Heavy Scalar, Vector and Axial-Vector Mesons in Hot and Dense Nuclear Medium

In this work we shall investigate the mass modifications of scalar mesons $\left( D_{0}, B_{0}\right)$,vector mesons $\left( D^{\ast}, B^{\ast}\right)$ and axial-vector mesons $\left(D_{1}, B_{1}\right)$ at finite density and temperature of the nuclear medium. The above mesons are modified in the nuclear medium through themodification of quark and gluon condensates. We shall find the medium modification of quark and gluon condensates within chiral SU(3) model through the medium modification of scalar-isoscalar fields $\sigma$ and $\zeta$ at finite density and temperature. These medium modified quark and gluon condensates will further be used through QCD sum rules for the evaluation of in-medium properties of above mentioned scalar, vector and axial vector mesons. We shall also discuss the effects of density and temperature of the nuclear medium on the scattering lengths of above scalar, vector and axial-vector mesons. The study of the medium modifications of above mesons may be helpful for understanding their production rates in heavy-ion collision experiments. The results of present investigations of medium modifications of scalar, vector and axial-vector mesons at finite density and temperature can be verified in the Compressed Baryonic Matter (CBM) experiment of FAIR facility at GSI, Germany.


I. INTRODUCTION
The motive behind the heavy-ion collision experiments at different experimental facilities is to explore the different phases of QCD phase diagram. These experiments help us to understand the nuclear matter properties for different values of temperatures and densities. The hadronic matter produced in heavy-ion collisions may undergo different phase transitions e.g. liquid-gas phase transition, the kaon condensation, the restoration of chiral symmetry and may be the formation of quark gluon plasma [1][2][3][4]. The Compressed Baryonic Matter (CBM) experiment of the FAIR project at GSI, Germany may explore the phase of hadronic matter at high baryon densities and moderate temperatures. These kind of phases may exist in the compact astrophysical objects e.g. neutron stars. The property of restoration of chiral symmetry is closely related to the medium modifications of hadrons [4]. The medium modifications of Kaons, D mesons and light vector mesons had been studied using different theoretical approaches e.g. chiral model [5][6][7][8][9][10][11][12], QCD sum rules [13][14][15][16][17][18] and coupled channel approach [19][20][21]. Due to interactions the properties of hadrons in the medium are found to be different as compared to their free space properties.
The medium modifications of heavy scalar, vector and axial-vector mesons at finite density and temperature of the medium had been studied very rarely [22][23][24][25]. In the present investigation we shall study the mass modifications of heavy scalar mesons (D 0 , B 0 ), vector mesons (D * , B * ) and axial vector mesons (D 1 , B 1 ) at finite densities and temperatures.
The study of in-medium properties of scalar, vector and axial-vector mesons will be helpful to understand their experimental production rates. The medium modification of charmed mesons may modify the experimental production of ground state charmonium J/ψ and the excited charmonium states (ψ ′ and χ c ). The charmonium, J/ψ may be produced due to the decay of the higher charmonium states. However the vacuum threshold value of heavy meson pairs lies above the vacuum mass of the excited charmonium states. Now if these heavy mesons get modified (undergo mass drop in the medium) then the excited charmoium states may decay to the open charmed meson pairs instead of decaying to the ground state charmonium. Thus to understand the production of charmonium states in heavy-ion collisions it is very necessary to study the medium modification of the heavy scalar, vector and axial-vector mesons. The medium modifications of heavy vector mesons may also help us in understanding the dilepton spectra produced in heavy-ion collision experiments [26][27][28].
The dileptons are considered as interesting probe to study the evolution of matter produced in heavy ion collision experiments as they do not undergo strong interactions in the medium.
In ref. [29] the production of open charm and charmonium in hot hadronic medium had been investigated using the statistical hadronization model at SPS/FAIR energies. In this work it was observed that the medium modifications of charmed hadrons do not lead to appreciable changes in cross-section for D mesons production. This is because of large charm quark mass and different times scales for charm quark and charm hadron production. However, the charmonia yield is effected appreciably due to in-medium modifications.
The properties of scalar charm resonances D s0 (2317) and D 0 (2400) and hidden charm resonance, X(3700) had been studied in ref. [24] using coupled channel approach. In these studies the D s0 (2317) and X(3700) were found to undergo a width of about 100 and 200 MeV respectively at nuclear matter density. However, for the D 0 (2400) mesons there was already large width of resonance in the free space and the medium effect were found to be weak as compared to D s0 (2317) and X(3700). In ref. [17] the mass splitting of D-D and B-B mesons had been studied using the QCD sum rules in the cold nuclear matter and the calculated values of mass splitting at nuclear saturation density were 60 and 130 MeV respectively. The Borel transformed QCD sum rules had also been used to study the properties of pseudoscalar D mesons [18] and vector mesons, ρ, ω and φ [16]. The properties of the scalar mesons (D 0 , B 0 ) in the cold nuclear matter using QCD sum rules have been investigated in ref. [22]. The vectors mesons (D * , B * ) and axial vector mesons (D 1 , B 1 ) had also been studied using QCD sum rules in cold nuclear matter in ref. [23]. Note that in ref. [22] and [23] the properties of the meson were investigated at zero temperature and at normal nuclear matter density. However in the present investigation we shall find the inmedium masses of the scalar (D 0 , B 0 ) and vector (D * , B * ) and axial vector (D 1 , B 1 ) mesons at finite temperatures as well as at the densities greater than the nuclear saturation density.
In the present work to investigate the properties of scalar, vector and axial-vector mesons we shall use the QCD sum rules and chiral SU(3) model [5]. Within QCD sum rules, the in-medium properties of mesons are related to the in-medium properties of quark and gluon condensates. We shall investigate the in-medium properties of quark and gluon condensates using the chiral SU(3) model. Using chiral SU(3) model we shall find the values of quark and gluon condensates at finite values of temperatures and baryonic densities. These values of condensates will further be used to find the medium modification of mesons using QCD sum rules. The chiral SU(3) model along with QCD sum rules had been used in the literature to investigate the in-medium modification of the charmonium states J/ψ and η c [30].
The present article is organized as follows: In section II we shall give a brief review of chiral SU(3) model. Then in section III we shall discuss that how we will evaluate the inmedium modifications of the scalar, vector and axial-vector mesons within QCD sum rules and using the properties of quark and gluon condensates as evaluated in the chiral SU (3) model. In section IV we shall discuss the results of the present investigation and finally in section V we shall give a brief summary of present work.

II. CHIRAL SU(3) MODEL
In this section we shall briefly review the chiral SU(3) model used in the present investigation for the in-medium properties of heavy mesons. The chiral SU(3) model is based on the broken scale invariance and non-linear realization of chiral symmetry [31][32][33][34][35]. The model involve the Lagrangian densities describing e.g. kinetic energy terms, baryon-meson interactions, self interactions of scalar mesons, vector mesons, symmetry breaking terms and also the scale invariance breaking logarithmic potential terms.
For the investigation of hadron properties at finite temperature and densities we use the mean field approximation. Under this approximation all the meson fields are treated as classical fields and only the scalar and the vector fields contribute to the baryon-meson interactions. From the interaction Lagrangian densities, using the mean-field approximation, we derive the equations of motions for the scalar fields σ and ζ and the dilaton field, χ in isospin symmetric nuclear medium. We solve these coupled equations to obtain the density and temperature dependence of scalar fields σ and ζ and the dilaton field, χ in isospin symmetric nuclear medium [11]. The concept of broken scale invariance leading to the trace anomaly in (massless) QCD, where G a µν is the gluon field strength tensor of QCD, is simulated in the effective Lagrangian at tree level [36] through the introduction of the scale breaking terms [11]. Within chiral SU(3) model the scale breaking terms are written in terms of the dilaton field χ and also the scalar fields σ and ζ. From this we obtain the energy momentum tensor and this is compared with the energy momentum tensor of QCD which is written in terms of gluon condensates. In this way we extract the value of gluon condensates in terms of the scalar fields σ and ζ and the dilaton field, χ and is given by the following equation [11], where the value of parameter d is 0.064 [9], m π and m K denote the masses of pions and kaons and have values 139 and 498 MeV respectively. f π and f K are the decay constants having values 93.3 and 122 MeV respectively. The symbols σ, ζ and χ denote the nonstrange scalar-isoscalar field, strange scalar-isoscalar field and the dilaton field respectively. In this section we shall discuss the QCD sum rules [22,23] which will be used later along with the chiral SU(3) model for the evaluation of in-medium properties of scalar, vector and axial vector mesons. To find the mass modification of above discussed heavy mesons we shall use the two-point correlation function Π µν (q), In above equation J µ (x) denotes the isospin averaged current, x = x µ = (x 0 , x) is the four coordinate, q = q µ = (q 0 , q) is four momentum and T denotes the time ordered operation on the product of quantities in the brackets. From above definition it is clear that the two point correlation function is actually a Fourier transform of the expectation value of the time ordered product of two currents. The two-point correlation function for the scalar mesons is defined as, For the scalar, vector and axial vector mesons isospin average currents are given by the expressions and respectively. Note that in above equations q denotes the light u or d quark whereas c denotes the heavy charm quark. Note that in the present work, instead of considering the mass splitting between particles and antiparticles, we emphasize on the mass shift of iso-doublet D and B mesons as a whole and therefore we consider the average in the definitions of scalar, vector and iso-vector currents which is referred as centroid [18]. To find the mass splitting of particles and antiparticles in the nuclear medium one has to consider the even and odd part of QCD sum rules [17]. For example, in ref. [17] the mass splitting between pseudoscalar D andD mesons was investigated using the even and odd QCD sum rules whereas in [18,22,23] the mass-shift of D mesons was investigated under centroid approximation.
The two point correlation function can be decomposed into the vacuum part, a static one-nucleon part and pion bath contribution i.e. we can write where In above equation |N(p) denotes the isospin and spin averaged static nucleon state with the four-momentum p = (M N , 0). The state is normalized as N(p)|N(p ′ ) = (2π) 3 2p 0 δ 3 (p−p ′ ).
The third term, Π P.B. µν (q) in equation (7) gives the contribution from pion bath at finite temperature. Note that in the present work instead of considering the contribution of pion bath the effects of finite temperature of the nuclear matter on the properties of D and B mesons will be evaluated through the temperature dependence of scalar fields σ, ζ and χ.
The temperature dependence of scalar fields σ, ζ and χ modify the nucleon properties in the medium and these modified nucleons further modify the in-medium properties of D and B mesons at finite temperature and density. In literature the properties of kaons and antikaons, D mesons and charmonium had been studied at finite temperature of the nuclear matter using the above mentioned scalar fields σ, ζ and χ [7,11,12,30].
As discussed in Ref. [23], in the limit of the 3-vector q → 0, the correlation functions T N (ω, q ) can be related to the D * N and D 1 N scattering T-matrices. Thus we write [23] In above equation a D * and a D 1 are the scattering lengths of D * N and D 1 N respectively.
Similarly, we can also write the scattering T matrix corresponding to D 0 N (D 0 is a scalar meson) in terms of the scattering lengths [22], Near the pole positions of the scalar, vector and axial vector mesons the phenomenological spectral densities can be parameterized with three unknown parameters a, b and c i.e. we write [18,22,23] The term denoted by ... represent the continuum contributions. The first term denotes the double-pole term and corresponds to the on-shell effects of the T-matrices, Now we shall write the relation between the scattering length of mesons and their inmedium mass-shift. For this first we note that the shift of squared mass of mesons can be written in terms of the parameter a appearing in equation (11) through relation [16], where in the last term we used equation (12). The mass shift is now defined by the relation The second term in equation (11) denotes the single-pole term, and corresponds to the off-shell (i.e. ω 2 = M 2 D 0 /D * /D 1 ) effects of the T -matrices. The third term denotes the continuum term or the remaining effects, where, s 0 , is the continuum threshold. The continuum threshold parameter s 0 define the scale below which the continuum contribution vanishes [37].
It can be observed from equations (13) and (14) that if we want to find the value of mass shift of mesons then we first need to find the value of unknown parameter a. For this we proceed as follows: we note that in the low energy limit, ω → 0, the T N (ω, 0) is equivalent to the Born term T Born D 0 /D * /D 1 N (ω, 0). We take into account the Born term at the phenomenological side, with the constraint a M 4 Note that in Eq. (15) the phenomenological side of scattering amplitude for q µ = 0 is not exactly equal to Born term but there are contributions from other terms. However, for ω → 0, T N on left should be equal to T Born on right side of Eq. (15) and this requirement results in constraint given in Eq. (16). As we shall discuss below the constraint (16) help in eliminating the parameter c and scattering amplitude will be function of parameters a and b only. The Born terms to be used in equation (15) for scalar, vector and axial-vector mesons are given by following relations [22,23] T Born In Note that in equation (18) we have two unknown parameters a and b. We differentiate equation (18) w.r.t. 1 M 2 so that we could have two equations and two unknowns. By solving those two coupled equations we will be able to get the values of parameters a and b. Same procedure will be applied to obtain the values of parameters a and b corresponding to vector and axial-vector mesons. For vector meson, D * , the Borel transformation equation is given by [23], where, B = . For the axial-vector mesons, D 1 , the Borel transformation equation is given by [23], where, C = In the above equations m 2 c = m 2 c x . As discussed earlier, in determining the properties of hadrons from QCD sum rules, we shall use the values of quark and gluon condensates as calculated using chiral SU(3) model.
Any operator O on OPE side can be written as [16,37,38], In the quark and gluon condensates can be expressed in terms of scalar fields σ, ζ and χ. As discussed earlier, the finite temperature effects in the present investigation will be evaluated through the scalar fields and therefore contribution of third term will not be considered.
However, for completeness we shall compare the temperature dependence of scalar quark and scalar gluon condensates at zero baryon density as evaluated in the present work with the situation when the temperature dependence is evaluated using only pion bath contribution [37]. Thus within chiral SU(3) model, we can find the values of O ρ B at finite density of the nuclear medium and hence can find O N using The quark condensate,qq, can be extracted from the explicit symmetry breaking term of the Lagrangian density and is given by, In our present investigation of hadron properties, we are interested in light quark condensates,ūu anddd, which are proportional to the non-strange scalar field σ within chiral SU (3) model. Considering equal mass of light quarks, u and d i.e. m u = m d = m q = 0.006 GeV, we can write, The condensate qg s σGq ρ B is given by the equation [39], Also we write [39], As discussed above the quark condensate, qq ρ B , can be calculated within the chiral SU (3) model. This value of qq ρ B can be used through equations (25) and (26) to calculate the value of condensates qg s σGq ρ B and qiD 0 iD 0 q ρ B within chiral SU(3) model. The value of condensate q † iD 0 q is equal to 0.18GeV 2 ρ B [39].

IV. RESULTS AND DISCUSSIONS
In this section we shall present the results of our investigation of in-medium properties of scalar (D 0 , B 0 ), vector (D * , B * ) and axial vector mesons (D 1 , B 1 ). The nuclear matter saturation density used in the present investigation is 0.15 fm −3 . The values of various [22]. The respectively [22,23]. As discussed earlier the mass-shift of scalar, vector and axial-vector mesons is calculated through the parameter a which is related to scattering length through equations (12) to (14). This parameter a, for example, for D 0 is calculated by solving the coupled equations as discussed after Eq. (18) and is subjected to the medium modifications through the medium dependence of condensates. The medium dependence of condensates is further evaluated through the scalar fields σ, ζ and χ. The various coupling constants At zero baryon density, we observe that the magnitude of the scalar fields σ and ζ and the dilaton field χ decreases with increase in the temperature. However, the change in the values of scalar fields with temperature of the medium is observed to be very small.
The reason for the non zero values of scalar fields at finite temperature and zero baryon density of the medium is the formation of baryon-antibaryon pairs [32,40]. At finite baryon densities, the magnitude of the scalar fields increases with increase in the temperature of symmetric nuclear medium. This also leads to increase in the masses of the nucleons with the temperature of the nuclear medium for finite baryon densities [11]. At nuclear saturation density, ρ 0 , the magnitude of the scalar fields, σ and ζ and the dilaton field χ increases by 5.47, 1.28 and 0.96 MeV respectively as we move from T = 0 to T = 150 MeV respectively.
In figures (1) and (2) we show the variation of the light scalar quark condensateqq, given by equation (24) and the scalar gluon condensate, G 0 = αs π G a µν G aµν , given by equation (1), respectively as a function of density of the symmetric nuclear medium. We show the results for temperatures, T = 0, 50, 100 and 150 MeV respectively. From equation (24) we observe that the value of the scalar quark condensateqq is directly proportional to the scalar-isoscalar field, σ. Therefore, the behavior of theqq as a function of temperature and density of the nuclear medium will be same as that of σ field. gluon condensates at finite temperature are evaluated due to contribution from pion bath using equations (27) and (28) respectively [37]. Using m q qq 0 = -(0.11 GeV) 4 and αs π G 2 = 0.005 GeV 4 [37] we calculate the quark and gluon condensates at finite temperature and zero baryon density using equations (27) and (28)  MeV respectively. From above discussion we observe that as a function of temperature there are very small variations in the values of scalar quark and gluon condensates in the nuclear medium. This observation agree e.g. with ref. [41] where gluon and chiral condensates were studied at finite temperature with an effective Lagrangian of pseudoscalar mesons coupled to a scalar glueball. The gluon condensates were found to be very stable up to temperatures of 200 MeV, where the chiral sector of the theory reaches its limit of validity [41]. Actually scalar quark and gluon condensates are found to vary effectively with temperature above critical temperature but in hadronic medium, for zero baryon density, these are not much sensitive to temperature effects [42,43].
Note that in the above discussion of scalar gluon and quark condensates, calculated using equations (1) and (24) respectively, we considered the vacuum values of decay constants f π and f K as well as masses m π and m K of pions and kaons respectively. Now we shall discuss the effect of medium modified values of f π , f K , m π and m K on the scalar condensates. In the chiral effective model the pion and kaon decay constants are related to the scalar fields σ and ζ through relations [5,11] f π = −σ (29) and respectively. From the above relations it is clear that the medium dependence of scalar fields σ and ζ can be used to study the density and temperature dependence of decay constants of pions and kaons. For example, using equation (29) respectively which were calculated without the medium modification of f π , f K , m π and m K .
We conclude that the medium modification of f π , f K , m π and m K causes more decrease in the values of scalar quark and gluon condensates at finite baryonic density. We also calculate the values of scattering lengths of scalar mesons using equation (12) for are observed to be 1.23 (4.40) and 0.66 (2.28) fm respectively. We note that the value of scattering length decreases as a function of density and temperature of the nuclear medium.
As discussed above the scattering lengths are evaluated using equation (12). In this equation we have the parameter a which is directly proportional to the scattering length of mesons.
As discussed earlier the value of parameter a is evaluated by solving simultaneously equation (18)   Similarly, "Quark 1 ", "Quark 2 ", "Quark 4 " and "Gluon 1 " are denoting the contribution of qq , q † iD 0 q , qiD 0 iD 0 q and αsGG π respectively to the in-medium properties of mesons.          for the vector mesons a negative value of scattering lengths were observed. Also it was found that the magnitude of the scattering lengths of scalar, vector and axial-vector mesons decreases as we move from low to high value of density or temperature of the nuclear medium.
The present investigation of medium modification of scalar, vector and axial vector mesons will be helpful for understanding their production rate and also the phenomenon of J/ψ suppression in the Compressed Baryonic matter experiment at FAIR, GSI.