Standard Model Proton Decay Rate

Question

The electro-weak force is known to contain a chiral anomaly that breaks $B+L$ conservation. In other words, it allows for the sum of baryons and leptons to change, but still conserves the difference between the two. This means that the standard model could have a channel for protons to decay, for example into a pion and a positron. Does anyone know what the total proton decay rate through standard model channels is?

Answer 1

Electroweak instantons violate baryon number (and lepton number) by three units (all three generations participate in the 't Hooft vertex). This is explained in 't Hooft's original paper. As a result, the proton is absolutely stable in the standard model. The lightest baryonic state that is unstable to decay into leptons is $^3$He. The deuteron is unstable with regard to decay into an anti-proton and leptons.

The rate is proportional to $[\exp(-8\pi^2/g_w^2)]^2$, which is much smaller than the rates for proton decay that have been discussed in extensions of the standard model. Note that the decay $^3\mathrm{He}\to$ leptons involves virtual $(b,t)$ quarks, and the rate contains extra powers of $g_w$ in the pre-exponent (which does not matter much, given that the exponent is already very big).

Just to give a rough number, the lifetime is a typical weak decay lifetime (say, $10^{-8}$ sec), multiplied by the instanton factor $$ \tau = \tau_w \exp(16\pi^2/g_w^2)=\tau_w\exp(4\pi\cdot 137\cdot\sin^2\theta_W) = \tau_w\cdot 10^{187}\sim 10^{180}\, sec \sim 10^{173}\, yrs $$ where I have neglected many pre-exponential factors which can be calculated, in principle, in the standard model.

Answer 2

As far as I know, the standard model is assumed to have vanishing anomalies, i.e. that the proton does not decay in the standard model. See page 5 in this reference.

You are asking for this calculation. I do not know if one can keep calling it "the standard model".

Here is a strong statement, at the end of chapter 7.3.1:

Thus all possible anomalies cancel for every generation of the standard model. If in one generation a quark (or any other particle) were missing, one would get non-vanishing anomalies (not for SU(3)SU(3)SU(3), but for the three other combinations)

This was for the unbroken phase, but it continuous to make the same statement for the broken phase.

So the answer is that there should be an extension of the standard model to study B+L conservation effects.

Answer 3

There have been several attempts to measure proton decay. So far, all have been unsuccessful. Various calculations give estimates ranging from $10^{30}$ to $10^{36}$ years.

Knowing the sensitivity of the experiments, we can set limits for the proton half-life. The current best measurements indicate it is $10^{34}$ years or more. For example, a 2014 publication from the Super-Kamiokande neutron detector in Japan gives a minimum of $5.9 × 10^{33}$ years.