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 1997͒ 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 mass is also calculated including such vacuum polarization effects. ͓S0556-2813͑97͒02509-0͔
PACS number͑s͒: 21.65.ϩf, 21.30.Ϫx
I. INTRODUCTION to be developed to consider nuclear many-body problems. The present work is a step in that direction including vacuum Quantum hadrodynamics ͑QHD͒ is a general framework polarization effects. for the nuclear many-body problem ͓1–3͔. 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 ͓12,13͔ 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-I͒ with nucleons interacting with scalar ͑͒ and variational ansatz for the vacuum structure and does not use vector ͑͒ 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 ͓14͔ as obtained by summing an infinite series tor couplings ͓4͔. 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 ͓15͔ as mation, several groups have investigated the effects of scalar obtained through the solution of the Schwinger-Dyson equa- self-interactions in nuclear matter ͓5͔ and finite nuclei ͓6͔ 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 ͓7͔. 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 ͓8͔, 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 ͓9͔ 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 condensates also, ͓10͔, which one encounters while calculating the meson which are favored with a quartic term in the 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 ͓16͔. The quartic coupling is tions ͓11͔. 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 field. We also calculate the expansion. This might be because the couplings involved effective mass arising through such quantum corrections as here are too large ͑of order of 10͒ 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.
0556-2813/97/56͑3͒/1380͑9͒/$10.0056 1380 © 1997 The American Physical Society 56 STRUCTURE OF THE VACUUM IN NUCLEAR MATTER:... 1381
II. VACUUM WITH BARYON AND ANTIBARYON ͑kជ ͒ CONDENSATES cos 2 We start with the Lagrangian density for the linear Wa- U ͑kជ ͒ϭ u , r ជ Ir lecka model given as ͑k͒ ជ kˆ sin ͩ • 2 ͪ 1 ץץ ϪMϪg Ϫg ␥ ͒ϩ ץϭ¯͑i␥ L 2 ͑kជ ͒ Ϫជ kˆ sin • 2 1 2 1 2 1 Ϫជ ϭ Ϫ m 2ϩ m Ϫ , ͑1͒ Vs͑ k͒ vIs . ͑8͒ 2 2 4 ͑kជ ͒ cos ͩ 2 ͪ with For free massive fields cos(kជ)ϭM/⑀(kជ) and sin(kជ)ϭ͉kជ͉/⑀(kជ), . ͑2͒ ץ Ϫ ץ ϭ with ⑀(kជ)ϭͱkជ 2ϩM 2. The above are consistent with the equal-time anticommu- In the above, , , and are the fields for the nucleon, , tator algebra for the operators c and ˜c as given by and mesons with masses M, m , and m , respectively. We use the mean-field approximation for the meson fields c kជ ,c† kជ ϭ kជϪkជ ϭ ˜c kជ ,˜c † kជ . and retain the quantum nature of the fermion fields ͓14͔. This ͓ Ir͑ ͒ Is͑ Ј͔͒ϩ ␦rs␦͑ Ј͒ ͓ Ir͑ ͒ Is͑ Ј͔͒ϩ amounts to taking meson fields as constant classical fields ͑9͒ with translational invariance for nuclear matter. Thus we The perturbative vacuum, say ͉vac͘, is defined through shall replace † cIr(kជ)͉vac͘ϭ0 and ˜c Ir(kជ)͉vac͘ϭ0. g ͗g ͘ϵg , ͑3a͒ To include the vacuum-polarization effects, we shall now → 0 consider a trial state with baryon-antibaryon condensates. We thus explicitly take the ansatz for the above state as g ͗g ͘ϵg ␦0ϭg , ͑3b͒ → 0 † where ͗•••͘ denotes the expectation value in nuclear matter ͉vacЈ͘ϭexp dkជf͑kជ͒cIr͑kជ͒ars˜cIs͑Ϫkជ͒ϪH.c. ͉vac͘ and we have retained the zeroth component for the vector ͫ ͵ ͬ field to have a nonzero expectation value. ϵUF͉vac͘. ͑10͒ The Hamiltonian density can then be written as Here a ϭu† (ជ kˆ)v and f (kជ) is a trial function associated ϭ ϩ ϩ , ͑4͒ rs Ir • Is H HN H H with baryon-antibaryon condensates. We note that with the above transformation the operators corresponding to ͉vacЈ͘ with are related to the operators corresponding to ͉vac͘ through the Bogoliubov transformation ϭ†͑Ϫi␣ជ ٌជ ϩM ͒ϩg ¯, ͑5a͒ HN • ជ ជ ˆ ជ ជ dI͑k͒ cosf ͑k͒ Ϫជ •k sinf ͑k͒ cI͑k͒ 1 ϭ , 2 2 ˜ ˆ ϭ m , ͑5b͒ ͩ d ͑Ϫkជ ͒ͪ ͩ ជ k sinf ͑kជ ͒ cosf ͑kជ ͒ ͪͩ˜c ͑Ϫkជ͒ͪ H 2 I • I ͑11͒
1 for the nucleon. ϭg †Ϫ m2 2. ͑5c͒ H 0 2 0 We then use the method of thermofield dynamics ͓18͔ 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. ͑10͒ to finite tem- perature and density as ͓13͔ † ͓␣͑xជ,t͒,͑yជ,t͔͒ϩϭ␦␣␦͑xជϪyជ͒, ͑6͒ ͉F͑͒͘ϭU͉͑͒vacЈ͘ϵU͑͒UF͉vac͘. ͑12͒ where ␣ and  refer to the spin indices. We may now write down the field expansion for the nucleon field at time tϭ0 The temperature-dependent unitary operator U() is given as given by ͓17͔ as ͓18͔ U͑͒ϭexp͓B†͑͒ϪB͔͑͒, ͑13͒ 1 ជ ͑xជ ͒ϭ ͓U ͑kជ ͒c ͑kជ͒ϩV ͑Ϫkជ͒˜c ͑Ϫkជ͔͒eik•xជdkជ, ͑2͒3/2 ͵ r Ir s Is with ͑7͒ B†͑͒ϭ dkជ͓ ͑kជ,͒d† ͑kជ͒dគ† ͑Ϫkជ͒ with cIr and ˜cIs as the baryon annihilation and antibaryon ͵ Ϫ Ir Ir creation operators with spins r and s, respectively. In the ˜ ˜ above, Ur and Vs are given by ϩϩ͑kជ,͒dIr͑kជ͒dគ Ir͑Ϫkជ͔͒. ͑14͒ 1382 MISHRA, PANDA, SCHRAMM, REINHARDT, AND GREINER 56
The underlined operators are the operators corresponding to contributions to the entropy density from and mesons, the doubling of the Hilbert space that arises in thermofield respectively. It may be noted here that these are independent dynamics method. We shall determine the condensate func- of the functions f (kជ), Ϯ(kជ,) associated with the nucleons tion f (kជ), and the functions Ϫ(kជ,) and ϩ(kជ,) of the and hence are not relevant for the nuclear matter properties at thermal vacuum through minimization of the thermodynamic zero temperature. Extremizing the thermodynamic potential potential. To evaluate the expectation value of the energy ⍀ with respect to the condensate function f (kជ) and the func- density with respect to the thermal vacuum, we shall use the tions ϯ yields formula
g0͉kជ͉ 1 ជ ជ † Ϫik•͑xជϪyជ͒ tan2 f ͑k͒ϭ 2 ͑22͒ ͗ ͑xជ ͒ ͑yជ ͒͘ ϭ ⌳Ϫ͑kជ,͒ e dkជ, ⑀͑k͒ ϩMg ␥ ␦  ͑2͒3 ͵ „ …␦␥ 0 ͑15͒ and where 1 sin2 ϭ , ͑23͒ 1 ϯ exp͕͓⑀*͑k͒ϯ*͔͖ϩ1 ⌳ ͑kជ,͒ϭ ͕͑cos2 ϩsin2 ͒Ϫ͓␥0cos ͑kជ ͒Ϫ2 f ͑kជ ͒ Ϫ 2 ϩ Ϫ „ … 2 2 1/2 with ⑀*(k)ϭ(k ϩM* ) and *ϭϪg0 as the effec- ˆ ជ ជ 2 2 ϩ␣ជ •k sin„͑k͒Ϫ2 f ͑k͒…͔͑cos ϩϪsin Ϫ͖͒. tive energy density and effective chemical potential, where the effective nucleon mass is M*ϭMϩg . ͑16͒ 0 Then the expression for the energy density becomes We now proceed to calculate the energy density ⑀ϭ⑀Nϩ⑀ϩ⑀ , ͑24͒ ⑀ϵ͗ ͘ϭ⑀Nϩ⑀ϩ⑀ , ͑17͒ H with with Ϫ3 ជ 2 *2 1/2 2 2 ␥ g0 ⑀Nϭ␥͑2͒ ͵ dk͑k ϩM ͒ ͑sin ϪϪcos ϩ͒, ⑀NϭϪ 3 dkជ ⑀͑kជ͒cos2 f ͑kជ ͒Ϫ ͓M cos2 f ͑kជ ͒ ͑2͒ ͵ ͭ ⑀͑k͒ ͑25a͒
ϩ͉kជ͉sin2 f ͑kជ ͔͒ ͑cos2 Ϫsin2 ͒, ͑18a͒ 1 ϩ Ϫ ⑀ ϭ m2 2 , ͑25b͒ ͮ 2 0 1 ⑀ ϭ m2 2 , ͑18b͒ and 2 0 1 and ⑀ ϭg ␥͑2͒Ϫ3 dkជ͑sin2 ϩcos2 ͒Ϫ m2 2 . 0 ͵ Ϫ ϩ 2 0 1 ͑25c͒ ⑀ ϭg ␥͑2͒Ϫ3 dkជ͑cos2 ϩsin2 ͒Ϫ m2 2 . 0 ͵ ϩ Ϫ 2 0 We now proceed to study the properties of nuclear matter ͑18c͒ at zero temperature. In that limit the distribution functions The thermodynamic potential is then given as for the baryons and antibaryons are given as
1 sin2 ϭ⌰͓*Ϫ⑀*͑kជ ͔͒; sin2 ϭ0. ͑26͒ ⍀ϭ⑀Ϫ Ϫ , ͑19͒ Ϫ ϩ  S B The energy density after subtracting out the pure vacuum with the entropy density contribution then becomes
⑀ ϵ⑀͑ , f ͒Ϫ⑀͑ ϭ0,f ϭ0͒ ϭϪ␥͑2͒Ϫ3 dkជ͓sin2 ln͑sin2 ͒ 0 Ϫ Ϫ S ͵ Ϫ Ϫ ϭ⑀MFTϩ⌬⑀, ͑27͒ 2 2 2 2 ϩcos Ϫ ln͑cos Ϫ͒ϩsin ϩ ln͑sin ϩ͒ with ϩcos2 ln͑cos2 ͔͒ϩ ϩ ͑20͒ ϩ ϩ S S 1 and the baryon density Ϫ3 2 2 1/2 2 2 ⑀MFTϭ␥͑2͒ dkជ͑k ϩM* ͒ ϩ m 0 ͵ ជ ͉k͉ϽkF 2 Ϫ3 2 2 Bϭ␥͑2͒ dkជ͑cos ϩϩsin Ϫ͒. ͑21͒ 1 ͵ ϩg Ϫ m2 2 ͑28͒ 0 B 2 0 In the above, ␥ is the spin isospin degeneracy factor and is equal to 4 for nuclear matter. Further, and are the and S S 56 STRUCTURE OF THE VACUUM IN NUCLEAR MATTER:... 1383
through summing tadpole diagrams for the baryon propaga- Ϫ ⌬⑀ϭϪ␥͑2͒ 3 ͵ dkជͫ͑k2ϩM*2͒1/2Ϫ͑k2ϩM 2͒1/2 tor in the relativistic Hartree approximation ͓7͔.
g0M III. ANSATZ STATE WITH BARYON-ANTIBARYON Ϫ . ͑29͒ ͑k2ϩM 2͒1/2ͬ AND CONDENSATES We next consider the quantum corrections due to the sca- The above expression for the energy density is divergent. It lar mesons as arising from a vacuum realignment with is renormalized ͓7͔ by adding the counterterms condensates. This means that the field is no longer classi- 4 cal, but is now treated as a quantum field. As will be seen n later, a quartic term in the field would favor such conden- ⑀ctϭ ͚ Cn0 . ͑30͒ nϭ1 sates. Self-interactions of scalar fields with cubic and quartic terms have been considered earlier ͓19͔ in the no-sea ap- The addition of the counterterm linear in 0 amounts to nor- proximation ͓6͔ as well as including the quantum corrections mal ordering of the scalar density in the perturbative vacuum arising from the fields ͓1,20,21͔. They may be regarded as and cancels exactly with the last term in Eq. ͑29͓͒7͔. The mediating three- and four-body interactions between the first two terms of the same equation correspond to the shift in nucleons. The best fits to incompressibility in nuclear matter, the Dirac sea arising from the change in the nucleon mass at single-particle spectra and properties of deformed nuclei are finite density when acquires a vacuum expectation value, achieved with a negative value for the quartic coupling in the and consequent divergences cancel with the counter terms of field. However, with such a negative coupling the energy Eq. ͑30͒ with higher powers in 0 ͓7͔. Then we have the spectrum of the theory becomes unbounded from below ͓22͔ expression for the finite renormalized energy density for large and hence it is impossible to study excited spectra or to include vacuum polarization effects. ⑀renϭ⑀MFTϩ⌬⑀ren , ͑31͒ Including a quartic scalar self-interaction, Eq. ͑5b͒ is modified to where 1 1 2 2 4 ϩ m ϩ , ͑36͒ ץץ M* ϭ ␥ ⌬⑀ ϭϪ M*4ln ϩM 3͑MϪM*͒ H 2 2 ren 162 ͫ ͩ M ͪ with m and being the bare mass and coupling constant, 7 13 Ϫ M 2͑MϪM*͒2ϩ M͑MϪM*͒3 respectively. The field satisfies the quantum algebra 2 3 ͓͑xជ ͒,˙ ͑yជ ͔͒ϭi␦͑xជϪyជ). ͑37͒ 25 Ϫ M͑MϪM*͒4 . ͑32͒ 12 ͬ We may expand the field operators in terms of creation and annihilation operators at time tϭ0as For a given baryon density as given by 1 dkជ ជ † ik•xជ ͑xជ,0͒ϭ 3/2 ͓a͑kជ͒ϩa ͑Ϫkជ͔͒e , Ϫ3 ͑2͒ ͵ ͱ2 kជ͒ Bϭ␥͑2͒ dkជ⌰͑kFϪk͒, ͑33͒ ͑ ͵ ͑38a͒ the thermodynamic potential given by Eq. ͑19͒ is now finite i kជ ͑ ͒ † ikជ x and is a function of 0 and 0 . This when minimized with ˙ ͑xជ,0͒ϭ dkជ ͓Ϫa͑kជ͒ϩa ͑Ϫkជ͔͒e •ជ. ͑2͒3/2 ͵ ͱ 2 respect to 0 gives the self-consistency condition for the effective nucleon mass ͑38b͒
2 In the above, (kជ) is an arbitrary function which for free g ␥ M* M*ϭMϪ dkជ ⌰͑k Ϫk͒ϩ⌬M*, ជ ជ 2 2 2 3 ͵ * F fields is given by (k)ϭͱk ϩm and the perturbative m ͑2͒ ⑀͑k͒ ͑34͒ vacuum is defined corresponding to this basis through a͉vac͘ϭ0. The expansions ͑38͒ and the quantum algebra where ͑37͒ yield the commutation relation for the operators a as
2 ͓a͑kជ ͒,a†͑kជЈ͔͒ϭ␦͑kជϪkជЈ͒. ͑39͒ g ␥ M* ⌬M*ϭ M*3ln ϩM 2͑MϪM*͒ m2 ͑2͒3 ͫ ͩ M ͪ As seen in the previous section a realignment of the ground state from ͉vac͘ to ͉vacЈ with nucleon condensates 5 11 ͘ Ϫ M 2͑MϪM*͒2ϩ M͑MϪM*͒3 . ͑35͒ amounts to including quantum effects. We shall adopt a 2 6 ͬ similar procedure now to calculate the quantum corrections arising from the field. We thus modify the ansatz for the We note that the self-consistency condition for the effec- trial ground state as given by Eq. ͑10͒ to include conden- tive nucleon mass as well as the energy density as obtained sates as ͓13͔ here through an explicit construct of a state with baryon- antibaryon condensates are identical to those obtained ͉⍀͘ϭUUF͉vac͘, ͑40͒ 1384 MISHRA, PANDA, SCHRAMM, REINHARDT, AND GREINER 56 with 1 1 dkជ ⑀ ϵ͗⍀͉ ͉⍀͘ϭ H 2 ͑2͒3 ͵ 2͑k͒ UϭUIIUI , ͑41͒ ϫ͓k2͑sinh2gϩcosh2g͒ϩ2͑k͒͑cosh2gϪsinh2g͔͒ † where Uiϭexp(Bi ϪBi), (iϭI,II). Explicitly the Bi are given 1 1 as ϩ m2 Iϩ62Iϩ3I2ϩ m2 2ϩ4 . ͑45͒ 2 0 2 0 0
͑kជ͒ Extremizing the above energy density with respect to the B†ϭ dkជ f ͑kជ͒a†͑kជ͒ ͑42a͒ I ͵ ͱ 2 function g(k) yields
2 and 6Iϩ60 tanh2g͑k͒ϭϪ 2 2 . ͑46͒ ͑k͒ ϩ6Iϩ60 1 B† ϭ dkជg͑kជ͒aЈ†͑kជ͒aЈ†͑Ϫkជ͒. ͑42b͒ II 2 ͵ It is clear from the above equation that in the absence of a quartic coupling no such condensates are favored since the condensate function vanishes for ϭ0. Now substituting this ជ ជ Ϫ1 ជ ជ ជ In the above, aЈ(k)ϭUIa(k)UI ϭa(k)Ϫͱ(k)/2f(k) value of g(k) in the expression for the -meson energy den- corresponds to a shifted field operator associated with the sity yields coherent state ͓13͔ and satisfies the same quantum algebra as given in Eq. ͑39͒. Thus in this construct for the ground state 1 1 1 we have two functions f (kជ) and g(kជ) which will be deter- 2 2 4 2 2 1/2 2 ⑀ϭ m ϩ ϩ 3 dkជ͑k ϩM ͒ Ϫ3I , mined through minimization of energy density. Further, 2 0 0 2 ͑2͒ ͵ since ͉⍀͘ contains an arbitrary number of aЈ† quanta, ͑47͒ aЈ͉⍀͘Þ 0. However, we can define the basis b(kជ), b†(kជ) where corresponding to ͉⍀͘ through the Bogoliubov transformation as 2 2 2 M ϭmϩ12Iϩ120 ͑48͒ b͑kជ ͒ a ͑kជ͒ Ј Ϫ1 ϭUII UII with ͩ b†͑Ϫkជ ͒ͪ ͩaЈ†͑Ϫkជ͒ͪ
coshg Ϫsinhg aЈ͑kជ ͒ 1 dkជ 1 ϭ . ͑43͒ Iϭ ͑49͒ ͩϪsinhg coshg ͪͩ aЈ†͑Ϫkជ ͒ͪ ͑2͒3 ͵ 2 ជ2 2 1/2 ͑k ϩM͒
It is easy to check that b(kជ)͉⍀͘ϭ0. Further, to preserve obtained from Eq. ͑44c͒ after substituting for the condensate translational invariance f (kជ) has to be proportional to ␦(kជ) function g(k)asinEq.͑46͒. The expression for the ‘‘effec- 3/2 tive potential’’ ⑀ contains divergent integrals. Since our ap- and we take f (kជ)ϭ0(2) ␦(kជ). 0 will correspond to a classical field of the conventional approach ͓13͔. We next proximation is nonperturbatively self-consistent, the field- calculate the expectation value of the Hamiltonian density dependent effective mass M is also not well defined for the meson given by Eq. ͑36͒. Using the transformations because of the infinities in the integral I given by Eq. ͑49͒. ͑43͒ it is easy to evaluate that Therefore we first obtain a well-defined finite expression for M by renormalization. We use the renormalization pre- scription of Ref. ͓23͔ and thus obtain the renormalized mass ⍀͉͉⍀ ϭ ͑44a͒ ͗ ͘ 0 mR and coupling R through but 2 2 mR m ϭ ϩ12I1͑⌳͒, ͑50a͒ 2 2 R ͗⍀͉ ͉⍀͘ϭ0ϩI, ͑44b͒ where 1 1 ϭ ϩ12I2͑⌳,͒, ͑50b͒ R 1 dkជ Iϭ ͑cosh2gϩsinh2g͒. ͑44c͒ where I1 and I2 are the integrals ͑2͒3 ͵ 2͑k͒
1 dkជ Using Eqs. ͑36͒ and ͑44͒ the energy density of with I1͑⌳͒ϭ 3 , ͑51a͒ H ͵ ជ respect to the trial state becomes ͓13͔ ͑2͒ ͉k͉Ͻ⌳ 2k 56 STRUCTURE OF THE VACUUM IN NUCLEAR MATTER:... 1385
1 dkជ 1 1 The expectation value for the energy density after sub- I ͑⌳,͒ϭ Ϫ , tracting out the vacuum contribution as given by Eq. ͑27͒, 2 2 3 2 2 ͵͉kជ͉Ͻ⌳ ͑2͒ ͩ2k 2ͱk ϩ ͪ now with condensates is modified to ͑51b͒ ⑀ ϭ⑀finiteϩ⌬⑀, ͑56͒ with as the renormalization scale and ⌳ as an ultraviolet 0 0 momentum cutoff. It may be noted here that with the use of where the above renormalization prescription the effective mass M and the energy density ultimately become independent finite Ϫ3 2 2 1/2 of ⌳ and stay finite in the limit ⌳ ϱ. Using Eqs. ͑50a͒ and ⑀0 ϭ␥͑2͒ dkជ͑k ϩM* ͒ ϩg0B → 2 ͵ ជ ͉k͉Ͻk ͑50b͒ in Eq. ͑48͒, we have the gap equation for M in terms F of the renormalized parameters as 1 2 2 Ϫ m0ϩ⌬⑀ , ͑57͒ 2 2 2 2 M ϭmRϩ12R0ϩ12RI f͑M ͒, ͑52͒ with ⌬⑀ given through Eq. ͑55͒ and ⌬⑀ is the divergent part where of the energy density given by Eq. ͑29͒. We renormalize by adding the same counter terms as given by Eq. ͑30͒ so that as 2 2 M M earlier the renormalized mass and the renormalized quartic I ͑M ͒ϭ ln . ͑53͒ f 162 ͩ 2 ͪ coupling remain unchanged ͓1͔. This yields the expression for the energy density Then using the above equations we simplify Eq. ͑47͒ to ob- ⑀ ϭ⑀finiteϩ⌬⑀ , ͑58͒ tain the energy density for the in terms of 0 as ren 0 ren
2 2 4 2 with ⌬⑀ren given by Eq. ͑32͒. As earlier the energy density is mR M M 1 ⑀ ϭ3 2ϩ ϩ ln Ϫ Ϫ3 I2 to be minimized with respect to 0 to obtain the optimized Rͩ 0 12 ͪ 642 ͫ ͩ 2 ͪ 2ͬ R f * R value for 0 , thus determining the effective mass M in a self-consistent manner. Ϫ24 . ͑54͒ 0 The energy density from the field as given by Eq. ͑55͒ is still in terms of the renormalization scale which is arbi- We might note here that the gap equation given by Eq. ͑52͒ trary. We choose this to be equal to the renormalized mass is identical to that obtained through resumming the daisy and m in doing the numerical calculations. This is because superdaisy graphs 16 and the effective potential includes R ͓ ͔ changing would mean changing the quartic coupling , higher-order corrections from the meson fields. This is an R and here enters as a parameter to be chosen to give the improvement over the one-loop effective potential calculated R incompressibility in the correct range. earlier ͓21͔. The expression for the energy density given by For a given baryon density , the binding energy for Eq. 54 is in terms of the renormalized mass m and the B ͑ ͒ R nuclear matter is renormalized coupling R except for the last term which is still in terms of the bare coupling constant and did not get E ϭ⑀ / ϪM. ͑59͒ renormalized because of the structure of the gap equation B ren B ͓16͔. However, from the renormalization condition ͑50b͒ it is The parameters g , g, and R are fitted so as to describe easy to see that when R is kept fixed, as the ultraviolet the ground-state properties of nuclear matter correctly. We cutoff ⌳ in Eq. ͑51b͒ goes to infinity, the bare coupling discuss the results in the next section. 0Ϫ . Therefore the last term in Eq. ͑54͒ will be neglected in→ the numerical calculations. After subtracting the vacuum contribution, we get IV. RESULTS AND DISCUSSIONS We now proceed with the numerical calculations to study ⌬⑀ϭ⑀Ϫ⑀͑0ϭ0͒ the nuclear matter properties at zero temperature. We take 4 2 the nucleon and -meson masses to be their experimental 1 M M 1 values as 939 and 783 MeV. We first calculate the binding ϭ m2 2ϩ3 4ϩ ln Ϫ Ϫ3 I2 2 R 0 R 0 642 ͫ ͩ 2 ͪ 2ͬ R f energy per nucleon as given in Eq. ͑59͒ and fit the scalar and 4 2 vector couplings g and g to get the correct saturation M ,0 M ,0 1 properties of nuclear matter. This involves first minimizing Ϫ ln Ϫ ϩ3 I2 , ͑55͒ 642 ͩ 2 ͪ 2 R f 0 the energy density in Eq. ͑58͒ with respect to to get the ͫ ͬ 0 optimized scalar field ground state expectation value min . where M ,0 and I f 0 are the expressions as given by Eqs. ͑52͒ This procedure also naturally includes obtaining the in- and ͑53͒ with 0ϭ0. medium -meson mass M through solving the gap equation In the limit of the coupling Rϭ0, one can see that Eq. ͑52͒ in a self-consistent manner. Obtaining the optimized ͑55͒ reduces to Eq. ͑25b͒ as it should. Also, we note that the min amounts to getting the effective nucleon mass sign of R must be chosen to be positive, because otherwise M*ϭMϩgmin . We fix the meson couplings from the the energy density would become unbounded from below saturation properties of the nuclear matter for given renor- with vacuum fluctuations ͓3,21,22͔. malized mass and coupling mR and R . Taking 1386 MISHRA, PANDA, SCHRAMM, REINHARDT, AND GREINER 56
FIG. 1. The binding energy for nuclear matter as a function of FIG. 3. The scalar potential US ͑negative values͒ and vector Fermi momentum kF corresponding to the no-sea and the relativis- potential UV ͑positive values͒ for nuclear matter as functions of tic Hartree approximations and the approach including quantum Fermi momentum kF . corrections from baryon and meson given by Eq. ͑59͒. It is seen that the equation of state is softer with such quantum corrections. 2 ⑀ ץ 2 KϭkF 2 ͑60͒ kFץ mRϭ520 MeV, the values of g and g are 7.34 and 8.21 for Rϭ1.8, and are 6.67 and 7.08 for Rϭ5, respectively. evaluated at the saturation Fermi momentum. The value of K Using these values, we calculate the binding energy for is found to be 401 MeV for Rϭ1.8 and 329 MeV for nuclear matter as a function of the Fermi momentum and Rϭ5. These are smaller than the mean-field result of 545 plot it in Fig. 1. In the same figure we also plot the results for MeV ͓4͔, as well as that of relativistic Hartree of 450 MeV the relativistic Hartree and for the no-sea approximation. ͓7͔ and are similar to those obtained in Ref. ͓21͔ containing Clearly, including baryon and -meson quantum corrections cubic and quartic self-interaction of the meson. leads to a softer equation of state and the softening increases In Fig. 2 we plot the effective nucleon mass M* for higher values of R . The incompressibility of the nuclear ϭMϩgmin as a function of Fermi momentum with min matter is given as ͓24͔ obtained from the minimization of the energy density in a self-consistent manner. At the saturation density of
FIG. 4. The in-medium -meson mass M of Eq. ͑52͒ as a FIG. 2. The effective nucleon mass for nuclear matter as a func- function of density, which is seen to increase with density. How- tion of the Fermi momentum, kF . ever, the change is seen to be rather small. 56 STRUCTURE OF THE VACUUM IN NUCLEAR MATTER:... 1387
FIG. 5. The incompressibility K versus the quartic coupling R FIG. 6. The effective nucleon mass as a function of R for for various values of m , which is seen to decrease with . The R R different values of mR . It is seen to decrease with increase in the values of K are higher for larger values of the mass. coupling.
Ϫ1 kFϭ1.42 fm , we get M*ϭ0.752M and 0.815M for meson quantum corrections as done here in a self-consistent Rϭ1.8 and Rϭ5, respectively. These values may be com- manner includes multiloop effects. This leads to a further pared with the results of M*ϭ0.56M in the no-sea approxi- softening of the equation of state. The value for the incom- mation and of 0.72M in the relativistic Hartree. pressibility of nuclear matter is within the range of 200–400 In Fig. 3 we plot the vector and the scalar potentials as MeV ͓25͔. It is known that most of the parameter sets which functions of kF for self-coupling Rϭ1.8 and 5. At satu- explain the ground-state properties of nuclear matter and fi- ration density the scalar and vector contributions are nite nuclei quite well are with a negative quartic coupling. USϵgminϭϪ232.7 MeV and UVϵg0ϭ163.4 MeV But the energy spectrum in such a case is unbounded from for Rϭ1.8 and are Ϫ173.14 and 107.74 MeV for Rϭ5, below ͓22͔ for large thus making it impossible to include respectively. These give rise to the nucleon potential vacuum polarization effects. We have included the quantum (USϩUV)ofϪ69.3 and Ϫ65.4 MeV and an antinucleon effects with a quartic self-interaction through condensates potential (USϪUV)ofϪ396.1 and Ϫ280.9 MeV for taking the coupling to be positive. We have also calculated Rϭ1.8 and 5. Clearly the inclusion of the quantum correc- the effective mass of the field as modified by the quantum tions reduces the antinucleon potential as compared to both corrections from baryon and fields. The effective mass is the relativistic Hartree (Ϫ450 MeV) ͓7͔ and the no-sea re- seen to increase with density. sults (Ϫ746 MeV) ͓4͔. We have also looked at the behavior of the incompress- In Fig. 4 we plot the in-medium -meson mass M of Eq. ibility as a function of the coupling R for various values of ͑52͒ as a function of baryon density for Rϭ1.8 and 5. M mass, which is seen to decrease with the coupling. Finally, increases with density as R is positive and the magnitude of we have looked at the effect of the quartic coupling on the min increases with density too. However, the change in M effective nucleon mass which grows with the coupling. Gen- is rather small. erally, higher values of the quartic term in the potential of In Fig. 5 we plot the incompressibility K as a function of the meson tend to reduce the large meson fields and thus the quartic coupling R for different values of mR , the renor- the strong relativistic effects in the nucleon sector. Clearly, malized mass in vacuum. The value of K decreases with the approximation here lies in the specific ansatz for the increase in R similar to the results obtained in Ref. ͓21͔.In ground-state structure. However, a systematic inclusion of Fig. 6 we plot the effective nucleon mass versus the self- more general condensates than the pairing one as used here coupling for various values of mR . The value of M* in- might be an improvement over the present one. The method creases with R , which is a reflection of the diminishing can also be generalized to finite temperature as well as to nucleon- coupling strength for larger values of the quartic finite nuclei, e.g., using the local density approximation. self-interaction. Work in this direction is in progress. To summarize, we have used a nonperturbative approach to include quantum effects in nuclear matter using the frame- work of QHD. Instead of going through a loop expansion and summing over an infinite series of Feynman diagrams ACKNOWLEDGMENTS we have included the quantum corrections through a realign- ment of the ground state with baryon as well as meson con- The authors would like to thank J. M. Eisenberg for ex- densates. It is interesting to note that inclusion of baryon- tensive discussions, many useful suggestions, and for a criti- antibaryon condensates with the particular ansatz determined cal reading of the manuscript. A.M. and P.K.P. acknowledge through minimization of the thermodynamic potential yields discussions with H. Mishra during the work. They would the same results as obtained in the relativistic Hartree ap- also like to thank the Alexander von Humboldt-Stiftung for proximation. This results in a softer equation of state as com- financial support and the Institut fu¨r Theoretische Physik for pared to the no-sea approximation. The calculation of scalar hospitality. 1388 MISHRA, PANDA, SCHRAMM, REINHARDT, AND GREINER 56
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