Recursive construction of $\mathrm{\text{Higgs}}+\mathrm{\text{multiparton}}$ loop amplitudes: The last of the “$\varphi$-nite” loop amplitudes

We consider a scalar field, such as the Higgs boson $H$, coupled to gluons via the effective operator $H\mathrm{tr}{G}_{\mu \nu }{G}^{\mu \nu }$ induced by a heavy-quark loop. We treat $H$ as the real part of a complex field $\varphi$ which couples to the self-dual part of the gluon field-strength, via the operator $\varphi \mathrm{tr}{G}_{\mathrm{SD}\mu \nu }{G}_{\mathrm{SD}}^{\mu \nu }$, whereas the conjugate field ${\varphi }^{†}$ couples to the anti-self-dual part. There are three infinite sequences of amplitudes coupling $\varphi$ to quarks and gluons that vanish at tree level, and hence are finite at one loop, in the QCD coupling. Using on-shell recursion relations, we find compact expressions for these three sequences of amplitudes and discuss their analytic properties.

Figure 1

(a) An $L$ type primitive amplitude, in which the fermion line turns left on entering the loop (following the arrow). (b) In an $R$ type primitive amplitude, the fermion line turns right.

Figure 2

(a) Graphs in which the external fermion line passes to the left of the loop are assigned to $L$ type. (b) Graphs in which it passes to the right are called $R$ type. Gluons, fermions or scalars run in the loop. The same decomposition is used when gluons are emitted from the external fermion line.

Figure 3

Diagram corresponding to the recursion relation (4.25) for $A_n^{(0)}(\varphi ,1^{-},2^{+},\dots ,j^{-},\dots ,n^{+})$.

Figure 4

Two cuts of a one-loop $\varphi$ amplitude which become the same cut in the soft limit $k_{\varphi }\to 0$: (a) the cut in the $s_{i\cdots j}$ channel, and (b) the cut in the $s_{\varphi ,i\cdots j}$ channel.

Figure 5

Diagram corresponding to the recursion relation (6.3) for $A_n^{(1)}(\varphi ,1^{+},2^{+},\dots ,n^{+})$.

Figure 6

Diagrams corresponding to the terms in the recursion relation (7.8) for $A_n^{L-s}(\varphi ,j_f^{+})$, for $j\le n-2$. The second term contains an unreal pole. In the last term, the sum over $l$ runs from $j+1$ to $n$, inclusive.

Figure 7

Diagrams corresponding to the terms in the recursion relation (7.30) for $A_n^s(\varphi ,j_f^{+})$. The second diagram contains a double pole, and an unreal pole underneath it. In the last diagram, the sum over $l$ runs from $j+1$ to $n-2$, inclusive.