Massive quarks and large transverse momenta

Nuclear PhysicsB80 (1974) 299-321. North-HollandPublishing Company

MASSIVE QUARKS AND LARGE TRANSVERSE MOMENTA (I). Large-angle t w o - b o d y scattering G. PREPARATA CERN, Geneva

Received 27 March 1974 (Revised 4 June 1974)

Abstract: We develop the ideas of the massive quark model into an effective computational scheme for large-angle two-body hadronic scattering. Our results resemble somewhat those of the interchange model of Blankenbecler,Brodsky and Gunion, but are derived in a different physical context, which does not suffer from the well-known difficulties of the standard parton model.

1. Introduction The massive quark model (MQM) has been so far developed [1,2] to cope with the problems of electromagnetic and weak interactions in the deep inelastic region. Its main ideas, which we shall bri~fly discuss in the next section, describe the hadrons as composite structures made out of a "quark stuff", which has the crucial property of only appearing in nature in zero-triality conglomerates. This remarkable and mysterious fact requires quite peculiar propagation properties of the quark matter, of such a character which seems definitely extraneous to present day field theoretical ideas. Nevertheless we think it useful to proceed further with ~ese ideas, which are not obviously unfounded, and try to discuss a wider class of physical problems, with the hope that this may lead tls to understand the hadronic world at a more fundamental level. This paper is the first attempt to use this description ofhadronic matter to calculate properties of purely hadronie scattering. The lack of a simple and neat probe like an electromagnetic or weak current forces us to develop the perturbative scheme which was briefly sketched in ref. [ 1]. This is done in sect. 3; we emphasize that the analysis carried out there is relevant not only for the problem of large angle scattering which is the object of this paper, but can be applied to the forward and the backward directions as well. In the last year or so.we have seen a considerable interest, both experimental and theoretical in hadronic interactions with large transverse momenta. The theoretical problem has been neatly discussed by Berman, Bjorken and Kogut [3] and b y

G. Preparata, Massive q u a r k s

300

Blankenbecler, Brodsky and Gunion (BBG) [4]. The spirit of our approach is very close to the one of the latter authors, who in a series of extremely interesting papers have advanced a very suggestive interpretation of large angle scattering in terms of a quark interchange force. As emphasized by Bjorken [5] the problem is to find a mechanism which is responsible for fairly hard collisions between hadrons, but not as hard as those (electromagnetic and weak) which exhibit scaling. From the experimental analysis, one begins to unravel a number of power laws, obeyed by the two-body differential cross sections of the type d__~a, l__f(co s 0c m ) dt

sn

•

•

(1.1) '

where n and the function f (cos 0c.m.) depend on the particular scattering process. The power law (1.1) has been discussed by a host of authors [5-7], and in particular by BBG [5] who are able to compute n and to obtain approximate expressions for f(cos 0can.) for a number of processes. Although our analysis is quite different we obtain very similar results, which compare quite satisfactorily with the experimental data, i.e., for meson-baryon scatterhag we derive do MB

s \ ?1 1og2 I[7) IMB
<12)

with [(z) a very simple function; for baryon-baryon scattering doBB 1 1 2 s

(1.3)

and similarly for BB. Another interesting aspect of the existence of a hard component in hadronic scattering is the possibility of learning about the phases and the structure of the Regge amplitudes by looking at the interference of the Regge terms with the large angle amplitude. In fact, the experimental data on p~- and lrp scattering show very suggestive interference patterns. This may represent a welcome shot in the arm to the phenomenology of two-body and quasi two-body scattering. Our analysis can be also extended to photon induced processes and to inclusive hadronic processes. This shall be carded out in future papers. The plan of the paper is the following: sect. 2 deals with the ideas of the MQM and the development of the perturbative scheme. In sect. 3 the analysis of the important diagrams is carded out; sect. 4 contains the study of the basic ingredients of the calculation. Sect. 5 is devoted to a discussion of the computational problems, sect. 6 to spinology, and f'mally, the results for two-body hadronie scattering comprise sect. 7.

G. Preparata, Massive quarks

301

2. The basic assumptions of the model. The perturbative scheme The MQM, in the present phenomenological state, revolves around a number of quite general ansatze which have been abstracted from our experimental and theoretical knowledge of the properties of hadrons. This is a list [1,2] of the basic assumptions: (i) quarks are a relevant degree of freedom of hadronic matter; (ii) The quanta associated with such fields are very massive (MQ ~ oo is also conceivable); (iii) hadronic states can be described as bound states of zero triality conglomerates of strongly interacting quarks; (iv) quark Green functions display Regge behaviour in the high energy regions and have a fast decrease in the "masses" of the external quark legs; (v) one can construct the hadronic amplitudes in terms of the quark Green functions by following an iterative procedure in the number of the intermediate quark legs. The first four assumptions have been discussed and motivated at length elsewhere and we refer the reader to the published literature [1, 2]; the fifth, being crucial to the developments reported in the present paper, deserves particular attention and discussion. In the first paper on the MQM [1] we pointed out that a number of known facts seemed to suggest that hadronic dynamics, even in its multiparticle aspects, attributes to the quarks a good degree of individuality, a situation which can be described as the absence of the "quark soup". It was natural, based on this suggestion, to try to develop a perturbative approach to the calculation of hadronic amplitudes, by expanding on a small parameter associated with the number of quark legs needed to construct such amplitudes in terms of the more elementary building blocks which are the quark Green functions. In order to develop a consistent scheme for calculating hadronic amplitudes, which makes use of the assumed saturation properties of quark forces, we shall quark tire(q)

Meson w.f.

diquarkline(d)

Bory(~ w.f.

q~

qd

-

~

dd

0(xz) Ve

Fig. I. The elementary building blocks of the perturbative expansion.

302

G. Preparata, Massive quarks P

0

0

0

Fig. 2. Lowest order approximation to the pp scattedng amplitude at high energy. P

P

o

o

P

P

÷

P p Fig. 3. t-channel iteration of the pp amplitude. The square box represents the amplitude of fig. 2.

proceed as follows. The elementary building blocks of our diagrammatic expansion are shown in fig. 1. They are: (a) quark and diquark lines; the diquark lines are useful (as suggested by the baryon spectrum) when dealing with baryons, they should be looked at, however, as a compound field; (b) meson and baryon wave functions; (c) elementary scattering amplitudes : quark-antiquark, quark-diquark and diquark-antidiquark, their magnitude is associated to a coupling constant ~'I which is not much smaller than 1 (see later); (d) irreducible vertex functions, they describe the coupling of three meson like systems (V6) and two baryon- and one meson-like system (V8). They are associated with a couplingconstant ~2 ~< ?'1, as suggested by the small ratios between particle widths and masses. ~ The diagrammatic expansion b u t t out of these four ingredients will thus obey crossing and unitarity order by order in ?'1 and ~'2 provided the elementary amplitudes do so. We recall that in order to obtain amplitudes with the correct analyticity properties, the loop integrals involving quark lines must be defined as principal value integrals, as discussed elsewhere [2]. An estimate of the value of X1 is quite problematic at this stage; in fact one should keep clearly in mind that the basic ingredients of the perturbative expansion vary with energy in an important way. However, just for orientation purposes, the total high energ~ pp inelastic cross section is 0(X4) (fig. 2) while the elastic cross section is 0Q,~). The experimental ratio between elastic and inelastic is approximately 0.2, thus X4,,. 1 - 0.2, a not terribly convergent expansion parameter. In view of this it is convehient, whenever possible, to sum classes of diagrams which only involve X1 through well-known iteration schemes (see, for example, fig. 3). As for the coupling ~'2 we can proceed perturbatively in the usual fashion.

G. Preparata,Massive quarks

=(p~)

~t

c~)

303

s:(p~ +%)2=(p3+p* )2

s

d (p~)

t= C~ -IO31 = (P4- P21 ,,: (q -P~ )2 =(p2 -% )2

Fig. 4. The large angleamplitude and the relevant kinematics. ~t o

C

b

(s)

+

d

+ ....

d

(t)

(u)

Fig. 5. First approximation to the scattering amplitude for the process in fig. 4.

3. Large-angle scattering. Analysis of the important diagrams Let us consider the scattering process shown in fig. 4 for large s and 0can. fixed. Following the perturbative strategy expounded in sect. 2, the first approximation 0(?~2) is given by the diagrams in fig. 5 (scalar case). Each one bf the three diagrams in fig. 5 gives rise to two distinct topologies by decomposing the kernels V6 into their Mandelstam components, i.e., we have in order the topologies (st), (su), (ts), (tu), (us) and (ut). These diagrams are the same as those considered by BBG [4] and describe what they call the "interchange" force. However, it is easy to convince oneself that these diagrams in our picture cannot give rise to the relatively slow fall-off of the cross-sections at large angle. In fact the asymptotic behaviour of this amplitude for large values of s and fixed c.m. angle 0c.rn" is determined by the off-sheU behaviour of the quark legs of the kernels appearing in fig. 5; it can be power-like only if the fall-off is like a power as it appears clearly in the BBG calculation. This, however, is inconsistent with the assumed fast decrease of the quark Green function in the external masses, and with the related and more important physical requirement that the constituent~ shall not appear as real intermediate states. This is not surprising, since the BBG approach is really a parton model calculation. Incidentally, the suppression of the first approximation diagrams in the large angle region illustrates an aspect of the expansion discussed which should always be borne in mind, i.e., the important rble played by the energy range in determining the relevant diagrams. However, the diagrams in fig. 5 may play an interesting r61e at low energies in the forward and backward regions, as we shall see later.

G. Preparata,Massive quarks

304

~ Q ..........J c 4. ~

c

(st)

(ut)

Fig. 6. Second approximation to the scattering amplitude for the process in fig. 4. tl

t2 Od;"-"

,.A,2t P+A/2

f ,/,'-w2 --"t

}....

P-A/2

Fig. 7. The basic kernel of large angle scattering. Having disposed of the first approximation, we shall consider the diagrams to order k4 (fig. 6). They are two box-like diagrams corresponding to the topologies (st) and (ut). Corrections of order kl have the effect of yielding rescattering corrections which will have no effect on the structure of the amplitudes but changing only the normalization (which cannot be determined at this stage). In fig. 6 we have encircled with dashed lines portions of the diagrams which we call kernels which can be discussed globally according to point iv) in sect. 2, without resolving them into their elementary components. The determination of these diagrams at large s and fixed 0e.m. requires an accurate study of the properties of the kernels involved, this shall form the object of sect. 4. However, it should already be clear that we cannot exclude a priori a power law in s; in fact we can route large momenta through q~- channels, which do not decrease fast due to their non-exoticity.

4. The basic kernel For 0can. fixed and large s, t must also be large and this can only be accomplished if at least one of the momentum transfers tl, 2, say t 1 , is large. Thus for tl, t large and t2, t' = (l_/)2 fixed we obtain for the relevant fictitious production amplitude the Regge limit exhibited in fig. 8 *. Proceeding in a standard fashion and neglecting any Toller angle dependence the leading Regge contribution is, for fixed M 2 > 0, given by (we parametrize the various cut-off functions as exponentials)

[ --t~~]q ( O ) K R (M 2, t, tl, t 2 . . . ) ~ ~

f(M2, ot)ebt'e-blK21 e- b 2K21, 2

(4.1)

where KI± and K2z are the transverse momenta of the legs P - ~A and ½A _ l' respectively, and are given by (neglecting all masses) * For a discussion of similar amplitudes, se~ thethlrd paper of ref. [2].

G. Preparata, Massive quarks

r

p+AI2

o÷r

,

tk./o.t'

t~.~_.._

M2= (p+Q)2

~

t' = (I-[') 2

t"

I+A/2

Fig. 8. The Regge limit of the basic kernel.

K21.~M2

t tl- t '

305

ptA/2= ~q= I+A/2

Fig. 8a. The double Regge limit of the basic kernel.

(4.2)

K2± ~ _ t2 oe;

finally a (0 ~
K ( M 2, t, t 1, t 2 . . . . ) = ~1 of M ' 2dM - M'22 - i e diSCM2K(M, 2 . . . . ).

(4.3)

One should notice that the integral extends only over positive values of M '2, because for M '2 ~< 0 we cannot have any discontinuity (superexotic channel). In the region • of interest we can compute dlSCM2 K(M ' 2 . . . . ) from K R in (4.1), and using (4.2) we readily obtain ~-t) ~q (0)e m.... e b.t~a z z K(M 2, t, t 1, t2", . . . )M-+ 2 largeM 21 + ie

G(a,

a),

(4.4)

where a = (tit l - t ) and G(a, a) is given by the integral representation: 1 G(a, a)= - ~- ? dM2 7 (M2, a) (M2)-aq(O)e -blM2a. 0

(4.5)

Eqs. (4.4) and (4.5) are the central results of this section and the basic ingredients of the calculations reported in this work. Notice that the assumed dispersion relation (3.3) implies that the kernel K has a behaviour (powerlike) at large M 2 which is very different from the one exhibited by its discontinuity discM2 K (exponential); this is analogous to what happens for

G. Preparata, Massive quarks

306

Mueller-Regge six-point functions where the strong cut-off in transverse momentum is not a property of the full amplitudes but only of their discontinuities. Of importance for the following is to determine the behaviour of G(a, or) when a ~ 0. It is clear from the representation (4.5) that such limit is determined by the behaviour o f T ( M 2, a) when M 2 ~ oo. This limit is governed by the double Regge limit of fig. 8a, and we get easily the result:

G(a, ct) ~

g(ot) (a) aq(0)-l.

(4.6)

a~0 In the following we shall need G(a, a) also for negative values of a, the integral representation (4.5) defines an analytic function with a cut running from - ~ to 0, whose behaviour around a = 0 is determined by (4.6). We shall also assume that the limit a ~ 0 is harmless *. Thus according to (4.4), for large M 2 and t the kernel K exhibits a pole located at J = - 1 ** ( J is the angular momentum associated with the partial-wave expansion in the t channel) and its residue is given by a function of t which falls with a power ( - a q ( 0 ) ) determined by the electromagnetic form factor ***. Furthermore the exponential factors in (4.4) require that t 2 and t' remain small. The origin of this pole is obvious by looking at eqs. (4,3) and (4.5); in order to remove it one needs the breakdown of the dispersion relation and the related exponential behaviour at large mass, which does occur in the Veneziano-like amplitudes. Therefore the fixed pole is a general property of all theories which have nonexponential behaviour at fixed angle and leading Regge behaviour at small M 2 [eq. (4.1)]. In sect. 5 we shall use these results to compute large angle scattering according to the approximation expressed by the diagrams of fig. 6.

5. Computational procedures Before applying the results of the previous sections to compute the physically interesting processes, we shall calculate one of the diagrams in fig. 6 (see fig. 9) for the fictitious case of scalar particles. The reason for doing this is to exhibit in a simple situation some of the tricks and short-cuts which render calculations with many loops manageable. Firstly we attribute the G 2 amplitudes to one of the kernels,this will only modify the normalization of the kernel but not its behaviour given by (4.4). The amplitude in fig. 9 is then given by * Recall that the Hmit a ~ 0 with a f'Lxedcorresponds to the double Regge limit in which a "quark" trajectory ~q(t') is exchanged in the t' channel. ** Thus, this is not the fixed pole of Compton scattering. *** This connection is also present in the work of BBG, ref. [4]; but it is somewhat different, hence the difference between their and our predictions (see later).

307

G. Preparata,Massivequarks

A (s, c°s O) -- f

d41

f

d41'

f

d4Q

(5.1)

K a4 K23"

In the large angle scattering region large momenta will go through the quark legs; the assumed fast decrease in the "masses" can then be expressed via 6 functions of the masses [1]. We shall also replace all exponentials with 5 functions whenever their arguments are integration variables, an obviously legitimate procedure. Thus we write _ . . 2 ~ . ( 0 ) £ d4Q f d4/ A(s, cos 0) ~ 2 ( - 0 ~ J c ~ ) 4 J ~ J - ~ -

fd4l ' ~ 6 [(l + ½A)2] 8 [(a + t')2].

1

X 1----~-62 2 [(1'-~a)2] b61 [(l_l,)2] f da 6(t2) ot b 2 M1 M2 0

f d a G14 (a, or) G23 (a, or)

where M2 = (Pl + Q - ~ A)2 and M 2 = (P2 - Q + ½A)2; and the factor of 2 comes from the identical contribution from the region [t21 ~" It 1 I. The string of 6 functions appearing in (5.2) can be substantially simplified by taking into account the identity 6(p 2) 6(q 2) 6

[(p_q)2] = ~rfdX64

(q -~W) 6 (P2),

(5.3)

which is quite easy to derive. We get 6 [(Q + 1') 2] 6 [(1'-½A) 2] 6(t2)6 [(Q+/)2] 6 [(•_•,)2] = ~Tr2 f d X 6 4 (a÷ 1 - X (a + l'))fda 8 4 (½A - l' -- ct ( a + l')) 6 [(a + l,)2], (5.4) and inserting this expression into (4.2) we can perform the l and l' integrations iramediately, obtaining

t r~Cp2)

n: (P4)

Fig. 9. A typical diagram contributing to large angle scattering.

G.Preparata,Massivequarks

308

t 2~q(O) 1 A(s'c°sO)"212~)312(-)b-~2 O f. a(lda+ot)2

xdFd4Q(2rr) 4M 21---~f d ~ ' ' ( t 2 ) ' ( t l ~ - a ) ' I ( 1 - - ~

d4e XjT~8(t2)

6

8 !~1-

fdaG14(a, oOG23(a,a) (Q+½A)-(Q-½A))2

t a + 1~ 1 --a-]M21M2 "

(5.5)

Let us now discuss briefly the main features of (5.5). Counting powers of momenta we readily conclude that (within power of log s)

A(s, cos O) ~-~-/(cos 0),

(5.6)

where m = 2-2 aq(0), thus obtaining a power law for large angle scattering of the well-known type do -dt

~

1 - s2m+2

F(cos

0).

(5.7)

The actual form off(cos 0) depends on the structure of the G's and on the particular form of the Q integrand in (5.5). We report in the appendix the evaluation of the integral in (5.5) in the case of the st topology. From those results, we get (- t) 2~q(O)-1 12 A (s, cos O) -~ K I2(~n)312 bb

s

log(-~-2) ,

(5.8)

where 1

d~ (a, c,) ~). K = - f d a f a(1 +a) G14 G23(a' 0 is a finite constant according to the discussion in the appendix. We shall next apply these procedures to the evaluation of the realistic case where spinning particles are involved.

309

G. Preparata, Massive quarks

6. Spinology Our task is now to apply the developments discussed in sects 4 and 5 to the physically interesting case of spinning particles. We shall treat mesons and baryons separately. 6.1. Mesons

We wish to extend the result (4.4) to the case Where the hadronic legs are rr's (or K's) and the quark legs carry spin ~. We are only interested in the legs comprising the t 1 channel; the t 2 channel is in fact at finite mass and no important spin factor can arise from there. Thus we are led to consider the simplified amplitude in fig. 10 where the dashed line represents the "quark-like" Regge trajectory aq(t') describing the r¢ form factor [2].

Khlh2 (M'2, . . . ) = -%

(I+~A)

%

(6.1)

( I - ~ A ) K ( M '2 . . . . )

where u and ff are Dirac spinors of zero mass and represent the quark legs in the asymptotic region. Thus we compute K++ = K_ _ = 0,

(6.2)

K+_ = K_+ = ( - t ) ½ K ( M ' 2 , . . . ),

(6.3)

where K ( M '2 . . . . ) is given by (4.1), following the dispersive procedure we finally get [see (4.4)] K+_(M 2, t, tl, t 2 . . . . )

_

. .~q(O)+½ ~-t) ebt'e b2t2a G (a, a) large M 2 M 2 + ie ~

(6.4)

1 ebt'e b2t2a G ( a , ~) , M 2 +ie

where use has been made Ofaq(0) = -½ [2]. 6.2. Baryons

For baryons the problem is a bit more complicated; we have to take care also of the spin of the hadrons. The simplified diagram is drawn (fig. 11); note that the

h1~*~/2 . t,.y---5. .... rt(P+AI2) q

h2~ °M2 ~,~ 'ix(p-A/2)

Fig. 10. Simplified diagram for the basic mesonic kernel, exhibiting the spin structure.

(7,.Preparata,Massivequarks

310

hq1~)t+l~l 2 hq~ p(p~) hl--'"'"o.';q'"'"'Vp(p2 ) hz Fig. 11. Simplifieddiagramfor the basic baryonic kernel. •

.

t

•

•

dotted line represents the "diquark-like" Regge singularity a2q(t ) dominating the baryon form factor [2].The helicity amplitudes now depend on the coupling structure at the hadronic vertices; the experimental indication that the GE form factor is suppressed at large q2 [ 12] suggests that the coupling Bqa2q is helicity conserving *, i.e.,

Khlhql, h2hq2= U--hqi(l+ 1A)'yc~Uhl(Pl)~h2(P2)"/c,Uhq2 (t--½A) 1 K(M,2 .... ) ; (6.5)

the extra factor lit comes from the helicity flip factor in the Regge expansion. From (5.5) we compute

Khlhql,h2hq2

1 ( - t ) ~2q(0) -+Shlhql ~h2hq2 2 M 2

ebt,eb2t2~GB(a, or)

l (1)ebt'eb2t2a G(B) (a, ct) = 8hlhqlSh2hq 2 -~

(6.6)

where we have set t~2q (0) = -1 [2] From (5.4)and (5.6)we can already determine the power law of large angle scattering for meson-baryon and baryon-baryon scattering. The angular dependence requires an explicit evaluation of the integrals (5.5) which shall be done in sect. 7. We have (within log s) for meson-baryon scattering AMB(s, cos/9) = l___~f

smMB

MB

(cos 0),

(6.7)

where mMB = 3, as it is immediate to check from (5.5), (6.4) and (6.6); according to (5.7) we have do MB 1 dt (s, cos 0) ~ ~ FMB (cos 0).

(6.8)

With analogous notation, we compute mBB = 4,

andtherefore * This is also assumed by BBG but with different conclusions.

(6.9)

G. Preparata,Massive quarks

311

de BB 1 F dt (cos 0 ) ~ s l 0 BB (cos 0).

(6.10)

These results coincide with those of ref. [14], but not with those of BBG whose power law is s -12.

7. Two-body hadronic scattering In this section we shall complete the analysis of two-body hadronic scattering.

7.1. Elastic pp scattering The topologies one must consider are shown in fig. 12. Other topologies must be excluded because, as it should be clear from figs. 7 and 8, we must require that from a proton (antiproton) vertex come out only p or n quarks (~-, ~ antiquarks) but not X, p, n quarks (~-, p, n quarks *). We shall make use of the results of sect. 6 and of the appendix to compute the (ut) topology; the other diagram shall follow by a simple substitution. By substituting in (5.5) the expression (6.6) we obtain:

Ahlh2,h3h 4 (u, t) = ~hlh4Sh2h3 1

4

bb 2

--(27r)"rd4Q,(t2),(ta_t~_~_)

! t~(ldtZ+a)

1 22' M1M ~ (7.1)

using (A.4) we get: fd4Q

5(,2)8(tl_tl+____aa~ 1

lallog(_t/la2)"

1 _ 1 [ ~

As w.e can see from the calculation in the appendix the origin of the log is to be traced to the integration region t 1 large t 2 small, M 2 large and M 2 small (or.equivalently M 2 large M12 small).

(ut)

(tu)

Fig. 12. The topologies contributing to pp elastic scattering.

* This argument also appears in the BBG work.

312

G. Preparata, Massive quarks

Therefore we can write, by following the same procedure as before:

(ut) (ut) 1 -(__~) Ahlh2,h3h 4 = c 6hlh4 6h2h3 ~t31og ,

(7.2)

l

where c is a real constant which depends on the host of normalization factors which we cannot fully handle at this stage. Notice again that the amplitude is real. The (tu) topology can be obtained by making the substitution (u ~+t), and we get

Ahlh2,h 3h4(t,u) = c ~hlh48h2h3 ~-3 log --

.

(7.3)

To get the full amplitude, according to Fermi statistics, we have to antisymmetrize over the final protons and we get *

Ahlh2,h3h 4 ~ c log

(~-)[u~- ~] +

(~hlh4~h2h3 - ~hlh3~h2h4 ) ,

(7.4)

from which we compute the unpolarized cross section: d o P P c2 s ~ [u 4 + 2u2t 2 + t4]. dt s~- l°g2/a2 (ut)6

(7.5)

This result can be cast in the following power law form: dcrPP dt

-1- log2 ( ~ 2 sl0

Fpp (sin 0),

(7.6)

where F p (sin 0)

1 I16-16 sin 2 0 + 4 sin4 0] (sin 0) 12

(7.7)

Landshoff and Polkinghorne (LP) [8] have carried out an analysis of experimental data and shown that a power law d__~_o-+ 1 f (sin 0)

dt

sn

does indeed hold, with n = 9.7 + 0.5 and the function f(sin 0) being very close to (sin 0) -14 . In view of this (7.6) and (7.7) appear as quite successful. (In fig. 13 we plot the calculated Fpp (sin 0) together with (sin 0) -14 and the experimental points as deduced from the LP analysis.) We should also mention that the actual form (7.7) is quite insensitive to the assumption of helicity conservation made in sect. 6. • I thank Professors S. Brodskyand P. Landshofffor pointing out a mistakein an earlier version of this work.

(7.Preparata,Massivequarks \

\

313

\

4-

A

3-

2-

1 X

I

0.5

I

I

0.6

I

07

0.8

I

0.9

I

1

sing

Fig. 13. Plot of log F pp (sin 0) (eq. (7.7)), the dashed line corresponds to the curve: (sin 0)-14; the experimental pbin~s are taken from the analysis of ref. [8]. P~)

ct2q _ P(P3)

P(R)~

P(P3)

%qi (st)

i~2. (ts)

Fig. 14. Topologies contributing to pp" elastic scattering.

7.2. p~ elastic scattering The interesting topologies are in fig. 14. The calculation of the cross section is straightforward, we need only substitute in the amplitudes (7.4) u with s. We have daPPdt ~ s 2c2-:1°g2(--~)

[s4+2s2t2+t4].-~ 1

(7.8)

We should also notice that the amplitude is asymptotically real with the ratio p = (Im/Re) -> O(1/ln s). A comparison with the 5 GeV data of Chabaud et al. [9] is shown in fig. 15 *. We notice the striking difference between the pp and the pp- data. Whereas the pp cross section is smooth the p~ shows a typical interference pattern. * Recall that (7.8) should not be trusted for Itt < 2 GeV2.

G. Preparata, Massive quarks

314

e~ o\ • N\ •

10-I

~eo

\

I/ \

\

\\%

///

\

!

/ PP

/

10`2

E

~I~ IO"

10-~.

i0-s

I

1

I

2

I

3 -t

I

4

t

5

It

6

(GeV2)

Fig. 15. Plot of (dtr/dt)PP [eq. (7.8)] atp = 5 GeV/c. The data points are from ref. [9]; the dashed line is calculated from (7.7). We can qualitatively understand * this by noticing that in p~ we have a strong Regge amplitude with a rotating phase combining with a "large angle" amplitude with a fairly small imaginary part. In fact, with a linear meson trajectory a(t) ~- ½ + t ~ e would expect constructive interference to occur at t ~- 2.5 GeV2. Here a pronounced structure is observed of the right shape and size. For pp scattering exchange degeneracy leads to a contribution which is purely real, as in the case of the large angle amplitude and no interference pattern should be observed. We believe that two or quasi two-body scattering phenomenology could greatly benefit from the consideration of "large angle" amplitudes with so well defmed properties. Another interesting aspect of p~ scattering [9] is the backward peak. Its energy dependence is quite step (s -9.4) and cannot be generated by usual Regge exchange (exotic channel). The diagrams of fig. 16 can be relevant here. In fact in the backward direction and at fairly low energy, the diagram in fig. 5 is likely to dominate. We easily compute: ---+--d° 1 (S)2a4q(O) e2bU du

s6

• Similar observations appear in ref. [21 ].

G. Preparata, Massive quarks

315

S--~ |

U

Fig.16. Diagramcontributingin backwardpff elasticscattering. et ~t rc(r~) c~ n'(;h) n'(p,) n'(~) P(Pz) P(P~) (ut) (st) Fig. 17. Topologies contributing to *T+p elastic scattering.

where a4fl(0) is the trajectory associated with the exchange of four quarks. We know that Ot4q(0) < a2q(0 ) = - I , and we could speculate that Ot4q(0) = -2. This would lead to

do ~ l__l_eT.bu du

sl0

perfectly consistent with the experiments. Finally the expression (7.8) fits quite well the energy dependence s -4"2 observed for - t = 2.5 GeV2 [9], where the Regge contributions begin to be unimportant.

Z3. Irp scattering The relevant diagrams for ~r+p scattering are in fig. 17. The amplitudes are given by (the indices +, - refer to the proton helicities) 1 2 , A(~)= 0, A(~t) -+Ap log (~)Ut-----

(7.9)

A(+SO_~-An log(~2)l----~, A~)= O.

(7.10)

Thus .4+__ 0r+p)~t~-log C~22) l a p

;hi.

(7.11,

For rr-p scattering we have analogous diagrams and simply obtain A+_ (rr-p)-+

t~ log ( ~- ) [ 4Ap An ] , s

and finally from isospin invariance

(7.12)

316

G.

Preparata, Massive quarks

10 -I

rt* 0 T~•

10-4

10-s I

I

I

1

2

3

[

I

I

I

4

5

6

7

-t (GeV~)

16ig. 18. The predictions (7;14) and (7.15) compared with the 5 GeV data from ref. [10]. The forward and backward straight lines are extrapolations of the forward and backward Regge contributions. ~" at 10 GeV.

I0-3

e~t

/ ~0-~

.o E

~1~ 1°-~

10 -6

10.7

,

I

4

,

I

6

,

I

,

I

8 10 -t (GeV2)

m

I

12

~

I

F i g . 19. S a m e as i n f i g . 18 f o r t h e 10 G e V d a t a .

i

I

16

317

G. Preparata, Massive quarks 10 ~

•

rt"10 GeV

o

K" 10 GeV

K-

10'

" ~

/

'° -°'.6

-&

-o~2

•

cosOcM

I

°.2

I

o.,

[

0.6

Fig. 20. Plots of R(cos 0) = [do/dt (cos O)]/[do/dt (0)]. 7r± full curve; K~ . . . . curve and K- dot and dash curve.

-

dashed

A+_0r-p -~ nOn) = ~/½ [A+_ (rr+p) -A+_(rr-p)] (7.13) In the asymptotic region we have (define z = cos 0can) dcr (7r+p)_ s8llog2/ ~J) ,

1

1

( l - z ) 4 (1 + z) 2 [2ap +An (I + z)] 2

(7.14)

do s~ (~_~) 1 1 [2An +A p (1 +z)] 2 d--/-0r-p) -+ log2 ( l - z ) 4 (1 + z) 2

(7.15)

do 1 log2 (~_~) 1 (Ap_An)2 [2 +(1 +z)] 2 d-'t-- (rr-p ~ irOn) ~ s 8 (l-z)4(1 + z) 2 "

(7.16)

d7

A look at the data [10] in the large angle region shows that the Ir+ and rr- cross sections are very close. This would in turn require A n ~ A p , and imply the suppression 6f the charge exchange cross section. This seems to be experimentally the ease [11 ], in fact the 90 ° charge exchange cross section shows a decrease with energy with an exponent n ~ 10.7 + 1.8. By setting Ap -~A n we compare with the rr data at 5 and 10 GeV [10] (see figs. 18-20). Again one can notice the pronounced structure at t = -2.6 GeV 2, which can be interpreted as a destructive interference between the Regge term and the real "large angle" amplitude. It can also be qualitatively understood why such structure tends to disappear with increasing energy: the large angle amplitudes decreases like I/s, while the Regge one falls offlike sa(-2-5) = s 2.

G.Preparata,Massivequarks

318

Z 4. Kp scattering The contributing topologies are (ut) for K+p ~ K+p; (st) for K - p -~ K - p and for KI, p ~ KsP. We easily compute

½ ((ut)-(st))

A (K+p) -+ X A p log (-t/la 2) -~-t2,

(7.17)

A (K-p) -+ -• A p log (-t//a 2) slt2 ,

(7.18)

A (KLP -~ KsP)-~ ;

An - T log ( - t / / 2 ) ( 1 + sl-)2,

(7.19)

where we have allowed for a factor X to take into account possible SU 3 violation (X = 1 in the SU 3 limit). From (7.17), (718) and (6.11), we have do + do + X2 d-)- (K p ) / ~ (rr p) = (l_u/s)2,

(7.20)

do (K-p)/ (K+p) = ?7a-/-

(7.21)

The comparison with the data [13] is shown in fig. 21. 10"~I

p:5 GeV. o

K" • K"

10-~

10"~ 10 -s

/ I

2

I

3

I

I

4 5 -t (GeV=)

t

I

6

Fig. 21. The predictions (?.17) and (?.18) (~ = 1.3) compared with the 5 GeV data of ref. [13],

The straight lines are extrapolations of the forward and backward Reggeterms.

319

G. Preparata, Massive quarks

We end here the description of hadronic large-angle two body scattering. We stress that no systematic phenomenological analysis has been presented here. We have simply shown that the ideas described in the first sections of this paper are mdeed capable of yielding results which compare favourably with experiments. Some of these results were already obtained by BBG [4], and in ref. [14], but we stress that this approach is quite different and does not have the difficulties of the standard parton model. Finally we believe that this way of looking at two-body scattering has a corisiderable phenomenological potentiality. We look forward to a thorough analysis which uses the interference patterns mentioned above to learn about the details both of the Regge terms and of the large angle amplitudes and to a vigorous effort to understand the non-asymptotic corrections to the characteristic power law behaviours.

Appendix

We evaluate here the integrals appearing in (5.5) in the st topology (fig. 22). We can change variables and introduce the variable 1 as in fig. 22, we have

sd4e...=s-

d41

(W4 X

S(1*)8 (Z+ iA)* -tv

.

(W4 1

1

@1_()2_p2 t ie (p2t I)*

(A-1)

- p* + ie

where we have introduced a typical hadron mass /.L*in the propagators to avoid possible infra-red troubles. It is easy to check that for a finite the s discontinuity of (A.l) vanishes (just apply the Cutkosky rules), thus we can define the integrals involved as principal value integrals and the amplitude will be real. We work in the c.m. frame where

0, 0, l),

PI = f&L

p2 = iG(l,

0, 0, -l),

A = i&(0,

sin 8,0, cos e-l),

and defining I, = 1, f I3 and II we have 1 f d416

(Z+l_ - z;, 6 2Al- ; ( 1 -a_ (27r)4

z-1- - l

- 1-1*&i+-

b’,dZ_ dl: dq 6 (I+“_ - 1:) 6

4 (2794

x (I-

cos e) 1

1

1

v+-1_3>’ sQ+l_- cj.12/ds)

P*

- 4 IL sin 0 cos cp-4

$ (l+- I_)

G. Preparata, Massive quarks

320

q

r~

~-t

~

t,

r~*t

o~

t ,.° m

q

Fig. 22. The st topology in the scalar ease. We trivially carry out the integrations over l? and ~, and we get _

1

1

1

2 (27r)4 2s~

....

fdl+dl_

1

O(Q)

l+t_ - Oa2/~/s) (t+-t_)

--~/J'

(A.2)

where

E,l+l 1+cos0 1-cosO

Q=

( l + - - I - V~-/a)2].

We now change variables and set l+-I

= x,

=y;

l+l

we then get (A.2)-

1 1 1 f 2 (2~.14 2(-t-)x/~ ** dx f dy Y0 1

X

1 + COS0

1

( x - v q / a ) 2 ~' fy + ~x 2) ,~

1

(A.3)

- Oa2/

where 1 l_cosO (x ~ _ ) 2 . YO-4 l+cosO The integral in (A.3) is of the form (A'3) =

2 ( -10

f

dx G(x)

with oo

1 a(x) - - (21r)4

X

f x 2 1-cos 0 -4--

[(X + ~ / a ) 2 + 4 y ] 2t"

dy

1 ~2 Y-~7~ (x+V~/a)

1 1 +cos 0 _ x

1-cos 0

G. Preparata,Massivequarks

321

F o r lxl ~>V~ IG(x)[ --> 1Ix 2, thus from such a region we obtain a term only o f the order l/s; which can be neglected compared with the contribution from [xl < Vs. In fact we have

-x/s

-x/s

S F~,2

4 . 2 1 + cos 0 7 }

C

._[1 (-¢)

,.._ 1 lal 2 (2rr)3 s

og

+ finite term

,]

.

(h.4)

On carrying out the principal value integral on a (which, according to (4.6) around a = 0 is o f the t y p e ~C(da/a) and therefore finite) we obtain the result in (5.8).

References [1] G. Preparata Phys. Rev. D7 (1973) 2973. [2] R. Gatto and G. Preparata, Nucl. Phys. B67 (1973) 362; Phys. Roy. D9 (1974) 2104; G. Preparata, Exclusive eleetroproduction in a massive quark model; the production of vector mesons, to be published in Phys. Letters. [3] S.M. Berman, J.D. Bjorken and J.B. Kogut, Phys. Rev. D4 (1971) 3388. [4] R. Blankenbecler, S.L Brodsky and LF. Gunion, Phys. Letters 39B (1972) 649; Phys. Rev. D6 (1972) 2652; D8 (1973) 287. [5] J.D. Bjorken, Rapporteur talk at the Aix-en-Provence Conf. September 1973, [6] J.M. Cornwall and D.J. Levy, Phys. Rev. D3 (1971) 712; D.Horn and M. Moshe, Nucl. Phys. B48 (1972) 557. [7] P.V. Landshoff and J.C. Polkinghome, Phys. Rev. D, to be published. [8] P.V. Landshoff and J.C. Polkinghorne, Phys. Letters B44 (1973) 293. [9] V. Chabaud et aL, Phys. Letters 38B (1972) 449. [10] V. Chabaud et al., Phys. Letters 38B (1972) 441; C. Baglin et al., Phys. Letters 47B (1973) 85. [11] A.S. Carroll et al., Phys. Rev. 177 (1969) 2047; J.E. Nelson et al, LBL-1027 (1972); R.K. Yamamoto et al., MIT Preprint (197~); W.S. Brockett et al., Phys. Rev. Letters 26 (1971) 527. [12] J.D. Bj0rken, Talk at the 1971 CorneU Conf. [13] V. Chabaud et a1., Phys. Letters 38B (1972) 445; C. Raglin et al., Phys. Letters 47B (1973) 89. [14] V.A. Matveev, R.M. Muradyan and A.N. Tavkhelidze, Nuovo Cimento Letters 5 (1972) 907; S.J. Brodsky and G. Farrar, Phys. Rev. Letters 31 (1973) 1153. [15] A. Donnachie and P.R. Thomas, Daresbury preprint DNPL/P149 (1973).