Amplitude determination for $$M M \rightarrow M M$$, $$M = \pi , K$$ and cross-sections for $$\gamma \gamma \rightarrow \pi ^+ \pi ^-, \pi ^0 \pi ^0, \pi ^0 \eta $$ in a chiral model

Abstract Dai and Pennington have performed a comprehensive analysis of essentially all pion and kaon pair production data from two-photon collisions below 1.5 GeV, including all high statistics results from Belle, as well as the older data from Mark II at SLAC, CELLO at DESY, and Crystal Ball at SLAC. Imposing the basic constraints required by analyticity, unitarity, and crossing symmetry and making use of Low’s low-energy theorem for QED, they were able to extract the final-state, strong-interaction scattering amplitudes for the intermediate $$\pi \pi \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> and $$\pi \pi \rightarrow K \overline{K}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> reactions in a model-independent fashion. In addition, they provided good fits to the respective $$\gamma \gamma \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> cross-sections that are known in the low-energy sector in the restricted angular range, $$| \cos \theta | &lt; 0.6 - 0.8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>cos</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>0.6</mml:mn> <mml:mo>-</mml:mo> <mml:mn>0.8</mml:mn> </mml:mrow> </mml:math> . Using the parameters obtained in this fashion, these authors constructed the $$\gamma \gamma \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> cross-sections integrated over the full angular range. In the present work, we use a version of chiral perturbation theory developed by Oller and Oset to evaluate the final-state, strong-interaction amplitudes theoretically, and we compare our low-energy QCD-based results with the amplitudes extracted by Dai and Pennington. We also calculate the $$\gamma \gamma \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> cross-sections (integrated over the full angular range) and compare them with those obtained by Dai and Pennington. These calculations give a more detailed insight into the fit of chiral perturbation theory, not just to the measured $$\gamma \gamma \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> cross-sections, as is usually presented, but rather to a higher level of detail through the available analysis of the experimental data for the underlying final-state, strong-interaction, meson–meson scattering amplitudes $$\pi \pi \rightarrow \pi \pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> and $$\pi \pi \rightarrow K \overline{K}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> themselves. The fits appear to be sensible over the energy range considered. The detailed calculations of strong-interaction transition matrices, as presented in this paper, also pave the way to address the possible presence of the postulated kaonium atom $$K^+ K^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> in the cross-section.