Higgs decay into diphoton in the Composite Higgs Model
aa r X i v : . [ h e p - ph ] A ug Higgs decay into diphoton in theComposite Higgs Model
Haiying Cai
Department of Physics, Peking University, Beijing 100871, China
Abstract
We explore the Higgs couplings to gauge bosons in the minimal SO (5) /SO (4)4D composite Higgs model. The pion scatterings put unitary constraints on thecouplings and therefore determine the branching ratios of various Higgs decays.Through fine-tuning the parameters, enhancement of Higgs to diphoton rate ispossible to be achieved with the existence of vector meson fields. Introduction
The composite Higgs model provides an alternative solution to the little hierarchyproblem compared with the well-known supersymmetric models, since the economicformulation of the standard model is impossible to explain the lightness of Higgs mass.The composite Higgs boson emerges as a pseudo-Nambu-Goldstone boson (pNGB)from a spontaneously broken global symmetry, therefore its mass is much lighter thanthe other resonances from the strong dynamic sector. The original minimal compositemodel is realized in the five-dimensional Randall Sundrum model, and the Higgs is thefifth component of the broken gauge bosons [1]. Using the holographic approaching,the effective Lagrangian is gained after integrating out the bulk field with the UV branevalue fixed. The potential for the holographic composite Higgs could be calculated inthe form of brane to brane 5D propagators [2]. In the last few years increased attentionhas been focused on the deconstruction version of the 5D theory, which leads to varietiesof 4D composite Higgs models, assuming the existence of one elementary sector andone strong interaction sector [3]. Without the presence of additional composite fields,the composite Higgs has a reduced coupling with the gauge bosons, which may leadto the violation of unitary in the pion scatterings before the cutoff scale is reached.The method to restore the perturbative unitary is to introduce vector resonances. Theunitary requirement will correlate the global symmetry breaking scale f with the mass m ρ of the vector resonance. It is interesting that the presence of the vector resonancewill also modify the Higgs coupling, with its deviation parametrized by ξ = v /f ,which in turn changes the branching ratio of various Higgs decay. Another crucialingredient in the composite Higgs model is the partial compositeness of gauge bosonsdue to the nonlinearity, with the degree of compositeness mainly controlled by thegauge couplings. In this paper we first review a simple model setup of 4D compositeHiggs and show that it is capable to accommodate the 125 GeV resonance with theappropriate properties recently discovered at the LHC [4].1 Lagrangian of the sigma model
Let us start with the basic model setup. Our Higgs is realized as one pNGB from astrong interacting sector using the nonlinear sigma model. We formulate the effectiveLagrangian for those pNGBs via the Callan-Coleman-Wess-Zumino (CCWZ) prescrip-tion [5]. In the following, we are going to review the nonlinear realization of compositeHiggs and capture the necessary ingredients for our calculations. Considering the globalsymmetry breaking pattern SO (5) → SO (4), there are four pNGBs which fit a basicrepresentation of the SO (4) symmetry group. The first three, i.e. π , , , are eaten bythe W, Z bosons, with the remaining one, π , identified as the Higgs. Denoting theGoldstone bosons as U = exp ( i √ π ˆ a T ˆ a /f ), the sigma field would transform nonlinearlyunder the full global symmetry as U → gU h † ( g, π ), and one can calculate the structureof iU † ∂ µ U = d ˆ aµ T ˆ a + E a L µ T a L + E a R µ T a R , where T ˆ a , ˆ a = 1 , , , SO (5) /SO (4), and T a L ( a R ) , a L ( a R ) = 1 , , SO (4) ≃ SU (2) L × SU (2) R symmetry group. It should be noted thatall the generators in the SO (5) symmetry group are normalized as Tr( T a T b ) = δ ab .The building blocks of the CCWZ formalism are the variables d ˆ aµ and E a L ,a R µ decom-posed in the broken and unbroken generator directions respectively. Following theusual formulation, we gauge a subgroup SU (2) × U (1) in the global SO (4), resultingin an explicit breaking of the full global symmetry. The gauged CCWZ structuresare calculated in a similar approach by substituting ∂ µ with the covariant operator D µ = ∂ µ − ig W aµ T aL − iB µ T R . At the leading order of the chiral expansion, d ˆ aµ and E a L ,a R µ are expressed as: d ˆ aµ = − √ f D µ π ˆ a + √ f [ π, [ π, D µ π ]] ˆ a + · · · , (1) E aµ = g W aµ + g ′ B µ δ a + if [ π, D µ π ] a + · · · (2)under the local symmetry group, the corresponding transformation rules are: d µ → h ( g, π ) d µ h † ( g, π ) , E µ → h ( g, π ) E µ h † ( g, π ) + ih ( g, π ) ∂ µ h † ( g, π ) (3)2ince E µ behaves as a gauge field, the coupling of Goldstone bosons to the fundamentalfermions is via the covariant derivative ∂ µ − iE µ . In this paper we are only concernedwith the vector meson effects and would not explore too much into the fermion sector.We can conveniently calculate the mass terms for the W and Z gauge bosons afterelectroweak symmetry breaking through the kinetic terms: f d µ d µ = 12 2 m W v ( v + 2 ah + b h v ) W + µ W − µ + 12 m Z v ( v + 2 ah + b h v ) Z µ Z µ + O ( h ) (4) m W = g f sin θ , m Z = ( g + g ′ ) f sin θ , a = cos θ, b = cos θ − sin θ (5)where the parameters a and b , both of which are always less than one, indicate that theHiggs couplings are reduced as compared with the Standard Model. In the minimal SO (5) /SO (4) setup, θ is the misalignment of the true vacuum relative to the gauged SO (4) subgroup, with the VEV of the Higgs defined as v = f sin θ = 246 . ρ Lµ and ρ Rµ in the rep-resentations (3 , ⊕ (1 ,
3) of SU (2) L × SU (2) R , transforming under the local symmetrygroup as ρ → h ( g, π ) ρ µ h † ( g, π ) + ih ( g, π ) ∂ µ h † ( g, π ), needs to be added into the strongdynamic sector. The gauge invariant Lagrangian for the vector resonances consistingof kinetic terms and mass terms is formulated as: L ρL = −
14 Tr ( ρ L,µν ρ µνL ) + a ρ L f (cid:0) g ρ L ρ Lµ − E Lµ (cid:1) , (6) L ρR = −
14 Tr ( ρ R,µν ρ µνR ) + a ρ R f (cid:0) g ρ R ρ Rµ − E Rµ (cid:1) . (7)At the low energy scale, we are only interested in the interactions which are relevant tothe pion scatterings, that is the Goldstone bosons self-interactions and at most theirinteractions with the vector resonances. After a little bit of algebra, it is easy to reachthe explicit Lagrangian: L ρ L π + π = a ρL g ρL (cid:2) ε ijk π i ∂ µ π j ρ kLµ + (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kLµ (cid:3) − a ρL f h ( π a ∂ µ π a ) − (cid:0) π a ∂ µ π b (cid:1) i (8) L ρ R π + π = a ρR g ρR (cid:2) ε ijk π i ∂ µ π j ρ kRµ − (cid:0) π k ∂ µ π − π ∂ µ π k (cid:1) ρ kRµ (cid:3) − a ρR f h ( π a ∂ µ π a ) − (cid:0) π a ∂ µ π b (cid:1) i (9)3ince the hπ and h π interactions are determined by a and b , whereas the pion self-interaction and pion interaction with vector meson are related to a ρ and the globalsymmetry breaking scale f , the correlation between those parameters and the allowedparameter space information could be extracted from both the pion elastic and thepion inelastic scatterings. ππ → ππ scattering For the scattering π a π b → π c π d of the SU (2)-triplet Goldstones, the amplitude has thegeneral isospin structure: A ( π a π b → π c π d ) = A ( s, t, u ) ( ππ ) δ ab δ cd + A ( t, s, u ) ( ππ ) δ ac δ bd + A ( u, t, s ) ( ππ ) δ ad δ bc (10) A ( s, t, u ) ( ππ ) = sv − a ρL f " s + m ρL s − ut − m ρL + s − tu − m ρL ! − a ρR f " s + m ρR s − ut − m ρR + s − tu − m ρR ! − a v s s − m h (11)where the terms with dependence on the mass m ρ L,R comes from ρ L,R meson mediated t channel and u channel diagrams, and the last term comes from the light Higgs mediated s channel diagram, whereas the remaining terms come from the contact interaction.The amplitude can be decomposed into 1 , , T ( s, t, u ) = 3 A ( s, t, u ) + A ( t, s, u ) + A ( u, t, s ) , (12) T ( s, t, u ) = A ( t, s, u ) − A ( u, t, s ) , (13) T ( s, t, u ) = A ( t, s, u ) + A ( u, t, s ) . (14)It is then possible to transform these isospin amplitudes in terms of the partial wave(PW) decomposition T I ( s, t ) = ∞ X J =0 π (2 J + 1) P J (cos α ) a IJ ( s ) , (15)4ith the partial waves provided by a IJ ( s ) = 164 π Z +1 − d cos α P J (cos α ) T I ( s, t ( s, cos α )) , (16)and with cos α = 1 − t/s . In this normalization the partial waves can be writtenin the form a IJ ( s ) = i (1 − ηe iδ IJ ), with the inelasticity obeying the unitarity bound0 ≤ η ≤
1. This implies the constraints | Re a IJ ( s ) | ≤ , Im a IJ ( s ) ≤ , | a IJ ( s ) | ≤ . (17)We will make use of the first one in order to constrain our partial wave amplitudes.One must be aware of the slight arbitrariness of this choice, as we could also considerthe last constraint in (17) to determine when the theoretical determinations “violate”unitarity. The root of this ambiguity lies on the fact that the tree-level amplitude isnever truly unitary for s >
0, as the tree-level PW always lies out of the Argand circleand has inelasticity η >
1. Nevertheless, as far as | Re a IJ ( s ) | ≤ / m h ≪ | s | we get the partial wave powerexpansion, a ( s ) ( ππ ) = (4 − a ρL − a ρR ) s πf + (cid:20) a ρL m ρL m ρL s + 2 ! log sm ρL + 1 ! − ! πf + ( L ↔ R ) (cid:21) . (18)Notice that we have made use of the SO (5) /SO (4) relations v = f sin θ and a = cos θ .The first term in the r.h.s. of the equation diverges like ∼ O ( s ) at high energiesand spoils that unitarity bound very quickly. Hence, one usually requires the exactcancellation of the O ( s ) term in the high-energy ππ scattering [7, 8, 9], this is, a ρL + a ρR = 43 . (19)For the left-right symmetric case a ρL = a ρR = a ρ this turns into a ρ = 23 . (20)5 TeV 4 TeV3 TeV
800 1000 1200 1400 1600 1800 20000.700.750.800.850.900.951.00 m Ρ GeV c o s Θ H a L Π Π ® Π Π
800 1000 1200 1400 1600 1800 20000.700.750.800.850.900.951.00 m Ρ GeV c o s Θ H b L Π Π ® Ρ Ρ
Figure 1:
Parameter-space region where the unitarity bound | Re a ( s ) ( ππ ) | ≤ is violatedat energies s ≤ Λ , for Λ = 3 . . . g ρ = 2 .
0, i.e. only the regions above the lines arepermitted by perturbative unitary. The left panel is for the pion elastic scattering and theright panel is for the pion inelastic scattering.
It should be noticed that after imposing a ρ = 2 / a ( s ) ≃ m ρ πf (cid:18) sm ρ − (cid:19) at high energies. However, this mild ln ( s ) divergentbehavior at high energies will eventually exceed the “unitarity” bound. Since themass of vector resonance is m ρ = a ρ g ρ f , as we fix the coupling g ρ , two independentparameters are left. We are going to adopt another method to constrain the parameterspace of ( a, m ρ ) by demanding the unitary bound is satisfied below a fixed cutoff scaleΛ. In Fig. 1 (a), we have plotted the parameter-space region where the unitarity bound | Re a ( s ) ( ππ ) | ≤ is violated at energies s ≤ Λ , for Λ = 3 . .2 ππ → hh scattering For the inelastic scattering: A ( π a π b → hh ) = A ( s, t, u ) ( hh ) δ ab . The isospin structureis quite simple in this process and it gets a contribution from the contact interaction,along with π , ρ L , and ρ R exchanged t and u channels. The full prediction is: A ( s, t, u ) ( hh ) = − bv s − a v " ( t − m h ) t + ( u − m h ) u + a ρL + a ρR f ( − s + 2 m h ) − a ρL f m ρL s − ut − m ρL + s − tu − m ρL ! − a ρL f m h t − m ρL + 1 u − m ρL ! − a ρR f m ρR s − ut − m ρR + s − tu − m ρR ! − a ρR f m h t − m ρR + 1 u − m ρR ! (21)One may then perform a PW projection similar to that in Eq. (16) but with the effectof Higgs mass included in cos α = 2( t − m h + s/ / ( sβ h ( s )) and the phase-space factor β h ( s ) = p − m h /s . In the light Higgs limit m h ≪ | s | one gets a ( s ) ( hh ) = 12 a ( s ) ( ππ ) . (22)Here we made use of a = cos θ and b = cos 2 θ . With the existence of SO (5) globalsymmetry, we expect to get the same expectation as in ππ → ππ when we demand the O ( s ) term to exactly cancel out at high energies. Nonetheless, due to the extra factor , the violation of our “unitarity bound” by the linear s divergence and the residualln( s ) high energy divergence occurs later. ππ → ρρ scattering Now we come to consider the inelastic scattering π a π b → ρ cL ρ dL , where the longitudinalcomponent of the vector meson is parametrized as ǫ L ( k ) = (cid:18) √ s β ρ m ρ , √ s m ρ ~n k (cid:19) with β ρ = q − m ρ /s . As the Higgs is realized as one pNGB, the Higgs coupling c ρ f m ρ vhρ aµ ρ aµ comes from the mass term of the vector meson and the parameter c ρ is suppressedby g /g ρ . The isospin decomposition is similar to the elastic one but with two formfactors. For A ( s, t, u ) three diagrams contribute: the s channel h exchanged diagram, t channel and u channel π exchanged diagrams; whereas for B ( s, t, u ), there is one ρ µ s channel diagram, one π mediated u channel diagram, and one h mediated t channel diagram: A ( π a π b → ρ cL ρ dL ) = A ( s, t, u ) ( ρρ ) δ ab δ cd + B ( s, t, u ) ( ρρ ) δ ac δ bd + B ( s, u, t ) ( ρρ ) δ ad δ bc , (23) A ( s, t, u ) = ac ρ f s ( s − m ρ ) s − m h + a ρ f β ρ u (cid:16) s (cid:0) β ρ + 1 (cid:1) + t − m ρ (cid:17) + a ρ f β ρ t (cid:16) s (cid:0) β ρ − (cid:1) − t + m ρ (cid:17) , (24) B ( s, t, u ) = 14 f ( s + 2 m ρ )( t − u )( s − m ρ ) + a ρ f β ρ u (cid:16) s (cid:0) β ρ + 1 (cid:1) + t − m ρ (cid:17) + a ρ f β ρ t − m h ) (cid:16) s (cid:0) β ρ − (cid:1) − t + m ρ (cid:17) . (25)Since the vector resonance is introduced to restore the perturbative unitary, it is usuallydemanded that the cutoff scale satisfying 2 m ρ < Λ < πf . When the threshold effectof the final states could be ignored, i.e. m ρ ≪ Λ, there are only linear growing s term and constant term in the partial wave transformation. But as the mass m ρ iscomparable with Λ, logarithmic terms also appear: a ( s ) ( ρρ ) = (cid:18) ac ρ β ρ + 10 a ρ (cid:0) − β ρ (cid:1) β ρ + 5 a ρ (cid:0) − β ρ (cid:1) log (cid:18) (1 − β ρ ) s − m ρ (1 + β ρ ) s − m ρ (cid:19)(cid:19) s πf β ρ − (cid:18)(cid:0) ac ρ β ρ + 5 a ρ (cid:1) β ρ + 5 a ρ (cid:18)(cid:0) − β ρ (cid:1) − m ρ s (cid:19) log (cid:18) (1 − β ρ ) s − m ρ (1 + β ρ ) s − m ρ (cid:19)(cid:19) m ρ πf β ρ (26)In the limit m h , m ρ ≪ Λ, the partial wave displays a linear growing pattern: a ( s ) ( ρρ ) ≃ s πf (cid:18) ac ρ − a ρ (cid:19) , which pushes the partial wave to grow quickly after the twomesons threshold is reached, and this process provides a complementary constraint forthe parameter space. In Fig. 1 (b), with the c ρ term ignored, we show that the unitarybound actually imposes a more stringent constraint on the parameter space, that is itrequires one larger value of global symmetry breaking scale f (i.e. cos θ needs to bemore close to 1) for the same value of m ρ and Λ as compared with the elastic scattering.8 Higgs to diphotons from ρ mesons In this section, we discuss the resonance effects on the Higgs sector. The gauge bosonscouple to the vector resonances via the mass terms described in Eq.(6-7), since it is thecombination of ( g ρ ˜ ρ L,Rµ − E L,Rµ ) that transforms homogeneously under the symmetrygroup of SO (4). Due to the mixing between the gauge eigenstates of ˜ ρ aLµ , ˜ W aµ , ˜ ρ Rµ and ˜ B µ , the exact gauge bosons in the effective theory gain the property of partialcompositeness. At the leading order of ξ = v /f = (1 − cos θ ), the mass terms aresimplified as: L mρ = m ρ g ρ (cid:16) g ρ ˜ ρ aLµ − g ˜ W aµ (cid:17) + m ρ g ρ (cid:16) g ρ ˜ ρ Rµ − g ′ ˜ B µ (cid:17) (27)such that the gauge couplings for SU (2) L × U (1) Y in the standard model are determinedby the relations of: g = g g ρ g + g ρ , g = g ′ g ρ g ′ + g ρ (28)In the following analysis, we will take the same benchmark point g ρ = 2 . g and g ′ are fixed in order to reproduce the SM model couplings g and g at the electroweak scale. Including higher order expansion of the Higgs fields,the mixing would be further modified, as indicated by the following derivation. Inthe unitary gauge, all the pion fields are eaten and the Goldstone boson in the fourthdirection is the Higgs field, i.e. π ≃ h . There is one interaction term in the form of( h ) ( g ˜ W aµ − g ′ ˜ B µ δ a ) embedded in the connection E µ ’s explicit expression : E µ = (cid:16) g ˜ W aµ T aL + g ′ ˜ B µ T R (cid:17) + 1 f h h T ˆ4 , [ g ˜ W aµ T aL + g ′ ˜ B µ T R , h T ˆ4 ] i + · · · = (cid:16) g ˜ W aµ T aL + g ′ ˜ B µ T R (cid:17) − ( h ) f ( g ˜ W aµ − g ′ ˜ B µ δ a )( T aL − T aR ) + · · · (29)With the EW symmetry breaking, the second term in the above equation gives rise toone new interaction term between the Higgs fields and gauge bosons.Substituting both ( h ) by their VEVs would modify the mixing among the gaugebosons and vector mesons. Assuming the charged W ± µ gauge bosons are zero modes,9e will retain the correction occurring at the linear order of ξ but are justified to ignorecorrections at the order of m W /m ρ . The full rotation for the charged gauge bosons is:˜ W ± µ = g ρ W ± µ + g ρ ± Lµ ( g ρ + g ) / + ξg ρ g ( g W ± µ − g ρ ρ ± Lµ )4( g ρ + g ) / (30)˜ ρ ± Lµ = g W ± µ − g ρ ρ ± Lµ ( g ρ + g ) / − ξg ρ g ( g ρ W ± µ + g ρ ± Lµ )4( g ρ + g ) / − ξ ρ ± Rµ (31)˜ ρ ± Rµ = ρ ± Rµ + ξ ( g W ± µ − g ρ ρ ± Lµ )4( g ρ + g ) / (32)where ρ ± L and ρ ± R are mass eigenstates with corresponding masses of m ρ L = g ρ a ρ f and m ρ R = ( g ρ + g (1+ ξ/ a ρ f . The neutral gauge bosons mixing pattern is distinct fromthe charged ones as indicated by the E µ expression [see Eq.(29)]. The eigenstates forthe three massive neutral states are rather complicated. However it is easy to projectout the exact zero mode, i.e. the photon, which is the combination of the four neutralgauge eigenstates ˜ W µ , ˜ B µ , and ˜ ρ Lµ , ˜ ρ Rµ : A µ = g ′ g ˜ ρ Lµ + g ′ g ˜ ρ Rµ + g ρ ( g ′ ˜ W µ + g ˜ B µ ) q g ρ ( g ′ + g ) + 2 g ′ g (33)and for completeness the Weinberg mixing angle and the electromagnetic coupling aregiven as: c w = g ( g ′ + g ρ )2 g g ′ + ( g ′ + g ) g ρ ≈ g g ′ + g (34) s w = g ′ ( g + g ρ )2 g g ′ + ( g ′ + g ) g ρ ≈ g ′ g ′ + g (35) e = g g ′ g ρ q g g ′ + ( g ′ + g ) g ρ ≈ g g ′ g + g ′ (36)which are consistent with the SM formulas as we abandon the corrections at the orderof g ′ /g ρ and g /g ρ . We prefer to conduct the calculation with the mass eigenstate sincethe trilinear gauge interaction with one photon and the quartic gauge interaction withtwo photons are diagonal in that basis.On the other hand, with only one h gain VEV in Eq. (29) and the mixing massterm gives us the following Lagrangian for H - ˜ ρ - ˜ W and H - ˜ ρ - ˜ B interactions: L mix = m ρ ξ g ρ v h ˜ ρ aLµ ( g ˜ W aµ − g ′ ˜ B µ δ a ) − m ρ ξ g ρ v h ˜ ρ aRµ ( g ˜ W aµ − g ′ ˜ B µ δ a ) (37)10dapting it in terms of the mass eigenstates, we find positive shifts for the h W + µ W − µ and h Z µ Z µ vertices and a negative shift for the h ρ + Lµ ρ − Lµ at the leading order of ξ = v /f . It is convenient to parametrize the Higgs interactions with the gaugebosons adopting the effective theory approach: L eff = a W m W v h W + µ W − µ + a Z m Z v h Z µ Z µ + c ρ m ρ v h ρ + Lµ ρ − Lµ + c ρ R W m ρ v h (cid:0) W + µ ρ − Rµ + W − µ ρ + Rµ (cid:1) + c ρ L ρ R m ρ v h (cid:0) ρ + Lµ ρ − Rµ + ρ − Lµ ρ + Rµ (cid:1) + c f (cid:16) m f v ¯ f f (cid:17) h + c γ α πv h A µν A µν + c Zγ α πv h Z µν A µν (38) a W = (cid:18) g ρ g + g ρ + g g ρ ξ g + g ρ ) (cid:19) cos θ + g ξ g + g ρ ) m ρ m W (39) c ρ = (cid:18) g g + g ρ − g g ρ ξ g + g ρ ) (cid:19) m W m ρ cos θ − g ξ g + g ρ ) (40) a Z = cos θ + ( g + g ′ ) m ρ ξ g ρ m Z (41)where the third terms in a W and c ρ and the second term in a Z come from the mass mix-ing terms h ˜ ρ aL ( g ˜ W a − g ′ ˜ Bδ a ) and h ˜ ρ aR ( g ˜ W a − g ′ ˜ Bδ a ), and the c γ term is originatedthrough the loop contribution of heavy charged particles. Notice that only diagonalvertices are relevant to the branching ratio of Higgs decay into diphoton whereas thereare additional nondiagonal Higgs vertices along with nondiagonal trilinear and quarticgauge interactions which would contribute to h → Zγ . The latter process is correlatedto h → γγ due to the electroweak symmetry. The corrections to c γ come from thevector meson and its mixing with W , Z gauge bosons: c γ = c t N c (2 / F / (4 m t /m h ) + a W F (4 m W /m h ) + c ρ F (4 m ρ /m h ) (42) F / ( x ) = − x (cid:0) − x ) arcsin ( x − / ) (cid:1) (43) F ( x ) = 2 + 3 x + 3 x (2 − x )arcsin ( x − / ) (44)with x i = 4 m i /m W . For the large mass limit of ρ mesons and top quark mass, theasymptotic values for those form functions are: F ( x ) ≈ F / ( x ) ≈ − / ΓΓ =1.0 R ΓΓ =2.0 R ΓΓ =3.5 ΠΠ®ΡΡΠΠ®ΠΠ cos Θ m Ρ H G e V L Enhanced ratio: R ΓΓ = Br (cid:144) Br SM H h ®ΓΓ L Figure 2:
Contour plot for the R γγ in the gluon fusion channel assuming c t = 1. The blackdashed line is the unitary bound for the elastic pion scattering ππ → ππ and the orangedashed line is the unitary bound for the inelastic scattering ππ → ρρ with Λ = 5 TeV. Theregion approaching the cos θ = 1 direction is permitted. With the knowledge of those couplings, the partial width for Higgs to diphoton inthe composite Higgs model with respect to its prediction in the SM and the respectiveratios for the other two bosons channels are fixed to be:Γ / Γ( H → γγ ) sm = c γ c γ,sm , Γ / Γ( H → W W ∗ ) sm = a W a W,sm , Γ / Γ( H → ZZ ∗ ) sm = a Z a Z,sm (45)However at the LHC, only the product of σ × Br ( h → V V ′ ) is measurable.The variable,which indicates the deviation of composite Higgs models from the standard model, isthe so-called R parameter [10], i.e. the observing signal events divided by its corre-sponding SM expectation. For the diphoton process, the R γγ is defined as: R γγ = σ ( pp → h X ) σ sm ( pp → h X ) × Br ( h → γγ ) Br sm ( h → γγ ) (46)where σ is the production cross section for the Higgs boson and X denotes any particleassociatively produced with the Higgs boson. At the available energy scale, the main12roduction channels for the Higgs bosons are gluon fusions gg → h and vector bosonfusions q ¯ q → h jj . The modified cross sections for those two processes are [11]: σσ sm ( gg → h ) = c t , σσ sm ( gg → h jj ) = a W σ Wsm + a Z σ Zsm σ Wsm + σ Zsm . (47)For simplicity, in this paper we are going to assume that all the fermion couplings arethe same as they are in the SM , i.e. c f = 1, with the consequence that the top quarkinduced gluon fusion cross section is the same as in the SM. The observing ratios forthe diphoton process could be expressed in a more convenient form: R γγ = σ/σ sm · | c γ /c smγ | a W Br ( W W ∗ ) sm + a Z Br ( ZZ ∗ ) sm + | c γ /c smγ | Br ( γγ ) sm + · · · (48)The R γγ dependence on the (cos θ, m ρ ) for the gluon fusion channel is plotted in Fig. 2.We put the unitary bound on that plot by requiring that the perturbative unitaryis violated at Λ = 5 TeV. As we can see, if we demand that the composite Higgsmodel prediction does not give a significant deviation from the LHC measurement,the perturbative unitary is a very loose requirement for the allowed parameter space.To achieve a diphoton enhancement rate not larger than a factor of 1.5, we roughlyneed cos θ > .
97 and m ρ > . R γγ in the vector boson fusion process issimilar, but with a W , a Z >
1, a larger diphoton enhancement rate is encountered inthis channel.Adding new fermion resonances to the composite model would be quite interestingsince, under certain circumstance, it possibly enhances the production cross section ofHiggs bosons but at the same time it reduces the decay branching ratio into diphotons.The balanced effect might depend on the specific model details. Furthermore, thosecomposite fermions are introduced into the model as vector-like quarks, thus theirmixing with the SM quarks would inevitably modify the W - t - b and Z - b - b vertices andpossibly give a notable contribution to the oblique parameters [12]. Detailed studiesneed to be devoted to explore the influence of the third generation composite quarkson the Higgs sector [13, 14, 15]. 13 Conclusion
In summary, for a light composite Higgs boson which is realized as one pNGB from astrong interacting sector, ππ scatterings put some mild constraint on the (cos θ, m ρ )parameter space. We conduct a careful analysis for both the elastic and inelastic pionscatterings and the deviation of the Higgs to gauge couplings from the standard modeloccurring at the order of v /f is allowed as we reduce m ρ , the mass of composite mesonfield. The nonlinear realization enriches the Higgs interaction with SM gauge bosons.It is noticed that in the minimal SO (5) /SO (4) coset model, with the presence of vectormesons in the fundamental representation of SO (4), a new interaction originating fromthe strong interacting sector may shift the Higgs couplings a W and a Z in the positivedirection due to the partial compositeness of W and Z gauge bosons after electroweaksymmetry breaking. Therefore it is easy for us to accommodate an enhancement ofdiphoton rate which is observed at the LHC. It is believed that through extendingthe model structure (with effects on Higgs productions and decays) and fine tuningthe parameter space, the light composite Higgs probably could fit the experimentalmeasurements much better than the standard model. Note : It is interesting to observe that there is nondiagonal contribution to Higgscoupling to Z and photon. The calculation for this form factor is put in the appendix. Acknowledgments
The author is grateful for discussion of the manuscript with Juan Sanz Cillero. H.Caiis supported by the postdoc foundation under Grant No. 2012M510001.14 non-diagonal gauge boson contribution to H - Z - γ In this appendix, we are going to show that including non-diagonal couplings exclusivelyresults in a gauge-inviarant contribution for the form factor c Zγ . Similar nondiagonalccontribution from charginos in the MSSM is calculated in a reference [16]. Z h h Z h Z W W W W(a) (b) (c)
Figure 3: (a) triangle feyndiagram with one vector resonance, (b) triangle feyndiagram withtwo vector resonances , (c) quartic feyndiagram with one vector resonance.
Assuming there is one non-diagonal Higgs-gauge coupling c ρ R W m ρ v (cid:0) h ρ + R,µ W − µ + h.c. (cid:1) and we can express some generic non-diagonal trilinear and quartic gauge self couplingsin the following way: L Zρ R W = ( − i e cot θ w ) c Zρ R W (cid:18) ∂ µ Z ν ρ + R,µ W − ν + ∂ µ ρ + R,ν W − µ Z ν + ∂ µ W − ν Z µ ρ + R,ν + ∂ µ Z ν ρ − R,ν W + µ + ∂ µ ρ − R,ν W + ν Z µ + ∂ µ W + ν Z ν ρ − R,µ − ( µ ↔ ν ) (cid:19) (49) L AZρ R W = ( e cot θ w ) c Zρ R W (cid:18) Z µ A µ ρ + R,ν W − ν − Z µ A ν ρ + R,µ W − ν − Z ν A µ ρ + R,µ W − ν + 2 Z µ A µ ρ − R,ν W + ν − Z µ A ν ρ − R,ν W + µ − Z ν A µ ρ − R,ν W + µ (cid:19) (50)The amplitude for Higgs decay into Z and photon is adding up the three diagramsillustrated in Fig.[3] and it is necessary to times a factor of two to account for thecrossing symmetry. As we put all the external particles to be on shell, the amplitudecan be organized into a gauge invariant form: M ( h → Zγ ) = 2 · (cid:0) M ( a ) + M ( b ) + M ( c ) (cid:1) = − i e π v (2 c ρ R W c Zρ R W ) c (1) Zγ ( g µν k · k − k µ k ν ) ε ∗ µ ( k ) ε ∗ ν ( k ) (51)15t is convenient to express the form factor c (1) Zγ in terms of one-loop three-point scalar andvector functions, i.e. C ( m h , m Z , , m , m , m ) and C ( m h , m Z , , m , m , m ) definedin [17], with two different masses circulating in the loop. c (1) Zγ = cot θ w · m ρ m W · (cid:20) (cid:18)(cid:18) m h m ρ + m W m ρ + 1 (cid:19) (cid:0) m Z − m ρ − m W (cid:1) − m W (cid:19) · (cid:18) C (cid:0) m h , m Z , , m W , m ρ , m W (cid:1) + C (cid:0) m h , m Z , , m ρ , m W , m ρ (cid:1) (cid:19) + 2 (cid:18) m W m ρ (cid:0) m Z − m W − m ρ (cid:1)(cid:19) · C (cid:0) m h , m Z , , m W , m ρ , m W (cid:1) + 2 (cid:0) m Z − m W − m ρ (cid:1) · C (cid:0) m h , m Z , , m ρ , m W , m ρ (cid:1) (cid:21) (52)where the special combination of vector functions in the large bracket can be recastedinto Passarino-Veltman functions B and C : C (cid:0) m h , m Z , , m W , m ρ , m W (cid:1) + C (cid:0) m h , m Z , , m ρ , m W , m ρ (cid:1) = m Z ( m Z − m h ) (cid:18) B (cid:0) m Z , m ρ , m W (cid:1) − B (cid:0) m h , m ρ , m W (cid:1) (cid:19) + 1( m Z − m h ) + m W ( m Z − m h ) C (cid:0) m h , m Z , , m W , m ρ , m W (cid:1) + m ρ ( m Z − m h ) C (cid:0) m h , m Z , , m ρ , m W , m ρ (cid:1) (53)It should be noticed that in the limit of m ρ = m W , our new form factor c (1) Zγ will reduceexactly to the W gauge bosons mediated SM contribution[18].16 eferences [1] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B , 148 (2003)[hep-ph/0306259]; K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B ,165 (2005) [hep-ph/0412089].[2] A. Falkowski and M. Perez-Victoria, Phys. Rev. D , 035005 (2009)[arXiv:0810.4940 [hep-ph]]; H. Cai, H. -C. Cheng, A. D. Medina and J. Terning,Phys. Rev. D , 115009 (2009) [arXiv:0910.3925 [hep-ph]]; H. Cai, H. -C. Cheng,A. D. Medina and J. Terning, Phys. Rev. D , 015019 (2012) [arXiv:1108.3574[hep-ph]].[3] C. Anastasiou, E. Furlan and J. Santiago, Phys. Rev. D , 075003 (2009)arXiv:0901.2117 [hep-ph]; G. Panico and A. Wulzer, JHEP , 135 (2011)arXiv:1106.2719 [hep-ph]; S. De Curtis, M. Redi and A. Tesi, JHEP , 042(2012) arXiv:1110.1613 [hep-ph] .[4] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716 (2012) 1 arXiv:1207.7214[hep-ex]; S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716 (2012) 30arXiv:1207.7235 [hep-ex].[5] S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2239 (1969); C. G. Callan,Jr., S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969).[6] R. Contino, D. Marzocca, D. Pappadopulo and R. Rattazzi, JHEP , 081(2011) [arXiv:1109.1570 [hep-ph]].[7] D. Marzocca, M. Serone and J. Shu, JHEP 1208 (2012) 013 [arXiv:1205.0770[hep-ph]].[8] B. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, JHEP 1211 (2012) 003[arXiv:1205.4032 [hep-ph]].[9] Z.H. Guo, J.J. Sanz-Cillero and H.Q. Zheng, Phys.Lett. B661 (2008) 342-347[arXiv:0710.2163 [hep-ph]]; 1710] D. Barducci, A. Belyaev, M. S. Brown, S. De Curtis, S. Moretti and G. M. Pruna,arXiv:1302.2371 [hep-ph].[11] G. Cacciapaglia, A. Deandrea, G. D. La Rochelle and J. -B. Flament,arXiv:1210.8120 [hep-ph].[12] L. Lavoura and J. P. Silva, Phys. Rev. D 47 (1993) 2046; H. Cai, JHEP , 104(2013) arXiv:1210.5200 [hep-ph].[13] A. Falkowski, Phys. Rev. D , 055018 (2008) [arXiv:0711.0828 [hep-ph]] ;[14] J. Kearney, A. Pierce and N. Weiner, Phys. Rev. D , 113005 (2012)arXiv:1207.7062 [hep-ph].[15] A. Carmona and F. Goertz, arXiv:1301.5856 [hep-ph].[16] A. Djouadi, V. Driesen, W. Hollik and A. Kraft, Eur. Phys. J. C , 163 (1998)[hep-ph/9701342].[17] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B , 365 (1979); G. Passarinoand M. J. G. Veltman, Nucl. Phys. B , 151 (1979); A. Denner, Fortsch. Phys.41