Hadronic matrix elements of gluon operators in the instanton vacuum

TL;DR

This paper develops a comprehensive framework to compute hadronic matrix elements of gluon operators in the instanton vacuum by treating the interacting instanton ensemble in a grand canonical formalism and matching it to a Nambu–Jona-Lasinio–type effective fermion theory. By incorporating fermions and implementing a variational approach for instanton interactions, the authors derive an effective action that yields dynamical mass generation and preserves QCD renormalization properties, including the trace and U(1)_A anomalies. They introduce effective gluon operators within the fermionic theory, enabling calculation of nucleon matrix elements of F^2 and F tilde F, and show these reproduce known anomaly relations and the isosinglet axial coupling g_A^(0) in the large-Nc limit. The framework also provides a way to assess θ-dependence and topological fluctuations on the lattice, and yields practical routes to study nucleon structure and higher-twist gluonic effects in a nonperturbative QCD vacuum. Overall, the work establishes a self-consistent, anomaly-compatible method to relate instanton dynamics to hadronic observables mediated by gluon operators.

Abstract

We propose a method to evaluate hadronic matrix elements of QCD gluon operators in the instanton vacuum. We construct the ground state of the interacting instanton ensemble for non-zero $\vartheta$--angle using a variational principle. A method to study the $\vartheta$--dependence of observables on the lattice is suggested. We then derive the effective fermion action, which allows to calculate hadronic correlation functions in a $1/N_c$--expansion (Nambu--Jona-Lasinio type effective fermion theory). Gluon operators are systematically represented as effective fermion operators. Physical matrix elements are obtained after integrating the correlation functions over fluctuations of the numbers of instantons. The influence of the fermion determinant on the topological susceptibility is taken into account. Our effective description gives matrix elements fully consistent with the trace and $U(1)_A$ anomalies. The approach allows to consistently evaluate the nucleon matrix elements of various gluon and mixed quark--gluon operators in a chiral soliton picture of the nucleon.