\(B^*_{s,d} \rightarrow \mu ^+ \mu ^-\) and its impact on \(B_{s,d} \rightarrow \mu ^+ \mu ^-\)

The European Physical Journal C, Oct 2016

This study investigates \(B^*_{s,d}\rightarrow \mu ^+\mu ^-\) in the dimuon distributions and the hadronic contribution \(B_{s,d}\rightarrow B^*_{s,d}\gamma \rightarrow \mu ^+\mu ^-\). The \(\mu ^+\mu ^-\) decay widths of the vector mesons \(B^*_{s,d}\) are approximately a factor of 700 larger than the corresponding scalar mesons \(B_{s,d}\). The ratio of the branching fractions obtained, \(\frac{Br({B_{s,d}^*\rightarrow \mu ^+\mu ^-})}{{Br({B_{s,d}\rightarrow \mu ^+\mu ^-})}}\), is approximately \(\frac{0.3\times \mathrm{eV}}{{\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )}}\). The hadronic contribution \(B_{s,d}\rightarrow B^*_{s,d}\gamma \rightarrow \mu ^+\mu ^-\) is also estimated. The relative increase in the \(B_{s,d}\rightarrow \mu ^+\mu ^-\) amplitude is approximately \((0.01\pm 0.006) \sqrt{\frac{{\Gamma (B^*_{s,d}\rightarrow B_{s,d} \gamma )}}{{100~\mathrm{eV}}}}\). If we select \(\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=2~\)eV, then the branching fractions of the vector mesons to the lepton pair are \(5.3\times 10^{-10}\) and \(1.6\times 10^{-11}\) for \(B^*_{s}\) and \(B^*_{d}\), respectively. If we select \(\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=200~\)eV, then the updated branching fractions of the scalar mesons to the muon pair are \((3.78\,\pm \,0.25)\times 10^{-9}\) and \((1.09\,\pm \,0.09)\times 10^{-10}\) for \(B_{s}\) and \(B_{d}\), respectively. If we select the recent predicted M1 widths \(\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=313, 1230\) eV (arXiv:​1607.​02169), then the updated branching fractions are \((3.8\,\pm \,0.3)\times 10^{-9}\) and \((1.2\,\pm \,0.1)\times 10^{-10}\) for \(B_{s}\rightarrow \mu ^+\mu ^-\) and \(B_{d}\rightarrow \mu ^+\mu ^-\), respectively. Further studies on \(B^*_{s,d}\), including those on dielectron decay and two-body decay with \(J/\psi \), should be conducted.

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\(B^*_{s,d} \rightarrow \mu ^+ \mu ^-\) and its impact on \(B_{s,d} \rightarrow \mu ^+ \mu ^-\)

Eur. Phys. J. C Bs∗,d → μ+μ− and its impact on Bs,d → μ+μ− Guang-Zhi Xu 1 2 Yue Qiu 1 Cheng-Ping Shen 0 1 Yu-Jie Zhang 0 1 0 CAS Center for Excellence in Particle Physics , Beijing 100049 , China 1 Beijing Key Laboratory of Advanced Nuclear Energy Materials and Physics, and School of Physics, Beihang University , Beijing 100191 , China 2 Department of Physics, Liaoning University , Shenyang 110036 , China This study investigates Bs∗,d → μ+μ− in the dimuon distributions and the hadronic contribution Bs,d → Bs∗,d γ → μ+μ−. The μ+μ− decay widths of the vector mesons Bs∗,d are approximately a factor of 700 larger than the corresponding scalar mesons Bs,d . The ratio of Br(Bs∗,d →μ+μ−) the branching fractions obtained, Br(Bs,d →μ+μ−) , is approximately (Bs0∗,.d3→×eBVs,d γ ) . The hadronic contribution Bs,d → Bs∗,d γ → μ+μ− is also estimated. The relative increase in the Bs,d → μ+μ− amplitude is approximately (0.01 ± - 0.006) (Bs∗1,d0→0eBVs,d γ ) . If we select (Bs∗,d → Bs,d γ ) = 2 eV, then the branching fractions of the vector mesons to the lepton pair are 5.3 × 10−10 and 1.6 × 10−11 for Bs∗ and Bd∗, respectively. If we select (Bs∗,d → Bs,d γ ) = 200 eV, then the updated branching fractions of the scalar mesons to the muon pair are (3.78 ± 0.25) × 10−9 and (1.09 ± 0.09) × 10−10 for Bs and Bd , respectively. If we select the recent predicted M1 widths (Bs∗,d → Bs,d γ ) = 313, 1230 eV (arXiv:1607.02169), then the updated branching fractions are (3.8 ± 0.3) × 10−9 and (1.2 ± 0.1) × 10−10 for Bs → μ+μ− and Bd → μ+μ−, respectively. Further studies on Bs∗,d , including those on dielectron decay and twobody decay with J /ψ , should be conducted. 1 Introduction The leptonic decays of the Bs,d mesons play an important role in the standard model (SM) and the new physics (NP) [1,2]. The leptonic decays are highly suppressed in the SM because flavor-changing neutral current decays are generated through W-box and Z-penguin diagrams. Furthermore, the branching fractions of the leptonic decays of scalar meson go through an additional helicity suppression factor by m2μ/MS2, where mμ and MS denote the masses of the muon lepton and the scalar meson, respectively. The suppression can be removed in several NP models, such as the two-Higgs-doublet models [3], the minimal supersymmetric standard model [4], the next minimal supersymmetric standard model [5], the dark matter [6], the universal extra dimensional model [7], the lepton universality violation model [8], the fourth generation of fermions [9], and so on [10]. The branching fractions of Bs,d → μ+μ− measured by the CMS and LHCb Collaborations [2], and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections are presented in Table 1. On one hand, the experimental branching fractions of Bs,d → μ+μ− are measured from the dimuon distributions by the CMS and LHCb Collaborations [2]. Thus, the process Bs∗,d → μ+μ− will enhance the dimuon distributions for mass splitting between Bs,d and Bs∗,d at approximately 45 MeV. On the other hand, the hadronic contribution Bs,d → Bs∗,d γ → μ+μ− is missing in the theoretical prediction [1]. Therefore, this study focuses on Bs∗,d → μ+μ− and its impact on Bs,d → μ+μ− within SM. The Bs → Bs∗γ → μ+μ−γ process was considered in Ref. [13]. Bs∗,d → μ+μ− was recently considered in Refs. [14,15]. Moreover, Refs. [16,17]. also investigated the hadronic contribution of charmonium in B → K (∗) + − and B → Xs γ . 2 The Decay of Bs∗( Bd∗) → μ+μ− An effective Lagrangian related to bs¯ → μ+μ− within the SM is given in Refs. [18–20] L = N where N as follows: 2 = G√F2 Vtb Vt∗s 4eπ2 , and the operators O7,9,10 read Table 1 The branching fractions of Bs,d → μ+μ− measured by the CMS and LHCb Collaborations [2] and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections included (3.9+−11..64) × 10−10 (2.8+−00..76) × 10−9 (1.06 ± 0.09) × 10−10 (3.66 ± 0.23) × 10−9 where PL = (1 − γ5)/2 , PR = (1 + γ5)/2. The Wilson coefficients are C7e,ff9,10(μ f ) = (−0.316, 4.403 − 0.47i, −4.493) at μ f = mb = 4.5 GeV [15]. The superscripts γ , V , and A denote the contributions from photon, vector current, and axial vector current, respectively. The relationships between the quark level operators and the meson are described as follows: 0|s¯γ μb|Bs∗(q, ε) = m Bs∗ f Bs∗ εμ, 0|s¯σ μν b|Bs∗(q, ε) = −i f BT∗ (qμεν − εμqν ), s 0|s¯γ μγ5b|Bs (q) = i f Bs qμ, Afterward, the Bs∗(Bs ) → expressed as follows [13]: amplitudes are N M (Bs∗ → μ+μ−) = f Bs∗ 2 m Bs∗ μ¯ ε/ C Veff + C10γ5 μ, M (Bs → μ+μ−) = i f Bs N C10mμμ¯ γ5μ, C Veff = C9eff + 2 mb C7eff . m Bs∗ The helicity suppression factor m2μ/m2M in the decay width is removed in the vector meson decay. Then the Bs∗(Bs ) → μ+μ− decay widths are obtained: Vtb Vt∗s 2 Vtb Vt∗s 2 |C10|2 The decay width ratio is approximately 700 for Bs(∗) and Bd(∗) both. 3 The impact of Bs∗( Bd∗) → μ+μ− on Bs( Bd ) → Bs∗( Bd∗)γ → μ+μ− Furthermore, Bs∗,d will impact on the Bs,d leptonic decay through the loop contribution Bs,d → Bs∗,d γ → μ+μ−. The Feynman diagrams are shown in Fig. 1. This calculation is a part of EM corrections to Bs,d → μ+μ−. The NLO EW corrections of Bs,d → μ+μ− within the SM have been calculated in Refs. [1,12]. The hadronic contribution of Bs,d → Bs∗,d γsoft → μ+μ− + γsoft has been calculated in Ref. [13]. However, the contribution of Bs,d → Bs∗,d γ → μ+μ− is missing in the previous calculation. This calculation is incomplete. For instance, there is double counting between the NLO EW corrections Bs → b+s +γ → μ+μ− [12] and this calculation Bs → Bs∗γ → μ+μ−. We considered that the contribution of Bs → Bs∗γ → μ+μ− will be retained only when the Bs∗ is nearly on-shell. If Bs∗ is far away from Fig. 1 Feynman diagrams of Bs,d → Bs∗,d γ → μ+μ− mass shell, Bs → b+s +γ → μ+μ− is dominant. As is well known, the propagator of hadron will be modified duo to the off-shell of hadron [22], and the Wilson coefficients C7e,ff9,10 will be modified too [17,19]. Therefore, this treatment may be regarded as a crude estimate, and the error may be large in this treatment. The vertex of Bs,d → Bs∗,d γ is expressed as the following operator [23,24]: q=s,b q=s,b We can simplify the matrix element < Bs∗|q¯ ( p)σμν q( p)|Bs > with the procedure in Refs. [25,26],1 q=s,b q=s,b I is related to the wave functions of Bs∗ and Bs [25], which is I =< Bs | j0( pγ r )|Bs∗ >∼ 1 [27,28]. We can rewrite Eq. (12) as follows: m Bs∗ Here the dimensionless vector–scalar–photon coupling constant gBs Bs∗γ is related to the magnetic moments of b and s quarks; and the phase factor i is consistent with the amplitude of γ ∗ → V P in Ref. [29]. Ultraviolet (UV) logarithmic divergences are observed in the evaluation of loop integrals. In the NLO EW corrections of Bs,d → μ+μ− [12], the UV divergences are canceled by the renormalization of C10. Just as R− value of hadron production in e+e− annihilation, the hadronic contributions will return the quark contributions if the Bs∗ is far away from the mass shell. So that the UV part of loop integral will be suppressed due to avoid the double counting. We introduce a cutoff regularization scheme for the following UV divergence integral: 1 The decay constants defined of vector meson in Eq. (5) is different from Eq. (44) in Ref. [25] with an additional i. qi2 − (mi + q 2j − (m j + where i, j = Bs∗, γ or μ, and qi corresponds to the i momentum in the loop. MW for the amplitude is UV finite when W boson is involved. The hadronic contribution will be suppressed when q 2j−m j QC D, where is approximately several QC D. The cutoff regularization scheme is similar to Pauli–Villars regularization scheme; however, the cutoff regularization scheme acts on two propagators. The Pauli–Villars regularization scheme of the UV divergence integral is the same as the form factor F introduced in the Bs Bs∗γ vertex in Ref. [30], F = qB2s∗ − m2Bs∗ F = qB2s∗ − m2Bs∗ − qB2s∗ − 2 . (16) However, the cutoff regularization scheme acts on the UV divergence term, as well as the two propagators. Then the soft contribution will be maintained in our calculation. The amplitude from Bs → Bs∗γ → μ+μ− can be written as follows: = i eN gBs Bs∗γ R( )C Veff f Bs∗ mμμ¯ γ5μ, where C Veff = C9eff + 2mb/m Bs∗ C7eff and considered as a constant. mμ reappears in the amplitude of the leptonic decay of the scalar mesons. The R( ) factor serves as a function of the high energy cut, as shown in Fig. 2. Detailed information on the R( ) factor is provided in the appendix. Compared with Eq. (9), the previous amplitude added the factor F , F (Bs∗) = M (Bs → Bs∗γ → μ+μ−) M (Bs → μ+μ−) C Veff f Bs∗ egBs Bs∗γ R( ). We can estimate gBs Bs∗γ in several ways, including the heavy-quark and chiral effective theories [31,32] with the radiative and pion transition widths of D∗+, the light cone QCD sum rules [33,34], and the radiative M1 decay widths of Bs∗ → Bs γ from the potential model [27,35]. Fig. 2 R( ) of Bs → μ+μ− defined in Eq. (17) as a function of the cut off energy The heavy-quark and chiral effective theories yield the following expression [13–15]: The light cone QCD sum rules yield the following expression [33,34]: The parameters in the numerical calculation are selected as follows [37]: In the branch fraction of Bs∗,d , the weak decay is less than the M1 decay, tot(Bs∗,d ) ≈ (Bs∗,d → Bs,d γ ). The following ratio is obtained: The radiative M1 decay width of Bs∗ → Bs γ is derived as follows: The predicted M1 widths are 0.15–400 eV and 10–300 eV for Bs∗ → Bs γ and Bd∗ → Bd γ , respectively [24,26,27,35,36]. Recently new predicted M1 widths were given in Ref. [28]: Then we can get the following value: 4 Numerical result (19) (20) (21) The main uncertainty is derived from the f B∗s,d value. The dimuon invariant mass distribution measured by the CMS and LHCb Collaborations in Ref. [2] should include the Bs∗,d → μ+μ− contributions. If (Bs∗ → Bs γ ) = 1 eV [27], we can get BBrr((BBss∗→→μ++μ−−)) = 0.34 for = e, μ. Afterward, Bs∗ → e+e− may be searched by the CMS and LHCb experiments with larger data samples. Moreover, we observe that the amplitude of Bs,d → μ+μ− is modified by the contributions of Bs∗,d with a factor F (Bs∗,d ) = (0.011 ± 0.006) if (Bs∗,d → Bs,d γ ) ∼ 100 eV. The main uncertainty is caused by the value. The new predictions of (Bs,d → μ+μ−) are provided as follows: × 1 + (0.023 ± 0.012) × 1 + (0.023 ± 0.012) × 10−10, × 10−11. If (Bs∗,d → Bs,d γ ) = 200 eV, then this factor will increase the (Bs,d → μ+μ−) decay width by a factor of (3.3 ± 1.7)%, which is approximately a factor 10 times larger than the neglect NLO EW correction factor 0.3% at the decay width in Ref. [1]. In addition, the corresponding gBs,d Bs∗,d γ√= −1.5 about a factor 15 times larger than eq e = −1/3 4π αem = −0.10. (Bs∗,d → Bs,d γ ) may be measured through two-body decay Bs∗,d → J /ψ + M by CMS and LHCb. If (Bs∗,d → Bs,d γ ) = 313, 1230 eV [28], then we can New predictions of (Bs,d → μ+μ−) are provided as follows: Table 2 The branching fractions of Bs,d → μ+μ− measured by the CMS and LHCb Collaborations [2] and updated SM prediction with (Bs∗,d → Bs,d γ ) = 313, 1230 eV [28] Br(Bd →μ+μ−) Br(Bs →μ+μ−) Br ( Bs → μ+μ−) = (3.8 ± 0.3) × 10−9, Br ( Bd → μ+μ−) = (1.2 ± 0.1) × 10−10, The numerical results are shown in Table 2 too. 5 Summary In summary, this study investigated Bs∗,d → μ+μ− in the dimuon distributions and the hadronic contribution Bs,d → Bs∗,d γ → μ+μ−. The μ+μ− decay widths of the vector mesons Bs∗,d are approximately a factor of 700 larger than the corresponding scalar mesons Bs,d . The obtained ( Bs∗,d → the updated branching fractions are (3.8 ± 0.3) × 10−9 and (1.2 ± 0.1) × 10−10 for Bs → μ+μ− and Bd → μ+μ−, respectively. Further studies on Bs∗,d , including those on dielectron decay and two-body decay with J /ψ should be conducted. Acknowledgements We would like to thank K.T. Chao, Y.Q. Ma, C. Meng, Q. Zhao, and S.L. Zhu for useful discussions. Funding was provided by the Program for New Century Excellent Talents in University (Grant No. NCET-13-0030), the Major State Basic Research Development Program of China (Grant No. 2015CB856701), the Fundamental Research Funds for the Central Universities, and the National Natural Science Foundation of China (Grant Nos. 11375021, 11575017). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative The scalar functions B0 and C0 are given in Refs. [38–40]. As a numerical fit between 0.5 − 2 GeV, R( ) is obtained as follows: R( ) = 0.022 + 0.062 × ln (1.06 ± 0.09) × 10−10 (3.66 ± 0.23) × 10−9 (1.2 ± 0.1) × 10−10 (3.8 ± 0.3) × 10−9 0.031 ± 0.0036 Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: Appendix R( ) R( ) of Bs → Bs∗γ obtained as follows: → μ+μ− defined in Eq. (17) is × C0 m Bs 2, m2μ, m2μ, m Bs∗ 2, 0, m2μ + m Bs 2(m2μ − m Bs∗ 2) B0 0, m2μ, m Bs∗ 2 − B0 0, ( + m Bs∗ ) + m Bs∗ ) (3.9+−11..64) × 10−10 (2.8+−00..76) × 10−9 1. C. Bobeth , M. Gorbahn , T. Hermann , M. Misiak , E. Stamou , M. Steinhauser , Bs,d → l+ l− in the Standard Model with Reduced Theoretical Uncertainty . Phys. Rev. Lett . 112 , 101801 ( 2014 ). arXiv:1311.0903 2. LHCb , CMS Collaboration , V. Khachatryan et al., Observation of the rare B0 s → μ+μ− decay from the combined analysis of CMS and LHCb data . Nature 522 , 68 - 72 ( 2015 ). arXiv:1411.4413 3. X.-D. Cheng , Y.-D. Yang , X.-B. 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Guang-Zhi Xu, Yue Qiu, Cheng-Ping Shen, Yu-Jie Zhang. \(B^*_{s,d} \rightarrow \mu ^+ \mu ^-\) and its impact on \(B_{s,d} \rightarrow \mu ^+ \mu ^-\), The European Physical Journal C, 2016, DOI: 10.1140/epjc/s10052-016-4423-z