# Chiral effective Lagrangian for excited heavy-light mesons from QCD

Qing-Sen Chen  Center for Theoretical Physics, College of Physics, Jilin
University, Changchun 130012, China  Hui-Feng Fu<EMAIL_ADDRESS>Center
for Theoretical Physics, College of Physics, Jilin University, Changchun
130012, China  Yong-Liang Ma<EMAIL_ADDRESS>School of Fundamental Physics
and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS,
Hangzhou, 310024, China  International Center for Theoretical Physics Asia-
Pacific (ICTP-AP) (Beijing/Hangzhou), UCAS, Beijing 100190, China  Qing Wang
<EMAIL_ADDRESS>Department of Physics, Tsinghua University, Beijing
100084, China  Center for High Energy Physics, Tsinghua University, Beijing
100084, China

###### Abstract

We derive the chiral effective Lagrangian for excited heavy-light mesons from
QCD under proper approximations. We focus on the chiral partners with
$j_{l}^{P}=\frac{3}{2}^{+}$ and $j_{l}^{P}=\frac{3}{2}^{-}$ which amounts to
($1^{+},2^{+}$) and ($1^{-},2^{-}$) states respectively. The low energy
constants including the masses of the chiral partners are calculated. The
calculated spectrum for the excited mesons are found roughly consistent with
experimental data. In addition, our results indicate that quantum numbers of
$B_{J}(5970)$ can be identified with $1^{-}$ or $2^{-}$.

##  I Introduction

The heavy quark spin-flavor symmetry that exact under the
$m_{Q}\rightarrow\infty$ limit with $m_{Q}$ being the heavy quark mass plays
an important role in the hadronic systems containing one heavy quark/antiquark
Isgur and Wise (1990, 1989); Eichten and Hill (1990). Due to this symmetry,
the total angular momentum of a heavy-light meson has eigenvalues
$j_{\pm}=j_{l}\pm 1/2$ with $\vec{j}_{l}$ being the angular momentum of the
light degrees of freedom, and the corresponding eigenstates form a degenerate
doublet $(j_{-},j_{+})$ for each $j_{l}$. The angular momentum $\vec{j}_{l}$
can be decomposed as $\vec{j}_{l}=\vec{s}+\vec{l}$, where $\vec{s}$ is the
spin of light quark/antiquark and $\vec{l}$ is the orbital angular momentum.
For S-wave heavy-light mesons, $l=0$ and $j_{l}^{P}=\frac{1}{2}^{-}$
represents the degenerate doublet $(j^{P}_{-},j^{P}_{+})=(0^{-},1^{-})$. For
P-wave, $l=1$ and $j_{l}^{P}=\frac{1}{2}^{+}$ or $j_{l}^{P}=\frac{3}{2}^{+}$
represent two distinct doublets $(j^{P}_{-},j^{P}_{+})=(0^{+},1^{+})$ or
$(j^{P}_{-},j^{P}_{+})=(1^{+},2^{+})$, respectively. For D-wave , $l=2$ and
$j_{l}^{P}=\frac{3}{2}^{-}$ or $j_{l}^{P}=\frac{5}{2}^{-}$, we have two
distinct doublets $(j^{P}_{-},j^{P}_{+})=(1^{-},2^{-})$ and
$(j^{P}_{-},j^{P}_{+})=(2^{-},3^{-})$. The similar argument applies to other
excited states Yuan (1995).

For the light quarks, the chiral symmetry, which is dynamically (and
explicitly) broken, is essential. If the chiral symmetry were preserved, the
heavy-light mesons with the same $j_{l}$ but opposite parities would be
degenerated. We call them chiral partners. The mass splitting between the
chiral partners reflects the chiral symmetry breaking Nowak  _et al._ (1993);
Bardeen and Hill (1994).

Since heavy-light mesons capture the features of both heavy quark symmetry and
chiral symmetry, it is ideal to study heavy-light meson phenomena using the
chiral Lagrangian incorporating heavy quark symmetry. Since 2000, a series of
researches on deriving the chiral effective Lagrangian from QCD have been
carried out in Refs. Wang  _et al._ (2000a); Yang  _et al._ (2002); Jiang  _et
al._ (2010); Jiang and Wang (2010); Jiang  _et al._ (2013, 2015); Wang and
Wang (2000); Wang  _et al._ (2000b); Ren  _et al._ (2017). The advantage of
this method is that it can establish the analytic relationships between the
low energy constants (LECs) of the chiral effective theory and QCD. Recently,
we used the same methodology to derive the chiral Lagrangian incorporating
heavy quark symmetry from QCD to study heavy-light mesons Chen  _et al._
(2020a, b). In these works, we focused on the chiral partners
$j_{l}^{P}=\frac{1}{2}^{-}$ and $j_{l}^{P}=\frac{1}{2}^{+}$. The low energy
constants of the effective Lagrangian are expressed in terms of the light
quark self-energy which can be calculated by using Dyson-Schwinger equations
or lattice QCD. Numerical results of the low energy constants are globally in
agreements with the experiment data.

In recent years, more and more excited heavy-light mesons have been observed.
In the charm sector, many new excited states such as $D_{0}(2550)$,
$D^{\ast}_{J}(2680)$, $D(2740)$, $D^{*}_{2}(3000)$, $D_{J}(3000)$,
$D_{J}^{*}(3000)$, etc., have been announced by LHCb Aaij  _et al._ (2013,
2016, 2020) and BABAR del Amo Sanchez  _et al._ (2010). In the bottom sector,
new bottom states $B_{J}(5721)$, $B_{2}^{*}(5747)$, $B_{J}(5840)$ and
$B_{J}(5970)$ were observed by CDF Aaltonen  _et al._ (2014) and LHCb Aaij
_et al._ (2015). The properties of these hadrons have drawn extensive
attractions in recent years Godfrey and Moats (2016); Song  _et al._ (2015);
Chen  _et al._ (2017); Gupta and Upadhyay (2018); Kher  _et al._ (2017);
Gandhi and Rai (2019); Gupta and Upadhyay (2019); Godfrey and Moats (2019).

Here, we extend our approach to the effective field theory of excited heavy-
light mesons with chiral partner structure Nowak and Zahed (1993). Special
interests are given to the states with quantum numbers
$j_{l}^{P}=\frac{3}{2}^{+}$ and $j_{l}^{P}=\frac{3}{2}^{-}$ to lay the
foundation for researches on arbitrary spin heavy-light mesons.

The remaining part of this paper is organized as follows. In Sec. II, for
convenience, we give the general form of the chiral Lagrangian of the excited
heavy-light mesons. In Sec. III we derive the excited heavy-light meson
Lagrangian from QCD and determine the expression of low energy constants. The
numerical results calculated by using the quark self-energy obtained from a
typical Dyson-Schwinger equation and from lattice QCD are given in Sec. IV.
Section V is devoted to discussions. The expressions of the LECs with the
contribution from the renormalization factor of quark wave function are given
in Appendix A.

##  II chiral effective Lagrangian for excited heavy-light mesons

For the convenience of the following description, we present the chiral
effective Lagrangian for excited heavy-light meson doublets
$T^{\mu}=(1^{+},2^{+})$ and $R^{\mu}=(1^{-},2^{-})$ here. They correspond to
the $j_{l}^{P}=\frac{3}{2}^{+}$ in P-wave and $j_{l}^{P}=\frac{3}{2}^{-}$ in
D-wave, respectively. The excited heavy-light meson doublets $T^{\mu}$ and
$R^{\mu}$ can be expressed as Nowak and Zahed (1993)

$\displaystyle T^{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left\\{P^{\ast\mu\nu}_{2^{+}}\gamma_{\nu}-\sqrt{\frac{3}{2}}P^{\nu}_{1^{+}}\gamma_{5}[g^{\mu}_{\nu}-\frac{1}{3}\gamma_{\nu}(\gamma^{\mu}-v^{\mu})]\right\\},$
$\displaystyle R^{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left\\{P^{\ast\mu\nu}_{2^{-}}\gamma_{\nu}\gamma_{5}-\sqrt{\frac{3}{2}}P_{1^{-}}^{\nu}[g^{\mu}_{\nu}-\frac{1}{3}\gamma_{\nu}(\gamma^{\mu}+v^{\mu})]\right\\},$

where $(P^{\nu}_{1^{+}},P_{2^{+}}^{\ast\mu\nu})$ refer to
$J^{P}=(1^{+},2^{+})$ states, and $(P^{\nu}_{1^{-}},P_{2^{-}}^{\ast\mu\nu})$
refer to $J^{P}=(1^{-},2^{-})$ states, respectively. $v^{\mu}$ is the velocity
of an on-shell heavy quark, i.e. $p^{\mu}=m_{Q}v^{\mu}$ with $v^{2}=1$. Then
the chiral effective Lagrangian for excited heavy-light mesons can be written
as Kilian  _et al._ (1992); Nowak and Zahed (1993)

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\cal L}_{T}+{\cal
L}_{R}+{\cal L}_{TR},$ (2)

where

$\displaystyle{\cal L}_{T}$ $\displaystyle=$ $\displaystyle{}-i{\rm
Tr}\left(\bar{T}^{\mu}v\cdot\nabla T_{\mu}\right)-g_{T}{\rm
Tr}\left(\bar{T}^{\mu}T_{\mu}\gamma^{\nu}\gamma_{5}A_{\nu}\right)$
$\displaystyle{}+m_{T}{\rm Tr}\left(\bar{T}^{\mu}T_{\mu}\right),$
$\displaystyle{\cal L}_{R}$ $\displaystyle=$ $\displaystyle{}-i{\rm
Tr}\left(\bar{R}^{\mu}v\cdot\nabla R_{\mu}\right)+g_{R}{\rm
Tr}\left(\bar{R}^{\mu}R_{\mu}\gamma^{\nu}\gamma_{5}A_{\nu}\right)$
$\displaystyle{}+m_{R}{\rm Tr}\left(\bar{R}^{\mu}R_{\mu}\right),$
$\displaystyle{\cal L}_{TR}$ $\displaystyle=$ $\displaystyle g_{TR}{\rm
Tr}\left(\bar{R}^{\mu}T_{\mu}\gamma^{\nu}\gamma_{5}A_{\nu}\right)+h.c..$ (3)

The covariant derivative $\nabla_{\mu}=\partial_{\mu}-iV_{\mu}$ with
$V_{\mu}=\frac{i}{2}\left(\Omega^{\dagger}\partial_{\mu}\Omega+\Omega\partial_{\mu}\Omega^{\dagger}\right)$,
and
$A_{\mu}=\frac{i}{2}\left(\Omega^{\dagger}\partial_{\mu}\Omega-\Omega\partial_{\mu}\Omega^{\dagger}\right)$.
The $\Omega$ field is related to the chiral field $U(x)=\exp(i\pi(x)/f_{\pi})$
through $U=\Omega^{2}$. The parameters $g_{T},g_{R},g_{TR},m_{T}$, and $m_{R}$
are the LECs of the Lagrangian. They are free ones at the level of the
effective theory.

As mentioned previously, the states associated with $T^{\mu}$ and $R^{\mu}$
are called chiral partners, and their mass splitting arises from the chiral
symmetry breaking. In the $D$ meson family, $T^{\mu}$ may be associated to
$T^{\mu}=(1^{+},2^{+})=(D_{1}(2430),D_{2}^{*}(2460))$ and a possible
identification of the $R^{\mu}$ doublet may be
$R^{\mu}=(1^{-},2^{-})=(D_{J}^{\ast}(2600),D(2740))$ with respect that the
fact that the states in the $R^{\mu}$ doublet can decay to their chiral
partners in the $T^{\mu}$ doublet so that should have broad widths , so the
spin-averaged masses of the chiral partners are Zyla  _et al._ (2020); Aaij
_et al._ (2020)

$\displaystyle m_{T}\simeq~{}2448~{}{\rm MeV},\;\;\;m_{R}\simeq 2703~{}{\rm
MeV},$ (4)

which yields the mass splitting

$\displaystyle\Delta m$ $\displaystyle=$ $\displaystyle m_{R}-m_{T}\simeq
255~{}{\rm MeV}.$ (5)

In the $B$ meson family, $T^{\mu}$ may be associated to
$(B_{1}(5721),B_{2}^{*}(5747))$, so the spin-averaged mass is Zyla  _et al._
(2020)

$\displaystyle m_{T}$ $\displaystyle\simeq$ $\displaystyle~{}5735~{}{\rm MeV}$
(6)

For the $(1^{-},2^{-})$ doublet, the situation is subtle since the quantum
numbers of the possible candidates in PDG are not well determined. With
respect to the mass splitting between chiral partners in the charmed meson
sector, the masses of the states in the $R^{\mu}$ doublet should have masses
$\sim 6000~{}$MeV. This means that it is reasonable to identify the quantum
numbers of the state $B_{J}(5970)$ as $1^{-}$ or $2^{-}$ or maybe the $1^{-}$
and $2^{-}$ states have nearly degenerate masses.

##  III chiral effective lagrangian of excited heavy-light mesons from QCD

In this section, we follow our previous work Chen  _et al._ (2020a) to derive
the chiral effective Lagrangian for excited heavy-light mesons with
$j_{l}^{P}=\frac{3}{2}^{\pm}$ and its low energy constants.

The generating functional of QCD with an external source $J(x)$ is

$\displaystyle Z[J]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}q\mathcal{D}\bar{q}\mathcal{D}Q\mathcal{D}\bar{Q}\mathcal{D}G_{\mu}\varDelta_{F}(G_{\mu})$
$\displaystyle\times\exp\left\\{i\int
d^{4}x\left[{\mathcal{L}_{\mathrm{QCD}}(q,\bar{q},Q,\bar{Q},G_{\mu})+\bar{q}Jq}\right]\right\\},$

where $q(x),Q(x)$ and $G_{\mu}(x)$ are the light-quark, heavy-quark and gluon
fields, respectively.

By integrating in the chiral field $U(x)$ and the heavy-light meson fields,
and integrating out gluon fields and quark fields, we obtain the effective
action for the chiral effective theory asChen  _et al._ (2020a)

$\displaystyle S[U,\Pi_{2},\bar{\Pi}_{2}]$ $\displaystyle=$
$\displaystyle{}-iN_{c}\mathrm{Tr}\ln\left\\{\left(i\not{\partial}-\bar{\Sigma}\right)I_{1}+J_{\Omega}\right.$
(8)
$\displaystyle\left.{}\qquad\qquad\qquad+\left(i\not{\partial}+M\not{v}-M\right)I_{4}\right.$
$\displaystyle\left.{}\qquad\qquad\qquad-\Pi_{2}-\bar{\Pi}_{2}\right\\},$

where $\Pi_{2}$ and $\bar{\Pi}_{2}$ are the heavy-light meson field and its
conjugate, respectively. $M$ is the mass matrice of heavy quark $Q$.
$\bar{\Sigma}$ is the self-energy of the light quark propagator.
$I_{1}=\mathrm{diag}(1,1,0)$ and $I_{4}=\mathrm{diag}(0,0,1)$ are matrices in
the flavor space. $J_{\Omega}$ is the chiral rotated external source. To
obtain this effective action from QCD, we have taken the chiral limit, the
heavy quark limit and the large $N_{c}$ limit. More details can be found in
the Appendix A of Ref. Chen  _et al._ (2020a).

Since the introduced heavy-light meson field $\Pi_{2}$ is a bilocal field, we
need a suitable localization on $\Pi_{2}$ fields to get a local effective
Lagrangian. Here we follow the approach in Ref. Nowak and Zahed (1993), and
take the following localization conditions

$\displaystyle\Pi_{2}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\Pi_{2}(x_{1})\delta(x_{1}-x_{2})+\Pi^{\mu}_{2}(x_{1})\overrightarrow{\partial}_{\mu}\delta(x_{1}-x_{2}),$
$\displaystyle\bar{\Pi}_{2}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\bar{\Pi}_{2}(x_{1})\delta(x_{1}-x_{2})+\delta(x_{1}-x_{2})\overleftarrow{\partial}_{\mu}\bar{\Pi}^{\mu}_{2}(x_{1}).$

where

$\displaystyle\Pi_{2}(x)$ $\displaystyle=$ $\displaystyle H(x)+G(x),$
$\displaystyle\Pi^{\mu}_{2}(x)$ $\displaystyle=$ $\displaystyle
T^{\mu}(x)+R^{\mu}(x),$ (10)

Here, the fields $H$ and $G$ refer to $J^{P}=(0^{-},1^{-})$ states and
$J^{P}=(0^{+},1^{+})$ states, the fields $T^{\mu}$ and $R^{\mu}$ refer to
$J^{P}=(1^{+},2^{+})$ states and $J^{P}=(1^{-},2^{-})$ states respectively.
Since we are interested in the heavy-light meson doublets with
$j_{l}^{P}=\frac{3}{2}^{\pm}$ in this work, we focus on the chiral effective
Lagrangian for $\Pi_{2}^{\mu}$ alone. So the chiral effective action is
reduced to

$\displaystyle S[U,\Pi^{\mu}_{2},\bar{\Pi}^{\nu}_{2}]$ $\displaystyle=$
$\displaystyle{}-iN_{c}\mathrm{Tr}\ln\left\\{\left(i\not{\partial}-\bar{\Sigma}\right)I_{1}+J_{\Omega}\right.$
(11)
$\displaystyle\left.{}\qquad\qquad\qquad+\left(i\not{\partial}+M\not{v}-M\right)I_{4}\right.$
$\displaystyle\left.{}\qquad\qquad\qquad-\Pi^{\nu}_{2}\overrightarrow{\partial}_{\nu}-\overleftarrow{\partial}_{\mu}\bar{\Pi}^{\mu}_{2}\right\\}$

In order to obtain the chiral effective Lagrangian, we need expand the action
$S[U,\Pi^{\mu}_{2},\bar{\Pi}^{\nu}_{2}]$ with respect to the fields $U$,
$\Pi^{\mu}_{2}$ and $\bar{\Pi}^{\nu}_{2}$. The kinetic terms of the excited
heavy-light meson fields $T^{\mu}$ (a similar equation holds for $R^{\mu}$)
arise from

$\displaystyle S_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{\delta^{2}S}{\delta\bar{T}^{\mu}\delta
T^{\nu}}T^{\nu}\bar{T}^{\mu}$ $\displaystyle=$ $\displaystyle iN_{c}\int
d^{4}x_{1}d^{4}x_{2}$
$\displaystyle{}\times\mathrm{Tr}\Bigg{[}\left(i\not{\partial}-\bar{\Sigma}\right)^{-1}\delta(x_{2}-x_{1})\overleftarrow{\partial}_{\mu}\bar{T}^{\mu}(x_{1})$
$\displaystyle\qquad\qquad\times\left(iv\cdot\partial\right)^{-1}\overrightarrow{\partial}_{\nu}T^{\nu}(x_{2})\delta(x_{1}-x_{2})\Bigg{]},$

Taking the derivative expansion up to the first order, we obtain

$\displaystyle S_{2}$ $\displaystyle=$ $\displaystyle iN_{c}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[-\frac{p^{2}-(v\cdot
p)^{2}}{p^{2}-\Sigma^{2}}+\frac{p^{2}\Sigma}{(p^{2}-\Sigma^{2})v\cdot
p}\right]\mathrm{Tr}\left[\bar{T}^{\mu}(x)T_{\mu}(x)\right]$ (13)
$\displaystyle{}~{}~{}~{}~{}+iN_{c}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[-\frac{p^{2}}{(p^{2}-\Sigma^{2})v\cdot
p}-\frac{[2\Sigma+2\Sigma^{\prime}(p^{2}+\Sigma^{2})][p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right]\mathrm{Tr}\left[T_{\mu}(x)v\cdot\partial\bar{T}^{\mu}(x)\right]$
$\displaystyle{}~{}~{}~{}~{}+iN_{c}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[-\frac{2\Sigma^{\prime}(p^{2}+\Sigma^{2})[p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right]\mathrm{Tr}\left[T_{\mu}(x)v\cdot
V_{\Omega}\bar{T}^{\mu}(x)\right].$

In the calculation, we have used the relation $T^{\mu}\not{v}=-T^{\mu}$. It is
easy to identify the mass term and the kinetic term in Eq. (13). In addition,
an interaction term between the $T^{\mu}$ fields and $V_{\Omega}$ fields also
appear in Eq. (13) because we have taken
$\bar{\Sigma}(x-y)=\Sigma(\nabla^{2})\delta(x-y)$ in the calculation to retain
the correct chiral transformation properties in the theory Yang  _et al._
(2002).

The interaction between the heavy-light meson field $T^{\mu}$ and the light
Goldstone boson field $S_{3}$ can also be obtained by expanding the action
$S[U,\Pi^{\mu}_{2},\bar{\Pi}^{\nu}_{2}]$ with respect to the external
$J_{\Omega}$, $\Pi^{\mu}_{2}$ and $\bar{\Pi}^{\nu}_{2}$ as

$\displaystyle S_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{\delta^{3}S}{\delta J_{\Omega}\delta\bar{T}^{\mu}\delta
T^{\nu}}J_{\Omega}T^{\nu}\bar{T}^{\mu}$ (14) $\displaystyle=$
$\displaystyle{}-iN_{c}\int d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}$
$\displaystyle\qquad{}\times\mathrm{Tr}\left[(i\not{\partial}-\Sigma)^{-1}\delta(x_{2}-x_{3})J_{\Omega}(x_{3})(i\not{\partial}-\Sigma)^{-1}\delta(x_{3}-x_{1})\overleftarrow{\partial}_{\mu}\bar{T}^{\mu}(x_{1})\left(iv\cdot\partial\right)^{-1}\delta(x_{1}-x_{2})\overrightarrow{\partial}_{\nu}T^{\nu}(x_{2})\right].$

Upto the first order of the derivative expansion, one obtains

$\displaystyle S_{3}$ $\displaystyle=$ $\displaystyle{}-iN_{c}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}(\Sigma^{2}+\frac{2}{3}p^{2})}{(p^{2}-\Sigma^{2})v\cdot
p}-\frac{2\Sigma[p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right]\mathrm{Tr}\left[\bar{T}(x)^{\mu}T_{\mu}(x)\gamma^{\nu}\gamma_{5}A_{\nu}\right]$
(15) $\displaystyle{}-iN_{c}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}}{(p^{2}-\Sigma^{2})v\cdot
p}+\frac{2\Sigma[p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right]\mathrm{Tr}\left[T_{\mu}(x)v\cdot
V_{\Omega}\bar{T}^{\mu}(x)\right].$

Summing up Eqs. (13) and (15), we obtain the expressions of the constants
$m_{T}$ and $g_{T}$ as

$\displaystyle m_{T}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}}{Z_{T}}\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}-(v\cdot
p)^{2}}{p^{2}-\Sigma^{2}}-\frac{p^{2}\Sigma}{(p^{2}-\Sigma^{2})v\cdot
p}\right],$ $\displaystyle g_{T}$ $\displaystyle=$
$\displaystyle{}-\frac{iN_{c}}{Z_{T}}\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}(\Sigma^{2}+\frac{2}{3}p^{2})}{(p^{2}-\Sigma^{2})v\cdot
p}-\frac{2\Sigma[p^{2}-(v\cdot p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right],$

with $Z_{T}$ being the wave function renormalization factor

$\displaystyle Z_{T}$ $\displaystyle=$ $\displaystyle
iN_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\biggr{[}{}-\frac{p^{2}}{(p^{2}-\Sigma^{2})v\cdot
p}$
$\displaystyle{}\qquad\qquad\qquad-\frac{[2\Sigma+2\Sigma^{\prime}(p^{2}+\Sigma^{2})][p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\biggr{]}.$

Following the same procedure, one can get the LECs associated to $R^{\mu}$.
The only difference in the calculation is the relation
$R^{\mu}\not{v}=R^{\mu}$, which induces the differences between the LECs of
$R^{\mu}$ and those of $T^{\mu}$. The LECs associated to $T^{\mu}$ read

$\displaystyle m_{R}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}}{Z_{R}}\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}-(v\cdot
p)^{2}}{p^{2}-\Sigma^{2}}+\frac{p^{2}\Sigma}{(p^{2}-\Sigma^{2})v\cdot
p}\right],$ $\displaystyle g_{R}$ $\displaystyle=$
$\displaystyle{}-\frac{iN_{c}}{Z_{R}}\int\frac{d^{4}p}{(2\pi)^{4}}\left[\frac{p^{2}(\Sigma^{2}+\frac{2}{3}p^{2})}{(p^{2}-\Sigma^{2})v\cdot
p}-\frac{2\Sigma[p^{2}-(v\cdot p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\right],$
$\displaystyle Z_{R}$ $\displaystyle=$ $\displaystyle
iN_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\biggr{[}{}-\frac{p^{2}}{(p^{2}-\Sigma^{2})v\cdot
p}$
$\displaystyle{}\qquad\qquad\qquad+\frac{[2\Sigma+2\Sigma^{\prime}(p^{2}+\Sigma^{2})][p^{2}-(v\cdot
p)^{2}]}{(p^{2}-\Sigma^{2})^{2}}\biggr{]}.$

We also obtain the coupling constant between the chiral partner fields
$T^{\mu}$ and $R^{\mu}$ as

$\displaystyle g_{TR}$ $\displaystyle=$
$\displaystyle{}-i\frac{N_{c}}{\sqrt{Z_{T}Z_{R}}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggl{[}\frac{p^{2}(\Sigma^{2}+p^{2})}{(p^{2}-\Sigma^{2})^{2}v\cdot
p}\biggr{]}.$ (18)

In the above expressions, $\Sigma(-p^{2})$ stands for self-energy of light
quarks to be calculated from QCD. This is an apparent indicator that we have
established a connection between the low energy constants of the chiral
effective theory and the Green functions of QCD.

##  IV Numerical results and discussions

In order to obtained the explicit numerical values of the LECs, we first need
to obtain the light quark self-energy $\Sigma(-p^{2})$. As in our previous
works Chen  _et al._ (2020a, b), we use two method to determine
$\Sigma(-p^{2})$ for comparison, namely, the gap equation, i.e., the Dyson-
Schwinger equation for quarks, and the fittings from lattice QCD.

For the Dyson-Schwinger equation method, we use the differential form of the
gap equation Yang  _et al._ (2002):

$\displaystyle\left(\frac{\alpha_{s}(x)}{x}\right)^{\prime}\Sigma(x)^{\prime\prime}-\left(\frac{\alpha_{s}(x)}{x}\right)^{\prime\prime}\Sigma(x)^{\prime}$
$\displaystyle\qquad\qquad\qquad\quad{}-\frac{3C_{2}(R)}{4\pi}\frac{x\Sigma(x)}{x+\Sigma^{2}(x)}\left(\frac{\alpha_{s}(x)}{x}\right)^{\prime
2}=0,$

where the boundary conditions are

$\displaystyle\Sigma^{\prime}(0)$ $\displaystyle=$
$\displaystyle{}-\frac{3C_{2}(R)\alpha_{s}(0)}{8\pi\Sigma(0)},$
$\displaystyle\Sigma(\Lambda^{\prime})$ $\displaystyle=$
$\displaystyle\frac{3C_{2}(R)\alpha_{s}(\Lambda^{\prime})}{4\pi\Lambda^{\prime}}\int^{\Lambda^{\prime}}_{0}dx\frac{x\Sigma(x)}{x+\Sigma^{2}(x)},$
(19)

with $\alpha_{s}$ being the running coupling constant of QCD.
$\Lambda^{\prime}$ is an ultraviolet cutoff regularizing the integral, which
is taken $\Lambda^{\prime}\rightarrow\infty$ eventually. To solve the Eq.
(IV), we take a model description for $\alpha_{s}$ given in Ref. Dudal  _et
al._ (2012)

$\alpha_{s}(p^{2})=\alpha_{0}p^{2}\frac{p^{2}+M^{2}}{p^{4}+(M^{2}+m^{2})p^{2}+\lambda^{4}}.$
(20)

For convenience, we call it Gribov-Zwanziger (G-Z) Model. The parameters are
$M^{2}=4.303~{}{\rm GeV}^{2},(M^{2}+m^{2})=0.526~{}{\rm GeV}^{2}$ and
$\lambda^{4}=0.4929~{}{\rm GeV}^{4}$ Dudal  _et al._ (2012). $\alpha_{0}$ is a
model parameter to be determined. Although the UV behavior of G-Z formalism is
inconsistent with QCD, it should not have a sizable impact on our results
because the LECs are mostly controlled by its low energy behavior.

Solving the gap equation, we obtain $\Sigma(-p^{2})$. Then the LECs are
calculated according to Eqs. (LABEL:eq:mHgH)-(18). It is clear that the
integrals of the LECs have a physical ultraviolet cutoff $\Lambda_{c}$ which
should be of the order of the chiral symmetry breaking scale and serves as
another parameter in our calculations. Since we are studying excited states,
the cutoff $\Lambda_{c}$ should take a bit larger value than 1 GeV. For the
excited states we are considering now, the energy of the light quark in the
chromoelectrical fields generated by the heavy quark could be up to $\sim
1.45$ GeV, so $\Lambda_{c}$ taking a value $\gtrsim 1.5$ GeV would be more
appropriate for the present situation. For the G-Z model, we take
$\Lambda_{c}=1.6$ GeV and determine the model parameter $a_{0}$ by requiring
the calculated $f_{\pi}$ be consistent with the corresponding experimental
value. We find that using $a_{0}=0.52$ we can obtain the well-established
quantity $f_{\pi}$. The results are listed in Table 1. For comparison, we also
list the numerical results obtained with different $a_{0}$’s in the table. To
give an intuitive impression, we draw the running coupling constant
$\alpha_{s}$ calculated with the G-Z formalism at $a_{0}=0.52$ in Fig. 1. The
light quark self-energy $\Sigma(-p^{2})$ solved by the gap equation (IV) is
shown in Fig. 2.

Table 1: The heavy-light meson masses and the coupling constants calculated from the G-Z Model with $\Lambda_{c}=1.6$ GeV.  $a_{0}$ |  $\Sigma(0)$ (GeV) |  $f_{\pi}$(GeV) |  ${}-\langle\bar{\psi}\psi\rangle$ (GeV3) | $g_{R}$ | $g_{T}$ | $g_{TR}$ |  $m_{R}$ (GeV) |  $m_{T}$ (GeV) |  $\Delta m$ (GeV)
---|---|---|---|---|---|---|---|---|---
0.50 | 0.158 | 0.073 | $(0.328)^{3}$ | 0.882 | -0.479 | -0.990 | 1.365 | 1.104 | 0.261
0.51 | 0.193 | 0.084 | $(0.350)^{3}$ | 0.936 | -0.442 | -0.987 | 1.411 | 1.091 | 0.321
0.52 | 0.226 | 0.093 | $(0.368)^{3}$ | 0.991 | -0.409 | -0.984 | 1.459 | 1.080 | 0.379
0.53 | 0.259 | 0.102 | $(0.385)^{3}$ | 1.047 | -0.377 | -0.981 | 1.510 | 1.071 | 0.439
0.54 | 0.291 | 0.109 | $(0.399)^{3}$ | 1.107 | -0.347 | -0.979 | 1.564 | 1.064 | 0.500

Since there are more and more excited states observed experimentally whose
masses and quantum numbers have not been confirmed yet, we focus on the masses
of the chiral partners in our method in this work and hopefully could help
identifying some excited mesons. The mass splitting $\Delta m$ is a direct
indicator of the chiral symmetry breaking. In our calculations, it is
completely originated from the non-vanishing of the quark running mass, which
can be reflected by the quark condensate or the value of $\Sigma(0)$. From
Table 1, one can see that $\Delta m$ increases as the quark condensate or
$\Sigma(0)$ increases. At $a_{0}=0.52$, we find $\Delta m=379$ MeV which is a
bit larger than the expected value $\sim 255$ MeV as given in (4). This is not
surprising because on one hand the measured masses of the relevant excited
mesons are still not accurate, and on the other hand our results are suffering
from uncertainties due to the ignorance of the $1/m_{Q}$ contributions, etc..

The masses $m_{T}$ and $m_{R}$ listed in Table 1 are the “residue masses” with
the heavy quark mass rotated away. The physical masses (denoted as
$\tilde{m}_{T}$ and $\tilde{m}_{R}$) can be obtained by simply adding back the
heavy quark mass to $m_{T}$ and $m_{R}$. For the $D$ meson sector, using
$m_{c}\approx 1.27$ GeV Zyla  _et al._ (2020), we obtain the physical masses
to be $\tilde{m}_{T}\approx 2.35$ GeV and $\tilde{m}_{R}\approx 2.73$ GeV with
$a_{0}=0.52$. Keeping in mind the large uncertainties from both the
theoretical side and the experimental side, these values are roughly
comparable to the corresponding experimental data (4). For the $B$ meson
family, using $m_{b}\approx 4.66$ GeV Zyla  _et al._ (2020), we obtain the
physical masses to be $\tilde{m}_{T}\approx 5.74$ GeV and
$\tilde{m}_{R}\approx 6.12$ GeV. Comparing with the experimental data for the
bottom mesons in (6), the numerical result of $\tilde{m}_{T}$ is close to the
corresponding experimental data. And we expect $(1^{-},2^{-})$ $B$ meson
states appear at around 6.12 GeV. Actually, the excited state $B_{J}(5970)$
reported in Refs. Aaltonen  _et al._ (2014); Aaij  _et al._ (2015) with the
mass $\sim 5970$ MeV could be the candidate for the $1^{-}$ or $2^{-}$ states.
Up to now, experiments have not yet determined the $J^{P}$ quantum numbers for
$B_{J}(5970)$. We expect this could be done in the near future.

Now, let us turn to the results using the quark self-energy fitted from
lattice QCD. We use the lattice fittings for the quark wave function
renormalization $Z(-p^{2})$ and the running quark mass $M(-p^{2})$ given in
Ref. Oliveira  _et al._ (2019). The formula of LECs shown in Eqs.
(LABEL:eq:mHgH)-(18) can be directly extended to incorporate the contributions
from $Z(-p^{2})$. The relevant expressions are given in Appendix A. The
functions $Z(-p^{2})$ and $M(-p^{2})$ are also plotted in Fig. 2. The
numerical results of LECs calculated by using these lattice fittings are given
in Table 2. There is no free parameters in the functions of $M(-p^{2})$ and
$Z(-p^{2})$. We display results from various values of the cutoff
$\Lambda_{c}$ for comparison.

Figure 1: Running coupling constant of the G-Z model with $a_{0}=0.52$.  Figure 2: The lattice fittings of $M(-p^{2})$ and $Z(-p^{2})$ given in Ref. Oliveira  _et al._ (2019) and $\Sigma(-p^{2})$ from the gap equation with G-Z Model.  Table 2: LECs calculated from lattice fittings given in Ref Oliveira  _et al._ (2019).  $\Lambda_{c}$(GeV) |  $M(0)$(GeV) |  $f_{\pi}$ (GeV) |  ${}-\langle\bar{\psi}\psi\rangle$ (GeV3) |  $g_{R}$ |  $g_{T}$ |  $g_{TR}$ |  $m_{R}$ (GeV) |  $m_{T}$ (GeV) |  $\Delta m$ (GeV)
---|---|---|---|---|---|---|---|---|---
1.4 | 0.344 | 0.095 | $(0.341)^{3}$ | 1.101 | $-0.398$ | $-1.031$ | 1.349 | 0.948 | 0.401
1.5 | 0.344 | 0.096 | $(0.351)^{3}$ | 1.054 | -0.432 | -1.041 | 1.405 | 1.021 | 0.384
1.6 | 0.344 | 0.096 | $(0.359)^{3}$ | 1.015 | -0.462 | -1.050 | 1.462 | 1.095 | 0.368
1.7 | 0.344 | 0.096 | $(0.367)^{3}$ | 0.983 | -0.488 | -1.058 | 1.522 | 1.169 | 0.352
1.8 | 0.344 | 0.096 | $(0.375)^{3}$ | 0.956 | -0.511 | -1.064 | 1.582 | 1.245 | 0.337

From Table 2, one can see that the numerical results of LECs are comparable to
the results from the G-Z model when $\Lambda_{c}=1.6$ GeV, which implies that
$Z(-p^{2})$ does not have significant impacts on the excited states. The
physical masses are $\tilde{m}_{T}\approx 2.37$ GeV and $\tilde{m}_{R}\approx
2.73$ GeV for excited $D$ mesons, and $\tilde{m}_{T}\approx 5.76$ GeV and
$\tilde{m}_{G}\approx 6.12$ GeV for excited $B$ mesons. The conclusions could
be drawn from these results are the same as we have in the G-Z model.

##  V Conclusions

In this paper, we derive the chiral effective Lagrangian for excited heavy-
light mesons from QCD. We focus on the chiral partners with
$j_{l}^{P}=\frac{3}{2}^{+}$ and $j_{l}^{P}=\frac{3}{2}^{-}$ which amounts to
($1^{+},2^{+}$) and ($1^{-},2^{-}$) states respectively. The low energy
constants are expressed as momentum integrals of the light quark self-energy,
which in turn are obtained by using the gap equation or lattice QCD. The
relevant LECs in the chiral Lagrangian are calculated, and the resulted masses
for the excited $D$ mesons are found roughly consistent with experimental
data. For the excited $B$ mesons, we obtain $\tilde{m}_{T}\approx 5.74$ GeV
and $\tilde{m}_{R}\approx 6.12$ GeV. We find that $\tilde{m}_{T}$ are
consistent to the masses of $(B_{1}(5721),B_{2}^{*}(5747))$, and the result of
$\tilde{m}_{R}$ suggests that $B_{J}(5970)$ could be a good candidate for the
states in the doublet $(1^{-},2^{-})$.

As more and more excited states of charmed mesons and bottom mesons are
discovered in experiments, it will be an interesting and important research to
extend our method to obtain the chiral effective Lagrangian of heavy-light
mesons with arbitrary spins. Since this extension is straightforward, we will
not go to details here.

We finally want to say that, so far, we did not discuss the chiral partner
structure of the heavy-light mesons including a strange quark. One of the
reasons is that the quark content of some of these mesons is still under
debate. For example, the mesons $D_{s0}(2317)$ and $D_{s1}(2460)$ which can be
arranged to the $G$ doublet with quantum numbers $G=(0^{+},1^{+})$ are
regarded as the chiral partners of $D$ and $D^{\ast}$, respectively, in the
$H$ doublet with quantum numbers $H=(0^{-},1^{-})$ Nowak  _et al._ (2004). The
molecular state interpretation of these mesons and their bottom cousins cannot
be ruled out Faessler  _et al._ (2007a, b, 2008). We leave the discussion of
these strange mesons as future works.

###### Acknowledgements.

The work of Y. L. M. was supported in part by National Science Foundation of
China (NSFC) under Grant No. 11875147 and No.11475071. H.-F. Fu was supported
by NSFC under Grant No.12047569. Q. W. was supported by the National Science
Foundation of China (NSFC) under Grant No. 11475092.

##  Appendix A FORMULA FOR LECS WITH $Z(-p^{2})$

The quark propagator can be generally written as

$\displaystyle S(p)$ $\displaystyle=$ $\displaystyle\frac{i}{A(-p^{2})\not{p}\
-B(-p^{2})}$ $\displaystyle=$ $\displaystyle i\,Z(-p^{2})\,\frac{\not{p}\
+M(-p^{2})}{p^{2}-M^{2}(-p^{2})},$

where $Z(-p^{2})=1/A(-p^{2})$ stands for the quark wave function
renormalization and $M(-p^{2})=B(-p^{2})/A(-p^{2})$ is the renormalization
group invariant running quark mass. After a series of calculations, we can get
the LECs with $Z(-p^{2})$ as follows:

$\displaystyle m_{T}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}}{Z_{T}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggr{[}{}\frac{p^{2}-(v\cdot
p)^{2}}{p^{2}-M^{2}}-\frac{p^{2}M}{(p^{2}-M^{2})v\cdot p}\biggr{]}Z,$
$\displaystyle g_{T}$ $\displaystyle=$
$\displaystyle-\frac{iN_{c}}{Z_{T}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggl{[}\frac{p^{2}(M^{2}+\frac{2}{3}p^{2})}{(p^{2}-M^{2})^{2}v\cdot
p}-\frac{2M[p^{2}-(v\cdot p)^{2}]}{(p^{2}-M^{2})^{2}}\biggr{]}Z^{2},$
$\displaystyle m_{R}$ $\displaystyle=$
$\displaystyle\frac{iN_{c}}{Z_{R}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggr{[}{}\frac{p^{2}-(v\cdot
p)^{2}}{p^{2}-M^{2}}+\frac{p^{2}M}{(p^{2}-M^{2})v\cdot p}\biggr{]}Z,$
$\displaystyle g_{R}$ $\displaystyle=$
$\displaystyle-\frac{iN_{c}}{Z_{R}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggl{[}\frac{p^{2}(M^{2}+\frac{2}{3}p^{2})}{(p^{2}-M^{2})^{2}v\cdot
p}+\frac{2M[p^{2}-(v\cdot p)^{2}]}{(p^{2}-M^{2})^{2}}\biggr{]}Z^{2},$
$\displaystyle Z_{T}$ $\displaystyle=$ $\displaystyle
iN_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\Biggr{\\{}\biggr{[}-\frac{p^{2}}{(p^{2}-M^{2})v\cdot
p}-\frac{[2M+2ZM^{\prime}(p^{2}+M^{2})][p^{2}-(v\cdot
p)^{2}]}{(p^{2}-M^{2})^{2}}\Biggr{]}Z-\frac{2Z^{\prime}M}{p^{2}-M^{2}}\biggr{\\}}$
$\displaystyle Z_{R}$ $\displaystyle=$ $\displaystyle
iN_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\Biggr{\\{}\biggr{[}-\frac{p^{2}}{(p^{2}-M^{2})v\cdot
p}+\frac{[2M+2ZM^{\prime}(p^{2}+M^{2})][p^{2}-(v\cdot
p)^{2}]}{(p^{2}-M^{2})^{2}}\Biggr{]}Z+\frac{2Z^{\prime}M}{p^{2}-M^{2}}\biggr{\\}},$
$\displaystyle g_{TR}$ $\displaystyle=$
$\displaystyle-i\frac{N_{c}}{\sqrt{Z_{T}Z_{R}}}\int\frac{d^{4}p}{(2\pi)^{4}}\biggl{[}\frac{p^{2}(M^{2}+p^{2})}{(p^{2}-M^{2})^{2}v\cdot
p}\biggr{]}Z^{2}.$ (21)

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