Inclusive cross sections for pairs of identified light charged hadrons and for single protons in $e^{+}{e}^{-}$ at $\sqrt{s}=10.58\mathrm{GeV}$

We report the first double differential cross sections of two charged pions and kaons ($e^{+}{e}^{-}\to hhX$) in electron-positron annihilation as a function of the fractional energies of the two hadrons for any charge and hadron combination. The dependence of these dihadron cross sections on the topology (same, opposite hemisphere or anywhere) is also studied with the help of the event shape variable thrust and its axis. The ratios of these dihadron cross sections for different charges and hadron combinations directly shed light on the contributing fragmentation functions. For example, we find that the ratio of same-sign pion pairs over opposite-sign pion pairs drops toward higher fractional energies where disfavored fragmentation is expected to be suppressed. These dihadron results are obtained from a $655{\mathrm{fb}}^{-1}$ data sample collected near the $\mathrm{\Upsilon }(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. Extending the previously published single-pion and single-kaon cross sections, single-proton ($e^{+}{e}^{-}\to pX$) cross sections are extracted from a $159{\mathrm{fb}}^{-1}$ data subsample.

Figure 1

Illustration of dihadron fragmentation where the final-state hadrons are depicted as red arrows, the incoming leptons as blue arrows, and the event plane—spanned by leptons and thrust axis—is depicted as a light blue plane. In this case, both hadrons are found in opposite hemispheres defined by the thrust axis, and generally out of the plane, as indicated by the cones.

Figure 2

Illustration of dihadron fragmentation where the final-state hadrons are depicted as red arrows, the incoming leptons as blue arrows and the event plane—spanned by leptons and initial quarks/thrust axis—is depicted as a light blue plane. In this case, both hadrons are found in the same hemisphere as defined by the thrust axis, and generally out of the plane, as indicated by the cones.

Figure 3

Ratio of yields after and before applying the PID correction for various hadron combinations in any topology. For brevity, only diagonal ($z_1=z_2$) entries in each two-dimensional matrix are shown. Empty bins are visible where the yields become zero, especially for high-$z$ bins.

Figure 4

Statistical (full lines), upper (dashed lines) and lower (red, dotted lines) systematic uncertainties relative to the PID-corrected dihadron yields in any topology as a function of ($z_1=z_2$); the diagonal bins of each two-dimensional matrix is shown for brevity. Empty bins are visible where the yields become zero, especially for high $z$ bins.

Figure 5

Ratio of yields after and before applying the smearing correction for various hadron combinations without hemisphere assignment. For brevity, only diagonal ($z_1=z_2$) entries in each two-dimensional matrix are shown.

Figure 6

Fraction of hadron pairs in any topology as a function of $(z_1,z_2)$ originating from various subprocesses. The individual relative contributions are displayed from top to bottom for $uds$ (red filled area), charm (blue, dotted area), mixed [$\mathrm{\Upsilon }(4S)\to B^0{\overline{B}}^0$, dark-green, hatched area] and charged [$\mathrm{\Upsilon }(4S)\to B^{+}{B}^{-}$, violet, horizontally hatched area], $\tau$ pair (light green, scaled area), $eeu\overline{u}$ (purple, starred area), $ees\overline{s}$ (light blue, dotted area) and $eec\overline{c}$ (orange hatched area) events. Also, for comparison, the continuum (green, solid lines) and on-resonance (orange, dotted lines) data are shown. For brevity, only diagonal ($z_1=z_2$) entries in each two-dimensional matrix are shown.

Figure 7

Yield ratios after successively applying all acceptance and efficiency corrections (labeled AccI to AccIII as the three acceptance corrections discussed in the text) relative to the reference yields before (non-$q\overline{q}$ removal) for various hadron combinations without hemisphere assignment. For brevity, only diagonal ($z_1=z_2$) entries in the two-dimensional matrices are shown.

Figure 8

Fraction of hadron pairs in the any topology as a function of $(z_1,z_2)$ originating from weak and strong decays. The individual relative contributions are displayed from top to bottom for strong $uds$ decays (purple, dark filled area), weak charm decays (blue dotted area), strong charm decays (dark-green, negative hatched area) and weak $uds$ decays (red, horizontal striped area). The strong decay fractions are also displayed as dashed magenta lines. Also the weak decay fractions for $udsc$ MC data using the default pythia settings are indicated by the dark-green, solid lines. For brevity, only diagonal ($z_1=z_2$) entries in each of the two-dimensional matrices are shown.

Figure 9

Relative fractions of hadron pairs in the any topology as a function of $(z_1,z_2)$ originating from ISR or non-ISR events. The individual relative contributions are displayed from top to bottom for non-ISR events (energy loss less than 0.5%, filled purple area) and ISR events (blue, dotted area) from the generated generic $udsc$ MC data. The non-ISR fraction is also shown (red, dashed line). The relative ISR fraction for the default pythia MC data is also shown (orange, solid lines). For brevity, only diagonal ($z_1=z_2$) entries in each of the two-dimensional matrices are shown.

Figure 10

Ratio of yields after various corrections relative to the raw yields for the main hadron combinations without hemisphere assignment. From top to bottom, the ratios after acceptance correction (purple, dash-dotted line), initial-state radiation (red, long dash-dotted line), PID correction (dark-green, full lines), smearing correction (blue, dashed line) and non-$q\overline{q}$ removal (magenta, dotted lines) are shown. For brevity, only diagonal ($z_1=z_2$) entries in each of the two-dimensional matrices are shown.

Figure 11

Relative (asymmetric) systematic (lower uncertainties: dashed lines; upper uncertainties: dash-dotted lines) and statistical uncertainties (full lines) for the most relevant hadron pairs in the any topology as a function of $(z_1,z_2)$. For brevity, only the diagonal bins ($z_1=z_2$) are shown.

Figure 12

Differential cross sections for ${\pi }^{+}{\pi }^{-}$ (black circles), ${\pi }^{+}{\pi }^{+}$ (blue squares), ${\pi }^{+}{K}^{-}$ (green triangles), ${\pi }^{+}{K}^{+}$ (purple diamonds), $K^{+}{K}^{-}$ (red crosses) and $K^{+}{K}^{+}$ (violet downward triangles) pairs in the any topology as a function of $z_2$ for the indicated $z_1$ bins. The error boxes represent the systematic uncertainties.

Figure 13

Differential cross sections for ${\pi }^{+}{\pi }^{-}$ (black circles), ${\pi }^{+}{\pi }^{+}$ (blue squares), ${\pi }^{+}{K}^{-}$ (green triangles), ${\pi }^{+}{K}^{+}$ (purple diamonds), $K^{+}{K}^{-}$ (red crosses) and $K^{+}{K}^{+}$ (violet downward triangles) pairs in the same topology (including $T>0.8$ selection) as a function of $z_2$ for the indicated $z_1$ bins. The error boxes represent the systematic uncertainties. The vertical, dashed green line represents the $z_1+z_2=1$ limit in each panel.

Figure 14

Ratios of differential cross sections for ${\pi }^{+}{\pi }^{+}$ (blue squares), ${\pi }^{+}{K}^{-}$ (green triangles), ${\pi }^{+}{K}^{+}$ (purple diamonds), $K^{+}{K}^{-}$ (red crosses) and $K^{+}{K}^{+}$ (violet downward triangles) pairs relative to the ${\pi }^{+}{\pi }^{-}$ pairs in the any topology as a function of $z_2$ for the indicated $z_1$ bins. The shaded areas correspond to the systematic uncertainties under the assumption that they are independent between each two dihadron combinations.

Figure 15

Differential cross sections for the main hadron pairs within opposite hemispheres (including thrust selection $T>0.8$) as a function of $(z_1,z_2)$ for diagonal bins only. The error boxes represent the systematic uncertainties.

Figure 16

Differential cross sections for the main hadron pairs within the same hemisphere (including thrust selection $T>0.8$) as a function of $(z_1,z_2)$ for diagonal bins only. The error boxes represent the systematic uncertainties.

Figure 17

Differential cross sections (top panels) and ratios to the same and opposite topology sum (bottom panels) for the main hadron combinations, stacking same (gray filled areas, including thrust selection $T>0.8$) and opposite (blue hatched areas, including thrust selection $T>0.8$) hemisphere data and comparing to those without hemisphere assignment (red curves). For visibility, only the diagonal $(z_1,z_2)$ bins and statistical uncertainties are displayed.

Figure 18

Differential cross sections (top panels) and ratios to the data (bottom panels) for the main hadron pairs in the any topology as a function of $(z_1,z_2)$ for diagonal bins only. Various pythia tunes are also displayed as described in the text. For comparison, the relative statistical and systematic uncertainties are shown for the data as well.

Figure 19

Differential cross sections for ${\pi }^{+}{K}^{-}$ pairs in the same hemisphere as a function of $(z_1,z_2)$. Various pythia tunes are also displayed as described in the text.

Figure 20

Ratio of yields after various corrections relative to the raw yields for the main single hadrons are shown as a function of $z$. From top to bottom, the ratios after acceptance correction (purple, dash-dotted line), initial-state radiation (red, long dash-dotted line), PID correction (dark-green, full lines), smearing correction (blue, dashed line) and non-$q\overline{q}$ removal (magenta, dotted lines) are shown.

Figure 21

Relative fractions for pions, kaons and protons as a function of $z_1$ originating from ISR or non-ISR events. The individual relative contributions are displayed from top to bottom for non-ISR events (energy loss less than 0.5%, purple, filled area) and ISR events (blue, hatched area) from generated generic $udsc$ MC data. The relative fractions are also shown for the non-ISR fraction (green, dashed lines).

Figure 22

Single-pion (black circles), -kaon (blue squares) and -proton (green triangles) cross sections from top to bottom, as a function of $z$.

Figure 23

Differential cross sections (top panels) and ratios to the data (bottom panels) for the main single hadrons as a function of $z$. Various MC tunes are also displayed as described in the text. For comparison, the relative statistical and systematic uncertainties are shown for the data as well.