# Thermal field theory in heavy-ion collisions and cosmology

Jacopo Ghiglieri

► To cite this version:

Jacopo Ghiglieri. Thermal field theory in heavy-ion collisions and cosmology. Rencontres de physique des particules, Feb 2025, Annecy (France), France. hal-04931887

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Submitted on 6 Feb 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Thermal Field Theory
in Heavy-Ion Collisions
and Cosmology

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agence nationale
de la recherche

AU SERVICE DE LA SCIENCE

NUCLÉAIRE
& PARTICULES

Jacopo Ghiglieri, SUBATECH, Nantes

RPP 2025, Annecy, February 5th 2025
# Thermal Field Theory

* TFT: many-body relativistic QFT at

* nonzero temperatures and densities (in equilibrium)

* more-generally, with a non-vacuum, time-dependent density operator $\hat{\rho}$

* From $\langle \hat{O} \rangle = \langle 0 | \hat{O} | 0 \rangle$ to $\langle \hat{O} \rangle = \text{Tr}[\hat{\rho}\hat{O}]$.

$$ \hat{\rho}_{\text{vac}} = |0\rangle\langle 0|, \hat{\rho}_{\text{eq}} = e^{-\beta(H-\mu N)}/Z $$

* Can describe thermodynamics and kinetics
* Concentrate on kinetics: vaste programme, with important applications in heavy-ion collisions and in cosmology

* In particular, focus on

* perturbative and non-perturbative determinations of production and equilibration rates for light weakly-coupled states

* photons in heavy-ion collisions

* light relics (axions, gravitons, sterile neutrinos, etc..) in cosmology
# Rates in heavy-ion collisions

* Thermalisation rates

* for the QGP itself

* For *hard probes*: energetic modes out of equilibrium: jets, heavy flavors

* competing against the expansion rate of the fireball

* Production rates for photons and dileptons, who can then escape the QGP

<figure><img src="image_6.png" /></figure>
# How electromagnetic probes are made

(and what they tell us)

<figure><img src="image_7.png" /></figure>

* Hard partonic processes within colliding nucleons produce *prompt photons*, *dileptons*, W and Z bosons. Learn about nPDFs, photon-tagged hadrons and jets

* At a later stage, quarks and gluons form a plasma

* (pre)thermal production of QGP photons and dileptons. thermalisation, T, hydro

* A jet traveling can radiate jet-thermal photons. Jet quenching

* Later on, hadronization. hadron gas photons and dileptons. T, T_c, hydro

* (Some) hadrons decay into decay photons and dileptons
<figure><img src="image_8.png" /></figure>

Garcia-Montero Mazeliauskas Plaschke Schlichting 2308.09747

<figure><img src="image_9.png" /></figure>

2308.09747
# Rates in the early universe

<figure><img src="image_10.png" /></figure>

* Over the long thermal history, many phenomena enter and/or leave equilibrium

* DM candidates

* Mechanisms for the BAU

* Thermal relics

* ...

<figure><img src="image_11.png" /></figure>

* governed by rates (production, equilibration, interaction) competing with *H*
* Factor the system into “**fast**” and “**slow**” modes, and integrate out the former to obtain evolution eqs. for the latter

<figure><img src="image_12.png" /></figure>

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$$1/t_{\text{QCD}} \sim \Lambda_{\text{QCD}}$$
# Production and equilibration

* A particle $\phi$ is weakly coupled (coupling $h$) to an equilibrated bath with its internal couplings $g$

J built of bath fields, one can prove to first order in $h$ and all orders in $g$
Bödeker Sangel Wörmann PRD93 (2016)

$$ \begin{gathered} \dot{f}_{\phi}(t, \mathbf{k}) = \Gamma(k) [f_{\text{eq}}(k^0) - f_{\phi}(t, \mathbf{k})] + \mathcal{O}(h^4) \\ \Gamma(k) = \frac{h^2}{2k^0} \int d^4 X e^{iK \cdot X} \langle [\mathbf{J}(X), \mathbf{J}(0)] \rangle \end{gathered} $$

* Single-particle phase-space distribution: $f(t, \mathbf{k}) = (2\pi)^3 dN_{\phi}/d^3\mathbf{x}d^3\mathbf{k}$, sensible only for sufficiently weakly interacting particles

* $\langle \hat{O} \rangle$ ensemble average
$$ \dot{f}_{\phi}(t, \mathbf{k}) = \Gamma(k) [f_{\text{eq}}(k^0) - f_{\phi}(t, \mathbf{k})] + \mathcal{O}(h^4) \qquad \Gamma(k) = \frac{h^2}{2k^0} \int d^4 X e^{iK \cdot X} \langle [J(X), J(0)] \rangle $$

* When using these equations in cosmology, the l.h.s is modified to include Hubble expansion
$$ \dot{f}_{\phi}(t, \mathbf{k}) \rightarrow (\partial_t - H\mathbf{k} \cdot \nabla_{\mathbf{k}})f_{\phi}(t, \mathbf{k}) $$
and often (number, energy) densities are the quantity of interest, e.g. $n_{\phi} = \int_{\mathbf{k}} f_{\phi}$

$$ \dot{n}_{\phi} + 3Hn_{\phi} = \int_{\mathbf{k}} \Gamma(k) [f_{\text{eq}}(k^0) - f_{\phi}(t, \mathbf{k})] $$

* If scale separation is present and $g \ll 1$, perturbative expansion of $\Gamma(k \ge T)$ can reproduce standard Boltzmann. But **quasiparticle picture is not necessary**!
# How to compute rates: the photon example

* At leading order in QED and to all orders in QCD the photon rate is given by
  $$ \dot{f}_{\gamma}(t, \mathbf{k}) = \Gamma_{\gamma}(k) [f_{\text{eq}}(k^0) - f_{\gamma}(t, \mathbf{k})] + \mathcal{O}(e^4) $$

$$ \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle [J^{\mu}(X), J_{\mu}(0)] \rangle $$

$$ \Gamma_{\gamma \text{ prod}} = n_B(k) \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle J^{\mu}(0) J_{\mu}(X) \rangle $$

## J electromagnetic current

* $\alpha \ll 1$ implies that photon production is a rare event and that rescatterings and back-reactions are negligible: medium is transparent to/not cooled by photons
# How to compute rates: the photon example

$$ \Gamma_{\gamma} = -\frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle [J^{\mu}(X), J_{\mu}(0)] \rangle $$

$$ \Gamma_{\gamma \text{ prod}} = n_B(k) \Gamma_{\gamma} = -\frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle J^{\mu}(0) J_{\mu}(X) \rangle $$

* In equilibrium $\langle \dots \rangle = \text{Tr}[\hat{\rho} \dots]$ can be rewritten as a path integral with Euclidean time evolution from 0 to $\beta = 1/T$

* These quantities however need to keep track of the Minkowski time separation between the two currents

* Path integral, known operators and action: either expand $e^{iS} = 1 + iS + \dots$ or discretize (Euclidean) path integral $\int \mathcal{D}\phi \rightarrow \sum_i \phi_i$
# How to compute rates: the photon example

$$ \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle [J^{\mu}(X), J_{\mu}(0)] \rangle $$

$$ \Gamma_{\gamma \text{ prod}} = n_B(k) \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle J^{\mu}(0) J_{\mu}(X) \rangle $$

* Perturbation theory (in g):

* can keep both real and imaginary times with the Schwinger-Keldysh contour. Analytical continuation before path-integral evaluation

* How to extrapolate from $g \ll 1$ (more later) to $g \ge 1$? $\alpha_s \sim 0.3$ typical value close to QCD transition

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# How to compute rates: the photon example

$$ \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle [J^{\mu}(X), J_{\mu}(0)] \rangle $$

$$ \Gamma_{\gamma \text{ prod}} = n_B(k) \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle J^{\mu}(0) J_{\mu}(X) \rangle $$

* Lattice:

* Possible for QCD, not necessarily for (B)SM theories in cosmology

* Analytical continuation **after** numerical path-integral evaluation: inverse problem
$$ \Gamma_{\gamma} = - \frac{e^2}{2k} \int d^4 X e^{iK \cdot X} \langle [J^{\mu}(X), J_{\mu}(0)] \rangle \qquad J^{\mu} = \sum_{q=uds} e_q \bar{q} \gamma^{\mu} q : \gamma \langle $$

* What it means: cut two-point function with thermal propagators. At one loop:

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Kinematically forbidden. Need to kick one of the quarks off-shell.

* Leading order photon is $\alpha_{EM} g^2$

* Strength of the kick (virtuality) naturally divides the calculation in the distinct 2↔2 processes and collinear processes

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* Cut two-loop diagrams ($\alpha_{EM} g^2$)

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2 ↔ 2 processes (with crossings and interferences):

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$$ \int_{\text{ph. space}} f(p) f(p') (1 \pm f(k')) |\mathcal{M}|^2 \delta^4(P + P' - K - K') $$

* Equivalence with kinetic theory: distributions x matrix elements

* IR divergence (Compton) when $t$ goes to zero
* Long-wavelength ($\gg 1/T$) intermediate quark can no longer resolve individual thermal constituents. Emergence of **collectivity** at the scale $gT$: Debye screening, plasmons, Landau damping

* **Hard Thermal Loop** resummation correctly describes LO collective effects and cures IR divergence Braaten Pisarski NPB337 (1990)

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* Long-wavelength ($\gg 1/T$) intermediate quark can no longer resolve individual thermal constituents. Emergence of **collectivity** at the scale $gT$: Debye screening, plasmons, Landau damping

* **Hard Thermal Loop** resummation correctly describes LO collective effects and cures IR divergence Braaten Pisarski NPB337 (1990)

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* In the end one obtains the result
$$ \left. \frac{d\Gamma_{\gamma}}{d^3k} \right|_{2\leftrightarrow 2} \propto e^2 g^2 \left[ \log \frac{T}{m_{\infty}} + C_{2\leftrightarrow 2} \left( \frac{k}{T} \right) \right] $$
Collinear processes

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* These diagrams contribute to LO if small (g) angle radiation/annihilation Aurenche Gelis Kobes Petitgirard Zaraket 1998-2000

* Photon formation times is then of the same order of the soft scattering rate $\Rightarrow$ interference: LPM effect

* Requires resummation of infinite number of ladder diagrams

<figure><img src="image_25.png" /></figure>

AMY (Arnold Moore Yaffe) JHEP 0111, 0112, 0226 (2001-02)
QGP photons

<figure><img src="image_26.png" /><figcaption>Thermal photon rate, $\alpha_s=0.2$</figcaption></figure>

LO: AMY (2001-02) NLO: JG Hong Kurkela Lu Moore Teaney JHEP0503 (2013)
Intermediate summary

* Photon rate in QCD: important observable in heavy ions

* In two decades we went from partial to full LO to NLO (relative $O(g)$)

* Can we test this against lattice results?

* What other lessons can we apply to cosmological rates?
# The photon rate on the lattice

* What is measured directly is the Euclidean correlator

$$G_E(\tau, k) = \int d^3x J_\mu(\tau, \mathbf{x}) J_\mu(0, 0) e^{i\mathbf{k} \cdot \mathbf{x}}$$

* Analytical continuation $G_E(\tau, k) = G^{<}(i\tau, k)$

$$G_E(\tau, k) = \int_0^\infty \frac{dk^0}{2\pi} \rho_V(k^0, k) \frac{\cosh(k^0(\tau - 1/2T))}{\sinh(\frac{k^0}{2T})} \qquad G^{<}(K) = n_B(k^0)\rho_V(k^0, k)$$

* It contains much more info (full spectral function), but hidden in the convolution. Inverse problem: ill-posed, discrete dataset with errors
# The photon rate on the lattice

* What is measured directly is the Euclidean correlator

$$G_E(\tau, k) = \int d^3x J_\mu(\tau, \mathbf{x}) J_\mu(0, 0) e^{i\mathbf{k} \cdot \mathbf{x}}$$

* Analytical continuation

$$G_E(\tau, k) = G^<(i\tau, k)$$

$$G_E(\tau, k) = \int_0^\infty \frac{dk^0}{2\pi} \rho_V(k^0, k) \frac{\cosh(k^0(\tau - 1/2T))}{\sinh(\frac{k^0}{2T})}$$

$$G^<(K) = n_B(k^0)\rho_V(k^0, k)$$

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# The photon rate on the lattice

* If $k>0$ spf describes DIS ($k^0<k$), photons ($k^0=k$) and dileptons ($k^0>k$)

<figure><img src="image_28.png" /></figure>
* If $k>0$ spf describes DIS ($k^0<k$), photons ($k^0=k$) and dileptons ($k^0>k$)

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* Plots and spectral function from Jackson Laine JHEP1109 (2019)
  Quenched lattice from JG Kaczmarek Laine F.Meyer PRD94 (2016)
# The photon rate on the lattice: fit reconstruction

* Main idea: assume a fitting form for the spf, relying on Ansätze

* Get the Euclidean correlator from this ansatz spf and fit the spf coefficients to the lattice data

* Two approaches so far

<figure><img src="image_31.png" /></figure>

* Quenched, continuum extrapolated lattice data, standard vector spf $\rho_V=2\rho_T+\rho_L$

* Convolution dominated by (well-understood) vacuum physics at $\omega \gg k$

JG Kaczmarek Laine F.Meyer PRD94 (2016)
# The photon rate on the lattice: fit reconstruction

* Main idea: assume a fitting form for the spf, relying on Ansätze

* Get the Euclidean correlator from this ansatz spf and fit the spf coefs to the lattice data

* Two approaches so far

* $N_f=2$ continuum extrapolated, modified spf
  $\rho_{\text{Mainz}}=2\rho_T-2\rho_L$

* Vacuum contribution vanishes identically
  (Lorentz invariance). $\rho_{\text{Mainz}}(\omega=k)=\rho_V(\omega=k)$
  Brandt Francis Harris H.Meyer Steinberg
  1710.07050 Cè Harris H.Meyer Steinberg Toniato
  PRD102 (2020)

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# The photon rate on the lattice: fit reconstruction

$$ \bullet \text{ Define } D_{\text{eff}}(k) \equiv \begin{cases} \frac{\rho_V(k, \mathbf{k})}{2\chi_q k}, & k > 0 \\ \lim_{\omega \to 0^+} \frac{\rho^{ii}(\omega, \mathbf{0})}{3\chi_q\omega}, & k = 0 \end{cases} $$

<figure><img src="image_33.png" /></figure>

JG Kaczmarek
Laine F.Meyer

$$ \bullet \text{ In the hydro limit } k \ll T D_{\text{eff}} \rightarrow D \quad \sigma = e^2 \sum_{f=1}^{N_f} Q_f^2 \chi_q D $$

<figure><img src="image_34.png" /></figure>

Cè Harris H.Meyer
Steinberg Toniato
# The photon rate on the lattice: moments

* New idea: introduce $H(\omega_n) = -\int_0^\beta d\tau \int d^3x e^{i\omega_n\tau} e^{-\omega_n z} \langle J_x(\tau, \mathbf{x}) J_x(0) \rangle$

Fourier-transform of Euclidean transverse correlator with imaginary spatial momentum. $\omega_n = 2\pi Tn, n \in \mathbb{Z}$ Matsubara frequency
Meyer 1807.00781

* Dispersive relation to a moment of the spectral function

$$H(\omega_n) - H(\omega_r) = \int_0^\infty \frac{d\omega}{\pi} \omega \rho_T(\omega, k = \omega) \left[ \frac{1}{\omega^2 + \omega_n^2} - \frac{1}{\omega^2 + \omega_r^2} \right]$$

* L.h.s. can be directly computed on the lattice, it is UV finite and not vacuum-dominated
# The photon rate on the lattice: moments

$$H(\omega_n) - H(\omega_r) = \int_0^\infty \frac{d\omega}{\pi} \omega \rho_T(\omega, k=\omega) \left[ \frac{1}{\omega^2 + \omega_n^2} - \frac{1}{\omega^2 + \omega_r^2} \right]$$

<figure><img src="image_35.png" /></figure>

* Lattice determination of $H(\omega_1)$ and $H(\omega_2) - H(\omega_1)$ is compared with value from LO pQCD $\rho_T \propto \Gamma_\gamma$.

* pQCD larger than lattice for $|H(\omega_1)|$, comparable to lattice for $H(\omega_2) - H(\omega_1)$. Transition from $k \ge T$ photons to $k \ll T$ transport physics related to electrical conductivity probably overestimated by pQCD
* Photon rate in QCD: important observable in heavy ions

* In two decades we went from partial to full LO to NLO (relative $O(g)$)

* Lattice determinations are far from trivial, reasonable agreement for $k \gtrsim T$ with at most a factor of 2 effect

* What other lessons can we apply to cosmological rates? Let's consider the axion
# Cosmology: the axion case

* **Strong CP problem:** a term $\propto \epsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma}$ in the QCD Lagrangian is allowed by gauge symmetry and renormalizability. It would violate CP.

* Unobserved neutron electric dipole moment implies $|\theta| < 10^{-10}$. Why so small?

* A mechanism of spontaneous symmetry violation would explain this, forcing $\theta \to 0$ with the **axion** as its light (would-be) Goldstone boson and “cleaning up” this problem

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# Cosmology: the axion case

* It turns out the axion can also “clean up” another problem: when $T \lesssim T_{QCD}$ chiral symmetry breaking “tilts” the potential, giving a small mass to the axion and forcing a CP conserving vacuum

* The cosmological axion field would start oscillating around this new non-degenerate minimum, creating a cold, matter contribution to the energy density of the universe

* **Cold dark matter** from ultralight bosons in the form of a *Bose condensate*

<figure><img src="image_38.png" /></figure>
# Cosmology: the axion case

* This is not the only expected cosmological population of axions. Consider the so called invisible (KSVZ) axion [Kim Shifman Vainshtein Zakharov](https://arxiv.org/abs/1107.3644) $\mathcal{L}_{\text{int}} = \frac{\alpha_s}{8\pi} \frac{a}{f_{\text{PQ}}} \epsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma}$, with $a$ the axion field and $f_{\text{PQ}}$ the symmetry-breaking scale, $f_{\text{PQ}} > 4 \times 10^8$ GeV from astrophysics

* At sufficiently high temperatures *hot axions* would be in thermal equilibrium. They would later freeze out and contribute to *dark radiation*. $\Gamma_{\text{int}} \sim T^3/f_{\text{PQ}}^2$, Hubble rate $H \sim T^2/m_{\text{Pl}}$. Hence $T_{\text{f.o.}} \sim f_{\text{PQ}}^2/m_{\text{Pl}}$

* This is constrained by BBN and CMB determinations of the so-called *effective number of neutrinos*
# Cosmology: the axion case

$$e_{\text{rad}} = \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N_{\text{eff}} \right] e_{\gamma}, \quad N_{\text{eff}} = \overbrace{N_{\text{eff}}^{\nu}}^{\text{SM}} + \overbrace{\Delta N_{\text{eff}}}^{\text{BSM}} \\ \Delta N_{\text{eff}} = \frac{8}{7} \left( \frac{11}{4} \right)^{4/3} \frac{e_a}{e_{\gamma}} \bigg|_{\text{CMB}}$$

* The smaller $f_{\text{PQ}}$, the later the freeze out, the larger the contribution to $\Delta N_{\text{eff}}$

* At $T > T_{\text{EW}}$ the axion is one in $O(100)$ light d.o.f.s in equilibrium. At $T < T_{\text{QCD}}$ one in $O(10)$

<figure><img src="image_39.png" /></figure>

* Future measurements will probe higher and higher freeze-out temperatures
# Cosmology: the axion case

* Invisible (KSVZ) axion $\mathcal{L}_{\text{int}} = \frac{\alpha_s}{8\pi} \frac{a}{f_{\text{PQ}}} \epsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma}$

* To study its freeze out we then need

$$ \dot{f}_a(t, \mathbf{k}) = \Gamma_a(k) [f_{\text{eq}}(k^0) - f_a(t, \mathbf{k})] + \mathcal{O}(T^2/f_{\text{PQ}}^2) $$

$$ \Gamma_a = \frac{\alpha_s^2}{2k(8\pi f_{\text{PQ}})^2} \int d^4 X e^{iK \cdot X} \langle [\textcolor{red}{J(X)}, \textcolor{red}{J(0)}] \rangle \qquad J = \epsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma} $$

* Conceptually very similar to photons, but lattice much more complicated: discretizing $FF$, finding a vacuum-free operator and dealing with topology are major hurdles. What does pQCD tell us?
# The axion rate

$$ \Gamma_a = \frac{\alpha_s^2}{2k(8\pi f_{PQ})^2} \int d^4 X e^{iK \cdot X} \langle [J(X), J(0)] \rangle \qquad J = \epsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma} $$

* For $k \ge T, 2 \leftrightarrow 2$ processes and HTL resummation for soft $t$ gluon

<figure><img src="image_40.png" /></figure>

$$ \Gamma_a(k) = \frac{\alpha_s^3 T^3}{f_{PQ}^2} \left[ c_{LL} \ln \frac{k}{m_D} + f(k/T) \right], \quad \text{with } m_D \sim gT $$

Braaten Yuan PRL66 (1991) Graf Steffen 1008.4528

* No collinear emission (suppressed by derivative coupling)
# The axion energy density

$$ \dot{e}_a + 4He_a = \int_k k \Gamma_a(k) [f_{\text{eq}}(k) - f_a(t, \mathbf{k})] \qquad \Gamma_a(k) = \frac{\alpha_s^3 T^3}{f_{\text{PQ}}^2} \left[ c_{LL} \ln \frac{k}{m_D} + f(k/T) \right], \quad \text{with } m_D \sim gT $$

* To obtain the energy density one integrates over the $k \ll m_D$ region, where the rate extrapolates to unphysical, negative values

* Idea 1: resum one-loop Feynman-gauge self-energies for all loop momenta, rather than gauge-invariant HTL for soft momenta Rychkov Strumia hep-th/0701104, Salvio Strumia Xue 1310.6983

* Claimed price was a relative $\mathcal{O}(\alpha_s)$ gauge dependence. Recent careful numerical implementation found a divergent rate: divergent gauge-dependent sensitivity to the $g^2T$ chromomagnetic scale Bouzoud JG 2404.06113

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>
# The axion rate, redux

<figure><img src="image_43.png" /></figure>

* New tuned resummation to avoid negative rates: $1/t \rightarrow q^2/(q^2 + \xi^2 m_D^2)t$, with $\xi$ tuned to HTL result. Differs from strict LO by $\mathcal{O}(g)$
Bouzoud JG 2404.06113
# The axion energy density, redux

* Solving the Boltzmann equation for the energy density

* First estimate of NLO $O(g)$ by assuming they are equal to those to $\hat{q}$, a jet quenching quantity sharing the same soft-gluon sector

* A 7% effect on the observable at the smallest $f_{PQ}$

<figure><img src="image_44.png" /></figure>
# Summary

* Photon rate in QCD: important observable in heavy ions

* In two decades we went from partial to full LO to NLO (relative $O(g)$)

* Lattice determinations are far from trivial, reasonable agreement for $k \gtrsim T$ with at most a factor of 2 effect

* Axion rate: quantification of theory uncertainty from different ways of handling collective effects at $T \gtrsim T_{QCD}$, crucial for future CMB precision measurements
<figure><img src="image_45.png" /></figure>
# The QCD transition

<figure><img src="image_46.png" /></figure>

Review: JG Kurkela Strickland Vuorinen Phys. Rep. 880 (2020)

Lattice: Budapest-Wuppertal, Borsanyi et al JHEP1011 (2010)
# A way to large lepton asymmetries?

* Right-handed neutrino interaction rate $\Gamma$ for a specific seesaw parametrisation

<figure><img src="image_47.png" /></figure>