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Parton model for hA and AA collisions at high energies
Nuclear Physics North-Holland
AS23 (1991) 694-714
PARTON
MODEL
FOR hA AND HIGH
AA COLLISIONS
AT
ENERGIES
M.A. BRAUN Departament
E.C.M.,
Facultat de Fisica, Universitat de Barcelona, 08028 Barcelona, Spain*
Diagonal
647,
Received 16 January 1990 (Revised 5 July 1990) The parton model for hA and AA interactions is developed for arbitrary dependence of parton amplitudes on energy. Conditions are studied under which the Glauber formula results for total cross sections. The fulfillment of the AGK rules is shown for all energies and registered particle momenta. Inclusive A’A cross sections in the forward hemisphere prove to be A’ times larger than for NA collisions.
Abstract:
1. Introduction The study of hA and AA collisions at the microscopic level is usually done within the framework of the Glauber multiple scattering model. Proposed originally for nonrelativistic scattering, it may be generalized to high energies under some not very convincing assumptions ie3). In the parton model the derivation of the Glauber formula uses the assumption that hN cross sections do not depend on energy. Experimentally this is wrong, since the cross sections rise with energy. The dependence on energy requires one to consider the mechanism of energy division among partons during the interaction. This dependence has been previously taken into account when studying inclusive spectra to satisfy the energy conservation law. A phenomenological view on the energy division was taken: the initial energy is somehow divided among physical intermediate states, i.e. among cut pomerons in the Regge model ‘) or cut amplitudes in the parton model ‘). It remains unclear which part of the energy goes to virtual intermediate states, i.e. uncut pomerons or amplitudes. This problem is of importance in view of the well-known AGK cancellations in inclusive cross sections “) and therefore has a direct relation to calculations of observable spectra in hA and AA collisions. In this paper we generalize the parton model introduced in ref. 3, for hA and AA collisions to take into account the energy dependence of elementary cross sections. As expected, in the general case we do not obtain simple Glauber-like formulas. Nevertheless, the singularity structure of amplitudes remains the same which results in the AGK rules being exactly satisfied. For inclusive cross sections this means l On leave of absence from Leningrad University. Permanent Physics, Institute of Physics, Leningrad University, Uljanovskaya USSR.
0375-9474/91/%03.50
@ 1991 - Elsevier
Science
Publishers
address: Department of Theoretical 1, Petrodvoretz, 198904 Leningrad,
B.V. (North-Holland)
M.A. Brawn / Parton model
695
that contributions from physical and virtual intermediate states cancel each other and only the impulse approximation survives. This contribution, however, does not exhaust the total inclusive cross section. The diffractive cont~bution should be added to it, which originates from observation of spectator partons. The sum appears to be much less than according to the AGK rules and satisfies the energy conservation sum rule. A specific prediction of our model for A’A scattering is that the inclusive cross sections in the forward hemisphere are A’ times larger than for hA scattering. To avoid misunde~tanding we stress that by no means do we pretend to give a proof for the validity of the Glauber approximation ,at high energies. In fact we have little doubt that the Glauber formula can give, at most, a very crude description of the collision. Nevertheless, experimentally the Glauber predictions for total cross sections are not at all bad. Our aim is therefore to build a model ‘which would lead to Glauber-like formulas under reasonable assumptions about the space-time picture of the process. In the literature, a few theoretical papers based on parton ideas are well-known, which in the framework of the Regge theory effectively sum the so-called “fan” diagrams and lead to essentially non-Glauber results 7V8).However, their conclusions can be questioned and are not confirmed experimentally 9). 2. Model; total cross sections In the parton model for hA collisions the incoming hadron breaks into a number N of partons, a part of which, n, interact with nucleons of the target nucleus (fig. 1). Let us study the simplest possibility when the vertices yN for the transition of the hadron into partons are real and practically independent of longitudinal momenta, so that all the dependence on the latter is in pat-ton propagators as shown in fig. 1. The nuclear part of the hA scattering amplitude can be separated in the standard manner. Thus the amplitude a(b) in the impact parameter representation normalized to satisfy &‘=
2 Irn~(~)
d’b,
has the structure i&l(b) = C iW,,C:T”,
e
(2-l)
I I
I
I
t
Fig. 1. The high-energy part of tbe hA amplitude in the parton model. The Small circles correspond to parton-nucleon scattering ampiitudes. The horizontal external lines describe the intout) going hadron. The nuclear part of the hA amplitude is not shown. The dashed vertical line corresponds to taking the absorptive part.
696
MA. Braun / Parton model
where W,,
is the high-energy part corresponding T(b) =
I
to fig. 1,
dzp(b, z)
(2.2)
and p is the nucleon density of the target normalized to 1. Explicitly [see ref. ‘) for details]
~fi (-i(kf-p2+i0))-1
i (ia(si)(-i(k~2-~2+i0))-1). 1
.l
(2.3)
Here kj = ki + qi; qi+ = qil= 0. In contrast to ref. 3, we here admit that partonnucleon amplitudes a may depend on Si = 2k,p,JA, where PA is the momentum of the target nucleus. Limiting ourselves with fast partons in the lab frame we take si = fi mk,, . Performing integrations over minus components of ki, k: in (2.3) and denoting (Y~= ki+/p+ we obtain (2.4)
Here we have introduced pat-tons:
PnN(~*,...,%)=
the n-fold distribution N!
(N-n)!
in (Y for the configuration
of N
(2.5)
with
%v(~1,.~.,%4)=
(2.6)
The stability of the incoming hadron requires that the denominator in (2.6) never vanish, so that w,,, and pnhr are positive. The sum rules are obeyed by pnhr : da ~p”+,.~(a,,...,a,,a)=(N-n)p,,(a,,...,a,),
(2.7)
Summation over N gives the total high-energy part W, for the n-fold interaction with the target: (2.9)
MA.
697
Braun j Parfon model
where pn is the total n-fold distribution in a: pn = CNan pnhl. Note that the introduced symbol ( ) implies taking the mean value over (Ywith probability distributions pn depending on the number of amplitudes. We assume by definition (1) = 1. Then from $21) and (2.9) we find i&(b)=((l+i~~T(b))~-1)
(2.10)
i.e. the Glauber formula inside the mean value 0. Co~espondingly cross section we have o’,“‘=-2Re
I
d2b((l+iaT(b))A-1).
for the total
(2.11)
Eq. (2.10) shows that id(b) is a real function of elementary amplitudes ia. Therefore, the AGK rules 6, will be exactly satisfied in spite of the energy dependence of the amplitudes a, i.e. their dependence on cy.Indeed let us consider the contribution at) to the total cross section coming from cutting r parton amplitudes (see fig. 1). Cutting of an amplitude results in the substitution of 2 Im U(ai) for ia in the integral over LY’Sin eq. (2.10). Each uncut amplitude may be both on the right-hand side and on the left-hand side of the cut, which, summed, results in the substitution of -2 Im a(#,) for ia( Therefore, a(,“=(-l)“-‘C:C:
J (1
d’b fi 2 Im O(b)
> ,
(2.12)
which exactly corresponds to the AGK rules. If parton amplitudes are weakly dependent on the energy they can be taken out of the integral (2.9) at some intermediate value of (Yi= 5. Then iW,=(LI)‘&
(2.13)
with d = a(&) and
zn=J (I-&h...,*“) I
being proportional to the probability to find not less than n partons in the incoming hadron. If 4, factorize, i.e. & = [;, then (2.13) will lead to the ordinary Glauber formula in which the hN amplitude a,, is given by &a. According to (2.5) &, = &~n~ONN!f(Nn)!, where pON has the meaning of the total probability for the incoming hadron to break precisely into N partons. If we assume that the distribution porv is poissonian: ponr =exp (-A)A~/~!
(2.15)
then & indeed factorize: & = A“. So in this case we come to the Glauber formula with ah = Au, where A is the mean number of partons ‘).
698
M.A. Braun / Parton model
Another possibility to obtain the Glauber formula is the factorized approximation for the n-fold probability: (2.16) Then iWn = (i(a))” with(a)=l (da/4&a)p(a)a(a) beingthemeanamplitude. From this we get the Glauber formula with ah = (a). However, assumption (2.16) violates the energy conservation law 1; tyi s 1. A better assumption seems to be
Pn(%,..., (Y,)‘e
(
l-~ai 1
>
(2.17)
up. 1
Representing the &function by the Fourier integral we obtain the Glauber formula under the sign of an integral dz/(2~i(z-i0))((1+i(a(z))7’(b))A-1),
(2.18)
where
(u(z)>=j
e
p(a)a(cu) exp (-iaz).
(2.19)
3, Inclusive cross sections; direct contribution For simplicity we shall study pat-tonic inctusive cross sections. Hadronic ones can be obtained from these by covoluting with appropriate fragmentation functions. Contributions to inclusive cross sections may be divided into direct and spectator parts. The direct contribution corresponds to intermediate states in cut parton amplitudes. The spectator contribution corresponds to cutting parton propagators, In this section we treat the direct contribution, which originates from absorptive parts of the amplitude of the type shown in fig. 2a. We limit ourselves with cross sections integrated over transverse momenta and depending only on the part x of the longitudinal momentum carried by the registered particle. Let such a cross section for scattering of an incoming parton with a part Q of the longitudinal momentum off a nucleon at rest be,j(,, x). Then as follows from fig. 2a, for r cut par-ton amplitudes,out of their total number n, the contribution J’,“(x) to the inclusive hA cross section will be obtained by a substitution of rj(,, x) for one of absorptive parts 2 Im U(ai) in (2.12):
Jz)= (-l)"-'rC~C~ A remarkable
property
d2b( TT (2 Im U(b)) I
of eq. (3.1) is that for arbitrary
j) . (Y dependence
(3.1) of
M.A. Braun / Parion model
699
I -p
(cl
Fig. 2. Diagrams for the inclusive cross section; (a) the direct part, (b) and (c) the spectator part. The blobs correspond either to parton-nucleon scattering amplitudes (with four legs) or to inclusive production in parton-nucleon collisions (with six legs). The horizontal external lines describe the in(out)going hadron.
Im 4~),i(~, xl and dal,. . . , uy,) the AGK cancellations take place: C:=r J!,” = 0, n 2 2 [ref. “)I. Therefore the direct contribution is nonzero only for n = I = 1: J/,(x) = A
Ie
daMa,
x) = AJ,(x) .
(3.2)
It is A times larger than for a collision on a single nucleon. Standard derivations of this result are related to the asymptotical conditions s + 00 or/and x + 0 (central region). In our.model (3.2) is exactly fulfilled at any finite s and x. In the literature the AGK cancellations (3.2) have been extensively discussed 4*9). A stimulating circumstance has been that experimentally the multiplicity on a nucleus does not grow as A 1’3 for finite x as follows from (3.2), but is of the order or even less than on a single nucleon. In’ref. “) this fact was explained by the dependence of the elementary cross section j(q x) on the initial energy, i.e. on CLWith r cut amplitudes, each of them on the average cannot have the energetic variable greater than s/r. Therefore particles in intermediate states in them have x ( l/r. At x > l/r the diagram with r cut amplitudes gives no contribution. In that region according to ‘) the AGK cancellations do not take place and the multiplicity becomes less than (3.2). At x + i by this reasoning only one amplitude can be cut (r = 1, arbitrary n) which gives the multiplicity on a nucleus less than on a nucleon. Comparing this argument with the results of our model we find that the conclusions of ref. ‘) are
700
M.A. Braun / Parton model
not justified. The dependence of elementary amplitudes on energy does not change the AGK cancellations. The reason is that with the growth of the energy carried by the registered particle the energy is decreased which goes not only into cut amplitudes but also into uncut amplitudes and exactly in the same degree (both appear as Im a in cr’). Therefore at, say, x+ 1 not only the contribution from one cut amplitude is left (r = 1) but also screening vanishes (i.e. the contribution from uncut amplitudes). So we are left with r = n = 1, which corresponds to (3.2). For illustration consider the simplest case when the energy is equally divided and A = 2 (deutron). For n = 1 we find J\” = 2j(l, x), For pl= 2 two contributions arise with the second amplitude uncut (r = 1) and cut (r = 2): Jg’ = 2( -l)‘g(f) j (i, x)A with A = I d26T2(b). Clearly the sum Jy’+ Ji*’ = 0, so that only J’,” survives. On the other hand, in the picture of ref. ‘) based on purely probabilistic considerations the energy should be divided among r cut amplitudes. Following ref. “) one should take the quantities ETand j in J$” at the whole energy so that ..Jy’ = -2a( 1)j (1, x)A. Now the AGK cancellation does not occur and the total inclusive cross section J = 2j( 1, x)( 1 - V( 1)A) + 2j(f, x)&A. In this approach, indeed, at x c f corrections to the impulse approximation cancel provided w is weakly dependent on energy, and at x > 1 the last term vanishes and a screening factor (1 - CTA) appears which lowers the multiplicity. This example clearly shows that the argument in ref. “) based on the purely probabilistic picture does not correctly reflect the structure of Feynman diagrams, in which the energy cannot be divided differently depending on whether an amplitude is cut or uncut*. Of course this leaves open the questions why experimentally the multiplicity falls in the diffraction region and how the conservation of energy is achieved. In our model the answer lies in an additional contribution to the cross section coming from observation of spectators. 4. Spectator contribution
Fig. 1 shows that in the total cross section, inte~ediate spectator partons appear which correspond to inclusive cross sections of two types shown in fig. 2b, c. Unfortunately even in this simplest model the total spectator part is not rigorously exhausted by these two contributions. At high energies, the cross section for particle production is obtained from amplitudes of three different kinds which correspond to fixing the registered particle in one of the three branches of the diagram shown in fig. 3. Respectively, apart from diagrams shown in fig. 2, the inclusive cross section will also contain a contribution from intermediate one-particle states in the diagram of fig. 4a. This planar diagram by itself goes to zero at high energies and its l As revealed during the discussion with one of the authors (A.B.K.), in ref. 4, it was supposed that the total inclusive cross section is given by its direct part. This is equivalent to the assumption that the spectator part has the same form as the direct one in the appropriate kinematical region. Therefore their results should rather be compared with our total inclusive cross sections, which may Iead to eliminating the disagreement.
M.A. Braun / Parion model
701
Fig. 3. A production amplitude in the parton model. Observed particles may correspond either directly to partons (the upper and lower lines) or to secondaries produced in a parton-nucleon collision (the middle line).
contribution to the total cross section vanishes. However, the mentioned one-particle contribution does not vanish. As is well-known it is cancelled by the contribution of multiparticle intermediate states schematically shown in fig. 4b. Inclusion of such states (“inelastic screening”) cannot be realized by any reliable method (see, however ‘)). Nevertheless it is known that the inelastic screening is small. As we shall not attempt to calculate the ‘inelastic screening, we shall also neglect the contribution of fig. 4a, which is integrally cancelled by the inelastic screening. ‘so finally we are left with the contributions of diagrams of figs. 2b, c. These contributions can be calculated in the same manner as when studying the total cross section. For the total number of partons N (registered included) and n active partons we get the high-energy part W!,‘,.$corresponding to fig. 2b:
The contribution
Wri corresponding
to fig. 2c is given by
iW~~(X)=.i.(X)l~(~i~(,i)),.,(,,,...,~”_~,X).
(4.2)
To go over to inclusive cross sections we have to take absorptive parts of these
Fig. 4. Intermediate states which give contribution to the inclusive cross section apart from those shown in fig. 2.
M.A. Braun / Partan model
702
amplitudes. For Wf$ it is sufficient to simply take the imaginary part: 21m W~:?(x)=-ZRel~(~ju(nr))p.+*,N(a,,...,a.,x).
(4.3)
For W!$, the argument is somewhat more complicated. Its absorptive part consists of two contributions, one with ia another with its complex conjugate. Consider the part with &z(x). Making all possible cuts in the rest part of the diagram we shah get the total contribution for it equal to the right-hand side of eq. (4.2) with the product n iU((Wi)substituted by -2 Re n ia( (Ui)(i.e. the total discontinuity). However there is one more contribution corresponding to the cut which leaves all the amplitudes ia on one side of the cut and iafx) on the other. This contribution is equal to the right-hand side of (4.2) with the change of all ia to complex conjugates. Joining this contribution with the found earlier and also with the conjugate contributions we finally get 21m W~~(x)=-nZReia(x)l~‘(~io(s))p.N(a,,...,~~-,,~).
(4.4)
Note an important remark. The contribution Wiz exists only for n 3 2 as for n = 1 in one of the parts of the cut diagram there is no interaction with the target. For n 3 2 the total spectator contribution is given by DnN = 2 Im ( W’,‘,‘,+ Wi*A). According to (2.7)-(2.8) DnN obeys the sum rules
J J e
DnN(x) = 2N Im W,,
ED,,(x)=2Im
WnN.
+niu(x)p,(a,,.
. . , a,-‘,X)
,
(4.5)
Let us now sum over all N:
>
.
(4.7)
The spectator inclusive cross section 1, is obtained from this by attaching the proper nuclear factor and summing over all II: I, = Ig’-Aa(x)pl(x),
(4.8)
where (4.9)
M.A. Braun / Parton model
703
The subtracted term in (4.8) serves to eliminate the extra contribution Im W’,“; a(x) = 2 Im a(x) is the total partonic cross section. From (4.6) we find a sum rule for Z(Al)
Ie
z:)(x) = ay ,
(4.10)
which implies that particles described by Z2’ carry all the energy. The rest part of the spectator cross section proportional to A can be taken together with the direct contribution, so that the total inclusive cross section
Let us find sum rules for I(‘). Using sum rules for j
$$a, x) = dab(a),
ej(a,x)=ao(a),
(4.12)
where Z.Lis the partonic multiplicity we get
I dx ~I,
(2)
(x)=0,
f$ Zjf'(x) =A IT I
e
+)P*(x)(P(x)
- 1).
(4.13)
In agreement with (4.10), the first sum rule shows that the spectator part included in Zg’ completely cancels the energy carried by particles produced by the direct mechanism. In this sense the interference between the spectator and direct mechanisms restores the conservation of energy and leads to a decrease of multiplicity in the region where the spectator contribution is different from zero, i.e. in the projectile fragmentation region. To study the x dependence of the inclusive cross section we have to make some assumptions about the structure of pn. It would be natural to propose a factorized form for pn (2.16) or (2.17). However, it can be seen from the sum rule (2.7) that the one-parton distribution p(a) has to depend on the number n of active partons. This is natural, of course, as the total number of partons N 2 n and the average parton momentum should decrease with n. To simplify the problem let us use the weak dependence of a’s on energy and take them out of the integral (4.7). Then using (2.7) DnN(x)=-2Re(id)“-‘p$!,(x)((l-n/N)iti+(n/N)ia(x)),
(4.14)
where p’,,&= P,~N!/( N - n) !. The sum rules for pIN involve pON on the right-hand side. Assuming (2.15) we can perform summation over N to get for p’,” =CNrn p’.::
I
Epy)(x)=h”(h+n),
I dx GPn
(1)
(x)=A”.
(4.15)
Thus a simple factorization property of pn(al,. . . , a,) cannot be true, since p’.” as a function of x should depend on n. The absorptive part DnN contains one more
704
M.A. Braun f Parton model
function pizA= p$$/ TV.One can obtain sum rules also for it. But in (4.14) it is multiplied by a small factor a(x)-n and we shall neglect this contribution. We introduce the normalized distribution p”‘,”= p(n’h-” satisfying the sum rules
I
dx -/$‘(x)=A+n, 4VX
E&l”(X)
(4.16)
= 1.
Then we obtain D,(x) = -2 Re
(id,Jn&‘)(x) ,
(4.17)
where &, = Ad is the averaged hadronic amplitude. With the necessary nuclear factors and after summing over n Ik”(X) = -2 Re
d*b f
(4.18)
Ci( id,, T( b))“$,“(x) .
The cross section (4.18) satisfies (4.10) and also $1$‘(x)
= AuSqOf - 2 Re
d2bAi&T(b)(l+ia”,T(b))A-*.
(4.19)
The second term on the right-hand side is negligible for A> 1, so that the total multiplicity generated by 12) on heavy nuclei ~2) = A. For A = 1 (a single nucleon) &‘=A+l. Summation over n in (4.18) can be performed if we choose a concrete parametrization of p’,“(x). We assume (4.20)
p”,“= c,x@(l -x)“,
where the exponent yn is taken to be n-dependent to satisfy (4.16). From (4.16) we find m=np+/IA-p-1, c;‘= B(P + 1, yn + 1). Now we can explicitly perform the summation over n in (4.18) using the integral representation for the B-function. If we denote y = Gh T( b)( 1 - x)~ then xP(l -x)‘o *~)(X)=-r(p+l)r(-B_l)2Re x
I
dt t-e-‘( 1 - t)P’
(4.21)
d2b((l+y(l-t)P)A-1).
For large A we have Ay + -co for b inside the nucleus. Therefore inner integral is equal to -aRf, = -ivy. From this we get rk”(x)-X~(l-x)%3-‘(p+l,
3/*+1)aaO’,
for AS=-1 the
A>l.
(The correct derivation of (4.22) involves a nontrivial analytic continuation exponents in (4.21)). For a single nucleon I(,“(x)=x~(l-x)Y’B-‘(p+l,
y,+l)a:ot.
(4.22) in some (4.23)
M.A. Bmun / Parion model
705
Since y1 = -yO+ /3 and p 3 0 the particle spectrum generated by 1(l) on a nucleus is somewhat more energetic than on a single nucleon, though the total multiplicity is less by 1. To describe in this framework the spectator part of I(‘) we first perform summation over N of the sum rules for pIN with the use of (2.15) to get
1em=“,
I
2
p,(x) = 1-exp (-A)
(4.24)
Separating the factor [ = 1 -exp (-A), we get for b,(x) = p,(x){-’
dx
w
$?rLJdx)=l. ~ompa~ng this with (4.16) we find that one may choose p’i(x) =$‘(x) where A + Y= A/f: Including a factor A into cr and j to go over to hadronic cross sections a,, and j, we obtain finally
We expect that for x + 0 p’l”+ 0 so that in the central region the total contribution reduces to the direct one. With the growth of x the spectator part of 12’ must cancel the direct contribution to satisfy the first sum rule (4.13). This implies that starting from some x 3 x0 the cross section 12) becomes negative. The total cross section is, of course, positive. So we expect x0 to lie in the region where I:’ and Ia” are of (If t he cross section 1%’ contains a the same order. Note also that as compared to IA factor J/A depending explicitly on the mean number of partons and falling like l/A for A )z 1. Therefore we expect I g’ to be considerably less than 12’ for finite X. These qualitative considerations can be confirmed by numerical calculations of the multiplicity &&._, = I,/ aho using (4.18), (4.20) and (4.26), which are illustrated in figs. 5-7. For ji, the simplest form has been chosen ~,(cx, x) = rr,@(tu -x). In fig. 5 the contributions p(i) and p(2) corresponding to Iti) and I(‘) are shown separately as functions of the antilab rapidity y = In l/x for A = 64. Parameter /3 governing the low x behaviour has been chosen f (continuous curves) or 0 (dash-dotted curves). In the latter case results depend on the projectile rapidity which here and elsewhere was put equal to 10. The mean number of partons A was put equal to 6. As can be seen *(*’ indeed becomes negative at small enough x, although the total p = pU)+pL(2) remains positive. Fig. 6 illustrates the dependence of p on the mean number of partons A for A = 64 and /3 =i (a) and /I =0 (b). Evidently the Adependence is much more pronounced for /3 = i than for /I = 0 in which case it is quite weak. For fl= f with the growth of A the Plato in the central region goes over into a well. The A-dependence is shown in fig. 7 for p = 4 (a) and p = 0 (b). A has been chosen equal to 6. It is clearly seen that at low y the multiplicity decreases
M.A. Braun / Parton model
706
2.0 1.8_
Fig. 5. Multiplicities
I
11111111 A=64 A=6
curve) and p = 0 (dash-dotted pA(I) and ~2’ for p = f (continuous ofy=Inl/xforA=64andh=6.
I
2.6
curve) as functions
I I111111
1.8 _
2.4
A=64
1.6-
2.2
#3=0
1.4 -
2.0
1.2 -
1.8
b)
1.6 1.4 1.2 I
I
0.6
l/x Fig. 6. Total multiplicities
pcLAas functions
0.1
0.2
of y for different
I 1111111 0.5
I 1.0
2.0
I 11111. 5.0 y = In l/x
A and p =i (a) or p = 0 (b); A=64.
707
M.A. Braun / Parton model P
-(a)
3.0 -
I
’ f3 = ’l/2 “““’
I
I lllll
A=6
l/x
lb)
’ p=o’ ’ ’ ’ ’ ” ’
I
I
Illlf
i
cc-------:
A=6
Fig. 7. Total multiplicities
I
pcLAas functions
of y for different A and B = 4 (a) or p = 0 (b); A = 6.
with the naive AGK with A, although for x-0 it behaves as A1’3 in accordance rules. For comparison the calculations of p according to the recipe of ref. “) are shown by dashed curves. The normalization is taken to be by ay rather than by a: so as to correspond to our results. The difference is mostly felt at small y in the region 0.2 s x =S1 where ref. “) gives a much less multiplicity than our calculations.
5. Other models Results obtained so far are based on simplest parton diagrams of fig. 1. However, one can see that they do not practically depend on a particular structure of pax-ton
M.A. Braun / Parton model
708
Fig. 8. Consecutive decay into partons.
diagrams.
Consider
for comparison
a different
mechanism
of breaking
into partons,
a consecutive decay, illustrated in fig. 8. Elementary calculations give in this case the same formulas (2.4)-(2.6), in which only the structure is specified of parton vertices depending now also on all (Y’S:
YN = gW-O
N-Z
fl
Ci;‘(tTt~-C t?l~*/OZi-tif~/CS[)-’ . 1
/=I
(5.1)
constant. The square Here 67, = CL’+ (1: kil)2, &, = 1-C: (Y~,g is the interaction brackets mean symmetrization over all (Y~,. . . , ffN. Clearly the probability (Y,) remains positive. Therefore the AGK rules remain fully satisfied as &I(%,.-., they follow only from the fact that pn are real lo). From this the AGK cancellations result for inclusive cross sections and consequently the direct part of the cross section will be given by the same eq. (3.2). Furthermore, it is not difficult to see that in the spectator part an irregularity occurs for n = 1: cutting of the propagator of the active parton need not be taken into account in this case. This results in separating from the spectator contribution a term proportional to A, which can be joined with .I to form I(‘) with the same structure as (4.11) and obeying the same sum rules (4.13). From the latter it follows that the bulk of the spectator part Z(l) satisfies (4.10). From the sum rules for Z(l) and Zc2) one can conclude that general properties
of inclusive
cross sections
remain
in fig. 8. The contribution Zc2) proportional decreasing as x grows until it becomes
the same for parton
diagrams
shown
to A will dominate in the central region negative and compensated by the main
spectator part Z(l) proportional to A2’3. This argument makes us believe that the described picture of particle production in hA collisions is true in a parton model with an arbitrary mechanism of breaking into partons. Another line of generalization is inclusion of different sorts of partons. Let, for simplicity, there are partons only of two sorts, 1 and 2. Then instead of (2.1) we shall find i&(b)=
1 ~INI”ZNZ
CiCi1iWn,N,n2N2T”(b),
n=n,+n,,
(5.2)
M.A. Braun / Parton model
709
where ni and Ni are the number of active partons and their total number for the ith sort. The high energy part W will be given by the formula
where ai is the parton-nucleon amplitude for the ith sort. The probability P,,,~,_~~ describes the distribution of partons of both sorts in parts of the longitudinal momentum and can be expressed through vertices for the transition of the incoming hadron into partons similarly to (2.5) and (2.6). Summation over N1 and N2 gives iW.,.,=($io,$io,).
(5.4)
Here the average is implied over parts ai and Pk of the longitudinal momentum carried by partons with the probability pn,_ which results after summation over N1 and Nz. From (5.2) and (5.4) we find for the amplitude is8(b)=((l+iT(b)(a,+~,))~-1).
(5.5)
If we assume that both the amplitudes Ui depend weakly on energy and parton numbers for both sorts are distributed independently according to Poisson’s law, then (5.5) goes over into the Glauber formula with the hadron amplitude a,,= h,a,+h2az where hi is the mean number of partons of the ith sort. Less trivial in the case when partons of one sort (say, the 2nd) are distributed according to Poisson’s law and the number of the partons of the 1st sort is rigidly fixed. Let Ni =3 in correspondence with the number of valence quarks in a baryon. Assuming again ai weakly dependent on energy we find iJ8(b)=(1+iaJ)A-3(1+iT(a,+Aa,))3-1.
(5.6)
Eq. (5.6) is different from the Glauber formula. However, the difference is in fact very small for A s 1. The influence of the partons of the 1st sort is negligible and the total contribution is determined by the first factor which originates from the interaction of the partons of the 2nd sort and turns the first term in (5.6) into zero. Inclusive cross sections can be studied in total analogy with the case of one sort of partons and with the same results. For the direct contribution the AGK cancellations can be established: if J”n;n: ’ ) IS * its part with ni partonic amplitudes of the ith sort out of which ri are cut, then (5.7) Thus the direct contribution
reduces to the two terms of the impulse approximation (5.8)
M.A. Braun / Parton model
710
This is true for arbitrary (5.8) is fulfilled contradiction
behaviour
when the number
to a naive
of U,(a) of partons
probabilistic
be multiplied by 3. One can also study more complicated
and P,,,“, (ai, &).
eq.
of some sort is fixed, e.g. N1 = 3, in
interpretation models
In particular,
according
of parton
to which
interactions
J, should
like the dual
amplitude refers to an interaction of par@ model ‘**‘*), in which the elementary two different partons from the incoming hadron with the target nucleon (formation of two “strings”).
For this model our approach
also leads to the rigorous
fulfillment
of the AGK rules for the direct part of the inclusive cross section: JA = AJ,, valid for arbitrary partonic distributions and amplitudes. In ‘*) where the probabilistic picture and the Glauber probabilities for multiple collision were used, expressions for JA were obtained which do not obey AGK: JA = ;AJ, . The factor i came from the ratio of the number of diquark strings in an NA collision (A+ 1) to that in an NN collision (2). Our calculations based on Feynman diagrams clearly speak against these simple
conclusions.
6. AA collisions The proposed parton model can be directly generalized to AA collisions. Nuclear factors can be separated from the amplitude by the standard manner for both colliding nuclei. We shall briefly discuss only the high-energy part of the amplitude. It can be represented by a sum of diagrams for interactions of nucleons from the projectile and target of the type shown in fig. 9. In the parton model wavy lines denote par-ton-parton amplitudes. Vertices for the projectile and target have the same structure as for hA collisions (fig. 1) and are given by the same expression W,, (eq. (2.4)). The only difference is that a now refers to a parton-parton interaction and depends not on (Y but rather on the product CYPwhere /3 is the part of energy carried by the second parton taking part in the interaction. Besides, a may now depend on a nonzero momentum transfer over which one has to integrate in loops in the diagrams of fig. 9. Summation over the total parton number N leads to the vertex W,, (eq. (2.9)). We can proceed in this manner with every vertex and represent the whole high-energy part as an average of parton amplitudes with the probability given by the product of all parton probabilities for all participant
Fig. 9. Interactions of nucleons from the projectile and target in an AA collision. Wavy lines denote parton scattering amplitudes. The nuclear part of the AA amplitude is not shown.
M.A. Braun / Parton model
711
nucleons both from the projectile and from the target. Supplying the necessary nuclear factors, similar to (2.10) we obtain the Glauber formula for AA collisions under the sign of the average over partonic probabilities: id(b)=
fi (d*xiTA(xi)) fi (d*xSTA,(x$-b) i=l
x
fi (
i=lj=l
j=l
(l+ia(x,-xi))-1
fi
(6.1)
'
The amplitudes a are taken here in the impact parameter representation. Eq. (6.1) turns into the ordinary Glauber formula if (n ia) =n (iu). Since the average over parton distributions is taken independently in different nucleons, this condition will be satisfied if it is fulfilled for hA scattering (sect. 2). In particular, for weak dependence of a on energy we shall find the Glauber result if in a nucleon partons are distributed according to Poisson’s law with an average number h. The NN amplitude is then equal to A*a. Since id is a real function of iu, the AGK rules will be rigorously satisfied for arbitrary a and P,,. Therefore, the direct contribution to the inclusive cross section for the production of a parton with a given part x of the longitudinal momentum in the c.m. system of colliding nucleons (-1 s x d 1) will be given by the impulse approximation: JAnA
= A’A
~p,(u)p:(B)j(aa,x)=A’AJ,(x).
(6.2)
The spectator part of the inclusive cross section is described by the same diagrams as shown in Fig. 2b, c, which however may also include some interactions of other nucleons. One has to take into account that for x > 0 a nonzero contribution will come only from partons originating from the projectile nucleons and for x < 0 only from partons from the target nucleons. For definiteness we consider x > 0. Using the results of sect. 4 for hA collisions we find that the spectator contributions corresponding to fig. 2b or fig. 2c are obtained by first substituting W’,l’ or W’,z’ for W,, in one of the projectile vertices, second, taking the imaginary part according to (4.3) or (4.4) and, finally, multiplying the result by the number of nucleons in the projectile A’. There is, though, a difference as compared to hA scattering. One has to take not the total imaginary part but only contributions from cuts drawn through the distinguished vertex. The sum of all these contributions can be easily calculated using the A.GK rules. One finds that only contributions from diagrams will be left which do not contain interactions not involving the distinguished vertex. As a result the spectator part IXA for x > 0 proves to be A’ times larger than the corresponding cross section IA for NA collisions:
LA(X) = A’L(x) ,
x=-o.
(6.3)
M.A. Braun / Parion model
712
will be given by the same formulas (4.8) and (4.9), the cross section a(x) of the parton-nucleon interaction being now expressed through the pure partonic cross section I,
The subtraction term in (4.8) can be joined with the direct part into the cross section Ik”!, proportional to A’A: IfjA = A’AI’,2’,
(6.4)
where 1’1” is given by (4.30). According to (4.13) we have
from which we conclude that, as for hA collisions, the total energy is carried away by particles corresponding to the bulk of the spectator cont~buti~n I:&. According to (6.3) we find la’!* = A’Iz’+ A@ .
(6.6)
The first term is nonzero for x > 0, the second for x < 0. Integrating (6.6) and using (4.10) gives
to be compared with the energy conservation
sum rule
The difference is due to the presence in the inclusive cross section of an additional contribution ft3) with terms proportional to 6(x - 1) and 6(x + 1) corresponding to the direct observation of spectator nucleons. These terms can be seen from diagrams studying the contributions from cuts to the total cross section which cancel when forming the inclusive cross section IA’A. The magnitude of b-terms can be established from (6.7) and (6.8): I(A3?A = A’( oj,o!, --a;$?(~-l)+A(tr::A-a::)S(x+l).
(6.9)
As a result we find that in the proposed parton model the inclusive AA cross sections are related in a simple manner to the inclusive NA cross sections. From (6.2) and (6.3) it follows that in the whole forward hemisphere in the c.m. system
M.A. Bmun / Parton
model
713
of the colliding nucleons for x > 0, x # 1 I:!A(x) = A'ly(x)
.
(6.10)
In the backward ,hemisphere the projectile and target interchange. All nontrivial dependence on both A and A’ appears to be limited to the region of spectator nucleons with x = *l. One has to note that eq. (6.10) immediately follows from a naive application of the AGK-rules for AA scattering. Our achievement is to sho& that this simple result persists even when conservation of energy is taken into account at finite energies in contrast to conclusions made in the context of a pure probabilistic approach in ref. 13). Relating (6.10) with the experimental data one has to keep in mind that our model, as practically all other models for high-energy AA-scattering, does not take into account particle production resulting from a cooperative interaction of several nucleons of the projectile (target) either at short distances (merging of parton clouds belonging to, different nucleons, presence of 3n -q-bags) or at long distances (rescattering and cascade proliferation of produced particles). These effects obviously dominate at 1x1close to or more than 1 in the so-called cumulative region where the right-hand side of eq. (6.10) is found equal to zero. For 0 s x < 1 they should be comparatively small for a light projectile. Indeed it has long been known
24
16
12
8
Fig. 10. The experimental ratio of pseudorapidity distributions of charged secondaries in central collisions of 200 GeV/nucleon I60 and p with Ag/Br taken from the data of ref. I’). The shaded area corresponds to 10% statistical errors. The arrow marks the division between the forward and backward hemispheres in the c.m. system.
714
MA.
Braun f Parion model
that, for such x, eq. (6.10) is fulfilled with a good accuracy for the scattering of 1.05 GeV/nucleon and 2.1 GeV/nucleon deuterons and alphas from various nuclear targets 14). Turning to more recent experiments and heavier projectiles we present in fig. 10 the experimental ratio of pseudorapidity distributions of charged secondaries for central collisions of 200 GeV/nucleon I60 and protons from Ar/Br taken from the data of ref. “). The experimental distributions are normalized to unity so that eq. (6.10) predicts an unknown constant for this ratio for q B 3.3, which corresponds to the forward hemisphere. As one observes in fig. 10 the experimental ratio grows with 7. This growth is not very pronounced for 3.3 < q d 4.7, i.e. for particles produced with comparatively high velocities in the antilab system. For larger v the growth becomes steeper. This effect may be partially attributed to the mentioned cooperative interactions, in particular to cascade interactions of particles with small momenta in the antilab system. For fast particles in this system such interactions become ineffective due to the finite formation time and somewhat better agreement is observed. Another reason for the deviation of experimental data from eq. (6.10) may be the selection of central collisions in ref. I’), which results in a deformation of spectra in the projectile fragmentation region. More definite conclusions require information on the complete particle spectra and their dependence on the atomic number A’ of the projectile. The author expresses his gratitude to A.B. Kaidalov, E.M. Levin and Yu.M. Shabelsky for helpful and stimulating discussions. He is also most thankful to R. Tarrach and E. Elizalde for their kind hospitality and interest in this study during his stay at Barcelona University where this work was completed. He is also greatly indebted to E. Gaztaiiaga for his help in preparing this paper for publication.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS)
V.N. Gribov, JETP 56 (1969) 892; 57 (1969) 1306 L. Caneshi and A. Schwimmer, Nucl. Phys. B133 (1978) 408 M.A. Braun, Yad. Fiz. 47 (1988) 262 A. Capella and A.B. Kaidalov, Nucl. Phys. Blll (1976) 477 A. Capella and A. Krzywicky, Phys. Lett. B67 (1977) 84 V.A. Abramovsky, V.N. Gribov and O.V. Kancheli, Yad. Fiz. 18 (1973) 595 O.V. Kancheli, Pis’ma JETP 18 (1973) 465 J. Koplic and A.H. Mueller, Phys. Rev. Dl2 (1975) 3638 V.M. Braun and Yu.M. Shabelsky, IJMP A3 (1988) 2417 M.A. Braun, Yad. Fiz. 48 (1988) 409 A.B. Kaidalov, Phys. Lett. B116 (1982) 459 A. Capella, Preprint LPTHE ORSAY (1988)# 88/67 Yu. M. Shabelsky, Preprint LNPI (1988) # 1433 J. Papp et at, Phys. Rev. Lett. 34 (1975) 601 L.M. Barbier ez at, Phys. Rev. Len. 60 (1988) 405
AS23 (1991) 694-714
PARTON
MODEL
FOR hA AND HIGH
AA COLLISIONS
AT
ENERGIES
M.A. BRAUN Departament
E.C.M.,
Facultat de Fisica, Universitat de Barcelona, 08028 Barcelona, Spain*
Diagonal
647,
Received 16 January 1990 (Revised 5 July 1990) The parton model for hA and AA interactions is developed for arbitrary dependence of parton amplitudes on energy. Conditions are studied under which the Glauber formula results for total cross sections. The fulfillment of the AGK rules is shown for all energies and registered particle momenta. Inclusive A’A cross sections in the forward hemisphere prove to be A’ times larger than for NA collisions.
Abstract:
1. Introduction The study of hA and AA collisions at the microscopic level is usually done within the framework of the Glauber multiple scattering model. Proposed originally for nonrelativistic scattering, it may be generalized to high energies under some not very convincing assumptions ie3). In the parton model the derivation of the Glauber formula uses the assumption that hN cross sections do not depend on energy. Experimentally this is wrong, since the cross sections rise with energy. The dependence on energy requires one to consider the mechanism of energy division among partons during the interaction. This dependence has been previously taken into account when studying inclusive spectra to satisfy the energy conservation law. A phenomenological view on the energy division was taken: the initial energy is somehow divided among physical intermediate states, i.e. among cut pomerons in the Regge model ‘) or cut amplitudes in the parton model ‘). It remains unclear which part of the energy goes to virtual intermediate states, i.e. uncut pomerons or amplitudes. This problem is of importance in view of the well-known AGK cancellations in inclusive cross sections “) and therefore has a direct relation to calculations of observable spectra in hA and AA collisions. In this paper we generalize the parton model introduced in ref. 3, for hA and AA collisions to take into account the energy dependence of elementary cross sections. As expected, in the general case we do not obtain simple Glauber-like formulas. Nevertheless, the singularity structure of amplitudes remains the same which results in the AGK rules being exactly satisfied. For inclusive cross sections this means l On leave of absence from Leningrad University. Permanent Physics, Institute of Physics, Leningrad University, Uljanovskaya USSR.
0375-9474/91/%03.50
@ 1991 - Elsevier
Science
Publishers
address: Department of Theoretical 1, Petrodvoretz, 198904 Leningrad,
B.V. (North-Holland)
M.A. Brawn / Parton model
695
that contributions from physical and virtual intermediate states cancel each other and only the impulse approximation survives. This contribution, however, does not exhaust the total inclusive cross section. The diffractive cont~bution should be added to it, which originates from observation of spectator partons. The sum appears to be much less than according to the AGK rules and satisfies the energy conservation sum rule. A specific prediction of our model for A’A scattering is that the inclusive cross sections in the forward hemisphere are A’ times larger than for hA scattering. To avoid misunde~tanding we stress that by no means do we pretend to give a proof for the validity of the Glauber approximation ,at high energies. In fact we have little doubt that the Glauber formula can give, at most, a very crude description of the collision. Nevertheless, experimentally the Glauber predictions for total cross sections are not at all bad. Our aim is therefore to build a model ‘which would lead to Glauber-like formulas under reasonable assumptions about the space-time picture of the process. In the literature, a few theoretical papers based on parton ideas are well-known, which in the framework of the Regge theory effectively sum the so-called “fan” diagrams and lead to essentially non-Glauber results 7V8).However, their conclusions can be questioned and are not confirmed experimentally 9). 2. Model; total cross sections In the parton model for hA collisions the incoming hadron breaks into a number N of partons, a part of which, n, interact with nucleons of the target nucleus (fig. 1). Let us study the simplest possibility when the vertices yN for the transition of the hadron into partons are real and practically independent of longitudinal momenta, so that all the dependence on the latter is in pat-ton propagators as shown in fig. 1. The nuclear part of the hA scattering amplitude can be separated in the standard manner. Thus the amplitude a(b) in the impact parameter representation normalized to satisfy &‘=
2 Irn~(~)
d’b,
has the structure i&l(b) = C iW,,C:T”,
e
(2-l)
I I
I
I
t
Fig. 1. The high-energy part of tbe hA amplitude in the parton model. The Small circles correspond to parton-nucleon scattering ampiitudes. The horizontal external lines describe the intout) going hadron. The nuclear part of the hA amplitude is not shown. The dashed vertical line corresponds to taking the absorptive part.
696
MA. Braun / Parton model
where W,,
is the high-energy part corresponding T(b) =
I
to fig. 1,
dzp(b, z)
(2.2)
and p is the nucleon density of the target normalized to 1. Explicitly [see ref. ‘) for details]
~fi (-i(kf-p2+i0))-1
i (ia(si)(-i(k~2-~2+i0))-1). 1
.l
(2.3)
Here kj = ki + qi; qi+ = qil= 0. In contrast to ref. 3, we here admit that partonnucleon amplitudes a may depend on Si = 2k,p,JA, where PA is the momentum of the target nucleus. Limiting ourselves with fast partons in the lab frame we take si = fi mk,, . Performing integrations over minus components of ki, k: in (2.3) and denoting (Y~= ki+/p+ we obtain (2.4)
Here we have introduced pat-tons:
PnN(~*,...,%)=
the n-fold distribution N!
(N-n)!
in (Y for the configuration
of N
(2.5)
with
%v(~1,.~.,%4)=
(2.6)
The stability of the incoming hadron requires that the denominator in (2.6) never vanish, so that w,,, and pnhr are positive. The sum rules are obeyed by pnhr : da ~p”+,.~(a,,...,a,,a)=(N-n)p,,(a,,...,a,),
(2.7)
Summation over N gives the total high-energy part W, for the n-fold interaction with the target: (2.9)
MA.
697
Braun j Parfon model
where pn is the total n-fold distribution in a: pn = CNan pnhl. Note that the introduced symbol ( ) implies taking the mean value over (Ywith probability distributions pn depending on the number of amplitudes. We assume by definition (1) = 1. Then from $21) and (2.9) we find i&(b)=((l+i~~T(b))~-1)
(2.10)
i.e. the Glauber formula inside the mean value 0. Co~espondingly cross section we have o’,“‘=-2Re
I
d2b((l+iaT(b))A-1).
for the total
(2.11)
Eq. (2.10) shows that id(b) is a real function of elementary amplitudes ia. Therefore, the AGK rules 6, will be exactly satisfied in spite of the energy dependence of the amplitudes a, i.e. their dependence on cy.Indeed let us consider the contribution at) to the total cross section coming from cutting r parton amplitudes (see fig. 1). Cutting of an amplitude results in the substitution of 2 Im U(ai) for ia in the integral over LY’Sin eq. (2.10). Each uncut amplitude may be both on the right-hand side and on the left-hand side of the cut, which, summed, results in the substitution of -2 Im a(#,) for ia( Therefore, a(,“=(-l)“-‘C:C:
J (1
d’b fi 2 Im O(b)
> ,
(2.12)
which exactly corresponds to the AGK rules. If parton amplitudes are weakly dependent on the energy they can be taken out of the integral (2.9) at some intermediate value of (Yi= 5. Then iW,=(LI)‘&
(2.13)
with d = a(&) and
zn=J (I-&h...,*“) I
being proportional to the probability to find not less than n partons in the incoming hadron. If 4, factorize, i.e. & = [;, then (2.13) will lead to the ordinary Glauber formula in which the hN amplitude a,, is given by &a. According to (2.5) &, = &~n~ONN!f(Nn)!, where pON has the meaning of the total probability for the incoming hadron to break precisely into N partons. If we assume that the distribution porv is poissonian: ponr =exp (-A)A~/~!
(2.15)
then & indeed factorize: & = A“. So in this case we come to the Glauber formula with ah = Au, where A is the mean number of partons ‘).
698
M.A. Braun / Parton model
Another possibility to obtain the Glauber formula is the factorized approximation for the n-fold probability: (2.16) Then iWn = (i(a))” with(a)=l (da/4&a)p(a)a(a) beingthemeanamplitude. From this we get the Glauber formula with ah = (a). However, assumption (2.16) violates the energy conservation law 1; tyi s 1. A better assumption seems to be
Pn(%,..., (Y,)‘e
(
l-~ai 1
>
(2.17)
up. 1
Representing the &function by the Fourier integral we obtain the Glauber formula under the sign of an integral dz/(2~i(z-i0))((1+i(a(z))7’(b))A-1),
(2.18)
where
(u(z)>=j
e
p(a)a(cu) exp (-iaz).
(2.19)
3, Inclusive cross sections; direct contribution For simplicity we shall study pat-tonic inctusive cross sections. Hadronic ones can be obtained from these by covoluting with appropriate fragmentation functions. Contributions to inclusive cross sections may be divided into direct and spectator parts. The direct contribution corresponds to intermediate states in cut parton amplitudes. The spectator contribution corresponds to cutting parton propagators, In this section we treat the direct contribution, which originates from absorptive parts of the amplitude of the type shown in fig. 2a. We limit ourselves with cross sections integrated over transverse momenta and depending only on the part x of the longitudinal momentum carried by the registered particle. Let such a cross section for scattering of an incoming parton with a part Q of the longitudinal momentum off a nucleon at rest be,j(,, x). Then as follows from fig. 2a, for r cut par-ton amplitudes,out of their total number n, the contribution J’,“(x) to the inclusive hA cross section will be obtained by a substitution of rj(,, x) for one of absorptive parts 2 Im U(ai) in (2.12):
Jz)= (-l)"-'rC~C~ A remarkable
property
d2b( TT (2 Im U(b)) I
of eq. (3.1) is that for arbitrary
j) . (Y dependence
(3.1) of
M.A. Braun / Parion model
699
I -p
(cl
Fig. 2. Diagrams for the inclusive cross section; (a) the direct part, (b) and (c) the spectator part. The blobs correspond either to parton-nucleon scattering amplitudes (with four legs) or to inclusive production in parton-nucleon collisions (with six legs). The horizontal external lines describe the in(out)going hadron.
Im 4~),i(~, xl and dal,. . . , uy,) the AGK cancellations take place: C:=r J!,” = 0, n 2 2 [ref. “)I. Therefore the direct contribution is nonzero only for n = I = 1: J/,(x) = A
Ie
daMa,
x) = AJ,(x) .
(3.2)
It is A times larger than for a collision on a single nucleon. Standard derivations of this result are related to the asymptotical conditions s + 00 or/and x + 0 (central region). In our.model (3.2) is exactly fulfilled at any finite s and x. In the literature the AGK cancellations (3.2) have been extensively discussed 4*9). A stimulating circumstance has been that experimentally the multiplicity on a nucleus does not grow as A 1’3 for finite x as follows from (3.2), but is of the order or even less than on a single nucleon. In’ref. “) this fact was explained by the dependence of the elementary cross section j(q x) on the initial energy, i.e. on CLWith r cut amplitudes, each of them on the average cannot have the energetic variable greater than s/r. Therefore particles in intermediate states in them have x ( l/r. At x > l/r the diagram with r cut amplitudes gives no contribution. In that region according to ‘) the AGK cancellations do not take place and the multiplicity becomes less than (3.2). At x + i by this reasoning only one amplitude can be cut (r = 1, arbitrary n) which gives the multiplicity on a nucleus less than on a nucleon. Comparing this argument with the results of our model we find that the conclusions of ref. ‘) are
700
M.A. Braun / Parton model
not justified. The dependence of elementary amplitudes on energy does not change the AGK cancellations. The reason is that with the growth of the energy carried by the registered particle the energy is decreased which goes not only into cut amplitudes but also into uncut amplitudes and exactly in the same degree (both appear as Im a in cr’). Therefore at, say, x+ 1 not only the contribution from one cut amplitude is left (r = 1) but also screening vanishes (i.e. the contribution from uncut amplitudes). So we are left with r = n = 1, which corresponds to (3.2). For illustration consider the simplest case when the energy is equally divided and A = 2 (deutron). For n = 1 we find J\” = 2j(l, x), For pl= 2 two contributions arise with the second amplitude uncut (r = 1) and cut (r = 2): Jg’ = 2( -l)‘g(f) j (i, x)A with A = I d26T2(b). Clearly the sum Jy’+ Ji*’ = 0, so that only J’,” survives. On the other hand, in the picture of ref. ‘) based on purely probabilistic considerations the energy should be divided among r cut amplitudes. Following ref. “) one should take the quantities ETand j in J$” at the whole energy so that ..Jy’ = -2a( 1)j (1, x)A. Now the AGK cancellation does not occur and the total inclusive cross section J = 2j( 1, x)( 1 - V( 1)A) + 2j(f, x)&A. In this approach, indeed, at x c f corrections to the impulse approximation cancel provided w is weakly dependent on energy, and at x > 1 the last term vanishes and a screening factor (1 - CTA) appears which lowers the multiplicity. This example clearly shows that the argument in ref. “) based on the purely probabilistic picture does not correctly reflect the structure of Feynman diagrams, in which the energy cannot be divided differently depending on whether an amplitude is cut or uncut*. Of course this leaves open the questions why experimentally the multiplicity falls in the diffraction region and how the conservation of energy is achieved. In our model the answer lies in an additional contribution to the cross section coming from observation of spectators. 4. Spectator contribution
Fig. 1 shows that in the total cross section, inte~ediate spectator partons appear which correspond to inclusive cross sections of two types shown in fig. 2b, c. Unfortunately even in this simplest model the total spectator part is not rigorously exhausted by these two contributions. At high energies, the cross section for particle production is obtained from amplitudes of three different kinds which correspond to fixing the registered particle in one of the three branches of the diagram shown in fig. 3. Respectively, apart from diagrams shown in fig. 2, the inclusive cross section will also contain a contribution from intermediate one-particle states in the diagram of fig. 4a. This planar diagram by itself goes to zero at high energies and its l As revealed during the discussion with one of the authors (A.B.K.), in ref. 4, it was supposed that the total inclusive cross section is given by its direct part. This is equivalent to the assumption that the spectator part has the same form as the direct one in the appropriate kinematical region. Therefore their results should rather be compared with our total inclusive cross sections, which may Iead to eliminating the disagreement.
M.A. Braun / Parion model
701
Fig. 3. A production amplitude in the parton model. Observed particles may correspond either directly to partons (the upper and lower lines) or to secondaries produced in a parton-nucleon collision (the middle line).
contribution to the total cross section vanishes. However, the mentioned one-particle contribution does not vanish. As is well-known it is cancelled by the contribution of multiparticle intermediate states schematically shown in fig. 4b. Inclusion of such states (“inelastic screening”) cannot be realized by any reliable method (see, however ‘)). Nevertheless it is known that the inelastic screening is small. As we shall not attempt to calculate the ‘inelastic screening, we shall also neglect the contribution of fig. 4a, which is integrally cancelled by the inelastic screening. ‘so finally we are left with the contributions of diagrams of figs. 2b, c. These contributions can be calculated in the same manner as when studying the total cross section. For the total number of partons N (registered included) and n active partons we get the high-energy part W!,‘,.$corresponding to fig. 2b:
The contribution
Wri corresponding
to fig. 2c is given by
iW~~(X)=.i.(X)l~(~i~(,i)),.,(,,,...,~”_~,X).
(4.2)
To go over to inclusive cross sections we have to take absorptive parts of these
Fig. 4. Intermediate states which give contribution to the inclusive cross section apart from those shown in fig. 2.
M.A. Braun / Partan model
702
amplitudes. For Wf$ it is sufficient to simply take the imaginary part: 21m W~:?(x)=-ZRel~(~ju(nr))p.+*,N(a,,...,a.,x).
(4.3)
For W!$, the argument is somewhat more complicated. Its absorptive part consists of two contributions, one with ia another with its complex conjugate. Consider the part with &z(x). Making all possible cuts in the rest part of the diagram we shah get the total contribution for it equal to the right-hand side of eq. (4.2) with the product n iU((Wi)substituted by -2 Re n ia( (Ui)(i.e. the total discontinuity). However there is one more contribution corresponding to the cut which leaves all the amplitudes ia on one side of the cut and iafx) on the other. This contribution is equal to the right-hand side of (4.2) with the change of all ia to complex conjugates. Joining this contribution with the found earlier and also with the conjugate contributions we finally get 21m W~~(x)=-nZReia(x)l~‘(~io(s))p.N(a,,...,~~-,,~).
(4.4)
Note an important remark. The contribution Wiz exists only for n 3 2 as for n = 1 in one of the parts of the cut diagram there is no interaction with the target. For n 3 2 the total spectator contribution is given by DnN = 2 Im ( W’,‘,‘,+ Wi*A). According to (2.7)-(2.8) DnN obeys the sum rules
J J e
DnN(x) = 2N Im W,,
ED,,(x)=2Im
WnN.
+niu(x)p,(a,,.
. . , a,-‘,X)
,
(4.5)
Let us now sum over all N:
>
.
(4.7)
The spectator inclusive cross section 1, is obtained from this by attaching the proper nuclear factor and summing over all II: I, = Ig’-Aa(x)pl(x),
(4.8)
where (4.9)
M.A. Braun / Parton model
703
The subtracted term in (4.8) serves to eliminate the extra contribution Im W’,“; a(x) = 2 Im a(x) is the total partonic cross section. From (4.6) we find a sum rule for Z(Al)
Ie
z:)(x) = ay ,
(4.10)
which implies that particles described by Z2’ carry all the energy. The rest part of the spectator cross section proportional to A can be taken together with the direct contribution, so that the total inclusive cross section
Let us find sum rules for I(‘). Using sum rules for j
$$a, x) = dab(a),
ej(a,x)=ao(a),
(4.12)
where Z.Lis the partonic multiplicity we get
I dx ~I,
(2)
(x)=0,
f$ Zjf'(x) =A IT I
e
+)P*(x)(P(x)
- 1).
(4.13)
In agreement with (4.10), the first sum rule shows that the spectator part included in Zg’ completely cancels the energy carried by particles produced by the direct mechanism. In this sense the interference between the spectator and direct mechanisms restores the conservation of energy and leads to a decrease of multiplicity in the region where the spectator contribution is different from zero, i.e. in the projectile fragmentation region. To study the x dependence of the inclusive cross section we have to make some assumptions about the structure of pn. It would be natural to propose a factorized form for pn (2.16) or (2.17). However, it can be seen from the sum rule (2.7) that the one-parton distribution p(a) has to depend on the number n of active partons. This is natural, of course, as the total number of partons N 2 n and the average parton momentum should decrease with n. To simplify the problem let us use the weak dependence of a’s on energy and take them out of the integral (4.7). Then using (2.7) DnN(x)=-2Re(id)“-‘p$!,(x)((l-n/N)iti+(n/N)ia(x)),
(4.14)
where p’,,&= P,~N!/( N - n) !. The sum rules for pIN involve pON on the right-hand side. Assuming (2.15) we can perform summation over N to get for p’,” =CNrn p’.::
I
Epy)(x)=h”(h+n),
I dx GPn
(1)
(x)=A”.
(4.15)
Thus a simple factorization property of pn(al,. . . , a,) cannot be true, since p’.” as a function of x should depend on n. The absorptive part DnN contains one more
704
M.A. Braun f Parton model
function pizA= p$$/ TV.One can obtain sum rules also for it. But in (4.14) it is multiplied by a small factor a(x)-n and we shall neglect this contribution. We introduce the normalized distribution p”‘,”= p(n’h-” satisfying the sum rules
I
dx -/$‘(x)=A+n, 4VX
E&l”(X)
(4.16)
= 1.
Then we obtain D,(x) = -2 Re
(id,Jn&‘)(x) ,
(4.17)
where &, = Ad is the averaged hadronic amplitude. With the necessary nuclear factors and after summing over n Ik”(X) = -2 Re
d*b f
(4.18)
Ci( id,, T( b))“$,“(x) .
The cross section (4.18) satisfies (4.10) and also $1$‘(x)
= AuSqOf - 2 Re
d2bAi&T(b)(l+ia”,T(b))A-*.
(4.19)
The second term on the right-hand side is negligible for A> 1, so that the total multiplicity generated by 12) on heavy nuclei ~2) = A. For A = 1 (a single nucleon) &‘=A+l. Summation over n in (4.18) can be performed if we choose a concrete parametrization of p’,“(x). We assume (4.20)
p”,“= c,x@(l -x)“,
where the exponent yn is taken to be n-dependent to satisfy (4.16). From (4.16) we find m=np+/IA-p-1, c;‘= B(P + 1, yn + 1). Now we can explicitly perform the summation over n in (4.18) using the integral representation for the B-function. If we denote y = Gh T( b)( 1 - x)~ then xP(l -x)‘o *~)(X)=-r(p+l)r(-B_l)2Re x
I
dt t-e-‘( 1 - t)P’
(4.21)
d2b((l+y(l-t)P)A-1).
For large A we have Ay + -co for b inside the nucleus. Therefore inner integral is equal to -aRf, = -ivy. From this we get rk”(x)-X~(l-x)%3-‘(p+l,
3/*+1)aaO’,
for AS=-1 the
A>l.
(The correct derivation of (4.22) involves a nontrivial analytic continuation exponents in (4.21)). For a single nucleon I(,“(x)=x~(l-x)Y’B-‘(p+l,
y,+l)a:ot.
(4.22) in some (4.23)
M.A. Bmun / Parion model
705
Since y1 = -yO+ /3 and p 3 0 the particle spectrum generated by 1(l) on a nucleus is somewhat more energetic than on a single nucleon, though the total multiplicity is less by 1. To describe in this framework the spectator part of I(‘) we first perform summation over N of the sum rules for pIN with the use of (2.15) to get
1em=“,
I
2
p,(x) = 1-exp (-A)
(4.24)
Separating the factor [ = 1 -exp (-A), we get for b,(x) = p,(x){-’
dx
w
$?rLJdx)=l. ~ompa~ng this with (4.16) we find that one may choose p’i(x) =$‘(x) where A + Y= A/f: Including a factor A into cr and j to go over to hadronic cross sections a,, and j, we obtain finally
We expect that for x + 0 p’l”+ 0 so that in the central region the total contribution reduces to the direct one. With the growth of x the spectator part of 12’ must cancel the direct contribution to satisfy the first sum rule (4.13). This implies that starting from some x 3 x0 the cross section 12) becomes negative. The total cross section is, of course, positive. So we expect x0 to lie in the region where I:’ and Ia” are of (If t he cross section 1%’ contains a the same order. Note also that as compared to IA factor J/A depending explicitly on the mean number of partons and falling like l/A for A )z 1. Therefore we expect I g’ to be considerably less than 12’ for finite X. These qualitative considerations can be confirmed by numerical calculations of the multiplicity &&._, = I,/ aho using (4.18), (4.20) and (4.26), which are illustrated in figs. 5-7. For ji, the simplest form has been chosen ~,(cx, x) = rr,@(tu -x). In fig. 5 the contributions p(i) and p(2) corresponding to Iti) and I(‘) are shown separately as functions of the antilab rapidity y = In l/x for A = 64. Parameter /3 governing the low x behaviour has been chosen f (continuous curves) or 0 (dash-dotted curves). In the latter case results depend on the projectile rapidity which here and elsewhere was put equal to 10. The mean number of partons A was put equal to 6. As can be seen *(*’ indeed becomes negative at small enough x, although the total p = pU)+pL(2) remains positive. Fig. 6 illustrates the dependence of p on the mean number of partons A for A = 64 and /3 =i (a) and /I =0 (b). Evidently the Adependence is much more pronounced for /3 = i than for /I = 0 in which case it is quite weak. For fl= f with the growth of A the Plato in the central region goes over into a well. The A-dependence is shown in fig. 7 for p = 4 (a) and p = 0 (b). A has been chosen equal to 6. It is clearly seen that at low y the multiplicity decreases
M.A. Braun / Parton model
706
2.0 1.8_
Fig. 5. Multiplicities
I
11111111 A=64 A=6
curve) and p = 0 (dash-dotted pA(I) and ~2’ for p = f (continuous ofy=Inl/xforA=64andh=6.
I
2.6
curve) as functions
I I111111
1.8 _
2.4
A=64
1.6-
2.2
#3=0
1.4 -
2.0
1.2 -
1.8
b)
1.6 1.4 1.2 I
I
0.6
l/x Fig. 6. Total multiplicities
pcLAas functions
0.1
0.2
of y for different
I 1111111 0.5
I 1.0
2.0
I 11111. 5.0 y = In l/x
A and p =i (a) or p = 0 (b); A=64.
707
M.A. Braun / Parton model P
-(a)
3.0 -
I
’ f3 = ’l/2 “““’
I
I lllll
A=6
l/x
lb)
’ p=o’ ’ ’ ’ ’ ” ’
I
I
Illlf
i
cc-------:
A=6
Fig. 7. Total multiplicities
I
pcLAas functions
of y for different A and B = 4 (a) or p = 0 (b); A = 6.
with the naive AGK with A, although for x-0 it behaves as A1’3 in accordance rules. For comparison the calculations of p according to the recipe of ref. “) are shown by dashed curves. The normalization is taken to be by ay rather than by a: so as to correspond to our results. The difference is mostly felt at small y in the region 0.2 s x =S1 where ref. “) gives a much less multiplicity than our calculations.
5. Other models Results obtained so far are based on simplest parton diagrams of fig. 1. However, one can see that they do not practically depend on a particular structure of pax-ton
M.A. Braun / Parton model
708
Fig. 8. Consecutive decay into partons.
diagrams.
Consider
for comparison
a different
mechanism
of breaking
into partons,
a consecutive decay, illustrated in fig. 8. Elementary calculations give in this case the same formulas (2.4)-(2.6), in which only the structure is specified of parton vertices depending now also on all (Y’S:
YN = gW-O
N-Z
fl
Ci;‘(tTt~-C t?l~*/OZi-tif~/CS[)-’ . 1
/=I
(5.1)
constant. The square Here 67, = CL’+ (1: kil)2, &, = 1-C: (Y~,g is the interaction brackets mean symmetrization over all (Y~,. . . , ffN. Clearly the probability (Y,) remains positive. Therefore the AGK rules remain fully satisfied as &I(%,.-., they follow only from the fact that pn are real lo). From this the AGK cancellations result for inclusive cross sections and consequently the direct part of the cross section will be given by the same eq. (3.2). Furthermore, it is not difficult to see that in the spectator part an irregularity occurs for n = 1: cutting of the propagator of the active parton need not be taken into account in this case. This results in separating from the spectator contribution a term proportional to A, which can be joined with .I to form I(‘) with the same structure as (4.11) and obeying the same sum rules (4.13). From the latter it follows that the bulk of the spectator part Z(l) satisfies (4.10). From the sum rules for Z(l) and Zc2) one can conclude that general properties
of inclusive
cross sections
remain
in fig. 8. The contribution Zc2) proportional decreasing as x grows until it becomes
the same for parton
diagrams
shown
to A will dominate in the central region negative and compensated by the main
spectator part Z(l) proportional to A2’3. This argument makes us believe that the described picture of particle production in hA collisions is true in a parton model with an arbitrary mechanism of breaking into partons. Another line of generalization is inclusion of different sorts of partons. Let, for simplicity, there are partons only of two sorts, 1 and 2. Then instead of (2.1) we shall find i&(b)=
1 ~INI”ZNZ
CiCi1iWn,N,n2N2T”(b),
n=n,+n,,
(5.2)
M.A. Braun / Parton model
709
where ni and Ni are the number of active partons and their total number for the ith sort. The high energy part W will be given by the formula
where ai is the parton-nucleon amplitude for the ith sort. The probability P,,,~,_~~ describes the distribution of partons of both sorts in parts of the longitudinal momentum and can be expressed through vertices for the transition of the incoming hadron into partons similarly to (2.5) and (2.6). Summation over N1 and N2 gives iW.,.,=($io,$io,).
(5.4)
Here the average is implied over parts ai and Pk of the longitudinal momentum carried by partons with the probability pn,_ which results after summation over N1 and Nz. From (5.2) and (5.4) we find for the amplitude is8(b)=((l+iT(b)(a,+~,))~-1).
(5.5)
If we assume that both the amplitudes Ui depend weakly on energy and parton numbers for both sorts are distributed independently according to Poisson’s law, then (5.5) goes over into the Glauber formula with the hadron amplitude a,,= h,a,+h2az where hi is the mean number of partons of the ith sort. Less trivial in the case when partons of one sort (say, the 2nd) are distributed according to Poisson’s law and the number of the partons of the 1st sort is rigidly fixed. Let Ni =3 in correspondence with the number of valence quarks in a baryon. Assuming again ai weakly dependent on energy we find iJ8(b)=(1+iaJ)A-3(1+iT(a,+Aa,))3-1.
(5.6)
Eq. (5.6) is different from the Glauber formula. However, the difference is in fact very small for A s 1. The influence of the partons of the 1st sort is negligible and the total contribution is determined by the first factor which originates from the interaction of the partons of the 2nd sort and turns the first term in (5.6) into zero. Inclusive cross sections can be studied in total analogy with the case of one sort of partons and with the same results. For the direct contribution the AGK cancellations can be established: if J”n;n: ’ ) IS * its part with ni partonic amplitudes of the ith sort out of which ri are cut, then (5.7) Thus the direct contribution
reduces to the two terms of the impulse approximation (5.8)
M.A. Braun / Parton model
710
This is true for arbitrary (5.8) is fulfilled contradiction
behaviour
when the number
to a naive
of U,(a) of partons
probabilistic
be multiplied by 3. One can also study more complicated
and P,,,“, (ai, &).
eq.
of some sort is fixed, e.g. N1 = 3, in
interpretation models
In particular,
according
of parton
to which
interactions
J, should
like the dual
amplitude refers to an interaction of par@ model ‘**‘*), in which the elementary two different partons from the incoming hadron with the target nucleon (formation of two “strings”).
For this model our approach
also leads to the rigorous
fulfillment
of the AGK rules for the direct part of the inclusive cross section: JA = AJ,, valid for arbitrary partonic distributions and amplitudes. In ‘*) where the probabilistic picture and the Glauber probabilities for multiple collision were used, expressions for JA were obtained which do not obey AGK: JA = ;AJ, . The factor i came from the ratio of the number of diquark strings in an NA collision (A+ 1) to that in an NN collision (2). Our calculations based on Feynman diagrams clearly speak against these simple
conclusions.
6. AA collisions The proposed parton model can be directly generalized to AA collisions. Nuclear factors can be separated from the amplitude by the standard manner for both colliding nuclei. We shall briefly discuss only the high-energy part of the amplitude. It can be represented by a sum of diagrams for interactions of nucleons from the projectile and target of the type shown in fig. 9. In the parton model wavy lines denote par-ton-parton amplitudes. Vertices for the projectile and target have the same structure as for hA collisions (fig. 1) and are given by the same expression W,, (eq. (2.4)). The only difference is that a now refers to a parton-parton interaction and depends not on (Y but rather on the product CYPwhere /3 is the part of energy carried by the second parton taking part in the interaction. Besides, a may now depend on a nonzero momentum transfer over which one has to integrate in loops in the diagrams of fig. 9. Summation over the total parton number N leads to the vertex W,, (eq. (2.9)). We can proceed in this manner with every vertex and represent the whole high-energy part as an average of parton amplitudes with the probability given by the product of all parton probabilities for all participant
Fig. 9. Interactions of nucleons from the projectile and target in an AA collision. Wavy lines denote parton scattering amplitudes. The nuclear part of the AA amplitude is not shown.
M.A. Braun / Parton model
711
nucleons both from the projectile and from the target. Supplying the necessary nuclear factors, similar to (2.10) we obtain the Glauber formula for AA collisions under the sign of the average over partonic probabilities: id(b)=
fi (d*xiTA(xi)) fi (d*xSTA,(x$-b) i=l
x
fi (
i=lj=l
j=l
(l+ia(x,-xi))-1
fi
(6.1)
'
The amplitudes a are taken here in the impact parameter representation. Eq. (6.1) turns into the ordinary Glauber formula if (n ia) =n (iu). Since the average over parton distributions is taken independently in different nucleons, this condition will be satisfied if it is fulfilled for hA scattering (sect. 2). In particular, for weak dependence of a on energy we shall find the Glauber result if in a nucleon partons are distributed according to Poisson’s law with an average number h. The NN amplitude is then equal to A*a. Since id is a real function of iu, the AGK rules will be rigorously satisfied for arbitrary a and P,,. Therefore, the direct contribution to the inclusive cross section for the production of a parton with a given part x of the longitudinal momentum in the c.m. system of colliding nucleons (-1 s x d 1) will be given by the impulse approximation: JAnA
= A’A
~p,(u)p:(B)j(aa,x)=A’AJ,(x).
(6.2)
The spectator part of the inclusive cross section is described by the same diagrams as shown in Fig. 2b, c, which however may also include some interactions of other nucleons. One has to take into account that for x > 0 a nonzero contribution will come only from partons originating from the projectile nucleons and for x < 0 only from partons from the target nucleons. For definiteness we consider x > 0. Using the results of sect. 4 for hA collisions we find that the spectator contributions corresponding to fig. 2b or fig. 2c are obtained by first substituting W’,l’ or W’,z’ for W,, in one of the projectile vertices, second, taking the imaginary part according to (4.3) or (4.4) and, finally, multiplying the result by the number of nucleons in the projectile A’. There is, though, a difference as compared to hA scattering. One has to take not the total imaginary part but only contributions from cuts drawn through the distinguished vertex. The sum of all these contributions can be easily calculated using the A.GK rules. One finds that only contributions from diagrams will be left which do not contain interactions not involving the distinguished vertex. As a result the spectator part IXA for x > 0 proves to be A’ times larger than the corresponding cross section IA for NA collisions:
LA(X) = A’L(x) ,
x=-o.
(6.3)
M.A. Braun / Parion model
712
will be given by the same formulas (4.8) and (4.9), the cross section a(x) of the parton-nucleon interaction being now expressed through the pure partonic cross section I,
The subtraction term in (4.8) can be joined with the direct part into the cross section Ik”!, proportional to A’A: IfjA = A’AI’,2’,
(6.4)
where 1’1” is given by (4.30). According to (4.13) we have
from which we conclude that, as for hA collisions, the total energy is carried away by particles corresponding to the bulk of the spectator cont~buti~n I:&. According to (6.3) we find la’!* = A’Iz’+ A@ .
(6.6)
The first term is nonzero for x > 0, the second for x < 0. Integrating (6.6) and using (4.10) gives
to be compared with the energy conservation
sum rule
The difference is due to the presence in the inclusive cross section of an additional contribution ft3) with terms proportional to 6(x - 1) and 6(x + 1) corresponding to the direct observation of spectator nucleons. These terms can be seen from diagrams studying the contributions from cuts to the total cross section which cancel when forming the inclusive cross section IA’A. The magnitude of b-terms can be established from (6.7) and (6.8): I(A3?A = A’( oj,o!, --a;$?(~-l)+A(tr::A-a::)S(x+l).
(6.9)
As a result we find that in the proposed parton model the inclusive AA cross sections are related in a simple manner to the inclusive NA cross sections. From (6.2) and (6.3) it follows that in the whole forward hemisphere in the c.m. system
M.A. Bmun / Parton
model
713
of the colliding nucleons for x > 0, x # 1 I:!A(x) = A'ly(x)
.
(6.10)
In the backward ,hemisphere the projectile and target interchange. All nontrivial dependence on both A and A’ appears to be limited to the region of spectator nucleons with x = *l. One has to note that eq. (6.10) immediately follows from a naive application of the AGK-rules for AA scattering. Our achievement is to sho& that this simple result persists even when conservation of energy is taken into account at finite energies in contrast to conclusions made in the context of a pure probabilistic approach in ref. 13). Relating (6.10) with the experimental data one has to keep in mind that our model, as practically all other models for high-energy AA-scattering, does not take into account particle production resulting from a cooperative interaction of several nucleons of the projectile (target) either at short distances (merging of parton clouds belonging to, different nucleons, presence of 3n -q-bags) or at long distances (rescattering and cascade proliferation of produced particles). These effects obviously dominate at 1x1close to or more than 1 in the so-called cumulative region where the right-hand side of eq. (6.10) is found equal to zero. For 0 s x < 1 they should be comparatively small for a light projectile. Indeed it has long been known
24
16
12
8
Fig. 10. The experimental ratio of pseudorapidity distributions of charged secondaries in central collisions of 200 GeV/nucleon I60 and p with Ag/Br taken from the data of ref. I’). The shaded area corresponds to 10% statistical errors. The arrow marks the division between the forward and backward hemispheres in the c.m. system.
714
MA.
Braun f Parion model
that, for such x, eq. (6.10) is fulfilled with a good accuracy for the scattering of 1.05 GeV/nucleon and 2.1 GeV/nucleon deuterons and alphas from various nuclear targets 14). Turning to more recent experiments and heavier projectiles we present in fig. 10 the experimental ratio of pseudorapidity distributions of charged secondaries for central collisions of 200 GeV/nucleon I60 and protons from Ar/Br taken from the data of ref. “). The experimental distributions are normalized to unity so that eq. (6.10) predicts an unknown constant for this ratio for q B 3.3, which corresponds to the forward hemisphere. As one observes in fig. 10 the experimental ratio grows with 7. This growth is not very pronounced for 3.3 < q d 4.7, i.e. for particles produced with comparatively high velocities in the antilab system. For larger v the growth becomes steeper. This effect may be partially attributed to the mentioned cooperative interactions, in particular to cascade interactions of particles with small momenta in the antilab system. For fast particles in this system such interactions become ineffective due to the finite formation time and somewhat better agreement is observed. Another reason for the deviation of experimental data from eq. (6.10) may be the selection of central collisions in ref. I’), which results in a deformation of spectra in the projectile fragmentation region. More definite conclusions require information on the complete particle spectra and their dependence on the atomic number A’ of the projectile. The author expresses his gratitude to A.B. Kaidalov, E.M. Levin and Yu.M. Shabelsky for helpful and stimulating discussions. He is also most thankful to R. Tarrach and E. Elizalde for their kind hospitality and interest in this study during his stay at Barcelona University where this work was completed. He is also greatly indebted to E. Gaztaiiaga for his help in preparing this paper for publication.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS)
V.N. Gribov, JETP 56 (1969) 892; 57 (1969) 1306 L. Caneshi and A. Schwimmer, Nucl. Phys. B133 (1978) 408 M.A. Braun, Yad. Fiz. 47 (1988) 262 A. Capella and A.B. Kaidalov, Nucl. Phys. Blll (1976) 477 A. Capella and A. Krzywicky, Phys. Lett. B67 (1977) 84 V.A. Abramovsky, V.N. Gribov and O.V. Kancheli, Yad. Fiz. 18 (1973) 595 O.V. Kancheli, Pis’ma JETP 18 (1973) 465 J. Koplic and A.H. Mueller, Phys. Rev. Dl2 (1975) 3638 V.M. Braun and Yu.M. Shabelsky, IJMP A3 (1988) 2417 M.A. Braun, Yad. Fiz. 48 (1988) 409 A.B. Kaidalov, Phys. Lett. B116 (1982) 459 A. Capella, Preprint LPTHE ORSAY (1988)# 88/67 Yu. M. Shabelsky, Preprint LNPI (1988) # 1433 J. Papp et at, Phys. Rev. Lett. 34 (1975) 601 L.M. Barbier ez at, Phys. Rev. Len. 60 (1988) 405