PH YSICAL RK VIEW 0 VOLUME 24, NUMBER 7 1 OCTOBER 1981

Cloudy bag model: Convergent perturbation expansion for the nucleon

L. R. Dodd

Adelaide, Adelaide, SA 5001, Australia ~

A. W. Thomas
TRIUMF, 4004 8'esbrook Mall, Vancouver, British Columbia, Canada V6T2A3

R. F. Alvarez-Estrada
Departamento de Fi'sica Teorica, Universidad Complutense de Madrid, Madrid-3, Spain

(Received 26 February 1981)

A previously published bound on the probability of finding n pions in the dressed nucleon in Chew-Low theory is

improved. The proof is then extended to the recently derived cloudy-bag-model Hamiltonian. Together with a
bound on the average number of pions (0.9+1.0), our result strongly suggests a rapid convergence of the

perturbation expansion in the cloudy bag model.

I. INTRODUCTION

The concept of chiral symmetry has been of
great importance in elementary-particle physics
for many years. ' ' In the context of massless
quarks and quantum chromodynamics it is, of
course, an exact symmetry, and should survive
the proof of confinement in some way. It is not

surprising, therefore, that immediately after the
presentation of the original MIT bag model4 Chodos
and Thorn' attempted to repair its obvious vio-
lation of chiral symmetry. Their method relied
on introducing massless, elementary o and m

fields which coupled to the quarks, only at the
bag surface, in such a way as to restore exact
chiral symmetry.

In the past year or so, interest in this problem
has been dramatically revived. For a longer re-
view of these developments we refer to the dis-
cussion of Ref. 6, but a few brief comments will
be useful here. Brown and collaborators have
argued that the w field is actually a crucial aspect
of the confinement process for the nucleon. ' That
is, chiral symmetry should be manifest in the
Wigner-Weyl mode inside (no pions), and in the
Nambu-Goldstone mode outside (the pion is the
Goldstone boson). ' In their purely classical model
this external pion exerts a large pressure on the

bag, resulting in a confinement volume of a few
tenths of a fermi for the nucleon. They argue
further that such a picture (the "little bag") would

be more consistent with classical nuclear physics.
On the other hand, Jaffe' and others' have de-

veloped classical models of a bag surrounded by
a pion field which merely acts as a small pertur-
bation on the usual MIT ground state. Once again
the pion field appears as a Goldstone boson, ex-
cluded from the interior of the bag. Recent work

by Johnson has also shown the importance of col-
lective qq excitations in the volume about the MIT
bag." However, the phenomenological replace-
ment of such excitations by a pion field has not
yet been clarified.

At the same time as these developments using
a classical pion field were taking place, the
TRIUMF-University of Washington group has con-
structed a quantized version of the theory —the
"cloudy bag model" (CBM)."" In order to avoid
technical problems, the CBM (like that of Chodos
and Thorn) allows the pion field to penetrate the
bag. By working only to lowest order in the pion
field, which is assumed to be small, it was pos-
sible to derive a Hamiltonian (see Sec. II) des-
cribing an interacting-system of (bare) nucleons,
deltas, and pions. In Ref. 12, hereafter CBM-1,
this Hamiltonian was used to settle the longstand-
ing problem of the nature of the (3, 3) resonance.
In CBM-2 (Ref. I3) this model has been used, with
considerable success, to calculate pionic correc-
tions to the MIT model of the nucleon (g„,mag-
netic moment, and charge radii).

In the CBM work the contribution from the pion
field inside the bag was rather small, and could
be justified as a crude approximation to the effect-
of virtual qq pairs inside the bag. Indeed it is
just this point which was made recently by De-
Tar. ' " His work provides some formal link be-
tween the theory of Jaffe' with the pion excluded
from the bag and the CBM Hamiltonian, which he
also used in a calculation of nucleon properties.

There is a great deal of interesting physics in
these developments, but for our present purpose
we note only that all groups, except Stony Brook,
rely on a perturbative treatment of pionic effects.
This perturbative treatment has two aspects.
First, the exponential coupling at the bag surface

1961



1962 R. DODD, A. W. THOMAS, AN D R. F. AI VARKZ-ESTRADA

q exp(if' ' $y, /f)q is replaced by q(1+if ~ Pp, /f)q,
and the covariant derivative of the pion field re-
duces to a normal derivative (D„$-8„$).Second,
the resulting linear Hamiltonian describing the
coupling of a pion to a baryon is used only in low-
er-order perturbation theory to obtain the pionic
corrections for nucleon observables. In this pa-
per we shall only address the second aspect of
this problem.

In his classical treatment of the problem Jaffe
extracted a parameter E, rel.ated to the strength
of the pion field at the bag surface [e =g„/(8vf'R')],
which should be small. if perturbation theory is to
work. For the usual MIT parameters his a is
quite small, but it certainly is not small for the
"little bag". In the calculations using a quantized
pion field, that is the CBM Hamiltonian, only the
one- and two- pion terms have been retained.

Until now there has been no rigorous proof of
convergence in any of these calculations. This
paper takes the CBM Hamiltonian as given, and
provides such a proof. Of course, if the AÃg

coupling were omitted from this model it would
be identical in form to the static Chew-Low mod-
el' s' for the pN system For that model a great
deal of formal work has been done to establish
convergence properties. For example, Alvarez-
Estrada" "has proven rigorously that the per-
turbation expansion of the physical nucleon state
in the Chew-Low model does converge, in the
sense that a rigorous least upper bound (LUB) P„
can be placed on the probability of finding n pions
in it. P„does tend to zero as n goes to infinity,
but the convergence is very sloze. For example,
his LUB on P„is not a useful limit, that is, it is
not less than one, until n=5.

Henley and Thirring were aware of this prob-
lem": "For a long time it has been one of the
main goals of meson theory to analyze the physi-
cal nucleon in terms of the bare nucl. eon and its
surrounding meson cloud. This problem led into
a dead-end road. . . . The reason is that the. . .
resonant state of the nucleon is not important for
the ground state. " It is exactly on this point that
the CBM has something new to say. As stressed
in CBM-1, the quark model has an elementary ~
which carries most of the strength of the P33 scat-
tering. Therefore, we do not need such a large
bare coupling constant, or such a high cutoff mo-
mentum. " Consequently, one is led to hope that
the theory may be more convergent.

In this paper we first improve the original bound
of Alvarez-Estrada (for the Chew-Low model) by
a factor of 4, corresponding to the spin-isospin
degeneracy of the nucleon. The proof is also gen-
eralized to the CBM Hamiltonian by extending the
space of bare baryon states. For the parameters

II. THE CLOUDY-BAG-MODEL HAMILTONIAN

The Hamiltonian of the cloudy bag model (CBM)
of Ref. 12 takes the form

K =HO+K

H, =g m Nt N + g ~~aiba~,

(2.1a)

(2.1b)

and

HI = v, ~N~N~a~+ H.c. (2.1c)

Here N (Ni) are annihilation (creation) opera-
tors for the static baryon bag states ~a) of bare
mass m . In our application the states ~a) in-
clude the single-particle states n, s, f) of the nuc-
leon with spin s„=&and isospin t„=2,and the
single-particle states ~@st) of the 4 with spin s~
= & and isospin t~ = ~. The .labels s and t are the
spin and isospin projections, respectively. The
sum over the index k represents the integration
over the momentum k and the sum over isospin
projections j of the pion,

gals2s3

Since there is no renormalization of the pion in
the theory, the rest mass p. and the bare mass of
the pion are identical, and the pion energies in Eq.
(2.1b) are given by

(k2 ~ @2)l/2

of the CBM (Refs. 12 and 13)—or indeed any rea-
sonable parameters near those of the MIT bag
model —this leads to a remarkable proof of con-
vergence of the perturbation expansion of the
dressed nucleon state. Indeed we find that the
probability of finding three pions about the (bag)
core of the nucleon is strictly less than 12%. In
view of the weakness of the bound, the real prob-
ability is almost certainly a factor (2-3) smaller.
Even more impressive is the bound and standard
deviation on the mean number of pions in the phys-
ical nucleon. For the CBM (Ref. 13) these numbers
are 0.9 and 1.03, in comparison with the Chew-Low
values of 2.16 and 2.22, respectively.

This rapid convergence of perturbation theory
for strong interactions is a novel feature of the
CBM. It comes about because of the .large size of
the pion source. As we point out in the final sec-
tion, this rapid convergence has important con-
sequences not only for the calculation of nucleon
properties'"" and the N-N force,"but also for
such exotic questions as the proposed tests of
grand unified theories in the search for proton de-
cay.



CLOUDY BAG MODEL: CONVERt ENT PERTURBATION. . . 1968

The interaction Hamiltonian (2.1c) allows tran-
sitions between a nucleon and a & with the emis-
sion or absorption of a pion. An important feature
of the cloudy bag model is that the interaction ma-
trix elements n ~ are highly constrained by the
underlying quark structure of the baryons, and are
determined by a single coupling strength f and a
single form factor u(kR). Explicitly,

Since the interaction (2.1c) conserves baryon num-
ber, the expansion of this state in terms of the
eigenstates of the bare Hamiltonian (2.1b) may be
restricted to states containing a single baryon

~
a& and arbitrary numbers of the field quanta:

with

and

t~u(kR}
(2v)3/2(2~ ))./2 /

8:"" 8:~), )' 5 +4M
cf/).d]l 5 f l(4~ 10

(2 .2a)

(2.2b)
with

and

+ g g g c„(a;k.. .k„;nst)
Pal c ~l

x( )/2 a~ a e ~ ka ~Q)
(3.2)

6...5„,Z, ' '=(nst ~ns't'& (3.3)

)
3f, (kR) (2.2c) c„=( „,(a)a, a, ~ ~ ~ a, ~nst&.

1
(3.4)

pn4 phd phd, p (2.4)

The parameter 8 is the bag radius and j, is a
spherical Bessel function of order one.

The Hermitian transition spin operator% of Eq.
(2.2a), which acts in the spin subspace of the bary-
on states, is defined by

(s s~I ~ )s)sss') = P (-1)' ' '
( )S„.(2.8)

Here, 0 is the spherical component of k, and the

3j symbol couples the spins s and sz (nucleons
or &'s) to the angular momentum of the pion. The
isospin transition operator T&, coupling the iso-
spins of a, p, and the pion, is defined similarly. "

The standard Hamiltonian of the Chew theory
may be recovered from Eqs. (2.1) by taking f as
the unrenormalized pseudovector coupling con-
stant"s" and restricting the sum over 0. to the nuc-
leon states only or, alternatively, by setting

The bare & (with no pions) does not appear on the
right-hand-side of (3.2), since it has a different
total spin and isospin from the nucleon,

(est inst& = 0. (3.5)

~c~(a'k&. . .k„'nst)~2.

The normalization condition from Eqs. (3.2),
(3.4), and (3.6) is

(3.6)

The matrix element e„is the probability amplitude
for finding r pions with momenta k„k„.. . , k„
and isospin projections j„j„.. . ,j„surrounding
the bag state a (either a nucleon or a & with spin
s' and isospin t', depending on the index a) in the
physical nucleon with spin s and isospin t.

The probability of finding r pions of any momen-
ta and isospin surrounding the bag state n is
then

(2.5)

so in this case we obtain the standard interaction
of the Chew-Low theory,

The transition operators for n-n transitions are
proportional to the usual Pauli spin and isospin
operators

s=a,

(3."t)

The probability of finding r pions in and around
the cloudy bag is then

(3.3)

if u(k)
(2 )3/s) (2 )1/2

III. BOUNDS

(2.6)
In order to construct bounds on P„,it is useful to
define" "a state ~)t „&by removing r pions of pre-
scribed momenta and isospin from the physical
nucleon,

The physical nucleon of mass A with isospin
projection t and spin projection s is described in
the model by a state (nst&-=~n& which is a solu-
tion of »~6~7

) ( ) / a ( t)

Then, from Eq. (3.4), c„=(a~)t)„)and

(3.9)

a)n& =tk )n&. (3.1) (3.10)



1964 L. R. DODD, A. W. THOMAS, AND R. F. ALVARKZ-KSTRADA

- Z (3.11)

Interchanging the sums over pion states and bag
states, and using the completeness of the state
~a) in the single-baryon subspace, we have

p„= g (y„~o)&o(y„)
ky. . . k

(3.12)
~ P,) =a, ~R) = [a~, H~] IPg)

n k1

is easily established using Eq. (3.1) and the re-
»tion [H„aJ= —(o,a,. For brevity let us intro-
duce the notation 4, =-l, and denote the commu-
tator in Eq. (3.12) by C, =[a„,Hzj. For the par-
ticular interaction (2.lc), we have

k~ k„ ky. . .k

where ~0)(0
~

is the projector for the pion vac-
uum times the unit operator in the baryon sub-
space.

Our aim is to find simple, explicit expressions
for IP,I in order to place upper bounds on the
probabilities P„using Eq. (3.11). First, consider

The identity

, = QNtg~a
ag

(3.13)

Hy applying a, to Eq. (3.12) and using the identity

1 1 1
ak a+ &k-kz-H z —&ok-H k z —ar -H kz-H '

k

(s.i4)
we find that

I ca&.+c. c, I n&
1 !

A„-QP&- (d2 —H II, Q —~ H j

c,+c,! 1
+„—&g- +2-H & '%„-(o, -H 0,) s)=W pg. (3.16)

Repeated application of the identity (3.15) yields the following result: Let Y,y, . .'.p'„be an arbitrary per-
mutation of 1, 2, . . .x, then

C„... 1 c„R).r-0 r3 ~ + H tr
(3.16)

Taking norms throughout Eq. (3.16), we have our key result:

llln&ll (v'I)&h Zs=, ~s,, r„Zi,~sZs=, ~s
'''

&ar„
Ilc„II Ilc„II

' . Ilc„II

"r

="- Ilc, ll

(r! )~» '" (3.1V)

In deriving (3.1V), we have assumed that the spectrum of the total Hamiltonian H begins at A„,the physi-
cal mass of the nucleon, so that for any ~& 0 the inequality

1 1
Hl~ —(0 —H Q7

(s.i6)
where

(S.ie)

holds.
From the inequalities (3.11) and (3.1V), we

obtain the central result of this section: The prob-
ability of finding r pions in the pion cloud is
bounded by

(s.2o)

Consequently an upper bound for the mean number
of pions present in the pion cloud is



CLOUDY BAG MODEL: CONVERGENT PERTURBATION. . . 1965

(r)= g rP„(Ae~. (3.21)

A much tighter bound on the mean number of
pions present is given by considering the expecta-
tion value of the number operator directly,

1

II ~S) II* ( a

ll IR) II'
(3.22)

It is shown in Appendix A that the uncertainty in
the number of pions in the cloud (4r)'= (r ) —(r)'
is bounded by

Ar
~

A2+-'
~

(3.23)

With the specific interaction of the CBM, Eq. (2.2),
we find that

A =~f'18),
where

3 t"kS(kR)'
„

p'(2v )' , (o,'

(3.24)

(3.25)

Some details of the evaluation of A are presented
in Appendix B.

IV. NUMERICAL RESULTS.AND DISCUSSION

First let us consider the case where the & is ex-
cluded from the single-particle space, i.e., the
system is described by the Chew Hamiltonian
(2.6). With the coupling matrix (2.4) Eg. (84) of
Appendix B gives

A= f2I(R) . (4.1)

[A bound of the same form was derived originally

f& pR, (4.2)

where R=k '. In the Chew-Low theory the "ra-
dius" of the nucleon is small, R = 0.28 fm, and the

by Alvarez-Estrada. "2' However, A of (4.1) is
improved by a factor of 4—through the use of com-
pleteness in Eq. (3.11)—corresponding to the spin
and isospin degeneracy of the nucleon. ]

We also note that A of Eq. (4.1) occurs in the

expression for the probability Z, of Eg. (3.3) when

it is evaluated to second order in perturbation
theory. In the Chem-Lom theory, unlike the cloudy
bag model, the functional form of the factor u(kR)
is not well determined, and often a simple step
function with a cutoff k=k

„

is adopted. As des-
cribed by Henley and Thirring, analysis of ex-
perimental data leads (with some ambiguity) to
values of about f'/4@= 0.22 (compared with f'/4v
=0.08 for the renormalized coupling constant) and

k,„-5p,. For these values 4=2.16, which, ac-
cording to Eq. (3.22), is also a bound on the mean
number of pions in the nucleon. The corresponding
bounds on the probabilities P„arelimited to small
values only for x) 6, and the uncertainty in the
number of pions present in the cloud is from Eq.
(3.23) bounded by 2.22. Also listed in Table I are
numerical results for the Lorentzian form factor
u(k)- P/(P+k') used by Fubini and Thirring. ~c

For the Chew Hamiltonian our bounds give no

reason to expect that perturbation theory is valid
for the values of the coupling constant and form
factors required by experiment. Defining a di-
mensionless parameter X by X=k,„/p, we find I(R)
=X'/2 for reasonable form factors, and if we take
from Eq. (4.1) Xf&1 as the criterion for the va-
lidity of perturbation theory, the unrenormalized
coupling constant is restricted to values

TABLE I. Upper bounds for the probabilities P„offinding r pions surrounding the nucleon in

the Chew-Low and cloudy bag models. 'The column headed (r) gives the mean value of the

number of pions present, calculated using Eq. {3.22), while the column headed &r lists the un-

certainties in pion numbers, Eq. f3.23). The values a for the Chew-Low model were calculated

with a step-function form factor, and for case b a Lorentzian form factor was used. In the

cloudy bag model, the results labeled c correspond to the values of the coupling constant f and

bag radios R determined in Ref. 13. Those labeled d are constrained, as in Ref. 13 (Theberge

et al.) to yield in perturbation theory the renormalized valuef„2/4m= 0.08.

Theory

Chew- Low a 0.22
b 022

0.28 9.80
0.30 9.03

2.16
1.99

2.33 1.67
1.97 1.31

0.90
0.65

0.39
0.26

0.14
0.09

2.16 2.22
1.99 2.05

CBM c 0.078
& 0.113

fi

0.109

i

0.100

i,
0.096

&0.093

0.82, 5.04
0.6 10.44
0.7 7.16
0.8 -5.26
0.9 ' 3.98
1.0 3.08
1.1 2.43

0.90
2.69
1.78
1.24
0.91
0.68
0.52

0.40,
3.62
1.58
0.76
0.41
0.23
0.14

0.12
2.73
0.94
0.31
0.12
0.05
0.023

0.03
1.55
0.56
0.10
0.03
0.009
0.003

0.005
0.71
0.20
0.024
0.005
0.001

0.27
0.06
0.005
0.001

0.90 1.03
2.69 2.74
1.78 1.85
1.24 1.34
0.91 1.04
0.68 0.85
0.52 0.72



1966 L. R. DODD, A. %. THOMAS, AND R. F. ALVAREZ-KSTRADA

unrenormalized coupling constant f far too large
to satisfy the criterion (4.2}.

Turning now to the cloudy bag model, we note
that there is an additional factor —'„'in Eq. (3.23)
due to the presence of the 4, which tends to in-
crease the bound. However, in the CBM, smaller
values of the unrenormalized coupling constant f
are needed to fit the observed quantities. In the
calculations of Ref. 13, it was found that, with
the form factor (2.2c} fixed by the bag model,
there was very little freedom in fitting the P33
phase shift through the 4 resonance. A bag radius
of 0.82 fm and a coupling constant f '/4v = 0 078 w. ere,
determined. Evaluating A for these values and the
form factor (2.2c}we find that the mean number
of pions present in the nucleon is bounded by A
=0.9, and P tends to zero quite rapidly, the root-
mean-square fluctuation in the pion number being
bounded by 1.03. The repormalized coupling con-
stant calculated with these values of f and It, using
perturbation theory, is f„'/4m= 0.071, somewhat
less than the accepted value of f„'/4v= 0.080.

In Table I under the entries labeled d, we have
also listed bounds for the values of f and R taken
in the perturbative calculations of Theberge et al. "
of the static properties of the nucleon in the CBM.
Here, f and 8 are constrained to produce the value
f„'/4w= 0.080 for the renormalized coupling con-
stant.

It is seen from Table I that for reasonable values
of the bag radius R™0.9 fm, the use of perturbation
theory, or other approximate methods which trun-
cate the number of pions in the pion cloud, is
much more acceptable in the case of the CBM
than in the Chew-Low theory. The criterion (4.2)
is much closer to being satisfied, and the bound on
the mean number of pions in the cloud is about
unity.

Our bounds are simple, but quite crude, and

probably overestimate P„significantly. The prob-
ability of finding one pion in the physical nucleon
takes the value 35% in the perturbation calcula-
tions of Ref. 13. This value may be compared with
our bound of 0.9.

V. CONCLUSION

Within the framework of a static source theory,
we have established an improved, rigorous bound
on the probability of finding the physical nucleon to
contain n pions. For the recently developed Ham-
iltonian of the cloudy bag model, this bound goes
rapidly to zero as n goes to three or more pions.
In this model the mean number of pions about the
nucleon is less than about 0.9, and the standard
deviation is less than 1.0. This represents a re-
markable improvement in convergence over earl-
ier models such as the Chew-Low model —essen-

tially because of the inclusion of the bare 4 iso-
bar in the CBM.

It is certainly true that the calculation of pionic
corrections to nucleon properties such as mag-
netic moments and charge radii is more compli-
cated than simple probabilities. This is because
of the interference between amplitudes with dif-
ferent numbers of pions. Thus, even though the
probability of finding three pions is very small,
it is conceivable that the three-pion terms could
alter the calculations of Refs. 13 and 15 at a no-
ticeable level. Nevertheless, the convergence
properties of the CBM seem to be so good that we
do not expect any major change in their conclu-
sions.

Not only do our results give great support to
the perturbative approach to single-baryon prop-
erties, but one may hope for new insight in sev-
eral other areas. For example, one might now
expect to make progre'ss in the understanding of
the long- and intermediate-range N-N force using
similar techniques. " We might also mention the
proposed tests of the various grand unified theo-
ries." In particular, there are many experiments
under way which look for proton decay modes,
such as P - e+v'. With few exceptions (e.g. , Ref.
26), the assumption is usually made that the nuc-
leon consists of just three quarks, two of which
annihilate to an antiquark and a lepton. If the
dressed nucleon actually had a cloud of pions
like that in the Chew-Low model, the theoretical
predictions based on the three-quark picture would
be quite unreliable, because of the dominance of
multipion decay modes. However, within the CBM
our bounds strongly suggest that decays to a lep-
ton and one or two pions will dominate. Detailed
calculations on this problem would be very useful.

Our purpose in this paper has been to put bounds
on the pion content of the dressed nucleon, within
the framework of the linearized equations (2.1).
This is a worthwhile exercise in itself, in view of
the interest in such Hamiltonians in low- and me-
dium-energy nuclear physics. However, we did
remark in the Introduction that Eqs. (2.1) are an
approximation to a highly nonlinear, exactly
chiral-symmetric theory. '"' Unlike the truncated
version discussed here that theory is not renorm-
alizable, and discussions of it (e.g. , the nonlinear
a model) usually rely on the tree approximation.
It is worth observing though, that the reason for
this problem is the treatment of the pion as an
elementary, pointlike object. Our underlying mo-
tivation for introducing the pion is that one expects
in the limit of exact SU(2) x SU(2) symmetry, that
the pion should appear as a massless Goldstone
boson associated with the dynamical breaking of
the symmetry of the vacuum. Once the pion has



CLOUDY BAG MODEL: CONVERGENT PERTURBATION ~ ~ ~ I967

some internal structure the pion sector of the
theory will have a natural cutoff too, and one might
expect a fairly rapid truncation of the higher-order
terms (in f '), required formally for exact chiral
symmetry. Thus, although our results may at
first appear to be of somewhat limited interest
because they rely on a linearized version of the
equations, they may be rather cl.ose to reality.

In conclusion we must remark that the conver-
gence of this expansion in number of pions is es-
sential to the internal consistency of the CBM.
At present, the internal structure of the pion is
ignored in our model, and therefore we should
only expect to describe the long-range piece of
the pion field about the nucleon —that is, the one-
and two-pion pieces. By the time we get to three
or more pions we are probing phenomena within
one- or two-tenths of a fermi of the bag surface—
where the bag model itself, and particularly the
static cavity approximation, is probabl. y unreal-
istic.

n a, a, n = n akak, akak Q

+ n atkak n

=2+(Q, ~P, &+(n ~n&(r), (A1)
kk',

where we have used the commutator [a~a, aa, j = 6»,
and (r) is the mean number of pions present.

Consequently for (&r}'=(r&—(r)', we have the
express ion

(A2)
aa

Since the maximum value of (r) -(r)' is —,', and
from Eq. (3.17}and definition (3.20},

~aa. II I n& II' (A3)

the uncertainty in the number of pions in the cloud
is bounded by

ACKNOWLEDGMENTS ~r c (A'+-'}" (A4)

It is a pleasure to acknowledge many helpful.

discussions with Dr. G. A. Miller and Mr. S.
Theberge. One of the authors (L.R.D.) wishes
to acknowledge the hospitality of the TRIUMF
theoretical group. We would also )ike to thank

the following organizations for their financial
support: National Sciences and Engineering Re-
search Council (A.W.T.), Junta de Energia Nuc-

lear, Instituto Estudios Nucleares, Madrid, and

Comision Asesara de Investigacion Cientffica y
Tecnica (R.F.A. -E.).

APPENDIX A
/

The expectation value of the square of the num-

ber operator for the pion field is related to the
vector ~Q, ) of Eq. (3.9) in the following way:

APPENDIX B' EVALUATION OF A

To evaluate A, we seek the maximum value of
the magnitude of the vector C, ~g&, C, given by
Eq. (3.13), as the normalized vector ~g& ranges
over the complete single-particle space, i.e., if

) ands. ld. l'=1, t e expanstonco-
efficients d must be chosen to maximize the
quantity

k k k k t o

After introducing spin and isospin labels explicitly
by setting (o'&= (ast), (p&= (ps"t"&, and (y&

= ~ys't'), and substituting the interaction of Eq.
(2.2a}, expression (Bl}becomes

d'k —. . .Q d,*„d„.,„QV"S'" Q&s s(5 K(s„s&&s„s )5 k(s, s )
k

('dk' f

xg&t. t (r, (t„t&&t„t(r, (t,t &. (B2)

The evaluation is simplified by integrating over
the angles of k first, using

The spin and isospin sums may then be perform-
ed with the help of the identities

g g(s s ~8, )s„s'&(s„s'(S,(sas~&=6...„5„(2s yl} '

and

gg(t t(T&(t„t'&(t„t'(Tz (tat"& =0„,6 „(2ta+1}',

which follow from the definition (2.3) of the spin



1968 L. R. DODD, A. W. THOMAS, AND R. F. ALVARKZ-KSTRADA

and isospin transition operators. The result is

~~ 2(2w)3p, '(2s +l)(2t +l)

X gF "Pt"O——3u(kR)md'. (B4)
4~ u'
3 co~

%ith the specific values of the coupling matrix of
Eg. (2.2b), we find the maximum value of (4.4) is
attained when d~„=0,giving the results, Ejs.
(3.24) and (3.25), of the text.

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