On the momentum broadening of in-medium jet evolution using a light-front Hamiltonian approach

Following the non-perturbative light-front Hamiltonian formalism developed in our preceding work [Phys.Rev.D 104 (2021) 5, 056014], we investigate the momentum broadening of a quark jet inside a SU(3) colored medium. We perform the numerical simulation of the real-time jet evolution in Fock spaces of a single quark, a quark-gluon state, and coupled quark- and quark-gluon states at various jet momenta $p^+$ and medium densities. With the obtained jet light-front wavefunction, we extract the jet transverse momentum distribution, the quenching parameter, and the gluon emission rate. We analyze the dependence of momentum broadening on $p^+$, medium density, color configuration, spatial correlation, and medium-induced gluon emission. For comparison, we also derive analytically the expectation value of the transverse momentum of a quark-gluon state in any color configuration and in an arbitrary spatial distribution in the eikonal limit. This work can help understand jet momentum broadening in the non-eikonal regime.


Introduction
In heavy-ion collisions, energetic quarks and gluons are produced at early stages, propagating through the dense and hot medium.Similar processes happen in deeply inelastic scattering where quark and gluon jets traverse cold nuclear matter.We developed a non-perturbative computational method, the time-dependent Basis Light-Front Quantization (tBLFQ) [1], for simulating the evolution of a quark jet inside a classical color background field, first for a | state [2], then also including | components [3,4].Unlike pQCD-based approaches, tBLFQ calculates the evolution process at the amplitude level, and enables relaxation of the eikonal and collinear radiation approximations.

Methodology
We consider a high-energy quark jet moving in the positive  direction, traversing a medium moving in the negative  direction.We treat the quark as a quantum state and the medium as an external background field, with the interaction occurring over a finite distance 0 ≤  + ≤   .

The light-front Hamiltonian in the |𝑞 + |𝑞𝑔 space
The Lagrangian for the process being considered is the QCD Lagrangian with an external field, where , and   =   + A  is the sum of the quantum gauge field   and the background gluon field A  .The light-front Hamiltonian is obtained through Legendre transformation in the light-cone gauge  + = A + = 0.In the truncated Fock space | + | , it consists of three parts,  − ( + ) =  −   +   +  A ( + ), the kinetic energy, the interaction between the quark and the dynamical gluon, and the medium interaction, The background field A  accounts for the target, and we describe it using the McLerran-Venugopalan (MV) model [5].It is a classical field satisfying the reduced Yang-Mills equation, The saturation scale is related to μ as

Evolution of the state in a basis representation
The evolution of quantum states is governed by the time-evolution equation on the light front.In the interaction picture (denoted by the subscript ), the equation reads with the interaction Hamiltonian   ( The interaction picture state is related to the Schrödinger picture state by |;  +  =   1 2  −    + |;  + .We implement a non-perturbative treatment by decomposing the time-evolution operator into many small steps of the light-front time  + , then solving each timestep in the sequence numerically, with T + denoting light-front time ordering.The step size is  + ≡  + /, and the intermediate time is  +  =  + ( = 0, 1, 2, . . ., ) with  + 0 = 0 and  +  =  + .We use a lattice with periodic boundary conditions in the transverse dimensions ì  ⊥ , ranging in [− ⊥ ,  ⊥ ] with 2 ⊥ sites such that  ⊥ =  ⊥ / ⊥ is the lattice spacing, and a loop with (anti-)periodic boundary condition in the  − direction, of length 2, for the gluon(quark).The lattice introduces infrared (IR) and ultraviolet (UV) cutoffs in the transverse momentum space ì  ⊥ ,    =   = / ⊥ and    = / ⊥ .The longitudinal momentum  + is quantized in units of 2/, and the gluon(quark) is allowed to take a positive (half-)integer number in this unit.The total momentum is denoted as  + = 2/ with  a half-integer.Then the longitudinal momentum fraction of the gluon,  ≡  +  / + , has a resolution of 1/.In the chosen discrete basis representation, the state is a column vector of basis coefficients, and the Hamiltonian is in the matrix form.The numerical method for this specific problem is optimized in Ref. [3].In short, within each small time step  + , we treat  −   and  A as time-constant and carry out matrix exponentiation in the momentum and coordinate space, respectively; the operation with   uses the fourth-order Runge-Kutta method in the momentum space.

Result
In studying the phenomenon of jet momentum broadening inside a medium, we examine the expectation value of the transverse momentum square  2 ⊥ ( + ) , and the related quenching parameter defined as q = Δ  2 ⊥ ( + ) /Δ + .

Eikonal analytical result
In the eikonal limit of  + = ∞, only the  A term survives in the Hamiltonian, and the evolution operator reduces to the Wilson line, then  2 ⊥ ( + ) and q can be derived analytically using the Wilson line correlators.Here, we present the derivation result for the single quark/gluon state, and a novel derivation for the quark-gluon state following Ref.[4].

Single-particle state
The Wilson line of a quark is   (0,  + ; ì is the SU(3) generator in the fundamental representation.Replacing   by the generators in the adjoint representation,   , one gets the adjoint Wilson line for the gluon,   (0,  + ; ì  ⊥ ).The momentum transfer can be evaluated from the Wilson line correlator Meĳian Li The quenching parameter q follows as, In analogy, one gets the gluon q replacing   by   =   in Eq. (8).

Quark-gluon state
The quark-gluon Wilson line is built as the tensor product of a quark and a gluon Wilson line, The probability distribution of the quark-gluon state is then given by Wilson line correlator  †    , and the state's transverse momentum square can be calculated accordingly.It contains three terms, The first two terms are the same as Eq. ( 7) with the corresponding Casimir, and the third term depends on the initial color configuration and the quark-gluon separation [4], in which  represents the initial color configuration of the state, and The quantity   (ì  ⊥ ) is the distribution function of the quark-gluon relative coordinate ì  ⊥ = ì  ,⊥ − ì  ,⊥ , and it can be obtained by integrating the wavefunction square over the center-of-mass coordinate ì

Non-eikonal numerical result
We perform the simulations in the | + | space, with the initial state as a single quark of ì  ⊥ = ì 0 ⊥ at a finite  + .To quantify the medium-induced gluon emission, we define  | as the difference of the probability of the quark jet in the | sector in the medium and that in the vacuum, The result is shown in Fig. 1.In the left panel, we see that the  | curve forms a slight dip at an early time, then after around the point  + = 12 GeV −1 , grows linearly in time.Additionally, a We then analyze the non-eikonal and radiative correction to the momentum broadening.We define   2 ⊥ and  q as the difference of the quantity that is calculated from the total momentum of the quark jet in the | + | space, and the eikonal result of a bare quark [as in Eq. (8)], The results are shown in Fig. 2:   2 ⊥ increases over the evolution time at various   , and  q extracted from the final state   2 ⊥ increases non-trivially when   increases.

Summary
We present a study on the momentum broadening of in-medium jet evolution using the tBLFQ approach [3,4], a non-perturbative light-front Hamiltonian formalism.We first provide a novel analytical derivation of the eikonal expectation value of the quark-gluon state's transverse momentum for any color and spatial distribution.We then perform the numerical simulation of the real-time jet evolution in the Fock space of | + | at various medium densities.With the obtained jet light-front wavefunction, we extract the gluon emission rate and the quenching parameter.We find their non-eikonal contributions sizable, time-dependent, and associated with saturation scale.