# Hadronic molecule model for the doubly charmed state $T^{+}_{cc}$

S. S. Agaev  Institute for Physical Problems, Baku State University, Az–1148
Baku, Azerbaijan  K. Azizi  Department of Physics, University of Tehran, North
Karegar Avenue, Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Dudullu-Ümraniye, 34775 Istanbul, Turkey  H. Sundu  Department of
Physics, Kocaeli University, 41380 Izmit, Turkey

###### Abstract

The mass, current coupling, and width of the doubly charmed four-quark meson
$T_{cc}^{+}$ are explored by treating it as a hadronic molecule
$M_{cc}^{+}\equiv D^{0}D^{\ast+}$. The mass and current coupling of this
molecule are calculated using the QCD two-point sum rule method by including
into analysis contributions of various vacuum condensates up to dimension
$10$. The prediction for the mass $m=(4060\pm 130)~{}\mathrm{MeV}$ exceeds the
two-meson $D^{0}D^{\ast+}$ threshold $3875.1~{}\mathrm{MeV}$, which makes
decay of the molecule $M_{cc}^{+}$ to a pair of conventional mesons
$D^{0}D^{\ast+}$ kinematically allowed process. The strong coupling $G$ of
particles at the vertex $M_{cc}^{+}D^{0}D^{\ast+}$ is found by applying the
QCD three-point sum rule approach, and used to evaluate the width of the decay
$M_{cc}^{+}\to D^{0}D^{\ast+}$. Obtained result for the width $\Gamma=(3.8\pm
1.7)~{}\mathrm{MeV}$ demonstrates that $M_{cc}^{+}$ is wider than the
resonance $T_{cc}^{+}$.

##  I Introduction

Recently, the LHCb collaboration informed about observation, for the first
time, of a doubly charmed axial-vector state $T_{cc}^{+}$ composed of four
quarks $cc\overline{u}\overline{d}$ Aaij:2021vvq ; LHCb:2021auc . This state
was fixed in $D^{0}D^{0}\pi^{+}$ mass distribution as a narrow peak with the
width $\Gamma=(410\pm 165\pm 43_{-38}^{+18})~{}\mathrm{keV}$, which means that
it is longest living exotic meson discovered till now. The mass of
$T_{cc}^{+}$ is very close to the two-meson $D^{0}D^{\ast+}$ threshold
$3875.1~{}\mathrm{MeV}$, but is smaller than this limit by an amount of
$\delta m_{\exp}=(-273\pm 61\pm 5_{-14}^{+11})~{}\mathrm{keV}$. These features
of $T_{cc}^{+}$, in particular its narrow width, made the doubly charmed
exotic meson $T_{cc}^{+}$ an object of intensive studies Agaev:2021vur ;
Feijoo:2021ppq ; Yan:2021wdl ; Fleming:2021wmk ; Azizi:2021aib ; Meng:2021jnw
; Ling:2021bir ; Chen:2021vhg ; Xin:2021wcr .

It is worth emphasizing that doubly charmed tetraquarks attracted already
interests of researchers. This is connected with estimated stability some of
tetraquarks containing heavy diquarks $bb$, $bc$ and $cc$ against strong and
maybe electromagnetic decays. If exist, such particles can transform to mesons
only through weak decays, and have mean lifetimes which would be considerably
longer than that of conventional mesons Agaev:2018khe ; Agaev:2019lwh ;
Agaev:2020zag ; Agaev:2019kkz . There is growing conviction that tetraquarks
built of $bb$ diquarks are stable particles, whereas the situation with ones
composed of $bc$ and $cc$ diquarks is still remaining controversial
Karliner:2017qjm ; Eichten:2017ffp ; Agaev:2020zad .

Because the present work is devoted to investigation of doubly charmed states,
below we restrict ourselves by analyses of problems and achievements connected
only with these particles. Thus, tetraquarks
$cc\overline{q}\overline{q}^{\prime}$ were theoretically studied using
different methods of the high energy physics. In the framework of the QCD sum
rule method they were analyzed in Refs. Navarra:2007yw ; Du:2012wp . In the
first article the authors explored the axial-vector tetraquark
$cc\overline{u}\overline{d}$. Prediction for its mass $(4000\pm
200)~{}\mathrm{MeV}$ implies that the axial-vector tetraquark
$cc\overline{u}\overline{d}$ is unstable and readily decays to mesons
$D^{0}D^{\ast}{}^{+}$. Four-quark exotic mesons of general
$cc\overline{q}\overline{q}^{\prime}$ content and quantum numbers
$J^{\mathrm{P}}=0^{-},~{}0^{+},~{}1^{-}$ and $1^{+}$ were investigated in Ref.
Du:2012wp . In accordance with results of this analysis, masses of tetraquarks
$cc\overline{q}\overline{q}$, $cc\overline{q}\overline{s}$, and
$cc\overline{s}\overline{s}$ are above corresponding thresholds for all
explored quantum numbers. In other words, a class of tetraquarks composed of a
diquark $cc$ and a light antidiquark does not contain strong-interaction
stable particles.

Discovery of doubly charmed baryon $\Xi_{cc}^{++}=ccu$ by the LHCb
collaboration Aaij:2017ueg , and extracted experimental information stimulated
relatively new studies of heavy tetraquarks. A reason was that, these
experimental data were employed as new input parameters in a phenomenological
model to estimate masses of the axial-vector tetraquarks
$T_{bb;\overline{u}\overline{d}}^{-}$ and
$T_{cc;\overline{u}\overline{d}}^{+}$ Karliner:2017qjm ; Eichten:2017ffp . In
these articles it was demonstrated that $T_{cc;\overline{u}\overline{d}}^{+}$
has the mass $(3882\pm 12)~{}\mathrm{MeV}$ and $3978~{}\mathrm{MeV}$,
respectively, which are above thresholds for both $D^{0}D^{\ast+}$ and
$D^{0}D^{+}\gamma$ decays. Other members of $cc\overline{q}\overline{q}$, and
$cc\overline{q}\overline{s}$ families were considered in Ref. Eichten:2017ffp
: None of them were classified as a stable state. Similar conclusions about
properties of $T_{cc;\overline{u}\overline{d}}^{+}$ were drawn in Refs.
Wang:2017dtg ; Braaten:2020nwp ; Cheng:2020wxa  as well. Contrary to these
studies, in Ref. Meng:2020knc  the authors calculated the mass of
$T_{cc;\overline{u}\overline{d}}^{+}$ using a constituent quark model and
found that it is $23~{}\mathrm{MeV}$ below the two-meson threshold. Stable
nature of $T_{cc;\overline{u}\overline{d}}^{+}$ was demonstrated also by means
of lattice simulations Junnarkar:2018twb , in which its mass was estimated
$(-23\pm 11)~{}\mathrm{MeV}$ below the two-meson threshold.

Detailed studies of pseudoscalar and scalar exotic mesons
$cc\overline{u}\overline{d}$ were done in Ref. Agaev:2019qqn . Analysis
performed there, demonstrated that these particles are strong-interaction
unstable structures, and fall apart to conventional mesons. Full widths of
these tetraquarks were evaluated by utilizing their decays to
$D^{+}D^{\ast}(2007)^{0}$, $D^{0}D^{\ast}(2010)^{+}$, and $D^{0}D^{+}$ mesons,
respectively. It was found, that these structures with widths $\sim
130~{}\mathrm{MeV}$ and $\sim 12~{}\mathrm{MeV}$ are relatively wide
resonances. Structures $cc\overline{s}\overline{s}$ and
$cc\overline{d}\overline{s}$ form another interesting subgroup of doubly
charmed tetraquarks, because they are also doubly charged particles. Masses
and widths of such pseudoscalar tetraquarks were evaluated in Ref.
Agaev:2018vag .

Doubly charmed four-quark structures were studied also in the context of the
hadronic molecule picture, i.e., they were modeled as molecules of
conventional mesons. It is worth noting that charmonium molecules are not new
objects for investigations: Problems of such compounds were addressed in
literature decades ago Novikov:1977dq . As a hadronic molecule
$M_{cc}^{+}\equiv D^{0}D^{\ast+}$ built of ordinary mesons $D^{0}$ and
$D^{\ast+}$, the axial-vector state $cc\overline{u}\overline{d}$ was
considered in Refs. Dias:2011mi ; Li:2012ss . The mass of $M_{cc}^{+}$ was
estimated in Ref. Dias:2011mi  using the QCD spectral sum rule approach.
Obtained prediction $(3872.2\pm 39.5)~{}\mathrm{MeV}$ shows that this molecule
cannot decay to mesons $D^{0}$ and $D^{\ast+}$, but its mass is enough to
trigger the strong decay $M_{cc}^{+}\rightarrow D^{0}D^{0}\pi^{+}$.

In our recent article, we treated $T_{cc}^{+}$ as an axial-vector diquark-
antidiquark (tetraquark) state with quark content
$cc\overline{u}\overline{d}$, and calculated its spectroscopic parameters and
full width Agaev:2021vur . Computations performed in the context of the QCD
two-point sum rule method led for the mass of this state to the result
$(3868\pm 124)~{}\mathrm{MeV}$, which is consistent with the LHCb
measurements. This means that $T_{cc}^{+}$ does not decay to a meson pair
$D^{0}D^{\ast+}$. Therefore, we evaluated full width of $T_{cc}^{+}$ by
considering its alternative strong decay channels. In fact, production of
$D^{0}D^{0}\pi^{+}$ can run through decay of $T_{cc}^{+}$ to a scalar
tetraquark $T_{cc;\overline{u}\overline{u}}^{0}$ and $\pi^{+}$ followed by the
process $T_{cc;\overline{u}\overline{u}}^{0}\rightarrow D^{0}D^{0}$. The
process $T_{cc}^{+}\rightarrow\widetilde{T}\pi^{0}\rightarrow
D^{0}D^{+}\pi^{0}$ is another decay mode of $T_{cc}^{+}$. Here,
$\widetilde{T}$ is the scalar exotic meson with content
$cc\overline{u}\overline{d}$. This means, that in our analysis decays to
scalar tetraquarks $T_{cc;\overline{u}\overline{u}}^{0}$ and $\widetilde{T}$
was considered as a dominant mechanism for transformation of $T_{cc}^{+}$ .
Full width of $T_{cc}^{+}$ estimated in Ref. Agaev:2021vur  equals to
$\Gamma=(489\pm 92)~{}\mathrm{keV}$ which nicely agrees with the experimental
data.

As is seen, an assumption about the diquark-antidiquark structure of
$T_{cc}^{+}$ gives for its mass and width results compatible with the LHCb
data Agaev:2021vur . In accordance to Ref. Dias:2011mi , the molecule model
for the mass of $M_{cc}^{+}$ leads to almost the same prediction.
Unfortunately, in this paper the authors did not compute width of the molecule
$M_{cc}^{+}$, therefore it is difficult to declare a full convergence of
results for $T_{cc}^{+}$ and $M_{cc}^{+}$ obtained in the framework of the QCD
sum rule method. The reason is that masses of $T_{cc}^{+}$ and $M_{cc}^{+}$
were extracted, as usual, with theoretical uncertainties, and due to
overlapping of relevant regions, this information is not enough to distinguish
diquark-antidiquark and molecule states. To make reliable statements about
internal organization of the four-quark state seen by LHCb, it is necessary to
investigate decay modes of this particle, and calculate its full width.

The program outlined above was realized in the diquark-antidiquark picture in
our article Agaev:2021vur . In the present work, we consider this problem in
the framework of the hadronic molecule model, and calculate the mass and width
of $M_{cc}^{+}$. We wish to answer a question whether both the mass and width
of $M_{cc}^{+}$ agree with new LHCb data. For these purposes, we calculate the
spectroscopic parameters of $M_{cc}^{+}$ using the QCD two-point sum rule
method Shifman:1978bx ; Shifman:1978by . Our analysis proves that the mass of
$M_{cc}^{+}$ exceeds the LHCb data, which makes the process
$M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ kinematically allowed one. The width of
this decay channel is found by means of the three-point version of QCD sum
rule approach: It is used to extract the strong coupling $G$ at the vertex
$M_{cc}^{+}D^{0}D^{\ast+}$.

This article is structured in the following manner: In Sec. II, we compute the
mass $m$ and coupling $f$ of the molecule $M_{cc}^{+}$ in the context of the
QCD two-point sum rule method. In these calculations, we take into account
various vacuum condensates up to dimension $10$. In Sec. III, we consider the
decay mode $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$, find the strong coupling
$G$ and evaluate the width of this process. We reserve Sec. IV for discussion
and conclusions.

##  II Spectroscopic parameters of $M_{cc}^{+}$

The sum rules necessary to evaluate the spectroscopic parameters of the
molecule $M_{cc}^{+}$ can be derived from analysis of the correlation function

$\Pi_{\mu\nu}(p)=i\int d^{4}xe^{ipx}\langle
0|\mathcal{T}\\{J_{\mu}(x)J_{\nu}^{{\dagger}}(0)\\}|0\rangle,$ (1)

where $J_{\mu}(x)$ is the interpolation current for the axial-vector state
$M_{cc}^{+}$. In the hadronic molecule model the current $J_{\mu}(x)$ is given
by the expression

$J_{\mu}(x)=\overline{d}_{a}(x)\gamma_{\mu}c_{a}(x)\overline{u}_{b}(x)\gamma_{5}c_{b}(x),$
(2)

where $a$ and $b$ are color indices.

To find the sum rules for $m$ and $f$, we express the correlation function
$\Pi_{\mu\nu}(p)$ in terms of $M_{cc}^{+}$ molecule’s physical parameters.
Because $M_{cc}^{+}$ is composed of ground-state mesons $D^{0}$ and
$D^{\ast+}$, it can be treated as lowest lying system in this class of
particles. Therefore, in the correlation function
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$, we write down explicitly only first term
that corresponds to $M_{cc}^{+}$

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{\langle
0|J_{\mu}|M_{cc}^{+}(p,\epsilon)\rangle\langle
M_{cc}^{+}(p,\epsilon)|J_{\nu}^{\dagger}|0\rangle}{m^{2}-p^{2}}+\cdots.$ (3)

The $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ is obtained by inserting into the
correlation function Eq. (1) the full set of states with spin-parities
$J^{\mathrm{P}}=1^{+}$, and carrying out integration over $x$. The dots in Eq.
(3) denote contributions coming from higher resonances and continuum states.

To derive Eq. (3), we assume that the physical side of the sum rule can be
approximated by a single pole term. In the case of the multiquark systems
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ receives contribution, however, also from
two-meson reducible terms Kondo:2004cr ; Lee:2004xk . That is because the
current $J_{\mu}(x)$ interacts not only with a molecule $M_{cc}^{+}$, but also
with the two-meson continuum with the same quantum numbers and quark content.
Effects of current-continuum interaction, properly taken into account,
generates a finite width $\Gamma(p^{2})$ of the hadronic molecule and leads to
the modification in Eq. (3) in accordance with the prescription Wang:2015nwa

$\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p^{2})}.$
(4)

The two-meson contributions can be included into analysis by rescaling the
coupling $f$ of $M_{cc}^{+}$, and keeping untouched its mass. Calculations
demonstrated that these effects are small and do not exceed uncertainties of
sum rule calculations. Indeed, in the case of the doubly charmed pseudoscalar
tetraquark $cc\overline{s}\overline{s}$ with the mass
$m_{T}=4390~{}\mathrm{MeV}$ and full width $\Gamma_{T}\approx
300~{}\mathrm{MeV}$, two-meson effects lead to additional $\approx 7\%$
uncertainty in the current coupling $f_{T}$ Agaev:2018vag . For the resonance
$Z_{c}^{-}(4100)$ these ambiguities amount to $\approx 5\%$ of the coupling
$f_{Z_{c}}$ Sundu:2018nxt . As we shall see below, the molecule $M_{cc}^{+}$
has the width $(3.8\pm 1.7)~{}\mathrm{MeV}$. Therefore, aforementioned effects
are negligible, and in $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ it is enough to
employ the zero-width single-pole approximation.

The function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ can be presented in a more
compact form. To this end, we introduce the matrix element

$\langle 0|J_{\mu}|M_{cc}^{+}(p,\epsilon)\rangle=fm\epsilon_{\mu},$ (5)

where $\epsilon_{\mu}$ is the polarization vector of the molecule
$M_{cc}^{+}$. It is not difficult to demonstrate that the function
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ in terms of $m$ and $f$ has the simple form

$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)=\frac{m^{2}f^{2}}{m^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m^{2}}\right)+\cdots.$
(6)

The QCD side of the sum rules $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ has to be
calculated in the operator product expansion ($\mathrm{OPE}$) with some fixed
accuracy. To find $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$, we calculate the
correlation function using explicit form of the current $J_{\mu}(x)$. As a
result, we express $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$ in terms of heavy and
light quark propagators

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p)=i\int d^{4}xe^{ip\cdot
x}\left\\{\mathrm{Tr}\left[\gamma_{5}S_{c}^{bb^{\prime}}(x)\gamma_{5}S_{u}^{b^{\prime}b}(-x)\right]\right.$
$\displaystyle\times\mathrm{Tr}\left[\gamma_{\mu}S_{c}^{aa^{\prime}}(x)\gamma_{\nu}S_{d}^{a^{\prime}a}(-x)\right]-\mathrm{Tr}\left[\gamma_{\mu}S_{c}^{ab^{\prime}}(x)\gamma_{5}\right.$
$\displaystyle\left.\left.\times
S_{u}^{b^{\prime}b}(-x)\gamma_{5}S_{c}^{ba^{\prime}}(x)\gamma_{\nu}S_{d}^{a^{\prime}a}(-x)\right]\right\\}.$
(7)

In Eq. (7) $S_{q}^{ab}(x)$ and $S_{c}^{ab}(x)$ are propagators of $q(u,d)$ and
$c$-quarks, formulas for which are collected in Appendix.

The QCD sum rules can be derived using the same Lorentz structures in
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p)$ and $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. For
our purposes, the structures proportional to $g_{\mu\nu}$ are appropriate,
because they are free of contributions of spin-$0$ particles. To obtain a sum
rule, we equate invariant amplitudes $\Pi^{\mathrm{Phys}}(p^{2})$ and
$\Pi^{\mathrm{OPE}}(p^{2})$ corresponding to these structures, and apply the
Borel transformation to both sides of the obtained expression. The last
operation is necessary to suppress contributions stemming from the higher
resonances and continuum states. At the following phase of manipulations, we
make use an assumption about the quark-hadron duality, and subtract from the
physical side of the equality higher resonances’ and continuum contributions.
By this way, the final sum rule equality acquires a dependence on the Borel
$M^{2}$ and continuum threshold (subtraction) $s_{0}$ parameters. This
equality, and second expression obtained by applying the operator
$d/d(-1/M^{2})$ to its both sides, form a system which is used to find sum
rules for the mass $m$ and coupling $f$

$m^{2}=\frac{\Pi^{\prime}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (8)
$f^{2}=\frac{e^{m^{2}/M^{2}}}{m^{2}}\Pi(M^{2},s_{0}),$ (9)

where $\Pi^{\prime}(M^{2},s_{0})=d\Pi(M^{2},s_{0})/d(-1/M^{2})$.

In Eqs. (8) and (9) the function $\Pi(M^{2},s_{0})$ is Borel transformed and
continuum subtracted invariant amplitude $\Pi^{\mathrm{OPE}}(p^{2})$. We
calculate $\Pi(M^{2},s_{0})$ by taking into account quark, gluon and mixed
vacuum condensates up to dimension $10$. It has the following form

$\Pi(M^{2},s_{0})=\int_{4m_{c}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$
(10)

where $\rho^{\mathrm{OPE}}(s)$ is the two-point spectral density. The second
component of the invariant amplitude $\Pi(M^{2})$ contains nonperturbative
contributions calculated directly from $\Pi_{\mu\nu}^{\mathrm{OPE}}(p)$. The
explicit expression of the function $\Pi(M^{2},s_{0})$ is removed to Appendix.

The quark, gluon and mixed condensates which enter to the sum rules (8) and
(9) are universal parameters of computations:

$\displaystyle\langle\overline{q}q\rangle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3},\ $ $\displaystyle\langle\overline{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\overline{q}q\rangle,\ m_{0}^{2}=(0.8\pm
0.1)~{}\mathrm{GeV}^{2},\ $
$\displaystyle\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle=(0.012\pm
0.004)~{}\mathrm{GeV}^{4},$ $\displaystyle\langle
g_{s}^{3}G^{3}\rangle=(0.57\pm 0.29)~{}\mathrm{GeV}^{6},$ $\displaystyle
m_{c}=1.275\pm 0.025~{}\mathrm{GeV}.$ (11)

The correlation function $\Pi(M^{2},s_{0})$ depends on the $c$ quark mass,
numerical value of which is shown in Eq. (11) as well. Contrary, the Borel and
continuum threshold parameters $M^{2}$ and $s_{0}$ are auxiliary quantities of
calculations: Their choice depends on the problem under consideration, and has
to meet restrictions imposed on the pole contribution ($\mathrm{PC}$) and
convergence of $\mathrm{OPE}$.

To estimate the $\mathrm{PC}$, we use the expression

$\mathrm{PC}=\frac{\Pi(M^{2},s_{0})}{\Pi(M^{2},\infty)}.$ (12)

The convergence of the operator product expansion is checked by means of the
formula

$R(M^{2})=\frac{\Pi^{\mathrm{DimN}}(M^{2},s_{0})}{\Pi(M^{2},s_{0})},$ (13)

where $\Pi^{\mathrm{DimN}}(M^{2},s_{0})$ is the contribution of the last three
terms in $\mathrm{OPE}$, i.e., $\mathrm{DimN=Dim(8+9+10)}$ .

In the current investigation, we use a restriction $\mathrm{PC\geq 0.2}$ which
is typical for multiquark hadrons. We also consider $\mathrm{OPE}$ as a
convergent provided at the minimum of the Borel parameter the ratio $R(M^{2})$
is less than $0.01$. Calculations confirm that the working windows that
satisfy these requirements are

$M^{2}\in[4,6]~{}\mathrm{GeV}^{2},\ s_{0}\in[19.5,21.5]~{}\mathrm{GeV}^{2}.$
(14)

In fact, within these regions $\mathrm{PC}$ changes on average in limits
$0.20\leq\mathrm{PC}\leq 0.61,$ and at the minimum
$M^{2}=4~{}\mathrm{GeV}^{2}$, we get $R(M^{2})\leq 0.01$. In general, sum
rules’ predictions should not depend on the choice of $M^{2}$, but in real
analysis there is a undesirable dependence of $m$ and $f$ on the Borel
parameter $M^{2}$. Therefore, the window for $M^{2}$ should minimize this
dependence as well, and the region from Eq. (14) obeys this condition.

To extract the mass $m$ and coupling $f$, we calculate them at different
choices of the parameters $M^{2}$ and $s_{0}$, and find their values averaged
over the working regions Eq. (14)

$\displaystyle m$ $\displaystyle=$ $\displaystyle(4060\pm
130)~{}\mathrm{MeV},$ $\displaystyle f$ $\displaystyle=$ $\displaystyle(5.1\pm
0.8)\times 10^{-3}~{}\mathrm{GeV}^{4}.$ (15)

These results correspond to the point $M^{2}=5~{}\mathrm{GeV}^{2}$ and
$s_{0}=20.4~{}\mathrm{GeV}^{2}$ which is approximately at middle of the
regions Eq. (14). The pole contribution computed at this point is equal to
$\mathrm{PC}\approx 0.53$, which guarantees credibility of obtained
predictions, and the ground-state nature of $M_{cc}^{+}$ in its class of
particles.

The mass $m$ of the molecule $M_{cc}^{+}$ as a function of $M^{2}$ is plotted
in Fig. 1. Here, we show dependence of $m$ on the Borel parameter in a wide
range of $M^{2}$. One can see, that predictions obtained at values of $M^{2}$
from Eq. (14) are relatively stable, though residual effects of $M^{2}$ on $m$
is evident in this region as well: This is unavoidable feature of the sum rule
method which limits its accuracy. At the same time, this method allows one to
estimate ambiguities of performed analysis which is the case only for some of
nonperturbative QCD approaches.

The second source of uncertainties is the continuum threshold parameter
$s_{0}$, that separates a ground-state term from contributions of higher
resonances and continuum states. It carries also physical information about
first excitation of the $M_{cc}^{+}$, meaning that $\sqrt{s_{0}}$ should be
smaller than a mass of such state. Parameters of excited conventional hadrons
are known from theoretical studies or were measured in numerous experiments.
Therefore, a choice of the scale $s_{0}$ in relevant studies does not create
new problems. The mass spectra of multiquark hadrons, in general, may have
more complex structure. Additionally, there are only a few resonances, which
can be considered as radially or orbitally excited exotic hadrons. Thus, the
resonances $Z_{c}(3900)$ and $Z_{c}(4430)$ with a mass gap $\approx
530~{}\mathrm{MeV}$ may be treated as the ground-state and first radially
excited axial-vector tetraquark $[cu][\overline{c}\overline{d}]$, respectively
Maiani:2014 . This conjecture was later confirmed by the sum rule calculations
in Refs. Wang:2014vha ; Agaev:2017tzv . The mass spectra of the doubly heavy
tetraquarks were analyzed in Ref. Kim:2022mpa  in the framework of a chiral-
diquark picture. The difference between masses of doubly charmed $1S$ and $2S$
axial-vector tetraquarks was found there equal to $\approx
400~{}\mathrm{MeV}$. In light of these investigations, $\sqrt{s_{0}}$ may
exceed $m$ approximately $0.4-0.6~{}\mathrm{MeV}$. In our case, this gap is
$\sqrt{s_{0}}-m\approx 450~{}\mathrm{MeV}$ which is a reasonable estimate for
an exotic state composed of mesons $D^{0}$ and $D^{\ast+}$ and containing two
$c$ quarks. Dependence of $m$ on the continuum threshold parameter $s_{0}$ is
shown in Fig. 2.

Figure 1: The mass of the hadronic molecule $M_{cc}^{+}$ as a function of the
Borel parameter $M^{2}$ at fixed $s_{0}$. Vertical lines show boundaries of
working region for $M^{2}$ used in numerical computations.  Figure 2:
Dependence of $m$ on the continuum threshold parameter $s_{0}$ at fixed
$M^{2}$.

The central value of the mass $m$ is above the two-meson
$D^{0}D^{\ast}(2010)^{+}$ threshold $3875.1~{}\mathrm{MeV}$, and exceeds the
datum of the LHCb collaboration. Even the low estimate for the mass
$3930~{}\mathrm{MeV}$ overshoots this boundary. In other words, the dominant
decay channel of the molecule $M_{cc}^{+}$ is the process
$M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$. In the next section, we are going to
calculate the width of $M_{cc}^{+}$ using this decay.

##  III Width of the decay $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$

The four-quark state $T_{cc}^{+}$ was observed in $D^{0}D^{0}\pi^{+}$ mass
distribution, and therefore it decays strongly to these mesons. The hadronic
molecule $M_{cc}^{+}$ has the same quantum numbers and quark content,
therefore the process $M_{cc}^{+}\rightarrow D^{0}D^{0}\pi^{+}$ is among
possible decay modes of $M_{cc}^{+}$. This decay may proceed through two
stages: the process $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ followed by the
decay $D^{\ast+}\rightarrow D^{0}\pi^{+}$. Our calculations show that the mass
of $M_{cc}^{+}$ is enough to generate this chain of transformations.

In this section, we study the decay $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ and
find the strong coupling $G$ of particles at the vertex
$M_{cc}^{+}D^{0}D^{\ast+}$. The QCD three-point sum rule for this coupling can
be derived from analysis of the correlation function

$\displaystyle\Pi_{\mu\nu}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\langle
0|\mathcal{T}\\{J_{\nu}^{D^{\ast}}(y)$ $\displaystyle\times
J^{D}(0)J_{\mu}^{\dagger}(x)\\}|0\rangle,$ (16)

where $J_{\mu}(x)$, $\ J_{\nu}^{D^{\ast}}(x)$ and $J^{D}(x)$ are the relevant
interpolating currents. For the molecule $M_{cc}^{+}$ the current $J_{\mu}(x)$
is given by Eq. (2). The $J_{\nu}^{D^{\ast}}(x)$ and $J^{D}(x)$ are currents
of the mesons $D^{\ast+}$ and $D^{0}$, which have the following forms

$J_{\nu}^{D^{\ast}}(x)=\overline{d}_{i}(x)\gamma_{\nu}c_{i}(x),\
J^{D}(x)=\overline{u}_{j}(x)i\gamma_{5}c_{i}(x),$ (17)

where $i$ and $j$ are color indices. The 4-momenta of the particles
$M_{cc}^{+}$ and $D^{\ast+}$ are $p$ and $p^{\prime}$, respectively, hence the
momentum of the $D^{0}$ meson is $q=p-p^{\prime}$.

We continue using standard prescriptions of the sum rule method and, first
calculate the correlation function $\Pi_{\mu\nu}(p,p^{\prime})$ in terms of
physical parameters of involved particles. Isolating in Eq. (16) a
contribution of the ground-state particles, we get

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=\frac{\langle
0|J_{\nu}^{D^{\ast}}|D^{\ast+}(p^{\prime},\varepsilon)\rangle\langle
0|J^{D}|D^{0}(q)\rangle}{(p^{2}-m^{2})(p^{\prime
2}-m_{D^{\ast}}^{2})(q^{2}-m_{D}^{2})}$ $\displaystyle\times\langle
M_{cc}^{+}(p,\epsilon)|J_{\mu}^{\dagger}|0\rangle\langle
D^{0}(q)D^{\ast+}(p^{\prime},\varepsilon)|M_{cc}^{+}(p,\epsilon)\rangle+\cdots,$
(18)

which is the physical side of the sum rule. In Eq. (18) $m_{D^{\ast}}$ and
$m_{D}$ are the masses of the $D^{\ast+}$ and $D^{0}$ mesons PDG:2020 ,
respectively:

$\displaystyle m_{D^{\ast}}$ $\displaystyle=$ $\displaystyle(2010.26\pm
0.05)~{}\mathrm{MeV},$ $\displaystyle m_{D}$ $\displaystyle=$
$\displaystyle(1864.84\pm 0.05)~{}\mathrm{MeV}.$ (19)

For our purposes, it is necessary to employ the $D^{\ast+}$ and $D^{0}$
mesons’ matrix elements, and, by this way to find compact expression for the
function $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$. This can be achieved by
using the matrix elements

$\displaystyle\langle
0|J_{\nu}^{D^{\ast}}|D^{\ast+}(p^{\prime},\varepsilon)\rangle=f_{D^{\ast}}m_{D^{\ast}}\varepsilon_{\nu},\
$ $\displaystyle\langle 0|J^{D}|D^{0}(q)\rangle=\frac{f_{D}m_{D}^{2}}{m_{c}},$
(20)

where $f_{D^{\ast}}$ and $f_{D}$ are their decay constants, whereas
$\varepsilon_{\nu}$ is the polarization vector of the meson $D^{\ast+}$. We
model the vertex $\langle
D^{0}(q)D^{\ast+}(p^{\prime},\varepsilon)|M_{cc}^{+}(p,\epsilon)\rangle$ by
the expression

$\displaystyle\langle
D^{0}(q)D^{\ast+}(p^{\prime},\varepsilon)|M_{cc}^{+}(p,\epsilon)\rangle=G(q^{2})\left[(p\cdot
p^{\prime})\right.$
$\displaystyle\left.\times(\varepsilon^{\ast}\cdot\epsilon)-(p\cdot\varepsilon^{\ast})(p^{\prime}\cdot\epsilon)\right],$
(21)

with $G(q^{2})$ being the strong coupling at the vertex
$M_{cc}^{+}D^{0}D^{\ast+}$. Then, it is not difficult to show that

$\displaystyle\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=G(q^{2})\frac{fmf_{D^{\ast}}m_{D^{\ast}}m_{D}^{2}f_{D}}{m_{c}(p^{2}-m^{2})(p^{\prime
2}-m_{D^{\ast}}^{2})}$
$\displaystyle\times\frac{1}{(q^{2}-m_{D}^{2})}\left(\frac{m^{2}+m_{D^{\ast}}^{2}-q^{2}}{2}g_{\mu\nu}-p_{\nu}p_{\mu}^{\prime}\right)+\cdots.$
(22)

The double Borel transformation of
$\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$ over variables $p^{2}$ and
$p^{\prime 2}$ yields

$\displaystyle\mathcal{B}\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})=G(q^{2})\frac{fmf_{D^{\ast}}m_{D^{\ast}}m_{D}^{2}f_{D}}{m_{c}(q^{2}-m_{D}^{2})}e^{-m^{2}/M_{1}^{2}}$
$\displaystyle\times
e^{-m_{D^{\ast}}^{2}/M_{2}^{2}}\left(\frac{m^{2}+m_{D^{\ast}}^{2}-q^{2}}{2}g_{\mu\nu}-p_{\nu}p_{\mu}^{\prime}\right)+\cdots.$

The function $\mathcal{B}\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$ is the
sum of two terms $\sim g_{\mu\nu}$ and $\sim p_{\nu}p_{\mu}^{\prime}$, which
may be utilized to obtain the required sum rule. For further studies, we
choose the invariant amplitude $\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2},q^{2})$
corresponding to the structure $\sim g_{\mu\nu}$. The Borel transformation of
this amplitude constitutes the physical side of the sum rule.

To determine the QCD side of the three-point sum rule, one should compute
$\Pi_{\mu\nu}(p,p^{\prime})$ using quark propagators. As a result, one gets

$\displaystyle\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})=i^{2}\int
d^{4}xd^{4}ye^{i(p^{\prime}y-px)}\left\\{\mathrm{Tr}\left[\gamma_{\nu}S_{c}^{ia}(y-x)\right.\right.$
$\displaystyle\left.\times\gamma_{\mu}S_{d}^{ai}(x-y)\right]\mathrm{Tr}\left[\gamma_{5}S_{u}^{bj}(x)\gamma_{5}S_{c}^{jb}(-x)\right]$
$\displaystyle\left.-\mathrm{Tr}\left[\gamma_{\nu}S_{c}^{ib}(y-x)\gamma_{5}S_{u}^{bj}(x)\gamma_{5}S_{c}^{ja}(-x)\gamma_{\mu}S_{d}^{ai}(x-y)\right]\right\\}.$
(24)

The correlation function $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$ is
computed by taking into account terms up to dimension $6$, and has structures
identical to ones from $\Pi_{\mu\nu}^{\mathrm{Phys}}(p,p^{\prime})$. The
explicit expression of $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$ is rather
lengthy, therefore we do not provide it here.

The double Borel transform of the invariant amplitude
$\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$ which corresponds to the term
$\sim g_{\mu\nu}$ forms the QCD side of the sum rule. By equating the Borel
transforms of the amplitudes $\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$
and $\Pi^{\mathrm{Phys}}(p^{2},p^{\prime 2},q^{2})$, and carrying out the
continuum subtraction, one gets the sum rule for the coupling $G(q^{2})$.

The amplitude $\Pi^{\mathrm{OPE}}(p^{2},p^{\prime 2},q^{2})$ after the Borel
transformation and subtraction can be expressed in terms of the spectral
density $\rho(s,s^{\prime},q^{2})$ which is determined as a relevant imaginary
part of $\Pi_{\mu\nu}^{\mathrm{OPE}}(p,p^{\prime})$,

$\displaystyle\Pi(\mathbf{M}^{2},\mathbf{s}_{0},q^{2})=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{m_{c}^{2}}^{s_{0}^{\prime}}ds^{\prime}\rho(s,s^{\prime},q^{2})$
$\displaystyle\times e^{-s/M_{1}^{2}}e^{-s^{\prime}/M_{2}^{2}}.$ (25)

Here, $\mathbf{M}^{2}=(M_{1}^{2},M_{2}^{2})$ and
$\mathbf{s}_{0}=(s_{0},s_{0}^{\prime})$ are the Borel and continuum threshold
parameters, respectively. The sum rule for $G(q^{2})$ is given by the
following expression

$\displaystyle
G(q^{2})=\frac{2m_{c}}{fmf_{D^{\ast}}m_{D^{\ast}}m_{D}^{2}f_{D}}\frac{q^{2}-m_{D}^{2}}{m^{2}+m_{D^{\ast}}^{2}-q^{2}}$
$\displaystyle\times
e^{m^{2}/M_{1}^{2}}e^{m_{D^{\ast}}^{2}/M_{2}^{2}}\Pi(\mathbf{M}^{2},\mathbf{s}_{0},q^{2}).$
(26)

The coupling $G(q^{2})$ is a function of $q^{2}$ and parameters
$(\mathbf{M}^{2},\mathbf{s}_{0})$: the latter, for simplicity, are not written
down in Eq. (26) as its arguments. In what follows, we use a new variable
$Q^{2}=-q^{2}$ and fix the obtained function by the notation $G(Q^{2})$.

The sum rule Eq. (26) depends on the mass and coupling of the hadronic
molecule $M_{cc}^{+}$, which are original results of the current work and have
been presented in Eq. (15). The equation (26) also contains the masses and
decay constants of the mesons $D^{\ast+}$ and $D^{0}$. The masses of these
mesons have been written down in Eq. (19), whereas for their decay constants,
we employ

$\displaystyle f_{D^{\ast}}$ $\displaystyle=$ $\displaystyle(223.5\pm
8.4)~{}\mathrm{MeV},$ $\displaystyle f_{D}$ $\displaystyle=$
$\displaystyle(212.6\pm 0.7)~{}\mathrm{MeV}.$ (27)

Besides these parameters, for computation of $G(Q^{2})$ one should choose
working windows for $\mathbf{M}^{2}$ and $\mathbf{s}_{0}$ as well. The
restrictions used in such analysis are standard ones for sum rule computations
and have been considered above.

Figure 3: The sum rule results and functions $F(Q^{2})$ (FF 1) and
$\overline{F}(Q^{2})$ (FF 2) for the strong coupling $G(Q^{2})$. The red
diamond fixes the point $Q^{2}=-m_{D}^{2}$.

The windows for $M_{1}^{2}$ and $s_{0}$ correspond to the $M_{cc}^{+}$ channel
and are given by Eq. (14). The parameters $(M_{2}^{2},s_{0}^{\prime})$ for the
$D^{\ast+}$ meson’s channel vary inside the intervals

$M_{2}^{2}\in[2,4]~{}\mathrm{GeV}^{2},\
s_{0}^{\prime}\in[5.5,6.5]~{}\mathrm{GeV}^{2}.$ (28)

We calculate $G(Q^{2})$ at fixed $Q^{2}=1-6~{}\mathrm{GeV}^{2}$ and plot
obtained results in Fig. 3. It is worth noting that at each $Q^{2}$
computations satisfy constraints imposed on parameters $\mathbf{M}^{2}$ and
$\mathbf{s}_{0}$ by the sum rule analysis. Thus, in Fig. 4 the coupling
$G(Q^{2})$ is depicted as a function of the parameters $M_{1}^{2}$ and
$M_{2}^{2}$ at $Q^{2}=1~{}\mathrm{GeV}^{2}$ and middle of the regions $s_{0}$
and $s_{0}^{\prime}$. A relative stability of $G(1\ \mathrm{GeV}^{2})$ upon
changing of $\mathbf{M}^{2}$ is evident: Variations of $M_{1}^{2}$ and
$M_{2}^{2}$ within explored regions do not exceed $30\%$ of the central value
for $G(1\ \mathrm{GeV}^{2})$. Numerically, we find

$G(1\ \mathrm{GeV}^{2})=0.48_{-0.09}^{+0.14}~{}\mathrm{GeV}^{-1}.$ (29)

Figure 4:  The strong coupling $G=G(1~{}\mathrm{GeV}^{2})$ as a function of
the Borel parameters $M_{1}^{2}$ and $M_{2}^{2}$ at
$s_{0}=20.5~{}\mathrm{GeV}^{2}$ and $s_{0}^{\prime}=6~{}\mathrm{GeV}^{2}$.

The width of the process $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ is determined
by the coupling $G$ at the mass shell $q^{2}=m_{D}^{2}$ of the meson $D^{0}$,
which cannot be calculated directly using the sum rule method. To avoid this
difficulty, we introduce fit functions $F(Q^{2})$ and $\overline{F}(Q^{2})$
that for the momenta $Q^{2}>0$ give results identical to QCD sum rule’s ones,
but can be extrapolated to the region of $Q^{2}<0$ to fix $G$. We employ the
fit functions $F(Q^{2})$ and $\overline{F}(Q^{2})$ given by the expressions

$F(Q^{2})=F_{0}\mathrm{\exp}\left[c_{1}\frac{Q^{2}}{m^{2}}+c_{2}\left(\frac{Q^{2}}{m^{2}}\right)^{2}\right],$
(30)

and

$\overline{F}(Q^{2})=\frac{\overline{F}_{0}}{\left(1-\frac{Q^{2}}{m^{2}}\right)\left(1-\sigma_{1}\frac{Q^{2}}{m^{2}}+\sigma_{2}\left(\frac{Q^{2}}{m^{2}}\right)^{2}\right)},$
(31)

where $F_{0}$, $c_{1}$ and $c_{2}$ and $\overline{F}_{0}$, $\sigma_{1}$ and
$\sigma_{2}$ are fitting parameters. From numerical computations, it is not
difficult to find that $F_{0}=0.39~{}\mathrm{GeV}^{-1},$ $c_{1}=3.29$ and
$c_{2}=-1.95$. Similar analysis gives
$\overline{F}_{0}=0.39~{}\mathrm{GeV}^{-1},$ $\sigma_{1}=2.11$ and
$\sigma_{2}=2.97$. In Fig. 3, along with the sum rule results for $G(Q^{2})$,
we plot also the functions $F(Q^{2})$ and $\overline{F}(Q^{2})$. It is seen,
that there are nice agreements between the fit functions and QCD data. Their
predictions for the coupling $G$ are also very close to each other, and
generate only small additional uncertainties $\pm 0.01$ in $G$.

The functions $F(Q^{2})$ and $\overline{F}(Q^{2})$ at the $D^{0}$ meson’s mass
shell lead to the average result

$G=(0.192\pm 0.061)~{}\mathrm{GeV}^{-1},$ (32)

where the theoretical errors are the sum (in quadrature) of uncertainties
coming from sum rule computations $\pm 0.06$ and ones due to fitting
procedures. The width of decay $M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$ is
determined by the expression

$\displaystyle\Gamma\left[M_{cc}^{+}\rightarrow
D^{0}D^{\ast+}\right]=G^{2}\frac{m_{D^{\ast}}^{2}\lambda\left(m,m_{D^{\ast}},m_{D}\right)}{24\pi}$
$\displaystyle\times\left(3+\frac{2\lambda^{2}\left(m,m_{D^{\ast}},m_{D}\right)}{m_{D^{\ast}}^{2}}\right),$
(33)

where

$\lambda\left(a,b,c\right)=\frac{1}{2a}\sqrt{a^{4}+b^{4}+c^{4}-2\left(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}\right)}.$
(34)

Using the strong coupling from Eq. (32), one can evaluate width of the process
$M_{cc}^{+}\rightarrow D^{0}D^{\ast+}$

$\Gamma\left[M_{cc}^{+}\rightarrow D^{0}D^{\ast+}\right]=(3.8\pm
1.7)~{}\mathrm{MeV}.$ (35)

There are also other decay modes of the molecule $M_{cc}^{+}$, which produce
mesons $D^{0}D^{0}\pi^{+}$ or $D^{0}D^{+}\pi^{0}$. They run through creation
of intermediate scalar tetraquarks followed by their decays to a pair of
conventional mesons Agaev:2021vur . These modes establish the main mechanism
for strong decays of $T_{cc}^{+}$, but are subdominant processes for the
molecule $M_{cc}^{+}$, and therefore can be neglected. Our prediction for the
width of $M_{cc}^{+}$ demonstrates that it is a relatively wide resonance.

##  IV Discussion and conclusions

We have calculated the mass and width of the doubly charmed axial-vector state
with quark content $cc\overline{u}\overline{d}$ by modeling it as the hadronic
molecule $M_{cc}^{+}=D^{0}D^{\ast+}$. Predictions obtained for the mass
$m=(4060\pm 130)~{}\mathrm{MeV}$ and width $\Gamma=(3.8\pm
1.7)~{}\mathrm{MeV}$ of this molecule exceed the LHCb data Aaij:2021vvq ;
LHCb:2021auc . In our previous work, we carried out similar analysis by
treating the axial-vector $T_{cc}^{+}=cc\overline{u}\overline{d}$ state in the
context of the tetraquark model Agaev:2021vur . Parameters of the exotic meson
$T_{cc}^{+}$ agree nicely with data of the LHCb collaboration. Comparing with
each another results extracted from the sum rules in tetraquark and molecule
models, we see that the molecule $M_{cc}^{+}$ is heavier and wider than the
tetraquark structure.

Actually one might expect such outcome, because colored diquark and
antidiquark compact to form tightly-bound state, whereas interaction of two
colorless mesons is less intensive. Decays of tetraquarks and molecules to
conventional mesons also differ from each other. Indeed, in the case of a
tetraquark these processes require reorganization of its quark structure.
Contrary, a hadronic molecule’s dissociation is free of such obstacles. Hence,
hadronic molecules are usually heavier and wider than their tetraquark
counterparts.

In this regards, it is instructive to recall a situation with the resonance
$X_{0}(2900)$ discovered also by the LHCb collaboration. In Ref. Agaev:2020nrc
, we investigated $X_{0}$ and evaluated its parameters. Results obtained for
the mass and width of $X_{0}$ allowed us to interpret it as the hadronic
molecule $\overline{D}^{\ast}K^{\ast}$. We argued additionally that a ground-
state $1S$ scalar tetraquark with the same content should have considerably
smaller mass. This conclusion was supported by analysis of Ref. He:2020jna ,
in which $X_{0}$ was considered as the radially excited $2S$ tetraquark
$[ud][\overline{s}\overline{c}]$. The mass difference between $1S$ and $2S$
particles equals there to $\approx 500~{}\mathrm{MeV}$. Because the molecule
$\overline{D}^{\ast}K^{\ast}$ and $2S$ tetraquark
$[ud][\overline{s}\overline{c}]$ have approximately equal masses, the same
estimate is valid for a mass gap between the molecule
$\overline{D}^{\ast}K^{\ast}$ and ground state tetraquark. In the case of the
exotic mesons $M_{cc}^{+}$ and $T_{cc}^{+}$ this mass difference amounts to
approximately $200$ $\mathrm{MeV}$ being in a qualitative agreement with the
above analysis.

The molecule $M_{cc}^{+}$ was considered using the QCD sum rule method also in
other articles. Thus, in Ref. Dias:2011mi  the mass of $M_{cc}^{+}$ was
estimated indirectly using the spectral sum rule prediction for the ratio
between the masses of $M_{cc}^{+}$ and resonance $X(3872)$. Let us note that
relevant calculations were carried by taking into account condensates up to
dimension-$6$. The prediction $m=(3872.2\pm 39.5)~{}\mathrm{MeV}$ obtained
there for the mass of $M_{cc}^{+}$ is below two-meson threshold and close to
the LHCb data.

Direct sum rule computations of parameters of doubly charmed axial-vector
states $cc\overline{u}\overline{d}$ were performed in Refs. Xin:2021wcr ;
Wang:2017dtg . In the tetraquark model the mass of such state was predicted
within the range Wang:2017dtg

$\widetilde{m}=(3900\pm 90)~{}\mathrm{MeV}.$ (36)

This is higher than the LHCb data, and exceeds also our result $(3868\pm
124)~{}\mathrm{MeV}$ for this model from Ref. Agaev:2021vur .

The axial-vector isoscalar and isovector molecules built of mesons $D^{0}$ and
$D^{\ast+}$ were explored in Ref. Xin:2021wcr , in which their masses were
found equal to

$\displaystyle m_{\mathrm{I=0}}$ $\displaystyle=$ $\displaystyle(3880\pm
110)~{}\mathrm{MeV},$ $\displaystyle m_{\mathrm{I=1}}$ $\displaystyle=$
$\displaystyle(3890\pm 110)~{}\mathrm{MeV},$ (37)

respectively. The isoscalar molecule was interpreted as the LHCb resonance
$T_{cc}^{+}$, or its essential component. Of course, comparing Eqs. (15) and
(37) one sees overlapping regions for the mass of $M_{cc}^{+}$, but there are
essential differences between relevant central values.

It is interesting that masses of the tetraquark and molecule states from Eqs.
(36) and (37) coincide with each other. This fact may be explained by
different choices for the renormalization/factorization scale $\mu$ used to
evolve vacuum condensates and quark masses. Indeed, $\widetilde{m}$ was
obtained at $\mu=1.3~{}\mathrm{GeV}$, whereas $m_{\mathrm{I}}$ extracted by
employing the scale $\mu=1.4~{}\mathrm{GeV}$. Different scales presumably
eliminate a typical mass gap between tetraquark and molecule structures. The
choice for the scale $\mu>1\ \mathrm{GeV}$ may also generate the discrepancy
between Eq. (37) and our result for the molecule $D^{0}D^{\ast+}$. It is worth
to note that Eq. (15) have been obtained at $\mu=1~{}\mathrm{GeV}$, which is
necessary for leading-order QCD calculations.

To fix the scale $\mu$ unambiguously and damp sensitivity of results against
its variations, one needs to find physical quantities with the next-to-leading
order (NLO) accuracy: The NLO results have enhanced predictive power, and
their comparisons with data lead to more reliable conclusions. The factorized
NLO perturbative corrections to doubly heavy exotic mesons’ masses and
couplings were computed in Ref. Albuquerque:2022weq  in QCD Inverse Laplace
sum rule approach. These corrections are important to legitimate the choice of
heavy quark masses used in relevant studies, though in the
$\overline{\mathrm{MS}}$ scheme NLO effects themselves are small
Albuquerque:2022weq . The authors explained by this fact success of
corresponding leading-order QCD analyses. It is possible to carry out similar
NLO computations in the context of the two-point sum rule method to remove
ambiguities in the choice of the scale $\mu$ while calculating parameters of
$T_{cc}^{+}$ and $M_{cc}^{+}$, which, however, are beyond the scope of the
current article.

Summing up, investigations of $T_{cc}^{+}$ and $M_{cc}^{+}$ in the framework
of QCD sum rule method lead for these states to wide diversity of predictions.
The sum rule method relies on fundamental principles of QCD and uses universal
vacuum condensates to extract parameters of various hadrons, nevertheless it
suffers from theoretical errors which make difficult unambiguous
interpretation of obtained results. Thus, predictions for the mass and width
of the hadronic molecule $M_{cc}^{+}$ obtained in the current article differ
from relevant LHCb data, but these differences remain within $1.5$ and $2$
standard deviations, respectively. Therefore, though our studies demonstrate
that a preferable assignment for the LHCb resonance is the tetraquark model
$T_{cc}^{+}$, within the theoretical uncertainties, they also do not rule out
the molecule picture $M_{cc}^{+}\equiv D^{0}D^{\ast+}$.

Controversial predictions for parameters of the molecule $M_{cc}^{+}$ were
made in the context of alternative methods Meng:2021jnw ; Ling:2021bir ;
Chen:2021vhg  as well. Results for the full width of the $M_{cc}^{+}$ obtained
in papers Meng:2021jnw ; Ling:2021bir  are rather small compared with the LHCb
data. At the same time, a nice agreement with recent measurements was declared
in Ref. Chen:2021vhg . Moreover, in this work the authors predicted existence
of another doubly charmed resonance with the mass $m=3876~{}\mathrm{MeV}$ and
width $\Gamma=412~{}\mathrm{keV}$.

As is seen, even in the context of same models and methods, theoretical
investigations sometimes lead to contradictory predictions for the parameters
of the molecule $M_{cc}^{+}$: New efforts are required to settle existing
problems. Additionally, more accurate LHCb data are necessary for the full
width of the doubly charmed state $T_{cc}^{+}$ to compare with different
theoretical results.

*

##  Appendix A  The propagators $S_{q(Q)}(x)$ and invariant amplitude
$\Pi(M^{2},s_{0})$

In the current article, for the light quark propagator $S_{q}^{ab}(x)$, we
employ the following expression

$\displaystyle
S_{q}^{ab}(x)=i\delta_{ab}\frac{\not{x}}{2\pi^{2}x^{4}}-\delta_{ab}\frac{m_{q}}{4\pi^{2}x^{2}}-\delta_{ab}\frac{\langle\overline{q}q\rangle}{12}+i\delta_{ab}\frac{\not{x}m_{q}\langle\overline{q}q\rangle}{48}-\delta_{ab}\frac{x^{2}}{192}\langle\overline{q}g_{s}\sigma
Gq\rangle$
$\displaystyle+i\delta_{ab}\frac{x^{2}\not{x}m_{q}}{1152}\langle\overline{q}g_{s}\sigma
Gq\rangle-i\frac{g_{s}G_{ab}^{\alpha\beta}}{32\pi^{2}x^{2}}\left[\not{x}{\sigma_{\alpha\beta}+\sigma_{\alpha\beta}}\not{x}\right]-i\delta_{ab}\frac{x^{2}\not{x}g_{s}^{2}\langle\overline{q}q\rangle^{2}}{7776}$
$\displaystyle-\delta_{ab}\frac{x^{4}\langle\overline{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}{27648}+\cdots.$ (A.38)

For the heavy quark $Q=c$, we use the propagator $S_{Q}^{ab}(x)$

$\displaystyle
S_{Q}^{ab}(x)=i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\Bigg{\\{}\frac{\delta_{ab}\left({\not{k}}+m_{Q}\right)}{k^{2}-m_{Q}^{2}}-\frac{g_{s}G_{ab}^{\alpha\beta}}{4}\frac{\sigma_{\alpha\beta}\left({\not{k}}+m_{Q}\right)+\left({\not{k}}+m_{Q}\right)\sigma_{\alpha\beta}}{(k^{2}-m_{Q}^{2})^{2}}$
$\displaystyle+\frac{g_{s}^{2}G^{2}}{12}\delta_{ab}m_{Q}\frac{k^{2}+m_{Q}{\not{k}}}{(k^{2}-m_{Q}^{2})^{4}}+\frac{g_{s}^{3}G^{3}}{48}\delta_{ab}\frac{\left({\not{k}}+m_{Q}\right)}{(k^{2}-m_{Q}^{2})^{6}}\left[{\not{k}}\left(k^{2}-3m_{Q}^{2}\right)+2m_{Q}\left(2k^{2}-m_{Q}^{2}\right)\right]\left({\not{k}}+m_{Q}\right)+\cdots\Bigg{\\}}.$

Here, we have used the short-hand notations

$G_{ab}^{\alpha\beta}\equiv G_{A}^{\alpha\beta}\lambda_{ab}^{A}/2,\ \
G^{2}=G_{\alpha\beta}^{A}G_{A}^{\alpha\beta},\
G^{3}=f^{ABC}G_{\alpha\beta}^{A}G^{B\beta\delta}G_{\delta}^{C\alpha},$ (A.40)

where $G_{A}^{\alpha\beta}$ is the gluon field strength tensor, $\lambda^{A}$
and $f^{ABC}$ are the Gell-Mann matrices and structure constants of the color
group $SU_{c}(3)$, respectively. The indices $A,B,C$ run in the range
$1,2,\ldots 8$.

The invariant amplitude $\Pi(M^{2},s_{0})$ obtained after the Borel
transformation and subtraction procedures is given by Eq. (10)

$\Pi(M^{2},s_{0})=\int_{4m_{c}^{2}}^{s_{0}}ds\rho^{\mathrm{OPE}}(s)e^{-s/M^{2}}+\Pi(M^{2}),$

where the spectral density $\rho^{\mathrm{OPE}}(s)$ and the function
$\Pi(M^{2})$ are determined by formulas

$\rho^{\mathrm{OPE}}(s)=\rho^{\mathrm{pert.}}(s)+\sum_{N=3}^{8}\rho^{\mathrm{DimN}}(s),\
\ \Pi(M^{2})=\sum_{N=6}^{10}\Pi^{\mathrm{DimN}}(M^{2}),$ (A.41)

respectively. The components of $\rho^{\mathrm{OPE}}(s)$ and $\Pi(M^{2})$ are
given by the expressions

$\rho^{\mathrm{DimN}}(s)=\int_{0}^{1}d\alpha\int_{0}^{1-a}d\beta\rho^{\mathrm{DimN}}(s,\alpha,\beta),\
\ \rho^{\mathrm{DimN}}(s)=\int_{0}^{1}d\alpha\rho^{\mathrm{DimN}}(s,\alpha),$
(A.42)

and

$\Pi^{\mathrm{DimN}}(M^{2})=\int_{0}^{1}d\alpha\int_{0}^{1-a}d\beta\Pi^{\mathrm{DimN}}(M^{2},\alpha,\beta),\
\
\Pi^{\mathrm{DimN}}(M^{2})=\int_{0}^{1}d\alpha\Pi^{\mathrm{DimN}}(M^{2},\alpha).$
(A.43)

depending on whether $\rho$ and $\Pi(M^{2})$ are functions of $\alpha$ and
$\beta$ or only of $\alpha$. In Eqs. (A.42) and (A.43) variables $\alpha$ and
$\beta$ are Feynman parameters.

The perturbative and nonperturbative components of the spectral density
$\rho^{\mathrm{pert.}}(s,\alpha,\beta)$ and
$\rho^{\mathrm{Dim3(4,5,6,7,8)}}(s,\alpha,\beta)$ have the forms:

$\displaystyle\rho^{\mathrm{pert.}}(s,\alpha,\beta)=\frac{\Theta(L_{1})}{49152\pi^{6}L^{2}N_{1}^{8}}\left[m_{c}^{2}N_{2}-s\alpha\beta
L\right]^{2}\left\\{s^{2}\alpha^{2}\beta^{2}L^{3}\left[18\beta^{2}+18\alpha(\alpha-1)-\beta(18+325\alpha)\right]+m_{c}^{4}N_{1}^{2}\left[18\beta^{5}\right.\right.$
$\displaystyle\left.+18\alpha^{3}(\alpha-1)^{2}+\beta^{4}(37\alpha-36)+\beta^{2}\alpha(54-110\alpha+29\alpha^{2})+\beta^{3}(18-91\alpha+29\alpha^{2})+\beta\alpha^{2}(54-91\alpha+37\alpha^{2})\right]$
$\displaystyle-2m_{c}^{2}s\alpha\beta\left[18\beta^{7}+18\alpha^{3}(\alpha-1)^{4}-3\beta\alpha^{2}(\alpha-1)^{3}(18+13\alpha)-3\beta^{6}(24+13\alpha)+\beta^{5}(108+63\alpha-296\alpha^{2})\right.$
$\displaystyle-2\beta^{2}\alpha(\alpha-1)^{2}(-27-3\alpha+148\alpha^{2})+\beta^{4}(-72+45\alpha+598\alpha^{2}-571\alpha^{3})+\beta^{3}\left(18-123\alpha-254\alpha^{2}+930\alpha^{3}\right.$
$\displaystyle\left.\left.\left.-517\alpha^{4}\right)\right]\right\\},$ (A.44)
$\rho^{\mathrm{Dim3}}(s,\alpha,\beta)=\frac{m_{c}\Theta(L_{1})}{512\pi^{4}N_{1}^{5}}\left[m_{c}^{2}N_{2}-s\alpha\beta
L\right]\left\\{2\langle\overline{d}d\rangle(11\beta-\alpha)\left[m_{c}^{2}N_{2}-2s\alpha\beta
L\right]-\langle\overline{u}u\rangle(\beta-11\alpha)\left[m_{c}^{2}N_{2}-3s\alpha\beta
L\right]\right\\},$ (A.45)
$\displaystyle\rho^{\mathrm{Dim4}}(s,\alpha,\beta)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle\Theta(L_{1})}{73728\pi^{4}(\beta-1)L^{2}N_{1}^{6}}\left\\{-s^{2}(\beta-1)\alpha^{2}\beta^{2}L^{2}\left[18\beta^{4}+6\alpha(\alpha-1)^{2}(3\alpha-2)+3\beta^{3}(417\alpha-16)\right.\right.$
$\displaystyle\left.+\beta^{2}(42-1521\alpha-91\alpha^{2})-3\beta(4-94\alpha-333\alpha^{2}+423\alpha^{3})\right]-m_{c}^{4}N_{1}^{2}\left[18\beta^{8}+\beta^{7}(611\alpha-108)\right.$
$\displaystyle+\beta^{6}(228-2053\alpha+1918\alpha^{2})-6\alpha^{3}(\alpha-1)^{2}(2-9\alpha+3\alpha^{2}+2\alpha^{3})+\beta^{5}(-216+2567\alpha-5375\alpha^{2}+2181\alpha^{3})$
$\displaystyle+\beta^{4}(90-1455\alpha+5518\alpha^{2}-5083\alpha^{3}+857\alpha^{4})+\beta\alpha^{2}(-36+342\alpha-555\alpha^{2}+248\alpha^{3}+23\alpha^{4}-22\alpha^{5})$
$\displaystyle-\beta^{3}(12-366\alpha+2565\alpha^{2}-4315\alpha^{3}+1686\alpha^{4}+94\alpha^{5})-\beta^{2}\alpha\left(36-540\alpha+1743\alpha^{2}-1306\alpha^{3}+16\alpha^{4}\right.$
$\displaystyle\left.+105\alpha^{5}\right)+2m_{c}^{2}s\alpha\beta\left[15\beta^{10}+13\beta^{9}(74\alpha-9)+\beta^{8}(372-5060\alpha+3754\alpha^{2})-3\alpha^{3}(\alpha-1)^{4}(4-14\alpha+5\alpha^{2}+2\alpha^{3})\right.$
$\displaystyle+\beta^{7}\left(-630+10956\alpha-17152\alpha^{2}+6047\alpha^{3}\right)-\beta\alpha^{2}(\alpha-1)^{3}(-36+348\alpha+174\alpha^{2}-439\alpha^{3}+11\alpha^{4})$
$\displaystyle+\beta^{6}\left(615-12500\alpha+31809\alpha^{2}-23517\alpha^{3}+4514\alpha^{4}\right)+\beta^{5}\left(-345+7970\alpha-30691\alpha^{2}+36238\alpha^{3}-13408\alpha^{4}+160\alpha^{5}\right)$
$\displaystyle-\beta^{2}\alpha(\alpha-1)^{2}\left(36-672\alpha+1809\alpha^{2}+1220\alpha^{3}-2788\alpha^{4}+538\alpha^{5}\right)+\beta^{4}\left(102-2772\alpha+16465\alpha^{2}-28426\alpha^{3}+14658\alpha^{4}\right.$
$\displaystyle\left.\left.\left.+2648\alpha^{5}-2675\alpha^{6}\right)-\beta^{3}\left(12-480\alpha+4893\alpha^{2}-12403\alpha^{3}+7946\alpha^{4}+6091\alpha^{5}-8153\alpha^{6}+2094\alpha^{7}\right)\right]\right\\},$
(A.46) $\displaystyle\rho^{\mathrm{Dim5}}(s,\alpha,\beta)$ $\displaystyle=$
$\displaystyle-\frac{m_{c}\Theta(L_{1})L}{1024\pi^{4}N_{1}^{4}}\left\\{-\langle\overline{u}g_{s}\sigma
Gu\rangle(\beta-11\alpha)\left[m_{c}^{2}N_{2}-2s\alpha\beta L\right]\right.$
(A.47) $\displaystyle\left.+\langle\overline{d}g_{s}\sigma
Gd\rangle(11\beta-\alpha)\left[m_{c}^{2}N_{2}-3s\alpha\beta
L\right]\right\\},$
$\displaystyle\rho_{1}^{\mathrm{Dim6}}(M^{2},\alpha,\beta)=\frac{\langle
g_{s}^{3}G^{3}\rangle\Theta(L_{1})}{45\cdot
2^{19}\pi^{6}L^{2}N_{1}^{7}}\left\\{36m_{c}^{2}N_{1}^{2}\left[12\beta^{9}-5\beta^{4}\alpha^{5}-4\beta^{3}\alpha^{5}(\alpha-1)+12\beta\alpha^{5}(\alpha-1)^{3}+12\alpha^{6}(\alpha-1)^{3}\right.\right.$
$\displaystyle+4\beta^{8}(8\alpha-9)+\beta^{2}\alpha^{5}(13-18\alpha+5\alpha^{2})+\beta^{7}(36-76\alpha+45\alpha^{2})+3\beta^{5}\alpha(-4+11\alpha-12\alpha^{2}+5\alpha^{3})+2\beta^{6}\left(-6\right.$
$\displaystyle\left.\left.+28\alpha-39\alpha^{2}+18\alpha^{3}\right)\right]-s\alpha\beta
L^{2}\left[216\beta^{9}+216\alpha^{6}(\alpha-1)^{3}-24\beta\alpha^{5}(\alpha-1)^{2}(3+73\alpha)-24\beta^{8}(27+73\alpha)\right.$
$\displaystyle+\beta^{2}\alpha^{4}(\alpha-1)^{2}(144+2773\alpha)+\beta^{3}\alpha^{4}(-543-1759\alpha+2302\alpha^{2})+\beta^{7}(648+3432\alpha+2611\alpha^{2})$
$\displaystyle-3\beta^{4}\alpha^{2}(-48+235\alpha-544\alpha^{2}+357\alpha^{3})+2\beta^{6}(-108-804\alpha-2539\alpha^{2}+989\alpha^{3})$
$\displaystyle\left.-\beta^{5}\alpha(72-2323\alpha+1273\alpha^{2}+1233\alpha^{3})\right]+72m_{c}^{2}\left[2\beta^{13}+2\alpha^{8}(\alpha-1)^{5}-\beta\alpha^{7}(\alpha-1)^{4}(6+\alpha)-\beta^{12}(10+\alpha)\right.$
$\displaystyle+\beta^{11}(20-2\alpha-5\alpha^{2})-\beta^{2}\alpha^{6}(\alpha-1)^{3}(-6+10\alpha+5\alpha^{2})+\beta^{3}\alpha^{5}(\alpha-1)^{3}(2-21\alpha+8\alpha^{2})+\beta^{4}\alpha^{5}(\alpha-1)^{2}$
$\displaystyle\times(12-46\alpha+31\alpha^{2})+\beta^{10}(-20+18\alpha+5\alpha^{2}+8\alpha^{3})+\beta^{5}\alpha^{3}(\alpha-1)^{2}(-2+8\alpha-30\alpha^{2}+41\alpha^{3})+\beta^{9}\left(10-32\alpha\right.$
$\displaystyle\left.+21\alpha^{2}-45\alpha^{3}+31\alpha^{4}\right)+\beta^{8}(-2+23\alpha-43\alpha^{2}+89\alpha^{3}-108\alpha^{4}+41\alpha^{5})+\beta^{7}\alpha\left(-6+28\alpha-77\alpha^{2}+135\alpha^{3}\right.$
$\displaystyle\left.\left.\left.-112\alpha^{4}+32\alpha^{5}\right)+\beta^{6}\alpha^{2}(-6+27\alpha-70\alpha^{2}+109\alpha^{3}-92\alpha^{4}+32\alpha^{5})\right]\right\\},$
(A.48)
$\displaystyle\rho_{1}^{\mathrm{Dim7}}(M^{2},\alpha,\beta)=\frac{m_{c}\langle\alpha_{s}G^{2}/\pi\rangle\Theta(L_{1})}{4608\pi^{2}N_{1}^{4}}\left\\{2\langle\overline{d}d\rangle\left[25\beta^{5}-2\beta^{4}(14+19\alpha)+\beta^{3}(3+65\alpha-68\alpha^{2})\right.\right.$
$\displaystyle\left.+\beta^{2}\alpha(-27+84\alpha-56\alpha^{2})+\beta\alpha^{2}(-27+53\alpha-26\alpha^{2})+\alpha^{3}(3-4\alpha+\alpha^{2})\right]$
$\displaystyle+\langle\overline{u}u\rangle\left[-5\beta^{5}+\beta^{4}(8+22\alpha)+\beta\alpha^{2}(27-37\alpha+10\alpha^{2})+\alpha^{3}(-3-16\alpha+19\alpha^{2})\right.$
$\displaystyle\left.\left.+\beta^{2}\alpha(27-72\alpha+34\alpha^{2})+\beta^{3}(-3-49\alpha+46\alpha^{2})\right]\right\\},$
(A.49)
$\rho_{1}^{\mathrm{Dim8}}(M^{2},\alpha,\beta)=-\frac{\langle\alpha_{s}G^{2}/\pi\rangle^{2}}{24576\pi^{2}N_{1}^{3}}\Theta(L_{1})\alpha\beta
L.$ (A.50)

The components $\rho^{\mathrm{Dim6(7,8)}}(s,\alpha)$ are given by the formulas

$\rho_{2}^{\mathrm{Dim6}}(s,\alpha)=\frac{\langle\overline{d}d\rangle\langle\overline{u}u\rangle}{192\pi^{2}}\Theta(L_{2})\left[13m_{c}^{2}+3s\alpha(\alpha-1)\right],\
\ $ (A.51)
$\rho_{2}^{\mathrm{Dim7}}(s,\alpha)=\frac{m_{c}\langle\alpha_{s}G^{2}/\pi\rangle}{9216\pi^{2}}\Theta(L_{2})\left[\langle\overline{u}u\rangle(1-12\alpha)+\langle\overline{d}d\rangle(-22+24\alpha)\right],\
$ (A.52)

and

$\rho_{2}^{\mathrm{Dim8}}(s,\alpha)=\frac{\langle\overline{d}d\rangle\langle\overline{u}g_{s}\sigma
Gu\rangle}{48\pi^{2}}\Theta(L_{2})\alpha(\alpha-1).\ $ (A.53)

Components of the function $\Pi(M^{2})$ are:

$\displaystyle\Pi^{\mathrm{Dim6}}(M^{2},\alpha,\beta)=\frac{m_{c}^{4}\langle
g_{s}^{3}G^{3}\rangle}{45\cdot
2^{18}\pi^{6}\alpha\beta(\beta-1)L^{3}N_{1}^{4}}\exp\left[-\frac{m_{c}^{2}N_{2}}{M^{2}\alpha\beta
L}\right]\left[29\beta^{11}-(\alpha-1)^{3}\alpha^{7}(99+70\alpha)\right.$
$\displaystyle+\beta^{10}(-186+113\alpha)+\beta^{9}(384-447\alpha-50\alpha^{2})-\beta(\alpha-1)^{2}\alpha^{6}(-108+212\alpha+111\alpha^{2})$
$\displaystyle+\beta^{8}(-326+663\alpha+200\alpha^{2}-447\alpha^{3})+\beta^{6}\alpha(108+340\alpha-1886\alpha^{2}+2485\alpha^{3}-1042\alpha^{4})$
$\displaystyle+\beta^{7}(99-437\alpha-370\alpha^{2}+1510\alpha^{3}-880\alpha^{4})+\beta^{5}\alpha^{2}(-120+1030\alpha-2607\alpha^{2}+2577\alpha^{3}-740\alpha^{4})$
$\displaystyle+\beta^{4}\alpha^{3}(-207+1209\alpha-2358\alpha^{2}+1593\alpha^{3}-377\alpha^{4})+\beta^{3}\alpha^{4}\left(-270+943\alpha-1143\alpha^{2}\right.$
$\displaystyle\left.\left.+527\alpha^{3}-120\alpha^{4}\right)+\beta^{2}\alpha^{5}(-120+112\alpha+179\alpha^{2}-74\alpha^{3}-97\alpha^{4})\right],$
(A.54)
$\displaystyle\Pi^{\mathrm{Dim7}}(M^{2},\alpha,\beta)=\frac{m_{c}\langle\alpha_{s}G^{2}/\pi\rangle}{4608\pi^{2}M^{2}\alpha\beta
L^{2}N_{1}^{4}}\left\\{M^{2}\alpha^{2}\beta^{2}L^{3}[2\langle\overline{d}d\rangle(11\beta^{2}-\alpha^{2})+\langle\overline{u}u\rangle(11\alpha^{2}-\beta^{2})]\right.$
$\displaystyle+m_{c}^{4}\langle\overline{u}u\rangle
N_{1}^{2}\exp\left[-\frac{m_{c}^{2}N_{2}}{M^{2}\alpha\beta
L}\right]\left[10\beta^{5}+10\alpha^{4}(\alpha-1)+\beta^{4}(-10+19\alpha)+\beta^{3}\alpha(-20+29\alpha)\right.$
$\displaystyle\left.+\beta\alpha^{3}(-20+31\alpha)+\beta^{2}\alpha^{2}(-20+41\alpha)\right]+m_{c}^{2}\langle\overline{d}d\rangle\exp\left[-\frac{m_{c}^{2}N_{2}}{M^{2}\alpha\beta
L}\right]\left[m_{c}^{2}N_{1}^{2}\left(10\beta^{5}+10\alpha^{4}(\alpha-1)\right.\right.$
$\displaystyle\left.+\beta\alpha^{3}(-20+19\alpha)+\beta^{2}\alpha^{2}(-20+29\alpha)+\beta^{4}(-10+31\alpha)+\beta^{3}\alpha(-20+41\alpha)\right)$
$\displaystyle-M^{2}\alpha\beta\left(10\beta^{7}+10\alpha^{4}(\alpha-1)^{3}+\alpha^{3}\beta(\alpha-1)^{2}(-20+29\alpha)+\beta^{6}(-30+41\alpha)+2\beta^{5}(15-51\alpha+41\alpha^{2})\right.$
$\displaystyle+\beta^{2}\alpha^{2}(-20+99\alpha-137\alpha^{2}+58\alpha^{3})+\beta^{3}\alpha(-20+111\alpha-180\alpha^{2}+89\alpha^{3})$
$\displaystyle\left.\left.\left.+\beta^{4}(-10+81\alpha-173\alpha^{2}+101\alpha^{3})\right)\right]\right\\},$
(A.55)

$\displaystyle\Pi^{\mathrm{Dim8}}(M^{2},\alpha,\beta)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle^{2}}{27\cdot
2^{13}\pi^{2}M^{4}\alpha^{2}\beta^{2}L^{4}N_{1}^{3}}\left\\{3M^{4}\alpha^{3}\beta^{3}L^{5}+m_{c}^{2}\exp\left[-\frac{m_{c}^{2}N_{2}}{M^{2}\alpha\beta
L}\right]\left\\{8m_{c}^{4}\alpha^{2}\beta^{2}N_{1}^{2}\right.\right.$
$\displaystyle\times\left[2\beta^{3}+2\alpha^{2}(\alpha-1)+\alpha\beta(-4+5\alpha)+\beta^{2}(-2+5\alpha)\right]-8m_{c}^{2}M^{2}\left[3\beta^{11}+3\alpha^{6}(\alpha-1)^{5}+3\beta^{10}(-5+7\alpha)\right.$
$\displaystyle+3\beta\alpha^{5}(\alpha-1)^{4}(-4+7\alpha)+\beta^{9}(30-96\alpha+74\alpha^{2})+\beta^{2}\alpha^{4}(\alpha-1)^{3}(21-84\alpha+74\alpha^{2})$
$\displaystyle+3\beta^{8}(-10+58\alpha-102\alpha^{2}+57\alpha^{3})+\beta^{3}\alpha^{3}(\alpha-1)^{2}(-24+153\alpha-295\alpha^{2}+171\alpha^{3})$
$\displaystyle+\beta^{4}\alpha^{2}(\alpha-1)^{2}(-21+159\alpha-385\alpha^{2}+286\alpha^{3})+\beta^{7}(15-156\alpha+495\alpha^{2}-637\alpha^{3}+286\alpha^{4})$
$\displaystyle+\beta^{5}\alpha(-12+147\alpha-625\alpha^{2}+1215\alpha^{3}-1091\alpha^{4}+366\alpha^{5})+\beta^{6}\left(-3+69\alpha-389\alpha^{2}+914\alpha^{3}\right.$
$\displaystyle\left.\left.-957\alpha^{4}+366\alpha^{5}\right)\right]+M^{4}\alpha\beta\left[48\beta^{9}+48\alpha^{4}(\alpha-1)^{5}+3\beta^{8}(-80+71\alpha)+3\beta\alpha^{3}(\alpha-1)^{4}(-23+71\alpha)\right.$
$\displaystyle+2\beta^{2}\alpha^{2}(\alpha-1)^{3}(21-180\alpha+239\alpha^{2})+\beta^{7}(480-921\alpha+478\alpha^{2})+6\beta^{6}(-80+259\alpha-299\alpha^{2}+120\alpha^{3})$
$\displaystyle+3\beta^{3}\alpha(\alpha-1)^{2}(-23+116\alpha-317\alpha^{2}+240\alpha^{3})+3\beta^{5}(80-422\alpha+852\alpha^{2}-797\alpha^{3}+287\alpha^{4})$
$\displaystyle\left.\left.+\beta^{4}(-48+489\alpha-1684\alpha^{2}+2970\alpha^{3}-2588\alpha^{4}+861\alpha^{5})\right]\right\\}.$
(A.56)

The dimension $9$ contribution to the correlation function is equal to zero.
The $\mathrm{Dim10}$ term is exclusively of the type (A.43) and has two
components $\Pi_{1}^{\mathrm{Dim10}}(M^{2},\alpha,\beta)$ and
$\Pi_{2}^{\mathrm{Dim10}}(M^{2},\alpha)$

$\displaystyle\Pi_{1}^{\mathrm{Dim10}}(M^{2},\alpha,\beta)=\frac{\langle\alpha_{s}G^{2}/\pi\rangle\langle
g_{s}^{3}G^{3}\rangle}{135\cdot
2^{16}\pi^{2}M^{8}\alpha^{4}\beta^{4}(\beta-1)^{2}L^{6}N_{1}^{4}}\exp\left[-\frac{m_{c}^{2}N_{2}}{M^{2}\alpha\beta
L}\right]\left\\{36M^{8}\alpha^{3}\beta^{3}L^{6}R_{1}(\alpha,\beta)\right.$
$\displaystyle-
m_{c}^{8}\alpha\beta(\beta-1)^{2}N_{1}^{4}R_{2}(\alpha,\beta)+m_{c}^{6}M^{2}(\beta-1)^{2}N_{1}^{3}R_{3}(\alpha,\beta)-2m_{c}^{4}M^{4}\alpha\beta(\beta-1)^{2}N_{1}^{2}R_{4}(\alpha,\beta)$
$\displaystyle\left.-2m_{c}^{2}M^{6}\alpha^{2}\beta^{2}L^{2}R_{5}(\alpha,\beta)\right\\},$
(A.57)

and

$\displaystyle\Pi_{2}^{\mathrm{Dim10}}(M^{2},\alpha)=\frac{m_{c}^{2}\langle\alpha_{s}G^{2}/\pi\rangle\langle\overline{d}d\rangle\langle\overline{u}u\rangle}{432M^{4}\alpha^{3}(\alpha-1)^{3}}\exp\left[-\frac{m_{c}^{2}}{M^{2}\alpha(\alpha-1)}\right]\left[m_{c}^{2}+2M^{2}\alpha(\alpha-1)\right]$
$\displaystyle\times\left(1-2\alpha+2\alpha^{2}\right),$ (A.58)

where the functions $R_{i}(\alpha,\beta)$ are:

$\displaystyle
R_{1}(\alpha,\beta)=\beta^{5}-4\beta^{6}+6\beta^{7}-4\beta^{8}+\beta^{9}+\beta^{4}\alpha^{5}+4\beta^{3}\alpha^{5}(\alpha-1)+6\beta^{2}\alpha^{5}(\alpha-1)^{2}+4\beta\alpha^{5}(\alpha-1)^{3}$
$\displaystyle+\alpha^{5}(\alpha-1)^{4};$ $\displaystyle
R_{2}(\alpha,\beta)=2\beta^{9}+16\beta\alpha^{6}(\alpha-1)^{2}+2\alpha^{7}(\alpha-1)^{2}+4\beta^{8}(4\alpha-1)+\beta^{5}\alpha^{2}(12+64\alpha-111\alpha^{2})$
$\displaystyle+\beta^{6}\alpha(16-30\alpha-35\alpha^{2})+\beta^{3}\alpha^{4}(-30+64\alpha-35\alpha^{2})+6\beta^{2}\alpha^{5}(2-5\alpha+3\alpha^{2})+2\beta^{7}\left(1-16\alpha\right.$
$\displaystyle\left.+9\alpha^{2}\right)-3\beta^{4}\alpha^{3}(10-44\alpha+37\alpha^{2});$
$\displaystyle
R_{3}(\alpha,\beta)=27\beta^{13}+27\alpha^{8}(\alpha-1)^{5}+75\beta\alpha^{7}(\alpha-1)^{4}(2\alpha-1)+15\beta^{12}(-9+10\alpha)+\beta^{11}\left(270-675\alpha\right.$
$\displaystyle\left.+376\alpha^{2}\right)+\beta^{2}\alpha^{6}(\alpha-1)^{3}(42-333\alpha+376\alpha^{2})+3\beta^{10}(-90+400\alpha-487\alpha^{2}+139\alpha^{3})$
$\displaystyle+3\beta^{3}\alpha^{5}(\alpha-1)^{2}(25-23\alpha-130\alpha^{2}+139\alpha^{3})-\beta^{4}\alpha^{4}(\alpha-1)^{2}(-138+741\alpha-1056\alpha^{2}+326\alpha^{3})$
$\displaystyle+\beta^{9}(135-1050\alpha+2169\alpha^{2}-1224\alpha^{3}-326\alpha^{4})+\beta^{8}\left(-27+450\alpha-1501\alpha^{2}+1128\alpha^{3}\right.$
$\displaystyle\left.+1708\alpha^{4}-1960\alpha^{5}\right)+\beta^{5}\alpha^{3}(75-1017\alpha+4264\alpha^{2}-7734\alpha^{3}+6372\alpha^{4}-1960\alpha^{5})$
$\displaystyle-\beta^{6}\alpha^{2}(42+219\alpha-2676\alpha^{2}+7734\alpha^{3}-8756\alpha^{4}+3438\alpha^{5})$
$\displaystyle-\beta^{7}\alpha(75-459\alpha+177\alpha^{2}+3179\alpha^{3}-6372\alpha^{4}+3438\alpha^{5});$
$\displaystyle
R_{4}(\alpha,\beta)=6\beta^{13}+6\alpha^{7}(\alpha-1)^{6}-12\beta\alpha^{6}(\alpha-1)^{5}(-6+5\alpha)-12\beta^{12}(3+5\alpha)+\beta^{11}(90+372\alpha-552\alpha^{2})$
$\displaystyle-3\beta^{2}\alpha^{5}(\alpha-1)^{4}(75-257\alpha+184\alpha^{2})-\beta^{3}\alpha^{4}(\alpha-1)^{3}(-273+1914\alpha-3696\alpha^{2}+2087\alpha^{3})$
$\displaystyle-\beta^{10}(120+960\alpha-2979\alpha^{2}+2087\alpha^{3})+\beta^{9}\left(90+1320\alpha\right.$
$\displaystyle\left.-6621\alpha^{2}+9957\alpha^{3}-4948\alpha^{4}\right)-\beta^{4}\alpha^{3}(\alpha-1)^{2}(273-2490\alpha+8327\alpha^{2}-11038\alpha^{3}+4948\alpha^{4})$
$\displaystyle-\beta^{5}\alpha^{2}(\alpha-1)^{2}(225-2283\alpha+8789\alpha^{2}-14735\alpha^{3}+8390\alpha^{4})-\beta^{8}\left(36+1020\alpha-7734\alpha^{2}+19263\alpha^{3}\right.$
$\displaystyle\left.-20934\alpha^{4}+8390\alpha^{5}\right)+\beta^{7}(6+420\alpha-4986\alpha^{2}+19190\alpha^{3}-35351\alpha^{4}+31515\alpha^{5}-10793\alpha^{6})$
$\displaystyle+\beta^{6}\alpha(-72+1671\alpha-10257\alpha^{2}+30182\alpha^{3}-46649\alpha^{4}+35918\alpha^{5}-10793\alpha^{6})$
$\displaystyle
R_{5}(\alpha,\beta)=99\beta^{15}+99\alpha^{7}(\alpha-1)^{6}+\beta^{14}(-792+591\alpha)-3\beta\alpha^{6}(\alpha-1)^{5}(116-263\alpha+66\alpha^{2})$
$\displaystyle+3\beta^{13}(924-1495\alpha+615\alpha^{2})+3\beta^{12}(-1848+4949\alpha-4384\alpha^{2}+1279\alpha^{3})$
$\displaystyle+3\beta^{2}\alpha^{5}(\alpha-1)^{4}(154-926\alpha+1274\alpha^{2}-460\alpha^{3}+33\alpha^{4})+\beta^{11}\left(6930-27993\alpha+40629\alpha^{2}\right.$
$\displaystyle\left.-25599\alpha^{3}+6053\alpha^{4}\right)+3\beta^{3}\alpha^{4}(\alpha-1)^{3}(-113+1295\alpha-3950\alpha^{2}+4306\alpha^{3}-1722\alpha^{4}+199\alpha^{5})$
(A.59)
$\displaystyle+\beta^{4}\alpha^{3}(\alpha-1)^{3}(-339+3231\alpha-13948\alpha^{2}+21305\alpha^{3}-11307\alpha^{4}+1809\alpha^{5})$
$\displaystyle+\beta^{10}(-5544+32865\alpha-70902\alpha^{2}+73401\alpha^{3}-37829\alpha^{4}+7829\alpha^{5})+\beta^{5}\alpha^{2}(\alpha-1)^{2}\left(462-3732\alpha\right.$
$\displaystyle\left.+15793\alpha^{2}-38400\alpha^{3}+43898\alpha^{4}-21541\alpha^{5}+3639\alpha^{6}\right)+\beta^{9}\left(2772-24591\alpha+76245\alpha^{2}\right.$
$\displaystyle\left.-117654\alpha^{3}+100903\alpha^{4}-46182\alpha^{5}+8648\alpha^{6}\right)+\beta^{8}\left(-792+11445\alpha-51540\alpha^{2}+114630\alpha^{3}\right.$
$\displaystyle\left.-149274\alpha^{4}+116108\alpha^{5}-48984\alpha^{6}+8414\alpha^{7}\right)+\beta^{6}\alpha\left(348-4854\alpha+24609\alpha^{2}-72793\alpha^{3}\right.$
$\displaystyle\left.+137991\alpha^{4}-162173\alpha^{5}+110796\alpha^{6}-39539\alpha^{7}+5615\alpha^{8}\right)+\beta^{7}\left(99-3027\alpha+21267\alpha^{2}\right.$
$\displaystyle\left.-68907\alpha^{3}+133130\alpha^{4}-162246\alpha^{5}+118955\alpha^{6}-46584\alpha^{7}+7313\alpha^{8}\right).$
(A.60)

In expressions above, $\Theta(z)$ is Unit Step function. We have used also the
following short-hand notations

$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\beta^{2}+\beta(\alpha-1)+\alpha(\alpha-1),\ \ \ \ \
N_{2}=(\alpha+\beta)N_{1},\ L=\alpha+\beta-1,\ $ $\displaystyle,\ \ \ L_{1}$
$\displaystyle\equiv$ $\displaystyle
L_{1}(s,\alpha,\beta)=\frac{(1-\beta)}{N_{1}^{2}}\left[m_{c}^{2}N_{2}-s\alpha\beta
L\right],\ L_{2}\equiv L_{2}(s,\alpha)=s\alpha(1-\alpha)-m_{c}^{2},\ $ (A.61)

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