LOWER BOUNDS ON CHARGED HIGGS BOSONS FROM LEP AND THE TEVATRON

We point out that charged Higgs bosons can decay into final states different than $\tau^+ \nu_\tau$ and $c \bar{s}$, even when they are light enough to be produced at LEPII or at the Tevatron, through top-quark decays. These additional decay modes are overlooked in ongoing searches even though they alter the existing lower bounds on the mass of charged Higgs bosons that are present in supersymmetric and two Higgs doublets models.

The discovery of a charged Higgs boson would be an unambiguous signal of an extended Higgs sector and possibly of supersymmetry. In supersymmetric models, at least two Higgs doublets are needed to give mass to all fermions: one is coupled only to down-type quarks and leptons; the other, only to up-type quarks. A Two Higgs Doublet Model (2HDM) is said of Type II if the doublets are coupled as in supersymmetric models with minimal particle content. It is said of Type I if one Higgs doublet does not couple to fermions at all and the other couples as the Standard Model (SM) doublet.
After electroweak symmetry breaking, five physical states remain: two CP-even Higgs bosons h and H (with m h < m H ), a CP-odd Higgs boson A, and two charged states H ± . The charged-Higgses-fermions interactions, can then be comprehensively expressed as: where V is the CKM matrix. The equality X = Z = 1/Y = tan β, with tan β the ratio of the two vacuum expectation values, identifies 2HDMs of Type II and supersymmetric models; Y = −X = −Z = cotβ, identifies 2HDMs of Type I. Besides the mass of h, H, A, and H ± , two additional parameters are needed to describe the Higgs sector in 2HDMs of Types I and II: tan β and the mixing angle α. In supersymmetric models, the Higgs sector is more constrained, and only two free parameters are needed at the tree level, m A and tan β. Supersymmetry induces a relation between tan 2β and tan 2α and the well-known tree-level sum rule m 2 H ± = m 2 W + m 2 A , which is only mildly altered by one-loop corrections [1]. Together with the experimental lower bound on m h , m A > 92 GeV, for tan β > 1 [2], this sum rule makes the supersymmetric charged Higgs bosons possible candidates for discovery at the Tevatron, but not at LEP II.
Strong constraints on charged Higgs bosons come from searches of processes where H ± is exchanged as a virtual particle. Among them, the measurement of the inclusive decayB → X s γ [3] excludes charged Higgs bosons in a 2HDM of Type II up to ∼ 165 GeV [4]; however it is, in general, inconclusive for supersymmetric models [5] and 2HDMs of Type I [6,4]. Other indirect bounds on the ratio m H ± /tan β come from inclusive semileptonic b-quark decays B → Dτ ν τ , m H ± ∼ > 2.2 tan β GeV [7] and from τlepton decays, m H ± ∼ > 1.5 tan β GeV [8]. They apply to charged Higgs bosons of Type II in 2HDMs and supersymmetric models. In the former, however, they are non-competitive with the stronger lower bound due to the measurement ofB → X s γ; in the latter they are already saturated by the above sum rule and the lower bound on m A . Constraints on the low-tan β region and light H ± in Type I models come from the measurement of Z → bb and B 0 -B 0 mixing (see discussion in [6]).
It is possible that the 2HDMs described above are only "effective" models, i.e. the low-energy remnant of Multi-Higgs-Doublets models, with the same number of degrees of physical states non-decoupled at the electroweak scale. In this case, more freedom remains in the possible values that X, Y, and Z can acquire. For X = −1/Y = −a, with a ≥ 2, for example, charged Higgs bosons with m H ± = 100 GeV can escape theB → X s γ constraint [4], while having widths for decays into light fermions substantially coinciding with those obtained in a 2HDM of Type II. Moreover, lepton and quark couplings in (1) may be unrelated, thus rendering the indirect bounds from b-quark and τ -lepton decays independent of that coming fromB → X s γ. Indirect and direct bounds are, therefore, all equally necessary in providing the complementarity that allows the exclusion of certain ranges of m H ± in supersymmetric models, in Type I and Type II 2HDMs, and in those models that may counterfeit them in one specific search.
Charged Higgs bosons are searched for at LEP II, above the LEP I limit, in the range 45 ∼ < m H ± ∼ < 100 GeV and at the Tevatron in the range m H ± < m t − m b , i.e. when produced by a decaying t-quark. Searches at LEP II rely on the assumption that no H + decay mode, other than cs and τ + ν τ , is kinematically significant; they give a limit m H ± ∼ > 78. 6 GeV [9], which applies to 2HDMs of Types II and I. Indeed, within the assumption BR(H + → cs, τ + ν τ ) ≃ 100%, in Type I models the two branching ratios are tan β-independent and approximately equal to those obtained in Type II models with tan β = 1.
At the Tevatron, searches of an excess of tt events in the τ channel provide a tan β-m H ± exclusion contour that constrains the very-large-tan β region in supersymmetric models and 2HDMs of Type II [10], for which the rate of t → H + b is large. Similarly large is this rate in the region of low tan β (tan β ∼ < 1), for Type II Yukawa couplings. Searches of H + apply in this region to the non-supersymmetric case. They are carried out, specifically for this type of couplings, looking for: i) a deficit in the e, µ channels, due to H + → cs, for m H ± ∼ < 130 GeV, ii) a larger number of taggable b-quarks due to H + → t * b →bbW for m H ± ∼ > 130 GeV [11,12]. Given the limited luminosity at present available at the Tevatron (∼ 1 fb −1 ), there is no sensitivity to the intermediate range of tan β where the rate t → H + b becomes low. This region, partially accessible at the upgraded Tevatron, will be fully covered at the LHC [13].
The aim of this letter is to show that there exist additional decay modes, which are overlooked in ongoing searches of H ± within 2HDMs and supersymmetric models, and which alter the existing lower bounds on m H ± . In the following, the considered type of weak scale supersymmetry has minimal particle content and R-parity conservation. No specific assumption is made on the superpartner spectrum and on the scale/type of messengers for supersymmetry breaking. All branching ratios presented for supersymmetric models are calculated using HDECAY [14].
In 2HDMs, these modes are H + → AW + and/or hW + (HW + ). They produce mainly the same final statebbW + , as the above-mentionedbt * mode and, to a lesser extent, the state τ + τ − W + . Our statement is based on the fact that there is no stringent lower bound on m A and/or m h coming from LEP [15]. Indeed, since the mixing angle α is, in this case, a free parameter, one can think of a scenario in which the coupling ZhA vanishes. This coupling being proportional to cos(β − α), the required direction is α = β ± π/2. In this case, the process Z * → hA does not occur and the LEP II bound m A > 92 GeV obtained for supersymmetric models does not hold. Nevertheless, the cross section for the process e + e − → Z * → hZ, proportional to sin 2 (β − α), is not suppressed with respect to that for the corresponding production mechanism of the SM Higgs boson, and the LEP II bound m h > 114 GeV [2] applies to our case. The coupling ZHA, still proportional to sin(β − α), has also full strength, whereas HZZ vanishes. The process Z * → HA could in principle provide a bound on m A depending on m H and tan β. For large m H , however, no real lower bound can be imposed on m A . Conversely, even without making specific choices on the angle α, one can assume h to be heavy enough to render impossible any significant lower bound on m A . The other two production mechanisms possible at LEP I (they require larger numbers of events than LEP II can provide) are the decay Z → Aγ and the radiation out of bb and τ + τ − pairs [16]. The first is mediated only by fermion loops, unlike the decay Z → hγ, which has additional contributions from W -boson loops. The corresponding rate is about two orders of magnitude smaller than that for Z → hγ and therefore too small to allow for a visible signal [17]. The second process allows for sizeable rates only for very large values of tan β. No bound can be obtained for nonextreme values of tan β and for 2HDMs of Type I. In general, therefore, one remains with the rather modest bound from the decay Υ → Aγ, which has been searched for by the Crystal Ball Collaboration [18], m A > 5 GeV.  If one recalls that the interaction term H + W − A is weighted by a gauge coupling, unsuppressed by any projection factor, it is clear that the decay H + → AW + can be rather important for Type I models, or for models of Type II with small tan β. This remains true even for an off-shell W -boson, in spite of the additional propagator and weak coupling that are then required. For a 2HDM of Type I and Type II with tan β = 1, the branching ratios BR(H + → AW + ) are shown in Fig. 1 as functions of m A for different values of m H ± (solid lines). Already for m H ± = 70 GeV, roughly the lower bound obtained at LEP II when BR(H + → cs, τ + ν τ ) ≃ 100% is assumed, the branching ratio is 50%-20% for m A = 10-30 GeV. More strikingly, for heavier H ± , when the W -boson is not too far from being on shell, this decay mode becomes the dominant one. We also show in Fig. 1 the branching ratios for this decay mode in a Type II model with a higher value tan β = 2 (for Type I model, the situation does not change). BR(H ± → W * A) is of course smaller because the competing decay mode, H − → τ − ν τ , has an enhanced decay width. This is more striking for low m H ± values when the H ± → AW * decay channel occurs only at the three-body level. For a heavier H ± boson, values of tan β slightly larger than unity do not change the main trend. This is particularly true when the W boson is on-shell as in the example with M H ∼ 150 GeV and a light pseudoscalar A boson. In this case, only for much larger tan β values that the H − → τ ν decay mode becomes dominant and then, the search for the H ± boson at LEP II will be the standard one and the limit m H ± ≥ 78.6 GeV form τ ν and cs decay [9] will hold (for intermediate tan β values, one has to take into account simultaneously all decay modes, rendering the analysis more complicated).
Since the two modes hW + and HW + are forbidden respectively by our choice of α and the requirement of a very heavy H, the other competing channels are τ + ν τ , cs for m H ± in the LEP II range, and τ + ν τ , cs, andbt * in the Tevatron searches. In Fig. 2, the final branching ratio BR(H + →bbW + ) is shown as a function of m H ± in a 2HDM of Type II, with our choice of α, for different values of tan β and of m A . For the larger m A , the mode AW + is forbidden. Indeed, above m H ± = 130 GeV the mode cs is quickly taken over bybt * , with the same tan β dependence, but much larger Yukawa couplings, which can compensate the virtuality of the t-quark. The deviations from this pattern become striking when the mode AW + starts being allowed.  The situation described here corresponds to a particular direction of parameter space. One could have similarly allowed decays into hW + and HW + . For instance, a search strategy based on tagging three b-quarks for each produced t-quark at the Tevatron (one b-jet coming form the t → bH + decay and two b-jets coming from H + → W + + h, H, A with the Higgs bosons decaying into bb pairs) would then sum over all these decays. The corresponding theoretical branching ratio, however, becomes a function of m A , m h , m H and α, in addition to m H ± and tan β. Searches at LEP II and the Tevatron aimed at constraining 2HDMs of Type II in the low tan β regime and/or 2HDMs of Type I will have to be modified accordingly. Constraints in the region of very large tan β for Type II couplings, when only the mode τ + ν τ survives, remain unchanged.
In supersymmetric models, and in particular in the minimal version (MSSM), since m A cannot be much smaller than m H ± and the angle α is not an independent parameter, a non-trivial role is played only by the mode H + → hW + * . However, the branching ratio is large only for small values of the parameter tan β, tan β ∼ < 2, for which the h boson is constrained to be rather heavy form LEP data [2] [in fact, such a low tan β scenario is by now excluded]. For larger values of tan β, the H + W h coupling is suppressed [and the H + τ ν coupling is enhanced], making the branching ratio for this decay mode rather small, not exceeding ∼ 5% over the LEP allowed region. [Note that the situation might be different in extensions of the MSSM, such as in the case of additional singlet fields, the NMSSM, where m H ± and m A are not as strongly related as in the MSSM and the present LEP constraints on m h and m A do not hold; in this case BR(H + → hW * , AW * ) might be rather large.] In general, however, decays into the lightest chargino χ + 1 and neutralino χ 0 1 as well as decays into sleptons are still allowed by present experimental data, and they dominate when they occur. (The importance of the channel χ + 1 χ 0 1 for a constrained minimal supersymmetric model was already discussed in [19]; for decays of MSSM Higgs bosons into supersymmetric particles, see also Ref. [20].) The latest lower bounds on χ + 1 from LEP II, m χ + 1 ∼ > 103. 6 GeV, rely on the assumption of very heavy sleptons and/or a relatively large mass splitting with the lightest neutralino [21]. For large values of the Higgs-higgsino mass parameter µ, the lighter chargino and neutralino states χ + 1 and χ 0 1 are respectively wino-and bino-like, with masses ∼ M 2 and ∼ M 1 . In this case, even assuming gaugino mass universality at the very high scale: M 1 = 5 3 tan 2 θ W M 2 ∼ 1 2 M 2 , the decay channel H + → χ + 1 χ 0 1 is possible for m H ± > 165 GeV. It gives rise to jets or leptons and missing energy and to τ 's and missing energy. The branching ratio BR(H + → χ + 1 χ 0 1 ) is shown in Fig. 3 as a function of m H + , for tan β = 4, M 2 = 150 GeV and µ = 200 GeV (solid line). [Here, and in the example for tan β = 4 in the next discussion, we have set the sfermion masses at ∼ 1 TeV and the trilinear stop coupling A t at √ 6 TeV (the so-called maximal mixing scenario) to evade the experimental bound [2] on the h bound mass.] For these values of parameters, χ + 1 and χ 0 1 have respectively masses of 107 and 60 GeV.
The LEP II limits on χ + 1 and χ 0 1 become weaker if the assumption on very heavy slepton masses and/or gaugino mass universality is relaxed. In both cases, the channel χ + 1 χ 0 1 becomes kinematically allowed for lighter H ± 's. As an example, we show in Fig. 3 the branching ratio in a direction of supersymmetric parameter space with M 1 disentangled from M 2 (dotted line). While keeping all other parameters fixed to the previous values, M 1 is set to 25 GeV, which induces a mass for χ 0 1 of ≃ 19 GeV. The mode χ + 1 χ 0 1 opens now already at ∼ 125 GeV. Figure 3 clearly shows that, in the region of moderate tan β, if no other decay of H + into superpartners is possible, the mode χ + 1 χ 0 1 can be dominant if it is kinematically allowed. For m H ± ≃ 170 GeV and tan β = 4, the contribution of the χ + 1 χ 0 1 mode to the H ± 's total decay width, indeed, is respectively 78% and 92% GeV. An increase of tan β reduces the branching ratio BR(H + → χ + 1 χ 0 1 ), while a smaller value of tan β, if allowed, would make this decay mode even more dominant, in particular in the case of non-unified gaugino masses.
The existing lower bounds on the charged slepton masses from LEP II, are respectively 95, 88, and 76 GeV forẽ,μ,τ when the mass difference with the lightest neutralino is rather large (∆M ∼ > 15 GeV) and the sleptons are assumed to decay exclusively into ℓ ± χ 0 1 final states [22]. These bounds, in particular in the case ofτ , can be much weaker if they are nearly degenerate with the LSP neutralino. For sneutrinos, an absolute bound ∼ > 45 GeV comes from the measurement of the invisible Z boson decay width. Hence, the decay H + →τ +ν τ is therefore kinematically allowed and produces a final τ + + missing energy, but with a softer τ + than that coming from the direct decay H + → τ + ν τ . We show in Fig. 4 the relative branching ratio for two choices of input parameters: a) tan β = 4, M 2 ∼ 2M 1 = 120 GeV, µ = −500 GeV, ml L = ml R = ml = 90 GeV and A τ = 0 (small or moderate mixing scenario). This leads to a slepton spectrum: mν ∼ 66 GeV, mẽ ∼ mμ ∼ 100 GeV and the twoτ masses ∼ 20 GeV below and above this value (the lightest chargino and neutralino masses are m χ + 1 ∼ 123 GeV and m χ 0 1 ∼ 60 GeV). b) tan β = 25, M 2 ∼ 2M 1 = µ = 150 GeV, ml = 100 GeV and A τ = −800 GeV (the large A τ value is chosen to maximize the H ±τν τ coupling as will discussed later). This leads to the following spectrum: mν ∼ 76 GeV, mẽ ∼ mμ ∼ 110 GeV and theτ 1 mass mτ 1 ∼ 63 GeV almost degenerate with the lightest neutralino mass m χ 0 1 ∼ 61 GeV (therefore the decayτ 1 → χ 0 1 τ gives very soft τ leptons, which will be overwhelmed by the γγ background and the LEP II lower limit on mτ 1 does not hold in this case).
Below the threshold for scenario a) with tan β = 4, the dominant decays are τ + ν τ and hW * , while AW + and cs are below the percent level. Above the threshold, the branching ratio for the decay H ± →τ ±ν τ can become rather sizeable, possibly reaching the level of ∼ 30%. For large enough H ± masses, the channels H ± →μ ±ν µ andẽ ±ν e , open up, leading to an increase of BR(H ± →lν) up to ∼ 80%. In scenario b) with tan β = 25 and a large A τ value, τ + ν τ decays are by far dominant below the threshold. When the decay H ± →τ ±ν τ opens up, the branching ratio quickly reaches the level of ∼ 75%. The prominence ofτ +ν τ decays observed above threshold is explained by the H ± coupling to sleptons. For small stau mixing and small tan β values, the Lagrangian term H +ν * Ll L , −(g/ √ 2)M W sin 2β, is very large with respect to the Yukawa coupling −(g/ √ 2)(m τ /M W ) tan β. Owing to the sin 2β dependence, this term quickly dies off for increasing tan β. In this case, however, there exists other directions of parameter space where this decay mode still has a branching ratio ∼ 100%. For instance, when A τ and tan β are large, since the coupling of the Lagrangian term H +ν * Lτ R : −(g/ √ 2)(m τ /M W )(µ+ A τ tan β) becomes very strong, the decay rate is enhanced as shown in Fig. 4 (note that for A τ ∼ µ tan β, the left-right mixing in the slepton mass matrix tends to vanish). Summarizing, at very large tan β, a possible excess of τ 's softer than those predicted by a 2HDM of Type II may indicate the presence of a heavier H ± decaying intoτ +ν τ . Searches in the region of tan β ∼ > 1 should already consider multi-b signals coming from hW + * ,bbW + as well as τ -signals with a wide momentum distribution coming from χ + 1 χ 0 1 , τ +ν τ , and τ + ν τ and jets/leptons + missing energy signals from χ + 1 χ 0 1 . It is needless to say that all these modes will play an important role in future searches and will not be blind to the intermediate range of tan β. This would be particularly the case at the Tevatron Run II where the H ± bosons, if light enough, can be produced copiously in top quark decays (other production channels would have much smaller rates) [23]. While it would be always possible to detect them in a "disappearance" search (i.e. by looking at one top quark decaying into the standard mode, t → W + b, which should have a relatively large branching ratio, and ignoring the decay products of the other) [23], the direct search for 2HDMs H ± bosons decaying into W bb final states would be in principle relatively easy with high enough luminosity, since the performances of the CDF and D0 detectors for b-quark tagging are expected to be rather good. In the case of SUSY models, where the H ± should be tagged though the leptonic decays of charginos or τ sleptons, the detection might be more challenging because of the softness of these particles. A detailed Monte-Carlo analysis, which is beyond the scope of this letter, will be needed to assess the potential of the Tevatron to search for the H ± bosons in these new decay channels.
Note Added: After the first submission of this paper, a search for 2HDM charged Higgs bosons decaying into AW * finals states has been performed by the OPAL collaboration [24]; constraints in the (m A , m H ± ) plane for various tan β values have been set. In addition, the decay mode H ± → bbW ± has been taken into account in simulations of H ± searches at the upgraded Tevatron [23] and at the LHC [25]. Some of the decays modes discussed here have been also revisited in theoretical papers in the context of 2HDM [26] and the MSSM [27].