Theory overview: electroweak emission from heavy-ion collisions

TL;DR

This paper reviews the current theoretical understanding of electroweak emissions, such as photons and weak bosons, in heavy-ion collisions, emphasizing their potential as probes unaffected by strong interactions.

Contribution

It provides an overview of the emerging theoretical framework and discusses uncertainties in electroweak emission predictions in heavy-ion collisions.

Findings

Current models describe electroweak emissions with notable uncertainties

Electroweak probes offer a unique window into collision dynamics

Theoretical developments are ongoing to improve predictions

Abstract

An important class of observables in the heavy-ion collision programme concerns probes which are not sensitive to the prevailing strong interactions of QCD. The emission of photons, weak gauge bosons, and leptons fall into this category. Here I will describe the current status of such investigations, focusing on the emerging theoretical picture and its uncertainties.

Equations

$$\Gamma_{\gamma}\;\sim\;e^{2}\,\big{\langle}\,{{J_{\mbox{\scriptsize em}}\,J_{\mbox{\scriptsize em}}}}\,\big{\rangle}+{\cal O}(e^{4})\,,$$

$$\frac{{\rm d}N^{\mbox{\scriptsize DY}}_{{\mbox{\scriptsize AB}}\to\ell\bar{\ell}}}{{\rm d}M^{\,2}\,{\rm d}y}\bigg{|}_{\mbox{\scriptsize LO}}\;=\;T_{\mbox{\scriptsize AB}}\,\frac{e^{4}\,{\textstyle{\sum_{i}}}Q^{2}_{i}}{36\pi\,M^{\,2}s}\ f_{i/{\mbox{\scriptsize A}}}(x_{1})\,f_{\bar{i}/{\mbox{\scriptsize B}}}(x_{2})\;,$$

$$\frac{{\rm d}\Gamma_{\gamma}}{{\rm d}^{3}\text{\boldmath$k$}}\bigg{|}_{2\to 2}\ =\ \frac{1}{4k(2\pi)^{3}}{\textstyle{\sum\limits_{a,b,c}}}\int\!{\rm d}\Omega_{2\to 2}\,{{f_{a}}}\,{{f_{b}}}\,[1\pm{{f_{c}}}]\,|{\cal M}_{ab\to c\gamma}|^{2}\;,$$

$$\rho^{\mu\nu}(\omega,\text{\boldmath$k$})=-{\rm Im}\big{[}\textstyle\int_{0}^{1/T}\!\!{\rm d}\tau\;e^{i\,k_{n}\tau}G^{\mu\nu}(\tau,\text{\boldmath$k$})\big{]}_{ik_{n}\to\omega+i\,0^{+}}\ ;\quad G^{\mu\nu}\equiv\textstyle\int_{\text{\boldmath$x$}}e^{-i\text{\boldmath$k$}\cdot\text{\boldmath$x$}}\,\big{\langle}\,V^{\,\mu}(\tau,\text{\boldmath$x$})V^{\,\nu}(0)\,\big{\rangle}\,$$

$$\frac{\mathrm{d}{\Gamma }_{\gamma }}{{\mathrm{d}}^3\text{boldmathk}}$$

$$\frac{\mathrm{d}{\Gamma }_{\ell \bar{\ell }}}{\mathrm{d}\omega {\mathrm{d}}^3\text{boldmathk}}$$

$$G^{\mu\nu}(\tau,\text{\boldmath$k$})\;=\;{\textstyle{\int}}_{0}^{\infty}{\rm d}\omega\,\rho^{\,\mu\nu}\big{(}\omega,\text{\boldmath$k$}\big{)}\,\cosh[(\tfrac{1}{2T}-\tau)\omega]\big{/}\big{(}\pi\sinh[\tfrac{\omega}{2T}]\big{)}\;.$$

$$\sigma_{\mbox{\scriptsize el}}\ =\ \chi_{\mbox{\scriptsize q}}\,D\ =\ \tfrac{1}{2}\,{\textstyle{\lim\limits_{k\to 0}}}\ \rho_{\mbox{\scriptsize V}}(k,\text{\boldmath$k$}\,)\,\big{/}\,k\ =\ \tfrac{1}{3}\,{\textstyle{\lim\limits_{\omega\to 0}}}\ \rho_{\mbox{\scriptsize V}}(\omega,\text{\boldmath$0$})\,\big{/}\,\omega\;.$$

$$\frac{\mathrm{d}{N}_{\ell \bar{\ell }}}{\mathrm{d}M\mathrm{d}y\mathrm{d}{\Omega }_{\ell }}\propto 1+{\lambda }_{\theta }{\cos }^2{\theta }_{\ell }+{\lambda }_{\varphi }{\sin }^2{\theta }_{\ell }\cos 2{\varphi }_{\ell }+\dots ,$$

Full text

1 Introduction

For weak probes of strongly interacting systems, the hierarchy $\alpha_{\mbox{\scriptsize em}}\ll\alpha_{s}$ allows the calculation to be conveniently organised by first expanding the result in $\alpha_{\mbox{\scriptsize em}}=e^{2}/(4\pi)\,$. For instance, photons are generated (via $H_{\mbox{\scriptsize int}}=i\,e\,A_{\mu}J^{\,\mu}_{\mbox{\scriptsize em}}\,$) at a rate given by Laine:2016hma ; Kapusta:2006pm

[TABLE]

where the average is calculated from the density matrix \langle...\rangle={\rm Tr}\big{[}\,\hat{\rho}\,(...)\,\big{]}\,. In heavy-ion collisions the system is too short-lived for photons or other weak probes to equilibrate, so they “escape” basically unmodified, unlike jets or soft hadrons (see refs. Gale:2025ome and Geurts:2022xmk for recent overviews). Produced at all stages and throughout the volume, they are most copious when the medium is hottest, posing the challenge of disentangling the various sources.

The theoretical agenda is twofold. It involves $i$) the microscopic calculation of production rates, as well as $ii$) the macroscopic embedding of those rates in simulations of heavy-ion collisions. Task $i$) requires field theory calculations of eq. (1), such as by expanding in $\alpha_{s}$, using effective hadronic or QCD based theories, or from non-perturbative lattice studies. Predictions, e.g. for the $p_{\mbox{\tiny\rm{T}}}$ spectrum of photons and the invariant mass $M$ distribution of dileptons, are obtained from $ii$) by integrating over the spacetime history. This second step must be done numerically using multi-stage frameworks, which typically involve hydrodynamic codes such as MUSIC Gale:2021emg or Trajectum Massen:2024pnj .

In these proceedings, I will highlight recent theoretical progress on electroweak probes as presented at the conference. Let me draw attention to the excellent similar reports in refs. Du:2025ith ; Vujanovic:2024bby ; Paquet:2023vkq ; Tripolt:2020dac , from the past few years. For an update of experimental developments, see ref. Sebastian . Here I will focus on photons and dileptons, using a notation where $\Gamma_{\gamma}$ and $\Gamma_{\ell\bar{\ell}}$ are the respective differential rates per time and volume. The dilepton pair is produced through a virtual photon, of energy \omega\geq({4m_{\ell}^{2}+\text{\boldmathk}^{2}})^{1/2}, where $k$ is the photon’s momentum and $m_{\ell}$ is the mass of each lepton. For real photons, \omega=k\equiv|\text{\boldmathk}|\,.

2 Main stages and their emission rates

A source of electromagnetic probes which is common to both pp and AA collisions, originates from the first instant of the collision where eq. (1) involves the initial hadronic wavefunction of the projectiles. Such photons and dileptons are called prompt and can be calculated within the QCD factorization framework, involving the parton distribution functions (PDFs, $f_{i/{\mbox{\scriptsize A}}}$ below) Arleo:2011gc . For dileptons this is called the Drell-Yan process, and at leading order (LO) it is 222 Equation (2) only includes the $\gamma^{*}$ intermediate state, but for large $M\,$, the $Z$ boson must also be taken into account.

[TABLE]

the geometric overlap $T_{\mbox{\scriptsize AB}}=N_{\rm coll}/\sigma^{\mbox{\scriptsize inel}}_{\rm pp}$ is an effective luminosity, the quark charge fractions are Q_{i}=\big{\{}\tfrac{2}{3},-\tfrac{1}{3}\big{\}}\,, and nuclear PDFs (per nucleon) are evaluated at the longitudinal momentum fractions $x_{1,2}=Me^{\pm y}/\sqrt{s}$ ($y$ is the rapidity of the dilepton). At higher order in perturbation theory, eq. (2) generalises to a convolution of the nuclear PDFs with a hard scattering kernel and the kinematics is more complicated.333 The formalism introduces a renormalisation scale $\mu_{\mbox{\tiny\rm{R}}}$ (e.g. in $\alpha_{s}$) and a factorization scale $\mu_{\mbox{\tiny\rm{F}}}$ (e.g. in $f_{i/{\mbox{\scriptsize A}}}$) which are typically evaluated at $\mu_{\mbox{\tiny\rm{R}}}\,,\mu_{\mbox{\tiny\rm{F}}}=M\,$. This choice is varied by a factor of $2$ to estimate the uncertainty.

The Drell-Yan signal is dominant for large $M$ and allows one to study isospin content (via $Q_{i}$), sea quark asymmetry, and possibly the small-$x$ behaviour of nuclear PDFs (at large/small $y$).

Quite rapidly (within a time $\tau_{\mbox{\scriptsize eq}}\simeq 1$ fm/$c$), the longitudinal initial momentum distributions of partonic degrees of freedom approach local thermal equilibrium. Considerable progress in understanding how this occurs has been made with kinetic transport theory. In this approach, Boltzmann equations describe the evolution of quark and gluon distribution functions (denoted $f_{q,\bar{q},g}$ below) and can similarly track photon emission Garcia-Montero:2023lrd . The rate from $2\to 2$ processes is

[TABLE]

where at LO one has $q\bar{q}\to g\gamma$ and $qg\to q\gamma$. Also included are $1+n\leftrightarrow 2+n$ processes, as mandated by the Landau-Pomeranchuk-Migdal (LPM) effect. The rate $q\bar{q}\to\gamma^{*}$ has been used to describe pre-equilibrium dilepton yields.444 Such a LO dilepton rate is similar to eq. (2), with the nPDFs replaced with general distribution functions.

Those yields have a particular scaling in time which involves the viscosity to entropy density ratio Garcia-Montero:2024lbl . At LHC energies, the initial wavefunction is dominated by gluons and it takes some time to populate the quark and antiquark distributions. This implies a suppression of very early electroweak emissions, which It has been suggested may influence the dilepton invariant mass spectrum Coquet:2021lca ; Wu:2024pba , see fig. 1.

In thermal equilibrium, eq. (1) is evaluated with a thermal density matrix characterized by a temperature $T$ (and chemical potentials). The differential rates are proportional to the vector spectral function, given formally by analytic continuation of the Euclidean correlator:

[TABLE]

where \int_{\text{\boldmathx}}\equiv\int{\rm d}^{3}\text{\boldmathx} and $V^{\,\mu}\equiv\bar{\psi}\gamma^{\mu}\psi\,$. The photon and dilepton rates are given, respectively by

[TABLE]

Here $\omega$ and $k$ are the photon energy and momentum in the medium’s local rest frame. However, eqs. (5) and (6) apply cell-by-cell when simulating a heavy ion collision. The temperature T(t,\text{\boldmathx}) varies in space and time, while $\omega$ and $k$ need to be boosted to the lab frame according to the local fluid velocity u^{\,\mu}(t,\text{\boldmathx})\,. Hydrodynamical evolution governs these macroscopic quantities, and hence the observed yields {\rm d}N=\int_{t,\text{\boldmathx}}{\rm d}\Gamma are sensitive to viscous corrections (as are the rates themselves) Shen:2014nfa ; Vujanovic:2017psb ; Vujanovic:2019yih . New results for finite $\mu_{\mbox{\scriptsize B}}$ were presented at the conference Xian-Yu .

As the system cools and expands, partons confine into a gas of hadrons. Particles are generated on a freeze-out surface, ending the thermal phase, and undergo hadronic re-scatterings and resonance decays. It is speculated that the hadronization process itself may entail photon production Fujii:2022hxa . The subsequent hadronic dynamics is usually treated with transport codes (e.g. SMASH or UrQMD). Photons are produced by individual scatterings in these approaches, requiring cross sections as input Gotz:2021dco . For lower collisional energies (e.g. the RHIC beam energy scan), the bayon-rich region of the phase diagram is explored. A “cocktail” of hadronic reactions show up in the dilepton invariant mass spectrum, see e.g. ref. Jorge:2025wwp . Other novel phenomena may also occur as $\mu_{\mbox{\scriptsize B}}$ increases, for example a phase transition (rather than crossover) Savchuk:2022aev or a possible “moat” regime Nussinov:2024erh could lead to enhanced low mass dileptons.

3 Perturbation theory and lattice

The spectral function in eqs. (5) and (6) may be evaluated from first principles either by a weak coupling expansion (in $\alpha_{s}$) or by non-perturbative lattice QCD. Much of the phenomenology in sec. 2 is currently based on perturbation theory.555 The fixed value $\alpha_{s}\approx 0.3$ ($g\approx 2$) is commonly used, although not always stated in the literature.

In equilibrium, the nature of the strict perturbative calculation of $\rho_{\mbox{\scriptsize V}}$ (which has a long history) depends upon the kinematic domain. To compare with lattice data, the strict NLO result (applicable for M\mathrel{\hbox to0.0pt{\lower 2.5pt\hbox{\sim}\hss}\raise 1.99997pt\hbox{>}}T) is combined with the regime of LPM resumation (for $M\sim gT$), see ref. Jackson:2019yao . If $\rho^{\mu\nu}$ is known,

[TABLE]

The reverse, i.e. extracting the frequency dependence of $\rho$ from measurements of $G\,$, is an ill-conditioned operation for which there are a variety of strategies Francis:2025elc . Studies were first made at zero momentum (\text{\boldmathk}=0) Ding:2010ga and then non-zero momentum Ghiglieri:2016tvj . Recent progress enables us to scrutinize pQCD and make quantitative estimates of non-perturbative contributions.666 A different approach, free of the inverse problem, utilises imaginary spatial momenta Meyer:2018xpt ; Ce:2023oak .

The tensor structure has two independent components, transverse (T) and longitudinal (L) w.r.t. $k$ . Aligning the momentum with the $z$-axis: \text{\boldmathk}\to k\,\text{\boldmathe}_{z}\,, they can be expressed as $G_{\mbox{\scriptsize T}}\equiv\tfrac{1}{2}{\textstyle\sum_{i=1}^{2}}G^{ii}$ and $G_{\mbox{\scriptsize L}}\equiv G^{33}-G^{00}\,$. Another basis is $G_{\mbox{\scriptsize V}}\;\equiv\;2\,G_{\mbox{\scriptsize T}}+G_{\mbox{\scriptsize L}}$ and G_{\mbox{\scriptsize H}}\;\equiv\;2\,\big{(}G_{\mbox{\scriptsize T}}-G_{\mbox{\scriptsize L}}\big{)}. For an $N_{s}^{3}\times N_{\tau}$ lattice, the momenta $k=2\pi n/(aN_{s})\,$, where $a$ is the lattice spacing and $n$ is an integer. The correlators are periodic in the imaginary-time direction and symmetric on the interval $\tau\in[0,1/T]\,$, although cut-off effects are difficult to control for \tau T\mathrel{\hbox to0.0pt{\lower 2.5pt\hbox{\sim}\hss}\raise 1.99997pt\hbox{<}}\frac{1}{4}\,.

Note that for $\omega=k\,$, the Ward identity implies $\rho^{00}=\rho^{33}$ and hence $\rho_{\mbox{\scriptsize L}}=0\,$. Therefore either $G_{\mbox{\scriptsize V}}$ or $G_{\mbox{\scriptsize H}}$ may be analyzed for the photon rate. Dileptons however require $\rho_{\mbox{\scriptsize V}}$ in the timelike domain, where it receives a large contribution $\propto M^{2}=\omega^{2}-k^{2}$ from vacuum physics. This reflects itself in $G_{\mbox{\scriptsize V}}\sim 1/\tau^{3}$ for small $\tau\,$, making it difficult to constrain the thermal information in $\rho_{\mbox{\scriptsize V}}\,$. On the other hand, $\rho_{\mbox{\scriptsize H}}$ vanishes exactly in vacuum (as well as for \text{\boldmathk}=0) and satisfies the sum rule $\int{\rm d}\omega\,\omega\rho_{\mbox{\scriptsize H}}=0\,$. There is now a wealth of information on Euclidean correlators and spectral functions for $T>T_{c}$ Ce:2020tmx ; Ce:2022fot ; Ali:2024xae . Phenomenological models should therefore be tested on these lattice data before making predictions. So far these studies seem to indicate, see fig. 2, that pQCD alone slightly overestimates the photon rate at large $k/T$.

4 What we can learn from EM probes

4.1 Temperature of the QGP

The thermal distribution function in eqs. (5) and (6) implies that the integrated rates scale like $T^{4}$, i.e. hotter regions radiate the most. For this reason photons and dileptons can act as “thermometers” of the medium. If the measured spectra are approximately exponential, an effective temperature $T_{\mbox{\scriptsize eff}}$ can be defined from the inverse logarithmic slope, e.g. ${{\rm d}N_{\gamma}}/({{\rm d}y\,p_{\mbox{\tiny\rm{T}}}{\rm d}p_{\mbox{\tiny\rm{T}}}})\;\propto\;\exp(-{p_{\mbox{\tiny\rm{T}}}}/{T_{\mbox{\scriptsize eff}}})$ and ${{\rm d}N_{\ell\bar{\ell}}}/({{\rm d}y\,{\rm d}M})\;\propto\;(MT_{\mbox{\scriptsize eff}})^{3/2}\,\exp(-M/T_{\mbox{\scriptsize eff}})\,$. However, as clear from sec. 2, the value of $T_{\mbox{\scriptsize eff}}$ derives from a superposition of sources. It is important to understand which factors control this value, in order to interpret the experimental findings Sebastian . In the case of photons, $p_{\mbox{\tiny\rm{T}}}$ is blue shifted relative to its value in the local rest frame due to radial flow Du:2024pbd ; Massen:2024pnj . For dileptons $M$ is an invariant quantity and the slope parameter is found to correlate with the initial mean temperature in simulations Churchill:2023zkk ; Churchill:2023vpt ; Renan .

4.2 Electrical conductivity

The limit $\omega,k\ll T$ is one in which charge fluctuations are classical in nature and hence the spectral function takes the “universal” form: \rho_{\mbox{\scriptsize V}}(\omega,\text{\boldmathk}\,)\;=\;\big{(}\tfrac{\omega^{2}-k^{\,2}}{\omega^{2}+{D}^{\,2}k^{\,4}}+2\big{)}\ \chi_{\mbox{\scriptsize q}}\,D\,\omega, where $D$ is the electrical diffusion coefficient and \chi_{\mbox{\scriptsize q}}=\frac{1}{T}G^{00}(\tau,\text{\boldmath0}) is the quark susceptibility. The classical (hydrodynamic) description involves parameters which need to be matched to the microscopic theory, e.g. the electrical conductivity $\sigma_{\mbox{\scriptsize el}}\,$ defined by \langle\text{\boldmathJ}_{\mbox{\scriptsize em}}\rangle=\sigma_{\mbox{\scriptsize el}}\text{\boldmathE}\,. One can obtain the value of the latter from certain limits, as given by the Kubo formulae

[TABLE]

For this reason, is has been suggested that $\sigma_{\mbox{\scriptsize el}}$ can be obtained from the appropriate limit of photon and dilepton spectra Floerchinger:2021xhb . For dileptons, the hadronic component is paramount in this limit and in-medium modifications to vector mesons need to be treated accurately Rapp:2024grb ; Atchison:2024lmf .

4.3 Dilepton polarization

The largest theoretical uncertainty in the succession of models described in sec. 2 is the earliest timespan where the system is highly anisotropic. This far from equilibrium state is difficult to treat rigorously, but impacts the polarization of photons Hauksson:2023dwh . The angular distribution of dileptons has drawn attention recently, being sensitive to polarization Speranza:2018osi , and is usually parametrized (by the leading, frame dependent, coefficients $\lambda_{\theta}$ and $\lambda_{\phi}$) by

[TABLE]

where the angles ${\rm d}\Omega_{\ell}={\rm d}\cos\theta_{\ell}\,{\rm d}\phi_{\ell}$ refer to one of the final leptons, measured w.r.t. a $z$-axis chosen conventionally in the virtual photon’s rest frame.

In the helicity frame (HX), $\lambda^{\mbox{\scriptsize HX}}_{\theta}\simeq(\rho_{\mbox{\scriptsize T}}-\rho_{\mbox{\scriptsize L}})/(\rho_{\mbox{\scriptsize T}}+\rho_{\mbox{\scriptsize L}})\,$ which has been studied in both hadronic Seck:2023oyt and QGP Wu:2024vyc phases. The coefficient $\lambda^{\mbox{\scriptsize HX}}_{\theta}$ receives a large NLO correction for M\mathrel{\hbox to0.0pt{\lower 2.5pt\hbox{\sim}\hss}\raise 1.99997pt\hbox{<}}2 GeV (fig. 3). In the Collins-Soper (CS) frame, $\lambda^{\mbox{\scriptsize CS}}_{\theta}=3Q/(\frac{2}{5}-Q)$ where $Q$ is the quadrupole moment of the lepton distribution, which was calculated for the LO pre-equilibrium process $q\bar{q}\to\ell\bar{\ell}$ Coquet:2023wjk . When the initial quarks have large longitudinal momenta: $\lambda^{\mbox{\scriptsize CS}}_{\theta}=1\,$At the other extreme, when the initial quarks carry purely transverse momentum: $\lambda^{\mbox{\scriptsize CS}}_{\theta}=-\frac{1}{3}\,$. A sign change of $\lambda_{\theta}^{\mbox{\scriptsize CS}}$ would thus distinguish these scenarios, see fig. 3.

5 Summary

Electroweak probes are valuable tools for diagnosing the medium in heavy ion collisions. Perturbative and lattice approaches now complement each other, enabling quantitative estimates of non-perturbative effects in emission rates. New insights are being provided by differential observables, like dilepton polarization. Yet some puzzles remain, for example: Measurements of the $v_{2}$ for direct photons suggests that current predictions are underestimating thermal photons. This is in tension with fig. 2, where lattice analyses indicate that perturbation theory is overestimating thermal photons. Another questions is the scaling of direct photons with $({\rm d}N_{\mbox{\scriptsize ch}}/{\rm d}y)^{\alpha}$, although there is some disagreement about the value of $\alpha$ on the experimental side Sebastian . Diverse as the theoretical progress has been, there is still room (and even need) for contributions on both formal aspects and phenomenology.

{acknowledgement}

I am very grateful to F. Arleo, D. Bala, M. Coquet, L. Du, C. Gale, J. Ghiglieri and M. Laine for their valuable comments. The author is supported by the Agence Nationale de la Recherche, under grant ANR-22-CE31-0018.