# On -broadening of high energy partons associated with the LPM effect in a finite-volume QCD medium

## Abstract

We study the contributions from radiation to -broadening of a high energy parton traversing a QCD medium with a finite length . The interaction between the parton and the medium is described by decorrelated static multiple scattering. Amplitudes of medium-induced gluon emission and parton self-energy diagrams are evaluated in the soft gluon limit in the BDMPS formalism. We find both the double-logarithmic correction from incoherent scattering, which is parametrically the same as that in single scattering, and the logarithmic correction from the LPM effect. Therefore, we expect a parametrically large correction from radiation to the medium-induced -broadening in perturbative QCD.

## I Introduction

-broadening of high energy quarks and gluons is directly related to the properties of the background medium in relativistic heavy ion collisions. At leading order in perturbative QCD, -broadening in decorrelated multiple scattering takes the following form (1)

(1) |

where is the length of the medium and the transport coefficient is a characteristic property of the background medium, which can be expressed in terms of the gluon distribution function(1); (2). Moreover, in the soft gluon limit is found to be closely related to radiative energy loss, that is,(3); (1); (4)

(2) |

By determining from experimental data related to -broadening and radiative energy loss, one can acquire information about the QCD matter created in relativistic heavy ion collisions.

Leading-order results of -broadening and radiative energy loss in perturbative QCD lead to paradoxical results when confronted with RHIC’s data(6); (7); (5). In Ref. (5), it is shown that RHIC’s data about suppression indicate that at the thermalization time is less than /fm in most central Au+Au collisions if one uses Equ. (1)(8); (9). In contrast, in Ref. (6) the values of the time-averaged are found to exceed /fm by fitting high- hadron spectra with radiative energy loss. Besides, such paradoxical results have been already obtained in different analyses about jet quenching((7) and references therein). Could such contradictory results be due to that leading order results in perturbative QCD are not so reliable at RHIC energy? To answer this question, one has to calculate corrections of higher orders in to the above results. In Ref. (10), corrections of to have been calculated in an effective theory of QCD. It is found that even at relatively weak coupling the corrections are already numerically large.

On the theoretical side, it is also intriguing to calculate the contributions to -broadening from radiation in perturbative QCD, which is of . By parametric analysis, the medium-induced -broadening from gluon emission in perturbative QCD must have the following form

(3) |

with a dimensionless coefficient and the saturation momentum squared. In Ref. (11), it is found that in terms of the appropriate saturation momentum -broadening in the strongly coupled SYM plasma takes the same parametric form as Equ. (3) if one replaces with . There are no contributions from multiple scattering in SYM plasma. Parametrically, -broadening is multiple scattering dominated in perturbative QCD and radiation dominated in SYM theory(11). However, the numerical value of has not been evaluated in perturbative QCD in the literature. Therefore, it will help us understand better about these two theories by evaluating to see the limitations of the above parametric conclusion.

In single scattering, the contribution to -broadening of a high energy quark associated with an uncorrelated gluon emission is double-logarithmically enhanced in the infrared regime(12), that is,

(4) |

where is the mean free path of the parton, is the typical transverse momentum squared transferred from the medium to the parton in single scattering, is the screening mass of the constituent particles(scatterers) of the medium and the transport coefficient . Here and in the following, physical quantities of different partons are followed by a subscript with or respectively standing for quark jets or gluon jets. Since such a double-logarithmic behavior is well known already, we focus on the contributions to from radiation in the LPM regime in this paper.

Our calculations are carried out in the BDMPS formalism(3); (1); (4), in which the interaction between high energy partons and the medium is described by decorrelated static multiple scattering. The complete version of the BDMPS formalism in Ref. (4) gives only details about the calculation of the medium-induced gluon spectrum. We need first to generalize it to the case of the -differentiated gluon spectrum, that is, . Besides, the complete corrections also include the contributions from medium-induced self-energy diagrams. Unfortunately, the evaluation of those diagrams in the LPM regime had not been done in the literature. Therefore, we also need to find a way to calculate the contributions from parton self-energy in the LPM regime inside the medium.

The paper is organized as follows. In Sec. II, we derive the general formula of the final-state distribution function associated with the LPM effect. We especially focus on the general formula related to the medium-induced self-energy diagrams. The details about the evaluations of all those medium-induced amplitudes defined in Sec. II are shown in Sec. III. In Sec. IV, we give the complete results of -broadening from medium-induced gluon emission diagrams as well as medium-induced parton self-energy diagrams. The physical interpretation of our results is presented in Sec. V.

## Ii Medium-induced gluon emission and self-energy diagrams

When a high energy parton travels in the medium, it can lose its energy due to the medium-induced gluon emission. In the meanwhile the final-state distribution of the parton in transverse momentum space is modified by the gluon recoil. Due to the conservation of probability, one also needs to consider the contributions from the medium-induced self-energy diagrams, which have not been evaluated in the literature. Therefore, in this section we focus on the general formula for the corrections from medium-induced self-energy diagrams in the BDMPS formalism(3); (1); (4). Before that, we first give a brief review on the medium-induced gluon emission.

### ii.1 Review of the medium-induced gluon emission

Inspired by the discovery of jet quenching at RHIC, radiative energy loss has been extensively studied in the last decade(3); (1); (4); (13); (14); (15); (16) (see(17) for a complete review). A typical medium-induced gluon emission process is illustrated in Fig. 1. The multiple scattering of the parton-gluon pair in region II is responsible for the so-called LPM effect. In the BDMPS formalism, one denotes the contribution from region II by the full amplitude and the contribution from the last scattering vertex in this region by . To calculate broadening of the leading parton, we introduce the transverse coordinate conjugate to the parton’s transverse momentum . In the soft gluon limit, the amplitudes and depend only on and , the gluon’s transverse momentum. The contributions from multiple scatterings in regions I and III are those from the multiple scattering of a single parton. Putting the contributions in the three regions together, one has the -differentiated gluon spectrum as follows

(5) | |||||

where , is the transverse coordinate conjugate to ,

(6) |

and

(7) |

If one integrates over and , the above formula reduces to the medium-induced gluon spectrum in Refs. (3); (4).

### ii.2 The medium-induced self-energy of high energy partons

In this subsection, we will derive the general formula for the corrections from medium-induced self-energy diagrams to the distribution function in the BDMPS formalism. As illustrated in Fig. 2, the screening length provides a natural scale that separates physics at long coherent times from that at short times in decorrelated multiple scattering. If the virtual gluon is emitted and absorbed within a time with only one scatterer involved, the amplitude of the corresponding diagram is the same as that in single scattering. Since such a virtual gluon does not have enough time to dip into the medium, the amplitude does not carry the information about the medium through the LPM phases. In this section, we only deal with self-energy diagrams with virtual gluons that live much longer than , which are called medium-induced self-energy diagrams. In such diagrams, the emitted gluons are allowed only to travel forwards along the positive -direction and one can identify time variables with longitudinal coordinates in the old-fashioned perturbation theory(3); (4).

As illustrated in Fig. 3, a medium-induced self-energy diagram is characterized by two time variables: the emission time and the absorption time . One can separate the whole process into three different regions in . In region I , a high energy parton is randomly kicked by the scatterers. In region II , the parton emits a virtual gluon at , which, after being subjected to multiple scattering together with the leading parton, is absorbed at a later time . We will account the opposite sequence of times by multiplying a factor of 2 in our final formula. Then, in region III, the leading parton alone continues to accumulate its -broadening from multiple scattering until traveling outside of the medium. Combining the contributions from the three regions, we can write the corrections from medium-induced self-energy diagrams to the distribution function in the following form

(8) | |||||

where is the contribution from multiple scatterings in regions I and III, is the full amplitude in the LPM region (region II), which is the same as that used in (5), and is the amplitude of the gluon absorption vertex at . Here,

(9) |

In terms of these amplitudes we define the distribution function from radiation as

(10) |

The conservation of probability requires that

(11) |

## Iii Calculations of medium-induced gluon emission/absorption amplitudes

In this section, we calculate the gluon emission amplitudes , and following Ref. (4). Among them, evaluations of , and involve only the gluon emission/absorption in single scattering. In contrast, the calculation of is more involved and can be obtained by solving the Schrödinger-like evolution equation in the BDMPS formalism.

### iii.1 The medium-induced gluon emission/absorption amplitudes in single scattering

We first evaluate . Here, one needs to include all the gluon emission diagrams with only one scatterer at involved. All of such diagrams are shown in Fig. 4, which are the same as those in Ref. (4). The color factors and the gluon emission vertices are already known. The difference is that we need to put in a phase factor with in the amplitudes of Fig.s 4(a), 4(b) and 4(c) to record the information about the transverse momenta transferred from the medium to the parton-gluon system. In this way, we get the following amplitudes from the gluon emission vertex in each diagram

(12) |

Summing over all the contributions above, we get

(13) | |||||

where the differential cross-section is defined as

(14) |

In impact parameter space, it takes the following form

(15) | |||||

Here, we have used

(16) |

Next, let us calculate . One should take into account all the gluon emission diagrams with only one scatterer at involved in the conjugate amplitude. All the relevant diagrams are shown in Fig. 5. After being scattered at the parton-gluon system behaves like a single parton, and the transverse momenta of the emitted gluon will not change any more. Therefore, only those diagrams with transverse momenta transferred from the medium to the leading parton in the amplitude are given a common phase factor . The amplitude of each diagram in Fig. 5 is as follows

As a result, we have

(17) | |||||

and

(18) |

As shown in Fig. 5, at the diagrams with the gluon emitted inside the medium in the amplitude and emitted outside of the medium in the conjugate amplitude are already accounted in our calculation of .

As shown in Fig. 6, diagrams contributing to have a one-to-one correspondence with those contributing to , whose amplitudes are given by

(19) |

Accordingly,

(20) | |||||

and

(21) |

Taken in Fig. 6, one can easily see that the diagrams with the virtual gluon emitted in the medium and absorbed outside of the medium are already accounted in our calculation of .

### iii.2 The medium-induced gluon emission amplitude at time

To get at any time , we first find the evolution equation. Given at , one can choose an infinitesimal time interval such that the difference between and can be calculated by considering at most one extra scattering. In transverse momentum space we have

(22) |

where the first term on the right hand side results from the free evolution of the parton-gluon system(4) and the second term comes from the contributions of one extra scattering shown in Fig. 7. There is an extra phase factor in the terms corresponding to Fig.s 7(a) and 7(b) because transverse momenta are transferred from the medium to the leading particle in their conjugate amplitudes. In the limit , the above equation reduces to Equ. (18) in Ref. (4) in the soft gluon limit. In impact parameter space, we have

(23) |

with

(24) |

After changing variable , we have

(25) |

Let us define

(26) |

with

(27) |

According to (25), satisfies

(28) |

with

(29) |

In the following, we redefine the following dimensionless variables

(30) |

and accordingly redefine the dimensionless gluon emission/absorption amplitudes by replacing all the variables in their expressions with the corresponding dimensionless variables.

In terms of the above dimensionless variables, the Schrödinger-type evolution equation (28) is the same as that for a two-dimensional harmonic oscillator with mass and the angular frequency . The amplitude is given by

(31) |

where

(32) | |||||