Which $W$ boson is involved in electron capture?

Question

I am learning about Electron Capture.

$$p+e\to n+\nu_e$$

My question is whether the $W^+$ boson or the $W^-$ boson is involved in this transfer. I can consider this two ways:

1) Since the electron is losing a negative charge in this situation, this would indicate that this should be mediated by the $W^-$ boson.

2) Since inside the proton (uud), there is a change from an up quark $\left(+\frac{2}{3}\right)$ to a down quark $\left(-\frac{1}{3}\right)$ in the neutron (udd), we are losing a positive charge of $+1$, suggesting that this reaction should be mediated by the $W^+$ boson to 'carry' this charge away.

My question is which boson is the correct boson that is involved in this transfer?

Answer 1

In time-ordered diagrams, you would have to deal with both:

1) $e^- \rightarrow \nu_e+W^-$ followed by $W^- + u \rightarrow d$

2) $u \rightarrow d + W^+$ followed by $W^+ + e^-\rightarrow \nu_e$

but those aren't covariant. In the modern language of Feynman diagrams as $W$ boson is exchanged. There is no time ordering to the vertices (Feynman diagrams are manifestly covariant). Four momentum is conserved at all vertices, so if you look at scattering processes, the square for momentum of the exchanged boson (called $Q^2$) is less than zero: it is space-like.

Answer 2

In electron capture, ignoring the quark level,this is the diagram:

Conservation of charge needs a positron, which means the intermediate virtual boson exchanged going from left to right has to carry a + charge. Usually one plots the arrow from left to right. BUT the Feynman diagram rules can read it as W- from right to left.

We are in the quantum mechanical regime,for a proton to turn into a neutron a quark interaction has to happen, again from conservation of charge .

Looking at the quark content , proton uud and neutron udd, an up quark (charge +2/3) has to turn into a down quark (charge -1/3) , which happens with the an up -> d +W*+, the star because the W is in a virtual state, off mass shell. When one writes the quark diagrams for electron capture, the reverse can also happen , within the Uncertainty principle of quantum mechanics, depending on the time evolution, there are two diagrams:

Both of these reactions occur superposed. The uncertainty principle means that you don't know exactly when the particles react. The existence of one diagram implies the other. This is why some books show W+ going one way and others show a W- going the other.

So the answer is , both of these diagrams have to be included in calculating the crossection, so sometimes it is a W- one way and/or a W+ the other, due to the uncertainty principle in time ordering.