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Light Heavy MSSM Higgs Bosons at Large tan_beta

The region of MSSM Higgs parameter space currently excluded by the CDF Collaboration, based on an analysis of ~1 fb^-1 of integrated luminosity, is less than the expected sensitivity. We analyze the potential implications of the persistence of this discrepancy within the MSSM, assuming that the soft supersymmetry-breaking contributions to scalar masses are universal, apart from those to the Higgs masses (the NUHM model). We find that a light heavy MSSM Higgs signal in the unexcluded part of the sensitive region could indeed be accommodated in this simple model, even after taking into account other constraints from cold dark matter, electroweak precision observables and B physics observables. In this case the NUHM suggests that supersymmetric signatures should also be detectable in the near future in some other measurements such as BR(B_s ->mu+ mu-), BR(b ->s gamma) and (g-2)_mu, and M_h would have to be very close to the LEP exclusion limit. In addition, the dark matter candidate associated with this model should be on the verge of detection in direct detection experiments.

The searches for the bosons appearing in its extended Higgs sector are among the most promising ways to search for evidence of supersymmetry (SUSY) at the Tevatron collider.
The CDF and D0 Collaborations have already established important limits on the (heavier) MSSM Higgs bosons, particularly at large tan β [1][2][3][4][5]. Recently the CDF Collaboration, investigating the channel has been unable to improve these limits to the extent of the sensitivity expected with the analyzed integrated luminosity of ∼ 1 fb −1 [4], whereas there is no indication of any similar effect in D0 data [5]. 1 Time will tell whether the CDF effect persists. Within the MSSM the channel (1) is enhanced as compared to the corresponding SM process by roughly a factor of tan 2 β /((1 + ∆ b ) 2 + 9) [7], where tan β is the ratio of the two vacuum expectation values, and ∆ b includes loop corrections to the φbb vertex (see Ref. [7] for details) and is subdominant for the τ + τ − final state. Correspondingly, the unexpected weakness of the CDF exclusion might be explicable within the MSSM if M A ≈ 160 GeV and tan β > ∼ 45 and, doubtless, also within other theoretical frameworks.
In this paper we investigate whether light heavy Higgs bosons just beyond the region currently excluded by CDF could be accommodated within GUT-inspired MSSM scenarios, and what the possible consequences would be. We consider the constraints imposed by other measurements, such as the limits on BR(B s → µ + µ − ), BR(b → sγ), (g − 2) µ and M h , assuming that R parity is conserved and that the lightest neutralinoχ 0 1 constitutes the astrophysical dark matter [8]. Whereas we find no solution within the constrained MSSM (CMSSM), in which all soft SUSY-breaking contributions to scalar masses are assumed to unify at the GUT scale, we find that all the constraints may be satisfied in the case that universality at the GUT scale is relaxed for the scalar Higgs mass parameters (the NUHM model [9][10][11]). However, we point out that any interpretation of the CDF effect within the NUHM would be tightly constrained by the other measurements. Specifically, the constraints are so tight that one or more of these measurements should display a discrepancy with the SM, either now or in the near future.
The essence of the argument is as follows. The absence of exclusion by CDF, compared to their expected sensitivity, as mentioned above, would correspond to M A (≈ M H ) ∼ 160 GeV, and a value of tan β ∼ 45 or greater. Since the H/A contribution to BR(B s → µ + µ − ) ∼ tan 6 β, values of tan β > ∼ 45 are already excluded for this value of M A for substantial portions of the NUHM parameter space, depending largely on the values of m 1/2 and m 0 . The parameter space is so constrained that, in areas which are still allowed, we expect that a SUSY signal should appear very soon, as we show below. We assume R-parity conservation, and restrict our attention to the NUHM with values of the relic CDM density Ω CDM that fall within the range favoured by WMAP and other astrophysical and cosmological observations. This restriction imposes important constraints on m 1/2 and µ, the values of m 0 and A 0 being less essential. As for BR(b → sγ), it is well known that the world average experimental value currently agrees well with the SM [12]. In the MSSM, there are two important contributions with opposite signs, due to H ± and chargino exchanges, respectively. In order for the net MSSM contribution to be unnoticeable so far, the H ± and chargino exchanges must cancel to a great extent, imposing a relation between the H ± and chargino masses. Since the H ± mass is very similar to the H, A masses, this yields a preferred range of relatively small values of the chargino mass and hence the soft supersymmetry-breaking gaugino mass m 1/2 , favouring in turn a non-negligible contribution to (g − 2) µ . The combination of a preferred value for m 1/2 and the WMAP constraint then limits the possible range of µ. The preference for relatively light sparticles (see also Ref. [13]) translates into a relatively small value for M h , and compatibility between the LEP limit on M h [14,15] and the upper limit on BR(B s → µ + µ − ) [16,17] selects a limited range of A 0 .
In preparation for our survey of the NUHM parameter space, we first recall that in the CMSSM, the electroweak vacuum conditions determine |µ| and M A in terms of tan β and the input soft supersymmetry-breaking parameters. In the NUHM, these are the scalar mass m 0 (which is assumed to be universal, except for the Higgs multiplets) 2 , the gaugino mass m 1/2 , the trilinear coupling A 0 and the degrees of non-universality of the soft supersymmetrybreaking contributions to the masses of the two Higgs doublets. In our analysis, we invert the electroweak vacuum conditions, treating |µ| and M A as free parameters and adjusting the non-universal Higgs mass inputs accordingly. In the cases of interest this difference in the scalar masses at the GUT scale is no more than 50% for low values of m 0 , and of order 10% for higher values of m 0 . The values M A = 160 GeV and tan β ≥ 45 are chosen to match the excess of signal-like events observed in CDF, and we assume that µ > 0 so as to avoid severe problems with (g − 2) µ and BR(b → sγ). Hence our four free parameters are m 1/2 , m 0 , A 0 and µ > 0. However, these are tightly constrained by other phenomenological limits, as we now discuss.
Since we assume that R parity is conserved, and that the lightest neutralinoχ 0 1 constitutes the astrophysical dark matter [18], we impose the requirement that the relic neutralino density falls within the range allowed by WMAP and other observations: 0.085 < Ω CDM h 2 < 0.119 [8]. As we see later, requiring the relic density to fall within this narrow range effectively reduces the dimensionality of the NUHM parameter space. In order to apply the constraints on the NUHM parameter space that are provided by BR(b → sγ), BR(B s → µ + µ − ), (g − 2) µ and M h , we use the following experimental values and theory evaluations (for more details, specifics on error treatments and an extended list of references see Ref. [19]): BR(b → sγ) exp = (3.55 ± 0.24) × 10 −4 [12], where the theory evaluation is based on Refs. [20,21]; BR(B s → µ + µ − ) exp < 10 −7 [16,17], where details about the theory evaluation can be found in Ref. [ We consider first the (m 1/2 , m 0 ) plane shown in panel a) of Fig. 1, which has tan β = 45, µ = 370 GeV and A 0 = −1800 GeV, as well as M A = 160 GeV. 4 The dark (brown) shaded region at low m 0 is forbidden, because there the LSP would be the lighter stau. The WMAP cold dark matter constraint is satisfied only within the lighter (turquoise) shaded region. To the left of this region, the relic density is too small, due to s-channel annihilation through the Higgs pseudoscalar A. As m 1/2 increases away from the pole, the relic density increases toward the WMAP range. However, as m 1/2 is increased, the neutralino acquires a larger Higgsino component and annihilations to pairs of W and Z bosons become enhanced. To the right of this transition region, the relic density again lies below the WMAP preferred value. The shaded region here is therefore an overlap of the funnel and transition regions discussed in Ref. [10].
The BR(B s → µ + µ − ) constraint is satisfied between the outer black dash-dotted lines, labelled 10 −7 , representing the current limit on that branching ratio 5 . Also shown are the contours where the branching ratio is 2 × 10 −8 , close to the sensitivity likely to be attainable soon by CDF and D0. Between these two contours, there is a strong cancellation between the flavor-violating contributions arising from the Higgs and chargino couplings at the one-loop level and the Wilson coefficient counterterms contributing to BR(B s → µ + µ − ). 3 Consequences for other observables such as BR(B u → τ ν τ ) and the W -boson mass are discussed further below. 4 Here and elsewhere, we assume m t = 171.4 GeV and m b (m b ) = 4.25 GeV, but our results are insensitive to the exact values of m t and m b , and take into account their current uncertainties, δm t = 1.8 GeV and δm b (m b ) = 0.11 GeV. 5 A slightly more stringent upper limit of 0.93 × 10 −7 at the 95% C.L. has been announced more recently by the D0 Collaboration [25]. However, applying this limit would have only a minor impact on our analysis.
The dash-dotted (red) line shows the contour corresponding to M h = 114 GeV, and only the region to the right of this line is compatible with the constraint imposed by M h (it should be kept in mind that there is still a ∼ 3 GeV uncertainty in the prediction of M h [24]). Also shown in pink shading is the region favoured by (g − 2) µ at the two-σ level. The oneand two-σ contours for (g − 2) µ are shown as elliptical dashed and solid black contours, respectively. The region which is compatible with the WMAP relic density and M h , and is also within the two-σ (g − 2) µ experimental bound, has BR(B s → µ + µ − ) > 2 × 10 −8 . The measured value of BR(b → sγ) is in agreement with the theory prediction only to the left of the solid (green) region. We see, in the space between the M h and b → sγ exclusions, a slightly diagonal allowed strip of width < ∼ 200 GeV in m 1/2 . We see that there is a narrow wedge of allowed parameter space in Fig. 1(a), which has m 1/2 ∼ 600 GeV and m 0 ∼ 700 to 1100 GeV. The BR(b → sγ) constraint is satisfied easily throughout this region, and (g − 2) µ cuts off the top of the wedge, which would otherwise have extended to m 0 ≫ 1500 GeV. Within the allowed wedge, M h is very close to the LEP lower limit, and BR(B s → µ + µ − ) > 2 × 10 −8 . If M A were much smaller (< 130 GeV), there would be no wedge consistent simultaneously with the Ω CDM , M h and BR(B s → µ + µ − ) constraints (the M h bound, however, could then be relaxed due to a weaker hZZ coupling). At higher values of tan β, the allowed region drifts up to higher values of m 0 , as is shown in We have also considered the potential impact of direct searches for supersymmetric cold dark matter in the NUHM. As has been pointed out previously [26], these searches are potentially important at small M A and large tan β, as required in the scenario discussed here (see also Ref. [27]). However, the interpretation of the search limits is less precise than for the other constraints, for two reasons. One is the local density of supersymmetric cold dark matter, which is usually estimated as 0.3 GeV/cm 3 . This estimate is subject to a systematic uncertainty that is itself uncertain, but might be ∼ 50% or so. The second significant uncertainty is in the hadronic scattering matrix element of the local operator generated by the short-distance supersymmetric physics. The dominant contribution to the scattering is spin-independent, and given by the matrix elements in the nucleons of the scalar quark densitiesqq: q = u, d, s. These may be determined from the octet matrix element σ 0 , which is estimated to be 36 ± 7 MeV, and the π-N scattering matrix element Σ πN = 45 ± 8 MeV. These give a range y ≡ 2 N|ss|N / N|(ūu +dd)|N = 0.2 ± 0.2 [28].
Generally speaking, the dark-matter scattering cross section increases with y.
Given the matrix element uncertainties summarized above, we show in Fig. 1a) the 1-σ lower limit on the calculated value of the elastic cross section as compared to the CDMS  upper limit [29]. In the portion of the plane to the left of the (orange) dotted line, the lower limit on the calculated spin-independent elastic cross section is smaller than the CDMS upper bound, assuming the canonical local density. Whilst we have assumed Σ πN = 45 MeV, the calculated lower limit effectively assumes zero strangeness contribution to the proton mass, i.e., y = 0. In the region of interest, the lower limit on the calculated cross section is about 80% of the CDMS upper bound, whereas with a strangeness contribution of y = 0.2, the cross section would exceed the CDMS bound by a factor of ∼ 3. Thus, if Nature has picked this corner of the NUHM parameter space, we expect direct detection of dark matter to be imminent. We note that the XENON Collaboration has recently announced a stronger upper limit on the elastic scattering cross section [30]. Consistency with this limit would further require a reduction in the local dark matter density (to its lower limit). Intriguingly, the XENON10 experiment has seen some potential signal events that are, however, interpreted as background. In the following we will show the limits obtained by CDMS, but the potentially somewhat stronger limits from XENON10 should be kept in mind.
This  Fig. 1(a). In this case, the region allowed by the BR(B s → µ + µ − ) constraint is below the upper dash-dotted black line, and the LEP M h constraint is satisfied only above the dash-dotted red line. We see that only a restricted range 360 GeV < µ < 390 GeV is compatible with the dark matter constraint. This corresponds to the transition strip where the neutralino is the appropriate bino/Higgsino combination. To the left of this strip, the relic density is too small and to the right, it is too large. Only a very restricted range of A 0 ∼ −1600 GeV is compatible simultaneously with the M h and BR(B s → µ + µ − ) constraints. Very large negative values of A 0 are excluded as the LSP is the lighter stau.
On the (µ, A 0 ) plane, the elastic scattering cross section is a rapidly decreasing function of µ and is almost independent of A 0 . Indeed, NUHM points excluded by CDMS (or XENON10) generally have low values of µ and M A [26]. Values of µ > 355 GeV are compatible with CDMS if the strangeness contribution to the proton mass is negligible. For this choice of parameters, the entire displayed plane is compatible with BR(b → sγ) and (g − 2) µ .
Panel b) of Fig. 2 shows what happens if m 1/2 is increased to 700 GeV, keeping m 0 and the other inputs the same. Compared to Fig. 2(a), we see that the WMAP strip becomes narrower and shifts to larger µ ∼ 400 GeV, and that BR(b → sγ) starts to exclude a region visible at smaller µ. If m 1/2 were to be increased much further, the dark matter constraint and BR(b → sγ) would no longer be compatible for this value of m 0 . We also Nevertheless, a viable portion of parameter space remains in tact at lower m 1/2 and higher m 0 . At higher tan β, the CDMS bound also becomes stronger and restricts m 1/2 < ∼ 560 GeV in panel c) and < ∼ 700 GeV in panel d). Thus in panel c), we are forced into a small triangular region centered at m 1/2 ∼ 530 GeV and m 0 ∼ 1350 GeV bounded by CDMS, M h , and (g − 2) µ . There, the branching ratio, BR(B s → µ + µ − ) is near its minimum value of 2 × 10 −9 due to the strong cancellation discussed above. In panel d) we are left with a significantly larger quadrilateral region.
The analogous (µ, A 0 ) planes are shown in Fig. 2c) and d). In panel c), only a small portion of the WMAP strip around A 0 ∼ −1200 GeV is compatible with both BR(B s → µ + µ − ) and M h , while the range in panel d) is considerably larger. In both cases, the CDMS limit is stronger and begins to cut into the WMAP strip.
We have surveyed systematically the allowed region of parameters in the (m 1/2 , m 0 ) plane for tan β between 45 and 55, varying the inputs µ and A 0 . To summarize the regions in the NUHM parameter space which are compatible with CDM and all phenomenological constraints and a light heavy Higgs, we first recall that the values of m 1/2 compatible with WMAP depend on µ, but depend less on m 0 . Specifically, as µ increases, the preferred region of the (m 1/2 , m 0 ) plane is a near-vertical strip that moves to larger m 1/2 , which is truncated at low m 0 just above the region where the lighter stau is the LSP. For µ < 350 GeV, there is no WMAP-compliant region compatible with the LEP lower limit on M h , but a small allowed region appears for µ slightly below 370 GeV. Typical values for m 1/2 are 500 − 600 GeV at this value of µ and A 0 must be below −1300 GeV. When µ = 400 GeV, the WMAPcompatible strip moves to larger m 1/2 , and there are allowed regions of the (m 1/2 , m 0 ) plane for A 0 ∼ −1600 GeV to ∼ −2400 GeV. As µ increases further, the WMAP-compatible strip moves to even larger m 1/2 , and the only allowed region is a small piece of coannihilation strip close to the boundary with the stau LSP region. This region is negligible for µ > 500 GeV. Combining all the allowed values of µ and A 0 , we find that only a small portion of the (m 1/2 , m 0 ) plane can ever be compatible with all the constraints. It is roughly triangular in shape, with vertices (m 1/2 , m 0 ) = (500, 1400), (700, 700) and (800, 900) GeV. We now analyse the dependence of the results on the assumption of scalar-mass universality. The assumptions that all squarks with the same electroweak quantum numbers have identical input soft supersymmetry-breaking scalar masses is motivated in general by the suppression of flavour-changing neutral interactions. Specifically, the upper limit on BR(B s → µ + µ − ) and the agreement of BR(b → sγ) with the SM would require a certain degree of fine-tuning in the presence of large squark mass non-universality. On the other hand, the relative locations of both the WMAP strip and the (g − 2) µ constraint depend on the relationship between the soft SUSY-breaking squark and slepton scalar masses. If the slepton masses are decreased relative to the squark masses (which we continue to denote by m 0 ), the lower ends of the WMAP strips in Fig. 1 will rise (due to relatively ligher staus), as will the (g − 2) µ contours, raising the preferred range of m 0 . Conversely, if the slepton masses are increased relative to the squark masses, the preferred range of m 0 will be lower.
Thus, neglecting the universality between squark and slepton masses in general enlarges the allowed region of parameter space that is compatible with the experimental constraints.
We now discuss the possible phenomenological signatures of a scenario with M A = 160 GeV and tan β > ∼ 45 within the NUHM. The interplay of the various constraints in Fig. 1 implies: (i) The predicted value of BR(B s → µ + µ − ) in the allowed region is generally > 2 × 10 −8 . Thus, this channel may offer good prospects within the near future for either supporting or contradicting the NUHM interpretation of the weaker CDF tan β bound, as compared to the expected sensitivity.
(ii) We find that M h must be very close to the LEP lower limit, i.e. in the range where LEP observed a couple of Higgs-like events [15]. This part of parameter space could be probed at the Tevatron with 8 fb −1 of integrated luminosity. Due to the large value of tan β and the small mass of the A boson, the rates of h decays into bottom quarks and tau leptons are enhanced as compared to the SM.
(iii) The predicted value of BR(b → sγ) in the allowed region is ∼ 4.6×10 −4 , which is about one σ above the current experimental value (if the experimental and theory errors are added linearly). Consequently, an improvement in the present theoretical uncertainty might enable a discrepancy to appear between BR(b → sγ) and the SM value.
(iv) The discrepancy between the experimental measurement of (g − 2) µ and the SM calculation can easily be explained in this scenario, although the (g − 2) µ discrepancy should be somewhat smaller than the current central value. However, a much smaller discrepancy (corresponding to larger m 0 values) could also be accomodated.
(v) Confronting the prediction for BR(B u → τ ν τ ) [31] with the measurement from Belle and BABAR [32] already yields interesting constraints on the charged Higgs-boson mass as a function of tan β, although the present experimental errors are still very large, see e.g. Ref. [33]. In the scenario considered here BR(B u → τ ν τ ) is predicted to be relatively low as compared to its SM value, where the ratio of MSSM/SM is ∼ 0.33. This is about one σ below the current central value.
(vi) The ratio of the B s mass difference to the SM prediction is close to unity, ∼ 0.91 [31]. In view of the theoretical uncertainties, it will be difficult to establish such a small deviation.
(vii) The W boson mass is predicted to be [34,35] relatively low, M W ∼ 80.367 GeV, i.e. again about one σ below the current central experimental value.
(ix) Direct detection experiments should see a signal at current sensitivities. The lack of a signal in the CDMS experiment could be explained by a reduced strangeness contribution to the proton mass. The reported XENON10 limit would further require a reduction in the local halo density (to its lower limit).
Performing a χ 2 fit for nine precision and B-physics observables along the lines of Ref. [19] yields a total value of χ 2 tot ∼ 9 − 10 in the allowed part of the NUHM parameter space, where even slightly smaller values can be found for M h < ∼ 114 GeV. 6 Very likely the weaker CDF tan β bound from the search for heavy Higgs bosons compared to its expected sensitivity is due to a statistical fluctuation that will eventually evaporate. Nevertheless, it is interesting to know whether a signal at this level could be accommodated within the MSSM. We have shown that M A ∼ 160 GeV is possible for tan β > ∼ 45 within the NUHM, though it stretches various experimental constraints to their limits. Correspondingly, small improvements in some of these measurements, e.g., of BR(B s → µ + µ − ) or of the dark matter scattering rate, would either exclude such a signal in the NUHM framework, or else provide supporting evidence. One way or another, it should be possible soon to cast light on the interpretation of the CDF search.