IL N UOVI~ CI~[FNTO
VOL. XVI, N. 3
1o 5[uggio 1960
The Inelastic Scattering of Elementary Particles-II. G. FELDiVIAN(*), P. T. ~'IrATTH:E~VSand A. ~ALAM Imperial College - Lo~don
(rieevuto il 15 Febbraio 1960)
An approximate scheme for calculating the inelastic scattering of elementary paxtieles is proposed. The scheme fully incorporates the requirements of unitarity and partially includes causality. When applied to the simple problems of =-p and rc-r~ scattering it reproduces very simply some well known results.
Summary.
1.
-
Introduction.
I n a previous paper (~) it has been shown how the requirements of unitarity in m~my channel se~ttering ma,y be very simply incorporated in the inverse T-matrix. I n this paper we consider the ~malytie properties of this matrix, when only two p~rtiele channels a,re included. We conjecture t h a t in this ease the matri:c elements of T -~ for partieul~r energ'y and angular m o m e n t u m states satisfy dispersion relations, which depend on Born ~pproximation and discontinuities on <(left >~ and ~ right hand >>cuts. The inte~zral over the ri~th hand cut is completely determined b y unit:~rity. I f the ~pproximation is made of ueglectin~z the left haud cuts, an explicit solution is obtained, which depends only on ~ limited number of renormMized coupling constants. When applied to ~-p ~md ~-= scattering the m e t h o d very simply reproduces the essential features of the approximat, e solutions due to (?nEw and L o w (~) ~md CHFW and MANDEI~ST,~M (3). This a pprox-
(') On leave of ~bsenee from The Johns Hopkins University, Baltimore, Md. (1} l) T. ~IATTnEWSand A. SALAM:NUOVO Cimetdo, 13, 381 (1959), referred t(, as ([). (2) F. Low and G. F. CHEW: Phy.~. Ray., 1 0 1 , 1570 (1956). (a) G. F. C4tmv and ,~. MANI)ELSTA~: Phys. l;ec. (to be published).
•~ 5 ( )
G.
FELDMAN,
P.
T.
MATTIIE~,VS
and
A.
SALAM
imation has been developed with ~ view to expressing all the s-waw: d a t a on K - . V and i~-~N~ scattering in t e r m s of the four coupling constants gKA, gKZ, g~A, .%Z" This will be discussed in detail in a later paper.
2. - U n i t a r i t y .
I n (r) |he unit arity condition was given Imn-eovariantly for s-wave seattering only. W e now generalize the result. We define a T inatrix in t e r m s of the X - m a t r i x b y the relation (2.1)
N =: [ -'- i T .
F r o m the unitarity of ~_~, it follows i m m e d i a t e l y t h a t (2.2)
T-- T~= iTT t .
Multiplying left and right b y T -~ and (Tt) - ' respectively gives (2.3)
2 I m T -~ := - - 1
n
when tn')- is a complete set of states. ~7ow if P H is the e n e r g'y - m o m e n t u m oper a t o r a a d m a t r i x elements of T - ' are t a k e n in the space of this operator (in which T -~ is diagonal) (2.4)
:'p i"T-~ p > :=:~(2~)~ (5(p - - p ~)T~,-~
and
(2.5)
p In: = (2vp b ( p - - p ~ ) .
Thus, f r o m (2.'¢)
n
F o r further considerations we work in the bnryeentric system for which p
:=0~
po-E.
The st~tes !n} are defined as eigenstates of P , and of some conserved operators (.', which include the angular m o m e n t u m J, and some other operators ~, which, with P~, and C, m a k e up a complete c o m m u t i n g set. The
THE
INELASTIC
SCATTERING
OF
ELEMENTARY
PARTICLES
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551
m a t r i x T -1 is diagonal in the variables C, so we m a y define
(2.7) Then (2.8)
2 I m T~ ~ = (2z) 4 ~ ] ~ ( d ) > 5(p -- p~)(a(c') I,
where the s u m m a t i o n is over a set of states belonging to the eigenvalues C ~. Thus T~ ~ is a m a t r i x in the space of the operators ~, only. I f states with m o r e t h a n t w o particles are involved in the s u m m a t i o n the space ~ is still infinitedimensional and the relation between T [ ~ and T~ is a a integral equation of the first kind. l=lowever if the s u m m a t i o n is restricted to two particle states~ either exactly or as a n a p p r o x i m a t i o n , the sp~ee ~ is finite-dimensional and simply specifies the nature of the particles in the channel. The m a t r i x T is related to T -~ b y a straightforward inversion of the m a t r i x in channel space. To obtain T71 f r o m T ; ~, the integration over the k i n e m a t i c factors in a n y t)artieular channel is, d4k~ d4k2
(2.9)
k
• (2~)4(~ (k~ ~ k 2 - - E ) ( k ~ , k~, s p i n t J ) where:
E
~]~k~ O(E - - E ~ ) , (2~)~E
is the threshold energy for t h e channel, for 2 boson s t a t e s , m
2/ ~
mt~l, 2
for 1 boson, 1 fermion s t a t e s , for 2 fermion s~ates.
k s is the centre of mass m o m e n t u m and m is the fermion mass. I f we r e n a m e Tc b y Tj, we h a v e finally (2.10)
I m T ; 1 := ~ i a >
~?~k~
3. - D i s p e r s i o n r e l a t i o n s lor T -1.
l~or two-particle channels the M a n d e l s t a m conjecture leads to the conclusion (3) t h a t Tj(E) are themselves b o u n d a r y values of analytic functions so
552
and
G. F:ELDMAN, P. T. MATTIIEVVS
A. SALAM
t h a t dispersion relations of the following form Call be written (3.1)
Re T/,s')
B '~"
1 j i m Tj(~%'r)
%,% where S - - E 2 and Bj(S) is Born approximation. Consider first the case where there is only one channel, so t h a t
(3.2)
T~(~) = (T~(~))-~,
and suppose t h a t at some point '~s (3.3)
Bj(SB) = ~ .
We then conjecture t h a t T -~ is also an analytic function with singularities i) along the same cuts, UR, C~ as T~; ii) corresponding to zeros of T~, (which we ignore), and iii) d~le to poles and branch points arising from B(S). We suppose further t h a t T+(S) is equal to B+(S) whenever the latter is infinite. This leads to the equation (3.4)
T~,~Bj(S)
I + N~,q, /
I m {TjXB/b'r)}
. (N"-- N +- iv) (S:-- N,)
d,S".
CL,OR
Note t h a t Bj(S) is real in the physical region, so t h a t this equation ensures t h a t the imaginary part of T j ~ is just as required by unitarity. F u r t h e r the integral over the right hand cat is c o m p l e t d y determined by (2.10) since I m T-~(S) involves a factor O(E--Eth ) where Eth is the threshold energy, there is never any (( nnphysical ~>region over the right h a n d cut. If Born approximation is zero or a constant we m a y express the amplitude in terms of its v~hle at some unphysieal point, Eo, where the amplitude is purely real: T0%) = ~. (rt m a y be possible to interpret i as a renormalized coupling constant associated with the process). Then the dispersion relation for T -1 becomes (3.5)
~T;~(N) :-:l + '%'-- S o /
Im {XY;~(~r)] (Nr - ~S' i~) (~q~-- ~o) d.~". -
-
'l'II]~
INELASTI(!
SCATTERING
~)1" E L E M E N T A R Y
PARTICI,ES
- I1
553
~f Born approximation does not become infinite at any finite point we m a y 7 assume t h a t a no-subtra(¢ion relation holds for 2 j and so T., -+ I;~ as £~ --->oo thus we write "
(3.6)
jr~
l
*r
1 i I'm t2a /~.,('S )}
Ti~(N) B , ( S ) = 1 -? n ]
X'
,'~ -i~"
d£'~'
and proceed as before. We now proceed to the generalization of the m e t h o d to mnlti-charmel systems. We require the following lemma. Lemm~.
If an element B,,(N) of a matrix is infinite at No..,
(3.7)
Ba.(S,a.) = c o ,
then for ,%,, where either I
lit =: ]~ 2~
i~
...~ tt
or m
Proo].
(3.s)
=
k,
I
" 1, 2, ...~ n
.
Let ; ~ , be the minor of B,~.. Then
Bik'
= B n . ~ i , + B..,.~.~ _L. ... + I,,,.~,; ~
"
~fow B a , B,~, ..., B , , are not contained in ,°A,, so that if any of these n u m b e r s is infinite at a particular energy, B ~ 1 is zero at this energy. A similar argument applies if the determinant is expanded along the k-th column. Using" this Lemma, the generalization of (3.4) is immediate. ])efine n
~
(3.9)
' ' (,S' - - ~s,,.)(,S " , - ~Sse)
J:=l
s=l
then
(3.10)
[,Tj ~ I ¢{N 1] ,, -- b,,-L
F ~S,I" Im{~"?B,(S~)},, ~*' '1 (S'
N--it')F,x(N')
(l,%
Cb,C R
ff the approxim~tion is made of neglecting tile integral over the left hand out, whieh should be reasonable for energies E which are close to threshold, the above is a completely explicit expression in terms of (2._10) and the appropri{~te Born aoproximation expression in terms of re-normalized coupling constants.
,~54
G. FELDMAN, P. T. MATi'IIEWS ~ t l d A. SALAM
4.-., A p p l i c a t i o n s .
4"1. "z,-3~ scattering. - Borer ~ p p r o × i m a t i o n for tim /" = :~ J = ~ a m p l i t u d e is
(4.~)
B(E)
_fl~ ( ~ +
4~
2~,,, -::~+ ,2k2 _ 4k ~ ~ (2e:,, --/.t") l o g 2~-.(,, /.~2 _ 2k 2 '
N) (E -- N!
8k 4 E
where N a n d ff are the nucleon a n d pion m:~sses, e a n d t,) the corresponding energ'ies and k the centre of m a s s m o m e n t u m . I n the low e n e r g y limit this redll(;es
to
4k 2 B(o,) :=/2 3 . , '
(4.2)
12
g~ 1 4n 4N:'"
Also ~ =.: N ~ + if2 + 2No>. B y (3.4) "rod (2.10), neglecting the left h~nd cut, 4]2 k'3 ctg b
(4.3)
....... 3
1
4]~ [+ U a d ' ' ' ~ ~'~!., ' ~(~,,' o,) '
~,~
which is the ( ' h e w - L o w (2) expression (neglecting crossing) for the resonance, t h e position of which is d e t e r m i n e d b y the cut-off which h~s to be i n t r o d u c e d in the final integr~fl. 4"2. =-= Scattering. - For A'-wave pion-pion scattering, r e n o r m a l i z e so t h a t T,t~_o) ,4, = 2a(4z0 ~
I n t h e not,~t,ion i n t r o d u c e above. S : : 4(o 2 ,
'~'o = 4~,;~
thus, b y (3.5) ~md (2.10) with neglec.t of left, h a n d cuts k ctg 5 O)
1
--~(I
0) 2 - - 0)2 f
=
~]~t do)/
ji+,i '2
Sntroduoin~ k2 =
a n d puttino"
~t
~,,~)(.,'~-+,,~)
T t I E I N E L A S T I C ,~('AT'I'ERIK(I O]" E L E M I ' i N T A R Y
I'ARq'I(TI~E~ - II
555
this reduces to
(,)° -,-]- 1
l'
ct, gzr5 ::--a d
--
~'o [ '
,pr½(l~/
J
,,)'
which is the explicit s<)lution derived I)y ('[[EW and MANDELSTAM (a). We have thus shown t h a t the proposed a p p r o x i m a t i o n reproduces v e r y simply the essential features of previous solutions of the pion-nucleon and pionpion interactions. The m e t h o d is not cf much significance for these problems since it is not at all clear how the a p p r o x i m a t i o n could be improved upon. [ t has been developed primarily with a view to application to the K-nucleon ~md K-nucleon system for which the requirements of u a i t a r i t y arc already so complicated, that, it seems improb~ble that, ~ny p r o g r a m m e , even if it sels out to be more sophisticated, will be able, in fact, to include more t h a n has been included here. An analysis of this interaction in terms for our coupling constants gKA, g,~x, ga=, g"~ will be the subject of a separate publication.
One of us (G.F.) would like to t h a n k the others and the l).8.I'.R, for the hospitality of Imperial College.
R[AsSI'NT~)
(*)
Si propone uno schema approssimatt~ per calcolare lo scattering anelastico delle par|icelle elementari. Lo schema ineorpora pienamente le esigenze di unitarieth e eomprende in part e le eausalith. Applicalo a semplici problemi dello scattering =-p e T:-= riproduee in maniera, semplice Mcuni ben noi.i risult.fli.
{*) T r a d u z i o n e
a e l t r a dcll¢t R e d a z i ( m r .