The non-linear evolution of jet quenching

We construct a generalization of the JIMWLK Hamiltonian, going beyond the eikonal approximation, which governs the high-energy evolution of the scattering between a dilute projectile and a dense target with an arbitrary longitudinal extent (a nucleus, or a slice of quark-gluon plasma). Different physical regimes refer to the ratio $L/\tau$ between the longitudinal size $L$ of the target and the lifetime $\tau$ of the gluon fluctuations. When $L/\tau \ll 1$, meaning that the target can be effectively treated as a shockwave, we recover the JIMWLK Hamiltonian, as expected. When $L/\tau \gg 1$, meaning that the fluctuations live inside the target, the new Hamiltonian governs phenomena like the transverse momentum broadening and the radiative energy loss, which accompany the propagation of an energetic parton through a dense QCD medium. Using this Hamiltonian, we derive a non-linear equation for the dipole amplitude (a generalization of the BK equation), which describes the high-energy evolution of jet quenching. As compared to the original BK-JIMWLK evolution, the new evolution is remarkably different: the plasma saturation momentum evolves much faster with increasing energy (or decreasing Bjorken's $x$) than the corresponding scale for a shockwave (nucleus). This widely opens the transverse phase-space for the evolution and implies the existence of large radiative corrections, enhanced by the double logarithm $\ln^2(LT)$, with $T$ the temperature of the medium. This confirms and explains from a physical perspective a recent result by Liou, Mueller, and Wu (arXiv:1304.7677). The dominant corrections are smooth enough to be absorbed into a renormalization of the jet quenching parameter $\hat q$. This renormalization is controlled by a linear equation supplemented with a saturation boundary, which emerges via controlled approximations from the generalized BK equation alluded to above.


A ubiquitous transport coefficient
In pQCD, all such phenomena find a common denominator: incoherent multiple scattering off the medium constituents L k random kicks leading to Brownian motion in k ⊥ : k 2 ⊥ q ∆t acceleration causing medium induced radiation (BDMPSZ, LPM) multiple branchings leading to many soft quanta at large angles Will this universality survive the quantum ('radiative') corrections ? if so, how will these corrections affect the value ofq ?
An energetic quark acquires a transverse momentum p ⊥ via collisions in the medium, after propagating over a distance L Quark energy E typical p ⊥ =⇒ small deflection angle θ 1 The quark transverse position is unchanged: eikonal approximation The quark is a 'right mover' : x + ≡ (t + z)/ √ 2 √ 2t is its LC time An energetic quark acquires a transverse momentum p ⊥ via collisions in the medium, after propagating over a distance L Quark energy E typical p ⊥ =⇒ small deflection angle θ 1 The quark transverse position is unchanged: eikonal approximation The quark is a 'right mover' : x + ≡ (t + z)/ √ 2 √ 2t is its LC time Transverse momentum broadening (2) Direct amplitude (DA) × Complex conjugate amplitude (CCA) : The p ⊥ -spectrum of the quark after crossing the medium (r = x − y) Average over A − a (the distribution of the medium constituents)

Dipole picture
Formally, S xy is the average S-matrix for a qq color dipole 'the quark at x' : the physical quark in the DA 'the antiquark at y' : the physical quark in the CCA Quark cross-section ←→ dipole amplitude The dipole S-matrix also controls the rate for medium-induced gluon branching (energy loss, jet fragmentation) The tree-level jet quenching parameter n : density of the medium constituents; m D : Debye mass The cross-section for p ⊥ -broadening : The physical jet quenching parameter :q 0 (Q 2 s ) ∝ ln L Radiative corrections to p ⊥ -broadening The quark 'evolves' by emitting a gluon ('real' or 'virtual') The 'evolution' gluon is not measured: one integrates over ω and k All partons undergo multiple scattering: non-linear evolution dω ω d 2 k k 2 The emission requires a formation time τ 2ω/k 2 ⊥ For our present purposes, better use τ instead of ω τ can take all the values between λ ∼ 1/T and L For a given τ , k 2 ⊥ should be larger thanqτ (multiple scattering) but smaller than Q 2 s =qL (dipole resolution r ∼ 1/Q s ) The phase space The radiative corrections are suppressed by powers if α s ...
... but can be enhanced by the phase-space for gluon emissions A 'naive' argument: bremsstrahlung in the vacuum τ can take all the values between λ ∼ 1/T and L For a given τ , k 2 ⊥ should be larger thanqτ (multiple scattering) but smaller than Q 2 s =qL (dipole resolution r ∼ 1/Q s ) The phase space The radiative corrections are suppressed by powers if α s ...
... but can be enhanced by the phase-space for gluon emissions A 'naive' argument: bremsstrahlung in the vacuum τ can take all the values between λ ∼ 1/T and L For a given τ , k 2 ⊥ should be larger thanqτ (multiple scattering) but smaller than Q 2 s =qL (dipole resolution r ∼ 1/Q s )

Non-linear evolution
The previous argument is 'naive' as it ignores multiple scattering Non-linear evolution is well understood for a shock-wave target proton-nucleus collisions at RHIC or the LHC Generalization of the JIMWLK (or BK) equations to an extended target ('medium') (E.I., arXiv: 1403.1996) The BK equation for jet quenching × S L,t 2 (x, y) S t 2 ,t 1 (x, r(t)) S t 2 ,t 1 (r(t), y)S t 1 ,0 (x, y) − S L,0 (x, y)  Very different from the respective evolution for a shock wave: stronger dependence of Q 2 s upon τ (or 1/x) To DLA, the dipole S-matrix S L (r) preserves the same functional form as at tree-level, but with a renormalizedq : Universality :q 0 (L) →q(L) in all the quantities related to S p ⊥ -broadening, radiative energy loss, jet fragmentation ...

BK equation reduces to a relatively simple, linear, equation forq(L)
The double logarithmic approximation To DLA, the dipole S-matrix S L (r) preserves the same functional form as at tree-level, but with a renormalizedq : Universality :q 0 (L) →q(L) in all the quantities related to S p ⊥ -broadening, radiative energy loss, jet fragmentation ... Multiple scattering is tantamount to gluon saturation in the target Q 2 s (x) is proportional to the width of the region where a gluon (with longitudinal fraction x) can overlap with its sources for a shockwave, this region is the SW width L (fixed and small) for a gluon in the medium, this is the gluon longitudinal wavelength: The x-dependence of Q 2 s (x) is further amplified by the evolution

Fixed coupling
Use logarithmic variables, as standard for BFKL, or BK: Y ≡ ln τ λ ('rapidity') and ρ ≡ ln Not the standard DLA (as familiar from studies of DGLAP, or BFKL) ! saturation boundary: ρ 1 ≥ Y 1 (multiple scattering) Straightforward to solve via iterations (Liou, Mueller, Wu, 2013) The standard artifact of using a fixed coupling (recall e.g. BK) Running coupling (E.I. Triantafyllopoulos, arXiv:1405.3525) One-loop QCD running coupling :ᾱ →ᾱ(ρ 1 ) ≡ b The standard DLA with RC (no saturation boundary) would givê The actual solution is very different (and much more complicated !) Once again, the cross-section can be related to (adjoint) dipoles: However, the radiated gluon is relatively hard, k + ∼ ω c , so the hierarchy is preserved between radiation and fluctuations: ω k + during the relatively short lifetime t 2 − t 1 = τ of the fluctuation (ω), the radiated gluon (k + ) can be treated as eikonal Then the same arguments apply as in the case of p ⊥ -broadening: q (0) →q τ f (k 2 ⊥ ) ... in agreement with J.-P. Blaizot and Y. Mehtar-Tani