A Divergent contributions to the diagrams of Figs. 2 and 3

Renormalisability of noncommutative GUT inspired field theories with anomaly safe groups


FTI/UCM 101-2009

Renormalisability of noncommutative GUT inspired field theories with anomaly safe groups

C.P. Martín1 and C. Tamarit2 Departamento de Física Teórica I, Facultad de Ciencias Físicas

Universidad Complutense de Madrid, 28040 Madrid, Spain Kavli Institute for Theoretical Physics, University of California

Santa Barbara, CA, 93106-4030, USA

We consider noncommutative GUT inspired field theories formulated within the enveloping-algebra formalism for anomaly safe compact simple gauge groups. Our theories have only gauge fields and fermions, and we compute the UV divergent part of the one-loop background-field effective action involving two fermionic fields at first order in the noncommutativity parameter . We show that, if the second-degree Casimir has the same value for all the irreps furnished by the fermionic multiplets of the model, then, that UV divergent part can be renormalised by carrying out multiplicative renormalisations of the coupling constant, and the fields, along with the inclusion of -dependent counterterms which vanish upon imposing the equations of motion. These -dependent counterterms have no physical effect since they vanish on-shell. This result along with the vanishing of the UV divergent part of the fermionic four-point functions leads to the unexpected conclusion that the one-loop matter sector of the background-field effective action of these theories is one-loop multiplicatively renormalisable on-shell. We also show that the background-field effective action of the gauge sector of the theories considered here receives no -dependent UV divergent contributions at one-loop. We thus conclude that these theories are on-shell one-loop multiplicatively renormalisable at first order in .

PACS: 11.10.Gh, 11.10.Nx, 11.15.-q, 12.10.-g.
Keywords: Renormalization, Regularization and Renormalons, Non-commutative geometry.

1 Introduction

Noncommutative gauge theories with simple groups can only be formulated with the so called enveloping algebra approach, which makes use of Seiberg-Witten maps to relate noncommutative gauge orbits to ordinary ones [1]. Since the Seiberg-Witten maps are generically obtained perturbatively in the noncommutativity parameters , the resulting theories, which are invariant under ordinary gauge transformations, involve interaction terms at all orders in . This, and the fact that has negative mass dimensions, seems to suggest that the theories are only meaningful as effective theories. However, some intriguing results seem to point towards a perturbative self-consistency of the theories: it could well be that for some models the structure imposed by the Seiberg-Witten maps survives quantum corrections, so that the divergences can be absorbed by both multiplicative renormalisations and by physically irrelevant counterterms (e.g., couterterms which vanish on-shell). The first one of these results concerns the fact that the gauge anomaly cancellation conditions have been shown to be, to all orders in , equal to their commutative counterparts [2]; this allowed to formulate noncommutative extensions of the Standard Model [3], and GUT theories [4]. Other results concern the renormalisability of the gauge sector at one-loop, observed for a variety of models independently of the matter content [5, 6, 7, 8, 9, 10, 11, 12]; in fact, the matter determinants contributing to the one-loop gauge effective action are known to yield renormalisable contributions to all orders in , at least for non-chiral theories [13].

Despite these auspicious results, the matter sector –matter in the fundamental representation– of the theories studied so far –having U(1) and SU(2) as gauge groups– is nonrenormalisable [5, 7, 11] and the lack of renormalisability can be traced back to problematic divergences in four point functions of the matter fields . There are, however, promising exceptions: on the one hand, supersymmetric (S)U(N) theories with adjoint Majorana fermions in a vector multiplet have been shown to be one-loop renormalisable [13], and, on the other, noncommutative GUT inspired theories with arbitrary groups and representations have been shown to be free of the unwelcomed four fermion divergences just mentioned [14, 15].

In this paper we continue the study of the renormalisability of noncommutative GUT inspired theories with no scalar fields, by computing the UV divergent part of the effective action involving two fermion fields. Here, we do it for theories with anomaly safe compact simple gauge groups –groups for which the anomaly coefficient vanishes in all representations– since among these groups one finds the phenomenologically promising SO(10) and . These models have, as a consequence of the anomaly cancellation condition, no vertices of order one in in the bosonic part of the classical action, and hence are not sensitive to ambiguities in the trace over bosonic fields [4]. We use the background field method in the Feynman-background-field gauge in conjunction with dimensional regularisation to reconstruct, at first order in , the full one-loop UV divergent contribution involving two fermion fields. We do so by using gauge invariance and working out the pole part of the two- and three-point Green functions involving, respectively, two fermion fields and one gauge field and two fermion fields. The result is the following: whenever all the irreducible representations carried by the fermion multiplets of the theory share the same second-degree Casimir, the UV divergences can be renormalised by using multiplicative renormalisation of the coupling constant, the noncommutative matrix parameter and the fields, and by adding -dependent counterterms which vanish on-shell, i.e., upon imposing the equation of motion. These -dependent counterterms which vanish on-shell have, of course, no physical effect. If one combines this result with the absence of UV divergent contributions to the fermionic four-point function of these theories –see ref. [15]–, one concludes that the one-loop matter sector of the theory is renormalisable on-shell at first order in ; this is the first time that such property is shown to hold in a noncommutative theory with nonmajorana fermions. The requirement of a common second-degree Casimir for the matter representations can be fulfilled by using a single irreducible representation –as is commonly done in ordinary GUTs such as SO(10) and –, though our renormalisability result is valid for any choice of representation– or combining a representation with its conjugate. Finally, once the matter sector has been seen to be renormalisable, we show by using formal arguments that there are no UV divergent contributions to the gauge sector which are of order one in . We thus put forward, for the first time in the literature, a huge family of noncommutative theories with chiral fermionic matter and GUT gauge groups which are one-loop renormalisable at first order in , in the physical sense that only the ordinary renormalisation of the coupling constant and a new multiplicative renormalisation of the noncommutative matrix parameter are needed to workout UV finite S-matrix elements: the counterterms –in particular, the a priori problematic -dependent counterterms– which are not given by the renormalisations of and the coupling constant vanish on-shell. Recall that the free parameters of our classical noncommutative field theories are the coupling constant and .

Now, to make sure that the Hamiltonian formulation of our theories is the elementary one –only one canonical momenta per generalised coordinate–, we shall choose a noncommutative matrix parameter such that , . Hence, without lose of generality one can say that is characterized by a single noncommutative parameter, say, .

The paper is organised as follows. The theory is defined in section 2, where the computation by means of the background field method is also outlined. Section 3 includes the results of the computations of the UV divergent part of the effective action involving two fermion fields, whose renormalisability is discussed in section 4. Section 5 is dedicated to argue in favour of the renormalisability of the gauge sector. Conclusions are presented in section 6. We also include two appendices: appendix A provides the results for the divergent contributions to the Feynman diagrams involved in the computations of section 3, while appendix B gives the results for the beta functions of the physical parameters of the theory, and .

2 The theory and their background field method quantisation

We shall consider a general four-dimensional noncommutative GUT inspired theory with an arbitrary anomaly safe [2] compact simple gauge group and no scalar fields as formulated in ref. [4]. We thus define the theory by means of a noncommutative left-handed chiral multiplet in an arbitrary representation of the gauge group, and an enveloping-algebra valued gauge field with action


where, at first order in , the noncommutative fields are defined in terms of the ordinary ones by the following standard Seiberg-Witten maps,


Note that denotes an arbitrary unitary representation, which can be expressed as a direct sum of irreducible representations, . Accordingly, the fermion fields can be expressed as a direct sum of irreducible multiplets, , .

Upon substituting eq. (2.2) in eq. (2.1) and, then, expanding up to first order in , one obtains a classical action for the ordinary fields and . Within the enveloping-algebra formalism, the quantisation of the theory defined by this classical action defines the corresponding noncommutative field theory at first order in . It has been shown in ref. [4] that for compact simple gauge groups the anomaly cancellation condition [2] makes the first order in contribution coming from the noncommutative Yang-Mills action in eq. (2.1) vanish. So for the family of theories studied in this paper, and at first order in , the only classical noncommutative corrections to the ordinary classical action come from the fermionic action in eq. (2.1).

Since we shall formulate the Feynman rules of our theory in terms of ordinary Dirac fermions, we include in it a spectator right-handed fermion, as done in ref. [15]

Again as in ref. [15], we shall regularise the theory by means of dimensional regularisation in dimensions, using the BMHV scheme for defining [16, 17]. In this scheme there is an infinity of dimensionally regularised actions which reduce to (2.1) in the limit , and which differ from one another by evanescent operators [18]. Following [18] we will keep all the vector indices in interaction vertices “four-dimensional”, i.e., contracted with the “barred” metric ; we shall also define the dimensionally regularised as being “four-dimensional”. Furthermore, in our computations we shall discard any contribution which has a pole in but whose numerator is an evanescent operator. Since we shall be dealing with an anomaly free theory, these contributions involving evanescent operators have no physical effects at the one-loop level [18, 19]–although they are needed at two loops and beyond [20] – and are mere artifacts of the regularisation procedure. It is not difficult to convince onself that the famous one-loop contributions to Green functions are uniquely fixed by the pole contributions to the effective action with no evanescent operators.

Our aim is to compute the one-loop UV divergent part of the effective action involving two fermion fields and no evanescent operator in a manifestly covariant approach, which allows to reconstruct the full contribution to the effective action from a minimum number of diagrams, as was done in ref. [12]. For this we use the background field method [21]. This method amounts to split the gauge field in a background part and a quantum part


and choose a gauge fixing which preserves background gauge transformations

This gauge fixing is

Adapting to our case the discussions in refs. [21] and [12], introducing the classical fields , the 1PI functional is given by


The previous effective action is gauge invariant under gauge transformations of the classical fields . The dimensionally regularised version of the effective action above is not strictly speaking gauge invariant, i.e, it is gauge invariant modulo an evanescent operator which as we have argued above can be dropped for anomaly- free theories in UV divergent one-loop computations.

Let us notice that is equivalent to a background 1PI diagram with background gauge field legs, fermionic legs and anti-fermionic legs. (Note that our definitions do not involve any symmetrisation over the background gauge fields). The vertices relevant to our calculations and their associated Feynman rules for are given in Fig. 1. In the Feynman rules, the background field legs are denoted with an encircled “b”; the rules are defined without symmetrising over background field legs, in accordance with eq. (2.4).


p \psfragm A \psfragn B \psfragr C \psfragm C \psfragi A \psfragj B \psfragq\psfragk \psfragi s\psfragj t\psfragk,a,m \psfragk1\psfragk2\psfragk3 \psfragm1\psfragm2  

Figure 1: Feynman rules of the noncommutative interactions relevant to our calculations, involving the Dirac fermion .

3 Computation of the UV divergent part of the effective action involving two fermion fields

In this section we shall compute the UV divergent contributions (not involving evanescent operators) to the effective action involving two fermion fields, at one-loop and first order in , by calculating the background field 1PI diagrams with no external quantum field legs of eq. (2.4), using the Feynman rules in Fig. 1.

To ease the computation we consider the following simplifications, which do not mean a loss of generality of the results:

  • Choice of gauge . This choice greatly simplifies the gauge propagator and since the on-shell effective action is independent of the gauge-fixing term –see [22] and references therein–, the conclusions we shall draw from our explicit computations upon taking them on-shell will also be gauge independent.

  • Computing a minimum number of diagrams. Since the use of the background field method ensures gauge invariance (modulo one-loop irrelevant evanescent operators) of the result for an anomaly free theory, the full gauge invariant contribution to the UV divergent part with no evanescent operator of the effective action –which is local in the fields– can be reconstructed from a reduced number of 1PI diagrams . These UV divergent contributions to the effective action can be expanded in a basis of independent gauge invariant terms. If their contributions with a given number and types of fields are also independent, then the coefficients in the expansion can be fixed by computing the 1PI diagrams with the same number and types of fields.

In order to identify the diagrams that must be computed, we should start by choosing a basis in 4 dimensions of gauge invariant terms whose integrals are independent. Since it was shown in ref. [15] that noncommutative GUT inspired theories such as the ones under consideration have no four fermion divergences, a little power-counting takes us to the conclusion that we only need to consider terms with two fermion fields. We choose the following ones, for each flavour :


In the formulae above, and are shorthands for and . We will omit explicit indications of the representations in future formulae; it will be assumed that a Lie-algebra valued field or generator acting on a fermion does so in the representation . Note that there are other admissible gauge invariant terms, involving symmetric invariant tensors of the gauge group, such as ; however, these terms can be seen not to appear in the UV divergent part of the effective action, and may be ignored (also, recall that we are dealing with anomaly safe theories with ). Schematically, the terms that appear in the UV divergent part of the effective action are of the form –spanned by , which involve at least two fermion fields and have independent two-field contributions– and , spanned by , which involve at least two fermion fields and a gauge field, in such a way that these contributions are independent of each other. From the discussions above, it is clear that the coefficients of the expansion of the UV divergent contributions to the effective action involving two fermion fields can be obtained by computing only 1PI diagrams with two fermion fields, , and with one gauge field and two fermion fields, . The diagrams are shown in Figs. 2 and 3. Note that there is a subtlety in the computation: the diagrams of Fig. 2 yield the contribution to the effective action (see eq. (2.4)), which fixes the coefficients of and in the expansion of the effective action in terms of the basis of gauge invariant terms. In turn, the diagrams of Fig. 3 contribute as to the effective action, and this will be a sum of three-field terms coming from both the combination fixed beforehand and from the terms. In order to get the coefficients of the latter, the three-field contributions of and have to be subtracted.



Figure 2: Diagrams contributing to at order .

b1\psfragb2\psfragb3\psfragb4\psfragb5\psfragb6 \psfragb7\psfragb8\psfragb9

Figure 3: Diagrams contributing to at order .

The results for the UV divergent contributions of the diagrams of figs. 2 and 3 are shown in appendix A. The final result for the gauge invariant contributions to the divergent part of the one-loop effective action is the following,


where all covariant derivatives and field strengths are evaluated on the background field , and we have suppressed the hats on the classical fields as well as explicit indications of the representations of the field strength and its covariant derivatives to ease the notation. In the formulae above, represents the second Casimir of the representation , corresponding to the adjoint representation. It is defined as . In terms of the basis of eq. (3.5)


4 Analysing renormalisability of the matter sector

The objective of this section is to check whether the noncommutative divergences involving two fermion fields of eq. (3.6) can be subtracted by means of multiplicative renormalisation of the coupling constant, the noncommutative parameter and fields plus counterterms which vanish on-shell.

First, the the one-loop divergences at order zero in are, as is well known, renormalisable by means of multiplicative renormalisations of fields and parameters. These multiplicative renormalisations take the form


with . The parameter is the dimensional regularisation scale. Note that in the background field method there is no need to renormalise the quantum field of eq. (2.3). It is easily seen that gauge invariance forces , while the divergences in the ordinary theory yield


where is the index of the representation ; its relation with is given in eq. (B.16). Let us introduce the following type of counterterms


which vanish on-shell due to the equations of motion

In order to preserve gauge symmetry, and have to transform in 4 dimensions under gauge transformations as follows

We consider the following and


which have the appropriate behaviour under gauge transformations. Note that we only considered a left-handed part for , since all the divergences in eq. (3.6) involve left-handed projectors .

The counterterm action involving two fermion fields obtained by considering the multiplicative renormalisations of eq. (4.8) and the tree-level contributions of the counterterms vanishing on-shell of eqs. (4.10) and (4.11), has the following expansion in the basis of terms :


The of eq. (4.11) also generate, to and at tree-level, terms involving four fermion fields,


As seen in ref. [15], four-fermion divergences are absent in noncommutative GUT compatible theories at one loop, . Renormalisability of the divergences involving two and four fermions amounts to demand

where and are given, respectively, by eqs. (4.13), (4.12) and (3.7) The first equation is solved by choosing to be flavour independent (). Solving the second identity in the basis of independent terms and projecting the resulting equations into their real and imaginary parts, one gets

The equations are compatible, and we find the following family of solutions