Module 1 · Week 2 · Graduate
Sakharov's Three Conditions
In a prescient 1967 paper, Andrei Sakharov enumerated three necessary conditions for any dynamical mechanism to generate a matter-antimatter asymmetry starting from a symmetric initial state. The three conditions are elegant, rigorous, and surprisingly hard to satisfy simultaneously.
1. Historical Context
In 1967 Andrei Sakharov published a three-page paper in JETP Letters titled “Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe”. At the time, CP violation had just been discovered in neutral kaon decays (Cronin & Fitch, 1964), and the hot Big Bang was still being worked out (Dicke, Peebles, Roll & Wilkinson, 1965).
Sakharov made the then-radical connection: the very same symmetry violation observed in kaons could, combined with other dynamical ingredients, explain why there is anything at allin the universe instead of just equal parts matter and antimatter annihilating to photons.
“I am proposing a set of three conditions which appear to be necessary for the occurrence of a nonvanishing baryonic charge in the universe.” — A. D. Sakharov, 1967
2. The Three Conditions
(1) Baryon number violation
Without processes of the form \(X \to q\,q\) (with \(\Delta B \neq 0\)), no net baryon number can ever develop — baryon number is exactly conserved, so if it starts at zero it stays at zero.
Formally: some Lagrangian term, perturbative or non-perturbative, must have \(\Delta B \neq 0\).
(2) C and CP violation
Even if B violation is present, the rates for \(X \to qq\) and \(\bar X \to \bar q \bar q\) would be equal if C and CP were exact. Both must be violated so that the decays into baryons and antibaryons have different probabilities.
C alone is insufficient: under CP, \(X\to qq\) maps to \(\bar X \to \bar q \bar q\). CP conservation forces equal rates.
(3) Departure from thermal equilibrium
In thermal equilibrium, detailed balance plus CPT forces the expectation value \(\langle B \rangle = 0\). Proof below.
Departure from equilibrium can be achieved via (a) an out-of-equilibrium heavy particle decay ( \(\Gamma < H\)), or (b) a first-order phase transition (bubble nucleation).
3. Why Equilibrium Implies <B> = 0
The CPT theorem, a bedrock consequence of Lorentz invariance and locality, guarantees that particles and antiparticles have identical masses. In thermal equilibrium at temperature T, the expectation value of the baryon number charge B is
Under CPT, \(H\) is invariant and \(B \to -B\). Since the trace is unchanged under a similarity transformation,
where \(\Theta\) is the antiunitary CPT operator. This forces \(\langle B\rangle_{eq} = 0\).
In other words, once the plasma reaches equilibrium, any previously generated asymmetry is washed out to zero. Baryogenesis must therefore occur out of equilibrium.
Detailed balance derivation
Detailed balance states that, in equilibrium, forward and reverse reaction rates are equal. For a B-violating process \(X \to f\) with partial rate \(\Gamma_f\) and CP conjugate \(\bar X \to \bar f\) with \(\bar\Gamma_f\), equilibrium implies
CPT guarantees \(n_X^{eq} = n_{\bar X}^{eq}\), and unitarity ( \(\sum \Gamma_f - \sum \bar\Gamma_f = 0\)) ensures no net asymmetry accumulates.
4. Diagram: Three Conditions Interlocking
The intersection of all three conditions is non-empty; dropping any of the three forces \(\eta = 0\).
5. The CPT Theorem
The CPT theorem (Lüders 1954; Pauli 1955; Jost 1957) states: any relativistic, local, Lorentz-invariant QFT with a Hermitian Hamiltonian is invariant under the combined action of charge conjugation C, parity P, and time reversal T.
Immediate consequences crucial for baryogenesis:
- Equal masses: \(m_X = m_{\bar X}\)
- Equal lifetimes: \(\tau_X = \tau_{\bar X}\) (equivalently, total widths)
- Equal but opposite magnetic moments: \(\mu_X = -\mu_{\bar X}\)
- No zero-temperature asymmetry: partial widths to CP-conjugate final states may differ, but the total must agree.
Key subtlety: partial vs total rates
CPT does not force \(\Gamma(X\to f) = \Gamma(\bar X\to \bar f)\). It only forces the totals summed over final states to be equal: \(\sum_f \Gamma(X\to f) = \sum_f \Gamma(\bar X\to \bar f)\). This is precisely the loophole that allows CP-violating partial-rate differences to generate baryogenesis, while still being consistent with CPT.
6. A Minimal Toy Model
Consider a heavy scalar X with two decay modes, one into a pair of quarks ( \(\Delta B = 2/3\)) and one into an anti-quark and an anti-lepton ( \(\Delta B = -1/3\)):
Its CP conjugate: \(\bar X \to \bar q\bar q\) (branching \(\bar r\)), \(\bar X \to q\ell\) (branching \(1-\bar r\)). CPT forces \(r + (1-r) = \bar r + (1-\bar r) = 1\) (trivial!), but does notforce \(r = \bar r\). If \(r \neq \bar r\), the net baryon number per X-plus-Xbar pair is
So \(\epsilon_{CP} \equiv r - \bar r\) characterizes the CP asymmetry. It arises from interference between tree-level and one-loop diagrams, typically giving \(\epsilon_{CP} \sim (g^2/16\pi^2) \sin\phi_{CP}\) where \(\phi_{CP}\) is the complex phase.
Condition 3 is satisfied if the X decay rate \(\Gamma_X < H(T = m_X)\), so the X population does not reach equilibrium before annihilating away. The final yield is then approximately \(Y_B \approx \epsilon_{CP}\, Y_X^{eq}/g_*\) where \(Y_X^{eq} = 45/(2\pi^4 g_*)\) for a relativistic species.
Simulation: Symmetry Violation is Necessary
This simulation explicitly demonstrates the Sakharov argument: a simple two-state model of X and Xbar decay is evolved with various combinations of the three conditions turned on or off. Only when all three are simultaneously active does a net baryon number accumulate.
Click Run to execute the Python code
Code will be executed with Python 3 on the server
7. Variants and Loopholes
The Sakharov conditions are necessary, but are they truly the tightest possible? Several subtleties and proposed evasions are worth noting:
(a) CPT violation
The equilibrium argument relies on the CPT theorem. If CPT were somehow violated, particle and antiparticle chemical potentials could differ in equilibrium, producing an asymmetry. However, CPT is tested to \(|m_K - m_{\bar K}|/m_K < 10^{-18}\)— an extraordinarily tight bound. Models with spontaneous CPT violation (e.g. Cohen & Kaplan 1987) exist but are disfavored.
(b) Asymmetric initial conditions
The Sakharov conditions assume symmetric initial conditions. A universe that simply started baryon-asymmetric would not need any of them. But inflation dilutes pre-existing asymmetries: the post-inflation universe must be produced dynamically. So Sakharov's conditions apply to any inflationary cosmology.
(c) Spontaneous baryogenesis
Cohen, Kaplan & Nelson (1987) showed that a time-varying scalar coupling to the baryon current ( \(\partial_\mu\phi\, J^\mu_B\)) can generate an effective chemical potential for baryon number while in equilibrium. Technically this is consistent with all Sakharov conditions once rewritten in proper variables; it simply relaxes condition (3) by using a field expectation value as the non-equilibrium background.
(d) Asymmetric dark matter
If dark matter carries a conserved charge related to B, the asymmetry could be shared between sectors. Then \(\Omega_{DM}/\Omega_B\) \u2248 5 becomes a prediction rather than an accident, and Sakharov conditions apply to the combined system.
8a. Why Both C and CP Must Be Violated
The weak interaction violates C maximally: only left-handed neutrinos and right-handed antineutrinos participate. But maximal C violation alone is consistent with exact CP. To see why both C and CP must be violated for baryogenesis, consider a decay \(X\to f_L\) (left-handed final state). Under:
- C alone: Maps to \(\bar X \to \bar f_L\) (also left-handed). Rate is zero in SM, so this is not useful for asymmetry.
- P alone: Maps to \(X \to f_R\) (right-handed). Not in SM interactions.
- CP: Maps to \(\bar X \to \bar f_R\). If CP is exact, rates are equal; no asymmetry.
Therefore, even if C is maximally violated by the weak interaction, CP conservation would still force particle/antiparticle decay rates to be equal when summed over chiralities. Both symmetries must fail for a net asymmetry to develop.
8b. The Cronin-Fitch Experiment in Context
The 1964 discovery of CP violation in neutral kaon decays was a crucial ingredient in Sakharov's thinking. The experiment measured the branching fraction \(K_L \to \pi^+\pi^-\) as (2.0 \u00B1 0.4) \u00D7 10 \(^{-3}\)— astonishing because this decay is CP-forbidden if CP were exact. The discovery earned Cronin and Fitch the 1980 Nobel Prize.
For baryogenesis, CP violation alone is insufficient (the Jarlskog quark-mass suppression makes it far too small in the SM). The lesson of Sakharov is that some CP violation must exist, and it must be amplified by out-of-equilibrium dynamics and B-violating couplings.
9. Historical Context: Sakharov and Beyond
Sakharov's 1967 paper went largely unnoticed for over a decade. The Pati-Salam model (1974), Georgi-Glashow SU(5) (1974), and the discovery of asymptotic freedom (1973) transformed the landscape. By 1978\u20131979, Yoshimura and Weinberg realized that GUT baryogenesis could quantitatively implement Sakharov's ideas. Then in 1985, Kuzmin, Rubakov, and Shaposhnikov showed sphaleron-driven baryon violation would erase any pure-B asymmetry in the SM electroweak era, forcing attention toward B−L-producing mechanisms. In 1986 Fukugita and Yanagida proposed leptogenesis; by 2002, Davidson and Ibarra established the quantitative lower bound on right-handed neutrino masses.
Modern research (2020s) focuses on electroweak baryogenesis with BSM Higgs sectors, flavored and resonant leptogenesis, Affleck-Dine scenarios, and dark baryogenesis — all direct descendants of Sakharov's three conditions.
Worked Example: Minimal Toy Model Calculation
Consider a heavy scalar X with mass M, coupling to quarks and leptons with a complex Yukawa coupling \(y\,e^{i\phi}\). Suppose X has two decay modes with partial widths \(\Gamma_1, \Gamma_2\). CPT forces \(\Gamma_1 + \Gamma_2 = \bar\Gamma_1 + \bar\Gamma_2\). The CP asymmetry
A simple tree\u2013one-loop interference calculation (Kolb-Wolfram 1980) gives, for a typical model,
where \(\alpha = y^2/4\pi\) and m is a loop mass. For \(\alpha \sim 0.1\), \(\phi = \pi/4\) and modest log: we get \(\epsilon \sim 10^{-3}\). Together with out-of-equilibrium X decay ( \(K \sim 1\)), this produces a yield \(Y_B \sim \epsilon Y_X^{eq}/g_* \sim 10^{-7}\).
8. Mathematical Bestiary of C, P, T Operators
For reference, the action of C, P, T on a Dirac fermion \(\psi\):
Under the combined CPT:
The CPT theorem follows from combining Lorentz invariance, locality, and Hermiticity of the Hamiltonian; see Greenberg's proof (2006) for the cleanest modern treatment.
References
- Sakharov, A. D., “Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe”, JETP Lett. 5, 24 (1967).
- Lüders, G., “On the equivalence of invariance under time reversal and under particle-antiparticle conjugation for relativistic field theories”, Dan. Mat. Fys. Medd. 28, 5 (1954).
- Greenberg, O. W., “Why is CPT fundamental?”, Found. Phys. 36, 1535 (2006).
- Kolb, E. W. & Turner, M. S., The Early Universe, Chapter 6.
- Riotto, A. & Trodden, M., “Recent progress in baryogenesis”, Ann. Rev. Nucl. Part. Sci. 49, 35 (1999).