II , NUOVO CIMENTO VoL. 60A, N. 3 1 Dicembre 1980

Large Transverse Momentum in the Symmetric-Group Model.

I. Exclusive Scattering (*).

~ . QuIR6s

Iust i tuto de Estructura de la Materia - Serrauo, 119, ]lad~'id-6 Departam,enIo de Fisica, Universidad de A~caId - Alcald de t lenares

tr icewlto il 24 Settembre 1980)

S u m m a r y . --- The inabi l i ty of the Veneziano model to define a reasonable fixed-angle behaviour is a challenge to the applica.tion of the dual approach, t towever, the high-energy, fixed-angle l imit of the ghost-free F rampton dual model, which is a par t icular case of the symmetric-group model, is in good agreement with the predictions of the s-channel Regge- pole model of Schrempp and Schrempp. The par t icular shape of the fixed-angle l imit depends on the unitarizat ion of the model. In a realistic model with unit pomeron intercept, the differential cross-section behaves as exp [--h(0)pT], in agreement with recent results on pp elastic scat- tering, and the function h(O) is f i t ted with experimental data, at angles between 50 ~ and 90 ~ with an error less than 10%.

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

W h e t h e r or n o t l a r g e - t r a n s v e r s e - m o m e n t u m h a d ron ic p rocesses a r e sens i t i ve

to t h e quaxk s t r u c t m ' e s t i l l r e m a i n s an open ques t ion . The k i n e m a t i c a l reg ion

r l ook for i n d i c a t i o n s of p o i n t l i k e c o n s t i t u e n t s in exc lus ive s c a t t e r i n g m e a s -

u r e m e n t s is t h e h i g h - e n e r g y f ixed -ang le region . H o w e v e r , t h e e x p e r i m e n t a l

o b s e r v a t i o n b e i n g t h a t c ross - sec t ions a t f ixed ang les fa l l r a p i d l y , t h e d e t a i l e d

c o m p a r i s o n of t h e o r y w i t h e x p e r i m e n t , wh ich is poss ib le in t h e p e r i p h e r a l

p e a k s , is r u l e d out .

(*) Presented at the European Sympos ium on Few-Body Problems in Nuclear and Particle Physics, June 3-6, 1980, Sesimbra, Portugal.

185

186 ~. qwn6s

Theoret ical models describing, in ~ re la t ively successful way, large-angle scat ter ing processes c~n be divided into two classes (1): a) pa r ton modcls~ where the in teract ion is point l ike as in deep inelastic ep scattering, and b) non- pa r ton models, where the in teract ion is governed by a length scale R ~_ 1 fro.

Pa r t on models predic t for the ampl i tude A - ] - B - + C + D , in the region s -+ 0% 0 fixed, the general behaviour

(1) [A(s, t)[ ~ s(~-'~)J2/(O) .

The exponent n ~nd the funct ion / are given by par t icu la r model.~. Thus

the const i tuent count ing rules (2) predic t for n the value

(2) n = % + % + n o § % ,

% being the n u m b e r of valence quarks in par t ic le I , while the ,regular de- pendence ](0) is predic ted by the const i tuent interchange model (~), whose s- dependence is only in rough agreement with (2). In fact , the value of n pre- dieted b y the con~,tituent in terchange model seems to be slightly dependent on the considered energy range, in good agreement wi th exper imenta l data .

As representa t ive of nonpar ton models we shall stress the geometr ical model of Chu and Hend ry (~), Schrempp and Schrempp (5) and Kondo, Shimizu and Sugawara (~). The s-channel Regge-pole model of Schrempp and Schrer~pp predicts for the ampl i tude, in the large-angle region, the behaviour

(3) [A(s, t ) [ ~ R(s){2~ (s) sin (0/~)} -1/2 exp [-- 2 sin (0/2) In , %(s)] ,

where their periphera.1 complex-:Regge t ra jec to ry ~(s) behaves as

(a) ~,(s) ~ i(R/27~) s ~/2 In (-- s / s o ) .

The const i tuent count ing rules have been fitted by LANDSHOFF ~uid POL- KI~-GItOR~E (7) in pp elastic scattering, at several angles, with a reason,~ble

good agreement . However , ,'~t higher energies, s > 2 5 (GeV) ~, the behaviour of

(1) I). SIVE~S: Ann. Phys. (N. Y.), 90, 71 (1975). (2) V.A. ~IATVE~V, R. M. MURADYAN and A. N. TAVKIII~LIDZE: Left. Nuovo Ci~e~to, 7, 719 (1973); S. ~BRODSKY and G. FAn~AI~: Phys. Rev. Left., 31, 1153 (197:3). (a) R. BLAXK~XB~CJ,ZR, S. J. BRODSKY and J. F. GUNION : Phys. Lett. B, 59, 649 (1972); Phys. Rev. D, 6, 2652 (1972); 8, 287 (1973). (4) S. Y. CHu and A. W. HEND:RY: Phys. Rev. D, 6, 190 (1972); 7, 86 (1973). (s) B. Scmc~Pp and F. Scnl~E~lPe: h~ucl. Phys. B, 54, 525 (1973); 60, 110 (1973); see also in Proceedings of the I V G I F T Seminar in Theoreticol Physics (Barcelona, 1973). (6) T. KONDO, Y. SHImZU and H. SUGAWARA: Prog. Theor. Phys., 50, 1916 (1973). (~) P. V. I,AXDSn()I~'V and J. C. POLXINGHORNE: Phys. Rev. D, 8, 927 (1973).

LA~G:E TRA:KSVERSE MOMENTUM IN TIlE SYMMeTRIC-GROUP I',IODEL - I ]-~7

t he ampl i tude for pp sca t t e r ing seems to be g iven by (1) wi th q~ ~ 1 4 . This

fea ture is p red ic ted b y more deta i led cons t i tuen t i n t e rchange models and canno t be considered as ~ sho r t coming of p a r t o n models.

:Based on the i r s -channel Regge ~pproaeh, S C ~ E ~ P P and SCttlgE~IPP ]lave p roposed (s) a genera l behav iou r of the h igh-ene rgy f ixed-angle ampl i tude as

a func t ion of p~ = q sin O, q being the cent re-of -mass m o m e n t u m of the scat- t e red par t ic les :

(5) IA(s, t)l ~ exp [-- Rg(O) pT] ,

in qua l i t a t ive ag reemen t wi th the original Orear fo rmula (9) and also in agree- m e n t wi th tlle behav iou r p red ic ted b y the s ta t i s t ica l model (lO) where the l eng th

scale is 1/m~. Behav iou r (5) has been f i t ted b y SCH~E~IPP and SC~RE3~PP (s)

in pp elast ic sca t t e r ing at fixed angles 0 = 90 ~ 68 ~ and 55 ~ wi th a ve ry good

agreement , and by SIV]~RS (~) a t 0 = 90 ~ On the o ther hand , HOJVAT and OnnaIr (11) have looked at new d a t a for

pp elastic sca t te r ing a t 0 = 4.85 ~ and found the behav iou r da/dt~., s -6~ in con t ra s t to the behav iou r s -Jo p red ic ted by the cons t i tuen t coun t ing rules.

However , t h e y find t h a t t he d a t a over a wide range of s axe f i t ted b y exp [ - - 7pT] in a s imple Pv plot . This behav iou r is in good a g r e e m e n t wi th the mos t r ecen t experimenta.1 d a t a (x~) on pp elast ic sca t ter ing.

Jn shor t we can say t h a t n o n p a r t o n models are in good a g r e e m e n t wi th expe r imen ta l da t a on h igh-ene rgy large-angle elast ic pp scat ter ing, while p a r t o n

models m u s t be ruled ou t for v e r y small (but finite) angles.

2. - F ixed-ang le a s y m p t o t i c behav iour o f the V e n e z i a n o mode l .

The Venez iano mode l (,3) is a n o n p a r t o n mode l where the length scale R gove rn ing the in t e rac t ion is re la ted to the universa l slope ~' ~ 1 (GeV) -2 of the

(s) B. SC~InEMPP and F. SCHREMPP: Phys. Lett. t~, 55, 185 (1975). (9) J. OREAR: Phys. Rev. Lett., 12, 112 (1964). (10) S. FRAUTSCm: 2r Cime~tto A, 12, 133 (1972). (11) C. HOJVAT and J. OR]~AR: Cornell University Report CLNS-346 (1976); J. ],AeIl: in Proceedings of the Twel]th Ren(ontre de Motioned, Vol. I I , Deep S(atte~i~g ~ d Had~o~ic Structure, edited by J. ThAN THANH VAN (1977). (12) j . L . HARTMANN, J. OREAR, J . VRIESLANDER, S. CO~]~TTI, C. HOJVAT, D. G. RYAN, K. SHAHBAZIAN, D. G. STAIRS, J. TRISCHUK, W. FAISSLER, ~[. GETTNER, J. R. JOHNSON, T. KEPHART, ]~. I)OTHIER, D. POTTER, M. TAUTZ, I ). BARANOV and S. RUSAKOV: Phys. Rev. Lett., 39, 975 (1977); S. CON:ETTI, C. HOJVAT, D. G. RYA~, K. StIAHBAZlAN, D. G. STAIRS, J. TRISCItUK, W. FAISSL:ER, }I. CxETTN:ER, J . R. JOHNSON, T. KEFItART, E. POTHIER, D. POTTER, M. TAUTZ, ]). BARANOV, J. L. HARTMA):N, J. OREAR, S. RU- SAKOV and J. VRIESLAXDER: Phys. Rev. Lett., 41, 924 (1978). (13) For ~ general review of the Veneziano model, see V. ALESSANDRINI, D. AMATI, M. LE BELLAC and D. OLIVE: Phys. Rep. C, 1, 269 (1971).

188 ~. Qu~nSs

Regge trajectories by ~'~--(R/2z) 2. In this section we shall briefly review the main features of the high-energy behaviour of the four-point dual ampli tude

(6) V(s, t) = ~ ( - ~(s)) r ( - - ~(t)) r ( - - ~(s) - - ~(t)) '

where ~(x)----~(0)-~ a ' x are l inear l~egge trajectories. In the Regge limit, s --~ c~, t fixed, the ampli tude behaves as (~)

(7) - - n~(s)~(t)(F(1 + ~(t)))- '{ctg n~(s) ~- ctg 7r:c(t)}.

I f Im ~ is s tr ict ly zero, eq. (7) gives just poles in s and Im V is a sequence of 5-functions. However, we expect t ha t for large s the narrow-resonance ap- proximat ion is very poor, and Im a(s) --~ c~ as s -> c~. In tha t case ctgTe~(s) --> - ~ - i and ampli tude (6) will finally tend to

(8) - - n exp [-- in~(t)] {F(1 ~- ~(t)) sin ~ ( t ) } - ~ ( s ) ~(~)

as in the Regge theory. Fur the rmore , it is usually assumed (~5) tha t Im c((s)/ /ln (~('s) -+ oo as s -~ oo in order to obtain an exponential decrease of V(s, u) in the limit s - ~ oo, t fixed. Let us remark tha t this is not an unfounded conjecture because, when dual models are unitarized through loop diagrams (~e), they not only produce normal threshold singularities demanded by uni t~ri ty (~7), bu t ~lso determine the renormalizat ion of the trajectories (~s) with Im ~( s ) r 0, ~s we will discuss in detail in the appendix.

In the fixed-angle limit, the ampli tude V(s, t) behaves in the following way (14,19):

F ( - - b l~ea(s)/(1 + b)) H(s, b) (9)

w h e ~'e

(10) H(s, b) = (--2,~b Re ~(s)/(1 + b))"". �9 exp [ i ( - - b= Re~(s) + I m :r In (l-l- b))]-

[Re a(s)(-- b In (-- b) ~- (1 -~ b) In (1 -~ b)) ~- b ~ X p

L

b(Ima(s))2 ] _L ( 1 + b) Re~(s) ~- O(Im~(s)/Re~(s)) .

(14) G. VE-~EZlA):O: Nuovo Cimento A, 57, 190 (1968). (15) j . A. SIIAP1RO: Phys. Rev., 179, 1345 (1969). (16) K. KIKAV~*A, B. SAKITA and M. A. VIRASORO: Phys. Rev., 184, 1701 (1969); K. KIKAWA, S. KLEIn% ]~. SAKITA and M. A. VIUASORO: Phys. Rev. D, 1, 3258 (1970). (17) D. J. GROSS, A. N Ev]~tT, J. SCUERK and J. H. SCttWARZ: Phys. Rev. D, 2, 697 (1970). (is) A. NEv]~u and J. SCHERK: Phys. Bey. D, 1, 2355 (1970). (~9) A. KRZYWICKI: Nucl. Phys. B, 32, 149 (1971).

LARGE TRANSVERSE MOMENTUM IN THE SYMMETRIC-GROUP MODEL - I 189

I n this way the fixed-angle behaviour of the Veneziano ampl i tude is governed b y Re a(s), so t h a t i t is not sensit ive to any renormal iza t ion of the Regge

t ra jec tory . Two main comments can be made abou t behaviour (9).

a) The fixed-angle behaviom' of the Veneziano ampl i tude violates the Cerulus and Mart in (~o) lower bound exp [--e(O)s~121ns]. I t can be argued t h a t the Cerulus-Martin bound is based on the usual h[andels tam aua ly t ic i ty in the t-plane, while in the Veneziano model the cuts on the real axis are re-

placed b y an infinite series of poles. However , when the theory is uni tar ized b y means of loop diagrams, the normal threshold singularities of the lV[andelstam ana ly t i e i ty do appear , bu t ELL:IS and FREUND (~) claim tha t the a sympto t i c expression (9) remains valid. We are t e m p t e d to in te rpre t the conjecture of Ellis and Freund in the sense t h a t the fixed-angle behaviour is controlled b y Re a(s), as we have poin ted out above, so t h a t it is insensitive to any uni ta- r izat ion of the model. I n any way, behaviour (9) has been proved b y AL]~S- SANDI~I1NI~ ~kl~IATI and MOREL tO hold a t the one-loop level (22).

The Cerulus-Martin bound was extended by C~[Iu and TA~- (~a) to the more general value, exp [--%(0)8 v In 8], 1/2 < 7 < 1 , and they p roved t h a t Y = 1 is ~p- propr iu te for a l inear Regge t ra jec tory . However , as CHru and TA~ have poin ted out, consideration of ana ly t ic i ty in the s-channel might raise the lower bound

obta ined b y them.

b) This fas t fall-off of the fixed-angle cross-section is ne i ther in agreement with the successful models reviewed in sect. 1, nor suppor ted by exper imenta l

data . The fixed-angle behaviour of other dual-resonance models, as the ~ e v e u -

Schwarz nmdel or the Shapiro-Virasoro model, follow closely behaviour (9), so t h a t the usual (~ addi t ive ~) dual models seem unable to describe high-energy fixed-angle scattering, and this is a serious objection to these dual models in

spite of the i r o ther theoret ical successes. The only way of overcolning this difficulty is to sum an infinite number of

Veneziano-like t e rms in such a way t h a t dual i ty and Regge behaviour should be preserved, bu t such t h a t the fixed-angle behaviour of the series would be qual i ta t ive ly different f rom the behaviour (9) of each term. :in par t icu lar we

shall res tr ic t ourselves to absolutely convergent Veneziano series as

(11) co

A ( 8 , t) = e r ( n - - r ( n - - - - - -

n = O

(~o) F. CV.RULUS a n d A. MARTIn: P h y s . Le t t . , 8, 80 (1964). (21) S. D. ELLIS a n d P. G. O. FREtTND: N A L R e p o r t No. N A L - T H Y - 8 2 (unpub l i shed) . (o~) V. ALESSAI~'DRINI, D. AMATI a n d B. MOREL: N u o v o C i m e n t o A, 7, 797 (1972). (23) C. B. C a l u a n d C.-I. TAN: P h y s . R ev . , 162, 1701 (1967).

14 - I1 Nuovo Cimento A.

190 ~L QUIRES

GARDINER and FRAMPTO]N proved (24) t h a t Regge behaviour as s --> c~, t fixed, and exponent ia l decrease off the real axis as s --> c~, u fixed, for the absolutely convergent Veneziano series (11), necessari ly required t ha t the funct ion

(12) F(z) ---- ~ c~z ~

be ana ly t ic everywhere in the complex z-plane, except perhaps when z is real and 1 < z ~ oo. I n the following section we shall consider a par t icu lar Vene- ziano series and analyse its fixed-angle behaviour .

3. - Fixed-angle asymptotic behaviour of the symmetric-group model.

I n this section we shall analyse the fixed-angle behaviour of the symmet r ic - group model of F r a m p t o n (25) whose four-point funct ion can be wri t ten as

(13) 1

A4(8 , ~) = f d ~ x - ' ~ ( ' ) - ' ( 1 - - x ) - ~ ' t ) - l { ( x - - e x p [ / ;T~/3])(X - - e x p [ - - i~ /3] ) } :''2- o

�9 q~,(~(s), ~(t), ~(u); x ) ,

where y ~ :~(s) § a(t) § :r + 1 and the funct ion ~4 satisfies an a t t r ac t ive invar iance under the $3 p e r m u t a t i o n group. The symmetr ic -group ampl i tude (13) has the proper t ies of dua l i ty and Regge behaviour wi th absence of odd- daughter te rms, wi thout the requi rement of uni t in tercept as in the Veneziano model. The original u model, as well as the lqeveu-Schwarz model, are par t i cu la r cases of (13) for cer ta in values of the funct ion ~a and the inter- cepts xp(0) and an(0 ). The same is t rue for earlier absolutely convergent Ve- neziano series as the model of ]Kandelstam (26), which was originally mo t iva t ed to avoid odd-daughter trajectories, bu t which is plagued with ghosts. To

our knowledge, the only ghost-free absolutely convergent Veneziano series is de termined by the function (25)

( 1 4 ) q~,-- ( 1 - x § x~)-'(x~(s) § (1 - x ) : z ( t ) - x ( 1 - x)~(u)}

with the physical in tercepts a~(0) = J/2, an(0 ) = 0. Thus the ghost-free model of F r a m p t o n

1

( 1 5 1 A(s, t) - - - - x ( 1 - -

o

�9 { x ~ ( s ) + (1 - x ) ~ ( t ) - x ( 1 - x ) ~ ( u ) }

(~4) p. H. FRA]~IPTON and C. W. GARDI~n: Phys. Rev. D, 2, 2378 (1970). (25) p. H. FRAMI~TOZ~T: Phys. Rev. D, 7, 3077 (1973); 9, 487 {1974). (z6) S. MANDELSTAI~I: Phys. Rev. Lett., 21, 1724 (1968).

L A R G / 5 T R A N S V E R S E M O M E N T U M I N T H E S Y M M ] g T R I C - G R O U P I~,IODEL - I 191

with ~--~ 1 - - % ( 0 ) has both theoretical and phenomenological advantuges.

There is no tachyon on the leading positive-intercept trajectory, all the levels

lying lowest in mass are ghost free W) (although there is not an operatorial

ghost-killing mechanism) and the a.mplitude is genera]iza,ble to a factorizable (~8)

N-point function. F rom the phenomenological point of view the spectrum of daughter resonances agrees ra ther well with the predictions of the F rampton

model: "~bsence of z(600) resonance, an r s-wave resonance strongly coupled to ~rc and absence of {~(1250) p-wave resonance. On the other hand, a recent ; ~ phase-shift analysis (29) strongly favours the s model over

other usual phenomenological dual nmdels.

Let us remark tha t amplitude (16) falls into the category of (11), in the

borderline case in which F(z) has a branch point ut z ---- 1. This feature is essential to determine the fixed-angle behaviour of the amplitude.

I n the fixed-nngle limit integral (15) can be writ ten as

(16)

1

A( ,{ t , t ) ~ _ o : ( 8 ) f d x o ( x ) - a { s ) - l ( 1 - - x ) ~ - - x ( 1 - - . ~ ) } ) , , 2 .

o �9 {x -H b(1 - - x) -H (b -H 1)x(1 - - x ) } ,

where {o(x)= x ( 1 - x) b and a and b depend on the centre-of-mass scattering

angle 0 as

(17) b -~ - - (1 - - cos 0)/2 ---- - - sin 2 (0/2), a -- %(0}{1 - - b) -H 4a~(0) b.

At this point it is convenient to change variables in (16) from x to z as given

by exp [ - -z ] = w(x), so tha t integral (16) cun be writ ten as

(18)

where

-t- co

A(s, t) ~ ~(~)fd~ exp [(~(s) + 1) z] ](z), --co

(19) I(~) = ,~,(exp r - ~]){1 - v ( e ~ p [ - - ~])}~176 - - (b + 1 ) ~ ( e x p E-- ~])}-1-

�9 {1 - - v(e~p [-- ~])(1-- ~(exp [-- ~])YfT(e~P E-- z]) + b(1 -- ~(exp [-- ~]) +

+ (1 + b ) ~ ( e ~ p E-- ~])(1 - - ~,(e~p E-- ~)}

and v is the inverse function of oJ. The properties of this change of variables

(27) p. H. FRAMPTON and K. A. GEE]~: Phys. Rev. D, 10, 1284 (1974). (2s) p. H. FRAMPTON: Phys. Rev. D, 9, 2861 (1974). (29) C. D. FROGGATT, H. B. NIELSEN and J. L. PETERSEN: Phys. Rev. D, 18, 4094 (1978).

192 M. QuxR6s

have been studied in great detai l by GARDINER (3o) and we shall use his results in the following. The singularit ies of the in tegrand (19) are of two kinds:

i) The funct ion { 1 - - x ( 1 - - x ) } a/2, when 2 is not an even number , has two cuts wi th branch points a t x = exp [~i~/3], so tha t , af ter the change of variables, the corresponding funct ion of z has cuts wi th b ranch points at

wi th Z = Z(1,n )

(20a) ~ Im z~,~) 1) 2 n z (n = • Re z(~,,,)= O, = • (7t/3)(b - - -~ O, • ...).

ii) The funct ion v(y) is ana ly t ic th roughout the complex y-plane, apa r t f rom two cuts along arg (y) = • wi th b ranch points at lY[ = ]bb( b ~- 1)-b-ll. I n this way the funct ion v(exp [ - -z ] ) has cuts in the complex z-plane, wi th

where b ranch points a t z = z(2,~),

(20b) Rez~,, ,~=--]lnbb(l~-b)-l-b[, Imz~,,,)----- • ( n = 0 , 4-1, • ...).

Le t us note t ha t R e z ~ < 0 when b = / : 0 , - - 1 (apar t f rom the forward a, nd (2,~)

backward directions). I n short, in tegrand (19) is ana ly t ic th roughout the complex z-plane apa r t

f rom cuts, paral lel to the real axis, f rom - - c~ to the b ranch points z = z~,,) (i : 1~ 2; n : 0, ~ 1 , ~ 2 , ...).

Le t us suppose for the m o m e n t t ha t I m a ( s ) > 0. Then we can close the real axis (-- co, + c ~ ) b y a semicircle F of radius ~o ~ c~, having its centre a t

the origin, above the real axis. The new contour C can be deformed to a new contour which is composed of the sum of pa ths C ~ (~,~ as given by

(21) C = C ~- i~,~ Jr C~,~) , i = 1 = n=O

where C~,n) is the p.~th which encloses the cut wi th b ranch point a t z,,,,) • s ta r t ing a t (( minus infinity ~) under the cut, encircling the b ranch point once counter- clockwise and re turn ing to the s ta r t ing point above the cut. Thus, in thef ixed- angle l imit, the ampl i tude can be decomposed as the sum

(22) A(s, t) = ~ {A+(s, t) + AT(s , t)}, i = l

where A~(s, t) is the sum of contr ibut ions f rom the contours C ~ in (21). ( i , n )

Let us first compute the cont r ibut ion f rom the b ranch points z(1,~ ) ~: , Ax(s , t). The integrals round the contours C(1,~ ) differ only b y phases coming f rom

(no) C. W. GAI~DINnR: Phys. Rev. D, 9, 2340 (1974).

L A I 4 G E T R A N S V E R S E M O M E N T U M I N TIt~E S Y M M E T R I C - G R O U P M O D i ~ L - I 1 ~ 3

exp [ ( a ( s ) § 1 )z ] which, once e x t r a c t e d a n d s u m m e d over n, can be fac to r i zed ou t of t he in teg ra l s as

(23)

where

(24)

A+-(s, t) ~__ ~(s){(if2) exp [4- izca(s)] cosec ~ ( s ) } 1 •

~• =j'd~ cxp [(~(,,) § 1)q l(z). 6'~1,0)

The leading c o n t r i b u t i o n of (24) as s -~ c~ can be e x t r a c t e d b y e x p a n d i n g the

• the e x p o n e n t i a l in (24) i n t e g r a n d ](z) a round z(~,o ) * , because for ]~e z ~ Re z~i,o ) goes v e r y qu ick ly to zero. A s t r a i g h t f o r w a r d ca lcu la t ion shows t h a t the in- t eg ra l I~ of eq. (24) can be cas t in the l imi t s -+ c~ as

(25)

wi th

(26)

I~_ "~ exp[ •247 { exp[~iz~(b--1)/3] ( • 2isinT~/3)} ~'~. -~ - - 1 - - (b § 1) exp [ 4- i7e/3] 1 -- (b § 1) exp [4- i:n/3]

�9 {(1 + b) + exp [ 4- i~/3] § b exp [ - - ( • i~/3)]} J +

J~ = f d z exp [(~(s) + 1 )z ] {exp [ - - z ] - exp [ ~ i (z /3)(1 b)]} ~/2

0~ ,0 )

whose asympto t ic , b e h a v i o u r is eas i ly c o m p u t e d b y e x p a n d i n g cxp [ - - z ] in

a powcr series, so t h a t one ob ta ins

(27)

a.nd

(28)

J ~ ~_ exp [ ~ (~(s) § 1) iTe(b - - 1)/3] exp [ ~ i,~(1 - - b)/3. (,t/2)] I f

K fdz exp [ - (~(s) + ~)z] z ~'~ c

where C is t he I )a th of integr%tion which s t a r t s a t << inf in i ly )~ on the real axis, encircles the or igin in the pos i t ive d i rec t ion and r e tu rn s to the s t a r t i n g poin t . F ina l l y t he a s y m p t o t i c b e h a v i o u r of K can be e v a l u a t e d as

(29) K ~_ (2~i)(-- :<(,))-~,~-,(r(- ~/2))-1.

Dne to the i n v e r s e / " - f u n c t i o n , this t e r m van i shes when ), is an even n u m b e r . This is the case, for ins tance , of t he usua l Venez iano mode l or the Neveu - Schwarz model , where ~p(0) = 1 and 2. = 0. I n t he F r a m p t o n mode l ~ ( 0 ) = 1/2 and 2 = 1/2, so t h a t (29) gives a f n i t e con t r ibu t ion .

As can be seen f r o m (23)-{29), t he f ixed-angle b e h a v i o m ' of A l(s, t) is s t rong ly d e p e n d e n t on I m a(s). W e shal l come b a c k to this po in t la ter .

194 ~. quin6s

The contr ibut ion f rom the branch points z ~ (~,.) can be computed in essentia~lly

the same way, ge t t ing the following a sympto t i c behavionr :

(3o) A2(s, t) _~exp [ - - :r ]bb(1 q- b)-~-b]~ i~b)] ,

which is essential ly the same behaviour of the Veneziano ampl i tude (10). I n th is way 1A~(s, t)[ is governed b y Re a(s) and thus insensitive to any uni ta- r izat ion of the model. As we shall see, it will be negligible, as compared to

A~(s, t), and we shall not dwell upon it any longer. Le t us now re turn to the a sympto t i c behaviour of A~. As we have remarked ,

it depends on I m a(s) and we expect , as poin ted out in sect. 2, t ha t for large s the narrow-resonance approx ima t ion is very poor and I m ~(s) -~ c~ in the l imit s -~ o0. This is closely re la ted to the p rob lem of the uni tar izat ion of the theory

and the renormal iza t ion of Regge trajectories, as we shall discuss in the ap- pendix. ~[u the l imit I m ze(s) --> oo the ra t io [A+I[/[A~I behaves as

(31) IA+(s, t)!/IA[(s , t)l _~exp [-- (2z~/3)(2 q- b) l m ~(s)] -~ O,

so t ha t

(32) IAI(S, t) I ~ [AU(s, t)l

_~ (C /F ( - - 2/2))(3 + cos ~ O)-Z~'l~z(s)l-~' exp [--1(0) h n ~(s)] ,

where C ---- 2z(3)v2{2(3)1~2} ~/2 and the funct ion ](0) is given b y

(33) (1 + sin ~- (0/2)), i(o) =

which is in good qual i ta t ive agreement wi th predictions (3) and (4) of the s-channel l~egge-pole nmdel of Scbrempp and Schrempp (5). I f we normal ize

the funct ion ] a s / ( 0 ) = fi](O) such t h a t ](0) --~ 2 sin (0/2) a t 0 ~- 90 ~ (or equi- va len t ly Imp(s ) - - - - f l Im~p(s) , where ~(s) is the per ipheral t ra jec tory of

Schrempp and Schrempp) , we obtain

(34) ](0) - - 2 sin (0/2) [(0) < 0.15

for angles 60 ~ ~<0 ~90 ~ For small angles the relat ive difference (34) becomes

large. The shape of the fixed-~ngle behaviour (32) depends on the behaviour of

I m a(s) in the l imit s -~ oo. In the following we shall use

(35) I m ~(s) ~ A(~ 's ) in ,

L A R G E T R A N S V E R S E M O M E N T U M I N T t I E S Y M M E T R I C - G R O U P M O D E L - I l ~

where A is some dimensionless constant . We shall ground t h a t behaviour on the following points :

a) The calculation of the wid th of low-energy resonances by LOVELAC]~ (3~) in the Veneziano model using an <( effective >> t ra jec tory wi th ] m a(s) --~ 0.28-

�9 ( s - - 4 f f ~ ) ~ / ~ .

b) Behaviour (35) is s t rongly suggested b y the uni tar izat ion of a <( realistic ~) Veneziano model wi th a~(0)~--1/2 and g r (0 ) -~ 1. Since the

F r a m p t o n model has a realist ic p-intercept ~(0)---~ 1/2 and the pomeron s ingular i ty comes f rom nonplanar or ientable diagrams, the p rob lem of the uni tar iz~t ion of the model is ~n ex t remely interest ing one, in connection with the fixed-angle behaviour of the ~mpli tude, and wor thy of fu r ther invest igat ion. The lmitar iz~t ion of the Veneziano model wi th uni t pomeron in tercept will be analysed in more detail in the appendix.

c) Behaviour (35) is also suggested by the per iphera l t r a jec to ry of Schrempp ~nd Schrempp, (4), where the (( d iamete r )) of the hadron bo

----R In (S/So) is roughly a constant .

Thus, using (35) in (32), we get

(36) [A(s, t)[ ~ exp [ - - h(O)(o~')i/~pT ] ,

where the a.ngular-distribution function h(O) is given b y

3 - - c o s 0 (37) h ( O ) : A

3 sin0 '

which is in qual i ta t ive agreement wi th behaviour (5) proposed b y SCUREMPP a.nd SCmCEM~e (5). These authors predic t a smooth behaviour of the i r func- t ion g(O), eq. (5), based on a fit of pp elastic scat ter ing at large ~ngles. ~ o r e

precisely, they find exp [-- 4.06pT] , exp [-- 4.21p~] and exp [ - - 4.47pT] a t 0----90 ~ 68 ~ and 55~ respectively. Thus, if we normalize, for instance, the cons tan t A in (35) so as to have h(35 ~ z 4.21, we get A ~ 0.28 fm/g '1t2 ~nd h(90 ~ --~ 4.46, wi th a relat ive error, wi th respect to the Schrempp and Schrempp value, of - -0 .09 , and h(55 ~ ~-4.40, with a relat ive error of 0.01. Thus the simple formul~ (36) is able to fit the exper imenta l dat~ a t angles between 90 ~ and 55 ~ with an error bound of 9%. On the other hand, a simple analysis shows t h a t the funct ion h(O) has a relat ive min imum at 0m~ , ~ 70.52 ~ and tends to infinity at 0 ~ 0 ~ 180 ~ Around 0m~ , the funct ion has ~ smooth var ia t ion which

(31) C. LOVELACE: Phys. LeLt. B, 28, 264 (1968); D. SIVERS and J. YELLIN: Rev. Mod. Phys., 43, 125 (1971).

196 ~. quI~ds

can be me~sured b y the ra t io

(38) R(O)-- h(O)--h(O~,,d h(0mi~)

I t is s t ra ight forward to prove t h a t IR(0)]<0.08 for 5 0 o < 0 < 9 0 ~ Since the Schrempp and Schrempp ' s fit (s), as well as recent exper imenta l results (1~), suggest t ha t the angular dis t r ibut ion in (36) should be of smooth variat ion, the angular region where formula (36) is expected to hold, within an accuracy of 10%, is 4 5 o < 0 < 9 0 ~ However , a t smaller angles, the F r a m p t o n model uni tar ized in the way we have used th roughout this section, eq. (35), is in agreement nei ther wi th the geometricM models nor wi th the exper imenta l

PT distr ibutions. ~ e could also imagine t h a t h n a(s) behaves as B In (~'s) in the l imi t s ~- c~.

However :

a) This behaviour is not suppor ted by the uni tar izat ion of a <, realistic ~

dual model wi th uni t pomeron intercept , us we shall see in the appendix.

b) I t does not give the proper Regge behaviour of the to ta l ampl i tude (15) A(s, t) -[- A(s, u) @ A(u, t) at t fixed.

c) I t should give the fixed-angle behaviour

(39) da/dt ~ s -'~(~ ,

where n(O) is the following funct ion:

(40) 0)

n ( 0 ) = l ~ - s i n ~ @ ) . @ 2

and ~ = 3 a ( 0 ) - - 1 @ 4:r m being the p ro ton mass. Taking the experi- men ta l values ~(0) = 0.5, ~'---- 0.9 (GeV) -~ and m~ = 0.88 (GeV) 2, we ge t

~t ---- 3.668.

Equa t ion (40) is in d isagreement wi th the Landshoff and Polkinghorne fit of pp elastic scat ter ing a t fixed angles 3 0 0 < 0 < 9 0 ~ where n = 9.7 :L 0.5 is consistent wi th the exper imenta l datu. Nevertheless, more recent experi- men ta l da ta (11) show a decrease of n a t smaller angles, in qual i ta t ive agree- m e n t wi th (39). ] f we normalize n(O) at n(90 ~ = 12, we get n(68 ~ = 11.2 and n(55 ~ = 10.7, which are consistent wi th the exper imenta l da ta p lo t t ed in ref. (s). At smaller angles, there is a very recent fit by I-IARTI~IAS"N et al. (1~) at 0 = 15 ~ as s -" wi th n = 9.7 • which is in good agreement with the v~lue

obta ined f rom eq. (40), n(15 ~ = 9.9.

I,A]r T R A N S V E R S ] ~ M O M E N T U M I N TII]~ S Y M M E T R I C - G R O U P M O D E L - I 197

4 . - C o n c l u s i o n .

I n sect. 1 we have briefly reviewed pa r ton and nonpar ton approaches which have been re la t ively successful describing large-angle scat ter ing processes. For present da ta nonpar ton ~pproaches are a t least as successful as const i tuent

models. However , as w~s po in ted out in sect. 2, the usual Venezian0 model, including un i t a r i ty corrections, has a fixed-angle behaviour , decreasing too fas t a t s --> 0% which is not suppor ted by exper imenta l d~ta and violates the Cerulus-Martin bound. The inabi l i ty of dual models to define a reasonable fixed-angle behaviour mus t be considered as a. challenge to the appl icat ion of the du~l approach. One remaining possibil i ty is to consider absolutely con- vergent Veneziano series.

In sect. 3 we have studied the fixed-angle behaviour of the ghost-free Fr,~mpton dual model, which is a par t icular case of the symmetr ic -group model, wi th lhe following proper t ies : a) it has a realistic p-intercept ~(0)- - - -1 /2 , b) there is no tachion on lhe leading posi t ive- intercept t ra jec tory , c) it is ghost- free up to ~(s) ~ 30, a l though the absence of ghosts have not been proved in general because an operator ia l ghost-kill ing mecha.nism is still lacking~ d) i t is genera.]izable to a ft~ctorizable N-po in t function, and e) it has "~ reasonable

low-energy spec t rum with absence of odd-daughter t rajectories. The fixed-angle behaviour of the F r a m p t o n model is governed by l m ~(s),

in agreement with the s-channel Regge-pole model of Sehrempp and Sehrempp. I n any dual model wi th linea.r Regge t rajectories and narrow resonances, the width of resonances and unit~trity thresholds is provided by loop di,~grams which renormalize the Regge trajectories giving to t hem "m imaginary pa r t . Thus the par t icular shape of l m a(s) a t s --> c<) s trongly depends on the unita- r izat ion procedure. In this way the rigorous uni tar izat ion of the F r a m p t o n model is an ex t remely interest ing problem for fur ther invest igation. Guided by the uni tar izat ion of the Veneziano model in the ideal case of uni t pomeron intercept , we have used the behaviour I m ~- - - (~ ' s ) ~/2 as s - + oo. The result is in agreement wi th a fit of pp elastic scat ter ing a t 0 ~ 90 ~ by SIVEns, and a t 0 = 90 ~ 68 ~ and 55 ~ by SCHREMPP and SCHREMPP with an error less t h a n 10~ The model is expected to hold at large angles, be tween 45 ~ and 90 ~ with an accuracy of ] 0 % , while a t small angles the predictions of the model fall too rapidly with PT and are not suppor ted by exper imenta l data .

Closely related to the fixed-angle exclusive scattering, which has been con- sidered th roughout this paper , there is another interest ing case: th e Iargc-pT one-part icle inclusive cross-section E d3a/dap, which behaves exper imenta l ly as

p ~ e x p [ - - b x T ] , where n ~ 8 , b_~13 and X~.~-2PT/S ~/~. This is a h~rder problem, because the usual geometr ical models (as the statistie,~l models) give

a too fast fall-off wi th PT, as exp [-- CP.r] , in clear disagreement wi th experi- men ta l data , while only nonpar ton models containing ad hoc features can

198 ~ . Qui1%5s

describe inclusive cross-sections a t large PT (3~). The behaviour of inclusive cross-sections in the Veneziano model is even worse. The predic t ion of the dual-resonance model is exp [-- 4:r in the centra l region (as), and this be-

haviour remains essential ly unchanged when the P o m e r a n c h u k d iagram is considered (34). Recent calculat ions in the Shapiro-Virasoro model wi th po- merons as in te rmedia te states give the same qual i ta t ive behaviour (ss), so t h a t the only open possibi l i ty seems to lie, as for fixed-angle exclusive scattering, in an absolutely convergent Veneziano series. The F r a m p t o n model being a good candida te for the reasons we have ment ioned above, an invest igat ion on this line is being carried on.

The au thor would like to t h a n k the hosp i ta l i ty of the C E R N Theoret ical

Division where p a r t of this work was carr ied out.

A P P E N D I X

Loop diagrams in dual-resonance models have the effect of renormal iz ing the l inear Regge t ra jec tory as (1~)

(A.1) ~ ( t ) = ~(o) § ~'t + g~_r(t) .

I n this appendix we shall compute the a sympto t i c behaviour , as t - - > - c~, of the contr ibut ion of one-loop p lanar diagrams to 2:(t), which can be writ- ten as (18)

(A.2) N ( t ) = (--1)~12~ -~+2)~ ~- (~ ~f(~o)-12~(~ (~-2~/~ dOJ(q, O, t) D(~)(O, o)) ,

o o

where ae(0) is the Pomeron intercept , which we shall take as a p a r a m e t e r here, eo----exp[2n2/lnq], 1(o~) is the usual par t i t ion funct ion appear ing in dual loop calculations (~7), and

(A.3) ](q, 0, t) = w(0, o~)=,'~{ - ln -~oD(1)(0 , (,~)}~",

(32) For a review see D. SIx~RS, S. J. BRODSKY and R. BLANKENBEKLER: Phys. Rep. C, 23, 1 (1976). (33) M. A. VIRASORO: Phys. Rev. D, 3, 2834 (1971); D. GORDON and G. V]CN]~ZIA~O: Phys. Rev. D, 3, 2116 (1971); C. E. DE TAR, K. K,aNG, C.-I. TAN and J. H. ~VEIs: Phys. Rev. D, 4, 425 (1971). (34) V. ALESSANDR~N1 and D. A~fATI: Nuovo Cimento A, 13, 663 (1973); J. P. ADER and L. CLAV~LLI: Nucl. Phys. B, 133, 327 (1978). (as) j . t). ADE~ and L. CLAVELLI: Phys. Rev. D, 18, 1295 (1978); L. CLAVELLI: Nucl. Phys. B, 154, 47 (1979).

L A R G E T R A N S V E R S E M O M E N T U M I N T I l E S Y M M e T R I C - G R O U P M O D E L - I 199

where ~v(0, oJ)~---lno~O~(lnx/ln~o)/O;(O) and n (1) is the der iva t ive of the Jacobi funct ion 01 with respect to lnx / ln~o. Using the explicit expressions of ~p and D(1)~ one can prove tha t

(A.4) t(q, 0, t) = exp [a' t In (: § 0(q~))],

so tha t , in the l imit t --> - - oo, integral (A.2) i~ domina ted by values of q such t ha t q ~< (Et) -~/2. Then, expanding the in tegrand around q = 0 and using the following est imat ions a t q--> 0:

(A.5) o~ ~ 1 , ]-'(oJ) ~ q-1/12, D(1)(0, vJ) _~ - - ~ / s i n : 0 ,

we c~n cast the a sympto t i c behav iour of X(t) in the following fo rm:

(A.6)

co

- - J 2x o

gt

�9 {1 In (x/-- o ) ( - 0

zt2/sin: 0) ,

where we h~ve changed var iables as - - ~'tq 2 = x ~nd 1(0, x) is defined by

(A.7) ](x, O) = l i m ](q = (x / - - s ~-, O, t ) . t - - - > - - ~

Thus in a realistic dual theory where a,(0)--~ l ~ l e t us note tha t the v~lue of the pomeron in tercept is de termined by nonplanar orientable d i a ~ ' a m s - - w e h ~ v e

(A.8) Z( t ) = ( - -1 ) ' (2~) <"-2'!2{ln(--s t)}~o-~-)'~-(s t):n I +O( ( - - tW21n( - - t ) (o -* '~ ' ) ,

where I is the integral

(A.9) I=fdxx, ,fdof(x,o) sin- 0 o o

whose singulari ty ~t x = 0 does not mat te r , bectmse it is cancelled b y the Neveu-Scherk regularization.

Express ion (A.8) is compat ib le wi th the behaviour i m p ( s ) ~ ( ~ ' s ) ~/2 we have used in sect. 3 of this paper . However , if the space- t ime dimensional i ty is D = 4, as is s t rongly suggested in the symmetr ic -group model, which contains negat ive probabil i t ies for ~ny space- t ime dimensionMity sat isfying D > 5 (~), then Z ( t ) _ ( - - s ( - -~ ' t ) , in good agreement wi th the imagi- nary pa r t of the per ipheral t ra jec tory of Schrempp and Schrempp, eq. (4), a n d leading to Kinosh i ta ' s minimal- in terac t ion principle applied to the Cerulus- Mart in bound (3~).

(36) p. H. FRAMPTON: Phys. Rev. D, 11, 953 (1975). (aT) T. KIN()SIIITA: Phys. Rev. Lett., 12, 256 {1964).

2 0 0 .~. QUZR~Ss

�9 i { I A , q S I ' N ' I ' ~ ) ( ' )

L'inc:~paci th dcl mod~.llo di Vcneziano di dei ini re un rag ionevole c.onq~ortanwnto a d angolo fissato m(.t.tc in discuss ione l ' app l i eaz ione del l 'approc, : io duale. ( ' o n t u n q u e il l imi te ad a l t a ~.nergia ,, ad angolo fissato del model lo dua le di F r a m p t o n senza f an t a smi , t h e 6 |ill {yH[4o pa r t i co la re del model lo dci g rupp i simmet.rici , 6. in buon accordo con lc p rcv is ion i del model lo del polo di Regge con cana le s di S c h r e m p p e Schren ,pp . l , ' anda - t ,wnto part . icolarc del l imi te ad angolo fissato d ipende dai f a t t o t h e si r ende u n i t a r i o il modello. In un n,od{.llo rcal is t ico con int.er, .etta u n i t a r i a del p o m e r o n e la sezion, ' d ' u r t o differenziale si , .ompor ta cozno exp[---h(O)PTI, in a,.cordo con recent i r i su l t a t i su]lo sca~t.ering ,.lastico pp. e la funz ione b(O) si a d a t t a ai da t i spe r imen t a l i pe r antzoli t ra .St) ~ e 90 ~ ,.,m un er rore minoro d,.1 10~ .

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