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Pairing in multiparticle amplitudes: Inclusive distributions
Nuclear Physics B55 (1973) 132
156. North-ttolland Publishing C o m p a n y
P A I R I N G IN M U L T I P A R T I C L E A M P L I T U D E S : INCLUSIVE DISTRIBUTIONS* John STEINHOFF** 1)epartment o f Physics and the Enrico Fermi Institute, The University o f Chicago, Chicago, Illinois 6063 7 and ltigh Energy Ph),sics Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 25 August 1972 Abstr,'tct: A model for multi-pion p r o d u c t i o n in the central region in high-energy collisions is studied which describes factorizable emission of pion pairs. A m a t h e m a t i c a l identification between the exclusive cross section for pion emission in our mode l (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the a s y m p t o t i c single-particle inclusive distribution, the two-particle inclusive correlation function, and the e x p o n e n t of s in the total cross section by means of cluster diagrams. An integral equation is ex hibited for summing the terms corresponding to the cluster diagrams. A specific model is then considered, which we call "s-channel pole d o m i n a n c e " . In this model the amplitude is assumed to be large only when the subenergies of pairs of pions are near the mass of a low-lying two-pion resonance, and the transverse m o m e n t u m of each resonance is small. The dependence of the a m p l i t u d e on other variables is ignored, so that we effectively have independent emission of two-pion resonances with non-zero width. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, as s ~ ~, and that the correlation function can have an exponential " t a i l " m rapidity of qualitatively longer range than the resonance. A crude numerical sinmlation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, and to have the same e x p o n e n t i a l " t a i l " as the I = 0 correlation function.
1. Introduction Classical statistical-mechanics techniques have been exploited in the past several years to describe multiparticle production processes in high-energy hadron-hadron collisions [ 1], and quantities such as the single- and two-particle inclusive spectra * Thesis su bmitted in partial fulfilhnent of requirements for the P h.D . degree at the University of Chicago. ** A.U.A.-A.N.L. Fellow. Present address: Department of Physics, McGill University, Montreal, P.Q., Canada.
J. Steinhoff, Multiparticle amplitudes
13 3
have been calculated for various models. The validity of a cluster expansion in a q~3 field theory in the ladder approximation has been verified [2]. Specific use of calculational methods of statistical mechanics has been made in the application of the "molecular-field" approximation to the calculation of the effects of final-state multiparticle absorption on inclusive spectra [3]. We use here a cluster-diagram technique to compute the behaviour of the total cross section and the single- and two-particle inclusive distribution functions for a class of models describing pion production in the central region. The models that we consider take into account correlation of the pions in pairs, but are otherwise similar to independent emission (IE) models. IE of pions in high-energy collisions is not a new idea [4]. Recently, IE models describing production of particles with small transverse momentum but longitudinal momentum increasing with energy have been constructed, that describe experimentally-observed single-particle spectra quite well [5]. When energy, momentum and charge conservation are taken into account [6], particles cannot be produced independently, but we will call such a model IE if the amplitude for particle production is completely factorizable. Inclusive two-particle correlation functions are of course identically zero in an IE model without conservation constraints; and numerical calculations done for a model with constraints do not give appreciable correlation in the region where both particles have small rapidity [7]. On the other hand, this correlation function has been measured experimentally for 7r n pairs in this region [8], and is seen to be positive and fairly large. Thus, IE models do not seem to be adequate to explain experimental data. Theoretically, one would expect the production amplitude to become large when the subenergies of distinct pairs of pions (with correct spin and isospin) are close to the mass of a low-lying two-pion resonance*. This property is of course not consistent with IE. In this paper we discuss a pion-production amplitude with pairwise correlations, that can incorporate this property. For simplicity we go to asymptotic total collision energies, ignore energy, momentum and charge conservation constraints, and consider only the central region in rapidity. These constraints should not be important, in this limit, for the quantities that we will compute. A mathematical identification between the exclusive cross section for pion emission in our model (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the single-particle inclusive distribution, the twoparticle correlation function, and the contribution of the central region to the s de-
* The amplitude is expected to have poles in the complex sij plane, where sij is the subenergy of particles i and/'. However there should not be simultaneous poles in sij and Sik, J 4=k, as explained in, e.g., ref. [9]. Also these poles should produce large isolated peaks (resonances) at low si/, hut smooth behaviour at high si/,.
134
.L Steinhofj~ Multiparticle amplitudes
pendence of the total cross section by means of cluster diagrams. An integral equation is exhibited for sunnning the terms corresponding to the cluster diagrams. A specific mode] is then considered, which describes independent emission of two-pion pairs, each pair being correlated in an I = 0 or I = I resonance with nonzero width. This amplitude is large only when the subenergies of distinct I = 0 or 1 = 1 pairs of pions are near the resonance mass, and the transverse m o m e n t u m of each resonance is small; the dependence on other variables is ignored. We call this the "s-channel pole dominance" approximation. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, regardless of the form taken for the resonance, and that the correlation function can have an exponential "tail" in rapidity of qualitatively longer range than the resonance. A crude numerical simulation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, as suggested by experiment
181. 2. The model The class of models that we consider are characterized by a partially-factorized form for the amplitude for production o f N pions in the central region, and a set of particles in the fragmentation regions: A ( s { r f } ; r l , r 2 ..... r N ) = A f ( s , { r f } ) A N ( r l
,r 2 ..... r N ) ,
where N is restricted to be even,
. . ~¢" A N ( r l , r 2 ..... r N ) -. . .g~.
~
A2(rl,r2)A2(r3,r4)...A2(rN
l,rN)"
(1)
distinct permutations S is tire total c.m. energy squared, and r i = ( y i , q i ) labels particles in the central region with c.m. rapidity v. and transverse momentum qi. The set of particles in tire fragmentation regions is specified by {rt, }. This particular method of separating out the fragmentation dynanlics was chosen for simplicity: the only property that we require is the lack of long-range correlations between particles in the central region an0 fragmentation regions. We do not include charge indices for the pions, or indices describing production of other particles in the general treatment; they can be included in r, without altering airy of our formulae, as seen in the numerical exmnple that we consider. One of the central features of multiperipheral [11] and current 1E models is the existence of a region in rapidity that includes all of the kinematically allowed values except for finite (fragmentation) regions at each end, as s -~ oo, where (finite) multiparticle distribution functions depend only on relative rapidities, charge states •
I'
J. Steinhof]; Multiparticle amplitudes
13 5
and transverse momenta of the pions. We assume these features, which are believed to be more general than the above models [1]. Hence, we define A2(ri,r/) to be a function only of qi , q] andYi - -vs, where A 2(ri ,r/) ~ 0 when ly i Y/I ~ oo, or q2 or q2 becomes large, so that the particles are effectively emitted in independent pairs, with amplitude A 2(ri ,ri). This defines the class of models under consideration, which includes as a special case that of independent p or e meson emission. The generalization to include single-particle production in the amplitude (unpaired pions) is straightforward, and the corresponding formulae are given in appendix A. Since we only consider inelastic events that result in produced pions in the central region, diffractive dissociation and elastic events are not described by our model, and our distribution functions will not include these processes. We shall only be concerned with the central region, and we assume the mean number of particles produced in this region increases linearly with Ins as s -~ oo, while the number in the fragmentation regions remains finite. We accordingly define an "exclusive" N-particle differential cross section to be the probability of exactly N particles being produced, with specific molnenta, in the central region and anything in the fragmentation regions, ttence, all quantities are summed over (rf}. The "exclusive" N-particle differential cross section is then written
d3NoN(s;r 1..... r N) 3 3 = °o(S)QN(rl"'rN)' dr I ...dr~
where o0(s ) is the integral of IA f(s;{rf})l 2 over (rf} with energy and longitudinalnlomentum conservation, divided by the flux factor, and our "exclusive probability function" is 2
QN(r 1..... rN ) = g~N
~
distinct permutations
A2(rl,r2)A2(r3,r4)... A2(r N_ l,rN)
(2,)
We do not consider the dynamics in the fragmentation region, nor do we explicitly discuss the dependence of our amplitude on s, and we write the total cross section, and single- and two-particle inclusive cross sections in the form OT(S ) =
Oo(s)QO(s),
d3o l (s;r) _ on(s)Q 1(s;r) , dr 3
d6o2(s;rl,r2) dr~ dr 3
- oo(s)Q2(s;rt,r2).
In the above equations
136
QK(s;rl ..... rK)=
J. SteinhoJ]~ Multiparticle amplitudes ~ z2NQK2N(S;rl ..... r2N) N=~E(K) (2N-K)! '
(3)
where E(k) = K ( K + I ) , K even (odd), and
K Q2N(S;rl ..... rN)= f d3rK+l...d 3r2NQZN(rl ..... r2N ) .
(4)
We have introduced a formal parameter Z to be used in solving our equations and then set equal to 1. The dependence of a quantity on Z will be suppressed unless needed. We will see that (this part of our discussion parallels much of that in ref. [2] for a 4~3 field-theory model) as s ~ ~, QO(y) = const, e c~Y, where Y = lns is written as the energy-dependent variable in the QK functions, and we will compute the (energy-independent) quantities p l ( r ) _ QI(Y;r) _ d3ol(s;r) 1 Q0(y) dr 3 OT(S) ,
(5)
Q2(y;rl,r2) d6°Z(s;rl,r2) 1 p2(rl'r2) = QO(y) = dr~ dr32 °T(S)"
(6)
These quantities can be completely determined with our model amplitude (1) once the solution to a linear integral equation is known. This integral equation sums a series of terms obtained by a cluster-expansion technique.
3. Cluster-diagram expansion The exclusive probability distribution o f N pions is QN(rl,...,rN). The properties of the functions A2(r l,r2) contained in this function are such that the pions are distributed along t h e y axis in clusters, and confined to a finite region in q. We can define an "analog system" of classical nonrelativistic particles in equilibrium interacting in pairs [ 12] such that the configurational probability function for this system is mathematically isomorphic in terms of structure to QN(rl,...,rN), where the r i describe the positions of these particles in ordinary three-dimensional space. Since * each r i appears in QN(rl ..... rN) in the function g 2A2(ri,rj)A2(ri,rk);f, k 4= i, each of these particles is bound to one or two others, and to the line q = 0, which can be thought of as a thin wire running along the y axis. As in a system of molecules interacting with each other and an external object through two strong "chemical bonds" per molecule, we can form rings, or "polymers" with no open bonds. Also, these two bonds are different, one being represented by gA~(ri,rk), and the other by
gA2(r i ,r]). The terms contained in QN(rl ..... rN) in eq. (2) are easily enumerated using ele-
J. Steinhoff, Multiparticle amplitudes
13 7
mentary cluster-diagram techniques [13]. A "diagram" consists of N points, labeled 1 through N (we do not rigorously adhere to the terminology of graph theory) with two lines attached to each point, one solid and the other dashed, connecting it with one or two other points. The term corresponding to a particular diagram is obtained by writing a factorgA2(ri,r/) for a solid line between point i and point/, gA~(ri,r/) for a dashed line, and forming the product of such factors for each line in the diagram. For N points we thus have a product of N factors, where N is restricted to be even. Corresponding to each distinct diagram we draw in this way, there is a distinct term, and the sum of all such terms is equal to QN(rl ..... rN). As an example, f o r N = 8 one term would be
g8 [A2(rl,r2)A2(r3,r4)A2(rs,r6)A2(r7,r8)] X [A2(rl,r6)A2(r2,r5)A2(r3,r4)A2(r7,r8)]*~ and the corresponding diagram is drawn in fig. 1. The amplitude to produce an unspecified group of particles in the fragmentation regions and 8 particles in the central region is pictorially represented in fig. 2, where gA2(ri,r/) is represented by a vertex with lines labeled, i and/, and a heavy line. The particular interference term represented by the diagram of fig. 1 is depicted in fig. 3. We see that for our model amplitude of eq. (1) all the terms are products of factors that can be represented by "ring" diagrams. For instance the term described in the example above contains the factors
8
3
7
4 6
5
Fig. 1. A diagram representing a term in Qs(rl,...,rs).
"•1"•
l fragmentation
~4 distinct ~.J permutatt0~7
;~
}fragmentation
Fig. 2. A pictorial representation of A (s, {rf} ;r I ,...,r8 ).
.l. SteinhofJ; Multiparticle amplitudes
138
~ dp•!uClatio•
l:ig. 3. An interference term contributing to Qs(rl,...,?8).
g4A 2(r l,r 2 ) A ~(r 2,r 5 ) A 2(r5 ,r6 )A ~(r 6 , r l ) , g2A2(r3,r 4)A ~(r4,r 3 ) , g2A2(r 7,rs)A~(r8,r7 ) . Once we have the forms of the cluster diagrams we can consult the literature to get Q0(Y), P l(r) and C(rl,r2) in terms of integrals over the factors corresponding to ttle ring diagrams, as shown in appendix B. We derive the results here since the arguments are rather simple. We first compute Q lv(r ) and Q2(rl,r2)m terms of Q ° ( Y ) , K < N and the cluster integrals Iy(r r ) = gA 2(r,r ), •
")
r
t
l~(r,r')=gg f d3r2d3r3...d3rKA2(r, r2)A;(rz,r3)...A(2*)(rK,r' ) , K > I ,
(7)
where the last (*) applies i l K is even, and
IlK(r) =12K(r,r) ,
K even.
(8)
We always assume we are far from the end regions i n y so that the limits of inte2 ,) depends graticn can be taken t o y = _+0%and IlK(r) is independent of y , and IK(r,r only o n y y',qandq'. To compute Ql(r 1) we consider all distinct N-point diagrams and integrate the corresponding functions over r/, 1 < 1"~< N. We order the terms according to the number of points 2K that are on the ring which contains point 1. There are, for each K, (N--1)!/[(N 2 K ) ! ( 2 K - 1 ) ! ] ways of choosing the 2 K - 1 points out of the N 1, and there are ( 2 K - l ) ! distinct ring diagrams, each of which results in the same function [ll2K(r 1)] when the corresponding factor is integrated over ri, 1 < i ~< 2K. The integral over the remaining set o f N - 2K coordinates represented by points not connected to 1 results in a factor Q°_2K(Y ). Hence, we have, defining QO = 1,
J. SteinhofJ~Multiparticleamplitudes -~N
(N 1)!
QO-2K(Y)I1K(rI'(N
Q1N(rl)= ~
139
2K)!
K=I
By fixing coordinates r 1 and r 2 we similarly have
"N
Q2(rl,r2)= ~ Qo K=I
2K(Y)K ~K(rl,r2)( N 9(~-.) N "(-_~K ~,1 9)1'
where K~K(rl,r2) is represented by the set of diagrams with 2K 2 points connected directly or indirectly to either point 1, point 2, or both. We have diagrams with both point 1 and 2 on the same ring, and on separate rings. Summing the different terms, we get 2K
"> KSK(rl,r2) = (2K
2)!
1
[~ /=1
~ "~* lT(rl,r2)15 K /(rl,r2)
K 1
+ 2
/=I
where the second term is absent ifK = 1. We can sum over N now, and from eq. (3) we get
QI(y;rl)= ~ Z 2N Q~A'(Y) N=0
(2N)! K=I ~
z2KI~K(rl)'
or from eq. (5), dropping the subscript,
(9) K=I
Also, we get
Q2(y;rlr2) = Qo(y)
z2KIIK(rl) K=I
+ Q°(Y)[lc(rl,r2t2+lc'(rl,r2)121 , or
~ K=I
z2KIlK(r2)
J. Steinhoff, Multiparticleamplitudes
140
p2(rl,r2) = o l ( r l ) p l ( r 2 ) + C(rl,r2), (10) , 2 C(rl,r 2) = lc(rl,r2)l 2 + tc(rl,r2)l ,
where
c(rl,r2) = ~
Z2Kl2K(rl,r2),
(11)
K=I
c'(rl,r 2) = ~-J Z 2K-II2K_l(rl,r2).
(12)
K=I From eqs. (8) and (9), we see that
pl(r) = c(r,r),
(13)
and, from eq. (7),
c'(rl,r 2) = ZgA2(rl,r 2) + Zg f d3rc(rl,r)A2(r, r2).
(14)
Note that C(rl,r2) is always positive in our model, regardless of the form of A2(rl,r2). By including charge indices in r, we will see that C(rl, r2) is positive for pions of the same or opposite charge, regardless of the isospin properties of
A2(rl,r2). The equations for o l(r) and C(rl,r2) have a very simple interpretation. Each distinct configuration of unlabeled points that lie on a ring with one point fixed contributes equally to p l(r) (before integration over the coordinates represented by the points). Since there is only one distinct configuration of points for each value of K, the number of points on the ring, we have eq. (9). If we fix two points on the ring, we have effectively two open chains between the two points, and after enumerating the distinct configurations of points and doing the integrals, we have eqs. (10). The restriction that we have only an even number of points on the complete ring means that both chains have either an odd or an even number of points, and we have two separate terms in eqs. (10). Some contributions to p l(r), C(rl,r2) , c(r],r2) and c'(rl,r2) are depicted in fig. 4, where integration is ilnplied for an unlabeled point. We can now solve for QO(y). From eqs. (3) and (4) we see that (writing the Z dependence explicitly) z F dZ 1
QO(z,Y) = 1 + J -~1 0 But, from eq. (5),
d3rQl(Zl'Y;r)"
J. Steinhoff, Multiparticle amplitudes 1
p}'(r~)• ~
+
+
1
_
:
=
c1(r ~ 2' ) r l =
:
_...
1
2
~.
r. + r.
Fig. 4. D i a g r a m s r e p r e s e n t i n g
+
2
1
+
2
1 c(r.,r2)
1
(_...__..~+
•
141
i
+
-
--
2
1
+ ~.
2
I
:_
:
:
--
1
_~---o
+
....
2
~ -
-+
....
t e r m s in p J (r), C(rl,r2 ), c ( r l , r 2 ) a n d c ' ( r l , r 2 ) .
QI (Z, Y;r) =Q°(Z,y)p l (Z,r) . Hence Z dZl o Q0(z,y)-- + f y][Q (Zl,Y) fd3rpl(z ,r), 0 so Z
Qo(Z,Y)=exp f ~ 0
fd3rpl(Zl,r)
(15)
1
Now, except near the ends of the y region, p l (Z,r) is independent of y , so
Qo(y) = exp (~Y+/3), where/3 is a constant and 1
f -dZ ~ - fd2qpl(Z,q )
OL=
0 or, from
eq. (9),
(16)
142
.1. Steinhoff, Multiparticle amplitudes
Z;
(17)
K= I 2 K '
where
I~K =f d2ql~K(q). From eqs. (10), (13) and (16) we see that once we have c(Z, rl,re) we can obtain c~, o l ( r ) and C(rl,r2). Also, as shown in appendix C, we can then easily obtain higher correlation functions.
4. T h e integral e q u a t i o n
From eqs. (7) and ( 1 1) we can obtain an integral equation for
c(Z,r l,r2) : Z2K(rl,r2) + Z 2
c(Z, rl,r2):
fd3rK(rl,r)c(Z,rr2) ,
(18)
where
K(rl.r 2) = g2 fd3rA 2(rl,r)A~(r, r2)
(19)
depends only o n y =Yl v-~ 0 = q51 02 (tile angle about the), axis between ql and q 2), Pi = q~ (i= 1,2) and-isospin indices of the pions. Defining Fourier transforms:
2rr
"~m(Z,co,l,l,P2 ) = f
+~
dq5 e im¢, f
d3, e-icoy c(Z,y,dAPl,t)2) '
we have only the square of the transverse momentum left as a variable in our integral equation:
. '=,,z (oO,Pl,P2)+ c~m.(Z,co,pl,p.~j=Z-K X "~m(Z, co,p,p2) .
1Z2 f dpi,n(co,l)l,t)) 0 (20)
By eq. (19)K(I)I.P2) is Hermitian (we shall suppress Z, m and co dependence unless needed), and we expand it in eigenfunctions with real eigenvalues:
/~(PI'P2) = ~ K
~VKk'K(Pl)~'*K(P2)"
J. Steinhof]i Multiparticle amplitudes
143
We then have
c'(Pl,P2) = ~
UK~K(Pl)k~:(P2) ,
K
where
x~(co)
~ ( z co)
2
Z
1
m
--~XK(co)
Inverting the m and co transforms, we have +~
+~
c(Z,¢,)',q~,q2) = ~
2
e imO ---~ . _~
K m =-~ m "1'
2
+~
d Y 2 k ~ ( Y l , q 2) dYl f _oo
,m
(21)
X k K (y2,q2)¢k ( Z , y - y l Y2), where +~
k2O',q2) --
1
c~((Z,y) =
f
m dco e +ico y ~kK (co,q2),
dco e +lwy Z -2 C
~ m
-
:~'X( c o )
'
(22)
and the contour c includes all poles of the integrand (co=co/) such t h a t y lm co/> 0 (y4=O) (see ref. [12]).
5. P h y s i c a l results
We first discuss some general properties of C(rl,r2) that can be obtained from our equations, with somewhat more detailed considerations of A2(rl,r2). Then, as an example, we assume a factorizable form for A2(rl,r2) that approximates an I = 0 spinless resonance, and numerically compute C(rl,r2).
5.1.1. Isospin dependence We first consider n+n - resonances. (We do not include charge indices in r here.) The amplitude A2(rl,r2) is then zero if particles 1 and 2 are both n + or both rr . The kernel K(rl,r2), defined by eq. (19) is then zero for n+n - pairs, but non-zero for n+lr+ or rr n - pairs. It then follows from eq. (18) that c(rl,r2) is non-zero only for rr+rr+ or rr n (•=2) pairs, and from eq. (14) that c(rl,r2) is non-zero only for n+n - pairs.
144
J.
SteinhofJ~ Multiparticle amplitudes
The correlation function C(rl,r2), from eq. (10), can be seen to separate into Ic(rl,r2)l 2 for rr+rr+ or 7r rr- pairs, and ic'(rl,r2)l 2 for rr+Tr pairs. If rr°Tr0 pairs are also produced, as in an I = 0 resonance, then both c(rl,r2) and c'(rl,r 2) contribute to the 7r°rr0 correlation function. In this case, though, both c(rl,r2) and c'(rl,r2) are zero for rr+rr° or rr rr0 pairs, and hence there is no rr-+rr0 correlation. In the context of our model amplitude, defined by eq. (1), all the resonances are in the same quantum state. Though the states are not necessarily eigenstates of total isospin or charge, we assume that the expectation value of the charge is zero. This property is implied when we comment on the diminishing effect of charge-conservation constraints at high energy. Thus for I = I resonance production, we can consider not only zero-charge states (rr+rr - pairs), but a coherent superposition of + 1, 0 and 1 charge states. In this case we would have the possibility of (positive) correlations between pions of any charge.
5.1.2. C(rl,r2) at large y Although the equations for C(rl,r2) in sect. 4 are valid for a general amplitude we are particularly interested in an amplitude of very short range in y and even in ~ and v. The first iteration ofA2(rl,r2), K(rl,r2) will then still be of short range. In general the functions k~(y,q 2) will then vanish for large 3'. When we 2 look at the function c(Z,~,y,ql,q~ ) at large y (large compared to the range of A2(rl,r2) ), from eq. (21) we see that it will be determined by the function cry(z,3 ,) with the longest range. But from eq. (22) we have the expansion ( 3 ' > 0 )
A2(rl,r2),
= 4i ~ e z2
w}y eiCORy p X ~ ( c o )
] 1,
(23)
I CO=CO/
j
where the sun, runs over the (K-, m-dependent) values of co/= coR + ico~ such that ~ ( c o / ) = , Z --. Finding the coj's with the smallest imaginary part, and calling them co(), -co~ with correspondingK 0 and m0, we have, at large lYl, , 7 ~ ~ ~4 c(Z,~,),q]',qs-)
v 2 B(coo,y,q'f,q2)
cos moo e-COollyl ,
where "~ eiCORlyl - , n o ,n~ B(coo,.v,q ~,q 5) = d kKo (COO,q 2l -)kK o (coo,q 22)
+e-iCO°RlYi'*Tm°" •
2,~m o, * 2, a KKo ~.CO0,ql)KKo (CO0,q2),
•
mo
1
d=[-laXX°(co)L, 3co
] tO=CO 0
"
(24)
J. Steinhoff, Multiparticle amplitudes
145
I! coR = 0 or m 0 = 0 we would have ½ the above result. We see from eq. (14) that
c ((~,y,q~,q2) will have the same asymptotic form ifA2(rl,r2) is of short range, and we have the two-particle correlation function:
C((9,y,q~,q 2) ~ e 2wollyl [a 1+a 2 cos(2coRlYl+01)] . A short-range basic two-pion correlation in the amplitude, such as would result from low-mass resonance decays, should thus develop a longer range "tail" in the correlation function. It is interesting that precisely the same phenomenon is seen in liquids with short-range two-body forces [14]. Whether this effect is pronounced or not depends on the specific form of A2(rl,r2) and on the strength g. To solve for p l ( r ) and ~ we need the functions cr~(Z,y) f o r y = 0. We then have
ol (Z,q 2) = e(Z, O, O,q2,q2) , and the expansion derived above may not be useful. A direct numerical solution of eq. (20) by iteration may then be the best method. We can then obtain c~ from pl(Z, q2) by eq. (16). In practice, we may have an idea what A2(rl,r2) should be, but not know the strength g. We then would fix g by fitting some experimental data with our model. If we choose to fit the observed multiplicity, then g becomes exactly analogous to the fugacity in statistical mechanics.
5.2. A numerical example We first briefly discuss some properties of an e-meson emission amplitude, and then discuss a crude simulation which should give the essential features of the more literal representation. This example is uncomplicated by spin and isospin. If eacb e meson were produced with transverse momentum q and amplitude gAc(q), we would have, igncring threshold behaviour, 2
me
g(A2)ulu2(rl'r2) =gMulu2 Ae(ql+q2) Sl 2 _ me2 + ikm2e '
(25)
where we used a pole approximation with half-width 2kin 2 and Mulu2 , to be given below, describes the dependence of the amplitude on the charge states of the emitted pions. The energy squared of the two-pion system is s12 = 2(m2+q2)~(m2+q2) ~- c ° s h ( Y l - Y 2 ) - 2ql "q2 + 2 m 2 , where m is the pion mass, and m e is the mean e mass. The quantities [gA2(rl,r2)l and Im [gA2(rl,r2)] are plotted in fig. 5 for ql = - q 2 and q l = +q2, [qll = 300 MeV, for various values o f y = Y l - Y2. In fig. 6 the above quantities are plotted w i t h y = 1 for IqlJ = 300 MeV and q2 = xql,
146
J. SteinhofJ] Multiparticle amplitudes
1.2
"~',
IgA21~°-~
1.0
;aussian ~ k \ \ ' ? A2 5 .8
gA2
.6
.4
.2
0
---J7 _
\. [gA21 q]°+q,
"NN\
\~', • ~%~ .,~ \ \ \ , /
0
1
- , t ~ ~ -~ev
2
3
4
l:ig. 5. Plot o f g A 2 ( r l , r x ) ; q l = +-q2, ]qll = 300 MeV.
1.0
sian / / / gA2~/
~~.~"
,~
-3
-2
1
I
-i
+i
I
+2
+3
l:ig. 6. Plot ofgA2(rbr2); IqlJ = 300 MeV, q2 = x q h y = 1.
J. Steinhof¢; Multiparticle amplitudes
14 7
2.0
1.5
Gau~sian
gA2
1.0 ImgA2J " .5
200
400
600
800
I000
I qAI (M~vl Fig. 7. Plot of gA2(rbr;);ql = O, y = 1. - 3 ~
g(A2)ulu2(rl,r2) =gMulu2 e -aq(ql+q2)z e aq(ql q2)2 e ~yy2 , where
M=
1
,
0 for an 1 = 0 resonance, and V = 1, 2.or 3 denotes rr-, n 0 or n + respectively. This
J. Steinhoff, Multiparticle amplitudes
148
quantity is plotted in figs. 5, 6 and 7 for C~q = 3 GeV -2, Oe.v= 0.25. It can be seen that it represents a crude approximation to (25). We easily obtain, from eqs. ( 19)-(22),
8 vq~z2 eicoy c~,u,(a'-v'ql'q2) : ~-~T-~2k ( q 2 ) k ( q 2) f dco , ~ 2 exp [co2/2e~y]/a - 1
(26)
where we only have one eigenfunction. In the above equation
k(q 2) = ~ a-
e
2c~qq 2
2g2rr 3 c(v(4%)2
We treat a as a parameter to be determined by the (inclusive single-particle) density of observed pions. Setting er = 2/a, assuming3' > 0, the poles of our integrand are at co/I. = l~co~ i 2+ ~/(~%) 1 2 2+ ( -3~ % • , ) 4 1"4-
,
l 2 I 2 )2 +(_lrJOey) -~ - 4 ] ~~ s i g n ( j ) , COR = [ 7coa+X/{~coa
where coa = ~ 7 . The smallest 1 = wh.ere coR We have col() = coa, COO R = 0 and ( v 4 : 0 ) co/ occurs a t / 0, = 0. then
CU1,u2(.I ' ,q l,q 2 ) =
40~qOCy -~ 2 2, -aq(ql+q2) ~" e(~/., i j.l 2
_ooaly I ~ e_OO~l.vI XL- ~ +2/~ 1 ~ 2 X(co) c°scoR[yl
fe
coRsincoRly,)].
(27)
w/ +%.
For v = 0 we have, from eqs. (13) and (26)
p l ( q 2 ) = cm,(O,q,q) = ~
4aq e 4~qq 2 F(1,7) 772
where
1:
r(o,~) = _-r=, Ira)
x °-I
d x - -
e x+y - 1
is an important function in the statistical mechanics of Bose gases. This specific form arises only because of the Gaussian-separable assumption for A2(rl,r2) in our numerical compu ration.
J. St einhoff , Multiparticleamplitudes
149
We fix the density of pions of each charge per interval in y:
r~= fd2qpl(q2), and compute 7 from the equation for F(½,7) given in ref. [16], and the relation
For 2 V ~ y r~ >~ 1, we get, from ref. (16), x/~ ~ 0.8 +
77.
The correlation length from eqs. (10) and (27) isy c = (2we0 -1 = (2 2V~yT) 1. Hence, we have 0.4 Yc
~
r/ + 2ay " t
For r/= 0.5, ay = 0.25, we g e t y c ~ 1.6. culu2(y,ql,q2 ) has the same asymptotic form as eUlU2(y,q 1,q 2), with dependence on indices given by Muau2, instead of 6u1~2 .The longest range part of c(y,q l,q2 ) is
C~lu2(y,ql,q2 ) ~ 0.8
4aq 7"/"
exp [
~
2+ 2
_aq(q I q2)]
× exp(-lYl/3.2)6UlU2 , and we get
4aq e 4o~qq~ 4O~qe_4e~qq~ CUluz (y,ql,q2) ~ 0.6 7r rr × e-[Y[/1.6 (6ul,2 + M , , , z ). This separation of C(rl,r2) into a part describing rr+n- and nOn ° correlation, and a part describing lr+n*, n n , and n0n 0 correlation was discussed in subsect. 5.1.1. The positive definite form of eq. (10) suggests that this positive n+n +, n n - correlation function, comparable in magnitude to the n+n- correlation function, is not a result of the particular form we have chosen for A2(rl,r2), but results from the general broad resonance decay amplitude that our form approximates. Also, we see the exponential asymptotic behaviour i n y that was discussed in subsect. 5.1.2. This, too, should be independent of the precise form of A2(rl,r2). We see that in our factorizable model Culu2 (y,q 1,q 2) has an exponential "tail" of correlation length ~ 1.6 independent of pion charge if interference terms are taken into account, but a Gaussian shape which decreases to 1/e of its y = 0 value at
J. Steinhoff, Multiparticle amplitudes
150
3' ~ 1.4 in the I = 0 channel, if no interference terms are considered, with 11o correlation in the I = 2 channel. The correlation f u n c t i o n for rr+rr pairs, integrated over q l and q2, which results from solving eq. (26) numerically is plotted in fig. 8, together with the correlation function when interference terms are neglected (the normalization is changed to obtain the same value of 7-/). Also, the correlation function for the I = 2 channel (rr+rr +, rr r r - ) pairs is plotted in fig. 9. In our model, it can be seen that the I = 0 and I = 2 correlation functions only differ qualitatively at small rapidities, where they are still comparable in magnitude. l f w e had used the form (25) for an e-meson decay amplitude, rather than our factorizable form, we would not have gotten quite so large a value for the correlation function for I --- 2 pairs, or for large v. In general any correlation in gA2(rl,r2) should decrease the long range part and the 1 = 2 part ofC(y,rl,r2)(for fixed r0. Since Yc, the correlation length, depends only on one particular eigenvalue of K(rl,r2), as we get more correlation in gA2(rl,r2) we have more eigenfunctions which are important, and the average magnitude of each eigenvalue should decrease (to satisfy the constraint on r/). In particular, as X -+ 0 in eq. (25), we would need an infinite n u m b e r o f eigenfunctions and each eigenvalue would ~ 0 . Thus, the kernel and m h o m o g e n e o u s term in our integral equation for c ( r l , r 2 ) ; K(rl,r2) -->O. Also, in this limit our I = 2 correlation function would vanish. For comparison, we do a calculation in this, the narrow-resonance approxilnation. We get in that approximation, with eqs. (7), (11) and (12), ¢
Cgl**2 ( r l , r 2 ) = g(A 2 )**1,,2 (rl 'r2 ) ' .5
.4
_
C(y) .3
-
C(y)
.Z !
Co(Y) : interference terms neglected
CoIY)
.1
0
I
1
2
3
4
5
L
6
--
?
y Fig. 8. Correlation function for I = 0, integrated over ql and q2, with and without interference ternls.
J. Steinhoff, Multiparticle amplitudes
151
.5
.4
C(y)
.2
.1
- - 1 -
1
I
I
I
~
i
2
3
4
5
6
y Fig. 9. Correlation function for I = 2, integrated over ql and q2.
C~lu2(rl,r2)
= 0, r 1 4=r 2 .
Hence
pl = ~ 1
4aq e_aaqq2 -2a --~
a),a 4aq 4aq C~lu2(y,ql,q2)-~-~ -~ e 4aqq2 ~. e-4aqqZ2e 2~yy2 M~zl~2
,
or
Culu2 (y,ql,q2) ~
0.2
4aq e_4aqq2 4C~qe_4aqq ~ e {y2 7r rr Mu~u2 "
6. Discussion A model for multiparticle production in the central region at high energies has been considered. An i n d e p e n d e n t emission model was modified to accomodate strong correlations in the p r o d u c t i o n amplitude between distinct pairs o f particles at low subenergy, expected from theoretical considerations. Damping in transverse m o m e n t u m was assumed, and dependence of the amplitude on other variables was ignored.
15 2
J. St einho]]] Multiparticle amplitudes
A mathematical identification between the exclusive cross section for pion emission (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium was exploited to obtain an expansion for the asymptotic single-particle inclusive distribution, the two-particle inclusive correlation function, and the contribution of the central region to the exponent of s in the total cross section by means of cluster diagrams. An integral equation was exhibited for summing the terms corresponding to the cluster diagrams. A specific model was then considered, which described independent production of two-pion resonances with small transverse m o m e n t u m and non-zero width. In this, the "s-channel pole dominance" approximation, the inclusive correlation functions were shown to be non-negative between any types of pions, regardless of the spin, isospin or subenergy dependence of the resonance. For I = 0 resonance production, the ~+n+, n n , n o n ° and n+n - correlation functions were seen to be positive at small rapidity while correlations were absent for n+n ° and n n O pairs. For neutral I = 1 resonance production, there were positive correlations for n+n +, n n and n+n pairs. A coherent mixture of charged I = 1 resonances was seen to allow the possibility of positive correlations between any types of pions. The positive correlations in all of the above cases were seen to have an exponential "tail" at large rapidities, for sufficient strength of the coupling constant. A numerical example was then discussed, which gave a crude simulation of a broad I = 0 spinless resonance. The features of the more realistic anrplitude that were retained were the short range in rapidity, the limited transverse m o m e n t u m and the isospin dependence. The resulting n+n + and n n correlation functions were seen to be quite large: conrparable to the rr+zr- correlation functions at small rapidities, and to have the same exponential "tail" at large rapidities. These correlation functions are certainly larger than would be obtained from a more realistic resonance decay amplitude, where threshold effects and other features are considered, but may be qualitatively similar. We see that if the most important features of the production amplitude are transverse m o m e n t u m damping and 1 = 0 or I = 1 resonance production, we can expect a rather long range inclusive correlation, positive in n+n +, n n and n+n - channels. The significance of this study is, perhaps, to point out that under the stated assumptions, the scale of the correlation function in the I = 2 channel is quite broad. From fig. 9 it can be seen that the correlation function in this channel decreases to half of i t s y = 0 value at y ~ 2.25. Available data analyses [8] do not allow enough of a rapidity variation in the central region to determine the relevant features of the I = 2 correlation function. The analyses do show, however, that there is a positive correlation there, and that correlations in the production amplitude are called for. To attempt even a crude comparison with the above data, the strong effects of energy-momentum conservation in the fragmentation regions would have to be included in our calculations.
J. Steinhoff, Multiparticle amplitudes
153
It is a pleasure to thank Dr. Richard Arnold for patiently explaining many of the aspects of high-energy phenomenology and classical fluid theory upon which this work is based, and for providing a great deal of guidance during the past few years. Also, I would like to thank Drs. P.G.O. Freund and D. Sivers for reading the manuscript and making many helpful suggestions.
Appendix A If unpaired pions are produced with amplitude g'Al(r ), together with two-pion resonances, our amplitude of eq. (1) is modified: '2E'(N)
A'N(rl,r 2 ..... rN)=
~
~
gKA2K(rl,r 2 ..... r2K)
distinct K=1 permutations
X g'A 1(r2K+ 1)'"g'A 1(rN), where A2K(rl ..... r2K ) is defined in eq. (1) and E'(N) = (N 1)N, N odd (even). The single-particle inclusive distribution and the correlation function are now
p 1(r) = c(rlr ) + glco(r)l 2 , r
C(rl,r2) = tc(rl,r2)l 2 + Ic (rl,r2)l
2
*
+ ZC(rl,r2jco(rl)co(r2)
Zc (rl,r2)co(rl)co(r2) + Zc (rl,r2)co(rl)co(r2) + Zc'* (r l, r 2 )co(r 1)Co(r 2), where * ,)] co(r ) = g'Al(r) + g' ,, ' d3 , r [c(r,r ,)Al(r ,)+ c ,(r,r ,)Al(r and c(rl,r2) and c'(rl,r2) are given by eqs. (11) and (12). The exponent a can again be determined from P 1(r) with eq. (16). It is perhaps interesting that, because of the terms in C(rl,r2) linear in c(rl,r2) and c'(rl,r2) , the long range part of C(rl,r2) will now have the same range as c(rl,r2). Thus, if there is a small unpaired pion component (small so that the normalization o f g is not effected), then there will be a small exponential "tail" in the correlation function of twice the range of the "paired pion" correlation function. This part need not be positive.
J. Steinhoff, Multiparticle amplitudes
154
Appendix B We show how eqs. (17), (9) and (10) could have been obtained from published work in elementary statistical mechanics. To construct a probability distribution function for a system of molecules that is equal to QN(rl ..... rN), we first consider a system interacting through two different external and two-body potentials. The probability function is then
QCN(r 1..... rN) : exp
/3 ~
,
(u (r i, r/) + u *(r i, r]))
where fi is a constant, and u(ri,r/) -+ 0 when Iri r/l -+ ~. This serves as a generating function for QN(rl ..... rN). Defining Mayer functions
f(ri,r])=
e ~u(ri'r/)
-1 ,
we have
Q~f(r I ..... r N ) = [ 1 [1 f(ri,r]) ] [ [ [1 f * ( r i , r / ) ] . i
pcK(Z;r I ..... r K ) -
QcK(Zzrl ..... r K) QCo(Z)
where
QcK(Z:rl ..... rK)= ~
ZN
(,V
K)! f d3rK<"d3rNQ~'(rl
..... r ¥ ) .
N = 1
The cluster expansions o f l n Q c° , p 0 ( r 1) and CC(rl,r2 ) = fic2(rt,r 2) fiO(rl)ficl(r2) are given in ref. [13], eqs. (4-3), (4-71 and (4-5) respectively. Each term in these expansions consists of integrals of products off(ri,rj) functions. The general cluster expansion becomes quite complicated as we consider integrals over increasingly larger numbers of coordinates. In our special case, however, when we operate with
3".Steinhoff, Multiparticle amplitudes
155
P before doing the integrals, we only have a simple subset of these terms, represented by "ring" diagrams. When we set f(ri,rj) = gA2(ri,ri), we obtain
In Qo : ~ Z2K g2K f d3rl...d3r2KA2(rl,r2)A~(r2,r3)...A~(r2K,rl), K=I VOK z2K g2K f d3r2...d3r2KA2(r, r2)A~(r2,r3)...A~(r2K,r) K=I V1K
pl(r ) = ~
and
z 2K C(r,r,)= ~ _ _ g 2 K K=I V2K
2K ~ /=2
fd3r2...d3rj ld3rj+l...d3r2K
X A2(r, r2)...Af2*)(r] 1,r')...A2(r2K,r), where the (*) applies i f / i s odd, and the VmK are "symmetry numbers", described m ref. [13]. We have for our set of cluster terms, V° K = 2K, V1K = 1 and V2K = 1. These above equations are equivalent to eqs. (17), (9) and (10). Note that f(ri,rj) , and not e ~u(ri'ri) describes the probability function for two molecules interacting through a bond. Sincef(ri,ri) -+ 0 when lYi-Y/I -+ 0% a potential describing the interaction of the molecules would become infinite in this limit.
Appendix C We can easily get higher correlation functions, in our models, by considering rings with more particles fixed. We first redefine some quantities: c+ (rl,r2)
= c(rl,r2),
c +(rl,r2) = c*(rl,r2) : c(r2,r 1), c++(rl,r2) = c'(rl,r2) : c'(r2,rl ) , c
(rl,r2) : c'*(rl,r2).
For the N-particle correlation function we draw N points, and join them as in the cluster diagrams described above. However, now we have four types of lines, labeled with a (+) or ( - ) at each end, and we must sum over all allowed combinations. The rule is that at each point a (+) and ( - ) have to meet. Thus, for N = 2,
C(rl,r2) = c+_(rl,r2)c + (r2,r 1) + c++(rl,r2)c _ ( r 2 , r l ) ,
J. Steinhof]] Multiparticleamplitudes
156 and for N = 3, if
p3(rl,r2,r3) = pl(rl )pl(r2)pl (r3) +
pl(rl)C(r2,r 3) cyclic permutations
+ C(3)(rl,r2,r3), C(3)(rl,r2,r3)=c+_(rl,r2)c+ (r2,r3)c+ + c +(rl,r2)c_+(r2,r3)c+(r3,rl) +
( r 3 , r 1)
~
c+_(rl,r2)c++(r2,r3)c__(r3,rl)
permutations
References [ 1] [2] [3] [4] [5]
[6] [7] [8]
[9] [10] [ 11 ] [12] [ 131 [ 14] [15] [16]
K. Wilson, Cornell preprint (1970), unpublished. D. Campbell and S.J. Chang, Phys. Rev. D4 (1971) 1151;ibid D4 (1971) 3658. R.C. Arnold and J. Steinhoff, Argonne preprint (1972). E. l:ermi, Phys. Rev. 81 (1951) 683. R.C. Arnold, Phys. Rev. D4 (1971) 3488; P.E. Heckman, Phys. Rev. D1 (1970) 934; D. Horn and R. Silver, Ann. of Phys. 66 (1971) 509. E. Predazzi and G. Veneziano, Nuovo Cimento Letters 2 (1971) 749; J. Ellis, J. Finkelstein and R.D. Peccei, SLAC preprint 1020 (1972). D. Sivers and G. Thomas, Argonne preprint 7218 (1972). W.D. Shepard et al., Phys. Rev. Letters 28 (1972) 703; W. Ko, Phys. Rev. Letters 28 (1972) 935. E. Berger, B. Oh and G. Smith, Argonne preprint 7209 (1972). tt. Satz and G. Thomas, University of ltelsinki preprint 7 - 7 1 (1971). D. Sivers and G. Thomas, Argonne preprint 7228 (1972). D. Amati, S. Fubini and A. Stanghellini, Nuovo Cimento 26 (1962) 896; C. Detar, Phys. Rev. D3 (1971) 128. R.C. Arnold, Phys. Rev. D5 (1972) 1724. G. Steil, in The equilibrium theory of classical fluids, eds. H. Frisch and J. Lebowitz (Benjamin, New York, 1964). S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience, New York, 1965) especially ch. 2. E.I. Shibata, D.H. Frisch and M.A. Wahlig, Phys. Rev. Letters 25 (1970) 1227. J.E. Robinson, Plays. Rev. 83 (1951) 678.
156. North-ttolland Publishing C o m p a n y
P A I R I N G IN M U L T I P A R T I C L E A M P L I T U D E S : INCLUSIVE DISTRIBUTIONS* John STEINHOFF** 1)epartment o f Physics and the Enrico Fermi Institute, The University o f Chicago, Chicago, Illinois 6063 7 and ltigh Energy Ph),sics Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 25 August 1972 Abstr,'tct: A model for multi-pion p r o d u c t i o n in the central region in high-energy collisions is studied which describes factorizable emission of pion pairs. A m a t h e m a t i c a l identification between the exclusive cross section for pion emission in our mode l (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the a s y m p t o t i c single-particle inclusive distribution, the two-particle inclusive correlation function, and the e x p o n e n t of s in the total cross section by means of cluster diagrams. An integral equation is ex hibited for summing the terms corresponding to the cluster diagrams. A specific model is then considered, which we call "s-channel pole d o m i n a n c e " . In this model the amplitude is assumed to be large only when the subenergies of pairs of pions are near the mass of a low-lying two-pion resonance, and the transverse m o m e n t u m of each resonance is small. The dependence of the a m p l i t u d e on other variables is ignored, so that we effectively have independent emission of two-pion resonances with non-zero width. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, as s ~ ~, and that the correlation function can have an exponential " t a i l " m rapidity of qualitatively longer range than the resonance. A crude numerical sinmlation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, and to have the same e x p o n e n t i a l " t a i l " as the I = 0 correlation function.
1. Introduction Classical statistical-mechanics techniques have been exploited in the past several years to describe multiparticle production processes in high-energy hadron-hadron collisions [ 1], and quantities such as the single- and two-particle inclusive spectra * Thesis su bmitted in partial fulfilhnent of requirements for the P h.D . degree at the University of Chicago. ** A.U.A.-A.N.L. Fellow. Present address: Department of Physics, McGill University, Montreal, P.Q., Canada.
J. Steinhoff, Multiparticle amplitudes
13 3
have been calculated for various models. The validity of a cluster expansion in a q~3 field theory in the ladder approximation has been verified [2]. Specific use of calculational methods of statistical mechanics has been made in the application of the "molecular-field" approximation to the calculation of the effects of final-state multiparticle absorption on inclusive spectra [3]. We use here a cluster-diagram technique to compute the behaviour of the total cross section and the single- and two-particle inclusive distribution functions for a class of models describing pion production in the central region. The models that we consider take into account correlation of the pions in pairs, but are otherwise similar to independent emission (IE) models. IE of pions in high-energy collisions is not a new idea [4]. Recently, IE models describing production of particles with small transverse momentum but longitudinal momentum increasing with energy have been constructed, that describe experimentally-observed single-particle spectra quite well [5]. When energy, momentum and charge conservation are taken into account [6], particles cannot be produced independently, but we will call such a model IE if the amplitude for particle production is completely factorizable. Inclusive two-particle correlation functions are of course identically zero in an IE model without conservation constraints; and numerical calculations done for a model with constraints do not give appreciable correlation in the region where both particles have small rapidity [7]. On the other hand, this correlation function has been measured experimentally for 7r n pairs in this region [8], and is seen to be positive and fairly large. Thus, IE models do not seem to be adequate to explain experimental data. Theoretically, one would expect the production amplitude to become large when the subenergies of distinct pairs of pions (with correct spin and isospin) are close to the mass of a low-lying two-pion resonance*. This property is of course not consistent with IE. In this paper we discuss a pion-production amplitude with pairwise correlations, that can incorporate this property. For simplicity we go to asymptotic total collision energies, ignore energy, momentum and charge conservation constraints, and consider only the central region in rapidity. These constraints should not be important, in this limit, for the quantities that we will compute. A mathematical identification between the exclusive cross section for pion emission in our model (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium is exploited to obtain an expansion for the single-particle inclusive distribution, the twoparticle correlation function, and the contribution of the central region to the s de-
* The amplitude is expected to have poles in the complex sij plane, where sij is the subenergy of particles i and/'. However there should not be simultaneous poles in sij and Sik, J 4=k, as explained in, e.g., ref. [9]. Also these poles should produce large isolated peaks (resonances) at low si/, hut smooth behaviour at high si/,.
134
.L Steinhofj~ Multiparticle amplitudes
pendence of the total cross section by means of cluster diagrams. An integral equation is exhibited for sunnning the terms corresponding to the cluster diagrams. A specific mode] is then considered, which describes independent emission of two-pion pairs, each pair being correlated in an I = 0 or I = I resonance with nonzero width. This amplitude is large only when the subenergies of distinct I = 0 or 1 = 1 pairs of pions are near the resonance mass, and the transverse m o m e n t u m of each resonance is small; the dependence on other variables is ignored. We call this the "s-channel pole dominance" approximation. It is seen that an I = 0 or I = 1 resonance results in a positive two-particle inclusive I = 2 correlation function at small rapidity separations, regardless of the form taken for the resonance, and that the correlation function can have an exponential "tail" in rapidity of qualitatively longer range than the resonance. A crude numerical simulation of a broad I = 0 spinless resonance is discussed, and the resulting I = 2 inclusive correlation function is seen to be quite large at small rapidity separations, as suggested by experiment
181. 2. The model The class of models that we consider are characterized by a partially-factorized form for the amplitude for production o f N pions in the central region, and a set of particles in the fragmentation regions: A ( s { r f } ; r l , r 2 ..... r N ) = A f ( s , { r f } ) A N ( r l
,r 2 ..... r N ) ,
where N is restricted to be even,
. . ~¢" A N ( r l , r 2 ..... r N ) -. . .g~.
~
A2(rl,r2)A2(r3,r4)...A2(rN
l,rN)"
(1)
distinct permutations S is tire total c.m. energy squared, and r i = ( y i , q i ) labels particles in the central region with c.m. rapidity v. and transverse momentum qi. The set of particles in tire fragmentation regions is specified by {rt, }. This particular method of separating out the fragmentation dynanlics was chosen for simplicity: the only property that we require is the lack of long-range correlations between particles in the central region an0 fragmentation regions. We do not include charge indices for the pions, or indices describing production of other particles in the general treatment; they can be included in r, without altering airy of our formulae, as seen in the numerical exmnple that we consider. One of the central features of multiperipheral [11] and current 1E models is the existence of a region in rapidity that includes all of the kinematically allowed values except for finite (fragmentation) regions at each end, as s -~ oo, where (finite) multiparticle distribution functions depend only on relative rapidities, charge states •
I'
J. Steinhof]; Multiparticle amplitudes
13 5
and transverse momenta of the pions. We assume these features, which are believed to be more general than the above models [1]. Hence, we define A2(ri,r/) to be a function only of qi , q] andYi - -vs, where A 2(ri ,r/) ~ 0 when ly i Y/I ~ oo, or q2 or q2 becomes large, so that the particles are effectively emitted in independent pairs, with amplitude A 2(ri ,ri). This defines the class of models under consideration, which includes as a special case that of independent p or e meson emission. The generalization to include single-particle production in the amplitude (unpaired pions) is straightforward, and the corresponding formulae are given in appendix A. Since we only consider inelastic events that result in produced pions in the central region, diffractive dissociation and elastic events are not described by our model, and our distribution functions will not include these processes. We shall only be concerned with the central region, and we assume the mean number of particles produced in this region increases linearly with Ins as s -~ oo, while the number in the fragmentation regions remains finite. We accordingly define an "exclusive" N-particle differential cross section to be the probability of exactly N particles being produced, with specific molnenta, in the central region and anything in the fragmentation regions, ttence, all quantities are summed over (rf}. The "exclusive" N-particle differential cross section is then written
d3NoN(s;r 1..... r N) 3 3 = °o(S)QN(rl"'rN)' dr I ...dr~
where o0(s ) is the integral of IA f(s;{rf})l 2 over (rf} with energy and longitudinalnlomentum conservation, divided by the flux factor, and our "exclusive probability function" is 2
QN(r 1..... rN ) = g~N
~
distinct permutations
A2(rl,r2)A2(r3,r4)... A2(r N_ l,rN)
(2,)
We do not consider the dynamics in the fragmentation region, nor do we explicitly discuss the dependence of our amplitude on s, and we write the total cross section, and single- and two-particle inclusive cross sections in the form OT(S ) =
Oo(s)QO(s),
d3o l (s;r) _ on(s)Q 1(s;r) , dr 3
d6o2(s;rl,r2) dr~ dr 3
- oo(s)Q2(s;rt,r2).
In the above equations
136
QK(s;rl ..... rK)=
J. SteinhoJ]~ Multiparticle amplitudes ~ z2NQK2N(S;rl ..... r2N) N=~E(K) (2N-K)! '
(3)
where E(k) = K ( K + I ) , K even (odd), and
K Q2N(S;rl ..... rN)= f d3rK+l...d 3r2NQZN(rl ..... r2N ) .
(4)
We have introduced a formal parameter Z to be used in solving our equations and then set equal to 1. The dependence of a quantity on Z will be suppressed unless needed. We will see that (this part of our discussion parallels much of that in ref. [2] for a 4~3 field-theory model) as s ~ ~, QO(y) = const, e c~Y, where Y = lns is written as the energy-dependent variable in the QK functions, and we will compute the (energy-independent) quantities p l ( r ) _ QI(Y;r) _ d3ol(s;r) 1 Q0(y) dr 3 OT(S) ,
(5)
Q2(y;rl,r2) d6°Z(s;rl,r2) 1 p2(rl'r2) = QO(y) = dr~ dr32 °T(S)"
(6)
These quantities can be completely determined with our model amplitude (1) once the solution to a linear integral equation is known. This integral equation sums a series of terms obtained by a cluster-expansion technique.
3. Cluster-diagram expansion The exclusive probability distribution o f N pions is QN(rl,...,rN). The properties of the functions A2(r l,r2) contained in this function are such that the pions are distributed along t h e y axis in clusters, and confined to a finite region in q. We can define an "analog system" of classical nonrelativistic particles in equilibrium interacting in pairs [ 12] such that the configurational probability function for this system is mathematically isomorphic in terms of structure to QN(rl,...,rN), where the r i describe the positions of these particles in ordinary three-dimensional space. Since * each r i appears in QN(rl ..... rN) in the function g 2A2(ri,rj)A2(ri,rk);f, k 4= i, each of these particles is bound to one or two others, and to the line q = 0, which can be thought of as a thin wire running along the y axis. As in a system of molecules interacting with each other and an external object through two strong "chemical bonds" per molecule, we can form rings, or "polymers" with no open bonds. Also, these two bonds are different, one being represented by gA~(ri,rk), and the other by
gA2(r i ,r]). The terms contained in QN(rl ..... rN) in eq. (2) are easily enumerated using ele-
J. Steinhoff, Multiparticle amplitudes
13 7
mentary cluster-diagram techniques [13]. A "diagram" consists of N points, labeled 1 through N (we do not rigorously adhere to the terminology of graph theory) with two lines attached to each point, one solid and the other dashed, connecting it with one or two other points. The term corresponding to a particular diagram is obtained by writing a factorgA2(ri,r/) for a solid line between point i and point/, gA~(ri,r/) for a dashed line, and forming the product of such factors for each line in the diagram. For N points we thus have a product of N factors, where N is restricted to be even. Corresponding to each distinct diagram we draw in this way, there is a distinct term, and the sum of all such terms is equal to QN(rl ..... rN). As an example, f o r N = 8 one term would be
g8 [A2(rl,r2)A2(r3,r4)A2(rs,r6)A2(r7,r8)] X [A2(rl,r6)A2(r2,r5)A2(r3,r4)A2(r7,r8)]*~ and the corresponding diagram is drawn in fig. 1. The amplitude to produce an unspecified group of particles in the fragmentation regions and 8 particles in the central region is pictorially represented in fig. 2, where gA2(ri,r/) is represented by a vertex with lines labeled, i and/, and a heavy line. The particular interference term represented by the diagram of fig. 1 is depicted in fig. 3. We see that for our model amplitude of eq. (1) all the terms are products of factors that can be represented by "ring" diagrams. For instance the term described in the example above contains the factors
8
3
7
4 6
5
Fig. 1. A diagram representing a term in Qs(rl,...,rs).
"•1"•
l fragmentation
~4 distinct ~.J permutatt0~7
;~
}fragmentation
Fig. 2. A pictorial representation of A (s, {rf} ;r I ,...,r8 ).
.l. SteinhofJ; Multiparticle amplitudes
138
~ dp•!uClatio•
l:ig. 3. An interference term contributing to Qs(rl,...,?8).
g4A 2(r l,r 2 ) A ~(r 2,r 5 ) A 2(r5 ,r6 )A ~(r 6 , r l ) , g2A2(r3,r 4)A ~(r4,r 3 ) , g2A2(r 7,rs)A~(r8,r7 ) . Once we have the forms of the cluster diagrams we can consult the literature to get Q0(Y), P l(r) and C(rl,r2) in terms of integrals over the factors corresponding to ttle ring diagrams, as shown in appendix B. We derive the results here since the arguments are rather simple. We first compute Q lv(r ) and Q2(rl,r2)m terms of Q ° ( Y ) , K < N and the cluster integrals Iy(r r ) = gA 2(r,r ), •
")
r
t
l~(r,r')=gg f d3r2d3r3...d3rKA2(r, r2)A;(rz,r3)...A(2*)(rK,r' ) , K > I ,
(7)
where the last (*) applies i l K is even, and
IlK(r) =12K(r,r) ,
K even.
(8)
We always assume we are far from the end regions i n y so that the limits of inte2 ,) depends graticn can be taken t o y = _+0%and IlK(r) is independent of y , and IK(r,r only o n y y',qandq'. To compute Ql(r 1) we consider all distinct N-point diagrams and integrate the corresponding functions over r/, 1 < 1"~< N. We order the terms according to the number of points 2K that are on the ring which contains point 1. There are, for each K, (N--1)!/[(N 2 K ) ! ( 2 K - 1 ) ! ] ways of choosing the 2 K - 1 points out of the N 1, and there are ( 2 K - l ) ! distinct ring diagrams, each of which results in the same function [ll2K(r 1)] when the corresponding factor is integrated over ri, 1 < i ~< 2K. The integral over the remaining set o f N - 2K coordinates represented by points not connected to 1 results in a factor Q°_2K(Y ). Hence, we have, defining QO = 1,
J. SteinhofJ~Multiparticleamplitudes -~N
(N 1)!
QO-2K(Y)I1K(rI'(N
Q1N(rl)= ~
139
2K)!
K=I
By fixing coordinates r 1 and r 2 we similarly have
"N
Q2(rl,r2)= ~ Qo K=I
2K(Y)K ~K(rl,r2)( N 9(~-.) N "(-_~K ~,1 9)1'
where K~K(rl,r2) is represented by the set of diagrams with 2K 2 points connected directly or indirectly to either point 1, point 2, or both. We have diagrams with both point 1 and 2 on the same ring, and on separate rings. Summing the different terms, we get 2K
"> KSK(rl,r2) = (2K
2)!
1
[~ /=1
~ "~* lT(rl,r2)15 K /(rl,r2)
K 1
+ 2
/=I
where the second term is absent ifK = 1. We can sum over N now, and from eq. (3) we get
QI(y;rl)= ~ Z 2N Q~A'(Y) N=0
(2N)! K=I ~
z2KI~K(rl)'
or from eq. (5), dropping the subscript,
(9) K=I
Also, we get
Q2(y;rlr2) = Qo(y)
z2KIIK(rl) K=I
+ Q°(Y)[lc(rl,r2t2+lc'(rl,r2)121 , or
~ K=I
z2KIlK(r2)
J. Steinhoff, Multiparticleamplitudes
140
p2(rl,r2) = o l ( r l ) p l ( r 2 ) + C(rl,r2), (10) , 2 C(rl,r 2) = lc(rl,r2)l 2 + tc(rl,r2)l ,
where
c(rl,r2) = ~
Z2Kl2K(rl,r2),
(11)
K=I
c'(rl,r 2) = ~-J Z 2K-II2K_l(rl,r2).
(12)
K=I From eqs. (8) and (9), we see that
pl(r) = c(r,r),
(13)
and, from eq. (7),
c'(rl,r 2) = ZgA2(rl,r 2) + Zg f d3rc(rl,r)A2(r, r2).
(14)
Note that C(rl,r2) is always positive in our model, regardless of the form of A2(rl,r2). By including charge indices in r, we will see that C(rl, r2) is positive for pions of the same or opposite charge, regardless of the isospin properties of
A2(rl,r2). The equations for o l(r) and C(rl,r2) have a very simple interpretation. Each distinct configuration of unlabeled points that lie on a ring with one point fixed contributes equally to p l(r) (before integration over the coordinates represented by the points). Since there is only one distinct configuration of points for each value of K, the number of points on the ring, we have eq. (9). If we fix two points on the ring, we have effectively two open chains between the two points, and after enumerating the distinct configurations of points and doing the integrals, we have eqs. (10). The restriction that we have only an even number of points on the complete ring means that both chains have either an odd or an even number of points, and we have two separate terms in eqs. (10). Some contributions to p l(r), C(rl,r2) , c(r],r2) and c'(rl,r2) are depicted in fig. 4, where integration is ilnplied for an unlabeled point. We can now solve for QO(y). From eqs. (3) and (4) we see that (writing the Z dependence explicitly) z F dZ 1
QO(z,Y) = 1 + J -~1 0 But, from eq. (5),
d3rQl(Zl'Y;r)"
J. Steinhoff, Multiparticle amplitudes 1
p}'(r~)• ~
+
+
1
_
:
=
c1(r ~ 2' ) r l =
:
_...
1
2
~.
r. + r.
Fig. 4. D i a g r a m s r e p r e s e n t i n g
+
2
1
+
2
1 c(r.,r2)
1
(_...__..~+
•
141
i
+
-
--
2
1
+ ~.
2
I
:_
:
:
--
1
_~---o
+
....
2
~ -
-+
....
t e r m s in p J (r), C(rl,r2 ), c ( r l , r 2 ) a n d c ' ( r l , r 2 ) .
QI (Z, Y;r) =Q°(Z,y)p l (Z,r) . Hence Z dZl o Q0(z,y)-- + f y][Q (Zl,Y) fd3rpl(z ,r), 0 so Z
Qo(Z,Y)=exp f ~ 0
fd3rpl(Zl,r)
(15)
1
Now, except near the ends of the y region, p l (Z,r) is independent of y , so
Qo(y) = exp (~Y+/3), where/3 is a constant and 1
f -dZ ~ - fd2qpl(Z,q )
OL=
0 or, from
eq. (9),
(16)
142
.1. Steinhoff, Multiparticle amplitudes
Z;
(17)
K= I 2 K '
where
I~K =f d2ql~K(q). From eqs. (10), (13) and (16) we see that once we have c(Z, rl,re) we can obtain c~, o l ( r ) and C(rl,r2). Also, as shown in appendix C, we can then easily obtain higher correlation functions.
4. T h e integral e q u a t i o n
From eqs. (7) and ( 1 1) we can obtain an integral equation for
c(Z,r l,r2) : Z2K(rl,r2) + Z 2
c(Z, rl,r2):
fd3rK(rl,r)c(Z,rr2) ,
(18)
where
K(rl.r 2) = g2 fd3rA 2(rl,r)A~(r, r2)
(19)
depends only o n y =Yl v-~ 0 = q51 02 (tile angle about the), axis between ql and q 2), Pi = q~ (i= 1,2) and-isospin indices of the pions. Defining Fourier transforms:
2rr
"~m(Z,co,l,l,P2 ) = f
+~
dq5 e im¢, f
d3, e-icoy c(Z,y,dAPl,t)2) '
we have only the square of the transverse momentum left as a variable in our integral equation:
. '=,,z (oO,Pl,P2)+ c~m.(Z,co,pl,p.~j=Z-K X "~m(Z, co,p,p2) .
1Z2 f dpi,n(co,l)l,t)) 0 (20)
By eq. (19)K(I)I.P2) is Hermitian (we shall suppress Z, m and co dependence unless needed), and we expand it in eigenfunctions with real eigenvalues:
/~(PI'P2) = ~ K
~VKk'K(Pl)~'*K(P2)"
J. Steinhof]i Multiparticle amplitudes
143
We then have
c'(Pl,P2) = ~
UK~K(Pl)k~:(P2) ,
K
where
x~(co)
~ ( z co)
2
Z
1
m
--~XK(co)
Inverting the m and co transforms, we have +~
+~
c(Z,¢,)',q~,q2) = ~
2
e imO ---~ . _~
K m =-~ m "1'
2
+~
d Y 2 k ~ ( Y l , q 2) dYl f _oo
,m
(21)
X k K (y2,q2)¢k ( Z , y - y l Y2), where +~
k2O',q2) --
1
c~((Z,y) =
f
m dco e +ico y ~kK (co,q2),
dco e +lwy Z -2 C
~ m
-
:~'X( c o )
'
(22)
and the contour c includes all poles of the integrand (co=co/) such t h a t y lm co/> 0 (y4=O) (see ref. [12]).
5. P h y s i c a l results
We first discuss some general properties of C(rl,r2) that can be obtained from our equations, with somewhat more detailed considerations of A2(rl,r2). Then, as an example, we assume a factorizable form for A2(rl,r2) that approximates an I = 0 spinless resonance, and numerically compute C(rl,r2).
5.1.1. Isospin dependence We first consider n+n - resonances. (We do not include charge indices in r here.) The amplitude A2(rl,r2) is then zero if particles 1 and 2 are both n + or both rr . The kernel K(rl,r2), defined by eq. (19) is then zero for n+n - pairs, but non-zero for n+lr+ or rr n - pairs. It then follows from eq. (18) that c(rl,r2) is non-zero only for rr+rr+ or rr n (•=2) pairs, and from eq. (14) that c(rl,r2) is non-zero only for n+n - pairs.
144
J.
SteinhofJ~ Multiparticle amplitudes
The correlation function C(rl,r2), from eq. (10), can be seen to separate into Ic(rl,r2)l 2 for rr+rr+ or 7r rr- pairs, and ic'(rl,r2)l 2 for rr+Tr pairs. If rr°Tr0 pairs are also produced, as in an I = 0 resonance, then both c(rl,r2) and c'(rl,r 2) contribute to the 7r°rr0 correlation function. In this case, though, both c(rl,r2) and c'(rl,r2) are zero for rr+rr° or rr rr0 pairs, and hence there is no rr-+rr0 correlation. In the context of our model amplitude, defined by eq. (1), all the resonances are in the same quantum state. Though the states are not necessarily eigenstates of total isospin or charge, we assume that the expectation value of the charge is zero. This property is implied when we comment on the diminishing effect of charge-conservation constraints at high energy. Thus for I = I resonance production, we can consider not only zero-charge states (rr+rr - pairs), but a coherent superposition of + 1, 0 and 1 charge states. In this case we would have the possibility of (positive) correlations between pions of any charge.
5.1.2. C(rl,r2) at large y Although the equations for C(rl,r2) in sect. 4 are valid for a general amplitude we are particularly interested in an amplitude of very short range in y and even in ~ and v. The first iteration ofA2(rl,r2), K(rl,r2) will then still be of short range. In general the functions k~(y,q 2) will then vanish for large 3'. When we 2 look at the function c(Z,~,y,ql,q~ ) at large y (large compared to the range of A2(rl,r2) ), from eq. (21) we see that it will be determined by the function cry(z,3 ,) with the longest range. But from eq. (22) we have the expansion ( 3 ' > 0 )
A2(rl,r2),
= 4i ~ e z2
w}y eiCORy p X ~ ( c o )
] 1,
(23)
I CO=CO/
j
where the sun, runs over the (K-, m-dependent) values of co/= coR + ico~ such that ~ ( c o / ) = , Z --. Finding the coj's with the smallest imaginary part, and calling them co(), -co~ with correspondingK 0 and m0, we have, at large lYl, , 7 ~ ~ ~4 c(Z,~,),q]',qs-)
v 2 B(coo,y,q'f,q2)
cos moo e-COollyl ,
where "~ eiCORlyl - , n o ,n~ B(coo,.v,q ~,q 5) = d kKo (COO,q 2l -)kK o (coo,q 22)
+e-iCO°RlYi'*Tm°" •
2,~m o, * 2, a KKo ~.CO0,ql)KKo (CO0,q2),
•
mo
1
d=[-laXX°(co)L, 3co
] tO=CO 0
"
(24)
J. Steinhoff, Multiparticle amplitudes
145
I! coR = 0 or m 0 = 0 we would have ½ the above result. We see from eq. (14) that
c ((~,y,q~,q2) will have the same asymptotic form ifA2(rl,r2) is of short range, and we have the two-particle correlation function:
C((9,y,q~,q 2) ~ e 2wollyl [a 1+a 2 cos(2coRlYl+01)] . A short-range basic two-pion correlation in the amplitude, such as would result from low-mass resonance decays, should thus develop a longer range "tail" in the correlation function. It is interesting that precisely the same phenomenon is seen in liquids with short-range two-body forces [14]. Whether this effect is pronounced or not depends on the specific form of A2(rl,r2) and on the strength g. To solve for p l ( r ) and ~ we need the functions cr~(Z,y) f o r y = 0. We then have
ol (Z,q 2) = e(Z, O, O,q2,q2) , and the expansion derived above may not be useful. A direct numerical solution of eq. (20) by iteration may then be the best method. We can then obtain c~ from pl(Z, q2) by eq. (16). In practice, we may have an idea what A2(rl,r2) should be, but not know the strength g. We then would fix g by fitting some experimental data with our model. If we choose to fit the observed multiplicity, then g becomes exactly analogous to the fugacity in statistical mechanics.
5.2. A numerical example We first briefly discuss some properties of an e-meson emission amplitude, and then discuss a crude simulation which should give the essential features of the more literal representation. This example is uncomplicated by spin and isospin. If eacb e meson were produced with transverse momentum q and amplitude gAc(q), we would have, igncring threshold behaviour, 2
me
g(A2)ulu2(rl'r2) =gMulu2 Ae(ql+q2) Sl 2 _ me2 + ikm2e '
(25)
where we used a pole approximation with half-width 2kin 2 and Mulu2 , to be given below, describes the dependence of the amplitude on the charge states of the emitted pions. The energy squared of the two-pion system is s12 = 2(m2+q2)~(m2+q2) ~- c ° s h ( Y l - Y 2 ) - 2ql "q2 + 2 m 2 , where m is the pion mass, and m e is the mean e mass. The quantities [gA2(rl,r2)l and Im [gA2(rl,r2)] are plotted in fig. 5 for ql = - q 2 and q l = +q2, [qll = 300 MeV, for various values o f y = Y l - Y2. In fig. 6 the above quantities are plotted w i t h y = 1 for IqlJ = 300 MeV and q2 = xql,
146
J. SteinhofJ] Multiparticle amplitudes
1.2
"~',
IgA21~°-~
1.0
;aussian ~ k \ \ ' ? A2 5 .8
gA2
.6
.4
.2
0
---J7 _
\. [gA21 q]°+q,
"NN\
\~', • ~%~ .,~ \ \ \ , /
0
1
- , t ~ ~ -~ev
2
3
4
l:ig. 5. Plot o f g A 2 ( r l , r x ) ; q l = +-q2, ]qll = 300 MeV.
1.0
sian / / / gA2~/
~~.~"
,~
-3
-2
1
I
-i
+i
I
+2
+3
l:ig. 6. Plot ofgA2(rbr2); IqlJ = 300 MeV, q2 = x q h y = 1.
J. Steinhof¢; Multiparticle amplitudes
14 7
2.0
1.5
Gau~sian
gA2
1.0 ImgA2J " .5
200
400
600
800
I000
I qAI (M~vl Fig. 7. Plot of gA2(rbr;);ql = O, y = 1. - 3 ~
g(A2)ulu2(rl,r2) =gMulu2 e -aq(ql+q2)z e aq(ql q2)2 e ~yy2 , where
M=
1
,
0 for an 1 = 0 resonance, and V = 1, 2.or 3 denotes rr-, n 0 or n + respectively. This
J. Steinhoff, Multiparticle amplitudes
148
quantity is plotted in figs. 5, 6 and 7 for C~q = 3 GeV -2, Oe.v= 0.25. It can be seen that it represents a crude approximation to (25). We easily obtain, from eqs. ( 19)-(22),
8 vq~z2 eicoy c~,u,(a'-v'ql'q2) : ~-~T-~2k ( q 2 ) k ( q 2) f dco , ~ 2 exp [co2/2e~y]/a - 1
(26)
where we only have one eigenfunction. In the above equation
k(q 2) = ~ a-
e
2c~qq 2
2g2rr 3 c(v(4%)2
We treat a as a parameter to be determined by the (inclusive single-particle) density of observed pions. Setting er = 2/a, assuming3' > 0, the poles of our integrand are at co/I. = l~co~ i 2+ ~/(~%) 1 2 2+ ( -3~ % • , ) 4 1"4-
,
l 2 I 2 )2 +(_lrJOey) -~ - 4 ] ~~ s i g n ( j ) , COR = [ 7coa+X/{~coa
where coa = ~ 7 . The smallest 1 = wh.ere coR We have col() = coa, COO R = 0 and ( v 4 : 0 ) co/ occurs a t / 0, = 0. then
CU1,u2(.I ' ,q l,q 2 ) =
40~qOCy -~ 2 2, -aq(ql+q2) ~" e(~/., i j.l 2
_ooaly I ~ e_OO~l.vI XL- ~ +2/~ 1 ~ 2 X(co) c°scoR[yl
fe
coRsincoRly,)].
(27)
w/ +%.
For v = 0 we have, from eqs. (13) and (26)
p l ( q 2 ) = cm,(O,q,q) = ~
4aq e 4~qq 2 F(1,7) 772
where
1:
r(o,~) = _-r=, Ira)
x °-I
d x - -
e x+y - 1
is an important function in the statistical mechanics of Bose gases. This specific form arises only because of the Gaussian-separable assumption for A2(rl,r2) in our numerical compu ration.
J. St einhoff , Multiparticleamplitudes
149
We fix the density of pions of each charge per interval in y:
r~= fd2qpl(q2), and compute 7 from the equation for F(½,7) given in ref. [16], and the relation
For 2 V ~ y r~ >~ 1, we get, from ref. (16), x/~ ~ 0.8 +
77.
The correlation length from eqs. (10) and (27) isy c = (2we0 -1 = (2 2V~yT) 1. Hence, we have 0.4 Yc
~
r/ + 2ay " t
For r/= 0.5, ay = 0.25, we g e t y c ~ 1.6. culu2(y,ql,q2 ) has the same asymptotic form as eUlU2(y,q 1,q 2), with dependence on indices given by Muau2, instead of 6u1~2 .The longest range part of c(y,q l,q2 ) is
C~lu2(y,ql,q2 ) ~ 0.8
4aq 7"/"
exp [
~
2+ 2
_aq(q I q2)]
× exp(-lYl/3.2)6UlU2 , and we get
4aq e 4o~qq~ 4O~qe_4e~qq~ CUluz (y,ql,q2) ~ 0.6 7r rr × e-[Y[/1.6 (6ul,2 + M , , , z ). This separation of C(rl,r2) into a part describing rr+n- and nOn ° correlation, and a part describing lr+n*, n n , and n0n 0 correlation was discussed in subsect. 5.1.1. The positive definite form of eq. (10) suggests that this positive n+n +, n n - correlation function, comparable in magnitude to the n+n- correlation function, is not a result of the particular form we have chosen for A2(rl,r2), but results from the general broad resonance decay amplitude that our form approximates. Also, we see the exponential asymptotic behaviour i n y that was discussed in subsect. 5.1.2. This, too, should be independent of the precise form of A2(rl,r2). We see that in our factorizable model Culu2 (y,q 1,q 2) has an exponential "tail" of correlation length ~ 1.6 independent of pion charge if interference terms are taken into account, but a Gaussian shape which decreases to 1/e of its y = 0 value at
J. Steinhoff, Multiparticle amplitudes
150
3' ~ 1.4 in the I = 0 channel, if no interference terms are considered, with 11o correlation in the I = 2 channel. The correlation f u n c t i o n for rr+rr pairs, integrated over q l and q2, which results from solving eq. (26) numerically is plotted in fig. 8, together with the correlation function when interference terms are neglected (the normalization is changed to obtain the same value of 7-/). Also, the correlation function for the I = 2 channel (rr+rr +, rr r r - ) pairs is plotted in fig. 9. In our model, it can be seen that the I = 0 and I = 2 correlation functions only differ qualitatively at small rapidities, where they are still comparable in magnitude. l f w e had used the form (25) for an e-meson decay amplitude, rather than our factorizable form, we would not have gotten quite so large a value for the correlation function for I --- 2 pairs, or for large v. In general any correlation in gA2(rl,r2) should decrease the long range part and the 1 = 2 part ofC(y,rl,r2)(for fixed r0. Since Yc, the correlation length, depends only on one particular eigenvalue of K(rl,r2), as we get more correlation in gA2(rl,r2) we have more eigenfunctions which are important, and the average magnitude of each eigenvalue should decrease (to satisfy the constraint on r/). In particular, as X -+ 0 in eq. (25), we would need an infinite n u m b e r o f eigenfunctions and each eigenvalue would ~ 0 . Thus, the kernel and m h o m o g e n e o u s term in our integral equation for c ( r l , r 2 ) ; K(rl,r2) -->O. Also, in this limit our I = 2 correlation function would vanish. For comparison, we do a calculation in this, the narrow-resonance approxilnation. We get in that approximation, with eqs. (7), (11) and (12), ¢
Cgl**2 ( r l , r 2 ) = g(A 2 )**1,,2 (rl 'r2 ) ' .5
.4
_
C(y) .3
-
C(y)
.Z !
Co(Y) : interference terms neglected
CoIY)
.1
0
I
1
2
3
4
5
L
6
--
?
y Fig. 8. Correlation function for I = 0, integrated over ql and q2, with and without interference ternls.
J. Steinhoff, Multiparticle amplitudes
151
.5
.4
C(y)
.2
.1
- - 1 -
1
I
I
I
~
i
2
3
4
5
6
y Fig. 9. Correlation function for I = 2, integrated over ql and q2.
C~lu2(rl,r2)
= 0, r 1 4=r 2 .
Hence
pl = ~ 1
4aq e_aaqq2 -2a --~
a),a 4aq 4aq C~lu2(y,ql,q2)-~-~ -~ e 4aqq2 ~. e-4aqqZ2e 2~yy2 M~zl~2
,
or
Culu2 (y,ql,q2) ~
0.2
4aq e_4aqq2 4C~qe_4aqq ~ e {y2 7r rr Mu~u2 "
6. Discussion A model for multiparticle production in the central region at high energies has been considered. An i n d e p e n d e n t emission model was modified to accomodate strong correlations in the p r o d u c t i o n amplitude between distinct pairs o f particles at low subenergy, expected from theoretical considerations. Damping in transverse m o m e n t u m was assumed, and dependence of the amplitude on other variables was ignored.
15 2
J. St einho]]] Multiparticle amplitudes
A mathematical identification between the exclusive cross section for pion emission (with all interference terms) and the configurational probability distribution function for a classical system of interacting molecules in equilibrium was exploited to obtain an expansion for the asymptotic single-particle inclusive distribution, the two-particle inclusive correlation function, and the contribution of the central region to the exponent of s in the total cross section by means of cluster diagrams. An integral equation was exhibited for summing the terms corresponding to the cluster diagrams. A specific model was then considered, which described independent production of two-pion resonances with small transverse m o m e n t u m and non-zero width. In this, the "s-channel pole dominance" approximation, the inclusive correlation functions were shown to be non-negative between any types of pions, regardless of the spin, isospin or subenergy dependence of the resonance. For I = 0 resonance production, the ~+n+, n n , n o n ° and n+n - correlation functions were seen to be positive at small rapidity while correlations were absent for n+n ° and n n O pairs. For neutral I = 1 resonance production, there were positive correlations for n+n +, n n and n+n pairs. A coherent mixture of charged I = 1 resonances was seen to allow the possibility of positive correlations between any types of pions. The positive correlations in all of the above cases were seen to have an exponential "tail" at large rapidities, for sufficient strength of the coupling constant. A numerical example was then discussed, which gave a crude simulation of a broad I = 0 spinless resonance. The features of the more realistic anrplitude that were retained were the short range in rapidity, the limited transverse m o m e n t u m and the isospin dependence. The resulting n+n + and n n correlation functions were seen to be quite large: conrparable to the rr+zr- correlation functions at small rapidities, and to have the same exponential "tail" at large rapidities. These correlation functions are certainly larger than would be obtained from a more realistic resonance decay amplitude, where threshold effects and other features are considered, but may be qualitatively similar. We see that if the most important features of the production amplitude are transverse m o m e n t u m damping and 1 = 0 or I = 1 resonance production, we can expect a rather long range inclusive correlation, positive in n+n +, n n and n+n - channels. The significance of this study is, perhaps, to point out that under the stated assumptions, the scale of the correlation function in the I = 2 channel is quite broad. From fig. 9 it can be seen that the correlation function in this channel decreases to half of i t s y = 0 value at y ~ 2.25. Available data analyses [8] do not allow enough of a rapidity variation in the central region to determine the relevant features of the I = 2 correlation function. The analyses do show, however, that there is a positive correlation there, and that correlations in the production amplitude are called for. To attempt even a crude comparison with the above data, the strong effects of energy-momentum conservation in the fragmentation regions would have to be included in our calculations.
J. Steinhoff, Multiparticle amplitudes
153
It is a pleasure to thank Dr. Richard Arnold for patiently explaining many of the aspects of high-energy phenomenology and classical fluid theory upon which this work is based, and for providing a great deal of guidance during the past few years. Also, I would like to thank Drs. P.G.O. Freund and D. Sivers for reading the manuscript and making many helpful suggestions.
Appendix A If unpaired pions are produced with amplitude g'Al(r ), together with two-pion resonances, our amplitude of eq. (1) is modified: '2E'(N)
A'N(rl,r 2 ..... rN)=
~
~
gKA2K(rl,r 2 ..... r2K)
distinct K=1 permutations
X g'A 1(r2K+ 1)'"g'A 1(rN), where A2K(rl ..... r2K ) is defined in eq. (1) and E'(N) = (N 1)N, N odd (even). The single-particle inclusive distribution and the correlation function are now
p 1(r) = c(rlr ) + glco(r)l 2 , r
C(rl,r2) = tc(rl,r2)l 2 + Ic (rl,r2)l
2
*
+ ZC(rl,r2jco(rl)co(r2)
Zc (rl,r2)co(rl)co(r2) + Zc (rl,r2)co(rl)co(r2) + Zc'* (r l, r 2 )co(r 1)Co(r 2), where * ,)] co(r ) = g'Al(r) + g' ,, ' d3 , r [c(r,r ,)Al(r ,)+ c ,(r,r ,)Al(r and c(rl,r2) and c'(rl,r2) are given by eqs. (11) and (12). The exponent a can again be determined from P 1(r) with eq. (16). It is perhaps interesting that, because of the terms in C(rl,r2) linear in c(rl,r2) and c'(rl,r2) , the long range part of C(rl,r2) will now have the same range as c(rl,r2). Thus, if there is a small unpaired pion component (small so that the normalization o f g is not effected), then there will be a small exponential "tail" in the correlation function of twice the range of the "paired pion" correlation function. This part need not be positive.
J. Steinhoff, Multiparticle amplitudes
154
Appendix B We show how eqs. (17), (9) and (10) could have been obtained from published work in elementary statistical mechanics. To construct a probability distribution function for a system of molecules that is equal to QN(rl ..... rN), we first consider a system interacting through two different external and two-body potentials. The probability function is then
QCN(r 1..... rN) : exp
/3 ~
,
(u (r i, r/) + u *(r i, r]))
where fi is a constant, and u(ri,r/) -+ 0 when Iri r/l -+ ~. This serves as a generating function for QN(rl ..... rN). Defining Mayer functions
f(ri,r])=
e ~u(ri'r/)
-1 ,
we have
Q~f(r I ..... r N ) = [ 1 [1 f(ri,r]) ] [ [ [1 f * ( r i , r / ) ] . i
pcK(Z;r I ..... r K ) -
QcK(Zzrl ..... r K) QCo(Z)
where
QcK(Z:rl ..... rK)= ~
ZN
(,V
K)! f d3rK<"d3rNQ~'(rl
..... r ¥ ) .
N = 1
The cluster expansions o f l n Q c° , p 0 ( r 1) and CC(rl,r2 ) = fic2(rt,r 2) fiO(rl)ficl(r2) are given in ref. [13], eqs. (4-3), (4-71 and (4-5) respectively. Each term in these expansions consists of integrals of products off(ri,rj) functions. The general cluster expansion becomes quite complicated as we consider integrals over increasingly larger numbers of coordinates. In our special case, however, when we operate with
3".Steinhoff, Multiparticle amplitudes
155
P before doing the integrals, we only have a simple subset of these terms, represented by "ring" diagrams. When we set f(ri,rj) = gA2(ri,ri), we obtain
In Qo : ~ Z2K g2K f d3rl...d3r2KA2(rl,r2)A~(r2,r3)...A~(r2K,rl), K=I VOK z2K g2K f d3r2...d3r2KA2(r, r2)A~(r2,r3)...A~(r2K,r) K=I V1K
pl(r ) = ~
and
z 2K C(r,r,)= ~ _ _ g 2 K K=I V2K
2K ~ /=2
fd3r2...d3rj ld3rj+l...d3r2K
X A2(r, r2)...Af2*)(r] 1,r')...A2(r2K,r), where the (*) applies i f / i s odd, and the VmK are "symmetry numbers", described m ref. [13]. We have for our set of cluster terms, V° K = 2K, V1K = 1 and V2K = 1. These above equations are equivalent to eqs. (17), (9) and (10). Note that f(ri,rj) , and not e ~u(ri'ri) describes the probability function for two molecules interacting through a bond. Sincef(ri,ri) -+ 0 when lYi-Y/I -+ 0% a potential describing the interaction of the molecules would become infinite in this limit.
Appendix C We can easily get higher correlation functions, in our models, by considering rings with more particles fixed. We first redefine some quantities: c+ (rl,r2)
= c(rl,r2),
c +(rl,r2) = c*(rl,r2) : c(r2,r 1), c++(rl,r2) = c'(rl,r2) : c'(r2,rl ) , c
(rl,r2) : c'*(rl,r2).
For the N-particle correlation function we draw N points, and join them as in the cluster diagrams described above. However, now we have four types of lines, labeled with a (+) or ( - ) at each end, and we must sum over all allowed combinations. The rule is that at each point a (+) and ( - ) have to meet. Thus, for N = 2,
C(rl,r2) = c+_(rl,r2)c + (r2,r 1) + c++(rl,r2)c _ ( r 2 , r l ) ,
J. Steinhof]] Multiparticleamplitudes
156 and for N = 3, if
p3(rl,r2,r3) = pl(rl )pl(r2)pl (r3) +
pl(rl)C(r2,r 3) cyclic permutations
+ C(3)(rl,r2,r3), C(3)(rl,r2,r3)=c+_(rl,r2)c+ (r2,r3)c+ + c +(rl,r2)c_+(r2,r3)c+(r3,rl) +
( r 3 , r 1)
~
c+_(rl,r2)c++(r2,r3)c__(r3,rl)
permutations
References [ 1] [2] [3] [4] [5]
[6] [7] [8]
[9] [10] [ 11 ] [12] [ 131 [ 14] [15] [16]
K. Wilson, Cornell preprint (1970), unpublished. D. Campbell and S.J. Chang, Phys. Rev. D4 (1971) 1151;ibid D4 (1971) 3658. R.C. Arnold and J. Steinhoff, Argonne preprint (1972). E. l:ermi, Phys. Rev. 81 (1951) 683. R.C. Arnold, Phys. Rev. D4 (1971) 3488; P.E. Heckman, Phys. Rev. D1 (1970) 934; D. Horn and R. Silver, Ann. of Phys. 66 (1971) 509. E. Predazzi and G. Veneziano, Nuovo Cimento Letters 2 (1971) 749; J. Ellis, J. Finkelstein and R.D. Peccei, SLAC preprint 1020 (1972). D. Sivers and G. Thomas, Argonne preprint 7218 (1972). W.D. Shepard et al., Phys. Rev. Letters 28 (1972) 703; W. Ko, Phys. Rev. Letters 28 (1972) 935. E. Berger, B. Oh and G. Smith, Argonne preprint 7209 (1972). tt. Satz and G. Thomas, University of ltelsinki preprint 7 - 7 1 (1971). D. Sivers and G. Thomas, Argonne preprint 7228 (1972). D. Amati, S. Fubini and A. Stanghellini, Nuovo Cimento 26 (1962) 896; C. Detar, Phys. Rev. D3 (1971) 128. R.C. Arnold, Phys. Rev. D5 (1972) 1724. G. Steil, in The equilibrium theory of classical fluids, eds. H. Frisch and J. Lebowitz (Benjamin, New York, 1964). S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience, New York, 1965) especially ch. 2. E.I. Shibata, D.H. Frisch and M.A. Wahlig, Phys. Rev. Letters 25 (1970) 1227. J.E. Robinson, Plays. Rev. 83 (1951) 678.