Proceedings of the Second Hawaii Topical Conference in Particle Physics (1967) 9780824885588

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PROCEEDINGS OF THE SECOND HAWAII TOPICAL CONFERENCE IN PARTICLE PHYSICS

(1967)

Edited by

S. Pakvasa and S. F. Tuan University of Hawaii-Honolulu

CONTRIBUTORS

R. H. Dalitz

G. Goldhaber

V. L. Fitch

T. D. Lee

UNIVERSITY OF HAWAII PRESS HONOLULU, HAWAII

Copyright 1968 by The U n i v e r s i t y of Hawaii Library of Congress Card Catalog Number:

Press

68-20138

Printed in the United States of A m e r i c a

PREFACE

The Second Hawaii Topical Conference in Particle Physics took place from August 14th to 25th, 19^7 at the Manoa Campus of the University of Hawaii, Honolulu.

It is the second of a proposed series of

Pacific Summer Conferences, the idea of which originated two years ago with Professor Vincent Z. Peterson who heads the University of Hawaii high energy physics group.

The conference was supported

directly by the University of Hawaii and the National Science Foundation, and we are very much indebted to Dean Wytze Gorter, Professor Peterson, and Dr. Arnold Feingold of the National Science Foundation for their encouragement and wholehearted support.

The interest

in this Conference of the U. S. Atomic Energy Commission, which sponsors the program of research in high energy physics at the University of Hawaii, is also gratefully

acknowledged.

The present Conference focussed on the strong, electromagnetic, and weak interactions of elementary particles.

Principal lectures were given by T. D. Lee,

R. H. Dalitz, V. L. Fitch, and G. Goldhaber and are recorded in the Proceedings here. prepared the final manuscripts

The Lecturers

themselves.

D e t a i l e d surveys were g i v e n o n

field-current

identity, the CP v i o l a t i o n p r o b l e m , a n d the q u a r k m o d e l on the t h e o r e t i c a l side.

These were m a t c h e d

well by experimental lecture series on the status of CP v i o l a t i o n as well as a c o m p a r i s o n of the

established

h a d r o n resonances w i t h the p r e d i c t i o n s of the q u a r k theory.

The Principal lectures were s u p p l e m e n t e d by

seminars and d i s c u s s i o n sessions where the 70 odd p a r t i c i p a n t s all j o i n e d in. One of us (S. P. T.) in his capacity as D i r e c t o r , w o u l d like to thank some of his

Conference

colleagues,

especially Department C h a i r m a n John H o l m e s , Peter D o b s o n , and Vic Stenger, for their

enthusiastic

assistance during the Conference period. we are deeply g r a t e f u l to the Conference

Above

all

Secretary,

Mrs. Caroline Chong, who was indefatigable in the s u p e r v i s i o n of arrangements b o t h p r i o r , during, and after the Conference, and thus is in large m e a s u r e r e s p o n s i b l e for the success

achieved. S. P a k v a s a S. F. Tuan Editors

D e c e m b e r 15, 1967 Honolulu, Hawaii

CONTENTS T. D. Lee Page ALGEBRA OF THE OBSERVED CURRENT OPERATORS I.

Review

13

1.

Introduction

13

2.

Current Algebra

14

3-

Spectral Representation and the Schwinger Term

II.

III.

17

Field-Current Identity

24

1.

General Discussion

24

2.

Algebra of Fields

25

3.

Propagator

28

Application to (Jy)i=i

32

1.

Spectral Function

32

2.

Lagrangian

34

3.

Feynman Graphs and p-propagator

38

4.

Commutation Relations and Sum Rules ....

40

5.

Photon Propagator

42

6.

Remarks

44

7.

Form Factor

46

8.

Algebra of (J^)I= 1

48

Page IV.

V.

Iso-Scalar Electromagnetic Current

49

1.

General Discussions

49

2.

-a) Mixing

52

3.

Special Models

55

A l g e b r a of the Observed Currents

59

1.

U-l Algebra

59

2.

S U 2 Algebra

59

C, P, T SYMMETRIES I.

General Discussion 1.

67

Symmetry Principles and Their Violations

II.

67

2.

Time Reversal Symmetry

69

3.

Implication of Non-Conservation

71

4.

Group Extension

72

Possible Existence of

and T g t

Violating

Electromagnetic Interaction

76

1.

General Discussion

76

2.

Some Necessary Properties of K^

77

3.

Classification

80

4.

Some Further Properties of Q K

82

5.

Experimental Consequences of K y

85

6.

Application of Field Current Identity

...

88

V. L. Fitch

Page

WEAK INTERACTIONS EXPERIMENTAL ASPECTS I. II. III. IV. V. VI. VII. VIII. IX. X.

Introduction

93

Phase of n+_ from K° - K^ Interference

99

Phase from Coherent Regeneration

100

Charge Asymmetry

Ill

The Decay of Kg to Two Neutral Pions

113

Specific Tests of T-noninvariance

121

The AS = AQ Rule

134

The Mass Difference, 6 = m.1-m2

137

Comments on Coherent Regeneration

141

Summary

144 G. Goldhaber STRONG INTERACTIONS EXPERIMENTAL ASPECTS

I.

The L = 1 qq Nonets

165

1.

The K*'s of the L = 1 Nonets

167

2.

The Structure of the Kttit Mass Distribution

3.

169

Interference Phenomena in the K-tttt Mass Distribution

4.

Further Remark on K* Mixing

5.

The irp Boson Cluster in the A-j_, A 2 Region

179 185

186

Page 6. II. III.

What about the J P C = 0 + + States

3

P0?

... 189

Higher Mass Boson Resonances

236

The Search for Higher Boson and Baryon Multiplets

248

1.

The Exotic Particles to Look For

2.

The Evidence on the Existence of Exotic Particles:

3.

IV.

251

The Evidence on the Existence of Exotic Particles:

4.

PRO

CON

255

Conclusion

257

Various Topics on Strong Interactions 1.

249

275

New Determination of the Low EnergyScattering Length and Coupling Constants

2.

275

SU(3) Decay Constants for B a r y o n Resonances

3.

277

Elastic Scattering and Charge Exchange Cross Sections 3.1

The Coulomb Interference

278 Region,

9 ( V f )

S=0, 1

and

(A^)

S=0,1

These current operators are direct observables. For example, through the use of the Maxwell Equation 9F — ^ 9x H

=

e rj L

y V

+

(jy) V

+

I=1

(jy) V

(1.1) 1=0 _

where e = charge of the electron (e < 0) , the totality of our experimental efforts in studying electromagnetic processes can be simply reduced to a table of the various matrix elements of these current operators; the same is also true for all leptonic weak interaction processes. If the nonleptonic weak interaction is of the (current X current) form, then the same

14

can also be true for the non-leptonic weak interaction processes. A remarkable fact about these current operators is that their matrix elements exhibit symmetry properties which are, nevertheless, v i o lated by the entire physical system. Both the leptonic currents

and

wk satisfy p - e symmetry. Y e t , the physical muon weighs far more than the electron. Similar examples can also be cited for the hadrons, such as the symmetry between ir and K , the nucleon and other hyperons, etc. It is, therefore, natural to investigate the possibility that these observed current operators may satisfy a set of simple and exact algebraic relations. 2. Current Algebra An important attempt was made by G e l l - M a n n , ' who suggested that al observed hadronic current operators

satisfy the same algebraic relations

as the corresponding quadratic expressions of some spin 5 field operators. For example, one may assume the existence of three spin 5 quark fields p , n , X,

and identify

(JX>I=1 = + ^ ( p V A P - n V x n ) s=0

= +i n

( A

=

Mk)s_0

< VxP>

+ i ( n t y

4r*r5

P )

( L 2 )

15

etc., where the sign corresponds to our choice e < 0 in (1.1). Under the S U g transformation of the fields p , n and A , one generates a total of 2 X 8 = 16 quadratic expressions of the quark fields, such as those given by the right-hand side of (1.2). It is convenient to denote the corresponding set on the left-hand side, which includes all observed hadron currents, as

(1° OH

where a = 1, 2, • • - , 16 . These expressions, then, lead to the following algebraic relations: [ J 4 V 0 ,

[ J 4 V O ,

}

4

V,t)]

|jb(r',t)]

=

=

C

Q b c

C

a b c

J4C(r,f) 6 3 ( r - L " )

JjC(i>0 6 3 ( r - r ' )

+

S

(1.3)

a b

V.63(r-r') (1.4)

and D;a(r,0,

¿ 0

(1.5)

where the subscripts i , j denote always the space components,

is

a b totally antisymmetric with respect to its superscripts and S The antisymmetric part C

a b c

is symmetric.

is identical with the structure constant of the

Lie algebra satisfied by the generators of the S U ^ X S U ^ group.

[ If one c o n -

siders only the three I = 1 vector currents, then the superscript a varies only from ea'3C

1 to 3 , the group becomes the usual isospin S U j group, and - C

a b c

which is + 1 , - 1 or 0 depending on whether ( a , b , c) is an even

permutation of (1, 2, 3) , an odd permutation, or otherwise.] The symmetric

=

16

part

S

qb

is usually called the Schwinger term.

2

As we shall see, from

the positive definiteness of probabilities it follows that, independently of the quark (or any other) model,

.ab

t 0•

If one uses the free qudrk fields, then -ab S

=oo

To derive these equal-time commutators, we note that (1.2) can be written as jpx)

=

q'(x) 0 ° q(x)

(1.6)

where

and 0 °

is the direct product of a (3 X 3) matrix in the SU^-space times

the usual Dirac matrix

y^ y^ or y^ y^ y^ . By using the equal-time anti-

commutator [q(r,t),

q ^ t ) }

=

s \ -

it is easy to show in a purely heuristic (though, in fact, incorrect) way that [ V ( J i f 0 °Ha

0 °vb q ( j L ' - » ) ]

0 * =

q

f

(r,t) [o°,

0vb] q ( L , t ) S 3 ( r - ^ ) .

(1.7)

17

This h e u r i s t i c expression leads d i r e c t l y to (1.3). w i t h o u t the S c h w i n g e r term'.

It also g i v e s (1.4), but

W e w i l l now prove the necessity of the

S c h w i n g e r term w h i c h shows that the a b o v e h e u r i s t i c a p p r o a c h does not g i v e the c o r r e c t result.

3.

S p e c t r a l Representation and the S c h w i n g e r Term To demonstrate the necessity of the S c h w i n g e r term, let us c o n -

sider the v a c u u m e x p e c t a t i o n

=

< v a c | | 5T(x) , | K M '

( T (X1) | | v a c > } > > ] |

where i ;

-

1=1,0

- -

S = 1,0

. o,

wk ( ( Ha - ) S = l, . For clarity,

we may write >

=

lT,k>

=

I ••m , J , h , k > r r r '

(1.10)

where T denotes all other internal characteristics that are necessary to specify the state, such as helicity h

f

mass m ^ , spin J ^ ,

charge

Q ^ , etc.,

where m

2 _ =

r

, 2 _ - k = m

P2

E

,2 - k

In (1.9), the sum extends over all possible eigenstates, including those which are related by Lorentz transformations. More explicitly, one may write for the sum in (1.9)

2 ••• -

/ d M

2 S(M - m ) r T Sir3

/ d3 k • • •

(1.11)

By recalling that under a Lorentz transformation parallel to the direction of

19

so that

A

h

, k

X

r ~

h

r

, k

-

is an invariant, one can easily perform the k integration in (1.11) by transforming all states to their respective rest systems. In addition, one notes that under a 3-dimensional rotation, the space components of transform like those of a 3-vector and its time component like a scalar. Consequently,

< vac I

I T, k = 0 > = 0

Therefore, for fixed

r

if

f v

^ or, J

r

= 1 , h = 4 . (1.12)

[or, J

r

= 0,

p/4

but summing over the different possible

hel¡cities, one has

£ < vac I a Q ( 0 ) I T , k > < T , k I a b ( 0 ) I vac > helicity W ~ ~ °V , # k k \ m if J =0 r m

r «¿W-^i-E ,

b r

#

k

K < >k v + ^m

k

\ m

r

if

J

r

= l

if

J

r

> l

(1.13)

20

where „ > )

' ¡ V )

= | < J r = 0 , k = 0 | | ° ( 0 ) | vac >

=

h E

| |

5

a b

.

(1.14)

In deriving (1.13) and (1.14), we have neglected the weak and electromagnetic interaction.

By using (1.8) one sees that different members of

carry different quantum numbers, such as I , ab therefore

aj

or

J^1

, or S , or parity;

ab ( T ) = c r j ( T ) = 0 if a / b . C If one includes the weak

and electromagnetic interactions, then it is convenient to use the stationarystate-representation for I r , k > . From CPT invariance, the matrix e l e ment < vac I | °(0) I r , > is real for p / 4 and imaginary for p = 4 . valid Thus, (1.13) remains valid,

ab ab cr^ and cr^ remain both real and symmetric,

but no longer diagonal. J Upon substituting these expressions into (1.9), one obtains

C ^ x - x ' )

=

1

/d(M2)

M

3

2

9x

H

^ab(M) 9x

AM(x-x')

v

o" o a b (M) A M ( x - x ' ) • (1.15)

+ /d(M^) M

9x

H

9x

V

J

21

In (1.15), the spectral functions (TQ^M) and crj3 ^(M) are given by

^b(M) = E ß ( M - m

) and

I B > . The left-hand side of (2.3), or (2.4), or (2.5)

becomes 2 < A I < I J ^ ( x ' ) I B >

-

(exchange

with J ^ ) ,

(2.6)

27

and the right-hand side consists of

< A I

J I B>

and/or some known

constants. The measurements of these matrix elements < A I (L ° I >
give, then, a direct test of these commutation relations

through the sum (2.6). The only practical question is to select the most appropriate states

I A > and

I B > , so that the resulting sum rule becomes

more accessible to the presently available experimental facilites. This new set of commutation relations is sometimes called "the algebra of fields," or simply "field algebra," since in the present view the observed current operator

^

is the basic meson field. The field algebra

also represents the simplest possible equal-time commutation relations that these currents

can satisfy. The structure constant C

0

^

is deter-

mined by the corresponding Lie algebra satisfied by the spatial integrals

1 qQ

W

=

S

f4a(r,t)d3r

.

(2.7)

The Schwinger term cannot be simpler than a finite c. number times

8°^,

and the commutator between the space-components cannot be simpler than zero. The identification that the observed current operator is the meson field will be referred to simply as the field-current identity.

A s we shall

see later, this identification naturally leads to the idea of "vector domi n a n c e " of the electrodynamics and to the consequence that the unrenormalized form of electrodynamics is finite.

28

3.

Propagator The concept of "field-current 1 identity" implies that the total

number of

1 + and

1-

meson fields

J^(x)

' s determined by the number

of the observed current operators on the one hand, and by the particular group algebra on the other; it is not determined by the observed number of

1 + and

1-

meson resonances. These resonances merely appear as the

different singularities of the same set of propagators of these meson fields '

L'n

a s m

' ''ar

wa

Y>

0-meson states would appear in the approp-

riate longitudinal parts of these propagators, or in the propagator of the »

d

appropriate derivative

9x

a

Jv

v

Let us consider the vacuum expectation value of the time-ordered product

• By following similar steps to those used in

deriving the spectral representation (1.15) of the vacuum expectation value of the commutator, it is straightforward to show that

) /

- 2 ^ - 2 q + M - ie

o d M

'

31

these poles are on the real axis for i r ,

K and rj (if the electromagnetic

and weak interactions are neglected), they are away from the real axis for the higher resonances of 0 - mesons.

32

III.

(Jr)

Application to

H

1= 1

1. Spectral Function For clarity, we consider first only the iso-vector hadron current 2 (Jr) and note that, to the lowest order in e , H 1=1 9

J7 H

dx

For any state

1=1 — -

=

0

.

(3.1)

H

I T > with k = 0 and J = 0 , one has, on account of * r

(1.12), < v a c I J^ r (x) l m

,

r

k-0,

J

r

= 0>

= 0

if

p / 4 .

(3.2)

Thus, < vac I J J (x)

I m

=

( I m

k = 0, J

1

1=1

_ "J r

)

=

£

3 T < H

r

= 0 > 1

~

v a c l J

i M

M

lm 1= 1

k = 0 , J

r

0 .

r

=0>

(3.3)

The propagator is then, according to (2.12), of the form

°

HV>

"

/

2m

2

IT

d

q + M n

2

2

- ie

(V VH V

+

^

M

7

'

J ( M )

( 3

'4)

33

where

J o (M)

||j = Jp

=

ab is the same as cr^ ( M ) ^

in (2.10) by setting

; the corresponding

is 0 on account of

(3.3). The function D ^ (q) is, by definition, directly related to the scattering amplitude of e+ + e

e+ + e

.

(3-5)

i 2

At ( - q ) e+ e

2

near the observed mass of the

p

o

resonance, the amplitude of

scattering amplitude is expected to consist of a resonant part

e+ + e

e+ + e

p°

(3.6)

which is proportional to

4

5

,2

J

f

2m

q

25

+ M

5 2

a

- ie

J / Mm ^ ( )

and a non-resonant part due to, e.g., the ¡so-scalar hadron current

(J^) H

y and the lepton current j

ht\ i3-7)

— , etc.

[The complete expression for e

1=0

e

scattering depends on the entire photon propagator which will be given later. ] In obtaining (3.7), we have used the property that Y + J q < v a c I i ' I e e > = 0 : therefore, the nq q term in D (q) does not "»p fi pv contribute.

34

At the resonance, the resonant amplitude should be purely imaginary; i.e., the real part of (3.7) is zero. One has, then,

dM2 - 0

I f

at

q + M

q2=-m2 P

(3.8)

where IP denotes the principal value. Equation (3.8) gives the precise condition of the p-resonance and the exact determination of its mass. If (3.7) has more than one complex pole, then there can be more than one resonance. One notes that in a Breit-Wigner type approximation, one ex2 ~ 2 2 near - m , (3.7) = (real constant) • (q + m - i y m ) , P P P 2 2 which also requires the vanishing of the real part of (3.7) at q = - m . pects, at q

2

2. Lagrangian The field-current identity implies that jj. ° is the meson field. For simplicity we consider again only the iso-vector current

and inquire the general form of the Lagrangian £ = C ( ,

A ^ , t//) so

that 9F -

M

( s f

}v

" i f

V ( " a l ^ + r x2

"

P

-

}

P

M

)

' y4 M

P

2 +

-

'

x

_ i m

* IF)*]

'

2

(3

•

-13)

We note that in the standard treatment of field theory the dimension of a field is usually set to be (length) ^ , while the dimension of

* Y^ Y^ ty + (w term) . From (3.21), one

sees that the unrenormalized mass m^

9

0

=

is given by 2 m -=-£1 + r?

.

v(3.23)

'

38

3. Feynman Graphs and

p-propagator

The Fourier transform of the covariant

where o-^(M)

is related to o ^ M )

p-propagator is

[see ( 3 . 4 ) ]

by

2 (3.25) m

P

The same propagator can also be e x p l i c i t l y evaluated by using, e.g., the Lagrangian (3.16). We note that the free Lagrangian (3.17) is chosen to give the correct spectrum i f the neutral

p-meson were stable.

By using (3.18) as a perturbation,

one finds that the zeroth order propagator is

1

q

m P

'

as the second order term in g

P

is, on account of

ip = p or ir

9x

= 0 , of the form V

39

and the - z rt G

2 |1V

term gives

(q2 5

=

- q q ) rj p v

jiv

.

Thus, x(

Dp) pv

( 1q 2 S

=

(IV

-q

q ) [ 1 + ^ 7 + s 9 2 K q 2 ) J] + m 2 6 H*1 p P |iV

(3.26)

where the sum extends over all proper self-energy diagrams. If

ip=spinO,

or spin 5 fields, then Ffa 2 )

is logarithmically

divergent; thus, by choosing

- g 2 Re 2 F ( - m 2 ) P P

V =

(3.27)

one finds r /

O and

2m IT

(3.38)

2 V

= - 1

=

finite

.

(3.39)

This implies that in (3.27) the entire sum g 2 S F(q2) P although each term in the sum is not.

= finite,

(3.40)

42

5. Photon FVopagator We now consider the case e / 0 and start again with the Lagrangians (3.11) and (3.13).

By using (3.15), we find

2 m

£

=

- -=£- rp 2 v

9

9

/s 9

- i F - i G (1 + tj) + C , (p, «) pv pv free x r '

- i9

P*r4 y P P +

( « - P term)

(3.41)

where p

=

H

p

- 9

H

A

(3.42)

H

and G

=

pv

- 1 3x p

Thus, there is a gauge invariant p in addition to the conventional

A

H H

rp

v

- A

- - J 9x v

r$

.

v

(3.43)

^ 2 coupling term in - i ( 1 + tj) G M H^

- p and A

P

- ir couplings. The photon

propagator would contain, among others, the self-energy term

e2(q28pv

-q

H

qv)2F(q2)

(3.44)

2 where

2F(q

) is the same one in the p-propagator (3.26). From (3.26), one

has (3.44) = [ ( D p ) " 1 - ( q L (iv

2

8

pv

-q

2

q ) (1+tj)-m2s 1 p nv p pv J

2 9

Since

. / m +

=

\jf-J N

n2

A

o '

m




= if

J

-D^(q

where f j is the J ^ - c o u p l i n g constant, defined at q

2

2

= 0)Iz(A)

(3.54)

= 0 . Since the left-

45

hand side is, by definition, i I

z

(A) , one finds

fj

In the definition of p

M

such that the rp-propagator

and its value at q

2

=0

p

pv

(3.55)

(Vm)(Ji) P H j=i

(3-56)

is related to D"^ by pv

fJV

D

.

, we introduce

=

Pu M

X2

=

(X2/m2)DJ p |iV

-

(3.57)

becomes

D

HPV^

= 0)

(3

= Q - J '

-58)

From (3.56), it follows that /p4d3r

=

(iX/m

)I

z

.

(3.59)

2 Thus, its coupling g ^ I z ( A )

or

to any state A , defined at q

< A I / p 4 d3r I A > = i g g

P

=

Iz(A) D £ ( q

X m

P

2

= 0 , satisfies

= 0) ,

(3.60) (3.61)

46

which implies (3.20) m (Jy) H

2

= -E- P 9

1=1

P

H

From this discussion, it is clear that had we used, instead of p^ , a p^ which corresponds to, say, the higher resonance, then at m^

by

equating (Jy)

=

K ) — P

'

,

(3.62)

2 g^

would correspond to the p'-coupling constant defined at q

furthermore,

g'

g

J m P

D

= 0;

=

TS 6 - • m P

P (q = 0) = (

1 P '

(3.65)

by

KV

- J - F 9x uv M S i m i l a r l y , the form factor for the

A

= e J

y

.

v

v(3.66)

'

p - t r a n s i t i o n (real or v i r t u a l )

—

B + p°

(3.67)

is d e f i n e d by FPg(q)

=

(3.68)

where J L G - m 2 p 9x pv P v H

= -g

p

J

p

v

,

(3.69)

and where

uv r

9x

r

H

v

9x

r

v

ur

Thus,


=

+ -5-^-5-

r

q

m 9

.

(3.70)

+ m P

The i d e n t i t y (Jy) = ^1=1

FPB(q2)

2

p

P H

P

(3-71)

48

implies that for any states A

JJ

If F ^ g ( q ^ ) — q ^ as

and

B m - e - r 2 q + m P

1r = 1i

_ FAp0(q2) A B

- q^ — oo , then -4

(3.72)

1=1 as - q

-* oo . (3-71) is quite consistent with the present experimental

situation.

8. Algebra of

(Jy) M

1=1

We note that from (3.71) and (3.44), the observed current

(J^) M

1=1

satisfies [jj(r,0

J

4

°

I

_

1

]

=

[Ji

y ( r / t )

I

_

1

'

J

/

( r ,

'°i-l]

= 0

(3.73) and

2 r-

Jj(r,t) L

4

, J . y ( r ' , t) ] i=i J i=iJ

/m \ =(-£-) V v

_ 6 (r-r1)

V J

.

(3.74)

49

IV.

Iso-scalar Electromagnetic Current

1. General Discussions The iso-scalar current ( J ^ ) is conserved if the effects of the * 1=0 weak interaction are neglected. The spatial integral of its fourth-component gives the hypercharge Y .

For convenience, we shall denote

2(J^)

* by Y

M

1=0

:

2(J?) *

Y

=

1=0

(4J)

„

where /

Y,d3r

=

iY

.

(4.2)

The propagator of Y ^ is known to carry, at least, two poles whose real parts are, respectively, m. =



(ii)

=

< v a c I u^(x) I 4>> = 0

.

(4.22)

In the limit of perfect S U ^ symmetry, < vac I N (x) I > < H

I Y

v

(x) I vac >

for any state I > . Thus, S is diagonal and G y =

=

0

= 0 in the

SU^

limit. (¡ii) In general, ©y

Oj^ - The numerical values of g ^ , Q y and

Qj^l can be determined (at least approximately) from the observed rates of

° —

I and

K K

a o — IL. + i

Assuming the pole dominance approximation in the resonance region 2

2

m^ < - q

2 < m^

2

_ Y

one finds that the angle Q y can be determined by &) ~ 4 m i ) Rqte(" - & 2, 2 _ 2 W 2 . 2 j m^ 2 (m u 2 + 2 m 2 ) ( m * - 4 m £ 2 ) 5 Rate (°- £ + + 2m

. (4.23) f )

If, in addition, one assumes that the vertex functions for these decays do not vary too much from q

2

coupling constants g Y

2 2 2 = 0 to q = - m^ and -m^

C notice that the

2 are defined at q = 0 , and not at the poles 2

and g ^

one has

2

2

(m^ - 4 m K )

Rate ( f - K ' K " ,

or K ° K ° )

=

2

2

48* m 2

gy

2

cos

0N

Ccos(0y-0n)D2

'

(4.24) q

Rate (° - i

+

£")

=

2

-1 , 2 — T2

(gy/4ir)

and „ t , o n+ Rate ( u - £ i

. 2 1 a ) = -jj —^

(9 Y /4ir)

+ 2m2 _

f~2~~2

2

_4m

Ä

2Q Y

COS

(4.25) 2 2 m + 2m. /-r 2 u 1 / 2 m - 4m„ ^ y/n u f i . mu

sin

2

0., .

Y

(4.26) From the known rate of "

- K + K~ = 1.7 ± 0.4 M e V , g y cos 0

N

•

cos ( 0 Y - 0 N )

1ÌT

= 1.4 ± 0 . 3

.

(4.27)

3. Special Models W e assume the pole dominance approximation in the resonance region, which implies that Re TT (- m 2 )

= Re TT ( - m 2 ) .

(4.28)

56

Two special models are of interest: 2 2 2 (i) TTt is diagonal, but Re TT(- m^ ) = Re 1T(- m^ ) is not. This model is called the "current-mixing model." 2 2 2 (ii) TTt ' s n ° t diagonal, but Re TT(- m ) = Re 1T(- m ) is diagonal.

1=1

requires

ir H

under the strong interaction. Theorem.

=

0

< 5- 5>

In order that (5.5) is a consequence of the equation of motion,

the strong interaction Lagrangian C ^

£ . = - i m 2 ( p ^st p

)2

-

must be of the form

¿ ( 1 + n ) ? 2 + C x( ty, D pv m v

pv

) (5.6)

where f

Mv

=

-

(x s r

oV ^

fi

Pv" i f

= \( drxv

+

v

0-

B

P

9p ^ - P V

( ?

H

X

•

can be any other f i e l d and T is the representation of iso-spin generator on 4>. Due to the l i m i t a t i o n of time, we w i l l not give the proof of this g theorem. (See the paper by Lee and Zumino. )

61

From this theorem, it follows that the generalized coordinates p. , ^ J

and their conjugate momenta, defined respectively by

Tf.= I

-i J4

and P,

- i —

=

,

(5.7)

are related to p ^ by

- - f pr [ y

ff

j>

•

1

»p < V

r

V

•

(5 8)

'

By using (3.11), (3.12) and (3.20), one can easily extend the theorem to include the electromagnetic effects. As a result, the field current identity implies that to all orders of e , the observed currents

V

w k

= (Vwk)

*

H

S = 0

and J7 M

=

(Jr) r J= ]

=

[V.wk(r,t),

satisfy

CV.wk(r,t),

Jy(r',t)]

Vwk(r',t)a

J

J

=

[ V . w k ( r , t) , V w k ( r ' , t ) f 3 J

= 0

(5.9)

62

CVjV,'),

= - CjJ(r, t) , V^V,t)] = Ï S3(r-r') V*k(r',t)

(5.10)

[Vjk(r,t), Vwk(r',t)3 = 0

CV^t), V^V^/D -

2i53(r-r')

t)

(5.11)

and [V4wk(r,0,

VWk(r',t)f] = -2i63(r-r') J V J

J

,

0

+ 2 (m /g )2(V. + ie A.(r',t)) S3(r-r') . P

P

J

J

(5.12) These commutation relations can also be extended to include other hadron currents ( MV ; , (M A ) , etc., as well as to the various S^O S=0, 1 lepton currents.

63

References

1. M . Gel I - M a n n , Phys. R e v . J 2 5 , 1067(1962); Physics J , 63 (1964). 2. T. Goto and T. Imamura, Progress Theoret. Physics (Kyoto) 14, 396 (1955). J . Schwinger, Phys. Rev. Letters 3, 296 (1959). For an earlier discussion of a related term see R. Serber, Phys. Rev. 49, 545 (1936). 3. S. L. Adler, Phys. Rev. Letters J 4 , 1051 (1965); W . I. Weisberger, Phys. Rev. Letters ] 4 , 1047(1965). 4. S. Weinberg, Phys. Rev. Letters J 8 , 507 (1967). 5.

N . M . K r o l l , T. D. Lee and Bruno Zumino, Phys. Rev. J 5 7 , 1376 (1967).

6. T. 0 . Lee and Bruno Zumino, Nuovo Cimento (to be published). 7. T. D. Lee, S. Weinberg and Bruno Zumino, Phys. Rev. Letters 18, 1029 (1967). 8. T. D. Lee and Bruno Zumino, Phys. Rev. (to be published). 9. See Appendix A of T. D. Lee and C . N . Yang, Phys. Rev. 128, 885 (1962).

64

65

C ,

P, T

SYMMETRIES '

T. D. Lee Columbia University, New York, N . Y .

This research was supported in part by the U. S. Atomic Energy Commission.

66

67

I.

General

Discussion

1. Symmetry Principles and Their Violations Symmetry principles in physics are directly connected with "nonobservables."

For example, the impossibility of measuring the absolute space-

time position leads to the four-dimensional translational symmetry, which, in turn, implies the conservation of energy and momentum. The impossibility of measuring the absolute spatial direction leads to the three-dimensional rotational symmetry and, therefore, the conservation law of angular momentum. W i t h our growing experimental techniques, more physical quantities become measurable. W h a t were thought to be non-observables might turn out to be actually measurable, and hence the breakdown of symmetry principles. The discrete symmetry operators C ,

P and T are related, respec-

t i v e l y , to the supposed non-measurability of particle-antiparticle, right-left, and the absolute sign of t . As examples of the violations of these symmetry principles, we may recall:^ (i)

0 - T

Puzzle 0 - mode / + O o. (or ir ir t )

From TT° -• 2Y

or from the p-state capture IT + N ^

P(w°)

= - 1 •

T - mode .

N

one infers that

(1)

68

Since

K+

has spin 0

and since the three pions in the

T - m o d e are found

to be (predominantly) in the S-states, one concludes that the parity of and

0

T-modes must be different; i . e . ,

P(r)

=

P(9) • P(ir°)

=

- P(6)

(2)

which shows the violation of parity conservation. (ii)

(J-decay In

p+-decay u

+

r

ail

e+

—

e

+

v

+

p

v

(3)

e

w

are observed to have their spins parallel to their momenta, thus

showing the separate violation of (iii)

+

C

and

P symmetries.

In a similar way, by comparing the final states in

K°

ir + it

and

2ir°

and l and where r is the 2x2 unitary decay matrix and M is the corresponding unitary mass matrix.

We keep the

(mass-decay) matrix in a general form to allow for the possibility of external fields or TCP Violation or for the presence of material if we write -ài dt

=

_d_ dt

/ai lb'

=

R P \q2 s

2

which has eigen values r1>2 =

5±s

±

ViRz§li

+

p2q2

(3)

When R = S, these reduce to the familiar results first derived by Lee, Oehme, and Yang.

For any initial

state of the K-K system characterized by

95

• = (b(o)j ia(0> ' ^ h e r e

ex

i s t standing ivave solutions to

the equation such that

• - (bI

- IDI

with A = [(r 1 -s)a(o) + p 2 b ( o ) ] / ( r 1 - r 2 ) b = [q 2 a(o) + (r 1 -R)b(o)]/(r 1 -r 2 ) c = [( r 1 -R)a(o) - p 2 b(o)]/(r 1 -r 2 )

(it)

D = [-q 2 a(o) + (r 1 -S)b(o)]/(r 1 -r 2 ) The rJ- > ^are, of course, identified with the mass and lifetime of the K r

i =

1 m

i

+

-L >

viz. r

IT x

2

=

1 m

2

+

h

(5)

2

r

We further note that if R-S L 2 ) e - ( i 6 A(2) = ^

+

l/2)(L1+g>

k (16+1/2) To t h e s e two v e c t o r s one adds a f i x e d n +

t o get t h e

t o t a l amplitude immediately behind r e g e n e r a t o r ( 1 ) as seen i n F i g .

11.

141

At first i n s p e c t i o n it w o u l d appear that the is sensitive to the sign of

r > c/> C/> m S CD 5

hj-

162

STRONG INTERACTIONS EXPERIMENTAL ASPECTS

G. Goldhaber University of California, Berkeley, Calif.

164

165

LECTURE I The L = 1 qq nonets I want to begin my talk by reviewing very briefly for you the present status of the well known Boson Resonances and what octets and singlets (or nonets) they fit into.

This will be just a very brief survey

so that I can define more precisely what I want to discuss later.

The main point is that the Bosons

which have been discovered so far fit into a definite pattern as has been pointed out by Gell-Mann and Ne'eman.

Their spin-parities and charge conjugation

quantum numbers are such as to correspond to some of the simplest states expected on the quark-antiquark model of Gell-Mann and Zweig.

If we treat the quark-

antiquark system as a fermion anti-fermion pair, then the parity of the system is given by P = ( - 1 ) L + 1 and L+S the charge conjugation by C = (-1)

.

Here L is

the angular momentum in the qq system and S the total quark spin. to the

SQ}

The first three Boson nonets correspond and

J

P 2 quark-antiquark systems.

This

In these lectures I concentrate on new data obtained since the 1966 Berkeley Conference. However, the selection is entirely on the basis of data available to me, and no attempt at completeness has been made.

166

1 2 is illustrated in the first figure. ' While the p s e u d o s c a l a r and vector nonets are in g o o d accord w i t h the q q system in++ an S state, in the pp case of the tensor nonet arises.

=2

a new

situation

Now we can no longer consider the q u a r k -

antiquark system to be in the L = 0 state, but m u s t go to the L = 1 3 p 2 state. However, once we admit the •3 P 2 state we m u s t ask about the other possible P "3 "3 I states, the

PQj

P

!>

a n d

states.

Figure 2 i n d i -

cates what the spectrum m i g h t look like. are the two S-wave nonets.

First

there

(In this figure the SU(3)

splitting is not shown and are r e p r e s e n t e d 3 1 the nonets by just one line.)

The

S q and

S

states, the tt and

p nonets, are split by a s p i n - s p i n coupling.

Then

there is a centrifugal potential w h i c h comes into play as L increases and gives a considerable b e t w e e n the L = 0 and L = 1 states.

splitting

These3 various

splittings have been e s t i m a t e d by Dalitz.

The L = 1

state is the one we are i n t e r e s t e d in for this d i s c u s sion and this is split into four distinct nonets by a spin orbit force.

If we look around for possible

bosons to c o r r e s p o n d to these states we know that the A 2 corresponds to the

J

P 2 state, as shown in Fig. 1,

a3 n d the B a n d the A^ might correspond to the 1 P ; L and P, states respectively. The q u e s t i o n m a r k s in the

167

figure indicate that in the case of B the spin parity h a s not yet b e e n r e l i a b l y m e a s u r e d and in the case of the A

we believe that we k n o w the spin parity but

are not so sure w h e t h e r the particle exists!

I will

present some arguments later favoring the existence of an A.^ resonance.

If we accept these assignments

porarily, t h e n we notice that there is roughly mass or r a t h e r mass squared splitting.

tem-

equal

This w o u l d

indicate spin-orbit splitting as opposed to a tensor force, as p o i n t e d out by D a l i t z .

Our job t h e n is to

find the nine members of each of these nonets.

As

indicated in Fig. 2, this m o d e l can continue to L = 2 and higher, and some evidence o b s e r v e d in the CERN m i s s i n g - m a s s - s p e c t r o m e t e r w o r k indicates that

such

h i g h e r levels m a y exist.

subject

I will discuss this

later. 1.

The K*'s of the L = 1 Nonets Let us now consider the K 's belonging to the four

L = 1 nonets.

I w i l l just consider the simplest

m o d e s , namely K u, Kp, and Kir.

The p r o p e r t i e s of

these four K * 1 s are g i v e n in Table I. already i d e n t i f i e d the we are

Of these we have

K * ( 3 P 2 ) as the K*(l420). ft

decay

Since

"I

suggesting that K ( P^) corresponds to the B *

o

m e s o n and K ( P ) to the A

m e s o n , let us call them

TABLE I The simplest decay modes, and names, of the four = 1 K*'s.

Decay ^\.Mode K

3

P

P

I

K*(

3

P

K*(

3

P

K*( K V

K*(890)TT

or Kp

)

/

)

/

1

)

•

0

)

2

—

KIR

/

Name

K*(1920)

—

Kb

—

kAi

/

169

K B and

•

There is h o w e v e r a delicate point here.

E a c h qq structure goes w i t h a definite value of C. q u e s t i o n is what does .this m e a n for K 's?

The

The m e a n i n g

for C here is supposed to represent the type of q u a r k , anti-quark structure we are dealing with.

(Incidental-

ly, there is a generalization of charge conjugation in SU(3) called unitary parity first i n t r o d u c e d by Dothan but I do not want to go into that subject h e r e . 4 )

If

we consider C just as an indication of the quark £

structure, t h e n because a K

is not an eigenstate of C

there can be particle mixing b e t w e e n the two K*'s. This is a new situation.

We are already well

acquaint-

ed w i t h m i x i n g b e t w e e n the two singlets in a nonet. See Pig. 3-

Here we have two n o n e t s , each presumably

h a v i n g m i x i n g b e t w e e n its own singlets.

But since the

nonets have the same spin-parity another type of m i x i n g , b e t w e e n the K 's of the two n o n e t s , can also occur.

Of course the isovectors A-j_ and B and the

isoscalars, w h i c h have not yet b e e n all i d e n t i f i e d w i l l not mix b e t w e e n nonets since for these G and C respectively are good quantum numbers.

Thus the

A1

decays into irp and finally to three pions, while the B decays into 2.

TTCJ

and finally to four pions.

The Structure of the

KTTTT

Mass

Enhancement

A very large enhancement in the

KTTTT

system has

170 been observed in a series of experiments in both K + p and K~p reactions for incident momenta from 2.6 to 13 GeV/c' ta ~ 12 . data.

Figures 4 and 5 show a sample from K + p , * In Fig. 4 we show our own data with K or p £

selection.

In Fig. 5 a K

selection is made and two

other momenta available to us are included.

The

enhancement lies roughly in the mass region 1.1 to 1.5 GeV.

Most experiments agree that the enhancement

consists of at least two phenomena: (a)

The K*(l420) in the K ™

decay mode.

This has

been clearly resolved from the main enhancement in a number of experiments. 8 ' 9

(See Fig. 5.)

As noted by

Morrison 13 the cross section for this resonance, which can be most readily ascertained from its KIT decay mode, decreases with P l a b the incident laboratory momentum as a ^ Piab

(b)

wi_t

h

n

=

2,2

±

0'2

A large enhancement roughly in the mass

region 1.1 to 1.4 GeV has been recently 6 observed and is called the "Q enhancement".

The structure of this

has shown distinct, statistically significant, differences for various incident momenta.

It is the latter

feature we wish to discuss here. In our earlier work, Shen et al. 8 with K + p at 4.6 BeV/c we noted that the K*(l420) stood out clearly and that the Q enhancement has definite structure,

171

namely at least one resonant p e a k — K wide b a c k g r o u n d — p r e s u m a b l y or "diffraction

(1320)—and a

due to the Deck effect

dissociation".

In other experiments the degree of structure varied. (a)

The 9 BeV/c K + p

experiment

In a very recent experiment by A. F i r e s t o n e , B. C. Shen and m y s e l f 5 we have o b s e r v e d a distinctly different structure of the Q peak.

See Figs. 4 and 5-

One thus has to try and e x p l a i n this v a r i a t i o n in structure w i t h incident momentum.' (i)

The experimental

resolution

Before g o i n g into the d e t a i l e d i n t e r p r e tations one must first convince oneself that the m e a s u r e m e n t s — p e r f o r m e d on the Berkeley Flying D i g i t i z e r ( F S D ) — h a v e the resolution to observe

Spot such

effects. To obtain an independent check of our knowledge of the bubble chamber optics and the m a g n e t i c field we have s t u d i e d separately the 2 - p r o n g e d events w i t h an a s s o c i a t e d vee. y

We fit the vees in these events to a

As was brought out in the discussions at this conference in p a r t i c u l a r the nature of the "diffraction dissociation" or Pomeranchuk exchange is not at all u n d e r s t o o d on a fundamental theoretical basis. On the other h a n d there is no doubt that experimentally some such process exists.

172

special two constraint fit using the measured variables of the two charged decay tracks interpreted as pions and the measured direction of the neutral decaying track, but leaving unspecified both the momentum and mass of the neutral track.

Fig. 6 shows the fitted

mass of the neutral track from this 2 constraint fit for the 2-pronged events with vee.

The central value

of the mass peak is calculated to be 497.88 ± 0.43 which is to be compared to the known mass of the K° meson of 497.87 ± 0.16 as compiled by Rosenfeld, et al. 2

The distribution is Gaussian and has a full

width at half maximum of 12 MeV.

The error for these

2C fits is thus ±6 MeV. (ii)

Identification and resolution for

1-constraint fits As an illustration of the reliability of our identification of events and the mass resolution, we show in Fig. 7(a) the ir+Tr-Tr° mass spectrum in 20 MeV intervals for events of the reaction: K+p

K+iT+Tr~iT0p

2196 events

(1)

We find that the w meson is clearly observed at 788 ± 5 MeV.

The observed experimental width of the

a) meson is approximately 35 MeV.

The shaded area £

corresponds to the distribution after events in the N band ( 1 . 1 6 BeV < M(PTT + ) < 1 . 3 2 BeV) have been removed.

173

In the investigation of the

K T T T T

system, we have

studied primarily the reactions: K+p

K+ir~ir+p

2572 events

(2)

K+p

K°Tr°Tr+p

4l6 events

(3)

The typical errors of the

K T T T T

mass region

1

.

1

-

1.5 BeV are 7 MeV for 4c - fit events of reaction (2) and 15 MeV for lc - fit events of reaction (3)(iii)

The momentum-transfer distribution

Figure 7b shows the distribution without £ any cuts and with N out (shaded). Figure 8 shows the y jf distribution with K in or p in and N out. The shaded K T T T T

£

part in Figure 8a represents the estimate of K (1420) contribution in the

K T T T T

mode based on the observed

K*(l420) decay in the KTT mode from the reaction K p -»• K u p. Two peaks centered at 1250 and 1400 MeV above a broad background can be clearly distinguished.

These

two peaks, are well resolved by a 60 MeV wide dip of about 3-2 standard deviations, and are respectively 7 and 6 standard deviation effects above background. We have further observed that the Kirir mass distribution appears to be a sensitive function of the momentum transfer to the recoil proton. is illustrated in Figures 8(b-d).

This feature

For events with

174 2

very small momentum transfer to the proton, A '

This implies that the

interference effects can vary w i t h t. (b)

(Here t=A

o P

.)

The K* Alignment

One other feature which follows is that the alignment of the two K*'s may also change with incident momentum as the relative strength of A p and A^ change. Thus at low momentum we would expect an alignment characteristic of a considerable component due to meson exchange which as the momentum increases should go over to an alignment characteristic of Pomeron exchange. There is experimental evidence for a change in the alignment of the structure in the Q enhancement.

Thus

185

in the K + p experiment 5 at 4.6 BeV/c it was observed that the K*(1320) peak corresponds primarily to a £

K.1II K out. scattering angle a in the K (890) cm system in the equatorial region, i.e., |cos a| < 0.8. Figure 20.)

(See

The same result can be noted for the peak

at 1270 MeV observed in a K~p experiment at 3.8 BeV/c by Field et al. 1 6

(See Figure 21.)

These results are o

thus indicative of an isotropic or sin a distribution for the respective peaks observed in the Q enhancement at these momenta.

On the other hand in the K + p

experiment at 9 BeV/c both peaks observed in the Q enhancement occur both in the equatorial and polar regions in cos a (see Figure 22). This corresponds to 2 + a cos a distribution, which, is indicative of 1 resonance formation by Pomeron exchange. 4.

Further Remark on K* Mixing As C is not a good quantum number for K*'s the

possibility of mixing exists. 1 7 » 1 8 the pure K*'s K A

If we again call

and Kg, the two physical particles,

say K x and KJJ, may each be a mixture of the 1

P 1 states. 17

These are related by 1 = K.

KT

K-J.-J. = - K a

cos 6 + lU -tl sin 8 sin 3 + Kg cos 3

J

P 1 and

186

where 3 is the m i x i n g angle b e t w e e n the quark singlet and triplet states.

spin

Until the KITTT mass

spectrum is fitted w i t h unique values for E 1 and E 2 we w i l l not have reliable values for MCKj) = E 1 M(K I ] : ) = E 2 , w h i c h are n e e d e d to give 6.

and

Present

indications are that g is small. 5.

The irp B o s o n Cluster in the A^, A 2 (a)

Region

Possible Interference P h e n o m e n a R e l a t e d w i t h

Aj F o r m a t i o n In Figures 23-28 we see the irp mass

distribution

for TT±P experiments at a series of incident

momenta.1

We note that in the low m o m e n t u m region an A-^ signal is clearly visible.

H o w e v e r , this appears to v a n i s h above

5 B e V / c and reappear possibly at 8 but certainly at 11 BeV/c.

If this p h e n o m e n o n represents a real

p h y s i c a l effect (rather t h a n statistical

fluctuations)

we may be observing an interference effect here as well. One other p h e n o m e n o n w h i c h is beginning to show up is a s u g g e s t i o n of m o r e s t r u c t u r e — t h e the r e g i o n 1160 to 1200 MeV.

so called A, —in -L » J

Some such

effect—although

not always at exactly the same m a s s 1 9 ' 2 0 — c a n be n o t e d in Figures 25, 26 and 27.

The origin of this effect

is completely obscure at p r e s e n t .

If b o t h A 1 and A 1 ^

t u r n out to be distinct resonance effects then we may have contributions from other than the L = 1 qq state

187

in this boson cluster (see next lecture for definition). (b)

The A^-Background Interference Calculation

To gain some insight into the question, what interference effects with background would be like, I have computed the following model: dM

a

( I G + B 1- . E 1 * | 2 +

|BP|2)P

,

where G represents a uniform background term.

X have

performed the numerical evaluation of this expression with E 1 = 1080 MeV, r 1 = 130 MeV, G = 1, E £ = 1300 MeV, and r 2 = 80 MeV. 1080 MeV.

For = 0 there is an "A1 peak" at

The peak wanders down to 960 MeV for

4> % 3w/5; and disappears completely for 4» Ri 4ir/5.

It

then reappears at 1 1 ^ 0 MeV for 4» ^ IT and moves back to 1080 MeV as -»- 2IT.

(See Figure 29.)

If this is

indeed the correct interpretation of the A^ behavior with suitable variation of ^ as a function of incident momentum, we must still ask about the origin of G. The background irp production is presumably due to Pomeron exchange (or Deck effect) which should lead to

188

a predominantly J p = 1 + (c)

state.*

Possible Structure in the A ?

Peak

A p h e n o m e n o n that is just as o b s c u r e — a n d

again

does not fit into the L=1 B o s o n cluster i d e a — i s

the

splitting of the A 2 peak o b s e r v e d by Chicovani et al. in the CERN m i s s i n g m a s s s p e c t r o m e t e r The experimental evidence looks

experiment.21 good—they

observe a 6 standard d e v i a t i o n dip in the center of the A 2 peak.

There is one small point one

perhaps worry about.

should

The runs from different

years

were combined by super-imposing the respective central mass values.

A2

H o w e v e r , as the authors point

out, since the absolute value of the b e a m m o m e n t u m is only k n o w n to 2%,

the apparent shifts

appear

*This has b e e n r e - e m p h a s i z e d by M. Ross and Y. Y. Yam [Phys. Rev. Letters 1£, 5^6 (1967)3 who stress that the b a c k g r o u n d should consist of three effects: (a) The Deck effect; i.e., p i o n scattering at the n u c l e ó n vertex. (b) The inverse Deck effect; i.e., p scattering at the n u c l e ó n vertex. (c) P o m e r o n exchange. They find -from their calculations that these effects could be comparable and that the three amplitudes m u s t be added to give the kinematic background. My assumption is that the A^ resonance p h e n o m e n o n i n t e r feres w i t h this rather complex J p = 1+ b a c k g r o u n d effect. The phase angle $ m a y then be a function of the incident m o m e n t u m as w e l l as t. It m a y also be i n f l u e n c e d by other r e s o n a n c e s p r e s e n t such as p o s sible "catalytic" effects due to the N* band. (See Ref. 1 for definition.)

189

reasonable. The combined data is shown in Figure 30.

The

authors interpret this data as: (a)

two incoherent resonances at

or (b)

M x = 1.275 (±0.016)GeV T ± = 0.020

(±0.01)GeV

M 2 = 1.318 (±0.016)GeV r 2 = 0.020

(+0.01)GeV

two coherent r e s o n a n c e s — o r a "dipole,"

i.e., two resonances with equal amplitudes and widths N(M) -x,

r

{(M-M0)

with M 0 = 1.296 r

= 0.028

X 2

V + 3ur 2 } 2

(±0.0l6)GeV (±0.003)GeV.

The single resonance hypothesis which ignores the dip o gives only a 1% x

probability.

In this connection M o r r i s o n 1 3 has concluded primarily from a study of the production cross section of A 2 — w h i c h he finds follows a a ^ P^ab n = .5

±

law

.2 that these may be two distinct objects

in this p e a k — a 2 + resonance as well as another one with 2" or 1 + . 6.

What about the J p c = 0 + + states (a)

3

P0?

The Isoyector

Here we have the threshold KK I = 1 state as a candidate.

The decay angular distribution indicates

190 J P =0 + .

(See Figure 31.)

The CERN-PARIS Group 22 finds

three possible solutions for this enhancement. Figure 32.)

(See

At present they all fit about equally

well.

a = 2.5

±

(i)

A positive (real) scattering length

IF.

This implies a non resonant nature for

the enhancement. (ii)

A complex scattering length a + ib

which turns out essentially real and negative a = -2.3F + •-> and b = 0 QF. This implies a Dalitz-Tuan type bound +lc> state at 975 ^Q MeV which is very narrow because of the small imaginary part.

It is tempting to associate this

with the 6(96^1) which could then decay via TTTI.

Here,

however, some caution about the 6 as an established phenomenon is still in order. (iii)

A resonance just above threshold in

the KK system E = 1016 ± 10 MeV. Thus for cases (ii) or (iii) this phenomenon could represent the 0 + + isovector.*

Very similar results

were observed earlier by the Yale Group. 23

(See

*It was emphasized by L. Kirsch (Brandeis) during the discussion, however, that aside from the K* enhancement the Dalitz plot has two symmetric enhancements at low Kir masses. Thus the possibility of a specific matrix element being responsible for all the low mass enhancements cannot be completely ruled out.

191

Figure 33.) (b)

The K*

If we assume the same isodoublet-isovector (mass)2 difference as observed for the K*(l420) and

(i.e.,

the well-known member of the L = 1 multiplets), %

2

A = 0.29 (BeV) , then we can predict the location of the K*(3p Q ) which should decay via KIT. for case (ii) we get M(K*( 3 P 0 )) = 1105 MeV for case (iii) we get M(K*( 3 P Q )) = 1150 MeV So far no such peak has been observed, although a very slight enhancement at 1080 MeV has been reported.1 At any rate, no relation to k(725). (c)

The Singlets (i)

There is a mounting body of evidence

that the asymmetry in the p° decay is indeed a J p c = 0 + + resonance. 24 ' 25

See Figures 3^ to 37•

Here

Figure 31* shows the analysis of Walker et al. for the •mr system.

They get one set of phase shifts which

passes through 90° at around 750 MeV. this set to be unlikely. at 850-950 MeV.

They consider

Another set approaches 90°

They favor this solution.

Figures

35 to 37 show the data of Malamud and Schlein.

They

have carried out a moments analysis, and again find two phase shift solutions which pass through 90°, now, however, in the region of 730 MeV and one set of

192

solutions w h i c h hovers in the r e g i o n of 90° throughout and w o u l d thus not c o r r e s p o n d to a resonance.

Thus in

b o t h cases there appears to be evidence for a resonance somewhere in the 730-900 M e V r e g i o n , a l t h o u g h the individual analyses differ in the exact location.

Just

in general it may be well to b e a r in m i n d that we should still keep some reservations about the validity of off-shell phase shift (ii) again 1

analyses.

The K K I = 0 t h r e s h o l d state.

Here

it is a q u e s t i o n of e i t h e r a b o u n d state, or a

resonance in the p h y s i c a l r e g i o n w i t h ECK-j^) = 1068 M e V T i K j K ^

= 80 MeV.

Thus while there are still many doubts about most of these states at least we know w h i c h avenues to pursue in identifying a n d e s t a b l i s h i n g the 0 + +

nonet!

REFERENCES Gerson Goldhaber, Rapporteur Talk, "Boson Resonances," in Proceedings of the Xlllth International Conference on High Energy Physics, Berkeley, September 1966 (University of California Press, Berkeley and Los Angeles, 1967) page 103• This article contains references to the original papers.

No attempt has been made to duplicate

these here. Arthur H. Rosenfeld, Angela Barbaro-Galtieri, William J. Podolsky, LeRoy R. Price, Paul Soding, Charles G. Wohl, Matts Roos, and William J. Willis, Rev. of Mod. Phys. 39, No. 1 ( 1 9 6 7 ) . R. H. Dalitz, Rapporteur Talk, "Symmetries and the Strong Interactions," in Proceedings of the Xlllth International Conference on High Energy Physics, Berkeley, September 1966 (University of California Press, Berkeley and Los Angeles, 1967)> page 215.

See also in this volume.

The precise definition and references are given by B. W. Lee in High Energy Physics and Elementary Particles, 1965 Trieste Seminars

(International

Atomic Energy Agency, Vienna, 1 9 6 5 ) . G. L. Kane, Phys. Rev. 156, 1738

See also

(1967).

194

4a.

Jerome H. Friedman and Ronald R. Ross, Bull. Am. Phys. Soc. II 12, 539 (1967).

4b.

C. Y. Chien, P. M. Dauber, D. J. Mellema, P. Schreiner, W. E. Slater, D. H. Stork, and H. K. Ticho, Bull. Am. Phys. Soc. 12, 506 ( 1 9 6 7 ) , and Preprint.

4c. 5.

R. Zdanls, Bull. Am. Phys. Soc. 12, 567 (1967). G. Goldhaber, A. Firestone, B. C. Shen, Phys. Rev. Letters 19, 972 (1967).

6.

J. Berlinghieri, M. S. Farber, T. Ferbel, B. Forman, A. C. Melissinos, T. Yamanouchi, and H. Yuta, Phys. Rev. Letters 18, 1087 (1967).

7.

T. Ludlam, J. Kim, J. Lach, J. Sandweiss, and H. Taft, Bull. Am. Phys. Soc. 12, 540 ( 1 9 6 7 ) .

8.

B. C. Shen, I. Butterworth, C. Fu, G. Goldhaber, S. Goldhaber, and G. H. Trilling, Phys, Rev. Letters 17, 726 (1966).

9.

Drs. Y. Goldschmidt-Clermont, R. Jongejans and V. Henri - CERN - Private Communication on the data from the Bruxelles-CERN Collaboration.

10.

J. Bartsch, et al. (Aachen - Berlin - CERN Imperial College (London) - Vienna Collaboration) - Phys. Letters 22, 357 (1966).

11.

T. P. Wangler, M. Derrick, L. G. Hyman, J. G. Loken, F. Schweingruber, R. G. Ammar,

R. E. P. Davis, W. Kropac, J. Mott, and J. Park, Bull. Am. Phys. Soc. 12, 5^0 (1967). 12.

P. J. Dornan, V. E. Barnes, B. B. Colwick, P. Guidoni, G. R. Kalbfleisch, R. B. Palmer, D. Radojicic, D. C. Rahm, R. R. Rau, C. R. Richardson, N. P. Samios, I. 0. Skillicorn, B. Goz, M. Goldberg, T. Kikuchi, and J. Leitner, Bull. Am. Phys. Soc. 11, 3^2 (1966).

13.

D. R. 0. Morrison, CERN Preprint D. Ph. II/Physics 67-15» Submitted to Physics Letters.

14.

C. Y. Chien, UCLA Physics Note (1967) (unpublished) .

15.

K. Barnham, D. Colley, L. Riddiford, J. Beaney, I. Hughes, I. McLaren, R. Jennings, M. G. Bowler, R. J. Hemingway, E. J. Sumner, Preprint submitted to 1967 Heidelberg International Conference on elementary particles.

16.

J. Field, T. Hendricks, 0. Piccioni and P. Yager, Phys. Letters 24B, 639 (1967).

17.

This was first pointed out by R. Gatto, L. Maiani and G. Preparata, Nuovo Cimento 39, 1192 (1965).

18.

Here it must be stressed that in view of the new experimental evidence (Ref. 5) and the present arguments, the earlier assignment (Ref. 1) of

A

and K (1320) to a single nonet is no longer

valid. 19.

N. M. C a s o n et al., Notre Dame, Phys. Rev. Letters 18, 880

20.

(1967).

C. Baltay et al., C o l u m b i a - R u t g e r s , Bull. Am. Phys. Soc. - 1967 New York m e e t i n g .

21.

G. Chikovani et al., CERN, Phys. Letters 44

22.

25B,

(1967).

A. A s t i e r , J. C o h e n - G a n o u n a , M. D e l l a Negra, B. M a r é s h a l , L. M o n t a n e t , M. T h o m a s , M. B a u b i l l i a , J. Duboc, CERN, P r e p r i n t , June

23.

1967.

C. Baltay et al., Yale, Phys. Rev. 142, 932 (1966).

24.

W. D. W a l k e r , J. C a r r o l l , A. G a r f i n k e l , and B. Y. Oh, Phys. Rev. Letters 18, 6 3 0

25.

(1967).

E. M a l a m u d and P. E. S c h l e i n , Phys. Rev.

Letters 19, 1056 ( 1 9 6 7 ) .

197

J =0" Nonet ' S o ( q q ) C =+l,0o=IO° 1.0 r

Vector J =1" Nonet (q q), C=-l, 0| ? 4 0 *

V*

% 0.5 DD

*

^> 1.0 a> CD - 0.5

V

K

K* a>

7T -1/2 0 Iz

0 Iz

1/2

J p = 2 + Nonet 3 P 2 (q q ) , C = + l, 0= 3 0 e 2.5 CVI > a> 2.0 GO

1.5

f =

= — ~~ 0 Iz

Figure 1

(1500)

K (1415) — +1

A2(I320) f ° (1254)

198

q q Distributions

model of

nonets

( Not to scale ) S U 3 s p l i t t i n g not shown. Nonets

L = 2

Assignments

E3!

R.

, 2 D2

R,

3

R,

D

Spin-orbit repulsive potential

D,

Centrifugal p o t e n t ia I

L=I

A'

j'

A 2 ( 1310) B (1210)? A , ( 1080)?

3p

. Spin-orbit splitting

Centrifugal potentia I

L=0 7T

Spin- spin splitting

XBL677-3466

Figure

2

199

K*

MIXING

BETWEEN

NONETS

S

l+ "

P,

THE

TWO

'P,

l + NONETS

Figure 3

M

(

K t t t t

)

i n

B e V

Figure 4 X B L 6 7 B - 377I

201

F 1 — 1 K 7T° 7T+ p

K*

in , N* + + out Berkeley - L R L

K + p at 4.6 BeV/c ( 865 events)

2.0

2.4

Bruxelles - C E R N K + p at (2116

1.6

5 BeV/c events)

2.0 2.4 Johns Hopkins K + p at 5-5 BeV/c K*°in , N* ++ out (535 events)

iApA

2.0 2.4 Berkeley — L R L K + p at 9 B e V / c ( 7 1 8 events )

0.8

1.2

1.6 M

2.0

( K tttt)

Figure 5

in

2.4 BeV

2.8

202

1 1 1 1 1 1 1 11 | l

300

K^p —

i 1 i

2 Prong + Vee

| i i _

9BeV/c

1956 events (a)

250

—

MKo=497.88-0.43 MeV > a>

200

-

evi 150

—

c a >> UJ

12 MeV -> Experimental width

100

—

50

0 0.40

1 i 0.50

0.45 MÌtt+tt-)

IVw-I^i—u_ 0.55

(BeV) XBL678-3699

Figure 6

203

1 1 I I r K+p—-V+Tr+TT-Tr0 p 2196 events 9 BeV/c out , 1575 events

0.8

1.2

1.6

M(Tr+7r-7r0)

2.0

2.8

(BeV)

T

(b)

0.8

K+p — K* wz 7T+ p 9BeV/c 2988events 0N*++out, I 946 events

2.4

M(Kirir)

3.2

(BeV)

XBL677 - 3662

Figure 7

9 BeV/c K+p —-K*irsu-+p

p or K * ( 8 9 0 )

in,N*++out

XSL «77-36 5 »

Figure 8

205

K+p

— ^ K ^ t t ' t t + P

9 BeV/c

XBL677-3664

Figure 9

9 BeV/c

40

K+p — k V t t + p IV II III 1 i i+ 1 K , K 1 Kinc ,

-(c)'

i^i^cm

49

130

v/ V V a

Cos

- Kiric

b

Oh 5

|

i

p

i

(e)

L

m

J

71

I

T

i

i

e

cos

1

|

K

/k*°(8 90)c.m. 7r

20

0

•

, N*++out

m.

^K*°(890)

20

i

52 events

123

(d)

V

K*° (890) ir

-

20

i

i

i 0

hsJ1

o

i

r-i o Cos £

Figure 10

" V v u i i

r

o

WL i'-i o

i

207

/K> «i (¿VV*

208

UCLA

k+p

7.3 BeV/c

K

+

p - K V - 7 r

+

p

co UJ

ir u co

ir o

co UJ

oc ILI co

cc o z

CD
bJ

AT 8 . 4 1 ALL

j.1280 l

BeV/c T EVENTS

3849 EVENTS

À

60

LpiIWI

ni

40 20

500

1000

ce. ui >a 3

+ TT~

M

80

2

+

r 1090 lj>90

120

ì

P+7T++7T

.

Columbia - Rutgers

I

I

1500

2000

MASS 100

T b)

eo

1090

2500

3000

OF77- + 7r + 7r"MeV X

.1280

II

N * + + REMOVED pSELECTED 1169 EVENTS

60 40

20 r-Tlrui n-n.-

0 500

1000

1500

2000

MASS OF p° 77"+ MeV

Figure 26

2500

3000

225

C a s o n et al.

(Notre Dame)

8.0 BeV/c

566 EVENTS

Figure 27

226

7r7p "l

11 GeV/c

Genoa - Hamburg - Milan - Saclay

rr~p — > TT~p t t * it" 758 EVENTS

p° IN

3A

3.0

M {Y^TTTf]

3S GEV/c

T t o -oTÌD + NEUTRALS

1491 EVENTS

finn j

1.0

»

•

14

«.„ « 18

«

•

2.2

Figure 28

I

*_ 26

I

I 30

I

1

• In in.n.flI 3.4 3ß

M ^rr+NEUTRAl.s"] CEV

2

227

A, - Background

Interference

t,

10 so, Et=. I loo / i> So ff-l

HCxrr;« XBL 677-4283

Figure 29

228

CERN

Missing Mass Spectrum

300-

TOTAL A 2 280May 1968 ( 6 G e v / c ) Oct. 1965 (7 Gev/c) Jan. 1967 (7 Gev/c)

260-

240-

220-

5 LO

200-

180-

ff)

Ld > UJ

160-

140h

1.150

1.200

1.250

1.300

Figure 30

1350

1.400 M x 1 . 4 5 0 GeV

229


• A + A

31

p

+ p

32

TT+

+ p

33

1,-

+ p

->• N * °

34

K+

+ p

-

35

K"

+ p

->-

36

K"

+ p

-

+ p

p

-I- P7I + 7T°

+

N*(1238)

+ p -*• A + K ° O r +

37

K"

38

( , "

+ p - > E

+

+

+ p

-0.7

5.3

+0.6 •0.4

0.78

1.3±0. 6

0 . 90 + 1 . 9 -0.61

-0.05

1.6±1.4

O.23 + 0 . 8 7 -O.18

2.08

8.0

1.0

+ 1.2 -0.6

2.l±0.5

0.10 + 0 . 1 3

2.6+0.5

0 . 0 4 ^

+1.0 -0.6

1.9+0.3

0.15

+2.4

2.8±0.3

0

+2.4 -1.1

2.0+0.5

0.24

2.2±0.2 5

0.11 +0.09 0.05

1.9±0.3

0.11

1.8+0.4

0.09

3.8+0.8

0 . 0 2 8 « ; ° 3 7

4.1+0.7

°-028_S:gIi

3.5+0.3

°-o26-o:O°?2

+0.6

3.1

2 . 24

3-5

2.2

2.24

10.1

1.2

3.0

6.94

0.9

3.0

6.94

0.6'

2.24

3.5

1.9

1.65

10.1

2.5

5.0

+1.0 1.2

+

1.8+0.2

+ 5.0 1.0

10.1

AT°

-0.17

+0.06

1.3

2.24

ti"

0.30 + 0 . 3 9

4.65

u

+

1.3±0.6

3.0

1.2

""

0. 34

15

10.1

K+

+0.32 -0.17

1. 1±0.2

6

2.24

0.3

-1.1

+1.0 -0.5

+1.0 -0.5

+ 0.8 -0.4

-0.05

+ 0.14 -O.O7

14+0"04 -0.03

+0.26 0.13

+ 0.12 0.06 + 0.12 - 0 . 05

P° p°

+

K*°

ir +

Y*"(1385)

+

, - )

t

4

2.24

-i- H" + K + +

K°

5.0

1.2 + 1 . 7

7T°

+

+

8.0

0.8

or

N*++

1.0

1.1 -0.5

1 +

8.0

+

(1385)

N *+ +

8.0

cont'd

2.08

+

•>• I l °

+ p

S°+

K+

Y* ( 1 3 8 5 )

+

2.08

1.73

K°

V

i g o 0

+ 2.4

1.1 + 3.0 -1.3 -0.5

325

Hadron Spectroscopy

R. H. Dalitz Department of Theoretical Physics, Oxford, England

326

327

1. I n t r o d u c t i o n - the Arguments for Hadronic Substructure. The r a p i d g r o w t h of our list of

"strongly-

interacting p a r t i c l e s " , or "hadrons", as they have come to be called, is one of the m o s t striking of high-energy physics at present.

aspects

This g r o w t h is

w e l l i l l u s t r a t e d by reference to the

successive

editions of the table of elementary particles

and

their p r o p e r t i e s , p r o v i d e d by A. H. R o s e n f e l d and his collaborators^, w h i c h have n e e d e d

substantial

r e v i s i o n and e x t e n s i o n annually.

We are now

faced

w i t h the p r o b l e m of establishing or r e c o g n i z i n g

some

order governing the existence and p r o p e r t i e s of this increasingly complex array of hadronic

states.

Over m o r e than a d e c a d e , we have learned to expect that hadronic states should appear as isospin m u l t i p l e t s , groups of (21+1) charge states w i t h essentially the same m a s s , for total isospin I. This is the property of charge independence,

first

e s t a b l i s h e d for the strong interactions in n u c l e i , and now e s t a b l i s h e d r a t h e r generally for all strong interactions.

328

In recent y e a r s , since about 1963, we have l e a r n e d further that the i s o s p i n m u l t i p l e t s w i t h the same spin-parity

(J,P) appear g r o u p e d in larger,

t w o - d i m e n s i o n a l patterns w h i c h c o r r e s p o n d to the SU(3) unitary symmetry.

E a c h unitary m u l t i p l e t

consists of a c h a r a c t e r i s t i c array of isospin m u l t i p l e t s whose i s o s p i n is I and whose

hypercharge

is Y, in correspondence w i t h a p a t t e r n of combinations w h i c h is t y p i c a l of some r e p r e s e n t a t i o n of the group of SU(3)

(I,Y)

definite transformations.

The mass values for these i s o s p i n m u l t i p l e t s are not the same, but generally span a range of about 0.5 GeV; this already indicates the existence of m o d e r a t e l y strong forces w h i c h are charge independent but w h i c h do not obey this SU(3)

symmetry.

Each SU(3) r e p r e s e n t a t i o n is

characterized

uniquely by two integers (p,q), w i t h p ^ O , q*0. The net m u l t i p l i c i t y

a of p h y s i c a l l y distinct

charge

states contained w i t h i n this SU(3) m u l t i p l e t is g i v e n by a = (p+1) (q+1) (p+q+2)/2,

(1.1)

and it is customary to characterize each r e p r e s e n t ation simply by the n o t a t i o n {a}, at least for the simpler representations

since each of these

corresponds to a different value for a.

The only

329

confusion is between the representations

(i,j) and

(j,i), since these have the same value for a; the notation {a} is used for the representation with q>p,{ci} for the representation with q Air), and there are likely to be

selecti o n r u l e s , a c c o r d i n g to w h i c h some

are f o r b i d d e n

transitions

(as a p p e a r s to be the case f o r A(1236) ->

332

N(938)y(E2), and

pir).

If the hadron states are

found to be grouped into larger supermultiplets, this may lead to some quantitative understanding of the systematics of these transition processes. for two decay processes A ->• BC and A'

For example,

B'C', where A

and A' belong to a supermultiplet a, B and B' to supermultiplet

and C and C' to supermultiplet

will generally be some their amplitudes.

th.ere

numerical relationship between

In this way, the number of indepen-

dent decay amplitudes may be greatly reduced, in consequence of the dynamical symmetries expressed most directly by the existence of these supermultiplets. The present data on the low-lying hadronic levels leads to the qualitative conclusion that the hypothesis of SU(6) symmetry for the strong interactions (due first to Gursey and Radicati

8

9 and to Sakita ) provides

quite a good first approximation to the facts.

For

example; (i) the patterns for the established levels fit SU(6) symmetry quite naturally.

The low-lying baryon

states with positive parity are the octet of 3 baryons and the decuplet of

excited baryons:

altogether, these multiplets include 56 particle states with definite quantum numbers (Y,l3,J z ), which correspond precisely with the basis for the ^-representation

333

of SU(6) symmetry.

Th.e low-lying meson states of

negative parity are a mixed singlet and octet of (1-) mesons, and an octet of (0-) mesons, together with a further (0-) meson n'(959) standing rather apart from the (0-) octet; the first multiplets fit directly into interpretation as a

representation of SU(6) symmetry,

the n'(959) being interpreted as an SU(6) singlet. (ii) the electromagnetic properties of these hadron states agree qualitatively with the expectations from SU(6) symmetry.

For example, the proton and

neutron magnetic moments were predicted by Beg et al.

10

to be in the ratio y /y = - 3/2, in good accord with P n the facts; the M 1 amplitude for the transition yp -> A + (1236) was expressed 10 in terms of y , in rough accord with the photo-production data 1 1 , while the corresponding E2 amplitude was predicted to be zero by 12

Becchi and Morpurgo

, as is found experimentally.

This agreement provides us with a puzzle.

SU(6)

symmetry is not based on the total spin J , but on the internal spin S z , whose group symmetry is that of the Pauli spin, the SU(2) group.

The SU(6) symmetry

combines the two-dimensional space of Pauli spin with the three-dimensional space of SU(3)j to form a sixdimensional space of SU(3), to form a six-dimensional space in which the SU(6) transformations operate.

334

SU(6) symmetry requires invariance w i t h respect to all unitary t r a n s f o r m a t i o n s involving simultaneous

changes

in F.^ (the 8 - d i m e n s i o n a l "unitary spin", consisting the g e n e r a t o r s of the SU(3) group) and i n a^ 3-dimensional

of

(the

"Pauli spin", consisting of the g e n e r a -

tors of the SU(2) group in spin space).

The

difficulty

is that the cr-transformations c o r r e s p o n d only to r o t a t i o n a l t r a n s f o r m a t i o n s in 3 - d i m e n s i o n a l space; they do not correspond to the w i d e r group of Lorentz

trans-

formations, w h i c h is a p p r o p r i a t e only to the n o n relativistic

regime.

It was an early hope that SU(6) symmetry t u r n out to be a n o n - r e l a t i v i s t i c

would

limit of some w i d e r

exact symmetry corresponding to invariance w i t h respect to some group of t r a n s f o r m a t i o n s w h i c h contains all the t r a n s f o r m a t i o n s L(4) x SU(3), where L(4) denotes the i n h o m o g e n e o u s Lorentz group of space-time tions.

transforma-

It has now become r e a l i z e d that this is not

easily p o s s i b l e . b r i e f l y , as

The difficulties m a y be

summarized

follows: 12,13

(i) the M i c h e l - 0 ' H a i f e r t a i g h t h e o r e m

states

that invariance w i t h respect to any symmetry w h i c h contains L(4) x SU(3) as a subgroup

group

necessarily

requires that the e n e r g y - m o m e n t u m v e c t o r have more t h a n four

components;

335 3.1+ 15 (ii) the Unitarity theorem.

'

The

(i) can be avoided by combining the four spinor indices w i t h the SU(3) indices

difficulty

relativistic

(the n a t u r a l

attempt to use r e l a t i v i s t i c spin) and by

confining

a t t e n t i o n to those t r a n s f o r m a t i o n s for this

combined

index w h i c h are independent of the m o m e n t a . a.

However,

invariance w i t h respect to all the (11(12) or SL(6c)) t r a n s f o r m a t i o n s of this 12-dimensional index

leads

n e c e s s a r i l y to a conflict w i t h the unitarity

condition,

unless the p a r t i c l e m u l t i p l e t s all have dimensionality.

infinite

l6

(iii) the Causality theorem.

If these

dimensional m u l t i p l e t s are d e g e n e r a t e

(i.e. the group

symmetry holds exactly) t h e n the requirements causality forbid F e r m i statistics

infinite-

(i.e.

of

anti-commutation

relations for the particle fields) for all p a r t i c l e s , irrespective of their

spin.

These difficulties suggest that there is no relati.yis^ tic v e r s i o n of SU(6) symmetry w h i c h is a c c e p t a b l e terms of these basic p h y s i c a l principles.

in

It appears

that only symmetry group we can insist on in the relativistic regime is the p r o d u c t group L(4) x SU(3). This s i t u a t i o n leads r a t h e r directly to the n o t i o n that the SU(6) symmetry o b s e r v e d m a y be, instead, an approximate symmetry w h i c h happens to be v a l i d only

336

for the low-lying hadronic states, these being

composed

of some subunits w h i c h are tightly b o u n d together but w h i c h move w i t h n o n - r e l a t i v i s t i c i n t e r n a l m o t i o n s .

The

m o t i v a t i o n s u n d e r l y i n g this picture are as follows: 1. The h i g h m u l t i p l i c i t y o b s e r v e d for the states.

hadronic

This m a y be a c h i e v e d most readily by the

e x c i t a t i o n of some internal degrees of freedom, corresponding to some substructure for these

states

(just as the h i g h density of m o l e c u l a r levels, for e x a m p l e , arises from the r o t a t i o n a l and v i b r a t i o n a l excitations). 2. SU(3) symmetry appears to h o l d r a t h e r w e l l , despite the fact that there are large

SU(3)-breaking

m a s s differences o b s e r v e d for the low-lying

hadronic

states, especially for the m e s o n s P (where m ^ / m ^ ^ 3.5, the m a s s difference being ( m K - m w ) % 350 MeV) and the baryons B (where ( m _ - m N ) £ 380 MeV).

In fact:

(a) the m a s s e s of the i s o s p i n m u l t i p l e t s

which

m a k e up a g i v e n SU(3) m u l t i p l e t appear to follow quite well the r e g u l a r i t i e s of the G e l l - M a n n - O k u b o

mass

IT,18 , formula

(which is based on a first-order

treatment

of an SU(3)-breaking i n t e r a c t i o n L M g w h i c h is the I=Y=0 m e m b e r of an octet), as far as this can be t e s t e d at present

(i.e. for the B octet and the D d e c u p l e t ,

and for the P and V(vector) m e s o n s ) .

To judge by the

337

corrections second-order in L M g w h i c h are n e e d e d to fit the B and D m a s s e s a c c u r a t e l y , the p a r a m e t e r m e a s u r e s the effect of this S U ( 3 ) - b r e a k i n g

which

interaction

appears to be no more than 1/10. (b) the p a r t i a l decay widths for hadronic decayprocesses

(such as the decays V ->- PP and D -* BP) fit

the SU(3) formulae quite w e l l , after allowance is m a d e for the large differences in final phase space and in centrifugal b a r r i e r p e n e t r a t i o n effects w h i c h result from the SU(3)-breaking m a s s differences b e t w e e n the various m e m b e r s of e a c h unitary m u l t i p l e t . that the interior w a v e f u n c t i o n s for these

This m e a n s particle

states m u s t be in quite g o o d agreement with the ture p r e d i c t e d by SU(3)

struc-

symmetry.

These features of agreement w i t h SU(3) can be u n d e r s t o o d r a t h e r directly if the

symmetry

observed

states result from the tight binding of very m a s s i v e sub-units, t h r o u g h super-strong interactions w h i c h are S U ( 3 ) - i n v a r i a n t and of short range.

In this

situation,

the m o d e r a t e l y - s t r o n g S U ( 3 ) - b r e a k i n g interactions be a small p e r t u r b a t i o n relative to the

would

super-strong

i n t e r a c t i o n s , w h i c h are r e q u i r e d to produce s u c h high binding as to cancel almost all of the rest m a s s of the constituent sub-units. It w o u l d t h e n be j u s t i f i e d to treat the SU(3)-breaking interactions by

first-order

338

perturbation theory, which would account success of the G e l l - M a n n - O k u b o mass In this situation, the

for the

formula.

interior w a v e f u n c t i o n ,

the region of the super-strong

interactions,

then still correspond to definite S U ( 3 )

in

would

representations.

In the exterior region, the wavefunctions

necessarily

violate S U ( 3 ) symmetry b a d l y , owing to the large P and B m a s s differences. channels may be closed

Some of the

SU(3)-equivalent

at the resonance energy,

while

others may have high Q - v a l u e , comparable w i t h the m a s s of the initial state.

For example, the

A ( 1 2 3 6 ) has two SU(3)-equivalent ttN and KZ. threshold

The ttN threshold

meson-baryon

at 1685 M e V ; at 1236 M e V , the exterior

3

1=2", Y=+l decuplet

discussed

channels,

is at 1080 M e V , the KL

function necessarily has the structure

for exact

resonance

19

irN, w h e r e a s

state has the structure

SU(3) symmetry.

As Oakes and Yang

in the exterior region may

m i x i n g of different

SU(3)-representations

interior w a v e f u n c t i o n .

the

(-ttN + K £ ) / / 2 , have

, the question is whether or not this

SU(3)-breaking

wave-

strong

lead to

strong

in the

We shall consider this

question

20 here in terms

of a simple m o d e l

Let us suppose that the super-strong

interactions

are confined to an interior region, of radius a. interior w a v e f u n c t i o n s may be characterized

by the

The

339

boundary

condition: Ka)

= aR*(a)

(1.5)

where R is a m a t r i x in channel space.

The m a t r i x R

characterizes completely the p r o p e r t i e s of the interior system.

Since SU(3) holds there, R is d i a g o n a l

w h e n the r e p r e s e n t a t i o n basis is changed from the channel states to SU(3) e i g e n s t a t e s , the e i g e n v a l u e the r e p r e s e n t a t i o n

for

{a} b e i n g d e n o t e d by R(a), the

corresponding e i g e n f u n c t i o n by (p')dV

2M2 (E2-p2-HM2)

^(¿lM 2 +p 2 )

^

/ V(p,£')i(p')dV-

2 T a k i n g the n o n - r e l a t i v i s t i c

2

limit p /M

f i n a l l y the a p p r o x i m a t e t w o - p a r t i c l e

2 G e V for this case.

Of course, these

limit

experiments

are severely l i m i t e d in the energy available and w o u l d be quite ineffective if the 3 m a s s were m u c h above 5 GeV. Cosmic ray experiments have the advantage

of

looking at the p r o d u c t s of collisions at m u c h h i g h e r energies flux).

(but w i t h a very m u c h smaller incident M o s t of these experiments are simply

proton

ionization

m e a s u r e m e n t s for the cosmic ray p a r t i c l e s incident the apparatus, and give an estimate

(or upper

on the flux of such p a r t i c l e s w i t h some charge.

on

limit)

specified

Some results from recent experiments are now

27

listed

.

The first four results r e f e r to the

g e n e r a l cosmic radiation; the experiment of Peters et al. is t r i g g e r e d by large air showers, and is sensitive to particles of charge

|e| arriving at the

d e t e c t i o n system w i t h a time delay (of order 100 ns)

,

348

Charge Value

Flux Limit ptles/ 2 cm.sr.sec.

2e/3j

• UTT width requires f = 5G/MU. Mitra and Srivastava^^ have fitted the L = 1 i p meson widths by including a form factor exp(—i- ak ) and by treating the s- and d- radial integrals as

arbitrary parameters.

This fit is possible

because

the k^ dependence can be b a l a n c e d by a strong

form

factor to fit b o t h A-j_ and A 2 widths at the same time. H o w e v e r , this fit requires /a = 4.0 (GeV)

corres-

ponding to an r.m.s. radius of 1.4 fm for the pion (this remark assumes that the form factor results

from

the G a u s s i a n form calculated for the w a v e f u n c t i o n

for

a harmonic qq potential). the value

This is m u c h larger t h a n

we o b t a i n e d from the L/(Mass)

p l o t ; this

led to a form factor e x p ( - k 2 / 4 B ) , w i t h l/2g = 0.4(Gev)" ten times smaller than the value r e q u i r e d by M i t r a and Srivastava.

Their formula predicts a w i d t h of

about 3 M e V for the { > nn decay. The process "IT + Nucleus

A-^ + N u c l e u s " is

o b s e r v e d to be diffractive, i.e. c o n c e n t r a t e d very m a r k e d l y in the f o r w a r d direction.

It is often said

that such a process results from the exchange of a P o m e r a n c h u k o n , a m y t h i c a l p a r t i c l e w i t h the same quantum numbers as the vacuum.

We should point out

here that this description is not consistent w i t h the internal structure required by our m o d e l for the A 1 m e s o n , even though it has the appropriate quantum numbers.

(external)

An object of spin 0 + , interacting at

low m o m e n t u m transfer, does not allow spin flip: h o w e v e r the t r a n s i t i o n TT

A-, involves S = 0

S = 1

390

and is therefore forbidden at momentum transfer t

2

= 0.

S-wave pion interactions can also be induced by an S-wave term which must be added to make the q -* q + it coupling Galilean-invariant; for free quarks, the Galilean-invariant interaction would have the form, G^a • (k - ^

(P + P'))t-tt

(4.29)

where P and P' denote the initial and final quark momenta and w denotes the energy of the emitted pion. The complete S-wave term thus has the form (the bar denotes anti-quark terms), G

12M~

+ P')) + T a a-(P + P ' ) ) .

This has order of magnitude (gjjf'g)

an

s

(4.30)

~wave term,

2

as compared with k R from the p-wave interaction term. For k

500 MeV, and R ^ 0.2fm ^ (1000 M e V ) - 1 , the

Galilean term is smaller by a factor ^ 1/20, so it appears implausible that this term (4.30) should be important here.

However, it is far from clear that

the s-wave term should have the coefficient (t-^—) for tightly bound quarks, and this question is at present under investigation.

For the moment, this s-wave

interaction term could be adopted with an arbitrary coefficient, to be adjusted to fit the experimental

391

data, as an ad hoc model for the s-wave decay processes. However, the amplitudes for B ->• TTU>, A]_

irp and 5

nir

are all comparable (each with spin Clebsch-Gordon coefficient of order unity), and the processes involve comparable c.m. momenta (k = 3^0, 2^5, and 300 MeV/c, respectively).

As a result, it would only be possible

to account for the narrow width r(6) < 5 MeV at the same time as r(A x ) ^ 100 MeV and r(B) ^ 120 MeV by a fortuitous cancellation between the s- and p-interaction terms in the 6

p

amplitude.

We have to conclude

here that we do not really understand the widths of the + parity bosons. Heavier mesons.

Obviously it is not worthwhile for

us to discuss the decay rates for the heavier bosons at the present stage.

Even where their total widths

are known, their branching ratios for individual decay modes are not; also, their spin-parity values are not yet established. Here we shall remark only that, for large L, the centrifugal barrier penetration factor can have a strongly suppressive effect on the decay rates.

Its

amplitude is measured roughly by j L (kr), where R denotes the r.m.s. radius of the decaying meson; this may be approximated by

392

J (kR) „ L

1.3.5...C2L+1)

{1

_ iM)! UL+6

+

(H.3!)

o valid as long as (kR) point is the large the denominator.

(5 0.7 fm.

This does not give a pleasing picture of the baryon.

The quark charge and magnetic moment clouds

must overlap significantly and this could mean

411

substantial distortions of these clouds, with the possibility of exchange currents between the quarks. Konno and Kobayashi^f have examined the latter possibility on the basis of meson theory and have concluded that the effects on the magnetic amplitudes need not be at all large.

We emphasize that the data

on the proton e.m. structure requires extended quarks, with r.m.s. radius R^ 'v* 0.75 fm, whether parafermi statistics or Fermi statistics hold; the data on meson e.m. structure also requires extended quarks, with, about the same radius.

This e.m. structure for the

quark could well be mediated by the low-lying vector mesons, a situation which would make intelligible such a large size as is required for the quark by the above considerations. 2.

Unwanted States.

It has been pointed out

by Thirring^® that the configuration i 1 2

=

=

1

which we have considered above for L = 0+ can also lead to an L = 1+ antisymmetric state, given by •Space(L=l+) =

1

, 2

2

2V N

(r 1 xr 2 +r 2 xr 3 +r 3 xr 1 )exp(-2BCr 1 +r 2 +r 3 )). (5.41)

In this state, each qq pair has p-wave relative motion. However, the kinetic energy in this state is lower than for the L = 0+ state since the structure of th.e

412

nodal planes Is m u c h less complicated.

In shell-model

terms, w i t h harmonic forces, this w a v e f u n c t i o n o corresponds to the c o n f i g u r a t i o n

(ls)(2p)

.

(5.41)

This

state corresponds to a two-quantum e x c i t a t i o n , whereas the L = 0+ state (5-19) corresponds to a three-quantum excitation. It is possible that the qq p o t e n t i a l has

form

more complex than h a r m o n i c , and that the qq p o t e n t i a l energy gives less a t t r a c t i o n in the L = 1+ t i o n than in the L = 0+ configuration.

configura-

H o w e v e r , we

can safely conclude that, w i t h forces appropriate to make the L = 0+ c o n f i g u r a t i o n the g r o u n d state of the three quark system, there should be a low-lying ¿6 supermultiplet w i t h L = 1+ c o r r e s p o n d i n g to the c o n f i g u r a t i o n (5.41). This w o u l d give rise to octets 1 3 1 3 with J = and and to decuplets w i t h J = , , 5 and —+.

* It is conceivable that N (1400) may belong to

1 * 3 a (£+) octet, and that A (1688) may belong to a (jrO decuplet

69

, but there is no i n d i c a t i o n of the other

m u l t i p l e t s w h i c h this c o n f i g u r a t i o n w o u l d require. 70 Morpurgo

has p o i n t e d out that there are some

very restrictive s e l e c t i o n rules w h i c h g o v e r n the decay of the states w i t h L = 1+.

Their decay by m e s o n

e m i s s i o n to the octet and decuplet states of the (L = 0+, 5_6) supermultiplet are forbidden, as long as

413

this m e s o n emission takes place through

one-quark

interactions.

transitions

The amplitudes for these

are all p r o p o r t i o n a l to the m a t r i x - e l e m e n t (L = 0 + | E . e x p ( i k - r , ) | L = 1+), w h i c h n e c e s s a r i l y j J vanishes since it must be an axial vector, whereas the only vector available is the p o l a r vector k.

Their

decay of Y - e m i s s i ° n to these L = 0+ states is also strongly suppressed, since the large

electromagnetic

i n t e r a c t i o n of the q u a r k s , the coupling w i t h their anomalous m a g n e t i c m o m e n t s , similarly leads to a vanishing space factor in the amplitude.

These L = 1+

states can decay through the e m i s s i o n of u n c o r r e l a t e d m e s o n s , and the channels ttitB(56_, L = 0 + ) are likely to be their most probable decay modes.

W i t h these

s e l e c t i o n rules, the direct f o r m a t i o n of these L = 1+ states is likely to be quite difficult; they w o u l d probably be formed m o s t readily as a decay x* resulting from the decay of h e a v i e r B

product,

resonance

states. 6.

The L = 1 - B a r y o n

Supermultiplet

There are a considerable n u m b e r of b a r y o n i c resonance states w i t h negative parity now in the mass range 1^00 - 1800 MeV. known

states1'69>715^2

established

We first list the

414

{1}

Y*(1405)

{8}

N * ( 1 5 3 5 ) , Y * ( l 6 7 0 ) , Y* (17^0),.

{8}

N*(1710),

{10} < f >

{1}

Y*(1520)

{8}

N * ( 1 5 2 0 ) , Y * ( l 6 9 0 ) , Y*(l662),.

{10} d o

A(l640), Y * ( 1 7 3 0 ) ? ,

{8}

A(l690), H*Cl8l5),• N* (1680) , Y* C1770), Y * CI827), •

The next highest levels w i t h negative parity are the (J--) levels, N*(2190) and Y * ( 2 1 0 0 ) , and two

further

levels, the (|-) level A(1954) and the (§-) N*C2057) 69 recently r e p o r t e d by Donnachie et al.

These

levels

are all w e l l s e p a r a t e d in mass from the levels

listed

above, and they are all compatible w i t h assignment to the next supermultiplet w i t h negative p a r i t y , as w e shall discuss later (cf. Sec. 7)P h e n o m e n o l o g i c a l Discussion.

It is reasonable to

expect the lowest c o n f i g u r a t i o n w i t h negative parity to correspond to an orbital

e x c i t a t i o n to L = 1-.

The angular f u n c t i o n for an L = 1 - state has p e r m u t a t i o n symmetry M , and has the form:

G)

C6.1)

415

where \ = ( r ^ r ^ r

)//5"

(6.2a)

P = (r-, i-i -r,)//2~ i-2

(6.2b)

We recall that the s y m m e t r i c a l part of the vector r-|_ is zero matrix

(since i ^ + r g + r ^ O

in the c.m. frame); the

(6.1) is w r i t t e n s u c h as to provide a s t a n d a r d

basis for the M r e p r e s e n t a t i o n of the p e r m u t a t i o n group P^All of the unitary m u l t i p l e t s listed above are contained in the supermultiplet

(TO, L = 1-).

the 70_ r e p r e s e n t a t i o n has M symmetry, the wave function must also have M symmetry. the general form for

Since

spaceWith

(6.1),

„„_ is

S p a C c

(6.3)

space

where the matrix the M representation.

This w a v e f u n c t i o n

(6.3) must

then be combined w i t h spin and u n i t a r y - s p i n w a v e f u n c tion with symmetry M to give an overall Fermi symmetry in the p a r t i c l e labels 1,2,3-

anti-

F i n a l l y , L and

S must be coupled to give the total a n g u l a r m o m e n t u m J; the final splitting of the states then depends on the details of the spin orbit

forces.

We note here t h a t , w i t h p a r a f e r m i statistics

and

416

harmonic force wavefunctions, the first excited level for the three-quark system has the configuration p

(Is) (2p).

This configuration necessarily has L = 1~

and M symmetry and therefore requires the 70_ representation; its space wavefunction corresponds to expression (6.3), with = = =0. Mass Formula.

With SU(6)-breaking forces of the form,

V' = V a a(l)-a(2) + V p F(l)-F(2),

(.6.4)

direct calculation leads to the following expression for the masses of the submultiplets, M = Mq + msS2 + mpF2. This expression splits the

k

(6.5)

P states from the

o

P o

states, and gives equal spacing for the states 2

P({8}) and V

SO

2

P({1}),

P({10}), in that order.

We consider next a spin-orbit force of the form* = ( a + bP(l)-F(21)(a(l) + -a(2))-L w(r ). (6.6) ~

—L2

12

This form has the following motivation. The form (l+o_( 1) • o_(2)) (1+F( 1) »P( 2)) appropriate to a spaceexchange interaction will result from the exchange of both the P nonet and V nonet (as a (35+1) representation) between the quarks, as pointed out by R a d i c a t i ^ , The spin-orbit terms would then arise from the V-exchange terms, through the non-relativistic reduction, and therefore have the unitary-spin dependence (1+F(1)-F(2)), corresponding to (6.6) with b=a. In a more phenomenological spirit, it is certainly possible to consider more general forms than (6.6) (e.g., with an additional term ca_(l)*a_(2) in the first factor) but we shall not consider these possibilities in detail here.

417

The a d d i t i o n of (6.6) gives rise to a spin-orbit additional to M = M

O

term

(6.5), + m Q S 2 + m „ P 2 + (A+BF2/l8)(S-L). q— r — —

The coefficients A, B are g i v e n by the

(6.7)

expressions

A = (a-4b) / p 2 w ( r 1 2 ) U 2 + < t > 2 ) d v , A + 2B = (a+2b ) / p 2 w ( r

(6.8a)

) (2+2)dv. 12

S

(6.8b)

2

We note that the choice b = 0 w o u l d require the

spin-

orbit splitting to have the same sign for e a c h of the unitary m u l t i p l e t s .

This can be seen ab initio

from

the fact that (a_(l) + a_(2) ) is p a r a l l 1 e l to the 3total S, w i t h the same sign for b o t h S = that, in (6.8),

w

(r12)

&

spin

and S = ^ , and

l w a y s m u l t i p l i e d by a

positive quantity in the e x p e c t a t i o n values.

On the

other hand, F(1)*F(2) can take positive or n e g a t i v e values for the qq system. The spin-orbit i n t e r a c t i o n (6.6) also mixes ^P({8}) and

2

P({8}) multiplets.

the

H e r e , we shall give

the energy m a t r i c e s for the special case (A + 2B) = 0 (appropriate to the case calculated by Greenberg and o Resnikoff ), just for the purpose of illustration. We also include the effect of a tensor force of the form V

T

= (a(l)-r a(2)-r ~ 12~12

- 4 a ( l ) - a ( 2 ) r 2 )v (r ), (6.9) 12 T 12

as c a l c u l a t e d to first-order in V T by Karl^^.

The

418

tensor force has no diagonal matrix-element for the o P states. J =

For the {8} states, we have, then: / e + |s + 1 1 + T 2

+|s + T 1 - T 2 '

+|s + T 1 - T 2 J -

6

+

4

s

J =

e

s

+

- &

/

+S

- K

4

s

+

/

k r 1 - /roT;

I0 T 1 - /rtrT2

_ |g + |(T 1 +T 2 )

Here the symbols ^

and T 2 refer to the integrals

T. = c /^i(3Z^ 2 -r^ 2 )v T (r 1 2 )dv

(6.10)

where tj^, ip2 are the upper and lower elements of the matrix (6.3)k

The symbol e denotes the energy of the

P octet relative to the

2

P octet, before the spin-

orbit interaction is turned on.

In terms of ( 6 . 7 ) , we

have e = 3mg and S = (A+B). We have not yet calculated the wavefunction tp appropriate to Fermi statistics.

The functions s,

A, and ( N*(l680,|-)

is k n o w n to be very weak

(at m o s t , several percent

of the w e l l k n o w n yp -»- N (1690,2+)

transition),

consistent w i t h its forbiddenness according to the above s e l e c t i o n rules. We now consider the s i t u a t i o n several p a r t i c u l a r states of the

concerning

(70_, L = 1-)

supermultiplet. (a)

H*(l8l5).

been a puzzle.

The assignment of H*Cl8l5) has

The direct evidence from the AK

decay mode allows the p o s s i b i l i t i e s

or

its mass value lies low relative to the other

426

(|+) baryonic states (N*(l690), Y*(l8l5), Y*(1910)), so that the latter assignment appears rather unlikely. If S*(l8l5) belongs the same octet as N*(1520), Y*(l662) and Y*(l690), then the data on their partial widths requires a large £

ratio^' ^

EK/AK

~ (1815)> typically of order 10.

for

In fact, the upper

limit now given empirically for this ratio 1 is EK/AK

Mitra and Ross 5 ^ have

1 0.03 for =*(1815).

pointed out that a decuplet assignment for E (1815) would require wavefunction

(KE+TTH+KA-UH)/2

for this

state; with the larger phase space and centrifugal barrier factor for the AK channel relative to the EK channel, the predicted branching ratio for this * — — E

state would then be

EK/AK

=

0.3.

However, they

also point out that quite a small admixture (say 10% FT O

in intensity) of an octet 5

) state could lower

the predicted ratio to be consistent with the empirical value.

This suggests that H (1815) may £

belong to the same decuplet as A (1691); an equal spacing rule for this decuplet would require further (|-) states at Y-^1750) and a (1880), for which, there is no evidence at present. For the A (1690), the width given by the Mitra-Ross calculation is about 100 MeV, with inelasticity about 0.2, to be compared with the

427

empirical estimates of 270 MeV and 0.14, 72

respectively. 1

#

The width given for E (1815) is

10 MeV, with h*tt: Htt:AK: SK in the ratio 4:1:5:0 (including some octet mixing-^), to be compared with 16±8 MeV and branching ratio 2.5:1:6.5:0. (b)

the ^P octets.

predicts that the J =

-i

The Mitra-Ross calculation ^ ) N from the 4 P configuration

should be a very broad resonance (r^600 MeV).

This

is in accord with the assignment to this configuration of the broad (r^300 MeV) s-wave ttN resonance placed 72

at 1710 MeV in the recent phase-shift analyses. The J = (|-) octet, which includes N (1680), necessarily belongs to the ^P configuration, and the data available on their partial decay rates is in good general agreement with the Mitra-Ross calculation. Th.e •3 # k J = ) N from the P configuration would be expected *

to lie more or less between these two states N (1710) and N (1680), if the spin-orbit splitting is due only to an L.S interaction (this remark neglects the possibility of ^P -

2

P mixing for the J = 1 and

and the possibility of SU(3)-breaking spin-orbit couplings).

The predicted properties of this N

are:

it should be relatively narrow (r£90 MeV) and very inelastic (x = r e l / r t mode ttA(1236) .

t -0.05),

with dominant decay

As mentioned earlier, there are

428

some preliminary indications that there may be some TTN resonance effect in the D-^ wave at about 1700 MeV, which is strongly inelastic in character. Parastatistics Calculation.

Greenberg and

R e s n i k o f f

2

> 3

have quite recently carried out an analysis of the low-lying states of the three-quark system, using shell-model configurations and assuming symmetry S for the over-all wavefunction of the system (which can be the case for the low-lying states, with parafermi statistics for quarks).

This work repre-

sents an extension of Greenberg's earlier SU(6)ftn symmetric calculations0

which classified the states

contained within these configurations.

The recent

calculations include the effect of the SU(_6)-breaking interactions, assuming these to be represented as a sum of simple SU(6) tensors. The configuration appropriate to one-quantum excitation with negative parity is uniquely (Is)2(2p) .

This configuration has L = 1-, and

necessarily has M symmetry, as we have discussed earlier (cf. Sec. 5), so that it corresponds to a 70-representation of SU(6)-symmetry•

Hence, this

model leads directly, without any special assumptions, to the configuration (J0_, L = 1-) which is required by the data on the lowest SU(6) x 0(3) supermultiplet

429

w i t h negative

parity.

The q - q system belongs to the SU(6)

representa-

tions , 6 x 6 = -

(21) + (15) , — S — A

where the suffix denotes the p e r m u t a t i o n of the r e p r e s e n t a t i o n .

(6.13) symmetry

Explicitly the tensor base

of these r e p r e s e n t a t i o n s is g i v e n by T

(AB) =

T

[AB]

=

q

q

A(1)qB(2) A(1)qB(2)

+

q

B(1)qA(2)'

" V

1 ) q

A

( 2 )

(6.14a) (6

-

W i t h S-symmetry for the t w o - q u a r k system, the

'l4b)

suffix

also distinguishes the symmetry of the orbital m o t i o n w h i c h is appropriate to each

representation;

h e n c e , i,12 = e v e n for (£1) and z 1 2 = odd for (15) • For the q - q i n t e r a c t i o n s , then, the

following

reductions are of interest: 2 1 x 2 1 = 1 + 3 5 + 1 5 x 1 5 = 1

405,

(6.15a)

+ 35 + 189..

(6.15b)

The S U ( 6 ) - i n v a r i a n t forces b e t w e e n two quarks 1 and 2 are spin and u n i t a r y - s p i n scalars w h i c h have the character i 1 2 = 1.

21

T ^ for i ± 2 = 0 ,

This n o t a t i o n

b

Tc

2, and

for

specifies a tensor

430

depending on the labels and co-ordinates

,

F i , a) of two q u a r k s , w h i c h belongs to an SU(3)~ r e p r e s e n t a t i o n {c} contained w i t h i n the

SU(6)-

r e p r e s e n t a t i o n a, acting in a q - q state w h i c h is appropriate to the S U ( 6 ) - r e p r e s e n t a t i o n b.

There

are also one-body operators effective, w h i c h are r e p r e s e n t e d simply by tensors T^, and whose

effects

cannot be d i s t i n g u i s h e d from those for a certain c o m b i n a t i o n of the two-body

operators.

The S U ( 3 ) - i n v a r i a n t forces w h i c h violate SU(6) are limited to the spin scalars c o n t a i n e d in 35, 189, and 405.

—-

-1*0 5

There are only two, T1,,

189

for a = even.

for I = odd and

There is also the

force, denoted by T?" . L•S

spin-orbit

These terms are

together

equivalent to the explicit forces i n t r o d u c e d in our discussion

above.

The S U ( 3 ) - v i o l a t i n g terms are as follows. one-body operator is are

21

T^5,

15T

35*

t

®05

Q

a n d

T h e T

i89'

two-body 0 n l y

is d i s t i n g u i s h a b l e

The

operators

t h e

d i f f e r e n c e

(21^8

_ 15t^)

from the one-body

term.

G r e e n b e r g a n d R e s n i k o f f also include

an

S U ( 3 ) - b r e a k i n g spin-orbit t e r m T® . L •S These forces have m a t r i x - e l e m e n t s which, are o c o m m o n to the d i s c u s s i o n of the 56 (Is) the 70 ( I s ) 2 ( 2 p ) c o n f i g u r a t i o n and the 20

configuration, (ls)(2p)2

c o n f i g u r a t i o n w i t h L = 1+.

Hence Greenberg

and

Resnikoff carried out their analysis, taking the (56, L = 0 + ) d a t a and the (70, L = 1") d a t a together.

They made a least-squares fit to the

data, with ten adjustable p a r a m e t e r s in all, fitting ten e s t a b l i s h e d odd-parity resonances

and

the b a r y o n and decuplet states k n o w n for the (56, L = 0 + ) supermultiplet.

The parameter

w h i c h they found a p p r o p r i a t e to the

values

(J0_, L = 1 )

were as follows: Strong forces

(SU(3), but not SU(6)

symmetry):

2 1 M 1 _ 15m1 = - 2 9 1 . 5 M e V M*

= 199 M e V , M^-

M

=

£.S

Moderately-strong 21

M^5 -

3

5

M

e

forces 15

M^5

= 36.8 M e V

V

(breaking SU(3)

symmetry)

= -22.9 MeV

M^ q 5 = 121 MeV, m8Q9 = -103 MeV M8 = - 1 8 9 MeV. L.S We note that all of these S U ( 6 ) - b r e a k i n g terms are of the same order of m a g n i t u d e . strong forces do not appear to be

The m o d e r a t e l y significantly

w e a k e r than the strong forces in these states; the absolute locations of the supermultiplets

are

432

d e t e r m i n e d by the s u p e r - s t r o n g

forces, of

course.

The f i n a l s y s t e m of (7£, L = 1-) l e v e l s r e s u l t s f r o m the c a l c u l a t i o n of G r e e n b e r g

which

and

R e s n i k o f f are l i s t e d i n the a c c o m p a n y i n g T a b l e .

The

l e v e l s u n d e r l i n e d i n d i c a t e the s t a t e s w h i c h w e r e u s e d as i n p u t d a t a .

The m a s s v a l u e s i n b r a c k e t s

are g i v e n w h e r e n e c e s s a r y

to i d e n t i f y the

input

l e v e l s u s e d , since the f i n a l fit does n o t all t h e i n p u t levels p r e c i s e l y , of

reproduce

course.

T h e r e are s e v e r a l p o i n t s of i n t e r e s t to

note

here. 1.

Decuplet Levels. p

guration of t h e

(Is)

W i t h the p a r t i c u l a r

confi-

_

(lp), the L = 1

space w a v e f u n c t i o n is

form (6.16)

This m a y be c o m p a r e d w i t h o u r e x p r e s s i o n the case of F e r m i s t a t i s t i c s . the fact that Space

h a s

T a k i n g into

Space

for

account

to be c o m b i n e d w i t h the

70 SU(6) w a v e f u n c t i o n to g i v e o v e r - a l l here, whereas

(6.3)

symmetry

to be c o m b i n e d w i t h th.e

70 S U ( 6 ) w a v e f u n c t i o n to g i v e o v e r - a l l

antisymmetry

in the case of F e r m i s t a t i s t i c s , we f i n d that forms for the e x p e c t a t i o n v a l u e s

calculated

the

433

,

CM \ rH

CT\ cr\

rH

CM OO rH

o vo rH

O LPi rH w

Ln o rH

on V£> LT\ rH

Pk CM

CM rH

CM O rH

CM on

o on in rH

\

II

CM \ on

C\J — i(

on on rH

CM m

\

CM \ Ln

O CO rH

^J

*

\

PL, -4-

\

*

LPl rH

OO m rH

LT\ on rH

rH O rH

C\J rH VO rH

Ln rH CO rH

Ln O C— rH

on Ln oo rH

>"5 O CM LT\ H

«

co rH

OO

\o t>-

rH

«

on MD rH

rH

O rH

w -P 0 -P o O

C\J \ on

rH o on m o oo CM rH

CM vo o vo vo rH rH

rH rH on Ln oo o CM rH

CTS UD in t-VO rH rH

il

II 1-3

m •p (D •H hO G P-f •H CM W

«

CM CO t- O m CO rH rH


•Ö rH cd G II 0 i-q > 0 rH CQ rH CDrH « T3 0 vol ^ O > ml 1 S 0 w rH rH rH •Ö II rH 0 0 CD.G Xi l-q £ -P CQ m •H O rH bO-P .Q cd 1 C t-l - — ' •H -P CQ 0 CQ CD 3 -P 0 •G -P 1 -P G •H rH O 'H rl CH •H 0 O -p 0 5 cd > rH rH cd 0 H cd O -p — i 1 co hO Cd—/

+

1—1

>

+

enaj iricvj V-/ ,

/

+

S

+

OTM . HOJ en eu

G* ft 0 PH vo D 0Q

*—'

,+ s

hicm

•—•

n

i—i — .> + OTCVJ

•—-, .—^ + + i—ICM — iC |M

^

^—-,

+

iHICM

•

+

omcM

i—i

-P 0 0 Tl -G S 0 0 -P S M -p ,g >s O CO H Oc cd cdC •H 0 Ph -p O M •H Ch S

1 1

vol Lnl

i—i y—N + LTiOJ

••—*

+

i—i /-S + oniCM

/—N + CO CM s-' 1 1

en cm w .—. y—N + + — i 1 CM i—ICM 1 1 i i

M

CXI 1

1 1

+

xn

HI C\J N—' i i

-—•

°Ml

i^l

•—/

t-3

N

i—i + in cm

m

-p 0 rH bO c •H co

+

mevj

LTMCM w /-N + COICM

+

C\J

+

CM

+

H

+

H

+

O

G o •H -P cd X Ph PH 0 3 cd M faO 3 iH •H cr Ph

i 1 CTCVJ LfNCM

*

1 e n c\i * n

,—.

1—1 w •p (D rH hO ß •H CO

1 MOJ

—>

— .^ •>

1 Lnoj ^—••

i

i

1 i

irJoj i—i

i—1

1 LTMOJ

1 UNOJ

-—'

1,J

^

1 m—'

i

i

i cHCM —1 1

i-HIOJ

i—i s. 1 ncM

^ H 1 LP£M y—-, 1 erra

i—i 1 nevi V—•

r*

1 1 i—ICVJ m i cm

^—

1 1

1—1 —N

rs

'

1 1

-—-

o •—s 1 H I CM ^

j

I m|C\J

i

ß

ft

0 cc 3 • o rH co 0) rH On -p c d - — s í cd

O CM 3 ß cm — o^o; ^ i M CO CO 0) Q) rH 0) m -p • ^ a) -p 0 ß a x : 0 • H OJ - P Ph H cd ft ft— CO l—J P-l •H o co a) co cd o cd a 1 EH O 214 (1966).

31

R. Hagedorn, Statistical Thermodynamics of Strong Interactions at High Energies - III Heavy Pair (Quark) Production Rates, CERN preprint TH. 751 (April, 1967).

32

F. Chilton, D. Horn and R. Jabbur, Phys. Letters 22, 91 (1966).

33

M. Gell-Mann, Phys. Letters 8, 214 (1964).

34

G. Zweig, An SU(3) Model for Strong Interaction Symmetry and its Breaking, CERN preprint

455

References

(cont.)

8419/TK. 412 (February, 1964), u n p u b l i s h e d .

See

also G. Zweig, In Symmetries in Elementary Particle Physics

(ed. A. Zichichi, Academic

Press,

New Y o r k , 1965), p. 192. 35-

0. G r e e n b e r g , P a r a f i e l d T h e o r y , Ch.ap. 3 in Conf. on Math. Theory of E l e m e n t a r y

Particles

(ed. R. G o o d m a n and I. Segal, M . I . T . Press,

1966),

p. 29. 36.

H. Green, Phys. Rev. £0, 270

(1953).

37.

0. Greenberg and A. M e s s i a h , Phys. Rev. 138, B1155 (1965) •

38.

M. H a n and Y. N a m b u , Phys. Rev. 139, B1006

(1965).

39.

J. B j o r k e n and S. G l a s h o w , Phys. Letters 1^, 255 (1964).

40.

T. Lee, Nuovo Cimento 35., 933

(1965).

41.

J. S c h w i n g e r , Phys. Rev. 135,

B8l6

42.

M. F o c a c c i , W. K i e n z l e , B. L e v r a t , B. M a g l i c , and

(1964).

M. M a r t i n , Phys. Rev. Letters 17., 890 43.

(1966).

B. Levrat, C. T o l s t r u p , P. S c h u b e l i n , C. Nef, M. M a r t i n , B. M a g l i c , W. K i e n z l e , M. Focacci., L. Dubai, and G. Chikovani, Phys. Letters 714

44.

22,

(1966).

L. Dubai, M. F o c a c c i , W. K i e n z l e , C. L e c h a n o i n e , B. L e v r a t , B. M a g l i c , M. F i s c h e r , P. G r i e d e r ,

456

References

(cont.)

H. Neal, and C. Nef, Evidence for Fine Structure in Boson Mass-spectrum at 1700 MeV, to be published (November, 1967). 45.

G. Goldhaber, Proc. X I I I Intl. Conf. on High Energy Physics

(Univ. California Press,

1967),

p. 103. 46.

G. Goldhaber, "Experimental Aspects of Strong Interactions", Proc. Second Hawaii Topical Conf. on Particle Physics

(Univ. Hawaii Press, Honolulu,

1968). 47.

C. Akerlof, W. Ash, K. Berkelman, C. Lichtenstein, A. Ramanauskas, and R. Seemann, Measurement of the Pion Form Factor (Cornell University

preprint,

1967)• 48.

G. Chikovani, M. Focacci, W. Kienzle, C. Lechanoine, B. Levrat, B. Maglic, M. Martin, P. Schubelin, L. Dubai, M. Fischer, P. Grieder, H. Neal and C. Nef, Phys. Letters 25B, 44

(1967),

(1967).

49.

D. Morrison, Phys. Letters 25B, 238

50.

W. Kienzle, B. Maglic, B. Levrat, F. Lefebres, D. Freytag and H. Blieden, Phys. Letters 19, 438 (1965)•

51.

I. Butterworth, Proc. Heidelberg Intl. Conf. on Elementary Particles

(September, 1 9 6 7 ) ,

to be

457

References

(cont.)

pub]ished. 52.

J. Danysz, B. Freuch, and V. Simak, CERN preprint Phys.

53.

67.1.

P. Baillon, D. Edwards, B. Marechal, L. Montavet, M. Tomas, C. d'Andlau, A. Astier, J. CohenGanouna, M. Delia Nigra, S. Wojcicki, M. Baubillier, J. Duboc, P. James and F. Levy, Nuovo Cimento 50, 393

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54.

A. Mitra and M. Ross, Phys. Rev. 158, 1630

55.

H. Lipkin, Phys. Rev. 159, 1303 (1967).

56.

J. Uretsky, On the Bosons of Zero Baryon Number as Bound States of Quark-Anti-Quark Pairs, in High Energy Theoretical Physics (ed. H. Aly, Univ. of Beirut, 1 9 6 7 ) .

57.

R. Van Royen and V. Weisskopf, Nuovo Cimento 51, 583 (1967). (1966).

58.

Becchi and G. Morpurgo, Phys. Rev. 149, 1284

59.

A. Mitra and P. Srivastava, Quark Model Couplings and Strong Decays of Tensor and Axial Mesons, Intl. Centre Theor. Phys. preprint I C / 6 7 / 3 1

CMay,

1967). 60.

M. Bolsterli and E. Jezak, Phys. Rev. 135, B510 (1964).

458

References (cont.) 61.

S. B. Gerasimov, Journ. Nucl. Ph.ys. (U.S.S.R.), C 1 9 6 7 ) ; circulated as Dubna preprint D-2439.

62.

A. Mitra and ft. Majumdar, Ph.ys. Rev. 150, 119^

(1966). 63.

R. Dalltz and D. Dorren, quoted on p. 85 of ref. 64.

64.

R. Dalltz, Quark Substructure for Mesonic and Baryonlc States, In 1966 Tokyo Lectures In Theoretical Physics : part II, Elementary Particle Physics (Syokabo, Tokyo, and Benjamin, New York, 1967), p. 56.

65.

D. Harrington and A. Pagnamenta, Phys. Rev. Letters 18, 1147

66.

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R. Kreps and J. de Swart, Phys. Rev. 162, 1729 (1967).

67.

K. Konno and T. Kobayashi, Progr. Theor. Phys. 38, 671 (1967).

68.

W. Thirring, "Triplet Model of Elementary Particles", Lectures at Intl. University Week in Nuclear Physics (.Vienna, March., 1 9 6 6 ) .

69.

A. Donnachie, R. Kirsopp, and C. Lovelace, Evidence from Trp Phase Shift Analyses for Nine More Possible Nucleon Resonances, CERN preprint TH. 838 (October, 1967).

459

References

(cont.)

70.

G. M o r p u r g o , Phys. L e t t e r s 22, 214

C1966).

71.

R. A r m e n t e r o s , M. P e r r o - L u z z l , D. Leith, R. L e v i - S e t t i , A. M l n t e n , R. Tripp, H. Fi.lth.uth, V. H e p p , E. K l u g e , H. S c h n e i d e r , R. B a r l o u t a u d , P. G r a n e t , J. M e y e r and J. P o r t e , Phys. 24B, 198

72.

Letters,

(1967).

C. L o v e l a c e , irN Phase-Shift A n a l y s i s , Proc. ttN Scattering Conf. at Univ. C a l i f o r n i a Irvine

(Dec.,

1967). 73«

L. R a d i c a t i , 1965 Cargese Lectures in Theoretical 1966).

Physics

(ed. M. Levy, B e n j a m i n Inc.,

74.

G. Karl

(Oxford), private

75.

P. B a r e y r e , C. B r i c m a n and G. V i l l e t , Phase shift

communication.

analysis of p i o n - n u c l e o n elastic scattering b e l o w

1.6 GeV, Saclay preprint 76.

(1967).

C. J o h n s o n , P. Grannis, M. H a n s r o u l , 0.

Chamber-

lain, G. Shapiro, and H. S t e i n e r , Lawrence Lab. Rept. in p r e p a r a t i o n

(1967).

77.

R. M o o r h o u s e , Phys. Rev. Letters l6_, 771

78.

M. G o l d b e r g , J. L e i t n e r , R. Musto and L. 0 ' R a i f e a r t a i g h , Nuovo Cimento 4_5, 1 6 9

79.

Rad.

(1966).

(1966).

M. M a r i n o v , Sov. Journ. of Nucl. Phys. 2, 228 (1966).

460

References

(cont.)

80.

0. Greenberg, Phys. Rev. Letters

81.

A. M i t r a , V. B h a s i n and D. Katyal, Phys. Rey. l6l, 1546

(1967).

598

(1964).

461

M£OSON

Fig. 1.

(GEV) 2

A plot of the dominant mesonic excitations observed in the reaction ir~+p ->• p+(missing mass) studied with the missing mass spectrometer by Pocacci et al. The peaks are numbered in order of increasing mass, starting with the p-meson, and the peak number is plotted against (mass)2 for each mesonic excitation.

462

Fig. 2.

Internal orbital angular m o m e n t u m in the three-quark system, w i t h £ J 2 + = L, for total orbital angular m o m e n t u m L.

463

Pig. 3.

When particle 3 is ro = 0 in the c.m. system, then the distances r^ = I2I23I and r 2 = I£31) are equal. For L = 0, a the space wave function s p a.e e scalar and depends only on r l 5 r 2 and r3 = |r,-L2|j therefore, for the configuration depicted above, the wavefunction cannot change when labels 1 and 2 are interchanged (since = r 2 , and r^ is symmetrical in 1, 2). But Space is antisymmetric in the labels 1 and 2j hence its value must be zero in this configuration.

464

S 11 (1710) (1-) 2

D13(1700)?

(1-)

D 15 (1680)

(i-)

{8}

D 33 (l690)

(1-)

(10) Son(1640) _

(i-)

'1 _L8J_

D 13 C1520)

Y"(1520) JJLL Y*(l405)

Fig.

The pattern of submultiplets for the (70, L = 1-) Supermultiplet.

(±_)1 V2

465

Fig. 5.

The N* and A* States for the S u p e r m u l t i p l e t (56, L = 2 + ) .

466

2P

r

37

(I-)

fio> 035(195^)

D13(2057)

(I-)

(f-)

D

G17(2200?)

15

(¡-)

G1t(2200?)

{8}

{8} (5-)

2

17

(2100) {1}

Fig. 6.

T h e N * a n d A* (and s i n g l e t Y * ) S t a t e s for the S u p e r m u l t i p l e t (70, L = 3-).

Errata and addenda to th Second Hawaii Topical Confer p. 106 line 3j "regeneration" should p. Ill RHS of Eq. (21) should read 1

p. 113 line 2, "The experiments" shou p. 135 line 2 should read: In one ex found where 49 K + -»• tt+tt~£.+v e p. 137 RHS of the equation should rea p. 137 line 15, x should read Im x. p. 137 line 16 and footnote 3, 3 Re e p. 143 line 2 from bottom, Debye temp from the scattering angle e. read 0 . p. 144 line 2, When W = 12 should rea p. 2 6 1 top row in the 1 0 , 2 / 3 should p. 262 second row in the 10_, x should p. 263 Note that û' is not an exotic Y = 2 states have not been mar p. 333 delete all of line 2 from bott p. 333 line 17 should read Becchi and p. 334 lines 10 and 11 should read: invariance in space-time. In only to non-relativistic regim p. 339 line 2 from bottom, "experimen p. 3^2 h (appearing in any algebraic p. 344 line 15, "radiative" should re p. 3^9 line 6 should read: Chilton e p. 369 in Eq. (4.7) -2/2/3A should re p. 372 line 4 from botton, a(o) shoul line 2 from bottom, 1.05p(Gev) p. 382 line 2, r should read r. p. 385 line 1, L(Mass) 2 should read L p. 386 lines 9, 10, and 17 from botto: p. 396 line 2 from bottom, (Is)2 shou p. 397 line 4 from botoom, "line S" s p. 399 line 2 from bottom, (Is)2 shou p. 400 line 3, (Is)2 should read (Is) line 7, "and" should read "whi p. 401 Eq. (5.16), the second bracket p. 406 Eq. (5.30), the second S + shou p. 412 line 18, "a" should read "this p. 413 line 7, "of" should read "thro p. 416 footnote, the last sentence is p. 430 line 10, "forces" should read line 11, "scalars" should read p. 440 Table I, column 5, line 2, the should be (7/2+). p. 445 Drs. G. Karl and E. Obryk have incomplete and in error. The (ls) 2 (3p) should be included o already listed. The following 3-

56

3-

20

[(3_),(5_),(2_)5(2_)

1-

20

[(£-),(|-),(|-)]

the Proceedings of the mference in Particle Physics >uld read "regenerator." id 1 + if Re e (1 - Ixl2) U - ¿|2 should read "two experiments." le experiment no K + -»• IT+TT+h-O events were 'v events were r e c o r d e d ^ . 1 read as in the correction to p. Ill above, x. Re e should read 3-7 Re e. temperature should be 0 to distinguish it e. All subsequent 6 preceded by k should 1 read When W = 1/2. >uld read 3/2. lould read >tic state. Also the positions of the l marked. bottom, after the word "dimensional." . and Morpugo^o,... id: "formations, which is appropriate to In other words, Pauli spin is appropriate 'egime." 'imentally" should read "exponentially." •aic expression) should read "ft on this page. .d read "nuclear recoil." ;on et al. 3 2 . . . .d read -2/2/3A$- . ihould read ap(0). Gev) 2 should read 1.05(Gev)2. >ad L/(Mass) 2 , (ottom, m should read m g . should read (Is) S" should read "line A." should read (Is)3. (ls)3.

"which." icket should read (1 + P(1).P(2)). should read S_. 'this" both times, 'through. " :e is incorrect and should be deleted, 'ead "central forces." read "and unitary spin scalars." the last entry inside square bracket have pointed out that Table III is both The configurations (ls)(2p)(2s) and led on the same footing as the configurations >wing multiplets should then be added: [ ( § - ) , [ ( ¡ - ) , ( | - ) , ( | - ) , ( | - ) 3

(§-)] I

[(l-),(l-)] C(J-),(|-)3

- 2 -

The supennultiplet (7£, L • 1-) lis in Table III is mistaken. States w forbidden for any three particle sy described by two polar vectors (r^ impossible to construct a pseudosca p. 447 The statement that the first radial as the first p-excitation is incorr oscillator state (ni.) is E(nfc) = ftu what surprising to find the (1S)2(2 but a q-q s-wave interaction more s can lead to this situation, p. 448 The two configurations (ls)(2s) an The first excited (56, L = 0+) stat position of them; the other (56., L is spurious. Hence the possibiliti not distinct, p. 449 The Pii(1751) level may be generate (1S)2(3S) and (Is)(2s)2, together w as needed to eliminate spurious com

list occurs twice. The L = 0- entry es with L = 0 and negative parity are e system; the internal motion -Is . + £2 ~ 2£.3> Hi ~ £2^ a n d ^ i s oscalar from any two such vectors. dial excitation has the same energy correct. The energy of the harmonic = ftu(2n-A-i). It is therefore some) 2 (2S) state lying below (ls) 2 (2p), re singular than a harmonic potential ) and (ls)(2p) 2 must be taken together, state consists of a linear super, L * 0+) state which can be formed lities (i) and (ii) given here are rated by four-quantum excitations er with other N = U configurations, components of its wavefunction.