The BFKL Pomeron Calculus in the dipole approach
In this paper we continue to pursue a goal of finding an effective theory for high energy interaction in QCD based on the colour dipole approach, for which the BFKL Pomeron Calculus gives a low energy limit. The key problem, that we try to solve in this paper is the probabilistic interpretation of the BFKL Pomeron Calculus in terms of the colourless dipoles and their interactions. We demonstrate that the BFKL Pomeron Calculus has two equivalent descriptions : (i) one is the generating functional which gives a clear probabilistic interpretation of the processes of high energy scattering and also provides a Hamiltonian-like description of the system of interacting dipoles; (ii) the second is the Langevin equation with a specific noise term which is rather complicated. We found that at high energies this Langevin equation can be reduced to the Langevin equation for directed percolation in the momentum space if the impact parameter is large, namely, , where is the transverse momentum of a dipole. Unfortunately, this simplified form of Langevin equation is not applicable for summation of Pomeron loops, where one integrates over all possible values of impact parameter. We show that the BFKL Pomeron calculus with two vertices ( splitting and merging of Pomerons ) can be interpreted as a system of colourless dipoles with two processes: the decay of one dipole into two and the merging of two dipoles into one dipole. However, a number of assumptions we have to make on the way to simplify the noise term in the Langevin equation and/or to apply the probabilistic interpretation, therefore, we can consider both of these approaches in the present form only as the QCD motivated models.
July 30, 2019
The simplest approach that we can propose for high energy interaction is based [1, 2] on the BFKL Pomeron  and reggeon-like diagram technique for the BFKL Pomeron interactions [4, 5, 6, 7]. This technique, which is a generalization of Gribov Reggeon Calculus , can be written in the elegant form of the functional integral (see  and the next section). It is a challenge to solve this theory in QCD finding the high energy asymptotic behaviour. However, even this simple approach has not been solved during three decades of attempts by the high energy community. This failure stimulates a search for deeper understanding of physics which is behind the BFKL Pomeron Calculus. On the other hand, it has been known for three decades that Gribov Reggeon Calculus has intrinsic difficulties  that are related to the overlapping of Pomerons. Indeed, due to this overlapping we have no hope that the Gribov Reggeon Calculus could be correct in describing the ultra high energy asymptotic behaviour of the amplitude. The way out of these difficulties we see in searching for a new approach which will coincide with the BFKL Pomeron Calculus at high, but not very high, energies (our correspondence principle) but it will be different in the region of ultra high energies. In a spirit of the parton approach we believe that this effective theory should be based on the interaction of ‘wee’ partons. We consider, as an important step in this direction, the observation that has been made at the end of the Reggeon era [10, 11, 12], that the Reggeon Calculus can be reduced to the Markov process  for the probability of finding a given number of Pomerons at fixed rapidity . Such an interpretation, if it would be reasonable in QCD, can be useful, since it allows us to use powerful methods of statistical physics in our search of the solution.
The logic and scheme of our approach looks as follows. The first step is the Leading Log (1/x) Approximation (LLA) of perturbative QCD in which we sum all contributions of the order of . In the LLA we consider such high energies that
It is well known that the LLA approach generates the BFKL Pomeron (see  and the next section) which leads to the power-like increase of the scattering amplitude ( with ).
The second step is the BFKL Pomeron Calculus in which we sum all contributions of the order of
The structure of this approach as well as its parameter has been understood before QCD  and was confirmed in QCD (see [1, 2, 4, 5, 6, 7, 15, 16]). This calculus extends the region of energies from of LLA to . The BFKL Pomeron Calculus describes correctly the scattering process in the region of energy:
For higher energies the corrections of the order of should be taken into account making all calculations very complicated.
Our credo is that we will be able to describe the high energy processes outside of the region of Eq. (1.3), if we could find an effective theory which describes the BFKL Pomeron calculus in the kinematic region given by Eq. (1.3), but based on the microscopic degrees of freedom and not on the BFKL Pomeron. In so doing, we hope that we can avoid all intrinsic difficulties of the BFKL Pomeron calculus and build an approximation that will be in an agreement with all general theorems like the Froissart bounds and so on. Solving this theory, we can create a basis for moving forward considering all corrections to this theory due to higher orders in contributions, running QCD coupling and others.
The goal of this paper is to consider the key problem: the probabilistic interpretation of the BFKL Pomeron Calculus based on the idea that colourless dipoles are the correct degrees of freedom in high energy QCD . We believe that colourless dipoles and their interaction will lead to a future theory at high energies which will have the BFKL Pomeron Calculus as the low energy limit (see Eq. (1.3)) and which will allow us to avoid all difficulties of dealing with BFKL Pomerons at ultra high energies.
Colourless dipoles play two different roles in our approach. First, they are partons (‘wee’ partons) for the BFKL Pomeron. This role is not related to the large approximation and, in principle, we can always speak about probability to find a definite number of dipoles instead of defining the probability to find a number of the BFKL Pomerons. The second role of the colour dipoles is that at high energies we can interpret the vertices of Pomeron merging and splitting in terms of probability for two dipoles to merge into one dipole and of probability for decay of one dipole into two ones. It was shown in  that splitting can be described as the process of the dipole decay into two dipoles. However, the relation between the Pomeron merging () and the process of merging of two dipoles into one dipole is not so obvious and it will be discussed here.
This paper is a next step in our programme of searching the simplest but correct approach to high energy scattering in QCD in which we continue the line of thinking presented in [18, 19, 20, 21]. The outline of the paper looks as follows.
In the next section we will discuss the BFKL Pomeron Calculus in the elegant form of the functional integral, suggested by M. Braun about five years ago . In the framework of this approach we find a set of equations for the amplitude of -dipole interaction with the target. We show that the recent intensive work on this subject [22, 23, 20] confirms the BFKL Pomeron Calculus in spite of the fact that these attempts were based on slightly different assumptions.
In section three we demonstrate statistical interpretation of the theory with interacting Pomerons. The one-to-one correspondence between BFKL Pomeron calculus and Langevin theory is found by showing how full Lagrangian generates stochastic Langevin equation with a peculiar noise term. In toy model with zero transverse dimensions this noise is reduced to one typical for directed percolation. Unfortunately the complexity of the noise term restricts the practical use of Langevin equation in this form, and one should look for further simplifications. One of them is to assume that impact parameter is much larger than any dipole size in the system (see Eq. (3.58)). Using this assumption and going to momentum space we rewrite our theory in the form of Langevin equation with a noise term proportional to the field (directed percolation universality class). It should be mentioned that large impact parameter approximation is unapplicable for summation of Pomeron loops, where we integrate over all possible values of the impact parameter.
Next, we discuss an approach based on generating functional. We show the equivalence between generating functional approach and the BFKL Pomeron calculus in the kinematical region Eq. (1.3) that leads to a clear interpretation of the BFKL Pomeron calculus as an alternative description of the system of interacting colourless dipoles. The interrelation between vertices of the Pomeron interactions and the microscopic dipole processes is considered. It is instructive to notice that the generating functional approach leads to a feedback to the BFKL Pomeron Calculus restricting the integration over Pomeron fields in the functional integral by the range .
In the fourth section we suggest a practical way of building the Monte Carlo code to solve the equation for the generating functional which can be a basis for consideration of the multiparticle production processes.
In conclusion we are going to compare our approach with other approaches on the market.
2 The BFKL Pomeron Calculus
2.1 The general structure of the BFKL Pomeron calculus
We start with a general structure of the BFKL Pomeron calculus in QCD. The BFKL Pomeron exchange can be written in the form (see Fig. 1-1)
with , and denotes the all needed integrations.
It is easy to understand the main parameters of the BFKL Pomeron calculus by comparing the contributions of the first ‘fan’ diagrams of Fig. 1-2 with the one BFKL Pomeron exchange.
This diagram has the following contribution
where and are the sizes of the projectile and target dipoles while denotes all dipole variables in Pomeron splitting and/or merging.
One can see that the ratio of this two diagrams is proportional to which is the parameter given by Eq. (1.2). When this ration is about 1 we need to calculate all diagrams with the Pomeron exchange and their interactions (see Fig. 1-a - Fig. 1-f ). All vertices, that are shown in Fig. 1, has been calculated in [4, 5] and they have the following order in 111In Eq. (2.6) we use the normalizations of these vertices which are originated from calculation of the Feynman diagrams. In the dipole approach we use a different normalization (see below section 3 and 4) but all conclusions do not depend on the normalization.:
As we have mentioned in the introduction the BFKL calculus sums all diagrams at such a high energy that parameter is of the order of 1 (see Eq. (1.2)). In this kinematic region we need to take into account the diagrams of Fig. 1-1 , Fig. 1-2 and Fig. 1-3 (see Eq. (2.4),Eq. (2.1) and Eq. (2.7)). Indeed, diagrams of Fig. 1-4 and Fig. 1-6 ( see Eq. (2.7) and Eq. (2.8)) are small since in the kinematic region of Eq. (1.3), while the diagrams of Fig. 1-5 (see Eq. (2.8)) are small at .
The first conclusion that we can derive from this analysis that in the kinematic region where we need to take into account all diagrams with and vertices while the diagrams with and vertices give small, negligible contributions.
It is interesting to notice that the vertex can be neglected even at such large values of .
Finally, we can conclude that the first step of our approach can be summing of the diagrams with and vertices in the kinematic region or .
However, we would like to stress that we need to make an additional assumption inherent for the BFKL Pomeron calculus: the multi-gluon states in -channel of the scattering amplitude lead to smaller contribution at high energies than the exchange of the correspondent number of the BFKL Pomerons (see more in ). This statement is supported by the fact that numerous attempts to find the intercept of these states being larger than the intercept for multi-Pomeron exchanges have failed.
2.2 The path integral formulation of the BFKL calculus
The main ingredient of the BFKL Pomeron calculus is the Green function of the BFKL Pomeron describing the propagation of a pair of gluons from rapidity and points and to rapidity and points and 222Coordinates here are two dimensional vectors and, strictly speaking, should be denoted by or . However, we will use notation hoping that it will not cause difficulties in understanding.. Since the Pomeron does not carry colour in -channel we can treat initial and final coordinates as coordinates of quark and antiquark in a colourless dipole. This Green function is well known, and has a form
where vertices are given by
where , 333 and are components of the two dimensional vector on -axis and - axis , ; and . The energy levels are the BFKL eigen values
where and is the Euler gamma function. Finally
The interaction between Pomerons is depicted in Fig. 2 and described by the triple Pomeron vertex which can be written in the coordinate representation  for the following process: two gluons with coordinates and at rapidity decay into two gluon pairs with coordinates and at rapidity and and at rapidity due to the Pomeron splitting at rapidity . It looks as
and arrow shows the direction of action of the operator . For the inverse process of merging of two Pomerons into one we have
The theory with the interaction given by Eq. (2.13) and Eq. (2.15) can be written through the functional integral as was proposed and developed by Braun in . We include a discussion of the basics of this approach for the sake of completeness of our presentation.
For we have local interaction both in rapidity and in coordinates, namely,
where () stands for the projectile and target, respectively. The form of functions depend on the non-perturbative input in our problem and for the case of nucleus target they are given in .
For the case of projectile being a dipole that scatters off a nucleus the scattering amplitude has the form
where extra comes from our normalization and we neglect term with in Eq. (2.19).
Generally, for the amplitude of interaction of dipoles at rapidity we can write the following expression 444Starting from this equation we use notations for the coordinates of quark while denote the coordinates of antiquarks. For rapidity we will use .
The extra factor is due to the fact that in the source for both projectile and target, has extra minus sign.
It is useful to introduce the Green function of the BFKL Pomeron that includes the Pomeron loops. This function has the form
For further presentation we need some properties of the BFKL Green function :
2. The initial Green function () is equal to
This form of has been discussed in . In appendix A we demonstrate that this expression for stems from term in sum of Eq. (2.9). Only this term is essential at high energies since all other terms lead to contributions decreasing with energy.
3. It should be stressed that
4. In the sum of Eq. (2.9) only the term with is essential for high energy asymptotic behaviour since all with are negative and, therefore, lead to contributions that decrease with energy. Taking into account only the first term one can see that is the eigen function of operator , namely
The last equation holds only approximately in the region where , but this is the most interesting region which is responsible for high energy asymptotic behaviour of the scattering amplitude.
2.3 The chain of equations for the multi-dipole amplitudes
Substituting and expanding this equation to first order in , we find
We redefine the integration variables in the third term as follows
We denote the new variation parameter by and use the property of the initial Green function Eq. 2.2 to rewrite in terms of as follows
Thus, Eq. 2.30 can be written as
Noting that the r.h.s. of Eq. (2.32) should vanish for any possible variation of we obtain
where we interchanged . We notice that the third and last terms are independent of rapidity and can be absorbed in the initial condition. This is obvious for the last term which represents the target source. To show this for the third term we use the property of the Casimir operator at high energies ()
and the definition of the Green function (see Eq. 2.22). This equation will be discussed in the next section in more details. We see that the third term results into the product of two initial Green functions which are independent of rapidity.