Measurement of $\gamma {\gamma }^{*}\to {\pi }^0$ transition form factor at Belle

We report a measurement of the process $\gamma {\gamma }^{*}\to {\pi }^0$ with a $759{\mathrm{fb}}^{-1}$ data sample recorded with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The pion transition form factor, $F(Q^2)$, is measured for the kinematical region $4{\mathrm{GeV}}^2\lesssim Q^2\lesssim 40{\mathrm{GeV}}^2$, where $-Q^2$ is the invariant-mass squared of a virtual photon. The measured values of $Q^2|F(Q^2)|$ agree well with the previous measurements below $Q^2\simeq 9{\mathrm{GeV}}^2$ but do not exhibit the rapid growth in the higher $Q^2$ region seen in another recent measurement, which exceeds the asymptotic QCD expectation by as much as 50%.

Figure 1

A Feynman diagram for $\gamma {\gamma }^{*}\to q\overline{q}\to {\pi }^0$ in $e^{+}{e}^{-}$ collisions.

Figure 2

Two-dimensional angular regions in $(\cos {\theta }_e,\cos {\theta }_{\gamma \gamma })$ defined as the Bhabha mask (see Appendix 1 for the details). Events in the yellow (white) regions are selected (rejected). The gray region is outside of the angular acceptance. The mask is symmetric for $e$ and $\gamma \gamma$. The scattered dots in the figure show signal Monte Carlo simulation events without the Bhabha veto applied.

Figure 3

Energy asymmetry [$E_{\mathrm{asym}}$, defined in selection criterion (10)] for the two-photon candidates for events in which $p_e$, $E_{\gamma \gamma }$, and ${\theta }_{\gamma \gamma }$ satisfy the criteria for (a) experimental data and (b) signal Monte Carlo samples. The dashed lines show the selection boundaries. Arrows indicate the accepted regions.

Figure 4

Scatter plots for $E_{\mathrm{ratio}}$ versus $M_{\gamma \gamma }$ for (a) experimental data and (b) signal MC samples. See the text for the selection conditions. The horizontal band between the dashed lines in (b) is the accepted region.

Figure 5

Scatter plots for $E_{\mathrm{ratio}}$ versus $|\Sigma {\mathit{p}}_t^{*}|$ for the (a) experimental data and (b) signal MC samples. See the text for the selection requirements. The solid box in (b) shows the selection region.

Figure 6

Scatter plots for $E_{\gamma \gamma }\Delta \theta$ versus $M_{\gamma \gamma }$ for (a) the experimental data and (b) signal MC samples. This distribution is for the final samples in the range $15{\mathrm{GeV}}^2<Q^2<40{\mathrm{GeV}}^2$ where only the $E_{\gamma \gamma }\Delta \theta$ criterion is not applied. The dashed lines show the selection boundaries. Arrows show the accepted regions.

Figure 7

Calculated $2A(Q^2)$ function for $\sqrt{s}=10.58\mathrm{GeV}$.

Figure 8

The total selection efficiencies as a function of $Q^2$ determined by MC for runs at the ${\rm Y}(4S)$ (solid histogram, where the error bars show statistical errors) and ${\rm Y}(5S)$ energies (dashed histogram).

Figure 9

The trigger efficiencies determined from MC for a typical run at the ${\rm Y}(4S)$ energy. The error bars show statistical errors.

Figure 10

Comparison between the ${\pi }^0$ angular distributions in signal candidate events in data (circles) and signal MC (histograms) for different $Q^2$ regions. Closed (open) circles and solid (dashed) histograms are from events of p-tag (e-tag). In the data, the background contribution is subtracted. The MC events are normalized to the experimental yield after the background subtraction in each $Q^2$ region.

Figure 11

Scatter plots of $Q^2$ versus $M_{\gamma \gamma }$ for the final experimental samples, shown separately for the p-tag (left) and e-tag (right) events.

Figure 12

$M_{\gamma \gamma }$ distributions for the signal MC events for the selected $Q^2$ ranges for the p-tag (left, $7{\mathrm{GeV}}^2<Q^2<8{\mathrm{GeV}}^2$) and e-tag (right, $20{\mathrm{GeV}}^2<Q^2<25{\mathrm{GeV}}^2$) samples. The curves are fits with a double-Gaussian function without background terms.

Figure 13

Results of the fit to the experimental $M_{\gamma \gamma }$ distributions for representative $Q^2$ bins. The caption in each subfigure shows the type of tag (p- or e-tag), the required trigger, whether or not the Bhabha mask was applied (upper line), and whether or not the mass and resolution parameters ($m$ and $\sigma$) are fixed in the fit and the $Q^2$ range (lower line).

Figure 14

(a) The $E_{\mathrm{ratio}}$ distribution for signal candidates for right-sign (RS, dashed line) and wrong-sign (WS, solid line) events. (b) The difference between RS events and WS events. (c) The $E_{\mathrm{ratio}}$ distribution for the signal MC sample. (d) The $M_{\gamma \gamma }$ distribution of wrong-sign events with the best fit superimposed (solid line) where the ${\pi }^0$ peak and the background contribution (dashed line) are taken into account in the fit.

Figure 15

The $E_{\gamma \gamma }\Delta \theta$ distributions for the (a), (d) experimental, (b), (e) signal MC, and (c), (f) radiative-Bhabha background MC samples, for two different $Q^2$ regions indicated in each subfigure. The dashed line shows the selection boundary. Arrows show the accepted regions. The background MC sample corresponds to an integrated luminosity 140 times smaller than the experimental data.

Figure 16

Two-dimensional plots of $|\cos {\theta }^{*}|$ versus invariant mass of the ${\pi }^0{\pi }^0$ system for the selected $(e)e{\pi }^0{\pi }^0$ events (a) for the experimental data and (b) for the $(e)e{\pi }^0{\pi }^0$ MC samples.

Figure 17

Scatter plot for the selected $(e)e{\pi }^0\gamma$ events: $|\cos {\theta }^{*}|$ versus invariant mass of the ${\pi }^0\gamma$ (see the text for the definition of the ${\theta }^{*}$).

Figure 18

The number of events for the signal candidates in $Q^2$ bins before and after the unfolding. Note that the yield is not normalized to the bin width, and the background yield is not subtracted. The open squares (dots with error bars) are the yields before (after) the $Q^2$ unfolding. The vertical error bars are statistical and are only shown for the plots after the unfolding. The data points at $Q^2<6{\mathrm{GeV}}^2$ for the e-tag are the results from the $\mathrm{HiE}+50*\mathrm{CsiBB}$ sample.

Figure 19

Results for the $e^{+}{e}^{-}\to (e)e{\pi }^0$ differential cross section. “Combined” corresponds to twice the weighted average of the p-tag and e-tag results. Error bars show statistical uncertainties only. The horizontal positions in the p-tag and e-tag plots are shifted slightly relative to each other to make it easier to see the sizes of the statistical uncertainties.

Figure 20

Differences between the $e^{+}{e}^{-}\to (e)e{\pi }^0$ differential cross section measured by the p-tag (squares) and e-tag (circles) from their weighted mean (equal to a half of the combined cross section). The vertical scale is normalized by the error of the weighted mean (which is the same as half of the error for the combined cross section). The horizontal positions of the p-tag and e-tag plots are displaced intentionally so that the error bars are visible.

Figure 21

Comparison of the results for the $e^{+}{e}^{-}\to (e)e{\pi }^0$ differential cross sections among different experiments. The vertical error bars are statistical only. The algorithms used to measure the cross section differ between the experiments.

Figure 22

Results for the ${\pi }^0$ transition form factor from the present measurement multiplied by $Q^2$. The error bars are statistical only. The dashed line shows the asymptotic limit from pQCD ($\sim 0.185\mathrm{GeV}$).

Figure 23

Distribution of the ISR energy fraction ($r_k$) calculated from observables in data (solid dots) and in the signal MC samples (histogram). The data distribution is shown after background subtraction. The MC events are normalized to the experimental yield after background subtraction. Note that there are uncertainties in the value of $r_k$ originating from the measurement and kinematical approximation.

Figure 24

Comparison of the results for the product $Q^2|F(Q^2)|$ for the ${\pi }^0$ from different experiments. The error bars are a quadratic sum of statistical and systematic uncertainties. For the Belle and BaBar results, only a $Q^2$-dependent systematic-error component is included. The two curves denoted fit(A) use the BaBar parametrization while the curve denoted fit(B) uses Eq. (23) (see the text). The dashed line shows the asymptotic prediction from pQCD ($\sim 0.185\mathrm{GeV}$).

Figure 25

The threshold behavior for the four selected Logics of the CsiBB trigger (identified in each subfigure) obtained from the experimental samples (left column) of radiative-Bhabha events and from the MC events, where the trigger simulator is tuned (right column). The dashed lines show the position of the nominal trigger thresholds.

Figure 26

Comparison of the experimental radiative-Bhabha samples (dots) and the corresponding MC (Rabhat, histogram) normalized to the integrated luminosity, triggered by HiE or CsiBB, where the Bhabha mask is not applied. The distributions are (a) polar angle of the photon, (b) polar-angle of the electron, (c) sum of the photon and electron energies, and (d) identification number of the cell in the $(\cos {\theta }_e,\cos {\theta }_{\gamma })$ plane.

Figure 27

(a), (b) Comparison of the experimental radiative-Bhabha samples (dots) and the corresponding MC (Rabhat, histogram) normalized to the integrated luminosity, triggered by HiE or CsiBB, where the Bhabha mask is not applied, for the $Q^2$ distributions of the detected electron, where the two selection criteria related to the $p_t$ balance are loosened (see the text). (c), (d) The ratio of the experimental yield to the normalized MC yield. The sizes of the vertical error bars correspond to the statistical errors in both experimental and MC samples. The horizontal lines at 1.0 show the $Q^2$ region where the measurement of the ${\pi }^0$ TFF is performed.

Figure 28

Comparison of the experimental radiative-Bhabha samples (dots) and the corresponding MC (Rabhat, histogram) normalized to the integrated luminosity, triggered by HiE, where the Bhabha mask and veto are applied. See the caption of Fig. 26 for more information.

Figure 29

Comparison of the experimental radiative-Bhabha samples (dots) and the corresponding MC (Rabhat, histogram) normalized to the integrated luminosity, triggered by HiE, where the Bhabha mask and veto are applied. See the caption of Fig. 27 for each figure. The horizontal lines at 1.0 in (c), (d) show the $Q^2$ region where the measurement of the ${\pi }^0$ TFF is performed using HiE-trigger data.

Figure 30

(a), (b) Comparison between the efficiency of the “$\mathrm{\text{Bhabha}}\mathrm{\text{−}}\mathrm{\text{mask}}\times \mathrm{\text{Bhabha}}\mathrm{\text{−}}\mathrm{\text{veto}}$” for the radiative-Bhabha samples [dots, $x_{\mathrm{ratio}}(\mathrm{data})$ described in the text] and the corresponding MC [Rabhat, histogram, $x_{\mathrm{ratio}}(\mathrm{MC})$] for (a) p-tag and (b) e-tag events. (c), (d) Ratio (${ϵ}_{\mathrm{DR}}$) of the efficiencies which are plotted in (a), (b) for the radiative-Bhabha samples and the corresponding MC for (c) p-tag and (d) e-tag events. The vertical error size is derived from the statistical errors in both experimental and MC samples. The horizontal lines at 1.0 show the $Q^2$ region where the measurement of the ${\pi }^0$ TFF is performed using the Bhabha mask and Bhabha veto.