Abstract

We calculate the soft gluon radiation spectrum off heavy quarks (HQs) interacting with light quarks (LQs) beyond small angle scattering (eikonality) approximation and thus generalize the dead-cone formula of heavy quarks extensively used in the literatures of Quark-Gluon Plasma (QGP) phenomenology to the large scattering angle regime which may be important in the energy loss of energetic heavy quarks in the deconfined Quark-Gluon Plasma medium. In the proper limits, we reproduce all the relevant existing formulae for the gluon radiation distribution off energetic quarks, heavy or light, used in the QGP phenomenology.

1. Introduction

High energy heavy-ion collision (HIC) programs have put quantum chromodynamics (QCD), the quantum field theory of strongly interacting matters, to test in an ambiance of high temperature () and density (). It is of paramount importance to measure quantities which will delineate the attributes of Quark-Gluon Plasma (QGP), a medium of deconfined quarks and gluons, expected to be materialized in HIC in RHIC and LHC. One needs a probe to look into the characteristics of this medium. Heavy quarks (HQs), in this context, are believed to be very clean probes because they come to existence well before the advent of QGP and hence they are able to watch the whole evolution of QGP. Notwithstanding the fact that the softer part of the HQ spectrum gets thermalized owing to its interaction with bath particles, the high frequency counterpart sheds considerable bulk of energy which influences the experimental observables like nuclear suppression factor (), azimuthal asymmetry (), and so forth. Heavy quarks interact with thermal light quarks (LQs)/antiquarks and gluons () mainly through elastic and/or inelastic scattering. Between the two principal modes of energy loss, the elastic energy loss succumbs to the radiative one in high momentum region. That is why, with increasing colliding energies, a surge of studies in the radiative domain has been seen in the past few years [1–18].

One of the main ingredients to calculate the radiative energy loss of HQs inside QGP is the radiation spectrum. For single scattering, for example, the radiation spectrum can be obtained by scaling the inelastic amplitude by the elastic amplitude. Quantum chromodynamics based analytical computations of radiation spectrum have so far assumed “soft-eikonal-collinear” limits of parton kinematics and there is a constant endeavour to remove the approximations. The phrase “soft-eikonal-collinear” briefly conveys the following:(1)Soft gluons from hard partons: the energy, , of the emitting parton is much larger than that of the emitted gluon, .(2)Eikonal propagation of hard jets:(a) there is no recoil of both the projectile and the target parton, that is, , where is the transverse momentum transfer due to scattering (the Eikonal I approximation);(b) recoil effect on the leading parton due to emission of radiative gluon is being neglected, that is, , where is the transverse momentum of the emitted gluon (the Eikonal II approximation); however, this is no additional approximation since soft gluon emission, , already encompasses it.(3)Collinear emission of soft gluons: according to this assumption, gluons are predominantly emitted collinearly with the parent parton, that is, .Hence, the “soft-eikonal-collinear” approximation assumes the following hierarchy of different scales: where is the thermal mass of quarks/gluons and is the scale of QCD theory.

The radiation distribution for heavy quarks assuming some/all of the above-mentioned approximations has been a subject matter of. [19–21]. Reference [20] shows that the HQ radiation spectrum () is related to LQ spectrum () by where is the radiation angle and is the ratio of mass of heavy quark () to its energy (). However, (2) assumes small angle () limit of the relation , where and are the transverse momentum and energy of bremsstrahlung gluon, respectively. The factorin (2) is the celebrated “dead-cone” factor.

References [19, 20] used soft-eikonal-collinear approximations while finding out the dead-cone factor. Reference [21] has attempted to find out the gluon radiation spectrum and hence the heavy quark dead-cone factor in collision removing the collinearity approximation. They have obtained a collinearity removed “dead-cone” factor which is given bywhere the subscript “NC” denotes “noncollinear.”

It is believed that, due to the presence of the “dead-cone” around the direction of propagation heavy quark, the energy loss of heavy quarks inside medium becomes different from those of light quarks. Since heavy quark energy loss is related with experimental observables characterizing the QGP, a precise estimate of the heavy quark radiation distribution; and hence, the dead-cone factor is essential.

Earlier the radiative energy loss of heavy quarks considering the Gunion-Bertsch formula [22] and modified kinematics for heavy quarks has been calculated in [4]. The dead-cone factor in (2) has been used while finding out the heavy quark energy loss inside QGP medium [23, 24]. Very recently, the noncollinear soft gluon radiation distribution containing the factor has been used while calculating the heavy quark energy loss inside QGP in [17, 25].

So, the latest calculation of heavy quark radiative energy loss is free from noncollinear approximation of the emitted gluon. Though the radiative energy loss calculation is free from the assumption of collinearity, the Eikonal I approximation, which is neglecting recoil of heavy quarks due to scattering with medium particles, still lingers. The Eikonal I approximation will be removed once we consider the nonnegligible value of transverse momentum transfer with respect to the energy of the incident heavy quark. In the calculations in centre of momentum frame (COM frame), is related to the Mandelstam variable and the energy is related to Mandelstam variable . Hence, the consideration of the terms in the matrix elements calculated in [21] will enable us to remove the Eikonal I approximation.

The present manuscript attempts to revisit the calculations of soft gluon () radiation spectrum off heavy quarks () scattering with light quarks () when the recoil of heavy quark due to scattering is not negligible which is when the Eikonal I approximation is not applicable any more. The hierarchy of energy scales used is the following: With the help of this calculation, (a)we generalize the noneikonal soft gluon radiation spectrum already existing in [26] for light quarks where the effect of the removal of Eikonal I approximation is expected to be more pronounced;(b)we show that the eikonal formula (in (4)) in Eikonal I limit of heavy quark is reproduced;(c)we get back the Gunion-Bertsch radiation distribution formula for massless quarks [22];(d)we get back the Dokshitzer-Kharzeev formula (in (3)) in soft-eikonal-collinear limit;(e)we provide an estimate of the effect of the large-angle scattering on the energy loss.

This manuscript is organized as follows: In the next section we describe in detail the Feynman diagrams we use and the kinematic variables necessary for describing our calculations. To compare with the previous works, we consider Feynman diagrams, where and is the strong coupling. The kinematic approximations will also be discussed at length. In Section 3 we write down the possible Feynman amplitudes for the process in terms of the kinematic variables discussed in Section 2, derive the amplitude in terms of them, and find out the noneikonal gluon radiation spectrum. In Section 4, we show the plots of the radiation distribution function and show the effect of noneikonality on radiation spectrum. In Section 5 we demonstrate that the present formula generalizes all the existing heavy quark single scattering radiation distribution formulae [19–22, 26] used so far by taking relevant kinematic limits. In Section 6 we calculate energy loss of heavy/light quarks undergoing large-angle scattering while interacting with other (light) quarks in the medium and make a comparison with those obtained using the results available in the literature. In the last section we summarize, draw conclusions, and attempt to mention some applications of the results obtained.

2. Notations and Approximations

It is well known that the gluon radiation spectrum in process is given by the ratio of radiative amplitude square to the collisional amplitude square. So our aim will be to calculate relaxing the eikonal approximation due to scattering. The Feynman diagrams contributing to the radiative process are shown in Figure 1.

Figure 1 

Feynman diagrams corresponding to the process . Double line denotes heavy quarks. are all quark colours. are gluon colours and Greek indices denote gluon polarizations.

For the process obeying the four-momentum conservation relation , we have six Mandelstam variables, , , , , , and , where subject to the constraint equation Hence, we need five variables for 3-body phase space. At this point, we may assume the four-momentum of the emitted gluon, , to be small enough so that the corresponding kinematics reduces to one due to scattering. This approximation is called the “soft gluon emission approximation.” The simplification of kinematics due to soft gluon emission () approximation has been discussed in detail in [27–29]. In approximation, , , and , which lead to Hence, the kinematics we are dealing with is approximately similar to the two-body kinematics which needs two Mandelstam variables, and (say), square of COM scattering energy and COM scattering angle, respectively, to be specified. We may write down and , in COM frame in terms of mass () and energy () of heavy quark; and is the COM scattering angle between the incoming HQ (momentum ) and the scattered HQ (momentum ). We can form, for 2-body scattering processes, two dimensionless variables from the available quantities of our present problem. One is and the other is . Besides, there may be another quantity, , which reminds us of the fact that we are dealing with a 3-body phase space, in reality. Now, ; and from the previous section we know that , where is the angle the radiation makes with the parent quark. Also, for on-shell radiated gluon. Consequently, all the components of are now expressible in terms of ; and the third dimensionless quantity becomes proportional to . Assuming , we consider the soft limit of emitted gluon. Under this approximation, we explore the effect of noneikonal contributions, which is terms and higher in Feynman amplitude.

All our calculations are done in the COM frame. We hereby specify our choice of four momenta of interacting particles. Assuming that the incoming particles have no transverse momentum (i.e., they are travelling along the -axis), say, we stick to the following choice of four momenta , :The scattered particles are assumed to acquire a transverse momentum . Since we are working in COM frame in the soft gluon radiation limit, we may approximately assume and , where approximation sign is replaced by equality for case.

3. Radiative Matrix Elements of HQs

There are five Feynman diagrams pertaining to the process under discussion, . Obeying the standard practice [21], we denote a generic matrix element, Clearly, (or ) denotes the Feynman diagram being indicated among the five diagrams in Figure 1. Below, we list down the matrix elements, , up to terms with all large corrections in . For with we jot down , , where . , in point of fact, and hence ,; and , do not contribute to . The definitions of the quantities used in describing the matrix elements in (11) are written below: In the COM frame, Now, to define the total matrix element, , we need the following functions obtainable from (12):where , , and are given byUsing gluon rapidity and the light cone variable , we can getwhere we use is related with the radiation spectrum off HQs when the Eikonal I approximation is removed.

4. The Noneikonal Radiation Spectrum off Heavy Quarks

In Figure 2 we show the variations of the noneikonal spectra () scaled by the eikonal spectrum () with respect to the gluon transverse momentum . We see that for soft approximation and for comparatively less noneikonality () the contribution due to noneikonality may be 50% more than that due to eikonality. This excess may reach up to ~30% (~15%) for ().

Figure 2 

Variation of the noneikonal radiation spectrum scaled by the eikonal spectrum () off heavy quark () with gluon transverse momentum. Red (dashed): ; brown (dotted): ; blue (dot-dashed): .

In Figure 3 we plot the noneikonal radiation spectrum off heavy quarks, , with varying of gluons for different values. signifies the extent of transverse momentum transferred to the heavy quark due to scattering with light quarks. Hence, can be treated as the noneikonality parameter in our calculation.

Figure 3 

Variation of gluon spectrum off heavy quark (, ) with gluon transverse momentum for different extents of recoil of heavy quarks. Green (solid): ; red (dashed): ; brown (dotted): ; blue (dot-dashed): .

If we want to calculate the energy loss and its effect on the nuclear suppression factor we have to consider the scattering which will have more cross-sections than , too. While, in the eikonal case [21], the matrix element differs from that of just by a number due to colour factor, the noneikonal case is not going to be so simple and we have to calculate the Feynman amplitudes of a lot more diagrams. The present calculation may, in principle, be useful when quarks dominate in the medium. But that needs a consistent treatment of the multiple scattering process. Once that is done, we can easily find out the effect of noneikonality in energy loss.

5. Behaviour of Noneikonal Heavy Quark Spectrum at Different Kinematic Regions

5.1. Region I: Massless Quark with Noneikonal Trajectory

In the massless limit of (15), we obtain the noneikonal gluon radiation spectrum off light quarks. Below we jot down the forms of the functions , , , and , , , when we take massless limit, that is, :

Hence,

If we retain the terms up to of , , and and put , we get In the limit (21) boils down to the light quark noneikonal (up to ) matrix element obtained in [26].

5.2. Region II: Massive Quark with Eikonal Trajectory

This region considers Hence, from (12), From (14) we get, in the same limit, HenceUsing (23), (24), and (25), we get with ; and the expression embraced by the curly braces is the radiated gluon spectrum (~) for this case. Evidently, the present calculation yields the calculation in [21] in the small angle scattering limit (in (26)).

5.3. Region III: Massless Quark with Eikonal Trajectory

Now we explore the behaviour of the radiation spectrum in the following limits: The above limits force (26) to take the form given below: which in the limit can be written aswhere , are two different light quark flavours. The part within the square braces can very well be identified with the celebrated Gunion-Bertsch gluon spectrum [22] emitted from light quarks.