Constructing the Tree-Level Yang-Mills S-Matrix Using Complex Factorization

TL;DR

The paper shows that the tree-level Yang-Mills S-matrix can be constructed from purely on-shell, complex-momentum consistency conditions, with BCFW recursion arising as an algorithm for assembling higher-point amplitudes from fundamental three-point data. By enforcing complex factorization on four-point amplitudes, it derives nontrivial constraints on couplings, including the Jacobi identity and the equivalence-principle-like universality for gravity, without relying on a Lagrangian. The authors demonstrate inductively that BCFW constructions with allowed shifts yield correctly factorizing higher-point amplitudes, while invalid shifts produce unphysical poles, thereby identifying the permissible shifts for gauge theories and outlining the gravity case. The work links large-$z$ scaling to factorization and suggests a deep S-matrix origin for gauge and gravitational interactions, with potential extensions to higher dimensions and one-loop structure.

Abstract

A remarkable connection between BCFW recursion relations and constraints on the S-matrix was made by Benincasa and Cachazo in 0705.4305, who noted that mutual consistency of different BCFW constructions of four-particle amplitudes generates non-trivial (but familiar) constraints on three-particle coupling constants --- these include gauge invariance, the equivalence principle, and the lack of non-trivial couplings for spins >2. These constraints can also be derived with weaker assumptions, by demanding the existence of four-point amplitudes that factorize properly in all unitarity limits with complex momenta. From this starting point, we show that the BCFW prescription can be interpreted as an algorithm for fully constructing a tree-level S-matrix, and that complex factorization of general BCFW amplitudes follows from the factorization of four-particle amplitudes. The allowed set of BCFW deformations is identified, formulated entirely as a statement on the three-particle sector, and using only complex factorization as a guide. Consequently, our analysis based on the physical consistency of the S-matrix is entirely independent of field theory. We analyze the case of pure Yang-Mills, and outline a proof for gravity. For Yang-Mills, we also show that the well-known scaling behavior of BCFW-deformed amplitudes at large z is a simple consequence of factorization. For gravity, factorization in certain channels requires asymptotic behavior ~1/z^2.

Figures (6)

• Figure 1: The two diagrams in the BCFW construction of a four-point amplitude $A(1,2,3,4)$ using a $[12\rangle$ shift. Both $t$- and $u$-channel poles are exposed in the left and right pieces respectively.
• Figure 2: A diagrammatic picture for the requirement of complex factorization.
• Figure 3: A graphical classification of poles in a general amplitude. The BCFW construction manifestly has proper factorization on the poles $(a)$ to the left of the thick line. The three classes of diagrams to the right of the line must be checked explicitly. We discuss the "unshifted" poles $(b)$ in Sec. \ref{['sec:bigunshifted']} (the small diagram below is a special case). The remaining two-particle poles correspond to invariants that involve one or more BCF-shifted legs, but nonetheless are not altered by BCF shifts. The "unique diagram" poles shown in $(c)$ are discussed in Sec. \ref{['sec:12pole']}, and the "wrong-helicity" poles $(d)$ in Sec. \ref{['sec:wrongfac']}. Notable features of each pole are discussed in the text.
• Figure 4: A graphical summary of the argument for factorization of the BCFW amplitudes in unshifted poles. The top line identifies the subset of terms in the BCFW recursion relation that are singular as $P_I^2 \rightarrow 0$ (those in which all legs of $I$ are on the same side). Their singularities are shown on the second line; pulling out a common factor $A(-P_I^{-h_I}, I)$, we recognize the sum as the BCFW-recursed expression for the second factor in \ref{['unshifted-pole-ans']}.
• Figure 5: Left: factorization limit for the "unique diagram" pole $[12]\rightarrow 0$. Right: in each of the BCFW terms that contribute to this singularity, $\hat{1}$ is a soft line attached to one of the unshifted legs.