Higgs Production from Gluon Fusion in Warped Extra Dimensions
aa r X i v : . [ h e p - ph ] J u l UMD-PP-10-010
Higgs Production from Gluon Fusion in Warped ExtraDimensions
Aleksandr Azatov ∗ , Manuel Toharia † , and Lijun Zhu ‡ Maryland Center for Fundamental Physics,Department of Physics, University of Maryland,College Park, MD 20742, USA.
Abstract
We present an analysis of the loop-induced couplings of the Higgs boson to the massless gaugefields (gluons and photons) in the warped extra dimension models where all Standard Model fieldspropagate in the bulk. We show that in such models corrections to the hgg and hγγ couplings arepotentially very large. These corrections can lead to generically sizable deviations in the productionand decay rates of the Higgs boson, even when the new physics states lie beyond the direct reachof the LHC.
PACS numbers: ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Warped extra dimensions, `a la
Randall-Sundrum model (RS) present one of the mostelegant solutions to the Standard Model (SM) hierarchy problem [1]. Placing SM fields inthe bulk of the extra dimension can simultaneously explain the hierarchies of the SM fermionmasses [2–4]. Such models provide a very attractive way to suppress flavor violation by theso called RS Glashow-Iliopoulos-Maiani(GIM) mechanism [2, 5, 6]. The electroweak preci-sion tests put important bounds on the scale of new physics, but by introducing custodialsymmetries [7, 8] one can have it around few TeV [7–9].In this paper, we will analyze the Higgs couplings to massless vector bosons in RS modelswhere all SM fields are in the bulk, and the modification to the hgg and hγγ couplings arisesfrom integrating out Kaluza-Klein (KK) partners of the SM fields. Previous works on thistopic for RS models have been done in [10–14] . These effects were also studied in modelsof warped extra dimensions in which the Higgs arises as Pseudo-Nambu-Goldstone boson(PNGB) [15] and within the effective theory formalism [16, 17]. The studies of the Higgsproduction in flat extra dimensions in the models with gauge Higgs unification were carriedout in [18]. We will stick to the models with flavor anarchy [5, 6] in which the hierarchiesin masses and mixings in the the fermion sector are explained by small overlap integralsbetween fermion wave functions and the Higgs wave function along the extra dimension.Previous studies of this framework have mainly focused on bounds on the KK scale comingfrom new flavor violating sources. In spite of the RS-GIM mechanism, it was still foundthat ∆ F = 2 processes mediated by the KK gluon push the mass of the KK excitations tobe above ∼
10 TeV [19–21], making them very hard to produce and observe at the LHC[22]. These bounds coming from flavor violation in low energy observables can be relaxedby introducing additional flavor symmetries [20, 23–25], or by promoting the Higgs to be a5D bulk field (instead of being brane localized) [26, 27]. A similar tension was found in thelepton sector in [28], making scale of O (5) TeV still compatible with experiments. LowerKK scales can be achieved by changing the fermion representations [29] or by introducingflavor symmetries [24]. It is interesting to point out that flavor violating effects can also bemediated by the radion [30], a graviscalar degree of freedom which might be generically thelightest new physics state and therefore may lead to important phenomenological bounds.More recently, it has also been pointed out that models with fermions in the bulk give rise One of the main differences between our work and previous analysis is that we present analytical resultsfor the contribution of the full KK fermion tower. Other subtle differences are discussed in the main text.
2o flavor violation in the couplings of Higgs to SM fermions [31, 32], leading to interestingconstraints from ∆ F = 2 processes and to flavor violating collider signatures such as h → tc (see also the most recent analysis of [12, 33] for further details). Other interesting collidereffects like rare top decays t → cZ were discussed in [34].The outline of the paper is as follows: in section II, we consider the effect of just twovector-like heavy fermions, one singlet under SU (2) L and one doublet. This simple casehelps us understand in simple terms the effects caused by the full tower of KK fermions ina realistic 5D setup. In section III we present a calculation of the hgg and hγγ couplingsfor the simple model where all the fermions are in a doublet representation of SU (2) L or SU (2) R . In this section and in Appendix A we also present a simple way to evaluate thecomplete KK fermion tower contribution to hgg and hγγ couplings. Having explained andderived the new contributions to the Higgs couplings caused by the heavy KK fermions, weproceed in section IV to study quantitatively the main phenomenological effects and outlineour conclusions in section V. II. WARM-UP: NEW VECTOR-LIKE FERMIONS
We begin by computing the new contribution to the hgg coupling using effective theorywith just the zero and first KK modes, where we only consider one family of light quarks(say, up and down quarks) augmented by the presence of two heavy vector-like fermions, onein doublet representation of SU (2) L and the other in singlet representation. This effectivetheory description has the advantage of being economical and gives lucid physical intuitionof the source of new physics contribution. Therefore, we adopt this approach in this sectionjust to illustrate the essential points of our calculation. Moreover, the calculation is moregeneral is the sense that it applies to any Beyond Standard Model (BSM) model in whichthere exist extra vector-like fermions which mix with SM fermions (see [35] for a similardiscussion). The full calculation of the hgg coupling in the 5D warped extra dimensionmodel will be carried out in the next section.To start, we review here the Higgs boson production through gluon fusion in SM. Thecoupling between gluon and Higgs mainly comes from top quark loop (See Fig. 1). Thepartonic cross section for gg → h is [36] σ SMgg → h = α s m h π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X Q y Q m Q A / ( τ Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ (ˆ s − m h ) , (1)where the sum is for all SM fermions, ˆ s is invariant mass squared of the two incoming3 IG. 1: hgg coupling induced by fermion loop. gluons, τ Q ≡ m h / m Q , y Q and m Q are Yukawa couplings and masses of the quarks, and theform factor for fermion in the loop is A / ( τ ) = 32 [ τ + ( τ − f ( τ )] τ − , (2)where f ( τ ) = [arcsin √ τ ] , ( τ ≤ − (cid:20) ln (cid:18) √ − τ − − √ − τ − (cid:19) − iπ (cid:21) , ( τ > . (3)We note that for τ Q → m h ≪ m Q , the form factor tends to be unity, while for τ Q → ∞ i.e. m h ≫ m Q , the form factor tends to zero. For reference, we consider a Higgsboson with mass 120 GeV, then for c-quark, we have A / ( τ c ) ≈ .
01; and for a KK fermionwith mass 2000 GeV, we have A / ( τ kk ) ≈ . q L , u R ) andfirst KK fermions ( Q (1) L , Q (1) R , U (1) L , U (1) R ), where q, Q denote the up-type quark from SU (2) L doublet, and u, U denote the up-type quark from SU (2) L singlet. Then we have the followingmass matrix: (¯ q L , ¯ Q (1) L , ¯ U (1) L ) Y qLuR ˜ v √ Y qLUR ˜ v √ Y QLuR ˜ v √ M Q Y
QLUR ˜ v √ Y ULQR ˜ v √ M U u R Q (1) R U (1) R + h.c , (4)where Y q L u R etc. are the Yukawa couplings between the corresponding chiral fermions, and˜ v is the Higgs VEV (note that it is not the same as v SM ). The Yukawa couplings matrix isgiven by (¯ q L , ¯ Q (1) L , ¯ U (1) L ) Y qLuR √ Y qLUR √ Y QLuR √ Y QLUR √ Y ULQR √ u R Q (1) R U (1) R h + h.c . (5)4o calculate these fermion contributions to the hgg coupling, we assume that the massesof the KK fermions ≫ m h , and therefore their form factors are approximately unity. Beforeproceeding let us classify different effects contributing to the shift of hgg coupling from thatof the SM: • relation between mass and Yukawa coupling of the lightest state (SM fermion) ismodified from the SM value y light RS = m f v SM ; • we have loop of KK fermion running in the triangle diagrams (see Fig. 1).So we should calculate y light RS m light A / ( τ light ) + X heavy Y i M i = Tr( ˆ Y ˆ M − ) + y light RS m light (cid:0) A / ( τ light ) − (cid:1) , (6)where ˆ M and ˆ Y are the fermion mass and Yukawa matrices given in Eq. (4) and (5) . Thefirst term on the LHS of the above equation gives the contribution from the SM fermion(lightest mass eigenstate), and the second term comes from the contributions of heavy KKfermions. Note that ˆ Y = ∂ ˆ M∂ ˜ v , therefore, we can use the following trick to calculate the trace[37]: Tr( ˆ Y ˆ M − ) = Tr ∂ ˆ M∂ ˜ v ˆ M − ! = ∂ ln Det( ˆ M ) ∂ ˜ v , (7)we also haveDet( ˆ M ) = Y q L u R M Q M U ˜ v √ Y Q L u R Y U L Q R Y q L U R (cid:18) ˜ v √ (cid:19) − Y q L u R Y Q L U R Y U L Q R (cid:18) ˜ v √ (cid:19) . (8)Now we expand to first order in ˜ v M Q M U :Tr( ˆ Y ˆ M − ) ≈ v (cid:20) (cid:18) Y Q L u R Y U L Q R Y q L U R Y q L u R − Y Q L U R Y U L Q R (cid:19) ˜ v M Q M U (cid:21) . (9)Note that the masses and Yukawa couplings of the SM fermions are also modified (see [32]for details), y light RS m light ≈ v (cid:18) Y Q L u R Y U L Q R Y q L U R Y q L u R ˜ v M Q M U (cid:19) , (10) Note that the real part of the Yukawa coupling will lead to the operator hG µν G µν , and the imaginarypart will lead to the operator hG µν ˜ G µν . For simplicity in this paper we everywhere will assume that wehave only hG µν G µν operator. Y q L u R ≪ Y q L U R , Y Q L u R ≪ Y Q L U R ). This assumption is true the quarks of the first two generations,and the extra contribution which is important for the quarks of the third generation will bepresented in the next section. Now Eq. (6) reduces to y light RS m light A / ( τ light ) − ˜ v Y Q L U R Y U L Q R M Q M U . (11)We can see that for the light generation quarks, A / ( τ light ) ≈
0, we get − v Y Q L U R Y U L Q R ˜ v M Q M U ,which is just the contribution coming from the KK modes. Note that this contribution isproportional to Y Q L U R Y U L Q R , which is the product of Yukawa couplings of the KK fermionsof opposite chiralities, this structure of the contribution will become essential in calculatingthe effects in realistic warped model in the next section. It is interesting to see that eventhough the light SM quarks give negligible contribution to hgg coupling, their KK partnerscan give sizable new contributions. In addition, there would be an multiplicity enhancementof these KK contributions due to the number of flavors.The analysis above showed that additional vector-like fermions which mix with SMfermions can alter the hgg coupling significantly. In warped extra dimension models with 5Dfermions propagating in the bulk, these extra vector-like fermions naturally come up as theKK towers of fermions. Therefore, we expect generically sizable new physics contributionsto hgg coupling in this class of models. We carry out the detailed calculations in warpedextra dimension in the next section. III. MINIMAL WARPED EXTRA DIMENSION MODEL WITH CUSTODIALPROTECTION
In this section, we first calculate the KK fermion contributions to hgg coupling in warpedextra dimensions (RS). We then apply similar techniques to calculate both KK fermion andKK gauge boson contributions to hγγ coupling. We show that simple analytical formulascan be obtained for these new physics contributions. A. hgg coupling in RS In this subsection, we consider the effect of the full KK fermion tower on hgg coupling.We consider models with bulk gauge group SU (2) L ⊗ SU (2) R , which is motivated to ease thebound from electroweak precision test [7]. We consider here just a single family of quarks for6he sake of simplicity. A generalization to 3 generation quarks can be easily applied later.For the quark fields, we consider the simple spinorial representation with the following fieldcontents: Q uL (+ , +) Q uR ( − , − ) Q dL (+ , +) Q dR ( − , − ) ! , U ′ R ( − , +) U ′ L (+ , − ) D R (+ , +) D L ( − , − ) ! , U R (+ , +) U L ( − , − ) D ′ R ( − , +) D ′ L (+ , − ) ! . (12)The first multiplet is a doublet of SU (2) L and the last two are doublets of SU (2) R . Theboundary conditions are denoted for the corresponding chirality. They have the followingYukawa couplings Y u √ R ( ¯ Q uL U R + ¯ Q dL D ′ R ) H + Y d √ R ( ¯ Q uL U ′ R + ¯ Q dL D R ) H + ( L ↔ R ) + h.c. (13)Note that Y u , Y d are dimensionless and order one, and 1 /R = k is the curvaturescale. After KK decomposition in the basis where Higgs vev is zero, we have zero modes q u, (0) L , q d, (0) L , d (0) R , u (0) R and the KK modes Q u, ( i ) L,R , Q d, ( i ) L,R , D ( j ) L,R , U ( j ) L,R , U ′ ( k ) L,R , D ′ ( k ) L,R . For up-typequarks, we have the following infinite dimensional mass matrix(¯ q u, (0) L , ¯ Q u, ( i ) L , ¯ U ( j ) L , ¯ U ′ ( k ) L ) Y uqu ˜ v √ Y uqUb ˜ v √ Y dqU ′ c ˜ v √ Y uQiu ˜ v √ M Q Y uQiUb ˜ v √ Y dQiU ′ c ˜ v √ Y u, ∗ UjQa ˜ v √ M U Y d, ∗ U ′ kQa ˜ v √ M U ′ u (0) R Q u, ( a ) R U ( b ) R U ′ ( c ) R + h.c , (14)where i, j, k, a, b, c are KK indices. The Yukawa couplings matrices are defined e.g. by Y uQ i U b = Y u √ R Z dz (cid:18) Rz (cid:19) h ( z ) q u, ( i ) L ( z ) u ( b ) R ( z ) , (15)i.e. it is an integral of product of Higgs and fermion wavefunctions, where h ( z ) is a profileof the Higgs field normalized in the following way1 = Z R ′ R dz (cid:18) Rz (cid:19) h ( z ) . (16)The KK mass matrices are diagonal, e.g. M Q = diag( M Q , M Q , · · · ). One naively mightthink that the couplings Y U j Q a vanish in the limit of brane Higgs due to the odd boundaryconditions of U L and Q uR , so it is safe to ignore them in this matrix. But these are preciselythe Z odd operators described in detail in [32] (detailed analysis without these operators We consider here a general bulk Higgs [38] with vector-like Yukawa coupling for simplicity. hgg coupling . To avoid subtletieswith wave function being discontinuous at IR brane we will assume that the Higgs is 5Dbulk field and only at the end we will take a brane Higgs limit.Now we can use the same determinant trick, the determinant of the mass matrix to theorder of ˜ v is Det( ˆ M ) = Y i,j,k M Q i M U j M U ′ k ! × " Y uqu ˜ v √ − Y uqu (cid:18) ˜ v √ (cid:19) X a,b Y dQ a U ′ b Y d, ∗ U ′ b Q a M Q a M U ′ b + Y uQ a U b Y u, ∗ U b Q a M Q a M U b ! + (cid:18) ˜ v √ (cid:19) X a,b Y uqU b Y u, ∗ U b Q a Y uQ a u M Q a M U b + Y dqU ′ b Y d, ∗ U ′ b Q a Y uQ a u M Q a M U ′ b ! . (17)Now we getTr( ˆ Y ˆ M − ) = ∂ ln Det( ˆ M ) ∂ ˜ v = 1˜ v " − ˜ v X a,b Y dQ a U ′ b Y d, ∗ U ′ b Q a M Q a M U ′ b + Y uQ a U b Y u, ∗ U b Q a M Q a M U b ! + ˜ v Y uqu X a,b Y uqU b Y u, ∗ U b Q a Y uQ a u M Q a M U b + Y dqU ′ b Y d, ∗ U ′ b Q a Y uQ a u M Q a M U ′ b ! . (18)Again, for the light generation quarks there are corrections to the SM fermion masses andYukawa couplings [32] m light = Y uqu ˜ v √ X a,b Y uqU b M U b Y u, ∗ U b Q a M Q a Y uQ a u (cid:18) ˜ v √ (cid:19) (19)+ X a,b Y dqU ′ b M U ′ b Y d, ∗ U ′ b Q a M Q a Y uQ a u (cid:18) ˜ v √ (cid:19) ,y light RS = Y uqu √ √ X a,b Y uqU b M U b Y u, ∗ U b Q a M Q a Y uQ a u (cid:18) ˜ v √ (cid:19) (20)+ 3 √ X a,b Y dqU ′ b M U ′ b Y d, ∗ U ′ b Q a M Q a Y uQ a u (cid:18) ˜ v √ (cid:19) . Therefore y light RS m light ≈ v X a,b Y uQ a u Y u, ∗ U b Q a Y uqU b ˜ v M Q a M U b Y uqu + X a,b Y uQ a u Y d, ∗ U ′ b Q a Y dqU ′ b ˜ v M Q a M U ′ b Y uqu ! . (21) These operators can be mimicked by higher dimensional derivative operators [32], which shows UV sen-sitivity of the effect.
8o the total contribution to hgg coupling by light generation quarks and their KK partnersis (see Eq. 6) − ˜ v X a,b Y dQ a U ′ b Y d, ∗ U ′ b Q a M Q a M U ′ b + Y uQ a U b Y u, ∗ U b Q a M Q a M U b ! + y light RS m light A / ( τ light ) . (22)Note that this result is very similar to the one we obtained in the last section (Eq. 11),except for an extra term corresponding to the contribution of extra states in the doubletrepresentation of SU (2) R . For light generations, the last term is negligible, and we are leftwith first two terms. The first two terms can be written as − ˜ v X a,b h Y u Y u, ∗ R Z dzdz ′ (cid:18) Rz (cid:19) (cid:18) Rz ′ (cid:19) q ( a ) L ( z ) q ( a ) R ( z ′ ) M Q a u ( b ) R ( z ) u ( b ) L ( z ′ ) M U b h ( z ) h ( z ′ ) ! (23)+ Y d Y d, ∗ R Z dzdz ′ (cid:18) Rz (cid:19) (cid:18) Rz ′ (cid:19) q ( a ) L ( z ) q ( a ) R ( z ′ ) M Q a u ′ ( b ) R ( z ) u ′ ( b ) L ( z ′ ) M U ′ b h ( z ) h ( z ′ ) ! i . Now we have to evaluate the following sums X a> q ( a ) L ( z ) q ( a ) R ( z ′ ) M Q a , X b> u ( b ) R ( z ) u ( b ) L ( z ′ ) M U b , X b> u ′ ( b ) R ( z ) u ′ ( b ) L ( z ′ ) M U ′ b . (24)We can calculate them by using equations of motion for fermion wavefunctions (see discussionin the Appendix A). From the forms of these sums (see Eq. (A10)), we see that we needto evaluate the integrals of Higgs wavefunction times θ ( z − z ′ ) and θ ( z − z ′ ) . This can bedone for general bulk Higgs. But for illustration purpose we take the brane Higgs limit ofbulk Higgs. Then we get Z dzdz ′ θ ( z − z ′ ) h brane ( z ) h brane ( z ′ ) = 12 , (25) and Eq. (23) now reduces to 12 (cid:0) Y u Y u, ∗ + Y d Y d, ∗ (cid:1) ˜ vR ′ . (26)Therefore, for light generations, the contribution to hgg coupling is (cid:0) Y u Y u, ∗ + Y d Y d, ∗ (cid:1) ˜ vR ′ /
2, which comes just from KK fermions and is independent offermion bulk mass parameters. To evaluate this integral we have to somehow regularize the wavefunction of the brane Higgs( δ function),we used bulk Higgs inspired regularization of the delta function h brane ( z ) = lim β →∞ βR ′ (cid:16) zR ′ (cid:17) β . One canalso use a rectangular regularization of brane Higgs wavefunction which will lead to the same result. − ∆ t,b m ˜ v ) (see Appendix C for details). This givesus additional contribution relative to (Eq. 22)∆ t m t ˜ v + ∆ b m b ˜ v . (27)Also in this case contributions of the SM bottom and top qaurks are no longer negligible,so we have to include them y RSb m b A / ( τ b ) + y RSt m t A / ( τ t ) . (28)Note that now Yukawa couplings of the top and bottom quarks are shifted( see discussionin Appendix C).It is simple to generalize the above result to three generations. The KK towers of thequarks give a contribution proportional to Tr( Y u Y † u + Y d Y † d ), and we have to combine themwith the effect coming from top and bottom quarks. To summarize, compared with SM, theHiggs production cross-section from gluon fusion in RS is σ RSgg → h σ SMgg → h = (cid:16) v SM ˜ v (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr( Y u Y † u + Y d Y † d )˜ v R ′ + ∆ t m t + ∆ b m b + x t A / ( τ t ) + x b A / ( τ b ) A / ( τ t ) + A / ( τ b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (29)where x t = y RSt ˜ vm t and x b = y RSb ˜ vm b , with y RSt , y RSb the shifted top and bottom Yukawa couplingsin RS (reference [12] presented numerical results for the analysis of the brane Higgs modelincluding Z odd operators, however, it is hard to compare it with our result due to differentparticle content of the models). We consider here the ratio σ RSgg → h σ SMgg → h in order to reduce theuncertainty coming from higher order QCD corrections. It is also important to notice thatin the case when the couplings of the SU (2) L and SU (2) R are not equal the ratio (cid:0) v SM ˜ v (cid:1) might be quite significant, see discussion and analysis in [13]. In the rest of the paper we willassume that SU (2) L and custodial SU (2) R have the same gauge couplings (see appendix Bfor discussion of VEV shift in this case).It is also interesting to point out that the same diagrams that contribute to the gluonfusion will also contribute to the modification of the di-Higgs production. This might becomean interesting option to disentangle new physics contribution (see discussion in the effectivefield theory approach in [35]). 10 . hγγ coupling in RS The calculation of the hγγ coupling comes from similar diagrams as the one for the hgg coupling, the only difference now is that we have to take into account contributions ofthe towers of charged KK gauge bosons and KK leptons. We will again use the simplestcustodial model where leptons are in the doublet representation of SU (2) L or SU (2) R . Wecan calculate their contribution in the same way as we did for the quarks. Contribution ofthe KK tower of the W ± was presented in [13], so here we just quote their results and thereader can find more details about the derivation in the Appendix B. The contribution ofthe tower of the KK W ± is given by X n ≥ C ndiag M n A ( τ n ) = C hww M w ( A ( τ w ) + 7) − v , (30)where C ndiag is coupling between Higgs field and the n-th KK modes (mass eigenstates) ofthe W ± , and C hww is coupling between SM W and the Higgs. A ( τ w ) is the form-factor forthe gauge bosons (see Eq. (B7)). Including the modification of the coupling between SM W and Higgs, this sum can be expresssed in the following way: X n ≥ C ndiag M n A ( τ n ) = g ˜ v M w (cid:18) − ˜ v R ′ ( g D + ˜ g D )4 R (cid:19) ( A ( τ w ) + 7) − v ≈ v (cid:20)(cid:18) − ˜ v R ′ ( g D + ˜ g D )8 R (cid:19) A ( τ w ) −
78 ˜ v R ′ ( g D + ˜ g D ) R (cid:21) , (31)where g D and ˜ g D are the 5D gauge couplings of SU (2) L and SU (2) R respectively. Addingboth fermion and gauge boson contributions together, now we can present our results forthe ratio of Γ( h → γγ ) between RS and SM:Γ RS ( h → γγ )Γ SM ( h → γγ ) = (cid:16) v SM ˜ v (cid:17) | A ( τ w ) + A / ( τ t ) + A / ( τ b ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − v SM R ′ ( g D + ˜ g D )8 R (cid:19) A ( τ w ) − v SM R ′ ( g D + ˜ g D )8 R + 169 x t A / ( τ t )+ 49 x b A / ( τ b ) + 12 v SM R ′ Tr (cid:20) (cid:16) Y † u Y u + Y † d Y d (cid:17) + 43 Y † l Y l (cid:21) + 16∆ t m t + 4∆ b m b (cid:12)(cid:12)(cid:12)(cid:12) . (32) IV. PHENOMENOLOGY
In this section, we discuss the phenomenology of the Higgs boson in warped extra dimen-sions. We focus our study on the Higgs production through gluon fusion and the branching11 óó ó óóó ó óóóó óó óó ó óóó óóó óó óóóóó óó óóó óó óó ó óóóó ó óóóó óóó ó óó ó óóóó ó óó óóóó óó óó óó ó óóó óóóó óó ó ó óó óóó óóóó óóó óóó ó óóó óó ó ó ó óó óóó ó óó óó óó óó ó ó óóó ó ó óó óóó óóóóó óóó óóóó ó óó + + ++++++++ + + ++++ +++++ ++++ + +++ ++++ ++ ++++ +++++ +++++++ ++ +++ +++ +++ ++ ++++ ++ ++ ++ ++++ +++ ++++ ++++ ++ +++ ++ ++++ + ++++ + ++ + ++ ++ ++ ++++ + ++++ ++ + ++ ++ + +++ ++ + ++ ++++ ++ ++++´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´ ´´´´´´´´´´ ´´´´´ ´´´´´´´´´´ ´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´´ ´´´´´´´ ø SM €€€€€€€€€€€€€€€€€€€€€€€€€€€€Σ gg ® h RS Σ gg ® h SM €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ Br H h ® ΓΓ L RS Br H h ® ΓΓ L SM FIG. 2: Scattered plot of σ RSgg → h σ SMgg → h and Br( h → γγ ) RS Br( h → γγ ) SM , for bulk Higgs with vector-like Yukawa couplings( Y = Y ). The dimensionless 5D Yukawa couplings are varied between Y ∈ [0 . ,
3] and m h = 120GeV. The black “ × ” corresponds to the KK scale R ′− = 5 TeV, green “+” to R ′− = 2 TeV, andred “ △ ” to R ′− = 1 . fraction of h → γγ decay. We will compare our results with that of holographic PNGBHiggs model studied in [15].To get a handle on the size of new physics contributions, we scan the parameter space ofRS with the assumption of flavor anarchy, i.e. the 5D Yukawa matrices are order one anduncorrelated. We find the set of 5D Yukawa couplings and fermion zero mode wavefunctionswhich give the correct SM quark masses and CKM mixing. We then calculate σ ( gg → h )and Br( h → γγ ) using Eq. (29) and (32), and find the ratio with that of SM. The result ofthe scan for bulk Higgs is shown in Fig. 2.We can see from the plot in Fig. 2 that the new physics contribution to σ ( gg → h )tends to be positive and gets larger for lower KK scale. Also the new physics contributionto σ ( gg → h ) and Br( h → γγ ) are correlated: an increase in σ ( gg → h ) is accompaniedby a decrease in Br( h → γγ ). Before proceeding further let us stop and see whether wecan understand these results intuitively. First let us focus on the enhancement of the Higgsproduction due to gluon fusion. As we argued in the sections III A these effects come mainlyfrom the modification of the top Yukawa coupling and from the loop with KK fermions. Aswas shown in [32] top Yukawa coupling is reduced compared to the SM value, so naively one12hould expect the reduction of the Higgs production. But let us now look on the contributionof the KK modes. One can see from (Eq. 29) that this contribution is proportional toTr( Y u Y † u + Y d Y † d ) which is always positive, so the sign of this contribution is fixed. Also thetypical size of this term will be roughly equal to N ¯ Y where N is number of SM familiesand ¯ Y is an average size of the Yukawa couplings, so adding both up and down quark KKtowers will lead to an overall enhancement factor of 18. Therefore KK fermions give a largepositive contribution to σ ( gg → h ). Reduction of the Br( h → γγ ) can be understood fromthe fact that in the SM the dominant contribution comes from the loop with W ± , and thefermion contribution has an opposite sign, thus enhancement of the fermion contributionseffectively decreases the overall coupling.This implication is two-fold. First, it means that even with a KK scale out of the reach ofthe LHC ( & σ ( gg → h ) can be used to distinguish between RSwith bulk Higgs and holographic PNGB Higgs model (or gauge-Higgs unification). In thelatter model, a reduction is usually expected, which can be contrasted with our results forbulk Higgs. Note that the difference in these two models comes from the extra symmetry inPNGB Higgs, which constrains the Higgs interactions (see discussion in [16, 17]).To study the dependence of new physics contributions on the Higgs boson mass, we plotin Fig. 3 the ratio σ RSgg → h σ SMgg → h vs. m h for various KK scales. We can see that the new physicscontribution decreases as m h increase from 100 to ∼
360 GeV. This can be understoodfrom the fact that in SM, the form factor for the top quark attains its largest value when m h ≈ m t . Since in RS with bulk Higgs, the top quark Yukawa coupling is reduced comparedto that of SM, there is a larger negative new physics contribution to hgg coupling when m h ≈ m t , leading to a smaller total new physics contribution.In Fig. 4, we plot the dependence of the ratio σ RSgg → h σ SMgg → h on the average size of the 5D Yukawacouplings. We can see quite clearly that the size of new physics contribution increasesas the 5D Yukawa couplings increases. This is expected from the fact that KK fermioncontributions are proportional to Tr( Y u Y † u + Y d Y † d ). In the framework of flavor anarchy, the One can see that for sufficiently large Yukawa couplings our expansion in powers of
Y Y † v R ′ mightbecome ill defined, and also contribution of the higher order loops with KK fermions and Higgs mightbecome important, so the one loop result becomes not reliable if the new physics contribution is muchlarger than that of the SM. At the same time we would like to note that our result even for the large 5DYukawa couplings will give a typical size of the expected correction to the SM coupling. ´ ´ ´´´ ´ ´ ´ ´ ´´´ ´´ ´´ ´´ ´ ´´´ ´´´´ ´ ´ ´´´ ´ ´´´´´ ´´ ´´ ´´ ´ ´´ ´´ ´´ ´´´ ´ ´´´´ ´ ´´´ ´ ´´ ´´´ ´´´ ´´´´ ´´ ´´ ´´ ´ ´´´ ´ ´ ´´ ´´ ´´´ ´ ´´ ´´´ ´´´ ´´ ´´ ´ ´´´´ ´´´ ´´ ´´´ ´´ ´´ ´ ´ ´´´ ´ ´´ ´´´ ´´ ´´´ ´´ ´ ´´ ´´ ´´ ´´ ´ ´´ ´´ ´´´ ´´´ ´´´ ´´ ´ ´´´ ´´ ´´´´ ´´ ´´´´ ´ ´´´ ´´ ´ ´´´ ´´ ´´´´ ++ + +++ + + ++ +++ + ++ + +++ +++ +++ + +++ +++ ++ +++ +++ ++ +++ + ++ +++++ ++++ ++ ++++ ++ +++ + ++ ++ +++ + +++ ++ +++ +++ + ++++ ++++ ++ + ++ +++ ++ + + +++ + +++ ++ ++ ++++ +++ ++ +++++ + ++ ++ ++ ++ ++ ++ + ++ + +++ +++ ++ ++++ + + ++ ++ +++ ++ + ++ +++ + ++ +++ +++ ++++ ++ ++++ ó ó ó óóóó ó ó óóóó ó ó óó óó óóóó óó óó óó óó óóó ó óó óó óóó ó óó óóóó óóó ó óó óó ó ó óó óó óó ó óóóó ó óó ó óó óó ó óóóó óó óóó óó óó ó óó óó óó ó ó óóó óóó ó ó ó óóó ó óó óó óóó óó óóó óó ó óóó óó óóó óó óóó óó óó ó óó óó óó ó óóó óóó ó ó óóó ó ó óó óó óóóó óó óó óó ó ó óó óó ó óóó óóó ó óó óó ó
100 200 300 400 500 600 m h €€€€€€€€€€€€€€€€€€€€€€€€€€€€Σ gg ® h RS Σ gg ® h SM FIG. 3: Dependence of σ RSgg → h σ SMgg → h on the Higgs mass for different values of R ′− in bulk Higgs scenariowith vector-like Yukawa couplings ( Y = Y ). The dimensionless 5D Yukawa couplings are variedbetween Y ∈ [0 . , × ” corresponds to KK scale R ′− = 5 TeV, green “+” to R ′− = 2TeV, and red “ △ ” to R ′− = 1 .
5D Yukawa couplings are order one. We can see from Fig. 4 that for order one Yukawacouplings, we have sizable new physics contributions to σ ( gg → h ).So far we have been assuming that the Higgs is the bulk field and 5D Yukawa couplingsare vector-like i.e. L = Y ¯ Q uL U R H + Y ¯ U L Q uR H with Y = Y . (33)In the case where the Higgs is a 5D bulk field this condition of Y = Y is forced by the5D Lorentz symmetry. But the Higgs can be brane localized or even a bulk Higgs mighthave brane localized couplings and these couplings do not have to respect 5D bulk Lorentzsymmetry. So generally speaking Y = Y , and they could be independent of each other. Letus see how this might modify our results. The first thing to notice is that the contributionof the tower of KK modes now has the following structure Y Y † . Before proceeding furtherwe immediately see that the overall sign of the contribution is not fixed any more! So wecannot predict in generic RS model the sign of the effect: whether it is enhancement orsuppression for both hgg and hγγ couplings. This is shown in Fig. 5. We can see that the14 + ++ ++ + ++++ + + + +++ + ++ ++ +++ ++ ++ +++ + +++ + ++ + ++ ++ +++ +++ ++ + + ++ + ++ +++ ++ + ++ +++ + +++ ++ + +++ + +++ ++ ++ + ++ + + + +++ +++ ++ + ++ ++ + ++ ++ ++ + + +++ +++ +++ ++ + +++ + + ++ + + ++ ++++ + +++ ++ + ++ ++ ++ ++ + +++ ++ +++ ++ +++ ++ + +++ +++ ++ + y (cid:143)(cid:143) €€€€€€€€€€€€€€€€€€€€€€€€€€€€Σ gg ® h RS Σ gg ® h SM FIG. 4: Dependence of σ RSgg → h σ SMgg → h on the average size of dimensionless 5D Yukawa couplings ¯ Y , for theHiggs mass m h = 120 GeV and KK scale R ′− = 2 TeV. size of new physics contribution is generically large for moderate KK scale, but now its signcan be both positive and negative. V. CONCLUSIONS
In conclusion, we summarize the results presented in the paper. We calculated the cor-rections to the hgg and hγγ couplings in RS at one loop order. We have found that the newphysics states can modify significantly these couplings. We have shown that the dominantcontribution to these coupling comes from the towers of KK fermions running inside trianglediagrams. We have shown that the KK towers of the light fermions do contribute signifi-cantly to these couplings, contrary to the models with Higgs being a PNGB boson wherethis contribution is sub-leading. We have shown that in the models with the Higgs in thebulk and Yukawa couplings being vectorlike ( Y = Y ), hgg coupling becomes enhanced and hγγ coupling suppressed compared to that of SM, even though the top Yukawa couplingis suppressed compared to the SM value. This naively counterintuitive result is explainedby the fact that the contribution of the KK towers of all SM fermions is so strong that itovercomes the effect from suppression of the top Yukawa coupling. Modification of the Higgs15 ó ó óó óóóóóóó óóó óó óóó ó óóóó ó óó ó óó ó óó óó ó óó óóó óó óóóó óóó ó ó óóó óóóóóó óó ó óó óóó óóó óó ó óó ó óó óó óó óó ó ó óóó óóóó óóó óó óó óóó óó óó óóó óó óó ó ó ó óó ó óó ó óó óó óóó ó óóó óóóó óó óó ó óóó +++ ++ ++ +++ +++ ++++ ++++ +++ +++ +++ ++++ + +++ + + ++ +++ + +++ + ++++ ++++ ++ + +++ +++ + +++ + ++ ++ ++ + +++ +++++ + +++++ + ++ ++ ++ ++ + ++++++ ++++ ++ ++ + +++ +++ + + ++ + ++ ++ + ++ + + +++ ++ + ++ +++ + ´´´´´´´´´´ ´´´´´ ´´´´´´´´´´´´´´´´´´ ´´´´´ ´´´´´´ ´´´´´´´´´´´´´´ ´´´´´´´´´´´´´´´´´´´´´ ´´´ ´´´´´´´´´´´´´´ ´´´´´´´´´´´´´´´ ´´´´´´´´´´ ´´´´´´´´´´´´´´´´´´´´´´´´´´ ´´´ ø SM €€€€€€€€€€€€€€€€€€€€€€€€€€€€Σ gg ® h RS Σ gg ® h SM €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ Br H h ® ΓΓ L RS Br H h ® ΓΓ L SM FIG. 5: Scattered plot for the modification of
Br( h → γγ ) RS Br( h → γγ ) RS and σ RSgg → h σ SMgg → h for brane Higgs with Y independent of Y , where 5D Yukawa couplings are varied between Y ∈ [0 . ,
3] and m h = 120 GeV.The black “ × ” corresponds to the KK scale R ′− = 5 TeV, green “+” to the R ′− = 2 TeV, andred “ △ ” to the R ′− = 1 . production cross-section remains significant even for a KK scale far from LHC accessibility.Specifically, we can get order one corrections even with lightest KK modes above 5 TeV. Forthe generic models with Higgs on the brane or bulk Higgs with brane Yukawa couplings thesign of the effect remains unpredictable. We might have enhancement as well as suppression,but the parametric size of the effect remains the same. The total effect comes from collectivecontributions of the KK partners of all generations. Therefore, the size of these new physicscontributions is large, even if the KK fermions are heavy. This shows us that in the absenceof new resonances an analysis of the Higgs couplings might become a very important toolin understanding the structure of BSM physics. Acknowledgements
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In this section we will present a way of efficiently performing KK sums for the fermions (such as Eq. (24)). Let us look at the equations of motions for the fermions in the absence ofthe Higgs vev. In the absence of the Higgs vev we can always choose a basis where 5D bulkmasses are diagonal, and so we can ignore all the mixings. Let us concentrate on the KKdecomposition of the SU (2) L doublet Q L,R with boundary conditions ( ± , ± ). The equationsof motion of the KK wavefunctions are − m n q ( n ) L − ∂ z q ( n ) R + c q + 2 z q ( n ) R = 0 , (A1) − m ∗ n q ( n ) R + ∂ z q ( n ) L + c q − z q ( n ) L = 0 . (A2) Similar tricks were discussed in [39]
19e take the first equation and rewrite it as: − m n q ( n ) L − z c q +2 ∂ z (cid:16) q ( n ) R z − c q − (cid:17) = 0 . (A3)We now multiply by z − c q − and integrate between R and z : − m n Z z R dzz − c q − q ( n ) L ( z ) = q ( n ) R z − c q − (cid:12)(cid:12)(cid:12)(cid:12) z R , Z z R dzz − c − q ( n ) L ( z ) = − m n q ( n ) R ( z ) z − c − . (A4)We now use the completeness relation ∞ X n =0 q ( n ) L ( z ) q ( n ) L ( z ) = z R δ ( z − z ) (A5) ⇒ ∞ X n =1 q ( n ) L ( z ) q ( n ) L ( z ) = z R δ ( z − z ) − q L ( z ) q L ( z ) . (A6)Based on (Eq A4) we will get − Z z R dz z − c q − ∞ X n =1 q ( n ) L ( z ) q ( n ) L ( z ) = z − c q − ∞ X n =1 q ( n ) R ( z ) q ( n ) L ( z ) m n , (A7)where we have explicitly extracted the zero mode contribution from the sum. Let us notethat q L ( z ) = N L z − c q with N L = r − c q ǫ c q − − R c q − / , (A8)and where we have defined the warp factor ǫ = RR ′ ∼ − .Now we can finally write: ∞ X n =1 q ( n ) R ( z ) q ( n ) L ( z ) m n = − z c +21 Z z R dz z − c − (cid:16) z R δ ( z − z ) − q L ( z ) q L ( z ) (cid:17) = z c q z − c q R " − θ ( z − z ) + (cid:0) z R (cid:1) − c − (cid:0) R ′ R (cid:1) − c − . (A9)Similarly we can calculate the sum for the other three possible boundary conditions : ψ L (+ , +) : X q ( n ) R ( z ) q ( n ) L ( z ) m n = z c z − c R " − θ ( z − z ) + (cid:0) z R (cid:1) − c − (cid:0) R ′ R (cid:1) − c − ,ψ L (+ , − ) : X q ( n ) R ( z ) q ( n ) L ( z ) m n = − z c z − c R θ ( z − z ) ,ψ L ( − , +) : X q ( n ) R ( z ) q ( n ) L ( z ) m n = z c z − c R θ ( z − z ) ,ψ L ( − , − ) : X q ( n ) R ( z ) q ( n ) L ( z ) m n = z c z − c R " θ ( z − z ) − (cid:0) z R (cid:1) c − (cid:0) R ′ R (cid:1) c − . (A10)20sing these relations we can now perform all the necessary sums to calculate the KK fermioncontribution to hgg coupling. Appendix B: Gauge boson couplings and contribution to hγγ coupling
In this section just for the sake of the completion we present analysis for the modificationof the gauge boson coupling to the Higgs boson, and their contribution to the hγγ coupling.We start from the modification of the Higgs vev v SM ≈ ˜ v − ˜ v R ′ R (cid:0) g D + ˜ g D (cid:1) , (B1)where v SM = 246 GeV, ˜ g D is five dimensional gauge coupling of the custodial SU (2) R , so˜ v ≈ v SM (cid:18) R ′ v SM R ( g D + ˜ g D ) (cid:19) . (B2)This effect will lead to the overall modification of the SM hgg and hγγ coupling by thefactor 1 − R ′ v SM R ( g D + ˜ g D ) ≈ .
95 for ( R ′− = 1500 TeV , g D = ˜ g D ).
1. Couplings of W ± to Higgs in RS To calculate modification of the hγγ coupling we also have to calculate contributioncoming from the W boson. From the Lagrangian (see [7]) L = g (cid:18) h + ˜ v √ (cid:19) − R ′ ( g D + ˜ g D )4 R (cid:18) h + ˜ v √ (cid:19) ! W + µ W − µ , (B3)one can immediately deduce coupling between Higgs and W . L = C hww hW + µ W − µ ,C hww = g ˜ v (cid:20) − R ′ ( g D + ˜ g D )˜ v R (cid:21) . (B4)
2. Contribution of the KK tower of W ± to the hγγ In this subsection we derive the contribution of the W ± KK modes to the hγγ coupling(we will closely follow discussion presented in [13]). First let us denote by M the masssquared matrix of the charged gauge bosons, then the coupling to the Higgs boson will begiven by the matrix C = ∂M ∂ ˜ v , M is diagonal we will get C diag = U ∂M ∂ ˜ v U † , (B5)where U is a unitary matrix that diagonalizes M . We can parameterize the contribution ofthe gauge boson KK modes to the hγγ coupling in the following way: X n ≥ C ndiag M n A ( τ n ) = C hww M w A ( τ w ) + X n> C ndiag M n A ( τ n ) , (B6)where A ( τ ) is the form factor for vector bosons in the loop [36] ( τ = m h / M n ) A ( τ ) = − [2 τ + 3 τ + 3(2 τ − f ( τ )] τ − , (B7)where f ( τ ) is given by Eq. (3). For KK gauge bosons τ n →
0, and A ( τ n ) ≈ −
7, so we get C hww M w A ( τ w ) − X n> C ndiag M n = C hww M w ( A ( τ w ) + 7) − X n ≥ C ndiag M n . (B8)To evaluate X n ≥ C ndiag M n we can use the following trick [37] X n ≥ C ndiag M n = Tr h(cid:0) M diag (cid:1) − C i = Tr (cid:20) ∂M ∂ ˜ v (cid:0) M (cid:1) − (cid:21) = ∂∂ ˜ v ln (cid:0) Det M (cid:1) . (B9)Let us see how the determinant of the gauge boson mass matrix depends on ˜ v . For simplicitywe assume that the Higgs is localized on the IR brane. We denote by f ( i ) , ˜ f ( j ) values of theprofiles on the IR brane for KK modes of SU (2) L and SU (2) R gauge bosons respectively.Then the mass matrix will look like: M = g D f v f (0) f (1) g D ˜ v f (0) ˜ f (1) g D ˜ g D ˜ v ...g D f (0) f (1) ˜ v M + f (1)2 g D ˜ v f (1) ˜ f (1) g D ˜ g D ˜ v ...g D ˜ g D f (0) ˜ f (1) ˜ v f (1) ˜ f (1) g D ˜ g D ˜ v ˜ M + ˜ f ˜ g D ˜ v ... ... ... ... . . . . (B10)One can see from the structure of the matrix that the determinant is equal toDet M = g D f ˜ v Y i,j M i ˜ M j . (B11)We have checked that for generic bulk Higgs Det M ∝ ˜ v + O (˜ v ), one can calculate itusing mixed position momentum propagators. So the results presented in this section areapproximately independent of the Higgs localization. Now we can proceed to the evaluationof the sum in Eq. (B8) and substituting result for the determinant we get X n ≥ C ndiag M n A ( τ n ) = C hww M w ( A ( τ w ) + 7) − v . (B12)22 ppendix C: Review of Higgs Flavor violation In this appendix we present general formulas for the misalignment between SM fermionmasses and Higgs Yukawa couplings in RS(see for details[32]). We define the followingquantity to parameterize the misalignmentˆ∆ = ˆ m − ˜ v ˆ y, (C1)where ˆ m, ˆ y are mass matrix and Yukawa couplings of the SM fermions. Then it can be splitinto two parts ˆ∆ = ˆ∆ + ˆ∆ , (C2)where ˆ∆ is the main contribution for the light generations and ˆ∆ becomes important onlyfor the third generation of quarks. Then calculations show that ˆ∆ for the up type quarksis equal to ˆ∆ u = ˜ v √ (cid:18) ˜ v R ′ (cid:19) ˆ F ( c q ) h Y u Y † u Y u + Y d Y † d Y u i ˆ F ( − c u ) (C3) where c u , c q are bulk mass parameters for the multiplets containing zero modes of theSM right-handed and left-handed up quarks respectively. ˆ F ( c ) is a diagonal matrix withelements given by the profiles of the corresponding quarks respectively F ( c ) ≡ s − c − (cid:0) RR ′ (cid:1) − c . (C4)One can get these expressions by evaluating the sum (Eq. 19) directly using the rules of(Eq. A10) or by solving for the exact wavefunctions profiles as described in [32]. For theother contribution ˆ∆ we will get the following expressionˆ∆ u = R ′ h ˆ m u (cid:16) ˆ m † u ˆ K ( c q ) + ˆ K ( − c u ) ˆ m † u (cid:17) m u + ˆ m d ˆ˜ K ( − c d ) ˆ m † d ˆ m u i (C5)where˜ K ( c ) ≡ − (cid:0) R ′ R (cid:1) c − − c − (cid:0) R ′ R (cid:1) − c − c ,K ( c ) ≡ − c − (cid:0) RR ′ (cid:1) c − − (cid:0) RR ′ (cid:1) c − − (cid:0) RR ′ (cid:1) (cid:16)(cid:0) RR ′ (cid:1) c − − (cid:17) (3 − c ) + (cid:0) RR ′ (cid:1) − c − (cid:0) RR ′ (cid:1) (1 + 2 c ) (cid:16)(cid:0) RR ′ (cid:1) c − − (cid:17) (C6)Note that subdominant contribution ∆ is only important for the third generation, and inthe text we denote ∆ t,b to be equal to ( ˆ∆ u,d ) . We assume here that Yukawa couplings are vectorlike Y = Y1