Colour in production on nuclei

Volume 78B, number 2,3

PHYSICS LETTERS

25 September 1978

COLOUR IN PRODUCTION ON NUCLEI Z.J. REK and G. WILK Institute of Nuclear Research, 00-681 Warsaw,Poland Received 8 June 1978

A picture of multiparticle production on nuclei based on the previously conjectured role of colour in the low PT multiparticle production on nucleons is presented. Possible links with other models are discussed.

Quantum chromodynamics (QCD) [1], so successful in the description of short-distance phenomena, has also been applied to the low PT production on nucleons (h + N ~ h ' + anything) [ 2 - 4 ] . The rising of hadron multiplicity was attributed to the necessity of colour confinement. The process itself occurs in two stages: (a) The exchange of a coloured object (in what follows we will take it to be a gluon) between h and N, producing a coloured hadron h c and nucleon N c moving apart with the rapidity separation Y ~ In(sire2). In principle this part can be calculated in detail in QCD. (b) A kind of a colour compensating flow (CCF) between h c and N c resulting from the necessity of colour confinement. So far QCD was not able to describe this stage adequately. One can argue that it results in a chain of finally observed colourless hadrons somehow distributed in rapidity. In a more model dependent approach one can estimate that the hadron multiplicity n ~ y k and that this stage starts at the distance of several fermis [2]. In the present note we would like to extend this picture to the low PT production on nuclei (h + A -+ h ' + anything). Our picture will be the following. As in the hN case, production on nuclei occurs in two stages: (A) The propagation of an originally colourless hadron h through the nucleus which finally gives a coloured hadronic state h c with rapidity (in the LAB frame)y v = Y~--ln(s/m 2 ) moving apart from the coloured nucleons N c with rapiditiesy i ~ 0 spread over the nucleus (see fig. 1). This is because the second

(b) y Rapidity axis

(a)

[Nuclearmatterc c c h

hI

h2

h.v_I

I h~

h~ yh=Y

Fig. 1. The scheme of the first stage of hA scattering (a) and the resulting situation in rapidity space (b). For the notation consult the text.

stage of each individual hN scattering will probably not occur inside the nuclear matter. Due to the long distance character of colour confinement it occurs after the excited coloured hadronic state h e passed a distance of several fermis. Tile presence of other nucleons on the way will then act in favour of repeating the first stage again rather than starting the CCF. This, we assume, will delay the beginning of CCF. The situation resembles in a sense that of the sequential decay of heavy clusters in the nuclear matter [5]. (b) The CCF between h e and all N c. Assuming, like in hN, that the production of chain from h e results in its decolorization, one is forced to accept the necessity of the splitting of that chain at some rapiditiesy i into chains going on to the coloured nucleons N c. This is because these nucleons are at the distinct space positions inside the nucleus and, although their net colour is such that it would compensate that o f h c , it is impossible that they can be decolourized by only one 333

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chain, as one needs one chain per excited nucleon. The notion of one chain per nucleon is a simplification made for the sake of clarity of presentation. It results from the assumed one gluon exchange type of each individual excitation in (A) and from the assumption that h c are all colour octets. We believe that the abandoning of these assumptions would not spoil the spirit of our picture. Apart from the colour, the state (A) looks like a kind of "elastic" propagation o f h through the nucleus. One could think of making use of a Glauber-like method to calculate this part of the process but because of colour we do not think such approach would be justifiable. On the other hand, the diagrammatic calculation of (A) based on QCD, being extremely complicated, would also be premature in view of our inadequate knowledge of part (b) in hN process. Hence, we decided to describe (A) in a simplest way. We took the nucleus to be a spherical bag (with radius RA) of colourless nucleons distributed with uniform density p. (Thus we neglect the possibility of existence of multinucleonic colourless structures inside the nuclei [6] .) Following [7], the probability Pv of a given number v of colour excitations inside the nucleus will be

25 September 1978

depends on several factors: - the shape of each chain which can be taken from hN scattering; - the way of splitting; we assume fk(v) = fk(Y) - a distribution for a hN chain of length Y; - a weight function gv (Y;Yv-1 ..... Yl) describing the probability distribution of the positions of chain splitting, y / ( i = 1, ..., v - 1). In (2) the ordering yi %Yi+l was assumed. The effect of dynamical average over all possible colour splittings is included in the functions gv(Y; Yi)" The present stage of development of QCD does not allow to calculate it explicitly. Instead we have chosen two simplest possibilities for further study: a constant weight - each distribution of splitting points equally probable:

_ ( v - 1)!. gl

(3)

yv- 1 '

the weight favouring the short chains and thus amplifying the colour compensation between excited nucleons:

1

Z

P=~fv. f

xV+t e-X dx

(1)

0 where c is the normalization factor, z = 2RA/X and X = 1/po the "mean free path" of the hadronic state h~ between the subsequent exchanges of colour; a is then a "cross section" for such process. In the part (B) the splitting of chains is thought of as a reflection of the 3- and 4-gluon couplings in QCD. Hence we expect two modes of splitting: 1 ~ 2 and 1 ~ 3, for simplicity only the former will be dealt with in detail. The rapidity distribution of particles resulting f~om the v-fold colour excitation (see fig. 2) dN

Y P

•. . f

0

~YlV • =3

"N ...........

<-.

dYlgv(Y;Yv_ 1..... Yl )

~yl~ =3

(a)

v oo, k - y ) ; k ( y ) k=l

(b)

,~ "x..~.

~.

! • .i ...............

Y._I Y

(2)

334

With the hN rapidity distribution of the form f l y ) = z~n=0 akY k we get

,,

y~ •

(4)

...........i...... i t

Yv-1

-f0 <-,f0

g2 : ~.yv_ 1 ""Y2 "

•

Y._2 Y

I. 0 y" ¥

Yl ¥

.

.~,~ ~

Y_2 Y

1.0 _Y Y

Fig. 2. The examples of expected rapidity distributions of finally produced particles resulting from the v = 3-fold colour excitation in the nucleus for given rapidities Yl, Y2 of the chain splitting. ( a ) f i y ) = 1; (b) f(y) = 1/2 + 2 y / Y - 2(y/Y) 2. Solid lines correspond to Z~c=1 ofY k - y) f(Y), dashed lines to dN3/dy for gl and dash-dotted lines to dN3/dy for g2.

Volume 78B, number 2,3

PHYSICS LETTERS

dN dy = v - (v - 1)

f(y)

for g l '

(5)

Table 1 The values of R for different nuclei and different weight functions in the case f(y) = 1. Parameters used in eq. (1) are: p = 0.14 and cr = 32 mb which gives z = 1.1A I/3 [7[. Weight function

dy

v

_

(v-

_Z

y ~

v-l-1

1) y - - ~ - / = 1

In-f-

l]

for g2"

The behaviour of dNv/dy for the case v = 3 and two simple choices o f f @ ) is shown in fig. 2. Passing on to the quantity

R=LG f n

v

dy

o

where n = f f f ( y )

dy, we obtain

m R=I~ n k=0

k+l

+~-

yk+l

( k + l ) ( k + 2 ) ak > ½(1 + i f )

ai=O,i> l

forgl,

m

R

(7) oo

1

n k=0 (X + l) 2

akyk+lE +2

(X

for g2"

A 12

69

132

208

gl

1,59

2.16

2.49

2.78

g2

1.42

1.63

1.71

1.77

f(y) (6)

(8)

= Ev vPv is the mean number of colour excitations in the nucleus. Note that for only a 0 ~ 0 eq. (7) is identical to the results o f other approaches based on completely different dynamical inputs as the two-phase model [8], the "wounded nucleons" concept [9] or the parton-like models [10,11], ifY is to be treated as the number of inelastic scatterings inside the nucleus: Y = A(a~N/a~nA). One can then argue that with gl we have a contact with these models - if not for a rather obscure origin of the g function. Both of them give identical results for v ~< 2, the difference starts at v ~> 3 and grows rather fast with v. This is best seen in the different A dependence of R (see table 1) from R ~ A 1/3 f o r g 1 to R ~ A 'x, a < 0.1 (in both cases only a 0 :~ 0). With Pv calculated more precisely the A dependence of R(and also dN/dy = ~v Pv dNv/dY not shown here) will impose constraints on the pos-

25 September 1978

sible form o f g . This, of course, would be of some importance in QCD but only after a method of calculating o f g ' s is developed. To summarize, we presented a picture of production on nuclei based o n the already conjectured role o f colour in the low PT production processes. The final results, eqs. (5)-(8), seem rather trivial, being identical to other approaches (for gl)- One should bear in mind, however, that now these results follow from a substantially different physical picture. The crucial point, as we stressed in (B), rests in the necessity of splitting the chains of particles produced in order to confine the colour. Although formaly similar to the picture with interacting reggeons, the splitting of chains is inherent in our picture and enters on the basic level of production, not as a high energy correction as in the case of reggeons [11 ]. The different choice of the weight function, g2, spoils this similarity. The results are now quite different. To investigate this problem in more detail one would have to enter into more model dependent calculations, like [2], which are outside the scope of the present note. As for comparison with experiment, it is obvious that we can fit the data on R and dN/dy with gl (as other models do) but we may have some troubles with other choices of the weight function g. In our picture we have among others a possibility of simultaneous production from the v chains of length Y each (all Yi = Y). These are chains joining the h e state directly with all excited nucleons N e . In a sense it is in the spirit o f the Coherent Tubelike Models [13]. Although in our case the multiplicity coming from this event is ~vfoYf(y) dy and in CTM it is ~fYo'fly)dy, r' = ln(v S]m2), this may be due to the incomplete knowledge o f underlying dynamics. (For example, the simultaneous production from the v chains can result in a kind o f final state interaction connected with a redistribution of energy and thus a change in the final 335

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PHYSICS LETTERS

energy dependence of the multiplicity.) Whether this could be treated as an indication of the field theoretic origin of CTM (as we would like to believe) remains to be checked.

References [ 1] For an updated review of QCD see W. Marciano and H. Pagels, Phys. Rep. 36 (1978) 137. [2] F.E. Low, Phys. Rev. D12 (1975) 163. [3] S. Nussinow, Phys. Rev. Lett. 34 (1975) 1286;Phys. Rev. D14 (1976) 246. [4] S.J. Brodsky and J.F. Gunion, Phys. Rev. Lett. 37 (1976) 402; S.J. Brodsky, SLAC-PUB-1937 (1977). [5 ] M. Anselmino and G. Wilk, Acta Phys. Polon. B9 (1978) 365.

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25 September 1978

[6] V.A. Matveev and P. Sorba, Lett. Nuovo Cim. 20 (1977) 435. [7] K. Gottfried, in: Proc. of Vth Intern. Conf. on High Energy Physics and Nuclear Structure, Uppsala, 1973. [8] P.M. Fishbane and J.S. Trefil, Phys. Lett. 5iB (1974) 139. [9] A. Bialas, M. Bleszyfiskiand W. Czyz, Nucl. Phys. Bl19 (1976) 461. [10] S.J. Brodsky, J.F. Gunion and J.H. K/ihn, Phys. Rev. Lett. 39 (1977) 1120. [11] A. Capella and A. Krzywicki, Phys. Lett. 67B (1977) 84; Orsay preprint LPTPE 77/31 (1977). [12] M. Moshe, Phys. Rep. 37 (1978) 255. [13] G. Berlad, A. Dar and G. Eilam, Phys. Rev. DI3 (1976) 161; for a review and comparison with other approaches see S. Fredriksson, talk given at the Triangle Seminar, Campione d'Italia, October 1977, CERN TH 2423.