Dark Matter and Higgs Mass in the CMSSM with Yukawa Quasi-Unification
We present an updated analysis of the constrained minimal supersymmetric standard model with supplemented by an ‘asymptotic’ Yukawa coupling quasi-unification condition, which allows an acceptable -quark mass. Imposing constraints from the cold dark matter abundance in the universe, physics, the muon anomalous magnetic moment, and the mass of the lightest neutral CP-even Higgs boson, we find that the lightest neutralino cannot act as a cold dark matter candidate. This is mainly because the upper bound on the lightest neutralino relic abundance from cold dark matter considerations, despite the fact that this abundance is drastically reduced by neutralino-stau coannihilations, is incompatible with the recent data on the branching ratio of . Allowing for a different particle, such as the axino or the gravitino, to be the lightest supersymmetric particle and, thus, constitute the cold dark matter in the universe, we find that the predicted ’s in our model favor the range .
The well-known constrained minimal supersymmetric standard model (CMSSM) [1, 2, 3, 4], which is a highly predictive version of the minimal supersymmetric standard model (MSSM) based on universal boundary conditions for the soft supersymmetry (SUSY) breaking parameters, can be further restricted by being embedded in a SUSY grand unified theory (GUT) with a gauge group containing and . This can lead  to ‘asymptotic’ Yukawa unification (YU) , i.e. the exact unification of the third generation Yukawa coupling constants , , and of the top quark, the bottom quark, and the tau lepton, respectively, at the SUSY GUT scale . The simplest GUT gauge group which contains both and is the Pati-Salam (PS) group [7, 8] – for YU within , see Refs. [9, 10].
However, given the experimental values of the top-quark and tau-lepton masses (which, combined with YU, naturally restrict ), the CMSSM supplemented by the assumption of YU yields unacceptable values of the -quark mass for both signs of the parameter . This is due to the presence of sizable SUSY corrections  to (about 20), which arise [11, 12] from sbottom-gluino (mainly) and stop-chargino loops and have the sign of – with the standard sign convention of Ref. . The predicted tree-level , which turns out to be close to the upper edge of its confidence level (c.l.) experimental range, receives, for , large positive [negative] corrections which drive it well above [a little below] the allowed range. Consequently, for both signs of , YU leads to an unacceptable with the case being much less disfavored.
The usual strategy to resolve this discrepancy is the introduction of several kinds of nonuniversalities in the scalar [9, 10] and/or gaugino [14, 15] sector of MSSM with an approximate preservation of YU. On the contrary, in Ref. , concrete SUSY GUT models based on the PS gauge group are constructed which naturally yield a moderate deviation from exact YU and, thus, can allow acceptable values of the -quark mass for both signs of within the CMSSM. In particular, the Higgs sector of the simplest PS model [7, 8] is extended so that the electroweak Higgs fields are not exclusively contained in a bidoublet superfield, but receive subdominant contributions from other representations too. As a consequence, a moderate violation of YU is naturally obtained, which can allow an acceptable -quark mass even with universal boundary conditions. It is also remarkable that the resulting extended SUSY PS models support new successful versions  of hybrid inflation based solely on renormalizable superpotential terms.
These models provide us with a set of ‘asymptotic’ Yukawa quasi-unification conditions, which replace exact YU and can be applicable [16, 18, 19] for both signs of the MSSM parameter . We focus here on the case since is strongly disfavored by the constraint arising from the deviation of the measured value of the muon anomalous magnetic moment from its predicted value in the standard model (SM). Indeed, is defended only at 3 by the calculation of based on the -decay data which is presented in Ref. , whereas there is a stronger and stronger tendency at present to prefer the -annihilation data for the calculation of , which favor the regime. Moreover, in Ref. , it was claimed that, after some improvements, the -based result shifts considerably towards the -based one.
This relation depends on a single parameter , which is taken, for simplicity, to be real and lying in the range . With fixed masses for the fermions of the third generation, we can determine the parameters and so that Eq. (1) is satisfied. In contrast to the original version of the CMSSM [2, 3, 4], therefore, is not a free parameter, but can be restricted, within our set-up, via Eq. (1) to relatively large values. The remaining free parameters of our model are the universal soft SUSY breaking parameters defined at , i.e.
where the symbols above denote the common gaugino mass, scalar mass, and trilinear scalar coupling constant, respectively. These parameters can be restricted by employing a number of experimental and cosmological requirements as in Refs. [16, 18, 19] and most recently in Refs. [22, 23]. In this talk, we review the results of Ref.  implementing the following improvements:
2 Cosmological and Phenomenological Constraints
In our investigation, we integrate the two-loop renormalization group equations for the gauge and Yukawa coupling constants and the one-loop ones for the soft SUSY breaking parameters between and a common SUSY threshold ( are the stop mass eigenstates) determined in consistency with the SUSY spectrum. At , we impose radiative electroweak symmetry breaking, evaluate the SUSY spectrum employing the publicly available calculator SOFTSUSY , and incorporate the SUSY corrections to the and mass . The corrections to the -lepton mass (almost 4) lead [16, 18] to a small decrease of . From to , the running of gauge and Yukawa coupling constants is continued using the SM renormalization group equations.
The parameter space of our model can be restricted by using a number of phenomenological and cosmological constraints. We calculate them using the latest version of the publicly available code micrOMEGAs . We now briefly discuss these requirements – for similar recent analyses, see Ref.  for the CMSSM or Refs. [14, 29] for the MSSM with YU.
SM Fermion Masses.
The masses of the fermions of the third generation play a crucial role in the determination of the evolution of the Yukawa coupling constants. For the -quark mass, we adopt as an input parameter in our analysis the -quark mass, which at 1 is 
with the central value being . For the top-quark mass, we use the central pole mass () as an input parameter :
with being the running mass of the quark. We also take the central value  of the tau-lepton mass at .
Cold Dark Matter Considerations.
According to the WMAP results , the c.l. range for the CDM abundance is
In the context of the CMSSM, the lightest neutralino can be the lightest supersymmetric particle (LSP) and, thus, naturally arises as a CDM candidate. In this case, the requirement that the relic abundance does not exceed the c.l. upper bound from Eq. (6), i.e.
strongly restricts the parameter space of the CMSSM. This is because increases, in general, with and so an upper bound on can be derived from Eq. (7). The calculation of in micrOMEGAs includes accurately thermally averaged exact tree-level cross sections of all the possible (co)annihilation processes [3, 34], treats poles [4, 16, 35] properly, and uses one-loop QCD and SUSY QCD corrections [11, 16, 36] to the Higgs decay widths and couplings to fermions. It should, though, be noted that the restrictions induced by Eq. (7) can be evaded if we adopt one (or a combination) of the following scenarios:
The cosmological evolution before Big Bang nucleosynthesis (BBN) deviates from the standard one [37, 38]. Since within the CMSSM is essentially a pure bino, the scenario which fits better this case is the low reheat temperature scenario with the decoupling of occurring before reheating. This scenario, however, is disfavored since it requires a very low reheat temperature . We will, thus, assume that the decoupling of the neutralino from the cosmic fluid occurs during the conventional radiation dominated era.
The lightest neutralino is not the LSP and, thus, the relic density of another SUSY particle , which is the LSP, is to account for . This particle could be the gravitino  or the axino [41, 42]. The case of gravitino is tightly restricted in the CMSSM due to the BBN constraints imposed during the decay of the lightest ordinary supersymmetric particle (LOSP) to it. On the other hand, axino CDM  is, in general, possible once its mass and the reheat temperature are chosen appropriately. In such a case, may play the role of the LOSP and can contribute to the non-thermal production of the LSP. In particular, its contribution to the relic density of the LSP is equal to times the ratio of the LSP mass to and, thus, ’s exceeding the bound in Eq. (7) can be perfectly acceptable.
The Branching Ratio of .
The most recent experimental world average for is known  to be and its updated SM prediction is . Combining in quadrature the experimental and theoretical errors involved, we obtain the following constraints on this branching ratio at c.l.:
The computation of in the micrOMEGAs package presented in Ref.  includes  next-to-leading order (NLO) QCD corrections to the charged Higgs boson () contribution, the enhanced contributions, as well as resummed NLO SUSY QCD corrections. The contribution interferes constructively with the SM contribution, whereas the SUSY contribution interferes destructively with the other two contributions for . The SM plus the and SUSY contributions initially increases with and yields a lower bound on from the lower bound in Eq. (8). (For higher values of , it starts mildly decreasing.)
The Branching Ratio of .
The rare decay occurs via penguin and box diagrams in the SM and, thus, its branching ratio is highly suppressed. The SUSY contribution, though, originating [46, 47] from neutral Higgs bosons in chargino-, -, and -mediated penguins behaves as ( is the mass of the CP-odd Higgs boson ) and hence is particularly important for large ’s, especially after the new more stringent c.l. upper bound
recently reported by CMS and LHCb . This new bound significantly reduces the previous bound , which we had adopted in Ref. . The bound in Eq. (9) implies a lower bound on since decreases as increases.
The Branching Ratio of .
The purely leptonic decay proceeds via - and -mediated annihilation processes. The SUSY contribution, contrary to the SM one, is not helicity suppressed and depends on the mass of the charged Higgs boson since it behaves [47, 49] as . The ratio of the CMSSM to the SM branching ratio of increases with and approaches unity. It is to be consistent with the following c.l. range :
A lower bound on can be derived from the lower bound in this inequality.
Muon Anomalous Magnetic Moment.
The quantity , which is defined in Sec. 1, can be attributed to SUSY contributions arising from chargino-sneutrino and neutralino-smuon loops. The relevant calculation is based on the formulas of Ref. . The absolute value of the result decreases as increases and its sign is positive for . On the other hand, the calculation of is not yet completely stabilized mainly because of the ambiguities in the calculation of the hadronic vacuum-polarization contribution. According to the evaluation of this contribution in Ref. , there is still a discrepancy between the findings based on the -annihilation data and the ones based on the -decay data – however, in Ref. , it was claimed that this discrepancy can be considerably ameliorated. Taking into account the more reliable calculation based on the data and the experimental measurements  of , we obtain the following c.l. range – cf. Ref. :
A lower [upper] bound on can be derived from the upper [lower] bound in Eq. (11). As it turns out, only the upper bound on is relevant in our case. Taking into account the aforementioned computational instabilities and the common practice , we consider this bound only as an optional constraint.
For our analysis, the only relevant collider bound is the c.l. LEP bound  on the lightest CP-even neutral Higgs boson mass
which gives a lower bound on . However, we should keep in mind that recent data from ATLAS and CMS  provide a c.l. upper bound and a hint in favor of the range . Allowing for a theoretical error of and adding in quadrature the experimental and theoretical uncertainties, we construct the 1 range of interest :
The calculation of in the package SOFTSUSY  includes the full one-loop SUSY corrections and some zero-momentum two-loop corrections . The results are well tested  against other spectrum calculators.
3 Restrictions on the SUSY Parameters
Imposing the requirements described above, we can delineate the allowed parameter space of our model. Throughout our investigation, we consider the central values for the SM parameters , , , and . We adopt the following conventions for the various lines and regions in the relevant figures (Figs. 2 and 3) – see Fig. 1:
On the solid black line, Eq. (7) is saturated.
The horizontally hatched region is allowed by Eq. (7).
In the light gray region, the lightest stau is lighter than .
The dark gray region is excluded by the lower bound in Eq. (8).
The gray region is excluded by Eq. (9).
The yellow region is excluded by the lower bound in Eq. (10).
The vertically hatched region is favored by the lower bound in Eq. (11).
The red region is excluded by Eq. (12).
Note that the upper bounds in Eqs. (8), (10), and (11) do not restrict the parameters of our model. The region with lighter than can not be excluded if the LSP is a neutral sparticle other than . One should, though, make sure that the decay of to the LSP does not destroy the predictions of the standard BBN [40, 41].
We present the restrictions from all the requirements imposed in the plane for , , , and in Fig. 2. From the relevant data, we observe that the lower bound in Eq. (10) is fulfilled for the mass of the CP-odd Higgs boson and almost independently of the other parameters. Note also that, for and , the bound in Eq. (12) is violated for and, consequently, does not appear in the relevant diagrams. It is obvious that, for all the ’s considered in Fig. 2, we are left with no region allowed by all the restrictions of Sec. 2. This is due to the fact that the constraint in Eq. (7), which necessarily holds if is the LSP, is nowhere fulfilled simultaneously with the bound in Eq. (9). Note, finally, that the upper bound on from the lower bound in Eq. (11) is also nowhere satisfied simultaneously with the bound in Eq. (9) for the values taken for in Fig. 2. However, this conflict is less serious since, as we already explained in Sec. 2, the lower bound in Eq. (11) is considered here only as an optional constraint.
The region to the left of the almost vertical part of the line corresponding to the upper bound on from Eq. (7), where the neutralino annihilation via the -channel exchange of a CP-odd Higgs boson is by far the dominant (co)annihilation process. However, this region is excluded by the constraints in Eqs. (9) and (10). On the other hand, it is well known – see e.g. Refs. [4, 16] – that this region is extremely sensitive to variations of . Indeed, we find that, as decreases, the -boson mass increases and approaches . The -pole neutralino annihilation is then enhanced and is drastically reduced causing an increase of the upper bound on . However, even if we reduce , we do not find any -pole neutralino annihilation region which is allowed by the requirements in Eqs. (9) and (10).
The narrow region which lies just above the light gray area with lighter than the neutralino, where bino-stau coannihilations [3, 34] take over leading to a very pronounced reduction of . A large portion of this region survives after the application of all the other requirements of Sec. 2 except for that in Eq. (9). To get a better understanding of this region, we can replace the parameter by the relative mass splitting between and the lightest stau, which controls the strength of bino-stau coannihilations. The coannihilation region then approximately corresponds to . It is evident from Fig. 2 that the slope of the boundary line with increases as moves away from zero in both directions. Note that this slope in our model turns out to be larger than the one obtained in other versions of the CMSSM – cf. Ref.  – with lower values of . As a consequence, small variations of or lead, in our model, to more drastic variations in .
To investigate further whether the incompatibility between the constraints in Eqs. (7) and (9) extends to all possible ’s, we focus on the coannihilation regime and construct the regions allowed by all the restrictions of Sec. 2 in the plane for . We depict our results in Fig. 3. The choice ensures the maximal possible reduction of due to the coannihilation. So, for a given value of , the maximal or allowed by Eq. (7), which holds under the assumption that the neutralino is the LSP, corresponds to . We find that, for , processes with in the initial state and , in the final one become more efficient (with a total contribution to the effective cross section of about to as decreases from to ) and so coannihilation is strengthened and ’s larger than in the case are allowed by Eq. (7). The overall maximal or allowed by Eq. (7) is encountered at yielding . Comparing the above upper bound on with the corresponding one in Fig. 3 of Ref.  (represented by a solid and two dotted black lines), we observe that here the bound is considerably enhanced in the region of low as well as the region of large values of since we do not consider the constraint from the lower bound on from the -based calculation. However, it always remains smaller than the lower bound on derived from Eq. (9) – note that for the overall lower bound on is derived from Eq. (10). Indeed, the smallest lower bound on or is found at yielding . Note that increasing within the range does not alter the boundaries of the various constraints in any essential way, except the solid line which is displaced to the left shrinking, thereby, the area allowed by Eq. (7) considerably. Needless to say that the more stringent optional upper bound on from the lower bound in Eq. (11) is also not compatible with the constraint in Eq. (9). Consequently, for every , there is no range of parameters simultaneously allowed by all the constraints and, therefore, can be now excluded as a CDM particle in our model.
The exclusion, in our model, of as a CDM candidate, resulting from the incompatibility between Eqs. (7) and (9), is further strengthened if one tries to reconcile Eqs. (7) and (13). Indeed, the tension between the neutralino CDM and the new data on is quite generic within the CMSSM since the fulfillment of Eq. (13) requires a very heavy SUSY spectrum, which leads to conflict with Eq. (7) – cf. Refs. [54, 57]. It would be interesting to investigate this issue in our model, which yields large values of and so Eq. (13) can be possibly satisfied with a lighter SUSY spectrum relative to other versions of the CMSSM with lower ’s. Our results are presented in Fig. 4, where we draw (bold lines) versus for and , , , , and . In the table included in Fig. 4, we also list the minimal ’s, , for which the inequality in Eq. (9) and the lower bound in Eq. (10) are satisfied for given values of as well as the corresponding ranges of as varies from to about . Since we take , the derived takes its minimal possible value. In this plot, we also depict by a dotted line the value of for all possible ’s and . This line terminates at since beyond this value the stability of the electroweak vacuum fails. The overall minimal is encountered at and . It is interesting to note that increases with (or ) and as decreases and, eventually, this mass enters inside the gray region in Fig. 4, which is preferred, at 1, by the recent LHC searches. However, it is obvious from Fig. 4 and the values of in the table included in this figure that Eqs. (7) and (13) are incompatible even in our model.
Departure from is not expected to alter drastically our predictions as regards the value of since depends crucially on , but only mildly on . Moreover, one can deduce from the slop of the left boundary of the gray regions in Fig. 2 that, increasing , smaller ’s and, therefore, slightly lighter Higgs masses are permitted by Eq. (9). Note that, since cannot be the LSP, both signs of are possible. In particular  corresponds to  being the LOSP. The critical case gives the minimal possible relic abundance of the LOSP in both cases due to the coannihilation effect and, therefore, the maximal possible mass of the LSP if this is produced mainly non-thermally – see Refs. [40, 41, 42].
Having in mind mostly the latter possibility, we proceed in the presentation of our predictions for the sparticle and the Higgs boson spectrum of our model, which may be observable at the LHC. In Table 1, we list the model input and output parameters, the masses in GeV of the sparticles – neutralinos , , , , charginos , , gluinos , squarks , , , , , , , , and sleptons , , , , , – and the Higgs bosons (, , , ), and the values of the various low energy observables for , , , , , and and for the minimal allowed by Eq. (9) in each case. We consider the squarks and sleptons of the two first generations as degenerate. From the values of the various observable quantities, we see that the bound in Eq. (7) and the optional lower bound in Eq. (11) are violated. So, the lightest neutralino cannot be the LSP. It is also very interesting to observe that the predicted values of lie close or even inside the range in Eq. (13) favored by Ref.  – cf. Refs. [54, 57].
The deviation from YU can be estimated by defining  the relative splittings and at through the relations:
Along the dotted line of Fig. 4, the ranges of the parameters , and are
Let us underline that, although the required deviation from YU is not so small, the restrictions from YU are not completely lost since remains large – close to 60 – and that the deviation from exact YU is generated within well-motivated SUSY GUTs described in Ref. .
It is worth emphasizing that our results do not invalidate the candidacy for a CDM particle in all versions of the CMSSM with Yukawa quasi-unification. This is because the (monoparametric) condition of Eq. (1), which we considered here, is only a simplified case of the Yukawa quasi-unification conditions shown in Eq. (15) of Ref. , which depend on one real and two complex parameters. Actually, the investigation of the viability of as a CDM candidate within the CMSSM with Yukawa quasi-unification conditions more complicated than the one in Eq. (1) derived from the GUT models of Ref.  is under consideration. Alternatively, our present model may be perfectly consistent with data if we avoid the restriction from Eq. (7) by assuming that the LSP is the axino [41, 42] with mass a little lower than and that the reheat temperature is adequately low.
|\br Input parameters|
|\mr Output parameters|
|\mr Masses in of sparticles and Higgs bosons|
|\mr Low energy observables|
We performed a revised scan of the parameter space of the CMSSM with applying a suitable Yukawa quasi-unification condition predicted by the SUSY GUT model of Ref. , which has been constructed in order to remedy the -quark mass problem arising from exact YU and universal boundary conditions. We took into account updated constraints from collider and cosmological data. These constraints originate from the CDM abundance in the universe, physics (, , and ), , and . Although the neutralino-stau coannihilations drastically reduce the neutralino relic abundance and, thus, enhance the upper bound on implied by the assumption that the neutralino is a CDM particle, they do not quite succeed to bring it to an acceptable level compatible with the lower bound on induced by . Therefore, – contrary to our findings in Ref.  – is excluded as CDM particle by the combination of the constraints from and CDM. As a consequence, the model can become consistent with observations only if the LSP is a SUSY particle other than the neutralino. This could be the axino or the gravitino and can account for the present CDM abundance in the universe. It is interesting to note that, in this case, the lowest predicted is enhanced and gets closer to the range favored by the recent preliminarily results announced by LHC.
We would like to thank A. Djouadi, N. Mahmoudi, K.A. Olive, P. Paradisi, Q. Shafi, and J. Wells for useful discussions. This work was supported by the European Union under the Marie Curie Initial Training Network ‘UNILHC’ PITN-GA-2009-237920 and the Greek Ministry of Education, Lifelong Learning and Religious Affairs and the Operational Program: Education and Lifelong Learning ‘HERACLITOS II’.
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