Abstract
QFTs with supersymmetry (SUSY) have some remarkable properties that make them particularly interesting to study from the point-of-view of the RG.
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Notes
- 1.
The kinetic term can be generalized to all the terms with two derivatives:
$$\begin{aligned} \fancyscript{L}_\mathrm{kin }=-\frac{\partial ^2K}{\partial \phi _i\partial \phi _j^*} \partial _\mu \phi _i\partial ^\mu \phi _j^*+\text{ terms } \text{ involving } (\psi ,F), \end{aligned}$$(5.3)determined by a real function \(K(\phi ,\phi ^*)\), the Kähler potential.
- 2.
It is a key aspect of the RG that the scaling dimension of a general composite operator \({\fancyscript{O}}={\fancyscript{O}}_1\cdots {\fancyscript{O}}_p\) is usually not the sum \(\varDelta _{\fancyscript{O}}\ne \varDelta _{{\fancyscript{O}}_1}+\cdots +\varDelta _{{\fancyscript{O}}_p}\). For instance, for \(\phi ^4\) theory around the Wilson-Fisher fixed point one has \(\varDelta _{\phi ^4}\ne 2\varDelta _{\phi ^2}\).
- 3.
There is a caveat to this in that it depends on the fact that kinetic terms are suitably well behaved at the minima.
- 4.
The \(\theta \) angle multiplies a term \(\frac{1}{64\pi ^2}\int d^4x\,\varepsilon _{\mu \nu \rho \sigma } F^{a\mu \nu }F^{a\rho \sigma }\) in the action. This integral, which computes the 2nd Chern Class of the gauge field, is topological, in the sense that for any smooth gauge configuration it is equal to \(2\pi k\), for \(k\) an integer. Furthermore, this term does not contribute to the classical equations-of-motion. In the quantum theory which involves the Feynman sum over configurations, \(\theta \) becomes physically meaningful and should be treated as another coupling in the theory.
- 5.
The following discussion is taken from Arkani-Hamed and Murayama (1998).
- 6.
The calculation is explained in detail by Arkani-Hamed and Murayama (1998).
- 7.
The coupling here and in the following is generally the canonical gauge coupling, however, we will not distinguish between \(g\) and \(g_c\) from now on.
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Hollowood, T.J. (2013). RG and Supersymmetry. In: Renormalization Group and Fixed Points. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36312-2_5
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