Radiative transitions and magnetic moments of the charmed and bottom vector mesons in chiral perturbation theory

TL;DR

This paper uses chiral perturbation theory to systematically analyze radiative decays and magnetic moments of charmed and bottom vector mesons, providing theoretical decay rates and magnetic moments that align with experimental data.

Contribution

It presents the first comprehensive calculation of radiative decay widths and magnetic moments of heavy vector mesons using chiral perturbation theory up to one-loop level, including SU(2) and SU(3) cases.

Findings

Decay widths for D* and B* mesons are calculated and agree with experimental data.

Full widths of specific D* and D_s* mesons are estimated.

Analytical expressions for magnetic moments are derived for future lattice QCD extrapolations.

Abstract

In this work, we systematically study the radiative decays and magnetic moments of the charmed and bottom vector mesons with chiral perturbation theory up to one-loop level. We present the results in SU(2) and SU(3) cases with the mass splitting in loop diagrams kept and unkept, respectively. The obtained decay rates for $D^\ast$ and $B^\ast$ mesons in SU(3) case with the mass splitting kept are: $\Gamma_{\bar{D}^{\ast 0}\to \bar{D}^0\gamma}=16.2^{+6.5}_{-6.0}$ keV, $\Gamma_{D^{\ast-}\to D^-\gamma}=0.73^{+0.7}_{-0.3}$ keV, $\Gamma_{D_s^{\ast-}\to D_s^-\gamma}= 0.32^{+0.3}_{-0.3}$ keV, and $\Gamma_{B^{\ast+}\to B^+\gamma}=0.58^{+0.2}_{-0.2}$ keV, $\Gamma_{B^{\ast0}\to B^0\gamma}=0.23^{+0.06}_{-0.06}$ keV, $\Gamma_{B_s^{\ast0}\to B_s^0\gamma}=0.04^{+0.03}_{-0.03}$ keV. The decay width for $D^{\ast-}\to D^-\gamma$ is consistent with the experimental measurement. As a byproduct, the full widths of $\bar{D}^{\ast0}$ and $D_s^{\ast-}$ are $\Gamma_{\mathrm{tot}}(\bar{D}^{\ast0})\simeq77.7^{+26.7}_{-20.5}~\mathrm{keV}$ and $ \Gamma_{\mathrm{tot}}(D_s^{\ast-})\simeq0.62^{+0.45}_{-0.50}~\mathrm{keV}$, respectively. We also calculate the magnetic moments of the heavy vector mesons. The analytical chiral expressions derived in our work shall be helpful for the extrapolations of lattice QCD simulations in the coming future.

Tables

Table 1. Table 1: The possible flavor structures of the 𝒪 ​ ( p 4 ) 𝒪 superscript 𝑝 4 \mathcal{O}(p^{4}) Lagrangians that contribute to the magnetic moments.

Group representations	$1\otimes 1\to 1$	$1\otimes 8\to 8$	$8\otimes 1\to 8$	$8\otimes 8\to 1$	$8\otimes 8\to 8_1$	$8\otimes 8\to 8_2$
Flavor structures	$\mathrm{Tr}\left({\chi }_{+}\right)\mathrm{Tr}\left(f_{\mu \nu }^{+}\right)$	$\mathrm{Tr}\left({\chi }_{+}\right){\tilde{f}}_{\mu \nu }^{+}$	${\tilde{\chi }}_{+}\mathrm{Tr}\left(f_{\mu \nu }^{+}\right)$	$\mathrm{Tr}\left({\tilde{\chi }}_{+}{\tilde{f}}_{\mu \nu }^{+}\right)$	$[{\tilde{\chi }}_{+},{\tilde{f}}_{\mu \nu }^{+}]$	$\{{\tilde{\chi }}_{+},{\tilde{f}}_{\mu \nu }^{+}\}$

Table 2. Table 2: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = a , … , d ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑎 … 𝑑 \mathcal{C}_{\phi}^{(x)}~{}(x=a,\dots,d) in Eqs. ( 40 )-( 43 ) for the D ¯ ∗ superscript ¯ 𝐷 ∗ \bar{D}^{\ast} mesons.

Decay modes	${𝒞}_{\pi }^{\left(a\right)}$	${𝒞}_K^{\left(a\right)}$	${𝒞}_{\pi }^{\left(b\right)}$	${𝒞}_K^{\left(b\right)}$	${𝒞}_{\pi }^{\left(c\right)}$	${𝒞}_K^{\left(c\right)}$	${𝒞}_{\pi }^{\left(d\right)}$	${𝒞}_K^{\left(d\right)}$	${𝒞}_{\eta }^{\left(d\right)}$
${\overline{D}}^{\ast 0}\to {\overline{D}}^0\gamma$	$2$	$2$	$-4$	$-4$	$4$	$4$	$24a$	$\frac{8}{3}\left(6a-\tilde{a}\right)$	$\frac{8}{9}\left(3a+\tilde{a}\right)$
$D^{\ast -}\to D^{-}\gamma$	$-2$	$0$	$4$	$0$	$-4$	$0$	$4\left(6a+\tilde{a}\right)$	$\frac{8}{3}\left(6a-\tilde{a}\right)$	$\frac{4}{9}\left(6a-\tilde{a}\right)$
$D_s^{\ast -}\to D_s^{-}\gamma$	$0$	$-2$	$0$	$4$	$0$	$-4$	$0$	$\frac{8}{3}\left(12a+\tilde{a}\right)$	$\frac{16}{9}\left(6a-\tilde{a}\right)$

Table 3. Table 3: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = e , … , j ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑒 … 𝑗 \mathcal{C}_{\phi}^{(x)}~{}(x=e,\dots,j) in Eqs. ( 44 )-( 47 ) for the D ¯ ∗ superscript ¯ 𝐷 ∗ \bar{D}^{\ast} mesons.

Decay modes	${𝒞}_{\pi }^{\left(e\right)}$	${𝒞}_K^{\left(e\right)}$	${𝒞}_{\eta }^{\left(e\right)}$	${𝒞}_{\pi }^{\left(h\right)}$	${𝒞}_K^{\left(h\right)}$	${𝒞}_{\eta }^{\left(h\right)}$
${\overline{D}}^{\ast 0}\to {\overline{D}}^0\gamma$	$12a$	$\frac{4}{3}\left(6a+\tilde{a}\right)$	$\frac{4}{9}\left(3a-\tilde{a}\right)$	$2\left(\tilde{a}-3a\right)$	$\frac{4}{3}\left(\tilde{a}-3a\right)$	$\frac{2}{9}\left(\tilde{a}-3a\right)$
$D^{\ast -}\to D^{-}\gamma$	$2\left(6a-\tilde{a}\right)$	$\frac{4}{3}\left(6a+\tilde{a}\right)$	$\frac{2}{9}\left(6a+\tilde{a}\right)$	$-\left(\tilde{a}+6a\right)$	$-\frac{2}{3}\left(\tilde{a}+6a\right)$	$-\frac{1}{9}\left(\tilde{a}+6a\right)$
$D_s^{\ast -}\to D_s^{-}\gamma$	$0$	$\frac{4}{3}\left(12a-\tilde{a}\right)$	$\frac{8}{9}\left(6a+\tilde{a}\right)$	$0$	$-\frac{4}{3}\left(\tilde{a}+6a\right)$	$-\frac{4}{9}\left(\tilde{a}+6a\right)$

Table 4. Table 4: The contribution of each loop diagram to the transition magnetic moment of D ¯ ∗ 0 → D ¯ ​ γ → superscript ¯ 𝐷 ∗ absent 0 ¯ 𝐷 𝛾 \bar{D}^{\ast 0}\to\bar{D}\gamma in different cases (in units of μ N subscript 𝜇 𝑁 \mu_{N} ).

Cases		$\left(a\right)$	$\left(b\right)$	$\left(c\right)$	$\left(d\right)$	$\left(e\right)$	$\left(f\right)$	$\left(g\right)$	$\left(h\right)$	$\left(i+j\right)$
SU(2)	$\Delta =0$	$0.21$	$-0.085$	$0.062$	$0.012$	$0.006$	$0$	$0$	$-0.045$	$-0.053$
	$\Delta \ne 0$	$0.29$	$-0.085$	$0.062$	$-0.0016$	$0.0021$	0	0	$0.073$	$-0.0082$
SU(3)	$\Delta =0$	$0.71$	$-0.37$	$0.27$	$0.088$	$-0.00017$	0	0	$-0.13$	$-0.19$
	$\Delta \ne 0$	$0.81$	$-0.37$	$0.27$	$0.033$	$-0.0038$	0	0	$0.13$	$-0.19$

Table 5. Table 5: The masses of the constituent quarks adopted in different works (in units of GeV).

	$m_u$	$m_d$	$m_s$	$m_c$	$m_b$
Kamal Kamal:1992uv	$0.34$	$0.34$	$0.55$	$1.8$	$\cdots$
Ebert Ebert:2002xz	$0.33$	$0.33$	$0.5$	$1.55$	$4.88$
Cheng Cheng:1992xi	$0.338$	$0.322$	$0.51$	$1.6$	$5.0$
Eichten Eichten:1979ms	$0.335$	$0.335$	$0.45$	$1.84$	$5.17$

Table 6. Table 6: The radiative decay widths for V → P ​ γ → 𝑉 𝑃 𝛾 V\to P\gamma (in units of keV). Br expt and Γ expt subscript Γ expt \Gamma_{\mathrm{expt}} denote the branching ratio and decay width measured in experiments. Γ 1 , … , 4 subscript Γ 1 … 4 \Gamma_{1,\dots,4} are the model predictions.

Decay modes	SU(2)		SU(3)		Experimental data and model predictions
	$\Delta =0$	$\Delta \ne 0$	$\Delta =0$	$\Delta \ne 0$	Brexpt$|{\Gamma }_{\mathrm{expt}}$ Tanabashi:2018oca	${\Gamma }_1$ Choi:2007se	${\Gamma }_2$ Priyadarsini:2016tiu	${\Gamma }_3$ Ebert:2002xz	${\Gamma }_4$ Zhu:1996qy
${\overline{D}}^{\ast 0}\to {\overline{D}}^0\gamma$	${30.0}_{-6.6}^{+7.3}$	${23.9}_{-6.3}^{+5.0}$	${22.9}_{-7.0}^{+8.2}$	${16.2}_{-6.0}^{+6.5}$	$\left(38.1\pm 2.9\right)\%|\cdots$	$20.0\pm 0.3$	26.5	$11.5$	$12.9\pm 2$
$D^{\ast -}\to D^{-}\gamma$	${1.0}_{-0.6}^{+0.9}$	${0.5}_{-0.4}^{+0.5}$	${1.8}_{-0.9}^{+1.3}$	${0.73}_{-0.3}^{+0.7}$	$\left(1.6\pm 0.4\right)\%|1.33\pm 0.33$	$0.9\pm 0.02$	0.93	$1.04$	$0.23\pm 0.1$
$D_s^{\ast -}\to D_s^{-}\gamma$	$\cdots$	$\cdots$	${0.15}_{-0.1}^{+0.5}$	${0.32}_{-0.3}^{+0.3}$	$\left(94.2\pm 0.7\right)\%|\cdots$	$0.18\pm 0.01$	$0.21$	$0.19$	$0.13\pm 0.05$
$B^{\ast +}\to B^{+}\gamma$	${0.75}_{-0.2}^{+0.2}$	${0.71}_{-0.2}^{+0.2}$	${0.63}_{-0.2}^{+0.2}$	${0.58}_{-0.2}^{+0.2}$	$\cdots |\cdots$	$0.4\pm 0.03$	0.58	$0.19$	$0.13\pm 0.03$
$B^{\ast 0}\to B^0\gamma$	${0.19}_{-0.05}^{+0.05}$	${0.18}_{-0.05}^{+0.05}$	${0.25}_{-0.06}^{+0.06}$	${0.23}_{-0.06}^{+0.06}$	$\cdots |\cdots$	$0.13\pm 0.01$	0.18	$0.07$	$0.38\pm 0.06$
$B_s^{\ast 0}\to B_s^0\gamma$	$\cdots$	$\cdots$	${0.05}_{-0.03}^{+0.03}$	${0.04}_{-0.03}^{+0.03}$	$\cdots |\cdots$	$0.068\pm 0.017$	$0.12$	$0.05$	$0.22\pm 0.04$

Table 7. Table 7: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = a , c , d , e ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑎 𝑐 𝑑 𝑒 \mathcal{C}_{\phi}^{(x)}~{}(x=a,c,d,e) in Eqs. ( 76 )-( 80 ) for the D ¯ ∗ superscript ¯ 𝐷 ∗ \bar{D}^{\ast} mesons.

States	${𝒞}_{\pi }^{\left(a\right)}$	${𝒞}_K^{\left(a\right)}$	${𝒞}_{\pi }^{\left(c\right)}$	${𝒞}_K^{\left(c\right)}$	${𝒞}_{\pi }^{\left(d\right)}$	${𝒞}_K^{\left(d\right)}$	${𝒞}_{\pi }^{\left(e\right)}$	${𝒞}_K^{\left(e\right)}$	${𝒞}_{\eta }^{\left(e\right)}$
${\overline{D}}^{\ast 0}$	$-\frac{1}{2}$	$-\frac{1}{2}$	$2$	$2$	$-2$	$-2$	$6a$	$\frac{2}{3}\left(6a+\tilde{a}\right)$	$\frac{2}{9}\left(3a-\tilde{a}\right)$
$D^{\ast -}$	$\frac{1}{2}$	$0$	$-2$	$0$	$2$	$0$	$6a-\tilde{a}$	$\frac{2}{3}\left(6a+\tilde{a}\right)$	$\frac{1}{9}\left(6a+\tilde{a}\right)$
$D_s^{\ast -}$	$0$	$\frac{1}{2}$	$0$	$-2$	$0$	$2$	$0$	$\frac{2}{3}\left(12a-\tilde{a}\right)$	$\frac{4}{9}\left(6a+\tilde{a}\right)$

Table 8. Table 8: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = g , l + m ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑔 𝑙 𝑚 \mathcal{C}_{\phi}^{(x)}~{}(x=g,l+m) in Eqs. ( 82 )-( 84 ) for the D ¯ ∗ superscript ¯ 𝐷 ∗ \bar{D}^{\ast} mesons.

States	${𝒞}_{\pi }^{\left(g\right)}$	${𝒞}_K^{\left(g\right)}$	${𝒞}_{\eta }^{\left(g\right)}$	${𝒞}_{\pi }^{\left(lm\right)}$	${𝒞}_K^{\left(lm\right)}$	${𝒞}_{\eta }^{\left(lm\right)}$
${\overline{D}}^{\ast 0}$	$6a$	$\frac{2}{3}\left(6a-\tilde{a}\right)$	$\frac{2}{9}\left(3a+\tilde{a}\right)$	$\left(\tilde{a}+3a\right)$	$\frac{2}{3}\left(\tilde{a}+3a\right)$	$\frac{1}{9}\left(\tilde{a}+3a\right)$
$D^{\ast -}$	$6a+\tilde{a}$	$\frac{2}{3}\left(6a-\tilde{a}\right)$	$\frac{1}{9}\left(6a-\tilde{a}\right)$	$\frac{1}{2}\left(6a-\tilde{a}\right)$	$\frac{1}{3}\left(6a-\tilde{a}\right)$	$\frac{1}{18}\left(6a-\tilde{a}\right)$
$D_s^{\ast -}$	$0$	$\frac{2}{3}\left(12a+\tilde{a}\right)$	$\frac{4}{9}\left(6a-\tilde{a}\right)$	$0$	$\frac{2}{3}\left(6a-\tilde{a}\right)$	$\frac{2}{9}\left(6a-\tilde{a}\right)$

Table 9. Table 9: The magnetic moments of the charmed and bottom vector mesons (in units of nucleon magnetons μ N subscript 𝜇 𝑁 \mu_{N} ), and a comparison with the Bag model (Bag), extended Nambu-Jona-Lasinio model (NJL) and extended Bag model (Extended Bag) predictions.

States	SU(2)		SU(3)		The results from other theoretical works
	$\Delta =0$	$\Delta \ne 0$	$\Delta =0$	$\Delta \ne 0$	Bag Bose:1980vy	NJL Luan:2015goa	Extended Bag Simonis:2018rld
${\overline{D}}^{\ast 0}$	${1.38}_{-0.25}^{+0.25}$	${1.60}_{-0.25}^{+0.25}$	${1.18}_{-0.25}^{+0.25}$	${1.48}_{-0.38}^{+0.22}$	$0.89$	$\cdots$	$1.28$
$D^{\ast -}$	$-{1.14}_{-0.15}^{+0.15}$	$-{1.39}_{-0.15}^{+0.15}$	$-{1.31}_{-0.15}^{+0.20}$	$-{1.62}_{-0.08}^{+0.24}$	$-1.17$	$-1.16$	$-1.13$
$D_s^{\ast -}$	$\cdots$	$\cdots$	$-{0.62}_{-0.15}^{+0.15}$	$-{0.69}_{-0.10}^{+0.22}$	$-1.03$	$-0.98$	$-0.93$
$B^{\ast +}$	${1.86}_{-0.25}^{+0.25}$	${1.90}_{-0.20}^{+0.20}$	${1.71}_{-0.25}^{+0.25}$	${1.77}_{-0.30}^{+0.25}$	$1.54$	$1.47$	$1.56$
$B^{\ast 0}$	$-{0.75}_{-0.11}^{+0.11}$	$-{0.78}_{-0.11}^{+0.11}$	$-{0.87}_{-0.11}^{+0.13}$	$-{0.92}_{-0.11}^{+0.15}$	$-0.64$	$\cdots$	$-0.69$
$B_s^{\ast 0}$	$\cdots$	$\cdots$	$-{0.25}_{-0.11}^{+0.11}$	$-{0.27}_{-0.10}^{+0.13}$	$-0.47$	$\cdots$	$-0.51$

Table 10. Table 10: The transition magnetic moments and magnetic moments of the charmed and bottom vector mesons calculated in the SU(2) case order by order (in units of μ N subscript 𝜇 𝑁 \mu_{N} ).

Physical quantity	$\Delta =0$				$\Delta \ne 0$
	${𝒪}_{\mu }\left(p^1\right)$ Tree	${𝒪}_{\mu }\left(p^2\right)$ Loop	${𝒪}_{\mu }\left(p^3\right)$ Loop	Total	${𝒪}_{\mu }\left(p^1\right)$ Tree	${𝒪}_{\mu }\left(p^2\right)$ Loop	${𝒪}_{\mu }\left(p^3\right)$ Loop	Total
${\mu }_{{\overline{D}}^{\ast 0}\to {\overline{D}}^0\gamma }$	$-2.24$	$0.21$	$-0.10$	$-2.13$	$-2.24$	$0.29$	$0.04$	$-1.91$
${\mu }_{D^{\ast -}\to D^{-}\gamma }$	$0.55$	$-0.21$	$0.05$	$0.39$	$0.55$	$-0.29$	$0.02$	$0.28$
${\mu }_{B^{\ast +}\to B^{+}\gamma }$	$-1.80$	$0.16$	$-0.09$	$-1.73$	$-1.80$	$0.19$	$-0.07$	$-1.68$
${\mu }_{B^{\ast 0}\to B^0\gamma }$	$0.99$	$-0.16$	$0.046$	$0.88$	$0.99$	$-0.19$	$0.04$	$0.84$
${\mu }_{{\overline{D}}^{\ast 0}}$	$1.48$	$-0.21$	$0.11$	$1.38$	$1.48$	$0.07$	$0.05$	$1.60$
${\mu }_{D^{\ast -}}$	$-1.31$	$0.21$	$-0.05$	$-1.14$	$-1.31$	$-0.07$	$-0.007$	$-1.39$
${\mu }_{B^{\ast +}}$	$1.93$	$-0.16$	$0.09$	$1.86$	$1.93$	$-0.13$	$0.09$	$1.90$
${\mu }_{B^{\ast 0}}$	$-0.86$	$0.16$	$-0.05$	$-0.75$	$-0.86$	$0.13$	$-0.05$	$-0.78$

Table 11. Table 11: The transition magnetic moments and magnetic moments of charmed and bottom vector mesons calculated in the SU(3) case order by order (in units of μ N subscript 𝜇 𝑁 \mu_{N} ).

Physical quantity	$\Delta =0$				$\Delta \ne 0$
	${𝒪}_{\mu }\left(p^1\right)$ Tree	${𝒪}_{\mu }\left(p^2\right)$ Loop	${𝒪}_{\mu }\left(p^3\right)$ Loop	Total	${𝒪}_{\mu }\left(p^1\right)$ Tree	${𝒪}_{\mu }\left(p^2\right)$ Loop	${𝒪}_{\mu }\left(p^3\right)$ Loop	Total
${\mu }_{{\overline{D}}^{\ast 0}\to {\overline{D}}^0\gamma }$	$-2.24$	$0.71$	$-0.34$	$-1.86$	$-2.24$	$0.81$	$-0.13$	$-1.57$
${\mu }_{D^{\ast -}\to D^{-}\gamma }$	$0.55$	$-0.21$	$0.19$	$0.54$	$0.55$	$-0.29$	$0.08$	$0.34$
${\mu }_{D_s^{\ast -}\to D_s^{-}\gamma }$	$0.20$	$-0.50$	$0.15$	$-0.15$	$0.20$	$-0.51$	$0.10$	$-0.21$
${\mu }_{B^{\ast +}\to B^{+}\gamma }$	$-1.80$	$0.55$	$-0.34$	$-1.58$	$-1.80$	$0.58$	$-0.30$	$-1.52$
${\mu }_{B^{\ast 0}\to B^0\gamma }$	$0.99$	$-0.16$	$0.17$	1.0	$0.99$	$-0.19$	$0.14$	$0.95$
${\mu }_{B_s^{\ast 0}\to B_s^0\gamma }$	$0.65$	$-0.39$	$0.13$	$0.38$	$0.65$	$-0.39$	$0.11$	$0.36$
${\mu }_{{\overline{D}}^{\ast 0}}$	$1.48$	$-0.71$	$0.40$	$1.18$	$1.48$	$-0.40$	$0.40$	$1.48$
${\mu }_{D^{\ast -}}$	$-1.31$	$0.21$	$-0.21$	$-1.31$	$-1.31$	$-0.07$	$-0.24$	$-1.62$
${\mu }_{D_s^{\ast -}}$	$-0.96$	$0.50$	$-0.16$	$-0.62$	$-0.96$	$0.47$	$-0.21$	$-0.69$
${\mu }_{B^{\ast +}}$	$1.93$	$-0.55$	$0.34$	$1.71$	$1.93$	$-0.52$	$0.36$	$1.77$
${\mu }_{B^{\ast 0}}$	$-0.86$	$0.16$	$-0.17$	$-0.87$	$-0.86$	$0.13$	$-0.19$	$-0.92$
${\mu }_{B_s^{\ast 0}}$	$-0.51$	$0.39$	$-0.13$	$-0.25$	$-0.51$	$0.38$	$-0.14$	$-0.27$

Table 12. Table 12: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = a , … , d ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑎 … 𝑑 \mathcal{C}_{\phi}^{(x)}~{}(x=a,\dots,d) in Eqs. ( 40 )-( 43 ) for the B ∗ superscript 𝐵 ∗ B^{\ast} mesons.

Decay modes	${𝒞}_{\pi }^{\left(a\right)}$	${𝒞}_K^{\left(a\right)}$	${𝒞}_{\pi }^{\left(b\right)}$	${𝒞}_K^{\left(b\right)}$	${𝒞}_{\pi }^{\left(c\right)}$	${𝒞}_K^{\left(c\right)}$	${𝒞}_{\pi }^{\left(d\right)}$	${𝒞}_K^{\left(d\right)}$	${𝒞}_{\eta }^{\left(d\right)}$
$B^{\ast +}\to B^{+}\gamma$	$2$	$2$	$-4$	$-4$	$4$	$4$	$-12a$	$-\frac{8}{3}\left(3a+\tilde{a}\right)$	$-\frac{4}{9}\left(3a-2\tilde{a}\right)$
$B^{\ast 0}\to B^0\gamma$	$-2$	$0$	$4$	$0$	$-4$	$0$	$-4\left(3a-\tilde{a}\right)$	$-\frac{8}{3}\left(3a+\tilde{a}\right)$	$-\frac{4}{9}\left(3a+\tilde{a}\right)$
$B_s^{\ast 0}\to B_s^0\gamma$	$0$	$-2$	$0$	$4$	$0$	$-4$	$0$	$-\frac{8}{3}\left(6a-\tilde{a}\right)$	$-\frac{16}{9}\left(3a+\tilde{a}\right)$

Table 13. Table 13: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = e , … , j ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑒 … 𝑗 \mathcal{C}_{\phi}^{(x)}~{}(x=e,\dots,j) in Eqs. ( 44 )-( 47 ) for the B ∗ superscript 𝐵 ∗ B^{\ast} mesons.

Decay modes	${𝒞}_{\pi }^{\left(e\right)}$	${𝒞}_K^{\left(e\right)}$	${𝒞}_{\eta }^{\left(e\right)}$	${𝒞}_{\pi }^{\left(h\right)}$	${𝒞}_K^{\left(h\right)}$	${𝒞}_{\eta }^{\left(h\right)}$
$B^{\ast +}\to B^{+}\gamma$	$-6a$	$-\frac{4}{3}\left(3a-\tilde{a}\right)$	$-\frac{2}{9}\left(3a+2\tilde{a}\right)$	$3a+2\tilde{a}$	$\frac{2}{3}\left(3a+2\tilde{a}\right)$	$\frac{1}{9}\left(3a+2\tilde{a}\right)$
$B^{\ast 0}\to B^0\gamma$	$-2\left(3a+\tilde{a}\right)$	$-\frac{4}{3}\left(3a-\tilde{a}\right)$	$-\frac{2}{9}\left(3a-\tilde{a}\right)$	$3a-\tilde{a}$	$\frac{2}{3}\left(3a-\tilde{a}\right)$	$\frac{1}{9}\left(3a-\tilde{a}\right)$
$B_s^{\ast 0}\to B_s^0\gamma$	$0$	$-\frac{4}{3}\left(6a+\tilde{a}\right)$	$-\frac{8}{9}\left(3a-\tilde{a}\right)$	$0$	$\frac{4}{3}\left(3a-\tilde{a}\right)$	$\frac{4}{9}\left(3a-\tilde{a}\right)$

Table 14. Table 14: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = a , c , d , e ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑎 𝑐 𝑑 𝑒 \mathcal{C}_{\phi}^{(x)}~{}(x=a,c,d,e) in Eqs. ( 76 )-( 80 ) for the B ∗ superscript 𝐵 ∗ B^{\ast} mesons.

States	${𝒞}_{\pi }^{\left(a\right)}$	${𝒞}_K^{\left(a\right)}$	${𝒞}_{\pi }^{\left(c\right)}$	${𝒞}_K^{\left(c\right)}$	${𝒞}_{\pi }^{\left(d\right)}$	${𝒞}_K^{\left(d\right)}$	${𝒞}_{\pi }^{\left(e\right)}$	${𝒞}_K^{\left(e\right)}$	${𝒞}_{\eta }^{\left(e\right)}$
$B^{\ast +}$	$-\frac{1}{2}$	$-\frac{1}{2}$	$2$	$2$	$-2$	$-2$	$-3a$	$-\frac{2}{3}\left(3a-\tilde{a}\right)$	$-\frac{1}{9}\left(3a+2\tilde{a}\right)$
$B^{\ast 0}$	$\frac{1}{2}$	$0$	$-2$	$0$	$2$	$0$	$-\left(3a+\tilde{a}\right)$	$-\frac{2}{3}\left(3a-\tilde{a}\right)$	$-\frac{1}{9}\left(3a-\tilde{a}\right)$
$B_s^{\ast 0}$	$0$	$\frac{1}{2}$	$0$	$-2$	$0$	$2$	$0$	$-\frac{2}{3}\left(6a+\tilde{a}\right)$	$-\frac{4}{9}\left(3a-\tilde{a}\right)$

Table 15. Table 15: The flavor-dependent coefficients 𝒞 ϕ ( x ) ​ ( x = g , l + m ) superscript subscript 𝒞 italic-ϕ 𝑥 𝑥 𝑔 𝑙 𝑚 \mathcal{C}_{\phi}^{(x)}~{}(x=g,l+m) in Eqs. ( 82 )-( 84 ) for the B ∗ superscript 𝐵 ∗ B^{\ast} mesons.

States	${𝒞}_{\pi }^{\left(g\right)}$	${𝒞}_K^{\left(g\right)}$	${𝒞}_{\eta }^{\left(g\right)}$	${𝒞}_{\pi }^{\left(lm\right)}$	${𝒞}_K^{\left(lm\right)}$	${𝒞}_{\eta }^{\left(lm\right)}$
$B^{\ast +}$	$-3a$	$-\frac{2}{3}\left(3a+\tilde{a}\right)$	$-\frac{1}{9}\left(3a-2\tilde{a}\right)$	$\frac{1}{2}\left(2\tilde{a}-3a\right)$	$\frac{1}{3}\left(2\tilde{a}-3a\right)$	$\frac{1}{18}\left(2\tilde{a}-3a\right)$
$B^{\ast 0}$	$-\left(3a-\tilde{a}\right)$	$-\frac{2}{3}\left(3a+\tilde{a}\right)$	$-\frac{1}{9}\left(3a+\tilde{a}\right)$	$-\frac{1}{2}\left(3a+\tilde{a}\right)$	$-\frac{1}{3}\left(3a+\tilde{a}\right)$	$-\frac{1}{18}\left(3a+\tilde{a}\right)$
$B_s^{\ast 0}$	$0$	$-\frac{2}{3}\left(6a-\tilde{a}\right)$	$-\frac{4}{9}\left(3a+\tilde{a}\right)$	$0$	$-\frac{2}{3}\left(3a+\tilde{a}\right)$	$-\frac{2}{9}\left(3a+\tilde{a}\right)$

Equations

$$⟨P(p^{\prime })|J_{em}^{\mu }(q^2)|V(p,{\epsilon }_V)⟩=e{\mu }^{\prime }(q^2){ϵ}^{\mu \nu \alpha \beta }{p}_{\nu }{q}_{\alpha }{\epsilon }_{V\beta },$$

$$H_{\mathrm{int}}=\int d^{3}xe{A}_{\mu }{J}_{em}^{\mu },$$

$$|\mathcal{H}(p)⟩=\sqrt{m_H}\left[|\mathcal{H}(v)⟩+\mathcal{O}(1/m_H)\right].$$

$$⟨P(p^{\prime })|J_{em}^{\mu }|V(p,{\epsilon }_V)⟩=e\sqrt{m_V{m}_P}{\mu }^{\prime }(q^2){ϵ}^{\mu \nu \alpha \beta }{v}_{\nu }{q}_{\alpha }{\epsilon }_{V\beta },$$

$$\Gamma \left[V\to P\gamma \right]=\frac{1}{3}\int d{\Omega }_{\hat{q}}\frac{1}{32{\pi }^2}\frac{|\mathbf{q}|}{m_V^2}\sum |\mathcal{M}{|}^2,$$

$$\Gamma \left[V\to P\gamma \right]=\frac{\alpha }{3}\frac{m_P}{m_V}{\left|{\mu }^{\prime }(0)\right|}^2|\mathbf{q}{|}^3,$$

$${\mu }_{V\to P\gamma }=\frac{e}{2}{\mu }^{\prime }(0).$$

$${\mathcal{G}}^{\mu }(q^2)$$

$$q^{\mu }=(p-p^{\prime }{)}^{\mu }=(0,\mathbf{Q}),\mathbf{Q}=Q\hat{z},p^{\mu }=(p^0,\frac{1}{2}\mathbf{Q}),$$

$$p^{\prime \mu }=(p^0,-\frac{1}{2}\mathbf{Q}),-q^2=Q^2\ge 0,p^0=\sqrt{m_V^2+\frac{1}{4}{Q}^2}.$$

$${\mathcal{G}}^0(Q^2)$$

$$\mathcal{G}(Q^2)$$

$${\mathcal{G}}_C$$

$${\mathcal{G}}_Q$$

$${\mathcal{G}}_M$$

$$\phi=\left(\begin{array}[]{ccc}{\pi^{0}+\frac{1}{\sqrt{3}}\eta}&{\sqrt{2}\pi^{+}}&{\sqrt{2}K^{+}}\\ {\sqrt{2}\pi^{-}}&{-\pi^{0}+\frac{1}{\sqrt{3}}\eta}&{\sqrt{2}K^{0}}\\ {\sqrt{2}K^{-}}&{\sqrt{2}\bar{K}^{0}}&{-\frac{2}{\sqrt{3}}\eta}\end{array}\right),$$

$${\Gamma }_{\mu }$$

$$u_{\mu }$$

$$u^2=U=\exp \left(\frac{i\varphi }{f_{\varphi }}\right),r_{\mu }=l_{\mu }=-eQ{A}_{\mu },$$

$$J_{\mu }^{\ell }=\frac{2}{3}\bar{u}{\gamma }_{\mu }u-\frac{1}{3}\bar{d}{\gamma }_{\mu }d-\frac{1}{3}\bar{s}{\gamma }_{\mu }s.$$

$${\mathcal{L}}_{\varphi \gamma }^{(2)}=\frac{f_{\varphi }^2}{4}\mathrm{Tr}\left[{\nabla }_{\mu }U{\left({\nabla }^{\mu }U\right)}^{†}\right],$$

$${\nabla }_{\mu }U={\partial }_{\mu }U-i{r}_{\mu }U+iU{l}_{\mu }.$$

$$\mathcal{H}$$

$$\bar{\mathcal{H}}$$

$$P=({\bar{D}}^0,D^{-},D_s^{-}),P^{\ast }=({\bar{D}}^{0\ast },D^{\ast -},D_s^{\ast -}),$$

$$P=(B^{+},B^0,B_s^0),P^{\ast }=(B^{\ast +},B^{\ast 0},B_s^{\ast 0}).$$

$${\mathcal{L}}_{H\varphi }^{(1)}=-i⟨\bar{\mathcal{H}}v\cdot \mathcal{D}\mathcal{H}⟩-\frac{1}{8}\Delta ⟨\bar{\mathcal{H}}{\sigma }^{\mu \nu }\mathcal{H}{\sigma }_{\mu \nu }⟩+g⟨\bar{\mathcal{H}}/u{\gamma }_5\mathcal{H}⟩,$$

$${\mathcal{L}}_{H\gamma }^{(2)}=\tilde{a}⟨\bar{\mathcal{H}}{\sigma }^{\mu \nu }{\tilde{f}}_{\mu \nu }^{+}\mathcal{H}⟩+a⟨\mathcal{H}{\sigma }^{\mu \nu }\bar{\mathcal{H}}⟩\mathrm{Tr}(f_{\mu \nu }^{+}),$$

$$f_{\mu \nu }^R$$

$$f_{\mu \nu }^{\pm }$$

$${\tilde{f}}_{\mu \nu }^{\pm }$$

$${\mathcal{L}}_{H\varphi \varphi }^{(2)}$$

$$3\otimes \bar{3}$$

$$8\otimes 8$$

Full text

I introduction

Electromagnetic form factors play a very important role in mapping out the internal structures of nucleons, which offer valuable information about the distribution of the constituent quarks and gluon degree of freedom in nucleons Gao:2003ag ; Arrington:2006zm ; Pacetti:2015iqa ; Chupp:2017rkp . Probing the shape and inner structure of hadrons still remain an intriguing and challenging topic. Especially, in recent decades, a large number of exotic states were observed in experiments, many of which cannot be readily reconciled with the predictions of the conventional quark models Chen:2016qju ; Guo:2017jvc ; Liu:2019zoy .

Magnetic moments can be related to the form factors by extrapolating the form factor $\mathcal{G}_{M}(q^{2})$ to zero moment transfer Peskin:1995ev . Unlike proton and neutron, vast majority of hadronic states are unstable against strong interactions Tanabashi:2018oca . Thus, their magnetic moments cannot be directly measured with the conventional ways due to their very short lifetime. Therefore, the radiative transition becomes a very effective way to help us catch a glimpse of quark dynamics in the hadrons. In addition, the quark model cannot give the nonanalytic dependence of the magnetic moments, such as the $\log X$ term. These terms are much more difficult to naively estimate and may be sometimes singular to give the much enhanced contributions which cannot be predicted accurately unless carefully calculated.

In this work, we focus on the charmed and bottom vector mesons, i.e., ($\bar{D}^{\ast 0}$, $D^{\ast-}$, $D_{s}^{\ast-}$), and ($B^{\ast+}$, $B^{\ast 0}$, $B_{s}^{\ast 0}$). As a consequence of heavy quark spin symmetry, the mass shifts between these spin triplets and singlets are generally small. Because of the small phase space, the dominant decay channels are one-pion emission transitions and radiative decays for the charmed vector mesons, while only radiative decays are allowed for the bottom vector mesons.

From Review of Particle Physics (RPP) Tanabashi:2018oca , only the width of $D^{\ast\pm}\to D^{\pm}\gamma$ is known by combining the decay branching ratio and the total width of $D^{\ast\pm}$. For the other radiative decay modes, only the branching ratios are available, and the absolute widths are still absent in experiments. Even worse, there is no experimental information for the radiative transitions of the $B^{\ast}$ mesons.

Many theoretical methods have been applied to study the radiative decays of the $D^{\ast}$ and $B^{\ast}$ mesons, such as various quark models Sucipto:1987qj ; Kamal:1992uv ; Ivanov:1994ji ; Jaus:1996np ; Priyadarsini:2016tiu ; Choi:2007se , heavy quark effective theory and vector meson dominance model Colangelo:1993zq , quark-potential models Godfrey:1985xj ; Simonis:2016pnh ; Ebert:2002xz ; Bose:1980vy ; Goity:2000dk ; Simonis:2018rld , QCD sum rules Dosch:1995kw ; Zhu:1996qy ; Aliev:1994nq , lattice QCD simulations Becirevic:2009xp , constituent quark-meson model Deandrea:1998uz , chiral effective field theory Cho:1992nt ; Cheng:1992xi ; Amundson:1992yp ; Casalbuoni:1996pg , extended Nambu-Jona-Lasinio model Deng:2013uca ; Luan:2015goa , and so on.

Here, we adopt the $\mathrm{SU}(3)$ chiral perturbation theory ($\chi$PT) to investigate the radiative decay properties and magnetic moments of the $D^{\ast}$ and $B^{\ast}$ mesons. The framework of $\chi$PT has been widely used to study the radiative decays and magnetic moments of the charmed and bottom vector mesons111In Refs. Cho:1992nt ; Cheng:1992xi , Cho et al and Cheng et al calculated the decay widths of $D^{\ast}\to D\gamma$ and $B^{\ast}\to B\gamma$ at the tree level in the heavy hadron chiral theory, respectively. Our Lagrangians are the same with Refs. Cho:1992nt ; Cheng:1992xi at the leading order. In Ref. Amundson:1992yp , Amundson et al investigated the same process with the same framework to the next-to-leading order. But the heavy quark spin symmetry breaking effect is ignored. Cho:1992nt ; Cheng:1992xi ; Amundson:1992yp ; Casalbuoni:1996pg , the octet baryons Jenkins:1992pi ; Meissner:1997hn , the doubly charmed and bottom heavy baryons Li:2017cfz ; Li:2017pxa ; Meng:2017dni ; Blin:2018pmj , the singly heavy baryons Jiang:2015xqa ; Wang:2018xoc ; Meng:2018gan ; Wang:2018cre , as well as the related chiral quark-soliton model for singly heavy baryons Yang:2018uoj ; Kim:2018nqf . In our calculations, we construct the effective Lagrangians with chiral symmetry and heavy quark symmetry up to $\mathcal{O}(p^{4})$. There are two independent low-energy constants (LECs) at the leading order, which correspond to the contributions from the light quark and heavy quark electromagnetic currents, respectively. These two LECs can be estimated with quark model or other theoretical methods. The contributions from the tree diagrams at the next-to-leading order can be absorbed into the ones from the leading order. At the next-to-next-to leading order, the tree diagram incorporates three independent LECs, which cannot be determined due to lack of experimental data. We present our numerical results up to $\mathcal{O}(p^{4})$, and consider the contributions from $\mathcal{O}(p^{4})$ tree diagrams as errors.

Our numerical results are calculated both in SU(2) and SU(3) cases with the mass splitting kept and unkept in loop diagrams. The partial decay widths of $D^{\ast-}\to D^{-}\gamma$ predicted in different scenarios are consistent with the experimental data.

This paper is organized as follows. The definitions of the electromagnetic form factors and magnetic moments are given in Sec. II. The effective Lagrangians are constructed in Sec. III. The analytical expressions and numerical results for the transition magnetic moments and magnetic moments are presented in Sec. IV and Sec. V, respectively. A summary is given in Sec. VI. Some supplemental materials for the $B^{\ast}$ mesons, the loop integrals and an estimation of the light quark mass with the vector meson dominance model are collected in Appendices A, B and C, respectively.

II electromagnetic form factors and magnetic moments

We first consider the radiative transition process $V\to P\gamma$, where $V$ stands for the vector mesons ($D^{\ast}$ or $B^{\ast}$), and $P$ denotes the pseudoscalar mesons ($D$ or $B$). The M1 transition form factor $\mu^{\prime}(q^{2})$ can be defined through a covariant expression of the hadronic matrix elements Casalbuoni:1996pg ,

[TABLE]

where $J_{em}^{\mu}$ is the electromagnetic current at hadronic level. $q_{\alpha}=(p-p^{\prime})_{\alpha}$ is the transferred momentum, and $\varepsilon_{V\beta}$ denotes the polarization vector of the initial vector meson.

The interaction Hamiltonian can then be written as

[TABLE]

where $A_{\mu}$ is the photon field.

For a heavy meson $M$ that is composed of a heavy antiquark $\bar{Q}$ and a light quark $q$, the ground spin doublet $(P,P^{\ast})$ can be represented by a $4\times 4$ Dirac-type matrix $\mathcal{H}$. We use the $\mathcal{H}(p)$ and $\mathcal{H}(v)$ to denote the heavy meson fields in relativistic and heavy meson effective theory (HMET) convention, respectively. They can be related with each other by

[TABLE]

Then, in the framework of HMET, Eq. (1) can be reexpressed as

[TABLE]

where the recoil effect is negligible in the above equation.

With the above preparation, one can easily get the expression of the decay rate,

[TABLE]

where $\mathcal{M}$ represents the transition amplitude, and a sum over the final-state photon polarization and an average over the initial $V$ polarization has been performed.

Explicitly, we have

[TABLE]

where $\alpha=1/137$ is the fine-structure constant. The transition magnetic moment $\mu_{V\to P\gamma}$ can be defined as

[TABLE]

In the following, we derive the magnetic moment of a vector meson. The matrix element of $J_{em}^{\mu}(q^{2})$ are defined in terms of the standard Lorentz covariant decomposition Arnold:1979cg ,

[TABLE]

This expression can be simplified under Breit frame. In our calculations, we define

[TABLE]

A straightforward derivation under Breit frame gives the time component of $\mathcal{G}^{\mu}(q^{2})$ as

[TABLE]

where $\mathcal{G}_{C}$ and $\mathcal{G}_{Q}$ represent charge and quadrupole form factors, respectively. In deriving Eq. (9), we have used the transverse condition of the initial and final state polarization vectors, i.e., $p\cdot\varepsilon=0$, and $p^{\prime}\cdot\varepsilon^{\prime\ast}=0$.

Similarly, the space component of $\mathcal{G}^{\mu}(q^{2})$ is

[TABLE]

where $\mathcal{G}_{M}$ is the magnetic dipole form factor. The expressions of $\mathcal{G}_{C}$, $\mathcal{G}_{Q}$ and $\mathcal{G}_{M}$ read

[TABLE]

where $\eta={Q^{2}}/{(4m_{V}^{2})}$.

III effective Lagrangians

III.1 The leading order chiral Lagrangians

We first introduce the Lagrangian of Goldstone bosons and photon. The octet of the light pseudoscalar field is represented by the field $U(x)=e^{i\phi/f_{\phi}}$ with

[TABLE]

where the $\eta$ field denotes the octet $\eta_{8}$. In the SU(3) quark model, the $\eta$ meson is regarded as the mixing of the octet $\eta_{8}$ and singlet $\eta_{0}$ with $|\eta\rangle=\cos\theta|\eta_{8}\rangle-\sin\theta|\eta_{0}\rangle$ Wang:2016qmz , where $\theta\simeq-19.1^{\circ}$ is determined by the experimental measurements Coffman:1988ve ; Jousset:1988ni . Because the mixing angle is not very large and the $\eta$ field only serves as the quantum fluctuations in the loops, so the mixing effect is ignored in our calculations.

The definitions of the chiral connection and axial-vector current are

[TABLE]

where

[TABLE]

and $Q=Q_{l}=\mathrm{diag}(2/3,-1/3,-1/3)$ represents the electric charge matrix of the light current $J^{\ell}_{\mu}$,

[TABLE]

$f_{\phi}$ is the decay constant of the light pseudoscalar mesons. The experimental value of $f_{\phi}$ for $\phi=\pi,K,\eta$ are $f_{\pi}=92.4$ MeV, $f_{K}=113$ MeV, and $f_{\eta}=116$ MeV, respectively.

The leading order $[\mathcal{O}(p^{2})]$ Lagrangian for the interaction of the light pseudoscalars and photon read Li:2017cfz ; Li:2017pxa ; Meng:2017dni

[TABLE]

where

[TABLE]

We use $\operatorname{Tr}(X)$ and $\langle X\rangle$ to denote the trace for $X$ in flavor space and spinor space, respectively.

We construct the effective Lagrangian for the heavy mesons with the superfield $\mathcal{H}$. For a heavy meson composed of a heavy antiquark $\bar{Q}$ and a light quark $q$, the superfield $\mathcal{H}$ is defined as

[TABLE]

where for the charmed mesons

[TABLE]

and for the bottom mesons

[TABLE]

The leading order Lagrangians for describing the interactions between the heavy matter field and light pseudoscalars are Wise:1992hn ; Manohar:2000dt

[TABLE]

where the covariant derivative $\mathcal{D}_{\mu}=\partial_{\mu}+\Gamma_{\mu}$. Here, the electric charge matrix in the $\Gamma_{\mu}$ should be replaced by those corresponding to the heavy mesons. For instance, $Q=Q_{D}=\mathrm{diag}(0,-1,-1)$ for $(\bar{D}^{\ast 0},D^{\ast-},D_{s}^{\ast-})$, and $Q=Q_{B}=\mathrm{diag}(1,0,0)$ for $(B^{\ast+},B^{\ast 0},B_{s}^{\ast 0})$, respectively. The second term in Eq. (22) is due to the mass difference between $P$ and $P^{\ast}$, and $\Delta=m_{P^{\ast}}-m_{P}$ stands for the mass splitting. $g$ represents the axial coupling constant. For the $D$ meson, its value can be extracted by the partial decay width of $D^{\ast+}\to D^{0}\pi^{+}$ Tanabashi:2018oca ; Ahmed:2001xc , while for the $B$ meson, $g$ can only be determined via the theoretical method, such as the quark model Casalbuoni:1996pg and lattice QCD Ohki:2008py ; Detmold:2012ge .

We also need the Lagrangians to describe the (transition) magnetic moments at the tree level, which can be written as Li:2017cfz ; Li:2017pxa ; Meng:2017dni

[TABLE]

where $\tilde{a}$ and $a$ are two LECs. The first and second terms correspond to the contributions from the light quark and heavy antiquark, respectively. The field strength tensor $\tilde{f}_{\mu\nu}^{+}$ and $f_{\mu\nu}^{+}$ are defined as

[TABLE]

where $Q=Q_{D}$ for the $D$ mesons and $Q=Q_{B}$ for the $B$ mesons, respectively. From Eq. (III.1) we can see that $\tilde{f}_{\mu\nu}^{+}$ is proportional to $Q_{l}$ and traceless. $f_{\mu\nu}^{+}$ is not traceless because it contains the electric charge matrix of the heavy mesons. One can also understand Eq. (23) from the standpoint of group representation theory. Recall that $3\otimes\bar{3}=1\oplus 8$, and the operator $\tilde{f}_{\mu\nu}^{+}$ transforms as the adjoint representation. Thus the two terms in Eq. (23) correspond to $8\otimes 8\to 1$ and $1\otimes 1\to 1$, respectively.

In the following, we construct the Lagrangian for the interactions of the heavy mesons and light pseudoscalar mesons at $\mathcal{O}(p^{2})$, which will contribute to the $\mathcal{O}(p^{4})$ magnetic moment at the one-loop level Li:2017cfz ; Li:2017pxa ; Meng:2017dni ,

[TABLE]

Actually, the tensor structure sandwiched between $\bar{\mathcal{H}}$ and $\mathcal{H}$ in Eq. (25) can also be $\{u_{\mu},u_{\nu}\}$ and $\mathrm{Tr}(u_{\mu}u_{\nu})$. For the $\mathrm{SU}(3)$ group representations,

[TABLE]

The axial-vector current $u_{\mu}$ (or $u_{\nu}$) transforms as the adjoint representation, thus $\mathrm{Tr}(u_{\mu}u_{\nu})$, $[u_{\mu},u_{\nu}]$ and $\{u_{\mu},u_{\nu}\}$ belong to $1$, $8_{1}$ and $8_{2}$ flavor representations, respectively. But $\mathrm{Tr}(u_{\mu}u_{\nu})$ and $\{u_{\mu},u_{\nu}\}$ would vanish when they are contracted with $\sigma^{\mu\nu}$ because of the symmetric Lorentz indices $\mu$ and $\nu$. Therefore, only one independent term containing $[u_{\mu},u_{\nu}]$ survives in Eq. (25).

III.2 The next-to-leading order chiral Lagrangians

The electromagnetic chiral Lagrangians at $\mathcal{O}(p^{3})$ read Wang:2018cre

[TABLE]

The structure is similar to those in Eq. (23). The possible contributions that include covariant derivative $\mathcal{D}_{\mu}$ can be absorbed into the LECs $\tilde{c}$ and $c$ with the equation of motion of the heavy mesons. Meanwhile, the contributions from Eq. (III.2) can be absorbed into Eq. (23) by renormalizing the LECs $\tilde{a}$ and $a$, i.e.,

[TABLE]

III.3 The next-to-next-to-leading order chiral Lagrangians

At this order, we also employ group representation methods to construct the electromagnetic chiral Lagrangians (one can find the possible flavor structures in Table 1), the detailed form reads Li:2017cfz ; Li:2017pxa ; Meng:2017dni

[TABLE]

where a spurion $\chi_{\pm}$ is introduced as

[TABLE]

At the leading order,

[TABLE]

In principle, there should be six independent terms in Eq. (29) as the possible flavor structures listed in Table 1. However, the terms $\mathrm{Tr}(\chi_{+})\mathrm{Tr}(f_{\mu\nu}^{+})$ and $\mathrm{Tr}(\chi_{+})\tilde{f}_{\mu\nu}^{+}$ can also be absorbed into Eq. (23) by renormalizing $\tilde{a}$ and $a$, respectively. Another term $[\tilde{\chi}_{+},\tilde{f}_{\mu\nu}^{+}]$ vanishes since both $\tilde{\chi}_{+}$ and $\tilde{f}_{\mu\nu}^{+}$ are diagonal matrices at the leading order. Therefore, only three terms are retained in Eq. (29).

IV Radiative transitions

IV.1 Power counting and analytical expressions for the transition from factors

The standard power counting scheme gives the chiral order of a Feynman diagram as

[TABLE]

where $N_{L}$, $I_{M}$ and $I_{H}$ are the numbers of loops, internal light pseudoscalar lines and internal heavy meson lines, respectively. $N_{n}$ is the number of vertices governed by the $n$th order Lagrangians. Usually, the order of the (transition) magnetic moment is

[TABLE]

Therefore, the transition form factors of $V\to P\gamma$ can be expressed as follows,

[TABLE]

where the numbers in the parentheses are the chiral order $\mathcal{O}_{\mu}$.

We first study the $V\to P\gamma$ transitions. The tree diagrams are illustrated in Fig. 1. Expanding the Lagrangians in Eqs. (23) and (29) we can easily get the transition amplitudes of Figs. 1$(a)$ and 1$(b)$, respectively. We can extract the $q^{2}$-independent form factor $\mu^{\prime}$ at the tree level by comparing the transition amplitudes with Eqs. (1) and (4). The expressions read

[TABLE]

We show the analytical expressions for the $D$ mesons, and display the expressions for the $B$ mesons in Appendix A.

The one-loop Feynman diagrams that contribute to the transition processes are shown in Fig. 2. Here, we need to deal with the loop integrals when extracting the $q^{2}$-dependent form factors from the transition amplitudes. Various types of loop integrals $\mathcal{J}$ have been defined and given in Appendix B. In the following, we list the transition form factors of Figs. 2$(a)$-2$(j)$ with a compact form, correspondingly.

[TABLE]

where the summations over $\phi$ denote the possible contributions from the light pseudoscalars ($\phi$ could be $\pi$, $K$, $\eta$) in the loops. $\mathcal{C}_{\phi}^{(x)}~{}(x=a,\dots,j)$ are the flavor-dependent coefficients, and their values are given in Tables 2-3. In the $\mathcal{J}$ functions, $m_{\phi}$ is the mass of the corresponding particle in the loop. $\mathcal{E}$ is the residual energy of heavy mesons, which is defined as $\mathcal{E}=E_{D^{(\ast)}}-m_{D^{(\ast)}}$. $\mathcal{E}$ is set to be zero in our calculations. $q$ denotes the transferred momentum carried by the photon. $D$ is the dimension in dimensional regularization. $\{X\}_{r}$ represents the finite part of $X$, which is defined in Appendix B. The coefficients $\mathcal{C}_{\phi}^{(ij)}$ can be obtained via the relation

[TABLE]

IV.2 Estimation of the leading order LECs

In $\mu^{\prime(1)}_{\mathrm{tree}}$, there exist two $\mathcal{O}(p^{2})$ LECs $\tilde{a}$ and $a$ (see Eq. (23)). Another $\mathcal{O}(p^{2})$ LEC $b$ (see Eq. (25)) resides in $\mu^{\prime(3)}_{\mathrm{loop}}$. In the following, we estimate the values of $\tilde{a}$, $a$ and $b$ with the quark model and resonance saturation model, respectively. It is hard to determine the other higher-order LECs ($\tilde{d}$, $\bar{d}$, and $d$) in $\mu^{\prime(3)}_{\mathrm{tree}}$ for the moment because of very limited experimental data. Therefore, we consider the contributions from $\mu^{\prime(3)}_{\mathrm{tree}}$ as errors of our numerical results.

We first demonstrate how to determine $\tilde{a}$ and $a$ from the scenario of constituent quark model. In this model, the transition matrix element of $V\to P\gamma$ in the rest frame of the initial state can be written as Cheng:1992xi

[TABLE]

where $e_{i}$ and $m_{i}$ are the electric charge and mass of $i$th quark in the heavy meson, $\bm{\sigma}$ and $\mathbf{B}$ are the Pauli matrix and magnetic field, respectively. For simplicity, we choose the direction of the magnetic field $\mathbf{B}$ along the $z$ axis. In order to work out Eq. (49), we need the flavor-spin wave functions of $V$ and $P$, which read

[TABLE]

Inserting Eqs. (50) and (51) into Eq. (49), one can obtain

[TABLE]

where $\mu_{i}=e_{i}/(2m_{i})$. Matching Eq. (52) with the leading order transition amplitudes (i.e., replacing the $\mu^{\prime}(q^{2})$ in Eq. (4) with the expressions in Eqs. (34)-(36), and making use of $B^{k}(\bm{q})=-i\epsilon^{ijk}q^{i}A^{j}(\bm{q})$), one can easily get

[TABLE]

where $m_{q}$ and $m_{\bar{Q}}$ are the masses of light constituent quark and heavy antiquark in heavy mesons (in Appendix C we also give an estimation of the light quark mass with vector meson dominance model), respectively.

Next, we evaluate the value of LEC $b$ in Eq. (25) using the resonance saturation model Epelbaum:2001fm ; Ecker:1988te . A diagrammatic presentation of the resonance saturation scheme is illustrated in Fig. 3. We need the interaction Lagrangians for $VP\rho$ and $\rho\pi\pi~{}(\phi KK)$. The $VP\rho$ Lagrangian can be obtained with local hidden symmetry Casalbuoni:1996pg , which reads

[TABLE]

where

[TABLE]

and

[TABLE]

The $\rho\pi\pi~{}(\phi KK)$ Lagrangian reads Casalbuoni:1996pg

[TABLE]

where the expression of $\Gamma_{\mu}^{(0)}$ can be extracted from the chiral connection defined in Eq. (13) by omitting the photon field.

With the above preparations, we use the amplitude of Fig. 3$(a)$ governed by Lagrangians in Eqs. (54) and (60) to match the amplitude of Fig. 3$(b)$ depicted by the Lagrangian in Eq. (25). We can get the $b$ explicitly

[TABLE]

where $g_{v}=5.8$, $\lambda=0.56~{}\mathrm{GeV}^{-1}$ Li:2012ss . $m_{v}$ is the mass of the exchanged light vector meson, such as $m_{\rho}=0.77~{}\mathrm{GeV}$, and $m_{\phi}=1.02~{}\mathrm{GeV}$ Tanabashi:2018oca . The sign of $b$ is determined with the quark model.

The numerical values of the parameters are Tanabashi:2018oca ; Li:2017cfz ; Li:2017pxa ; Meng:2017dni ; Ahmed:2001xc ; Ohki:2008py

[TABLE]

Since the masses of the mesons have been precisely measured in experiments Tanabashi:2018oca , so we don't quote their minor errors. The masses of the constituent quarks are adopted from previous works Li:2017cfz ; Li:2017pxa ; Meng:2017dni . Generally, it's hard to give the errors of the masses of the constituent quarks, because these values used in different quark models vary a lot some times. In this work, we try to give a conservative estimation by setting the $10\%\times m_{q}$ as the parameter errors. The axial constant $g$ for $D^{\ast}D\pi$ coupling is extracted from the experimental result of the CLEO Collaboration Ahmed:2001xc . The $B^{\ast}B\pi$ coupling is quoted from the unquenched lattice result Ohki:2008py .

IV.3 Numerical results and discussions

With the parameters listed above, we first show the transition magnetic moments of $V\to P\gamma$ calculated under SU(2) and SU(3) symmetries222Here, SU(2) and SU(3) symmetries only imply the effective Lagrangians are constructed under these two symmetries. The SU(3) breaking effect is included explicitly in our calculations. For example, we use the $m_{u,d,s}$ and the physical masses of $\pi$, $K$ and $\eta$ in Eq. (IV.2) as inputs. in the upper half parts of Tables 10 and 11, correspondingly. In Tables 10 and 11, the transition magnetic moments $\mu_{V\to P\gamma}$ are given order by order. As expected, the convergence of the chiral expansion in the SU(2) case is better than that in SU(3). Besides, we also calculate the $\mu_{V\to P\gamma}$ with the mass splitting $\Delta$ in the propagators of the loop diagrams kept and unkept. The influence of $\Delta$ in the charm sector is more significant than that in the bottom sector because the mass difference of the charmed mesons is larger than that of the bottom mesons.

In the SU(2) case, the mass splitting $\Delta$ only appears in the loop diagrams. The transition magnetic moments at $\mathcal{O}_{\mu}(p^{1})$ remain unchanged no matter we choose $\Delta=0$ or $\Delta\neq 0$. At $\mathcal{O}_{\mu}(p^{2})$, the correction from the finite mass splitting ($\Delta\neq 0$) is about $40\%$ and $20\%$ for $\mu_{\bar{D}^{\ast}\to\bar{D}\gamma}$ and $\mu_{B^{\ast}\to B\gamma}$, respectively. Such a correction is also significant at $\mathcal{O}_{\mu}(p^{3})$. Similar behavior is observed in the SU(3) case at each order. In Table 4, we show the contribution of each loop diagram to the transition magnetic moment of $\bar{D}^{\ast 0}\to\bar{D}\gamma$ in different cases. The contributions of the diagrams 2$(f)$ and 2$(g)$ vanish in the heavy quark limit. Except for the diagrams 2$(b)$ and 2$(c)$, other diagrams that contain the heavy meson internal line are effected by the mass splitting $\Delta$. For the charmed vector mesons, $\Delta>m_{\pi}$, so the loop integrals with the nonanalytic structures $\log\frac{y^{2}+m_{\pi}^{2}-\Delta^{2}-i\varepsilon}{\lambda^{2}}$ and $\sqrt{m_{\pi}^{2}-\Delta^{2}-i\varepsilon}$ (see Appendix B) would largely impact the numerical result. This is vividly reflected in Table 4. However, for the bottom vector mesons, $\Delta\simeq 1/3m_{\pi}$, so the influence of $\Delta$ on the bottom sector is not so obvious.

The corresponding decay widths evaluated in different cases are illustrated in Table 6. The errors in our calculations can stem from many sources, such as quark masses, hadron masses, coupling constants, higher order contributions and so on. As shown in the RPP Tanabashi:2018oca , the errors of the hadron masses appeared in this work are very small, so we ignore their effects. Meanwhile, the axial coupling constant extracted from the experiments and lattice QCD are also very small. Furthermore, the convergence of chiral expansion works very well in our calculations. Therefore, we consider two main error sources. The first one is the contribution of the $\mathcal{O}(p^{4})$ Lagrangians (see Eq. (29)). Since the LECs in Eq. (29) cannot be fixed at present, we adopt the nonanalytic dominance approximation to give an estimation of the $\mathcal{O}(p^{4})$ tree diagram Wang:2018atz . The second one is the uncertainty from the quark models. For example, the masses of constituent quarks are different in various models (see Table 5). We take this uncertainty into account. The change of the quark masses would lead to a $10\%$ variation of the leading order LECs.

From Table 6, we see that the decay rate for $D^{\ast-}\to D^{-}\gamma$ calculated in different scenarios all agrees with the experimental data. The branching ratios for the other decay channels cannot be obtained due to the absence of the total widths of these states in experiments at present. We also compare our results with other model predictions, such as light-front quark model Choi:2007se , relativistic independent quark model Priyadarsini:2016tiu , relativistic quark model Ebert:2002xz and QCD sum rules Zhu:1996qy . The results in these literatures are consistent with our calculations. Furthermore, the results from the extended Bag model Simonis:2016pnh ; Simonis:2018rld , lattice QCD simulations Becirevic:2009xp and extended Nambu-Jona-Lasinio model Deng:2013uca are also compatible with ours.

Up to now, only the full width of $D^{\ast\pm}$ and the branching ratio of $D^{\ast\pm}\to D^{\pm}\gamma$ are available in RPP Tanabashi:2018oca . The life time of $\bar{D}^{\ast 0}$ and $D_{s}^{\ast-}$ has not been measured yet. The convergence of the chiral expansion for transition magnetic moments calculated in SU(3) case with $\Delta\neq 0$ is very reasonable. Therefore, as a byproduct, we use the following relation with our results in SU(3) and $\Delta\neq 0$ as inputs to estimate the full widths of these two states,

[TABLE]

where the total width $\Gamma_{\mathrm{tot}}(\bar{D}^{\ast 0})$ in the above equation can be extracted with the predicted $\Gamma(\bar{D}^{*0}\rightarrow\bar{D}^{0}\gamma)$. Analogously, $\Gamma_{\mathrm{tot}}(D_{s}^{\ast\pm})$ can also be calculated with the same way as in the case of $\bar{D}^{\ast 0}$. The full widths of $\bar{D}^{\ast 0}$ and $D_{s}^{\ast-}$ are estimated to be

[TABLE]

respectively.

V Magnetic moments

The anomalous magnetic moments of nucleons reveal that the proton and neutron are not elementary particles and they have internal substructures. As in the case of nucleons, the magnetic moments of $D^{\ast}$ and $B^{\ast}$ also encode important information of their underlying substructures.

V.1 Analytical expressions for the magnetic moments

We have studied the radiative transitions $V\to P\gamma$ in previous section. The decay rate for $D^{\ast-}\to D^{-}\gamma$ is consistent with the experimental data. So we adopt the same set of parameters to calculate the magnetic moments of the $D^{\ast}$ and $B^{\ast}$ mesons. The $\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$ tree level Feynman diagrams that contribute to the magnetic moments are displayed in Fig. 4.

In the following, we write out the magnetic moments of the $D^{\ast}$ mesons from Figs. 4$(a)$ and 4$(b)$,

[TABLE]

The magnetic moments from the one-loop diagrams in Fig. 5 are given as

[TABLE]

where the values of the coefficients $\mathcal{C}_{\phi}^{(x)}~{}(x=a,\dots,o)$ for the $D^{\ast}$ mesons are listed in Tables 7-8. In Eqs. (76) and (77), we have used the relation $\mathcal{J}_{31}^{T}=-\frac{1}{2}\mathcal{J}_{21}^{T}$ when $q^{2}=0$ Wang:2018atz . The unlisted coefficients $\mathcal{C}_{\phi}^{(b)}$ and $\mathcal{C}_{\phi}^{(f)}$ can be obtained by the relation

[TABLE]

Analogous to the transition form factors $\mu_{V\to P\gamma}^{\prime}$ in Eq. (33), the magnetic moments $\mu_{V}$ can be written as

[TABLE]

where $\mu^{(1)}_{\mathrm{tree}}$, $\mu^{(2)}_{\mathrm{loop}}$ and $\mu^{(3)}_{\mathrm{loop}}$ can be calculated by using the parameters in Eq. (IV.2) as inputs.

V.2 Numerical results and discussions

The numerical results for the magnetic moments $\mu_{V}$ calculated in the SU(2) and SU(3) cases are given order by order in the lower half parts of Tables 10 and 11, respectively. We see that the convergence of the chiral expansion in the SU(2) case remains very good and the convergence is also reasonable in SU(3).

In the SU(2) case, the magnetic moments at $\mathcal{O}_{\mu}(p^{1})$ are independent of $\Delta$. The $\Delta\neq 0$ correction reduces the $\mu_{V}$ at $\mathcal{O}_{\mu}(p^{2})$ and $\mathcal{O}_{\mu}(p^{3})$. Consequently, the total results are increased. In the heavy quark limit, there exists a strict relationship between $\mu_{V}$ and $\mu_{V\to P\gamma}$ at each order, i.e., $|\mu_{V}|=|\mu_{V\to P\gamma}|$ when we take $D=4$ and $\Delta=0$ in the loop functions. Both the radiative transitions and magnetic moments of the heavy vector mesons are solely governed by the light quark since the heavy quark decouples completely.

In the SU(3) case, one notices the similar variation trend at $\mathcal{O}_{\mu}(p^{2})$ as in SU(2). At $\mathcal{O}_{\mu}(p^{3})$, there is a moderate increasement when the mass splitting is included. The total results are enhanced in the $\Delta\neq 0$ case. It's interesting to diagnose the convergence of the chiral expansion for magnetic moments from a straightforward dimensional analysis.

The magnetic moments $\mu_{V}$ at the leading order (LO), next-to-leading order (NLO), and next-to-next-to-leading order (NNLO) can be parameterized as follows,

[TABLE]

where the coefficients $A,\dots,E$ are order-one dimensionless constants. $\Lambda_{\chi}\sim 1$ GeV denotes the chiral breaking scale.

For the $D^{\ast-}$ and $B^{\ast 0}$ mesons, the internal light pseudoscalar lines in the $\mathcal{O}(p^{3})$ loop diagrams (Figs. 5$(a)$ and 5$(b)$) can only be the charged pions. But for the $\mathcal{O}(p^{4})$ wave function renormalization diagrams (Figs. 5$(l)$-5$(o)$), $K$ and $\eta$ would contribute to the loops. Since $m_{K}/m_{\pi}\simeq 3.5$ and $m_{\eta}/m_{\pi}\simeq 4.0$, the $\mathcal{O}(p^{4})$ contribution would be enhanced to the same magnitude as the $\mathcal{O}(p^{3})$ correction from the SU(3) violation effect. Let's take the $D^{\ast-}$ meson as an example. In the strict heavy quark limit, the contributions of Figs. 5$(a)$ and 5$(b)$ are equal. However, for the charmed mesons, the mass splitting $\Delta>m_{\pi}$. Hence the amplitudes of Figs. 5$(a)$ and 5$(b)$ are of similar size but with opposite sign, which makes the contributions of these two diagrams for $D^{\ast-}$ largely cancel with each other. This effect does not contribute to the transition magnetic moments, because there is only a single one-loop diagram with $\Delta=0$ at $\mathcal{O}(p^{3})$ level (see Fig. 2$(a)$). Moreover, the influence of the mass splitting on the magnetic properties of the $B^{\ast}$ is not obvious due to $\Delta\ll m_{\phi}$ in the bottom sector.

The magnetic moments for the $\bar{D}^{\ast}$ and $B^{\ast}$ mesons calculated in different cases are shown in Table 9, where the errors also stem from $\mu^{(3)}_{\mathrm{tree}}$, i.e., $\mathcal{O}(p^{4})$ Lagrangians and quark models. The magnetic moments of the vector $\bar{Q}u$, $\bar{Q}d$ and $\bar{Q}s$ states given by the bag model Bose:1980vy ; Simonis:2018rld and Nambu-Jona-Lasinio model Luan:2015goa are compatible with our predictions.

VI Summary

For the ground vector $\bar{Q}q$ states, heavy quark spin symmetry implies the mass splitting between the spin triplets $V$ and spin singlets $P$ is very small, which is of the same order as the pion mass $m_{\pi}$. Thus the decay modes of $V$ are largely restricted. For the ground-state charmed vector mesons, the dominant decay channels are $V\to P\pi$ and $V\to P\gamma$. For the $b\bar{q}$ states, the only dominant decay modes are $V\to P\gamma$.

In this work, we calculate the decay rates of $V\to P\gamma$ for the charmed and bottom vector mesons. Our result for $D^{\ast-}\to D^{-}\gamma$ is in accordance with the experimental measurement. We also investigate the convergence of the chiral expansion of the transition magnetic moments in the SU(2) and SU(3) cases with the mass splitting kept and unkept. The results indicate that the convergence in SU(2) case is very good, and it is reasonable for SU(3) likewise. The effect of the mass splitting for the charmed mesons is more significant than that for the bottom mesons. The radiative decay widths of the $D^{\ast}$ and $B^{\ast}$ mesons from other theoretical models and lattice QCD simulations also are consistent with ours. As a byproduct, the full widths of $\bar{D}^{\ast 0}$ and $D_{s}^{\ast-}$ are estimated to be $77.7^{+26.7}_{-20.5}$ keV and $0.62^{+0.45}_{-0.50}$ keV, respectively.

In this work, we also calculate the magnetic moments of the $D^{\ast}$ and $B^{\ast}$ mesons. Our results agree with the predictions of bag model Bose:1980vy ; Simonis:2018rld and NJL model Luan:2015goa . The magnetic moments of heavy vector mesons are good platforms to probe their inner structures. For example, the magnetic moment of $\bar{D}^{\ast 0}$ should be zero if we use the classical formula $\bm{\mu}=\frac{e}{2m}\mathbf{S}$ (where $e$, $m$ and $\mathbf{S}$ denote the charge, mass and spin, respectively). The large anomalous magnetic moment of $\bar{D}^{\ast 0}$ clearly demonstrates that it is not a point particle.

In summary, we have systematically studied the radiative transitions and magnetic moments of charmed and bottom vector mesons with $\chi$PT up to $\mathcal{O}(p^{4})$. Our numerical results are presented up to this order with different scenarios. The LECs $\tilde{a}$, $a$ and $b$ in the $\mathcal{O}(p^{2})$ Lagrangians are estimated with the quark model and resonance saturation model, respectively. We notice the one-loop chiral correction plays a very important role in mediating the (transition) magnetic moments. Our result indicates the quark model prediction is not enough to describe the magnetic properties of the charmed and bottom vector mesons. The quark dynamics of the light degree of freedom that is related with the spontaneous breaking of chiral symmetry is non-negligible.

The present investigations of the radiative decays of $D^{\ast}$ and $B^{\ast}$ shall be helpful to the future measurement at facilities such as BelleII and LHCb. Furthermore, the analytical expressions derived in $\chi$PT shall be helpful for the chiral extrapolations of lattice QCD simulations on the electromagnetic transitions and magnetic moments of heavy vector mesons.

Acknowledgments

B. W is very grateful to X. L. Chen and W. Z. Deng for helpful discussions. This project is supported by the National Natural Science Foundation of China under Grants 11575008, 11621131001 and National Key Basic Research Program of China(2015CB856700).

Appendix A Some supplemental materials for the $B^{\ast}$ mesons

The transition form factors from Figs. 1$(a)$ and 1$(b)$ for the $B^{\ast}$ mesons read

[TABLE]

The magnetic moments from Figs. 4$(a)$ and 4$(b)$ for the $B^{\ast}$ mesons read

[TABLE]

The flavor dependent coefficients $\mathcal{C}_{\phi}^{(x)}$ in Eqs. (40)-(47) and Eqs. (76)-(84) for the $B^{\ast}$ mesons are listed in Tables 12-13 and Tables 14-15, respectively.

Appendix B Loop integrals

Here, we show the detailed forms of the $\mathcal{J}$ functions used in the text. One can find the complete forms in Ref. Wang:2018atz .

[TABLE]

where $q\vee v\equiv q^{\alpha}v^{\beta}+q^{\beta}v^{\alpha}$. The $\mathcal{J}$ functions defined above can be calculated with the dimensional regularization in $D$ dimensions. In the following, we write out the expressions of the used $\mathcal{J}$ functions.

[TABLE]

where $\tilde{\Delta}=y^{2}+\tilde{A},~{}\tilde{A}=m^{2}-\omega^{2}-i\varepsilon$; $\bar{\Delta}=x(x-1)q^{2}+m^{2}-i\varepsilon$; $\Delta=y^{2}+A$, $A=x(x-1)q^{2}+m^{2}-(\omega-xq_{0})^{2}-i\varepsilon$.

$L$ is defined as

[TABLE]

where $\gamma_{E}$ is the Euler-Mascheroni constant $0.5772157$. We adopt the $\overline{\mathrm{MS}}$ scheme to renormalize the loop integrals, which is equivalent to making use of the following relation,

[TABLE]

where $\{X\}_{r}$ represents the finite part of $X$.

Appendix C Estimating the light quark mass with vector meson dominance model

In general, the transition form factor of $V\to P\gamma$ at the leading order can be parameterized as follows,

[TABLE]

where $\mathcal{Q}_{\bar{Q}}$ and $\mathcal{Q}_{q}$ denote the charge matrices of $\bar{Q}$ and $q$, respectively. $\Lambda_{\bar{Q}}$ and $\Lambda_{q}$ are the mass parameters that can be understood as the masses of the constituent quarks in the quark model. Heavy quark symmetry guarantees $\Lambda_{\bar{Q}}\approx m_{\bar{Q}}$ (see the discussions in Ref. Manohar:2000dt ). However, the photon coupling to the light quark part of the electromagnetic current is not fixed by the heavy quark symmetry, thus the $\Lambda_{q}$ is not a ``well-defined" constant, its value is largely model dependent to some extent. Here, we adopt the vector meson dominance (VMD) model Pacetti:2015iqa ; Colangelo:1993zq to estimate the value of $\Lambda_{q}$.

In the VMD model, the light quark part of the electromagnetic current $\langle P|J_{\mu}^{\ell}|V\rangle$ can be expressed as follows by inserting the light vector resonance $\mathcal{V}$,

[TABLE]

where the $\langle P_{a}\mathcal{V}|V_{a}\rangle$ vertex is given in Eq. (54) (A diagrammatic presentation of Eq. (C) is shown in Fig. 6). The matrix element $\langle 0|\bar{q}_{a}\gamma_{\mu}q_{a}|\mathcal{V}(q,\varepsilon_{\mathcal{V}}^{\lambda})\rangle$ can be calculated by assuming the SU(3) symmetry with

[TABLE]

where $f_{\mathcal{V}}$ and $\varepsilon_{\mathcal{V}}$ denote the decay constant and polarization vector of the light vector meson, respectively. $(T^{a})_{lm}=\delta_{al}\delta_{am}$, and $a=1,2,3$ for $u,d,s$, respectively. The $f_{\mathcal{V}}$ can be determined by the electromagnetic decay $\mathcal{V}\to e^{+}e^{-}$. $f_{\rho}=0.17$ GeV2 for the $\rho$ meson, and $f_{\phi}=0.25$ GeV2 for the $\phi$ meson Colangelo:1993zq .

Following the same procedure in obtaining Eq. (53), one can get

[TABLE]

where the values of $g_{v}$ and $\lambda$ are the same as those in Eq. (61). One can obtain $\Lambda_{q}$ by considering the SU(3) breaking effect in Eq. (117), eventually,

[TABLE]

These values are very close to the $m_{u}$, $m_{d}$ and $m_{s}$ given in Eq. (IV.2).