Improved Opacity Expansion in QCD Media

• Improved opacity expansion is a systematic method that resums multiple soft scatterings into a harmonic oscillator framework, achieving rapid convergence.
• It seamlessly interpolates between dilute single-hard and dense multiple-soft regimes, providing analytic control over jet quenching observables in heavy-ion collisions.
• Its expansion structure, organized as LO, NLO, and NNLO corrections, yields accurate and efficient predictions for medium-induced radiation in QCD media.

The improved opacity expansion (IOE) is a systematic expansion technique for computing medium-induced parton radiation and splitting rates in dense and finite QCD media. It generalizes the standard opacity expansion by resumming the leading logarithms associated with multiple soft scattering into a harmonic oscillator baseline, then treating the residual hard tail perturbatively. This approach provides analytic control and rapid convergence for both dilute (single-hard) and dense (multiple-soft) regimes, and has wide applicability to jet quenching, transverse-momentum broadening, and radiative energy loss in heavy-ion collisions and related phenomena.

1. Formalism and Mathematical Structure

The IOE is constructed by splitting the in-medium potential for transverse parton dynamics into two pieces: $$v(\mathbf{r}, t) = v_{\mathrm{HO}}(\mathbf{r}, t) + \delta v(\mathbf{r}, t),$$ where $v_{\mathrm{HO}}$ is a harmonic oscillator (HO) term containing the full soft multiple-scattering physics,

$$v_{\mathrm{HO}}(\mathbf{r}) = \frac{1}{4} \mathbf{r}^2\, \hat q_{\mathrm{eff}}(Q_{\mathrm{sub}}^2, t),$$

with the effective jet-quenching parameter,

$$\hat q_{\mathrm{eff}}(Q_{\mathrm{sub}}^2, t) = \frac{\hat q_0(t)}{2}\left[1 + \left(\frac{2C_R}{N_c} - 1\right)z^2 + (1-z)^2\right] \ln\left(\frac{a Q_{\mathrm{sub}}^2}{\mu^2}\right),$$

and the subtraction scale $Q_{\mathrm{sub}}^2$ self-consistently set by the typical transverse momentum exchange during formation,

$$Q_{\mathrm{sub}}^2 \simeq \sqrt{\omega\, \hat q_{\mathrm{eff}}(Q_{\mathrm{sub}}^2)}.$$

The remainder, $\delta v(\mathbf{r}, t)$, encodes the non-Gaussian hard tail,

$$\delta v(\mathbf{r}) = \frac{1}{4} \hat q_0 \mathbf{r}^2 \ln\left(\frac{1}{\mathbf{r}^2 Q_{\mathrm{sub}}^2}\right).$$

The in-medium splitting spectrum is written as

$$z\frac{dI_{b\leftarrow a}}{dz} = \frac{\alpha_s\,z\,P_{b\leftarrow a}(z)}{[z(1-z)E]^2} \,2\Re\int_0^\infty dt_2 \int_0^{t_2} dt_1 \left[\mathcal{K}_{ba}(\mathbf{r}=0, t_2; \mathbf{r}=0, t_1) - \mathcal{K}_0(\mathbf{r}=0, t_2; \mathbf{r}=0, t_1)\right],$$

where $\mathcal{K}$ is the full Green's function in the combined potential.

The IOE consists in expanding $\mathcal{K}$ in powers of $\delta v$ around the exact HO result, organizing the series as

$$\mathcal{K} = \mathcal{K}_{\mathrm{HO}} - \int d^2u \int_{t_0}^{t_2} ds\, \mathcal{K}_{\mathrm{HO}}(0, t_2; \mathbf{u}, s)\, \delta v(\mathbf{u}, s)\, \mathcal{K}_{\mathrm{HO}}(\mathbf{u}, s; 0, t_1) + \mathcal{O}(\delta v^2).$$

This functionally corresponds to summing all soft exchanges and perturbing by hard kicks.

2. Physical Motivation and Regimes of Applicability

The main theoretical motivation for the IOE is the failure of the standard (GLV) opacity expansion for dense media ($L/\ell_{\mathrm{mfp}}\gtrsim 1$), where multiple soft scatterings and Landau-Pomeranchuk-Migdal (LPM) interference effects dominate. While GLV is suited to dilute regimes (single hard scattering), the BDMPS-Z formalism resums Gaussian soft scatterings in a harmonic oscillator approximation, valid for high opacity (dense media).

The IOE interpolates between these two analytic regimes:

• Multiple-soft (BDMPS-Z/HO): spectrum exhibits LPM suppression, $$\omega\frac{dI}{d\omega} \sim 2\sqrt{\omega_c/(2\omega)}$$ for $\omega\ll\omega_c=\frac{1}{2}\hat q L^2$.
• Single-hard (GLV): spectrum is dominated by one hard scattering, giving $$\omega\frac{dI}{d\omega} \sim \frac{\pi}{4} \frac{L^2}{2\omega}$$ for $\omega\gg\omega_c$.

In the IOE, the leading order (LO) resums the soft sector (HO/BDMPS-Z), and the perturbative terms (NLO, NNLO, etc.) systematically restore the correct hard tail.

3. Explicit Expansion and Convergence Properties

The IOE is organized as a rapidly converging series: $$\omega \frac{dI}{d\omega} = \omega \frac{dI^{\mathrm{LO}}}{d\omega} + \omega \frac{dI^{\mathrm{NLO}}}{d\omega} + \omega \frac{dI^{\mathrm{NNLO}}}{d\omega} + \cdots$$

• LO (HO): Captures the Gaussian broadening, LPM effect, and soft multiple scatterings.
• NLO: First hard insertion, restores the $1/\omega$ GLV tail and provides leading Coulomb logarithmic corrections.
• NNLO: Second hard insertion, gives subleading corrections ($\sim 1/\omega^2$), typically a few percent or less for realistic parameters.

For example, for gluon radiation in a QCD medium of length $L$, the NLO correction is given by

$$\omega \frac{dI^{\mathrm{NLO}}}{d\omega} = \frac{\bar\alpha\,\hat q_0}{2\pi} \Re \int_0^L ds\, \ln\left(\frac{1}{Q^2 s^2}\right) \int d^2z\, e^{k^2(s) z^2},$$

where $\bar\alpha=\alpha_s C_R/\pi$, and $k^2(s)$ encodes the HO propagator.

Numerical studies demonstrate rapid convergence: at LHC-inspired parameters, NNLO corrections are a few percent or less compared to LO+NLO across relevant $\omega$ (Barata et al., 2020). Truncation at NLO provides accurate results, and matching to single-inclusive GLV and fully resummed solutions is robust.

4. Matching, Separation Scales, and Uncertainties

The IOE requires the specification of matching/separation scales for the HO resummation ($Q_{\mathrm{sub}}^2$), set self-consistently by

$$Q^2_{\mathrm{sub}} \simeq \sqrt{\omega\, \hat q_0\ln(a\,Q_{\mathrm{sub}}^2/\mu^2)}.$$

Varying the subtraction scale within reasonable ranges yields uncertainty bands—tens of percent near the crossover ($\omega\sim\omega_c$), but much less at large or small $\omega$. For soft gluons ($\omega \ll \omega_c$), the choices matter little (single-hard kicks are subdominant); at high frequency, the tail is robust (hard emission dominates).

The significance of this matching lies in the clean interpolation between dilute and dense regimes, providing analytic results without ad hoc matching prescriptions.

5. Generalizations, Sub-Eikonal and Flow Corrections

Recent extensions of IOE incorporate sub-eikonal corrections to account for medium motion, flow, and gradients (Sadofyev et al., 2022, Sadofyev et al., 2021). At first sub-eikonal order ($\mathcal{O}(1/E)$), the medium-induced emission kernel acquires corrections proportional to local flow velocity and gradients,

$$\frac{dN^{(1)}}{dx\,d^2k_\perp} = \frac{\alpha_s\,C_R}{\pi^2} \int_0^L dz \int d^2q_\perp\, \bar\sigma(q_\perp)\, P_{gq}(x)\, [ \mathcal{K}_0 + \vec u_\perp(z)\cdot \Gamma_{\mathrm{rad}} ],$$

where $\Gamma_{\mathrm{rad}}$ encodes anisotropic flow effects.

These corrections lead to anisotropic diffusion of jet broadening and medium-induced radiation aligned with the medium flow, enabling event-by-event "jet tomography" of QGP velocity fields and cold nuclear matter structures, and can be coupled directly to hydrodynamic simulations.

6. Practical Implementation and Phenomenological Applications

The IOE is designed for straightforward embedding in radiative transport, jet quenching, and Monte Carlo event generators:

• Inputs: medium profiles ($T[x, t]$, $n[t]$), jet energy $E$, coupling $\alpha_s$, Debye mass $\mu$.
• For each candidate splitting, solve for $\hat q_{\mathrm{eff}}(\omega, t)$ and matching scale.
• Compute emission rates using the IOE master formulas, sampling recoil according to HO + NLO kernels.
• No explicit separation between "soft" and "hard" kicks; IOE provides smooth interpolation.
• Closed-form LO+NLO expressions ensure computational efficiency.

Phenomenologically, the IOE methodology enables high-precision calculations for jet quenching observables, broadening, radiative energy loss, and sub-jet structure in heavy-ion collisions. It also extends to evolving media, e.g., in deep inelastic scattering off cold nuclei.

7. Limitations, Outlook, and Open Problems

The IOE shows rapid convergence for all practical applications ($\omega \gtrsim \omega_{\mathrm{BH}}$), and truncation at NLO suffices for single-inclusive observables. Remaining limitations include:

• Near or below the Bethe-Heitler regime ($\omega\sim T^4/\hat q_0$), the approach loses control and further resummations become necessary.
• Inclusion of finite length, finite $x$, and full angular distributions demands further development.
• Dynamical media, running coupling, and heavy-quark mass effects are open extensions.

Ongoing work includes systematic treatment of higher-order corrections, Monte Carlo implementation guidelines, and application to full in-medium parton showers.

Table: IOE vs. Standard Opacity Expansion

Approach	Expansion Parameter	Soft Resummation	Hard Tail Matching
Standard Opacity (GLV)	$\chi = L/\lambda$	No	Manual matching; poor convergence
IOE (Improved)	$1/\ln(Q_{\mathrm{sub}}^2/\mu^2)$	All orders (HO)	Systematic (NLO, NNLO, fast convergence)

The IOE thus provides a unified, fast-converging analytic expansion for medium-induced radiation, systematically connecting the BDMPS-Z multiple-soft regime and the GLV single-hard limit, with broad utility for jet quenching phenomenology in QCD media (Mehtar-Tani et al., 2019, Mehtar-Tani, 2020, Barata et al., 2020, Barata, 2021, Barata et al., 2021, Sadofyev et al., 2022, Sadofyev et al., 2021).