Pre-equilibrium Energy Loss

• Pre-equilibrium energy loss is the process of early energy degradation and redistribution before a many-body system reaches full equilibrium.
• In heavy-ion collisions, it affects observables such as jet quenching, collective flow, and particle suppression by altering the interaction period prior to hydrodynamics.
• In nuclear and astrophysical contexts, it governs phenomena like pre-equilibrium nucleon emission, dipole radiation, and weak energy-loss via neutrino emission, influencing overall reaction outcomes.

Pre-equilibrium energy loss denotes energy degradation, dissipation, or redistribution that occurs before a many-body system reaches the equilibrium or hydrodynamic regime. In relativistic heavy-ion physics, the term covers both early-time quenching of hard probes and pre-hydrodynamic transport of the bulk energy-momentum tensor; in low-energy nuclear reactions it refers to particle emission and collective radiation before compound-nucleus equilibration; in pre-supernova matter it refers to neutrino and antineutrino energy loss from thermally excited nuclei before collapse (Li et al., 2020). Across these settings, the defining feature is temporal ordering: the relevant energy-transfer mechanism acts while the medium is still far from equilibrium, anisotropic, or only partially thermalized.

1. Conceptual scope and regimes

In the heavy-ion literature, the pre-equilibrium interval is the stage before hydrodynamics is assumed applicable. One heavy-flavor study takes this as $0<\tau<\tau_0$ with $\tau_0=0.6~\mathrm{fm}$ in the default setup, and explicitly asks when heavy-quark–medium interaction begins and what temperature profile should be used before $\tau_0$ (Li et al., 2020). Weak-coupling kinetic descriptions instead formulate the same interval as evolution from a saturated initial state at $\tau_0\sim Q_s^{-1}$ to a later $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$ when viscous hydrodynamics becomes applicable (Mazeliauskas, 2016). In Bayesian jet-quenching phenomenology, hydrodynamics starts at $\tau_{\rm hyd}=0.4~\mathrm{fm}$, and pre-equilibrium energy loss is represented by allowing quenching to turn on at an earlier onset time $\tau_{\min}$ (Pablos et al., 23 Sep 2025).

Domain	Carrier of energy loss or redistribution	Representative observables
Relativistic heavy-ion hard probes	Heavy quarks, jets, partons	$R_{AA}$, $v_2$, jet $R_{AA}$, jet $\tau_0=0.6~\mathrm{fm}$0
Relativistic heavy-ion bulk medium	$\tau_0=0.6~\mathrm{fm}$1, entropy, preflow	Green functions, hydrodynamic initial conditions
Low-energy heavy-ion reactions	Dynamical dipole radiation, nucleon emission	$\tau_0=0.6~\mathrm{fm}$2-ray spectrum, emitted $\tau_0=0.6~\mathrm{fm}$3, angular distributions
Pre-supernova stellar matter	Weak processes on hot nuclei	Neutrino spectra, antineutrino spectra, energy-loss rates

The phrase does not denote a single microscopic mechanism. In some works it means collisional or radiative parton energy loss before equilibration (Carrington et al., 2011); in others it means bulk redistribution of energy and momentum before hydrodynamic matching rather than loss from the system (Kurkela et al., 2018). A frequent source of confusion is the use of free streaming as a proxy for pre-equilibrium evolution. One isobar-collision study is explicit that free streaming is an ad hoc extreme scenario, that the pre-equilibrium stage is poorly understood, and that free streaming with velocity $\tau_0=0.6~\mathrm{fm}$4 should be interpreted as an upper bound on smearing of the initial geometry (Gardim et al., 2023).

2. Early-time quenching of hard probes in heavy-ion collisions

For heavy flavor, pre-equilibrium energy loss has been analyzed in a modified Langevin framework coupled to hydrodynamics. The heavy-quark momentum obeys

$\tau_0=0.6~\mathrm{fm}$5

with drag, white-noise diffusion,

$\tau_0=0.6~\mathrm{fm}$6

and recoil from medium-induced gluon radiation. The fluctuation-dissipation relation gives

$\tau_0=0.6~\mathrm{fm}$7

so the interaction strength is characterized by $\tau_0=0.6~\mathrm{fm}$8. Radiation is incorporated probabilistically using a higher-twist gluon-emission spectrum with dead-cone suppression and a low-energy cutoff $\tau_0=0.6~\mathrm{fm}$9 below which radiation is switched off (Li et al., 2020).

The pre-equilibrium stage enters in two ways: by changing the time interval over which the Langevin evolution acts, and by changing the temperature history that controls drag, diffusion, and radiation rates. Four pre-equilibrium temperature profiles were tested for $\tau_0$0: free streaming, $\tau_0$1; linear rise, $\tau_0$2; constant temperature, $\tau_0$3; and Bjorken-like evolution, $\tau_0$4. An important implementation point is that the same modified Langevin equation is used before $\tau_0$5 whenever a temperature profile is supplied (Li et al., 2020).

The phenomenological consequence is that suppression and flow do not respond identically. After retuning $\tau_0$6 so that different scenarios yield similar $\tau_0$7, delaying the interaction onset from $\tau_0$8 to $\tau_0$9 requires about $\tau_0\sim Q_s^{-1}$0 smaller $\tau_0\sim Q_s^{-1}$1, i.e. stronger effective coupling, to reproduce the same $\tau_0\sim Q_s^{-1}$2. Under the same retuning, low-$\tau_0\sim Q_s^{-1}$3 $\tau_0\sim Q_s^{-1}$4-meson $\tau_0\sim Q_s^{-1}$5 increases by $\tau_0\sim Q_s^{-1}$6 at RHIC and $\tau_0\sim Q_s^{-1}$7 at the LHC. For pre-equilibrium temperature profiles, the free-streaming assumption yields about $\tau_0\sim Q_s^{-1}$8 larger low-$\tau_0\sim Q_s^{-1}$9 $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$0 at RHIC and $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$1 larger at the LHC than constant-temperature and Bjorken-like evolution. The paper’s broader result is that different modeling of initial conditions, pre-equilibrium evolution, and in-medium interaction can each induce about $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$2 uncertainties in low-$\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$3 heavy-flavor suppression and flow (Li et al., 2020).

A conceptually distinct early-time mechanism attributes azimuthal anisotropy itself to pre-equilibrium energy loss in the strong color field of overlapping strings or string clusters. In that picture, a parton traverses an anisotropic color field before the medium equilibrates, radiates, and loses energy by an amount that depends on the path length $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$4. The resulting elliptic-flow scaling law is

$\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$5

with $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$6 as the central theoretical expectation for $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$7. The fitted scaling curve is $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$8 with $\tau_{\rm hydro}\sim 1~\mathrm{fm}/c$9 and $\tau_{\rm hyd}=0.4~\mathrm{fm}$0, and the model interprets the near-universality across centralities, species, and RHIC/LHC energies as evidence that early-time quenching can generate much of the observed $\tau_{\rm hyd}=0.4~\mathrm{fm}$1 scaling (Andrés et al., 2016).

More recent jet phenomenology introduces pre-equilibrium loss through the hydrodynamic attractor. In that framework, the quenched jet spectrum is a convolution of an nPDF-modified vacuum cross section with an energy-loss probability distribution, quenching is applied only for $\tau_{\rm hyd}=0.4~\mathrm{fm}$2, and the same energy-loss machinery is used in the pre-equilibrium stage with an attractor-extrapolated temperature and preflow field. A Bayesian analysis using jet $\tau_{\rm hyd}=0.4~\mathrm{fm}$3 and jet $\tau_{\rm hyd}=0.4~\mathrm{fm}$4 finds that jet $\tau_{\rm hyd}=0.4~\mathrm{fm}$5 alone leaves $\tau_{\rm hyd}=0.4~\mathrm{fm}$6 poorly constrained because earlier onset can be compensated by smaller coupling, but including jet $\tau_{\rm hyd}=0.4~\mathrm{fm}$7 constrains $\tau_{\rm hyd}=0.4~\mathrm{fm}$8, well before $\tau_{\rm hyd}=0.4~\mathrm{fm}$9. The same study reports good simultaneous description of jet $\tau_{\min}$0, jet $\tau_{\min}$1, RHIC jet suppression, and charged-hadron $\tau_{\min}$2, with the main tension in low-$\tau_{\min}$3 hadron $\tau_{\min}$4, which is underpredicted (Pablos et al., 23 Sep 2025).

3. Initial-value formulations and unstable plasma dynamics

A separate line of work formulates pre-equilibrium parton energy loss as an initial-value problem in a weakly coupled, anisotropic, unstable quark-gluon plasma. The energetic parton is treated as a classical colored particle obeying the Wong equations, and the instantaneous energy loss is

$\tau_{\min}$5

Because the medium is not stationary, the analysis uses a one-sided Fourier transform,

$\tau_{\min}$6

so that the energy transfer retains explicit dependence on initial chromodynamic fields and on unstable collective modes (Carrington et al., 2011).

In equilibrium or any stable plasma, all collective poles lie in the lower half of the complex $\tau_{\min}$7-plane, their contributions decay, and the familiar steady collisional energy-loss formula is recovered. In unstable plasmas, however, poles of the retarded propagator enter the upper half-plane,

$\tau_{\min}$8

producing factors $\tau_{\min}$9 and hence explicitly time-dependent energy transfer. This is the origin of exponentially growing contributions to pre-equilibrium loss in the weak-coupling unstable-plasma literature (Carrington et al., 2011).

Two benchmark anisotropic systems have been studied in detail. For a two-stream plasma with

$R_{AA}$0

the instability is longitudinal and chromoelectric. The computed energy loss per unit length oscillates strongly in time, has an amplitude that grows with time, and depends strongly on the angle between the parton velocity and the stream direction (Carrington et al., 2012). For an extremely prolate plasma, the momentum distribution is infinitely elongated along one direction, unstable Weibel-like modes appear, and the energy loss is strongly time dependent, strongly directional, and can become much larger than the equilibrium loss; in the numerical setup discussed in the paper, the corresponding equilibrium benchmark is about $R_{AA}$1 for $R_{AA}$2, $R_{AA}$3, $R_{AA}$4 (Carrington et al., 2013).

The ultraviolet structure is also nontrivial. For the extremely prolate case, the soft contribution is logarithmically ultraviolet divergent, and a dedicated analysis argues that a good approximation is obtained by cutting the divergence off with the parton energy, $R_{AA}$5 (Carrington et al., 2013). Another extension shows that the sign of the energy transfer can depend on initial conditions. With uncorrelated initial chromodynamic fields, the parton typically loses energy and the magnitude is comparable to equilibrium; with maximally correlated initial fields induced by the parton, the parton can either gain or lose energy, the transfer grows exponentially in time, and the magnitude can far exceed equilibrium loss. The effect is maximal for $R_{AA}$6 in extremely prolate plasmas and for $R_{AA}$7 in extremely oblate plasmas (Carrington et al., 2015).

Strong-coupling holography yields a different nonequilibrium pattern. In collisions of gravitational shock waves dual to colliding sheets of energy, the drag force on a heavy quark moving through far-from-equilibrium matter is not dramatically enhanced relative to equilibrium estimates with a local effective temperature or pressure, but it turns on only after a delay. For zero rapidity the force rises after the collision with a delay roughly of order $R_{AA}$8 at low velocity and later becomes semi-quantitatively describable by equilibrium drag based on the local hydrodynamic temperature. At larger rapidity, gradients in the fluid velocity produce qualitatively new effects: the force required to hold the quark can point in the same direction as the quark velocity, and a substantial component perpendicular to the velocity can appear even in the local fluid rest frame (Chesler et al., 2013). This establishes an important contrast: unstable weak-coupling plasmas can generate exponential enhancement, whereas a strongly coupled far-from-equilibrium collision does not automatically imply extra drag, but does imply delay and gradient corrections.

4. Bulk pre-equilibrium transport, preflow, and medium background

In weak-coupling kinetic theory, pre-equilibrium “energy loss” often refers not to hard-probe quenching but to transport and redistribution of the bulk energy-momentum tensor. The central object is the Boltzmann equation with leading-order QCD processes,

$R_{AA}$9

where the collision kernel includes elastic $v_2$0 scattering and inelastic $v_2$1 splittings with screening and Landau-Pomeranchuk-Migdal suppression. The purpose is to propagate a far-from-equilibrium, gluon-dominated, boost-invariant system from $v_2$2 to a time when viscous hydrodynamics becomes valid (Keegan et al., 2016).

The practical output is a Green-function map from initial perturbations to hydrodynamic initial data. In one formulation, the evolved tensor at $v_2$3 is decomposed into a local homogeneous background plus linear response to initial energy and momentum perturbations,

$v_2$4

This event-by-event, causal construction propagates local energy density and momentum perturbations only within the causal neighborhood $v_2$5 (Kurkela et al., 2018).

The background evolution exhibits a transition from approximately free streaming, with $v_2$6 and $v_2$7, toward viscous hydrodynamics. One 2016 study finds that by $v_2$8 the constitutive relations are already close to the kinetic-theory result, and by $v_2$9 they are closer still (Keegan et al., 2016). In an event-by-event framework calibrated for central Pb-Pb at $R_{AA}$0 with $R_{AA}$1, the typical time scale when viscous hydrodynamics becomes applicable is $R_{AA}$2 (Kurkela et al., 2018).

Pre-equilibrium transport generates transverse flow before hydrodynamics. In the long-wavelength, conformal limit, the coordinate-space relation

$R_{AA}$3

emerges as the preflow relation (Keegan et al., 2016). The same framework shows that pre-equilibrium evolution is not a passive interpolation: it redistributes energy, relaxes pressure anisotropy, and produces entropy. One estimate gives

$R_{AA}$4

so the final gluon multiplicity can be more than twice the initial one (Keegan et al., 2016). An event-by-event implementation further reports that over $R_{AA}$5 of final entropy per rapidity is produced by the end of the pre-equilibrium stage, that gluon number density roughly doubles from $R_{AA}$6 to $R_{AA}$7, and that hydrodynamic observables at freeze-out become much less sensitive to the choice of hydro starting time than in simplistic free-streaming or Bjorken-rescaling prescriptions (Kurkela et al., 2018).

This distinction between bulk redistribution and hard-probe loss is essential. The kinetic-theory initialization papers do not compute jet quenching; they propagate how initially deposited energy and momentum are broadened, attenuated locally, and partially converted into collective flow and entropy before hydrodynamics (Mazeliauskas, 2016). A plausible implication is that any extraction of pre-equilibrium hard-probe energy loss inherits systematic uncertainty from how the evolving medium itself is modeled in the same early-time window.

The geometry of that background is also modified during pre-equilibrium evolution. In isobar collisions, an extreme free-streaming stage reduces small-scale structure, makes the energy density more rounded, decreases spatial anisotropies $R_{AA}$8, and increases the rms radius $R_{AA}$9. The ordering of sensitivity is

$\tau_0=0.6~\mathrm{fm}$00

For ratios between isobar configurations, $\tau_0=0.6~\mathrm{fm}$01 and $\tau_0=0.6~\mathrm{fm}$02 are usually altered by less than about $\tau_0=0.6~\mathrm{fm}$03, whereas higher harmonics and especially the Pearson correlators $\tau_0=0.6~\mathrm{fm}$04 and $\tau_0=0.6~\mathrm{fm}$05 are appreciably sensitive to the duration of free streaming (Gardim et al., 2023). This shows that pre-equilibrium smearing need not erase geometry information uniformly, and it clarifies why early-time quenching observables that depend on path-length anisotropy can be especially sensitive to the assumed pre-hydrodynamic evolution.

5. Pre-equilibrium dissipation in low-energy nuclear reactions

In low-energy heavy-ion reactions, pre-equilibrium energy loss is primarily a problem of early dissipation before formation of a fully equilibrated compound nucleus. For the charge-asymmetric reaction $\tau_0=0.6~\mathrm{fm}$06 at $\tau_0=0.6~\mathrm{fm}$07, two linked mechanisms are identified: pre-equilibrium dipole oscillations and pre-equilibrium nucleon emission. The initial isovector dipole moment at contact is

$\tau_0=0.6~\mathrm{fm}$08

and the emitted $\tau_0=0.6~\mathrm{fm}$09-ray probability is

$\tau_0=0.6~\mathrm{fm}$10

At $\tau_0=0.6~\mathrm{fm}$11, the dipole motion shows damped oscillations exhausted in about $\tau_0=0.6~\mathrm{fm}$12, with damping driven by both mean-field and two-body collisional effects (Zheng et al., 2016).

This dissipation is sensitive to nuclear-matter properties below saturation. The same study concludes that both pre-equilibrium dipole oscillations and nucleon emission are sensitive to the symmetry energy in the range $\tau_0=0.6~\mathrm{fm}$13, to the effective mass, and to the nucleon-nucleon cross section. Momentum-dependent interactions produce larger centroid energies and larger spectral strength than momentum-independent interactions. Increasing the $\tau_0=0.6~\mathrm{fm}$14 cross section leaves the dipole centroid nearly unchanged but decreases the magnitude by enhancing damping. Early emitted nucleons are identified in regions with density $\tau_0=0.6~\mathrm{fm}$15 at $\tau_0=0.6~\mathrm{fm}$16; their total number and emitted $\tau_0=0.6~\mathrm{fm}$17 ratio serve as direct measures of particle-emission energy loss and isospin distillation (Zheng et al., 2016).

A different manifestation of pre-equilibrium loss appears in light-ion reactions that populate the same compound nucleus but through different entrance channels. In the comparison of $\tau_0=0.6~\mathrm{fm}$18 at $\tau_0=0.6~\mathrm{fm}$19 and $\tau_0=0.6~\mathrm{fm}$20 at $\tau_0=0.6~\mathrm{fm}$21, both reactions form $\tau_0=0.6~\mathrm{fm}$22 at essentially the same excitation energy, allowing the equilibrium evaporation component to be benchmarked cleanly. The backward-angle $\tau_0=0.6~\mathrm{fm}$23 spectra are described as essentially pure compound-nucleus evaporation and are reproduced by Hauser-Feshbach calculations with the Gilbert-Cameron composite level density, while the $\tau_0=0.6~\mathrm{fm}$24-induced reaction contains a significant pre-equilibrium component at all angles (Voinov et al., 2011).

The angular distributions show that pre-equilibrium emission is not necessarily identical with a purely direct, forward-peaked mechanism. For $\tau_0=0.6~\mathrm{fm}$25, the pre-equilibrium component is forward-peaked as expected, but the measured angular distribution falls more steeply with angle than Kalbach systematics predict and exhibits a slight rise above about $\tau_0=0.6~\mathrm{fm}$26. After subtracting the equilibrium baseline, the remaining pre-equilibrium distribution integrated over $\tau_0=0.6~\mathrm{fm}$27–$\tau_0=0.6~\mathrm{fm}$28 proton energies is strongly forward-peaked, steeper than predicted, nearly flat above $\tau_0=0.6~\mathrm{fm}$29, and slightly increasing at $\tau_0=0.6~\mathrm{fm}$30. The interpretation advanced is that a non-negligible multi-step compound contribution exists within the pre-equilibrium sector itself, so that part of the energy-dissipation chain has already lost memory of the projectile direction before emission occurs (Voinov et al., 2011).

6. Hot nuclei and pre-supernova energy-loss rates

In astrophysical pre-supernova matter, pre-equilibrium energy loss refers to weak energy-loss channels operating before core collapse in a medium of hot, thermally populated nuclei. For hot $\tau_0=0.6~\mathrm{fm}$31 in a MESA pre-supernova model of a $\tau_0=0.6~\mathrm{fm}$32 star at the onset of collapse, the relevant conditions span $\tau_0=0.6~\mathrm{fm}$33 to $\tau_0=0.6~\mathrm{fm}$34, $\tau_0=0.6~\mathrm{fm}$35 to $\tau_0=0.6~\mathrm{fm}$36, and electron chemical potential $\tau_0=0.6~\mathrm{fm}$37 to $\tau_0=0.6~\mathrm{fm}$38. Charged-current processes include electron capture, positron capture, $\tau_0=0.6~\mathrm{fm}$39-decay, and $\tau_0=0.6~\mathrm{fm}$40-decay; neutral-current de-excitation produces $\tau_0=0.6~\mathrm{fm}$41 pairs from thermally excited states (Dzhioev et al., 2023).

The microscopic input is the thermal Gamow-Teller strength,

$\tau_0=0.6~\mathrm{fm}$42

which incorporates thermal occupation of excited states, thermal unblocking, and downward transitions. The total emission and energy-loss rates are

$\tau_0=0.6~\mathrm{fm}$43

The central result is that hot nuclei lose significantly more energy by neutrino and antineutrino emission than nuclei in their ground state, because finite temperature opens additional transition channels, shifts the GT resonance downward, and generates negative-energy transitions associated with de-excitation (Dzhioev et al., 2023).

The neutral-current $\tau_0=0.6~\mathrm{fm}$44-pair channel is particularly important. The study concludes that de-excitation via $\tau_0=0.6~\mathrm{fm}$45-pair emission is presumably a dominant source of antineutrinos under pre-supernova conditions. Near the stellar center, pair emission dominates because $\tau_0=0.6~\mathrm{fm}$46-decay is blocked by the high electron chemical potential; farther outward, decreasing $\tau_0=0.6~\mathrm{fm}$47 unblocks $\tau_0=0.6~\mathrm{fm}$48-decay and makes the low-energy antineutrino contributions comparable. The resulting antineutrino spectrum exhibits a low-energy peak around $\tau_0=0.6~\mathrm{fm}$49–$\tau_0=0.6~\mathrm{fm}$50 from thermally unblocked low-energy GT$\tau_0=0.6~\mathrm{fm}$51 transitions and a broader peak around $\tau_0=0.6~\mathrm{fm}$52 from de-excitation of the thermally populated GT$\tau_0=0.6~\mathrm{fm}$53 resonance. As temperature drops outward, the high-energy antineutrino component decreases by more than two orders of magnitude (Dzhioev et al., 2023).

The neutrino sector is dominated by electron capture, and the average neutrino energy remains roughly stable around

$\tau_0=0.6~\mathrm{fm}$54

At the same time, the paper emphasizes that the single-state approximation for neutrino spectra may fail under certain pre-supernova conditions, especially when the electron chemical potential is not high enough for electron capture to excite the GT$\tau_0=0.6~\mathrm{fm}$55 resonance and when negative-energy transitions from thermally excited states contribute significantly (Dzhioev et al., 2023). This suggests that, in the stellar context, pre-equilibrium energy loss is controlled by finite-temperature nuclear structure as much as by macroscopic thermodynamic conditions.

Taken together, these literatures show that pre-equilibrium energy loss is a family of early-time transport phenomena rather than a single universal process. In relativistic heavy-ion collisions it may refer to quenching before hydrodynamization, to unstable-plasma amplification, or to bulk redistribution of $\tau_0=0.6~\mathrm{fm}$56; in low-energy reactions it includes dipole radiation, nucleon emission, and multi-step pre-equilibrium emission; in stellar matter it denotes thermally enhanced weak energy-loss channels. The common technical problem is that observables are sensitive not only to the strength of the interaction but also to the onset time, the evolving background, and the nonequilibrium degrees of freedom retained in the model.