Bootstrapping one-loop QCD amplitudes with general helicities

The recently developed on-shell bootstrap for computing one-loop amplitudes in nonsupersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from nonstandard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through the use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining $n$-gluon amplitudes in QCD. We give a recursive solution for the one-loop $n$-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity and the remaining ones of positive helicity. We provide an explicit analytic formula for the QCD amplitude $A_{6;1}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$, as well as numerical results for $A_{7;1}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+},7^{+})$, $A_{8;1}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+},7^{+},8^{+})$, and $A_{8;1}(1^{-},2^{-},3^{-},4^{-},5^{+},6^{+},7^{+},8^{+})$. We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.

Figure 1

Schematic representation of recursive contributions. The labels ‘$T$’ and ‘$L$’ refer to tree and loop vertices. The multiparticle factorization-function contribution (c) does not appear for MHV amplitudes.

Figure 2

The recursive diagrams arising from a $[3,4⟩$ shift in $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+})$. Diagram (b) has a nonstandard complex singularity.

Figure 3

The recursive diagrams arising from a $[1,2⟩$ shift in $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+})$. As discussed in the text, only diagram (d) is nonvanishing.

Figure 4

The overlap diagrams arising from an auxiliary $[3,4⟩$ shift in $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+})$.

Figure 5

The recursive diagrams arising from an auxiliary $[3,4⟩$ shift in $A_{6;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$. Diagram (c) has nonstandard complex singularities.

Figure 6

Some vanishing recursive diagrams for the $[1,2⟩$ shift of $A_{6;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$.

Figure 7

Nonvanishing recursive diagrams. Diagram (i) is the factorization-function contribution.

Figure 8

The overlap diagrams for the $[1,2⟩$ shift. Diagram (a) vanishes.

Figure 9

A configuration of poles and branch cuts for a term in a one-loop amplitude, with a branch-cut-hugging contour.

Figure 10

A nonstandard factorization using an auxiliary $[a,b⟩$ shift is assumed to be suppressed in the large-$z$ limit of the primary $[j,l⟩$ shift, if the tree amplitude naively appearing in the factorization contains both legs $j$ and $l$ and is suppressed.

Figure 11

For a $[j,l⟩$ shift, the four potential channels with nonstandard complex singularities. Whether these are actually present in a given amplitude depends on the helicities of the legs nearest to $j$ and $l$ in the color ordering.

Figure 12

Recursive diagrams for $A_{6;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+},6^{+})$, for the auxiliary $[6,4⟩$ shift. We drop the diagrams for which the tree vertex vanishes. Diagram (e) has nonstandard complex singularities. Only diagrams (k) and (l) contribute to the large-$z$ limit of the $[1,2⟩$ shift. Diagrams (c) and (f) represent the factorization-function contribution.

Figure 13

Nonvanishing recursive diagrams for $A_{6;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{-},5^{+},6^{+})$, for the auxiliary $[4,5⟩$ shift. We use this recursion to determine the large-$z$ behavior under the primary $[1,2⟩$ shift. Only diagrams (a) and (b) contribute to the large-$z$ limit.

Figure 14

Nonvanishing diagrams of the $[4,5⟩$ shift recursion relation for $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{-},3^{-},4^{+},5^{+})$. Only diagram (a), which has legs 1 and 2 on opposite sides of the pole, contributes to the large-$z$ limit of the $[1,2⟩$ shift.

Figure 15

The overlap diagrams of the $[4,5⟩$ shift.

Figure 16

The recursive diagrams of the $[1,2⟩$ shift for an $n$-point helicity amplitude. Some of the diagrams may vanish depending on the helicity choices.

Figure 17

Overlap diagrams corresponding to the $s_{2\dots j}$, $s_{2\dots r}$, and $s_{2\dots (r+1)}$ channels.