X(3872): propagating in a dense medium

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ScienceDirect

Nuclear Physics B 936 (2018) 151–168

www.elsevier.com/locate/nuclphysb

X(

3872):

propagating

in

a

dense

medium

K. Azizi

a,b,∗

_,

_{N. Er}

c

a_School_of_Physics,_Institute_for_Research_in_Fundamental_Sciences_(IPM),_P._O._Box_19395-5531,_Tehran,_Iran b_Department_of_Physics,_Doˇgu¸s_University,_{Acıbadem-Kadıköy,}₃₄₇₂₂_Istanbul,_Turkey

c_Department_of_Physics,_Abant_˙Izzet_Baysal_University,_Gölköy_Kampüsü,₁₄₉₈₀_Bolu,_Turkey

Received 29July2018;receivedinrevisedform 27August2018;accepted 18September2018 Availableonline 21September2018

Editor: Hong-JianHe

Abstract

Incoldnuclearmatter,theshiftsinthemassandcurrent-mesoncouplingaswellthevectorselfenergy oftheexotic X(3872) arecalculatedusingthediquark–antidiquarkcurrentwithintheframeworkofthe in-mediumtwo-pointQCDsumrule.Attherestframeofthemedium,thethreemomentumoftheconsidered particleisfixedtoremovethecontributionsoftheparticleswithnegativeenergy.Inthecalculations,we includethein-mediumcondensatesofquark–quark,gluon–gluonandquark–gluon.Itisobservedthat,the shiftduetothenuclearmatterisnegativeandisabout25% whenthesaturationdensityisused.Suchshift isconsiderablylargeandcomparablewiththenucleon’massshiftduetothenuclearmedium.Thenegative shiftinthecurrent-mesoncouplingduetonuclearmatterisapproximately10%.Atthesaturationdensity, thevectorselfenergyoftheexotic X(3872) stateisfoundtobe υ= 1.31 GeV.Itisshownthatthemass, current-mesoncouplingandvectorselfenergyof X(3872) stronglydependonthedensityofcoldnuclear matter.

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

In recent years, the new hadronic states such as meson–meson, meson–baryon and baryon– baryon molecules, tetraquarks, pentaquarks, hybrids and glueballs have been in the focus of much attention. These exotic states can not be considered as the usual quark–antiquark or

three-* _{Corresponding}_author.

E-mailaddress:kazizi@dogus.edu.tr(K. Azizi).

https://doi.org/10.1016/j.nuclphysb.2018.09.014

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quark form of the well-known hadronic spectroscopy. Both the quark model and QCD do not exclude the existence of such states. Hence, the investigation of such states can play essential role in understanding of their structures and gaining useful knowledge on the perturbative and non-perturbative natures of QCD.

Among the exotic states, the tetraquark state X(3872) is one of the most interesting parti-cles. Firstly, the Belle Collaboration [1] detected X(3872) as a narrow charmonium-like state produced in the exclusive decay process B±→ K±π+π−J /ψ, which decays into π+π−J /ψ, with a mass of mX= 3872.0 ±0.6(stat) ±0.5(syst) MeV. It was confirmed later in some other

ex-periments: CDF II Collaboration [2] via ¯pp collisions with mX= 3871.3 ± 0.7(stat) ±0.4(syst)

MeV, D0 Collaboration [3] again in ¯pp collisions with the mass difference between the X(3872) state and J /ψ as 774.9 ± 3.1 (stat)±3.0 (syst) MeV and BABAR detector [4] at the PEP-II e+e− asymmetric-energy storage ring with the mass 3873.4 ± 1.4 MeV. Belle Collaboration also as-signed the quantum numbers of X(3872) as JP C= 1++[5]. The decay width of X(3872) state was estimated as X(3872)<1.2 MeV [6,7].

Following the discovery of X(3872) both the theoretical and experimental researches on the non-conventional particles gained an acceleration. Such that, many tetraquark and pentaquark states have been discovered in the experiments and a rush of theoretical papers on the struc-tures of the discovered and possible exotic particles appeared. Despite a lot of theoretical and experimental studies; the structure, nature and quark organisation of the newly founded states have not been understood exactly, yet. There are a lot of suggestions on the nature of the ex-otic X(3872) states, for instance, see Refs. [8–38]. In Ref. [28], for example, it is assumed as a diquark–antidiquark states with hidden or open charm of the forms: [cq][¯c ¯q_{] and [cq][¯s ¯q}_{] with} q, q= u, d. In Ref. [29], it was suggested that it might be a JP C= 1++cusp due to the D ¯D∗ threshold. The author in Ref. [30] presented a study on the possibility of X(3872) to be consid-ered as a hybrid state of c¯cg. In another study, Ref. [31], it is proposed that X(3872) can be a vector glueball mixed with neighbouring vector states of charmonium. Despite a lot of studies on the structure of X(3872), it’ nature remains unclear and needs to be investigated further.

Almost all studies on the properties of X(3872) and other exotic states have been performed in vacuum and there is almost no study on the in-medium properties of non-conventional par-ticles, except Ref. [39] which investigates the thermal properties of X(3872). Investigation of the in-medium properties of exotic states, besides the standard hadrons, can play crusial role in the analysing of the results of the heavy-ion collision experiments. Many experimental col-laborations, such as PANDA, CBM at FAIR and NICA are planning to study the in-medium properties of hadrons including the newly discovered non-conventional states [40–47]. For in-stance, PANDA have focused on the search of the not-yet discovered glueballs and on the already discovered but not-yet understood X, Y, Z states, [40]. Therefore investigation of exotic particles in medium becomes an attractive and exciting subject.

In the present study, we are going to evaluate the spectroscopic parameters such as the mass, current-meson coupling constant and vector self energy of the exotic X(3872) at cold nuclear medium within the in-medium two-point QCD sum rule method and considering a diquark– antidiquark picture for this particle. We will look for the shifts on the quantities under considera-tion due to the nuclear matter at a saturaconsidera-tion density. We will discuss the variaconsidera-tions of parameters of X(3872) with respect to the variations of the density, as well.

This work is organised in the following way. In section2, the in-medium QCD sum rules for the mass and current-meson coupling of the exotic X(3872) (in what follows denoted as X) are obtained. In section3, the numerical analysis of the obtained in-medium sum rules are performed. We investigate the variations of the quantities with respect to the variations of the density and

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calculate the shifts on the quantities under consideration due to the nuclear matter in this section. The last section includes our concluding remarks.

2. In-medium QCD sum rules for the spectroscopic properties of the exotic X(3872)

In the framework of the QCD sum rules, for the calculation of the mass and current coupling constant of X, we start with the following in-medium two-point correlation function:

μν(p)= i

d4xeip·xψ0|T [ημ(x)η†ν(0)]|ψ0, (1) where p is the four momentum of X, |ψ0 is the parity and time-reversal symmetric ground state of the nuclear matter and ημ(x)is the interpolating current of X. In diquark–antidiquark model,

the interpolating current of X with the quantum numbers JP C= 1++is written as [36] ημ(x)= iabcdec √ 2 q_aT(x)Cγ5cb(x) ¯qd(x)γμC¯cTe(x) +q_aT(x)Cγμcb(x) ¯qd(x)γ5C¯cTe(x) , (2)

where a, b, c, d and e are color indices, q represents u or d light quark and C is the charge con-jugation matrix. To obtain the QCD sum rules for the mass and current coupling constant of X, the aforementioned correlation function is calculated in two different representations: hadronic (Had) and QCD with the help of operator product expansion (OPE). These two representations are equated through a dispersion relation to get the desired sum rules. To suppress the contribu-tions of the higher states and continuum, a Borel transformation as well as continuum subtraction are applied to both sides of the obtained sum rules.

2.1. Hadronic representation

In the hadronic side, the correlation function is obtained in terms of the hadronic degrees of freedom. By performing integration over four-x, we get

H ad_μν (p)= −ψ0|ημ|X(p)X(p)|η †

ν|ψ0

p∗2− m∗2_X + ..., (3)

where p∗is the in-medium momentum, m∗_Xis the modified mass of the X state due to the cold nuclear medium and . . . is used for contributions of the higher resonances and continuum. The decay constant or current-meson coupling is defined through the following matrix element in terms of the polarisation vector εμof X state:

ψ0|ημ|X(p) = fX∗m∗Xεμ, (4)

where f_X∗is the in-medium current coupling constant of the X state. After inserting Eq. (4) into Eq. (3), the hadronic side of the correlation function is obtained as

H ad_μν (p)= − m ∗2 XfX∗2 p∗2− m∗2_X − gμν+ p∗_μp_ν∗ m∗2_X , (5)

where the summation over spins of the Dirac spinors has been applied. To proceed, the in-medium momentum is represented in terms of the self energy μ,ν as p∗μ= pμ− μ,v. The

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μ,v= vuμ+ vpμ, (6)

where v is the vector self energy of X and uμ is the four velocity of the nuclear medium. In

mean-field approximation, the momentum independent scalar and vector self energies are taken to be real and the ν is identically zero [48,49]. In this context, particles of any three-momentum

appear as stable quasi-particles with self energies that are roughly linear in the density up to nu-clear matter density [49,50]. The calculations are performed in the rest frame of the nuclear medium, i.e. uμ= (1, 0) and at fixed three-momentum of X state, | p| = 0.27 GeV (i.e.

approx-imately the Fermi momentum). Note that in the vacuum, where the invariant functions depend only on p2the separation of p0and p dependence is not necessary. At finite density, however, we will keep the dispersion relations as integrals over p0with the three-momentum held fixed. This provides a clean identification of the intermediate quasiparticles. By this way, the contributions coming from the negative energy quasiparticles is clearly separated, which lets us isolate the contributions of the positive energy quasiparticles by adopting an appropriate weighting func-tion. After the replacement of the in-medium momentum in Eq. (5), the hadronic side becomes,

H ad_μν (p)= − m ∗2 XfX∗2 (p2_{− 2} υp0+ 2υ)− m∗2X − gμν +pμpν− υpμuν− υpνuμ+ 2υuμuν m∗2_X , (7)

where p0= p · u is the energy of the quasi-particle. Using the positive energy pole Ep= v+

| p|2_{+ m}∗2

X and the negative energy pole ¯Ep= v−

| p|2_{+ m}∗2 X, we get H ad_μν (p)= − m ∗2 XfX∗2 (p0− Ep)(p0− ¯Ep) − gμν +pμpν− υpμuν− υpνuμ+ 2υuμuν m∗2_X . (8)

In terms of spectral densities, the hadronic correlation function can be written in an integral form as H ad_μν (p0,p) = 1 2π i ∞ −∞ dωρ H ad μν (ω,p) ω− p0 , (9)

where the spectral densities ρμνH ad(ω, p) which are defined as

ρH ad_μν (ω,p) = Lim→0+

H ad_μν (ω+ i, p) − H ad_μν (ω− i, p)

, (10)

are obtained in the following form: ρH ad_μν (ω,p) = m ∗2 XfX∗2 2 | p|2_{+ m}∗2 X − gμν +pμpν− υpμuν− υpνuμ+ 2υuμuν m∗2_X ×δ(ω− Ep)− δ(ω − ¯Ep) . (11)

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It is necessary to remove the negative-energy pole contribution. For this purpose the correlation function is multiplied by the weight function (ω− ¯Ep)e

−ω2

M2 _{and the integral over ω is performed}

from −ω0to ω0, H ad_μν (p0,p) = ω0 −ω0 dωρ_μνH ad(ω,p)(ω − ¯Ep)e− ω2 M2. (12)

Here, M2is the Borel mass parameter and ω0is the threshold parameter of the subtraction. These parameters will be fixed in numerical analyses. As a result of these calculations, the final form of the hadronic side of correlation function is obtained in terms of the corresponding structures as

H ad_μν (p0,p) = m∗2XfX∗2e −E2p M2 − gμν +pμpν− υpμuν− υpνuμ+ 2υuμuν m∗2_X . (13) 2.2. OPE representation

The correlation function in QCD side is calculated in terms of QCD degrees of freedom in deep Euclidean region by the help of the operator product expansion. We can decompose the correlation function in QCD side in terms of different structures as

OP E_μν (p0,p) = OP E1 gμν+ OP E2 pμpν+ OP E3 pμuν

+ OP E

4 pνuμ+ OP E5 uμuν. (14)

The coefficient of each structure can be represented in terms of the corresponding spectral density as OP E_i (p0,p) = 1 2π i ∞ −∞ dωρ OP E i (ω,p) ω− p0 , (15)

where ρ_iOP E(ω, p) are the imaginary parts of the correlation functions OP E_i (p0, p). For the calculation of spectral densities, the explicit form of the interpolating current is substituted into the correlation function in Eq. (1) and all quark pairs are contracted by applying the Wick’s theorem. As a result, we get

OP E_μν (p)= −i 2εabcεabcεdecεdec × d4xeipxψ0| T r γ5˜S_qaa(x)γ5S_cbb(x) × T rγμ˜See c (−x)γνSdd q (−x) + T r[γμ˜See c (−x) × γ5Sdd_q (−x) T r γν˜Saa q (x)γ5Sbb c (x) + T rγ5˜Saa q (x)γμSbb c (x) T r γ5˜See c (−x)γνSdd q (−x) + T rγν˜Saa q (x)γμSbb c (x)

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× T rγ5˜See c (−x)γ5Sdd q (−x) |ψ0, (16)

in terms of the light and heavy quark propagators. The abbreviation ˜Sq(c)= CST_q(c)C has been

used in the above equation. The light and heavy quark propagators in coordinate space and in the fixed point gauge are written in mq→ 0 limit as

Sqij(x)= i 2π2δ ij 1 (x2₎2x/+ χ i q(x)¯χ j q(0) − igs 32π2F ij μν(0) 1 x2[/xσ μν_{+ σ}μν_x_/_{] + · · · ,} ₍₁₇₎ and Scij(x)= i (2π )4 d4ke−ik·x δij / k− mc −gsFμνij(0) 4 σμν(/k+ mc)+ (/k + mc)σμν (k2_{− m}2 c)2 +π2 3 αsGG π δijmc k2+ mc/k (k2_{− m}2 c)4 + · · · , (18) where χqi and ¯χ j

q are the Grassmann background quark fields. We use the short-hand notation

F_μνij = F_μνAtij,A, A= 1, 2, ..., 8, (19)

where, F_μνA are classical background gluon fields, and tij,A=λij,A₂ with λij,Abeing the standard Gell-Mann matrices. Using Eqs. (17) and (18) in Eq. (16), we get the products of the Grass-mann background quark fields and classical background gluon fields which correspond to the ground-state matrix elements of the corresponding quark and gluon operators [49,51],

χaαq (x)¯χ q bβ(0)= qaα(x)¯qbβ(0)ρN, F_κλAF_μνB = GA_κλGB_μνρN, χaαq ¯χ_bβq FμνA = qaα¯qbβGAμνρN, and χaαq ¯χ_bβq χcγq ¯χ_dδq = qaα¯qbβqcγ¯qdδρN, (20)

where ρNis the nuclear matter density.

To proceed; the quark, gluon and mixed condensates are needed to be defined in nuclear matter. The matrix element qaα(x)¯qbβ(0)ρN can be parameterized as [49]

qaα(x)¯qbβ(0)ρN= − δab 12  ¯qqρN+ x μ_¯qD μqρN +1 2x μ_xν_¯qD μDνqρN+ ... δαβ +  ¯qγλqρN+ x μ_¯qγ λDμqρN +1 2x μ_xν_¯qγ λDμDνqρN+ ... γ_αβλ . (21)

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The quark–gluon mixed condensate can also be defined as gsqaα¯qbβGAμνρN = −t A ab 96 gs¯qσ · GqρN σμν+ i(uμγν− uνγμ)/u αβ +gs¯q/uσ · GqρN σμν/u+ i(uμγν− uνγμ) αβ −4  ¯qu · Du · DqρN+ imq ¯q/uu · DqρN × σμν+ 2i(uμγν− uνγμ)/u αβ , (22)

where Dμ=1₂(γμD/+ /Dγμ)is the four-derivative. The matrix element of the four-dimension

gluon condensate can also be parametrized as GA κλG B μνρN= δAB 96 G2 ρN(gκμgλν− gκνgλμ) + O(E2_{+ B}2 ρN) , (23)

where the last term in this equation has a negligible contribution, so it can be ignored safely. The different operators in Eqs. (20)–(22) are defined as [49,52]:

 ¯qγμqρN= ¯q/uqρNuμ,

 ¯qDμqρN= ¯qu · DqρNuμ= −imq ¯q/uqρNuμ,

 ¯qγμDνqρN= 4 3 ¯q/uu · DqρN(uμuν− 1 4gμν) +i 3mq ¯qqρN(uμuν− gμν),  ¯qDμDνqρN= 4 3 ¯qu · Du · DqρN(uμuν− 1 4gμν) −1 6gs¯qσ · GqρN(uμuν− gμν),  ¯qγλDμDνqρN= 2 ¯q/uu · Du · Dq ρN uλuμuν− 1 6(uλgμν+ uμgλν+ uνgλμ) −1 6gs¯q/uσ · GqρN(uλuμuν− uλgμν), (24)

where, in their derivations, the equation of motion has been used and the terms O(m2_q)have been neglected due to their ignorable contributions [49].

After using the above in-medium operators, the correlation function in the coordinate space is derived. In order to transfer the calculations to momentum space, we use

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1 (x2₎m = _dD_k (2π )De −ik·x_i(₋₁₎m+1 2D−2mπD/2 ×[D/2 − m] [m] − 1 k2 D/2−m , (25)

and replace xν by i_∂p∂_ν. The following formula is used to perform the resultant four-integrals:

d4 ( 2₎m (2_{+ )}n = iπ2(−1)m−n[m + 2][n − m − 2] [2][n](−)n−m−2 . (26)

By the help of the following relation, we extract the imaginary parts corresponding to different structures: _D 2 − n − 1 D/2_−n =(−1)n−1 (n− 2)!(−) n−2_ln_[−]. (27) To get rid of the contributions of the negative energy particles, the expressions in the OPE side are multiplied by the weight function (w− ¯Ep)e−

w2

M2_{, as well and the following integrals are}

performed over w: OP E_i (w0,p) = w0 −w0 dwρ_iOP E(w,p)(w − ¯Ep)e− w2 M2_. ₍₂₈₎

The obtained results are functions of w0and M2. Applying the continuum subtraction procedure (for details see for instance [53]) with the aim of more suppression of the contributions of the higher states and continuum we obtain the following representation in the Borel scheme:

OP E_i (s₀∗, M2)= s₀∗ 4m2 c dsρ_iOP E(s)eM2−s, (29)

where we have used w0=

s₀∗with s₀∗being the continuum threshold in nuclear matter. Here, ρ_iOP E(s)are the new spectral densities and they are given in terms of the perturbative (pert), two-quark condensate, two-gluon and mixed quark–gluon condensates parts as

ρOP E(s)= ρpert(s)+ ρq ¯qρN(s)+ ρGGρN(s)+ ρqG ¯qρN(s). (30) As examples, we present the explicit forms of the above spectral densities corresponding to the structure gμν in Appendix.

After getting the hadronic and the OPE sides of the correlation function and matching the coefficients of the selected structures form both sides, the following sum rules are obtained

−m2 XfX∗2e −E2p M2 = OP E 1 , f_X∗2e− E2_p M2 = OP E 2 , −υfX∗2e− E2_p M2 = OP E 3 , −υfX∗2e− E2p M2 = OP E 4 , 2_νf_X∗2e− E2p M2 = OP E 5 , (31)

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Table 1

Numericalinputs[54–56].

Input Value Unit

ρsat_N 0.113 GeV3 q†_q ρN 3 2ρN GeV3  ¯qq0 (−0.241)3 GeV3 mq 0.00345 GeV σN 0.059 GeV  ¯qqρN  ¯qq0+ σN 2mqρN GeV 3 q†_iD 0qρN 0.18ρN GeV4 _α s πG2 0 (0.33± 0.04) 4_ρ N GeV4 _α s πG2 ρN _α s πG2 0− 0.65 GeVρN GeV 4  ¯qgsσ Gq0 0.8 ¯qq0 GeV5  ¯qgsσ GqρN  ¯qgsσ Gq0+ 3GeV 2_ρ N GeV5  ¯qiD0iD0qρ_N 0.3ρN−18 ¯qgsσ GqρN GeV 5 3. Numerical analyses

For the numerical analyses of the physical quantities under consideration in nuclear matter and vacuum, we need the value of the saturated nuclear matter density and the in-medium and vacuum operators, whose numerical values are presented in Table1.

Besides these input parameters, the obtained QCD sum rules are also functions of two aux-iliary parameters: the Borel mass parameter M2and the continuum threshold s₀∗. The working intervals for these auxiliary parameters are found based on the standard prescriptions of the method used. The standard criteria for the calculation of working intervals for these parameters are weak dependence of the results on these parameters, achieving to the maximum pole contri-butions (PC), convergence of the series of the OPE and exceeding of the perturbative parts over the non-perturbative contributions. As a result, we get

3 GeV2 M2 5 GeV2, (32)

for M2and

4.12GeV2 s₀∗ 4.32GeV2, (33)

for s₀∗. Considering these intervals of the auxiliary parameters we plot the pole contribution versus M2in Fig.1at different fixed values of the in-medium continuum threshold. From this figure we extract the average value of PC which is 56% of the total contribution. This is a good amount of pole contribution in a tetraquark channel.

Now, to see how the results vary with respect to the Borel mass M2, in Fig.2, we plot the ratio of the mass in nuclear matter to the mass in vacuum, m∗_X/mX, (top-panel), the ratio of

the vector self energy to vacuum mass, υ/mX (mid-panel) and the ratio of the in-medium

current-meson coupling to it’ vacuum value, f_X∗/fX, (bottom-panel) versus M2for three different

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Fig. 1.PCversusM2atdifferentfixedvaluesofthein-mediumcontinuumthreshold.(Forinterpretationofthecolours inthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

As is seen, the results very weakly depend on the variations of the Borel parameter M2. We also see very small variations of the results when we change the value of the in-medium threshold in its working region. Obtained from our analyses, we find an average %25 negative shift in the mass of X particle due to the cold nuclear medium when the saturation density is used. The shift in the meson-current coupling is about %10 and is negative. Such shifts in the mass and residue of X are comparable with those of the nucleon obtained in Ref. [51]. These considerable shifts in the values of the parameters of X can be attributed to its up/down quark content which strongly interacts with the nucleons in the medium. We find the average value of the vector self energy of Xparticle as υ= 1.31 GeV at saturation density, ρNsat= 0.113.

It is also possible to investigate the effects of nuclear matter density to the shifts on the mass, vector self energy and the current-meson coupling. For this purpose, in Fig.3, we plot m∗_X/mX

(top-panel), υ/mX(mid-panel) and f_X∗/fX(bottom-panel) versus ρN/ρ_Nsat at the average value

of the Borel parameter and at fixed values of continuum thresholds. As is seen from these figures, the ratios m∗_X/mX, υ/mXand f_X∗/fXstrongly depend on the density. It is worth noting that

the dependencies of these ratios on the nuclear matter density are roughly linear.

4. Conclusion

In this study, we used the interpolating current in a diquark–antidiquark form to investigate the spectroscopic parameters of the famous X(3872) tetraquark in the cold nuclear matter. This is the first attempt to calculate the properties of non-conventional particles in nuclear matter. The properties of standard hadrons in nuclear medium were previously investigated extensively in the literature (see for instance Refs. [49,57–68] and references therein). We derived the in-medium sum rules to evaluate the shifts in the mass and current-meson coupling of the X(3872) as well as found its vector self energy. The obtained results reveal that the mass of X(3872) is considerably affected by the nuclear matter. We found a negative 25% shift in its mass when the saturation density was used. The shift in the current-meson coupling is also negative and it is roughly 10% of its vacuum value. We found the vector self energy of X(3872) at the saturated density to be υ= 1.31 GeV. The order of shifts in the mass and residue of this particle is comparable with

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Fig. 2.m∗_X/mX,υ/mXandfX∗/fXasfunctionsofM2atthesaturatednuclearmatterdensity,ρsatN = 0.113GeV3

andatfixedvaluesofthecontinuumthreshold.

We also investigated the dependence of the ratios m∗_X/mX, υ/mXand f_X∗/fXon the nuclear

matter density. We observed that, although the dependencies are linear the results strongly depend on the nuclear matter density.

The results obtained in the present work may be used in analyses of the heavy ion collision experiments as well as those which are aiming to investigate the properties of the standard and non-conventional hadrons in nuclear medium such as PANDA Collaboration at FAIR.

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Fig. 3.m∗_X/mX,υ/mXandfX∗/fXversusρ/ρNsatatdifferentvaluesofthethresholdparameters0∗andaveragevalue

ofM2.

Any experimental results on the in-medium properties of X(3872) and comparison of those with the results of the present study can increase our knowledge of the exotic states and help us gain useful information on the not well-known structures of the exotic states, specially the newly discovered tetraquarks.

K. A. thanks Doˇgu¸s University for the partial financial support through the grant BAP 2015-16-D1-B04.

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Appendix A. The spectral densities used in the calculations

In this Appendix, we collect the spectral densities corresponding to the gμνstructure used in

the calculations: ρpert(s)= − 1 3072π6 1 0 dz 1−z 0 dy 1 κ8 yz (syz(y+ z − 1) − m2 c (y3+ y2(2z− 1) + 2y(z − 1)z + (z − 1)z22 3m4c y3+ y2(2z− 1) + 2y(z − 1)z + (z − 1)z2 2 − 26m2 csyz y4+ y3(3z− 2) + y2(4z2− 5z + 1) + yz(3z2_{− 5z + 2) + (z − 1)}2_z2 + 35s2_y2_z2_(y_{+ z − 1)}2 [L(s, z, y)], (A.1) ρq ¯qρN(s)= 1 4π4 1 0 dz 1−z 0 dy mc(y+ z) 4κ5 3m4_c y3+ y2(2z− 1) + 2y(z − 1)z + (z − 1)z22_{− 10m}2 csyz y4+ y3(3z− 2) + y2(4z2− 5z + 1) + yz(3z2_{− 5z + 2) + (z − 1)}2_z2_{+ 7s}2_y2_z2_(y_{+ z − 1)}2  ¯uuρN +mqmc s₀∗yz(y+ z) κ5 5m2_c(y3+ y2(2z− 1) + 2y(z − 1)z + (z − 1)z2₎_{− 7syz(y + z − 1)} u†_u ρN [L(s, z, y)], (A.2) ρGGρN(s)= 1 96π4 1 0 dz 1_−z 0 dy − imqmc κ4 m2_c 4y6z+ y5(24z2− 8z + 1) + y4₍ 52z3− 44z2+ 6z − 1) + y3z(60z3− 80z2+ 55z − 2) + y2_z2₍_40z3_{− 68z}2_{+ 95z − 34)} + 6yz3₍ 2z3− 4z2+ 13z − 11) + 33(z − 1)z4 + yz(y + z − 1)4s₀∗4y3z+ y2(16z2− 4z + 1) + 12y(z − 1)z2_{+ z}2_{− 3s} 4y3z+ y2(16z2− 4z + 1) + 12y(z − 1)z2_{+ 17z}2 _]u†_iD 0uρN

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+ 1 1536(y− 1)2_κ6 m4_c(y− 1) y2+ y(z − 1) + (z − 1)z224y8z+ 12y7(11z2− 8z + 3) + 3y6₍_112z3_{− 156z}2_{+ 45z − 36)} + 2y5₍ 258z4− 516z3+ 332z2− 201z + 54) + y4₍ 516z5− 1320z4+ 1574z3− 1198z2 + 471z − 36) + y3_z( 336z5− 1032z4+ 1707z3− 1769z2+ 858z − 132) + y2_z2₍ 132z5− 468z4+ 1122z3− 1253z2+ 807z − 180) + 4yz3₍_6z5_{− 24z}4_{+ 152z}3_{− 155z}2_{+ 96z − 27)} + 8z4₍ 23z3− 29z2+ 9z − 3) − m2 qs(y− 1)yz 24y10z+ 12y9(13z2− 12z + 3) + y8₍ 492z3− 852z2− 89z − 180) + y7(984z4− 2424z3+ 283z2 + 672z + 360) + y6₍ 1368z5− 4296z4+ 2149z3+ 1121z2− 270z − 360) + y5₍ 1368z6− 5160z5+ 4057z4− 420z3+ 275z2− 676z + 180) + y4₍_984z7_{− 4296z}6_{+ 4151z}5_{− 809z}4_{+ 1804z}3_{− 1901z}2_{+ 615z − 36)} + y3_z( 492z7− 2424z6+ 2485z5+ 834z4+ 43z3− 2396z2+ 1098z − 132) + y2_z2₍_156z7_{− 852z}6_{+ 898z}5_{+ 1887z}4_{− 2785z}3_{− 27z}2_{+ 903z − 180)} + 12y(z − 1)2_z3₍ 2z5− 8z4+ 18z3+ 89z2− 12z − 9) + 8(z − 1)3_z4₍ 23z2+ 30z + 3) − 12syz2_(y_{+ z − 1)}3_y5₍₍ 17s+ 4)z − 4) + y4₍ 5(3s+ 2)z2− 2(17s + 4)z + 2) + y3z(s(32z2− 13z + 17) + 4(3z2_{− 4z + 1)) + 2y}2_z2_(s(_16z2_{− 32z − 1) + 7z}2_{− 10z + 3)} − 8y(z − 1)z3₍ 4s− z + 1) + 2y6+ 4(z − 1)2z4 _α_s πG 2 ρN [L(s, z, y)], (A.3) ρqG ¯qρN_(s)= 1 96π4 1 0 dz 1−z 0 dy −mc κ6 × m2_c 4y6z+ y5(24z2− 8z + 1) + y4₍ 52z3− 44z2− 2z − 1) + y3_z( 60z3− 80z2+ 31z + 6) + y2z2(40z3− 68z2+ 71z − 18) + 2yz3₍ 6z3− 12z2+ 35z − 29) + 33(z − 1)z4 y2+ y(z − 1) + (z − 1)z 3 + yz(y + z − 1)4s₀∗(4y9z+ y8(28z2− 16z + 1) + 3y7₍_28z3_{− 36z}2_{+ 25z − 1) + y}6₍_160z4_{− 300z}3_{+ 371z}2_{− 169z + 3)} + y5₍ 208z5− 508z4+ 838z3− 694z2+ 157z − 1)

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+ y4_z( 192z5− 564z4+ 1116z3− 1276z2+ 567z − 51) + 2y3_z2₍_62z5_{− 210z}4_{+ 463z}3_{− 642z}2_{+ 409z − 82)} + y2_(z_{− 1)}2_z3₍_52z3_{− 92z}2_{+ 239z − 164)} + 3y(z − 1)3_z4₍ 4z2− 4z + 17) + (z − 1)3z5) − s12y9z+ y8(84z2− 48z + 3) + y7(252z3− 324z2+ 121z − 9) + y6₍ 480z4− 900z3+ 713z2− 195z + 9) + y5₍_624z5_{− 1524z}4_{+ 1730z}3_{− 986z}2_{+ 159z − 3)} + y4_z(_576z5_{− 1692z}4_{+ 2500z}3_{− 2060z}2_{+ 709z − 49)} + 2y3_z2₍ 186z5− 630z4+ 1125z3− 1218z2+ 635z − 98) + y2(z− 1)2 × z3₍ 156z3− 276z2+ 589z − 292) + y(z − 1)3z4(36z2− 36z + 193) + 51(z − 1)3_z5  ¯uiD0iD0uρN + mc 4κ7 yz(y+ z − 1) 2s₀∗(4y9z+ y8(28z2− 16z + 1) + 3y7₍_28z3_{− 36z}2_{+ 25z − 1) + y}6₍_160z4_{− 300z}3_{+ 371z}2_{− 169m} cz+ 3) + y5₍_208z5_{− 508z}4_{+ 838z}3_{− 694z}2_{+ 157z − 1)} + y4_z( 192z5− 564z4+ 1116z3− 1276z2 + 567z − 51) + 2y3_z2₍ 62z5− 210z4+ 463z3− 642z2+ 409z − 82) + y2_(z_{− 1)}2_z3₍ 52z3− 92z2+ 239z − 164) + 3y(z − 1)3_z4₍ 4z2− 4z + 17) + (z − 1)3z5) + s(4y9_z_{+ y}8₍_28z2_{− 16z − 59) + y}7₍_84z3_{− 108z}2_{− 401z + 177)} + y6₍ 160z4− 300z3− 1197z2+ 1259z − 177) + y5₍ 208z5− 508z4− 2022z3+ 3534z2− 1271z + 59) + y4_z( 192z5− 564z4− 2204z3+ 5368z2− 3185z + 425) + 2y3_z2₍ 62z5− 210z4− 751z3+ 2380z2− 1945z + 464) + y2_(z_{− 1)}2_z3₍_52z3_{− 92z}2_{− 861z + 760)} + y(z − 1)3_z4₍_12z2_{− 12z − 173) + 25(z − 1)}3_z5₎_{− m}2 c(y2+ y(z − 1) + (z − 1)z)3 4y6z+ y5(24z2− 8z − 35) + y4(52z3− 44z2− 170z + 35) + y3_z(_60z3_{− 80z}2_{− 353z + 174) + y}2_z2₍_40z3_{− 68z}2_{− 301z + 234)} + 2yz3₍ 6z3− 12z2− 37z + 43) + 9(z − 1)z4 _¯ug sσ GuρN [L(s, z, y)], (A.4) where

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= (y + z − 1), κ= (y2+ y(z − 1) + (z − 1)z), L[s, y, z] = ((−1 + y)(−(syz(−1 + y + z)) + mc2(y3+ 2y(−1 + z)z + (−1 + z)z2+ y2(−1 + 2z)))) (y2_{+ y(−1 + z) + (−1 + z)z)}2 , (A.5) and is the usual step function.

References

[1]S.K.Choi, etal.,BelleCollaboration,Observationofa narrowcharmonium–likestate inexclusive B+→ K+π+π−J /ψdecays,Phys.Rev.Lett.91(2003)262001,arXiv:hep-ex/0309032.

[2]D._√Acosta,etal.,CDFCollaboration,ObservationofthenarrowstateX(3872)→ J/ψπ+π−in ¯pp collisionsat s= 1.96 TeV,Phys.Rev.Lett.93(2004)072001,arXiv:hep-ex/0312021.

[3]V.M.Abazov,etal.,D0Collaboration,ObservationandpropertiesoftheX(3872) decayingtoJ /ψ π+π−inp¯p collisionsat√s= 1.96 TeV,Phys.Rev.Lett.93(2004)162002,arXiv:hep-ex/0405004.

[4]B.Aubert,etal.,BaBarCollaboration,StudyoftheB→ J/ψK−π+π−decayandmeasurementoftheB→ X(3872)K−branchingfraction,Phys.Rev.D71(2005)071103,arXiv:hep-ex/0406022.

[5]K.Abe,etal.,BelleCollaboration,ExperimentalconstraintsonthepossibleJP CquantumnumbersoftheX(3872), arXiv:hep-ex/0505038.

[6]M.Nielsen,F.S.Navarra,S.H.Lee,NewcharmoniumstatesinQCDsumrules:aconcisereview,Phys.Rep.497 (2010)41,arXiv:0911.1958 [hep-ph].

[7]S.-K.Choi,etal.,Boundsonthewidth,massdifferenceandotherpropertiesofX(3872)→ pi + pi − J/psi decays,Phys.Rev.D84(2011)052004,arXiv:1107.0163 [hep-ex].

[8]P.G.Ortega,D.R.Entem,F.Fernandez,E.RuizArriola,Countingstatesandthehadronresonancegas:doesX(3872) count?arXiv:1707.01915 [hep-ph].

[9]G.L.Yu,Z.G.Wang,Z.Y.Li,Theanalysisofthecharmonium-likestatesX∗(3860),X(3872),X(3915),X(3930) andX(3940) accordingtoitsstrongdecaybehaviors,arXiv:1704.06763 [hep-ph].

[10]Z.Y.Zhou,Z.Xiao,UnderstandingX(3862),X(3872),andX(3930) inaFriedrichs-model-likescheme,Phys.Rev. D96 (5)(2017)054031,arXiv:1704.04438 [hep-ph].

[11]W.Wang,Q.Zhao,Deciphertheshort-distancecomponentofX(3872) inBcdecays,Phys.Lett.B755(2016)261,

arXiv:1512.03123 [hep-ph].

[12]J.Ferretti,E.Santopinto,H.García-Tecocoatzi,Highercharmoniaandbottomonia.NatureoftheX(3872),arXiv: 1510.00433 [hep-ph].

[13]C.Peña,L.Jacak,AbraidmodelfortheparticleX(3872),arXiv:1411.1574 [hep-ph].

[14]F.K.Guo,C.Hidalgo-Duque,J.Nieves,A.Ozpineci,M.P.Valderrama,Detectingthelong-distancestructureofthe X(3872),Eur.Phys.J.C74 (5)(2014)2885,arXiv:1404.1776 [hep-ph].

[15]Z.G.Wang,T.Huang,AnalysisoftheX(3872),Zc(3900) andZc(3885) asaxial-vectortetraquarkstateswithQCD

sumrules,Phys.Rev.D89 (5)(2014)054019,arXiv:1310.2422 [hep-ph].

[16]V.Baru,E.Epelbaum, A.A.Filin,C.Hanhart,U.-G.Meissner,A.V.Nefediev,Quarkmassdependenceofthe X(3872)bindingenergy,Phys.Lett.B726(2013)537,arXiv:1306.4108 [hep-ph].

[17]P.Wang,X.G.Wang,StudyonX(3872)fromeffectivefieldtheorywithpionexchangeinteraction,Phys.Rev.Lett. 111 (4)(2013)042002,arXiv:1304.0846 [hep-ph].

[18]C.Y.Cui,Y.L.Liu,G.B.Zhang,M.Q.Huang,CanX(3872)beaJP= 2−tetraquarkstate?Commun.Theor.Phys. 57(2012)1033,arXiv:1112.5976 [hep-ph].

[19]M.Takizawa,S.Takeuchi,X(3872)asahybridstateofcharmoniumandthehadronicmolecule,PTEP2013(2013) 093D01,arXiv:1206.4877 [hep-ph].

[20]S.Coito,G.Rupp,E.vanBeveren,X(3872)isnotatruemolecule,Eur.Phys.J.C73 (3)(2013)2351,arXiv: 1212.0648 [hep-ph].

[21]J.Ferretti,G.Galatà,E.Santopinto,QuarkstructureoftheX(3872) andχb(3P ) resonances,Phys.Rev.D90 (5)

(2014)054010,arXiv:1401.4431 [nucl-th].

[22]M.Karliner,J.L.Rosner,X(3872),Xb,andtheχb1(3P ) state,Phys.Rev.D91 (1)(2015)014014,arXiv:1410.7729

(17)

[23]N.N.Achasov,E.V.Rogozina,X(3872),IG(JP C)= 0+(1++),astheχ1c(2P ) charmonium,Mod.Phys.Lett.A

30 (33)(2015)1550181,arXiv:1501.03583 [hep-ph].

[24]A.B.Larionov,M.Strikman,M.Bleicher,DeterminationofthestructureoftheX(3872) in ¯pA collisions,Phys. Lett.B749(2015)35,arXiv:1502.03311 [nucl-th].

[25]A.Larionov,M.Strikman,M.Bleicher,TestoftheX(3872) structureinantiproton-nucleuscollisions,AIPConf. Proc.1735(2016)060005,arXiv:1512.05536 [nucl-th].

[26]C.Meng,J.J.Sanz-Cillero,M.Shi,D.L.Yao,H.Q.Zheng,RefinedanalysisontheX(3872)resonance,Phys.Rev. D92 (3)(2015)034020,arXiv:1411.3106 [hep-ph].

[27]W.Chen,H.y.Jin,R.T.Kleiv,T.G.Steele,M.Wang,Q.Xu,QCDsum-ruleinterpretationofX(3872)withJP C= 1++mixturesofhybridcharmoniumandDD∗ molecularcurrents,Phys.Rev.D88 (4)(2013)045027,arXiv: 1305.0244 [hep-ph].

[28]L.Maiani,F.Piccinini,A.D.Polosa,V.Riquer,Diquark-antidiquarkswithhiddenoropencharmandthenatureof X(3872),Phys.Rev.D71(2005)014028,arXiv:hep-ph/0412098.

[29]D.V.Bugg,Reinterpretingseveralnarrow‘resonances’asthresholdcusps,Phys.Lett.B598(2004)8,arXiv:hep -ph/0406293.

[30]B.A.Li,IsX(3872)apossiblecandidateofhybridmeson,Phys.Lett.B605(2005)306,arXiv:hep-ph/0410264.

[31]K.K.Seth,AnalternativeinterpretationofX(3872),Phys.Lett.B612(2005)1,arXiv:hep-ph/0411122.

[32]E.S.Swanson,ShortrangestructureintheX(3872),Phys.Lett.B588(2004)189,arXiv:hep-ph/0311229.

[33]Z.G.Wang,T.Huang,PossibleassignmentsoftheX(3872),Zc(3900) andZb(10610) asaxial-vectormolecular

states,Eur.Phys.J.C74 (5)(2014)2891,arXiv:1312.7489 [hep-ph].

[34]E.Braaten,D.Kang,J /ψ ωdecaychanneloftheX(3872)charmmesonmolecule,Phys.Rev.D88 (1)(2013) 014028,arXiv:1305.5564 [hep-ph].

[35]H.X.Chen,W.Chen,X.Liu,S.L.Zhu,Thehidden-charmpentaquarkandtetraquarkstates,Phys.Rep.639(2016) 1,arXiv:1601.02092 [hep-ph].

[36]R.D.Matheus,S.Narison,M.Nielsen,J.M.Richard,CantheX(3872)bea1++four-quarkstate?Phys.Rev.D75 (2007)014005,arXiv:hep-ph/0608297.

[37]X.W.Kang,J.A.Oller,DifferentpolestructuresinlineshapesoftheX(3872),Eur.Phys.J.C77 (6)(2017)399, arXiv:1612.08420 [hep-ph].

[38]H.Sundu,Themassandcurrent-mesoncouplingconstantoftheexoticX(3872)statefromQCDsumrules,SDUJ. Nat.Appl.Sci.20 (3)(2016)448.

[39]E.V.Veliev,S.Gunaydin,H.Sundu,ThermalpropertiesofexoticX(3872) stateviaQCDsumrule,arXiv:1707. 03714 [hep-ph].

[40]F.Giacosa,Non-conventionalmesonsatPANDA,J.Phys.Conf.Ser.599 (1)(2015)012004,arXiv:1502.02682 [hep-ph].

[41]E.Prencipe,etal.,PANDACollaboration,AIPConf.Proc.1735(2016)060011,arXiv:1512.05496 [hep-ex].

[42]S.Biswas,etal.,MeasurementofthesparkprobabilityofaGEMdetectorfortheCBMmuonchamber(MuCh), Nucl.Instrum.Methods,Sect.A800(2015)93,arXiv:1504.00001 [physics.ins-det].

[43]http://www.gsi.de/fair/experiments/CBM/index.e.html. [44]http://www-panda.gsi.de/auto/phy/home.htm.

[45]B.Friman,etal.,TheCBMPhysicsBook:CompressedBaryonicMatterinLaboratoryExperiments,Springer, Heidelberg,2011.

[46]M.F.M.Lutz,etal.,PANDACollaboration,PhysicsperformancereportforPANDA:stronginteractionstudieswith antiprotons,arXiv:0903.3905 [hep-ex].

[47]http://nica.jinr.ru/innovation.php.

[48]R.Thomas,T.Hilger,B.Kampfer,Four-quarkcondensatesinnucleonQCDsumrules,Nucl.Phys.A795(2007) 19,arXiv:0704.3004 [hep-ph].

[49]T.D.Cohen,R.J.Furnstahl,D.K.Griegel,X.m.Jin,QCDsumrulesandapplicationstonuclearphysics,Prog.Part. Nucl.Phys.35(1995)221,arXiv:hep-ph/9503315.

[50]B.D.Serot,J.D.Walecka,Therelativisticnuclearmanybodyproblem,Adv.Nucl.Phys.16(1986)1.

[51]K.Azizi,N.Er,Propertiesofnucleoninnuclearmatter:oncemore,Eur.Phys.J.C74(2014)2904,arXiv:1401.1680 [hep-ph].

[52]X.m.Jin,T.D.Cohen,R.J.Furnstahl,D.K.Griegel,QCDsumrulesfornucleonsinnuclearmatter.2,Phys.Rev.C 47(1993)2882.

[53]S.S.Agaev,K.Azizi,H.Sundu,ApplicationoftheQCDlightconesumruletotetraquarks:thestrongvertices XbXbρandXcXcρ,Phys.Rev.D93 (11)(2016)114036,arXiv:1605.02496 [hep-ph].

[54]T.D.Cohen,R.J.Furnstahl,D.K.Griegel,Quarkandgluoncondensatesinnuclearmatter,Phys.Rev.C45(1992) 1881.

(18)

[55]V.M. Belyaev, B.L. Ioffe, Determination of the baryon mass and baryon resonances from the quantum-chromodynamicssumrule.Strangebaryons,Sov.Phys.JETP57(1983)716,Zh.Eksp.Teor.Fiz.84(1983)1236.

[56]B.L.Ioffe,QCDatlowenergies,Prog.Part.Nucl.Phys.56(2006)232,arXiv:hep-ph/0502148.

[57]K.Azizi,N.Er,H.Sundu,Scalarandvectorself-energiesofheavybaryonsinnuclearmedium,Nucl.Phys.A960 (2017)147,Nucl.Phys.A962(2017)122,Erratum,arXiv:1605.05535 [hep-ph].

[58]A.Hosaka,T.Hyodo,K.Sudoh,Y.Yamaguchi,S.Yasui,Heavyhadronsinnuclearmatter,Prog.Part.Nucl.Phys. 96(2017)88,arXiv:1606.08685 [hep-ph].

[59]Z.G.Wang,AnalysisofthedoublyheavybaryonsinthenuclearmatterwiththeQCDsumrules,Eur.Phys.J.C72 (2012)2099,arXiv:1205.0605 [hep-ph].

[60]Z.G.Wang,AnalysisoftheQbaryonsinthenuclearmatterwiththeQCDsumrules,Phys.Rev.C85(2012)

045204,arXiv:1109.2180 [hep-ph].

[61]Z.G.Wang,AnalysisoftheQbaryonsinthenuclearmatterwiththeQCDsumrules,Eur.Phys.J.C71(2011)

1816,arXiv:1108.4251 [hep-ph].

[62]R.J.Furnstahl,X.m.Jin,D.B.Leinweber,NewQCDsumrulesfornucleonsinnuclearmatter,Phys.Lett.B387 (1996)253,arXiv:nucl-th/9511007.

[63]E.G.Drukarev,M.G.Ryskin,V.A.Sadovnikova,QCDsumrulesdescriptionofnucleonsinasymmetricnuclear matter,Phys.Rev.C70(2004)065206,arXiv:nucl-th/0406027.

[64]E.G.Drukarev,M.G.Ryskin,V.A.Sadovnikova,QCDcondensatesandhadronparametersinnuclearmatter: self-consistenttreatment,sumrulesandallthat,Prog.Part.Nucl.Phys.47(2001)73,arXiv:nucl-th/0106049.

[65]Y.Koike,A.Hayashigaki,QCDsumrulesforrho,omega,phimeson–nucleonscatteringlengthsandthemass shiftsinnuclearmedium,Prog.Theor.Phys.98(1997)631,arXiv:nucl-th/9609001.

[66] A. Hayashigaki,Phys. Lett. B 487(2000)96, https://doi.org/10.1016/S0370-2693(00)00760-7, arXiv:nucl-th/ 0001051.

[67]K.Ohtani,P.Gubler,M.Oka,Negative-paritynucleonexcitedstateinnuclearmatter,Phys.Rev.C94 (4)(2016) 045203,arXiv:1606.09434 [hep-ph].

[68]K.Suzuki,P.Gubler,M.Oka,Dmesonmassincreasebyrestorationofchiralsymmetryinnuclearmatter,Phys. Rev.C93 (4)(2016)045209,arXiv:1511.04513 [hep-ph].