Vector-Boson Production of Light Higgs Pairs in 2-Higgs Doublet Models
M. Moretti, S. Moretti, F. Piccinini, R. Pittau, J. Rathsman
aa r X i v : . [ h e p - ph ] D ec SHEP-07-11FNT/T 2007-04October 23, 2018
Vector-Boson Production ofLight Higgs Pairs in 2-Higgs Doublet Models
M. Moretti
Dipartimento di Fisica, Universit`a di Ferrara andINFN - Sezione di Ferrara, Via Paradiso 12, 44100 Ferrara, Italy
S. Moretti
School of Physics & Astronomy, University of Southampton,Highfield, Southampton SO17 1BJ, UK, andLaboratoire de Physique Th´eorique, Universit´e Paris–Sud, F–91405 Orsay Cedex, France
F. Piccinini
INFN - Sezione di Pavia, Dipartimento di Fisica Nucleare e Teorica,Via Bassi 6, 27100 Pavia, Italy
R. Pittau Dipartimento di Fisica Teorica, Universit`a di Torino andINFN - Sezione di Torino, Via Giuria 1, 10125 Torino, Italy, andDepartamento de F´ısica Te´orica y del Cosmos,Centro Andaluz de F´ısica de Part´ıculas Elementales (CAFPE),Universidad de Granada, E-18071 Granada, Spain
J. Rathsman
High Energy Physics, Uppsala University, Box 535, 751 21 Uppsala, Sweden
Abstract
At the Large Hadron Collider, we prove the feasibility to detect pair production of thelightest CP-even Higgs boson h of Type II 2-Higgs Doublet Models through qq ( ′ ) → qq ( ′ ) hh (vector-boson fusion). We also show that, through the hh → b decay channel inpresence of heavy-flavour tagging, further exploiting forward/backward jet sampling, onehas direct access to the λ Hhh triple Higgs coupling – which constrains the form of theHiggs potential. Present address:
Institute of Nuclear Physics, NCSR ”DEMOKRITOS”, 15310, Athens, Greece. Introduction
If only a light Higgs boson (with mass M h < ∼
140 GeV) is found at the Large Hadron Collider(LHC), it may be difficult to tell whether it belongs to the Standard Model (SM) or indeeda model with an enlarged Higgs sector. For example, in the case of a CP-conserving TypeII 2-Higgs Doublet Model (2HDM) [1]–[4] , possibly in presence of minimal Supersymmetry(SUSY) – the combination of the two yielding the so-called Minimal Supersymmetric StandardModel (MSSM) – this happens in the so-called ‘decoupling region’, when M H , M A , M H ± ≫ M h ,for suitable choices of the other MSSM and 2HDM parameters, where - for the same mass - the h couplings to ordinary matter in the SM are the same as in both the 2HDM and MSSM. Evenin these conditions, however, it has been proved that one could possibly establish the presenceof an extended Higgs sector by determining the size of the trilinear Higgs self-coupling λ hhh [5].If the extended model is not in a decoupling condition, then it is generally possible toestablish the presence of additional Higgs signals, H, A and/or H ± [6, 7]. However, even whenthis is the case, it may be difficult to distinguish, e.g., between a generic Type II 2HDM andthe MSSM (unless, of course, one also detects the SUSY partners of ordinary matter and Higgsbosons). In fact, despite there exist well establish spectra among the four different masses in theMSSM (for fixed, say, M h and tan β , the ratio of the vacuum expectation values of the two Higgsdoublets in either model), it may well be possible that the additional 2HDM parameters arrangethemselves to produce an identical mass pattern. However, such a degeneracy between the twomodels would not typically persist if one were able to also measure certain Higgs couplings,chiefly those among the Higgs bosons themselves (involving two or more such particles). Infact, while the measurement of only two among the four Higgs boson masses ( M h , M H , M A and M H ± ) – or, alternatively, one such masses and tan β – would fix (at tree-level) all Higgs massesand couplings in the MSSM, this is no longer true in a generic Type II 2HDM [1], because of thefreedom in selecting the free additional parameters. For example, the general CP-conservingType II 2HDM that we are going to consider can be specified uniquely by seven parameters: M h , M H , M A , M H ± , β , α (the mixing angle between the two CP-even neutral Higgs states) and λ (see eq. (2) later on). It may then happen that the first six of these are measured and foundto agree with the MSSM pattern, but one would still need to measure λ to verify that it is theHiggs sector of the MSSM that is present. One way to do so would be by measuring trilinearHiggs self-couplings, such as λ hhh and λ Hhh . Alternatively, the measurement of the latter twocouplings would constitute a test of the MSSM relations if one knew M h and tan β but not α .In this paper, we make the assumption that only one parameter is known, M h , as maywell happen at the LHC after only a h resonance is detected. We further imply that all Of the initial eight degrees of freedom pertaining to the two complex Higgs doublets, only five survive asreal particles upon Electro-Weak Symmetry Breaking (EWSB), labelled as h, H , A (the first two are CP-even or‘scalars’ (with M h < M H ) whereas the third is CP-odd or ‘pseudoscalar’) and H ± , as three degrees of freedomare absorbed into the definition of the longitudinal polarisation for the gauge bosons Z and W ± , upon theirmass generation after EWSB. h pair production, thereby possibly also distinguish between, e.g.,a generic Type II 2HDM and the MSSM?It is the purpose of this paper to show that this is the case, so long that enough luminositycan be accumulated at the LHC, also in view of the Super-LHC (SLHC) option [8]. We willillustrate how we have come to this conclusion, i.e., after investigating the process [9] qq ( ′ ) → qq ( ′ ) hh (vector − boson fusion) , (1)with q ( ′ ) referring to any possible (anti)quark flavour combinations . The relevant Feynmandiagrams corresponding to process (1) in both the MSSM and 2HDM considered here can befound in Fig. 1. In our selection analysis, we will resort to the extraction of two h → b ¯ b resonances, in presence of the following signature: • ‘four b -quark jets and two forward/backward-jets’.This signature was already considered in Ref. [5] in the SM context (from which we will importsome of the results).Our paper is organised as follows. In the next section, we outline the computational proce-dure. Sect. 3 presents our numerical results and discusses these in various subsections. Sect. 4contains our conclusions. We have assumed √ s = 14 TeV for the LHC energy throughout. Our numerical results areobtained by setting the renormalisation and factorisation scales to 2 M h for the signal while forthe QCD background we have used the average jet transverse momentum ( p T = P n p T j /n ).Both Higgs processes and noise were estimated by using the Parton Distribution Function(PDF) set MRST99(COR01) [18]. While the background calculations were based on exact tree-level Matrix Elements (MEs) using the ALPGEN program [19], all signal rates were obtained The gluon-gluon production mode [10] was considered in Refs. [11] and [12] (see also [13]), and later on[14, 15], where – despite significant kinematic differences exist between signal and QCD noise – it was eventuallyshown that the extraction of the gg → hh → b ¯ bb ¯ b signal is essentially impossible at the (S)LHC because of theoverwhelming QCD noise, both reducible and irreducible. Recently, encouraging results on the cross-section formulti-Higgs boson production in the gluon-gluon production mode has been obtained in models beyond the SMand MSSM [16]. The possibility of using Higgs boson pair production more generally to access trilinear Higgscouplings has also been studied on the level of total cross-sections in [17]. M A and tan β . Through higher orders, we have considered the so called ‘Maximal Mixing’scenario ( X t = A t − µ/ tan β = √ M SUSY ) [23], wherein we have chosen for the relevant SUSYinput parameters: µ = 200 GeV, A b = 0, with M SUSY = 5 TeV, the latter – as already intimated– implying a sufficiently heavy scale for all sparticle masses, so that these are not accessible atthe LHC and no significant interplay between the SUSY and Higgs sectors of the model cantake place . Masses and couplings within the MSSM have been obtained by using the HDECAYprogram [24].Before giving the details of the 2HDM setup we are using, let us recall the most generalCP-conserving 2HDM scalar potential which is symmetric under Φ → − Φ up to softlybreaking dimension-2 terms (thereby allowing for loop-induced flavour changing neutral cur-rents) [1], V = m Φ † Φ + m Φ † Φ − n m Φ † Φ + h.c. o + 12 λ (cid:16) Φ † Φ (cid:17) + 12 λ (cid:16) Φ † Φ (cid:17) ++ λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + (cid:26) λ (cid:16) Φ † Φ (cid:17) + h.c. (cid:27) . (2)In the following, the parameters m , m , m , λ , λ , λ and λ are replaced by v , M h , M H , M A , M H ± , β and α (with v fixed). Hence, as intimated already, the CP-conserving 2HDMpotential is parameterised by seven free parameters. Notice that from the scalar potential allthe different Higgs couplings needed for our study can easily be obtained. (See [2, 3] for acomplete compilation of couplings in a general CP-conserving 2HDM.)In our 2HDM, we will fix M h and M H to values similar to the ones found in the MSSMscenario we are considering, by adopting three different setups:1. M h = 115 GeV, M H = 300 GeV,2. M h = 115 GeV, M H = 500 GeV,3. M h = 115 GeV, M H = 700 GeV.We always scan over the remaining parameters in the ranges − π/ < α < π/ , − π < λ < π, < tan β < , The only possible exception in this mass hierarchy would be the Lightest Supersymmetric Particle (LSP),whose mass may well be smaller than the lightest Higgs mass values that we will be considering. However, wehave verified that invisible h decays (including the one into two LSPs) have negligible decay rates.
400 GeV < M A < ,
100 GeV < M H ± < . In order to accept a point from the scan we also check that the following conditions arefulfilled: the potential is bounded from below, the λ i fulfill the tree-level unitarity constraintsof [25] and yield a contribution to | ∆ ρ | < − . In short the unitarity constraints amountsto putting limits on the eigen values of the S matrices for scattering various combinations ofHiggs and electroweak gauge bosons. We have followed the normal procedure [1] of requiringthe J = 0 partial waves ( a ) of the different scattering processes to fulfill | Re(a ) | < /
2, whichcorresponds to applying the condition that the eigenvalues Λ Z Y σ ± of the scattering matrices (ormore precisely 16 πS ) fulfill | Λ Z Y σ ± | < π [26]. In other words we allow parameter space pointsall the way up to the tree-level unitarity constraint | Re(a ) | < /
2. In order to investigatethe sensitivity to this upper limit we will also report results as a function of the value of themaximal eigenvalue, Λ max . The spectrum of masses, couplings and decay rates in our 2HDMis the same as in Ref. [27], obtained by using a modification of HDECAY [24] (consistent witha similar manipulation of the program used in Ref. [28]). For each accepted point in the scanthe partial decay rates for the different Higgs bosons are then calculated using HDECAY andalso taking possible additional partial widths of the H into account.While the parameter dependence of the MSSM Higgs sector renders the computation of thetree-level MSSM cross-sections rather straightforward (as the latter depends on two parametersonly, M A and tan β ), the task becomes much more time-consuming in the context of the 2HDM.In order to calculate the cross-sections in this scenario, they are schematically written as acombination of couplings and kinematic factors in the following way: σ tot = Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 g i M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d LIPS = X i =1 5 X j = i g i g j σ ij , (3)where all the explicit dependence on α , β , λ Hhh and λ hhh is contained in the couplings g i : g = sin ( β − α ), g = cos ( β − α ), g = cos( β − α ) λ Hhh , g = sin( β − α ) λ hhh , and g = 1,whereas the dependence on masses and other couplings is in the factors σ ij = 11 + δ ij Z (cid:16) M † i M j + M † j M i (cid:17) d LIPS . (4)Note that the sum over subamplitudes M i also contains all interference terms and that colourfactors etc. are included properly. The σ ij are then calculated numerically for fixed masses.We can then get the cross-section in an arbitrary parameter space point by multiplying thekinematic factors with the appropriate couplings. However, there is a slight complication sincethe kinematic factor for the H → hh contribution depends on the width Γ H if there is a s-channel resonance and the width in turn depends on the couplings. In this case the kinematicfactor scales as 1 / Γ H which is accounted for by assuming a fixed value for the width when the Here, Z refers to the Z symmetry, Y is the hypercharge, and ~σ is the total weak isospin We have carefully verified the integrity of our procedure. M A and M H ± . The contributions of main interest, which containthe λ Hhh and λ hhh couplings, only depend on these masses indirectly through the unitarityconstraints. At the same time there are other contributions to the cross-section which dependexplicitly on these masses. However, these contributions are very small in the parts of parameterspace of interest and can thus be safely neglected. In our investigation of the emerging hadronic final state, we will assume that b -quark jetsare distinguishable from light-quark and gluon ones and neglect considering b -jet charge de-termination. Finite calorimeter resolution has been emulated through a Gaussian smearingin transverse momentum, p T , with ( σ ( p T ) /p T ) = (0 . / √ p T ) + (0 . , for all jets. Thecorresponding missing transverse momentum, p miss T , was reconstructed from the vector sum ofthe visible momenta after resolution smearing. Finally, in our parton level analysis, we haveidentified jets with the partons from which they originate and applied all cuts directly to thelatter, since parton shower and hadronisation effects were not included in our study. In this section, after a preliminary analysis of the Higgs mass and coupling spectra in theMSSM and a general Type II 2HDM, we will start our numerical analysis by investigating themodel parameter dependence of the Higgs pair production process in (1) at fully inclusive level,in presence of the decay of the latter into two b ¯ b pairs, with the integration over the phasespace being performed with no kinematical restrictions. This will be followed by an analysis ofthe production and decay process pertaining to the Higgs signal of interest at fully differentiallevel, in presence of detector acceptance cuts and kinematical selection constraints. Finally, wewill compare the yield of the signal to that of the corresponding background and perform adedicated signal-to-background study including an optimisation of the cuts in order to enhancethe overall significance. We will treat the MSSM and 2HDM in two separate subsections. As representative of the low and high tan β regime, we will use in the remainder the values of 3and 40. We have instead treated M A as a continuous parameter, varying between 100 and 700GeV or so . Before proceeding with the numerical analysis of the signal, it is worthwhile toinvestigate both the Higgs mass and coupling dependence in the MSSM with respect to the twoinput parameters M A and tan β . This is done in Figs. 2 and 3, respectively. In the latter, we Values of M A below 90 GeV or so are actually excluded by LEP for the lower tan β value: see [29]. G hV V and G HV V ), wherein V refers to either a W ± or a Z . In the same figure, the symbol φ refers tothe SM Higgs boson, with mass identical to that of the lightest MSSM Higgs state ( M φ = M h ).While the pattern of masses has been well established in past literature, it is interesting to noticehere that the product of the MSSM couplings entering process (1) is always smaller than inthe SM case. However, in the MSSM, resonance enhancements can occur (such as in H → hh ),so that the actual MSSM production rates can in some cases overcome the corresponding SMones (for M φ = M h ).Fig. 4 presents the fully inclusive MSSM cross-section for the process of interest, as definedin (1), times (effectively) BR( hh → b ¯ bb ¯ b ). The shape of the curves is mainly dictated by theinterplay between phase space (see Fig. 2) and coupling (see Fig. 3) effects, with the exceptionof the region M A > ∼
220 GeV and tan β = 3, where the onset of the H → hh resonance isclearly visible. Cross-sections are generally sizable, particularly at low tan β . The displayedrates however coincide to the ideal situation in which all final state jets are detected with unitefficiency and the detector coverage extend to their entire phase space, so that they only serveas a guidance in rating the phenomenological relevance of the process discussed.A more realistic analysis is in order, which we have performed as follows. The four b -jetsemerging from the decay of the hh pair are accepted according to the following criteria: p bT >
30 GeV , | η b | < . , ∆ R bb > . , (5)in transverse momentum, pseudorapidity and cone separation, respectively. Their taggingefficiency is taken as ǫ b = 50% for each b satisfying these requirements, ǫ b = 0 otherwise . Inaddition, to enforce the reconstruction of the two Higgs bosons, we require all such b ’s in theevent to be tagged and that at least one out of the three possible double pairings of b -jetssatisfies the following mass preselection:( m b ,b − M h ) + ( m b ,b − M h ) < σ m , (6)where σ m = 0 . M h . We further exploit ‘forward/backward-jet’ tagging, by imposing that thenon- b -jets satisfy the additional cuts p fwd / bwd T >
20 GeV , . < η fwd < , − . > η bwd > − . (7)Tab. 1 shows the rates of the signal after the implementation of the constraints in eqs. (5)–(7)(hereafter, referred to as ‘acceptance and preselection cuts’ or ‘primary cuts’). While our processdoes yield non-negligible rates after the latter, it turns out that it is of no phenomenologicalrelevance, even assuming very high luminosity. Firstly, in view of the fact that b -taggingefficiencies are not taken into account in this table: for the ‘4 b -jet’ tagging option, one shouldmultiply the numbers in Tab. 1 by ǫ b , that is, 1/16. (Alternative approaches requiring a lesser Here and in the remainder, the label b refers to jets that are b -tagged while j to any jet (even thoseoriginating from b -quarks) which is not. β = 3 M A (GeV) M h (GeV) σ ( qq ( ′ ) → qq ( ′ ) hh ) [fb] σ (background) [fb]160 108 0.19 218200 112 0.23 232240 114 0.46 229tan β = 40 M A (GeV) M h (GeV) σ ( qq ( ′ ) → qq ( ′ ) hh ) [fb] σ (background) [fb]160 129 0.26 224200 129 0.20 224240 129 0.17 224Table 1: Cross-sections for Higgs pair production via vector-boson fusion, process (1), afterHiggs boson decays (relevant BRs are all included) and the acceptance and preselection cutsdefined in (5)–(7), for two choices of tan β and a selection of M A values, assuming the MSSM inMaximal Mixing configuration (the corresponding values of M h are also indicated in brackets).No b -tagging efficiencies are included here.number of b -jets to be tagged as such were not successful either.) Secondly, the backgroundrates, after the same cuts in eqs. (5)–(7), are always overwhelming the signal, despite our effortsin further optimising the cuts. For this reason, rather than dwelling upon the latter now, wepostpone their discussion to the next subsection and simply conclude here that our channel isaltogether inaccessible at both the LHC and SLHC in the context of the MSSM. As already alluded to, the parameter space of the general CP-conserving Type II 2HDM weare considering is quite large as it depends on seven unknown parameters. In order to get afeel for the dependence of the signal cross-section for the process qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b we therefore present in Figs. 5 through 7 the results of our three selected scenarios, wherein wescan the allowed parameter space over 10000 randomly chosen points. (Note that similarly tothe MSSM case we have included the BR( hh → b ¯ bb ¯ b ) but not any 4 b -jet tagging efficiency.)Comparing with the cross-sections in the MSSM the main differences are due to the follow-ing: • the triple Higgs couplings λ Hhh and λ hhh are not related to the gauge couplings; • the different parameters can vary independently of each other.Conversely, the kinematic factors in the two models will be the same for a given set of massesand widths of the different Higgs bosons. Therefore, in those cases, many features of the signal, We use the same definitions of these couplings as in [2]. H (GeV) σ ( qq ( ′ ) → qq ( ′ ) hh ) [fb] with different cutsinclusive primary optimal optimal, H → hh
300 1453 71.9 31.2 25.8500 396 25.3 11.4 7.7700 80 7.1 3.3 2.0Table 2: The maximal cross-sections in the 2HDM under consideration for M h = 115 GeV, and M H = 300, 500 and 700 GeV, respectively, with the following different cuts: inclusive, withprimary cuts in eqs. (5)–(7), and with optimised cuts of eq. (8) in the latter case also whenonly considering the H → hh resonant contribution.such as the differential distributions, will be similar to those of the MSSM even though thenormalisation can be completely different. In fact, comparing Fig. 4 with 5 through 7 we seethat in the more general 2HDM the cross-sections can be more than two orders of magnitudelarger than in the MSSM thus rendering a much larger potential for a detectable signal (asit will be discussed below). To be more quantitative on this we give in Tab. 2 the maximalinclusive cross-sections obtained in the scans for M h = 115 GeV and M H = 300, 500 and700 GeV.In order to study the potential signal in more detail we first of all apply the same primarycuts as in the case of the MSSM, those listed in eqs. (5)–(7). The resulting cross-sectionsare given Tab. 2. Comparing with the cross-section without the primary cuts we see thatthe reduction is substantial, but even so the signal cross-section can still be more than twoorders of magnitude larger than in the MSSM scenario considered in subsection 3.1.1 and it iscomparable to the background (see Tab. 1, specifically for low tan β , where the M h values inthe two models are very similar). In this section, we will continue the discussion of our numerical analyses limitedly to the TypeII 2HDM considered so far. In order to enhance the statistical significance S/ √ B we studiedseveral differential distributions for signals and background with the event selection of eqs. (5)–(7), with the aim of introducing optimised cuts, allowing at the same time to keep the signalevent numbers at a reasonable level. To begin with, for simplicity, we have limited ourselvesto use the contribution from the H → hh resonance to the signal for M H = 300 GeV in ascenario where the cross-section is close to maximal, with cos( β − α ) = 1, λ Hhh = 1000 GeVand Γ H = 30 GeV, when comparing with the background.The most sensitive distributions, able to discriminate between the signal and background,turn out to be the minimum transverse momentum of the forward/backward jets and the next-to-minimum invariant mass of the b ¯ b pairs, which we show in Fig. 8. (Although to a some more9imited extent, also the minimum b ¯ b invariant mass is useful.) Before selecting a specific setup, we performed also a systematic analysis of the significance for different combination of cuts(30 GeV ≤ m min bb ≤ m next − to − min bb ≤
100 GeV, 20 GeV ≤ p fwdT ≤
60 GeV). The best optimisedcuts, on top of the basic ones of eqs. (5)–(7), that we found are : p fwd / bwd T >
40 GeV , m min bb >
40 GeV , m next − to − min bb >
80 GeV . (8)We show in Fig. 9 the 4 b invariant mass distribution for three signals ( M H = 300, 500 and700 GeV with the widths Γ H = 30, 50 and 200 GeV, respectively) and the background afterthe optimised cuts of eq. (8) have also been imposed. For each of the three signals shown inthe figure we have used the parameter space point which gives the maximal signal cross-sectionfrom the resonant H → hh contribution when restricting the width Γ H to be less than 30, 50and 200 GeV, respectively. In this context we note that there are two effects which mainlydetermine the width of the signal distribution. On the one hand, the smearing of momenta weuse gives a contribution to the measurable width of about 30 GeV. On the other hand, one ofcourse has the intrinsic width of the H .Taking suitable mass windows around the peaks for the different Higgs mass values il-lustrated in Fig. 9, we obtain the maximal signal cross-sections, event numbers and statis-tical significances quoted in Tab. 3. In order to calculate the signal cross-sections in therespective windows for different parameter space points, taking the actual width of the H into account, we rescaled the contribution from the H → hh resonance with a factor c M H ≡ (arctan [2( M H − m L ) / Γ i ] + arctan [2( m U − M H ) / Γ i ]) where m L and m U are the lower and up-per limits of the signal window, Γ i is the width of the signal distribution in parameter space point i estimated from Γ i = q Γ H i + Γ with Γ = 30 GeV being the width of the m b -distributionsfrom finite detector resolution. (Notice then that c M H is a normalisation determined from sce-narios with Γ H i = 30, 50 and 200 GeV for the different H masses.) Thus we approximatethe cross-section in the m b window as σ peak = c M H σ H → hh . As a further requirement we alsoimposed that at least 50% of the signal cross-section after the optimal cuts comes from the H → hh resonance such that the would-be-signal would not be obscured by other non-resonantcontributions.The distributions of the signal cross-sections obtained in this way are given in Fig. 10. Forillustration, the 5 σ limits at LHC, assuming an integrated luminosity of 300 fb − , σ peak > , 1.5 (1.3-1.8) and 0.8 (0.7-1.0) fb, for M H = 300, 500 and 700 GeV respectively, are Note that the efficiency of these is rather insensitive to the actual Higgs mass values, so that we have usedthe same set for any choice of the latter. The ranges given within parenthesis in this paragraph have been obtained by varying the factorisa-tion/renormalisation scale for the background by a factor of two around the default value, which makes thecorresponding cross-section decrease by 30 % or increase by 50 % respectively. The reason for this is that, beingessentially a six-jet cross-section, the background rate is proportional to α s and it is therefore quite sensitive tothe renormalisation scale. We also note that our default scale ( p T = P n p T j /n ) has conservatively been chosento be small so, if anything, our estimate of the final signal-to-background rates should be regarded as conserva-tive. In a real experiment one should of course attempt to use the sidebands for background normalisation. b window B events σ maxpeak [fb] S events S/ √ B @LHC S/ √ B @SLHC280 – 340 (GeV) 102 15.1 283 28 89460 – 540 (GeV) 30 3.8 71 13 41660 – 740 (GeV) 8 0.35 6.6 2.3 7.4Table 3: Number of events and significances for M h = 115 GeV and M H = 300, 500, 700 GeV inthe respective best case scenarios, for a 4 b -tagging efficiency of (50%) and after the optimisedcuts of eq. (8). The assumed integrated luminosity at LHC and SLHC are 300 fb − and3000 fb − , respectively.also illustrated and the fractions of parameter space points which gives cross-sections largerthen this are 27 (24-30), 8 (5-12) and 0%, respectively. The corresponding numbers for theSLHC with 3000 fb − are 43 (41-45), 31 (28-33) and 2 (0-2)%, respectively. Thus even atthe SLHC we find no scope of observing a M H = 700 GeV resonance in the channel underinvestigation.Finally we have also investigated the effects of restricting the allowed parameter space fromtree-level unitarity by putting harder constraints on the maximal eigenvalue of the scatteringmatrices, Λ max . For this purpose, Fig. 11 shows the signal cross-sections obtained in the scan asa function of Λ max . From the figure it is clear that the results (at least for M H = 300 and 500GeV) are not sensitive to the precise value used for applying the unitarity constraint. On theother hand, applying a much harder constraint of the order Λ max < ∼ max < π )essentially leads to that the sensitivity for detection at the LHC is more or less washed out for M H = 300 (500) GeV. The same also holds at the SLHC assuming an integrated luminosity of3000 fb − . We would like to conclude our paper by stating that, at both the LHC and SLHC, there existsa great potential to extract a H → hh → b resonance when M h is constrained in the vicinity of115 GeV. This is a crucial result if one recalls that the detection of a sole Higgs resonance andconsequent extraction of an M h value may not point unambiguously to the underlying modelof EWSB, not even in presence of further measurements of the heavier Higgs masses, M H , M A and/or M H ± .For example, the 2HDM considered here may be realised in a configuration wherein allvisible Higgs masses are degenerate with those of the MSSM. Under these circumstances, wehave proved that • it is not possible to extract an H → hh → b resonance from vector-boson fusion in theMSSM (not even if M H is known) whilst11 the opposite case is true in a substantial fraction of the parameter space of our 2HDM(even if M H is not known), thereby enabling one to possibly measure the triple-Higgscoupling λ Hhh .The latter is a Lagrangian term, which is different between these two models even when theirpatterns of Higgs masses and couplings to SM objects are the same, that would give a uniqueinsight into the underlying EWSB mechanism.To be more specific our results show that in the most favourable scenario with M H = 300GeV up to 27 (43) % of the parameter space would give a 5 σ signal at the (S)LHC assuming anintegrated luminosity of 300 (3000) fb − when using the standard tree-level unitarity require-ment on the J = 0 partial waves, Re( a ) < /
2. These results are not sensitive to the precisevalue used for applying the unitarity constraint, albeit for very strong constraints the sensitivityfor detecting the signal goes away. In the case of M H = 500 GeV the fraction of parameterspace probed is smaller with up to 8 (31) % giving a 5 σ signal, whereas for M H = 700 GeVthere is essentially no sensitivity at all.Despite we lack a full Monte Carlo simulation we believe to have incorporated the mostcritical aspects of the latter so that we do not expect more realistic studies (including partonshower, hadronisation, heavy hadron decays and detector effects) to affect too strongly ourconclusions.Finally, we are currently pursuing other work along the directions outlined here, coveringthe case of lightest (neutral) Higgs boson pair production in the case of Higgs-strahlung and inassociation with heavy quarks [30]. Acknowledgments
We are all grateful to the (formerly) CERN Theory Division for hospitality when this work wasstarted. Several discussions with M.L. Mangano are acknowledged. SM and FP thank JR forhis kind hospitality in Uppsala in September 2006. FP thanks SM for his stay in Southamptonin March and June 2007. RP acknowledges the financial support of the MIUR under contract2006020509 004 and of the RTN European Programme MRTN-CT-2006-035505 (HEPTOOLS,Tools and Precision Calculations for Physics Discoveries at Colliders). SM acknowledges thelatter too for partial funding. RP’s research was partially supported by the ToK Program”ALGOTOOLS” (MTKD-CT-2004-014319). FP thanks the CERN Theory Unit for partialsupport.
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Figure 1: Feynman diagrams for q q → q q h h . Depending on the (anti)quark flavour combi-nation, the W ± - and Z -mediated graphs may not interfere. Besides, for final state (anti)quarksof different flavours, only half of the diagrams survive.15igure 2: The masses of the neutral CP-even Higgs bosons as a function of the CP-odd one,for two choices of tan β , assuming the MSSM in Maximal Mixing configuration.16igure 3: The relevant couplings of the neutral CP-even Higgs bosons entering the productionprocess in (1) as a function of the CP-odd Higgs boson mass, for two choices of tan β , assumingthe MSSM in Maximal Mixing configuration. 17igure 4: The inclusive cross-sections (as defined in the text) for vector-boson fusion in (1),followed by hh → b ¯ bb ¯ b decays, as a function of the CP-odd Higgs boson mass, for two choicesof tan β , assuming the MSSM in Maximal Mixing configuration.18 -1 -0.5 0 0.5 1 sin as [ fb ] -1 tan bs [ fb ] -10 -5 0 5 10 l s [ fb ] s [ fb ] m A [ GeV ] s [ fb ] m H+ [ GeV ] s [ fb ] G H [ GeV ] qq → qqhh → qqbbbb in 2HDM, m h =115 GeV, m H =300 GeV Figure 5: The dependence of the inclusive cross-section qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b in the2HDM under consideration on the different parameters when scanning over 10000 parameterspace points for M h = 115 GeV and M H = 300 GeV. (Note that M A and M H ± are freeparameters.) 19 -1 -1 -0.5 0 0.5 1 sin as [ fb ] -1 -1 tan bs [ fb ] -1 -10 -5 0 5 10 l s [ fb ] -1 s [ fb ] m A [ GeV ] -1 s [ fb ] m H+ [ GeV ] -1 s [ fb ] G H [ GeV ] qq → qqhh → qqbbbb in 2HDM, m h =115 GeV, m H =500 GeV Figure 6: The dependence of the inclusive cross-section qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b in the2HDM under consideration on the different parameters when scanning over 10000 parameterspace points for M h = 115 GeV and M H = 500 GeV. (Note that M A and M H ± are freeparameters.) 20 -1 -1 -0.5 0 0.5 1 sin as [ fb ] -1 -1 tan bs [ fb ] -1 -10 -5 0 5 10 l s [ fb ] -1 s [ fb ] m A [ GeV ] -1 s [ fb ] m H+ [ GeV ] -1 s [ fb ] G H [ GeV ] qq → qqhh → qqbbbb in 2HDM, m h =115 GeV, m H =700 GeV Figure 7: The dependence of the inclusive cross-section qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b in the2HDM under consideration on the different parameters when scanning over 10000 parameterspace points for M h = 115 GeV and M H = 700 GeV. (Note that M A and M H ± are freeparameters.) 21igure 8: The distribution of the next-to-minimum b ¯ b invariant mass (left) and of the minimumtagging jet transverse momentum (right) for the signal (cross section for qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b in a close to best-case scenario for M H = 300 GeV) and the background. The basiccuts of eqs. (5)–(7) are imposed. 22igure 9: The differential cross-section dσ ( qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b ) /dm b in the best casescenarios for M H = 300, 500 and 700 GeV obtained when scanning over the available param-eter space restricting the width Γ H to be less than 30, 50 and 200 GeV, respectively. Whencalculating the signal distributions the actual widths have been assumed to be Γ H = 30, 50and 200 GeV, respectively. 23 -3 -2 -1 -3 -1 n s [ fb ] total with optimal cutsH → hh contribution s peak , H res-contr > ← s -limit27 %qq → qqhh → qqbbbb in 2HDM, M h =115 GeV , M H =300 GeV -3 -2 -1 -3 -1 n s [ fb ] total with optimal cutsH → hh contribution s peak , H res-contr > ← s -limit8 %qq → qqhh → qqbbbb in 2HDM, M h =115 GeV , M H =500 GeV -3 -2 -1 -3 -1 n s [ fb ] total with optimal cutsH → hh contribution s peak , H res-contr > ← s -limitqq → qqhh → qqbbbb in 2HDM, M h =115 GeV , m H =700 GeV Figure 10: Distributions of the resulting cross-sections qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b in the 2HDMunder consideration using the optimal cuts obtained in a scan over 10000 parameter space points(the area is normalised to 1 for the cross-section with optimal cuts) for three different sets ofHiggs boson masses as indicated in the respective plots. The solid line shows the results withoptimal cuts, the dashed line shows the resonant contribution from the H → hh processes andthe long dashed line shows the resonant contribution in the respective signal windows requiringthat at least 50% of the cross-section comes from the H → hh resonance. The vertical linecorresponds to the 5 σ -limit at LHC assuming 300 fb − and the integral of the curves to theright of it gives the percentage of parameter space points where the resonant cross-section islarger than this. 24 -3 -2 -1
110 0 5 10 15 20 25 30 s [ fb ] L max qq → qqhh → qqbbbb in 2HDM, M h =115 GeV , M H =300 GeV -3 -2 -1
110 0 5 10 15 20 25 30 s [ fb ] L max qq → qqhh → qqbbbb in 2HDM, M h =115 GeV , M H =500 GeV -3 -2 -1
110 0 5 10 15 20 25 30 s [ fb ] L max qq → qqhh → qqbbbb in 2HDM, M h =115 GeV , M H =700 GeV Figure 11: The distributions in resulting signal cross-sections qq ( ′ ) → qq ( ′ ) hh → qq ( ′ ) b ¯ bb ¯ b inthe 2HDM under consideration using the optimal cuts obtained in a scan over 10000 parameterspace points as a function of the maximal eigenvalue Λ max of the scattering matrix for threedifferent sets of Higgs boson masses as indicated in the respective plots. The upper (lower)horizontal line corresponds to the 5 σ -limit at (S)LHC assuming 300 fb − (3000 fb −1