Radiative energy loss and radiative p_T-broadening of high-energy partons in QCD matter

We study the connection between radiative energy loss and radiative p_T-broadening of a high-energy quark or gluon passing through QCD matter. The generalized Baier-Dokshitzer-Mueller-Peigne-Schiff-Zakharov (BDMPS-Z) formalism is used to calculate energy loss due to multiple gluon emission. With L the length of the matter and l_0 the size of constituents of the matter we find a double logarithmic correction proportional to ln^2(L/l_0) to parton energy loss due to two-gluon emission. We also show that the radiative energy loss per unit length -dE/dz = alpha_s N_c/12 by carrying out a resummation of the double logarithmic terms. Here, the transverse momentum broadeningis obtained by resumming terms proportional to alpha_s ln^2(L/l_0) in Ref. [11]. Our result agrees with that by the renormalization of qhat proposed in Refs. [13,14].


I. INTRODUCTION
Jet quenching is an important signal of the production of dense QCD matter in the ultrarelativistic heavy-ion collisions at RHIC [1,2] and LHC [3][4][5]. Parton energy loss in QCD matter is believed to be responsible for such a suppression of the yields of large p ⊥ hadrons and jets with respect to p + p collisions (see [6] for a recent review).
The radiative energy loss and the p ⊥ -broadening of a high-energy parton in QCD matter are closely related to each other. Energy loss is dominated by radiating a gluon with the maximum energy ω c . The gluon picks up a transverse momentum broadening p 2 ⊥ = ωc tc within the coherent (formation) time t c . As a result, the energy loss of the parton per unit length is given by [7] − In a medium of length L the p ⊥ -broadening due to multiple scattering is given by p 2 ⊥ =qL withq being the transport coefficient. If one only considers multiple scattering, radiative energy loss is dominated by one gluon with t c ≃ L and ω c ≃qL 2 . In this case one has [8,9] − dE dz ∼ α s N cq L.
The radiative p ⊥ -broadening of a high-energy parton in QCD matter is first studied in Ref. [10]. Double logarithmic terms due to the recoil effect of one-gluon emission are found in the kinetic region of single scattering. The complete result of such a double logarithmic correction is obtained in Ref. [11], which takes the form where l 0 is the size of constituents of the matter. Moreover, the resummation of the double logarithmic terms can be carried out to give [11] The transport coefficientq can be written as the expectation value of a gauge-invariant operator, which is proportional to the gluon distribution function of the medium [7]. A renormalization ofq, based on the DGLAP evolution of the gluon distribution, has been proposed in Ref. [12]. More recently, Refs. [13,14] propose another evolution equation for the renormalization ofq, valid in the double logarithmic approximation. This equation applies in the regime of multiple soft scattering while the former one in Ref. [12] is better suited for the study of the high-momentum tail of the p ⊥ -broadening associated with a single hard scattering [14]. The solutionq ren (L) to the equation in [13,14] is consistent with the result in eq. (4) provided it is rewritten as p 2 ⊥ =q ren L. Based on such a proposal, eq. (1) is expected to hold for radiating an arbitrary number of gluons in the double logarithmic approximation.
In this paper we give a detailed calculation of the double logarithmic correction to the energy loss of a high-energy parton by radiating two or more gluons in QCD matter. Our aim is to go beyond the parametric estimate in eq. (1) and to show explicitly how the double logarithmic correction to radiative energy loss is related to the radiative p ⊥ -broadening in eq. (3).
The paper is organized as follows. In Sec. II, we first give the general formalism, as a generalization of that by BDMPS-Z [8,9,15,16], for calculating parton energy loss due to multiple gluon emission. Then, we give all the diagrams relevant for energy loss due to two-gluon emission. The time-evolution of two gluons together with a dipole in the medium is studied in Sec. III. In Sec. IV we evaluate the double logarithmic correction to radiative energy loss in details. Our conclusion is presented in Sec. V.

II. MEDIUM-INDUCED ENERGY LOSS DUE TO TWO-GLUON EMISSION
A. Review of the generalized BDMPS-Z formalism for radiative energy loss In this subsection we first give the general formalism for medium-induced energy loss of a high-energy parton due to multiple gluon emission. The formalism, used to calculate the radiative p ⊥ -broadening in Refs. [10,11], is a slight extension of the BDMPS-Z formalism [8,9,15,16] 1 by including virtual gluon emission. In this paper we choose to use the path integral representation in the transverse coordinate space [11,15,16].
To calculate the medium-induced energy loss of a high-energy parton, one needs the spectrum of n soft gluons. Since the gluons can be real or virtual, the spectrum shall be 1 The interested reader is referred to Refs. [17,18] and references therein for different approaches used to calculate parton energy loss. denoted by ω 1 ω 2 · · · ω n dI (m,n−m) dω 1 dω 2 · · · dω n (5) with m (≤ n) real gluons and (n − m) virtual gluons. The spectrum can be calculated using the following steps.
1. Draw all the relevant graphs with m real gluons and (n − m) virtual gluons.
Inside the medium it is usually more convenient to draw graphs as a product of amplitudes and their complex conjugates.
2. Obtain the contributions of each graph from the following Feynman rules ω, c, B where T a R is the SU(N c ) matrix in the representation R corresponding to the highenergy parton (solid line) and the gluon transverse polarization vector ǫ λ⊥ satisfies In this paper vectors in the transverse plane are denoted by bold letters.

Integrate out the background medium.
The background medium is modelled by the background gluon field A a− (t, x) with a the color index. The ensemble average over the background field is defined by which is related to the transport coefficientq by where n(t) is the number density of the scatterers and σ R (x) is defined by [10] with C R = N c for gluons and C R = C F ≡ N 2 c −1 2Nc for quarks. In this subsection we give all the graphs for the energy loss due to two-gluon emission, which is denoted by ∆E 2 . In these graphs at least one of the two gluons is real. In terms of the spectra defined in the previous subsection, ∆E 2 is given by All the graphs contributing to ∆E 2 can be obtained by cutting through the forward amplitudes in Fig. 1. The cuts of Fig. 1 (h) are not relevant for medium-induced energy loss and shall be ignored. Besides, we shall also ignore the graphs obtained by cutting Fig. 1 (g) because the gluon-quark transition is suppressed compared to the soft gluon emission at high energies. Then, we are left with 17 possible cuts of Fig  Here, the time variables satisfy 0 < z 1 < z 2 < z 3 < z 4 < L.
of the other. The distribution of the uncorrelated soft gluons is the QCD analog of that of the soft photons in QED, which has the form of a Poisson distribution after all the uncorrelated multiple gluon emissions are included [19]. And there is no double logarithmic correction proportional to ln 2 L l 0 2 from these diagrams.
• 26 fully-overlapping emissions: In these diagrams both of the two emission times of one gluon lie in between those of the other gluon. In the following sections we shall show that there is a double logarithmic correction to ∆E 2 from these diagrams, which is the same as that to p 2 ⊥ in Ref. [11].
• 40 partially-overlapping emissions: In these diagrams only one of the two emission times of one gluon lies in between those of the other gluon. The evaluation of those diagrams is the most difficult part to obtain the complete result of ∆E 2 . Fortunately, unlike the fully-overlapping emissions the diagrams contributing to the double logarithmic terms ∝ ln 2 L l 0 of p 2 ⊥ do not show up as subdiagrams of these diagrams. Therefore, there is no double logarithmic correction the same as that in p 2 ⊥ from these diagrams.

III. THE TIME EVOLUTION OF TWO GLUONS IN THE MEDIUM
In this paper we use a dipole-like picture 3 to describe the whole process of two-gluon emission in the QCD medium [11]. In our calculation we shall only include the fully-overlapping emissions as illustrated in Fig. 2: the first gluon with energy ω 1 emitted off the quark at t = z 1 lives until t = z 4 ; and the second gluon with energy ω 2 is emitted at t = z 2 > z 1 and absorbed at t = z 3 < z 4 either by the first gluon or by the dipole. In our case the energy of the high-energy parton E ≫qL 2 and, therefore, the change of the transverse coordinates of the dipole can be neglected. Due to the homogeneity of the medium in the transverse plane, our result should be independent of the transverse coordinates of the dipole, which are chozen to be 0.
The time evolution of one gluon with energy ω 1 together with the dipole inside the medium is described by G (3) [8,9,15,16] For the medium-induced energy loss the harmonic oscillator approximation is justified [8] and one has where and the propagator of the harmonic oscillator During z 3 > t > z 2 one needs to understand how the two gluons together with the dipole evolve in the medium. Nc 2 (T d A ) ce T e R for each diagram in the figure, which gives the color factor Nc 2 for each diagram. As a result the potential of the evolution Hamiltonian of the 4-body system can be easily calculated, which takes the form Let us assume that the first gluon is emitted off the quark of the dipole. If the second gluon is emitted off the first gluon or the quark, the propagator of the 4-body system is given by Otherwise, the propagator takes the form In the harmonic oscillator approximation one has where the new coordinates are defined as  and In the case ω 1 ≫ ω 2 one has

LOSS
In this section we calculate the double logarithmic correction to radiative energy loss.
The double logarithmic correction comes from the diagrams with fully-overlapping emission defined in Sec. II. All these diagrams can be constructed either from Fig. 4 or its complex conjugate. First, we give all the contributions of these diagrams to ∆E 2 in a compact form in terms of G (3) and G (4) (Ḡ (4) ) defined in the previous section. Next, we shall show that they give a double logarithmic correction to radiative energy loss, which is the same as that to the radiative p ⊥ -broadening in Ref. [11].

A. Contributions from diagrams with fully-overlapping emission
In total, there are 26 diagrams with fully-overlapping emission. Let us denote the energies of the two gluons respectively by ω 1 and ω 2 and their formation times respectively by t 1 ≡ z 4 − z 1 and t 2 ≡ z 3 − z 2 . And we assume that ω 1 ω 2 and t 1 t 2 . As illustrated in Fig. 4, there are respectively 4 or 9 diagrams in which the gluon with energy ω 1 is virtual or real.
The 26 fully-overlapping emissions include these 13 diagrams and their complex conjugates.
And one can obtain the corrections to the energy loss from fully-overlapping emissions by taking 2 times the real part of the contributions of these 13 diagrams.
There are some cancellations between the contributions from different diagrams. As a consequence of the conservation of probability, moving one gluon emission (absorption) vertex from the quark (antiquark) line to the antiquark (quark) line in a diagram only changes the overall sign of the contribution of the diagram [10]. To see such a cancellation easily, we use a diagramatic representation in which the integrations over all the time variables and transverse coordinates in Fig. 4 are omitted. It is easy to see that we have the following Therefore, only these 9 diagrams with the gluon of energy ω 1 being real contribute to ∆E 2 .
The contributions to ∆E 2 from these 9 diagrams and their complex conjugates can be classified as follows It is easy to show that the overall color factors for e 1 , e 2 and e 3 are respectively given by N c C R , N c C R /2 and N c C R /2. In terms of G (3) , G (4) andḠ (4) , we get, from the Feynman rules in Sec. II, the following compact expressions where the short-hand notation And the contributions from all the fully-overlapping emissions to the energy loss are given by It is still too complicated to be evaluated analytically even in the harmonic oscillator approximation. In the next subsection, we shall evaluate it in the double logarithmic approximation following Ref. [11].

B. ∆E 2 in the double logarithmic approximation
The calculation of ∆E 2 in eq. (37) is simplified in the double logarithmic region. The first gluon with energy ω 1 , similar to that in the case with one-gluon emission, typically has t 1 ≡ z 4 − z 1 ≃ L, ω 1 ≃qL 2 and |x| 2 ≃ 1 qL . In the double logarithmic region [11], the second gluon with energy ω 2 typically has |B| 2 |x| 2 , ω 2 ω 1 , and In this region one can use the following approximation By inserting (14), (39) and (40) into (37), we have In the double logarithmic region one has t 1 t 2 . Therefore, we can neglect the difference between z 3 and z 2 and write Now we are ready to show how the double logarithmic correction to radiative p ⊥broadening shows up in the calculation of radiative energy loss. By dropping terms in-dependent ofq 5 and keeping only the double logarithmic terms proportional to x 2 2 , we have where t 2 and ω 2 have been integrated over the double logarithmic region in Fig. 5 with L replaced by z 3 − z 1 . As we shall show below, one can simply replace z 3 − z 1 in (44) by L in the double logarithmic approximation. Therefore, such a double logarithmic result is exactly the same as that of radiative p 2 ⊥ and eq. (44) is the same as eq. (26) in Ref. [11] divided by L.
Let us evaluate the double logarithmic correction to radiative energy loss. Inserting (44) into (42) and integrating out x 2 gives Since the integrand on the right-hand side of the above equation is proportional to z 3 − z 1 as z 3 → z 1 , the leading double logarithmic term from the integration over z 3 can be obtained simply by integration by parts, that is, Similarly, the leading double logarithmic term from the integration over t 1 is of the form 5 In this way one calculates the medium-induced energy loss as explained in Refs. [8,9,15,16]. A consequence of such a subtraction is that one drops the vacuum diagrams (including the UV divergent ones) and, therefore, ignores the effects of running coupling. The consequences of running coupling to p 2 ⊥ or the renormalizedq is studied in [20], which is beyond the scope of this paper.
whereω ≡ 1 2 q A L 2 ω and recall that radiative energy loss due to one-gluon emission is given by [9] ∆E 1 =2α s C R Re dω 1 ω 2 The resummation of the double logarithmic correction in eq. (47) can be carried out in exactly the same way as that in the calculation of p 2 ⊥ . For (n + 1)-gluon emission, one has Therefore, the total energy loss is given by V. CONCLUSIONS In this paper we calculate the double logarithmic correction to radiative energy loss of a high-energy parton in the generalized BMDMPS-Z formalism [10,11]. Radiative energy loss per unit length due to one-gluon emission is given by [9] − In this case it is dominated by radiating one gluon with the maximum energy ω c ≃qL 2 and the formation time t c ≃ L. And within t c the gluon is of a typical size x 2 ∼ 1 qL . The double logarithmic correction comes from the diagrams by adding a second gluon with the two emission times both lie in between those of the first gluon of energy ω c . In the kinetic region for the double logarithmic correction this second gluon has a smaller energy, a shorter formation time and a larger size than the first gluon. It modifies the transverse momentum broadening of the first gluon and, therefore, contributes to the radiative energy loss according to the parametric estimate in eq. (1). Our detailed calculation confirms this picture and