Scattering in Instanton Backgrounds

Abstract: 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.

• The paper demonstrates that one-loop all-plus helicity gluon amplitudes factorize into a Parke-Taylor structure multiplied by the Fourier transform of the instanton density.
• It employs twistorial methods to regulate fermion zero modes and cancels mixed gauge-flavour anomalies via a Green-Schwarz mechanism.
• The study reveals that flavour instantons induce corrections to the soft gluon theorem through a nonzero Kac-Moody level shift proportional to the instanton number.

One-Loop All-Plus Gluon Amplitudes in Flavour Instanton Backgrounds

Overview and Main Results

The paper "Scattering in Instanton Backgrounds" (2602.17538) establishes exact analytic expressions for one-loop all-plus helicity gluon amplitudes in $SU(N_c)$ gauge theories with $N_f$ fundamental fermions, specifically in the presence of an %%%%2%%%% flavour instanton background. The work leverages the framework of self-dual gauge theory, exploiting its simplifications at the one-loop level, and employs twistorial methods to regulate fermion zero modes and resolve anomalies through a Green-Schwarz mechanism.

A principal outcome is the demonstration that the trace-ordered all-plus amplitude, in such a background, factorizes into a Parke-Taylor denominator and the Fourier transform of the instanton density evaluated at the total external momentum: $$A^\text{trace-ordered}(1^+,\dots,n^+;A_f) = - 2 \frac{D(A_f,P)}{\langle12\rangle\dots\langle n1\rangle}$$ where $D(A_f,P)$ is the Fourier transform of the SU($N_f$) instanton density and $P$ is the sum of all external momenta. This structure generalizes in a straightforward way to disconnected amplitudes and to the case where fundamental scalars replace fermions, with appropriate sign changes.

The most novel physical implication is the explicit correction to the leading soft gluon theorem in the presence of a flavour instanton, yielding a nonzero Kac-Moody level shift proportional to the instanton number.

Technical Approach

The analysis is conducted within self-dual Yang-Mills theory, where only one-loop all-plus amplitudes are nontrivial due to self-duality constraints. The presence of fermion zero modes, customary in instanton backgrounds, is resolved by introducing a chiral mass term $m\tilde\psi \tilde\psi$, which regulates the Grassmannian integration and enables the computation of nontrivial scattering amplitudes without explicit zero mode insertions.

A key conceptual step utilizes the twistorial formulation of gauge theory in which anomalies in the holomorphic twistor action manifest as 't Hooft anomalies on spacetime. The mixed gauge-flavour anomaly associated with the fermion content in the instanton background is shown to be cancelled by a Green-Schwarz term, introducing a dimension-zero axion $\rho$ coupled to the topological density difference between dynamical and background gauge fields: $$\frac{1}{2}\int d^4x (\Box\rho)^2 + \mu \int d^4x\, \rho \big( \operatorname{tr}(F(A)^2) - \operatorname{tr}(F(A_f)^2)\big)$$ The amplitude computation thus reduces to evaluating tree-level axion exchange diagrams, sourced by the instanton density. This technique yields the stated Parke-Taylor $\times$ instanton Fourier structure, with normalization fixed by matching to singular (zero-size) instanton configurations via the ADHM construction.

Implications for Soft Theorems and Symmetries

A notable achievement is the connection between instanton backgrounds and corrections to soft theorems. In both non-Abelian (gluon) and Abelian (photon) cases, the leading soft theorem receives a correction term, altering the algebraic structure of asymptotic charges. Specifically, the commutator of soft gluon currents, encodable in the Kac-Moody algebra, acquires a level shift of $2k$, with $k$ the instanton charge: $$\lim_{p_i \to 0} A^\text{disconnected}(1^+,\dots,n^+;A_f) = \langle J_{a_1}(z_1) \cdots J_{a_n}(z_n)\rangle_{2k}$$ This is an explicit, nonperturbative modification to the leading soft theorem, contrasting with loop-induced (infrared) corrections in perturbation theory. Conceptually, this ties instanton physics to celestial Kac-Moody symmetries in a quantifiable manner.

For dynamical instantons, the total amplitude is integrated over the instanton moduli space with the appropriate one-loop measure, including the effects of the chiral mass term on the $\beta$-function and ultraviolet behaviour. The transition from flavour instantons to dynamical instantons is handled via integration over moduli, preserving the basic amplitude structure.

Ultraviolet and Infrared Behaviour

A detailed investigation into ultraviolet (UV) and infrared (IR) divergences associated with instanton contributions is carried out. For fermionic matter, the structure of the moduli space measure ensures the absence of UV singularities from small instantons, provided no scalars are present and $\tilde{b}>4$. However, the inclusion of bifundamental scalars leads to circumstances where UV divergences cannot be removed by local counter-terms—signalling a form of non-perturbative non-renormalizability. These divergent contributions are associated with singular (zero-size) instantons and cannot be subtracted via standard renormalization, instead requiring the addition of generalized, dimensionful $\delta$-function measures on instanton moduli space.

For spacelike external momenta, the integral over instanton moduli space converges absolutely due to exponential decay of the Fourier-transformed instanton density, as shown using explicit Bessel function representations. The analysis is refined by supplementing the large-instanton asymptotics using results for higher-charge instantons due to Osborn [Osborn:1981yf].

Theoretical and Practical Consequences

This work solidifies several important aspects for both quantum gauge theory and amplitude computations:

1. Explicitly quantifies instanton corrections to scattering amplitudes at the loop level, potentially impacting precision QCD predictions in backgrounds with nontrivial topology.
2. Demonstrates that instanton effects induce higher-level Kac-Moody symmetry structures in soft limits, which may have implications for celestial holography and the classification of asymptotic symmetries beyond perturbation theory.
3. The approach—combining twistor-space anomalies, Green-Schwarz mechanism, and explicit moduli space integration—offers a transferable blueprint for computing loop amplitudes on broader class of self-dual or curved (Einstein) backgrounds.
4. The results clarify the renormalization structure of quantum gauge theories in backgrounds with bifundamental scalars, raising questions about the viability of certain theories once nonperturbative effects are accounted for.

Future research directions include generalization to non-self-dual backgrounds, computation of higher-loop and non-MHV amplitudes using similar twistorial/anomaly-based methods, and exploring the role of modified soft theorems in black hole physics or gravitational wave memory.

Conclusion

The paper provides a rigorous and explicit prescription for evaluating one-loop all-plus gluon amplitudes in the presence of flavour instantons, clarifying the impact of instantonic gauge backgrounds on both amplitude structure and asymptotic symmetries. The results enable precise identification of instanton-induced corrections to soft gluon and photon theorems, enhancing the theoretical foundation for the study of nonperturbative effects in gauge theory amplitudes and their associated symmetry algebras. This work broadens the toolkit for the systematic incorporation of topological nontrivial backgrounds in high-energy scattering and paves the way for further advances in both amplitude computations and gauge-gravity correspondences.