Scalar Leptoquark Effects in the Lepton Flavor Violating Exclusive $b \to s {\ell }_{i}^{-} {\ell }_{j}^{+}$ Decays

Abstract

Leptoquarks have been suggested to solve a variety of discrepancies between the expected and observed phenomenon. In this paper, we investigate the effects of scalar leptoquarks on the lepton flavor violating B meson rare decays which involve the quark level transition $b \to s {\ell }_{i}^{-} {\ell }_{j}^{+} (i \neq j)$. The leptoquark parameter spaces are constrained by using the recently measured upper limits on $\mathcal {B}({B}_{s}^{0} \to {\ell }_{i}^{-} {\ell }_{j}^{+})$ and $\mathcal {B}(B \to K^{(*)} {\ell }_{i}^{-} {\ell }_{j}^{+})$. Using such constrained leptoquark parameter spaces, some relevant physical quantities are predicted and we find that the constrained new physics parameters in the leptoquark model have very obvious effects on the relevant physical quantities. With future measurements of observables in $B \to K^{(*)} {\ell }_{i}^{-} {\ell }_{j}^{+}$ decays at the LHCb, more and more differentiated from the other new physics explanations could be tested.

Appendix

From (2) and (4) in Section 2, we get the following expressions about the square matrix element for B → K^(∗)ℓi−ℓj+ processes

$$\begin{array}{@{}rcl@{}} |{\mathcal{ M}}(B \to K^{(*)} \ell^{-}_{i} \ell^{+}_{j})|^{2}&=& |G_{LQ}|^{2} |\langle K^{(*)}|\bar{s}\gamma^{\mu}(1+\gamma_{5}){b}|{B} \rangle \bar{\ell_{i}}\gamma_{\mu}(1-\gamma_{5})\ell_{j}|^{2}\\ &\equiv& |G_{LQ}|^{2} L_{\mu\nu}H^{\mu\nu}, \end{array} $$

(14)

with G_LQ is $G_{LQ}^{7/6}$ and $G_{LQ}^{1/6}$ for model A and model B, respectively.

1.1 A: Formulae of the $B \to K \ell _{i}^{-} \ell _{j}^{+}$ Decays

Because the axial-vector current matrix elements $\langle K(p_K)|\bar s \gamma ^{\mu } \gamma _5 b| B(p_B)\rangle = 0$, the hadronic helicity amplitudes are same with each other in model A and model B. The non-vanishing helicity amplitudes $H_0(q^2)=\frac {2 m_B |\vec {\textbf {p}}|}{\sqrt {q^2}}f_+(q^2)$ and $H_{t}(q^2) =\frac {m_B^2 - m_K^2}{\sqrt {q^2}} f_0(q^2)$ with $|\vec {\textbf {p}}|=\frac {\sqrt {\lambda (m^2_B,m_K^2, s)}}{2 m_B}$. The form factors f₊(q²) and f₀(q²) in H_0(t)(q²) are taken from Refs. [43,44,45].

Using the similar method in Refs. [49, 50], we obtain the lepton helicity amplitudes in the LQ model

$$\begin{array}{@{}rcl@{}} &&h^{7}_{\mp\frac{1}{2},\mp\frac{1}{2}}=\bar{u}_{\ell_{i}}\left( \pm{\frac{1}{2}}\right)\gamma^{\mu}(1+\gamma_{5})v_{\ell_{j}}\left( \pm{\frac{1}{2}}\right) \left\{{\epsilon_{\mu} (\pm1) \atop \epsilon_{\mu} (t),\epsilon_{\mu} (0)} \right\}\ , \\ &&h^{1}_{\mp\frac{1}{2},\mp\frac{1}{2}}=\bar{u}_{\ell_{i}}\left( \pm{\frac{1}{2}}\right)\gamma^{\mu}(1-\gamma_{5})v_{\ell_{j}}\left( \pm{\frac{1}{2}}\right) \left\{{\epsilon_{\mu} (\pm1) \atop \epsilon_{\mu} (t),\epsilon_{\mu} (0)} \right\}\ , \end{array} $$

(15)

where the superscripts of $h^{7,1}_{\lambda _{\ell _i},\lambda _{\ell _j}}$ represent model A X = (3, 2, 7/6) and model B X = (3, 2, 1/6). In model B, the lepton helicity amplitudes are same with h¹ listed in Eq. (28) due to squark exchange in RPV SUSY model [39]. In model A, the lepton helicity amplitudes are

$$\begin{array}{@{}rcl@{}} |h^{7}_{~~\frac{1}{2},~~\frac{1}{2}}|^{2} &=& \frac{1}{s}\left[s^{2}-\left( m^{2}_{\ell_{j}}-m^{2}_{\ell_{i}}-\sqrt{\lambda_{L}}\right)^{2}\right], \\ |h^{7}_{~~\frac{1}{2},-\frac{1}{2}}|^{2} &=& 4\left( s-m^{2}_{\ell_{i}}-m^{2}_{\ell_{j}}+\sqrt{\lambda_{L}}\right), \\ |h^{7}_{-\frac{1}{2},~~\frac{1}{2}}|^{2} &=& 4\left( s-m^{2}_{\ell_{i}}-m^{2}_{\ell_{j}}-\sqrt{\lambda_{L}}\right), \\ |h^{7}_{-\frac{1}{2},-\frac{1}{2}}|^{2} &=&\frac{1}{s}\left[s^{2}-\left( m^{2}_{\ell_{i}}-m^{2}_{\ell_{j}}-\sqrt{\lambda_{L}}\right)^{2}\right], \end{array} $$

(16)

with $\lambda _L\equiv \lambda (m^2_{\ell _i},m^2_{\ell _j},s)$.

For B → Kℓi−ℓj+ processes, the double differential decay rates with $\lambda _{\ell _{i,j}}$ can be represented as

$$\begin{array}{@{}rcl@{}} \frac{d^{2} {\Gamma}^{K}[\lambda_{\ell_{i}} = \frac{1}{2},\lambda_{\ell_{j}} = \frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{|h_{\frac{1}{2},\frac{1}{2}}|^{2} |H_{t}(q^{2})- H_{0}(q^{2})\cos\theta |^{2} \right\}, \\ \frac{d^{2} {\Gamma}^{K}[\lambda_{\ell_{i}} =-\frac{1}{2},\lambda_{\ell_{j}} = \frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ \frac{1}{2}|h_{-\frac{1}{2},\frac{1}{2}}|^{2} |H_{0}(q^{2})|^{2} \sin^{2}\theta\right\}, \\ \frac{d^{2} {\Gamma}^{K}[\lambda_{\ell_{i}} =\frac{1}{2},\lambda_{\ell_{j}} =-\frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ \frac{1}{2} |h_{\frac{1}{2},-\frac{1}{2}}|^{2} |H_{0}(q^{2})|^{2} \sin^{2}\theta \right\}, \\ \frac{d^{2} {\Gamma}^{K}[\lambda_{\ell_{i}} = -\frac{1}{2},\lambda_{\ell_{j}} = -\frac{1}{2}]}{ds d \cos\theta}\! &=&\! \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ |h_{-\frac{1}{2},-\frac{1}{2}}|^{2} |H_{t}(q^{2}) - H_{0}(q^{2})\cos\theta |^{2} \right\},\\ \end{array} $$

(17)

where h_{m, n} is $h_{m,n}^7$ and $h_{m,n}^1$ (G_LQ is $G_{LQ}^{\frac {1}{6}}$ and $G_{LQ}^{\frac {7}{6}}$) for model A and model B, respectively.

1.2 B: Formulae of the $B \to K^{*} \ell _{i}^{-} \ell _{j}^{+}$ Decays

Unlike the process B → K, the hadronic helicity amplitudes of B → K^∗ are different from each other between model A and model B, since the B → K^∗ matrix element for the axial-vector current don’t vanishes, i.e., $\langle K^{*}(p_{K^{*}})|\bar s \gamma ^{\mu } \gamma _5 b| B(p_B)\rangle \neq 0$. In model B, the hadronic helicity amplitudes are same with H¹(q²) in Ref. [39]. In model A, the helicity amplitudes can be written as

$$\begin{array}{@{}rcl@{}} H^{7}_{\pm \pm}(q^{2}) &=&-(m_{B} + m_{K^{*}}) A_{1}(q^{2}) \mp \frac{2 m_{B}}{m_{B} + m_{K^{*}}} |\vec{\textbf{p}}| V(q^{2}), \\ H^{7}_{00}(q^{2}) &=&-\frac{1}{2 m_{K^{*}} \sqrt{q^{2}}} \left[({m_{B}^{2}} - m_{K^{*}}^{2} -q^{2}) (m_{B} + m_{K^{*}}) A_{1}(q^{2}) - \frac{4 {m_{B}^{2}} |\vec{\textbf{p}}|^{2}}{m_{B} + m_{K^{*}}} A_{2}(q^{2}) \right], \\ H^{7}_{0t}(q^{2}) &=&-\frac{2 m_{B} |\vec{\textbf{p}}|}{\sqrt{q^{2}}} A_{0}(q^{2}), \end{array} $$

(18)

with $|\vec {\textbf {p}}|=\frac {\sqrt {\lambda (m^2_B,m_{K^{*}}^2,s)}}{2 m_B}$. Noted that the hadronic helicity amplitudes in model A and model B are opposite.

For B → K^∗ℓi−ℓj+ processes, the double differential decay rates with $\lambda _{\ell _{i,j}}$ can be represented as

$$\begin{array}{@{}rcl@{}} \frac{d^{2} {\Gamma}^{K^{*}}[\lambda_{\ell_{i}} = \frac{1}{2} ,\lambda_{\ell_{j}} = \frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ |h_{\frac{1}{2},\frac{1}{2}}|^{2} \left[|H_{0t}- H_{00}\cos\theta|^{2}\right.\right.\\ && \left.\left. +\frac{1}{2}|H_{++}|^{2}\sin^{2}\theta + \frac{1}{2}|H_{--}|^{2}\sin^{2}\theta \right] \right\}, \\ \frac{d^{2} {\Gamma}^{K^{*}}[\lambda_{\ell_{i}} =-\frac{1}{2},\lambda_{\ell_{j}} = \frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ |h_{-\frac{1}{2},\frac{1}{2}}|^{2} \left[\frac{1}{2} |H_{00}|^{2}\sin^{2}\theta\right.\right.\\ && \left.\left. + \frac{1}{4}|H_{++}|^{2}(1-\cos\theta)^{2} + \frac{1}{4}|H_{--}|^{2}(1+\cos\theta)^{2} \right] \right\}, \\ \frac{d^{2} {\Gamma}^{K^{*}}[\lambda_{\ell_{i}} =\frac{1}{2},\lambda_{\ell_{j}} =-\frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ |h_{\frac{1}{2},-\frac{1}{2}}|^{2} \left[\frac{1}{2} |H_{00}|^{2} \sin^{2}\theta\right.\right.\\ && \left.\left. + \frac{1}{4}|H_{++}|^{2}(1+\cos\theta)^{2} + \frac{1}{4}|H_{--}|^{2}(1-\cos\theta)^{2} \right] \right\}, \\ \frac{d^{2} {\Gamma}^{K^{*}}[\lambda_{\ell_{i}} =-\frac{1}{2},\lambda_{\ell_{j}} =-\frac{1}{2}]}{ds d \cos\theta} &=& \frac{u(s)|G_{LQ}|^{2}}{2^{9} \pi^{3} {m^{3}_{B}} s} \left\{ |h_{-\frac{1}{2},-\frac{1}{2}}|^{2} \left[|H_{0t}- H_{00}\cos\theta|^{2}\right.\right. \\ && \left.\left. + \frac{1}{2}|H_{++}|^{2}\sin^{2}\theta + \frac{1}{2}|H_{--}|^{2}\sin^{2}\theta \right] \right\}. \end{array} $$

(19)