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REFERENCEIC/68/46

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

A SIMULTANEOUS PARTIAL-WAVEEXPANSION IN THE MANDELSTAM

VARIABLES

THE GROUP SU(3)

A. P. BALACHANDRAN

W. J. MEGGS

J. NUYTS

A N D

P. RAMOND

1968

MIRAMARE - TRIESTE

IC /68 /46

IHTERMTIOML ATOMIC ENERGY AGEWCY

IUTERffATIOFAL CENTRE FOR THEORETICAL PHYSICS

A SIMULTANEOUS PARTIAL-tfAYE EXPANSION" IN THE MANDELSTAM VARIABLES

THE GROUP S U ( 3 ) *

A.P. Baleohandran **

International Centre for Theoretical Physio3

and Physios Department, Syracuse University,

Syracuse, New York, USA

W.J. Meggs **

Physios Department, Syracuse University, Syracuse, New York, USA

J. Nuyts

La"boratoire de Physique Theorique et Hautes Energies, 91 Orsay, Prance

and

P, Ramond1"

Physios Department, Syracuse University, Syracuse, New York, USA

TRIESTE

June 1968

* To "be submitted for publication.

** Supported in part by the US Atomic Energy Commission,

* Supported by ITDEA Fel lowship.

ABSTRACT

The elastic scattering amplitude of two spinless particles

of equal mass •§• was expanded elsewhere in a double series of eigen-

functions whioh "displayed" its dependence on all the Mandelstam

variables s,t,u ( s + t + u = l ) # The expansion was then used to

investigate the crossing properties of partial-wave amplitudes. We

show in this paper that these eigenfunctions are certain base

vectors of the representations (a,a) of a suitably defined SU(3).

The unequal mass problem is also discussed.

-1-

A SIMULTANEOUS PARTIAL-WAVE EXPANSION I F THE MA1TDELSTAM VARIABLES

THE GROUP 5 U ( 3 )

I. IFTRODUCTIOIT

In a previous paper , we proposed an eigenfunction

expansion for the elastic scattering amplitude F of two spinless

particles of equal mass. These eigenfunotions formed a complete

set for a certain class of functions of the Mandelstam variables

s,t,u and were associated with well-defined values of angular

momenta. They were generated "by a partial differential operator (9

whioh commuted with the angular momentum in the three channels and

which was invariant under s,t,u permutations. The expansion

coefficients were shown to satisfy an infinite sequence of finite-

dimensional crossing relations due to the crossing symmetry of F.

Indications for extending the approaoh to systems with internal

symmetry or spin were also given. In a second paper , the

eigenvectors of the crossing matrices were constructed.

In the present work, we formulate a plausible group-

theoretical basis for this expansion. In Sec.II, the operator &

is identified with the quadratic Casimir operator of a certain

SU(3) and the eigenfunotions of (9 with a specific subset of base

vectors of its irreducible representations (cr,o"). The partial—wave

crossing matrices are just ffeyl reflections in these representations.

In Sec.Ill, the difficulties enoountered by this method

when the particles do not have the same mass are discussed. Some

reasonable hypotheses for the action of SU(3) on the scattering

variables are made in order that (Q may have the requisite

features. The resultant constraints are fulfilled only when the

particles are degenerate in mass.

In Appendix A, the unequal mass system is considered once

more and a few rather remarkable properties of an associated

Gram determinant are mentioned. These suggest possible generaliza-

tions of our equations to such systems, but alBO raise many unsolved

questions.

In Appendix B, we try to identify the new variables we

were naturally led to introduce in Sec.II.

The discussion is specialized in much of what follows to

the situation where the eigenvalue problem associated with (V is

solved on the Mandelstam triangle. (The "boundaries of this triangle

-2-

are s • 0, i » 0, u a d ) As indioated elsewhere , however,

the problem can tie solved equally well in the physical region. There

should be no difficulty in recognizing that the group which underlies

the corresponding eigenfunotions is SU(2,l) rather than SU(3). The

necessary modifications in .the analysis will also "be indioated in

the text.

II. THE GROUP

We first recall a few pertinent facts from our previous

paper. The partioles are supposed to have the same mass %•* If s,t,uare

the Mandelstam variables, -fche Casimir operator of the s-ohanriel little

group when it acts on a function of s and t has the form

(a) X2 - - • (l-zj)

(II.D

The eigenfunctions of X are the Legendre polynomials ?/?(z ) »

In writing (II.lb), we treat s,t,u as independent variables. This

is permissible since (9. ~ 3 ) (B + t + u) *» 0. Only spinlesst u

systems are considered. The corresponding t and u channeloperators Y and Z are obtained from (II.lb) by cyclio permutations

2 2 2of s,t,u. The operator (0 is construoted in terms of X , Y , Zby the definition

(D = X 2 + Y 2 + Z2 . (II.2)

It is required to identify iD with the Casimir operator of

sorae group of transformations *\ . We first enumerate the

properties which are expected to characterize Xt t

i) Jci must leave the surface s + t + u = 1 invariant. For

if it did not, the kinematical constraints on the system would not "be

maintained under the action of •& . The form (II.2) for £0 also

suggests the following:

-3-

ii) iL must contain three SU(2_) subgroups such that thea ;T 2 2

corresponding Casimir operators reduce to X , Y , Z when restrictedto functions of s and t,

iii) The quadratic Casimir operator of Qi niust reduce to (f?

under a corresponding restriction.

Next, we show that it is possible to identify iL- with the

group SU(3). Let us implement the transformations of this group on

the complex three-vector (z , 2O, Z ) of unit length |_ £/z. | •= l j ,

Let IjU,V be the generators of the three distinguished SU(2) sub-

groups which act on the pairs (z ., zo^* ^ 22* 1)1 (-i» ^i) • -^

I-, U-, V- are the diagonal generators in this basis (I, + U^ + V,=> 0)

and if f is a function of 2. such that

(a) I 3 f - U f » 0 ,

then

(b) C2 f - (?2 + U 2 + ? 2 ) f (II.3)

where C_ is the quadratic Caaimir operator of SU(3). (The action of

an element oL 6 SU(3) on a function h of ,Z. is taken to be

standard".

(ah) ^ 2 ^ x 2 3

(H.4)

The problem is therefore solved if the dependences of z^ on

s,t,u are such that I,, U- are zero on these latter variables and,

moreover,if

, v _2 "i2 T2 ^2 72 ^2

(b) I ( zt I 2 = B + t + u (II.5)

The restriction to functions of s,t,u is understood in (a), (b)

suggests that we set '

^ - . * e ^ , z^t^e1^, ^ - ^ e ^ * (11.6)

-4-

i , - . ^ . . , . , , : ™ . ( . a p , 1 « ,,. . . .

Next, we note that due to (11,4),the generators of SU(3)can be written in the form

.7)

where the relationship, "between Br and I,U,V is given in footnote 2 ,

Change variables from z , Z , Z_ to s,t,u, V!. , V* , "A and use.1 t J> X £- j

results like

to verify that I-, TJ_ annihilate s,t,u and that (ll,5a) is

satisfied. Thus, the (£? of (II.2) is just the quadratic Casimir

operator of this SU{3) when restricted to functions of s and t '.

(The scattering amplitude, of course, is a function of this sort.) It

is easily shown from what follows that the second Casimir operator

is not an independent one in the representations associated with 19 ,

This completes our identification of %i * We note here an

identity which was vital in the preceding construction. A3 the

cosines of the scattering angles in the three channels may be defined

to be

2t 2u 2scos0 o 1 + —- , cos0. • 1 + r - ; » COS0 a 1 + —r /T T Q \

0 S""l t t~i U U—i ' \LL,yJ

we have,

tan— , (S). „ t a n ^ , (f) . tan -Ji (11.10)

The eigenfunctions of 0 are contained in the base vectors of

a certain class of irreducible representations of SU(3)» The

harmonic functions of SU(3) have been constructed by BEG and HUEGG ',la)

A comparison of their results with our basis '

(11.11)

-5-

shows that these are the central elements in the weight diagram of

the representations (<r,cr). Here, the axes in the weight space are

related in a well-known way to the operators J

• -i - 4>, h - ( + -z A

say. When acting on F(s,t), these operators are clearly zero. Hence

the "basis vectors S » "belong only to the centre of the diagram.

It may be observed that the elements of the Weyl group on the

three variables (z , Z , 2" ) induce permutations of s,t,u in the

arguments of the amplitude F. The partial-wave crossing matrices

which were evaluated in our last paper are thus the matrix elements

of the Weyl operators in the representations (cr,cr) .

In the discussion of this section^it was necessary to

introduce three arbitrary angles V, , ^» % in order to identify

our operators with functions of SU(3) generators* In Appendix B,

we attempt an interpretation of these variables.

III. THE UNEQUAL MASS PROBLEM

The operator (Q was singled out to generate eigenfunctions

for the expansion of F in the equal mass configuration due to its

following featurest

i) It commuted with the total angular momentum in the

s,t,u channels (when restricted to functions of s and t).

ii) It was invariant under s,t,u permutations.

Now, the Casimir operators of any SU(3) have to commute with

the Casimir operators of its SU(2) subgroups and the group elements

which permute these sub/groups. Therefore, to obtain a generalization

of {Q when the particles are not of the same mass, one may try to

fit the three little groups of the scattering process together into

a suitable SU(3). We discuss the simplest of these possibilities

here and show that it does not work. (See also Appendix A*) The

considerations are not general enough to rule out SU(3) altogether.

-6-

Let ("z, 2 2, Z^) be the labels of a vector in the (3,0)

representation of this SU(3). Experience with the equal mass system

and, in particular, eq.. (11.10), suggests that we set

• tan

where s' t' u

*

tan —

are the three scattering angles. But (Ill.l)

(III.l)

implies the identity

+tan tan tan u (III.2)

It may he verified that this identity is fulfilled if and only if

all the four particles are of equal mass.

It is also of interest to attempt to realize only two of the

three little groups correctly rather than all three. (Consider, for

instanoe, pion-nucleon scattering.) So, we may try, instead of

(III.l),

fitan (III.3)

This means that

J2 ICOt - ^ t a n (III.4)

There are two other constraints to he considered. The partial-wave

expansion of P in the s-channel, for example, is its expansion in a

series of Pjf (cos0 ) when the variable s is held fixed. (This is

not the same as its expansion in terms of P,g(cosff ) when s(l+cos0 )

and cos6 are regarded as independent variables, say,)5

must be a function of s alone and that of

Therefore,

t alone;

(t) (III.5)

First set t = 0 and then set s = 0 in (III.4) to solve for

-7-

and Vz I » Then, verify that these solutions are inconsistent

with (III.4) for arbitrary s and t in the unequal mass problem.

ACKNOWLEDGMENTS

Conversations with Drs, A Bohm, A.M. Gleeson, J.J. Loeffel,

J. Schecter, N.J, Papastamatiou, E.C.G. Sudarshan and F. Zaccaric

are gratefully acknowledged. Special thanks are due to

Drs. Papastamatiou and Zaccaria for their suggestions on the

manuscript. One of us (APB) wishes to thank Professors Abdus

Salam and P. Budini and the IAEA for hospitality at the International

Centre for Theoretical Physics, Trieste,

-8-

APPENDIX A

SOME PROPERTIES OP A GRAM DETERMINANT FOR THE UNEQUAL MASS SYSTEM

Consider the scattering amplitude for the process 1 + 2 —* 3 + 4

which is characterized "by the momenta p^ ' (l « 1,2,3»4)« The p

satisfy the identities

ED: (A.I)

- P(3) + (4) (A.2)

We can associate a Gram matrix g with this system as followst

Define first the three four-vectors

(a) S,, - P

P

(1)

.(4) _ B (2)'At

(o) (A. 3)

The Handelstam variables are

s - S , t T2, u (A.4)

while

(a) (S.T) - m

(T.U) - + m2 - - m.

(c) (U.S) ') (A.5)

The Gram matrix g is'defined to he the real symmetric matrix

ohtained from the scalar products of S,T,U ;

B -

(S.T)

(S.T) (T.U)

(S.U) (T.U) u (A.6)

-9-

When the masses are equal, (S.T) * (T.U) - (U.S) » 0 and g is

diagonal,

The matrix g has some remarkable properties of which we

now list a few*

With every vector R . of the form H SL + £ TM + y U^ ,

we can identify a row vector *r » (of, £, y ) in the space of g such

that

R^ R^ « T g T (A.7)

Let N.H be the normal to the scattering plane :

The square of the norm of IT is proportional to the determinant

of g ;

fy u^ . - det g . (A.9)

The equation

det g - 0 (A.10)

describes the boundaries of the physical region . Further,

2 2 2 2Traoe g = s + t + u -> m. + m + m« ••+ m. (A,11)

Let s,t,u be the three eigenvalues of g and let cj, Tj , i be the

correspnnding real orthonormal eigenvectors. As explained above,

there is a natural mapping of ^, ,^ into the space of the four-

vectors. It is given by the equation

and similar ones for T, U . ¥e have,

(a) (S.T) - (T.U) - (U.S) - 0 ,

(b) "S2 - I,"?2 - ^ U 2 - u (A.13)

In terms of the variables s,t,u the physioal regions are bounded by

-10-

det g • 8 t u • 0, that i s , "by the three straight lines

s - 0 , t a O , u = 0 . (A. 14).

If the masses are equal, s" = s, If « t, u = u.

The sign ambiguity of , * ] , f can "be resolved where desired"by requiring that S,T,U reduce to S,T,U when the masses are the

same.

The change of variables s,t,u ->s,t,u involves square and

cube roots and is not trivial. But it has the "beauty of formally

mapping the unequal mass scattering regions, into the corresponding

equal mass ones. As such, it suggests the following solution to the

unequal mass problem: replace the variables s,t,u in the equal

mass results ty s,t^u. We shall not discuss here the physical

implications of suoh an expansion and its relation to the analyticity

properties of the scattering amplitude.

-11-

APPENDIX B

THE INTERPRETATION- OF THE ANGLES S£ , f^ f^

If we are to find a physical interpretation of the group- 2 ~*2 " 2

generators and not just of operators like I , U , V , it is

necessary to specify the action of the group on the four-momenta

of the system. In particular, the dependences of the momenta on

the angles f, , which were introduced in the main part of the

text, have to be identified. In the familiar theory of angular

momentum analysis where only one of the three channels is relevant

at a time, Sp - % > for instance, can be thought of as the azimuthal

angle of the spatial momentum in the s-channel final state in

a special sort of orthogonal co-ordinate system. This system has one

of its axes in the direction of the incident momentum. The polar

angle of the final momentum is defined by this exceptional axis and

coincides with the scattering angle d , We refer always to thes

centre-of-mass of the process. Such a choice of coordinates is not

unique. In our problem, the three channels are simultaneously

involved and a consistent interpretation of the tys along these

lines has proved to be difficult. We therefore present two

alternative explanations of these variables in what follows.

(&) .We shall continue to work within the Mandelstam triangle.

The spatial parts of the four—momenta are pure imaginary here. It is

understood until (Q>) below that an i has been factored out of the

space parts of the momenta and hence that all four-vectors are

Euclidean, The requisite modifications for the physical region

and Minkowski metric will be occasionally indicated within parentheses.

We use the notation of Appendix A with the additional proviso

that the masses are equal. The sixteen variables p^ are constrained

by the eight equations (A.l), (A.2). In the absence of spin, the

"scattering manifold" is thus characterized by eight independent

variables. The first suggestion that comes to mind is to imbed

^1* 2* ^ ^n ^^a manifold in a suitable manner. We shall now

characterize the latter in terms of familar geometrical entities.

If S,T,U are introduced as in (A.3), (A.2) is fulfilled

without further demands since

-12-

(a)

p ( 2 )

(o) P ( 3 ) * i ( 5 + U - T),

(d> p( 4 > - *(S + T - U), (B.I)

•while (A.l) assumes the form

(a) s + t + u = 4DI J

00 (S.T) . (T.U) . (U.S) . 0 . (B.2)

2In the body of the paper, we set m =*•£••

A A ALet us next define the normalized vectors S,T,U through

(a) 3s2

, . A T(b) T = 4

(o) U - ^ • (B.3)u2

•A A A

The vectors S,T,U and the unit vector

^ ?> > . V V - +1 (B-4)

normal to the plane of the momenta p ' form an orthonormal tetrad

in the four-dimensional Euclidean space. (For the physical region,

S =, s/e*f T = T/(-t)*, U - U/(-uJ*, N = V^P ^ ^ *'A A u \

form an ortbonormal tetrad in Mlhkowski space with L M• • -l.jA A '^A A

Such a tetrad depends on six parameters. Let SQ, TQ, UQt N O M ~

A A A= t \ o n\p ^o > ^no "be a given fixed set of orthonormal

vectors (the reference coordinate system)* Then,there is a uniqueA A A A A A A A

element g of S0(4) which maps SQ, TQ, UQ, N Q onto S.T,!!,!?:

(B.5)

-13-

Conversely, every g £ SO(4) defines a unique orthonormal set

S,T,U,N through (B.5). Thus the set (S,T,U,IT) is in one-to-one

correspondence with the group manifold of SO(4). (Similarly,A A A A

in the physical region, the set (S,T,U,]f) is in one-to-one

correspondence with the group manifold of SO(3,1).)We have proved that the scattering manifold -within the

(in the physical region).Mandelstam trianglejcan DB laenxified with the product of the

surface 3 + t + u = 4^ and the group manifold of 5.0(4,) (30 (3,1),) *

The remaining problem is to imbed the topological product

of the circles labelled by the angles i: (a torus) in the SO(4)

manifold. There are clearly an infinite number of distinct ways of

realizing such a map. Uone of them, however, Be^ms very satisfactory.

*- We shall finally mention a rather amusing though far-fetched

possibility. Some of the ideas sketched here will be explained

more completely in another article .

A particle in a relativistic theory is abstractly associated

with a basis of a representation of the Poinoare" group. In a

scattering problem with four particlesj there is a separate Poincare

group for each separate particle. Let the corresponding Poincare

generators be labelled as P^1 \ M^\j (i » 1,2,3,4). Let f be a

function of the four-monventa p^"' . If the particles are spinless^

the action of the generators on f can be taken to be

(S.6)

If f is the scattering amplitude P, then

(a) P^F - 0,

(b) M^ VP - 0 (B.7)

where

(a) p _

(b) M u v , Z . M ^ (*.8)1

-14-

.,., , i . . 1 . f ^ t l l ( j j , ^ . j S 1 . . »f- 4 * ; i! .•• . • •,, i . j

As a consequence, F ie a funotion of s and t alone.

The operators P ^ + p j 4 ) , M/^) + M^4) g e n e r a i ; e m

isomorphic to that of the Poincarfe group. This algebra has two

Casimir invariants. The first is the operator ( P ^ + P^4^) with

eigenvalue (pt3^ + p W ) and the second is the square of the

s-channel Pauli-Lubanski operator ^

(B.9)

•which acts only on part icles 3 and 4* When restr icted to a scattering

amplitude, X is invariant tinder 3—*1, 4 - * 2. Let Yu , Z^ "be

defined analogously in the t - and u-channels. I t is immaterial for

us here whether they act on the i n i t i a l or the final partioles in the

channels. Finally, define

\3.) X a Xu X , Y = T T , 2 B Z J , Z ,

It is easily checked that:

i) when restricted to a function of s and t alone

(that is, a function which satisfies (B,7)) , the primed operators

in (B.10) reduce to the corresponding unprimed operators in the text,

ii) their commutators with P^ , M^y vanish identically and

without any such restriction.

It follows from (ii) that the vector

r (1) (2) _ (3) _ (4) ,B

(the eigenvalue of the operator Pu_ of eq.. (B,8a)] can "be regarded

as a fixed non-zero vector in the problem (fixed,that is,under the

little group operations') which is allowed to vanish at the end

of the calculations. The Mandelstam invariants in such a case are

six in number and are given by

-15-

and similar equations for t, t1, u, u'.They are subject to the on©

constraint

s + s' + t + t' + u + u1 - P 2 + 8m2 (B.13)

The angles y. can now "be imbedded in this system if we setv

2i = A (B* + is'*) - ~9i (s + B ' ) * eitfl (B.H)

and likewise for "Z, 2, , The group SU(3) is allowed to act on

fc , -z , 2 ) in the standard way (cf. eg.. (II.4) and Ref.4) . When

£ goes to zero, c, goes to zero > s1, t', u' go respectively to

s,t,u and (B.13) reduces to the familiar oonstraint (B,2a).

The situation oan "be expressed in a picturesque form by

saying that our group SU(3) is the remaining ghost group of the

five-particle problem when one of the particles collapses to the

vacuum•

We conclude with a minor explanation. The three differential

operators in (B.lOa) are a priori defined in their corresponding

physical regions and the latter are mutually disjoint. In writing

(B.lOb) (and in many other remarks in this paper) the analytic

continuation of these operators to suitable domains is understood.

-16-

REFERENCES ATJD POOTHOTES

1) a) A.P. BALA.GHMDRM and J . MHTS, "A simultaneous par t i a l -wave

expansion i n t he Mandelstam v a r i a b l e s , c ross ing symmetry for p a r t i a l

waves", Syracuse Univers i ty P rep r in t (1968) and Phys. Rev. (in press )*

See a lso:

*0 A.P. BALACHANDRAtt and J . miYTS, to be publishedj

o) A.P. BALACHANDRAN, ¥ . J . ICEGGS and P . RAMOMD, "Eigenvectors

for the partial-wave 'crossing matrices1*, ICT?, Trieste , preprint

IC/68/44.

- * • - » —

2) The generators I , U, V are formally related to the X of

M. Oell-Mann, Phys. Rev. 12^., 1067 (1962) through

The generators B £ of eq..(ll.7) are related to I, XJ, V through

where the equations for U, v are obtained hy permuting the

indices 1,2,3 of B? ,

We emphasise that the internal symmetry group SU(3) and -the

SU(3) of this paper act on different physical variables and should

not be identified except by an isomorphism* The correspondence

introduced above is purely abstraot,

3) As we have mentioned in the Introduction, the discussion in

the body of the paper is largely oonfined to the interior of the

-17-

Mandelatam triangle where the relevant group is SU(3). We shall

here give an example of the modifications necessary for the

physical region £s £. 1* l - s ^ t ^ O j where the group becomes

SU(2,l). The constraint equation

s - (-t) - (-u) » 1

is identified with the invariant quadratio form

of SU(2,1) by the relations

—> -% —> -> . -*As regards the generators I, TJ, V , U refers to the SU(2; and I

~>and V to the SU(l,l) subgroups (the latter being locally isomorphic

to the 2 + 1 Lorenta group S0(2,l)) , Some information regarding

SU(2,l) can be found in L.C. Biedenharn, J. Nuyts and N. Straumann,

Annales de l'Institut Henri Poincare _, 13 (1965)* See also

Appendix B of Eef,1a .

4) M.A.B. BEG and H. RUEOG, J. Math. Phys. 6_, 677 (19^5). The

d-functiona in their eq.(3«24) are related to Jacobi polynomials*

See, for example, A,R, Edmonds, "Angular momentum in quantum

meohanios", Princeton University Press (1957)» V>3^*

5) See, in this connection, the discussion of tfeyl reflections

in SU(3) by A.J, Macfarlane, E.C.G.Sudarshan and C. Dullemond,

Huovo Cimento ^0, 845 (1963); N. 1TOKD¥DA and L.K. PANDIT, J, Math.

Phys. 6», 746 (1965) and K.J. LEZUO, J. Math. Phys. §,, 1163' (1967).

Further references may also be found there.

6) T.tf.B. KIBBLE, Phya. Rev. Ill, n 5 9 (i960).

7) See, for example, J. Strathdee, J.F. Boyce, R. Delbourgo

and Abdus Sal am, "Partial wave analysis'* (Part I),IOTP, Trieste,

10/67/9 (1967).-18-

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