CP Violation in Heavy MSSM Higgs Scenarios

We introduce and explore new heavy Higgs scenarios in the Minimal Supersymmetric Standard Model (MSSM) with explicit CP violation, which have important phenomenological implications that may be testable at the LHC. For soft supersymmetry-breaking scales M_S above a few TeV and a charged Higgs boson mass M_H+ above a few hundred GeV, new physics effects including those from explicit CP violation decouple from the light Higgs boson sector. However, such effects can significantly alter the phenomenology of the heavy Higgs bosons while still being consistent with constraints from low-energy observables, for instance electric dipole moments. To consider scenarios with a charged Higgs boson much heavier than the Standard Model (SM) particles but much lighter than the supersymmetric particles, we revisit previous calculations of the MSSM Higgs sector. We compute the Higgs boson masses in the presence of CP violating phases, implementing improved matching and renormalization group (RG) effects, as well as two-loop RG effects from the effective two-Higgs Doublet Model (2HDM) scale M_H+ to the scale M_S. We illustrate the possibility of non-decoupling CP-violating effects in the heavy Higgs sector using new benchmark scenarios named CPX4LHC.


Introduction
Supersymmetry (SUSY) remains one of the best-motivated extensions of the Standard Model (SM), despite the current lack of evidence for supersymmetric partner particles at the LHC. In particular, the discovery of a light Higgs boson in Run I of the LHC, is in agreement with the predictions from SUSY. Supersymmetric theories provide a viable mechanism for stabilizing the electroweak vacuum [1] and require a restricted range for the mass of the lightest Higgs boson [2][3][4] that contains the measured value [5]. Moreover, minimal low-energy SUSY models with masses of the additional Higgs bosons and supersymmetric particles larger than the weak scale lead to values of the lightest Higgs couplings that are close to the SM ones [6], as suggested by current LHC experiments [7].
On the other hand, the non-discovery of SUSY in Run I of the LHC has disproved benchmark scenarios proposed previously [8], and motivates the consideration of new benchmarks that can be tested in future runs of the LHC. Specifically, it is plausible to consider the case that the common soft SUSY-breaking scale M S is > ∼ 2 TeV, whereas the mass scale M H of the heavy MSSM Higgs bosons, as determined by the charged Higgs boson mass M H + , could be somewhat lower, in the few to several hundred GeV range. In relation to this, we recall that future runs of the LHC at 13/14 TeV are expected to be sensitive to squarks and gluinos weighing < ∼ 3 TeV, and heavy MSSM Higgs bosons weighing < ∼ 2 TeV depending on the value of tan β. Accordingly, in this paper we introduce and explore MSSM Higgs boson benchmark scenarios with 100 GeV ≪ M H ≪ M S > ∼ 2 TeV. Our principal interest in new heavy Higgs boson benchmark scenarios is the possible manifestation of observable CP violation in the Higgs sector of the MSSM. It is well known [9][10][11][12][13][14][15][16][17] that such a possibility arises in the MSSM Higgs potential beyond the treelevel approximation, predominantly from CP phases in the soft SUSY-breaking trilinear couplings of stops and sbottoms, but also from CP phases in the gaugino masses. However, experimental upper limits on electric dipole moments (EDMs) severely constrain the size of such CP-violating parameters as predicted in the MSSM at one-, two-and higher loops [18]. In particular, in the absence of cancellations between these different contributions [19,20] as occur along specific directions in the space of CP-odd phases [21,22], the EDM constraints effectively preclude the observation of CP-violating effects in the couplings of the Higgs boson discovered at the LHC [23]. However, the observation of CP violation effects elsewhere, notably in the heavy MSSM Higgs sector [10,11,14] or B-meson decays [24,25] is not excluded. These CP-violating effects have often been studied in the framework of the CPX scenarios proposed previously [8], but in light of the LHC Run-I limits on supersymmetric particle masses and the observed Higgs boson properties, the CPX benchmarks should be revisited.
With the above motivations in mind, in this paper we present new precision calculations of the MSSM Higgs spectrum in the presence of CP violation, which are suitable for scenarios in which the SUSY scale M S is (far) beyond the TeV region. To this end, we solve the twoloop RGEs of the two-Higgs-doublet model (2HDM) in the range M S > Q > M H , as well as the two-loop SM RGEs in the range M H > Q > m pole here are being implemented in a new version of the public code CPsuperH [26][27][28], namely CPsuperH3.0 * . The full description with all the detailed information about CPsuperH3.0 will be presented in a future publication. Section 2 of this paper reviews the conventions and notations of CPsuperH that we use for our analysis, as well as some basic formulae for the Higgs boson self-energies. Section 3 specifies the matching conditions and the RG running effects that we incorporate. In section 4 we present some numerical results for the Higgs spectra. In section 5 we introduce our new CP-violating benchmark scenarios (CPX4LHC) for the MSSM heavy Higgs sector, and present the results for the CPX4LHC benchmarks. Our conclusions are summarized in Section 6. The main text of the paper is accompanied by Appendices containing detailed formulae: Appendix A contains the relevant SM RGEs, Appendix B contains the one-loop 2HDM RGEs, Appendix C contains the two-loop 2HDM RGEs, and Appendix D summarizes the threshold corrections to quartic couplings at the scale M S .

The CP-Violating MSSM Higgs Sector
In this section we review the computation of the Higgs boson self-energies and pole masses and record the basic expressions used in CPsuperH3.0, that underlie our present analysis. We follow the conventions and notations of CPsuperH [26][27][28], unless stated otherwise explicitly.

The Two-Higgs-Doublet Model (2HDM)
The tree-level 2HDM Higgs potential can be written as [11]: . The relations between these and the conventional MSSM parameters are The doublet Higgs fields may be decomposed as follows: where the charged and neutral Goldstone bosons, G ± and G 0 , are determined through the relations: with s β ≡ sin β, c β ≡ cos β and tan β = v 2 /v 1 .
To make contact with the notations used in [31], we make the following identifications: Moreover, the kinematic parameters as defined in [31] are related to ours as follows: The one-loop 2HDM RGEs are given in Appendix B † , and the two-loop 2HDM RGEs are given in Refs. [32], [33] and Appendix C.

Charged Higgs Bosons
In the {φ ± 1 , φ ± 2 } basis, the RG-improved charged Higgs-boson self-energy matrix can be found in Eq. (2.6) of Ref. [16]: The first term, M 2 ± , is the two-loop Born-improved squared-mass matrix, expressed in terms of relevant parameters such as the real part of the soft bilinear Higgs mixing, ℜem 2 12 , and the quartic coupling λ 4 . The bar on these parameters indicates the sum of the tree-level and of the one-and two-loop leading logarithmic contributions. When solving the 2HDM RGEs,λ 4 is to be estimated at the scale M H where the heavy Higgs bosons decouple, and ℜem 2 12 is fixed when the charged-Higgs-boson pole mass is given as an input, as shown below.
The second term in (6) describes the threshold effects of the sfermions (top and bottom squarks) and is the product of two quantities: (i) the anomalous dimension factors ξ i (9) † We note that the RGE running parameter used in Ref. [31] is related to ours by t → 2t.
In the above, the SUSY-breaking scale M S is used to decouple the heavy sfermions. Moreover, the superscript "(1)" in λ (1) 4 and ℜem 2(1) 12 indicates that these quantities contain the one-loop leading logarithmic contributions and they can be obtained from Eqs. (3.6) and (3.7) of [14] by choosing Q = M S .
We note that the vacuum expectation values (VEVs) v 1,2 of the Higgs doublets Φ 1,2 , and hence tan β, evolve with the wave-function renormalization factors ξ 1,2 of the corresponding neutral Higgs bosons: where tan β(M H ) is the input value of tan β, i.e. at the scale Q = M H . Consequently, the SM VEV v is related to the Higgs VEVs v 1,2 through: The SM VEV v is fixed at the RG scale Q = m t , by virtue of the relation: Here γ(t) is the anomalous dimension of the SM Higgs doublet, which is given in (A.12) in the one-loop approximation.
Finally, the last terms on the RHSs of (9) and (6), namely Π ± f and Π ± f , can be expressed as follows, with all quantities in the RHS of (13) computed in the MS scheme. Explicit one-loop calculations yield with T a = T a 2 /c β = −T a 1 /s β and where Π ± (a) , Π ± ,(b) , Π ± (c) , T In the {G ± , H ± } basis, the inverse-propagator matrix of the charged Higgs bosons is given by where we have defined In (15), the {22} matrix element of the second term is given by with This yields the pole mass condition which may be used to eliminate ℜem 2 12 in favor of the charged-Higgs boson pole mass M 2 H ± .
The quantities ∆Π S and ∆Π P may be written as ∆Π S Here, Π S f and Π S f are given by which are specified in Eqs. (B.5), (B.6), and (B.14) of [16]. In addition, Π P f and Π P f are given by which can be obtained from Eqs. (B.5), (B.6) by replacing φ i → a i and from Eq. (B.14) of [16]. Moreover, the CP-violating self-energies Π SP f and Π SP f may be expressed as The non-zero self-energy Π SP ,(a) is given by Eq. (B.11) of [16].
Finally, the inverse propagator matrix of the neutral Higgs bosons in the {φ 1 , φ 2 , a, G 0 } basis is given by and the physical masses can be obtained from the pole-mass conditions. We should reiterate here that the parameter tan β is defined at s = 0. In this kinematic limit, the Goldstone boson G 0 decouples from the 4 × 4 propagator matrix, independently of the presence of explicit CP violation in the theory [9], as a consequence of the Goldstone theorem.

Matching Conditions and RG Running Effects
Here we detail the MS renormalization group approach that we follow for the computation of the masses and mixings of the neutral and charged Higgs bosons in the CP-violation case.
In particular, we state explicitly our matching conditions at the relevant threshold scales. Given these matching conditions, we compute the RG running effects to the relevant gauge, Yukawa and quartic couplings between the different threshold scales.
To start with, we define the SUSY-breaking scale M S by which acts as the SUSY threshold scale. For the purposes of this study, we ignore possible hierarchies between the third-generation sfermions, by assuming they are small as compared to the other two hierarchical scales: (i) the heavy Higgs-sector scale M H ≡ M H + ; (ii) the top-quark mass m t .
The matching conditions for the quartic and Yukawa couplings at the threshold M S are as follows:λ where , and ∆h f and δh f are the supersymmetric threshold corrections to the third generation Yukawa couplings [14,34]. The difference between δh f and ∆h f is that δh f is a radiative correction to the supersymmetric h MSSM f coupling of up-quarks, down-quarks and leptons. The coupling ∆h f , instead, is a loop-induced coupling of the fermions to the Higgs doublet to which they do not couple in the supersymmetric limit. Therefore, below the scale M S the theory becomes a general 2HDM, with up-quarks coupled to Φ 2 and down-quarks and leptons coupled to Φ 1 , with couplings given by h MSSM f (1 + δ f ), respectively, but with additional loop-induced couplings ∆h f to the other Higgs doublet.
The couplings h 2HDM f are the combinations of these Yukawa couplings related to the running fermion masses in the same way as in a Type-II 2HDM. Notice that in the present approach, we treat the loop-induced couplings ∆h f as small departures from a Type-II 2HDM. Hence, we are working in a Type-II approximation to a general 2HDM.
The RGEs for the 2HDM used for M S > Q > M H are described in Appendices B and C. At the heavy Higgs threshold M H ≡ M H ± , the following matching conditions are employed: where M EP H 1 denotes the effective potential mass of the lightest neutral Higgs boson calculated in the limit of zero external momentum s = 0. In the above, we have ignored the small effects due to scheme conversion from dimensional regularization to dimensional reduction [35]. In practice, while evaluating the evolution of the gauge, Yukawa and quartic couplings, at M H < Q < M S , we have assumed an effective Type-II 2HDM, in which the Yukawa couplings are given by h 2HDM f with the matching condition, Eq. (31) given at the scale M H . As already mentioned above, this amounts to an approximate treatment of the loop-induced ∆h f effects on the computation of the Higgs boson masses and mixing angles.
At scales below the heavy Higgs scale M H , the only physical degrees of freedom are the SM ones. The RGEs for the SM used for Q < M H are described in Appendix A. We define the SM Higgs potential as Note that the quartic coupling of Φ is defined with a factor (-2) difference compared to the quartic couplings of Φ 1,2 in Eq. (1). In order to compare with the experimental results, it is important to define the SM boundary conditions for the gauge, Yukawa and quartic couplings. In our work we use [36] y t = 0.93697 + 0.00550 The pole mass-squared of the lightest Higgs boson is then given by [35]: We take the renormalization group scale Q RG = m pole t , and the function B 0 used in (33) is defined in [16].

Numerical Results for the MSSM Higgs Sector
We first illustrate the effects of the RG running in the range of scales Q > M H using a specific scenario with universal SUSY parameters fixed to be 1 TeV: where ρQ = MQ 1,2 /MQ 3 , ρŨ = MŨ 1,2 /MŨ 3 , etc, and we have assumed no hierarchy between the three generations of sfermion masses. In the same context, it is important to comment that, for hierarchical scenarios with ρ > 1, where ρ ≡ max (ρQ ,Ũ,D,L,Ẽ ), the proper matching conditions should be imposed at the highest soft SUSY-breaking scale M ′ S = ρM S , rather than M S . As an illustrative example, we consider another scenario with various different values for the soft SUSY breaking parameters: In this scenario, the matching conditions are imposed at the highest soft SUSY-breaking scale M ′ S = 10 5 GeV instead of M S = 10 4 GeV, as illustrated in Figure 2. We observe that the running between M S and M ′ S changes the size of the quartic couplings by an amount of ∼ 2% for ρ = 10, which results in a less than 1 GeV increase in the mass prediction for the H 1 boson. Even though such changes may not appear too significant for scenarios with The other input parameters are given in Eq. (34). The thin red lines show the running of −(g 2 + g ′2 )/8, −(g 2 − g ′2 )/4, and g 2 /2 in the panels forλ 1,2 ,λ 3 , andλ 4 , respectively.
mass spectrum hierarchies of ρ < ∼ 10, they are nevertheless accurately described within our multi-threshold RG approach that we follow here for the computation of the Higgs-boson masses and mixing angles. Figure 2: The same as in Fig. 1, but for an hierarchical scenario with ρ = 10 and input parameters given in Eq. (35).
to the relation between the pole and running masses [36], [37] and, as stressed before, it is used as the standard value for CPsuperH3.0.
From Fig. 3 we see that in the MHMAX scenario, the mass of the lightest Higgs boson calculated using CPsuperH3.0 is ∼ 1 GeV smaller than that obtained using CPsuperH2.3  given by Eqs. (3.4) and (3.10) of Ref. [14] includes the threshold corrections, we have not included these corrections in CPsuperH3.0. These Yukawa thresholds are still included in the relevant computation of the threshold corrections to the quartic couplings, which lead to the asymmetry between positive and negative values of X t = A t − µ * / tan β in the CP-conserving limit of the theory. This small difference between the CPsuperH3.0 and CPsuperH2.3 is rapidly compensated by RG effects and, as expected, the mass difference changes sign when M S ∼ 2 TeV, and

CP-Violating Heavy Higgs Scenarios
We now consider various CPX4LHC benchmark scenarios for showcasing the effect of CP violation in the MSSM heavy Higgs sector and their possible signatures. We assume a common CP-violating phase Φ A = arg(A t ) = arg(A b ) = arg(A τ ), set with M 2 = 2 M 1 = 200 GeV and M 3 = 2 TeV, and vary tan β, M H ± , and M S . We do not include gaugino phases in this analysis, as they enter the Higgs sector only through the threshold corrections to the MSSM top-, bottom-, and tau-Yukawa couplings. Instead, we include the CP-conserving leading-log enhanced contributions due to gauginos to the self-energies Π ±,S,P . In our CPX4LHC scenarios, since we fix M 2 = 2M 1 = 200 GeV and M 3 = 2 TeV, and do not increase them as M S increases, the gaugino phase effects are relatively insignificant.
We first present in Figure 8 [35].
eigenstates H i (with i = 1, 2, 3): which characterize the mixtures between the CP-odd state a and the CP-even states φ 1,2 . For instance, such CP-violating expressions occur when studying CP violation in Higgs-boson decays to fermions [23,40]. For a recent analysis of CP violation in the decays H 1,2,3 → τ + τ − , see [41].  Similar results for the second mass eigenstate H 2 are shown in Figs. 12 and 13, and for the third mass eigenstate H 3 in Figs. 14 and 15. We see here that the mixing quantities (37) can be much larger for the heavy mass eigenstates H 2,3 than for the lightest mass eigenstate H 1 , attaining unity for many of the values of M S and tan β studied. This suggests, a priori,  Figure 17. In the case of H 1 , we see that the mixing quantities → 0 at large M H ± , as expected, whereas in general the corresponding coefficients for the heavy neutral Higgs mass eigenstates H 2,3 do not vanish, and retain large values even for very large M H ± . Thus, large CP-violating effects in the couplings of these states are a robust signature of the CP-violating scenarios discussed in this paper.

Conclusions
We present new MSSM scenarios with explicit CP violation that contain heavy Higgs bosons in the few to several hundred GeV range and are consistent with constraints from Run I of the LHC. The scenarios suggested here are similar in spirit to the CPX scenarios previously proposed , and have phenomenological implications that can be tested during Run II of the LHC. In light of this, we call them CPX4LHC benchmark scenarios.
In this work we explicitly demonstrate that, although CP violation and other newphysics effects decouple from the lightest Higgs boson sector for sufficiently large charged Higgs boson masses and soft SUSY-breaking scales M S , they can still be significant in the MSSM heavy Higgs boson sector. Large masses of the supersymmetric particles also help to maintain agreement with limits on EDMs and other low-energy observables. We consider scenarios in which the charged Higgs bosons H ± and the two heavier neutral Higgs bosons H 2,3 could be much lighter than all third generation supersymmetric scalar fermions, which are assumed to have masses M S > ∼ 2 TeV. In light of this possibility, we have revisited previous calculations by considering improved matching and renormalization group (RG) effects, specifically including two-loop RG effects in the two-Higgs-doublet model (2HDM) that is effective between the heavy Higgs scale M H ± and the SUSY scale M S .
We compare our new results with those obtained with the previous code version CPsuperH2.3. We also discuss the specific CPX4LHC benchmark scenarios relevant for the analysis of Higgs physics at the LHC, with particular emphasis on the masses of the heavier neutral Higgs bosons H 2,3 and on the CP-violating effects they may manifest. These offer interesting prospects for future runs of the LHC and future colliders.
All the improvements discussed in this study are being incorporated in a new version of the public code CPsuperH, called CPsuperH3.0. The numerical results presented here have  Figure 11: As in Figure 10, but showing the CP mixing quantity for the lightest mass eigenstate H 1 .  Figure 10, but showing the CP mixing quantity

M SUSY = 1 TeV
for the third mass eigenstate H 3 .  Figure 15: As in Figure 10, but showing the CP mixing quantity for the third mass eigenstate H 3 .

Appendices
In the appendices that follow, we present the relevant Renormalization Group Equations (RGEs) that are applicable above the three typical thresholds: (i) the top-quark mass m t , (ii) the heavy Higgs mass M H ≡ M H + , and (iii) the soft SUSY-breaking scale M S .
It is convenient to write the RGEs in the form where c stands for any kinematic parameter, such as quartic, gauge and Yukawa couplings, anomalous dimensions, and tan β. In addition, we use the abbreviations: t ≡ ln Q and κ = 1/(4π) 2 .

A SM RGEs
For scales of Q, for which M H 1 ∼ m t < Q < M H , one needs to consider the RGEs of the SM.
To properly take into consideration intermediate particle threshold effects, we introduce the short-hand notation for the step function Upon neglecting the Yukawa couplings of the first two generations of quarks and leptons, the one-loop SM RGEs that we use [35,42] read:

B One-Loop 2HDM RGEs
For RG scales Q between M H and M S , the effective theory becomes a general 2HDM, whilst for Q > M S the theory becomes fully supersymmetric and the quartic couplings λ 5,6,7 do not run. Here, we give the RGEs of the general 2HDM at the one-loop level, and relegate to Appendix C the presentation of the two-loop results.