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B meson Dileptonic Decays in NMSSM with a Light CP-odd Higgs Boson

In the next-to-minimal supersymmetric model (NMSSM) a light CP-odd Higgs boson is so far allowed by current experiments, which, together with a large tan-beta, may greatly enhance the rare dileptonic decays B->X_s l^+ l^- and B_s->l^+ l^- gamma. We examine these decays paying special attention to the new operator allowed by the light CP-odd Higgs boson. We find that in the parameter space allowed by current experiments like LEP II and b->s gamma, the branching ratios of these rare decays can be greatly enhanced and thus the existing experimental data on B->X_s mu^+ mu^- can further stringently constrain the parameter space (especially the region with a super-light CP-odd Higgs boson and large tan-beta). In the surviving parameter space we give the predictions for other dileptonic decay branching ratios and also show the results for the forward-backward asymmetry.


I. INTRODUCTION
Recently some non-minimal supersymmetric models such as the next-to-minimal supersymmetric model (NMSSM) have attracted much attention [1] since these models can solve the µ-problem and alleviate the little hierarchy. In the NMSSM, for example, the µ-term in the superpotential is forbidden by imposing a discrete Z 3 symmetry and instead it is generated through the coupling between the two Higgs doublets and a newly introduced gauge singlet scalar which develops a vacuum expectation value of the order of the SUSY breaking scale. In this way, the µ parameter at the weak scale can be naturally explained.
The NMSSM can ameliorate the little hierarchy by either tuning the parameters to enhance the theoretical upper bound for the mass of the lightest CP-even Higgs boson or relaxing the LEP2 bound of 114 GeV through allowing for a light CP-odd Higgs boson (A 1 ) with mass below 2m b [2].
It is interesting to note that in the NMSSM the lightness of such a CP-odd Higgs boson can be naturally predicted in the enlarged parameter space, and is also allowed by the LEP II data [1]. This light Higgs boson can not only alleviate the little hierarchy, but also can help to explain the observed anomaly in the decay Σ + → pµ + µ − [3]. On the other hand, if a Higgs boson is indeed so light, its effects in some low energy processes may be sizable and thus are necessary to check [4,5]. For a super light A 1 the decay b → A 1 s is open and an analysis has been performed in [4] (note that as analysed in [4], a small CP-odd Higgs mass is only protected from RGE-effects in the limit of large tan β). In this work we consider the full possible mass range of A 1 (heavy, intermediately heavy and light) and check the NMSSM effects in the rare B meson dileptonic decays B → X s ℓ + ℓ − and B s → ℓ + ℓ − γ [6].
These rare dileptonic decays are induced by the flavor-changing neutral-current (FCNC) b → s transition and are of special interest due to their relative cleanness and high sensitivity to new physics. In the Standard Model (SM) such FCNC processes are suppressed and have very small branching ratios [7] but can be greatly enhanced in some new physics models [8,9,10,11,12,13]. Since experimental data on B → X s µ + µ − is available and the future LHCb or super B factory will further scrutinize B meson decays, these dileptonic decays serve as a good probe of new physics.
In supersymmetric models these dileptonic decays can be drastically enhanced by large tan β since both the b → s transition loops (such as the charged Higgsino loops) and the Higgs couplings in the Higgs-propagated diagrams are proportional to tan β. It has been shown (see e.g. [12]) that in the minimal supersymmetric model (MSSM) great enhancements are possible for these decays. In the context of NMSSM, in addition to the tan β enhancement, the presence of a light CP-odd Higgs boson could further enhance these dileptonic decays. Thus, the parameter space, especially the region with a super light CP-odd Higgs boson and a very large tan β, is constrained by the existing data on B → X s µ + µ − .
In our analysis we will examine the NMSSM effects in these dileptonic decays by scanning over the parameter space allowed by the LEPII experiments and the data on b → sγ. We will show the 2σ constraints from B → X s µ + µ − on the parameter space and then give the predictions for other dileptonic decay branching ratios and forward-backward asymmetry.
A key point in our calculations is the presence of a new operator due to the light CP-odd Higgs boson. In contrast to the MSSM, where all Higgs bosons and sparticles are so heavy that they can be integrated out at the weak scale, the light CP-odd Higgs boson A 1 in the NMSSM cannot be integrated out at the weak scale and thus a new operator O A describing the interaction A 1 bs must be treated carefully.
This work is organized as follows. In Sec. II a brief description of the NMSSM is presented. In Sec. III we calculate the Wilson coefficients paying special attention to the new operator O A . In Sec. IV we evaluate the NMSSM effects on the dileptonic decay branching ratios and the forward-backward (FB) asymmetry, and present some numerical results. The conclusion is given in Sec. V and analytical expressions from our calculations are presented in the Appendix.

II. A BRIEF DESCRIPTION OF NMSSM
In the NMSSM a singlet Higgs superfieldŜ is introduced. A discrete Z 3 symmetry is imposed and thus only the cubic and trilinear terms are allowed in the superpotential. The Higgs terms in the superpotential are then given by Note that there is no explicit µ-term and an effective µ-parameter is generated when the scalar component (S) ofŜ develops a vacuum expectation value s/ √ 2: 3 The corresponding soft SUSY-breaking terms are given by The scalar Higgs potential is then given by where G 2 = g 2 1 + g 2 2 with g 1 and g 2 being the coupling constants of U Y (1) and SU L (2), respectively.
The scalar fields are expanded as follows: The mass eigenstates can be obtained by unitary rotations where H 1,2,3 and A 1,2 are respectively the CP-even and CP-odd neutral Higgs bosons, G 0 and G + are Goldstone bosons, and H + is the charged Higgs boson. It is clear that the charged Higgs sector is the same as in the MSSM, while the neutral Higgs sector contains one more CP-even and one more CP-odd Higgs boson. U A and U H are given by where C X = cos X and S X = sin X (X = θ A , θ H ). The mixing angles are given by [14] θ with v ≃ 246 GeV and 4 The Lagrangian of the Higgs couplings to quarks for a large tan β are given by Here one can see that the coupling of the neutral CP-odd Higgs A i with up-type quarks are suppressed by a large tan β and thus can be neglected in the large tan β limit.
Since one more Higgs superfieldŜ is introduced in the NMSSM, we have a new neutral Higgsino ψ S . So the neutralino sector is composed of U Y (1) gaugino λ 1 , SU L (2) gaugino λ 2 , and the Higgsinos ψ 1 H d , ψ 2 Hu and ψ S . The corresponding mass terms are given by where The neutralinos are obtained by the unitary rotation ψ 0 Similar to the charged Higgs sector, the chargino sector of the NMSSM is the same as in the MSSM with µ replaced by µ ef f . The chargino masses are obtained by the diagonalization of the mass matrix with two unitary matrices Z − and Z + :

III. CALCULATIONS OF WILSON COEFFICIENTS
In our calculations we consider the flavor mixing betweenb ands, which make contributions to the dileptonic B meson decays through gluino or neutralino loops. Following the analysis in [15], we assume the flavors are diagonal at tree level and the mixings are induced at loop level. Such mixings can be parameterized by a small mixing parameter ǫ 1 which is dependent on some soft-breaking mass parameters [15]. In our numerical calculations we input ǫ 1 = 0.1 for illustration. We perform the calculations in the Feynman gauge and thus the Goldstone bosons will be involved in the loop diagrams.
Since in the NMSSM the lighter mass eigenstate, the CP-odd neutral Higgs boson A 1 , can be rather light, with a mass ranging from 100 MeV to the weak scale [1,2] ( using the package NMHDECAY [16], we checked that such a light CP-odd Higgs boson A 1 is indeed allowed by the LEP II data), in our calculations we pay special attention to this wide mass range of A 1 and discriminate three cases.
(i) Case A: Heavy A 1 .

For a heavy
where O i and Q i are operators listed in [8,12], and C i and C Q i are respectively their Wilson coefficients, and µ r is the renormalization scale. Note that the most general Hamiltonian in low-energy supersymmetry also contains the operators O ′ i and Q ′ i which respectively are the flipped chirality partners of O i and Q i . However, they give negligible contributions and thus are not considered in the final discussion of physical quantities [17].
In this case, there are no new operators and we only need to calculate the NMSSM contributions to these coefficients C i and C Q i at the scale of m W . For the processes in our analysis, only C 7,9,10 and C Q 1,2 are relevant. The NMSSM contributions to C 7,9,10 are the same as in the MSSM, which are computed in [18]. For C Q 1,2 the NMSSM contributions are different from the MSSM contributions [12] and thus need to be calculated here. The Feynman diagrams we need to calculate are shown in Fig.1, where the loops respectively involve the charged Higgs bosons, charginos, gluinos and 6 neutralinos. From the calculations of these diagrams we obtain the Wilson coefficients at m W scale, which are presented in the Appendix.
For the calculation of the dileptonic B meson decays we need to know the Wilson coefficients at the m b scale, which can be obtained from the running of the coefficients at the m W scale down to the m b scale. Such a running is governed by the anomalous dimension which can be found in [12]. 7 From the calculations of the corresponding diagrams in Fig. 1 we obtain the Wilson coefficient C A (m W ). It is composed of the charged Higgs loop contribution, the chargino loop contribution, the neutralino loop contribution and the gluino loop contribution, whose analytic expressions are given in the Appendix.
For the running of the Wilson coefficients including C A from the m W scale to the m A 1 scale we work out the anomalous dimensions by calculating one-loop diagrams with operator insertions. We find that all the Wilson coefficients in Eq. (19) run in the same way as in the MSSM [12], while the new coefficient C A is not changed, i.e., Finally, for the running of the Wilson coefficients from the m A 1 scale to the m b scale the anomalous dimensions are the same as in the MSSM [12].
In this case we retain A 1 as an active field in the entire analysis. At the m W scale we integrate out all heavy particles and obtain the Wilson coefficients including C A .
Then we run the coefficients down to the m b scale. At the m b scale the effects of O A are represented by a change in C Q 2 where p is the momentum transfer and Γ A 1 is the total width of A 1 . Since A 1 can be on-shell in this case, the effects of A 1 can be sizable even if without tan β enhancement.
Note that the chargino loop contributions to C A were also calculated in [4] where the corresponding diagrams induced by the A 1 -squark-squark vertex are neglected since the author considered the large tan β limit. In our numerical calculations we used the full results by keeping all terms and thus we also included the diagrams induced by the A 1 -squark-squark coupling although they contain no leading tan β terms. Except for the case of a large tan β, such diagrams induced by the A 1 -squark-squark coupling should be included since the A 1squark-squark coupling can arise from the F-term of the superpotential and not suppressed 8 by the singleness of A 1 . We checked that in the large tan β limit we can reproduce the analytical result given in [4] for the chargino-loop contributions.

IV. DILEPTONIC B-MESON DECAYS IN NMSSM
With the effective Hamiltonian and the running of the Wilson coefficients presented in the preceding section we calculate the inclusive decays B → X s ℓ + ℓ − and their forward-backward (FB) asymmetry, as well as the exclusive decays B s → ℓ + ℓ − γ. The formulas in terms of the Wilson coefficients can be found in [8,12].
Note that our supersymmetric contributions to the Wilson coefficients are given at oneloop level (next-to-leading order), while the SM contributions are known at two-loop level (next-to-next-to-leading order) [7]. In our numerical calculations we consider the one-loop results for the NMSSM, while for the SM we also include the two-loop results.
For the inclusive decays B → X s ℓ + ℓ − we exclude the resonances J/Ψ and Ψ ′ contributions by using the same cuts as in the experiments [19], i.e., the invariant dilepton mass in the so that our results can be compared with the experimental measurements.
For exclusive decays B s → ℓ + ℓ − γ we follow [10,12] and consider the photon in B s → ℓ + ℓ − γ as a hard photon by imposing a cut on the photon energy E γ , which means that the radiated photon can be detected in the experiments. This cut requires E γ ≥ δ m Bs /2 with δ = 0.02. (Note that for a soft photon both processes B s → ℓ + ℓ − γ and B s → ℓ + ℓ − must be considered together and in this case the infrared singular terms in B s → ℓ + ℓ − γ are be In our numerical calculations we perform a scan over the NMSSM parameter space with fixed parameters for the sfermion and gaugino sector (500 GeV for all sfermions and the gluino, and 200 GeV and 100 GeV for SU (2) and U(1) gaugino masses M 2 and M 1 , respectively). In our scan we consider the following constraints:
Among the relevant dileptonic decays experiment data is only available for Br(B → X s µ + µ − ), which is given by [21] Br In displaying our numerical results we will show this bound and use it to constrain the parameter space. For other dileptonic decay branching ratios, with no experiment data available, we will compare the NMSSM predictions with the SM values given by Br(B s → γµ + µ − ) = 1.33 × 10 −8 , Br(B s → γτ + τ − ) = 1.35 × 10 −8 .
Note that the SM prediction for Br(B → X s τ + τ − ) was also given in [22]. But our result is different from theirs because it is very sensitive to the cuts around the resonances J/ψ and ψ ′ . While our cuts are chosen as in Eq. (23), we cannot find the corresponding cuts used in [22]. We can easily reproduce the result in [22] by varying the cuts.
In Fig.2 we show the scatter plots of the branching ratio for B → X s µ + µ − versus tan β.
Here we present the results for the three cases: a super light A 1 (m A 1 <5GeV), an interme- Let us take a look on the constraints from the process B s → µ + µ − , whose branching ratio is given by [23] Br We see that the contributions are from C 10 and C Q 1,2 . While the contribution from C 10 is suppressed by the factor m µ /m Bs , the contributions from C Q 1,2 can be enhanced by large tan β (C Q 1,2 contain terms which are proportional to tan 3 β, as shown in the Appendix).
This feature can be seen from Fig.3 in which we set aside the b → sγ constraints and illustrated the B s → µ + µ − constraints. We see that, similar to the b → sγ constraints, the B s → µ + µ − constraints are stringent for a large tan β. If we impose the b → sγ constraints which exclude a very large tan β, then the further constraints from B s → µ + µ − are stringent only for the parameter space with a very light A 1 . The results for other dileptonic decay branching ratios, for which no experiment data are available, are presented in Fig.4. To see how stringent the constraints from B → X s µ + µ − are we display the scatter plots with and without such constraints (all the points satisfy b → sγ and B s → µ + µ − ).

10
From Fig.4 we see that under the constraint from B → X s µ + µ − , the branching ratio of B → X s τ + τ − does not deviate significantly from the SM value. The reason is that these two decays are highly correlated except for the contributions of C Q 1,2 and C A which are dependent on the lepton mass. If the contributions of C Q 1,2 and C A are dominant, then Br(B → X s τ + τ − ) should not be severely constrained by B → X s µ + µ − . As discussed below Eq.(31), the contributions of C Q 1,2 and C A are important only for very large tan β which is not allowed by b → sγ. As a result, the contributions of C Q 1,2 and C A are not dominant and thus B → X s τ + τ − is highly correlated to B → X s µ + µ − .
Note that for B → X s τ + τ − it may be rather challenging to disentangle the NMSSM effects from the SM value in future experiments. One reason is that, as discussed above, the NMSSM effects are no longer so sizable under the constraint of B → X s µ + µ − . The other reason is that the SM prediction has its own uncertainty. If we consider the uncertainty of the input SM parameters, we can obtain the uncertainty (about 20% as found in [22]) of the Finally, in Fig.5 we show the results for the forward-backward asymmetry in B → X s µ + µ − under the constraints from B → X s µ + µ − .

V. SUMMARY
In the framework of the NMSSM we examined the rare dileptonic decays B → X s ℓ + ℓ − and B s → ℓ + ℓ − γ paying particular attention to the light CP-odd Higgs boson. We found that in the parameter space allowed by current experiments, such as LEP II and b → sγ, the branching ratios of these rare decays can be greatly enhanced and thus the experimental data on B → X s µ + µ − further stringently constrains the parameter space, especially with a super light CP-odd Higgs boson and large tan β. In the surviving parameter space we gave the NMSSM predictions for other unmeasured dileptonic decays which may hopefully be measured at the future LHCb or super B factory.

Physics under No. DE-FG02-91-ER4086 and by the JSPS invitation program under No.
S-08034.

APPENDIX A: WILSON COEFFICIENTS
The Wilson coefficients C 7 , C 9 and C 10 in the NMSSM are the same as in the MSSM [18]. Here we present the new coefficient C A , and C Q 1 and C Q 2 , whose predictions in the NMSSM are different from the MSSM. We checked that we can analytically reproduce the MSSM results [12,13,18] (However, the NMSSM results can not explicitly reduce to the MSSM results by simply dropping out the singletŜ (say setting λ = κ = 0) because the µ-term is generated byŜ).
Although in our numerical calculations we used the complete results by keeping all terms, here, for simplicity, we only present the terms which can be enhanced by large tan β. At the m W scale each Wilson coefficient is composed of the charged Higgs loop contribution from Fig.1(a-c), the chargino loop contribution from Fig.1(d-g), the neutralino and gluino loop contribution from Fig.1(h-k).
For the charged Higgs contributions: Note that although H ± W ± A 1 vertex has 1/ tan β suppression by the singleness of A 1 , H ± G ± A 1 vertex which comes from the soft term A λ λSH 1 u H 2 d does not have such 1/ tan β suppression. Thus the contribution to C A from the loop involving H ± and G ± with H ± G ± A 1 coupling (in the Feynman gauge) is proportional to tan β.

15
For the chargino contributions: C e Note that here C Q 1,2 contain terms which can be enhanced by tan 3 β (the overall factor tan 2 β multiplied by 1/ cos β gives tan 3 β in large tan β limit).
For neutralino contributions: For gluino contributions: In the above expressions the constants and functions are defined by 2