$V{V}^{\prime }P$ form factors in resonance chiral theory and the $\pi -\eta -\eta \prime$ light-by-light contribution to the muon $g-2$

The description of the $V{V}^{\prime }P$ form factors ($V,V^{\prime }$ stands for vector particles and $P$ for a pseudoscalar meson) for different particles virtualities remains a challenge for the theory of strong interactions. While their chiral limit is well understood, recent measurements of the ${\gamma }^{*}\omega {\pi }^0$ and ${\gamma }^{*}\gamma {\pi }^0$ form factors at high photon virtualities seem to depart from the simplest scaling behavior suggested by QCD. Here we attempt to describe them in their whole measured energy regimes within the resonance chiral theory, a framework which naturally incorporates the chiral limit constraints and extends to higher energies by including the resonances as active fields. Specifically, we obtained an accurate description of the data up to 2 GeV on the former form factor by including three multiplets of vector resonances. Good agreement with measurements of the latter was possible even in the single resonance approximation, although we propose to measure the $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}{\pi }^0$ cross section and dimuon invariant mass distribution to better characterize this form factor. We have then evaluated the pion exchange contribution to the muon $g-2$ obtaining $(6.66\pm 0.21)\times 10^{-10}$ with an accurate determination of the errors. We have also recalled that approximating the whole pion exchange by the pion-pole contribution underestimates the corresponding result for the anomaly [by (15, 20)%)]. Based on these results, we have predicted the ${\eta }^{(\prime )}$ transition form factors obtaining good agreement with data and obtained their respective contributions to the muon anomaly. In this way, the contribution of the three lightest pseudoscalars to it yields $(10.47\pm 0.54)\times 10^{-10}$, in agreement with previous evaluations but with smaller error.

Figure 1

SND [13, 14] and isospin-rotated CLEO data [17] for $\sigma (e^{+}{e}^{-}\to \omega {\pi }^0)$ are confronted to the best fit results including two (blue dashed-dotted line) or three resonance multiplets (solid purple line). Fits obtained including data in the charmonium region (orange dashed line) as well as in the bottomonium region (grey dotted line) are also displayed. Higher-energy data can be seen in Fig. 2.

Figure 2

SND [13, 14], isospin-rotated CLEO data [17], CLEO data in the charmonium region [15, 16], and Belle [8] data in the bottomonium region for $\sigma (e^{+}{e}^{-}\to \omega {\pi }^0)$ are confronted to the best fit results including three resonance multiplets (solid purple line) as well as data in the charmonium region (orange dashed line) and also in the bottomonium region (grey dotted line).

Figure 3

SND [13, 14] and isospin-rotated CLEO data [17] for $\sigma (e^{+}{e}^{-}\to \omega {\pi }^0)$ below 2 GeV are confronted to our best fit results. The form factor includes the contribution of three $\rho$-like resonances and the continuum. The data in the quarkonium region, which can be seen in Fig. 4, are taken into account in the fit.

Figure 4

SND [13, 14], isospin-rotated CLEO data [17], CLEO data in the charmonium region [15, 16], and Belle [8] data in the bottomonium region for $\sigma (e^{+}{e}^{-}\to \omega {\pi }^0)$ are compared with our best fit results. The form factor includes three $\rho$-like resonances plus a continuum contribution.

Figure 5

CELLO [63], CLEO [64], Babar [7], and Belle [65] data for the $\pi \mathrm{TFF}$ are compared to our best fit result using the form factor in Eq. (38) as explained in the main text. The error band associated with the 10% variation of $F_V$ cannot be appreciated.

Figure 6

Our predictions for $\sigma (e^{+}{e}^{-}\to {\pi }^0{\mu }^{+}{\mu }^{-})(s)$ are plotted using the values of the couplings in Eq. (48). The very small error band cannot be appreciated.

Figure 7

Our predictions for ${\mu }^{+}{\mu }^{-}$ distribution at $s=(1.02\mathrm{GeV}{)}^2$ are plotted using the values of the couplings in Eq. (48). The error band cannot be appreciated.

Figure 8

Our predictions for $e^{+}{e}^{-}\to {\eta }^{(\prime )}{\mu }^{+}{\mu }^{-}$ cross section are plotted using the values of the couplings in Eq. (48) and Eqs. (54, 55, 56) for the $\eta$-${\eta }^{\prime }$ mixing.

Figure 9

Our predictions for the ${\mu }^{+}{\mu }^{-}$ distribution in the $e^{+}{e}^{-}\to {\eta }^{(\prime )}{\mu }^{+}{\mu }^{-}$ processes are plotted at $s=4{\mathrm{GeV}}^2$ using the values of the couplings in Eq. (48) and Eqs. (54, 55, 56) for the $\eta$-${\eta }^{\prime }$ mixing.

Figure 10

Our predictions for the $\eta \mathrm{TFF}$ using the values of the couplings in Eq. (48) and the $\eta$-${\eta }^{\prime }$ mixing in Eq. (55) are compared to Babar [95], CELLO [63], and CLEO [64] data. The error band is completely dominated by the uncertainty on the $\eta$-${\eta }^{\prime }$ mixing.

Figure 11

Our predictions for the ${\eta }^{\prime }$TFF using the values of the couplings in Eq. (48) and the $\eta$-${\eta }^{\prime }$ mixing in Eq. (55) are compared to Babar [95], CELLO [63], and CLEO [64] data. The error band is completely dominated by the uncertainty on the $\eta$-${\eta }^{\prime }$ mixing.