Impact of CP phases on a light sbottom and gluino sector

We study a scenario in which light bottom squarks and light gluinos with masses in the range $2-5.5$ GeV and 12-16 GeV, respectively, can coexist in the MSSM, without being in conflict with flavor-conserving low-energy observables. We find that in such a scenario, the anomalous magnetic moment of a muon could be as large as $10^{-9}$, if the theory conserves CP. However, if the theory violates CP, we conclude that not both, the gluino and bottom squark, can be light at the same time, after the neutron electric dipole moment constraint on Weinberg's 3-gluon operator has been taken into account.

Inspired by the unforeseen excess of the bottom-quark production observed at the hadronic collider of Fermilab [1], Berger et al. [2] proposed a solution with light sbottomb 1 (mb 1 ≃ 2.0 − 5.5 GeV) and light gluinog (mg ≃ 12 − 16 GeV). Interestingly, various problems can easily be avoided, such as by adopting the proper mixing angle of two sbottomsb L andb R , the Z-peak constraint can be evaded; and also, the R b contribution, arisen from sbottom-gluino loop, can be suppressed by considering the second sbottomb 2 being lighter than 180 GeV for the CP conserved case [3], while the mass ofb 2 can be heavier in the CP violating one [4]. The amazing thing is that such light supersymmetric particles, so far, haven't been excluded by experiments, even after including the data of precise measurements [5]. Moreover, for searching the signals of the light sbottom and gluino, many testable proposals are raised, e.g., the rate of χ b decaying to a pair of light bottom squarks [6], radiative B meson decays [7], the decays Z → bb * 1g +bb 1g followed byg → bb * 1 /bb 1 and e + e − → qqgg [8], as well as the running of strong coupling constant α s [9].
To further explore more impacts on other processes, it is necessary to investigate different systems instead of those in which the final states are directly associated with the light sbottom and gluino. It is known that supersymmetric models not only supply an elegant mechanism for the electroweak symmetry breaking and a solution to the hierarchy problem, but also guarantee the unification of gauge couplings at the scale of GUTs [10]. Therefore, besides the effects mentioned above, other contributions will also appear when considering different phenomena. Inevitably, new parameters will come out. To avoid introducing the irrelevant parameters, such as the mixing angles among different flavors of squark, we have to consider the processes in which the dependent parameters are still concentrated on the minimal set. The best candidates are the flavor-conserving processes.
One of the mysteries in the standard model (SM) is whether the Higgs mechanism plays an essential role for the symmetry breaking and the resultant of Higgs particle can be captured in future colliders. Based on the same philosophy of the symmetry breaking, the minimal supersymmetric standard model (MSSM) needs the second Higgs doublet field to balance the anomaly of quantum corrections, i.e., there are three neutral Higgs particles in MSSM, one of them is CP-odd (A 0 ) and the remains are CP-even (h and H). It is obvious that besides the mixing angle of sbottoms and the masses of sbottom (gluino), the essential parameters in MSSM are A t(b) , the trilinear SUSY soft breaking terms, µ, the mixing parameter of two Higgs superfields, m A 0 (h,H) , the masses of corresponding Higgses, and tan β, defined by the ratio of v u to v d in with v u(d) being the vacuum expectation of the Higgs field that couples to up (down) type quarks. And also, unlike the non-SUSY two-Higgs-doublet model, the mixing angle of two neutral Higgs fields, denoted by α, is not an independent parameter and can be related to tan β [11].
The activity of searching for the Higgs particle and studying its properties is proceeding continuously [12,13]. In particular, the remarkable results with a large tan β have been investigated enormously because the exclusive characteristic can give the unification of bottom and tau Yukawa couplings and the realization of the top to the bottom mass ratio in GUTs [14]. It has been found that if the SUSY soft breaking terms carry the explicit CP violating phases, due to the enhancement of large A t(b) and µ, radiative effects can induce a sizable mixing between scalar and pseudoscalar such that the lightest Higgs boson could be 60 − 70 GeV and thus escape the detection of detectors [15]. Moreover, with the requirements satisfied with electroweak baryogenesis, a novel prediction on the muon electric dipole moment (EDM) of 10 −24 e cm can be reached by the proposed experiment [16]. In sum, it will be more exciting that if the light sbottom and gluino can be compatible with the Higgs physics with or without CP violation (CPV).
In order to further pursue the implications of the light sbottom and gluino, in this paper, we concentrate on two flavor-conserving processes: one is anomalous magnetic moment of muon, ∆a µ , and the other is EDMs of muon and neutron. The former corresponds to CP conservation (CPC) while the latter is related to CPV. Since the results must be proportional to the mixing ofb L andb R , for generality, we describe the relationship between weak and physical eigenstates as where δ b is the CP violating phase which could arise from the off-diagonal mass matrix To suppress the coupling Zb 1b1 , we set sin 2θ b = 0.76 in our discussions. In the following analyses, we separate the problem into CPC and CPV cases.
Unlike non-SUSY two-Higgs doublet models, it is well known that besides the enhancement of large tan β or 1/ cos β, there exists another enhanced factor A b in the couplings of S = h and H to bottom squarks [11]. As described early, some novel consequences based on both large factors have been displayed on the Higgs hunting. With these factors, we examine the implication on ∆a µ by considering the case of a light sbottom. In addition, for convenience, we adopt the decoupling limit with m A 0 >> m Z so that tan 2α ≈ tan 2β. The effective interaction hγγ is induced from the radiative effects in which sbottoms are the internal particles of the loop, illustrated in Fig. 1(a), and the gauge invariant form of the coupling can be obtained as where N c = 3 is the color number, α em is the fine structure constant, Q b = −1/3 is the charge of sbottom, and m b (Mb i ) is the mass of bottom quark (squarks). Because we concentrate on the light sbottom case, we do not discuss the contributions of stops by setting their masses being heavy. Since h couples to different sbottomsb L andb R but γ couples to the same sbottoms, it is clearly inevitable to introduce the mixing angle θ b . In Eq. (2), we already neglect the smaller contribution related to µ cos α/ cos β. By using the result of Eq. (2), via the calculation of the loop in Fig. 1(b), the anomalous magnetic moment of muon from two-loop Higgs-sbottom-sbottom diagrams is given by [17] Immediately, we see that although this is a two-loop effect and there appears one suppressed factor m b /v·(m u /M h ) 2 , the ∆a µ could be enhanced in terms of ( If we take Mb 2 = 180 GeV and assume the lightest Higgs boson 100 < M h < 140 GeV, one gets that  CP problem has been investigated thoroughly in the kaon system since it was discovered in 1964 [18]. Now, CPV has been confirmed by Belle [19] and Babar [20] with high accuracy in the B system. Although the mechanism of Kobayashi-Maskawa (KM) [21] phase in the SM is consistent with the CP measurements, the requirement of the Higgs mass of 60 GeV for the condition of the matter-antimatter asymmetry has been excluded by LEP. One of candidates to deal with the baryogenesis is to use the SUSY theory. As known, any CP violating models will face the serious low energy constraints from EDMs of lepton and neutron, which are T and P violating observables and at elementary particle level usually are defined by d ff σ µν γ 5 f F µν , with F µν being the electromagnetic (EM) field tensor. We note that not only the EM field but also the chromoelectric dipole moment (CEDM) of gluon for colored fermions will contribute. Therefore, we have to examine the implication of the light sbottom on EDMs while δ b is nonzero, ı.e., A b and µ are complex.
Inspired by the previous mechanism for ∆a µ , the similar effects with pseudoscalar A 0 instead of scalar h, shown in Fig. 1(a), will also contribute the EDMs of leptons and quarks.
Since the effects have been analyzed by Refs. [16,22], we directly summarize the formalisms for the fermion EDM and CEDM as respectively, with R f = tan β (cot β) for T f . For simplicity, we set the effects of stops to be negligible. According to Eq. (4), we see that the lepton EDM of d γ ℓ is proportion to the m ℓ . Due to the ratio m µ /m e ≈ 205, the strictest limit comes from the EDM of electron, with the current upper bound being 1.6 × 10 −27 e cm [23]. Although the renormalization factor (g s (M W )/g s (Λ)) 32/23 for the EDM of neutron is around one order larger than (g s (m W )/g s (Λ)) 74/23 for the CEDM contribution, due to α s and 1/(Q 2 b Q q ) enhancements, the CEDM of quark is much larger than the EDM of quark. Hence, only the effects of the CEDM on neutron are considered. Since the unknown parameters for the EDM of electron and CEDM of neutron are the same, the values of parameters are taken to satisfy with the bound of the electron EDM. We present the results as a function of Mb 1 with tan β = 10 (20) and A b = 500 (200) GeV in Fig. 3, in which the solid, dashed and dashed-dotted lines correspond to M h = 100, 120 and 140 GeV, respectively. We note that the origin of CP violation comes from Im(A b e iδ b )/|A b | and it is set to be O(10 −1 ). From the figure, we see clearly that without fine tuning the CP phase to be tiny, the EDM of electron can be lower than the experimental bound and the CEDM of neutron is also not too far away from current limit.
The effects become testable in experiments. It seems that so far the scenario of the light sbottom with the mass of few GeV could give interesting results in ∆a µ , d γ e and d C N . However, what we question is whether both light sbottom and gluino can coexist when the CP phases are involved. We find that the possibility encounters fierce resistance of the Weinberg's 3-gluon operator, defined by [24] with G αµρ and f αβγ being the gluon field-strength tensor and antisymmetric Gell-Mann coefficient, respectively. With the bottom, sbottom and gluino being the internal particles of the two-loop mechanism, the contribution of the Weinberg's operator is given by [25] where function H is a two-loop integration, Here, we have rotated away the phase of gluino. With naive dimentional analysis, the neutron EDM could be estimated by d G N = (eµ X /4π)η G d G , with µ X ∼ 1.19 GeV and η G ∼ 3.4 being the chiral symmetry breaking scale and renormalization factor of the Weinberg's operator, respectively. Although the estimation still has a large theoretical uncertainty, our concern is on the problem of order of magnitude. Due to the result being proportional to 1/M 3 g , we see that the lighter Mg is, the larger d G . It is worth mentioning that the similar tendency can be also found in Ref. [26], in which the considered situation is via the gluino−stop−top-quark two-loop. For an illustration, we present the results in Table I    have also found that except extreme fine tuning, the light sbottom and gluino cannot coexist after the neutron electric dipole moment constraint on the Weinberg's 3-gluon operator has been taken into account. We note that it is possible that there exist some cancellations while we consider all possible contributions to the neutron EDM [27]. Nevertheless, it will depend on how to fine tuning the involved parameters and that is beyond our scope of the current paper.