74 JIN Meng and LI Jia-Rong Vol. 39 From the path integral principle, the partition function can be written in the following form [13] = [d ][d ][d][d

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1 Commun. Theor. Phys. (Beijing, China) 39 (23) pp. 73{77 c International Academic Publishers Vol. 39, No. 1, January 15, 23 Inuence of Vacuum Eect on Behavior of Hot/Dense Nulcear Matter JIN Meng y and LI Jia-Rong Institute of Particle Physics, Huazhong Normal University, Wuhan 4379, China (Received April 23, 22) Abstract From the Lagrangian density of QHD-I model, we study the properties of hot and dense nuclear matter when the zero-point correction is considered and nd that the inuence of zero-point correction is very important at high temperatrure. PACS numbers: f, 11.1.Wx, 64.7.Fx Key words: QHD-I model, nuclear matter, vacuum eect 1 Introduction The studies on the properties of hot and/or dense nuclear matter have received a remarkable progress recently. In the aspect with the phase transition of nuclear matter, the liquid-gas phase transition at low temperature is con- rmed by dierent theoretical model. [1;3] Furthermore, a new rst order phase transition at high temperature is also found in QHD-I model. [4] Another important aspect of the investigations on the hadron level is the eective hadronic mass in hot/dense medium. It is widely accepted that the eective mass will decrease with the increasing temperature (T ) or density () in spite of the quantitatively various or even opposite conclusion obtained from dierent models. [5;9] Some authors combine the studies of these two topics and nd that the eective hadronic mass do not decrease monotonically with T or when the phase transition is considered. [1 11] It is worth while to point out that most of these existent results are obtained in the mean-eld approximation (MFA). During the calculation of thermodynamics of a system in MFA, one often encounters a divergent term which similar to the zero-point energy of vacuum in the zero-temperature eld theory. Unfortunately, it is dropped simply by the regularization procedure as done in the zero-temperature case. This is imcomplete when we handle a hot system in nite-temperature eld theory for that the zero-point correction will play a role due to the thermal eect. The nite contribution from vacuum energy should be departed by proper renormalization. Through calculation we nd this vacuum contribution is not very large at low temperature. This may be the reason why it is neglected by many existent works. With the progress of heavy-ion collision experiments, much attention is paid to the properties of hadronic matter at high temperature. The consideration of vacuum correction in a hot nuclear system becomes important and necessary. From the QHD-I model, we discuss the inuence of zero-point correction to the eective nucleon mass, pressure and other quantities of a nuclear system in a wide temperature range. The paper is arranged as follows. In the second section, we analyze the thermodynamics foundation of QHD-I model from path integral. From the numerical computation in the third section, the inuence of zero-point correction on many properties of nuclear matter, such as eective mass (M ), energy density ("), pressure (P ), heat capacity (C V ), are discussed. The last section is the results and discussions. 2 Thermodynamics of QHD-I Model The QHD-I model was rst presented by J.D. Walecka in [12] Under the MFA, it gives a good description about the ground properties and the EOS of nuclear matter. As the basic discussion later, the thermodynamics foundation of the model would be given at rst place. The Lagrangian density of QHD-I model is L = ; M) ; m 2 2 ) ; 1 4 F F m2!v V ; g v V + g s + L (1) where, F V ;@ V, L is the renormalization counterterm,, V,, and are the elds of,!, anti-nucleon (n) and nucleon (n) separately, M, m s, and m v are the mass of nucleon, meson and! meson, and g s, g v are the coupling constants correspondingly. Under the formalism of nite-temperature eld theory, we consider the partition function of grand canonical ensemble = Tre ;( ^H; ^N) (2) where ^H and ^N are the Hamiltonion and baryon number operator of the system, is the chemical potential, and (= 1=T ) is the inverse of temperature. The project supported by National Natural Science Foundation of China under Grant No y jinm@iopp.ccnu.edu.cn

2 74 JIN Meng and LI Jia-Rong Vol. 39 From the path integral principle, the partition function can be written in the following form [13] = [d ][d ][d][dv ] n exp d d 3 x(l E + o 4 ) (3) where L E is the Lagrangian density (1) in Euclidean space. With the aid of MFA, the partition function can be solved easily. The fundamental ideal of MFA is that when the source of the eld equation gets large enough, the meson eld operators can be replaced by their expectation values, while the nucleon eld operator keeps unchanged. After replacing the quantum operators of mesons with their expectation values and setting ^! h ^i = ^V! h ^V i = V (4) M = M ; g s = ; g V V (5) the function integral of Eq. (3) can be rewritten into a Gaussian-type one, = [d n ][d ] exp d d xh 3 + M ) ; 4 ; 1 2 m2 s m2 v V 2 io : (6) Removing the part which does not relate to the nucleon eld out o the integral, the part involving the nucleon eld can be integrated out by the Gaussian functional integration formula. From the standard procedure of the imaginary-time formalism of nite-temperature eld theory, the thermodynamic potential can be given = ;T ln = ; (2) 3 d 3 k[! + T ln(1 + e ;(! ; ) ) + T ln(1 + e ;(! + ) )] m2 s 2 ; 1 2 m2 vv 2 (7) where! = p k 2 + M 2. (= 4) is the spin{isospin degeneracy of nuclear matter. The rst term in the square bracket of Eq. (7) is the vacuum contribution of hot/dense nuclear matter. It is similar to the zero-point energy in the zero-temperature eld theory. In MFA, this term is dropped by regularization procedure. The other thermodynamics quantities, such as pressure P and energy density ", can be deduced from thermodynamic potential through general thermodynamics relations P MFA = ; 1 2 m2 s m2 vv 2 + 3(2) 3 d 3 k 2 k p k2 + M (n 2 F + n F ) (8) " MFA = 1 2 m2 vv m2 s 2 + (2) 3 d 3 k p k 2 + M 2 (n F + n F ) (9) where the distribution functions are 1 n F = e (! ; ) + 1 n 1 F = e (! + ) + 1 : (1) The mean elds and V are determined by minimizing the thermodynamic potential with respect to the two elds respectively and then equation (5) is changed to M = M ; g2 s m 2 s (2) 3 d 3 M k p k2 + M (n 2 F + n F ) (11) = ; g2 v m 2 v (2) 3 d 3 k(n F ; n F ) : (12) Equation (11) is the self-consistent equation for the effective nucleon mass. With Eqs. (8) (12), we can discuss the properties of nuclear matter in MFA. In the MFA, the contribution of vacuum energy is dropped simply. This is unreasonable when we treat the problem at nite temperature. The proper method is that this term should be regularized rst and then the counterterm be introduced to cancel the divergence. After these steps, we get the nite result which depends on the temperature and chemical potential. It stands for the contribution arising from the change of vacuum in nuclear medium at the nite temperature. We dene the contribution of zero-point correction (PC) in thermodynamic potential as = ; (2) 3 d 3 k p k 2 + M 2 : (13) The contribution of the vacuum part at zero temperature must be dropped rst, = ; (2) 3 d 3 k[ p k 2 + M 2 ; p k 2 + M 2 ] : (14) We nd this contribution is identical to that of summing over the baryon tadpole diagrams of relativistic Hartree approximation (RHA). We follow the same method as in Ref. [14] and get the nite result for the vacuum correction in hot nuclear system, PC = ; 1 h 4 2 M 4 ln M M + M 3 (M ; M ) ; 7 2 M 2 (M ; M ) M(M ; M ) 3 ; (M ; M ) 4i : (15) Now, the thermodynamic potential and other quantities including the vacuum corrections should be changed into the following expression = MFA + PC (16) P = P MFA ; PC (17) " = " MFA + PC : (18)

3 No. 1 Inuence of Vacuum Eect on Behavior of Hot/Dense Nulcear Matter 75 The eld will be changed by the variation of vacuum, but the V eld keeps unchanged. So the expression for is not changed whereas the self-consistent equation for M is converted into M = M ; g2 s m 2 s (2) 3 M d 3 k p k2 + M 2 + g2 s m 2 s 2 h M 3 ln M M ; M 2 (M ; M) ; 5 2 M(M ; M) 2 ; 11 6 (M ; M) 3i : (19) Equations (12) and (19) form another set of self-consistent equations which include the contribution of the zero-point 3 The Inuence of Vacuum Correction on Properties of Hot and/or Dense Nuclear Matter Before the calculation, we give the parameters in QHD- I model as in Ref. [6], through tting the saturation properties with energy density "= B ; M = ;15:75 MeV and the Fermi momentum k F = 1:3 fm ;1 : Parameter g 2 s g 2 v ms (MeV) mv (MeV) M (MeV) MFA 19:6 19: MFA+PC 54:3 12: With the parameters in the table, we solve the selfconsistent Eq. (11) or (19) in the whole range of temperature through the following steps regions of T and in which three values of M correspond to one. This non-trivial feature is the multi-solution phenomenon. [1 11] 1. Give a set of value to T and 2. Solve the self-consistent Eq. (11) or (19) to give the value of M 3. Put the values of (T,, M ) into Eq. (12) to determine the chemical potential 4. Change the value of, repeat the above steps and get the dierent values of M and corresponding to various T. It is necessary to point out that the data obtained from this approach is the same as the one that was given through xing T and in the rst step. For comparison, we include the results of MFA (without the zero-point correction) which have been obtained in many other works. At rst, we give the variation curve of M vs. at dierent temperatures, as shown in Fig. 1. Fig. 2 The eective nucleon mass vs. chemical potential for various values of temperature including zero-point From Eq. (9), the variationcurve of energy density versus chemical potential at dierent temperatures is given in Fig. 3. The multiple values of energy density are found both at low temperatures and high temperatures. Fig. 1 The eective nucleon mass vs. chemical potential for various values of temperature in MFA. From Fig. 1, we see the eective nucleon mass decreases with the increase of T or. But there are two Fig. 3 The energy density vs. chemical potential for various values of temperature in MFA.

4 76 JIN Meng and LI Jia-Rong Vol. 39 Fig. 4 The energy density vs. chemical potential for various values of temperature including the zero-pint In the high temperature region, the similar feature to the low temperature phase transition is found in Fig. 6. This indicates that there is a rst order phase transition around this region. [4] When the phase transition takes place, the heat capacity of the system appears a discontinuous property. The heat capacity can be obtained by the following C V = V where " is the energy density of the system and V is the volume. From Eq. (9) of " and the self-consistent Eq. (11) of M, C V can be obtained and its variation curves versus temperature are given in Figs. 7 and 8 (the solid line). The variation curve of pressure vs. baryon number density is also given, as shown in Figs. 5 and 6 (the solid line). Fig. 7 The heat capacity Cv vs. T at = 914 MeV. zero-point Fig. 5 The pressure P vs. B at low temperatures. zero-point Fig. 8 The heat capacity Cv vs. T at = MeV. zero-point Fig. 6 The pressure vs. B at high temperatures. The solid (dashed) line denotes the result without (with) zeropoint correction, where the temperatures are 15 MeV, 195 MeV, and 2 MeV from top to bottom for the dashed line, respectively. Figure 5 is the well-known phase transition diagram at low temperatures [1] and gure 6 depicts the variation of P vs. tot at high temperatures, where tot is dened as follows: tot = (2) 3 d 3 k(n F + n F ) : (2) Figure 7 is given at the xed chemical potential = 914 MeV and the gure 8 is given at the xed chemical potential = MeV. From these two gures, we can see that the heat capacity peaks present at T 15 MeV and T 186 MeV. The occurrence of peaks indicates that there are phase transitions in these two regions. Now, we are in the position to discuss the eect of vacuum Following the steps in Sec. 2 and using Eq. (19), we give out the variation curve of M versus in Fig. 2.

5 No. 1 Inuence of Vacuum Eect on Behavior of Hot/Dense Nulcear Matter 77 The multi-solution phenomenon remains in lowtemperature region but disappears when the temperature is high. This indicates that vacuum correction is small at low temperatures, but becomes very remarkable at high temperatures. Comparing the curve of T = 2 MeV in Figs. 1 and 2, M equals 2 MeV at zero chemical potential in MFA but nearly 8 MeV when the vacuum correction is included. The eect of zero-point correction heighten the eective nucleon mass in a very large scale at high temperature. The diagram of " vs. of a dierent temperatures including the zero-point correction is depicted in Fig. 4. It has been proved in Ref. [15] that the zero-point correction is negligible even at high density. From Figs. 3 and 4, our calculation supports this result at low temperature but we nd that it is not the case at high temperatures. Let us compare the curve of T = 2 MeV in Fig. 3 with that in Fig. 4, when equals MeV, the energy density is nearly 7 MeV/fm 3 in MFA but less than 2 MeV/fm 3 when including the zero-point The non-trivial feature of " is also found at low temperatures but disappears at high temperatures. This change is caused by the contribution from large vacuum correction at high temperatures. The variation curve of pressure vs. baryon number density at dierent temperatures including the zero-point correction is depicted by the dashed line of Figs. 5 and 6. At low temperatures (Fig. 5), the correction does not change the feature of the diagram but only causes the curve to drop slightly. This means the critical temperature in MFA is a little smaller than that when the zero-point correction is included. At high temperatures (Fig. 6), the feature of phase transition disappears and the curve becomes monotonic. This indicates that the phase transition at high temperaturedisappears when we consider the zeropoint The variation curve of heat capacity C V versus temperature T including vacuum correction is given by the dashed line in Figs. 7 and 8. We see that the peak of C V at low temperature region still exists (Fig. 7), but at high temperatures, the peak disappears and the curve of C V becomes continuous (Fig. 8). This phenomenon also indicates the disappearance of the phase transition at high temperatures. 4 Results and Discussions In summary, we discussed the properties of nuclear matter in hot and dense surroundings from QHD-I model without zero-point correction and with it, respectively. The results indicate that the zero-point correction is very important at high temperature. It changes many properties of nuclear matter to a large extend and even its phase structure at high temperatures. This correction is also considered in Ref. [16] when discussing the thermodynamics of Nambu{Jona{Lasinio (NJL) model. These results suggest that the zero-point correction cannot be neglected when we studying the behavior of nuclear matter at high temperatures. References [1] R.J. Furnstahl and B.D. Serot, Phys. Rev. C41 (199) 262. [2] M. Malheiro, A. Delno, and C.T. Coelho, Phys. Rev. C58 (1998) 426. [3] GUO Hua, LIU Bo, and M. Di Toro, Phys. Rev. C62 (2) [4] J. Theis, H. Stocker, and J. Polonyi, Phys. Rev. D28 (1983) [5] G.E. Brown and M. Rho, Phys. Rev. Lett. 66 (1991) 272. [6] H.C. Jean, J. Piekarwicz, and A.G. Williams, Phys. Rev. C49 (1994) [7] H. Shiomi and T. Hatsuda, Phys. Lett. B334 (1994) 281. [8] A. Delno, C.T. Coelho, and M. Malheiro, Phys. Rev. C51 (1995) [9] A. Bhattcharyya, J. Alam, S. Raha, and B. Sinha, Int. J. Mod. Phys. A12 (1997) [1] Bei-Wei hang, De-Fu Hou and Jia-Rong Li, Phys. Rev. C61 (2) [11] SHU Song, JIN Meng, and LI Jia-Rong, Mod. Phys. Lett. A16 (21) 221. [12] J.D. Walecka, Ann. Phys. 83 (1974) 491. [13] J.I. Kapusta, Finite-Temperature Field Theory, Cambrige Univ. Press (1989). [14] J.D. Walecka, Theoretical Nuclear and Subnuclear Physics, Oxford Univ. Press (1995). [15] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [16] T.M. Schwarz, S.P. Klevansky, and G. Papp, Phys. Rev. C6 (1999) 5525.