IL 7NUOV0 CIMENT0 VOL. XXXVIII , N. 1 lo Luglio 1965

S-Matrix Elements in the Two-Particle Unitarity Approximation for a World of One Space Dimension.

]~. DELBOURGO

International Centre /or Theoretical Physics - Trieste

(ricevuto 1'8 Gennaio 1965)

S u m m a r y . - - For a world having a single spatial dimension we apply the unitarity equations in the two-particle approximation to deduce the crossing-symmetric scattering amplitude and thereby determine the form factor and propagator of the theory; comparison of the full amplitude with the (( Born approximation )) reveals their quite different asymptotic characters. The solution is ~lso used to discuss the problem of overlapping final-state interactions and it is proved that the enhancement is multi- plicative in form.

1. - I n t r o d u c t i o n .

The elimination of angular variables which results f rom narrowing space-

t ime to two 4imensions is a major simplification when one comes to analyse the equations of mot ion in fiel4 theories, an4 as such a mo4el for electrodynamics

is completely soluble (1) it has arouse4 much interest of late, especially as i t elucidates the connection between gauge properties an4 massless particles.

Al though a worl4 of this type confers decide4 advantages such as superrenor-

malizability, re4uction in the number of relativistic invariants nee4e4 to de- scribe any process, 4isappearance of par t ia l waves, etc., a complete solution

is still lacking for a model which possesses no gauge invariance. Nevertheless, LA~D~E~ (2) has obtained an explicit expression for the elastic scattering am-

(1) j . SCn~WINGER: Phys. Rev., i28, 2425 (1962); L. S. BROWN: 2(uovo Cimento, 29, 617 (1963). A difficulty encountered with this theory has recently been pointed out by B. ZU•INO: Phys. Lett., 10, 224 (1964).

(2) j. C. L~RDNER: NUOVO Cimento, 28, 1375 (1963).

~--~r :EL:E1VIENTS IN THE TWO-PARTICLE UIqlTARITY APPROXIiVIATIOI~" ~TC. 411

pli tude by adopt ing the uni t~r i ty equations gn4 retaining in termedia te states of up to two particles. Hopeful ly the solution is a good approximat ion to the real state of things because of the strong convergence of phase-space integrals.

Using this solution as a convenient s tar t ing point we wish to show how one may set out a general program for calculation of S-mat r ix elements up to any desired accuracy (at least in principle) along the lines presented in a pre- vious paper (~). This me thod is explained in Sect. 2; however the immedia te aims of this paper are more modest bu t none the less interesting. Thus, af ter discussing in detail a speci~A class of Lardner solutions (~) (fixed C.D.D. (4) properties) in Sect. 3, the implications on the behavior of the ve r t ex funct ion and propagator are presented in Sect. 4. In this way we are enabled to make the comparison between the <~ Born ampli tude ~> and the proper un i t a ry ampli- tude. F i n a l l y Sect. 5 is devoted to the problem of overlapping final-state in teract ions in a decay process and the fac t t ha t the solution corresponds to mult ipl icat ive enhancement , as in the s t a t i c model of PEIE~LS and TARSKI (5)~ cannot bu t exclude addi t ive enhancement in the three-dimensional counter- pa r t problem.

2. - Interact ion ca lculat ion of S -matr ix e lements .

Although eminent ly suitable for describing processes in a space-time of two dimensions because of the highly convergent na ture of the F e y n m a n integrals tha t enter, we will not envisage a pe r tu rba t ion theory because we 40 not know a priori what is the fundamenta l in terac t ion Lagrangian; i t might conceivably consist of a s tr ing of terms of the type ~n~ ~. Ins tead we propose to employ the equations given to us by the un i t a r i ty of the S-mat r ix and to derive correspondence with renormalized per tu rba t ion theory by imposing on our so- lutions the appropr ia te boundary conditions. F i r s t ly we shall cut down the infinite number of equations which un i ta r i ty p rov ides - -o r for t ha t m a t t e r any other complete theory - - to just a set of N--3 coupledintegrat equations in the following manner~ by what we call the (( N-point approximat ion >>:

In terms of the T [ = - - i ( S - - 1 ) ] - n ~ t r i x , un i ta r i ty m ay be expressed in the figurative form

T ~ (1) 2 I m = 5

Taking' into account the effects of all the channels (to main ta in crossing sym-

(s) A. SALA~ and R. D]~LBOURCO: Phys. Rev., 135, B 1398 (1964). (4) L. CASTILLEJO, 1~. H. DALITZ and F. J. Dyso~: Phys. t~ev., 101, 453 (1956). (5) •. F. PEIERLS and J. TARSXI: Phys. Rev., 129, 981 (1963).

41~ R. DELBOURGO

m e r r y for instance) we set down the equat ions for the 4-to-~Y-point mass-shell functions, re ta ining no more in te rmedia te states t h a n will give rise to n-poin t funct ions wi th n ) N . ~ s ment ioned above, we are lef t wi th N - - 3 equa- t ions which can be solved in principle with bounda ry condit ions t h a t serve to

fix among other things the posi t ions and residues of poles, the n u m b e r and posit ions of C.D.D. (4) zeros, etc.

The connec te r n-point functions, n ~ - N ~ - l , ..., 2N, m a y then be di rect ly eva lua ted f rom the lower n-point funct ions by comput ing the discont inui t ies across the i r singulari t ies v ia eq. (1). This done, we will know all the first 22/- point funct ions to a first approx ima t ion . These can now be re in t roduced on the r igh t -hand side of eq. (1) in a <~ 2N-poin t a p p r o x i m a t i o n )> to compute the first 4N-poin t functions. Similar i tera t ions will yield the full S-matr ix . Ob- viously the whole p rog ram is meaningfu l only if i t converges, a t ]east for a l imi ted set of k inemat ica l regions. Pe r tu rba t i on theory and the <~ dominance of the neares t singularity)> phi losophy suggests t h a t i t will in the weak-coupling" l imi t and/or a t low energies.

Of course, prac t ica l calculat ion is quite another proposi t ion and i t seems a lmost impossible to car ry out th is p rog ram except b y s t a r t ing out on the 4-point approx imat ion , the ve ry simplest possible; or even so t h a t we will be able to de te rmine much more t han the 5-point fu~ct io~ b y this means . Our a ims in this paper are therefore modest bu t not devoid of in teres t as i t hap- pens. We shall ca r ry out the ini t ia l step of calculat ing the 4-point funct ion following LARD~NEIr (2), and in the same spiri t will evaluate the ve r t ex funct ion and p ropaga to r for the purpose of compar ing the Born t e r m (one-particle exchange in the crossed channel) wi th the sca t te r ing ampl i tude . As a fu r the r appl ica t ion we shall derive the effect of f inal-state in terac t ions on the decay spec t rum of three final pgrticles.

A na tu ra l quest ion comes to mind. <( W h a t is the comleet ion or even the bear ing of these results on our actual space- t ime world? ~> I t is difficult to give a precise ~nswer, bu t the resul ts mus t in a sense correspond to the S-wave dominan t solutions of the physical world because the s ingular i ty s t ruc ture is much the same. However this ana logy mus t not be pushed ve ry far . Thus one obvious difference is t h a t in two dimensions the scat ter ing ampl i tude be- eome~ absorp t ive and vanishes a t threshold as a direct consequence of uni ta- r i ty, whereas in four dimensions i t becomes real and constant .

3. - The scat ter ing ampl i tude .

We will select the easiest possible model by cons t ruc t ing a world of neut ra l (scalar) par t ic les hav ing a common mass which we set a t un i ty for convenience. The elastic scat ter ing ampl i tude is s imply a funct ion of u single var iable by

S - ) $ s ELEMENTS IN THE TWO-PARTICLE UN!TARITY A~'PROX]IVIs ETC. 4 1 3

the cons t ra in t t ha t all m o m e n t a are collinear. I n t e rms of the ~r variables, the s t a t emen t t h a t there is only forward or backward sca t te r ing in any par t icu la r channel, reads

(2) s t u = O , s + t §

I f s is the energy~-variable we m a y select t = 0 and u = 4 - - s as the one- p a r a m e t e r descr ipt ion of the sca t te r ing ampl i tude M. I t has been shown b y LAlCDNEI~ (2) tha t , subject to the - usual assumpt ions abou t rea l i ty and anal- y t ic i ty , M is bes t expressed as a funct ion of the var iable

( 3 ) z = - - su = s(s - - 4 ) = u ( u - - 4 ) .

I n the app rox im a t i on of re ta ining two-par t ic le in te rmedia te s ta tes the general solution is

(4) - - M - I ( z ) ---- ~ ( - - z ) - ~ + ~ ( z ) , ~ ( z ) = _ ~ * ( z * ) ,

where R(z) is n ra t ional po lynomia l reflecting the arbi t rar iness in M to the ex ten t t h a t h igh-energy propert ies , C.D.D. zeros and poles of M have not ye t been specified. I t will be sufficiently general for our purposes to focus ~tten- t ion on a class of R funct ions t h a t can s imula te m a y different behaviors

( 5 ) R!z ) -= a z + b + c(z - - Zo) -1 .

Here a, b, c, z0 are real constants , Zo< 0, and in all eases we have

l im Iz-1M(z) ] ---- 0 .

We sham now examine the solution in detai l for var ious possibili t ies.

3"1. s one-particle pole. c : O. - To ensure t h a t M-~(z) does no t van ish for negat ive z (ghost state) we mus t t ake a<~O, b>~3a/4:

- - ICe M - l ( z ) = � 8 8 z ) ( - - z) -~ + az + b ,

(6) I - - I m M-~(z) : ~O(z)z -~ , ! [ - t g v ( z ) = _ ~ O ( z ) z - ~ ( a z + b) -~ ,

U represent ing the phase of M. The ex t reme case a = b : 0 when U remains fixed a t z]2 is uninteres t ing and will not be considered. The behav iour of the ampli - tude in the physical region (z>~0) is mos t easily demons t r a t ed b y p lo t t ing i ts

414 R. DELBOURGO

r ea l a n d i m a g i n a r y p a r t s in t h e c o m p l e x p l a n e w i t h z as p a r a m e t e r . Th is i s

shown i n F ig . l a a n 4 lb . O u r p h a s e c o n v e n t i o n is to fix ~ = z /2 a t th resho ld~

so t h a t t h e p r e d o m i n a n c e of I m M -~ ove r R e M -~ j u s t a b o v e t h r e s h o l d g u a r -

Fig.

J a)

1. - a) Complex M(z) plane, a < O. B) b = O ;

b<0 I

b)

Arrows denote increasing z. A) b > 0; C) b < 0. b) Complex M(z) plane, a = 0 .

a n t e e s as a g e n e r a l f e a t u r e ( t rue a lso in p o t e n t i a l t h e o r y ) t h a t t h e a m p l i t u d e

becomes a b s o r p t i v e t h e r e a n d b e h a v e s l i n e a r l y in m o m e n t u m , i.e., u p to

m u l t i p l i c a t i v e c o n s t a n t w h i c h d e p e n d s on t h e n o r m a l i z a t i o n c o n v e n t i o n f o r

s t a t e s we m a y w r i t e

(7) M(k) = k exp [i~] s in V, k = �89 - - 4) �89

w i t h ~ ( k ) - - > z / 2 as k - ~ 0 . R e p l a c i n g z b y 16k ~ we r ecogn ize in ou r s o l u t i o n

--iM-~(k) : b + 16ak2+ i/16k

a n (( e f f ec t i ve - r ange e x p a n s i o n )). W h e n a a n 4 b a r e b o t h n e g a t i v e u r e s o n a n c e

wi l l d e v e l o p a n d i f ]bl<<]a ] i t occurs a t k ~( - -b /16a ) �89

3"2. One-particle pole. c = 0. The i n c l u s i o n of a s i n g l e - p a r t i c l e po l e in t h e

d i r e c t a n d c rossed channe l s a t s = l a n d u = l , r e s p e c t i v e l y , w i t h r e s i d u e g~

m e a n s t h a t we m u s t i m p o s e t h e c o n d i t i o n s

(9)

(8) ~ - 1 ( _ 3)~= o , _ ~ - r ( _ 3) = :t/2g~.

This u n i q u e l y fixes a a n d b, a n 4

--:M-l(z) = �89 z)-~ + ~(z -f- 3) - - ~ ( 3 ) ~ ,

1 1

2g 2 24(3)F

. ' . t g ~(z) ---- - - O(z)/4z~{~(z + 3) - - ~(3)�89

Fig. 2. - Complex M(z} plane: A) 4 ~ / 3 < g2< 12%/3;

~-~r ] ~ L E ~ ] ~ T S IN THE TWO-t~AI:tTICLE U~ITARITY 2LPPICOXI~2kTION ~TC. 4 1 5

We sh~ll no t envisage the possibi l i ty t h a t a < 0, i.e., f > 1 2 ( 3 ) �89 because an- o ther pole appears on the nega t ive z axis con t ra ry to the original assumpt ion . Figure 2 summarizes the results. Observe t h a t a resonance occurs when 4(3)+< g~< 12(3) +.

3"3. One-part ic le pole. a -~ O. - Equa t ions (5) and (8) yield

so t h a t

~c = - - (b ~- 1/4(3)�89 2 , ~(Zo + 3) = - - (b + U4(3) +)

( lo) - M-~(~) = ~( - - ~)-+ + b - - (b + 1/4(3)+)/{~(z + 3) + b + ~/4(3)+}.

Since M-l(oo) = - - b -1, - - b -~ is d i rec t ly p ropor t iona l to the coupling cons tan t of a ~o 4 in te rac t ion Lagrangian, i t mus t be noted. The condi t ion t h a t there is no more t h a n the single pole a t z = - - 3 res t r ic t s the coupling constants to the range b~>0, g~<12(3) �89 We m a s t dis t inguish careful ly be tween the two possibil i t ies b = 0, bve 0, because the a s y m p t o t i c behav iour differs in an es- sent ia l way. Thus as z - + c % for b = O , t g ~ ( z ) ~ z +, while for b # 0 ,

A

a) b)

Fig. 3. - Complex M(z) plane: a) b=0; b) b > 0 : A) f < g ~ ; B ) f > g ~ .

tg~(z) ~ z -�89 These si tuat ions are depicted in Fig. 3a and 3b. When b > 0, a resonance will (or will not) exis t according as to whether ~ < (or > ) {1+b-1/4(3)�89 �89 i.e., g~< (or > ) some g~.

The preceding" is more by way of i l lus t ra t ing how a par t i cu la r class of so- lutions sat isfying two-par t ic le un i t a r i ty can be constructed. However , in t h a t t ---- 0 has been chosen beforehand, i t is not obviously crossing symmet r ic . This slight deficiency is easily rectified if we re in te rp re t z as the cross ing-symmetr ic var iable

(ii) z ---- - - (su A- ut q- ts) .

A more convenient way of mak ing the results crossing symmet r i c comes b y

416 R. DELBOURGO

eonsi4ering the phase represen ta t ion (9

co

(12) M(z) ~: exp 3 x - - z J 0

with U(z) comple te ly known al rea4y. Recognizing t h a t wi th z = - - s u an4 t----0

co

ax=f ( _) J x - - z ~(x) dx 1 + 1 X - - 8 X - - U

o 4

where ~(s)=~(z(s)), we can finally s ta te

1 1 1 (13) M ( s , t , u ) ocexp ~ ~(x) dx ~ +~+

4

up to some ra t iona l polynomial . For cases 3"2 an4 3"3 where the re exists a s ingle-part icle pole this po lynomia l m u s t contain the fac tor g2 [ ( s - -1 ) -~+ + ( u - - 1 ) - ~ + ( t - - 1 ) - ~ ] . I n any case i t will Mways conta in the fac tor ( s - - 4 ) . �9 (t - - 4 ) ( u - - 4) to ensure the correct threshol4 behav iou r in each channel.

4 . - The form factor and propagator .

A r m e 4 wi th the knowle4ge of the elastic sca t te r ing ampl i tu4e i t becomes a s imple m a t t e r to evalua te the fo rm fac tor of the theory, a lways wi th in the two-par t ic le approx imat ion . We are concerne4 wi th the en t i t y

(1~) <0 I~(o)I 2> = A t ,

where P i s the proper ve r t ex funct ion. Wat son ' s f inal-state in te rac t ion theorem, the expression of un i ta r i ty , tells us t h a t

Arg[A (s) F(s)] = Arg M(s) = ~(s),

indee4 exac t ly for s < 9. The normal iza t ion con4i t ion P ( 1 ) : g allows us to wri te the fun4amen ta l solution

(15) A ( s ) F ( s ) - -

co

g(4-~) [(~-1) ( ~/~)a~ ] 3 ( s - - l ) exp [ ~ J (x-- z) (x-- s)

4

(8) ]~. 0~Nfis: Nuovo Cimento, 8, 316 (1958); N. MUSKHELISEVILI: Singular Integral Equations (Amsterdam, 1946).

~-M'&TRIX :EL:E:3~ENTS :iN TI~IE TWO-PART]:CL~] UN:[TARITu A.PPROX~MAT~ON ]~T(~. 417

This sub t rac t ion ut s ~ 1 has ensured t h a t the dispersion integral will a lways converge. Note Mso tha t , because $-----~/2 a t threshold,

h(s) F ( s ) ~ i ( s - - 4) -~

in the expected manner near s ~ 4. The two-par t ic le a p p r o x i m a t i o n a t the same t ime allows us to infer the

behav iour of the p ropaga to r A since

(16) I m A (8) = - ~ I A ( s ) /~ (~ ) l ~ (8(8 - - ~ ) } -~

a n 6

1 +! (ImAIx)dx (17) A(s) - - s - - 1 z j x - - s

Equat ions (15), (16) an4 (17) then provide us wi th all the in fo rmat ion we can ex t rac t f rom the scat ter ing ampl i tude, and the compar ison be tween Born ampl i tude A ( u ) P ( u ) and proper un i t a ry ampl i tude can then be made. This is of especial in teres t a t h igh energies. Grea t caut ion mus t be appl ied before t rans la t ing our conclusions to those of physical sca t ter ing ampli tudes , because in four dimensions we have only one cons t ra in t ins tead of two on our ~V~an- de ls tam variables, and i t is no t inconceivable for the Born t e r m to a p p r o x i m a t e to the full sca t te r ing ampl i tude in the region of large energy ancl m o m e n t u m transfer .

Le t us tu rn to the specific models in Sect. 3 to see the repercussions.

1) I n this case we ~re asser t ing t h a t there is no th ree-poin t funct ion.

2) A glance a t Fig. 2 shows t h a t $ approaches z a t infinity.

(18) .'. A(s) F(s ) ~ g / s , I m A ( s ) ~ - - g 2 / s S ,

A(s) ~ 1/s , a n d l"(s) ~ g .

F r o m the above we conclude t h a t bo th Z~ 1 and Z : l ~ - l + O ( g ~) for the renor- real izat ion constants . The Born t e r m

F ( u ) A (u) F ( u ) ~ - - g2/s

is to be cont ras ted wi th the complete ampl i tude

M(s) ~ - - 1/o~s 2 .

2 7 - I1 Nuovo Gimento.

418 ~. D~L~OU~GO

3) Figures 3a and 3b depict the two cases to be dis t inguished

A) When b = O, ~ -+z~/2~ as s -->- c~,

(19) [ .'. A(s) F(s) ~ ig/s �89 ,

[ .'. A(s) ~ 1/s and

I m A (s) ~ g2/s2,

F(s) ~ i g s �89

Now Z ~ = co but once again Z ~ - - l + O ( g 2 ) , and the Born te rm

F(u) A (u) F(u) ~ g~

behaves quite differently f rom the un i t a ry ampl i tude

M(s) ~ is -1

in the a s y m p t o t i c region.

B) When b V=0, $(s)-+xr as s - . ~ . Consequently all our conclu- sions for the p ropaga to r and ve r t ex funct ion ~ollow the same p a t t e r n ~s in (12). The Born ampl i tude, being o f order s -~ behaves quite differently f rom M(s) which is of order 1. All these examples seem to indicate t h a t the a s y m p t o t i c characters of Born umpl i tude and scat ter ing ampl i tude are always different, for two dimensions a t least.

5. - O v e r l a p p i n g - f i n a l - s t a t e i n t e r a c t i o n s .

An i m p o r t a n t p rob lem t h a t has no as ye t been solved exact ly , is the in- fluence of finM-state in teract ions be tween final pairs of par t ic les on the spec- t r u m of a par t ic le which decays into more t h a n two others (7). Many of the difficulties s tem f rom the angular in tegra t ions due to var ious ways of coupling

angular momen ta . Therefore by s tudying the coun te rpa r t s i tuat ion in two d imendons we will be dispensing wi th these par t icu la r t roubles and be be t t e r able to ar r ive a t the essential fo rm of the resul t even if the final pa i r forces happen to be strong. The p rob lem which we s tudy then is the following: g iven

mass m which decays into three par t ic les of uni t mass, wha t is the fo rm of the ampl i tude if we neglect all bu t elastic scat ter ing of final pairs?

F i r s t ly we will suppose t h a t m ~ 3 and solve for the ampl i tude T(s, t, u, m2). Then we will assume t h a t the case m~>3 can be reached by sui table anMytic

(7) A great deal of literatm'e has appeared on this particular topic. Most of the references are to be found in the following articles: I. J. R. AITCHISO~: Phys. gev., 133, B 1239 (1964); C. LOV~LAC]~: Phys. Rev., 135, Bi225 (1964).

~-~/KTRIX ELE~IENTS IN THE TWO-P&RTICLE U~ITA.RITY /KPPI~OXIRLd.TIO~ ]~TC. 419

continuat ion in m 2. I n any event we have our two constraints

(20) s t u = (m2--1) ~, s q - t - { - u = m 2 q - 3

which allow us to express t and u, say, in terms of s an4 m2:

(21)

with

(22)

2u(s, m ~) = m ~ q- a - - s q- Q(s , m ~ ) l s ,

2t (s, m 2) = m ~ q- a - - s - - Q(s , m 2 ) / s

O2(s, m ~) = s [ s - - 4 ] [ s - - ( m - - 1 ) ~ ] [ s - - (mq-1)2] .

i) m < 3 .

Q(s) is real for ( m - - 1 ) 2 < s < 4 and consequently there is e~ closed curve

in the s - - t - - u t r iangular plot on which s, t and u arc simultaneously real. Nevertheless this is an ~mphysieal region. Let M represent the elastic-scattering

ampli tude for two of our final particles (s), which we have already determined in the elastic approximation. Uni ta r i ty in the s-channel takes the form

] ' I n 5 ( 8 , gYb 2) = 1 ~ $ ( 8 , m 2) _/]/i(8)0(8 - - 4 ) / [ 8 ( 8 - - 4 ) ] ~ .

We then conclude tha t Y and M have the same phase ~ for s > 4 and can write

the usual phase representation. Extending to all channels, it is easy to verify tha t the correct crossing-symmetric solution is

r

s x - - - - - - t + ~ " 4

Then let us examine the possibility of various denominators in (23) to vanish;

take s > 4, say. F rom eqs. (20) we conclude tha t t an4 u are complex conjugate and if real will not exceed 4; hence only one denominator ca~ vanish at a t ime

an4 the usual + i s prescription can be used to specify the physical sheet.

if) m > 3 .

Complications set in when m s increases an4 exceeds 9 owing to the opening

of the 4ecay channel: this can be seen f rom eq. (22) in the way t h a t the branch points of Q(s , m ~) a t 4 and ( m - - l ) ~ cross one another, and will be reflected

as a branch point i n m ~ o f T ( m ~, s) . The curve bounded by

4 < s , t, u < ( m - - l ) ~

(s) All questions concerning complex singtflarities will be avoided if we consider the class of solutions M which do not have one-particle poles, e.g., in Sect. 3"1.

420 R. DELBOURGO

will now f o r m p a r t of t he phys ica l region. To see how an ana ly t i c a l cont i - n u a t i o n of (23) is to be achieved, viz., how we will define the phys ica l sheet,

we fol low s t a n d a r d p rac t i ce b y e x a m i n i n g the dependence of T on m 2 and one of the subenergies~ says s. Outs ide the decay region, i.e., for s < 4 the re

De- is no diff iculty in the i n t e r p r e t a t i o n of (23), so we shall fix s, rev~l ~ 4.

f ining

(2~) I(mo, + (x--u)-lJ =

f [2~ + s - - m ~ - - 3] ~(x) = d X x 2 + x ( s _ m : 3 ) • '

we see t h a t the d e n o m i n a t o r van i shes a t ra ~ m + ( , x ) , where

2m~(s, x) = sx + 2 • { s x ( s - - 4 ) ( x - - 4)}~.

F o r s = 4 the re is t he re fo re a cu t in m 2 e x t e n d i n g f r o m 2 s + 1 a long the real

axis and the d i s c o n t i n u i t y across i t is g iven b y

1 I m Imp(s) =fdxe(x) s ( 2 z + s - 3 - m 2) 6[(m 2 - - m~_) ( m ~ - - m ~ ) ] = 2~

= f dx~(x)s(2x + s - - 3 - - m 2) 6{Ix - - t(s, m2)][x - - u (s, m2)]}= ss(t - - u)[~(t) - - ~(u)] .

F ina l ly , if t he phys ica l sheet is defined as the l imi t on the cu t f r o m above

(s--l) 2

f s~(t-- u) {~[t(s, t,~)]-- ~ [ u ( s , t,~)]} (25) I(m2~ s) ~ d# ~ # 5 _ m 2 _ i s

2 s + l

We h a v e cu t off t he in teg ra l a t ( s - - 1 ) ~ because the 6 f u n c t i o n occu r r ing in

the a b s o r p t i v e p a r t will van i sh w h e n t or u exceeds (/~--1) 2. Thus for m 2 > 9

[lf lxldx (26) T(m ~ , s ) = P ( m 2,s) exp ~ x - - s - - i s +

4

(s--l) ~

2 s + l

~ o t i c e once aga in tha t , w h e n s jus t exceeds 4, t he d ispers ion in teg ra l in m 2

becomes negligible and the in teg ra l over the s -channel cu t p roduces the fa-

mi l ia r t h r e sho ld f ac to r ik because ~(4)~--~/2 and P ( m ~, s) conta ins the f ac to r (8--4).

and s~>4 our so lu t ion is

I~-]~A.TRIX :EL:E~ENTS IN TI tE TWO-P&RTICLE URTITARITY APPROXIMATION ETC. 4 2 1

W e wi l l n o t p u r s u e t h e consequences of eqs. (23) a n d (26) a n y f u r t h e r here ,

e x c e p t to a d 4 t h a t , s ince our m o 4 e l is s u g g e s t i v e l y s i m i l a r in i t s a n a l y t i c a l

p r o p e r t i e s to t h e S - w a v e d o m i n a n t so lu t ions of 4 - 4 i m e n s i o n a l s c a t t e r i n g a m p l i -

tudes , t h e e n h a n c e m e n t of t h e d e c a y s p e c t r u m clue to ~ v e r l a p p i n g f i na l - s t a t e

i n t e r a c t i o n s cannot be o] the additive type (9). Th i s is th~ m a i n c onc lu s ion to

be d r a w n f r o m our work , b e c a u s e a l t h o u g h a P e i e r l s - T a r s k i f o r m of s o l u t i o n

is f avou re4 , t h e c o u p l i n g of a n g u l a r m o m e n t a in t h e 9 a r i o u s f ina l channe l s

c a n n o t b u t m u d d l e t h e i n t e r p r e t a t i o n in t h e r ea l p h y s i c a l wor ld .

I t h a n k Dr . J . 1~. GILLESPIE fo r i n f o r m a t i v e d i scuss ions ~ t t h e W i s c o n s i n

I n s t i t u t e for T h e o r e t i c a l P h y s i c s w h e n t h e b u l k of t h i s w o r k was c a r r i e d ou t .

I w o u l d a lso l i ke to exp re s s m y t h a n k s to P ro f . A. SALA~ for t h e h o s p i t a l i t y

e x t e n d e d to m e a t t h e I n t e r n a t i o n a l Cen t re .

(9) Of course when the final-state interactions are small there is nothing much between mult ipl icat ive and addi t ive enhancements. No~e tha t we are defining enhance- meat to be addit ive in ]orm if T(s, t . . . . ) = A + F ( s ) + E ( t ) + . . . and mult ipl icat ive in form if T(s, t . . . . )=AF(s)F(t)... irrespective of the constraints or the kinematic variables s, t . . . . .

R I A S S U X T O (*)

Ad un monde con una sola dimensione spaziale, si applicano le equazioni di unitariets nell 'approssimazione a due partieelle per dedurre l 'ampiezza di scattering nella simmetria incrociata e quindi determinate il fa t tore di forma e il propagatore della teoria; un confronto fra l'amloiezza intera e la (( approssimazione di Born )) mostra la grande diversits del lore eomportamento asintotico. Ci si serve inoltre della soluzione per discu- tere i l problema delle interazioni fra gli s ta t i finali sovrapponentisi e si prova che il rafforzamento 6 di ripe moltiplicativo.

(*) T r a d u z i o n e a c u r a de l la R e d a z i o n e .