Is the $p + p \to K^+ + K^+ + n + n$ process allowed in the SM?

Question

In an exercise it asks to prove why certain processes cannot happen in the Standard Model. One such process is the following: $$ p + p \to K^+ + K^+ + n + n $$

This process conserves baryonic number and charge, which are the main laws dictating whether a process involving mesons and baryons can happen. It doesn't conserve strangeness (nor $I_3$, third component of isospin) which mean it should be able to happen.

Since it's a collision it's not really constrained by energy, the only thing I haven't checked is whether CPT is conserved but I haven't really had the need to do this in previous exercises and would not know how to go about it for such a complicated process.

Is the exercise itself wrong or am I missing something (which is probably the case)?

Answer 1

What you are virtually certainly missing is the $80^{-8}\sim 10^{-15}$ suppression with respect to a strong cross-section, like that for $pp\to pK^{+}\Lambda$. Yours is a doubly-weak process (see below) and any head-scratching about absolutist principles of the SM is greatly misplaced. 15 orders of magnitude suppression in a pp melee is a no-go.

You may start by writing down the valence quarks of the incoming particles, uuduud, and those of the outgoing ones, which are the same, plus the extra $dd\bar{s}\bar{s}$. The strange antiquarks were not pair produced, so a $W^+$ must have produced each, e.g. by $u\to d W^+\to d u\bar{s} $, repeated twice.

So there are necessarily two virtual Ws sparking around, each propagator for which has a $1/m_W^2$ suppression, so the amplitude has a $1/m_W^4$, and the cross section a $1/m_W^8$, to be matched to a dimensionless number by a basic strong scale of 1 GeV, as I did above, overgenerously, whence the $80^{-8}\sim 10^{-15}$ suppression. This is the first thing that should go through your brain at the ganglionic level, before CPT, etc...