Collider Bounds on Lee-Wick Higgs Bosons
ZZU-TH-06/11
Collider Bounds on Lee-Wick Higgs Bosons
Ezequiel Alvarez
CONICET, IFIBA, INFAP and Departamento de F´ısica,FCFMN, Universidad Nacional de San Luis,Av. Ejercito de Los Andes 950, San Luis, Argentina
Estefania Coluccio Leskow
Departamento de F´ısica, FCEyN, Universidad de Buenos Aires,Ciudad Universitaria, Pab. 1, (1428) Buenos Aires, Argentina
Jos´e Zurita
Institut f¨ur Theoretische Physik, Universit¨at Z¨urich,Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland.
Abstract
We study the constraints on the Lee-Wick Higgs sector arising from direct collider searches. Wework in an effective-field theory framework, where all of the Lee-Wick partners are integrated out,with the sole exception of the Lee-Wick Higgs bosons. The resulting theory is a two-Higgs doubletmodel where the second doublet has wrong-sign kinetic and mass terms. We include the boundscoming from direct Higgs searches at both LEP and Tevatron using the code
HiggsBounds , andshow the currently excluded parameter space. We also analyze the prospects of LHC Run-I, findingthat with a total integrated luminosity of 5 fb − and a center-of-mass energy of 7 TeV, most ofthe parameter space for the SM-like CP-even Higgs will be probed. PACS numbers: 14.80.Cp a r X i v : . [ h e p - ph ] J u l . INTRODUCTION The Lee-Wick Standard Model (LWSM) [1] is a recent proposal to solve the hierarchyproblem. The model is based on ideas of Lee and Wick [2, 3] which were originally conceivedfor QED regularization in the late 60’s. In the most simple picture, the LWSM key featureconsists in adding a Lee-Wick (LW) partner for each Standard Model (SM) field, but withlarger mass and wrong sign kinetic term. Henceforth, without modifying the low energyphenomenology, partial cancellations in loop corrections eliminate quadratic divergences inthe Higgs mass.One of the main controversies in the Lee-Wick ideas is, at the same time, one of themost challenging and attractive points of the theory: the LW-partners fields seem to violatemicroscopic causality. Despite the alarm this statement may provoke at a first glance,the theory has remained robust during these last years. As a matter of fact, the theoryrespects macroscopic causality, since LW resonances are only intermediate unstable statesand therefore the resulting theory is unitary at least at tree level [1]. Moreover, it has beenproved that a O ( N ) Lee-Wick theory is unitary at all orders [4], and it has also been shownthat violation of microscopic causality does not imply any paradox as far as this violation ismicroscopic enough [5]. It is fair also to recall at this point that there has never been foundany inconsistency in the original Lee-Wick ideas, but, instead, they lost the due attentionbecause of the seminal paper of ’t Hooft and Veltman which finally solved the regularizationand renormalization of gauge theories [6].Several phenomenological and theoretical aspects of the LWSM have been studied in thelast few years [4, 7–27]. In particular, electroweak precision measurements [19–23] force theLW gauge bosons and quark masses (assuming Minimal Flavor Violation) to be around afew TeV. The LW leptons are only constrained by direct detection through pair production,and thus the bound is about 100 GeV [28].On the other hand, the scalar sector of the theory has been indirectly constrained ina practical and efficient article by Carone and Primulando [25]. In this paper the authorsobtain bounds on the LW Higgs sector through the one-loop contributions of Higgs bosons toprocesses such as B -meson mixing, b → X s γ and Z → b ¯ b . Despite being formally one-loopsuppressed, these kind of observables are enhanced by large Yukawa couplings and are verywell measured. Their combined analysis yields a bound to the non SM-like Lee-Wick Higgs2cale of around 400 GeV.In this article, as a complement of the indirect analysis performed in Ref. [25], we studythe constraints to the LW Higgs sector arising from direct searches at LEP, Tevatron andLHC. Although this work achieves better present bounds than the indirect study only ina minor part of the phase space, it settles the perspectives for direct detection of the LWHiggs sector. Moreover, we show that after LHC run I, the non SM-like Higgs parameterspace should be mostly probed up to the TeV scale, which will improve considerably thepresent indirect limits.On a first step, we implement the present collider direct constraints from LEP and Teva-tron with the help of the code HiggsBounds [29–33], and study the exclusion of the parameterspace. This computer code allows to analyze simultaneously several LEP and Tevatron di-rect search signals and choose the best one to constrain a given point in parameter spaceat a 95 % C.L. . In a second step, we include the exclusion projections from the ATLAScollaboration, finding that the LHC with a center-of-mass energy of 7 TeV and an integratedluminosity of 5 fb − (expected to be achieved by the end of 2012) would be able to probemost of the parameter space for the lightest Higgs mass. We also focus on the heavy Higgsbosons, showing the prospect for their detectability at the LHC Run I.This paper is organized as follows: In Section II we review the LWSM Higgs sector, focus-ing on the couplings and relevant branching fractions of the Higgs bosons. We summarizeand analyze the input needed to run HiggsBounds . In Section III we obtain the main directbounds to the Higgs sector coming from LEP and Tevatron signal analysis. Afterwards,we study the LHC perspectives for direct search in the
W W and γγ channels. Section IVcontains our conclusions and final remarks. II. THE LEE-WICK STANDARD MODEL HIGGS SECTOR
In this section we write down and discuss the LWSM Higgs sector in the physical basis andcompute all the collider-relevant effective couplings which involve scalars. These effectivecouplings are used in next section to constrain the parameter space of the model using allthe suitable results from LEP and Tevatron with the aid of
HiggsBounds .3 . Theoretical setup As mentioned in the Introduction, the LW Lagrangian can be written in two equivalentways. One of them introduces none additional fields, but involves higher derivatives (HD)which give additional degrees of freedom to the theory. The other one introduces the newdegrees of freedom explicitly as new partner fields for each SM field. This second formulation–the LW picture– is more clear in dealing with collider experiments, since the contribution ofthese new fields to the processes can be understood in terms of the usual Feynman diagrams.Along this work the LW formulation will be employed.We now focus on the LWSM Higgs sector Lagrangian. (The Lagrangian and details of thefull LWSM can be found elsewhere [1, 21, 22].) Given the stringent bounds on the LW gaugebosons, M , M (cid:38) . (cid:38) m ˜ h ± (cid:38)
450 GeV, coming from b → sγ bounds [25], it is suitable to decouple from the theorythe LW gauge bosons and quarks and to keep only the LW scalars. Notice that although theLW leptons cannot be constrained from EWPT and their only mass limit comes from directsearch, m leptons >
100 GeV [28], they can be safely decoupled due to their tiny Yukawacouplings to the Higgs sector. We may also neglect mixing between SM and LW leptons.Within these assumptions the LWSM Higgs sector reads before symmetry breaking and inthe flavor basis, L Higgs = ( D µ H ) † D µ H − ( D µ ˜ H ) † D µ ˜ H + M ˜ H † ˜ H − V ( H − ˜ H ) + (cid:16) g iju ¯ u iR ( H − ˜ H ) (cid:15)Q jL − g ijd ¯ d iR ( H † − ˜ H † ) Q jL − g ije ¯ e iR ( H † − ˜ H † ) L jL + h . c . (cid:17) , (1)where V ( X ) = − m X † X + λ X † X ) , (2)the Q jL and E jL are the SU (2) L doublets for the j th family of quarks and leptons, respectively,and g ijX the corresponding Yukawa coupling before diagonalization. Since we have decoupledthe LW gauge bosons from the theory, D µ is the usual SM SU (2) × U (1) covariant derivative.The Lagrangian L Higgs corresponds to a two-Higgs doublet model (2HDM) which is dif-ferent from the usually found in the literature, mainly due to the opposite sign in the kineticterms of the second Higgs. Notice also that contrary to the usual type I and type II models,here both Higgs bosons couple to all fermions. However, as first observed in Ref. [25], after4lectroweak symmetry breaking it is possible –and useful– to identify some pieces of this2HDM Lagrangian with the MSSM model for different values of tan β .The diagonalization to the physical basis is driven in two steps after electroweak symmetrybreaking. First, one diagonalizes the scalar fields in the quadratic part of the kinetic pluspotential Lagrangian (first line in Eq. (1)), and second, the fermion basis is diagonalized inthe Yukawa interactions (second line in Eq. (1)).Spontaneous symmetry breaking is understood as usual through the Higgs potential (lasttwo terms in the first line of Eq. (1)). In this case, the minimization conditions forbid ˜ H to acquire a vacuum expectation value, leaving the usual SM relation m = λv / v ≈
246 GeV is the vacuum expectation value of H . The expressions for the electroweakgauge boson masses are exactly the same as in the SM. In the unitary gauge one can write H = 1 √ v + h , ˜ H = 1 √ √ h + ˜ h + i ˜ P . (3)The spectrum consists of five Higgs fields: two neutral CP-even Higgs, ( h and ˜ h ), one neutralCP-odd Higgs ˜ P and two charged bosons, ˜ h ± .After electroweak symmetry breaking one can diagonalize the scalar sector. The neutralCP-even bosons mix with each other via a symplectic transformation h ˜ h = cosh θ sinh θ sinh θ cosh θ h ˜ h , (4)where the subscript 0 is reserved for the mass eigenstates. The mass eigenvalues are givenby m h , ˜ h = M (cid:0) ∓ (cid:114) − m M (cid:1) . (5)and the mixing angle factors bycosh θ = 1(1 − r ) / , sinh θ = − r (1 − r ) / , r ≡ m h m ˜ h . (6)At tree level, both ˜ P and ˜ h ± are degenerate in mass, and the following sum rule holds m h + m h = m P = m h ± = M . (7)Note that the model implies the following hierarchy in the spectrum: m h ≤ m ˜ h < m ˜ P .5n order to diagonalize the Yukawa interactions in Eq. (1), since we have decoupled LWfermions, the usual SM rotation in the quark fields shall be performed. Henceforth, theresulting Yukawa fermion-scalars interacting Lagrangian may be divided in a neutral scalarand charged scalar Lagrangian. The neutral scalar Lagrangian is as the usual SM Lagrangianreplacing h → h − (˜ h + i ˜ P ) . (8)The charged scalar Lagrangian for the quark fields is L ˜ h ± f ¯ f = √ v (cid:104) ˜ h + (¯ u R M u V d L − ¯ u L M d V d R ) + ˜ h − (cid:0) − ¯ d R V † M d u L + ¯ d L V † M u u R (cid:1) , (cid:105) , (9)where V is the SM CKM matrix and M u,d are the diagonal mass matrices. Notice thatEq. (9) is up to an overall minus sign, since ˜ h ± comes from ˜ H doublet, the same result thatone obtains in a general type II 2HDM for tan β = 1. To obtain the physical basis interaction h and ˜ h shall be rotated to their physical counterparts h and ˜ h through Eq. (4).Having diagonalized the Higgs sector we now write down the relevant effective couplingsof these scalars to fermions, to gauge bosons, and between them. The customary notation issuch that g φXY denotes the effective coupling of any Higgs boson φ to the fields specified inthe string XY , normalized to the SM case for neutral Higgs bosons, and to charged would-beGoldstone bosons for the charged Higgs. If the process does not have an analog in the SM,then the normalization will be defined in each case.
1. The Kinetic and potential couplings
When the kinetic and potential terms (first line in Eq. (1)) are rotated to the physicalbasis, gauge bosons to scalar and trilinear scalar couplings, among others, show up.We first turn our attention to the interactions between scalars and gauge bosons. Sincequartic interactions are not relevant for the LHC collider phenomenology, we are left toconsider trilinear couplings. One has to keep in mind that in the interaction basis only H isable to couple to a vector boson pair since v (cid:54) = 0 . Using h = cosh θh + sinh θ ˜ h , one getsthat the ratio of the LWSM coupling to the usual SM coupling is g h V V = cosh θ , g ˜ h V V = sinh θ . (10)6he two other relevant interaction are x ˜ P Z and x ˜ h ± W ∓ , where x is a CP-even Higgs boson.They come entirely from ˜ H , and therefore one gets g h ˜ P Z = g h ˜ h ± W ∓ = − sinh θ , g ˜ h ˜ P Z = g ˜ h ˜ h ± W ∓ = − cosh θ , (11)where, in the absence of a SM reference process, we have normalized to g /c W = e/c W s W .Given the hierarchy of the mass spectrum, the only kinematically allowed decay modesinvolving these couplings are ˜ P → h / ˜ h Z and ˜ h ± → h / ˜ h W ± . The corresponding decaywidths read Γ( ˜ A → V x ) = G F √ πm A g x ˜ AV λ / ( m x , m V , m ˜ A ) , (12)where the kinematic factor λ is given by λ ( m , m , m ) = ( m + m − m ) − m m . (13)and ˜ A = ˜ P , ˜ h ± , V = Z, W ± and x = h , ˜ h .For the trilinear Higgs interactions, all the couplings depend only on the single factor λv r ) m h v = r (1 + r ) m h v . (14)The final expression for the absolute trilinear coupling of three generic Higgs bosons xyz isgiven by g xyz = − λv c g xf ¯ f g yf ¯ f g zf ¯ f , (15)where c = 2 if there are charged Higgs bosons, c = 3 for h h ˜ h and h ˜ h ˜ h , c = 1 for allother cases, and g xf ¯ f are given below. The CKM matrix shall be set to unity in this formula.For instance, g h ˜ h + ˜ h − = λv (sinh θ − cosh θ ). The only possible decay involving three Higgsbosons, due to the hierarchy in the spectrum and the CP quantum numbers, is ˜ h → h h .The decay width in this channel readsΓ(˜ h → h h ) = 18 πm ˜ h √ − r g h h h , with g ˜ h h h = 3 λ v g h f ¯ f , (16)where the corresponding symmetry factors due to the indistinguishability of the h bosonshave been added.Quartic Higgs interactions are not relevant for collider phenomenology. However, the h ˜ h + ˜ h − and ˜ h ˜ h + ˜ h − couplings are a necessary ingredient for the decay of neutral Higgsbosons into γγ and γZ , as will be explained in Subsection II A 3. . The Yukawa couplings The Yukawa couplings for the neutral Higgs are easily derived by performing the replace-ment of Eq. (8) in the usual SM Lagrangian and then rotating the fields through Eq. (4).One gets g h f ¯ f = − g ˜ h f ¯ f = cosh θ − sinh θ = 1 + r √ − r , g ˜ P f ¯ f = − . (17)In the charged Yukawa interactions the most relevant coupling occurs in the third gener-ation where the new channel t → ˜ h + b opens for the top quark decay in case of a light Higgs.Neglecting the bottom mass, the width for this decay isΓ( t → ˜ h + b ) = G F √ π m t (1 − m h ± m t ) . (18)
3. The loop-mediated effective couplings
The aim of this subsection is to compute the relevant information concerning the loop-mediated effective couplings of Lee-Wick Higgs bosons to two gluons, two photons, and aphoton and a Z . Effective couplings of h to gluons and photons were already presented in[14]. • Effective coupling of Higgs and two gluons
The gluonic case is straightforward, because it involves a single diagram: the trianglefermion loop. Since we are decoupling the LW top quark and the other fermions maybe ignored due to their suppressed Yukawa coupling, the only fermion we keep in theloop is the top. Therefore, the only modification with respect to the SM case is simplythe change in the top Yukawa coupling, which can be read from Eq. (17). Hence, g h gg = g h f ¯ f = − g ˜ h gg = − g ˜ h f ¯ f .For ˜ P the loop function for fermions is different with respect to the one for CP-evenHiggs bosons (see Appendix A) due to the CP-odd nature of ˜ P . Therefore one has g P gg = σ ( gg → ˜ P ) σ SM ( gg → H ) = | g ˜ P t ¯ t F ˜ P / ( β t ˜ P ) F / ( β t ˜ P ) | , (19)where β t ˜ P = (2 m t /m ˜ P ) , the SM cross-section is evaluated at m H = m ˜ P , and the loopfunctions are defined in Appendix A. 8 Effective coupling of Higgs and two photons
In the photon case, in addition to the top quark loop discussed for the gluon case, onehas to consider also W ± and charged Higgs in the loops, including the proper relativescaling factors for each amplitude.For the top amplitude, one has to multiply the loop function F / by N c Q top and bythe relative coupling g xt ¯ t , where x = h , ˜ h , ˜ P . The W loop amplitude has to be simplymultiplied by g xV V . Finally, the charged Higgs loop contribution will include the F function, an extra factor of m W /m h ± , and the g x ˜ h + ˜ h − coupling. This yields g xγγ = g xt ¯ t N c Q t F x / ( β tx ) + g xV V F ( β Wx ) + g x ˜ h +˜ h − m W /v m W m h ± F ( β h ± x ) N c Q t F / ( β tx ) + F ( β Wx ) , (20)where N c = 3, Q t = 2 / F h , ˜ h / = F / . Notice that since g ˜ P V V = g ˜ P ˜ h + ˜ h − = 0 thenonly the top loop contributes to the numerator of the g ˜ P γγ coupling. Our expressionfor g h γγ agrees with the ones in Refs. [14] and [34] . • Effective coupling of Higgs to Z and photon
The last of the loop-mediated interactions is the decay of a neutral Higgs boson into aphoton and a Z boson. The corresponding MSSM formulae can be found in AppendixC. of Ref. [35]. Since this process can be obtained simply by replacing a photon bya Z boson with respect to the γγ channel, from our previous discussion one alreadyknows all the required relative factors. However, it is worth noting that the presenceof the Z yields rather cumbersome expressions which read g xZγ = g xt ¯ t A xt ( β tx , β tZ ) + g xV V A W ( β Wx , β WZ ) + g x ˜ h +˜ h − m W /v A ˜ h ± ( β h ± x , β h ± Z ) A h t ( β tx , β tZ ) + A W ( β Wx , β WZ ) . (21)Again, notice that for ˜ P only the top loop contributes in the numerator. B. Phenomenological qualitative analysis
In this section we analyze some qualitative features of the LWSM Higgs sector abovepresented in order to understand how the different Higgs bosons will face the direct searches These two groups do not agree between them at the NLO level, however, since we are decoupling the LWpartners and working at LO, we do agree with both of them at this level.
HiggsBounds g h V V , g ˜ h V V and g h f ¯ f (which is equal to g h gg = − g ˜ h gg = − g ˜ h f ¯ f ) as a function of their only variable, r . Both g h gg and g h V V are greater than one, and this has direct implications in colliderphysics bounds: at LO, the SM-like Higgs production cross sections in all relevant channelsat LEP, Tevatron and LHC (gluon fusion, vector boson fusion, Higgs-strahlung, associatedproduction with gauge bosons and/or heavy quarks, bottom fusion) are always larger thatthe SM ones. In particular, this implies that the LEP 114 . h , as argued in Ref. [25]. (a) (b)FIG. 1. (a) From top to bottom, relative couplings of the neutral SM-like Higgs to fermions g h f ¯ f (which is equal to g h gg = − g ˜ h gg = − g ˜ h f ¯ f ), to gauge bosons g h V V , and of the LW CP-even Higgsto gauge bosons g ˜ h V V , as a function of the ratio of the physical masses, r . Notice the increasein the couplings for r →
1, while for r → h behaves as the SM Higgs. We also note that g h f ¯ f > g h V V and g h V V < g h V V hold. (b) g gg ˜ P as a function of m ˜ P in GeV. Notice that the σ ( gg → ˜ P ) cross section is always greater than the corresponding value for a SM (CP-even) Higgsof the same mass, and that this relative coupling peaks at the value 2 m t . The effective couplings in Fig. 1a are monotonic functions of the variable r . Moreover,10heir values blow in the limit r →
1. However, in this limit one enters in the non perturbativeregime: the strongest constrain comes from requiring that the top Yukawa coupling squaredis smaller than 4 π , which yields r < . r , the effective couplingsare g h f ¯ f = 3 . g h V V = 1 .
91 and g ˜ h V V = − . g h V V − g h V V = 1 , (22)while the general 2HDM result involves a + sign. This means that in the LW model both CP-even Higgs bosons can have sizable couplings to the gauge bosons (and even be largerthan one), in contrast with the most general 2HDM result. In the limit r →
0, the LW Higgsdoublet ˜ H decouples. Hence, it is not a surprise to see that, in this region, the h couplingstend to the SM value, while ˜ h is gaugephobic.Another interesting point to note is that the following inequality holds g h gg > g h V V > g h V V . (23)This result has a strong implication for hadron-collider phenomenology. At the Tevatronand the LHC, the different production modes will scale with either the fermion/gluon (gluonfusion, associated top production, bottom fusion) or the vector boson coupling (VBF, Hig-gstrahlung, etc). Given that the gluon fusion mode has the largest cross section, this impliesthat the effective-gluon coupling squared is actually an upper bound for the enhancementfactor of the total production cross section.In Fig. 1b we show the relative effective coupling of the CP-odd LW Higgs to gluons asa function of m ˜ P . We see that, as expected, the relative cross section peaks at m ˜ P = 2 m t .Although we do scan this region in parameter space in next section, this peak is alreadyoutside of the allowed region in Ref. [25]. We find that values for g P gg are 4 . .
6) for m ˜ P = 463 (1000) GeV.As for the γγ and Zγ decay modes we will only consider in this discussion the case of h , since the corresponding branching fractions are only relevant in the 120 −
160 GeV massrange, where according to Ref. [25] neither ˜ P nor ˜ h lie. In Fig. 2 we show contour plots ofthe absolute squared value of g h γγ and g h Zγ in the m h , m ˜ h plane. For the case of γγ , wesee that the coupling squared varies by, at most, 10 %. We have explicitly checked that in11he region of parameter space not excluded from the analysis of Ref. [25], the actual value isalways smaller than one. The Zγ presents a variation of, at most, 1% in this same region. m h H GeV L m h Ž H G e V L (a) m h H GeV L m h Ž H G e V L (b)FIG. 2. Contour plots of | g h γγ | (a) and | g h Zγ | (b) in the m h , m ˜ h plane. Notice that bothquantities are generally below 1. For the γγ case, one finds that in the region of parameter spacenot excluded by the analysis of Ref. [25], the relative coupling squared is strictly below one, whilefor the Zγ case the value would not depart much from unity. From the previous paragraphs, concerning the neutral Higgs effective couplings, one canwithdraw important information about the branching ratios. It can be seen that from thedifferent partial widths which scale with g h gg , g h V V , g h γγ and g Zγ , the f ¯ f and gg modesscale with a larger factor than the other ones. Therefore, the branching fraction in the f ¯ f and gg channels will be larger than in the SM, while the remaining will have a branchingratio lower than the SM value. Hence, the observable quantity production cross sectiontimes branching ratio (henceforth called rates ) can be, in principle, either suppressed orenhanced with respect to the SM and each rate should be studied separately.The charged LWSM Higgs phenomenological analysis for the colliders search is not asappealing as the neutral one due to the conjunction of the Higgs sum rule in Eq. 7 and theLEP and Tevatron processes included in the HiggsBounds routine. At LEP, one considersthe process e + e − → ˜ h + ˜ h − , where m ˜ h ± ≤
94 GeV. At the Tevatron, the charged Higgsis produced via t → ˜ h + b , with ˜ h ± decaying into either τ ν τ or c ¯ s . For these analyses,12 ˜ h ± ≤
155 GeV, and one needs the branching ratios of the top for the ˜ h + b and W + b channels. The region of parameter space being tested is m h ≤
110 GeV, due to the Higgsmasses sum rule. We note that, despite the aforementioned region is excluded by both theanalysis of Ref. [25] and the LEP bound on m h , we still implement our charged Higgs sectorinto HiggsBounds , in order to provide a cross-check on our results.One of the main features of a light charged Higgs sector is the opening of a new decaychannel for the top quark, t → h + b . In order to analyze the available experimental dataon charged Higgs searches one needs the branching ratios of the top to the h + b and W + b channels. In the LWSM it is easy to see that the former channel is only important for Higgsmasses below ∼
90 GeV which is already ruled out by direct LEP searches [36].The charged Higgs decays can be divided into three groups, according to whether thedecay is leptonic ( τ + ν τ , µ + ν µ ), hadronic ( u i ¯ d j ) or into another Higgs boson plus a gaugeboson.The leptonic case is the easiest, since we are ignoring neutrino masses, and as such flavormixing in the lepton sector. The partial width into this channel is given byΓ( H + → (cid:96) + ν (cid:96) ) = G F √ π m h + x (1 − x ) . (24)where (cid:96) is the leptonic family index and x = m (cid:96) /m h + .The hadronic case is slightly different due to the presence of the CKM matrix and to thefact that both the up and down-type fermions have non-zero masses. The LO partial widthis given byΓ( H + → u i ¯ d j ) = 3 G F √ π | V ij | m h + (cid:112) (1 − x − y ) − xy [(1 − x − y )( x + y ) − xy ] . (25)where x = m u i /m h + , y = m d j /m h + . The square root term comes from the 1 → W , is givenby Γ(˜ h ± → xW ± ) = g πm W m h ± g x ˜ h ± W ∓ λ / ( m W , m x , m ˜ h ± ) . (26)Before closing this section, we would like to stress that since the one-loop corrections tothe production and decay of the SM Higgs boson, due to the Higgs boson itself, are known13o be very small, and these corrections are not much different to those in the LWSM withinthe approximations used in this work, we expect our tree expressions to hold with sufficientaccuracy at higher orders. III. NUMERICAL RESULTS
In this section we use the above results in order to run the computer code
HiggsBounds (HB) to confront the LWSM Higgs sector against the available LEP and Tevatron data onHiggs search.The code HB, roughly speaking, uses the information from the previous section to com-pute, for each point in parameter space, the LWSM cross-section of the different Higgs searchsignal topologies. Since HB is an exclusion code, the program chooses, for each point in pa-rameter space, the LEP or Tevatron channel in which the ratio between the predicted LWSMcross-section and the background cross-section for the signal is maximized. Therefore, thischannel is the most sensitive in order to exclude the model in that point. If the predictedLWSM cross-section of the selected process is greater than the observed cross-section thenHB decides that the given point in parameter space is excluded at a 95% C.L.. A detailedexplanation on the HB code may be found elsewhere [29–33].Since HB works under the narrow width approximation, there are two requirements tobe fulfilled by the points in parameter space to be tested. On one hand, all Higgs bosonsmass and width shall fulfill the narrow width approximation, which technically limits itsapplicability to Higgs bosons not heavier than ∼
600 GeV. This upper value, which is thehighest value quoted in the experimental MC studies of both ATLAS and CMS, gives awidth to mass ratio ∼ .
2. On the other hand, in order to have the signals disentangled, theHiggs bosons have to be separated in mass much more than the maximum of their width.In the LWSM we find that this condition is fulfilled if m h ≤ . m ˜ h . This requirementautomatically satisfies the non-perturbativity constraint of y t ≤ π .A third requirement to run HB is that the model does not change the expected backgroundsubstantially, which is accomplished in the LWSM.14 . Impact of collider bounds: LEP and Tevatron We will start by examining the impact of the direct searches of LW Higgs boson. Alongthis subsection we will not include the constraints on the charged Higgs sector, just to assessthe coverage of the existing exclusions from both LEP and Tevatron.In Fig. 3 we show the collider bounds on the model, in the m h , m ˜ h plane. We employ thefollowing color code: points in green (red) are excluded by LEP (Tevatron), and the magenta(blue) are those where the most sensitive channel comes from LEP (Tevatron) data. In theleft panel, we select a mass range where both h and ˜ h can be excluded. In the rightpanel, we extend the range of m ˜ h up to 1 TeV. The SM reference values are taken fromthe internal subroutines of HiggsBounds . The region to the left of the solid line is excludedby the analysis of Ref. [25], and in the one above the dashed line the top Yukawa couplingbecomes non-perturbative. The inclusion of the newer version of
HiggsBounds excludesa significant portion of the parameter space, due to the fact that the gg → H → W W exclusion is stronger, since it incorporates the most recent data [38]. This study does notonly has a stronger bound due to the increased luminosity, but also extends the publishedkinematical range of the exclusion from 200 GeV to 300 GeV, and includes other analyses,that can probe Higgs masses up to 320 GeV.As we have predicted, the LEP bound of 114 . e + e − → hZ, h → b ¯ b . For high values of r , where both CP-even Higgs bosons are closein mass, one finds that the exclusion can reach up to 120 GeV, which is the maximum massvalue published by the LEP collaboration in this channel. As for the Tevatron exclusions,they come entirely from the latest dedicated study of the gg → h / ˜ h → W W channel[38],where both CDF and D0 results have been combined, with each experiment contributingwith a total luminosity of 4.8 and 5.4 f b − respectively. It is clear that the horizontal stripearound m h ∼
165 GeV correspond to the exclusion due to h . What might seem slighlty lessclear, is that the vertical stripe around m ˜ h ∼
165 GeV correspond to the same experimentalsearch, but this time with ˜ h . The blue regions around those stripes do also make sense:they correspond to those points where the reach in the h → W W channel is not enough toexclude points, but, however, it is still more sensitive than LEP and other Tevatron searches.The horizontal exclusion band becomes narrower as soon as r decreases (or, equivalently, the15 a) (b)FIG. 3. m h as a function of m ˜ h , obtained with HiggsBounds π in the below (above) the dashed line. mass of ˜ h increases), since the enhancements on the couplings tend to be smaller. On theother hand, the vertical exclusion band becomes much narrower as r decreases (or the massof h decreases), since in this case g ˜ h V V →
0. We stress the fact that the collider boundsare able to exclude a portion of the parameter space not probed before by the constraintscoming from Ref. [25]. Let us note that we can make the m ˜ h -independent exclusion of the163 −
166 GeV range for m h , which comes from analyzing the large m ˜ h limit, while for thelower allowed values of m ˜ h , the exclusion covers the 160 −
175 GeV range.It is worth noticing that this limit, which should correspond to the SM case, does notretrieve the well known CDF and D0 combined analysis [39] which excludes the 158 − HiggsBounds does not combine analysis. This is due to the fact that
HiggsBounds works on a channel-by-channel basis, and therefore it does not perform anycombination of channels for the same Higgs (it does, however, combine channels if theycorrespond to different Higgs bosons). It is expectable, given the increasing behaviour ofthe effective couplings g h V V and g h f ¯ f , that a full combined analysis of the LWSM HiggsSector would exclude a much larger region of h masses as far as m ˜ h is not too large.16 . Impact of collider bounds: projection and perspectives for LHC We analyze in this section the potential discovery of LHC for the LW Higgs sector.In particular we focus on a SM-like Higgs boson with mass between 110 GeV and 200GeV and we restrict to points with m ˜ h below 1 TeV and not excluded by the b → sγ (420 GeV < m ˜ h < r to the interval0 . − . (a) (b)FIG. 4. g h f ¯ f (a) and g h γγ (b) as a function of m h , obtained with HiggsBounds
At this point it is worth looking at the effective couplings again, now restricted to thisregion of parameter space, in order to asses the impact on LHC phenomenology. In Fig. 4we show the coupling of the lightest Higgs boson to fermions and photons. The increase in g h f ¯ f is at most of 30% for Higgs masses below 160 GeV, and can go up to 60% for Higgsmasses between 180 and 200 GeV. As discussed in subsection II B this constitutes an upperbound for the enhancement in the total cross section, and also for the rates in the channelsinvolving electroweak gauge bosons, due to the reduction of the branching fractions. Fromthe right panel we see that the coupling to photons is reduced with respect to the SM.However, this reduction is not dramatical: at most, 5 %.We stress the fact that, since one has to recover the SM result in the r → r , or,correspondingly, to a not too heavy ˜ h boson. In this region we may expect to be able todifferentiate the LWSM from the SM.We note that the relative coupling of the CP-even Higgs bosons to EW gauge bosons (notshown here) presents a similar shape to the fermionic one, but its value (as mentioned before)is always below than the one for fermions for the same Higgs mass. More concretely, lookinginto the currently not excluded points, for g h V V we find a maximum value of 1 .
02 (1 .
06) for m h in the 110 −
160 GeV (180 −
200 GeV) , and for g h V V one has 0 .
02 (0 .
06) in the samerange of masses. Evidently, in this region the heavy CP-even Higgs in gaugephobic, andthus the scenario where both CP-even Higgs couple stronger than the SM Higgs to gaugebosons is not achievable: g h V V > r > . m h , m ˜ h plane, and briefly analyzed our expecta-tions for the rates, now we move into the study of the LW Higgs sector at the LHC. Here wefocus on the Run I (until the end of 2012), using the expected exclusion limits provided bythe ATLAS collaborations [40, 41]. The aforementioned studies include the γγ and W W decay modes for masses below 200 GeV, and also the production of the Higgs by vectorboson fusion and the decay into either bottoms or taus, in the same mass range. For heavyHiggs bosons (with masses below 600 GeV) they also include the exclusion limits in the ZZ channel. The SM reference values for cross sections and branching ratios at the LHC areobtained from the LHC Higgs Working Group report [43].In Fig. 5 we show the cross section times branching ratio for the decay modes of thelightest CP-even Higgs into W W and γγ as a function of the Higgs mass, at the LHC andfor a center-of-mass energy of 7 TeV. We also show the reach of each channel and assumethree different integrated luminosities: 1 (end of 2011), 5 (end of 2012, realistic) and 10 f b − (end of 2012, optimistic). The exclusion limits for the reference luminosity of 1 fb − are taken from Refs. [40, 41], and for the other two scenarios, we scale the expected resultby the square root of the luminosity: this procedure yields the right result if one assumes abackground dominated regime and neglects all systematics errors.From the figure we see that the γγ channel has actually a rate lower than the SM for m h ≤
128 GeV. For masses below 140 GeV the difference between the SM and LWSM rates The choice of the ATLAS studies over the corresponding ones from CMS [42] is only due to their morestringent exclusion limits. a) (b)FIG. 5. Total cross section times branching ratio of the lightest CP-even Higgs boson h in the(a) γγ and (b) W W channels. The red points are excluded by current collider data, while theblue points are not. We only show here points satisfying the b → sγ constraint. The LHC reach isshown for three different integrated luminosities: 1 fb − − − is at most 5 %. This means that, at the LHC, it would be impossible to distinguish thisscenario from the SM in the diphoton channel, since in the 120 −
140 GeV range a changeof at least 20% is necessary [44].As for the gauge boson channel, we see that the increase is of at most 30% (60 %) fora 160 (200) GeV Higgs. In this case one can distinguish the LW Higgs from the SM if theenhancement is larger than 10 %, with 300 fb − of data [45].From Figure 5 it can be seen that, with 1 fb − of data, the W W channel would alreadybe able to exclude most of the points, except for bosons with masses above ∼
180 GeV, orbelow 130 GeV. For the heavy mass range, the ZZ decay mode can exclude an importantfraction of points already with this luminosity, and for higher luminosities both channelscompletely cover this area. For an integrated luminosity of 5 fb − , one probes masses largerthan 122 GeV, while in the optimistic scenario of 10 fb − the exclusion extends to 120 GeVmasses.The 120 GeV limitation is simply due to the fact that both CMS [42] and ATLAS [41]cut off the analysis of these channels at 120 GeV. However, the recent update of ATLAS for19he SM Higgs boson sensitivities [ ? ], where different channels are combined, reports that atotal luminosity of 4 . − is required in order to exclude a 115 GeV SM Higgs. Therefore itis rather likely, that a combination of all possible channels (and maybe also a combinationof both collaborations) would be able to fully test the lightest LW Higgs boson at the LHCRun I, even without assuming the optimistic 10 fb − scenario.Now we briefly comment on the prospects for the other LW Higgs bosons. In increasingorder of its masses, the next boson is ˜ h . Since, as we stated before, this boson is gaugepho-bic, and its mass is above the top pair production threshold (due to the masses sum rule),we find the t ¯ t channel to be its main decay mode with a branching fraction always higherthan 60%. The second decay mode is actually the ˜ h → h h , whose branching fractionvaries, roughly speaking, in the 1 −
30 % range. This provides a very interesting decay modeof either four fermions, or two fermions plus two photons; however this signature will onlybe at the reach of LHC with very high luminosities (see Ref. [46]).As for the remaining Higgs bosons, we find that the charged Higgs decays into top-bottomwith a ratio larger than 0.95. Together with the ˜ h ± → h W ± channel, they comprise thetwo observable decay channels of the charged Higgs, since all the others have branchingfractions below 10 − . The CP-odd Higss ˜ P decays predominantly into tops; the ˜ P → hZ channel account for, at most, 10%, and the gluon-gluon and b ¯ b channels account for a 1 %and 0.1 % of the decays, respectively. It is therefore evident that to directly probe theseheavy Higgs bosons at the LHC high luminosities are required: these channels also appearin the MSSM, and typical studies are done with, at least, 30 fb − [46].To summarize this subsection, we have found that the lightest LW Higgs boson can beexcluded at the LHC Run I, while the existence of other Higgs bosons can not be directlyprobed. If a LW Higgs signal is seen at Run I, it will look like a SM Higgs: a very highluminosity would be required to rule out the SM case, and it might indeed not be enough[45,46]. If the measure of the W W channel is higher than the SM cases, then one expects theremaining Higgs bosons to be not too heavy. If the
W W channel measurement is compatiblewith the SM Higgs, then that could mean that the remaining LW Higgs bosons are heavier.In either case, one would still need to directly probe the other Higgs bosons. We have alsofound that the decay modes of the heavy Higgs bosons is rather restricted, mainly due to thehierarchy in the Higgs spectrum that does not allow, for instance, one of the heavier Higgsbosons to decay into another one plus a gauge boson ( interesting decays like ˜ h ± → W ± ˜ h ,20r ˜ P → Z ˜ h are kinematically forbidden). IV. CONCLUSIONS
In this work we have studied the bounds on the LW Higgs sector coming from directsearches at both the LEP and Tevatron collider, using the code
HiggsBounds . We have alsoanalyzed the prospects for direct detection at the LHC Run I. This work complements thestudy of the indirect constraints performed by Carone and Primulando [25] and settles downthe state of the art and perspectives for direct search of LW Higgs Bosons in colliders.While direct collider searches place strong constraints mostly on m h , the observablesconsidered in Ref. [25] are mostly sensitive to the loop contribution of the charged Higgsboson. These indirect constraints rule out a great portion of the parameter space ( m (cid:101) h ± >
463 GeV), while we have shown that the current Tevatron result is only able to constrainta small region of parameter space. A light SM-like LW Higgs boson with a mass in the163 −
166 GeV range is forbidden (independently of the non SM-like LW mass scale), as canbe seen from Fig. 3. If ˜ h is not too heavy ( ∼
450 GeV) then the exclusion for m h extendsto the 150 −
175 GeV. However, by the same token, as soon as the LHC starts to probe awider range of masses for h , the exclusion will become stronger, as shown in Fig. 5: alreadywith 1 fb − of data h masses between 130 and 180 GeV can be ruled out.In this model, all the production modes of h are always enhanced with respect to the SM,and also its branching fraction into fermions. Conversely, the decay modes involving gaugebosons ( ZZ, W W, Zγ, γγ ) are always reduced with respect to the SM case. When analyzingthe rates for the
W W and γγ channel, we have found that in the W W channel one alwaysobtains an enhancement of the total rate, (it can go up to 50 % for m h ∼
200 GeV), whilein the photon channel suffers a slight reduction (at most, 5 %) in the interesting light massrange ( m h ≤
128 GeV), and it can be enhanced by at most 4 %, for heavier masses.The magnitude of these two enhancements provides important messages. On one hand,the
W W channel enhancement means that the LHC will be able to test a wider range ofmasses for h than for the SM Higgs. With only 1 fb − of data, it would be already able toprobe a significant portion of parameter space, which can be further extended into the lightermass region with increased luminosity. Taking into account the current projections for theSM Higgs, one can state that the h can be fully probed by the LHC Run I. Moreover, if h
21s not too light, the LHC Run II (around 300 fb − of data) will be able to tell apart the LWscenario from the SM. The mild variation (with respect to the SM) of the γγ channel doesalso have phenomenological implications for LHC Run I. On one side, this channel alone isnot a discovery channel for LHC Run I, not even if a total integrated luminosity of 10 fb − (optimistic scenario) were to be achieved. Moreover, the LHC Run II will not be able to usethe diphoton channel to tell apart h from the SM Higgs.One important feature of the LW Higgs sector, is the fact that both CP-even Higgs bosonscan couple stronger to the W and Z than the SM Higgs. While this is a very appealingpossibility, we have also shown that, in its minimal realization, this scenario does not occur.However, a non-minimal LW Higgs sector might be able to accommodate this interestingphenomenological possibility.As for the remaining Higgs bosons, the main decay channel is t ¯ t (top bottom) for theneutral (charged) bosons. However, for the heavy CP-even Higgs, the detection through˜ h → h h → γγb ¯ b constitutes a promising channel, and it might be worth fully exploringthe consequences of this decay mode. Similar Higgs chain decays modes do not occurwith a substantial rate, mainly due to kinematically closure of these channels, due to thestrict hierarchy in the Higgs spectrum. It would also be interesting to analyze if radiativecorrections can alter the spectrum hierarchy. ACKNOWLEDGMENTS
We thank Marc Gillioz and Alejandro Szynkman for reading the manuscript. We alsothank Roman Zwicky for useful discussions. The work of J.Z. is supported by the SwissNational Science Foundation (SNF) under Contract No. 200020-126691.
Appendix A: Loop functions
In this appendix we collect all of the relevant formulae for the loop functions that appearin Subsection II A 3. The expressions are adapted from [35].For h , ˜ h , ˜ P → γγ one defines the function f ( x ) as f ( x ) = arcsin (1 / √ x ) if x ≥ − [log( √ − x −√ − x ) − iπ ] if x < . (A1)22e recall that f ( x ) ≈ x + x in the x → ∞ limit. The loop functions for bosons (1) andscalars (0) for the γγ case are simply given by F ( x ) = 2 + 3 x + 3 x (2 − x ) f ( x ) , (A2) F ( x ) = x (1 − xf ( x )) . (A3)For the fermions (1 /
2) one has to distinguish between CP-even and CP-odd Higgs bosons.As such, we define F / ( x ) = − x (1 + (1 − x ) f ( x )) , (A4) F ˜ P / ( x ) = − xf ( x ) , (A5)where F ˜ P / is the function for the ˜ P , and, for simplicity, we omit the superscript in F / forthe case of the CP-even Higgs bosons. Note that Eq. (19) is derived simply by taking theabsolute squared value of the ratio of F ˜ P / and F / .For the h , ˜ h , ˜ P → γZ decays, one has to define the following functions, A xt ( β tx , β tZ ) = N c ( − Q t ) s W c W g tV (cid:0) ζ x I ( β tx , β tZ ) − I ( β tx , β tZ ) (cid:1) , (A6) A W ( β Wx , β WZ ) = − t − W (cid:8) − t W ) I ( β Wx , β WZ ) + (cid:2) (1 + 2 β Wx ) t W − (5 + 2 β Wx ) (cid:3) I ( β Wx , β WZ ) (cid:9) , (A7) A ˜ h ± ( β h ± x , β h ± Z ) = 1 − s W c W s W I ( β h ± x , β h ± Z ) m W m h ± , (A8)where s W = sin θ W , c W = cos θ W , t W = tan θ W , Q t = 2 / N c = 3, g tV = I Lt − Q t s W is thevectorial coupling of the top quark and ζ x is 1 (0) for CP-even (odd) Higgs bosons. Theindex x refers to which Higgs ( h , ˜ h or ˜ P ) is the effective coupling sought. The functions I , read I ( a, b ) = ab a − b ) + a b a − b ) [ f ( a ) − f ( b )] + a b ( a − b ) [ g ( a ) − g ( b )] , (A9) I ( a, b ) = − ab a − b ) [ f ( a ) − f ( b )] , (A10)with f given by Eq. (A1) and g by g ( x ) = √ x − / √ x ) if x ≥ √ − x (cid:2) log( √ − x −√ − x ) − iπ (cid:3) if x < . (A11) Notice that F differs from F η of Ref. [14] by an overall minus sign.
1] B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev. D , 025012 (2008) [arXiv:0704.1845[hep-ph]].[2] T. D. Lee and G. C. Wick, Nucl. Phys. B (1969) 209.[3] T. D. Lee and G. C. Wick, Phys. Rev. D (1970) 1033.[4] B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev. D (2009) 105019 [arXiv:0805.2156[hep-th]].[5] S. Coleman, In *Erice 1969, Ettore Majorana School On Subnuclear Phenomena*, New York1970, 282-327. [6] G. ’t Hooft, M. J. G. Veltman, Nucl. Phys.
B44 (1972) 189-213.[7] B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev. D (2008) 065010 [arXiv:0710.5528[hep-ph]].[8] Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LAT2007 (2007) 056[arXiv:0710.3151 [hep-lat]].[9] F. Knechtli, N. Irges and M. Luz, J. Phys. Conf. Ser. (2008) 102006 [arXiv:0711.2931[hep-ph]].[10] B. Grinstein and D. O’Connell, Phys. Rev. D (2008) 105005 [arXiv:0801.4034 [hep-ph]].[11] T. R. Dulaney and M. B. Wise, Phys. Lett. B (2008) 230 [arXiv:0708.0567 [hep-ph]].[12] F. Wu and M. Zhong, Phys. Lett. B (2008) 694 [arXiv:0705.3287 [hep-ph]].[13] J. R. Espinosa, B. Grinstein, D. O’Connell and M. B. Wise, Phys. Rev. D (2008) 085002[arXiv:0705.1188 [hep-ph]].[14] F. Krauss, T. E. J. Underwood and R. Zwicky, Phys. Rev. D (2008) 015012 [arXiv:0709.4054[hep-ph]].[15] C. D. Carone and R. F. Lebed, JHEP (2009) 043 [arXiv:0811.4150 [hep-ph]].[16] T. G. Rizzo, JHEP (2007) 070 [arXiv:0704.3458 [hep-ph]].[17] T. G. Rizzo, JHEP (2008) 042 [arXiv:0712.1791 [hep-ph]].[18] C. D. Carone, Phys. Lett. B (2009) 306 [arXiv:0904.2359 [hep-ph]].[19] C. D. Carone and R. F. Lebed, Phys. Lett. B (2008) 221 [arXiv:0806.4555 [hep-ph]].[20] E. Alvarez, L. Da Rold, C. Schat and A. Szynkman, JHEP (2008) 026 [arXiv:0802.1061[hep-ph]];
21] E. Alvarez, C. Schat, L. Da Rold and A. Szynkman, arXiv:0810.3463 [hep-ph].[22] T. E. J. Underwood and R. Zwicky, Phys. Rev. D (2009) 035016 [arXiv:0805.3296 [hep-ph]].[23] R. S. Chivukula, A. Farzinnia, R. Foadi et al. , Phys. Rev. D81 , 095015 (2010).[arXiv:1002.0343 [hep-ph]].[24] B. Fornal, B. Grinstein and M. B. Wise, Phys. Lett. B (2009) 330 [arXiv:0902.1585[hep-th]].[25] C. D. Carone and R. Primulando, Phys. Rev. D , 055020 (2009) [arXiv:0908.0342 [hep-ph]].[26] R. S. Chivukula, A. Farzinnia, R. Foadi et al. , Phys. Rev. D82 , 035015 (2010).[arXiv:1006.2800 [hep-ph]].[27] J. R. Espinosa, B. Grinstein, [arXiv:1101.5538 [hep-ph]].[28] K. Nakamura et al. (Particle Data Group), J. Phys. G , 075021 (2010) 1; P. Achard et al. [L3 Collaboration], Phys. Lett. B (2001) 75.[29] P. Bechtle, O. Brein, S. Heinemeyer et al. , Comput. Phys. Commun. , 138-167 (2010).[arXiv:0811.4169 [hep-ph]].[30] P. Bechtle, O. Brein, S. Heinemeyer et al. , [arXiv:0905.2190 [hep-ph]].[31] P. Bechtle, O. Brein, S. Heinemeyer et al. , AIP Conf. Proc. , 510-513 (2010).[arXiv:0909.4664 [hep-ph]].[32] P. Bechtle, O. Brein, S. Heinemeyer et al. , [arXiv:1012.5170 [hep-ph]].[33] P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein and K. E. Williams, arXiv:1102.1898 [hep-ph].[34] G. Cacciapaglia, A. Deandrea, J. Llodra-Perez, JHEP , 054 (2009). [arXiv:0901.0927[hep-ph]].[35] J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson, “The Higgs Hunter’s Guide,” Front.Phys. , 1-448 (2000), arXiv:hep-ph/9302272.[36] [LEP Higgs Working Group for Higgs boson searches and ALEPH Collaboration and DELPHICollaboration and L3 Collaboration and OPAL Collaboration], arXiv:hep-ex/0107031.[37] M. Spira, Fortsch. Phys. , 203 (1998) [arXiv:hep-ph/9705337].[38] T. Aaltonen et al. [CDF and D0 Collaboration], arXiv:1005.3216 [hep-ex].[39] T. Aaltonen et al. [ CDF and D0 Collaboration ], [arXiv:1103.3233 [hep-ex]].[40] ATLAS collaboration, ATLAS-CONF-2011-004[41] ATLAS collaboration, ATL-PHYS-PUB-2010-015
42] CMS collaboration, CMS NOTE 2010/008.[43] LHC Higgs Cross Section Working Group, S. Dittmaier, C. Mariotti, G. Passarino, R. Tanaka(Eds.), et al. , arXiv:1101.0593 [hep-ph] .[44] M. Duhrssen, S. Heinemeyer, H. Logan, D. Rainwater, G. Weiglein and D. Zeppenfeld, Phys.Rev. D , 113009 (2004) [arXiv:hep-ph/0406323].[45] M. D¨uhrssen, ATL-PHYS-2003-030.[46] V. Buescher, K. Jakobs, Int. J. Mod. Phys. A20 , 2523-2602 (2005). [hep-ph/0504099]., 2523-2602 (2005). [hep-ph/0504099].