The Analysis of b → sℓ⁺ℓ⁻ in the Family Non-Universal $Z^{\prime }$ Model

Abstract

Motivated by recent experimental results on $R_{K^{(*)}}$ and $\mathcal {P}^{\prime }_{5}$ in B → K^(∗)ℓ⁺ℓ⁻ and the latest improved measurement result in Λ_b →Λℓ⁺ℓ⁻ processes. We will investigate the rare decay on the flavor-changing neutral current process of Λ_b baryon and B meson in the family non-universal $Z^{\prime }$ model, which is one of the well motivated extensions of the Standard Model. We have obtained the upper limits on the NP coupling parameters from the recent experimental measurements of the decay processes B_s → ℓ⁺ℓ⁻ B → K^(∗)ℓ⁺ℓ⁻, B → X_sℓ⁺ℓ⁻ and Λ_b →Λℓ⁺ℓ⁻. Then we analyze the branching ratio, the normalized forward-backward asymmetries and a series of angular observables in the Standard Model and family non-universal $Z^{\prime }$ model. We find that the constrained NP coupling parameters have few obvious effects on the process b → sμ⁺μ⁻, nevertheless, NP coupling parameters have very large effects on the process b → sτ⁺τ⁻. In the future we expect the precision measurements of these observables will be researched by LHCb and Belle-II.

Appendices

Appendix A: Helicity-Based Form-Factor Parametrization for Λ _b →Λ ℓ ⁺ ℓ ⁻

In the following, we will provide some useful definitions for Λ_b →Λ from factors which aim to improve precious definitions [66]. For the vector currents, we have

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda}\left( k,s_{k}\right)|\bar{s}\gamma^{\mu} b |{\varLambda}_{b}\left( p,s_{p}\right)\rangle &=& \bar{u}(k,s_{k})\left[f_{0}\left( q^{2}\right)\left( m_{{\varLambda}_{b}}-m_{{\varLambda}}\right)\frac{q^{\mu}}{q^{2}}\right.\\ && + f_{+}\left( q^{2}\right) \frac{m_{{\varLambda}_{b}}+m_{{\varLambda}}}{s_{+}} \left\{p^{\mu} + k^{\mu} - \frac{q^{\mu}}{q^{2}}\left( m_{{\varLambda}_{b}}-m_{{\varLambda}}\right) \right\}\\ &&\left.+ f_{\perp}\left( q^{2}\right) \left\{ \gamma^{\mu} - \frac{2m_{{\varLambda}}}{s_{+}}p^{\mu} - \frac{2m_{{\varLambda}_{b}}}{s_{+}}k^{\mu} \right\} \right]u(p,s_{p}) , \end{array} $$

(29)

where the variables s_± are defined as $s_{\pm }=(m_{{\varLambda }_{b}} \pm m_{{\varLambda }})^{2}-q^{2}$

For the axial-vector currents we get

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda}(k,s_{k})|\bar{s}\gamma^{\mu}\gamma_{5} b |{\varLambda}_{b}\left( p,s_{p}\right)\rangle &=& - \bar{u}(k,s_{k}) \gamma_{5} \left[ g_{0}\left( q^{2}\right) \left( m_{{\varLambda}_{b}}+m_{{\varLambda}}\right) \frac{q^{\mu}}{q^{2}} \right.\\ && + g_{+}\left( q^{2}\right) \frac{m_{{\varLambda}_{b}}-m_{{\varLambda}}}{s_{-}} \left\{p^{\mu} + k^{\mu} - \frac{q^{\mu}}{q^{2}} \left( m_{{\varLambda}_{b}}^{2}-m_{{\varLambda}}\right)^{2} \right\} \\ &&\left. + g_{\perp}\left( q^{2}\right) \left\{\gamma^{\mu} + \frac{2m_{{\varLambda}}}{s_{-}}p^{\mu} - \frac{2m_{{\varLambda}_{b}}}{s_{-}}k^{\mu} \right\} \right] u\left( p,s_{p}\right) . \end{array} $$

(30)

From (29) and (30), the matrix elements for the scalar and the pseudo-scalar currents can be written

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda}(k,s_{k})|\bar{s} b |{\varLambda}_{b}(p,s_{p})\rangle =& f_{0}\left( q^{2}\right) \frac{m_{{\varLambda}_{b}}-m_{{\varLambda}}}{m_{b}-m_{s}} \bar{u}(k,s_{k})u\left( p,s_{p}\right) , \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda}(k,s_{k})|\bar{s} \gamma_{5} b |{\varLambda}_{b}(p,s_{p})\rangle =& g_{0}\left( q^{2}\right) \frac{m_{{\varLambda}_{b}}+m_{{\varLambda}}}{m_{b}+m_{s}} \bar{u}(k,s_{k}) \gamma_{5} u\left( p,s_{p}\right) , \end{array} $$

(32)

where we do not neglect the mass of strange quark in the denominator. For the dipole operators we have

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda} |\bar{s}i q_{\nu}\sigma^{\mu\nu}b|{\varLambda}_{b}\rangle &=& -\bar{u}(k,s_{k})\left[ h_{+}\left( q^{2}\right) \frac{q^{2}}{s_{+}}\left( p^{\mu} + k^{\mu} - \frac{q^{\mu}}{q^{2}}\left( {m}_{{\varLambda}_{b}}^{2} - {m}_{{\varLambda}}^{2}\right) \right) \right.\\ &&\left.+ h_{\perp} (m_{{\varLambda}_{b}} + m_{{\varLambda}}) \left( \gamma^{\mu} - \frac{2m_{{\varLambda}}}{s_{+}}p^{\mu} - \frac{2m_{{\varLambda}_{b}}}{s_{+}}k^{\mu} \right) \right]u\left( p,s_{p}\right), \end{array} $$

(33)

and

$$ \begin{array}{@{}rcl@{}} \langle {\varLambda} |\bar{s}i q_{\nu}\sigma^{\mu\nu}\gamma_{5} b|{\varLambda}_{b}\rangle &=& -\bar{u}(k,s_{k})\gamma_{5} \left[\tilde h_{+} \frac{q^{2}}{s_{-}} \left( p^{\mu} + k^{\mu} - \frac{q^{\mu}}{q^{2}}\left( m^{2}_{{\varLambda}_{b}}-m^{2}_{{\varLambda}}\right) \right)\right. \\ &&\left.+ \tilde h_{\perp} \left( m_{{\varLambda}_{b}} - m_{{\varLambda}}\right)\!\left( \!\gamma^{\mu} + \frac{2m_{{\varLambda}}}{s_{-}}p^{\mu} - \frac{2m_{{\varLambda}_{b}}}{s_{-}}k^{\mu} \!\right)\!\right] u\left( p,s_{p}\right) \end{array} $$

(34)

Significantly, our notations of these form factors are same with Refs. [59, 67] and we slightly changed notation compared to Refs. [37, 39].

Appendix B: Transverse Amplitudes for Λ _b →Λ ℓ ⁺ ℓ ⁻

The definitions and the spinor matrix elements are worked out in Appendixes in Refs. [37, 38]. For the VA operators the non-vanishing hadronic helicity amplitudes are gotten

$$ \begin{array}{@{}rcl@{}} {H}_{\text{VA,0}}^{L(R),+\frac{1}{2},+\frac{1}{2}}&=&{f}_{0}^{V}(m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{\frac{s_{-}}{q^{2}}} \mathcal{C}_{\text{VA}}^{L(R)} {-} {f}_{0}^{A}(m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{\frac{s_{+}}{q^{2}}} \mathcal{C}_{\text{VA}}^{L(R)} \\ &+& \frac{2m_{b}}{q^{2}} \left( {f}_{0}^{T} \sqrt{q^{2} s_{-}} {-} {f}_{0}^{T5} \sqrt{q^{2} s_{+}} \right) \mathcal{C}_{7}^{\text{eff}}, \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} {H}_{\text{VA,0}}^{L(R),-\frac{1}{2},-\frac{1}{2}}&=&{f}_{0}^{V}(m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{\frac{s_{-}}{q^{2}}} \mathcal{C}_{\text{VA}}^{L(R)} {+} {f}_{0}^{A}(m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{\frac{s_{+}}{q^{2}}} \mathcal{C}_{\text{VA}}^{L(R)} \\ &+& \frac{2m_{b}}{q^{2}} \left( {f}_{0}^{T} \sqrt{q^{2} s_{-}} {+} {f}_{0}^{T5} \sqrt{q^{2} s_{+}} \right) \mathcal{C}_{7}^{\text{eff}}, \end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} {H}_{\text{VA,+}}^{L(R),-\frac{1}{2},+\frac{1}{2}} &=&{-f}_{\perp}^{V} \sqrt{2s_{-}} \mathcal{C}_{\text{VA}}^{L(R)} {+} {f}_{\perp}^{A} \sqrt{2s_{+}} \mathcal{C}_{\text{VA}}^{L(R)}\\ &-&\frac{2m_{b}}{q^{2}} \left( {f}_{\perp}^{T}(m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{2s_{-}} {-} {f}_{\perp}^{T5} (m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{2s_{+}} \right)\mathcal{C}_{7}^{\text{eff}}, \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} {H}_{\text{VA,-}}^{L(R),+\frac{1}{2},-\frac{1}{2}} &=& -{f}_{\perp}^{V} \sqrt{2s_{-}} \mathcal{C}^{L(R)}_{\text{VA}} {-} f^{A}_{\perp} \sqrt{2s_{+}} \mathcal{C}^{L(R)}_{\text{VA}} \\ &-&\frac{2m_{b}}{q^{2}} \left( f^{T}_{\perp}(m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{2s_{-}} {+} f^{T5}_{\perp} (m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{2s_{+}} \right)\mathcal{C}_{7}^{\text{eff}}, \end{array} $$

(38)

$$ \begin{array}{@{}rcl@{}} {H}_{t}^{+\frac{1}{2},-\frac{1}{2}}&=&~~ 2\left( {f_{t}^{A}} (m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{\frac{s_{-}}{q^{2}}} {-} {f_{t}^{V}} (m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{\frac{s_{+}}{q^{2}}} \right) \mathcal{C}_{10}^{\text{tot}}, \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} {H}_{t}^{+\frac{1}{2},-\frac{1}{2}}&=&{-} 2\left( {f_{t}^{A}} (m_{{\varLambda}_{b}} + m_{{\varLambda}}) \sqrt{\frac{s_{-}}{q^{2}}} {+} {f_{t}^{V}} (m_{{\varLambda}_{b}} - m_{{\varLambda}}) \sqrt{\frac{s_{+}}{q^{2}}} \right) \mathcal{C}_{10}^{\text{tot}}, \end{array} $$

(40)

where $\mathcal {C}_{\text {VA}}^{L(R)}=\mathcal {C}_{9}^{\text {tot}} \mp \mathcal {C}_{10}^{\text {tot}}$. It is clearly to find that our result about $\mathcal {C}_{\text {VA}}^{L(R)}$ is same with the $\mathcal {C}_{\text {VA,+(-)}}^{L(R)}$ listed in the (4.17—4.18) of Ref. [37] when $\mathcal {C}_{\mathrm {V}}^{(^{\prime })}=\mathcal {C}_{\mathrm {A}}^{(^{\prime })}=0$.

Using the representations of the lepton spinors which are given in Appendix of Ref. [37]. The author derived the expression of $L^{\lambda _{1},\lambda _{2}}_{L(R)}$ and $L^{\lambda _{1},\lambda _{2}}_{L(R),\lambda }$ which are shown in Eq. (4.25) of Ref. [37] for the limit m_ℓ = 0. Similar to the steps in Ref. [37], when m_ℓ≠ 0, we obtain the non-zero results of the leptonic helicity amplitudes for different λ₁ and λ₂.

$$ \begin{array}{@{}rcl@{}} {L}_{L,+1}^{+\frac{1}{2},+\frac{1}{2}}&=& L_{R,+1}^{+\frac{1}{2},+\frac{1}{2}}= {L}_{L,-1}^{-\frac{1}{2},-\frac{1}{2}}=L_{R,-1}^{-\frac{1}{2},-\frac{1}{2}}=\sqrt{2}m_{\ell}\sin{\theta_{\ell}}, \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} L_{L,+1}^{-\frac{1}{2},-\frac{1}{2}}&=& {L}_{R,+1}^{-\frac{1}{2},-\frac{1}{2}}= L_{L,-1}^{+\frac{1}{2},+\frac{1}{2}}= {L}_{R,-1}^{+\frac{1}{2},+\frac{1}{2}}=-\sqrt{2}m_{\ell}\sin{\theta_{\ell}}, \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} {L}_{L,+1}^{+\frac{1}{2},-\frac{1}{2}}&=&{-L}_{R,-1}^{-\frac{1}{2},+\frac{1}{2}}=-\sqrt{\frac{q^{2}}{2}}(1-{\beta}_{\ell})(1-\cos{\theta_{\ell}}), \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} {L}_{L,+1}^{-\frac{1}{2},+\frac{1}{2}}&=&{-L}_{R,-1}^{+\frac{1}{2},-\frac{1}{2}}=\sqrt{\frac{q^{2}}{2}}(1+{\beta}_{\ell})(1+\cos{\theta_{\ell}}), \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} {L}_{R,+1}^{+\frac{1}{2},-\frac{1}{2}}&=&{-L}_{L,-1}^{-\frac{1}{2},+\frac{1}{2}}=-\sqrt{\frac{q^{2}}{2}}(1+{\beta}_{\ell})(1-\cos{\theta_{\ell}}), \end{array} $$

(45)

$$ \begin{array}{@{}rcl@{}} {L}_{R,+1}^{-\frac{1}{2},+\frac{1}{2}}&=&{-L}_{L,-1}^{+\frac{1}{2},-\frac{1}{2}}=\sqrt{\frac{q^{2}}{2}}(1-{\beta}_{\ell})(1+\cos{\theta_{\ell}}), \end{array} $$

(46)

$$ \begin{array}{@{}rcl@{}} -L_{L,0}^{+\frac{1}{2},+\frac{1}{2}}&=&{L}_{L,0}^{-\frac{1}{2},-\frac{1}{2}}=-L_{R,0}^{+\frac{1}{2},+\frac{1}{2}}=L_{R,0}^{-\frac{1}{2},-\frac{1}{2}}=2m_{\ell}\cos{\theta_{\ell}}, \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} L_{L,0}^{+\frac{1}{2},-\frac{1}{2}}&=&{L}_{R,0}^{-\frac{1}{2},+\frac{1}{2}}=\sqrt{q^{2}}(1-{\beta}_{\ell})\sin{\theta_{\ell}}, ~{L}_{L,0}^{-\frac{1}{2},+\frac{1}{2}}={L}_{R,0}^{+\frac{1}{2},-\frac{1}{2}}=\sqrt{q^{2}}(1+{\beta}_{\ell})\sin{\theta_{\ell}}, \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} {L}_{L,t}^{+\frac{1}{2},+\frac{1}{2}}&=&{L}_{L,t}^{-\frac{1}{2},-\frac{1}{2}}=-L_{R,t}^{+\frac{1}{2},+\frac{1}{2}}=-L_{R,t}^{-\frac{1}{2},-\frac{1}{2}}=2m_{\ell}. \end{array} $$

(49)

The leptonic helicity amplitudes which are not shown are zero for other combinations of λ₁ and λ₂.

Appendix C: Constraints on $Z^{\prime }$ Couplings for B _s → ℓ ⁺ ℓ ⁻ and B → X _s ℓ ⁺ ℓ ⁻

The full expression for the branching ratio of ${B_{s}^{0}}\to \ell ^{+}\ell ^{-}$, due to the non-universal Z’ couplings, can be written [36]

$$ \begin{array}{@{}rcl@{}} \mathcal{B}({{B}_{s}^{0}}\to\ell^{+}\ell^{-}) &=&\tau_{B_{s}}\frac{{{G}_{F}^{2}}}{4\pi}{f}_{{B}_{s}}^{2}m_{\ell}^{2} m_{B_{s}}\sqrt{1-\frac{4m_{\ell}^{2}}{m_{B_{s}}^{2}}}|V_{tb}V_{ts^{*}}|^{2} \\ &&\times \left|\frac{\alpha}{2\pi \sin^{2}\theta_{W}}Y(x_{t})-2\frac{{B}_{sb}^{L} (B_{\ell\ell}^{L}-{B}_{\ell\ell}^{R})}{V_{tb}V_{ts^{*}}}\right|^{2}. \end{array} $$

(50)

For the process b → sℓ⁺ℓ⁻, introducing the normalized dilepton invariant mass $\hat s=(p_{l^{+}}+p_{l^{-}})^{2}/{{m}_{b}^{2}}$, the differential decay rate is shown

$$ \begin{array}{@{}rcl@{}} R(\hat s) \equiv \frac{\frac{d}{d\hat s}\mathcal{B}(b\to s \ell^{+}\ell^{-})}{\mathcal{B}(b\to c \ell^{-}\bar{\nu})} =\frac{\alpha^{2}}{4\pi}\frac{|V_{ts}^{*}{V}_{tb}|^{2}}{|V_{cb}|^{2}}\frac{(1-\hat s)^{2}}{f((\chi))\kappa(\chi)}\sqrt{1-\frac{4t^{2}}{\hat{s}}}D(\hat s), \end{array} $$

(51)

with

$$ \begin{array}{@{}rcl@{}} D(\hat{s})&=&(1+2\hat{s})\left( 1+\frac{2t^2}{\hat{s}}\right)\left|{C}_9^{\text{tot}}\right|^2 + 4\left( 1+\frac{2}{\hat{s}}\right)\left( 1+\frac{2t^2}{\hat{s}}\right)|{C}_7|^2\\ &&+ \left[(1+2\hat{s})+\frac{2t^2}{\hat{s}}(1-4\hat{s})\right]\left| {C}_{10}^{\text{tot}}\right|^2+12\left( 1+\frac{2t^2}{\hat{s}}\right){C}_7 {\text{Re}}\left( {C}_9^{\text{tot}\ast}\right) , \end{array} $$

where t = m_l/m_b, χ = m_c/m_b and ${\mathscr{B}}(B\to X_{c} l^{-}\nu _{l})=(10.65\pm 0.16)\%$ ¹. The phase-space factor f(χ) and the 1-loop QCD correction factor κ(χ) for the process B → X_cl⁻ν_l are given in Refs. [36, 68]

$$ \begin{array}{@{}rcl@{}} f(\chi)&=&1-8\chi^{2}+8\chi^{6}-\chi^{8}-24\chi^{4}\ln\chi , \\ \kappa(\chi)&=&1-\frac{2\alpha_{s}(\mu)}{3\pi}\left[\left( \pi^{2}-\frac{31}{4}\right) (1-\chi)^{2}+\frac{3}{2}\right] . \end{array} $$

The normalized forward-backward (FB) asymmetry and CP-violation for B → X_cl⁻ν_l can be parameterized as

$$ \begin{array}{@{}rcl@{}} A_{FB}({s})=\frac{{{\int}_{0}^{1}}dz\frac{d^{2}{\varGamma}}{d\hat{s}dz} -{\int}_{-1}^{0}dz\frac{d^{2}{\varGamma}}{d\hat{s}dz}}{{{\int}_{0}^{1}}dz\frac{d^{2}{\varGamma}} {d\hat{s}dz}+{\int}_{-1}^{0}dz\frac{d^{2}{\varGamma}}{d\hat{s}dz}} =-3\sqrt{1-\frac{4t^{2}}{\hat{s}}}\frac{E(\hat{s})}{D(\hat{s})}, \end{array} $$

(52)

with $E(\hat {s})={\text {Re}}({C}_9^{\text {tot}} {C}_{10}^{\text {tot}\ast }\hat {s}+2{C}_7^{\text {eff}}{C}_{10}^{\text {tot} \ast })$.

$$ \begin{array}{@{}rcl@{}} A_{CP}(s)&=&\frac{d\mathcal{B}/ds(B\to X_s\ell^+\ell^-)-d\mathcal{B}/ds(\bar{B}\to \bar{X_s}\ell^+\ell^-)}{{d\mathcal{B}/ds(B\to X_s\ell^+\ell^-)+d\mathcal{B}/ds(\bar{B}\to \bar{X_s}\ell^+\ell^-)}}. \end{array} $$

(53)