Ordered exponentials and the effective currents of high energy gravity
Abstract
We derive a Lipatov type effective action for the Reggeized gravitons. In place of the order by order construction of the effective currents from diffeomorphism invariance, we propose a current which is a single ordered exponential Wij along the light cone, generated by the optical tidal matrix integrated once. The nested ordering of it reproduces the known perturbative answer with some new terms added. These new terms are non-local along the light cone and have a structure which the diffeomorphism construction cannot produce, the form of the third order contribution to the current is also calculated. The effective action, the current to Reggeon vertex and the current to current interaction are obtained in closed form, in three equivalent writings, one of them covariant. Generally defined Wij reduces to four parameters of the ray bundle, its density, its shear, the orientation of the shear and a rotation of the image. The effective current uses three of them, being the second light cone derivative of a single scalar, it is not sensitive to the orientation. Comparing with the standard eikonal gravitational Wilson line, which is the phase, Wij is the Hessian of the same eikonal: it carries the shear and the focusing that the Abelian phase does not, and it is inverse to the Jacobi propagator of the congruence at first order in the field. A generalization of the proposed ordered operator is discussed as well as applications of the gravitational effective action.
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BibTeX
@misc{bondarenko2026ordered,
title = {{Ordered exponentials and the effective currents of high energy gravity}},
author = {Sergey Bondarenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26831},
eprint = {2609.26831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26831},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ordered exponentials and the effective currents of high energy gravity
AU - Sergey Bondarenko
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.26831
UR - https://arxiv.org/abs/2609.26831
AB - We derive a Lipatov type effective action for the Reggeized gravitons. In place of the order by order construction of the effective currents from diffeomorphism invariance, we propose a current which is a single ordered exponential W_ij along the light cone, generated by the optical tidal matrix integrated once. The nested ordering of it reproduces the known perturbative answer with some new terms added. These new terms are non-local along the light cone and have a structure which the diffeomorphism construction cannot produce, the form of the third order contribution to the current is also calculated. The effective action, the current to Reggeon vertex and the current to current interaction are obtained in closed form, in three equivalent writings, one of them covariant. Generally defined W_ij reduces to four parameters of the ray bundle, its density, its shear, the orientation of the shear and a rotation of the image. The effective current uses three of them, being the second light cone derivative of a single scalar, it is not sensitive to the orientation. Comparing with the standard eikonal gravitational Wilson line, which is the phase, W_ij is the Hessian of the same eikonal: it carries the shear and the focusing that the Abelian phase does not, and it is inverse to the Jacobi propagator of the congruence at first order in the field. A generalization of the proposed ordered operator is discussed as well as applications of the gravitational effective action.
ER -