Multiplicity distribution of dipoles in QCD from Le, Mueller and Munier equation

In this paper we derived in QCD the BFKL linear, inhomogeneous equation for the factorial moments of multiplicity distribution($M_k$) from LMM equation. In particular, the equation for the average multiplicity of the color-singlet dipoles($N$) turns out to be the homogeneous BFKL while $M_k \propto N^k$ at small $x$. Second, using the diffusion approximation for the BFKL kernel we show that the factorial moments are equal to: $M_k=k!N( N-1)^{k-1}$ which leads to the multiplicity distribution:$ \frac{\sigma_n}{\sigma_{in}}=\frac{1}{N} ( \frac{N\,-\,1}{N})^{n - 1}$. We also suggest a procedure for finding corrections to this multiplicity distribution which will be useful for descriptions of the experimental data.


I. INTRODUCTION
During the past several years a robust relation between the principle features of high energy scattering and entanglement properties of the hadronic wave function have been in focus of the high energy and nuclear physics communities [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this paper, we continue to explore the relation between the entropy in the parton approach [20][21][22][23] and the entropy of entanglement in a proton wave function [5]. In Ref. [5], it is proposed that parton distributions can be defined in terms of the entropy of entanglement between the spatial region probed by deep inelastic scattering (DIS) and the rest of the proton. This approach leads to a simple relation S = ln N between the average number of color-singlet dipoles and the entropy of the produced hadronic state S. This simple relation shows that a proton becomes a maximally entangled state in the region of small Bjorken x. All these conclusions were made from estimates in the simple, even naive model for QCD cascade of color-singlet dipoles. However, it has been demonstrated in Refs. [6,10,11,16,19] that these ideas are in qualitative and, partly, in quantitative agreement with the available experimental data. Actually, it is shown in Ref. [5] that the simple cascade of color-singlet dipoles leads to the multiplicity distribution: (1) where N is the average number of dipoles. The goal of this paper is to study the multiplicity distribution and the entanglement entropy in the effective theory for QCD at high energies (see Ref. [24] for a general review). We have approached this problem in Refs. [5,18] and have demonstrated that Eq. (1) arises in QCD cascades. In this paper we analyze the multiplicity distribution for Balitsky-Kovchegov (BK) cascade [25] in which one dipole at low energy generates a large number color-singlet dipoles at high energy. The equation for such a cascade are known (see Refs. [24,[27][28][29] ) and the first try to solve them have been undertaken in Ref. [18]. However, in this paper we return to this problem and study the multiplicity distribution using the new equation (Le, Mueller and Munier (LMM) equation) for the probability generating function that has been derived in Ref. [30].
In the next section we derive LMM equation from the equation for the BK parton cascade. In the rest of the paper we discuss the equations for the factorial moments that follow from the LMM equation. We show that every factorial moment satisfies the linear but inhomogeneous equation with the Balitsky, Fadin, Kuraev and Lipatov (BFKL) kernel [31,32]. We attempt to solve these equations and demonstrate that in the diffusion approximation to the BFKL kernel factorial moments are equal to: We show that Eq. (2) leads to Eq. (1). In section VI we suggest an approach to go beyond diffusion approximation, which cannot give a reliable description of the experimental data even in the leading order of perturbative QCD. In this approach we propose to solve exactly the equations for the factorial moments and using the difference between the exact solution and Eq. (2) (∆M k = M k (exact) − M k (Eq. (2))) we develop the way how to estimate the multiplicity distributions beyond diffusion approximation. In conclusion section we summarize our results.

II. GENERAL FEATURES OF THE CASCADE OF COLOR-SINGLET DIPOLES IN QCD
In QCD at large number of colors N c (N c 1) the color-singlet dipoles play the role of partons (see Ref. [24] for review). As discussed in Refs. [24,[27][28][29] for them we can write the following equations: is the probability to have n-dipoles of size r i , at impact parameter b i and at rapidity is a typical cascade equation in which the first term describes the reduction of the probability to find n dipoles due to the possibility that one of n dipoles can decay into two dipoles of arbitrary sizes, while the second term describes the growth due to the splitting of (n − 1) dipoles into n dipoles.
The initial condition for the DIS scattering is which corresponds to the fact that we are discussing a dipole of definite size which develops the parton cascade. Since P n (Y ; {r i }) is the probability to find dipoles {r i }, we have the following sum rule i.e. the sum of all probabilities is equal to 1. This QCD cascade leads to Balitsky-Kovchegov (BK) equation [24,25] for the amplitude and gives the theoretical description of the DIS. We introduce the generating functional [27] where u (r i b i ) ≡ u i is an arbitrary function. The initial conditions of Eq. (4) and the sum rules of Eq. (5) take the following form for the functional Z: Multiplying both parts of Eq. (3) by n i=1 u (r i b i ) and integrating over r i and b i we obtain the following linear functional equation [29]; Searching for the solution of the form Z ([u(r i , b i , Y )]) for the initial conditions of Eq. (7a), Eq. (8a) can be re-written as the non-linear equation [27]: Therefore, the QCD parton cascade of Eq. (3) takes into account non-linear evolution.

III. DERIVATION OF LE, MUELLER AND MUNIER (LMM) EQUATION
In this section we derive the LMM equation which is proposed in Ref. [30]. First, we introduce the same notations as in Ref. [30]:w One can see thatw n (r, b, y) is the probability that the dipole with size r produces n dipoles with all possible sizes. Eq. (7b) reads as ∞ n=1w n (r, b, y) = 1 Taking all u i (r i , b i ) = λ one can see that we can re-write Eq. (6) in the form: Plugging Eq. (17) into Eq. (9) we obtain the LMM equation in the form: In addition, discussing multiplicity distribution P n = σn σtot , where σ n is the cross section for production of n colorsinglet dipoles, we need to integrate w n (r, b, y) over b. In this case the initial condition for the dipole cascade takes the form: which leads to the probabilities, that do not depend on impact parameters. Since in Eq. (3) b enters as a parameter, P n (Y ; r i ), which does not depend on b i , is also a solution to Eq. (3), which satisfies Eq. (14). Eq. (13) reduces to Eq. (15) is a particular case of the general equation that has been derived in Ref. [30]. In this paper instead of Eq. (10) the more general form of this equation is proposed, viz.: where S (r i , b i , y 0 ) is the scattering S-matrix for elastic interaction of the dipole with size r i at rapidity Y 0 (y 0 =ᾱ S Y 0 ) and at impact parameter b i with the target at Y = 0. Since S is a unitarity matrix, w n (r, b, y; y 0 ) is the probability that the dipole with size r produces n dipoles with all possible sizes, which interact with the target. Bearing this in mind, we see that Eq. (11) holds for w λ (r, b, y; y 0 ), which is defined as For the case of P n which do not depend on b i , inserting in Eq.
we see that we obtain the LMM equation in its original form (see Ref. [30]):

IV. AVERAGE NUMBER OF COLOR-SINGLET DIPOLES
The average number of dipoles can be calculated using the following formula: Differentiating Eq. (13) with respect to λ we obtain that Eq. (20) shows that the average number of dipoles satisfies the linear BFKL [31,32] equation and increases in the region of small x (large y). Therefore, we see that the general QCD cascade reproduces the main observation of Ref. [5] which was made in the oversimplified model for the QCD cascade. In this model the dependence on the size of the dipoles were neglected.
The general solution takes the following form: where ξ = ln 1 r 2 and χ(γ) is the BFKL kernel: where ψ(z) is the Euler ψ-function (see Ref. [36] formula 8.36). n in (γ) has to be found from the initial condition N (r, b, y = 0, y 0 = 0) = 1 (see Eq. (4) and Eq. (7a)). It gives Introducing multiplicity in the momentum representation: we can re-write Eq. (20) in the form: where K k T , k T is the BFKL kernel in momentum representation: Solution to this equation has the same form of Eq. (21) but with the replacement of ξ → ξ = ln k 2 T and Eq. (27) reproduces the value of N (k T , y = 0, y 0 = 0), since Eq. (24) at y = 0 leads to N (k T , y) = ln k 2 T + O 1/k 2 T . It is worth mentioning that (1) in the double log approximation n in (γ) = 1/γ 2 , which leads to N (k T , y) = ln k 2 T and (2) in semi-classical approximation, which we will use below, we can neglect all corrections of the order of 1/k 2 T .

V. EQUATIONS FOR MOMENTS OF THE MULTIPLICITY DISTRIBUTION
A. The second moment

Equation
We start a derivation of the evolution equation for the moments of the multiplicity distributions considering the second moment, which has the following form: Taking the second derivative with respect to λ from Eq. (18) we obtain the equation for M 2 (r, y, y 0 ) Eq. (29) is a linear but inhomogeneous equation with the inhomogeneous term, which is determined by the multiplicity of the dipoles.
The initial conditions (see Eq. (4) and Eq. (7a)) for this equation is First, lets us start to solve Eq. (29) making first iteration at small y = ∆y. For y = 0 N (r, y = 0) = 1 and M 2 (r, y = 0) = 0, and hence where ω is given by Eq. (8b). The first iteration of Eq. (20) leads to N (1) = 1 + ∆y ω G (r). Comparing these two estimates one can see that the first iteration can be written as the expansion of the solution M 2 (r, y) = 2 N 2 (r, y) − N (r, y) with respect to ∆ y. Hence, we see, that at small ∆y we obtain the simple expression for M 2 , which turns out to be the same as for the multiplicity distribution of Eq. (1) for the simple toy model [5]. We will try to prove this equation below, but we have succeeded only in the semi-classical approximation.

General solution
The general solution to Eq. (29) we can obtain going to momentum representation: Eq. (29) takes the form: Taking the Mellin transform: from both part of Eq. (33) we obtain: where χ (γ) is given by Eq. (8b) and n 2 denotes the Mellin image of N 2 (k T , b, y, y 0 ).
where m BFKL 2 (γ, y, y 0 ) is a solution to the homogeneous linear BFKL equation with the initial condition of Eq. (30). In the following, we neglect the contribution of this term.

Semi-classical solution
For large y and ξ we can use the semi-classical approximation( SCA, see Refs. [24,33] and references therein) to take the integral over y in Eq. (36). In this approximation we are searching for N = e S N = e ω(ξ ,y) y+γ(y,ξ )ξ (37) where ω (ξ , y) and γ (y, ξ ) are smooth functions of y and ξ : ∂ ω (ξ , y) /∂ y ω 2 (ξ , y) , ∂ ω (ξ , y) /∂ ξ ω (ξ , y) γ (y, ξ ) , ∂ γ (ξ , y) /∂ ξ γ 2 (ξ , y) , ∂ γ (ξ , y) /∂ ξ ω (ξ , y) γ (y, ξ ). Such form of N stems from Eq. (21) if we use the method of steepest descent for calculating the integral over γ. Indeed, using this method one can see that In the SCA the Mellin image of N 2 can be written as follows: Indeed, taking the integral by the method of steepest descent we obtain the following equation for the saddle point (γ with the solution γ (2) SP = 2 γ SP , where γ SP is given by Eq. (38). Plugging this solution into Eq. (39) we see that The integral over ∆γ leads to a smooth function, which in the SCA can be considered as a constant. Therefore, comparing Eq. (39) and Eq. (41) one can see that Eq. (39) is correct. We can derive Eq. (39) using a more general consideration. Actually, the expression for the Mellin transform of N 2 (ξ , y, y 0 ) is the convolution in γ of Mellin images of N , which has the following form: Taking the integral over γ using the method of steepest descent one can see that the equation for the saddle point has the following form: with the solution γ SP = 1 2 γ. Plugging this solution in Eq. (42) we reduce it to: which reproduces the Mellin transform of Eq. (39). Plugging in Eq. (36) n 2 = e 2χ( 1 2 γ) y H (γ) we can take the integral over y and the solution has the form: Note, that in Eq. (45) we neglected the contribution of m BFKL 2 (γ, y, y 0 ) in Eq. (36). Before fixing H (γ) we need to go back to coordinate representation. Indeed, in this representation we have simple initial conditions for M 2 of Eq. (30). Since all solutions are solutions of the linear equations and γ SP 1, we can replace ξ = ln k 2 T by ξ = ln 1 r 2 . Bearing this in mind, we can reduce the solution to the form: First, we note, that taking the integral over γ using the method of steepest descent, we reproduce Eq. (45) with the particular choice of H (γ) = χ (γ) /γ, which has been discussed in Eq. (27). Second, one can see that at y → 0 this solution coincides with Eq. (31).
In DLA this solution takes the form (see Eq. (22): However, it turns out that Eq. (47a) reproduces at least two terms of the expansion at small values of y of the following relation: Concluding, we see that Eq. (2) does not hold in the DLA, but Eq. (48) gives us some hope to find an approach in which it will be correct.

Diffusion approximation
Actually, the most adequate approach at high energies is the diffusion one (see Eq. (22)). Plugging in Eq. (46) the BFKL kernel in the form χ (γ) = ω 0 + D γ − 1 2 2 and taking the integral using the method of steepest descent we obtain the saddle point We can obtain the solution of Eq. (50) directly from the equation for M 2 (see Eq. (29)), if we note that the BFKL kernel has maxima at r → 0 and |r − r | → 0. In Fig. 1 we plot the term of Eq. (29) , which is proportional to N 2 : d 2 r K (r , r − r |r) 2 N (r , y, y 0 ) N (r − r , y, y 0 ) ∝ d 2 r K (r , r − r |r) ( r 2 (r − r ) We can see from Fig. 1, that I(τ ) has a maximum at τ = 1. Note, that in Eq. (51) we introduce γ = 1 2 , which corresponds to the DA, to estimates the value of this contribution.
It is worth noting that Eq. (48) follows from the multiplicity distribution, which is given by Eq. (1). We will concentrate our efforts on DA in our presentation below.

B. The third moment
The third moment can be found from the generating functionw λ (r, b, y, y 0 ) in the following way: Rewriting Eq. (56) in momentum representation(see Eq. (32)) we reduce this eqaution to the form: where we use lowcase letters denoting the moments in the momentum representation. The Mellin image of Eq. (56) has the form: where n 2 n is the Mellin image of m 2 (k T , y, y 0 ) N (k T , y, y 0 ) which has the general form: where n 2 (γ, y) is determined by (see Eq. (46)) Plugging Eq. (21) and Eq. (59) into Eq. (58) one can see that n 2 n (γ, y) is a sun of two terms. Taking the integral over γ in each of them in the saddle point approximation (see Eq. (42) -Eq. (44)) we obtain two equations for the saddle points: Hence, n 2 n (γ, y) takes the form: The solution does not depend on the form of H 2 (γ) , which we will specify below. Plugging this equation in Eq. (57) we obtain the solution: Going to the coordinate representation as was discussed above and choosing H 2 (γ) = χ (γ) /γ, which reproduces the correct initial conditions we obtain Using the method of steepest descent and neglecting contributions of the order of ξ 2 D y in all pre-exponential factors, we see that M 3 (ξ, y, y 0 ) = 6 N (ξ, y, y 0 ) (N (ξ, y, y 0 which is the same as for the multiplicity distribution of Eq. (1).

C. General approach
The equation for a general moment we can obtain by differentiating Eq. (18) with respect to λ. It has the simple form in the momentum representation (see Eq. (32)): As we have discussed, we can go back to coordinate representation in Eq. (66), since m k are solution to the linear equations, and, therefore, have the Mellin image In coordinate representation we have On the other hand, in SCA m k−i (k T , y, y 0 ) m i (k T , y, y 0 ) has the image in γ-representation, which is equal to Const exp kχ 1 k γ y m k−i,0 γ k m i,0 γ k H k (γ) (see Eq. (45) and Eq. (61)) 2 .
Using this image, one can see, that the coordinate representation for m k−i (k T , y, y 0 ) m k (k T , y, y 0 ) can be reduced to d 2 r K (r , r − r |r) M k−i (r , y, y 0 ) M i (r , y, y 0 ), which means, that H k (γ) = χ (γ) as we expect from Eq. (46) and Eq. (63).
Assuming that for all i ≤ k − 1 which follows from Eq. (1), we will prove that for i = k we have the same expression. Plugging Eq. (70) in Eq. (69) we get the inhomogeneous term in the form k! (k − 1) N 2 (N − 1) k−2 . In the following we will use that the Mellin image of N i (r, y, y 0 ) ( n i ) is equal to which can be derived using the method of steepest descent in the estimates of the integrals over γ's (see Eq. (42) for example).
In the double Mellin transform Eq. (69) takes the form: Hence from Eq. (72) we have: In Eq. (73) we calculate the integrals over ω closing the contour of integration on poles. Taking the integral over γ using the method of steepest descent in the diffusion approximation and neglecting the corrections of the order of ξ /Dy (see discussions above) we reduce Eq. (73) to the following expression: Since we have obtained Eq. (70) for i = 2 and i = 3 , we prove this equation for any value of i.

VI. FINDING CORRECTIONS AND COMPARISON WITH EXPERIMENTS
Eq. (70) generates the multiplicity distribution of Eq. (1). However, several questions arise, when we wish to compare this distribution with the experimental data, let say with DIS. The average multiplicity of the color-singlet dipoles is equal to the sea quark structure function xΣ sea x, Q 2 [19]. On the other hand, in Eq. (70) the multiplicity enters in the coordinate representation. In diffraction approximation the momentum and coordinate representations are related by the replacement ln k 2 T → − ln r 2 . Therefore, my suggestion is to use Eq. (1) with N = xΣ sea x, Q 2 but to calculate the corrections to this distribution. The multiplicity distribution can generally be written, using the cumulant generating function f (λ) as follows [34,35]: where the contour of integration is the circle around the point λ = 0 and f (λ) is the cumulant generating function, which is defined as where κ n are cumulants. Generally speaking, we have the following definition for the cumulants: where M k are the factorial moments, that we have discussed above (see Eq. (70)). In our case we can view f (λ) as a sum of f (λ) = f Eq. (1) (λ) + ∆f (λ). f Eq. (1) (λ) generates the multiplicity distributions of Eq. (1), which includes the most dominant contributions and ,in particular, the average number of color-singlet dipoles is taken into account exactly. We suggest to introduce function ∆f (λ) in the following way: one can see that the resulting multiplicity distribution takes the form σ n σ in = n k=0 n! (n − k)! k! P n−k (N ) P k (N ) In the case of ∆κ 2 = 0 but ∆κ n = 0 for n > 2 the distribution P n (N ) has been found in Ref. [34] and it has the following form in our notations: where H n is Hermite polynomial (see formula 8.95 in Ref. [36]).

VII. CONCLUSIONS
This paper has two main results. First, we derived the BFKL linear, inhomogeneous equation for the factorial moments of multiplicity distribution (M k ) from LMM equation. In particular, the equation for the average multiplicity of the color-singlet (N ) turns out to be the homogeneous BFKL equation which leads to the power-like growth in the region of small x. From these equations it follows that M k ∝ N k at small x.
Second, using the diffusion approximation for the BFKL kernel, which is generally considered to be responsible for the small x behaviour, we show that the factorial moments satisfy Eq. (2), which reproduces the multiplicity distribution of Eq. (1). This result is in agreement with the attempts [18] to find solutions to the equations for the cascade of color-singlet dipoles (see Eq. (3).
We also suggest a procedure for finding corrections to this multiplicity distribution, which, we believe, will be useful for descriptions of the experimental data.
In general, the multiplicity distribution, that has been discussed in the paper, confirms the result of Ref. [5], that the entropy of color-singlet dipoles is equal S = ln N in the region of small x, and gives the regular procedure to estimate corrections to this formula.