1 Introduction
Abstract

We elaborate on the structure of the graviton-gauge-gauge vertex in the electroweak theory, obtained by the insertion of the complete energy-momentum tensor () on 2-point functions of neutral gauge currents (). The vertex defines the leading contribution to the effective action which accounts for the conformal anomaly and related interaction between the Standard Model and gravity. The energy momentum tensor is derived from the curved spacetime Lagrangian in the linearized gravitational limit, and with the inclusion of the term of improvement of a conformally coupled Higgs sector. As in the previous cases of QED and QCD, we find that the conformal anomaly induces an effective massless scalar interaction between gravity and the neutral currents in each gauge invariant component of the vertex. This is described by the exchange of an anomaly pole. We show that for a spontaneously broken theory the anomaly can be entirely attributed to the poles only for a conformally coupled Higgs scalar. In the exchange of a graviton, the trace part of the corresponding interaction can be interpreted as due to an effective dilaton, using a local version of the effective action. We discuss the implications of the anomalous Ward identity for the correlator for the structure of the gauge/gauge/ effective dilaton vertex in the effective action. The analogy between these effective interactions and those related to the radion in theories with large extra dimensions is pointed out.

The Conformal Anomaly and the Neutral Currents Sector of the Standard Model



Claudio Corianò, Luigi Delle Rose, Antonio Quintavalle and Mirko Serino



Departimento di Fisica, Università del Salento

and INFN-Lecce, Via Arnesano 73100, Lecce, Italy111claudio.coriano@unisalento.it, luigi.dellerose@le.infn.it, antonio.quintavalle@le.infn.it, mirko.serino@le.infn.it

1 Introduction

Gravity couples to the Standard Model, in the weak gravitational field limit, via its energy momentum tensor (EMT) . This interaction is responsible for the generation of the radiative breaking of scale invariance [1, 2, 3], which is mediated, at leading order in the gauge coupling (), by a triangle diagram: the vertex (see [4, 5, 6, 7]), where denote two gauge bosons. The computation of the vertex is rather involved, due to the very lengthy expression of the EMT in the electroweak theory, but also not so obvious, due to the need to extract the correct external constraints which are necessary for its consistent definition.

The constraints take the form of 3 Ward identities derived by the conservation of the EMT and of (at least) 3 Slavnov-Taylor identities (STI’s) on the gauge currents. All of them need to be checked in perturbation theory in a given regularization scheme, in order to secure the consistency of the result. In the case under exam they correspond to the and vertices, where is the photon and the neutral massive electroweak gauge boson. We will be stating these identities omitting any proof, since the details of the derivations are quite involved.

The explicit computation of these radiative corrections (i.e. of the anomalous action) finds two direct applications. The first has to do with the analysis of anomaly mediation as a possible mechanism to describe the interaction between a hypothetical hidden sector and the fields of the Standard Model, as shown in Fig. 1 (a). One of the results of our analysis, in this context, is that anomaly mediation is described by the exchange of anomaly poles in each gauge invariant sector of the perturbative expansion, as shown in Fig. 1 (b). This feature, already present in the QED and QCD cases, as we will comment below, is indeed confirmed by the direct computation in the entire electroweak theory. One of the main implications of our analysis, in fact, is that this picture remains valid even in the presence of mass corrections due to symmetry breaking, for a graviton of large virtuality and a conformally coupled Higgs sector. We will comment on this point in a separate section (section 5) and in our summary before the conclusions.

A second area where these corrections may turn useful is in the case of an electroweak theory formulated in scenarios with large extra dimensions (LED), with a reduced scale for gravity. In this case the virtual exchanges of gravitons provide sizeable corrections to electroweak processes - beyond tree level - useful for LHC studies of these models, as illustrated in Fig. 2 in the case of the annihilation channel. In these extensions a graviscalar (radion) degree of freedom is induced by the compactification, which is expected to couple to the anomaly () as well as to the scaling-violating terms, as we are going to clarify, by an extra prescription. This prescription is based on the replacement of the classical trace of the matter EMT by its quantum average. A rigorous discussion of the fundamental anomalous Ward identity for the correlator will clarify some subtle issues involved in this prescription. We will show, in parallel, that the anomalous effective action induces in the 1-graviton exchange channel a similar interaction. This interaction can be thought as being mediated by an effective massless dilaton, coupled to the trace anomaly equation (and to its mass corrections).

2 Definitions and Ward and Slavnov-Taylor identities

We start with few definitions, focusing our discussions only on the case of the graviton/photon/photon () and graviton/Z/Z () vertices.

We recall that the fundamental action describing the gravity and the Standard Model is defined by the three contributions

(1)

where , with being the four dimensional Newton’s constant and is the Higgs doublet. We have denoted with the contribution from gravity (Einstein-Hilbert term) while is the Standard Model (SM) quantum action, extended to curved spacetime. denotes the term of improvement for the scalars, which are coupled to the metric via its scalar curvature . The factor should be recognized as giving a conformally coupled Lorentz scalar, the Higgs doublet. The EMT in our conventions is defined as

(2)

and around a flat spacetime limit

(3)

with the symmetric rank-2 tensor accounting for the metric fluctuations.

Figure 1: Gravitational interaction of the Standard Model fields with a hidden sector (H.S.), at leading order in the gravitational constant (a). The interaction in perturbation theory responsible for the trace anomaly is illustrated in (b) via the exchange of an anomaly pole.
Figure 2: Typical leading order () contributions to the production of two gauge bosons with gravitational mediation. Not included are the initial state (Standard Model) corrections on the /graviton vertex and the loops of gauge bosons and Higgs mediating the decay of the graviton. The latter contribute to the conformal anomaly.

We denote with the complete (quantum) EMT of the electroweak sector of the Standard Model. This includes the contributions of all the physical fields and of the Goldstones and ghosts in the broken electroweak phase. Its expression is uniquely given by the coupling of the Standard Model Lagrangian to gravity, modulo the terms of improvements, which depend on the choice of the coupling of the Higgs doublets. As we have mentioned, we have chosen a conformally coupled Higgs field. Our computation is performed in the gauge. The expression of the EMT is symmetric and conserved. It is therefore given by a minimal contribution (without improvement) and the improvement EMT, , with

(4)

where the minimal tensor is decomposed into

(5)

The various contributions refer, respectively, to the gauge kinetic terms (field strength, ), the fermions, the Higgs, Yukawa, gauge fixing contributions () and the contributions coming from the ghost sector. As we have already mentioned, in order to fix the structure of the correlator one needs to derive and implement the necessary Ward and STI’s. Their derivation is quite lengthy as is their implementation in perturbation theory, given the sizeable number of diagrams involved in the expansion and the very long expression of the vertex extracted from the EMT.

We obtain:
1) A Ward identity related to the conservation of the EMT in the flat spacetime limit (i.e. ), which takes the form

where we have introduced the off-diagonal 2-point function

(6)

where denotes amputated external gauge lines. Notice that the gravitational field, in this computation, is just an external field and the 1PI conditions apply only to the external gauge lines. This point emerges from a closer investigation of the defining STI’s of the correlator. This Ward identity applies to any gauge boson in the neutral sector.

2) A STI for the vertex. Specifically, introducing the photon gauge-fixing function

(7)

we obtain the relation

(8)

with denoting the gauge-fixing parameter.

Figure 3: Amplitudes with the triangle topology for the two correlators and .
Figure 4: Amplitudes with t-bubble topology for the correlators and .
Figure 5: Amplitudes with s-bubble topology for the correlators and .
Figure 6: Amplitudes with the tadpole topology for the correlators and .
Figure 7: Amplitudes with the triangle topology for the correlator .
Figure 8: Amplitudes with the t-bubble topology for the correlator .
Figure 9: Amplitudes with the s-bubble topology for the correlator .
Figure 10: Amplitudes with tadpole topology for the correlator .

3) A STI for the correlator. Introducing the gauge-fixing function of the gauge boson

(9)

where is the Goldstone of the , this takes the form

(10)

4) A STI for the vertex

(11)

We illustrate the overall structure of the results for the and vertices, focusing on the essential parts, and in particular on those form factors which contribute to the trace part, since they are simpler. The complete result is indeed quite involved and some details can be found in [8].

3 Results for the case

In the case, we introduce the notation to denote the one-loop amputated vertex function with a graviton and two on-shell photons.

In momentum space we indicate with the momentum of the incoming graviton and with and the momenta of the two photons. In general, the correlator is defined as

(12)

In the 2-photon case is decomposed in the form

(13)

as a sum of a fermion sector (F) (Fig. 3(a), Fig. 4(a)), a gauge boson sector (B) (Fig. 3(b)-(g), Fig. 4(b)-(g), Fig. 5, Fig. 6) and a term of improvement denoted as . The contributions to the (F) and (B) sectors are obtained by the insertion of . The contribution from the term of improvement is given by diagrams of the same form of those in Fig. 3(c), 3(e) and Fig. 5(b), but now with the graviton - scalar - scalar vertices determined only by the energy momentum tensor .

The tensor basis on which we expand the vertex is given by four independent tensor structures

(14)
(15)
(16)

where has been defined as

(17)

among which only and show manifestly a trace, the remaining ones being traceless. A complete computation gives for the various gauge invariant subsectors

(18)
(19)
(20)

The first three arguments of the form factors stand for the three independent kinematical invariants , while the remaining ones denote the particle masses circulating in the loop. We use the on-shell renormalization scheme.

As already shown in the QED and QCD cases [5, 6, 7], in an unbroken gauge theory the entire contribution to the trace anomaly comes from the first tensor structure .

In the vertex, the contribution to the trace anomaly in the fermion sector comes from which is given by

(21)

Here we have introduced the QED coupling and the function

(22)

obtained from the scalar triangle integral, with

(23)

The sum is taken over all the fermions () of the Standard Model. As one can immediately realize, this form factor is characterized by the presence of an anomaly pole

(24)

which is responsible for the generation of the anomaly in the massless limit. To appreciate the significance of this ”pole contribution” one needs special care, since a computation of the residue (at ) shows that this is indeed zero in the presence of mass corrections. However, this leading behaviour in the trace part of the amplitude, as we are going to show, is clearly identifiable in an (asymptotic) expansion (), and is corrected by extra terms, where denotes generically any fermion of the SM. In other words, this component is extracted in the UV limit of the amplitude even in the massive case and is a clear manifestation of the anomaly.

The other gauge-invariant sector of the vertex is the one mediated by the exchange of bosons, Goldstones and ghosts in the loop. We will denote with and the masses of the W’s and gauge bosons and the Higgs mass respectively. In this sector the form factor contributing to the trace is

(25)

which multiplies the tensor structure , responsible for the generation of the anomalous trace.

In this case the anomaly pole is easily isolated from (25) in the form

(26)

The term of improvement is responsible for the generation of two form factors, both of them contributing to the trace. They are given by

(27)
(28)

the first of them being characterized by an anomaly pole

(29)

Our considerations on the UV behaviour of the a) radiative plus the b) explicit mass corrections to the anomalous amplitude are obviously based on an exact computation of the correlator.

In the asymptotic limit (), the expansions of the three form factors contributing to the trace part can be organized in terms of the ”pole component” plus mass corrections, which are given by

(30)
(31)
(32)

The energy suppressed terms (, ) take the typical form , with denoting, generically, any explicit mass term generated in the broken phase of the theory. This separation of the radiative from the explicit contributions to the breaking of conformal invariance, due to the tree-level mass terms, is in agreement with the obvious fact that in the UV limit, masses can be dropped. At the same time the (radiative) breaking of the conformal symmetry remains, with no much surprise.

Preliminarily, we recall that in the scheme the functions of the Standard Model are given by

(33)

for the hypercharge, weak and strong interactions respectively, and is the number of generations. The expression of the function of the electromagnetic coupling, , is given in the same scheme by

(34)

At this point, the residue of the anomaly pole which appears in the form factors , and is uniquely determined by the beta function of the electromagnetic coupling constant. Indeed we have

(35)

where we have used the fact that .

4 Results for the case

Moving to the vertex with two massive gauge bosons, one discovers a similar pattern. Also in this case, as before, we introduce the notation to describe the corresponding correlation function. We have several contributions appearing in the global expression of the correlator:

(36)

, for on-shell bosons, can be separated into three contributions obtained using the insertion of (sectors and ) and a fourth one coming from the term of improvement. In this case the gravitational interaction is mediated by .

The labelling of the first three is inherited from the types of particles (and corresponding masses) that circulate in the loops. Beside the fermion sector (F) with diagrams depicted in Figs. 3(a) and 4(a)), the other contributions involve a gauge boson (sector (W)), with diagrams Fig. 3(b)-(g), Fig. 4(b)-(g), Fig. 5 and Fig. 6), and the mixed /Higgs bosons sector with contributions shown in Figs. 7, 8, 9 and 10). There is also a diagram proportional to a Higgs tadpole (Fig. 10(a)) which vanishes in the on-shell renormalization scheme. Finally there is a contribution from the term of improvement (I). This is given by the diagrams depicted in Fig. 3(c), (d), 5(b), together with those of Figs. 7(b), (c), (d) and Fig. 9. In this case, however, the graviton - scalar - scalar vertices is generated by .

As we have already mentioned, we take the two gauge bosons on the external lines on-shell, and an insertion of at a nonzero momentum transfer . The four contributions can be expanded on a tensor basis given by 9 tensors, and corresponding form factors as

(37)
(38)
(39)
(40)

where the first three arguments of the ’s are the virtualities of the external lines , while the last two give the masses in the internal lines. 7 of the 9 tensor structures are traceless, while the only two responsible for the breaking of scale invariance are

(41)

The four form factors responsible for generating a pole term are those accompanying the tensor structure , while the form factors , corresponding to the tensor structure , show no pole. The latter give contributions which are suppressed as . Therefore, as for the vertex, the trace parts show a distinctive contribution plus corrections of , as we have specified above. Being the complete result of this vertex quite lengthy, we omit details and just focus our attention on the pole terms extracted from each sector. These are summarized by rather simple expressions. We obtain

(42)

for the fermion sector of the vertex, where and are short notations for and . Similarly, in the other sectors we have

(43)

for diagrams involving ’s, while the diagrams with and gauge bosons give

(44)

The term of improvement contributes to two tensor structures but only one of the two form factors from this sector has a pole term. We have, in this case

(45)

while asymptotically.

The coefficient of the anomalous pole contribution is fixed by the beta functions of the theory. In this case it is proportional to a linear combination of the beta functions of the couplings and of hypercharge and . Indeed we have

(46)

5 The coupling of the radion/dilaton beyond tree level and the effective dilaton

One of the most significant applications of the results of the previous sections concerns the study of the coupling of the radion/dilaton to the fields of the Standard Model in theories with LED. We will use the term radion () to denote the fundamental scalar introduced in the usual compactifications of theories with LED, and reserve the name of ”effective dilaton” () for the scalar interaction dynamically induced by the anomaly. As we are going to show, the radion has interactions with matter which are quite similar to those allowed to the effective dilaton, although the latter shows up in a different channel (the 1-graviton exchange channel). We proceed first with a rigorous discussion of the interaction of the LED radion and then illustrate the analogies between the two states to clarify these points.

We recall that in models with LED, with matter on the brane and gravity in the bulk, the compactification of the extra dimensions gives rise in the 4 dimensional effective field theory to towers of Kaluza-Klein gravitons and dilatons. For definiteness we consider a theory compactified on a torus and consider the zero modes of the 4D graviton field and of the dilaton generated by this procedure. These two fields will couple, via their lowest Kaluza-Klein modes, to the EMT with the interaction Lagrangian [9]

(47)

where is the number of extra dimensions. To understand the main features of the dilaton interaction at 1-loop level we proceed as follows.
We first recall that the structure of the anomaly equation in the presence of a classical trace in a certain theory takes the form

(48)

where we have taken the quantum average of each term. is the operator describing the anomalous behavior of the fields under scale transformations while the operator is the non-amomalous contribution to the trace of the EMT. This second term vanishes in the conformal limit (i.e. before electroweak symmetry breaking) using the equations of motion of the fields. In an exact gauge theory the expected structure of the anomaly is given by the relation

(49)

where and are the field strengths and the gauge couplings of the gauge fields in the unbroken phase, corresponding to the Standard Model gauge group . For a theory in a broken phase, and in the photon case, the anomaly is again proportional to . By taking two functional derivatives of the trace identity (48) with respect to the sources and of the gauge fields and , we obtain the anomalous identities on the correlation functions analyzed in this work

(50)

The first term on the right-hand side of the equation above defines the residue of the anomaly pole that we have already discussed and isolated in the previous sections. The second term, instead, is the correlation function obtained by inserting the trace of the EMT on the two point functions (with the inclusion of terms of gauge fixings and ghosts). This would be the only contribution describing the explicit breaking of the conformal symmetry - in the absence of an anomalous breaking induced by the radiative corrections -. It is also evident from the structure of Eq. (49) that the complete anomalous effective action takes contributions from vertex functions with two and three gauge bosons on the external lines, due to the and field strengths .

In the context of theories with extra dimensions, the correlator obtained by the insertion of the trace plays a key role in describing the radiative corrections to the tree-level coupling of to matter. We present here the explicit form of this vertex when the radion couples to on-shell external photons. It is defined as

(51)

(with an amputated correlation function) and can be decomposed in the form

(52)

where

(53)
(54)
(55)

correspond to the contributions coming from the insertion on the photon 2-point function of the trace of the EMT, as specified in (51). These correspond to fermion () and boson () loops, together with terms of improvement (). A description of these terms can be found in [10].
Note that these expressions are ultraviolet finite and do not need any renormalization counterterm. One can also observe the presence of two scaleless terms in Eq. (54) and Eq. (55), (the terms), which do not depend on any mass parameter but only on . These are not part of the anomaly - since the ’s correspond to explicit breaking of the conformal symmetry - and seem to invalidate our argument about the pole origin of the entire anomaly for a spontaneously broken theory. However these extra scaleless contributions, as one can easily check, cancel in (52) if the Higgs scalar is conformally coupled, since they appear with the opposite sign.
We can summarize this analysis by saying that is zero for a conformal theory (e.g. QED with massless fermions) and it is expected to be proportional to any mass parameter of the theory otherwise. For instance it is nonzero for QCD and QED when the quarks are massive. Indeed one can explicitly check, for example, that in the QCD case the corresponding amplitude , coming from the insertion of the trace of the EMT on the gluon 2-point function, even if not zero, does not contribute any scaleless term on the right-hand-side of the anomaly equation (Eq (48) or (50)). The same is true in the electroweak theory only if the Higgs doublet is conformally coupled to gravity. In our case this is guaranteed - by construction - due to the specific choice of the coefficient in front of the term of improvement. If the improvement had not been included in the EMT, then this would have implied that extra scaleless contributions had to combine with the pole term in Eq. (26) to saturate the anomaly. This is equivalent to saying that the pole term in the correlator, in this specific case, would not be entirely responsible for the generation of the anomaly. Indeed, for a conformally coupled Higgs only the sum of Eq. (26) and (29) encloses the entire contribution to the anomaly, which thus can be entirely attributed to the pole part.

It is important to observe that if the definition of the coupling of the dilaton to the trace of the EMT is 4-dimensional, then there is no tree level coupling of the same state to the anomaly. This is indeed the content of Eq. (52), which does not include any anomalous term of the form generated by the classical Lagrangian geometrically reduced on the brane. For this reason, the coupling of the dilaton to the anomaly is obtained only if we make one extra assumption.
For instance, in our formulation we need to replace the vertex appearing in (47) with the vertex at the onset, and then use Eq. (48). Notice that in this expression the EMT does not need to be renormalized. In fact, one can show explicitly that the renormalization does not affect the trace of the same tensor, being the counterterm vertex in proportional to a traceless form factor [4, 7]. For this reason the operation of trace on (i.e. ) can be computed by the insertion of the bare EMT in 2-point functions.
In other approaches the same coupling requires a redefinition of the trace of the EMT from 4 to dimensions. In this second case the renormalization of the trace operator is essential in order to generate the coupling of the dilaton to the complete scale violations (anomaly plus explicit terms) present in the anomaly equation. This is obtained by the replacement in (47) of (in 4 dimensions) with ), where is the trace of the renormalized EMT, computed in dimensions [11, 12].

We are now going to briefly discuss and compare the structure of the effective scalar interactions obtained from the trace anomaly pole against those coming from the exchange of a fundamental radion introduced by a generic extra dimensional model. An effective degree of freedom in the form of a dilaton () interacting both with the anomaly and with the (explicit) scale violating terms is induced by the effective action generated by the anomaly loop. This effective interaction can be carefully identified in the 1-graviton exchange channel not only for a massless theory [4] but also in the presence of explicit scale non-invariant terms. One can investigate the salient features of these interactions by a direct computation.

5.1 Amplitudes for graviton/radion exchange in the production of two gauge bosons

For example, let’s consider the production of two photons by a gravitational source characterized by a certain EMT . The tree-level amplitudes with the exchange of the first modes of the KK towers, namely a massless graviton and a massless dilaton can be formally written as

(56)

where the are the polarization vectors of the two final state photons and is the graviton - two photons vertex ()