Anomaly Actions and the Dilaton/Axion Multiplet

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Local and Nonlocal Chiral, Conformal and Superconformal AnomalyGauging Stuckelberg actionsAxions : the Axi-Higgs and the Dilaton/Axion Multiplet Claudio(Coriano’(( ( Claudio Coriano` Dipar&mento,di,Matema&ca,e,Fisica,“Ennio,De,Giorgi”,, Universita’,del,Salento,, Is&tuto,Nazionale,di,Fisica,Nucleare,,Lecce,,Italy,

Universidad Autonoma de Madrid Oxford University 21/1/2016 26 May 2014

Thursday, May 29, 14 Abstract

Variants of the usual Peccei-Quinn axion theory for the solution of the strong CP problem allows to generate more general axion-like terms in an effective Lagrangean beyond the Standard Model. One of these extensions involves Stuckelberg axions and (gauged) anomalous abelian symmetries. Similar interactions are generated by other methods, for instance by a decoupling of chiral fermions from the low energy spectrum in an anomaly-free theory. A third possibility is encoded in a scale invariant theory, where an axion, a dilaton and a dilatino are the anomaly multiplet of an N=1 Superconformal theory, in a nonlinear realization. General Results

Effective actions of Stuckelberg-type: SU(3)xSU(2)XU(1)_Y x U(1)’ Generalising a PQ global symmetry to a local U(1) symmetry (Stuckelberg axion models). Predict a fundamental axion (gauged axion) (the axi-Higgs) of a generic mass.

The mass is related to a misalignment potential which is generic. It can cover the TeV region. Obviously, the misalignment has to be strong For an axion at the Ter ascale.

Two models: MSLOM (Irges, Kiritsis, C.C.) USSM-A (Lazarides, Irges, Mariano, C.C.) (Stuckelberg supermultiplet)

These models are built using a Wess-Zumino Lagrangean with an asymptotic and elementary axion

Decoupling of a Heavy fermion and a gauged (anomaly free U(1) symmetry Can also also be described by this class of models ALTERNATIVE PATHS

AXIONS, DILATONS AS COMPOSITE

Conformal/superconformal anomaly

Dilaton interactions and the anomalous breaking of scale invariance of the Standard Model Delle Rose, Quintavalle, Serino, C.C. JHEP 1306 (2013) 077

Superconformal sum rules and the spectral density flow of the composite dilaton (ADD) multiplet in N=1 theories Delle Rose, Costantini, Serino, C.C. JHEP 1406 (2014) 136

Work to appear soon: Bandyopadhyay, Irges, Guzzi, Delle Rose, C.C. “Heavy Axions and Dilatons” A superconformal theory can generate these states due to the alignment of The anomaly multiplet. Nonlinear realization of the superconformal symmetry ) ) ) ) ) ) Axions'emerge'as'a'candidate'soluMon'of'the'strong'CP'problem'

The'well'known'soluMon'of'the'strong'CP'problem'is'due'to'' R.'Peccei'and'H.'Quinn'(PQ)'' It)is)based)on)the)introducKon)of)an)extra)U(1))global)symmetry)of)the) SM)broken)by)an)anomaly.)) ))

GeneralizaMon'of'the'PQ'proposal'

Irges,)Kiritsis,)CC,)2005))))))))))) ))U.)of)Crete,)U.)of)Salento))

Anomalous)U(1))extension)of)the)Standard)Model) (N.)Irges,)S.)Morelli,)C.C.)) ) Phenomenology:)M.'Guzzi'(Manchester)U.),)R.)Armillis,)C.C.) ) Susy)extensions:)Irges)(Athens)TU),)A.'Mariano'(Salento)U.),)C.C.) ) Cosmology:)G.'Lazarides'(Thessaloniki)U.),)A.)Mariano)(Salento),)C.C.) The role played by anomalies and anomaly actions in QFT can be hardly underestimated.

Anomalies describe the radiative breaking of a certain classical symmetry and theorists have tried to use anomaly actions as a way to show the effect of the anomaly (example: chiral dynamics and the pion, AVV anomaly) but also have tried to cancel anomalies when these symmetries are gauged

Anomaly cancellation (for a gauge symmetry):

1. by charge assignment in gauge theory (Standard Model): in the exact (unbroken ) phase of the theory, choose the representation in such a way that anomalous chiral interactions cancel

2. by the introduction of extra sectors (axions, dilatons) in the form of local actions (Wess Zumino actions)

3. More complex mechanisms such as “anomaly inﬂows”

Thursday, May 29, 14 Anomaly inflow on branes

Hill, Phys. Rev. D74 (2006)

(RobertaCallan and Harvey)Armillis – Axions in some anomalous extensions of the SM - Mytilene, 27 settembre 2007 5 Anomalies from triangle Feynman diagrams

In the previous section, we have established precise relations, in the form of anomalous Ward identities, between (functional derivatives of) the anomaly and certain proper vertex (one-particle irreducible) functions. The relevant proper vertex functions in four dimensions are triangle diagrams. In this section we will very explicitly evaluate such triangle Feynman diagrams. We first do the computation for the abelian anomaly and confirm our results of section 3. Then we compute the triangle diagram for chiral fermions coupled to non-abelian gauge fields, thus establishing the non-abelian gauge anomaly. We will do the computation in Pauli-Villars regularization so that one can very explicitly see how and where the anomalies arise. We will provide many computational details. The reader who is less interested in these details may safely skip most of the calculations and directly go to the results (5.13) and (5.16) for the abelian anomaly, and (5.43) as well as (5.44) for the non-abelian gauge anomaly.

5.1 The abelian anomaly from the triangle Feynman diagram: AVV

We will now compute the anomalous triangle diagram with one axial current and two vector currents (AVV) and probe it for the conservation of the axial current. In the above language, we will do µ⌫⇢ a Feynman diagram computation of 5 . In the next subsection, we will be interested in a very similar computation. In order to be able to easily transpose the present computation, we will replace µ ¯ µ j5 = i 5t by the more general µ ¯ µ j5↵ = i 5t↵ , (5.1) 8.1 Weak coupling evolution for a general U(1)0 charge assignment µ⌫⇢ and instead compute 5↵. We may then replace the non-abelian generator t↵ by the abelian generator t in the end. SU(3) SU(2) U(1) U(1) é é c w Y 0 Perturbativity regions with g '= 0.1, g= 0 Perturbativity regions with g '= 0.2 , g= 0 10 QL 3 2 1/6 zQ p p 4 ν ν uR 3 1 2/3 zu j j β β 5 dR 3 1 -1/3 2zQ zu k−p k+p 2 L 1 2 -1/2 3zQ µ µ k u 0 j j u 0 z eR 1 1 -1 2zQ zu 5α k 5α z −p−q −p−q H 1 2 1/2 z H k−q k+q - 2 ⌫R,k 1 1 0 zk - 5 ρ ρ j γ j γ 1 1 0 z q - 4 q

- 10 Table 1: Charge assignment of fermions and scalars in the U(1)0 SM extension.- 6 - 4 - 2 0 2 4 6 - 3 - 2 - 1 0 1 2 3

Figure 2: The two triangle diagramszQ contributing to the abelian anomalyzQ

(a) µ⌫⇢ (b) The cancellation of the gravitational anomalies, can be imposed,As shown in general, in Fig. 2, at there inter-generational are two diagrams contributing to i5 , corresponding to the two ways level. In the presentConstraints analysison weAbelian will opt, however,Extensions for ato of the completely contract the fermion symmetric ﬁelds contained(family inde- in the currents. As explained above, the vertices contribute µ ⌫ ⇢ 5t↵ and t, resp. t, while the fermion loop contributes an extra minus sign. A propagator pendent) assignmentStandard Model from of the RH neutrinos chargesTwo-Loopz1 = zVacuum2= z3, which allow to reduce the corresponding parameterStability space.and U(1)B−L With this choice, the U(1)0GG constraint from the gravitational anomaly reduces to a single equation for just one charge. On the other hand, the cancellation 28 of the analogue gravitational anomalies in the SM, obtained from the U(1)Y GG sector, is a natural consequence of the hypercharge assignments of the same model, and does not generate any additional constraint. As we have shown above, the solutions of the anomaly cancellation conditions are deﬁned in terms of the two free U(1)0 charges, zQ and zu, of theDelle Rose, Marzo, C.C. LH quark doublet QL and of the RH up quark uR. Notice also that the generators of the U(1)0 gauge group can be re-expressed, in general, as a linear combination of the SM hypercharge, Y , and the B-L quantum number, (c)

YB L.Indeed,wehave Figure 3: Values of the free U(1)0 charges zQ and zu for which the perturbativity constraint is z = ↵Y Y + ↵B LYB L. satisﬁed up to 105 GeV (blue(6) region), 109 GeV (green region), 1015 GeV (yellow region) and 19 10 GeV (red region) for g0 =0.1 (a) and g0 =0.2 (b). In (c) it is shown the same study for the In Eq. (6) the coecients ↵Y and ↵B L are functions of the two independent charges and are charges ↵Y =2zu 2 zQ and ↵B L =4zQ zu and g =0.1. The shadowed areas are excluded explicitly given by ↵Y =2zu 2zQ and ↵B L =4zQ zu. In the B-Lregions case, corresponding we set ↵Y =0(i.e. to M =2.5 TeV (pink shadows) and M = 5 TeV (green shadows) Z0 Z0 zu = zQ). respectively. The black thick dot indicates the B-L charge assignment. The charges of the two scalars can be ﬁxed from the requirement of gauge invariance of the

Yukawa interactions. From the Yukawa coupling of the electron LHe¯ R we have

(3z )+z +( 2z z )=0, (7) Q H Q u which gives z = z z , implying that the ordinary Higgs ﬁeld is singlet respect to B-L. 21 H u Q c Concerning the charge of the scalar ﬁeld , the Majorana mass term ⌫R⌫R, in the case of family independent (symmetric) charge assignment z = z 4z , we get the condition z = 2z . ⌫ u Q ⌫ For a B-L charge assignment we obtain z = 2, zu = zQ =1/3.

Irges,)Kiritsis,)C.C.)) On)the)effecKve)theory)of)lowTscale)) OrienKfold)vacua) The)study)the)effecKve)field)theory)of)) a)class)of)models)containing)a)gauge)structure)of)the)form))The)study)the)effecKve)field)theory)of)) )))))))))))))))))a)class)of)models)containing)a)gauge)structure)of)the)form)) ))))))))))))))))))))))))))))))))))))))))SM)x)U(1))x)U(1))x)U(1)) )))))))))))))))))))))))))))))))SU(3))x)SU(2))x)U(1)))))))))))SM)The)study)the)effecKve)field)theory)of))x)U(1))x)U(1))x)U(1)))x)U(1)…..) a)class)of)models)containing)a)gauge)structure)of)the)form))Y from)which)the)hypercharge)is)assigned)to)be)anomaly)free)))))))))))))))))))))))))))))))))))))SU(3))x)SU(2))x)U(1)Y)x)U(1)…..) These)models)are)the)object)of)an)intense)scruKny)by))from)which)the)hypercharge)is)assigned)to)be)anomaly)free)))))))))))))))))))))))))SM)x)U(1))x)U(1))x)U(1)) These)models)are)the)object)of)an)intense)scruKny)by)) )))))))))many)groups)working)on)intersecKng)))))))))))))))))))SU(3))x)SU(2))x)U(1)Y)x)U(1)…..)branes)in)the)past.)) )))))))))many)groups)working)on)intersecKng)from)which)the)hypercharge)is)assigned)to)be)anomaly)free))branes)in)the)past.)) Antoniadis, Kiritsis, Rizos, Tomaras AntoniadisThese)models)are)the)object)of)an)intense)scruKny)by)), Kiritsis, Rizos, Tomaras Antoniadis)))))))))many)groups)working)on)intersecKng), Leontaris, Rizos branes)in)the)past.)) Antoniadis Antoniadis, Leontaris, Kiritsis, Rizos, Rizos , Tomaras IbanezIbanez, Marchesano, MarchesanoAntoniadis, , RabadanLeontaris, Rabadan, Rizos, , GhilenceaGhilencea, IbanezIbanez, Ibanez, , IrgesMarchesano, Irges, Quevedo, ,Rabadan Quevedo, See. E. KiritsisGhilencea’ ’ ,review Ibanez, onIrges Phys., Quevedo Rep . See. E. KiritsisSee review. E. Kiritsis on’ review Phys. on Phys.Rep .Rep . ) ) ) The)analysis)is)however)quite)general:))The)analysis)is)however)quite)general:))The)analysis)is)however)quite)general:)) What)happens)if)you)to)have)an)anomalous))What)happens)if)you)to)have)an)anomalous)) What)happens)if)you)to)have)an)anomalous))U(1))at)low)energy?)What)is)its)signature?) U(1))at)low)energy?)What)is)its)signature?) U(1))at)low)energy?)What)is)its)signature?) Gauged'Stuckelberg'axions:'field'theory'realizaMon'of'the'' '''''''Green6Schwarz'mechanism'of'string'theory'

The'gauging'procedure'requires'an'anomalous'abelian'symmetry'' (an'anomalous'U(1))'and'a'periodic'potenMal'in'order'to'' make'the'axion'physical.'' ' '

But)ﬁrst)we)are)going)to)review)the)PQ)axion) Axions and the Strong CP Problem

Axions have appeared in physics in an attempt to solve the strong CP problem of QCD.

Why is the ✓GG˜ term so small? Consider an SU(2) gauge theory

G a = @ Aa @ Aa + g✏abc Ab Ac µ⌫ µ ⌫ ⌫ µ µ ⌫

G = @ A @ A +[A , A ] G = G a T a µ⌫ µ ⌫ ⌫ µ µ ⌫ µ⌫ µ⌫ 1 1 A UA U + U@ U µ ! µ µ 1 G UG U µ⌫ ! µ⌫ We look for minima of the Euclidean action 1 S = d 4xTrG G 2g 2 µ⌫ µ⌫ Z In a nonabelian theory a vanishing ﬁeld strength is possible with

1 Aµ = U@µU (pure gauge). Solutions of this condition are instanton conﬁgurations, characterised by a topological number. 2 16⇡2Q(x)=Tr [G G˜ ]=Tr [✏ [2@ (A @ A + A A A )], µ⌫ µ⌫ µ⌫↵ µ ⌫ ↵ 3 ⌫ ↵ 1 1 2 G˜ = ✏ G ↵, Q(x)=@ J , J = ✏ A (@ A + A A ) 2 µ⌫↵ µ µ µ 8⇡2 µ⌫↵ ⌫ ↵ 3 ↵ For an SU(3) gauge theory such as QCD, similarly, the Lagrangean then allows a total derivative term ✓GG˜ which is a boundary term, but cannot be neglected. For instantons

G = G˜, d 4xGG˜(x) = 32⇡2n, Z Therefore There is a dimension-4 operator that we can write ! down in the Standard Model (SM)

✓0GG˜

(violates Parity and Time reversal, CP is broken) It is a total derivative term and as such it does not contribute in perturbation theory Adding a total derivative term gives a zero momentum vertex in perturbation theory, but it contributes non-perturbatively How? If we consider an instanton (Euclidean) conﬁguration, then the contribution to the path integral is

1 4 8⇡2 2 d xFF 2 eS0 = e 4g = e g ⇠ R

I These conﬁgurations, at small coupling, give a negligible contribution

I They are solutions of the classical eq. of motion of QCD, which is scale invariant at classical level However, the solution of the equation G = G˜ involves an integration constant, the size of the instanton.

I The solution breaks scale invariance, because of the integration constant, which remains arbitrary. It tells us where the energy of the conﬁguration is localized. At tree level g is constant, but at 1-loop it runs. Scale invariance is broken by renormalization. I The running is controlled by the size of the instanton, g = g()

I The saddle point approximation is not valid any more since the action is O(1).

In the functional integral we need to sum over all these conﬁgurations. Small instantons (R)

I large scale 1/R ! ⇠ I small coupling g() 1 ! ⌧ 8⇡2 2 I large suppression in e g () . The contribution is ! perturbative, since g is small, but it is negligible. The instanton contribution to the QCD action is dominated by large instantons (g() large). Unfortunately the contribution is non-perturbative. I The running is controlled by the size of the instanton, g = g() In the functional integral we need to sum over all these conﬁgurations. Small instantons (R)

I The saddle point approximation is not valid any more since the action is O(1). The partition function can be written in the form

8⇡2/g 2() i✓ e 0 ⇠ and summing over instantons/anti instantons

8⇡2/g 2() e cos ✓ ⇠ 0 ¯ XI I ✓0 is not directly observable. One expects the energy density to dependen on ✓0 Notice, however, that QCD has a U(1)A anomaly, due to fermions. There is an axial symmetry

q qei5↵ ! and the integration measure is not invariant

i ↵ F Fd˜ 4x DqDq¯ DqDqe¯ 16⇡2 ! R Therefore ✓0 is not physical because it can be shifted by a ﬁeld redeﬁnition ✓ ✓ +2↵ 0 ! 0 But also the quark mass term gets a phase under the chiral transformation

q¯ Mq + h.c. q¯ Mq e2i↵ + h.c. L R ! L R therefore argM argM +2↵ ! and ✓ ✓ argM ⌘ 0 is invariant under field redefinitions. If we have fermions in complex representations of the gauge group, ✓0 is a↵ected by field redefinitions and is not physical,but✓ is physical. This can be generalized to nf fermions.

✓ ✓ +2n ↵, ArgdetM ArgdetM +2n ↵ 0 ! 0 f ! f ✓ ✓ ArgdetM ⌘ 0 is physical. But also the quark mass term gets a phase under the chiral transformation

✓ ✓ +2n ↵, ArgdetM ArgdetM +2n ↵ 0 ! 0 f ! f ✓ ✓ ArgdetM ⌘ 0 is physical. Experimentally ✓ is very small. We can set this value to zero assuming a cancellation between

I ✓0 ( reated to gluon dynamics ) I ArgDetM (relatedtotheelectroweaksector,Yukawasand Higgs ) We can easily derive some properties of the vacuum energy as a function of ✓.

VE(✓) S[] i ✓ F Fd˜ 4x e = De 32⇡2 | | Z R

S[] i ✓ F Fd˜ 4x VE(✓=0) D e 32⇡2 = e  | | Z R E(✓) E(0) It is also even in ✓: E(✓)=E( ✓). Periodic of period 2⇡. We can eliminate the ✓0 term and bring it completely into the fermion Mass matrix.

+i✓ /2 i✓ /2 q q e 0 q q e 0 L ! L R ! R Then i✓ /2 i✓ /2 M e 0 Me 0 ! It can be generalized to

f +iQ ✓ /2 f iQ ✓ /2 q q e f 0 q q e f 0 L ! L R ! R as far as TrQf =1

(global phase is ✓0). QCD with light quarks has a chiral symmetry (u,d)

U(2) U(2) = SU(2) SU(2) U(1) U(1) L ⇥ R L ⇥ R ⇥ V ⇥ A broken by quark condensates and anomalies to

SU(2) U(1) V ⇥ V 0 with U(1)V =baryon number. Three NG-models ⇡±, ⇡ of the broken chiral symmetry. We try to ﬁx the low energy e↵ective action using the left-over global symmetries

[J]= DeiSQCD ()+J = D⇡eiS(⇡,J) Z Z Z ⇡0 p2⇡+ p2⇡ ⇡0  i⇡ T /f⇡ U = e · 2 f⇡ µ = Tr @ U†@ U +2B Tr MU† + M†U L 4 µ 0 2⇣ h i h i⌘ f m 0 i ⇡0 0 E(✓, ⇡)= ⇡ 2B 2ReTr u ei✓/2Exp 4 0 0 m f 0 ⇡0 ✓ d ⇡  ◆ ✓ m m 2 ✓ ⇡0 = m2 f 2 cos2 + u d sin2 cos( (✓)) ⇡ ⇡ 2 m + m 2 f s ✓ u d ◆ ⇡ where m m ✓ sin()= d u sin2 md + mu 2 0 A minimum is obtained for (vev) ⇡ = f⇡(✓)(with 2 m⇡ = B0(mu + md )) Then

2 2 4mumd 2 ✓ E(✓)= m⇡f⇡ 1 2 sin s (mu + md ) 2 when the mass of any of the quarks goes to zero, the ✓-dependence disappears. For ✓ =0 E(0) = m2 f 2 ⇡ ⇡ Possible solutions. Can we use any existing SM symmetry? After we turn on the Yukawa’s only B and L are left as global symmetries of the SM. In the SM we have an anomalous symmetry B, baryon number and L, lepton number (B-L is anomaly free). But B is not anomalous respect to SU(3)c , whence it cannot produce a Fg F˜g (gluon). We then require an extra U(1)PQ global symmetry. There is another solution: if Yu = 0 then we could rotate:

u ei↵u R ! R . This symmetry would be anomalous under SU(3)c and we could erase the ✓Fg F˜g term. Notice that in the electroweak case we could also consider a ”weak CP” problem ✓ F F˜ ⇠ W W W In fact B is anomalous under SU(2)L, electroweak quark doublets therefore could be redeﬁned under U(1)baryon,cancelingthe corresponding weak-CP violating term. A second type of protection from ✓W contributions come from the fact that the theory is in a Higgs phase. The contribution is 8⇡2 e gw (W )2 which are screened due to the masses of the W’sand Z. KSVZ axion(Kim, Shifman, Vainshtein, Zakharov) A pseudoscalar a(x) that shifts under a global U(1)PQ symmetry (NG mode) a(x) a(x)+↵f can do the job. ! a Use the Lagrangean

1 a(x) @ a@µa + F F˜ + iQ¯µ@ Q + Q¯ Q 2 µ 32⇡2 µ L R

has a typical mexican-hat potential, with = vPQ . Then a(x) h i v +⇢ i (x)= PQ e vPQ and p2

a i v Q¯LQR vPQ e PQ Q¯LQR p2 ⇠ We perform now a chiral ﬁeld redeﬁnition

a i 2v 5 Q Q0 = e PQ Q vPQ Q¯0 Q0 ! p2 L R a ˜ . We will generate a term = 2 F F ,sincetheﬁeld S 32⇡ vPQ redeﬁnition is anomalous under U(1)PQ . Now we can integrate out Q and ⇢. We are left with an interaction

a(x)N ˜ a(x) ˜ 2 F F = 2 F F 32⇡ vPQ 32⇡ fa

vPQ for N quarks Q,with N = fa. DFSZ axion (PQ), (WW). This is generated using only scalars.

Hu, Hd ,

Up to dimension-4 involves three mexican-hat types of potentials for Hu, Hd and , and an extra contribution V 0 which depends on 2 2 2 2 2 2 H , H , , H H† , H H , H H . | u| | d | | | | u d | | u · d | u · d Collecting the phases, one can identify the NG mode of the U(1)PQ using the condition that it has to be orthogonal to the hypercharge There are 3 phases. One of them will identify the Goldstone mode. Orthogonality respect to the Goldstone of the Z boson is found by µ looking at the bilinear mixing MZ Zµ@ GZ

q a(x) qua(x) qd a(x) vu i v vd i v v i v Hu = e PQ Hd = e PQ = e PQ p2 p2 p2 v 2 v 2 q = 1 q =2 d q =2 d u v 2 d v 2 2 2 2 v = vu + vd absence of mixing with G : q2v 2 q2v 2 = 0. v is the electroweak Z u u d d vev (246 GeV). By requiring that a(x) is canonically normalized: 2 2 vu vd vPQ = v + v sin 2,withsin = v and cos = v . Notice that a(x) is associated mostly to . From the Yukawa couplings one gets

Y q¯ H q Y q¯ H d u L u R d L d R 2 a 2 a vu 2ia sin v vd 2ia cos v Yuu¯L e PQ uR Yd d¯L e PQ dR p2 p2 Doing a chiral redeﬁnition

6 ˜ µ c µ = 2 aF F q¯L DµqL @µaq¯ 5q L 32⇡ vPQ ! vPQ 1 a 1 1 = (✓ = 0) + @ Jµ + ✓ F F˜ + @ a@µa L LQCD f µ f 32⇡2 2 µ a ✓ a ◆ we can clearly redeﬁne a(x)inordertoabsorbe✓. Since fa is very large, then we can treat a(x) as an external source. To determine its potential, we can then take V (✓)with✓ a/f ! a a m m a V ( )= m2 f 2 1 u d sin2 f ⇡ ⇡ (m + m )2 2f a s u d ✓ a ◆ from which one can extract the axion mass

2 2 m⇡ 2 mumd ma = 2 f⇡ 2 fa (mu + md ) A"periodic"poten,al"is"generated"at"the"QCD"hadron"transi,on"

Clear analogy with the Peccei The"breaking"of"the"PQ"symmetry"" Quinn mechanism takes"place"at"a"large"scale"f_a,"but"" The"wiggling"of"the"PQ"poten,al"" Occurs"much"later,"at"the"QCD"phase"" transi,on"""

Compared to a Peccei-Quinn axion, the new axion is gauged

For a PQ axion a: m = C/fa, while the aFF interaction is also suppressed by : a/fa FF with fa = 10^9 GeV

In the case of these models, the mass of the axion and but with 2 its gauge interactions are unrelated

1. tilting of the potential, as we are going to show, in our case may occur at important the mass is generated by the combination of the Higgs and different stages (sequentialthe Stuckelberg misalignments) mechanisms combined differences 2. mass and coupling of the axion to gaugeThe interaction ﬁelds are UNRELATED is controlled by the Stuckelberg mass (M1)

The axion shares the properties of a CP odd scalar

Thursday, May 29, 14 Thursday, May 29, 14 2 – Elettrodinamica in presenza di assioni

L’analisi dell’attivitàottica nel background assionico giàa↵rontata puòessere utiliz- zata nel caso in cui gli assioni, o piùin generale le particelle molto leggere (o di massa nulla), a spin zero, inducono dei piccoli cambiamenti nello stato di polarizzazione di un fascio di laser che viaggia in un campo magnetico esterno. Possiamo trovare un riscontro interessante di questo risultato in letteratura [2], che ci permette di stabilire dei limiti per la costante di accoppiamento e la massa di queste ipotetiche particelle a partire dalle misure di ellitticitàe rotazione del piano di polarizzazione. Le proprietàmacroscopiche del vuoto possono essere legate a vari processi subnucleari, quali la di↵usione fotone-fotone (prevista dalla QED ma mai osservata sperimental- mente) e la produzione di particelle leggere (particelle simili ad assioni, come vedremo in seguito) che si accoppiano in due fotoni. L’analisi di questo esperimento richiede alcuni concetti di ottica che sono stati trattati nell’Appendice A e che hanno interessanti impieghi nel caso del vuoto quantistico.

2.7.1 PVLAS: funzionamento e risultati

L’esperimento PVLAS, in linea di principio,Experimental consiste nell’inviaresignatures un fascio di laser attraverso un forte campo magnetico trasverso Bext, localizzato in due polarizzatori incrociati, il tutto contenuto in una cavit`avuota.

Ovviamente se il fascio non interagisce con Bext non riuscir`aad uscire dal secondo

PVLAS)(INFN))

Figura 2.5. Idea alla base di PVLAS polarizzatore, pertanto non verr`arilevato dal fotodiodo. Al contrario se il vuoto quantistico perturbato dal campo magnetico presenta propriet`aottiche quali il dicroismo e

CAST)(Cern)) 2 – Elettrodinamica in presenza di assioni questa equazione ènota come equazione iconale dell’ottica geometrica,eW èdetto di conseguenza iconale. L’approssimazione dell’ottica geometrica saràvalida quando

2A 2⇡ 2 r n2k2 = . (2.48) A2 – Elettrodinamica⌧ in presenza di assioni ✓ ◆

Per le ondeche diventa piane, in cui l’iconale ècostante, questa disuguaglianza èsempre valida, @ B @' per cui la scelta dell’approssimazioneJ j = g ' geometrica+ E nel nostrog B caso ègiustificata.E ' , (2.35) @t r⇥ @t ⇥r 2A contiene le derivate seconde✓ di A, che scriviamo◆ approssimativamente✓ ◆ r dove i due termini nellae prima parentesi sono nullie per la (2.24). Inoltre si ha che d2f f(x + h) 22 –f Elettrodinamica(x)+f(x h) in presenza di assioni , (2.49) dx2 ⇠ µj 0j h2 ij 0j ijk k @µF = @0F + @iF = @0F + ✏ @jB (2.36) 2.2.1 Equazioni del moto in presenza dell’assione se poniamocheh corrisponde= /2⇡ avremo a grazie alla (2.48) j µj @E J = @µF = B , (2.37) ModifichiamoA(x + h nuovamente) 2A(x)+ laA lagrangiana(x r⇥h) aggiungendo1@t un termine cinetico per il cam- , (2.50) che unita alla (2.35) dà h2A ⌧ h2 po ' e un termine di interazione di ' con il campo elettromagnetico di background. Abbiamo @E @' semplificandola diventa 2 – ElettrodinamicaB in= presenzagB di assioni+ gE '. (2.38) Optical r⇥activity @t @t ⇥r 1 µ⌫ 1 µ 1 µ⌫ che svolgendo il prodottoA(x + diventah) 2A(xL)+= A(xFµ⌫ehF) + A@(eµx'@) . ' + g ' Fµ⌫F (2.51), (2.11) | 4 | ⌧ |2 | 4 Raccogliendo questi risultati,1 le equazioni 1 D(0) + g'(0)H(0) = E(0) + g2'(0)2E, e (2.88) 2 4 2 – Elettrodinamicae in presenza di assioni Questo significa che la derivata secondaB =0 di A, deve essere molto piccola rispetto ad A, dove g èuna costanter · di accoppiamento. Indicando i tre addendi della lagrangiana nell’intervalloeliminando della da questa lunghezza i termini d’onda.e quadratici@B Applichiamo in ['(0)g(0)] adessoe si ha questi nuovi risultati nel 8 + 2.4.3E =0, Rotazione del piano diPVLAS polarizzazione-type nel background rispettivamente con@t Lr⇥1, L2, L3 ed applicando a questi tre termini le (2.7), risulta nostro caso di un background> assionico. Supponiamo1 di avere un’onda elettromagne- e D>(0)⇤' ==E(0)g E gB'(0), assionicoH(0). (2.89) > 2· e tica che si propaga nel campo<> di sottofondoE = g ''.B Usando, l’approssimazione dell’otti- r · Mostreremor · ora che la risoluzione delle equazioni (2.52) e (2.53) ci permette di dare La forma di questa equazione ci mostrae anche@E come @'L si1 possa L prendere1 come iconale ca geometrica si ottengono, dal set diB equazioni=e ottenuto@gB + nellagE sezione=0', , precedente, le > una@t stimaµ della@t rotazione del piano di polarizzazione di un’onda che viaggia in un la quantitàdipendente dal> campor⇥ assionico,e ed in@ riferimentoµ' '⇥r alla (2.48), questa è seguenti > un’ulteriore conferma della ponderatezza:> background della scelta assionico. dell’approssimazione dell’ottica sono quelle che ci danno i campi in presenzae dell’assione.e Le primeL. Carcagni due’, C.C. equazioni di geometrica. 1 Iniziamo1 definendo due nuove quantità, corispondono con le⇤ equazioni(E g di'B Maxwell)= (1.1)g' ⇤B ed, (1.2) in unitànaturali(2.52) e ci1 indicano Un ragionamento del tutto analogo 2 puòessere 2 e↵ettuato per z = L e quindiD E possia-g'B, (2.81) che il campo B rimane solenoidale1 e le1 proprietàrotazionali del campo⌘E si conservano2 mo calcolare la variazione⇤ del(B campo+ g'EElungo)= g l’asse' ⇤Ez., che saràsicuramente non nulla(2.53) L2 L22e 2 e1 1 1 in presenza del@ backgroud assionico.=@ @ '@µ' = @ H @B'+g↵eg@'E'. = (2.82) poichè E non soddisfaµ all’equazione@ ' ' di D’Alembertµ @ ' 2 nonµ omogenea,µ @ ' 2⌘ ↵ 2 µ µ ✓ ◆ µ ✓ ◆ Dedichiamo i seguenti due paragrafie alla dimostrazioneSostituendoe la (2.52) di queste all’interno relazioni della a (2.53), partire ed approssimando al primo ordine, no- 1 µ ↵1 µ ↵ 1 µ e ↵µ E E(L)= E@µ(0)g↵ =g g@''H+(0)g. g @↵' = @µ(g (2.90)@' + g @↵')= dalle equazioni2.4 dei Attivitàottica campi in presenza⌘ dell’assione2 neltiamo background che2 viste questi nel nuovi precedente vettori assionico paragrafo. soddisfanoOptical 2 activity all’equazione di D’Alembert omogenea, che 1 significa µ cheµ µ La rotazione del piano di polarizzazione= @✓µ(che@ ' ci+e proponiamo@ ')=@µ@ di' trovare= ⇤', èin genere 232 D = 0 (2.83) moltoIn piccola, ottica classica, data la natura si definiscono del campo sostanze assionico,otticamente per cui la attive scelta, quelle di approssimare particolari⇤ sostan- quest’angoloze che hanno con il la suo proprietàdi seno ègiustificata. ruotaree sostituendo il Ricordando piano di la (2.53) polarizzazione che all’internoE èdefinito delladi un da(2.52), fascio una alla di stessa luce maniera avremo linearmente polarizzato che le attraversa. E’ possibile trovare una trattazione piùap- di↵erenza di vettori, si ha ⇤H =0. (2.84) E profondita di questo fenomeno✓ L nell’Appendice3sin ✓=L3 , A. Ci1 proponiamoµ⌫ di dimostrare1 (2.91)µ⌫ che lo @µ ' AssumiamoE=(0) che i campig' F sianoFµ⌫ funzioni= gF di z,F cioèscrivamoµ⌫, E = E(z), B = B(z)e spazio permeato da un campo@µ assionico' ' presenta' questa4 medesima proprietà.4 e sostituendo le espressioni (2.86) e (2.90),' = trascurando'(z). L’onda✓ i termini piana ètale del◆ secondo per cui in ordine unitànaturali per z = 0 possiamo scrivere e e e e rispetto a g', abbiamo E(0) =30B(0) per cui la (2.81) diventa 1 e sommando questi risultati si ha la prima equazione del moto in presenza1 dell’assione ✓ = g', D(0) = E(0)(2.92)1 g'(0) , (2.85) e 2 2 che esprime la rotazione del piano di polarizzazione in funzione di ' '(L) '✓(0). ◆ mentree la (2.82)g diventa ⌘ e Ciòimplica che ad ogni cambiamento di ' di una' quantitàF µ⌫F',=0 lungo. la traiettoria (2.12) ⇤ 4 µ⌫ 1 di un’onda elettromagnetica, E e B ruotano di un angolo ✓ dato dallaH(0) (2.92). = E(0) Valori1+ g'(0) . (2.86) e 2 positivi di ✓ corrispondono a rotazioni orarie guardando lungo26 e il percorso del fotone.✓ ◆ Le equazioni (2.83) e (2.84) ci dicono inoltre chee D ed H rimangono invariati per Questo e↵etto èapprossimativamente indipendente dalla frequenza. traslazioni di una quantitàarbitraria L lungo l’asse z, per cui risulta D(L)=D(0) e Le implicazioni cosmologiche della (2.92)H sono(L)= notevoli.H(0). Infatti le onde piane, per la loro proprietàdi rappresentare la radiazioneUtilizzando emessa da le sorgenti relazioni puniformi date dalla (2.81)poste all’in-e dalla (2.82) si ha finito, si prestano in maniera del tutto naturale allo studio della radiazione proveniente1 1 D(0) = E(0) g'(0) H(0) g'(0)E(0) , (2.87) 2 2 37 ✓ ◆ e 36 e Gauging axionic symmetries The chain of anomalous U(1) symmetries require

- One Stuckelberg termfor each anomalous symmetry - The U(1)’s are in a massive (Stuckelberg phase) - One linear combination of them generates the anomaly free hypercharge

Possibility of describing axion-like particles.

Such types of particles have been conjectured in several phenomenological analysis. The mass of the particle and its interactions with the photons are independent quantities.

This brings us to a mechanism of cancelation of the gauge anomalies of “Green-Schwarz” type Compared to a Peccei-Quinn axion, the new axion is gauged

For a PQ axion a: m = C/fa, while the aFF interaction is also suppressed by : a/fa FF with fa = 10^9 GeV

In the case of these models, the mass of the axion and its gauge interactions are unrelated

the mass is generated by the combination of the Higgs and the Stuckelberg mechanisms combined

The interaction is controlled by the Stuckelberg mass (M1)

The axion shares the properties of a CP odd scalar

Thursday, May 29, 14 Asymptotic axions for Wess Zumino actions and gauge invariance

1 = F 2 + i ¯µ(@ + ig B ) L 4 B µ B 5 µ 1 1 = F 2 F 2 + i ¯µ(@ + ig A + ig 5B ) L 4 B 4 A µ A µ B µ

Using a Stuckelberg axion and the inclusion of local counterterms

Bµ Bµ @µ✓ b b ! aBBB FB FB + aBAA FA FA M ^ M ^ b b + M✓ ! 1 (@ b + MB )2 2 µ µ

Thursday, May 29, 14 One then considers the effective action

1 1 1 b = F 2 + (B + @ b)2 + i ¯µ(@ + ig ) + a F F L 4 B 2 µ M µ µ B 5 n M B ^ B where the anomaly generated at one loop level by the fermion is removed by the Wess-Zumino counterterm

b a F F n M B ^ B Somehow, this mechanism is viewed, from the point of view of QFT, as the mechanism of “Anomaly Cancellation”

But anomalies are not cancelled by local One could go counterterms. One should notice that the mechanism of “anomaly cancellation”, in this case, is to a gauge where based on introducing an extra ﬁeld degree of freedom (b(x)) b(x)=0. In what sense, then we cancel the anomaly?

Thursday, May 29, 14 Thursday, May 29, 14 Variants: Higgs-axion mixing There are some variants of this Lagrangian which may help us clarify this issue In this case we consider a model with 2 U(1)’s. The two gauge ﬁelds are A and B. The fermion has axial vector couplings to B and is vector coupled to A. We have BBB and BAA anomalies. Vector ﬁeld B is massive, A is massless

B mass generated via a combination of the Stuckelberg + Higgs mechanisms.

is the Higgs ﬁeld

B field massive by the Higgs and Stuckelberg mechanism

Goldstone mode is a combination of Stuckelberg field and CP odd part of the Higgs

Thursday, May 29, 14 The mass of the B gauge boson is a combination 1 physical axion (axi-Higgs) B of the Higgs and the Stuckelberg mechanism 1 Higgs h1 1 massive gauge boson Bµ

The Stuckelberg has a gauge invariant physical component, B A massive axi-Higgs (periodic potential) ordinary Higgs potential V

V 0 extra potential allowed by the symmetry massive axi-Higgs

Thursday, May 29, 14 Generic extension

Gauge kinetic Stuckeberg mass terms Chern Simons abelian interactions

Stuckelberg axions

Abelian CS terms Higgs sector

Typical mass terms for the gauge bosons are generated both from the Higgs and the Stuckleberg contributions There will be bilinear mixings in the broken (electroweak) phase

We can extract the NG modes by a rotation, identifying 1 single physical axion

The scalar potential has an ordinary 2-Higgs doublet part and an extra contribution set limits on their parameter space.

Acknowledgements We thank Pierre Sikivie and Nikos Irges for discussions. C.C.thanksthePhysicsDepartmentat Thessaloniki for hospitality. This work is supported in partbytheEuropeanUnionthroughtheMarie Curie Research and Training Network UniverseNet (MRTN-CT-2006-035863).

The Standard Model with 1 extra anomalousA Appendix.U(1) and an Theaxion model, deﬁnitions and conventions

f Q uR dR L eR We summarize in this section some results concerning the model with a single anomalous U(1) discussed qB qB qB qB qB qB Q uR dR L eR in the main sections. The effective action has the structure given by f SU(3)C SU(2)L U(1)Y U(1)B B = 0 + Yuk + an + WZ + CS (55) Q 3 2 1/6 qQ S S S S S S B B uR 3 1 2/3 qQ + qu where 0 is the classical action. It contains the usual gauge degrees of freedom of the Standard Model d 3 1 1/3 qB qB S R − Q − d plus the extra anomalous gauge boson B which is already massive, before electroweak symmetry L 1 2 1/2 qB − L breaking, via a Stückelberg mass term, reviewed in Sec. 2. Its complete expression is given in [17]. e 1 1 1 qB qB R − L − d Here we briefly describe the structure of the anomalous contributions and of the induced counterterms B Hu 1 2 1/2 qu for the restoration of gauge invariance in the 1-loop effectiveaction. B Hd 1 2 1/2 qd In Eq. (55) the anomalous contributions coming from the 1-loop triangle diagrams involving abelian

Table 1: Charges of theY fermion and of theB scalar fieldsand non-abelianB gauge interactionsSU(3) are summarizedSU(2) by the expression 1 1 1 B B B B = T B BWW + T BGG + T BBB 3TheelectroweakpotentialformasslessfieldsSeff =S0 + + + + San 2!⟨+BWW ⟩ 2!+⟨ BGG ⟩ 3!⟨ BBB ⟩ 1 1 Y Y B + T BY Y + T YBB , (56) As in previous works [15], in the construction of the effective action we follow a bottom-up approachSU2!(3)⟨ BY Y ⟩ 2!SU⟨ (2)YBB ⟩ with general charge assignments parameterized just by the set of free charges of U(1) .Theseare where the symbolsB denote integration. For instance, the contributions in configuration space are shown in Fig. 1, together with the fundamentalY gauge structuB re of the StandardB Model. The⟨⟩ SU scalar(3) SU(2) sector of the anomalous abelian models that we are interestedinischaracterizedbyaratherstandardgiven explicitly by electroweak potential involving,b in the simplest+ b formulation,+ b two Higgs doublets+ b V (H ,H )plus+ b PQ u d λµν,ij λ µ ν TBWW BWW dx dy dzTBWW (z,x,y)B (z)Wi (x)Wj (y)(57) one extra contribution, denoted as VP/Q/ (Hu,Hd,b)orV ′,[17]whichmixestheHiggssectorwiththe⟨ ⟩≡ Stückelberg axion b,neededfortherestorationofthegaugeinvarianceoftheeY Y B ffective LagrangianSU(3) ! SU(2) and so on, where TBWW denotes the anomalous triangle diagram with one B field and two W ’s external

VFigure= VPQ 1:(H Anomalousu,Hd)+V contributionsP/ Q/ (Hu,Hd,b) to. thegauge Lagrangian lines. Theand gluonsWZ counte are(10) denotedrterms by G.TheWess-Zumino(WZ)countertermsaregivenby

The appearance of the physical axion in the spectrum of the model takes place after that the phase- WZ = CBB bFB FB + CYY bFY FY + CYB bFY FB dependent terms,briefly here assumed comment to on be the of list non-perturbative of the charge ori assignmentsgin and generated of the si atngle the extra electroweakS U(1) model,⟨ which∧ ⟩ is given⟨ ∧ ⟩ ⟨ ∧ ⟩ W W G G phase transition,in find Table their (1). way in the dynamics of the model and induce a curvature on the scalar+F bTr[F F ] + D bTr[F F ] , (58) B B ⟨ ∧ ⟩ ⟨ ∧ ⟩ potential. The mixingSpecifically, induced inq theL ,q CP-oddQ denote sector the determin chargeses of the the presence left-handed of a linear lepton combination doublet (L)andofthequark B B B while the gauge dependent Chern-Simons (CS) abelian and non abelian counterterms [40] needed to of the Stückelbergdoublet field b (andQ), of while the Goldstonesqur ,qdr ,qeR ofare the the CP-odd charges sector, of the called right-handedχ,whichischaracterizedSU(2) singlets (quarks and leptons). B B B cancel the mixed anomalies involving a B line with any other gauge interaction of the SM take the by an almost flatWe direction. denote with To better∆q illustrate= qu qd thisthe point, differencewe begin between our analysis the two by charges turning of to the the up and down Higgses B B − ordinary potential(qu of,q 2d Higgs)respectively.Thetrilinearanomalousgaugeinteraction doublets, sinducedbytheanomalousU(1) and the relative counterterms, which are all parts of the 1-loop effective action, are illustrated in Fig.24 1. 2 2 2 2 T 2 V = µ H†H + µ H†H + λ (H†H ) + λ (H†H ) 2λ (H†H )(H†H )+2λ′ H τ H PQ u u Theu numericald d d valuesuu u ofu the countertermsdd d d − appearingud u u on thed d secondud line| u of2 Fig.d| 1 are fixed by the conditions of gauge invariance of the Lagrangian and are summarized by the(11) following relations 1 4 1 1 C = qB + qB + qB qB + qB , BY Y −6 Q 3 uR 3 dR − 2 L eR C = 9 (qB)2 +2(qB )2 (qB )2 +(qB)2 (qB )2, YBB − Q ur − dR L − eR C = 6(qB)3 +3(qB )3 +3(qB )3 2(qB)3 +(qB )3, BBB − Q uR dR − L eR 1 C = ( 2qB + qB + qB ), Bgg 2 − Q dR uR 1 C = ( qB 3qB). (7) BWW 2 − L − Q 2 They are, respectively, the counterterms for the cancellation of the mixed anomaly U(1)B U(1)Y and 2 3 U(1)Y U(1)B ;thecountertermfortheBBB anomaly vertex or U(1)B anomaly, and those of the 2 2 U(1)B SU(3) and U(1)B SU(2) anomalies. They are defined in the appendix. From the Yukawa

couplings we get the following constraints on the U(1)B charges

qB qB qB =0 qB + qB qB =0 qB qB qB =0. (8) Q − d − dR Q u − uR L − d − eR

In Tab. (1) we also show the expressions of the free U(1)B charges appearing on each generation, having

7 set limits on their parameter space.

A Appendix. The model, deﬁnitions and conventions

We summarize in this section some results concerning the model with a single anomalous U(1) discussed in the main sections. The eﬀective action has the structure given by

= + + + + (55) S S0 SYuk San SWZ SCS where is the classical action. It contains the usual gauge degrees of freedom of the Standard Model S0 plus the extra anomalous gauge boson B which is already massive, before electroweak symmetry breaking, via a Stückelberg mass term, reviewed in Sec. 2. Its complete expression is given in [17]. Here we briefly describe the structure of the anomalous contributions and of the induced counterterms for the restoration of gauge invariance in the 1-loop effectiveaction. In Eq. (55) the anomalous contributions coming from the 1-loop triangle diagrams involving abelian and non-abelian gauge interactions are summarized by the expression 1 1 1 = T BWW + T BGG + T BBB San 2!⟨ BWW ⟩ 2!⟨ BGG ⟩ 3!⟨ BBB ⟩ 1 1 + T BY Y + T YBB , (56) 2!⟨ BY Y ⟩ 2!⟨ YBB ⟩ where the symbols denote integration. For instance, the contributions in configuration space are ⟨⟩ given explicitly by

T BWW dx dy dzT λµν,ij (z,x,y)Bλ(z)W µ(x)W ν (y)(57) ⟨ BWW ⟩≡ BWW i j ! and so on, where TBWW denotes the anomalous triangle diagram with one B ﬁeld and two W ’s external gauge lines. The gluons are denoted by GAxionic.TheWess-Zumino(WZ)countertermsaregivenbycontributions

= C bF F + C bF F + C bF F SWZ BB⟨ B ∧ B⟩ YY⟨ Y ∧ Y ⟩ YB⟨ Y ∧ B ⟩ +F bTr[F W F W ] + D bTr[F G F G] , (58) ⟨ ∧ ⟩ ⟨ ∧ ⟩ whileform the gauge dependent Chern-SimonsAbelian (CS)/non abelian-abelian andChern non Simonsabelianterms counterterms [40] needed to cancelform the mixed anomalies involving a B line with any other gauge interaction of the SM take the =+d BY F + d YB F SCS 1⟨ ∧ Y ⟩ 2⟨ ∧ B⟩ CS =+d1 BYµνρσFY +SUd2(2)YB FBµνρσ SU(3) S +c⟨1 ϵ ∧ Bµ⟩Cνρσ ⟨ +∧c2 ϵ⟩ BµCνρσ . (59) ⟨µνρσ SU(2) ⟩24 µ⟨νρσ SU(3) ⟩ +c1 ϵ BµCνρσ + c2 ϵ BµCνρσ . (59) The non-abelian CS forms given by ⟨ ⟩ ⟨ ⟩ The non-abelian CS forms given by SU(2) 1 i W 1 ijk j k Cµνρ = 1 Wµ Fi, νρ +1 g2 ε Wν Wρ + cyclic , (60) CSU(2) = 6 W i F W + g3 εijkW jW k + cyclic , (60) µνρ 6 ! µ "i, νρ 3 2 ν ρ # $ SU(3) 1! a" G 1 abc b #c $ Cµνρ = 1 Gµ Fa, νρ +1 g3 f GνGρ + cyclic . (61) CSU(3) = 6 Ga F G + g3 f abcGb Gc + cyclic . (61) µνρ 6 ! µ "a, νρ 3 3 ν ρ # $ χ ! " # $ The structure of gχγγ • The structure of gγγ • The coefficients in front of the WZ counterterms are determinedbyrequiringgaugeinvarianceofthe The coefficients in front of the WZ countertermsχ are determinedbyrequiringgaugeinvarianceofthe effective action. We outlineWith a single the caseanomalous of χgγγ and itsU(1) relationthese toterms thecare fundamentalnot essential parameters/scales. of effective action. We outline the case of gγγ and its relation to the fundamental parameters/scales of the theory. Among these are the Stückelberg mass M,thehyperchargeandweakcouplingsgY and the theory. Among these are the Stückelberg mass M,thehyperchargeandweakcouplingsgY and g2 and the charges of the fermion running inside the anomaly loops. These fix the coefficient of the g2 and the charges of the fermion running inside the anomaly loops. These fix the coefficient of the 2 2 anomalies CBY Y and CBWW (for the U(1)B U(1)2 and SU2(2) U(1)B anomalies) and the rotation anomalies CBY Y and CBWW (for the U(1)B U(1)Y Yand SU(2) U(1)B anomalies) and the rotation A χ matricesmatrices of the neutral neutral gauge gauge bosons bosonsOOA andand of of the the CP-odd CP-odd sector sectorOχ,definedinEq.(16).ThisisO ,definedinEq.(16).Thisis defineddefined as as in Eq. (20) (20) in in terms terms of of the the counterterms counterterms g g a a FF==B igB 2ig2n Cn C , , (62) (62) MM 2 22 2BWWBWW withwith 1 1 B B CBWWC = = qfLq, , (63) (63) BWW −8 fL −8f %%f ii withwith aan == 2π2 beingbeing the theAVAV V Vanomaly,anomaly, and and n − 2π2 gBg 2 ana CCYY == igBYig 2 CnBYC Y , , (64) (64) YY MM Y2 2 BY Y which is defined by the charges which is defined by the charges

1 B Y 2 B Y 2 CBY Y = 1 q (q ) q (q ) . (65) C =8 fRqB fR(qY −)2 fLqBfL(qY )2 . (65) BY Y 8f fR fR − fL fL %f& ' % & ' Explicit expressions for CBY Y and CBWW are given in Eq. (9). Explicit expressions for CBY Y and CBWW are given in Eq. (9). Fermion interactions • Fermion interactions • The covariant derivatives are deﬁned as The covariant derivatives are deﬁned as i i D = ∂ + ig T aGa + ig τ aW a + g qY Y + g qBB , (66) µ µ s µ 2 µ 2 Yi µ 2 Bi µ D = ∂ + ig T aGa + ig τ aW a + g qY Y + g qBB , (66) µ µ s µ 2 µ 2 Y µ 2 B µ 25 25 f Q uR dR L eR qB qB qB qB qB qB Q uR dR L eR

f Q uR dR L eR qB qB qB qB qB qB f SU(3)C SU(2)L U(1)Q Y uR U(1)dR BL eR

f SU(3) SU(2) BU(1) U(1) Q 3 2 1/C6 LqQ Y B B Q 3 2 B 1/B6 qQ uR 3 1 2/3 q + qu Q B B uR 3 1 2/3 q + q B B Q u dR 3 1d 3 1/3 1qQ q1d/3 qB qB R − −− Q − d B B L 1 2L 1 1/2 2 q 1/2 qL − L− e 1 1 1 qB qB R to whichB we add−B a secondL − d term eR 1 1 1 qL qd H 1 2 1/2 qB u − − u b b b B igB(qu qd) M igB(qu qd) M 2 igB(qu qd) M VP/ Q = λ0(Hu†Hde− B − 2 )+λ1(Hu†Hde− − 2 ) + λ2(Hu†Hu)(Hu†Hde− − 2 )+ Hu 1 2Hd 11/2/ 2 qu 1/2 qd b igB(qu qd) λ3B(H†Hd)(Hu†Hde− − 2M )+h.c., (12) Hd 1 Table2 1: Charges1/ of2 the fermionqd andd of the scalar ﬁelds

These terms are allowed by the symmetry of the model and are parameterized by one dimensionful (λ0) Table3Theelectroweakpotentialformasslessfields 1: Charges of the fermion and of the scalar fields and three dimensionless constants (λ1, λ2, λ3). They are assumed to be generated at the electroweak As in previous works [15], in the constructionphase of transition the effective non-perturbatively,action we follow a bottom-up and as approach such their values are related to an exponential factor with general charge assignments parameterized just by the set of free charges of U(1) .Theseare 3Theelectroweakpotentialformasslessfieldscontaining as a suppression the instantonB action. In the equations below we will rescale λ0 by the shown in Fig. 1, together with the fundamental gauge structure of the2 Standard2 Model.¯ The scalar electroweak scale v = vu + vd (λ0 λ0v)sotoobtainahomogeneousexpressionofthemassofχ as sector of the anomalous abelian models that we are interestedinischaracterizedbyaratherstandard≡ As in previous works [15], in the construction of the effectiveafunctionoftherelevantscalesofthemodelwhichare,besiaction we follow! a bottom-up approach de the electroweak vev v,theStückelberg electroweak potential involving, in the simplest formulation, two Higgs doublets VPQ(Hu,Hd)plus mass M and the anomalous gauge coupling of the U(1)B , gB. with general charge assignmentsone extra parameterized contribution, denoted just as byVP/Q/ the(Hu,H setd,b)or of freeV ′,[17]whichmixestheHiggssectorwiththe charges of U(1)B .Theseare to which we add a second term The physical axion χ emerges as a linear combination of the phases of the various terms, which shown in Fig. 1, together withStückelberg the fundamental axion b,neededfortherestorationofthegaugeinvarianceofthee gauge structure of the Standard Model.ffective The Lagrangian scalar b areb either due to the componentsb of the Higgs sector or to the Stückelberg field b.Toillustratethe igB(qu q ) igB(qu q ) 2 igB(qu q ) V = λ (H†H e− − d 2M )+λ (H†H e− − d 2M ) + λ (H†H )(H†H e− − d 2M )+ sector of the anomalousP/ Q/ 0 u abeliand models that1 weu d areV intereste= VPQappearance(Hdinischaracterizedbyaratherstandardu,Hd)+2 Vu ofP/ Q/u a(H physicalu,Hu dd,b). direction in the phase of(10) the extra potential, we focus our attention just on b igB(qu q ) λ (H†H )(H†H e− − d 2M )+h.c., (12) electroweak potential3 involving,d dTheu appearanced in the of simplest the physical formulat axion intheion, the CP-odd spectrum two Higgs sector of the moof doublets thedel takes total placeV potential,PQ( afterHu,H that whichd)plus the is phase- the only onethatisrelevantforourdiscussion.The dependent terms, here assumed to be ofexpansion non-perturbative of this ori potentialgin and around generated the at electroweak the electroweak vacuum isgivenbytheparameterization one extra contribution,These terms are denoted allowed by as theV symmetryP/Q/ (Hu,H of thed,b model)or andV ′,[17]whichmixestheHiggssectorwiththe are parameterized by one dimensionful (λ0) phase transition, find their way in the dynamics of the model and induce a curvature on the scalar and three dimensionlessto which constants we add a second (λ term, λ , λ ). They are assumed to be generated at the electroweak Stückelberg axion b,neededfortherestorationofthegaugeinvarianceofthee1 2 3 ffective LagrangianH+ H+ potential. The mixingig (q q ) inducedb in theig CP-odd(q q ) b 2 sector determinesig the(q presenceq ) b of a linearu combination d V = λ (H H e B u d 2M )+λ (H H e B u d 2M ) + λ (H H )(H H e B u d 2M )+ phase transition non-perturbatively,P/ Q/ 0 u† d − and− as such1 u† theird − val− ues are2 relatedu† u tou† d an− exponential−Hu = factor 0 Hd = 0 . (13) of the Stückelberg fieldig (bq andq ) b of the Goldstones of the CP-odd sector, called χ,whichischaracterizedvu + H vd + H λ (H†H )(H H e B u d 2M )+h.c., (12)" u # " d # containing as a suppression3 thed d instantonu† d − action.− In the equations below we will rescale λ0 by the by an almostV = V flatPQ direction.(Hu,Hd To)+ betterV / illustrate(Hu,Hd this,b) point,. we begin our analysis by turning(10) to the 2 2 ¯ P Q/ electroweak scale vThese= termsvu + arev allowed(λ0 byλ the0v symmetry)sotoobtainahomogeneousexpressionofthemassof of the model andThis are papotentialrameterized is by characterizedone dimensionful (λ0) by twoχ as null eigenvalues corresponding to two neutral Goldstone modes ordinary potentiald ≡ of 2 Higgs doublets, afunctionoftherelevantscalesofthemodelwhichare,besiand three! dimensionless constants (λ1, λ2, λ3). They arede assumed1 the2 electroweak to be generated vev at thev electroweak,theStückelberg (G0,G0)andaneigenvaluecorrespondingtoamassivestatewithanaxion component (χ). In the The appearance of the physicalphase transition axion2 non-perturbatively, in the2 spectrum and as such of their the val2ues mo aredel related takes to2 an exponential place after factor that the phase-T 2 mass M and the anomalousVPQ = gaugeµuHtou†H coupling whichu + µdH of wed†H thed add+Uλuu(1) a(HB second,u†HgBu.) +0 termλdd(Hd†0Hd) 2λud(Hu†Hu)(Hd†Hd)+2λud′ Hu τ2Hd containingto which as a wesuppression add a secondthe instanton term action. In(Im the equHationsd , Im belowHu,b we)CP-oddbasiswegetthefollowingnormalizedeigenstates will− rescale λ0 by the | | dependent terms, here assumed to be of2 non-perturbative2 ¯ origin and generated at the electroweak The physical axionelectroweakχ emerges scale v = asvu a+ linearvd (λ0 combinationλ0v)sotoobtainahomogeneousexpressionofthemassof of the phases of the various terms,χ as which (11) ≡ b b b afunctionoftherelevantscalesofthemodelwhichare,besi! ig (q q ) b igBde(q theu electroweakigqd)(1q q ) vevb v,theStückelberg2 igB(qu qigd)(q q 2) b igB(qu qd) are either due to the components of the HiggsB u sectord 2M or to the1 StückelbergB 2uM d field2M b.Toillustratethe B 2Mu d 2M 2M phase transition, find their wayVP/ Q/ in= theλ0V(H/ dynamicsu†Hde=− λ0−( ofH† theH)+deλ model−1(Hu†Hde a−−nd induce− )+λ) 1 a+(Hλ curvature2†(HHu†dHeu−)(Hu†H onde−− the scalar−) +)+λ2(H†Hu)(H†Hde− − )+ mass M and the anomalousP Q/ gauge coupling ofu the U(1)BG, g0B.= (vd,vu, 0)u u u appearance of a physical direction in the phase of the extrab potential,2 we focus2 our attention just on The physical axion χ emerges as a linearigB( combinationqu qd) 2M of the phasesvu + of thev variousb terms, which potential. The mixing induced in theλ3( CP-oddH†Hd)(Hu† sectorHde− determin− )+h.c.es the, igB presence(qud qd) of a linear combination(12) d λ (H†H )(H†H e− − 2M )+h.c., (12) the CP-odd sectorare of eitherthe total due to potential, the components which of the3 is Higgs thed onlysectord onor toethatisrelevantforourdiscussion.Theu thed S!tückelberg9 field b.Toillustratethe of the Stückelberg field b andappearance of the of a physical Goldstones direction in ofthe phase the of CP-odd the extra potential, sector, we focus called our attentionχ,whichischaracterized1 just on g (q q )v v2 g (q q )v2v expansion of this potentialThese around terms are the allowed electroweak by the symmetry vacuum isgivenbytheparameterization of2 the model and are parameterized by one dimensionfulB (λ0d) u d u B d u d u √ 2 2 the CP-odd sector of the total potential, which is the onlyG on0 ethatisrelevantforourdiscussion.The= − , − , 2M vu + vd 2 2 2 ⎛− 2 2 ⎞ by an almost flat direction.expansionand To three of better this dimensionless potentialThese illustrate around terms the constants electroweak this are allowed point, (λ vacuum1, λ2, λ iwesgivenbytheparameterization by3). Theythe begin symmetry are our assumed analysis2 of to2 bethe generated by model2 turning atand the2 are toelectroweak pa therameterized2 by one dimensionful2 (λ0) + + gB(qd qu) vdvu +2M vd + vu vu + vd vd + vu ! phase transitionHu non-perturbatively, and asHd such their val− ues are related to an exponential factor Hu = and three dimensionless+ Hd = constants+ . (λ , λ , λ ). They(13) are assumed⎝ to be generated at the electroweak ⎠ ordinary potential of 2 Higgs doublets, 0 Hu Hd!0 1 2 1 3 ! ! v H+uH= H = v + H . (13) $ % containing" asu a suppressionu # 0 the instanton"d dχ action.= d0 # In the equations below we will rescale√λ2Mvby theu, √2Mvd,gB (qd qu)vdvu (14) " vu + Hu # " vd + H # 0 phase transition non-perturbatively,d 2 and2 2 as2 such2 their2 2 values are− related to an− exponential factor electroweak scale v = v2 + v2 (λ λ¯ v)sotoobtainahomogeneousexpressionofthemassofgB(qd qu) vuvd +2M (vd + vu) χ as 2This potential2 is characterizedThis potential is bycharacterized two null by2 eigenvalues twou null eigenvaluesd 0 corresponding cor0 2responding to two two neutral− neutral Goldstone Goldstone modes modes T * 2 + † †≡ † VPQ = µuHu†Hu + µdHd(GH1,Gafunctionoftherelevantscalesofthemodelwhichare,besid2+)andaneigenvaluecorrespondingtoamassivestatewithanaλuu(Hcontainingu†Hu) !+ λ asdd( aH suppressiondHd) 2!λud the(xionH instantonu† componentHdeu)( theH (dχ electroweak).H action. Ind)+2 the λ vev Inud′ v theH,theStückelbergu τ equ2Hationsd below we will rescale λ0 by the (G1,G2)andaneigenvaluecorrespondingtoamassivestatewithana0 0 xion componentχ (χ). In the 0 0 0 0 and− we indicate with O the orthogonal| matrix which| allows to rotate them on the physical basis (ImHd , ImHu,b)CP-oddbasiswegetthefollowingnormalizedeigenstates2 0 0 mass M andelectroweak the anomalous scale gaugev coupling= v of2 the+ vU(1)(λB , gB. λ¯ v)sotoobtainahomogeneousexpressionofthemassofχ as (ImHd , ImHu,b)CP-oddbasiswegetthefollowingnormalizedeigenstatesu d 0 0 (11) 1 The1 physical axion χ emerges as a linear combination≡ of the phases of the various1 terms, which 0 G0 = (vd,vu, 0) G ImH 2 2 afunctionoftherelevantscalesofthemodelwhichare,besi0 de the electroweakd vev v,theStückelberg 1 vu + v ! G1 = (v ,vare, 0) eitherd due to the components of the Higgs sector or to the Stückelberg field b.Toillustratethe2 χ 0 0 d u ! G0 = O ImHu , (15) 2 2 mass1 M and theg anomalous(q q )v v2 g gauge(q q )v2 couplingv of the U(1)⎛ ⎞, g . ⎛ ⎞ vu + v 2appearance of a physical direction inB thed u phased u ofB thed extrau d u p√otential,2 we2 focus our attentionB B just on d G0 = − , − , 2M vu + vd g2 (q q )2v2v2 +2M 2 v2 + v2 ⎛− v2 + v2 v2 + v2 ⎞ χ b the CP-oddB d u sectord u of thed totalu potential,u whichd is thed onlyu onethatisrelevantforourdiscussion.The! ⎜ ⎟ ⎜ ⎟ ! − The physical axion χ emerges as2 a linear combination of the phases of the various terms, which !1 1 g ⎝(q !q )v v2 g (!q q )v v ⎠ ⎝ ⎠ ⎝ ⎠ 2 χexpansion= of this potential$ around9B% √2dMv the,u electroweak√2dMvu ,gB(q d vacuumq )v uv d isgivenbytheparameterizationu (14)2 2 G0 = − u d, B d −u d u , √2M vu + v g2 (q q are)2v2v2 either+2M 2(v2 due+ v2) to the− components− of the Higgs sectord or to the Stückelberg field b.Toillustratethe 2 2 2 2B d u2 u 2d 2 ⎛d − u * 2 2 2 +2 ⎞ g (qd qu) v vu +2−M v + vu vu + v v + vu B − d! d + d d + ! and we indicate with Oχ the orthogonal matrix whichHu allows to rotate them on the physHdical basis ! 1 appearance⎝Hu of= a! physical direction!Hd = in the phase. of the⎠ extra potential,(13) we focus our attention just on χ = $ % √2Mv ,v √+2HMv0 ,g (q q )v vv + H0 (14) 1 "u u 0u d# B d u "d ud d # 2 2 2 2 2 2 2 G0 − ImH − g (qd qu) v v +2Mthe(v + CP-oddv ) sector of thed total potential, which is the only onethatisrelevantforourdiscussion.The10 B u d d u * 2 = Oχ 0 , + (15) − This potential is characterized⎛G0⎞ by two⎛Im nullHu⎞ eigenvalues corresponding to two neutral Goldstone modes ! expansion ofχ this potentialb around the electroweak vacuum isgivenbytheparameterization χ1 2 ⎜ ⎟ ⎜ ⎟ and we indicate with (OG0,Gthe0)andaneigenvaluecorrespondingtoamassivestatewithana orthogonal matrix⎝ which⎠ allows⎝ ⎠ to rotate them on the physicalxion basis component (χ). In the (ImH0, ImH0,b)CP-oddbasiswegetthefollowingnormalizedeigenstates d u 1 0 + + G0 ImHd Hu Hd χ Hu = Hd = . (13) 1 1 G2 = O ImH10 0 , 0 (15) 0 G0 = ⎛(vd0,v⎞u, 0) ⎛ u⎞ " vu + Hu # " vd + Hd # v2 + v2 χ b u d⎜ ⎟ ⎜ ⎟ ! ⎝ ⎠ ⎝ ⎠ This potential1 is characterizedg (q by twoq )v v null2 g eigenvalues(q q )v2v corresponding to two neutral Goldstone modes 2 B d u d u B d u d u √ 2 2 G0 = 1 2 − , − , 2M vu + vd g2((Gq 0,Gq )02)andaneigenvaluecorrespondingtoamassivestatewithanav2v2 +2M 2 v2 + v2 ⎛− v2 + v2 v2 + v2 ⎞ xion component (χ). In the B d − u d u d u u d d u ! 0 0 ⎝ ⎠ ! (ImHd , Im1Hu,b)CP-oddbasiswegetthefollowingnormalizedeigenstates$ % ! ! χ = 10 √2Mvu, √2Mvd,gB (qd qu)vdvu (14) g2 (q q )2v2v2 +2M 2(v2 + v2) − − B d − u u d 1 d u * + ! G1 = (v ,v , 0) and we indicate0 with Oχ the orthogonald matrixu which allows to rotate them on the physical basis 2 2 vu + vd 1 0 ! G0 ImHd 2 χ 0 2 2 G0 =1 O ImHu , g (q q )v v (15)g (q q )v v 2 ⎛ ⎞ ⎛ ⎞ B d u d u B d u d u √ 2 2 G0 = χ b − , − , 2M vu + vd g2 (q q⎜)2v⎟2v2 +2⎜M 2 ⎟v2 + v2 ⎛− v2 + v2 v2 + v2 ⎞ B d − ⎝u ⎠d u ⎝ ⎠d u u d d u ! ⎝ ⎠ ! 1 $ % ! ! χ = √2Mvu, √2Mvd,gB (qd qu)vdvu (14) g2 (q q )2v2v2 +2M 2(v2 + v2) − − B d − u u d 10 d u * + ! and we indicate with Oχ the orthogonal matrix which allows to rotate them on the physical basis

1 0 G0 ImHd 2 χ 0 ⎛G0⎞ = O ⎛ImHu⎞ , (15) χ b ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

10 which is given by which is given by vd vu 0 vd v vu v 2 2 0 which is given by v v gB(q qu)v vu 2 gB(qd qu)vdvu 2 d d √2Mv χ gB(qd qu)vdv − gB(qd qu)v vu − √2Mv χ whichO is given= ⎛ by− 2u 2 2 2 2 −2 d 2 2 2 2 2 2 2 2 2 2 2 2 ⎞ (16) Ovd = ⎛ 2 −2v√2 g2vBu (qd 2 q2u) vdvu2+2M v 2 2 2v√gB2(q2d qu) 2vdvu+2M2 2 v2 √2 2gB⎞(qd qu(16)) vdvu+2M v − v√gB(qd qu) vdvu+2M−v v√gB (qd qu) 0vdvu+2M v − √gB (qd qu) vdvu+2M v − v − vvd √ − vu √2Mv − gB(qd qu)vdvu 2 √2Mvu 2 2Mvu √2Mvd gBd0(qd qu)vdvu gB(qd qu)vd⎜vu ⎜ gB(qd quv)vdvu √2vMv ⎟ − ⎟ χ − 2 22− 2 2 2 2 2 2 2 22 2 2 2 2 − 2 22 2 2 2 2 2 2 O = ⎛ 2 2 2 2 2 2 ⎜2 2 2 ggB(q(d2qd2qu2)2vqd2uvu) 2v 2 v +2M2 gBv(qd 2q2u)22vd2vu g22(2q2⎞d qu) (16)v√2vMv+22M2 v 2 2g (qd qu) v v +2M v ⎟ v√g (qd qu) v v⎜u+2M√vgχB(qdv√qug ) (qv√duvqdBu+2) −vMvuv+2M vu−d√√gBg(q(dqd qquu−)) vvu−vud+2√MMBvv √gB (qud dqu) vuvd+2M√v B⎟ u d − B d O = ⎛− B 2 d2 −2 2 2 2 B2 − 2 2d 2 2 2 −2 2−2 2 2 2 ⎞ (16) − − ⎜ ⎜v√g −(qd qu) v vu+2M v v√g (qd− qu) v vu+2M v √g (qd qu) v vu+2M v ⎟ ⎟ √2Mvu − B √−2Mvdd B gB−(qd qud)vdvu B − d ⎜ ⎝ ⎝ √2Mv √−2Mv ⎟ gB(qd qu)vdvu ⎠ ⎠ 2 2 2 2 2 2 ⎜ 2 2 2u 2 2 2 2 2 2d 2 2 2 − ⎟ ⎜ √gB(qd qu) vuvd+2M v −√gB2(qd qu) 2vu2 v2d+2M2 v2 √gB2(qd qu) 2vu2 v2d+2M2 v2 ⎟ 2 2 2 2 2 2 − 2 2 ⎜ √gB(qd− qu) vuvd+2M v √gB (qd− qu) vuvd+2M v √gB (qd qu) vuvd+2M v ⎟ ⎜ where v = v + v . ⎜ 2 − 2 − − ⎟ − ⎟ ⎝ whereu v =d ⎝ vu + vd. ⎠ ⎠ where v = v2 +χv2inherits. WZ'where interactionv = v2 + v since2. b can be related to the physical axion χ and to the Goldstone modes u d χ inherits WZ'u interactiond since b can be related to the physical axion χ and to the Goldstone modes χ inherits WZ' interactionvia this matrixsinceχbinheritscan be WZ' related interaction to the since physicalb can axion be relatedχ and to the to thephysical Goldstone axion χ modesand to the Goldstone modes via this matrix via this matrix via this matrix b = Oχ G1 + Oχ G2 + Oχ χ, (17) b = 13Oχ0G1 + O23χ G02 + Oχ 33χ, (17) b = Oχ G1 + Oχ G2 + Oχ χ, 13 Stuckelberg0 23 0 χ axion33 1 χ(17)2 χ 13 0 23 0 33 b = O13G0 + O23G0 + O33χ, (17) or, conversely,or, conversely, or, conversely, Physical axi-Higgs (gauged axion) or, conversely, χ χ χ χ =χ O31ImHd + Oχ32ImHu + O33b.χ (18) χ χ = O χImHd + O ImHu + O b. (18) χ = O ImHd + O ImHu + 31O b. 32 33 (18) 31 32 33 χ Notice that the rotation of b into the physical axion χ involvesχ a factor O33 whichχ is of order v/Mχ .This χ = O ImHd + O χImHu + O b. (18) Notice that thecarries rotation as a consequence of b into that theχ physicalinherits from axionbχan interactionχ involves31 with a factor the gaugeO32 fieldswhich which is is of suppressed33 order v/M.This Notice that the rotation of b into the physical axion χ involves a factor O33 which is of order v/M.This33 by a scale M 2/v.Thisscaleistheproductoftwocontributions:a1/M suppression coming from the carries as a consequencecarries that as aχ consequenceinherits from thatb an interactionχ inherits with from theb an gauge interaction fields which with is suppressed the gauge fields which is suppressedχ Noticeoriginal that Wess-Zumino the rotation counterterm of b ofinto the Lagrangian the physical (b/MF axionF˜)andafactorχ involvesv/M obtained a factor byO the which is of order v/M.This by a scale M 2/v.Thisscaleistheproductoftwocontributions:a1by a scale M 2/v.Thisscaleistheproductoftwocontributions:a1/M suppression coming from/M thesuppression coming33 from the projection of b into χ due to O . original Wess-Zumino countertermcarries of as the a consequence Lagrangian (b/MFχ thatF˜)andafactorχ inherits fromv/M obtainedb an˜ interaction by the with the gauge fields which is suppressed original Wess-ZuminoMore details counterterm on the structure of of the the Lagrangian various operators (b/MF appeaFring)andafactor in this model havev/M beenobtained included by the projection of b into χ due to Oχ. 2 projectionby ofin ab an scaleinto appendixχMdue in/v order to.Thisscaleistheproductoftwocontributions:a1O toχ. make our treatment self-contained.Wehaveincludedalsoabriefdiscussion/M suppression coming from the More details on the structure of the various operators appearing in this model have been included Moreoriginal detailsof the on construction Wess-Zumino the structure of gχγγ of,whichisthefactorinfrontofoneofthemostimportantcoun counterterm the various operators of the appea Lagrangianring in this (b/MF modelF˜ have)andafactorterterms been includedv/M obtained by the in an appendix in order to makeneeded our treatment in our numerical self-contained analysis.Wehaveincludedalsoabriefdiscussion and which controls the decayoftheaxionintophotons.Webriefly in an appendix in order to make our treatment self-contained.Wehaveincludedalsoabriefdiscussion of the construction of gχγγ,whichisthefactorinfrontofoneofthemostimportantcounprojectioncomment on of itsb structure.into χ due to Oχ. terterms needed in our numericalof the constructionanalysisMore andThe final which details of couplinggχγγ controls,whichisthefactorinfrontofoneofthemostimportantcoun on appears the the deca as structure a coeyoftheaxionintophotons.Webrieflyfficient of in the the interaction various ofthephysicalaxionwithtwophotons operators appearing in thisterterms model have been included comment on its structure.needed in our numerical analysis and which controls the˜ decayoftheaxionintophotons.Webriefly in an appendix in order to make ourgχγγ treatmentχFγ Fγ self-contained.Wehaveincludedalsoabriefdiscussion(19) The final couplingcomment appears on as its a coe structure.fficient in the interaction ofthephysicalaxionwithtwophotons of the construction of g ,whichisthefactorinfrontofoneofthemostimportantcounterterms The finaland coupling is given byappears as a coeχγγfficient in the interaction ofthephysicalaxionwithtwophotons gχγγχFγ F˜γ (19) needed in our numericalχ analysisA A and whichA controlsA χ the decayoftheaxionintophotons.Webriefly gγγ = FOW3γOW3γ + CYYOY γ OY γ O33. (20) g χF F˜ (19) and is given by comment on its structure.( χγγ γ γ ) It is defined by a combination of matrix elements of the rotation matrices OA and Oχ,togetherwith A χ someThe counterterms finalA A couplingF and C appearsYYA . OA is as theχ a matrix coeffi thatcient rotates in the the neutral interaction gauge bosons ofthephysicalaxionwithtwophotons from the and is givengγγ = by FOW γOW γ + CYYOY γ OY γ O . (20) interaction3 to the3 mass eigenstates after electroweak33 symmetry breaking and has elements which are ( ) It is defined by a combinationO of(1), matrix being elements expressedχ of in the terms rotati ofA ratioson matricesA of couplingOA constantandA Oχs.A,togetherwith Theyχ correspond˜ to mixing angles. gγγ = FOW3γOW3γ + CYYOY γ OgYχγγγ Oχ33Fγ. Fγ (20) (19) The coeA fficients F and CYY are the WZ counterterms for cancelling the anomalies emerging from the some counterterms F and CYY. O is2 the matrix that2 rotates the neutral gauge bosons from the SU(2)U(1)B and U(1)B U(1)(Y sectors and can be found in the appendix.) They areA both suppreχ ssed interaction to theIt mass is defined eigenstatesand by is a after given combination electroweak by of symm matrixetry elements breaking and of the has rotati elementson matrices which areO and O ,togetherwith O(1), being expressed in terms of ratios of coupling constantA s. They correspond to mixing angles. some counterterms F and CYY. O is the matrix11 that rotates the neutral gauge bosons from the The coefficients Finteractionand CYY are to the theWZ masscounterterms eigenstates for after cancelling electroweakχ the anomalies symmA emergingetryA breaking from th andeA hasA elementsχ which are 2 2 gγγ = FOW3γOW3γ + CYYOY γ OY γ O33. (20) SU(2)U(1)B andOU(1),(1)B U being(1)Y sectors expressed and incan terms be found of ratios in the appendix. of coupling They constant are boths. suppre Theyssed correspond to mixing angles. ( ) A χ The coeffiItcients is definedF and C byYY aare combination the WZ counterterms of matrix for elements cancelling of the the anomalies rotation emerging matrices fromO thande O ,togetherwith 2 112 SU(2)U(1) and U(1)B U(1) sectors and can beA found in the appendix. They are both suppressed someB countertermsY F and CYY. O is the matrix that rotates the neutral gauge bosons from the interaction to the mass eigenstates after electroweak symmetry breaking and has elements which are 11 O(1), being expressed in terms of ratios of coupling constants. They correspond to mixing angles.

The coeﬃcients F and CYY are the WZ counterterms for cancelling the anomalies emerging from the 2 2 SU(2)U(1)B and U(1)B U(1)Y sectors and can be found in the appendix. They are both suppressed

11 Thursday, May 29, 14

Replaces f_a of Peccei Quinn

Thursday, May 29, 14 The$PQ$axion$remains$a$massless$Goldstone$mode$un4l$the$QCD$phase$transi4on,$$ a9er$which$it$develops$a$small$mass.$The$size$of$the$periodic$poten4al$$ is$important$in$order$to$ﬁx$the$size$of$the$mass.$$$$ PQ axion. Vacuum misalignment at the QCD phase transition

If#an#axion#has#charges#both## under#SU(3)#and#SU(2)## we#could#consider#the## possibility#of#sequen>al## misalignments.#The#dominant## misalignment#clearly#comes## from#the#largest#poten>al###

Thursday, May 29, 14 In the neutral sector both a CP-even and a CP-odd subsectors are present. The CP-even sector is described by which can be diagonalized by an appropriate rotation matrix in terms of CP-even N2 mass eigenstates (h0,H0)as

ReH 0 sin α cos α h0 u = − , (15) ReH 0 cos α sin α H0 ! d " # $ ! " with (1, 1) (2, 2) √∆ tan α = N2 − N2 − (16) 2 (1, 2) N2 and + ∆ =( (1, 1))2 2 (2, 2) (1, 1) + 4 ( (1, 2))2 +( (2, 2))2 . (17) N2 − N2 N2 N2 N2 The deﬁnition of these matrix elements( a ) is left to an appendix( b .Theeigenvaluescorrespondingtothe) physical neutral Higgs ﬁelds are given by Figure 2: Contributions to the χ γγ decay. → 2 1 √ mh0 = 2(1, 1) + 2(2, 2) ∆ gB = 0.1, M1 = 1 TeV,2 MLSOMN charge assignment =N f(-1,-1,4) − gB = 0.1, M1 = 1 TeV, MLSOM charge assignment = f(-1,-1,4) gB = 0.1, M1 = 1 TeV, MLSOM charge assignment = f(-1,-1,4) gB = 0.1, M1 = 1 TeV, MLSOM charge assignment = f(-1,-1,4) 1e-20 1 %g = 0.1, M = 1 TeV, MLSOM charge assignment = f(-1,-1,4) & 1e-30 g = 0.1, M = 1 TeV, MLSOM charge assignment = f(-1,-1,4) 1e-20 2 B 1 mχ = 0.001 eV 1e-30 B 1 gauged axion m 0 = (1, m1)χ = 0.001 + eV (2, 2) + √∆ . gauged(18) axion H 2 mχ = 0.012 eV PQ axion the contribution from 1e-25 the Wess-Zumino 1e-20 interactionsmχ = 0.01 eV from those which are obtained 1e-30 from the loopPQ axion 1e-25 2 N mχ N = 0.1 eV mχ = 0.001 eV gauged axion mχ = 0.1 eV m = 0.001 eV mχ = 0.01 eV PQ axion 1e-30 1e-30 1e-25 % ma = 0.001a eV 1e-40& 1e-40 corrections. We obtain m = 0.01 eV mχ = 0.1 eV ma = 0.01a eV We refer to [36] for a more detailed discussion of them scalar = 0.1 eV ma s =ector 0.001 eV of the model 1e-40 with more than one 1e-35 1e-35 1e-30 ma = 0.1a eV 2 ma = 0.01 eV 1e-35 ⃗ m⃗a = 0.1 eV 1e-50 1e-50 extra U(1). 1e-40 1e-40 spin PQ dk1 dk2 4 (4) ΓPQ(a γγ)= 1e-40 |M | (2π) δ (k 1e-50k1 k2), (35)

1e-45 3 0 3 0 [GeV] χ

1e-45 [GeV] χ [GeV] χ → 2m (2π) k (2π) k Γ − −

[GeV] χ a Γ Γ 1 2 1e-60 [GeV] χ Γ 1e-45! 1e-60

1e-50 [GeV] χ 1e-50 Γ The CP-odd sector Γ 1e-60 • 1e-55 1e-50 where the squared amplitude 1e-55 is given by 1e-55 1e-70 1e-60 1e-70 1e-60 1e-70 The symmetric matrix describing 1e-60 the mixing of the CP-odd Higgs sector with the axion field b is 1e-65 2 2 1e-65PQ = 1e-65 point like + loop χ given by 3.Afterthediagonalizationwecanconstructtheorthogonal− 1e-05 matrix0.0001 0.001O that 0.01 rotates 0.1 1 |M | 5 10 |M 15 20 25 M 30 | 35 40 1e-05 1e-05 0.0001 0.0001 0.001 0.001 0.01 0.01 0.1 0.1 1 1 N spin spin mχ [eV] 5 10 15 5 tan 1020β 15 25 20 30 25 35 30 40 35 40 the Stückelberg" field and the" CP-odd phases of the two Higgs doublets into the mass eigenstatesmχ [eV]mχ [eV] tanβ tanβ 2 0 0 a 2 2 2 (χ,G1 ,G2 ) cγγ e 4 1 τf f(τf ) 2 2 mf =8 a 2 ma + Nc(f)i 2 e Qf gf +interf., Figure 3:F Total decay32π rate of the2 axi-Higgs% for several4π mf mass values.vPQ Here,% for the PQ axion, we have chosen # Figure$ # 3: Total$ decay0 rate% f of the axi-Higgs for several# mass$ value% s. Here, for the PQ axion, we have chosen Figure 3:10 Total decay rateImH ofu the%" axi-Higgsχ for several mass values.% Here, for the PQ axion, we have chosen fa =10 GeV. 10 % % 10 fa =10 GeV. 0 % χ 0 % (36) fa =10 GeV. ⎛ ImHd ⎞% = O ⎛ G1 ⎞ . % (19) b G0 where, inIn the Fig. second 2a we haveterm, isolatedNc⎜(f)isthecolorfactor,andthefunction the massless⎟ contribution⎜ 2 ⎟ to the decayτf ratef(τf coming)isafunctionofthe from the WZ counterterm In Fig. 2a we⎝ have isolated⎠ the⎝ massless⎠ contribution to the decay rate coming from the WZ counterterm mass of the fermions˜ circulating in the loop. We have introduced the function f(τ), defined in any The mass matrixInχ Fig.Fγ Fγ of 2awhose this we sectorhave expression˜ isolated exhibits is the two massless zero eigenvalues contributioncorresponding to the d toecay the rate Goldstone coming modes from the WZ counterterm kinematic domain, whoseχFγ realFγ whose part is expression given by is G0,G0 and a mass˜ eigenvalue, that corresponds to the physical axionfieldχ,withavalue 1 2 χFγ Fγ whose expression is 3 mχ χ 2 3 2 mχ B B(arcsin 1/Γ√WZτ2 )(χ γγ)= if τ(gγγB1) . B 2 χ2 22 (40) 1 q q v sin 2β →1ΓWZ(χ4π γγ(≥q)=3 q )(gγγv )v . (40) 2 2 Re[f(τ)]u = d 1 2 1+√1 τ 22 → muχ 4dπ u d (37) mχ = cχ v 1+ − log =− cχπv 1+if τ < 1 − χ 2 , (20) −2 M& 4 2 1Γ√WZ1 τ−(χ2 γγ)= M(gγγ2 ) . v2 (40) Combining- also# in this1 − case the$ tree−. level− decay−→ / with the4π 1-lo1 op amplitude,0 we obtainG. forLazaridesχ γγ , A.Mariano, C.C. Combining also' in this( case the) tree* level decay with the 1-loop amplitude, we obtain→ for χ γγ while its imaginarythe amplitude part is → with the coefficientCombiningthe amplitude also in this case the tree level decay with the 1-loop amplitude, we obtain for χ γγ → µν µν the amplitude 0ifµν b1 2 τ 1 µν µν c =4 4λ + λ cot β + (χ γγ)=µ+ν λ tanWZβ+ . f , (21) (41) χ Im[f(τ)]1 = 3 M 2 √→ (χM2 ≥γγ)=M WZ + f , (38) (41) π log v1+ sin1 2Mτβ if τ→< 1 M M # & 2 1 √1−τ $ µ−ν − µν µν shown in Fig.shown 2. In in this Fig. case 2. In the this' rates case( are the(χ derived rates)*γγ are)= from derived theWZ exp from+ression thef , expression (41) 2 2 M → M M where τ =4mf /mχ.Inourcasewetakethebranch9 τ > 1.

As weshown move to in compute Fig. 2. the In decaythis case of χ theand rates assume are a derivedfree varying from mass the for exp thisression particle, the2 WZ 3 2 interaction (Fig. 2a) is given by mχ 3 1 τf f(τf ) χ m2χ χ 2 1 2 τf2 fχ(,fτf ) 2 2 χ,f Γχ Γ(χ γγ)= 8(gγγ) + Nc(f)i e Qf c Γχ Γ(χ γγ)= 8(gγγ) + 2Nc(f)i 2 e Qf c ≡ → ≡ →32π ⎧ 32π ⎧2 $ 2 $4π mf 4π m$f $ µν χ ⎨ $ f $ f $ 2 $ WZ(χ γγ)=4gγγε3[µ, ν,k1,k2⎨]. $% $% $ (39) $ M → mχ χ 2 $ 1 $ τf f(τf ) $2 2 χ,f $ Γχ Γ(χ γγ)= 8(gγγ) $+ τ fN($τc()f)i e$ Q c $ ⎩ χ ⎩ $ χ f f 2 τf2 fχ2(,fτf ) 2 $ 2 fχ,f ≡ → 32π+4⎧ g N+4c(f2g)$i N$ (fe)iQ4cπ mfe Q c $ $ (42) (42) γγ γγ$ 4fπ2mc f 2 ⎫ f $ ⎨ f $% f 4π mf ⎫ $ 16 % $ %f ⎬ ⎬ $ ⎩ $ τf f(τf ) $ χ $ χ 2 χ2⎭ 22 χ,f $ χ χ and are shownand in are Fig.3. shown In in the Fig.3. equation In the above equation both+4 above thegγγ direct bothN (c the(f()g directiγγ) )andtheinterference(2 ( (geγγQ) f)andtheinterference(c ⎭ gγγ) (42)gγγ) ∼ 4π m∼f ⎫ ∼ ∼ contributionscontributions are suppressed are suppressed as inverse powers as inverse of the powers Stüc%f kelberg of the Stüc mass.kelberg We show mass. the We results⎬ show the of this results of this comparativecomparative study in Fig. study 3, where in Fig. in 3, the where left panel in the we left presen paneltresultsforthedecayratesof we presentresultsforthedecayratesofχ 2 χ γγ χ χγγ and are shown in Fig.3. In the equation above both the direct ( (gγγ) )andtheinterference(⎭ → →gγγ) for severalfor values several of the values axion of mass the axion as a function mass as ofa function tan β = ofvu/v tand.Theplotsindicateaverymildβ∼= vu/vd.Theplotsindicateaverymild∼ contributions are suppressed as inverse powers of the Stückelberg mass. We show the results of this dependencedependence of the rates of on the this rates parameter, on this parameter, even for rather even la forrge rather variations. large variations. In the same In plot the the same rates plot the rates comparative study in Fig. 3, where in the left panel we presentresultsforthedecayratesofχ γγ for the PQfor case the are PQ shown case as are constant shown as lines, constant just for lines, comparis just foron. comparis Noticeon. that Notice we have that chosen we have a rather chosen a→ rather for several values of the axion mass as a function of tan β = vu/vd.Theplotsindicateaverymild dependence of the rates on this parameter, even for rather large variations. In the same plot the rates for the PQ case are shown as constant lines, just17 for comparis17 on. Notice that we have chosen a rather

17 2 2 where τ =4mf /mχ. xFinally,the1-loopdecayχ γγ is given by the following amplitudes → µν (χ γγ)= µν + µν (100) M → MWZ Mf and the rate computed from the two contributions shown in Fig.1is

2 m3 1 τ f(τ ) Γ(χ γγ)= χ 8(gχ )2 + N (f)i f f e2Q2 cχ,f → 32π ⎧ γγ 2 $ c 4π2m f $ $ f f $ ⎨ $% $ $ $ $ $ ⎩ χ $ τf f(τf ) 2 2 χ,f $ +4gγγ Nc(f)i 2 e Qf c . (101) 4π mf ⎫ %f ⎬ In Fig. 1a we have isolated the massless contribution to the decay rate coming from⎭ the WZ counterterm

χFγ Fγ whose expression is

m3 Γ (χ γγ)= χ (gχ )2. (102) WZ → 4π γγ Since themass is an independent parameter, you can also Consider the axi-Higgs tobe in the GeV range. +

( a ) ( b ) tanβ = 40, gB = 0.2, M1 = 1 TeV, f(-1,-1,4) tanβ = 40, gB = 0.6, M1 = 1 TeV, f(-1,-1,4)

tanβ = 40, gB = 0.2, M1 = 1 TeV, f(-1,-1,4) tanβ = 40, gB = 0.6, M1 = 1 TeV, f(-1,-1,4) 1 1 Figure 1: Massless plus massive contributions to the bbχ γγ process. bb 1 tanβ = 40, gB = 0.2, M1 = 1 TeV, 1 f(-1,-1,4) tanβ = 40, gB = 0.6,0.1 M1 = 1 TeV, f(-1,-1,4) → 0.1 bb bb ττ ττ 1 1 0.1 0.1 bb 0.01 γγ bb gg 0.01 γγ gg ττ ττ

0.1 0.1 ) ) 0.01 0.01 χ 0.001 ss χ 0.001 ss γγ gg ττ γγ We should noticegg that the massive contribution fromττ amplitude (92) is completely independent BR( BR( ) 0.01 γγ ) gg 0.01 γγ gg µµ µµ χ χ 1e-04 1e-04 0.001 ss 0.001 ss χ,f ) ) BR( BR( χ 0.001 ss of the anomalousχ coupling 0.001 gB,whichdoesnotappearinthecoess ﬃcients c ,ascanbeseenfrom 1e-04 µµ 1e-04 µµ 1e-05 1e-05 BR( 1e-04 Eq. (88).µµ For theBR( decay 1e-04 into two gluons we proceed in a similarµµ manner (see Fig. 8) and the amplitude 1e-05 1e-05 1e-06 cc Z γ 1e-06 cc Z γ 1e-05 1e-05 1e-06 cc Z γ 1e-06 is given bycc Z γ 1e-07 1e-07 1 10 100 1 10 100 1e-06 cc Z γ 1e-06 cc Z γ 1e-07 1e-07 mχ [GeV] mχ [GeV] 1 10 1e-07 100 1 10 1e-07 100 µν χ 1 10 100 1 (gg 10 χ)=4gggε[µ, ν,k 1001,k2], (103) mχ [GeV] mχ [GeV] WZ M → tanβ = 10, gB = 0.2, M1 = 1 TeV, f(-1,-1,4) tanβ = 10, gB = 0.6, M1 = 1 TeV, f(-1,-1,4) mχ [GeV] mχ [GeV] tanβ = 10, g = 0.2, M = 1 TeV, f(-1,-1,4) tanβ = 10, g = 0.6, M = 1 TeV, f(-1,-1,4)χ 1 1 B 1 B 1 bb bb tanβ = 10, gB = 0.2, M1 = 1 TeV, f(-1,-1,4)where the coeﬃcient g gg is giventanβ in = 10, Eq. gB = 0.6, (192). M1 = 1 TeV, The f(-1,-1,4) second amplitude (Fig. 8b) is a pure massive 1 1 0.1 0.1 1 bb bb 1 ττ ττ bb contribution bb 0.1 0.1 0.01 γγ gg 0.01 γγ gg 0.1 ττ ττ0.1

ττ ) ττ ) 0.01 γγ gg 0.01 γγ µν gg χ 2 0.001 a b χ,q ss χ 0.001 ss 0.01 γγ gg q (gg 0.01χ)= γγ iC0(mχ,mq)Tr[Tgg T ]cgg ε[µ, ν,k1,k2],q= u, d (104)

BR( cc BR( cc ) ) M → µµ { } µµ χ χ 0.001 ) ss 0.001 ) ss q 1e-04 1e-04 χ 0.001 ss χ 0.001 ss

BR( cc BR( cc % BR( µµ cc BR( µµ cc 1e-05 1e-05 1e-04 1e-04 1e-04 µµ 1e-04 µµ

1e-05 1e-05 1e-05 1e-05 1e-06 Z γ 1e-06 Z γ

1e-06 1e-06 Z γ 1e-06 Z γ 1e-06 Z γ 1e-07 Z γ 1e-07 1 10 100 1 10 100

1e-07 1e-07 1e-07 1e-07 mχ [GeV] mχ [GeV] 1 10 1 100 10 1 100 10 1 100 10 28 100

mχ [GeV] mχ [GeV] mχ [GeV] mχ [GeV] Figure 2: Study of the branching ratios of the axi-Higgs. We analyze thedependenceonthefreeparameters

Figure 2: Study of theFigure branching 2: ratiosStudy of of the the axi-Higgs. branching We analyze ratios of thedependenceonthefreeparameters the axi-Higgs. We analyze thedependenceonthefreeparametersgB, tan β. g , tan β. gB, tan β. B with u = u, c, t and d = d, s, b ,andthecoefficients cχ,q are defined in relations (88). The decay Axions from Intersecting Branes and Decoupled Chiral Fermiχ,qons { } { } with u = u, c, t andwith d = ud =, s,ub, ,andthecoec, t and dffi =cientsd, scχ,,qb are,andthecoe defined in relationsfficients (88).c Theare decay definedrate in is relations then given (88). by The decay { } { { } at the} Large Hadron{ } Collider rate is then given byrate is then given by M. Guzzi, C.C. 3 2 2 mχ χ 2 1 Ncτf f(τf ) χ,q 3 2 Γ(χ gg)= 8(g ) + i 4πα c 3 mχ 1 Ncτf f(τf ) gg 2 s m 1 N τ f(τ ) χ 2 χ,q → 16π ⎡ 2 4π mf χ Γ(χχ 2 gg)=c f f 8(ggg)χ,q+ i 2 4παsc # q # Γ(χ gg)= 8(ggg) →+ i 16π ⎡ 4παsc 2 4π mf #$ # → 16π ⎡ 2 4π2m # q # # # # q f # #$ # ⎣ Claudio Corianòand# Marco Guzzi # # # #Ncτf f(τf ) # $ ⎣ # # +4gχ #i 4πα cχ,q , # (105) ⎣ # χ # Ncτf f(τf ) χ,q gg 2 s #N τ f(τ ) +4g # #i 4πα c , # (105) 4π mf χ # c f f χ,q gg# 4π2m s q % +4ggg i 2 4παsc , q f (105) % $ 4π mf $ q % while the expression of the isolated contribution from the corresponding WZ counterterm is instead while theDipartimento expression ofdi Fisica, the$ isolated Universitàdel contribution Salento from the corresponding WZ counterterm is instead while the expression of the isolated contribution from the corresponding WZ counterterm is instead given by givenand by INFN Sezione di Lecce, Via Arnesano 73100 Lecce, Italy given by m3 m3 χ χ 2 χ χ 2 ΓWZ(χ gg)= (ggg) . (106) m3 ΓWZ(χ gg)= (ggg) . (106) → 2π Γ (χ gg)= χ (gχ )2. → 2π (106) WZ → 2π gg The decay χ γ The decay χ γ • → Z • → Z The decay χ γ Abstract • → Z 29 We present a study of a class of effective actions which show typical axion-like29 interactions, and of their possible effects at the Large Hadron29 Collider. One important feature of these models is the presence of one pseudoscalar which is a generalization of thePeccei-Quinnaxion.Thiscanbevery light and very weakly coupled, with a mass which is unrelated to its couplings to the gauge fields, described by Wess Zumino interactions. We discuss two independent realizations of these models, one derived from the theory of intersecting branes and the second one obtained by decoupling one chiral fermion per generation (one right-handed neutrino) from an anomaly-free mother theory. The key features of this second realization are illustrated using a simple example. Charge assignments of intersecting branes can be easily reproduced by the chiraldecouplingapproach,whichremains more general at the level of the solution of its anomaly equations. Using considerations based on its lifetime, we show that in brane models the axion can be darkmatteronlyifitsmassisultralight ( 10−4 eV), while in the case of fermion decoupling it can reach the GeV region, due to the ∼ arXiv:0905.4462v1 [hep-ph] 27 May 2009 absence of fermion couplings between the heavy Higgs and the light fermion spectrum. For a GeV axion derived from brane models we present a detailed discussion of its production rates at the LHC.

1 new heavier gauge boson modifies the invariant mass distribution also on the Z peak due to the small new heavier gauge boson modifies the invariant mass distribution also on the Z peak due to the small modifications induced on the couplings and to the Z Z′ interference. In the anomalous model that we modifications− induced on the couplings and to the Z Z′ interference. In the anomalous model that we − have investigated, though based on a specific chargehave assignm investigated,ent, though we find based larger on a rates specific for charge these assignm distributionsent, we find larger rates for these distributions ′ both on the peak of the Z and of the Z comparedboth to on the the other peak of models the Z and investigated, of the Z′ compared if the toextra the other resonance models investigated, if the extra resonance ′ is around 1 TeV. This correlation is expected to dropis around as the 1 TeV. ma Thisss of correlation the extra is expectedZ′ increases. to drop In as our the ma case,ss of as the extra Z increases. In our case, as we will specify below, the mass of the extra resonance is givenbytheStückelberg(M1)mass,whichappears we will specify below, the mass of the extra resonance is givenbytheStückelberg(M1)mass,whichappears also (as a suppression scale) in the interaction of the physical axion to the gluons and is essentially a free also (as a suppression scale) in the interaction of the physical axionˆ toj theˆ j gluons and is essentially a free ig2 µ Z′,jparameter.ig2 1 2 L,j 2 YL YR gz zˆL,j zˆR,j µ − γ gV = − εcwT3 + εsw( + )+ cw( + ) γ new heavier gauge boson modifies theparameter. invariant mass distribution also on4 thecw Z peakIncw due DY,2 !− the to investigation the small2 of2 the NNLOg2 2 hard scatterings2 " goes back to [38], with a complete computation of ′ j j ig ′ ig 1 Yˆ Yˆ gz zˆR,j zˆL,j modifications induced on the couplingsIn and DY, to the the investigationZ Z interference. of the NNLO In− the hard2 γµ anomalousγ5 scatteringsg Zthe,j = invariant− 2 model goesεc mass2 T L,j ba that+ distributions,ckεs2 to( weR [38],L )+ with madec abefore( complete that) theγ computationµγ NNLO5, corrections of to the DGLAP evolution had been − 4c A c 2 w 3 w 2 2 g w 2 2 w fully completed.w ! In our analysis− we will2 compare− three" anomaly-free models against a model of intersecting have investigated, though based on athe specific invariant charge mass assignm distributions,ent, we find made larger before rates that for the these NNLO distributionsnewcorrections heavier gauge to the boson DGLAP modifies evolution the invariant had(8) been mass distribution also on the Z peak due to the small ′ brane with a single anomalous U(1). The anomaly-free charge assignments come from a gauged B L both on the peak of the Z and of thefullyZ compared completed. to In the our other analysis models we will investigated, compare three if the anoma extramodificationsly-free resonance models induced against on the a model couplings of intersectingand to the Z Z′ interference. In the anomalous− model that we where j is an index which represents the quark or the lepton and we havesetsinθW = sw, cos θW = cw for ′ abelian symmetry, a “q + u”model-bothdescribedin[39]-andthefreefermionicmodel− analyzed in [40]. brevity. is around 1 TeV. This correlation isbrane expected with to a drop single as theanomalous mass ofU the(1). extra TheZ anomaly-freeincreases.We start Inby charge summarizing ourhave case, investigated, assignments as our definitions though come and based from conventions. on a a gauged specific chargeB L assignment, we find larger rates for these distributions ′ − both on the peak of the Z and of the Z compared to the other models investigated, if the extra resonance we will specify below, the mass of theabelian extra resonance symmetry, is a give “q nbytheStückelberg(+ u”model-bothdescribedin[39]-andthefreefermionicmodelM1)mass,whichappearsIn′ the anomaly-free case we address abelian extensionsanalyzed of in th [40].egaugestructureoftheformSU(3) 2.1 An anomalous extra Z ′ 3 µ µ × SU(2) U(1)isY aroundU(1)z,withacovariantderivativeinthe 1 TeV. This correlation is expectedW to,B drop,Bz as(interaction) the mass of basis the defined extra Z as increases. In our case, as also (as a suppression scale) in the interactionWe start by of summarizing the physical our axionIn definitions presence to the of gluonsanomalous and conventions. and interactions is essentially× we can× use the a free same formalism developed so far for anomaly-freeµ Y we will specify below, the mass of the extra resonance is givenbytheStückelberg(M )mass,whichappears models with some appropriate changes. Since the effective lagrangean of the class of the anomalous models 1 parameter. In the anomaly-free case we address abelian extensions of thˆ egaugestructureoftheform1 1 2 2 3 3 SU(3)gY ˆ µ gz µ that we are investigating includes both a Stückelbergalso (asDµ a= and suppression∂ aµtwo-Higgsig2 W doubletµ scale)T + W insector,µ theT the+ interactionW massesµ T of thei of theYBY physii calzBˆ axionz to the gluons and(1) is essentially a free 3 µ µ − − 2 × − 2 In DY, the investigation of the NNLOSU(2) hardU scatterings(1)Y U(1) goesz,withacovariantderivativeinthe back to [38], with a complete computationW ,B ,B ofz (interaction) basis defined as × × neutral gauge bosons are provided by a combinationparameter.µ Y of these! two mechanisms." In this case we# take as $ free parameters the Stüeckelbergwhere mass weM denoteand the with anomalousg2,gY ,g couplingz the couplings constant g of,withtanSU(2), Uβ (1)as inY theand U(1)z,withtanθW = gY /g2.Afterthe the invariant mass distributions, made before that the NNLO corrections to the DGLAP evolution1 hadIn DY, been the investigation of theB NNLO hard scatterings goes back to [38], with a complete computation of ˆ remaining anomaly-free1 1 models.2 diagonalization2 As we3 have3 already of thegY stressed massˆ µ,theanalysisdoesnotdependsignificantlyon matrixgz we haveµ Dµ = ∂µ ig2 Wµ T + Wµ T + Wµ T i YB i zBˆ z (1) fully completed. In our analysis we will compare three anomaly-freethe modelschoice of this against parameter. a modelThe value of of the intersectingthe Stückelberg invariant mY massass M distributions,is loosely constrained made by before the D-brane that the NNLO corrections to the DGLAP evolution had been − − 2 − 12 3 j j Aµ sin θW cos θW 0 W ! model in terms of suitable′ wrappings (n)ofthe4-braneswhichdefinethechargeembedding[41,14]reˆ ˆ $ ported µ brane with a single anomalous U(1). The anomaly-free charge assignments" ig2 µ comeZ ,j fromig2 1 a2 gaugedL,j#fully2 completed.BYL YLR gz In ourzˆL,j analysiszˆR,j µ we will compare three anomaly-free models against a model of intersecting where we denote with g ,gY ,gz the couplings− γ gV = of−SU(2),εcwUT3(1)+Yεswand( +U(1))+z,withtancw( + θW) γ= gY /g .Afterthe Y 2 in Tabs. 1,2,34cw and 4. cw 2 !− 2 − 2⎛ Zgµ2 ⎞ =2 ⎛ 2cos"θW sin2 θW ε ⎞ ⎛ Bµ ⎞ (2) brane with a single′ anomalous U(1).− The anomaly-free charge assignments come from a gauged B L abelian symmetry, a “q + u”model-bothdescribedin[39]-andthefreefermionicmodelThe mass-matrix in the neutral gaugeanalyzed sector is in givenj [40]. byj z ′ ˆ ˆ Z ε sin θW ε sin θW 1 B − diagonalization of the mass matrix weig have2 µ 5 Z ,j ig2 1 2 L,j 2 YR Y⎜L µgz ⎟ zˆR,j⎜ zˆL,j µ 5 ⎟ ⎜ µ ⎟ − γ γ gA = − εcwT3 abelian+ εsw( symmetry,)+ cw a( “q +−u”model-bothdescribedin[39]-andthefreefermionicmodel) γ γ , analyzed in [40]. We start by summarizing our definitions and conventions. 4cw cw 2 2 − 2⎝ Wg2 ⎠ 2 ⎝− 2 ⎠ ⎝ ⎠ ! 3 " −3 where ε is a perturbativeWe start by parameter2 summarizing which our is definitions around 10 andfor conventions. the models analyzed, introduced in [39] and mass =(W ,Y,B) M 3Y , (8) In the anomaly-free case we address abelian extensions of thAegaugestructureoftheformµ sin θW cosL θW 3SU0 (3) W⎛µ ⎞ 2 2 2 2 2 2 [40]. It is definedIn as the× anomaly-freeB caseg wev address1 abeliang xB extensions1 g ofxB thegaugestructureoftheform2 2 SU(3) 3whereµj is anµ index which represents the quark or the lepton and we⎜Y havesetsin⎟ θ = s , cos θ = c for SU(2) U(1) U(1) ,withacovariantderivativeintheW ,BZµ,B =(interaction)cos θW basissin definedθW asε B⎝µ ⎠ W w W w (2)+ 3 µ(NBBµ g v ), × Y z µ⎛ Y ⎞z ⎛ SU(2)⎞ ⎛U(1) ⎞U(1) ,withacovariantderivativeinthe2 2 4 W ,B ,B (interaction) basis defined as where B is the Stückelberg field and− the mass matrix is definedY as z 2 ′ 4 µ8 Y z × × brevity.′ × z × ≃ δ−MZZM1 M1 − Zµ Anomalousε sin θW ε sinextra θW2 2Z prime 1 2 Bµ ε = 2 2 (3) ⎜ g ⎟ ⎜ g g2 v g⎟2 gY⎜v g2⎟xB M ′ M 1 1 2 2 3 3 Y µ −z µ ′ 1 − − Z 1 1Z 2 2 3 3 gY µ gz µ ˆ ⎝ ˆ⎠ ⎝ 2 2 ⎠2 ⎝2 ⎠ˆ − ˆ Dµ = ∂µ ig2 Wµ T + Wµ T + Wµ T 2.1i AnYB anomalousY i zBˆ extraz ZM = g2 gY v gY v (1) gY xBDµ = ∂µ ig2 Wµ T + Wµ(9)T + Wµ T i YBY i zBˆ z (1) 4 ⎛ −3 2 ⎞ 1− ′ 2 2 2 − 2 − 2 2 − where ε is a perturbative− parameter2 − which2 is aroundwhile 10 the massfor of the the modelsZ boson2 analyzed,and′ of the extra introducedZ are 2 in2 [39] and2 2 2 2 2 2 2 2 2 g2 xB gY xB 2MM1 +ZNBB =! 2M1 +g vg v 1+gNxBB +1 g x2M g v +$ N +4g x (12) ! In presence of anomalous interactions$ we⎝ can− use the same formalism developed⎠ so far for anomaly-free" B # B 1 2 2 BB B " # where we denote with g ,g 4,g the couplings of SU2 (2), +U(1)4 and (UN(1)BB−,withtang v ), θ = g /g .Afterthe where we denote with g2,gY ,gz the couplings[40]. It is ofdefinedSU(2), as U(1)Ymodelsandwith withU(1) somez,withtan appropriate changes.θW = SincegY /g the2 e.Aftertheffective lagrangean of the class2 ofgY the2 anomalousz ≃ models2 − M 4 M Y 8 z− W Y 2 M 2 = 2 !(v2 + v2 ) 1+O1(ε2) 1" % Z 2 H1 H2 # $ that we are investigating includes bothB a2 Stückelberg2 Bdiagonalization2 2 and2 a two-Higgs ofB doublet the2 massB4cos sector,2 matrixθW the2 masses weNBB have of the diagonalization of the mass matrix we have NBB = qu vu + qd vd gB,xB = qu vu + qd vd gB . (10) 2 M 2 + 1 . 2 2 2 2 neutral gauge bosons are providedδ byM a combination′ of these two mechanisms.2 In this1 case′ we take as+ , 2 2 2 2 2 ZZ 2 gz 2 M2Z =2 2 2 2M12 +2 g v + NBB2 + 2M1 g v + NBB +4g xB (12) ε = ) * ) M ′ =≃B (*z Bv + z v4 + z v )(3)1+O(ε ) 3 free parametersHere vu and thev Stüeckelbergd denote the mass vevs2 M of1 theand2 two the Higgs anomalous fields couplingHu,Hd while constantZ qu andgB,withtanHqA1d µHare1 theβ Has Higgs2 inH2 thesin chargesθφWφ cos θW 0 W−µ 3 M ′ M 4 ! " % Aµ sin θW cos θW 0 Wµ Z Z 2 2 2 2 N # Y $ remainingunder anomaly-free the extra anomalous models. AsU(1) weB− have.Wehavealsodefined already stressed,theanalysisdoesnotdependsignificantlyonv = vu + v2d and g =Zg2gg2z+ g=Y .Themassless2 22cos θ2BB2 sin θ ε B 2 (2) ′ ⎛ µ ⎞ ⎛M + W .+ W, ⎞ ⎛ µ ⎞ Y The mass of the Z δgaugeMZZ = boson gets(zH1 vH corrected11 + zH2 vH2 )−. by terms of the order v (4)/M1,seeFig.1,convergingtothe Zµ = cos θW sin θW theε choiceeigenvalue ofB thisµ of parameter. the mass matrixThe value′ is associated of the Stückelberg to the photon mass(2)MA1γ+is,whilethetwonon-zeromasseigenvalues loosely constrained−4cos+′ θ byW the≃ D-brane 2 z ⎛ ⎞ while⎛ the mass of the Z boson⎞ and⎛ of the⎞ extra Z are ′ Zµ ε sin θW ε′sin θW 1 Bµ ′ − modeldenote in terms the of masses suitable of wrappings the Z and (n of)ofthe4-braneswhichdefinethechargeembedding[41,14]re the Z vector bosons. These are given by ⎜ ⎟ ⎜ported− ⎟ ⎜ ⎟ z In thisSM class value of models as M we1The have mass two,whilethemassofthe Higgs of the⎝ doubletZ gauge⎠ H ⎝ bosonand H gets,whosevevsare correctedZ gauge by termsv boson⎠and⎝ of thev can order⎠and grow anv2/M extra large,seeFig.1,convergingtothe with M1.Thephysicalgauge Zµ ε sin θW ε sin θW in1 Tabs. 1,2,3B andµ 4. 1 2 − H1 H2 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 1 →∞ 2 3 − 2 gM22 = 2 2M 2 +2g2v2 +whereN ε is22M a2 perturbativeg2v2 + N parameter+4g2x2 which is(11) around 10 A′ for the models analyzed, introduced in [39] and ⎝ ⎠ ⎝ The⎠M mass-matrix⎝ = ⎠ in theZ neutral(v4SU gauge(2)+fields1W sectorv singlet) is can1+ givenφBBwhose beO− by(ε obtained) vev1SM− is valuevφ.Theextra from asBBM1 theUB(1) rotation,whilethemassofthez charges matrixof the HiggsOZ doubletgauge are boson respectively can growzH large1 and with M1.Thephysicalgauge −3 Z 2 H,1 H2 →∞- where ε is a perturbative parameter which is around 10 for the models4cos analyzed,θW introduced[40]. in [39] It+ is and) definedfields as can be* obtained from the rotation matrix OA W 2 7 3 g + 2 , 5 Aγ 2 W3 [40]. It is defined as 2 z 2 2 2 mass2 2 =(2W23,Y,B2 2) M2 2 2 Y , 2 ′ ′ g v 1 g xB 1 g x⎛B ⎞ 2 2 δMZZAγ W3 MZ = (zH1 vH1 +L zH2 vH2 ++zφvφ) 1+(N O(gεv)), A Y 4 2 4 BBB ε = 2 2 A Y (3) ≃ 2 − M1 4 M1 8⎜ ⎟− ZM ′ Z=M O= O AA (13) (13) 2 ⎝ ⎠ Z Z δM ′ 2 g2gz 2 2 2 2 ⎛ ′ ⎛⎞− ′ ⎞ ⎛⎛ ⎞⎞ ZZ where BδisM the Stückelberg′ = field and the(z massv matrix+ z is definedv + ) as. , (4)Z B ε = ZZ 2 1 H21 H2 12 H2 H2(3) 2 2 2 2 2 2 Z′ B 2 2 M4cosZ′ = θ 2M1 + g v +whileNBB + the mass2M1 ofg v the+ NZBBboson+4g andxB of the extra(12)Z are M ′ M − W4 g 2v2 g g v2 g x− ⎝ ⎠ ⎝ ⎠ Z Z ! 2 − 2 Y " − 2 B % ⎝ ⎠ ⎝ ⎠ − 2 1 N 2 2 2 # which can be$ approximated at the2 first order as M = 2 g2 BBgY v gY v gY xB (9) ′ M4 1 + . 2 g2 2 2 2 In this class of models we have two Higgs doublet⎛H−which1 and H can2,whosevevsare be approximated2 ⎞ vH1 and at thevH2 firstand order an extra as while the mass of the Z boson and of the extra Z are ≃ g2x g x 2M + N MZ = 2 (vH1 + vH2 ) 1+O(ε ) − 2 B Y B 1 BB 4cos θW SU(2)W singlet φ whose vev is vφ.TheextraU(1)⎝z charges of the Higgs doublet⎠ 2 are respectively zH andg g 2 with The mass of the Z gauge boson gets corrected by terms of the order v /M1,seeFig.1,convergingtothe2 1 Y + 2 , 0 ′ 2 gz 2 2 g 2 2 2 g2 2 g ′ 2 2 2 2 SM2 value as M1 ,whilethemassoftheZ gauge boson can grow large with MM1.Thephysicalgauge= A(zH vHg2 + zH vH +gzYφvφ) 1+O(ε g) M = (v + v ) 1+O(ε ) →∞ B 2 2 B 2 2 2 B 2 B 2 Z g1 1 22 2 g 2 Z 2 H1 H2 NBB = qu vu + qd vd gB,xA B = qu vu + qd vd gB . O4 (10) Y g + O(ϵ1) g 2+ O(ϵ1) 2 ϵ1 (14) 4cos θW fields can be obtained from the rotation matrix O ≃ ⎛ g − g 0 ⎞ 5 2 gg2gz 2 2 2 g2Y 2+ , 2 2 ) * ) B * B δMZZ′ = (2zϵH1 vHg + zH2 vϵ1H ). 1+O(ϵ1) (4) + Here vu and,vd denote the vevs of the two HiggsAγ fields Hu,HWd while qu and q areA the Higgsg charges2 −2 1 1Y 2 22 g 2 gz 2 2 2 2 2 2 2 3 d O −+4cos⎝O(θϵW) + O(ϵ ) ϵ ⎠ (14) ′ A Y2 2 2 2 g 1 g 1 2 1 MZ = (zH1 vH1 + zH2 vH2 + zunderφvφ) the1+ extraO anomalous(ε ) U(1)B .WehavealsodefinedZ = Ov = Av + v and g = g + g .Themassless (13) 4 ⎛ ⎞ ⎛ uwhich⎞ d is the analogue2 Y≃ of⎛ the matrixg in Eq. (2)− forg the anomaly-free models, but⎞ here the role of the mixing In′ this class of models we have two Higgs doublet2 H and HY,whosevevsarevH 2and vH and an extra eigenvalue of the mass matrix is associated to theZ photon Aγ+,whilethetwonon-zeromasseigenvaluesB + ϵ 1 2 ϵ 1+O(1ϵ ) 2 2 g2gz 2 2 2 2 parameter ϵ1 is taken by the expression2 1 2 1 1 ′ + , ′ ⎝ ⎠ ⎝ ⎠ − δMZZ = (zH1 vH1 + denotezH2 vH the2 ) masses. of the Z and of the Z vector bosons.SU(2) TheseW singlet(4) are givenφ bywhose vev is vφ.Theextra⎝ U(1)z charges of the Higgs doublet are respectively⎠ zH1 and −4cosθW which can be approximated at the first order as x 1 B 2 2 2which2 is the2 analogue2 2 of2 the2 2 matrix in Eq. (2) for theϵ1 = anomaly-. free models, but here the role(15) of the mixing MZ = 2M1 + g v + NBB 2M1 g v + NBB +4g xB (11) 2 4 M. Guzzi, C.C.g − g − M1 In this class of models we have two Higgs doublet H1 and H2,whosevevsarevH,1 and vH2 Yand+ an extra2 0 - 5 parameterg ) ϵ1g is taken* by the expression A g2 2 gY 2 g O + O(ϵ1)7 + O(ϵ1A) relationϵ1 between the two expansion(14) parameters can be easilyobtainedinanapproximatewaybyadirect SU(2)W singlet φ whose vev is vφ.TheextraU(1)z charges of the Higgs doublet are respectively≃ ⎛ g z−Hg1 and 2 ⎞ g2 gY 2 2 ϵ1 2 ϵ1 comparison.1+O(ϵ1) For simplicity we take all the charges to be O(1) in all the models obtaining ⎝ − ⎠ xB which is the analogue of the matrix in Eq. (2) for the anomaly-free models, but here the role of the mixing ϵ1 = 2 . (15) 5 M1 parameter ϵ1 is taken by the expression 2 2 2 xB MZ g2v A relationϵ1 = between. the two expansion parameters(15) can∼ be easilyobtainedinanapproximatewaybyadirect M 2 2 2 2 2 1 M ′ M g v Z − Z ∼ z φ A relation between the two expansioncomparison. parameters can be For easilyobtainedinanapproximatewaybyadirect simplicity we take all the charges to2 be O(1)2in all the models obtaining δMZZ′ g2gzv (16) comparison. For simplicity we take all the charges to be O(1) in all the models obtaining ∼ giving 2 2 2 2 v MZ g v 2 ϵ 2 2 , (17) ∼ 2 M 1g v 2 2 2 2 2 Z ∼2 M1 MZ′ MZ gz vφ ∼ − ∼ 2 2 2 2 2 which2 is the analogue of Eq. (15), having identified′ the Stückelberg mass with the vev of the extra singlet δMZZ′ g2gzv (16) MZ MZ gz vφ ∼ Higgs, M g v .ThisisnaturalsincetheStückelbergmechanismcanbetho− ∼ ught of as the low energy 1 z φ 2 2 giving ∼ ′ 2 remnant of an extra Higgs whose radial fluctuationsδMZZ haveg2 beengzvfrozen and with the imaginary phase surviving (16) v ∼ ϵ1 2 , (17) ∼ M1 at low energy as a CP-odd scalar [18]. which is the analogue of Eq. (15),giving having identified the Stückelberg mass with the vev of the extra singlet Higgs, M gzvφ.ThisisnaturalsincetheStückelbergmechanismcanbethought of as the low energy 2 1 ∼ v8 remnant of an extra Higgs whose radial fluctuations have been frozen and with the imaginary phase surviving ϵ1 2 , (17) at low energy as a CP-odd scalar [18]. ∼ M1

which is the8 analogue of Eq. (15), having identified the Stückelberg mass with the vev of the extra singlet Higgs, M gzvφ.ThisisnaturalsincetheStückelbergmechanismcanbethought of as the low energy 1 ∼ remnant of an extra Higgs whose radial fluctuations have been frozen and with the imaginary phase surviving at low energy as a CP-odd scalar [18].

8 The UV/IR conspiracy of the anomaly

The possibility that axions are associated to a rearrangement of the vacuum of a gauge theory cannot be excluded.

A similar rearrangement is possible for the conformal anomaly.

Follow the case of the QCD dynamics The QCD U(1) anomaly responsible for the pionà gamma gamma decay

Nonlinearly realised Lagrangean in the IR.

TRACE ANOMALYIn the UV the AND MASSLESSaction SCALARexhibits DEGREESan OFanomaly... pole PHYSICAL REVIEW D 79, 045014 (2009) in terms of the linearly independent and complete basis set of only the four tensors, (1, (2, (4, and (5. Indeed this has been the general practice in the literature on the axial anomaly [11,16,19]. Eliminating these or any of the other two tensors is not necessary for our purposes, and we choose instead to work with the overcomplete set of six tensors listed in Table II. This will have the consequence that the coefficient functions fi are determined only up to the freedom to choose arbitrary coefficients of the linear combinations (17), i.e. to FIG. 2. Discontinuity or imaginary part of the triangle diagram shift each of the coefficients fi by an arbitrary scalar obtained by cutting two lines. function h via the rule,

f k2; p2;q2 f k2; p2;q2 h k2; p2;q2 ; (18a) By taking general linear combinations of the eight pseu- 1ð Þ ! 1ð Þþ ð Þ f k2; p2;q2 f k2; p2;q2 h k2; p2;q2 ; (18b) dotensors in Table I, we find then that there are exactly six 2ð Þ ! 2ð Þþ ð Þ third rank pseudotensors, (" p; q , i 1; ...; 6 which f k2; p2;q2 f k2; p2;q2 h k2; p2;q2 ; (18c) i 3ð Þ ! 3ð ÞÀ ð Þ can be constructed from them toð satisfyÞ the¼ conditions (9), with the shift in f , f , f determined by (15) by the " " 4 5 6 2 2 p(i p; q 0 (i p; q q 0;i1; ...; 6; interchange of p and q . The arbitrary function h drops ð Þ¼ ¼ ð Þ ¼ ¼ (12) out of the final amplitude by use of (17). The computation of the finite coefficients given in the given in Table II. Hence we may express the amplitude (7) literature [11,16] amounts to a specific choice of the arbi- satisfying (9) in the form, trary function h (by the order in which the matrix traces 6 are performed), and yields " " À p; q fi( p; q ; (13) i 2 ð Þ¼i 1 ð Þ e 1 1 x xy X¼ À f1 f4 2 dx dy ; (19a) where f f k2; p2;q2 are dimension 2 scalar func- ¼ ¼ % 0 0 D i i Z Z ¼ ð Þ 2 2 À2 2 1 1 x tions of the three invariants, p , q , and k . e À x 1 x f2 2 dx dy ð À Þ ; (19b) We note also that the full amplitude (7) must be Bose ¼ % 0 0 D Z Z symmetric, 2 1 1 x e À y 1 y " " f5 2 dx dy ð À Þ ; (19c) À p; q À q; p : (14) ¼ % 0 0 D ð Þ¼ ð Þ Z Z " " f3 f6 0; (19d) Since (i 3 p; q (i q; p for i 1, 2, 3, it follows ¼ ¼ þ ð Þ¼ ð Þ ¼ that the six scalar coefficient functions fi also fall into where the denominator of the Feynman parameter integral three Bose conjugate pairs, i.e. is given by f k2; p2;q2 f k2; q2;p2 ; (15a) 1ð Þ¼ 4ð Þ D p2x 1 x q2y 1 y 2p qxy m2 f k2; p2;q2 f k2; q2;p2 ; (15b) ð À Þþ ð À Þþ Á þ 2ð Þ¼ 5ð Þ p2x q2y 1 x y xyk2 m2; (20) f k2; p2;q2 f k2; q2;p2 ; (15c) ¼ð þ Þð À À Þþ þ 3ð Þ¼ 6ð Þ strictly positive for m2 > 0, and spacelike momenta, k2, p2, related by interchange of p2 and q2. q2 > 0. Thus each of the dimension 2 scalar coefficient Actually, owing to the algebraic identity obeyed by the functions f in (19) are finite and freeÀ of any UV regulari- symbol, i zation ambiguities, and the full amplitude À" p; q sat- g"#&' g"#&' g#&'" g&'"# isfying ð Þ þ þ þ g'"#& 0; (16) þ ¼ (i) Lorentz invariance of the vacuum, in four dimensions, the six tensors (i are not linearly (ii) Bose symmetry (14), independent, and form an overcomplete basis. The identity (iii) vector current conservation (9), (16) leads to the relations, (iv) unsubtracted dispersion relation of real and imaginary parts, (" p; q (" p; q (" p; q ; (17a) 3 ð Þ¼ 1 ð Þþ 2 ð Þ (" p; q (" p; q (" p; q : (17b) with the finite imaginary parts determined by the cut 6 ð Þ¼ 4 ð Þþ 5 ð Þ triangle diagram of Fig. 2, is given by (13) and (19), Thus the tensors (3 and (6 could be eliminated completely without any need of regularization of ultraviolet divergent by means of (17), and the full amplitude expressed entirely loop integrals at any step.

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A A 0 λ 1 1 µ 0 ,k ,k 6 3 4 Armillis ν 1 ffi ( ( .T h e s ea r eg i v e na sp a r a m e t r i ci n t e g r a l sa n da r ee a s i l yc o m p Rosenberg, 1963 n NNSzoed ec,VaAnsn 30 ec,Italy Lecce, 73100 Arnesano Via Lecce, di Sezione INFN and k 2 2 k dw = ute ihuiaiyo h ehns finflo of mechanism the of unitarity with culties A ) ) ragedarmwt nailvco urn ( current axial-vector an with diagram Triangle 1 1 with [12] h orivratamplitudes invariant four The iho with hs xlctfr ilb okdotblw Both below. out worked be will form explicit whose hr h eea massive general the where ntetovco ie,giving lines, vector two o finite the rendered on be can which integrals, divergent formally h oesmer ntetovco etcswt indices with vertices vector two the amplitude on divergent symmetry formally Bose the The re-express to allow which relations ε ε ,k ,k A A [ [ (2 ! k k 1 2 2 2 ∆ 0 1 1 i π ( ( )= )= 3 ,k ,µ, 1 I k k ) − V V 0 λ st 4 I 1 1 iatmnod iia nvri` e Salento Università del Fisica, di Dipartimento w 2 µ ,k ,k st , Delle Rose, ,µ, ! ν nerli endby defined is integral ν dzw ff A U , ( 2 2 seletra ie hw nFg1i hrfr wri therefore is Fig.1 in shown lines external -shell λ )= )= k d i − − (1) λ ff ]+ A A 4 for 1 + = n hi nml Poles Anomaly their and s cieato n t opeinby completion its and action ective ] q A A ,k k 6 5 z Tr ( ( 6 5 2 ν t A k k ( ( i " oesi oradFv Dimensions Five and Four in Models 2 + k k 1 1 z 2 A A ≥ k k )= " ,k ,k 2 2 ( (1 γ 1 1 2 A ,k ,k k 4 1 3a r efi n i t ea n dg i v e nb ye x p l i c i tp a r a m e t r i ci n t e g r a l s λ 2 2 A · 1 q 5 ( ( ∆ − )= )= γ 1 1 ,k k ( 2 k k 5 )= )=16 5 k 2 ! ( A A ( 0 λ 1 1 z ( 2 1 k q A µ q 0 / ,k ,k ) ) ,k 1 3 4 ε k − 6 ν 1 Abstract 3 ,k = − − ( ( [ ( 1 2 2 Guzzi k k 2 2 k dw k = − − ) k 16 2 + with [12] h orivratamplitudes invariant four The iho with hs xlctfr ilb okdotblw Both below. out worked be will form explicit whose hr h eea massive general the where ntetovco ie,giving lines, vector two o the finite rendered on be can which integrals, divergent formally h oesmer ntetovco etcswt indices with vertices vector two the amplitude on divergent symmetry formally Bose the The re-express to allow which relations 2 k ) ) π .it,[email protected],[email protected]. ε /) 1 1 1 ) )+ ,µ, A A ε ε ,k [ 1 2 2 ,k ,k π k w A A γ [ [ (2 ∆ [ 3 4 ( ! 1 2 k k ff I 2 ν ( ( q (1 ,k I 1 2 ν 2 2 k 20 )+ sele -shell k k ( 0 0 λ 1 1 i I π 11 − ( ( , )= )= q 1 / 3 A 2 2 µ ,k ,µ, 1 ( st , C.C. 2 − I λ k k ,k ,k ) ( − ν · k ff , − 1 k st k ]+ ( 4 1 1 k k 1 ν w w 2 seletra ie hw nFg1i hrfr wri therefore is Fig.1 in shown lines external -shell tnacrigt 1]i h form the in [12] to according tten g 1 1 1 k 1 l yipsn h adidentities Ward the imposing by nly ,k ,k ,k 2 2 2 , and k ) ,µ, ) )(11) ! / ,k k ν si nt e r m so ft h ec o n v e r g e n to n e s . 1 ) + = λ nerli endby defined is integral − A A 2 dzw . 1 ff A A k 2 ,k nadtelongitudinal/transverse the and on µ , ] ) 2 2 2 ) 4 6 ec k 2 2 q γ 2o ft h em u o na n di nt h ep r o o f λ 3 ) )= )= d ( ( i 1 − − µ − and ( , 2 +2 A λ µ k k ieato.Teoii fthese of origin The action. tive − k ]+ A A A A k 4 )= for wi ne x t r ad i m e n s i o n a lm o d e l s q + iae ypromn com- a performing by cidated h aclaino hs polar these of cancellation the / 1 1 m rvdsantrlbridge natural a provides rms A q s euetoidpedn (but endent indep two use We ] 2 q A A 1 I 1 k 2 ,k ,k # k 4 1 6 5 z ,k 10 wz an o n l o c a lt h e o r y .S o m ea d - A Tr e c .(4) +exch. ( ( ∆ r nta ersne by represented instead are ( ( 6 5 2 ν t ν 2 2 k k ( A k k 2 = ( ( 6 i ) ) ! caoaosvrie,as vertices, anomalous ic k " A A 0 λ oe xrce rmthe from extracted poles ) + 1 1 ( ( , , k k sflle hnst the to thanks fulfilled is 1 1 V V 1 z ε µ 2 0 k k ,k ,k ≥ k k econcludethatconsis- " ltoso ntrt in unitarity of olations ,k ,k 3 4 ,k 2 2 [ ν ( 1 1 (1 1 oaypls r also are poles, nomaly k ( ( γ 1 1 5 2 A ,k ,k ,k k k k k 2 2 k dw 1 2 = 3a r efi n i t ea n dg i v e nb ye x p l i c i tp a r a m e t r i ci n t e g r a l s λ 2 2 2 A · ,k ) ) 1 q )] 2 5 1 1 2 ) 2 − )= )= γ 1 1 ε ε ,k k g hoyb dis- by theory uge ,k ,k ( 2 ) .W ilr-nlz h eiaino the of derivation the re-analyze will We ). 5 A A − , )= )=16 2 [ [ (2 ε 5 k ! 2 k k ( ,µ, ( q [ z ( 1 2 2 2 2 1 k k q 0 1 1 i m A π q / ( ( )= )= k ) ) − ,k 3 k 1 ,k ,µ, 1 1 I k k ε − k ) 6 ,k c Guzzi rco 2 − 1 3 λ 2 ,k k − st − 4 # 1 1 [ ( 1 2 w ] 2 2 − k ,k ,k k 2 k 1 − − ,µ, ! ) k ff 16 ν , 1 2 nerli endby defined is integral + 1 dzw 2 k π ε /) A 1 ν ν , ) )+ , 2 2 ,µ, rn t erent A A ,k λ [ , 2 2 λ )= )= d π i k w − − λ γ λ o ie n[1 wyfo h chiral the from away [11] in given ion )a n dt w ov e c t o rc u r r e n t s( nrbto switnhr na explicit an in here written is ontribution [ 3 4 ]+ A A ( 4 for 1 2 ] I 2 ν s ] ( ( q q A A k (1 ,k I k ν k 6 5 20 )+ z k k ( 2 it µ (10) (12) Tr 11 ( ( 6 5 2 − ν .W okudrtems general most the under work We e. , t q 1 / . (5) (8) (6) (7) (9) A 2 2 A 1 k k ( ( ( 2 − . . i tpiti rvdb okn ta at looking by proved is point st λ " ,k ,k + eappears. The + ( · k k k 1 1 et h r e e - p o i n tc o r r e l a t o ri nt e r m s , − z 1 2 ≥ k k k k ]+ " Va n dw h i c ha l l o wt od e t e c tt h e ,k ,k k 2 2 k 1 ν ( (1 w noilsrcue,dntdby denoted structures, ensorial tg e n e r a lc a s e ,b yf o c u s i n go u r γ tnacrigt 1]i h form the in [12] to according tten 1 1 2 1 A 1 1 ,k ,k 1 l yipsn h adidentities Ward the imposing by nly ,k k 2 2 2 , and k ) )o ihsmerco symmetric with or s) ) )(11) / 3a r efi n i t ea n dg i v e nb ye x p l i c i tp a r a m e t r i ci n t e g r a l s ,k λ 2 2 A · si nt e r m so ft h ec o n v e r g e n to n e s . 1 q ) λ 5 et h / decomposition, L/T the to ve A A 2 . − )= )= ino naoaypl from pole anomaly an of tion γ 1 1 1 ,k k A k ( 2 2 µ 5 ] )= )=16 ) muain,uigtetwo the using omputations, icto fteindependent the of tification 5 2 k ) 2 4 6 k 2 ( 2 ( γ 3 z ( ) 2 1 ( ( k q A 1 µ − and q ( / , +2 ) ) ,k A µ k k 1 k ε − k 6 3 q ,k + / 1 1 − − 2 [ ( 1 1 I 2 2 ,k ,k k # k ,k − − tbeol nfwcases, few in only utable ) k 10 k 16 2 + wz 2 k A π e c .(4) +exch. r nta ersne by represented instead are ε /) 1 ) )+ ν ,µ, 2 2 A A ,k ueit longitudinal into tude ( [ 2 2 q 2 π 6 k ) ) w k γ ) [ 3 4 ( ( ( , , sflle hnst the to thanks fulfilled is 1 − 2 I 2 1 ν ε ( ( q (1 k k ,k I q ν A k 20 )+ ,k k k ( k [ 1 11 1 − , k q 1 / A 2 2 ( ,k k 2 − λ ,k ,k 1 ( · 2 k , − 1 2 k ,k k ]+ )] k k 1 ν ) 2 V w VV AV V tnacrigt 1]i h form the in [12] to according tten 1 1 1 1 l yipsn h adidentities Ward the imposing by nly ,k 2 2 2 , and ) k ) ) )(11) / ,k − , 2 si nt e r m so ft h ec o n v e r g e n to n e s . ) λ ε A A 2 . 1 ,µ, A k 2 µ [ ] ) 2 ) 4 6 k k 2 2 m γ 3 ) ( ( 1 µ − 1 and ( , +2 A µ k k ,k 2 k λ q + / 1 1 k µ k # 2 1 I ,k ,k ] # amplitude 2 − ,k 1 10 k 2 , wz A e c .(4) +exch. r nta ersne by represented instead are , 1 ν 1 2 2 ( 2 ν ν 6 ν ) ) , k ) ( ( , , sflle hnst the to thanks fulfilled is , .The ). ff 1 ε k k λ ,k [ 1 1 -shell k ] ,k k k 1 2 2 ,k )] 2 µ (10) (12) ) 2 . (5) (8) (6) (7) (9) ) − , 2 ε ,µ, [ k m 1 ,k 2 λ # ] − k 2 , 1 1 ν ν , , λ ]

2 µ

(10) (12)

. (5)

(8) (6) (7)

(9) 2.3 Explicit expressions in the massless case

To extract the explicit form of the parametric integrals given by Rosenberg, we proceed with adirectcomputationoftheinvariantamplitudesoftheparameterization using dimensional reduction. We perform the traces in 4 dimensions and the loop tensor integrals in D dimensions, using the common2.3 Explicit techniques expressions of tensor reduction. in the We massless use dime casensional regularization with minimal subtractionTo extract and the explicit find, as form expected, of the parametricthe cancellation integralsof the given dependence by Rosenberg, of the we proceed result with on the renormalization scale. Therefore, the parametric integral I11 and the combinations 2.3 Explicitadirectcomputationoftheinvariantamplitudesofthepara expressions in the massless case meterization using dimensional I20 I10 arereduction. trivially identified We perform at the the traces end of in the 4 dimensions computation. and the The loop resulttensor is expressed integrals in inD termsdimensions, − To extract the explicit form of the parametric integrals given by Rosenberg, we proceed with of elementaryadirectcomputationoftheinvariantamplitudesoftheparausing functions, the common except techniques for the functionof tensormeterizationΦ reduction.(x, y)[13],whichisrelatedtooneofthe using We dimensional use dimensional regularization with 2.3 Explicit expressions in the massless case two masterreduction.minimal integrals We perform subtraction of the the traces decomposition, in 4 and dimensions find, and as the the expected, loop scalartensor theintegralsmassl cancellationess in D triangle.dimensions,of We the obtain dependence for generic of the result Tousing extract the common the explicit techniques form of tensorof the reduction. parametric We useintegrals dimensional given regularization by Rosenberg, with we proceed with virtualitiesminimalon of subtraction the the external renormalization and find, lines as expected, scale. the cancellation Therefore,of the the dependence parametric of the in resulttegral I11 and the combinations adirectcomputationoftheinvariantamplitudesoftheparameterization using dimensional on theI renormalization20 I10 are triviallyscale. Therefore, identified the parametric at the end integral of theI11 and computation. the combinations The result is expressed in terms − i i s1s2 (s2 s1) s1 Areduction.(Is,20 sI10,sare)= We trivially perform identified the+ at traces the end in ofΦ 4 the( dimensionss computation.,s ) and The the− result loop is+ expressedtensors (s integrals in termss )log in D dimensions, 1 − 1 of2 elementary−4π2 functions,8π2σ except1 for2 the functions Φ(x,1 y)[13],whichisrelatedtooneofthe2 − 12 s usingof elementary the common functions, techniques except for the of! function tensorΦ reduction.(x, y)[13],whichisrelatedtooneofthe We use dimensional regularization" # with two mastertwo integrals master of integrals the decomposition, of the the decomposition, scalar massless triangle. thes scalar2 We obtain massl for genericess triangle. We obtain for generic minimal subtraction and find, as expected,s2 (s1 thes12 cancellation)log of, the dependence of the result (13) virtualitiesvirtualities of the external of the lines external lines− − s on the renormalizationi scale. Therefore, the parametric" in#$tegral I and the combinations i i s 2s (s s ) s 11 A3(s,A s1(s,,s s2,s)=)= + sΦ1(s2,s4) s1122 +3(2 − 1s1++s (ss2) ss12)log+2s11s2 Φ(s1,s2) 1 1 2 22 2 2 i1 2 i 1 2 s112s2 (s2 s1) s1 I20 I10 are trivially−84ππ identifiedsσ8π σ− at the ends of the computation.− Thes result is expressed in terms A1(s, s1,s2)=! + Φ(s1,s2) − + s1 (s2 s12)log − 2 2 s2 " # s % −4sπ (&s 8sπ)logσ , s ' (13) − 1 s of elementary functions, except− for2 1 the−212ss function12σ!s ssΦ1([2x,s y1)[13],whichisrelatedtooneofthes2 + s12 (3s2 + s12)] log − − s2 s " # i 2 " #$ twoA master3(s, s1,s2 integrals)= of thes1s decomposition,2 4s12 +3(s1 + s2) thes12 +2 scalarss21s(2sΦ1 massl(s1,ss122ess) )log triangle. Wes, obtain for generic (13) 8π2sσ2 − 2 − − s 2 " # ss2 s12 + s1 (2s2 +3s12s) log , (14) virtualities of the external% lines& 2iss σ− ss [2s s + s (3's + s )] log 1" #$ s 12 1 1 2 212 2 12 s A3(s, s1,s2)= − − s1s2 4s12 +3(s1 + s2) s12 +2s1s2 Φ(s1,s2) i 2 2 s2 " # " #$ 8π sss3σ s2 +−s (2&s +3s ) log2 , ' (14) A4(s, s1,s2)= i s1 i4s212 12+2(1 s1 2s+21s2s(122s)2 s12s1+2) s1s2s12 + s1 (s1 s1 s2) s2 Φ(s1,s2) A1(s, s1,s2)= 2 2 + − Φ(s1,s2) − s + s1 (s2 s12)log s1 8πi sσ2 2 % & 2ss σ" #$ss [2s s + s (3's− + s )] log A (s, s ,s )= −4πs 4s83 π+2(σ s& +2s ) s2 +2s s ss'12+ s (s 1s ) s 1 Φ−2(s ,s )12 s2 s 12 4 1 2 2 2 1% 12& !1 2 12 −1 2 12 1− 1 2 22 1 2 2 ' s 8π sσ +2ss1σ + s (s1 + ss212)− 2s + s1s2 log" # s2 12 s2 " # % & +2ss σ + s2(s(s+1 s )s122s)log2 + s2s log , ' s (13) 1 − 1 −12 ss122 s112s2 + s1 (2s2 +3s12) log , (14) − s s1 " s# 2 s1 ( ) i +2 2ss1 4s ( s1 ("s2)#$3"s12# ) log , (15) +ssi 1 4s12 s1 (s2 123s12) log , (15) " #$ A3(s, s1,s2)= 2 2 s1s2 4s12−+3(3s−1 +−s2)&s12 s+2− s1s22 Φ(s1,s2s) ' A4(s, s1,s82π)=sσ − s1 4s +2(s1 +2s2) s +2s1s2s12 + s1 (s1 s2) s2 Φ(s1,s2) 2 &2 12 ' " #$ 12 2 2 2 8π sσ 2 " #$ s − where s = kNothing, s1 = k1,specifics2 = k2, emergess12 = k1 k2from with σ&this= scomputation12 s1s2 and thein functionthis'parameterizationΦ(x, y)is 1 2 2 % 2 ·& 2ss σ ss− [2s s + 2s (3's + s )] log s where s = k , s1 = k1, s2 = k2, s12 =% k1&12 k2 with1 σ1 =2 s1212 s21s2 and12 2 the function 2Φ(x,' y)is defined as [13] − · −+2ss1σ + s (s1−+ s12) 2s12 + s1ss2 log defined as [13] 2 s2 " # s 1 ss2 ys12 +1+sρ1y(2s2 +3s12) logπ2 , s1 " (14)# Φ(x, y)= 2[Li ( ρx)+Li ( ρy−)] + ln ln +ln(2ρx)ln(ρy)+ s,( (16) ) λ 2 − 2 − x 1++ssρx1 4s12 s1 (s2 3 3s12) log , (15) i " #$ s 2 1* 3 & 2 y −1+'ρy −$ π A4(s, s1,s2)= 2 2 s1 4s12 +2(s1 +2s2) s12 +2s1s2s12 + s1 (s1 s2) s"2 Φ#$(s1,s2) withΦ(x, y)= 82[π Lisσ2( ρx)+Li2( ρy)] + ln& ln +ln(ρ−x')ln(ρy)+ , (16) where s =λk2, s =−k2, s = k2,−s = k kx with1+σρx= s2 s s and the3 function Φ(x, y)is 1 % 1& 2 2 12 1 2 2 12 1s2 ' * λ(x, y)=√∆, +2ss1σ∆+=(1s (s1x+· sy)122 ) 42xy,s12 + s1s2 log−(17) $ defined as [13] − − − s with −1 s1 s2 ρ(x, y)=2(1 x y + λ) ,x=2 ,y=( . s)1 (18)" # − − +ss1 4s12s s1 (s2 3ss12) log , (15) 1 − − y 1+s ρy π2 Φ(x, y)= 2[Li2(√ ρx&)+Li2( ρy)] + ln' ln" #$2 +ln(ρx)ln(ρy)+ , (16) 2 2 λ(λx,2 y)= −∆, −∆ =(12 xx 1+y) ρx 4xy, 3(17) where s = k , s1 = k1, s2 = k2, s12 =7k1 k2 with σ = s12 s−1s2 −and the− function Φ(x, y)is * · s1− s2 $ defined asρ [13](x, y)=2(1 x y + λ)−1,x= ,y= . (18) with − − s s 1 y 1+ρy π2 Φ(x, y)= 2[Li2( ρx)+Li2( ρy)] +√ ln ln +ln(ρx)ln(ρy)+2 , (16) λ − λ(x,− y)= ∆x, 1+ρx∆ =(1 x y) 3 4xy, (17) * 7 s − − − $s ρ(x, y)=2(1 x y + λ)−1,x= 1 ,y= 2 . (18) with − − s s

λ(x, y)=√∆, ∆ =(1 x y)2 4xy, (17) s7 − − − s ρ(x, y)=2(1 x y + λ)−1,x= 1 ,y= 2 . (18) − − s s

7 is respected. As mentioned above, the diﬀerence between the massless and the massive decomposition of the triangle amplitude lies in the particularsetofscalarintegralsinvolvedin

the tensor reduction. Here we deﬁne D1 and D2 as a combination of two-point scalar massive

integrals (B0)ofdiﬀerent internal momenta a +1 a +1 D (s, s ,m2)=B (k2,m2) B (k2,m2)=iπ2 a log i a log 3 i =1, 2 i i 0 − 0 i i a 1 − 3 a 1 ! i − 3 − " (34)

in which the dependence on the regularization scheme disappears in the diﬀerence of the two

scalar self-energies involved in (34). The expression of C0 can be given explicitly in various forms [14], for instance as

3 2 2 1 bi 1 bi 1 bi +1 bi +1 C0(s, s1,s2,m )= iπ Li2 − Li2 − − + Li2 − Li2 − 2√σ ai + bi − ai bi ai bi − ai + bi i=1 # ! − − " (35)

with

2 4m si + sj + sk ai = 1 ,bi = − , (36) $ − si 2σ true presence of the pole in thetrue IR has presence to be of checked the pole by in taking the IR the has corresponding to be checked limit, by taking since the corresponding limit, since where s3 = s and in the last equation i =1, 2, 3andj, k = i.Otherexpressions,suitablefor the Schouten relations allow the extraction of̸ a pole in the IRregionatthecostofextra numerical implementations, are giventhe in [15]. Schouten The region relationsin which all these allow functions the extraction have real of a pole in the IRregionatthecostofextra singularitiesarguments inand the do not parameterization. need any analyticsingularities continuations For this in are reason the those parameterization. discussed we start in section by recalling 2.3, For for this the reason structure we s oftart the by recalling the structure of the true presence of the poleL/T inthe parameterization, themassless IR case. has In to general, be which checked the prescription separates by takingfor theiη in longitudinal the the presence corresponding of fr aom mass the in the transverse limit, internal since components of the loop - in the fermion propagator - isL/T taken as parameterization,m m iη.Wehavecheckednumericallythe which separates the longitudinal from the transverse components of the → − the Schouten relationsanomaly allowagreement vertex, the between extraction which the expressions is given of a presented pole by in above the and IRregionatthecostofextra thosegiveninparametricform. anomaly vertex,as s1 = s2 =0. which is given by singularities in the parameterization. For this reason we startIn the by L/T recalling parameterization the we structure expect a similar of polar the result,aftersummingoverthecontri- butions coming both from the longitudinal and transverse tensors. In this case, the only two λµν 1 L λµν T λµν L/T parameterization, which4Thevertexinthelongitudinal/transverse(L/T)for- separates the longitudinalW = from theW transverseW components, (+) of1 the (37) TRACE ANOMALY AND MASSLESS SCALAR DEGREESnon-vanishing2 OF ... coefficients arePHYSICALwL andλwµTν REVIEW D 79, 045014L λµν (2009) T λµν 8π − W = 2 W W , (37) (+)8π 4i − anomaly vertex, whichLagrangian, is givenmulation by and comparisons wL(s, 0, 0) = w (s, 0, 0) = , (84) where the longitudinal component ! " T − s where the longitudinal component(−) (−) ! " " wT (s, 0, 0) =w ˜T (s, 0, 0) = 0 (85) iThec second$@" parameterizationieA" c mc of the three-point correlator function that we are going to discuss ð À Þ À L λµν λ is the one" $λ presentedµν in1 [11].5 OneL λ ofµν theW featuresT λ=µ ofandν thisw the residuesk paramε[µ,eterization mustν,k be computed,k is](38) the combining presence them of with the corresponding tensor structures. ic W@" =ieA" igWB" c mccW (45) ,L 1 2 L λµν (37) λ ! ð À À2 Þ À (+) W = wL k ε[µ, ν,k1,k2](38) alongitudinalcontributionforgenericvirtualitiesofth8π − It is wortheexternalmomentaandnotjustin noticing that tλµν (k1,k2)=0fors1 = s2 =0.Thiscanbeimmediatelychecked so that the variation of the corresponding action withstarting from its definition given in Eq. (39) and with the aid of the two Schouten identities respectthe to B specificgives configuration! under which it appears in Rosenber" g’s formulation. Of course, the where the longitudinal(with componentwL" = 4i/s)describestheanomalypole,whilethetransversecontribushown in Eqs.(22,23), which in this case become tions take the − (with wL = 4i/s)describestheanomalypole,whilethetransversecontributions take the S " 12 −FIG. 3. Tree diagram of the effective action (53s) showing the form kλ ε[k ,k ,µ,ν]= kν ε[k ,k , λ,µ]+ ε[k , λ,µ,ν], (86) g J5 A: (46) massless propagator1 1 2D1 representing1 1 the2 massless 01À state in LBλ"µν¼ h i λ form − 2 W = wL k ε[µ, ν,k1,k2](38)the triangle amplitudeλ (43) to physicalµ photons whens the fermion k2 ε[k1,k2,µ,ν]=k2 ε[k1,k2, λ, ν] ε[k2, λ,µ,ν], (87) Henceforth weW shallT set(k the,k axial)= vectorw(+) couplingk2,kg 2,k12. t(+)mass(km ,k0. )+ w(−) k2,k2,k2 t(−−)2 (k ,k ) λµν 1 2 T 1¼ 2 T λµν 1¼ 2 T(+) 2 1 2 2 2 λµ(+)ν 1 2 (−) 2 2 2 (−) Next let us decompose the axial vector B into its trans-so that the unique contribution to the residue for s 0comesfromthelongitudinalpart " − W λµν (k−1,k2)=wT k ,k1,k2→ tλµν (k1,k2)+ wT k ,k1,k2 tλµν (k1,k2) (with wL = 4i/s)describestheanomalypole,whilethetransversecontribuverse and longitudinal parts, ( ) 2 2 2 Then( one) can verify thattions the local take quadratic the action func- + wT# k ,k1,k$ 2 tλµν (k1,k2), #− 1 $ L λµν − (39) − tional, lim sWµνλ(s, 0, 0)( =) 2lim sW2 2 ( ) s→0 + wT# 8πk2 s→,k0 1,k$ 2 tλµν (k1,k2), # $ (39) form B " B"? @"B (47) 1 ¼ þ # $ 4= " lim swL(s, 0, 0) kλ"ε[µ, ν,k1,k2] S eff ; 1; A; B d x 8@π2 s→0 @"1 1@ B" withwith@" the transverse0 and a pseudoscalar. tensors given Then,% by by an inte- % ½ ¼ ð# Þð ÞÀ$ B"? B i λ ¼(+) (+) with the transverse(−) tensors(− given) Z = by k ε[k1,k2,µ,%ν]. (88) T gration by parts inThe the2 anomaly action2 2 correspondingis associated to (to a 45),longitudinal we 2 2component, 2 which%e2 −has2π2 a pole: W λµν (k1,k2)=wT k ,k1,k2 tλµν (k1,k2)+ wT k ,k1,k2 tλµν (k1,k2)F F~"# (53) have (+) the anomaly pole (1/s). The transverse sector does not contribute2 "#to the anomaly. t (k ,k )=k ν ε[µ, λ,k ,k ] Wek concludeµ ε[ν, thatλ,k the,k pole] is indeedþ(k8 present% k ) inε the[µ, L/Tν, amplituλ, (kde if thek conditions)] s1 = s2 =0 λµν (1−) 2 2 2 12 (−) 1 2 (+) 2 1 2 1 2 1 2 −with s =0aresimultaneouslysatisfied− · − + wT# k" ,k1,k$ 2S tλµν2(k1,k2 2t)λ,µν2(k1,k#yields̸ 2)= backk the$1ν ε equations[µ, λ,k1,k of2 motion] (39)k2 (µ52ε)[ν when, λ,k freely1,k2] (k1 k2) ε[µ, ν, λ, (k1 k2)] @" J5 A : k1 + k2 (48)k varied with respect to 1 and , respectively,− λ while evalu- − · − h i ¼À B + − kλ ε[µ, ν,k1λ,kµν 2] , 2 2 2 i k In the on-shell case (two photons2 on shell) ∆ (s, 0, 0) =kWµ+νλ(s,k0, 0) =k ε[k1,k2,µ,ν]. (89) k ating to (50) upon using1 these2 equations2π2 ofs motion to solve Thus the axial anomaly# (5) implies$ that there is a term in + − − kλ ε[µ, ν,k1,k2] , with the transverse tensors given% by % 2 2 for and eliminate the auxiliaryk2 fields. In the auxiliary field the one-loop(−) effective action in a background A" andk1B" kAnother2 interesting case is represented by a symmetric kinematical configurations in which t (k1,k2)= (k1 k2)λ − formkλ ofε[ theµ, effectiveν,k1,k2 action,] (53) the2 quantum2 expectation field, linearλµ inν of the form, − the2 external particles are massive gauge bosonsk of massk M.Thiswillturnusefulinthenext B − − ( k) value of the chiral current (46) is1 given by2 (+) tλµν (k1,k2)= (k1 k2)λ − kλ ε[µ, ν,k1,k2] 2 & sections, when' we will discuss the behaviour of a BIM2 amplitude with massive external lines tλµν (k1,k2)=k1ν ε[µ,(−)λ,k1,ke 2] k2µ ε[ν, λ,k1,k2] (k1 k2) ε[µ, ν, λ−, (k1 −k22 )] k t (k1,k2)=4 k1"#&'ν ε[µ, λ,k1,k2]+atk high2µ ε energy,[ν", λ,k showing,1,kS2eff] also in( thisk1 case,k2e) itsε[ poleµ, ν dominanc, λ,k]e.. There are some(40) conclusions Sλeffµν 2 −d x F"#F&'B; (49)− · & " − "h 1 &'' ¼À2 162% 2 (−) J5 1 −@ 1 · 2 @ À FF&' k1 + k2 Zk tλµνthat(k1 we,k can2)=½ draw¼ fromkB1ν this"ε¼[ studyµ, λ,k which¼1,k16 are%2]+ importantk2µ forε[ theν, λ an,kalysis1,k of2 the] next(k sections.1 k2) ε[µ, ν, λ,k]. (40) +! k ε[µ, ν,k ,k ] , − · or since @ B! hB,2 − λ 1 2 Notice that in all the cases that we have discussed it is possible to isolate(54) a 1/s contribution The form% factors¼ k wT (s, s1,s2)arealldefinedinthefollowingEqs.(50-52). in w for any kinematical configurations other than the massless (s 0) one, where the e2 2 2 L at least insofar as its anomalous divergence is concerned.→ (−) S k1 k2 The form% factors wT (s, s1,s2)arealldefinedinthefollowingEqs.(50-52). t (k ,k )= eff(Noticek k that)2 in this representationk ε[µ, ν,k the,k presence] The effective of massless action (53)poles reproduces is explicitthe anomalous for diver- any kine- λµν 1 2 ¼À1 16%2 λ −2 λ 1 2 " 20 − − k Notice thatgence in ofthisJ5 representation, but not necessarily the the nonanomalous presence of parts massless poles is explicit for any kinematical configuration4 4 "#&' and not just1 in! the massless collinearh i limit," where the diagram takes the & d x d y F"#F&''xhxyÀ @ B! y; of the tensor amplitude À p; q . The and 1 fields and (−) Â ½ ½ ð Þ t (k1,k2)=k1ν ε[µ,Zλ,k1Z,k2]+k2µ ε[ν, λmatical,k1,k2] configuration(k1theirk2 propagator) ε[ andµ, ν, notareλ,k a justlocal]. field in the representation massless(40) of collinear a mass- limit, where the diagram takes the λµν Dolgov-Zakharov form. A second observation(50) concerns the presence of other pole-like singulari- − less· 0À state propagating in the physical matrix element 1 Dolgov-Zakharov form.2 A second2 2 observation concerns the presence of other pole-like singulari- whereties inhxyÀ theis the transverse Green’s function invariant for the amplitude massless scalar and( tensor43) with strucp tures.q m It is0, whichthen mayobvious be represented that one has ¼ ¼ ¼ "# The form% factors wT (s, s ,s )arealldefinedinthefollowingEqs.(50-52)." by the effective tree diagram with source F F~ on one wave1 operator2 h @"@ . Thus from (46),ties this in nonlocal the transverse invariant amplitude and tensor"# structures. It is then obvious that one has to wonder whether¼ the pole present in wL is balanced,end and @!B awayon the from other, the as in collinear Fig. 3. region, by other Notice that in thisaction representation gives [25] the presence of massless poles is explicit! for any kine- to wonder whetherThe the same pole diagram present also represents in wL is the balanced,vector current away from the collinear region, by other contributions which2 are also singular. Indeed, as we are going" to show, this is the case. In fact, " e " 1 &' expectation value J B in the presence of a background matical configuration and notJ5 justA in2 the@ h masslessÀ FF collinear&'; (51)limit, where the diagramh i takes the due toh thei Schouten¼ 16% relations,contributions we are always whichaxial allowed field areB to! alsoand intro singular. gaugeduce field new Indeed,A", polar also implied as amplitudes we byare the going and to show, this is the case. In fact, original triangle diagram (Fig. 1) upon reversing the roles Dolgov-Zakharov form.whichbalance A explicitly second them exhibits observation with the additional massless concerns scalar contributionsdue pole the to in the pr theesence Schouten on the of remainin other relations, pole-likegtensorstructures.Infactweare we are singulari- always allowed to introduce new polar amplitudes and massless limit of (54), and which agrees with the explicit of the axial vertex and one of the vector vertices, i.e. " ties in the transversecalculation invariant of amplitude the physical 0 andJ p; tensor q trianglebalance struc amplitudetures. them with It is then additional obvious contributionsS that onee2 has on the remainingtensorstructures.Infactweare h j 5 j i " eff ~#" to two photons (43) in the previous section for p2 q2 J 2 F @# 13 ½ ¼ A" ¼À4% to wonder whether them2 pole0. present in wL is balanced, away¼ ¼ from the collinear region, by other The¼ nonlocal action (50) can be recast into a local form e2 13 F~#"@ h 1@!B : (55) contributions which areby the also introduction singular. of two Indeed, pseudoscalar as weauxiliary are fields going to show, this is¼ the4%2 case.# InÀ fact,! due to the Schoutenand relations,1 satisfying we the are second always order allowed linear equations to intro of duceThis crossing new polar symmetry amplitudes or equivalently, and the fact that the motion, nonlocal action (49) involves a mixed term involving both F F~"# and @!B is the reason why two massless pseu- balance them with additionalh contributions@!B ; on the remainin(52a)gtensorstructures.Infactweare"# ! ¼À ! doscalar auxiliary fields rather than just one are required to e2 e2 h1 F F~"# "#&'F F : (52b) describe the amplitude correctly through a local effective ¼ 8%2 "# ¼ 16%2 13 "# &' action. A single auxiliary field would necessarily produce

045014-9 Conformal anomalies share the same behaviour

✪ Anomaly poles as the common signature of chiral and conformal anomalies Phys.Lett. B682 (2009) 322-327 Armillis, Delle Rose, C.C.

Conformal Anomalies and the Gravitational Eﬀective Action: ✪ The TJJ Correlator for a Chiral Fermion

Phys.Rev. D81 (2010) 085001 (a)Roberta Armillis, (a,b)Claudio Corian`oand (a)Luigi Delle Rose 1

Giannotti and Mottola Phys.Rev. D79 (2009) 045014 (a)Dipartimento di Fisica, Universit`adel Salento The Trace and INFN SezioneAnomaly di Lecce,and ViaMassless Arnesano 73100Scalar Lecce, Italy ✪ Degrees of Freedom in Gravity (b) Department of Physics, University of Crete Heraklion, Crete, Greece

v The general conformalAbstract bootstrap in momentum space for 3-point functions We compute in linearizedconfirms gravitythese all the contributionsresults (Bzowski to the gravitat, McFaddenional effective, actionSkenderis due ) to a virtual Dirac fermion, related to the conformal anomaly. This requires, in perturbation theory, the identification of the gauge-gauge-graviton vertex off mass shell, involving the correlator of the energy-momentum tensor and two vector currents (TJJ), which is responsible for the generation of the gauge contributions to the conformal anomaly in gravity. We also present the anomalous effective action in the inverse mass of the fermion as in the Euler-Heisenberg case. arXiv:0910.3381v2 [hep-ph] 3 Feb 2010

[email protected], [email protected], [email protected]

1 2Theconformalanomalyandgravity2Theconformalanomalyandgravity

In thisIn section this section we briefly we briefly summarize summarize some somebasic andbasic well and known well knownaspectsaspects of the of trace the anomaly trace anomaly in in quantumquantum gravity gravity and, in and, particular, in particular, the identification the identification of thenon-localactionwhosevariationgenerates of thenon-localactionwhosevariationgenerates agiventraceanomaly.agiventraceanomaly. We recallWe recall that the that gravitational the gravitational trace anomaly trace anomaly in 4 spacetim in 4 spacetimedimensionsgeneratedbyquantumedimensionsgeneratedbyquantum effectseff inects a classical in a classical gravitational gravitational and electromagnetic and electromagnetic background background is given is by given the by expression the expression

µ µ1 12 ′2 ′ 2! 2 2 2 Tµ = T =2bC +22bCb +2E b ER +2!RcF+2cF (1) (1) −µ8 −8 − 3 − 3 2Theconformalanomalyandgravity! ! " " # # $ $ ′ ′ wherewhereb, b andb, b′candarec parametersare parameters that for that a single for a single fermion fermion in the in theory the theory result resultb =1/b320=1π2/,320b π=2, b′ = 2 2 2 22 2 2 2 11/576011/π5760,andπ c,and= ec =/24 πe /;furthermore24 π ;furthermorep C denotesC denotes the Weyl the tensor Weyl tensor squared squared and E andis theE Euleris the Euler In this section we briefly− summarize− some− basic− and well known aspects of the trace anomaly in density given by p + l p quantum gravity and, in particular,densityk given the identificationby k of thenon-localactionwhosevariationgeneratesk p l l l − q = + exch. ++exch.R2 2 2 2 λµνρ λµνρ λµνρ λµνρ µν µν R agiventraceanomaly. C =C Cλ=l −µ qνρCCλµνρC= Rλµ=νρRRλµνρR 2Rµν R2Rµ+ν R + (2) (2) q − − 3 3 ∗ ∗∗ qλµνρ∗ λµνρ λµνρ λµνρ µν µ2ν 2 We recall that the gravitational trace anomalyE =E R=λ inµνρR 4Rλ spacetimµνρ R= Rλµ=νρedimensionsgeneratedbyquantumRRλµνρR 4qRµν R4Rµ+ν RR . + R . (3) (3) (a) (b) (c) − − effects in a classical gravitationalThe effTheective and effective action electromagnetic action is identified is identified by bacsolving bykground solving the following the is following given variational by variational the equation expression equation by inspection by inspection Figure 1: The complete one-loop vertex (a) given by the sum of the 1PI contributions called V µναβ(p, q)(b) µναβ 2 δΓ and W (p, q)(c). 2 δΓµ µ µ 1 2 ′ 2 gµν gµν= Tµ2. = T . Armillis et al (4) (4) T = 2bC +2b E − √!gR− δg+2µν cFδg µ (1) µ −8 − 3 √ µν Giannotti and Mottola Itsand solutionIts solution is well! is known well known and is and given" is bygiven the by non-local the# non-local expre$ssion expression ′ ′ ′ 2 where b, b and c are parameters that for a single fermiond4l tr W inµνα the(l/ + m theory)γβ(l/ q/ + resultm) b =1/320 π , b = SanomS[g,anom AW]=[µg,ναβ A(]=p, q)= (ie2)i2 − , (33) (5) (5) 2 2 2 2 4 2 2 2 2 1 − (2π2) " [l m ][(l q) m ] # 2 11/5760 π ,andc = e /24 π ;furthermore41 4 ′ C ′′denotes! ! the2 Weyl− ′ tensor− −′ 2 squared′ ′ and! 2E is theµ Eulerν d x√ d4gx√d gx d4xg E g′ E R !GR4(x, xG)(x,2bC x ) +2bCb 2 E+ b ER +2!RcFµ+2ν FcF F. µν . − − so that the8 one-loop− amplitude in Fig.− 1 results− 3 4 − 3 µν′ 8 − " − −#3x x ! " −#3 $x x′ density given by % % % %& " # ! " # $ µναβ µναβ &2 µνβα µναβ µνβα NoticeNotice thatΓ we that are(p, we q)= omitting areV omitting(p,√ qgR)+ V√termsgR(2q,pterms which)+W which are(p, not q)+ are necessaryW not necessary(q,p) at. one at loop one(34) level. loop level. The notation The notation ′ 2 G24(x, x )denotestheGreen’sfunctionofthedi′ λµνρ λµνρ fferentialµ operatorν R defined by CThe bare=G4 Ward(x,C xλ identityµ)denotestheGreen’sfunctionofthediνρC which allows= R toλµ defineνρR the divergent2 amRµplitudesνffRerential that+ contribute operator to de thefined anomaly by (2) in Γ in terms of the remaining finite ones is obtained by− re-expressing the classical3 equation µ ν µν 2 µν2 2 µν 1 µ 1 2 2 ∆∗ 4 µ∗ λµνρ +2µ Rν µRgν λµνρ µνν= ! +2µRν2 µ µ2νν + ( R) µµ R! (6) ∆4 µ +2R Rg ν = ! +2R µ ν + ( R) µ R! (6) E = Rλ≡∇µνρ R∇ ∇ = Rλµ−νρ3µRν µ∇ν 4Rµν R +∇R∇ . 3 ∇ ∇ − 3 (3) ≡∇" ∇ ∇ ∂ν T =− 3F# Jν ∇ ∇ ∇ 3 (35)∇ ∇ − 3 " ph − − # with G theand gravitational requiresand requires constant. some boundary some The boundary non-local conditions action conditions reduces to be specified. to to be specified. This o Thisperator operator is conformally is conformally covariant, covariant, in fact in fact The effective action isas identified an equation of generating by solving functionals the in following the background var electromagneticiational equation field by inspection under a rescaling of thec metric4 one4 can′ show′ (1) that−1 αβ underS [ ag, Arescaling]= ofd x the√ g metricd x oneg canR show! ′ [F thatF ] ′ , (9) anom µν x µν x,x αβ x −6 − ∂ν T A−= F Jν A, Exchange of an (36)anomaly pole ! 2 ! ⟨δphΓ⟩ −σ ⟨ ⟩ −2σ ¯ gµ"ν = e g¯µµν σ ∆4 = e ∆4−. 2σ ¯ (7) valid for a weak gravitational field. In this caseg =gµTν =. e→g¯µν ∆4 = e ∆4. (4) (7) which can be expanded perturbativelyµν as µ → − √g δgµν (1) x x µν ! µν Notice that the generalRx ∂µ solution∂ν h ofh, (4) involves, h = ηµν h in. principle, also a conformally(10) invariant part that is Notice that the≡ generalµν solution− µν of (4)4 involves, in principle, also a conformally invariant part that is ∂ν T A = F Jν d w(ie)J A(w) + ... . (37) 2 not identified by this method.⟨ ph ⟩ As− in ref.⟨ [16], we concentrate· ⟩ on the contribution proportional to F 2 Its solution is wellThe known presence and ofnot the is Green’s identified given function by by thethis of the method. non-local! operator As in in expre! Eq. ref. (9) [16],ssion is the we clear concentrate indicationon that the the contribution proportional to F and perform an expansion of this term for a weak gravitationalfieldanddropfromthisactionallthe solution ofNotice the anomaly that we equation have omitted is characterized the first term by in an the anom Dyson’saly pole. seriesof In theJν nextA,shownonther.h.sof(37) sections we are and perform an expansion of this term for a weak gravitationa⟨ ⟩ lfieldanddropfromthisactionallthe going to performtermssince J which aν direct=0.ThebareWardidentitythentakestheform are diagrammatic at least quadratic computation in the of this deviation action and of the reobtain metric from from it the flat pole space Sanom[g, A]= ⟨terms⟩ which are at least quadratic in the deviation of the metric from flat space (5) contribution identified in the dispersive analysis of [16] and the conformal invariant extra terms which 2 µλ 2 1 ′ 2 µναβg =δ ηF′ +(z)κJhλ(z) A ′κ =16π 2G, (8) 4 are not present4 in (9).′ We start with! an∂ analysisν Γ µ=ν of theµ correlν ⟨atorµν following2⟩ an approach2 ! which is (38) µν d x√ g d x g E R G (x, xgδµAν)α=(x2)ηδbCAµν (+y)κ+hµbν E κ =16Rπ G, +2cF F . (8) 4 $ β %&A=0 µν 8 −close to that followed− in ref. [16].− 3 The crucial point of the derivation presented& in− that3 work is the ′ x & x % % which takes contribution" only from# the first term on! the r.h.s4of Eq.& (37)." This relation can# be written $ imposition of the& Ward identity for the TJJ correlator (see Eq. (42) below) which4 allows to eliminate Notice that we areall the omitting Schwingerin momentum√ (gradients)gR space.2 terms terms For this which which we useotherwise the are definition plague not ofany necessary the derivation vacuum polarization based at one on the loop canonical level. The notation ′ formalism and are generated by the equal-timeαβ commutator of2 the energy momentum tensor with the Π (x, y) ie Jα(x)Jβ (y) , (39) G4(x, x )denotestheGreen’sfunctionofthedivector currents. In reality, this approach can beff bypassederential≡− byjustimposingatadiagrammaticlevelthe operator⟨ ⟩ defined by validity of an operatorial relation for the trace anomaly, evaluated at a nonzero momentum transfer, 8 togetherµ withν the conservationµν 2 of the vectorµν currents on the2 other twoµν vector vertices1 of theµ correlator. 2 ∆ µ +2R Rg ν = ! +2R µ ν + ( R) µ R! (6) 4 ≡∇ ∇ ∇ − 3 ∇ ∇ ∇ 3 ∇ ∇ − 3 3Theconstructionofthefullamplitude" # Γµναβ(p, q) and requires some boundary conditions to be specified. This operator is conformally covariant, in fact We consider the standard QED lagrangian under a rescaling of the metric one can show that 1 µν µ = Fµν F + i ψγ¯ (∂µ ieAµ)ψ mψψ¯ , (11) L −4 − − σ −2σ ¯ with the energy momentum tensorgµν = splite intog¯µ theν free∆ fermioni= cparte T∆f ,theinteractingfermion-photon. (7) → 4 4 part Tfp and the photon contribution Tph which are given by ↔ ↔ µν (µ ν) µν λ Notice that the general solution of (4)T = involves,iψγ¯ ∂ ψ in+ g princip(iψγ¯ ∂ λψle,m alsoψψ¯ ), a conformally invariant(12) part that is f − − 2 µν (µ ν) µν λ not identified by this method. As in ref.T [16],= eJ weA concentrate+ eg J Aλ , on the contribution(13) proportional to F fp − and perform an expansionand of this term for a weak gravitationalfieldanddropfromthisactionallthe µν µλ ν 1 µν λρ T = F F g F Fλρ, (14) terms which are at least quadratic in theph deviationλ − of4 the metric from flat space where the current is defined as J µ(x)=ψ¯(x)γµψ(x) .2 (15) gµν = ηµν + κ hµν κ =16π G, (8) In the coupling to gravity of the total energy momentum tensor

T µν T µν + T µν + T µν (16) ≡ f 4 fp ph 5 1 form factor in the TVV is responsible for the anomaly

Appearance of sum rules as a signature of the anomaly away from the conformal point

This is a strong indication of compositeness in the IR realization of the anomaly action

(Serino, Marzo, Delle Rose, C C.)

Dilaton Interactions and the Anomalous Breaking of Scale Invariance of the Standard Model γ Claudio Corian`o, Luigi Delle Rose, Antonio Quintavalle and Mirko Serino

µ JDipartimentoD di Matematica e Fisica ”Ennio De Giorgi” Universit`adel Salento and INFN-Lecceγ Via Arnesano 73100, Lecce, Italy1

Figure 5: Exchange of a dilaton pole mediatedAbstract by the JDVV correlator.

We discuss the main features of dilaton interactions for fundamental and effective dilaton fields. In with the tensor basis givenparticular, by we elaborate on the various ways in which dilatons can couple to the Standard Model and on the role played by the conformal anomaly as a way to characterize their interactions. In the case of a dilaton t µναβ(p, q)=(s ηµν kµkν ) uαβ (p, q), 1 derived from a− metric compactification (graviscalar), we present the structure of the radiative corrections µναβ to its decay intoαβ two photons,µν a photonµ andν a Zµ,twoν Z gaugeµ bosonsν andµ ν two gluons, together with their t2 (p, q)= 2 u (p, q)[s η +2(p p + q q ) 4(p q + q p )] , renormalization− properties. We proves that, in the electroweak− sector, the renormalization of the theory is t µναβ(p, q)= pµqν + pνqµ gαβ + ηαν ηβµ + ηαµηβν 3 guaranteed only if the Higgs is conformally2 coupled. For such a dilaton, its coupling to the trace anomaly is quite general, ands determines, for# instance, an enhancement$ of its decay rates into two photons and two ! ηµν ηαβ " qαpβ ηβνpµ + ηβµpν qα ηαν qµ + ηαµqν pβ, (80) gluons.− We then2 turn our− attention− to theories containing a non-gr− avitational (effective) dilaton, which, in our perturbative analysis, manifests as a pseudo-Nambu Goldstone mode of the dilatation current (JD). The ab # $ # $2 ! 2 " where δ is the diagonalinfrared matrix coupling in color of such space. a state Again to the we two-photons have s = andk to=( thep tw+o-gluonsq) ,withthevirtualitiesofthe sector, and the corresponding 2 anomaly2 enhancements of its decay rates in these channels, is critically analyzed. two gluons being p = q =0.Noticethatinthemasslessfermionlimitonlythefirst(t1)ofthese3form factors contributes to the anomaly. The corresponding on-shell form factors with massive quarks are n g2 g2 f 1 1 4m2 Φ (s;0, 0) = (2n 11N )+ m2 (s, 0, 0,m2,m2,m2) 1 i , 1 −72π2 s f − C 6π2 i s2 − 2sC0 i i i − s i=1 & ' ()

arXiv:1206.0590v2 [hep-ph] 24 Apr 2013 % g2 Φ (s;0, 0) = (n N ) 2 −288π2 s f − C n g2 f 1 3 1 2m2 m2 + (s, 0, 0,m2,m2)+ (s, 0, 0,m2,m2,m2) 1+ i , − 24π2 i s2 s2 D0 i i sC0 i i i s %i=1 & ' () g2 g2 N 11 Φ (s;0, 0) = (11n 65N ) C (s, 0, 0) (0, 0, 0) + s (s, 0, 0, 0, 0, 0) 3 288π2 f − C − 8π2 6 B0 − B0 C0 ' ( n g2 f 1 1 5 2m2 + (s, m2,m2)+m2 + (s, 0, 0,m2)+ (s, 0, 0,m2,m2,m2) 1+ i , 8π2 3B0 i i i s 3sD0 i C0 i i i s i=1 & ' ' (() %1 [email protected], [email protected], [email protected], [email protected] (81)

1 where mi denotes the quark mass, and we have summed over the fermion ﬂavours (nf ), while NC denotes the number of colors. Notice the appearance of the 1/s pole in Φ1,whichsaturatesthecontributiontothetrace anomaly in the massless limit which becomes g2 Φ (s;0, 0) = (2n 11N ) . (82) 1 −72π2 s f − C As for the QED case, this is the only invariant amplitude whichcontributestotheanomaloustracepartofthe correlator. The pole completely accounts for the trace anomaly and is clearly inherited by the QCD dilatation

20 where the symbol on the right indicates that the quantity is evaluated at ✓ = ✓¯ = 0. | Notice that in the above equations the F - and D-terms have been removed exploiting their equations of motion. Having deﬁned the model, we can introduce the Ferrara-Zumino hypercurrent

V V 1 V V V =Tr W¯ e W e ¯ ¯ e e D¯ + ¯ D e , (21) JAA˙ A˙ A 3 rA˙ rA A˙ rA rA˙ A  where is the gauge-covariant⇥ derivative⇤ in the superﬁeld formalism whose action on chiral superﬁelds is rA given by

V V V V = e D e , ¯ ¯ = e D¯ e ¯ . (22) rA A rA˙ A˙ The conservation equation for the hypercurrent is JAA˙ 2 Superconformal˙ 2 g Sum Rules and the1 Spectral Density Flow@ () D¯ A = D (3T (A) T (R)) TrW 2 D¯ 2(¯eV )+ 3 () W , (23) of theJAA˙ Composite3 A 16⇡2 Dilaton (ADD) Multiplet 8 in =1W Theories @  N✓ ◆ where is the anomalous dimension of the chiral superfield. The first two terms in Eq. (23) describe the quantum anomaly of the hypercurrent, while the last is of classicalDelle Rose, Costantini, Serino, C.C. JHEP 2014 origin and it is entirely given by the superpotential. In particular, for a classical scale invariant theory, in which is cubic in the superfields or identically zero, this term identically vanishes. If, on Claudio Corianò,AntonioW Costantini, Luigi Delle Rose and Mirko Serino the other hand, the superpotential is quadratic the conservation equation of the hypercurrent acquires a non-zero contribution even at classical level. This describes the explicit breaking of the conformal symmetry. We can now project the hypercurrent defined in Eq.(21) onto its components. The lowest component JAA˙ Dipartimento diµ Matematica e Fisica ”Ennio De Giorgi”, Universitàdelµ Salento¯ and is given by the R current, the ✓ term is associated with the supercurrent SA,whilethe✓✓ component contains the energy-momentum(a) INFN-Lecce, tensor T Viaµ⌫.Inthe Arnesano,(b) = 1 73100super Yang-Mills Lecce,(c) Italy theory⇤ described by the Lagrangian N Figurein Eq. 1: The (17), triangle these diagram three currents in the fermion are defined case (a), as the collinear fermion configuration responsible for

the anomalyµ (b) and aa diagrammaticµ a 1 representationµ of theµ exchange viaµ an intermediate state (dashed line) R = ¯ ¯ + ¯ ¯ +2i†Abstract 2i( )† , (24) (c). 3 i i i Dij j Dij j i µ ⌫⇢ µ¯a ⇣ a µ ⌫ µ⌘ We discussS the= signaturei( of)AF the⌫⇢ anomalousp2(⌫¯ breakingi)A( ijj of)† theip superconformal2( ¯i) †(†) symmetry in =1super A D Wi N vertex at nonzero momentuma transfersµ a in QEDµ [19][20] and QCD [21]. The case of a general non-abelian Yang Mills theory, mediatedig(†T j by)( the¯ ) Ferrara-Zumino+ S , hypercurrent ( ) with two vector ( )supercurrents(25) theory, with the inclusioni ofij scalars, is newA andIA is discussed - in the masslessJ case - in Section 3.V This ( ) and its manifestation in thei anomaly action, in the form of anomaly poles. This allows to inves- JVVmay serveT toµ⌫ highlight= F someaµ⇢F specifica ⌫ + properties¯a¯µ( ofac these@!⌫ typesgtabc ofA verticesb ⌫)c + which¯a¯µ have( ac not@ received⌫ gtabc suAbcient⌫)c +(µ ⌫) tigate in a unified way both conformal⇢ 4 and chiral anomalies. The analysis is performed in parallel$ to the attention in the past.  StandardThen Model, we turn to for a perturbativecomparison.µ ⌫ study We of investigate, the⌫ e↵ectiveµ action in particular, ini the caseµ massive of supersymmetric!⌫ deformationsa a ⌫ = 1 theories, of the = 1 theory +( j)†( k)+( ijj)†( k)+ ¯i¯ (ij @ +igTijA N)j N Dij Dik D Dik 4 andfocusing the spectral on the components densities of the the FZ anomaly multiplet form and on factors the corresponding which are anomalies. extracted This from is followed the components by a of this study of the spectral densities of the relevant diagrams which are responsible for the superconformal anomaly. correlator. In this+¯ extended ¯µ( framework@ ⌫ +igT a A ita is⌫) shown +(µ that⌫ all) the⌘µ anomaly⌫ + T µ⌫ form, factors are characterized by (26) We show the existencei of a uniqueij sum ruleij for eachj component$ of the multiplet,L I and of a spectral density spectral densities which flow with the mass deformation. In particular, the continuum contributions from flow driven by the mass perturbations. Asµ the deformationµ⌫ (mass) parameter turns to zero, restoring the the two-particlewhere is given cutsof in theEq.(17) intermediate and S and statesT turnare the into terms into of poles improvement in the zero in massd = 4 limit, of the with supercurrent a single and superconformalL symmetry, the flow gets localizedI at zeroI invariant mass, signalling the exchange of a massless of the EMT respectively. As in the non-supersymmetric case, these terms are necessary only for a scalar sumpole rule in the satisfied anomaly by e↵ eachective component. action. We will Non compare anomalous non supersymmetric form factors, and instead, supersymmetric in the realizations, same anomalous correlators,highlightingfield are and, characterized the therefore, di↵erences byreceive between non-integrable contributions the two cases. spectral In only particular from densities. the we will chiral These show multiplet. in tend detail to how They uniform the arecancellation distributions explicitly given as by one of the extra poles of non anomalous form factors is realized in supersymmetric theories. Finally, we present moves towards the conformal point, with aµ clear dual4p2 behaviour.µ⌫ As in a previous analysis of the dilaton the structure of the anomaly action as a combinationSIA = of the polei contributions,@⌫(ii†) plus, the non anomalous (27) pole of the Standard Model, also in this case the poles3 can be interpretedA as signaling the exchange of a (logarithmic) terms. In superspace, the first hadµ⌫ been identified1 µ⌫h 2 in theµ past⌫ i relying on supersymmetric composite dilaton/axion/dilatino (ADD)T multiplet= in⌘ the@ e↵ective@ @ Lagrangian.† . The pole-like behaviour (28) arguments [2]. We will conclude our analysisI with some3 comments on the possiblei i implications of our of the anomaly form factors is shown to be a global feature of the correlators, present at all energy scales, results about the physical manifestation of anomaly poles - for global anomalous symmetries - and their duecancellations to the sum in rules. the case A of similar their superconformal behaviour is gaugings. shown to be11 present in the Konishi current, which identifies additional composite states. We conclude that global anomalous currents characterized by a single flow in arXiv:1402.6369v3 [hep-th] 2 Jun 2014 the2 perturbative Sum rules picture always predict the existence of composite interpolating fields. In case of gauging of these currents, as in superconformal theories coupled to gravity, we show that the cancellation of the As pointed out long ago in the literature on the chiral anomaly [22], its perturbative signature is in the corresponding anomalies requires either a vanishing function or the inclusion of an extra gravitational appearance of a massless pole (an anomaly pole) in the spectrum of the AV V diagram. The pole is present, h i sectorin perturbation which e↵ectively theory, only sets in the a specific residue kinematic at the configuration,anomaly poles namely of the at zero gauged fermion currents mass and to vanish. with on-shell vector lines. The intermediate state which is exchanged in the e↵ective action, see Fig. (1), and which is mediated by the AV V diagram, is characterized by two massless and collinear fermions moving h i on the light cone. It is rather compelling to interpret the appearance of this intermediate configuration - within the obvious limitations of the perturbative picture - as signalling the possible exchange of a bound state in the quantum e↵ective action.

⇤[email protected], [email protected], [email protected], [email protected]

1 we are going to show next, share a similar behaviour. We would expect, though, that the breaking of a symmetry should manifest in the apperance of a massless state in the spectrum of the e↵ective theory, and in this respect the saturation of the spectral density with a single resonance in an anomaly form factor acquires a special status. Elaborating on Eq. (1), one can show that the e↵ect of the anomaly is, in general, related The terms of improvement are automatically conserved and guarantee, for () = 0, upon using the W equations ofto motion, the the behaviour vanishing of the classical of the trace spectral of T µ⌫ and of density the classical gamma-trace at any of values the of s, although, in some kinematical limits, it is the supercurrent Sµ . regionA around the light cone (s 0) which dominates the sum rule, and amounts to a resonant contribution. The anomaly equations in the component formalism, which can be⇠ projected out from Eq. (23), are In a phenomenological context, what gives a far broader significance to this kinematical2 mechanism is In fact, the combinationg2 1 of the scaling behaviour of the corresponding form factor F (Q ) (equivalently of @ Rµ = T (A) T (R) F aµ⌫Fã , (29) µ 16⇡2 3 µ⌫ ✓ ◆ Suoerconformalanomaly the appearanceits density ⇢ of) with a3 g certain2 the requirement1 kinematic of integrabilityduality, which of the accompanies spectral density, any essentially perturbative fix f to anomaly. be a constant In this case ¯ Sµ = i T (A) T (R) ¯a¯µ⌫ F a , (30) µ A 8⇡2 3 A µ⌫ multiplet ✓ 2 ◆ it is betterand the known sum rule as3 g (1)2 Q to-duality be1 saturated , relating by a the single resonance massless resonance. and the asymptotic Obviously, a region superconvergent of a certain sum correlator rule, in ⌘ T µ⌫ = T (A) T (R) F aµ⌫F a . (31) µ⌫ 32⇡2 3 µ⌫ obtained for f = 0,✓ would not◆ share this behaviour. At the same time, the absence of subtractions in the Thea first nontrivial and the last equations way are [23].respectively This extracted property, from the imaginary in and general,the real part of the finds✓ its justification in the existence of a sum rule for the Ρ !Π component ofdispersion Eq.(23),Χ while the relationsgamma-trace of2 the guarantees supercurrent comes the from significancethe lowest component. of the sum rule, being this independent of any ultraviolet spectral1.0 density ⇢(s, m ). Generically, it is given in the form cuto↵. 5 The perturbative expansion in the component formalism 0.8 It is quite straightforward to show that1 Eq.1 (1) is a constraint2 on asymptotic behaviour of the related In this section we will present the one-loop perturbative analysis of the one-particle irreducible correlators,⇢(s, m )ds = f, (1) built with a singleform0.6 current factor. insertion The contributing proof - at leading is obtained order in the gauge by coupling observing constant⇡ 0 - that to the the dispersion relation for a form factor in the spacelike anomaly equations previously discussed. Z 2 2 We define theregion three correlation (Q functions,= k(R)>, (S0)) and (T ) as with the0.4 constant f independent of any mass (or other) parameter which characterizes the thresholds or the 2 ab µ↵(p, q) Rµ(k) Aa ↵(p) Ab (q) RV V , 1 1 ⇢(s, m ) (R) ⌘h ih i 2 2 ab µ↵ µ a ↵ ¯b F (Q ,m )= ds , (2) strengths0.2 of the(p, resonant q) S (k) A states(p) (q) eventuallySV F , present in the integration2 region (s>0). It should be stressed (S) AB˙ ⌘hA B˙ ih i ⇡ 0 s + Q ab µ⌫↵(p, q) T µ⌫(k) Aa ↵(p) Ab (q) TVV , (32) Z that sum rules(T ) formulated⌘h for theih studyi 2 of the1 structure1 1 of1 the resonances, i.e. their strengths and masses, once we expand the denominator in Q ass 2 = 2 2 s 2 + ... and make use of Eq. (1), induces the with k = p + q and where0 we have factorized,2 for the4 sake of simplicity,6 the Kronecker8 deltas on+ theQ adjointQ Q Q indices.as in These the correlation QCD functions case, have been involve computed at a one-loop parameterization order in the dimensional2 2 reduction of the resonant behaviour of ⇢(s, m) at low s values, via a following asymptotic behaviour(n) on F (Q ,m ) Anomaly form factors ⇢ 2 schemeFigure (DRed) 6: Representatives using the Feynman of the family rules listed of spectral in Appendix densities E. We⇡ (s recall) plotted that versus in thiss schemein units the of tensorm .The andphenomenologicalfamily scalar ”flows” loop integrals towards are the computeds = 0 region approach,in thebecoming analytically a (s) continued function with as spacetimem the2 goes towhile inclusion zero. the sigma algebra of is the asymptotic behaviour of the correlator, amenable to restricted to four dimensions. lim Q2F (Q2,m2)=f. (3) InperturbationFrom order to the provide two branches more details, encountered theory, we will withpresent forthe thei results✏ largerprescriptions, for thes matter. the For discontinuity chiral and this gauge is then significant vector2 present multiplets only for interplay between the infrared (IR) and the ultraviolet 3 SUM RULES for the ±anomalies (chiral and conformalQ ) separately,k2 > 4m for2, as on-shell expected external from unitaritygauge lines. arguments, The chiral and contribution the result for will the be discontinuity, discussed first, obtained!1 and the using result the will(UV)definition be given in with regions, Eq. the (76), inclusion clearly the of agrees the corresponding withterm2 Eq. (77),duality mass computed corrections. insteadis indeed by the cutting2 quite rules. appropriate to qualify the implications of a given sum rule. NoticeThe thatdispersive theThe matter representationF chiral superfield off/Q(k2,m belongs2)behaviour in this to a case certain is written representation at as largeR ofQ the,with gauge group.f If theindependent of m, shows the pole dominance of F for C0 representation is complex, for⇠ instance the fundamental of SU(N), then the superfield must be accompanied It was2 pointed out, some time2 ago, that one specific tract of the chiral anomaly is in the existence Q . The UV/IR2 2 1 conspiracy1 ⇢(s, m ) of the anomaly, discussed in [31, 32, 20], is in the reappearance of the pole by another superfield (eventually with the0(k same,m mass))= belongingds to the2 , complex-conjugate representation(79) C ⇡ 2 s k ¯ !1¯a ¯ 4m ¯a a T a R. In this case, the generator T of R are related to thoseZ of R by the equation T = (T ) = (T )⇤. For2 of a sumcontribution2 rule for at the veryAV large V valuediagram of the [24],invariant laterQ extended, even for a to nonzero a similar mass studym. In of fact, the as trace we are of going the energy simplicity,which, for ink the< following0 gives the we identitywill consider just theh case of a singlei chiral superfield in a real representation of the gauge group. The extension to a complex representation2 amounts just2 to a2 factor of 2 in front of momentumto show1 tensor inds the following(tr1+T⌧)(s, m for) sections, the1 tr2 TVV⌧ the(k ,m spectral)+1 in QED density (with hasV supporta vector around current), the s = at 0 region zero momentum (⇢(s) (s)), transfers 2 log = 2 log , (80) 2 2 2 2 ⇠ 4m (s k )s 1 p⌧(s, m )! 2k h p⌧(k ,m ) i1 [25, 26].as in SimilarZ the massless analysis (m12= were 0) case. performed This point on is the quiteTT subtle,correlator since the in flows 2-dimensional of the spectral gravity densities [27], with which is with ⇢(s, m2) given by Eqs. (75) and (77). Thep identity in Eq. (80) allowsp to reconstruct the scalar integral 2 2 h i 0(k ,m )m fromshow its dispersive the part. decoupling of the anomaly pole for a nonzero mass. Here, the term decoupling will be used C a↵Havingected determined by the the spectral trace function ofanomaly. the scalar integral The(k2,m2), analysis we can extract the brought spectral substantial evidence that the sum rule, combined to refer to the non resonant behaviourC0 of ⇢. Therefore, the presence of a 1/Q2 term in the anomaly form withdensity associated the originalwith the anomaly identification form factors in Eqs. (61), (62), of (63) the and (74), anomaly which is given by pole from the perturbative spectral density [22], were two factors is a property(k2,m2) of the(k2,m2 entire)/k2, flow which a) converges(81) to a localized massless state (i.e. ⇢(s) (s)) important and related aspects⌘ 1 of the anomaly phenomenon. We recall that the study of⇠ these types of and whichas can bem computed0, as while b) the presence of a non vanishing sum rule guarantees the validity of the asymptotic ! 2 2 correlators2 2 has2 a quite2 2 long2 history1 [28,2 29,0(k ,m 30].) Discconstraint(k ,m )=(k + illustratedi✏,m ) (k i✏,m in)= Eq.Disc (3). Notice2m Disc C that although. (82) for conformal deformations driven by a single mass k2 k2 ✓ ◆ ✓ ◆ Moreparameter recently, the independence very general of perturbative the asymptotic analysis value f on of them is aTVV simplecorrelator consequence (or of the graviton-gauge-gauge scaling behaviour ver- 22 h i tex), performedof F (Q2,m2), o↵ it-shell holds and quite at generally nonzero even momentum for a completely transfer, o↵-shell have kinematics shown that [19]. the general features observed in the AVIn summary,V and tr inTVV completecases agreement where with preserved a previous [19, analysis 20, 21, by 31]. Giannotti and Mottola [19], we are going toh verifyi that forh a generici supersymmetric = 1 theory, the two basic features of the anomalous behaviour A specific feature of the spectral density of theN chiral and conformal anomalies is that the pole is introduced of a certain form factor responsible for chiral or conformal anomalies are: 1) the existence of a spectral flow in the spectrum in a specific kinematical limit, as a degeneracy of the two-particle cut when any second scale which turns a dispersive cut into a pole as m goes to zero and 2) the existence of a sum rule which relates (for instancethe asymptotic the fermion behaviour mass) of the is sent anomaly to zero. form The factor property to the strength of the cut of the turning pole resonance. into a pole is peculiar to finite (non superconvergent) sum rules. It is related to a spectral density which is normalized by the sum rule just like an ordinary weighted distribution, and whose support6 is located at the edge of the allowed phase space (s = 0) as the conformal deformation turns to zero. This allows to single out a unique interpolating state out of all the possible exchanges permitted in the continuum, i.e. for s>4m2, as the theory flows towards its conformal/superconformal point.

2.1 Sum rule and the UV/IR conspiracy of the anomaly

As we have just mentioned, the existence of a sum rule for the form factor responsible for a certain anomaly indicates a UV/IR connection manifested by the corresponding spectral density, but it is not exclusively related, obviously, to the anomaly phenomenon. In fact, non anomalous form factors, in some cases, as

5 bound states. For convenience, this point has been brieﬂy reviewed by us, in the case of three-point correlators, in Appendix D, to which we refer for further detais. Here, instead, we just present the expression of the quantum e↵ective action obtained from the three-point correlation functions that we have previously discussed. We consider the massless case for the chiral supermultiplet and on-shell external gauge bosons and gauginos. The anomalous part is given by the three terms Away from the conformal limit (nonzero mass) Sanom = Saxion + Sdilatino + Sdilaton (156) Appearance of sum rules and spectral density flows which are given, respectively, by

g2 T (R) 1 1 S = T (A) d4zd4x @µB (z) F (x)F˜↵(x) (157) axion 4⇡2 3 µ 4 ↵ ✓ ◆ Z ⇤zx g2 T (R) @ 1 S = T (A) d4zd4x @ (z)µ⌫⇢ ⇢ ¯↵¯(x) F (x)+h.c. (158) dilatino 2⇡2 3 ⌫ µ 2 ↵ ✓ ◆ Z  ⇤zx g2 T (R) 1 1 S = T (A) d4zd4x ( h(z) @µ@⌫h (z)) F (x)F ↵(x). (159) dilaton 8⇡2 3 ⇤ µ⌫ 4 ↵ ✓ ◆ Z ⇤zx We show in Figs. 10 the three types of intermediate states which interpolate between the Ferrara-Zumino hypercurrent and the gauge (A) and the gaugino () of the ﬁnal state. The axion is identiﬁed by the collinear exchange of a bound fermion/antifermion pair in a pseudoscalar state, generated in the RV V correlator. Aa ↵(p) Aa ↵(p) h i In the case of the SV F correlator, the intermediate state is a collinear scalar/fermion pair, interpreted as h µi µ a dilatino. In the TVVR (kcase,) the collinear exchange is aS linearA(k) combination of a fermion/antifermion and h i b ¯b (q) scalar/scalar pairs. A (q) B˙

a ↵ a ↵ The non-anomalous contribution is associatedA with(p) the extra term S0 which is givenA (p) by

2 T µ⌫(k) T µ⌫(k) g 4 4 ˜ + ˜ µ⌫ S0 = 2 d zd xhµ⌫(z) T (R) 2(z x)+T (A) V (z x) Tgauge(x) 16⇡ Ab (q) Ab (q) Z g2 ⇣ ⌘ + d4zd4x i (z) T (R) ˜ (z x)+T (A) V˜ (z x) Sµ (x)+h.c. , (160) Figure 10: The64⇡ collinear2 diagramsµ corresponding to2 the exchange of a composite axiongauge (top right), a dilatino (top left) and theZ two sectors of an intermediate⇣ dilaton (bottom). Dashed lines⌘ denote intermediate scalars. where ˜ (z x) and V˜ (z x) are the Fourier transforms of (k2, 0) and V (k2) respectively. Their contri- 2 2 We recall that in the = 4 theory the spectrum contains a gauge field Aµ, four complex fermions i butions in position space correspondN to nonlocal logarithmic terms. (i =1, 2, 3, 4) and six real scalars = (i, j =1, 2, 3, 4). All fields are in the adjoint representation of The relation between anomaly poles,ij spectral ji density flows and sum rules appear to be a significant the gauge group. feature of supersymmetric theories a↵ected by anomalies. It is then clear that supersymmetric anomaly-free From the point of view of the = 1 SYM, this theory can be interpreted as describing a vector and N theories shouldthree massless be free chiral of such supermultiplets, contributions all in in the the adjoint anomaly representation. e↵ective Therefore action. the In thisTVV respect,correlator it in natural h i to turn to the= 4 can= be 4 easily theory, computed which from is the free general of anomalies, expressions in in order Eqs (51) to and verify (59) and which validate give this reasoning. N N Indeed the function of the gauge coupling constant in this theory has been shown to vanish up to three g2 T (A) g2 T (A) µ⌫↵(p, q)= V (k2)+3 (k2, 0) tµ⌫↵(p, q)= k2 (k2, 0) tµ⌫↵(p, q) . (161) loops [37, 38, 39],(T ) and there16 are⇡2 several arguments2 about2S its vanishing 8⇡2 toC0 all the2 perturbativeS orders. As ⇥ ⇤ a consequence,One can the immediately anomaly observe coecient from inthe the expression trace ofabove the the energy-momentum vanishing of the anomalous tensor, form being factor proportional propor- to µ⌫↵ the function,tional to must the tracefull vanish identically tensor structure andt1 theS . same The partial occurs contributions for the other to the anomalous same form component, factor, which related can to the R andbe to computed the S currents using Eqs. in (51) the and Ferrara-Zumino (59) for the various supermultiplet. components, are all a↵ected by pole terms, but they add up to give a form factor whose residue at the pole is proportional to the function of the = 4 theory. N It is then clear that the vanishing of the conformal36 anomaly, via a vanishing function, is equivalent to the cancellation of the anomaly pole for the entire multiplet. Notice also that the only surviving contribution in Eq. (161), proportional to the traceless tensor structure µ⌫↵ 2 t2S , is finite. This is due to the various cancellations between the UV singular terms from V (k ) and 2 2(k , 0) which give a finite correlator without the necessity of any regularization. We recall that the cancellation of infinities and the renormalization procedure, as we have already seen in the = 1 case, involves only the form factor of tensor tµ⌫↵, which gets renormalized with a counterterm N 2S proportional to that of the two-point function AA , and hence to the gauge coupling. For this reason the h i finiteness of the second form factor and then of the entire TVV in = 4 is directly connected to the h i N vanishing of the anomalous term, because its non-renormalization naturally requires that the function has to vanish.

37 µ−ϵ µ−ϵ 1 ddx gˆ Gˆ = ddx √gG+ d4x √g τ G 4 R !τ +8Rαβ ∂ τ − ϵ − ϵ Λ − − ∇β α ! ! ! # $ % " 2 + 2 R τ !τ + R ∂λτ∂ τ +4Rαβ ∂ τ∂ τ τ ∂ τ +6(!τ)2 6 β∂ατ ∂ τ Λ2 λ α β − ∇β α − ∇ ∇β α & $ % ' 4 2 Rαβ τ∂ τ∂ τ +2τ (!τ)2 +5∂λτ∂ τ !τ +6∂ατ∂βτ ∂ τ 2 τ β∂ατ ∂ τ − Λ3 α β λ ∇β α − ∇ ∇β α $ % 2 2 + 4 τ∂λτ∂ τ !τ +3 ∂λτ∂ τ +8τ∂ατ∂βτ ∂ τ . (44) Λ4 λ λ ∇β α $ %* ( ) Notice that the renormalization scale µ is not Weyl gauged due to the presence of the d-dimensional (rather than 4-dimensional) integration measure on the left-hand side of (43) and(44). The expressions above can be simpliﬁed using integrations by parts and the identity for the commutator of covariant derivatives of a vector [ , ] v = Rλ v . (45) ∇µ ∇ν ρ ρµν λ After these manipulations we ﬁnd that the Weyl-gauging of the counterterms gives µ−ϵ µ−ϵ τ 2 2 R ddx gˆ Fˆ = ddx √gF + d4x √g F !R ∂λτ∂ τ +(!τ)2 − ϵ − ϵ − Λ − 3 − Λ2 3 λ ! ! ! & $ % $ % " 4 2 2 + ∂λτ∂ τ !τ ∂λτ∂ τ , (46) Λ3 λ − Λ4 λ ' µ−ϵ µ−ϵ τ 4 ( R ) ddx gˆ Gˆ = ddx √gG+ d4x √g G + Rαβ gαβ ∂ τ∂ τ − ϵ − ϵ − Λ Λ2 − 2 α β ! ! ! & $ % " 4 2 2 + ∂λτ∂ τ !τ ∂λτ∂ τ . (47) Λ3 λ − Λ4 λ ' ( ) Obviously, a Weyl variation applied to (46) and (47) gives zero by construction. It follows that the Wess-Zumino action can be extractedThe IR from (12)stricture of the N=1

Γ [g,τ]=Γ [g,τ] Γˆ [g,τ] (48) Resort to the anomaly actionWZ in the ren Wess− Zumino ren form and thus takes the form

τ 2 2 R 4 2 2 Γ [g,τ]= d4x √g β F !R + ∂λτ∂ τ +(!τ)2 ∂λτ∂ τ !τ + ∂λτ∂ τ WZ a Λ − 3 Λ2 3 λ − Λ3 λ Λ4 λ ! # & $ % $ % ' τ 4 R 4 2 (2 ) + β G Rαβ gαβ ∂ τ∂ τ ∂λτ∂ τ !τ + ∂λτ∂ τ . (49) b Λ − Λ2 − 2 α β − Λ3 λ Λ4 λ & $ % '* ( ) Notice that the ambiguity in the choice of the Weyl tensor discussed above - i.e. between F and Fd - implies that no

dilaton vertex is expected to emerge from the gauging of the Fd-counterterm, being the latter conformal invariant. This is indeedConformal the case Trace and Relations we find the from relation the Dilaton Wess-Zumino Action −ϵ −ϵ µ d µ d 4 τ d x gˆ Fˆd = d x √gFd d x √g F, (50) Claudio Corianò,− Luigiϵ Delle Rose, Carlo− Marzoϵ and Mirko Se−rino Λ ! " ! ! that modifies the structure of the Wess-Zumino action and it allows to eliminate all the terms proportional to βa in (49) exceptDelle Rose, Marzo, Serino, C.C. for (τ/Λ) F . Dipartimento di Matematica e Fisica Finally, we remark that in the caseUniversitàdel in which Salento a finite counterterm of the kind (37) is present, the formulae of this section are modified according to the simpleand prescription (see Eq. (40)), INFN Lecce, Via Arnesano 73100 Lecce, Italy β β 18 β , (51) a → a − fin Abstract

We use the method of Weyl-gauging in the determination of the Wess-Zumino9 conformal anomaly action to show that in any even (d =2k)dimensionsallthehierarchyofcorrelationfunctionsinvolving traces of the energy-momentum tensor is determined in terms of those of lower orders, up to 2k.Weworkoutexplicitly the case d = 4, and show that in this case in any conformal field theory onlythefirst4tracedcorrelatorsare independent. All the remaining correlators are recursivelygeneratedbythefirst4.Theresultisaconsequence of the cocycle condition which defines the Wess-Zumino actionandofthefiniteorderofitsdilatoninteractions. arXiv:1306.4248v2 [hep-th] 20 Jun 2013

1 momenta (k1,k2,k3,k4,k5) the available pairs are

5, (k ,k ) = (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ) T{ i1 i2 } { 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 } (63) while the possible triples are

5, (k ,k ,k ) = (k ,k ,k ), (k ,k ,k ), (k ,k ,k ), (k ,k ,k ), (k ,k ,k ), T{ i1 i2 i3 } { 1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 (k ,k ,k ), (k ,k ,k ), (k ,k ,k ), (k ,k ,k ), (k ,k ,k ) . (64) 1 4 5 2 3 4 2 3 5 2 4 5 3 4 5 } As we move to higher orders, the description of the momentum dependence gets slightly more involved and we need to distribute the external momenta into two pairs. The notation 5, [(k ,k ), (k ,k )] denotes the set of T{ i1 i2 i3 i4 } independent paired couples which can be generated out of 5 momenta. Their number is 15 and they are given by

5, [(k ,k ), (k ,k )] = [(k ,k ), (k ,k )], [(k ,k ), (k ,k )], [(k ,k ), (k ,k )] T{ i1 i2 i3 i4 } { 1 2 3 4 1 2 3 5 1 2 4 5 [(k1,k3), (k2,k4)], [(k1,k3), (k2,k5)], [(k1,k3), (k4,k5)], [(k1,k4), (k2,k3)], [(k1,k4), (k2,k5)], [(k1,k4), (k3,k5)],

[(k1,k5), (k2,k3)], [(k1,k5), (k2,k4)], [(k1,k5), (k3,k4)], [(k2,k3), (k4,k5)], [(k2,k4), (k3,k5)], [(k2,k5), (k3,k4)] . (65) momenta (k1,k2,k3,k4,k5) the available pairs are }

5, (k ,k ) = (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ), (k ,k ) It is obvious that a direct computationT{ i1 i2 of} 2{, 13 2and1 3 4 from1 4 the1 5 anomaly2 3 2 4 action2 5 (59)3 4 allows3 5 4 to5 } extract the explicit I I I (63) structure of these vertices in momentum space while the possible triples are 4 5, (ki1 ,ki2 ,k4i3 ) = (k1,k2,k3), (k1,k2,k4), (k1,k2,k5), (k1,k3,k4), (k1,k3,k5), 2(k1, k1)= T{2 βa k1 ,} { Λ (k1,k4,k5), (k2,k3,k4), (k2,k3,k5), (k2,k4,k5), (k3,k4,k5) . (64) I − − } As we move to8 higher orders, the description2 of the momentum2 dependence gets2 slightly more involved and we 3(k1,k2,k3)= βa + βb k k2 k3 + k k1 k3 + k k1 k2 need to distribute3 the external momenta into1 two pairs. The2 notation 5, [(ki 3,ki ), (ki ,ki )] denotes the set of I Λ · · T{ 1 2 · 3 4 } independent paired! couples which"! can be generated out of 5 momenta. Their number is 15 and" they are given by 16 5, [(k ,k ), (k ,k )] = [(k ,k ), (k ,k )], [(k ,k ), (k ,k )], [(k ,k ), (k ,k )] (k ,k ,k ,k )= i1 i2 i3 βi4 + β 1 k2 k3 4k 1k 2 +3k 5 k1 k2 4k 5 + k k k k , (66) I4 1 2 3 4 T{ −Λ4 a } b { 1 · 2 3 · 4 1 · 3 2 · 4 1 · 4 2 · 3 [(k1,k3), (k2,k4)], [(k!1,k3), (k2,k5)],"![(k1,k3), (k4,k5)], [(k1,k4), (k2,k3)], [(k1,k4), (k2,k5)], [(k1,k4), (k3,k5)], " [(k1,k5), (k2,k3)], [(k1,k5), (k2,k4)], [(k1,k5), (k3,k4)], [(k2,k3), (k4,k5)], [(k2,k4), (k3,k5)], [(k2,k5), (k3,k4)] . (65) (with kn (k2)n/2). These relations can be used together with (57) in order to extract the structure} of the 2- 3- i i It is obvious that a direct computation of , and from the anomaly action (59) allows to extract the explicit ≡ I2 I3 I4 and 4-point functions of thestructure traced of correlators, these vertices in momentum which space are given by 4 (k , k )= β k 4 , I2 41 − 1 −Λ2 a 1 T (k1)T ( k1) = 4 βa k1 , 8 (k ,k ,k )= β + β k2 k k + k2 k k + k2 k k ⟨ − ⟩ − I3 1 2 3 Λ3 a b 1 2 · 3 2 1 · 3 3 1 · 2 ! "! " 3 16 (k ,k ,k ,k )= β + β k k k k + k k k k + k k k k , (66) 4 T (k1)T (k2)T (k3) =8I4 1 β2a +3 4βb −fΛ34 (ka1,kb2,k13·)+2 3 f· 34(k21 ,k· 312,k· 34 )+1 · f43(2 k· 33,k1,k2) + βa ki , ⟨ ⟩ − ! "! " (with kn #(k2)n/!2). These relations"! can be used together with (57) in order to extract the structure of the" 2- 3- i=1 % i ≡ i $ and 4-point functions of the traced correlators, which are given by T (k1)T (k2)T (k3)T (k4) =86 βa + βb kii ki2 ki3 ki4 T (k )T ( k ) = 4 β k 4 , ⟨ ⟩ ⟨ 1 − 1 ⟩ − a 1 · · T 3 & ! "# 4,[(ki1 ,ki2 ),(ki3 ,ki4 )] T (k )T (k )T (k ) =8 β + β f$(k ,k ,k )+f (k ,k ,k )+f (k ,k ,k ) + β k4 , ⟨ 1 2 3 ⟩ − { a b 3 1 2 3 3 }2 1 3 3 3 1 2 a i i=1 # ! "! " $ % T (k +)T (fk 4)T(k(k1)Tk(2k,k) 3=8,k4)+6 βf+4(βk2 k1,k3,k4)+fk4(kk3 kk1,kk 2,k4)+f4(k4 k1,k2,k3) ⟨ 1 2 3 4 ⟩ a b ii · i2 i3 · i4 & ! "#T 4,[(k ,k ),(k ,k )] i1 $i2 i3 i4 % { 4 } + f4(k1 k2,k3,k4)+f4(k42 k1,k3,k4)+f4(4k3 k1,k2,k4)+f4(k4 k1,k2,k3) βa (ki1 + ki2 ) +4 ki , % (67) − 4 4 i=1 4 ! T 4,(k ,k βa) (ki1 + ki2 ) +4 ki "', (67) $i1 −i2 $ ! T 4,(ki ,ki ) i=1 "' { } { $1 2 } $ where we have introduced thewhere compact we have introduced notation the compact notation f (k ,k ,k )=k2 k k , 3 a b c a b · c 2 2 f (k ,k ,k )=f4(ka,kb,kkc,kdk)=kka,(kb kc + kb kd + kc kd) . (68) 3 a b c a b · c · · · Thef third(k and,k fourth,k order,k results,)= in particular,k2 (k werek established+ k ink [12]+ viak the explicitk ) . computation of the ﬁrst (68) three functional4 a derivativesb c d of the anomalya [g] andb exploitingc b recursivelyd thec hierarchicald relations (8). A · · · The third and fourth order results, in particular, were established12 in [12] via the explicit computation of the ﬁrst three functional derivatives of the anomaly [g] and exploiting recursively the hierarchical relations (8). A Delle Rose, Marzo, Serino 12 Conclusions

Stuckelberg models have introduced the concept of gauging of an anomalous U(1) symmetry.

They predict an axion and an anomalous U(1) In their susy version they predict a supersymmetric multiplet. (fundammental)

But there are variants

Anomaly actions seem to indicate that we coudl also take a different route. All the states that we come from the breaking of a dilatation current, assuming A conformal symmetry in the UV are the anomaly poles of the supersymmetric current. The presence of sum rules in N=1 theories away from the conformal point seem to provide this indication

The symmetry should also appear as broken (spontaneously?)