Duality diagrams and Regge cuts in meson-baryon scattering: Description of πN and KN scattering

Nuclear Physics B69 (1974) 107-141. North-Holland Publishing Company

DUALITY DIAGRAMS AND REGGE CUTS IN MESON-BARYON SCATTERING: D E S C R I P T I O N O F 7rN A N D K N S C A T T E R I N G G. GIRARDI, R. LACAZE and R. PESCHANSKI* Service de Physique Thdorique, Centre d'Etudes Nucldaires de Saclay, BP 2, 91190 Gif-sur-Yvette, France

Received 16 July 1973 (Revised 10 August 1973) Abstract: We propose a reggeized absorption model including lower-lying Regge cuts and satisfying duality requirements. It is shown that the introduction of these singularities, especially the reggeon-pomeron-reggeon (RPR) cuts, provides a solution to many puzzles in meson-baryon scattering. Theoretical evidences for the cuts, using duality diagrams and s-channel unitarity are given. These theoretical constraints reduce strongly the flexibility of the model and give it a good predictive power. With only twelve parameters, we obtain an excellent description of observables in ~z-nucleon and K-nucleon forward scattering in a large energy range (5 ~
1. Introduction A s far as t w o - b o d y reactions are concerned, the idea o f a b s o r p t i o n a n d the c o n c e p t o f duality have p r o v e n to be very useful tools. H o w e v e r a b s o r p t i v e models with exchange degenerate poles have run into difficulties b o t h o f theoretical a n d p h e n o m e n o l o g i c a l nature. On pure theoretical grounds, a l t h o u g h the reggeon calculus [I-3] projects some light on Regge cuts, the cut discontinuity is given by an integral over an off-shell reggeon particle scattering a m p l i t u d e , a b o u t which physical i n f o r m a t i o n is lacking. F u r t h e r m o r e the role o f Regge cuts in duality is still unclear, indeed duality was derived u n d e r the crude a s s u m p t i o n that they can be neglected. O n the other hand, severe p h e n o m e n o l o g i c a l difficulties a p p e a r e d out o f a m o r e detailed analysis. F o r instance all simple a b s o r p t i o n m o d e l s failed in predicting the n N charge exchange p o l a r i z a t i o n at large values o f lt l. M o r e o v e r precise a m p l i t u d e analyses have shown t h a t exchange degeneracy is definitely b r o k e n in the space like region ( K N a n d K N charge exchange reactions, h y p e r c h a r g e exchange reactions, etc.), for instance see ref. [4].

* In partial fulfillment of a "Th6se de Doctorat d'l~tat 6s-Sciences Physiques", Facult6 des Sciences, Universit6 Paris VI, 1973.

108

G. Girardi et aL, Duality diagrams

Despite these troubles, on the basis of the impressive qualitative success of absorption and duality, we claim that all the troubles can be cured by the introduction of lower-lying singularities*. In a previous paper [5], with such a point of view, we introduced Regge-pomeron-Regge cuts, in zcN and K N charge exchange. In the present paper we generalize previous arguments to include an extensive study of all high energy data for 7rN, K N and I~N scattering. Moreover we present an argument based on s-channel unitarity to exhibit R P R contributions. Duality makes possible to consider secondary Regge cuts in such a way that their parametrization, or their existence itself in the studied channels, are submitted to strong selection rules. This reduces strongly the number of free parameters and gives to our model a great predictive power. In this respect we differ strongly of some other models [6, 9]. In sect. 2 starting from the Regge pole duality diagrams, we build the expansion of the high energy amplitude and find (already known and new) selection rules for Regge cuts in meson baryon scattering (subsect. 2.1). A perturbative view of n particle production amplitudes makes it possible to use unitarity and to go deeper into the physical understanding of the nature and properties of the new proposed singularities - - t h e R • P * R cuts--(subsect. 2.2). The description of ~N and K N reactions, which is proposed as a practical application of the model (sect. 3), uses as much as possible the theoretical framework. In particular the parametrization of the s-channel helicity amplitudes (subsect. 3.1) involves only twelve parameters for eight studied reactions (z~- p ~ zc- p, n+ p --, ~+ p, g - p ~ z c ° n , ~ - p---, ~/n, K+ p-* K+ p, K - p-~ K - p, K - p ~ K°n, K+ n ~ K ° p ) . A fit, performed on forward elastic pion-nucleon data, leads to a very good description of available data of all the reactions considered in the range - 2 . 5 G e V / c < _ P ~ a b <_ _< 40 GeV/c, and 0 _< - t _< 1 (GeV/c) 2 (subsect. 3,2). At last (sect. 4) the conclusions are drawn, and some possible experimental tests and some theoretical open problems are discussed.

2. Derivation of the model

Our starting point is the expansion of the amplitude in terms of Regge poles and their iterations M = P+R+P,R+R,R+R,R,P+R,P,R+

....

(1)

Cut contributions are built following Gribov's prescription [2] (fig. 2.1). In fact we shall only consider all Regge cuts lying half a unit below the non-pomeron dominant pole.

* As correcting terms to a weak absorption model with exchange degenerate poles

G. Girardi et aL, Duality diagrams o

109

c

(o)

(hi

(c)

(d)

Fig. 2.1. Regge-Regge cut (a) with the sub-processes particle-reggeon I and J of sub-energies s~ and ul (b and c) and the corresponding dual representation (d).

Careful study of physical amplitudes has shown that it is enough to add to exchange degenerate poles corrected by absorption a contribution with well defined properties [4]: (i) existence in exotic s and u channel, (ii) contribution to negative signature with central real parts, (iii) violation of exchange degeneracy. Dual prescriptions enable us to obtain Regge cuts with such properties. Duality diagrams for pole exchange are given in fig. 2.2 (strongly exchange degenerate Regge poles), and cut diagrams in fig. 2.3 (avoiding exotic intermediate

(a)

(b)

(c)

(d)

Fig. 2.2. Duality diagrams for Regge poles (a and b) and for the Pomeranchuck singularity (c and d).

G. Girardi et al., Duality diagrams

110

(a)

(b)

(c)

(a) Fig. 2.3. Duality diagrams for Regge cuts: R * P ( a ) , R * R ( b ) , R * R * P ( c ) and R * P * R ( d ) .

states). This leads to selection rules given in table 1 obtained by looking at the quark path. Table 1 says that a Regge cut diagram exists whenever the corresponding pole diagram can be drawn. Table 1 Classification and selection rules o f duality diagrams Duality diagram

R

R*P

(s, t)

e -~=

e-~='*P

(u, t)

1

I*P

(s, u)

R*R

R*R*P

R*P*R

e-i"=*P* e - l~= I,P*I e-~=* 1 1,e-in=

e-in=* 1, P 1,e-in=,p

2.1. Phenomenoloyical properties of cuts These properties can be immediately read off in table 1,

G. Girardi et al., Duality diagrams

111

2.1.1. R * R and R * R * P cuts

These cuts do not exist in exotic s or u channel and they tend to cancel in negative signature. Indeed if we consider two pairs of exchange degenerate poles R1i and R+,2 because of the rule 1 • e - i ~ , we obtain M- = (R+I-R;)*(Rf

+R2)-(R+ +R;)*(R~-R2)

= 2(R~-*R2-R I*R~).

(2) Also they do not give a violation to exchange degeneracy because, with a ~ ½, 1 • e - i ~ is mainly real. We note that the first two properties were already quoted respectively by Finkelstein [12] and Worden [13], hence such contributions cannot give the required effect. 2.1.2. R * P * R cuts

These cuts contribute in exotic channels, and they contribute in negative signature: M - = (R;- +R~-) * P * (R~- + R 2 ) - ( R + - R 1 - ) * P * (R~- - R2) = 2(n~+ * n * R 2 + R ; * P * R ~ ) .

(3)

As a double convolution of poles, their contribution is central. Also they give a violation of exchange degeneracy because in an exotic channel 1 • P , 1 is mainly imaginary. These properties show that the R . P . R cuts make it possible to describe the physical amplitudes in the framework of an absorption model with lower lying Regge cuts satisfying duality requirements. This makes it differ from a previous model [14], for which R * R cuts played the main role. 2.2. Regge cuts f r o m s-channel unitarity

Concerning R , P , R cuts, which are the main new singularities we have introduced, arises the same question as for the absorption corrections: what are the physical arguments for introducing such terms and what can be said about their sign and their strength with respect to the other singularities? Actually, in its present status dual theory of two-body scattering does not answer these questions. It is tempting to refer to the same physical picture as is relevant for absorption: the reflection of the many body production channels in the two body scattering by means of unitarity. In the following we propose a generalization of the shadow scattering picture which, in addition to the usual absorption correction provides us with secondary contributions in agreement with dual selection rules. The Pomeranchuck singularity can be described as the shadow scattering of the many particles channels opened at very high energy, through the unitarity equation Z AJ AJ t

.lf,.l,i = I m

n

PS•fi.

(4)

G. Girardi et al., Duality diagrams

112

Eq. (4) exhibits the t-channel partial waves of the Pomeranchuck amplitude (P J) and multiproduction amplitudes (A J). Here we have implicitly used the coherence condition of Van Hove [15], that is only diagonal terms (the initial and final states are the same, i - - f ) which are real positive, give a non-vanishing contribution. In the following, we consider that A, is described by the multiperipheral model (fig. 2.4).

Ani

(ReA)ni

Fig. 2.4. Multiperipheral graph and its Regge correction.

In order to study, through shadow scattering, the cuts in which the Pomeranchuck appears, we propose a correction to the T.i amplitudes, such that (see fig. 2.4)

(5)

T.i = A,i + {R * A}ni .

This form is nothing else than the two first terms of the Regge perturbative expansion of the production amplitudes. This is particularly true in dual perturbative theory where the Pomeranchuck singularity is built by dual multi-Regge diagrams [10]. Substituting T,i for A,i in the unitarity equation (4) gives new contributions to the imaginary part of the two body amplitudes, with different order in the R interaction, of which we shall give a more explicit form

~ TrS TS.i* = ~ [A[.+(R * A)L][AS~ +(R * A).i]s* n

n

j Jt = [~AfnAni]+[E(R,A)Sfn An isf+Afn(R.I n

* A)nij~f-]

(6)

n

+ [ ~ (R * A)S.(R * A)Si*] = Yp+ YRv+ YI~eR• n

Here Yp is the imaginary part of the Pomeranchuck amplitude and we will show that YRP and YReR are associated respectively with absorption corrections and the R P R cut. Using the convolution formula (see appendix A) with one intermediate state, one can calculate YRP and YRPR. One finds

(R * A)[.

( R * a)a.*i

=

=

ip

87~x/; -

s Ah.,

Rfh

ip c - As, Rj? .k ki

8n,,/s

(7) .

G. Girardi et aL, DuaRty diagrams

I 13

Using eqs. (4) and (7), YRe can be interpreted as absorption corrections:

YRP _ 8 in~P~s[Rfh(E A hs . A ,stI ) _ ( Z

-

~s ast~os% -"3tfnZanklJt~kil

ip s st 8 ~ - / ~ [R[h Im PS6hl--Im p 6tk Rki ].

-

(8)

"v-

With the approximation ReP = 0 we obtain

YRP =

lp

Im

PJ(RJfi-R~i f) =

2p

Im p s l m R [ i

- lm (2 i(P * R)Si). Similarly we rewrite YneR:

YRPR

=

ip 2 a a s* *J 8~ ~-s Rfh(Z Ah"A"k)Rki "

(9)

In exotic reactions YReR corresponds to the Regge-Pomeranchuck-Regge duality diagram: 1 . P * 1, since the Regge pole contribution is of (u, t) type in these reactions. Using the definition of the convolutions (eq. (7)), one can transform (9) to obtain

Ym,R =

lSnImPl•i

= -Ira(1 .P.

1)/i.

(10)

In the non-exotic reactions, the Regge pole contributions are of (s, t) type, thus not real in the s-channel. So that the preceding trick cannot be used. However s - u crossing relates exotic and non exotic channels, i.e. 1 • P • 1 and e - i ~ . p . e-i"~ contributions. Then, once existing in an exotic channel, the R • P • R cut also exists in non-exotic channels, with the form predicted by duality selection rules. With the proposed correction to T.i (eq. (5)) we only obtain part of the total two body amplitude, since I m R and some contributions to the R * P * R cuts are lacking. Indeed, we never claimed to get directly a solution of the whole unitarity problem. In particular, we did not pay a detailed attention to the contributions of the two body channels themselves to unitarity. Our statement is that solving exactly the unitarity equation will not change the singularities which are determined by the inelastic channels, but only give the lacking contributions. The effect of production channels may be well understood when it contributes to background, say in exotic channels. In this precise case, eq. (6) can be interpreted as a genuine solution of unitarity since R and R • P do not contribute to the discontinuity of the amplitude, likewise for R . R and R * R , P in agreement with duality prescriptions (see subsect. 2.1). In non-exotic reactions where background and resonances occur, the situation is less clear, since I m R itself, is not

I 14

G. Girardi et al., Duality diagrams

an output of our calculations. The difficulty for exhibiting exactly the R • P • R term seems related to the peculiar nature of such a contribution which is dual of both resonances and background (fig. 2.3d). Our approach is quite similar to Caneschi's who first introduced the same kind of corrections in many body scattering [16]. We differ in the application, since no use is made of unitarity as an iteration procedure. This allows us to stay compatible with duality and the physical picture of our cut corrections. Among the differences, let us note that we do not take care of the elastic part of unitarity, and of P • A, contributions. In our scheme they would correspond to P * P cuts in the two-body amplitude and only modify the Pomeranchuck singularity. A new feature of R • P • R cuts appears in eq. (10): their negative sign. We point out that this result is quite interesting when related to the continuous war between the "plus sign" and "minus sign" of Regge cuts. In our derivation, R • P cuts appear with the "plus sign" in agreement with the "eikonale" or " a b s o r p t i o n " picture, but R * P • R cuts have the "minus sign", in agreement with the "multiperipheral" or "rescattering" picture. This behaviour is well justified by the phenomenological analysis of sect. 3 in pion-nucleon and kaon-nucleon scattering. Such a feature rules out a general eikonale expansion, even modified to take into account duality, since it would predict always the plus sign. The strength of R • P and R • P • R cuts can be discussed by taking into account all intermediate states h or k which can appear in the convolutions (eq. (7)). Contrary to what is sometimes obtained in other absorption models [9], the introduction of intermediate excited states does not change the strength of absorption in our unitary framework. The summation over these states in equation (8) for YRe is cancelled by the g-functions (6hl and 6~0, which are there because of the coherence condition (4). Only the fundamental states (h = i, k = f), remain, thus giving an absorption strength equal to 1, in agreement with dual reggeon calculations [4]. The result for the strength of R * P • R cuts is somewhat different. Since taking into account intermediate states involves a double summation on h and k in eq. (9), the coherence condition (4) gives non vanishing contributions for all diagonal terms (h = k). Therefore the strength of R • P • R cuts can be modified, in particular when the couplings of both reggeons to intermediate states are of the same sign. In this case (for instance for two identical R-exchanges) the intensity of R * P * R cuts can be notably increased. This makes it possible to consider that R . P * R cuts are strongly coupled to the physical channels.

3. Application to nN and KN scattering

3.1. Parametrization of the amplitudes We have to parametrize the different contributions of the Regge expansion (1), in agreement with the corresponding duality diagrams for r~N and K N scattering.

G. Girardi et al., Duality diagrams

115

3.1.1. Regge poles (R) As previously mentioned the use of duality diagrams implies the existence of pairs of exchange degenerate Regge poles, with natural parity in ~rN and K N reactions. The parametrization we choose is in agreement with the high energy limit of a Veneziano formula. The residues are constant, up to an overall scale parameter vo

(Vo # 1/c~')

\Vo/ where v = ½ ( s - u ) , i is the t-channel isospin, and n the total helicity flip in the s-channel. Writing down the Regge poles and their combinations corresponding to the (u, t) and (s, t) duality diagrams, we define Regge poles: R~- = P ' , R o = co, R +

=

A2,

R~- = p,

and we obtain for (u, t) diagrams: 1"o = ½(P'+co),

rt = ½(A2+p),

1 t (s, t) diagrams : ~oo = ~(P -co),

(Pl = ½(A2-p).

(13)

One also defines the inverse formulae p' = ro+~Oo,

co

=

ro--Cpo

,

A2 = r l + ~ o t ,

P = rl-qo 1.

Exchange degeneracy has three important consequences. (i) A unique Regge trajectory,

ocv(t) = ~A2(t) = ~W(0 = ~o(t) = ~ 0 + e ' t . (ii) Exchange degeneracy of intermediate mesons (n, B)(K, Q) in Gribov vertices. Since Regge poles have different G-parity in a same degenerate pair, exchange degeneracy cannot give a relation between Regge pole couplings in nN--* nN, where positive G-parity is selected. One has to relate lrN ~ nN to the reaction teN ~ BN, where B stands for the Bouddha resonance, the degenerate partner of the pion. In fact, using (s, u) terms to describe the Gribov vertices (fig. 2.1a) also implies degenerate intermediate particles. Therefore we consider the degenerate pairs of mesons with unnatural parity (~, B), (K, Q). (iii) Exchange degeneracy and SU(3) relations. The couplings of Regge poles to external particles are related by exchange degeneracy and SU(3) invariance. One

116

G. Girardi et aL, Duality diagrams

obtains (p, A2) / gpNrl I,gonn [ (co, P') {gvNr~

gA2 N~I

gA2nB = 2 g o ~ = 2gpQ~, = 2qA2KIE

=

2gA2Q~. , (14)

goNN

With these relations, only two residues in 7rN scattering are to be determined, gt and gO: gl = goNNgp~,

gO = gl"NNgP'n~.

Since there are two s-channel helicity amplitudes M+ and M_ (see appendix A)the number of independent parameters is actually four. The Pomeranchuck singularity is taken as a simple pole with a linear trajectory p = ia~o veat VCq,(t)- 1 e-½1~ (~p(t)- 1).

(15)

3.1.2. R e g g e expansion in zcN scatterin9

The amplitudes for rcN scattering are defined in terms of t channel isospin, M ° and MI: rc-p~z-p:

M°+M

1,

rc+p~n+p:

M°-M

1,

r~-p--.rc°n :

- ~ / 2 M 1,

r~-p--*r/n

x/~M'l-

:

The amplitude M 'x is deduced from M 1 by changing the signature of all contributions. Having parametrized the input Regge poles, one obtains the cuts corresponding to the diagrams of fig. 2.3 by convolution. The different contributions of the expansion (1) are listed in table 2. Introducing some arbitrary weights in order to take into account the unknown strength of Regge cuts, we parametrize our amplitudes as follows: M ° = P+P',(l+2aP)+b{(P'*P'-og*w)+z(A2*A2-p*p)}*(l+2aP)

+d°{(P ',P,P'+co,P*og)+z(A

2 , P * A2 + p * P * p) },

M ~ = p,(l+2aP)+2b(p,P'-co,A2),(l+2aP)+2d~(p*P*P'+cg*P*A2),

(16) M ,1 = A 2 , ( l + 2 a P ) + 2 b ( A 2 , P ' - c o , p ) , ( l + 2 a P ) + 2 d X ( A 2 , P , P ' + o ) , P , p ) .

G. Girardi et aL, Duality diagrams

117

Table 2 Duality diagrams contributions in nN and KN amplitudes :~N scattering

M° M ~ M'I

P

R

R*P

R*R

R*R*P

R*P*R

p

r°+tp ° r~-q~t rl+tpl

(r°+~o°),p (rl-~gm),p (rt+cpl),p

frO,cO frO,~O,p [rl,tpl [rl,epl,P r ° , ~ - ~ o ° , r I rO, q ~ , p - ~ O , r l , p r°*tpl+~°*r ~ r°,¢l,p+~oO,rl,p

f r O , p , r O + q~o, p , ¢po [rl*p*rl+69~*P,¢~ r~,P,r°-¢~,p,q~ ° r~,P,r°+~ol,p,q °

KN scattering M°x

P

r°

rO,p

M,O,

p

tpo

~po,p

fr° , P , r [rl,P,r ~ tpo, p , [ ~°I*P*

r~ ~ol

r~ * P tpl , p

r °* P * r l ¢o , p , q~l

Mix M~½~

o 1 ~o ~pt

In expression (16), one did not take into account helicity. The complete expressions for helicity amplitudes are given in the appendix B. Let us discuss now the physical meaning of the parameters we choose. (i) Absorptive cuts. In agreement with the most commonly used reggeized absorption models, we use only two parameters: the strength of absorption in nonflip (a+) and flip amplitudes (a_ = / m + ) . It is the same for different isospin amplitudes and different absorptive cuts (R • P and R • R • P). The weak absorption model corresponds to a+ = 1. This choice is made in our fit, in agreement with our theoretical arguments (see subsect. 2.2). (ii) R • R cuts. In eq. (16), we have assumed that all different R * R duality diagrams have the same weight: we obtain the same contribution in flip and non-flip helicity amplitudes, a factor two due to the symmetrization R~ * R2+R2 * R1 in M 1 and finally a determination of z (z =-~-: see appendix C). R • R cuts cancel in the amplitude M 7, but not in M [ : this important point for the determination of the parameter b in the fit is shown as follows. From table 2, we get R *R o¢ r ° , ~pl

¢po, r ' = g ° g 1[(4--'~)" 1 *e - i ~ - l * e -i'~ (4-----;)n3,

where we used an abbreviate notation: r i = g i l , q~i=gie-i~. In the non-flip amplitude (n---0), the result is zero. In the flip amplitude (n = 1), there is only a partial cancellation, as can be shown from the convolution formula of two different exponentials in the t variable:

(x/~tt)e #t*e#''-e#~*ep''(x/'~) - ~ / - ~ e #p't/¢p÷#'' fl-fl-------~'# O, 16gs

(fl+fl,)2

118

G. Girardi et aL, Duality diagrams

with here fl = const. + ~' log s, fl' = const. + ~' (log s - in). In practical calculations we take [bl = ½, which corresponds to the strength _+ 1 for each duality-diagram contribution. The sign, undetermined f r o m any of our theoretical arguments is determined from the phenomenological analysis. (iii) R . P . R cuts. O u r parameters are d I and d o , the strength of R * P . R cuts in isospin-one and isospin-zero amplitudes. In the flip amplitudes, no new parameter is introduced, the same ratio as for absorption corrections having been choosen for flip over non flip strengths of the cut.

(a:)

d°

O u r phenomenological analysis shows that d o and d ~ cannot be taken equal, in particular as concerns the cross-over in ~N and K N reactions. It appears that the same a m o u n t of 0 9 , P , P ' and p , P , P ' is obtained only with d ° = 2 d 1 (see appendix B). We considered it as an empirical rule, justified by a remark made at the end of subsect. 2.2: diagonal R , P , R diagrams, such as the diagram (~o + P ' ) * P * (09 + P'), can be strongly enhanced with respect to non diagonal ones, such as ( p + A 2 ) * P * ( P ' + 0 9 ) . We are then left with only one new parameter d = d 1 = ½d °. 3.1.3. Regye expansion in K N scattering The main difference is the absence of R * R and R • R * P cuts (see subsect. 2.1). In table 2 are shown the contributions o f the other Regge cuts for the following amplitudes describing all the reactions we consider: K+p~K+p,

M=M°x

+

M~x,

K-p~K-p,

M = M .oe x +

K+n~K°p,

M=

2M~x,

K - p--*K°n,

M=

2M~x,

a Mnex,

the amplitudes o f exotic (Mex) and non-exotic channels (M,,x) are related by s - u crossing. SU(3) and exchange degeneracy relations (14) allow to express directly the amplitudes of K N scattering in terms of the parameters of rcN scattering, if we take the same parameters d o and d 1. We get M~° = P + ½(P' + 09)*(1 + 2 aP) + l d o [(p, + og)*P*(P' + 09) + z ' ( a z + p ) . P * ( A 2 + p)], 0

t

r

M , ex = P + ½(P' - 09)*(1 + 2 aP) + ½d o [(P' - 09)*P*(P - 09) + z (A2 - p)*P*(A2 - p)], Mix =

½(A2+p)*(l+ZaP)+dl(P'+09)*P*(A2+p),

M~n~=

½(Az-p)*(l+2aP)+dl(P'-09)*P*(Az-p).

(17)

G. Girardi et al., Duality diagrams

119

In the expressions (17), all reggeons (except P) have the same couplings as in n N ~ n N reactions. The only change with relations (16) is z', equal to 5 (see appendix C). We note that the strength of R • P • R cuts, remains the same (with respect to Regge poles): as an example compare M~x (formula (17)) to the combination ½ ( M I + M '1) (formula (16)). This is not an obvious result, since it comes from a cancellation of two factors: The SU(3) factor ½ which relates nn couplings to K K couplings (formula (14)) and a factor 2 due to the contribution of K and Q intermediate states in all R P R cuts. 3.2. Results The parameters of the model are given in table 3. We have twelve free parameters, including those of Regge trajectories. The starred parameters in the table are fixed a priori from theoretical considerations. Table 3 P a r a m e t e r s o f the m o d e l . T h e s t a r r e d n u m b e r s c o r r e s p o n d to the p a r a m e t e r s we d o n o t v a r y in the fit a) R e g g e p o l e s

P

Helicity

Couplings

t-dependence

Regge trajectories

M+

(o'~)~N = 21.2 m b (tr~o)tN = 16.9 m b

ann = 2.75 aKN = 2.2

Ctp-- I = 0.2 t

So = .275

ct = 0 . 5 + 0 . 7 6 t

M+

gO = _ 19, g l = - - 4 . 5

M_

gO = - 1 , g l , - 18

R

b) R e g g e cuts

M+ M_

R*P

R*R

R*R*P

R*P*R

a

b

c ( = a)

d

1" 0.5

0.5* 0.5*

1" 0.5*

-3 - 1.5"

3.2.1. Calculations and results The twelve parameters are determined, using a set of data on pion-nucleon reactions. The fit leads to excellent results for other pion-nucleon observables and all kaon-nucleon data in the usual range of validity for reggeized models (5
120

G. Girardi et al., Duality diagrams

The first step was to fix all parameters of Regge poles (table 3a) with a fit to the total cross sections (a.~, fig. 3.1) and the differential cross section ( d a ~ , da x, fig. 3.3, 4) in the energy range 5 _
(mb)

x-p

27 25 !

/.0 P~bGeV/c 5(] aFig. 3.1. nN total cross sections (data from ref. [31]).

The next step consisted in fixing the Regge cut parameters, p and d (table 3b), using all our set of data, including elastic polarizations (P±, fig. 3.5) and imposing the position of the cross-over point, as determined by a recent accurate measurement of d a ~ at low energy [17]. Note that there is another "free" parameter namely the sign of dual R • R cuts. The sign is determined from elastic polarizations which are sensitive to such a small contribution near the symmetrical zero ( - t "-- 0.6 ( G e V / c ) Z ) . The curves for the other observables in pion-nucleon scattering are predictions of the model: the rotation parameters (R, A, fig. 3.6), the charge exchange polarization parameter (px, fig. 3.8). It is well-known that absorption corrections give a non-zero px, but negative after the transfer momentum value of the crossover [18]. In our model, we obtain a polarization staying positive at this point, in

(Re/Ira) It =0

10

'

20

'PLab

GeV/c x-p

-.1 t

.~+p

-.2

Fig. 3.2. Ratio (Re M + / I m M+)t =o for n + p elastic scattering. The data are from ref. [32] (crosses for n+p and circles for n - p ) .

G. Girardi et al., Duality diagrams

121

D. I.

I

Fig. 3.3. n ± p elastic differential cross section for different value o f PJab varying f r o m 2.5 GeV/c to 50 G e V / c (data f r o m ref. [33]).

(GeV/c)2 1

Fig. 3.4. Charge exchange differential cross section for Plab varying from 2.5 GeV/c to 50 GeV/c. T h e d a t a are f r o m ref. [34].

good agreement with data [19]*. More generally, the results of our fit in pion-nucleon scattering are excellent, showing that we reach a good description of the amplitudes themselves. This is well confirmed by the comparison of our amplitudes with the results of the amplitude analysis at P~ab = 6 GeV/c (figs. 3.9, 10) [20]. The reaction n - p ~ qn is predicted, using SU(3) invariance and the experimental value of the FQI--.27)/F(q ) branching ratio [21]: The differential cross section predictions (da ~ fig. 3.7) and the polarization (P~, fig. 3.8) are in satisfactory agreement with data. In the K N reactions we have to fix the two parameters of the Pomeranchuck singularity (table 3a). A simple geometrical relation reduces this number to one parameter

(O'~)~tN

((Yoe)KN

alrN

aKN

O u r a g r e e m e n t is better with the C E R N data t h a n with the A r g o n n e data. It s h o u l d be r e m a r k e d that this is n o t a test for choosing the best data or the best m o d e l since the s a m e qualitative features are present in both sets o f data: a non-negative polarization, with a variation o f the peak with incident energy. In particular a smaller value o f d (not c h o o s e n by the minimization) would agree with the A r g o n n e data.

G. Girardi et al., Duafity diagrams

122

i

0

.5+~

1.

1.5

J-

I

I

6 GeM~

~

~

T

-t

-"

II--

p

-s" I,E.J.L'I" I'

-t

p'_

P-

-.5

,p

1/, GeV/¢ '~p

.5

0

.5

1.

1.5

t_...-- P + t -'i ~...~ p_ It

-t P

-.5

-.5-

10 GeV/c

40 GeV/c

Fig. 3.5. Elastic polarization at Plab = 6, 10, 14, 40 GeV/c from ref. [35].

o

6 GeVlc

4A +

4R + •

.

~5

~R0.

-1 1! RX

r

-t

I"A~

I

-t 1 AX

Fig. 3.6. R and A parameters for elastic and charge exchange n N scattering. The data are taken from ref. [36].

G. Girardi et al., Duality diagrams

123

It~.L

.5

1.

-t

Fig. 3.7. Differential cross section for n - p - - + r / n f r o m Plab = 5.9 GeV/c up to P~ab = 18.2 GeV/c. The data are from ref. [21]. A pX

,

1.

1.

0

t.~

-.5 Dx .-=

=11

. _t =

- .5 /..9 G e V / c

.5

-.S 7.85 GeV/c

Fig. 3.8 Polarization for ~-p----~z°n and 7r-p--+v/n and the comparison with the C E R N data [19] at Plab = 4.9 GeV/c and 7.85 GeV/c.

G. Girardi et al., Duality diagrams

124

++1 4.

2.

0

.5

FL//

1.

! 1.

-t F +0 _ /

I

-.4 Fig. 3.9. Isospin zero amplitudes (~/m-b/GeV/c) versus - t in (GeV/c) 2. The flip amplitude is defined with respect to M ° , F , being the parallel component and Fj. the orthogonal one to M°+. Data are from ref. [20].

.4

FI+//

FI++ / .2

.2

5

1 i-~~ffFU'-~

1.

1.-'-t

2" .5

1.

-t

Fig. 3.10. Isospin t amplitudes with the same definition as in fig. 9.

G. Girardi et al., Duality diagrams

125

Then, with only one parameter, assuming SU(3) invariance and exchange degeneracy, all is predicted in the four studied reactions: the total cross sections (fig. 3.11), the elastic (fig. 3.12) and charge exchange differential cross sections (fig. 3.13) and polarizations (fig. 3.14a, b). The good results we obtain are confirmed by the analysis of K N amplitudes [22] (fig. 3.16). Note in particular the good results for the

r 0 T

(rob) 25

1

20 K+p

I

I

I

I

5

10

15

20

- b~Pta_neV/e

Fig. 3.11. Total cross sections for K + p (data from ref. [37]).

1 -t

1. - t

Fig. 3.12. K + p elastic differential cross sections (data f r o m ref. [38]).

G. Girardi et al., Duality diagrams

126

~'t (K+n--~K'P) P'b/(GeV/c)2

5

i:d° (K~-PK~)[~b/(OeV/e)2

.01 12." -t .5

1.

Fig. 3.13. KN and K N charge exchange differential cross section. Data from ref. [39].

polarization in K-p--* K°n (fig. 3.15). This is a pure prediction since the (preliminary) data at P~ab = 8 GeV/c have just now become available [23] after we obtained our results.

3.2.2. Sensitivity of the fit and the role of secondary cuts The sensitivity of the fit to the Regge-poles parameters is strong, due to their small number. Only slight changes of (Go~)~N, (a~)KN, gO, gl+, and ~o are allowed the parameters ~', ctp, a~N and v0 can vary a little more, since they are important only in the t-dependence of the cross sections. The parameter Vo, in particular, is fixed by the differential cross section d~rx at larger transfer. It corrects the behaviour of all Regge residues together. The less important parameter is go, limited to small values. t

G. Girardi et al., Duality

diagrams

P+

?K

.5

.5

6 GeV/c

10 GeV/c

Fig. 3.14b

Fig. 3.14a Fig. 3.14. Kfp

P’K

127

elastic polarization.

Data from ref. [35a].

In particular,

no clear choice for its sign seems to be indicated by data (we differ on that point from amplitude analysis. See fig. 3.9). The Regge cut parameters (table 3b) are less precisely fixed by the data, since the gross features of the data are well reproduced by the Regge poles. Nevertheless their order of magnitude and their sign are well fixed by the role they play as far as the detailed structures of the data are concerned, making it possible to go from a qualitative to a quantitative understanding of the experimental results.

6 GeV/c

Fig. 3.15. K-p+K”n

polarization

at Plab = 8 GeV/c from ref. [23].

128

G. Girardi et al., Duality diagrams

(i) The absorption strength. The parameter a+ was fixed at one in our fit, as previously indicated. But relaxing this constraint does not change the final result: a minimization gave a+ = 0.99. Let us insist on this point, which is an important feature of the model. It is k n o w n that the physical amplitudes with negative signature have peripheral imaginary parts [24]. This is also true for our amplitudes as shown in the impact parameter representation (fig. 3.17). In our " w e a k " absorption model, R * P cuts alone are not strong enough to obtain peripherality, indicated for instance by the position of the zero in I m M~+ (~- F/l, see fig. 3.10). The R * P • R cut gives in the imaginary parts the a m o u n t of absorption that is lacking. This is opposite to what is called the strong absorption mechanism [9] for which all the peripheral character is due to R • P cuts. The value of # (/~ = a_/a+) which is mainly sensitive to ax and P ± at larger m o m e n t u m transfer ( - t > - 0.5 (GeV/c) 2 demonstrates t h a t absorption corrections have to be weaker in flip amplitudes. We now turn to the role played by the different cut contributions. (ii) The R * P * R cut. The R • P • R cut plays a key role in the non flip amplitudes (.the R . P . R cuts are also depressed by the factor /~, as well as absorption corrections). In negative signature amplitudes (p, ~ exchange) it increases the absorption in the imaginary part, as previously mentioned and decreases the

\ 20(

I

M:

100

-t

x

x x

A/

/ o¥

M'_

i// /

/

__

Real

Parts

___ lmon. Ports

Fig. 3.16. M ° and M x_in ~v/p-b-/GeV/cversus - t in (GeV/e) 2 for KN scattering. The solid (dashed) line represents real (imaginary) part. Data from ref. [22].

G. Girardi et al., Duality diagrams

//

129

1.

b fern~

b fermi

M'_

1 H+ Rear Ports ----

Imag.l~ts

Fig. 3.17. Hankel transform of isospin 1 amplitudes in arbitrary units versus the impact parameter in fm. The solid (dashed) line represents the real (imaginary) part.

Ax

L

.5i

1. i

T

r

]

v

AK

-t

Fig. 3.18. Difference between elastic scattering of particles and antiparticles from proton. We have quoted the quantity: dx =

dcr/dt ( X - p ) - d~/dt (X + p) ~/8 [dcr/dt ( X - p ) + dtr/dt (X + p)

where X is ~ or K. Data from ref. [17].

in ~ ¢ / ~ / G e V / e

130

G. Girardi et aL, Duality diagrams

Im (I)) ....

\

-~

....

++

I m (F~ ++

__1,..- . . . . . .

_~

Fig. 3.19. Comparison of imaginary parts of isospin 1 amplitude and non-pomeron isospin 0 amplitude (non-flip parts) normalized to one in the forward direction.

Fig. 3.20. Comparison of exotic (solid line) and non exotic (dashed line) K N charge exchange difforential cross sections for Ptab varying from 1.2 GeV/c up to 100 GeV/c.

G. Girardi et aL, Duality diagrams

131

absorption in the real part, introducing a noticeable central contribution (fig. 3.17). This can be seen in another way through the zero structure of the amplitude M+~ , where, contrary to the usual absorption and other peripheral models, Re M~+ vanishes only after Im m~+ [5] (compare F) to F/) in figs. 3.10). This leads to the good sign of the polarization P~ (fig. 3.8) and the correct cross-over at low energy in n + p ~ n + p and K - + p ~ K + p as shown by the prediction for the quantity A at Plab = 5 GeV/e (fig. 3.18) A =

[(da)_-(da)+]

~/8 [(d.)_ + (da) +] In positive signature amplitudes, we have the opposite effect: the real part is more absorbed and the imaginary part less. This gives a striking violation of exchange degeneracy. In particular one may note the double zero in Im P', in contrast with the single zero in I m p (fig. 3.19) and obtained without there being any zero in the reggeon residue [25]. In the same way, a violation of exchange degeneracy appears in the low energy extrapolation for the K N and K N charge exchange differential cross sections (fig. 3.20). In fact, in qualitative agreement with the data [26], the exotic cross section ( K + n ~ K ° p ) is larger than the non exotic one ( K - p ~ K ° n ) . Note also the similarity with the hypercharge exchange cross sections [27], for which the absorption model gives the opposite prediction. Another effect of R • P * R cuts, is to give a small break in the elastic differential cross sections, since it corresponds to a small, but slowly decreasing function of t. This effect near - t = 0.8 (GeV/c) 2 is too small to agree with the data but its energy dependence is in agreement with them, showing that a better treatment of large transfer in the parametrization of the Regge poles for instance, can easily improve the fit. (iii) The R * R cuts. As previously mentioned R * R cuts contribute in the M~ amplitude but not in M~+. In fact, they play a role in px, avoiding a zero at - t - - 0 . 6 (GeV/c) z, which would exist if the Regge poles only or with weakly coupled R , P and R * P , R cuts were present. This fixes their sign but their strength is not accurately determined. 3.2.3. Predictions and limitations o f the model

Our set of results shows a good quantitative agreement with the data over the range in energy and transfer which is usually considered in reggeology. In particular, when compared with the usual absorption models, our result is that the phases and the zeros of the amplitudes, that is the t-dependence, are essentially well reproduced in the domain - t _ < 1 (GeV/c) 2. The question of s-dependence is more controversal and is to be discussed now in more detail. At very high energy, (P~ab- 20--30 GeV/c) one expects that the R and R , P contributions should dominate the secondary contributions R * R, R * R * P and

132

G. Girardi et al., Duality diagrams

i

I0I

, 20

PLob GeVTel._

I0

2toO PLQb GeV/e

Fig. 3.21. Variation of the cross-over point with energy.

R * P * R, due to the difference in intercept (at least at small t). This means that all features of a pure absorption model will appear at high enough energy. In particular the cross-over point is expected to go to larger value of t (fig. 3.21) and the charge exchange polarization to get a zero at small transfer (fig. 3.22). We believe that the second effect is a better test of the validity of our model than the first one, due to the fact that it depends on the real part of the amplitude, which is less affected than the imaginary part by possible changes in the Pomeranchuck amplitude or in the

p(1~-p.-~ItOn)

|

--_, I.

_

PtobV

Fig. 3.22. Variation of the polarizations with energy (n-p-->~°n and n-p--+r/n).

G, Girardi et al., Duafity diagrams

133

strength of absorptive cuts. We then propose to experimentalists measurements of px at an impulsion in the laboratory higher than 20 GeV/c.

!

P(K+n-'-~K°P

j

~

5

>'

,~

.~

"~~.'5

,,:,o ~"""

20

1I.

-'t

1.

;20

//10

-~

,'/,s

/'//

-1

:,(K-p.._~.KOnI --

Fig. 3.23. Variation of the polarizations with energy ( K - p-->K°n and K + n--+K°p).

At small energy (P~b < 6 GeV/c) one expects the effects of the secondary cuts to increase. Almost qualitative agreement is obtained with nN differential cross sections at Piab = 2.5 GeV/c (fig. 3.3 and 4). In K N scattering, one obtains a qualitative agreement with the low energy data on charge exchange differential cross sections, as already mentioned (fig. 3.20). Nevertheless, one problem arises about the question of s-dependence: at low energy and near the forward direction, the R • P • R cut seems to give a too large contribution to the K + p total cross section under P~,b = 8 GeV/e (fig. 3.11). This fact is related to the detailed F.E.S.R. analysis of Worden [28], which shows that the imaginary part of the non flip amplitude is too much absorbed at low energy, due to the stronger effect of the R * P * R cut. In fact all models which propose to modify the absorption model in order to improve it have the same difficulty. That is the reason why we think that the problem is not due to the R • P • R cut but to other singularities contributing to the imaginary part, probably the Pomeranchuck singularity. Other results (very high energy differential and total pp cross sections, for instance) show that the Pomeranchuck singularity is more complicated than a simple Regge pole. In fact a dual Pomeranchuck singularity corresponding to Regge cuts in the s-channel, would have to be smaller than the actual pole, at low energy,

134

G. Girardi et al., Duality diagrams

since the cuts give an increasing contribution to the background at small s, instead of a constant one. An explicit example of such a Pomeranchuck singularity is given in Lacaze's thesis [29]. Then the R * P * R cut is not obliged to be small, and could give the precise contribution which has to be added at low energy to an increasing but small Pomeranchuck contribution.

4. Conclusion and outlook

Though it is based on a small modification of the reggeized absorption models since it consists in taking into account secondary cuts, our model leads to some interesting conclusions as far as the theory and the phenomenology related to duality problems is concerned. 4.1. Phenomenological conclusions and tests o f the model

First, the smallness of the modification itself makes one believe in the basic validity of the absorption idea, as applied to Regge poles. R • R and R • P * R cuts are naturally obtained as corrections to the first dominant Regge singularities (P, R, R * P) which make up the reggeized absorption model. Secondly, duality makes it possible to find selection rules which help in reducing the number of free parameters. Let us recall here the main features of R * P * R cuts: (i) they contribute to exotic channels, (ii) they strongly contribute in negative signature amplitudes, increasing the absorption in the imaginary part, and decreasing it in the real part of the amplitude, (iii) they give a violation of exchange degeneracy, since in the positive signature amplitude they reduce the absorption in the imaginary part contrary to what happens in the negative signature amplitude. R • R cuts, for which duality selection rules forbid those same effects, nevertheless play a role in some amplitudes in particular in that with s-channel helicity flip and negative signature. All the secondary cut contributions modify the amplitudes of the absorption model in such a way as to give a quantitative description of all observables in 7oN and K N scattering, in particular the charge exchange and elastic polarizations and the cross-over effect. The question arises of finding a test of the model, especially of the secondary Regge cuts we introduce. One test is given by the energy variation of the model amplitudes: at high energy, a typical property of the model is to tend to the " w e a k " absorption itself, with a charge exchange polarization changing sign at some high energy (see fig. 3.22). A second test can be provided by studying other reactions where phenomenological problems appear. In particular, the violation of exchange degeneracy in hypercharge exchange reactions [27] ( K - p ~ T z - Z +, ~ + p - ~ K + Z +) gives identical effects as in low energy K N and K N reactions ( K - p - ~ K°n, K + n ~ K ° p ) . The comparison with data in these reactions, by a straightforward

G. Girardi et al., Duality diagrams

135

application of our model, will then be very useful. The extension of our model to meson-meson and baryon-baryon reactions (with different duality diagrams and selection rules) is also under study. A third test concerns more particularly R , R cuts, which are less sensitive to the fit in the reactions we have studied. R,_R cuts intervene in the description of forbidden peaks or t-channel exotic reactions. Our selection rules (see table l) forbid R * P * R cuts in such reactions, but not R * R cuts. Therefore we consider it very important to study these reactions in connection with the particular selection rules of R * R cuts. 4.2. Duality, unitarity, and a theoretical outlook The Regge cuts play in our model a very important role within the framework of duality. The R , P , R cut gives a natural mechanism for breaking exchange degeneracy, in contrast with a model involving Regge poles only as in the first versions of duality. Exchange degeneracy appears as an exact symmetry property of Regge poles, with a " w e a k " violation due to absorption corrections (if Re P # 0) and a " m a x i m a l " violation by R , P * R cuts (the R * P * R contribution in an exotic s-channel is nearly imaginary). Moreover incorporating R * P * R and R * R cuts into phenomenological duality relation between resonances and t-channel singularities (now Regge poles and cuts) leads to a new aspect of this relation which can be called "duality for real parts". Indeed it is possible to associate the central contribution in the real parts of the amplitudes - characteristic of R * P * R and R * R cuts (fig. 3.17) - with similar features of low energy amplitudes due to resonances. We know that the main resonances are peripheral, since their physical spin J and masses M obey the empirical relation b=J+l~ M

lfm,

where b is the impact parameter. In fact the imaginary parts of the peripheral resonances are strong near the corresponding energy (x/s = M), but the real parts vanish at this point and have extrema at lower and larger v/s (cf. the Breit-Wigner formula). In terms of the impact parameter, this gives respectively larger and lower values than 1 fm, that is non peripheral real parts. It is possible to show by comparison of the amplitude analysis in the resonance domain (P~R~= 2.5 GeV/c) and the Regge domain (P~ab = 6 GeV/c), that the two effects are related [20]. The use of s-channel unitarity (cf. subsect. 3.2) provides theoretical arguments for R • P and R * P * R cuts. It connects them and gives satisfactory results for their signs and strengths except for the flip amplitudes, for which an explanation of why they are practically not absorbed is still lacking (cf. however Lovelace [30]). The modification proposed to the multiperipheral graph is one of the first corrections which appear in a dual perturbative expansion of many particle production

136

G. Girardi et al., Duality diagrams

amplitudes. Such an a p p r o a c h to m a n y b o d y production processes, together with the same for two-body scattering, seems to be an interesting open field for phenomenology. We would like to thank F. Hayot, H. Navelet and particularly G. CohenTannoudji, and the whole p h e n o m e n o l o g y group at Saclay (DPh-T) for their constant help and enlightening discussions. We are grateful to P. Le D u for c o m m u nicating us recent experimental data on the polarization in K - p ~ K ° n at 8 GeV/c, and J.P. Merlo, for the polarization in n - p ~ n - p at 40 GeV/c. We are also grateful to E. Cotteverte for her careful typewriting.

Appendix A. Kinematics and notations A . 1 . K i n e m a t i c variables

We consider two-body reactions, which are described by amplitudes depending on the variables s, t, u: (fig. A.1) s -----(Pl-t-P2) 2 , p2x = p2 = m z

t = (pl--P3) 2 , p2 = p2 = M z

u --- ( p l - - P 4 ) 2 ,

s+t+u

= 2MZ+2m

2

•

Fig. A.1. Scattering 1 +2--+3 +4.

One also uses the variable v = 1 ( s - u ) . We use kinematical variables defined in the c.m. frame, the m o m e n t u m p, the scattering angle 0, or in the lab frame, the energy and m o m e n t u m of the incident particle E1ab, Plab. We have the relations:

1

p = - -

2,/;

El~b =

[(s-(M--m)2)(s-(M+m)2)]

s_M2_m

2M

2

,

1/2,

Plab ---- ( E ~ b - - m 2 ) 1/2 •

t

cos 0 = 1 + - 2p 2'

G. Girardi et aL, Duality diagrams

137

A. 2. Amplitudes and observables The reactions we consider are of the type (spin 0 + s p i n ½ ~ s p i n 0 + s p i n ½). The scattering amplitudes we use are s-channel helicity amplitudes. We have four amplitudes M~,~, depending on the helicities of the nucleon 2, 2'=_+½, among which only two are independent, M÷ and M_: M 1 / 2 , 1/2 = M - l ~ 2 , M1/2,

-1/2

---

-1/2

-M-1/2,

:

M+

1/2 : M _

(no helicity flip), (helicity flip).

The observables are obtained using the following formulae total cross section:

1 atot = - -

differential cross section: polarization:

Im M+ (s, 0),

da _ 1 (IM+ (s, 012 + IM- (s, t)]2), dt 647rsp 2

P = 2 Im M+M*_/(IM+I2+IM_I2).

A.3. Partial-wave expansion and convolution

The partial-wave expansion is written as follows M~,~ = Z ( J + 1/2) d~,~ ~ (0) M~,~, J J

where d~,2 are the rotation matrix elements. We define the convolution operator • on the helicity amplitudes

(M1,M2)~,~= :P-_ Y. (M1)~,,(M2)~. 8~ ~/s ,=±1/2

Appendix B. Helicity amplitudes of the model In this appendix, we establish the list of all helicity amplitudes for 7rN and K N scattering, in their definitive form for practical use. We consider the Regge pole amplitudes with their couplings to 7rN reactions (formula (14)) and the five Regge cut parameters a, b, d °, d 1, I~ (z ----~--, 11 z' = 5, see appendix C). For helicity amplitudes, one must take into account the summation over intermediate helicity in the computation of Regge cuts. The Pomeranchuck pole giving no helicity flip, only Regge poles are considered in the following formula ( A . B ) + =(A.B)I/21/2 =A1/21/2"B1/21/2 +A1/2-1/2 * B - 1/2 1/2 =A+*B+ - A _ . B _ (A • B)_ = (A • B)l/2 _ 1 / 2 = A1/2 1/2*B1/2_1/2+A1/2_1/2*B_l/2_l/2

= A+ *B_ + A _ * B + .

,

138

G. Girardi et al., Duality diagrams

The nN amplitudes are written as follows: M ° = P+P'+ *(1 + 2aP)+ b {(P+ *P+ -co+ *co+) +'c(A2 + * A2+ --P+ *P+) -(P'_ , P'_ -co_ *og_)-z(A2_ * A 2 - - P - *P-)} (1 +2aP)

+ d ° { ( P + , P , P ' + +co+*P*co+)+z(A2+*P*A2+ + p + * P * p + ) _p2 (p,_ , p , p , _ +co_ ,p,co_)_zl~2(A2_ *P* A2- + p - * P ' P - ) } ,

M °_ = P'_ ,(1 + 2 a p P ) + Z b {(P+ *P'_ -co+ *co_) +z(A2+ * A z _ - p + * p - ) } * ( I + 2 a p P )

+2d°p{(P+ , P , P ' _ +co+ , P , c o _ ) - z ( A : + * P * A2- +p+ * P * P-)}, M~+ = p+ ,(1 + 2 a P ) + 4 d ' (p+ *P*P'+ -I-t:p_ * P* P'-), M 1 = p_ *(1 + 2 a p P ) + 2 b { ( p + *P'_ -co+ *A2-) +(p_ , P + -09_ * A2+)}*(I + 2 a p P ) + 2 d l # { ( p + , P * P _ +t c o + * P

,

A z _ ) + ( p - * P * P + +t C O - * P * A z + ) } ,

M~ = A2+*(1 +2aP)+Zb{(Aa+*P+ - c o + * p + ) - ( A z - * P ' - -co_*p_)}*(1 +2aP) + 2 d x{(A2+ * P ' P + +co+ * P * P + ) - # Z ( A 2 - *P* P'- +co- *P* P-)}, M'-~ = A z - *(1 + 2 a # P ) + 2b{(A2+ * P'- -co+ * p - ) + ( A z - *P+ - c o - *P+)} *(1 + 2 a#P)

+ 2 d 1 tt{(Az+ * P , P _t +co+ * P * p - ) + ( A z - * P ' P + +co- *P'P+)}. The KN and I~N amplitudes are expressed in terms of the same Regge pole amplitudes: 0 M~+ = PKN + z1 ( P i+ +CO+)*(1 + 2 a P ) + i d° {(P + +co+)*P*(P+t +co+)

+z'(A2+ +p+)*P*(A2+ + p + ) _ # Z ( p , +co_),p,(p,_ +co_) -I.tZz'(A:_ + p _ ) * P * ( A a - +P-)}, 0 Mex= I(P'- +co-)*( 1+ 2al~P)+pd o {(P+ , +co+)*P*(P-, +co-)

+z'(A2+ + p + ) , P , ( A : _ +p_)}, o = PnN+½(P+ - c o + ) , (1 +2aP)+½d°{(P+ -co+)*P*(P+ -co+) mnex+ + z' (Az + - p +)* P *(A: + - p +) - I~z(P '- -CO-)* P *(P'- - c o _ ) -I~zz'(Az_ - p _ ) * P * ( A 2 _ - p-)},

G. Girardi et al., Duality diagrams 0

Mnex_

=

139

½(P'_ -6o_)*(1 +2alaP)+lad°{(P'+ - c o + ) , P , ( P ' _ - c o _ )

-}-'c'(A2 + - p + ) * P * ( A 2 _ -- p_)}, 1 Mex+ =½(A2++p+)*(l+eaP)+d

I{(P++a)+)*P*(A2++p+)

- / t 2 ( P '_ +co_)* P*(A 2 _ + p_)}, 1 Mex= ½(A2- + p _ ) * ( 1 +2a~n)+21td I {(n+ +og+)*n*(A2_ + p _ )

+(P'_ + o)_), P , ( A 2 + +p+)}, 1 M.e.+ = ½(A~+-p+)*(1 + 2 a P ) + d I { ( P + - c o + ) * P*(A2 + - p + )

--/22 (P '_ - co_), P , (A 2 - - p_)}, 1 Mnex= l ( A z - - p _ ) * ( 1 +2at~n)+21td 1 {(P+ - c o + ) * n * ( A 2 _ - p _ )

+(P'_ - co_), P , (A2 + - p + ) } . Let us check the empiral rule proposed in the text, relating d o and d t. The contribution to negative signature in 7rN elastic scattering (non-flip amplitude) is: p+ ,(1 + 2 a P ) + 4 d ~(p+ ,P*P'+ _ # 2 p _ , p , p ,

).

The same contribution in K N elastic scattering is mainly due to ~o exchange; we get from the negative signature contribution to M°x: ½~o+ *(1 + 2 u P ) + d°(a~+ *P*P'+ + z ' p + * P * A2+

-1~2o9 , p , p ' _ -t~2.r'p_ , P , A z _ ). The main contributions of both amplitudes are

p + * ( l + 2 a P ) + 4 d l p + * P * P + ,'

½[co+*(1 + 2 a P ) + 2 d o~o+,P,P+]; ,

d ° = 2d 1, is then the condition to have the same correction (up to an overall) coupling): this is taken as an empirical rule in our fit.

Appendix C. Calculation of the factors x and x'

We compute the factors z and z', which give the relative weight of the following contributions to isospin 0 amplitudes: double isospin 1 exchange and double isospin 0 exchange, z and z' are obtained by drawing all different R * R duality diagrams (see fig. C.1, the diagrams are drawn in a simplified but convenient form). SU(2) and SU(3) Clebsh-Gordan coefficients make it possible to relate all the

I40

G. Girardi et al., Duality diagrams

contributions of the diagrams. Applying the rule of the same weight for all diagrams, we get for the pion-nucleon case (fig. C.la, b, c, d) "f =

(a)

1 YE-

~-

~-

=0

I=I

I=oI P

P

P

P

(o)

~-

3 ~-

~-

I=I

I=I

P

P

P

(b) K- K'~Q- K-

I=O

I=O

~0

I

I=1

P

n

P

(c) K'- K~Q-

K-

K-

K-

£=I]

I=I

I=I

I=I

P

P

I

n

(d)

I

P

P

P

P

(o')

P

n

P

(c')

(b')

Fig. C.1. Duality diagrams with isospin 0 exchanged.

In the kaon-nucleon case, we get (fig. C.la', b', c')

,

(b') + (c')

1 + (x/2) 4

(a')

1

(The same values • and ~' were also used for R * R * P and R * P • R cuts in the amplitudes see appendix B). Note that in order to ensure correct signature properties of the Regge cuts, we are obliged to introduce crossed diagrams for some of them when there are neutral intermediate states (fig. C.2). ~-

~0

~-

~+

~0

~+

I Icr='X P

n

P

P

~+ ~0

~+

" l n

P

P

A+ +

P

Fig. C.2. s-u crossed diagrams in the case of neutral intermediate states.

References [1] J.C. Polkinghorne, J. Math. Phys. 4 (1963) 503. [2] V.N. Gribov, JETP (Sov. Phys.) 26 (1963) 414. [3] C. Lovelace, Phys. Letters 34B (1971) 500 and Proc. of the Amsterdam Conf. (North-Holland, 1971) p. 141. [4] A.V. Efremov and R. Peschanski, J I N R preprint E2-6.350 (1972), unpublished. [5] G. Girardi, R. Lacaze, R. Peschanski, G. Cohen-Tannoudji, F. Hayot and H. Navelet Nucl. Phys. B47 (1972) 445.

G. Girardi et al., Duality diagrams

141

[6] N. Barik, B.R. Desai, P. Kaus and R.T. Park, Phys. Rev. D4 (1971) 2933. [7] V. Barger and R.J.N. Phillips, Phys. Rev. Letters 24 (1970) 291 ; E. Leader and B. Nicolescu, Westfield college preprint (April 1972). [8] G.A. Ringland, R. G. Roberts, D. P. Roy and J. Tran Than Van, Nucl. Phys. B44 (1972) 395. [9] B.J. Hartley and G.L. Kane, Rutherford Laboratory preprint (January 1973); F. Henyey, G.L. Kane, J. Pumplin and M. Ross, Phys. Rev. 182 (1969) 1579. [10] G. Cohen-Tannoudji, G. Girardi, F. Hayot, R. Lacaze, A. Morel and R. Peschanski, P6riphe. risme, absorption et dualit6, l~cole d'6t~ de physique des particules 616mentaires, Gif-sur-Yvette, 1972; G. Veneziano, Phys. Letters 43B (1973) 418; HH. Lee, Phys. Rev. Letters 30 (1973) 719. [11] S. Mandelstam, Nuov. Cimento 30 (1963) 1148. [12] J. Finkelstein, Nuov. Cimento 5A (1971) 413. [13] R.P. Worden, Phys. Letters 40B (1972) 260. [14] R. HHong Tuan, J. Kaplan and G. Sanguinetti, Nucl. Phys. B32 (1971) 655. [15] L. Van Hove, Lectures at the Carg6se summer school, 1963; A. Bialas and L. Van HHove; Nuovo Cimento 38 (1965) 1385. [16] L. Caneschi, Phys. Rev. Letters 23 (1969) 254. [17] A.B. Wicklund, I. Ambats, D.S. Ayres, R. Diebold, A.F. Greene, S.L. Kramer, A. Lesnik, D.R. Rust, C.E.W. Word and D.D. Yovanovith; Phys. Rev. Letters 29 (1972) 1416. [18] J.M. Drouffe and H'. Navelet, Nuovo Cimento 2A (1971) 39. [19] P. Bonamy et al., Nucl. Phys. B52 (1973) 392; D. Hill et al., Phys. Rev. Letters 30 (1973) 239. [20] G.A. Ringland and D.P. Roy, Phys. Letters 36B (1971) 110; F. Halzen and C. Michael, Phys. Letters 36B (1971) 367; G. Cozzika et al., Phys. Letters 40B (1972) 281. [21] O. Guisan et al., Phys. Letters 18 (1963) 200. [22] F. Hayot and HH. Navelet, Phys. Letters 37B (1971) 424. [23] CERN-Saclay-Ecole Polytechnique-lmperial College Collaboration, preliminary data. [24] H. Harari, Phys. Rev. Letters 26 (1971) 1400. [25] H. Harari and Y. Zarmi, Phys. Rev. 187 (1969) 230. [26] H. Harari, in summer School in elementary particle physics, B. N. L. 5.0212 (C. 59) (1969). [27] A.C. Irving, A.D. Martin and C. Michael, CERN Th/1304 (1971); B. Sadoulet, thesis CERN/DPhT.II Phys. 71-39 (1971); G. Girardi, CNRS-Marseille preprint 71/P.410 (October 1971). [28] R.P. Worden, Michigan preprint (1973). [29] R. Lacaze, thesis, Saclay report DPhT/72-34 (June 1972). [30] C. Lovelace, Rutgers preprint (June 14, 1972). [31] K.J. Foley et al., Phys. Rev. Letters 19 (1967) 330; Denisov et al., Phys. Letters 36B (1971) 528, 415. [32] K.J. Foley et al., Phys. Rev. 181 (1969) 1775. [33] A. Giacomelli, P. Pini and S. Stagni, CERN compilation H E R A 69-1 (1967). [34] Stirling et al., Phys. Rev. Letters 14 (1965) 763; Sonderegger, Phys. Letters 20 (1966) 75; Yvert, private communication (1968); Wahlig, Phys. Rev. 168 (1968) 1515. [35] P. Laurelli, NP internal report 71-3, CERN-preprint (1971); C. Bruneton et al., Saclay Serpukov-Dubna-Moscow Collaboration, to appear in Phys. Letters. [36] A. De Lesquen et al., Phys. Letters 40B (1972) 277. [37] E. Flaminio, J.D. Hansen, D. R. O. Morrison and N. Tovey; CERN compilations H E R A 70-4, 70-6. [38] K.J. Foley et al., Phys. Letters 11 (1963) 503; R.J. Miller et al., Phys. Letters 34B (1971) 230. [39] P. Astbury et al., Phys. Letters 23 (1966) 396; D. Cline, J. Penn and D.D. Reeder, Nucl. Phys. B22 (1970) 247.