# Medium-induced coherent gluon radiation for $2 \to 2$ processes with general kinematics

Greg Jackson, Stéphane Peigné, Kazuhiro Watanabe

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Greg Jackson, Stéphane Peigné, Kazuhiro Watanabe. Medium-induced coherent gluon radiation for $2 \to 2$ processes with general kinematics. 12th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions: Hard Probes 2024 (HP2024), Sep 2024, Nagasaki, Japan. pp.04010, 10.1051/epjconf/202533904010. [hal-05404155](https://hal.science/hal-05404155v1)

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# Medium-induced coherent gluon radiation for $2 \to 2$ processes with general kinematics

Greg Jackson¹,*, Stéphane Peigne¹,**, and Kazuhiro Watanabe²,***

¹SUBATECH UMR 6457 (IMT Atlantique, Université de Nantes, IN2P3/CNRS), 4 rue Alfred Kastler, 44307 Nantes, France

²Department of Physics, Tohoku University, Sendai 980-8578, Japan

**Abstract.** High-energy proton-nucleus (pA) collisions offer valuable insight into the role of cold nuclear matter effects in hadron production. In particular, multiple rescatterings of an incoming parton by the nuclear target are known to induce the radiation of many soft gluons. Hadron production is affected by fully coherent energy loss (FCEL), owing to those radiated gluons with a long formation time. Here we present a recently derived formula for the induced single soft gluon radiation spectrum beyond leading logarithmic accuracy, whose main features are demonstrated with the example of $qg \to qg$ scattering.

## 1 Introduction

Heavy flavor production in high-energy nuclear collisions has been an intriguing probe of QCD medium interactions at different stages of the collision [1]. Experimental data accumulated at collider facilities like the LHC and RHIC heralds a precision era for studying physics in high-energy nuclear collisions, addressing QCD dynamics at high temperature and ‘cold’ nuclear matter effects, which were the main topics discussed in the conference.

Proton-nucleus (pA) collisions have provided valuable opportunities to study various effects of cold nuclear matter on particle production. Compared to proton-proton (pp) collisions, partons participating in hard scatterings in pA collisions will likely undergo multiple interactions in nuclear targets. Such multiple rescatterings induce the emission of many soft gluons that could have a formation time $t_f \sim \omega/k_\perp^2$ with $\omega$ and $k_\perp$ the energy and transverse momentum of the radiated gluon, respectively, larger than L, the length traversed through the medium [2]. This results in the modification of hadron production rates due to fully coherent energy loss (FCEL).

The parametric dependence of the corresponding average energy loss is $\Delta E_{FCEL} \sim E$ with E being the energy of the incoming parton from the projectile. Of particular importance is that medium-induced coherent energy loss overwhelms the QCD Landau-Pomeranchuk-Migdal (LPM) effect, valid for $t_f < L$, when $E \to \infty$ because $\Delta E_{LPM}/E \to 0$ [3–6]. Therefore, medium-induced FCEL has proven to be crucial in explaining heavy meson ($J/\psi$, D) and light hadron nuclear suppression in pA collisions in a wide range of collision energies from FNAL, RHIC, to LHC [7–11]. Understanding both the qualitative and quantitative role of FCEL in suppressing hadron production over a broad kinematic range is a crucial task. Addressing this requires rigorous calculations of the medium-induced soft gluon radiation spectrum.

<footnote>*e-mail: jackson@subatech.in2p3.fr</footnote>

<footnote>**e-mail: peigne@subatech.in2p3.fr</footnote>

<footnote>***e-mail: kazuhiro.watanabe.b8@tohoku.ac.jp (speaker)</footnote>
We have recently derived new results for this spectrum (for all $2 \to 2$ scatterings in QCD) beyond leading logarithmic accuracy, enhancing the predictive power of FCEL estimations in the phenomenology of hadron production in pA collisions [12]. The general formula is valid in the full kinematic range of the underlying scattering processes. It should provide an important baseline to evaluate more accurately FCEL effects on hadron production. This paper aims to review the medium-induced soft gluon spectrum and illustrate how the new formula for $2 \to 2$ scatterings can be 'matched' to the original ones obtained for $2 \to 1$ forward processes.

## 2 Recap of FCEL at LLA

Let us begin with the induced soft gluon spectrum for $2 \to 1$ scattering, such as $qg \to q$ and $gg \to g$. We shall consider forward scattering of an asymptotic fast parton (labelled 1) with large $E$, traversing the target medium in the target rest frame. The fast parton from the projectile acquires (due to multiple soft scatterings in the medium) an averaged momentum kick $l_{\perp}^2 \propto L$ in addition to a single hard scattering (by parton labelled 2) which delivers a transverse momentum $q_{\perp}$ to the final parton (labelled 3, and 'tagged' with a transverse mass $m_{\perp} = \sqrt{m^2 + q_{\perp}^2}$). It then induces soft gluon emission far before and after the scatterings. The induced soft-gluon spectrum $dI/d\omega$ is obtained from an interference between initial state gluon radiation and final state gluon radiation. In the leading-log approximation (LLA) [13], it reads

$$ \omega \frac{dI}{d\omega} \Big|_{2 \to 1} \approx F_c \frac{\alpha_s}{\pi} \left[ \ln \left( 1 + \frac{l_{A\perp}^2 E^2}{\omega^2 m_{\perp}^2} \right) - (l_{A\perp} \to l_{p\perp}) \right]. \quad (1) $$

Here, $F_c$ is a color factor given by a combination of Casimir operators associated with the partons involved in the hard scattering. A general rule for $2 \to 1$ processes is: $F_c = C_1 + C_3 - C_2$, where $C_{1,2}$ are the Casimirs of the incoming partons and $C_3$ is that of the final-state parton. For example: In $gg \to g$, $F_c = N_c > 0$ (energy loss) and in $qg \to q$, $F_c = -1/N_c < 0$ (energy gain).

Next, we consider the induced soft gluon spectrum for $2 \to 2$ partonic process in the LLA, which is relevant for dijet and open heavy flavor production. Allowing both final partons to have mass $m$, we introduce the usual Mandelstam variables; $s = (p_1 + p_2)^2$, $t = (p_1 - p_3)^2$ etc., and define: $\xi \equiv (m^2 - t)/s$ and $\tilde{\xi} \equiv (m^2 - u)/s = 1 - \xi$. Then $\xi \in [0, 1]$ and the dijet invariant mass in the final state is $M_{\xi}^2 = m_{\perp}^2/(\xi(1-\xi))$. In the LLA, the radiated soft gluon sees only the global color of the final state (the final state is viewed as a "point-like dijet" by the radiation) [14], and we have:

$$ \omega \frac{dI}{d\omega} \Big|_{2 \to 2} = \sum_{\alpha} \rho_{\alpha} F_{\alpha} \frac{\alpha_s}{\pi} \left[ \ln \left( 1 + \frac{l_{A\perp}^2 E^2}{\omega^2 M_{\xi}^2} \right) - (l_{A\perp} \to l_{p\perp}) \right], \quad (2) $$

where $\alpha$ labels an irreducible representation (irrep) of the color state of the final parton pair, with associated probability $\rho_{\alpha}$ and Casimir $C_{\alpha}$. Here $F_{\alpha} = C_1 + C_{\alpha} - C_2$ with $C_{\alpha}$ the global color charge of the final parton pair and $C_{1,2}$ the Casimir of the initial partons. Comparing with eq. (1), we see that $m_{\perp}$ is generalized to $M_{\xi}$, which now depends on the scattering kinematics via $\xi$. The LLA above requires that $\xi \sim \tilde{\xi} \sim 1/2$, because the logarithm in eq. (2) becomes small for $\xi \to 0$ or $1$.

Let us note that the spectrum (2) arises from the logarithmic domain $\omega M_{\xi}/E \ll k_{\perp} \ll l_{A\perp}$ with $k_{\perp}$ the radiated gluon transverse momentum, allowing to interpret physically the LLA as the limit of purely non-abelian medium-induced radiation (see Ref. [14] for a discussion). In the next section, we will explain how eq. (2) may be generalized beyond this limit, to the full kinematic range $\xi \in [0, 1]$.
follows:

$$ \left. \frac{dI}{dx} \right|_{\xi = \frac{1}{2}} = \frac{\alpha_s}{\pi x} \sum_{\alpha^s} \Phi_{\alpha^s \alpha^s}^{s} (C_1 + C_{\alpha^s} - C_2) \mathcal{L}_{\xi=1/2}, \qquad (9) $$

$$ \left. \frac{dI}{dx} \right|_{\xi=0} = \frac{\alpha_s}{\pi x} \sum_{\alpha^t} \Phi_{\alpha^t \alpha^t}^{t} (C_1 + C_3 - C_{\alpha^t}) \mathcal{L}_{\xi=1}, \qquad (10) $$

$$ \left. \frac{dI}{dx} \right|_{\xi=1} = \frac{\alpha_s}{\pi x} \sum_{\alpha^u} \Phi_{\alpha^u \alpha^u}^{u} (C_1 + C_4 - C_{\alpha^u}) \mathcal{L}_{\xi=1}, \qquad (11) $$

where the superscript s, t, or u indicates the channel in which the sum over available irreps α is to be considered, and in which the matrix Φ is expressed.

## 4 Illustration: quark-gluon scattering

Here, we will consider the case of $qg \to qg$ scattering as an illustration, where in accordance with the definitions below eq. (2), $\xi$ corresponds to the energy fraction of the final gluon in the nucleus rest frame. The amplitude for this process reads

$$ M_{qg \to qg} = \mathcal{F} \left[ \begin{tikzpicture}[baseline=(current bounding box.center)] \draw (-0.5,0) -- (0.5,0); \node[above] at (-0.5,0) {$\xi$}; \draw (-0.5,-0.5) -- (-0.5,0.5); \node[below] at (-0.5,-0.5) {$-\xi$}; \draw (-0.5,-0.5) -- (0.5,0.5); \node[above] at (0.5,0.5) {$\xi$}; \draw (0.5,0) -- (0.5,-0.5); \node[below] at (0.5,-0.5) {$-\xi$}; \end{tikzpicture} \right], \qquad (12) $$

where the coefficient $\mathcal{F}$ contains most of the Lorentz structure, and the diagrams represent the color.

In the preferred basis for each of the limiting cases $\xi = \frac{1}{2}$, $\xi = 0$, and $\xi = 1$, the spectrum takes the form specified by eq. (9), eq. (10) and eq. (11), where the corresponding available irreps are $\alpha^s = \{3, \bar{6}, 15\}$, $\alpha^t = \{1, 8\}$, and $\alpha^u = \{3, \bar{6}, 15\}$ in the s, t or u-channel, respectively. To be specific,<fnref n="2" /> we have

$$ \left. \frac{dI}{dx} \right|_{\xi = \frac{1}{2}} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1/2} [\rho_3 (2C_F - N_c) + \rho_6 (2C_F - 1) + \rho_{15} (2C_F + 1)], \qquad (13) $$

$$ \left. \frac{dI}{dx} \right|_{\xi \to 0} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1} [\rho_1^t (2C_F) + \rho_8^t (2C_F - N_c)], \qquad (14) $$

$$ \left. \frac{dI}{dx} \right|_{\xi \to 1} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1} [\rho_3^u N_c + \rho_6^u - \rho_{15}^u]. \qquad (15) $$

It is obvious from eq. (12), that for $\xi \to 0$ the t-channel exchange is forced into an octet, i.e. $\rho_1^t = 0$ and $\rho_8^t = 1$. Using color conservation, $\begin{tikzpicture}[baseline=(current bounding box.center)] \draw (-0.5,0) -- (0.5,0); \node[above] at (-0.5,0) {$\xi$}; \draw (-0.5,-0.5) -- (-0.5,0.5); \node[below] at (-0.5,-0.5) {$-\xi$}; \draw (-0.5,-0.5) -- (0.5,0.5); \node[above] at (0.5,0.5) {$\xi$}; \draw (0.5,0) -- (0.5,-0.5); \node[below] at (0.5,-0.5) {$-\xi$}; \end{tikzpicture}$, one observes that for $\xi \to 1$, the u-channel exchange must occur in the triplet, i.e. $\rho_3^u = 1$ and $\rho_6^u = \rho_{15}^u = 0$. For $\xi = \frac{1}{2}$, the s-channel probabilities are $\rho_3 = \frac{(N_c^2+1)^2}{(N_c^2-1)(5N_c^2-1)} = \frac{25}{88}$, $\rho_6 = \frac{2N_c^2(N_c-2)}{(N_c-1)(5N_c^2-1)} = \frac{9}{44}$, and $\rho_{15} = \frac{2N_c^2(N_c+2)}{(N_c+1)(5N_c^2-1)} = \frac{45}{88}$, satisfying $\rho_3 + \rho_6 + \rho_{15} = 1$.

On the other hand, we may directly evaluate eq. (3) (using the matrices provided in Ref. [12]), to find, for all $\xi$,

$$ \frac{dI}{dx} = \frac{\alpha_s}{2\pi x} \frac{1}{C_F \xi^2 + N_c \bar{\xi}} \left[ (N_c^2 - \xi^2) \mathcal{L}_\xi + \left( \frac{\xi^2}{N_c^2} - 2\bar{\xi} \right) \mathcal{L}_{\bar{\xi}} \right]. \qquad (16) $$

<footnote n="2">²Let us recall the needed Casimir s: $C_3 = C_F$, $C_6 = C_F + N_c - 1$, $C_8 = N_c$, $C_{15} = C_F + N_c + 1$.</footnote>
From the above expression one can readily obtain the spectra for $\xi = \frac{1}{2}$, $\xi = 0$, and $\xi = 1$:

$$ \left. \frac{dI}{dx} \right|_{\xi=\frac{1}{2}} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1/2} \frac{(N_c^2 - 1)(4N_c^2 - 1)}{N_c(5N_c^2 - 1)}, \qquad (17) $$

$$ \left. \frac{dI}{dx} \right|_{\xi \to 0} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1} \left( -\frac{1}{N_c} \right), \qquad (18) $$

$$ \left. \frac{dI}{dx} \right|_{\xi \to 1} = \frac{\alpha_s}{\pi x} \mathcal{L}_{\xi=1} N_c . \qquad (19) $$

We note that eq. (18) is negative, which may be attributed to fully coherent (medium-induced) energy gain. Indeed, it turns out that significant regions of the phase space exhibit a negative spectrum. This is demonstrated in Figure 1, which displays a chart of the sign of $dI/dx$ in the $(x, \xi)$-plane, with a boundary indicating where the induced radiation spectrum vanishes.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Regions in the $(x, \xi)$-plane, corresponding to energy-loss (solid gray) or energy-gain (hatched purple). In this figure, $N_c = 3$, $l_{\perp A} = \frac{1}{4}m_{\perp}$ and $l_{\perp P} = \frac{1}{10}m_{\perp}$.</figcaption></figure>

We stress that such a negative medium-induced spectrum is not unphysical, owing to the prescription which defines $dI/d\omega$, namely by a difference between the pA and pp situations. Be that as it may, for a projectile to incur less radiation within a nuclear target seems counter-intuitive. Let us appeal to the $2 \to 1$ spectrum for insight. Indeed, the prefactor in eq. (1) contains a (strictly positive) part $\sim C_1 + C_3$ of the total spectrum which originates from the abelian-like radiation. The latter is focussed within a cone circumscribed by the scattering angle $\theta_s \equiv q_{\perp}/E$. Large-angle radiation $\theta > \theta_s$ is mainly associated to the non-abelian charge carried by parton 2 (which delivers $q_{\perp}$) and the corresponding part of the total spectrum is thus proportional to $C_2$. The total spectrum is an increasing function of $q_{\perp}$, but it would be incorrect to equate this with the additional soft rescatterings provided by the nucleus.

The medium-induced spectrum arises from an integration over the $k_{\perp}$ of the radiated gluon, or equivalently the angle $\theta \approx k_{\perp}/\omega$. However, the abelian and non-abelian radiation cones are not affected in the same way by the presence of the medium. Soft rescatterings do effectively grow $\theta_s$ (which increases the abelian radiation) but the non-abelian cone is ultimately capped at $\theta < q_{\perp}/\omega$, being fixed by the hard process and unchanged by the presence of the medium. Thus, the non-abelian component of the allocated radiation must actually shrink. In this way, we heuristically understand that $F_c \propto C_1 + C_3 - C_2$ in eq. (1) contains a positive part $C_1 + C_3$ but also the negative part $-C_2$. Care is evidently needed when defining medium-induced observables.
# 5 Summary

In order to disentangle parton energy-loss from other nuclear effects, such as nuclear parton distribution functions, it is important to control energy-loss calculations in a wide kinematic range. For FCEL, this is particularly crucial for channels in forward open heavy flavor production in pA collisions. The effect should not be neglected when extracting precise information about shadowing of the gluon distribution in target nuclei. Such studies, useful in a high-precision era with rich experimental programs, ought to include various observables like single hadron and dihadron/dijet production, which could probe different energy-loss patterns. Building a quenching weight from the new medium-induced spectrum would be required in forthcoming phenomenological applications. We leave it for future work.

# Acknowledgements

This work is supported by JSPS KAKENHI Grant No. JP25K07286 and in part by the Agence Nationale de la Recherche (ANR) under grant ANR-22-CE31-0018.

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