Soft Gluon Radiation off Heavy Quarks beyond Eikonal Approximation

We calculate the soft gluon radiation spectrum off heavy quarks (HQs) interacting with light quarks (LQs) beyond small angle scattering (eikon- ality) 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 im- portant 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.

Stunning testimonials of jet quenching coming from dedicated heavy-ion collider experiments viz., STAR, PHENIX/RHIC/BNL [1], ALICE/LHC/CERN [2] strongly articulated the fact that, the primordial hot soup of nuclear matter, produced in those experiments, indeed contain partonic degrees of freedom before freezing out to hadrons. A typical jet event at an ion collider generally goes through three main stages of development. Firstly, an early time parton level hard scattering event produce a few energetic partons (quarks and gluons). Then, the developement of shower in which each parton, via branching, broadens into a 'jet' is observed. Finally comes the hadronization stage in which the colored particles in the jet recombine into color-neutral mesons and baryons via fragmentation (at hard sector) and via coalescence (at soft sector). Quantitative understanding of all the three stages is required in order to expose the jet event completely. Strong modification of internal architecture of the jet shower is generally governed by both elastic (associated with non-radiative scattering) and inelastic (associated with radiative scattering) processes. This modifications may either be in the form of energy degradation of the most energetic leading parton of the jet and broadening effects on the transverse momentum or can be change in in-jet gluon multiplicity leading to jet quenching. Observation of strong suppression of inclusive yields of high momentum hadrons and semi-inclusive rate of azimuthal back-to-back high momentum hadron pairs relative to p-p collisions, are the expectations from jet quenching [3].
Quantum chromodynamics based analytical computations of quenching of jets has, so far, been clogged up with 'soft-eikonal-collinear' limits of parton kinematics. According to 'soft-eikonal-collinear' kinematic approximation, energy of the leading projectile parton E is taken to be quite large compared to the energy of the radiated gluon ω which, again, is supposed to be much larger than both the transverse momentum carried by the gluon k ⊥ , as well as recoiling momentum (exchanged momentum) of scattering centre q ⊥ ; and hence we can write, The phrase 'soft-eikonal-collinear' briefly conveys the following, (1) Soft gluons from hard jets : The staunch hierarchical ordering of the energy, E ≫ ω, is a hitch for treating all daughter partons and study the full internal architecture of the jet shower.
(2) Eikonal propagation of hard jets : (a) Once there is no recoil of both the projectile as well as of the target parton, it is hard to have a transverse deviation of the straight trajectory of the leading jet. Any scope of diffusion for jets in hot-dense quark matter can then be ruled out. On the other hand, (b) Recoil effect on the leading jet due to emission of radiative gluon is being neglected, i.e. E ≫ k ⊥ . However, this is not an additional approximation since soft gluon emission, E ≫ ω, already encompasses it.
(3) Collinear emission of soft gluons : Here predominantly gluons are not emitted collinearly with the heavy flavor jet unlike the light one due to dead cone effect. the very assumption, ω ≫ k ⊥ , makes it hard to study the cloud of gluons away from the forward direction,which is essential as far as the architectural study of jet is concerned.
The foundation analysis of radiative parton energy loss [4] are based on this 'soft-eikonal-collinear' approximations. Same legacy of this clutchy approximations is also being carried out by the next generation prevailing models of heavy flavor jet quenching [5]. Extrapolating computations of parton energy loss, valid only in certain kinematic domain [ given by (1)], to the full phenomenologically relevant kinematic range induces lots of unavoidable uncertainties [6]. This may also lead to oversight of qualitatively new affair of medium radiation mechanisms for heavy flavor jet.
The main sprite of eikonality is propagation along straight trajectory. In heavy flavor jet models these constraints are implemented both at the level of single emission kernel calculations and in multiple gluon emission estimation schemes. To study quenching of any phenomenologically peppy theory of QCD jet in hot and dense deconfined quark-gluon plasma medium, one requires understanding of medium induced parton energy loss mechanisms beyond the limit of eikonal kinematics. In this article we remove the eikonal jet trajectory restriction (2(a)) partially, and small angle/collinear emission approximation completely, for the radiative process most extensively used light cone gauge with light cone variables 1 . However, we have used the soft gluon emission which automatically include the rest of eikonal parton trajectory (2(b)) assumption 2 . It is known from the previous study with soft-eikonal-collinear approximation that soft gluon emission by bremsstrahlung off heavy flavor jet is considerably suppressed compared to that from a light quark in the forward direction due to the mass effect (the 'dead cone effect' [7,8]). Our study confirms the bremsstrahlung dead cone in eikonal limit. However once the jet is allowed to bend, additional gluons pop up within the so called 'depopulated' dead cone region. As this emission is exclusively a phenomenon due to bending of jet by recoil effect and lies around the direction of outgoing jet, one may reasonably identify it as QCD analog of Quantum Electrodynamic (QED) synchrotron radiation. In QED, (electrodynamically) charged particles moving in curved orbits emit photons within tangential cones of emission, called synchrotron radiation. Similar effect is occuring due to bending of (chromodynamically) charged particles in an ambiance of color field where the interaction is much stronger than that in electromagnetism 3 . Resulting emission pattern of gluons looks surprisingly similar to the standard synchrotron radiation. Till date, though a lot is known about jet showering, yet some puzzles remain, e.g. about the angular energy distribution in the jet close to or away from the trajectory of the leading parton. Our study shows that when recoil is sufficient and heavy flavor jet is able to bend, in addition to conic bremstrulung emission, additional synchrotron like radition starts streaming the relatively depopulated region of dead cone dwelling along the direction of motion of outgoing heavy flavour jet [ Fig. 2].
Among the interactions that a heavy flavor jet goes through as it traverses a dense medium, inelstic(radiative) scattering is, certainly, the most interesting one. The process Q(k 1 )q(k 2 ) → Q(k 3 )q(k 4 )g(k 5 ) at O(α 3 s ), which we have considered in the present work, is represented by five t channel diagrams. In the entrance channel, k 1 and k 2 are momenta of the heavy projectile quark and light target quark respectively. k 3 and k 4 are corresponding outgoing momenta; and k 5 is fourmomentum of the emitted gluon [see Fig. 3]. Scattering angle between three-vectors k 1 and k 3 is θ q , whereas θ g is the angle between the three-vectors k 1 and k 5 . The hierarchy among various scales of momenta, considered herein, is, We note that the above relaxed hierarchy [compared to (1)] among various momenta partially removes the eikonal parton trajectory approximation and completely removes the small angle/collinear approximation. In doing so, it, however, incorporates an additional crappy hierarchy (q ⊥ ≫ k ⊥ ) which was not there in (1). In jet studies with light cone gauge and in light cone coordinate, variable x is defined as the ratio of light cone positive momentum carried by the emitted gluon to that of the parent heavy quark, i.e. x = k + /p + = k ⊥ e ηg / √ s. For sake of brevity we define, ∆ M = M/ √ s, z 2 = t/s, and ζ = q ⊥ / √ s, where t and s are the Mandelstam variables and z is dependent on ζ. The differential gluon emission spectrum containing the recoil effect and applicable in full rapidity range is the following (for details see [12]), where, Here we have used, q = k 1 − k 3 and q ⊥ = q sin θ q . Eq.(3) contains coefficients C 1 , C 0 and C −1 that eventually contain z, and hence the non-eikonal effects which introduces bending in the heavy-flavor jet propagation profile. Also, the expression contains rapidity η g explicitly and the emission angle (hence rapidity η g , again) through the factors F 35 , F 45 and G implicitly. As emission angle θ g can vary from 0 to π (η g runs from −∞ to +∞) 4 , Eq.(3) is completely free from collinear approximation and is pertinent for the full gluon rapidity range. The standard LPM suppression factor is taken inside S x ,k ⊥ . In the eikonal limit, Eq.(3) reproduces the standard result ∝ k 2 ⊥ /(k 2 ⊥ + x 2 M 2 ) 2 , producing a dead cone in forward region [8].
In Figs. [4][5][6][7] we have presented polar plots of W[ω, θ g ] (alternatively W[x, k 2 ⊥ ]) at ζ = 0.0, 0.15, 0.30 and 0.45. It is easy to espy that once ζ is non-zero, so that the heavy flavor leading the jet bends, additional gluons pop-up along the direction of propagation of the outgoing heavy flavor. More the recoil more is this exclusive emission as well as the opening angle defined by that between the two outermost arms in Figs. [4][5][6][7]. In the eikonal limit (z = 0) a 10 GeV Charm produces an opening angle roughly of 24 0 (∼ 2M/E). At ζ = 0.15, 0.30 and 0.45 the opening gradually widens and becomes roughly ∼ 36 0 , ∼ 60 0 and ∼ 72 0 respectively. Fig. [8] depicts the variation of W(x, k ⊥ ), which , in turn, reflects the emission spectrum, with k ⊥ at different ζ(= q ⊥ / √ s) values. The green curve represents the conventional eikonal spectrum of jet models. The red, blue and brown curves are spectrum profile correspond to ζ = 0.15, 0.30 and 0.45 respectively. It can be shown that spectrum integrated over k ⊥ decreases slowly with increasing non-eikonality. On the other hand, k ⊥ is blue-shifted with increasing ζ values.   One can reasonably expect that bending effects might play important role in explaining the emergence of correlations and ridge formations. Two-particle correlations have revealed a long range pseudo-rapidity correlation called the "ridge" in heavy-ion collisions. The ridge is present irrespective of whether a high transverse momentum trigger particle is required or not. In contrast to observations in p+p collisions exhibiting di-jet peaks, twoparticle correlations with a high transverse momentum trigger particle exhibit a broadened and double-peaked structure on the away side in heavy-ion collisions. Location of the away side symmetrical double-peaked structure is offset by an angle 0.3π from the exact rear side. One possible explanation of this phenomenon may lie in the fact that as long as the away side jet traverses the dense medium it may experience sufficient kick from medium partons which might result in the deviation from the straight eikonal trajectory. So, instead of reaching the exact rear side it is deviated yielding the away side double peak. A quantitative estimation of this aspect, embedded into a hydrodynamically evolving density distribution, is underway.
One of the main challenges of contemporary jet study initiatives [13] is to extend the theoretical framework for jet-medium interaction in hot-dense QCD ambiance beyond soft and collinear approximations and reduce uncertainties intrinsic to the current theoretical studies. In that march, the present work removes the collinear approximation completely in single emission kernel calculation thereby probing the cloud of gluon away from the forward direction. This endeavour also removes the eikonal approximation and hence shows the advent of a treatment which allows the jet to bend with non-negligible recoil effect from the scattering in chromo-magnetic ambiance enabling one to treat the color synchrotron radi-ation of color charges.