The Googly Amplitudes in Gauge Theory

TL;DR

The paper demonstrates that googly amplitudes in gauge theory can be efficiently computed within the off-shell MHV-vertex framework of CSW-Witten. It derives explicit off-shell amplitude formulas for configurations with one or two positive-helicity gluons and proves a central spinor identity via pole analysis, enabling a clean calculation of the adjacent-googly amplitude. The resulting googly amplitude matches the known results and is the complex conjugate of the corresponding MHV amplitude, providing strong support for the CSW-Witten approach and contributing to a deeper understanding of amplitude structure in gauge theory. The methods leverage spinor helicity formalism, 3-gluon vertices, and inductive diagrammatic decompositions, with broader implications for perturbative QCD techniques.

Abstract

The googly amplitudes in gauge theory are computed by using the off shell MHV vertices with the newly proposed rules of Cachazo, Svrcek and Witten. The result is in agreement with the previously well-known results. In particular we also obtain a simple result for the all negative but one positive helicity amplitude when one of the external line is off shell.

Figures (7)

• Figure 1: The 3 gluon vertex. It is not important which gluon has positive helicity.
• Figure 2: A multi-gluon amplitude with only one positive helicity can be decomposed into two multi-gluon amplitudes with less gulons. The two sub-multi-gluon amplitudes also have only one positive helicity. A summation over $i$ should be understood.
• Figure 3: The decomposition of the googly amplitude $A_n(1+,2-,\cdots,(n-1)-,n+)$. We note that there is only one 4 gluon vertex.
• Figure 4: The decomposition for the generic googly amplitude. We note that the helicity of the internal line can be different, depending where we put the positive helicity gluon.
• Figure 5: The pole terms $\langle v_1,v_2\rangle$ from the factor $\langle V_1, V_2\rangle$. There is a summation over $k$.