A dispersive analysis of $$\varvec{\eta '\rightarrow \pi ^+\pi ^-\gamma }$$ and $$\varvec{\eta '\rightarrow \ell ^+\ell ^-\gamma }$$
Simon Holz, C. Hanhart, Martin Hoferichter, Bastian Kubis
Abstract
Abstract We present a dispersive representation of the $$\eta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> transition form factor that allows one to account, in a consistent way, for the effects of $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> – $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> mixing in both the isoscalar and the isovector contributions. Using this formalism, we analyze recent data on $$\eta '\rightarrow \pi ^+\pi ^-\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <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:mi>γ</mml:mi> </mml:mrow> </mml:math> to constrain the isovector part of the form factor, individually and in combination with data for the pion vector form factor, which suggests a tension in the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> – $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> mixing parameter. As a first application, we use our results, in combination with the most recent input for the isoscalar part of the form factor, to predict the corresponding spectrum of $$\eta '\rightarrow \ell ^+\ell ^-\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mo>-</mml:mo> </mml:msup> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> , in particular we find the slope parameter $$b_{\eta '}=1.455(24)\,\text {GeV}^{-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.455</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>24</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:msup> <mml:mtext>GeV</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . With forthcoming data on the latter process, our results establish the necessary framework to improve the evaluation of the $$\eta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -pole contribution to the anomalous magnetic moment of the muon using experimental input from both $$\eta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> decay channels.