Correlation coefficients in the constituent interchange model

Volume 55B, number 1

PHYSICS LETTERS

CORRELATION

COEFFICIENTS

IN THE CONSTITUENT

20 January 1975

INTERCHANGE

MODEL

D. SCHIFF, A.P. CONTOGOURIS * and J.L. ALONSO **

Laboratoire de Physique Thdorique et ParticulesEldmentaires, Orsay, France Received 4 December 1974 Correlation coefficients (CC) for two large transverse momentum hadrons are studied in the constituent interchange model. For p + p ---,n° + n° + X when the two n° 's are produced in opposite directions with momenta PTC, PTD the model predicts significant structure of the CC at PTC ~' PTD; and when the two n° 's are produced in the same direction it predicts a very small CC. Comparison with existing data is also given. Among several parton models for hadronic phenomena at large transverse momenta PT the constituent interchange mode (CIM) [1,2] is generally believed to offer today the simplest and most complete account o f a number of experimental facts. In this model production of large PT particles is assumed to proceed via a restricted number of underlying parton-hadron scattering subprocesses; direct parton-parton interaction is postulated to be unimportant. In this paper, we calculate correlation coefficients (CC) in the CIM for two large PT hadrons produced either in opposite directions or in the same direction. We show that the structure o f the inclusive cross-section for two hadrons in opposite directions is a direct and sensitive test o f the dynamics at the parton level. Also we predict that, in general, the CC for two pions in the same side is much smaller than the CC for two pions in opposite sides. Some experimental data on CC at large PT have already been reported [3] ; and they appear to be in contradiction with the predictions of CIM, in particular concerning the magnitude o f the CC for mr in the same side. We start with p + p ~ lr + X and assume, as usual, that it is dominated by the irreducible subprocesses 7r + q --, n + q and q + ~ --, Ir + M (q denotes quark and M meson); this allows for a good description of the one-pion inclusive production [2]. We shall first consider the contribution of 7r + q --, n + q to the inclusive cross-section for p + p ~ n ° + rf° + X with the 7r°'s produced at 90 in opposite directions (fig. l(a)). Following ref. [2] we denote by Gno/p(Xl) and Gq/p(X2) the probabilities that the Ir° and the quark emitted via hadronic bremsstrahlung by the protons A and Bhave the longitudinal momentum fractions x I and x 2 respectively. We call y the longitudinal fraction taken by the rr°(C) emitted from the scattered parton and define in the same way Gno/q(y). The first step is to consider the general case where the two 7r°'s have transverse momenta PTC and PTD and are emitted in the directions 0C and 0 D with respect to the initial proton in the proton CM frame (fig. l(b)). The contribution to the 2-particle inclusive cross-section is then d2a

o do = G n/p(xl)Gq/p(X2) ~ (s,i) Gno/q(y ) dx 1 dx 2 d i d y .

(1)

The invariants which characterize the quark-hadron subprocess are s = (Pl + P 2 ) 2 ( ~ XlX2S), i = (Pl - p~)2 and tt = (Pl - P D ) 2. There is a one-to-one correspondence between the variables which describe the quark-hadron scattering and those which specify the 2-particle inclusive distribution (0 c , 0 D , PTC, PTD). The following relations can be derived

t= --p2D(l +tg~OctgJ$OD) , Y =PTC/PTD ,

Xl =~-sD (Ctg 21Oc + tg21OD) , X2=~DsD(ctg2X0D+tg2X0C)

On sabbatical leave f r o m Mc Gill University, Montreal, Canada. ** Now at SLAC, Stanford University, Stanford, California, U.S.A. *

87

V o l u m e 55B, n u m b e r 1

PHYSICS LETTERS

20 J a n u a r y 1975

C(~*)~,

~¢(q)

V

o,

~ P2

IID(~') (o)

(b}

(c)

Fig. 1. N o t a t i o n f o r t h e b a s i c s u b p r o c e s s 7r° + q -~ ~r° + q. a) D y n a m i c s o f p + p -* Ir° + ~ + x Or° ' s in t h e o p p o s i t e d i r e c t i o n ) pro-

ceeding via ~r° + q -~ ~r° + q. b) Kinematicsfor two ~r°'s in the opposite direction, c) Kinematics for two ~r° 's in the same direction. so that d3Pc d3pD tg½0 C tg½0 D EC ED = 27t2pTCPTD [(1 + tg•0 C tg ½0D)PTD/X/~]2 d i dy dx 1 dx 2 . When 0 C = 0 D = 90 ° this gives:

X2

EcE D d2°(°PP)

- l p T ~ - P T D G~to/p(XTD) Gq/p(XTD) G¢/q d3pcd3PD 2rt2

(XTC.1 do \XTD] -~

(2)

with xTC = 2PTC/X/~, XTD = 2PTD/V~and da/di calculated at s = 4p2TD and i = --2P2TV; throughout this derivation: PTC ~
G¢/p(X)

=

(1 -x)5/x,

Gq/p(X) = (1 -x)3/x,

G~r,/q(y) = (1 -y)/y.

(3)

We write do/d i under the general form

do/d i =f(x 1, x 2)/;4

,

(4)

and assume, as usual, that f(Xl, x2) is a slowly varying function o f x l , x 2. Under such a general form the contribution of the graph deduced from fig. l(b) by interchanging x I ~ x2, i+-~ t~ is also included. Eq. (2) then gives

EcE D d 2 o ( o p p ) _ d3pcd2pD

1 1 (I_XTD)S(I_y)f(XTD,XTO) 27r2p2 C (2PTD) $

(5)

For the one-particle inclusive distribution at ISR good agreement with the data has been obtained [2] by retaining the contribution of fig. l(a) as the dominant one, i.e. by using •

E do/d3p = C(1 --XT)9/p~,

(6)

C is normalization coefficient. Then the CC for rr°lr ° at 90 ° in opposite directions is

R(opp)-

°inEcEDd2°(°PP)/d3pcd3pc (Ecd°/d3pc)(EDdo/d3pD)

°in - 2~t2

f(XTD'XTD) s3 C2

X6TC

214 (1--XTC)9(1--XTD)

(1 - x T C ]

(7)

XTDJ"

An important feature of (7) is the threshold factor 1 - y = 1 - XTc/XTD. As a result the model predicts dips of R(opp) at xTC ~ XTD. Away from these values, for fixed XTD (7) predicts that R (opp) increases quite fast with 88

Volume 55B, number 1

PHYSICS LETTERS

20 January 1975

108

10e

R (opp)

x~. =0.5

10 7

10"~

I0 s

i0 s

I"

0.185

%

10 s

/

105

/

,I

o.~ej

o 10 '~

10 4

0.12

10 3

10 3

# . 0 7 5 < x ~- < . I 0 5 ~, .135<~<.165

102

II

10 2

*

Ii

I i

Ill

xT i

.1

, .lO5
.2

,

i

.3

,

I

.4

i

I

.5

i

l

.6

i

I

.7

x~

I

.B

.9

.7

Fig. 2. The correlation coefficient R (opp) for two n°'s in the opposite direction at 90 ° as a function of their transverse fractions l 1 1 x 0T and with xTC and XTD usedin the text is as follows:n When x 0T < x T : x T~ r = x 0T , x T_ n = XT; I x T . The, correspondence ~ _ whenx'~ ^ > XT: xTC ~x~£,XTD ---x~'. The expetirnenta[ data ate taken from ref. [3]. a) x~- is fixed equal to 0.09, 0.15 and 0.30. b) x~ is fixed equal to 0.12, 0.185 and 0.50. xTC; also for fLxed xTC, R (opp) is predicted to increase with XTD. To compare with the data [3] we use the fact that f(XTD, XTD)/C2is an adjustable constant. Then we see (fig. 2) that the predicted increase o f R (opp) with xTC and XTD is in fair agreement with experiment. However, the data show no evidence of dips at xTC ~ XTD. To examine the effect o f the subprocess q + q ~ ~t° + M we have considered the extreme situation where this completely dominates the one- and two-pion inclusive cross-sections. We obtain, instead of (7):

°inf(XTD'XTD) s3

R(opp)

X6c

=

(1

xTC) 3 -

27t2

C2

214 ( 1 - x T C ) 1 1 ( I --XTD )

.

(S)

XTD

We have found that the comparison o f ( 8 ) with the data is less satisfactory (see e.g. brokenline in fig. 2(b)). However, (8) and (7) have the same qualitative features. In particular notice the threshold factor (1 _ y ) 3 in (8), which again predicts dips at xTC ~ XTD. A special contribution to R (opp) will also arise from the subprocess q + V~-~ n ° + n ° with the two It°'s produced in opposite directions. It is important that this contributes exactly at xTC = xTV (due to m o m e n t u m conservation), where the dips of (7) and (8) are predicted. The shape o f this special contribution depends on the magnitude o f the experimental resolution with respect to, say, xTC; and the size depends on the relative magnitude o f q + 7= l -+ It° + n ° versus q + ~l -+ n° + M. Detailed calculations require additional assumptions; nevertheless, it can be said that, 89

Volume 55B, number 1

PHYSICS LETTERS

20 January 1975

in general, this special contribution will produce a strong variation ofR (opp) around XTC~-, XTD. Thus anyway, we conclude that CIM predicts significant structure and strong variation of R (opp) in the region XTC ~ XTD. Such a structure does not seem to be supported by present experiment. It might be argued that in most cases the data of fig. 2 do show some structure at the upper end of the available XTD range. If this is evidence of dips (and not simple statistical effects) these dips must be displaced from the positions predicted by (7) (or (8)). Such a displacement could arise e.g. by introducing some PT dependence in our probability functions (i.e. using G~ro/q(y,PT)), as seems also to be required by the lack of complete coplanarity of the away jet observed in association with pp ~ zr°X [4]. Anyway, more experimental work is necessary in order to decide whether there is indeed some structure in R (opp). Next we turn to the CC for 7r°Tr° produced in the same direction. We follow a similar procedure and examine the subprocess illustrated in fig. l(c). Now the problem is simplified from the start by restricting our considerations to particles C and D emitted in the same direction. If G(Yl,Y2) denotes the probability that the scattered quark emits two particles (in the same direction) with longitudinal fractionsYl,Y2(.y 1 = P~/P~',Y2 =PD/Pl* / *' defined in the hadron-parton CM frame) then we consider do ~ ~ d2°oC=OD=O=Gn°/p(X1)Gq/p(X2)~t (s, t)G(Yl,Y2) dXl dx2 dt dy I dy 2 ,

(9)

with PTC (tg½0+ ctg½0~ Yl = ~ - s \ X2 Xl ] '

Y2 =

PTD[tg~O -~SS ~ X2

+

ctg½0~ , Xl ]

~=_SXltgX20(tg½0

ctg~01-1 \ X2 + ' Xl ] .

(10)

Thus dYldY2d[ = s i2~ dPTcdPTD dO, and the inclusive cross-section at 0C = 0 D = 0:

ECED

d2o(same)

1

sin0

-

d3pc d3pD

2rr2 PTCPTD

Using the expressions (3) for

eCeD d2o(same)

d3p Cd3pD

ffa,,O/p(Xl)Cq/p(X2)i)¢(.vl,Y2)dxxdx2 do

G~ro/p,Gq/p and eq. (4) we have at 0c = 0 D = 90°:

_ ,1 1 2rr 2 PTCPTD

JJ££(1-Xl)5( 1 - x 2 ) 3 f ( x l , x 2 ) ~ ( , y l , Y 2 ) d x l d X 2 Xl

x2

,

(11)

whereYl andY2 are given by (10) with 0 = 90 °. The integration domain is determined by the condition Yl + Y2 ~< 1. Now, under the assumption that f(Xl, x 2) varies slowly in this domain, we notice that for x 1, x 2 not very near 1 the function

K(x 1, x2) ---(1 -

x 1)5 (1 - x2)3f(x 1, x2)

is also a slowly varying function (in comparison with the rest of the integrand). Thus we simplify as follows

eceo

d2o(same) 1 d3pc'd3p D ~271" ~

K(xo,xo) rrG(YI,Y2) PTC~TD I1..,.,, ~

dxt dx2 Xl X2

(12)

It can be easily seen that a reasonable value for x 0 is x 0 = xTC + XTD. Detailed (but straightforward) analysis shows that (1 2) is a good approximation provided that x 0 ~ 0.6. In general, the exact form of G(Yl,Y2) is unknown. We shall consider two obvious choices by analogy with the single-particle probability distribution: 90

Volume 55B, number 1

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XTC = 0.40 100

0.20

kd

0.13

.=_ 0

==

/ I I

Ii / i I /

f

. . . ~ -'=

~r XTC=0.13 * 0.11

/ OE

0

I

I

I

0.1

0.2

0-3

0.09

I

0.4 XTD

Fig. 3. The correlation coefficient R (same) for two n° 's in the same direction at 90° . Experimental data from ref. [3]. l - Y 1 -Y2 , ii) G(Yl,Y2) = (1 - y l ) ( 1 -Y2) Yl +Y2 YlY2 '

i) G(Yl,Y2) =

(12a, b)

the second form has also been used in ref. [5]. Then straightforward integration leads to the expression (0C = 0 D = 90°): EcED d2o(same)~, 1 K(xo'xo) BSZ d3pcd3pD 2rt2 PTCPTD s4 ' where B = 2Ix 0 - 1 and corresponding to the above choices i) and ii): = 1__~+1

1 1 1 1 1 B 9 8 0 +B-22--~ + B 3 140

i) Z

7840

ii)Z~

4 (1 B B2XCTXDT B+I 2x 0

17 1 3 120+B20 1 1 + 1

1 11 + 1 17 t/4 336 B5420

(13)

1 41 B6280

1 143

1 1/4127 + 2 ,

B72940 +B-8'~2~

B+I~

7m---~] '

1 1+ 1 1 ) B28 B 3 6 - " "

3 +12~+...)+1(2_~0+1

1+1

1 +l

1 ~+...).

BXCTXDT

In this approximate treatment where x 0 is relatively small, we have kept only the first few terms of the power series in lIB. Using again eq. (6) for the single~ ° inclusive distribution we obtain the CC for two rr°'s at 90 ° on the same side: R (same)

.

°in K(xo'xo) s 3 (XTCXTD)7 " . . . . . BsZ . 27r2 C2 214 (1 --xTC)9(1 -- XTD)9

(14)

The expression (14) is proportional to the constant f(x 0, xo)C-2. In view of the slow variation o f f ( x , x ) we 91

Volume 55B, number 1

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may fix this constant to the value used in the calculation of R ( o p p ) . The resulting R (same) as a function o f x T D is given in fig. 3 for a number o f values o f x T c and for each o f the choices i), ii)*. Now fig. 3 also shows the present data on R (same); and, although these contain large errors, it is clear that they exceed the predicted values by more than one order o f magnitude. The fact that CIM predicts very small R (same) might have been anticipated: When two hadrons o f large m o m e n t a PTC, PTD are produced in the same side, energy-momentum conservation requires that the subprocess I + 2 ~ 1 ' + 2 ' (fig. l(c)) proceed with a rather large total energy, (~)1/2/> 2 (PTC+FTD). Since d o / d t ~ ~ - 4 one expects a strong suppression o f the corresponding 2-hadron inclusive distribution. More generally, on the same side with a large PT hadron CIM cannot tolerate additional hadrons in large numbers o f large PT" The simplest form o f CIM considered above can be modified e.g. by the inclusion o f 2 - 3 b o d y subprocesses, or b y introducing appropriate final state interactions; the latter are usually believed to be unimportant, at least concerning the leading asymptotic behaviour. Anyway, if experiment definitely establishes important correlations for large PT hadrons on the same side *, drastic modifications of XIM will be necessary. We would like to thank R. Blankenbecler for initiating us to CIM and for helpful discussions. * Notice that R(same) calculated with the choice ii) exceeds that of i) by roughly one order of magnitude. This is plausible by the fact that with ii) G(yl,Y2) has a product of the two y's in the denominator and that most of the contribution to the integrals of (12) comes from smaUy's. Additional evidence in support of important correlations between a ~r° and a charged hadron of large PT on the same side can be found in preliminary data of the CERN - Columbia - Rockefeller - Saclay collaboration (see Landshoff [4] ).

References [1] R. Blankenbecler, S.J. Brodsky and J.F. Gunion, Phys. Letters 39B (1972) 649 and 42B (1973) 461; Phys. Rev. D6 (1972) 2652 and D8 (1973) 287. [21 R. Blankenbecler and S,J. Brodsky, SLAC-PUB-1430 (1974). [3] F.W. Biisser et al. Phys. Letters 51B (1974) 311. [4] Pisa, Stony Brook collaboration, in P.V. Landshoff, report to the XVII Intern. Conf. on High-energy Physics, London (July 1974). [5] S.D. Ellis and M.B. Kislinger, Phys. Rev. D9 (1974) 2027.

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