Probing warped extra dimension via gg -> h and h -> gamma gamma at LHC
aa r X i v : . [ h e p - ph ] A p r SINP/TNP/2009/06
Probing warped extra dimension via gg → h and h → γγ at LHC Gautam Bhattacharyya and
Tirtha Sankar Ray
Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, India
Abstract
The processes gg → h and h → γγ are of paramount importance in the context of Higgs searchat the LHC. These processes are loop driven and hence could be sensitive to the presence of any newcolored fermion states having a large coupling with the Higgs. Such a scenario arises in a warpedextra dimensional theory, where the Higgs is confined to the TeV brane and the hierarchy of fermionmasses is addressed by localizing them at different positions in the bulk. We show that the Yukawacoupling of the Higgs with the fermion Kaluza-Klein (KK) states can be order one irrespective oftheir zero mode masses. We observe that the gg → h and h → γγ rates are substantially alteredif the KK states lie within the reach of LHC. We provide both intuitive and numerical comparisonbetween the RS and UED scenarios as regards their quantitative impact in such processes. PACS Nos: 04.50.Cd, 11.10.KkKey Words: Higgs boson, Warped Extra Dimension
Introduction : For an intermediate mass ( <
150 GeV) Higgs boson, the relevance of its productionat the CERN Large Hadron Collider (LHC) via gluon fusion ( gg → h ) and its subsequent decay intotwo photons ( h → γγ ) cannot be over-emphasized. Since these are loop induced processes, a naturalquestion arises as how sensitive these processes are to the existence of new physics. In this paper,we explore such a possibility by embedding the Standard Model (SM) in a Randall-Sundrum (RS)warped geometry [1], where the bulk is a slice of Anti-de Sitter space (AdS ) accessible to some or allSM particles [2, 3]. The virtues of such a scenario include a resolution of the gauge hierarchy problemcaused by the warp factor [1], and an explanation of the hierarchy of fermion masses by their respectivelocalizations in the bulk keeping the Higgs confined at the TeV brane [4]. Besides, the smallness of theneutrino masses could be explained [5], and light KK states would lead to interesting signals at LHC[6]. We demonstrate in this paper that the loop contribution of the KK towers of quarks and gaugebosons emerging from the compactification would have a sizable numerical impact on the gg → h and h → γγ rates. This happens because the Higgs coupling to a pair of KK fermion-antifermion is notsuppressed by the zero mode fermion mass and can easily be order one [7]. The underlying reason issimple. Although the zero mode wave-functions of different flavors have varying overlap at the TeVbrane depending on the zero mode masses, the KK profiles of all fermions have a significant presenceat the TeV brane where the Higgs resides. As a result, the KK Yukawa couplings of different flavorsare not only all large, they are also roughly universal. This large universal Yukawa coupling in the RSscenario constitutes the corner-stone of our study. On the contrary, in flat Universal Extra Dimension(UED) only the KK top Yukawa coupling is large, others being suppressed by the respective zero modefermion masses. We provide comparative plots to demonstrate how the warping in RS fares againstthe flatness of UED for the processes under consideration.1 arped extra dimension : The extra coordinate y is compactified on an S / Z orbifold of radius R , with − πR ≤ y ≤ πR . Two 3-branes reside at the orbifold fixed points at y = (0 , πR ). Thespace-time between the two branes is a slice of AdS geometry, and the 5d metric is given by [1], ds = e − σ η µν dx µ dx ν + dy where σ = k | y | . (1)Above, 1 /k is the AdS curvature radius, and η µν = diag( − , , , y = 0 brane is the Planck scale ( M P ), while the effective mass scale associated with the y = πR brane is M P e − πkR , which is of the order of a TeV for kR ≃
12. To address the fermion masshierarchy, the Higgs boson has to be confined to the TeV brane, thus solving the gauge hierarchyproblem in the same stroke. The bulk contains the fermions and gauge bosons. After integrating outthe y -dependence, the 4d Lagrangian can be written in terms of the zero modes and their KK towers.A generic 5d field can be decomposed as (only fermions and gauge bosons are relevant for us) [3]Φ( x µ , y ) = 1 √ πR ∞ X n =0 Φ ( n ) ( x µ ) f n ( y ) , where f n ( y ) = e sσ/ N n h J α ( m n k e σ ) + b α ( m n ) Y α ( m n k e σ ) i , (2)with s = (1 ,
2) for Φ = { e − σ Ψ L,R , A µ } . Above, b α = − ( − r + s ) J α ( m n k ) + m n k J ′ α ( m n k )( − r + s ) Y α ( m n k ) + m n k Y ′ α ( m n k ) , and N n ≃ p π R m n e − πkR , (3)where r = ( ∓ c,
0) and α = ( c ± ,
1) for Z even/odd modes. By imposing boundary conditions on f n ( y ) in Eq. (2) and in the limit m n ≪ k and kR ≫
1, one obtains the KK masses as (for n = 1 , , . . . ), m n ≃ (cid:18) n + 12 | c − | − (cid:19) πk e − πkR (fermions) ; m n ≃ (cid:18) n − (cid:19) πk e − πkR (gauge bosons) . (4)Now, the Yukawa part of the action with two 5d Dirac fermions Ψ iL ( x, y ) and Ψ iR ( x, y ) for each flavor i is given by [3] S y = Z d x Z dy √− g λ ij (5 d ) H ( x ) (cid:16) ¯Ψ iL ( x, y )Ψ jR ( x, y ) + h . c . (cid:17) δ ( y − πR ) . (5)For simplicity we ignore flavor mixing, and further assume c iL = c iR = c i . The Yukawa coupling ofthe zero mode fermions turns out to be [3] λ i = λ i (5 d ) k (1 / − c i ) (cid:16) − e (2 c i − πkR (cid:17) − . (6)Assuming the 5d coupling λ i (5 d ) k ∼
1, one can trade the zero mode fermion masses in favor of thecorresponding c i ( c q = 0 . , . , . , . , − . , .
48 for q = u, d, c, s, t, b ). This is how the fermionmass hierarchy problem is addressed. Next, we derive the Yukawa coupling of the n th KK fermion for m n ≪ k ∼ M P , kR ≫ λ i (5 d ) k ∼ λ ( n ) i ∼ cos (cid:16) " n − | c − | − | c ∓ | − π (cid:17) , (7)where ∓ correspond to Z odd/even KK modes. Thus the KK Yukawa couplings for Z even KKmodes, regardless of their flavors and KK numbers, are roughly equal to unity for the values of c q quoted above. 2 ontribution of KK states to σ ( gg → h ): The process gg → h proceeds through fermiontriangle loops. The SM expression of the cross section is given by ( τ q ≡ m q /m H ) σ SM gg → h = α s πv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X q A q ( τ q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where A q ( τ q ) | SM = 2 τ q [1 + (1 − τ q ) f ( τ q )] , (8)with f ( τ ) = arcsin (cid:18) √ τ (cid:19) for τ ≥ , and f ( τ ) = − (cid:20) ln (cid:18) √ − τ − √ − τ (cid:19) − iπ (cid:21) for τ < . (9)Above, α s is the QCD coupling at the Higgs mass scale, v is the Higgs vacuum expectation value and A q is the loop amplitude from the q th quark. In the SM, the dominant contribution comes from thetop quark loop. Now, there will be additional contributions from the KK quarks. Importantly, dueto the large universal KK Yukawa couplings, not only the KK top but also the KK modes of otherquarks would have sizable contribution. Indeed, the lightest modes ( n = 1) would have dominantcontributions. Setting the KK Yukawa couplings to unity, as suggested by Eq. (7), we derive theamplitude of the n th KK mediation of the q th flavor, with the same normalization of Eq. (8), as A q ( τ q n ) | KK = 4 v m h [1 + (1 − τ q n ) f ( τ q n )] . (10)In 5d the sum over n yields a finite result. Eq. (10) is different from the UED result [8] in two ways:(i) we have set the KK Yukawa coupling to unity irrespective of quark flavors, while in UED the KKYukawa coupling is controlled by zero mode masses; (ii) in UED there is an additional factor of 2because both Z even and odd KK modes contribute, while in RS the odd modes do not couple tothe brane-localized Higgs. In Fig. 1, we have plotted the variation with m h of the deviation of theproduction cross section σ RS ( gg → h ) from its SM expectation σ SM ( gg → h ) normalized by the SMvalue. The dominant QCD correction cancels in this normalization. We have chosen four referencevalues of m KK (= 1 . , . , . . m KK is the KK mass of the n = 1 gauge bosons,which also happens to be the lightest KK mass in the bulk (corresponding to the conformal limit, c = 1 / m h below 150 GeV, the deviation is quite substantial (close to 45%) for m KK = 1 TeV. For larger m KK = 1.5 (3.0) TeV, the effect is still recognizable, around 18% (5%). Inthe inset, we exhibit a comparison between RS and UED contributions to the same observable, wherethe KK mass scales of the two scenarios, namely m KK for RS and 1 /R for UED, have been assumedto be identical (= 1 TeV). For m h <
150 GeV, the RS contribution is about 2.5 times larger than theUED contribution, while the margin slightly goes down with increasing m h . This factor 2.5 can beunderstood in the following way: In RS, five n = 1 KK flavors (except the KK top) have mass around m KK with order one Yukawa coupling. So naively we would expect a factor of 5 enhancement relativeto UED. But in UED both Z even and odd modes contribute. This reduces the overall enhancementfactor in RS over UED to about 2.5. Contribution of KK states to Γ( h → γγ ): The h → γγ process proceeds through fermiontriangles as well as via gauge loops along with the associated ghosts. The decay width in the SM canbe written as Γ h → γγ = αm h π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X f N fc Q f A f ( τ f ) + A W ( τ W ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11)where α is the electromagnetic coupling at the Higgs mass scale. The expression for A f is givenin Eq. (8), and the dominant SM contribution to A f comes from the top quark loop. The W -loop3 ( σ R S gg → h - σ S M gg → h ) / σ S M gg → h m h [GeV]1 TeV1.5 TeV2 TeV3 TeV UEDRSm kk = 1/R = 1 TeV Figure 1:
The fractional deviation (from the SM) ofthe gg → h production cross section in RS is plottedagainst the Higgs mass. The four curves correspond tofour different choices of m KK . In the inset, we havecompared the UED contribution for /R = 1 TeV withthe RS contribution for m KK = 1 TeV. -0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 100 150 200 250 300 350 400 450 500 ( Γ R S h → γγ - Γ S M h → γγ ) / Γ S M h → γγ m h [GeV] RSUEDm kk = 1/R = 1 TeV Figure 2:
Same as in Fig. 1, except that the fractionaldeviation in h → γγ decay width is plotted. amplitude in the SM is given by A W ( τ W ) | SM = − [2 + 3 τ W + 3 τ W (2 − τ W ) f ( τ W )] . (12)We derive the KK contribution of the gauge sector as A W ( τ W n ) | KK = − [2 + 3 τ W + 3 τ W (2 − τ W n ) f ( τ W n ) − τ W n − τ W ) f ( τ W n )] . (13)Again, the sum over n yields finite result and in the limit of large KK mass the KK contributiondecouples. Our Eq. (13) is very different from the corresponding UED expression [8], primarily becausethe Higgs is confined at the brane in the present scenario while it resides in the bulk in UED. In Fig. 2,we have plotted the decay width Γ( h → γγ ) in RS relative (and normalized as well) to the SM. Again,the four choices of m KK are 1.0, 1.5, 2.0 and 3.0 TeV. There is a partial cancellation between quarkand gauge boson loops, both in real and imaginary parts, not only for the zero mode but also for eachKK mode. The meeting of the four curves just above the m h = 2 m t threshold is a consequence of theabove cancellation and at the meeting point the SM contribution overwhelms the KK contribution.Unlike in Fig. 1, we witness both suppression and enhancement with respect to the SM contribution.The inset carries an illustration how RS fares against UED for identical KK masses.Next we construct a variable R = σ gg → h Γ h → γγ . In Fig. 3, we have studied variation of ( R RS − R SM ) /R SM with m h . For m KK = 1.0, 1.5, 2.0 and 3.0 TeV, the fractional changes in R are 30%,14%, 8% and 4%, respectively, for m h <
150 GeV. The comparison shown in the inset shows that RSwins over UED roughly by a factor of 2 for identical KK scale for m h <
150 GeV. Incidentally, ourUED plots in the insets of all the three figures are in complete agreement with [8]. See also [9] for anumerical simulation of the Higgs signal at LHC in the UED context.
Conclusions : In conclusion, we highlight the core issues: In the RS scenario, the brane-boundHiggs can have order one Yukawa coupling with the KK fermions of all flavors. Such large KKYukawa couplings can sizably enhance the Higgs production via gluon fusion and alter the Higgs decay4 ( R R S - R S M ) / R S M m h [GeV]1 TeV1.5 TeV2 TeV3 TeV UEDRSm kk = 1/R = 1 TeV Figure 3:
Same as in Fig. 1, except that the fractional deviation in R = σ gg → h Γ h → γγ has been plotted. width into two photons, provided the KK masses are in a regime accessible to the LHC. Because ofthe proactive involvement of more flavors inside the loop the effect in RS is significantly stronger(typically, by a factor of 2 to 2.5) than in UED for similar KK masses. Admittedly, this advantagein RS is somewhat offset by the fact that the lightest KK mass in UED can be as low as 500 GeVthanks to the KK-parity, while in RS a KK mass below 1.5 TeV would be difficult to accommodate(see below). However, attempts have been made to impose KK parity in warped cases as well [10].Electroweak precision tests put a severe lower bound on m KK ( ∼
10 TeV) [11]. To suppressexcessive contribution to T and S parameters the gauge symmetry in the bulk is extended toSU(2) L × SU(2) R × U(1) B − L , and then m KK as low as 3 TeV can be allowed [12, 13]. A further discretesymmetry L → R helps to suppress Zb L ¯ b L vertex correction and admits an even lower m KK ∼ . T and S by partial cancellation, m KK ∼ m KK inthe range of 1-3 TeV chosen for illustration may arise in the backdrop of such extended symmetries.Furthermore, if the b ′ quark, present in the case of left-right gauge symmetry, weighs around 1 TeV,one obtains an additional ∼
10% contribution to σ ( gg → h ) [15].A very recent paper [16] lists the relative contribution of different scenarios (supersymmetry, flatand warped extra dimension, little Higgs, gauge-Higgs unification, fourth generation, etc.) to gg → h and h → γγ for some benchmark values. A comparison between their work and ours in order. Asregards the RS scenario, the authors of [16] consider the region of parameters where the zero modequarks mix with their KK partners. Additionally, their choice of c L is substantially different from c R ,where they observe large destructive interference in the effective ggh coupling. On the other hand, ourworking hypothesis is based on: c ≡ c L = c R (see Eq. (6)), and we assume KK number conservationat the Higgs vertex. We observe that the Higgs coupling to KK quarks is large for any flavor (seeEq. (7)), and the (direct) loop effects of the KK quarks (which carry the same quantum numbers astheir zero modes) do enhance the effective ggh vertex (like the enhancement observed for the fourthfamily contribution [16], or the b ′ quark contribution [15], or the UED contribution [8, 16]), and themagnitude is rather insensitive to the value of c as long as | c | ∼ > .
5. The authors of [17] also calculate5he KK-induced effective ggh vertex, but they rely on the gauge-Higgs unification set-up, and hencean efficient numerical comparison of their work with ours is not possible.
Acknowledgments:
TSR acknowledges the support (S.P. Mukherjee fellowship) of CSIR, India.GB’s research is partially supported by project no. 2007/37/9/BRNS (DAE), India. We thank G.Cacciapaglia for making valuable comments.
References [1] L. Randall and R. Sundrum, Phys. Rev. Lett. (1999) 3370 [arXiv:hep-ph/9905221].[2] W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. (1999) 4922 [arXiv:hep-ph/9907447];W. D. Goldberger and M. B. Wise, Phys. Lett. B (2000) 275 [arXiv:hep-ph/9911457];H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Lett. B (2000) 43[arXiv:hep-ph/9911262]; A. Pomarol, Phys. Lett. B (2000) 153 [arXiv:hep-ph/9911294];S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, Phys. Rev. D (2000) 084025.[arXiv:hep-ph/9912498]; A. Pomarol, Phys. Lett. B (2000) 153 [arXiv:hep-ph/9911294].[3] T. Gherghetta and A. Pomarol, Nucl. Phys. B (2000) 141 [arXiv:hep-ph/0003129].[4] S. J. Huber and Q. Shafi, Phys. Lett. B (2001) 256 [arXiv:hep-ph/0010195].[5] Y. Grossman and M. Neubert, Phys. Lett. B (2000) 361 [arXiv:hep-ph/9912408].[6] K. Agashe, S. Gopalakrishna, T. Han, G. Y. Huang and A. Soni, arXiv:0810.1497 [hep-ph];K. Agashe et al. , Phys. Rev. D (2007) 115015 [arXiv:0709.0007 [hep-ph]]; K. Agashe,A. Belyaev, T. Krupovnickas, G. Perez and J. Virzi, Phys. Rev. D (2008) 015003[arXiv:hep-ph/0612015]; K. Agashe, G. Perez and A. Soni, Phys. Rev. D (2007) 015002.F. Ledroit, G. Moreau and J. Morel, JHEP (2007) 071 [arXiv:hep-ph/0703262]; A. Djouadi,G. Moreau and R. K. Singh, Nucl. Phys. B (2008) 1 [arXiv:0706.4191 [hep-ph]].[7] G. Bhattacharyya, S. K. Majee and T. S. Ray, Phys. Rev. D (2008) 071701 [arXiv:0806.3672[hep-ph]].[8] F. J. Petriello, JHEP (2002) 003 [arXiv:hep-ph/0204067].[9] S. K. Rai, Int. J. Mod. Phys. A (2008) 823 [arXiv:hep-ph/0510339].[10] K. Agashe, A. Falkowski, I. Low and G. Servant, JHEP (2008) 027.[11] J. L. Hewett, F. J. Petriello and T. G. Rizzo, JHEP (2002) 030; C. Csaki, J. Erlich andJ. Terning, Phys. Rev. D (2002) 064021.[12] K. Agashe, A. Delgado, M. J. May and R. Sundrum, JHEP (2003) 050[arXiv:hep-ph/0308036].[13] C. Bouchart and G. Moreau, Nucl. Phys. B (2009) 66 [arXiv:0807.4461 [hep-ph]].[14] M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Phys. Rev. D (2007) 035006[arXiv:hep-ph/0701055]; M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Nucl. Phys.B (2006) 202 [arXiv:hep-ph/0607106]. 615] A. Djouadi and G. Moreau, Phys. Lett. B (2008) 67 [arXiv:0707.3800 [hep-ph]].[16] G. Cacciapaglia, A. Deandrea and J. Llodra-Perez, arXiv:0901.0927 [hep-ph].[17] A. Falkowski, Phys. Rev. D (2008) 055018 [arXiv:0711.0828 [hep-ph]]; N. Maru and N. Okada,Phys. Rev. D77