# Renormalization of Unitarized Weinberg-Tomozawa Interaction without On-shell Factorization and $I=0$ $\bar K N$ - $\pi \Sigma$ Coupled Channels

## TL;DR

This paper develops a renormalization method for the unitarized Weinberg-Tomozawa interaction in coupled channels without relying on on-shell factorization, revealing small differences near the $ar K N$ threshold but significant variations in the second pole position.

## Contribution

It introduces a renormalization scheme that accounts for finite and infinite parts without on-shell factorization, providing new insights into the $ar K N$-$\pi \Sigma$ scattering and the $ ext{Lambda}(1405)$ resonance.

## Key findings

- The scattering $T$-matrix has two poles similar to previous models.
- Differences between approaches grow with energy distance from the renormalization point.
- The second pole position varies significantly between methods.

## Abstract

We calculate the scattering $T$-matrix of $I=0$ $\bar K N-\pi \Sigma$ coupled channels taking a ladder sum of the Weinberg-Tomozawa interaction without on-shell factorization, regularizing three types of divergent meson-baryon loop functions by dimensional regularization and renormalizing them by introducing counter terms. We show that not only infinite but also finite renormalization is important in order for the renormalized physical scattering $T$-matrix to have the form of the Weinberg-Tomozawa interaction. The results with and without on-shell factorization are compared. The difference of the scattering $T$-matrix is small near the renormalization point, close to the observed $\Lambda$(1405). The difference, however, increases with the distance from the renormalization point. The scattering $T$-matrix without on-shell factorization has two poles in the complex center-of-mass energy plane as with on-shell factorization, the real part of which is close to the observed $\Lambda$(1405). While the difference is small with and without on-shell factorization in the position of the first pole, closer to the observed $\Lambda$(1405), the difference is considerably large in the position of the second pole: the imaginary part of the center-of-mass energy of the second pole without on-shell factorization is as large as or even larger than twice that with on-shell factorization. Also, we discuss the origin of the contradiction about the second pole between two approaches, the chiral unitary approach with on-shell factorization and the phenomenological approach without on-shell factorization.