One-loop W L W L and Z L Z L scattering from the Electroweak Chiral Lagrangian with a light Higgs-like scalar
aa r X i v : . [ h e p - ph ] J a n Prepared for submission to JHEP
One-loop W L W L and Z L Z L scattering from theElectroweak Chiral Lagrangian with a lightHiggs-like scalar Rafael L. Delgado, Antonio Dobado and Felipe J. Llanes-Estrada
Departamento de Física Teórica I, Universidad Complutense de Madrid, 28040 Madrid, Spain
Abstract:
By including the recently discovered Higgs-like scalar ϕ in the ElectroweakChiral Lagrangian, and using the Equivalence Theorem, we carry out the complete one-loop computation of the elastic scattering amplitude for the longitudinal components ofthe gauge bosons V = W, Z at high energy. We also compute ϕϕ → ϕϕ and the inelasticprocess V V → ϕϕ , and identify the counterterms needed to cancel the divergences, namelythe well known a and a chiral parameters plus three additional ones only superficiallytreated in the literature because of their dimension 8. Finally we compute all the partialwaves and discuss the limitations of the one-loop computation due to only approximateunitarity. ontents ωω scattering 53.2 Scattering amplitudes involving the new ϕ scalar boson 6 m ϕ The LHC directly probes for the first time the sector of the Standard Model responsible forElectroweak Symmetry Breaking. Two-particle invariant mass spectra of the longitudinalcomponents of gauge boson pairs W L W L and Z L Z L are not yet at hand, but expected in thenext years. As the remainder of the Goldstone bosons of electroweak symmetry breaking,the scattering of the longitudinal bosons at high-energy (high compared with M W , but notlarger than about πv ≃ ) is predicted by theory through the equivalence theorem [1],even in the presence of strong interactions that may make other predictions doubtful.The finding that the LHC collaborations ATLAS [2] and CMS [3] have published [4] isa boson with scalar quantum numbers and couplings compatible with those of a StandardModel Higgs. This might bring the Minimal Standard Model (MSM) to closure.Most interestingly, no further new particle has been sighted [5] in the first run of theLHC, up to an energy of 600-700 GeV (and higher yet for additional vector bosons). Thismass gap in the spectrum also naturally suggests that the Higgs is an additional Goldstoneboson, perhaps a dilaton from spontaneous breaking of scale invariance, or from a compositeHiggs model based on SO (5) /SO (4) or any other coset. The effective Lagrangian approachincludes these cases as we will point out, but irrespective of the true nature of the scalar– 1 –oson, it encodes its interactions with the rest of the symmetry breaking sector. One featurethat we will adopt from these models, though, also shared by the Standard Model, is thatthe Higgs-potential self-couplings are of order M ϕ , and thus negligible for s ≫ M ϕ . Apartfrom this assumption, that covers all models of interest at the present time, our discussionwill remain general and uncommitted to a particular new physics scenario.Several groups[6–14] are studying in detail the formulation of effective Lagrangians forthe four visible particles, their scattering amplitudes at low-energy and the unitarization ofthose amplitudes to reach higher energies. These studies extend traditional effective-theoryapproaches [15] to the Electroweak Symmetry Breaking Sector (EWSBS) modeled in totalanalogy to Chiral Perturbation Theory in QCD [16].In a recent work [17] we have shown that, for essentially any parameter choice exceptthat of the Standard Model and perhaps other very carefully tuned sets, the interactionswill generically become strong at sufficiently high energy, and have argued that a second,very broad scalar pole is expected.In this article we complete the one-loop computation of the two-body scattering am-plitudes among the ω Goldstone bosons and the ϕ scalar with such a generic effectiveLagrangian, in the kinematic regime M ϕ ≪ s < πv ≃ . The Lagrangian densityis exposed in section 2 and the scattering amplitudes derived therefrom, in dimensionalregularization, are given in section 3. The calculation has been performed both analyti-cally and also with standard one-loop automated computer tools, and the results agree.The Feynman diagrams resulting from the effective Lagrangian are delayed to the appendixgiven their large number.Renormalization is carried out in section 4. Five NLO coefficients are necessary, the wellknown a and a from the Electroweak Chiral Lagrangian, and three less studied ones, alsomultiplying operators of dimension 8, that renormalize the Higgs self-interactions and thechannel coupling between scalar and longitudinal vector bosons. We do not assess operatorsthat are unnecessary to carry out the renormalization of the one-loop computation, withthe exception of ( ϕ∂ µ ϕ )( ϕ∂ µ ϕ ) that we examine in subsection 6.3; the interested readercan find a table of the 59 dimension-6 operators that extend the SM in [11].In section 5 we provide the partial-wave projections of all three two-body amplitudes,that will prove necessary in future work to examine the possible existence of new resonanceschannel by channel.Section 6 shows a numerical computation of the various partial waves to gain a feelingfor their behavior and sensitivity to the unknown parameters that carry the theory beyondthe Standard Model, and also to expose the violation of unitarity in perturbation theory,since in the effective Lagrangian approach, amplitudes grow like a power of Mandelstam- s .Our findings are summarized in section 7.– 2 – The Electroweak Chiral Lagrangian with a Light Scalar
One of the lowest-order equivalent forms of the universal Electroweak Chiral Lagrangianwith the known particle content is a gauged SU (2) L × SU (2) R /SU (2) C = SU (2) ≃ S Non-linear Sigma Model (NLSM) coupled to a scalar field ϕ as L = v g ( ϕ/f ) ( D µ U ) † D µ U + 12 ∂ µ ϕ∂ µ ϕ − V ( ϕ ) (2.1)where U is a field taking values in the SU (2) coset that can be parametrized for exampleas U = p − ˜ ω /v + i ˜ ω/v ; ˜ ω = ω a τ a being the would-be Goldstone boson (WBGB)field [22]. The SU (2) L × U (1) Y subgroup is gauged as usual through the covariant derivatives D µ U = ∂ µ U + W µ U − U Y µ , W µ = − giW iµ τ i / , Y µ = − g ′ iB iµ τ / . In terms of Fermi’s weakconstant, v := 1 / ( √ G F ) = (246 GeV) , while f is an arbitrary, new-physics energy scalecontrolling the generic dynamics of the EWSBS. The scalar field interacts through g ( x ) , anarbitrary analytical functional; in the effective-theory approach only the first terms of itsTaylor expansion are probed g ( ϕ/f ) = 1 + 2 α ϕf + β (cid:18) ϕf (cid:19) + .. (2.2)Here we have introduced two parameters α and β instead of the more common a and b in [18],but clearly we have a = αv/f and b = βv /f . With this natural but maybe unconventionalchoice, having f instead of v in the denominators, the value of the adimensional vacuum-tiltparameter ξ ≡ v /f that corresponds to the Standard Model is ξ = 1 .Our philosophy here is to weigh the WBGB field intensity against the EWSB scale v and the scalar field ϕ against the possible new scale f . Of course f = v is a particularpossibility corresponding to α = a and β = b (see [19] for some recent experimental boundson the a and b that we have also briefly discussed in [17]). Finally V is an arbitraryanalytical potential for the scalar field, V ( ϕ ) = ∞ X n =0 V n ϕ n ≡ V + M ϕ ϕ + λ ϕ + λ ϕ + ... (2.3)At the the next to leading order in the chiral expansion one should add the four derivativeterms L = a ( trV µ V ν ) + a ( trV µ V µ ) (2.4) + γf ( ∂ µ ϕ∂ µ ϕ ) + δf ( ∂ µ ϕ∂ µ ϕ ) tr ( D ν U ) † D ν U + ηf ( ∂ µ ϕ∂ ν ϕ ) tr ( D µ U ) † D ν U + ... where V µ = D µ U U † . We have written explicitly only the five terms strictly needed for therenormalization of the one-loop elastic WBGB scattering amplitudes (for s ≫ M W ) andthe unitarity-related processes ωω → ϕϕ and ϕϕ → ϕϕ . These terms produce additionalcontributions to the amplitudes which are of order s .The chiral parameters a and a (multiplying the operators O D and O D in the clas-sification of [13]) and the new ones γ , δ and η depend on whatever unknown underlying– 3 –ynamics responsible for the spontaneous symmetry breaking of electroweak interactionsmight exist. They all vanish in the MSM. The operators with coefficient δ and η are iden-tified as O and O in the classification of Azatov et al. [7] while they are P and P in [8] and are NLO equivalent to O D , O D in [13]. The operator associated with γ isdenoted as P H in [8] and O D in [13]. None of these authors give much detail on the useor scale-dependence of these operators, important to this work.The two operators multiplying δ and η are apparently of dimension 6. But this leadingdimension affects only transverse gauge-boson inelastic scattering W T W T → ϕϕ and notthe longitudinal ones. When expanding U , the relevant ωω → ϕϕ terms are of dimension 8as shown shortly in Eq. (2.6). Thus, they are apparently of a high order in the classificationof all operators beyond the Standard Model, but as we will see they are necessary alreadyin one-loop renormalization.The Lagrangian in Eq. (2.1) and Eq. (2.4) is able to reproduce the low-energy physicsof this sector of the SM for any possible dynamics having at least an approximate SU (2) custodial isospin symmetry in the limit g = g ′ = 0 . For example the MSM corresponds tothe parameter selection α = β = ξ = 1 and a = a = γ = δ = η = 0 . The Higgs field H isjust the scalar field ϕ so that M H = M ϕ = 2 λv , and the scalar self-couplings are λ = λv , λ = λ/ (both proportional to M ϕ ) and λ i = 0 for i ≥ .In dilaton models [20] ϕ would represent the dilaton field, α = β = 1 as in the MSMbut ξ is arbitrary, f being the scale of the symmetry breaking. The potential and NLOparameters depend on the particular dilaton model but in any case λ i is of order M ϕ forany i .Third, we also have the example of the SO (5) /SO (4) Minimally Composite HiggsModel [21] where α = cos θ/ √ ξ , β = cos(2 θ ) /ξ , sin θ = √ ξ and a , a and the scalar-bosoncouplings depend on the particular details of the model, but it can be assumed that the λ i are of order M ϕ too.Finally it is also possible to reproduce the old Higgsless Electroweak Chiral Lagrangian(EWChL) in [15] by the simple parameter choice α = β = γ = δ = η = 0 .As discussed in the introduction, we pursue the elastic scattering of the longitudinalcomponents of the electroweak bosons at high energies, i. e., for √ s ≫
100 GeV . In thiscase, we can apply the Equivalence Theorem: T ( ω a ω b → ω c ω d ) = T ( W aL W bL → W cL W dL ) + O (cid:18) M W √ s (cid:19) , (2.5)and thus we will be probing the WBGB dynamics. This theorem applies for any renor-malizable gauge, but the Landau gauge (where there remain massless WBGB) turns outto be particularly useful. Therefore, in the following we will set g = g ′ = 0 and the onlydegrees of freedom to be considered will be the massless (in the Landau gauge) WBGB andthe Higgs-like scalar ϕ . Moreover, according to ATLAS and CMS M ϕ ≃ GeV. Then M ϕ ∼ M W ∼ M Z ∼
100 GeV . As a consequence it is a perfectly consistent approximationto consider the massless ϕ limit, i.e., M ϕ ≃ if one is only interested in the energy regionwhere the ET can be applied. Therefore we will concentrate on the WBGB scattering for– 4 – ϕ , M W , M Z ≃ ≪ s < Λ where Λ is some ultraviolet (UV) cutoff of about 3 TeV, settingthe limits of applicability of the effective theory.As in the three particular models just mentioned, we will also assume that the λ i pure-scalar potential parameters are of order M ϕ so that we can neglect the scalar potentialaltogether. In this kinematic regime, the relevant Lagrangian, derived from Eqs. (2.1)through (2.4) is L = 12 α ϕf + β (cid:18) ϕf (cid:19) ! ∂ µ ω a ∂ µ ω b (cid:18) δ ab + ω a ω b v (cid:19) + 12 ∂ µ ϕ∂ µ ϕ + 4 a v ∂ µ ω a ∂ ν ω a ∂ µ ω b ∂ ν ω b + 4 a v ∂ µ ω a ∂ µ ω a ∂ ν ω b ∂ ν ω b + γf ( ∂ µ ϕ∂ µ ϕ ) + 2 δv f ∂ µ ϕ∂ µ ϕ∂ ν ω a ∂ ν ω a + 2 ηv f ∂ µ ϕ∂ ν ϕ∂ µ ω a ∂ ν ω a . (2.6)Notice also that by rescaling f and redefining β it is possible to set α = 1 in Eq. (2.2)without losing generality, g ( ϕ/f ) = 1 + 2 ϕf ′ + β ′ (cid:18) ϕf ′ (cid:19) + . . . (2.7)This leaves as free parameters in the above Lagrangian in our energy region of interest theredefined f and β , the chiral parameters a and a , and the three γ , δ , η ones involving thenew scalar boson. However, in the following we will still keep the explicit α -dependence inour formulae so that we can easily trace for comparison with previous works. In particular,as already pointed out, the old EWChL without any Higgs-like light resonance correspondsto α = β = 0 (and vanishing higher order inelastic couplings). ωω scattering In this section we compute the scattering amplitudes using the Landau gauge and dimen-sional regularization. We start by elastic WBGB scattering. Due to the custodial symmetryof the SBS of the SM in the limit g = g ′ = 0 the WBGB amplitude ω a ω b → ω c ω d can bewritten as A abcd = A ( s, t, u ) δ ab δ cd + A ( t, s, u ) δ ac δ bd + A ( u, t, s ) δ ad δ bc (3.1)because the four particles are identical, the amplitude has to be crossing-symmetric andexpressible in terms of only one amplitude A . This, we conveniently expand following thechiral counting, and also separately quote the NLO tree-level and 1-loop subamplitudes as A = A (0) + A (1) · · · = A (0) + A (1)tree + A (1)loop . . . (3.2)Then, from the Lagrangian in Eq. (2.6) the following tree-level amplitude results A (0) ( s, t, u ) + A (1)tree ( s, t, u ) = (1 − α ξ ) sv + 4 v (cid:2) a s + a ( t + u ) (cid:3) . (3.3)– 5 –t the one-loop level, a lengthy computation of the Feynman diagrams in the appendixgives A (1)loop ( s, t, u ) = 136(4 π ) v [ f ( s, t, u ) s + ( α ξ − ( g ( s, t, u ) t + g ( s, u, t ) u )] (3.4)where we have defined auxiliary functions f ( s, t, u ) := [20 − α ξ + ξ (56 α − α β + 36 β )]+ [12 − α ξ + ξ (30 α − α β + 18 β )] N ε + [ −
18 + 36 α ξ + ξ ( − α + 36 α β − β )] log (cid:18) − sµ (cid:19) + 3( α ξ − (cid:20) log (cid:18) − tµ (cid:19) + log (cid:18) − uµ (cid:19)(cid:21) (3.5) g ( s, t, u ) := 26 + 12 N ε − (cid:20) − tµ (cid:21) − (cid:20) − uµ (cid:21) (3.6)and in dimensional regularization D = 4 − ǫ the pole is contained as usual in N ǫ = 2 ǫ + log 4 π − γ . (3.7)Because of the factors of ξ , the amplitude in Eq. (3.4) contains terms proportional to /v , / ( v f ) , and / ( f ) , reflecting the various possible intermediate states in the one-loop computation. We have checked also that our results agree with those found in [6] inthe limit of vanishing light scalar mass. ϕ scalar boson The next two-body processes to consider are the channel coupling ω a ω b → ϕϕ between two ω WBGB and a scalar boson pair and ϕϕ → ω a ω b , that are needed to obtain one-loopunitarity in ωω scattering. Obviously both processes have the same amplitude because oftime reversal invariance. Since ϕ is an isospin singlet, the amplitude can be expressed as M ab ( s, t, u ) = M ( s, t, u ) δ ab . (3.8)Performing the chiral expansion as in Eq. (3.2), we find at tree level, M (0)tree ( s, t, u ) + M (1)tree ( s, t, u ) = ( α − β ) sf + 2 δv f s + ηv f ( t + u ) (3.9)that takes a one-loop correction: M (1)loop ( s, t, u ) = α − β π f (cid:20) f ′ ( s, t, u ) s v + α − βf [ g ( s, t, u ) t + g ( s, u, t ) u ] (cid:21) (3.10)where f ′ ( s, t, u ) = − − ξ (11 α − β )] − N ε [ − ξ (7 α − β )] (3.11) + 36( α ξ −
1) log (cid:20) − sµ (cid:21) + 3 ξ ( α − β ) (cid:18) log (cid:20) − tµ (cid:21) + log (cid:20) − uµ (cid:21)(cid:19) – 6 –nd the function g is as defined in Eq. (3.6).Finally we have the amplitude for the elastic scattering ϕϕ → ϕϕ , T ( s, t, u ) = T (0) + T (1)tree + T (1)loop . . . (3.12)The tree amplitude is T (0) ( s, t, u ) + T (1)tree ( s, t, u ) = 2 γf ( s + t + u ) (3.13)and the one-loop piece can be written in terms of only one function T ( s ) = 2 + N ε − log (cid:18) − sµ (cid:19) (3.14)as T (1)loop ( s, t, u ) = 3( α − β ) π ) f (cid:2) T ( s ) s + T ( t ) t + T ( u ) u (cid:3) . (3.15) Comparing the tree-level amplitudes in Eqs. (3.3), (3.9), (3.13) with the loop ones inEqs. (3.4), (3.10), (3.15) we see that the divergences in the one-loop pieces can be absorbedjust by redefining the couplings a , a , γ , δ and η from the NLO tree-level Lagrangian.Therefore no α , β , v , f , wave-function nor mass renormalization is needed to obtain a finiteamplitude (a pleasant feature of dimensional regularization).We proceed by choosing the modified minimal-substraction or M S scheme, so therenormalized couplings are given by a r = a + N ǫ π (1 − ξα ) a r = a + N ǫ π (2 + 5 ξ α − ξα − ξ α β + 3 ξ β ) γ r = γ + 3 N ǫ π ( α − β ) δ r = δ − N ǫ π ( α − β )(7 ξα − ξβ − η r = η + N ǫ π ξ ( α − β ) . (4.1)Some limits of these renormalization relations can be easily checked. For example in thecase of the MSM ( α = β = ξ = 1 ) we see that none of these five couplings is strictlyneeded because of the renormalizability of the model. The case of the Higgsless EWChLcorresponds to α = β = 0 and consistently we find that γ, δ and η do not need anyrenormalization and also we reproduce the well known results for the constants a and a [15] in this case. Finally we have checked also that the renormalization a and a agree withthe corresponding ones found in [6]. In terms of these renormalized couplings the elastic– 7 –BGB amplitude reads A ( s, t, u ) = sv (1 − ξα ) + 4 v [2 a r ( µ ) s + a r ( µ )( t + u )] (4.2) + 116 π v (cid:18)
19 (14 ξ α − ξα − ξ α β + 9 ξ β + 5) s + 1318 ( ξα − ( t + u ) −
12 (2 ξ α − ξα − ξ α β + ξ β + 1) s log − sµ + 112 (1 − ξα ) ( s − t − u ) log − tµ + 112 (1 − ξα ) ( s − t − u ) log − uµ (cid:19) . The inelastic ωω → ϕϕ amplitude is correspondingly M ( s, t, u ) = α − βf s + 2 δ r ( µ ) v f s + η r ( µ ) v f ( t + u )+ ( α − β )576 π v f (cid:26)(cid:20) − ξα + 16 ξβ + 36( ξα −
1) log − sµ + 3 ξ ( α − β ) (cid:18) log − tµ + log − uµ (cid:19)(cid:21) s + ξ ( α − β ) (cid:18) − − tµ − − uµ (cid:19) t + ξ ( α − β ) (cid:18) − − uµ − − tµ (cid:19) u (cid:27) (4.3)and finally the ϕϕ → ϕϕ amplitude may be written as T ( s, t, u ) = 2 γ r ( µ ) f ( s + t + u ) (4.4) + 3 ξ ( α − β ) π v (cid:20) s + t + u ) − s log − sµ − t log − tµ − u log − uµ (cid:21) . Apparently, the amplitudes in Eqs. (4.2), (4.3) and (4.4) depend on the arbitrary scale µ through the log terms. However they also depend on µ implicitly through the renormalizedcouplings a . . . η .However, as there is no wave or mass renormalization, the amplitudes must be observ-able and therefore µ -independent; then we may require that their total derivatives withrespect to log µ vanish. Integrating the resulting (very simple) differential equations, wefind the renormalization-group evolution equations for the different couplings which turnout to be – 8 – r ( µ ) = a r ( µ ) − π (1 − ξα ) log µ µ a r ( µ ) = a r ( µ ) − π (2 + 5 ξ α − ξα − ξ α β + 3 ξ β ) log µ µ γ r ( µ ) = γ r ( µ ) − π ( α − β ) log µ µ δ r ( µ ) = δ r ( µ ) + 1192 π ( α − β )(7 ξα − ξβ −
6) log µ µ η r ( µ ) = η ( µ ) − π ξ ( α − β ) log µ µ . (4.5)These equations allow to reexpress the amplitudes at any second scale. They are diagonal, sothat the various coefficients do not enter the evolution equation for any other ones, a featurethat will not persist at higher orders in perturbation theory. Since no resonance beyondthe Standard Model is presently known, there is no particularly natural renormalizationscale µ , so we will arbitrarily employ µ = 1 TeV. All NLO numerical couplings quoted insection 6 below are to be understood as taken at this scale.
The unitarity properties of the three scattering amplitudes are best exposed in terms ofthe isospin- and spin-projected partial waves. For elastic WBGB scattering there are threecustodial-isospin A I amplitudes ( I = 0 , , ) analogous to those in pion-pion scattering inhadron physics, A ( s, t, u ) = 3 A ( s, t, u ) + A ( t, s, u ) + A ( u, t, s ) (5.1) A ( s, t, u ) = A ( t, s, u ) − A ( u, t, s ) A ( s, t, u ) = A ( t, s, u ) + A ( u, t, s ) . We can then project them over definite orbital angular momentum (the WBGBs carry zerospin), and choose the normalization as A IJ ( s ) = 164 π Z − d (cos θ ) P J (cos θ ) A I ( s, t, u ) . (5.2)These partial waves also accept a chiral expansion A IJ ( s ) = A (0) IJ ( s ) + A (1) IJ ( s ) + ..., (5.3)where A (0) IJ ( s ) = KsA (1) IJ ( s ) = s (cid:18) B ( µ ) + D log sµ + E log − sµ (cid:19) . (5.4)– 9 –he constants K , D and E and the function B ( µ ) depend on the different channels IJ =00 , , , as shown below and we will use the same notation for the inelastic and pure- ϕ scattering reactions.As A IJ ( s ) must be scale independent we have B ( µ ) = B ( µ ) + ( D + E ) log µ µ ; (5.5)For elastic ωω scattering, this B function is linear in the NLO chiral constants (with certainproportionality coefficients p and p that can be read off Eq. (5.7) and following) B ( µ ) = B + p a ( µ ) + p a ( µ ) , (5.6)where from now on we omit the superindices r on the renormalized coupling constants forsimplicity.A direct evaluation of the integral in Eq. (5.2) substituting the renormalized amplitudeobtained in Eq. (4.2) for the ωω → ωω process produces the following auxiliary K , D , E constants and B ( µ ) functions.For the scalar-isoscalar channel with IJ = 00 , K = 116 πv (1 − ξα ) B ( µ ) = 19216 π v [101 + 768(7 a ( µ ) + 11 a ( µ )) π + ξ (169 α ξ + 68 β ξ − α (101 + 68 βξ ))] D = − π v [7 + ξ (10 α ξ + 3 β ξ − α (7 + 3 βξ ))] E = − π v [(1 − ξα ) + 34 ξ ( α − β ) )] . (5.7)For the vector isovector IJ = 11 amplitude, K = 196 πv (1 − ξα ) B ( µ ) = 1110592 π v [8 + 4608( a ( µ ) − a ( µ )) π − ξ (67 α ξ + 75 β ξ + 2 α (8 − βξ ))] D = 19216 π v [1 + ξ (4 α ξ + 3 β ξ − α (1 + 3 βξ ))] E = − π v (1 − ξα ) . (5.8)For the scalar isotensor IJ = 20 : K = − πv (1 − ξα ) B ( µ ) = 118432 π v [91 + 3072(2 a ( µ ) + a ( µ )) π + 7 ξ (17 α ξ + 4 β ξ − α (13 + 4 βξ ))] D = − π v [11 + ξ (17 α ξ + 6 β ξ − α (11 + 6 βξ ))] E = − π v (1 − ξα ) (5.9)– 10 –nd finally for the tensor isoscalar IJ = 02 , K = 0 B ( µ ) = 1921600 π v [320 + 15360(2 a ( µ ) + a ( µ )) π + ξ (397 α ξ + 77 β ξ − α (320 + 77 βξ ))] D = − π v [10 + ξ (13 α ξ + 3 β ξ − α (10 + 3 βξ ))] E = 0 . (5.10)Since the “Higgs” boson has zero custodial isospin, the ωω → ϕϕ and ϕϕ → ϕϕ reactions only proceed in the isospin zero channel I = 0 . In the first, inelastic, case wehave M ( ωω → ϕϕ ) = √ M ( s, t, u ) and for the scalar-scalar interaction, T ( ϕϕ → ϕϕ ) = T ( s, t, u ) . The chiral expansions equivalent to the ωω elastic one in Eq. (5.4) are now M J ( s ) = K ′ s + s (cid:18) B ′ ( µ ) + D ′ log sµ + E ′ log − sµ (cid:19) . . .T J ( s ) = K ′′ s + s (cid:18) B ′′ ( µ ) + D ′′ log sµ + E ′′ log − sµ (cid:19) . . . (5.11)(with J subindex omitted in the constants). The functions B ′ ( µ ) and B ′′ ( µ ) are in allanalogous to B ( µ ) as defined in Eq.(5.6), but with the constants a , a renormalizing theelastic ωω channel being substituted by δ , η (for B ′ ) and γ (for B ′′ ) involving the ϕ boson.In consequence we find for the ωω → ϕϕ partial waves M J , starting by the scalar one, K ′ = √ πf ( α − β ) B ′ ( µ ) = √ πv f (cid:18) δ ( µ ) + η ( µ )3 (cid:19) − √ α − β )18432 π v f (71 ξα + ξβ − D ′ = − √ α − β ) π f E ′ = − √ α − β )512 π v f (1 − ξα ) (5.12)while for the tensor M channel K ′ = 0 B ′ ( µ ) = η ( µ )160 √ πv f + 83( α − β ) √ π f D ′ = − ( α − β ) √ π f E ′ = 0 . (5.13)Finally for the ϕϕ → ϕϕ reaction the T ( s ) scalar partial-wave amplitude is given by theset of constants – 11 – ′′ = 0 B ′′ ( µ ) = 10 γ ( µ )96 πf + ( α − β ) π f D ′′ = − ( α − β ) π f E ′′ = − α − β ) π f (5.14)and the tensor T in turn by K ′′ = 0 B ′′ ( µ ) = γ ( µ )240 πf + 77( α − β ) π f D ′′ = − ( α − β ) π f E ′′ = 0 . (5.15)The µ -invariance of all the above partial waves is easy to check by substituting the µ -evolution of the renormalized couplings in Eq. (4.5).The partial-wave amplitudes A IJ ( s ) , M J ( s ) and T J ( s ) are all analytical functions ofcomplex Mandelstam- s , having the proper left and right (or unitarity) cuts, shortened toLC and RC respectively. The physical values of their argument are s = E CM + i ǫ (i.e. onthe upper lip of the RC), where E CM is the total energy in the center of mass frame. Forthese physical s values, exact unitarity requires a set of non-trivial relations between thedifferent partial waves that we now spell out.For I = 0 and either of J = 0 , J = 2 , where channel coupling is possible, Im A J = | A J | + | M J | (5.16) Im M J = A J M ∗ J + M J T ∗ J Im T J = | M J | + | T J | . These relations are not exactly respected by perturbation theory, but are instead satisfiedonly to one less order in the expansion than kept in constructing the amplitude. At theone-loop level one has Im A (1)0 J = | A (0)0 J | + | M (0) J | Im M (1) J = A (0)0 J M (0) J + M (0) J T (0) J Im T (1) J = | M (0) J | + | T (0) J | . For the remaining channels with I = J = 1 and I = 2 , J = 0 the ωω → ωω reaction iselastic and the unitarity condition is just Im A IJ = | A IJ | I = 0 (5.17)– 12 –nd at the NLO perturbative level, Im A (1) IJ = | A (0) IJ | I = 0 . (5.18)There are in all eight independent one-loop perturbative relations, that can also beobtained by applying the Landau-Cutkosky cutting rules and directly checked in each ofthe partial waves for the three reactions, providing a very good, non-trivial check of ouramplitudes. In this section we evaluate all partial waves with the constants in Eq. (5.7) and followingand expose their dependence on the LO parameters that separate them from the StandardModel, and on the NLO parameters as well. Generically, the partial waves (whether weplot the real, the imaginary part or the modulus) will correspond to the OY axis and bedenoted by t , while the OX axis is the squared physical cm energy s = E .In figure (1) we have plotted the elastic ωω → ωω amplitude without NLO constants,by setting a = a = 0 at µ = 1 TeV (scale also chosen in all examples to follow). Also f has been chosen at 500 GeV, which is about v , and to avoid channel coupling we havekept α = β = 1 (Standard Model values) so that all the ( α − β ) factors vanish.First we observe the real part of the amplitude (top plot in the figure). We concludethat just like in low-energy hadron physics, the IJ = 00 wave is strongly attractive, the IJ = 11 (without NLO constants) mildly attractive, the wave negligible in the low-energyregion, and the IJ = 20 wave is actually repulsive.Next we turn to the imaginary part (middle plot) and modulus (bottom plot). It isplain that unitarity is badly violated at a scale between 2 and 3 TeV (for this modest valueof f ) because the modulus of A exceeds 1, which is not possible according to Eq. (5.17).But moreover, the equation is not well satisfied even for much smaller scales. This is ahandicap of perturbation theory.We now switch-on a and a within the range of values explored in reference [23], andplot the results in figure 2.In agreement with that reference, we find that positive values of a or a enhance the IJ = 00 channel at low energy, while negative values suppress it. The vector amplitudeis enhanced by positive a and negative a , while the isotensor one is larger for negative a or a . The IJ = 02 amplitude seems too small in the low-energy region to be of muchuse in early experiments for small a and a , but it is sensitive to the NLO terms.We now fix a and a to 0.0025 at the same scale of 1 TeV and note the variation ofthe amplitudes respect to f in figure 3. – 13 – )-0.400.40.81.2 R ea l ( t ) I=0, J=0I=1, J=1I=2, J=0I=0, J=2 )00.40.8 I m a g i n a r y ( t ) I=0, J=0I=1, J=1I=2, J=0I=0, J=2 )00.40.81.2 | t | I=0, J=0I=1, J=1I=2, J=0I=0, J=2
Figure 1 . From top to bottom: real part, imaginary part, and modulus of the elastic ωω → ωω scattering amplitude to one loop. Here f = 500 GeV (approximately 2 v ), and the NLO constantsare chosen to be a = a = 0 at a scale µ = 1 TeV. We show the four NLO non-vanishing partialwaves A IJ . | A | , in the top left plot, is seen to shoot more rapidly for larger f , implying thegeneric ωω interaction will be stronger yet. The effect is opposite for the tensor amplitudein the bottom right panel, | A | , presenting a smaller modulus. The other two amplitudesare initially larger at smaller energies, but as s increases the trend changes and they becomeless prominent for larger f . – 14 – )00.20.40.60.81 | t | a = a = 0a = +0.005 a =0a = -0.005 a = 0a = 0 a = +0.005a = 0 a = -0.005 0 1 2 3 4 5s (TeV )00.20.40.60.81 | t | a = a = 0a = +0.005 a =0a = -0.005 a = 0a = 0 a = +0.005a = 0 a = -0.005 )00.20.40.60.81 | t | a = a = 0a = +0.005 a =0a = -0.005 a = 0a = 0 a = +0.005a = 0 a = -0.005 )00.20.40.60.81 | t | a = a = 0a = +0.005 a =0a = -0.005 a = 0a = 0 a = +0.005a = 0 a = -0.005 Figure 2 . For f = 500 GeV, we take a non-zero (positive or negative) a or a at µ = 1 TeV andplot the modulus of the partial wave amplitudes for elastic ww → ww scattering. In clockwise sensefrom the top left, we show | A | , | A | , | A | , | A | . Figure 4 shows the inelastic ωω → ϕϕ and elastic ϕϕ → ϕϕ for parameters f = 350 GeV, α = 1 . (perfectly allowed by current LHC bounds [19]), β = 2 (unconstrained atthe LHC) and all three NLO parameters γ , δ , η set to zero at 1 TeV. Because α − β is negative with this parameter choice, the real part of M is also negative, while ReT remains positive due to the factor appearing squared. More than in elastic ωω scattering,the J = 2 amplitudes are completely negligible.This is also the case for the ϕϕ → ϕϕ tensor amplitude in figure 5 showing the sen-sitivity to including the γ parameter with a small value of ± . ; obviously if a tensorresonance exists that couples to this T channel, it will entail a large value of γ . The scalaramplitude is more commensurate with others, yet keeping in mind that it is very dependenton ( α − β ) . The effect of a positive γ is to enhance (negative γ , to decrease) the amplitudeat very low scales.In figure 6 we plot the moduli of the ωω → ϕϕ scalar and tensor amplitude showing theeffect of adding either δ = ± . or η = ± . . The tensor piece M is only affected by η as per Eq. (5.13). The scalar amplitude on the left panel is on the other hand influencedby both, and becomes larger for either of δ or η taking a negative value.– 15 – )00.40.81.2 | t | f=v=246 GeVf=500 GeVf=1 TeV )00.40.81.2 | t | f=v=246 GeVf=500 GeVf=1 TeV0 1 2 3 4 5s (TeV )00.40.81.2 | t | f=v=246 GeVf=500 GeVf=1 TeV 0 1 2 3 4 5 6 7 8 9s (TeV )00.40.81.2 | t | f=v=246 GeVf=500 GeVf=1 TeV Figure 3 . For fixed a = a = 0 . at µ = 1 TeV, we vary f as indicated and plot the modulusof the perturbative partial wave amplitudes for elastic ww → ww scattering. In clockwise sensefrom the top left, we show | A | , | A | , | A | , | A | . In view of these results, an experimental programme to measure the parameters of theEWSBS and check them against the Minimal Standard Model from “low” energy data inthe TeV region or below would start by a partial-wave analysis of W L W L spectra, where onewould hope to be able to fit a , a , f and β (setting α = 1 ). The tensor wave being verysmall a priori, one would resort to the scalar-isoscalar, vector-isovector, and scalar-isotensorfinal states, selected by the charge combinations of the W ’s. If A is nevertheless found tobe large, this would immediately point out to important NLO contact terms.To proceed, one would first attempt an extraction of the leading order ( ∝ s ) scalaramplitude (see Eq. (5.7) ) whose slope gives access to f . Then the three NLO ( ∝ s )partial waves in Eqs.(5.7), (5.8), (5.9) give access to different linear combinations of a , a and β , so obtaining the slopes of the s terms in the spectra allows their isolation.In the absence of a ϕϕ spectrum, a unitarity analysis of the partial W L W L wavesmay reveal the leak of probability to that unmeasured channel. The slope of the LO term( ∝ s ) is an independent measure of β/f . The slope of the NLO s term in Eq. (5.12)then gives access to the combination δ + η/ . Separately measuring η (and thus δ ) seems– 16 – )-0.4-0.200.2 R ea l ( t ) ωω−>ϕϕ J=0 ωω−>ϕϕ
J=2 ϕϕ−>ϕϕ
J=0 ϕϕ−>ϕϕ
J=2 )-0.2-0.100.10.2 I m a g i n a r y ( t ) ωω−>ϕϕ J=0 ωω−>ϕϕ
J=2 ϕϕ−>ϕϕ
J=0 ϕϕ−>ϕϕ
J=2 )00.10.20.30.4 | t | ωω−>ϕϕ J=0 ωω−>ϕϕ
J=2 ϕϕ−>ϕϕ
J=0 ϕϕ−>ϕϕ
J=2
Figure 4 . From top to bottom: real part, imaginary part, and modulus of the elastic ϕϕ → ϕϕ and cross-channel ωω → ϕϕ scattering amplitude to one loop. Here f = 350 GeV (somewhat largerthan v ), α = 1 . , β = 2 , and the NLO constants are chosen as γ , δ , η ( µ = 1TeV) = 0 . quite hopeless because it requires to separate the tiny tensor M channel. Unless the BSMspectrum contains a tensor resonance in the TeV scale, this will be heroic.One can do little else unless a ϕϕ two-scalar boson spectrum becomes available. Insuch case, one may access the γ parameter directly from the scalar amplitude at NLO, whilefitting simultaneously δ and η . – 17 – )00.10.20.30.4 | t | T γ=0 T γ=0.005 T γ=−0.005 )00.010.020.03 | t | T γ=0 T γ=0.005 T γ=−0.005 Figure 5 . ϕϕ elastic scattering in the presence of the NLO γ parameter with µ = 1 TeV. Left:modulus of the scalar partial-wave. Right: modulus of the tensor partial-wave. Note the verydifferent scale. )00.10.20.30.40.50.60.70.8 | t | M δ=η=0 M δ=0.005 M δ=−0.005 M η=0.005 M η=−0.005 )00.010.020.03 | t | M η=0 M η=0.005 M η=−0.005 Figure 6 . ωω → ϕϕ channel-coupling amplitude in the presence of the NLO δ and η parameterstaken at µ = 1 TeV, alternatively. Left: modulus of the scalar partial-wave. Right: modulus of thetensor partial-wave. Note the very different scale.
It remains to comment that, although we have been speaking of “strong” interactionsin case the parameters of the low-energy Lagrangian density separate from the MSM, thecross-sections are rather small because of the /s flux factor. For example, the ωω → ωω cross-section can be expressed as σ ( s ) = 64 πs X IJ (2 I + 1)(2 J + 1) | A IJ | ; (6.1)the /s factor sets the scale at 1 TeV − ≃ . nbarn (increased to a meager 78 nbarnwith the π factor). For example, we can use the amplitudes from figure 2 with the sameparameters there indicated to plot the elastic ωω → ωω cross section in figure 7, that shootsup rapidly for essentially any a or a enhancing BSM physics, but remains relatively smallwhen compared with hadronic cross sections (of order 70 mbarn at the LHC, five orders ofmagnitude larger). – 18 – )050100150200 σ ( nb a r n ) a = a = 0a = +0.005 a =0a = -0.005 a = 0a = 0 a = +0.005a = 0 a = -0.005 Figure 7 . Cross-section for ωω → ωω resulting from evaluating Eq. (6.1) with the amplitudes infigure 7 and the parameters there described (taken again at µ = 1 TeV).
What “strong interactions” means in this context is that the A IJ amplitudes have mod-uli of order 1. Then, for example, Watson’s final state theorem applies due to rescattering,and the phases of the W L W L or ϕϕ production amplitudes should be the same as the phasesof the elastic amplitudes (that are not directly accessible since we do not have asymptoticbeams of these unstable particles). m ϕ We have been working in the chiral limit with m ϕ ≃ m W ≃ m Z ≃ . This is a theoreticallimit that may bear resemblance with reality in the energy region s ≃ TeV where squaredmomenta are significantly larger than masses, but it is perhaps useful to briefly assess thesize of the terms neglected.We focuse on ϕϕ → ϕϕ elastic scattering, because our chiral amplitude vanishes at O ( s ) with the series starting at O ( s ) (see Eq. (5.14) where K = 0 ) so one expects maximumsensitivity to the correction.Our amplitude, at a simple reference point such as µ = s = 1 TeV can be written as T (cid:0) s = 1 TeV (cid:1) = 1 TeV πv ξ (cid:18) γ (1TeV) + α − βπ (cid:19) (6.2) = 0 . ξ (cid:18) γ + α − βπ (cid:19) In the first place, if we consider instead the Higgs self-coupling potential in the StandardModel, V self = m ϕ v ϕ + m ϕ v ϕ , (6.3)– 19 –n the absence of new physics, the amplitude (heretofore vanishing due to our taking thechiral limit) would be, after projecting over J = 0 , T self = − m ϕ πv m ϕ s − m ϕ + 2 × m ϕ s − m ϕ log m ϕ s − m ϕ !! . (6.4)At s = 1 TeV , this is numerically equal to − . .
048 + 2 × . − .
12 10 − .Comparing with Eq. (6.2), we see that the Standard Model Higgs self-couplings arenegligible respect to the BSM ones at a scale of 1 TeV when ξ ( α − β ) ≫ . . (6.5)As discussed in [17], this is phenomenologically viable (essentially, β is unconstrained todate).Even when α = β = 1 , the other term in eq. 6.2 can also dominate the scattering if π ξ γ ≫ . , (6.6)as Standard Model Higgs self-couplings would then be negligible. In this case α = β = 1 (see sec. 5), so the channels decouple; still, the interactions are strong.Nevertheless, if the separation from the SM is small so that α ≃ β , ξ ≃ and γ . . · − , a phenomenological analysis should keep the SM couplings (this is akinto keeping the pion-mass terms in chiral perturbation theory for low-energy QCD, androutinely done).As we proceed to consider operators beyond the standard model in the Higgs sector, weencounter two more of dimension six [11], that involve two derivatives of the Higgs doubletfield, Q H ✷ := (cid:16) H † H (cid:17) ✷ (cid:16) H † H (cid:17) (6.7) Q HD := (cid:16) H † D µ H (cid:17) ∗ (cid:16) H † D µ H (cid:17) . After substituting the real physical field ϕ and neglecting the coupling to the transversegauge bosons, they reduce to Q ϕ ✷ = ϕ ✷ ϕ (6.8) Q ϕ∂ = ( ϕ∂ µ ϕ ) ( ϕ∂ µ ϕ ) . These last two operators are related by use of Green’s first identity, Q ϕ ✷ = − Q ϕ∂ + boundary term . (6.9)In this paragraph we explore the addition of a term proportional to Q ϕ∂ to the effectiveLagrangian density in Eq. (2.1).One can discuss the normalization of the operator, whether γ f Q ϕ∂ or v γ f Q ϕ∂ shouldbe taken in the effective Lagrangian, with a certain coefficient γ . Irrespective of this, thescattering amplitude ϕ ϕ → ϕ ϕ can be easily calculated, yielding iT Q ϕ∂ = − i γ f (cid:0) ξ (cid:1) ( p p − p p − p p − p p − p p + p p ) (6.10)– 20 –here the prefactor of 4 is combinatoric (hence the 1/4 in the normalization) and ξ dependson how one decides to normalize the operator. Eliminating the momenta in terms of theMandelstam variables, iT Q ϕ∂ = − i γ f (cid:0) ξ (cid:1) (cid:0) s + t + u − m ϕ (cid:1) (6.11)we see that T Q ϕ∂ = γ m ϕ f (cid:0) ξ (cid:1) . (6.12)Nominally, the operator is zero in the chiral limit m ϕ → and thus negligible in theTeV region unless the unknown coefficient γ is not of natural size (in the Standard Model,of course, γ = 0 ).In conclusion, because the operator Q ϕ∂ is of one more order (at least) in the /f counting than the SM Higgs self-couplings, it can be neglected to a first approximationin the 100 GeV region respect to the SM ones; because it is of the same order in the m ϕ counting, it can be consistently neglected against the other beyond SM operators in dealingwith TeV-scale ϕϕ scattering. With the present experimental situation, the Electroweak Symmetry Breaking Sector mightbe completely described by the Glashow-Weinberg-Salam Standard Model [24], with 3 lon-gitudinal ω L | z L bosons and the potential finding of its Higgs boson on the table. If BeyondSM physics exists, the mutual couplings of these four bosons will separate from the SM.The most interesting feature is the absence of any new particles below about 600-700 GeVimplying that a separation from the SM in the couplings will lead to strong interactions.We have calculated and renormalized the 1-loop amplitudes in Electroweak ChiralPerturbation Theory supplemented by the new scalar boson, a natural alley of investigationbased on a low-energy Effective Lagrangian that other groups are also pursuing. In doingso we have found that 3 dimension-8 derivative operators, not analyzed in depth in previousliterature, are necessary in addition to the standard ones associated with a and a . Wehave shown sensitivity of WBGB scattering [25] including now the new scalar boson, to allthe LO and NLO parameters in the Lagrangian density.Strong interactions and unitarity violations in perturbation theory appear as soon as v = f ( ξ = 1 ), α = 1 as seen in Eq. (3.3) or β = 1 , or finally any of η , γ , δ , a , a = 0 .As usual in an effective Lagrangian, the tree-level amplitudes present polynomial behaviorand the one-loop diagrams bring in standard left and right cuts into the partial waves.We have found that, without the NLO constants, the amplitudes behave (unsurpris-ingly) just like in hadron physics, with strongly attractive I = J = 0 elastic ωω scattering,not so strong I = J = 1 scattering, small (vanishing at LO) J = 2 scattering, and repulsive I = 2 scattering.The latter one leaves us wondering. In QCD π + π + scattering is naturally repulsive dueto the Pauli exclusion principle operating at the quark level, with u ¯ d − u ¯ d blocking-off partsof the ground state wavefunctions. From the point of view of the effective Lagrangian, this– 21 –epulsion is built into the flavor structure, inherited by the Electroweak Chiral Lagrangian.But it is odd, since there is no known nor necessary fermion constitution of the W + bosons,that W + W + “exotic” scattering should be repulsive. In this respect it is relieving to find,as we did in figure 2, and in agreement with [23], that the addition of either a or a NLOterms with a negative sign makes the tensor amplitude much stronger at low energy. Exoticresonances are possible and leave a low-energy footstep in the EWChL (and are natural inextensions of the MSM that need a Higgs multiplet with charged members).We wish to remark also that the ϕϕ → ϕϕ scattering occurs via a non-diagonal ωω loop if the interactions indeed become strong through mismatches of α and β as well as f and v to their Standard Model values, quite irrespective of the value of the Higgs self-couplings λ , λ . These couplings do not need to be known to high precision [26] if stronginteractions beyond the Standard Model operate in the TeV region, as they would have anegligible effect anyway. It appears that the Run II of the LHC should be quite conclusiveas respects further strong interactions in EWSB, as long as W L W L can be separated.In all the amplitudes studied, the tensor J = 2 projections computed from the LO andloop-NLO (no NLO counterterms) are totally negligible in the few-TeV regime: two-bodyscattering in the electroweak symmetry breaking sector is dominated by J = 0 , with anon-negligible contribution of J = 1 in ωω → ωω (that because of Bose symmetry, cannotbe present in the channels involving ϕϕ ).In future work we intend to study unitarization methods that render these one-loopamplitudes more theoretically sensible, and see what resonances appear in the differentspin-isospin channels for physically acceptable values of the parameters. Acknowledgments
AD thanks useful conversations with D. Espriu, M. J. Herrero and J. J. Sanz-Cillero. Thework has been supported by the spanish grant FPA2011-27853-C02-01 and by the grantBES-2012-056054 (RLD).
References [1] J.M. Cornwall, D.N. Levin and G. Tiktopoulos, Phys. Rev. D (1974) 1145; C.E. Vayonakis,Lett. Nuovo Cim. (1976) 383; B.W. Lee, C. Quigg and H. Thacker, Phys. Rev. D (1977)1519; M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. (1985) 379; M. S. Chanowitz, M.Golden and H. Georgi, Phys. Rev. D (1987) 1490; A. Dobado J. R. Peláez Nucl. Phys.B (1994) 110; Phys. Lett.B329 (1994) 469 (Addendum, ibid, B (1994) 554.[2] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B , 1 (2012).[3] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B , 30 (2012).[4] G. Aad et al. (ATLAS Collaboration), Report No. ATLAS-CONF-2012-168; S. Chatrchyanet al. (CMS Collaboration), Report No. CMS-HIG-12-015.[5] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B , 123 (2011) [arXiv:1107.4771[hep-ex]]; G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 22 (2012)[arXiv:1112.5755 [hep-ex]]; G. Aad et al. [ATLAS Collaboration], Phys. Lett. B , 305(2013) [arXiv:1301.5272 [hep-ex]]. – 22 –
6] D. Espriu, F. Mescia and B. Yencho, Phys. Rev. D , 055002 (2013) [arXiv:1307.2400[hep-ph]].[7] A. Azatov, R. Contino and J. Galloway, JHEP , 127 (2012) [Erratum-ibid. , 140(2013)] [arXiv:1202.3415 [hep-ph]].[8] I. Brivio et al. arXiv:1311.1823 [hep-ph].[9] R. Alonso, et al. Phys. Lett. B , 330 (2013) [arXiv:1212.3305 [hep-ph]].[10] A. Pich, I. Rosell and J. J. Sanz-Cillero, arXiv:1307.1958 [hep-ph].[11] E. E. Jenkins, A. V. Manohar and M. Trott, JHEP , 087 (2013) [arXiv:1308.2627[hep-ph]].[12] C. Degrande, N. Greiner, W. Kilian, O. Mattelaer, H. Mebane, T. Stelzer, S. Willenbrockand C. Zhang, Annals Phys. , 21 (2013) [arXiv:1205.4231 [hep-ph]].[13] G. Buchalla, O. Cata and C. Krause, arXiv:1307.5017 [hep-ph].[14] G. Buchalla and O. Cata, JHEP , 101 (2012) [arXiv:1203.6510 [hep-ph]].[15] T. Appelquist and C. Bernard, Phys. Rev. D22, 200 (1980). A. Longhitano, Phys. Rev. D22,1166 (1980), Nucl. Phys. B188, 118 (1981). A.Dobado, D. Espriu, M.J. Herrero, Phys. Lett.B255, 405 (1991). B.Holdom and J. Terning, Phys.Lett. B247 (1990) 88. A. Dobado, D.Espriu and M.J. Herrero, Phys.Lett. B255 (1991) 405. M. Golden and L. Randall, Nucl.Phys. B361 (1991) 3.[16] S Weinberg, Physica A 96 (1979) 327 J.Gasser and H.Leutwyler, Ann. of Phys. 158 (1984)142, Nucl. Phys. B250 (1985) 465 y 517[17] R. L. Delgado, A. Dobado and F. J. Llanes-Estrada, arXiv:1308.1629 [hep-ph].[18] R. Contino et al. , JHEP (2010) 089 [arXiv:1002.1011 [hep-ph]]; R. Contino,arXiv:1005.4269 [hep-ph]; R. Grober and M. Muhlleitner, JHEP (2011) 020[arXiv:1012.1562 [hep-ph]].[19] G. Belanger et al. , Phys. Rev. D , 075008 (2013) [arXiv:1306.2941 [hep-ph]]; T. Corbett, etal. , Phys. Rev. D , 075013 (2012) [arXiv:1207.1344 [hep-ph]]; ibid. arXiv:1306.0006[hep-ph]; J. Ellis and T. You, JHEP , 103 (2013) [arXiv:1303.3879 [hep-ph]];P. P. Giardino et al. , arXiv:1303.3570 [hep-ph]; A. Falkowski, F. Riva and A. Urbano,arXiv:1303.1812 [hep-ph].[20] E. Halyo, Mod. Phys. Lett. A (1993) 275; W. D. Goldberger, B. Grinstein and W. Skiba,Phys. Rev. Lett. (2008) 111802 [arXiv:0708.1463 [hep-ph]].[21] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B , 165 (2005)[arXiv:hep-ph/0412089]. R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D , 055014(2007) [arXiv:hep-ph/0612048]. D. Barducci et al. JHEP , 047 (2013) [arXiv:1302.2371[hep-ph]].[22] D. B. Kaplan and H. Georgi, Phys. Lett. B (1984) 183; S. Dimopoulos and J. Preskill,Nucl. Phys. B , 206 (1982); T. Banks, Nucl. Phys. B , 125 (1984). D. B. Kaplan; H.Georgi and S. Dimopoulos, Phys. Lett. B , 187 (1984); H. Georgi, D. B. Kaplan and P.Galison, Phys. Lett. B , 152 (1984); H. Georgi and D. B. Kaplan, Phys. Lett. B , 216(1984); M. J. Dugan, H. Georgi and D. B. Kaplan, Nucl. Phys. B , 299 (1985); G. F.Giudice, et al. , JHEP (2007) 045 [arXiv:hep-ph/0703164].[23] D. Espriu and B. Yencho, Phys. Rev. D , 055017 (2013) [arXiv:1212.4158 [hep-ph]]. – 23 –
24] S. L. Glashow, Nucl. Phys. 22 (1961) 579 S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264 A.Salam, Proc. 8th Nobel Symp., ed. N. Svartholm, p. 367, Stockholm, Almqvist y Wiksells(1968)[25] A. Dobado and M.J. Herrero, Phys. Lett.B228 (1989) 495 and B (1989) 505. J. Donoghueand C. Ramirez, Phys. Lett.B (1990) 361.[26] R. S. Gupta, H. Rzehak and J. D. Wells, Phys. Rev. D , 055024 (2013) [arXiv:1305.6397[hep-ph]].[27] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, arXiv:1310.1921 [hep-ph].[28] T. Hahn, Comput. Phys. Commun. , 418 (2001) [hep-ph/0012260].[29] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. , 153 (1999) [hep-ph/9807565].[30] J. Kuipers, T. Ueda, J. A. M. Vermaseren and J. Vollinga, Comput. Phys. Commun. ,1453 (2013). A Feynman diagrams
In this appendix we present the Feynman diagrams that we have employed to generate theone-loop parts of the amplitudes written down in Eq. (3.4), (3.10), (3.15). The diagramshave been automatically generated with FeynRules [27] and FeynArts [28], and evaluatedwith FormCalc [29, 30], though the whole computation has also been carried out analytically,since it is of moderate difficulty, being a one-loop evaluation in the massless limit. Both waysof computing the amplitudes agree, thus checking the output of the computer programs.The only minor issue is that the automated tools seem to have problems with Einstein’ssummation convention when confronting the counterterm ( ∂ µ ϕ∂ µ ϕ ) that needs to be typed-in as ( ∂ µ ϕ∂ µ ϕ )( ∂ ν ϕ∂ ν ϕ ) .Figure (9) shows the ωω → ωω one-loop contributions that build up Eq. (3.4). Thereone can easily identify vertex corrections (such as Feynman diagrams 1 to 10), s and t/u channel bubbles (diagrams 13-15) or box diagrams (11 and 12).The same kinds of diagrams (but in smaller numbers) can be identified in figure 8 thatshows what needs to be computed for ϕϕ → ϕϕ elastic scattering in Eq. (3.15).Finally, figures 10 and 11 contain the Feynman diagrams necessary to compute theinterchannel amplitude ωω → ϕϕ in Eq. (3.10).– 24 – igure 8 . Feynman diagrams corresponding to ϕϕ elastic scattering. – 25 – igure 9 . Feynman diagrams corresponding to ωω elastic scattering. – 26 – igure 10 . Feynman diagrams corresponding to ωω → ϕϕ channel coupling. – 27 – igure 11 . Further Feynman diagrams corresponding to ωω → ϕϕ channel coupling.channel coupling.