Abstract
We explore the prospects for observing CP violation in the minimal supersymmetric extension of the Standard Model (MSSM) with six CPviolating parameters, three gaugino mass phases and three phases in trilinear soft supersymmetrybreaking parameters, using the CPsuperH code combined with a geometric approach to maximise CPviolating observables subject to the experimental upper bounds on electric dipole moments. We also implement CPconserving constraints from Higgs physics, flavour physics and the upper limits on the cosmological dark matter density and spinindependent scattering. We study possible values of observables within the constrained MSSM (CMSSM), the nonuniversal Higgs model (NUHM), the CPX scenario and a variant of the phenomenological MSSM (pMSSM). We find values of the CPviolating asymmetry \(A_\mathrm{CP}\) in \(b \rightarrow s \gamma \) decay that may be as large as 3 %, so future measurements of \(A_\mathrm{CP}\) may provide independent information about CP violation in the MSSM. We find that CPviolating MSSM contributions to the \(B_s\) meson mass mixing term \(\Delta M_{B_s}\) are in general below the present upper limit, which is dominated by theoretical uncertainties. If these could be reduced, \(\Delta M_{B_s}\) could also provide an interesting and complementary constraint on the six CPviolating MSSM phases, enabling them all to be determined experimentally, in principle. We also find that CP violation in the \(h_{2,3} \tau ^+ \tau ^\) and \(h_{2,3} {\bar{t}} t\) couplings can be quite large, and so may offer interesting prospects for future \(pp\), \(e^+ e^\), \(\mu ^+ \mu ^\) and \(\gamma \gamma \) colliders.
Introduction
The minimal supersymmetric extension of the Standard Model (MSSM) contains many possible sources of CP violation beyond the Kobayashi–Maskawa phase of the Standard Model and the strong CP phase. These additional sources of CP violation arise from the soft supersymmetrybreaking terms in the lowenergy effective Lagrangian, and include phases in the gaugino masses, the trilinear scalar couplings and the sfermion mass matrices. However, the Kobayashi–Maskawa phase accounts very well for the CPviolating effects seen in the \(K^0\) system and in \(B\) meson decays, and no other violations of CP have been observed despite, for example, sensitive experimental searches for electric dipole moments (EDMs). Thus, it might be tempting to suggest that the extra MSSM sources of CP violation are absent. On the other hand, experimental upper limits still allow considerable scope for additional CPviolating effects in, for example, \(B^0_s\) mixing, and some additional source of CP violation is needed to explain the cosmological baryon asymmetry, which might be due to these MSSM phases. For these reasons, there have been many studies of possible MSSM CPviolating effects in experimental observables, and powerful phenomenological tools have been developed for calculating these effects.
In view of the success of the Cabibbo–Kobayashi–Maskawa (CKM) model in describing flavour mixing and CP violation in the quark sector, it is often assumed that the strong CP phase is negligibly small for some reason, and that flavour and CP violation for squarks is generated by the CKM mixing in the quark sector, the hypothesis of minimal flavour violation (MFV). However, even in this case there remain several additional sources of CP violation in the MSSM, namely the phases in the gaugino masses and the trilinear couplings. One is thus led to consider the maximally CPviolating, minimal flavourviolating (MCPMFV) model that contains six CPviolating phases beyond the Kobayashi–Maskawa phase: three phases \(\Phi _{1,2,3}\) in the masses of the U(1), SU(2) and SU(3) gauginos, and three phases \(\Phi _{A_{t,b,\tau }}\) in the trilinear soft supersymmetrybreaking couplings \(A_{t,b,\tau }\) of the thirdgeneration stop, sbottom and stau sfermions, respectively.^{Footnote 1} In this study, we allow the six CPviolating phases to vary independently in all the scenarios considered. Predictions of the MCPMFV scenario for CPviolating observables such as the CPviolating asymmetry in \(b \rightarrow s \gamma \) decay, \(A_\mathrm{CP}\), the CPviolating phase in \(B_s\) mixing, \(\phi _s\), and EDMs have been considered in [1, 2], and possibilities for probing these CPviolating phases through the polarisation of thirdgeneration fermions, \(t\) and \(\tau \), produced in the decays of the corresponding sfermions have also been explored [3].
It might be thought that the MSSM phases \(\Phi _{1,2,3,t,b,\tau }\) must necessarily be small, in view of the stringent upper limits on several EDMs shown in Table 1. However, this is not necessarily the case, since there are four main independent EDM constraints on what is, a priori, a 6dimensional space of CPviolating MSSM phases, so there are in principle ‘blind directions’ corresponding to combinations of phases that do not ‘see’ the EDM constraints. In principle, individual phases could be large along these directions, as discussed in [4] for example, and could have significant effects on other CPviolating observables such as \(A_\mathrm{CP}\) and \(\phi _s\).^{Footnote 2}
A brute force way to study this possibility would be to sample randomly the 6dimensional space of CPviolating MSSM phases, but this is not the most efficient procedure to explore the possible magnitudes of CPviolating effects in the MSSM. If one wishes to generate a large sample of parameter sets that respect other phenomenological constraints such as those from the flavour, Higgs and dark matter sectors, one would prefer to optimise the search for MSSM scenarios with maximal CP violation. A geometric approach to this problem was proposed in [2] and used to analyse the impacts of three EDM constraints in certain specific benchmark MSSM scenarios.
In this paper we adapt and extend this geometric approach to study systematically the possible magnitudes of CPviolating effects in light of the updated EDM constraints shown in Table 1. The inclusion of a fourth EDM constraint requires a slight extension of the analysis based on three EDMs made in [2], as we discuss in Sect. 2. Also, the geometric approach was originally formulated as a linear expansion around the CPconserving limit, whereas we are interested in the largest possible values of the CPviolating phases. Accordingly, here we extend the approach using an iterative procedure, finding an initial ‘blind direction’ as in [2], then choosing a CPviolating point along that direction with nonzero phases, and then repeating the geometrical optimisation in a new linear approximation around this CPviolating point, as also discussed in Sect. 2. In Sect. 3 we then apply the geometric approach to four variants of the MSSM, a bestfit scenario [11, 12] within the constrained MSSM (CMSSM) in which the soft supersymmetrybreaking parameters are constrained to be universal at the GUT scale (apart from the CPviolating phases), a generalisation of this model in which the soft supersymmetrybreaking contributions to the two Higgs doublet masses are allowed to vary independently (NUHM2), a version of the CPX scenario defined in [13] that is modified to be in agreement with the LHC results, and the phenomenological MSSM (pMSSM) [14], in which extrapolation to the GUT scale is ignored and universality is not imposed.^{Footnote 3} In each case, in addition to the EDM constraints in Table 1, we also consider the relevant constraints from flavour physics, from the measured properties of the known Higgs boson and searches for other MSSM Higgs bosons, and upper limits on the cosmological density of dark matter and the direct detection of dark matter via scattering on nuclei.
We focus, in particular, on four possible signatures of MSSM CP violation: the possibility that there might be another neutral Higgs boson lighter than the one already discovered by ATLAS and CMS, the CPviolating asymmetry in \(b \rightarrow s \gamma \) decay, \(A_\mathrm{CP}\), and the nonStandardModel contribution to the \(B_s\) meson mixing parameter, \(\Delta M_{B_s}\), and CPviolating couplings of the heavier neutral Higgs bosons. We find that, although a neutral MSSM Higgs boson lighter than that discovered would be consistent with the EDM constraints, it is excluded by the available limits on other Higgs bosons, notably the absence of a light charged Higgs boson. Secondly, we find that values of \(A_\mathrm{CP} \lesssim 3\) % are allowed by the EDMs and other constraints in some of the MSSM scenarios studied. This opens up the possibility that \(A_\mathrm{CP}\) could be significantly larger than in the Standard Model, providing a signature of CPviolating MSSM. Conversely, if a nonzero value of \(A_\mathrm{CP}\) were not to be found in future experiments, this could provide a constraint on CP violation in the MSSM that is independent of, and complementary to, those from EDMs. Thirdly, in the case of \(\Delta M_{B_s}\), we find that it could also provide an independent constraint on the CPviolating MSSM if the theoretical uncertainties could be reduced, thereby enabling in principle a complete determination of all the phases for fixed values of the CPconserving MSSM parameters. Fourthly, we also find that CP violation in the \(h_{2,3} \tau ^+ \tau ^\) and \(h_{2,3} {\bar{t}} t\) couplings can be quite large, and may offer interesting prospects for future \(pp\), \(e^+ e^\) and \(\mu ^+ \mu ^\) experiments.
Method
In this section we outline our approach to sampling the parameter spaces of MSSM scenarios while respecting the four EDM constraints in Table 1. Since the EDM constraints are quite strong, they effectively reduce the dimensionality of any MSSM scenario by four. The challenge is to sample efficiently this subspace of codimension four, so as to assess how large any other CPviolating observable may be. Moreover, the thorium monoxide EDM constraint on the electron EDM is now so strong that we have designed a new method to sample effectively the parameter space. We do this by adapting and extending the geometric approach proposed in [2]. In the first subsection we discuss how the approach may be modified to take into account four EDM constraints, in the following subsection we describe an extension of the analysis beyond the smallphase approximation, and in the third subsection we summarise our sampling algorithm.
Geometric approach to maximizing a CPviolating observable with four EDM constraints
Initially, we consider the four EDMs \(E^{a,b,c,d}\) of Table 1 in the smallphase approximation,^{Footnote 4} where
with \({\varvec{\Phi }}\equiv \Phi _\alpha = \Phi _{1,2,3,t,b,\tau }\) and \({\mathbf E}^i \equiv \partial E^i/\partial {\varvec{\Phi }}\) (i.e., \(E^i_\alpha \equiv \partial E^i/\partial \Phi _\alpha \)). The \({\varvec{\Phi }}\) subspace of codimension four is spanned by the following quadruple exterior product:
where the symbols \([\ldots ]\) denote antisymmetrisation of the enclosed indices. This subspace is a 2dimensional plane, as in the simple example in Section 2.1 of [2]. We now consider some CPviolating observable \(O\) whose dependence on the phases \(\Phi _\alpha \) is given in the smallphase approximation by \({\mathbf O} \equiv \partial O/\partial {\varvec{\Phi }}\) (i.e., \(O_\alpha \equiv \partial O/\partial \Phi _\alpha \)). One can then define the vector
that characterises a direction in the space of CPviolating phases where there is no contribution to the observable \(O\), nor to the EDMs. The EDMfree direction that optimises \(O\) is clearly orthogonal to \(B_\mu \) as well as to the EDM vectors \(E^{a,b,c,d}_\alpha \). As such, it is characterised by the sixvector
with an unknown normalisation factor.
Iterative geometric approach
The linear geometric approach described above and used in [2] entails choosing a sample of points in the MSSM scenario of interest, fixing the phases to \(0^{\circ }\) or \({\pm }180^{\circ }\) for each scan point. Next one computes the optimal direction using the above geometric approach, and then one chooses randomly sets of phases along this direction. This is suitable as long as the phases are small, but we are also interested in the possibilities for large phases.
Here we use an iterative approach to extend and improve the efficiency of the linear geometric approach. After fixing the phases to \(0^{\circ }\) or \({\pm }180^{\circ }\) and computing the favoured direction with the geometric approach as discussed above, we move by \(20^{\circ }\) along the favoured direction, and then recompute the favoured direction at this new position. This procedure is then iterated up to \(100^{\circ }\).
Sampling strategy
We have generated several million points in each of the MSSM scenarios studied in the next section. Among those points, we have retained only those for which one of the neutral Higgs bosons has a mass in the range 121–129 GeV (corresponding to the measured value \({\simeq }125\) GeV with a generous theoretical uncertainty), and we require the LSP to be the lightest neutralino. In addition, we impose the LEP and Tevatron SUSY mass limits and require squarks and the gluino to have masses above 500 GeV as a conservative implementation of the LHC SUSY limits. Although the LHC SUSY search limits are stronger in more constrained MSSM scenarios, they become weaker in more general scenarios such as the pMSSM [16–18]. For consistency, here we apply the same loose constraints on the squark and gluino masses in all studied scenarios. Other constraints, such as those imposed by heavyflavour, Higgs and direct dark matter measurements, are imposed at later stages in the analyses.
The SUSY mass spectra and couplings, as well as the EDM constraints, are computed with CPsuperH [19–21]. The thorium monoxide EDM is calculated using the following formula [22]:
where \(d_e\) is the electron EDM and \(C_S\) the coefficient of the CPodd electron nucleon interaction, which is also present in the thallium EDM. The left hand side of Eq. (5) is the quantity on which experimental constraints are provided currently [22]. Flavour constraints are calculated with SuperIso [23, 24] and CPsuperH. For the calculation of the dark matter relic density we used SuperIso Relic [25] and micrOMEGAs [26–28], and the later is also used for the calculation of scattering cross sections for dark matter direct detection. Finally, we use HiggsBounds [29] to assess the viability of the model points in view of the Higgs constraints.
Studies of MSSM scenarios
We now apply the approach described above to several representative MSSM scenarios.
The CMSSM
We first consider the CMSSM, in which the soft supersymmetrybreaking parameters \(m_0, m_{1/2}\) and \(A\) are each constrained to have universal values at an input grandunification scale. This model is often analysed assuming some fixed value of \(\tan \beta \), the ratio of Higgs v.e.v.s. Generalizing the usual CMSSM setup, here we vary the 6 MSSM CP phases independently in order to allow more flexibility and a closer comparison with the other MSSM scenarios. Our startingpoint here is one of the bestfit CMSSM points found recently in a global analysis [11] of the \(m_0, m_{1/2}, A, \tan \beta \) parameter space for the Higgsino mixing parameter \(\mu > 0\), neglecting all the possible MSSM sources of CP violation.^{Footnote 5} This point has
We use this point as a base for the geometric approach using the EDM limits in Table 1, treating the CP asymmetry in \(b \rightarrow s \gamma \), \(A_\mathrm{CP}\), as the observable to be maximised, in a followup of the study presented in [15]. We have generated more than 600000 sets of phases along the favoured direction, and have found that about half of them pass the EDM constraints, which shows that the method is very efficient.
Figure 1 shows the distributions of the six CPviolating phases \(\Phi _\alpha \) obtained from our sampling. The reader should bear in mind that these distributions have no ‘probability’ or ‘likelihood’ interpretation but only indicate how our iterative geometric procedure samples large values of the phases. We see that the effectiveness of the procedure differs significantly for different phases. We see that large values of \(\Phi _{A_b}\) are relatively well sampled, whereas only intermediate \(\Phi _{A_t}\) and \(\Phi _{A_\tau }\) values can be reached, and we find no parameter sets with \(\Phi _{1,2,3}\) substantially different from zero. This is because for the CMSSM bestfit point (6) it is not possible to cancel the contributions of the phases to all the EDMs simultaneously.
Figure 2 displays the results of this scan of the CPviolating CMSSM for the masses of the three neutral Higgs bosons \(M_{h_1},M_{h_2},M_{h_3}\). The Higgs masses all lie in narrow ranges around their nominal values at the bestfit point in the CPconserving CMSSM, namely \(M_h = 123\) GeV, \(M_A \simeq M_H = 1410\) GeV. In view of the theoretical uncertainties in calculating the Higgs masses for any specific set of CMSSM inputs, measuring Higgs masses would not constrain usefully the CPviolating parameters at the CMSSM bestfit point.
Figure 3 displays the results of this CMSSM scan for the CP asymmetry in \(b\rightarrow s\gamma \), \(A_\mathrm{CP}\), (left) and the spinindependent neutralino–proton scattering cross section \(\sigma _\mathrm{SI}^p v\) (right). We find in this model values of \(A_\mathrm{CP} \ll 10^{3}\), which are considerably below the current and prospective experimental sensitivities. We conclude that the prospects for discovering the \(A_\mathrm{CP}\) signature of CP violation in this particular CMSSM scenario are not good. Also, the spread in the values of \(\sigma _\mathrm{SI}^p v\) is quite small, and much smaller than the theoretical uncertainties related to hadronic matrix elements and the astrophysical uncertainties in the local dark matter density, so this observable is also not a promising one for the CPviolating CMSSM.
We have also studied the possibility of a signature in \(B_s\) meson mass mixing, with discouraging results. We have found that the new physics contribution, \(\Delta M^\mathrm{NP}_{B_s}\) is always very small, namely \({\sim } 0.1\)/ps, which is far below any prospective reduction in the uncertainty in the theoretical calculation of the contribution from the Standard Model [30]. Moreover, after applying the EDM constraints the CPviolating CMSSM contribution is forced to be exceedingly close to the value in the CPconserving CMSSM.
In Fig. 4 we show scatter plots of \(h_1\) signal strengths \(\mu _X\) (normalised relative to the Standard Model values) in the bestfit CMSSM scenario (6) with nonzero CPviolating phases before (green dots) and after (blue dots) the EDM constraints. We see that the CPviolating case expands the ranges of these observables found already in the CPconserving case, in particular after imposing the EDM constraints. However, these expanded ranges all lie well within the current experimental uncertainties. In the left panel we see a strong, almost linear correlation between \(\mu _{\gamma \gamma }\) and \(\mu _{gg}\), which becomes milder in the right panel, between \(\mu _{VV}\) and \(\mu _{{\bar{b}}b}\). The signal strengths are close to but smaller than unity.
We emphasise that the Higgs couplings to fermions provide the most unambiguous probe of its CP properties, even more so when there is CP mixing, as the Higgs may couple to the CPeven and CPodd fermion states in a democratic manner. In cases of the \(h_{i}VV\) couplings, there are two effects of CP mixing in the Higgs sector. One is a reduction in the strength of coefficient of the \(g_{\mu \nu }\) term in the \(h_{i} VV\) vertex and thus in the rates. The second is the simultaneous presence of the CPeven and CPodd tensor structures in the vertex. The coefficient of the CPodd term in the \(h_{i} VV\) vertex involving the \(\epsilon _{\mu \nu \rho \lambda }\) tensor is by necessity small, as it is always loopinduced. Reduction in the production rates is reflected in signal strengths, but these, while currently providing the best available information, are necessarily ambiguous, as there are other mechanisms that may lead to the rate modification. On the other hand, since the fermions couple democratically to the CPeven and CPodd parts of the Higgs couplings, ascertaining the simultaneous presence of \({\bar{f}} f h_{i}\) and \({\bar{f}} \gamma _5 f h_{i}\) terms in the vertex through various angular distributions and kinematic variables is unambiguous.
We have therefore analysed the prospects for CP violation in the couplings of the neutral Higgs bosons to \(\tau ^+ \tau ^\) and \({\bar{t}} t\), by calculating the quantities \(\phi ^{h_i}_\tau \) and \(\phi ^{h_i}_t\) for \(i = 1, 2, 3\), which are expressed in terms of the corresponding pseudoscalar and scalar couplings by
After imposing the EDM constraints, we find that the phases for the \(h_1\) couplings are very small, \({\lesssim } 0.02\) radians. On the other hand, the phases for the \(h_2\) and \(h_3\) couplings may be quite large, as seen in Figs. 5 and 6, respectively. The \(h_2\) couplings have phases close to \({\pm } \pi \), corresponding to a mainly CPodd state, while the \(h_3\) couplings are close to 0 corresponding to a mainly CPeven state. A detailed discussion of the prospects for measuring these phases at the LHC and/or future colliders lies beyond the scope of this work. Clearly, any such future analysis would need to take into account the neardegeneracy of the \(h_2\) and \(h_3\) bosons, as seen in Fig. 2, whose implications would be different for \(pp\), \(e^+ e^\), \(\mu ^+ \mu ^\) and \(\gamma \gamma \) colliders.
We limit ourselves to pointing out a few of these. In the case of the light Higgs, associated production of Higgs with a \({\bar{t}} t\) pair or a single \(t\) or \({\bar{t}}\) can be used for this [31, 32]. However, in our case since it is the heavier Higgses that have the larger CP violation, associated production may not be the best way, but decays of the Higgs into a \(\tau \) pair (or even into a \({\bar{t}} t\) pair if the Higgs is heavy enough) and analysis of the spins of the decay \(\tau /t\) can be used at \(\gamma \gamma \) colliders [33–36] and even at the LHC [37–39]. The method of Ref. [37] is particularly promising when the \(h_{2,3}\) are degenerate.
NUHM2
We now use the iterative geometric approach with four EDM constraints to analyse CP violation in the NUHM2 scenario, in which the gaugino masses, trilinear couplings and soft supersymmetrybreaking contributions to the squark and slepton masses \(M_{\tilde{f}}\) are universal, but those to the two Higgs doublets are allowed to vary independently. The freedom in these two parameters can be traded via the electroweak vacuum conditions for free values of \(\mu \) and a heavy Higgs mass parameter: to avoid complications with the threeway CPviolating mixing in the neutral sector, we take this second free parameter to be \(m_{H^\pm }\). We perform a random scan over the following ranges of the NUHM2 mass parameters:
with \(\tan \beta \in [1,60]\) and varying the six phases \(\Phi _\alpha \) independently as before, using the geometrical approach to seek maximal values of \(A_\mathrm{CP}\).
Figure 7 displays the samples of the six CPviolating phases \(\Phi _\alpha \) obtained in our analysis. We see that our iterative geometrical approach enables us to sample effectively large values of \(\Phi _1, \Phi _{A_t}\) and \(\Phi _{A_b}\), whereas large values of \(\Phi _3\) and \(\Phi _{A_\tau }\) are sampled less effectively, and we do not find large values of \(\Phi _2\). Again, we emphasise that these distributions do not have any ‘probability’ or ‘likelihood’ interpretation. However, the absence of large values of \(\Phi _2\) indicates that there is no way to cancel the contributions of this and the other phases to all the EDMs simultaneously.
Figure 8 provides a visualisation of the cancellations that are required to respect the EDM constraints. In the left panel we see the correlation these constraints impose between \(\Phi _3\) and \(\Phi _{A_t}\), and in the right panel the correlation between \(\Phi _3\) and \(\Phi _{A_b}\). In both cases we see diagonal features corresponding to close correlations, but we also see populations of points with large phases, e.g., in the neighbourhood of \((\Phi _{A_t}, \Phi _3) \sim (90^{\circ }, 90^{\circ })\) in the left panel, and extending to \((\Phi _{A_b}, \Phi _3) \sim (90^{\circ }, 90^{\circ })\) in the right panel. These examples serve as reminders that the EDM constraints do not require all the CPviolating phases to be small simultaneously.
The iterative geometric approach was designed to find the maximal values of the CPviolating asymmetry in \(b \rightarrow s \gamma \) decays, \(A_\mathrm{CP}\), that are compatible with the EDM constraints. We see in the left panel of Fig. 9 that values of \(A_\mathrm{CP} \lesssim 2\) % can be found in the NUHM2 for values of the \(b \rightarrow s \gamma \) branching ratio lying within the experimentally allowed range. The right panel of Fig. 9 displays a histogram of these results for the NUHM2 (grey: full sample, black: points satisfying the EDM constraints). The present experimental constraints on \(A_\mathrm{CP}\) are shown as vertical red dashed lines [40], and the vertical green dashed lines represent the possible future improvement in the experimental sensitivity by a factor of 10, corresponding to the prospective Belle II sensitivity [41]. We see that there are CPviolating NUHM2 models that could be explored with such an improvement: the EDM constraints do not exclude an observable value of \(A_\mathrm{CP}\), and such a measurement would provide additional information on CP violation within the NUHM2.
We have also calculated the possible new physics contribution to \(B_s\) meson mass mixing, \(\Delta M^\mathrm{NP}_{B_s}\), in the NUHM2 scenario, as shown in Fig. 10. The grey histogram is for the full sample of NUHM2 points satisfying the Higgs mass and other constraints, and the black histogram is for points that also satisfy the EDM constraints. The present experimental upper limit on \(\Delta M^\mathrm{NP}_{B_s}\) is shown as the vertical red dashed line [40]. The vertical yellow dashed line in Fig. 10 represents the possible sensitivity if the theoretical uncertainty in the Standard Model contribution to \(B_s\) mixing could be reduced by a factor of 10 thanks to improved lattice calculations. In this case, many of the viable NUHM2 models (indicated by the black histogram) could be explored.
We have not imposed a priori consistency with the cosmological constraints on the relic LSP density \(\Omega _{\chi } h^2\) and the spinindependent dark matter scattering cross section \(\sigma _\mathrm{SI}^p v\). As we see in the left panel of Fig. 11, the values of the relic density for the CPviolating NUHM2 (green points) are very similar to those in the CPconserving version (blue points), and they are generally within the range allowed for a supersymmetric contribution to the dark matter density. The right panel of Fig. 11 shows that the values of \(\sigma _\mathrm{SI}^p v\) are also rather similar, with some differences for low crosssection values well below the experimental upper limit from LUX [42], which is shown as the black solid line.
Figure 12 shows scatter plots of values of \(h_1\) branching ratios in the NUHM2 scenario. The left panel displays \((R_{\gamma \gamma }, R_{gg})\) and the right panel displays \((R_{VV}, R_{{\bar{b}}b})\). The blue dots are CPconserving parameter choices with \(\Phi _\alpha = 0\), and the green dots are from a scan of CPviolating points with \(\Phi _\alpha \ne 0\). We note in the left panel a strong correlation between \(R_{\gamma \gamma }\) and \(R_{gg}\), which may be either much smaller than in the Standard Model or somewhat larger, which is due to the variation of the Higgs width induced by a modification of the Higgs to \({\bar{b}}b\) branching fraction.^{Footnote 6} We see in the right panel that a large reduction in \(R_{VV}\) is also possible, which may be accompanied by values of \(R_{{\bar{b}}b}\) that are either larger or smaller than in the Standard Model. The branch with larger values of \(R_{{\bar{b}}b}\) is also related to the variation of the Higgs width, while the points corresponding to a decrease of both ratios are due to an enhancement of decays to light SUSY particles [43–45].^{Footnote 7}
Scatter plots of \(h_1\) signal strengths \(\mu _X\) in the NUHM2 scenario with the CPviolating phases \(\Phi _\alpha =0\) (blue dots) and \(\ne 0\) (green dots) are shown in Fig. 13.
We see a strong, almost linear correlation between \(\mu _{\gamma \gamma }\) and \(\mu _{gg}\) in the left panel, and in the right panel we see a correlation between \(\mu _{VV}\) and \(\mu _{{\bar{b}}b}\) that is bimodal for small values of \(\mu _{VV}\). No significant difference is observed between the CPconserving and the CPviolating cases.
The prospects for CP violation in the couplings of the heavy neutral Higgs bosons to \(\tau ^+ \tau ^\) and \({\bar{t}} t\) in the NUHM2 scenario (8) are shown in Figs. 14 and 15. As in the CMSSM case discussed previously, we find that after imposing all the constraints the phases for the \(h_1\) couplings are small, namely \({\lesssim } 0.02\) radians. On the other hand, \(h_{2,3}\) decays may provide interesting prospects for probing CP violation also in this NUHM2 scenario.
CPX
We now apply the iterative geometric approach with four EDMs and one CPviolating observable described earlier to a CPX scenario in which
performing random scans over the following parameter ranges:
with the six CPviolating phases of the MCPMFV model being considered independent, as before.
Figure 16 displays the distributions of the six CPviolating phases \(\Phi _\alpha \) sampled in our analysis. We emphasise that these distributions do not have any ‘probability’ or ‘likelihood’ interpretation. Rather, they serve to indicate how well our iterative geometric procedure gives access to large values of the phases that are difficult to sample in a simple random scan, because of the cancellations required to bring the EDMs within the allowed ranges shown in Table 1. We see that the effectiveness of the procedure differs significantly for different phases. For example, in the case of \(\Phi _{A_b}\) our procedure yields almost as many parameter sets with \(\Phi _{A_b} \sim {\pm } 90^{\circ }\) as with \(\Phi _{A_b} \sim 0^{\circ }\) or \(180^{\circ }\), and actually yields more parameter sets with intermediate values of \(\Phi _{A_b}\). In the case of \(\Phi _{A_t}\), the procedure yields a factor \({\sim } 100\) lower sampling density for \(\Phi _{A_b} \sim {\pm } 90^{\circ }\) than for \(\Phi _{A_b} \sim 0^{\circ }, 180^{\circ }\), and larger factors for \(\Phi _2\), \(\Phi _1\) and \(\Phi _{A_\tau }\). Finally, we find no parameter sets for \(\Phi _3 \sim \pm 90^{\circ }\): this is because (for the choices of soft supersymmetrybreaking parameters in (9)) there is no way to cancel the contributions of this and the other phases to all the EDMs simultaneously.
In the CPX scenario we do not find values of \(A_\mathrm{CP}\) that are large enough to be observable in the foreseeable future. However, we do find a possible signature in the new physics contribution to \(B_s\) meson mass mixing, \(\Delta M^\mathrm{NP}_{B_s}\), as shown in Fig. 17. The grey histogram is for CPX points satisfying the Higgs mass and other constraints, and the black histogram is for points that also satisfy the EDM constraints, including the present experimental upper limit on \(\Delta M^\mathrm{NP}_{B_s}\), which is shown as the vertical red dashed line. The magnitude of this upper limit is largely due to the theoretical uncertainty in the Standard Model contribution to \(B_s\) mixing, which is in turn associated with lattice calculations. If this uncertainty could be reduced by a factor of 10, the sensitivity to new physics in \(B_s\) mixing would become that indicated by the vertical yellow dashed line in Fig. 17, which could explore many of the CPX models indicated by the black histogram.
We display in Fig. 18 scatter plots of values of branching ratios in the CPX scenario of the lightest Higgs boson, \(h_1\), normalised relative to the Standard Model values. The left panel shows \((R_{\gamma \gamma }, R_{gg})\) and the right panel shows \((R_{VV}, R_{{\bar{b}}b})\) in the limits where the phases \(\Phi _\alpha = 0\) (blue dots) and scanning over the values of \(\Phi _\alpha \ne 0\) allowed by the EDMs (green dots). There are very small differences between the values of these quantities found in the CPconserving and CPviolating samples. In both cases, correlated substantial reductions in \(R_{\gamma \gamma }\) and \(R_{gg}\) are possible, as is a large reduction in \(R_{VV}\) relative to the Standard Model value. On the other hand, the ‘Cuba’shaped plot in the right panel shows that \(R_{{\bar{b}}b}\) is anticorrelated with \(R_{VV}\), and may be enhanced to \({\sim } 1.3\) times the Standard Model value.
Figure 19 shows scatter plots of \(h_1\) signal strengths \(\mu _X\) in the CPX scenario with the CPviolating phases \(\Phi _\alpha =0\) (blue dots) and \({\ne } 0\) (green dots): again only very small differences are seen. In the left panel we see a strong, almost linear correlation between \(\mu _{\gamma \gamma }\) and \(\mu _{gg}\), and in the right panel we see a nonlinear correlation between \(\mu _{VV}\) and \(\mu _{{\bar{b}}b}\).
As already mentioned, our results for \(A_\mathrm{CP}\) in the CPX scenario are very small, so we do not display them. Taken together with the results shown in Figs. 18 and 19, where no distinctive signatures of nonzero phases \(\Phi _\alpha \ne 0\) are visible, our results suggest that one should look elsewhere for probes of CP violation in the CPX scenario.
We have also analysed the prospects for CP violation in the couplings of the neutral Higgs bosons to \(\tau ^+ \tau ^\) and \({\bar{t}} t\) in the CPX scenario (9, 10), as given by the phases \(\phi ^{h_i}_\tau \) and \(\phi ^{h_i}_t\) for \(i = 1, 2, 3\) defined in (7). As in the CMSSM case discussed previously, we find that after imposing the EDM constraints the phases for the \(h_1\) couplings are small, \(\phi ^{h_i}_\tau \lesssim 0.1\) radians and \(\phi ^{h_i}_t \lesssim 0.02\) radians. On the other hand, the phases for the \(h_2\) and \(h_3\) couplings may again be quite large, as seen in Figs. 20 and 21, respectively. Thus \(h_{2,3}\) decays may also provide interesting prospects for probing CP violation in this CPX scenario.
Phenomenological MSSM (pMSSM)
We now consider the MCPMFV version of the phenomenological MSSM (pMSSM), which has 25 parameters: the 19 real parameters
and the six phases \(\Phi _\alpha \) discussed previously. We perform a scan of the pMSSM parameter space using the iterative geometric approach described in Sect. 2. We first generated about 40 million points, and then kept only points with a neutral Higgs boson with a mass in the range 121–129 GeV (thereby allowing for a conservative theoretical uncertainty in the Higgs mass calculation), and with a neutralino LSP. These requirements reduced the number of points to about 1 million. Imposing the EDM constraints then left about 150000 valid points. In the following plots, in addition to these constraints, we also impose flavour constraints, the cosmological upper bound on the dark matter density, the LUX direct upper limit on spinindependent dark matter scattering (except when the same observable is plotted), and we require squarks and the gluino to have masses above 500 GeV.
Figure 22 shows the samplings of the phases \(\Phi _\alpha \) obtained after imposing these constraints. We see that values of \(\Phi _{A_{t,b}}\) and \(\Phi _1 \sim {\pm } 90^{\circ }\) are quite well sampled, as are values of \(\Phi _{A_\tau } \sim 90^{\circ }\). On the other hand, large values of \(\Phi _3\) are less well sampled, and the range of \(\Phi _2\) is very restricted with only small deviations from the CPconserving cases being allowed.
We see in Fig. 23 the extent to which the EDM constraints impose cancellations \(\Phi _3\) and \(\Phi _{A_t}\) (left panel) and between \(\Phi _3\) and \(\Phi _{A_b}\) (right panel). We see that large values of \((\Phi _{A_t, A_b}, \Phi _3) \sim ( {\pm }90^{\circ }, {\pm }90^{\circ })\) are allowed, and we also see diagonal features corresponding to correlations. As in the NUHM2, it is apparent that the EDM constraints do not require all the CPviolating phases to be small simultaneously.
The left panel of Fig. 24 displays a scatter plot of the values of \(A_\mathrm{CP}\) found in the pMSSM using the iterative geometric approach. We see that values \({\lesssim }3\) % are possible for values of the \(b \rightarrow s \gamma \) branching ratio lying within the experimentally allowed range. The right panel of Fig. 24 shows a histogram of \(A_\mathrm{CP}\) values, imposing only the Higgs mass and EDM cuts. Here we see tails extending to larger values of \(A_\mathrm{CP}\) that lie outside the experimentally allowed range when the EDM constraints are not applied (grey histogram), whereas the black histogram is for points satisfying the EDM constraints. The vertical red dashed lines show the present experimental constraints on \(A_\mathrm{CP}\), and the possible future improvement in the experimental sensitivity by a factor of 10 is indicated by vertical green dashed lines. As in the NUHM2, there are CPviolating pMSSM parameter sets that could be explored with such an improvement: it would provide additional information on CP violation within the pMSSM.
The possible new physics contribution to \(B_s\) meson mass mixing, \(\Delta M^\mathrm{NP}_{B_s}\), in the pMSSM scenario is shown in Fig. 25. As in the previous cases studied, the grey histogram is for the full sample, and the black histogram is for points that also satisfy the EDM constraints. If the theoretical uncertainty in the Standard Model contribution to \(B_s\) mixing could be reduced by a factor of 10 thanks to improved lattice calculations, the sensitivity to \(\Delta M^\mathrm{NP}_{B_s}\) would become that indicated by the vertical yellow dashed line in Fig. 25. In this case, many of the pMSSM models that are currently viable (indicated by the black histogram) could be explored.
In Fig. 26, we show in the left panel the values of the relic LSP density \(\Omega _{\chi } h^2\) that we find in our pMSSM scan, and in right panel we show values of the spinindependent dark matter scattering cross section \(\sigma _\mathrm{SI}^p v\). We see that values of \(\Omega _{\chi } h^2\) considerably above the cosmological upper limit are possible in both the CPconserving (blue dots) and the CPviolating cases (green dots). We also see in the right panel of Fig. 26 that values of \(\sigma _\mathrm{SI}^p v\) above the LUX upper limit are also possible. In both panels, there are no large differences between the CPconserving and CPviolating cases.
Scatter plots of values of \(h_1\) branching ratios in the pMSSM scenario are in Fig. 27, the left panel displaying \((R_{\gamma \gamma }, R_{gg})\) and the right panel displaying \((R_{VV}, R_{{\bar{b}}b})\). As previously, the blue dots are CPconserving parameter choices with \(\Phi _\alpha = 0\), and the green dots are from a scan of CPviolating points with \(\Phi _\alpha \ne 0\).
As in the NUHM2 scenario, we note in the left panel a strong correlation between \(R_{\gamma \gamma }\) and \(R_{gg}\), which may be either much smaller than in the Standard Model or somewhat larger, and we also see in the right panel that a large reduction in \(R_{VV}\) is possible. Also as in the NUHM2 scenario, the reduction in \(R_{VV}\) may be accompanied by values of \(R_{{\bar{b}}b}\) that are either larger or smaller than in the Standard Model, the latter possibility arising when the Higgs boson can decay into light sparticles.
Figure 28 displays scatter plots of \(h_1\) signal strengths \(\mu _X\) in the pMSSM scenario in the CPconserving case with phases \(\Phi _\alpha =0\) (blue dots) and in the CPviolating case where the \(\Phi _\alpha \ne 0\) (green dots). As in the NUHM2 case, we see a strong correlation between \(\mu _{\gamma \gamma }\) and \(\mu _{gg}\) in the left panel, and in the right panel we see a correlation between \(\mu _{VV}\) and \(\mu _{{\bar{b}}b}\) that becomes bimodal for small values of \(\mu _{VV}\).
We have also studied whether the Higgs boson discovered at the LHC might be one of the heavier Higgs bosons in the pMSSM, with or without CP violation. As seen in the left panel of Fig. 29, if the known Higgs boson is identified with the \(h_2\), it is not possible to satisfy the Higgs signal strength constraints. This is possible if the discovered Higgs boson is identified with the \(h_3\), as seen (green dots) in the right panel of Fig. 29, in which case the \(h_1\) mass is about 60–80 GeV. Figure 30 displays these points in both the CPconserving case (blue dots) and the CPviolating case (green dots), which are quite similar. On the other hand, none of these points survive the charged Higgs and \(A/H \rightarrow \tau \tau \) constraints, nor the flavour constraints. We therefore conclude that the pMSSM does not provide a way to conceal a neutral Higgs boson that is lighter than the one discovered, even if CP is violated.
Assuming that the Higgs boson discovered at the LHC is indeed the lightest MSSM Higgs boson \(h_1\), we now assess the prospects for CP violation in the couplings of the heavy neutral Higgs bosons to \(\tau ^+ \tau ^\) and \({\bar{t}} t\) in the pMSSM scenario (11) which are shown in Figs. 31 and 32. We see that, as in the CMSSM, CPX and NUHM2 cases discussed previously, \(h_{2,3}\) decays may provide interesting prospects for probing CP violation also in this pMSSM scenario. On the other hand, we again find that after imposing all the constraints the phases for the \(h_1\) couplings are small, namely \(\phi ^{h_1}_\tau \lesssim 0.03\) radians and \(\phi ^{h_1}_t \lesssim 0.02\) radians, respectively.
Conclusions
The geometrical approach to implementing EDM constraints and maximizing other CPviolating observables proposed in [2] provides a suitable way to explore the possibilities for CP violation in variants of the MSSM, which we have applied in this paper to explore the CMSSM, the CPX scenario, the NUHM2 and the pMSSM. We have adopted an iterative extension of the geometric approach, which is suitable for exploring larger values of the CPviolating phases. Our explorations have been within the maximally CPviolating, minimal flavourviolating (MCPMFV) framework with six CPviolating phases, of which two combinations are unconstrained a priori by the four EDM constraints. The following are our principal results:

In the CMSSM we have explored CPviolating generalisations of the lowmass bestfit point (6) that was identified in [11, 12], where we found relatively little scope for large deviations from the CPconserving case, e.g., in the masses of the Higgs bosons and the spinindependent dark matter scattering cross section. Moreover, we found that only very small values of \(A_\mathrm{CP} \lesssim 0.001\) would be possible in this case, and the new physics contribution to \(B_s\) meson mixing, \(\Delta M^\mathrm{NP}_{B_s}\), would not be observable.

We have then explored the CPX scenario (9), where we also found no scope for measurable values of \(A_\mathrm{CP}\). On the other hand, we found in this model that \(\Delta M^\mathrm{NP}_{B_s}\) could be large enough to provide a possible signature if the current lattice theoretical uncertainty in the Standard Model contribution to \(B_s\) mixing could be reduced by a factor of 10, as seen in Fig. 17.

The situation in the NUHM2 scenario (8) is rather more favourable for observable signals of CP violation. In this case, \(A_\mathrm{CP}\) could be as large as \({\sim } 2\) % and hence lie well within the reach of experiment, as seen in Fig. 9, and \(\Delta M^\mathrm{NP}_{B_s}\) might also be large enough to provide a possible experimental signature, as seen in Fig. 10.

A similar situation was found in the pMSSM scenario (11), in which case \(A_\mathrm{CP}\) could be as large as \({\sim } 3\) %, as seen in Fig. 24, again within the reach of experiment. We also find in this scenario that \(\Delta M^\mathrm{NP}_{B_s}\) could be large enough to be observable with a prospective reduction in the theoretical uncertainty in the Standard Model calculation of \(B_s\) mixing, as seen in Fig. 25.

In all the scenarios studied, the CPviolating phases in the \(h_1 \tau ^+ \tau ^\) and \(h_1 {\bar{t}} t\) couplings are small. On the other hand, the phases in the \(h_{2,3} \tau ^+ \tau ^\) and \(h_{2,3} {\bar{t}} t\) couplings can be quite large, and may present interesting prospects for future \(pp\), \(e^+ e^\) and \(\mu ^+ \mu ^\) experiments, though their detailed study lies beyond the scope of this work.
Our analysis serves as a reminder that the EDM constraints do not force all the six nonKM CPviolating phases in MCPMFV to be small, and that in some variants of the MSSM there could be observable signatures of CP violation beyond the Standard Model, e.g., \(A_\mathrm{CP}\) in \(b \rightarrow s \gamma \) decay. We look forward to a generation of \(A_\mathrm{CP}\) measurements, and also to improved theoretical calculations of the Standard Model contribution to \(B_s\) meson mixing, which might enable a new physics contribution \(\Delta M^\mathrm{NP}_{B_s}\) to be isolated. If enough soft supersymmetrybreaking parameters could be measured, and both \(A_\mathrm{CP}\) and \(\Delta M^\mathrm{NP}_{B_s}\) could be shown to have measurable deviations from the Standard Model, one might finally be able to fix all the six nonKM CPviolating phases in MCPMFV.
Notes
We assume that the strong CP phase is negligible, and also neglect the phases in the trilinear couplings of the sfermions in the first and second generations, which are much less important for phenomenology.
We note in passing that there are also wellmotivated supersymmetric models in which the phases are naturally small, so that the EDM bounds are not very constraining, see [5] for example.
Partial results in the case of CMSSM were presented in [15].
We use Latin indices \(i,j,\ldots \) for the EDMs, and Greek indices \(\alpha ,\beta ,\ldots \) for the CPviolating phases.
This analysis also found a highmass bestfit point with a slightly lower value of the global \(\chi ^2\) function. However, the ATLAS jets + missing transverse energy constraint has subsequently been revised, and the lowmass point is now the global minimum of the \(\chi ^2\) function [12].
Small values of \(R_{\gamma \gamma }\) and \(R_{gg}\) are disfavoured by LHC Higgs measurements.
Small values of \(R_{VV}\) are disfavoured by LHC Higgs measurements, whereas a relatively large range of \(R_{{\bar{b}}b}\) is still allowed.
References
J.R. Ellis, J.S. Lee, A. Pilaftsis, BMeson observables in the maximally CPviolating MSSM with minimal flavour violation. Phys. Rev. D 76, 115011 (2007). arXiv:0708.2079 [hepph]
J. Ellis, J.S. Lee, A. Pilaftsis, A geometric approach to CP violation: applications to the MCPMFV SUSY model. JHEP 1010, 049 (2010). arXiv:1006.3087 [hepph]
T. Gajdosik, R.M. Godbole, S. Kraml, Fermion polarization in sfermion decays as a probe of CP phases in the MSSM. JHEP 0409, 051 (2004). arXiv:hepph/0405167
K.A. Olive, M. Pospelov, A. Ritz, Y. Santoso, CPodd phase correlations and electric dipole moments. Phys. Rev. D 72, 075001 (2005). arXiv:hepph/0506106
S.A.R. Ellis, G.L. Kane, Theoretical prediction and impact of fundamental electric dipole moments. arXiv:1405.7719 [hepph]
B.C. Regan, E.D. Commins, C.J. Schmidt, D. DeMille, New limit on the electron electric dipole moment. Phys. Rev. Lett. 88, 071805 (2002)
W.C. Griffith, M.D. Swallows, T.H. Loftus, M.V. Romalis, B.R. Heckel, E.N. Fortson, Improved limit on the permanent electric dipole moment of Hg199. Phys. Rev. Lett. 102, 101601 (2009)
C.A. Baker, D.D. Doyle, P. Geltenbort, K. Green, M.G.D. van der Grinten, P.G. Harris, P. Iaydjiev, S.N. Ivanov et al., An Improved experimental limit on the electric dipole moment of the neutron. Phys. Rev. Lett. 97, 131801 (2006). arXiv:hepex/0602020
J. Baron et al. (ACME Collaboration), Order of magnitude smaller limit on the electric dipole moment of the electron. Science 343(6168), 269 (2014). arXiv:1310.7534 [physics.atomph]
G.W. Bennett et al. (Muon (g2) Collaboration), An improved limit on the muon electric dipole moment. Phys. Rev. D 80, 052008 (2009). arXiv:0811.1207 [hepex]
O. Buchmueller, R. Cavanaugh, A. De Roeck, M.J. Dolan, J.R. Ellis, H. Flacher, S. Heinemeyer, G. Isidori et al., The CMSSM and NUHM1 after LHC run 1. Eur. Phys. J. C 74, 2922 (2014). arXiv:1312.5250 [hepph]
O. Buchmueller, R. Cavanaugh, M. Citron, A. De Roeck, M.J. Dolan, J.R. Ellis, H. Flaecher, S. Heinemeyer et al., The NUHM2 after LHC run 1. arXiv:1408.4060 [hepph]
M.S. Carena, J.R. Ellis, A. Pilaftsis, C.E.M. Wagner, CP violating MSSM Higgs bosons in the light of LEP2. Phys. Lett. B 495, 155 (2000). arXiv:hepph/0009212
A. Djouadi et al. (MSSM Working Group Collaboration), The minimal supersymmetric standard model: group summary report. arXiv:hepph/9901246
G. Brooijmans et al., Les Houches 2013: physics at TeV colliders: new physics working group report. arXiv:1405.1617 [hepph]
A. Arbey, M. Battaglia, F. Mahmoudi, Constraints on the MSSM from the Higgs sector: a pMSSM study of Higgs searches, \(B^0_s \rightarrow \mu ^+ \mu ^\) and dark matter direct detection. Eur. Phys. J. C 72, 1906 (2012). arXiv:1112.3032 [hepph]
A. Arbey, M. Battaglia, F. Mahmoudi, Complementarity of WIMP sensitivity with direct SUSY, monojet and dark matter searches in the MSSM. Phys. Rev. D 89, 077701 (2014). arXiv:1311.7641 [hepph]
M. CahillRowley, J.L. Hewett, A. Ismail, T.G. Rizzo, Lessons and prospects from the pMSSM after LHC run I: neutralino LSP. arXiv:1407.4130 [hepph]
J.S. Lee, A. Pilaftsis, M.S. Carena, S.Y. Choi, M. Drees, J.R. Ellis, C.E.M. Wagner, CPsuperH: a computational tool for Higgs phenomenology in the minimal supersymmetric standard model with explicit CP violation. Comput. Phys. Commun. 156, 283 (2004). arXiv:hepph/0307377
J.S. Lee, M. Carena, J. Ellis, A. Pilaftsis, C.E.M. Wagner, CPsuperH2.0: an improved computational tool for Higgs phenomenology in the MSSM with explicit CP violation. Comput. Phys. Commun. 180, 312 (2009). arXiv:0712.2360 [hepph]
J.S. Lee, M. Carena, J. Ellis, A. Pilaftsis, C.E.M. Wagner, CPsuperH2.3: an updated tool for phenomenology in the MSSM withexplicit CP violation. Comput. Phys. Commun. 184, 1220 (2013). arXiv:1208.2212 [hepph]
K. Cheung, J.S. Lee, E. Senaha, P.Y. Tseng, Confronting Higgcision with electric dipole moments. JHEP 1406, 149 (2014). arXiv:1403.4775 [hepph]
F. Mahmoudi, SuperIso: a program for calculating the isospin asymmetry of \(B \rightarrow K^* \gamma \) in the MSSM. Comput. Phys. Commun. 178, 745 (2008). arXiv:0710.2067 [hepph]
F. Mahmoudi, SuperIso v2.3: a program for calculating flavor physics observables in supersymmetry. Comput. Phys. Commun. 180, 1579 (2009). arXiv:0808.3144 [hepph]
A. Arbey, F. Mahmoudi, SuperIso relic: a program for calculating relic density and flavor physics observables in supersymmetry. Comput. Phys. Commun. 181, 1277 (2010). arXiv:0906.0369 [hepph]
G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, MicrOMEGAs 2.0: a program to calculate the relic density of dark matter in a generic model. Comput. Phys. Commun. 176, 367 (2007). arXiv:hepph/0607059
G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, Dark matter direct detection rate in a generic model with micrOMEGAs 2.2. Comput. Phys. Commun. 180, 747 (2009). arXiv:0803.2360 [hepph]
G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, micrOMEGAs 3: a program for calculating dark matter observables. Comput. Phys. Commun. 185, 960 (2014). arXiv:1305.0237 [hepph]
P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, K.E. Williams, \({\sf HiggsBounds4}\): improved tests of extended Higgs sectors against exclusion bounds from LEP, the Tevatron and the LHC. Eur. Phys. J. C 74, 2693 (2014). arXiv:1311.0055 [hepph]
A. Lenz, U. Nierste, Numerical updates of lifetimes and mixing parameters of B mesons. arXiv:1102.4274 [hepph]
See for example, J. Ellis, D.S. Hwang, K. Sakurai, M. Takeuchi, JHEP 1404, 004 (2014). arXiv:1312.5736 [hepph]
F. Boudjema, R.M. Godbole, D. Guadagnoli, K. Mohan, in Ref. [15]
See for example, J.R. Ellis, J.S. Lee, A. Pilaftsis, Nucl. Phys. B 718, 247 (2005). arXiv:hepph/0411379
S.Y. Choi, J. Kalinowski, Y. Liao, P.M. Zerwas, Eur. Phys. J. C 40, 555 (2005). arXiv:hepph/0407347
J.R. Ellis, J.S. Lee, A. Pilaftsis, Phys. Rev. D 72, 095006 (2005). arXiv:hepph/0507046
R.M. Godbole, S. Kraml, S.D. Rindani, R.K. Singh, Phys. Rev. D 74, 095006 (2006). arXiv:hepph/0609113 [Erratumibid. D 74, 119901 (2006)]
J.R. Ellis, J.S. Lee, A. Pilaftsis, CERN LHC signatures of resonant CP violation in a minimal supersymmetric Higgs sector. Phys. Rev. D 70, 075010 (2004). arXiv:hepph/0404167
S. Berge, W. Bernreuther, B. Niepelt, H. Spiesberger, How to pin down the CP quantum numbers of a Higgs boson in its tau decays at the LHC. Phys. Rev. D 84, 116003 (2011). arXiv:1108.0670 [hepph]
A. Chakraborty, B. Das, J.L. DiazCruz, D.K. Ghosh, S. Moretti, P. Poulose, The 125 GeV Higgs signal at the LHC in the CP violating MSSM. Phys. Rev. D 90, 055005 (2014). arXiv:1301.2745 [hepph]
K.A. Olive et al. (Particle Data Group Collaboration), Review of particle physics. Chin. Phys. C 38, 090001 (2014)
T. Aushev, W. Bartel, A. Bondar, J. Brodzicka, T.E. Browder, P. Chang, Y. Chao, K.F. Chen et al., Physics at super B factory. arXiv:1002.5012 [hepex]
D.S. Akerib et al. (LUX Collaboration), First results from the LUX dark matter experiment at the Sanford Underground ResearchFacility. Phys. Rev. Lett. 112, 091303 (2014). arXiv:1310.8214 [astroph.CO]
D. Albornoz Vasquez, G. Belanger, R.M. Godbole, A. Pukhov, The Higgs boson in the MSSM in light of the LHC. Phys. Rev. D 85, 115013 (2012). arXiv:1112.2200 [hepph]
A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi, The Higgs sector of the phenomenological MSSM in the light of the Higgs boson discovery. JHEP 1209, 107 (2012). arXiv:1207.1348 [hepph]
A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi, An update on the constraints on the phenomenological MSSM from the new LHC Higgs results. Phys. Lett. B 720, 153 (2013). arXiv:1211.4004 [hepph]
Acknowledgments
The work of J.E. was supported in part by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research Council via the Advanced Investigator Grant 267352 and from the UK STFC via the research Grant ST/J002798/1. The work of A.A. was supported in part by the Fédération de Recherche A.M. Ampère de Lyon. R.M.G. wishes to acknowledge support from the Department of Science and Technology, India under Grant No. SR/S2/JCB64/2007 under the J.C. Bose Fellowship scheme and hospitality in the CERN theory division.
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Arbey, A., Ellis, J., Godbole, R.M. et al. Exploring CP violation in the MSSM. Eur. Phys. J. C 75, 85 (2015). https://doi.org/10.1140/epjc/s100520153294z
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DOI: https://doi.org/10.1140/epjc/s100520153294z
Keywords
 Dark Matter
 Higgs Boson
 Neutral Higgs Boson
 Standard Model Contribution
 Heavy Neutral Higgs Boson