Radion-Higgs Mixing in 2HDMs
PPrepared for submission to JHEP
Radion-Higgs Mixing in 2HDMs
Marco Merchand a , Marc Sher a and Keith Thrasher a a High Energy Theory Group, College of William and Mary, Williamsburg, VA 23187, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the custodial Randall-Sundrum model with two Higgs doubletslocalized in the TeV brane. The scalar potential is CP- conserving and has a softly broken Z symmetry. In the presence of a curvature-scalar mixing term ξ ab R Φ † a Φ b the radion thatstabilizes the extra dimension now mixes with the two CP-even neutral scalars h and H .A goodness of fit of the LHC data on the properties of the light Higgs is performed on theparameter space of the type-I and type-II models. LHC direct searches for heavy scalarsin different decay channels can help distinguish between the radion and a heavy Higgs.The most important signatures involve the ratio of heavy scalar decays into b quark pairsto those into Z pairs, as well as the decay of the scalar (pseudoscalar) into a Z plus apseudoscalar (scalar). a r X i v : . [ h e p - ph ] S e p ontents The electroweak scale set by the vacuum expectation value (VEV) v ≈
246 GeV of the Higgsfield is very sensitive to physics at high scales. This sensitivity appears in loop correctionsto the Higgs mass and is known as the hierarchy problem. Randall and Sundrum [1]proposed a solution to this puzzle by considering an extra dimensional model with theextra dimension being spatial in nature and compactified into a S /Z orbifold. In thismodel there are two 4D manifolds, called “3-branes”, separated by a distance y c = πr c in the extra dimension where r c is the ”radius of compactification”. The brane at y = y c is called the TeV-brane or IR-brane and the brane at y = 0 is usually called the UV- orPlanck brane. A fine tuning is required between the 5 D cosmological constant and thebrane tensions in order to achieve a static flat solution which corresponds to a vanishingeffective 4 D cosmological constant. The solution to Einstein equations gives the 5 D metric ds = e − A η µν dx µ dx ν − dy , (1.1)– 1 –here A = k | y | is the warp factor and k is the AdS curvature scale. This solution corre-sponds to a slice of
AdS space between the two branes. The result of their seminal workcan explain the hierarchy of scales by warping down the Planck scale to the TeV scale,i.e. M TeV = M P l e − ky c , therefore requiring that ky c ≈ O (1) [5, 8]. Several works with bulk fermions have appeared [9–16].One inconvenience in RS models with gauge and matter fields propagating in the bulkare large contributions to electroweak precision observables (EWPO) [17] that push theKK scale far beyond the reach of accelerators. A possible cure can be implemented byimposing a gauge SU (2) L × SU (2) R × U (1) X symmetry in the bulk that is spontaneouslybroken to provide custodial protection [18] for the S and T parameters and this reducesthe bound on the KK scale to m KK (cid:38) Zb ¯ b vertex from large corrections [19].Scalar fluctuations in the RS metric give rise to a massless scalar field called theradion and in order to fix the size of the extra dimension, the radion needs to have a mass.Goldberger and Wise [20] were the first to consider a model with a scalar field propagatingin the bulk of AdS and solved for its profile functions and KK masses. Later they showedin [21] that by choosing appropriate bulk and boundary potentials for the scalar one cangenerate an effective 4 D potential for the radion and therefore were able to stabilize itwithout requiring fine tuning of the parameters. This became known as the GoldbergerWise (GW) mechanism. However in the GW mechanism they used an ansatz for themetric perturbations that do not satisfy Einstein equations and did not include the radionwavefunction and the backreaction of the metric due to the stabilizing field. In the paperof Csaki et al [22] these effects were included by using the most general ansatz [23] and thesuperpotential method [24] to solve for the backreaction. Then they considered the smallbackreaction approximation to solve for the coupled scalar-metric perturbation system andfound the radion mass to be m r ∼ l TeV where l parametrizes the backreaction and itsvalue is model dependent on the specifics of the scalar VEV profile. Therefore the radioncould have a mass of few hundred of GeV and is the lightest particle in the RS model.Since the radion field emerges as the lightest new state the possibility of being ex-perimentally accessible and its effects on physical phenomena must be investigated. Ingeneral, when a scalar is propagating on the brane one can include, by arguments of gen- We use the value M pl = 10 GeV – 2 –ral covariance, in the four dimensional effective action terms involving the Ricci scalar
L ⊇ M R ( g ) φ − ξ R ( g ) φ . In this way a scalar can couple non-minimally to gravity. Ifthe brane scalar is a Higgs boson, gauge invariance implies M = 0 and from dimensionalanalysis one expects ξ to be an O (1) number with unknown sign. Particular attention hasbeen placed on the curvature-Higgs term R Φ † Φ since after expanding out the radion fieldaround its VEV this term induces kinetic mixing between the radion field and the Higgs,therefore requiring a non-unitary transformation to obtain the canonically normalized de-grees of freedom. After diagonalization the physical fields become mixtures of the originalnon-mixed radion and Higgs boson. The phenomenological consequences of a non-zeromixing ξ (cid:54) = 0 have been studied extensively in the literature [22, 25–36]The radion interacts with matter via the trace of the energy-momentum tensor and theform of these interactions is very similar to those of the SM Higgs boson but are multipliedby v/ Λ where Λ ∼ O (TeV) is a normalization factor. In the case ξ = 0, there is no Higgs-radion mixing and the branching ratios of the radion become very similar to those of theSM in the heavy mass region, being dominated by vector bosons while for the low massregion the gg mode is dominant. Due to its large, anomaly induced, coupling to two gluonsa radion can be produced through gluon fusion.The parameter space coming from the curvature-Higgs mixing scenario consists offour parameters, viz., the bare mass terms m h and m r , the mixing parameter ξ and thenormalization scale Λ. However in some of the above references, the Higgs boson had beendiscovered [37, 38] and their parameter space is reduced to ( m r , ξ, Λ). The ξ − m r parameterspace is very constrained by direct searches for additional scalars at the LHC [35] leavingonly small experimentally and theoretically allowed windows for Λ = 3 TeV and thesewindows open up as one increases Λ. The bounds on the parameter Λ are dependent themass the first KK excitation m KK and the curvature scale k as was shown in [39].Despite the model differences in the analyses that have appeared on Higgs-radionmixing, the overall conclusion is that there is possibility that the measured Higgs bosoncould be in fact a mixture of the radion with the Higgs doublet that is consistent withexperimental data. However the constraints mentioned in the previous paragraph will bepushed further if a radion signal is not seen in the coming future and it would be interestingto look at possible ways to relax these constraints.In addition to the RS model, several Beyond the Standard Model (BSM) scenarioshave appeared in the last several decades as promising candidates for new physics. One ofthe most studied and simplest extensions is the Two-Higgs-Doublet Model (2HDM) wherea second Higgs doublet is added to the electroweak sector. The 2HDM was primarilymotivated by minimal supersymmetry [40] and it has also been studied in the context ofaxion models [41], the baryon asymmetry of the universe [42, 43], the muon g − L ⊇ ξ ab R ( g ind ) Φ † a Φ b where a, b = 1 , We first review the RS model with a custodial [18] gauge symmetry SU (2) L × SU (2) R × U (1) X × P LR in the bulk where P LR is a parity symmetry that makes left and rightgauge groups equal to each other. In our notation Latin letters denote 5 D indices M =( µ,
5) and Greek letters denote 4 D indices µ = 0 , , ,
3. The background metric is thatof equation (1.1) and we use the convention for the flat space Minkowski tensor η µν = diag (+1 , − , − , − D action of the model is given by S = (cid:90) d x √ g (cid:2) − M R ( g ) + L φ + L gauge + L fermion (cid:3) + (cid:90) d x (cid:112) g ind ( y = y c ) [ L H + L Y − V IR ( φ )] − (cid:90) d x (cid:112) g ind ( y = 0) V UV ( φ ) (2.1)where the first term corresponds to the Einstein-Hilbert action where M is the 5 D Planckscale and R the Ricci scalar and L Y and L H are the SM Yukawa and Higgs Lagrangiansrespectively. The stabilization mechanism is contained in L φ together with its brane po-tentials V IR and V UV . We do not discuss this sector and simply assume that stabilizationis performed as in [22]. The gauge sector is given by L gauge = − g MO g NP (cid:20)
12 Tr { L MN L OP } + 12 Tr { R MN R OP } + 14 X MN X OP (cid:21) (2.2)where L MN , R MN and X MN are the gauge bosons associated with SU (2) L , SU (2) R and U (1) X respectively. In the Planck-brane the symmetry is broken SU (2) R × U (1) X → U (1) Y by appropriate BC’s of the gauge fields to generate the SM gauge group. This BC’s aregiven by [46] ∂ L aµ ( x,
0) =0 , a = 1 , , ,R iµ ( x,
0) = 0 i = 1 , g X ∂ R µ ( x,
0) + g R ∂ X µ ( x,
0) = 0 − g R R µ ( x,
0) + g X X µ ( x,
0) = 0 (2.3)where g L , g R and g X are the 5 D gauge couplings associated with the gauge fields L aµ , R aµ and X µ respectively. The SM gauge bosons W ± , Z and the photon are embedded into the– 4 – D gauge bosons. Calculation of the spectrum and profiles was performed in Ref. [46, 47]with different KK basis.Boundary mass terms are generated by the Higgs VEV’s L mass = v + v g L L aµ − g R R aµ ) δ ( y − y c ) , (2.4)where v and v are the vevs of the Higgs doublets. Therefore in the TeV brane the gaugesymmetry is spontaneously broken down by the Higgs VEV’s to the diagonal group, i.e. SU (2) L × SU (2) R → SU (2) V so that SU (2) V generates custodial protection for the T parameter. The extra parity symmetry P LR : SU (2) L ↔ SU (2) R was introduced to protectthe Zb L ¯ b L vertex from non universal corrections [19].In the fermion sector all three generations are embedded in the same representation ofthe gauge group with the following transformation properties [47, 48] Q L ∼ ( , ) , (2.5) u R ∼ ( , ) , (2.6) d R ∼ ( , ) ⊕ ( , ) , (2.7)and this choice guarantees custodial protection for the Zbb coupling and for flavor violatingcouplings Zd iL d jL as well. Using appropriate BC one can ensure that only the SM quarksappear in the low energy theory.The motivation for the custodial symmetry came from requiring corrections to EWPO,parametrized by the Peskin-Takeuchi parameters S and T , be sufficiently small. The cor-rections have contributions from the KK excitations of the fermions and gauge bosons,from the 2HDM sector and from the radion. As discussed in the introduction, an extendedgauge custodial symmetry in the bulk keeps the corrections from the KK excitations undercontrol [18]. In the absence of mixing, a custodially symmetric 2HDM potential has van-ishing contributions to the T parameter [49] and the contributions of the radion are alsosmall (see Csaki et al. [22]). However when one includes mixing, the radion and Higgsscalar couplings are modified and could result in large corrections depending on the valuesof the mixing parameters and masses. The contributions in this model are discussed inSection 4. In this work we consider two Higgs doublets living in the visible brane. The most generalparametrization for the scalar potential [42, 50] is given by V (Φ , Φ ) = ¯ m Φ † Φ + ¯ m Φ † Φ − (cid:16) ¯ m Φ † Φ + H.c. (cid:17) + λ † Φ ) + λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ (cid:20) λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + H.c. (cid:21) , (2.8)– 5 –here m , m , and λ , , , are real by hermiticity and m and λ , , are in generalcomplex. In this expression there are fourteen parameters, however the freedom in thechoice of basis can be used to reduce this number down to eleven degrees of freedom thatare physical.To provide custodial protection for the T parameter we promote the Higgs fields tobi-doublets M i = ( ˜Φ i , Φ i ) (with ˜Φ i = iσ Φ ∗ i ) of the gauge group SU (2) L × SU (2) R thattransform in the representation ( , ¯ ) [51] M i → U L M i U † R , i = 1 , . (2.9)where U L ∈ SU (2) L , U R ∈ SU (2) R . (2.10)Using the three independent invariant quadratic forms Tr[ M † M ], Tr[ M † M ] and Tr[ M † M ] the most general expression that has all possible combinations of traces invariants is givenby V ( M M ) = ¯ m M † M ] + ¯ m M † M ] − ¯ m Tr[ M † M ] + λ M † M ] + λ M † M ] + λ M † M ]Tr[ M † M ] + λ (cid:48) M † M ] + λ (cid:48) M † M ]Tr[ M † M ] + λ (cid:48) M † M ]Tr[ M † M ] (2.11)where all the parameters are real and the correspondence with the potential of equation(2.8) is λ (cid:48) ≡ λ = λ , λ (cid:48) ≡ λ , λ (cid:48) ≡ λ . (2.12)Thus by imposing the gauge SU (2) L × SU (2) R symmetry one immediately reduces thenumber of free parameters in the scalar potential down to nine. Also a custodially protected2HDM potential is automatically CP conserving.The kinetic terms for the Higgs bi-doublets are given by L H ⊇ (cid:88) i =1 , g µνind
12 Tr[( D µ M i ) † D ν M i ] (2.13)where g µνind is the induced metric on the TeV brane and the covariant derivative is D µ M i = ∂ µ M i − ig L L µ M i + ig R M i R µ (2.14)and L µ = L aµ τ aL is the gauge boson associated with SU (2) L . Therefore under the custodialgauge symmetry the gauge bosons transform as L µ → U L L µ U † L − ig L ∂ µ U L U † L , (2.15) R µ → U R R µ U † R + ig R U R ∂ µ U † R . (2.16) For a basis independent treatment see Ref. [49] – 6 –f course one needs to also include the term corresponding to the gauge group U (1) X which violates the custodial symmetry.In conventional 2HDM’s one can avoid the presence of potentially dangerous flavorchanging neutral currents (FCNC) by imposing a discrete Z symmetry Φ → Φ , Φ →− Φ , on the Higgs doublets. The fermion mass in (2.51) is generated either by Φ or Φ since the discrete Z symmetry is extended to the fermion sector. This results in fourdifferent types of Yukawa interactions [52]. In the type-I model all fermions couple to asingle Higgs doublet, usually chosen to be Φ . In the type-II model up-type quarks coupleto Φ and d-type quarks and leptons couple to Φ . In the lepton-specific model all leptonscouple to Φ and all quarks couple to Φ . Finally in the flipped model up-type quarksand leptons couple to Φ and d-type quarks couple to Φ . In general, radion mediatedFCNC can be present and this was analyzed in [53]. For simplicity we don’t consider flavormixing in the bulk mass parameters, i.e., c i,jL,R = c i,iL,R since we want to achieve minimalflavor violation [54] in the Yukawa sector.In terms of bi-doublets this symmetry reads M → M , M → − M , (2.17)and implies λ (cid:48) = λ (cid:48) = 0 with ¯ m (cid:54) = 0 remaining as a soft-violating term. The Higgsdoublets can be expressed as Φ a = (cid:32) φ + a ¯ v a + ρ a + iη a √ (cid:33) , a = 1 , v a are the VEV of the scalars. The VEV’s satisfy the relation ¯ v = ¯ v + ¯ v with ¯ v thelocalized Higgs VEV and should not be confused with the SM value v = ¯ ve − ky c = 246 GeVsince we still need to canonically normalize the Higgs doublets .The fields appearing in the expression of the Higgs doublets (2.18) are not the physicalscalars. To obtain the physical eigenstates one has to diagonalize the mass matrices thatare constructed using equation (2.11) with the appropriate imposed symmetries. For acustodial and Z symmetric scalar potential the mass matrix for the CP-odd state and forthe charged Higgs fields are equal (cid:32) ¯ m + ¯ v λ +¯ v λ − ¯ m + ¯ v ¯ v λ (cid:48) − ¯ m + ¯ v ¯ v λ (cid:48) ¯ m + ¯ v λ +¯ v λ (cid:33) = (cid:32) ¯ m
212 ¯ v ¯ v − λ (cid:48) ¯ v − ¯ m + ¯ v ¯ v λ (cid:48) − ¯ m + ¯ v ¯ v λ (cid:48) ¯ m
212 ¯ v ¯ v − λ (cid:48) ¯ v (cid:33) (2.19)where in the last equality ¯ m and ¯ m were eliminated using the minimization conditionsof the potential. The matrix above has a zero eigenvalue corresponding to the Goldstonebosons G and G ± and the nonzero mass eigenvalue is given by¯ m A = ¯ m H ± = ¯ m ¯ v ¯ v ¯ v − λ (cid:48) ¯ v . (2.20)The fact that the CP-odd field mass is degenerate with the charged Higgs bosons is adirect consequence of imposing a custodial symmetry in the scalar potential however this We put a bar on mass parameters that are not yet redshifted down to the EW scale. – 7 –ymmetry is not respected by the hypercharge gauge and Yukawa interactions, so we canonly expect the masses to be approximately degenerate. The diagonalization of the CPodd fields (as well as the charged scalars) is carried out by the orthogonal transformation (cid:32) η η (cid:33) = (cid:32) c β − s β s β c β (cid:33) (cid:32) G A (cid:33) (2.21)where c β = cos β , s β = sin β and tan β = v /v . G is the neutral Goldstone boson and A is the physical pseudoscalar.The physical CP even scalars are obtained by the rotation (cid:32) ρ ρ (cid:33) = (cid:32) c α − s α s α c α (cid:33) (cid:32) Hh (cid:33) (2.22)where h ( H ) corresponds to the lighter (heavier) scalar.Notice that there were 7 real parameters in the Higgs potential to start with, namely { ¯ m , ¯ m , ¯ m , λ (cid:48) , λ (cid:48) , λ (cid:48) , λ (cid:48) } . Using the two minimization conditions we can trade ¯ m and ¯ m for v and v and then use the relations v = v + v and tan β = v /v to trade v and v for v and β . Finally we can trade the soft breaking parameter and three lambdasfor the three scalar masses and α ending up with the set { β, α, m h , m H , m A , λ } (noticethat λ = λ (cid:48) ) where we fixed v = 246 GeV therefore we only have to specify 6 parameters. ξ uh ξ dh ξ lh ξ uH ξ dH ξ lH ξ uA ξ dA ξ lA Type-I c α /s β c α /s β c α /s β s α /s β s α /s β s α /s β cot β − cot β -cot β Type-II c α /s β − s α /c β − s α /c β s α /s β c α /c β c α /c β cot β tan β tan β Table 1 : Scalar couplings to pairs of fermions.The couplings of the scalars with the fermion fields can be written as [52] L ffφ = (cid:88) f = u,d,l m f v (cid:16) ξ fh ¯ f f h + ξ fH ¯ f f H − iξ fA ¯ f γ f A (cid:17) , − (cid:40) √ V ud v ¯ u ( m u ξ uA P L + m d ξ dA P R ) dH + + √ m l ξ lA v ¯ ν L l R H + + h.c. (cid:41) , (2.23)where the mixing factors are summarized in Table 1. Here the gauge bosons and fermionsare the zero modes of the 5D bulk fields. Non-zero KK modes are presumed to be sufficientlyheavy that they will not have a phenomenological impact.The couplings of the scalars to a pair of gauge bosons are given by L W W,ZZφ = ( h sin ( β − α ) + H cos ( β − α )) (cid:18) m W v W + µ W µ − + m Z v Z µ Z µ (cid:19) , (2.24) L gg,γγφ = (cid:88) φ = h,H,A − φ v (cid:110) α s π b φQCD G aµν G aµν + α EM π b φEM F µν F µν (cid:111) , (2.25)– 8 –here b φQCD = ξ tφ × (cid:40) F f , φ = h, H,f ( τ t ) τ t , φ = A, (2.26) b hEM = (cid:18) ξ th F f − sin( β − α ) F W + g h F H (cid:19) , (2.27) b HEM = (cid:18) ξ tH F f − cos( β − α ) F W + g H F H (cid:19) , (2.28) b AEM = 83 ξ tA f ( τ t ) τ t , (2.29)The form factor for the charged Higgs in the loop is [55, 56] F H = − τ H (1 − τ H f ( τ H ))and has limiting behaviors F H → / τ > F H → τ <
1. The couplingsmultiplying the form factor are given by g φ = − m W gm H ± g φH + H − with g φH + H − the tree levelcoupling that arises from the 2HDM potential. For the background metric solution in the RS model, given by equation (1.1), any valueof the radius dimension y c is equally acceptable. Therefore a mechanism is needed to fixthe value y c ∼ /k so that the EW hierarchy is explained and this must be accomplishedwithout severe fine tuning of parameters. Here we simply assume that a GW bulk scalaris responsible for the stabilization and that the bulk and brane potentials are chosen byapplying the method of the superpotential of Ref.[24]. This method has the advantage ofreducing the coupled non-linear second order Einstein equations to simple ordinary differ-ential equations for a simple choice of superpotential. The backreaction of the backgroundmetric due to the scalar can be solved directly using this method.After the extra dimension is stabilized the radion field arises from the scalar fluctua-tions of the metric given by the general ansatz [22, 23] ds = e − A − F ( x,y ) η µν dx µ dx ν − (1 + G ( x, y )) dy , (2.30)and since the background VEV for the bulk scalar also depends on the extra dimensionone also has to include the fluctuations in the GW scalar namely: φ ( x, y ) = φ ( y ) + ϕ ( x, y ) where φ is the background VEV and ϕ denotes the fluctuation. By evaluatingthe linearized Einstein equations one is able to derive G = 2 F . To solve the system onelinearizes the Einstein and scalar field equations to obtain coupled relations for ϕ and F . In particular, by integrating the ( µ,
5) component of the linearized Einstein equations δR µ = κ δT µ with κ = 1 / M , one obtains φ (cid:48) ϕ = 3 κ ( F (cid:48) − A (cid:48) F ) (2.31)where the prime indicates d/dy and this equation implies that the fluctuations ϕ and F will have the same KK eigenstates but with different profiles. Using the Einstein equationstogether with (2.31) a single differential equation in the bulk for F can be obtained [22]: F (cid:48)(cid:48) − A (cid:48) F (cid:48) − A (cid:48)(cid:48) F − φ (cid:48)(cid:48) φ (cid:48) F (cid:48) + 4 A (cid:48) φ (cid:48)(cid:48) φ (cid:48) F = e A (cid:50) F (2.32)– 9 –upplemented by the boundary conditions( F (cid:48) − A (cid:48) F ) | y =0 ,y c = 0 , (2.33)where the boundary conditions are simplified in the limit of stiff boundary potentials of thebulk stabilizer ∂ V i /∂φ (cid:29) ϕ | y = y i = 0. In the system there are two integrationconstants and one mass eigenvalue (cid:50) F n ( x, y ) = − m n F n ( x, y ). One integration constantcorresponds to an overall normalization while the other constant and the mass eigenvalueare determined by the boundary conditions. In Ref [22] this differential equation wassolved in a perturbative approach in the limit of small backreaction of the metric dueto the stabilizing scalar, and it was found to zero-order in the backreaction that the KKzero-mode can be approximated by F ( x, y ) ≈ e k | y | R ( x ) + O ( l ) , (2.34)where R ( x ) is the radion field. Using the boundary conditions the radion mass is [22] m r ≈ . l ke − ky c (2.35)where l ≡ φ P / M is the backreaction and φ P is the VEV of the bulk stabilizer field on thePlanck brane. It should be noted that generically, the radion mass is always proportionalto the backreaction independently of the stabilization mechanism. From the expressionabove, the radion mass is expected to be of O (TeV) scale. The canonical normalization ofthe radion comes from integrating out the extra dimension in the Einstein-Hilbert action M (cid:90) dy √ g R (¯ g ) ⊇ M k e ky c ( ∂ µ R ( x )) (2.36)therefore a canonically normalized radion is obtained by writing R ( x ) = r ( x ) e − ky c √ M P l . (2.37)It is explicitly proved in [22] that the normalization is dominated by the gravitationalcontribution coming from the Einstein-Hilbert action against that coming from the kineticterm of the bulk stabilizer.We now proceed to present the radion interactions with the SM fields. The inducedmetric on the TeV brane is given by¯ g indµν ( x ) = e − A ( y c ) e − e kyc R ( x ) η µν ≡ e − ky c Ω( r ) η µν , (2.38)where we use ¯ g MN to denote the metric with scalar perturbations included. After rescalingof the doublets Φ a → e ky c Φ a , the radion couplings to the Higgs sector are obtained from(including the possibility of adding extra scalars in the sum) S H = (cid:90) d x (cid:88) a =1 , η µν
12 Tr[( D µ M a ) † D ν M a ]Ω( r ) − V ( M , M )Ω( r ) , (2.39)– 10 –nd all mass terms are redshifted accordingly. Expanding to linear order in the radion fieldΩ( r ) ≈ − r γv , with γ ≡ v/ Λ and Λ ≡ √ M P l e − ky c , a straightforward calculation yieldsthe coupling of the radion with the trace of the energy-momentum tensor γv r T µµ ⊇ − (cid:88) γv r (cid:2) ( ∂ µ φ ) − m φ φ (cid:3) , (2.40)with the sum performed over all physical scalars.The couplings to the EW gauge sector are obtained from the kinetic terms of the Higgsdoublets expanding to linear order in the perturbations S H ⊇ − (cid:90) d x γv r ( x ) η µν (cid:110) m W W (0)+ µ ( x ) W (0) − ν ( x ) + m Z Z (0) µ ( x ) Z (0) ν ( x ) + ... (cid:111) (2.41)where the dots represent higher KK excitations. In addition to the boundary terms thereare tree level couplings of the radion coming from the kinetic term of the bulk gauge bosons[10] S gauge ⊇ − (cid:90) d x γv r ( x ) (cid:26) ky c η µν η αβ V (0) µα ( x ) V (0) νβ ( x ) + m V k e ky c ky c η µν V (0) µ ( x ) V (0) ν ( x ) (cid:27) . (2.42)where V MN = ∂ M V N − ∂ N V M is the usual field strength and V = {√ W ± , Z, A } and m V = { m W , m Z , } . The coupling to the field strengths above becomes significant formomentum transfer much larger than the EW scale and the second term constitutes acorrection of about 20% to the dominant TeV-boundary coupling. In the case of thephoton only the first term is present. A similar expression for gluons should be included.Overall we can write L W W,ZZr = γv r (cid:26) m W (cid:18) − m W ky c Λ (cid:19) W + µ W µ − + m Z (cid:18) − m Z ky c Λ (cid:19) Z µ Z µ (cid:27) . (2.43)For massless gauge bosons we have to include the contributions coming from the lo-calized trace anomaly and from loop triangle diagrams in which the W gauge boson andfermions in the case of the photon and only fermions in case of the gluons that inducecouplings to the radion.All these contributions can be written as [10, 25, 31, 35] L gg,γγr = − γ v r (cid:26)(cid:18) ky c + α s b rQCD π (cid:19) G µν G µν + (cid:18) ky c + α EM b rEM π (cid:19) F µν F µν (cid:27) , (2.44)with α s ( α EM ) being the strong (electroweak) coupling constant and b rQCD = 7 + F f , (2.45) b rEM = −
113 + 83 F f − F W , (2.46) F f = τ f (1 + (1 − τ f ) f ( τ f )) , (2.47) The Lagrangian takes into account only the leading order mass effects for the radion coupling to exactlytwo gauge bosons. – 11 – W = 2 + 3 τ W + 3 τ W (2 − τ W ) f ( τ W ) , (2.48) f ( τ ) = Arcsin ( 1 √ τ ) τ ≥ , (2.49) f ( τ ) = − (cid:18) log 1 + √ − τ − √ − τ − iπ (cid:19) , τ < , (2.50)and τ i = ( m i m r ) , m i is the mass of the particle going around the loop. An importantproperty of the kinematic functions is their saturation F f → / F W → τ f ( τ ) → τ > F f,W → τ < c = m/k which specifies their location in the bulk. In addition, the boundary conditions of theirprofiles at the location of the branes force either the left- or the right-handed zero modesto be zero [6]. Therefore for each SM fermion we need to introduce two different bulkfermions, one with bulk mass parameter c L and for which the right-handed zero modevanishes and the other with a bulk mass parameter c R and for which the left-handed zeromode vanishes.The couplings of the radion to SM fermions can be simplfyfied as [35] S ⊇ (cid:90) d x (cid:88) f = u,d,e γv r ( x ) m f ¯ f f × (cid:40) P lanck ( c L − c R ) TeV . (2.51)with the lower option if the zero-mode profile is peaked towards the TeV brane c L < / , c R > − / The most general term that will give rise to kinetic mixing between the Higgs doublets andthe radion field is given by L ξ = √ ¯ g ind ξ ab R (¯ g ind ) 12 Tr[ M † a M b ] (3.1)where the indices a, b = 1 , ξ = ξ and thus thepseudoscalar does not mix with the radion. Evaluation of the Ricci scalar is straightforwardand yields the following expression [22] L ξ = − ξ ab Ω (cid:2) (cid:50) ln Ω + ( ∇ ln Ω) (cid:3)
12 Tr[ M † a M b ] (3.2)– 12 –he warp factor disappears after we make the rescaling of the Higgs doublets. Using theexpression for the Higgs mass eigenstates (2.22) and expanding to linear order in the fieldswe can write L ξ ⊇ − (cid:20) − γv (cid:50) r + γ v r (cid:50) r (cid:21) (cid:20) v K r + v K h h + v K H H (cid:21) , (3.3)where γ ≡ v/ Λ and we define the mixing parameters by K r = ξ c β + ξ s β + 2 ξ s β c β , (3.4) K h = 2( ξ s β c α − ξ c β s α ) + 2 ξ cos( α + β ) , (3.5) K H = 2( ξ c β c α + ξ s β s α ) + 2 ξ sin( α + β ) . (3.6)Adding the kinetic and mass terms of each field, the mixing Lagrangian can be expressedas L = −
12 (1 + 6 γ K r ) r (cid:50) r − m r r + (cid:88) φ = h,H (cid:26) γK φ φ (cid:50) r − φ ( (cid:50) + m φ ) φ (cid:27) (3.7)The kinetic terms can be diagonalized by performing the transformation r → r (cid:48) Z , φ → φ (cid:48) + 3 γK φ Z r (cid:48) (3.8)with φ = h, H and Z = 1 + 6 γ K r − γ ( K h + K H ) , (3.9)is the determinant of the kinetic mixing matrix and therefore should always satisfy Z > α , β and γ . Thistransformation induces mixing in the mass terms. The mass matrix obtained can be writtenas M = ω rr ω rh ω rH ω rh m h ω rH m H , (3.10)where ω rr = m r Z + 9 γ Z (cid:0) K h m h + K H m H (cid:1) , (3.11) ω rφ = 3 γZ K φ m φ . (3.12)The physical eigenstates are obtained by performing a three dimensional rotation r (cid:48) h (cid:48) H (cid:48) = U r D h D H D . (3.13)The relation between the gauge eigenstates and the mass eigenstates can be written as r = U Z r D + U Z h D + U Z H D , (3.14)– 13 – = (cid:18) U + 3 γ K h Z U (cid:19) r D + (cid:18) U + 3 γ K h Z U (cid:19) h D + (cid:18) U + 3 γ K h Z U (cid:19) H D , (3.15) H = (cid:18) U + 3 γ K H Z U (cid:19) r D + (cid:18) U + 3 γ K H Z U (cid:19) h D + (cid:18) U + 3 γ K H Z U (cid:19) H D . (3.16)For later convenience we name the coefficients of this transformation as U rr = U Z , U rh = U Z , U rH = U Z , (3.17) U hr = U + 3 γK h U Z , U hh = U + 3 γK h U Z , U hH = U + 3 γK h U Z , (3.18) U Hr = U + 3 γK H U Z , U Hh = U + 3 γK H U Z , U HH = U + 3 γK H U Z , (3.19)which will be used in the next section for the predictions of the electroweak precisionobservables.The Higgs scalars-radion system is determined by the three mixing parameters ofequation (3.1), the two mixing angles of the Higgs sector, the scale γ and the three scalarmasses, giving a total of nine parameters. However one of the physical masses will be setto the Higgs mass value and only the set ( ξ , ξ , ξ , α, β, γ, λ r , λ H ) needs to be specified.Another important parameter in the study of RS models with bulk gauge bosons is theKK scale defined to be the mass of the first excited state of the gauge bosons. Recall thatthis parameter is independent of the gauge symmetry and gauge couplings and is universalfor all gauge bosons that satisfy the same BCs. In particular, for gauge bosons satisfyingNeumann BCs at both branes it is given by [27] m KK = 2 . k √ M P l Λ , (3.20)so any bound on the KK scale will directly affect the allowed values of the curvature scale k and Λ.In Higgs-radion mixing scenarios there is a particular point in the parameter spacecalled the “conformal point” [25, 35, 36], usually around ξ = 1 / gg decay mode dominates even in the large radion mass limit. In this work we do not attemptto calculate a conformal point due to the large number of parameters.In what follows we will sometimes reduce the parameter space by assuming that thediagonal elements of the curvature-scalar mixing matrix are equal to each other, ξ = ξ ≡ ξ and for simplicity we will refer to the off diagonal as ξ ≡ ξ . Relaxing thisconstraint will not radically alter the numerical results in the following sections. However,we will primarily focus on K r , K h , K H , which is independent of this assumption.From now on we will drop the subindex D for the diagonal eigenstates and simplywrite them as r , h and H . Whenever we need to distinguish between the non-diagonal andphysical states a clarification will be made.– 14 – Model Predictions
The motivation for the custodial symmetry came from requiring corrections to EWPO,parametrized by the Peskin-Takeuchi [57] parameters S and T , be sufficiently small. Thecorrections have contributions from the KK excitations of the fermions and gauge bosons,from the 2HDM sector and from the radion. As discussed in the introduction, an extendedgauge custodial symmetry in the bulk keeps the corrections from the KK excitations undercontrol [18]. In the absence of mixing, a custodially symmetric 2HDM potential has van-ishing contributions to the T parameter [49] and the contributions of the radion are alsosmall (see Csaki et al. [22]).However when one includes mixing, the radion and Higgs scalar couplings are modifiedand could result in large corrections depending on the values of the mixing parameters andmasses. This was first discussed by Csaki, et al.(CGK) [22] and a paper dedicated entirelyto electroweak precision constraints was written by Gunion et al. (GTW)[58], we followthe notation of the latter. Both showed that there are three types of contributions to the S and T parameters: (1) with each scalar eigenstate going through the loop of the vacuumpolarization graph of the vector bosons, (2) anomalous terms coming from the conformalcouplings of the radion when the theory is regulated and (3) higher dimensional operatorswhich arise after integrating out the heavy degrees of freedom, e.g. spin-2 graviton states.The first contribution comes from vacuum polarization graph loops. Let us first con-sider the single Higgs case in which there is one ξ term. As shown above, this leads tokinetic mixing between the radion and Higgs. Diagonalizing the kinetic mixing terms andthen further diagonalizing the mass matrix gives [22, 58](with h and φ being the masseigenstates and h and φ the geometric eigenstates): h = c φ + d h φ = a φ + b h (4.1)where a = − cos θZ b = sin θZ c = (cid:18) sin θ + 6 ξvZ Λ φ cos θ (cid:19) d = (cid:18) cos θ − ξvZ Λ φ sin θ (cid:19) (4.2)The terms with an explicit ξ are obtained when the kinetic terms are diagonalized and theothers arise when rotating to the mass basis from the geometric basis. Here Z = 1 + 6 ξ (1 − ξ ) v / Λ φ tan 2 θ = 12 γξZ m h m φ − m h ( Z − ξ γ ) (4.3)Here, m h and m φ are the Higgs and radion masses when the mixing vanishes. Note that γ = v/ Λ φ is very small.As shown in [22, 58], the contributions of the radion and Higgs to the electroweak S and T parameters are S i = − g i π (cid:18) B ( M Z , m i , M Z ) − B ( M Z , m i , M Z M Z − B (0 , m i , M Z ) − B (0 , m i , M Z ) M Z (cid:19) (4.4)– 15 – i = − g i π sin θ W (cid:18) B (0 , m i , M W ) − B (0 , m i , M W M W − B (0 , m i , M Z )cos θ W − B (0 , m i , M Z ) M W (cid:19) (4.5)where the B -functions are the Passarino-Veltman functions [59]. There are contributionsfrom the Higgs and the radion and the couplings are given by g h = d + bγ and g φ = c + aγ .In the 2HDM-radion model, the expressions in the above paragraph for S i and T i arestill present, but now one includes contributions from the radion and both h and H (note,additional contributions from charged Higgs masses and neutral scalar mass splittings willbe discussed later). However, the three g i are now different. The diagonalization of thekinetic terms was given in Eqs. (3.14)-(3.16). The mass matrix in Eq. (3.13) is a 3 × g i can be read off from Eqs.(3.14)-(3.16). In addition, thereare contributions from loops with two scalars, diagram (b) in Fig. 1, including the chargedscalar and the pseudoscalar with a neutral scalar. The charged scalar can also contributeto Π Zγ and Π γγ . In this model, the contributions involving the physical fields r , h and H for diagrams of the type (b) and (c) in Figure 1 are listed in Appendix C. Figure 1 : Feynman diagrams relevant for the contributions of the scalar sector to theoblique parameters.There are two other contributions in the Higgs-radion mixing case. There are anoma-lous terms where the linear (cid:15) -terms in dimensional regularization in the radion-matterinteractions are elevated to finiteness by 1 /(cid:15) poles in the radion loops. These are calcu-lated in both CGK and GTW and turn out to be negligibly small, as noted explicitly inCGK. In GTW, an additional term is shown to be present, but this term (as they state)makes a negligible contribution for a Higgs mass of 120 GeV. There are extra parameters– 16 –n this model, but barring very unnatural values of these parameters, we expect the contri-bution to be completely negligible. The other contribution comes from non-renormalizableoperators which come from integrating out heavy states above the scale Λ φ . These canaffect the T parameter since they can break isospin symmetry. However, the value of theseparameters at the scale Λ φ is completely arbitrary. Although CGK discussed these terms,they didn’t include them in their calculation. GTW assume they vanish at Λ φ , and con-sider the running of these terms down to M Z . In this section we are concerned with howthis model differs from the single Higgs case, and since the terms are arbitrary (and likelyto be different in the single and two Higgs cases), we will not include these terms here.As a first case example, Case 1, we consider the case of the exact alignment limit wherecos( β − α ) = 0 and ξ = 0. In this simplified scenario we only have one mixing parametersince the mixing coefficients of equations (3.4)-(3.6) reduce to K r = ξ , K h = 2 ξ and K H = 0. Then the field H doesn’t mix with the radion and has vanishing tree-levelcouplings to pairs of gauge bosons therefore diagrams of type (c) with the H running inthe loop are absent however diagrams of type (a) and type (b) are still present. This doesn’tquite reduce to the single Higgs radion mixing case because of the presence of H − A loopin diagram (b).The S and T parameters in the Case 1 scenario are shown in fig. 2. As can benoticed the constraints from the T on the mixing parameter become more stringent withincreasing radion mass. The combined constraints are − . < K r < . m r = 200 GeV, − . < K r < . m r = 400 GeV and − . < K r < . m r = 600 GeV. To comparewith the single Higgs case, we include in the S-parameter plot the calculation without theadditional H − A loop (due to the custodial symmetry, the T parameter is not changed).We see immediately that the additional Higgs bosons increase the S parameter (this alsooccurs in the conventional 2HDM, of course), but the model is still acceptable. Now,however, we relax the ξ = 0 assumption (which gave K H = 0) and see how the EWPOcontributions change.We first will continue to set cos( β − α ) to zero, since it must be small, as shown bythe fit of the model to the LHC Higgs data in the next subsection. As ξ is now nonzero,we will have K r , K h and K H all nonzero. The results will be plotted as curves in the K H , K h plane, and we will see that the results are very insensitive to K r . The region ofparameter-space in this plane allowed by the positivity of the determinant of the kineticmatrix, Z >
0, is a circle in this plane (since the K r term is multiplied by γ and thus isvery small) as shown in Eq. (3.9).For radion masses of 200, 400 and 700GeV, the allowed region is shown in Fig. 3. Inthese figures, the x-axis is K h and the y-axis is K H . We see that the bounds for S arefairly mild, but are much stronger for the T parameter. Note that the K H = 0 value givesresults identical to the earlier result for K h (which in that limit is twice K r ). We see thatthe largest allowed values occur when either K h or K H is small. One can see that theparameter space does get squeezed for higher radion masses. It turns out the results arealmost unchanged if one chooses a nonzero value of K r .We thus see that, in the alignment limit of cos( β − α ), the parameter space in the2HDM is restricted in a manner similar to the single Higgs case (with the exception of the– 17 – igure 2 : S and T parameter curves as a function of the mixing parameter K r for differentvalues of the radion mass. The solid horizontal black line represents the 2 σ upper boundfrom current value. The rest of the parameters chosen were cos( β − α ) = 0, ξ = 0,tan β = 1, m h = 125 GeV, m H = 1000 GeV, Λ = 5 TeV and m A = 500 GeV. The dottedline in the S-plot corresponds to the single Higgs limit, without the H − A loops included,for a radion mass of 200 GeV (it is very insensitive to the radion mass).increase in S due to heavy Higgs loops) but is a two dimensional restriction, rather thana restriction on the single mixing parameter. If either K h or K H is near zero, the resultsare similar, but differ if they are both nonzero.– 18 – igure 3 : Constraints on K H (y-axis) and K h (x-axis) from the S and T parameters.The circle is the theoretically allowed region, the blue dots are allowed by the S and T parameter bounds, and the red dots are disallowed. The other parameters chosen are listedat the top of the figures with m A = 500 GeV and Λ = 5TeV.– 19 –hat if one moves away from the alignment limit? Using cos ( β − α ) = 0.2, we findthe results in Fig. 4. The S parameter constraints are similar, but the T parameterconstraints become much more restrictive. These features do not change much as theradion mass changes from 200 GeV to 700 GeV. As we will see in the next section, though,the Type II model does not allow cos( β − α ) much larger than 0 .
1, and thus this restrictionwill not be relevant, although it will be for the type I model.
Figure 4 : Constraints on K H (y-axis) and K h (x-axis) from the S and T parameters, inthe case in which one moves away from the alignment limit. The circle is the theoreticallyallowed region, the blue dots are allowed by the S and T parameter bounds, and the reddots are disallowed. The other parameters chosen are listed at the top of the figures with m A = 500 GeV, Λ = 5TeV.One should consider these results with caution. We have not included the non-renormalizable contributions since they are arbitrary at the cutoff scale, and those couldaffect the T parameter, which gives the strongest constraints. As shown in GTW, givencertain assumptions, these can be substantial for large mixing and could broaden the pa-rameter space. In addition, it is quite possible that the custodial symmetry will be brokenon the Higgs brane, in which case the charged Higgs and pseudoscalar masses will not bedegenerate. Depending on which is heavier, the T parameter can be substantially increasedor decreased, which would drastically affect the bounds (this arbitrariness, of course, is notrelevant in the single Higgs case). In the 2HDM the interactions of all the scalars to the SM fields are completely determinedby the two mixing angles of the scalar sector β and α . In addition, the alignment limit is– 20 –efined to be the limit in which one of the CP-even scalars has exactly the same interactionsas the SM Higgs and corresponds to cos( β − α ) = 0.In this section we perform an analysis on the effects Higgs-radion mixing has on the2HDM parameter space, cos( β − α ) and tan β . We use a chi-square test to fit the modelto the data presented in Appendix B and find the region in the 2HDM parameter spaceallowed by current LHC data on the SM-like Higgs boson, h . By definition the chi-squarefunction to be minimized is written as χ = (cid:88) i ( R pi − R mi ) ( σ i ) , (4.6)where R Pi is the signal strength predicted by the model, R mi is the measured signal strengthand σ i is the corresponding standard deviation of the measured signal strength. Asymmet-ric uncertainties are averaged in quadrature σ = (cid:113) σ + σ − . The expected signal strengthsare defined as the production cross section times branching ratio of a particular decaychannel f f normalized to the standard model prediction, i.e., R pf ≡ σ ( pp → h ) BR ( h → f f ) σ ( pp → h SM ) BR ( h SM → f f ) . (4.7)Directly obtaining analytical expressions for the mass eigenstates is challenging thereforewe resort to numerical techniques. The analysis was carried out using two benchmarksfor the radion vev, Λ = 3 , α, β ), the curvature scalar couplings ( ξ , ξ ) and the scalar mass parameters before radionmixing ( m h , m H , m r ) amounting to seven degrees of freedom. By imposing the field h hasa mass of 125 . ± . , γK φ /Z ∼ / ξ ∼ O (1) and therefore the unitarymatrix that diagonalizes (3.10) is nearly diagonal which implies that the couplings of theSM-like Higgs to a pair of gauge bosons and fermions receive very small corrections andare nearly given by the corresponding couplings in the 2HDM, i.e., g hV V = U hh sin( β − α ) + U Hh cos( β − α ) + U rh γ (1 − m V ky c Λ ) ≈ sin( β − α ) , (4.8) g hff = U hh ξ fh + U Hh ξ fh + U rh γ ( c L − c R ) ≈ ξ fh , (4.9)where U ij are the elements of the non-unitary transformation. The general shape of theregions is understood by looking at the behavior of the couplings. In the type-I model– 21 – igure 5 : The top plots show the allowed regions for the type-I model and the bottomplots show the allowed regions in the type-II model. The blue (red, black) points shownare used for the Λ = 3(5 , ξ , ξ were allowed to range between [ − , ξ th = cos α/ sin β and in the large tan β limit the production cross section is suppressed,allowing the parameter space to grow. For type-II model the coupling to a pair of b quarksis ξ bh = − sin α/ cos β and therefore the production cross section is enhanced by the b quarkloop squeezing the parameter space. Figure 6 : Theoretically allowed ξ - ξ parameter space for different values of tan β . Theblue (red) region is for Λ = 3(5)TeV.The allowed region of the curvature-scalar parameter space is constrained by the re-quirement that the determinant of the kinetic mixing matrix, Eq. (3.9), be positive. Thiscondition was discussed in the last section. We can examine the constraint in the ξ − ξ – 22 –lane. This depends only on tan β and γ and is given, for Λ = 3 , ξ i can require some fine-tuning, and we have found that thedensity of points in a scatterplot drops substantially once ξ i is greater than 4 and less than-4. As a result, restricting the mixing parameters to the range between − ≤ ξ i ≤ K h , K H parameters and the ξ , ξ parameters depends on α and β , the S, T constraints in the last subsection will notsubstantially reduce the allowed region (especially in view of the cautionary remarks at theend of the last subsection).
Figure 7 : The parameter space of ξ and ξ allowed by the chi-square goodness of fit. Theblue and red points correspond to Λ = 3 TeV and Λ = 5 TeV respectively. Let us now consider some predictions of this model accessible to the LHC and how one maydistinguish this model from some other multi-Higgs model. One feature of a multi-Higgsmodel is that the sum of the CP-even scalar couplings to Z bosons in quadrature shouldtotal to the square of the SM Higgs coupling to the Z bosons, namely g − h SM ZZ n (cid:88) i g φ i ZZ = 1 . (4.10)Due to the bulk couplings of the radion to the bulk gauge bosons we find that the sumof the neutral scalar couplings in quadrature normalized to the h SM ZZ coupling gives1 + γ (1 − m Z ky c / Λ ) being bounded from below by 1 and setting it apart from othermulti-Higgs models. However, this deviation from unity may be quite small. For Λ φ = 3TeV one finds Eq. 4.10 gives 1.0054 and the deviation from unity vanishes in the limitΛ φ → ∞ . It is unlikely that the LHC will be able to measure such a small deviation, butsuch a measurement may be possible at the future ILC.– 23 –nother strategy to distinguish the heavy scalar state H from a radion is to measurethe ratio of the widths of the heavy scalars to b ¯ b and ZZ pairs, R Φ bb/ZZ ≡ Γ(Φ → ¯ bb )Γ(Φ → ZZ ) , for Φ = r, H. (4.11)The mass eigenstates, H and r are primarily aligned with the unmixed states. This meansthat couplings of H to the Z boson and b quark should be dominated by the correspondingexpressions in a 2HDM . Then for H , R Hbb/ZZ should mostly scale like (cid:16) sin α sin β β − α ) (cid:17) for the type-I model and (cid:16) cos α cos β β − α ) (cid:17) for the type-II model. In either case this ratiobecomes quite large in the neighborhood of cos( β − α ) = 0. For the radion, in the limitthat its fully aligned with the unmixed radion, R rbb/ZZ ∝ ( c L − c R ) (cid:18) − m Zkyc Λ2 (cid:19) ≈ ( c L − c R ) . Thisis typically less than one and thus measurement of this ratio might distinguish r from H .As an example, consider the benchmark point with tan β = 1, cos( β − α ) = 0 . ξ = 2 and ξ = −
3. The values of the masses beforemixing are fixed to m r = 540 GeV, m h = 125 GeV and m H = 600 GeV which yield the masseigenvalues m r ≈ m H ≈
600 GeV, m h = 125 GeV and R rbb/ZZ ≈ . R Hbb/ZZ ≈ The radion interactions with the scalar sector come from the following sources:1 The quartic interactions in the 2HDM potential V (Φ , Φ ) ⊇ λ † Φ ) + λ † Φ ) + λ Φ † Φ Φ † Φ + λ † Φ + Φ † Φ ) . (4.12)2 The coupling of the radion with the trace of the energy momentum tensor L ⊇ − r Λ (( ∂ µ h ) − m h h + ... ) . (4.13)3 The curvature-scalar mixing term L = − ξ ab R Φ † a Φ b , where we expand the Ricci scalarup to second order in γ : R ⊇ − γv (cid:50) r + 2 γ v r (cid:50) r + γ v ( ∂ µ r ) + O ( γ ) . (4.14)4 There is a model dependent contribution coming from the potential of the GW scalarfield that one can consider however we will assume this interaction to be small as itis proven in [27] that addition of this extra term doesn’t affect the phenomenology.5 Non-zero mixing will also induce tree-level interactions of the radion with a gaugefield and a scalar, namely rW ± H ∓ and rZA coming from a direct expansion of thekinetic term in equation (2.13). – 24 –n this model the amount of kinetic mixing between the Higgs field and the radion isparametrized by the parameter K h of equation (3.5). Similarly the amount of kinetic mix-ing between the heavy Higgs state and the radion is encoded in the parameter K H given inequation (3.6). We use the most recent LHC direct searches for a heavy scalar decaying intoa pair of SM Higgs bosons [60, 61], into W W bosons [62] and into a pair of ZZ bosons [63] tofind bounds on the amount of mixing. The most relevant decay channels, when kinemat-ically accesible, are φ i → hh, φ j φ j , hφ j , bb, tt, W W, ZZ, gg, AA, H + H − , ZA, W ± H ∓ with φ i = r, H . The trilinear interactions coming from the 2HDM potential have a dependenceon the pseudoscalar mass m A and on the quartic coupling of the potential λ .– 25 – igure 8 : Scatter plots of the amount of mixing between the Higgs and the radion, K h defined in equation (3.5), as function of the radion mass for the type-I 2HDM. The blackregion is theoretically allowed and the points colored yellow, green and red are forbidden byheavy scalar searches in the W W , ZZ and hh channels respectively. The benchmark pointΛ = 3(5)TeV was used on the left (right). Due to the custodial symmetry, the chargedscalar mass is identical to the pseudoscalar mass, whose value is given above each figure.The heavy neutral Higgs mass, m H , is varied from 200 to 1000 GeV.We scanned over all the parameters and chose as benchmark values Λ = 3 , m A = 200 , ,
700 GeV and fixed λ = 0 .
1. Changing the value of the quartic couplingdoes not affect significantly the results. The results are presented as scattered plots in– 26 –gures 8 and 9 where we show the allowed region in m r - K h and m H - K H parameter spacefor the type-I 2HDM (for the type-II the results are not dramatically different and thereforewe do not show them here). In those figures the background black points correspond to thepoints that are both theoretically allowed and that survived the chi-square analysis of theprevious subsection while the points colored yellow, green and red correspond to regionsthat are forbidden by LHC searches of a heavy scalar decaying in the W W , ZZ and HH channels respectively. No bounds were found from Higgs resonant production searches in[61]. One can immediately notice that direct searches in the W W and ZZ channel forbidmainly the low mass region m r = 200 −
400 GeV with the bounds from thee
W W beingweaker than those from the ZZ channel and no bounds at all from the W W channel werefound for the heavy Higgs. The di-Higgs search channels put constraints mostly in theintermediate mass region m r/H = 300 −
800 GeV.From the figure we can notice that as the pseudoscalar mass increases the boundscoming from the di-Higgs boson and ZZ channels become more stringent. This is rea-sonable since an increase in the pseudoscalar mass corresponds, via the 2HDM potential,to an increase in the trilinear coupling of the radion to a pair of SM Higgs fields and thebranching fraction becomes bigger.The LHC has also searched for a CP-odd Higgs scalar in the processes pp → H/A → ZA/H [64–66] where the final state Z boson decays into two oppositely charged electronsor muons and the scalar, either H or A , is assumed to decay into a pair of b quarks. Thesefinal states were motivated by the large branching fractions predicted in a 2HDM with type-II Yukawa structure and the benchmark values tan β = 0 . . β − α ) = 0 .
01 areused in those references. In those papers, the charged Higgs boson masses were kept equalto the highest mass involved in the benchmark signal, namely m H ± ≈ m H for H → ZA or m H ± ≈ m A for A → ZH .Due to the custodial symmetry imposed in the 2HDM potential we can only accountfor the latter triplet mass degeneracy but we can consider both decay topologies. To thebest of our knowledge there has been no search for the signal H → ZA with m H ± ≈ m A .If such a search appears in the literature we would expect more stringent bounds since thebranching fraction BR ( H → ZA ) would be reduced by the opening of the channels H + H − and W ± H ∓ . – 27 – igure 9 : Scatter plots of the amount of mixing between the heavy Higgs and theradion, K H defined in equation (3.6), as function of the heavy Higgs mass for the type-I2HDM. The black region is theoretically allowed and the points colored yellow, green andred are forbidden by heavy scalar searches in the W W , ZZ and hh channels respectively.The benchmark point Λ = 3(5)TeV was used on the left (right). Due to the custodialsymmetry, the charged scalar mass is identical to the pseudoscalar mass, whose value isgiven above each figure. The radion mass, m r , is varied from 200 to 1000 GeV.In figure 10 we show the production cross section, via gluon fusion, for A times thebranching fractions BR ( A → ZX ) BR ( Z → l + l − ) BR ( X → b ¯ b ) in the type-I (top) and– 28 –ype-II model (bottom) as a function of the mass m X where X = H (red), r (blue). Thevalues m A = 700 GeV and λ = 0 . Figure 10 : The observable σ ( gg → A → ZX ) BR ( Z → l + l − ) BR ( X → b ¯ b ) as a function ofthe resonance mass with X = H (red), r (blue) for type-I (top) and type-II (bottom) models.We fixed Λ = 3 TeV, m A = 700 GeV and λ = 0 .
1. Due to the custodial symmetry, thecharged scalar mass is identical to the pseudoscalar mass, whose value is given above eachfigure. The heavy neutral Higgs (radion) mass is varied from 200 to 1000 GeV in the right(left) figures and the values of α and β are chosen to be consistent with the constraints ofFigure 5. The solid lines represent current and future upper bounds at the LHC.The 95% CL upper limits from ATLAS [66], after multiplying by BR ( Z → l + l − ) ≈ . m A = 700 GeV are shown in Fig. 10. We have also shown the expectedlimits for 300 fb − and 3000 fb − . It is clear that the LHC will only be able to cover asmall range of parameter space, however discovery of the process for m H >
400 GeV inthe near future would rule out the model. In any event the hadronic decay mode ( b ¯ b or t ¯ t )will dominate the pseudoscalar decays.In figure 11 we show the production cross section via gluon fusion of a heavy Higgsboson (red) and a radion (blue) times the branching fractions BR ( X → ZA ) BR ( Z → l + l − ) BR ( A → b ¯ b ) as a function of the mass m X and with X = H , r for the type-I (top) Since the limits are background limited, we are assuming in Figs. 10 and 11 that the bounds will scaleas 1 / √ N . – 29 –nd type-II (bottom) models. For type-I model we fixed m A = 200 GeV and in the type-II,due to lower bounds on the charged Higgs [68], we fixed m A = 500 GeV. Figure 11 : The observable σ ( gg → X → ZA ) BR ( Z → l + l − ) BR ( A → b ¯ b ) as a functionof the resonance mass with X = H (red), r (blue) in the type-I (top) and type-II (bottom)models. We fixed Λ = 3TeV, m A = 200GeV( m A = 500GeV) on top (bottom) and λ = 0 . α and β are chosento be consistent with the constraints of Figure 5. The solid lines represent future upperbounds at the LHCCurrent upper limits from CMS [64, 65] are out of the range of the figures. Extrapo-lations of the expected reach for 300 fb − and 3000 fb − are given by the brown and greenlines, respectively, in figure 11.We can see from figure 11 that for this decay our predictions are not in reach forthe LHC except at the very edge of the parameter space in the type-I 2HDM. Note thatdiscovery of this decay mode in the near future would rule out these models. The primarydecays of the radion would be into pairs of Higgs bosons or Z’s depending on its mass andscalar trilinear coupling. The decays of H might also be into these final states as well as b ¯ b and t ¯ t depending on its mass and scalar trilinear coupling.– 30 – Conclusions
In this work we considered two Higgs doublets coupling to the Ricci scalar in the TeV-brane of an RS model. Assuming CP-conservation, the inclusion of this term causes kineticmixing between the CP-even scalars of the 2HDM and the radion field of the RS model.The most up to date LHC measurements of the signal strengths of the SM Higgs bosonwere used to fit the model and the allowed cos( β − α )-tan β parameter space for type-I andtype-II 2HDM were presented.We have discussed two possible ways to differentiate this model from other scenarioswith similar scalar states. One possibility is to look at the sum of squared couplings of thescalars to gauge bosons. This model predicts a small deviation of about 0 .
5% from the SMvalue which could be measured at a future ILC. The other possibility is to look at the ratioof decay widths to a pair of b quarks and Z bosons for both scalars. Future experimentsmight distinguish the scalars by determining the value of the mixing angles α and β .Throughout this work we have taken the mass of the extra scalars to be in the rangeof 200-1000 GeV and we study the constraints that LHC searches of heavy resonancesimpose on the amount of mixing. The most stringent bounds arise if we take Λ = 3 TeVand m A = 700 GeV where a radion is disfavored in the mass range m r <
780 GeV while aheavy Higgs is disfavored in the mass range 300 GeV < m H <
750 GeV and m H <
250 GeVand kinetic mixing for both, radion and Higgs, is constrained to − < K h , K H <
4. Theseconstraints relax significantly by reducing m A and increasing the value Λ.Finally we showed how improvements of the experimental analysis for the decay topolo-gies X → ZA and A → ZX where X = r or H could further constrain the parameterspace of, or possibly eliminate, the model Acknowledgments
This work was supported by the National Science Foundation under Grant PHY-1519644.MM also acknowledges support from CONACYT.
Appendix A Scalar Couplings After Mixing
The interactions of the physical scalars to SM fields can be obtained by substituting thetransformation of equation (3.13) into the unmixed couplings. A summary is given by g φV V = U φ sin( β − α ) + U φ cos( β − α ) + U φ γ (cid:18) − m v ky c Λ (cid:19) φ = r, h, H, (A.1) g φff = U φ ξ fh + U φ ξ fH + U φ γ ( c fL − c fR ) , φ = r, h, H, (A.2) g φgg = (cid:18) πα s ky c + 7 (cid:19) U φ γ + (cid:88) q F q ( ξ qh U φ + ξ qH U φ + γU φ ) φ = r, h, H. (A.3)– 31 –he trilinear interactions between scalar eigenstates r , h , and H are given by L ⊇ y r∂ µ h∂ µ H + y r (cid:50) hH + y rh (cid:50) H + g rhH rhH, (A.4)where y = 2 v γ {− γ [ ξ sin( β − α ) + ξ cos( α + β )] ( U U U + U U U + U U U ) − γ [ ξ cos( β − α ) + ξ sin( α + β )] ( U U U + U U U + U U U )+ 6 ξ U [sin(2 α )( U U − U U ) + cos(2 α )( U U + U U )] + 6 ξ U ( U U + U U ) − U U U − U U U + U U U + U U U + U U U + U U U } , (A.5) y = 2 v γ { U ( U U + U U ) ξ + U ( U U + U U )(1 + 3 ξ )+ 3( U U U + U U U + U U U + U U U ) ξ cos(2 α ) − U ( U U + 2 U U ) γξ cos( α − β ) − U ( U U + 2 U U ) γξ cos( α + β )+ 3( − U U U − U U U + U U U + U U U ) ξ sin(2 α )+ 6 U ( U U + 2 U U ) γξ sin( α − β ) − U ( U U + 2 U U ) γξ sin( α + β ) } , (A.6) y = 2 γv ( U ( U U + U U ) + 3( U U U + U U U + U U U + U U U ) ξ + 3( U U U + U U U + U U U + U U U ) ξ cos(2 α ) − U U U + U U U + U U U ) γξ cos( β − α ) − U U U + U U U + U U U ) γξ cos( α + β ) + 3( − U U U − U U U + U U U + U U U ) ξ sin(2 α ) + 6( U U U + U U U + U U U ) γξ sin( α − β ) − U U U + U U U + U U U ) γξ sin( α + β )) . (A.7)The tree-level coupling has two contributions, one from the trace of the energy-momentum tensor and another one from the 2HDM potential, i.e. g rhH = g tracerhH + g HDMrhH – 32 –here g HDMrhH = 12 v (cos β ( U cos α − U sin α )( U cos α + U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α )+ cos β ( U cos α − U sin α )( U cos α + U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α )+ cos β ( U cos α − U sin α )( U cos α + U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α )+ ( U cos α − U sin α )( U cos α − U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α ) sin β + ( U cos α − U sin α )( U cos α − U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α ) sin β + ( U cos α − U sin α )( U cos α − U sin α )( U cos α + U sin α )( m A − v λ − ( m h − m H ) cos α csc β sec β sin α ) sin β − U cos α + U sin α )( U cos α + U sin α )( U cos α + U sin α )(( m A + v λ ) cot β − csc β ( m h cos α + m H sin α )) sin β + 6 sec β ( U cos α − U sin α )( U cos α − U sin α )( U cos α − U sin α )( m H cos α + m h sin α − ( m A + v λ ) sin β ) + 2 v λ (( U U + U U ) cos(2 α ) + ( − U U + U U ) sin(2 α ))( U cos( α + β ) + U sin( α + β )) + 2 v λ (( U U + U U ) cos(2 α ) + ( − U U + U U ) sin(2 α ))( U cos( α + β )+ U sin( α + β )) + 2 v λ (( U U + U U ) cos(2 α )( − U U + U U ) sin(2 α )( U cos( α + β ) + U sin( α + β ))) , (A.8) g tracerhH =4 γv ( m h ( U U U + U U U + U U U )+ m H ( U U U + U U U + U U U )) . (A.9)The other interactions like rhh , rHH , etc. can be similarly obtained and are not illustratedhere. – 33 – ppendix B LHC Data Decay Production Measured Signal Strength R m γγ ggF+tthVBF +VhggFVBFVh 1 . +0 . − . [CMS] [69]1 . +0 . − . [CMS] [69]0 . +0 . − . [ATLAS] [70]2 . +0 . − . [ATLAS] [70]0 . +0 . − . [ATLAS] [70]WW* ggFVBFggFVBFWh 1 . +0 . − . [ATLAS] [71]1 . +0 . − . [ATLAS] [71]0 . ± .
21 [CMS] [72]1 . +1 . − . [ATLAS] [73]3 . +4 . − . [ATLAS] [73]ZZ* ggFVBF + VhggFVBF 1 . +0 . − . [ATLAS] [74]0 . +1 . − . [ATLAS] [74]1 . +0 . − . [CMS] [75]0 . +1 . − . [CMS] [75]bb VBFVhVh − . +2 . − . [CMS] [76]1 . +0 . − . [ATLAS] [77]1 . ± . τ τ VBFggFVBF + VhWHtth 1 . ± . . +1 . − . [ATLAS] [80]1 . +0 . − . [ATLAS] [80]2 . ± . . +1 . − . [ATLAS] [82] Table 2 : Measured Higgs Signal Strengths
Appendix C Contributions to S and T from the scalar sector.
The relevant contributions are below. (b) and (c) refer to the diagrams of Fig. 1– 34 –n any new physics model (NP), any field that couples to the SM gauge bosons γ , W ± and Z will contribute to their vacuum polarization diagrams and will generate the tensorstructure Π µνV V = Π V V ( p ) η µν + ˜Π V V ( p ) p µ p ν (C.1)where p µ is the 4-momentum of the gauge boson.These corrections can be parametrized by the oblique parameters S and T [57] S ≡ c W s W α (cid:18) Π ZZ ( m Z ) m Z − Π ZZ (0) m Z − c W − s W c W s W Π Zγ ( m Z ) m Z − Π γγ ( m Z ) m Z (cid:19) (C.2) αT ≡ Π W W (0) m W − Π ZZ (0) m Z (C.3)which are defined relative to the SM contributions so that S = T = 0 in the SM for somereference value of the Higgs mass.Π new ( b ) ZZ ( m Z ) = g π c W (cid:88) φ = r,h,H (cid:110) [ γU rφ + s β − α U hφ + c β − α U Hφ ] B ( m Z ; m Z , m φ )+ [ s β − α U Hφ − c β − α U hφ ] B ( m Z ; m A , m φ ) (cid:111) (C.4)Π new ( c ) ZZ ( m Z ) = − g m Z π c W (cid:88) φ = r,h,H (cid:20) c β − α U Hφ + s β − α U hφ − γ (cid:18) − m Z ky c Λ (cid:19) U rφ (cid:21) × B ( m Z ; m Z , m φ ) (C.5)Π new ( b ) W W (0) = g π (cid:88) φ = r,h,H (cid:110) [ c β − α U Hφ + s β − α U hφ − γU rφ ] B (0; m W , m φ )+ [ c β − α U hφ + s β − α U Hφ ] B (0; m A , m φ ) (cid:111) (C.6)Π new ( c ) W W (0) = − g m Z π (cid:88) φ = r,h,H (cid:20) c β − α U Hφ + s β − α U hφ − γ (cid:18) − m W ky c Λ (cid:19) U rφ (cid:21) × B (0; m W , m φ ) (C.7) References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999)doi:10.1103/PhysRevLett.83.3370 [hep-ph/9905221].[2] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. , 2080 (2000)doi:10.1103/PhysRevLett.84.2080 [hep-ph/9909255]. – 35 –
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