Chiral symmetry and nucleon structure: Low energy aspects

7c

Nuclear Physics A497 (1989) 7c-22c North-Holland, Amsterdam

CHIRAL

SYMMETRY

Wolfram

WEISE

AND NUCLEON

STRUCTURE:

Institute of Theoretical Physics, University D-8400 Regensburg, W.Germany

LOW

ENERGY

ASPECTS*

of Regensburg,

The symmetries and currents of QCD at low energy and long wavelength are realized in the form of mesons, rather than quarks and gluons. In this talk I summarize the merits, but also the limits, of chiral non-linear meson theories and their soliton solutions, in descriptions of nucleon structure and the nucleon-nucleon interaction.

1. INTRODUCTION Yukawa’s works

with

very good

remarkable

up to about

and gluons.

Nuclear

Theory.

and that energies

There

quarks

“see”

and < G,,G””

established

domains

is general

analogies

and mesons, (rather

that

a

and momentum

rather than quarks

than just a few GeV) undressed

that QCD

are the fundamental

which govern the physics

of

hosts condensates,

is the theory

degrees

calculations; of finding

the relevant

long wavelength according

*Work supported in part by BMFT meinschaft, grant We 655/9-2

B.V.

of freedom

(bosonic)

at low

values

existence

is shown

in condensed

collective

to symmetry

grant MEP

037Sp9474/89/$03.S0 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

degrees

However,

is non-perturbative;

expectation

Their

example

strategies

QCD

interactions

of these condensates

phenomena.

a representative

with corresponding

Lagrangian

i.e. non-zero

of modern

of strong

of freedom.

The appearance

hadronic

of hadrons

one of the key problems

and large length scales of order lfm,

find the leading low frequency, build an effective

of mesons,

well established2)

at energies

q uarks as bare constituents,

d-)

has become

consensus

of low energy

that the program many

in terms of nucleons

> of quark pairs and gluons.

by lattice

phenomena

the relevant degrees of freedom

(the vacuum)

to our understanding

It is in fact

of nuclear

light (u-and

kinematic

and gluons

state

physics.

by the exchange

clouds.

(1 GeV and below)

its ground

are mediated

that one has to go to the multi-GeV

about

in various

forces

in nuclear

1 GeV is achieved

polarization

The question

shows

success

It appears

and nuclei

that nuclear

of a wide variety

in order to actually

their mesonic

natural

idea’)

description

transfers

region

original

< 44 > is crucial

is quite

in Fig.1.

for low-energy

matter

physics

well

It is then QCD

: try to

modes of the system and

principles.

0234 REA

and by Deutsche

Forschungsge-

W. Weise / Chiral symmetry and nucleon structure

,

CHRAL CONDENSATE on ttie LATTICE I I

<@l>

Nf = 2 flavours

0.4 [arb. units 1

0.0.

’ 5.2

’

’

’

’

’ 5.3

’

’

’

’

’ ’ 5.4

’

FIGURE 1 The chiral order parameter, or quark condensate < gq > for two flavours in lattice QCD as function of the inverse coupling strength. The quark msss here is chosen m = 2.5 x 10-2a-' in terms of the lattice spacing a. From ref. 22).

14

72 IO 8 6 4 2 0 -0.4

-----_

0.8 0.6 CENTER OF MASS ENERGY

1.0 W [GeVI

FIGURE 2 Dominance of vector mesons in the cross section u(efe- -+ hadrons) normalized to a(e+e- --t p+p-) below W = 4 < 1 GeV. Also shown for comparison is the corresponding cross section ratio for production of quasifree u- and d-quarks (dashed line).

9c

W. Weise / Chiral symmetry and nucleon structure

In this presentation physics

is governed

the extreme The

of a collective

approach,

conjecture3) theory

Chiral that,

&CD.

in which

solitons,

(J”

=

section

i.e. classical,

as anticipated limitations

+

long ago by Skyrme

CHIRAL

2.1 Symmetries Consider chirality,

QCD

in Fig.2.

solutions

symmetry

breaking

mesons

vacuum.

chiral

enter They

Baryons

of the non-linear

in terms of

is the quark

quite totally

naturally dominate

emerge

as topological

meson

field equations,

there

quark flavors.

is an underlying u-and

are the isovector

In addition

THEORY

of QCD

QCD with two massless

so that

symmetry

global

d-quarks.

vector

symmetry,

quarks

sum

The two conserved

and axial

have a good helicity,

x SlJ(2)n currents

or

corresponding

associated

with this

currents,

U(1) current,

there is an isoscalar

Massless

J,(z)

= q(z)rrq(z)

which carries

the conserved

number.

Let us construct

an effective

the basic symmetries

of QCD

Such an effective

in terms

Lagrangian

of the lowest

For massless

u-and

N 140 MeV current

d-quarks,

masses

hadron

Goldstone i.e.

m,

for low energy

hadron

for its non-pertubative

will not be expressed

mass composite

(a small number

quark

Lagrangian

but accounts

pion which is the .7” = O-, I=1

m,

as shown

symmetry

In the

entirely

4,. The purpose of this paper is to discuss the merits and

MESON

and Currents

to left- and right-handed

small

of the

models.

interesting

are realized.

broken

is

to a non-linear

is expressed

Vector

3.

quark

reduces

theory

This

of such an approach.

2. NON-LINEAR

ties.

chiral

f 25)MeV)

and stable

QCD

of the spontaneously

excitations

hadrons)

localized

N,,

hadron

freedom.

to the popular

Lagrangian

spontaneous

(-(225

non-strange

degrees

by the conceptually

of the original

effective

boson

opposite

is guided of colors

all the symmetries

dipole

o(e+e-

quite

of a large number

dd >= l-)

as the relevant

Theory,

which “measures”

< Qu >=<

lowest

baryon

apparently

the corresponding

The parameter

the cross

mesons

Meson

the pion is the Goldstone

condensate as the

model,

Non-linear

limit,

the point of view that low-energy

and vector

in the limit

of mesons

long wavelength of pions;

I will examine

by pions

states.

Boson

in the chiral on hadronic

limit,

which preserves

of quarks

The one of leading

of spontaneously

scales)

N md N 10 MeV.

in terms

physics

long-wavelength

importance

broken

chiral

the pion has zero mass. is understood

Current

2m$f,f= -(mu + md) <

algebra51

qq >

proper-

and gluons,

in terms

but is the

symmetry. The

actual

of non-zero

gives the relation

but

1oc

W. Weise f Chiral symmetry and nucleon structure

which

involves

0 as mu,d +

the pion

decay

2.2 Chiral

Effective

w) vector

limit.

the pion is the most important

Next to the pion in importance,

mesons

represent

the lowest

then to build low energy hadron partly

motivated

mentioned.

limit

realizations

current

that

m,

-+

degree of freedom

isovector

and isoscalar

of the QCD

vacuum.

in which QCD currents

turns

of QCD

into a meson have their

The

mesons.

theory,

direct

in the (p and idea is This is

as already

correspondence

at low energies:

(see eq. lb)

is expressed

in terms

of the pion field aa

to leading

as

so that

the PCAC

relation

B. The vector for example

PAZ

currents

correspondence

current

(la)

while the isoscalar

in the vector

which is in leading

meson

fields.

order related

Consider

to the p meson

electromagnetic

current

-~,p-)+.

and we have assumed ties such as eqs.(3-5)

a universal illustrate

coupling

is related

fermionic

conserved

resent

efficiently

the low energy

more

bosonization

of QCD involves

We only mention or schematic The leading

here that

models part

based

in terms

becomes

where

effective

valued na(z)

7,. Altogether,

which we cannot

effective

Lagrangian

chiral

The

in recent meson

years

is the non-linear

which rep-

systematic

elaborate

theories

identi-

the replacement

field variables

currents.

has been made

Current-field

namely

bosonic

of these

formalism

much progress

field via

p and w meson.

of “bosonization”,

by appropriate

on it, with equivalent

of the SU(2)

as pion composites

g for both

properties

substantial

of the resulting

= ezp [i?‘. ?(z)/fir],

currents

to the w meson

-$d,(2T),

the basic concept

of originally

(4

..)

J,‘=“(z) =

npw-system

(3)

by

V/3”) =

expressed

+ .... . . . . results.

= -f,m$&‘(z)+

find their

the isovector-vector

field p;(z)

urally

l-

in terms of pions and vector

A;(z) = f&&=(z)

U(Z)

collective

the J”=

excitations

entirely

the conserved

bosonic

A. The axial

dipole

physics

by the large-N,

Furthermore,

in the following

order

the fact

Lagrangian

Due to its small mass, low energy

fir= 93.3 MeV and reflects

constant

0.

partial

at this point.

to connect

&CD,

at low energy5B6). o model,

field

is the pion field.

Vector

the non-linear

effective

mesons

are introduced

Lagrangian

nat-

of the coupled

llc

W. Weise / Chiral symmetry and nucleon structure

L eff

= L0 -

$tr

[a,xJ&F

+ ;gw’BP

f a,&7w7

+ $n;f;tr(U

+ (p and w kinetic (For a detailed non-strange mesons

discussion,

mesons:

chiral

8,9).

+ rrpw couplings). This

symmetry

where

g is the a+~-

pna

decay;

to the Vector

coupling finally,

Meson

The Lagrangian

constant

with very special

derives

algebra

the relevant

physics

constraints;

the vector

of

L =ff,

from

principle

the coupling

and

which

still

related

to this current

ago41 and later the baryon

has only

The

non-linear

three

of bosons

quantity should

remnant

fermions (mesons).

(winding

= 0, irrespecThis

of which

(quarks) The charge

number).

be identified

meson

(the

theory

even

current

is part though

of the

B = s d3s B,(z)

It was conjectured

with the baryon

pion

decay

as represented constant

current,

AS SOLITONS

interesting localized

point

ansatz

AND THEIR

is now that

and stable

long

i.e. B is

of maximal

this purely

as pointed

symmetry

bosonic They

(“hedgehog”

of the solution.

The boundary

B = 1, as can be seen from from eq. (9)

:

vector

meson

values.

theory should

has soliton be identified

out long ago by Skyrme4).

pionic

content

the universal

Lagrangian

SIZES

field configurations.

quantized,

by the effective

fK,

empirical

radial forms for the p and w meson fields.

derived

PB,

it is conserved,

corresponding

number

(4,5).

(9)

form of the Lagrangian.

the fundamental

B,

chiral

parameters

once they are properly the

identities

field to a current

the SU(2)

g and the pion mass mrr), all fixed at their

3. NUCLEONS

classical,

leads

number.

In summary,

coupling

action”),

in terms

that

from

necessarily

by the current-field

of the c-tensor,

is a topological

confirmed”)

g N 6 determined

interactions

of the w meson

of the detailed

Wess-Zumino

entirely

value

[U+d”UU+dxUU+~“U]

By virtue

“remembers”

is now expressed

empirical

as expressed

= $$tl.

i.e. independent

the so-called

(8)

xhfd,

the

(7) involves

physics

with

incorporates

of electromagnetic

Dominance

properties.

tive of any dynamics,

=

with

the introduction

BP(z)

(18)

L,.f

and current

have masses

mp = m,

p” +

2

- l)+

terms

see e.g. refs.

it satisfies

- ig(r’. & + Q)]

conditions

the explicit

form

ansatz)

U(?)

= z,

with

i.e.

baryons

One looks for solitons =

exp[ir’.

The radial profile F(r) F(0)

solutions,

F(oo)

of the baryon

iF(r)], represents

and the

= 0 imply baryon

number

distribution

W. We&e / Chiral symmetry and nucleon structtire

12c

B,(r) with B = 4rJom dr r2B,(r)

=

-‘FI(‘)sin2F(r), 2x2 r2

= (2/n) s,” dF sin2F

00)

= 1. Collective quantization following

adiabatic rotation of the soliton gives the Hamiltonian IT = MO + I(1 + 1)/20,

where MO is

the classical soliton mass, 0 the moment of inertia, and I = J = l/2, 3/2 refer to nucleon and A(1232), respectively. One should note that nucleon properties emerge as true predictions in this approach. There is no freedom in the choice of parameters (f,,m,,g) and forever in the meson sector. An import~t

which have been fixed once

outcome is the predicted size of the baryon

number distribution inside the nucleong): < r”B >= 4m

I0

O”dr r4B&)

This value reflects roughly the scale < ri >li2-

u (0.5j+‘.

(gf,)-’

at which the soliton is stabilized

by the presence of the vector mesons. The connection between the distributions of baryon number and charge turns out to be precisely of the form suggested by the Vector Meson Dominance principle. To see this, consider the isoscalar charge distribution of nucleon. The current-field identity(5) implies that the probing isoscalar photon sees the nucleon through the w meson content of the soliton. The source of the static w meson field is the baryon number density B,(r):

(f2 - mE)w(r)=

$53,(r).

(12)

Let FB(q2) be the corresponding form factor with the baryon mean square radius < rb >= 6 d~~(qz)/dqz

&s=~. Then

and the mean square isoscalar radius becomes (14) Thus for m,

= fi

gf= u 780MeVwith

charge radius < ri >Sl/2? 0.8

g = 5.9 and < rb >‘i2=

0.5 fm the isoscalar

fm resuIts , very close to the empirical vaIue. In this picture,

the baryonic charge carried by the three valence quarks in the nucleon is then concentrated within l/2 fm, whereas the electric charge radius induced by the meson cloud is considerably larger. We show in Fig. 3 typical results for the nucleon charge and magnetic form factors as obtained in the chiral soliton approach.

It is seen that the pattern of electromagnetic

sizes of the proton and neutron comes out quite well ( although the negative slope of the neutron charge form factor turns out to be somewhat too large).

13c

W. Weise / Chiral symmetry and nucleon structure

'ROTON MAGNEK FORM FACTORG:(qkG;( 0)

PROTON CHARGE FORM FACTOR G;(q2

01

0

NEUTRONCHARGEFORM FACTORG;lq',

02

03

0.4

05 Iq'llGeV'l

NEUTRONMAGNETICFORMFACKR G;Cs'YG;COl

0.08 p---j--

0

Nucleon electromagnetic (taken from ref. 9).

The

r-space

together

with

baryon

number

number

interesting

soliton

modelg)

density.

is shown in Fig.5.

< r2A >li2=

0.64 fm, to be compared (0.63 f 0.03)

>,‘g=

An important to be seen in close rize,

one finds

that

baryon

charge

form

this

are shown

illustrates

related

inside

once

empirical

to the

the nucleon.

The corresponding

with the recent

model with vector

mesons

in Fig. more

4

that

of charge.

factor

distribution

0.5 Iq'llGeV21

densities

For the proton

axial

OL

axial value

radius

nucleon The

axial

result

is predicted

of as

13)

fm.

observation relation

is the

the spin-isospin

<

r;

and neutron

well inside the distribution

quantity

It measures

0.3

02

FIGURE 3 calculated in a chiral soliton

of proton

is concentrated

A;(z).

the chiral

distributions

the baryon

A further current

form factors

01

from

these

to the probing number

results

is that

fields by which

is concentrated

within

the concept they < ri

of nucleon

are measured. >‘/‘=

sizes has

To summa-

0.5 fm, while

the

14c

W. Weise / Chiral symmetry and nucleon structure

NUCLEON AXIAL FORM FACTOR G,(d)/G,(O)

2.0

-z 0

-2

5 1.0

o_ 0 0.5

1.0

2.0 0 r[fml

1.0

2.0

0 FIGURE

4

Baryon number and charge distribution of proton and neutron in a chiral soliton model (taken from ref. 9).

empirical

05 FIGURE

1.0

5

1.5

ILi’IlGeV ’I

The nucleon axial form factor in a chiral soliton model (taken from ref. 9).

predicted [ref.Q]

pion charge radius 0.66 f 0.01

0.63

0.63 + 0.03

0.64

< r; >;!02tCVz if6

0.86 * 0.01

0.92

If4

0.86 f 0.07

0.84

[fm2]

-0.12 f 0.01

-0.22

If4

0.88 f 0.07

0.85


[fm]

nucleon axial radius < ri >l/’

[fm]

nucleon el.-magn. radii

< %I >;!02ton < r; >neutron < G4 >zron

TABLE 1: Pion and nucleon radii; empirical values are from ref. 12) (pion e.m. radius), ref. 13) (nucleon axial radius) and ref.14) ( nucleon electromagnetic radii). The predicted values are taken from ref. 9) (“minimal” model).

1%

W. Weise / Chiral symmetry and nucleon structure

larger

electromagnetic

surrounding between

radii

the baryon

an object

has different

The different than

The

baryon

baryon

apparent

number

the charged

the axial

current

we are quite familiar

on which probing

of the polarization

distribution

domain

inside

which

here in terms

the

hosts

the

of non-linear

phenomenologically

meson

cloud

is intermediate

with the idea that

field is used to explore

phenomena

surrounding

it.

the “core”,

in simple

nucleon valence

with

its small

quarks.

meson

theory,

constituent

quark

Its

size can

radius

is actually models

<

quite

roughly

ri

>‘/‘zY

close to the

in order

to reproduce

spectroscopy.

An objection

frequently

raised against

The electromagnetic

are altogether

not very

nucleon

question.

Let us therefore

for example

(see table

sizes out of pointlike examine

the pion form factor

form of a once-subtracted

dispersion

The

imaginary

can be excited

t > 4mz.

part

This

mean

square

This

relation l-

implies

hadron

with

m

of charge

= 1+

O” &

$

Im

s 477%:

t(t

-

is measured

-+ A+A-

by the e+e-

derived

the pion charge

(1 -

electromagnetic a large extent

in particular, q2/mz)-’

fact

does

(for small

close to the empirical and axial form factor the mass spectra

in ref.151.

closer.

Consider

it in the familiar

(15) of physical

with

squared

cross section.

hadron

states

four-momenta

Consider

now the

from eq.(15):

radius

that

has given rise to the Vector

&,

which

is a serious

*r(t) q2 - it).

photons

The

is

mesons,

relation:

distribution

that

This

by writing

the spectral

spectrum.

vector

an approach

radii somewhat

and its low q2 behaviour

by isovector

radius

trust

meson theory

interactions?

is determined

this

spectrum

Meson

by the low-lying

is visibly

Dominance

not

enter

spacelike

in this

1. Analogous

of the nucleon;

of multi-pion

states

isovector by the p

principle

mentioned

p meson mass spectrum;

consideration.

q2 < 0) leads

value in table

dominated

(VDM)

. Hence the pion radius (15) is just given by the physical

0.63 f m, quite

as shown

and probably

represents

pion charge

the p meson =

F,(t)

of non-linear

and pions,

out of the vacuum

spectrum

(see Fig.2)

previously

Im

as solitons

1); how can one then

mesons

the meaning

F1,(q2)

F=(q2)

that

nucleons

sizes of nucleons

different

calculates

FT(q2)

systems

radii depending

the following.

meson

with

of the core itself.

as the

fm, expressed

=

f m reflect

(0.8 - 0.9) associated

with many-body

a reflection

one required

J”

>lf2= radius

sizes are characteristic

be interpreted l/2

The

these.

From the experience

rather

< T;,~

core.

The

to < ri

>i{zn=

considerations

at low q2 these

VMD film,

N

hold for the

form factors

with the appropriate

result

quantum

reflect

to

numbers,

W. We&e / Chi~al~.v~~~~ry and nucleon structwe

16C

One should note that all nucleon radii fbaryon, isoscalar charge, magnetic and axial) can actually be expressed in terms of one single scale parameter:

A = 2gfr = fi with the vector meson mass mv = fi

mv -

(17)

gfir. As a rough rule, the baryon radius is &/A.,

while the proton charge and magnetic radii are S&/A charge radius &/‘?/A.

1 GeV,

and the axial radius equals the pion

Note that from the point of view of the large N, expansion, the scale

A in (17) is of order Nz, i.e. independent of Arc, just like any meson mass.

4. MESON-NUCLEON

FORM FACTORS AND NN-INTERACTION

We can now establish contacts with the successful one-boson exchange (OBE) phenomenology ‘s) of the nucleon-nucleon interaction.

Meson exchange in lowest order can be

discussed entirely at a classical level; in fact, Yukawa’s original idea was phrased in terms of classical meson fields. Let us recall the basic steps. For each meson of mass m described by a field #J, (the pion and the p and w mesons in our case), there exists a static field equation of the form

(d” -

rn2)dx(q

= .Jx(q.

(18)

Here this equation is derived from the effective Lagrangian (7). The sources Jx are nonlinear functions of all the coupled meson fields involved. The Fourier transforms of the .7x(fl correspond to the meson-nucleon form factors. Consider now two nucleons 1 and 2 at positions 7; and Fa and at a sufficiently large distance / Fl - F’21so that their baryon number distributions do not overlap and the mutual pofarization of the two nucleons can be neglected. Under these conditions, the NN-potentiai mediated by the meson fields can be written as

(19) in terms of the source function of nucleon 1 and the field generated by nucleon 2. We will now discuss those aspects of the NN interaction which can be developed within this classical framework. We begin with the pion field surrounding a static nucleon’51. The proper pion field PO”(?) is related to the chiral soliton profile F(r) by ~“(3 field equation (?2-m:)p”(?).

= ~S~P^PsinF(r)

and satisfies the static

The corresponding source function J,“(q = -a’.? PJlr(r) has

the radial distribution .7,(r), shown in Fig.6 together with the Fourier transform G,NN(q2). It has a characteristic range < r2 >t;/N”N 0.56 fm which measures the size of the pion-nucleon interaction region and which is close to the baryon radius < r$ >1/2= l/2 fm.

W. Weise / Chiral symmetry and nucleon structure

PION SOURCE FUNCTION rfm+l

1

17c

nNN FORM FACTOR

J”(r) 20-

--

Gn,,,, (q')

4

2

L

-J

0

1.0 r [fml

0.5

II

I

I

-9mi 0.

0.5

FIGURE 6 The radial source function and the pion-nucleon form factor calculated model defined by the effective Lagrangian (7) (taken from ref. 15).

I

I

10 zi2[GeVz1 in the chiral

soliton

NN ISOVECTORTENSOR INTERACTION ,'V, (OPE) /I

60

0

The isovector

tensor

0

NN-potential

1

FIGURE in momentum

2

3 l~l[fm~'l

7 space:

with nNN and pNN form factors calculated in the chiral soliton model (see ref. 17), in units MeV . fm3. Dashed curve: one-pion exchange with pointlike nucleons; long-dashed curve: one-pion exchange with TNN form factor included; solid curve: additional p meson exchange included. The “data” are deduced from NN phase shift analysis.

18c

W. Weise / Chiral symmetry and nucleon structure

Similar considerations15)

lead to the p meson-nucleon

form factor Gp~~(q2)

which

turns out to be dominated by the pNN tensor coupling proportional to 3 x < (after nonrelativistic

reduction), just as required in phenomenological

OBE models.

The predicted

size of the p-nucleon interaction region in r-space is again about l/2 fm. A monopole form factor of this range corresponds to a cutoff h N 1 GeV. The role of the extended structures of meson-nucleon source functions is best illustrated by examining the isovector tensor NN interaction in momentum space, which is known to play a very important role in nuclear physics. The resuit in Fig.7 shows how T and p meson exchange cooperate,

together with the finite sizes of their soliton source distributions,

to

reproduce the main features of the empirical tensor force. This should be considered as a success of non-linear meson theory, in the sense that a chiral effective Lagrangian, based on the symmetries of QCD with a minimal set of parameters fixed entirely in the meson sector, is able to reproduce a particularly important part of the nucleon-nucleon

interaction up to

momentum transfers / 4;I< 600 MeV/c. On the other hand, the classical mesonic approach has its obvious limits. In phenomenological OBE potentials the short range repulsion is generated by an effective w meson with a large wNN coupling gzNN /4x N 10 - 11 (normalized at q2 = 0). Somewhat more than half of this repulsion can be generated by the w meson induced potential derived from the Lagrangian (7). The important central attraction in the NN force, commonly associated with two-pion exchange, does not exist at the classical level. The missing short-range and the missing intermediate-range

repulsion

attraction indicate the necessity of a detailed treatment

of second order polarization effects related to KP- and 7rp-exchange’6~1s).

5. MESON EXCHANGE

CURRENTS

. A COMMENT

The Skyrme soliton model with vector mesons offers a natural approach to meson exchange two-body the Wess-Zumino

currents.

The isoscalar exchange current is uniquely determined by

term and has been applied successfully in a calculation of the deuteron

magnetic. form factor”).

We comment here briefly on the isovector exchange current which

is pion dominated. The usual approach is to evaluate the Feynman diagrams for the KrollRuderman

(or pair, or seagull) current,

Fig.Ba, and the pionic current,

Fig8b,

first for

pointlike nucleons, and then introduce ad hoc form factors to account for the nucleon and pion sizes. The derivation of the isovector exchange current in a chiral soliton model”), shows that the separation into pair and pionic terms is generally not meaningful for nucleon sources with finite size, unless the two nucleons are at a large distance. In this limiting case, the model makes unique assignments of form factors to the relevant vertices as follows20): in the Kroll-Ruderman GA(q2).

current, the 7rN vertex is to be associated with the &

This is a necessary consequence of the underlying chiral symmetry.

couplings are governed by their form factors GrrNH(q2) the photon-pion

as usual.

coupling involves the pion form factor Fz(q2).

form factor The rNN

In the pionic current,

Current conservation then

19c

W. Weise / Chiral symmetry and nucleon structure

requires meson

that

(see table

the empirical are indeed

should

GA(~‘)

theory

radii

equal

l),

deduced

within

Th’ IS is exactly

equal F,(q’).

from

error

a.

Chiral

AND

the pion electromagnetic

posedly proof

from partial

of freedom

tum transfers with

the

al

suggested b.

pions

fi

times

the mass

to non-perturbative,

Lagrangian

and vector

hadron

scale is close to the chiral

as solitons

the r.m.s.

to be &/A Vector

of the

(although

mesons

physics

symmetry

in this approach.

radius

Meson

in this

-

and supa rigorous

are the collective

at energies 1GeV

and momen-

which coincides

p and w mass

breaking

ward way to explain

of their

interaction

approach:

of the chiral

half of the short-range

scale,

as given

by

Achirar N &fir

size is determined

distribution

form factors

currents.

inside

exchange

original

by

the nucleon

are more extended

Meson-nucleon

and related

Yukawa’s

currents

idea of meson

in

form factors, find their

exchange

does

&CD. soliton

repulsion

the NN attraction

of the nuclear

characteristic

number

Electromagnetic

Dominance

in non-perturbative

the limitations

Their

of the baryon

N l/2 f m.

nucleon-nucleon

explanation

elements

of the QCD

approach

The scale is set by A = 2gfT

and equals

have a foundation

about

form factors

the l/NC expansion

1 GeV.

long-range

However,

through

up to about

This

size, through

c.

bosonization In this

as an approximation

foundation

non-strange

such that

natural

axial

also

current: Kroll-Ruderman or pair are the relevant form factors at the

govern

emerge

is predicted

the

chiral

Note that

in ref.19).

Nucleons A-l,

and nucleon

which

mass

figf,.

=

is interesting

It has a conceptual

does not yet exist).

degrees

my

meson theory

QCD.

results

result.

CONCLUSIONS

non-linear

low energy

non-trivial

bars.

FIGURE 8 Basic processes contributing to the one-pion exchange current (left), and pionic current (right). Also indicated photon-coupling vertices.

6. SUMMARY

what is found in non-linear

q2, a highly

at least for small

force are missing.

picture

are obvious.

in the NN interaction. at intermediate This

It fails to account There

distances.

is probably

Thus

not a fault

for

is no straightfortwo important

of the underlying

2oc

W. Weise / Chiral symmetry and nucleon structure

effective

Lagrangian,

but rather a deficiency

the only one so far which scattering

including

properly

is reasonably

A-excitation

described

for the core size of nucleons, with nuclear

descriptions

beyond

successes,

the relevant

“mesonic”

to “quark-gluon” problem,

s-wave

p-wave

nN

method,

pion-nucleon

interactions

one of them being that it predicts

rather than the 1 fm scale of the MIT

phenomenology.

It might

by adding

would be quite inefficient.

&~e visibly

challenging

is not a problem,

q2 > A2 - 1 GeV’,

but such a procedure

quantization

Although

are not

(see ref. 23).

Meson theory has its obvious

incompatible

of the semiclassical

manageable.

degrees

more mesons

bag which

to continue

fm scale

is evidently

with mesonic

of higher masses and spins,

Clearly, at very large q2, quarks and gluons

Precisely

of freedom.

degrees

be possible

a l/2

of freedom

where

should

and how the transition

be made,

from

is still an unsolved

and

at least for light quarks.

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