On the Higgs mass in the CMSSM

CERN-PH-TH/2005-173, UMN–TH–2416/05, FTPI–MINN–05/43, ACT-09-05, MIFP-05-23 On the Higgs Mass in the CMSSM John Ellis1 , Dimitri Nanopoulos2 , Keith A. Olive3 and Yudi Santoso4 1 TH Division, PH Department, CERN, Geneva, Switzerland P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA; Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA; Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece 3 William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA 4 Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada; Perimeter Institute of Theoretical Physics, Waterloo, ON, N2J 2W9, Canada arXiv:hep-ph/0509331 v1 28 Sep 2005 2 George Abstract We estimate the mass of the lightest neutral Higgs boson h in the minimal supersymmetric extension of the Standard Model with universal soft supersymmetry-breaking masses (CMSSM), subject to the available accelerator and astrophysical constraints. For mt = 174.3 GeV, we find that 114 GeV < mh < 127 GeV and a peak in the tan β distribution ≃ 55. We observe two distinct peaks in the distribution of mh values, corresponding to two different regions of the CMSSM parameter space. Values of mh < 119 GeV correspond to small values of the gaugino mass m1/2 and the soft trilinear supersymmetry-breaking parameter A0 , lying along coannihilation strips, and most of the allowed parameter sets are consistent with a supersymmetric interpretation of the possibly discrepancy in gµ − 2. On the other hand, values of mh > 119 GeV may correspond to much larger values of m1/2 and A0 , lying in rapid-annihilation funnels. The favoured ranges of mh vary with mt , the two peaks being more clearly separated for mt = 178 GeV and merging for mt = 172.7 GeV. If the gµ − 2 constraint is imposed, the mode of the mh distribution is quite stable, being ∼ 117 GeV for all the studied values of mt . CERN-PH-TH/2005-173 September 2005 1 Introduction One of the characteristic predictions of the minimal supersymmetric extension of the Standard Model (MSSM) is the mass of the lightest neutral Higgs boson h, which is expected to be mh < ∼ 150 GeV [1]. This is very consistent with the range mh < ∼ 200 GeV that is favoured by global analyses of the available precision electroweak data [2]. Various studies have shown that the lightest neutral MSSM Higgs boson is very likely to be discovered at the LHC, and possibly at the Fermilab Tevatron collider. It is therefore interesting to attempt to refine the MSSM prediction for mh , and to consider what one would learn from a measurement of the h mass [3]. We study these questions within the constrained version of the MSSM (CMSSM), in which the soft supersymmetry-breaking scalar masses m0 and gaugino masses m1/2 are each assumed to be universal at some GUT input scale, as are the trilinear soft supersymmetrybreaking parameters A0 . We impose on the CMSSM the available phenomenological constraints from accelerator experiments, astrophysics and cosmology [4, 5], treating the supersymmetric interpretation of the anomalous magnetic moment of the muon, gµ − 2, as an optional constraint, and interpreting the WMAP range for the cold dark matter density [6] as an upper bound: Ωχ h2 < 0.129. We base our study on a statistical sampling of the CMSSM parameter space that is uniform in the (m1/2 , m0 ) plane for 100 GeV < m1/2 < 2 TeV, m0 < 2 TeV, |A0 /m1/2 | < 3, 2 < tan β < 58 and µ > 0, assuming initially that mt = 174.3 GeV [7] and discussing later other possible values of mt . We began with a random sample of over 320,000 CMSSM points: requiring the lightest supersymmetric particle (LSP) to be a neutralino brought the number down to somewhat over 260,000. As seen in Fig. 1(a), before imposing the various phenomenological constraints we find that mh 1 is distributed between very low values < 110 GeV and an upper limit ∼ 128 GeV, with a single peak at ∼ 120 GeV. The drop-off in the count at low mh is mainly related to our choice of a uniform measure in the CMSSM input parameters: because of the logarithmic dependence of mh on m1/2 and m0 , low values of mh only occur at low values of m1/2 , m0 and tan β. (We recall that mh evolves quickly as m1/2 is increased at low m1/2 and more slowly at large m1/2 .) The fall-off at large mh is largely due to our choice of 2 TeV as the upper limit on m1/2 . Extending this upper limit would slowly push the peak in Fig. 1(a) to the right, and the count at the peak would grow rapidly. Once again, because of the logarithmic dependence of mh on m1/2 , even a modest change in the position of the peak would require increasing the upper limit on m1/2 1 We use Fortran code FeynHiggs [8] to calculate mh . 1 300 16000 All points after cuts Points satisfying g-2 14000 250 12000 200 Count Count 10000 8000 6000 150 100 4000 50 2000 0 0 109 112 115 118 121 124 127 130 mh (GeV) 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 mh (GeV) Figure 1: The distribution of the mass of the lightest neutral Higgs boson, mh , in the CMSSM (a) before applying the accelerator cuts and the WMAP relic density constraint, and (b) after applying these constraints. In the latter case, the red (light) histogram shows the points favoured by the optional gµ − 2 constraint. We assume mt = 174.3 GeV in both panels. substantially. We next apply a series of constraints, including the LEP lower limit on the chargino mass of 104 GeV, b → sγ [9] and the limit on Ωχ h2 that was discussed above, as well as the the direct experimental limit on mh of ∼ 114 GeV [10] 2 . The most severe cut on the sample, by far, is that due to the relic density, which for most points exceeds the WMAP upper limit. When all cuts are applied our sample is reduced to 3075 points, which are plotted in Fig. 1(b). Most of the range in mh is still available after imposing the various phenomenological constraints, as seen in Fig. 1(b). However, we see that the distribution of mh within this range exhibits significant structures, with peaks at mh ≃ 121 and 117 GeV, and a dip at mh ≃ 119 GeV. In the rest of this paper, we explain the origins of these features, describe the domains of the CMSSM parameter space that populate these peaks in the mh distribution, and discuss the effects of imposing the optional gµ − 2 constraint [12] and varying mt . The peaked structures in tan β and mh reflect different processes that might reduce the density of supersymmetric relics χ into the range allowed by WMAP and other observations 3 : 2 We recall that, although this lower limit may be relaxed in some variants of the MSSM, its value does not change for the CMSSM studied here [11]. 3 Note that although we consider A0 6= 0, the the stop-coannihilation region [13] is beyond the range we scan. 2 either coannihilations with sleptons, most importantly the lighter stau: τ̃1 [14], or rapid annihilations: χχ via the heavier neutral Higgs bosons A, H [15], or (exceptionally) rapid annihilations: χχ via the lightest neutral Higgs bosons h [16]. The structures in the mh distribution imply that, once mt is better known from Tevatron and/or LHC measurements and assuming the CMSSM framework 4 , a measurement of mh at the LHC or Tevatron might enable one to estimate ranges for the values of m1/2 , A0 and tan β, even if sparticles themselves are not yet discovered. If sparticles are discovered, confronting their masses with the ranges inferred from mh will be a crucial test of the CMSSM framework. 2 Effects of Phenomenological Constraints on the CMSSM Parameter Space As already mentioned, we have sampled uniformly the (m1/2 , m0 ) plane for m1/2 , m0 < 2 TeV, |A0 /m1/2 | < 3, 2 < tan β < 58 and µ > 0 5 , assuming mt = 174.3 GeV as our default 6 . As tan β is increased, there is an increasing fraction of sample points that do not yield consistent electroweak vacua. Nevertheless, the consistent solutions are distributed quite smoothly in tan β before applying the accelerator and cosmological cuts, as seen in panel (a) of Fig. 2. However, after applying the cuts, the distribution in tan β is far from uniform, as seen in panel (b) of Fig. 2. The distribution of allowed models is sharply peaked towards large tan β, with a relatively small tail surviving below tan β ∼ 20. This observation holds for both the general sample and the (gµ − 2)-friendly subsample 7 , shown as the light (red) shaded histogram in panel (b). The preference for large tan β is an understandable consequence of the interplay of the various accelerator and cosmological constraints. For example, the cosmological upper limit on the supersymmetric relic density in the coannihilation region imposes an upper limit on m1/2 that is significantly relaxed at large tan β, in particular by rapid χχ → A, H annihilation. Moreover, the funnels due to the rapid-annihilation processes χχ → H, A are broader than the coannihilation strips that define the acceptable cosmological regions at lower tan β. We also note that the range of small values of m1/2 that is excluded by the experimental lower limit on mh diminishes as tan β increases, and recall that the predominance of high 4 We also assume that theoretical errors in the CMSSM calculation of mh can be reduced along with the experimental error. 5 This sign of µ is suggested by even a loose interpretation of gµ − 2. 6 We discuss later the effect of varying mt in our analysis. 7 We assume here gµ − 2 range from 6.8 to 43.6 × 10−10 [12]. 3 7000 700 6000 600 5000 500 Count Count All points after cuts 4000 400 3000 300 2000 200 1000 100 0 Points satisfying g-2 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 tan β tan β Figure 2: (a) The distribution in tan β of the sample points shown in Fig. 1(a), before the accelerator and WMAP constraints are applied, and (b) their distribution after applying these phenomenological constraints. In the latter, we see that the distribution of gµ − 2-friendly points (coloured red/grey) is similar to that of the total sample. tan β in satisfying constraints was clearly seen in a likelihood analysis [17] when comparing regions of high likelihood for tan β = 10 and 50. Panel (a) of Fig. 3 displays the allowed points in the (A0 , tan β) plane. We see that they gather into three clusters: one centered around A0 = 0 that extends to small values of tan β, and two at large values of |A0 | that are concentrated at larger tan β, particularly for A0 < 0. As seen in panel (b) of Fig. 3, these accumulations populate different regions of mh . Specifically, the points with mh ∈ (114, 119) GeV, which populate the low-mass peak in Fig. 1(b), have relatively low values of A0 , most of which are negative. On the other hand, points with A0 < −2 TeV generally have mh ∈ (119, 122) GeV and points with A0 > 1 TeV have mh ∈ (122, 127) GeV. Between these wings, there are addditionally some low-|A0 | points with mh ∈ (119, 124) GeV. Thus, the higher-mass peak in Fig. 1(b) receives contributions from all regions of A0 . We also see in Fig. 1(b) that essentially all the low-mass points are (gµ − 2)-friendly (shaded red/grey), that only some of the high-mass points with A0 > −2 TeV are (gµ − 2)-friendly, and that none of the points with A0 < −2 TeV are (gµ − 2)-friendly. 4 Figure 3: (a) The distribution in the (A0 , tan β) plane of the sample points shown in Fig. 1(b) that survive the accelerator and WMAP constraints, and (b) a scatter plot of these points in the (A0 , mh ) plane. The (gµ − 2)-friendly points are coloured red (grey). 3 Interpretation of Features in the mh Distribution The origins of many of these features can be understood qualitatively by referring to the various (m1/2 , m0 ) planes displayed in Fig. 2 of [18] for different values of tan β and A0 . Updated planes for the case tan β = 55, whose importance can be seen from panel (b) of Fig. 2 and panel (a) of Fig. 3, are shown in Fig. 4. When tan β < ∼ 45 and µ > 0 as assumed here, the regions allowed by WMAP and the other constraints are essentially narrow coannihilation strips that decrease in width as m1/2 increases, terminating when m1/2 ∼ 1000 GeV. In most of these regions, |A0 | < ∼ 1000 GeV also, so these points populate the central A0 ∼ 0 region in Fig. 3(b). Therefore, they provide the majority of the models in the low-mass peak in Fig. 1(b), but also a tail extending under the higher-mass peak, as seen in panel (b) of Fig. 3. These coannihilation strips are also the dominant features for tan β = 55 when A0 > 0, as seen in panel (a) of Fig. 4 by the two examples for A0 = +m1/2 , +2m1/2 . However, a second class of features is visible in Fig. 2 of [18] when tan β > ∼ 50, namely rapid-annihilation funnels at large m1/2 , as updated in panels (a, b) of Fig. 4 for A0 = 0, −m1/2 , −2m1/2 . These funnel regions populate the high-mass peak in Fig. 1(b). The funnels are typically broader than the coannihilation strips, and therefore have a larger weighting in the constant-density sampling of the (m1/2 , m0 ) plane that we have made in this paper. As a result, the larger values of tan β have a strong weight in the sample of models surviving the accelerator and WMAP constraints that we showed in Fig. 2. We recall that we retain 5 tan β = 55 , µ > 0 1000 1000 500 0 100 tan β = 55 , µ > 0 1500 m0 (GeV) m0 (GeV) 1500 500 A 0 = + 2 m1/2 500 1000 1500 A 0 = - 2 m1/2 A 0 = + m1/2 A 0 = - m1/2 A0 = 0 A0 = 0 2000 2500 m1/2 (GeV) 0 100 500 1000 1500 2000 2500 m1/2 (GeV) Figure 4: Regions in the (m1/2 , m0 ) planes for tan β = 55, as calculated for mt = 174.3 GeV using the latest version of the SSARD code [19]. Note the narrow coannihilation strips for A0 > 0 in panel (a) and the broader rapid-annihilation funnels for A0 < 0 shown in panel (b). in our analysis points whose relic density Ωχ h2 falls below the range favoured by WMAP [6], which typically have slightly lower values of m0 than along the coannihilation strips, while remaining within the region where the LSP is the lightest neutralino χ, and lie inside the rapid-annihilation funnels. Restricting our plots to points with Ωχ within the WMAP range would reduce the statistics in our plots, but not alter their basic features. The weight of the rapid-annihilation points could be diminished if one used a different sampling procedure, e.g., if one gave less weight to regions of parameter space with large m1/2 and/or m0 , and hence |A0 |, as might be motivated by fine-tuning considerations. However, the ‘twin-peak’ structure of the mh distribution would survive any smooth reweighting of parameter space. The rapid-annihilation funnels are responsible for the dense cluster of models at large tan β and A0 < −2 TeV in panel (a) of Fig. 3, which have mh > 119 GeV as seen in panel (b) of Fig. 3, and hence populate the higher peak in the Higgs mass distribution in Fig. 1. It is also clear from Figs. 1 and 3 that the basic feature of a doubly-peaked Higgs mass distribution linked to different ranges of A0 would also survive any smooth reweighting of the parameter space. As discussed in [18] and seen in Fig. 4, the locations of the rapid-annihilation funnels are very sensitive to A0 , reflecting the sensitivity of mA,H to this parameter (among others). Starting from the A0 = 0 case where the funnel extends above m1/2 ∼ 1000 GeV for mt = 6 174.3 GeV, the funnel moves to smaller m1/2 as A0 decreases and merges progressively with the coannihilation strip. On the other hand, no funnels are visible for A0 sufficiently > 0. The WMAP regions for A0 = ±2m1/2 provide points in Fig. 3 with extreme positive and negative values of A0 , respectively. The different breadths of these regions explain the asymmetry in panel (b) of Fig. 3, in particular. Indeed, for A0 > 0 these points are simply continuations of the coannihilation strips. As seen in Fig. 3, some of these points are (gµ − 2)-friendly, and provide the tail under the high-mass peak in Fig. 1. We now consider the (m1/2 , A0 ) planes shown in Fig. 5, where panel (a) shows the combination of all vales of tan β, and panel (b) shows only models with tan β > 50. These panels update analogous plots in [18], and display significant differences due to the reduction in mt from 178 GeV to 174.3 GeV and improvements in the treatment of vacuum stability requirements. Previously, we had seen a clear separation between ‘fins’ at A0 ∼ ±1.5m1/2 and a ‘torso’ at A0 ∼ 0, which has vanished apart (possibly) from a vestigial fin at A0 > 2 TeV that is more visible at large tan β. We also note the appearance of a small ‘head’ with a ‘tooth’ at m1/2 < ∼ 150 GeV and A0 ∼ 0, which is due to points with mχ ∼ mh /2, whose relic density falls within the WMAP range thanks to rapid annihilation through the light CMSSM Higgs pole [16]. These points have mχ± very close to the LEP lower limit, and might be accessible to the Tevatron. We see in panel (a) of Fig. 5 that the points with mh < 119 GeV (darker blue/black and red/grey colours) cluster at m1/2 , A0 < 1 TeV and A0 > −3 TeV. Almost all these points make a supersymmetric contribution to gµ − 2 that could explain the possible discrepancy between experiment and the Standard Model calculation based on e+ e− data (indicated in red/grey). On the other hand, only a small fraction of the models with mh > 119 GeV (pale colours) are compatible with this supersymmetric interpretation of gµ − 2 (pink/light grey). As seen in panel (b) of Fig. 5, all the parameter sets with tan β > 50 have mh > 119 GeV. The (gµ − 2)-friendly points are concentrated at m1/2 < ∼ 1 TeV. This analysis can be used as a diagnostic tool when the Higgs boson is discovered at the Tevatron or the LHC, at least within the CMSSM framework and assuming that mt = 174.3 GeV. This framework would be invalidated if mh > 127 GeV. On the other hand, if the Higgs boson is discovered with a mass mh < 119 GeV, one can infer from Fig. 5(a) that both m1/2 and A0 must be small, and that supersymmetry is likely to lie along a coannihilation strip. On the other hand, if mh > 119 GeV, supersymmetry may well have chosen a rapid-annihilation funnel. 7 Figure 5: (a) The distribution in the (m1/2 , A0 ) plane of the points shown in Fig. 1(b) that satisfy the accelerator and WMAP constraints, separated into the low-mh region (blue points) and the high-mh region (red points). (b) The distribution in the (m1/2 , A0 ) plane of points with tan β > 50. 4 Potential Impact of gµ − 2 We now comment further on the potential impact of imposing the gµ − 2 constraint [12], which we treat as optional. We see in Fig. 1(b) that this constraint would suppress the high-mh peak, while retaining most of the low-mh models. The suppression of the high-mh peak is a consequence of the removal of points with large m1/2 and/or m0 that would make a very small contribution to gµ − 2, many of which are in the rapid-annihilation funnels. A similar effect reduces also the upper part of the low-mh peak, but the coannihilation strips would be less affected by the gµ −2 constraint. On the other hand, as seen in Fig. 2, imposing the gµ − 2 constraint would not alter the statistical preference for large tan β. As we see in Fig. 3, imposing the gµ − 2 constraint would disfavour models with large negative A0 , as well as many with large positive A0 , but some models with large tan β and a small A0 would survive. 5 Dependence on mt In all the above, we have assumed that mt = 174.3 GeV [7]. The central value was formerly 178 GeV [20], and the current central value is mt = 172.7±2.9 GeV [21], following significant evolution during recent months. In view of this and the remaining experimental uncertainty, 8 60 200 All points after cuts All points after cuts Points satisfying g-2 Points satisfying g-2 50 150 Count Count 40 100 30 20 50 10 0 0 114 116 118 120 122 124 126 128 114 130 mh (GeV) 116 118 120 122 124 126 128 130 mh (GeV) Figure 6: As for panel (b) of Fig. 1, but assuming (a) mt = 178 GeV and (b) mt = 172.7 GeV. we have also considered the dependence of the above analysis on mt . We recall that ∆mh ∼ 1 GeV for ∆mt = 1 GeV in theoretical calculations, and that the parameter regions allowed by WMAP vary quite considerably with mt , particularly in the rapid-annihilation funnel region, as seen in Fig. 1 of [18]. Specifically, this region moves to smaller (larger) m1/2 for smaller (larger) mt . As was already mentioned, the coannihilation strips mainly populate the low-mass peak in Fig. 1 whereas the high-mass peak is largely due to the rapid-annihilation funnels. Accordingly, we would expect these peaks to be more separated at large mt than at smaller values. Precisely this effect is seen in the two panels of Fig. 6. We see in panel (a) that the upper peak in Fig. 1(b) shifts upwards by ∼ 4 GeV if mt = 178 GeV [20], and is very clearly separated from the low-mass peak. Correspondingly, the upper limit on mh increases to 130 GeV. On the other hand, we see in panel (b) of Fig. 6 that the two peaks merge for mt = 172.7 GeV [21], and the upper limit on mh decreases to 126 GeV. Likewise, many of the other features discussed previously in this paper become more (less) pronounced for larger (smaller) mt . By the time the Higgs boson is discovered, we expect mt to be known with considerably greater precision than the present uncertainty δmt ≃ 2.9 GeV. Once mt is known with an accuracy δmt < ∼ 2 GeV, and assuming that the accuracy of theoretical calculations in the CMSSM can be brought to the same level, there will be no theoretical ambiguity between the Higgs mass peaks, and diagnosis of the supersymmetric parameters will indeed be possible along the lines discussed in the previous Section. 9 As in the case mt = 174.3 GeV shown in panel (a) of Fig. 1, for mt = 178 GeV the effect of imposing the gµ − 2 constraint would also be to remove the high-mass peak, leaving a plateau extending from mt ∼ 118 GeV to ∼ 127 GeV. The low-mass-peak would also be reduced, but most of the intermediate plateau for mt = 178 GeV would survive the gµ − 2 constraint. In the case of mt = 172.7 GeV shown in panel (b), there is a more pronounced peak at mt ∼ 117 GeV and a tail extending to ∼ 124 GeV. It is striking that, whatever the value of mt , the mode of the mh distribution is relatively stable at ∼ 117 GeV and that the upper limit on mh also remains relatively stable around 125 GeV, if the gµ − 2 constraint is imposed. 6 Conclusions We have shown that the available experimental and cosmological constraints on the CMSSM allow only a limited range of mh . If mt = 174.3 GeV, this is < 127 GeV without the gµ − 2 constraint and < 124 GeV if it is imposed. If gµ − 2 is not imposed, we find twin peaks in the mh distribution at mh ∼ 117, 121 GeV. The upper bound and the lower peak are quite insensitive to variations in mt , whereas the upper peak is sensitive, and merges with the lower peak for low mt . Large values of tan β ∼ 55 are favoured by our analysis, whether the gµ − 2 constraint is applied, or not. We have also shown in this paper that the mass of the lightest CMSSM Higgs boson may be a useful diagnostic tool for identifying the most likely regions of the CMSSM parameter space, even if sparticles are not (yet) discovered. This is because the CMSSM is divided into distinct coannihilation and rapid-annihilation regions, and measuring mh could provide us with a hint which alternative is chosen by Nature. Acknowledgments The work of D.V.N. was supported in part by DOE grant :DE-FG03-95-ER-40917. The work of K.A.O. was supported in part by DOE grant DE–FG02–94ER–40823. The work of Y.S. was supported in part by the NSERC of Canada, and Y.S. thanks the Perimeter Institute for its hospitality. 10 References [1] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. 85 (1991) 1; Phys. Lett. B 262 (1991) 54; H. E. Haber and R. Hempfling, Phys. Rev. Lett. 66 (1991) 1815; J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B 257 (1991) 83; Phys. Lett. B 262 (1991) 477; A. Yamada, Phys. 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