Relativistic Chiral Hartree-Fock description of nuclear matter with constraints from nucleon structure and confinement
We present a relativistic chiral effective theory for symmetric and asymmetric nuclear matter taken in the Hartree-Fock scheme. The nuclear binding is insured by a background chiral invariant scalar field associated with the radial fluctuations of the chiral quark condensate. Nuclear matter saturation is obtained once the scalar response of the nucleon generating three-body repulsive forces is incorporated. For these parameters related to the scalar sector and quark confinement mechanism inside the nucleon we make use of an analysis of lattice results on the nucleon mass evolution with the quark mass. The other parameters are constrained as most as possible by standard hadron and nuclear phenomenology. Special attention is paid to the treatment of the propagation of the scalar fluctuations. The rearrangement terms associated with in-medium modified mass and coupling constants are explicitly included to satisfy the Hugenholtz -Van Hove theorem. We point out the important role of the tensor piece of the rho exchange Fock term to reproduce the asymmetry energy of nuclear matter. We also discuss the isospin dependence of the Landau nucleon effective mass.
pacs:24.85.+p 11.30.Rd 12.40.Yx 13.75.Cs 21.30.-x
A fundamental question of present day nuclear physics is to relate low energy non perturbative QCD and in first rank chiral symmetry and confinement to the very rich structure of the nuclear many-body problem. However it is presently hopeless to derive the observed nuclear properties from the underlying QCD and a more modest ambition is to put some constraints on the modelling of nuclear matter properties not only from hadronic phenomenology but also from lattice QCD data. This may also constitute a starting point to elucidate an old and central question of strong interaction physics, namely the interrelation between the many-body effects governing the equation of state of nuclear matter and the nucleon substructure response to the nuclear environmment.
A first attempt to go beyond the standard non relativistic treatment of nuclear matter is the relativistic mean field approach initiated by Walecka and collaborators SW86 (). In this framework the nucleons move in an attractive background scalar field and in a repulsive vector background field. This provides a very economical saturation mechanism and a spectacular well known success is the correct magnitude of the spin-orbit potential since the large vector and scalar fields contribute to it in an additive way. Another successful modern attempt is based on in-medium chiral perturbation theory where a pion loop expansion is performed on top of scalar and vector background scalar fields using a density functional formulation FKVW06 (). Now the question of the very nature of these background fields has to be elucidated or said differently it is highly desirable to clarify their relationship with the QCD condensates and in particular the chiral quark condensate.
To address this question we take the point of view that the effective theory has to be formulated, as a starting point, in term of the fields associated with the fluctuations of the chiral quark condensate parametrized in a matrix form . The sigma and the pion, associated with the amplitude and phase fluctuations of this condensate are promoted to the rank of effective degrees of freedom. Their dynamics are governed by an effective potential, , having a typical mexican hat shape associated with a broken (chiral) symmetry of the QCD vacuum. Explicit construction of such an effective theory for the description of nuclear matter can be performed for instance within the NJL model BT01 ().
As proposed in a previous paper CEG02 () an alternative and very convenient formulation of the resulting sigma model is obtained by going from cartesian to polar coordinates i.e., going from a linear to a non linear representation, according to : . The new pion field corresponds to an orthoradial soft mode which is automatically massless (in the absence of explicit chiral symmetry breaking) since it is associated with rotations on the chiral circle without cost of energy. The new sigma meson field, , which is a chiral invariant, describes a radial mode associated with the fluctuations of the “chiral radius” around its vacuum expectation value, . It can be associated with the ordinary sigma meson which gets a very large width from its strong decay into two pions. Since it has derivative couplings to the pion field, it decouples from low energy pions whose dynamics is described by chiral perturbation theory. The evolution of the expectation value of is related to the non pionic contribution to the in-medium chiral condensate CEG02 (); EC07 (). This expectation value plays the role of a chiral order parameter around the minimum of the effective potential and the medium can be seen as a shifted vacuum. With increasing density, its fluctuations are associated with the progressive shrinking of the chiral circle and it governs the evolution of the nucleon mass. Here comes our main physical assumption proposed in ref. CEG02 (). We identify this chiral invariant field with the sigma meson of nuclear physics and relativistic theories of the Walecka type, or, said differently, with the background attractive scalar field at the origin of the nuclear binding. This also gives a plausible answer to the long-standing problem of the chiral status of Walecka theories.
One motivation of the present work is to study in some details whether this hypothesis yields a viable description of nuclear matter. It is nevertheless well known that in such chiral theories, independently of the details of the modelling, tadpole diagrams associated with the mexican hat potential automatically generate attractive three-body forces destroying saturation KM74 (); BT01 (). The origin of this failure can be attributed to the neglect of the effect of nucleon substructure linked to the confinement mechanism as already pointed out in some of our previous works CE05 (); CE07 (); EC07 ().
Our article is organized as follows. The second section, which is in some sense a brief summary of our previous works, is devoted to the constraints brought by lattice data for the description of the nucleon and nuclear matter. In section III we present the chiral lagrangian and section IV is devoted to the construction of the hamiltonian in the static approximation; we also give a detailed description of the treatment of the propagation of the in-medium modified scalar field. The Hartree-Fock approach including rearrangement terms is presented in section V and is applied to the case of infinite matter in section VI. Finally in section VII numerical results are given and the results discussed.
Ii Constraints on the chiral effective theory from QCD susceptibilities
ii.1 Tests of the effective theory with a chiral invariant scalar field
Once the appropriate couplings of the chiral fields to the baryons are introduced one can build an effective lagrangian to describe nuclear matter. Vector mesons ( and ) must be also included to get the needed short range repulsion and asymmetry properties (see sections III and VII). At the Hartree level, the pion and the rho do not contribute for symmetric nuclear matter whose energy density written as a function of the order parameter is :
is the energy of an effective nucleon with the effective Dirac mass . is the scalar coupling constant of the model; in the pure linear sigma model it is . The effective potential when reexpressed in term of the new polar representation has the typical form :
is obtained by minimization of the energy density and is given at low density by : . Its negative value is at the origin of the binding but the presence of the term (tadpole) has very important consequences as already mentioned in the introduction. This tadpole is at the origin of the chiral dropping HKS99 () of the sigma mass (a effect at ) and generates an attractive three-body force which makes nuclear matter collapse and destroys the Walecka saturation mechanism. Hence the chiral theory does not pass the nuclear matter stability test.
This failure, which is in fact a long-standing problem KM74 (); BT01 (), is maybe not so surprising since the theory, as it is, also fails to describe some nucleon structure aspects as discussed below. The nucleon mass, as well as other intrinsic properties of the nucleon (sigma term, chiral susceptibilities), are QCD quantities which are in principle obtainable from lattice simulations. The problem is that lattice calculations of this kind are not feasible for quark masses smaller than MeV, or equivalently pion mass smaller than MeV, using the GOR relation. Hence one needs a technics to extrapolate the lattice data to the physical region. The difficulty of the extrapolation is linked to the non analytical behaviour of the nucleon mass as a function of (or equivalently ) which comes from the pion cloud contribution. The idea of Thomas et al TGLY04 () was to separate the pion cloud self-energy, , from the rest of the nucleon mass and to calculate it in a chiral model with one adjustable cutoff parameter . They expanded the remaining part in terms of as follows :
At this point it is important to stress that the above expansion is in reality an expansion in terms of the current quark mass which is the genuine parameter occuring in the lattice calculation. The pion mass appearing in eq. (2) is just the pion mass deduced from the quark mass assuming the GOR relation . This reparametrization has been adopted only for convenience. The best fit value of the parameter shows little sensitivity to the shape of the form factor, with a value while (see ref. TGLY04 ()). The small value of reflects the fact that the non pionic contribution to the nucleon mass is almost linear in (i.e., in ). Taking successive derivatives of with respect to (i.e., to ), it is possible to obtain some fundamental chiral properties of the nucleon, namely the pion-nucleon sigma term and the scalar susceptibility of the nucleon. The non pionic pieces of these quantities are given by :
In the above equations the first equalities correspond to the definitions, the second equalities make use of the GOR relation and the last ones come from the lattice QCD analysis. With typical cutoff used in this analysis, GeV, which yields MeV, the total value of the sigma term is MeV, a quite satisfactory result in view of the most recent analysis. It is interesting to compare what comes out from the lattice approach with our chiral effective model. At this stage the only non pionic contribution to the nucleon mass comes from the scalar field, or more microscopically the nucleon mass entirely comes from the chiral condensate since the nucleon is just made of three constituent quarks with mass MeV. Hence the results for the non pionic sigma term and scalar susceptibility are identical to those of the linear sigma model :
The identification of with of our model fixes the sigma mass to a value MeV, close to the one MeV that we have used in a previous article CE05 (). As it is the ratio which is thus determined this value of is associated with the coupling constant of the linear sigma model . Lowering reduces . Similarly, the identification of with the lattice expression provides a model value for . The numerical result is while the value obtained in the expansion is only .
ii.2 Nucleon structure effects and confinement mechanism
The net conclusion of the above discussion is that the model as such fails to pass the QCD test since the coefficient is much larger in the chiral model than the one extracted from the lattice data analysis. In fact this is to be expected and even gratifying because it also fails the nuclear physics test as discussed in subsection II.1. We will see that these two important failures may have a common origin. Indeed an important effect is missing, namely the scalar response of the nucleon, , to the scalar nuclear field, which is the basis of the quark-meson coupling model (QMC) introduced in ref. G88 (). The physical reason is very easy to understand: the nucleons are quite large composite systems of quarks and gluons and they should respond to the nuclear environment, i.e., to the background nuclear scalar fields. This response originates from the quark wave function modification in the nuclear field and will obviously depend on the confinement mechanism. This confinement effect is expected to generate a positive scalar response , i.e., it opposes an increase of the scalar field, a feature confirmed by the lattice analysis (see below). This polarization of the nucleon is accounted for by the phenomenological introduction of the scalar nucleon response, , in the nucleon mass evolution as follows :
This constitutes the only change in the expression of the energy density (eq. 1) but this has numerous consequences. The effective scalar coupling constant drops with increasing density but the sigma mass gets stabilized :
The non-pionic contribution to the nucleon susceptibility is modified, as well CE07 () :
We see that the effect of confinement () is to compensate the pure scalar term. Again comparing with the lattice expression one gets a model value for the parameter :
where is the dimensionless parameter . Numerically gives , implying a large cancellation. As discussed in ref. CE05 (); CE07 (); EC07 () such a significant scalar response will generate other repulsive forces which restore the saturation mechanism. At this point it is important to come again to the underlying physical picture implying that the nucleon mass originates both from the coupling to condensate and from confinement. In the original formulation of the quark coupling model, nuclear matter is represented as a collection of (MIT) bags seen as bubbles of perturbative vacuum in which quarks are confined. Thus in such a picture the mesons should not appear inside the bag and should not couple to quarks as in the true non perturbative QCD vacuum. Consequently the bag picture is at best an effective realisation of confinement which must not to be taken too literally. Indeed, QCD lattice simulations strongly suggest that a more realistic picture is closer to a shaped color string (confinement aspect) attached to quarks B05 (). Outside this relatively thin string one has the ordinary non perturbative QCD vacuum possessing a chiral condensate from which the quarks get their constituent mass.
Iii The chiral Lagrangian
To get a more complete description of symmetric and asymmetric nuclear matter we complete the lagrangian used in ref. CEG02 () essentially by the introduction of the rho meson. We also consider the possibility of incorporating the scalar isovector meson. Written with obvious notations, it has the form
As discussed previously the form of (eq. 6) reflects the internal nucleon structure and contains a quadratic term involving the scalar response of the nucleon which is constrained by lattice data. However the nucleon mass may very well have higher order derivatives with respect to the scalar field. In practice, as in our previous works CE05 (); CE07 (), we introduce a cubic term :
Hence the scalar susceptibility becomes density dependent
and vanishes at full restoration, , where is the expectation value of the field. Hidden in the above Lagrangian is the explicit chiral symmetry breaking piece
which generates the pion mass term with the identification . It is thus implicit that neglecting the higher order terms in the exponent, the self-interactions of the pions are omitted. Notice that the only meson having a self-interacting potential is the scalar meson . We take it in practice as in the linear sigma model with the inclusion of the explicit chiral symmetry breaking piece :
The other parameters ( and the meson masses) will be fixed as most as possible by hadron phenomenology. It is in principle also possible to calculate or at least to constrain these parameters in an underlying NJL model. One specific comment is in order for the tensor coupling of vector mesons. The pure Vector Dominance picture (VDM) implies the identification of with the anomalous part of the isovector magnetic moment of the nucleon, i.e., . However pion-nucleon scattering data HP75 () suggest (strong rho scenario). We will come to this point later on in the discussion of the results (section VII). The omega meson should also possess a tensor coupling but, according to VDM the corresponding anomalous isoscalar magnetic moment is . Since it is very small we neglect it here. For completeness we also add a delta meson which may generate a splitting between the proton and neutron masses but with the chosen coupling constant, , its influence is in practice negligible.
Iv Construction of the hamiltonian
The conjuguate momenta of the various mesonic fields are :
The hamiltonian is obtained by the usual generalized Legendre transformation with the result :
iv.1 Static approximation
We will first formulate the Hartree-Fock approach in the static approximation, i.e., neglecting the retardation effects. In such a case the conjuguate momenta are :
The various pieces of the hamiltonian simplify according to :
This hamiltonian can be rewritten as :
The first term is the nucleonic piece including the Yukawa coupling of the nucleons to the meson fields :
The second piece os of purely mesonic nature :
iv.2 Equation of motion for classical and fluctuating meson fields
In the static approximation the equation of motion for each meson field can be written formally as , where is the conjugate momentum of the field . This gives :
Following the method used in ref. GMST06 () we now assume that it makes senses to decompose each meson field as :
where denotes the ground state expectation value of the meson field and corresponds to its fluctuation considered as a small quantity.
Since the case of the scalar field is the most delicate one we treat it below with some details. The equation of motion for the field can be expanded in according to :
Explicitly we have :
We now develop the source term of the equation of motion according to :
and we consider the fluctuating term as small and of same order than . In the same spirit we replace the second derivative of by its expectation value :
The equation of motion will be solved order by order :
In the equation for the fluctuating part it appears the effective scalar mass :
already introduced in our previous work CE05 (). This is the physical in-medium scalar mass propagating the quantum fluctuations of the scalar field, i.e., the quantum fluctuations of the chiral condensate in a non-trivial way. In particular we will see below that it is the mass appearing in the Fock term of the scalar exchange at variance with appearing in the Hartree scalar exchange. In that sense the treatment of the self-interacting scalar field deviates from the one of ref. BERNARD93 ().
iv.3 Kinetic, Hartree and exchange hamiltonians
We now develop the hamiltonian (eq. 20) to second order in the fluctuations, limiting ourselves to the nucleon field and the field, the generalization to the other mesons being straightforward.
Using the classical equation of motion and the one for the fluctuating field and replacing again the second derivative of by its expectation value, we obtain :
The approach can be extended to the other mesons. Only the delta field and the time component of the rho and omega mesons have a non zero expectation value, solution of the classical equations :
The fluctuating fields are solutions of :
Generalizing the result of eq. (LABEL:HS) to all the mesons, the full hamiltonian, in the static approximation, can be written in a form reminiscent of the density functional theory :
The first term is a one-body operator containing the kinetic energy hamiltonian of the nucleons and the other pieces of the hamiltonian contributing to the Hartree energy. Its explicit form is :
Considering only this part of the hamiltonian we come to the conclusion that symmetric and asymmetric nuclear matter are seen as an assembly of nucleons, i.e., of Y shaped color strings with massive constituent quarks at the end getting their mass from the chiral condensate. This nucleons move in self-consistent scalar () and vector background fields (). The scalar field which we associate with the radial mode of the condensate modifies the nucleon mass according to . describes the in-medium dropping of the nucleon mass from its coupling to the in-medium modified chiral condensate but there is another term, , which corresponds to the response of the nucleon to the background scalar field. The scalar response depends on the structure of the nucleon and takes into account the modification of the quark wave functions inside the nucleon and obviously depends on the confinement mechanism. The nucleon is also submitted to an isovector field, () and in asymmetric nuclear matter to an isovector vector field () or even an isovector scalar field ().
The second piece of the hamiltonian, incorporates the exchange term mediated by the propagation of the fluctuations of the meson fields. In particular the scalar fluctuation, i.e., the fluctuation of the chiral condensate propagates, as already stated, with an in-medium modified sigma mass . Its explicit form is :