Implication of Z-mediated FCNC on semileptonic decays B_s→φl⁺l^–and B⁺→K⁺l⁺l^–

Abstract

Rare B meson decays mediated by flavour changing neutral current (FCNC) transition play interesting role to probe the flavour sector of the standard model (SM). Generally at the tree level, FCNC processes are not allowed in the SM but occurs at the loop levels. This gives an excellent hunting ground for new physics (NP). From various experimental studies it is found that the FCNC processes having quark level transition b → s are challenging. Here, we investigate different kinematic observables like forward-backward asymmetry, differential branching ratio and lepton polarization asymmetry for semileptonic rare B decay modes B_s → φl⁺l⁻ and B⁺ → K⁺l⁺l⁻ (l = μ, τ) considering the contribution of Z-mediated FCNC. A noticeable deviation of the observables for these decay channels from the SM value is found because of non-universal Z − bs coupling.

Appendices

Appendix 1

Table 3 Input parameters for form factor calculations

Appendix 2

1.1 Form factors for B_s → φ transition

The form factors of B_s → φ transition are given as [24, 102].

$$ g\left({q}^2\right)=\frac{V\left({q}^2\right)}{M_B+{M}_{\varphi }} $$

(70)

$$ f\left({q}^2\right)=\left({M}_B+{M}_{\varphi}\right){A}_1\left({q}^2\right) $$

(71)

$$ {a}_{+}\left({q}^2\right)=\frac{A_2\left({q}^2\right)}{M_B+{M}_{\varphi }} $$

(72)

$$ {a}_{-}\left({q}^2\right)=-\frac{2{M}_{\varphi }}{q^2}\left({A}_3\left({q}^2\right)-{A}_0\left({q}^2\right)\right) $$

(73)

$$ \kern0.75em {g}_{+}\left({q}^2\right)=-{T}_1\left({q}^2\right) $$

(74)

$$ \kern0.5em {g}_{-}\left({q}^2\right)=-\frac{{M_B}^2-{M_{\varphi}}^2}{q^2}\left({T}_1\left({q}^2\right)-{T}_2\left({q}^2\right)\right) $$

(75)

$$ {g}_0\left({q}^2\right)=-\frac{2}{q^2}\left({T}_2\left({q}^2\right)-{T}_1\left({q}^2\right)+\frac{q^2}{{M_B}^2-{M_{\varphi}}^2}{T}_3\left({q}^2\right)\right) $$

(76)

The definition of V(q²), A₀(q²) and T₁(q²) can be parameterized as

$$ \kern2.75em F\left({q}^2\right)=\frac{r_1}{1-{q}^2/{m}_R^2}+\frac{r_2}{1-{q}^2/{m}_{fit}^2} $$

(77)

Similarly A₂(q²), $\overset{\sim }{T_3}$ and A₁(q²), T₂(q²) are defined by (B9) and (B10) respectively

$$ \kern2.75em F\left({q}^2\right)=\frac{r_1}{1-{q}^2/{m}^2}+\frac{r_2}{{\left(1-{q}^2/{m}^2\right)}^2} $$

(78)

$$ \kern2.75em F\left({q}^2\right)=\frac{r_2}{1-{q}^2/{m}_{fit}^2} $$

(79)

T₃(q²) and A₃(q²) can be parameterized as

$$ \kern3.25em {T}_3\left({q}^2\right)=\frac{{M_B}^2-{M_{\varphi}}^2}{q^2}\left(\overset{\sim }{T_3}\left({q}^2\right)-{T}_2\left({q}^2\right)\right) $$

(80)

$$ \kern4em {A}_3\left({q}^2\right)=\frac{M_B+{M}_{\varphi }}{2{M}_{\varphi }}{A}_1\left({q}^2\right)-\frac{M_B-{M}_{\varphi }}{2{M}_{\varphi }}{A}_2\left({q}^2\right) $$

(81)

Appendix 3

1.1 Form factors for B⁺ → K⁺ transition

The form factors defined for B⁺ → K⁺ transition are represented as follows [103].

$$ {f}_{-}\left({q}^2\right)=\frac{{M_B}^2-{M_{K^{+}}}^2}{q^2}\left({f}_0\left({q}^2\right)-{f}_{+}\left({q}^2\right)\right) $$

(82)

$$ t\left({q}^2\right)=\frac{f_T\left({q}^2\right)}{M_B+{M}_{K^{+}}}. $$

(83)

Here f₊(q²), f_T(q²) and f₀(q²) and can be expressed by following functions which are represented by eqs. (C3) and (C4)

$$ {f}_{+(T)}\left({q}^2\right)=\frac{r_1}{1-{q}^2/{m}_1^2}+\frac{r_2}{{\left(1-{q}^2/{m}_1^2\right)}^2} $$

(84)

$$ {f}_0\left({q}^2\right)=\frac{r_2}{1-{q}^2/{m}_{fit}^2}.\kern0.5em $$

(85)

In the above equations, the parameters used in the fit functions of the form factors are taken from [24, 101] and encapsulated in the following table.