Measurement of azimuthal asymmetries in inclusive production of hadron pairs in $e^{+}{e}^{-}$ annihilation at $\sqrt{s}=10.58\mathrm{GeV}$

The Collins effect connects transverse quark spin with a measurable azimuthal asymmetry in the yield of hadronic fragments around the quark’s momentum vector. Using two different reconstruction methods we measure statistically significant azimuthal asymmetries for charged pion pairs in $e^{+}{e}^{-}$ annihilation at center-of-mass energies of 10.52 GeV and 10.58 GeV, which can be attributed to the fragmentation of primordial quarks with transverse spin components. The measurement was performed using a data set of $547{\mathrm{fb}}^{-1}$ collected by the Belle detector at KEKB improving the statistics of the previously published results by nearly a factor of 20.

Figure 1

Definition of the azimuthal angle $\varphi$ in the fragmentation of a transversely polarized quark ${\mathbf{S}}_q$ with momentum $\mathbf{k}$ into an unpolarized hadron with transverse momentum ${\mathbf{P}}_{h\perp }$.

Figure 2

Definition of the azimuthal angles ${\varphi }_1$ and ${\varphi }_2$ of the two hadrons, between the scattering plane and their transverse momenta ${\mathbf{P}}_{hi\perp }$ around the thrust axis $\widehat{n}$. The angle $\theta$ is defined as the angle between the lepton axis and the thrust axis.

Figure 3

Definition of the azimuthal angle ${\varphi }_0$ formed between the planes defined by the lepton momenta and that of one hadron and the second hadron’s transverse momentum $P_{h1\perp }^{\prime }$ relative to the first hadron.

Figure 4

Side view of the Belle detector.

Figure 5

Simulated thrust distributions for selected 2-pion pairs at $\sqrt{s}=10.58\mathrm{GeV}$, for $e^{+}{e}^{-}\to B^{+}{B}^{-}$ events (open diamonds), $e^{+}{e}^{-}\to B^0{\overline{B}}^0$ events (open circles), $e^{+}{e}^{-}\to c\overline{c}$ events (full triangles), and for light quark production $e^{+}{e}^{-}\to q\overline{q}$, $q\in uds$ (full squares) normalized to the total number of events in all channels. The vertical line represents the minimal thrust value selected for the analysis.

Figure 6

Angular deviation $\Delta \theta$ from the original quark-antiquark axis of the reconstructed thrust axis (solid line) and of the thrust axis calculated from generated events before detector simulation (dashed line). The distributions are based on a light quark MC simulation with a $T>0.8$ requirement for the generated events.

Figure 7

Hemisphere variable $W_{\mathrm{hemi}}$ compared to the product of energies of the two hadrons (top plot) and the transverse virtual photon momentum $Q_T$ (bottom plot) for pion pairs with $W_{\mathrm{hemi}}>-5{\mathrm{GeV}}^2$. The horizontal line corresponds to the maximum $Q_T$ value selected for the analysis.

Figure 8

Raw data. Top: Example of uncorrected unlike-sign (open circles) and like-sign (open squares) pion pair normalized rate $R_0$ vs $2{\varphi }_0$ in the bin $z_1(z_2)\in [0.5,0.7]$, $z_2(z_1)\in [0.3,0.5]$. Bottom: Pion pair double ratio $R_0^U/R_0^L$ vs $2{\varphi }_0$ in the same $z_1$, $z_2$ bin. The results of the fit described in the text (full and dashed lines) are also shown.

Figure 9

Raw asymmetry parameters $a_{\alpha }^U$ for unlike-sign pion pairs and $a_{\alpha }^L$ for like-sign pairs as a function of the transverse virtual photon momentum $Q_T$. Light quark MC generated (triangles), reconstructed (squares), and selected data sample (circles) for the $\alpha =0$, 12 reconstruction methods are shown.

Figure 10

Asymmetry parameters $A_0^{UL}$ (circles) and $A_{12}^{UL}$ (squares) as a function of $z_2$ for 4 $z_1$ bins obtained from MC double ratios. The results of fits to a constant are also displayed as dash-dotted lines. They are $(0.13\pm 0.09)\%$ for $A_0^{UL}$, $(0.08\pm 0.09)\%$ for $A_{12}^{UL}$, $(0.06\pm 0.04)\%$ for $A_0^{UC}$, and $(0.04\pm 0.04)\%$ for $A_{12}^{UC}$. The $UC$ data points are not shown.

Figure 11

Double ratio results $A_0^{UL}$ (circles) and $A_{12}^{UL}$ (squares) as a function of $z_2$ for 4 $z_1$ bins for reconstructed light quark MC, reweighted as described in the text. The generated asymmetry $0.1z_1{z}_2$ evaluated at the bin centers is also shown as a histogram. The dotted line corresponds to the zero asymmetry.

Figure 12

Angular distributions of $\cos {\theta }_{\mathrm{CMS}}$ (top plot) and $2{\varphi }_{\mathrm{LAB}}$ (bottom plot) for muon pairs.

Figure 13

Comparison of the $A^{UL}$ asymmetries for the published off-resonance data sample (triangles) and the on-resonance data sample (squares). The upper plot shows the $A_0$ asymmetries, the lower plot the $A_{12}$ asymmetries as a function of $z$.

Figure 14

${\chi }^2$ distribution per degree of freedom of the $A^{UL}$ (triangles) and $A^{UC}$ (circles) asymmetries as functions of the experiment number for the $A_0$ (top) and the $A_{12}$ (bottom) asymmetries. The open symbols represent the off-resonance data samples, the full symbols the on-resonance data samples.

Figure 15

${\chi }^2$ distribution per degree of freedom of the $A^{UL}$ (top plots) and $A^{UC}$ (bottom plots) asymmetries for the $A_0$ (left plots) and the $A_{12}$ method (right plots).

Figure 16

Squares of the contributions to the systematic error for the $A_{12}^{UL}$ asymmetries as a function of $z_1$ and $z_2$. The lowermost bars correspond to the contribution from the differences between the double ratio and subtraction reconstruction methods. The next bars correspond to the charged pion pair ratios, the next to the MC double ratios, then the contribution due to particle identification and the uppermost bar corresponds to the uncertainties in correcting the measured asymmetries for the quark-antiquark/thrust axis deviation.

Figure 17

Light quark ($uds$) $A_0$ asymmetry parameters as a function of $z_2$ for 4 $z_1$ bins. The $UL$ data are represented by triangles and the systematic error by the upper error band. The $UC$ data are described by the squares and their systematic uncertainty by the lower error band.

Figure 18

Light quark ($uds$) $A_{12}$ asymmetry parameters as a function of $z_2$ for 4 $z_1$ bins. The $UL$ data are represented by triangles and the systematic error by the upper error band. The $UC$ data are described by the squares and their systematic uncertainty by the lower error band.

Figure 19

Light quark ($uds$) $A_0^{UL}$ (top) and $A_{12}^{UL}$ (bottom) asymmetry parameters as a function of ${sin}^2\theta /(1+{cos}^2\theta )$, for ${\theta }_2$ (squares) and for ${\widehat{\mathbf{n}}}_z$ (triangles). Linear fits are also displayed as dashed and continuous lines, respectively. The systematic error for the ${\theta }_2$ case is represented by the lower, that for ${\widehat{\mathbf{n}}}_z$ by the upper error band.

Figure 20

Light quark ($uds$) $A_0^{UC}$ (top) and $A_{12}^{UC}$ (bottom) asymmetry parameters as a function of ${sin}^2\theta /(1+{cos}^2\theta )$, for ${\theta }_2$ (squares) and for ${\widehat{\mathbf{n}}}_z$ (triangles). Linear fit are also displayed as dashed and continuous lines, respectively. The systematic error for the ${\theta }_2$ case is represented by the lower, that for ${\widehat{\mathbf{n}}}_z$ by the upper error band.

Figure 21

Light quark ($uds$) $A_0^{UL}$ (top) and $A_{12}^{UL}$ (bottom) asymmetry parameters as a function of $Q_T$, for events with $T>0.8$ (triangles) and asymmetries for events with $T<0.8$ not corrected for heavy quark contributions (squares). The vertical line represents the main data selection $Q_T<3.5\mathrm{GeV}$.

Figure 22

Light quark ($uds$) $A_0^{UC}$ (top) and $A_{12}^{UC}$ (bottom) asymmetry parameters as a function of $Q_T$, for events with $T>0.8$ (triangles) and asymmetries for events with $T<0.8$ not corrected for heavy quark contributions (squares). The vertical line represents the main data selection $Q_T<3.5\mathrm{GeV}$.

Figure 23

Charm asymmetry parameters $A_0$ and $A_{12}$ as a function of the combined $z$. The $UL$ data are described by triangles, their systematic error being the top error band while the $UC$ data are described by the squares and their systematics by the lower error band.

Figure 24

${\chi }^2$ distribution as a function of the parameters $a$ and $b$ as specified in the text using the KKP parametrization of the unpolarized fragmentation functions for the $A_0$ data. The ${\chi }^2$ values above ${\chi }_{\max }^2≔10\times$ the minimum were set to 65% of ${\chi }_{\max }^2$ for better visibility. The minimum lies slightly below 100. The vertical and horizontal lines correspond to the parameters $a$ and $b$ of the lowest ${\chi }^2$.