# Analysis of the electromagnetic form factors and the radiative decays of the
vector heavy-light mesons

Jie Lu1,2 Guo-Liang Yu1,2<EMAIL_ADDRESS>Zhi-Gang Wang1
<EMAIL_ADDRESS>Bin Wu1 1 Department of Mathematics and Physics, North China
Electric Power University, Baoding 071003, People’s Republic of China
2 Hebei Key Laboratory of Physics and Energy Technology, North China Electric
Power University, Baoding 071000, China

###### Abstract

In this article, we analyze the electromagnetic form factors of the vector
heavy-light mesons to the pseudoscalar heavy-light mesons in the framework of
three-point QCD sum rules, where the contributions of vacuum condensate terms
$\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$,
$\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and
$\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ are considered.
With these results, we also obtain the radiative decay widths of the vector
heavy-light mesons and then compare our results with those of other
collaboration’s. The final results about the radiative decay widths are
$\Gamma(D^{*0}\to D^{0}\gamma)=1.74^{+0.40}_{-0.37}$ keV, $\Gamma(D^{*+}\to
D^{+}\gamma)=0.17^{+0.08}_{-0.07}$ keV, $\Gamma(D_{s}^{*}\to
D_{s}\gamma)=0.029^{+0.009}_{-0.008}$ keV, $\Gamma(B^{*0}\to
B^{0}\gamma)=0.018^{+0.006}_{-0.005}$ keV, $\Gamma(B^{*+}\to
B^{+}\gamma)=0.015^{+0.007}_{-0.007}$ keV and $\Gamma(B^{*}_{s}\to
B_{s}\gamma)=0.016^{+0.003}_{-0.005}$ keV.

###### pacs:

13.25.Ft; 14.40.Lb

##  I Introduction

As one of the most interesting research areas in particle physics, the
experimental and theoretical investigations about the heavy flavor hadrons
have been developing rapidly. This is due to their key roles in both
understanding of QCD long-distance dynamics, and the determination of the
fundamental parameters of the standard model (SM). Especially, the decay
behavior of heavy flavor hadrons is an important and interesting research
topic for testing the SM and finding new physics beyond the SM. These decay
processes can be classified into leptonic, semileptonic and nonleptonic decays
which involve both electroweak and strong interactions. However, theoretical
investigations about the heavy hadrons is very difficult because the QCD is
non-perturbative in low energy regions. Therefore, many phenomenological
approachesBeneke:1999br ; Muta:2000ti ; Lu:2000em ; Keum:2000ph ; Keum:2000wi
; Lu:2000hj ; Bauer:2000yr ; Bauer:2001yt ; FermilabLattice:2011njy ;
Yang:2005bv ; Aliev:2007rq ; Wang:2007ys ; Ghahramany:2009zz ; Wang:2008da ;
Colangelo:2010bg ; Sun:2010nv ; Han:2013zg ; Colangelo:1994jc ; Cho:1992nt ;
Ball:1997rj ; Casalbuoni:1996pg ; Deandrea:1998uz ; Orsland:1998de ;
Goity:2000dk ; Li:2008tk ; Chen:2007na ; Choi:2007se ; Jaus:1996np ;
Wang:2019mhm ; Li:2022vby ; Lu:2022kos ; Cui:2023jiw ; Ivanov:2022nnq ;
Tran:2023hrn  have been employed to analyze the decay processes of the heavy
flavor hadrons.

The QCD sum rules (QCDSR) is one of the most powerful non-perturbative
approach, and has been widely used to analyze the mass spectra and the decay
behavior of hadrons Lucha:2011zp ; Narison:2012xy ; Gelhausen:2013wia ;
Zhu:1996qy ; Zhu:2000py ; Wang:2010it ; Chen:2016phw ; Azizi:2020tgh ;
Duraes:2007te ; Liu:2007fg ; Zhang:2008pm ; Xin:2021wcr ; Li:2015xka ;
Yu:2019sqp ; Wang:2023sii ; Wang:2023kir . In recent years, some tasks were
carried out by three-point QCDSR, such as the analysis of electroweak and
electromagnetic form factorsWang:2007ys ; Aliev:1994nq ; Khosravi:2013lea ;
Peng:2019apl ; Shi:2019hbf ; Zhao:2020mod ; Xing:2021enr ; Zhang:2023nxl , and
the strong coupling constants Azizi:2010jj ; Sundu:2011vz ; Cui:2012wk ;
Bracco:2011pg ; Rodrigues:2017qsm ; Lu:2023gmd ; Lu:2023lvu ; Azizi:2014bua ;
Azizi:2015tya ; Yu:2018hnv ; Lu:2023pcg . These parameters are very important
to analyze the decay process of hadrons. In our previous work, the
$B_{c}^{*}\to B_{c}$ electromagnetic form factor were studied by three-point
QCDSR, and the corresponding radiative decay $B_{c}^{*}\to B_{c}\gamma$ was
calculatedWang:2013cha . In RefYu:2015xwa , we analyzed the momentum dependent
strong coupling constant $G_{D_{s}^{*}D_{s}\phi}(Q^{2})$. According vector
meson dominant(VMD), the electromagnetic coupling constant
$G_{D_{s}^{*}D_{s}\gamma}$ was also obtained by setting $Q^{2}=0$. And the
radiative decay width for $D_{s}^{*}\to D_{s}\gamma$ was determined to be
$0.59\pm 0.15$ keV. As a continuation of these works, we systematically
analyze the electromagnetic form factors of vector heavy-light mesons to
pseudoscalar heavy-light mesons. Based on these results, the decay widths
about vector heavy-light mesons to pseudoscalar heavy-light mesons plus a
photon are obtained. Although some radiative decays, such as $D^{*+}\to
D^{+}\gamma$, $D^{*0}\to D^{0}\gamma$ and $D_{s}^{*}\to D_{s}\gamma$ were
already analyzed by other research groupsCasalbuoni:1996pg ; Deandrea:1998uz ;
Orsland:1998de ; Goity:2000dk ; Aliev:1994nq . However, these results were
obtained by different methods and not consistent well with each other, which
needs further confirmation by other theoretical methods.

The experimental data for the radiative decay of the vector meson $D^{*+}$ in
Particle Data Group(PDG)ParticleDataGroup:2022pth  is
$\Gamma_{Total}(D^{*+})=83.4\pm 1.8$ keV, and $\frac{\Gamma(D^{*+}\to
D^{+}\gamma)}{\Gamma_{Total}(D^{*+})}\approx 1.6\pm 0.4\%$. The decay widths
for $D^{*0}$ and $D_{s}^{*}$ are given as $\Gamma_{Total}(D^{*0})<2.1$ MeV,
$\frac{\Gamma(D^{*0}\to D^{0}\gamma)}{\Gamma_{Total}(D^{*0})}\approx 35.3\pm
0.9\%$, and $\Gamma_{Total}(D_{s}^{*})<1.9$ MeV, $\frac{\Gamma(D_{s}^{*}\to
D_{s}\gamma)}{\Gamma_{Total}(D^{*}_{s})}\approx(93.5\pm 0.7)\%$, respectively.
However, more exact values about these decays have not been determined yet.
For the mesons $B^{*\pm(0)}$ and $B_{s}^{*}$, the radiative decays as their
main decay channels have not been determined in experiment. In general, more
exact experimental results and theoretical predictions need to be provided,
which can help us to understand the nature of the mesons and test the
theoretical models.

The layout of this paper is as follows, After introduction in Sec. I, the
radiative decays of the vector heavy-light mesons are analyzed in the
framework of SM in Sec. II, and the electromagnetic form factor is introduced.
In Sec. III, we systematically analyze the electromagnetic form factors of
vector heavy-light meson to pseudoscalar heavy-light meson by the three-point
QCDSR, where the contributions of perturbative part and vacuum condensate
including $\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma
Gq\rangle$, $\langle g_{s}^{2}G^{2}\rangle$, $\langle f^{3}G^{3}\rangle$ and
$\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ are considered in
QCD side. Sec. IV is employed to present the numerical results and
discussions. Sec. V is reserved as conclusions. Some important figures are
shown in Appendix.

##  II The radiative decay of vector heavy-light mesons

To analyze the radiative decay of vector heavy-light mesons($\mathbb{V}$) to
pseudoscalar heavy-light mesons($\mathbb{P}$) (
$\mathbb{V}\to\mathbb{P}\gamma$), we firstly write the following
electromagnetic lagrangian $\mathscr{L}_{EM}$,

$\displaystyle\mathscr{L}_{EM}=-e_{Q}\bar{Q}\gamma_{\mu}QA^{\mu}-e_{q}\bar{q}\gamma_{\mu}qA^{\mu}$
(1)

where $A^{\mu}$ is the electromagnetic field, $q$ and $Q$ denote the light (u,
d or s) and heavy (c, b) quarks. $e_{q}$ and $e_{Q}$ are electric charges
which are taken as $-\frac{1}{3}e$ for d, s and b quarks, and $\frac{2}{3}e$
for u and c quarks. The assignments of $\mathbb{P}$, $\mathbb{V}$, $Q$ and $q$
for different decay modes are shown in Table 1.

Table 1: The assignments of $\mathbb{P}$, $\mathbb{V}$, $Q$ and $q$ for different decay modes. Mode |  $\mathbb{P}$ |  $\mathbb{V}$ |  $Q$ |  $q$
---|---|---|---|---
$D^{*+}\to D^{+}\gamma$ |  $D^{+}$ |  $D^{*+}$ |  c |  d
$D^{*0}\to D^{0}\gamma$ |  $D^{0}$ |  $D^{*0}$ |  c |  u
$D_{s}^{*+}\to D^{+}_{s}\gamma$ |  $D_{s}^{+}$ |  $D_{s}^{*+}$ |  c |  s
$B^{*-}\to B^{-}\gamma$ |  $B^{-}$ |  $B^{*-}$ |  b |  u
$\bar{B}^{*0}\to\bar{B}^{0}\gamma$ |  $\bar{B}^{0}$ |  $\bar{B}^{*0}$ |  b |  d
$B_{s}^{*0}\to B_{s}^{0}\gamma$ |  $B_{s}^{0}$ |  $B_{s}^{*0}$ |  b |  s

From this lagrangian(Eq. (1)), the decay amplitude can be expressed as the
following form,

$\displaystyle\left\langle{\mathbb{P}(p^{\prime})\gamma(q)}\right|{\mathscr{L}_{EM}}\left|{\mathbb{V}(p)}\right\rangle$
$\displaystyle=\left\langle{\mathbb{P}(p^{\prime})\gamma(q)}\right|(-{e_{Q}}\bar{Q}{\gamma_{\mu}}Q-{e_{q}}\bar{q}{\gamma_{\mu}}q){A^{\mu}}\left|{\mathbb{V}(p)}\right\rangle$
$\displaystyle=-\left\langle{\gamma(q)}\right|{A^{\mu}}\left|0\right\rangle\left\langle{\mathbb{P}(p^{\prime})}\right|({e_{Q}}\bar{Q}{\gamma_{\mu}}Q+{e_{q}}\bar{q}{\gamma_{\mu}}q)\left|{\mathbb{V}(p)}\right\rangle$
(2)

with

$\displaystyle\left\langle{\mathbb{P}(p^{\prime})}\right|{e_{Q}}\bar{Q}{\gamma_{\mu}}Q+{e_{q}}\bar{q}{\gamma_{\mu}}q\left|{\mathbb{V}(p)}\right\rangle$
$\displaystyle={\varepsilon_{\mu\sigma\alpha\beta}}{\zeta^{\sigma}}{p^{\alpha}}p{{}^{\prime\beta}}\frac{{V({q^{2}})}}{{{m_{\mathbb{V}}}+{m_{\mathbb{P}}}}}$
(3)

where $V(q^{2})$ is electromagnetic form factor, $p^{\alpha}$ and
$p^{\prime\beta}$ are four momentum of the vector and pseudoscalar heavy-light
mesons, and $\zeta^{\sigma}$ is the polarization vector of vector heavy-light
mesons. According to Eqs. (II) and (II), the decay amplitude finally can be
written as,

$\displaystyle\left\langle{\mathbb{P}(p^{\prime})\gamma(q)}\right|{\mathscr{L}_{EM}}\left|{\mathbb{V}(p)}\right\rangle$
$\displaystyle={\varepsilon_{\mu\sigma\alpha\beta}}{\zeta^{\sigma}}{p^{\alpha}}p{{}^{\prime\beta}}\frac{{V(0)}}{{{m_{\mathbb{V}}}+{m_{\mathbb{P}}}}}(-i){\eta^{\mu}}$
(4)

where $\eta_{\mu}$ is the polarization vector of the photon. This decay
process can be explicitly illustrated by feynman diagrams in Fig. 1.

Figure 1: The feynman diagrams of radiative decay for
$\mathbb{V}\to\mathbb{P}\gamma$.

##  III QCDSR for the electromagnetic form factors

To study the electromagnetic form factor of the vector heavy-light meson to
the pseudoscalar heavy-light meson, we firstly write down the three-point
correlation function,

$\displaystyle{\Pi_{\mu\nu}}(p,p^{\prime})$
$\displaystyle={i^{2}}\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
(5) $\displaystyle\times\left\langle
0\right|T\\{{J_{\mathbb{P}}}(x){j_{\mu}}(y)J_{\nu}^{+}(0)\\}\left|0\right\rangle$

where $T$ denotes the time ordered product, $J_{\mathbb{P}}$ and $J^{+}_{\nu}$
are the interpolating currents of the pseudoscalar and vector heavy-light
mesons, and $j_{\mu}$ is the electromagnetic current. These currents are taken
as the following forms,

$\displaystyle{J_{\mathbb{P}}}(x)$ $\displaystyle=\bar{q}(x)i{\gamma_{5}}Q(x)$
$\displaystyle{j_{\mu}}(y)$
$\displaystyle={{e_{Q}}\bar{Q}(y){\gamma_{\mu}}}Q(y)+{{e_{q}}\bar{q}(y){\gamma_{\mu}}}q(y)$
$\displaystyle{J_{\nu}}(0)$ $\displaystyle=\bar{q}(0){\gamma_{\nu}}Q(0)$ (6)

In the framework of QCDSR, the correlation function will be calculated at two
sides which are called the phenomenological side and the QCD side. Then,
according to the quark hadron duality, the calculations of these two sides
will be coordinated and the sum rules for electromagnetic form factor in the
hadronic level can be obtained.

###  III.1 The Phenomenological side

In the phenomenological side, a complete sets of hadronic states with the same
quantum numbers as the interpolating currents $J_{\mathbb{P}}$ and
$J^{+}_{\nu}$ are inserted into the correlation function. By using the
dispersion relation, the correlation function can be written
asColangelo:2000dp ,

$\displaystyle{\Pi^{\mathrm{phy}}_{\mu\nu}}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J_{\mathbb{P}}}(0)\left|{\mathbb{P}(p^{\prime})}\right\rangle\left\langle{\mathbb{V}(p)}\right|J_{\nu}^{+}(0)\left|0\right\rangle}}{{(m_{\mathbb{V}}^{2}-{p^{2}})(m_{\mathbb{P}}^{2}-p{{}^{\prime
2}})}}$ (7)
$\displaystyle\times\left\langle{\mathbb{P}(p^{\prime})}\right|{j_{\mu}}(y)\left|{\mathbb{V}(p)}\right\rangle+h.c.$

where $h.c.$ represents the contributions coming from higher resonances and
continuum states. The hadronic matrix elements are defined by the following
parameterized equations,

$\displaystyle\left\langle
0\right|{J_{\mathbb{P}}}(0)\left|{\mathbb{P}(p^{\prime})}\right\rangle$
$\displaystyle=\frac{{{f_{\mathbb{P}}}m_{\mathbb{P}}^{2}}}{{{m_{q}}+{m_{Q}}}}$
$\displaystyle\left\langle{\mathbb{V}(p)}\right|J_{\nu}^{+}(0)\left|0\right\rangle$
$\displaystyle={f_{\mathbb{V}}}{m_{\mathbb{V}}}\zeta_{\nu}^{*}$
$\displaystyle\left\langle{\mathbb{P}(p^{\prime})}\right|{j_{\mu}}(y)\left|{\mathbb{V}(p)}\right\rangle$
$\displaystyle={\varepsilon_{\mu\sigma\alpha\beta}}{\zeta^{\sigma}}{p^{\alpha}}p{{}^{\prime\beta}}\frac{{V({q^{2}})}}{{{m_{\mathbb{V}}}+{m_{\mathbb{P}}}}}$
(8)

with $q=p-p^{\prime}$. In these above equations, $f_{\mathbb{P}}$ and
$f_{\mathbb{V}}$ are the decay constants of the pseudoscalar and vector heavy-
light mesons, $\varepsilon_{\mu\sigma\alpha\beta}$ is the Levi-Civita tensor,
and $V(q^{2})$ is the electromagnetic form factor. $\zeta_{\nu}$ is the
polarization vector of vector heavy-light meson with the following properties,

$\displaystyle\zeta_{\nu}^{*}{\zeta_{\sigma}}=-{g_{\nu\sigma}}+\frac{{{p_{\nu}}{p_{\sigma}}}}{{{p^{2}}}}$
(9)

With these above equations, the correlation function in phenomenological side
can be expressed as the following form,

$\displaystyle{\Pi^{\mathrm{phy}}_{\mu\nu}}(p,p^{\prime})$
$\displaystyle=-\frac{{{f_{\mathbb{P}}}m_{\mathbb{P}}^{2}{f_{\mathbb{V}}}{m_{\mathbb{V}}}}}{{({m_{q}}+{m_{Q}})({m_{\mathbb{V}}}+{m_{\mathbb{P}}})}}$
(10)
$\displaystyle\times\frac{{V({q^{2}}){\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}p{{}^{\prime\beta}}}}{{(m_{\mathbb{V}}^{2}-{p^{2}})(m_{\mathbb{P}}^{2}-p{{}^{\prime
2}})}}+h.c.$
$\displaystyle=\Pi^{\mathrm{phy}}(p,p^{\prime})\varepsilon_{\mu\nu\alpha\beta}{p^{\alpha}}p{{}^{\prime\beta}}+h.c.$

where $\Pi^{\mathrm{phy}}(p,p^{\prime})$ is called scalar invariant amplitude,
$\varepsilon_{\mu\nu\alpha\beta}p^{\alpha}p^{\prime\beta}$ is the
corresponding tensor structure.

###  III.2 The QCD side

In this part, we firstly contract the quark filed with Wick’s theorem, and
then do the operator product expansion(OPE). After contraction of the quark
filed, the correlation function in QCD side can be expressed as,

$\displaystyle{\Pi^{\mathrm{QCD}}_{\mu\nu}}(p,p^{\prime})$
$\displaystyle=-i\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
(11)
$\displaystyle\times\\{{e_{Q}}Tr[S_{Q}^{nk}(y){\gamma_{\nu}}S_{q}^{km}(-x){\gamma_{5}}S_{Q}^{mn}(x-y){\gamma_{\mu}}]$
$\displaystyle+{e_{q}}Tr[S_{Q}^{nk}(x){\gamma_{\nu}}S_{q}^{km}(-y){\gamma_{\mu}}S_{q}^{mn}(y-x){\gamma_{5}}]\\}$
$\displaystyle=e_{Q}{\Pi_{1\mu\nu}}(p,p^{\prime})+e_{q}{\Pi_{2\mu\nu}}(p,p^{\prime})$

where $S_{q}^{mn}(x)$ and $S_{Q}^{mn}(x)$ are the full propagators of light
and heavy quarks, and they have the following formsPascual:1984zb ;
Reinders:1984sr ,

$\displaystyle{S_{q}^{mn}}(x)$
$\displaystyle=\frac{{i{\delta^{mn}}x\\!\\!\\!/}}{{2{\pi^{2}}{x^{4}}}}-\frac{{{\delta^{mn}}{m_{q}}}}{{4{\pi^{2}}{x^{4}}}}-\frac{{{\delta^{mn}}\left\langle{\bar{q}q}\right\rangle}}{{12}}+\frac{{i{\delta^{mn}}x\\!\\!\\!/{m_{q}}\left\langle{\bar{q}q}\right\rangle}}{{48}}$
$\displaystyle-\frac{{{\delta^{mn}}{x^{2}}\left\langle{\bar{q}{g_{s}}\sigma
Gq}\right\rangle}}{{192}}+\frac{{i{\delta^{mn}}{x^{2}}x\\!\\!\\!/{m_{q}}\left\langle{\bar{q}{g_{s}}\sigma
Gq}\right\rangle}}{{1152}}$
$\displaystyle-\frac{{i{g_{s}}G_{\alpha\beta}^{a}t_{mn}^{a}(x\\!\\!\\!/{\sigma^{\alpha\beta}}+{\sigma^{\alpha\beta}}x\\!\\!\\!/)}}{{32{\pi^{2}}{x^{2}}}}-\frac{{i{\delta^{mn}}{x^{2}}x\\!\\!\\!/g_{s}^{2}{{\left\langle{\bar{q}q}\right\rangle}^{2}}}}{{7776}}$
$\displaystyle-\frac{{{\delta^{mn}}{x^{4}}\left\langle{\bar{q}q}\right\rangle\left\langle{g_{s}^{2}GG}\right\rangle}}{{27648}}-\frac{{\left\langle{{{\bar{q}}^{n}}{\sigma^{\mu\nu}}{q^{m}}}\right\rangle{\sigma_{\mu\nu}}}}{8}$
$\displaystyle-\frac{{\left\langle{{{\bar{q}}^{n}}{\gamma^{\mu}}{q^{m}}}\right\rangle{\gamma_{\mu}}}}{4}+...$
$\displaystyle{S_{Q}^{mn}}(x)$
$\displaystyle=\frac{i}{{{{(2\pi)}^{4}}}}\int{{d^{4}}k}{e^{-ik\cdot
x}}\\{\frac{{{\delta^{mn}}}}{{k\\!\\!\\!/-{m_{Q}}}}$ (12)
$\displaystyle-\frac{{{g_{s}}G_{\alpha\beta}^{a}t_{mn}^{a}}}{4}\frac{{{\sigma^{\alpha\beta}}(k\\!\\!\\!/+{m_{Q}})+(k\\!\\!\\!/+{m_{b}}){\sigma^{\alpha\beta}}}}{{{{({k^{2}}-m_{Q}^{2})}^{2}}}}$
$\displaystyle+\frac{{{g_{s}}{D_{\alpha}}G_{\beta\lambda}^{a}t_{mn}^{a}({f^{\lambda\beta\alpha}}+{f^{\lambda\alpha\beta}})}}{{3{{({k^{2}}-m_{Q}^{2})}^{4}}}}$
$\displaystyle-\frac{{g_{s}^{2}{{({t^{a}}{t^{b}})}_{mn}}G_{\alpha\beta}^{a}G_{\mu\nu}^{b}({f^{\alpha\beta\mu\nu}}+{f^{\alpha\mu\beta\nu}}+{f^{\alpha\mu\nu\beta}})}}{{4{{({k^{2}}-m_{Q}^{2})}^{5}}}}$
$\displaystyle+...\\}$

Here, $\langle g_{s}^{2}GG\rangle=\langle
g_{s}^{2}G^{a}_{\alpha\beta}G^{a\alpha\beta}\rangle$,
$D_{\alpha}=\partial_{\alpha}-ig_{s}G_{\alpha}^{a}t^{a}$,
$t^{a}=\frac{\lambda^{a}}{2}$. $\lambda^{a}$($a$=1,…,8) are the Gell-Mann
matrices, $m$ and $n$ are color indices, $q$ denotes the light quarks(u, d,
and s), $Q$ represents the heavy quarks(c and b),
$\sigma_{\alpha\beta}=\frac{i}{2}[\gamma_{\alpha},\gamma_{\beta}]$, and
$f^{\lambda\alpha\beta}$, $f_{\alpha\beta\mu\nu}$ have the following forms,

$\displaystyle{f^{\lambda\alpha\beta}}=(k\\!\\!\\!/+{m_{Q}}){\gamma^{\lambda}}(k\\!\\!\\!/+{m_{Q}}){\gamma^{\alpha}}(k\\!\\!\\!/+{m_{Q}}){\gamma^{\beta}}(k\\!\\!\\!/+{m_{Q}})$
(13) $\displaystyle{f^{\alpha\beta\mu\nu}}=$
$\displaystyle(k\\!\\!\\!/+{m_{Q}}){\gamma^{\alpha}}(k\\!\\!\\!/+{m_{Q}}){\gamma^{\beta}}(k\\!\\!\\!/+{m_{Q}})$
(14)
$\displaystyle{\gamma^{\mu}}(k\\!\\!\\!/+{m_{Q}}){\gamma^{\nu}}(k\\!\\!\\!/+{m_{Q}})$

From these above equations, the correlation function in QCD side(Eq. (11)) can
also be transformed into the following form,

$\displaystyle\Pi^{\mathrm{QCD}}_{\mu\nu}(p,p^{\prime})=\Pi^{\mathrm{QCD}}(p^{2},p^{\prime
2},q^{2})\epsilon_{\mu\nu\alpha\beta}p^{\alpha}p^{\prime\beta}$ (15)

where $\Pi^{\mathrm{QCD}}$ is the scalar invariant amplitude in QCD side, and
it can be divided into several parts according to different condensate terms,

$\displaystyle\Pi^{\mathrm{QCD}}$
$\displaystyle=\Pi^{\mathrm{pert}}+\Pi^{\langle\overline{q}q\rangle}+\Pi^{\langle
g_{s}^{2}G^{2}\rangle}+\Pi^{\langle\overline{q}g_{s}\sigma Gq\rangle}$ (16)
$\displaystyle+\Pi^{\langle
f^{3}G^{3}\rangle}+\Pi^{\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}+...$

Here, $\Pi^{\mathrm{pert}}$ represents the perturbative part, and
$\Pi^{\langle\overline{q}q\rangle}$, $\Pi^{\langle g_{s}^{2}G^{2}\rangle}$,
$\Pi^{\langle\overline{q}g_{s}\sigma Gq\rangle}$, $\Pi^{\langle
f^{3}G^{3}\rangle}$ and $\Pi^{\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}$ are the vacuum condensate terms with dimension 3, 4,
5, 6 and 7, respectively. The perturbative part and the gluon condensate terms
$\Pi^{\langle g_{s}^{2}G^{2}\rangle}$ and $\Pi^{\langle f^{3}G^{3}\rangle}$
can be written as the following form according to the dispersion relation,

$\displaystyle\Pi(p,p^{\prime})=-\int\limits_{s_{1}}^{\infty}{\int\limits_{u_{1}}^{\infty}{\frac{{{\rho}(s,u,{q^{2}})}}{{(s-{p^{2}})(u-p{{}^{\prime
2}})}}dsdu}}$

where $\rho(s,u,q^{2})$ is the QCD spectral density and it can be represented
as,

$\displaystyle\rho(s,u,{q^{2}})$
$\displaystyle={\rho^{\mathrm{pert}}}(s,u,{q^{2}})+{\rho^{\left\langle{g_{s}^{2}{G^{2}}}\right\rangle}}(s,u,{q^{2}})$
(17)
$\displaystyle+{\rho^{\left\langle{{f^{3}}{G^{3}}}\right\rangle}}(s,u,{q^{2}})$

with $s=p^{2}$, $u=p^{\prime 2}$ and $q=p-p^{\prime}$.

For the perturbative part, we firstly substitute the free propagator of heavy
and light quarks in the momentum space in Eq. (11). Then, the correlation
function can be expressed as the following form after performing the
integrations in the coordinate and momentum space,

$\displaystyle\Pi_{\mu\nu}^{\mathrm{pert}}(p,p^{\prime})=-\frac{3}{{{{(2\pi)}^{4}}}}\int{{d^{4}}k}$
$\displaystyle\times\\{{e_{Q}}\frac{{Tr[(\not{k}+\not{q}+{m_{Q}}){\gamma_{\nu}}(\not{k}-\not{p}^{\prime}+{m_{q}}){\gamma_{5}}(\not{k}+{m_{Q}}){\gamma_{\mu}}]}}{{[{{(k+q)}^{2}}-m_{Q}^{2}][{{(k-p^{\prime})}^{2}}-m_{q}^{2}]({k^{2}}-m_{Q}^{2})}}$
$\displaystyle+{e_{q}}\frac{{Tr[(\not{k}+\not{p}^{\prime}+{m_{Q}}){\gamma_{\nu}}(\not{k}-\not{q}+{m_{q}}){\gamma_{\mu}}(\not{k}+{m_{q}}){\gamma_{5}}]}}{{[{{(k+p^{\prime})}^{2}}-m_{Q}^{2}][{{(k-q)}^{2}}-m_{q}^{2}]({k^{2}}-m_{q}^{2})}}\\}$
$\displaystyle={e_{Q}}\Pi_{1\mu\nu}^{pert}(p,p^{\prime})+{e_{q}}\Pi_{2\mu\nu}^{pert}(p,p^{\prime})$
(18)

Then, the QCD spectral density for the perturbative contribution can be
obtained by putting all the quark lines on mass-shell using the Cutkoskys’s
rules(see Fig. 2),

$\displaystyle\rho_{1\mu\nu}^{\mathrm{pert}}(s,u,{q^{2}})=\frac{3}{{{{(2\pi)}^{3}}}}\int{{d^{4}}k\delta}[{(k+q)^{2}}-m_{Q}^{2}]$
$\displaystyle\times\delta[{(k-p^{\prime})^{2}}-m_{q}^{2}]\delta({k^{2}}-m_{Q}^{2})Tr[(\not{k}+\not{q}+{m_{Q}}){\gamma_{\nu}}$
$\displaystyle\times(\not{k}-\not{p}^{\prime}+{m_{q}}){\gamma_{5}}(\not{k}+{m_{Q}}){\gamma_{\mu}}]$
$\displaystyle\rho_{2\mu\nu}^{\mathrm{pert}}(s,u,{q^{2}})=-\frac{3}{{{{(2\pi)}^{3}}}}\int{{d^{4}}k\delta[{{(k-p^{\prime})}^{2}}-m_{Q}^{2}]}$
$\displaystyle\times\delta[{(k+q)^{2}}-m_{q}^{2}]\delta({k^{2}}-m_{q}^{2})Tr[(\not{k}-\not{p}^{\prime}-{m_{Q}}){\gamma_{\nu}}$
$\displaystyle\times(\not{k}+\not{q}-{m_{q}}){\gamma_{\mu}}(\not{k}-{m_{q}}){\gamma_{5}}]$
(19)

Figure 2: The feynman diagrams for $\Pi^{\mathrm{pert}}_{1}$(a) and
$\Pi^{\mathrm{pert}}_{2}$(b) in Eq. (III.2). The dashed lines denote the
cutkosky cuts.

After performing integrations of these Dirac delta functions in Eq. (III.2),
the spectral density can be expressed as,

$\displaystyle\rho_{1\mu\nu}^{\mathrm{pert}}(s,u,{q^{2}})=\frac{3}{{{{(2\pi)}^{3}}}}\frac{\pi}{{2\sqrt{\lambda(s,u,{q^{2}})}}}Tr\\{[({C_{p}}+1)\not{p}$
$\displaystyle+({C_{p^{\prime}}}-1)\not{p}^{\prime}+{m_{Q}}){\gamma_{\nu}}[{C_{p}}\not{p}+({C_{p^{\prime}}}-1)\not{p}^{\prime}+{m_{q}}]{\gamma_{5}}$
$\displaystyle\times({C_{p}}\not{p}+{C_{p^{\prime}}}\not{p}^{\prime}+{m_{Q}}){\gamma_{\mu}}\\}$
$\displaystyle\rho_{2\mu\nu}^{\mathrm{pert}}(s,u,{q^{2}})=-\frac{3}{{{{(2\pi)}^{3}}}}\frac{\pi}{{2\sqrt{\lambda(s,u,{q^{2}})}}}Tr\\{[C{{}^{\prime}_{p}}\not{p}$
$\displaystyle+(C{{}^{\prime}_{p^{\prime}}}-1)\not{p}^{\prime}-{m_{Q}}){\gamma_{\nu}}[(C{{}^{\prime}_{p}}+1)\not{p}+(C{{}^{\prime}_{p^{\prime}}}-1)\not{p}^{\prime}$
$\displaystyle-{m_{q}}]{\gamma_{\mu}}(C{{}^{\prime}_{p}}\not{p}+C{{}^{\prime}_{p^{\prime}}}\not{p}^{\prime}-{m_{Q}}){\gamma_{5}}\\}$
(20)

where,

$\displaystyle{C_{p}}=\frac{{(u+m_{Q}^{2}-m_{q}^{2})(s+u-{q^{2}})-2u(u-{q^{2}}+m_{Q}^{2}-m_{q}^{2})}}{{\lambda(s,u,{q^{2}})}}$
$\displaystyle{C_{p^{\prime}}}=\frac{{(u-{q^{2}}+m_{Q}^{2}-m_{q}^{2})(s+u-{q^{2}})-2s(u+m_{Q}^{2}-m_{q}^{2})}}{{\lambda(s,u,{q^{2}})}}$
$\displaystyle
C{{}^{\prime}_{p}}=\frac{{(u+m_{q}^{2}-m_{Q}^{2})(s+u-{q^{2}})-2u(u-{q^{2}}+m_{q}^{2}-m_{Q}^{2})}}{{\lambda(s,u,{q^{2}})}}$
$\displaystyle
C{{}^{\prime}_{p^{\prime}}}=\frac{{(u-{q^{2}}+m_{q}^{2}-m_{Q}^{2})(s+u-{q^{2}})-2s(u+m_{q}^{2}-m_{Q}^{2})}}{{\lambda(s,u,{q^{2}})}}$
$\displaystyle\lambda(s,u,{q^{2}})={(s+u-{q^{2}})^{2}}-4su$ (21)

As for the gluon condensate terms $\langle g_{s}^{2}G^{2}\rangle$ and $\langle
f^{3}G^{3}\rangle$, the following integral will be encountered,

$\displaystyle
I_{ijk}=\int{{d^{4}}k\frac{1}{{{{[{{(k+q)}^{2}}-m_{1}^{2}]}^{i}}{{[{{(k-p^{\prime})}^{2}}-m_{2}^{2}]}^{j}}{{({k^{2}}-m_{3}^{2})}^{k}}}}}$
(22)

This integral can also be calculated by Cutkosky’s rules according to the
following transformations,

$\displaystyle
I_{ijk}=\frac{1}{{(i-1)!(j-1)!(k-1)!}}\frac{{{\partial^{i-1}}}}{{\partial{A^{i-1}}}}\frac{{{\partial^{j-1}}}}{{\partial{B^{j-1}}}}\frac{{{\partial^{k-1}}}}{{\partial{C^{k-1}}}}\int{{d^{4}}k}$
$\displaystyle\times\frac{1}{{[{{(k+q)}^{2}}-A][{{(k-p^{\prime})}^{2}}-B]({k^{2}}-C)}}{|_{A\to{m_{1,}}B\to{m_{2,}}C\to{m_{3}}}}$
$\displaystyle\to\frac{{{{(-2\pi i)}^{3}}}}{{{{(2\pi
i)}^{2}}}}\frac{1}{{(i-1)!(j-1)!(k-1)!}}\frac{{{\partial^{i-1}}}}{{\partial{A^{i-1}}}}\frac{{{\partial^{j-1}}}}{{\partial{B^{j-1}}}}\frac{{{\partial^{k-1}}}}{{\partial{C^{k-1}}}}$
$\displaystyle\times\int{{d^{4}}k}\delta[{(k+q)^{2}}-A]\delta[{(k-p^{\prime})^{2}}-B]$
$\displaystyle\times\delta({k^{2}}-C){|_{A\to{m_{1,}}B\to{m_{2,}}C\to{m_{3}}}}$
$\displaystyle=\frac{{{{(-2\pi i)}^{3}}}}{{{{(2\pi
i)}^{2}}}}\frac{1}{{(i-1)!(j-1)!(k-1)!}}\frac{{{\partial^{i-1}}}}{{\partial{A^{i-1}}}}\frac{{{\partial^{j-1}}}}{{\partial{B^{j-1}}}}\frac{{{\partial^{k-1}}}}{{\partial{C^{k-1}}}}$
$\displaystyle\times\frac{\pi}{{2\sqrt{\lambda(s,u,{q^{2}})}}}{|_{A\to{m_{1,}}B\to{m_{2,}}C\to{m_{3}}}}$
(23)

Besides of these above contributions, the condensate terms of
$\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$ and
$\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ are also taken into
account in this work. The feynman diagrams of these condensate terms can be
classified into two groups(see Fig. 3 and Fig. 4) which correspond to the
correlation functions $\Pi_{1}$ and $\Pi_{2}$ in Eq. (11), respectively.

Figure 3: Feynman diagrams of the vacuum condensate terms
$\langle\overline{q}q\rangle$(a), $\langle g_{s}^{2}G^{2}\rangle$(b-g),
$\langle f^{3}G^{3}\rangle$(h-q), $\langle\overline{q}g_{s}\sigma
Gq\rangle$(r-t) and $\langle\overline{q}q\rangle\langle
g_{s}G^{2}\rangle$(u-x) for $\Pi_{1}$ in Eq. (11). Figure 4: Feynman diagrams
of the vacuum condensate terms $\langle g_{s}^{2}G^{2}\rangle$(a-f) and
$\langle f^{3}G^{3}\rangle$(g-p) for $\Pi_{2}$ in Eq. (11).

###  III.3 The electromagnetic form factor

We take the change of variables $p^{2}\to-P^{2}$, $p^{\prime 2}\to-P^{\prime
2}$ and $q^{2}\to-Q^{2}$ and perform double Borel transformsIoffe:1982ia ;
Ioffe:1982qb  for the variables $P^{2}$ and $P^{\prime 2}$ to both
phenomenological and QCD sides. And then $P^{2}$ and $P^{\prime 2}$ will be
replaced by $T_{1}^{2}$ and $T_{2}^{2}$ which are called Borel parameters.
After matching the phenomenological and QCD side by using quark-hadron
duality, the sum rule for the electromagnetic form factor is obtained as,

$\displaystyle V({Q^{2}})$
$\displaystyle=\frac{{({m_{q}}+{m_{Q}})({m_{\mathbb{P}}}+{m_{\mathbb{V}}})}}{{{f_{\mathbb{P}}}m_{\mathbb{P}}^{2}{f_{\mathbb{V}}}{m_{\mathbb{V}}}}}{e^{m_{\mathbb{P}}^{2}/T_{1}^{2}}}{e^{m_{\mathbb{V}}^{2}/T_{2}^{2}}}$
(24)
$\displaystyle\times\Big{\\{}{e_{Q}}\Big{[}-\int\limits_{{s_{1}}}^{{s_{0}}}{\int\limits_{{u_{1}}}^{{u_{0}}}{{\rho_{1}}(s,u,{Q^{2}}){e^{-u/T_{1}^{2}}}{e^{-s/T_{2}^{2}}}duds{|_{|{b_{1}}(s,u,{Q^{2}})\leq
1|}}}}$
$\displaystyle+\mathscr{BB}[\Pi_{1}^{\langle\overline{q}q\rangle}+\Pi_{1}^{\langle\overline{q}g_{s}\sigma
Gq\rangle}+\Pi_{1}^{\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}]\Big{]}$
$\displaystyle+{e_{q}}\Big{[}-\int\limits_{{s_{1}}}^{{s_{0}}}{\int\limits_{{u_{1}}}^{{u_{0}}}{{\rho_{2}}(s,u,{Q^{2}}){e^{-u/T_{1}^{2}}}{e^{-s/T_{2}^{2}}}duds{|_{|{b_{2}}(s,u,{Q^{2}})\leq
1|}}}}\Big{]}\Big{\\}}$
$\displaystyle={e_{Q}}{V_{1}}({Q^{2}})+{e_{q}}{V_{2}}({Q^{2}})$

where,

$\displaystyle{b_{1}}(s,u,{Q^{2}})=$
$\displaystyle\frac{{(u+{Q^{2}}+m_{Q}^{2}-m_{q}^{2})(s+u+{Q^{2}})-2s(u+m_{Q}^{2}-m_{q}^{2})}}{{\sqrt{{{(u+{Q^{2}}+m_{Q}^{2}-m_{q}^{2})}^{2}}-4sm_{Q}^{2}}\sqrt{\lambda(s,u,{Q^{2}})}}}$
$\displaystyle{b_{2}}(s,u,{Q^{2}})=$
$\displaystyle\frac{{(u+{Q^{2}}+m_{q}^{2}-m_{Q}^{2})(s+u+{Q^{2}})-2s(u+m_{q}^{2}-m_{Q}^{2})}}{{\sqrt{{{(u+{Q^{2}}+m_{q}^{2}-m_{Q}^{2})}^{2}}-4sm_{q}^{2}}\sqrt{\lambda(s,u,{Q^{2}})}}}$

In Eq. (24), $\mathscr{BB}[~{}]$ denote the double Borel transforms. It needs
to be explained that the vacuum condensate terms
$\langle\overline{q}q\rangle$, $\langle\overline{q}g_{s}\sigma Gq\rangle$ and
$\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ for the
electromagnetic form factor $V_{2}(Q^{2})$ in Eq. (24) will vanish after
taking double Borel transforms. The threshold parameters $s_{0}$ and $u_{0}$
are introduced in dispersion integral to eliminate the contributions of higher
resonances and continuum states. They should satisfy the relations
$m^{2}_{\mathbb{V}}\leq s_{0}\leq m^{\prime 2}_{\mathbb{V}}$ and
$m^{2}_{\mathbb{P}}\leq u_{0}\leq m^{\prime 2}_{\mathbb{P}}$, where
$m_{\mathbb{V}[\mathbb{P}]}$ and $m^{\prime}_{\mathbb{V}[\mathbb{P}]}$ are the
masses of the ground and the first excited state of the vector (pseudoscalar)
heavy-light mesons. The masses of the ground and excited states commonly
fulfill the relation
$m^{\prime}_{\mathbb{V}[\mathbb{P}]}=m_{\mathbb{V}[\mathbb{P}]}+\Delta$, where
$\Delta$ is usually taken as the value of $0.4\sim 0.6$ GeVBracco:2011pg .

##  IV Numerical results and Discussions

The masses of the heavy and light quarks, and the values of vacuum condensate
terms are energy-scale dependent, which can be expressed as the following
forms according to the re-normalization group equation(RGE),

$\displaystyle{m_{b}}(\mu)$
$\displaystyle={m_{b}}({m_{b}}){[\frac{{{\alpha_{s}}(\mu)}}{{{\alpha_{s}}({m_{b}})}}]^{\frac{{12}}{{33-2n_{f}}}}}$
$\displaystyle{m_{c}}(\mu)$
$\displaystyle={m_{c}}({m_{c}}){[\frac{{{\alpha_{s}}(\mu)}}{{{\alpha_{s}}({m_{c}})}}]^{\frac{{12}}{{33-2n_{f}}}}}$
$\displaystyle{m_{s}}(\mu)$
$\displaystyle={m_{s}}(2\mathrm{GeV}){[\frac{{{\alpha_{s}}(\mu)}}{{{\alpha_{s}}(2\mathrm{GeV})}}]^{\frac{12}{33-2n_{f}}}}$
$\displaystyle\left\langle{\bar{q}q}\right\rangle(\mu)$
$\displaystyle=\left\langle{\bar{q}q}\right\rangle(1\mathrm{GeV}){[\frac{{{\alpha_{s}}(1\mathrm{GeV})}}{{{\alpha_{s}}(\mu)}}]^{\frac{12}{33-2n_{f}}}}$
$\displaystyle\left\langle{\bar{s}s}\right\rangle(\mu)$
$\displaystyle=\left\langle{\bar{s}s}\right\rangle(1\mathrm{GeV}){[\frac{{{\alpha_{s}}(1\mathrm{GeV})}}{{{\alpha_{s}}(\mu)}}]^{\frac{12}{33-2n_{f}}}}$
$\displaystyle\left\langle{\bar{q}{g_{s}}\sigma Gq}\right\rangle(\mu)$
$\displaystyle=\left\langle{\bar{q}{g_{s}}\sigma
Gq}\right\rangle(1\mathrm{GeV}){[\frac{{{\alpha_{s}}(1\mathrm{GeV})}}{{{\alpha_{s}}(\mu)}}]^{\frac{2}{{33-2n_{f}}}}}$
$\displaystyle\left\langle{\bar{s}{g_{s}}\sigma Gs}\right\rangle(\mu)$
$\displaystyle=\left\langle{\bar{s}{g_{s}}\sigma
Gs}\right\rangle(1\mathrm{GeV}){[\frac{{{\alpha_{s}}(1\mathrm{GeV})}}{{{\alpha_{s}}(\mu)}}]^{\frac{2}{{33-2n_{f}}}}}$
$\displaystyle{\alpha_{s}}(\mu)$
$\displaystyle=\frac{1}{{{b_{0}}t}}[1-\frac{{{b_{1}}}}{{b_{0}^{2}}}\frac{{\log
t}}{t}$ (26) $\displaystyle+\frac{{b_{1}^{2}({{\log}^{2}}t-\log
t-1)+{b_{0}}{b_{2}}}}{{b_{0}^{4}{t^{2}}}}]$

where $t=\mathrm{log}\frac{\mu^{2}}{\Lambda_{QCD}^{2}}$,
$b_{0}=\frac{33-2n_{f}}{12\pi}$, $b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda_{QCD}=210$ MeV, 292 MeV and 332 MeV for the flavors $n_{f}=$5, 4 and
3, respectivelyParticleDataGroup:2022pth . The $\overline{\mathrm{MS}}$ masses
of the heavy and light quarks are adopted from the
PDGParticleDataGroup:2022pth , where $m_{c}(m_{c})=1.275\pm 0.025$ GeV,
$m_{b}(m_{b})=4.18\pm 0.03$ GeV, $m_{u(d)}(\mu=1\mathrm{GeV})=0.006\pm 0.001$
GeV and $m_{s}(\mu=2\mathrm{GeV})=0.095\pm 0.005$ GeV. As for the values of
other input parameters, they are all listed in Table 2, where the quark and
quark-gluon condensate parameters take their values in the energy scale
$\mu=1$ GeV. Based on our previous researches about the heavy-light
mesonsWang:2015mxa , the values of decay constants are selected by following
energy-scale, where $\mu=1$ GeV for $D^{*}\to D$ electromagnetic form factor,
$\mu=2.5$ GeV for $B^{*}\to B$, $\mu=1.1$ GeV for $D^{*}_{s}\to D_{s}$ and
$\mu=2.6$ GeV for $B^{*}_{s}\to B_{s}$.

Table 2: Input parameters (IP) in this work. IP | values(GeV) | IP | values
---|---|---|---
$m_{D}$ | 1.86ParticleDataGroup:2022pth  | $f_{B}$ |  $0.192\pm 0.013$ GeVWang:2015mxa
$m_{D^{*}}$ | 2.01ParticleDataGroup:2022pth  | $f_{B^{*}}$ |  $0.213\pm 0.018$ GeVWang:2015mxa
$m_{D_{s}}$ | 1.97ParticleDataGroup:2022pth  | $f_{B_{s}}$ |  $0.230\pm 0.013$ GeVWang:2015mxa
$m_{D_{s}^{*}}$ | 2.11ParticleDataGroup:2022pth  | $f_{B^{*}_{s}}$ |  $0.255\pm 0.019$ GeVWang:2015mxa
$m_{B}$ | 5.28ParticleDataGroup:2022pth  | $\langle\overline{q}q\rangle$ |  $-(0.23\pm 0.01)^{3}$ GeV3Shifman:1978by ; Reinders:1984sr
$m_{B^{*}}$ | 5.32ParticleDataGroup:2022pth  | $\langle\overline{q}g_{s}\sigma Gq\rangle$ |  $m_{0}^{2}\langle\overline{q}q\rangle$Shifman:1978by ; Reinders:1984sr
$m_{B_{s}}$ | 5.37ParticleDataGroup:2022pth  | $\langle\overline{s}s\rangle$ |  $-(0.8\pm 0.1)\langle\overline{q}q\rangle$Shifman:1978by ; Reinders:1984sr
$m_{B^{*}_{s}}$ | 5.42ParticleDataGroup:2022pth  | $\langle\overline{s}g_{s}\sigma Gs\rangle$ |  $m_{0}^{2}\langle\overline{s}s\rangle$Shifman:1978by ; Reinders:1984sr
$f_{D}$ |  $0.210\pm 0.011$ Wang:2015mxa  | $m_{0}^{2}$ |  $0.8\pm 0.1$ GeV2Shifman:1978by ; Reinders:1984sr
$f_{D^{*}}$ |  $0.236\pm 0.021$ Wang:2015mxa  | $\langle g_{s}^{2}G^{2}\rangle$ |  $0.88\pm 0.15$ GeV4Narison:2010cg ; Narison:2011xe ; Narison:2011rn
$f_{D_{s}}$ |  $0.259\pm 0.010$Wang:2015mxa  | $\langle f^{3}G^{3}\rangle$ |  $(8.8\pm 5.5)\langle\alpha_{s}G^{2}\rangle$ Narison:2010cg ; Narison:2011xe ; Narison:2011rn
$f_{D_{s}^{*}}$ |  $0.308\pm 0.021$Wang:2015mxa  |  |

In the framework of QCDSR, two criteria should be satisfied, which are the
pole dominance and convergence of operator product expansion (OPE). The pole
and continuum contribution can be defined asBracco:2011pg ,

$\displaystyle\mathrm{Pole}=\frac{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T_{1}^{2},T_{2}^{2})}{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T_{1}^{2},T_{2}^{2})+\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T_{1}^{2},T_{2}^{2})}$
$\displaystyle\mathrm{Continuum}=\frac{\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T_{1}^{2},T_{2}^{2})}{\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T_{1}^{2},T_{2}^{2})+\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T_{1}^{2},T_{2}^{2})}$
(27)

with

$\displaystyle\Pi^{\mathrm{OPE}}_{\mathrm{pole}}(T_{1}^{2},T_{2}^{2})=-\int_{s_{1}}^{s_{0}}\int_{u_{1}}^{u_{0}}\rho^{\mathrm{OPE}}(s,u,Q^{2})e^{-\frac{s}{T_{1}^{2}}}e^{-\frac{u}{T_{2}^{2}}}dsdu$
$\displaystyle\Pi^{\mathrm{OPE}}_{\mathrm{cont}}(T_{1}^{2},T_{2}^{2})=-\int_{s_{0}}^{\infty}\int_{u_{0}}^{\infty}\rho^{\mathrm{OPE}}(s,u,Q^{2})e^{-\frac{s}{T_{1}^{2}}}e^{-\frac{u}{T_{2}^{2}}}dsdu$
(28)

It can be seen from Eqs. (24) and (IV) that the results of QCDSR depend on
input parameters such as the Borel parameters $T_{1}^{2}$ and $T_{2}^{2}$, the
continuum threshold parameters $s_{0}$ and $u_{0}$, and the square momentum
$Q^{2}$. In this article, we take $T_{1}^{2}=T^{2}$ and
$T_{2}^{2}=kT_{1}^{2}=kT^{2}$. $k$ is a constant which is related to the ratio
of vector and pseudoscalar heavy-light mesons and can be expressed as
$k=\frac{m^{2}_{\mathbb{V}}}{m^{2}_{\mathbb{P}}}$. The threshold parameters
are defined as $s_{0}=(m_{\mathbb{V}}+\Delta_{\mathbb{V}})^{2}$ and
$u_{0}=(m_{\mathbb{P}}+\Delta_{\mathbb{P}})^{2}$. The values of
$\Delta_{\mathbb{V}}$ and $\Delta_{\mathbb{P}}$ should be smaller than the
experimental value of the distance between the ground and first excited state.
Taking the electromagnetic form factor $V_{1}$ of $D^{*}\to D$ as an example,
we simply discuss how to select $s_{0}$ and $u_{0}$. Fixing $Q^{2}=1$GeV2 in
Eq. (IV), we firstly plot the electromagnetic form factor $V_{1}$ on Borel
parameter $T^{2}$ in Fig. 5, where different values of $s_{0}$ and $u_{0}$ are
adopted. It is indicated that the results have good stability when $s_{0}$ is
taken to be $6.30$ GeV2($\Delta_{\mathbb{V}}$=0.5 GeV) and $u_{0}$ is $5.57$
GeV2($\Delta_{\mathbb{P}}$=0.5 GeV). Thus, the final results will be obtained
by taking $\Delta_{\mathbb{V}}=\Delta_{\mathbb{P}}=0.4$, 0.5 and 0.6 GeV,
where 0.5 GeV is used to determine the central values of the electromagnetic
form factors, and 0.4, 0.6 GeV are for the lower and upper bounds of the final
results.

An appropriate region of the Borel parameter $T^{2}$ needs to be chosen to
make the final results stable and reliable. This region is commonly called
’Borel platform’. After repeated trial and contrast, we finally determine the
Borel platforms which are shown in Fig. 8 in Appendix. From Fig. 8, we can see
that the values of the form factors have good stability in the Borel regions,
which means the convergence of OPE is well satisfied.

Figure 5: The $D^{*}\to D$ electromagnetic form factor $V_{1}$ on Borel
parameter $T^{2}$ with different $s_{0}$(a) and $u_{0}$(b).

In addition, the pole and continuum contributions in the Borel platforms are
plotted in Fig. 9. We can see from Fig. 9 that the central values of pole
contribution in the Borel platform satisfy the condition of pole dominance
($>50\%$). That is to say, two criteria of pole dominance and convergence of
OPE are both satisfied. The Borel platform, pole contributions in the Borel
platform, and the values of electromagnetic form factors in $Q^{2}=1$ GeV2 are
listed in Table 3.

Table 3: The Borel platform, pole contributions (Pole) and the values of electromagnetic form-factors in $Q^{2}=1$ GeV2 for different radiative decay modes. Mode | Form factor | Borel platform | Pole(%) |  $V_{i}(Q^{2}=1$ GeV${}^{2})$
---|---|---|---|---
$D^{*}\to D\gamma$ | $V_{1}$ | $3\sim 5$ | $60\sim 44$ | $0.97^{+0.10}_{-0.09}$
$V_{2}$ | $3.5\sim 5.5$ | $68\sim 44$ | $0.69^{+0.17}_{-0.16}$
$D_{s}^{*}\to D_{s}\gamma$ | $V_{1}$ | $4.5\sim 6.5$ | $58\sim 42$ | $0.98^{+0.12}_{-0.11}$
$V_{2}$ | $5\sim 7$ | $64\sim 46$ | $0.95^{+0.20}_{-0.17}$
$B^{*}\to B\gamma$ | $V_{1}$ | $11\sim 13$ | $53\sim 46$ | $1.42^{+0.13}_{-0.12}$
$V_{2}$ | $12\sim 14$ | $58\sim 49$ | $2.54^{+0.46}_{-0.47}$
$B_{s}^{*}\to B_{s}\gamma$ | $V_{1}$ | $13\sim 15$ | $52\sim 45$ | $1.30^{+0.12}_{-0.13}$
$V_{2}$ | $15\sim 17$ | $57\sim 50$ | $3.00^{+0.45}_{-0.48}$

By taking different values of $Q^{2}$, the electromagnetic form factors
$V_{i}(Q^{2})$ in space-like regions ($Q^{2}>0$) can be obtained, where
$Q^{2}$ is in the range of $0.5\sim 5.5$ GeV2 for $V_{1}(Q^{2})$, and $1\sim
6$ GeV2 for $V_{2}(Q^{2})$. For the values of the electromagnetic form factors
at $Q^{2}=0$, which are used to analyze the radiative decay of the vector
heavy-light mesons, they can be obtained by extrapolating the results
$V_{i}(Q^{2})$ into $Q^{2}=0$. This process is realized by selecting
appropriate analytical functions to fit the results $V_{i}(Q^{2})$ in space-
like region. In general, the single pole fitting or exponential fitting are
usually used to roughly parameterize the electroweak form factors. More
precise and physical method is the $z-$ series expansion which is used to
extrapolate the heavy-to-light electroweak form factors to large momentum
transfer regions$(q^{2}=-Q^{2}>>0)$Boyd:1994tt ; Wang:2015vgv ; Cui:2022zwm .
For the electromagnetic form factors of vector heavy-light mesons to
pseudoscalar heavy-light mesons, both the initial and final states are heavy
mesons, and we only take care of the values at $Q^{2}=0$. Thus, we do not
directly employ the $z-$ series expansion method to fit the electromagnetic
form factors. After repeated trial and error, we find that the results in
space-like regions ($Q^{2}>0$) can be well fitted by the combinations of the
single pole, exponential and polynomial functions. In addition, we know that
the series expansion of the exponential function and $z-$ series expansion are
essentially consistent with each other in their mathematical form. Thus, the
simple parameterizations method is adopted to fit the form factors in this
work, and it has the following form,

$\displaystyle V_{i}(Q^{2})=\frac{A}{1+BQ^{2}}Exp(-CQ^{2})+D$ (29)

where $A$, $B$, $C$ and $D$ are the fitting parameters and their values are
listed in Table 4. The fitting results are also explicitly shown in Figs. 6
and 7. By setting $Q^{2}=0$ in the analytical functions(Eq. (29)), the values
of electromagnetic form factors in $Q^{2}=0$($V_{i}(0)$) are obtained, which
are also listed in the last column of Table 4.

Table 4: The parameters for the analysis function and the values of electromagnetic form factors in $Q^{2}=0$ for different radiative decay modes. Mode | Form factor | $A$ | $B$ | $C$ | $D$ | $V_{i}(0)$
---|---|---|---|---|---|---
$D^{*}\to D\gamma$ | $V_{1}$ | $1.01$ | $1.18$ | $-0.0006$ | $0.53$ | $1.54^{+0.08}_{-0.11}$
$V_{2}$ | $1.33$ | $0$ | $0.56$ | $-0.04$ | $1.29^{+0.23}_{-0.21}$
$D_{s}^{*}\to D_{s}\gamma$ | $V_{1}$ | $0.70$ | $0$ | $0.40$ | $0.53$ | $1.23^{+0.19}_{-0.18}$
$V_{2}$ | $1.62$ | $0$ | $0.57$ | $0.04$ | $1.65^{+0.26}_{-0.24}$
$B^{*}\to B\gamma$ | $V_{1}$ | $1.62$ | $0$ | $1.93$ | $1.22$ | $2.85^{+0.67}_{-0.64}$
$V_{2}$ | $5.00$ | $0$ | $0.67$ | $-0.04$ | $4.96^{+0.54}_{-0.46}$
$B_{s}^{*}\to B_{s}\gamma$ | $V_{1}$ | $1.54$ | $5.46$ | $0.0002$ | $1.09$ | $2.62^{+0.31}_{-0.65}$
$V_{2}$ | $5.19$ | $0$ | $0.57$ | $0.01$ | $5.20^{+0.61}_{-0.56}$

According to the heavy-hadron chiral perturbation theory, the relation of the
form factors $\frac{V_{1}(0)}{V_{2}(0)}<1$ should be satisfiedCheng:1992xi .
From Table 4, we find the values of $\frac{V_{1}(0)}{V_{2}(0)}$ for
$D_{s}^{*}\to D_{s}$, $B^{*}\to B$ and $B_{s}^{*}\to B_{s}$ are all lower than
1, which means the results are consistent well with the conclusion of heavy-
hadron chiral perturbation theory. For the form factor of $D^{*}\to D$, the
central value of $V_{1}(0)$ is slightly larger than that of $V_{2}(0)$, which
looks as if these results are contradict with the conclusion of chiral
perturbation theory. Actually, if we consider the error bar of the results,
the relation $\frac{V_{1}(0)}{V_{2}(0)}<1$ is still roughly satisfied. It is
known that the next-to-leading order contribution is important in small
$Q^{2}$ regions. However, the calculation of this diagram is very complicated
for three-point QCDSR, thus we do not consider this contribution, which may
lead to errors in the final results. In addition, the parametrization of the
form factors is model dependent, it may also lead to some errors in the
results. According to these above discussions, we think there are no
contradiction between the results of QCDSR and that of the heavy-hadron chiral
perturbation theory.

Basing on these above analysis about the electromagnetic form factors, we can
now discuss the radiative decays of vector heavy-light mesons. The standard
form of two body decay width can be written as,

$\displaystyle\Gamma$ $\displaystyle=\frac{1}{{2J+1}}\sum{\frac{p}{{8\pi
M_{i}^{2}}}|T{|^{2}}}$ $\displaystyle p$
$\displaystyle=\frac{{\sqrt{[M_{i}^{2}-{{(m+{M_{f}})}^{2}}][M_{i}^{2}-{{({M_{f}}-m)}^{2}}]}}}{{2{M_{i}}}}$
(30)

where $M_{i}$ and $M_{f}$ represent the masses of initial and final mesons,
$J$ is the total angular momentum of the initial meson, $\Sigma$ denotes the
summation of all the polarization vectors, and $T$ is the scattering
amplitude.

Figure 6: The fitting results for $D^{*0(\pm)}\to D^{0(\pm)}$ (a,b) and
$D_{s}^{*\pm}\to D_{s}^{\pm}$ (c,d). The blue circles are the form factors in
space-like regions($Q^{2}>0$), the red solid lines are the fitting curves, and
the red bands represent the error regions.

Figure 7: The fitting results for $B^{*0(\pm)}\to B^{0(\pm)}$ (a,b) and
$B_{s}^{*0}\to B_{s}^{0}$ (c,d). The blue circles are the form factors in
space-like regions($Q^{2}>0$), the red solid lines are the fitting curves, and
the red bands represent the error regions.

With Eqs. (II) and (IV), the decay width of vector heavy-light meson to
pseudoscalar heavy-light meson and photon can be expressed as,

$\displaystyle\Gamma=\frac{{\alpha|Q_{Q}V_{1}(0)+Q_{q}V_{2}(0){|^{2}}({m_{\mathbb{V}}}+{m_{\mathbb{P}}}){{({m_{\mathbb{V}}}-{m_{\mathbb{P}}})}^{3}}}}{{24m_{\mathbb{V}}^{3}}}$
(31)

Here, $Q_{Q[q]}$ denote the electric charges of quarks, which are taken as
$\frac{2}{3}$ for u and c quarks, $-\frac{1}{3}$ for d, s and b quarks.
$\alpha$ is the electromagnetic fine structure constant whose value is
$\frac{1}{137}$. Using the values about electromagnetic form factors
$V_{i}(0)$ in Table 4, we can obtain the decay widths of radiative decay for
vector heavy-light mesons. The results of present work and those of other
collaboration’s are all listed in Table 5.

According to Particle Data Group(PDG)ParticleDataGroup:2022pth , the
experimental data of the total decay width for $D^{*+}$ is $83.4\pm 1.8$ keV,
and the ratio of the decay width of $D^{*+}\to D^{+}\gamma$ to the total width
is about $(1.6\pm 0.4)\%$. The results of present work about this radiative
decay is $0.17^{+0.08}_{-0.07}$ keV, which is about 0.20% of the total width
$83.4\pm 1.8$ keV and is smaller than the experimental data. In
RefAliev:1994nq , the radiative decay for $D^{*+}$ was also analyzed by QCDSR,
and their result is compatible with ours, which is also smaller than the
experimental data. The total widths for $D^{*0}$ and $D_{s}^{*}$ are given as
$\Gamma(D^{*0})<2.1$ MeV and $\Gamma(D_{s}^{*})<1.9$ MeV, and the ratios of
the radiative decays are about $(35.3\pm 0.9)$% and $(93.5\pm 0.7)$% of the
total widths. However, the exact values of these decay widths have not be
determined yet. Our results for $D^{*0}$ and $D_{s}^{*}$ are
$1.74^{+0.40}_{-0.37}$ keV and $0.029^{+0.009}_{-0.008}$ keV, which are
compatible with the experiment data. In addition, the electromagnetic form
factor which has the same expression as ours was firstly analyzed by Lattice
QCDDonald:2013sra , their radiative decay width about $D_{s}^{*}\to
D_{s}\gamma$ is given as $0.066\pm 0.026$ keV. Considering the uncertainties,
our result is in agreement well with theirs. We also notice that the radiative
decays of the heavy-light mesons were analyzed in the framework of light-cone
QCD sum rules(LCSR) by consedering the subleading power corrections in recent
yearsLi:2020rcg ; Pullin:2021ebn . In RefLi:2020rcg , the electromagnetic
coupling constants of the heavy-light mesons were obtained by considering the
contributions from twist-two photon distribution amplitude at next-to-leading-
order in $\alpha_{s}$. For the convenience of comparing the results, we use
their electromagnetic coupling constants to get the corresponding decay widths
which are shown in Table. 5. As for the results of mesons $B^{*\pm(0)}$ and
$B_{s}^{*}$, it can be seen from Table 5 that the theoretical results of
different collaborations are not consistent well with each other, which needs
to be further testified by experiments in the future.

##  V Conclusions

In this article, the electromagnetic form factors in space-like regions
($Q^{2}>0$) for vector heavy-light meson to pseudoscalar heavy-light meson are
firstly analyzed within the framework of three-point QCDSR. Then, the
electromagnetic form factors in $Q^{2}=0$ are obtained by fitting the results
into analytical functions in space-like regions. Based on these results, the
radiative decays of the vector heavy-light mesons are systematically analyzed.
We hope the results about electromagnetic form factors and radiative decays
can help to shed more light on the nature of the hadrons.

## Acknowledgements

This project is supported by National Natural Science Foundation, Grant Number
12175068 and Natural Science Foundation of HeBei Province, Grant Number
A2018502124.

Table 5: The radiation decay width of vector heavy-light mesons, all values are in units of keV. Method | Ref | $\Gamma(D^{*0}\to D^{0}\gamma)$ | $\Gamma(D^{*+}\to D^{+}\gamma)$ | $\Gamma(D_{s}^{*}\to D_{s}\gamma)$ | $\Gamma(B^{*0}\to B^{0}\gamma)$ | $\Gamma(B^{*+}\to B^{+}\gamma)$ | $\Gamma(B_{s}^{*}\to B_{s}\gamma)$
---|---|---|---|---|---|---|---
QCDSR | present work | $1.74^{+0.40}_{-0.37}$ | $0.17^{+0.08}_{-0.07}$ | $0.029^{+0.009}_{-0.008}$ | $0.018^{+0.006}_{-0.005}$ | $0.015^{+0.007}_{-0.007}$ | $0.016^{+0.003}_{-0.005}$
Aliev:1994nq  | $2.43\pm 0.21$ | $0.22\pm 0.06$ | $0.25\pm 0.08$ | - | - | -
LCSR | Li:2020rcg  | $13.70^{+5.88}_{-4.55}$ | $0.14^{+0.27}_{-0.12}$ | $0.041^{+0.137}_{-0.041}$ | $0.17^{+0.05}_{-0.05}$ | $0.42^{+0.14}_{-0.11}$ | $0.15^{+0.03}_{-0.04}$
Pullin:2021ebn  | $27.83^{+9.23}_{-9.50}$ | $0.96^{+0.58}_{-0.62}$ | $2.36^{+1.49}_{-1.41}$ | $0.16^{+0.06}_{-0.06}$ | $0.45^{+0.17}_{-0.16}$ | $0.24^{+0.08}_{-0.08}$
Aliev:1995zlh  | 14.40 | 1.50 | - | 0.16 | 0.63 | -
Lattice QCD |  Becirevic:2009xp ∗, Donald:2013sra † | $27\pm 14^{*}$ | $0.8\pm 0.7^{*}$ | $0.066\pm 0.026^{\dagger}$ | - | - | -
Salpter-like model | Colangelo:1994jc  | 20.8 | 0.46 | 0.38 | 0.092 | 0.243 | 0.080
$\chi$-loop model | Casalbuoni:1996pg  | 10.3 | 0.16 | 0.06 | 0.09 | 0.14 | 0.03
VMD model | 18.0 | 0.64 | 0.35 | 0.12 | 0.37 | 0.09
Chiral quark model | Deandrea:1998uz  | 13.11 | 0.25 | - | 0.05 | - | -
Goity:2000dk  | 32.02 | 1.42 | 0.3 | - | - | -
Bag model | Orsland:1998de  | 7.18 | 1.72 | - | 0.064 | 0.272 | 0.051
Light front model | Choi:2007se  | $20.0\pm 0.3$ | $0.90\pm 0.02$ | $0.18\pm 0.01$ | $0.13\pm 0.01$ | $0.40\pm 0.03$ | $0.068\pm 0.017$
Jaus:1996np  | 21.69 | 0.56 | - | 0.142 | 0.429 | -
GI quark model | Li:2022vby  | 97 | 9.9 | - | - | - | -
Covariant confined quark model |  Ivanov:2022nnq ∗, Tran:2023hrn † | $18.4\pm 7.4^{\dagger}$ | $1.21\pm 0.48^{\dagger}$ | $0.55\pm 0.22^{\dagger}$ | $0.126\pm 0.019^{*}$ | $0.372\pm 0.056^{*}$ | $0.090\pm 0.014^{*}$

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Appendix: The pole and continuum contributions, and the contributions of
different vacuum condensate terms.

Figure 8: The contributions of different vacuum condensate terms with
variation of the Borel parameter $T^{2}$ for $D^{*0(\pm)}\to D^{0(\pm)}$
(a,b), $D_{s}^{*\pm}\to D_{s}^{\pm}$ (c,d), $B^{*0(\pm)}\to B^{0(\pm)}$ (e,f),
and $B_{s}^{*0}\to B_{s}^{0}$ electromagnetic form-factors, where A-G
represent the total, perturbative term, $\langle g_{s}^{2}G^{2}\rangle$,
$\langle f^{3}G^{3}\rangle$, $\langle\overline{q}q\rangle$,
$\langle\overline{q}g_{s}\sigma Gq\rangle$, and
$\langle\overline{q}q\rangle\langle g_{s}^{2}G^{2}\rangle$ contributions, here
the $q$ denotes the light quarks $u(d)$ or $s$.

Figure 9: The pole and continuum contributions with variation of the Borel
parameter $T^{2}$ for $D^{*0(\pm)}\to D^{0(\pm)}$ (a, b), $D_{s}^{*\pm}\to
D_{s}^{\pm}$ (c, d), $B^{*0(\pm)}\to B^{0(\pm)}$ (e, f), and $B_{s}^{*0}\to
B_{s}^{0}$ electromagnetic form-factors.