Long range rapidity correlations in soft interaction at high energies.

# Long range rapidity correlations in soft interaction at high energies.

## Abstract:

In this paper we take the next step (following the successful description of inclusive hadron production) in describing the structure of the bias events without the aid of Monte Carlo codes. Two new results are presented :(i) a method for calculating the two particle correlation functions in the BFKL Pomeron calculus in zero transverse dimension; and (ii) an estimation of the values of these correlations in a model of soft interactions. Comparison with the multiplicity data at the LHC is given.

Soft Pomeron, BFKL Pomeron, Diffractive Cross Sections, Survival Probability
12

## 1 Introduction

The goal of this paper is twofold: to consider the two hadron long range rapidity correlations in the BFKL Pomeron Calculus in zero transverse dimensions; and to calculate these correlations in a model of soft interactions at high energy. The BFKL Pomeron Calculus in zero transverse dimension describes the interaction of the Pomerons through the triple Pomeron vertex () with a Pomeron intercept and a Pomeron slope . The theory that includes all these ingredients can be formulated in a functional integral form [1]:

 Z[Φ,Φ+]=∫DΦDΦ+eSwithS=S0+SI+SE, (1.1)

where, describes free Pomerons, corresponds to their mutual interaction and relates to the interaction with the external sources (target and projectile). Since , has the form

 S0=∫dYΦ+(Y){−ddY+Δ}Φ(Y). (1.2)

includes only triple Pomeron interactions and has the form

 SI=G3IP∫dY{Φ(Y)Φ+(Y)Φ+(Y)+h.c.}. (1.3)

For we have local interactions both in rapidity and in impact parameter space,

 SE=−∫dY2∑i=1{Φ(Y)gi(b)+Φ+(Y)gi(b)}, (1.4)

where, stands for the interaction vertex with the hadrons at fixed .

At the moment this theory has two facets. First, it is a toy-model describing the interaction of the BFKL Pomerons in QCD. Many problems can be solved analytically in this simple model leading to a set of possible scenarios for the solution in BFKL Pomeron calculus[1, 2, 3, 4, 5, 6, 7]. Our first goal is to find an analytical solution for the correlation function in rapidity defined as

 R(y1,y2)=1σind2σdy1dy21σindσdy11σindσdy2, (1.5)

where, , and are inelastic, double and single inclusive cross sections. We consider this problem as the most natural starting point to search for a solution for , in a more general and more difficult approach based on high density QCD.

On the other hand, recent experience in building models for high energy scattering [8, 9, 10, 11, 12, 13] shows that a Pomeron with can describe the experimental data including that at the LHC. It also appears in N=4 SYM [14, 15, 16, 17, 18] with a large coupling, which at the moment, is the only theory that allows one to treat the strong interaction on a theoretical basis. Therefore, our second goal is to evaluate the correlation function in our model for soft high energy interactions (see [8, 9, 10]).

## 2 Correlation function in the BFKL Pomeron Calculus in zero transverse dimensions

### 2.1 General approach

It is well known[19] that the most appropriate framework to discuss the inclusive processes has been developed by A.H. Mueller[20] (Mueller diagrams). In Fig. 1 we show the most general Mueller diagram for the double inclusive cross section (see also Fig. 2). From Fig. 1-a one can see that it is necessary to calculate the amplitudes of the cut Pomeron interaction with the hadrons, denoted by and .

The fact that we can reduce the calculation of the double inclusive production to an evaluation of and stems from the AGK cutting rules[21] which state that the exchanges of the Pomerons from the top to the bottom of the Mueller diagram cancel each other leading to the general structure of Fig. 1-a. Recall that the AGK cutting rules are violated in QCD due to the emission diagrams from the triple Pomeron vertex (see Fig. 1-b) (see Ref. [22]). In our treatment we neglect such a violation since turns out to be smaller at high energy than . Indeed, in the first approximation while and at large values of .

Analyzing the diagrams one can see that their contributions are proportional to two parameters which are large at high energy:

 L(Y)=g(b)G3IPΔeΔY;andT(Y)=G23IPΔ2eΔY. (2.6)

Note that in the dominator stems from the integration over internal rapidities of the triple Pomeron vertices.

We consider the first three diagrams (see Fig. 3) for (see Fig. 1) to illustrate how these two parameters appear in the calculations. For the diagrams of Fig. 3-a, Fig. 3-b and Fig. 3-c we have, respectively,

 A(Pomeron) = g(b)eΔ(Y−y); (2.7) A('fan' diagram) = −g2(b)G3IP∫Y0dy′e2Δ(Y−y′)eΔy′ (2.8) = −g(b)eΔ(Y−y)(L(Y−y)−g(b)G3IPΔ) Y−y≫1−−−−−→ −A(Pomeron)L(Y−y); A(enhanced diagram) = −g2(b)G23IP∫Y0dy′∫y′0dy′′eΔ(Y−y′)e2Δ(y′−y′′)eΔy′′ (2.9) = −g(b)eΔ(Y−y)(T(Y−y)−g(b)G23IPΔ2(1+Δ(Y−y)) Y−y≫1−−−−−→ −A(Pomeron)T(Y−y);

At high energy both and and we can neglect other contributions in each diagram. In this kinematic region each Pomeron diagram is proportional to powers of and . Therefore, the first approximation is to sum the largest contributions at high energies in every Pomeron diagram. Such an approach to high energy scattering was proposed by Mueller, Patel, Salam and Iancu (MPSI approximation[23]). It turns out that the value of is rather small (see discussion below). Based on this fact we propose that the leading approximation shall be to sum all contributions proportional to having in mind the following kinematic region:

 L(Y−y)≥1;T(Y−y)≪1;g(b)≪1;G3IP≪1. (2.10)

For the scattering with nuclei , and in this region which covers all reasonable energies, the main contribution emanates from ’fan’ diagrams (see Fig. 4 and Fig. 3- b for the first diagram of this kind). The expression for is known [24, 25]:

 Missing or unrecognized delimiter for \right (2.11)

As we shall see below the factor 2 stems from the initial cut Pomeron. Below we shall obtain these expressions using a more general technique in which we find the sum of the diagrams in a more general kinematic region:

 L(Y−y)≥1;  T(Y−y)≥1;   g(b)≪1;  G3IP≪1, (2.12)

selecting contributions of the order of . i.e. we shall find the scattering amplitude in the kinematic region of Eq. (2.12) using MPSI approximation.

The most important diagrams for are shown in Fig. 3-a. One can see that the kinematic region of Eq. (2.10) is:

 N(Y−y1,Y−y2)=Γ(Y−y1)Γ(Y−y2). (2.13)

### 2.2 Generating function approach

We believe that the method of a generating function (functional) is the most appropriate method for summing Pomeron diagrams. In the MPSI approach, one can explicitly see the conservation of probability (unitarity constraints) in each step of the evolution in rapidity. This method was proposed by Mueller in Ref.[4] and has been developed in a number of publications(see Ref.[26] and references therein). In Ref.[27] it was generalized to account for the contribution to the inelastic processes by summing both cut and uncut Pomeron contributions. For completeness of the presentation, in this section we shall discuss the main features of this method, referring to Refs.[27, 9, 10] for essential details. Following Ref.[27], we introduce the generating function

 Z(w,¯w,v|Y)=∑k=0∑l=0∑m=0P(k,l,m|Y)wk¯wlvm, (2.14)

where, stands for the probability to find uncut Pomerons in the amplitude, uncut Pomerons in the conjugate amplitude and cut Pomerons at some rapidity . and are independent variables. Restricting ourselves by taking into account only a Pomeron splitting into two Pomerons, we can write the following simple evolution equation:

 ∂Z∂Y=−Δ{w(1−w)∂Z∂w−¯w(1−¯w)∂Z∂¯w}−Δ{2w¯w−2wv−2¯wv+v2+v)∂Z∂v}. (2.15)

Fig. 5 illustrates the two steps of evolution in rapidity for . The general solution to Eq. (2.15) has the form

 C1Z(w)+C1Z(¯w)+C2Z(w+¯w−v), (2.16)

where, and are constants and is the solution to the equation:

 ∂Z∂Y=−Δξ(1−ξ)∂Z∂ξ. (2.17)

The particular form of and the values of are determined by the initial condition at .

### 2.3 Amplitude in the MPSI approach: general formula

The general formula for the amplitude in the MPSI approach has the form (see Ref.[27])

 NMPSI(γ,γin|Y) = (exp⎧⎨⎩−γ∂∂γ(1)∂∂γ(2)−γ∂∂¯γ(1)∂∂¯γ(2)+γin∂∂γ(1)in∂∂γ(2)in⎫⎬⎭−1) (2.18) Z(γ(1),¯γ(1),γ(1)in|Y−Y′)Z(γ(2),¯γ(2),γ(2)in|Y′)|γ(i)=¯γ(i)=γ(i)in=0,

where, , and .

Eq. (2.18) has a very simple meaning which is clear from Fig. 6. The derivatives of the generating functional determine the probability to have cut and uncut Pomerons at , while the derivatives of lead to the probabilities of the creation of cut and uncut Pomerons from two initial cut Pomerons at rapidity . Two uncut Pomerons interact with the amplitude at rapidity and with the amplitude in the case of cut Pomerons. The phases of the amplitude are given by related signs in Eq. (2.18): minus for and plus for . In addition, we assume that the low energy at which the wee partons from two Pomerons interact is large enough to assume that and are purely imaginary. We denote the imaginary part of the amplitude, by ’s. It follows from the AGK cutting rules that

 γin=2γ. (2.19)

According to Eq. (2.18), the contribution to the scattering amplitude of one Pomeron exchange is equal to 3

 ~geΔ(Y−Y′)γeΔY′~g. (2.20)

For the first ’fan’ diagram, Eq. (2.18) leads to the following contribution:

 ~g∫YY′dy′eΔ(Y−y′)Δe2Δ(y′−Y′)γ2,e2Δ(Y′)~g2, (2.21)

while the first enhanced diagram can be written as

 ~g∫YY′dy′eΔ(Y−y′)Δe2Δ(y′−Y′)γ2∫Y′0dy′′e2Δ(Y′−y′′)ΔeΔy′′~g. (2.22)

Comparing these expressions with the Pomeron diagrams (see Eq. (2.7),Eq. (2.8) and Eq. (2.9)), we have the correspondence between these two approaches,

 ~g=g/√γ;γ=G23IPΔ2. (2.23)

### 2.4 MPSI approximation: instructive examples

#### Glauber-Gribov formula

The pattern of calculation of Glauber-Gribov rescatterings due to Pomeron exchanges is shown in Fig. 7-a. The forms of the generating functions and are simple,

 Z(γ(1),¯γ(1),γ(1)in|Y−Y′) = e~geΔ(Y−Y′)(w(1)+¯w(1)−v(1)−1)=e~geΔ(Y−Y′)(γ(1)+¯γ(1)−γ(1)in); (2.24) Z(γ(2),¯γ(2),γ(2)in|Y′) = e~geΔ(Y′)(w(2)+¯w(2)−v(2)−3)=e~geΔ(Y−Y′)(γ(2)+¯γ(2)−γ(2)in). (2.25)

These generating functions describe the independent (without correlations) interaction of Pomerons with the target and the projectile. In the case of nuclei, Pomerons interact with different nucleons in the nucleus, and the correlations between nucleons in the wave function of the nucleus are neglected. Note that Eq. (2.18) with ’s from Eq. (2.24) and Eq. (2.25) do not depend on the sign of (). However, we shall see below that the choice of the above equation is correct since it reproduces Eq. (2.11), which has been derived by summing the Pomeron diagrams.

Using Eq. (2.18), we can calculate the inelastic cross section requiring that at rapidity we have at least one cut Pomeron (one ). The result is:

 σin=1−e−γin~g2eΔY=1−e−2g2eΔY. (2.26)

which reproduces the well known expression for the inelastic cross section in the Glauber-Gribov approach.

We can also calculate the contribution which has no cut Pomeron at rapidity (elastic cross sections). It has the form

 Missing or unrecognized delimiter for \Big (2.27)

The total cross section is given by:

 σtot=σel+σin=2(1−e−g2eΔY). (2.28)

#### Summing ’fan’ diagrams

As one can see from Fig. 7-b, the form of is the same as in the previous problem. It is given by Eq. (2.24). To obtain an expression for , we need to find ’s and in Eq. (2.16) with the initial condition

 Z(γ(2),¯γ(2),γ(2)in|Y′=0)=v. (2.29)

The resulting solution is of the form (see more details in Ref.[27])

 Z(w,¯w,v;Y′)=Zel(w,¯w;Y′)+Zin(w,¯w,v;Y′); (2.30) Zel(w,¯w;Y′)=we−ΔY′1+w(e−ΔY′−1)+¯we−ΔY′1+¯w(e−ΔY′−1)−(w+¯w)e−ΔY′1+(w+¯w)(e−ΔY′−1); (2.31) Zin(w,¯w,v;Y′)=(w+¯w)e−ΔY′1+(w+¯w)(e−ΔY′−1)−(w+¯w−v)e−ΔY′1+(w+¯w−v)(e−ΔY′−1). (2.32)

Substituting for in Eq. (2.18) we obtain for the inelastic part of (see Fig. 1-b),

 Γin(Y−y)=2~gγeΔ(Y−y)1+2~gγeΔ(Y−y)=2L(Y−y)1+2L(Y−y). (2.33)

Eq. (2.33) has been derived from the direct summation of the Pomeron diagrams in Ref.[25]. The fact that we reproduce the results of Ref.[25] , vindicates our choice of the generating functions in Eq. (2.24) and Eq. (2.25).

Using we obtain the elastic contribution which is intimately related to the processes of diffraction production:

 Missing or unrecognized delimiter for \left (2.34)

The resulting is given by:

 Γ(Y−y)=2L(Y−y)1+L(Y−y). (2.35)

Actually Eq. (2.35) gives the same expression as Eq. (2.11). The difference in an extra factor, , stems from the fact that, we need to take rather than in the vertex for the Pomeron-hadron interaction.

#### Single inclusive production in MPSI approximation

As one can see from Fig. 1-c, to evaluate the single inclusive cross section, we need to calculate . We have done so in the previous section, however, we now want to take into account both and contributions. From Fig. 7-c we see that has the form given in Eq. (2.30). However, in , we need to take into account that each Pomeron at , creates a cascade of Pomerons that is described by Eq. (2.15). In other words, we need to replace , and in Eq. (2.24) by

 w(1)→w(1)e−Δ(Y−Y′)1+w(1)(e−Δ(Y−Y′)−1);                                  ¯w(1)→¯w(1)e−Δ(Y−Y′)1+¯w(1)(e−Δ(Y−Y′)−1); (2.36)
 v(1)→w(1)e−Δ(Y−Y′)1+w(1)(e−Δ(Y−Y′)−1)+¯w(1)e−Δ(Y−Y′)1+¯w(1)(e−Δ(Y−Y′)−1)−(w(1)+¯w(1)−v(1))e−Δ(Y−Y′)1+(w+¯w−v)(e−Δ(Y−Y′−1). (2.37)

Using these substitutions we obtain

 Z(w(1),¯w(1),v(1);Y−Y′)=exp((w(1)+¯w(1)−v(1))e−Δ(Y−Y′)1+(w+¯w−v)(e−Δ(Y−Y′−1)). (2.38)

Using the generating function for Laguerre polynomials (see Ref.[29] formula 8.973(1)),

 (1−z)−α−1exp(xzz−1)=∞∑n=0Lαn(x)zn. (2.39)

We obtain for Eq. (2.38)

 Z(w(1),¯w(1),v(1);Y′)=−∞∑n=0L−1n(~gi)(−(γ(1)+¯γ(1)−γ(1)in)eΔ(Y−Y′))n. (2.40)

From Eq. (2.18) using

 ∂l∂lγ(1)∂m∂m¯γ(1)∂n−l−m∂n−l−mγ(1)in(γ(1)+¯γ(1)−γ(1)in)n=(−1)n−l−mn!. (2.41)

We obtain

 Γ(Y−y)=∞∑n=1L−1n(~gi)n!(−γeΔY)n=∞∑n=1L−1n(~gi)n!(−1)nTn(Y−y). (2.42)

Introducing we reduce Eq. (2.42) to the form

 Γ(L(Y−y),T(Y−y))=∫∞0dξe−ξ⎛⎜⎝e−ξ~gγeΔ(Y−y)1+ξγeΔ(Y−y)−1⎞⎟⎠=∫∞0dξe−ξ(e−ξL(Y−y)1+ξT(Y−y)−1). (2.43)

Using Eq. (2.43) we obtain the following result for the single inclusive cross section:

 dσdy=aIPΓ(L(Y−y),T(Y−y))Γ(L(y),T(y)), (2.44)

where, denotes the vertex of emission of the hadron from Pomeron (see Fig. 1 and Fig. 2-b).

### 2.5 The Correlation function in MPSI approximation

Calculating (see Fig. 1-a) we use , given by Eq. (2.38), as one can see from Fig. 6. However, is different from the expression which has been used in the calculation of the single inclusive cross section, and it can be written as:

 Z(γ(2),¯γ(2),γ(2)in|Y′−y1,Y′−y2)= (2.45) Z({Eq.~{}(???)}|γ(2),¯γ(2),γ(2)in|Y−y1)Z({Eq.~{}(???)}|γ(2),¯γ(2),γ(2)in|Y−y2).

First, we calculate at . Using Eq. (2.40) and Eq. (2.41) we obtain from Eq. (2.18) that

 N(Y−y1,Y−y1) = ∞∑n=1L−1n(~gi)n!(n−1)(−1)nTn(Y−y) (2.46) = T2ddT(1/T){∞∑n=1L−1n(~gi)n!(−1)nTn(Y−y)} = ∫∞0dξe−ξ{1+e−ξL(Y−y1)1+ξT(Y−y1)(−1−ξT(Y−y1)−ξL(Y−y1)(1+ξT(Y−y1))2)}. (2.47)

At we expand Eq. (2.47) to estimate the importance of the the correction depending on . The first four terms are given by:

 N(L(Y−y1),T(Y−y1);L(Y−y1),T(Y−y1))=L2(Y−y1)(1+L(Y−y1))2 (2.48) Missing or unrecognized delimiter for \right =L2(Y−y1)(1+L(Y−y1))2−2∑n=2n!Ln(Y−y1)Tn−1(Y−y1)(1+L(Y−y1))2n. (2.49)

Note that all corrections have minus signs and the function of Eq. (2.47) gives the analytical summation of the asymptotic series of Eq. (2.48). For we have a more complex answer, namely,

 N(L(Y−y1),T(Y−y1);L(Y−y2),T(Y−y2))= (2.50) Missing or unrecognized delimiter for \left

The double inclusive cross section can be written as (see Fig. 1-a)

 d2σdy1dy2= (2.51) a2IPN(L(Y−y1),T(Y−y1);L(Y−y1),T(Y−y1))N(L(y1),T(y1);L(y1),T(y1)).

## 3 Correlations in a model for soft interactions

Recently considerable progress has been achieved in building models for soft scattering at high energies[8, 9, 10, 11, 12, 13]. The main ingredient of these models is the soft Pomeron with a relatively large intercept and exceedingly small slope . Such a Pomeron appears in N=4 SYM [14, 15, 16, 17, 18] with a large coupling. This is, at present , is the only theory that allows us to treat the strong interaction on the theoretical basis. Having , the Pomeron in these models has a natural matching with the hard Pomeron that occurs in perturbative QCD. Therefore, these models could be a first step in building a selfconsistent theoretical description of the soft interaction at high energy, in spite of its many phenomenological parameters (of the order of 10-15) in every model.

In this section we shall discuss the size of the correlation function in our model[8, 9, 10]. This model describes the LHC data (see Refs.[30, 31, 32, 33]), including the single inclusive cross section. Thus our next step is to try, to understand the predicted size of the long range rapidity correlations in this model.

### 3.1 Estimates of the rapidity correlation function

In Table 1 we present the main parameters of our model. The parameter is small in our model reaching about 0.3 at the LHC energies. However, is large (see Ref.[8, 9]).

 gi(b)=giSi(b)=gi4πm3ibK1(mib). (3.52)

One can see that is as large as 25 at . Therefore, we can evaluate the influence of the corrections with respect to , by calculating the contributions of two diagrams: Fig. 4-a (the main contribution) and Fig. 4 - b (the corrections ).

We need to use the first two terms of Eq. (2.48) to calculate while being careful to account for the correct dependence.

Introducing two functions,

 Γ(1)(Li(Y−y;b))=ΔIPLi(Y−y;b)1+Li(Y−y;b);   Γ(2)(Li(Y−y;b))=ΔIPLi(Y−y;b)(1+Li(Y−y;b))2. (3.53)

We can see that Fig. 4-a has the following contributions:

 d2σ(0)dy1dy2 = ∫d2b{∫