Hypothetical proton decay and the stability of matter

Matter stability is an input to a remote-future scenario

A remote-future account of ordinary matter depends on which particles remain stable and which decays the underlying theory permits. An assumed proton lifetime is therefore a physical hypothesis, rather than a consequence of continued cosmic expansion.

Compare a stable hypothesis, with decay rate Γ = 0, with an explicitly hypothetical exponential law Γ = 1/τ_*. Here τ_* is an arbitrary positive proper mean lifetime; no measured lifetime is supplied.

Begin with 80 independent identical protons in a closed ideal cohort. For the decay hypothesis, define u = t/τ_*. Each proton survives with probability exp(−u), giving expected survivors 80exp(−u).

These expectations describe repeated realizations of a supplied probability model. They are not observed proton decays, a detector forecast or a determination of how ordinary matter will actually end.

A decay need not produce a selected event

Supply three disjoint hypothetical decay branches: an electron–pion channel with fraction B_e = 1/2, a muon–pion channel with B_μ = 1/4, and other channels with the remaining 1/4.

Assign selection efficiencies ε_e = 3/4 and ε_μ = 1/2 for the first two channels. These are stipulated probabilities of selecting an event after its decay channel occurs, not physical branching fractions.

The expected selected counts are 80[1−exp(−u)]B_eε_e and 80[1−exp(−u)]B_με_μ. The total selection probability conditional on a decay is 3/8+1/8 = 1/2.

The unselected category includes decays in other branches and missed events in the first two branches. Set backgrounds to exactly zero for this finite example; a real experiment needs its own background and response model.

Zero selected events remains a probabilistic outcome

For one proton, the probability of any selected event by u is p_sel = [1−exp(−u)]/2. Independence of the 80 supplied protons makes the total selected count binomial with parameters 80 and p_sel.

Its probability of zero selected events is [1−p_sel]⁸⁰. At u = ln 2 this is (3/4)⁸⁰ ≈ 1.011349×10⁻¹⁰; at u = ln 4 it is (5/8)⁸⁰ ≈ 4.681676×10⁻¹⁷.

These are exact finite-cohort probabilities, rather than an automatic Poisson approximation. Expected counts such as 22.5 do not imply fractional observed events; realized counts remain integers.

A null result can constrain a rate under a stated model, but it does not logically prove that the rate is zero. Nor does this table provide an actual observed null result, confidence construction or posterior probability for proton stability.

Specify the branch before quoting its lifetime

If the total mean lifetime is τ_* and a branch has fraction B, its partial decay rate is B/τ_*. The corresponding partial lifetime is τ_*/B.

The supplied electron branch therefore has partial lifetime 2τ_*, and the muon branch 4τ_*. Selection efficiency changes the expected observed count, but does not change the underlying physical partial lifetime.

A channel search connects a lifetime or rate hypothesis to exposure, efficiencies, backgrounds and uncertainties. Its inference is conditional on that channel and response rather than an unconditional measurement of all possible proton decay.

The cited historical Super-Kamiokande search illustrates channel-specific methodology. No numerical limit from it is adopted here as a current bound, and no claim about the latest detection status is needed for the finite calculation.

A search constraint and an assumed cosmic fate are different

Under the stable ideal hypothesis, all 80 protons remain in this cohort and no decay-selected events occur. Under the stipulated nonzero rate, the expected survivor inventory declines with proper time.

Changing branches and efficiencies changes what a particular search can select. Changing the underlying rate changes the physical survival law. These operations should not be conflated.

Real matter also involves nuclear binding and other particles, reactions and environments. The finite independent-proton cohort does not calculate the complete fate of atoms, stellar remnants or all ordinary matter.

A remote-future scenario should state whether it assumes stability, which decay law and channels it uses, and what evidence constrains those assumptions. A conditional pathway is not converted into a detection by drawing its expected future curve.

Worked reasoning

Try the reasoning

1. What is the physical partial lifetime for the B_e = 1/2 branch?

   Show the answer and reasoning

   2τ_*.

   Partial lifetime is total mean lifetime divided by branching fraction, independently of selection efficiency.

2. Is exp(−20) the exact probability of zero selected events at u = ln 2?

   Show the answer and reasoning

   No: the exact probability is (3/4)⁸⁰.

   The supplied finite cohort follows a binomial law with selection probability 1/4 per proton.

3. Does 22.5 expected selected events mean a detector records half an event?

   Show the answer and reasoning

   No.

   An expectation averages integer-valued outcomes over repeated realizations of the model.

What to carry forward

• A hypothetical decay rate is an additional physical assumption in a future scenario.
• Physical decays, branches and detector selections require separate accounting.
• Lifetime constraints and null observations remain conditional on channel and probability models.