octet enhancement ik hadronic interactions by choi...
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OCTET ENHANCEMENT IK HADRONIC INTERACTIONS
BY
C H O I - L A I CHAN
B . S c , U n i v e r s i t y o f Hong Kong, 1964-
A THESIS SUBMITTED,IN PARTIAL FULFILLMENT OF
THE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN THE DEPARTMENT
OF
PHYSICS
We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o the r e q u i r e d s t a n d a r d
THE UNIVERSITY OP BRITISH COLUMBIA
J U L Y , 1968
In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s
f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree
t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and
Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s
t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my
Department or by fills r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g
or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d
w i t h o u t my w r i t t e n p e r m i s s i o n .
Department o f
The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada
Date 5"^ Aft-. iWS
ABSTRACT
In t h i s thesis, a detailed study i s made of the
phenomenon of Octet Enhancement i n hadronic i n t e r a c t i o n s .
A f t e r a survey of the experimental consequences of Octet
Enhancement i s made i n Chapter II, a review and discussion
of the various theories of Octet Enhancement i s given i n
Chapter I I I . In Chapter IV, a general group t h e o r e c t i c a l
discussion, based on an extension of Glashow's Method (S.
L. Glashovr, 1963)» of spontaneous symmetry breaking i s pre
sented. A general theorem i n connection with spontaneous
symmetry breaking i s stated and proved. The theorem lends
I t s e l f to a number of i n t e r e s t i n g a p p l i c a t i o n s . Among these
i s a demonstration that i f a unitary t r i p l e t exists i n nat
ure which i s n o n - t r i v i a l l y coupled to the res t of the had-
rons, then Octet Enhancement follows without recourse to
any d e t a i l e d dynamical assumption. In p a r t i c u l a r , the t r i
p l e t need not be quarks i n the sense of fundamental b u i l d
ing blocks of a l l matter. I t i s al s o demonstrated group
t h e o r e t i c a l l y that mixing can i n p r i n c i p l e occur as a
p a r t i c u l a r form of spontaneous symmetry breaking, i n exact
l y the same way that a spontaneous mass s p l i t t i n g can occur.
Assuming Octet Dominance i n the symmetry breaking of a de
generate nonet of vector mesons, i t follows quite generally
that the formulae (i) )n£ & m$ = 2^* and ( i i ) + 3
( C(nZ0 -f su?Q rnjl,)-4-mj!» = Q must hold. Here again, no
- i i -
- i i i -
dynamical d e t a i l i s required. In Chapter V, a bootstrap
model consisting of an octet of vector mesons i s construct
ed with which i t i s demonstrated step by step how Octet
Enhancement follows from the i n t r i c a t e working of two dyna
mical p r i n c i p l e s — (a) the bootstrap hypothesis and (b)
that the mesons p a r t i c i p a t i n g i n the bootstrap must be
phy s i c a l p a r t i c l e s and not ghost states. In Chapter VI,
the model i s extended to study <fi-cv mixing. A phenomenon
which can be interpreted as a "sponteneous" mixing i s
found to i n f a c t occur. Furthermore, formulae (i) and ( i i )
al s o emerge from t h i s model. The only extra assumption be
sides (a) and (b) that we have to make here i s the mere
"existence" of a unitary s i n g l e t vector meson whose "bare"
mass (the ma.ss of the p a r t i c l e before i t i s embroiled i n the
bootstrap) i s equal to average mass of the octet.
TABLE OF CONTENTS
Abstract . . . . i i
Table of Contents . . . i v
Acknowledgement v i
Chapter I -- Introduction . . . . . . . . . . . . . 1
Chapter II — Experimental Consequences of Octet Enhancement . . . . . . . . . . . . . . 9 1. What i s Octet Enhancement? . . . . 9
2. Octet Enhancement i n Electromagnetic Interactions 13
3* Data from Nuclear Physics 20
Chapter III — Various Theories of Octet Enhancement 25
1. The Quark Model . . . . . . . . . . 25
2. The Mixing Model . . . . . . . . . 2?
3. The Tadpole Model . . . . . . . . . 33
k. The Bootstrap Model 37
Chapter IV — Group Theoretical Treatment of Spontaneous Symmetry Breaking ' k6
1. Spontaneous Symmetry Breaking . . . 4 6
2. Group Theoretical Treatment of Spontaneous Symmetry Breaking i n an S.U (3) Symmetric Bootstrap Containing Eight Baryons 4 8
3. Generalized Group Theoretical Treatment of Spontaneous Symmetry Breaking 50
4 . Applications of the General Results 62
Chapter V Octet Enhancement i n a Bootstrap Model 85
1. The Bootstrap Hypothesis 8 5
2. Basic Assumptions i n S-Matrix Theory 90
i v
- V -
3. Octet Enhancement i n a Bootstrap
Model of Vector Mesons 105
4. Miscellaneous Remarks . . . . . . 115
Chapter VI — 4-co Mixing i n a Bootstrap Model of
Vector Mesons 124
1. Introduction 124
2. The Vector Nonet Model 125 Appendix 1 — Crossing Matrices 1 39 Appendix 2 — Comments on the V a l i d i t y of the Para
metrized One Pole Approximation . . . 143
Bibliography . • 148
ACKNOWLEDGEMENT
I am deeply g r a t e f u l to Prof. M. McMillan f o r
suggesting the t r u l y rewarding research topic i n thi s thesis
and f o r h i s invaluable advice and encouragement.
I would l i k e to take t h i s opportunity to express
my sincerest gratitude to Prof. W. Opechowski f o r his kind
ness and help during my graduate years i n U.B.C.
I am al s o g r a t e f u l to Prof. E. Vogt f o r a conversa
t i o n i n connection with the charge dependence of nuclear
forces and to Prof. S. Coleman f o r several conversations
held a t Erice i n connection with the present status of the
theories of Octet Enhancement.
As regards f i n a n c i a l support, I am gr a t e f u l to the
National Research Council of Canada f o r a Studentship.
- v i -
CHAPTER I
INTRODUCTION
At the present time, whatever l i t t l e that one under
stands about Hadronic Physics comes from the ex p l o i t a t i o n of
two mathematical techniques, one an a l y t i c and the other
algebraic.
The use of the algebraic techniques started from
the discovery of Isospin i n nuclear physics (W. Heisenberg,
1 9 3 2 ; B. Cassen and E. U. Condon, 1 9 3 6 ; E. P. Wigner, 1937)
which was l a t e r extended to F i e l d Theory (N. Kemmer, 1 9 3 8 ) .
So f a r as electromagnetic and weak interactions can be neg
lected, i t was found that the strong i n t e r a c t i o n i s invariant
under a group SU ( 2 ) , which leads to the conservation of
is o s p i n . Later, the discovery of strange p a r t i c l e s ( G . D.
Rochester and C. C. Butler, 1947) l e d subsequently to the
discovery of another good quantum number i n strong i n t e r
actions, strangeness (M. Gell-Mann, 1 9 5 3 . 1 9 5 6 ; T. Nakano
and K. Nishijima, 1 9 5 3 ; K. Nishijima, 1 9 5 5 ) . the conserva
t i o n of which depends on the Invariance under a c e r t a i n gauge
transformation. In the l a t e r half of the f i f t i e s , many
v a l i a n t e f f o r t s were made to uncover an underlying inexact
symmetry group that would contain the above two groups as
subgroups (M. Gell-Mann, 1 9 5 7 ; J . Schwinger, 1 9 5 7 )• The
1
f i n a l answer came i n 1961 i n the form of the "Eight-fold Way"
or octet v e r s i o n of SU(3) symmetry (M. Gell-Mann, 1 9 6 l ;
Ne'eman, 1 9 6 l ; M . Gell-Mann and Y . Ne'eman, 1 9 6 4 ) . Later came
the discovery of higher symmetry groups such as SU(6) (see F.J.
Dyson, 1966 f o r reference) and the use of the current algebras
(see S. L. Adler and R. Dashen, 1968 f o r reference). We are
not going to dwell on the l a s t two developments.
The a n a l y t i c technique i s usually used i n the form
of d i s p e r s i o n r e l a t i o n s f o r S-matrix elements and these, when
combined with unitary (conservation of probability) and cross
ing symmetry (concept of an a n t i - p a r t i c l e ) provide a most con
venient framework f o r some of the r e l a t i v e l y f r u i t f u l ideas
ofhydronic dynamics, among these the bootstrap hypothesis, (see
Chapter v)• There i s no reason why the bootstrap hypothesis
cannot be b u i l t on a framework of Lagrangian F i e l d Theory;
that i s , provided a reasonably usable such theory e x i s t s . In
the S-matrlx Theory, where the a n a l y t i c approach i s f u l l y im
plemented, most of the divergence d i f f i c u l t i e s i n a Lagrangian
f i e l d theory can be circumvented. Hence, i t i s owing to such
theories that p h y s i c i s t s were able to reap most of the q u a l i
t a t i v e or semi-quantitative r e s u l t s i n hadronic physics.
Some of the recent works i n p a r t i c l e physics are
connected with the der i v a t i o n of algebraic symmetries using
the a n a l y t i c technique (E. Abers, F. Zachariasen and C.
Zernach, 1 9 6 3 ; R. H. Capps, 1 9 6 3 a ; Hong-Mo Chan, P. C.
3
D e C e l l e s and J . E . P a t o n , 1 9 6 3 , 196k; A . W. M a r t i n a n d K . C . W a l i ,
1963» 196^°, R. B l a n k e n b e c l e r , D . D . Coon a n d S . M. Boy, 1 9 6 7 ) .
The a c h i e v e m e n t s , t h o u g h a s y e t modest , a r e e n c o u r a g i n g .
I t i s a l m o s t a n i n e v i t a b l e f a c t a b o u t s y m m e t r i e s
t h a t t h e y a r e made t o be b r o k e n . I t has b e e n c l e a r f r o m t h e
b e g i n n i n g , i n t h e c a s e o f SU(3) f o r example, t h a t t h e symmetry
b r e a k i n g e f f e c t s a r e g e n e r a l l y l a r g e a n d must be i n c l u d e d i n
o r d e r t o a c h i e v e a c o r r e c t d e s c r i p t i o n o f any b u t t h e c r u d e s t
f e a t u r e s o f t h e h a d r o n s p e c t r u m . T h u s , s u c h a symmetry w o u l d
have b e e n a l m o s t c o m p l e t e l y u s e l e s s i f i t i s b r o k e n i n a h a p
h a z a r d f a s h i o n . F o r t u n a t e l y , i t i s a l w a y s b r o k e n i n a
r e m a r k a b l y e l e g a n t way. As a m a t t e r o f f a c t , t h e "medium
s t r o n g " a n d t h e e l e c t r o m a g n e t i c v i o l a t i o n s o f SU (3) a p p e a r t o
f o l l o w a c h a r a c t e r i s t i c p a t t e r n , i n t h a t the v i o l a t i o n s w h i c h
t r a n s f o r m l i k e components o f a n o c t e t ( " O c t e t Enchancement")
seem t o p r e d o m i n a t e i n n a t u r e ( S . Coleman and S . L . Glashow,
1 9 6 ^ ) . * We c a n see t h i s more e x p l i c i t l y i n a s p e c i f i c example;
by l o o k i n g a t t h e mass s p l i t t i n g m a t r i x ( S . L . Glashow, 1963)
o f a h a d r o n m u l t i p l e t . I t t u r n s out t h a t t h e e x p e r i m e n t a l -
v a l u e s o f t h e masses a r e a l w a y s w e l l f i t t e d i f we assume t h a t
t h i s mass s p l i t t i n g m a t r i x t r a n s f o r m s l i k e t h e e i g h t h component
o f a n o c t e t , w h i c h i n f a c t l e a d s t o t h e G e l l - M a n n Okubo
F o r m u l a (M. G e l l - K a n n , 1 9 6 1 ; S . Okubo, 1 9 6 2 ) . T h i s f a c t c a n
* A r e v i e w o f t h e e x p e r i m e n t a l e v i d e n c e from O c t e t Enhancement w i l l be g i v e n i n C h a p t e r I I o f t h i s t h e s i s .
4
be summarized phenomenologically by w r i t i n g down an e f f e c t i v e
Hamiltonian that contains a mass term transforming l i k e the
eighth component of an octet ( J . J . DeSwart, 1963). There i s
no a p r l o i i r e a s o n f o r t h i s to be true. If we believe i n a
dynamical basis f o r the mass spectrum, i t evidently comes from
the strong i n t e r a c t i o n s . I f we believe i n a Lagrangian f i e l d
theory, we could, f o r example, follow Sakural ( J . J . Sakural,
I960. ; S. L. Glashow and H. Gell-Mann, 196l) by saying that
there i s a fundamental coupling between the hadronlc currents
and the vector mesons. One could then b u i l d i n a symmetry
breaking by assuming, f o r example, a coupling between the hyper-
charge current and the unitary s i n g l e t vector meson, or a
coupling between the baryon current and the T=0, Y=0 octet
vector meson, e i t h e r of which gives r i s e to a fundamental
SU(3) v i o l a t i n g i n t e r a c t i o n that transforms l i k e the T=0, Y=0
(eighth) component of an octet. However, one sees immediately
that t h i s i s hardly an answer to the question at hand, even
when one concedes to the ad hoc nature of the assumption. This
i s because the contribution to mass w i l l necessarily be of
second order i n such Interactions, and because of the Clebsch-
Gordan (C-G) decomposition ( J . J . DeSwart, 1963) of the d i r e c t
product of tvro octets as shovm i n (2 . 9 ) • There must be con
t r i b u t i o n s to mass s p l i t t i n g s that transform l i k e c e r t a i n
components of the 2 7 - p l e t . Such contributions are almost
t o t a l l y absent experimentally. Another current approach to
strong i n t e r a c t i o n dynamics i s to assume that the fundamental
equations are SU(3) invariant and that the v i o l a t i o n s to SU(3)
set i n as a p a r t i c u l a r but stable sol u t i o n to those equations
(M. Baker and S. L. Glashow, 1 9 6 2 ; M. Suzuki, 1964-a, 1964-b;
fi. E. Cutkosky and P. Tarjanne, 1 9 6 3 ; D. Y. Wong, 1 9 6 5 ; R.
Dashen and S. Frautschi, 1 9 6 5 a ) * Here, the question why
"spontaneous" symmetry breaking should emerge In the octet
pattern again presents I t s e l f . Fortunately, as we s h a l l see
i n l a t e r chapters, the condition f o r Octet Enhancement i n
symmetry breakings that are spontaneous or i n those that are
induced by a fundamental symmetry breaking i n t e r a c t i o n , Is
almost I d e n t i c a l to each other. I t w i l l a l s o be cl e a r that
the same mechanisms that account f o r octet enhancement i n med
ium strong v i o l a t i o n s would a l s o account f o r octet enhancement
i n electromagnetic e f f e c t s . Hence, i n our studies of Octet
Enhancement, we s h a l l always confine ourselves to spontaneous
symmetry breaking In the strong i n t e r a c t i o n s ; and we believe
that t h i s i s done without loss of generality.
Before plunging i n t o dynamical d e t a i l s , we have, i n
Chapter IV, c a r r i e d out a group t h e o r e t i c a l analysis of spon
taneous symmetry breaking i n an extension of Glashow's work
(S. L. Glashow, 1 9 6 3 ). We do t h i s i n order to f i n d out, as
comprehensively as possible, how much can be said about
spontaneous symmetry breaking that i s Independent of s p e c i f i c
models and dynamical d e t a i l s . A f t e r s t a t i n g and proving a
general theorem i n connection with spontaneous symmetry
6
b r e a k i n g , s e v e r a l i n t e r e s t i n g r e s u l t s emerge a s a p p l i c a t i o n s
o f t h e t h e o r e m . Among t h e s e a r e :
(a) A s t u d y o f t h e ways i n w h i c h SU(3) symmetry c a n be s p o n
t a n e o u s l y b r o k e n i n a b o o t s t r a p i n v o l v i n g t h e b a r y o n
o c t e t a l o n e . (A r e - d e r i v a t i o n o f G l a s h o w ' s r e s u l t ) .
(b) I n b o o t s t r a p s i n v o l v i n g more t h a n one m u l t i p l e t , a s t u d y
I s made o f t h e r e l a t i o n s between the p a t t e r n s o f symmetry
b r e a k i n g i n t h e v a r i o u s m u l t i p l e t s .
(c) I t i s f o u n d t h a t i f a h a d r o n i c t r i p l e t e x i s t s i n n a t u r e
w h i c h i s n o n - t r i v i a l l y c o u p l e d t o t h e r e s t o f t h e h a d r o n s ,
t h e n o c t e t enhancement i m m e d i a t e l y f o l l o w s i n d e p e n d e n t l y
o f any d y n a m i c a l d e t a i l . I n p a r t i c u l a r , t h i s t r i p l e t
n e e d n o t be q u a r k s i n t h e sense o f f u n d a m e n t a l b u i l d i n g
b l o c k s o f a l l m a t t e r .
(d) I n m o d e l s c o n t a i n i n g t h e v e c t o r n o n e t , i t i s f o u n d g r o u p -
t h e o r e t i c a l l y t h a t <j) - CO m i x i n g (M. G e l l - M a n n , 1962b;
S . L . Glashow, 1962; J . J . S a k u r a i , 1962) c a n o c c u r a s a
form o f s p o n t a n e o u s symmetry b r e a k i n g . Making t h e s o l e
a s s u m p t i o n t h a t t h e n o n e t " m a s s - s p l i t t i n g m a t r i x " t r a n s
forms a s a n a r b i t r a r y l i n e a r c o m b i n a t i o n o f o c t e t
components ( t h a t a r e a l l o w e d by c o n s e r v a t i o n o f i s o s p i n
a n d h y p e r c h a r g e ) . The v e r y a c c u r a t e mass f o r m u l a
i s e s t a b l i s h e d w i t h o u t r e c o u r s e t o any d y n a m i c a l d e t a i l .
A f u r t h e r r e l a t i o n i s f o u n d r e l a t i n g the masses o f the
7
n o n e t w i t h t h e m i x i n g a n g l e , w h i c h i s ,
'4w£*-3(covze n<? + Ac*zd wZ>)~r»f = o ( 1 . 2 )
I n p a r t i c u l a r , i f we p u t fry ~ tYi& , we g e t Cta 7B ~ % i n
agreement w i t h r e s u l t s g i v e n by q u a r k models ( s e e , e . g . ;
F. G u r s e y , T . D . Lee a n d M. Na.uenberg, 1 9 6 4 ; G . z w e i g ,
1 9 6 5 ) .
(e) I t i s a l s o shown e x p l i c i t l y how mass s p l i t t i n g s i n t h e
o c t e t p a t t e r n a n d (J) - CO m i x i n g t e n d t o enhance e a c h
o t h e r . T h u s , we would r a t h e r f a v o u r t h e i d e a t h a t t h e r e
i s a " b o o t s t r a p p i n g " o f t h e s e e f f e c t s , t h a n t h e a s s e r t i o n
t h a t one i s t h e cause o f t h e o t h e r a s i n S a k u r a l ' s (j) - CO
m i x i n g model ( J . J . S a k u r a l , 1 9 6 2 , 1 9 6 3 ) .
T h e r e e x i s t now a t l e a s t f o u r d i s t i n c t d y n a m i c a l
t h e o r i e s o f O c t e t Enhancement, w h i c h we a r e g o i n g t o r e v i e w I n
C h a p t e r I I I . Among t h e s e , t h e f i r s t t h r e e — t h e q u a r k m o d e l s ,
t h e $ - OJ m i x i n g model a n d t h e T a d p o l e Model ( S . Coleman and
S . L . Glashow, 1 9 6 4 ) — have b u i l t i n O c t e t Enhancement a t t h e
v e r y o u t s e t . The l a s t m o d e l , the B=A b o o t s t r a p model (R.
Dashen a n d S . F r a u t s c h i , 1 9 6 5 & ) » w h i c h employed t h e a n a l y t i c
t e c h n i q u e t o s t u d y a g r o x i p - t h e o r e t i c a l p r o b l e m , v i z ; t h e way
i n w h i c h a symmetry b r e a k s down, i s t h e o n l y model t h a t s u c c e e d
ed i n g i v i n g O c t e t Enhancement a d y n a m i c a l e x p l a n a t i o n . S i n c e
a l l t h e dynamics have b e e n fed: i n a t t h e v e r y o u t s e t , i t i s
d i f f i c u l t i n t h i s model t o g a i n much i n s i g h t I n t o t h e mechan-
8
isms that bring about Octet Enhancement. In Chapter V, we
s h a l l construct another bootstrap model which, though admit
tedly l e s s r e a l i s t i c when compared with the B-A model,
enables us to see step by step how Octet Enhancement follows
from the i n t r i c a t e working of two simple dynamical require
ments, which are (a) the bootstrap condition and (b) that the
p a r t i c l e s involved i n the bootstrap must be physical p a r t i c l e s
and not ghost st a t e s . In Chapter VI, the model i s extended
to study a phenomenon which can be interpreted as "spontaneous"
0-U) mixing. This, probably, i s the f i r s t model In which
0 - Oi mixing comes out as a dynamical consequence. In addi
t i o n , we again obtained the mass rel a t i o n s (1.1) and (1.2).
Note that i n t h i s dynamical c a l c u l a t i o n , i t i s not necessary
to make assumptions about the octet transformation property
of the symmetry breaking as we have done i n the group theore
t i c a l treatment. This, Octet Enhancement, comes out now as
a dynamical consequence of the c a l c u l a t i o n .
CHAPTER II
EXPERIMENTAL CONSEQUENCES OP OCTET ENHANCEMENT
1. What Is Octet Enhancement?
The remarkable success of SU(3) symmetry, apart from
c l a s s i f i c a t i o n of elementary p a r t i c l e s and resonances into
multiplets, l i e s i n the p a r t i c u l a r l y elegant way In which i t
Is broken. One of the f i r s t successes of the "Eight-fold Way"
i s the so-called Gell-Mann Okubo Formula (GMO Formula) which
was f i r s t derived by Gell-Mann (M. Gell-Mann, 1961) f o r a
unitary octet and subsequently generalized by Okubo (S. Okubo,
1962) to any unitary m u l t i p l e t . I t i s well-known that the
r e l a t i o n s
2W(N)+2rr>(B) = 3tn(A) + rn(z) (2.1)
4 rr>(kf = J w ^ y V ^ r V O 2 (2.2)
m ( Y * ) - t » ( N $ ) = n(s£)-*(Y,*)=»'(n')-»>(Sj£) (2.3)
which follow from the more general formula
{ 1+ *Y+b[l(M~JrY2]} (2.10
where m denotes masses, 1 hypercharge and I t o t a l isospin, are
s a t i s f i e d to a very remarkable degree of accuracy. As a matter
of f a c t , the existence and the mass of the XT was predicted
9
10
( M . Gell-Mann, 1962a) before i t was discovered (Barnes et. a l ,
1964) and i s often c i t e d as the major success of the unitary
symmetry scheme.
I t i s important to r e c a l l , however, that the formulae
( 2 . l ) - ( 2 . 4 ) , the GMO Formulae, are not consequences of SU(3)
symmetry. F u l l unitary symmetry i n f a c t requires members of
the same multipl e t to have the same mass. The GMO Formula a r i s e s
by r e q u i r i n g that SU(3) symmetry i s only approximate and that
i t Is broken i n a p a r t i c u l a r simple way. As we s h a l l see, even
when one requires Isospin and hypercharge conservation to be
preserved, SU(3) symmetry can be broken i n more than one way,
gi v i n g r i s e to formulae quite d i f f e r e n t from the GMO Formulae.
We s h a l l i l l u s t r a t e t h i s as follows. Neglecting the
e f f e c t of electromagnetic and weak inte r a c t i o n s on p a r t i c l e
masses, we s h a l l assume that the strong i n t e r a c t i o n can be
represented by a Hamiltonian Hs^, and that t h i s Hamiltonian
can be e f f e c t i v e l y broken i n t o two parts,
H s t = H s t + H s t m ( 2 * ^
where H g ° i s the so-called very strong part which Is Invariant
under SU(3) transformations, and H ^ i s the so-called medium
strong part which breaks SU(3) symmetry and s t i l l preserves
i s o s p i n and hypercharge conservation. Hence, the mass d i f f e r
ences within an SU(3) multiplet can only come from H^t. We
s h a l l not concern ourselves here with the question whether the
11
medium strong i n t e r a c t i o n s must be introduced at the outset,
or whether they a r i s e by some kind of spontaneous symmetry
breakdown. Let us consider the masses of an octet of baryons,
f o r Instance. We denote the r e s t states of these p a r t i c l e s
by > *~ l ' 2 ' ' We s h a l l make the usual assumption (see, , . m
e.g., J. J. DeSwart, 1963) that transforms as a single
i r r e d u c i b l e tensor I *» where labels the i r r e d u c i b l e
representation of SU(3) and Y, I, I z denotes the hypercharge,
t o t a l i s o s p i n , Z-component of Isospin of the p a r t i c u l a r mem-T- f/O y,l,l^ trans
forms. m
Since E s t must s t i l l be invariant under i s o s p i n
rotations and hypercharge gauge transformations, we must have
Y=I=I z =0, f o r a i r r e d u c i b l e tensor with non-zero hypercharge
w i l l change i t s phase under hypercharge gauge transformations
and the only i s o s p i n invariant object i s an i s o s c a l a r . Hence
one must have,
rts> / 0,0,0 Now, the mass of the baryons are
*Our notations as regards SU(3) follows DeSwart's review a r t i c l e ( J . J . DeSwart, 1963) as c l o s e l y as possible. Readers are r e f e r r e d to the said a r t i c l e f o r matters concerning SU(3) symmetry scheme.
12
— .-f <5"mt- ( 2 e ? )
where ^ i s Independent of i , and
swi = l «r/*> ~ ^ I r4» l*> . (2-8)
We assume that transforms as { 8 J . For S^i to
be non-zero, we know from the Wigner-Eckart Theorem that the
representation must be contained i n the decomposition
W*<?W = \>)®\s\@\8\'®\<o)®\>o}*'®\2i\ (2 .9)
Now the representations \±o\ and { lOJ* contain no
neutral l s o s l n g l e t and hence must be rejected as candidates
f o r I 0 j 0 0 . The only a l t e r n a t i v e s l e f t are
(a) HsT ^ ~To)o)o
w MT ~ T o t
HsT ~. T i t ( o ) Hst- o,o,o
(a), being invariant under SU(3), does not give r i s e to any
mass s p l i t t i n g within the octet. I f one assumes (b), one ends
up with the GMO formula ( 2 . 1 ) , which, as we have said, i s well
s a t i s f i e d experimentally. If one assumes (c), we get the
formula ( J . J . DeSwart, 1963)
yynN + m~ = St*-?- (2.10)
13
which i s not correct experimentally.
Similar situations p r e v a i l , not only i n the baryon
octet, but also i n meson octet and i n other multiplets such as
the baryon decuplet. The correct mass formulae ( 2 . 1 ) - ( 2 . 4 )
emerge as long as we assume that the medium strong i n t e r a c t i o n
transforms as the i s o - s i n g l e t component of an octet. The
phenomenon that the octet transformation property i s p r e f e r r
ed over the 2 7-plet transformation property f o r the i n t e r a c t i o n
Hamiltonian i s known as Octet Enhancement.
2. Octet Enhancement i n Electromagnetic Interactions
only i n medium strong i n t e r a c t i o n s . I t has been observed by
Coleman and Glashow (S. Coleman and S. L. Glashow, 1964) that
electromagnetic mass s p l i t t i n g s a l so show such enhancement.
Let us look at the simple case of electromagnetic mass s p l i t -
The e f f e c t of Octet Enhancement manifests i t s e l f not
t i n g i n the i s o t r i p l e t (2*s 2 ^ J "0 . To preserve charge
conservation, vie know that the mass matrix
Mj* o o
o Mz° o
o o
( 2 . 1 1 )
must be expandable i n the form
14
tensor with I»I Z denoting t o t a l i s o s p i n and the 3 r d component
of i s o s p i n r e s p e c t i v e l y . Terms higher than "]~2 do not appear
on account of the Wlgner-Eckart Theorem.
The matrices ~Jj° are normalised properly as follows:
I t follows from (12) and (13) that
( 2 . 1 3 )
(2.14)
The matrices can e a s i l y be calculated from SU(2)
Clebsch-Gordan (C-G) c o e f f i c i e n t s , y i e l d i n g
T,° = JL ~ l o o POO o o *\
V J L l o o 0 -2 o O O I
( 2 . 1 5 )
( 2 . 1 6 )
From ( 2 . 1 4 ) , ( 2 . 1 5 ) and ( 2 . 1 6 ) we have,
fa-.-Mi*
AM 0 ) =
AM® -
( 2 . 1 7 )
( 2 . 1 8 )
Before we look at the experimental value of A M 0 )
and A M ® t l e t us examine what vie would expect from a crude
t h e o r e t i c a l estimation. We make the usual assumption that the
photon i s coupled to the e l e c t r i c charge of the p a r t i c l e . In
the case of the2"-hyperon, which has zero hypercharge, the
1 5
b a s i c electromagentic Interaction transforms l i k e
charge ^/ T^ 0 ( 2 . 1 9 )
The electromagnetic self-energy of the p a r t i c l e must at l e a s t
be of second order i n electromagnetic Interaction since a photon
emitted must be reabsorbed. Because of the weakness of the
i n t e r a c t i o n , we expect the second order e f f e c t to dominate.
Hence the self-mass transforms as
2 o 2 (charge) ~ (T ) ( 2 . 2 0 )
• i ? T * ' * k l ? ( 2 - 2 i )
Form ( 2 1 ) , one would expect that the electromagnetic mass s p l i t s
t i n g that transforms l i k e T^ 0 would dominate over that trans
forming l i k e Tj 0, since the l a t t e r can only come from higher
order e f f e c t s . That i s , one expects
A M ^ / A M ^ 0 ( 2 . 2 2 )
We have not looked, however, into the d e t a i l e d structure of the
X-nyperons which may somehow suppress the contribution to AM® .*
* I t i s i n t e r e s t i n g to note, i n conjunction with ( 2 . 2 2 ) , that the electromagnetic self-mass of the % -hyperon, due to emission and absorption of a single photon, can be computed unambiguously. Of course, f o r such computations one needs the Z electromagentic form factors which are not a v a i l a b l e experimentally; but these can be deduced from the well-known nucleon form-factors using unitary symmetry. This kind of calculations have been c a r r i e d out by Coleman and Schnitzer (see R» Socolow, 1 9 6 5 ) and they found 1* = -0-7, 2 _-J°= 1-4- . From t h i s we can c a l c u l a t e the r a t i o of AM0> to AWZ) due to a photon exchange
W> I _ 0 59
This i s very f a r from the experimental value given i n ( 2 . 2 5 ) ,
16
In any case, I f experimental r e s u l t s depart too s i g n i f i c a n t l y
from ( 2 . 2 2 ) , we have some explaining to do.
Taking the experimental masses of the ]T-hyperons
and s u b s t i t u t i n g i n t o (2.17) and ( 2.18), we get
A M 0 ) = 5C MeV ( 2 . 2 3 )
= OH bfeV (2.24)
so that
AM(0/AM(Z) S" 8 ( 2 . 2 5 )
This i s very f a r from our expectation ( 2 . 2 2 ) .
Since the strong i n t e r a c t i o n i s mainly responsible
f o r the structure of the hadrons, i t seems l i k e l y , i f there Is
an explanation f o r t h i s kind of anomaly, that t h i s explanation
should come from the strong Interactions. That i s , there i s
something i n the strong interactions that enhances the part of
electromagnetic e f f e c t s that transforms l i k e an i s o t r i p l e t i n
preference to that part which transforms l i k e an isoquintet.
I f we r e c a l l that Coleman and Glashow (S. Coleman and S. L.
Glashow, 196l) derived sum rules f o r electromagnetic mass-
s p l i t t i n g s within unitary multiplets by ignoring medium strong
i n t e r a c t i o n and got very good agreement with experiment, we
showing that there must be something that enhances the A\?S~^ part and suppresses the part i n the electromagnetic s e l f -energy.
17
would be surprised i f the enhancement e f f e c t comes from the
medium strong part of the strong Interactions. So Coleman and
Glashow (S. Coleman and S. L. Glashow, 1964) were forced to
conjecture that the enhancement e f f e c t comes from the very-
strong part of the strong i n t e r a c t i o n s . Now the very strong
i n t e r a c t i o n i s completely invariant under SU(3), so that i n
the absence of symmetry breaking interactions l i k e the medium
strong i n t e r a c t i o n , we have complete isotropy within the same
unitary m u l t i p l e t . In p a r t i c u l a r , one would not be able to
t e l l which member belongs to which is o m u l t l p l e t as long as
they belong to the same unitary m u l t i p l e t . Hence the very
strong i n t e r a c t i o n w i l l be u t t e r l y incapable to enhance the
i s o t r i p l e t over the isoquintet i f they both belong to the same
unitary m u l t i p l e t . But i t can enhance the unitary octet over
the 2 7 - p l e t o In that case, the i s o t r i p l e t w i l l be automati
c a l l y enhanced since the octet contains only i s o t r i p l e t but no
isoquintet although the unitary 2 7-plet contains both. We see
here that the same kind of mechanism operates here as i n the
medium strong i n t e r a c t i o n , v i z ; Octet Enhancement. This was
why Coleman and Glashow coined the name "Universal Octet
Enhancement" f o r such e f f e c t s .
At t h i s juncture one i s tempted to look into other
i s o t r i p l e t s to see whether s i m i l a r anomalies occur. The re
s u l t of such a search i s not very f r u i t f u l due to d i f f e r e n t
reasons f o r baryons and mesons.
18
(a) Baryons; I f we look at other baryon i s o t r i p l e t s
such as the , the experimental error Involved In the measure
ment of t h e i r masses i s so large that sensible values f o r the
masses of d i f f e r e n t members of the t r i p l e t are simply not
a v a i l a b l e . In some cases, the charged members of the t r i p l e t
have r e l a t i v e l y well-defined mass but the mass of the neutral
component has not been determined by any experiment.
(b) Mesons: Let us look at the pseudoscalar t r i p
l e t ( TT+j Tf°* TT~ ). i n t h i s case, the two charged members of
the t r i p l e t are related by charge conjugation and due to CPT
invariance, they must have the same mass. Consequently,
A M 0 ^ i s compelled by a very stringent "kinematic" symmetry to
be zero. Thus i t seems quite Impossible to make a sensible
comparison between A M 0 > and that could i n any way re
f l e c t the i n t e r n a l symmetry or lack of i t i n the mesons. How
ever, b e l i e v i n g that SU(3) Is s t i l l a f a i r l y good symmetry, one
could compare the A M 0 ) as measured i n the K-meson doublet and
the A M ® i n the 7T - t r i p l e t . As suggested by Feynmann, (see
DeSwart, 19&3) o n e should generally use the squared mass f o r
mesons. Then one gets,
which d i f f e r s appreciably from (2.22).
19
S i m i l a r s i t u a t i o n s as i n (a) p r e v a i l as one goes to
higher i s o m u l t i p l e t s such as the/S -quartet. The mass resolu
t i o n between members of the same iso-multiplet i s too poor to
give us a meaningful comparison between the However,
we s h a l l work out the various formulae f o r f o r the cases
1=3/2 and 1=2 f o r future reference. For a r b i t r a r y I, we s h a l l (A)
define the A M 's by
M x ~ i f 4M< ° 7 i ' < 2-2?>
where M i s the mass matrix of the lsomultiplet with i s o s p i n I
and 12 i s the (21+1 ) x (21+1) matrix representation of the i r
reducible tensor with t o t a l i s o s p i n i and the t h i r d component
of i s o s p i n zero. These matrices are normalised as
tr (fc°T/) - S§ (2 . 28 )
Then, i t follows from ( 2 . 2 ? ) and (2.28),
= tr(MZfp ( 2 . 2 9 )
Using these formulae, and employing SU(2) C-G co-
e f f i c i e n t s to work out the various matrices , we have, f o r
I = 3/2 l s o m u l t i p l e t
AM(Z)= ^{(M% + M-#)-(M%+M-%)l ( 2 . 3 0 )
2 0
1 = 2 Isomultlplet
A /vf (*> = j=r ^ M 2 + M-2 + & M 0 - 4-M, - 4 M-i}
J Jio 1
\ . ( 2 . 3 D
Now, I f the notion of Octet Enhancement i s c o r r e c t ,
^ M ^ < £ 2 S M f 0 f o r a l l J l a r g e r than one. Unfortunately, we are
unable to t e s t t h i s r e l a t i o n i n the so-called elementary p a r t i
c l e s . The only 1=3/2 iso=,multiplet we know up to now, the A ,
has mass re s o l u t i o n so poor that i t i s impossible to compute
the / ^ M ^ ' s . And to t h i s date, no 1=2 multiplet has yet been
i d e n t i f i e d .
3» Data from Nuclear Physics
We can look to nuclear physics f o r a d d i t i o n a l data.
We have no reason to believe that there should be any funda
mental difference betv^een what we usually r e f e r to as hadrons
i n p a r t i c l e physics and nuclides i n nuclear physics. Iso-
multiplets of i s o s p i n as high as 3/2 and 2 (see, e.g., Butler
e t . a l , 1 9 6 7 ; G. T. Garvey, J . Cerny and R. Pehl, 1964) have
recently been i d e n t i f i e d i n nuclear physics with f a i r l y w e l l-
defined masses. We could calculate the various A H ^ 1 s by
using ( 2 . 1 7 ) , ( 2 . 1 8 ) , ( 2 . 3 0 ) and ( 2 . 3 1 ) and see whether A M ( 3 }
are r e a l l y small f o r j>l, as they should be i f one requires
21
consistency with Octet Enhancement. We must bear i n mind, how
ever, that the smallness of may be explainable by some
de t a i l e d dynamical calculations i n nuclear physics based on
s p e c i f i c models. In any case, these are necessary consequences
of Octet Enhancement and hence we f i n d i t necessary to check
them. Apart from the f a c t that the v a l i d i t y of the s p e c i f i c
models i s highly suspect, the a p p l i c a b i l i t y of these models
i s a l so of l i m i t e d generality and we f i n d i t somewhat useful
to f i n d a general p r i n c i p l e to explain the l s o m u l t i p l e t mass
s p l i t t i n g In nuclear physics transforms mainly as an i s o t r i p
l e t . We have c o l l e c t e d some of the experimental re s u l t s f o r
isomu l t i p l e t s , calculated the numbers A M & and tabulated
them as follows:
Table I; 1 = 1 isomultiplets
A W 1 * (MeV) ^M ( 2 ) (MeV) | 4 M ( 1 ) / * M ( 2 ) |
10 -2.4 0.4 6
12 -2.655 -0 .1615 .16 .4
14 -2.828 0.245 11
16 -2 .61 0.04 65
20 -4 .15 0 .71 6
22
Table I I ; I = 3/2 Isomultiplets
A AM*1* (MeV) AM ( 2 )(MeV)
7 - 1 . 2 0 - . 8 1 7 14 . 7
9 - 1 . 3 3 2 . 2 2 7 5 . 9
13 -2.180 .208 . 1 0 . 5
21 - 3 - 5 4 5 .126 28 .2
37 - 6 . 1 7 6 .144 42 . 9
Table I I I ; 1 = 2 isomultiplets
A AM ( 1 )(MeV) AM ( 2 )(MeV) |4M ( 1W 2 )| 16
20
- 2 . 9 3 "
- 3 . 6 9
.286
.286
1 0 . 3
1 2 . 9
We see from Tables I, II and I I I A (2) that AM i s
s i s t e n t l y about one order of magnitude smaller than A M ( ;
as required by Octet Enhancement. The A M ^ and /\M 4) , wher
ever they are calculable from the av a i l a b l e experimental data,
are a t l e a s t an order of magnitude smaller than A M ® .
I t i s al s o useful to note that, even i n some of the
simpler nuclides where a f a i r l y r e l i a b l e ' c a l c u l a t i o n of the
coulomb energy can be performed, the coulomb energy so calcu
l a t e d (Ee) f a l l s short by 40 - 5 0 $ of the experimental value A .
23
An example of such a c a l c u l a t i o n i s the one done by Okamoto
on the mass difference of He^ and (K. Okamoto, 1 9 6 4 ) . The
calculated coulomb energy Is found to be E c = 0 . 5 2 ~ 0 . 5 7 MeV,
whereas the experimental mass difference i s A= 0 . 7 6 4 MeV.
Similar r e s u l t s are shown to hold a l s o f o r heavier n u c l e i with
A > 3 » * I f such r e s u l t s are r e a l l y r e l i a b l e , then there i s
charge dependence i n the interactions between the nucleons
that cannot be accounted f o r by a straightforward c a l c u l a t i o n
with electromagnetic i n t e r a c t i o n alone. In other words, there
Is a small (apparent) charge dependence i n the nuclear force
i t s e l f . I t was suggested that, e.g., vector mixing between
J°6and oo' would give r i s e to diagrams l i k e F i g . I which would
give a charge dependent force that i s short ranged and that
w i l l not be incorporated by doing a simple coulomb energy c a l
c u l a t i o n .
F i g . I
Calculations with such potentials have a c t u a l l y been
*For reference see: K. Okamoto i n "Isobaric Spin i n Nuclear Physics" - ed. Fox-Eobson, Pg 659-676. The table given i n Pg 675 of the said a r t i c l e shows consistently that Ec i s smaller than A by a sizable f r a c t i o n (about 20 - JQ%).
24
performed by Stevens (M. S t . J . Stevens, 1965) who attempted
to explain the discrepancy between A and Ec basing on TT 0-^
mixing. The r e s u l t was considerably improved by taking into
account U)'-f° mixing (B. W. Downs and Y. Nogami, 1967) •
Okamoto and Lucas (K. Okamoto and C. Lucas, 1 9 6 ? ) estimated
the e f f e c t of the above mechanisms on nucleon - nucleon
scattering and found that i t i s consistent with the near
equality of p-p and n-n scattering lengths found experiment
a l l y . Now, as we s h a l l see i n l a t e r chapters, to'-y^mixing
can both be regarded as cause and consequence of Octet
Enhancement i n electromagnetic i n t e r a c t i o n . Consequently,
the above r e s u l t s i n nuclear physics lend further support to
the idea of Octet Enhancement i n hadronic Interactions.
There are further consequences of Octet Enhancement
i n electromagnetic i n t e r a c t i o n s . For example, r e l a t i o n s can
be found between mass s p l i t t i n g s i n d i f f e r e n t isomultiplets
within the same unitary m u l t i p l e t . For a discussion of t h i s ,
readers are referred to the a r t i c l e by Coleman and Glashow
(S. Coleman and S. L. Glashow, 1 9 6 4 ) .
When the idea of Octet Enhancement was f i r s t pro
posed (S. Coleman and S. L. Glashow, 1 9 6 4 ) , i t was hoped that
i t would help one understand the|All = §• rule (N. Cabbibo,
1964) i n non-leptonic weak i n t e r a c t i o n s . I t was found l a t e r
that t h i s leads to contradictions (see S. Coleman, 1 9 6 6 ) .
We have therefore concentrated on medium strong and e l e c t r o magnetic v i o l a t i o n s .
CHAPTER III
VARIOUS THEORIES OF OCTET ENHANCEMENT
In t h i s Chapter, the e s s e n t i a l ideas of e x i s t i n g
theories of Octet Enhancement are~reviewed and discussed.
These include (1) the quark model, (2) the mixing model,
(3) the tadpole model and (4) the bootstrap model.
1 . The Quark Model
The quark model (see, e.g. M. Gell-Mann, 1 9 6 4 ; F.
Gursey, T. D. Lee and M. Nauenberg, 1 9 6 4 ; G. Zweig, 1965)
provide easy "derivations" of the Gell-Mann Okubo Formulae.
We s h a l l describe the simplest of these as an example (G.
Zweig, 1 9 6 5 ) » In t h i s p a r t i c u l a r model, a l l the hadrons are
supposed to be b u i l t up of 3 quarks a l t a 2» a 3 and t h e i r a n t i -
p a r t i c l e s a 1 , a 2 , a^. The quarks a.^ transform according to the
defining representation J 3 | of SU(3) and the anti-quarks a 1
i n accordance with ^3 } ' *« Let us consider the simple case of
the octet of pseudoscalar mesons, which are considered to be
quark-antiquark bound states, with
|7r*>= l^a,} . J T T - > =
25
26
To derive the "medium strong" mass s p l i t t i n g , one assumes
sfri(at) = rn(a') = fn^) ~ r"(az) - m0 ( 3 . 2 )
yfyt(as) - rvi(o}) =(l + A)rn0 ( 3 . 3 )
where m denotes the mass.
Now,
where E 1 ^ i s the binding energy between a* and a^. Making the
assumption that a l l mass s p l i t t i n g flows from the assumed mass
difference between a^ and the other two quarks, so that a l l
dependence of on 1 and j comes from A, one gets
Ey = E0+A(8C
3+&\)E' + 0(A>) ( 3 . 4 )
( 3 . 2 ) and ( 3 . 3 ) can be summarized by
4n(a*)= 0 + S{3A)ni0 ( 3 . 5 )
m(*f) = (l+fyA)w0 ( 3 . 6 )
Hence m (a
caj) = 2 ^ 0 - E 0 + (Sc
g + 6*d) (M0-E')A + 0(1?) ( 3 - 7 )
Using ( 3 . 1 ) and (3«7) i we can calculate the masses of 7 T , K and ^
(m(rr) = r»(n+) = 2^0 -EP *- 0(A2)
ry)(k) =M(kt)= 2 m 0 - E 0 + A(mp-E')+0(Az)
= 2m0-Eo +jA(n0"E') +0CA 2)
27
From ( 3 « 8 ), one gets the r e l a t i o n
<n>(rr) + StnCn) = £<m(k) ( 3 . 9 )
which i s the Gell-Mann Okubo Relation. GMO equations f o r
other hadronlc multiplets are also obtainable from the quark
models. D e t a i l s are a v a i l a b l e i n the Interesting a r t i c l e by
G. Zweig (G. Zweig, 1 9 6 5 ) . Similar conclusions about octet
enhancement from quark model were obtained recently under less
r e s t r i c t i v e conditions (see, e.g., P. Feldman, H. R. Rubinstein
and I. Talml, 1 9 6 6 ; H. R. Rubinstein, I 9 6 6 ) . In Chapter IV,
we s h a l l show that i f there exists i n nature a hadronic t r i p
l e t which i s n o n - t r i v l a l l y coupled to the rest of the hadrons,
then Octet Enhancement immediately follows. The conclusion i s
based on a group t h e o r e t i c a l argument and i s independent of
any dynamical d e t a i l . In p a r t i c u l a r , t h i s t r i p l e t need not be
quarks In the sense of fundamental b u i l d i n g blocks of a l l
matter.
Unless and u n t i l t r i p l e t s are discovered experiment
a l l y , the foregoing cannot be considered a serious theory of
Octet Enhancement.
2. The Mixlna; Model
If SU(3) symmetry i s exact, the quantum numbers
l a b e l l i n g the various unitary multiplets — the eigenvalues
of the Caslmir Operators (see DeSwart, 1963) — w i l l be conserved and no mixing between the various multiplets can occur.
28
In p a r t i c u l a r , the two i s o s l n g l e t vector mesons known experi
mentally should be I d e n t i f i e d as a unitary s i n g l e t and a u n i t
ary octet component respe c t i v e l y . But i n the case when SU(3)
symmetry i s only approximate, the two physical i s o s l n g l e t
vector mesons can be l i n e a r combinations of the two states with
d e f i n i t e SU(3) transformation properties. This phenomenon i s
c a l l e d 0~UJ mixing. This may happen when there i s a fundamental
SU(3) breaking i n t e r a c t i o n or t h i s can happen spontaneously.
As we s h a l l show i n a general group t h e o r e t i c a l discussion i n
Chapter IV, mixing can i n p r i n c i p l e occur as one of the
possible forms of spontaneous symmetry breaking. That i t
a c t u a l l y comes out of a dynamical c a l c u l a t i o n w i l l be demons
trated i n a s p e c i f i c model i n Chapter VI.
I t was observed by Sakurai ( J . J . Sakural, 1 9 6 2 , 1963)
that neither the 6J(780) or the 0(1020) vector meson possesses a
mass s a t i s f y i n g the GMO Formula when one takes the 'p and the
K* mesons to complete the octet. This l e d him to suggest that
the U) and the fi mesons are l i n e a r superpositions of the form
' \4>y ~ c^e\co8} + s^Qlco'} ( 3 . 1 0 )
where [OJ'y i s a pure unitary s i n g l e t , and \C0*y i s the 1=0 mem
ber of a pure unitary octet.
Since ( 0y and f oS? are physical states, the mass matrix
29
with these two states as basis w i l l be diagonal. Because of
( 3 « 1 0 ) , t h i s w i l l not be true f o r the mass matrix with respect
to the b a s i s { |o>8>, l b ) 1 } } . In fa c t , i t w i l l be of the form
M = ( 3 . 1 D
8 1
The f i e l d operators of the "vector mesons" 0d and 60
w i l l s a t i s f y a Proca Equation (see P. Roman, 1 9 6 4 , pg 105-106)
[cv'j [T«). (3-12)
where CO^, 00^ j J J . - tj2,34, are the f i e l d operators and Jjf'*, t h e i r sources. Sakural assumed SU(3) symmetry apart
from 0-&> mixing, so that J^ a } and j£ S i transform l i k e a
unitary s i n g l e t and the 1=0 component of a unitary octet re
spec t i v e l y . The e f f e c t i v e Langrangian that gives r i s e to
( 3 . 1 2 ) w i l l contain a term,
This i s the only part of the f u l l Langrangian that i s non-
invariant under SU(3) and since C0^ i s SU(3) invariant and
OJpi transforms l i k e the 1=0 component of an octet, the
Hamiltonian EA w i l l transform l i k e the 1=0 component of an
octet. H^ w i l l give r i s e to t r a n s i t i o n s between and
\tiS*~y which are represented by the diagram shown i n F i g . I .
30
F i g . I
To obtain Octet Enhancement, Sakural ( J . J . Sakurai,
1963) assumed that the p r i n c i p a l contribution to symmetry
breaking i s due to Interactions involving F i g . I. For ex
ample, the mass s p l i t t i n g s of the baryons, pseudoscalar mesons
and vector mesons were assumed to be due mainly to self-energy
graphs of the form shown i n F i g . I I .
FIG. II
The v e r t i c e s represented by the b i g c i r c l e s O a^e SU(3) i n
va r i a n t . The heavy l i n e s connecting two such v e r t i c e s stand
f o r a l l strongly i n t e r a c t i n g states with the correct quantum
numbers. Each of the matrix element represented by diagrams
i n F i g . II has octet transformation properties. Hence the
self-mass matrix of any of these multiplets w i l l transform
l i k e the 1=0 component of an octet. And t h i s , as we have
seen, gives r i s e to the GMO Formula.
31
We s h a l l I l l u s t r a t e t h i s l a s t point by a s p e c i f i c
example. Let us take the diagram i n F i g . II (a) f o r the
baryon octet and consider the contribution to s e l f mass due
to the baryon octet only f o r s i m p l i c i t y . Assume that the BBV
coupling i s cx times D-coupling and (/-oO F-coupllng (see S.
Gasiorowlcz, 1966, pg. 281). The relevant part of t h i s SU(3)
invariant Interaction i s
and the mixing e f f e c t i v e Lagrangian i s
(3.14)
XA 00s +• cosco'j ( 3 . 1 5 )
We have l e f t out the spinor matrices and coordinate indices
f o r s i m p l i c i t y . With these, the self-mass to the baryon octet
can be calculated, r e s u l t i n g i n the self-mass matrix. 0 0 »-*
StnA
3-4*
3-4*
2a 2a
2a -3+2*
<-3+2cx
-2*
( 3 . 1 6 )
32
The mass matrix on the right-hand side of ( 3 . 1 6 ) transforms as
the 1=0 component of a unitary octet. As a matter of f a c t , i t
can be e a s i l y checked that the mass-splittings obey the GMO
Formula
2(SnN+&ns) ** 3*MA + Z>r>2 ( 3 . 1 7 )
To account f o r Octet Enhancement i n electromagnetic
mass s p l i t t i n g (L. E. Picasso, L. A. E a d i c a t t i , D. P. Zanello
and J . J . Sakurai, 1 9 6 5 ), i t i s only necessary to assume that
there i s a mixing between f° and CA)' , which w i l l be described >
by the e f f e c t i v e Hamiltonian,
fls = £msi'(vrfM 0+fr o*>r) ( 3 . 1 8 )
Since t h i s i n t e r a c t i o n v i o l a t e s not only SU(3) sym
metry but also charge independence, the coupling tV^/ i s 2 2
expected to be of order 6 , where e i s the e l e c t r i c charge
of the electron.
One d i f f i c u l t y about t h i s model i s that i t i s necess
ary to assume that the mixing between f b and CO8 i s very small
since otherwise i t would be d i f f i c u l t to understand why the
Coleman-Glashow sum rule (S. Coleman and S. L. Glashox?, 1 9 6 l ,
1 9 6 4 ) , i s so well s a t i s f i e d . If f°-C0 8 mixing i s not n e g l i g i b l e ,
then there w i l l be a term i n the Hamiltonian
( 3 . 1 9 )
33
with transformation property given by
where the states are l a b e l l e d asj{n},Y,I,Iz^> . Hence, i f an
i n t e r a c t i o n l i k e ( 3 . 1 9 ) e x i s t s , there w i l l be terms i n the
electromagnetic mass s p l i t t i n g that transforms l i k e {27} as
well as terms l i k e .{lO} and {lOj*. (The " i s o t r i p l e t enhance
ment" pattern, however, i s s t i l l I n t a c t ) .
One simple-minded way to explain t h i s small mixing,
i s to note that a f t e r medium strong mass s p l i t t i n g has set i n , 1 8 the mass of 00 l i e s c l o s e r to j> than does (A> (see F i g . 3 of
J . J . Sakurai, 1 9 6 3 ), rendering a mixing between 601 and o 0
easier than between 60 and f *
3 . The Tadpole Model
To account f o r Octet Enhancement, Coleman and
Glashow (S. Coleman and S. L. Glashow, 1964) postulated the
existence of an octet of s c a l a r mesons, an i s o s l n g l e t ,
an i s o t r i p l e t TC* and a doublet K/ with a n t i - p a r t i c l e s K ' .
As we s h a l l see, the so-called "scalar mesons" need not be
r e a l physical p a r t i c l e s i n order to produce the desired
e f f e c t . The s c a l a r i s o s l n g l e t , except f o r the f a c t that
i t transforms l i k e a member of an octet and that i t may have
non-zero mass, has exactly the same quantum number as the
34
vacuum, i.e Q=Y=I=Iz=0, J = 0 + , G=l and so on.
Hence, i f an e f f e c t i v e SU(3) breaking medium strong
i n t e r a c t i o n exists (this i n t e r a c t i o n may be fundamental or
may come from a spontaneous symmetry breaking) which contains
a part transforming l i k e the 1=0 member of an octet, there i s
no a p r i o r i reason to suppose that a v i r t u a l /^ / carrying zero
4-momenta cannot be created from vacuum. Subsequently there
e x i s t s the p o s s i b i l i t y of diagrams of the form shown i n
F i g . I l l , which, f o r obvious reasons, are usually referred
to as tadpole diagrams.
I f such diagrams e x i s t , then there i s a class of
Feynmann diagrams that contribute to symmetry v i o l a t i n g pro
cesses. These diagrams are shown i n F i g . IV.
F i g . I l l
(a) (b) (c) FIG. IV
35
The tadpoles are connected to the oval-shaped vertex
parts, which are assumed to be SU(3) Invariant. Because of
the appearance of , the whole amplitude transforms l i k e the
1=0 member of an octet. The fundamental assumption made by-
Col eman and Glashow i s that symmetry v i o l a t i n g processes are
dominated by symmetry breaking tadpole diagrams.* Evidently,
Octet Enhancement i n electromagnetic i n t e r a c t i o n can be ac
counted f o r by replacing the ^ ' l i n e i n F i g . IV by Tt'° meson.
There i s one advantage here of the tadpole model
over the mixing model. As we have seen, to account f o r
Octet Enhancement i n electromagnetic i n t e r a c t i o n using the
mixing model, i t i s necessary to assume, with no convincing o S
argument, that the mixing between j> and CO i s n e g l i g i b l e .
The same problem does not a r i s e here i n the Tadpole Model,
f o r the vacuum i s always a unitary s i n g l e t , and the only "mix
ing" that could occur i s that between the vacuum and a v i r t u a l
octet meson.
One condition f o r the tadpoles to dominate i s -
that the B B / r f / » PP^'and V V * l ' v e r t i c e s as represented by the
oval-shaped blobs i n F i g . IV must be large functions of t near
t=o, where t i s the square of the 4-momentum transferred to
the 7^ i n more general processes. In p a r t i c u l a r , the T T T T ^ '
vertex must be large near t=o. Now i n the Tt-TC scattering
*That t h i s mechanism can be responsible f o r the success of the GMO Formula has been suggested previously by J . J . Sakurai ( J . J . Sakurai, 1963)
36
problem, t h i s means that the S-wave scattering length must be
l a r g e . This, however, w i l l jeopardize the usual procedure i n
s o f t pion c a l c u l a t i o n s , (see M. Suzuki, 1965. H. Sugawara,
1965; C. G. Callen and S. B. Trleman, 1966; K. Kawarabayashi
and M. Suzuki, I 9 6 6 ; S. K. Bose and S. U. Biswas, 1966; Y.
Hara, Y. Nambu and J . Schechter, 1966; V. S. Mathyr, S. Okubo
and L. K. Pandit, 1966; M. Baker, 1966) which are nowadays
believed to be so successful, of extrapolating the pion mass
to zero. (I am indebted to Prof. S. Coleman f o r pointing t h i s
out to me i n a conversation held at Erice, 1967) . (See also
S. Weinberg, 1966). The case Is not completely l o s t yet f o r
the Tadpole Model. As a matter of f a c t , one could twist the
argument around and to use i t as a p l a u s i b i l i t y argument why
the asymmetric s o l u t i o n i n a theory with spontaneous symmetry
breaking can be more stable than the symmetric s o l u t i o n . Let
us denote the f i c t i t i o u s pions i n the symmetric s o l u t i o n by
7T^ and the r e a l physical pions by Tl . I f i n the symmetric
world, the TT(S* has the average mass of the physical pseudo-
s c a l a r mesons, we s h a l l have hnnco ^ ^rr . For Octet
Enhancement i n spontaneous symmetry breaking, we assume tad
pole dominance i n the symmetric solution, which i n turn implies
a large TTts)-TC^ s c a t t e r i n g length i n the S-state. Now a
large s c a t t e r i n g length means the existence of e i t h e r a bound
or anti-bound state or a resonance near threshold. Let us
take the case of a resonance f o r s i m p l i c i t y and without loss
of generality. Let us assume also that the s h i f t i n p o s i t i o n
37
of the resonance i s small when symmetry breaking sets i n .
Then we have i n the phys i c a l world a resonance which i s f a r
above the TV-TV threshold and hence there w i l l be no large
s c a t t e r i n g length which w i l l i n v a l i d a t e the soft-pion extra
p o l a t i o n procedure. Furthermore, due to the smallness of the
scattering length, any perturbation, including those trans
forming l i k e a member of an octet, w i l l not be s i g n i f i c a n t l y
enhanced. In p a r t i c u l a r , an "octet perturbation" i n the op
posite d i r e c t i o n , v i z . , the "symmetry re s t o r i n g " d i r e c t i o n ,
w i l l not be enhanced. Hence, we have a s i t u a t i o n where, s t a r t
ing from a symmetric solution, a perturbation transforming
l i k e an octet w i l l be enhanced by the tadpole mechanism.
But once the symmetry breaking has gone f a r enough, the enhance
ment e f f e c t w i l l fade out, so that any further symmetry break
ing or symmetry restoring w i l l not be boosted. Thus, we end
up with a stable configuration that i s not SU(3) symmetric.
I t must be emphasized that above arguments are extremely
speculative and must not be taken too seriously at t h i s stage.
3 . The Bootstrap Model*
The problem of the emergence of the GMO Formula was
f i r s t discussed by Cutkosky and Tarjanne (R. E. Cutkosky and
P. Tarjanne, 1963) i n the context of bootstrap dynamics.
*For a b r i e f introduction to the idea of a "Bootstrap", see Section 1 of Chapter V.
38
This was l a t e r taken up by Dashen and S. Frautschi, ( 1964a ;
1965a) and discussed very extensively i n the p a r t i c u l a r case
of the 8-A s t a t i c boostrap model. (See Chapter V, $ 4 , ( i l l ) )
Consider a bootstrap system of N supermultlplets
(SU(3) multiplets) of hadrons, l a b e l l e d by h, , h 0 , h , ..... x * 3
h N r e s p e c t i v e l y . Let the dimensions of the SU(3) Irreducible
representations corresponding to these supermultlplets be d- ,
dg. , d N r e s p e c t i v e l y . We write down a set of bootstrap
equations that possess SU(3) symmetry and assume that there
ex i s t s a s o l u t i o n which r e f l e c t s the f u l l symmetry of the
system, by which we mean that each of the supermultlplets has
degeneral mass, and that the coupling constants are related
by SU(3) C-G c o e f f i c i e n t s . Let us consider now the case where
another s o l u t i o n exists with masses and coupling constants
s l i g h t l y d i f f e r e n t from the symmetric ones and expand our
equations i n terms of these small s h i f t s from the symmetric
values. We s h a l l neglect coupling constant s h i f t s i n the
following discussion f o r s i m p l i c i t y . The mass s p l i t t i n g i n
each of the multiplet h^ can be represented by a d^ x d^
matrix
AM 1 ( 3 . 2 1 )
which i n general can be decomposed into d^ x d£ matrix repre
sentation of i r r e d u c i b l e tensors of SU(3)• (We s h a l l c a l l these
" i r r e d u c i b l e matrices" from now on f o r b r e v i t y ) . Thus we write
3 9
AM^'=^ AM±%f ( 3 . 2 2 )
where Ty^n i s "the x d^ i r r e d u c i b l e matrix that transforms
l i k e the YI*** component of the SU(3) i r r e d u c i b l e representation
j^tj and AMffn i s a numerical c o e f f i c i e n t . Since the super-
mul t i p l e t h^ transforms as ^di\ , the mass matrix AM^1 trans
forms l i k e jdij*® [dc] (See Chapter IV, 1). Hence, i n the
decomposition ( 3 . 2 2 ) , the summation should run over a l l the
i r r e d u c i b l e representations that appears i n the C-G series of
If a representation appears more than once
i n the C-G series f o r some h^, we s h a l l have to introduce an
extra l a b e l c*. Hence, we have to modify ( 3 . 2 2 ) &s follows:
In a bootstrap problem, the s h i f t s i n the bound
state p o s i t i o n AM^'^^^ depend on the mass s h i f t s of the
i n t e r a c t i n g and exchanged p a r t i c l e s /Sp/} ° . Hence, we
have a set of equations
For s e l f consistency, we have
AM Ah/1^n ( 3 . 2 5 )
40
Imposing SU(3) symmetry on ( 3 . 24 ) , we have
(i) The A matrix elements are independent
of m and n .
( l i ) The A matrix elements f o r V^p are zero
Hence we have
J M ^ = I ^ ^ ^ (3.26,
Dropping a l l indices i n ( 3 . 2 6 ) , we can write i t i n
the matrix form
« A,* - ( 3- 2 7 )
For spontaneous mass s p l i t t i n g to occur i n accord-
T' — ^ i fJ-jn f o r
each h i , i t i s necessary and s u f f i c i e n t f o r J\^.to have an
eigenvalue equal to one. The corresponding eigenvector i s of
course AM^,n
Thus f a r we have confined ourselves to spontaneous
mass s p l i t t i n g . Consider a s i t u a t i o n where we can introduce
an extraneous i n t e r a c t i o n other than those accounted f o r by
bootstrap dynamics. An example of t h i s i s the electromagnetic
i n t e r a c t i o n . We can again decompose t h i s contribution accord
ing to SU(3) transformation properties, then (3*26) must be
modified to
AM?;n = Z A f - ^ m f „ (3-28)
41
Or i n matrix form
AMf,» = A? • AM^,„ + Dy,n <3- 29>
D^u,n represents the e f f e c t s due to the extraneous i n t e r a c t i o n ,
and i s usually referred to as the " d r i v i n g term." (R. Dashen
and S. Frautschi, 1 9 6 4 a ,b 1 9 6 5 a ). Now, l e t us suppose that
Muhas an eigenvalue near one, then a small d r i v i n g term
w i l l give r i s e to a large mass s p l i t t i n g AM^n (with the same
fj. and n ). In that case, we say that there i s Enhancement.
Supposing i n p a r t i c u l a r that we have a problem where we can
write down a set of equations
AH2lm = (I-Ati)'1-])^,,, (3.30)
and a f t e r d e t a i l e d dynamical calculations, we f i n d that Ag has
an eigenvalue near unity, whereas Agr, and other A matrices
possess no such eigenvalue. Then perturbations Dg n and m
of comparable magnitude would produce very large mass s h i f t s
i n the octet pattern and small mass s h i f t s i n the 27-plet
pattern. This, suggested Dashen and Frautschi, might be the
reason f o r octet enhancement i n the electromagnetic i n t e r a c t
ion . Unfortunately, the SU(3) symmetric bootstrap c a l c u l a t i o n
42
cannot be performed to such a degree of p r e c i s i o n as to de- .
cide whether A 0 has an eigenvalue exactly equal to one, thus o making spontaneous "medium strong" mass s p l i t t i n g possible,
or that the eigenvalue i s only approximately unity so that a
fundamental medium strong i n t e r a c t i o n , f o r example Ne'eman*s
f i f t h i n t e r a c t i o n (Y. Ne'eman, 1964) , i s necessary f o r a
d r i v i n g term.
Using the above formalism, Dashen and Frautschi
calculated the A-matrlces i n the best known bootstrap system
to date, namely the B-A s t a t i c bootstrap and found that Ag
has an eigenvalue near unity, whereas has no such eigen
value. Thus they were able to e s t a b l i s h Octet Enhancement i n
t h i s p a r t i c u l a r bootstrap. Note also that i f there i s octet
enhancement f o r medium strong v i o l a t i o n s , there must also be
octet enhancement f o r the electromagnetic e f f e c t s , and vice
versa. This i s because the A matrices depend only on/*- and
not on li , i . e . , on the multiplet and not on a p a r t i c u l a r
member of the m u l t i p l e t . Since Ag has an eigenvalue near
unity, one sees from (3*30) that a d r i v i n g term D^ 3 trans
forming l i k e the 3 r d component of a unitary octet w i l l be
p r e f e r r e n t i a l l y enhanced over the corresponding component i n
the 2 7-plet. Thus the electromagnetic i n t e r a c t i o n , which pre
sumably gives r i s e to d r i v i n g terms transforming l i k e octet
and 2 7-plet of comparable magnitude, can only produce effects
transforming predominantly l i k e an octet component i n a
43
hadronic system. The source of t h i s enhancement e f f e c t l i e s
with the A-matrlx, which evidently depends only on the very
strong, SU(3) symmetric i n t e r a c t i o n . Similar, a medium
strong i n t e r a c t i o n ( i f i t exists) transforming l i k e the eighth
component of an octet w i l l be p r e f e r e n t i a l l y enhanced. In
the case where Ag has an eigenvalue exactly equal to one,
spontaneous mass-splitting w i l l occur according to the octet
pattern, g i v i n g r i s e to the GMO Formulae.
The bootstrap model enjoys at l e a s t one advantage
over the other models. In the boostrap model, octet dominance
comes out of a dynamical c a l c u l a t i o n ; whereas i n other theories
Octet Enhancement i s fed i n at the beginning. I t i s to a
large extent correct to say that both i n the $-10 mixing model
and the tadpole model, what has been achieved i s the formula
t i o n of the problem i n other terms. I t remains f o r us to
understand why 0-W t r a n s i t i o n diagrams i n one case and tadpole
diagrams i n the other should dominate symmetry breaking pat
terns. These explanations should come from a dynamical calcu
l a t i o n . Most people would agree that 0-W mixing does occur
and the doubtful point Is whether the whole pattern of SU(3)
symmetry breaking flows from i t . The c a l c u l a t i o n of Dashen
and Frautschi casts serious doubts on t h i s , because they
showed that pure bootstrap dynamics could at l e a s t be p a r t l y
responsible f o r Octet Enhancement. The Tadpole Model suffers
from the further d i f f i c u l t y that no sc a l a r meson has yet been
44
found experimentally* A counter point to t h i s i s that the so-
c a l l e d "scalar mesons" need not be r e a l physical p a r t i c l e s
manifesting themselves as resonances or bound states i n scatter
ing processes. They may be anti-bound states, f o r example,
which manifest themselves as poles i n the sc a t t e r i n g amplitude
on the unphysical sheet below threshold. The existence of
such states causes threshold enhancement, which i s the only
thing we need f o r tadpole dominance.
Enhancement should encompass a l l the above e f f e c t s , with any
one among them complementing and further enhancing the others.
I t has been shown by Dashen and Frautschi (R. Dashen and S.
Frautschi, 1965b) that i f an octet of sc a l a r mesons does exi s t
and tadpole diagrams involving the creation of these mesons do
dominate symmetry breaking, then the matrix Ag w i l l automati
c a l l y possess an eigenvalue near unity. I t has been shown by
Picasso et. a l (L. E. Picasso et. a l , 1965) that i f there
exists a strong i n t e r a c t i o n i n the 6J*-V channel (where V i s an
octet vector meson), producing a resonance i n a J p=0 state,
then 0-CO mixing would produce sca l a r tadpoles. Then domin
ance of self-energy diagrams involving 0-0) mixing reduces to
dominance of tadpole diagrams l i k e F i g . V.
I t seems probable that a true theory of Octet
F i g . V F i g . VI
4-5
Conversely, the existence of an l\ f tadpole can 1 8
bring about an 00 -CO t r a n s i t i o n . The dominance of such tad
poles increases the p r o b a b i l i t y of OJ-0 mixing. This i s i l l u s
t rated i n F i g . VI. One can already see how 0-0) mixing, tad
pole diagrams and bootstrap dynamics can conspire to give r i s e
to a large octet enhancement.
CHAPTER IV
GROUP THEORETICAL TREATMENT OF SPONTANEOUS SYMMETRY BREAKING
1. Spontaneous Symmetry Breaking
Let us suppose that f o r a s p e c i f i c problem i n Physics,
we can write down a basic set of equations which i s invariant
under a c e r t a i n symmetry group (Jj . Then we would expect that
solutions of these equations would r e f l e c t the f u l l symmetry
of the basic set of equations. I f f o r some reason, t h i s i s not
the case, i . e . , there exists a so l u t i o n which r e f l e c t s some
asymmetries with respect to the group (Jj , then we say that a
Spontaneous Symmetry Breaking has occurred.
Such a p o s s i b i l i t y was discussed by Heisenberg and
his co~workers (W. Heisenberg, 1 9 5 8 ; H. P. Durr, W. Heisenberg,
H. M i l t e r , S. Schlieder and R. Yamayaki, 1 9 5 9 . 1 9 6 l ) . They
pointed out that the equations of quantum f i e l d theory are non
l i n e a r operator equations. Since non-perturbative solutions to
non-linear equations do not i n general possess the f u l l symmetry
of the equations themselves, i t i s conceivable that the f i e l d
equations may be highly symmetric expressions, while t h e i r
solutions may r e f l e c t the asymmetries of nature. Several c a l
culations have been performed by various authors to confirm
the above conjecture of Heisenberg et. a l . . These calculations
have been done with s p e c i f i c models. Jona-Losino and Nambu
4 6
47
(G. Jona-Loslno and Y. Nambu, 1 9 6 l a , 1961b) considered a
theory with a Langrangian possessing Iff -invariance and found
that, although the basic Langrangian contains no mass term
tntyty since such terms v i o l a t e Iff -invariance, a solu t i o n
e x i s t s that admits fermions of f i n i t e mass. A c a l c u l a t i o n
with s i m i l a r conclusion was performed independently by
Goldstone ( J . Goldstone, 1 9 6 1 ). Baker and Glashow (M. Baker
and S. L. Glashow, 1962) considered a theory based on the
Dyson Equations (F. J . Dyson, 1 9 4 9 ; J . Schwinger, 1 9 5 1 ) , a
set of coupled non-linear equations r e l a t i n g the one-particle
Green's functions and the vertex functions f o r a set of p a r t i
c l e s , which were assumed to possess SU(3) symmetry. They
found that non-perturbative solutions e x i s t that contain
multiplets with respect to the symmetry group possessing non-
degenerate masses. Their formulation of the problem i s
e s s e n t i a l l y a bootstrap requirement since they required that
the physical masses are completely dynamical i n o r i g i n . (Tech
n i c a l l y t h i s means s e t t i n g the bare masses to zero). Their
conclusions, however, are not completely conclusive since to
overcome divergence d i f f i c u l t i e s , they had to employ cut-offs
In some of the i n t e g r a l s and assume the dominance of a c e r t a i n
class of Feynmann diagrams. Similar calculations have also
been performed by other authors, some i n the f i e l d t h e o r e t i c a l
framework (see, e.g., M. Suzuki, 1 9 6 3 , 1 9 6 4(a), 1 9 6 4(b)),
others i n terms of S-matrlx theory (see, e.g., D. Y. VJong,
1 9 6 5 ; Dashen and Frautschl, 1 9 6 5 ) .
48
2. Group T h e o r e t i c a l Treatment of Spontaneous Symmetry;
Breaking In an SU(3) Symmetric Bootstrap Containing
Eight Baryons
Glashow (S. L. Glashow, 1963) discussed group theo-
r e c t i c a l l y a bootstrap problem of eight baryons i n t e r a c t i n g
with each other with an Interaction invariant under SU (3)•
He found that, i f spontaneous mass-splitting does occur under
the condition that i s o s p i n and hypercharge are s t i l l conserv
ed, then mass s p l i t t i n g must occur according to one of the
f olloitfing patterns:
(a) The ma s s - s p l i t t i n g matrix transforms as a unitary
s i n g l e t : i n t h i s case we do not have any mass s p l i t t i n g .
The octet remains degenerate.
(b) The mass-splitting matrix transforms as the 1 = 0 , Y = 0
component of octet (the eight-dimensional i r r e d u c i b l e
representation of SU(3): i n t h i s case we have the G e l l -
Mann Okubo Formula (Eq. 2 . 1 ) .
(c) The mass s p l i t t i n g matrix transforms as the 1 = 0 , Y = 0
component of a 2 7-plet (the 27-dimenslonal i r r e d u c i b l e
representation of SU(3))s i n t h i s case we have a mass
s p l i t t i n g where the masses s a t i s f y the 2 7-plet Formula
(Eq. 2 . 1 0 ) .
I t i s Important to emphasize that Glashow fs treat
ment i s independent of the de t a i l e d dynamics of the system
49
and that
(i) I t does not show that spontaneous symmetry breaking
does i n f a c t occur, but only that i f i t does approxi
mate sum rules w i l l be s a t i s f i e d ,
( i i ) I t has not answered the question why symmetry breaking
i n accordance with (b) rather than (c) happens i n
nature. In other words, Octet Enhancement has not
been established.
No convincing answer seems to be a v a i l a b l e f o r ( i ) .
As f o r ( i i ) , the answer may l i e with the d e t a i l e d dynamics
as presented i n the various theories of Octet Enhancement i n
the previous chapter. In Chapter V, we s h a l l demonstrate
very v i v i d l y how Octet Enhancement follows from some we l l -
established dynamical requirements i n a simple bootstrap
model. As we have mentioned i n the f i r s t section of Chapter
III, an a l t e r n a t i v e , but less p l a u s i b l e answer to the above
question may l i e with the mere existence of a strongly
Interacting SU(3) t r i p l e t . This w i l l be discussed i n the
l a t e r part of t h i s Chapter.
In the r e s t of the Chapter we s h a l l present a gen
e r a l i z a t i o n of Glashow's method, which, as we s h a l l see, lends
i t s e l f to a number of i n t e r e s t i n g a p p l i c a t i o n s . Incidentally,
we s h a l l present, as a by-product of a more general theorem,
a de t a i l e d proof of Glashow's r e s u l t s , which i s almost t o t a l l y
50
absent i n Glashow fs extremely short a r t i c l e .
3« Generalized Group Theoretical Treatment of Spontaneous
Symmetry Breaking
Consider a s i t u a t i o n where the masses of a system
of p a r t i c l e s belonging to a number of SU(3) multiplets are
given dynamically by a set of equations. Such a set of equa
tions may, f o r instance, come out of a bootstrap c a l c u l a t i o n .
Let the set of equations be SU(3) i n v a r i a n t .
We s h a l l l i m i t ourselves to the case where there
are only two multiplets f o r s i m p l i c i t y . I f we expand the
equations i n terms of small deviations from SU(3) symmetry
and keep only l i n e a r terms, we have
smr = z: 3h< srf + z u« ( 4 . D
where $Mi (1=1, 2, ...,m) and (c*=l, 2, n) denote
mass s p l i t t i n g s of the members of the two m u l t i p l e t s * respect
i v e l y . A l l the d e t a i l e d dynamics of the system, such as the
coupling constants, has been relegated to the matrices (\7 t* )
and ( ) . Of course, to determine the various masses,
*We have used the term "multiplet" here i n a more general sense than usual i n that i t does not have to correspond to a i r r e d u c i b l e representation of SU(3). I t may correspond to a reducible representation such as the case of the vector "nonet". Our proof of the theorem i s independent of the i r -r e d u c i b i l i t y of the m u l t i p l e t s .
51
equations involving £tn& on the left-hand side are a l s o nec
essary and are also usually a v a i l a b l e i n dynamical c a l c u l a
t i o n s . For the following discussion we only have to consider
(4.1). To tre a t equation (4.1) group-theoretically, i t turns
out to be more convenient and f r u i t f u l to consider mass-splitt
ing matrices instead. So we write, instead of (4.1),
0 l4JZ,k*rt" ° fcfiifcn 0
To avoid using too many Indices, we s h a l l look upon ( SMy ) 2 2
and ( FlWeip ) as vectors In m and n dimensional vector spaces,
( \^ek,y ) a n& ( ^otfijij ) as m2 x m2 and n 2 x m2 matrices.
Then (4.2) can be written i n the symbolic form
Supposing that the two multiplets transform accord
ing to the SU(3) unitary representations {m} and jn} respect-
i v e l y , then i t i s easy to see that
J*M transforms as {m}* g> ^mj (4.4)
<Fm transforms as {nj* 0 {nj (4.5)
*The mass s p l i t t i n g matrix f o r a multiplet , i = 1, 2, m, i s defined as
where ESTt i s the mass s p l i t t i n g operator. Under an SU(3) transformation G
52
where <{mj* and .[nj are the adjoint representations of -{m} and
r e s p e c t i v e l y . The representations {m) , {n5-*@{n} are generally reducible. Let us assume, without l o s s of general
i t y , that they can be reduced into the following form:
f m f ® H = & £)<•» 0 S 0 )&> £ ) ( A ) & 3 ( " } ( 4 . 6 )
where the denote i r r e d u c i b l e representations of SU ( 3 ) .
The superscript (i) just l a b e l s the p a r t i c u l a r representation,
and has nothing to do with the dimensionality of the represen
t a t i o n . Then, i f equation ( 4 . 3 ) i s invariant under SU(3) and
equations ( 4 . 6 ) and ( 4 . 7 ) holds, then we can prove the follow
ing
Theorem: (I) I f i s not equivalent to a l l
1 * j , then
Hence, vre conclude that
$K transforms as jmJ1g>-{mJ-
(4.10)
53
( I D I f equivalent to , then
^ Pf3/0+Af4v?; P
( H I ) i f i s equivalent to ST- , then
To define the sumbols, l e t us denote the representation space
of fmj*g>{mj by S ^ , the representation space of {n)*®{nj by
S^ n^ f the subspace of that generates the representation
by S and the subspace of S ( n J that generates the represen-
t a t i o n ^ ' V S(N,L). Then P^(S°^-> Sfrn>) i s defined as the projec-
t i o n operator that projects the subspace S ' onto i t s e l f .
fjll\s<n)—> S(m)) i s defined as that operator which maps any vector
belonging to Sf"'° to that vector i n S(m'° with i d e n t i c a l trans
formation properties and maps that r e s t of S w to zero. In the
case when i s equivalent to P^J'; (S^-> Sc"°) i s the
operator that maps any vector i n S to that vector i n S
with i d e n t i c a l transformation properties and maps the rest of
S to zero. In the case when equivalent to
/ J i s the operator that maps any vector i n S
to that vector i n S with i d e n t i c a l transformation proper-
5 4
t i e s and the r e s t of S ("> to zero. A * , J^'*'J ^> f*m
are a l l numerical constants.
Proof t
Let us look at equation ( 4 . 3 ) under the SU(3) trans
formation
Sirj > S' = Sm (V* ® V) ( 4 . 1 5 )
where U and V are representation matrices i n \m) and j n | re
spectively induced by the same group element i n SU (3). For
( 4 . 3 ) to be invariant under SU ( 3 ) , we must have
= SM'-$-+fm'fi (^.16)
Substituting ( 4 . 1 4 ) , ( 4 . 1 5 ) into ( 4 . 1 6 ) , one gets
Comparing ( 4 . 3 ) and ( 4 . 1 7 ) ,
55
From ( 4 . 6 ) and ( 4 . 7 ) , we see that
u*® u - J"nx J (4.20)
Where J" and K are unitary matrices, and
( 4 . 2 2 )
( 4 . 2 3 )
I t should be emphasized a t t h i s point that J and K are independ
ent of the p a r t i c u l a r group element i n SU(3) which U and V
represent. Uj,, are representation matrices belonging to the
representations respectively. Substituting ( 4 . 2 0 )
56
and (4.21) Into (4.19), we have
where' $J ~ K J~'
Substituting (4.20) into (4.18), we have
11? = JU
where J J' J-J
Writing
u j o v,= !0 0 JLt
i
9
0 We can divide ,/•/ into appropriate blocks i n the form
-I — - -1— H/3 J
Hzi r H22 \ H.ZA
H31 ! Hn!
1 1 -
Hi , ' HJX Hxi \ 1
and into
•J = A ; F/3 j
_ . F*
^ 1 5, F 5 4 ;
T F« ; F52 F* !
T F« ; F*r
(4.24)
(4 . 2 5 )
(4.26)
( 4 . 2 7 )
(4.28)
( 4 . 2 9 )
( 4 . 3 0 )
57
We have, from (4 .24)
And a l s o from (4 .26)
UIRI UFJzj U F I 3
1 1 1
T _ _ _ + ( 1
— — 4 — 1 — + —
( 4 . 3 D
(4.32)
7 - - - r , ,
From ffy* l/p 8 3 (A' Hij t ] £ ^ 3 , i n (4.31), we have,
using Sehur's Lemma (see, e.g., M. Hamermesh, 1962, pg 98-IOI),
and from FM (Jji = Ufc Fk£ ; = JJ2/3,5",
^Mi~ $MX'A^OO MJ = 1,2,3,5- (4.34)
where Sry , <f/cf are kronecker delta functions; } )S® are
unknown numerical constants and 1(C)f 1(b) are i d e n t i t y matrices
of appropriate dimensions.
58
s h a l l now s p e c i a l i z e to the various cases.
i s not equivalent to <*J f o r a l l unequal I , j
From ( 4 . 3 1 ) , we have
Let \ be a vector that transforms l i k e Then, under
the group operation i n question, The vector
where we have made use of the f a c t that f-)ix does not trans
form under SU(3) and a l s o the equality ( 4 . 3 5 ) . Hence the
vector £ f/ix transforms according to «0 ( y ) - ® .
If ^H/x i s n o n - t r i v i a l , there exists a vector subspace
of the representation space of j£)0 ) that transforms l i k e
j Q ^ t which i s In contradiction with our assumption.
Hence £/-//x=0, f o r a l l % , which requires H/x=0 t
S i m i l a r l y we can prove Hxi = Hxz - Wz* - H<-3 - H?x ~ 0 .
A s i m i l a r argument can be applied to the equation Hxxl/x = \Jy Wxx , with the r e s u l t that Mxx = 0 .
Hence, vie have f o r t h i s case,
M ° % . ' \ 0 j 01 0 i
0 " 6 " ! 0 i 0
a ! 0 : 0 : 1
0
( 4 . 3 8 )
5 9
The e q u a l i t i e s i n ( 4 . 3 3 ) and ( 4 . 3 4 ) can be extended
i n t h i s case t o l ^ 4 i ^ 5 " , r e s u l t i n g i n
L V
0
o i i f 4 ) r i
( 4 . 3 9 )
Equations ( 4 . 3 8 ) and ( 4 . 3 9 ) established our assertions
( 4 . 8 ) and ( 4 . 9 ) f o r Case I.
) p<- i s equivalent to gc/
In t h i s case, J~ can be chosen so that U3 = U4 . Then
u = 0 ( 4 . 4 0 ) u = ( 4 . 4 0 )
From ( 4 . 3 5 ) , we have
- Hjx (4.41)
Write /-/jx = • u<" 1
( 4 . 4 2 )
Then from ( 4 . 4 l )
U3 Hix : U H£ H?x LA
Hence
sx t/3
6 0
A l l the other conclusions about the H-submatrices
are the same as i n Case ( I ) . Thus we have, i n t h i s case
41 = o
0 ^f%, o I 1 '
0 ! 0 0
(4.45)
We can also e a s i l y show that
c7 = --i 0
0 ( I I I ) s equivalent to
In t h i s case, the only equality obtainable from (4.31)
which w i l l give r i s e to conclusions d i f f e r e n t from Case (I)
i s
V i H (4.47)
Since now
Us 0 (4.48)
We s h a l l now write
6 1
Hx3 — H a) V3
nx3
and get from (4.48)
—
|V7 HS' Hence
HX5 = ^ -fe>
Therefore
0 (2)<
0 33
0
I t i s a l s o straightforward to show that
0 U._JJ
1 if^r i
0 i i
( 4 . 4 9 )
( 4 . 5 0 )
( 4 . 5 D
( 4 . 5 2 )
Q.E.D.
62
Applications of the General Besults
(i) Glashow !s Besults
We s h a l l reproduce Glashow 1s Besults (S. L.
Glashow, 1963) on the bootstrapping of an octet of
baryons here. There vie have the equation
k / = 2 ^ V ^ ( 4 . 5 3 )
A Comparing t h i s with ( 4 . 1 ) , we h&ve^ff - 0 . Now /m} = ;'•;
J8J , hence the mass matrix ( cTMy ) w i l l transform
according to
= jija- pj W ©j/4^{' °r®^ 7/ ( ^ . 5 5 ) .
were {<5P} and |<5V} are equivalent. Then using the
theorem f o r case (II), we have
J = f°> + pC'o) + ^ (">*) p (io»J + j^r) p(27)
( 4 . 5 6 )
where t h e ^ ' s are numerical constants and the P opera
tors have the meanings assigned to them i n the general
theorem.
We s h a l l now make the assumptions:
63
(A) T h a t t h e m a s s - s p l i t t i n g m a t r i x t r a n s f o r m s a s a s i n g l e
I r r e d u c i b l e t e n s o r o f SU(3)
(B) T h a t t h e s p o n t a n e o u s symmetry b r e a k i n g does n o t d i s r u p t
i s o s p i n a n d h y p e r c h a r g e c o n s e r v a t i o n .
A s we have s e e n , t h e m a t r i x (SM) t r a n s f o r m s a s
^8}*® -{8} w h i c h c a n be decomposed a c c o r d i n g t o ( 4 . 5 5 ) • From
( 4 . 5 5 ) we c a n now p i c k o u t a l l t h e p o s s i b l e s t a t e s t h a t a r e
I n v a r i a n t u n d e r i s o s p i n t r a n s f o r m a t i o n s and h y p e r c h a r g e
gauge t r a n s f o r m a t i o n . H a v i n g p i c k e d out t h e s e s t a t e s , w h i c h
a r e r e p r e s e n t e d by 8 X 8 m a t r i c e s , t h e s e m a t r i c e s c a n be c a l
c u l a t e d b y u s i n g t h e SU (3) C-G c o e f f i c i e n t s (De S w a r t , 1 9 6 3 ;
P . McNamee a n d F . C h i l t o n , 1 9 6 4 ) . These m a t r i c e s w i t h r e
s p e c t t o t h e b a s i s (/f?, n , 2*, 1° 5T, A , Z°, £~) a r e g i v e n a s
f o l l o w s :
(a)
(b)
0
0
-I
0
-I
0
+1
( 4 . 5 7 )
( 4 . 5 8 )
64
I f the mass s p l i t t i n g transforms as a l i n e a r super
p o s i t i o n of (a) and (b), I.e., I f i t transforms l i k e a
general octet component, then
-2d (4.59)
Eliminating p( and ft from (4.59), we can e a s i l y show that the
GMO formula
£(SmH+Sms) =£Sir,A+j:b~n>z (4.60)
must be s a t i s f i e d .
(c) SM ~ IfaLY^oy
I t can be s i m i l a r l y shown that the above mass s p l i t t i n g s a t i s
f i e s the formula
j r n N + S m E = 3Sr*z-SmA (ij.,62)
which was given by De Swart ( J . J . De Swart, 1963). I t i s well-
known that t h i s so-called 2?-plet formula (eq. 2.10) i s not
s a t i s f i e d experimentally. Also note that equation (4.62) i s
65
a weaker requirement than ( 4 . 6 l ) .
(d) SM~l4i},i=o,r**oy~Jji
0
0 (4 .63)
In t h i s case, we have no mass s p l i t t i n g .
( i i ) Bootstrap Involving the Baryon Decuplet and the
Baryon Octet (theB-A bootstrap)
Let $mt denotes the mass s p l i t t i n g of the decuplet,
denotes the -mass s p l i t t i n g of the octet.
Sr»t'*~ Z % &mi +2> Un'ftop i «**
Now since (see, e.g., De Swart, 1°63)
\iof® \w\ = {'}© j*}® M a {u\
(4.64)
(4.65)
We can quote case I I I of our general theorem to write
J = A 0>Pa) + A ^ + #iyP™ +" A^P^ (^.66)
(4.67)
66
where the notations are by now self-explanatory. Let
(<TM) denotes the 10 X 10 mass-splitting matrix of the
decuplet, the 8*8 mass-splitting matrix of the
octet. We make now the further assumption that:
(C) A l l multiplets that we have taken into our bootstrap
are n o n - t r i v i a l l y coupled. In other words, any mass
s p l i t t i n g i n one p a r t i c u l a r multiplet i s going to have
non-zero e f f e c t on the masses of the other multiplets
involved i n the bootstrap problem.
i d e n t i c a l transformation properties, we conclude that i f
(JVw ) transforms l i k e a c e r t a i n SU(3) i r r e d u c i b l e tensor,
then (SM ) w i l l a l s o transform l i k e the same i r r e d u c i b l e
tensor. In p a r t i c u l a r ,
(a) I f {Sm) transforms as a l i n e a r combination of ( ia )
and ( i t ), i . e . , i f the baryon octet s a t i s f i e s the
GMO Formula, then {SM) must be proportional to
Then, since J-f only connects mass matrices with
0 0
0
67
That i s , we have the equal-spacing rule f o r the decuplet
(Eq. 2 . 3 ) , which, as we have said, i s very well s a t i s f i e d
experimentally.
(b) I f [Sm) transforms as i n (/£), then (JM) must he pro
po r t i o n a l to
3
3 , 0
-5
0 '5
-3 -3
-1
which gives r i s e to the mass formula
s ~ 2 ~ G
which i s not at a l l i n agreement with experiment.
(c) I f (//») transforms as i n ( » d ) , i . e . , i f there i s no
mass-splitting i n the octet, then ($M) must be propor
t i o n a l to
I
0
0
68
In other words, there i s also no mass s p l i t t i n g i n the
decuplet.
( i l l ) Existence of a Unitary T r i p l e t and Octet Enhancement
Suppose there exists a t r i p l e t which i s non-
t r i v i a l l y coupled to the rest of the hadrons, by which
we mean that the mass s p l i t t i n g of the t r i p l e t has non
zero e f f e c t on the other m u l t i p l e t s . For s i m p l i c i t y ,
but without l o s s of actual generality, we s h a l l assume
that the t r i p l e t i s only coupled to an octet of baryons.
Then we have
Sto? = 2 55v srf + i fan ( 4 . 6 9 )
where Sm> denotes mass s p l i t t i n g within the octet, and
denotes the mass s p l i t t i n g i n the t r i p l e t respect
i v e l y . Then, because of
and the decomposition ( 4 . 5 5 )» we have
£ = y n r ° u (w'p°o)+x°°v>?v^ +jt *"°P7J
69
We see that tjri , apart from TT ' , which does not
connect ac t u a l mass s p l i t t i n g s among the multiplets,
involves only "projection operators" which connect
i r r e d u c i b l e matrices transforming l i k e a component of
{%\ . From our assumptions (C) that the t r i p l e t i s non-
t r i v i a l l y coupled to the octet and (A) that the mass-
s p l i t t i n g of each multiplet transforms l i k e a single
i r r e d u c i b l e tensor, i t obviously follows that the mass
s p l i t t i n g of the baryon octet must transform l i k e a
component of an octet, and hence must obey the G e l l -
Mann Okubo Formula.
Hence we have obtained Octet Enhancement inde
pendently of any d e t a i l e d dynamics, as soon as we assume
the existence of a t r i p l e t which i s n o n - t r i v i a l l y coupl
ed to the hadrons. In the above argument, we do not
have to assume that t h i s t r i p l e t of p a r t i c l e s are quarks,
i . e . , fundamental b u i l d i n g blocks of a l l matter. We
have said i n the f i r s t section of Chapter III that the
GMO Formula can be deduced from quark models (M. G e l l -
Mann, 1964; G. Zwelg, 1964a, 1964b, 1965; F. Gursey et.
a l . , 1964): but a l l these derivations depend on a great
er or l e s s e r extent to some d r a s t i c dynamical approxi
mations. The foregoing group t h e o r e t i c a l argument,
however, i s independent of any dynamical d e t a i l or
assumption.
70
Since t r i p l e t s have not yet been discovered, we
cannot a t t r i b u t e the phenomenon of Octet Enhancement
l i g h t l y to the influence of t r i p l e t s .
I t i s i n t e r e s t i n g to note that i f there are no
t r i p l e t s or other multiplets with non-zero t r i a l i t y ,
then we cannot carry out an argument s i m i l a r to the
above to account f o r Octet Enhancement, no matter how *
many " t e n s o r i a l " multiplets we put into our bootstrap.
This i s because f o r a l l t e n s o r i a l representations \n\
(except \ l j )
The {27} always appears i n the C-G s e r i e s .
* T r l a l l t y i s defined, f o r the representation {n} = D(p,q) as "t=(p-q) mod ( 3 ) . Tensorial representations are those with t=o. A l l known hadronic multiplets to date f a l l i n t o t e n s o r i a l representations.
**We can show t h i s most e a s i l y by using Speiser's Method f o r obtaining the C-G series (D. E. Speiser, 1 9 6 2 ; see a l s o J . J . De Swart, 1 9 6 3 , pg 3 2 6 - 3 2 7 ) . We have to f i n d the C-G series of D(q,p) <g> D(p,q) (since D(q,p) = D(p,q)*) f o r a l l p,q, such that p~q (mod3) and see whether they a l l contain the {27J = D ( 2 , 2 ) . I t turns out that i f we put the eigenvalue diagram of D(q,p) on top of the point (p,q) i n the l a t t i c e diagram ( F i g . 5 and 6 i n De Swart, 1963) i n the way s p e c i f i e d by the Speiser Method, we can show, by simple geometry, that f o r a l l p=q (mod 3) t "the eigenvalue diagram always covers the point ( 2 , 2 ) i n the 1 s t sextant of the l a t t i c e but never the image points of ( 2 , 2 ) i n the other sextants. By Speiser's rule, t h i s shows that D(q,p)® D(p,q) always contains the {27} = D ( 2 , 2 ) i n the C-G decomposition.
. 71
(iv) Spontaneous 0 - C O Mixing
Let us consider a "bootstrap problem where an octet
of vector mesons and a s i n g l e t vector meson are involved.
It may be necessary In practice to put i n more p a r t i c l e s
In order to complete the bootstrap, (see Chapter V I ) ,
say, the pseudoscalar octet: but the conclusions to be
discussed below w i l l be e s s e n t i a l l y unchanged. We must
bear i n mind a l s o that when we say "mass matrix" i t i s
a c t u a l l y the mass-squared matrix that we are r e f e r r i n g
to, since we are concerned here with bosons. Let us
write the mass matrix of the nonet i n the basis
( K*\ K**,?*, ?\ K*°, K*;to*,eo' ), where /fl'> trans
forms as the 8 t h component of an octet and ICO1} i s a
unitary s i n g l e t . The nonet of vector mesons transform as
( 4 . 5 8 )
Note that now we have a reducible representation of SU (3)•
This does not matter since, as we have stressed i n
Section 3• our theorem applies equally well to reducible
m u l t i p l e t s . Because of ( 4 . 5 8 ) , the mass s p l i t t i n g matrix
( SMij. ) transforms as
72
where and are equivalent to each other.
are a l l equivalent. I f mass
s p l i t t i n g should occur spontaneously, they would be
proportional to e i t h e r of the following matrices i f
i s o s p i n and hypercharge s t i l l concerves.
(a) -i 0
- I
0 -2 0
(b) JL 2 0
0
0
(c)
0
0
-a
-5 0
7 3
Cd) 0
0 0 0
0
0 D 0 I
1 0\
(e) 0 0 0
0 0 "L < 0
it)
J L 41 0
0
( s )
° . o 0
0 0
0 0 0 0
74
For our p a r t i c u l a r mesonic system, the p o s s i b i
l i t y of (b) can be ruled out due to charge conjugation
invariance. This i s because i f a mass s p l i t t i n g occurs
i n accordance with (b), then the K* and i t s a n t i - p a r t i c l e
K* w i l l have mass s h i f t s with opposite signs, r e s u l t i n g
i n m(K*)£ m(K*)t i n contradiction with charge conjuga
t i o n invariance or more generally with CPT invariance
(T. D. Lee, R. Oehme and C. N. Yang, 1957) .
We s h a l l make the assumption that the nonet i s
degenerate to s t a r t with. That i s , we s h a l l neglect
mass s p l i t t i n g s coming from (f) and (g).
We s h a l l i l l u s t r a t e how spontaneous mixing sets
i n by considering the very simple case where symmetry
breaking occurs In accordance with (d). Then to f i n d the
mass of the physical states, i t i s necessary to diagon-
a l i z e (d) by the orthogonal transformation
( 4 . 6 0 )
That i s , the physical states are, Instead of and
75
|to> =~^d(COsy j Co'} (4.61)
where 0 - 45°
Hence, the $ and£0 mesons are equal mixtures of jo fy
and fCO'y .
We s h a l l see now what happens when we Impose the
general condition of Octet Enhancement. That i s , we
s h a l l assume that there i s no symmetry breaking except
those which transform l i k e a component of an octet. In
other words, we s h a l l assume that the mass-splitting
matrix i s proportional to a general l i n e a r combination
of the various " 8 -matrices" (a), (d) and (e) and that
i n p a r t i c u l a r mass s p l i t t i n g i n accordance with (c) which
transforms l i k e a component of the 27-plet, cannot occur.
In that case the mass matrix looks l i k e ,
/ + /$
/+/5 (4.62)
76
This matrix can be diagonalized into the form
0 0
with the transformation
where
Ceo 9
I f we define
We have
(4 .63)
(4.64)
(4.65)
(4.66)
(4.67)
77
I f we have used /^c,'/> instead of a l l along i n our basis,
we would have avoided the appearance of 2 i n ( 4 . 6 2 ) . Since
we are free to do t h i s , we s h a l l assume that t h i s proper
choice of phase has been done and from now on we are going to
look at ( 4 . 6 2 ) , ( 4 . 6 3 ) and ( 4 . 6 5 ) with the understanding that
$= 0 .
Then we have, from ( 4 . 6 3 )
tog* = l + f i
l-2p
I t follows easy from ( 4 . 6 8 ) that
( 4 . 6 8 )
( 4 . 6 9 )
This r e s u l t i s independent of the mixing angle and i s good to
within 2% experimentally. From equations ( 4 . 6 5 ) and ( 4 . 6 8 ) ,
we can e s t a b l i s h the further r e l a t i o n
( 4 . 7 0 )
which as an afterthought i s hardly su r p r i s i n g since i t i s
exactly the GMO Formula i f one considers Cw 0 m<p + pu+ & mo>
to be the mass of the eighth component of an octet, whose other
components consist of the I s o t r i p l e t p , the doublet K* and i t s
a n t l p a r t l c l e s K*.
78
Let us summarize our res u l t s here. Sta r t i n g from a
nonet of vector mesons which we assume to possess (accidentally)
degenerate mass, we found that mixing between the two 1=0,
Y=0 mesons can i n p r i n c i p l e occur as a p a r t i c u l a r form of spon
taneous symmetry breaking. The mass-splitting matrices corres
ponding to such symmetry v i o l a t i o n s transform as the eighth
component of an octet, thus suggesting that <j>~Co mixing i s
just another manifestation of Octet Enhancement i n symmetry
breaking. Conversely, assuming Octet Enhancement i n symmetry
breaking i n i t s most general form, we found that spontaneous
symmetry breaking i n the vector nonet can only occur i n such
a way as to s a t i s f y equations (4.69) and (4.70).
We have no way here, i n t h i s group-theoretical d i s
cussion, to determine the mixing angle 0 . To get some f e e l
ing about the kind of mixing angle that would emerge from (4.70),
l e t us put
foco = (4.7D
which i s well s a t i s f i e d experimentally. We get, a f t e r sub
s t i t u t i n g t h i s into (4.69) and (4.70) that
C<^Q = 2/3= Zl% (4.72)
In other words, the <p meson i s a member of an octet 67% of the
time and a s i n g l e t 33$ of the time.
It i s i n t e r e s t i n g to compare our r e s u l t s with those
79
of other authors who "computed" mass formulae and mixing
angles i n more s p e c i f i c models.
(I) T r i p l e t Models (see F. Gursey, T. D. Lee and M. Nauenberg,
1964; and G. Zweig, 1965).
To be s p e c i f i c , we s h a l l describe Zweig's model.
In t h i s model, the vector mesons are again considered as
quark-antiquark bound states just as i n the case of the pseudo-
scal a r mesons presented i n Section One of Chapter I I I . The
spin and o r b i t a l angular momentum parts of the compound
states are of course d i f f e r e n t f o r the two kinds of mesons,
but the U-spin dependence i s i d e n t i c a l except f o r the two
i s o s l n g l e t vector mesons. More precisely, f has the same
U-spin dependence as and K* the same as K. But the <P and
to mesons are now defined as
/<£> = - \a3ai)
Now, we know that the states
transforms as the eighth component of an octet and a unitary
s i n g l e t respectively, as can be checked by d i r e c t computation
basing on the assumed transformation properties of f a 1 ^ and
\&y> . The states In (4.73) can be written as l i n e a r combina-
(4.73)
(4.74)
80
tlons of states i n ( 4 . 7 4 ) as follows,
( 4 . 7 5 )
where
( 4 . 7 6 )
Using the states ( 4 . 7 3 ) and the corresponding formulae f o r the
other vector mesons, the masses of the nine vector mesons can
be computed i n terms of the masses of t h e i r constituent quarks
i n exactly the same manner as was done i n Section 1 of Chapter
II I f o r the pseudoscalar mesons. This gives the formulae ( 4 . 7 1 )
and
which can be recognized as a s p e c i a l case of ( 4 . 7 0 ) , by s e t t i n g
Cov0 =Jj • ^ t i s important to emphasize that even i n t h i s
s p e c i f i c model of dubious v a l i d i t y (e.g., the existence of
quarks!), nothing more i s r e a l l y derived than i n the general
group t h e o r e t i c a l discussion. As we have shown, Octet
Enhancement i s a consequence of the "existence" of a unitary
t r i p l e t . The accidental degeneracy of the nonet i s i m p l i c i t
i n the d e f i n i t i o n of a l l the nine vector mesons as the same
kind of quark-antiquark states. The mixing angle i s a c t u a l l y
introduced already i n the d e f i n i t i o n of (^^ and j^°y as
4 - 2nip - m ^ - m y = o ( 4 . 7 7 )
81
shown i n (4 .73)• I t i s only a f t e r a l l these assumptions have
been made that equations (4.71) and (4.77) follows. In other
words, i n the quark model, equation (4.77) follows from an
assumption on the value of the mixing angle. This obviously
i s true also i n the general group t h e o r e t i c a l discussion —>
one simply has to substitute C&> Q into Eq. ( 4 . 7 0 ) .
By d e f i n i n g the mixing angle at the outset, the independence
of (4.69) on the amount of mixing was not recognized i n the
context of the quark model.
(II) Okubo's Model (S. Okubo, 1963)
Okubo wrote do\m an e s t h e t i c a l l y simple form f o r
the mass term of a Lagrangian involving the vector nonet which
treats the Co' on the same footing as the other vector mesons
(and of course assuming octet transformation property of the
Langrangian). He obtained (4.69) and ( 4 . 7 1 ) , but a d i f f e r e n t
value of the mixing angle. In his model,
the assumption about the mass term i n the Lagrangian i s of a
very ad hoc nature and the mass formulae and mixing angle are
immediate consequences of i t . So we tend to go along with
Gaslorowlcz (S. Gasiorowicz, 1966, pg. 327) i n be l i e v i n g that
(4.78)
the » observation of Okubo must be viewed as a c u r i o s i t y . it
Sakurai ( J . J . Sakurai, 1963) has studied phenomeno-
8 2
l o g i c a l l y the consequences of a mixing l i k e that shown i n
(4.?5) and found a connection between the masses of the phy
s i c a l vector nonet and the mixing angle. Although he has not
written down the connection i n a compact form, i t a c t u a l l y
can be summarized by Eq. ( 4 . 7 0 ). He substituted the experi-
mental masses of the vector nonet and found that Cm 0 = 60$ .
Since he has concentrated on the mixing angle, he has not
found ( 4 . 6 9 ) .
(v) Bootstrapping of 0-0) Mixing and Mass S p l i t t i n g s
In the Octet Pattern
Consider the s p e c i f i c example of a bootstrap i n
volving the vector nonet and the pseudoscalar octet, where
we can write down the equations,
where the S^t's are the mass s h i f t s of the vector nonet and
Sm^'s of the pseudoscalar octet. Since,
& \SY\ @\l0\®\\of® {21) (4.80)
83
the operators and / can be written, according to the gen
e r a l theorem, as
+ y p , x ) p(p , x > + y v , p ; p ( x ^ +
J T _ ^ o p a ; + >WpC0 + A ( « p W +
\&,P)D&/B> + ( 4 .82 )
PA ( P ^ ) p i * w , J i n ( 4 . 82 )
]-jrx,Dj A CPA) and / I i n ( 4 . 8 3 ) • r connects the mass s p l i t t i n g
A
mattix of the vector nonet that gives r i s e to <f>-(& mixing to a
mass s p l i t t i n g matrix that corresponds to mass s h i f t s i n accord-v fox)
ance with the GMO Formula. Thus i f A i s non-zero, any
<f>-to mixing that occurs i s going to further enhance the
"normal" mass-splittings i n the octet pattern. S i m i l a r l y , i f
y v i s non-zero, any "normal" mass-splitting i n the octet
pattern w i l l tend to give <fi-Co mixing a further boost. In an n-rx,p;
analogous manner, terms l i k e / I i n ( 4 .83 ) connect mass
s p l i t t i n g s of the pseudoscalar octet i n the octet pattern with
<fi-co mixing i n the vector nonet. Hence we see that a boots-
strapping between mixing and the "normal" mass-splittings
i n the octet pattern can i n p r i n c i p l e occur. Whether i t
r e a l l y does occur depends of course on the values of the co-e f f i c i e n t s \ { ™ , \<X»,J*M» M h l o h l n t u r n
depends on the det a i l e d dynamics. U n t i l we have some more
84
information about these c o e f f i c i e n t s , i t seems more natural
to consider (fi-co mixing i n the vector mesons and the mass
s p l i t t i n g according to GMO Formula i n the pseudoscalar octet,
say, on the same footing — that ei t h e r of these i s the cause
and consequence of the other. Dynamically, i t may turn out
to be more p r o f i t a b l e to look f o r both of these e f f e c t s i n a
bootstrap model instead of assuming that one i s more funda
mental than the other. We tend to believe, therefore,that
any attempt to "derive" mass-formulae from <fi-CO mixing can at
best be p a r t l y v a l i d .
CHAPTER V
OCTET ENHANCEMENT IN A BOOTSTRAP MODEL
1 . The Bootstrap Hypothesis
Stated b r i e f l y , the bootstrap hypothesis i s that a l l
p a r t i c l e s (or a t l e a s t the hadrons) are dynamical e n t i t l e s
composed of each other and bound by forces produced by the ex
change of the p a r t i c l e s themselves. The term "bootstrap" was
introduced i n 1959 by Chew and Mandelstam (G. P. Chew, 1 9 5 9 ;
G. F. Chew and S. Mandelstam, 1 9 6 l ) . The suggestion that a l l
hadrons with a l l the parameters characterizing them might be
determined by a c o l l e c t i v e bootstrap was made i n 1 9 6 l by Chew
and Frautschi (G. F. Chew and S. Frautschi, 1 9 6 1 ) .
I n t u i t i v e l y , the bootstrap hypothesis i s very appeal
ing, f o r i t allows us to understand the existence of the vast
number of p a r t i c l e s and resonances that are being discovered
a t such an enormous rate, without having to account f o r each
of them by b u i l d i n g i n a number of parameters (such as the
masses and coupling constants i n a Lagrangian theory) that can
only be determined by experiment. In p r i n c i p l e , i f the boot-
strap hypothesis i s correct at a l l , i t i s , i n t u i t i v e l y at
l e a s t , such a r e s t r i c t i v e condition, that probably a l l r e l e
vant parameters can be determined from i t . What we have to
do i s simply to make a guess at a set of hadrons with a r e l e
vant set of parameters such as coupling constants and masses,
85
86
throw i n a minimal amount of dynamical assumptions, and calcu
l a t e what comes out of i t such as the number of bound states
and t h e i r coupling constants and masses. If the output set
of information agrees with the input, then one says that one
has a complete bootstrap. Of course, one knows nothing to t h i s
date about the existence or uniqueness of the solution, but has
to hope f o r the best.
But the implementation of the bootstrap hypothesis
turned out to be f a r more e a s i l y said than done. One runs into
trouble at the f i r s t step already. The conjecture of the num
ber of hadrons and t h e i r c h a r a c t e r i s t i c s (such as spin) i s by
no.means an easy task. At t h i s stage one can again be hopeful
that a subset of the hadrons exists which forms an approximately
closed bootstrap. I f t h i s i s the case, the imposition of the
self-consistency condition (bootstrap condition) allows one to
determine a l l the parameters related to t h i s subset approxi
mately. I f t h i s can be done, i t i s by no means meagre achieve
ment considering the state of ignorance that we are i n concerning
the strong i n t e r a c t i o n s . To go any further, we mu.st again hope
that by allowing the subset of p a r t i c l e s to i n t e r a c t among
themselves by the exchange of each other, we can produce,
beside the p a r t i c l e s already i n the subset, some other p a r t i
c l e s . I f t h i s happens, one can then Incorporate these other
p a r t i c l e s with the subset and impose the bootstrap hypothesis
i n a second c a l c u l a t i o n . Thus we have a bigger bootstrap
87
problem that empasses more hadrons than before and one could
hope to gain i n t h i s manner more and more Information. This
idea has been t r i e d out In some simple cases with moderate
success; but of course, the computation becomes too complicat
ed a f t e r the f i r s t or second stage (Hong-Mo Chan, P.C. De
Celles and J . E. Paton, 1 9 6 3 . 1 9 6 4 ; Hong-Mo Chan, 1 9 6 4 ; Hong-
Mo Chan and C. Wilkin, 1 9 6 6 ) .
Most of the bootstrap calculations done to date
have been r e s t r i c t e d to the s e l e c t i o n of a subset of hadrons
and c a l c u l a t i n g the pertinent parameters by imposing the
self-consistency conditions. Usually these are done In the
S-matrix framework although there Is no reason why they cannot
be formulated i n a Lagrangian f i e l d t h e o r e t i c a l framework
(see A. Salam, i 9 6 0 ) . To i l l u s t r a t e more e x p l i c i t l y what i s
meant by a bootstrap l e t us return to the early work of Chew
and Mandelstam on 7T-7T s c a t t e r i n g . This i s one of the simpl
est systems believed to form an approximate bootstrap. The
most remarkable feature of 7T-TC scattering i s the appearance
of an 1=1, J=l" resonance f at energy 760 MeV with width 110
MeV. To look f o r a mechanism that gives r i s e to the binding
force between two pions, one notices at once that a single
pion cannot be exchanged due to conservation of angular mo
mentum and p a r i t y . Hence the simplest and l i g h t e s t system to
be exchanged i s a two-pion system. One knows experimentally
that the two pions tend to c l u s t e r together at low energies
I.
88
to form a p -meson. Forgetting f o r the time being that the
f-meson i s unstable, we could assume that the main binding
force comes from the exchange of a single f -meson, which i s
supposed to have a mass m^1 and a coupling constant Yp"n to
the two TT -mesons (corresponding to a width f > ). Suppos
ing now that there i s some dynamical scheme, which we s h a l l
not specify a t the moment, which allows one to calculate the
scattering amplitude of the 71-71 . Then, i f the bootstrap
assumption i s approximately correct, one would expect that a ,., out r-» out resonance with mass m ^ and width | f w i l l appear i n the
r> 14 4. * n *.* *n o u t r » i n r-»out P-wave amplitude. Puting m ^ =m^ , J p = \ ^ , one
has two equations to determine the two parameters m^ ~ m^ =
m p ° U t and ["J, ~ P fl n = rj,0ut' The simple model just describ
ed was worked out by Zachariasen,( 1 9 6 1 ). The values he got
were m, = 350 MeV and = 130 MeV. Similar calculations
have been performed by a number of authors. More s o p h i s t i
cated methods of dynamical approximation (see L.A.P. Balazs,
1 9 6 2 , 1963) and i n c l u s i o n of more channels (F. Zachariasen
and C. Zemach, 1962) both l e d to better agreement with
experiment.
I f the masses and coupling constants are r e a l l y
determinable from bootstrap dynamics, then i t i s obvious that,
if. one chooses the system of hadrons judiciously, the i n t e r n a l
symmetries such as SU(2) and SU(3) i n strong interactions with
t h e i r c h a r a c t e r i s t i c r e l a t i o n s between masses and coupling
89
constants, should a l s o follow from the bootstrap assumptions.
The p o s s i b i l i t y of t h i s has been demonstrated i n a large
number of c a l c u l a t i o n s (E. Abers, F. Zachariasen and C. Zemach,
1 9 6 3 ; B. E. Cutkosky, 1 9 6 3 a ; R. H. Capps, 1 9 6 3 a ; Hong-Mo Chan,
P. C. De Celles and J . E. Paton, 1 9 6 3 ; A. W. Martin and K. C.
Wall, 1 9 6 3 . 1 9 6 4 ; E. C. G. Sudarshan et. a l . , 1 9 6 4 ) .
I t i s well-known that some of the Internal symmetries
i n strong i n t e r a c t i o n s (such as SU(3)) are not exact, and th i s
f a c t , together with the p a r t i c u l a r ways i n which those symme
t r i e s are broken, should also be r e f l e c t e d i n a bootstrap
c a l c u l a t i o n . I t usually turns out to be too complicated to
calculate a s o l u t i o n with broken symmetry d i r e c t l y . Since an
exact symmetry would reduce the number of parameters to be
calculated enormously — because i n the symmetric solut i o n the
hadrons would f a l l into multiplets with degenerate mass and
the coupling constants would be related by the C-G c o e f f i c i e n t s
of the symmetry group —- i t i s customary to seek a s o l u t i o n
which possesses the f u l l symmetry as a f i r s t step and then
seek s e l f - c o n s i s t e n t perturbations from the symmetric values
of the parameters, hence obtaining a solu t i o n with broken
symmetry. In other words, one seeks spontaneous symmetry
breaking from the I n i t i a l symmetric solu t i o n . (R. E. Cutkosky
and P. Tarjanne, 1 9 6 3 ; R» Dashen and S. Frautschi, 1 9 6 5 a ;
D. Y.^Wong, 1 9 6 5 )• This i s the approach that we are going to
adopt when we study the cause of Octet Enhancement i n a
90
bootstrap model i n the l a t e r part of t h i s Chapter. We s h a l l
a l s o work i n the S-matrix framework which seems to be easier
to handle.
2 . Basic Assumptions i n S-Matrix Theory (See G. F. Chew,
1961; G. F. Chew and M. Jacob, 1964)
We s h a l l discuss here some of the basic assumptions
that have become the standard tools i n S-matrix Theory. As
t h i s i s intended only to serve as an introduction to the
method that we w i l l be using i n our bootstrap model, i t w i l l
be i n the form of a very b r i e f resume7 and makes no pretence to
completeness. We s h a l l confine ourselves here to spinless
p a r t i c l e s and sing l e channel processes, f o r reason of s i m p l i
c i t y . The generalization to multichannel cases involving
p a r t i c l e s with spin turn out to be rather straightforward and
involves no modification of matters of p r i n c i p l e . ( J . D.
Bjorken, i 9 6 0 ) .
The whole of S-matrix Theory rests on three general
properties of the S-matrix, namely
(i) The Substitution Law and Crossing Symmetry
( i i ) A n a l y t i c i t y
( i i i ) U n i t a r i t y
Consider a process where two p a r t i c l e s a and b come
i n with 4-momenta p1 and p 2, producing p a r t i c l e s c and d with
4-momenta p~ and pn respectively (Fig. I ) .
F i g . I
Invoking r e l a t i v i s t i c invariance, the scat t e r i n g
amplitude f o r the process can be expressed as a function of
invariants constructed from p- , p 2, Vy Plj,. Assigning con
servation of 4-momentum and putting the p a r t i c l e s onto t h e i r 2 2
respective mass s h e l l s —- that i s , requiring p^ = ma , p p 2 2 2 ? 2 02 2
p 2 % * p 3 = m c ' p 4 = m d ^ e r e P i = p ± - p ± only two Independent invariants can be constructed. I t turns out
2 2 to be most convenient to use s = ( p 1 + p 2) , t = ( p 1 - p^)
and f o r reason of symmetry which w i l l become evident l a t e r , 2
i t i s useful to add another i n v a r i a n t u = (p^ - p^) • These
three Invariants, which are usually referred to as the
Mandelstam variables (S. Mandelstam, 1 9 5 8 ; see al s o R. Hagedorn,
1963» f o r d e t a i l e d discussion), are related by the equation
(5.1)
where we have made the further assumption that the four parti
cles have equal mass, f o r s i m p l i c i t y . Let the amplitude of
92
the process be denoted by
>4(s,*>«0 = / \ ( f 3 f * ; - t O (5.2)
From the d e f i n i t i o n of s, t and-u, one can e a s i l y show that
the physical region f o r the variables s, t and U correspond
ing to the process shown i n F i g . I, i s r e s t r i c t e d by
f S > 4 m * > o
I tt <: 0
We s h a l l see that the same amplitude A(s,t,U) describes three
d i s t i n c t processes when the variables s, t and iA. are r e s t r i c t
ed to d i f f e r e n t regions. In the above-mentioned process, s i s
the square of the centre of mass (CM) energy. Consequently,
that process i s referred to as the s-channel of the amplitude
A.
Now i f one r e c a l l s the p r e s c r i p t i o n of Feynmann f o r
c a l c u l a t i n g s c a t t e r i n g amplitudes, one sees that the same
function A(s,t,*0 can be Interpreted as to represent quite
another process, i f one puts i n an appropriate range of values
f o r the arguments of the function. Feynmann has summarized
the s i t u a t i o n i n the rule that "an a n t i - p a r t i c l e can be look
ed upon as a p a r t i c l e propagated backwards i n time" (R. P.
Feynmann, 1962). Hence the amplitude fj ( f > f,} ) f o r
the process shown i n F i g . II i s given by
93
A
i i • I i t i i
F i g . II F i g . I l l
Writing , f / = - f * >
or ( 5 - 4 ) B (*,*,<0 = A (s,t,u) Since the process i n F i g . II i s described by B and t i s now
the square of CM-energy, the physical values f o r s, t, U are
r e s t r i c t e d by
jt } 4-mz > 0
5 ^ 0 ( 5 . 5 )
V- o
This i s c a l l e d the t-channel. S i m i l a r l y , the amplitude
A(s,t,u) can describe the process shown i n F i g . I l l , i f the
variables s,t,Utake on the values
S <: 0 ( 5-6)
94
That the same amplitude A(s,t,u) describes three d i s
t i n c t processes when i t s arguments take on d i f f e r e n t sets of
values i s c a l l e d the SUBSTITUTION RULE. As mentioned before,
i t follows from the Feynmann r u l e s . I t has also been proved
as one of the f i r s t theorems i n Axiomatic F i e l d Theory (see,
e.g., G. Barton, 1965, pg* 3 6 ) . In S-matrix theories, i t i s
u s u a l l y regarded as a fundamental postulate. I f the process
shown i n F i g . I i s i d e n t i c a l to that shown i n F i g . II, then
i t follows from ( 5 . 4 ) , that
A = A (*,s,u) ( 5 . 7 )
Then we say that there i s CROSSING SYMMETRY between the s and
t channels.
I t i s important to observe that the regions (5«3)»
(5*5) and (5*6) f o r the variables (s,t) are disconnected, so
that i t i s i n general meaningless to say that the processes i n
F i g . I, II and I I I are described by the same function. How
ever, i f one assumes enough a n a l y t i c i t y , then i t i s possible
to a n a l y t i c a l l y continue the function A(s,t) as defined i n
region (5«3) i n a unique way to the regions ( 5 . 5 ) and ( 5 . 6 ) ,
thus obtaining an a n a l y t i c function A(s,t) defined over a
region containing a l l these subregions ( 5 . 3 ) f ( 5 . 5 ) and ( 5 . 6 ) .
Then i t becomes meaningful to say that the 3 processes are
described by the same function A(s,t) and a complete know
ledge of the scattering amplitude i n one region (or a dense
subregion of i t ) would enable us to calculate the scattering
95
amplitude i n the other regions, and hence other channels.
This brings us to another property that the function A(s,t)
should have — ANALYTICITY.
The a n a l y t i c properties are described i n terms of
the Mandelstam Representation (S. Mandelstam, 1958) which i s
usually introduced as a postulate i n S-matrix Theory. The
Mandelstam Representation has been shown to hold f o r poten
t i a l s c a t t e r i n g (R. Blankenbecler et. a l , I 9 6 0 ; see also M.L.
Goldberger and K. M. Watson, 1964) f o r potentials of the form
and f o r a large class of Feynmann graphs (see R. J . Eden, P.V.
Landshoff, D. I. Olive and J . C. Polkinghorne, 1966).
The Mandelstam Conjecture i s usually used i n the •inform of p a r t i a l wave dispersion r e l a t i o n s or f i x e d momentum
trans f e r dispersion r e l a t i o n s , since no convenient method i s
yet a v a i l a b l e f o r using the double spectral i n t e g r a l s more
d i r e c t l y . I t can be shown from Mandelstam's Conjecture that
the X t h p a r t i a l wave amplitude, which i s a function of the
complex energy s, has the following a n a l y t i c properties.
The JL p a r t i a l wave amplitude A^(s) i s defined, f o r our simple case, by ^
where Pjz are the Legendre functions and a i s the angle between and sf? i n the CM system.
96
(See, e.g., G. F. Chew, 1 9 6 2 , Ch. 9 ) .
(1) I t i s a double valued function of the complex energy s
(we are neglecting i n e l a s t i c i t y which would give r i s e .
to extra complications), so that two Riemann sheets are
required to define a single-valued function. These
sheets pass into each other v i a branch cuts to be des
cribed l a t e r . Physical values f o r the scattering amp
l i t u d e are to be taken by allowing the complex energy s
to approach the r e a l axis on the f i r s t sheet from above.
Hence the f i r s t sheet i s usually referred to as the
"physical sheet." The amplitude i s an a n a l y t i c function
except f o r branch cuts and poles on or near the r e a l
a x i s .
(2) (a) I t possesses a left-hand cut extending from 5L to
s « - 0 0 . The p o s i t i o n of SL depends on the l i g h t
est physical system that i s exchanged (see F i g . IV).
Tm s
, bound stats pole
/
/ /
/
/ ReS
/ /
» / /
resonance pole (Oh 2nd sheet)
F i g . IV
97
(b) I t possesses a right-hand cut extending from the
threshold 5* to s = + 0 0 ,
(c) I t may have poles on the r e a l axis i n the gap bet
ween the l e f t and right-hand cut. Such poles
correspond to bound states. The mass of a bound
state i s given by the p o s i t i o n of the pole while
i t s coupling constant i s given by the pole residue.
(d) I t may have poles below but near to the r e a l axis
on the second sheet with Re s > 5 t • Such poles
can be interpreted as resonances.
(e) In the gap between the two cuts, the p a r t i a l wave
amplitude takes on r e a l values.
(3) From (1) and ( 2 e ) , i t follows from the Riemann-Schwarz
" p r i n c i p l e of r e f l e c t i o n " (see, e.g., E. C. Titchmarsch,
1939) i n the theory of functions of a complex variable,
where the a s t e r i s k denotes complex conjugation. Such
a n a l y t i c functions are usually simply referred to as
being " r e a l . "
For the r e a l a n a l y t i c function A , one could write
down the H i l b e r t Transform, which (provided that Aji converges
at i n f i n i t y ) i s an immediate consequence of Cauchy's Theorem
and (5*9) as follows
that
( 5 . 9 )
( 5 . 1 0 )
98
Now, /A^s) s a t i s f i e s the r e l a t i o n
where J}fs) i s a f a c t o r to remove kinematic s i n g u l a r i t i e s and
s,<o = e 2 i n m (5.12)
where i s the phase s h i f t . The statement of UNITARITY i n
the case of e l a s t i c s c a t t e r i n g that we are considering i s the
phase s h i f t s ^ f c ) are r e a l f o r Sy-$t* Then i t follows from
(5.11) and (5»12), by d i r e c t computation that
~ ftfs)/Ae<yJ? , s>St ( 5 . 1 3 )
or
• Im[Md]~l = ~j}(0 , s>s* (5.14)
This i s the right-hand s i n g u l a r i t y that i s usually inserted
i n t o the second i n t e g r a l of (5.10). On the other hand, the
left-hand cut i s connected with the forces responsible f o r the
scattering, i . e . with the p a r t i c l e systems to be exchanged. A
pole i n the t or-U va r i a b l e i n the function A(s,t,w), corres
ponding to the exchange of a p a r t i c l e with well defined mass,
would give r i s e to a cut f o r the function AJL(S) along the r e a l
axis of the complex s-plane, s t a r t i n g a t 5— S^-Me (where me
i s the mass of the exchanged p a r t i c l e ) and extending back to
S i - -co Hence the nearby part of the left-hand cut ari s e s from
the l i g h t e s t systems that can be exchanged and so corresponds
to the long-range forces of the problem. The f a r - o f f parts
99
a r i s e from the exchange of more massive systems and hence
correspond to short-range forces, which one knows very l i t t l e
about. Fortunately, i f one looks at the integrals on the
right-hand side of ( 5 . 1 0 ) , one sees that the contribution of
J t f »Aef r ;)is damped of f by a f a c t o r of , so that one
can hope that when one i s considering the amplitude Aj f o r
small s, the i n t e g r a l s are dominated by contributions from
nearby parts of the cuts, and the e f f e c t of the short-range
forces can, a t l e a s t i n a large class of situations, be ignored
to a good approximation.
Let us now take a b r i e f look at the so-called N/D
method (G. F. Chew and S. Mandelstam, i 9 6 0 ) That has been
employed to solve the i n t e g r a l equation ( 5 . 1 0 ) . The u n i t -
a r i t y condition ( 5 » 1 3 ) i s a non-linear equation i n Aj? » and
therefore, when i t i s substituted into ( 5 . 1 0 ) , gives a non
l i n e a r i n t e g r a l equation. The N/D method i s a device to con
vert t h i s into a p a i r of coupled l i n e a r equations. One writes,
Ajt(')*> NJL(S)/PJ(S) ( 5 . 1 5 )
where Nji has only the left-hand cut and Vji only the r i g h t -
hand cut. By using ( 5 . 1 5 )» one sees that
*This method has been widely used by p h y s i c i s t s during the past few years. I t was, however, introduced and extensively studied by Wiener and Hopf some f o r t y years ago. (See, e.g., E. C. Tltchmarsch, 1 9 4 8 ) .
100
Then, one can write down the H i l b e r t Transform
1/ ,s I (SL Sl* AjlO')- Jis'
Also, one has
Ir* Vj(s) = iV»[/v>C$> A/'fo]
Using ( 5 « l 4 ) , one gets
Performing a H i l b e r t Transform, one gets
( 5 . 1 6 )
( 5 . 1 7 )
(5.18)
^ * > S t ( 5 . 1 9 )
^T5V S" ( 5 . 2 0 )
( 5 . 2 1 )
To improve the convergence of the Integral on the right-hand
side, one writes down the corresponding expression f o r D[S0 ),
subtract i t from , and normalising D(X>) to unity, which i s
allowable since only the r a t i o of N to D i s the relevant function
to be calculated. Then one gets
( 5 . 2 2 )
101
The equations (5.18) and (5.22) are the basic set of
equations that we are going to work with. In the N/D method,
the bound states corresponds to zeros of the D-function, and
resonances to zeros of Re.D. The coupling constant squared i n
the f i r s t case and the reduced width i n the second case are
given by
evaluated a t the zero of D . The bound states do not go to
the N-function as input poles. In f a c t , since D vanishes at
a bound state, N/~ fyAjz does not have a pole at the bound
state, though A has i t .
Digression to multichannel case We s h a l l digress here very b r i e f l y to write down the
relevant formulae f o r the multichannel case since we w i l l be using them i n our model. The formulae usually used are rather obvious generalization of what vie had f o r the single channel case* ( J . B. Bjorken, i 9 6 0 ) . In the case of l\ coupled channels, the u n i t a r i t y condition f o r the p a r t i a l wave amplitude Ay , \& » , (we have omitted the l a b l e f o r the p a r t i a l wave f o r convenience) i s given by
JT (A<j® - AtfM) = Z ?i*®A*k(')A*j(s) ( 5 . 1 3 ' )
where fij i s the kinematic f a c t o r f o r the amplitude Ay, which i s written as
' cbdtp(s)
Then another relevant formula that we are going to use w i l l be
The existence of a single bound state In the coupled channels w i l l now be i d e n t i f i e d as a simple zero i n det D(s) and the
102
coupling constants of the bound states w i l l be given by
f "'fy (5.23')
evaluated a t the zero of det D. &j are the coupling constants of the bound state to the i t n and j t n channels r e s p e c t i v e l y .
A l l e x i s t i n g methods of solving (5.18) and (5 .22) to
date involve rather d r a s t i c approximations i n one form or
another. The d i f f e r e n t approximation procedures d i f f e r i n de
grees of complication, but not necessarily i n p r e c i s i o n . As
examples we s h a l l quote the determinental method (M. Baker,
1958; F. Zachariasen and C. Zemach, 1962) which i s sometimes
a l s o refe r r e d to as the Unitarised Born Approximation, The
Balazs Method (L.A.P. Balazs, 1 9 6 2 , 1963) and the single or
several poles approximation used, e.g., by Fautschi and
Walecka i n the T^H problem (S. C. Frautschi and J . D. Walecka,
I 9 6 0 ) . Among these, the simplest method to use seems to be
the pole approximation which consists of replacing the l e f t -
hand s i n g u l a r i t y by a pole, or i n the words of Chew (G. F.
Chew, 1 9 6 2 ) , replacing a l i n e charge by a point charge. We
s h a l l describe t h i s b r i e f l y i n the single channel case: the
extension to more general cases i s again quite s t r a i g h t f o r
ward .
Let us consider a s i t u a t i o n where two p a r t i c l e s , say,
two nucleons, s c a t t e r by exchange of a pion. Then the l e f t -
hand d i s c o n t i n u i t y w i l l be proportional to the square of the
103
coupling constant G and the di s c o n t i n u i t y w i l l be peak
ed around a point whose distance from the threshold i s pro
p o r t i o n a l to the mass of the exchanged pion. Hence, i f we wish
to replace the cut by a pole, we i^ould i n the most general case,
place the pole a t a p o s i t i o n
5 V = S > - 4 \ n z ( 5 . 2 4 )
2
and the residue of the pole should be proportional to G .
This corresponds to w r i t i n g Sr*Aji(s') (F-t8) as
where c and \ are two as yet unspecified constants, both of
which depend on the d e t a i l e d kinematics as well as dynamics.
Substituting ( 5 . 2 5 ) into ( 5 . 1 8 ) , one gets
where we have normalised Dj ( — 4-\rv? ) to unity, i . e . ,
we have put SQ = - 4-Ar*1 . Substituting (5.26) into (5.22),
we have
ft
I f we impose the bootstrap condition that the same pion with
mass M i s to be produced i n the Ji p a r t i a l wave as a bound
state, we have the two equations,
Vj>(m%) = o (5-28)
_ ^ 2 ( 5 . 2 9 )
104
I f we look back at equations ( 5 . 2 6 ) and ( 5 . 2 7 ) , we have four
unknown numbers — but two parameters c and X $ and the two 2 2
phy s i c a l quantities G and m — b u t only two equations to
r e l a t e them. To get a complete determination of the physical
2 2
quantities G and m , i t i s customary to assign some reason
able values to c and X . We s h a l l take as an example the
c a l c u l a t i o n done by Abers et. a l . (E. Abers, F. Zacharlasen
and C. Zemach, 1963)» They have retained the a r b i t r a r i n e s s
of c, but set \ - 1 and were able to draw some q u a l i t a t i v e
conclusion from the N-N-TT bootstrap; v i z . , SU(2) symmetry.
In the many channel problem that we are going to
t r e a t (which i n the l i m i t of complete SU(3) symmetry reduces
to a single channel problem), we s h a l l use the above single
pole approximation i n i t s most general form. We s h a l l c a l l
t h i s the "Parametrized One Pole Approximation," which, by
v i r t u e of the large number of parameters a v a i l a b l e , would
hopefully be adaptable to a large class of s i t u a t i o n s . What
we claim here i s that i t seems probable that the r e a l physical
left-hand s i n g u l a r i t i e s can be approximately by a single pole
i f we know how to choose the parameters c and \ j u d i c i o u s l y .
In that event, i f we can make some general statement inde
pendently of the choice of c and X • then that statement would
hold f o r the r e a l physical s i t u a t i o n .
105
3 . Octet Enhancement i n a Bootstrap Model of Vector Mesons
In t h i s section we discuss a bootstrap model of
vector mesons i n which Octet Enhancement ( i n t h i s case the
v a l i d i t y of the GMO Formula) a r i s e s from the condition that
the s e l f - c o n s i s t e n t values of the coupling constants be r e a l .
The model i s s i m i l a r i n s p i r i t to that of Cutkosky (R. E.
Cutkosky, 1963a) where an octet of vector mesons are two
body bound states of the same octet. As a matter of f a c t ,
t h i s model, which i s i n the f i e l d t h e o r e t i c a l language, has
been further developed f o r the study of Octet Enhancement
(R. E. Cutkosky and P. Tarjanne, 1 9 6 3 ) . But, as i n the words
of the authors — "We suggest, as the o r i g i n of the Gell-Mann
Okubo Rule, that j 1-K(8)| J l-K ( 2 7 ) | ( 3 1 ) . While our c a l
c u l a t i o n indicates the p l a u s i b i l i t y of obtaining (31) from
a more complete theory, we cannot claim to have established
i t . " — the r e s u l t s were not r e a l l y conclusive. We s h a l l see
that, i n our model here, which i s i n the S-matrix language,
we are not only able to e s t a b l i s h f i r m l y the v a l i d i t y of the
GMO Formula; but a l s o to trace the o r i g i n of Octet Enhance
ment through a v i v i d path so as to allow us to get more
Insight into the problem. Returning to a d e s c r i p t i o n of our
By "an octet of vector mesons" here we just mean a set of eight vector mesons, (an i s o t r i p l e t f , an i s o s l n g l e t COQ and a doublet K* with a n t i - p a r t i c l e s K*) . We are not assuming SU(3) symmetry or even SU(3) transformation properties of these p a r t i c l e s as a s t a r t i n g point.
106
model, the two body force i s produced by the exchange of the
same octet and i s assumed to conserve i s o - s p i n and hyper
charge. A f t e r bootstrap conditions are Imposed, we s h a l l f i n d
that a degenerate mass s o l u t i o n exists with coupling constants
r e l a t e d by SU(3) C-G c o e f f i c i e n t s . Similar r e s u l t s have been
obtained by various authors i n a number of models (R.E.
Cutkosky, 1 9 6 3 a ; R. H. Capps, 1 9 6 3 a ; H. M. Chan, P. C. De
Celles and J . E. Paton, 1 9 6 4 ) . Further we s h a l l f i n d that,
to f i r s t order i n mass s p l i t t i n g s , a non-degenerate mass
so l u t i o n e x i s t s i n which, i n agreement with the analysis of
Glashow (S. L. Glashow, 1 9 6 3 ; see a l s o Ch. IV, Section 4 ( 1 )
of t h i s t h e s i s ) , only two kinds of mass s p l i t t i n g can occur:
one according to the GMO Formula (4K* 2 = 5> 2 + 3&>g2) *, the
other according to "the 2 7-plet Formula" {^^>2=CJOQ
2 + 2K* 2).
In our model, we s h a l l f i n d that the type of asymmetric solu
t i o n that w i l l emerge depends on the det a i l e d dynamical
structure of the input force. As a f i r s t test to decide which
formula i s relevant, we s h a l l use the "Parametrized One Pole
Approximation" mentioned before f o r the input force. The
se l f - c o n s i s t e n t values of the coupling constants w i l l depend
on the pole p o s i t i o n parameterX • We s h a l l a l s o see that,
corresponding to two d i s t i n c t values of X » w e set the GMO
or the 2 7-plet formulae. (In the language of Frautschi and
We s h a l l use the l a b e l f o r the p a r t i c l e to denote i t s mass here.
10?
Dashen, one can say that corresponding to two d i s t i n c t values
of A , we have e i t h e r Ag = 0 or A ^ = 0. Our model can a c t u a l l y
be formulated i n such terms, although we choose not to do so) .
We s h a l l f i n d that the former value of A corresponds to rea
sonable values of the coupling constants, whereas the l a t t e r
value of X, corresponds to a "mathematical" s o l u t i o n where the
square of the i s o s c a l a r coupling constants are negative. We
s h a l l see that t h i s l a t t e r solution, which s a t i s f i e s the 27-
p l e t formula, i s p h y s i c a l l y inacceptable so that only the GMO
Formula remains.
Let us now construct our model. We write the
relevant J = 1 p a r t i a l s c a t t e r i n g aplitude T as usual i n the
form N/D, and we have as input
fjtVfy - itffts.k+ieluf) +itff(s.K'Mf) (5.30) .
(5-31)
ft There are ten independent amplitudes f o r the
scattering of two vector mesons. There may be one (or more) of these t h a t gives r i s e to bound state poles. We concentrate on such an amplitude and t e n t a t i v e l y assume i t to be a P-wave amplitude.
108
(5*32)
where f (s.^ s ^ . S y s^) i s some as yet unspecified function con
t a i n i n g the left-hand cut of the s c a t t e r i n g matrix, and where
i n 2 / s ^ i s the t o t a l outgoing mass; Z'/fs^ ' t n e t o t a l
incoming mass; jf&2 , the mass of the vector meson exchanged; 2
and s. (CM energy) . Further,
the numerical factors i n the N-matrices have been obtained
from SU(2) crossing matrices. For a discussion of crossing
matrices and a sample c a l c u l a t i o n f o r the N-functions, see
Appendix 1. We write the D-matrices with one subtraction
performed at $0 — £0 (s7j ^ s*) '
P«*)fy - / - i^ zX(S,k*l£fio<(s,K*ll<*lf') ( 5 - 3 4 )
( 5 . 3 5 )
109
where
(5.36)
/ J? (f'-S.Xs'-Sctf,,*,*)) ( 5 * 3 7 )
where 4M(s ,s ) and q'(s 9,s») are the threshold energy and CM
momentum f o r the process. Further,
4M(s 2,s 3) = 4 max ( s 2 , S 3 ) ( 5 - 3 8 )
We impose the bootstrap conditions and get the ten
equations,
D V " ( r f ^ ) = 0 ( 5 - 3 9 )
det D If) If) = 0 ( 5 . 4 0 )
(K*) o det Dv (K* ) = 0
6 ^ " ^
M(W-^jP ( p )(f)
(5.41)
(5.42)
( 5 . 4 3 )
110
( 5 . W
As i n several previous calculations of a s i m i l a r <
2 2
~ A,
nature by various authors, a f u l l y SU(3) symmetric so l u t i o n p p ? 2 e x i s t s i n the sense that (AQ = f = K* = m and
2 2 2
here g and the functions cX and f are related by
JL - 2* where cV<, ~ <fm> ^ £ = ™, *A ™ 3),
(5.46)
(5.47)
(5.48)
(5.49)
where the superscript (-) Indicates that the d e r i v a t i v e i s to 2
the l e f t . The value of m remains completely undetermined.
We are going to take the a t t i t u d e , however, that
t h i s mathematical s o l u t i o n i s ruled out p h y s i c a l l y by some
s t a b i l i t y argument which i s as yet unknown to us, and that the
physical world manifests i t s e l f as an asymmetric so l u t i o n to
the equations (5*39) - (5*44). To obtain t h i s asymmetric
sol u t i o n we resort to a s e l f - c o n s i s t e n t mathematical perturba-2 t i o n of the above so l u t i o n . Since the value of m i s
I l l
completely undetermined, and i n so f a r as we consider the
symmetric s o l u t i o n to be a mathematical f i c t i o n , i t does not
matter, when we put i n mass deviations, from which p a r t i c u l a r 2 2 m we st a r t , as long as the value of m i s reasonably close
to the actual masses of the physical p a r t i c l e s to enable an
expansion i n powers of "mass s p l i t t i n g " to make sense. In
p a r t i c u l a r , we can with no loss of generality, assume that the
"mass s p l i t t i n g s " are a l l negative, an assumption that circum
vents d i f f i c u l t i e s a r i s i n g from the lack of smoothness of the
function M. We assume further that the s p l i t t i n g s of the
coupling constants g^ from the SU(3) symmetric values are
n e g l i g i b l e compared with the mass s p l i t t i n g s from the degener
ate value, as seems to be j u s t i f i e d by several previous calcu
l a t i o n s (D. Y. Wong, 1 9 6 5 ; J . H. Fulco and D. Y. Wong, 19&5)
on s i m i l a r problems.
To f i r s t order i n the mass s p l i t t i n g we f i n d from
eqs. ( 5 . 3 9 ) , (5.40), ( 5 . 4 1 )
C * U*&x) CSk*x+ + ° < * 4 S K * X ) = 0
For ( 5 . 5 0 ) to have n o n - t r i v i a l solutions i n c*-, , & 0 , c*7»
(SJf7 - Sc*2-2Sk*)(4$r %-36(£-Sf)(4S^ .51)
112
Since <TK*2, <Tj>2, S~C0Q2< 0 by assumption,* the l a s t f a c t o r i s
always non-zero. Hence we have, f o r a solution, e i t h e r
3f 3-V/^ZK^O 2 7-plet Formula ( 5 - 5 2 )
or 4K* r~3&J-f i » 0 Gell-Mann - Okubo Formula ( 5 . 5 3 )
For ( 5 . 5 2 ) to hold,
( f o(}-&otz +5*u)(o f\ + 2offH>U) o ( 5 . 5 4 )
For ( 5 . 5 3 ) to hold
fa + o£)(o*i+20^ + 0(1.) ( 5 . 5 5 )
The discussion up to now has not been dependent on
the form of the input f o r f (s-^.s-.s ,s ). To see whether ^ j
( 5 . 5 4 ) or ( 5 . 5 5 ) w i l l be s a t i s f i e d , we use the following para
metrized one-pole approximation f o r f ,
One need not be unduly worried by the hideous i m p l i cations of the l a s t f a c t o r i n ( 5 . 5 1 )» about what would happen i f we had not made the assumption that the mass s p l i t t i n g s are a l l of the same sign. The reason i s that i f we had not made such an assumption, we would not get anything l i k e equation ( 5 . 5 0 ) . We ifould instead get 3 equations i n the 4 parameters e^.c^c*^ and/? g where the extra parameter ^ 2
i s t n e derivative to the r i g h t of P^(s-j_, S g , s^, s^) with respect to s 2 evaluated at s-1=s2=S2=Sij.=m2. We would then not be able to make simple algebraic statements such as we are able to do here. Note a l s o that i f we make the a l t e r n a t i v e assumption that all<fm s are p o s i t i v e , then exactly the same equations ( 5 . 5 0 ) and ( 5 « 5 D w i l l be obtained, only that now i n (5«50) a l l the a r e replaced b y £ 2 .
113
where c and \ are parameters. Further we take the subtraction
point at
S0 = 4 M ( ^ , ^ ) ( 5 . 5 7 )
I f we write t = 4 ( X - 1 ) , we see that (5«47) can now be written
as
Jl = C '
which equation should s t i l l be approximately true f o r the asym
metric s o l u t i o n .
We have calculated o^, <x2 * an&C*'^ numerically as a
function of t and found that,
(l ) there i s no s o l u t i o n force, + 2<*~ = 0 f o r 1 c. Hr
- 4 < t Q ^ 40 and there does not seem to be any tendency f o r the *
l e f t hand side to vanish when t —=>oot
o ( i i ) 9a\ - 6cv„ + $ocu = 0 f o r t = - 3 . 0 5 and there
JL c. o are no other solutions f o r -h<t^ 40.
( i i i ) o( . + oc^ - 0 f o r tQ = 1 .8 and there are no
other solutions f o r - 4 < t ^ 40. o
From ( i i ) and ( 5 . 5 4 ) , we see that i f t = - 3 . 0 5 , mass
s p l i t t i n g can occur according to the 2 7-plet Formula. Now, i f
we look at the function trf&i/c , which i s a dimensionless
Note that i n the symmetric l i m i t , the pole i n eq. ( 5 . 5 6 ) occurs at -m t , so that even f o r t D = 4 0 , the pole i s already r i d i c u l o u s l y f a r out to the l e f t .
114
i n t e g r a l independent of c, we can see that i t i s p o s i t i v e p
independently of t . Hence, according to ( 5 * 5 8 ) , S < 0 f o r
t Q = - 3 . 0 5 ; so that although there exists a mathematical
s o l u t i o n where the symmetry breaking occurs i n accordance
with the 2 7-plet Formula, the " p a r t i c l e s " i n t h i s s o l u t i o n
are ghost states and must be rejected as being unphysical.
On the other hand, f o r t Q = 1 . 8 i n ( i i i ) , which corresponds
to the v a l i d i t y of the GMO Formula, we found from ( 5 . 5 8 )
that ^Z/4-^ ^ 3«5 which i s reasonable f o r strong i n t e r
a c tions. Hence, i n t h i s rather simple model, we have traced
how Octet Enhancement In medium strong symmetry breaking
comes about. In f a c t , we have demonstrated how i t follows
from two general requirements, v i z . ,
(a) The bootstrap self-consistency condition, and
(b) That the "vector mesons" are r e a l physical
p a r t i c l e s and not ghost states which are
inacceptable as physical solutions.
At t h i s juncture one may ask whether our conclus
ion depends s e n s i t i v e l y on our p a r t i c u l a r way of approximating
the left-hand cut, v i z . the parametrized one-pole approxima-' i
The existence of ghost states leads to a v i o l a t i o n of u n i t a r i t y condition, i . e . , i t leads to non-conservation of p r o b a b i l i t y (G. Kallen and W. P a u l i , 1 9 5 5 ; S. Weinberg, 1 9 5 6 ; see also G. Kallen, 1 9 6 2 ) . This can be most e a s i l y seen as follows. We can write down f o r our problem an " e f f e c t i v e " Hamiltonian which i s proportional to g. Now an imaginary g(g <o) corresponds i n our p a r t i c u l a r case to a non-Hermitlan Hamiltonian and the ordinary proof of p r o b a b i l i t y conservat i o n In quantum mechanics no longer holds.
X 1 5
t i o n . We s h a l l take up t h i s question i n Appendix 2, where
two other methods of approximating the left-hand cut are given
which demonstrate a t l e a s t that our conclusion does not depend
s e n s i t i v e l y on the method of approximation.
4. Miscellaneous Remarks
(i) The D. Y. Wong Model (D. Y. Wong, 1965)
Wong has looked a t a model where an octet of vector
mesons are again considered as two-body bound states
of the same octet. The 2-body force, contrary to
what we have done here, was approximately by a pole
f i x e d a t the p o s i t i o n s = - s 0 , i r r e s p e c t i v e of the
mass of the vector meson that supposedly was exchang
ed, and furthermore, the residue of the pole was
given a t the outset by the product of the coupling
constants of the expected bound meson to the incom
ing p a r t i c l e s and the outgoing p a r t i c l e s respectively.
As we have seen, one of the general assumptions that
one has to impose to implement the boostrap hypothe
s i s i s Crossing Symmetry. To put i n crossing
symmetry properly, the 2-body force should be pro
p o r t i o n a l to the product of the coupling constants of
the p a r t i c l e s to be exchanged and the p o s i t i o n of the
pole should depend on the mass of the exchanged p a r t i
c l e . This we have ensured i n our model presented i n
Section 3«> -Wong has discovered that, i n his model,
116
p f o r s G = 1 .8 m , the Gell-Mann Okubo Formula comes
out exactly. We have extended the c a l c u l a t i o n i n p
Wong's model and found that f o r s 0 = - 2 . 3 m , spon
taneous mass s p l i t t i n g can also occur i n accordance
with the 2 7-plet Formula. However, equation (12)
i n Wong's a r t i c l e contains a r e l a t i o n between Si> * P
which was Interpreted as the square of the SU(3) re
duced coupling constant. We substituted s Q = - 2 . 3 m 2
i n t o t h i s equation and found that T< 0 . Hence that
p a r t i c u l a r s o l u t i o n corresponds to ghost states and
must be rejected as unphysical. Hence we see that
conclusions about Octet Enhancement can also be drawn
from Wong's Model.
(11) The Model of Fulco and Wong ( J . R. Fulco and D.Y".
Wong, 1964)
Fulco and Wong considered a model i n which the octet
of vector mesons are considered as two-body bound
states of an octet of pseudoscalar mesons. Since the
pseudoscalar mesons are not produced i n t h i s model,
one does not have enough equations to constitute a
bootstrap problem. That i s , the number of unknown
parameters f a r exceeds the number of equations a v a i l
able. However, one could s t i l l get some d e f i n i t e
q u a l i t a t i v e statements about the e f f e c t of mass
s p l i t t i n g i n one octet (the pseudoscalar octet) on
the mass s p l i t t i n g i n the other (the vector o c t e t ) .
117
Fulco and Wong found that i f the pseudoscalar octet
s p l i t i t s degeneracy i n accordance with the GMO
Formula, then the vector octet also s p l i t according
to the same r u l e . We have again ca r r i e d out the
c a l c u l a t i o n further i n t h i s case and found that t h i s
i s a l s o true f o r the 2 7-plet Formula, and also f o r a
t h i r d formula which states e s s e n t i a l l y that the
average mass s p l i t t i n g s i n the octets are zero. We
s h a l l present the d e t a i l e d c a l c u l a t i o n as follows.
Taking equations (15) (16) and (17) from Fulco and
Wong's a r t i c l e , we have
where R i s a constant depending on the d e t a i l e d dyna
mics of the problem.
Let us investigate what mass formula
&7T7 4- y £/<7-t-JuS'ty3— o would give r i s e to an analo
gous formula &f2 -f y fk* + /^S^ « O i n the other
octet, where y and JX> are constants. We write,
using ( 5 . 5 9 ) , ( 5 . 6 0 ) and ( 5 - 6 l )
( 5 . 5 9 )
( 5 . 6 0 )
( 5 . 6 1 )
118
Setting the right-hand side proportional to
/ j r ' t V S K z + ^ S y ] Z , we get JL V
( 5 . 6 3 ) / v A*'
Equation (5*63) y i e l d s the folloiving n o n - t r i v i a l
solutions f o r V and ^ .
(v,/0 = | (5.64)
Hence we have the following mass formulae f o r the
pseudoscalar octet that w i l l give r i s e to an analogous
formulae f o r the vector octet
Z Sk ? + Srf - 2 / r r '= 0 ( 5 . 6 6 )
4-£k r •*• S-y2+ 3$n 7= 0 ( 5 . 6 7 )
While ( 5 . 6 5 ) i s the GMO Formula, ( 5 . 6 6 ) the 2 7-plet
Formula, (5«67) i s just a formula that states that
the average mass s p l i t t i n g i s zero. We see here
again that there i s no preference of the GMO over the
2 7-plet Formula. Here, unlike i n Wong's Model, there
i s no s u f f i c i e n t l y d e t a i l e d dynamics to Indicate that
the l a t t e r s o l u t i o n i s unphyslcal.
119
(111) Comparison With Other Bootstrap Models of
Octet Enhancement
We have mentioned i n the "beginning of Section
3 "the work of Cutkosky and Tarjanne (E. E. Cutkosky
and P. Tarjanne, 1963) who i n i t i a t e d the idea of
f i n d i n g the reason f o r the v a l i d i t y of the GMO
Formula i n a bootstrap model. Their model i s simi
l a r to ours i n that they also consider an octet of
vector mesons sc a t t e r i n g among themselves and pro
ducing the same octet as bound states. The dynamic
a l framework they used, was, as d i s t i n c t from our
case, f i e l d t h e o r e t i c a l . In t h e i r model, as we have
pointed out i n Section 3» they were able to make
pl a u s i b l e , but not e s t a b l i s h unambiguously, the
v a l i d i t y of the GMO Formula. Hence t h e i r r e s u l t
could not be considered as conclusive.
Working with an extension to the multi-channel
case of t h e i r S-matrix Perturbation Method (R. Dashen
and S. Frautschi, 1 9 6 4 b ) , Dashen and Frautschi f o r
mulated (R. Dashen and S. Frautschi, 1965c) the
condition f o r spontaneous symmetry breaking i n accord
ance with an octet pattern i n bootstrap models —
namely, that a c e r t a i n matrix Ag, which depends on
the d e t a i l e d dynamics of the problem, should possess
120
an eigenvalue equal to unity (see Section 4 of
Chapter I I I ) . In a l a t e r paper (R. Dashen and S.
Frautschi, 1965a) they worked, with'a model i n wich
the J p = 2 + baryon octet (B) scatters with the
J p = 0~ pseudoscalar octet (P) by exchange of the p 3+
baryon octet and the J = baryon decuplet (A ),
producing the same baryon multiplets as bound states.
This model, which i s an escalation to SU(3) of the
r e c i p r o c a l bootstrap f o r the nucleon and the pion-
nucleon ( 3 . 3 ) resonance (G. F. Chew, 1 9 6 2 ) , has been
studied extensively i n the symmetric l i m i t by a
number of authors with s l i g h t a l t e r a t i o n s i n d e t a i l
(see, e.g., R. E. Cutkosky, 1 9 6 3 b ; R. H. Capps, 1 9 6 3 c ;
A. W. Martin and K. C. Wall, 1 9 6 3 , 1 9 6 4 ) . I t i s
considered to be the most r e a l i s t i c bootstrap model
known i n strong i n t e r a c t i o n physics to date. By
neglecting the contribution from vector meson ex
change to the binding force and the influence of
the mass s p l i t t i n g s of the pseudoscalar mesons and
taking parameters such as the F/D r a t i o from pre
vious works i n the symmetric l i m i t , Dashen and
Frautschi calculated the matrices Ag and A^y and
found that A Q possesses an eigenvalue near unity o whereas A has no such eigenvalues. Hence i t was
27 established rather unambiguously that Octet
Enhancement emerges from the model. Due to the
121
r e l a t i v e l y r e a l i s t i c nature of the model, a number of
p h y s i c a l quantities could, be computed, among these
the r a t i o of mass s p l i t t i n g s i n the baryon octet and
the decuplet. However, one cannot gain very much i n
sight i n t o the reason f o r Octet Enhancement through
t h i s model other than the bare f a c t that A has an o
eigenvalue near unity, since a l l the dynamical d e t a i l
has already been fed i n at the very outset. In our
model i n Section 3 of t h i s Chapter, we were able to
trace step by step how Octet Enhancement i s ensured
by the step by step imposition of dynamical require
ments o Our model, i s , of course, f a r l e s s r e a l i s t i c
than the B - A b o o t s t r a p . Due to the lack of experi
mental Information, i t i s d i f f i c u l t , as of to date,
to b u i l d up a more r e a l i s t i c and yet tangible, com
p l e t e l y self-determining bootstrap model involving,
say, the pseudoscalar and the vector mesons even i n
the SU(3) symmetric l i m i t . I t i s not even s e t t l e d
which channels are coupled to which (see, e.g.,
R. H. Capps, 1 9 6 3 b ; 1 9 6 5 ; M. Suzuki, 1 9 6 5 b ) .
Suzuki (M. Suzuki, 1964a) studied the problem
of i n f i n i t e s i m a l spontaneous mass s p l i t t i n g i n a
f i e l d t h e o r e t i c a l model. "Spontaneous mass s p l i t t
i ng" here means the appearance of non-degenerate phys i c a l masses i n the s o l u t i o n to a problem formulat e d symmetrically with degenerate bare mass. In
122
t h i s model, octet baryons and mesons Interact under
interactions of Yukawa type f u l l y symmetric under
SU(3) transformations. The relevant conclusions i n
Suzuki's c a l c u l a t i o n s are: (a) that the s p l i t t i n g
always occurs i n the form of a single i r r e d u c i b l e
representation and not of a l i n e a r combination of
d i f f e r e n t i r r e d u c i b l e representations and (b) any
pattern of v i o l a t i o n i s common to a l l multiplets of
hadrons involved i n the c a l c u l a t i o n . However,
Suzuki was not able to e s t a b l i s h the v a l i d i t y of the
GMO Formula i n preference, say, to the 2 7-plet
Formula which also appeared as a so l u t i o n i n his
model . Hence Octet Enhancement i s not a consequence
of his c a l c u l a t i o n . In a l a t e r paper (M. Suzuki,
1964b), Suzuki was able to show that the condition
f o r i n f i n i t e s i m a l spontaneous s p l i t t i n g i n the pattern
of a component of an i r r e d u c i b l e representation i s
nothing other than the condition f o r the existence of
a multip l e t of composite s c a l a r mesons transforming
i n accordance with the same i r r e d u c i b l e representation.
In p a r t i c u l a r , the v a l i d i t y of the GMO Formula depends
on the existence of a scalar meson transforming l i k e
the I = 0 , Y = 0 member of an octet. This finding,
See equations ( 4 . 5 ) and ( 4 . 1 7 ) i n Suzuki's a r t i c l e .
123
though i n t e r e s t i n g i n i t s e l f , cannot claim to have
explained Octet Enhancement, e s p e c i a l l y when no such
scal a r meson has yet been discovered.
CHAPTER VI
(j)-OJ MIXING IN A BOOTSTRAP MODEL OF VECTOR MESONS
!• Introduction
We have seen both i n equation (3.11) and equation
(4.62) that the phenomenon of <p~CO mixing would manifest i t
s e l f by the appearance of off-diagonal terms i n a mass matrix
ing to study the above said phenomenon i n a model with more
dynamical content. In t h i s model, however, no mass matrix
w i l l appear e x p l i c i t l y , so i t w i l l be useful to elucidate a t
the outset how <fi-CO mixing would manifest I t s e l f i n the new
language.
The model that we are going to use i s only a s l i g h t
extension of what we have used i n Chapter V. We have again an
octet of vector mesons (CO8, f, K* ) s c a t t e r i n g among them
selves and we s h a l l look a t the p o s s i b i l i t y of the unitary
s i n g l e t meson C0X coupling I t s e l f to the octet. Now t h i s vector
meson has 1 = 0 and consequently can only be coupled to the
two body states K*K», PP, CO^CO^ }CO8COhC0,0)t. But CO, has G = -1,
so that the l a t t e r four states have the wrong G-parity, which
i s conserved i n strong i n t e r a c t i o n s . Hence CO1 can only be
coupled to K*K*• Now i f exact SU(3) symmetry i s to hold, the
i s o s c a l a r coupling constants of CO1 to K*K* and pf , f o r
Instance, are rela t e d by
having (Co', CO* ) as basis vectors. In t h i s chapter we are go-
124
125
_ co, _ co i Now, we have seen that #p> = 0. Hence A m u s t also be
zero. Hence i n the SU(3) symmetric l i m i t , only one of the two
known vector i s o s l n g l e t s can be incorporated i n t o the boot
strap.
But i f SU(3) symmetry i s to be v i o l a t e d even s l i g h t
l y , the s i t u a t i o n may become d i f f e r e n t . In t h i s case, the
unitary s i n g l e t state \C0'? and the unitary octet state
may not be the i d e n t i f i e d physical states. The physical states
may be l i n e a r combinations of these two.
/OI> ~-u*elco*> + c*e lco'> ( 6 , 1 )
CO, Then even though = 0
jff-ffH'*" 0 . ffi'-Je**''** < 6- 2 )
may not be zero. I f <f>-(o mixing i s to occur, then one would
expect that two vector mesons (with a possible difference only
i n mass) should appear as bound states i n the K*K* channel,
with coupling constants r e l a t e d as shown i n ( 6 . 2 ) , where of C0G
course jK*j<y i s t n e calculated i n Chapter V.
2 . The Vector Nonet Model We assume now that there are nine vector mesons
scattering among themselves by the exchange of the same set of
vector mesons and producing themselves as bound states. Let
126
us l a b e l these nine mesons by CO1 ( i s o s l n g l e t ) , CO3 (isoslng
l e t ) , 5> ( I s o t r i p l e t ) , K* (isodoublet) and K* (isodoublet).
The relevant J = 1 p a r t i a l wave amplitude i s written i n the
form N/D, with the input N matrices
(6.4)
where we have written " _ a Cu,
to,'io><*¥$pF)\ K") &iric*m,(*¥f,x*) tfiomPfpp)1,**)
(6.5)
(6.6)
1 2 7
Then we can write down the corresponding D-matrices i n the same
way as we have done i n Chapter V f o r the octet case. Let us
look now f o r a degenerate solution, i . e . ,
^dtJbpCfyn,*)*: d**DM(hi*)=0 ( 6 . 7 )
Evaluating the various residues of the A~matrices X^n (m-i)A(s)
and equating them to the appropriate coupling constants as we
have done i n the octet case, we get a solut i o n
The l a s t equality i n ( 6 . 8 ) indicates that i n t h i s solution, one
of the i s o s l n g l e t s i s not coupled to the rest of the vector
mesons. The equations that we have are a l l invariant under
exchange of CO, and C0& . Consequently there w i l l he another
s o l u t i o n that looks exactly l i k e ( 6 . 8 ) but with the primed ^'s
interchanged with the double-primed ^ ' s . But since we have not
defined CO, and CO# i n our model, we can take equation ( 6 . 8 ) as
the s o l u t i o n without loss of generality. Motivated by the d i s
cussion i n Section 1, we c a l l t h i s uncoupled meson the unitary
s i n g l e t . Let us consider now th i s bootstrap of the subset
co n s i s t i n g of 8 mesons. As discussed i n the previous Chapter,
spontaneous mass s p l i t t i n g can again set i n , giving r i s e to a
non-degenerate octet obeying the GMO Formula
3 to/ + \ f 2 = (6 . 9 )
128
This equation can be written i n the parametric form
I k**= f n x ( i + p) (6.10)
v CO/ - FPiz(i+2p)
since
' 3/+ fm* (6.11) r- 2 m i s the average squared mass of the vector octet.
Of course one cannot be sure that (6.7) and (6.8),
even including the s o l u t i o n with spontaneous mass s p l i t t i n g
given by (6.9), are the only possible solutions. However, one
can be sure that f o r a large class of methods of approximating
the left-hand cut, i n c l u d i n g the one and many-poles approxima
tions and the s o - c a l l e d Unitarized Born Approximation, the
function <X which i s involved i n D i s a monotonlc function and
cannot have two zeros. Hence, i t i s impossible to have more
than one bound state coming out of the 1 = 0 , K*K* channel.
Resigning ourselves to the f a c t that two i s o s i n g l e t s
cannot be produced i n a bootstrap involving vector mesons alone,
l e t us s t r e t c h our imagination a l i t t l e b i t and consider a
bigger bootstrap problem that involves, i n a d d i t i o n to vector-
vector scattering, vector-pseudoscalar s c a t t e r i n g . Then even
i n the l i m i t of SU(3) symmetry, the unitary s i n g l e t CO1 i s
strongly produced. Due to the v i o l a t i o n of SU(3) symmetry,
t h i s meson, once produced, may couple i t s e l f weakly (this
129
coupling can be vanishingly weak f o r our discussion) to the
K*K* channel. Then, i n our truncated bootstrap problem con
s i s t i n g only of vector mesons, the existence of t h i s unitary
s i n g l e t i s not a consequence of our bootstrap dynamics and i n
t h i s sense i t can be regarded as an "elementary p a r t i c l e " In
our problem. Let us stress again, that t h i s so-called "ele
mentary p a r t i c l e " need not be a fundamental e n t i t y i n the
ordinary sense. I t may happen that i t can be incorporated i n
a more extensive bootstrap model. That i t has to be b u i l t
i n t o our truncated bootstrap just r e f l e c t s the f a c t that t h i s
subset of hadrons i s unable to produce i t s e l f - c o n s i s t e n t l y .
Now there i s an ambiguity i n the s o l u t i o n of d i s
persion equations by the N/D method that we can e x p l o i t . This
i s due to the observation that, with a given left-hand input,
the N/D s o l u t i o n to the equations i s by no means unique. This
i s most e a s i l y seen by noting that the replacement
DCs) — D ( s ) = vft -=r-
does not v i o l a t e the a n a l y t i c i t y conditions required of A ( s ) .
This pole term, which can be introduced into the D-function
almost at w i l l , i s c a l l e d the CDD pole (L. C a s t i l l e j o , R. H.
D a l i t z and F. J . Dyson, 1 9 5 6 ; M. Gell-Mann and F. Zachariasen,
1 9 6 l a , 1 9 6 l b ; see a l s o S. C. Frautschi, 1 9 6 3 , pg. 2 9 - 3 9 ) and
i s usually interpreted as an elementary p a r t i c l e . The funda
mental coupling of t h i s "elementary" p a r t i c l e i s proportional
to the residue, of the CDD pole that i s so introduced.
130
Let us assume now that the "tare" mass squared (the
mass of the "elementary" p a r t i c l e before i t i s embroiled i n
our bootstrap cal c u l a t i o n ) of the unitary s i n g l e t i s equal to
the average mass squared w 2 of the octet. Then we introduce
a CDD pole to the D-function of the 1 = 0 , K*K* channel as
follows T l 2 — 2
where P i s the "fundamental" coupling of the CO1 to K*K*.
The m u l t i p l i c a t i v e constant i s the l a s t term of (6.12) i s just to make P dimensionless. Now, as we have seen,
has a zero a t $ — in2+2j3ry>2
Hence near S — PPi 2 , we have 2__,
To f i n d the zeros of
where
* W ~ ~ [p^Ym'^pO]' < 6 , 1 5 >
The two roots f o r (6.14) are then
5 = n,2[ 1 + p ±j p7+r'l (6.16)
Hence we have two bound states now i n the 1 = 0 channel. We
s h a l l c a l l these two bound states \<py and \C0/> respectively
with masses given by
131
The states and \t»y are now the physical states,
We s h a l l write down t h e i r parametric mass formulae ( 6 . 1 0 ) and
( 6 . 1 7 ) together as follows (of. eq. ( 4 . 6 8 ) )
( 6 # 1 8 ) .
Equation ( 6 . 1 8 ) i s a set of 4 equations with 3 parameters fw2 ,
|S and . Eliminating these parameters, one gets
<
<$f+ of = 2 k*'- ( 6 . 1 9 )
This equation, as mentioned i n Chapter IV, Is good to about 2%,
To evaluate the coupling constants, we s h a l l neglect
spontaneous mass s p l i t t i n g , as we have consistently done i n
Chapter V. We s h a l l now proceed to calculate
<f> s 2**** =
co N ( V ;
( 6 . 2 0 )
where we have dropped the superscript (CO) to the N and D
functions.
( 6 . 2 1 )
Now,
cf = fz2d-r) ( 6 . 2 2 )
132
Hence
N(ffi*)+N'(f!i*)'*lr = NC»%)[ 1 (6-23)
From ( 6 . 1 3 ) , one sees that D and i t s derivative are very .~2 singular around w , so that i t i s appropriate to expand only
the f i r s t term i n powers of (S-ni2) and leave the second term
as i t i s
(6.24)
where we have made use of equation ( 6 . 1 5 ) .
(6.25)
Hence
ft (6.26)
S i m i l a r l y N(fP> z)
2P0 (6.27
Therefore we could write
(6.28)
where
i f (6.29)
133
Equation (6.28), as we have mentioned i n Section 1, can be
interpreted as <f>-Cti mixing.
Let us pause here to examine the physical s i g n i f i
cance of ( 6 . 2 9 ) . The important thing to note about t h i s
equation i s that as }f (and hence the "fundamental coupling"
P ) —=> 0 , C&Q and 4^*6 — ^ £ *. In other words,
a fundamental coupling of vanishing strength of an "elemen
tary p a r t i c l e " to the 1 = 0 channel would t r i g g e r o f f a very
s i g n i f i c a n t amount of configuration mixing between th i s p a r t i
cle and a bound state p a r t i c l e that i s already there. In
t h i s sense, the occurrence of the mixing depends on the mere
existence of the 60* p a r t i c l e . Hence we could look upon the
above phenomenon as a "spontaneous" <j>~CO mixing.
I t becomes imperative to check now that we s t i l l
have a s e l f consistent s o l u t i o n to the bootstrap equations.
I t i s straight-forward, although tedious, to check that to
lowest order i n P , the bootstrap equations are s t i l l
This i s not to say that i f we have put I = 0 at the beginning, we s t i l l have a half-and-half mixing. The point i s that i f p = 0 at the outset, we have only one bound state i n the 1 = 0 channel and mixing i s not defined at a l l . (By a coupling constant of vanishing strength we mean a quantity which can be made a r b i t r a r i l y small but s t i l l remains non-zero.) Mathematically t h i s i s quite a common occurrence when we take the double l i m i t of a function, say,
0 ( TfS ), i n two d i f f e r e n t orders. Since the function Q-i s singular around s - M 2 , i t i s not s u r p r i s i n g that
134
s a t i s f i e d by
f = k*7 = } <£2= m2(i+0 j 6)*=n>2(i-r) ( 6 . 3 0 )
^ i r (6.31)
where £ t f 7 $ and t> are given by (6.29).
With the coupling constants given as above, we look
a t the D-matrices. Abbreviating iu^Q by s and Ce^S by c,
we get
( 6 . 3 2 )
( 6 . 3 3 )
135
- c s f c t f r ^ & g r t
( 6 . 3 4 )
Requiring
We got the following equations i n terms of small mass devia
tions from the degenerate value.
£ = fi>&;V,*4^V2+<;^ ( 6 . 3 6 )
£ ! „ 2 Stfct, +4SK**</Z + (c*>a9S4>*+ Sv+ff)rtt] ( 6 . 3 7 ) J m L Eliminating P/ 2 from ( 6 . 3 6 ) and ( 6 . 3 7 ) , vxe get
Discarding the so l u t i o n Soo-Sty1 since i t would lead to the
unphysical conclusion of equal mass between the 60 and the <fi
mesons, we have
From the l a s t two e q u a l i t i e s i n ( 6 . 3 5 ) , we get
136
(l88f)ot, + (l2Sk*x+24£?)<X2+(3crieS$*+3s*6Stdx+ (6.40)
(6$k*l)cV, + (?fyl+6fr*+fc^z9S<f±3^QSco2)(Xz (6.41)
Now the c o e f f i c i e n t s of &, i n equations ( 6 . 3 9 ) , ( 6 . 4 0 ) and
( 6.4l) correspond to the influence of s h i f t s i n the bound
state poles on the bootstrap system. Now the p a r t i c u l a r channel
corresponding to ( 6 . 3 9 ) has one p e c u l i a r i t y i n that i t possess
es two bound states, so that one would expect that the i n f l u
ence of each of the bound states should be toned down by a
facto r proportional to the p r o b a b i l i t y with which that bound
state appears. Hence i t seems reasonable to replace fco7 by
'Sco*s**?Q and by S^C&^Q whenever these two quantities
appear before cx, . One might expect that t h i s should have
been taken care of automatically when we write down the various
equations. Hence the v a l i d i t y of t h i s procedure remains dub
ious. The answer to our doubts may come from further under
standing of the N/D method when CDD poles are involved.
Making the above-mentioned replacement, we get, i n
stead of ( 6 . 3 9 ) i
(6.42)
Comparing equations (6.40), (6.41) and (6.42) with ( 5 . 5 0 ) , we
see that to get the former from the l a t t e r i t i s only necessary
to replace $0)g by C**9&S4>*+ St»*$ ft*? . Since the CX^ 's are
137
the same as i n Chapter V, we know that the only p h y s i c a l l y
acceptable mass s p l i t t i n g must obey the formula p
(6.43)
Given <f , the angle G can i n p r i n c i p l e be c a l c u l a t
ed from equation (6.29); hut since the parameter V can only
come from a more extensive dynamical c a l c u l a t i o n (e.g., a
bigger bootstrap), we do not have enough information here to
c a l c u l a t e an exact value f o r 9 . We could mention, however,
that f o r vanishing V , C*v 2Q — > 5<0%, and that deviations
from th i s value should be of the order of the r a t i o of the
medium strong i n t e r a c t i o n to the very strong i n t e r a c t i o n . This
i s consistent with the experimental value of 60%.
In conclusion, apart from a technical assumption
that we have to make i n conjunction with the d e r i v a t i o n of
(6.43), the equations (6.19) and (6.43) are unambiguous conse
quences of our model. The only extra requirement that we have
imposed, beside the bootstrap hypothesis and that the p a r t i
c l e s must be physical, i s the existence of a unitary s i n g l e t
vector meson with mass equal to the average mass of the octet.
In our group t h e o r e t i c a l treatment of the problem i n Chapter
IV, we had to assume octet transformation property of the
mass s p l i t t i n g matrix. This requirement turns out to be a
consequence of our model here. Another i n t e r e s t i n g conse
quence of our model i s that the mere existence of the unitary
138
s i n g l e t would be s u f f i c i e n t to t r i g g e r o f f a substantial
(maximal) amount of mixing with the unitary octet i s o s l n g l e t
vector meson.
Hopefully a more d e t a i l e d dynamical c a l c u l a t i o n
l i k e a bigger bootstrap would allow us to calculate F , and
hence the angle 9 . In that event we would have two mass
formulae r e l a t i n g the masses of <P , CO , p and /<*.
APPENDIX I
CROSSING MATRICES
(J . J . De Swart, 1 9 6 4 ; D. E. N e v i l l e , 1 9 6 3 ; H. H. Capps, 1 9 6 4 ;
A. W. Martin and W. D. McGllhn, 1964) '
I t i s most convenient to Introduce the idea of a
crossing matrix by a simple example, (We s h a l l confine our
selves to SU(2) crossing matrices since these are the only
crossing matrices that we have used i n the t h e s i s ) . Let us
denote the KK sc a t t e r i n g amplitudes by A K$Kg ; fy where the
K's are taken to mean the kaons K+, K° and the K's t h e i r a n t i -
p a r t i c l e s K", K°. I f one assumes conservation of isospin,
the 16 amplitudes connecting |K 4'K J} J //^+/?°> , | K°> K"} and
can be summarized int o two independent amplitudes A,
and A 0 , where the subscripts l a b e l the t o t a l i s o s p i n of the
incoming as well as the outgoing system. Due to the s u b s t i
t u t i n g r u l e ,
AK*K*>K+K° fo-O- AK91C°;I:*K-(*>S) ( A I . I )
Now since
We have
\K->=-lh~i? ( A 1 - 2 )
\i>°> - fcaw-WK-y) . (A1.3)
139
140 J
From (A1.3).
Arjc'tK+K'fat-) = AtCs.-h) (A1.4)
AK°K° : Or* 0 = -£M*> 0 + j " * & s ) < A 1 «5)
Hence, subs t i t u t i n g (A1.4) and (A1.5) into ( A l . l ) ,
A,(s*) =-£A,(*,s) + ±A*(t,s) (A1'6)
Interchanging s and t i n (A1 . 6 ) , one gets another equation
A o O s t ) = £Ai6bs) + iA.fcs) ( A i . 7 )
We can write (A1 .6) and (A1 .7) i n the compact form
where the matrix
i s c a l l e d a Crossing Matrix.
I t turns out that the crossing matrix i s usefu l i n
another s i t u a t i o n . Supposing now that we are not concerned
with the whole K-K scattering amplitude, but with a subset of
the t o t a l i t y of Feynmann graphs that contribute to the ampli
tude ( as shown i n F i g . I a ) . Let us c a l l t h i s diagram B.
Now the diagram B i s related to the Diagram A (as shown i n
(A1.8)
141
A
a) P i g . I
F i g . 1 b) by crossing, i . e . ,
b)
BK+K°} K+K° = Aw; W(*js) (A1 .9)
Then we can retrace the arguments from ( A l . l ) on, and get
ft fr*) 1 = .2 j .
(ALIO)
This l a s t equation i s very useful when the amplitude A, and A 0
are e a s i l y c a l c u l a b l e . As a matter of fact, i n bootstrap c a l
culations, diagrams l i k e F i g 1 a) are supposed to give impor
tant contributions to the left-hand cut and hence are usually
used f o r the N-functions i n the N/D method. The diagrams A (
and A 0 are usually easy to c a l c u l a t e . Supposing that we are
considering a model where the main force i s due to vector meson
exchange, then i n the 1 = 0 , KK channel, the force a r i s e s from
the diagrams
d0 (^) ( A l . l l )
142
Now the terms on the right-hand side are d i f f i c u l t to calcu
l a t e d i r e c t l y . But the "crossed diagrams"
'5
\
(A1.12)
are expressible simply i n terms of i s o s c a l a r coupling const
ants ,
Using the corssing matrix elements i n (A1.10), we get
ft £.*-/• = §A,(hs) + fts)
Taking the i t h p a r t i a l wave of B . ( ^ t )
(A1.14)
^w = I>+iftH(*°> (A1-15)
where i n c i d e n t a l l y we have calculated the input N-function
N ^ 8 ^ ( s ) i n equation ( 5 * 3 0 ) . Of course i n equation ( 5 . 3 0 ) ,
the incoming p a r t i c l e s are vector mesons. This could only
change the kinematics of the c a l c u l a t i o n and has nothing to
do with the crossing matrix.
APPENDIX 2
COMMENTS ON THE VALIDITY OF THE
PARAMETRIZED ONE POLE APPROXIMATION
As we have mentioned, the parametrization of the de
pendence of pole p o s i t i o n and residue on the p a r t i c l e exchanged
by c and X presumably allows considerable freedom to encompass
a large class of situ a t i o n s , so that i t i s p l a u s i b l e that f o r
a suitable choice of c and X , the r e a l physical s i t u a t i o n i s
r e f l e c t e d to a good approximation. We s h a l l further t e s t the
dependence of our conclusion on the one pole approximation by
two examples i n th i s appendix.
(a) We s h a l l modify the one pole approximation by
replacing the single zero i n the denominator of the N-function
i n (5.26) by a zero of f r a c t i o n a l ord.er, i . e . , we s h a l l write
A/./k) = — (A2.1)
where (X(o) i s a new parameter with oi(o)= 0 corresponding to
the case we have considered. Note that f o r o^(o)^= O , we have,
instead of a simple pole, a branch point a t 5 - S*-4\n? and
we can define a left-hand cut along the negative r e a l axis
s t a r t i n g from S- and extending back tof=-co . We
One i s tempted at this juncture to look upon the parameter o<(o) as the value of the Regge trajectory of the vector meson at £ = o , by comparing the asymptotic behaviour of (A2.1) with Regge behaviour. This c o r r e l a t i o n , however, i s rather f a r fetched and hence we s h a l l r e s i s t the temptation.
143
l¥f
have made ca l c u l a t i o n s f o r - 1 &(p)£ 0 . 5 and found that the
conclusions about Octet Enhancement are e s s e n t i a l l y unchanged.
In other words, although mass s p l i t t i n g can occur mathemati
c a l l y both according to the GMO formula and the 2 7-plet formu
l a , only the former corresponds to p h y s i c a l l y acceptable (non-
ghost-state) so l u t i o n s . We have not made calculations f o r
value of o<(o) higher than 0 . 5 because of convergence d i f f i c u l
t i e s i n the numerical c a l c u l a t i o n s . However, we believe that
the range of values that we have allowed f o r o((o) has already
served to convince us that the conclusion about Octet Enhance
ment i s not s e n s i t i v e to the p a r t i c u l a r approximation that we
have made.
(b) We s h a l l further test how s e n s i t i v e l y our r e s u l t
depends on the parametrized one pole approximation by the f o l
lowing p a r t i c u l a r form of the two pole approximation. The
trouble with a two-pole approximation i s the enormous freedom
i t o f f e r s , so much so that almost any conclusion i s possible.
So we s h a l l seek a reasonable way to r e s t r i c t the positions of
the two poles. To do t h i s we calculate the Born scattering
amplitude of two s c a l a r mesons (since we s h a l l neglect the
spin of the incoming p a r t i c l e s ) of i d e n t i c a l mass unity, ex
changing a vector meson of the same mass. Evaluating the l e f t
hand d i s c o n t i n u i t y of the J = 1 p a r t i a l wave amplitude, which
s t a r t s at s = 3 , where s i s the CM energy squared, we have
I* B(S) ~ (2'-'->«-» ( A 2 . 2 )
145
which i s sketched i n F i g . I I .
-3 ~Z \ - I o St-4 . ,i
F i g . I I
Note that Xo i s the p o s i t i o n of the pole i n the one
pole approximation that gave r i s e to the GMO Formula, ^ P the
p o s i t i o n that gave r i s e to the 2 7-plet Formula. Evidently, i f
t h i s left-hand cut i s to be replaced by two delta functions
(two-pole approximation), i t would be reasonable to put one of
the poles i n the i n t e r v a l (-3>-l) and the other i n ( 2 , 3 ) . I t
turns out that i f we put the poles t h i s way, the conclusions
about Octet Enhancement are unchanged. Vfe s h a l l not give the
d e t a i l e d numerical r e s u l t s here, but the reason f o r the conclu
sion i s easy to see. Let us write the new ^ - f u n c t i o n that
appears i n N as
(A2.3)
where has a pole at x i n the i n t e r v a l ( - 3 t-l) and -f^ a pole
a t y i n the i n t e r v a l ( 2 , 3 ) . These functions are so normalised
146
so that i*» o r , when used alone, w i l l s a t i s f y the
bootstrap condition. Xy and are two constants. Prom
the d e f i n i t i o n of of, , tf} and £<* i n eq. (5.49),
0(C = A x ^ i * +X9<X$ ><L=\,2,4 (A2.4)
where the d e f i n i t i o n s of o<Vx and (X,'# are obvious. Now f o r
the 2 7 - p l e t formula to hold,
' q o t t — 6<X2 + 5"0C* » 0 (A2.5)
A f t e r s u b s t i t u t i n g (A2.4) into (A2.5), we have
X 6#S.+5rotf) + X% (<7c*,y~6cx!? + 5-c/J?) = 0 (A2.6)
Now, we know that ^c/,^- ^cv/* + 5"<?c 0 •= 0 \. and 'foT, - -h^c/J1
i s a slowly varying function of , so that f o r X i n the i n t e r
v a l (-3,-1) and y i n (2,3)
I 6*e*+srctf I > JW- 5V**| (A2.7)
Hence, f o r (A2.6) to hold
I Xy \ » Ux| (A2 .8)
Now, our bootstrap condition i n (5.47) says that
_L _ _ 2 (X* # * 2 X % & ? 2ottV
M?+X)tf ~ Xdtf = 1* But the l a s t term i n (A2.9) i s negative: the SU(3) reduced
coupling constant ^ i s imaginary and hence corresponds to
ghost states. Thus we see that, i n a two pole approximation,
(A2.9)
147
the r e s u l t o f Octet Enhancement s t i l l holds. This again
suggests that our conclusions on Octet Enhancement are i n
sen s i t i v e to the p a r t i c u l a r approximation that we have used.
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