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Structure of the Vacuum in Nuclear Matter: a Nonperturbative Approach

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Structure of the Vacuum in Nuclear Matter: a Nonperturbative Approach PHYSICAL REVIEW C VOLUME 56, NUMBER 3 SEPTEMBER 1997 Structure of the vacuum in nuclear matter: A nonperturbative approach A. Mishra,1,* P. K. Panda,1 S. Schramm,2 J. Reinhardt,1 and W. Greiner1 1Institut fu¨r Theoretische Physik, J.W. Goethe Universita¨t, Robert Mayer-Straße 10, Postfach 11 19 32, D-60054 Frankfurt/Main, Germany 2Gesellschaft fu¨r Schwerionenforschung (GSI), Planckstraße 1, Postfach 110 552, D-64220 Darmstadt, Germany ~Received 3 February 1997! We compute the vacuum polarization correction to the binding energy of nuclear matter in the Walecka model using a nonperturbative approach. We first study such a contribution as arising from a ground-state structure with baryon-antibaryon condensates. This yields the same results as obtained through the relativistic Hartree approximation of summing tadpole diagrams for the baryon propagator. Such a vacuum is then generalized to include quantum effects from meson fields through scalar-meson condensates which amounts to summing over a class of multiloop diagrams. The method is applied to study properties of nuclear matter and leads to a softer equation of state giving a lower value of the incompressibility than would be reached without quantum effects. The density-dependent effective s mass is also calculated including such vacuum polarization effects. @S0556-2813~97!02509-0# PACS number~s!: 21.65.1f, 21.30.2x I. INTRODUCTION to be developed to consider nuclear many-body problems. The present work is a step in that direction including vacuum Quantum hadrodynamics ~QHD! is a general framework polarization effects. for the nuclear many-body problem @1–3#. It is a renormal- The approximation scheme here uses a squeezed coherent izable relativistic quantum field theory using hadronic de- type of construction for the ground state @12,13# which grees of freedom and has quite successfully described the amounts to an explicit vacuum realignment. The input here is properties of nuclear matter and finite nuclei. In the Walecka equal-time quantum algebra for the field operators with a model ~QHD-I! with nucleons interacting with scalar ~s! and variational ansatz for the vacuum structure and does not use vector ~v! mesons, it has been shown in the mean-field ap- any perturbative expansion or Feynman diagrams. We have proximation that the saturation density and binding energy of earlier seen that this correctly yields the results of the Gross- nuclear matter may be fitted by adjusting the scalar and vec- Neveu model @14# as obtained by summing an infinite series tor couplings @4#. This was first done by neglecting the Dirac of one-loop diagrams. We have also seen that it reproduces sea and is called the no-sea approximation. In this approxi- the gap equation in an effective QCD Hamiltonian @15# as mation, several groups have investigated the effects of scalar obtained through the solution of the Schwinger-Dyson equa- self-interactions in nuclear matter @5# and finite nuclei @6# tions for the effective quark propagator. We here apply such using a mean-field approach. a nonperturbative method to study the quantum vacuum in To include the sea effects, one does a self-consistent sum nuclear matter. of tadpole diagrams for the baryon propagator @7#. This de- We organize the paper as follows. In Sec. II, we study the fines the relativistic Hartree approximation. There have also been calculations including corrections to the binding energy vacuum polarization effects in nuclear matter as simulated up to two-loops @8#, which are seen to be rather large as through a vacuum realignment with baryon-antibaryon con- compared to the one-loop results. However, it is seen that densates. The condensate function is determined through a using phenomenological monopole form factors to account minimization of the thermodynamic potential. The properties for the composite nature of the nucleons, such contribution is of nuclear matter as arising from such a vacuum are then reduced substantially @9# so that it is smaller than the one- studied and are seen to become identical to those obtained loop result. Recently, form factors have been introduced as a through the relativistic Hartree approximation. In Sec. III, we cure to the unphysical modes, the so called Landau poles generalize the vacuum state to include s condensates also, @10#, which one encounters while calculating the meson which are favored with a quartic term in the s field in the propagator as modified by the interacting baryon propagator Lagrangian. The effective potential as obtained here includes of the relativistic Hartree approximation. There have been multiloop effects and agrees with that obtained through the also attempts to calculate the form factors by vertex correc- composite operator formalism @16#. The quartic coupling is tions @11#. However, without inclusion of such form factors chosen to be positive, which is necessary to consider vacuum the mean-field theory is not stable against a perturbative loop polarization effects from the s field. We also calculate the expansion. This might be because the couplings involved effective s mass arising through such quantum corrections as here are too large ~of order of 10! and the theory is not a function of density. The coupling here is chosen to give the asymptotically free. Hence nonperturbative techniques need value for the incompressibility of nuclear matter in the cor- rect range. Finally, in Sec. IV, we summarize the results obtained through our nonperturbative approach and present *Electronic address: [email protected] an outlook. 0556-2813/97/56~3!/1380~9!/$10.0056 1380 © 1997 The American Physical Society 56 STRUCTURE OF THE VACUUM IN NUCLEAR MATTER:... 1381 II. VACUUM WITH BARYON AND ANTIBARYON x~kW ! CONDENSATES cos 2 We start with the Lagrangian density for the linear Wa- U ~kW !5 u , r W Ir lecka model given as x~k! sW kˆ sin S • 2 D 1 5¯c~igm] 2M2g s2g gmv !c1 ]ms] s L m s v m 2 m x~kW ! 2sW kˆ sin • 2 1 2 1 2 1 2W 5 2 m s21 m vmv 2 vmnv , ~1! Vs~ k! vIs . ~8! 2 s 2 v m 4 mn x~kW ! cos S 2 D with For free massive fields cosx(kW)5M/e(kW) and sinx(kW)5ukWu/e(kW), v 5] v 2] v . ~2! mn m n n m with e(kW)5AkW 21M 2. The above are consistent with the equal-time anticommu- In the above, c, s, and v are the fields for the nucleon, s, m tator algebra for the operators c and ˜c as given by and v mesons with masses M, ms , and mv , respectively. We use the mean-field approximation for the meson fields c kW ,c† kW 5 kW2kW 5 ˜c kW ,˜c † kW . and retain the quantum nature of the fermion fields @14#. This @ Ir~ ! Is~ 8!#1 drsd~ 8! @ Ir~ ! Is~ 8!#1 amounts to taking meson fields as constant classical fields ~9! with translational invariance for nuclear matter. Thus we The perturbative vacuum, say uvac&, is defined through shall replace † cIr(kW)uvac&50 and ˜c Ir(kW)uvac&50. g s ^g s&[g s , ~3a! To include the vacuum-polarization effects, we shall now s → s s 0 consider a trial state with baryon-antibaryon condensates. We thus explicitly take the ansatz for the above state as g v ^g v &[g v dm05g v , ~3b! v m→ v m v m v 0 † where ^•••& denotes the expectation value in nuclear matter uvac8&5exp dkWf~kW!cIr~kW!ars˜cIs~2kW!2H.c. uvac& and we have retained the zeroth component for the vector F E G field to have a nonzero expectation value. [UFuvac&. ~10! The Hamiltonian density can then be written as Here a 5u† (sW kˆ)v and f (kW) is a trial function associated 5 1 1 , ~4! rs Ir • Is H HN Hs Hv with baryon-antibaryon condensates. We note that with the above transformation the operators corresponding to uvac8& with are related to the operators corresponding to uvac& through the Bogoliubov transformation 5c†~2iaW ¹W 1bM !c1g s¯cc, ~5a! HN • s W W ˆ W W dI~k! cosf ~k! 2sW •k sinf ~k! cI~k! 1 5 , 2 2 ˜ ˆ s5 mss , ~5b! S d ~2kW !D S sW k sinf ~kW ! cosf ~kW ! DS˜c ~2kW!D H 2 I • I ~11! 1 for the nucleon. 5g v c†c2 m2 v2. ~5c! Hv v 0 2 v 0 We then use the method of thermofield dynamics @18# developed by Umezawa to construct the ground state for The equal-time quantization condition for the nucleons is nuclear matter. We generalize the state with baryon- given as antibaryon condensates as given by Eq. ~10! to finite tem- perature and density as @13# † @ca~xW,t!,cb~yW,t!#15dabd~xW2yW!, ~6! uF~b!&5U~b!uvac8&[U~b!UFuvac&. ~12! where a and b refer to the spin indices. We may now write down the field expansion for the nucleon field c at time t50 The temperature-dependent unitary operator U(b) is given as given by @17# as @18# U~b!5exp@B†~b!2B~b!#, ~13! 1 W c~xW !5 @U ~kW !c ~kW!1V ~2kW!˜c ~2kW!#eik•xWdkW, ~2p!3/2 E r Ir s Is with ~7! B†~b!5 dkW@u ~kW,b!d† ~kW!dI† ~2kW! with cIr and ˜cIs as the baryon annihilation and antibaryon E 2 Ir Ir creation operators with spins r and s, respectively. In the ˜ ˜ above, Ur and Vs are given by 1u1~kW,b!dIr~kW!dI Ir~2kW!#. ~14! 1382 MISHRA, PANDA, SCHRAMM, REINHARDT, AND GREINER 56 The underlined operators are the operators corresponding to contributions to the entropy density from s and v mesons, the doubling of the Hilbert space that arises in thermofield respectively.
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