Inclusive production of charged pion pairs in proton-antiproton collisions

In this study, we have considered the contribution of the higher-twist (HT) effects of the subprocesses to inclusive pion pair production cross section in the high energy proton-antiproton collisions by using various pion distribution amplitudes (DAs) within the frozen coupling constant approach and compared them with the leading-twist contributions. The feature of the HT effects may help the theoretical interpretation of the future PANDA experiment. The dependencies of the HT contribution on the transverse momentum $p_T$, the center of mass energy $\sqrt{s}$, and the variable $x_T$ are discussed numerically with special emphasis put on DAs. Moreover, the obtained analytical and numerical results for the differential cross section of the pion pair production are compared with the elastic backward scattering of the pion on the proton. We show that the main contribution to the inclusive cross section comes from the HT direct production process via gluon-gluon fusion. Also, it is strongly dependent on the pion DAs, momentum cut-off parameter $▵p$, and $\langle q_T^2\rangle$ which is the mean square of the intrinsic momentum of either initial parton.

Figure 1

QCD Feynman diagrams of the partonic process $gg\to M\overline{M}$ and $q\overline{q}\to M\overline{M}$ for direct meson pair production at leading order.

Figure 2

QCD Feynman diagrams of the partonic process $q\overline{q}\to M\overline{M}$ for semidirect meson pair production at leading level.

Figure 3

HT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}$ as a function of the transverse momentum $p_T$ for momentum cut-off parameter $▵p=0.3$ GeV/$c$, at $\sqrt{s}=15\mathrm{GeV}$ and $y=0$.

Figure 4

The sum of HT and LT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}+\mathrm{LT}}$ as a function of the transverse momentum $p_T$ for momentum cut-off parameter $▵p=0.3\mathrm{GeV}/c$, at $\sqrt{s}=15\mathrm{GeV}$ and $y=0$. Notice that curves for asy, hol, VSBGL, CZ, and BMS pion DA in the region $2\mathrm{GeV}/c<p_T<7\mathrm{GeV}/c$ completely overlap.

Figure 5

Ratio ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}/{\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{LT}}$ as a function of the transverse momentum $p_T$ of the pion pair for $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$, at the c.m. energy $\sqrt{s}=15\mathrm{GeV}$ and $y=0$.

Figure 6

HT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}$ as a function of the transverse momentum $p_T$ for momentum cut-off parameter $▵p=0.3$ GeV/$c$, at $\sqrt{s}=20\mathrm{GeV}$ and $y=0$.

Figure 7

The sum of HT and LT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}+\mathrm{LT}}$ as a function of the transverse momentum $p_T$ for momentum cut-off parameter $▵p=0.3\mathrm{GeV}/c$, at $\sqrt{s}=20\mathrm{GeV}$ and $y=0$. Notice that curves for asy, hol, VSBGL, CZ, and BMS pion distribution amplitudes in the region $2.5\mathrm{GeV}/c<p_T<7\mathrm{GeV}/c$ completely overlap.

Figure 8

Ratio ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}/{\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{LT}}$ as a function of the transverse momentum $p_T$ of the pion pair at the $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$, at the c.m. energy $\sqrt{s}=20\mathrm{GeV}$ and $y=0$.

Figure 9

HT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}$ as a function of the center-of-mass energy $\sqrt{s}$ at the $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$ and $y=0$.

Figure 10

The sum of HT and LT contribution to charged pion pair production $p\overline{p}\to {\pi }^{+}{\pi }^{-}X$ cross section ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}+\mathrm{LT}}$ as a function of the center-of-mass energy $\sqrt{s}$ for momentum cut-off parameter $▵p=0.3\mathrm{GeV}/c$ and $y=0$. Notice that curves for asy, hol, VSBGL, CZ, and BMS pion distribution amplitudes completely overlap.

Figure 11

Ratio ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}/{\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{LT}}$ as a function of the center-of-mass energy $\sqrt{s}$ at $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$ and $y=0$.

Figure 12

Ratio HT cross sections ${\mathrm{\Sigma }}_{{\pi }^{+}{\pi }^{-}}^{\mathrm{HT}}$ calculated with $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$ and $\langle q_T^2\rangle =1{\mathrm{GeV}}^2/c^2$ as a function of the pion pair transverse momentum $p_T$ at $\sqrt{s}=15\mathrm{GeV}$ and $y=0$.

Figure 13

HT $p\overline{p}\to {\pi }^{+}{\pi }^{-}$ pion pair production and $\pi p\to \pi p$ cross sections as a function of the transverse momentum $p_T$ of the pion for $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$, at the c.m. energy $\sqrt{s}=15\mathrm{GeV}$ and $y=0$.

Figure 14

HT $p\overline{p}\to {\pi }^{+}{\pi }^{-}$ pion pair production and $\pi p\to \pi p$ cross sections as a function of the transverse momentum $p_T$ of the pion for $\langle q_T^2\rangle =0.25{\mathrm{GeV}}^2/c^2$, at the c.m. energy $\sqrt{s}=15\mathrm{GeV}$ and $y=0$.

Figure 15

HT $p\overline{p}\to {\pi }^{+}{\pi }^{-}$ pion pair production cross section as a function of the variable $x_T$ for momentum cut-off parameter $\mathrm{\Delta }p=0.5\mathrm{GeV}/c$ at $p_T=6\mathrm{GeV}/c$ and $y=0$.

Figure 16

Ratio of HT to LT contributions as a function of the variable $x_T$ for momentum cut-off parameter $\mathrm{\Delta }p=0.5\mathrm{GeV}/c$ at $p_T=6\mathrm{GeV}/c$ and $y=0$.