#### Abstract

In this work we shall investigate the mass modifications of scalar mesons (; ), vector mesons (; ), and axial-vector mesons (; ) at finite density and temperature of the nuclear medium. The above mesons are modified in the nuclear medium through the modification of quark and gluon condensates. We will find the medium modification of quark and gluon condensates within chiral SU(3) model through the medium modification of scalar-isoscalar fields and 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 the above mentioned scalar, vector, and axial vector mesons. We will also discuss the effects of density and temperature of the nuclear medium on the scattering lengths of the above scalar, vector, and axial-vector mesons. The study of the medium modifications of the 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.

#### 1. 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, for example, the liquid-gas phase transition, the kaon condensation, the restoration of chiral symmetry, and maybe the formation of quark gluon plasma [1–4]. 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 kinds of phases may exist in the compact astrophysical objects, for example, neutron stars. The property of restoration of chiral symmetry is closely related to the medium modifications of hadrons [4]. The medium modifications of Kaons, mesons, and light vector mesons had been studied using different theoretical approaches, for example, chiral model [5–13], QCD sum rules [14–19], and coupled channel approach [20–22]. 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 [23–26]. In the present investigation we will study the mass modifications of heavy scalar mesons , vector mesons , and axial-vector mesons 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 and the excited charmonium states and . The charmonium, , 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 charmonium 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 [27–32]. 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 [33] 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 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 affected appreciably due to in-medium modifications.

The properties of scalar charm resonances and and hidden charm resonance, , had been studied in [25] using coupled channel approach. In these studies the and were found to undergo a width of about 100 and 200 MeV, respectively, at nuclear matter density. However, for the mesons there was already large width of resonance in the free space and the medium effects were found to be weak as compared to and . In [18] the mass splitting of - and - 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 mesons [19] and vector mesons, , , and [17]. The properties of the scalar mesons in the cold nuclear matter using QCD sum rules have been investigated in [23]. The vectors mesons and axial-vector mesons had also been studied using QCD sum rules in cold nuclear matter in [24]. Note that in [23, 24] the properties of the meson were investigated at zero temperature and at normal nuclear matter density. However, in the present investigation we will find the in-medium masses of the scalar and vector and axial-vector 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 will use the QCD sum rules and chiral SU 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 will investigate the in-medium properties of quark and gluon condensates using the chiral SU model. Using chiral SU model we will 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 model along with QCD sum rules had been used in the literature to investigate the in-medium modification of the charmonium states and [34].

The present paper is organized as follows. In Section 2 we will give a brief review of chiral SU model. Then in Section 3 we will discuss how we will evaluate the in-medium 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 model. In Section 4 we will discuss the results of the present investigation and finally in Section 5 we will give a brief summary of present work.

#### 2. Chiral SU(3) Model

In this section we will briefly review the chiral SU model used in the present investigation for the in-medium properties of heavy mesons. The chiral SU model is based on the broken scale invariance and nonlinear realization of chiral symmetry [35–40]. The model involves the Lagrangian densities describing, for example, 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 and and the dilaton field, , in isospin symmetric nuclear medium. We solve these coupled equations to obtain the density and temperature dependence of scalar fields and and the dilaton field, , in isospin symmetric nuclear medium [12]. The concept of broken scale invariance leading to the trace anomaly in (massless) QCD, , where is the gluon field strength tensor of QCD, is simulated in the effective Lagrangian at tree level [41] through the introduction of the scale breaking terms [12]. Within chiral SU model the scale breaking terms are written in terms of the dilaton field and also the scalar fields and . 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 and and the dilaton field, , and is given by the following equation [12]: where the value of parameter is 0.064 [10] and and denote the masses of pions and kaons and have values 139 and 498 MeV, respectively. and are the decay constants having values 93.3 and 122 MeV, respectively. The symbols , , and denote the nonstrange scalar-isoscalar field, strange scalar-isoscalar field, and the dilaton field, respectively. denotes the value of the dilaton field in vacuum. The vacuum values of , , and are −93.3, −106.6, and 409.8 MeV, respectively. Note that in above equation the gluon condensate is written considering finite quark masses. If we have massless QCD, then only first term written in terms of dilaton field contributes to the gluon condensates. Using the above equation we obtain the values of scalar gluon condensates at different values of densities and temperatures of the nuclear medium.

#### 3. QCD Sum Rules for Scalar , Vector , and Axial-Vector Mesons

In this section we will discuss the QCD sum rules [23, 24] which will be used later along with the chiral SU model for the evaluation of in-medium properties of scalar, vector, and axial-vector mesons. To find the mass modification of the above discussed heavy mesons we will use the two-point correlation function , In the above equation denotes the isospin averaged current, is the four coordinate, is four momentum, and 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 respectively. Note that in above equations denotes the light or quark whereas 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 and 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 [19]. 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 [18]. For example, in [18] the mass splitting between pseudoscalar and mesons was investigated using the even and odd QCD sum rules whereas in [19, 23, 24] the mass-shift of 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; that is, we can write where In the above equation denotes the isospin and spin averaged static nucleon state with the four-momentum . The state is normalized as . The third term, in (5) 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 and mesons will be evaluated through the temperature dependence of scalar fields , , and . The temperature dependence of scalar fields , , and modify the nucleon properties in the medium and these modified nucleons further modify the in-medium properties of and mesons at finite temperature and density. In literature the properties of kaons and antikaons, mesons and charmonium had been studied at finite temperature of the nuclear matter using the above mentioned scalar fields , , and [7, 8, 12, 13, 34].

As discussed in [24], in the limit of the -vector , the correlation functions can be related to the and scattering -matrices. Thus we write [24] In above equation and are the scattering lengths of and , respectively. Similarly, we can also write the scattering matrix corresponding to ( is a scalar meson) in terms of the scattering lengths [23], Near the pole positions of the scalar, vector, and axial-vector mesons the phenomenological spectral densities can be parameterized with three unknown parameters , and ; that is, we write [19, 23, 24] The term denoted by represents the continuum contributions. The first term denotes the double-pole term and corresponds to the on-shell effects of the -matrices, Now we will write the relation between the scattering length of mesons and their in-medium mass-shift. For this first we note that the shift of squared mass of mesons can be written in terms of the parameter appearing in (9) through relation [17], where in the last term we used (10). The mass shift is now defined by the relation The second term in (9) denotes the single-pole term and corresponds to the off-shell (i.e., ) effects of the -matrices. The third term denotes the continuum term or the remaining effects, where is the continuum threshold. The continuum threshold parameter define the scale below which the continuum contribution vanishes [42].

It can be observed from (11) and (12) that if we want to find the value of mass shift of mesons then we first need to find the value of unknown parameter . For this we proceed as follows: we note that in the low energy limit, , the is equivalent to the Born term . We take into account the Born term at the phenomenological side, with the constraint Note that in (13) the phenomenological side of scattering amplitude for is not exactly equal to Born term but there are contributions from other terms. However, for , on left should be equal to on right side of (13) and this requirement results in constraint given in (14). As we will discuss below the constraint (14) help in eliminating the parameter and scattering amplitude will be function of parameters and only. The Born terms to be used in (13) for scalar, vector, and axial-vector mesons are given by following relations [23, 24]: In the above equations , , and are the coupling constants. is the mass of the hadron; for example, corresponding to charm mesons we have and , whereas corresponding to bottom mesons we have the hadrons and . Corresponding to charm mesons we take the average value of the masses of and and it is equal to 2.4 GeV. For the case of mesons having bottom quark, , we consider the average value of masses of and and it is equal to 5.7 GeV.

Now we write the equation for the Borel transformation of the scattering matrix on the phenomenological side and equate that to the Borel transformation of the scattering matrix for the operator expansion side. For the scalar meson, , the Borel transformation equation is written as [23] where .

Note that in (16) we have two unknown parameters and . We differentiate (16) w.r.t. 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 and . Same procedure will be applied to obtain the values of parameters and corresponding to vector and axial-vector mesons. For vector meson, , the Borel transformation equation is given by [24] where . For the axial-vector mesons, , the Borel transformation equation is given by [24] where . In the above equations .

As discussed earlier, in determining the properties of hadrons from QCD sum rules, we will use the values of quark and gluon condensates as calculated using chiral SU model. Any operator on OPE side can be written as [17, 42, 43] In the above equation, gives us the expectation value of the operator at finite baryonic density. The term stands for the vacuum expectation value of the operator, gives us the nucleon expectation value of the operator, and denotes the contribution from the pion bath at finite temperature. Consider and are the thermal Boson and Fermion distribution functions. Within the chiral SU model the quark and gluon condensates can be expressed in terms of scalar fields , , and . 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 will 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 [42]. Thus within chiral SU model, we can find the values of at finite density of the nuclear medium and hence can find using The quark condensate, , 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, and , which are proportional to the nonstrange scalar field within chiral SU model. Considering equal mass of light quarks, and GeV, we can write, The condensate is given by the following [44]: Also we write [44] As discussed above the quark condensate, , can be calculated within the chiral SU model. This value of can be used through (23) and (24) to calculate the value of condensates and within chiral SU model. The value of condensate is equal to [44].

At finite temperature and zero baryon density we can write the expectation values of quark condensates and scalar gluon condensates as [42] respectively. In (25) and (26), . From (25) and (26) we observe that the contribution from pion bath to the expectation values of operator arises only at finite temperature. Note that in (16), (17), and (18) we need the nucleon expectation values of various condensates which can be evaluated in general using (20). In (1), (22), (23), and (24) the values of condensates are given at finite value of baryonic density. To find the corresponding nucleon expectation values of various condensates we use the values of condensates at finite baryonic density from (1), (22), (23), and (24) in (20). The equation (20) is then further used in (16), (17), and (18) for calculation of medium modifications of mesons.

It may be noted that the QCD sum rules for the evaluation of in-medium properties of scalar mesons, , vector mesons, , and axial-vector mesons, , can be written by replacing masses of charmed mesons, , , and , by corresponding masses of bottom mesons , , and in (16), (17), and (18), respectively. Also the bare charm quark mass, , will be replaced by the mass of bottom quark, .

#### 4. Results and Discussions

In this section we will present the results of our investigation of in-medium properties of scalar , vector , and axial-vector mesons . The nuclear matter saturation density used in the present investigation is 0.15 fm^{−3}. The values of various coupling constants are approximated to 6.74 [23]. The coupling constants are approximated to 3.86 [24]. The masses of mesons , , , , , and to be used in present investigation are 2.355, 5.74, 2.01, 5.325, 2.42, and 5.75 GeV, respectively. The values of decay constants , , , , , and are 0.334, 0.28, 0.270, 0.195, 0.305, and 0.255 GeV, respectively. The values of threshold parameters, , corresponding to , , , , , and mesons are 8, 39, 6.5, 35, 8.5, and 39 GeV^{2}, respectively [23, 24]. As discussed earlier the mass-shift of scalar, vector, and axial-vector mesons is calculated through the parameter which is related to scattering length through (10) to (12). This parameter , for example, for , is calculated by solving the coupled equations as discussed after (16) 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 , , and . The various coupling constants listed above, the decay constants of and mesons, and the threshold parameter are not subjected to the medium modifications. In the present work we will show the variation of mass shift as a function of squared Borel mass parameter, . The Borel window is chosen such that there is almost no change in the mass of and mesons w.r.t variation in Borel mass parameter. For the charmed scalar, , vector, , and axial-vector, mesons Borel windows are found to be (6.1–7.4), (4.5–5.4), and (6.5–7.6) GeV^{2}, respectively. The Borel windows for bottom scalar, , vector, , and axial-vector, mesons are (33–39), (22–24), and (34–37) GeV^{2}, respectively.

In the present work we are studying the in-medium masses of scalar, vector, and axial-vector mesons using QCD sum rules and chiral SU model. Since we are evaluating the quark and gluon condensates within the chiral SU model through the modification of scalar fields and and the scalar dilaton field , so first we will discuss in short the effect of temperature and density of the medium on the values of scalar fields and and the scalar dilaton field . We observe that as a function of density of the nuclear medium the magnitude of scalar fields decreases. For example, at nuclear saturation density, , the drop in magnitude of the scalar fields and and the dilaton field is observed to be and MeV, respectively. At baryon densities, , these values changes to and MeV, respectively.

At zero baryon density, we observe that the magnitude of the scalar fields and and the dilaton field decreases with the 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 nonzero values of scalar fields at finite temperature and zero baryon density of the medium is the formation of baryon-antibaryon pairs [36, 45, 46]. At finite baryon densities, the magnitude of the scalar fields increases with increase in the temperature of symmetric nuclear medium. This also leads to the increase in the masses of the nucleons with the temperature of the nuclear medium for finite baryon densities [12]. At nuclear saturation density, , the magnitude of the scalar fields, and , and the dilaton field increases by and MeV, respectively, as we move from = 0 to = 150 MeV, respectively.

In Figures 1 and 2 we show the variation of the light scalar quark condensate, , given by (22) and the scalar gluon condensate, , given by (1), respectively, as a function of density of the symmetric nuclear medium. We show the results for temperatures, , and 150 MeV, respectively. From (22) we observe that the value of the scalar quark condensate is directly proportional to the scalar-isoscalar field, . Therefore, the behavior of the as a function of temperature and density of the nuclear medium will be the same as that of field. For given value of temperature of the nuclear medium the magnitude of the light quark condensate decreases with the increase in the density. For example, at = 0, the values of light quark condensate are observed to be and GeV^{3} at baryon densities, and , respectively. At temperature, = 150 MeV, these values of quark condensates changes to and GeV^{3}, respectively. At zero baryon density the magnitude of the decreases with the increase in the temperature of the nuclear medium. At , the values of are observed to be , , , and GeV^{3} at temperatures = 0, 50, 100, and 150 MeV, respectively.

From Figure 2, we observe that the values of the scalar gluon condensates decrease with the increase in the density of the nuclear medium. At baryon density, , the values of are observed to be GeV^{4}, GeV^{4}, GeV^{4}, and 1.92 GeV^{4} for temperatures, = 0, 50, 100, and 150 MeV, respectively. For the same values of the temperature, in the absence of finite quark masses, the values of are observed to be 2.269 × 10^{−2} , 2.2771 × 10^{‒2} , 2.2857 GeV^{4}, and for . For baryon density, , the values of are given as (2.06 × 10^{−2} ), 1.74612 × 10^{−2} (2.0713 × 10^{−2} GeV^{4}), 1.7656 × 10^{−2} (2.094 × 10^{−2} ), and 1.78 × 10^{−2} (2.112 × 10^{−2} GeV^{4}) for values of temperature, = 0, 50, 100, and 150 MeV, respectively, for the cases of the finite (zero) quark masses in the trace anomaly.

It may be noted that in the above discussion the finite temperature effects of the nuclear medium on the values of quark and gluon condensates are evaluated through the temperature dependence of scalar fields and and the dilaton field . In literature the scalar quark and gluon condensates at finite temperature are evaluated due to contribution from pion bath using (25) and (26), respectively [42]. Using and = 0.005 [42] we calculate the quark and gluon condensates at finite temperature and zero baryon density using (25) and (26), respectively. The values of scalar quark condensates are observed to be , , and at temperatures = 50, 100, and 150 MeV, respectively. These values of scalar quark condensates can be compared to the values , , and at = 50, 100, and 150 MeV calculated in our present approach where finite temperature effects were considered through scalar fields , , and . Similarly, the values of scalar gluon condensates calculated using (26) are found to be , , and at = 50, 100, and 150 MeV, respectively. However, for zero baryon density, using the chiral SU model the values of scalar gluon condensates for finite (zero) quark mass term are observed to be (), (), and 1.94456 × 10^{−2} (2.3437 × 10^{−2}) at temperatures = 50, 100, and 150 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 agrees, for example, with [47], 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 [47]. 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 [48, 49].

Note that in the above discussion of scalar gluon and quark condensates, calculated using (1) and (22), respectively, we considered the vacuum values of decay constants and as well as masses and of pions and kaons, respectively. Now we will discuss the effect of medium modified values of , , , and on the scalar condensates. In the chiral effective model the pion and kaon decay constants are related to the scalar fields and through relations [5, 12] respectively. From the above relations it is clear that the medium dependence of scalar fields and can be used to study the density and temperature dependence of decay constants of pions and kaons. For example, using (27), at zero temperature the values of pion decay constant are observed to be , , and MeV at baryonic densities = 0, , and 2, respectively. The values of kaons decay constants, using (28), are observed to be , , and MeV at = 0, , and 2, respectively. In [50] the masses of pions were calculated in linear density approximation using the chiral perturbation theory. At baryon density = and zero temperature the mass of pion changes from the vacuum value MeV to MeV. The properties of kaons and antikaons had been investigated in the literature using the chiral model [9] and coupled channel approach [51–54]. In the present work, to study the effect of medium modified masses of kaons on the values of quark and gluon condensates we will use the chiral SU model [9]. In the nuclear medium the kaons feel repulsive interactions and their in-medium mass increases as a function of density whereas the antikaons feel attractive interactions and their in-medium masses drop as we move to higher baryon density. For the present purpose considering the average mass of kaons and antikaons, the in-medium mass at temperature is observed to be 494, 488.88, and 466.08 MeV at = 0, , and , respectively. Taking into account the above discussed in-medium properties of , , , and , at baryonic density and zero temperature, the values of scalar quark and gluon condensates are observed to be and , respectively. These values of scalar quark and gluon condensates can be compared to and , respectively which were calculated without the medium modification of , , , and . We conclude that the medium modification of , , , and causes more decrease in the values of scalar quark and gluon condensates at finite baryonic density.

Now we will calculate the in-medium masses of scalar, vector, and axial-vector mesons using the values of condensates from chiral SU model. In Figure 3, the subfigures (a), (c), and (e) show the variation of the mass shift of scalar mesons as a function of square of the Borel mass parameter, , at nuclear matter densities , , and , respectively. The subfigures (b), (d), and (f) show the variation of the scalar mesons as a function of square of the Borel mass parameter, , at nuclear matter densities , , and , respectively. In each subplot we have shown the results at temperatures = 0, 50, 100, and 150 MeV. We observe that for scalar mesons, , at temperature, = 0, the values of mass shift are found to be 76, 114, and 148 MeV at baryon densities , , and , respectively. For temperature = 50 MeV the values of mass shift are observed to be 71, 109, and 144 MeV at baryon densities , , and , respectively. At temperature = 100 MeV the above values of mass shift are observed to be 66, 103, and 139 MeV, whereas at = 150 MeV the values of mass shift changes to 58, 94, and 131 MeV at baryon densities , , and , respectively. From the above discussion we conclude that for a given value of temperature the mass shift of scalar mesons, , increases as a function of density of the nuclear medium. On the other hand as a function of temperature of the nuclear medium, for a constant value of density, the mass shift of scalar mesons, , decreases.

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As we can see from Figure 3 the values of mass shift of scalar mesons also increases as a function of density of the nuclear medium. At temperature, , the values of mass shift are observed to 224, 334, and 420 MeV at baryon densities , , and , respectively. At temperature = 50 MeV the above values of mass shift are found to be 211, 321, and 413 MeV at baryon densities , , and , respectively. For temperature = 100 MeV the values of mass shift are found to be 196, 302, and 399 MeV whereas at = 150 MeV these values of mass shift changes to 171, 274, and 372 MeV at , , and , respectively. For a given value of nuclear matter density the values of mass shift for the mesons are found to decrease with the increase in the temperature of the medium.

We also calculate the values of scattering lengths of scalar mesons using (10) for different values of density and temperature of the medium. At temperature = 0 and baryon densities, and , the values of scattering lengths for scalar mesons () are observed to be 1.42 (5.05) and 0.70 (2.41) fm, respectively. For temperature = 100 MeV and baryon densities, = and , the values of scattering lengths 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 (10). In this equation we have the parameter which is directly proportional to the scattering length of mesons. As discussed earlier the value of parameter is evaluated by solving simultaneously (16) and the equation obtained by differentiate (16) w.r.t. . The magnitude of parameter is found to decrease as we move from low to higher value of baryonic density or from zero to finite value of temperature of the medium. This behavior of parameter as a function of density and temperature of the medium results in the similar changes in the values of scattering lengths of scalar mesons.

Figure 4 shows the variation of the mass shift of vector mesons and as a function of square of the Borel mass parameter. Here also we have shown the results at nuclear matter densities , , and . We observe that at nuclear matter density the values of mass shift for vector mesons, are observed to be −76, −71, −65, and MeV at temperatures = 0, 50, 100, and 150 MeV, respectively. For baryon density the values of mass shift are found to be , and MeV, whereas for the values of the mass shift changes to , and MeV at temperatures = 0, 50, 100, and 150 MeV, respectively. Note that for a given value of density, the magnitude of the mass shift of vector mesons decreases as a function of temperature of the nuclear medium. On the other hand as a function of density of the medium the magnitude of the mass shift of the vector mesons increases. For nuclear matter saturation density the values of mass shift for the mesons are found to be , −344, −311, and −275 MeV at temperatures = 0, 50, 100, and 150 MeV, respectively. At baryon density the above values of mass shift changes to , −534, −498, and −447 MeV at temperatures = 0, 50, 100, and 150 MeV, respectively, whereas for baryon density the above values of mass shift are found to be , −689, −662, and −609 MeV at temperatures = 0, 50, 100, and 150 MeV, respectively. At temperature = 0 and baryon densities, = and , the values of scattering lengths for vector mesons are observed to be −1.31 (−7.72) and −0.54 (−3.58) fm, respectively. For temperature = 100 MeV and baryon densities, = and , the values of scattering lengths for are observed to be −1.12 (−6.72) and −0.51 (−3.54) fm, respectively. We observe that the magnitude of the scattering lengths of vector mesons decreases in moving from low to higher value of density or temperature of the medium.

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In Figure 5, for given values of temperatures and densities we have shown the variation of mass shift of axial-vector mesons and as a function of square of the Borel mass parameter. We observe that at baryon density, , the values of mass shift for axial-vector meson are observed to be , 69, 63, and 55 MeV at temperatures, = 0, 50, 100, and 150 MeV. At baryon density the values of mass shift are found to be 108 (131), 104(128), 97(123), and 87(113) at temperatures, = 0, 50, 100, and 150 MeV. For the axial-vector meson at baryon density the values of mass shift are found to be 267 (396), 251(381), 233(357), and 203(324) MeV at temperatures, = 0, 50, 100, and 150 MeV, respectively. At baryon density the values of mass shift are found to be 492, 485, 467, and 434 MeV at temperatures, = 0, 50, 100, and 150 MeV, respectively. The values of scattering lengths for axial-vector mesons () at temperature = 0 and baryon densities, = and , are observed to be 1.38 (6.02) and 0.62 (2.83) fm, respectively. For temperature = 100 MeV and baryon densities, = and , the values of scattering lengths for changes to 1.19 (5.25) and 0.58 (2.69) fm, respectively. From the above discussions we observe a positive value of mass shift for scalar (, ) and axial-vector mesons (, ) in the nuclear medium. However, the values of mass shift for vector mesons (, ) are found to be negative. It means the masses of above scalar and axial-vector mesons in the nuclear medium may be large compared to the value in free space and this may lead to a decrease in the yield of these mesons in heavy-ion collisions.

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Now we will discuss the effect of different individual condensates on the in-medium modification of scalar (, ), vector (, ), and axial-vector (, ) mesons. In Figures 6, 7, and 8 we compare the contributions of individual condensates to the mass shift of scalar mesons, , vector mesons, , and axial-vector mesons, , respectively. The subplots (a), (c), and (e) show the results at temperature = 0, whereas the subplots (b), (d), and (f) are plotted for temperature = 100 MeV. We have shown the results at baryon densities , , and . Note that in Figure 6 and in the subsequent figures of this paper the word “Total” is for the contribution of all condensates to the properties of mesons. Similarly, “Quark_{1}”, “Quark_{2}”, “Quark_{4,}” and “Gluon_{1}” are denoting the contribution of , , , and , respectively, to the in-medium properties of mesons.

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We observe that at temperature = 0; if we consider the contribution of scalar quark condensates, only then the values of mass shift for the scalar mesons are found to be and MeV at nuclear matter density = and , respectively. When we consider the individual contributions of , , and condensates then the values of mass-shift at () and = 0 are observed to be 4.01(16.07), 12.70(37.87), and 6.68(25.96) MeV, respectively. From above discussion we observe that the maximum contribution to the in-medium modification of scalar mesons is from light quark condensate . Note that leaving the quark condensate, , all the other condensates have been evaluated within chiral SU model in the present investigation. So in Figure 6 we also show the variation of the mass shift of scalar mesons as a function of squared Borel mass parameter when we neglect the contribution of condensate. We observe that if we neglect the contribution of condensate then the values of mass shift are found to be 78.70 and 158.99 MeV at densities and , respectively. Note that considering the contribution of all condensates, at temperature = 0, the values of mass shift were 75.88 and 148.05 MeV at densities and , respectively. So we observe that if we neglect the condensate then there is a percentage change of 4% and 7% in the mass shift of mesons at densities and , respectively. From Figure 7, we observe that for vector mesons, , at temperature = 0 and baryon density = (), the values of mass shift due to condensates , , , , and are observed to be −82.15(−151.15), 1.689(6.749), 18.95(57.16), −5.49(−6.16), and 3.54 (14.93) MeV, respectively. For temperature = 100 MeV the above values of mass shift at baryon density = () changes to −73.01 (−142.35), 1.689 (6.75), 17.85 (56.34), −4.53 (−5.43), and 3.43 (14.64) MeV, respectively. For the axial-vector mesons, , the values of mass shift due to individual condensates , , , , and are observed to be 79.31(152), 0.1516(0.607), −8.77(−22.63), 8.27(21.76), and 2.34 (10.08) MeV, respectively. For temperature = 100 MeV the above values of mass shift at baryon density = () changes to 70 (145), 0.151 (0.607), −7.99(−22.04), 7.58 (21.26), and 2.23 (9.81), respectively. In Figures 9, 10, and 11 we have shown the contributions of individual condensates to the mass shift of scalar mesons, , vector mesons, , and axial-vector mesons, , respectively. The subplots (a), (c), and (e) show the results at temperature = 0, whereas the subplots (b), (d), and (f) are plotted for temperature = 100 MeV. We have shown the results at baryon densities , , and . In Tables 1, 2, and 3 we have tabulated the values of mass shift for scalar mesons, , vector mesons, and axial-vector mesons, , respectively. The values of mass shift have been given at baryon densities and and temperatures = 0 and 100 MeV.

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In [23], the properties of scalar mesons and had been studied using QCD sum rules and the observed values of mass shift at nuclear saturation density were and MeV, respectively. The properties of mesons had also been studied in [25] using coupled channel approach. An extra widening from the already large width of the resonance in free space was observed for the meson. The properties of vector mesons ( and ) and axial-vector ( and ) had been studied using QCD sum rules in [24]. For vector mesons and the values of mass shift were and MeV, respectively, whereas for axial-vector mesons and these values change to and MeV, respectively. The observed positive values of mass shift for and mesons in our present investigation and also in earlier investigations indicate that these mesons feel repulsive interactions in the nuclear medium and their in-medium mass increases as function of baryonic density. This means the chances of decay of higher charmonium states to these heavy charmed mesons pairs are suppressed and hence these mesons may not cause a decrease in the production of mesons in heavy-ion collisions. However, for the vector mesons we observe the negative values of mass shift and hence we conclude that they feel attractive interactions in the nuclear medium. The decrease in the mass of vector mesons may cause the decay of higher charmonium states to pairs and hence it may be a cause of suppression in heavy-ion collision experiments.

#### 5. Summary

In the present paper we investigated the mass modifications of scalar mesons , vector mesons and axial-vector mesons at finite density and temperature of the nuclear medium. We used QCD sum rules along with chiral SU model to investigate the properties of above mentioned mesons. Using chiral SU model we found the values of quark and gluon condensates at finite density and temperature of the medium which were further used within QCD sum rules to find the in-medium masses of scalar, vector, and axial-vector mesons. We observed a positive value of mass shift for the scalar mesons ( and ) and axial-vector mesons ( and ); that is, their in-medium mass was found to be more than the vacuum mass. For a constant value of temperature the values of mass shift for these scalar and axial-vector mesons are found to increase as we increase the density of the nuclear medium. On the other hand for a constant value of density the temperature of the medium causes a decrease in the values of mass shift of scalar and axial-vector mesons. The vector mesons and are found to have negative values of mass shift. It means their in-medium masses are small as compared to vacuum masses. For a constant value of density, as a function of temperature, the magnitude of mass shift of and mesons decreases. However, for a constant value of temperature, as a function of density the magnitude of mass shift of vector mesons and are found to increase. We have also investigated the effects of individual terms on the mass shift of mesons. It was found that the scalar quark condensates, , have maximum contribution to the in-medium modification of scalar, vector, and axial-vector mesons. The effects of density and temperature of the medium on the scattering lengths of scalar, vector, and axial-vector mesons were also investigated. We observed the positive value of scattering lengths for scalar and axial-vector mesons, whereas 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 suppression in the compressed baryonic matter experiment at FAIR, GSI.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author is thankful to Amruta Mishra for fruitful discussions on chiral SU model and QCD sum rules. The financial support from the Department of Science and Technology (DST), India, for research project under fast track scheme for young scientists (SR/FTP/PS-209/2012) is gratefully acknowledged.