1 Introduction

(Non) Gauge Invariance of Wilsonian Effective Actions in (Supersymmetric) Gauge Theories : A Critical Discussion

LPTENS-07/18

April 2007


(Non) Gauge Invariance of Wilsonian Effective Actions

in (Supersymmetric) Gauge Theories :

A Critical Discussion
Adel Bilal

Laboratoire de Physique Théorique, École Normale Supérieure - CNRS1

24 rue Lhomond, 75231 Paris Cedex 05, France


Abstract

We give a detailed critical discussion of the properties of Wilsonian effective actions , defined by integrating out all modes above a given scale . In particular, we provide a precise and relatively convenient prescription how to implement the infrared cutoff in any loop integral that is manifestly Lorentz invariant and also preserves global linear symmetries such as e.g. supersymmetry. We discuss the issue of gauge invariance of effective actions in general and in particular when using background field gauge. Our prescription for the IR cutoff (as any such prescription) breaks the gauge symmetry. Using our prescription, we have explicitly computed, at one loop, many terms of the Wilsonian effective action for general gauge theories, involving bosonic and fermionic matter fields of arbitrary masses and in arbitrary representations, exhibiting the non-gauge invariant (as well as the gauge invariant) terms. However, for supersymmetric gauge theories all non-gauge invariant terms cancel within each supermultiplet. This is strong evidence that in supersymmetric gauge theories this indeed defines a Lorentz, susy and gauge invariant Wilsonian effective action. As a byproduct, we obtain the explicit one-loop Wilsonian couplings for all higher-derivative terms in the effective action of arbitrary supersymmetric gauge theories.

1 Introduction

The notion of an effective action plays a most important role in modern quantum field theory. While it had long been believed that a basic criterion for any quantum field theory is its renormalizability, over the years it has become increasingly clear that our preferred renormalizable theories are just to be considered as the infrared limits of some more fundamental theories. At any finite energy scale one should actually include higher dimension operators in the action and view these theories as effective field theories described by some effective action.

There are many quite different objects going under the name of effective action. The common feature is that they somehow describe the effective behavior of certain fields at low energy without having to worry in detail about the high energy physics that already has been ”integrated out”. Specifically, one may distinguish a set of heavy and a set of light fields and completely integrate out the heavy ones, obtaining an effective action for the light ones only. Another notion of effective action is that of the generating functional of one-particle irreducible (1PI) diagrams (proper vertices) where one already has computed all the loop-diagrams. A somewhat intermediate notion is that of Wilsonian effective action where for all fields one only integrates out the high momentum/high energy modes above some scale . For loop diagrams this means that all loop-momenta are only integrated down to some infrared cutoff . When this Wilsonian effective action is used to compute correlation functions one only needs to do the remaining integrations over the low momentum/energy modes, i.e. perform loop integrals now with a UV cutoff equal to . This last property is used to define the Wilsonian action in the context of the exact renormalization group (ERG) [1], where it is an effective action with a UV cutoff obeying a certain flow equation that guarantees that the correlation functions do not depend on the UV cutoff .

In the presence of massless fields, the 1PI effective action has infrared singularities, i.e. is non-analytic at zero momentum. On the other hand, the Wilsonian effective action allows an expansion in powers of the momenta divided by and thus is an (infinite) sum of local terms. It is this locality of the Wilsonian effective action that plays an important role in many places.

In supersymmetric theories there are important non-renormalization theorems for the superpotenial, or more generally for the -terms of the action. This has been shown in perturbation theory using the powerful supergraph techniques [2]. An alternative very elegant proof of these non-renormalization theorems was given by Seiberg [3] just based on the symmetries and holomorphy of the F-terms in the Wilsonian effective action. The proof deals with the Wilsonian action since locality is crucial in order to separate and -terms.2 The same symmetry arguments actually also constrain the non-perturbative corrections to the -terms. It is most important for the proof, and usually assumed to be true, that the Wilsonian effective action is supersymmetric and Lorentz invariant. In a supersymmetric gauge theory it should also be gauge invariant. However, these properties are by no means obvious.

With this motivation in mind, in this note we would like to discuss these questions in some detail: how exactly do we define the Wilsonian effective action with the infrared cutoff ? how do we make sure this definition and the introduction of is Lorentz invariant and does not break gauge invariance or supersymmetry?

In section 2, after recalling some important issues about 1PI effective actions, we will provide a detailed discussion of Wilsonian effective actions . In this note, we will define the Wilsonian action by starting from a “microscopic” theory and really integrate out all modes above a given scale . In particular, we will give a precise (and relatively convenient) prescription how to implement the finite infrared cutoff for any loop diagram that is Lorentz invariant and respects the various linear global symmetries, like e.g. supersymmetry. We will not use the flow equations of the ERG which are different in spirit. We discuss a simple one-loop example in scalar theory, as well as a two-dimensional example of chiral fermions coupled to an abelian gauge field, where one can explicitly see the transition in the Wilsonian effective action from a sum of local terms to a non-local expression as the ratio of momentum and is varied from to . The remainder of this section deals with the issue of gauge invariance of the effective actions. Here, we also discuss various approaches in the existing literature that are mainly concerned with the possibility to introduce invariant ultraviolet regularizations in the exact renormalization group and the corresponding flow equations. We further discuss the role of using background field gauge and the manifestation in the Wilsonian action of possible anomalies.

In section 3, using our prescription for the IR cutoff, we explicitly compute many one-loop terms of the Wilsonian effective action for general gauge theories involving bosonic and fermionic matter fields of arbitrary masses and in arbitrary representations. We find that the presence of the finite infrared cutoff explicitly breaks gauge invariance, as expected, and the Wilsonian effective action for a generic gauge theory contains infinitely many non-gauge invariant terms. (Nevertheless, it will be evident that the physical correlation functions computed from this Wilsonian effective action do satisfy the Ward identities.) However, we will also show, at least for those (infinitely many) terms of the Wilsonian effective action we explicitly computed, that in a supersymmetric gauge theory, when adding the contributions of all fields within any supermultiplet, the non-gauge invariant terms precisely cancel. We argue that this is strong evidence that in a supersymmetric gauge theory one can indeed introduce the infrared cutoff and still have a Lorentz, susy and gauge invariant Wilsonian effective action at any finite scale . We use our results to explicitly give the one-loop Wilsonian couplings for all higher-derivative terms in the Wilsonian effective action for arbitrary supersymmetric gauge theories.

In the appendix, we discuss in more detail how to implement the infrared cutoff for arbitrary -loop diagrams. To illustrate the procedure, we present a complete two-loop calculation in scalar theory of the Wilsonian . Although there are a few subtleties not present at one loop, in the end we will obtain a very explicit result.

2 The Wilsonian effective action

2.1 The 1PI effective action

To compute correlation functions in any quantum field theory it is most convenient to first obtain the effective action which is the generating functional of one-particle irreducible (1PI) diagrams (proper vertices). As is well-known, within perturbation theory one can then obtain all diagrams contributing to a given correlation function by summing all tree diagrams made up with the effective vertices (which are 1PI) and full propagators as given by . In this sense, already contains all effects from loops, and actually also includes contributions beyond perturbation theory. In particular, the whole issue of renormalization must be settled when computing the effective action . Also, all symmetries of the quantum theory are coded in . In particular, if the regulated functional integral measure does preserve any linear symmetry of the classical action (including possible gauge fixing terms), i.e. if these symmetries are non-anomalous, then also is invariant under these same symmetries. This is usually expressed by Ward or Slavnov-Taylor identities.

The issue of (non-abelian) gauge symmetries is more complicated since one has to add to the classical Lagrangian a gauge fixing term and a corresponding ghosts term which of course break the gauge symmetry. The gauge symmetry then is replaced by the BRST symmetry of the complete Lagrangian which is the sum of the three terms. The BRST symmetry acts nonlinearly and hence the effective action has no reason to be BRST invariant, and much less gauge invariant. Instead, one can show that the effective action obeys the Zinn-Justin equation [4] or in more modern terms the Batalin-Vilkoviski (quantum) master equation [5], which severely constrains the possible counterterms to be BRST invariant.

An alternative approach consists in employing the so-called background field method which explicitly introduces a background gauge field and computes from by treating as a gauge field and as transforming as a matter field in the adjoint representation. One can then introduce a gauge fixing for such that the effective action still is manifestly invariant under the gauge transformations of . Technically, integrating over necessitates the knowledge of the propagators and vertices in the presence of the background fields. They can be expanded in powers of reproducing the usual diagrammatic expansion with internal -lines and external -lines (without propagators). It clearly provides a method, at least in principle, to define a gauge invariant 1PI effective action .

In supersymmetric gauge theories one can use a somewhat modified version of the background field method directly in superspace. The necessary modification is due to the fact that the vector superfield which contains the gauge and gaugino fields transforms in a complicated way under the gauge symmetry and a linear split of the form is not appropriate. This is a slight complication only and this superspace background field method is well-known [2] (see also [6]). It guarantees manifest gauge and susy invariance of the effective action. Note that for extended supersymmetry the appropriate superspace is harmonic superspace and in this case there exists also a specific background field method which is even somewhat simpler [7]. In any case, we can define a susy and gauge-invariant 1PI effective action using these methods.

In general, the 1PI effective action is a complicated non-local functional of the fields . These non-localities are due to the momentum flow through the propagators in the loops. As long as no massless fields are present one can always expand these non-local terms in powers of (external) momenta over masses, resulting in a sum of local terms, although with arbitrarily many derivatives. If the theory contains massless fields the proper vertices exhibit singularities at zero momenta and such an expansion is not possible. One can trace the origin of these singularities as coming from the region of small loop momenta. To illustrate this, consider the one-loop contribution to in scalar -theory with mass . At one-loop, the two-point function only gets a momentum-independent constant contribution, while the four-point vertex function3 gets three contributions

(2.1)

where (cf. the left part of Fig. 1)

(2.2)

Here , and are the usual squares of sums of two external momenta , etc, is a Feynman parameter and is constant4 (in dimensional regularization e.g. with one has ). The function exhibits the usual unitary cuts for . Nevertheless, in a massive theory, for we can expand this in powers of . After Fourier transforming, the corresponding contribution to then is an (infinite) sum of terms that are local, i.e. involving a single integral over space-time , each one containing more and more derivatives. Of course, the effective action does not only contain 4-point vertices, but - a priori - all interactions that are consistent with the symmetries. For example, there is a six-point vertex function which gets contributions from a one-loop triangle diagram

(2.3)

where each is the sum of the two external momenta flowing into the triangle diagram at the vertex. Again, one can expand in for small enough obtaining a sum of local contributions to . Obviously, this is no longer true if : typically in a theory containing massless fields, the have branch cut singularities starting at zero momenta.

2.2 Defining the Wilsonian effective action

Since the singularities of in the presence of massless fields are due to the regions of small loop momenta, one way of avoiding them is to impose an IR cutoff in loop diagrams. This is exactly what one does when computing the Wilsonian effective action . It is computed just like but with the restriction that all loop momenta are only integrated down to some (large) “IR-cutoff” . This implies that, even for , the Wilsonian effective action is local in the following sense: As long as all external momenta are well below the scale one can safely expand in powers of momenta divided by obtaining an effective action that is an (infinite) sum of local terms. This is a most important difference with the 1PI effective action and one of the main reasons certain statements can be made about the Wilsonian action and not about the 1PI action. Note that the whole issue of UV-divergences and renomalization has to be dealt with when computing the Wilsonian effective action - just as for the 1PI action.

More generally, one may want to “integrate out” all high energy or high momentum modes. In particular in situations with a hierarchy of masses where heavy particles have masses and light particles have masses , one could as well completely integrate out the heavy fields5 and apply the IR cutoff only to loops with light fields. In practice though, this can be cumbersome as a loop could involve both light and heavy particles, and we will stick to the prescription that is an IR cutoff for all loops. For loops involving heavy particles the difference between both prescriptions clearly is suppressed by a factor , as one can also check on our explicit examples below.


Figure 1: Two contributions to a four-point function for scalars with a quartic interaction when using the Wilsonian effective action . The loop diagram on the left involves two vertices and is to be computed with a UV cutoff , while the tree diagram on the right involves the vertex of .

The Wilsonian is to be used as an effective action to compute correlation functions. Since only takes into account loop momenta above , one must now add tree diagrams and loop diagrams using the vertices and propagators from and integrate the loop momenta from 0 to which then serves as the UV-cutoff.6, see Fig. 1. This reconstitutes the full integration range for the loop momenta. It is this additional integration over the “low momenta” that reproduces the IR singularities of the 1PI action . In this two-step procedure - first compute and then use it to compute correlation functions - is only an arbitrary intermediate scale (it is a priori not the scale at which renomalization conditions are imposed), which should not affect the final answer for the correlation functions and drop out.

This cancellation of the dependence in the correlation functions is an important point. It is obviously true if one is only concerned with the one-loop approximation where there is a single loop momentum to be integrated, provided one uses the same implementation for the UV-cutoff as one used for the IR cutoff when computing . However, for multi-loop diagrams with two (or more) loops sharing a common momentum, this can become quite tricky. (The problem is somewhat similar to the usual difficulty associated with overlapping UV divergences.) At any rate, in order for the dependence to cancel one must have a precise universal prescription of how to implement the IR cutoff when computing and then use the same prescription for the UV cutoff when using as the action to compute correlators. Before turning to this issue, let us mention, however, that one can turn this argument around and define the Wilsonian effective action as containing all possible local interactions with -dependent coupling constants (and field normalization factors) such that

  • when computing correlation functions with a UV-cutoff all -dependence drops out,

  • when equals the UV-cutoff (in which case there are no loop integrals left) the couplings equal the bare coupling constants of the classical action (typically with only finitely many being non-zero).

Defining this way one does not really have to worry about complicated overlapping loops: they are still troublesome to evaluate in practice, but we don’t have to worry about them in principle. Also, this way of defining obviously is not restricted to perturbation theory or a diagrammatic expansion. It is the basis for the so-called exact renormalization group (ERG) [1], where the -independence of the correlators is the content of the flow equations. Of course, in many theories and in particular in gauge theories, we do not want to use an explicit UV cutoff , and then it is not so clear how to implement the second requirement.7 Also, this way of defining is somewhat less intuitive. For these reasons, we will not define this way, but instead keep with the first definition of explicitly integrating out all loop momenta above .

2.3 Explicit realization of the infrared cutoff

Let us now turn to the question of giving a precise explicit prescription how to implement the infrared cutoff on the loop momenta. A basic criterion is that it should not break Lorentz invariance. It also must be independent of the way we label the loop momenta, i.e. it should be insensitive to shifts of the loop momenta. Of course, it should also, as much as possible, preserve all other symmetries of the classical action.

First note that the problem is more complicated than the usual one of UV-regulating divergent diagrams. Indeed, many Lorentz-invariant ways are known to UV regulate diagrams with an explicit cutoff (e.g. by working with modified propagators). One does not have to be too specific since, in the end, one takes . Concerning the IR cutoff , however, we want to keep finite and make sure that one can implement the same prescription for the UV cutoff when using the action to compute correlators so that one reconstitutes the full momentum integrations (see Fig. 1) and the dependence really cancels. The required cancellation of the dependence excludes simple modifications of the propagators (like adding a mass term ) to implement the IR cutoff.8 We will now discuss two different ways how to separate the low and high momentum modes. They will provide explicit IR, resp. UV cutoffs which are manifestly Lorentz (and also susy) invariant.

One explicit way to separate high and low momentum modes (see e.g. ref.[12]) is to separate the Fourier expansion of each field (bosonic or fermionic) into two parts with vanishing for and vanishing for , being the Euclidean momentum. Since the free action is diagonal in momenta the propagators do not mix and . This splitting obviously is Lorentz invariant and also respects the global linear symmetries like e.g. supersymmetry: the off-shell algebra for global supersymmetry is linear in the fields and their derivatives and does not involve any explicit functions of space-time. Hence it commutes with the action of the projectors on low or high momentum modes, and the decomposition is susy invariant. Clearly, the same applies to any other global linear symmetry. One may then explicitly do the functional integral over the high momentum modes :

(2.4)

This can be evaluated, at least in principle, order by order in an expansion in -loops. Indeed, the play the role of external sources and only propagators ever appear in the expansion. Note that the expansion can also contain tree diagrams with propagators, see Fig. 2. Obviously, such tree diagrams arise if several low momenta of the external meeting at a vertex add up to produce a high momentum of a .


Figure 2: Shown are a few contributions to generated when integrating out in a theory with quartic interactions. The solid lines represent propagators, while the dashed ones represent external .

Obviously, for gauge theories, this separation into low and high momentum modes does not respect the gauge symmetry which is a local symmetry and hence non-diagonal in momentum: any (non-constant) gauge transformation like will mix and . Also, for practical calculations, this separation of modes quickly becomes very cumbersome.

Let us now describe an alternative method that (although not respecting gauge symmetry either) is more convenient for practical computations. It does not involve as sharp a momentum separation for each field as the method above, but instead separates loop momenta into two regions: larger and smaller than . Since one can always shift the loop-momenta one needs to give a specific prescription to avoid any ambiguities. To simplify the discussion, here we will only give the prescription for one-loop integrals. The generalization to any -loop integral is relatively straightforward and will be given in the appendix, where we also do an explicit two-loop computation to illustrate this prescription.

One proceeds in the following way: First evaluate all tensor and gamma matrix algebra and introduce Feynman parameters . Then any one-loop diagram depending on external momenta and having propagators takes the form

(2.5)

where the integrand is

(2.6)

Here is some polynomial in the momenta resulting from doing all the relevant spinor and tensor algebra, and it transforms in the appropriate representation of the Lorentz group. The bracket is a Lorentz scalar provided we also transform the loop momentum . Then to regulate any UV divergences we use dimensional regularization. Actually we have two options: Usually, in dimensional regularization, one starts with fully -dimensional Feynman rules and then the polynomial results from doing the tensor and -matrix algebra in dimensions. Alternatively, as is standard in supersymmetric theories, one can first do all tensor and -matrix algebra in four dimensions and only then do the dimensional regularization of the integrals. This latter procedure is known as dimensional reduction [13]. For the purpose of implementing the IR cutoff one can use either version, as long as one does so coherently throughout.

The loop integral then is convergent and we can shift the loop momentum from to . This allows us to put the one-loop integral into the following standard form:

(2.7)

It is on this standard form, after the usual Wick rotation, that we impose the IR cutoff:

(2.8)

Note that the cutoff is applied on which is the Euclidean version of the shifted loop momentum . The shift, and hence the actual cutoff, depend on the Feynman parameters and the external , but there is no arbitrariness and after integrating the we get a “standard” result, i.e. independent of the arbitrariness in assigning momenta to the internal lines.9

It should be perfectly clear that one can use exactly the same prescription to impose a UV cutoff on this standard form of any one-loop integral. Thus when using to compute correlation functions one just has to impose the UV cutoff with this same prescription and is is obvious that one exactly obtains the part of the momentum integral that was left out when computing . Let us mention that instead of Feynman parameters one could have used parametric representation of the propagators, with a very similar result.

For later reference, let us note that the relevant one-loop integrals with the IR cutoff ,

(2.9)

are given by (see appendix A.1)

(2.10)

where and , as well as

(2.11)

Of course, as already noted, the whole issue of UV-divergences and renormalization has to be addressed when computing the Wilsonian effective action just in the same way it had to be discussed when computing the 1PI effective action .

2.4 Examples

Scalar theory in 4 dimensions

As an explicit example, we apply this procedure to the one-loop diagram on the left of Fig. 1. Here we have two propagators, so that we just have a single Feynman parameter . Furthermore , and . Then . We are left with

(2.12)

Inserting (2.3) and yields

(2.13)

where . Note that in the limit, reproduces the standard one-loop contribution to the 1PI four-point vertex, cf. (2.2), as it obviously should. To remove the pole and make finite at this order one has to add the (-independent) counterterm , where the value of the finite constant depends on the renormalization condition. Then, up to order we have

(2.14)

where is still given by (2.13) but now with . This example clearly shows several of the general features discussed above:

  • In a massless theory is not just a fictitious mass: for is not the same as with simply replaced by .

  • It is obvious from (2.13) that the corresponding contribution to the Wilsonian effective action is indeed local, even for . Explicitly, for , one has with

    (2.15)

    which is free of singularities as long as .

  • As discussed below eq. (2.8), when computing correlation functions from the propagators and vertices given by the Wilsonian action (cf (2.14)) one has to implement the ultra-violet cutoff using exactly the same prescription. For the example of the 4-point vertex function at order one has to add the two contributions shown in Fig. 1 (for each of the , and -“channels”). The tree-level contribution corresponding to the vertex from (right part of Fig. 1) is just , while the contribution corresponding to the left part of Fig. 1 is the loop-diagram, now with a UV cutoff , involving two vertices from . Hence, the latter contribution is finite and can be computed directly in giving times

    (2.16)

    This contribution (2.16) and as given in (2.13) precisely add up to produce the order contribution to the 1PI four-point vertex given in (2.2), as promised.

Let us also note that, up to one loop, is given by

(2.17)

where the value of the finite constant depends on the renormalization conditions. In minimal subtraction e.g. . Note that, even for , there is a non-vanishing . One can now compute the 1PI up to order , starting from the Wilsonian action . It receives two contributions, a tree-level contribution with the 2-point vertex as given in (2.17) and a one-loop contribution with UV cutoff , involving the 4-point vertex . The latter gives

(2.18)

When added to the former, the -dependence disappears and one reproduces the one-particle irreducible .

In the appendix we compute the two-loop contributions to . This will turn out to be quite a non-trivial example.

Chiral fermions in 2 dimensions

It is instructive to look at another example: consider a massless chiral fermion coupled to a gauge field in two dimensions. At one loop, its contribution to the vacuum-polarization of the gauge field is given by the (anomalous) current two-point function . Its computation is straightforward and parallels e.g. the nice discussion of the two-point function of the energy-momentum tensor in ref. [14]. Here we want to implement the IR-cutoff on the momentum integral in the same way as we did above. The relevant momentum integral then is

(2.19)

where (so that ) and . The integral of vanishes by symmetry and the remaining integral is convergent for . One gets

(2.20)

where

(2.21)

Setting gives back the corresponding part of the 1PI effective action. Since this is

(2.22)

This clearly shows the non-local character of the corresponding contribution to the 1PI action and also exhibits the usual anomaly (which in turn, as always, is local). For non-vanishing , the integral is elementary (and similar to given above) and can be easily evaluated in the different regions and . However, it is immediately obvious that for large it has a series expansion . Thus for we get which clearly is local. It is also obvious that, as a function of the real variable , is everywhere decreasing since , and hence (since ) it must be singular somewhere. Indeed, is singular at : the expansion in inverse powers of ceases to converge at and one could say that it is at this point where undergoes the transition from an infinite sum of local terms to a non-local expression.

2.5 Symmetries of the Wilsonian action and (non)renomalization

As we have discussed at length, to compute we need to impose a UV regularization and to specify the IR cutoff . Just as for the 1PI effective action, any linear classical symmetry will be a symmetry also of the Wilsonian effective action if both, UV regularization and IR cutoff, preserve these symmetries. In particular, we have displayed cutoffs that preserve Lorentz invariance and, if present, supersymmetry. Consider first a non-gauge theory. The question of gauge invariance will be discussed below. The Wilsonian effective action can be expanded in a series of terms with increasing numbers of derivatives, each of them being local and invariant under the non-anomalous symmetries. The -dependent coefficients of these terms are the Wilsonian coupling constants . Accordingly, the Wilsonian -functions are defined as

(2.23)

If one can show, using the symmetries of , that certain couplings actually do not depend on at all, then these couplings equal their bare values. This means that the corresponding proper vertices do not receive any contributions from loop diagrams (with infrared cutoff ) or even non-perturbatively, i.e. they are not renormalized. This is typically the argument used in supersymmetric theories in [3] for the proof of the non-renormalization theorem for the -terms.

It is very important to realize that the Wilsonian couplings are different from the corresponding effective couplings in the 1PI action and that the corresponding -functions10 also are not the same. In theories involving massless fields, going from the Wilsonian couplings to the 1PI couplings one typically has to include terms that potentially receive infrared divergent contributions. These questions have been extensively discussed in ref. [9].

2.6 Gauge invariance of the Wilsonian effective action

Slavnov-Talor identities

Obviously, if we are dealing with a gauge theory and if the gauge symmetry is not anomalous, the 1PI effective action must reflect the gauge invariance. As already discussed above, this is encoded in the Zinn-Justin equations which are a reflection of the BRST invariance of the gauge-fixed action. We also noted that if one uses a background field gauge the 1PI effective action really is gauge invariant.

The gauge invariance of the Wilsonian effective action turns out to be a more complicated question. The basic point is that the introduction of the infrared scale a priori breaks gauge, resp. BRST invariance. For example, it is well-known from the one-loop computations of the vacuum polarization in gauge theories that the introduction of an explicit momentum UV-cutoff generates quadratic divergences that lead to non-gauge invariant mass terms for the gauge fields. Clearly the same happens with an explicit infrared momentum cutoff . Alternatively, consider a BRST transformation like . It is non-linear and hence is not diagonal in the momenta and the explicit introduction of the cutoff is not manifestly BRST invariant. Thus one cannot automatically conclude that the Wilsonian effective action satisfies the Zinn-Justin equation, or equivalently that the appropriate Slavnov-Taylor identities are satisfied.

The question of gauge invariance was much studied in the framework of the exact renormalization group (ERG) [1] using the flow equations. As already mentioned, in this context one computes with a UV cutoff and deals with effective actions that have -dependent interactions. The basic point then is how to guarantee that the physical correlation functions obey the Ward identities and that the S-matrix is unitary. Probably the first gauge invariant UV regularization scheme involving an explicit scale was constructed 11 by Warr [10] by adding ingeniously arranged higher covariant derivative terms to the action. This allowed him to obtain regularized Ward identities for the regularized correlation functions which reduce to the standard Ward identities for the (finite) correlation functions in the limit , thereby guaranteeing unitarity of the S-matrix. Although very interesting, this scheme is designed to study only physics at scales well below where one effectively can consider the limit. A somewhat different treatment was given by Bechi [15] who used a UV cutoff that breaks the gauge symmetry but showed that one can add appropriately fine-tuned non-invariant (-dependent) counterterms to the effective action in order to recover the Ward identities. A more modern treatment following the same idea can be found in [16]. These questions were also studied in detail in [17, 18] where it was shown that by exploiting the freedom in the choice of appropriate renormalization conditions, the Ward identities are recovered at the end of the renormalization group flow. Said differently, the Ward identities receive -dependent corrections which flow to zero. Similarly, ref. [19] showed that the effective action with IR cutoff obtained from the flow equations satisfies modified Slavnov-Taylor identities that reduce to the ordinary Slavnov-Taylor identities in the limit . Interesting as they are, these approaches only guarantee gauge invariance at the end point of the RG flow while we really would like to argue for gauge invariance at any finite scale . More recently, refs. [20] have formulated ERG flow equations for gauge theories in a manifestly gauge invariant way by realizing the cutoff via a spontaneously broken larger gauge invariance. Finally, we should mention that it is also possible to introduce explicit momentum cutoffs by using a lattice formulation even for chiral gauge theories without breaking gauge invariance [21] but, of course, the lattice breaks explicit Lorentz invariance.

As explained above, in this note we do not use the flow equations of the ERG and instead define the Wilsonian effective action , starting from ordinary microscopic Yang Mills theory, by explicitly integrating out all the modes above the scale . We want to see whether in some cases this could still lead to a gauge invariant for any finite .

Background field gauge

Again, in order to be able to argue for gauge invariance of , it is more convenient to work in background field gauge. This is the procedure adopted throughout refs. [9] for their study of supersymmetric gauge theories. As far as the UV regularization is concerned, these references use a combination of Pauli-Villars for the chiral multiplets and higher-derivative regularization for the vector multiplets. They do not, however, explicitly specify the way they implement the IR-cutoff . Note also that the above-mentioned gauge invariant regularization by Warr has been extended to background field gauge in ref. [22]. Here, we will use the explicit IR-cutoff introduced above which has the advantage of having a clear and intuitive interpretation, and which can take any finite value: has a well-defined meaning whether the external momenta satisfy or not. As already mentioned, this IR cutoff explicitly breaks gauge invariance by generating e.g. mass terms for the gauge fields. In the next section, we will proceed to an explicit one-loop computation of several terms in the Wilsonian effective action for general gauge theories. We will see that not only these mass terms are indeed present, but actually there are (infinitely) many other non-gauge invariant terms in the effective action for a generic gauge theory.12 We will give a complete one-loop computation of these terms that are bilinear in the gauge fields and involve arbitrarily many derivatives. However, we will also show that in a supersymmetric theory all these non-gauge-invariant terms in cancel within each supermultiplet. We take this as strong evidence that the same cancellation of the non-gauge invariant terms due to supersymmetry occurs for the full Wilsonian effective action which then is indeed Lorentz, susy and gauge-invariant for all , and can be expanded, as long as , as an (infinite) sum of local terms.

Anomalies

One more point we should discuss here concerns possible anomalies. In general we will be interested in theories that contain chiral fermions, potentially leading to gauge or global anomalies. A simple explicit example was discussed in section 2.4.2. Of course, gauge anomalies render the theory inconsistent and (as usual) we will suppose that the matter content is arranged in such a way that they cancel. However, anomalies in global symmetries often play an important role. An anomaly is a non-invariance of the effective action that cannot be removed by adding local counterterms to the classical action. As we have seen in the two-dimensional example above, the non-invariant terms in must be non-local since if they were local one could just subtract these terms from the classical action as local counterterms, and the new effective action would be invariant. At first sight it then seems as if no anomaly could manifest itself in the Wilsonian effective action (at least for large enough ) , and that it is only produced as an IR effect when going from to . This is not true, however, since although is a sum of local terms, even the non-invariant part a priori is an infinite sum, and so one would have to add infinitely many counterterms to the classical action. More important, these counterterms all have coefficients that depend on . For a fixed value of they would lead to an invariant new , but if we compute at another scale the non-invariant terms would no longer cancel. Thus there is no way to cancel the anomaly in for arbitrary by adding (-independent) local counterterms to the classical action, and it makes perfectly sense to discuss global (or gauge) anomalies at the level of the Wilsonian effective action.

3 One-loop Wilsonian action for gauge theories, non-gauge invariant terms and their cancellation in susy theories

We will now explicitly compute, at one loop, various terms of the Wilsonian effective action for general gauge theories. As discussed above, we will do this in background field gauge. It is by now a standard textbook computation using background field gauge to obtain the coefficient of the term in the 1PI effective action of gauge theories coupled to spin Dirac fields transforming in some representation of the gauge group, thereby deriving the celebrated -function. Here we will follow the presentation and computation of [23], and adapt it by introducing the explicit IR cutoff according to our prescription explained in section 2.3. We will first compute terms quadratic and quartic in a constant background gauge field and then quadratic terms in an arbitrary background gauge field. In all cases we will find many terms that are not gauge invariant. However, we will also see that, in supersymmetric gauge theories, these non-invariant terms cancel when adding the contributions of all fields in any supermultiplet. Using our results, we will give explicit formulae for the one-loop Wilsonian couplings for all higher-derivative terms in the Wilsonian effective action in arbitrary supersymmetric gauge theories.

3.1 Quadratic and quartic terms for constant background gauge fields

We will first compute the one-loop Wilsonian effective action up to quartic order in the (background) gauge fields at zero momentum, i.e. for constant fields , and at vanishing ghost and fermion field background. Then and . After going through the background gauge fixing procedure, the one-loop effective action is given by the logarithm of the product of determinants of the propagators, in the presence of the background fields, of the gauge (), ghost () and fermionic matter () fields.13 Since the latter are taken to be constant, the determinants are easily evaluated. Explicitly one has

(3.1)

where in Feynman gauge () [23]

(3.2)

Note that the traces in (3.1) are traces over Lorentz indices, Dirac matrices and Lie algebra generators. As usual, with the generators in the adjoint for the gauge and ghost fields and in some matter representation for the fermions. To compute the logarithms in (3.1), we split each as where is the free propagator and , resp. are linear, resp. bilinear in the background gauge field . Using the formula it is easy to pick out the contributions to the terms in the effective action involving a given number14 of gauge fields .

Quadratic terms

General gauge theories

First, any terms involving odd powers of obviously will vanish by Lorentz invariance (as we indeed use a Lorentz invariant UV and IR regularization) since at zero momentum there is no way to form a Lorentz scalar with an odd number of gauge fields . Next, we look at the term quadratic in . If present at zero momentum such a term clearly represents a mass term for the gauge field and breaks gauge invariance. The corresponding contribution to is

(3.3)

Here the first line contains the contributions from () and (), while the second line contains those of . Note that the factor of in the second line would be absent for Majorana fermions. In dimensional regularization without any IR cutoff one has

(3.4)

for all and in particular also for . This implies the vanishing of (3.3) and the absence of mass terms for the gauge field in the 1PI action .

In section 2.3, we discussed how to introduce the IR cutoff on the standard form of the loop integrals. To bring them into standard form one had to perform translations of the integration variables and in order to be able to do so we had to work with already convergent integrals. This is why we used dimensional regularization of the integrals. Here, however, since we work at vanishing external momentum, the integrals already are in standard form. Thus, alternatively, we can simply introduce a euclidean momentum UV cutoff and IR cutoff on the integrals in (3.3), working directly in 4 dimensions. It will be interesting to compare both UV regularizations. The relevant integrals (which we denote by to distinguish them from their dimensionally regularized cousins ) then are

(3.5)

and are given by

(3.6)

as well as

(3.7)

Using these integrals we get from (3.3)

(3.8)

with the first line coming from the gauge field and ghost loop and the second line from the fermion matter loop. Explicitly one has