Second Order Fermions in Gauge Theories

Abstract

The second order formalism for fermions provides a description of fermions that is very similar to that of scalars. We demonstrate that this second order formalism is equivalent to the standard Dirac formalism. We do so in terms of the conventional fermionic Feynman rules. The second order formalism has previously proven useful for the computation of fermion loops, here we describe how the corresponding rules can be applied to the calculation of amplitudes involving external fermions, including tree-level processes and processes with more than one fermion line. We comment on the supersymmetric identities relating fermions and scalars and the associated simplifications to perturbative calculations that are then more transparent.

Figures (4)

• Figure 1: The basic fermion unit.
• Figure 2: A generic fermion loop, as described in Eq. (\ref{['GenericLoop']}). We label the legs from $0$, such that the loop momentum flowing into the $0$-vertex along the fermion line is $k$, and $p_0 = - \sum_{i=1}^n p_i$.
• Figure 3: The second order rules for calculations of fermion-gauge interactions. Care should be taken when using these rules to ensure that the trace is taken over all terms and not just those containing a $\sigma^{\alpha\beta}$. All momenta are directed outwards, which accounts for the sign difference between the three point $\sigma^{\alpha\beta}$ contribution and that of the $B$ in Eq. (\ref{['DefBandC']}). When one traces over a closed fermion loop one should also multiply by $1/2$. For internal closed fermion loops there is the familiar additional factor of $-1$. See the text for the application of these rules to processes involving external fermions. We note that by setting $\sigma^{\alpha\beta}\rightarrow 0$ and $i \rightarrow -i$, these rules become those for a gauged-complex scalar field.
• Figure 4: This figure represents contributions to the MHV amplitudes which enjoy a close supersymmetric similarity. The text describes the simple relationship between the (a) second order fermionic and (b) scalar amplitudes.