another form, that will be usefull for following calculations,
(36) 
The ghost propagator obtained from the FaddeevPopov action is
(37) 
and its contribution only appears at the order, so that up to the order of approximation we are considering, its contribution will not appear.
iii.3 Is there an mixing?
From the action in Eq. (20), we see that an order process mixing and is in principle possible, what would result in a mixed propagator. The graph contributing to this process is represented in Fig. 5 and the corresponding Feynman amplitude,
(38) 
can be separated in two terms, , the first one giving
(39) 
while, for the second term, we found , so that the would be vanishes. Essential to this result was the choice of the order of the fields , instead of , in the action in Eq. (20). This choice, nevertheless, requires to change under gauge transformations, to keep the action gauge invariant, as we pointed out earlier. Clearly, one could choose the order to keep gauge invariant, but then the Moyal phase factors in the two vertices in Eq. (III.3) would not compensate each other and, as a consequence, and would not cancel. In this way, a mixed term would be generated in the effective action, making highly cumbersome the evaluation of propagators and quantum corrections.
Iv Renormalizability of the model
Let us now investigate the renormalizability of the model at the leading order. The power counting for the supersymmetric model, in the fundamental representation, is the same as in the adjoint representation that was studied in Asano:2004vq , so we just quote the result. The superficial degree of divergence of a given supergraph is
(40) 
where is the number of external legs of the field , and is the number of covariant derivatives acting on the external legs. We remark that, in order to have an isoscalar contribution to the effective action, these variables are subjected to the constraints that and the sum are even numbers.
From Eq. (40) we see that, apart from vacuum diagrams, the most divergent quantum corrections to the effective action are linearly ultraviolet divergent, and these can be dangerous to the renormalizability of the model since they can generate nonintegrable (linear) infrared UV/IR singularities. It is essential to secure that linear UV/IR singularities do not appear since they would invalidate the expansion at higher orders Minwalla:1999px . Some of the graphs with have already been analyzed and shown not to generate dangerous linear divergences: the ones corresponding to the spinorial superfield effective action (), calculated in Section III.2, and the graph with , which has been taken into account in the gap equations. The only remaining contributions with is the one with , which vanishes; in fact, it is proportional to . Since there is no corresponding counterterm in , the finiteness of this contribution is essential to the renormalizability of the model.
Now, we focus on graphs with logarithmic power counting. These generate integrable, and therefore harmless, UV/IR infrared singularities. However, some of these graphs are still dangerous because they can generate ultraviolet divergent contributions to the effective action which do not have corresponding counterterms in . Listing all possible graphs with , one finds several such potentially dangerous corrections. The contribution with has been already analyzed and found to be finite in Section III.1, whereas the one with (the mixing) yields a vanishing result, as shown in Section III.3. It still remains some harmful possibilities, namely for , , , and . However, after checking that the phase factor induced by the Moyal product is planar in all these cases, one can argue that the logarithmically divergent parts of the Feynman integrals will be proportional to , which vanishes due to the symmetric integration in the loop momentum.
As for the remaining logarithmically divergent supergraphs, they correspond to terms present in the counterterm action and are, therefore, in principle renormalizable. An explicit verification of the renormalizability of the supersymmetric noncommutative model would involve the calculation of the subleading corrections to several vertex functions, and one example of such a calculation is presented in the next section.
V Subleading corrections to the effective action
Let us calculate in detail the subleading contributions to the quadratic effective action of the superfield, which arise from the diagrams depicted in Fig. 6. Here the calculations become quite involved, and part of them were performed with the help of a symbolic computer program designed for superfield calculations susymath . The Feynman amplitude corresponding to the graph in Fig. (6a) is
(41) 
Integrating in and , renaming , and using that , we arrive at
(42) 
The amplitude corresponding to Fig. 6b is given by,
and finally the contribution of Fig. 6c is
(44) 
By adding the above contributions we get for the radiative corrections to the quadratic action in the expression
(45)  
which is planar and, therefore, do not generate UV/IR mixing. Despite being nonlocal, it still shares an overall factor with the piece of the classical action quadratic in . Separating its logarithmic divergent part,
(46)  