The radiative decays $B_{c}^{*\pm} \to B_{c}^{\pm} \gamma$ with QCD sum rules

Abstract

In this article, we calculate the $B_{c}^{*} \to B_{c}$ electromagnetic form-factor with the three-point QCD sum rules and then study the radiative decays $B_{c}^{*\pm} \to B_{c}^{\pm} \gamma$. Experimentally, we can study the radiative transitions using the decay cascades $B_{c}^{*\pm}\to B_{c}^{\pm} \gamma\to J/\psi\ell^{\pm}\bar{\nu}_{\ell} \gamma\to\mu^{+} \mu^{-} \ell^{\pm}\bar{\nu}_{\ell} \gamma$ in the future at the LHCb.

Appendix

The explicit expressions of the $I_{0}^{ijn}$, $I_{10}^{ijn}$, $I_{01}^{ijn}$, $J_{0}^{ijn}$, $J_{10}^{ijn}$, $J_{01}^{ijn}$,

$$\begin{aligned} iI_0^{ijn} =&B_{-p_1^2\rightarrow M_1^2}B_{-p_2^2\rightarrow M_2^2} \overline{I}_{ijn} \\ =&B_{P_1^2\rightarrow M_1^2}B_{P_2^2\rightarrow M_2^2}\frac {(-1)^{i+j+n}i}{\varGamma(i)\varGamma(j)\varGamma(n)} \\ &{}\times \int d^4K \int_0^\infty d\alpha\, d\beta\, d\gamma\, \alpha^{i-1}\beta^{j-1}\gamma^{n-1} \\ &{}\times\exp \bigl\{ -\alpha(K+P_1)^2- \beta(K+P_2)^2 \\ &{} -\gamma K^2-\alpha m_b^2-\beta m_b^2-\gamma m_c^2 \bigr\} \\ =&\frac{(-1)^{i+j+n}i\pi^2}{\varGamma(i)\varGamma(j)\varGamma (n)(M_1^2)^i(M_2^2)^j (M^2)^{n-2}} \\ &{}\times \int_0^1 d\lambda \frac{\lambda ^{1-i-j}}{(1-\lambda)^{n-1}} \\ &{} \times\exp \biggl\{ -\frac{(1-\lambda)Q^2}{\lambda (M_1^2+M_2^2)}-\frac{m_b^2}{\lambda M^2}- \frac{m_c^2}{(1-\lambda )M^2} \biggr\} \\ =&\frac{(-1)^{i+j+n}i\pi^2}{\varGamma(i)\varGamma(j)\varGamma (n)(M_1^2)^i(M_2^2)^j (M^2)^{n-2}} \\ &{} \times \int_0^\infty d\tau(\tau +1)^{i+j+n-4}\tau^{1-i-j} \\ &{}\times\exp \biggl\{ -\frac{ 1}{\tau} \biggl(\frac {Q^2}{M_1^2+M_2^2}+ \frac{m_b^2}{M^2} \biggr) \\ &{}-\frac{m_b^2+m_c^2}{ M^2}-\tau\frac{m_c^2}{M^2} \biggr\} , \end{aligned}$$

(A.1)

$$\begin{aligned} iI^\mu_{ijn} =&B_{-p_1^2\rightarrow M_1^2}B_{-p_2^2\rightarrow M_2^2} \overline{I}^\mu_{ijn} \\ =&\frac{(-1)^{i+j+n+1}i\pi^2}{\varGamma(i)\varGamma(j)\varGamma (n)(M_1^2)^{i+1}(M_2^2)^j (M^2)^{n-3}} \\ &{}\times \int_0^\infty d\tau(\tau +1)^{i+j+n-3}\tau^{1-i-j} \\ &{}\times\exp \biggl\{ -\frac{ 1}{\tau} \biggl(\frac {Q^2}{M_1^2+M_2^2}+ \frac{m_b^2}{M^2} \biggr) \\ &{} -\frac{m_b^2+m_c^2}{ M^2}-\tau\frac{m_c^2}{M^2} \biggr \}p_{1}^\mu \\ &{}+\frac{(-1)^{i+j+n+1}i\pi^2}{\varGamma(i)\varGamma(j)\varGamma (n)(M_1^2)^i(M_2^2)^{j+1} (M^2)^{n-3}} \\ &{}\times \int_0^\infty d\tau(\tau +1)^{i+j+n-3}\tau^{1-i-j} \\ &\times{}\exp \biggl\{ -\frac{ 1}{\tau} \biggl(\frac {Q^2}{M_1^2+M_2^2}+ \frac{m_b^2}{M^2} \biggr) \\ &{} -\frac{m_b^2+m_c^2}{ M^2}-\tau\frac{m_c^2}{M^2} \biggr \}p_{2}^\mu \\ =&iI_{10}^{ijn}p_{1}^\mu+iI_{01}^{ijn}p_{2}^\mu , \end{aligned}$$

(A.2)

$$\begin{aligned} \begin{aligned} &J_0^{ijn}=I_0^{ijn}\big|_{m_b\leftrightarrow m_c} , \\ &J_{10}^{ijn}=I_{10}^{ijn}\big|_{m_b\leftrightarrow m_c} , \\ &J_{01}^{ijn}=I_{01}^{ijn}\big|_{m_b\leftrightarrow m_c}, \\ &M^2=\frac{M_1^2M_2^2}{M_1^2+M_2^2} , \end{aligned} \end{aligned}$$

(A.3)

where we have used the Borel transform $B_{P^{2}\rightarrow M^{2}} \exp (-\alpha P^{2})=\delta(1-\alpha M^{2})$. Those analytical expressions are slightly different from that obtained in [11]; they are both correct.