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Collider Probes of the MSSM Higgs Sector with Explicit CP Violation

We investigate the hadron collider phenomenology of the Minimal Supersymmetric Standard Model (MSSM) with explicit CP violation for Higgs bosons that can be observed in Standard Model search channels: W/ZH(->b-bbar) at the Tevatron, and gg->H(->gamma-gamma), t-tbar-H(->b-bbar) and WW->H(->tau+tau-) at the LHC. Our numerical analysis is based on a benchmark scenario proposed earlier called CPX, which has been designed to showcase the effects of CP violation in the MSSM, and on several variant benchmarks. In most of the CPX parameter space, these hadron colliders will find one of the neutral MSSM Higgs bosons. However, there are small regions of parameter space in which none of the neutral Higgs bosons can be detected in the standard channels at the Tevatron and the LHC. This occurs because the neutral Higgs boson with the largest coupling to W and Z bosons decays predominantly into either two lighter Higgs bosons or a Higgs boson and a gauge boson, whilst the lighter Higgs boson has only small couplings to the W and Z bosons and the top quark. For other choices of CP-violating parameters, all three neutral Higgs bosons can have significant couplings to W and Z bosons, producing overlapping signatures: these may or may not be distinguishable from backgrounds. The existence of these regions of parameters provides a strong motivation for a detailed experimental simulation of these channels.


Introduction
Detailed studies of the mechanism of electroweak symmetry breaking top the agendas of operating particle accelerators, as well as those under construction or being proposed for the near future. One paradigm for many of these studies is provided by the minimal supersymmetric extension of the Standard Model (MSSM) [1]. There is increasing interest in the possibility that the MSSM includes explicit CP violation, which may provide opportunities to probe CP violating parameters through the Higgs sector [2,3,4,5,6,7,8,9,10,11,12,13].
Assuming universality of the soft supersymmetry (SUSY)-breaking gaugino masses m 1/2 , scalar masses m 0 and trilinear parameters A 0 at the Grand Unification Theory (GUT) input scale, the MSSM then provides only two new sources of CP violation in addition to the Kobayashi-Maskawa phase of the Standard Model (SM). In a suitable convention, these may be represented as the phases of complex parameters m 1/2 and A 0 . The values of these phases are constrained by the upper limits on the electric dipole moments (EDMs) of the electron, neutron [14,15] and mercury atom [16] and other measurements. The ongoing probes of CP violation at B factories can also test the consistency of the above minimal CP-violating supersymmetric scenario [17,18].
In the MSSM with explicit CP violation, the effect of CP violation on the Higgs sector enters beyond the Born approximation [2,3]. The effects [3,4,5,6,7] induced by arg(A 0 ) and arg(m 1/2 ) on the MSSM Higgs-boson masses have been studied in some detail [6,9]. Several interesting phenomenological implications for the Higgs bosons emerge in this minimal CPviolating framework of the MSSM. The three neutral Higgs bosons (h and H, which have scalar couplings to fermions at the tree level, and A with pseudoscalar tree-level couplings to fermions), all mix together in the presence of CP violation. The resulting three physical mass eigenstates have mixed CP parities. We denote these by H 1,2,3 in order of increasing masses, i.e., M H 1 ≤ M H 2 ≤ M H 3 . As a consequence of CP violation, all the three neutral Higgs bosons can now have tree-level couplings to pairs of W ± and Z bosons. The Higgs couplings to the gauge bosons obey the sum rules where g HW + W − and g HZZ are the Standard Model couplings. Additionally, there is an important complementarity relation between the H i ZZ and H i H j Z couplings: where ε ijk is the fully anti-symmetric Levi-Civita tensor. We have shown that the phenomenological consequences of (1.1) and (1.2) can be dramatic [6]. In particular, (1.1) implies that the H 1 ZZ coupling can be significantly suppressed, thus raising the possibility that a relatively light H 1 boson, with a mass even as low as 60 GeV, might have escaped detection at LEP 2. On the other hand, (1.2) leaves open the possibility of H 1 H 2 production at LEP 2, as discussed below. However, this brief discussion highlights the fact that a large part of the parameter space related to the Higgs sector must be re-explored in the presence of CP violation.
In this paper, we investigate systematically the physics potential of the Standard Model Higgs boson searches at the Tevatron collider and the LHC for observing neutral Higgs bosons in the MSSM with explicit CP violation. We focus on Standard Model search channels, since the existence of a light Higgs boson with a significant coupling to W and Z bosons is a prediction of weak-scale supersymmetry. The present work goes beyond previous work on this subject [10] by incorporating the most important radiative corrections to the Higgs sector and performing a numerical analysis based on the most realistic simulations of Standard Model Higgs boson searches at the Tevatron and LHC. Our phenomenological study is performed mainly within the context of the CPX scenario [8], which was chosen to showcase the possible effect of CP violation in the Higgs sector of the MSSM. Given the limits from the LEP collider and existing simulations of the capabilities of the Tevatron and the LHC, we show that there are small regions of parameters in which all three neutral Higgs bosons escape detection. In these regions of parameters, the heavier Higgs bosons with the dominant coupling to the W and Z boson decay into Higgs boson with a tiny coupling to the W and Z boson, or into a lighter Higgs boson and a gauge boson. Outside these small regions of parameters, our analysis indicates, even in the presence of CP violation, the discovery of at least one neutral MSSM Higgs boson in a set of complementary detection channels. This provides a strong motivation to extend the experimental simulations to Higgs boson decay channels containing lighter Higgs boson states. For other choices of CP-violating parameters, we find that all three neutral Higgs bosons can have significant couplings to W and Z bosons, while being closely spaced in mass. This situation also requires careful investigation, since some signals may be affecting the background estimates for other signals.
Our numerical analysis is performed using the code CPHDECAY [19], which is based on an extension of the code HDECAY [20] to calculate the Higgs boson decay properties, and of the code cph [6,21] to calculate the physical Higgs spectrum and mixing angles. The latter code includes the dominant one-and two-loop CP-violating contributions induced by the supersymmetric particles. At one loop, these contributions depend mostly on the phase of the trilinear mass parameters of the third-generation quarks, A t,b , relative to that of the supersymmetric mass parameter µ * . At the two-loop level, a significant dependence on the relative phase between µ * and the gluino mass parameter mg also appears. Electroweak corrections are incorporated at the leading-logarithmic level. In a phase convention where the µ parameter is positive, the input parameters for the code are the top-quark mass m t , the µ parameter, the soft supersymmetry-breaking masses M 2 Q , M 2 t,b of the third-generation squarks, the third-generation soft trilinear couplings A t,b together with their respective phases, the gaugino masses and the phase of the gluino mass parameter mg. Apart from its present application to hadron colliders, CPHDECAY may also serve as a useful tool for analogous studies at future linear e + e − colliders [11] or a muon collider [12], or for generic Higgs boson studies [13]. Section 2 contains a complete discussion of the effective couplings of the neutral and charged Higgs bosons to quarks and of the Higgs self-couplings; full expressions are given in Appendix A. The one-loop corrected Higgs couplings are essential, as they determine the branching ratios of the Higgs bosons in the presence of CP violation. Section 3 contains a discussion of the LEP results in the search for neutral Higgs bosons in the MSSM with explicit CP violation. Section 4 contains a numerical analysis of the different collider detection channels, showing graphically their complementary properties for the detection of Standard Model-like Higgs bosons in the MSSM with explicit CP violation. We restrict ourselves to the existing experimental simulations, identifying regions of parameters at small values of tan β and of the charged Higgs boson mass, for which none of the neutral Higgs bosons can be detected at the Tevatron or the LHC via the standard search channels. Our conclusions are presented in Section 5. Appendix A contains detailed formulae for couplings and Appendix B summarizes the information from detector simulations on which we base our analysis [22,24,23,26].

Effective Higgs-Boson Couplings
In order to study the collider phenomenology of the Higgs bosons in the MSSM with explicit CP violation, we must first calculate the couplings of the Higgs bosons to Standard Model and MSSM particles. In this section, we present explicitly those couplings which are the most important inputs to our numerical calculations. In particular, we review the effective couplings of the neutral Higgs bosons to quarks, and we present the effective Higgs selfcouplings. We also give the corresponding effective couplings of the charged Higgs boson to the up and down quarks, even though they are not needed for the present analysis.

Effective Higgs-boson couplings to quarks
Following our previous conventions throughout this paper [6], we define the physical scalar components of the Higgs superfields H 1 and H 2 as , respectively, where τ 2 is the usual Pauli matrix. The weak Higgs eigenstates are related to their mass eigenstates H + and H 1,2,3 through the orthogonal transformations: where G + and G 0 are the would-be Goldstone bosons associated with the W + and Z bosons, respectively, and O is a 3-by-3 orthogonal matrix that describes the mixing of the neutral Higgs states in the presence of CP violation [3].
Our starting point is the effective Lagrangian for the neutral Higgs-boson couplings to the u-and d-type quarks. We neglect quark-mixing effects, as their direct relevance to Higgs searches is not important. 1 Moreover, we assume that there is negligible mixing between the different generations of squarks, and we include weak-interaction effects only at the leading-logarithmic level.
Under these assumptions, the effective Lagrangian reads: where δh d,u /h d,u and ∆h d,u /h d,u are threshold radiative effects given by We would like to emphasize that no other work includes the ∆h b corrections, and that these have important phenomenological implications, as we discuss below.
In (2.4)-(2.7), α s = g 2 s /(4π) is the SU(3) c fine-structure constant, I(a, b, c) is the function , (2.8) and mq 1 ,q 2 (with q = u, d) are the squark masses, with In (2.3), the complex Yukawa couplings h d and h u are defined in a way such that the d-and u-quark masses are real and positive after the inclusion of radiative corrections, namely Substituting (2.2) and (2.10) into (2.3), the effective Lagrangian for the H iq q couplings can be written in the general form: 12) with [6,15] g S H id d = Re (2. 16) In the above, we have used the abbreviation: κ q = (∆h q /h q )/[1 + (δh q /h q )], for q = u, d, and the fact that u-and d-quark masses are real and positive, so that Im m d ∝ Im [h d + (δh d ) + (∆h d ) tan β] = 0 and Im m u ∝ Im [h u + (δh u ) + (∆h u ) cot β] = 0.
Though our phenomenological analysis is restricted to the neutral Higgs-boson sector, we consider for completeness the charged-Higgs-quark sector. The effective Lagrangian describing the interactions of the charged-Higgs fields φ + 1,2 with u and d quarks is given by The quantitiesδh u,d and∆h u,d contain the respective threshold radiative effects that affect the charged-Higgs couplings to quarks. Though they are related to the δh q and ∆h q given above for the neutral Higgs sector, they differ by SU(2) L -breaking terms.
To facilitate our presentation of the analytic forms ofδh q and∆h q , we first set up our conventions for the squark-mixing parameters and define some auxiliary functions. Thus, the left-and right-handed squark fieldsq L andq R are related to the physical fieldsq 1,2 through the transformations:q (2.18) where δ q = arg (A q − R q µ * ) and c q = cos θq and s q = sin θq are squark-mixing angles. Analytic expressions for c q and s q may be found in Appendix A of [9]. We introduce the following short-hand form for the squark-loop integrals: 19) where the indices i and j refer to the third-generation up-and down-squark mass eigenstates.
With the aid of (2.19), the following two auxiliary functions can be defined: Again, in the limit of neglecting mixing of generations, the threshold radiative effects related to the charged Higgs sector can be expressed in terms of (2.20) and (2.21) as follows: Then, the effective Lagrangian for the charged Higgs-boson couplings to u-and d-type quarks is given by (2.26) where P ∓ = [1 ∓ γ 5 ]/2, and (2.28) The quantities κ u,d are given after (2.16), whileκ u,d and ρ u,d in (2.27) and (2.28) are defined as follows:κ 29) with q = u, d, respectively.
It is apparent that the effective couplings of the neutral and charged Higgs bosons to quarks can be written in a form that depends entirely on the parameters κ u,d ,κ u,d and ρ u,d . It is technically interesting to note that all these quantities are renormalization-scale independent up to the one-loop order in our resummation approach. More explicitly, κ u,d andκ u,d are proportional to the non-holomorphic radiative corrections (∆h u,d /h u,d ) and (∆h u,d /h u,d ), respectively, which are manifestly scale-independent. The quantities ρ u,d are measures of SU(2) L breaking in the up-and down-Yukawa sectors. The parameters ρ u,d are also renormalization-scale independent, as a consequence of the SU(2) L gauge invariance of the original theory before spontaneous symmetry breaking. The scale independence of ρ u,d can be also understood by simply noticing that the ultra-violet infinities, e.g., of δh d andδh d are equal, as they emanate from the charged and neutral Higgs components of the same gauge-invariant operator, which is Q H 1 D in this case. The scale independence of these quantities has some analogies with that of Veltman's ρ parameter that characterizes the isospin breaking in the SM gauge sector through the difference between the W W and ZZ self-energies, that we do not develop further here.
Finally, we should bear in mind that the validity of the effective neutral and charged Higgs-boson couplings to quarks depend on two kinematic conditions: (i) the soft supersymmetry-breaking masses should be much larger than the electroweak scale and (ii) the external momenta of the quarks and Higgs bosons, e.g., p q,H , have to be sufficiently smaller than the soft supersymmetry-breaking mass scale M SUSY , so that they can be neglected when expanding the vertex functions in powers of p 2 q,H /M 2 SUSY .

Effective Higgs-boson self-couplings
To determine the effective Higgs self-couplings, one needs to know the analytic forms of both the proper vertex and self-energy graphs. So far, there is complete information only for the latter contributions in the MSSM with explicit CP violation [6,9], while the former effects have been computed in [3], in an expansion of the effective potential up to operators of dimension 4.
Here, we combine the above two pieces of information to obtain approximate analytic expressions for the trilinear and quadrilinear Higgs-boson self-couplings. 2 Our analytic expressions are not limited to the MSSM case, but can be applied equally well to the general Two-Higgs-Doublet Model (2HDM) with explicit CP violation.
To start with, we first write down the effective Lagrangian containing all operators of dimension 4: where Φ T 1,2 = (φ + 1,2 , φ 0 1,2 ) are the two Higgs doublets, with their individual charged and neutral components defined in (2.1) and (2.2). After (2.1) and (2.2) have been inserted into (2.30), the effective trilinear and quartic Higgs self-couplings may be cast into the form (in the unitary gauge): GeV is the SM vacuum expectation value. The couplings Γ 3H ijk and Γ 4H + are given by (2.33) characterize the proper vertex corrections to the corresponding Higgs self-couplings. We observe that, exactly as is the case for Γ 4H + in (2.34), all these newly-introduced couplings depend on the radiatively-corrected quartic couplings λ 1 , λ 2 , . . . , λ 7 . Their analytic forms are given in Appendix A.

Constraints on the CPX scenario from LEP searches
The OPAL Collaboration [29] has reported preliminary results on the search for Higgs bosons in the MSSM with explicit CP violation, taking the parameters to be those defined in the CPX scenario [6], where M SUSY is the characteristic third generation squark mass scale, and mB, mW are the bino and wino masses, respectively.
OPAL presented exclusion regions in the M H 1 -tan β plane for different values of the CP-violating phases, assuming arg A t = arg A b = arg mg = φ CPX , with φ CPX = 90 • , 60 • , 30 • , and 0 • . Quite generally, there is no reason to expect the equality of the phases of the trilinear scalar couplings and the gaugino masses, nor to assume that φ CPX ≤ 90 • , and our analysis departs from this assumption.
To reproduce the results of the OPAL analysis, we rely on the combined ALEPH-DELPHI-L3-OPAL (ADLO) results for the ZH SM (→ bb) channel [30] and the hA → 4b channel [31]. Although the experimental hA analysis has been done for approximately equal values of the masses of the neutral Higgs bosons h and A, there is no major loss in efficiency when the splitting between these masses becomes larger [32], and therefore it is safe to apply those limits to a more generic set of masses H i , H j in the CP-violating scenario.
Using the results of these experimental analyses, we have generated Figure 1, based on the CPX scenario with M SUSY = 0.5 TeV. In this plot, the light grey area covers the theoretically allowed region of parameter space (consistent with electroweak symmetry breaking), the medium grey region shows the exclusion from ZH i final states, the dark grey region is excluded by the search for Z * → H i H j → 4b final states, and the black region is excluded by both searches.
When comparing the results in Fig. 1 with the OPAL results [29], we note that our limits on tan β are somewhat stronger than the OPAL limits. This can be attributed to the fact that our analysis is based on combined ADLO limits obtained from all four experiments [31]. The results of our analysis show also a somewhat better reach than the OPAL one in the H i H j production channel. This can again be attributed to our use of the combined results of all four experiments and the fact that, following the above discussion, we have assumed no deterioration of the H i H j signal for M H i = M H j . We expect that our CPX estimates will be quite similar to the final LEP combined results.
We see from the cases displayed in Fig. 1 that the case of vanishing phases is most severely constrained by the LEP data. Most of the coverage arises from the ZH 1 or H 1 H 2 processes. The appearance of two 'fingers' in the ZH 1 coverage at large tan β for vanishing phases arises from the shape of the LEP exclusion curve, which is distorted by a marginal excess in the region m h ∼ 80 − 90 GeV [30]. The case of phase 30 • is very similar, and the 60 • case has analogous features.
For significant values of the phases, there are regions of parameters in which the lightest neutral Higgs boson is very weakly coupled to the Z gauge boson, and light enough for the heavier neutral Higgs states to decay into a pair of H 1 bosons. In these regions of parameters, the heavier Higgs states decay into a final state containing four b quarks. Therefore, the dominant production and decay modes contain 6 jets in the final state. The current experimental strategy is to employ the standard 4-b analysis by forcing (with suitable jet definitions) the 6-jet topologies into 4-jet ones. This leads to a very low efficiency for the real 6-jet signal, and hence a significant region of CPX parameter space where light Higgs bosons may exist but are not excluded by the current analyses. In our analysis of the MSSM parameter space excluded by Higgs boson searches at LEP, we have not attempted to simulate the signatures associated with the decay of heavy neutral Higgs bosons into lighter ones. Nonetheless, we are hopeful that a dedicated 6-jet analysis could cover this region. Returning to Fig. 1 Fig. 2 is a study of the interplay of phases for the trilinear parameters A t,b and the gluino mass parameter. The appearance of substantial phases to the parameters can significantly modify the collider phenomenology (the phase of A t providing the dominant effect). The addition of a phase to the gluino mass parameter, however contributes to CPviolating effects only at the two-loop level and its effect is comparatively smaller, as can be seen in Fig. 2. For the case of arg(A t,b ) = 90 • , it is clear from Fig. 2 that, for tan β ≃ 4-5, as the Higgs H 1 becomes lighter, it starts decoupling from the Z. There is a significant region of parameter space where, although heavier, H 2 is in the kinematic region accessible to LEP and couples relevantly to the Z gauge boson. For M H + < ∼ 130 GeV, H 1 becomes light enough that the decay H 2 → H 1 H 1 dominates over H 2 → bb, and therefore H 2 detection becomes difficult. Finally, for arg(A t,b ) = 135 • and arg(mg) = 90 • , and for moderate or large values of the charged Higgs mass, all neutral Higgs bosons rapidly become sufficiently heavy to be out of the reach of LEP for moderate and large values of tan β.
In Fig. 3 we display results for the CPX scenario, for equal values of the A t,b and gluino mass phases equal to 60 • and 140 • . The region of parameters left uncovered by LEP for low values of tan β and of the charged Higgs boson mass, for phases equal to 60 • , is seen more clearly than in Fig. 1. It is also apparent that, as happens for a CP-violating phase equal to 90 • , close to the region left uncovered by LEP, there is a large region of parameters covered by the ZH 2 channel.
For phases equal to 140 • , we find a peculiar behavior of the covered region of parameters for large values of the charged Higgs mass, which can be traced to the behavior of the H 1 mass shown in Fig. 4. For small and moderate values of tan β, it has the standard behavior of the MSSM without explicit CP violation, increasing with tan β. For tan β above about 10, however, it starts decreasing due to the effect of radiative corrections from the sbottom sector, which involve the bottom Yukawa coupling. The bottom quark Yukawa corrections are screened for sufficiently large values of tan β, causing the mass of the H 1 to increase again and eventually become larger than the LEP kinematic limit.
This screening phenomena occurs because the sbottom-induced radiative corrections are complex. At large values of tan β, the bottom-quark Yukawa coupling h b is approximately related to the bottom mass and to the Yukawa corrections ∆h b by The negative corrections to the Higgs mass squared coming from sbottom loops are proportional to the fourth power of the modulus of h b . For equal values of the phases of the gluino and of the trilinear mass parameter A t , i.e. arg(A t ) = arg(mg) = φ CPX , the modulus of h b is given by For cos(φ CPX ) ≥ 0, the above expression is a monotonically increasing function of tan β, meaning that the negative corrections induced by sbottom loops become larger for larger values of tan β.
On the other hand, for negative values of cos(φ CPX ), the expression for |h b | has a maximum at a value of (3.38) meaning that the negative effects on the Higgs mass induced by sbottom loops are most pronounced at intermediate values of tan β, and are reduced again at large tan β. For the particular case under consideration, we have |∆h b |/h b ≃ 1/20 and φ CPX = 140 • , with cos(φ CPX ) ≃ 1/ √ 2, while |h b | < ∼ 1 stays well within the perturbative range. Thus, lower values of the Higgs mass are obtained for tan β ≃ 25 than for either smaller or larger values of tan β. This feature is seen clearly in Fig. 4, and explains the stronger coverage of LEP at this intermediate region of tan β, as seen in panel (b) of Fig. 3.
Apart from the above property, the only salient feature is that in the region between the one covered by LEP in the ZH 1 channel and that covered by LEP in the H 1 H 2 channel, H 1 becomes light but couples very weakly to the Z boson.
The last two scenarios considered in Fig. 3 are interesting since, for moderate values of the charged Higgs mass of about 150 GeV, the three neutral Higgs bosons become quite similar in mass and share approximately equal couplings to the weak gauge bosons. We call these tri-Higgs-gauge coupling (3HGC) scenarios, which are denoted in short as 3HGC1 and 3HGC2. As we discuss below, the searches at hadron colliders become particularly difficult in 3HGC1 and 3HGC2. The excluded region for moderate and large values of the charged Higgs-boson mass and large values of tan β in the scenario 3HGC2 is related to the Higgs-sbottom coupling effect discussed above.
To summarize the results from LEP searches, the phenomenology is mainly sensitive to the values of g H i ZZ and the kinematic limit on the Higgs boson masses. The exceptions are the cases when the decay H 2 → H 1 H 1 becomes relevant at small tan β and when radiative corrections extend the LEP coverage for certain choices of phases φ CPX at large tan β. A Higgs boson with the Standard Model coupling can be excluded if M H i < 115 GeV, and even a low-mass Higgs boson can be excluded if |g H i ZZ | > ∼ .22 relative to the Standard Model coupling. Note that the application of the ZH i and H i H j results to the CPX scenarios treats each potential signature independently. Any possible complications from one potential signal effecting the background estimate for another (and vice versa) are ignored. We are not aware of any experimental analysis of such a situation.

Higgs Boson Searches at Hadron Colliders
Experiments at the Tevatron and LHC will probe the MSSM Higgs sector beyond the kinematic and dynamical reach of LEP. At these hadron colliders, we have considered several channels for Higgs boson H i searches, where H i stands for H 1,2,3 : In the absence of reliable experimental simulations, as in the case of LEP, we have not modified our analysis to account for the decay H j → H i H i → 4b (j > i), should it become relevant. The search channels (a) − (d) are considered to be the most promising for observing a Standard Model Higgs boson, and have been the most thoroughly analyzed and simulated. Since the MSSM Higgs sector has a decoupling limit, where there is a light Higgs boson with properties almost indistinguishable from a SM Higgs boson, it is appropriate to apply these SM analyses to the MSSM case. However, one would have to be careful in interpreting a signal in some of these channels. For example, signature (d) may arise when H i is replaced by a pseudo-Goldstone boson, as in Technicolor models. Signature (a) may arise, for example, when H i is replaced by a top-pion, as in a Topcolor model. Only observation of channels (b) and (c) would clearly indicate that the scalar H i is associated with EWSB. Signature (a) may indicate that the scalar responsible for EWSB also generates fermion masses. Nonetheless, if signature (a) or (d) were observed with the rate expected in a weakly-coupled theory, it could be considered strong evidence for a fundamental Higgs scalar or scalars. In the present work, we do not analyze those signatures associated with non-SM-like Higgs bosons, such as charged Higgs or pseudoscalar Higgs bosons, since their interpretation would be ambiguous. Figure 5 shows the coverage of the M H 1 -tan β plane by the Tevatron W/ZH i (→ bb) search (45 • lines) and the combined LHC coverage for the ttH i (→ bb), W W → H i (→ τ + τ − ) and gg → H i (→ γγ) searches (135 • lines) for the same values of the CP-violating phases as chosen in Fig. 1. For the Tevatron, we show the 3σ evidence coverage with 5 fb −1 , while, for the LHC, we show the 5σ discovery coverage for the γγ and bb channels with 100 fb −1 and the τ + τ − channel with 30 fb −1 . The previous LEP 95% C.L. exclusion is also included (medium grey) superimposed on the theoretically allowed region (light grey). While the Tevatron and LHC searches are adequate for extending the coverage into the large-tan β region, the region of small M H 1 and low tan β for φ CPX = 90 • , 60 • remains uncovered. Although the persistence of these regions can be clearly traced to the decay of the heavier Higgs bosons into lighter ones, one could inquire why the LHC cannot see the lighter Higgs boson in the ttH 1 and/or the gluon fusion channels. The reason is that, in the same region of parameters, H 1 couples weakly to the top quark, as well as negligibly to W and Z bosons. Therefore, not only is the ttH 1 production channel suppressed but the loop-induced gluon fusion production and decay into photons is suppressed as well. Furthermore, since tan β is small, H 1 would likely not be observed in the pseudoscalar Higgs channel bbH 1 (→ bb).
If, contrary to our expectations, a dedicated LEP analysis cannot cover this problematic region, there still remains the possibility of observing H 2 → H 1 H 1 decays (when H 2 is otherwise SM-like) at hadron colliders. Since H 2 is SM-like, there still is a substantial production cross section. Dedicated searches should be able to identify 4b final states with high efficiency and substantially less background than for the 2b case. Various cases of Higgs pair production, such as gg → H → hh/AA, have been considered in previous studies for the Tevatron and LHC, but mostly at the parton level [33,34]. The studies would have to be updated for the case of explicit CP violation. Searches for W/ZH 2 (→ 4b) or ttH 2 (→ 4b) may be more promising, if only because of the reduced backgrounds from requiring the leptonic decay of a W boson.
Outside these small regions of parameters, the search for neutral Higgs bosons at the Tevatron and LHC colliders are complementary and sufficient to cover the full parameter space in one or several channels. The Tevatron does not provide any additional coverage alone over all three LHC search channels. However, if we just focus on the W/ZH i and W W → H i channels -the only two which are dependent on the coupling g H i W W -then both are nearly sufficient to cover the entire parameter space: for 90 • , M H 1 ∼ 80 GeV and tan β ∼ 10, a small region is left uncovered without ttH i . This complementarity reflects the different production and decay channels that are being used in the search for neutral Higgs bosons, and is similar to that in the CP-conserving scenario [25]. For instance, there are regions of parameter space where either the bb or the τ + τ − decay branching ratio of the lightest neutral Higgs boson is suppressed with respect to the Standard Model ones, while the other neutral Higgs bosons are heavy and weakly coupled to the gauge bosons. These suppressions tend to occur for large values of tan β. The large values of µ chosen in our examples maximize the CP-violating effect, but also determine that for large values of tan β the quantum corrections to the bottom Yukawa coupling are large and sizeable, causing a visible displacement of the regions of suppressions of each coupling. Since, for a light enough Higgs, these two branching ratios tend to be very important, a suppression of one of them tends to enhance the other. We discuss specific examples below. For smaller values of µ, both the bb and τ + τ − couplings can be suppressed simultaneously and then the γγ decay branching ratio of the lightest Higgs boson tends to be enhanced. An analysis of these cases can be found in Ref. [25].

Detailed Analysis of the different scenarios
In Figs. 6-13, we present a detailed analysis of the regions covered by the relevant search channels at the Tevatron and at the LHC. In each of these Figures, the four panels display the regions for which Higgs bosons may be discovered at the 5-σ level at the LHC via the ttH i channel with 100 fb −1 (two darkest shades of grey) and 30 fb −1 (135 • lines) the weak-boson-fusion channel using 30 fb −1 (two darkest shades of grey) and the gluon fusion channel with H i → γγ using 100 fb −1 (two darkest shades of grey) and 30 fb −1 (135 • lines), as well as the region for which the Tevatron can provide 3-σ evidence with 5 fb −1 (two darkest shades of grey) and 2 fb −1 (135 • lines) of integrated luminosity. In each case, the darker shade of grey extends beyond the LEP exclusion. The 3-σ coverage with 5 fb −1 at the Tevatron is very similar to the region that would be obtained for 5-σ discovery with 15 fb −1 , or for 95% C.L. exclusion with 2 fb −1 . The reason why we have only analyzed 30 fb −1 in the case of weak-boson fusion is that a realistic analysis exists only for moderate instantaneous luminosity, and detection in this channel might be compromised in the different environment created by high-luminosity LHC running. For the Tevatron, we chose to present results that can simultaneously indicate the Higgs discovery potential under a conservative assumption of the total integrated luminosity (5 fb −1 ), for which only a 3-σ evidence may be obtained in the regions not excluded by LEP. This can be directly compared to a more optimistic assumption (15 fb −1 ), for which discovery may be possible well beyond the region of parameters covered by LEP. Finally, in all the scenarios studied, there are regions where at least two Higgs bosons are close in mass and have similar strength couplings to W/Z bosons. Our prescription is to add the signals from H i → bb and H i → τ + τ − with no degradation if the mass difference is less than 5 GeV (which is substantially smaller than the expected mass resolution), otherwise the signals are treated separately. We discuss this more later. Because of the precision electromagnetic calorimetry expected at the LHC, we do not combine H i → γγ signatures, though this is of little practical consequence. Figure 6 demonstrates the case of vanishing phases. Because the CPX scenario has large values for µ and m g , there is a large value for the supersymmetric loop corrections to the b-quark Yukawa coupling from (2.4, 2.5). Furthermore, since sgn(µm g ) > 0, the b quark Yukawa coupling is decreased relative to the τ lepton Yukawa coupling. The right lobe of the ttH i coverage (a) arises from H i = H 1 , with a transition to H 2 and then H 3 in the left lobe. The suppression of the bb coupling relative to τ + τ − , in conjunction with the sharing of the g H it t coupling between (in this case) two Higgs bosons, results in the dip near M H ± ∼ 150 GeV. For this case without explicit CP violation, that region corresponds exactly to M A ∼ m max h , where m max h is the maximal value of the Standard Model-like Higgs boson in the decoupling limit. Because of the enhanced τ + τ − coupling relative to bb, the W W → h channel (c) does not exhibit this phenomena. For tan β > a few, the W/ZH i channel (b) has a similar behavior as ttH i . The limited coverage from the h → γγ channel (d) is a common feature of all the scenarios we have studied. Even when g H i W W ∼ 1, a suppression of BR(h → γγ) occurs because g H ib b is enhanced over the SM value. The large values of |µ| and |A| necessary to cause substantial effects typically increased the offdiagonal elements of the 3×3 Higgs squared-mass matrix relative to the diagonal elements. Note that the W W → H i (→ τ + τ − ) channel with 30 fb −1 is sufficient alone to cover the M H ± -tan β plane. With the same integrated luminosity, neither the ttH i nor the H i → γγ channels even extend beyond the LEP coverage. Of course, the former is a discovery region, whereas the latter is only 95% C.L. exclusion.
Another possible variant on the CPX scenario is to introduce CP violation to the Higgs sector through a substantial phase for m g , φ g = 90 • (not shown). The primary effect is to decrease the suppression of g H ib b relative to g H i τ τ , but, as discussed before, the gluino phase effects are quite mild.
The introduction of a phase for A t = A b allows for CP-violation effects at small values of tan β, as shown in Fig. 7. As discussed previously, it is now possible for the lightest Higgs boson H 1 to become quite light and have a negligible coupling to vector bosons. Furthermore, the decay H 2 → H 1 H 1 becomes kinematically possible, introducing a region that remains uncovered by all experiments if one considers only the detailed analyses performed so far. Another region of difficult coverage appears in the ttH i (→ bb) channel, for a charged-Higgs mass close to 125 GeV. This did not occur for vanishing phases, and only arises now because of the transition away from the region where H 2 → H 1 H 1 occurs when H 2 has a substantial coupling to the Z boson. This does not occur for weak boson fusion in the H i → τ τ channel because of the enhancement of the τ coupling over that of the b. In this region of parameters, the coupling of H 1 and H 2 to the top quark and weak bosons is somewhat suppressed, though H 2 is relatively more strongly coupled to these particles than H 1 . Due to finite quantum corrections, the ratio of branching ratios BR(H 1 → bb)/BR(H 1 → τ τ ) is suppressed with respect to the Standard Model value. The substantial increase in BR(H 2 → τ τ ) means that H 2 → τ τ can still be observed, even though its vector-boson coupling is suppressed. Indeed, H 2 is also visible in the channel ttH 2 in a small region of parameters, for somewhat larger values of the charged Higgs mass.
We show in Fig. 8 the effect of including a gluino phase as well, which, as mentioned before, does not change significantly the picture, apart from the fact that small regions of uncovered parameter space appear. Those uncovered regions appear for a charged-Higgs mass around 130 GeV (160 GeV H 2 ) are separated in mass (greater than 5 GeV apart) and share equal couplings, of order a half of the SM values, to the weak bosons and the top quark.
An interesting feature appears for the case displayed in Fig. 9. For moderate and large values of the charged-Higgs mass and large values of tan β > ∼ 30, the lightest Higgs-boson mass is about 123 GeV. For these values of the Higgs mass, discovery in the weak-boson fusion process demands a slightly larger branching ratio into τ leptons than in the SM. This occurs for smaller values of the charged-Higgs mass, but as the charged-Higgs mass increases the branching ratio into τ leptons becomes smaller than the SM one, and remains like this until the decoupling limit. The bb decay branching ratio has a different behavior, as we see in the panel showing the expected Tevatron coverage. The mass M H 1 is about 123 GeV in the large-M H ± , large-tan β limit, but BR(H 1 → bb) is slightly larger than in the SM, explaining the appearance of coverage in that region; the lack of coverage starting near M H ± ∼ 235 GeV and tan β ∼ 20 reflects the difficulty of establishing evidence for a SM Higgs boson of mass 123 GeV. As usual, for larger values of the charged-Higgs mass, the γγ mode becomes relevant as a discovery channel, since the search for a Standard Model Higgs boson is efficient. Note also the disappearance of an uncovered region associated with H 2 → H 1 H 1 .
The case of phases equal to 60 • , displayed in Fig. 10, is similar to the case of 90 • , although the asymptotic value of the H 1 mass is somewhat lower. Most of the features of the Higgs searches in these two cases have similar behaviors, though occurring for different parameters, so we do not expand on them further.
Our final example for the CPX scenario has phases equal to 140 • . In this case, unlike the other examples, there is a relative suppression of the τ branching ratio of H 1 in large fractions of the parameter space, which explains the large regions of parameter space uncovered by the weak-boson-fusion process. This search channel becomes efficient whenever the Higgs mass is at the limit of the LEP reach, since in this case only a lower decay branching ratio is sufficient, as seen in Table 3 of Appendix B. For slightly shifted values of M H ± and tan β, instead the bb coupling is suppressed relative to τ + τ − . The complementarity of these channels is clearly illustrated in panels (b) and (d).
One interesting phenomena that arises in the CPX scenario, which is experimentally challenging, is the appearance of regions uncovered because of H 2 → H 1 H 1 decay. As noted previously, there are regions where two Higgs bosons share the coupling to W/Z bosons and tt, but are separated in mass. In this 'transition region' from one Higgs boson being most SM-like to another one, the effective signal is halved. In the MSSM without explicit CP violation, these Higgs bosons are nearly degenerate in mass, and they are most likely indistinguishable from two Higgs bosons with the experimental resolution. After including explicit CP violation, the mass splittings can be significantly larger, and all three Higgs bosons can share the W/Z coupling. We present here only two example scenarios that exhibit this phenomena -many more are possible. Under our prescription for adding signals occurring at masses within 5 GeV, the last two scenarios analyzed in Figs. 12 and 13 show small regions of parameters uncovered by any of the colliders. The size of these uncovered regions may be underestimated. Since there are potentially 3 Higgs bosons yielding a similar signature in these regions, and one may have to rely on experiment to normalize the background, the Higgs signal may be entirely washed out. Consider, for example, the expected number of signal and background events for the τ + τ − channel shown in Table 3. Since S/B ≫ 1, the accumulation of signals in nearby bins could substantially increase the background estimate. Clearly, the possibility that Higgs bosons may provide backgrounds for the other Higgs bosons should be understood and analyzed in more detail.
On the other hand, one consequence of sharing the coupling to W and Z bosons is that all three neutral Higgs bosons may be observable at hadron colliders in the W/ZH i and/or W W → H i channels. This is an exciting possibility, since it would indicate that a 2HDM without explicit CP violation is inadequate to describe the data. We find no such overlap of 5-σ signatures from all three neutral Higgs bosons in these channels for the scenarios presented here. Furthermore, using the expected reach of the ATLAS collaboration for the gg → H SM → ZZ * → 4ℓ channel [34], we have checked that there is almost no coverage beyond LEP for observing even a single Higgs boson.

Conclusions
We have presented in this article a phenomenological analysis of Higgs boson searches in Standard Model channels at the Tevatron collider and the LHC, for the case of the MSSM with explicit CP violation. We have also provided analytical expressions for the effective couplings of the neutral and charged Higgs bosons to fermions and to gauge bosons and the Higgs-boson self couplings within the MSSM with explicit CP violation. These expressions have been incorporated in the program CPHDECAY [19], used to perform our analysis.
After considering the LEP limits and comparing our results with the published LEP analyses, we have analyzed the reach of the Tevatron collider and the LHC in the Standard Model search channels for a neutral Higgs boson in the CPX scenario, as well as in other interesting scenarios. Our study was motivated by the fact that the latest LEP data prove insufficient to exclude a light MSSM Higgs boson in the CPX scenario, with a mass smaller than 60-70 GeV.       Figure 13: Same as Fig. 6, but for the 3HGC2 scenario defined in Fig. 3. As in Fig. 12, a region remains uncovered due to dilution of the g H i V V coupling among three Higgs bosons.
might still be possible with LEP data if searches for all the possible decay channels of the Higgs bosons were optimized. Our analysis for the Tevatron and the LHC is based on existing experimental simulations of the search for neutral Higgs bosons at these colliders, and could also be extended if more simulations were available.
We have shown that, in general, there is complementarity between the various searches at the two colliders for neutral Higgs bosons with Standard Model-like properties. However, there are still small regions of parameter space, for small values of the charged Higgs boson mass and moderate values of tan β, in which none of the three neutral Higgs bosons can be detected with a high statistical significance. In these regions of parameters, one of two phenomena occurs. First, the neutral Higgs boson with dominant couplings to the W and Z bosons can decay predominantly into channels which contain either two neutral Higgs bosons, or a neutral Higgs boson and a Z boson. The lighter Higgs boson has only feeble couplings to the W and Z bosons and top quarks, and escapes detection both at LEP and the hadron colliders. Secondly, all three neutral Higgs bosons can share the coupling to W and Z bosons and the top quark, resulting in three marginal signal excesses. The masses of the Higgs bosons may be uncomfortably close, leading to the possibility that one "signal" is a background to another. Detailed experimental simulations of these situations should be performed in order to decide the detectability of Higgs bosons in these regions of parameters.
Apart from these regions, the Standard Model search channels at the LHC are adequate for discovering a neutral Higgs boson over the range of supersymmetric parameters considered here. The ttH i (→ bb) and W W → H i (→ τ + τ − ) channels are most important. The gg → H i → γγ channel is greatly suppressed, and provides no additional coverage. This is a generic feature of the CPX and related scenarios where the b-quark Yukawa coupling can be enhanced over its Standard Model value. However, the Tevatron W/ZH i (→ bb) and the LHC W W → H i (→ τ + τ − ) channels fall in a special category for two reasons. First, the production of the Higgs boson through a tree-level coupling to the W and Z boson coupling demonstrates that this Higgs boson contributed to the Higgs mechanism. Secondly, radiative corrections can enhance or suppress the b-quark Yukawa coupling, while not affecting the τ -lepton Yukawa coupling, so that BR(H i → bb) and BR(H i → τ + τ − ) can be complementary to each other. We have presented results for 3-σ evidence with 5 fb −1 of luminosity at the Tevatron, and 5-σ discovery with 30 fb −1 of luminosity for the τ + τ − signature at the LHC and with 100 fb −1 for the bb and γγ channels. Our results indicate that the W/ZH i (→ bb) and W W → H i (→ τ + τ − ) channels alone do not cover the entire range of supersymmetric parameters, even excluding the difficult regions highlighted previously.
The restriction of W W → H i (→ τ + τ − ) results to only 30 fb −1 , which corresponds to accumulating data for a few years at a moderate instantaneous luminosity, is motivated by the potential deterioration of detection efficiency for this signal at a higher instantaneous luminosity. It is notable that, when comparing the reach with only 30 fb −1 of data for all search channels, the ttH i (→ bb) channel provides very little coverage beyond LEPthough the LHC coverage is for discovery, whereas the LEP coverage is for exclusion -and the gg → H i (→ γγ) channel provides no discovery potential. It is clear that the vectorboson fusion channel should be studied more thoroughly with detailed detector simulations, including the potential complications raised here regarding several marginal signals in a similar kinematic region. g 3H 133 = 2c β g 4H 1133 + s β g 4H 1233 , g 3H 233 = 2s β g 4H 2233 + c β g 4H 1233 , g 3H 333 = c β g 4H 1333 + s β g 4H 2333 . (A.6) The above relations are useful for checking the self-consistency of all the analytic expressions pertinent to the trilinear and quadrilinear Higgs self-couplings.

Experimental Simulations used in our Analysis
We summarize here the results from previous simulations of Higgs production at the LHC, upon which we have based our analysis of MSSM neutral-Higgs production in the presence of CP violation. In each Table, we provide the minimum ratio R, relative to the Standard Model product of cross section times branching ratio, for which a signal of the stated significance could be obtained with the quoted luminosity. We have provided more information regarding the expected number of signal S and background B events for those cases when S/B∼ 1.
M H (GeV) S (100 fb −1 ) B (100 fb −1 ) σ Gauss R 5σ (100 fb − Table 2: The expected signal, background, Gaussian significance and the R value for 5-σ discovery of a Higgs boson in the channel ttH(→ bb) at the LHC with 100 fb −1 of accumulated data [23]. These results are for the CMS experiment only. The Gaussian significance is based on a K factor of unity for the signal [35].  Table 3: The expected signal, background, Gaussian significance and the R value for 5-σ discovery of a Higgs boson in the channel W W → H(→ τ + τ − ) at the LHC with 30 fb −1 of accumulated data [24]. A detector-level analysis based on a fast simulation was presented in [36]. The results there for the combination of one hadronic and one leptonic decay of the τ + τ − pair are slightly less significant than the original analysis, but they have not been combined with the all leptonic decays. Therefore, we have chosen to use the original numbers, even though they are based on a parton-level analysis.  Table 4: The expected signal, background, Gaussian significance and the R value for 5-σ discovery of a Higgs boson in the channel gg → H(→ γγ) at the LHC with 100 fb −1 of accumulated data [26]. These apply to CMS only.