Electric corrections to π−π scattering lengths in the linear sigma model

In this article, we analyze the role of an external electric field, in the weak field approximation, on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>π</a:mi><a:mtext>−</a:mtext><a:mi>π</a:mi></a:mrow></a:math> scattering lengths. The discussion is presented in the frame of the linear sigma model. To achieve this, we take into account all one-loop corrections in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>s</c:mi></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>t</e:mi></e:math>, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>u</g:mi></g:math> channels associated with the insertion of a Schwinger propagator for charged pions, focusing on the region characterized by small values of the electric field. Furthermore, one of the novelties of our work is the explicit calculation of box diagrams, which were previously overlooked in discussions regarding magnetic corrections. It turns out that the electric field corrections have an opposite effect with respect to magnetic corrections calculated previously in the literature.