A probabilistic model of interacting strings

NUCLEAR P HV S I C S B

Nuclear Physics B390 (1993) 542—55 8 North-Holland

________________

A probabilistic model of interacting strings M. Braun and C. Pajares Department of Particle Physics, Physical Faculty, University of Santiago de Compostela, 15706 Santiago de Compostela, Spain Received 22 October 1991 (Revised 20 January 1992) Accepted for publication 4 September 1992

Interactions among colour strings produced in high-energy soft hadronic collisions are introduced via a probabilistic picture, in which strings fuse as soon as they overlap in the transverse area available. The possibility of formation of new strings, which have more colour and fragment with higher average transverse momentum and heavy flavour content, is studied. The obtained hadronic amplitude satisfies the s-channel unitarity. Predictions of the model for the main observables are compared with the experimental data and the results obtained in the framework of independent string fragmentation.

1. Introduction The success of semiphenomenological models for the soft hadron interactions, such as the Dual Parton Model [1,21or Fritiof model [31,seems to confirm the idea that soft processes are mediated by multiple exchanges of some localized objects, coloured strings. Actually it is the old pomeron exchange picture that is revitalized in these models with a new ingredient that the pomeron is made of two coloured strings. Technically this adds little to the description of the hadron amplitude in terms of the supercritical pomeron [4], the difference only coming when comparing to the lepton hadroproduction and counting contributions of different quark flavours in the hadrons. The actual number of strings seems to be rather small at present energies, but it rises with energy. Thus the problem of the interaction between strings starts to become of importance. Much attention has been given to this problem lately with the emphasis on the formation of strings with more colour [5,61. Such objects could at least partially account for the rise in multiplicity and K p~) together with the contribution coming from semihard processes. In this paper we present a very simple and intuitive picture of the interaction of strings and their fusion into strings with more colour based on probabilistic considerations. Although little, if any, dynamics is included in the model, it may 0550-3213/93/$06.00 © 1993

—

Elsevier Science Publishers By. All rights reserved

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describe in a crude qualitative manner what occurs if there indeed exists an attractive interaction between strings which binds them into multistring objects. The basis of the picture comes from the observation that the old Regge-eikonal picture of the pomeron exchange [4] may be reinterpreted in a probabilistic manner, similar to the Glauber theory of hA collisions. Suppose that in an hh collision a varying number N of some intermediary objects (pomerons, strings) are formed with the probability of the Poisson type p~=exp[(_Q)]qN/N!,

(1)

where Q is the mean value of N. Suppose also that the scattering matrix at fixed impact parameter for the configuration with a given number of pomerons is a product of elementary matrices for each pomeron SN(b) =S~v(b).

(2)

Then the expectation value of S is S(b)

=

(6) =exp(Q(s(b) —1)).

EPNS

(3)

N

Introducing the amplitudes S = 1

+

iA, s = 1

+

ia we get

iA=exp(iaQ—1)

(4)

which is the standard form for the hh amplitude in the Regge-eikonal model with the identification of the “renormalized” amplitude a~= Na as the contribution of a single pomeron exchange. Our basic idea is that the interaction between pomerons can be simulated by changing the poissonian distribution (1). Indeed suppose that, as a simple example, two pomerons can fuse into one, as accomplished by the 3-pomeron vertex in the Regge—Gribov theory. If this interaction is strong enough then two pomerons at the same point in the transverse plane will immediately go into one, so that the probability for them to exist separately in the total transverse volume V(2 dimensional) will be diminished by a factor (1 —x), where x = V0/V and V0 is the intrinsic volume of the pomeron determined by (pt). With three pomerons the inhibition factor becomes (1 x)(1 2x) and so on, until we come to the moment when 1 Nm5x <0 and no more pomerons can coexist in the volume V. This suggests a change from (1) to the distribution —

—

—

N-I

PN~(Q/N!)

fl

(1—kx)

k.~1

(5)

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which forms the basis of the following discussion. One may ask from where comes the total volume V? We take it to be characteristic for the pomeron distribution in the transverse plane, which, as we know, is governed by the hadron radius and the growing radius of the pomeron distribution proper, proportional to a’y with a’ the intercept and y rapidity. According to this picture the volume V extends with y, the intrinsic pomeron volume V0 staying stable. So x does down with y, but on the other hand, the mean number substantial.

Q

may rise. Thus the net effect may be

2. Fusion of strings One may imagine different physical scenarios for the interaction of strings. The simplest one has been mentioned in sect. 1, in which several strings merge into one of the same type, reminiscent of the multipomeron interaction in the Regge—Gribov theory. A richer picture follows if one assumes the existence of various types of strings with different colour content. At the same point in the transverse space, two strings may form a new object, a new string, which would correspond to a double colour exchange in the abelian case. More generally one can assume that any number of different strings characterized by their “colour numbers” n1, n2,... fuse into one with the colour number n ~n, when their transverse positions coincide. The ordinary strong corresponds to n 1. Such an interaction conserves the colour number. In fact, this picture of string interactions does not require that strings actually fuse into a new one, as a vortex line may do in superconductor-like models. The only thing that matters is that they fragment coherently, as a single object, in the manner discussed in ref. [7], even though their space configurations remain separate. Because of that the real character of their interaction, repulsion or attraction, does not seem important. Fusion of strings of a single type can be considered as a particular case of this more general scenario when strings with the colour number greater than one do not exist and the total interaction is provided only by fluctuations in the number of strings. So in the following we shall study this general case of the fusion of strings with various colour numbers. Consider first an event with the total colour number N given. Let the number of strings with the colour number n be i.’,,. Then N ~ and the total number of strings is M ~ n 1, 2 As explained we assume that if M strings are within the volume V0 they fuse to form a new string with the conservation of the colour number. With the total volume V this leads to the same inhibition factor as in (5) provided that the intrinsic volume V0 (or rather the volume within which the strings fuse) is the same for all types of strings. In the following we assume this =

=

=

=

=

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545

approximation. Simple probabilistic considerations then lead to the probability distribution PN(~n) for the numbers v~of strings of type n, with N fixed, M- I

PN(V~) =

(N!/fl~n!fl(fl!)””)X”

M

fl

(1—kr).

(6)

The overall distribution is obtained by multiplying (6) by the probability for the total colour number to have the value N. Taking it to be poissonian (1) we arrive at M- 1 =

fl (n!)”)x”~’~fl

c(Q”7JJ~~!

(1— ~)

(7)

which is our starting point. Note that (7) takes account of the colour number conservation in the process of string fusion. If only one sort of strings with n 1 exists then N M ~ and the distribution (7) becomes identical with (5). To normalize (7) we have to sum over all v~.To do this we first fix the value of M introducing the 6-factor =

=

=

=

f(dz/z)z~_M

the integral being taken in the complex plane around the origin. Then one can do all the sums over v,~with the result exp{z(exp(Qx

—

1))/x].

(8)

The integration over z picks up the Mth term in the expansion of (8) in powers of z. The final summation over M leads to c~ = {1

+g)~,

(9)

where ~ (Qx)~/n!=exp(Qx—1)

(10)

n=1

and (1 + Z}a denotes the sum of the first [a + 1] terms of the expansion of (1 + z)a with positive binomial coefficients. To calculate the hh amplitude we assume that for a given number of strings i.’,, of type n the scattering matrix is a product of individual matrices s,~for all strings, which depend on their colour number n. The total S-matrix results as an average S(b)

=

>~~P(Pn)11Sj~”(b).

(11)

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The sums over z.’,~ can be computed by the same technique as when normalizing the probability. We finally get S(b)

=

(1 +S(r)(b)}//{1

(12)

+g}’~

with the definition of the “renormalized” elementary S-matrix s(T)(b)

=

~ (Qx)~s~(b)/n!

(13)

n=1

which effectively takes into account contributions of strings of all colours. As we have noted, eqs. (12) and (13) remain valid for the case of only one type of string with a self-interaction, when one should keep only the terms with n 1 in the sums over n in (10) and (13). Eq. (12) substitutes (3) for interacting strings in our probabilistic treatment. One immediately sees a big difference between (3) and (12). Both give a unitary S if s is unitary (by construction). However the eikonal formula (4), as is known, gives a unitary S-matrix even for a not unitary, in particular, positive imaginary and large, as in the case of the supercritical pomeron. It thus possesses the property of unitarizing a nonunitary input. A new formula (12) lacks this feature. Unitarity therefore imposes a constraint on the possible values of s~(b).If, as is natural to assume, the elementary amplitudes a~(b)are imaginary and positive and fall with b then one should have 0 < —ia(0) < 2. Actually the S-matrix (12) involves the renormalized amplitude a~(b) defined like (12) with the substitution s,~—f a~. Introducing the corresponding renormalized elementary cross section o(b) 2 Tm a~(b)the unitarity gives 0 ~ u(0) < 4g. Actually this condition is too weak: we shall see that physical considerations in fact require a stronger limitation o a(0) ~ g. Denoting o-(0) ~g one transforms it into the requirement =

=

~

=

(14) Thus form the start the model based on the distribution (6), involves, apart of x, one more parameter 4, which indicates the degree of saturation of the unitarity with the maximum value ~ 1. =

3. Cross sections Starting from (12) one can separate the amplitude first into contributions from a given number M of exchanged strings of any sort and then, following the AGK rules, into contributions from k cut and M k uncut strings. Fixing k and —

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summing over all possible numbers of uncut strings one arrives at the cross section 0k for the k-fold inelastic collision (with k cut strings of any sort): Uk(b)

=

uk(b)cik/{1

+g

—

ff(b))~~/{l

~g}I/X

(15)

We observe that the right-hand side of (15) is positive if (14) is true, which is the reason why this somewhat stronger condition is needed. The sum of all 0~k gives the total non-diffractive inelastic cross section am: k max

(16)

um(b)= Eo~k(b) k=~1

with kmax [1/x + 1]. If we sum ka-k instead, then according to the AGK rules the resulting cross section may be interpreted as the one corresponding to a single pomeron exchange at,. From (15) we get =

ff~(b)=ff(b){1

(17)

+g}1~~/x{1+g)’~.

This equation demonstrates the change introduced by the new distribution (6) coupled with the unitarity requirement in terms of the properties of the pomeron. Taking b 0 and remembering that u(O) we have =

=

=

(~g/x)(1+g}1~1/{1 +g}~.

(18)

With ~ ~ 1 and g positive (18) means that in the model o-~(O)can grow with y only as 1/x, that is as y, and never as exp Lly as for a free supercritical pomeron. In this sense the interaction introduced by the distribution (6) transforms the pomeron from a supercritical into a nearly critical one. However this phenomenon may be felt only at very high values of y. If, for comparatively low y, g is small then a-~(0) may mainly grow because of the growth of g, which may well be exponential. The observed cross sections are obtained from (15) upon integration over b. We assume that the b-dependence of o-(b) is gaussian a-(b)

cr(0) exp(—b2/R2),

(19)

where R2 is the standard radius squared of the pomeron distribution R2=R~+4a’y.

(20)

One get for the cross section with k inelastic collisions N+ 1)/{1 where Bf is the incomplete B-function.

~g}l/X

(21)

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One should note that the cross sections (21) actually do not distinguish between the two scenarios of string interactions discussed above: (i) only one type of strings between themselves or (ii) various types with the conservation of colour. For the same set of parameters x, ~ and u(0) they produce identical results. The situation changes as we pass to a more detailed information about the scattering process, such as the multiplicity or the expectation value (p~). Then the distribution of strings in the colour number starts to play its role. Let us study the multiplicity of produced particles as a function of their momenta ~i(p). If a string of colour number n has multiplicity ~~(p) then the total multiplicity is ~k

=

1,k2,...

k1,k2,...

(22)

~knPn/U’~,

n

where the sum goes over all values of k1, k2,... of cut strings with colour numbers 0kk 1, 2 To compute (22) we have to know 2 For fixed b this can be done if we separate in (15) contributions of cut strings of different types combined in the factor u’~(b).Present o-(b) in the form ~(b)=

~(Qx)~cr~(b)/n!~ n=I

~

(23)

n=1

where a,~is the original cross section for the string of type n. Taking then the kth power of (23), summing over all k’s and separating the term with exactly k~ powers of ~(r) we get 6)

=

(k !/

fl kn!) fl

(~(r)(b)

)kxCk

(1 + g

—

u(b)} 1/x_k/{

Ukj k2,(

1

+ g}

l/x

(24)

where k Ekn. To integrate over b we assume 2. Introducing ratios that all form (19) with the same R =

zn=o;~T)(O)/o.(o),

~

~(r)(b)

Ez~=1

have the gaussian (25)

we finally get for the integrated cross sections 0k 1,k2,

=(k!/flkfl!)flzf~ak.

(26)

Returning to the expression for the multiplicity, the substitution of (26) into (22) leads to a simple factorized expression I.L(P)=
(27)

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where K k) is the mean number of cut strings (inelastic interactions) (28)

2crp(0)/om.

Kk)’= ~kuk/am=7rR k

The second form follows from the AGK rules. In the same manner one can study other properties, such as Kp~). To have definite predictions one has to make some assumptions as to the behavior of ~ )~for a single cut string of type n. We take a>0

KPj)nICfl,

where gives

K2

(29)

is K p~) for the ordinary string. Then the calculation of the total

K~~) =K2EZflfla.

K P~L) (30)

n

According to the conclusions of the study of the structure of multicolour strings in ref. [5] a 1. Formulas for ~a or Kp~~) essentially depend on parameters z,~describing the relative strength of the interactions with different type of strings according to eq. (25). With only one type of strings fusing into themselves we evidently have z 1 -1 and all others equal to zero. However, with different types of strings existing, all z,~ are different from zero generally. We know very little about their values a priori and have to recur to some simple assumptions. The simplest assumption, which we indicate as Version 1, is that u~(O)factorizes in the spirit of eikonal models: =

=

a-~(0)=o~(O). The cross section a~1 is determined from the condition ~ for Z’s 1/g~n!, u=log(1+g~)

(31) =

1. We then obtain (32)

Z~=u’

so that all quantities are determined through already introduced g and It is also natural to suppose that in (27) p,,~coming from the string of type n, at least in the central region are n times greater than for the ordinary string. Then with eq. (31) we obtain ~.

p~(y=O)=p~ 1(y=0)Kk>f(y),

(33)

where f(y) =u/(1 —exp(—u))

(34)

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and u is determined by eq. (32). Thus the multiplicity for many types of strings results as the one for a single type multiplied by a factor. With g rising with energy this factor is growing as log g. As to K p~),with a 1 in (29) we have likewise =

Kp2~=K2f(y).

(35)

Its behavior with energy is completely determined by the same factor that appear in (33). A different assumption (version 2), which leads to a faster growth of u and K~~) with energy, implies that is it not o-~(0)but rather Un(0)/fl! that factorize: O•n(0)/fl!

=

o-~(0).

(36)

Again u 1(0) is determined by the condition Z~=u~/g~,

The calculations of ji(y (35) with a new function

=

EZ~ =

(37)

u=g~/(1+g~).

K~~) now

0) and

1 to give

lead to the same formulas (33) and

f: (38)

f(y)=1+g~.

This factor rises with energy linearly in g~,that is much faster than in version 1. Going over to numerical estimates, we note that the proposed model involves three input parameters: the elementary cross section cr(0), the interaction probability x and the parameter ~ showing the degree of the saturation of unitarity, all of them depending on the lab rapidity y. As compared to the standard supercritical pomeron model of ref. [4]a new parameter x is introduced, the parameter ~ assuming the role of the analogous parameter C in ref. [4].In our model ~ naturally appears as energy dependent. As mentioned in sect. 1, we choose x as the ratio of the pomeron proper transverse area as measured by the mean value of K p ~> and the total interaction area determined from o~ The elementary cross section has been parametrized in the same manner as the one-pomeron contribution 2 exp /.ly (39) u(0) =C(R0/R) with the proton radius squared R~ 3.56 GeV2 and R2 defined by (20) with a’ 0.25 (see ref. [4]). It contains two numerical parameters: the intercept (minus one) 4 and normalization C. The function ~(y)has been determined adjusting the calculated for pp2. collisions to the experimental ones extrapolated higher It is remarkable that the dependence of the crosstosections energies linearly inony 4 and C is quite weak. With ~ fixed at its unitarity limit, and multiplicities 1, and with the variations 4 0.12—0.32 and C 0.25—8.0 the corresponding lfl

=

=

0.1fl

=

=

=

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0.8

0.6

04

0.2

0

8

10

12

I 14

I 16

I 18

20 y

Fig. 1. Values off adjusted to fit the experimental data for o’~with two sets of parameters I and 2.

change in a”~ stays below 10% up to energies as high as y 20. Upon adjusting ~(y) to the experimental ~ this dependence practically vanishes altogether, so that the curves presented below in reality correspond to all 4’s and C’s in the above-mentioned intervals. The fit to o~m and the corresponding behavior of ~(y) are shown in figs. 1 and 2 for two different sets of C and 4: 4 0.32, C 0.5 (set 1) and 4 0.145, C 2.0 (set 2). These particular values of 4 and C are favoured in the framework of the multistring picture. As mentioned, the curves for the two sets of 4 and C (with ~ adjusted) coincide in the scale of the picture. For comparison the unitarity limit 1 for ~ is also shown in fig. 2, as well as the predictions of the standard supercritical pomeron model of ref. [4]. As one can see, with the choice ~ I (unitarity limit) the cross sections come out significantly larger than the experimental ones. The relative values of Uk are presented in fig. 3. The global collision characteristics presented in figs. 1—3 are insensitive to the scenario of the string interactions and coincide for one type of strings and for multicoloured strings. Figs. 4 and 5 show the calculated values for charged multiplicities for the case of one type of strings with a self-interaction and for the fusion of strings of different =

=

=

=

=

=

=

M Braun, C. Pajares / Interacting colour strings

552

.0

E ~-120

-

0

6 0’

100

-

~

2) with adjusted

Fig. 2. The total p~inelastic cross sections as a function of y = In s (GeV and in the pomeron model of ref. [41(SP).

f, with f

=

colours respectively. To see qualitative features of the model we limit ourselves to the spectrum at y 0 in the center-of-mass system where it can be found directly from the AGK rules as corresponding to a single pomeron exchange (17). To avoid using any specific mechanism of hadronization we have normalized the multiplicities to the experimental data at 546 GeV. The model with a single type of strings leads to values of the multiplicity at y 0 which grow with energy essentially weaker than the free pomeron model and the experiment. This is of course to be expected, as configurations with higher numbers of strings are damped as a result of string fusion, which restrains the growth of the multiplicity. We may conclude that the introduction of a strong pomeron interaction as implied by (5) is not supported by experimental data, which is not surprising, since it has long been known that the 3 pomeron interaction is quite weak. In the scenario with the fusion of strings of different colour the rise of the central multiplicities gets more pronounced due to the factor f(y) in (33). This same factor produces the growth with the energy ofKp~)shown in fig. 6. The effect of f(y) depends strongly on the choice of the intercept 4 and very weakly =

=

=

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Interacting colour strings

553

0

00’ .1 0

£ 0.8

-

0’

0.6

-

0.4

-

SF, 546 0eV

0.2

-

546 0eV 53 0eV SF, 53GeV

0 I

I

I

I 1.5

I

I

2

I 2.5

I 3

_____

3.5

____

4

4.5

5

Fig. 3. The relative cross sections o-~/o~ in our model and in the pomeron model of ref. [4](SP).

on C and ~ in (39). To achieve the experimentally observed variation of ~.t(y 0) and K p~) with energy we were compelled to choose a rather high value 4 0.32 (set 1) in version 1. In version 2 the value of the intercept is lowered to 4 0.145 (set 2). From figs. 5 and 6 one can observe that with multistrings taken into account the multiplicities at y 0 improve and seem to grow very much like experimental ones. Also K p~) grows nearly as y in accordance with experimental facts. In version 2 the growth of the multiplicity and K p~) is stronger than in version 1, which is to be expected as mentioned, although the difference begins to show up at very high energies (y’s of the order of or larger than 15). =

=

=

=

4. Discussion We have introduced a general framework which makes it possible to deal with interactions among strings in a consistent way, giving a qualitative description of the main observables like cross sections, multiplicities and average transverse momentum in agreement with the experimental data. Other observables like

M Braun, C. Pajares / Interacting colour strings

554 8 0 >‘ .0 C ~0

-

6-

5-

0

8

10

12

14

16

18

20

y

2) in our model with interacting Fig. 4. Central charged multiplicities as a function of y = In s (GeV strings of a single type and in the pomeron model of ref. [4] (SP).

differential cross sections, multiplicity and K~~) distributions in rapidity can also be studied in our formalism. However our main goal was not to achieve a detailed fit for all possible observables but just to explore the possibility of introducing strong interactions of strings without spoiling the overall agreement with the experimental data which show the independent string models like [1—3]and also to study effects of such interactions that could be detected or have already been detected. In fact the dependence on the energy of the inelastic cross sections (fig. 2), the inclusive cross sections (fig. 5) and the average transverse momentum (fig. 6) are well described in our approach in the case when the fusion of strings gives rise to new string-like objects. At hadron supercollider energies and still higher energies the multiplicity is getting larger than for the standard supercritical pomeron model (SP). In relation to fig. 5 we would like to comment that the shown result of SP, normalized at y 12.6, does not include semihard processes. It is known that these produce a rise of the inelastic cross sections and multiplicities beginning at y 6—8. Its effect would be to describe better the experimental data within SP, so

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555

Interacting colour strings

8 C >.

>7-

6-

2 5-

4SF 3-

2

0

I

I

8

10

I

I 12

I

14

I 16

18

20 y

2) in our model with interacting Fig. 5. Central charged multiplicities as a function of y = In s (GeV multistrings for two sets of parameters 1 and 2 and in the pomeron model of ref. [4](SP).

that the difference with our approach would become not so large as in fig. 5 (see, for instance, the results for the inclusive cross sections of the Monte Carlo code of ref. [8]). As to the average transverse momentum, we obtain its rise with the energy (fig. 6), as well as with the multiplicity. Usually this latter is explained by semihard interactions [8—101.In our approach it is a direct consequence of our assumption (29). The new objects formed from fusion of several ordinary strings fragment into particles with more transverse momentum. This is what one expects in the fragmentation of colour ropes [11]. Strings formed from the fusion of two ordinary strings have a double colour (in the abelian approximation) giving rise to particles with a double average transverse momentum in the fragmentation. Fusion of three ordinary strings likewise leads to a triple average transverse momentum and so on. These new objects will also produce heavy flavour particles more efficiently. We have not studied this point here. The proper way of doing it would be to use some model, like the Dual Parton Model (DPM), as an underlying structure for the usual cut strings and their fragmentation into any kind of particles. Weights of each diagram of the DPM and those which come from the fusion of strings should

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Interacting colour strings

o 1.2 > 0 0’ 0.

-

0.8

:.:

III~II~

2) in our model with

Fig. 6. The mean transverse momentum (p~ function of y = In 1 sand (GeV interacting multistrings for as twoa sets of parameters 2.

then be taken from our formalism. Some work in this direction [5]points out that certain experimental data like the large average transverse momentum of ~ reached at 1.8 TeV and high multiplicities can be explained by the fusion of strings. Models which incorporate semihard interactions but no string fusion are incapable of describing such a large K~~) [81. An increase of strange and in general heavy flavour production is also expected at high energies and multiplicities. In our model to the usual eikonal series of soft pomerons (ordinary strings), which now contains a finite number of terms, an infinite tower of similar series of new pomerons has been added, each new pomeron being harder than the previous one. In this way transition from the soft region to the hard one is accomplished. It is smooth and independent of any explicit cutoff which might separate the two regions. Recently some work has been done to extrapolate QCD from the hard to soft region [121. Our attempt is in some way complementary although it seems hardly possible to establish a direct correspondence between the both models. The =

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rise of K p~)~with energy obtained in QCD can be accommodated in our approach with a suitable choice of the energy dependence of g in (34) or (38), although it is not clear that is would be the correct one. In the usual supercritical pomeron theory the bare pomeron violates unitarity, which is restored due to the infinite eikonal series. In our approach this mechanism does not work, since the series becomes finite because of the limited transverse area available. The unitarity thus leads to an explicit constraints ~ ~ 1. With this constraints satisfied, the sum of all different pomerons becomes unitary even if the initial bare pomeron was not. In fact, such a sum Up, given by (18) at b 0, may well rise with y exponentally if g is small, in seeming violation of the unitary. However at asymptotic energies (18) turns into a-~(0) ~g/x(1 + g) and behaves like ~/x. In this sense, as mentioned, the supercritical pomeron is transformed into a nearly critical one. The model presented here is similar to a gas of different kinds of particles with the same transverse size, which live in a limited area and can fuse into each other with the conservation of their (abelian) colour as soon as they overlap. The average number of particles grows with energy and so does the total area available. We have made a number of simplifying assumption to make it possible to obtain explicit formulas. One approximation is to take the same size for all strings. This is not strictly true: the new harder objects should be smaller. This could be incorporated in our formalism by assuming x to be dependent of the difference N M in (6) and falling as this difference rises. Depending on the concrete form of this relation, our formulas would then be changed. However we think that our main results would remain valid, the only modification concerning the dependence of g on x. Another assumption made is the one given by (31) or (36), which relates the cross sections for different strings. Eqs. (31) or (36) are to our mind most natural and simple and also span an interval of possibilities large enough. Study of the reggeon field theory with different kinds of pomerons interacting in the same way as in this work might shed some light on this problem. Work in this direction is in progress. To conclude we want to comment on the quark—gluon plasma (QGP) formation in this formalism. It is difficult to see how it could arise within the independent string fragmentation models like DPM or Fritiof. The same holds for the models which extrapolate QCD to the low p ~ region. In our approach, on the contrary, there exists a simple possibility to describe the QGP formation as a gas—liquid phase transition in the gas of pomerons in the transverse space. However to do it one has to assume that the proper transverse area of a string rises with its colour, so that fusion produces objects with a larger dimension. Then the phase transition would convert the whole transverse area into a very thick string, which is precisely the QGP. In such a picture there is no place to account for the growth of K p~> with new strings. This does not look so bad, since this growth may, after all, be due to semihard processes. However the realization of the described picture of the =

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QGP formation is technically very difficult, since it requires x to be dependent on the sort of strings which fuse, which complicates the formalism enormously. It is a pleasure to thank A. Capella, G. Gustafson and A. Kaidalov for helpful and stimulated discussions. The authors are also grateful to the General Director of the Scientific and Technical Investigation (DGICYT) of Spain for financial support.

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