Production of K+K− pairs through the decay of ϕ mesons

We develop a theoretical framework for the production of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>K</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>K</a:mi><a:mo>−</a:mo></a:msup></a:math> pairs through the decay of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ϕ</c:mi></c:math> mesons produced from a thermal background, based on the Nambu-Jona-Lasinio (NJL) model. The differential production rate of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>K</e:mi><e:mo>+</e:mo></e:msup><e:msup><e:mi>K</e:mi><e:mo>−</e:mo></e:msup></e:math> is related to the self-energy of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ϕ</g:mi></g:math> meson, incorporating the contributions of the quark loop at leading order and the kaon loop at next-to-leading order in the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>1</i:mn><i:mo>/</i:mo><i:msub><i:mi>N</i:mi><i:mi>c</i:mi></i:msub></i:math> expansion. We numerically evaluate the invariant mass spectrum of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>K</k:mi><k:mo>+</k:mo></k:msup><k:msup><k:mi>K</k:mi><k:mo>−</k:mo></k:msup></k:math> pair both in vacuum and at finite temperature. The inclusion of the kaon loop results in a finite width of the spectrum, improving agreement with experimental data. We also investigate the spin alignment of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>ϕ</m:mi></m:math> meson induced by its motion relative to the thermal background. In the NJL model with only chiral condensates, we find that deviations from the unpolarized limit of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mn>1</o:mn><o:mo>/</o:mo><o:mn>3</o:mn></o:mrow></o:math> are negligible.