The generalized hedgehog and the projected chiral soliton model

Nuclear Physics A481 (1988) 727-764 North-Holland, Amsterdam

THE GENERALIZED

HEDGEHOG

AND

SOLITON M. FIOLHAIS’*Z

K. GOEKE=,

THE PROJECTED

CHIRAL

MODEL

F. GRIMMER’

and

J.N.

URBANO’

’ Departamenfo de Fisica, Universidade de Coimbra, P-3000 Coimbra, Portugal ’ Institut fiir Kerphysik, Kernforschungsanlage Jiilich GmbH, D-51 70 Jiilich, West Germany ’ Institut fiir 7’heoretische Physik II, Ruhr-Universiilir Bochum, D-4630 Bochum, West Germany Received 5 October 1987 (Revised 20 November 1987) Abstract: The linear chirai soliton

model with quark fields and elementary pion and sigma fields is solved in order to describe static properties of the nucleon and the delta resonance. To this end a Fock state of the system is constructed which consists of three valence quarks in a 1s orbit with a generalized hedgehog spin-Aavour configuration cos q/ul) - sin nldf). Coherent states are used to provide a quantum description for the mesonic parts of the total wave function. The corresponding classical pion field also exhibits a generalized hedgehog structure. In a pure mean field approximation the variation of the total energy results in the ordinary hedgehog form (q = 45”). In a quantized approach, however, the generalized hedgehog baryon is projected onto states with good spin and isospin and then noticeable deviations from the simple hedgehog form occur (n=20’), if the relevant degrees of freedom of the wave functions are varied after the projection. Various nucleon properties are calculated. These include proton and neutron charge radii, and the magnetic moment of the proton for which good agreement with experiment is obtained. The absolute value of the neutron magnetic moment comes out too large, similarly as the axial vector coupling constant and the pion-nucleon-nucleon coupling constant. However, due to the generalization of the hedgehog, the Goldberger-Treiman relation and a corresponding virial theorem are fulfdled. Variation of the quark-meson coupling parameter g and the sigma mass m, shows that the g, is always about 40% too large compared to experiment. The concepts and results of the projections are compared with the semiclassical collective quantization method. it is demonstrated that noticeable deviations occur for the delta-nucleon splitting, the isovector squared charge radius and the axial vector coupling constant.

1. Introduction

Quantum chromodynamics (QCD) is currently considered as the fundamental theory of the strong interaction ‘). It is basically characterized by three properties, i.e. confinement, asymptotic freedom, and chiral symmetry. The first two are a consequence of the colour-W(3) gauge invariance, the third one holds for massless quarks and is hence to a great extent valid for systems involving up- and down-quarks only. The long distance, nonperturbative regime of QCD has so far not yet been solved. Only lattice gauge calculations ‘) have provided some information on this regime but the tremendous numerical difficulties place these calculations at the borderhne of the capacity of present day computers. This is why in recent years there is an increasing interest of theoreticians in effective phenomenological field theories which replace the complex interacting quark-gluon system by a simpler 0375-9474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

M. Fiolhais et al. / Generalized hedgehog

728

one involving the relevant Recently

quark and boson degrees there

fields. The hope is that by this one concentrates

of freedom

has been

in the nonperturbative

much

type 3-5) and its chiral invariant

interest

in soliton

generalization

models

suggested

and Levy “). The latter model allows for spontaneously involves

besides

quarks

a sigma field (scalar,

scalar, isovector). By a slight explicit breaking the hypothesis of the partial conservation of The models provide solitonic solutions from and delta can be extracted *-16). The quarks

on

regime. of the Friedberg-Lee

in fact earlier by Gell-Mann broken

isoscalar)

chiral symmetry

and

and a pion field (pseudo-

of the chiral symmetry one can fulfil the axial vector current 6Z7)(PCAC). which the properties of the nucleon are not confined absolutely and it is

hoped that the baryon properties in the low energy region are governed mainly by binding forces, describable in these models, rather than confining interactions. There are some attempts to incorporate confinement also into these models; however, so far the results are not conclusive ‘7,‘8). In addition, a generalization of the model to include vector mesons has been suggested I’). An alternative type of soliton model, also of high current interest, was proposed some time ago by Skyrme 20). In this approach the quarks are assumed to be integrated out and only boson fields of pion and sigma character occur. The baryon number is identified with a topological winding number. The soliton of this model, called skyrmion, is quantized in a semiclassical way 2’) in order to obtain observables of the nucleon and delta. Technically the skyrmion is easy to handle and many properties of this model have been studied by now, see e.g. for reviews refs. 22V23). In the present paper we consider the lagrangian of the linear chiral soliton model first suggested by Gell-Man and Levy “). It has been recently considered by several authors with special attention to providing solutions with proper angular momentum and isospin quantum numbers ‘07’2-‘6) in order to identify the nucleon and the delta isobar. Basically

two sorts of methods

have been suggested.

One consists

in assuming

hedgehog structures for the quarks and the pions and to apply projection techniques to the mean field solution ‘23’3). The other is the coherent pair approach 14,15) in which coherent states with suitable tensorial properties are constructed and coupled with bare nucleon

and delta quark states. There are also some suggestions

lo) for a

simplified projection technique in which the meson fields are considered as roughly classical, although this limit is not reached in the actual calculations. All the above approaches reproduce in some way the nucleon and delta properties. However, there is one common weakness in these solutions: They noticeably violate the Goldberger-Treiman relation and they do not give unique values for the pionnucleon-nucleon coupling constant, since they violate also an associated virial theorem. Since the Goldberger-Treiman relation is intimately connected with the chiral symmetry, i.e. the basic constituent of the chiral models, its violation is a serious shortcoming. In addition, the quality of the lagrangian as such cannot be judged since one does not know whether insufficiencies in the results should be attributed to the lagrangian, to the method to solve the equations of motion, or to both.

M. Fiolhais et al. / Generalized hedgehog

It is, therefore,

the first objective

the chiral

soliton

model

associated

virial

theorem.

structures,

which

of the present

fulfills

Since

model

consists

projections

on angular

technique,

than the coherent

of using a generalized

momentum

to construct

the Goldberger-Treiman

the projection

seems to be more appropriate

the present

paper

129

J and isospin

relation applied

of

and the

to hedgehog

pair approximation

hedgehog quantum

a solution

24*38),

ansatz 16) and separate

numbers

T of the nucleon

and the delta. Preliminary results of these calculations have already been published 16). In contrast to the ordinary hedgehog ansatz the generalized one also provides J # T states and hence allows for a more detailed study of the low-lying baryon spectrum. The second objective is to compare the projection theory with the standard semi-classical collective quantization method used e.g. in the Skyrme model. This will be done by a direct comparison

of our numbers

with those of Cohen

and

Broniowski 40). The various assumptions entering the collective quantization method will be evaluated by considering sum rules and narrow rotational overlaps. The paper is organized as follows: Sect. 2 introduces the lagrangian and reviews some of its general properties. Sect. 3 discusses the generalized hedgehog mean field solution, its formal background and some numerical results. Some remarks are made about the Dirac sea. The projection technique is presented in sect. 4, after the general outline and the review of the sum rules, the derivations of the various overlaps are sketched and the numerical procedure is shortly described. The general formalism to obtain observable quantities is presented in sect. 5. Numerical results concerning the projected soliton solutions are presented and discussed in sect. 6. This includes the Goldberger-Treiman relation, comparison with experimental data, an exposure of trends, and the presentation of excited baryon resonances. A conceptual and numerical comparison with the collective quantization method is performed in sect. 7. A summary,

general

discussions

and an outlook

in sect. 8, finalize

the paper.

2. The lagrangian Following Gell-Mann and Levy “) the lagrangian fields, q^(x), sigma and pion fields, G(x) and k(x),

involving operators of quark respectively, can be written as

~(x)=~(iyC”a,)~+~a”~a,~+_cia”~;ra,~-gg(~+ji7’my,)q*-U(&,,) with the self-interaction

(2.1)

potential U(~,,)=ah2(~*+~2-v2)2+C~++.

For c = 0 the lagrangian in the infinitesimal limit

is invariant

under

the chiral transformation,

(2.2) which reads

s*=(1++iy5~‘x)$, * 4 = $(l +;iy5T * x) , &=&‘+7;‘x $=&‘-&x,

3 (2.3)

M. Fiolhais

730

et al. / Generalized

where x is a real infinitesimal

isovector

and time (global

The corresponding

symmetry).

hedgehog

pseudoscalar

quantity, current

independent

on space

is the axial vector current

AI=~ty,Y57q*$.~a~~-~a”~, which

for

c = 0 is fully

spontaneously

broken

PA,

conserved,

with the vacuum

(2.4)

= 0. In nature

the chiral

symmetry

is

values

(Ol$( r)jO) = nrv = 0 , (Ol&( r)[O) = uv f 0,

Since

(2.5)

the corresponding

Goldstone boson is the pion, which has a finite mass symmetry is also explictly broken, which means that conservation of the axial current (PCAC), PA, = cn. Now,

m, = 0.1396 GeV, the chiral

we have only partial according

to the hypothesis

we should

have

of PCAC,

PA,

which

has been

experimentally

= fwrnin,

where fw = 0.093 GeV is the pion decay constant. To achieve theory we must identify c = f,mL. To fix the other parameters

confirmed,

(2.6) this in the present of the lagrangian it

is helpful to shift the sigma field, G(r) = G,,(r) + u, and to rewrite new field GO. One gets

-&JA~u~-A~v~~,+c)-A~~,,&,&~-[~A~(&v~)+cu,+~],

it in terms of the

(2.7)

We now demand that the vacuum nr, = (+,)= 0 is a solution of the lagrangian at the mean field level. From this we get A‘ov( at - v*) + c = 0. Furthermore we demand the pion field to have the proper tail associated to the physical pion mass. This yields A ‘( oz - v’) = rni . Together with c =fpmf, from the axial vector current we obtain uv = -f*. For the calculations and the following formulae we prefer to change the signs g+ -g,

Go-+ -Go,

n+

-n,

u,+

-(T, and we get altogether

2 ~2+$,

(2.8)

where mf, can be read off from the sigma mass term of the lagrangian. Due to the finite value a, = f, the quarks have an asymptotic mass of &. Thus confinement is not described for finite g. Whereas the mass of the pion is known it is not clear

M. Fiolhais et al. / Generalized hedgehog

which value to assume for rnrr since the physical The suggestions

reach from the m-meson

0.7 GeV) to the lowest glueballs the results

of the present

meaning

of the u-particle

in the nucleon-nucleon

(m, = 0.7 - 1.2 GeV). Actually,

model

731

are rather

insensitive

long as it is larger than 0.6 GeV. Hence throughout

is unclear.

interaction

(m, =

as we shall see later,

to the actual

value of m, as

the paper, if not otherwise

stated,

we use m, = 1.2 GeV. With the conjugate

momenta

i),(r)

and p,(r)

of the sigma and the pion field,

respectively, the hamiltonian density can be written in a way from which one can identify the kinetic energy of the quarks, the asymptotic quark mass, the quark-sigma interaction, the quark-pion interaction, the kinetic energy of the sigma and the pion fields, and the nonlinear sigma-pion energy.

with %!?Jr) = $(r)(-ia x,,(r)

* V)q^(r) ,

= &i(rMr),

Z&r) = gi(rMr)&(r) , ZJr)

= gi(r)[hT . %r)lq^(r),

XN,(r)=~A2[(&(r) +(2&i(r)+4f,Go(r))&(r)

* &(r))2+6~(r)+4f,G~(r) . G(r)],

%f~(r)=~[~~(r)+(V~o(r))2+m~(~o(r))2], %$(r)=t[P,(r)P,(r)+V&(r).V&(r)+mZ,&(r)*+(r)]. The formulation

of the projections

becomes

(2.10) simple

if one uses for &

and &

the

Fock space representation &=;

dkw,(k)(a+(k)a(k)++), I

fir=;

; ,=I

dkw,(k)(b:(k)b,(k)+;),

withw,(k)=(k2+m~)“2,w,(k)=(k2+m~)”2

(2.11)

and a(k) and b,(k) being the boson

annihilation operators for a sigma and a pion in the momentum t = 1,2,3 indicates the Cartesian isospin component.

state

I/r). The

3. The generalized hedgehog mean field soliton This section deals with the derivation soliton. The mean field approximation

and numerical calculation of the mean field consists in assuming a pure product Fock

M. Fiolhais et al. / Generalized hedgehog

732

state for the quarks equations

and classical

can be derived

states and the classical principle

states. This view is adopted

this procedure

is equivalent

Fock space if one describes

in the present

variational

The corresponding

of the static energy with respect

fields. Actually

in an full quark-boson

to nonrelativistic

fields for the bosons.

by variation

theories

static

to the quark

to a variational

the bosons

by coherent

paper since it allows easily to use analogies

like Hartree-Fock

and motivates

also the use

of projection techniques. Although coherent states are explicitly formulated in a given basis the final results in the mean field approach are independent on the basis chosen since only expectation values of normal ordered products of boson field operators are required. The latter feature is no longer true for projected will therefore be discussed at the beginning of sect. 4. We start the present

section

the free basis used throughout

3.1. COHERENT

by reviewing

some properties

results and

of coherent

states in

this paper.

STATES

In the following

we list some properties

of boson

sigma field as example. The G(r) and its conjugate be expanded in the free basis as dk (2w,(k))p”2(a(k)

G(r) = (2~)~~‘~

coherent

states taking

momentum

operator

eik.‘+u+(k)

e-ik’r),

the free g_(r)

(3.11

I Fu(r)

where

u(k)

is defined

= --i(277y3’2

dk ($w,(k))“2(a(k)

and w(k) have been already by a(k)

of the annihilation

= 0. A coherent

defined

eik”-a’(k)

e-i“‘r),

in sect. 2. The boson

state 1.X) is defined

can

(3.2) vacuum

(0)

such that it is an eigenstate

operator a(k)P)

=

rl(k)P) ,

(3.3)

yielding

12) = N-“2 exp

(3.4)

with N being a normalization constant such that (2 1.X) = 1. From the above definitions it follows directly that the coherent states have to a large extent classical properties as e.g. (~I:~n(r):I~)=(~I~(r)I~)“,

(q:P:(r):p)=o,

(3.5)

M. Fiolhais et al. / Generalized hedgehog

where

the normal

following coherent

ordering

expressions

is done

are useful

with respect

to evaluate

733

to the boson

matrix

elements

vacuum between

IO). The different

states:

with (I, 1I,) = N;“2N;“2 from which one gets immediately Analogous

formulae

exp

dk

rlfYk)rl2(k)

the normalization

factor of eq. (3.4).

exist for the pion field and the corresponding

coherent

state (3.8)

3.2. THE

GENERALIZED

HEDGEHOG

In the mean field approximation to have the form

SOLITON

the Fock state for the hedgehog

baryon

Ihl) = ldtl>lalh)l~)*

is assumed

(3.9)

Here Iq;J = c:c:c:lO), where the CT creates a quark of colour i in a space-spinflavour state which is the same for all colours and which is assumed to be nodeless Is valence orbital (rlc'l0) =

with Ix) being a spin-flavour

7k( ;)I’)

function.

If one assumes

circle, i.e. if (El(17,,162(r) +rr2(r)lflr,,,)lE) that Ix) has to be of hedgehog type

=ft,

the fields to be on the chiral

then it has been

IXhh)=4wHm and the pion coherent

state In,,)

(3.10)

iv(YiiJ.

shown

in ref. “)

(3.11)

must be such that

Ip

Q(r),

t=1

n,(r) = Ubhl~,(r)l4h) = 5 @i(r), I

t=2

!;ZWr),

t=3.

(3.12)

734

Actually

M. Fiolhais et al. / Generalized hedgehog

some recent investigations

is not necessary smallest

energy,

quark hedgehog

and

for quarks

the vacuum,

in s-states

is bound

of each component momentum

In a pure mean field solution

solution

with The

I$,,,,) both have the property

that

operators,

(J + T) I&,I,)= 0,

the solitonic

structure.

of the grand

and isospin

of the chiral circle

to have the hedgehog

Ixhh) and also the total hedgehog

they are eigenstates T are the angular

generally

besides

show 44) that the assumption

spin G = J+ T, where J and respectively.

(J+T)Ix,,d=O.

there is no need to generalize

(3.13) the hedgehog

structure.

However, in an angular momentum and isospin projection formalism, which goes beyond the mean field, the hedgehog is no longer distinguished and hence the spin-flavour function 1~) unknown. We do not try to generalize the considerations of ref. “) to include projected states. We prefer to use the convenient parametrization of Ix). Ix)=

R(P)@h[lul)cos 11-Id?) sin 71

(3.14)

as suggested in refs. 1’,27). Since R(p) and d( /?) are simple rotations around the y-axis in spin and isospin space, respectively, we consider only the n-degree of freedom explicitly and assume the generalized hedgehog structure r6)

IX& = Id>~0s77-Id?) sin 7,

(3.15)

which reduces to the ordinary hedgehog for n = 45”. The IX& is no longer eigenstate of all three components of the grand spin G, but only of the third one: (3.16)

(-J + TJ) I xgd = 0 .

One can show, by means of a simple variational calculation in the spirit of refs. ‘1X27) that in the mean field approximation the pion field compatible with eq. (3.15) yields a coherent

state Ifl,,)

with the property

(IIJ~,(r)lQJ

= f

@2(r)

,

If

t=2

(3.17)

I

Q%(r),

t=3

with a,(r) = &(r). Of course this reduces again to the ordinary hedgehog for Q,(r) = (P2( r) = 03(r). With this pion field and a spherical sigma field also the total generalized hedgehog I&h) = is eigenstate

of the third grand

Is;tl)ln,tl)l~)

(3.18)

spin component (-‘3 +

TdMgd= 0 .

(3.19)

M. Fiolhais et al. / Generalized 3.3. THE

EQUATIONS

Using

the ansatz

written

OF MOTION

(3.18)

FOR THE

hedgehog

GENERALIZED

for the generalized

hedgehog

735 HEDGEHOG

the total

SOLITON

energy

can be

with

dr*(r):IICTgd

&I, = (&hl:

(3.20)

yielding E,,,=h

j- drr’{$

[

+f

I

+fh’~~‘+:,(8o:+4~:o:+30:)+40’1:(20:+0:))~}. The equations

of motion

are obtained

(3.21)

by varying

O3 while treating r] as a fixed parameter of the quark wave functions,

Egh with respect

to be determined

drr2(u2+

to u, v, a, D1,

later. The normalization

v’) = 1 ,

(3.22)

I is incorporated E. The resulting

into the variational solitonic equations

principle are

by means

of the Lagrange

multiplier

dV

-=-2v+[~-g(~+~~)lu-g[S~,sin277+f~3]v, ar

r

a2a ar2=-Tar

3 ar2

2~+m~~+~g(u2-v2)+A2[cr3+3f~~2+f(2@~+@~)(c7+f~)],

a@, 2@, =-- 2 -+,+ rdr

r

m2, @, +&

sin (2v)guv

+A2[~~:+~~:+(T((++2~=)]~,) a2Q3 -= ar2

a& 20, -- 2 -+,+m7,sD3+&guv rar

r

+h2[~~:+~~:+(+(~+2fx)]~3.

(3.23)

M. Fiolhais

136

These equations

et al. / Generalized

hedgehog

have to be solved with the boundary

au -=O, ar and at r + 00 we have a(a)

v=o

@=O,

= 0 and @ stands

au

-= dr

-u

at r = 0

conditions

(3.24)

for @, and @,: 1

( > m,+-

r

,

r(l+m,r)~~=(2+2m,r+m2,r2)@, u[r(gf,+e)“2+

(gf= - s)-“~]

The third of the conditions

- r(gf,

(3.25) limits the quark -gf,G&S

+ s)“~v

eigenvalue

(3.25)

= 0.

to the range (3.26)

+gf,

For a given E the actual solutions of eq. (3.23) are obtained by the program COLSYS written by Ascher, Christiansen and Russel 25726).

3.4. THE

GENERALIZED

HEDGEHOG

SOLITON:

package

RESULTS

Some basic features of the solitonic solutions of eq. (3.23) are illustrated using the pure hedgehog. Due to the nonlinearity of the problem there exist two qualitatively different solutions. One is the vacuum, which is characterized by the absence of valence quarks, u(r) = v(r) = 0, by the absence of the pion field, Q(r) = 0, and by a constant Th is solution exists for any g and its energy is zero. sigma field, (El&(r)lE)=f=. with the critical g,-3.8, there occur two solutions (see fig. 1). One is For g>g,,

1 .so> C

c? ,l.OO0-l

,/ .'

: w

0.50-

0.00 3

I 4

I 5

I 6

I 7

1 a

9 Fig. 1. Hedgehog

energy

as function

of g. The total energy coupling constant

is shown g.

as a function

of the quark-meson

M. Fiolhais et al. / Generalized hedgehog

nonrelativistic

with E close to gf,, whose energy

of three free quarks, solution

i.e. 3gf,.

with E around

with increasing presented

as functions

approaches

for large g the limit

The other is the one of interest,

zero, it is localized

g. The reason

seen in fig. 2 where

131

around

for this particular

for various

g’s the norms

it is the relativistic

the origin and its energy decreases behaviour

of the solutions

of the quark

of the scaled single-particle

can be

wave functions

energy E’ = e/d,.

are

For sufficiently

large g there exist two solutions with norm equal to one converging both to a single solution at the critical g = g,. For g < g, no solution of norm one exists. 4.00 3.50 3.00 2.50 r g 2.00 i

-1.00

-.60

-30

-.40

-.20 quark

Fig. 2. Quark

.40

.60

.a0

norm as function of the quark eigenvalue. For various coupling norm N is shown as a function of the quark eigenvalue.

Fig. 3 shows the r-dependence solution for the hedgehog it is defined as (El6(r)lZ) one expects

.OO .20

1.00

eigenvalue

of the functions

constants

g the quark

u, v, @ and u of the relativistic

and for 77= 20”. The vacuum value of the u is zero because = a(r) +fr. The @(r) shows a p-wave Yukawa tail. As

for a relativistic

solution,

the lower component

of the Dirac spinor

is

appreciable. Qualitatively all the fields discussed in this paper are of this shape. It is interesting to see the effect of n on the pion field, displayed in fig. 4. For 71 =

0” the generalized

quark

hedgehog

is Ixe,,( n = 0’)) = 1u&) which yields

an axial

symmetric configuration in spin and isospin space. Hence only uncharged pions should constitute the cloud and indeed for n = 0” the @, = $ = 0. A similar effect happens at 77= 90” such that altogether the relevant interval for n is 0 s n s 45”.

3.5. POLARIZATION

OF THE

DIRAC

SEA

The calculations in this paper are performed by considering three valence quarks only, ignoring the polarization of the Dirac sea. The question arises how far such a procedure is justified. The only calculations with the present lagrangian involving

M.

738

-3

Fiolhais

et

al.

)

I

I

I

I

.oo

20

.40

.60

.80

/ Generalized

hedgehog

I

I

1.00

1.20

I

I

I

1.40

1.60

1.80

2.00

r (fm)

Fig. 3. The quark fields and the sigma field. The radial dependencies of the quark wavefunctions u and o and of the sigma field are plotted for various values of the mixing angle. The solid line corresponds to the hedgehog (7 = 45”) and the dotted line to n = 20”.

sea quarks have been published by Ripka and Kahana in a series of clearly written and assume the sigma papers 8335). They consider hedgehog mean field solutions and the pion fields to be on the chiral circle and to be given in a convenient parametrization. By this the extension R of the soliton can be varied from outside, the relevant parameter being the dimensionless quantity X = 2Rgf,/ AC. In fig. 6 of their paper 35) it is demonstrated that the contribution of the sea quarks to the total integrated

baryon

density

is negligible

for solitons

with X s 2. The calculations

in

__________________

‘\

\

\

-

PHI

\

\

i \

\

xy

‘:

‘\ \

iL/’ :

\ \<’

2.5

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

2.5

r W Fig. 4. Pion fields for various mixing angles. The radial dependencies the pion field are shown for various angles. The solid, dashed-dotted n = 45”, 20” and O”, respectively.

of the different components and dotted lines correspond

of to

M. Fiolhais et al. 1 Generalized hedgehog

our present

paper are self-consistent

corresponding the baryon

to X = 2.8. Hence number

contribution. following

is significant

A similar

ones and result all in solitons in our case the contribution but probably

feature

size, although

it may stabilize

quark

contributes

with R = 0.6 fm

of the sea quarks

not dominating

can be seen for the total

fig. 2 of ref. 35), the Dirac sea basically

of the present

139

to

over the valence energy

to which,

only to 20% for solitons

at a different

radius

if sea quarks

and

their polarization are taken into account. Altogether, there are indications, by no means a proof, that indeed the valence quarks are the dominating part for the solitons considered in this paper. It is, however, not known how other observables react to the polarization of the Dirac sea and this question certainly has to be studied in the near future. Such an investigation will not be simple because one has to combine

projection

techniques

with renormalization

schemes.

4. Projection theory The hedgehog baryonic mean field state II,&,) and its generalization I$& are no eigenstates of the angular momentum J2, J3 and of the isospin T2, T3. They are however

linear

combinations

of them, i.e.

I*)=JT E, cc;:;1J, a

T,

CY,

M, MT)

.

T

(4.1)

Here the (J, T, a, M, MT) are simultaneous eigenstates of J2, J3, T2, T3 and LYstands for an additional quantum number, which just distinguishes states of the same J, T, M, MT. The 1J, T, a, M, MT) are model states of the proton, the neutron, the A ++, etc., and contain therefore the physical information one is looking for. Hence they are the states one is interested in and the mean field state II/J) is only a means to construct them. They niques 28,29) in analogy techniques have been cloud 24,29,30)and to the

are extracted from I+) by Peierls-Yoccoz with the treatment of deformed nuclei.

applied by several authors to bag models with a pion chiral soliton model ‘2*‘3*‘6).In ref. ‘) a simplified procedure

is used, where the quantum In the present

projection techThese projection

formalism

fluctuations we expanded

of the pion field are not taken into account. the boson

fields into the corresponding

free basis and constructed a coherent Fock state in the same basis, see eqs. (3.1), (3.4) and (3.8). Whereas for a pure mean field approach all bases are equivalent this is no longer true for the projection. The pure mean field approach requires only the knowledge of the classical boson fields which are the expectation values of the field operators in any basis between the corresponding coherent states. In the projection formalism, however, overlaps are required, i.e. matrix elements of the field operators between rotated coherent states, and those overlaps depend on the basis chosen. A good basis would consist in the boson small amplitude eigenmodes around the mean field solution. The coherent state to be used in the projection would then be the boson vacuum associated with the eigenmodes. Such a basis

M. Fiolhais et al. / Generalized hedgehog

740

would

have the advantage

small amplitude

that the correlations

vibrations

would

other hand, such a calculation

be taken

not known

into account

would be extremely

and even then one does not know whether the most important

influences

to low energy

in the projection.

complicated

the choice

mode

On the

correlations

are

Actually

it is

like rotations.

of the basis

and

and time consuming

those small amplitude

ones for a large amplitude

at all what

corresponding

has on the projected

quantities, in particular if the projections are done before the variation. This is even not known in nonrelativistic nuclear physics where the analogues of the present calculations with the free basis and the eigenmode basis correspond to projected Hartree-Fock and projected RPA. In the present paper we ignore these problems and use the free plane wave basis, within which all properties are then determined variationally. It is understood, that the effects of the choice of the basis on projected properties require some detailed investigations.

4.1. GENERAL

PROJECTION

FORMALISM

The states IJ, T, a, M, MT) of eq. (4.1) can be obtained projection

from I+) by means

of the

operator (4.2)

and an analogous expression operators for spin and isospin

for P&,,.

The R(R)

R(CY,p, ?) = eieJ,

and

i(d)

are the rotation

,-‘@, emiYJ3,

ii(G, /j, +I = em;“r3 e-iET2 e-i+T,. The explicit

formula

for the normalized I J, T a, M MT) = ,;

and the projected

energy

(4.3)

states IJ, T, a, M, MT) reads 1T

g(K/=pT)Pk P;T,&)

(4.4)

E’ JTa) = (J, T, CY,M, MTjHl J, T, a, M, MT) is given by

where the properties

PJfK’M’ are used for the derivation. ation procedure C K’K:

P’MK

=

sJJf&,,Q,‘P;~K

The ggpT) are the results of a nonorthogonal

(h~~~,,K;-E’JT”‘n~~~K.K:)g’KJ~l=O

(4.6) diagonaliz-

(4.7)

741

M. Fiolhais et al. / Generalized hedgehog

with the kernels h’JT’

KKTK’K;=(~lHPJKK,P~TK;IICl) 9

(J-0

nKK,K,K;=(cCrlPJKK~P=TK;I~). The cr distinguishes

the various

solutions

T. The g(K2T’ can be normalized

for a given set of quantum

(4.8) numbers

J and

such that ( JTa” gK’K;

(JTn)*

c gKKT

(JT) nKKTK’K;=ka’.

(4.9)

As one sees the overlap kernels (4.8) are the relevant quantities for the diagonalization and their explicit expressions will be given in sects. 5.2 and 5.3. One of the interesting aspects of the projection formalism is that there exist sum rules. In the Hilbert

space spanned

by R(n)l?(d)l$) I =

the completeness

relation

reads

IJTcYMMT)( JTaMMTI ,

C

(4.10)

aJTMMT

and hence one obtains

immediately c’I~,!

=

(

(4.11)

JTcYMMT I+) ,

and with eq. (4.4) &TRT)= K; Thus the norm sum rule (NSR) JTAM, The probability

T

&:‘((cII

pLTK,I$)

(4.12)

*

is with (Ic,I $) = 1 1&

&~‘d%,K,

I* =

(4.13)

1 .

to find in I$) the state with a given J, T and (Y is given by P (“a) = A,

Similarly

&K

one obtains

an energy

(&

&l?$n!&!W,K,

sum rule (ESR)

(4.14)

I* .

involving

the mean

field energy

J%*=(~IffI(CI): 2 c JTaMMT

cg KKT

(JTm’ KKT

(JT’ n MKMTMT

ECJT”’

=

EMFA

.

(4.15)

The sums in eqs. (4.9)-(4.15) are in principle to be extended over all possible J, T, M and MT. Eqs. (4.13) and (4.15) provide an excellent check for the numerical accuracy of the calculation. A typical example is given in table 1, evaluated with g = 5.0, n = 20” and A = 1.3 (see sect. 5.4 for the definition of A). Listed are all states I JTa) with a probability p (ITa’ > 0.1% and J s G and T s $. These states exhaust the norm sum rule (4.13) to 90% and the energy sum rule (4.15) to 88%. This means roughly 10% of the wavefunctions contains states with J 25 or T ~4. In general the higher

M. Fiolhais et al. / Generalized hedgehog

742 J

or T the smaller

is the probability

amplitude.

Thus the method

leads to a natural

cutoff in the sense that the probability amplitude becomes negligible. In table 1 N yre*’ in the projected states. The larger the J

there is also listed the pion number or T the larger the pion number

4.2. INTRINSIC

QUANTITIES

For the evaluation quantities

becomes.

of the overlaps

required.

in the next subsection

The pion amplitude

for the generalized

hedgehog

intrinsic

is

t=1

&A,(k), 5fh(k) =

there are various

t=2

,

ik,,A,(k)

(4.16)

t=3

( ikA,(k), with, see eqs. (3.8), (3.12) for definition,

drr2j,(kr)@,(r). The intrinsic

number

of pions

t is defined

of isospin

N’,” = (&,I

dk TABLE

(4.17)

as

b:(k)b,(k)&t,)

(4.18)

1

For a given intrinsic solution, characterized by g, 7, A as given, the various projected states with their quantum numbers, energies and their probability amplitudes are listed J

T

(I

Energy

t

t

f ; z1 2 3 : I 1 f 4 I: I 2

+ f

1 2 1 2 1 1 2 3 1 2 1 2 3

0.938 1.599 1.669 2.069 2.110 1.099 1.692 2.161 1.866 2.490 1.455 2.254 2.822

f f 3 : I t + ; 2 + f

EWSR= 88% NEWSR = 90%

Prob. ampl.

%

12.6 1.9 2.2 0.3 0.2 35.2 6.4 0.6 2.4 0.2 19.7 2.1 0.2

NJT _ 1.01 1.48 1.64 1.66 1.54 1.23 1.52 1.60 1.95 2.39 1.98 2.48 2.89

g = 5.0 17=20” A =1.3

Only those are given with a probability amplitude than 0.1%. The NiT is the projected pion number.

of more

M. Fiolhais et al. / Generalized hedgehog

which yields after an elementary N;‘=

calculation

dktT(k)&(k)=4,rr I

Similarly

the intrinsic E’,“=(II,,l

dkk4A:(k).

(4.19)

I

kinetic

I

143

energy of the pion field is expressed

dkw,(k)b:(k)b,(k)ll7,,)=4~

Due to 0,(r) = @Jr) we have also A,(k) = A,(k), also needs the following mixed terms N”3’=$v Tr

,5”3’ Tr

=

I

as

dk k4w,(k)Af(k).

(4.20)

IV’,” = N’,Z’ and E’,” = E’,‘. One

dkk4A I(k)A,(k),

I

dk k4w,(k)A,(k)A3(k).

$

(4.21)

I

One can rewrite the kinetic

energies

dkk4w,(k)A:(k)

as =

(4.22)

For the mixed term this reads

4.3. THE

NORM

OVERLAP

In this section we derive the explicit the generalized hedgehog I$) = 1qbgh): (JT)

nK’K;KK,=

(25+ 1)(2T+

1)

(8~~)~

expressions

for the norm

dR dfi D~~K(fl)D~~K,(d)n(fi,

overlap

d)

(4.8) of

(4.24)

with n(fi, fi) =

(~,hl~(~)~(~)l&J.

(4.25)

For the spherical tensor algebra we use in the following the conventions of Brink and Satchler 3’) being identical to those of Messiah 32). The rotation operators R(0) and R(d) are products of operators acting in the quark-, pion- and sigma subspaces. Since, however, the coherent state 12) has (good) spin and isospin quantum numbers J = T = 0 its overlaps are always one. Thus one obtains with three quarks in identical states n(fl, h) = ni(fl, h)n,(.O,

f?) .

(4.26)

744

M. Fiolhnis

et al. / Generalized

hedgehog

The norm overlap of the single-quark wave function since R(R)[ai] = ai and the orbit is a 1s state:

n4(Q fi) is easy to evaluate

(4.27)

with L~,~lW~)~(~)lx,tJ g’tven in eq. (A.l) of the appendix.

For the overlap

pions

pion state of eq. (3.8)

one has to take into account

is again a coherent

that the rotated

coherent

of the

state 30):

R(fi)E(d)]n)

= N-l’* exp

i dk&(k, 1 ,=1 I

0, &b:(k)

I

(4.28)

IO)

with &(k, n, d) = ; &(d)&(R-‘(n)k) 1’=1 One obtains

the simple

.

(4.29)

expression n,(G

fi) =

(q&wm~~l~gtl) dk5T(k,

(4.30)

fl, &5,(k)

with (4.31) Using now the fact that for any function

I one obtains

f(k)

one has

dkf(k)kikj=f6,

(4.32)

the final expression

n,(~,d)=exp{(N’,“+N’,2’)m,(~,~)+N(,3’m3(~,ji)+N(,‘3’m,3(~,

d)} (4.33)

with m,(Q

4.4. THE

d),

m,(n,

d) and m,,(Q

HAMILTONIAN

In order integral

to evaluate

(2J+ h(JT) K’KkKKT =

d) given in eqs. (A.2)-(A.4)

of the appendix.

OVERLAP

the Hamilton

1)(2T+ (8~r*)* h(G

1) I

overlap

h’,Jz~,‘~~~ one has to calculate

dLi dfi D;&2)D;;,&?)h(f2,

fi) = &MW)~(fi)llClg,J.

fi) ,

the

(4.34) (4.35)

M. Fiolhais et al. / Generalized hedgehog

Following

the decomposition

of eq. (2.10) some terms of h(0,

745

d) are rather easily

obtained:

I

hqm(fl,d)=3gf,n(R,fi)

h,,(Q

d)=3gn(@fi)

drr2(u2-v*),

drr*(u*-v*)a, I

h,(R,~)=2lin(R,fi)~drr*[(~~+rn~~*]. For the evaluation

of the quark-pion

valid for the generalized

(4.36)

interaction

term one should

use the identity

hedgehog

(4.37) Using now the plane

wave expansion

(3.1) of the pion field and the fact that with

eq. (4.8) one has tfh(-k)* = &fh(k) one gets

I =

-gnt(R,

x ,=, i

I

fl)n,(Q

0)

@t(r)(xghi{

(+r7r,

dr r2w

R(n)&d))k,h>

The anticommutators are given in eq. (AS) of the appendix. the kinetic energy of the pions we need &(On, ii)

=($&hi

(4.38)

For the evaluation

of

: H,,:R(fl)&f%l+,h)

=il,gO,d) ;

,=,

Integrals of a similar sort have occurred can be written down as h,(R,fi)=n;(n,d)n,(n,

*

I

dkw.r(kMT(kMk 4 @.

already

(4.39)

in eq. (4.30) and thus the result

~)[(~',')+~',2))m,(n,~)+~(,))m~(n,~)

+EE(,'3bn,3(f2,~)]

(4.40)

M. Fiolhais et al. / Generalized

746

with m,, m3 and m,, given in eqs. (A.2)-(A.4) part of the analytics

is the evaluation

= n,(l2,

ti) f

hedgehog

of the appendix.

of the nonlinear

The most difficult

part of the hamiltonian

overlap:

dr {(n,,l:(n(r)n(*))‘:R(~)l?(~)ll7gh)

+(n,,l:~(r)~(r):R(R)~(~))I~~,)((+’+6f,a+8fZ,) +(T4+sf,(+3+18sf2,,~*+16~~~+5f~}. Its derivation

is rather

lengthy,

though

expression are given in ref. 43). A typical example for the hamiltonian

(4.35)

straightforward. overlap

is shown

Details

and

the

final

in fig. 5.

1

80

100

120

140

160

180

beta Fig. 5. Hamiltonian

4.5. PROJECTION

overlap

functions.

BEFORE

AND

The hamiltonian overlap are plotted.

AFTER

THE

functions

for various

sets of Euler angles

VARIATION

In the present paper we will consider four methods to obtain angular momentum and isospin projected Fock states of the chiral soliton lagrangian. They are characterized by the ansatz for the spin-flavour contents of the quark wave function, i.e. hedgehog or generalized hedgehog, and the method of varying the radial degrees of freedom. (i) Hedgehog, projection after the variation (HH-PAV): Here we assume an ordinary hedgehog structure, i.e. 77= 45” and 0, = a2 = Q3 = @. We solve first the mean field equation (3.23) and perform afterwards the projections. Thus first is evaluated. Calculations of (8+,,,,lHI1/,,,,)=0 is solved and then I$hh)=P,PTI$hh)

M. Fiolhais et al. / Generalized hedgehog

this sort have first been reported and Rosina

13). The present

(ii) Hedgehog, ordinary

projection

hedgehog

structure,

by Birse 12) and in some approximation

calculations

reproduce

before the variation however,

equations

PJP,I~,,).

modifies

variational

principle

the numbers (HH-PBV):

we perform

states, i.e. we solve the Euler-Lagrange This

741

of ref. ‘*).

Again we assume

the variation

for (&&~lI-II&~) the soliton

by Golli the

with projected = 0 with I+;;) =

equations.

Since

this

formalism for a pure hedgehog is given in detail in ref. I*) we omit it here completely, although we will present numerical results. (iii) Generalized hedgehog, projection after the variation (GH-PAV): While for the ordinary hedgehog the only degrees of freedom are the radial shapes of U, u, (T and @ the generalized hedgehog has two pion fields @, = 4pZ and Q3 and the additional parameter 7. Since a pure mean field solution as a function of n reduces always to the hedgehog, the generalization makes only sense if at least the n-degree of freedom is varied after the projection. Thus one produces several mean field solutions 1tjgh( 7)) by solving eq. (3.23) for various given n in the range 0 c n < 45”, constructs JI,!I~~(17)) and looks for the minimum of (1&!~(~)1Hl$~hT( 7)) with respect to 7. Since the projection is performed after the variation of the radial degrees of freedom this method is called GH-PAV. (iv) Generalized hedgehog, projection before the variation (GH-PBV): Basically in this method for each 17the radial shapes of U, v, a, @, = o2 and C& are determined by the variational principle (a$,‘;( n)l Hl&!hT( n)) = 0 and the minimum of (Jl~hTIHl&~) with respect to n is searched. This variational principle modifies the solitonic equations (3.23). We have derived in detail the full formalism which is rather lengthy and hence not presented in this paper. Actually there are several arguments that the unrestricted variation of all radial degrees of freedom can be replaced by a simpler, approximate one. If one compares HH-PBV and HH-PAV with each other and takes into account the volume element dr r2 then one realizes that u, z7 and (T are rather unchanged and that the radial dependence of Q(r) shows only little modification. Hence only the total strength of Q(r) is the relevant degree of freedom 12). The same feature occurs if one compares Q(r) from the hedgehog with Q,(r) = Q*(r) and Q3(r) from the generalized

hedgehog

for various

7. Again,

only

the overall strength changes. Thus we do not perform a full variation of all radial shapes for the generalized hedgehog, but introduce a parameter A by @r(r) + A@,(r) and @Jr) + A@*(r) and treat it as a variational parameter before the projection. An analogous variation of Q3 turns out to be superfluous since it lowers the total energy of the nucleon by merely 1 MeV. Thus our approximate GH-PBV method finding the minimum in the projected energy plane consists of (&l(n, A)IH[I,!J$(~, A)) with respect to 7~ and A. This energy plane is shown in figs. 6 and 7 for the case of the nucleon and the delta, respectively. The differences between the various methods can well be seen at table 2 if one evaluates for fixed values of g and m, the energies of the nucleon and delta and some relevant observable quantities, the deviation of which will be presented in

M. Fiolhais et al. / Generalized

748

hedgehog

1.7 1.6 1.5 1.4

\i

1.3 1.2 1.1 1.0 0.9 0.6 0.7

0

5

10

15

20 MIXING

25 ANGLE

30

35

40

45

ETA

Fig. 6. Total projected energy plane for the nucleon. The total projected energy for the nucleon displayed in form of equipotential lines in the plane spanned by the generalized hedgehog parameter and the scaling factor A.

is TJ

1.7 1.6 1.5 tx g 2 E

1.4 1.3

0.9

0.6

j

0.7 0

5

10

15

20

25

30

35

40

45

MIXING ANGLE ETA Fig. 7. Total projected in form of equipotential

energy plane for the delta. The total projected lines in the plane spanned by the generalized scaling factor A.

energy for the delta is displayed hedgehog parameter n and the

M. Fiolhais

et al. / Generalized

hedgehog

149

TABLE 2 For a given quark-meson

E,(GeV) Ed(GeV EC& (?$;rn’) (r!)(W pr(n.m.) p,(n.m.) g,lgv g = 5.3163

coupling constant g the various text, are compared “)

methods,

explained

in the

HH-PAV

HH-PBV

GH-PAV

GH-PBV

0.924 1.041 0.117 1.120 0.56 -0.08 2.48 -2.08 1.72

0.871 1.023 0.152 1.120 0.53 -0.08 2.51 -2.23 1.78

0.878

0.835 0.995 0.160 1.358 0.63 -0.11 2.83 -2.50 1.72

n = 20”

1.007 0.129 1.358 0.64 -0.12 2.54 -2.24 1.70

A4 = 1.15

AN= 1.3

“) Listed are the energy of the nucleon, of the delta, the delta-nucleon splitting, the mean field energy, the squared radii of proton and neutron, the magnetic moments of proton and neutron, and the ratio of the axial vector coupling constant to the vector coupling constant, g,/gv.

sect. 6. Altogether in improving the method from HH-PAV to GH-PBV one improves noticeably the squared charge radii and gains about 100 MeV in the nucleon energy. Compared to the simple hedgehog the projected state is lowered by about 285 MeV and compared to the mean field solution with n = 20” by about 500 MeV. One realizes here already that correlation energies, i.e. the difference between projected energies and the corresponding mean field ones, play a significant role.

4.6. THE

NUMERICAL

PROCEDURE

OF THE

PROJECTION

In the previous sections the general formalism of the angular momentum and isospin projections has been presented. However, due to the fact that the generalized hedgehog I+& obeys G,I$,,) = ( J3 + T,)I4,,) = 0 several simplifications occur as will be explained now. As example we use for this the expectation value of a scalar isoscalar

operator

S. We have:

The matrix element on the r.h.s. contains 0 and fi, which reads explicitly

a six-fold

integral

over the Euler angles

da dai dy dj sin p dp sin fi d/? eiKcl+iKT~+iK’y+iK’~dJK’K(P)d=;K,(p) I X (+ghl e-iaJJ-izT3 e-iPJ,-ifir,

e-iYJS-i?TJI+,h)s(fl,

fi) ,

(4.37)

M. Fiolhais

750

et al. / Generalized

hedgehog

Since JsI(Lgh)= -TJ,I$,,) to &-& coordinates depends

and

the matrix element on the r.h.s. of eq. (4.36) is proportional by defining new 8KC,_K;. Hence one can simplify the expression

4 = LY- c?, $ = y - f and 4 = (Y+ &, 6 = y + j? Since the overlap

also on the combinations

There remains

only a four-fold

4 and Cc,the integrations integral

~(0, fi)

over 6 and 6 are trivial.

which can be evaluated

by standard

discretiz-

ation techniques. Due to the Kronecker symbols we have immediately a reduction in the number of matrix elements. For the hamiltonian one has therefore /,‘JT’ 6 K. K, .h(KI? -6 K’.-K; (4.38) K’K;KK,, K’K,pK and an analogous expression II,$,,,) the expressions simplify

for the norm overlap. In case of the pure hedgehog even further I*) and one obtains eventually a twofold

integral and always J = T. Simplifications as described above for the case of a scalar-isoscalar operator occur for operators of all tensorial properties. Thus one always can reduce the six-fold integrations over the Euler angles to four-fold ones. In practice this can be done, besides simple normalization factors, by considering the original Euler angles but setting & = 7 = 0 and performing the integrations over a, P, Y and fi. At this point we should

mention

that the most efficient

way of performing

the

projection from the pure hedgehog soliton of this model is to use the technique suggested by Golli and Rosina 13). In the end the energy and other observables can be expressed in terms of the intrinsic number of pions and the norm kernel. The latter can be written in an analytic form involving modified Bessel functions 24). In case of the generalized however.

hedgehog

the analytic

approach

is no longer

feasible,

5. Observables The angular momentum and isospin projected Fock states of the previous section can be used to evaluate baryon properties. To this end various matrix elements of appropriate currents between projected states have to be evaluated, the details of which are presented in ref. 43). One needs T(hp) ($I(p&‘K

the general

of rank A between ,)+T(*p)p;‘,K,$)=

expression projected (2J+

for the matrix

element

of a tensor

operator

states 33) $J’+l)

(_l)f+K >

J’ XC -K’ W

A

J

p

K’-p

dR Dc,_P,K (0) >I

x ($1T(hP)R(fln)lG). The generalization of this formula to two projections is trivial. to be decomposed into their components with definite tensorial matrix elements have to be evaluated.

(5.1) The currents have character and the

751

M. Fiolhais et al. / Generalized hedgehog

Electromagnetic

properties

are evaluated

the electromagnetic

by considering

current (5.2)

j~,(r)=~(r)y~Q~(r)+Ejij7;;(r)i)~~j(r)

with the charge

projection

operator

for up and down quarks

Q+$+!+

The squared

charge radius

(5.3)

is then given by

(I’)‘,: = (JTaMM,I: In a similar

way the operator

of r with the 3-vector

dr r2JEM(r):l

for the magnetic

JTd4MT).

moment

(5.4)

is given by the cross product

part of j!&,,, c(r)

=;(rxj,,(r)).

(5.5)

The matrix elements of its z-component between the Fock states In?) and Ip?), integrated over all space, yield the magnetic moments of the proton and neutron, respectively. The axial vector coupling constant g, is the matrix element of the component A;(r) of the isovector axial current AC”(r) of eq. (2.4), integrated over all space. Explicitly one is interested in g,/gv, of the isovector vector current

where gv is the corresponding

matrix element

~“(*>=f~(*)r”7~(r)+~(r)Xr3~~(r). Since for the nucleon

the vector part yields just i one obtains

R =2(nTI: At 4-momentum

(5.6)

q2 = 0

altogether

drAf(r):lnr).

the pion-nucleon-nucleon

(5.7) coupling

drzj^‘,)‘(r):lnt).

constant

is given by (5.8)

Here MN is the nucleon mass andj,(r) the pion source current. The latter is obtained from the equations of motion derived from the lagrangian, i.e. (@“a, + m2,);r =j

(5.9)

and reads (5.10)

752

Besides

M. Fiolhais et al. / Generalized hedgehog

this “source

form”

for the pion-nucleon-nucleon

coupling

constant

there

is also a “field form” I

g,,, =4~

= mt(nt]:

dr z7;3(r):]n~)

.

(5.11)

The difference between both expressions corresponds to a virial theorem associated I with the operator 0 = j drzF3(r), where P(r) is the momentum field conjugate to m(r). A simple calculation shows that i&N,

g,NN

-=--i(ntl: 2MN 2MN

drz[ps(r),

fi]:]nT),

(5.12)

where fi is the hamiltonian of the system. Apparently the expectation value of the commutator vanishes between exact eigenstates of fi. Thus the extent to which an approximate used. Since

solution satisfies gmNN =gL,, provides a test of the approximation in nature the gWNN is measured at momentum transfer q* = rni the theoretical grNN values (5.8) and (5.11) have to be corrected for the finite pion mass to be compared with experiment. Birse 12) has calculated this correction factor as grrNN(q’)

=

g,rNN(“)(l

(5.13)

-brZ,q2)

with

(ntl: l dr r’j:“‘(r):lnr)

r2,=-

&N:(o)

.

(5.14)

This author has calculated r’, explicitly and found that g,NN(mi) is 5% larger than g,,NN(O). In this paper we accept his argument and multiply all theoretical values obtained from eqs. (5.8) and (5.11) by the factor 1.05. for &NN, Another check for the quality of the approximation is given by the GoldbergerTreiman relation. If we had a fully conserved axial vector current

PA,=0 one would

obtain

for the exact eigenstates gA -=-

fr

gv

MN

of fi the Goldberger-Treiman g,NN

(5.15) relation (5.16)

which provides another necessary condition to be fulfilled for a good approximate solution. In nature, however, the axial vector current is only partially conserved (PCAC) and hence the gzL\ obtained by eq. (5.16) from the experimental value of gA/gv is by 8% smaller than the experimental grrNN. Similarly to the pion-nucleon-nucleon coupling constant we consider also the pion-nucleon-delta coupling constant g,,, and the pion-delta-delta coupling constant g,,,. For the latter one we take the expectation values of the operators in

M. Fiolhais et al. / Generalized hedgehog

eqs. (5.8) and (5.11) between sponding

the states (J = T = z, A4 = MT = 1). The gxNA, corre-

to the decay A + N + v, is obtained

operator jdrzji(r) The derivations

153

in the source

form sandwiching

between (A:;\ and IN:,,). and the final expressions of the observables

the

are given in ref. 43).

6. Results 6.1. THE

GOLDBERGER-TREIMAN

As discussed

already in sect. 6 the satisfaction

and of the virial theorem the method

RELATION

associated

of the Goldberger-Treiman

with g,NN p rovide

used. Table 3 IiStS the V&eS

for

gA/gV,

relation

a check for the quality

&NN,

and for

&NN

gA

MN

of

(6-l)

gzk=fn-

using the present methods and values from the literature in comparison. The percentage shows the violation of the Goldberger-Treiman relation and of the virial theorem.

It is defined

as &NN

-

,

g,NN

+

2

l.o8gs;% - g:NN

(6.2)

+ gyNN ’

l.OSg:;;

g:NN

where gyNN = + (g,,N + g;NN). The factor 1.08 corrects for the slight explicit violation of the chiral symmetry in nature. It is chosen such that the violation is zero when using the experimental

numbers. TABLE 3

Values for the axial vector coupling constant g,/g,, constant in the source form g,,, and field form gk,, Method ref. 14) ref. lo) ref. “) HH-PAV HH-PBV GH-PAV GH-PBV Exp.

g,lg, 1.39 1.86 1.78 1.72 1.78 1.69 1.75 1.23

the pion-nucleon-nucleon

coupling

, and the value for gzzk = Mg,/ G,f,

g?riw

gkw

25.41 21.85 18.28 23.29 16.94 23.70 17.55 13.6

2.52 15.14 13.28 14.92 12.80 16.84 17.85 13.6

Violation 14.02 19.70 18.85 17.34 17.95 17.04 17.65 12.5

(%)

173 51 59 42 56 24 6 0

The methods used are described in the text. In the methods HH-PAV, HH-PBV, GH-PAV and GH-PBV the coupling constant g is adjusted in order to obtain a proper mass of the nucleon, M = 0.938 GeV. The coherent pair approach and the papers of Birse and Banerjee and of Birse are given in refs. ‘4S’o,‘2) respectively. The percentage of the violation of the Goldberger-Treiman relation and of the virial theorem is defined in eq. (6.2).

754

M. Fiolhais et al. / Generalized

The numbers

hedgehog

in table 3 show that the generalization

sable in order to fulfill the Goldberger-Treiman Due to the T-degree

of freedom

relation

the violation

of two to 24% and it is furthermore

reduced

after the projection.

Thus,

from this point

is the most reliable

one. The remaining

of the hedgehog

the pion fields and the quark

6.2. OBSERVABLES

OF NUCLEON

Table 4 lists the results

reduced

to only 6% if the variation of view the suggested

6% are probably

fields is not expected

AND

and the virial theorem

is immediately

perform the variation of the radial shapes of the fields reason could be that we ignore orbital deformations They should in principle occur since QX differs from deformation effects are estimated 39) to be small this

is indispen16).

by a factor is performed

GH-PBV

method

due to the fact that we

only approximately. Another of the quark wave functions. 0, and 02. Since, however, slight inconsistency between

to be important.

DELTA

of the “best”

of the methods

in comparison with the experimental data. As mentioned already, the GH-PBV procedure

studied,

fulfills

namely

GH-PBV,

the Goldberger-Treiman

relation and the pion-nucleon virial theorem in contrast to all the other approaches. Given are the total values and for some quantities the contributions coming from the quarks and from the mesons. While resulting

the nucleon energy is adjusted, the delta energy comes out too small in about half the experimental splitting. Since in the present model the TABLE 4

Results

of the

projection

before

the variation (GH-PBV) “) Quark

using

Meson

(GeV (GeV)

(GeV)

(W (fm*) (mm.) (nm.)

0.43 0.11 1.43 -1.08

0.21 -0.21 1.32 -1.32

0.96

0.79

the

generalized

hedgehog

Total

Exp.

0.938 1.091 0.154 0.64 -0.10 2.76 -2.40 1.15 1.75 17.50 1.1

0.938 1.232 0.294 0.65 -0.12 2.19 -1.91 1.46 1.23 13.60 1.5

“) Listed are the energy of the nucleon and the delta, the squared charge radii of the proton and the neutron, the magnetic moments of the proton and the neutron, the axial vector coupling constant g,/g,, the pion-nucleon-nucleon coupling constant g,,,, and the ratio of the pion-delta-delta coupling constant g,,, to g,,,. These final results are calculated with g = 5.0, m, = 1.2 GeV, m, = 0.1396 GeV,f, = 0.093 GeV and one obtains for the nucleon n = 21.6”, A = 1.288 and for the delta 7) = 23.4”, A = 1.183.

M. Fiolhais et al. / Generalized Ed -EN

originates

in the nucleon, splitting

from the different

this result

is in agreement

is made by one-gluon

The squared the proton

charge magnetic

kinetic

exchange

radii of proton moment.

such that the ration feature

155

of the pions

with the general contributions,

However,

come out properly a common

energies

and neutron

actually

hedgehog

belief

that half of the

not included

are well reproduced

the neutron

magnetic

pp/pUn is not correctly

to all present

in the delta and

static models.

Models

in our theory. similarly

moment reproduced.

as

does not This is

with none or with

a weak pion cloud yield wrong absolute values but proper ratios, just as the pure MIT bag model. On the other hand models with a strong pion cloud increase the absolute

values but destroy

the ratio. One guesses the addition

of vector mesons

to

the lagrangian will improve the situation. This is also hoped in case of the axial vector coupling constant and the pion-nucleon-nucleon coupling constant which are about 40% too large compared with experiment. Table 4 shows also the contributions of the quarks and of the mesons to the observable quantities. As already pointed out by Birse ‘*) both are equally important. This differs noticeably from the coherent pair approach ‘43’5), where only g,/gv was noticeably affected by the mesons. One should note, however, that the distinction is somewhat arbitrary, since e.g. in the case of the pion-nucleon-nucleon coupling constant one has two equivalent expressions (6.8) and (6.11), one of which is without quarks at all. One cannot exclude that similar virial theorems exist for the other observables. Some properties of the delta isobar are listed in table 5. It is interesting to note that we found for the ratio p(Ait)/pLp a value of 1.98 n.m. which is very close to the value of 2 in the naive quark model. Apparently the pion cloud does not have a big influence

on this number.

TABLES

The squared radii and magnetic moments of the proton (P), the neutron (N) and various delta states (A) with third spin component M are listed

P n A++ A++ A+ $ A0 AA-

M

(r2)(fm2)

t

0.64 -0.10 1.39 1.39 0.65 0.65 -0.10 -0.10 -0.84 -0.84

t S f ; t t f t t

The parameters table 3.

of the calculation

(b4n.m.) 2.76 -2.40 5.43 1.81 2.15 0.72 -1.12 -0.37 -4.40 -1.47 are those from

M. Fiolhais et al. / Generalized hedgehog

156 6.3. TRENDS

There are some simple to the variation similar

trends

to the one shown

simplicity

(HH-PAV).

and expecation

are exposed The quadratic

values

with regard

of the nucleon

mass on g and m, is

not presented

here. For the sake of

in ref. 15) and hence

the other trends

the variation

of the energies

of g and m,. The dependence

for the pure hedgehog

with projection

charge radius (r’,) shows a decrease

after of 20%

if one changes g from 4 to 6. With increasing coupling constant the pion field increases and together with it the pion pressure, which reduces the size of the system. The dependence of (T*)~ on m, is rather weak. If one changes m, from 0.6 GeV to 1.4 GeV the (r2)p decreases by only 5% supporting the present way of choosing a fixed sigma mass of 1.2 GeV for the bulk of the calculations. The pp is basically independent on g, it reduces its value by less than 10% if m, varies from 0.4 to 1.4. The absolute value of pu, behaves similarly. It is interesting to investigate g,/gv and glrNN a bit more in detail. As one can see at table 2 the g,/gv is only little modified if one introduces the generalized hedgehog and performs the variations after the projections. Since in the end the Goldberger-Treiman relation is fulfilled the g,/gv value is the most reliable one of all observables. Hence its dependence on g and rn_ tells something about the lagrangian rather than about the method to solve it. It is important to note that the value of the axial vector coupling constant is nearly independent on g and m,, and with no choice of these parameters the value of g,/gv can be made smaller than 1.65. If one assumes that sea quarks and centre-of-mass corrections do not change the above numbers, this is then an indication that the lagrangian has to be modified, possibly

by the introduction

and Banerjee 6.4. EXCITED

as suggested

already

by Broniowski

19). STATES

Fig. 8 shows the spectrum same isospin. The states

of vector bosons

are obtained

of excited from

the

baryonic

states grouped

diagonalization

(5.7)

into bands

where

of the

the fields

are

approximately optimized to the nucleon by means of the parameters 7 and A yielding g = 5.0, 17= 20.0” and A = 1.3. From all resulting states those with a percentage larger than 1% are selected and compared with the lowest experimental states of the same quantum numbers. The agreement between experiment and theory is not very good. The T = 3 states have a percentage of 20% and 2% in the state 1(Cl&, respectively (see table 1 for details), however in nature these states have not been seen yet. Also the order of the T = 1 states is not correctly reproduced. The energies of the other states are fairly well described. In the range of the Roper resonance we find also a corresponding state with the nucleon quantum numbers. Since in our approach concepts like “breathing” do not enter, this result is very interesting. Altogether, however, the excited spectrum needs further investigation. They have

M. Fiolhais et al. / Generalized hedgehog

T=1/2

(J")

T=3/2

751

(J-)

T=5/2

(J-)

i 2.5 I

2.0-

1/2+

5,2+

___---/I' I' 3/2+ -&______,_3/2+ .'

_*-m+ __** 3/2+--'

5/2+

1/2+

1/2+-._

-._,

1.5-

--1/2+

-s/2

,-3/2+ __e* 3/2+-'-

l.O-

________1/2+

1/2+-

Theor. Fig. 8. Excited

baryon

Exp. spectrum.

Theor.

The excited baryonic states below given isospin.

Exp. 2.5 GeV are shown

then to consider also the width of the states, their decay properties, have to be performed for each state individually. 7. Comparison

Besides the projection (CQM)

considerations been adapted involves

of

and the variations

with the collective quantization method

theory there exists also the collective

in order to quantize

in groups

collective

degrees

of freedom.

quantization

The method

method

is based on

of Bohr and Mottelson in connection with rotating nuclei and has to solitonic models by Adkins, Nappi and Witten 2’). The method

the use of collective

coordinates,

which parametrize

the collective

motion,

as e.g. Euler angles in case of rotations. These collective coordinates are allowed to become time dependent and one looks for stationary solutions to the EulerLagrange equations in the collectively moving frame (cranking). Having found these solutions one derives mean field expressions for the collective momenta which depend only on the collective coordinates chosen. This then gives rise to moments of inertia. The mean field expressions for the collective momenta are essentially classical in nature. The system is, however, quantized by demanding the proper commutation rules between the momenta and coordinates. Assuming adiabaticity, i.e. a decoupling between collective and intrinsic degrees of freedom, the hamiltonian can be integrated over the intrinsic coordinates and the remainder is the quantized collective hamiltonian whose eigenfunctions have to be evaluated. If one considers

M. Fiolhais et al. / Generalized hedgehog

758

only rotations

and uses Euler

wavefunctions

are the conventional

the resulting

spectrum

angles

has always

as collective Wigner

coordinates

D-matrices.

J = T and, as shown

then the collective

In case of the hedgehog in refs. 2’,40), the form

E (J,=EMFA

Other observables collective momenta of the system. It is interesting

(7.1)

can also be expressed in terms of collective coordinates and such that one obtains in the end a complete “collective image” to contrast

the simple

J( J+ 1) law of eq. (7.1) with the outcome

of the projection projected numbers quantized states In the projection which does not the mean field rotational and

theory. First of all, no J( Jf 1) pattern is encountered in the as to be seen in table 1. Second, the energies of the semiclassically E (J) of eq . (7 . 1) are always higher than the mean field energy E MFA. theory this cannot happen since one has the sum rule eq. (4.15) exist in the CQM. The lowering of the nucleon state compared to energy in the projection theory corresponds to subtracting the isorotational correlation energy. This correlation energy is about

200-300 MeV as one can see from table 1. Compared and delta this is a rather large number and hence

to the masses of the nucleon correlation energies of these

collective modes cannot be ignored. Actually the appearance of the correlation energy and also of the simple J( J + 1) law can easily be understood as a limiting case of the projection formalism. Consider a simplified angular momentum projection procedure applied to an axial state I+). In the limit of narrow overlaps h( /3) and n(p) the total energy simplifies after a lengthy

calculation

4’*42,45)to an expression

involving

the J( J + 1) law:

E’J)#‘A+&(/+Q_(/:)

20 The last term on the right-hand overlap

limit.

Comparison

side is the rotational

20. correlation

energy in the narrow

of eqs. (7.1) and (7.2) shows that in the semiclassical

collective quantization method the correlation energy is ignored and, in order to obtain the J( J + 1) law, narrow overlaps are assumed. Indeed ref. 42) shows explicitly for the rotational case that under those assumptions cranking is an approximation to projection. It is interesting to note, that for the present model the assumption of narrow overlaps is not justified. One can see this in fig. 5 where the relevant part of the overlap functions h(0, d) is displayed. Experience shows that the overlaps have to be very much narrower in order to justify a J( J + 1) law and hence the projected results of e.g. table 1 do not show this pattern. In addition, as mentioned already, one is not allowed to neglect the correlation energies. Thus one must conclude that a collective quantization method in the sense of Adkins, Nappi and Witten *‘) is not justified for the linear chiral soliton model.

M. Fiolhais et al. / Generalized hedgehog

Actually

Cohen and Broniowski

using the same lagrangian

comparison

from ref. 40), in comparison degrees

performed

of freedom

explicit calculations

and the same parameters

Hence a direct numerical Broniowski

4”) performed is possible.

as done in the present

paper. taken

first the mean field calculation in the generalization

with the CQM

Table 6 lists their numbers,

with ours using the GH-PBV

provided

759

method.

Since Cohen

and

and then the quantization of the hedgehog

the

are not utilized

in the CQM. The biggest difference in numbers occurs in the delta-nucleon splitting and in the isovector quadratic charge radius which both are in CQM by a factor of 2 larger than in the projection. The other expectation values are affected by about 10% except g,/g, and Pi=, which are lowered from their projection value by 25% due to the semiclassical CQM are not justified

assumptions. However, since the approximations made in and since in addition the virial theorem associated to the

pion-nucleon-nucleon

coupling

constant

is not checked

by Cohen

and Broniowski,

this seemingly better agreement with experiment and the better reproduction of the Goldberger-Treiman relation are of no importance. The masses of the nucleon and delta are about 250 MeV larger in the CQM compared to projection. Cohen and Broniowski corrected this by subtracting the correlation energy of the quarks AEo = 0.126 GeV yielding then the numbers in brackets. From the point of view of a projection theory it is certainly required to correct the semiclassical energies in a way like this. However, one should note that in the collective quantization methods this correction is ad hoc and, furthermore, the equally important contribution of the pions to the correlation energy is missing.

TABLE 6 Comparison

between

projection

GH-PBV g = 5.37

(MeV)

Ed-& EN- E&A

(MeV)

V)T=o (r%=,

Vm*) (fm*)

(MeV)

835 160 -285

and classical

CQM

quantization

methods

“)

ref. 40)

Skyrme ref. 2’)

Skyrme ref. 34)

1183

938

1425

(993) 253

294 73.5

286 71.5

Expt.

938 294

(-1;:)

Pr=1

(nm.) (n.m.)

gA/gv f,

(MeV)

cLT=O

0.52 0.74 0.33 5.33 1.72 93

0.49 1.23 0.38 4.00 1.42 93

0.47 1.07 0.73 3.21 0.61 65

0.45 4.41 1.23 93

0.52 0.77 0.88 4.70 1.23 93

“) Listed are the nucleon and delta masses, E, and Ed respectively, in comparison with the mean field hedgehog energy EKF,,. The squared isoscalar and isovector charge radii are given and also the isoscalar and isovector part of the magnetic moments. The parameters in the present GH-PBV calculation and the one of ref. 40) are both g = 5.37 and m, = 1.2 GeV. In the Skyrme calculation of ref. 2’) the E, and Ed were fitted, in the case of ref. 34) the values of g,/g, and f, have been fitted.

M. Fiolhais et al. / Generalized hedgehog

760

An approach In contrast degrees

which has received

to the present

of freedom

much attention

theory the Skyrme model

but only effective boson

a pion and a sigma meson. The stable solitons number

one are identified

the skyrmions

are subjected

with baryons.

is the Skyrme model *“).

does not include

fields having

explicit

the quantum

Adkins,

quantization

quark

numbers

of the model with topological

Following

to the semiclassical

recently

of

winding

Nappi

and Witten “)

procedure

which results

of course in eq. (7.1). The moment of inertia 0 is adjusted to the experimental delta-nucleon splitting yielding a value of (20))’ = 93.3 MeV. Hence the nucleon mass, E$/,“, is about 70 MeV larger than the mean field hedgehog mass. By now there are basically two ways to solve the skyrmion model. One, suggested by Adkins, Nappi and Witten *I), consists in fitting the mass of the delta and of the nucleon

and yields

rather

bad values

for g,/gv,

and the f,

experimental value. The other, suggested by Jackson and value of fn and fits g,/gv; however, then the mass of the out by far too large. These results are summarized in table 6 and compared calculations. It seems to be that the skyrmion cannot get and g,/gv procedure

must deviate

from its

Rho 34), uses the proper nucleon and delta come with the outcome of our simultaneously energies

and frr right. The reason lies probably in the semiclassical of the skyrmion which ignores the correlation energies.

quantization

8. Summary, discussion and outlook Assuming

spherical

symmetry

the linear

chiral

soliton

model,

involving

quark

fields and elementary pion and sigma fields, has been solved in order to obtain a description of static nucleon and delta properties. To this end a full quantum trial state of the quark-meson system has been constructed by using coherent states for the bosons and a product of single-particle states for the quarks. The good angular momentum and isospin quantum numbers have been restored by projection techniques. It has been shown that the ordinary hedgehog structure has to be generalized in order that the projected solutions fulfill the Goldberger-Treiman relation and a virial theorem associated to the pion-nucleon-nucleon coupling constant g,,, . The generalization

is simple and consists

on the quark level in the spin-flavour

function

luL> cos n -Id?) sin n and on the pion level in the isospin components (x@,(r), y@*(r), z@,(r)) with Q,(R) = QZ(r). Whereas the pure mean field soliton results always in the hedgehog (77 = 45”) the projected soliton requires q ^- 20” and noticeable deviations of 03(r) from Q,(r) = Q2( r). This reduces then the violation of the Goldberger-Treiman relation and the virial theorem from 50% to 6%. Since both relations must be fulfilled for the exact solution of the lagrangian the extent to which they are fulfilled by an approximate solution tells something about the quality of the approximation. It seems to be that projection techniques applied to generalized hedgehog Fock states provide the proper means to solve quantum field theoretical lagrangians of the considered kind. The projection methods, well known in nuclear

M. Fiolhais et al. / Generalized hedgehog

many-body methods

physics,

are therefore

in field theories

not too complicated,

an attractive

in particular,

lagrangian

mass is fitted by adjustment

nucleon-delta

splitting

coordinate

assumptions.

used are of mixed quality.

of the quark-meson

is about half the experimental

it leaves some room for colour-magnetic

to collective

since they are fully quantum-mechanical,

and do not rely on semiclassical

The actual results for the present nucleon

alternative

761

interactions

coupling

While the

constant

g, the

value. This is reasonable

since

which are generally

believed

to cause half of the splitting. The quadratic charge radii of proton and neutron come out rather well similar as the magnetic moment of the proton. The magnetic moment of the neutron is too small compared to experiment and so is the absolute value of pL,/pLI1. The latter feature of them reproduces both magnetic

seems to be common to all present models, none moments simultaneously. The axial vector coup-

ling constant g,/gv and, consistently, the g,,, come out by 40% too large. No parameter set within reasonable limits of g and rnrr and using the experimental value for f, was found which gave a lower value for these two quantities. This seems to indicate a general weakness of the lagrangian which might be overcome by the inclusion of vector mesons. Steps in this direction have already been done by Broniowski and Banerjee I’); however, their projection techniques need improvement. The nucleon expectation values are made to roughly equal parts from the quarks and the mesons. This corresponds to the number of pions in the cloud, increasing with increasing J and T, of IV,, = 0.8-2 for the low-lying baryon spectrum. This value of IV, and the fact that the number of colours IV, equals three causes rather broad norm- and Hamilton-overlaps n( 0, fi) and h(Q d), a feature which invalidates semiclassical approximations in the quantization of rotational and isospin degrees of freedom. Since these assumptions are made in the quantization skyrmion and since in addition the correlation energies, being important projection skyrmion

formalism, are neglected, model either the energies

groups of observables The present method

of the in the

we find this a reason for the fact that in the or g, and fr are reproduced but never both

simultaneously. allows also to evaluate

other baryonic

states than the nucleon

and the delta and in particular, due to the generalization of the hedgehog, states with J # T. Their energies are not in very good agreement with the experimental low-lying baryon spectrum. Generally states with J or T equal 2 come out too high which may be a consequence of the missing coupling to the continuum. States with J = T = 2 are found in the theory with a noticeable probability amplitude but they are not seen in nature. There are various assumptions made in the solution of the present lagrangian whose effects need to be discussed. There is first the effect of the polarization of the Dirac sea, which is ignored completely in this work. For the size of our solitons, R = 0.6-0.7 fm, one can infer from the calculations of Ripka and Kahana 35) that the polarization of the sea quarks does not dominate over the valence contribution,

M. Fiolhnis et al. / Generalized hedgehog

762

at least as far as the energy and the baryon is concerned.

It is not known,

more affected

or not. Second,

mass corrections investigations

to the solution of Liibeck

however,

number

whether

no attempts

density

as e.g. g,/gv

have been made, to calculate

by e.g. linear

momentum

et al. 36) show the correlation

of the same order of magnitude

of the mean field solution

other observables

as for the rotational

projection energies

and isospin

are

centre-of-

techniques.

As

for this mode are modes.

Thus one

expects some change of the numbers if this effect is taken into account. Third, the coherent states used are built up from the field quanta of the noninteracting pion basis. This is a somehow arbitrary choice which does not affect the classical fields, but the overlaps

n(0,

d)

and h(Q

fi).

It is not known

how far the use of boson

operators resulting e.g. from a one-loop calculation or, even simpler, the pion mass in w,(k) will affect our present results.

the change

of

In conclusion, some progress has been made to solve the chiral soliton model while fulfilling the Goldberger-Treiman relation and an associated virial theorem. Besides being of practical relevance this must also be seen under the aspect, how far conceptually clear microscopic many-body methods, well established in nuclear physics, can be applied to a relativistic quantum field theoretical system. Although it is evident that many challenging problems remain in the application of projection techniques combined with coherent states and mean field solutions, the success of the present calculations justifies continuing efforts in both the conceptual and technical aspects of the theory. It is hoped that the remaining problems will be tractable, so that eventually we will have an effective relativistic quantum field theoretical quark-meson theory of low energy nucleon and baryon statics and dynamics. It seems to be that this necessarily requires the inclusion of e.g. vector bosom. The authors

are indebted

to M. Harvey and G. Ripka for many useful discussions.

The work has been supported nologie, Bonn, by the JNICT,

by the Bundesministerium fiir Forschung und TechLisboa, and by the NATO Grant GR85/0217.

Appendix OVERLAP

KERNELS

For the following we need the matrix elements of the rotational matrix Rii(fi), which rotates a vector like Vi = cj Rii(0) 6. They are given in eq. (C.45) of ref. 32). The matrix elements (~~,,lR(fl)l?(h)(~~J are often needed. They can be evaluated as

M. Fiolhais et al. / Generalized hedgehog

763

+sint(a--)sin$(G--q)] +i(l-2cos*

~)cos~/3

cosfp[cosi(a+-y)sin:(cT++)

-sin+(a+r)cosj(G+Y)]. The coefficients

in the exponent

m,(G m,,(fi,

of the pion overlap

n,(L!, d) are

a =&,um,,u4, fi)=f

One needs the anticommutators (X.&V,,

(A.1)

I!I ,=I

[R,(~ii)R,j(~)+Rj,(~ii)Rj,(R)l.

(A.3) (A-4)

in eq. (4.38)

W)fi(d)Ilx,~J

= (Xphl{a, 71, W-‘)fiU-‘)llx,J = (xe,km,

Wn)~U%lxgJ

=-sinfPsintp[cost(a-r)cost(~-~)+sint((~--)sint(~-_)] -2sin~cos~co~~/3cos~~[cosf(cz+~)cos~(&+j) +sin+(cu+r)sint(G+f)], (x~,$(+sQ, which is explicitly

W)&@i))lxgd = -2(x,,lR(.n-‘)~(~-‘)IX,,),

(A.9

64.6)

given in eq. (A.l).

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