Conditions sought in baryogenesis mechanisms

Conditions identify a task for dynamics

In the conventional CPT-invariant equilibrium framework, baryogenesis mechanisms seek baryon-number violation, an asymmetry under charge-conjugation and CP transformations, and departure from thermal equilibrium. These ingredients have different jobs.

Baryon-number violation permits the number ledger to change. Asymmetric reaction probabilities can favor one sign, while nonequilibrium evolution avoids the cancellation that equilibrium forward and inverse processes enforce.

The conditions are framework dependent and do not by themselves calculate a surviving yield. Mechanisms with different foundational assumptions must state how their own source and consistency conditions work.

The supplied branching ledger and rate equations below illustrate production and survival. They are not a microscopic particle Lagrangian, computed reaction amplitudes or evidence that a particular history occurred.

Equal total widths can accompany unequal partial branching

Supply parents X and anti-X with baryon number zero and the same total decay rate Γ in s⁻¹. Each begins with one normalized excess-yield unit above equilibrium, so the initial net baryon number is zero.

Let X have probability p = 3/5 to produce final total baryon number +1, otherwise 0. Let anti-X have probability q = 2/5 to produce final total baryon number −1, otherwise 0.

The complementary probabilities complete each species’ total rate. Equal total widths do not require every charge-conjugate partial width to be equal; other charge assignments and microscopic consistency are not supplied by this small ledger.

For independently drawn outcomes of one parent pair, the mean final baryon number is 9/25−4/25 = 1/5. Dividing by two parents defines the source per excess parent ε = 1/10.

Use departure from equilibrium rather than all decays as a source

Define dimensionless elapsed rate-time x = Γ(t−t₀) and total excess-parent yield D = 2exp(−x). It obeys D′ = −D; the source is the excess above equilibrium, not the complete equilibrium parent population.

Let B be the normalized net baryon yield, initially zero. Supply B′ = εD−wB with constant nonnegative rate ratio w = W/Γ, where W describes baryon-violating relaxation or washout.

For w≠1, the solution is B = (1/5)[exp(−wx)−exp(−x)]/(1−w). The continuous w = 1 limit is B = (1/5)x exp(−x).

If the excess source D is zero, this model supplies no production term. A microscopic equilibrium treatment must include the inverse processes and detailed dynamics that justify the cancellation, rather than treating equilibrium decays alone as an asymmetry source.

The same source leaves different finite yields

At x = ln 4, the remaining parent excess is D = 1/2. Without washout, the net yield is B = 3/20.

With w = 1/2, the same source history instead gives B = (2/5)(1/2−1/4) = 1/10. With w = 1, it gives B = (ln 4)/20 ≈ 0.069315.

Thus a source with a specified positive mean does not by itself determine the surviving yield. Reaction history and washout can change the result substantially within the same declared model.

The table covers only a finite rate-time interval. If constant w>0 persisted indefinitely after the source faded, B would tend to zero; with w = 0 it would tend to 1/5. These are mathematical limits, not asserted cosmic endpoints.

A complete prediction requires more than an asymmetric parameter

Setting p = q makes ε = 0. With the stated zero initial B, no asymmetry is generated even when parents are out of equilibrium; the source must distinguish the relevant baryon-number outcomes.

Conversely, an asymmetric branching ledger alone does not establish the required nonequilibrium population or protect a generated yield from subsequent washout. The equations make those dependencies separate.

A physical baryogenesis calculation needs consistent particle dynamics, reaction rates, expansion and thermal history, plus later conversion and survival where relevant. These supplied effective rates do not determine them from nature.

A measured CP-violating quantity or a possible relic signature can constrain a specified mechanism through its predicted connections. Neither one alone computes the origin and size of the cosmic baryon excess.

Worked reasoning

Try the reasoning

1. Why do p = 3/5 and q = 2/5 not contradict equal total widths?

   Show the answer and reasoning

   They are partial branching probabilities, and complementary channels complete the same total rate Γ.

   The comparison distinguishes total decay rates from the probabilities of particular final baryon-number channels.

2. What if ε = 0 while the parents remain out of equilibrium?

   Show the answer and reasoning

   The net yield stays zero for the stated zero initial condition.

   The source εD vanishes and washout cannot create a nonzero B in this equation.

3. Is a nonzero CP parameter a complete prediction of the cosmic excess?

   Show the answer and reasoning

   No.

   The departure from equilibrium, baryon-number-changing dynamics, source history, washout and later processing still have to be calculated.

What to carry forward

• The conventional baryogenesis ingredients have distinct roles under stated assumptions.
• An asymmetric partial reaction ledger is different from a complete microscopic mechanism.
• Producing an asymmetry and preserving it require separate dynamical checks.