In-medium QCD splittings beyond the soft, large-$N_c$ and harmonic-oscillator approximations all at once

Technical Summary: In-medium QCD splittings beyond the soft, large-$N_c$ and harmonic-oscillator approximations

Problem Statement
The extraction of material properties of the quark-gluon plasma (QGP) relies heavily on the theoretical description of high-energy particle jets evolving within the medium. This evolution is governed by in-medium QCD splitting kernels, which encode the probability for a parton to branch into two daughter partons. The foundational BDMPS-Z formalism provides a path-integral formulation for these kernels but has historically been restricted to specific limiting cases due to the complexity of the resulting equations. Previous solutions typically relied on simultaneous simplifications: the soft limit ($z \to 0$ or $z \to 1$), the large-$N_c$ approximation (simplifying color algebra), and the harmonic-oscillator approximation (modeling parton-medium interactions via multiple soft momentum exchanges). While recent efforts have relaxed one or two of these assumptions individually, a complete numerical solution of the BDMPS-Z equations in their full generality—simultaneously incorporating finite-energy effects, subleading-color contributions, and realistic scattering potentials—remained an open problem.

Methodology
The authors present the first complete numerical solution to the BDMPS-Z equations for arbitrary energy fractions $z$ at finite $N_c$, fully resumming jet-medium interactions to all orders in opacity. The approach involves several key technical steps:

1. Formulation: Starting from the differential emission spectrum expressed as a path integral over Wilson lines, the authors decompose the cross-section into vacuum and medium-modified contributions. The medium modification factor, $R_{med}(z, p)$, is isolated by computing ratios relative to the vacuum splitting kernel.
2. Evolution Equations: The problem is recast into a system of non-Hermitian Schrödinger-like equations for Green's functions. Specifically, the authors solve for two objects:
   • $\mathbf{A}(t, p)$: Describing the evolution in the region where the splitting occurs partially inside and partially outside the medium ($t_1 < L < t_2$).
   • $\mathbf{B}(t, k, l)$: Describing the evolution where the splitting occurs entirely within the medium ($t_1, t_2 < L$). This requires solving a coupled system of differential equations for different color configurations ($\sigma$), with dimensions ranging from 2 to 6 depending on the splitting channel ($g \to q\bar{q}$, $q \to gq$, $g \to gg$).
3. Dimensionality Reduction: By exploiting the rotational symmetry of the medium and switching to variables $k = (p - \bar{p})/2$ and $l = (p + \bar{p})/2$, the dimensionality of the differential equations is reduced. The system is solved on a grid where only the subspace $l=0$ carries the physical information for the final spectrum.
4. Numerical Scheme: To overcome stability issues associated with explicit methods (like Runge-Kutta) for long propagation times and stiff equations, the authors employ a spectral method based on a Faber polynomial expansion of the evolution operator. This approach scales the Hamiltonian to ensure its spectrum lies within a complex ellipse, allowing for geometric convergence of the series. This method decouples expansion accuracy from the time step size, removing the stringent stability constraints of explicit schemes and enabling robust solutions across the entire kinematic phase space.
5. Medium Model: The calculations utilize the Gyulassy-Wang (GW) model, treating the medium as a collection of static scattering centers with a realistic scattering potential, rather than the simplified harmonic oscillator potential.

Key Results
The authors compute the medium modification factor $R_{med}(z, k_T)$ for various splitting channels ($q \to gq$, $g \to gg$, $g \to q\bar{q}$) using LHC-relevant kinematic parameters ($p^+ = 100$ GeV and $1$ TeV) and medium properties corresponding to central heavy-ion collisions.

• Kinematic Dependence: The results reveal a complex parametric dependence of $R_{med}$ on both medium properties and splitting kinematics. In the bulk of phase space, medium corrections exhibit non-trivial behavior, increasing significantly for unbalanced splittings (reaching factors of 3–3.5 as $z \to 0$ or $z \to 1$). Conversely, for high parent energies ($p^+$), $R_{med}$ can fall below unity, indicating medium-induced suppression.
• Finite-$N_c$ Corrections: The authors quantify deviations from the large-$N_c$ limit using the metric $\Delta N_c$. They find that while subleading color contributions are generally small, they are non-zero across all channels at finite $z$. In the soft limits ($z \to 0, 1$), color oscillations vanish, and deviations are driven solely by corrections to the Casimir factors ($C_F$ vs $N_c/2$). Notably, for $g \to gg$, $\Delta N_c$ vanishes in the soft limits, whereas for $g \to q\bar{q}$ and $q \to gq$, it remains non-zero due to the dominance of $C_F$ in specific limits.
• Comparison with Approximations: Comparing the exact numerical results with the Improved Opacity Expansion (IOE) semi-analytic approximation (which relies on the soft limit and harmonic oscillator potential), the authors find significant deviations. For $k_T \sim Q_s$ and $z \sim 0.5$, discrepancies reach up to 20%, highlighting the importance of finite-$z$ and realistic potential corrections.

Significance and Claims
The paper claims to establish a new baseline for all QCD in-medium splitting functions by overcoming long-standing analytical simplifications (soft, large-$N_c$, and harmonic-oscillator approximations) simultaneously. The authors assert that their numerical framework provides a precise determination of in-medium splitting functions that is significantly beyond the state-of-the-art.

The significance of this work lies in its ability to:

1. Quantify uncertainties associated with standard approximations in the literature, revealing substantial deviations across phase space.
2. Provide a robust, numerically efficient tool for calculating jet observables that is not limited by specific kinematic or color approximations.
3. Enable powerful new constraints on medium parameters from high-energy heavy-ion collider data by offering more precise theoretical predictions for jet quenching phenomena.

The authors conclude that this framework opens a path toward more precise calculations of jet observables, particularly those with reduced sensitivity to non-perturbative corrections (e.g., groomed substructure) and resummation effects, and facilitates the integration of exact splitting kernels into state-of-the-art Monte Carlo event generators for heavy-ion collisions.