HHIGGS PHYSICS IN WARPED EXTRA DIMENSIONS
FLORIAN GOERTZ
Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨atD-55099 Mainz, Germany
In this talk, I present results for the most important Higgs-boson production cross sections atthe LHC and the Tevatron as well as the branching fractions of the relevant decay channels inthe custodial Randall-Sundrum model. The results are based on a complete one-loop calcu-lation, taking into account all possible Kaluza-Klein particles in the loop. Due to the stronginfrared localization of the top quark and the Kaluza-Klein excitations, the SM predictionsreceive sizable corrections in the model at hand. This could effect Higgs searches significantly.
The Higgs boson represents the last missing ingredient of the Standard Model (SM) of ParticlePhysics. It offers the possibility to give masses to the weak gauge bosons and chiral fermionswithout breaking gauge invariance, which is important for a proper high energy behavior ofthe model. Electroweak precision measurements suggest that the SM Higgs boson is light, m h <
185 GeV at 95% C.L., including the direct Limit m h >
114 GeV from LEP2. Furthermore,theoretical arguments like unitarity, vacuum stability and triviality constrain the allowed rangefor the Higgs mass. In summary, we expect the SM Higgs boson to have a mass well below a TeVand to exhibit tree-level couplings to particles proportional to their mass. Imagine we do notdiscover the Higgs at the Large Hadron Collider (LHC) in the first years of running. Does thisalready mean that we have to abandon the corresponding mechanism of electroweak symmetrybreaking? The answer is certainly no. Beyond the SM physics could feature a standard Higgsmechanism that could be much harder to detect, even for a Higgs mass easily accessible atthe LHC. It is important to study Higgs physics in various models to be prepared for differentpossible scenarios. In this talk, I present results for Higgs production and decay within thecustodial Randall-Sundrum (RS) model with gauge and fermion fields in the (5D) bulk and aninfrared-brane Higgs sector. Here one expects big effects, due to the localization of the fields.
The RS model provides an elegant possibility to address the large hierarchy between theelectroweak scale M EW and the Planck scale M PL by means of a non-trivial geometry in a 5DAnti-de Sitter space. The fifth dimension is compactified on an S /Z orbifold. The RS metric ds = e − kr | φ | η µν dx µ dx ν − r dφ , (1) a r X i v : . [ h e p - ph ] M a y ith η µν = diag(1,-1,-1,-1) is such that length scales within the usual 4D space-time are rescaledby an exponential warp factor, depending on the position φ ∈ [ − π, π ] in the extra dimension. Thecurvature k and inverse radius r − of this dimension are of O ( M PL ). The Z fixed points at φ =0 , π correspond to boundaries: the ultraviolet (UV) and the infrared (IR) 3-branes. The modelsolves the gauge hierarchy problem by suppressing mass scales on the IR-brane. One achieves M IR ≡ e − L M Pl ≈ M EW (2)for L ≡ krπ ≈
36. The strong hierarchy between M PL and M EW is thus understood bygravitational red-shifting, if the Higgs field is localized on or close to the IR brane. The 5Dgauge and fermion fields are decomposed into infinite towers of (massive) 4D fields, featuringprofiles depending on φ , via a Kaluza-Klein (KK) decomposition. The massless zero modes canbecome massive via couplings to the Higgs sector and, given an appropriate setup of the model,they can be interpreted as the SM fields we observe in nature. The compactification of thefifth dimension leads to masses for the tower of KK excitations, which are set by the KK scale M KK ≡ k(cid:15) ∼ O (TeV). The warping of the fundamental Planck scale down to M IR ∼ O (TeV) onthe IR brane results in a cutoff for the RS model at several TeV for amplitudes calculated on thatbrane. At this scale the model is assumed to be UV completed by a theory of quantum gravity.This is important for Higgs physics, because it means that just the exchange of the first KKexcitations should be taken into account for the corresponding observables, while the effect ofthe higher modes is to be cut off. An attractive feature of RS models is the possibility to addressthe quark-mass hierarchies and the structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrixby localizing the fermion zero modes differently in the extra dimension, without any hierarchiesin the input parameters. This anarchic approach to flavor improves the predictivity of themodel, since the localizations of the quarks are now fixed to some extend by their masses andthe CKM parameters. The top quark, being the heaviest quark of the SM, has to reside closeto the IR brane, where also KK excitations tend to live. Due to the large overlap with theseexcitations, one expects the most interesting signatures of RS models in top and Higgs physics. Adirect consequence of the different localizations of fermions are flavor changing neutral currents(FCNCs). The non-universal couplings to massive gauge bosons lead to offdiagonal transitions,after going to the mass basis. Furthermore, the KK masses (which are due to compactification)lead to a misalignment between the mass and the Yukawa matrix which results in modified Higgscouplings for RS models including FCNCs. Our SM assumption for Higgs searches, a couplinggiven by the mass of the corresponding particle, is spoiled in this model. The most optimisticRS predictions for B ( t → cZ ) and B ( t → ch ) are both around 10 − . , Let me finally mentionthat in the minimal
RS model a leading order analysis of the electroweak S and T parametersfavors a heavy Higgs boson m h ∼ .The custodial RS model that provides the framework for the following analysis of Higgs physics,which is based on , features a protection for the T parameter as well as for Zb L ¯ b L couplings. The main production mechanism of the Higgs boson at hadron colliders is gluon-gluon fusion.In the SM, this process is dominated by a top-quark triangle loop. Within RS models, one hasto consider additionally the KK tower of the top quark as well as of all other flavors present inthe theory, which all contribute at the same order. The corresponding Feynman diagrams aregiven on the very left in the top row and bottom row in Figure 2. In order to obtain the gg → h production cross section in the custodial RS model, the SM prediction is rescaled according to σ ( gg → h ) RS = | κ g | σ ( gg → h ) SM , (3) = gg ® hqq ¢ ® Wh M KK =
150 200 250 3000.0010.010.1110 m h @ GeV D Σ @ pb D s =
10 TeV
LHC gg ® h qq H ¢ L ® qq H ¢ L h M KK =
200 300 400 500 6000.010.1110100 m h @ GeV D Σ @ pb D Figure 1: Main Higgs-boson production cross sections at the Tevatron (left) and the LHC (right). The dashedlines represent the SM predictions, while the solid lines correspond to the custodial RS model. See text for details. where κ g = (cid:88) i = t,b κ i A hq ( τ i ) + (cid:88) j = u,d,λ ν j (cid:88) i = t,b A hq ( τ i ) , (4)with τ i ≡ m i /m h . The first sum in the numerator contains top and bottom quark zero modesrunning in the loop with couplings (normalized to the SM values) given by κ t = Re[( g uh ) ] / (cid:18) m t v (cid:19) , κ b = Re[( g dh ) ] / (cid:18) m b v (cid:19) . (5)Here, v ≈
246 GeV is the Higgs vacuum expectation value and m t ( m b ) is the top (bottom)quark mass. The Higgs couplings ( g u,dh ) in the custodial RS model as well as the form factor A hq ( τ i ) can be found in . It is easy to show that in the RS model κ t,b <
1, independent ofthe input parameters, where κ t can become as small as 0.5 for M KK = 2 TeV, which we willalways employ in the following analysis. The second sum in (4) represents the contributionfrom the virtual exchange of KK excitations. The λ quarks, with electromagnetic charge 5 / Zb L ¯ b L vertex. Details on the sums over KK excitations are given in . Note that thecontributions of the first KK levels (after summing the different same-charge flavors within alevel) turn out to decrease quadratically. Thus the extrapolation from these levels to the wholetower, which actually should be cut off, does not change the results significantly. The results forthe Higgs-boson production cross sections at Tevatron and the LHC for center-of-mass energies √ s = 1 .
96 TeV and √ s = 10 TeV are shown in Figure 1. The solid red lines correspond tothe custodial RS expectations, whereas the SM predictions are indicated by dashed lines forcomparison. In addition to gluon-gluon fusion, the plots show (in blue) the predictions for weakgauge-boson fusion, qq ( (cid:48) ) → qq ( (cid:48) ) V ∗ V ∗ → qq ( (cid:48) ) h with V = W, Z , which is an important channelat the LHC, as well as for associated W -boson production, q ¯ q (cid:48) → W ∗ → W h , for the Tevatron.The results have been obtained by an averaging procedure over 10000 sets of input parameters,fitting the quark masses, CKM mixing angles and the phase within 1 σ . The plots show clearlythat the Higgs production cross sections in gluon-gluon fusion experience a strong reduction inthe custodial RS model. This depletion remarkably survives even for M KK = 5 TeV, whichcorresponds to a first KK-gauge boson mass of around 12 TeV, still reaching up to −
40% forboth colliders. The bump in the right plot is due to a destructive interference of the zero modeand KK contributions which becomes most effective for m h ≈ m t . g ht tt ! h b,tb,t ! h W,ZW,Z ! h γγtt t ! h γγWW W ! h Zγtt t ! ! h ZγWW W ! Figure 5: Examples of Feynman diagrams involving zero-mode fields only that con-tribute to the production and the decay of the Higgs boson at leading order of pertur-bation theory. Vertices indicated by a black square can receive sizable shifts in the RSmodel relative to the SM couplings. See text for details.can be parametrized by 1 − a t,b v /M with the coefficients a t,b given in Table 3. The quotedvalues of a t,b have been obtained from the best fits to the shown sample of scatter points.The suppression of the Yukawa couplings of the third-generation quarks, Re κ t,b ≤
1, aswell as the feature | Im κ t,b | m q /v ! (Φ q ) +(Φ Q ) " ≥ Φ q,Q introduced in (137) are absolutesquares. Second, the third term in (136) can be written in the ZMA as(∆˜ g uh ) = 4 m t vM j =1 m uj $ U † u diag % F − ( c Q i ) & U u ’ j $ W † u diag % F − ( c u ci ) & W u ’ j . (166)A similar formula applies to the case of (∆˜ g dh ) . Because the diagonal elements of the matrices U † u diag[ F − ( c Q i )] U u and W † u diag % F − ( c u ci ) & W u are absolute squares, the term with j = 3is obviously positive semi-definite. The terms with j = 1 ,
2, on the other hand, can havean arbitrary complex phase. Yet, due to the strong chiral suppression, m c /m t ≈ /
275 and m u /m t ≈ − , the imaginary part of (166) turns out to be negligibly small, leaving us with(∆˜ g uh ) ≥
0. The same holds true for (∆˜ g dh ) , although the chiral suppression is weaker in thiscase, m s /m b ≈ /
50 and m d /m b ≈ / g qh ) = m q /v ! (Φ q ) +(Φ Q ) " +(∆˜ g qh ) ≥ ht ¯ t and hb ¯ b couplings arepredicted to be suppressed relative to their SM values in both the minimal and the extendedRS models. We believe that this finding is model-independent and holds in a wide class of RSset-ups. The same conclusion has been drawn in the context of models where the Higgs arisesas a pseudo Nambu-Goldstone boson [52, 53].The second term in the numerator of (164) represents the contribution to the gg → h amplitude arising from the virtual exchange of KK quarks. The corresponding Feynman graphis shown on the very left in Figure 6. In the up-type quark sector the associated coefficient48 gg hq ( n ) q ( n ) q ( n ) h γ,Zγq ( n ) q ( n ) q ( n ) h γ,ZγW ( n ) W ( n ) W ( n ) Figure 6: Examples of one-loop contributions involving KK excitations that contributeto the production and the decay of the Higgs boson at leading order of perturbationtheory. See text for details.takes the form ν u = v ∞ ! n =4 ( g uh ) nn m un A hq ( τ un )= 2 π%L ∞ ! n =4 &a U † n C Un ( π − ) " − v M ˜ Y !u ¯ Y † !u S Un ( π − ) &a Un x un A hq ( τ un ) . (167)Similar relations hold in the sector of down-type and λ quarks. Since the mass of the firstKK up-type quark is already much larger than the Higgs-boson mass, m u /M KK = O (a few) " m h /M KK , it is an excellent approximation to replace the function A hq ( τ un ) by its asymptoticvalue of 1 obtained for τ un ≡ m un ) /m h → ∞ .Before presenting our numerical results for these contributions, we would like to add somecomments about the convergence of the sum in (167). In the SM, the top-quark contributionto the gg → h amplitude is proportional to y t /m t in the decoupling limit. In this limit theamplitude can be described by the effective operator h/v G aµν G aµν , whose Wilson coefficientis related to the QCD β -function. This relationship arises through low-energy theorems ap-propriate to external Higgs bosons with vanishing momentum [53–56], which apply to anyquantum field theory. In the context of the RS framework they imply that the sum in (167)must be convergent, because the running of α s can be shown to be logarithmic in warpedextra-dimension models [24, 57–63]. While the finiteness of the effective hgg coupling is thusguaranteed on general grounds, an explicit calculation of (167) in the KK decomposed 4Dtheory turns out to be non-trivial. This is due to the fact that the Higgs VEV induces O (1)mixings between the various modes of a single KK level [21]. For example, in the up-typequark sector there are five types of fields, namely u , u , u c , U , and U . Each of them exists inthree different flavors, so that there are altogether 15 KK modes of similar mass in each level.In the down-type quark sector, one instead ends up with nine modes, while in the minimalRS model one has six states per KK level in both the up- and the down-type quark sectors(corresponding to SU (2) L doublets and singlets). Finally, in the λ -type quark sector one againfaces nine KK excitations per level. In contrast, exotic matter is not present in the minimal With λ quarks we denote all fermionic KK excitations with electric charge 5 / M KK = gg tt bbZZWW ΓΓΓ Z - m h @ GeV D B H h ® f L Figure 2: left: Feynman diagrams for Higgs production and decay, right: Branching ratios for h → f as functionsof the Higgs-boson mass. The solid (dashed) lines indicate the custodial RS (SM) predictions. See text for details. Concerning the decay of the Higgs boson, processes with heavy quarks and gauge bosons in thefinal state can experience significant RS corrections. The corresponding Feynman diagrams aredepicted on the left of Figure 2, where vertices that receive non-negligible corrections are indi-cated by black squares. The analysis works in a similar way as that for Higgs production. Theresults are shown on the right of Figure 2, where the solid (dashed) lines correspond to the RS(SM) predictions. All final states that can feature non-negligible RS corrections and have branch-ing fractions above 10 − are considered. While for Higgs masses below the W W threshold theenhanced branching fraction into two photons could compensate the lower production cross sec-tion in gg → h → γγ , the discovery potential above this threshold is for all channels significantlyworse than in the SM. Most important, the golden channel gg → h → Z ( ∗ ) Z ( ∗ ) → l + l − l + l − suf-fers from the strong reduction in the production cross section. The presented results suggestthat a discovery of the Higgs boson, depending on its mass, could become more difficult in RSmodels. Existing SM bounds on the Higgs mass from the Tevatron and LEP are also altered ifwarped extra dimensions are realized in nature. Furthermore, the effects in Higgs physics shouldbe notable at the LHC, even for KK scales which are by far not directly accessible. Acknowledgments
I would like to thank the organizers of the
Rencontres de Moriond 2011 EW for the wonderfulatmosphere during the conference and the financial support. Furthermore, I want to thankSandro Casagrande, Uli Haisch, Matthias Neubert, and Torsten Pfoh for the nice collaborationon the subject and Julia Seng for carefully proofreading the manuscript.
References
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