# Can you predict cancellations by supersymmetry?

+ 4 like - 0 dislike
108 views

I'm studying supersymmetry breaking using Turing's textbook, Modern Supersymmetry (pg 91). He begins with the O'Raifeartaigh model,

W = g \Phi _1 \left( \Phi _3 ^2 - m ^2  \right) + M \Phi _2 \Phi _3

which gives the explicit Lagrangian,
\begin{align}
\Delta {\cal L} & = \sum _i ( i \bar{\psi}  _i \bar{\sigma}^{\mu}\partial_\mu \psi _i+ \partial _\mu \phi _i ^\dagger \partial ^\mu \phi _i )  -  M ( \psi _2 \psi _3 + h.c. ) \\  & - g ( \phi _1  \psi _3 \psi _3 + h.c. ) - 2g ( \phi _3 \psi _1 \psi _3 + h.c. )     \\  &  - M ^2 \phi _3 ^\ast \phi _3  - M ^2 \phi _2 ^\ast \phi _2  +g ^2 m ^2 (  \phi _3 ^2 + h.c. )  - 2 g  M ( \phi _1 \phi _2 \phi _3 + h.c. ) \\ & - 4 g ^2 \phi _1 \phi _1 ^\ast  \phi _3 \phi _3 ^\ast  \end{align}

where $\psi _1$ is the Goldstino. It's superpartner $\phi _1$ is also massless at tree level. Turing then goes on to look for its one loop mass correction which involves calculating the diagrams shown here (I can't figure out how to embed the image directly, if you know how feel free to edit the question).

He then says that The correction to the $\phi _1$ mass from the top three graphs... vanish by SUSY.'' Is this statement obviously true or just an interesting observation one could make after the fact? In other words, could I have made this claim without calculating it explicitly?

edited May 15, 2014

This is a general answer and not specific to the mentioned example. If one works with superfields and supergraphs, then bose-fermi cancellations are easy to see as several Feynman diagrams combine to form a single supergraph. Supergraphs are discussed in Wess & Bagger.

Thanks, I'll keep that in mind. I have yet to find the time to learn the supergraph formalism.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.