Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter

Abstract

Four-point one-loop nonsupersymmetric pure Yang-Mills amplitudes with the duality between color and kinematics manifest have been constructed in previous work. Here, we extend the discussion to fermions and scalars circulating in the loop with all external gluons. This gives another nontrivial loop-level example showing that the duality between color and kinematics holds in nonsupersymmetric gauge theory. The construction is valid in any spacetime dimension and written in terms of formal polarization vectors. We also convert these expressions into a four-dimensional form with explicit external helicity states. Using this, we compare our results to one-loop duality-satisfying amplitudes that are already present in literature.

Figures (6)

• Figure 1: The basic Jacobi relation for either color or numerator factors. These three diagrams can be embedded in a larger (loop) diagram.
• Figure 2: The labeling convention that we employ both for numerators with formal polarization vectors and for color-ordered amplitudes. The external legs have the ordering $(1, i_{2}, i_{3}, i_{4})$, with outgoing momenta $k_{1}$, $k_{i_{2}}$, $k_{i_{3}}$, $k_{i_{4}}$. The loop momentum is denoted by $p_{1}\equiv p$, while $p_{2}$, $p_{3}$, and $p_{4}$ are given by momentum conservation.
• Figure 3: The Jacobi relations determining either color or kinematic numerators of the four-point diagrams containing either a triangle or internal bubble.
• Figure 4: The color or kinematic Jacobi relations involving a bubble on an external leg or a tadpole. These diagrams have vanishing contribution to the integrated amplitude.
• Figure 5: The two two-particle unitarity cuts in which the exposed internal propagators are put on shell. The one-loop contributions to cut (a) come from Figs. \ref{['fig:ampLabel']}(a,b,d,f), and the one-loop contributions to cut (b) come from Figs. \ref{['fig:ampLabel']}(a,c,e,g). Diagrams with a bubble on an external leg and diagrams that contain a tadpole do not contribute to either cut.