Search for Lepton Flavor Non-universality with B→ (K, K*) τ⁺τ^- Decays

Abstract

The decay modes of B meson consisting tau leptons as final state particle are seemed to be the key feature for searching new physics. As the experimental measurements for b → sτ⁺τ⁻ processes are still not confirmed, the theoretical study of these decays is playing vital role in recent times. The lepton flavor universality violation gains much curiosity after the observation of R_K and ${R}_{K^{\ast }}$ anomalies in semileptonic $B\to \left(K,{K}^{\ast}\right)\mu \overline{\mu}$ channel which provides a hint towards μ − e non-universality in flavor sector. In this paper, we have devoted our concern to the decay mode B → (K, K^∗)τ⁺τ⁻ to predict the branching ratio. We have also searched for the τ − l (where l = e, μ) lepton flavor non-universality for neutral current transitions. We have estimated our results in the standard model as well as in Z′ model.

Appendices

Appendix A

The basis operators for the semileptonic decays B → (K, K^∗)τ⁺τ⁻ are given as

$$ {\displaystyle \begin{array}{l}{O}_1=\left({\overline{s}}_{\alpha }{\gamma}^{\mu}\left(1-{\gamma}_5\right){b}_{\alpha}\right)\left({\overline{c}}_{\beta }{\gamma}_{\mu}\left(1-{\gamma}_5\right){c}_{\beta}\right),\\ {}{O}_2=\left({\overline{s}}_{\alpha }{\gamma}^{\mu}\left(1-{\gamma}_5\right){b}_{\beta}\right)\left({\overline{c}}_{\beta }{\gamma}_{\mu}\left(1-{\gamma}_5\right){c}_{\alpha}\right),\\ {}{O}_{7\gamma }=\frac{e}{8{\pi}^2}{\overline{s}}_{\alpha }{\sigma}_{\mu \nu}\left[{m}_b\left(\mu \right)\left(1+{\gamma}_5\right)+{m}_s\left(\mu \right)\left(1-{\gamma}_5\right)\right]{b}_{\alpha }{F}^{\mu \nu},\\ {}{O}_{9V}=\frac{e}{8{\pi}^2}\left({\overline{s}}_{\alpha }{\gamma}^{\mu}\left(1-{\gamma}_5\right){b}_{\alpha}\right)\overline{\tau}{\gamma}_{\mu}\tau, \\ {}{O}_{10A}=\frac{e}{8{\pi}^2}\left({\overline{s}}_{\alpha }{\gamma}^{\mu}\left(1-{\gamma}_5\right){b}_{\alpha}\right)\overline{\tau}{\gamma}_{\mu }{\gamma}_5\tau .\end{array}} $$

The amplitudes for meson decays could be written as

$$ {\displaystyle \begin{array}{l}P\left({M}_2,{p}_2\right)\mid {V}_{\mu }(0)\mid P\left({M}_1,{p}_1\right)={f}_{+}\left({q}^2\right){P}_{\mu }+{f}_{-}\left({q}^2\right){q}_{\mu },\\ {}V\left({M}_2,{p}_2,\epsilon \right)\mid {V}_{\mu }(0)\mid P\left({M}_1,{p}_1\right)=2g\left({q}^2\right){\epsilon}_{\mu \nu \alpha \beta}{\epsilon}^{\ast \nu }{p}_1^{\alpha }{p}_2^{\beta },\\ {}V\left({M}_2,{p}_2,\epsilon \right)\mid {A}_{\mu }(0)\mid P\left({M}_1,{p}_1\right)=i{\epsilon}^{\ast \alpha}\left[f\left({q}^2\right){g}_{\mu \alpha}+{a}_{+}\left({q}^2\right){p}_{1\alpha }{P}_{\mu }+{a}_{-}\left({q}^2\right){p}_{1\alpha }{q}_{\mu}\right],\\ {}P\left({M}_2,{p}_2\right)\mid {T}_{\mu \nu}(0)\mid P\left({M}_1,{p}_1\right)=-2 is\left({q}^2\right)\left({p}_{1\mu }{p}_{2\nu }-{p}_{1\nu }{p}_{2\mu}\right),\\ {}V\left({M}_2,{p}_2,\epsilon \right)\mid {T}_{\mu \nu}(0)\mid P\left({M}_1,{p}_1\right)=\\ {}i{\epsilon}^{\ast \alpha}\left[{g}_{+}\left({q}^2\right){\epsilon}_{\mu \nu \alpha \beta}{P}^{\beta }+{g}_{-}\left({q}^2\right){\epsilon}_{\mu \nu \alpha \beta}{q}^{\beta }+{g}_0\left({q}^2\right){p}_{1\alpha }{\epsilon}_{\mu \nu \beta \gamma}{p}_1^{\beta }{p}_2^{\gamma}\right],\\ {}P\left({M}_2,{p}_2\right)\mid {T}_{\mu \nu}^5(0)\mid P\left({M}_1,{p}_1\right)=s\left({q}^2\right){\epsilon}_{\mu \nu \alpha \beta}{P}^{\alpha }{q}^{\beta },\\ {}V\left({M}_2,{p}_2,\epsilon \right)\mid {T}_{\mu \nu}^5(0)\mid P\left({M}_1,{p}_1\right)=\\ {}{g}_{+}\left({q}^2\right)\left({\epsilon}_{\nu}^{\ast }{P}_{\mu }-{\epsilon}_{\mu}^{\ast }{P}_{\nu}\right)+{g}_{-}\left({q}^2\right)\left({\epsilon}_{\nu}^{\ast }{q}_{\mu }-{\epsilon}_{\mu}^{\ast }{q}_{\nu}\right)+{g}_0\left({q}^2\right)\left({p}_1{\epsilon}^{\ast}\right)\left({p}_{1\nu }{p}_{2\mu }-{p}_{1\mu }{p}_{2\nu}\right),\end{array}} $$

where, q = p₁ − p₂, P = p₁ − p₂. We have used the following notations: γ⁵ = iγ⁰γ¹γ²γ³, ${\sigma}_{\mu \nu}=\frac{i}{2}\left[{\gamma}_{\mu },{\gamma}_{\nu}\right]$, ϵ⁰¹²³ =  − 1, ${\gamma}_5{\sigma}_{\mu \nu}=\frac{i}{2}{\epsilon}_{\mu \nu \alpha \beta}{\sigma}^{\alpha \beta}$ and Sp(γ⁵γ^μγ^νγ^αγ^β) = 4iϵ^μναβ.

Appendix B

The parameters used for calculating branching ratio and other observables of B → (K, K^∗)τ⁺τ⁻ decay processes

$$ {\displaystyle \begin{array}{l}{\beta}_P={\left|{C}_{9V}^{eff}\left({m}_b,{q}^2\right){f}_{+}\left({q}^2\right)+2\left({m}_b+{m}_s\right){C}_{7\gamma}\left({m}_b\right)s\left({q}^2\right)\right|}^2+{\left|{C}_{10A}\left({m}_b\right){f}_{+}\left({q}^2\right)\right|}^2\\ {}\hat{\varPi}=\left(\hat{t}-1\right)\left(\hat{t}-\hat{r}\right)+\hat{s}\hat{t}+\hat{m}\left(1+\hat{r}+\hat{m}-\hat{s}-2\hat{t}\right)\\ {}{\delta}_P={\left|{C}_{10A}\right|}^2\left\{\left(1+\hat{r}-\frac{\hat{s}}{2}\right){\left|{f}_{+}\left({q}^2\right)\right|}^2+\left(1-\hat{r}\right)\mathit{\operatorname{Re}}\left[{f}_{+}\left({q}^2\right){f}_{-}^{\ast}\left({q}^2\right)\right]+\frac{\hat{s}}{2}{\left|{f}_{-}\left({q}^2\right)\right|}^2\right\}\kern0.50em \ \kern0.50em \ \kern0.50em \ \kern0.50em \ \kern0.50em \end{array}} $$

with, $\hat{r}\equiv {\left({M}_K/{M}_B\right)}^2$ and $\hat{m}\equiv {\left({m}_l/{M}_B\right)}^2$.

$$ \Big[{\beta}_V^{(1)}=\left[\left(\hat{s}+2\hat{m}\right)\lambda \left(1,\hat{s},\hat{r}\right)+2\hat{s}\hat{\Pi}\right]{\left|G\left({q}^2\right)\right|}^2+\left[\hat{s}+2\hat{m}-\frac{\hat{\Pi}}{2\hat{r}}\right]{\left|F\left({q}^2\right)\right|}^2-\frac{\lambda^2\left(1,\hat{s},\hat{r}\right)}{2\hat{r}}\hat{\Pi}{\left|{H}_{+}\left({q}^2\right)\right|}^2+\frac{\hat{s}-1+\hat{r}}{\hat{r}}\hat{\Pi}R\left({q}^2\right) $$

$$ {\beta}_V^{(2)}=2\hat{s}\left[2\hat{t}+\hat{s}-\hat{r}-1-2\hat{m}\right]{R}_1\left({q}^2\right) $$

$$ {\left|G\left({q}^2\right)\right|}^2={\left|{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_Bg\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b+{m}_s\right)}{M_B}{g}_{+}\left({q}^2\right)\right|}^2+{\left|{C}_{10A}\left({m}_b\right){M}_Bg\left({q}^2\right)\right|}^2 $$

$$ {\left|F\left({q}^2\right)\right|}^2={\left|{C}_{9V}^{eff}\left({m}_b,{q}^2\right)\frac{f\left({q}^2\right)}{M_B}-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}\left(1-\hat{r}\right){B}_0\left({q}^2\right)\right|}^2+{\left|{C}_{10A}\left({m}_b\right)\frac{f\left({q}^2\right)}{M_B}\right|}^2 $$

$$ {\left|{H}_{+}\left({q}^2\right)\right|}^2={\left|{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_B{a}_{+}\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}{B}_{+}\left({q}^2\right)\right|}^2+{\left|{C}_{10A}\left({m}_b\right){M}_B{a}_{+}\left({q}^2\right)\right|}^2 $$

$$ R\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right)\frac{f\left({q}^2\right)}{M_B}-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}\left(1-\hat{r}\right){B}_0\left({q}^2\right)\right]{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_B{a}_{+}\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}{B}_{+}\left({q}^2\right)\right]}^{\ast}\right\}+{\left|{C}_{10A}\left({m}_b\right)\right|}^2\mathit{\operatorname{Re}}\left[{a}_{+}\left({q}^2\right){f}^{\ast}\left({q}^2\right)\right] $$

$$ {R}_1\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_Bg\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b+{m}_s\right)}{M_B}{g}_{+}\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right)\frac{f\left({q}^2\right)}{M_B}\right]}^{\ast}\right\}+\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right)\frac{f\left({q}^2\right)}{M_B}-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}\left(1-\hat{r}\right){B}_0\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right){M}_Bg\left({q}^2\right)\right]}^{\ast}\right\} $$

$$ {B}_0\left({q}^2\right)={g}_{+}\left({q}^2\right)+{g}_{-}\left({q}^2\right)\frac{\hat{s}}{1-\hat{r}};\cdot {B}_{+}\left({q}^2=-\hat{s}{M}_B^2\right)\frac{h\left({q}^2\right)}{2}-{g}_{+}\left({q}^2\right) $$

$$ {\delta}_V=\frac{{\left|{C}_{10A}\right|}^2}{2}\lambda \left(1,\hat{s},\hat{r}\right)\left\{-2{\left|g\left({q}^2\right){M}_B\right|}^2-\frac{3}{\lambda \left(1,\hat{s},\hat{r}\right)}{\left|\frac{f\left({q}^2\right)}{M_B}\right|}^2+\frac{2\left(1+k\right)-\hat{s}}{4\hat{r}}{\left|{a}_{+}\left({q}^2\right){M}_B\right|}^2+\frac{\hat{s}}{4\hat{r}}{\left|{a}_{-}\left({q}^2\right){M}_B\right|}^2+\frac{1}{2\hat{r}}\mathit{\operatorname{Re}}\left[f\left({q}^2\right){a}_{+}^{\ast}\left({q}^2\right)+f\left({q}^2\right){a}_{-}^{\ast}\left({q}^2\right)\right]+\frac{1-\hat{r}}{2\hat{r}}\mathit{\operatorname{Re}}\left[{M}_B{a}_{+}\left({q}^2\right){M}_B{a}_{-}^{\ast}\left({q}^2\right)\right]\right\} $$

$$ {\beta}_V=2\ \lambda \left(1,\hat{s},\hat{r}\right)\hat{s}{\left|G\left({q}^2\right)\right|}^2+\left[2\hat{s}+\frac{{\left(1-\hat{r}-\hat{s}\right)}^2}{4\hat{r}}\right]{\left|F\left({q}^2\right)\right|}^2+ $$

$$ \frac{\lambda^2\left(1,\hat{s},\hat{r}\right)}{4\hat{r}}{\left|H\left({q}^2\right)\right|}^2+\frac{\lambda \left(1,\hat{s},\hat{r}\right)}{2\hat{r}}\left(\hat{s}-1+\hat{r}\right)R\left({q}^2\right)\Big] $$

Here, $\hat{r}\equiv {\left({M}_{K^{\ast }}/{M}_B\right)}^2$.

$$ {\displaystyle \begin{array}{l}{R}_G\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_Bg\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b+{m}_s\right)}{M_B}{g}_{+}\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right){M}_Bg\left({q}^2\right)\right]}^{\ast}\right\}\\ {}{R}_F\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right)\frac{f\left({q}^2\right)}{M_B}-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}\left(1-\hat{r}\right){B}_0\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right)\frac{f\left({q}^2\right)}{M_B}\right]}^{\ast}\right\}\\ {}{R}_{H_{+}}\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_B{a}_{+}\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}{B}_{+}\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right){M}_B{a}_{+}\left({q}^2\right)\right]}^{\ast}\right\}\\ {}{R}_R\left({q}^2\right)=\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right)\frac{f\left({q}^2\right)}{M_B}-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}\left(1-\hat{r}\right){B}_0\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right){M}_B{a}_{+}\left({q}^2\right)\right]}^{\ast}\right\}\\ {}+\mathit{\operatorname{Re}}\left\{\left[{C}_{9V}^{eff}\left({m}_b,{q}^2\right){M}_B{a}_{+}\left({q}^2\right)-\frac{2{C}_{7\gamma}\left({m}_b\right)}{\hat{s}}\frac{\left({m}_b-{m}_s\right)}{M_B}{B}_{+}\left({q}^2\right)\right]{\left[{C}_{10A}\left({m}_b\right)\frac{f\left({q}^2\right)}{M_B}\right]}^{\ast}\right\}\end{array}} $$