Feynman Diagrams in 2 component notation

Question

When using two component notation people often prefer to refrain from using arrows in Feynman diagrams to denote charge flow as is done in four-component notation. Instead, if understand correctly, they use arrows to denote chirality. I'd like to know what is the prescription to draw out the diagrams. I have read here (pg. 39) that

arrows indicate the spinor index structure, with fields of undotted indices flowing into any vertex and field of dotted indices flowing out of any vertices

(see the reference above for many examples). However, trying this out on Majorana and Dirac mass terms, this doesn't seem to be correct. A Majorana mass term, $\psi ^\alpha \psi_\alpha +h.c.$, is thus composed only of undotted indices. With the reasoning above, it should have two arrows pointing into the vertex,

However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass. What am I missing?

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

Comments

Try this paper

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user Trimok

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@Trimok, thanks for the link. This is in fact the paper I referenced above. While they do a good job of explaining most things, I still don't quite understand why $m\psi \psi $ doesn't follow the rules they mention.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

Answer 1

When you say:

"However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass."

that's where you are confused. (Why did you think you were sure of this?) A Majorana mass has that form, and so does a Dirac mass. They have the exact same Feynman rule arrow structure when you use 2-component notation. It is just that for a Majorana mass, the 2-component fields being connected are the same, and for a Dirac mass they are different (typically with opposite charge under some gauge or global symmetry).

The answers about the Majorana-Weyl condition are not relevant. In 4 dimensions, a Majorana fermion is simply a 2-component Weyl fermion with a mass term by itself. A Dirac fermion is a pair of 2-component Weyl fermions with a mass term connecting them.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user Anna Nimity

Comments

I thought that was the case because I thought that the arrow direction indicated chirality of the fermion and a Dirac mass is a coupling between right and left handed fermions. Is this not true?

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

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Oh, I see the source of the confusion. A Dirac mass is not a coupling between a left-handed and a right-handed fermion. It is instead a coupling between a left-handed fermion and the conjugate of a right-handed fermion, which is another left-handed fermion.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user Anna Nimity

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Thanks that makes sense! So mass terms always couple left-to-left particles or right-to-right particles, the only difference between a Dirac and Majorana mass term is that in a Dirac mass term the "second" left-handed particle is the right-conjugate (or vice versa). Perfect!

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

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Yep! (Although by particles, really we mean specifically fermions. And, by the way, sorry if I sounded rude with "where you are confused" and "why were you sure of this". I really was just trying to sharply isolate the issues, but now rereading it made me wince. Sometimes things written on the internets sound kind of caustic even when not really intended that way.)

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user Anna Nimity

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Not at all! Thanks again.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

Answer 2

Majorana-Weyl spinors exist only in some dimensions. Let the signature of spacetime be $(p,q)$ -- M-W spinors exist only when $p-q=0\mod 8$. Thus, they don't exist in $(1,3)$ Minkowski spacetime. You can impose only the Majorana conditions or the Weyl condition in this case. The notation that you mention can only be applied to Weyl spinors. I can't think immediately of a reference for the existence statement -- it might be discussed in Sohnius' Physics Reports article on supersymmetry.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user suresh

Comments

The Majorana condition is a reality condition while the Weyl condition is a chirality condition. So a Weyl spinor is best thought of as two complex fields while a Majorana spinor is four real fields. A Majorana-Weyl if that could exist would then have two real fields. Look at the technical appendix of Sohnius paper that I mentioned. Look at this as well: people.maths.ox.ac.uk/daviesr/resources/notes/spinors.pdf

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user suresh

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So the answer is that when writing out Feynman diagrams in 2 component notation, the rule "arrows indicate the spinor index structure, with fields of undotted indices flowing into any vertex and field of dotted indices flowing out of any vertices", only applies if you are working with Weyl spinors?

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

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Yes! It just doesn't make sense for Majorana spinors. You are a hard person to convince. :-)

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user suresh

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Very strange! I'm just surprised that I've never seen that mentioned anywhere. Thanks!

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

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An interesting consequence of the existence of M-W spinors in (1,9) spatial dimensions leads to a unique theory $\mathcal{N}=1$ supersymmetry (a supercharge which is M-W): the ten-dimensional Supersymmetric Yang-Mills (SYM) theory which one dimensional reduction to (1,3) dimensions gives rise to the $\mathcal{N}=4$ SYM theory.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user suresh

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Thanks for your response. If Weyl spinors don't exist in the real (1,3) spacetime then how come people can use it to describe a supersymmetric Standard Model?

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user JeffDror

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I mean spinors that are Majorana and Weyl at the same time -- they are usually called Majorana-Weyl. You can have Majorana or Weyl but not both in (1+3) dimensions.

This post imported from StackExchange Physics at 2014-04-01 16:06 (UCT), posted by SE-user suresh