Two-loop QCD Amplitudes from the Chiral Algebra Bootstrap

TL;DR

This work shows that the chiral algebra bootstrap (CA) applied to twistorial theories can inform two-loop QCD amplitudes, by relating a hitherto unknown two-loop all-plus partial amplitude $A_n^{(-1,1);\text{all-plus}}$ to CA results via supersymmetry Ward identities. It further demonstrates that the full-color all-plus $n$-gluon QCD amplitude with $n_f$ quark flavors can be reconstructed from two-loop twistorial amplitudes along with one-loop and tree-level axion amplitudes for $N_c=2,3$, reducing the two-loop problem to tractable lower-loop data. A concrete all-plus partial-amplitude formula is provided, and a framework is outlined to determine the complete two-loop QCD amplitude by solving a linear system between twistorial components and axion contributions. The results highlight the potential of the CA bootstrap to illuminate the analytic structure of QCD amplitudes and enable precise predictions for collider phenomenology, while outlining avenues for extending to more general helicities.

Abstract

We show that the chiral algebra bootstrap, which computes form factors of twistorial theories, can help determine two-loop amplitudes in massless QCD. We give an $n$-gluon result for a previously unknown partial amplitude of the two-loop all-plus-helicity QCD amplitude by utilizing supersymmetry Ward identities and known chiral algebra bootstrap results. We then show that the full-color two-loop $n$-gluon amplitude of QCD with $n_f$ quark flavors can be obtained from certain two-loop form factors of twistorial theories and one-loop and tree-level amplitudes. Chiral algebra bootstrap results exist for these form factors when all gluons have positive helicities. Hence, the bootstrap simplifies the computation of these two-loop amplitudes by one loop level.

Figures (5)

• Figure 1: Two-loop topologies of diagrams contributing to $\mathcal{A}^{(2)}_{F}$ and $\mathcal{A}^{(2)}_{F^2}$. Planar diagrams are given by attaching tree graphs to any black line at a single leg. Non-planar diagrams are those with topologies $P_1$ and $P_3$ with trees that attach to the red propagator shared by the two loops. The partial amplitudes $A_n^{(1,1)}$, $A_n^{(-1,1)}$, and $A_n^{(0,2)}$ are completely built from the single-trace terms of planar graphs with topologies $P_1$ & $P_2$, $P_3$, and $P_4$, respectively.
• Figure 2: Examples of Feynman diagrams that contribute to the one-loop amplitude with a single axion propagator $\mathcal{A}^{(1)}_\text{ax}$. The left, middle, and right diagrams contribute to the amplitudes $\mathcal{A}^{[g\to ax]}_\text{ax}$, $\mathcal{A}^{[g]}_\text{ax}$, and $\mathcal{A}^{[f]}_\text{ax}$, respectively.
• Figure 3: An example of a Feynman diagram that contributes to the tree-level amplitude with two axion propagators $\mathcal{A}^{(0)}_{\text{ax}^2}$.
• Figure S1: Non-planar color diagrams contributing to $\mathcal{A}^{(2)}_F$ are of the form shown. The dots represent other external legs.
• Figure S2: The left side of the equation is a diagram representation of a term in the sum appearing in eq. \ref{['nonplanto1leg']}. The right side of the equation is the result of applying the commutation relation \ref{['commutator']}.