I am somewhat confused. Looking through these slides (especially the 11th), which show Feynman diagrams involving $W$-bosons, I can't figure out which way to draw the arrow near the $W$ boson? How do people determine if it is to the "right" or to the "left".

# [Physics] How to determine the direction of arrow on Feynman diagram for $W$ boson line

bosonsconventionsfeynman-diagrams

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Feynman diagrams are most definitely *not* a representation of what's going on between the particles. Feynman diagrams are simply a tool to help you remember formulas: if you want to calculate the probability that two electrons will scatter off each other in so-and-so angle, you draw all possible diagrams with two incoming electrons and two outgoing electrons (there are infinitely many diagrams!), and you translate each diagram into a mathematical expression.

When you do this it doesn't matter whether the lines look like they're attracting or not, because you can deform the diagram any way you like (as long as you keep the same external lines). The thing that tells you whether the force is attractive or repulsive is the math; if you use the electron-positron diagram to calculate the potential energy you will find that it corresponds to an attractive Coulomb potential; if you reverse the positron arrow so it now represents another electron (without moving the lines at all!), you will now find that the potential is repulsive.

The upshot here is that so-called "virtual particles", which are internal lines in a Feynman diagram (in your examples those would be the photon, the gluon and the pion), are not actual particles being exchanged. They're just a neat picture that helps visualizing the process, but in reality the particles are interacting through their quantum fields, and these fields are very hard (maybe even impossible) to understand intuitively. But remember that the diagrams in your post are what we call "tree level". They're the simplest diagrams for the given processes, but in reality there is an infinite number of them, with ever growing number of vertices and lines, and the more diagrams you calculate the more accurate your results will be.

I don't know what you mean with "rigorously" determine the Feynman diagrams that contribute to a process. In general from a theoretical point of view you can have two approaches

1)The hamiltonian approach in interaction picture, thorugh Dyson expansion

2)The path integral formalism. LSZ formula relates the transition amplitude to a green function. The latter can be calculated with the generating functional formalism through a perturbative expansion.

Both these methods, if carried on show, which Feynman diagrams are relevant, i.e. give a non null contribute to the transition amplitude. However that is not how things are done in practice. In practice you use the Feynman rules. To understand which diagram contributes, you just have to keep fixed the external states (ex, $ e^-(p_1)+e^+(p_2) \rightarrow e^-(p_3)+e^+(p_4)$, fixed to the angles of a rectangle) and look for which channels are permitted by the vertices of the theory. The fundamental point are the vertices rules, these shows which coupling are permitted and hence which Feynman diagrams contributes. For example from the covariant derivative term you can get a $e^2\chi^*\chi A_{\mu}A^{\mu}$ term, to which correspond the rule $ie^2g^{\mu\nu}$. So you can see that the seagull channel can contribute to the process with two final photons you wrote, but obviously not to your first process $e^-+e^+ \rightarrow e^-+e^+$. To sum up, check which vertices there are in your interaction lagrangian, fix the external legs of the process you desire, build all the Feynman diagrams with that external legs using the vertices the theory has.

## Best Answer

In general in Feynman diagrams an incoming particle can be read as an outgoing antiparticle and W+ is the antiparticle of W- and vice verso. Quantum number conservation holds at the vertices. (charge , lepton number..)

The reaction studied in 11 is a change of a proton to a neutron through the weak interaction. The charge of the proton has to go to the right . The diagram has a W- going to the left, i.e. a W+ which is what is necessary for charge conservation on the second vertex.

For 13, the reaction is a neutron turning into a proton by colliding with a neutrino. It has an arrow to the left and when read towards the lepton vertex it is a W- , mathematicaly, which is what is needed for charge conservation.

For 15, the reaction is antineutrino proton , turning into e+ neutron. The arrow correctly conserves charge at the vertices.

Seems to me whoever wrote the site has been playing games to make students think?