Nucleon-nucleon forces from bags, quarks and boson exchange

Nucleon-Nucleon Forces from Bags, Quarks and Boson Exchange

G. E. BROWN

NORDITA, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark and State University of New York, Stony Brook, N.Y. 11794, U.S.A.

ABSTRACT

The concept of chiral symmetry in QCD is developed, especially the two possible realizations: i) the Wigner mode, in which fermions are massless, ii) the Gold- stone mode, in which fermions acquire mass through spontaneous symmetry breaking. The former is used to describe quarks in the confinement region, the latter, to describe the external regions in which pions and o-mesons carry, in a phenomenological way, the relevant degrees of freedom.

A schematic treatment of interactions arising from quark-gluon exchange in region i), the perturbative region, is summarized and it is pointed out that in the primitive form given, perturbative QCD does not have much to say about the nucleon nucleon force, at least it does not describe known empirical features, such as the spin-orbit interaction and the behavior of the central repulsion with energy.

The chiral bag model, in which quarks are confined by fiat, is developed, starting from the M.I.T. bag model. Pions are coupled to the bag so as to conserve the axial-vector current. From this, boson exchange models can be built up.

The non-perturbative solution of the model indicates a possibility of obtaining the non-relativistic constituent quark model from the chiral bag. It is pointed out that little bags must be very deformed, and consequences of this for physical quantities are worked out.

KEYWORDS

Bag model; chiral invariance; nucleon-nucleon force; quark-gluon exchange; boson- exchange model; axial-vector coupling; magnetic moments; electromagnetic form

factors. INTRODUCTION

Traditionally, since Yukawa's classical work in 1937, nucleon-nucleon forces have been calculated in terms of boson exchange. Interactions from such calculations have done quite well in reproducing nucleon-nucleon scattering data, the deuteron binding energy, etc. In such calculations there is, however, not only some lati- tude in the choice of coupling constants, but the forces have to be cut off at short distances in an ad hoc way. Such a cut off is imposed, for example, in the

147

148 G.E. Brown

Paris potential, at a distance of ~ 0.8 fm. Already in the late 1940's, Gregory Breit proposed considering the nucleons to merge into a soup - he was thinking

about mesons, etc., at that time - when they come close together, and this idea was followed through by Feshbach and Lomon in their boundary condition model.

High energy experiments have shown nucleons to be made up out of quarks. The underlying theory of QCD, Yang-Mills theory, introduces gluons, which couple to quarks. These aspects will be adequately covered in the other lectures, so I sha~

only talk about the ingredients needed to make perturbative calculations in QCD. I shall, however, concentrate on the chiral aspects of the underlying theory,

which will be our chief guide in constructing a nucleon-nucleon interaction later.

In any case, once it was found out that nucleons consisted of quarks, it was only natural to consider the short-distance soup formed when nucleons merge to be a quark soup.

Like nucleons, quarks* possess spin and isospin (flavour); additionally, they possess colour. They come in three colours, say red, yellow and blue. Gluons are also coloured, and couple to quarks through both Dirac and colour matrices

~ = ~ ~ t(i) gp(i) (x) yP q(X) , (1)

but there is no isospin matrix. Here g is the colour coupling constant, g(x) is

the quark field (isospin dependence is here suppressed), gp(i) with i = 1 .. 8 are the eight vector gluon fields, and the I(i) are the eight colour matrices:

11 0 , 12 = 0 - , 13 = 1 i

14 0 , X 5 = 0 ' ~6 = 0 0

17 = 0 , 18 = ~ 0 0

0 0 1

0 0 -

'00 i) 00 ',10

(2)

We list also the Dirac matrices:

~k = k ; B = 0 -i

° k

5 1 2 3 4 (0 i0) Y = Y Y Y Y = - i

where o k are the two-by-two Pauli matrices

Our convention is the (x,y,z, ict) East Coast one.

*We deal in these lectures with only nonstran~quarks.

(3)

Nucleon-Nucleon Forces from Bags, Quarks and Boson Exchange 149

Quarks are confined; at least, this seems to be a reasonable assumption, since we don't see them. Nobody knows precisely how they are confined, presumably by some form of infrared slavery. Quarks at short distances inside the confinement region are supposed to interact perturbatively, by gluon exchange• Hopefully, nonlinear aspects of gluon interactions can be neglected in this region•

The interaction from exchange of a virtual gluon is, in momentum space,

2 (q i U q(xl))(~ i "~12 = ~4 11 Ylk 2 2 12 12 q(x2)) (4)

- k 0

where repeated indices denote a sum. The (k2 2

- k0) is the gluon propagator and q(x) is.the four-component Dirac spinor quark wave function. The dot product i~ y~ 15 y~ can be viewed as a general scalar product I~ • I~ of a 32-component "+! vector, although this may just ~omplicate matters. Operationally, the I s are similar to the isospin vectors T, since they refer to an intrinsic space of the particles, colour space. It is relatively easy to deal with the l's, because baryons and mesons are colourless; i.e., they are scalars in colour space.

Think of exchanging a n-meson between the neutron and proton in the deuteron which has total isospin I = O. One has an interaction involving T 1 • T2; this can be written,

-+ -+ 2 "+2 "+2 .+ .+ [(TI+T2) -TI-~ 2 T1 " T2 ~D (I=0) = L -2 ~D " (5)

Since ~D transforms as a scalar, any vector like T 1 + ~2 gives zero Therefore

'+2.'+2

T1 T2 TD (I=O) 2 ~D (I=0)

acting on it.

(5.1)

In a similar way,

( t1" t 2 + t1" t 3 + 12"13 ) T123(colour less )

(~i+~2+~3)2 "+2 "+2 #2 - 11-12-A 3

2 ~123 (5.2)

where we have now a three-quark wave function to describe a nucleon. From the 1's, eq. (2), it is easily found that

8 ~2 = [ hi hi = 16

3 i=l

(5.3)

• + + 8 (colourless), (5.4) • . 11.12 TI23 (colourless) = - ~ T123

~2 ~2 ~2 + ÷ + ÷ ~ ÷ where we have used the facts that I_ = 12 = 13 and I.'I^ = 12.13 = 12"I 3 when acting on colourless (colour-scalarJ objects. Thus,±th~ interaction from gluon exchange between two quarks in the same nucleon is given by

150 G.E. Brown

2 (q ~ _ 18 TI q(xl))(q ~2 q(x2)) _2 ~12 - - 4 3 k 2 2

- k 0 (5.5)

2. CHIRAL INVARIANCE

Chiral invariance is crucial as a guide in our construction of the nucleon-nucleon

force. With complete chiral invariance, the axial-vector current is conserved

3 Z = 0 , (6 )

just as ~ j = 0 for the electromagnetic current. Similar equations 3~ ~ = 0

hold for ~th~ strong interaction vector current: these were originally arrived

at by analogy with the electromagnetic current. The axial-vector and vector currents are, at this level, on the same footing.

At some stage chiral invariance has to be slightly broken, since complete in-

variance would imply that m~ = 0, where m~ is the pion mass. We shall assume that this can be accomplished without damage to the relations derived from

assuming complete chiral invariance; m~ is small compared with other hadron masses, and it does not seem to be a bad approximation to take it to be zero to

begin with. In practice, this means that in constructing interactions, we

approximate exp (-m~r) by I. The values of r we work with are ~ ~/mnC , so this is justified. Later, when reconstructing the pion cloud, where physical

quantities such as magnetic moments depend upon the long-range part of the pion

cloud, we shall have to reinstate the exp (-m~r).

In one way of realizing chiral symmetry, the Wigner mode, this symmetry results

only when fermions are massless. This is easily seen in the following way: Let us consider a Hamiltonian which contains a mass term

gH = m qq q

where q is a four-component Dirae wave function for a quark, m q

Under a global chiral transformation

iY5~ q ÷ q' = e q

iT5@

is the quark mass.

(6.1)

SO 2iY5~

6H ÷ ~H' = m q e q (6.2) q

and 6H' ~ ~H. The q is obtained from q+ by multiplying by g on the right-hand

side. This B anticommutes with T5 , and the result is as shown above.

The underlying Yang-Mills equations involve massless fermions and are completely

chiral invariant.

In order to construct the other mode of realizing chiral invariance, the Goldstone mode, we go over to a phenomenologi!al description of particles as fields. Al-

though we know the pion to have a qq-substructure, we now treat it as if it were


a fundamental field*; if the pion is small in size, this should be a good approximation, at least for wavelengths longer than its dimension. We also treat the axial-vector currents phenomenologically. They, together with the vector currents, form the generators of two SU(2)'s; the generators are

->" + X , ( 6 . 3 ) G± = ~P -

the arrow on top referring to isospin. The plus set of three vectors close on themselves under commutation, as do the minus set. (In neutrino physics, the operators 1 ± Y5 project out the two helicities; hence, the term chiral, when Y5's are involved in this way.)

There is an isomorphism of the product SU(2) x SU(2) and the Lorentz group, which is easily remembered by associating the axial-vector ~ and the electric vector ~; the vector V and the magnetic vectorS. (It may be confusing that the vector here for the A's and V's is in isospin space; we suppress the Lorentz index p, the above analogy holding for any U-) The electromagnetic current forms a 4- vector in the Lorentz group; the analogous 4-vector in the above SU(2) × SU(2) is formed by the three components of the pion field ~ and a neutral, scalar particle, the ~-meson. Under chiral (y58) rotations, the pion field is rotated into the o-field, and vice versa.

At this phenomenological level, chiral invariance was introduced long before it was known to apply to the underlying QCD equations, in fact, long before QCD was known. I Chiral invariance gained credence, as it was found to give an explanatio~ in terms of the symmetry, of relations which appeared accidental without it. 2 Thus, we have used it as a guiding principle in nuclear physics for a long time.

The immediate question coming to mind is how the underlying equations can describe massless particles, when we know the nucleon, etc., to be massive. This brings us to the mechanism of spontaneous symmetry breaking.

In order to illustrate spontaneous symmetry breaking, let's suppress kinetic energies, which are easily put in later, and use the Lagrangian of the linear o-model

_ + ÷ ¼ L = -g ~(o+i~'T y5 ) ~ - %2 {[o2+~2] _ f2}2

0 ~T ( 7 )

Here ~(x) refers to some Fermion field, say the nucleon. This Lagrangian is chiral symmetric, consisting of:

i) The scalar product of the two÷ chiral four-vectors: (i ~ ~ys~, ~ 4) and (~,o). (Remember that i Ty5~ transforms like a pion, ~ ~ as a scalar.)

+ ii) The square of the four-vector (~,o). In the above, f~ is at this state

just a parameter. It will be connected with the pion life time.

The term

_ -+ 2}2 i ~2 {[02 + ~T 2] -- f = V 4

(7.1)

*This is consistent with the notion of effective field theory relevant to the region in which perturbation theories do not exist. See Mannque Rho's lecture for further discussion on this point.

152 G.E. Brown

will be referred to as the potential. In terms of the o-field and a given com-

ponent of the pion field, it has the form of a roulette wheel, fig. i. For ~ = 0

the minimum in energy will clearly lie in the trough of the roulette wheel. Sinc(

Tf

( ~ ~ Physical world ,,, G

Fig. I. The potential V as function of o and T fields. The

radius of the circle giving the minimum in V is f .

the physical vacuum is scalar, not pseudoscalar, it is placed at ~ = 0. However,

as can be seen from fig. i, at this level it costs no energy to move in the ~- ÷ direction, so there will be vacua with T-fields present, degenerate with this

vacuum. At this stage, pionic excitations cost zero mass. At a later stage one

must add a term, such as c o , to the Lagrangian to favour the a-direction. It

then costs energy to move in the T-direction, which involves moving uphill, and

the pion gets a mass. We shall view such symmetry-breaking terms as small, and

neglect them here.

If we take a cross section of the potential, fig. i, in the o-direction, we find

the curve, fig. 2. The minimum in energy obviously comes when a = fT- Let us

V (G)

I

f~

Fig. 2. Potential energy V plotted as function of o.

assume the value of o to make only small fluctuations about this position. (Thes~

fluctuations will later be eliminated by letting ~2 grow, which makes the curva-

ture in V(o) sharper, limiting a to the value f~.) Then the term ~ o % takes on

the value fT ~' so that the nucleon field has effectively obtained a mass term

m 0~ = g f ~J (8) n T


Here we see the Goldberger-Treiman relation in simplest form

153

gA mn = g f~ ' (8.1)

gA = 1

although we have not yet shown that f is the pion decay constant. Thus, although the initial Lagrangian is chiral invariant and has no mass term, through coupling with the o-field, the fermion field acquires a mass. The above procedure is known as "spontaneous symmetry breaking", the vacuum taking on a particular value of the fields (o = f~, ~ = 0).

3 Now, however, one can see that if the density of fermions is increased, the term m n ~ grows. Beyond a certain value of the (scalar) fermion density, given by

m n ~ = V(o=O) - V(o=f ) (81 )

it will be energetically favourable to choose o = 0, and pay for increased field energy V(o=0) - V(o=f~), dropping the mass terms. At this point, the nucleon flips to a massless state. The same arguments are applied to quarks, except that quarks are confined, by a mechanism not yet understood, so that they exist only in massless state. (We consider only up and down quarks here; even these have

small bare masses, but, consistent with our neglect of the pion mass, we neglect these.)

Chiral invariance can then be realized in two modes:

I. The Wigner mode: o = 0, fermions are massless, there is no pion field. The vacuum is not degenerate.

II. The Goldstone mode: o = f , fermions are massive - here we talk of the phenomenological fermions, such as the nucleon. The quarks are, of course,

never "let out" of the Wigner mode. The vacuum is degenerate (with respect + to movement in the ~-direction).

Very important for our later considerations is that when quarks are present in

phase I, pions cannot be, simply because of the way in which chiral symmetry is realized. Nothing in the scheme outlined here would prohibit quarks from being

present in phase II, although they would then acquire mass through coupling to the o-field. Quarks are, however, presumably confined, because we don't see them, and we shall later accomplish this confinement through imposition of a boundary condition following the M.I.T. bag model; i.e., we shall confine them by fiat.

We shall be preoccupied with the conservation of the axial-vector current,

A = 0 .

This is an operator equation, and must be satisfied, whatever mode chiral symmetry is realized in; in particular, the axial current must be continuous at the junction between two modes. In the confinement region, the axial current must be carried by the quarks,

T ~(x) = i ~ ~5 Y~ 7 q(x). (9)

154 G.E. Brown

~n the external region, the axial current must be carried by the phenomenological and o-fields. Indeed, in the linear o model,

-+ + -+

A ( x ) = o D 7T - ~ ~ ~ . ( 9 . 1 )

+

This form is not convenient, involving both o- and r-fields. By letting %2 in

eq. (7) go to ~, equivalent to letting+the mass of the o-particle become infinite, we impose a constraint between ~- and ~-fields; namely,

+

Thus, o and ~ can be written

in terms of chiral angle 8.

+ O2 + ~2 = f 2 (9.2)

%

o = f cos8

+

= f ~ sin@ Z

Then ~ of eq . (9 .1 ) i s e a s i l y t r a n s f o r m e d i n t o

(9.3)

= f [~ $ @ + sin@ cos~ ~ ~]. z p P

( 9 . 4 )

Defining

we find

= f tan@

= f D p 7T

(9.5)

( 9 . 6 )

where D P

is the nonlinear derivative

/ i

Dp = { i +-- %2 -i

2 f

( 9 . 7 ) P

Thus, if chiral symmetry is to be realized in the Wigner mode in the internal

region, and the Goldstone mode in the external region, then the axial current in terms of quarks (9) must be made continuous with (9.6) at the interface.

In order to measure the pion decay, we use asymptotic, weak fields, so that the

nonlinear derivative (9.7) can be replaced by the linear term. From the

definition of f ,

<~i IA ]'[0> = f~ 6ij q~ (9.8)

we see that f is just the pion decay constant.

3. Nucleon-Nucleon Forces from Quark-Gluon Exchange

Colourless objects, like nucleons, cannot exchange coloured objects, like gluons, unless a quark is also exchanged. The analogy in nuclear physics is with the exchange of isospin. Two s-particles cannot interact by an isovector force, the

s-particles having isospin zero. In the language usual in many-body theory, which


Not Allowed

a l lowed k ~ k

g

I k k b)

a) Exchange of only Exchange of a quark

a)

Fig. 3. Mechanisms of gluon exchange. a gluon is not allowed, b) and gluon is allowed.

155

would be appropriate for nonrelativistic quarks, the self energy of particle k, in the process b) would be represented by fig. 4. This way of picturing the self

I

k

Fig. 4. Exchange energy in the electron gas.

energy is particularly useful for the colour-Coulomb interaction, and there is a close analogy in the nonrelativistic region, where we give quarks masses, with the electron gas representing the alkali metals.

In the electron gas, the direct term, fig. 5, is cancelled by the background uniform distribution of positive charge. The exchange term, fig. 4, has the valu~

Fig. 5. Direct interactions in the nonrelativistic electron gas. The wiggly line represents the Coulomb interaction.

156 G.E. Brown

2 e

M e x c h - I k-k' 12 , (i0)

the minus sign originating from the exchange. In practice, this interaction is

attractive, because it allows electrons of the same spin to get out of each

other's way, and, in this way, to minimize the repulsion.

In addition to this exchange interaction, the electrons have a kinetic energy

k 2

r : 2m (i])

The energy of the electron gas is usually plotted as function of r , the average s

distance between electrons, in units of the Bohr radius. Densities in different

alkali metals range from r s = 1 to 6. Since k ~ 9/rs, the kinetic energy

T ~ ~2/mr~. The exchange energy of an average electron comes from integrating (i) over electrons k' in the Fermi sea, and is ~ -e2/rs, the Fourier transform of

(k-k') -2 giving an r -I. Thus, the energy per electron is

2 . 2 ] 0 . 9 2 ( 1 2 ) E ( R y d ) = 2 r

r s s

The bubble sum plus 4th-order exchange energy contributes additional

6E(Ryd) : 0.62 in r - 0.142 (12.]) s

These are not very large in the region characterizing the alkali metals, and thei~

inclusion does not qualitatively change (12), from which one can easily calculate

the minimum in E as function of rs, i.e.,

dE 4.42 0.92 -- = 0 + (12.2) dr 3 2

r r s s

giving r s = 4.8, for a zero-order approximation. Note that although the contribution to the energy of the total bubble sum is not large, this summation must

first be carried through, and provides a screening for the effective interaction,

which goes now as

2 = e -pr

Veff (r) r e (12.3)

where i/2

n = ~e 2 kFm ) (12.4)

Perturbation theory can be used only after the screening is put in.

Gluon exchange, with nonrelativistic quarks, replaces e 2 by g2, and puts colour

matrices k i and Dirac matrices y~ at each vertex. For the colour Coulomb interaction, we use ~ = 4, and since a sum is carried out over al] quantum numbers of


the intermediate states labelled by k', we have a factor

8 ~2 16 i = 3- (i3)

i=l

Consequently, we find e 2 replaced by (16/3)g2/4 for the nonrelativistic colour

Coulomb gas.

Coulour magnetic interactions enter when we insert the Dirac three-vectors y at the vertices in fig. 4. They arise from the same graph, but it is convenient to consider them within another framework, which we shall outline later. At this stage,

-+

we remark that we will drop the v/c terms from matrix elements of y, but keep the spin currents, which have static limits; these give the colour magnetic interactions.

Already at this point, we can see a major difference between interactions arising from gluon-quark exchange, and those from boson exchange, the older, traditional model. The latter involve, in terms of particle exchange, both direct (t-channel) and exchange (u-channel) terms. With increasing energy of colliding particles, the t-channel exchange survives* (assuming one keeps the momentum transfer constant) whereas the u-channel exchange dies out. In gluon-gluon exchange models, one has only the u-channel processes. Therefore, if we keep the momentum transfer fixed, -u will increase with increasing energy, and these will die out rapidly. It is well known in nuclear physics that exchange processes do not survive substantial increases in bombarding energy, at least not in the forward direction.

4. Models of Quark-Gluon Exchange

4 Kislinger published an article "Quantum Chromodynamics Gives All the Nuclear Forces in Saturated Nuclei and Predicts the Saturation Itself." We shall review some of his conclusions, which indicate ways of looking at interactions arising from quark-gluon exchange.

Kislinger writes the exchange graph**, fi$. 3b as shown in fig. 6. The inverse of the gluon propagator is (p+q)~ _ (_p)Z, p and q here being four-vectors.

*With increasing energy,up in the multi GeV region, t-channel exchanges other than that of the pomeron, responsible for diffractive scattering, decrease with energy. This is described, at least phenomenologically, in the Regge picture, in terms of their trajectories a(t) intercepting t = 0 below unity. We shall try to stay below such high energy regions.

**My colleagues (L. Castillejo and A.D. Jackson; C. Carlson and M. Chachkhunashvili; C. DeTar) have been quick to point out to me that the graph, fig. 6, is just one out of many processes, such as given in fig. 6 bis.

a) Fig. 6 bis.

(footnote continued on next page l

b)

158 G.E. Brown

al

C

P q -p -q d

P -P b2

Fig. 6. Interaction of two quarks through quark-gluon

exchange. The a, b, c, d stand for colour indices.

Since we deal here with nonrelativistic quarks with mass, we can neglect the difference in fourth components, which is ~ v2/c 2 less than the difference in

three-vectors. The gluon propagating in the quark medium acquires an effective mass, usually called a plasmon mass, given by

2 m = ~t . T(w ÷ 0, k -~ 0) (14) P

where ~ is the gluon self energy***

%i %i n(p+k) (l-n(~))-n(p)(l-n(p+k)) 2 ab ba . . . . .

(14.1) = Tr g 2 2 w - (Ep+k-C p)

flavour p spin, colour ~ ~ ~

Fig. b) above represents the nucleon exchange of the process, fig. 6. In these additional processes, more of the momentum transfer must come from the quark wave functions than in the process, fig. 6, where it comes from the interaction. In two limits: i) Forward scattering at high energies; ii) The large-distance interaction between two clusters of three-quarks each, we believe the process, fig. 6, to give the dominant term.

It must be admitted that the following discussion, based on the behavior of Kislinger's process, fig. 6, is very incomplete, and the conclusions drawn should be qualified. In particular, as pointed out to me especially by L. Castillejo and A.D. Jackson, when the argument is made later that the contribution to, say, forward scattering of u-channel processes drops off with increasing energy, it is important to establish the range of energies over which this happens. This range of energies may be quite large, ~ 500 MeV - 1 GeV.

***In fact, this effective mass should be used only for Coulomb interactions arising from gluon exchange; the situation is different with the magnetic

components.


the n(p), n(p+k) being quark occupation numbers. becomes

The factor in the numerator

~n(p) n(p+k) - n(p) = k • ~p (14.2)

whereas ~Ep

- = k • - - gp+k E ~p ~ ~ ~

(14.3)

so that the last term can be replaced by (dn/de)s F. Consequently,

2 Bt. ~ = Tr f~_ xi xi f/dn\

0~+0 flavour 4 ab ba \dEj EF

k+0 spin colour

(14.4)

2 2 kf

re_ 2x2x2 4 22

(14.5)

where we have inserted (dn/ds)E F gas 5. Thus we find

= k2/2~ 2, appropriate for a relativistic Fermi

/4 mp = ~ ~ ~c kf . (14.6)

where 2 fL = (14.7)

c 4~

We can connect kf with the density,

2k~ PQ = 2 (14.8)

evaluated for two spin projections, two flavours and three colours. If we take the volume occupied per quark to have radius r0,

i

PQ (4~/3) r 0 (14.9)

and

1.06 (15) kf = ro

Thus, if the nucleon (three quarks) occupies a volume of radius I fm, r^ = i fm/3 I/~ U

.69 fm and kf = 1.54 fm -I = .30 GeV/c. Thus, even for relatively large volumes and low densities, the plasmon mass is large, when typical values of ~e' ~ i, are employed in (14.6).

160 G.E. Brown

Since gluon exchange is flavour independent, in isospin space there will be a factor

F = 612, 621,

which can be expanded as

F = a 611 , 622 , + b

(16)

f

i~2) (16.1) { !Jill, , 22'

Equating (16) and (16.1) and taking all four isospins in the + direction, we find

1 = a+b (16.2)

Taking 1 and i' in the + direction, 2 and 2' in the - direction,

0 = a-b (16.3)

1 thus giving a = b = ~ . Thus, the isospin dependence of the gluon exchange, con- verted to a direct-c~annel process, is

F = ~ (i + ~i " ~2 ) ; (16.4)

even though the gluon exchange is isospin independent, because it gives only an

exchange term, it gives an effective isospin dependence as a direct-channel process. The above rewriting of an exchange term as a direct-channel process can also be performed for the y-matrices, where it is known as a Fierz transformation. Although the exchange-channel interaction is only vector in nature, viewed in the direct channel, it includes also other invariants. Summing the isospin dependence F over the three quarks in each of the nucleons gives the factor

i -~ -+

F N = ~ (9 + TIN • T2N) (16.5)

to be used between the two nucleons, where tiN and T2N are the nucleonic isospins.

The above has accomplished crossing in isospin. We next deal with colour crossing.

(See the indices a, b, c, d in fig. 6.) We can write

8 A i A i = a 1 + b~Aiac Ab di (17) ad bc ac ibd .

i=l i

Only the colour singlet interaction 1 ibd will be useful, because we wish to C . . .

apply the crossed interactlon only between colou~less objects. Multlplylng both sides of eq. (17) by 6 ac 6 bd and summing over all indices, we find

16 = 9 a (17.1)

Nucleon-Nucleon Forces from Bags, Quarks and Boson Exchange 16]

or

16 = - -

a 9 ' (17.2)

where we have used eq. (13).

Finally, we have to perform the crossing in the Dirac matrices. procedure 6,7 and we find that

V A v = - s + ~ - ~ + P

This is standard

(18)

where S, V, A and P are the scalar, vector, axial-vector and pseudoscalar interactions. Putting factors together, we arrive at the formula

2 _ s+V A M Z_ 16 1 ÷ ÷ 2- 2+ P

= 4 --9 2-- (I+TI'T2) (q+2p)2 + p2 (i9)

as the basic quark-quark interaction.

In order to get a quick line on how matters go, let us look at the central, spin- independent interaction, in nonrelativistic reduction. This will come from the -S and from the y (i) y (2) in V, the Y4'S being replaced by unity in the nonrela- 4 4 tivistic approximation. From fig. 6 and from the way the momenta enter into the denominator in (19), it is clear that the Fourier transform of M 1 is a local potential, multiplied by the Majorana exchange operator pM. Thus, for the quark- quark central interaction, we have

2 -~r12

M1 = - f=--9 (I+TI'T2) ~ e PI2M (20) r12

where the Majorana exchange is defined by

M f f PI2 (Xl' x2) = (x2' Xl) " (21)

The approximation employed by Kislinger, taking some average value for p in (19) and dropping the angle dependence of the denominator with respect to p-q, is equivalent to limiting the interaction to only relative quark-quark s-states. In this approximation, the sum of (20) over quark isospins can be carried out, and one finds

2 TI'T 2 e~Ur V (r) = -g (i + ~ ) r c 4~ (22)

(relative quark-quark s-states)

The form of V is just that of a Yukawa potential resulting from the exchange C

between nucleons of a scalar particle of mass ~, coupled with constant g to the nucleon. The isospin dependence is weak, down by a factor of 9. Normally, the central interaction from vector exchange is repulsive; this interaction is attractive because we are working with the exchange term.

162 G.E. Brown

Let us next look at P = T5 (I) y5 (2), the quantum numbers of the pion. The nonrelativistic reduction gives, for the quark-quark interaction,

2 (l+Zl'S 2) ~2 q M = - f=- 16 1 ql'~ "~ (23)

2 2 p 4 9 2 (q+2p)2 + P 4M q

M To convert this to a local potential, we will again need a PI2' as in eq. (20). If, however, we agree to use the interaction in only relative even states of the quarks~

we can carry out the summation, obtaining

÷ ÷ ~IN "V q2N "V -Prl2 Vp 2 f~_ ( l + g ~ ~ - e (24)

= 9 4~ ZlN'~2N) 4M 2 r12 q

(quarks in relative even states)

where ON, T~ are now nucleon spins and isospins, r12 is the distance between nucleons. This integration has some similarity with that which would come from

pion exchange between nucleons. The latter contributes with ~l'W2, so the term independent of isospin would have to come from some other exchange, e.g., the ~.

In other ways, Vp has little to do with pion exchange. All color Coulomb terms

have equal mass parameter p, in the quark-gluon exchange model. From fits to data with boson exchange models, we know that the ranges in different channels are very different, that in the pion channel corresponding to a low mass, and that in the spin-orbit channel to masses mw, mp-

The spin-orbit interaction originates from vector exchange. Out of the amplitude (19) the quark-quark vector amplitude is

2 (I+TI'~ 2) ¥I'Y2 M = Z_ (25)

(q+2p) 2 2 v 9 + ~

Reduction of this to large components gives, as the part responsible for the spin-

orbit interaction

2 (I+TI'T2) (gl+~2)[qXp] M = f2_ 3i ~ ~ (26) so 9 (q+2p)2+p2 4M 2

q

In order to convert this into a nucleon-nucleon amplitude, we approximate

1 P = ] PN (26.1)

PN being the momentum of the nucleon. Neglecting p in the denominator of (26), which we understand from our preceding argument to'be equivalent to restricting the interaction to relative even states - specifically, setting P~2 = 1 - we find


2 i(3+g A -+ ÷ M = B__ <I'T2) so 9 2 2 (~° IN+~°2N) " [gxPN]

q + (27)

Since this amplitude has the same form and sign as that which would arise from the exchange of u-mesons and 0-mesons in the boson-exchange model, we can simply write the interaction in configuration space as

V l~c ~ 2mnh2 (V~)s0 so = ~ Y- ~-~-~ (g2/4~)

# (Vp)s0 (28)

(g02/4~)

(m = m O = p and quark-quark even states only).

In the boson-exchange model, some of the spin-orbit force arises from tensor couplings of the 0- and w-mesons; this is not included in (28), only the vector couplings.

A major problem arises from the limitation to quark-quark even states. At the lowest energies, when two nucleons collide, all quarks will be in relative s-states~ so the approximation of even quark-quark states necessary for the derivation of (28) is justified. On the other hand, the spin-orbit interaction is zero, because all quark angular momenta are zero. As the energy of relative motion of the two nucleons is increased, one and only one quark-quark pair will go into a relative P-state. But in this case, the Majorana exchange operator P~2'÷ which occurs in all of the quark-quark amplitudes in configuration space, will gzve an additional minus sign. Thus, in the initial energy region where the spin-orbit interaction enters, that from quark-gluon exchange will have opposite sign to that from boso m exchange. The spin-orbit interaction from boson exchange has, however, the same sign, both in the shell model and in the scattering at all energies analysed thus far, the sign being such that the interaction is most attractive in the states of highest J, for given L.

While agreeing that the spin-orbit potential from quark-gluon exchange has the wrong sign in nonrelativistic models with large quark masses Mq, (and is nearly zero for Mq ~ 300 MeV), Pirner 8 finds that for relativistic quarks with M_ ~ I00

H MeV the spin-orbit potential is large and has the correct sign in relative p- states. This is related to the change in sign of the central potential 5 (see § 6) in going from massive to massless quarks. Bag models, in dropping the fourth components of the momenta in the gluon propagator (see eq. (19)) behave like nonrelativistic models in the respects relevant here. We would expect them to give the wrong sign for the relative p-state spin-orbit interaction, although this has not yet been calculated, to our knowledge, in bag models.

As we go up in energy of the bombarding particle, keeping the invariant momentum transfer fixed, more and more states of relative motion enter into the scattering, even states giving one sign to the spin-orbit interactions, odd states the other sign. So the total interaction will die out quickly with increasing energy. All that we are saying is that the quark-gluon exchange will tend to give constant

164 G.E. Brown

polarization at constant u. Now

s + t + u = 4m 2 (29) n

or -u = s + t - 4 4 . For constant t, and increasing s, -u grows in magnitude, and

the propagator, which is essentially ~2 _ u, will cut off the interaction. Of

course, one could look at the polarization experimentally at constant u, as one

goes up in energy. However, we know from experiment that the spin-orbit inter-

action is a rather constant function of t. In fig. 7, from Love and Franey 9, we

I ° ~ - ' I v - i ' I -

I0 2

E i

~ 0 I ° ~ _ - - . . . . . . . . . 2 1 o

I , I I l , I I010 I 2 3

q ( f m -!)

- 4 2 5 ,--,2

Fig. 7. Effective nucleon-nucleon spin-orbit interaction,

as deduced via impulse approximation from nucleon-

nucleon phase shifts. (Love and Franey 9)


show the isospin-independent and isospin-dependent spin-orbit t-matrices, constructed via impulse approximation from the free nucleon-nucleon phase shifts.

the It~S(q) i, dominated in boson exchange models by the Especially m-meson exchange (The m-meson is coupled with g~/4~ ~ i0 in boson exchange models.) is constant, as function of energy, for given q. The higher values of q are probably not well-tested experimentally, but even considering q's ~ 2fm -I, this means that the description of the spin-orbit interaction in terms of t-channel exchange works down to distances of ~ 0.5 fm.

Don't forget that if the p-wave spin-orbit interaction from quark-gluon exchange has opposite sign to the boson-exchange interaction, then a dip in tLS(q), which would result from the quark-gluon picture taking over at large q, should be seen.

Whereas our simple result, that the quark-gluon exchange spin-orbit interaction has the wrong sign at low energies, just where it begins to come in, disposes only of the nonrelativistic constituent-quark model in the simple form used, the general point that the polarization depends simply on t, and not on u, makes it difficult for any quark-gluon exchange model to explain the polarization. It seems to us to be very important that the boson exchange model is needed here. In particular, the main contributions to the spin-orbit interaction come, in the boson-exchange model, at distances r = ~/mmc. This, and other phenomena which we shall discuss, demand that the connection between boson-exchange and quarks and gluons be made at these rather small distances, < 0.5 fm.

If one were to look at the polarization at fixed u, for a range of bombarding energies, in pn scattering, one might see some regularities stemming from quark- gluon exchange. At back angles, of course, the odd Legendre polynomials change sign, so contributions from different partial waves from the quark-gluon exchange should be coherent.

In this connection, it would be interesting to measure the NE spin-orbit splittin~ Arguments have been made I0 that whereas the nucleon is small, ~ 0.4 fm in radius, the pion field of the ~ is not sufficient to compress it, and the E will be large, R E ~ 1 fm. In this case there would seem to be a good chance that quark-gluon interactions would give most of the spin-orbit splitting.

Note that the spin-orbit splitting in the nucleon-nucleon interaction is an especially stringent test between quark-gluon exchange and boson-exchange models. As discussed earlier, the signs of the interactions produced in the two models are the same in the other channels, and one could imagine, by suitable choice of parameters, to get similar results, although one does not seem to have anything like effects from low-mass pion exchange in the quark-gluon exchange model.

5. Bag Model Calculations of Nucleon-Nucleon Forces*

From the preceding we see that nucleon-nucleon forces originate in two regions, an external one in which they are described by boson-exchange models, and an internal one, in which they are described by quark-gluon exchange. In §7, we shall show that chiral invariance shows us how to join the two regions.

It is convenient at this stage to introduce the M.I.T. bag model, in order to separate these two regions. The M.I.T. bag confines quarks by fiat, using a

*This chapter follows the discussion in the 1979 Nordita preprint by C. Carlson, F. Myhrer and G.E. Brown (unpublished).

166 G.E. Brown

boundary condition which makes the quark vector current zero at the bag surface. We shall develop this in detail later.

The most careful and detailed considerations of the nucleon-nucleon interaction within the confinement region have been carried out by DeTar II-13 We shall dis-

cuss his work, trying to make connections with the Fermi gas approach of the last section, because our main purpose is to make the joint between inside and outside and establish the boson exchange mechanism.

DeTar considers two clusters of three quarks each. Three quarks are in right- handed wave functions

qR = N(qs + ~ - qp) (30)

and three in left-handed wave functions

qL = N(qs - ~ - qp) ' <3o.z)

where s and p are the lowest quark orbitals in the bag, ~ is a parameter connected

with the distance between centers. In (30) the s- and p-radial functions add coherently on the right, where the p-wave is positive; they subtract on the left.

Thus, for % ~ i, the system consists of two separated clusters. DeTar finds the energy of the two-cluster system, as function of %, to be that shown in fig. 8.

DeTar allows the bag to deform, as quark clusters separate, but the deformation of the bag is unimportant for the energetics. We shall discuss here the reason for the maximum at zero separation, and the minimum at finite separation.

We first consider the colour electric energy. s-states, the colour electric energy is zero.

5 0 0 - - - -

2 0 0

I00

- I 0 0

- 2 0 0

For g = 0 where all quarks are in This comes about in the following

~AG + --

/ i

1 / , , | I 2

6 (Ira)

Fig. 8. Two-nucleon interaction energy (MeV) vs separation parameter 6(fm) for the six-quark system with I=0, S=I, Imsl=l (solid line with circles),com- puted variationally at fixed separation. Shown for comparison are the one-pion-exchange potential (solid line) and the interaction energy com- puted variationally at a fixed quadrupole moment (dashed line and plus signs).


way: For one of the three-quark clusters, the lowest-order gluon interactions are of the two types shown in fig. 9. The treatment of the self-energy term, fig. 9b, is delicate. The M.I.T. group have'~dapted the somewhat arbitrary pre- scription of including only that part of fig. 9(b) which must be added to fig. 9(a)

(a) Fig. 9.

(b) Lowest order (in ~c ) gluon interaction diagrams for a nucleon. (a) Gluon exchange. (b) Gluon self- energy.

to meet the gluon boundary conditions". In the case considered here, this means, when all quarks are in s-states, including only the same s-state as an intermediat~ state in fig. 9(b). This means effectively that each quark interacts with three quarks making up a colour singlet, so that the total colour electric energy is zero.

The positive energy at ~=0 stems from the colour magnetic energy. The colour magnetic energy from lowest-order gluon exchange is 14,15

AE 4 m=~c ~ [ ~ . ~J 1 i#j ~ 1

(31)

where ~ represents an integral involving magnetic fields, over the bag; we can consider it to be lumped with ~c as a parameter here. This leads to the supermultiplet formula for the energies of a system of n quarks, all in the same spatial state

4 ~ [n(n-6) + S(S+I) + 31(I+i)] (32) AEn = ~ ~c

where S is the spin of the total state, I, the isospin. The coefficient (4/3)~s(~/R) can be chosen from the energy difference of nucleon and 4(1230) isobar to be ~ 25 MeV. (The zero in energy in fig. 8 is chosen to be the energy of two separated nucleons; according to the above, each would have a colour magnetic energy of ~ -150 MeV. This explains the peak of ~ 300 MeV with DeTar's choice of origin.)

As the two clusters of three quarks each move to larger distances, described by % i, three effects come in:

(i) The colour magnetic energy drops by ~ 300 MeV, for the reasons just outlined. (2) The quark kinetic energies increase*, because p-state orbitals, which have

*Corrections should be made for spurious center of mass motion; DeTar has not made these, nor do we include them in this discussion.

168 G.E. Brown

(3)

higher energies in the bag, are introduced. For a bag of radius ~ i fm, effects (]) and (2) roughly cancel. Colour electric exchange terms, the largest one shown in fig. i0, come in.

$ P

Fig. i0. Colour electric exchange terms which enter when quarks are separated.

These colour electric exchange terms are attractive and grow larger in magnitude with increasing amount of p-orbital, reaching their largest value for % ~ i. They are of the type discussed earlier, where they produced the attraction in the even- state quark-gluon exchange central interaction. These explain the minimum in the curve, fig. 8. In fact, to the extent that the energies from (I) and (2) above cancel each other, the curve is just that of colour electric exchange energy, and the situation is like that of the Coulomb gas, eq. (12) and following. We do not carry our considerations out further beyond the minimum, because bag deformation energy begins to enter significantly. Note that to get the detailed shape of the curve from the minimum on in to r=0, we must include the particular self energies, discussed earlier, which make the colour electric energy zero for 6=0. These go beyond the present quark-sea considerations.

At first sight, DeTar's approach looks completely different from the work described in §3 and §4, involving nonrelativistic quarks with masses M , because q the nonstrange quarks in the M.I.T. bag model are nearly massless. However, in processes involving virtual gluon emission, such as those leading to the nucleon- nucleon force, energy denominators such as

- + ~i (Wg wf)

where wi, ~f are quark energies, w is the gluon energy, come in. In the M.I.T. bag the gluon eigenmodes have somewhat higher energies than the difference ~f-~i (which is, at most, ~p-w s in DeTar's work). Thus, wf-w i is neglected in the denominator, and one is left with only -~ . This approximation leads to denominators, for the gluon propagator, such as s~own in eq. (19), but without the plasmon mass term, ~2, which is not put in in the M.I.T. nor in DeTar's work. Thus, Yukawa potentials get replaced by i/r12.

The mechanism, fig. i0, giving rise to the attraction for % ~ 1 in DeTar's work, is the same as that, fig. 4, which gives rise to the attractive V c, eq. (22). We have explained that DeTar's colour electric interaction moves up to zero at %=0 because of the particular self-energy term, fig. 9b, which is included.

The M.I.T. bag model from massless (or nearly massless) up and down quarks. The spin-dependent part of the current J = e 4 + ~ in the M.I.T. model is


j e ^ d - -- r x (~+ o ~) (33) -- CO drr ~

where ~ are the large components of the wave function 4, and ~ is the quark eigen-

energy. This J has the same form as it would have for a nonrelativistic particle, but with m replaced by 2M in the latter case, Mq the mass of the particle. Thus, there seems to be somewha~ of a mapping of the bag model on to nonrelativistic quark models, with energies in the former mapping on to masses in the latter. What is usually called the constituent quark mass may not be a mass at all,

rather, an energy.

Let us see with the above rule for connecting the M.I.T. bag model and the Fermi sea calculations whether we can understand DeTar's detailed work 12 on the tensor

component of the two nucleon interaction in the quark-bag model, esp. his fig. 5 for the even-parity splittings. The central interaction (22) has values of

-(2/3)~ c and -(i0/9)~ c times a common Yukawa in isospin-zero and isospin-one nucleon-nucleon states, respectively. Let us look next at (23) and (24) in order to estimate the tensor interaction, which comes into the S=I state. From our pre- vious argument, we translate the magnitude of q/2Mg into unity to make contact with the M.I.T. bag model, and set gA = 1.09, the Nag model value. To get the sign of the tensor intraction, we look at (24), realizing that each V there will

> :~ o

I11

0

Sphere NN

3 0 G 1 : O , S = I,m~ : I

\ o -0 S:' m :O • - ,- ., s

200100 ~I ~ Even PorHy

-LO0

-200

-50O 0 8lira)

Fig. ii. Interaction energy for the two-nucleon configura-

tion in a spherical bag as a function of the con- strained separation parameter ~ for even-parity two-nucleon configurations.

be tur~ed into an ~12 after operating on the Yukawa. In any case, the dominant TIN • T2N term has the same sign and form as the usual OPEP, so we know that it

170 G.E. Brown

will favour prolate deformation with ms=l along the deformation axis. (The axis of separation is the z axis in DeTar's work.) Setting 191/2Mq = i, amounts to taking the magnitude of

~IN Y ~2~ " 2

4M q

to be unity, for the ms=l state (and zero for the ms=0 state). Thus, Vp has the 2 -> ~+ coefficient (2/9)e (i + g^ ~1~'To~) = -.57 ~c for the s=l, ms=0 state, and zero

for the other two. u Note t~at±~e ~n use the quark-gluon exchange tensor force of

Kislinger, even though his derivation has the wrong sign for odd-state quark bonds because it is applied to only nucleon-nucleon S- and D-states. The nucleon-nu-

cleon D-state can be made up out of either a single d-state quark bond, or two p-state bonds; in either case, the sign will come out to be that of Kislinger.

The other interaction which enters the nucleon-nucleon S-state is the Jl-O2 one. Here Kislinger has two different types of terms, one involving the quark momentum

explicitly. We are unable to evaluate the ~i'~2 interaction without considerably more calculation than for the terms above. At the present state, we shall simply hope that it is small.* To the extent that it is not, our rough agreement with DeTar for relative position of S=I and S=0 states in Table 1 will be destroyed.

Our predictions for the three states of DeTar are given in the table below. As one can see in comparison with fig. i0, our estimates produce sufficiently well

Table 1 Estimates of Energies of the Even States. The

Coefficients of ac e-Pr/r are given

State: S=I, I=O, m =i S=I, I=0, m =0 S=0, I=l s s

Central Pot. - 2/3 - 2/3 - 10/9

Tensor Pot. - .57 0 0

Total -1.24 - .67 - i.ii

DeTar's ratios so that we can claim to have made the connection between his model and the quark-gluon exchange model with Fermi seas. As to magnitude, equating

-~r e

(1.24 acp) p~ = 300 MeV

*The ~i'@2 interactions are small in the boson exchange model, where there is no such component in the long-range part of the force, since there is no spin-one, isoscalar particle. Of course, there is such a component from w-exchange, but it enters only in order m£/~ , and with a small coefficient (1/6 for the vector coupling of the w). Furthermore, its effects are cut down by the strong repulsion

from the central part of the w-exchange interaction, which has the same range. In effective nucleon-nucleon interactions, very little of this component is seen.


we see that there would be agreement for D the order of several hundred MeV,

depending upon ~c and the value of ~r. We have already explained that the rise in DeTar's curve as r + 0 results from the way in which the quark self energy is treated, and results from the fact that each of the separated nucleon has

-150 MeV colour magnetic self energy, the energy at 6 = 0 really being zero in this way of looking at matters.*

The agreement as to magnitude must be somewhat coincidental, since DeTar's calculation does not have plasma effects in it, and, therefore, no plasmon mass.

Having shown the similarities and differences of DeTar's calculation from the Fermi

sea one, we next argue that plasma effects should be taken into account, and that they make the calculation self limiting. Inside the bag, perturbative QCD is

supposed to apply, but as we have seen, the situation with respect to the colour electric forces is very analogous to that of the electron gas. In this case, there is a plethora of literature 17,18 documenting the fact that the bubble sum must first be carried out - i.e., the exchange term must first be screened, be-

fore perturbation theory can be applied, to the screened interaction. Taking DeTar's density of ~ 6 quarks/fm 3, and ~c ~ 2, one finds kf = 380 MeV from eq.(15) and m~ = 606 MeV. Thus,jat this density, the colour Coloumb interaction should have ~ factor of e -(3fm" )r in it, and his attraction will be considerably cut

down.

From the discussion of calculations in quark Fermi seas and of DeTar's work, which have many similarities, we see that many of the qualitative features of the nucleon-nucleon force are reproduced by quark-gluon exchange. Because of the strong screening of the central interaction, it seems unlikely that it will be able to produce enough attraction to reproduce the deuteron bound state and low-

energy scattering parameters, when the soft repuleive core of DeTar's more careful considerations is included. There is considerable uncertainty in the parameters, and there is some way to go before definitive conclusions can be drawn on this point.

It seems clear that none of the theories discussed can produce the long-range OPEP, the best tested part of the nucleon-nucleon interaction. All components have the

same range in these models. Furthermore, these models give a tensor force which oscillates in sign between even and odd angular momenta, whereas fits to nucleon-

nucleon data demand that the sign of the long-range tensor force be independent of parity. (Fits to phase shifts are consistently improved by taking phase shifts

for high ~ to come from OPEP.) On the other hand, the pion is brought in naturally when chiral invariance is restored to the bag model, as we shall discuss

in § 7.

The quark-gluon exchange gives the wrong sign for the spin-orbit interaction at low energies, at least in nonrelativistic models; nothing like the relatively simple t-channel type force seen at higher energies.

One should make two historical remarks at this point. (i) The spin-orbit

potential was first introduced 19 into the scattering problem in order to explain the nearby isotropic 300 MeV pp scattering. The spin-orbit amplitude has a sin coefficient, and if the interaction is sufficiently short-range, will build up the cross section towards 90 ° , whereas the other contributions tend to drop off

with increasing momentum transfer. (2) The vector mesons were predicted 20,

* This is not to say that DeTar's core is not real. The ~ -300 MeV colourmagnetic energy possessed by the separated nucleons disappears when all six quarks are forced on top of each other.

172 C.E. Brown

before observed, in a mechanism for the short-range spin-orbit potential, and it was realized that their masses had to be large. So there is a long history and

considerable phenomenology to support the view that the spin-orbit interactions are closely associated with vector-meson exchange.

Minimally, we invoke the boson exchange model for the longest-range part of the

spin-orbit interaction. None comes from pion exchange, so we must be seeing effects from other bosons. We shall show that once the pion is connected to the

bag, the (spin-zero, isospin-zero) o-meson is automatically coupled in by the nonlinear chiral theory. We shall show that construction of the vector-meson couplings is straightforward in principle, and that the effective u-meson couples strongly. As is well known, this will produce a repulsion between nucleons, which is ~ 40 MeV even at 1 fm separation and increases rapidly towards smaller distance

until cut off by the onset of the quark phase. The question is then, how often do nucleons at low energy come close enough together to see their quark-sea interior? Most estimates in the quark gas theories neglect the fact that low- energy nucleons will be kept apart by the above strong repulsive interactions. Thus, although the quark gas theories might furnish more or less the right amount of attraction for the nucleons when nucleon densities are freely averaged

over the volume, once the nucleons are properly anticorrelated, this can be cut down considerably.

ii .

DeTar's soft repulsive core is similar, for low bombarding energies, with that needed empirically in the Paris potential 21, but the latter needs this repulsion

to increase strongly (linearly) with energy in order to reproduce the nucleon- nucleon scattering data. We shall now show that DeTar's core has the opposite behavior, decreasing with increasing energy.

We earlier described the origin of DeTar's core, following the supermultiplet formula, eq. (32). (Remember that the coefficient (4/3)~ c ~/R ~ 25 MeV.) Two isolated nucleons each have colour magnetic energy of 150 MeV, which disappears when all six quarks are shoved on top of each other in the same orbital (when the n(n-6) term is zero). As the energy of the incoming nucleon increases, the three quarks in it no longer are in is states, even when the centers of mass of the nucleons are coincident- but they will be in multi-n s-states.

We learned early, figs. 3 and 4, that all interactions involving gluon exchange

also involve quark exchange. Thus, the gluon propagator, at least non-relativ- istically, will involve Ik'-kl 2 (see eq. i0), k' referring to the quark in the incident nucleons, k to the quark in the target nucleon. The numerator involves

matrix elements of either Dirac's ~ matrices or i, which cannot exceed unity. From the gluon propagator, we see that any interactions between the two nucleons will drop off as k -2, the inverse energy of the bombarding nucleon. Relativisti- cally 5, this drop~as E -I is also valid. Furthermore, the running coupling con-

stant also drops, so we have a net decrease as

(E ~n k2) -I

Once the interaction between the two nucleons drops out, we have just the sum of colour magnetic energies of the individual nucleons left, the energy they have when separated, so the repulsive core has disappeared.

Thus, although DeTar's repulsive core at low energies is not unlike that in the Paris potential, the dependence with energy is quite the opposite. In the several-hundred MeV region where a strong short-range repulsion is required to fit the data, the quark-gluon exchange interaction has little repulsion to offer.


Note that with a bag size of ~ i fm as used by DeTar, the repulsive core will dis- appear with a momentum scale ~ 1 fm -I.

In fig. 12 we show the growth, with incident energy, of the repulsion required to fit the nucleon-nucleon scattering data.

We see, then, from a most general feature of the quark-gluon interaction when treated perturbatively - that gluon exchange must involve quark exchange - that it is incapable of reproducing the strong short-range repulsion needed in the

)

E I

>

~ 0

-4 P_ ~'_~.:,-o 0 -J

- - 2 ~ ~ " 425 -~------ C-'Z'~

102 -- "~ -

- 2"

1_.1oo

~ }Lo

~_ ~,..-" ~.. -

; \~-" ,/,4o ;

; V --

I0 w I 1 I , , , i I i 2 :3

q (fro -I)

Fig. 12. Effective nucleon-nucleon central interaction, as deduced via impulse approximation from nucleon- nucleon phase shifts. (Love and Franey 9)

174 G.E. Brown

nucleon-nucleon interaction. This repulsion must come from some other agency; e.g., vector meson exchange. If this repulsion persists to short distances, as required above, quark seas will be strongly anticorrelated. To indicate just how strongly, let us remember that in the boson exchange models* g~/4~ ~ 10-12, and the a-exchange is cut off (phenomenologically) at ~ ~/mnC. At the cut-off point, the repulsion is several GeV in magnitude~

The above discussion suggests that the chief relevance of the quark substructure to the nucleon-nucleon force problem may he in providing a cut-off mechanism for quark seas to explain all of the features of the NN interaction.

6. Quark-Gluon Exchange Forces at Higher Energies

General features of the behaviour of the quark-gluon exchange forces derived in the nonrelativistic approximation, assuming that the quarks have masses, as one usually assumes for constituent quarks, are clear, and follow from the presence of the exchange operator P~2 in the quark-quark interaction. (See eq. (20), for example.) Whereas the interaction will have a given sign and be constant in magnitude at low energies where only relative s-states come in, as soon as relative p-states enter, they will come in with opposite signs, etc. Thus, the interaction will drop off in magnitude with increasing energy. This is a straightforward result following from the fact that the quark-gluon exchange gives a u-channel

force, not a t-channel one.

With increasing energy the nonrelativistic approximation will no longer be appli- cable (if it ever is~); the quark masses will become small compared with their momenta, and we shall have to go over to a relativistic treatment. This should occur in nucleon-nucleon scattering in the several-hundred MeV or early GeV region.

Fermi liquid theory 5 is a good vehicle for this discussion. The Landau Fermi- liquid interaction fpo,p,a,iS related to the two-particle forward scattering

amplitude via ~

M M

fp~,p,o, - P' o,p,~, (34)

where ~po,p, o, is the usual Lorentz-invariant matrix element, and ~p = p~ + M 2q •

The spin symmetric part of f is given by

M M s 1 ~q _Kq fpp' =7o,o,X ~p ~p,MKpo,p,o, • (35)

We first carry out this sum for an ordinary vector interaction, as in the relativistic electron gas, but keeping only the exchange contribution, which is the only one which survives as discussed earlier.

f~ e2 M M ~(p) ~{~ u(p') ~(p') y~ u(p) V,exch = - --4 ~ ~ ~q ~ ~ 2 ~ (36)

o,o' ~p ~p' (p, p)2 +

*This value of g~/4~ has independently been found in dispersion theoretical analyses of high energy forward angle nucleon-nucleon scattering by Riska and Verwest 22 and by Grein 23


where the u(p) are the Dirac spinors. The sum can be carried out:

175

V,exch

e 2 1 i 1 4 gp Ep,4 tr 7p('iy'p+Mq) y~(iy.p,+Mq)/[(p,_p)2 + p2]

2 2M 2 + p'p' e i i q 2 ep ep, (p,_p)2+ p2

(37)

where p-p' is now the scalar product of four-vectors. Changing from the relativistic electron gas to quark-gluon exchange is accomplished by:

(i) replacing ~by g2/4

(ii) introducing a factor of 1/2 for isospin crossing (the amplitude is averaged over isospin)

(iii) introducing a factor of 16/9 for crossing in colour

after which we end up with

2 2M 2 + p'p' fs = _ _i $ q pp' 9 e 2 ~~ PEp, (p,_p)2 + P

(38)

In the nonrelativistic approximation, this becomes

2 fs = _ f2_ 1

PP' nonrel 9 (p,_p)2 + 2 (39)

Fourier transformation of this would lead to the isospin independent part of eq. (20), as it should. In the relativistic approximation (neglecting the quark masses compared with their momenta) f becomes

2 fs = ~__ i (40)

pp' 18 IpIlp'l ~

where the IPl and Ip'I come from the E D and ep,. Thus, the spin-symmetric Fermi liquid ampl~tude which is attractive at low energies, becomes repulsive with increasing energy, but decreases in magnitude with the inverse of the momentum of the bombarding particle.

Let us come back to the analogy with the electron gas. M ~ of eq. (i0) would go of excn. as -e2/k 2 for large momenta of particle k. The change sign noted above comes

from transverse gluons taking over with increasing energy, and the change from k-2 to Ip1-1 is a relativistic effect.

7. The Chiral Bag Model

In the MIT bag, quarks are confined by fiat (which is a reasonable way to proceed, since the problem of confinement has not been solved, but they do seem to be confined). The boundary condition

176 G.E. Brown

~ ' ~ q = q [ s (41)

makes the normal component of the vector current zero on the surface of the bag. This is easily seen, because, from the hermitean conjugate of (41), one obtains

~'n = - ql S , (41.i)

+$ since q = q (see eq. (3)) and B antieommutes with y-n. Multiplying (41) on the left-hand side by q, and (41.1) on the right-hand side~by q, one finds that

~'~ q = qqIsl ' ~ ~'~ q = - qq[s'l respectively, so that

q ¥'n q [ = 0 , (42)

i.e., there is no normal component to the vector current at the bag surface. Therefore, no quarks will flow out.

The axial-vector current in the inside, quark domain is

T ~ = i q Y5 Y~ 2 q (x ) ( i n s i d e ) (43)

Using the boundary condition (41), one easily finds

->

e l 9"X= i ~ y s ~ q s , (44)

which is not zero; thus, the axial current isn't conserved unless it can continue ÷ outside the bag. Outside, one has mesons, but no quarks. Here A~ can be realized (see eq. (9.6)) in terms of the pion field

= f D ~ . (45)

Continuity of the normal component of the axial current at the surface, together with the fact that ~ obeys the (non-linear) free-field equations outside the bag, then determines $, and it is straightforward to show 25 that asymptotically in r,

$ is the usual Yukawa field with the usual coupling constant f. The connection between f~ and f is made by using the Goldberger-Treiman relations

f-I = 2f (46) gAm~

(8.1) to the case where gA # i, gA being the which is the generalization of eq. axial-vector coupling constant

_2 . - - = .08 (47) ~ 4

and the relation to g of eq. (8.1) is


i.e.

m

f = 2m g , (47.1) n

2 = 14 (47.2)

4~

The boundary conditions (41) and (41.1) can be included in the Lagrangian density

of a surface term

6 = qq 6 S • (48)

Such a term has the form of a mass term, and from the arguments in § 2, is obviously not chiral invariant. Within the framework of the linear o model, a chiral invariant boundary condition is 26

1 -> -~ ].n q = T- (o + i ~.~ y5)q . (49)

]I

One can now proceed in one of two ways:

(i) View the ~ in eq. (49) as small and develop solutions to the bag equations with the above boundary condition in perturbation theory26, 27

Transform this boundary condition to the nonlinear pion field ~ introduced in § 2, and try to solve the nonlinear equations 28.

(ii)

In the first way, all quantities are expanded in a power series in f~l , e.g.

ql q2

q = qo +7 +7 + ..

° 1 o = f~ +---~ + ..

f (50)

H I 72 = ~-- + ~-IF + ... etc.

Since the only bag quantity with dimensions of energy is R -I, the series is in of (f~R) -I. Much work has been proceeding in such perturbation theory. powers

Formulated in the above way, this procedure is similar to old-fashioned perturbation theory in powers of the pion-nucleon coupling, with an extended source of radius R. Because of the latter, form factors

3 Jl(kR) F(k) kR (50.1)

enter, since the pion wave function is zero inside of the bag. Because of these

178 G.E. Brown

form factors, nucleon self energies are finite.

The lowest-order pion self-energy is shown in fig. 13. straightforward fashion and gives 24 for the nucleon

It can be evaluated in

\

N,A ~Tr

/ N

Fig. 13. Lowest-order pion self-energy.

E(2) = _ i <Nucleon I Z 48~ f2 R3(2 2 )2 ~ i,j

0

o(i) -o(j)

r(i).T(j)[Nucleon> (51)

where ~ = 2.04. o

- 1 C o n v e r t i n g f t o f by t h e G o l d b e r g e r - T r e i m a n r e l a t i o n ( 4 6 ) , one f i n d s

T

E(2) ~ 25 mn (m R) 3 '

n

so that for a bag radius R ~ 1 fm, E (2) = - 0.2 m . n

c o n s i s t s , t h e n , o f t h e t e r m s :

(i)

(ii)

(iii)

(iv)

(52)

The energy of the nucleon

2.04 Kinetic energy T = 3 ----~ ; the value (2.04 follows from the

boundary condition (41).

47 R 3 (53) Bag energy B 3

Pion self energy E

Zero-point energies and corrections for spurious kinetic energy -Zo/R.

The bag energy (ii) arises because it costs energy to create a bubble in the physical vacuum; B is essentially the difference in energy, per unit volume, between perturbative (bubble) vacuum and the physical vacuum. The total energy of the nucleon consists of the sum of these energies, plus others, some of which we shall discuss. The bag constant B is not known, since the fundamental calculation


to determine the difference in energy of the perturbative (bubble) vacuum and

physical vacuum is not yet quantitative, so B is adjusted for each R in order to

make the bag energy an extremum.

When one adds the above energies up, one finds an energy, for R ~ 1 fm,of about

m n c 2, but the curve of energy vs R is quite flat, of the form Fig. 14. The

curve, fig. 14, involves different values of B for different R; B is chosen so that

E 2 113 C

I1

I R 1.0 fm

Fig. 14. Energy as function of bag radius R.

dE/dR = 0 for the given R.

which included only kinetic energy and bag energy. Then

It is instructive to run through the early MIT model,

2.04 4~R 3 (54) E = 3 • ----~ +--~---

and

2 = m c for R = R .

n mln

~E 6.12 + 4~R2B = 0 . (55) ~R R 2

The second equation sets

so that

4~R 3 " mln 2.04

B - (55.1) 3 R .

mln

2.04 E = 4 • - -

R . mln

R . being the R corresponding to minimum energy. mln

one sees that

h R . = 8.16 mln m c

n

(55.2)

Since E = mn c2 for the nucleon,

1.6 fm (55.3)

for this simple energy functional. (Introduction of the energy (iv), eq. (53)

brings Rmi n down to ~ 1 fm.) Viewed as function of R, E now is

6.12 2.04 R 3 E = R + ~ (56)

mln

180 G.E. Brown

where we have eliminated B in favour of Rmi n- form having a (shallow) minimum at R ~ 1.6 fm.

The curve of E vs R then has the

The difference between curves (]5)

E 2 m c fi

1.0 fin

J

I.G fm

Fig. 15. Curve of E vs. R, once B is set

and (14) is that in the latter B has been set to give an extremum at Rmi n. In the former, B is set to give an extremum at whatever value of R one wants it; E will

not necessarily be equal to mn c2 in this case. In fact, from a conceptual point of view, it would be better to replace fig. 14 by a plot of energy E vs bag con-

stant B, since the unknown quantity is B. Thus, the necessary value of B is

determined so as to make E(R) an extremum at the desired R.

The quantity responsible for the flattening of E(R) and decrease for small R is

the (negative) pion self energy E~. Whereas, once nonlinearities are included,

it does not increase in magnitude as rapidly as R -3 (see eq. (52)), it still

dominates the energy functional for small r. This has the result that for small R, the extremum in the energy functional is not a minimum, but either a point of

inflection or a maximum 28 and the bag is not stable.

It may be reasonable to restrict considerations to only pion self energies for

large bags; it would, however, be naive not to include effects from the heavier

mesons, for small bags. For example, coupling the w-meson gives a self-energy term 29

-2m R 1 gw 2 e w

E (l+m R) (57) w 2 4v R w

which becomes important at small R; being repulsive inclusion of E W stabilizes

the bag against collapse.

Inclusion of self-energy effects from scalar mesons is more tricky, because some of these effects are already included through the nonlinearities in the pion field

which result from the elimination of the scalar o-meson. There seems to be some

enhancement in the exchange of two-pion systems coupled to J = 0, I = 0, between nucleons, resulting in a scalar boson of m~ = 550 MeV, g2/4~ = 8 being needed in

boson exchange models 30. This c-meson mlght be convenlently thought of, for our purposes, as a 2q 2q bag 31, although, more generally, all we need is the enhance-

ment* of the two-pion s-wave interaction in this region of masses ~2.

*Since the w-meson couples so strongly to nucleons, it will hide effects of this

enhancement for invariant masses > 700 MeV.


Given the coupling of the scalar s-meson, there will be an attractive self energy

-2m R 1 gs e s

E - (i + msR) s 2 4~ R

(58)

so that the curve of E(R) vs R will have a well defined minimum in the region R ~ h/msc, before becoming highly repulsive at smaller R due to E~.

in the above there is considerable analogy to the nucleon-nucleon interaction. At large distances, only pion exchange is important; at smaller distances, effects from exchange of s- and u-mesons come in. In the case of the bag, the ~-meson plays the role of shrinking the bag; once R is small, the other mesons enter in. The way in which g-mesons couple in still needs to be worked out, but we believe that the above picture offers a mechanism for stabilizing small bags.

It should be realized that it is not easy to stabilize large bags against collapse

inwards. Although the MIT bag has a shallow minimum of the type shown in fig. 15, once the pion self energy and correction for spurious center of mass motion are included33, 34, this minimum nearly disappears. It is reestablished in refs. 33 and 34 by evaluating the correction for center of mass motion as a perturbation, not including it in the energy minimization (see eq. (55)). (Since this term is attractive, and goes as R -2, it tends to make the bag collapse when included in the minimization.)

A chief advantage of the chiral bag model is that it makes it possible to do dynamics. The pion field is zero inside the bag; thus, its normal derivative clearly has a discontinuity at the bag surface. The discontinuity in this normal derivative follows from eqs. (44) and (45):

6(n8 7)[ : 1 ~ I S ~ (~ Y5 2 q) S

(59)

where we have dropped nonlinearities in D . From this, it is easy to construct the pion current*

]~ = (-02 + m~)~ = i T T- (~ Y5 2 q) ~S (60)

showing that the bag surface is a source for pions. Given the pion current, the pion exchange between nucleons, the width for the transition A(1232)+N+~, etc. can be worked out. From the fact that ~ is asymptotically the usual Yukawa pseudoscalar field, it follows that the OPEP is the conventional one. In the simplest model, the isobar width turns out 25 to be ~ 95 MeV, not far from the empirical ii0 MeV. We shall discuss the discrepancy later. Quantization of the nonlinear equations we work with may be difficult. We assume here that fluctuations about minima in the mean pion field can be described in terms of j . This does not, however, establish what these minima are.

One can convert 25 j~-~ into the usual pion-nucleon interaction

= f--,~++o.v~ ~ (61) m~ ~ ~

*The pion mass is put in by hand.

182 G.E. Brown

where ~ refers to the nucleon.

We have been unable to solve the nucleon problem, even classically, because the

nucleon is necessarily deformed. The pion couples to the nucleon spin through o • k~, where k~ is the momentum of the pion; this means that if the nucleon spin direction is taken through the north pole of the nucleon, the pressure from the

pion cloud is large at the poles, small at the equator, producing a "squashed bag." Vento et al. 28 have been able to solve the classical problem of a spherical bag

with (nonlinear) pion cloud. Spherical symmetry is obtained by insisting that the pion pressure be uniform with angle; in practice this is achieved by choosing the expectation value of spin o to be in the radial direction r. This spin is always pointing radially inwards,~so this solution is called the "hedgehog" one.

In fig. 16 is shown the energy of this solution vs R. It will be noted that

acceptable solutions with energy near the nucleon mass exist for both large R ~ 1 fm, and small R, ~ .4 fm. At large R, the pion self-energy is small, so that the solution is essentially the M.I.T. bag. At small R the repulsive kinetic energy is large, but so is the attractive pion self energy, to compensate. The small -R solution is a nonperturbative one.

L.~

w

2.0!

~.0_

O.5 ~.0 R (fro)

Fig. 16. Energy of the hedgehog solution 28 as function of

bag radius R.

Continuity of the axial-vector current dictates that quarks couple to pions at the bag surface. As discussed earlier, this coupling gives the quarks an attractive

self energy, and it is interesting to see how the quark energy behaves with bag radius. In the original M.I.T. model 14 which does not have this pion coupling, the quark energy,

2.04 w = (62) o R

is all kinetic energy. In fig. 17 we plot ~ ~ ~R vs. R for the hedgehog solution. It will be seen that ~ decreases as R decreases.

Now in both the M.I.T. bag model and in the hedgehog solution, the large components

of the relativistic quark wave function are proportional to jo(~r), the small components to jl(wr), the j's being the spherical Bessel functions. The ratio of


small to large components jl(mr)/jo(~r) goes as ~/3 for small ~; i.e., if we move down in R into the small ~ region of fig. 17, the small components tend to drop out. Thus, in the region of small R, one recovers, in the case of the hedgehog solution, the constituent quarks, albeit with meson cloud.

This is an amusing situation. In the M.I.T. bag model, the up and down quarks are essentially massless. Small components of the relativistic quark wave

2.0

1.0.

0.5 1.0 R Ifml

Fig. 17. Plot of ~ = mR vs. R for the hedgehog solution

functions are appreciable; e.g., they reduce the gA from the constituent quark model value of 5/3 to 1.09. In the nonperturbative hedgehog solution, we find that the magnitude of the small components decreases with decreasing bag radius R, and that the constituent quark model formulae are reproduced for small R. The way that this comes about is illustrated by the quasilinear approximation which works quite well in the hedgehog case. Here, the quark energy goes as*

81 f2 n q m = m - - - - -

o 5 0 4 ~ m 2 R 3 'iI

(63)

where f is the strong interaction constant, f2/4~ = .08, m o is the M.I.T. quark eigenenergy, equal to 2.04/R, n is the number of quarks, 3 for the nucleon. One sees from this formula that~the pion coupling in this approximation produces a quark self-energy. Even though quark kinetic energies increase rapidly with decreasing bag radius, the pion self energy increases more rapidly in magnitude, bringing m down. In this way, the constituent quark model is recovered in the case of the hedgehog solution.

In the case of the nucleon, effects of pion coupling are larger, if anything, than in the hedgehog. We expect the same general feature; namely, that the quark eigenvalues m will decrease with decreasing R. This is our conjectured mechanism for making contact with the constituent quark model, which will be a good

*The total energy of the system, three quarks plus nonlinear pion cloud, has no simple expression in term of this w, however.

184 G.E. Brown

description if ~R is small. From the properties of the hedgehog solution, it loon to us as if this will result. This does not necessarily put us into a good posi-

tion. We shall not only have to correct for the known deficiencies in the constituent quark model, but we also do not have immediately its most notable

success, the ratio of - 3/2 for magnetic moments of the proton and neutron, since contributions to these still come from pion clouds.

Let us look at a mass fit from Glashow's article B5 "The Unmellisonant Quark" with-

in the framework of the constituent quark model. The masses of the ground state baryons are fit to the mass formula

~i 1 M = z M . + B Z M.~---] • J - ' i

i J i,j l ,Mj ] (64)

The first term is a sum of the constituent quark masses; the second is a sum of

the colour-magnetic "hyperfine" coupling of quark pairs. Choosing B : 50 MeV,

and up- and down-quark mass m u = 360 MeV and a strange quark mass m s = (3/2) mu, one has the following fit:

Table

Masses from the constituent quark model 35

Particle Prediction Observation

N 930 938

g 1230 1232

A iii0 1115

Z 1177 1185

Y* 1378 1385

1330 1323

-* 1530 1530

1687 1673

The fit is good. Fits to other baryonic properties have been developed in a similar vein by Harry Lipkin, within a general subject which could be termed:

"A Nuclear Shell-Model Theory of Quarks". Looking at the goodness of these fits, it is hard to dismiss them as coincidental; on the other hand, they cannot be

fundamental, because the up and down quarks certainly are not nonrelativistic objects. Since they are nearly massless at the level of current quarks, they must

behave in a highly relativistic fashion. We believe that in the foregoing, we

have given a possible way in which the relativistic SU(2) x SU(2) (chiral invariant model) can produce features of the constituent quark model. But we still have a long way to go[

8. The Nucleon

We take the bag radius to be small and, extrapolating from the nonperturbative

solution to the chiral bag model, we assume the small components of the quark


wave functions to be unimportant, except where needed to evaluate matrix elements

of operators which connect large and small components.

The first major problem we encounter is that gA' the axial-vector coupling, is

predicted to be 5/3 in the constituent quark model, whereas it is equal to 1.25 empirically.

The original M.I.T. bag model predicted gA = 1.09 which, with center-of-mass motion correction 36 came up to 1.25, and this looked like a great success.

Evaluation of gA amounts to taking the expectation value of

3 1 A 3 = ~ E o3(i) ~3(i) (65)

i

where the sum i is over quarks. In the relativistic description, 03 is not a good quantum number, the quark spin pointing down some of the time, even though

the nucleon spin points up, and this is why gA is brought down.

Jaffe 24 calculated the contribution to gA from the perturbative pion field for large bag radii, and found that it increased the M.I.T. value by a factor of

3/2. The pion cloud contribution is effectively described by the asymptotic pion field, so the contribution to it persists even as R ~ ~ Thus, the corrected

M.I.T. value is gA = 1.8. The meson cloud contribution will also increase the constituent quark value by about the same factor, so it would be raised to 2.5 Thus, a common feature of both the constituent quark model and all spherical bag models is, once the pion field is taken into account, that one obtains rather large values for the axial vector coupling constant.

Glashow 35 and Vento, Baym and Jackson 37 proposed a way out, pointing out that

introduction of D-state into the nucleon will decrease gA; in particular, for the constituent quark model (with meson cloud)

5 6 = ~ (i - ~ E 2)- (66)

gA

and we need ~ 40% D-state admixture.

This sounds wild, but let us look at other quantities, beginning with the SU(3) couplings. The D to F ratio given by the constituent quark model is such that

D+F D _--~-- ~ = 5 (67)

38 whereas the empirical value is 3.77. If the numerator, D + F, is determined,

as it is in practice, from the nucleon beta decay, then introduction of D-state into the nucleon will change this to

D+F 3 (68)

D - F ~A

so that if gA is obtained correctly, so is this ratio.

AN, N , where gAN is the axial vector matrix It is well known that the ratio of gA /gA element between nucleon and isobar, and N A gA is the nucleon axial vector, called gA above, is equal to

186 G.E. Brown

g = 72 N

gA

1.70 (69)

in the constituent quark model. The axial vector operator oT couples pions in the quark model, and the pion coupling between isobar and nucleon is known from the decay width of the isobar. Empirically 39

AN f~NA gA

f N ~NN gA

2.o (7o)

somewhat larger than the ratio in (69).

Effects of the introduction of D-states on this ratio have been worked out. Vento: Baym and Jackson 37 manufactured D-states for nucleon and isobar by multiplying S-states by the operator

B E S.. T.'T. i<j i] I ]

where 1 2

Sij = (oi'rij) (oj rij) - ~ (oi-g j) rij .

Some justification for this procedure will be given below, where it will be shown that this is the likely form of tensor interaction between quarks; it is, of course, the form of the tensor force between nucleons.

In the case of the A, there are two D-states, which can be characterized by spin- 3/2 and spin-i/2. The former has spatial symmetry, and does not connect with the nucleon D-state through the axial-vector operator. The phase of the latter state is such that the D-state axial-vector matrix element is coherent with the S-state one. The above operator produces these D-states with equal amplitude. Results, as function of D-state probability, are shown in fig. 18.

~ ~si~ ~ ~iy~t~%a~-i~i~ for g~ it is Agreement with experiment fo c achieved with 42% D-state. h i t b to ask for detailed agreement with experiment, it should be noted that the calculation of g~N described above assumes the nucleon and A to have identical radii R. All bag model calculations obtain somewhat different radii for the nucleon and the &, and correcting for this would involve cutting down the ratio gIN/g~. Thus, both this ratio and g~ are quite well fit with ~ 40% D-state admixture.

Results similar to those of Vento, Baym and Jackson 37 have been obtained, for small D-state probabilities, from perturbation theory by Hulthage and Wambach ~0. The first-order correction to the nucleon wave function in the chiral bag model contains both S- and D-states. Calculation of effects of the D-state admixture and extrapolation of the latter to the large values given in fig. 18 gives curves qualitatively similar to those shown here.

Glashow 35 does not offer a mechanism for inducing the large D-state admixtures. The interaction resulting from gluon exchange between quarks is known to contain some tensor force, but it is not strong enough to bring in probabilities of more

than a few percent.


2-

1.25

I_

_ _.~ .......... g ] ~ / g ~

and cloud

- - " qC~lfrom qu'~rl-s

I I __L - J _ _ _ L . . . .

~0 20 30 ~.0 5 0 %

Fig. 18. Axial-vector coupling as function of D-state admixtures. In both the case of the nucleon and the isobar, D-states were manufactured by multiplying S-states by the operator

6 E Sij Ti'T j , where Sij = (~i.rij) i<j

1 2 (~j'~ij)- 3 (~i'~j) rij"

The chiral bag certainly offers such a mechanism. As noted earlier, the pion couples to the spin of the nucleon. The pion pressure is strong at the poles (the north pole being given by the spin direction), weak at the equators. For small bags this inward pion pressure is very strong, nearly counterbalancing the pressure from the kinetic energy outwards; indeed, the bag constant can be treated as a perturbation here. Thus, it is clear that little chiral bags will be highly deformed.

Microscopically, D-state admixtures will be introduced by tensor interactions between quarks, induced by pion exchange. The form of this interaction between nucleons is known, and since nucleons are made up out of quarks, we can surmise that there is a tensor interaction between quarks.

Let us make a quick and ~ough estimate of the tensor interaction between quarks. Taking the pion current j~ from eq. (60), we can use the Dirac equation to eliminate the small components of the quark wave function in the evaluation of the matrix element of Y5' obtaining,

= f i X - X i ~--X ~ ~(r-R) d3r (71)

J

where × is the two-component (Pauli) wave function for the large component and is the quark eigenenergy; in principle, it could be different for initial and

final states. In our estimate, we replace 6(r-R) by R -1, and we then find the pion exchange potential between quarks to be

188 G.E. Brown

V 0o i ] + + -m~rl2

~o 4~(4f 2)~ (~R) 2 (TI.T2)(oI.VI) (~2.V2) e r12 (72) qq

Of course, the approximations made (choosing a common ~, replacing 6(r-R) by R -])

may not be valid, so this is hardly a derivation, but we believe it adequate as an estimate.

From the Goldberger-Treiman relation we have

f-i 2f

x gAmr (72.1)

and from the hedgehog solution for small bags (see fig. 16), we believe c)R to be l e s s t h a n u n i t y , so we f i n d *

- m r

f2 . -~ V > qq ~ (TI.T2) (Ol.V) (~2.V)_ e r (73)

where V = ½ (VI-V2). The right-hand side is just the one-pion-exchange potential between nucleons. It must be remembered that the pion current was originally accompanied by a 6(r-R), so this interaction is to be used only at the surface of the bag.

Rather than go through a pseudo-derivation, we could simply have said that the

one-pion-exchange potential between nucleons must arise from pion exchange between quarks and, on the basis of our earlier arguments, we propose to use this

down to small r, ~ ~/mnC. Once r is that small, ~ d~/mnC , one sees immediately that

f 2 F" mn~ 2 2 V ~-- -- i ir~ g 1 qq 47 ~-m ~ R 4~ R (74)

W

where R is again the bag radius. We encounter here our old friend

2

g= 14 4~

from the pion coupling in the linear o-model. This is hardly surprising, since the pion coupling to nucleons began as g2/4~ and only became f2/4~ = .08 because

we agreed to consider only pions of low momenta, k w ~ m r . If we go to high momenta, or shorter distances, the coupling is stronger.

The oI-V g2.V in eq. (73) can be written as

7 1 V 2 i~l'Y %'Y - ~ %'% b

i V2 + ~ ~1"~°2 •

The term in brackets will produce a tensor force. Thus, we see that (73), just as in pion exchange, produces a strong tensor force.

*In magnitude, of course.


-i Now unperturbed bag energies are ~ R so that, at least in our estimates, the tensor force is strong compared with these, and the nucleon ground state will

be a more or less equal mixture of S- and D-states, as required to explain the phenomena discussed above.

Note that the above discussion in no way contradicts the idea of asymptotic freedom. The pion-exchange interaction cuts off at r = R, and R, in the nature of things, is ~ i -I, where A is the scale-breaking momentum in QCD. The pion exchange has such strong effects because the pion-nucleon coupling is given by a numerically large coupling parameter,

2 ~-- = 14 4~

at these small distances. We have been working with just this coupling parameter for many years in the linear o model.

Let us return briefly to a discussion of magnetic moments; the constituent quark model result

= _--3

2 n

(75)

on the ratio of proton to neutron moments was one of the big initial successes of this model. With introduction of meson clouds and D-state admixtures, we have

complicated considerably what appeared to be a simple problem. However, the argu ments for the existence of the pion cloud are compelling, and we believe that one has to live with these complications.

Recent measurements 41 of the magnetic moments of the E +, E ° and E- particles are

not well fitted by the constituent quark model. Deviations from the latter can be described by introduction of pion cloud effects. These effects are qualita-

tively different for the nucleon and for the strange baryons. In particular, this is true for the two-body terms from the pion cloud (see fig. 19). These

terms contribute to the magnetic moment of the nucleon in the form

~(2) (1,2)= g(1) x [(2) i ~7(i) x 7(2 . (76)

Because this operator is antisymmetric under the interchange of spin (isospin)

indices, it has an expectation value only in the case of the nucleon, which has mixed symmetry spin and isospin states (although it also contributes to the E ° + i + y decay). The absence of this two-body contribution for the E- and E-

particles appears phenomenologically as a damping of nonstrange quark contributions for these particles, when discussed in the framework of the constituent quark model. Such effects appear to be seen (see the table) although the present value of the E--moment violates the picture badly. It is crucial for our picture

that the l--moment comes down by a factor of at least two in magnitude from its presently quoted value.

We come now naturally to the question of sizes: How large is the quark-gluon substructure of the nucleon and of the pion? In this connection, the Saclay experiment on electrodisintegration of the deuteron is most illuminating. By keeping the final neutron and proton at low relative energy, the interactions are forced far off shell, emphasizing pion-exchange currents, the dominant one

190 G.E. Brown

Fig. 19.

ql q2 q5 The two-body term in the coupling of the magnetic field to the pion cloud. The quarks are sources of the pion.

Table

Magnetic moments of the strange baryons (in nuclear magnetons)

Baryon Naive quark chiral bag Experiment model model*

E + 2.70 2.18 2.30 i 0.14

E -1.05 -0.54 -1.41 ± 0.25

_o -1.44 -1.31 -1.253 _+ .014

E- -0.50 -0.63 -0.75 ± .07

Values given in ref. 42 for cloud contribution 6 = 0.50.

being shown in fig. 20.

Fig. 20.

Comparison of theory and experiment is shown in fig. 2]

The pion-exchange current involving virtual nucleon pair production.


The good agreement between theory and experiment out to R2 ~ 16 fm -2 indicates

either that there are no large modifications in the simple theory due to nucleon

_ 16 2 "7

tL.

ld3 ..D C

"t-J \ o 10 L'

- - 1 - - ~ K

\ 0 (e,e') " , , ~ Be" : 155 °

',,, ~ [x(plal,:~n en~:rqy ' '~ X " X,~ I IO L..S HeV "

\\ •

¢ Rand etat I T S 0 ° ; \~"~--~-~" ~, Sa(lay

%

I I _ 4 _ _ _ _

5 10 15 20 q2 (fm-?)

Fig. 21. Electrodisintegration of the deuteron 43 leading to a low-energy final n-p system (so as to force the

process off shell).

or pion size out to such momenta, or, that an equivalent result comes from

another description, say, in terms of quarks. For simplicity, and because no one has constructed a description in terms of quarks which will do this, I prefer the

former alternative. This would indicate that the size of the pion-nucleon vertex is ~ 1/4 fm. This size is consistent with those arrived at from dispersion

theoretical calculations. In these, the size of the vertex comes mainly from ~-p intermediate states, as shown in fig. 22. In the nature of the calculations, the mass to be associated with the intermediate state is

= m + m + kinetic energy (77) mint ~ p

where "kinetic energy" is that of the ~ and p in the intermediate state. The

kinetic energy is of the order of one or two pion masses, so min t ~ mn, the nucleon mass. This means that the extent of the H-nucleon vertex function will

be ~/mnC, the nucleon Compton wavelength, which is completely consistent with the experiment, fig. 21.

The axial-vector vertex function determined from neutrino scattering has also a range 44 ~ ~/mnC. (More specifically, the mass in the cutoff in momentum space is ~ mn.) This is relevant because the pion is thought to couple through ~he

axial vector.

192 C.E. Brown

N N

Fig. 22. Structure of the wNN vertex function.

What about the nucleon electromagnetic form factor? It can be thought of as arising 45 in the way shown in fig. 23. Let us look at the isovector form factor,

¥ Ve~ctor meson

@ @

q

¥

Fig. 23. The electromagnetic form factor of the nucleon. a) Coupling of the y-ray directly to the bag. b) Coupling of the y-ray through vector mesons.

taking process b) of fig. 23 to be dominant. (Corrections for inclusion of process a) are given in ref. 45. This form factor is

2 m

_ e @ Fbag(q2) Fl(q2) ~ 2+m2

q P

where Fbag(q 2) is the form factor for tile 0-meson to couple to the bag.

The Fourier transform

(78)


2 of Fba~( q ) gives essentially the size of the bag. Now we know empirically that

F(q) iN well described by a dipole fit

e T-- M 2 -~

L- ~ empirical ~ -- (78.1)

where M is intermediate between m O and m n. Comparison with (78) shows that this can be achieved if

2

Fbag(q 2) mD q2+mD2 (79)

where m D ~ m n. Again we come to a mass of the order of the nucleon mass, indicat- ing that the bag has an extension ~ ~/mnC in configuration space. In fact, detailed fits using explicit bag wave functions 27 give bag radii of 0.3-0.4 fm.

We arrive in this way at a size for the nucleon which is as large as the vertex functions in the pion-nucleon interaction. This does not seem to leave much room

for any appreciable size for the pion. In fact, the pion may be a very small ob- ject. One has to be careful in terms here. The qq substructure of the pion may be very small in extent; the total extent of the physical pion, this substructure plus cloud, is known to have an rms radius of ~ .6 fm. A mechanism for the small

pion substructure will be discussed in the next section.

9. The Pion

Almost everyone agrees that the pion is a collective excitation of the non-pertur-

bative vacuum. This can be expressed in many different languages; here we use an analogy with the Anderson mode ~6 in superconductors. In this case, there is a

gap in the quasiparticle spectrum, see fig. 24, coming from the gap equation, so

Particle e)

Hole o @

Z~ Particle- Hole descripfion

A ~F

@ Fig. 24. Unperturbed particle-hole state going into the

collective Anderson Mode, which comes down to

zero energy as a result of the quasiparticle interactions.

that particle and hole excitation begin at an energy A. The particle-hole interaction then pulls a particular coherent superposition of particle-hole states down to zero energy (for a zero momentum mode). Since this amounts to building up, for the Anderson mode, a coherent wave packet with momentum spread given by

194 G.E. Brown

pap ~ Pf&P A (80)

m m

we find the particle and hole to be correlated to within distance £ given by

~5 46PF £ ~ (80. l) kp Am

which turns out to be the coherence length in superconductors. This coherent length is long, many interparticle spacings, in superconductors.

In the case of quarks, the gap A is replaced by quark mass Mq . This is the mass

that the quark would get, were it to venture outside the bag into the physical vacuum. The model is that it gets this mass through a dynamical symmetry break-

ing, in the same way that the quasiparticle gets a gap in the superconductor.

In the Princeton picture ~7, the interaction producing this is an instanton-induced

interaction (III). The mass would arise from a dynamical symmetry breaking, such

as portrayed in fig. 25. In the Princeton model the mass M obtained this way is q

large; in fact, it is taken to ~ for some purposes. The picture is, then, that

Fig. 25. Origin of the quark mass in the physical vacuum as a result of dynamical symmetry breaking. The box X

means that the quark in the interior line must be given a mass in order to give a mass to the quark

described by external lines.

the quark would acquire a large mass, were it to try to get out of the nucleon

bag; see fig. 26. If the quark gets a large mass in this way outside of the

bag, this makes it more difficult for the quark to get out, and helps to explain

why the quark wavefunction cuts off quickly with increasing r at the bag surface.

This part of the picture was already in the early calculation by Bogoliubov 48, who considered massless quarks confined to an infinite square-well potential.

The square well was motivated as an approximation to a self-consistent inter- quark potential.

In the case of the pion, described as a coherent superposition of qq-states (see fig. 24b), the gap becomes M . If a chiral-invariant interaction is em-

ployed, we know from the Goldston~ theorem that a state with quantum numbers of

*We avoid the question of how the pion gets its small final mass m .


Mq

I R

Fig. 26.

r

Variation of the quark mass with distance; R gives the bag radius.

the pion with zero mass* will result. The spread of the wave packet describing

this collective excitation will be

A ~ M (81) P q

so that the radius of this pion will be

r ~ d~ (82)

Mc q

If M is large, then r is small. Actually our analogy with the Anderson mode brea~s down here, since % of eq. (80.1) is large, many times the interparticle spacing. This results from the fact that A, the energy from symmetry breaking in our language, is small compared with the Fermi energy,

A << gF " (83)

The statement that M is large must mean q

M q

>> A (84)

since A is the only dimensional parameter available in QCD. For this very reason, M_ must be of the same order of magnitude as A" eq. (84) can be fulfilled by Mq being larger by a numerical factor. Such large numerical factors commonly appear in many-body problems when coherent states are formed.

This qq pion is not yet the physical pion. We know, for example, that the pion couples strongly to the p-meson; the p gets its width from decay into two pions.

*The mass Mq here is the real quark mass, equal to a few MeV inside the bag, not the constituent quark mass (which may be an energy) used in earlier sections

196 C.E. Brown

The electromagnetic rms radius of the pion is then given mainly by the extent of the 0-meson cloud. In fact, the rms radius of a charge cloud of form

e-mpr/r is ~/m ~ 0.6 fm, whereas the empirical pion rms radius is 0.57±.04 fm.

Thus, the pion ~as large qqqq, etc. components. Only the qq part of the pion couples to the axial current ~ , which is bilinear in quark fields. If we ex- pand the wave function of the p~ysical pion as

I ~ > p h y ~ i c a l = z lqq> + a2 lqqqq> + . . , (85)

our arguments indicate that

z << 1 , (85.1)

since the wave function must be normalized, implying that

Izl 2 + ia212 + .. = i (85.2)

and a number of the a's are appreciable in size. Now, the matrix element (9.8) will be, aside from numerical factors of ~ 1

<= IA [0> : 6 i j Z-R~ qlJ (86) physical

where R bag plu~ cloud). with f ,

then

so that

is the radius of the pion bag (not the radius of physical pion composed of If from (9.8) one connects the radius r of the physical pion

1 f ~ -- (86.1) r~

i f m Z -- (86.2)

R

R

r °J Z (86.3)

and for Z small, R can be small.

i0. The p-meson

Having a picture of the n-meson and a theory of its couplings, at least for long

wavelengths, we can set about constructing the interactions of the other mesons. Effects of the coupling of the S = 0 o-meson are already included in the theory, as discussed preceding eq. (58). So let us discuss how the p-meson couples. The "ur"-p-meson is viewed as a qq bag of small radius; it couples to the nucleon by exchange of two pions. We know that the width of the 0-meson from the decay

p * ~ + ~ (87)


will come out correctly in our theory, since with the restoration of chiral symmetry to the bag model, we have PCAC, which gives immediately the KFSR re-

lation 49 for the 0-meson width. This has been verified by explicit calculation. From continuity of the axial current,

T ~ = - i q Y~Y5 -2 q(x) e(R-r)

+ f D $ 0(r-R)

(88)

where the ~'s are step functions. In order to calculate transitions in the LSZ

formalism, we deal only with the fluctuations, and consequently need only the linear approximation to (88) so that D~ can be replaced by ~. Then

~ = f { $ 0(r-R) (88.1) P ~ ~ }

taking the quark masses to be zero. Once we have broken chiral symmetry so as to give the pion a mass, the right-hand side of (88.1) becomes f~m 2 ~ , dropping the 0(r-R) which may be justified if the bag is small. This gives us the operator

equation

~ = f m 2 ~ (89)

which is the "good" form 50 of PCAC.

It has been clear for many years that the 0-meson cannot be built up purely as a bound state of two pions; the J=l centrifugal barrier is simply not strong enough to contain the pions sufficiently to build up a resonance. There must be a more basic, underlying structure, here described as qq in nature. We do not have a theory, nor even a picture for the size of the p-meson. This size must be conditioned by coupling to two-pion states. However, the KFSR relation ~9 was de-

rived as a low-momentum theorem, and modifications must enter when the pion momentum q~ from decay of the 0 into two pions is ~ R -I. The goodness of the

0 KFSR relation would seem to indicate that the radius of the p-bag is < ~/mpC.

The interaction mediated by 0-exchange between two nucleons can be visualized as shown in fig. 27. The crossed-channel quantum numbers are, of course, J=l, I=l,

TF

TF

Fig. 27. The p-exchange interaction between nucleons described as three-quark states. The wavy lines here represent pions, connecting the basic p's, described as q~ states.

198 G.E. Brown

appropriate for p exchange. In between emitting the first and second pion, the nucleon may be in an excited state.

This figure may look somewhat picturesque. It is, however, easily understandable if we draw it as a t-channel diagram as shown in fig. 28. In this form it is seen to be the representation of the Amati-Leader-Vitale formalism 51 used in calculation of the Paris and Stony Brook potentials. The role of the underlying qq part of the p-meson is simply to provide a p-wave ~ scattering amplitude, roughly of the form

N N

N N t

Fig. 28. The diagram of fig. 27 redrawn as a t-channel diagram.

The shaded areas represent all possible excited states of the nucleon, etc. The dots replace the qq 0-meson, and represent here the p-wave pion-pion scattering amplitude.

F (90) al(w) . F

mp-~-z

where F is the width for the p-meson to decay into two pions (which results as

the empirical width in our model since PCAC is built in - see the earlier discussion) and mo is the mass of the o-meson. We thus see that the formalism for p- meson exchange in the little bag model maps neatly on to that presently in use for computation of nucleon-nucleon forces from meson exchange. In this model, nucleon interact with the basic qq o-mesons by the pions, which couple to the bag surfaces In this picture, the o-meson exists only outside of the nucleon bag, although qq fluctuations with quantum numbers of the @ can occur inside the bag, so that it is

to employ roughly the form factor Fbag(q 2) (see eq. (79) in coupling appropriate the 0-meson to the nucleon source).

Construction of the u-exchange interaction is somewhat more complicated. Durso


et al. 52 showed that most of the repulsion in the m-exchange channel (crossed- channel quantum numbers J=l, I=0) come from the processes shown in fig. 29. Of course, pion rescattering should be included. These processes gave an effective

A

Fig. 29.

P

. / / -

A

P

A

a) b)

Processes responsible for most of the repulsion in the m-exchange channel.

m coupling of

(g~)

eff 47

- - ~ 5-8 . (91)

In this work, vector dominance was used to relate the pNN tensor coupling to the vector coupling. HDhler and Pietarinen 53 find empirically that the square of this coupling f2NN is about 2.5 times the value given by vector dominance, so that a better value might be

gm 12-20 (91.1)

eff

Such processes are easily described in the formalisms depicted in figs. 28 and 29 by identifying one of each pair of wavy lines with the pion, the other with the p meson. It should be emphasized that the "SU(6) m", which would have a maximum coupling constant

2 2 g__~ = 9 gp

4~ 47 ~ 4.5 (92)

is not nearly large enough to describe the short-range repulsion in the nucleon- nucleon potential.

ii. Discussion

In the foregoing we have a picture of the boson-exchange model giving the nucleon- nucleon interaction down to relatively short distances R ~ 2 ~/mnC, the perturbative quark-gluon structure of the nucleon taking over at shorter distances. For the longer-range part of the interaction, once the quark-antiquark pairs are

200 G.E. Brown

arranged in colourless mesons, the interactions of the latter are numerically large, typically

2 i~ 14 .

47 (93)

This suggests a scheme for partial summations in QCD.

As Mannque Rho will discuss in his lectures, our development is essentially an expansion in chiral invariants, higher-order terms involving higher derivatives,

rather like a Taylor series. For a description of long-wavelength phenomena, only the lowest derivatives are necessary. The only scale in QCD is given by the scale- breaking parameter A as A -I, where A % 500 MeV. As Mannque Rho will develop, the series may converge faster than would be envisioned by this criterion, because of the short-range repulsions.

High energy physics, esp. deep inelastic scattering, goes at the problem from the other extreme, looking first at the point nature of quarks, introducing inter-

actions and structure as corrections. It is not clear that nuclear physics, beginning from an expansion in "smoothness", and particle physics, beginning from point structure, will be able to produce overlapping descriptions on a middle ground.

The common theme running through our expansion in smoothness is that chiral invariance provides us with stringent guidelines as to how to construct interactions

between mesons and nucleons, also mesons and mesons. This is not surprising, since chiral invariance is a property of the underlying QCD.

An important question is, how predictive is our description? If it is only a re-

phrasing, in different terms, of old-fashioned concepts, then it is not so interest ing, even though it is still useful to reconcile the quark-gluon description with the old-fashioned one.

We believe our scheme to have predictive power; e.g., see ref. 42. It is, however, hard to isolate those processes in low-energy nuclear physics which betray the quark-g]uon imprint, precisely because repulsions from meson exchange are so stron~

that they keep nucleons from coming close together much of the time. So we will have to work hard to find the imprint, lit took several decades to pin down the meson presence in nuclei, even though we saw copious numbers of mesons and nucleons, and had a reasonable understanding of their interactions.]

I would like to thank my close collaborators in the chiral bag model, especially Mannque Rho and Vincent Vento, and Carl Carlson and Fred Myhrer, for much con- structive criticism and many contributions which went into these lectures. I am particularly grateful to Carleton DeTar for tuition, criticism, and for providing me with the Table AI, pion-field corrections for the case where the pion field is limited to being only outside the bag. I would like to thank Ebbe Nyman for many discussions and for help with the manuscript.

This work was supported by the U.S. Department of Energy under Contract DE-ACO2- 76ER13001 with the State University of New York at Stony Brook.


Appendix A

We relegate to the appendix the discussion of nucleon size, because we believe the most important item on the agenda to be to get the chiral properties of the nucleon correct, and we have devoted most of the text - at least that part we believe to be most important - to that. We note here that there is no scale parameter in the fundamental Yang-Mills equation, so that a determination of scale can

only come from fits to empirical quantities (or a fundamental calculation of, say, the nucleon structure from first principles).

Arguments for the large, % 1 fm, radius used in the M.I.T. bag model usually proceed along the lines that, with this choice, the M.I.T. bag fits baryon spectros- copy and known properties of the nucleon well; therefore, a smaller radius will not do.

Firstly we shall show that without the pion cloud, the M.I.T. model does not do well; once the cloud is introduced, it does do well on most properties, with the glaring exception of gA (and we believe for the reasons outlined in § 7, that it will do poorly on the nucleon electromagnetic form factor for momentum transfers lqI~ m n c, although we have not been able to work out the form factors as function of q yet). Once the pion cloud is present, predicted magnetic moments, the rms cha~ge radius of the proton, etc., are insensitive to radius, being near to the empirical values over a wide range of radii. However, gA is much too large for all radii, and can only be brought down in value by introducing a large D-state

admixture into the nucleon (which is helpful, also, for other properties, as discussed in § 7).

Let us look* at properties of the proton calculated for R=I fm. Without pion

cloud, 2mUp is predicted to be 1.90 in the bag, is increased to 2.65 with introduction of corrections for center of mass motion, not far from the empirical 2.79.

The neutron moment is not reproduced well. The mean square proton charge radius is only 0.52 fm 2, to be compared with the empirical 0.77 fm 2, and DeTar's

theoretical value drops to 0.35 fm 2 with correction for spurious center-of-mass motion**. A complete calculation, within the Peierls-Yoccoz formalism 55, which removes center-of-mass motion gives <r2>p = 0.30 fm 2. So, even with such a large

radius, the model badly misses on the proton charge radius.

DeTar introduces pions both inside and outside the bag, an idea obviously abhorrent to us, following the discussion in § 2. If the pion field exists only

outside the bag, as we believe to be correct, the M.I.T. gX is increased by > 50% by contributions from the pion cloud; if the pion field is also allowed inside, it

makes essentially no contribution to gA' terms from inside and outside the bag cancelling. If the pion field is put also inside the bag, gl can then be brought down close to the empirical value without introduction of de@ormation, § 7. We believe this to be sacrificing principle for expediency, that chiral symmetry must be realized in either the Wigner or Goldstone mode, but not in both at the same time.

If the pion field exists only outside the bag, then it enters with amplitude 3/2 times that in DeTar's calculation. He has kindly repeated his calculation for us, limiting pions to only outside the bag; we reproduce some of his results in Table AI. The notation is the same as in his preprint. From these results, we

*Following

**Donoghue spurious

DeTar 54 .

and Johnson 36 are in error as to the sign of the correction for center-of-mass motion.

202 G.E. Brown

see that* 2m~ and <r2> are rather flat functions of R. (There are serious troubles with <r2>n, Shich is too large and leaves no room for further enlarge- ment by quark configuration mixing, the usual argument 56 for nonzero <r2>n. )

The sum of quark and pion contributions 2m~ o + 2m6~ (in DeTar's notation) has been calculated nonperturbatively by Logeais, Rho and Vento 57 and we give their results in Table A2. For the radii at which they can be compared, their sum is almost identical with that in DeTar's preprint (pion inside and outside the bag) Although 6~p is larger in Logeais et al., because of the larger pion field (only outside the-bag), 2m~o, the quark contribution, is smaller because of nonlinearities introduced by the pion field into the quark boundary condition.

Corrections must be applied to the results of Logeais et al., esp. for spurious center of mass motion, before comparison with experiment, but the table shows that the sum of pionic and quark contributions to magnetic moment is relatively insensitive to R.

Table AI

o (3~ Pion Outside Bag only gA = 1.09 , gA : ~2 1.09

Private communication from Carleton DeTar

R N ( GeV- l

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

proton 2m6~ z -1.17 -0.95 -0.79 -0.66 -0.56 -0.49 -0.42 -0.37 2m~HN,£ 0.39 0.31 0.25 0.20 0.17 0.14 0.12 0.ii 2m~H~ 1.21 0.96 0.77 0.62 0.51 0.43 0.36 0.30

0.43 0,31 0.22 0.16 0.12 0.09 0.06 0.04 2m6Hc.m" 0.84 0.67 0.55 0.46 0.40 0.35 0.31 0.27 2mP0 1.33 1.52 1.71 1.90 2.09 2.28 2.47 2.66 2mV 2.59 2.49 2.48 2.52 2.60 2.71 2.83 2.97

Empirical value: 2.79 neutron 2m~Hz 0.78 0.64 0.53 0.44 0.38 0.32 0.28 0.24 2m@~N,£ -0.42 -0.34 -0.29 -0.24 -0.21 -0.18 -0.16 -0.14 2m@~z -1.21 -0.96 -0.77 -0.62 -0.51 -0.43 -0.36 -0.30

-0.85 -0.66 -0.52 -0.42 -0.34 -0.28 -0.23 -0.20 [email protected]" -0.17 -0.15 -0.13 -0.12 -0.ii -0.i0 -0.09 -0.09 2m~0 -0.89 -i.01 -1.14 -1.26 -1.39 -1.52 -1.64 -1.77 2mH -1.90 -1.82 -1.80 -1.81 -1.84 -1.90 -1.97 -2.05

Empirical value: -1.91

*It has been noted in ref. i0 that some coupling through an isoscalar meson (the ~) must be introduced; this will increase 2m~p above its empirical value, decrease the magnitude of 2m~n.


Table A1 continued

proton 6<--~r_> z -0.22 -0.21 -0.19 -0.18 -0.17 -0.16 -0.15 -0.14 ~<r2>N,A 0.15 0.14 0.13 0.12 0.ii 0.i0 0.09 0.08 ~<r2>n 0.51 0.43 0.37 0.32 0.28 0.25 0.22 0.20

0.44 0.36 0.30 0.26 0.22 0.19 0.16 0.14 6<r2>c m -0.08 -0.ii -0.14 -0.17 -0.21 -0.25 -0.29 -0.34 <r2>0 " " 0.25 0.33 0.42 0.52 0.62 0.74 0.87 1.01 <r2> 0.60 0.58 0.58 0.60 0.64 0.68 0.75 0.82

Empirical value: -0.77±.05 fm 2

neutron 6~r2> N £ 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.05 6<r2>z ' -0.51 -0.43 -0.37 -0.32 -0.28 -0.25 -0.22 -0.20 <r2>0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 <r2> -0.44 -0.36 -0.30 -0.26 -0.22 -0.19 -0.16 -6.14

Empirical value: -0.12±.01 fm 2

Table A2

Contributions of the quarks and pion field to the proton magnetic moment following Logeais et al.

R(fm) 2m 6~ 2m ~ Sum 7T O

0.388 2.83 0.02 2.85

0.402 2.69 0.08 2.77

0.418 2.54 0.15 2.69

0.460 2.24 0.31 2.55

0.510 1.96 0.46 2.42

0.62 1.56 0.75 2.31

0. 72 1.31 0. 98 2. 29

0.85 1.10 1.22 2.32

0.90 1.03 1.32 2.35

1.16 0.79 1.78 2.57

1.31 0. 70 2.04 2. 74

I. 53 0. 60 2.41 3.01

204 G.E. Brown

The above indicates that the properties discussed are relatively insensitive to bag radius. We have listed in § 7 those properties of the nucleon which re- quire a small bag radius:

I) The behavior of the electromagnetic form factor of the nucleon for momenta

q ~mn c-

2) The necessity for large D-state admixtures, which can only be brought about by strong pion fields.

3) The need to keep conventional nuclear physics, at least its successes, down to radii such as it works.

REFERENCES

i. See J. Schwinger, "Chiral Dynamics", Phys. Lett. 24B (1967) 473 and earlier papers referred to there.

2. Many testimonies to the explanatory power of phenomenological chiral invariance are to be found in the three volumes "Mesons in Nuclei" by Mannque Rho and Denys Wilkinson, North-Holland Publ. Co., Amsterdam (1979).

3. Following T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291. 4. M.B. Kislinger, Phys. Lett. 79B (1978) 474. 5. Gordon Baym and Siu A. Chin, Nucl. Phys. A262 (1976) 527. 6. M. Fierz, Z. Phys.104 (1937) 553; W. Pauli, Proc. L'Inst. H. Poincar~ 6

(1936) 109. 7. See The Nucleon-Nucleon I n t e r a c t i o n by G.E. Brown and A.D. Jackson, Nor th -

Holland~American E l s e v i e r , Amsterdam, 1976, f o r a l i s t i n g o f fo rmulae. 8. H.J. Pirner, Phys. Lett 85B (1979) 190, private communication. 9. W.G. Love and M.A. Franey, to be published and private communication.

i0. G.E. Brown, M. Rho and V. Vento, Phys. Lett. 97B (1980) 423. ii. C. DeTar, Phys. Rev. 17 (1978) 302; ibid 17 (1978) 323. 12. C. DeTar, Phys. Rev. 19 (1979) 1451. 13. C. DeTar, Nucl. Phys. A335 (1980) 203. 14. T. De Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. DI2 (1975) 2060. 15. K. Johnson, Acta Physica Polonica B6 (1975) 865. 16. W.G. Love, Proc. of the Conf. on the (p,n) Reaction and Nucleon-Nucleon

Forces, Telluride, Colorado (Plenum, N.Y., 1980); W.G. Love and M.A. Franey, private communication.

17. A particularly good pedagogical discussion is in "The Theory of Fermi Liquids", D. Pines and P. Nozi~res, Benjamin, N.Y., 1966.

18. A discussion which should be palatable to field theorists is given by M. Gell- Mann and K. Brueckner, Phys. Rev. 106 (1957) 364.

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Phys. Rev. DI2 (1975) ]495. See W.N. Cottingham et al., Phys. Rev. D8 (1973) 800 for the calculation of the external potential, M. Lacombe et al., Phys. Rev. C21 (1980) 861 for a parameterization.

22. D.O. Riska and B. Verwest, Phys. Lett. 48B (]974) 17. 23. W. Grein, Nucl. Phys. BI31 (1977) 255. 24. R. Jaffe, 1979 Erice Summer School "Ettore Majorana" Lectures, Erice, Sicily. 25. G.E. Brown and Mannque Rho, Phys. Lett. 82B (1979) 177; G.E. Brown,

Mannque Rho and V. Vento, Phys. Lett. 84B (1979) 383. 26. A. Chodos and C.B. Thorne, Phys. Rev. DI2 (].975) 2733. 27. V. Vento, Thesis, Stony Brook, 1979.


28. V. Vento, Mannque Rho, E.M. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413.

29. V. Vento, private communication and to be published. 30. K. Holinde, Phys. Repts. 68 (1981) 122. 31. R.L. Jaffe, Phys. Rev. DI7 (1978) 1444. 32. Review of particle properties, Rev. Mod. Phys. 52 (1980) 51. 33. C. DeTar, University of Utah preprint, 1981. 34. F. Myhrer, G.E. Brown and Z. Xu, Nucl. Phys. A362 (1981) 317. 35. S.L. Glashow, Physica 96A (1979) 27. 36. J. Donoghue and K. Johnson, Phys. Rev. D21 (1980) 1975. 37. V. Vento, G. Baym and A.D. Jackson, Phys. Lett. I02B (1981) 97. 38. J. Linquist et al., Phys. Rev. DI6 (1977) 2104. 39. M.M. Nagels et al., Nucl. Phys. BI09 (1976) i. 40. I. Hulthage and J. Wambach, to be published. 41. R. Rameika et al., (to be published); results reported by R. Handler, XXth

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42. G.E. Brown, Mannque Rho and V. Vento, Phys. Lett. 97B (1980) 423. 43. M. Bernheim, E. Jans, J. Mougey, D. Royer, D. Tarnowski, S. Turch-Chieze,

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Nucleon-nucleon forces from bags, quarks and boson exchange