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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-
Available from the Office of the Scientific Information and DocumentationOfficer. International Centre for Theoretical Physics. 34100 TRIESTE, Italy
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