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Submitted on 1 Jan 1987
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PION CHIRAL AND FLUID DYNAMICS
M. Rosina
To cite this version:
M. Rosina. PION CHIRAL AND FLUID DYNAMICS. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-343-C2-346. �10.1051/jphyscol:1987252�. �jpa-00226521�
JOURNAL DE PHYSIQUE
Colloque C2, suppl6ment au n o 6, Tome 48, juin 1987
PION CHIRAL AND FLUID DYNAMICS
M. ROSINA
Faculty of Natural Sciences and Technology and J. Stefan Institute, E. Kardelj University, Jadranska 19, POI3 64, YU-61111 Ljubljana, Yugoslavia
R6sumQ. 2Jous decrivons le champ pionique cornme un champ moyen provenant des fluctuations chirales des quarks dsns la mer de Dirac, selon les id6es ae Nambu et Jona-Lasinio. Suivant la sug6stion de J.da Pros-idGncia, nos derivons la distribution des quarks dans l'espace le phase.
Abstract. An attempt is made to describe the pion field by a distribution function of Dirac sea quarks in the phase space. The formulation introduced by J. da Providencia is used.
1. Introduction
The aim of this contribution is pedagogical. I would like to show that in analog with the phase space approach to the dynamics of nucleons in the nucleus also e phase space approach to the chiral dynamics of quarks may become fruitful.
We shell follow the formulation introduced by J. da Providencia, sr.
We assume a system of u and d quarks interacting by some effective two-body poteatial the details of which we shall not need in our aualitative treatment.
We shall describe quarks with a distribution function in the phase space,including also the information on the polarization of spin and.c:?irsl angle. (The chiral angle expresses the ratio between the lower. and upper components of the Dirac bispinor).
In the vacuum, all the negative energy states are uniformly filled. The chiral angle fluctuations can change the mean field experienced by the quarks and can produce excitations with properties of a pion and sigma field.
In the zeroth approximation quarks are assumed to be massless and the quark-quark interaction to be chirally invariant. Therefore also the mean field appears as a chiral four-vector with three components corresponding to the three charges of the pseudoscalar pion field and the fourth component corresponding to the scalar sigma field. Actually, the symmetry and strength of interaction is so chosen as to break the chirel symmetry spontaneously : already in the vacuum there is a nonvanishing scalar mean field (sigma field) acting as a mass term in the single- particle Dirac Hamiltonian, thereby lowering the energy of the "negative Dirac seatt of the vacuum. The pion-type excitations appear as Goldstone bosons as if pions were massless. In next approximation one can break the chiral symmetry explicitly by a small m o u n t in order to bring the pion mass to the experimental value.
The motivation to develop the phase space approach is to treat more complex systems such as nuclei under extreme conditions. The work is still at a very preli- minary stage and it is aimed to describe,for example,the phase transition in which the valence quarks (clustered in nucleons) dissolve in some effective fields
("pion condensaten, "quark-gluon plasmaIt). Even in less extreme conditions such as Gww-Teller excitations at high energy, the pion field is expected to play an im- portant role and a simplified qualitative approach may be instructive.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987252
JOURNAL DE PHYSIQUE
2. The simplified model
For the pedagogical treatment we shall ignore the spin polarization so that the Dirac matrices can be written as 2x2 matrices. This approximation corresponds to having spins oriented in the direction of motion. The generalization is strai- ghtforward but tedious and shall not be done here. In ordeer to describe the chi- ral polarization we need a distribution function which is classical in the phase space and is a 2x2 density matrix in the chiral space. We shall normalize it so that there are only chiral fluctuations but no density fluctuations (the first term is 1/21 .
where the matrices
obey the same commutation relations as angular mornentum operators. They are genera- tors of chiral rotations.
The single particle Hamiltonian in the mean field approximation can be written as
where we used the symbols and C? to suggest the interpretation of the mean fields as pion and sigma fields :
For simplicity we assumed the system to be in a state with only neutral ''pion I " ~ ? l c l " . u(p) is a form. factor of the interaction (a :'cutoff'i).
The equation of motion can be written as
The commutator acts on Dirac inatrices and leaves a simple product of phase s p x e functions. The Poiszon bra.cket acts on t'ne phase space fuilctions and is acompanied by a symmetrized product of Dirac matrices ; in the spirit of t'ne semiclassicsl treatment we shall neglect this tern in the approxi2ate calculation since it is of first order in Planc?:'~ constant. By introducing the notation
f = (X,Y,Z) and h = ( p , r , G ) the equation of notion can be written elegantly as
f = 2 f : : h
One gets the well known stationary solution for a Dirac particle in a constant field
The symbol m was used in order to be suaestive of the mean field acting as a mass term.
-
In the vacuum (p = (p = 0 ; m 2: 2 = Z is the vacuum expectation value of the sigma field and it can be estimated @om lattice QCD calculations (or QCD sum rules) in conjunction with models for weak currents.
To get excited states, we can expand 7 and 6 around the vacuum values and.
keep only first order small terms in the equation of mo.tion.
We expand (for a relatively short range potential V(r) )
6 (x) = A Ju(p9) d3p' ~ r p f(x,p9) r f AR' u(p9) d3p7 irk v2f(x,p9) where A =
S
v d3r and A R ~ = J V r2d3r ,and a similar expansion for the pion field. Me shall write the equations of motion for m 1 = m -x 0 and rather than for X,Y and 2 . We get equations of motion for small oscillstions
ml = -
P
yielding the'pion - wave velocity
-
c
: = 2 E~ (1 - ~ ( p ) ) R2
Here w(p) is a function of the form factor u(p). The ve1oc;ty c n should be the same for all quark fluctuations of ciifferent p (and E ) ; due to the oversimplified treatment it is not so and we took an average. If we use a form factor
u(p) = @ (pma2 - p) with pmaX leading to a reasonable estimate of ZO we can get the estimgte R z 0.3 fm which seems to be a reasonable effective range of the quark-quark interaction (or maybe_slsize of the pionyt). Inorder to get this esti-mte we required c, = 1 , valid for massless pions of our chirally symmetric model.
I would like to thank J.da Providencia for the comment that finite range R is not a necessary condition for the propagation of pion (cz # 0 ) ; the Poisson bracket in the equation of motion (which we neglected) gives a contribu.tion to c, even for zero range forces.
3. Prospects
First the treatment has to be developped further in order to include spins and to 8nsure Lorentz invariance and correct pion properties. Tkien we would like to proceed to problems such as phase transition of nuclear matter.
In order to describe the nucleus we have to include also valence quarks
~rhich are clustereci in clusters of three, by taking a more general normalization for the distribution function F(x,p) . In ofader to describe a hot nucleus we have to allow a distribution function F(x,p) corresponding to the canonical ensemble.
bJe expect only qualitative results, but very instructive due to their simplicity.
I would like to note that the presented study of pion field as a mean field due to chiral fluctuations ?:as inspire6 by the ideas of Nmbu and Jona-Lasinio ( 1 )
(at the level of nucleons rather than quarks). Nore quantitative treatment of the pion was perforined by the RPA methods (2,3) and by the Thouless-Valatin and Peieris- Yoccoz methods (M.C.Ru.ivo, Coimbra preprint). Extensive study of the time depenaent
C2-346 JOURNAL DE PHYSIQUE
Ha-rtree-Fock and phase space description of the excitations of the Dirac sea of quarks is in progress in Coimbra (J.da ProvidGncia, M.C.3uivo and C.A.de Sousa, Coimbra preprint).
The author tioula like to acknocrledge the fine hospitality he enjoyed at the University of Coimbra where this ~ o r k wss initiated.
R e f e r e n c e s
1. Y.Nsmbu 3 r d G.Jona-Lasinio,Phys.Rev.122,345(1?.61) 2. J.R.Finger and J.E.blandula, Nucl.Phys. B199,168(1982) 3. R .Brockmann,\J.lleise and E-Ilerner ,Phys. Lett 122B,201( 1383)
V.Bernard et al.,Nucl.Phys.A412,349(1984)
V.Bernard,R.Brocl<mann and I.i.Ueise, Nucl.Phys.A440,605(1985) V.Bernard, Phys.Rev,D 34,1601(1985)
Appendix
I would like to point out the analogy between typical excitations in lox energy nuclear physics and in high energy nuclear (and hadron) physics by presenting the following VOCABULARY . This is already the third vocabulary presented in the today's session in order to cross-fertilize neighbouring branches of physics.
nucleons - quarks
closed shell - Dirac sea (of negative enersy states) giant resonances - mesons
spin or isospin - chiral angle
(isolspin wave - chiral angle wave propagation propagation
(iso)vector - chiral four-vector ,e .g. ( 6 ,Y',B~,T-)
