In supersymmetric gauge theories, the vector potential is a part of a vector supermultiplet which is represented by a real superfield $V$. Expanded out in components, the Lagrangian for such a field is given by

$$\mathcal{L}=-\frac{1}{4g^2}F_{\mu\nu}^a F^{\mu\nu a}+\cdots,$$

where the extra terms are in terms of the extra superpartners in this multiplet.

Upon gauge fixing, one introduces Faddeev-Popov ghosts to counter the overcounting in the path integral. The gauge-fixed Lagrangian in, say, the Lorenz gauge takes the form

$$\mathcal{L}_{\text{g.f.}}=\text{Tr}\left[-\frac{1}{4g^2}F_{\mu\nu} F^{\mu\nu}+\overline{c}D_{\mu}\partial^{\mu}c+\cdots\right].$$

Thus, since the vector potential $A$ transforms non-trivially under SUSY transformations, but the ghosts do not, it is clear that the introduction of ghost fields has broken supersymmetry at ghost level $n_g\neq 0$.

Is there a gauge-fixing method which circumvents this and allows for intrinsically supersymmetric perturbative calculations (for instance introducing a super partner for the ghosts and absorbing the broken supersymmetry into a $\mathbb{Z}_2$-graded version of extended BRST symmetry)?

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    $\begingroup$ Can you give a reference where I can read how the overcounting problem occurs? $\endgroup$ – my2cts Dec 18 '18 at 13:06
  • $\begingroup$ This paper might be useful. $\endgroup$ – Nogueira Dec 18 '18 at 13:20
  • $\begingroup$ @my2cts Any QFT book will discuss this in the quantization of non-abelian gauge theories in the path integral formalism (cf section 16.2 of Peskin and Schroeder). $\endgroup$ – Bob Knighton Dec 18 '18 at 14:13

In SUSY Yang-Mills the gauge parameter is not just a Lorentz-scalar, it is a full chiral superfield. So when we gauge fix, we don't just use a Lorentz-scalar ghost, we use a chiral superfield ghost. Likewise, the constraint isn't just a scalar, it needs to be some kind of supersymmetric object.


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