[Ergebnisse der Exakten Naturwissenschaften] Ergebnisse der exakten Naturwissenschaften Volume 36 ||...

Some Topics Related to Unitary Symmetry*

B y M i c h e l G o u r d i n

Univers i t6 de Pa r i s

Facu l td des Sciences - - O r s a y

Labora to i r e de P h y s i q u e Thdor ique e t

H a u t e s Energ ies

O r s a y S. e t O. (France)

Rece ived J a n u a r y 1964

C o n t e n t s

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I. U n i t a r y s y m m e t r i e s . . . . . . . . . . . . . . . . . . . . . . . 4 1. Special u n i t a r y g roup a n d isotopic sp in . . . . . . . . . . . . . . 4 2. U n i t a r y g roup a n d isotopic sp in . . . . . . . . . . . . . . . . . 8 3. Lie g roups a n d Lie a lgebra . . . . . . . . . . . . . . . . . . . 10 4. Charge c o n j u g a t i o n . . . . . . . . . . . . . . . . . . . . . . 14

I I . Par t ic le classif icat ion . . . . . . . . . . . . . . . . . . . . . . . 17 I. Genera l i t ies . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. B a r y o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3. Pseudo-sca la r m e s o n s . . . . . . . . . . . . . . . . . . . . . 18 4. Vector m e s o n s . . . . . . . . . . . . . . . . . . . . . . . . 19 5. M e s o n - b a r y o n resonances . . . . . . . . . . . . . . . . . . . 19 6. M e s o n - m e s o n r e sonances . . . . . . . . . . . . . . . . . . . . 22 7. Nuc lea r p h y s i c s . . . . . . . . . . . . . . . . . . . . . . . . 23

I I I . E x p e r i m e n t a l consequences of u n i t a r y s y m m e t r y . . . . . . . . . . 24 1. Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 25 3. I somet r i c s . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4. Appl ica t ion . . . . . . . . . . . . . . . . . . . . . . . . . 28 5. S t r o n g decays of exc i ted b a r y o n s . . . . . . . . . . . . . . . . 31 6. F in i t e g roup R . . . . . . . . . . . . . . . . . . . . . . . . 34 7. Sca t t e r ing of two oc tup le t s . . . . . . . . . . . . . . . . . . . 4 0

8. Pe r iphera l mode l . . . . . . . . . . . . . . . . . . . . . . . 44 9. Decup le t p roduc t ion . . . . . . . . . . . . . . . . . . . . . 46

I0. Two decuple t p roduc t ion . . . . . . . . . . . . . . . . . . . 51

* T h i s review art icle is t h e m a t t e r of va r ious series of lec tures g iven in M a y 1963 a t t h e B o r d e a u x Un ive r s i t y , in J u l y 1963 a t t h e Carg~se S u m m e r School a n d d u r i n g t h e academic y e a r 1963--1964 a t t h e Facu l t~ des Sciences of t h e U n i v e r s i t y of Par is ,

T h e a u t h o r apologizes for a v e r y incomple te l is t of references, in pa r t i cu la r wi th respec t to t h e m a t h e m a t i c a l p a r t w h i c h is a s s u m e d to be famil iar .

Erg, d. exakt. Naturw. 86 1

2 MICHEL GOURDIN :

I V . B r e a k d o w n of t h e u n i t a r y s y m m e t r y . . . . . . . . . . . . . . . . 5 2

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 2 . F i r s t o r d e r v i o l a t i o n . . . . . . . . . . . . . . . . . . . . . . 5 8 3 . V i o l a t i o n o f h i g h e r o r d e r . . . . . . . . . . . . . . . . . . . . 5 5 4 . M a s s f o r m u l a e . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 5 . co - - ~ m i x i n g . . . . . . . . . . . . . . . . . . . . . . . . . 5 9 6. E l e c t r o m a g n e t i c i n t e r a c t i o n s I y = 0 . . . . . . . . . . . . . . 61 7. E l e c t r o m a g n e t i c i n t e r a c t i o n s I y ~ 0 . . . . . . . . . . . . . . 71

C o n c l u d i n g r e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3

A p p e n d i x : U n i m o d u l a r u n i t a r y g r o u p S U (3) . . . . . . . . . . . . . . 7 5

R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0

Introduction

The possible existence of high symmetries in the elementary particle interactions is now a fashionable problem. This can be a reason to be interested in one of the most seductive realizations of this idea: the unitary symmetry.

The invariance of the physical laws with respect to the Lorentz group transformations has very important and general consequences as the energy-momentum and the total angular momentum conservation. On the other hand, the electric charge Q and the baryonic number, for instance, are well defined quantities for all physical states; they are conserved in all types of reactions and both Q and B occasion super- selection rules.

Beside the universal conserved quantities, one can define in some cases only approximately conserved quantities. For instance, if one forgets the so-called weak interactions, the parity and the charge conjugation invariance hold to a relatively high degree of precision from experiment. In weak interactions, both invariances are destroyed but their product appears as a fundamental universal invariance. More precisely three transformations, the space reflexion P, the time reversal T, the charge conjugation C are related in the P C T theorem from some general assumptions where the Lorentz invariance play, s a fundamental role.

Let us now consider the very interesting case of isotopic spin invariance. One can define a law of conservation if we restrict ourselves to strong interactions only. This concept has been useful in elementary particle physics and also in nuclear physics for it took into account the electromagnetic interactions assumed to be responsible for a small breakdown of the isotopic spin invariance. Actually, one introduces a hermitian space, called the charge space for the isotopic spin invariance, completely independent of the Minkowsld space of relativity. The connection between isotopic spin invariance described by the two- dimensional unimodular unitary group and the Lorentz invariance has been studied by many authors but a definitive and clear relation has not yet been given.

Some Topics Related to Unitary Symmetry 3

The starting point of all high symmetries research is the model of isotopic spin. From a mathematical point of view, one has to generaliz_~e the five elements algebra constituted by the isotopic spin rotat ions/ , the electric charge Q, the baryonic number B. Other familiar quantities such as the strangeness S and the hypercharge Y are related to the previous one by the usual expressions

(2 = 13+1/2 Y Y = S + B . Let us call G the high symmetry group assumed to be valid for strong

interactions. The usual assumption one does not intend to discuss is the following: G is a semi-simple compact Lie group for which all representations can be taken as unitary. The physical particles, elementary or not, bound states or resonances, are classified as weights of irreducible representations of G. Of course, the vectorial spaces of representations and the space time are independent; all members of a given supermultiplet must have the same spin and parity; the baryonic number must be the same for all partners of a supermultiplet and the B operator plays a privileged role in the Lie algebra of G.

The isotopic spin group has all the previous properties. In an exact symmetry theory, all properties for the particles of an isotopic spin multiplet are completely degenerated. For instance, all masses must be the same. Such a result is not true experimentally; the small observed mass-differences are generally explained as a violation of the isotopic spin invariance by the electromagnetic interactions.

For the supermultiplets of the invariance associated to G, the mass differences are ascribed, by analogy, to a breakdown of the symmetry due to a strong interaction which conserves, of course I, Q and B. The empirical results obtained in this domain remain yet unexplained theoretically.

In part I, we study the Lie algebra of the unitary symmetry and we define the maximal commutative subalgebra associated to the conserved quantities I~, Q and B. The relation between Lie algebra and Lie groups is discussed for the case of unitary symmetries. Finally, an extension of the group by the charge conjugation operation is studied.

In part II, we give the present classification of particles and resonances in the "eightfold way" model. In particular we discuss the predictions of the model in regard to spins and parities.

In part III, we study the experimental consequences of an exact unitary symmetry for the S-matrix elements. The products of representations are explicitly calculated in the more important cases and the expansion of various reaction amplitudes in eigenamplitudes of G is given. Some relations between cross-sections are established for the case of elastic and inelastic, meson-baryon, baryon-baryon, anti-baryon- baryon collisions.

In the last part, part IV, we are concerned with two breakdowns of unitary symmetry: the first one by some less strong interactions and the second one by the electromagnetic interactions. The famous mass formula is given and discussed for the different supermultiplets.

1"

4 MICHEL G O U R D t N :

Some problems related to unitary symmetry are not discussed here, essentially the application of unitary symmetry to weak interactions and the justification of unitary symmetry by dynamical considerations. Both questions are yet extremely model dependent and the conclusions one obtains in these domains seem to us at the same time very interesting and very contestable.

I. U n i t a r y s y m m e t r i e s

1. Special unitary group and isotopic spin

1.1. In its standard form, the A s Lie algebra is described by six operators, E~i, associated to the six non zero aq roots and three H i

operators of sum zero which gives

~23 12;13

~37 ~32

rise to the two-dimensional Cartan sub-algebra [1].

The isotopic spin sub-algebra of the A 1 type may be chosen in A ~ in an infinite number of ways, some of them being equivalent because of an inner automorphism. We shall make a choice such that the third component of the isotopic spin belongs to the Cartan sub-algebra. It is then easy to see that an angular momen-

.->.

tum J such that Fig. 1, Root Diagram

-~> -o- *---> j × ] = i ] (I, 1)

can be defined from three roots a, O, - a , in the following way

F 2 1 I/2

[__,_2 ],12n J - = L(~,~)J ---~

r(: , ' ) l Jo = L ~ J "

These relations are valid for any semi-simple group and the scalar products considered are taken in the weight space where the metric is defined by the Caftan tensor [1].

We shall define the isotopic spin in A ~ by the three generators

I+=V6EI~ I----]/6E21 Io = ~ (H1- H~) (I, 2)

Some Topics Related to Un i t a ry S y m m e t r y 5

1.2. The fundamental matr ix representation D3(1,0) of the A s Lie algebra allows us to write

i ° ! 1 i o i I+ = 0 I - = 0 Io = ~2- --1 0 0 0

an equivalent form of which is

T=±i 2t0 I t is well-known that in the Sakata model, for instance, this representation describes both the (p, n) doublet and the A singulet.

1.3. In the weight space, we shall use triangular coordinates X1X~X 3 of zero sum.

x3 Fig. 2. Weight Space

All the bases of the A s irreducible representations are built by means of a tensorial product from that of the fundamental representation D¢~)(1, 0) and that of its contragradient D(~}(0, 1). A vector of the DCN}(21, ~2) representation may be associated to a tensor with zero trace, symmetrized with respect to its contravariant indices 42 and with respect to its covariant indices 21 [21. Among the contravariant indices, we can find vl times index 1, v~ times index 2 and v3 times index 3

For the covariant indices, we shall have in the same way

We shall then consider an isotopic spin rotation associated to I o

A = exp (io~Io). I t is a unitary transformation, diagonal for all the irreducible representations of Aa. From the matr ix representation used above for the fundamental representation, we shall obtain in particular

x '1 = exp , -~-

x '2 = exp - ~ -if-

A; ~3 ~ :~ .

6 ]~ICHEL GOURDIN :

For the weight {v i, ~} of the D (N) (41, ~) representation, the A matrix has a diagonal element given by

{[ 4:]} A (v, ~) = exp i (vl - ~1) -~- - (v2 - ~)

Therefore the value of I o for that weight is

I o = 1/2 E(~l - ~1) - (v~ - ~2)]. (I, 3) I t is of course evident that the coordinates of a weight are known from the set {v, ~}

x ~ = (vj - ~j) - ~/3 G - ;:3) ( I , 4) Therefore the isotopic spin component I o associated with a weight X I X ~ X 3 is given by

Io = 1/, ( G - X , ) . (I, 5) 1.4. The Cartan sub-algebra contains two linearly independent

elements which can be chosen according to

V~ (H1 - H~) Io = -5 -

Y = H (HI + H~). The H 1 + H~ + H a - -0 condition involves an equivalent form for Y: Y = - V 6 H a , which makes the following commutation relations trivial

EI, Yl = 0. From the physical point of view, one operator is known: the hypercharge Y, which commutes w i t h the three components of the isotopic spin. The only relation between Y and ~ which can be deduced from this property is of the linear type

Y = a Y + 2fl (1,6) where ~ and /5 are constants which will be determined from physical requirements.

The hypercharge Y must have integer eigenvatues for all the physical states associated to the weights of a representation and also the differences of hypercharge between two arbitrary weights will be integer algebraic numbers: zJ Y = 0, + 1, ~: 2 . . . .

W e shall start by looking for the eigenvalue of Y associated to a weight X1X~X v The calculation method is identical to that of the above section and one immediately finds

We must now look for the form of the X i coordinates for the weights of an irreducible representation D (N) (~, ~) . The result is the following

X i -~ A (~x, 2~) (1) (I, 8) where (1) indicates modulo an integer and A (;t~, 2~) is now a function of the representation only, given by

A (21, ~) ~- ~ + 2 ~ (I, 9) 3

This result is obtained simply through tensorial product.


The Xj's being defined up to one integer, the only possible values for can be deduced to be ~: 1 and we shall choose + 1.

When A (21, 2~) is not an integer, it is impossible to choose fl = 0 if Y is to be integer. Unfortunately, the operator represented in all representations by a matrix proportional to the unity does not belong to the Ag. Lie algebra. We deduce the two following equivalent conclusions:

a) The eigenvalues of the algebra operator T are not integer for all representations when A (41, 23) * 0 (1).

b) The hypercharge operator Y for which, by definition, all eigenvalues are integer for all representations does not belong to the A s algebra.

On the other hand, when A(2 x, ~)------- 0(1), one can choose /5 = 0 and for these representations the physical hypercharge operator is given by

v = - (I , 10)

1.5. Similar and equivalent considerations can be made for the charge operator Q. From the definition

= / 3 + 1/2 7 (I, 11)

we obtain the value of ~ for a given weight

=xl.

The only representations for which the values of ~ -- which are then assi- milated to the charge Q - are integer, can be characterized by

2~i + 22~ ~= 0 (3). (I, 12)

The charge operator is then defined by

Q = I / 6 / / i . (I, 13)

1.6. Finally we shall give the description with respect to the isotopic spin and to the hypercharge of the representations we shall be using

D 1 (0, 0) • Y = 0 I = 0

Y = I

D s ( 1 , 1) ~ Y 0

2

_ Y = - - I

Fig.

I = x]~

8 MICHEL GOURDIN :

D *° (3, O)

Y = t I = 3 / ~

Y = 0 I = 1

Y = - - 1 I = '/z

Y = - - 2 I = 0 Fig, 4

D 1° (0, 3)

Y==2 I = 0

Y = 1 I = i/2

Y = 0 I = 1

. . . . Y = - - 1 I = "/~ Fig. 5

Y = O I = 0 , 1 , 2

r = - , x = 1 \ V / . . _ Y = - - 2 I = 1 .

Fig. 6

Remark. F o r the represen ta t ions in which e i ther ~ or 22 are zero, al l the weights a re s imple and there is a l inear re la t ion be tween the i sotopic spin a n d the hypercharge of a weight

2 ~ D (a, 0) Y = 2 I - - 3 -

2 2 ' ~ ' " ~'L,W,~ ) Y = - 2 I + 3 "

2 . U n i t a r y g r o u p a n d i s o t o p i c s p i n

2.1. The U(3) Lie a lgebra is no t a s imple one; i t is the d i rec t sum of the two s imple a lgebrae A 2 0 A0 where A o is the one-dimensional Lie algebra.


More precisely, the Lie algebra of U(3) contains a maximal commutative threedimensional sub-algebra d . The three generators oCt°i of d are such that the sum ~ = ~Yt°l + ~ s + Jr°3 commutes with all the elements of the U (3) Lie algebra and ~¢F can then be associated to the A 0 part previously introduced. The splitting A s ¢ A o is thus obtained in the well-known way

Hj = ~ - ~/~ ~ i = 1 , 2 , 3 .

As was expected, we then have in A s the zero trace relation

HI + H2 + H , = O .

2.2. A three-dimensional matrix representation of the commutative sub-algebra is then

i0i !0!, io! 1 , ' ~ 1 = o I ' ~ 2 = ~ V g # 3 = o . o o o

I t is interesting to work out the following change of basis

i ° ! Io = 1/s 1 / g ( ~ l - ~ s ) = 1/2 - 1 o

t'° i Y = ~ ( ~ 1 + Je:) = 0 ° o

i°i B = g g ( x e 1 + ~ 2 + x e ~ ) = 1 . o

The operators I+ and I - are always defined as above and we have the following commutation laws

[ I , Y ] = 0 [ I , B] = 0 [Y, B] = 0 (I, 14) . - - >

which allow us to interpret the five operators I , Y, B, as giving rise to a sub-algebra, isomorphic to that defined by the isotopic spin, the hypercharge and the baryonic number.

2.3. The irreducible representations of U(3) are characterized by three integers/1 > [2 > / 3 associated with the symmetry properties of a tensor of rank / = [hl + lhl + 1/31. I t is possible to set

& = h - t2 & = h - Is s = h

where n o w / I and ;;L~ are positive or zero integers. The representations of the unitary group U (3) corresponding to the

same couple 2142 and to different values of s are equivalent to D (z¢) (4x, 42) in S U (3) and simply denoted by ACN) (4~, 42); I t is then possible to classify the irreducible representations of U (3), starting from those of S U (3) in the following way [2]

(detA) ~ D (41, 4~.) s being an algebraic integer.

2.4. I t is obvious that the three generators I s, Y, B which generate the commutative sub-algebra can be diagonalized simultaneously and

10 MIC~mL GOURDIN :

that they are the only elements of the algebra which can commute with the three previous ones.

Let us now consider the transformation A which is a product of the three rotations 13, Y, B, the respective parameters being ~l, ~Y and ~B

A = e x p i [ ~ I 2 + o~yY + o~BB ] . (I, 15)

In all the representations, this transformation is represented by a diagonal matrix. For the particular representation [1, 0, 0] we have of course

x'i = exp [i ¢i] xi (I, 16) with

We can immediately calculate the determinant

detA = exp i[¢ 1 + ¢3 + Ca].

For the representation [3~ + s, s, - 22 + s] we therefore will have

A = exp i[(vl - ~ + s) Cx + (v~ - ~2 + s) ¢2 + (v3 - ~2 + s) ¢3]

which allows us to deduce the eigenvalues of 12, Y, B associated to a component (v i, ~7~, s)

12 = 11~ [ ( v~ - ~ i ) - ( v2 - ~ 2 ) ]

Y = (v~ - el) + (v2 - 'v~) + 2 s

B = ~ i - ~.~+ 3s=h+/2+/3 . We can again use triangular coordinates in the weight space and we obtain

13 = 1/2 (X 1 -- Z2) (I, 17) Y = - Xa + ~]a B .

We now understand how the S U(3) representations for which 21 + 222 * 0 (3) may be used. The hypercharge operator Y is constructed from Y and from the operator B which belongs now to the U(3) Lie algebra and possesses eigenvalues depending only on the representation and not on the particular weight of the representation.

3. L ie g r o u p s a n d Lie a l g e b r a [3]

3.1. The following argument being rank-independent, we shall use it for the simple Lie algebra A~.

Let us consider the direct sum

g = A~ ~ A o (I, 18)

and let us look for the connected Lie groups which are simply locally isomorphic among themselves and which admit g as Lie algebra. The general method [4] consists in looking for G* the universal covering


group of g and for the covering homomorphisms which have, as kernel, a subgroup of the center of G*.

3.2. The universal covering group of the A~ algebra is the unimodular unitary group S U(n + 1). The center of S U(n + t) is the set of the (n + 1) x (n + 1) unimodular unitary matrices which commute with all the others. From the Schur lemma, the only possible solution has the form o~I,+ 1 where I~+ 1 is the (n + 1) x (n + 1) unit matrix. The requirement of unimodularity is equivalent now to to "+I = 1. The center of S U(n + 1) is then isomorphic to the set of the (n + 1) m unity roots,

2ik~ . exp -n-+-l ' it is also isomorphic to the abelian additive group of integer

numbers modulo (n + 1). All these groups will be denoted by Z.+ v If (n + 1) is a prime number, Z.+ 1 does not admit any other sub-

group than itself and the unity. If (n + 1) is not a prime number, namely (n + 1) : mlmp, the groups Z ~ and Z,n~ are subgroups of Z,+ I.

3.3. The universal coveting group of A 0 is the abelian additive group R of the real numbers. An obvious subgroup of R is the set N of the integer numbers. The factor group R/N is isomorphic to the unitary one- dimensional group U(1) and admits also A 0 as Lie algebra. I t can be shown that all connected Lie groups of Lie algebra A o are isomorphic either to R or to U(1) [5].

3.4. Covering Homomorphism. The universal coveting group of a direct sum of algebrae is the direct product of the universal coveting groups [5]

G* = S U(n + 1) ® R . (I, 19)

Among the elements of the center Z of G* are of course direct products of elements of Z,+ 1 and elements of R. The kernel of the associated homomorphisms will have the structure of direct products and we have the following solutions

and eventually

if n + 1 = nhm v

S U (n + 1) ® R

s u(n + 1) ® u(1) SU(n + 1)/Z,+ 1 ® R

S U ( n + 1)/Z,,+~ ® U(1)

SU(n + i)/Z,~,® R

s u ( . + 1)lZ, , ® uo) S U(n + 1)/Z,,,. ® R

s u(n + 1)lZ ,. ® u(1)

Now, it is evident that the Lie algebra of theuni ta ry group U(n + 1) is g. The unitary group is connected and there must be a covering homomorphism h,+ 1 of G into U(n + 1)

G ~+~> U (n + 1). (I, 20)

12 M I C H E L G O U R D I N :

If we call 27,+ 1 an unitary unimodular matrix and ~ a real number, we have

The kernel Zl.+ 1 of h.+l must be represented by the identity in U (n + 1). I t is then easy to see that LI~+ 1 is generated by the following element of G*

- - n - T 1 ' e ~ / ( ' + l ? ' and we have

U(n + 1) ~ G*/A,+I. (I, 21)

I t is evident that the structure of zJn+ 1 is not a direct product. If (n + 1)= mlm ~, the kernels A,~ and LI,n, define two new

covering homomorphisms and the corresponding factor groups, G*/A,,, and G*]Am, do not have the structure of direct products and appear as two new solutions of our problem.

Let us remark the relation of isomorphism

U(n + 1)/U(1) ~ SU(n + 1)/Z~+ 1 . (I, 22)

3.5. Representations of g. a) The representations of the unitary unimodular group S U(n + 1)

are defined by n positive or zero integers 2 i. I t is possible to associate the D(21, ~ . . . . . 2n) representation to a completely contravariant tensor of rank p

- . S . p - k ) ~ (I, 23) I

2 i z , . . The element exp ~-~5-wmcn generates the center Z~+ I is represented

in D (21, h . . . . . 2 . ) by 2i7,p

exp n + 1 "

The representations of the factor group S U(n + 1)/Zn+~, generated by the tensorial powers of the adjoint representation are therefore characterized by

p-= 0 (n + 1) .

In particular, when n = 2, the representations of S U(3)/Zs are defined by the restriction 21 + 222~ 0(3). We are then able to interpret the results of the previous section.

Similar arguments can be applied to the other factor groups when they exist

SU(n + 1)]Z,~-+p-~ O(m) .

b) The group R possesses representations of the type :¢ --> e ~* where r is a real number. If r is restricted to an integer, they are also representations of U(1) [5].

c) We then can immediately characterize the representations of the Lie groups associated to the Lie algebra g. In the case of direct products,


the result is a simple consequence of the previous ones

S U(n + I) ® R

SU(n + 1) ® U(1)

S U(n + 1)/Zn+ 1 ® R

SU(n + 1)/Z~+l® U(1)

SU(n + 1)/Z~ ® R

SU(n + 1)/Z=® U(1)

p positive integer r real

p positive integer r integer

p ~ 0 (n + 1) r real

p ~ 0 (n + 1) r integer

p ~ 0 (m) r real

p ~ 0 (m) r integer

Let us now consider the factor groups which are not direct products, e.g. the unitary group U(n + 1), and if (n + 1) can be written as n + 1 = mira 2 also the groups G*/A~, and G*/A,,~. The kernel of the covering homomorphism must be represented by the unity in the factor groups and we obtain respectively

p ~ r ( n + l )

p ~ r (ml)

p ~ r (ms).

This relation between p and r defines the allowed values of r for a given p and can be ~a-itten as

r = p + ( n + 1) s

r = p + mltl

r = p + m~t,,

where s, t 1 and t~. are arbitrary algebraic integers. We remark that the representations of these groups are also representa-

tions of the direct product

SU(n + 1) ® U(1), (I, 24)

and we can use this product instead of the universal covering group to define the factor groups; for instance

SU(n + 1) ® U(1) ~ U(n+ 1). (1,25) An+1

Because of the relation (n+ I) = mlm 2, the representations of U(n + I) are also representations of the other possible factor groups G*/,dml and G*/A~,.

We now go back to the unitary group U(n + 1). Let us define (n+ 1) algebraic integers/i by

/j = 2 : k& + s . (I, 26) i

The 2k's being all positive, t h e / / s are ordered as follows

/1 = > / ~ - -> " ' " -~/~+1"

14 MICHEL GOURDIN :

The (n + 1) quantities /i characterize the irreducible representation []~, ]3 . . . . . /~+1] of U(n + 1) and we have the following well known relation

r = i: + 12 + " " + / n + t . (I, 27)

3.6. Special case n = 2. There are five connected Lie groups of the Lie algebra A 0 $ A 2. If we impose the following physical restrictions which must be satisfied for all representation weights

a) the baryonic number is an integer (r integer), b) the electric charge is an integer,

two possible groups only are left a) the unitary group U(3), used in the Sakata model, also called

the triplet model, b) the direct product U(1) ® S U(3)/Z a, which is the mathematical

support of the octuplet model introduced by NE'EMAN and by GELL- MANN.

4. Charge conjugation

4.1. We shall t ry to introduce the charge conjugation C defined as an abstract transformation which exchanges particles and anti-particles belonging, usually, to two contragradient representations. The method used is the following: one extends a group G by the two elements group { 1, C} isomorphic toZ~. The general solution has been given by MICHEL [5].

4.2. Let us call a the elements of group G. The couples {a, 1} and {~, C} are the two types of elements of the extensions of G~ by Z v In the particular case of an extension by Z 2, there exist two non-eqnivalent extensions which are defined by the following multiplication rules [s, s]

{~, 1} {fl, 1} = {aft, t} {~, 1} {~, C} = { ~ , C} {~, C} {~, 1} = {~L c}

where fl is the transform of fl by C. It is trivial to verify that these couples are elements of an extended

group and that {a, 1} is isomorphic to G and constitutes an invariant subgroup of the extension.

4.3. We shall study the case of a compact group G for which all the representations can be chosen as unitary. If R is an N-dimensional representation of G, its contragradient K' has the same dimension. We shall call E and E the N-dimensional vector space defined on the same field K as the Lie algebra of G. The three following equivalent results are well known

a) If DR (a) is a matrix representation of G in E, the imaginary con- jugate matrix D.~(~)= DR(~) is a matrix representation of G in E.


In other words, there exists a constant matrix UR such that

D ~ (o~) = DR(~ ) = URD~(a) U'~ 1 , (I, 28)

D being unitary, U R can be taken as unitary. b) If X, , (R) is a representation in E of the infinitesimal generators

of the Lie algebra, the matrix - X T (R) is a representation in E of the same generators and we have

-- X T (R) = URXa(_R) U-~ 1 . (I, 29)

c) In the tensorial product of the representations R and R enters always the scalar one-dimensional representation. Therefore, there exists a conserved bilinear form, constructed from elements of E and E, which is nothing else but

BT UR ( I , 30) with BE E and B E E .

Let us take the example of the unitary symmetry. In the Sakata model, the baryons are associated with the representation [1, 0, 0] of U(3) and the anti-baryons with its contragradient [0, 0 , - 1]. The conserved bilinear form is simply

~p + ~n + _&A. (I, 31)

In the octuplet version, the basic group is U(1)® S U(3)/Z~: baryons and antibaryons are in the adjoint representation of S U(3), which is equivalent to its contragradient but in two different representations of U(1), respectively defined by B = 1 and B = - 1. The vectorial spaces E and E are direct products and the representations R and R are not equivalent to one another but contragradient. The conserved bilinear form is then

~p + ~n + E- E - + E ° E ° + Z + Z + + Z - "~- + y,o y,o + AOA o " (I, 32)

4.4 We are then able to give the general form of the two possible extensions of G by Z~ we have previously defined. For {a, 1} we must find the representations of G in a reducible form

DR (a) 0 {~z, 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 D~ (a)

For {a,_C}, the form must be antidiagonal because of the connection of E and E by C. The solutions are the following [6]

o z~R(~) uR c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-4- D_~(a) U~ 1 0

The sign (_-k) which can be seen in {a, C} corresponds, of course, to that introduced in the multiplication table and distinguishes the two unequi- valent extensions.

16 MICHEL GOURDIN :

4.5. Let us now consider the general case where R and K are not equivalent representations, for instance when theydescribe baryons and anti-baryons. Let us call B the basis in E and B the basis in E. The bilinear form BTURB is invariant under the transformations of G. Let us perform a change of basis in E

B c= URn.

Now, the quanti ty BTB c is an invariant. To interpret B c we shall only consider the element {1, C} of the extension which is a simple charge conjugation

{1, C}IB = VRB I:t:: U ~ 1 B 1 "

With the new notation, we have

B --> B c (I, 33)

Bc'-~ _-LB.

I t is then easy, with these expressions, to interpret the two extensions. By using the bases B and B c, the representation of {e, C} takes the simple form

o ! DR(~) {~ , c } . . . . . . . . . . . . . . . . . . . . . . . . i ................. •

i D.,t. (.) ! 0

4.6. Let us now consider the case where R and R are equivalent and which usually occurs for the representations associated to the mesons. If M is the basis of E, we have the following relations

a) D~(~) = URDR(a ) U~ ~,

b) - X ~ = URXo U-: ,

c) M r URM invariant. I t can be immediately shown from a) or b) that UR can only be

either symmetrical or anti-symmetrical and its square is proportional to the uni ty N × N matrix. The matrix UR being also unitary, it is therefore either real or purely imaginary

Since UR is only defined up to a phase, we shall choose it real (e:~ = 1), that is, orthogonal. The symmetric or anti-symmetric character will depend, of course, on the representation R

UR r = ~R UR v ~ = ~ f l ut~ = u R . (I, as)

In the particular case of the unitary symmetry, the invariant form associated to the mesons is symmetrical (sR = + 1)

K + K - + K - K + + K ° K 0 + K - ' 6 K O + ~ + ~ - + = - ~ + + ~ o ~ ° + ~ ° ~ ° . (I, 36)


The extensions of G though Z 2 then take the following form

D R (~) 0 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

0 DR (~)

o DR(a) UR l

. . . . . . . . . . . . . o . . . . . . . . . . . "

If we introduce a direct product with the Pauli matrices, we can also write

{~, 1} = DR(~ ) ® 1

{~, C} = DR(~ ) UR® {(~+ 4- ~Rd-} .

II. P a r t i c l e c lass i f i ca t ion

1. Generalities

The octuplet model, as studied by NE'EMAN [7] and GELL-MANN [8] is associated to the symmetry group

U(1) ® S U(3)]Z 3 . (II, 1)

The baryonic number, B, is independent from the other quantum numbers and the same representation of the S U (3) group can be used to classify particles with different values of B such as baryons and mesons. All the representations of the factor group S U(3)/Za can be generated by tensorial powers of the adjoint representation 8. Thus considered, the octuplet model Call appear as a compound model constructed from eight baryons.

We must then arrange the particles as the irreducible representations of the group which allows us to define the supermultiplets of particles with the same spin and the same parity.

Experimentally, the spin and parity of all particles or resonances are not yet determined. The classification which is obtained through the unitary symmetry makes some predictions essential to the validity of the theory.

2. Baryons

The baryons are situated in the adjoint eight dimensional representation. This implies that all have a 1]~ spin and a positive parity. The only problem which has not yet been completely solved is that of the

Erg. d. exakt. Naturw. 36 2

18 MICHEL GOUEDIN :

E for which, experimentally, spin 1/z is favoured [9]

Y = 1 I = 1/2 N (938 MeV)

01} . ~X] (1193 MeV) Y = 0 I = • [ A ° (1115 MeV)

Y = - - 1 I = 1/z E (1321 MeV)

Fig. 7. Octuplet ] = 1/2 +

The antibaryons are also in an octuplet and the place of the weights is simply obtained by symmetry with respect to the origin in the weight space.

3. Pseudo-scalar mesons

The existence of a Yukawa type coupling between baryons and mesons allows us to consider, in a formal way, the mesons as bound states of the B B system and we must place them in one of the representations of the direct product of two octuplets

8@ 8 = 1 ~ 8 , @ 2 7 ~ 8 , ~ 1 0 5 1 0 . (II, 2)

I t is rather natural that we should choose again the adjoint representation ~or the pseudo-scalar mesons, M, and it will then comprise n-mesons, K-mesons and the resonance ~o the quantum numbers* of which, I -- 0, J = 0 -+ are in perfect agreement with those of the free weight in the center of the octuplet

Y = 1 I = I/2 K (494 MeV)

Y = o I = : (s48 MeV)

Y = - - 1 I = 1/~ K (494 M e V ) .

Fig. 8, Octuptet J ~ 0 -

* I f fo ta t ion: j P a : j = sp in , 2 ° = p a r i t y , G = G - p a r i t y


4 . V e c t o r M e s o n s

The adioint representation 8 belongs of course at the same time to the B B product and to the M M product and this allows us to consider the vector mesons V either as B B bound states or as M M resonances.

Experimentally, nine vector mesons have been observed, the p, the K* and its anti-particle K*, the ~ and the ?. I t is too much for an octuplet but one can think of adding a singlet as we have three weights in the center. The problem is then to know where to set the ¢o and the q~ which have both the same quantum numbers, I = 0, J = 1 - - and differ only by the mass. I t seems that the physical states experimentally observed, the co and the ~, belong to linear combinations of the two irreducible representations 1 and 8 which renders academic the choice of a name for the bare weights in the center of the octuplet and of the singlet. Such a mixture explains (or perhaps is explained by) a violation in strong interaction of the unitary symmetry, a violation exhibited by the large mass differences in a given supermultiplet between the isospin multiplets. By convention, we call ~o the octuplet weight and o~ ° the singulet vector meson.

Y = 1 I = 1/2 K* (888 MeV)

~} ~p (750 MeV) Y = 0 I = : [c~0

Y = - - 1 I = 1/z ~.* (888 MeV)

Fig. 9. Octuplet J = 1-

5 . M e s o n - b a r y o n r e s o n a n c e s

5.1. We shall work out again the product of two octuplets. The possible representations in the case of excited states decaying into baryon + mesons are then 1, 8, 8, 10, 10, 27.

5.2. The first ~-meson-nucleon resonance N,/z, at 1238 MeV has an isotopic spin of s] 2 and can therefore be classified only in a 10 or a 27 representation.

2*

20 MICHEL GOURDIN :

The simplest choice is, of course, the representation 10 and, at the time, 9 of the 10 partners of the J = 3/s+ decuplet are already known

Y ~ I I = a/~ ~ " ' * / N*/2(1238MeV )

Y = 0 I = 1 ~ -~ / Y~* (1385 MeV)

Y = --1 I=~/~ ~ ~112"a* (1532MeV)

Y = - - 2 I = 0 ~ - Fig. 10. Decuplet J = 3]2 +

The most recent measurements of the spin and parity of the Y* and the E , 1/2 seem to indicate that the value J = 3/~+ is favoured [10, 11 ].

Theory also predicts the existence of a new particle, called ~ - by GELL-MANN and Z- by SAKURAI which has a strangeness S = - 3 . A rough estimate of its expected mass through the Gell-Mann-Okubo formula (see detailed discussion in Par t IV) gives a value around 1680MeV. The threshold of the K E system is 1820 MeV and if mn < 1820 MeV, the ~ - can appear only as a K.~. bound state and not as a resonance. Therefore, the ~ - will be stable with respect to strong and electromagnetic interactions and the onty disintegration processes will be weak decays. For instance, one expects

f2- ~ X (A) + K

~2- @ 7~o + 1- + v 1 1 - = e ,~

The most natural production process of the ~ - seems to be the K - capture

K- + p ~ K + + K ° + ~o .

An experiment, performed with an incident K - beam of 3.5 GeV/c has not given any positive results, which means that the ~ - production cross section at this energy is lower than 3 ~ b [12]. One must first remark that the K - incident energy corresponding to a 1680 MeV ~ - threshold production is 3.2 GeV/c and the present experimental energy is not sufficiently far above this value to give any definite conclusion. On the other hand, there is no theoretical reason to believe this value of 1680 MeV for the ~ - , due to the different analytical properties of a resonance and of a bound state (see part IV) *.

* Note added to proof. The ~ - particle has recently been discovered at Brook- haven in the reaction K - + p ~ K + + K ° + ~ - ,

with a mass of 1686 =]= 12 MeV. There has been observed both production and weak decay of the ~ - into ~ - + .~o with a lifetime of the order of 10- xo s. The production cross section is, as was expected, of the order of 1 I~b [71].

Some Topics Related to Uni±azy Symmetry 21

In theory, it is possible to estimate an inferior limit for the production cross section by using unitary symmetry itself. Between the amplitudes, the following equality can be shown [13]

A (K-p ~ K+K ° f2-) + A (K-p ~ r:+K -~ Nff~) (n, 3)

---- ~/3-A (z~-p ~ K+K° E*/.:)

which leads to a triangular inequality between the corresponding cross- sections

a(•-) ~> a(K*) + 3a(E*) - 2 V3- ]/a(N*):(7:*) .

Such an inequality must of course be corrected so as to take into account the phase space which is very different for the three reactions. Unfortunat- ely, the actual data is too poor to deduce any useful estimate for the g~- production cross-section. Another reaction of the same type

K- + p ~ r: + + K ° + ~1/2

has been experimentally observed and has a cross section in the order of a few microbarns. So the statistics on the ~ - production experiment are not sufficient to allow a definite conclusion on the ~ - existence.

Let us point out that there is another possible interesting production mode with high energy antiproton beams

p + p ~ - + ~2-. (11,4) For the time being, only the pairs 1~/2, N--3"/2 and Y~' ~ have been observed in this type of experiment for which the background is extremely important of course.

Finally SAKURAI [14] has suggested that the event observed by EISENBERG in 1954 [15] and unexplained until now could be the weak disintegration ~ - ~ K - + Eo of this f~- particle; the mass calculated from this event was found to be of a correct order of magnitude.

5.3. The second re-meson-nucleon resonance, NI*/~ of 1512 MeV may be associated to a new octuplet where would take place a n - N resonance observed at 1520 MeV, the Y3*, and maybe the new resonance Y~* of 1660 MeV if its spin and its parity are J = a/2-. The octuplet would then appear as

Y = 1 I = t/~ NI! ~** (1512 MeV)

~} ~ YI** (1660 MeV) Y = 0 I = : ' (Y~* (1520 MeV)

Y = - - 1 I = 1/2 ~**/?)

Fig. 11. Octuplet g = 3/2-

22 ~/[ICHEL GOURDIN :

I f the spin and the par i ty of Y~* seem well-established, those of Y~* are not and the actual experimental data allow no conclusion to be drawn.

The model predicts the existence of a new 7~ - ~ resonance with also an isotopic spin I = 1/2. The mass formula gives an estimation of 1600MeV but up to now, there has been no experimental evidence of the existence of this resonance [12].

AT*** 5.4. The third 7:-meson-resonance, ~,1/2 at 1688 MeV can probably take place in an octuplet. If one follows the ideas of CHEW and FRAUTSCHI,

N * * * the nucleon Nit ~ and this resonance 112 belong to the same Regge trajectory and it is then natural to make them belong to representations of equivalent dimensionality which will then group a set of four trajectories - when the electromagnetic interactions are neglected, of course. The present octnplet would be the recurrence of the baryon octuplet. In the same way, the Y~** which was observed at 1815 MeV with a spin J > 3/2 would take place in this octuplet and on the Regge trajectory of the A °. There would remain to be discovered a third Y~** and a third ~*** 5/+ ~1/2 with both J = .

5.5. The fourth ~-meson-nucleon resonance, xT**** • ,a/2 of 1922 MeV would then, in a similar way, place itself on the N3"/2 Regge trajectory and one would have to deal with a new decuplet of J = ~/~+ spin. This predicts

,~**** the existence of a Y~'***, a ~112 and of a second S = - 3 particle, ~ .

5.6. Other =-meson-nucleon resonances have been observed, in parti- AT***** "N,T***** cular, a ~112 at 2190 MeV [16] and a ~,~12 at 2360 MeV [16].

Spin and par i ty are not yet known but certainly J is very large. I t has been suggested to interpret the 2190 MeV resonance as a second recurrence of the nucleon (Nl19, ~,T*** ~,T*****~ with a spin J = 9/2+ ±~1/2 , J-~I/2 ) and that of 2360 MeV as the second recurrence of the Na~2 isobar (Na/2,* N**** ~,T******~ with a spin J = n / + [17] 3/2 , x~3/2 ]

I t is evident that we shall need a new octnplet and a new decuplet to associate with these resonances.

For the time being, all these considerations are, of course, specula- tions and the only serious s tudy of meson-baryon resonances can be made on the multiplets associated to the three first 7~ - N resonances for which valid partners have been found in general.

5.7. Finally, there exists a 1405 MeV ~ : - ~ resonance, the Y* which has not been classified and which can be associated to a singlet. I t s spin and par i ty are not yet determined with certainty by the experiments (J = 1]~).

Let us remark tha t if the Y* is associated to a singlet, it can be used to test the consistency of certain assignments. In particular, the decay mode Y~* @ Y~ + 7~ can only take place if both YT* and the x belong to an octuplet [18].

6. Meson-meson resonances

In addition to the ~o and to the nine vector mesons we have just studied, there are some meson systems which appear to be resonating


at certain well defined energies. We shall only consider those of which the existence has been clearly established.

6.1. The f0 is a two n-meson resonance at 1250 MeV, with an isotopic spin I = 0 and probably jPG = 2++ [19] *.

I t can be associated with a singlet of the group. We note that the vacuum must be also associated with a singlet I = 0, J = 0 ++, B = 0. If the fo takes place on the vacuum Regge trajectory, it is, of course associated with a singlet.

6.2. A second K K resonance, decaying now through a K1 ° K ° system has been observed at 1020 MeV. The isotopic spin is I = 0 and the spatial parity, the G parity, and the spin must be even, the most probable assignment being jPG = 0++ [20]. In the absence of other scalar mesons, the most natural place is a singlet.

6.3. T h e ¢ : - K system exhibits a second anomaly at 725 MeV, oneinter- prets it as a resonance ×. The isotopic spin seems to be I = 1/2, but spin and pari ty have not yet been experimentally measured [21].

No definite solution has been given to place this resonance in the symmetry group. As suggested by NAMBU and SAKURAI [22] the × can exist only in a broken unitary symmetry and therefore, it will have no unitary partners.

As a last remark, if the × is a scalar meson, one can think of associating it with the previous K~°K [ resonating system in an octuplet, but we shall then meet with great difficulties in the mass formula because of the too large mass difference.

6.4. A rz - ¢o resonance, called ]3, at 1220 MeV, has been detected in =+p and 7:-p experiments [23]. Of course, the isotopic spin is I = 1 but the other quantum numbers are not yet defined, the two candidates being J = 1- or J = 1 +.

7. Nuclear physics

In nuclear physics, the classification of the nuclei was started a long time ago and the consideration of the isotopic spin has been in many cases of great interest for establishing symmetries between nuclei with the same total number of nucleons. Bound states in the hyperon-nucleon systems have been experimentally observed, the hyperfragments, and they yield precious information about the hyperon-nucleon and the hyperon-hyperon forces.

I t can be of interest to look for the consequences of a possible unitary symmetry in nuclear physics. We then define supermultiplets for systems with a defined baryonic number. But the unitary symmetry is violated because of mass differences between baryons; therefore, certain bound states which were predicted, will only appear as strong interaction effects [24]. Also, predictions on the mass spectrum can be made by using the Gell-Mann-Okubo mass formula (see part IV).

* s. page 18

M I C H E L GOURDIN :

7.1. B = 2. The deuteron I - - 0 , Y = 2 is a neutron-proton bound state which is possible to place in a 10 decuplet. No other bound state with B = 2 is known, but there is some evidence of a strong hyperon (A, Y~)-nucleon interaction, around 2130 MeV, in an isotopic spin state I - - 1/2 and ordinary spin J = 1 + [25, 26] which could be the counterpart of the deuteron,

7.2. B = 3. Two nuclei are known which form an isotopic spin doublet, He 8 and H 8 and also a hyperfragment AH s. The representation with lowest dimensionality is then D(S~l(1, 4) and we have t o consider a f = 1/2+ 35-plet.

In this hypothesis, the hyperfragment AH 3 belongs to an isotopic spin triplet in which the two other partners, AHe z and n z have not yet been found.

7.3. B = 4. The only bound state is the He*, of spin J = 0 + and isotopic spin I = 0. One can then think of using a representation D{2S) (0, 6) for which the hyperfragments AHe ~ and AH 4 would then form an isotopic spin doublet. I t seems that it is correct to assign a spin J = 0 + for the AHe 4.

By using the mass formula which, in this case, is an equal spacing rule (see part IV) one predicts, from unitary symmetry, the existence of an isotopic spin triplet Y = 0 with a binding energy of the order 2 MeV: AAHe 4, AAH 4, AAn 4 [24].

7.4. For the higher values of B, we can continue this kind of specula- tion, but the experimental information on hyperfragments is too poor for this type of classification to teach us anything. Let us note however tha t recently some double hyperfragments have been discovered [27] and that they will contribute to fill the lines Y = B - 2 of supermultiplets of unfortunately very high dimensionality associated to B = 8, 9, I0 and 11: AABel°, n , AALiS,9,1°.

III. E x p e r i m e n t a l c o n s e q u e n c e s o f u n i t a r y s y m m e t r y

1. Foreword

Unitary symmetry is not exactly a physical law for it leads to a complete degeneration of the mass spectrum of particles associated to a supermultiplet, which is in open contradiction with experiment.

I t is however possible, and maybe justified, to think that at high energy the S-matrix commutes with the unitary transformations and that such a symmetry may aH the same have a physical meaning when the energy concerned is large with respect to the masses of the interacting particles.

In this chapter, we shall set a working hypothesis, which is difficult to justify rigourously but has the advantage of being both simple and reasonable. We shall assume that the transition matrix elements, already


Lorentz invariant, are also invariant under unitary transformations. I t is evident that to go from T-matrix elements to cross section or to disintegration widths, one must introduce a certain number of well- known kinematical factors in which the masses of the particles concerned in tile reaction are taken with their experimental values. This allows to take into account, to a certain extent, the violation of unitary symmetry which is displayed by the mass differences.

2. Introduction

We are therefore going to look for the relations which exist between the amplitudes of reactions due to the invariance under the unitary transformations, or, in a more general way, under the transformations of a group which we shall call G. We must naturally expect to find again the equalities or unequalities due to the charge independence, as the isotopic spin is a sub-algebra of g.

The general method, fairly lengthy as far as practical calculations are concerned, is to develop the initial and the final states on the irreducible representations of G. The transition amplitude can then be written as a linear combination of all the scalars which can be constructed between the initial and the final state. We shall call this method the isometry method.

There is a faster though less powerful method which gives in certain particular cases the essentials of interesting physical results. In that case, a group of finite transformations R is considered, which plays the same role in relation to G as that of the charge symmetry in relation to the isotopic spin group.

Finally, we shall give the result of calculations made for peripherical models used in the framework of unitary symmetry using the coefficient tables established by many authors:

3. Isometries

3.1. Let us take as an example the "elastic" meson-baryon scattering

M + B :~ M + B . (III, !) The multiplets M and B are associated to irreducible representations of the group G, D i and D B. The transition amplitude

A = <M, B]TIM, B >

can be decomposed following the partial amplitudes A~} associated to a particular reaction

Ma + B e ~ Mb + B a (III, 2) and we can write

A = A ~ M a B ~ M b B a . (III, 3)

2 6 MICHEL GOURDIN :

3.2. The initial state ]MB) can be expanded on the irreducible representations of G following

IMB) = 22 a(~, Z)1~; Z). (III, 4) o g

The irreducible representations [or; X) are those which enter in the product of representations

DM ® DB = ~ v.Do . (III, s) e3

The coefficients v~ correspond to the multiplicity of the representations ~r differenciated with the index 2.

For instance in the product of two octuplets, the representation 8 appears twice, first in the symmetrical part of the product, 8 s and then in the antisymmetrical part 8~. The numbers a (~, •) are known quantities related to the Clebsch Gordan coefficients and to the is®scalar factors of the group [28].

3.3. The same reduction being made for the final state, we still consider a product of representations in G, but now to obtain the invariant representation, 1, associated to the transition amplitude. The orthogonality of the irreducible representations in G obliges us to combine only representations of same o. It follows that for a fixed a one expects, in general, (v,) ~ terms. Consequently, the maximum number of linearly independent amplitudes is 22 (v,)2.

a

Let us take as an example the meson-baryon scattering in the octuplet model. The product of two 8-dimensional representations is given by

8 ® 8 -- 1 $ 8, $ 27 * 8~ $ 1 0 $ 1 0 . (III, 6) There exist, in general, eight linearly independent amplitudes quoted as

A1, A~v Alo, An, As,s, As,~, As~,, A s ~ . (III, 7) When the time-reversal invariance holds, it is very easy to see that the two non-diagonal elements As~ s and Ass ~ are equal; the number of independent amplitudes becomes seven.

3.4. The same technique of calculation can be applied, of course, to the general case of different initial and final states. For a fixed value of o, the number of associated amplitudes is v~vj~ and the set of reaction is described by ~E' vi,vl , independent amplitudes.

ct

Let us consider, as an example, the production of excited baryons B* by meson-baryon collision

M + B ~ M + ]3*. (III, s) The M mesons and B baryons belong to two octuplets but now, the excited baryons B* belong to a decuplet. The reduction of the final state is performed following

8 ® 10 = 8 $ I0 $ 27 $ 35 (III, 9) and the reaction is determined by only /our independent amplitudes

Ass, Asa, Alo, A~7 . (III, 10)


3.5. We catl isometI T [29] an operator O(~,z) such that the quantity

transforms as t h e / , component of the irreducible representation IaZ) of G. The isometries are normalized in the following way

/£ a ( x •

3.6. From these isometries, we can define some operators P

which are projection operators due to the orthonormalization relations fulfilled by the f2 operators. We then have

pc.zz,,>~o~ p~v,,z,>~ = ~*~' ~z'x'" l:,~]z').~. (III, 13)

3.7. The amplitude for the reaction

is immediately decomposed in its invariant parts by using the projection operators

# t A~g = 27 2: A ~ , p~zz)~a (III, 14) a g g '

3.8. The three-body problems are very easy to formulate in terms of isometries. Let us consider, for instance, the Yukawa type coupling between a fermion field ~& associated with a representation D B and a meson field X i associated with a representation D u. Neglecting for the moment the space time properties, we can write the possible coupling in the following way

and there exists an independent coupling quoted by ~ for each time when the D i representation appears in the D B product through its contragradient D~.

In the octuplet model for instance, the pseudoscalar mesons M, the vector mesons V, and the baryons B are all associated to the adjoint 8-dimensional representation. We can then write two possible meson- baryon Yukawa type couplings

g21 s=)i~ ffl l B~ M i Qi **)ik B~ B~ M i

Qi s~)ik J~ B~ V i Qi s')i~ Bt B a V i .

In the Gell-Mann language [8], tile couplings using Q(s~) are said to be of the F type and those using ~Q(s,) of the D type.

Tile same situation cannot exist if we consider one vector meson and two pseudoscalar mesons. In fact, a generalized Panli principle can be applied to identical pseudoscalar mesons and excludes tile possibility of a symmetrical coupling. We only obtain

Q(s~)ia~ M~ M~ V i . (III, 15)


The situation however is quite contrary in the case of two vector mesons and one pseudoscalar meson. The same argument now excludes the possibility of an anti-symmetrical coupling

~9(8,~i~, v , v , M i. (III, 16)

4. Application

Yukawa t y p e coupl ing be tween th ree oc tuple ts . We consider the vectors in the space of the adjoint representation as mixed tensors of zero trace T~, according to the relation

3 x ~ = 8 + 1. ( I I i , 17)

For instance, the baryons B, the antibaryons B, the pseudoscalar mesons M and the vector mesons V have a matrix representation

-- A p n

B =

~ =

M=

V =

Eo

I AO 1

Z +

7 .=

K +

1 00 + 1 no

~+

K *+

1 ~o 1

p+

y,o y, -

1 1

~o

>2+

1 y~ 1 ~-~

Ko

I ~o l_l_~zo

K*° J

1 o 1 o

~,o

(III, 18)

The Yukawa type couplings are invafiant quantities constructed from three eight-dimensional representations and the matrix form will then be convenient. In fact, one has only to calculate the trace of a product of three matrices; taking into account the invariance of the trace with respect to a circular permutation of the factors, such a problem has only two independent solutions and this result is nothing else than the existence of two possible Yukawa type couplings between mesons and


baryons. One then obtains

H~----g~ T r ( B B M ) H ~ = g~ Tr (BBM)

Hs = gs [1/2 Tr (BBM) + ~/~ T r ( B B M ) ]

g~ = g~ ~x/~ T r ( B B M ) - x/~ T r ( B B M ) ] .

I t can be interesting to expand these hamiltonians on the basis of the 12 charge independent forms one can construct from the 8 mesons and the 8 baryons. In other terms, the unitary symmetry reduces the number of linearly independent coupling constants from 12 to 2 only. The explicit expression of the 12 charge independent forms is given in Table 1 and the expansion of the unitary symmetry hamiltonians in Table 2.

Table 1

AoZ • ZAO~

E ~o

AOA o.~o

~ A O K

AO.~. K

EAOK

~N~K

N,¢- ~ K

a--.. ZK

(pp -- ~n) =o + V~- (fip~- + pn =+) (k-~-a - - ~E°) ~° + ~tZ ( ~ a o ~ - + ~ - ~ + )

A°(X°~ ° + X+=-+ E-n+) E°A°~ ° + E-AO~-+ E+AO~+

(E~E- - X--~E +) ~° + (E°Z+ -- Z---: E °) ~ - + ( ~ X ° - ~ X - ) n+

(pp + ~n)'~° ( S - S - + E°S °) .~o AOA o ~o

(E+E+ + E-E- + E°E °) ~o X-~ (p K - + n I~)

pA ° K+ + nA ° K ° K~(~-K + + ~OKO)

~ - A ° K - + E°A ° K °

~'-~(pK--- n K °--) + }/2 (~- -nK- + E-TpK°) ~XoK+ -- nXoKo + 1,~- (fi F.- K+ + p Z + K °)

~ '~(~-K + _ ~,o KO) + }~-(~=.~.-Ko + ~-T~OK+ )

.~.~XOK- _ EOXOKO+ ]/2-(.~.-KOX- + ~ K - E + )

The more general form of the coupling between three octuplets is H~ + H 2 or, equivalently, H, + Ha. In the case of the M - B interaction, the n-meson-nucleon coupling constant is different from zero only in H 2 and it is convenient to set

g~ = 1/~g (III, 19) where, now, g is the well known r: - N coupling constant experimentally found to be

g2/4:~ ~-. 15.

The mixing between H 1 and H~ or between H s and H . is then described by a mixing coefficient a

g: = (1 -- 2a) g•.

30 MICHEL GOURDIN :

E'rEr:

A°'--~ • r: + XA°=

i(Z x Z). 7=

NN~ °

SS~ °

AoAo ~o

~ . Z~ °

A°NK + ~'A°K

A°EK + ~A°K

~N'~K + Nx~K

Tr (BBM)

0

1

V~ 1

1

V~

16'"

v:

0

1

Table 2

....... ~,!~'~!,,,,,~ - , ...... s

1 1

1 0

1 t

1 - - 0

V~- t 1

N 2V~

- 2 V ~ -

1 1

V~- V~- 1 1

- - 2V¢ 1 1

1 1

1 0

2V~-

1

2V~- 1 +

2Vg

1

3

3

2V~

3

21/~- 3

2V~ 1

1

2~

(1 - - 2 ,¢)g

2 (I - ~)g

- -2~g

1 - - - ~ - (1 - - 4 ~ ) g

1 - - - ~ (1 + 2o~)g

2 (1 - ~ ) g

2 (1 - ~ ) g

1 - - ~ - ( 1 +2~)g

1 - - ~ (1 - - 4 ~ ) g

V 3

(1 -- 2c*)g

The last column in Table 2 gives the expression of the coupling constant associated with each charge independent form. These results agree with those given by MARTIN and WALl E30] and those of de SWART [31] up to some arbitrary phases.

Let us point out that column S can be immediately applied for V V M couplings and column A to M M V couplings.

In order to facilitate the comparison with other calculations, we shall give the relation between ~ and the mixing angle 0 used by CUT- Kos~:Y Is2]

1 o~

1 3 " + V-V c tg o


5. Strong decays of excited baryons

5.1. We are first interested in the J = 3/9 meson-baryon resonances, respectively classified in a decuplet (J = 3/2+ ) and in an octuplet ( f = ~/F).

In the case J = ~]~+, only one matrix element describes all decays

B* ~ B + M ( i i i , 20)

because the excited baryons B* are in a 10-dimensional representation, which enters only one time in the product 8 ® 8.

In the case J = a/s-, one can define two reduced matrix elements for the decays

B** ~ B + M (III, 21)

and this result is analogous to the two Yukawa couplings one has defined between mesons and baryons.

5.2. The calculations have been performed by GLASHOW and ROSEN- FELl) [14] by using a phenomenological form factor where the kinematical coefficient are included

P c (p, M*) =

where p is the final momentum of the product of disintegration in the rest system of the excited baryon of mass M*, l the orbital angular momentum of the meson-baryon final state, and X a phenomenological quanti ty describing the strength of the interaction.

5.3. Of course, the most simple process is the decay of the decuplet because of the property that all amplitudes are proportional. The partial width for the disintegration

B* ~ Bj + M~ (iii, 22) then takes the following form

I'~; j,~ = c (p, M*) I(i k ] ~)l ~/'0 (III, 23)

where the Clebsch Gordan coefficient (i, k] ~) gives the projection of 8 ® 8 on 10. Taking into account the normalization condition

Z' I(ikl~>l ~= 1, j ,k

we have the set of coefficients shown in Table 3. In the last column, "Yes" or " N o " indicates whether the isobaric

mass M* is larger or not than the production threshold for the final state, in other words, whether the disintegration channel is open or closed.

We now calculate the coefficient C (p, M*). The parameter X is determined from calculations related to the B** disintegration and found to be

X = 350 MeV.

The same value has been used for the disintegration of the B*.

32 MICHEL GOURDIN :

Table 3

Excited state Final state l<iklc~>l ~

Ns*~ ( ross MeV)

YI* (1385 MeV)

(1532 MeV)

(1686 MeV)

rcN

K Z

=A rrZ

K N ~qZ K S

rrE

~A

mE

I/2

1/4

1/4

1/4

1/4 1/4

1

YES

NO

YES YES

NO NO NO

YES

NO NO NO

NO(?)

The last fac tor / 'o is adjusted so as to obtain the experimental value / ' = 100 MeV for the width of the 7: - N isobar.

The results are the following

Table 4

Excited state Total width Final state p (MeV]a) Measured width Calculated width

I00 MeV

50 MeV

< 7 MeV

A n g r :

E =

233

210 119

148

100

50 ~ 4

< 7

I00

35 5

12

5.4. The problem with the octuplet J = 8 / - is more difficult because of the presence of two linearly independent amplitudes. The values of the Clebsch Gordan coefficients can be deduced from the general tables 1 and 2. In this case, we have a mixing parameter a defined as previously, the case a = 0 corresponds to pure symmetrical coupling and the case

= 1 to purely antisymmetrical coupling. The results are given in Table 5. The experimental evidence of both decay modes

Y ~ * * ~ 7 : + A ° and Y~**~r~+Y,

shows clearly the necessity of using a mixing of both symmetrical and antisymmetfical coupling to describe the decay of the B** resonances (a 4= 0 and a =~ 1). The expression for the partial decay width

B~ ¢ ~ B i + M~


Table 5

Excited state

(1512 MeV)

y** (1660 MeV)

0520 ~ev)

~1/2 (~- 1600 MeV ?)

Finat s ta te

rcN ~N

KA o KZ

7~A o fez KN

~Z KE

KN

~A ° KE

KA K~,

3 1/~ (1 - 4a )*

1]a (1 + 2(z) ~ 3 (1 -- 2~) ~

4]3 (1 - - a)2 8~ 2 2 (1 - - 2~) ~

% (1 - ~)~ 2

4 (1 - - a)~ ~/3 (1 + 2~)~

4[a (1 -- a) 2 *h (1 - 4a)s

3 (1 - - 2a) ~ l / a (1 - - 4 ~ ) ~ 3 1/8 (1 + 20:) ~

YES YES

NO NO

YES YES YES

NO NO

YES YES

NO NO

YES ? ? ? ?

takes always the same form

I~,i,~ = C (p, M*) ](?'k]/~)]m To. ( I I I , 24)

The three free parameters X, *¢,/'o are de termined in order to reproduce the exper imental results for the three par t icular decays

Y o * * + = + E Y ~ * * + = + A Y P * ~ z ~ + Z .

The in terac t ion parameter X is equal to X = 350 MeV. The mix ing parameter ~ is found to be

--~ 0 ,34.

This value agrees with some previous calculations made b y CUTKOSK¥ [32] on the basis of a consistency requi rement in a dynamica l approach

- - 1 ~ 0.326. ( I I I , 25) ~ = V6- + 2 -

T h e results are the following

Table 6

Excited state Tota l width Final state p(MeV]c) Measured width Calculated width

xl*~* lOO

Y * * 40

Yo** 16

~ * * "~

Erg. d. exakt. Naturw+ 36

~N

rrA n Z ~ N ~Z KN

450

441 386 4O6

267 244

220

8O

11 13

~_3

9 5 ?

67

11 13 3

8 6

0.6


5.5. GLASHOW and ROSENFELD have also calculated together the partial widths for the decay of the excited baryons B*** of spin f = 5/2+ associated to a new octuplet. The calculations use the same form factor C (p, M*) with the same value of X and also the same mixing parameter c¢. The ouly free pa r ame te r / ' 0 is adjusted to give for the decay

N*** ~ :x + N 112

the experimental value F = 80 MeV. The results are the following

Table 7

Calculated Excited state Total width Final s ta te p(MeV/c) Measured width width

100 •*** 1/9

(1688 MeV)

Yo*** (1815 MeV)

y*** (188 ? MeV)

1t7 (197 ? MeV)

120

7zN ~N KA roe ~ N ~A ~A

~N

K~ ~E ~A

~E

572 387 235 504 538 345

595 548 586 322 208 531 540 479 290

80 < 20 < 2

<40 70

<1.3

80 0.5 1

29 41 3

15 20

5 2 1

5 1

41 2

The masses for Y*** and ~.*** 1/2

trajectories for the nucleon N, the rely parallel.

are estimated by assuming the Regge AO, the E and the .~. to be approxima-

6. Finite g roup R

6.1. We now consider a finite subgroup R of the special unitary group involving particular rotations defined in such a way that i t transforms a weight into another weight and not into a linear combination of weights as in the general case. The starting point is given by the isotopic spin group for which a rotation of angle ~ about the second axis in the 3-dimensional charge space exchanges the neutron and the proton up to a sign. More precisely, we have

J= i~= ~ and

J l # > = - in> J l ' > = 1#>.


In terms of infinitesimal generators, we have

i a~ = I + - I - . (III, 26)

6.2. By analogy, we consider the three elements of the As Lie algebra

(III, 27) 1, m , n = 1 , 2 , 3 .

The representations of J~ are the same for the two contragradient fundamental representations because of a possible correlation between the two bases such that E ~ . = - E , ~ which leaves f~ invariant. One then deduces easily an explicit representation for the Jz's [33]. The finite rotations associated to these generators are of the form

R~(a~) = exp a~J, . (III, 28)

Group R is generated from the three basic elements R, (a/2) by products. We use for convenience the following notations

where:

11 R x =

S I = !

0 R 2 = 1 R 3 = - - 0

- - 1 0 0

o _!o i -i °i - - 1 0 S 2 : 1 S 3 = - - 1 0 - - 0 - - 0

Z°i C+= o C - = o o 1

R~ = Rt (:z/2) St = R~(~)

C + C - --- 1 two circular permutat ions,

]By performing the products in all possible ways, we see that R contains 24 elements. An interesting sub-group S is defined by

S = {I , S1, 82, 83} ,

where I is the identity. The factor group T = R / S is the set of the classes of equivalence for which the elements can be chosen as: I , R 1, R~, R 8, C +, C-. We immediately verify the isomorphism between T and the pelTnutation group G 3 of three numbers and the isomorphism between S and the direct product Z 2 × Z v

6.3. From the 3-dimensional representation given above, one easily deduces the Iaw of transformation of the three basic vectors of a fundamental representation of A s which is the same, of course, for both representations D (a) (1, 0~ and D (3) (0, 1) [33].

6.4. In order to study the adjoint representation, we consider the direct product 3 @ 3. One can immediately see that the 8-dimensional representation of A s is reducible under the R group into 6 ~ 2. The two weights associated with root zero generate an invariant two-dimensional

3*

36 MICHEL GOURDIN :

sub-space. From now on, we shall be concerned only with the six weights lying on the six summits of the regular root hexagon

-7 2

3 - 3

Fig. 12

gy y~

g~ y ~ x y

I

(1) ( - 1 )

(2) (-2)

(3) (-3)

R1

- ( - 1 ) -(1)

- ( - 3 ) - (3) (-2)

(2)

Table 8

R2 t R3 C+

(-3) ~ ( -2 ) (2) (3) - ( 2 ) ( - 2 )

- - ( - -2) (--1) (3) -(2) (1) (-3) (-1) (-3) (1) -(1) -(3) (-1)

C'

(3) (--S)

(1) (--1)

(2) (--2)

S 1 S~ S,

+ +

6.5. In this section we are interested in two bodies giving two body reaction amplitudes. We introduce the particle systems [i, ?'] where i and i are components of two arbitrary octuplets associated, for instance, to mesons or baryons or ant±baryons.

The systems [i, j] and [j, i] are in general different from a physical point of view but they are very similar with respect to the group theory. In particular, they belong to the same weight of the product representation and have the same quantum numbers Q and Y. We restrict ourselves to the only interesting case for which i, j, = ± 1, ± 2, ± 3. I t is then possible to classify the systems i, j in irreducible classes with respect to the R transformations. We can define six classes with six systems according to

Class A [~', i]

Class B [i, - i ]

Class C [i, i] Class D [i, i]

Class E [ i , - i ] Class F [i, - i ] .


In a given class, the various systems It, {] are transformed by R of an arbitrary system. From Table 8, one deduces

Table 9

In Fig. 13 we give the positions of the weights for the various systems It, i].

A k-

AkcA° V \ D j

\ V ,4 K

Fig. 13

A

/ A

The conservation of Q and Y limit the possible physical reactions one can classify in the following way

a) Class A : only elastic reactions can occur and they are described by only one amplitude associated to the representation 27 in the product 8 ® 8

A~I= A{[ i , i ] ~ [j,i]} = A2~.

b) Class B: aI1 systems are characterized by Q = Y = 0 and one can define six independent reactions

B~t = A {[i, - i ] ~ [i, - i ] }

Box = A {[i, - i ] ~ I - i , i]}

B1 = A{[i , - i ] ~ [i, - i ] }

B lex = A{[i , - i ] ~ [ - i ,i]}

B 2 = A { [ - i , i] ~ [- / , i]}

B2~x = A { [ - / , i] ~ [i, - i ) } .

When the time reversal invariance holds, we obtain the supplementary restriction

B1 -- B 2 . (III, 29)

38 MICHEL GOURDIN :

c) Classes C and D: we have an elastic amplitude for each class and two possible inelastic reactions between systems of different classes

co, = A { [ i , j ] ~ [i, i]}

Det = A{[j', i] ~ [i, i]}

cox = A {[i, i] ~ [i, i]}

Dox = A{ [ j , i] ~ [i, j ] } .

The inelastic amplitudes are equal when the time reversal invariance holds

C.x = D~x. (III, 30)

d) Class E: We have an elastic amplitude and an inelastic amplitude, which, according to Fig. 13, are a mixing of AT6 and A2v. The symmetrical combination is A~ and the antisymmetrical one A10

Eet = A{[i, - i ] ~ [i, - ] ]} = 1/2(A~, + AlO )

Ecx = A {[i , - { J ~ [ - i , iJ} = 1/~(A27 - A10) .

e) Class F: we have an elastic amplitude and an inelastic amplitude, which, according to Fig. 13, are a mixing of A~ and A =v. The symmetrical combination is A~7 and the antisymmetrical one A~.

Fet = A { [ i, - i J ~ [i, - i ] } = 1/~(A27 + A-~)

F~x = A { [ i , - i ] ~ [ - i , i ] } = ~/~ (A~7 - A ~ ) .

We now define 15 amplitudes which, of course, are not linearly independent in the total group because of the property of the reactions 8 ® 8 = 8 ® 8 to be determined by 8 amplitudes only. We then must find 7 relations as consequences of invariances not contained in R.

If we apply the charge independence for particular members of the classes, we can immediately write 6 relations

C e l = B d - B tex Cex = Bex - B 1

Del = Bel - Bgex Dex = Bex - B 2 ( I I I , 31)

Eet + Eex =Aet = Fet + Fex-

The last relation can only be determined by using the Clebsch gordan coefficients but it is of very poor practical interest

Ee1+ Fe~=Aet+ Be t - Bex+ B 1 - Blex+ B 2 - B2ex. (111,32)

6.6. P h y s i c a l A p p l i c a t i o n s . We now consider the reaction where the target is a proton and the incident particle a charged particle. From this physical point of view, only the 11 reactions A, B, D, E are observable. The three supplementary constraints lead to triangular inequalities which are always very difficult to check experimentally.


Let us take first the elastic meson-baryon scattering: we obtain triangular relations in the three cases

a) a (K+p ~ K+p); e(7:+p ~ 7:+p); a(T:+p ~ K+E +) b) a ( K - p ~ K-p) ; a(r~-p ~ ~-p}; a ( K - p ~ n -X +) c) a ( K - p ~ K+,~,-); a ( ~ - p ~ K+E-) ; a ( K - p ~ ~+E-) .

In the case of elastic baryon-baryon scattering, some simplification occurs, the four octuplets being identical. We define symmetrical and anti-symmetrical amplitudes by

A t ( a + f l ~ y + 0 ) = A ( ~ + f l ~ y + 0 ) + A ( a + f l 4 0 + 7 ) . (III, 33)

and we obtain an equality

a (pp ~ pp) = as(Z+p ~ Z+p)

and two sets of triangular inequalities, which are quite impossible to check experimentally

a) as(E-p ~ E-p) , as(X-p ~ YTp), as(E-p ~ X-X +) b) a~(E-p ~ E-p), a~(E-p ~ E-p), a~(E-p ~ X-X +) .

When the time reversal invariance holds, the restriction B1 = B2 corresponds to an observable relation

A { [ - 2 , 2] ~ [ - 1, 1]} = A { [ - 2 , 2] ~ [3, -3 ]} (III, 34)

which gives, for instance

a ( K - p ~ K°E °) : a ( K - p ~ ~+E-) a(pp ~ E---~E °) = a (pp ~ E-rE+). (III, 35)

Of course, the actual experimental data on ~o production is very poor and the comparison with experiment is completely hopeless.

As a last remark, some particular reactions, for which the final state is a three body system, can be related by some particular transformation of the group R. For instance, by using RI, we obtain

a(E+p ~ E + + n + ~:+) = a(X+p ~ E ° + p + K +) . (III, 36)

6.7. The same technique can be used, of course, in the study of the strong decays as

B * 3 B + M B * * ~ B + M V ~ M + M

or in the production reactions

8 + 8 3 1 0 + 8

8 + 8 3 1 0 + 1 0 .

I t is easy to verify that, in R, the representation 10 is reducible into 1 + 3 + 6 where 1 is the zero weight, 3 are the three summits of the triangular diagram and 6 are the weights of the previous representation located on the root hexagon.

40 MICHEL GOURDIN :

One can follow the same approach and define independent amplitudes with respect to R. As previously, the charge independence allows to find physically interesting relations.

7. Scattering of two octuplets

7.1. In this section, we are interested in the symbolic reaction

8 ~ 8 ~ 8 ~ 8

which contains the set of physical processes

M + B ~ M + B

= > V + B

M + B**

~ V + B * *

B + B ~ B + B

--> B + B**

B** + B**

B + B ~ B + B

B** +

~ B + B**

B** + B**

~ M + M ~ M + V

~ V + V where

M is a pseudoscatar meson (J = 0-)

V is a vector meson (J = 1-)

]3 is a baryon (J = 1/2+ )

B** is an excited baryon ( j = 3/() .

As before, we are only concerned with the physically observable reactions where the target is a proton and the incident particle a charged particle such as the ~+- mesons, the K- + mesons, the protons, the antiprotons.

The general study has been made by FREUND, RUEGG, SPEISER and MORALES [34] in the case of meson-baryon scattering and we shall adopt their notations. We use the conventions given in Fig. t2 for the weights; for instance, a proton is associated to a weight 2.

The expansion of the scattering amplitudes on the complete basis of the invariant amplitudes previously defined, A ~ , is given in Table 10.

J I

] I

+ l

l l

I +

+ ~

~ ._

~ .

~a

~I

'

I t

1 I

+ 1

I +

f i

l I

+ I

1 +

I +

I I

+ I

+ I

I I

i I

I I

0

42 MICHEL GOURDIN:

We study 19 physical reactions. The transition amplitudes are linear combinations of the 8 invariant amplitudes A. z" One has to find 11 relations, some of which being simple consequences of the charge independence (S I). One of the more compact forms is the following

e4 : e2 + 156 e3 = el + •10 #8 = #2 -{- P l l

~ - P I 3 = - - ~ -~1- -~7 -+-1~12 / 3 - ~ 1 4 = - - ~ - j b3 - jD9Af - Jb12

3p15 = - 1/ Pl- + P, + +

If the time reversal invariance holds, the amplitudes Aa~ and A~ are equal and a new relation follows:

199 = Pll- (III, 39)

This equality and the three first ones of the above set have been given in the previous section as a consequence of the invariance by R and the isotopic spin.

7.2. If some reactions exhibit a peripherical aspect, it can be useful to analyze it, so as to know the expansion of the direct reaction amplitude

z +za.x +x, on the complete basis of eigen-amplitudes defined for the crossed reaction

X~+ X _ ~ X_~+ X~. The relation between the two channels can be described with an 8 × 8 crossing matrix connecting the eigen-amplitudes,

In order to obtain this matrix, it is sufficient to expand eight linearly independent amplitudes by using the properties contained in R and the expression given in Table 10, and to solve a simple system of eight linear equations. If we denote by d . z the eigen-amplitudes in the crossed channel, the crossing matrix is given in Table 1 I.

A 1

A,s

A27

A ~

Alo A~ Aas

Asa

1Is

z/s 1Is 1/s

1/s 1/s

1

--3/10 1/5

- - 2/5 - - 9/5

27/s 27/40

7/40 - - 9/8

-- 9/40 -- 0/4 0

Table 1 1

daa

--1]2

--1[12

i/4 /4

--15]2

15/2

d ~

5/4

--1/2

--1/12

;/4 1/4 15]2

--15/2

--1/3 0

1/3 0 - - 1 / 2

J~Csa

1/30 --1/30 --1/Z

--1/2


Independently, DE SWART has found the same result using a more general method and the properties of the Clebsch Gordan coefficients [35~.

As was expected, the square of this matrix can be verified to be one. If one can apply the time reversal invariance in the direct channel

we have Aas = A~, and from Table 11 there follows in the crossed channel the equality d~0 = ~¢~. In the same way, if the time reversal invariance holds in the crossed channel, we immediately deduce the relation in the direct channel A~o = A~. For instance, let us consider the direct reaction pF ~ 7:+7:- and the crossed reaction pr : - ~ p~:-; we have the equivalent relations

~ = ~ ' ~ ~- A~o = A ~ .

7.3. We are now able to expand the amplitudes considered in the above section on the complete basis ~/.z" The result is given in Table 12.

Process

Pl

P~

P3

P~

P~

~2

P3

~3

P7

P~ P, Plo Pn Px2

B4

I/8

1/.

i/s

l/s

--I/i 0

3 7 VY

3 -

3

sV~ --1/1 o

3/10

]1o

3/10

3/~ 0

3

1]2 0

1/5

--1/4 0

3

3

--1/4 0

1/5

7~o /5

1 1/2

1/2

7/2 0

3

N 2 3

9/20

7/4 0

Table 12 I

1/" 1 --i/la

--1 --1

1

1 - - - 2 ~

V l& G 1/.

1[~ 1/. 1/.

1 1

1 _ 1_1_

t/5- [!~ 1]4 I

-1/3 - / 1 ~

d ~

-i/1~

--1/2

1

1

i]i 2

-- i]6

i/12

i/6

116

I/a

--1/12

~/a s

--1/6 0

1

6oV~- 1

1

_~1/oo

],o

--1]B 0

!/60

120

1

"I/120

J~8 a

--1/6 0

3oV~-

1 60V~ 1

6oV~

-~-1/6 0

--1/6 0

--116 0

1/6 0

1/1£0

1

~2oV~-

40~/~ --1]120

44 MICHEL C-OURDIN:

8. Peripheral model

8.1. At very high energy, some scattering and production reactions exhibit a peripheral nature and it seems that the dominant mechanism, in this case, is the exchange of a small number of particles between the target and the incident particle. For instance, the one pion exchange model or the vector meson exchange are both peripheral.

Of course, the nature of such an exchanged particle can differ according to the various theories; we can use a classical field theory or the Regge pole theory to calculate the asymptotic behaviour of the cross sections.

In this model, we replace the notion of intermediate particle of Regge pole by the more general one of intermediate representation. We cannot give any prediction on the energy dependence of the cross section but only some relations between the cross-sections.

We expand the transition amplitude in eigenamplitudes, ~ x , of the crossed reaction for which the total baryonic number is zero. We assume that the processes are dominated by the amplitudes d 1 and d s and we neglect the other amplitudes associated to a higher dimensional representations such as d lo , ~ , ~'2~- Let us remark that the exchange of the Pomeranchukon is included in the ~1 amplitude and that of pseudoscalar and vector mesons in the d s z amplitudes. But the hypothesis is somewhat more general and permits the exchange of systems of particles in states ofdimensionali ty 1 and 8. As a second remark, the representations 10, 10 and 27 have not yet been used to classify resonating meson systems; for instance, experimentally no resonating = meson systems have been found in the isotopic spin state I = 2.

8.2. In the case of the reactions

8 + 8 o 8 + 8 ,

the crossed reaction is identical from the point of view of group theory. We have given in Table 12 the expansion of the physically observable reactions by the eigenamplitudes.

If now we assume dlo , d ~ , ~ negligible with respect to the amplitudes ~¢1 and ~s , we obtain three supplementary conditions

d~7 = 0 ~" P8 = 0

d~o = d ~ = 0 ~- P2 = P~ = P~ = O.

These constraints can be combined with the previous relations due to the unitary symmetry. We now find four "forbidden" reactions but we must, of course, interpret the term in an asymptotic sense and, at the actuM experimental energies, we must expect the "forbidden" reaction cross sections to be smaller than the "allowed" reaction cross-sections.


Let us take the case of meson-baryon scattering; we obtain

a(7:-p ~ K+E -) -~ 0 a ( K - p ~ z:+Y, -) ----- 0

a ( K - p ~ K°~ °) = 0 a ( K - p ~ K+~ -) ~ 0

a ( K - p ~ z:-Y, +) ~ 4 a ( K - p ~ 7:°~ °)

a(rc+p => K+E +) --~ 2 a ( ~ - p ~ I(°Y, °) .

Of course, the prediction of "forbidden reactions" is completely independent of the unitary symmetry and is only related to a double charge exchange (I > 1) in the crossed channel.

Another interesting example is the production of hyperon-anti- hyperon pairs by proton anti-proton collision. We obtain three "forbidden reactions"

a (~p ~ E - E - ) ~ a (pp ~ .~.o~0) ~ a (pp ~ ~-7~-) ~ 0

and one equality from unitary symmetry in this model

a(~p @ Y,+Y,+) --~ 4a(~p @ Eoy, o) .

From an experimental point of view, we have two qualitative results which permit us to use, with a reasonable argument, a peripheral model

a) The forbidden reactions have a smaller cross-section than the allowed reactions.

b) The allowed hyperon-anti-hyperon reactions described by the d s z amplitudes have a smaller cross-section than the elastic reaction where the d 1 amplitude plays an important role.

Quantitatively, the experimental data is the following [36]

a(~p ~ Y,-E-) ~_ (8 4- 3) [zb

a(~p =~ 7~-~-) _~ (4 4- 2.5) ~b

which one can compare with

a (~p =~ AA) ~ (87.5 ± 25) ~ b .

a (~p ~ pp) ~ (21.2 ± 1) m b .

Only in the last elastic reaction does the amplitude ~1 enter. 8.3. In the case of meson-baryon scattering, it is possible to obtain

more relations with a supplementary assumption equivalent to a vector meson exchange model for the intermediate representation 8 of the crossed channel ; the symmetrical coupling M M V is zero from the Pauli principle and the amplitudes ~'s as and ~¢s88 will be zero.

Now all physical amplitudes are linear combinations of d 1, d s ~ ~ and d s s a only; we can choose, as a new basis, the elastic amplitudes ei, related by the equality

e 1 + e 2 = e 3 + e~ . ( I I I , 40)


All inelastic amplitudes, of course, are proportional to the difference of two elastic amplitudes

V 2 - . ¢ ' 1 = - 2#~ = ~ - el = - ' / ~ d < , < , + 1/3o ~ / .

V 2 " ~ 3 = # 6 = - -~)10 = - - 2 ~ 1 2 = - - (21V3-)P14----- e l - - 63 = - -116 d~aa --ll60J~sa

V6ib4 = - (2/V3-) ~)13 ---- - - 2 # 1 5 -~ e l - e l = - 1/2 d a a + 116o ~ s a

A = A = A = /~9 = A1 = 0 .

We apply the optical theorem to the equality between elastic amplitudes and we obtain a relation between total cross-sections

(~T(r:-p) + aT(7:+p) = aT(K-p) + o'T(K+p) .

If we combine this result with the Pomeranchuk theorem, we show that at very high energy, the four cross sections become equal. This result is equivalent to assume that at very high energy tile amplitude d 1 dominates and it follows that all inelastic processes have a vanishing asymptotic cross-section.

9. Decuplet production

9.1. We are now interested in reactions where a baryon target is excited by collision with a meson, a baryon or an antibaryon. All these reactions have the symbolic form

8 + 8 ~ 10 + 8 (III, 41)

and are described by 4 invariant amplitudes only, Asa, Ass, Alo and A,~. I t follows that the number of relations between the physical amplitudes must be in a larger than in the previous case.

We use tile language of the meson-baryon collision

M + B ~ M + B *

but the results can be transposed immediately to any of the following cases by elementary substitution:

M + B ~ V + B *

B + B ~ B + B *

B** + B*

B + B ~ B + B *

~ B + B *

B** + B*

B** + B* .

Some Topics Related to Uni ta ry Symmetry 47

9.2. We restrict ourselves to a proton target and a charged incident particle. We then have to study 20 physical reactions, listed in Table 13 and we have to find 16 relations because of the unitary symmetry restrictions.

Reaction

~-p ~ - ,+

8¢2

7:o N~ °

~0 ~T*0

K+ Y*- K o y*o

,iX:+ ~*+ ~+P ~ ±'t~/~

~o XT*++ ~" 812

K + y*+

Ko 5I.o

Ko ~ ,o

=+ YI*- ~- y*+

=o ¥,o ~o y*o

K+p ~ K + N3~

K o XT*++

Notation

0¢ 2

~3

~4

~5

~6

~7 ~8

~9

~10

0~11

~12

0~13

~14

~15

~16

~17

0~12

0~20

Table 13

A2,

__1/~ [~/2o

3

2o V ~ I/4VE

1/20

- -3 V2/20 V,

-1/, ~% 3

41/~-

1/20

----1/2 o

1/20

__a/~

1/20

- -1 /4

1/1 o

t5/1o

t/2

-V3/2

1

- -1 /3

Vg/6

6V~ -V . V~

1/6

V~/6 1/2

-V~ VE 1

2V~ 1/2

- -1/6

+V6

1/6 1/6

--1/G

A**

--1/5

2 sV~

--1/5

V2/1o

- -1/5

+1/5

--1/5

--1/5

I/io |@1o

I -~ Aea

- -1 /3

-V~/s 2

3V ~

'-~/3

V~/6

1/3 --1/a

--1/a 2/3

--l/a

--2/3 1/2

-V~/6

The

V3-~1 + ~2+ V6~3 = 0

VY~+ V~s=O ce 5 + i ~ o% - ~Io = o

V ~ ~ 9 + 0~2o = o

charge independence alone gives 8 relations

~1 - ~ + l / 5 ~7 = o

V 3 0 ~ 4 - - ~ 9 = 0

0%1 + ~12 = 0

°q5 + ~16 + 2~n = 0.

In order to find new relations, characteristic of the unitary symmetry, we follow the method described by I~¢~ESHKOV, LEVINSON and LIPKIN [37] *.

* See also MESHXOV, S., G. A. SNow, and G. B. YODH: Phys. Rev. Left. 12, 87 (1964) (Ed.).

48 MICHEL GOURDIN :

Unitary symmetry implies, as a subgroup, isotopic spin invariance and also other 3-dimensional rotation subgroups: we consider one of these

--->.

subgroups, U, which conserves the charge Q, [U, Q] = 0, and for which the infinitesimal generators are defined by

- ( m , 42) -__>.

We remark that this group U will play a fundamental role in the study of electromagnetic interactions and we refer to the next section for a more systematic treatment.

The U invariance gives the last 8 relations

~15 -Ay 0~14 - - V 2 0~ 6 = 0 ~Z 3 - - V g 054 -- 2 0~ 6 = 0

~ - c ~ o - ~ = 0 o q - = n - ~qo = 0

v . ~ - ~ + ~ - V 3 - . ~ s = O .

We obtain the equality of four cross-sections and four sets of triangular inequalities

~l~a(=- p @ =+N~-) = a(=-p ~ K+Y *-) = a ( K - p ~ K+E*7) (III, 43)

= a ( K - p ~ =+YI*-)

a) a ( K - p ~ o ,0 K Nst2), b) a ( ~ - p ~ ~,oN,o~

c) ~(=+p ~ =+N*~+), ~-N.+~ d) a ( = - p ~ 3/2 j,

a(I~-p ~ K°E%°), 2 a ( = - p => K°Y *°)

3a(rc-p ~ '~°~'T*°~-'3m, 4a(rc-p ~ K°Y *°)

a(x+p ~ K+Y*+), a (K+p @ K+N~ +)

K - N *+~ a ( K - p ~ x - Y * - ) . a ( K - p ~ ~12 J,

The last relation, combined with the others gives an equality between seven cross-sections, of course, very difficult to check experimentally [37]

a( r : -p ~ ~o~T,o~ • - ~'3m + a (rc-p ~ .,'~°~'T*°~"~m +

+ ~r (K-p ~ K°N*~) + a ( K - p ~ ,~o=*°~,1,~, (III, 44)

= 2 Ea(K-p ~ r~°Y *°) + a ( K - p ~ woy~0) + a(rC-p ~ K°Y*°)].

Finally, we note that the substitution of a vector meson to the final pseudoscalar meson is trivial for all these relations.

9.3. In Table 13, we give the expansion of the physical amplitudes on the basis of the 4 invariant amplitudes of the direct channel. The calculations use the tables given by TARJANNE [38].

9.4. In the same way as in the previous case, we have calculated the 4 × 4 crossing matrix between the eigen-amplitudes A of the direct channel and the eigen-amplitudes d of the crossed channel

A M + B ~ M + B *

d M + M - - > B + B* .


The solution of a system of four linear equations gives the result of Table 14

A 27

A lolV~"

As~,lg~"

, , , , , ,

~7

1/10

9/20

--27/20

--9/~9,0

Table 14 H,,,

"~olV~

- -1 /2

- % 1/~

~/~

Jgs.g~-

- - 1

- - 1

0

9.5. Let us now study the consequences of a peripheral model used for these reactions a t very high energies. As previously, we obtain a set of "forbidden" reactions, associated to a double charge exchange and this result is independent of unitary sym- me t ry Reaction

~,~ = ~ 5 = ~ 1 ~ = ~ 4 = ~i, = 0 . ( I I I , 45)

All non-vanishing amplitudes are al linear combinations of two indepen- ~, dent amplitudes. We can choose as the 1/2 ~8 eigen-amplitudes zdsa and d s ~ in the ~g ~ crossed channel.

I f we associate now the inter- ~ mediate 8-dimensional state with some ~/2 ~ one-meson state, we immediately see ~7 tha t d s ~ corresponds to a vector 1/2//3~a meson exchange and d s ~ to a pseudo- ]/~ ~ scalar meson exchange.

9.6. I n c i d e n t a n t i p r o t o n b e a m . ~1o The reactions ~n

~ p ~ B B * ~

~p ~ BB* ~1~

have been observed experimentally. ~15 They are of the general form studied in ~e this section and in our notation are 2 ~7 described by the 8 physical amplitudes 21/'3 an, ~ m , . . . , 0qs. Beween these ampli- ~* tudes we have 2 relations due to I spin ~9 conservation and 2 relations due to 1/]/5~0 U spin conservation. For instance, as

Table 15

1

0 o 0

- -2

o

1 o 0

- -2

1

--1 1

o

0 o 1

1 --1

1

1 3 " ~ z¢,6

2

o

- -2

o

o

- -2 2

o

1

--1 o

o o 1

--1

- -3

- -1 1

a consequence of uni tary symmetry, we deduce the following equality between two cross-sections

a ( P p ~ ~o~,oxl,., = a ( P p ~ E - Y * - ) . ( I I I , 46) Erg. d. exakt. Naturw. 36 4

~ 0 M I C H E L GOURDIN :

If we believe a peripheral model is valid at high energy, some of these processes become "forbidden"

~(~p ~ E°E *°) ~- 0

~(pp ~ Z- v*-) ~ o .

An experiment has been performed with a 3.69 GeV]c anti-proton beam [39]. One observes a reactions such that:

~ + p ~ Y + Y + r : Y = A ° , E

and one looks for some resonant Y ~ or Y 7: systems. As a first result, the Y*+ production is more important than the Y*- production which is a "forbidden" reaction and the data seems to be in agreement with a peripheral mechanism with K and K* exchange. Unfortunately, the data is not sufficient to allow a quantitative comparison between theory and experiment.

9.7. I n c i d e n t K- + and ~+-beams. We now study the reactions

M + B ~ M + B * .

The experimental data is more copious but unfortunately the comparison with the predictions of the unitary symmetry is impossible.

The only apparent result seems to be the peripheral character of the reactions. The simplest test is of course the comparison between the cross-sections for "allowed" and "forbidden" reactions.

Let us take two examples, the first one in the N~*~ production, the second one in the Y* production

a) At :#I~ -- 2.24 GeV/c, the Y*- production is negligible with respect to the Y*+ production [40]

g(K-p ~ ~+Y*-) < g(K-p ~ =-Y*+).

b) At p= = 3.3 GeV/c, the N*~ production is not observed whereas the N *+ production is observable [41]

~r(=-p ~ T:+N~{) < ~(~-p ~ =-N *+~ a/2 1 •

The hypothesis of a peripheral model with exchange of a vector meson has also been proposed by STODOLSEY and SAKORAI [42]. By using an analogy between the photon and the p meson, they assume the vertex transition V + B ~ B* to be of the M1 ~ P3/~ type; there follows a particular form for the matrix element describing the final decay B* ~ B + M and the angular distribution of the final meson is predicted to be 1 + 3 (~, ~)" where ~ is a unit vector normal to the production plane and ~ a unit vector along the final meson, in the rest system of B*. Experimentally, the reactions:

K-p ~ ~-Y*+ [40] ~b K = 2.24 GeV/e ~+p ~ K+Y *+ [43] pn = 2.77 GeV/c

K+p @ K°N *++ [44] PK ---- .91 GeV/c

are in agreement with such an assumption.


10. Two decuplet production

10.1. We consider the symbolic reaction

8 + 8 ~ 1 0 + 10

experimentally observed, for instance, in baryon-antibaryon collisions

B + B ~ B * + B * .

The product of representations for the final state

1 0 ® 1 0 = 1 ~ 8 ~ 2 7 ~ 6 4

shows that all amplitudes are known from 4 eigen-amplitudes only that one can denote by A~, A8~, As,, A,7.

10.2. We now restrict ourselves to the only interesting physical case of ~p collisions. The number of physical amplitudes is 10 and we first apply the charge independence for the N*, and the Y~* reaction

A (N-) + A (N +) = 2A (N °)

A (N °) + A (N ++) = 2A (N +)

A (Y+) + A (Y-) = 2A (yo).

Analogous relations after rotation by 2r~]3 in the weight space can be deduced from the U spin invariance

A (N-) + A (E-) = 2A (Y-)

A (Y-) + A (a-) -- 2A (~-)

A (N °) + A (~.0) = 2A (yo). ,,xT,÷+8;2 N3;~*÷+

We can also consider a new 3- N"~ ~m N*o dimensional rotation invariance 3;2 ~;2

connected with the third diagonal 1~1~ of the root hexagon and we can y*- Y'*-' obtainthreerelations, but onlyone E.~ E.~ is independent oftheprevious ones ~_ ~ :

A (E-) + A (N +) = 2A (yo). E~ E~

It followsthat three amplitudes de. Y*÷ fine all ~p ~ B 'B* reactions and Y1 *° ~*'°x *° the expansion is given in Table 16.

Finai state

Table 16

AI AII

1 0

1/. ./. 0 1

o o 1/3 o

~1. 0 1/~ 1/3

AIII

o o o 0

1/. 2/s

1

1/3

We have seven sets of triangular inequalities and three only are consequences of the charge independence. But it is possible to obtain three sets of equalities between four cross section which follow from a particular 3-dimensional rotation subgroup

~N*++, - ~(N~r) (/~

3 [-(Yo*-) - a(a~-)] -~ . (N~-) - a (~-) (U -~) (III, 47)

a [ ~ ( E * ~ ) - ~ ( Y ~ + ) 3 = ~ ( f ~ - ) - ~ ( N * ~ + + ) •

The consequences of the U spin invariance have been given byLIPKIN [45]. 4*

52 MICHEL GOURDIN :

10.3. In a peripheral model, the double charge exchange is forbidden and AII= Am---0. All non-vanishing amplitudes are proportional as given in column AI of Table 16.

The experiments on N*2 production can be explained with a meson exchange [46] but the experiments on Y* production strongly disagree with a peripheral model [47]; more precisely, the "forbidden" reaction is favoured with respect to the allowed reaction for an incident energy of 3.69 GeV/c

a(~p ~ Y*-Y*-) =~ 7a(~p ~ Y*+Y*+).

IV. B r e a k d o w n o f the unitary s y m m e t r y

1. Introduction

1.1. The existence of relatively spread out mass spectra for the supermultiplets bound to the irreducible representations is the simplest manifestation of a breakdown of the unitary symmetry. The same phenomenon appears, but with a much smaller magnitude, in the isotopic spin multiplets, and these effects are then attributed to the existence of electromagnetic interactions which violate the charge independence.

We shall now make a similar hypothesis for strong interactions and we shall define three kinds of terms in the Lagrangian which are associated to these symmetry violations:

a) The dominant part I 0 is strictly invariant with respect to the entire group S U(3) and allows us to define supermultiplets where the particles have the same space-time properties and the same masses.

b) The middle strong part, Iy, which is not invariant by S U(3) but which ensures nevertheless tile conservation of I and Y, and is responsible for the separation inside a supermultiplet of the masses into isotopic multiplets.

c) Tile electromagnetic part I~, which conserves the Q charge, and is responsible for the separation of the masses inside an isotopic multiplet.

1.2. From the mathematical point of view, we shall make a hypothesis which will only be justified by its consequences, and which is not a consequence of the previous physical criterions but, of course, satisfies them. Iy and Iq are transformed, by S U (3), as the Y and Q of the adjoint representation [8, 48, 49]. We shall also assume that the physical quantities in which we are interested can be calculated in a perturbative way with respect to Iy and IQ. We shall not do this calculation which requires the use of dynamics, but it is possible to deduce some consequences which are simply due to the covariance properties of Iy and I O. And it appears that the first perturbation order leads to relations which seem to be verified in the physical field with a surprising precision. This experimental fact has not yet been satisfactorily explained.


1.3. The isometry theory will allow us to formulate the problem at any arbitrary perturbation order. In particular, it can be shown that, by assuming the Iy perturbation series to be convergent, we obtain for each representation a mass spectrum corresponding exactly to the isotopic spin multiplet decomposition of the supermultiplets. This result ensures the mathematical coherence of the method but, of course, has no new physical interest. In a similar way, the degenerecy completely disappears for all representations by using the complete series in I}, and _TO. But, in fact, we are only interested in small dimensionality representations for which the saturation is reached very rapidly at a low order perturbation.

2. First order violation

2.1. Let Mo be an operator which transforms as the component Q of the adjoint representation of a rank I semi-simple group. Let us consider the M e matrix elements between two states ~Px and ~0 2 bound to two irreducible representations, D.~ and D.~ of the group

A~ = <~ITM~t~2>. (IV, 1)

The quantity A o is different from zero only if the adjoint representation appears in the product D,~ ® D~ and we have an isometry problem to solve. The generalization of the Wigner-Eckart theorem allows us to write AQ in the following way

A - V ' F / . t D c " z ) r, \ (IV, 2) Q - - ~ a z X W l l q l~v2 / aZ

where ~(~x) is the .o component of the isometry, which allows us to go from the D~ I ® D~ product to the adioint representation DR. The number of terms of the preceding sum is equal to the number of times where the adjoint representation appears in the D~, ® D~, product.

When ~o 1 and ~ belong to the same representation described by a set of l positive or null integers, 2 i, the number of terms ~ of the previous sum is equal to the number of non-zero 2 i [50].

2.2. F i r s t o r d e r in Iy in u n i t a r y s y m m e t r y . For a given % there exist at the most two isometries which permit us to obtain the regular representation. The first one, which is always there, is given by the infinitesimal generators themselves. The second one exists if, and only if, ~u ~2 4=- 0 which is the case when, for instance, ,p belongs to an octuplet. We can then write the mass operator in the following way

m = m o ~(x) + ~q Q~ , + & ~ , , . (IV, 3)

The mo term is due to the invariant part by S U(3) and the terms a x und fix to the first order perturbation in Iy.

Up to normalization constants, g2 (1) is the identity operator and Q~) the Y hypercharge. For Q~*~ we will need the covering algebra as for all isometries, in general. But the zQ~ -~ case is particularly simple for it is

~4 MICHEL GOURDIN :

built from the symmetrized products of two generators; as the ~2~ s) operator must commute with the isotopic spin and the hypercharge, it is a three operator, 1, 12, y2, linear combination. A complete calculation gives [51]

~Q(1) = , 1

V~ = " 1 / ~ - x Y (IV, 4)

t i n } - ; ] t 2 } " = ~ / B x - 2 1 ( 1 + l) +l /~Y~ y y

The quantities x, y, z, N depend on the representation and are simple func- tions of 21 and 2~; in particular, N is the dimension of the representation and x the Casimir operator

y = 2 ( i l - 2,) [~t9(2~ + i~)2 + 1/0 212~ + i l + 2~ + 1] z = x* ( ' /~x + 1) - y * .

These quantities ensure the orthonormalization of the isometries. Finally, let us note that if we neglect these orthogonality and nor-

realization problems, we obtain the mass formula, at first order, under a simple form [8, 48]

m = a + b Y + c [ I ( I + 1) - ~I~Y 2] (IV, 5) but where the three coefficients a, b, c, contain first order perturbation contributions in Iv.

2.3. F i r s t o rder in I O. The assumptions made on I 0 are completely similar to that made for Iv and the operators are simply deduced by an inner automorphism, which can be represented graphically by a - ~ ] 3 rotation of the root diagram. The I 0 interaction leaves charge Q invariant, as well as the kinetic moment U, described by the three operators,

Y

! i?

i / ~ig 1,o. t~v~

l X r J


In the preceding formulae, which give the isometry expression, we shall proceed to the simple substitution

Y~Q, I ~ U . In particular, the electromagnetic vertices are bound among themselves by a formula of the preceding type. This gives a relation between the anomalous magnetic moments which is only valid in the order zero in Iv

3. Violation of higher order

3.1. P e r t u r b a t i v e se r ies in I~, [51]. For calculating the con- tribution of order n in Iy, we must look for the matrix elements of the tensorial power n of M. The elementary case of second order illustrates the method in a simple way. We know that

8 o S = l ~ 8 ~ e 8 s e 1 0 e 1 0 e 2 7 . Therefore in M~ ~ will appear the parts which transform by the preceding representations. But we are only interested in those which commute with the isotopic spin and the hypercharge. As the only representations which contain a weight I = Y = 0 are 1, 8, 27, the only parts of M~ ~ which contribute are those which transform as the components of 1,8, 27.

In a general way, we can show that in M ®" the only contributions h are those associated to t e symmetrical representations D(~, 2) with

~, = 0, 1 . . . . . n. We are then brought back to an isometry problem where we must extract the symmetrical representations of the q ® ~ product. We show that the maximum number of isometries, for a D (~, ~) representation, is 2 + 1. In particular, there exist at the most three 27 isometries but only one is apparent when ~0 is an octuplet. In this last particular case, we also see that the second order mass formula will comprise four terms, which is the number of isotopic multiplets of the 8 representation and saturation is therefore reached at second order; also, the parts of M ~ " associated to the D(~, ~) representations where ~ > 2 , ob- viously do not contribute and in this particular case we verify that the number of terms of the mass formula is equal to the number of isotopic spin multiplets.

3.2. P e r t u r b a t i v e se r ies in Iy. The same argument and the same results can of course be applied to the electromagnetic interactions when the Iy perturbation is neglected. For the infinite order of I V, one obtains the definition of U spin multiplets and the electromagnetic interactions are charge independent with respect to the U3 component of the multiplet.

Unfortunately, the physical meaning of this approximation is very debatable, for we mix in the same multiplet particles with masses as different as the neutron and the ~0 and we suppose that they have the same electromagnetic properties. For instance, a dynamical calculation

56 MICHEL GOURDIN :

o f anomalous magnetic momentas will use strong interactions and will have to take into account the difference of mass of the particles; it will certainly not give a similar result for the neutron and the E °.

4. Mass formulae*

4.1. B a r y o n s . We have placed the eight baryons in an octuplet containing the four isotopic spin multiplets. The first order formula in Iy contains 3 parameters and there exists between the masses the following relation

area + mz = 2(ran + rn_=). (IV, 6) This relation is experimentally verified with an accuracy better than 1%. To the second order in I y , four parameters appear and there is no longer a relation. The 8 isometries, written in the representation 8, take the form

ff2t~a ) -~ 1

.o<~,> = (~/~)v~ I1 -/(s + 1) + 'h Y ' ] The first order formula is thus written

a bl y m=l-~+__ V~. + b ~ , -~ [ l - I ( I + l ) + ~ / , Y ' 2 ] . (IV, 7)

The best fit is obtained for the following values of the parameters [51] (units MeV)

a = 3260 b t = - 4 6 8 b~ = - 121 while the exact values when the second order is included are

a = 3260 b 1-- - 4 6 8 b 2 = - t l l c~ = - 8 . We shall then make the following remarks

a) the agreement with the first order formula is excellent, b) the agreement is unexpected because of the importance of the

perturbation, which can roughly be evaluated to 15% (Ibllat ~ 0.14). The sum of all corrections of the second and higher orders appears

to be negligible. 4.2. P s e u d o - s c a l a r mesons . We have again an octuplet, but only

three isotopic spin multiplets because of the invariance by charge conjugation. For this reason, the antisymmetrical isometry disappears and we have again a relation between the masses.

FEYNMAN has suggested that these considerations should be applied to the squares of the masses when mesons are concerned. We then get

N + ~ [1 - I ( I + l! + lh Y,]

and the relation is ,mitten as

3m~o + m g = 4m~. (IV, 8) * See also TARJANNE, P., and R. E. C~J:rKos~¥: Phys . Rev. 133, B 1292 (1964)

(Ed.).

Some Topics Rela ted to Uni t a ry S y m m e t r y 57

Despite the very great mass differences between K, ~ and ~0 mesons this equality is satisfied to 6% which is remarkable and unexpected.

The best fit, at first order for Iy, is obtained for [51] (units MeV ~) = 467" 103 fl = 393" 10 a.

The calculation with 3 parameters deviates for x by 1.5 % and for fl by 3 % which shows that once more, the superior order contributions are extremely small. But this time, the agreement is still more difficult to explain than in the case of baryons, the first order perturbation appearing to be very important : fl/oc _~ 1.

4.3. J = 3/2 + r e s o n a n c e s . There is only one isometry which projects the 10 × 10 product on 8 and we have simply a first order formula which is a linear function of the hypercharge

a b m = ~ + - ~ Y. (IV, 9)

tn other words, the different isotopic spin multiplets must have an equal spacing. Experimentally we have

my* - - raN. ~ 147 MeV mz~ ~ - my, = 145-147 MeV, • ~ /~

A priori, we must be able to predict the mass of the last decuplet partner*, the isospin singulet of strangeness - 3 , the l i -

Ma- =~ 1679 MeV. (IV, 10) As in the case of the baryons, the ratio b/a is of the order of 13 % and the violation of unitary symmetry is strong.

Some problems connected with these meson-baryon resonances have recently been discussed from the point of view of the analytic properties of a transition amplitude in the complex plane of the energy variable s. The many channel case is interesting from the point of view of unitary symmetry and in an exact unitary symmetric theory, all poles in the second Riemann sheet associated to the weights of a supermultiplet are in coincidence.

Due to the large mass splitting, some complications apear; for instance, the various channels are no more all open and the position of the poles with respect to the physical thresholds is different in the various cases.

Using a particular model, OAKES and YANG [52] show that the motion of the poles associated to these resonances is very complicated and induces transitions between various unphysical sheets; it is then difficult to believe a first order mass formula is valid for these resonances.

The same problem has been discussed by EDEN and TAYLOR [53] and also by AMATI [54]; they show that the conclusions of OAKES and YANG are strongly model dependent and in general, when the reduced widths are relatively small, a first order mass formula can have some meaning.

The particular question of the f2- mass has been studied by FROIS- SART and JACOB [55]. They show that it is inconsistent, in general, to believe the Y2- theoretical mass predicted by the Okubo-Gell-Mann mass formula and one must probably expect an observed higher mass.

* The exper imen ta l value is 1686 :J: 12 MeV [71] (Ed.).


In our opinion, the main argument against the mass formula is the difficulty to understand it from a perturbative point of view; and in the particular case where in a supermultiplet we classify resonances and bound states, a new difficulty appears due to the different analytic properties of bound states, quasi-bound states and resonances.

4.4. J = 3]2- r e s o n a n c e s . Three of the four multiplets of isotopic ** the spin of this octuplet have been observed experimentally, the Nlj2,

Yo** and the Y~**. I t is therefore not possible to see if the mass formula is satisfied; but if we suppose that it is, we can predict the mass of an hypothetical ~ - E resonance of isospin 1/3 which remains to be discovered

ma~* ~ 1600 MeV. (IV, 11)

Because of the mass spectrum which is much less spread out, the perturbation appears in this case to be much less important (< 5 %).

4.5. V e c t o r mesons . The octuplet composed of the p(750MeV), the K* (888 MeV) and the c0 (782 MeV) does not satisfy at all the first order mass formula.

One thought of replacing the e-meson by the 9-meson (1020 MeV) but the disagreement remains.

Another solution was also to replace the K* by the K'* (725 MeV) if the spin is f = 1-, which makes the agreement a little better but it is not yet as good as in the preceding octuplet cases.

At the present time, it seems that the most reasonable hypothesis is that proposed by SAKURAI [56]. The co and the (~ have the same quantum numbers*, ]Pa = 1- - , I = 0 and SAKURAI supposes tha t the particles observed experimentally are two linear combinations of states belonging one, the ¢0 °, to the 1 representation, the other, the O °, to the 8 representation of the unitary group.

The question of whether interaction Iv is responsible for this phenomenon or is a consequence of this mixing is still open and maybe this relation exists only in a self-consistent mechanism of the bootstrap type [32, 57].

The ~ and 9 which were observed experimentally are linear combinations of the ~o which belongs to the octuplet and of the o~ o which belongs to the singulet ~ = cos 2" ~o ° - sin 2 . ? °

9 = sin J%. co ° + cos 2 . ?0.

In the o~ °, 9 ° basis, the mass matrix is not diagonal and it is the diagona- lization which provides the eigen physical states co and 9. The calculation gives [51] (unit = 1 MeV ~)

(¢0 ° tm~t o~ °) = 789" I0 a (90 lm"l ~o) = 863- 10 a (coo Im~ 1 90) _- (90 [m~l coo) = 2 1 1 . 1 0 a .

The diagonal matrix dement of ? ° is calculated from the first order mass formula which we suppose to be satisfied for the three multiplets.

m~,, = 929 MeV m~o. = 888 MeV. (IV, 12) As for the mixing parameter 2, it is defined by tg2 ~_ 0.841 or 2 ~ 41 °.

* s. p a g e 18.


5. t~-- tp mixing

5.1. The consequences of unitary symmetry are usually easy to express in terms of q)o and co ° because these non-physical states belong to irreducible representations of the group. Unfortunately, only ~ and o) are observed and the relations are then going to be more complicated because of the existence of a mixing parameter which permits the dia- gonalization of the matrices 2 x 2 which connect the states ~0 and co °.

5.2. V ~ M + M d i s i n t e g r a t i o n . We shall s tudy the strong disintegration of the vector mesons into two pseudo-scalar mesons. All the disintegration amplitudes associated to the octuplet are proportional to one another, the only invariant coupling allowed, M M V , being antisymmetrical in the exchange of the two pseudoscalar mesons. The proportionality coefficients have been given in the previous section.

The invariance of the M M V interaction under charge conjugation allows us to find immediately some results relative to o) and q~. The bare mesons q~0 and 6) 0 have the same quantum numbers as ~o and

I = 0 G = - 1 C = - 1 .

A first result is tha t the disintegration of (a and ~ into 2~ ° or 2 7: ° is forbidden by C.

Let us now consider the systems K K and K K which are linear superposition of eigenstates C = 1 and C = - 1

C = 1 K K - K K = K + K - + K - K ++K1KI° O_K2OK~O

C 1 K K + K K K + K - K - K + + ~ 1 K1K2. . . . . KOK o _ o 0

We shall therefore have

e a ~ K K - K K and ? = ~ K K - K K .

The system K K + I~K of isotopic spin I = 0 does not appear in representation 1 of the product 8 ® 8 but only in the 8~ part of this pro-

0 0 duct. As a result, o~ o cannot decay into a K1K2 system. In conclusion, the charge coniugation invariance forbids the o~ °

decay into two pseudoscalar mesons

t~ ° ~ M + M (IV, 13)

and we shall have in the t~ - (p mixing theory

A (~ ~ M~ + M2) = cos2 A (~o ~ Mx + M2)

Because of the real mass spectra of the vectorial mesons and of the pseudoscalar mesons, the only three types of disintegration V ~ M + M allowed are

f~ ~ 2T: K * ~ K + ~ ? ~ K + K ,

which are also the observation modes of the vectorial mesons as resonances of the filial system .The unitary symmetry binds the disintegration


widths in the following way

[A (p ~ 27:)12 = F o

]A (K* ~ Kx)l 2 = 3/4 F0

I A (~ ~ KK)I °" = 3/4 c°s~)~/'0.

These coefficients must be corrected because of the phase space which is very different in the three cases.

Let us consider the most simple V M 1 M ~ hamiltonian

H = i g V ~' [M 1 aM, OM1 ] -- Ms-b~-# ] + h,c.

The disintegration width V ~ M 1 + M s is given by (see for instance [56])

/ ' ( V * M 1 + M2) g2 2 p' 4~ 3 m~,

where p is the momentum of the final mesons in the rest system of the vector meson of mass mv.

We now apply the previous results of unitary symmetry to the effective coupling constants g with a phase space given by this local model. The only free parameter is this coupling constant and it has been adjusted so as to reproduce a p experimental width of 100 MeV. The results are given in column L of table 17 and the agreement with experiment is good [58, 59]. Another method of phase space calculation is the use of the form factor introduced by GLASHOW and ROSENFELD [14] in their calculation of excited baryon decay

C (p, mv) = p P* m v p* + x ~ "

We use the same X = 350 MeV as GLASHOW and ROSENFELD and the results are given in column E of Table 17; the agreement with experiment seems to be better for the K* width.

...... ,, , , , ,

R e a c t i o n

p ~ 2 r :

K* @ Kr:

=> K + K -

K 1Ku ~ @ o o

Table 17

T T o t a l (MeV) p (~,~eV)/c)

100 348 <- 50 283

127 ~3 ll0

/ ' c a l c u l a t e d ( M e V )

I00 I00

33 41 1.16 1.35 0.75 0.91

5.3. We can immediately give the relations between the various effective coupling constants of the V M M system. The couplings ~ o ~ o and ¢o~%q ° are forbidden by charge conjugation invariance and the coupling pr:~ ° is forbidden by G-parity conservation. We immediately

Some Topics Re la t ed to U n i t a r y S y m m e t r y 61

obta in the following results

~,~ =go- ~ x ~ 1 go- &,x~ 1 go- fk.x~ s 2 4~r 4~r - - 4 4 ~ - 4 4-------~---~--4 - g

I~K~ _ 3 cos~ / .gO. f ~ x ~ __ 3 s i n Z l . g ~ 4z 4 4~ 4 '

I f go- is chosen so t ha t it reproduces a 100 MeV width for the p decay, we obta in

go- ~ 2 . (IV, 14)

Let us finally r emark t ha t in the Gell-Mann nota t ion [60]

7p=,~ = Ih/p~-- (IV, 15)

5.4. The decay mode of ~ into p + ~, t hough i t is f avoured b y the phase space, is less f requent t han the K g. mode [59]. In this case, there exists a coupling be tween o~ ° and a neut ra l VM sys tem, bu t the mass spec t ra allow also a decay into p + re. In the octuplet case, the only V ~ V + M allowed decay reaction is 90 ~ p + r~. We therefore have

A (9 ~ pr:) = s i n / A (o~ ° ~ ,on) + cos~ A (90 ~ ,or~)

A (~ ~ ,o r~) = cos ), A (¢o ° ~ ,o =) - sin 2 A (9 ° ~ ,o ~) .

I t seems t h a t the decay mode ¢o ~ p= is impor t an t [60] and nei ther ampl i tude A (¢o ° ~ p re) nor ampl i tude A ($o ~ ,o r:) is small ; in the case of ~ decay, they par t ia l ly cancel each o ther in a ve ry mys te r ious way.

Exper imenta l ly , we have

A (~ --> ,o=) ~- O. I t follows t h a t

A %o ~ ~7:) ~- _ tg2 A (o ° ~ pro) - -

and therefore A (co ° ~ l:~) ~ A (9 0 ~ prr)

A ( o => ,or~) ~_ cos 2 sin

Such a result of course can only be tes ted b y models.

6. E l e c t r o m a g n e t i c in te rac t ions I r = 0

6.1. I f the m e d i u m st rong interact ions are neglected, i t is possible to define mul t ip le ts wi th respect to U, which are comple te ly degenerated.

Fo r the ba ryon octuplet ] = 1/o-+ for instance, we have

Q= I u = l h [X+p] Q=O U = l [E °,Bx,n3 Q=O U=O Bo

62 MICHEL GOURDIN :

where B o and B~ are two orthogonal combinations of A ° and Zo

Bo = ~/2 (~/3 Z° + A °) B, = 1/2 (Z o - ~/3 A°).

By a simple substitution, for the octuplet of pseudoscalar mesons jr = 0-, we obtain

with, this time

7~ + Q 1 u = 1/2 [ , K +]

Q=o U = l [K °,M 1,K °]

0 = o u = o [Mo] Q = - 1 U = 1/2 [K-, re-]

Mo = 1/2 ( l /g MO + ~0) M1 = 1/2 ( m - ~/~ ~o).

The case of the two octuplets J = 3/2- is then immediate. For the decuplet of excited baryons J = 3/2+ we obtain

Q = 2 u o N *++ = a/2

Q = 1 u = 1 /2 [Y1 ~ +, N*31=+na Q 0 U 1 [ = ~ o , y , o ,o

O = - i g = ~/~ [~-, 2~f, Y*-, N~:].

The electromagnetic properties of the partner of a given U multiplet are identical and independent of the U 3 component.

6.2. E l e c t r o m a g n e t i c p r o p e r t i e s of t h e b a r y o n s . We shall call ?" the electromagnetic current associated to a real or a virtual photon. (j)" therefore describes an electromagnetic interaction of order n, associated to n real or virtual photons.

a) U invariance. The U invariance gives us the following relations [61]

<Z+ l(/)"l Z+> = (P I(J)"i P> <r.-I(iPI ~-> = <2-1(/)"[ =7>

<2° l(i)"120> = (n I(i)~l n> . The three last relations

<g, [(i)~I B~> = <n l(i)'q n> <B, I(i)"l Bo> = <no I(i)~l BJ = o

can be split up into matrix elements between the physical A ° and ~o states. For instance, we shall be able to write

<Zo l(i)"l A°> -- < A° I(i)"l Zo>

],/~" <y~o [(j)~[ AO> = <y,o i(?')~l y~o> _ <n ](j)n I n>

a<AO i(i)~t Ao> = <No i(i)"i No> + 2@ l(i)~t n>.

The ten transition matrix elements are bound by six relations, as it was expected, the octuplet containing four U spin multiplets.


If n = 2, there is a simple application of the equalities to the Compton scattering on baryons. Unfortunately, an experimental check cannot be envisaged at the present time.

b) Form factors. If we only consider a first order electromagnetic interaction, two new

relations appear; in this case, we shall use a one-parameter formula

E=~=Q+ [2 - 2 V ( V + 1) + I/9 Q" ? (IV, 16)

and only two form factors are linearly independent, which can be chosen as that of the proton and that of the neutron. The two new equalities can be written as

(P 141 P) + (i1 14t n ) + (Z-14t Z - ) ---- 0 (IV, 17) ( A° ]/I A°I = 1/2 (n [/1 n ) ,

Experimental verification of such relations is very difficult and it seems that, at the present time, the most interesting is the last one written. Unfortunately, the experiments measuring the magnetic moment of the A ° are not precise enough to allow us to draw a conclusion. Let us note the value Table 18 predicted by the unitary symmetry ~p Fn

/ ~ 0 = 1 / 2 ~ n = - - 0.95 e 2raN " Fp 1 0

Also, we must not forget tha t the rela- F n 0 1 tions between form factors are only valid Fx + 1 0 for large values of the transfer momentum and that if we apply them to a zero transfer F~- - - 1 - - 1 in an approximation where all the masses F~o 0 --1/2 of the baryons are supposed to be equal, FA, 0 1/2 we have certainly a large risk of error. /7~OAO 0 --V~]2

We give, in Table 18 the expression F~o 0 1 of the baryon form factors as linear combinations of the proton and neutron F~- --1 - - i form factors.

6.3. E l e c t r o m a g n e t i c p r o p e r t i e s of t h e p s e u d o s c a l a r m e sons. a) We first have a set of relations due to the U conservation

< n+ I(J)fl n+> = ( K+ I(4)"I K+>

<n-I(i)'l n->-- <K- 1(/)~1 K-> (n° l(J)fl ~°5 -- (~° 1(4)~1 no>

1/3<n ° t(i)~l n°> = <n ° i(4)fl n°> - <KO t(j)~t KO> 3(~o l(?')~[ ~o) = (no ](j)n[ no) + 2 ( K 0 l(/)fl K°>

and the charge conjugation invariance will give a number of supplementary conditions for which we shall of course have to distinguish the pari ty of the perturbation order n, the value of C for a photon being C = - 1.

64 MICHEL GOURDIN :

Let us recall that for 7:o and ~o C = + 1.

@ + )(i)"l ~+> = ( - l)- <~- i(/),~i ~ - >

<K + l(i) ~l K +5 = ( - 1) n <K- I(]')~[ K->

<~o i ( i ) ~ + ~ [ ~o5 = o <,~o i ( i )~÷~ I ~o) _ o

<KO l(i)-~+~l KO> -_- O.

All the matrix elements are then known from three quantities only, as expected for an octuplet. In the case where n is odd, only the matrix elements between charged meson states are different from zero and therefore all proportional among themselves.

b) Form factors. The results obtained for first order are valid for any odd order. All

form factors of neutral mesons are zero and for charged mesons we have the evident relation

F~+ = - F ~ - = Fx+ = - F ~ - . (IV, 18) c) Even orders. All the results obtained for the second order are valid for any even

order and therefore also for the perturbation series summed on all even orders.

An interesting example of application is the electromagnetic ~i ° ~ ::o transition. I t seems that the decay

~0 ~ = + = - = o (IV, 19)

takes place mainly according to the sequence ~o=> ~ o ~ 3~, which is represented in Fig. 15 [62].

Fig. 15

The first transition of an electromagnetic nature may be calculated by using unitary symmetry. To each of the diagrams of this v} ° ~ ~o transition can be associated, in a one-to-one correspondance, an energy diagram of electromagnetic self-energy of the mesons. Using the previous relations

<~+ [(i)~n[ ~÷> = <K ÷ [(i)~'q K+>

]/3< r~° [(J)~"t ~o> = <r:o [(J)""t ~o> _ <K o [(J)'"[ K°>,

one obtains, after subtraction

Some Topics Related to Unitary Symmetry- 65

with [63]

- + - = - ( s 4 MeV) r =

The second transition has been calculated with a 4 z~ 2 (q)•. q~=)~ interaction. By using the value 2 = - 0 . 1 8 i 0.05, OKUBO and SAKITA have found [63]

/~(~o ~ ~+rCr~o) ~ 142L~ eV.

However, we think that the hypothesis Iy = 0 which is equivalent to m K = m= used in a problem where electromagnetic mass differences considerably weaker than the difference r n K - m~ are calculated, can be the origin of errors which will be very difficult to appreciate.

Using the same assumptions with the same restrictions on their validity, we can s tudy the radiative -c~ ° and rc ° decays. The charge conjugation invariance forbids the decay into an odd number of photons. As a consequence of U conservation, the transitions

M1 ~ 2m 7

are forbidden and we deduce a relation between the radiative decay matr ix elements for 7}o and 7z °

A (7: ° ~ 2m Y) - V ~-A (7}0 @ 2m y) = 0 . (IV, 20)

By introducing the phase space factors, the decay widths are

The experimental data on xo life-time are not in agreement between themselves

GLASSER [64] Z = (1.9 i 0.5) 10 -6 S

VON DARDEL [65] z = (1.05 ± 0.18) 10 - s s .

By using this last value, the calculated width for ~0 is

14 n+3° eV (IV, 21)

The experimental value of the ratio for these two decay modes we have just studied is [66]

/,@qo ~ 2 7) /,(~o ~ r~+r:-r:o) ----- 1.3 i 0.4 .

The agreement between theory and experiment is good but, of course, cannot be considered as a serious test of unitary symmetry, first because a rough model has been used for the decay 7}o ~ 7£~:-xo and secondly because the partial width for the radiative decay 7} o ~ 2 y may change by a factor 2 according to the experimental data of the x ° life-time one uses.

6.4. E l e c t r o m a g n e t i c p r o p e r t i e s of m e s o n - b a r y o n re- s o n a n c e s , a) We shall first consider the J = 3/+ excited baryons associated with a decuplet. The U rotation invafiance can be applied to the

Erg. d. exakt. Naturw. 36 5

66 MICHEL GOURDIN :

matrix elements of type (B* Iff)'q B*> and gives a set of evident relations

<N~ + I(i)"i N*+> = <Y*+ ](i)'q Y~*+>

<N~-](i)"[ Ng:;> = <Y*-I(i)~1Y*-> = <Y¢*; I(/)=1 ~N> -- <a-i(j)~i ~->

The ten series of matrix elements are related by six equalities as expected for a decuplet.

If now we study the B* form factors, we restrict ourselves to the first electromagnetic order and we then obtain a two parameter formula which describes all the form factors which are linear with respect to the electric charge. For instance, the four e.m. N~= form factors are related by two barycenttic relations

FN- + F~÷ = 2F~o Fro + Fro+ = 2 F r o .

b) Another interesting problem is that of radiative decay of the 13" particles according to the scheme B* ~ B + y; of course, this problem is equivalent to the study of the 13" B y vertex for real or virtual photons.

We first apply the U invariance to obtain relations which will be valid for all orders in f

(P I(/)"l N*+\ a/2 / =

, 0

<n I(/)=l N~°> =

<y7 I(i)~i y~,-> = Two transitions are forbidden at and the nonvanishing amplitudes

<z + I(i)~l Y**+> <zo I(i)~1-1,=~*°\, 2<2o i(/)~l y.o> = ~ <A o I(i)fl y,o>

o = (a - l(i)~i ~ ; > . all e.m. orders by U invariance only are known from two of them.

In the special case of first e.m. order, only one isometry exists and all matrix elements are proportional. This result can be shown as a result of first order violation of isotopic spin invariance which allows us to write

<p ](j)] N~ +> = <n [(i)[ N ~ ) . (IV, 22)

c) When the excited baryons belong to an octuplet, as for instance in the case of J = a/2- resonances, all the properties deduced in the case of the ordinary baryons J = I/2+ can be easily extended by elementary substitution to the various matrix elements

<B l(j)~[ B>; <B I(i)fl B**>; <B** t(i)~l B**>.

6.5. E l e c t r o m a g n e t i c p r o p e r t i e s of t h e m e s o n - m e s o n re - s o n a n c e s , a) All the properties we have found in the case of pseudoscalar mesons can be extended in a simple way to the vector meson octuplet. Let us simply remark that the charge conjugation acts in a different way and in particular, C = - 1 for p0 and ~o; also for co ° we have C = - 1.

The vector meson form factors, for instance, obey the following relations

Fo+ = - F o- = FK . . . . FE*- . (IV, 2a)


The radiative decay of po, ?o and co ° into an even number of photons is forbidden by charge conjugation invariance. The transition amplitudes into an odd number of photons are related by

A [?0 ~ (2m + 1) y~ - ~/3A [~o ~ (2m + 1) ~'J = 0

for any arbitrary integer m. Let us remark that the co ° ~ ~( transition is forbidden because the

photon is the U = Q = 0 component of an octuplet; the physical co ~ y and q~ ~ y transitions are then related by

cos2 A (c~ ~ y) + sin2 A (? ~ y) = 0 . (IV, 24)

After comparison of the two radiative decays

c o ~ e +e- and ? ~ e +e-

which occur essentially through one photon only, it seems possible to reach directly the mixing parameter ). [671.

For the matrix elements, we obtain

IA (po ~ e+e-)]2 = / ' o

IA (? ~ e+e-)l 2 = 1/3 cos~2 • F 0

IA (co ~ e+e-)[ 2 = 1/3 s i n ~ • 1~0.

Now, because of the large mass difference between vector mesons, we must make an estimate of the phase space corrections.

Let us introduce an effective hamiltonian so as to describe a direct coupling between the electromagnetic field Av and the vector boson field V.

H = eg V " A ,

and let us calculate the partial decay widths as indicated in Fig. 16.

V

Io* s ,'~"

Fig. 16

The result is the following

F(V e+e -) --3--~ .-mY-v. m~r PCM

where = is the fine structure constant, m e the electron mass, mv the vector boson mass, PcM the CM momentum of the final electron. I t is evident that for the electron case

me < mv PcM ~ 1/~ my, 5*

68 MICHEL GOURDII, r :

and the previous expression can be simplified into

/~(V ~ e+e -) 4~ ~2 g~ (IV, 25) - 5 - ~ "

An identical calculation gives the partial width for V ~ ~+~- which is of the same order of magnitude because of the small mJm V ratio

r ( v ~ e+e-) ~ - V / "

In the application of unitary symmetry, there exists a difficulty due to the dimension of the coupling constant g; one can consider either g or the dimensionless coupling constant glint. In the first case

F(co ~ e+e-) _~ tg22 (mm~_~) 3 -F'(9 ~ e+e - ) - -

and in the second case

F(co ~ e+e-) too, F ( ~ ~ e+e-) ~ tgZ 2 m~

Unfortunately, the experimental branching ratio is unknown. On the other hand, it is possible to give an estimate of the partial decay widths by calculating the p0_ y coupling constant from results on isovector nucleon form factors [60]. As a first crude approximation, the gp~ constant can be related to the effective/p ~, coupling constant used in the calculation of the p ~ 27: decay width

gpv 1

and we obtain the following branching ratio

/ ' ( p ~ e+e - ) - - ~ 0 . 6 6 ' 1 0 - 4 ,

/'(p ~ 2=)

corresponding for a p ~ 2r: width of 100 MeV to a partial p --> e+e - width of 6.6 keV.

The previous calculations allow us to estimate the partial widths for co ~ e+e - and ? ~ e+e - decays

F(co ~ e+e -) ~ (0.55 - 1.1) keV

/ ' ( ~ ~ e+e -) ~ (0.5 - 1.75) keV.

The first number corresponds to the application of unitary symmetry on g and the second one on g/m~ and for both cases we choose the value of ~ determined in the mass formuta problem.

I t seems to us that the theoretical uncertainty is too important to allow a quanti tat ive method of determination of the mixing parameter.

b) The radiative decay of vector mesons, V ~ M + Y, will give us an interesting set of relations. We first s tart by applying the U invariance


to the electromagnetic V - M transit ions

<=+ I(i)"l p+> = ( - I)- @ - I(i)"i p->

< K+ I(/)"1 K*+> = ( - 1)- <K- I(/)"1 K*->

(n0 [?~1 p0> = 0

(~o i?~1 oo) = o

(,:o l?~i ~o> = o

6 ¢ I?~1 ¢ > = o <i,:o i?ml I,:,o> = o .

The ten mat r ix elements are different f rom zero for the first order; they are proport ional to each other and have as propoi~ionality coefficient the only isometry which can contribute, f2 (ss), because of charge conjugation invari- Table 19 ance. We give in Table 19 the values of the coefficients of proport ional i ty [49J

I f now we consider the ra ° meson which appears in the singlet, the transit ion o ) °~

M l + y is forbidden b y U conservation, and it follows tha t

A (co°~ n ° Y) - ]/3A (co°~ ~0 Y) = 0. (IV, 26)

For the physical ampli tudes with 6) and % we obtain

cos ). A (? ~ ::o y) _ sin 2 A ((o ~ 7: ° y) = ] / 3 X s

cos), A (p ~ .~oy) _ s i n 2 A (co ~ v]°y) = - X s ,

where X 8 is the A (p ~ r:y) amplitude. After elimination of X 8

sin 2 [A (~ ~ 7: ° y) - ] / 3 A (o5 ~ ~0 y)] +

+ cos2 [A (co ~ ::oy) _ | / 3 A (o) ~ ~Oy)] = O.

6.6. P h o t o p r o d u c t i o n r e a c t i o n s .

a) Generalities.

The photoproduct ion reactions

V + X~ ~ X~ + X~ (IV, 27)

are first order electromagnetic reactions. The photon t ransforms like the Q charge operator of the Lie algebra and we are brought back to a four body reaction problem, where in the un i ta ry symmet ry framework, the photon plays a role similar to tha t of the M 0 combinat ion of mesons previously defined and characterized by Q = U = 0 in an octuplet.

A (p+ @ 7: + y) 1

A(p- @ =-~) I A (K*+ @ ~+ y) 1

A (K*-@ "~-'f) 1 A (K *° @ K ° y) - - 2 A (K*O~ K ° y) - - 2 A (po ~ r:o y) + 1

A (po~ ~° V) +V g A (~o ~ 7: o v) +V g A (~° ~'~°'~ " ) - -1

70 ~/~ICHEL GOURDIN :

As a result, the reactions

7 • 8 ~ 8 • 8 (I) (IV, 28)

are described by eight linearly independent amplitudes and the reactions

~ , , s ~ s • 10 (II) (IV, 29)

by four amplitudes only. The application of U invariance must give very useful relations be-

cause of the U = 0 spin for the photon: for instance, the reactions where the particles X~, X a, X v remain in their multiplets U~, U~, U v are all proportional to each other.

b) Let us first consider the particular case where the target is a #roton; there exist only six possible reactions of type t and the U invariance allows us to write two relations which lead to two sets of triangular inequalities

V2-A (yp ~ r:+n) -- ] /3A (yp ~ K+A °) - A (yp ~ K+Z °) (IV, 30)

V~ A ( y p ~ Koz+) = V~ A (yp ~ ~q0p) _ A (yp => zcOp).

Analogous equalities are valid for the type II reactions

A (vP ~ ~÷N*~) = - ]/~-A (~p ~ K÷Y *°) (IV, 31)

V~2-A (yp ~ K°Y *+) ~/3A (yp ~ ~0N.+~ A (yp ~ ~°N*+~ q ~ ' 3 / 2 I ~ 312 J •

The first one of each of these two series of relations has been pointed out by LIPKIN and coworkers [68] as a consequence of U invariance.

c) When the target is a neutron, relations of a new kind will appear due to the fact that U = x/~ for the proton and U = 1 for the neutron. For instance, one finds

]/-3-A (¥n ~ K°A °) - A (Tn ~ K°Z °) (IV, 32)

= ~/3A (yn ~ ~°n) - A (yn ~ ~°n)

A (yn ~ K+Y *-) = - [/3A (yn ~ ~+N~) (IV, ss)

2A (yn ~ K°Y *°) A (yn ~ 7:°N *°) ~/3-A (yn ~ ,, -.m~.

d) We shall have some supplementary relations which characterize the first order electromagnetic reactions; for instance, we have the weU- known equalities, completely independent of unitary symmetry

|/2A (yp ~ zPp) - A (yp ~ r:+n)

= ]/~-A (Tn ~ nOn) + A (~n ~ ~-p) (IV, 34)

~/2-A (yp ~ K+Z °) - A (yp ~ K°X +)

= V'2A(yn ~ K°E °) + A (yn ~ K+X-) .


7. Electromagnetic interactions I v * 0

In some problems where the strong interactions play an important role such as the electromagnetic mass differences or anomalous magnetic moments, it seems reasonable to take into account, to a certain extent, the Iy interactions.

7.1. The most trivial method which, however, can be contested from the mathematical point of view, is to consider separately Iv and t 0 and to neglect the violations of unitary symmetry of the form I v I O.

Such a rough approximation has been used in the electromagnetic mass difference problem and the s3n-nbolic formula is simply-

m = too(I, Y ) + m~(U, Q ) . (IV, 35)

This expression allows us, after elimination of m o and n h , to deduce some relations between the electromagnetic mass differences appearing inside a given isotopic spin muttiplet.

a) The case of the baryon octuplet has been studied by COLEMAN and GLASHOW [69]. They obtain one relation which allows us to calculate a theoretical value for the ~ - v 0 mass difference

m z - - m z . = rex- - mz+ + m p - m~ . (IV, 36)

The results are the following [70]

{ ( m z - - rnz,)ta = (7 ± 0.5) MeV

(m•- mzo)e~p (6.8 + 1.6) MeV.

The agreement with experiment is remarkable. b) The case of a decuplet is simpler because I and Y as well as U and Q

are related for a given weight. We can write four equations

(IV, 37) ~ N o - - r a N , - ~ ~ 4 y , o - - ~ ] / y * - ---~ m E , o - - ~ * - .

7.2. In the case of anomalous magnetic moments or from factors, we are restricted to first order electromagnetic violation and the formula becomes for a physical quanti ty E

E = Eo(I , Y) + y i O ~ ~ + y2O~ s' . (IV, 38)

a) Let us consider an octuplet; in general we have two relations between the b a r y o n / o r m / a c t o r s

Fz+ + F z - = 2Fzo

F z - - Fz+ + F v - F~ = F z - - F,~o.

The first one is independent of unitary symmetry and is due only to a first order electromagnetic violation of isotopic spin invariance.

The second one, as previously seen, holds at an arbitrary electromagnetic order under the assumption that the terms I r I e are neglected.

72 MICHEL GOURDIN :

b) For meson octulblets, the charge conjugation invariance gives the general relations

F.+ + F n - = 0 Fn0 = 0

Fx+ + Fx- = 0 Fx° + FX, = 0

F~, = 0 F~,=, = 0 .

Unitary symmetry, in the actual framework ( I v 4 : 0 at all orders, I r I o = 0 at all orders) relates the three independent form factors

F K + - - FK, = F ~ . ( I V , 39)

c) In the decuplet case. the first order electromagnetic approximation is very simple and reduces to

E = Eo(I , Y) + o~Q. (IV, 40)

From first order electromagnetic violation of isotopic spin only the four N1"2 form factors satisfy barycentric relations as noted previously and also the three Y* form factors.

The differences of form factors corresponding to different isotopic spin multiplets are now equal by unitary symmetry

F.~,, - F=_,- = Fy,0 - Fy , - = Fy,+ - Fy,° (IV, 41)

= FN,0 -- FN*- = FN,+ -- FN*0 = FN*++ -- FN*+.

7.3. I t seems more reasonable to take into account the crossed I r I o terms as well as I v terms. As an empirical result, the first order with respect to I v gives satisfactory results for the supermultiplet mass spectra calculation and it seems logical to keep only the first order in I y and the first order in I r I o at the same time as the first order in I O [491.

Let us call E a physical quanti ty: we have a general expression of the following type

E = E o + E~ ~Q~' + E 2 ~ s , + Ea ~O~, + E4 ~O~" + (IV, 42)

'~'1°~ E .O(~-~,+ ~" '~(~'~ ' ~ ' ~ ( ~ '~ D~,~- " + ~5~zeY+ ~ Q ~TazQy + ~s~QY + E o .

So, in general, when all isometrics are present, we have an 10 parameter formula. We shall look at the two cases of electromagnetic transitions for an octuplet (8 ~ 8) and for a decuplet (10 ~ 10).

a) In a particular case of the octuplet there exists only one 27 isometry and therefore the above formula has only 8 parameters. Between the 10 physically measurable quantities (8 diagonal and 2 non-diagonal) there exist therefore 2 relations.

I f we neglect the I v I Q terms, we obtain now 5 relations and if we forget completely the I v interactions, as in the previous sections, we have now 8 relations.

The two reactions which remain valid are of a very different nature and validity. The first one has been given already and is related to isotopic spin invariance only

Fx+ + Fx- = 2F~ , . ( I V , 43)


The second one allows for instance to express the 2 ° ~ A 0 + ~, transition momentum in terms of linear combinations of the anomalous magnetic moments for neutral baryons. Unfortunately this relation cannot be compared with experiment [49]

2 gg <a o 1il Xo> (IV, 44)

= <y~0 lil X°> + 3< A° li[A°> - 2 @ Ii[ n> - 2<~°1il~° > .

I t is interesting to note that at the limit I 0 = 0, we find again in the right hand side the Iy first order mass formula.

In the case of the meson octuplet, the only interesting application of this relation is the radiative decay of vector mesons [49]

}/5" [A (po => ~oy) + A (~o ~ =07) ] (IV, 45)

= A (p0 ~ =o3, ) + 3 A (~o ~ ~0T) _ 4A (K *° ~ K°¥)

At the present time, there is again no possibility of comparing these results with experiment.

b) For a decuplet, we have no 10 and 10 isometries and only one 8 isometry and one 27 isometry. The general formula depends only on four parameters

E = o~ + o~2Q + aay + ~ , O Y . (IV, 46)

The ten physically observable quantities are bound by six relations. If we neglect Iylo, we obtain one more relation (a4 = 0) and if we forget also I r , another supplementary relation (z¢ a = 0). So in the end we obtain the eight relations which were found before.

Three of the six relations are only consequences of first order violation of isotopic spin invariance as given previously; the three last ones are consequences of unitary symmetry in the actual approximation (first order in Iy, I y I O, Io)

FN*- + F~_,- = 2Fy,-

Fy, - + Fa*- = 2Fz, -

F•,o + Fz , o = 2 F y . . .

These relations between form factors seem also extremely hard to verify experimentally.

C o n c l u d i n g r e m a r k s

We have just studied, with some details, various aspects of the octuplet version of unitary symmetry. The classification of particles, bound states and resonances as weights of irreducible representations seems to us useful and non-trivial; we remark that, in some cases, the Regge trajectories theory leads to very similar conclusions and such a connection is certainly meaningful.

74 MICHEL GOURDIN :

Of course, the assumption of a unitary symmetry allows various processes to be related to each other and, in principle, one possesses in this way direct checks of unitary symmetry. But what is the domain of validity of unitary symmetry ? Such a question is very hard to answer and if one accepts the notion of asymptotic symmetry with respect to energy, it is therefore impossible to define in a precise way the beginning of the asymptotic energy region. For the particle masses, the unitary symmetry is strongly destroyed and, certainly, such a partial breakdown has an important effect in the actual experimental energy range. In our opinion it is impossible now to give definite conclusions on the existence of such symmetry in nature. This concept is certainly useful and we prefer to leave each reader to form a personal opinion on this question.

Let us accept for a moment this concept of high symmetry; what is the reason for choosing a particular mathematical group such as the physical symmetry group ? We consider that no satisfactory criterion has yet been used to establish the existence of unitary symmetry from the general principles. All attempts are practically of dynamical nature, and for instance, some people t ry to explain the existence of a unitary symmetry from bootstrap calculations; only very rough models can be used and the connection between assumptions and conclusions is not very clear in many cases. On the other hand, it is not evident, a priori, that the results obtained after various approximations can be extended to the general case of the exact problem, though it can be the case. And why not use an esthetic criterion of simplicity: the octuplet model of Lie algebra, A v is the simplest algebra which satisfies the physical requirements of isotopic spin and hypercharge invariance; its adjoint representation has eight dimensions and the number of fundamental baryons is precisely eight. Of course, this can be only a coincidence.

Let us now adopt, for a short moment, an optimistic position with respect to the existence of a unitary symmetry in strong and electromagnetic interaction physics as explained in the various sections of this paper. One of the most important problems is then to study the relation between weak interactions and unitary symmetry. Usually in weak interaction phenomenology one works with certain currents and some people have proposed to use for the strong interacting particle currents those given by the adjoint representation of S U(3). A stronger connection is very hard to define: for instance, it is difficult to use the irreducible representations of S U(3)/Z 3 to classify the leptons and the hypothetic intermediate vector bosons; the group must be enlarged but no definite solution has yet been given in the eightfold way scheme. In practice, the vectorial spaces for leptons and W bosons are completely independent of those associated to irreducible representations of unitary symmetry, but this fact seems very unsatisfactory and extremely difficult to understand. We think that the problem of a possible extension of unitary symmetry to weak interactions is difficult to solve before a better knowledge of the dynamics of weak interactions and of the selection rules is acquired. Both problems have not yet received a corn-


plete and satisfactory solution. Nevertheless, we think that the idea of a larger group including the S U(3)/Z~ ® U(1) group as a subgroup and allowing the classification of leptons and vector bosons, can be useful in the understanding of elementary particle phenomena.

Appendix

Unimodular unitary group S U(3)*

1. Generalities

1.1. L i n e a r g r o u p GL(3, R). The regular 3 x 3 matrices with real coefficients form the general linear group GL(3, R). This group depends, of course, on 9 real parameters. I t can be shown that the Lie algebra for the linear group is the Lie algebra for 3 X 3 matrices with real coefficients.

The commutation laws which define the Lie algebra on R are then

[Xrs, Xt.] = gstX,~ - g. ,Xts (A 1)

where g is a symmetrical linear inner connection of the 3-dimensional vector space on which the thus represented by

1.2. S p e c i a l l i n e a r

infinitesimal generators X~s act. These can be

a X,s = x, Gs " (A 2)

g r o u p SL(3, R). The unimodular 3 × 3 matrices with real coefficients form the special linear group S L (3, R) which depends on 8 parameters. The Lie algebra of S L (3, R) is defined from that of GL (3, R) by the equality

X' , = X,~ - I/a g,, g " X , , , , . (A 3)

The infinitesimal generator Y = g - - X.~,

commutes with all the infinitesimal generators of the linear group and is related geometrically to homotheties.

1.3. L i n e a r g r o u p GZ(3, C). The regular 3 × 3 matrices with complex coefficients form the linear group GL (3, C). This group depends on 18 real parameters. The Lie algebra of GL (3, C) is the complex form of that of GL (3, R) and it can be shown that it is isomorphic to that of the Lie algebra of the 3 × 3 matrices with complex coefficients.

1.4. S p e c i a l l i n e a r g roup . As before, we can define a two-parameter sub-group of GZ(3, C) associated to complex homotheties. The special linear group SL (3, C) is then the group of the unimodnlar 3 × 3 matrices with complex coefficients. I t depends of course on 16 parameters.

* References [1, 4, 71]. See also the article of FRONSDAL in Vol. 1 of the 1962 Brandeis lectures, Benjamin Inc., New York.

76 MICHEL GOURDIN :

1.5. U n i t a r y group . Let E(a)(C) be the 3-dimensional space on which the matrices of GL (3, C) act. Let g be a symmetrical antilinear inner connection which, if it is reduced to its diagonal form is the unify matrix; in other words, E l~) (C) is a hermitian space with a hermitian product g. The 3 x 3 matrices with complex coefficients, which leave the connection g invariant,

A * g A = g (A 4)

form the unitary group U (3) which is a sub-group of the linear group GL (3, C).

The infinitesimal generators of the unitary group are linear combinations of those of the complex linear group. These linear combinations being anti-hermitian, the Lie algebra of the unitary group has 9 dimensions.

1.6. S pe c i a l u n i t a r y g r o u p SU(3). There exists a linear combination, Y, of the infinitesimal generators commuting with all the generators which is bound to a one-parameter gauge group. This parameter multiplies the coordinates by a common phase.

The unimodular unitary matrices form then an 8 parameter group S V (3).

1.7. The unitary group U(3) and the special unitary group S U(3) are compact*. The Lie algebra of the special unitary group S U(3) is simple; in other words, the S U(3) group contains no proper invariant continuous sub-group.

Finally the group S U(3) is connected** and simply connected***. I t is therefore the universal covering group of its Lie algebra.

2. A2 algebra

2.1. In the Cartan notation, A~ is the Lie algebra of the special unitary group S U (n + 1).

The A 2 algebra is of rank 2, that is, the maximal abelian sub-algebra of A ~ is 2-dimensional: it is called the Cartan algebra.

The canonical basis of the algebra is obtained by solving a problem with proper values. One can show the existence of six non-zero roots distributed on a regular hexagon, and of a double zero root at the center of the hexagon. The infinitesimal generators associated to the non-zero roots a are written E~ and those associated to the zero roots and which yield the Caftan algebra, H i.

* a group G is compact if every infinite sequence in G has its limit in G.

** a group G is connected if it cannot be considered as the reunion of two not emp ty disjoint sub-groups.

*** a connected group G is s imply connected if all pa ths binding two points are equivalent. The S U (n) groups are s imply connected.


2.2. The roots of A e can be shown on a 2-dimensional diagram. I t is interesting, so as to make the most use of the symmetries of the problem, to introduce triangular coordinates, 2£1, X v Xa, of zero sum.

O~23 0~13

C¢31 e;~32

x~

Fig. 17

The ~ii roots, all of equal length, have then the following structure

in which the ei's are three orthogonal vectors of equal length in a 3-dimensional euclidean space.

2.3. When normalization must be taken into account, the length of the e~ vectors is 1/~/6. The commutation laws of the As Lie algebra are then written

[H~, Hi] = 0 Cartan algebra

[H~, E~] = ~E~

[E~,,E~]=N~aE~+ a if ~ + f l ~ 0 i s a r o o t (A6)

[E,, Ea] = 0 if a + fl ~ 0 is not a root

The covariant and contravariant components of the a roots are equal and given by the explicit form

[~.]~ = [~ j ]~ = O / l / g ) ( ~ - %~).

The quantities N~a which are the non-trivial structure constants are known from one of them through the general symmetry laws

N~a = - N p ~ = N_a_~ = N_(~+~) ~ .

We shall set, through phase convention

N~,~,, = C~',~,, = 1/~/6. (A 7)

78 MICHEL GOURDIN:

5. Representations of the A2 algebra

3.1. There exist two contragradient fundamental representations of dimension 3, which we shall call by convention

D3(1, 0) ~ 3

D"(0, 1)

A given representation can be obtained from the fundamentals by a tensorial product. I t is written D (21, 22) where 21 and 22 are two positive or null integers. The dimension of this representation is then given by the following formula

N(~, A,):(I + At)(1 + A~)(l + ~'+ I______A_~).

3.2. The representation weights are vectors of a 2-dimensional space - the same as that of the roots - the coordinates of which are the proper values of the Cartan sub-algebra operators.

The fundamental representations have three weights each. These weights are equivalent to each other for they are deduced from one another by symmetry with respect to a plane perpendicular to a root. Such geometrical transformations form the Weyl group, entirely defined by the knowledge of the roots.

3.3. The weights of two contragradient representations are deduced from one another by symmetry with respect to the origin. We then have the following situation for the two fundamental representations.

/ III ~%% ii / \\ /11

', ["-.. I / ,

3 Fig. IS

~ / /

3.4. If the weights of the fundamental representation 3 are written as ill Fig. 18 we see that the operator E12 makes us go from weight 2 to weight I and in a general way, the operator E~i from weight j to weight i. The matrix representation is then


and if we write it explicitly

ioi !o! V ~ E ~ = o V ~ = o V ~ E ~ = o o o o

i°! :° i i°! 0 ,0 1 0

[/6H1 = I/3 i We have used

i o o i o i - - t g H ~ = I/a 2 6 H ~ = 1/3 --1 . o 0 o

three H i generators which verify

H x + H~ + H a = 0 . (A8)

3.5. I t can be shown in a general way that if the X are a representation of the Lie algebra in a basis, we can find for the contragradient representation X~ a basis such that

If we choose for the representation 3 the basis deduced from that of representation 3 by symmetry with respect to the origin (Fig. 18) we immediately obtain

H ~ = - H j E ~ i = - E ~ i . (A 9)

3.6. The regular representation is always part of the product of two contragradient representations together with representation 1. This last representation corresponds to the symmetric sesquilinear form conserved by unitary transformations, and we shall write symbolically

3 ~ 3 = 1 . 8 . (A10)

As a convention, the first index is associated to representation 3, and the second one to its contragradient 3. The six following quantities

lij> = I0 ~ Ii> i . i are associated to the six non-zero roots 0qj of the adjoint representation. As to the three others, it is easy to verify that the completely s~anmetfical part corresponds to the conserved scalar product

3

ll> -- (1/V~) Z li> * 1i> (A ix) I

and that the weights of the adjoint representation are of the form 3

le> -- X e, li> ~ Ii> (A 12) 1

3 3

the Oi's verifying ~ ~ = 1 and ~ ei = O. 1 1

80 MICHEL GOURDIN :

This problem has two solutions which are linearly independent: [~) and I@, but there is of course a degeneracy in the ~, a plane. The choice of a particular basis will be made by considering certain physical criterions.

3.7. The infinitesimal generators can then be represented in the adjoint representation by the following matrices

b=3 ~/-6E, i = Z [lik> < i k [ - Iki> <k/l]

k = l k = s (A 13)

l/gH~ = Z flik> (ikl - N > <kill. k = l

An adjoint representation being equivalent to its contragradient, there exists a matr ix C which allows us to go from the Xo to the - X ~ . In the previous basis it is easy to see that this C matrix is given by

C = Z lii> <i¢1 + Iq> <el + la> <a[. (A 14)

All the basic vectors 1i/'>, [q>, ]@ being orthonormalized, we have

CC T = I C T = C . (A 15)

4. Representat ion product

The problem of a systematical s tudy of the irreducible representations product is very complex and we shall only give here several results*

3 ® S = 1 ~ 8 (A16)

8 ® 8 = 1 ~ 8 s e 2 7 e 8 , $ 1 0 • i 0 . (A17)

The adjoint representation can be found twice, first in a symmetrical combination of the indices, and then in a n antisymmetrical one

8 ® 1 0 = B e 1 0 e 2 7 e 3 5 . (A18)

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[40] BERTANZA, L., V. BRISSON, P. L. CONOLLY, E. L. HART, I. S. ~¢[ITTRA, G. C. MONETI, R. R. RAU, N. P. SAIvItOS, I. O. SKILLICORN, S. S. YAMAMOTO, M. GOLDBERG, J. LEITNER, S. LICHTMAN, and J. WESTGARD: Phys. Rev. Left. 10, 176 (1963)

[41] GUIZAGOSSlAN, Z. G. T.: Phys. Rev. Left. 11, 85 (1963) [42] STODOLSKY, L., and J. J. SAKURAI: Phys. Rev. Left. 11, 90 (1963) [43] YAMANATO, S. S.: quoted by SAKURAI, ref. 42 [44] KEHOE, t3. : Phys. Rev. Left. 11, 93 (1963) [45] LIPXtN, H. J. : Preprint (1963) [46] SMITH, G. A.: Phys. Rev. Left. 10, 138 (1963) [47] BALTAY, C., J. SANDWEISS, H. TAFT, B. ]3. CULWlCK, W. 13. FOWLER, J. K.

KoPP, R. I. LOUTIT, J. R. SAN'FORD, R. P. SHUTT, A, M. THORNDIKE, and M. S. WEBSTER: Phys. Rev. Left. 11, 32 (1963)

Erg. d. exakt Naturw. 36 6

82 MICHEL GOURDIN: Some Topics Related to Unitary Symmetry

[48] OKUBO, S.: Progr. Theor. Phys. 27, 949 (1962); 28, 24 (1962) [49] - - Phys. Lett. 4, 14 (1963) [50] GINIBRE, J . : j . Math. Phys. 4, 720 (1963) [51] - - Nuovo cimento 30, 407 (1963) [52] OAKES, R. J., and C. N. YANG: Phys. Rev. Lett. I1, 174 (1963) [53] EDEN, R. J., and J. R. TAYLOR: Phys. Rev. Left. 11, 516 (1963) [54] AMATI, D.: Phys. Lett. 7, 290 (1963) [55] FROISSART, M., and M. JACOB: Preprint Saclay (Nov. 1963) [56] SAKURAI, J. J. : Phys. Rev. Left. 9, 472 (1962) [57] KATZ, A., and H. J. LIPKIN: Phys. Lett. 7, 44 (1963) [58] GREGORY, B. : Proceedings of the 1962 International Conference on High

Energy Physics, CERN, p. 779 [59] CONOLLY, P, L., E. L. HART, K. W. LAI, G. LONDON, G. C. MONETI, a . R. I~AU,

IN'. P. SAMIOS, I. O. SKILLICORN, S. S. YAMAMOTO, M. GOLDBERG, M. GUNDZIK, J. LEITNER, and S. LICHTMAN: Phys. Rev. Lett. 10, 371 (1963)

[60] GELL-MANN, M., D. SHARP, and W. G. WAGNER: Phys. Rev. Left. 8, 261 (1962)

[61] CABIBBO, N., and R. GATTO: Nuovo cimento 21, 872 (1962) [62] BARTON, G., and S. P. ROSEN: Phys. Rev. Left. 8, 414 (1962) [63] OKUBO, S., and B. SAKITA: Phys. Rev. Left. 11, 50 (1963) [64] GLASER, R. G., et at. : Phys. Rev. 123, 1014 (1961) [65] VoN DARDEL, G:, et al.: Phys. Lett. 4, 51 (1963) [66] BACCI, C.: Phys. Rev. Left. 11, 37 (1963) [67] DALITZ, R. : Report at the Sienna Conference (1963) [68] LEVINSON, C. A., i . J. LIPKIN, and S. MEs~Kov: Phys. Left. 7, 81 (1963) [69] COLEMAN, S., and S. L. GLASHOW: Phys. Rev. Lett. 6, 1423 (1961) [70] ARMENTEROS, R. : Report at the Sienna Conference (1963)

F u r t h e r R e f e r e n c e s *

DE SWART, J. J. : The octet model and its Clebsch-Gordan coefficients. Rev. Mod. Phys. 35, 916 (1963)

NE'EMAN, Y. : The symmetry approach to particle physics, Israel Atomic Energy Commision I A-854 (1963)

Unitary Triplets ("quarks")

BACRY, H., J. NUYTS, and L. VAN HOVE: CERN preprint (Feb. 1964) GELL-MANN, M. : Physics Left. 8, 214 (1964) HARA, Y.: Preprint, Caltech, (Dec. 1963) ZWEIG, G. : CERN preprint (Feb. 1964)

Weak Processes

CABBIBO: Phys. Rev. Lett. 10, 531 (1963); 12, 62 (1964) GELL-i~fANN, M. : Phys. Rev. Lett. 12, 155 (1964) OI~UBO, S. : Physics Lett. 8, 362 (1964)

Concluded November 1963

Pro:[. MICHEL GOURDIN Universit4 de Paris, Facult4 des Sciences - - Orsay Laboratoire de Physique Th4orique et Hautes Energies ]3. P. Nr. 12 Orsay S. et O./France

* Added by the Editor

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