Study of ${\pi }^0$ pair production in single-tag two-photon collisions

We report a measurement of the differential cross section of ${\pi }^0$ pair production in single-tag two-photon collisions, ${\gamma }^{*}\gamma \to {\pi }^0{\pi }^0$, in $e^{+}{e}^{-}$ scattering. The cross section is measured for $Q^2$ up to $30{\mathrm{GeV}}^2$, where $Q^2$ is the negative of the invariant mass squared of the tagged photon, in the kinematic range $0.5\mathrm{GeV}<W<2.1\mathrm{GeV}$ and $|\cos {\theta }^{*}|<1.0$ for the total energy and pion scattering angle, respectively, in the ${\gamma }^{*}\gamma$ center-of-mass system. The results are based on a data sample of $759{\mathrm{fb}}^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The transition form factor of the $f_0(980)$ and that of the $f_2(1270)$ with the helicity-0, -1, and -2 components separately are measured for the first time and are compared with theoretical calculations.

Figure 1

Feynman diagram for the process $e^{+}{e}^{-}\to e(e){\pi }^0{\pi }^0$. $p_1$, $p_1^{\prime }$, $p_2$, and $p_2^{\prime }$ are the four-momenta of the incident and scattered electron or positron, $q_1$, $q_2$ are those of the virtual and real photons and $k_1$, $k_2$ are those of the produced ${\pi }^0$ mesons.

Figure 2

Definition of the coordinate system for ${\gamma }^{*}\gamma \to {\pi }^0{\pi }^0$. The incident ${\gamma }^{*}$ and $\gamma$ are along the $z^{*}$ axis, the tagged $e^{\pm }$ is in the $x^{*}{z}^{*}$ plane with $p_{x^{*}\text{tag}}>0$, and a ${\pi }^0$ is produced at angles $({\theta }^{*},{\phi }^{*})$.

Figure 3

(a) The experimental distributions for the $\gamma \gamma$ invariant mass forming $\pi 1$. The arrows indicate the selection range. (b) The distribution from the signal MC for the same variable. Statistics of the MC figure are arbitrary. The fit for the comparison is shown by a solid curve in each of the distributions (see the text). The background component in the experimental data from the fit is shown by the dashed line.

Figure 4

The distributions for $E_{\text{ratio}}$ for (a) the experimental data and (b) the signal-MC sample. The cross plots in (a) show the distribution for the wrong-sign events, and the arrows indicate the signal region. Statistics of the MC figure are arbitrary.

Figure 5

(a),(b),(c) The distributions for the $p_t$ balance for the data samples in three different $W$ ranges indicated in each panel. The arrows show the selection region. (d),(e),(f) The corresponding distributions from the signal MC. Statistics of the MC figures are arbitrary.

Figure 6

The experimental distributions for $W$ of the signal candidates for $3{\mathrm{GeV}}^2(5{\mathrm{GeV}}^2)<Q^2<30{\mathrm{GeV}}^2$ for the e-tag (p-tag) samples. Backgrounds are not subtracted. The cross and asterisk plots are for the e-tag and p-tag samples, respectively.

Figure 7

(a),(b),(c) The $Q^2$ distributions for the data samples in three different $W$ ranges indicated above each panel. The cross and asterisk plots are for the e-tag and p-tag samples, respectively. (d),(e),(f) The corresponding distributions from the signal MC, where the solid and dashed histograms are for the e-tag and p-tag samples, respectively. Statistics of the MC figures are arbitrary, but the scale is common for the e- and p-tags so their ratio could be compared between MC and data.

Figure 8

(a),(b),(c),(g),(h),(i) The distributions for the $\gamma \gamma$ invariant mass to construct $\pi 1$ (a),(b),(c) and $\pi 2$ (g),(h),(i) for the data samples in three different $W$ ranges indicated above each panel. (d),(e),(f),(j),(k),(l) The corresponding distributions from the signal MC for $\pi 1$ (d),(e),(f) and $\pi 2$ (j),(k),(l). The legend and comments are the same as those in Fig. 7.

Figure 9

(a),(b),(c) The distributions for the laboratory energy of $\pi 1$ and $\pi 2$ for the data samples in three different $W$ ranges indicated above each panel. (d),(e),(f) The corresponding distributions from the signal MC. (g),(h),(i) and (j),(k),(l) are the distributions for the laboratory angle of $\pi 1$ and $\pi 2$ for the data samples in the indicated $W$ ranges for the experimental samples and MC samples, respectively. The legend and comments are the same as those in Fig. 7.

Figure 10

The $|\cos {\theta }^{*}|$ distributions for (a) the experimental signal candidates, (b) the signal MC with the isotropic generation for $|\cos {\theta }^{*}|$, and (c) the background-MC events for the single-pion production process, where the yield is scaled to the expected contamination in the set of all signal candidates. The error bars are statistical from the MC samples but are not proportional to square root of the number of events because each event in the MC sample has a variable weight to reproduce the experimental $Q^2$ dependence. For (b), the normalization is arbitrary.

Figure 11

Scatter plot for the $\gamma \gamma$ invariant masses for $\pi 1$ and $\pi 2$ for (a) the experimental signal candidates for $|\cos {\theta }^{*}|>0.9$, (b) the signal-MC samples for $|\cos {\theta }^{*}|>0.9$, and (c) the experimental signal candidates for $|\cos {\theta }^{*}|<0.9$.

Figure 12

The distributions of $\mathrm{\Delta }\theta E_{\gamma \gamma }$ for (a) the experimental signal candidates in $|\cos {\theta }^{*}|>0.9$ and (b) in $|\cos {\theta }^{*}|<0.9$. Similar plots for (c) the signal-MC samples in $|\cos {\theta }^{*}|>0.9$ and (d) $|\cos {\theta }^{*}|<0.9$. The statistics of the MC figures are arbitrary.

Figure 13

The distribution of background events estimated from the MC for the process $e^{+}{e}^{-}\to e(e)\omega$, $\omega \to {\pi }^0\gamma$ in the $Q^2$ versus $W$ and $|\cos {\theta }^{*}|$ versus $W$ scatter plots. About three events in each plot correspond to a contamination of one event in the present signal-candidate sample.

Figure 14

Comparison between the experimental (left) and signal-MC (right) distributions for $E_{\text{ratio}}$ of events in the signal regions. Each row corresponds to the same $W$ and $Q^2$ regions indicated in the right panel. The dashed lines show the estimated nonexclusive background, which is assumed to distribute linearly with the horizontal variable. Statistics of the MC figures are arbitrary.

Figure 15

Comparison between the experimental (left) and signal-MC (right) distributions for the $p_t$ balance of events in the signal regions. Each row corresponds to the same $W$ and $Q^2$ regions indicated in the right panel. The dashed lines show the estimated nonexclusive background, which is assumed to distribute linearly with the horizontal variable. Statistics of the MC figures are arbitrary.

Figure 16

Trigger efficiency estimated by the signal-MC samples and trigger simulator. The solid (black) and dashed (red) histograms are for e-tag and p-tag events, respectively. The results are for the three $W$ points indicated above each panel: (a),(b),(c) the $Q^2$ dependence for the isotropically generated ${\pi }^0$ pairs in the ${\gamma }^{*}\gamma$ c.m. system; (d),(e),(f) the scattering- (polar-)angle dependence of ${\pi }^0$ in the ${\gamma }^{*}\gamma$ c.m. system. The $Q^2$ range $3{\mathrm{GeV}}^2(5{\mathrm{GeV}}^2)<Q^2<30{\mathrm{GeV}}^2$ is integrated for the e-tag (p-tag) plot.

Figure 17

Efficiency estimated from the signal-MC samples passed through all the selection and trigger conditions. Each panel provides the $W$ and $|\cos {\theta }^{*}|$ dependences of the efficiency for the four selected $Q^2$ bins for each of the p- or e-tags. The contour levels shown by different colors are not common across the panels.

Figure 18

The $W$ dependence of the $e^{+}{e}^{-}$-based cross section in each $Q^2$ bin. The full $|\cos {\theta }^{*}|$ range (0 to 1) is integrated. The closed circles (open circles) are for the e-tag (p-tag) measurements. The plotting location for each p-tag point is shifted slightly to improve the visibility of error bars. The efficiency corrections and background subtraction are applied.

Figure 19

The $|\cos {\theta }^{*}|$ dependence of the $e^{+}{e}^{-}$-based cross section in each $Q^2$ bin. The dependence for three different $W$ regions is shown above each panel, and the $W$ regions are 0.5–1.1 GeV, 1.1–1.6 GeV, 1.6–2.1 GeV from bottom to top within each panel, as indicated in the upper left panel. The closed circles (open circles) are for the e-tag (p-tag) measurements. The plotting location for each p-tag point is shifted slightly to improve the visibility of the error bars. The efficiency corrections and background subtraction are applied.

Figure 20

The ${\phi }^{*}$ distributions for $ϵ({\phi }^{*})\propto N_{\mathrm{MC}}({\phi }^{*})$ for the $W$ region 0.5–1.1 GeV (upper row) and 1.1–1.6 GeV (lower row), and $8{\mathrm{GeV}}^2<Q^2<12{\mathrm{GeV}}^2$. They are separately plotted for the five angular regions, whose central $|\cos {\theta }^{*}|$ value is indicated in each panel. The solid curve is the fit to the Fourier expansion described in the text.

Figure 21

The $|\cos {\theta }^{*}|$ dependence of the obtained parameters of $a$, $b$, and the efficiency correction factor $ϵ/{ϵ}^{\prime }$; $a$ and $b$ are the coefficients of $\cos {\phi }^{*}$ and $\cos 2{\phi }^{*}$ in $P({\phi }^{*})$ for each of the nine $(W,Q^2)$ regions (see the text). Thick lines indicate the efficiency correction factor, which is applied to obtain the ${\phi }^{*}$-integrated cross section.

Figure 22

Integrated cross section for ${\gamma }^{*}\gamma \to {\pi }^0{\pi }^0$ in nine $Q^2$ bins in ${\mathrm{GeV}}^2$ indicated in each panel.

Figure 23

$|\cos {\theta }^{*}|$ dependence in five $W$ bins: 0.9, 1.15, 1.25, 1.35, and 1.45 GeV from left to right and top to bottom and fitted results of the ${\phi }^{*}$ nominal fit. The legend of lines is: solid lines show the total (black), dotted $|S{|}^2$ (green), dashed $|D_0{|}^2$ (blue), long dashed $|D_1{|}^2$ (red), and dash-dotted $|D_2{|}^2$ (purple).

Figure 24

${\phi }^{*}$ dependence in five $W$ bins: 0.9, 1.15, 1.25, 1.35, and 1.45 GeV from left to right and top to bottom and fitted results of ${\phi }^{*}$ nominal fit. Legend of lines: solid lines for the total (black), dotted $t_0$ (red), dashed $t_1\cos {\phi }^{*}$ (blue), and dash-dotted $t_2\cos 2{\phi }^{*}$ (purple).

Figure 25

${\phi }^{*}$ dependence in five $|\cos {\theta }^{*}|$ bins: 0.1, 0.3, 0.5, 0.7, and 0.9 from left to right and top to bottom and fitted results of ${\phi }^{*}$ nominal fit. The lines are defined in Fig. 24.

Figure 26

$W$ dependence in five $|\cos {\theta }^{*}|$ bins: 0.1, 0.3, 0.5, 0.7, and 0.9 from left to right and top to bottom and fitted results of ${\phi }^{*}$ nominal fit. The lines are defined in Fig. 23.

Figure 27

$W$ dependence in nine ${\phi }^{*}$ bins: 10° to 170° in 20° step from left to right and top to bottom and fitted results of ${\phi }^{*}$ nominal fit. The lines are defined in Fig. 24.

Figure 28

Angular dependence of the cross section in the indicated $W$ bin and results of the $r_1$ fit at $Q^2=3.5{\mathrm{GeV}}^2$ (left) and $Q^2=4.5{\mathrm{GeV}}^2$ (right). Legend of lines: solid lines for the total (black), dotted $|S{|}^2$ (green), dashed $|D_0{|}^2$ (blue), and dash-dotted $|D_2{|}^2$ (purple).

Figure 29

Angular dependence of the cross section in the indicated $W$ bin and results of the $r_1$ fit at $Q^2=5.5{\mathrm{GeV}}^2$ (left) and $Q^2=7.0{\mathrm{GeV}}^2$ (right). The lines are defined in Fig. 28.

Figure 30

Angular dependence of the cross section in the indicated $W$ bin and results of the $r_1$ fit at $Q^2=9.0{\mathrm{GeV}}^2$ (left) and $Q^2=11.0{\mathrm{GeV}}^2$ (right). The lines are defined in Fig. 28.

Figure 31

Angular dependence of the cross section in the indicated $W$ bin and results of the $r_1$ fit at $Q^2=13.5{\mathrm{GeV}}^2$ (left) and $Q^2=17.5{\mathrm{GeV}}^2$ (right). The lines are defined in Fig. 28.

Figure 32

Angular dependence of the cross section in the indicated $W$ bin and results of the $r_1$ fit at $Q^2=25{\mathrm{GeV}}^2$. The lines are defined in Fig. 28.

Figure 33

Integrated cross section in the indicated $Q^2$ bin and results of the $r_1$ fit. The lines are defined in Fig. 28.

Figure 34

The measured helicity-2 TFF of the $f_2(1270)$ as a function of $Q^2$. Short (long) vertical bars indicate statistical (statistical and systematic combined) errors. The shaded area corresponds to the overall systematic uncertainty arising from that of ${\mathrm{\Gamma }}_{\gamma \gamma }$. Correlations of uncertainties between neighboring bins exist and are included in the long vertical bars. The solid line shows the predicted $Q^2$ dependence in Table 1 by Ref. [4] and those by Ref. [5]: Eq. (1) (dashed line) and Eq. (2) (dot-dashed line). The theoretical curves for the helicity-2 TFF are normalized to 1 at $Q^2=0$.

Figure 35

The measured helicity-0 TFF of the $f_2(1270)$ as a function of $Q^2$. Short (long) vertical bars indicate statistical (statistical and systematic combined) errors. The shaded area corresponds to the overall systematic uncertainty arising from that of ${\mathrm{\Gamma }}_{\gamma \gamma }$. Correlations of uncertainties between neighboring bins exist and are included in the long vertical bars. The solid line shows the predicted $Q^2$ dependence in Table 1 by Ref. [4].

Figure 36

The measured helicity-1 TFF of the $f_2(1270)$ as a function of $Q^2$. Short (long) vertical bars indicate statistical (statistical and systematic combined) errors. The shaded area corresponds to the overall systematic uncertainty arising from that of ${\mathrm{\Gamma }}_{\gamma \gamma }$. Correlations of uncertainties between neighboring bins exist and are included in the long vertical bars. Note that the helicity-1 TFF is calculated from $\sqrt{r_1(Q^2)}{F}_{f2}(Q^2)$, where $r_1(Q^2)$ is given by Eq. (43). The solid line shows the predicted $Q^2$ dependence in Table 1 by Ref. [4].

Figure 37

The measured $Q^2$ dependence of the TFF of the $f_0(980)$. Short (long) vertical bars indicate statistical (statistical and systematic combined) errors. The shaded area corresponds to the overall systematic uncertainty arising from that of ${\mathrm{\Gamma }}_{\gamma \gamma }$. Correlations of uncertainties between neighboring bins exist and are included in the long vertical bars. The solid curve is the prediction for a scalar particle given in Table 1 by Ref. [4].