###### Abstract

In this paper the semi-classical approach to the solution of non-linear evolution equation is developed. We found the solution in the entire kinematic region to the non-linear evolution equation that governs the dynamics in the high parton density QCD.

The large impact parameter () behavior of the solution is discussed as well as the way how to include the non-perturbative QCD corrections in this region of . The geometrical scaling behavior and other properties of the solution in the saturation (Color Glass Condensate) kinematic domain are analyzed. We obtain the asymptotic behavior for the physical observables and found the unitarity bounds for them.

TAUP-2732-2003

hep-ph/0305150

May 19, 2021

QCD Saturation in the Semi-classical Approach

S. Bondarenko
^{†}^{†}footnotetext: Email:.,
M. Kozlov ^{†}^{†}footnotetext: Email:
. and E. Levin ^{†}^{†}footnotetext: Email:
, .

HEP Department

School of Physics and Astronomy

Raymond and Beverly Sackler Faculty of Exact Science

Tel Aviv University, Tel Aviv, 69978, Israel

## 1 Introduction

High density QCD [1, 2, 3] which is dealing with the parton systems where the gluon occupation numbers are large, has entered a new phase of it’s development: a direct comparison with the experimental data. A considerable success [4, 6, 7, 8, 9, 10, 11] has been reached in description of new precise data on deep inelastic scattering [12] as well as in understanding of general features of hadron production in ion-ion collision [13]. The intensive theoretical [14, 15, 16] work lead to effective theory in the high parton density region with non-linear evolution equation for dipole-dipole amplitude that governs the dynamics in this region [17]. The practical application of the high density QCD approach to a comparison with the experimental data is based on analytic [18, 19, 20] and numerical [5, 8, 19, 21, 22] solution to the non-linear equation. In spite of understanding of the main qualitative and, partly, quantative properties of the non-linear dynamics, this equation has not been solved at arbitrary values of the impact parameters () and a number of “ad hoc” ansatzs were used for -dependence.

It turns out that -dependence is a challenging problem since the non-perturbative corrections are important in the large kinematic region. There are two different ideas on the market how to include the non-perturbative large behavior into high density QCD dynamics. The first one [23, 24, 25] claims that the non-perturbative corrections could be taken into account only in initial condition for the non-linear evolution equation, while the other idea demands the change of the kernel in the equation [26].

The objective of this paper is to find the semi-classical solution to the non-linear evolution equation with a special attention to the large behavior of the solution. The semi-classical approach has several advantages[1, 14, 15]. Firstly, it gives simple, transparent and analytic solution to the non-linear equation. Secondly, this solution has a good theoretical accuracy in the most interesting kinematic region: at low and small sizes of interacting dipoles. Thirdly, this approach leads to a natural definition of the new saturation scale which is the principle dimensional parameter that governs dynamics at high density QCD region [1, 2, 3]. It appears in the semi-classical approach as a critical line which divides the whole set of trajectories in two parts: inside and outside of the saturation region. In other words, this critical line is a separation line between two phase: color gluon condensate (CGC) and the gluon liquid and the saturation scale is the order parameter for this phase transition.

The shortcoming of the semi-classical approach is the fact that the experimental data are not in the kinematic region where this approach has a good theoretical accuracy. However, we believe, that we can develop the reasonable method for the numerical solution of the non-linear equation only after studying the property of the semi-classical approach.

In the next two sections we discuss the semi-classical approach for the dipole sizes smaller than the saturation scale ( where is the saturation momentum). We review the semi-classical solution which has been investigated in this region (see Refs. [1, 14, 15]) concentrating mostly on large -behavior of this solution.

Section 4 is devoted to the semi-classical approach inside of the saturation region (). We suggest a semi-classical solution which describes the behavior of the dipole-dipole amplitude deeply inside of this domain and discuss the matching of this solution with the semi-classical solution at small dipole sizes. It turns out that this matching occurs on the special trajectory of the non-linear equation which defines the saturation scale for dipole-dipole rescattering.

In section 5 we discuss the unitarity bound as well as energy behavior of the gluon structure function and the total cross section.

Section 6 gives the estimates of the accuracy of our approach. We calculate the first enhanced diagram as well as corrections to the non-linear equation that we neglected in our solution.

In the last section we summarize our results and discuss the possible practical applications.

## 2 General Approach

### 2.1 The non-linear evolution equation

The non-linear equation that governs the dynamics in the high parton density QCD domain can be written in the following form given by Balitsky and Kovchegov [17]:

(2.1) |

The meaning of Eq. (2.1) is very simple and can be seen in Fig. 1 and Fig. 2. It describes the process of dipole interaction as two stages process. The first stage is decay of the initial dipole with size into two dipole with sizes and with probability

In the second stage two produced dipoles interact with the target. The non-linear part of Eq. (2.1) takes into account the Glauber corrections for such an interaction ( see Fig. 1 ). The first term in Eq. (2.1) stands for possibility for the initial dipole to interact with the target without decaying into two dipoles.

The equation is written in the coordinate representation which has a certain advantage since where is the dipole amplitude. Directly from the unitarity constraint follows that giving the natural asymptotic behavior for , namely tends to unity at high energies. On the other hand the non-linear term in Eq. (2.1) contains the integration over sizes of produced dipoles. It means that we cannot conclude that the non-linear term is small even when the sizes of the initial dipoles are very small.

It turns out to be useful to consider Eq. (2.1) in a mixed representation,
fixing but introducing transverse momenta as conjugated variables
to dipole sizes. The relation between these two representations are given
by the following equations (see also Fig. 2)
^{1}^{1}1The extra factor in Eq. (2.2) makes
dimensionless.:

(2.2) | |||||

(2.3) |

In momentum representation the nonlinear term in Eq. (2.1), being the convolution in the coordinate representation, transforms to the product of two amplitudes, , at the same value of the initial transverse momentum (). Therefore, the large value of the initial transverse momentum guarantees the smallness of the non-linear corrections. One can actually see that the non-linear term in Eq. (2.1) is a convolution only at large value of the impact parameter . Indeed, let us rewrite the non-linear term in Eq. (2.1) in the momentum representation going also to momentum transfer instead of .

In this case

(2.4) |

The non-linear term reduces to the form

(2.5) |

At large values of and both and are small and of the order of . Neglecting terms and in -function in Eq. (2.5) we can reduce this equation to the product of .

However, one can see that there is a region of integration over , namely,

(2.6) |

with

Our strategy will be the following: we will solve the non-linear equation assuming that and and and, using this solution, we will come back to Eq. (2.1) and we will discuss the contribution of the region where

Finally, Eq. (2.1) can be written in the form (see Ref. [19] for details )

(2.7) |

where is an operator corresponding to the anomalous dimension of the gluon structure function and it is equal to

(2.8) |

The form of the first term on the r.h.s. of Eq. (2.7) as well as definition of the variable , we will discuss in the next section. Here, we would like only to draw your attention to the fact that the emission of the gluons in the BFKL equation is described by the first tern in the r.h.s. of Eq. (2.1) (see Fig. 2) which enters at the same value of the impact parameter as the l.h.s. of the equation.

Function is an eigen value of the BFKL equation [27]

(2.9) |

Eq. (2.7) we will solve in the semi-classical approach.

### 2.2 The solution to the BFKL equation at large

This solution has been discussed (see Refs. [29, 30, 31, 32, 25] ) and at large it has a form:

(2.10) |

where

It is easy to see that Eq. (2.10) is a solution to the Eq. (2.1) without the non-linear term. Since the form of solution given by Eq. (2.10) is very important both for a derivation of the linear part of Eq. (2.1) and for understanding the value of the typical dipole sizes in the non-linear part of the equation we will discuss this solution in some details follow Refs. [30, 31, 32]. The general solution to the BFKL equation in the coordinate representation was derived in Ref.[32], namely (see Fig. 5),

(2.11) |

with

(2.12) |

and

(2.13) |

The integration over in Eq. (2.11) was performed in Refs. [30, 31] with the result:

(2.14) |

where is hypergeometric function [28] and is equal to [30, 31]

(2.15) |

and and are function of .

At large values of and the argument is small, namely

(2.16) |

It is interesting to notice that turns out to be small and simple also for with . Indeed, for such values of

(2.17) |

Expanding Eq. (2.14) and rewriting the answer in the transverse momentum representation given by Eq. (2.2) and Eq. (2.3) we reduce the general solution of Eq. (2.14) to Eq. (2.10). Using Eq. (2.10) one can see that the linear part of Eq. (2.7) indeed gives the BFKL equation. Since all derivative in Eq. (2.7) are taken with respect to we can consider in Eq. (2.7) as an arbitrary function of .

The exact form of the initial condition depends on the particular reaction. We choose as an instructive example the virtual photon-photon scattering (see Fig. 3).

Fig. 4-a | Fig. 4-b | Fig. 4-c |

The advantage of this process is the fact that we know the initial condition which is under control of pQCD except large behavior. The problem of large behavior of the Born amplitude is addressed in Ref. [25] and it is shown that can be chosen in the form

(2.18) |

to satisfy both the of Born approximation in perturbative QCD, namely,

(2.19) |

and the non-perturbative behavior at large .

### 2.3 Semi-classical approach

In semi-classical approach we are looking for a solution in the form:

(2.20) |

where and are smooth functions of and : ,

Assuming Eq. (2.20) we can use the method of characteristic (see, for example, Ref. [33]) to solve the non-linear equation. For equation in the form

(2.21) |

where , we can introduce the set of characteristic lines on which are functions of variable (artificial time), which satisfy the following equations:

(2.22) |

where etc.

We can reduce the master equation (see Eq. (2.1) ) in semi-classical approach to the form of Eq. (2.21), namely,

(2.23) |

Using Eq. (2.20) and Eq. (2.23) we can write the set of equations (see Eq. (2.22)) in the form:

(2.24) |

We will solve these equations in the next section, but before doing this we are going to illustrate the method of characteristic solving the linear equation neglecting the non-linear tern in Eq. (2.1). In semi-classical approach the linear equation has a form:

(2.25) |

For linear equation the set of Eq. (2.24) reduces to two equations:

(2.26) |

with both and being constant as function of .

The solution is very simple, namely,

(2.27) |

with given by

(2.28) |

Comparing Eq. (2.27) and Eq. (2.28) with the solution to the BFKL equation of Eq. (2.10) one can see that the semi-classical approach is the exact solution of the linear (BFKL) equation in which the contour integral over is taken by the steepest decent method. Eq. (2.28) is the equation for the saddle point value of . The accuracy of the semi-classical approach is even worse than one for the steepest decent method since we cannot guarantee the pre-exponential factor in the semi-classical calculation which appears in the steepest decent method. However, the semi-classical approach has a great advantage since it allows us to treat the non-linear equation within the same framework as the linear one without major complications.

To solve Eq. (2.24) we need to find out the initial conditions for this set of equations, which we derive from the Glauber-Mueller formula for scattering ( see Ref. [25] ), namely:

(2.29) |

where is the dipole amplitude in the Born approximation. It has been calculated in Ref. [25]

(2.30) |

where the variable is defined as

(2.31) |

Comparing Eq. (2.31) with Eq. (2.16) and Eq. (2.17) we notice that . We put these values of just for the sake of simplicity. with

One can see that in the coordinate representation can be replaced by with

(2.32) |

At large value of we have the same definition of that has been used but Eq. (2.32) gives us a generalization which allows us to treat the case of .

Using Eq. (2.2) and Eq. (2.3) we can find the initial condition in the momentum representation:

(2.33) |

where is incomplete Euler gamma function [28] of zeroth order. The argument in Eq. (2.33) can be rewritten as

(2.34) |

with

(2.35) |

where we define as a value of at , see next section. The behavior of and at which follow from Eq. (2.33) is shown by Fig. 6 . on the critical trajectory for

As one will see below we need also to know a ratio

(2.36) |

for finding the solution. This ratio is shown in Fig. 7.

## 3 Solution at

### 3.1 Fixed QCD coupling

#### 3.1.1 Solution

As has been discussed, see [1, 14, 15], in this kinematic region we can safely use the semi-classical approach for sufficiently large values of . In this section we will find this semi-classical solution and will show, that the main qualitative feature of it, is the existence of special, critical trajectory which determines the saturation scale.

From equations 4 and 5 of Eq. (2.24) one can see that

(3.37) |

Eq. (3.37) has a very transparent physics since it means that the phase and group velocity of wave package, defined by Eq. (2.20), are equal.

Eq. (3.37) has an obvious solution

(3.38) |

where is a constant which has to be determined from the initial condition (see Fig. 7 and Eq. (2.36)). Using Eq. (3.38), Eq. (2.23) and Eq. (2.24)- 4 we obtain that

(3.39) |

However, before solving Eq. (3.39), we would like to draw your attention to the fact that Eq. (3.37) has itself a very interesting solution if we assume that non-linear corrections in Eq. (2.23) are small but they are valuable in Eq. (2.24)-4 and Eq. (2.24)-5. In other words, the non-linear term in the master equation is small in comparison with the linear one on this special trajectory but the non-linear contributions is essential in the equations for and dependence on . In this case Eq. (3.37) reduces to

(3.40) |

The solution to this equation is
^{4}^{4}4Eq. (3.40) has been derived in Ref.[1] and
has been discussed in detail in Refs. [34, 35, 36].
is called in Ref.[1], in Ref.
[34],
in Refs.[35, 36] and in Ref.
[25]..

The form of the trajectory is clear from Eq. (2.24)-1, namely

(3.41) |

The value of the dipole amplitude we can find using Eq. (2.24)-3 which can be reduced to

(3.42) |

which has the solution

(3.43) |

Since this solution falls down at large all our assumptions are self-consistent at least at . The trajectory at small values of should be found from Eq. (3.39).

#### 3.1.2 Numerical solution

To find the solution to the master equation we need to solve Eq. (3.39) and obtain as a function of , substitute this function into Eq. (2.23) and find out .These two functions and will determine the solution on the trajectory that is given by Eq. (2.24)-1.

One can see from Fig. 8 that we can divide all possible trajectories in two part with initial larger or smaller than . From starts the critical trajectory on which at large . The trajectories to the right of the critical one are close to the straight lines which are the trajectories of the linear equation. The trajectories to the left of the critical line differs significally from the trajectories of the linear equation. In this domain the non-linear corrections, induces by partons interaction in the parton cascade, are large and change considerably the physics of the QCD evolution traditionally related to the linear evolution equation.

The same physical picture we can see in Fig. 9 which shows the quite different behaviour of the anomalous dimension along trajectories. For the trajectories to the right of the critical line very rapidly reaches a constant value as it should be for the linear evolution. On the critical line approaches (see Eq. (3.40) ) but very slowly. For the trajectories to the left of the critical line we have a different behaviour and becomes larger than unity which is, of course, an indication that we cannot use a semi-classical approach, at least in the present form.

Fig. 10 shows the behavior of function ( ) versus . One can see that the dipole amplitude is decreasing slowly on the critical trajectory while it is rapidly falling down on the trajectories to the right of the critical line. It is worthwhile mentioning that at small () the - dependence stems mostly from the dependence of on ( see Fig. 9 ). We see the manifestation of the slow decrease of Eq. (3.43) only at .

#### 3.1.3 A general solution

Since trajectories cannot cross each other we see (i) that the domain to the left of the critical line does not know anything about the domain to the right of the critical line; and (ii) that the solution to the right of the critical line is very close to the solution of the linear evolution equation. Therefore, the semi-classical approach to the non-linear evolution equation leads to an idea[1] to solve the problem considering only the linear evolution equation but with initial condition not at fixed value of but on the critical line with is defined by Eq. (3.43). For fixed QCD coupling it is easy to find such a solution.

#### 3.1.4 Impact parameter dependence

Eq. (2.24) does not explicitly depend on . The entire dependence is hidden in initial condition (see Eq. (2.29)) both the initial dipole amplitude and the anomalous dimension depend actually on the variable as one can see from Eq. (2.29)), namely, and . For example, the critical line corresponds to and .

However, when we solve Eq. (2.24)-1 for the trajectories, we need to know the initial value of not only of . Indeed, the equation for the trajectories can be solved as

(3.45) |

It is obvious from Eq. (2.33) that

(3.46) |

It should be stressed that the dependence also enters the definition of . Substituting Eq. (3.46) into the equation for the critical line (see Eq. (3.41) ) one obtains the dependence of the saturation scale:

(3.47) |

This equation we can rewrite at large using the exact expression for (see Eq. (2.30)). It has the form

(3.48) |

In pQCD region Eq. (3.48) leads to

(3.49) |

As we will see below this expression for leads to the typical radius of interaction which increases as a power of energy, namely,

(3.50) |

in a full agreement with the analysis given in Ref. [26]. It gives the power-like increase of the total dipole cross section, namely

in an explicit violation of the Froissart theorem [45].

### 3.2 Running

#### 3.2.1 A general discussion

In this section we are looking for the semi-classical solution to our master equation (see Eq. (2.1)) considering running QCD coupling, namely, in Eq. (2.1) is equal to

It turns out that it is useful to search the semi-classical solution in the form

(3.51) |

where and are smooth function of and .

In this case the master equation has a form:

(3.52) |

Using Eq. (2.21), Eq. (2.22) and the explicit form of , namely

(3.53) |

we can derive the following set of equation.

(3.54) |

#### 3.2.2 Critical trajectory

There exists a critical trajectory in this case as well as in the case of fixed . To see it let us assume that and . For such we can neglect the non-linear term in Eq. (3.52) and Eq. (3.54)-3 has a form:

(3.55) |

One sees from Eq. (3.55) that there is a trajectory with the same equation for the anomalous dimension () as in the case of constant (see Eq. (3.40)) on which is constant. Eq. (3.54)-4 allows us to determine this constant. Indeed, we can see from Eq. (3.54)-4 that on this trajectory would be constant if [1]

(3.56) |