RPA method for quark-meson solitons

Nuclear Physics A458 (1986) 652-668 North-Holland, Amsterdam

RPA METHOD WOJCIECH

FOR QUARK-MESON

BRONIOWSKI’

and

THOMAS

SOLITONS D. COHEN

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, USA Received 9 December 1985 (Revised 20 May 1986)

Abstract: The quark-meson

RPA equations, which describe small oscillations of a bound quark-meson system about the stationary configuration, are derived through linearization of the classical timedependent Euler-Lagrange problem. The method has an immediate application in phenomenological quark-meson models for baryons. It provides a test of the classical stability of these systems. A number of measurable quantities, such as the spectrum of excited states and the meson-soliton phase shifts can be calculated. We demonstrate the QMRPA on a simple, 3+ 1 dimensional model of the nucleon - the chiral quark-meson model.

1. Introduction

Several authors have in the past few years considered phenomenological quarkmeson models of the nucleon le6). The approach is based on a belief that the QCD effects leading to the binding of baryons can be simulated by a simpler, effective theory of interacting valence quarks and mesons. At the classical level the models produce stable, localized objects, solitons, which are identified with baryons. The resulting phenomenologies were reasonably successful in reproducing a variety of baryon properties, thus making the approach physically interesting. Although the models deal with nucleon substructure, various concepts and techniques of traditional nuclear and many-body physics appear to be applicable and to yield good results. Such methods include the independent particle model, meanfield approximations, projection techniques for systems with broken symmetries and cranking ‘). This paper presents an adaptation of the RPA method to the case of quark-meson solitons. The technique allows for a straightforward calculation of a number of baryonic properties, such as the energies of baryon resonances and phase-shifts for meson-baryon scattering. It also provides a test of the classical stability of a soliton. The quark-meson models are based on quarks interacting via mesons; mesonmeson couplings are also included. Therefore one must deal with dynamical solitonic systems of both fermions and bosons. An elegant quantum-mechanical treatment of such systems has been developed by Dashen, Hasslacher and Neveu lo). Unfortunately, technical complications prevent a simple application of this method to I On leave from lnstitute 0375-9474/X6/$03.50 6 (North-Holland Physics

of Nuclear

Physics,

ul. Radzikowskiego

Elsevier Science Publishers Publishing Division)

B.V.

142, 31-342 Krakbw,

Poland.

W. Broniowski,

realistic problem.

quark-meson

models.

It is essentially

TD.

Instead,

Cohen / RPA method

we propose

a straightforward

653

a heuristic

generalization

approach

to the

of both the conventional

RPA method “) used in the many-fermion problem and of the semi-classical method used to treat soliton models with boson fields only “). We obtain the fermion-boson RPA equations through the linearization of the classical time-dependent EulerLagrange equations (sect. 2). Their algebraic structure is identical to that of the conventional RPA problem. Often in dealing with quark-meson models of baryons one considers only valence quarks and ignores the sea-quarks. It will be assumed in this work that sea-quark effects can be safely neglected. We shall not attempt to justify this assumption except to note that one can, in principle, test this assumption by doing quantum calculations similar to those of Kahana and Ripka 12). We show, however, how to interpret time-dependent valence quarks in a consistent manner. To illustrate the method we apply it to the case of the chiral quark-meson model (CQMM) of refs. 4,5). The CQMM, reviewed in sect. 3, is the simplest quark-meson theory which respects chiral symmetry, and as such is of particular physical interest. Sect. 4 presents the results of an RPA calculation for the CQMM. We consider a particularly interesting class of excitation - the breathing modes. It is verified that the solution identified with the ground state is classically stable with respect to such excitations. We also systematically study the dependence of the model on the quark-meson coupling constant. It is found that for values of the coupling constant which give good results for the nucleon properties, the RPA spectrum does not possess bound excited states. We note that such states are not observed experimentally. We also present an RPA calculation of the P,, pion-nucleon phase shift.

2. Small amplitude motion for coupled quark-meson In this section

we will extend

the RPA formalism

systems

to study interacting

fermion-

boson systems. While the formalism itself is general, we will have in mind a particular application to a model of interacting quarks and mesons. For the purposes of concreteness we will refer to the fermions as “quarks” and the bosons as “mesons”. We will call the resulting small amplitude theory the quark-meson RPA (QMRPA). The models

we consider

have lagrangians

of the general

form

(2.1) I

j

where j labels the species of meson. The vector cp represents the meson fields and the M’s are matrices in the Dirac x species spinor space which describe the Yukawa coupling of the mesons to the quarks. The mesons are treated as classical fields. The quark sector of a baryon state can be represented as a Slater determinant

I@= Ktbllvac>,

(2.2)

654

W. Broniowskl, T.D. Cohen / RPA method

where the subscript the creation structure

v indicates

operator

that the counting

associated

of v is contained

in spinor 4Jx)

The mean-field

or classical

is over valence

with the valence qU, defined

=

quark

states,

and +L is

v. The space-spin-species

by

(vacl+(x)+‘uIvac> .

(time-dependent)

equations

(2.3)

following

from (2.1) are

qcpi+(6v/GrPj)l,+Cqt,M,q”=o, 0 [ -icu *v + ~,M,cp,]q, = i&q,. Following problems, equations.

(2.4)

the treatment of small amplitude motion of the fermionic “) and bosonic ‘) we begin by finding stationary solutions to the coupled quark-meson For the static case eq. (2.4) become -V2Q~J-+(aV/aQ~~)l,,+C

”

4iv”j90u=0,

L-h! *v + qf]%, 1400= E”qo” 9

(2.5)

where the quark eigenvalues E, enforce the normalization of the spinors qov. The next step is to expand (2.4) about the static solution Qo, q. given by eqs. (2.5). Introducing aQj = QJ- ‘PO,, we obtain

the following

linearized

(2.6)

6q” = 4” - 40” 3

equations:

%sQ, = ar,, -a,ar,

=-V’SQ,

+(a2VIaQjaQk)l~O~Qk+ + MkaQk ,

id&I, = (h - au)% where

h is the single particle

(2.7)

. v

+

?t!fkQok,

over v and k are implicit.

Gq,(t)=[X,, 6~,(t)

=

exp(-iwt)+

[2-“‘(X,,+

&~,(t)=[-i~2-“~(X,,+ where A is an arbitrary dent of A. With ansatz

2

Dirac hamiltonian, h = -icU

and the summations

qLMj@,+&LMjqov

A harmonic

(2.8)

ansatz

can be made,

Y$exp(i~*Ft)]A~‘~,

Ym,) exp (-iwt)+c.c.]A, Y,,)exp(-iwt)+c.c.]A,

(2.9)

scale with units of energy. The physical results are indepen(2.9) eqs. (2.7) assume the standard RPA form (2.10)

W. Bromowski,

where X and

Y are defined

ZD.

Cohen / RPA method

655

as

x=[;:],Y=[;I], A and B can be represented

(2.11)

as (2.12)

and the subscripts elements

m and q indicate

quark and meson terms respectively.

of Agq, Aqm, A,,,,,, , B,, , B,, A qq(v.0’)= 6”Jh A 4m(v,J)

=

- GJl~

&f,qov/

A312

A mm(j,J’)=t[6~‘(-V2+A2)+

and B,,

are given

The matrix

by

9 ,

(a2V/a(pJ f3(p,f)),]/A2,

B,,=O, B w(u.j) =~M,q,,/A”“, B ,m(,,,‘,=~[SjJ,(-V2-A*)+(a2v/acp,

~~,~)Icpp,llA’.

(2.13)

The fact that B,, = 0 reflects the lack of two-body quark operators in the lagrangian. The preceding equations are the QMRPA equations. The solutions to the QMRPA equations have the obvious interpretation that one would obtain by analogy with the purely fermionic and bosonic systems. Namely, the classical modes of excitation correspond to quantum excitations of the and the existence of modes with imaginary frequency indicates instability soliton. The discrete part of the spectrum corresponds to excited states of the with excitation energies approximated by the vibrational frequency of the

system of the soliton mode.

The continuous part of the spectrum corresponds to meson-soliton or quark-soliton scattering states with the quantum phase shifts being given approximately by the classical phase shift for the mode. The notion of a theory of interacting meson and valence quarks is only sensible if there exists a concept of valence quark and if one can reasonably drop the contributions from the Dirac sea. For static mean meson fields there is a clear notion of “valence” based on an adiabatic mapping of the quark eigenstate to the eigenstate of quarks for a system with mesons at their vacuum values. The models discussed here do not possess confinement in the sense that they do not exclude plane-wave quark states. Therefore such a mapping is always possible since the Dirac spectra for mesons in both a soliton configuration and in the vacuum configuration have positive and negative energy continua. For our method to be sensible the notion of “valence” must be generalizable to a system with small amplitude motion.

W. Broniowski,

We note that the Dirac equation equations

has a time-dependent

T.D. Cohen / RPA method

656

in (2.4) used for the derivation

quark hamiltonian

of the QMRPA

if the self-consistent

meson fields

are used for vi+ With such a time-dependent hamiltonian, it is apparent that any plausible separation into “valence” and “sea” states will yield sea states as well as valence states which are time dependent. The easiest way to see that the sea in the time-dependent problem must be time dependent is to consider the Dirac equation for 6q, (2.6). For any reasonable q,(t), there will be no constraint which restricts Sq,( t) to be orthogonal to any infinite set of time-independent sea states (including the sea states for the time-independent problem). The notion of “valence” states based upon adiabatic mapping of eigenstates used in the time-independent problem can, however, be generalized to the small amplitude problem by noting that the mesons in any self-consistent solution of the RPA equations have only a periodic (in fact harmonic) time dependence. As has been observed in periodic TDHF theory 13) and in the formal treatment of solitons lo), a time analog of Bloch’s theorem exists which allows one to define time-periodic eigenstates. In particular, if a fermion state satisfies the time-dependent equation h(t)l$(t))= with h(f-t~)= h(t), then [$(t)) periodic “eigenstates”. Namely,

can be written

I+(r)> =C c, exp J where c, are time-independent which characterize the system

(2.14)

i&l+(t)>, as a linear

combination

(--iE,r)l$,(t)) ,

constants, the (I,!J,(t)) are time-dependent and are periodic with the same period I+#+

7)) = I+,(t)> *

of time-

(2.15) eigenstates as h(t) (2.16)

The e, are the “quasi-energies” which correspond to the mode j. The quasi-energies and time-periodic eigenstates can be found by study of the operator (i/ T) log U( 7,0), where U(t, ti) is the time evolution operator from t, to t. In particular, the quasienergies are the eigenvalues of (i/7) log U(T, 0), while the eigenvectors are the time-periodic eigenstates evaluated at t = 0. Thus, it is apparent that in a self-consistent mode of oscillation, the notion of a single quark eigenstate is valid, as long as one realizes that such an eigenstate is, in fact, a time-periodic function. Given a valid notion of eigenstate, one can see that the adiabatic mapping method may be used to distinguish “valence” from “sea” states. One simply considers the amplitude of oscillation as an adiabatic parameter which connects the eigenstates of the time-independent problem (for which “valence” and “sea” states are clearly distinguished) with the time-periodic eigenstates of the time-dependent problem. The theory outlined above clearly does not take into account the effects of the time-dependent sea and such time-dependent sea effects might, in principle, be important. On the other hand, we did not take sea effects into account when solving

W. Broniowski, T.D. Cohen / RPA method

the time-independent the time-dependent

problem problem.

and it would

not be consistent

We will consider

both

657

to include

time-dependent

them in and

time-

independent sea effects to be quantum corrections. Whether such corrections are small or not must be determined a posteriori. The question of whether these timedependent

sea effects are important

is beyond

the scope of the present

3. The chiral quark-meson

paper.

model

A simple, physically interesting example on which we can apply the QMRPA is the chiral quark-meson model, proposed by Birse and Banerjee4), and independently by Kahana, Ripka and Soni ‘). The model is based on chiral SU(2) x SU(2) symmetry. The CQMM, despite its simplicity, produced very encouraging results for the properties of N and A. We begin this section with a brief review of the model. It is based on the Gell-Mann-Levy a-lagrangian 14) with quarks of three colors interacting via exchange of a chiral quartet of meson fields: o and n. In the following the color index is suppressed. The lagrangian density is ~==[iBe+g(a+iY5?.p)]~++l(a~u)2+t(a~~)2-

U(a,7r),

(3.1)

with U(a,?r)=$A2[02+~2-V2]2+Cu, where +, (T and m represent interaction lead to a nonzero

(3.2)

the quark, sigma and pion fields. The meson-meson vacuum expectation value for (T given by = - Er I

(6, where F,, = 93 MeV is the pion-decay as

constant.

One can write the constants

(3.3) in (3.2)

A’=(wl2,-m2,)/2F2,,

v2= F~(m~-3m~)/(m~-m~), c=miF,.

(3.4)

where m, and mrr are the masses of the mesons. If the pion mass is non-zero, the term in (3.2) linear in @ explicitly breaks the chiral symmetry and leads to desired current-algebraic results (CVC, PCAC). The vacuum expectation value of (+ gives the free quark a dynamical mass of ms=gF,.

(3.5)

The model will be treated semiclassically, as described in sect. 2. To construct a baryon, one considers three quarks, coupled to a color singlet, in valence orbitals represented by time-dependent spinors q,( r, t), i = 1,2,3.

W. Broniowski,

658

Stationary

solutions

are of the hedgehog

T.D. Cohen / RPA method

to (5.1) have been found 4,5). The fields for these solutions form 15). The three quarks,

a single space-spin-isospin

state represented

coupled

to a color singlet,

occupy

by the spinor

(3.6) where x is the hedgehog

spin-isospin

spinor

x = The appropriate

mean-field

ansatz

(ui - d?)/fi.

for the mesons,

n:(r)

=

(3.7) consistent

with (3.6)~(3.7),

rr,(r)i".

is

(3.9)

The hedgehog ansatz (3.7)-(3.9) is the generalized spherical symmetry ansatz for the system. It breaks individually the spin and isospin invariances, however it remains invariant under the SU(2) group of combined space and isospin rotations. The conserved quantum number is the grand angular momentum K=J+l,

(3.10)

where J is the angular momentum and I the isospin. The system in ansatz (3.7)-(3.9) has Kp = O+, where P is the parity. Another useful symmetry respected by the hedgehog ansatz is the grand-reversal symmetry R, which is composed of time-reversal followed by an iso-rotation through rr about the 2-axis. Under the action of R the field operators way:

change in the following

44 t)+- dr, -t) ,

Hence, the time-independent spinor transforms as

‘iT(r, t)+7r(r,

-t).

mean

fields are unaffected

meson

4(r, t) + ~bw(r,

(3.11) by R. The quark

-t) ,

where X is the complex conjugation. We notice that the hedgehog spinor eigenstate of R. The described symmetries are helpful in practical calculations the model ‘).

(3.12) is an with

W Broniowski, T.D. Cohen / RPA method

For the stationary

hedgehog

fields the Euler-Lagrange

659

equations

are

(3.13) where we have introduced

0

the notation

Mcr=g

[I -1

0

0

1’

+ Mpo+

ho=

-1

M,=g

[ -1

1

0’

M,ro.

(3.14)

A fully self-consistent solution to eqs. (3.13), found numerically by Birse and Banerjee 4), is presented in fig. 1. It is localized near the origin, and asymptotically the fields approach their vacuum expectation values. As is typical for a ground state, the fields in fig. 1 are nodeless. To put the QMRPA analysis in prospective, we will first examine in some detail the dependence of the stationary solution on the coupling constant g. The topic is closely related to the question of the classical stability in the system. Our calculations were performed

with the values

of the meson

masses

used in ref. 4), namely

m,, = 139.6 MeV, m, = 1200 MeV .

(3.15)

We have verified numerically that the results presented below do not qualitatively depend on a particular choice on these parameters and hold even for the chiral limit (m, = 0) or the nonlinear version of the a-model (m, + co). We concentrate

on two characteristics

of the solution:

the quark

eigenvalue,

E,

and the hedgehog baryon mass, M, which is the total energy of the soliton. The results are shown in fig. 2. Such plots for CQMM were first obtained by KGppel and Harvey 16). Similar results were also found for the Friedberg-Lee model ‘). We note that for small values of the coupling constant, g g,, two stationary solutions of the type presented on fig. 1 exist for each value of the coupling constant.

660

W. Broniowski,

TD. Cohen / RPA method

\

\ 477r2(G2+F2) \ \

(a) _ I

2.0

I

I

I

I

I

2.5

I

I

I

I_

3.0

r In fm

r

in fm

Fig. 1. The hedgehog solution for g = 5.38. (a) Quark upper and lower components, G and F, in units of frnm3” (solid lines) and the radial quark density in units of fm-’ (dashed line). (b) Meson fields, r and S-, in units of F, (solid lines) and o*+ T? in units of Fi (dashed line).

Anticipating the result of the next section, we state that the solutions on the solid line of fig. 2 are classically stable, and those on the dashed line are classically unstable. We will refer to these lines as the stable and unstable branches. Since the quarks are bound, the eigenvalue E (fig. 2a) must lie between fmq, where m, is the quark mass (3.5). As the coupling constant is increased, E on the unstable branch asymptotically approaches mq. The slope, ds/dg, is negative on the stable branch; this reflects the fact that the interaction between the quarks and mesons is attractive. For the stable branch we have found that at the value g - 13.5 the eigenvalue merges with the negative continuum and the bound solution ceases to exist.

W. Broniowski,

T.D. Cohen / RPA method

661

E

-1000

1 (a)

-1500

’ 0

%r ’ 2

I

’ 4

I 6

I

I

I

I 8

I

I

I

I IO

I

I 12

I

I

I

14

gcr I

0

2

4

6

8

I

IO

I

I2

I4

Fig. 2. Dependence of (a) the quark eigenvalue E, in MeV, and (b) the hedgehog baryon mass A4, in MeV, on the coupling constant g. The solid (dashed) lines correspond to stable (unstable) solutions. The dotted line denotes (a) the limits for the bound state quark eigenvalue, fmq, and (b) the mass of three free quarks. The arrow points the critical value of g.

The plot of the hedgehog baryon mass M versus g (fig. 2b) develops a characteristic cusp at g,, [ref. ‘)I. The origin of this behavior is the fact that the quantity (3.16) is continuous even at the critical point. Thus the branches have to join with equal slopes and develop a cusp. The mass of the unstable solution is always bigger than the mass of three free quarks, 3m,, denoted on fig. 2b with a dotted line. Also a fragment of the stable branch near g,, has M > 3m,. Such states, although classically stable, are not

W. Broniowski,

662

T.D. Cohen / RPA method

energetically stable, i.e. they can leak out quantum-mechanically to a lower energy plane-wave solution. This remark has to be treated with some care, since the quantum corrections

could alter the effective potential

to a large extent. The part of the solid

line below 3m, represents the energetically stable solutions. If the coupling constant is sufficiently big (g > 1 l), another pair of solutions to eqs. (3.13) emerges. It corresponds to stationary radial excitation of the system and has extra nodes in the fields. At still higher values of g additional stationary radial excitations of the system can be found.

4. Breathing modes in CQMM In our application of QMRPA to the chiral quark-meson model we restrict ourselves to a class of excitations for which the quarks and mesons remain in the hedgehog ansatz modes in the system, with K ’ = O+. As we shall see, the breathing modes are responsible for the critical behavior described in the preceeding section, and are, from this point of view, most interesting. The meson fields are assumed to have the form (3.9), where now the radial functions u and 7r are time dependent. Similarly, the quark spinor (3.6)-(3.7) is now described by time-dependent radial functions G and F. In general, three time-dependent spinors are needed to describe the semiclassical oscillations of a three-quark system (sect. 2). For simplicity we restrict ourselves further and consider only coherent excitations, in which the motion of all three quarks is characterized by a single, time-dependent spinor. Such coherent excitations have maximal quark overlap, and on general grounds are expected to correspond to the lowest frequencies in the system. They in fact contain the decay mode of the unstable solutions and thus are important in the stability analysis. The restrictions we have made are of purely technical character. At the expense of numerical effort one could consider other Kp excitations, as well as non-coherent quark oscillations. In the language of the nuclear physics of deformed nuclei, the hedgehog excitations correspond to intrinsic excitations of the system. It is straightforward to construct the matrices A and B in eqs. (2.13) for the class of oscillations

described

above.

We obtain

(ho- &)/A’

2-1/2M&,/A3’2

A=

2-1/250+~+/~3/2

f(-v’+

B=

0 2--1/2&TMT/A’/2

2-‘i2M&,/A3’2 f(_v’+ v”_A2)/A2

where

-s2v

M=

j/n,=

v”+A2)/A2

Lb2 s2v &r&r

s2v

1

(4.la)

’

(4.lb)

’

I

1

so &?T a2v -&r2

.

I;X>O

(4.2)

W. Broniowski,

T.D. Cohen / RPA method

Entities

663

ho, M,, M, and &, are defined in (3.14). The vectors have components in the space of G, F, CTand r,

X and

Y in eqs. (2.8)

(4.3)

Since M, and M, are real, A and B are real and one may use reduced equations ‘) to find the spectrum, namely (A-

B)(A+

B)(X+

Y) = (w/A)‘(X+

dimensionality

Y) ,

(4.4a)

- Y) .

(4.4b)

or, equivalently, (A+ B)(A - B)(X

- Y) = (w/A)‘(X

The eigenmodes of the QMRPA equations appear in pairs conjugated by the grand-reversal symmetry. One can see that according to (3.11), (3.12) the action of R results in a flip of the sign of w and exchange of X and Y. A convenient feature of RPA is the separation of the spurious zero-frequency modes from the physical modes in systems with broken symmetries “). There is a pair of spurious zero-eigenvalue modes in our case associated with the U(1) symmetry, broken by the choice of phase of the ground-state quark spinor. Other spurious modes, related to the breaking of the translational and isospin invariance, are not contained in the Kp = O+ excitations, considered here. To find the spectrum numerically we have diagonalized eqs. (4.4) in the plane-wave basis of ref. I’). The basis functions are the spherical Bessel functions j,(kr), with k such that at some large cutoff radius R one hasjl( kR) = 0. Such basis is orthogonal in the truncated space (0, R). One also has to specify a cutoff in the momentum space, k,,, . If the cutoffs are sufficiently large, they do not affect the results. We have used R as large as 30 fm and k,,,

corresponding

to basis states with as many

as 50 modes. Our numerical calculation confirmed that the solutions on the solid line of fig. 2 are in fact stable. All eigenvalues were found to be real in this case. For the solutions on the dashed line one eigenvalue is imaginary, hence instability occurs. The situation is depicted in fig. 3, where the lowest nonspurious squared eigenvalue, WC,is plotted versus g. On the unstable branch CCJ~ < 0, and w0 is imaginary. As one moves toward the critical point wi gradually approaches zero; it crosses and becomes positive on the stable branch. The existence of a physical zero-eigenvalue mode at g = g,, is caused by the degeneracy at this point of the two solutions, one stable and one unstable one (fig. 2b).

W. Broniowski, T.D. Cohen

664

/ RPA method

\

2 ,Jrn,2

\ ‘\

\

-2

\ I

-\

---_ -I

%r

3.90

3.95

4.00

4.05

4.10

4.15

4.20

9 2

I

1

A---

i

%‘m7r 2

I

I

o2

/I

I \

-2 -

/’ \

_

/ L’

Ycr

(b) -4 3.5

1

“““““I”I’II’I”“““” 4.0

4.5

5.0

5.5

6.0

6.5

4 Fig. 3. (a) Dependence of the square of lowest RPA frequency, OJ& in units of M& on the coupling constant g. Solid (dashed) lines correspond to the stable (unstable) solutions. In the hatched region, representing CO’> mZ,, the pion is excited to scattering states. The arrow indicates the critical value of g. (b) Same as (a) for a different range of g.

The hatched region on fig. 3 denotes the eigenvalues in the continuum w2> m',. Whenever the eigenvalue is bigger than the pion mass, the smallest mass in the system, the pion is excited into a scattering state. From fig. 3 we can see that for the stable solutions with g > 3.98 there are no bound hedgehog excitations. Since there is no experimental evidence of bound excited states of the nucleon, realistic models should not predict them. This is the case of CQMM of ref. 4), which uses g = 5.38. It is interesting to look at the shape of the eigenmode responsible for the instability. Fig. 4 shows the quark and meson amplitudes of the mode corresponding to negative

W. Broniowski,

665

T.D. Cohen / RPA method

r

in fm

(b)

“““‘ll”““l’l(J”‘l(“” 0.0

0.5

1.0

1.5

2.0

2.5

3.0

r in fm

Fig. 4. The instability of F,, (b) amplitudes

mode of the unstable solution far g = 5.38: (a) Amplitudes of Re SC and Re SF in units of fm-3’2 and (c) amplitudes in units of fm-“‘. The overall scale of all fields is arbitrary.

of So and ST in units of Im 6G and Im 6F

W. Broniowski,

666 cd:

T.D. Cohen / RPA method

for g = 5.38. We plot Re 6G = 22”’ Re (X, + Yd) , Im SG = 2-l”

Im (X, - YG) ,

Re 6F=2-“2

Re (XF+

Im SF = 2-l”

Im (X, - YF) ,

Srr = 2-l”

Re (X,, + Y,) .

6a = 2-“2 Re (X, + Y,) ,

YF),

(6.4)

The real parts of the quark spinors develop a node, which is required by the orthogonality condition (4.7). The amplitudes of various fields are comparable, which reflects the strong coupling between the quarks and mesons in the system. It is possible to perform a pion-hedgehog baryon phase-shift calculation in the model. This is particularly simple if w2< rni and w2< rnt. Then the quark and o oscillations remain bound and only the pion can be in a scattering state. Analogous calculations I’) have been carried out by for the purely mesonic Skyrme model “) by Breit and Nappi. Matching asymptotically the pion excitation to the free solution, for a given momentum k = (co’- mt)“‘, &r(r) = a(k)j,(kr)+ one obtains

b(k)n,(kr),

(6.5)

the phase shift G(k)=tan~’

[-b(k)/a(k)].

(6.6)

The results for g = 5.38, the value in ref. 4), are shown on fig. 5. Experimental data show the Roper resonance at the energy of -500 MeV. Fig. 5 shows no evidence of a resonance. The phase shift peaks around 300 MeV, reaching barely 61”. We note, however, that the model should not be expected to give a realistic description of rrN scattering. Due to low quark mass, comparable to the position of the Roper

8(k)

0.4

0.6

0.8

1.0

k/mq

Fig. 5. The pion-hedgehog baryon phase shift. The hatched region indicates the values momentum k for which the calculation breaks down due to quark emission.

of the

W. Broniowski,

resonance,

661

T.D. Cohen / RPA method

the quarks can get excited into unbound

scattering

states at low energies.

This is unphysical and suggests that the model has to be supplemented confining mechanism to describe properly the nucleon excited states.

with some

5. Conclusion There are clear benefits of the quark-meson RPA method. The QMRPA equations are the classical stability equations for quark-meson systems, and as such can be used to determine the crucial question whether a soliton is stable or not. We have demonstrated that the solitons describing baryons in the chiral quark-meson model are in fact stable. The calculation revealed that there is a critical value of the coupling constant in the model, below which no solitonic solutions exist. Above this value the model develops a pair of solutions: one stable, identified with the baryon, and unstable, which is unphysical. The stable and unstable branches join at the critical point, forming a characteristic cusp in the dependence of the soliton mass versus the coupling constant. We should stress that such a behavior is typical for nontopological solitons in three space dimensions; it has been found for models with quarks ‘,19) and f or models with charged mesons 20). We have investigated small amplitude breathing vibrations of the soliton for both the stable and the unstable branch. For the unstable branch the QMRPA spectrum contains a mode with imaginary frequency. Hence the solitons on this branch decay; they do not correspond to physical states and their existence is just a formal consequence of the (nonlinear) model. On the stable branch the QMRPA spectrum is real and the solitons are classically stable. Continuity requires that at the critical point, where the two branches join, a zero-eigenvalue mode appeares. We observed this phenomenon in our numerical study. There is one important feature of the QMRPA spectrum for the stable branch. We have found that close to the critical point there is an excitation with frequency smaller

than

the mass of the lightest

particle

in the model

(the pion).

Hence

the

field configuration for this excitation is bound. The physical interpretation of such an excitation would be that the nucleon has a bound excited state with an appropriate RPA excitation energy. Experimentally, there is no evidence of such states; all known nucleon and delta resonances are observed as pion scattering states. As one departs from the critical point along the stable branch, the frequency of the bound oscillation increases rapidly and the mode becomes unbound with the pion in the continuum. We note that the value of the coupling constant which gives reasonable predictions for static properties of N and A [ref. “)I is in the region where no unphysical bound QMRPA excitations are predicted. On the other hand, the calculation of the P,, pion-nucleon phase-shift does not yield satisfactory results. This indicates that the chiral quark-meson model is not adequate for the description of physics at these energies. The reason is the low quark mass of -500 MeV, which allows for emission of quarks at energies of the

w. Broniowskz,

668

T.D. Cohen / RPA method

order of the Roper resonance. This suggests strongly the necessity of confinement for the description of baryon resonances in the framework of quark-meson models. One may in principle also calculate the QMRPA corrections to the ground state. Such a calculation must include what is commonly referred to as correlation energy effects in nuclear physics and the renormalization of the zero-point motion in soliton physics. The calculation would undoubtedly be quite difficult as the effects are quantum mechanical in nature and systematic techniques for dealing with quantum corrections for this sort of system have not been developed. One might also expect severe numerical problems to plague such a calculation. We have also alerted the reader to the problem of sea-quark effects. For oscillating quark-meson systems these effects are associated with the time-dependent Dirac sea. Since the sea effects have not been incorporated in the stationary case, their neglect in QMRPA was consistent. The effects of the Dirac sea and other quantum corrections can be important. It would therefore be of great interest if some feasible calculational

schemes

to describe

them were developed.

We wish to thank Manoj Banerjee for many useful discussions. US Department of Energy and the University of Maryland Computer is gratefully acknowledged.

Support of the Science Center

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

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