Can e^{+}e^{-}\rightarrow\mu^{+}\mu^{-} Happen?

[HairyFish]

I have spent a while trying to get to grips with the building blocks used for constructing feynman diagrams, below is my attempt at a set of reactions, how am I doing so far?

I don't think $$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$$ can happen since a gluon only interacts on particles with a colour charge, but am unsure

 

[vela]

I only looked at the top two so far, but they're both wrong. In the first one, you're changing the flavor of the neutrinos and the other leptons at each vertex, but the Z can't do that. The Z only couples to a particle and its antiparticle. In the second one, you have a photon coupling to a neutrino and a Z, neither of which has electric charge. Also, on the righthand side, you have four particles coming into a single vertex. You should only have three.

 

[diazona]

There's a certain amount of intuition involved when drawing out these diagrams - for starters, you really have to know well which elementary vertices are allowed, and only work with those. In most of your diagrams I see vertices which aren't allowed, so I'm guessing you're new to learning this stuff? (It took me a while to get used to this too)

The first thing I would suggest that you do when trying to draw the diagram for a reaction is figure out what kind of a process it is. For instance, any time neutrinos are involved, you know it's a weak interaction and there will most likely - no, wait, definitely - be a W boson exchanged. (Whenever you have a neutrino, it must go into or come out of a vertex that also has the corresponding lepton and a W boson.) Any time a quark changes flavor, or any time you produce (or destroy) two quarks with different flavors, again, there will be a W boson exchanged. As a matter of fact, weak interactions are the only ones that violate flavor conservation laws - that means they're the only ones that can turn a neutrino into a lepton or vice-versa, or that can turn a quark from one flavor into another, so they're often pretty easy to recognize.

If it isn't a weak interaction, or if it is but you think there's more going on than that, next consider electromagnetic interactions. Any time you have a particle and its antiparticle annihilating, they produce a photon. (Well, unless they're different-colored quarks, then you get a gluon) Also, if you have elastic scattering of charged particles, a photon is responsible for that.

Other than that, I guess you just have to get used to the rules... for what it's worth, the more practice like this you get, the better you'll know what is allowed and what isn't.

 

[diazona]

I couldn't resist checking so: beyond what vela said, your $d\nu_e \to u e^-$ is wrong - remember what I said about neutrinos. $gg\to W^+W^-$ I'm not sure about... I think I might have figured out a diagram for it, but it's more complicated than what you've got. I had to make some assumptions about the colors of the gluons involved since they're not specified. Remember that gluons only interact with colored particles, and the W bosons carry no color charge. And finally, for $K^+\to\pi^0e^+\nu_e$, there is a diagram for that one. (Strangeness is indeed not conserved, so what kind of interaction is it?)

The rest of them, as far as I can tell, seem right (or at least, I can't identify anything wrong with what you did).

 

[HairyFish]

Thank You for taking the time to respond, its really appreciated!

I am completely new to drawing Feynman diagrams and have spent ages trying to gather all the necessary rules/building blocks. I could not find a straight forward list of rules anywhere and what you guys have said is the best I have come across!

I have had another go at the ones you said I got wrong. Something I am unsure about is:

Whenever you have a neutrino, it must go into or come out of a vertex that also has the corresponding lepton and a W boson.

Does this mean you cannot have a W boson, an electron-neutrino and a positron all at the same vertex (as I have drawn in the second diagram but with a muon particles instead)

Also, I still don't know how to go about the bottom left one, should I be creating some quarks then destroying them forming the W bosons?

And the bottom right one, it is incomplete because I don't know how to form the pion or even what I have done is the right idea.

 

[vela]

Does this mean you cannot have a W boson, an electron-neutrino and a positron all at the same vertex (as I have drawn in the second diagram but with a muon particles instead)

That's what Diazona said you must have: a lepton, its corresponding neutrino, and the W. I don't see a contradiction between your drawing and what he said.

Also, I still don't know how to go about the bottom left one, should I be creating some quarks then destroying them forming the W bosons?

Yes. Gluons only couple to quarks, and quarks will couple to the W. You can probably draw it as a box diagram. You'll have a square loop with quarks along the edges, and the gluons and W connected at the corners.

And the bottom right one, it is incomplete because I don't know how to form the pion or even what I have done is the right idea.

You have the right idea. The strange antiquark turns into an up antiquark when it emits the W. The up quark and antiquark then join to form the pion.

 

[diazona]

That's what Diazona said you must have: a lepton, its corresponding neutrino, and the W. I don't see a contradiction between your drawing and what he said.

Yep, your second drawing looks correct. Keep in mind that in Feynman diagrams, the only difference between an electron and a positron (or generally, between any particle and its antiparticle) is which way the line points. So if you had a vertex involving an electron neutrino, the associated lepton could be either an electron or a positron depending on how the vertex is oriented. Similarly, the W boson could have either charge depending on the orientation.