In this letter we evaluate one-loop all-plus gluon amplitudes of $\mathrm{SU}(N_c)$ gauge theory with $N_f$ fundamental fermions in the presence of a flavour instanton background. Fermion zero modes are regulated with a chiral mass term. This computation is performed by cancelling a twistorial 't Hooft anomaly via the Green-Schwarz mechanism. We find that the trace-ordered amplitude has the form of a Parke-Taylor factor multiplied by the Fourier transform of the instanton density evaluated on the total momentum of the gluons. A background flavour instanton modifies the leading soft gluon and photon theorem, generating a level equal to twice the instanton charge in the soft Kac-Moody symmetry. We discuss the implications of our results for amplitudes in the presence of dynamical instantons.
An exact formula for the one-loop scattering of an arbitrary number of positive helicity gluons has been known since the classic works of [1][2][3]. In this paper we determine an exact formula for this scattering process for SU(N c ) gauge theory with N f fundamental fermions, in the presence of an arbitrary self-dual background gauge field for the SU(N f ) flavour symmetry. We will perform this computation in self-dual gauge theory where the only Feynman diagrams contributing at any loop order are such that each connected component has exactly one loop. Only the connected oneloop amplitude coincides with that of the non-self-dual gauge theory.
The presence of fermion zero modes makes amplitudes in an instanton background a little subtle. Perhaps the simplest way to interpret our computation is that we are calculating the effective measure on the space of fermion zero modes. Alternatively, as we will explain in more detail shortly, we can interpret our computation as giving the actual amplitude where the fermion zero modes have been lifted by introducing a chiral mass term mϵ β α ψα ψ β [4].
Our formula is very simple. Let A f denote a background gauge field for the SU(N f ) flavour symmetry group. We assume that
be the instanton density, normalized so that the integral over x of D(A f , x) is the instanton number, which is non-negative for self-dual instantons. Let D(A f , p) denote its Fourier transform. We are interested in the scattering in self-dual SU(N c ) gauge theory with N f Diracs, in the presence of a flavour symmetry instanton. The only non-trivial connected scattering process in self-dual gauge theory is the oneloop all-plus amplitude. We will consider both the connected trace-ordered partial amplitude, and the disconnected amplitude.
The connected, trace-ordered amplitude will be denoted A trace-ordered (1 + , . . . , n + ; A f )
and full amplitude will be denoted
Then:
Theorem 1.
A trace-ordered (1 + , . . . , n + ; A f ) = -2 D(A f , P) ⟨12⟩ . . . ⟨n1⟩ (4) where P is the total momentum of the n incoming gluons.
The disconnected amplitude is a sum of products of connected amplitudes, in the usual way.
The same formula holds, multiplied by a factor of -1 2 , if we scatter N s complex scalars charged under a background SU(N s ) instanton.
Full perturbative gauge theory is recovered by adding the term 1 2 g 2 YM tr(B 2 ). The one-loop all-plus amplitude is insensitive to the deformation term tr(B 2 ), and so can be computed in the self-dual theory; in fact it is the only amplitude of the self-dual theory [6].
Classically, self-dual Yang-Mills theory arises from a field theory on twistor space called holomorphic BF theory [7][8][9]. At loop-level, this generally fails to be true [10]; there is a one-loop gauge anomaly coming from a box diagram on twistor space. This anomaly can be removed by adding a counter-term on twistor space which is non-local, but which is local on space-time. From the space-time perspective, this is an anomaly to integrability [11].
A quantum field theory on space-time which arises from an anomaly free holomorphic theory on twistor space was called a twistorial theory in [10]. Twistorial theories have no scattering [12], which is why we can infer information about scattering from anomalies on twistor space.
Anomalies on twistor space can sometimes be cancelled by a Green-Schwarz mechanism [10] . On spacetime, the Green-Schwarz mechanism involves introducing a fourth-order scalar field ρ, with kinetic term ρ△ 2 ρ (where △ is the Laplace operator). Twistor anomalies, i.e., anomalies to integrability, are cancelled by coupling ρ to the topological charge tr(F(A) 2 ). This method successfully reproduces [13] the standard one-loop all-plus amplitude.
In our context, we want to cancel the mixed gaugeflavour anomaly and so introduce a term proportional to ρ(tr(F(A) 2 )tr(F(A f ) 2 )). We can view the ρ tr(F(A f ) 2 ) term as a source for ρ, since A f is not dynamical.
Anomaly cancellation implies that the one-loop amplitude coming from a loop of fermions in the presence of a flavour gauge field A f is cancelled by the treelevel diagram coming from gluons scattering of the axion field ρ sourced by tr(F(A f ) 2 ). As we explain in the supplementary material VII, this analysis leads to the formula (4).
The expression ( 4) is closely related to the Kac-Moody correlator at level 2k. Let
be the level k Kac-Moody correlator, where a 1 , . . . , a n are adjoint indices. Then, formula (4) implies that [14] lim
where the spinors λ i built from the null momenta p i are related to z i by λ i = (1, z i ).
This result implies a correction to the leading soft gluon theorem in an instanton background. To state the leading soft gluon theorem, consider a general perturbative amplitude A (1 + , t a 1 ), . . . , (n ± , t a n ) where we’ve included the dependence on the colour matrices t a i in the notation. The leading soft gluon theorem [15][16][17] predicts that lim
A (2 ± ,
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