Do the quarks in neutrons tend to have much larger momenta than the quarks in protons?

Question

Context

The Feynman diagram for beta decay of a neutron into a proton, electron, and electron antineutrino via an intermediate $W^-$ boson is given in many places, which includes [1]. I understand that in beta decay the weak force changes the flavour of a single quark.: $$ d\rightarrow u + W^-\tag{1}$$

By looking at the well-known graphic [2], which give the masses of the particles of the standard model, one observes that the mass of $W^-$ is about 10,000 times larger than the mass of the either the up or down quarks.

I understand that the masses of the reagents and products need not be equal. Yet, In Feynam diagrams, at each vertex there is 4-momentum conservation [3]. Thus, the vertex given in beta decay must have conservation of 4-momentum.

Q.1) Do the quarks in neutrons tend to have much larger momenta than the quarks in protons?

Q.2) Can we compute the probability that quarks in neutrons have sufficient momentum to undergo beta decay?

Q.3) Otherwise, how can one understand the ability of the reaction in Eq. 1 to take place?

Bibliography

[1] https://en.wikipedia.org/wiki/W_and_Z_bosons

[2] https://en.wikipedia.org/wiki/Standard_Model

[3] https://en.wikipedia.org/wiki/Feynman_diagram

Answer 1

Your initial premise is wrong because:

$$ d\rightarrow u + W^-\tag{1} $$

does not show the decay of a down quark. The Feynman diagram that you are thinking of:

does not show an actual physical process. It is a graphical representation of a calculation that gives the first term of an infinite sum used to calculate the decay probability. There is no $W^-$ boson present.

With a reaction like beta decay we know the initial state, i.e. a down quark, and we know the final state, i.e. an up quark, electron and antineutrino. However we cannot directly calculate the state of the system in between. Instead we approximate it using an infinite series, of which each term is represented by a Feynman diagram. The picture above shows the first term in this series - the tree level diagram. The $W^-$ in this diagram is a virtual particle that need not and generally does not have the same mass, energy and momentum as a real $W^-$.