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MSSM scenarios with a light CP-odd Higgs boson

The possibility of the presence of a light pseudoscalar with the mass of 28 GeV in the MSSM Higgs sector extended by dimension-six operators is analyzed. The perturbative unitarity and vacuum stability constraints are discussed. Tree-level production cross sections of a light pseudoscalar in the process gg→ μμbb̄, and cross sections of a light pseudoscalar production in the top quark decay are evaluated for suitable benchmark points.


Introduction
Light scalars are predicted in many extensions of the Standard Model (BSM) [1][2][3][4]. In light of these predictions an excess recently observed by CMS Collaboration in pp → b jµ + µ − channel at √ s= 8 and 13 TeV at the dimuon invariant mass m µµ ≈28 GeV in two event categories (with b-jet in forward region and without b-jet) [5] is of interest. The local significances are not large: 4.2σ and 2.9σ at √ s= 8 TeV and 2.0σ and 1.4σ (deficit) at √ s= 13 TeV. A similar search for a resonance decaying to two muons in the invariant mass range between 26 and 30 GeV produced in association with a b-jet and additional jet performed by ATLAS experiment yields no evidence of new signal [6]. However, the comparison of ATLAS and CMS reconstruction procedures is very difficult due to differences in the µ-trigger efficiency and b-tagging performance which suppress the signal/background ratio in the case of ATLAS and can be critical for such a weak signal. The situation is more intriguing in the light of a reanalysis of results of the ALEPH experiment at LEP where the excess has been found in the opposite-sign dimuon invariant mass spectra at m µµ =30.40 ± 0.46 GeV in events containing b-quarks with a local significance 5σ and a global significance 2.6σ [7].
In the most popular extension of the Standard Model (SM) -the Minimal Supersymmetric Standard Model (MSSM) -such a signal could be identified with a light pseudoscalar Higgs boson, but such possibility is closed in commonly used approximations. The MSSM extended by dimension-six operators in the Higgs sector [8], however, allows such a possibility [9]. Note that this excess can be also explained in the framework of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [10] and can be treated in a study of the muon anomalous magnetic moment [11].
In this proceeding MSSM scenarios under the assumption that a CP-odd Higgs boson has a mass of about 28 GeV are analyzed. Hereafter, an observed Higgs boson with a mass of 125 GeV [12][13][14][15] is identified with a light CP-even scalar 1 .

Higgs sector with dimension-six operators
The MSSM Higgs potential at the low energy scale is an effective two-Higgs doublet model (THDM) potential which in general can be expanded to all orders of perturbation theory where Φ i are the Higgs doublets Radiative corrections to the U eff = U (2) + U (4) have been analyzed in [16][17][18][19][20][21][22][23][24]. The term U (6) have been considered in [8] where the one-loop threshold corrections to parameters κ i were obtained by using the effective potential method 2 . After transformation to the mass basis, the potential (1) has the form where m h,H,A,H ± are Higgs boson masses, I 3,...6 denote the interaction terms of physical scalars. The mixing angle α of the CP-even Higgs sector is specified by where ∆M i j are radiative corrections to the mass matrix 3 , s 2β = sin 2β, c 2β = cos 2β, etc. This approximate model must satisfy the following requirements.

Existence of the Higgs mass basis
In the general case, radiative corrections to the Higgs sector may lead to spontaneous symmetry breakdown [25] when the local minimum of the Higgs potential can degenerate or even disappear. Such a situation might take place in calculations of loop diagrams when obtained radiative corrections and Higgs masses might not correspond to eigenvalues of mass matrices containing all factors in front of the lagrangian terms of dimension two in terms of S U(2) fields (as treated in effective field theory (EFT) approach). Under κ i =0 assumption, one can check the correspondence using the transformation (31)- (35), (40) in [22] which is valid for a general THDM regardless of the method of λ 1,...7 calculations. The mass basis is automatically ensured in the EFT approach with or without dimension-six operators taken into account.

Perturbative unitarity
The standard procedure to probe unitarity is to calculate two-particle scattering processes at tree level. All coupled channels for all particles should be considered, so the full treelevel analysis can be rather cumbersome. It was shown by Lee, Quigg and Thacker that at a very large scattering energy in the 't Hooft-Feynman gauge one can consider an amplitude for the scalar field theory instead of an amplitude for scattering of longitudinal intermediate bosons and physical Higgs particles with the accuracy of O(M i / √ s) (the Goldstone boson equivalence theorem) [30]. In other words, all diagrams with propagators can be neglected and only vertex type interactions should be analyzed. This simplification is usually used in the literature [31,32], however, as was noted in [33] scattering may never be in the regime with sufficiently large energies when particle masses in the propagators could be neglected. In this case, cubic couplings of large scalars must be taken into account which significantly modify the case.
For S 1 S 2 → S 3 S 4 scattering processes of scalars at finite √ s the perturbative unitarity constraints for the eigenvalues of S -wave partial amplitude matrix a 0 are [33] Re(a i 0 ) ≤ where the factor δ 12 (δ 34 ) is 1 if particles {1, 2} ({3, 4}) are identical, and zero otherwise, s = (p 1 + p 2 ) 2 is the Mandelstam variable, m 5 is the particle mass in a propogator, λ 1234 and κ i jk are quartic and trilinear couplings of scalars, are the centre of mass three-momenta, are the energies of the particles in the center of mass frame. It is obvious that unitarity constraints of the MSSM extension with dimension-six operators are different from ones with only renormalizable operators. Effective parameters κ i modify the Higgs masses and couplings [8]. An example of a constraint for heavy CP-even Higgs bosons scattering |a 0 (HH → HH)| as a function of a Higgs superfield mass parameter µ at tan β=2, M S =2 TeV, A t,b =6690 GeV, √ s=13 TeV is presented in figure 1. The red (blue) line corresponds to representation of the effective Higgs potential up to dimension-six (dimension-four) operators. As one can see the restrictions vary greatly in the range µ > 3 TeV.

Electroweak vacuum stability
The MSSM vacuum stability bounds are constructed under the assumption that among nonzero vacuum expectation values of SUSY fields the deepest minimum of the effective potential coincides with the electroweak (EW) minimum [34]. The case of stability of the local minimum of the Higgs potential with dimension-six operators [see eq. (1)] was considered in [35] where it was shown that for the majority of the MSSM benchmark scenarios [36,37] the allowed parameter regions coincide, but at the values of soft-SUSY breaking parameters of ∼5-10 TeV the parameter space is stronger restricted compared to four-dim operators decomposition. As a rule, perturbative unitarity constraints are weaker than the vacuum stability bounds [38].

The model features with a 28 GeV pseudoscalar
Following the MSSM approximation of [8] the considered model in the CP-conserving limit has the following free parameters: tan β, pseudoscalar mass m A , a SUSY scale M S , Higgs superfield mass parameter µ and trilinear couplings of the third generation of squarks A t , A b , where for simplicity A t = A b . Assuming m A =28 GeV, M S ={600, 1000, 2000, 3500, 5000} GeV, tan β={1, 2, 3, 5, 15, 20}, one can adjust parameters µ, A t,b in such a way that a light CPeven Higgs boson h has the mass 125 GeV at the alignment limit (7). This leads to stringent Table 1. Benchmark points under the assumption that m A =28 GeV, m h =125 GeV in the alignment limit BP tan β M S (GeV) A t,b (GeV) µ (GeV)  1  2  2000  8800  5320  2  3  2000  7820  6450  3  5  1000  3385  5040  4  5  2000 6690 7960 Note that the condition m A =28 GeV combined with h(125 GeV) in the alignment limit is possible only for Higgs potential decomposition up to dimension-six operators with large values of A t,b , µ that is rather unusual for common LHC scenarios [36,37] where A t,b , µ ∼ O (100 GeV). Nevertheless, for these BP's the Higgs potential minimum is stable, perturbative unitarity conditions for the probing scattering process HH → HH are also satisfied (see table 2 and figure 1) and the values of radiative corrections to λ i and κ j are below 1, as shown in figure 2 for the BP1. One can see from table 2 that the Higgs sector is nearly degenerate with masses at the electroweak scale.

Cross section of the process gg → µ + µ − bb
As was reported by the CMS Collaboration [5] an excess of the muon pair production in the mass range 12-70 GeV at about 28 GeV in association with a b-quark jet and an additional jet has been observed in two event categories denoted in the following by SR1 and SR2 at √ s=8 and 13 TeV. Reconstructed cross sections of the excess are 4.1 ± 1.4 fb (SR1) and 4.2 ± 1.7 fb (SR2) for √ s=8 TeV and 1.4 ± 0.9 fb (SR1) and −1.5 ± 1.0 fb (SR2) for √ s=13 TeV. Assuming that this excess is a light pseudoscalar one can evaluate production cross section at complete tree level for the process gg → Abb → µ + µ − bb, see the signal diagrams in figure 3. Corresponding to the CMS event selection [5]   Cross sections evaluated by CompHEP [39] package are presented in table 3. As expected, cross sections for complete sets of diagrams using phase space cuts A and B coincide with the production×decay approximation σ(gg → bbA)BR(A → µ + µ − ). One can see that the evaluated cross sections are smaller than the reconstructed cross sections of the CMS excess.

Top quark decay
The most sensitive way to search for exotic processes is to analyze the interactions with top quarks due to the large top quark Yukawa coupling. As was demonstrated above in the case of light pseudoscalar, the masses of Higgs bosons are of the order of EW scale. For all BP's m H ± < m top (see table 2), so in this case the exotic decay of the top quark t → H ± b should take place. The experimental constraints for this process are the following: (i) for tan β < 5 the ATLAS upper limit of BR(t → H + b) is 1-5% (95% CL) in the assumption that the dominant hadronic decay mode is H + → cs [40]; (ii) for tan β ≥ 5 the ATLAS and CMS upper limits on BR(t → H + b)BR(H + → τ + ν τ ) are 1.3-0.2% [40] and 1.2-0.5% [41].

Conclusion
The scenarios with a light pseudoscalar Higgs boson of the MSSM extended by dimension-six operators in the Higgs sector have been analyzed. It was shown that the CP-odd Higgs boson with mass of 28 GeV can be embedded in the model if the soft-SUSY breaking parameters are around 3-9 TeV and the SUSY mass scale M S is of a few TeV. Such an unusual parameter regime does not spoil the perturbative unitarity 4 and vacuum stability of the model. In such scenarios, the mass spectrum of Higgs bosons is splitted weakly at the EW scale. The top quark decay to charged Higgs boson and bottom quark as a sensitive way for a new physics search has been found promising. It was shown that model predictions do not contradict the direct experimental data. At the same time, estimated cross sections σ(gg → µ + µ − bb) are from several times to about an order of magnitude smaller than the cross sections experimenally reconstructed for the CMS excess.