Studying Radiative Baryon Decays with the SU(3) Flavor Symmetry

The light baryons octet $T_{8}$ and decuplet $T_{10}$ under the SU(3) flavor symmetry of $u,d,s$ quarks can be written as

	$\displaystyle T_{8}$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}\frac{\Lambda^{0}}{\sqrt{6}}+\frac{\Sigma^{0}}{\sqrt{2}}&\Sigma^{+}&p\\ \Sigma^{-}&\frac{\Lambda^{0}}{\sqrt{6}}-\frac{\Sigma^{0}}{\sqrt{2}}&n\\ \Xi^{-}&\Xi^{0}&-\frac{2\Lambda^{0}}{\sqrt{6}}\end{array}\right)\,,$		(4)
	$\displaystyle T_{10}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{3}}\left(\left(\begin{array}[]{ccc}\sqrt{3}\Delta^{++}&\Delta^{+}&\Sigma^{*+}\\ \Delta^{+}&\Delta^{0}&\frac{\Sigma^{*0}}{\sqrt{2}}\\ \Sigma^{*+}&\frac{\Sigma^{*0}}{\sqrt{2}}&\Xi^{*0}\end{array}\right),\leavevmode\nobreak\ \left(\begin{array}[]{ccc}\Delta^{+}&\Delta^{0}&\frac{\Sigma^{*0}}{\sqrt{2}}\\ \Delta^{0}&\sqrt{3}\Delta^{-}&\Sigma^{*-}\\ \frac{\Sigma^{*0}}{\sqrt{2}}&\Sigma^{*-}&\Xi^{*-}\end{array}\right),\leavevmode\nobreak\ \left(\begin{array}[]{ccc}\Sigma^{*+}&\frac{\Sigma^{*0}}{\sqrt{2}}&\Xi^{*0}\\ \frac{\Sigma^{*0}}{\sqrt{2}}&\Sigma^{*}{-}&\Xi^{*-}\\ \Xi^{*0}&\Xi^{*-}&\sqrt{3}\Omega^{-}\end{array}\right)\right).$		(14)

The single charmed anti-triplet $T_{c3}$ and sextet $T_{c6}$ can be written as

$\displaystyle T_{c3}$

$\displaystyle=$

$\displaystyle(\Xi^{0}_{c},\leavevmode\nobreak\ -\Xi^{+}_{c},\leavevmode\nobreak\ \Lambda^{+}_{c}),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T_{c6}=\left(\begin{array}[]{ccc}\Sigma^{++}_{c}&\frac{1}{\sqrt{2}}\Sigma^{+}_{c}&\frac{1}{\sqrt{2}}\Xi^{*+}_{c}\\ \frac{1}{\sqrt{2}}\Sigma^{+}_{c}&\Sigma^{0}_{c}&\frac{1}{\sqrt{2}}\Xi^{*0}_{c}\\ \frac{1}{\sqrt{2}}\Xi^{*+}_{c}&\frac{1}{\sqrt{2}}\Xi^{*0}_{c}&\Omega_{c}\end{array}\right)\,.$

(18)

The anti-triplet $T_{b3}$ and sextet $T_{b6}$ with a heavy b quark have a similar form to $T_{c3}$ and $T_{c6}$, respectively,

$\displaystyle T_{b3}$

$\displaystyle=$

$\displaystyle(\Xi^{-}_{b},\leavevmode\nobreak\ -\Xi^{0}_{b},\leavevmode\nobreak\ \Lambda^{0}_{b}),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ T_{b6}=\left(\begin{array}[]{ccc}\Sigma^{+}_{b}&\frac{1}{\sqrt{2}}\Sigma^{0}_{b}&\frac{1}{\sqrt{2}}\Xi^{*0}_{b}\\ \frac{1}{\sqrt{2}}\Sigma^{0}_{b}&\Sigma^{-}_{b}&\frac{1}{\sqrt{2}}\Xi^{*-}_{b}\\ \frac{1}{\sqrt{2}}\Xi^{*0}_{b}&\frac{1}{\sqrt{2}}\Xi^{*-}_{b}&\Omega_{b}\end{array}\right)\,.$

(22)

In the standard model, the weak radiative baryon decays $\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma$ with $q_{1}\to q_{2}\gamma$ transition can proceed via loop Feynman diagrams as shown in Fig. 1. The effective Hamiltonian for $q_{1}\to q_{2}\gamma$ transition shown in Fig. 1 can be written as Buchalla:1995vs

$\displaystyle\mathcal{H}_{eff}(q_{1}\to q_{2}\gamma)=-\frac{G_{F}}{\sqrt{2}}\frac{e\lambda_{q_{1}q_{2}}}{4\pi^{2}}C^{eff}_{7}m_{q_{1}}\Big{(}\bar{q}_{2}i\sigma^{\mu q}P_{R}q_{1}\Big{)}\epsilon_{\mu},$

(23)

where $P_{R}=(1+\gamma_{5})/2$, $\sigma^{\mu q}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})q_{\nu}$ with $q=p_{1}-p_{2}$, $\epsilon_{\mu}$ is the polarization vectors of photon, and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $\lambda_{q_{1}q_{2}}=V_{tb}V^{*}_{ts},\leavevmode\nobreak\ V_{tb}V^{*}_{td},\leavevmode\nobreak\ -V_{us}V^{*}_{ud},V^{*}_{cb}V_{ub}$ for $b\to s\gamma$, $b\to d\gamma$, $s\to d\gamma$, $c\to u\gamma$, respectively.

Then the decay amplitudes can be written as

	$\displaystyle\mathcal{M}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)$	$\displaystyle=$	$\displaystyle\langle\mathcal{B}_{2}\gamma|\mathcal{H}_{eff}(q_{1}\to q_{2}\gamma)|\mathcal{B}_{1}\rangle$		(24)
		$\displaystyle=$	$\displaystyle-i\frac{G_{F}}{\sqrt{2}}\frac{e\lambda_{q_{1}q_{2}}}{4\pi^{2}}C^{eff}_{7}m_{q_{1}}\epsilon_{\mu}q_{\nu}\langle\mathcal{B}_{2}|\bar{q}_{2}\sigma^{\mu\nu}P_{R}q_{1}|\mathcal{B}_{1}\rangle.$

The baryon matrix elements $\langle\mathcal{B}_{2}|\bar{q}_{2}\sigma^{\mu\nu}P_{R}q_{1}|\mathcal{B}_{1}\rangle$ can be parameterized by the form factors, but not all relevant form factors have been calculated and there is no very reliable method to calculate some form factors at present. Nevertheless, the baryon matrix elements also can be obtained by the SU(3) IRA. In terms of the SU(3) flavor symmetry, baryon states and quark operators can be parameterized into SU(3) tensor forms, while the polarization vectors $\epsilon_{\mu}$ are invariant under SU(3) flavor symmetry. The decay amplitudes in terms of the SU(3) IRA are given in later Tab. 1, Tab. 3, Tab. 4 and Tab. 5 for $T_{b3,b6}\to T_{8,10}\gamma$, $T_{b3,b6}\to T_{c3,c6}\gamma$, $T_{c3,c6}\to T_{8,10}\gamma$ and $T_{8,10}\to T^{{}^{\prime}}_{8,10}\gamma$ weak decays, respectively.

The branching ratios of the $\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma$ weak decays can be obtained by the decay amplitudes

$\displaystyle\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)=\frac{\tau_{\mathcal{B}_{1}}(m^{2}_{\mathcal{B}_{1}}-m^{2}_{\mathcal{B}_{2}})}{16\pi m^{3}_{\mathcal{B}_{1}}}\left|\mathcal{M}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)\right|^{2}.$

(25)

After extracting the masses, the Wilson Coefficients, etc, from $|\mathcal{M}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)|^{2}$, the branching ratios of the $T_{b3,c3,8}\to T^{\prime}_{c3,c6,8,10}\gamma$ and $T_{b6,c6,10}\to T_{c3,8}\gamma$ weak decays are Faustov:2017wbh ; Gutsche:2013pp

$\displaystyle\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)=\frac{\tau_{\mathcal{B}_{1}}\alpha_{e}}{64\pi^{4}}G^{2}_{F}m_{q_{1}}^{2}m^{3}_{\mathcal{B}_{1}}|\lambda_{q_{1}q_{2}}|^{2}|C^{eff}_{7\gamma}(m_{q_{1}})|^{2}\left(1-\frac{m^{2}_{\mathcal{B}_{2}}}{m^{2}_{\mathcal{B}_{1}}}\right)^{3}\left|A(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)\right|^{2},$

(26)

where $A(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)$ may be given by the form factors as $\left|A(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)\right|^{2}=K(|f_{2}^{TV}(0)|^{2}+|f_{2}^{TA}(0)|^{2})=K(|h_{\bot}(0)|^{2}+|\tilde{h}_{\bot}(0)|^{2})$ with $K=1$ for the $T_{b3,c3,8}\to T^{\prime}_{c3,c6,8,10}\gamma$ weak decays and $K=1/2$ for the $T_{b6,c6,10}\to T_{c3,8}\gamma$ weak decays. As for the $T_{b6,c6}\to T^{\prime}_{c6,10}\gamma$ weak decays, the branching ratios are Mannel:2011xg

$\displaystyle\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)=\frac{\tau_{\mathcal{B}_{1}}\alpha_{e}}{384\pi^{4}}G^{2}_{F}\frac{m_{q_{1}}^{2}m^{5}_{\mathcal{B}_{1}}}{m^{2}_{\mathcal{B}_{2}}}\left(1-\frac{m^{2}_{\mathcal{B}_{2}}}{m^{2}_{\mathcal{B}_{1}}}\right)^{3}|\lambda_{q_{1}q_{2}}|^{2}|C^{eff}_{7\gamma}(m_{q_{1}})|^{2}\left|A(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)\right|^{2}.$

(27)

For the electromagnetic $T_{10}\to T_{8}\gamma$ decays, the expressions of their branching ratios are different from Eq. (26), and the following relations will be used to obtain the results Junker:2019vvy

$\displaystyle\mathcal{B}^{E}(T_{10}\to T_{8}\gamma)\propto\frac{\tau_{T_{10}}(m^{2}_{T_{10}}-m^{2}_{T_{8}})}{m^{3}_{T_{10}}}\left(m_{T_{10}}-m_{T_{8}}\right)^{2}\left|A^{E}(T_{10}\to T_{8}\gamma)\right|^{2}.$

(28)

In addition, according to Refs. Verma:1988gf ; Lach:1995we ; Azimov:1996uf , other three kinds of Feynman diagrams might contribute to the weak baryon decays. The example for $s+u\to u+d+\gamma$ is displayed in Fig. 2. Fig. 2 (a-b) are two-quark and three-quark transitions with the $W$ exchange, which have been discussed, for examples, in Refs. Verma:1988gf ; Dubovik:2007qg . Since Fig. 2 (c) is suppressed by the two $W$ propagators, and its contribution can be safely neglected. We will consider the W-exchange contributions in Fig. 2 (a-b) in later analysis of SU(3) flavor symmetry.

The SU(3) flavor structure of the relevant $b\to s,d$ Hamiltonian can been found, for instance, in Refs. Zeppenfeld:1980ex ; Savage:1989ub ; Deshpande:1994ii . The SU(3) IRA decay amplitudes for $T_{b3,b6}\to T_{8,10}\gamma$ decays via $b\to s/d\gamma$ can be parameterized as

	$\displaystyle A(T_{b3}\to T_{8}\gamma)$	$\displaystyle=$	$\displaystyle a_{1}(T_{b3})^{[ij]}T(\bar{3})^{k}(T_{8})_{[ij]k}+a_{2}(T_{b3})^{[ij]}T(3)^{k}(T_{8})_{[ik]j},$		(29)
	$\displaystyle A(T_{b6}\to T_{8}\gamma)$	$\displaystyle=$	$\displaystyle a^{\prime}_{1}(T_{b6})^{ij}T(\bar{3})^{k}(T_{8})_{[ik]j},$		(30)
	$\displaystyle A(T_{b6}\to T_{10}\gamma)$	$\displaystyle=$	$\displaystyle a^{\prime\prime}_{1}(T_{b6})^{ij}T(\bar{3})^{k}(T_{10})_{ijk},$		(31)

with $T(\bar{3})=(0,1,1)$, which denotes the transition operators $(\bar{q}_{2}b)$ with $q_{2}=d,s$. The coefficients $a^{(^{\prime},^{\prime\prime})}_{i}$, which contain information about QCD dynamics, include the single quark emission contributions in Fig. 1 and $T_{b3}\to T_{8}\psi_{i}$ and $T_{b6}\to T_{8,10}\psi_{i}$ ($\psi_{i}$ are the set of all $J=1,l=0\leavevmode\nobreak\ (c\bar{c})$) long distance contributions Dery:2020lbc ; Deshpande:1994cn (the similar in following $b^{(^{\prime},^{\prime\prime})}_{i},\leavevmode\nobreak\ c^{(^{\prime},^{\prime\prime},^{\prime\prime\prime})}_{i},\leavevmode\nobreak\ d_{i}$ in Eqs. (36-47)), nevertheless, the long distance contributions in the b-sector are small and under control Golowich:1994zr ; Deshpande:1994cn . Noted that $T_{b3}\to T_{10}\gamma$ (and later $T_{c3}\to T_{10}\gamma)$ weak decays are not allowed by the quark symmetry. The SU(3) IRA amplitudes of $T_{b3}\to T_{8}\gamma$ and $T_{b6}\to T_{8,10}\gamma$ weak decays are given in Tab. 1. And the information of relevant CKM matrix elements $V_{tb}V^{*}_{ts}$ ($V_{tb}V^{*}_{td}$ ) for $b\to s\gamma$ ($b\to d\gamma$) transition is not shown in Tab. 1.

Now, we discuss the $T_{b3}\to T_{8}\gamma$ weak decays. Decays $\Lambda^{0}_{b}\to\Lambda^{0}\gamma$, $\Lambda^{0}_{b}\to\Sigma^{0}\gamma$, $\Xi^{0}_{b}\to\Xi^{0}\gamma$ and $\Xi^{-}_{b}\to\Xi^{-}\gamma$ ($\Lambda^{0}_{b}\to n\gamma$, $\Xi^{0}_{b}\to\Lambda^{0}\gamma$, $\Xi^{0}_{b}\to\Sigma^{0}\gamma$ and $\Xi^{-}_{b}\to\Sigma^{-}\gamma$) proceed via the $b\to s\gamma$ ($b\to d\gamma$) flavor changing neutral current transition. From Tab. 1, one can see that the 7 decay amplitudes of $T_{b3}\to T_{8}\gamma$ can be related by one parameter $A_{1}$. Noted that our $|A(T_{b3}\to T_{8}\gamma)|$ via the $b\to s/d\gamma$ transitions are consistent with ones of $T_{b3}\to T_{8}J/\psi$ in Ref. Fayyazuddin:2017sxq and ones of the CKM-leading part results $T_{b3}\to T_{8}J/\psi$ in Ref. Dery:2020lbc . Among these 7 decay modes, only $\mathcal{B}(\Lambda^{0}_{b}\to\Lambda\gamma)$ has been measured at present, which is listed in the second column of Tab. 3. Using data of $\mathcal{B}(\Lambda^{0}_{b}\to\Lambda\gamma)$ and the expression of $\mathcal{B}(\mathcal{B}_{1}\to\mathcal{B}_{2}\gamma)$ in Eq. (26) to get $|A_{1}|$, and then other 6 branching ratios are obtained, which are given in the third column of Tab. 3. Previous predictions for $\mathcal{B}(\Lambda^{0}_{b}\to\Lambda\gamma)$ in the light-cone sum rules and for $\mathcal{B}(\Lambda^{0}_{b}\to n\gamma)$ in the relativistic quark model and in the Bethe-Salpeter equation approach are listed in the last column of Tab. 3. Our SU(3) IRA prediction of $\mathcal{B}(\Lambda^{0}_{b}\to n\gamma)$ agrees with ones in the relativistic quark model or in the Bethe-Salpeter equation approach Liu:2019rpm ; Faustov:2020thr . More experimental data about the $T_{b3}\to T_{8}\gamma$ decays in the further could test the SU(3) flavor symmetry approach.

As given in Tab. 1, the decay amplitudes of the $T_{b6}\to T_{8}\gamma$ and $T_{b6}\to T_{10}\gamma$ weak decays can be parameterized by only one parameter $a^{\prime}_{1}$ and $a^{\prime\prime}_{1}$, respectively. And our $|A(T_{b6}\to T_{8}\gamma)|$ via the $b\to s\gamma$ transition are consistent with ones of $T_{b6}\to T_{8}J/\psi$ in Ref. Fayyazuddin:2017sxq . Unfortunately, none of the $T_{b6}\to T_{8}\gamma$ and $T_{b6}\to T_{10}\gamma$ weak decays has been measured at present. Any measurement of the $T_{b6}\to T_{8}\gamma$ ($T_{b6}\to T_{10}\gamma$) will give us chance to predict other 10 (11) decay modes.

In addition, $T_{b3,b6}$ baryons also can decay to $T_{c3,c6}$ by only $W$ exchange $b+u\to c+s/d+\gamma$ Cheng:1994kp . The SU(3) IRA decay amplitudes for the $T_{b3,b6}\to T_{c3,c6}\gamma$ decays are

	$\displaystyle A(T_{b3}\to T_{c3}\gamma)$	$\displaystyle=$	$\displaystyle\widetilde{a}_{1}V_{cb}V_{q_{j}q_{l}}(T_{b3})^{[ij]}(T_{c3})_{[il]},$		(32)
	$\displaystyle A(T_{b3}\to T_{c6}\gamma)$	$\displaystyle=$	$\displaystyle\widetilde{a}^{\prime}_{1}V_{cb}V_{q_{j}q_{l}}(T_{b3})^{[ij]}(T_{c6})_{il},$		(33)
	$\displaystyle A(T_{b6}\to T_{c3}\gamma)$	$\displaystyle=$	$\displaystyle\widetilde{a}^{\prime\prime}_{1}V_{cb}V_{q_{j}q_{l}}(T_{b6})^{ij}(T_{c3})_{[il]},$		(34)
	$\displaystyle A(T_{b6}\to T_{c6}\gamma)$	$\displaystyle=$	$\displaystyle\widetilde{a}^{\prime\prime\prime}_{1}V_{cb}V_{q_{j}q_{l}}(T_{b6})^{ij}(T_{c6})_{il}.$		(35)

Since the CKM matrix element $V_{cb}$ occur for all processes, we will absorb it in the coefficients $\widetilde{a}_{1},\widetilde{a}^{\prime}_{1},\widetilde{a}^{\prime\prime}_{1}$ as well as $\widetilde{a}^{\prime\prime\prime}_{1}$, and only keep $V_{ud}\approx 1$ and $V_{us}\approx\lambda=s_{c}\equiv\mbox{sin}\theta_{C}\approx 0.22453$ representing the Cabbibo angle $\theta_{C}$ PDG2020 . The SU(3) IRA amplitudes of $T_{b3,b6}\to T_{c3,c6}\gamma$ weak decays are summarized in Tab. 3, in which we keep $V_{ud}$ and $V_{us}$ information for comparing conveniently, and we may easily see the amplitude relations in this table. Just none of these decays has been measured yet.

$T_{c3,c6}\to T_{8,10}\gamma$ via $c\to u\gamma$ transition are similar to the $T_{b3,b6}\to T_{8,10}\gamma$ via $b\to s/d\gamma$ transition. Nevertheless, the short distance contributions from $c\to u\gamma$ could be negligible, and the dominant contributions in charmed baryon decays mainly from the W-exchange contributions similar as Fig. 2 (a-b) Singer:1995is . The SU(3) flavor structure of the relevant $b\to s,d$ Hamiltonian can been found, for instance, in Refs. Zeppenfeld:1980ex ; Savage:1989ub ; Deshpande:1994ii . The SU(3) IRA amplitudes of the $T_{c3,c6}$ baryon weak decays are

	$\displaystyle A(T_{c3}\to T_{8}\gamma)$	$\displaystyle=$	$\displaystyle b_{1}(T_{c3})^{[ij]}T^{\prime}(\bar{3})^{k}(T_{8})_{[ij]k}+b_{2}(T_{c3})^{[ij]}T^{\prime}(3)^{k}(T_{8})_{[ik]j}$		(36)
			$\displaystyle+\Big{(}\widetilde{b}_{1}H(\bar{6})^{lk}_{j}+\widetilde{b}_{4}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[il]k}+\Big{(}\widetilde{b}_{2}H(\bar{6})^{lk}_{j}+\widetilde{b}_{5}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[ik]l}$
			$\displaystyle+\Big{(}\widetilde{b}_{3}H(\bar{6})^{lk}_{j}+\widetilde{b}_{6}H(15)^{lk}_{j}\Big{)}(T_{c3})^{[ij]}(T_{8})_{[lk]i},$
	$\displaystyle A(T_{c6}\to T_{8}\gamma)$	$\displaystyle=$	$\displaystyle b^{\prime}_{1}(T_{c6})^{ij}T^{\prime}(\bar{3})^{k}(T_{8})_{[ik]j}$		(37)
			$\displaystyle+\Big{(}\widetilde{b}^{\prime}_{1}H(\bar{6})^{lk}_{j}+\widetilde{b}^{\prime}_{4}H(15)^{lk}_{j}\Big{)}(T_{c6})^{ij}(T_{8})_{[il]k}+\Big{(}\widetilde{b}^{\prime}_{2}H(\bar{6})^{lk}_{j}+\widetilde{b}^{\prime}_{5}H(15)^{lk}_{j}\Big{)}(T_{c6})^{ij}(T_{8})_{[ik]l}$
			$\displaystyle+\Big{(}\widetilde{b}^{\prime}_{3}H(\bar{6})^{lk}_{j}+\widetilde{b}^{\prime}_{6}H(15)^{lk}_{j}\Big{)}(T_{c6})^{ij}(T_{8})_{[lk]i},$
	$\displaystyle A(T_{c6}\to T_{10}\gamma)$	$\displaystyle=$	$\displaystyle b^{\prime\prime}_{1}(T_{c6})^{ij}T^{\prime}(\bar{3})^{k}(T_{10})_{ijk}+\widetilde{b}^{\prime\prime}_{1}H(15)^{lk}_{j}(T_{c6})^{ij}(T_{10})_{ilk},$		(38)

with $T^{\prime}(\bar{3})=(1,0,0)$ which denotes the transition operators $(\bar{q}_{2}c)$ with $q_{2}=u$. The $b^{(^{\prime},^{\prime\prime})}_{i}$ terms denote the contributions from $c\to u\gamma$ shown in Fig.1 and the long distance contribution from the real $q\bar{q}$ intermediate state $\rho$, $\omega$ and $\phi$. The $\widetilde{b}^{(^{\prime},^{\prime\prime})}_{i}$ terms denote the $W$ exchange contributions similar as Fig. 2. Noted that $H(\bar{6})^{lk}_{j}$ ($H(15)^{lk}_{j}$) related to $(\bar{q}_{l}q^{j})(\bar{q}_{k}c)$ operator is antisymmetric (symmetric) in upper indices. The non-vanish $H(\bar{6})^{lk}_{j}$ and $H(15)^{lk}_{j}$ for $c\to su\bar{d},du\bar{s},u\bar{d}d,u\bar{s}s$ transitions can be found in Ref. Wang:2017azm . Using $l,k$ antisymmetric in $H(\bar{6})^{lk}_{j}$ and $l,k$ symmetric in $H(15)^{lk}_{j}$, we have

$\displaystyle\widetilde{b}^{(^{\prime})}_{2}=-\widetilde{b}^{(^{\prime})}_{1},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \widetilde{b}^{(^{\prime})}_{5}=\widetilde{b}^{(^{\prime})}_{4},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \widetilde{b}^{(^{\prime})}_{6}=0.$

(39)