A Twistor Approach to One-Loop Amplitudes in N=1 Supersymmetric Yang-Mills Theory

TL;DR

The paper generalises twistor-string inspired loop techniques from ${\cal N}=4$ to ${\cal N}=1$ (and ${\cal N}=2$) SYM, delivering a dispersion-integral representation for amplitudes and validating it by explicitly deriving one-loop N=1 MHV amplitudes for arbitrary leg counts. By sewing MHV vertices and employing CSW off-shell continuation, the authors reproduce the known BD DK one-loop results for the ${\cal N}=1$ chiral multiplet, decomposed into finite box and triangle contributions (with bubbles as degenerate triangles). The approach relies on a dispersive treatment of the loop measure, a D=4-2ε regularisation, and a Passarino–Veltman reduction to connect tensor integrals to standard discontinuities. Collectively, the work extends the MHV-diagram framework to less supersymmetric theories and reinforces the twistor-space localisation picture at loop level, offering a complementary route to established cut-constructibility methods.

Abstract

We extend the twistor string theory inspired formalism introduced in hep-th/0407214 for calculating loop amplitudes in N=4 super Yang-Mills theory to the case of N=1 (and N=2) super Yang-Mills. Our approach yields a novel representation of the gauge theory amplitudes as dispersion integrals, which are surprisingly simple to evaluate. As an application we calculate one-loop maximally helicity violating (MHV) scattering amplitudes with an arbitrary number of external legs. The result we obtain agrees precisely with the expressions for the N=1 MHV amplitudes derived previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

Figures (6)

• Figure 1: The box function $F$ of \ref{['F']}, whose finite part $B$, Eq. \ref{['Bniceonecyril']}, appears in the ${\cal N}=1$ amplitude \ref{['BDDKNeq1']}. The two external gluons with negative helicity are labelled by $i$ and $j$. The legs labelled by $s$ and $m$ correspond to the null momenta $p$ and $q$ respectively in the notation of \ref{['Bniceonecyril']}. Moreover, the quantities $t_{m+1}^{[s-m]}$, $t_{m}^{[s-m]}$, $t_{m+1}^{[s-m-1]}$, $t_{s+1}^{[m-s-1]}$ appearing in the box function $B$ in \ref{['Neq1']} correspond to the kinematical invariants $t:=(Q+p)^2$, $s:=(P+p)^2$, $Q^2$, $P^2$ in the notation of \ref{['Bniceonecyril']}, with $p+q+P+Q=0$.
• Figure 2: A triangle function, corresponding to the first term $T_{\epsilon} ( p_m, q_{a+1,m-1}, q_{m+1, a})$ in the second line of \ref{['Neq1']}. $p$, $Q$ and $P$ correspond to $p_m$, $q_{m+1, a}$ and $q_{a+1, m-1}$ in the notation of Eq. \ref{['Neq1']}, where $j\in Q$, $i\in P$. In particular, $Q^2 \to t_{m+1}^{[a-m]}$ and $P^2 \to t_{m}^{[a-m+1]}$.
• Figure 3: This triangle function corresponds to the second term in the second line of \ref{['Neq1']} -- where $i$ and $j$ are swapped. As in Figure 2, $p$, $Q$ and $P$ correspond to $p_m$, $q_{m+1, a}$ and $q_{a+1, m-1}$ in the notation of Eq. \ref{['Neq1']}, where now $i\in Q$, $j\in P$. In particular, $Q^2 \to t_{a+1}^{[m-a]}$ and $P^2 \to t_{a+1}^{[m-a-1]}$.
• Figure 4: A one-loop MHV diagram, computed in \ref{['loopint']} using MHV amplitudes as interaction vertices, with the CSW off-shell prescription. The two external gluons with negative helicity are labelled by $i$ and $j$.
• Figure 5: A triangle function with massive legs labelled by $P$ and $Q$, and massless leg $p$. This function is reconstructed by summing two dispersion integrals, corresponding to the $P^2_z$- and $Q^2_z$-cut.