Abstract
In this article, a new perspective for obtaining the magnetic evolution of $\pi-\pi $ scattering lengths in the frame of the linear sigma model is presented. When computing the relevant one-loop diagrams that contribute to these parameters, the sum over Landau levels --emerging from the expansion of the Schwinger propagator-- is handled in a novel way that could also be applied to the calculation of other magnetic-type corrections. Essentially, we have obtained an expansion in terms of Hurwitz Zeta functions. It is necessary to regularize our expressions by an appropriate physical subtraction when $|qB| \rightarrow 0$ ($q$ the meson charge and $B$ the magnetic field strength). In this way, we are able to interpolate between the very high magnetic field strength region, usually handled in terms of the Lowest Landau Level (LLA) approximation, and the weak field region, discussed in a previous paper by some of us, which is based on an appropriate expansion of the Schwinger propagator up to order $|qB|^{2}$. Our results for the scattering lengths parameters produce a soft evolution in a wide region of magnetic field strengths, reducing to the previously found expressions in both limits.
Highlights
During recent years, the physics of strongly interacting hadron matter at high temperatures and densities, including the huge magnetic fields generated during peripheral heavy-ion collisions, has attracted the attention of the community, from both the experimental and theoretical points of view
When computing the relevant one-loop diagrams that contribute to these parameters, the sum over Landau levels—emerging from the expansion of the Schwinger propagator–is handled in a novel way that could be applied to the calculation of other magnetic-type corrections
It is necessary to regularize our expressions by an appropriate physical subtraction when jqBj → 0
Summary
The physics of strongly interacting hadron matter at high temperatures and densities (or baryonic chemical potential), including the huge magnetic fields generated during peripheral heavy-ion collisions, has attracted the attention of the community, from both the experimental and theoretical points of view. In the linear sigma model at the oneloop level, a new discussion on the magnetic dependence of the π-π scattering lengths, valid for arbitrary values of the magnetic field strength. III the detailed computation of the magnetic field contribution to the π-π scattering lengths is presented, including the regularization of a magnetic-dependent divergent term In this way, we are able to present our results for the magnetic evolution for the scattering lengths in both relevant isospin channels I 1⁄4 0, 2.
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