Subtleties in triangle loops for Ds+→ρ+η→π<…

We address a general problem in the evaluation of triangle loops stemming from the consideration of the range of the interaction involved in some of the vertices, as well as the energy dependence of the width of some unstable particles in the loop. We find sizeable corrections from both effects. We apply that to a loop relevant to the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msubsup><a:mrow><a:mi>D</a:mi></a:mrow><a:mrow><a:mi>s</a:mi></a:mrow><a:mrow><a:mo>+</a:mo></a:mrow></a:msubsup><a:mo stretchy="false">→</a:mo><a:msup><a:mrow><a:mi>π</a:mi></a:mrow><a:mrow><a:mo>+</a:mo></a:mrow></a:msup><a:msup><a:mrow><a:mi>π</a:mi></a:mrow><a:mrow><a:mn>0</a:mn></a:mrow></a:msup><a:mi>η</a:mi></a:mrow></a:math> decay, and find reductions of about a factor of 4 in the mass distribution of invariant mass of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mrow><d:mi>π</d:mi><d:mi>η</d:mi></d:mrow></d:math> in the region of the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>a</f:mi><f:mn>0</f:mn></f:msub><f:mo stretchy="false">(</f:mo><f:mn>980</f:mn><f:mo stretchy="false">)</f:mo></f:math>. The method used is based on the explicit analytical evaluation of the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi>q</j:mi><j:mn>0</j:mn></j:msup></j:math> integration in the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>d</l:mi><l:mn>4</l:mn></l:msup><l:mi>q</l:mi></l:math> loop integration, using Cauchy’s residues method, which at the same time offers an insight on the convergence of the integrals and the effect of form factors and cutoffs.