The $$\pi \eta $$ interaction and $$a_0$$ resonances in photon–photon scattering

Abstract

Abstract We revisit the information on the two lightest $$a_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> resonances and S -wave $$\pi \eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi><mml:mi>η</mml:mi></mml:mrow></mml:math> scattering that can be extracted from photon–photon scattering experiments. For this purpose we construct a model for the S -wave photon–photon amplitudes which satisfies analyticity properties, two-channel unitarity and obeys the soft photon as well as the soft pion constraints. The underlying I=1 hadronic T -matrix involves six phenomenological parameters and is able to account for two resonances below 1.5 GeV. We perform a combined fit of the $$\gamma \gamma \rightarrow \pi \eta $$ <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>η</mml:mi></mml:mrow></mml:math> and $$\gamma \gamma \rightarrow K_SK_S$$ <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:msub><mml:mi>K</mml:mi><mml:mi>S</mml:mi></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:mrow></mml:math> high statistics experimental data from the Belle collaboration. Minimisation of the $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> is found to have two distinct solutions with approximately equal $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> . One of these exhibits a light and narrow excited $$a_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> resonance analogous to the one found in the Belle analysis. This however requires a peculiar coincidence between the $$J=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and $$J=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> resonance effects which is likely to be unphysical. In both solutions the $$a_0(980)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>980</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> resonance appears as a pole on the second Riemann sheet. The location of this pole in the physical solution is determined to be $$m-i\varGamma /2=1000.7^{+12.9}_{-0.7} -i\,36.6^{+12.7}_{-2.6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mi>Γ</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>=</mml:mo><mml:mn>1000</mml:mn><mml:mo>.</mml:mo><mml:msubsup><mml:mn>7</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>0.7</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>12.9</mml:mn></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mspace/><mml:mn>36</mml:mn><mml:mo>.</mml:mo><mml:msubsup><mml:mn>6</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>2.6</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>12.7</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> MeV. The solutions are also compared to experimental data in the kinematical region of the decay $$\eta \rightarrow \pi ^0\gamma \gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>η</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mi>γ</mml:mi><mml:mi>γ</mml:mi></mml:mrow></mml:math> . In this region an isospin violating contribution associated with $${\pi ^+}{\pi ^-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> rescattering must be added for which we provide a dispersive evaluation.

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