Baryogenesis

Why an asymmetry at all?

A symmetric early universe — equal abundances of every particle and its antiparticle — is a perfectly good cosmology in every other respect except that nothing is left over. Particles and antiparticles annihilate; once the temperature drops below their rest mass and they can no longer be replenished by thermal collisions, the remaining gas is set by the freeze-out density. For nucleons that residue is calculably tiny: the ratio of surviving baryons to photons at T ~ 20 MeV in a symmetric scenario would be of order 10⁻¹⁸ — not enough to make a single galaxy, let alone the observed universe.

What the universe actually contains is one extra baryon for every roughly 1.6 billion photons. That number — the baryon-to-photon ratio η = n_B / n_γ ≈ 6.1 × 10⁻¹⁰ — is measured in two independent ways. The first is Big Bang nucleosynthesis: the relative abundances of D, ³He, ⁴He, and ⁷Li produced when the universe was three minutes old depend sensitively on η. Deuterium in particular, observed in metal-poor quasar absorption systems, fixes η at the 3 percent level. The second is the cosmic microwave background: the ratio of odd to even peak heights in the angular power spectrum probes the baryon density directly. Planck 2018 gives η = (6.12 ± 0.04) × 10⁻¹⁰. The two methods agree at the 1 percent level.

So this 10⁻¹⁰ is data. Baryogenesis is whatever produced it.

The Sakharov conditions

In 1967 Andrei Sakharov wrote a three-page paper that has framed the problem ever since. He showed that to dynamically generate a baryon asymmetry from an initially symmetric state, three conditions are individually necessary:

1. Baryon number violation. If B is exactly conserved by all interactions, the universe's initial B = 0 is also its final B. No process can generate a net baryon number.
2. C and CP violation. If charge conjugation symmetry C holds, then any reaction producing baryons has a matching reaction producing antibaryons at the same rate, and there is no net production. If only C is violated but CP is preserved, the same is true once you sum over all polarisations. Both C and CP must be violated.
3. Departure from thermal equilibrium. In full thermal equilibrium, CPT symmetry (which all local Lorentz-invariant QFTs respect) guarantees that any particle and its CPT conjugate have equal mass and therefore equal equilibrium abundance. So even if reactions violate B and CP, the equilibrium distributions enforce equal particle and antiparticle densities. You need a non-equilibrium state — typically the universe expanding faster than some relaxation rate — for the imbalance to be locked in.

All three are necessary; each is independently restrictive. Every working baryogenesis mechanism in the literature satisfies them in a specific way, and most disagreements among proposals are arguments about how to source them concretely.

Why the Standard Model alone doesn't do it

On a quick read, the Standard Model satisfies all three Sakharov conditions:

• (1) B-violation. The electroweak vacuum has infinitely many topologically distinct sectors labelled by Chern-Simons number. Transitions between them — mediated at finite temperature by sphalerons — violate B + L by Δ(B+L) = 6 per transition, while preserving B − L. So B is not exactly conserved in the SM.
• (2) CP violation. The CKM quark mixing matrix has a single physical CP-violating phase δ, observed in B-meson decays at LHCb and KEKB. So CP is violated.
• (3) Out of equilibrium. The universe is expanding, so any phase transition is potentially out of equilibrium.

Each condition fails quantitatively, though, and the failures compound. The CKM CP violation is parameterised by the Jarlskog invariant J = (3.2 ± 0.2) × 10⁻⁵. Weighted by ratios of quark masses (the relevant combinations are differences m_t² − m_c², etc., normalised by the temperature), the resulting baryon asymmetry from CKM-driven processes is of order

η_CKM ~ α_W⁵ × J × (m_q²/T²)² ~ 10⁻²⁰

which is ten orders of magnitude too small.

The third Sakharov condition fails even worse. For a Higgs mass below about 80 GeV the electroweak phase transition would have been first-order; for a Higgs mass at the measured 125 GeV it is a smooth crossover. There are no nucleating bubbles, no abrupt wall motion, no out-of-equilibrium thermal forces to convert CP violation into baryon number. Lattice simulations confirm this picture conclusively. So even the modest CKM CP violation has no out-of-equilibrium machinery to act on. The Standard Model under-produces η by roughly eight orders of magnitude. Baryogenesis is one of the most secure pieces of evidence that physics beyond the Standard Model exists.

Sphalerons — the SM's contribution to B-violation

The electroweak sphaleron is a static, unstable, saddle-point configuration of the SU(2) gauge field. Its energy is set by the Higgs vacuum expectation value:

E_sph(T) = (8π/g²) × v(T) × B(λ/g²)   ≈ 9 TeV at T = 0

where B is a numerical function of order unity. At zero temperature the rate of sphaleron-induced transitions is exponentially suppressed: Γ ∝ exp(−E_sph / T) → 0, which is why proton decay (which would violate B) is unobservably slow. At finite temperature T ≳ 100 GeV the suppression weakens; for T > T_EW ≈ 160 GeV the rate becomes

Γ_sph / V ~ α_W⁵ T⁴   (symmetric phase, high temperature)

which is fast compared with the Hubble rate, so B + L is in equilibrium. Sphalerons preserve B − L exactly. Two consequences follow:

• Any baryogenesis mechanism that produces only B + L charge gets washed out by sphalerons once the universe enters the equilibrium window. Successful B-violation must produce a net B − L.
• A primordial lepton asymmetry, with no initial baryon asymmetry, can be partially reprocessed by sphalerons into a baryon asymmetry. The exact partition in the Standard Model spectrum is B = −(28/79)(B − L), L = (51/79)(B − L), with about 36 percent of the original lepton asymmetry ending up in baryons. This is the engine of leptogenesis.

GUT baryogenesis — the earliest proposal

The first explicit baryogenesis scenarios, developed by Yoshimura (1978), Toussaint, Treiman, Wilczek, and Zee (1979), and Weinberg (1979), exploit Grand Unified Theories. A GUT like SU(5) or SO(10) embeds the SM into a single gauge group, with new gauge bosons X and Y that carry both colour and electroweak charges and have mass M_X ~ 10¹⁴–10¹⁶ GeV.

X bosons couple quarks to leptons. Their decays X → q + q (a quark pair, baryon number 2/3) or X → q̄ + ℓ⁺ (an anti-quark and lepton, baryon number −1/3) violate baryon number explicitly. CP violation in the GUT Yukawas makes the decay rates of X and X̄ slightly different, so a CP-violating phase ε generates a net baryon number per X-decay of order ε. If the X bosons are produced thermally in equilibrium and then their decay rate falls below Hubble (third Sakharov condition), the asymmetry freezes in.

The major problem with GUT baryogenesis is that SU(5) preserves B − L exactly, so any asymmetry it generates is in B + L only — and sphalerons subsequently wash it out. Models in SO(10), which contains the right-handed neutrino and naturally violates B − L through Majorana masses, evade this. But GUT baryogenesis sets the relevant scale at 10¹⁴–10¹⁶ GeV, where it is hard to constrain experimentally. Recent attention has therefore shifted toward lower-scale mechanisms.

Electroweak baryogenesis — making the SM transition first-order

If the electroweak phase transition were strongly first-order, the Higgs field would nucleate bubbles of broken phase at T ≈ 100 GeV. These bubbles would expand at near-light speed, sweeping the symmetric phase out of existence over Hubble time. The walls would be CP-asymmetric scattering surfaces: particles diffusing through them feel a different reflection probability than their antiparticles do (the CP violation), and the resulting current sources a chemical potential for left-handed quarks in the symmetric phase, where sphalerons are still active and turn the excess into a net B − L. Inside the broken phase sphalerons are exponentially shut off, so the asymmetry is locked in.

This scenario is conceptually beautiful and was the leading candidate through the 1990s. For the measured Higgs mass of 125 GeV the SM transition is a crossover, so EW baryogenesis requires beyond-SM physics that strengthens the transition. The standard additions are:

• Singlet scalars mixing with the Higgs and giving a tree-level barrier.
• Two-Higgs-doublet models with additional Higgs bosons whose loops modify the effective potential.
• MSSM with light stops (the supersymmetric partner of the top quark) — heavily disfavored after the LHC pushed stop masses above ~1 TeV.
• Strong sectors with new TeV-scale dynamics.

All these introduce additional CP-violating phases that contribute to electric dipole moments (EDMs) of the electron, neutron, mercury, and other systems. The 2023 ACME III bound |d_e| < 4.1 × 10⁻³⁰ e·cm essentially eliminates the simplest two-Higgs-doublet realisations, leaving narrow corners with cancellations or alignment limits. Electroweak baryogenesis is alive but on the back foot.

Leptogenesis — currently favored

Fukugita and Yanagida proposed leptogenesis in 1986, just after the seesaw mechanism was understood as the natural origin of neutrino masses. The argument is concise and economical:

1. The observed sub-eV neutrino masses are most easily explained by adding right-handed Majorana neutrinos N_i with masses M_i ≫ M_Z. The seesaw formula m_ν ≈ y² v² / M gives m_ν ~ 0.05 eV for y ~ 1 and M ~ 10¹⁵ GeV.
2. The Yukawa couplings y_iα connecting N_i to lepton flavour α generically contain CP-violating complex phases.
3. After inflation the N_i are produced thermally. They are unstable against decay N_i → ℓ_α + H or N_i → ℓ̄_α + H̄.
4. Interference between tree-level and one-loop self-energy and vertex diagrams gives different rates for the two channels: ε_i ≡ (Γ_i − Γ̄_i) / (Γ_i + Γ̄_i). This is a direct manifestation of Sakharov's second condition.
5. The N_i decays must be slow compared with the Hubble rate at T ~ M_1 (Sakharov's third condition). This requires the lightest right-handed neutrino mass M_1 ≳ 10⁹ GeV (the "Davidson-Ibarra bound" for hierarchical N_i).
6. The resulting lepton asymmetry is partially converted by sphalerons into a baryon asymmetry following B = −(28/79)(B − L).

The required ε_i for η ≈ 6 × 10⁻¹⁰ is naturally of order 10⁻⁶ for Yukawas of order one — fully consistent with what neutrino mass measurements imply. Leptogenesis is now the default scenario in cosmological model-building because it ties baryogenesis directly to the experimentally measured fact that neutrinos have mass, established by Super-Kamiokande, SNO, KamLAND, T2K, and now neutrino-mass-hierarchy constraints from IceCube atmospheric oscillations and PMNS phase measurements at T2K and NOvA. The same Yukawa matrix that gives the observed neutrino masses also gives — naturally, with no additional adjustable parameter — the right baryon-to-photon ratio.

The Affleck-Dine mechanism

In the MSSM, gauge-invariant combinations of squark and slepton fields parameterise flat directions: directions in field space along which the supersymmetric scalar potential vanishes identically. Examples include LH_u (a lepton doublet and the up-type Higgs) and udd (three quarks anti-symmetrised in colour). These directions are lifted only by non-renormalisable operators and by SUSY-breaking soft terms.

Ian Affleck and Michael Dine pointed out in 1985 that during inflation a flat-direction field φ can develop a large expectation value, much greater than the Hubble scale. When inflation ends and SUSY-breaking lifts the potential, φ starts oscillating; the oscillations are not radial (which would conserve baryon number) but angular in the complex plane, because soft A-terms and B-terms in the Lagrangian violate U(1)_B explicitly. The angular velocity is literally a baryon-number current. The condensate eventually decays to ordinary quarks, transferring its accumulated charge.

Affleck-Dine has several remarkable features. It can produce η of arbitrary size, including the small observed value or much larger asymmetries that are then partially diluted. It naturally fragments into stable, charged scalar condensate balls — Q-balls — which can themselves be a dark-matter candidate. It links baryogenesis to inflaton dynamics and supersymmetry-breaking phenomenology. Despite the LHC's lack of low-scale SUSY signals, the mechanism remains viable for high-scale supersymmetry.

Asymmetric dark matter

The cosmological data give Ω_DM ≈ 0.26 and Ω_B ≈ 0.049, so Ω_DM ≈ 5 Ω_B. From a particle-physics standpoint this is suspicious. Baryons are made of GeV-scale particles with η ~ 10⁻¹⁰. Standard WIMP dark matter is a thermal relic with mass at the weak scale and abundance set by annihilation cross-section — there is no a priori reason for the two abundances to be within a factor of 5 of each other.

Asymmetric dark matter models propose that the dark sector inherits an asymmetry from the same primordial physics that gave us baryogenesis. Either both sectors are produced by a single mechanism that distributes the asymmetry between them (sometimes called "co-genesis"), or one is generated and the other follows by a transfer operator at high temperature that subsequently shuts off. In either case the symmetric component of the dark sector annihilates away — just as for baryons — and the surviving asymmetry sets the relic density. Many concrete realisations predict n_DM ≈ n_B, which combined with Ω_DM ≈ 5 Ω_B forces

m_DM ≈ 5 m_p ≈ 5 GeV

This GeV-scale dark matter is exactly the window probed by sub-GeV direct detection experiments (SENSEI, DAMIC, SuperCDMS), and so far they have set increasingly strong bounds without a detection.

Comparison of mechanisms

Mechanism	Scale	B-violation source	CP source	Status (2026)
GUT baryogenesis	10¹⁴–10¹⁶ GeV	X, Y boson decay	GUT-scale phases	Constrained by sphaleron wash-out unless B − L is generated
Electroweak baryogenesis	10² GeV	Sphalerons in symmetric phase	New scalar phases	Squeezed by ACME III EDM bound, narrow corners alive
Leptogenesis (thermal)	10⁹–10¹⁵ GeV	Sphaleron reprocessing	Seesaw Yukawa phases	Currently favored; ties to neutrino masses
Resonant leptogenesis	TeV	As above	Quasi-degenerate N_i	Testable at colliders via heavy-N searches
Affleck-Dine	Inflation-scale	Flat-direction soft A-terms	Soft-term phases	Viable; ties to high-scale SUSY
Asymmetric DM	varies	Hidden sector	Hidden phases or transfer	Predicts m_DM ~ 5 GeV; under direct-detection test
Cold baryogenesis	Reheating	Sphaleron during preheating	BSM CP phases	Niche; depends on inflaton model

Worked example: leptogenesis arithmetic

Take a hierarchical right-handed neutrino spectrum with M_1 = 10¹⁰ GeV. The Davidson-Ibarra bound caps the CP asymmetry produced in N_1 decays:

|ε_1| ≤ (3/8π) (M_1 / v²) × m_max
        ≈ 10⁻⁶   for M_1 = 10¹⁰ GeV, m_max = 0.05 eV

The lepton asymmetry produced per N_1 decay is ε_1. The decay must occur out of equilibrium, characterised by the wash-out parameter K = Γ_1 / H(M_1). The "strong wash-out" regime K ≫ 1 is typical for masses pinned to observed neutrino masses; the efficiency factor κ ~ 0.01.

The lepton-to-entropy ratio is then

Y_L = κ × ε_1 / g*_∗ ~ 10⁻⁶ × 0.01 / 100 ≈ 10⁻¹⁰

Sphalerons convert about a third of this lepton asymmetry into a baryon asymmetry: Y_B ≈ −(28/79) Y_L ≈ 4 × 10⁻¹¹. Converting from entropy density to photons gives η ≈ 7 Y_B ≈ 3 × 10⁻¹⁰ — within a factor of two of the observed value, with no parameters fitted to the answer. The seesaw mass scale that matches observed neutrino masses also matches the observed baryon-to-photon ratio. That is the central elegance of leptogenesis.

Observational handles

• Neutrinoless double beta decay. If observed by KamLAND-Zen, LEGEND, or nEXO, it would establish the Majorana nature of the neutrino — the central ingredient of leptogenesis. The current limit on the effective Majorana mass is |m_ββ| < 36–156 meV depending on nuclear matrix elements.
• Electric dipole moments. An eEDM detection at ACME, JILA, or HfF⁺ experiments above ~10⁻³⁰ e·cm would point at electroweak-scale CP-violating physics consistent with EW baryogenesis. So far only upper bounds.
• Gravitational waves from a first-order EW transition. A strongly first-order EW transition would produce a stochastic GW background peaked near mHz, observable by LISA in the 2030s. A detection would be smoking-gun evidence for EW baryogenesis.
• Primordial isocurvature perturbations. Some Affleck-Dine variants generate baryon isocurvature; Planck constrains the baryon isocurvature fraction to below ~10⁻³, ruling out a class of models.
• Direct detection of asymmetric DM. Sub-GeV experiments probe the m_DM ~ 5 GeV window favored by asymmetric DM.

Common pitfalls

• Confusing baryogenesis with nucleosynthesis. Baryogenesis sets n_B at T ≫ 100 GeV; Big Bang nucleosynthesis fuses those baryons into nuclei at T ≈ 0.1 MeV. The two are sequential epochs, not the same process.
• Forgetting CPT. Without an out-of-equilibrium step, CPT enforces equal abundances of particles and antiparticles. Many would-be mechanisms fail because the proposed dynamics happen to occur near equilibrium.
• Ignoring sphaleron wash-out. An asymmetry in B + L only is destroyed by sphalerons. Mechanisms that violate B − L (right-handed Majorana neutrinos, B − L gauge boson decays) sidestep this, but pure SU(5) GUTs do not.
• Conflating CP violation with B violation. They are independent Sakharov conditions. The Standard Model has CP violation but generates no asymmetry because the out-of-equilibrium condition fails.
• Quoting Jarlskog J as the asymmetry. J is the leading CP-violating invariant in CKM. The baryon asymmetry depends on J multiplied by mass-suppression factors and powers of α_W. The naive number 3×10⁻⁵ overstates SM baryogenesis by many orders of magnitude.