The Conformal Anomaly and the Neutral Currents Sector of the Standard Model
Claudio Coriano, Luigi Delle Rose, Antonio Quintavalle, Mirko Serino
aa r X i v : . [ h e p - ph ] A p r The Conformal Anomaly and the Neutral Currents Sector of the Standard Model
Claudio Corian`o, Luigi Delle Rose, Antonio Quintavalle and Mirko Serino
Departimento di Fisica, Universit`a del Salentoand INFN-Lecce, Via Arnesano 73100, Lecce, Italy Abstract
We elaborate on the structure of the graviton-gauge-gauge vertex in the electroweak theory, obtainedby the insertion of the complete energy-momentum tensor ( T ) on 2-point functions of neutral gaugecurrents ( V V ′ ). The vertex defines the leading contribution to the effective action which accounts forthe conformal anomaly and related interaction between the Standard Model and gravity. The energymomentum 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 theprevious cases of QED and QCD, we find that the conformal anomaly induces an effective masslessscalar interaction between gravity and the neutral currents in each gauge invariant component of thevertex. This is described by the exchange of an anomaly pole. We show that for a spontaneouslybroken theory the anomaly can be entirely attributed to the poles only for a conformally coupled Higgsscalar. In the exchange of a graviton, the trace part of the corresponding interaction can be interpretedas due to an effective dilaton, using a local version of the effective action. We discuss the implicationsof the anomalous Ward identity for the T V V ′ correlator for the structure of the gauge/gauge/ effectivedilaton vertex in the effective action. The analogy between these effective interactions and those relatedto the radion in theories with large extra dimensions is pointed out. [email protected], [email protected], [email protected], [email protected] Introduction
Gravity couples to the Standard Model, in the weak gravitational field limit, via its energy momentumtensor (EMT) T µν . This interaction is responsible for the generation of the radiative breaking of scaleinvariance [1, 2, 3], which is mediated, at leading order in the gauge coupling ( O ( g )), by a trianglediagram: the T V V ′ vertex (see [4, 5, 6, 7]), where V, V ′ denote two gauge bosons. The computation ofthe 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 necessaryfor 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 inperturbation theory in a given regularization scheme, in order to secure the consistency of the result. Inthe case under exam they correspond to the T AA, T AZ and
T ZZ vertices, where A is the photon and Z 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 directapplications. The first has to do with the analysis of anomaly mediation as a possible mechanism todescribe the interaction between a hypothetical hidden sector and the fields of the Standard Model, asshown in Fig. 1 (a). One of the results of our analysis, in this context, is that anomaly mediation isdescribed 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 commentbelow, is indeed confirmed by the direct computation in the entire electroweak theory. One of the mainimplications of our analysis, in fact, is that this picture remains valid even in the presence of masscorrections due to symmetry breaking, for a graviton of large virtuality and a conformally coupled Higgssector. We will comment on this point in a separate section (section 5) and in our summary before theconclusions.A second area where these corrections may turn useful is in the case of an electroweak theory formulatedin scenarios with large extra dimensions (LED), with a reduced scale for gravity. In this case the virtualexchanges of gravitons provide sizeable corrections to electroweak processes - beyond tree level - usefulfor LHC studies of these models, as illustrated in Fig. 2 in the case of the q ¯ q annihilation channel. Inthese extensions a graviscalar (radion) φ degree of freedom is induced by the compactification, which isexpected to couple to the anomaly ( φT µµ ) as well as to the scaling-violating terms, as we are going toclarify, by an extra prescription. This prescription is based on the replacement of the classical trace ofthe matter EMT by its quantum average. A rigorous discussion of the fundamental anomalous Wardidentity for the T V V ′ correlator will clarify some subtle issues involved in this prescription. We willshow, in parallel, that the anomalous effective action induces in the 1-graviton exchange channel a similarinteraction. This interaction can be thought as being mediated by an effective massless dilaton, coupledto the trace anomaly equation (and to its mass corrections).2 a) (b) Figure 1: Gravitational interaction of the Standard Model fields with a hidden sector (H.S.), at leadingorder in the gravitational constant (a). The interaction in perturbation theory responsible for the traceanomaly is illustrated in (b) via the exchange of an anomaly pole.
We start with few definitions, focusing our discussions only on the case of the graviton/photon/photon(
T AA ) and graviton/Z/Z (
T ZZ ) vertices.We recall that the fundamental action describing the gravity and the Standard Model is defined bythe three contributions S = S G + S SM + S I = − κ Z d x √− g R + Z d x √− g L SM + 16 Z d x √− g R H † H , (1)where κ = 16 πG N , with G N being the four dimensional Newton’s constant and H is the Higgs doublet.We have denoted with S G the contribution from gravity (Einstein-Hilbert term) while S SM is the StandardModel (SM) quantum action, extended to curved spacetime. S I denotes the term of improvement for thescalars, which are coupled to the metric via its scalar curvature R . The factor 1 / SU (2) Higgs doublet. The EMT in our conventionsis defined as T µν ( x ) = 2 p − g ( x ) δ [ S SM + S I ] δg µν ( x ) , (2)and around a flat spacetime limit g µν ( x ) = η µν + κ h µν ( x ) , (3)with the symmetric rank-2 tensor h µν ( x ) accounting for the metric fluctuations.We denote with T µν 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 brokenelectroweak phase. Its expression is uniquely given by the coupling of the Standard Model Lagrangianto gravity, modulo the terms of improvements, which depend on the choice of the coupling of the Higgsdoublets. As we have mentioned, we have chosen a conformally coupled Higgs field. Our computation isperformed in the R ξ gauge. The expression of the EMT is symmetric and conserved. It is therefore givenby a minimal contribution T Minµν (without improvement) and the improvement EMT, T Iµν , with T µν = T Minµν + T Iµν , (4)where the minimal tensor is decomposed into T Minµν = T f.s.µν + T ferm.µν + T Higgsµν + T Y ukawaµν + T g.fix.µν + T ghostµν . (5)3 + . . . Figure 2: Typical leading order ( O ( κ )) contributions to the production of two gauge bosons with grav-itational mediation. Not included are the initial state (Standard Model) corrections on the q ¯ q /gravitonvertex and the loops of gauge bosons and Higgs mediating the decay of the graviton. The latter contributeto the conformal anomaly.The various contributions refer, respectively, to the gauge kinetic terms (field strength, f.s. ), the fermions,the Higgs, Yukawa, gauge fixing contributions ( g.f ix. ) 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 andimplement the necessary Ward and STI’s. Their derivation is quite lengthy as is their implementation inperturbation theory, given the sizeable number of diagrams involved in the expansion and the very longexpression 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. ∂ µ T µν = 0),which takes the form − i κ ∂ µ h T µν ( x ) V α ( x ) V ′ β ( x ) i amp = − κ (cid:26) − ∂ ν δ (4) ( x − x ) P − V V ′ αβ ( x , x ) − ∂ ν δ (4) ( x − x ) P − V V ′ αβ ( x , x ) + ∂ µ [ η αν δ (4) ( x − x ) P − V V ′ βµ ( x , x ) + η βν δ (4) ( x − x ) P − V V ′ αµ ( x , x )] (cid:27) , where we have introduced the off-diagonal 2-point function P − V V ′ αβ ( x , x ) = h | T V α ( x ) V ′ β ( x ) | i amp , (6)where amp denotes amputated external gauge lines. Notice that the gravitational field, in this computa-tion, is just an external field and the 1PI conditions apply only to the external gauge lines. This pointemerges from a closer investigation of the defining STI’s of the correlator. This Ward identity applies toany gauge boson in the neutral sector.2) A STI for the T AA vertex. Specifically, introducing the photon gauge-fixing function F A = ∂ σ A σ , (7)we obtain the relation1 ξ h T µν ( z ) F A ( x ) F A ( y ) i = − iξ (cid:26) η µν ∂ ρx h δ (4) ( z − x ) h A ρ ( x ) F A ( y ) i i − η µν ∂ ρz h δ (4) ( z − y ) h A ρ ( z ) F A ( x ) i i − (cid:18) ∂ xµ h δ (4) ( z − x ) h A ν ( x ) F A ( y ) i i − ∂ zµ δ (4) ( z − y ) h A ν ( z ) F A ( x ) i + ( µ ↔ ν ) (cid:19)(cid:27) , (8)4 f f (a) W ± W ± W ± (b) φ ± φ ± φ ± (c) W ± W ± φ ± (d) φ ± φ ± W ± (e) η + η + η + (f) η − η − η − (g) Figure 3: Amplitudes with the triangle topology for the two correlators
T AA and
T ZZ . f f (a) W ± W ± (b) φ ± φ ± (c) W ± φ ± (d) φ ± W ± (e) η + η + (f) η − η − (g) Figure 4: Amplitudes with t-bubble topology for the correlators
T AA and
T ZZ .with ξ denoting the gauge-fixing parameter.3) A STI for the T ZZ correlator. Introducing the gauge-fixing function of the Z gauge boson F Z = ∂ σ Z σ − ξM Z φ , (9)where φ is the Goldstone of the Z , this takes the form1 ξ h T µν ( z ) F Z ( x ) F Z ( y ) i = − iξ (cid:26) − iξ η µν δ (4) ( x − y ) δ (4) ( x − z ) + η µν ∂ ρx (cid:20) δ (4) ( x − z ) (cid:21) h Z ρ ( x ) F Z ( y ) i− ∂ xµ (cid:20) δ (4) ( x − z ) h Z ν ( x ) F Z ( y ) i (cid:21) − ∂ xν (cid:20) δ (4) ( x − z ) h Z µ ( x ) F Z ( y ) i (cid:21) + ∂ zµ (cid:20) δ (4) ( z − y ) (cid:21) h Z ν ( z ) F Z ( x ) i + ∂ zν (cid:20) δ (4) ( z − y ) (cid:21) h Z µ ( z ) F Z ( x ) i− η µν ∂ ρz (cid:18) δ (4) ( z − y ) h Z ρ ( z ) F Z ( x ) i (cid:19)(cid:27) . (10)5 ± W ± (a) φ ± φ ± (b) Figure 5: Amplitudes with s-bubble topology for the correlators
T AA and
T ZZ . W ± (a) φ ± (b) Figure 6: Amplitudes with the tadpole topology for the correlators
T AA and
T ZZ .4) A STI for the
T AZ vertex1 ξ h T µν ( z ) F A ( x ) F Z ( y ) i = − iξ (cid:26) − η µν ∂ σz (cid:20) δ (4) ( z − y ) h Z σ ( z ) F A ( x ) i (cid:21) + ∂ zν δ (4) ( z − y ) h Z µ ( z ) F A ( x ) i + ∂ zµ δ (4) ( z − y ) h Z ν ( z ) F A ( x ) i (cid:27) . (11)We illustrate the overall structure of the results for the T AA and
T ZZ vertices, focusing on theessential parts, and in particular on those form factors which contribute to the trace part, since they aresimpler. The complete result is indeed quite involved and some details can be found in [8].
T AA case
In the
T AA case, we introduce the notation Γ ( AA ) µναβ ( p, q ) to denote the one-loop amputated vertexfunction with a graviton and two on-shell photons.In momentum space we indicate with k the momentum of the incoming graviton and with p and q themomenta of the two photons. In general, the Γ ( V V ′ ) µναβ ( p, q ) correlator is defined as(2 π ) δ (4) ( k − p − q )Γ V V ′ µναβ ( p, q ) = − i κ Z d zd xd y h T µν ( z ) V α ( x ) V ′ β ( y ) i amp e − ikz + ipx + iqy . (12)In the 2-photon case ( AA ) is decomposed in the formΓ ( AA ) µναβ ( p, q ) = Γ ( AA ) µναβF ( p, q ) + Γ ( AA ) µναβB ( p, q ) + Γ ( AA ) µναβI ( p, q ) , (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 Γ µναβI . The contributions to the (F) and(B) sectors are obtained by the insertion of T Min . The contribution from the term of improvement is6
Z H (a) φφ H (b)
HH Z (c)
HH φ (d)
Figure 7: Amplitudes with the triangle topology for the correlator
T ZZ . Z H (a) φ H (b)
Figure 8: Amplitudes with the t-bubble topology for the correlator
T ZZ .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 T µνI .The tensor basis on which we expand the vertex is given by four independent tensor structures φ µναβ ( p, q ) = ( s η µν − k µ k ν ) u αβ ( p, q ) , (14) φ µναβ ( p, q ) = − u αβ ( p, q ) [ s η µν + 2( p µ p ν + q µ q ν ) − p µ q ν + q µ p ν )] , (15) φ µναβ ( p, q ) = (cid:0) p µ q ν + p ν q µ (cid:1) η αβ + s (cid:16) η αν η βµ + η αµ η βν (cid:17) − η µν (cid:16) s η αβ − q α p β (cid:17) − (cid:16) η βν p µ + η βµ p ν (cid:17) q α − (cid:0) η αν q µ + η αµ q ν (cid:1) p β φ µναβ ( p, q ) = ( s η µν − k µ k ν ) η αβ (16)where u αβ ( p, q ) has been defined as u αβ ( p, q ) ≡ ( p · q ) η αβ − q α p β , (17)among which only φ µναβ and φ µναβ show manifestly a trace, the remaining ones being traceless. Acomplete computation gives for the various gauge invariant subsectorsΓ ( AA ) µναβF ( p, q ) = X i =1 Φ i F ( s, , , m f ) φ µναβi ( p, q ) , (18)Γ ( AA ) µναβB ( p, q ) = X i =1 Φ i B ( s, , , M W ) φ µναβi ( p, q ) , (19)Γ ( AA ) µναβI ( p, q ) = Φ I ( s, , , M W ) φ µναβ ( p, q ) + Φ I ( s, , , M W ) φ µναβ ( p, q ) . (20)The first three arguments of the form factors stand for the three independent kinematical invariants k = ( p + q ) = s , p = q = 0 while the remaining ones denote the particle masses circulating in theloop. 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 entirecontribution to the trace anomaly comes from the first tensor structure φ .7 H (a) φφ (b) Figure 9: Amplitudes with the s-bubble topology for the correlator
T ZZ . H (a) φ (b) H (c) Figure 10: Amplitudes with tadpole topology for the correlator
T ZZ .In the
T AA vertex, the contribution to the trace anomaly in the fermion sector comes from Φ F whichis given byΦ F ( s, , , m f ) = − i κ α π s X f Q f (cid:26) −
23 + 4 m f s − m f C ( s, , , m f , m f , m f ) (cid:20) − m f s (cid:21)(cid:27) . (21)Here we have introduced the QED coupling α = e / (4 π ) and the function C ( s, , , m , m , m ) = 12 s log a + 1 a − a = r − m s . (23)The sum is taken over all the fermions ( f ) of the Standard Model. As one can immediately realize, thisform factor is characterized by the presence of an anomaly poleΦ F pole ≡ iκ α π s X f Q f (24)which is responsible for the generation of the anomaly in the massless limit. To appreciate the significanceof this ”pole contribution” one needs special care, since a computation of the residue (at s = 0) showsthat this is indeed zero in the presence of mass corrections. However, this leading 1 /s behaviour in thetrace part of the amplitude, as we are going to show, is clearly identifiable in an (asymptotic) expansion( s ≫ m f ), and is corrected by extra m f /s terms, where m f 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 caseand is a clear manifestation of the anomaly.The other gauge-invariant sector of the T AA vertex is the one mediated by the exchange of bosons,Goldstones and ghosts in the loop. We will denote with M W , M Z and M H the masses of the W’s and Z B ( s, , , M W ) = − i κ απ s (cid:26) − M W s + 2 M W C ( s, , , M W , M W , M W ) (cid:20) − M W s (cid:21)(cid:27) , (25)which multiplies the tensor structure φ , responsible for the generation of the anomalous trace.In this case the anomaly pole is easily isolated from (25) in the formΦ B,pole ≡ − i κ απ s . (26)The term of improvement is responsible for the generation of two form factors, both of them contributingto the trace. They are given byΦ I ( s, , , M W ) = − i κ α π s (cid:26) M W C ( s, , , M W , M W , M W ) (cid:27) , (27)Φ I ( s, , , M W ) = i κ α π M W C ( s, , , M W , M W , M W ) , (28)the first of them being characterized by an anomaly poleΦ I pole = − i κ α π s . (29)Our considerations on the UV behaviour of the a) radiative plus the b) explicit mass corrections to theanomalous amplitude are obviously based on an exact computation of the correlator.In the asymptotic limit ( s → ∞ ), the expansions of the three form factors contributing to the tracepart can be organized in terms of the 1 /s ”pole component” plus mass corrections, which are given byΦ ,F ( s, , , m f ) ≃ − i κ α πs X f Q f (cid:26) −
23 + m f s (cid:20) π − log ( m f s ) − iπ log( m f s ) (cid:21)(cid:27) , (30)Φ ,B ( s, , , M W ) ≃ − i κ απs (cid:26) − M W s (cid:20) π − log ( M W s ) − iπ log( M W s ) (cid:21)(cid:27) , (31)Φ ,I ( s, , , M W ) ≃ − i κ α πs (cid:26) − M W s (cid:20) π − i log( M W s ) (cid:21) (cid:27) , (32)The energy suppressed terms ( m f /s , M W /s ) take the typical form M /s , with M denoting, generically,any explicit mass term generated in the broken phase of the theory. This separation of the radiative fromthe explicit contributions to the breaking of conformal invariance, due to the tree-level mass terms, is inagreement 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 M S scheme the β functions of the Standard Model are given by β = g π (cid:20) n g + 16 (cid:21) , β = g π (cid:20) n g −
223 + 16 (cid:21) , β = g π (cid:20) −
11 + 43 n g (cid:21) , (33)for the hypercharge, weak and strong interactions respectively, and n g is the number of generations. Theexpression of the β function of the electromagnetic coupling, β e , is given in the same scheme by β e = c w β + s w β = e π (cid:20) n g − (cid:21) . (34)9t this point, the residue of the anomaly pole which appears in the form factors Φ ,F , Φ ,B and Φ ,I is uniquely determined by the beta function of the electromagnetic coupling constant. Indeed we haveΦ ,pole = Φ F ,pole + Φ B ,pole + Φ I ,pole = − i κ α πs (cid:20) − X f Q f + 52 + 1 (cid:21) = i κ s β e e , (35)where we have used the fact that P f Q f = n g . T Z Z case
Moving to the vertex with two massive Z gauge bosons, one discovers a similar pattern. Also in this case,as before, we introduce the notation Γ ( ZZ ) µναβ ( p, q ) to describe the corresponding correlation function.We have several contributions appearing in the global expression of the correlator:Γ ( ZZ ) µναβ ( p, q ) = Γ ( ZZ ) µναβF ( p, q ) + Γ ( ZZ ) µναβW ( p, q ) + Γ ( ZZ ) µναβZ,H ( p, q ) + Γ ( ZZ ) µναβI ( p, q ) . (36)Γ ( ZZ ) µναβ ( p, q ), for on-shell Z bosons, can be separated into three contributions obtained using the in-sertion of T Min (sectors
F, W and
Z/H ) and a fourth one coming from the term of improvement. In thiscase the gravitational interaction is mediated by T I .The labelling of the first three is inherited from the types of particles (and corresponding masses) thatcirculate in the loops. Beside the fermion sector (F) with diagrams depicted in Figs. 3(a) and 4(a)), theother contributions involve a W gauge boson (sector (W)), with diagrams Fig. 3(b)-(g), Fig. 4(b)-(g),Fig. 5 and Fig. 6), and the mixed Z /Higgs bosons sector ( Z, H ) 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 theon-shell renormalization scheme. Finally there is a contribution from the term of improvement (I). Thisis given by the diagrams depicted in Fig. 3(c), (d), 5(b), together with those of Figs. 7(b), (c), (d) andFig. 9. In this case, however, the graviton - scalar - scalar vertices is generated by T µνI .As we have already mentioned, we take the two Z gauge bosons on the external lines on-shell, and aninsertion of T µν at a nonzero momentum transfer k . The four contributions can be expanded on a tensorbasis given by 9 tensors, and corresponding form factors Φ i asΓ ( ZZ ) µναβF ( p, q ) = X i =1 Φ ( F ) i ( s, M Z , M Z , m f ) t µναβi ( p, q ) , (37)Γ ( ZZ ) µναβW ( p, q ) = X i =1 Φ ( W ) i ( s, M Z , M Z , M W ) t µναβi ( p, q ) , (38)Γ ( ZZ ) µναβZ,H ( p, q ) = X i =1 Φ ( Z,H ) i ( s, M Z , M Z , M Z , M H ) t µναβi ( p, q ) , (39)Γ ( ZZ ) µναβI ( p, q ) = Φ ( I )1 ( s, M Z , M Z , M W , M Z , M H ) t µναβ ( p, q )+ Φ ( I )2 ( s, M Z , M Z , M W , M Z , M H ) t µναβ ( p, q ) , (40)10here the first three arguments of the Φ i ’s are the virtualities of the external lines k = s, p = q = M Z ,while the last two give the masses in the internal lines. 7 of the 9 tensor structures are traceless, whilethe only two responsible for the breaking of scale invariance are t µναβ ( p, q ) = ( sg µν − k µ k ν ) h(cid:16) s − M Z (cid:17) g αβ − q α p β i ,t µναβ ( p, q ) = ( sg µν − k µ k ν ) g αβ . (41)The four form factors responsible for generating a pole term are those accompanying the tensor structure t , while the form factors Φ , corresponding to the tensor structure t , show no pole. The latter givecontributions which are suppressed as M /s . Therefore, as for the T AA vertex, the trace parts show adistinctive 1 /s contribution plus corrections of O ( M /s ), as we have specified above. Being the completeresult of this vertex quite lengthy, we omit details and just focus our attention on the pole terms extractedfrom each sector. These are summarized by rather simple expressions. We obtainΦ ( F )1 pole ≡ iα κ πc w s w s X f (cid:16) C f a + C f v (cid:17) (42)for the fermion sector of the vertex, where s w and c w are short notations for sin θ W and cos θ W . Similarly,in the other sectors we have Φ ( W )1 pole ≡ − i κ αs w c w π s (60 s w − s w + 81)72 (43)for diagrams involving W ’s, while the diagrams with W and Z gauge bosons giveΦ ( Z,H )1 pole ≡ iακ πsc w s w . (44)The term of improvement contributes to two tensor structures but only one of the two form factors fromthis sector has a pole term. We have, in this caseΦ ( I )1 pole = − i κ α π s w c w s (cid:0) − s w c w (cid:1) , (45)while Φ ( I )4 ∼ M W /s asymptotically.The coefficient of the anomalous pole contribution is fixed by the beta functions of the theory. Inthis case it is proportional to a linear combination of the beta functions of the couplings g and g ofhypercharge and SU (2). Indeed we haveΦ ,pole = Φ ( F )1 ,pole + Φ ( W )1 ,pole + Φ ( Z,H )1 ,pole + Φ ( I )1 ,pole = i κ s (cid:20) s w β g + c w β g (cid:21) . (46) One of the most significant applications of the results of the previous sections concerns the study of thecoupling of the radion/dilaton to the fields of the Standard Model in theories with LED. We will use the11erm radion ( φ ) to denote the fundamental scalar introduced in the usual compactifications of theorieswith LED, and reserve the name of ”effective dilaton” ( ϕ ) for the scalar interaction dynamically inducedby the anomaly. As we are going to show, the radion has interactions with matter which are quite similarto those allowed to the effective dilaton, although the latter shows up in a different channel (the 1-gravitonexchange channel). We proceed first with a rigorous discussion of the interaction of the LED radion andthen 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 com-pactification of the extra dimensions gives rise in the 4 dimensional effective field theory to towers ofKaluza-Klein gravitons and dilatons. For definiteness we consider a theory compactified on a torus andconsider the zero modes of the 4D graviton field and of the dilaton φ generated by this procedure. Thesetwo fields will couple, via their lowest Kaluza-Klein modes, to the EMT with the interaction Lagrangian[9] L int = − κ Z d x (cid:0) h µν T µν + ωφ T µµ (cid:1) , ω = s δ + 2) , (47)where δ is the number of extra dimensions. To understand the main features of the dilaton interaction at1-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 certaintheory takes the form η µν h T µν ( z ) i = hA ( z ) i + h T µµ ( z ) i , (48)where we have taken the quantum average of each term. A is the operator describing the anomalousbehavior of the fields under scale transformations while the T µµ operator is the non-amomalous contributionto the trace of the EMT. This second term vanishes in the conformal limit (i.e. before electroweaksymmetry breaking) using the equations of motion of the fields. In an exact gauge theory the expectedstructure of the anomaly is given by the relation A = X i β i g i F αβi F iαβ , (49)where F αβi and g i are the field strengths and the gauge couplings of the gauge fields in the unbroken phase,corresponding to the Standard Model gauge group SU (3) C × SU (2) L × U (1) Y . For a theory in a brokenphase, and in the photon case, the anomaly A is again proportional to β e . By taking two functionalderivatives of the trace identity (48) with respect to the sources J α and J ′ β of the gauge fields V α and V ′ β ,we obtain the anomalous identities on the correlation functions analyzed in this work η µν h T µν ( z ) V α ( x ) V ′ β ( y ) i = δ hA ( z ) i δJ α ( x ) δJ ′ β ( y ) + h T µµ ( z ) V α ( x ) V ′ β ( y ) i . (50)The first term on the right-hand side of the equation above defines the residue of the anomaly pole that wehave already discussed and isolated in the previous sections. The second term, instead, is the correlationfunction obtained by inserting the trace of the EMT on the two point functions h V α ( x ) V ′ β ( y ) i (with the12nclusion of terms of gauge fixings and ghosts). This would be the only contribution describing the explicitbreaking of the conformal symmetry - in the absence of an anomalous breaking induced by the radiativecorrections -. It is also evident from the structure of Eq. (49) that the complete anomalous effectiveaction takes contributions from vertex functions with two and three gauge bosons on the external lines,due to the SU (2) and SU (3) field strengths ( F , F ).In the context of theories with extra dimensions, the correlator obtained by the insertion of the trace T µµ plays a key role in describing the radiative corrections to the tree-level coupling of φ to matter. Wepresent here the explicit form of this vertex when the radion couples to on-shell external photons. It isdefined as(2 π ) δ (4) ( k − p − q ) D AAαβ ( p, q ) = − i κ Z d zd xd y h T µµ ( z ) A α ( x ) A β ( y ) i amp e − ikz + ipx + iqy (51)(with an amputated correlation function) and can be decomposed in the form D ( AA ) αβ ( p, q ) = D ( AA ) αβF ( p, q ) + D ( AA ) αβB ( p, q ) + D ( AA ) αβI ( p, q ) , (52)where D ( AA ) αβF ( p, q ) = − i κ απ X f Q f m f " s + 2 m f s − ! C (cid:0) s, , , m f , m f , m f (cid:1) u αβ ( p, q ) , (53) D ( AA ) αβB ( p, q ) = − i κ απ (cid:20) M W (cid:18) − M W s (cid:19) C (cid:0) s, , , M W , M W , M W (cid:1) − M W s − (cid:21) u αβ ( p, q ) , (54) D ( AA ) αβI ( p, q ) = − i κ απ (cid:2) M W C (cid:0) s, , , M W , M W , M W (cid:1)(cid:3) u αβ ( p, q )+ i κ απ M W s C (cid:0) s, , , M W , M W , M W (cid:1) η αβ (55)correspond to the contributions coming from the insertion on the photon 2-point function of the traceof the EMT, as specified in (51). These correspond to fermion ( F ) and boson ( B ) loops, together withterms of improvement ( I ). A description of these terms can be found in [10].Note that these expressions are ultraviolet finite and do not need any renormalization counterterm. Onecan also observe the presence of two scaleless terms in Eq. (54) and Eq. (55), (the ± i κ απs terms), whichdo not depend on any mass parameter but only on 1 /s . These are not part of the anomaly - since the D ’scorrespond to explicit breaking of the conformal symmetry - and seem to invalidate our argument aboutthe pole origin of the entire anomaly for a spontaneously broken theory. However these extra scalelesscontributions, as one can easily check, cancel in (52) if the Higgs scalar is conformally coupled , since theyappear with the opposite sign.We can summarize this analysis by saying that D AAαβ is zero for a conformal theory (e.g. QED withmassless 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 D ggαβ , coming from the insertion of thetrace of the EMT on the gluon 2-point function, even if not zero, does not contribute any scaleless term onthe right-hand-side of the anomaly equation (Eq (48) or (50)). The same is true in the electroweak theoryonly if the Higgs doublet is conformally coupled to gravity. In our case this is guaranteed - by construction13 due to the specific choice of the coefficient in front of the term of improvement. If the improvementhad not been included in the EMT, then this would have implied that extra scaleless contributions hadto combine with the pole term in Eq. (26) to saturate the anomaly. This is equivalent to saying thatthe pole term in the correlator, in this specific case, would not be entirely responsible for the generationof the anomaly. Indeed, for a conformally coupled Higgs only the sum of Eq. (26) and (29) encloses theentire 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 φ to the trace of the EMTis 4-dimensional, then there is no tree level coupling of the same state to the anomaly. This is indeedthe content of Eq. (52), which does not include any anomalous term of the form φF F generated by theclassical Lagrangian geometrically reduced on the brane. For this reason, the coupling of the dilaton tothe anomaly is obtained only if we make one extra assumption.For instance, in our formulation we need to replace the φT µµ vertex appearing in (47) with the vertex φg µν h T µν i at the onset, and then use Eq. (48). Notice that in this expression the EMT does not need tobe renormalized. In fact, one can show explicitly that the renormalization does not affect the trace of thesame tensor, being the counterterm vertex in T J J proportional to a traceless form factor [4, 7]. For thisreason the operation of trace on T µν (i.e. g µν h T µν i ) can be computed by the insertion of the bare EMTin 2-point functions.In other approaches the same coupling requires a redefinition of the trace of the EMT from 4 to D dimensions. In this second case the renormalization of the trace operator is essential in order to generatethe coupling of the dilaton to the complete scale violations (anomaly plus explicit terms) present in theanomaly equation. This is obtained by the replacement in (47) of φT µµ (in 4 dimensions) with φ h T rµµ i D ),where T rµµ is the trace of the renormalized EMT, computed in D dimensions [11, 12].We are now going to briefly discuss and compare the structure of the effective scalar interactionsobtained from the trace anomaly pole against those coming from the exchange of a fundamental radionintroduced by a generic extra dimensional model. An effective degree of freedom in the form of a dilaton( ϕF F ) interacting both with the anomaly and with the (explicit) scale violating terms is induced by theeffective action generated by the anomaly loop. This effective interaction can be carefully identified in the1-graviton exchange channel not only for a massless theory [4] but also in the presence of explicit scalenon-invariant terms. One can investigate the salient features of these interactions by a direct computation. For example, let’s consider the production of two photons by a gravitational source characterized by acertain EMT T ′ µν . The tree-level amplitudes with the exchange of the first modes of the KK towers,14amely a massless graviton and a massless dilaton can be formally written as M (0) grav = − κ (cid:20) T ′ µν P µνρσ ( k ) V ρσαβ ( p, q ) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q )= − κ k (cid:20) T ′ µν V µναβ ( p, q ) − n − T ′ µµ V ρραβ ( p, q ) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q ) , (56) M (0) dil = − κ ω (cid:20) T ′ µµ P ( k ) V ρραβ ( p, q ) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q ) = − κ ω k (cid:20) T ′ µµ V ρραβ ( p, q ) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q ) , (57)where the ǫ ( p ) , ǫ ( q ) are the polarization vectors of the two final state photons and − i κ V ρσαβ ( p, q ) is thegraviton - two photons vertex ( M V = 0) − i κ V ρσαβ ( p, q ) = − i κ (cid:26) ( k · k + M V ) C µναβ + D µναβ ( k , k ) + 1 ξ E µναβ ( k , k ) (cid:27) (58)with C µνρσ = g µρ g νσ + g µσ g νρ − g µν g ρσ ,D µνρσ ( k , k ) = g µν k σ k ρ − (cid:20) g µσ k ν k ρ + g µρ k σ k ν − g ρσ k µ k ν + ( µ ↔ ν ) (cid:21) ,E µνρσ ( k , k ) = g µν ( k ρ k σ + k ρ k σ + k ρ k σ ) − (cid:20) g νσ k µ k ρ + g νρ k µ k σ + ( µ ↔ ν ) (cid:21) . (59) P µνρσ ( k ) and P ( k ) are the (massless) graviton and dilaton propagators in the de Donder gauge whichare given by, in the framework of dimensional regularization ( n = 4 − ǫ ), iP µνρσ ( k ) = ik (cid:20) η µρ η νσ + η µσ η νρ − n − η µν η σρ (cid:21) , iP ( k ) = ik . (60)Now we consider the one-loop corrections to these expressions and introduce the notationΓ ( AA ) µναβ ( p, q ) = − i κ ( AA ) µναβ ( p, q ) , D ( AA ) αβ ( p, q ) = − i κ D ( AA ) αβ ( p, q ) , (61)in order to factorize the gravitational coupling constant. We obtain M (1) grav = − κ k (cid:20) T ′ µν ¯Γ ( AA ) µναβ ( p, q ) − n − T ′ µµ η ρσ ¯Γ ( AA ) ρσαβ ( p, q ) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q )= − κ k (cid:20) T ′ µν ¯Γ ( AA ) µναβ ( p, q ) − n − T ′ µµ (cid:16) ¯ D ( AA ) αβ ( p, q ) + A αβ ( p, q ) (cid:17) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q ) , (62) M (1) dil = − κ ω k (cid:20) T ′ µµ (cid:16) ¯ D ( AA ) αβ ( p, q ) + A αβ ( p, q ) (cid:17) (cid:21) ǫ ∗ α ( p ) ǫ ∗ β ( q ) . (63)The A αβ ( p, q ) term is the anomaly contribution generated by the pole terms in the Γ ( AA ) ( p, q ) vertex andit is given by A αβ ≡ − β e e u αβ ( p, q ) . (64)15otice that in (63) we have retained both the coupling of the dilaton φ to the explicit (non-conformal)and anomalous terms generated by the Ward identity of the trace anomaly. A similar scalar interactionappears in the graviton channel (proportional to T ′ µµ ), as one can easily infer from the right-hand-side ofEq. (62). Both interactions are, indeed, of dilaton type, being proportional to the complete ( D ( AA ) αβ + A αβ )trace of the anomaly loop. We will briefly comment on the origin of this effective dilaton interaction.For this purpose we recall [4, 7] that in the case of massless QED the effective interaction induced bythe trace anomaly takes the form S ∼ Z d xd y R (1) (cid:3) − ( x, y ) F µν ( y ) F µν ( y ) (65)where R (1) denotes the linearized scalar curvature and F µν is the abelian field strength. A similar resultholds for QCD. As shown in [4] this expression coincides with the long-known anomaly-induced actionderived by Riegert [13], which was derived for a generic gravitational field, after an expansion of itsexpression around the flat spacetime limit. Notice that in terms of auxiliary degrees of freedom (i.e.two scalar fields ( ϕ, ψ ′ )) which render the action (65) local [4], extra couplings of the form ϕF F areautomatically induced by the 1 / (cid:3) term. This interaction is indeed present in the equivalent Lagrangian S anom [ g, A ; ϕ, ψ ′ ] = Z d x √− g (cid:20) − ψ ′ (cid:3) ϕ − R ψ ′ + c F αβ F αβ ϕ (cid:21) , (66)( c = − β ( e ) / (2 e )) where ϕ and ψ ′ are the auxiliary scalar fields. This action does not account for anycorrection to the trace anomaly equation due to the appearance of mass terms. The presence of anexplicit breaking of scale invariance due to the D ( AA ) term, however, can be handled by a modificationof the ϕF F interaction present in (66), i.e. the anomaly term. In the presence of an explicit breakingof scale invariance, the effective dilation ϕ couples to the neutral currents of the final state just like thefundamental dilaton φ , which is the content of the Eq. (62). One can explicitly check the cancellation ofpossible ”double pole” contributions in the s-channel. These could be induced by the (single) pole of thegraviton propagator together with the anomaly pole coming from the triangle loop (present in ¯Γ ( AA ) µναβ ) (seeEq. 62). This cancellation holds under the condition that the source EMT T ′ µν is conserved, as expected.Indeed this is an additional check of the significance of this effective component of the 1-graviton exchangeamplitude generated in the presence of a trace anomaly vertex. There are some comments which are in order concerning the result of this analysis, which complete thoseobtained in the QED and QCD cases [4, 5, 7]. From all these investigations it seems clear that anomalymediation can be described, in a perturbative framework, as due to the exchange of effective masslessscalar degrees of freedom between gravity and the gauge sector. The physical interpretation of thesesingularities is probably easier to grasp by a dispersive analysis, at least in the massless case, as discussedin [4] for QED. In the QED case, in fact, this component is generated (diagrammatically) by a virtualgraviton decaying into two on-shell collinear (correlated) fermions, which later decay into two on-shellphotons. This interpretation follows from the fact that the spectral density of the fermion loop diagrams16 ρ ( s ) ∼ δ ( s )), which indeed generates a pole, is obtained by cutting the graviton → γγ amplitude in the s = ( p + q ) channel, thus setting two intermediate fermion lines in the triangle diagram on-shell. Theapproach is similar to that of the chiral anomaly [14]. The pole is found only in the massless case.Obviously, a similar interpretation of the origin of this singularity, which is present over the entirelight cone ( s ∼
0) of the
T V V ′ We have indeed seen, in combination with a previous study for QCD [7], that an explicit computation ofthe exact 1-loop effective action (at leading order in the combined gravitational ( κ ) and gauge couplingexpansion ( g )) shows two fundamental features:1) In a massless gauge theory the breaking of conformal invariance is characterized by a typical 1 / (cid:3) behaviour. Notice that this does not exclude the possible appearance of other nonlocal terms in the sameeffective action, such as those proportional to log( (cid:3) ) (see the discussion in [15]). These additional terms,at least in the case of the chiral anomaly, are generated by the insertion of the triangle diagram into agraph of higher perturbative order [16]. Therefore, in the chiral case, they are not part of the trianglediagram (i.e. of the lowest order contribution to the anomaly). The computations in QED and QCD ofthe trace anomaly are in line with this result and are in agreement with Riegert’s anomaly-induced action[13] in these two theories. This result of ours appears to be also in agreement with the observations in[15] to which we refer for further details.2) In a gauge theory in a spontaneously broken phase, such as the electroweak theory, the radiative andthe explicit breaking of conformal invariance are separately identifiable in the ultraviolet limit. This resultholds even if conformal invariance is broken by the Higgs vev already at tree-level.Therefore, the two (distinct) kinematical domains characterized by the dominance of the anomaly (viaits massless poles) are described, in our notations, by a single invariant, s , which denotes the virtualityof the external graviton. In particular, point 1), as we have already mentioned, is related to the s ∼ T V V ′ vertex (for a massless theory) and point 2) to the s → ∞ limit of the same correlator (in the massive case).Hence, in the massive case (point 2)), the 1 /s contribution appears only after an asymptotic expansionat large energy of the anomaly vertex, and should be interpreted as its dominant asymptotic componentin the trace part. As such, this component is not part of the amplitude in the infrared (i.e. it is notpresent at s = 0). In fact, the computation of the residue of the correlator at s = 0 shows that this indeedvanishes when masses are present. In the fermion case, for instance, this result is due to cancellations17etween the pole and the second and third terms of Eq. (21).There is no doubt, however, that the 1 /s term, present in the expansion of the anomalous diagram atlarge s , is a manifestation of the same ”anomaly pole” encountered in the infrared in the massless case,since its contribution is asymptotically corrected by mass effects ( M /s ) which become negligible in theUV limit. It seems obvious that we should recover the behaviour typical of a massless theory as we moveto high energy, and the pole term of the anomaly, as the expansions (32) suggest. We have seen thateffective dilaton interactions are automatically part of the effective action which parallel those introducedwithin models with large extra dimensions.Thus, it could be of interest, for instance, to explore the role that such contributions could play in theanalysis of perturbations in the early universe, for instance in the context of inflation driven by a vectorfield [17], where such interactions appear to be necessary. In this case this vertex would be a direct conse-quence of the conformal anomaly, without any need to resort to more complex scenarios for its generation.We have also rigorously shown that a pole term completely accounts for the anomaly, in the StandardModel, only if the Higgs scalar is conformally coupled. In the non-conformally coupled case, extra scale-less contributions appear in the anomalous Ward identity for the T V V ′ . For a conformally coupled scalarindeed there is a cancellation between scaleless contributions coming from the explicit breaking of theconformal symmetry and those generated by the terms of improvement (Eqs. (54) and (55)). Obviously,this picture is typical only of theories with a spontaneous breaking of the gauge symmetry. For unbrokengauge theories this subtlety disappears and the pole completely accounts for the anomaly, as found inprevious analysis of QED and QCD. The computation of the effective action describing the interaction of gravity with the Standard Model,related to the trace anomaly, is described by the diagrams that we have analyzed in this work. Thisapproach, even if rather laborious, allows to derive the exact expression of such an action at leadingorder, which is the starting point for further phenomenological analysis. As we have shown, this ischaracterized by the presence of effective massless degrees of freedom in two kinematical domains.One of the main phenomenological applications of these results is in theories with large extra dimensions.In this context, we have illustrated rather rigorously that the coupling of a radion to the anomaly requiresa specific prescription on the definition of the quantum trace of the energy-momentum tensor for thesetheories. We have discussed two different (but equivalent) ways to obtain this interaction using extradimensional models. We have also shown that the appearance of an effective dilaton - coupled both to theanomaly and to extra scale-dependent terms - is a generic feature of the effective action which accountsfor the trace anomaly. This effective interaction can be identified in the 1-graviton exchange channel whenwe couple Einstein gravity in 4 dimensions to the the Standard Model. We have illustrated the analogiesbetween the interaction of the radion and of the effective dilaton using some examples.
Acknowledgments
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