The form factors of the nucleon at small momentum transfer
Véronique Bernard, Harold W. Fearing, Thomas R. Hemmert, Ulf-G. Meißner
TL;DR
The paper investigates the low-$q^2$ structure of the nucleon's electroweak form factors using a pion–nucleon–$\Delta(1232)$ effective Lagrangian within the small scale expansion, up to ${\cal O}(\epsilon^3)$. It presents the complete ${\cal O}(\epsilon^3)$ renormalized framework, including Delta decoupling, and computes isovector and isoscalar vector as well as axial form factors, highlighting how Delta contributions modify radii and couplings. Key findings include improved agreement for the isovector Pauli radius due to $\Delta$–$\pi$ loops, a finite isoscalar radius determined by counterterms tied to vector meson physics, and the axial radius fixed by a finite counterterm $\tilde{B}_3$, with Delta effects confined to renormalization of $g_A$ at this order. The work also discusses the induced pseudoscalar sector and shows that Delta contributions at this order cannot explain the TRIUMF measurement of $g_P$, emphasizing the role of chiral dynamics and renormalization in shaping nucleon form factors at low energy.
Abstract
We study the low energy expansion of the nucleon's electroweak form factors in the framework of an effective chiral Lagrangian including pions, nucleons and the $Delta (1232)$. We work to third order in the so-called small scale expansion and compare the results with the ones previously obtained in the chiral expansion. In addition, these calculations serve as a first exploratory study of renormalization and decoupling within the small scale expansion.