Theoretical study of the ${\mathrm{\Delta }}^{++}-{\mathrm{\Delta }}^{-}$ configuration in the deuteron using an antiproton beam

We study the manifestation of the ${\mathrm{\Delta }}^{++}-{\mathrm{\Delta }}^{-}$ component of the deuteron wave function in the exclusive reaction $\overline{p}d\to {\pi }^{-}{\pi }^{-}{\mathrm{\Delta }}^{++}$. Due to the large binding energy the internal motion in the $\mathrm{\Delta }-\mathrm{\Delta }$ system is relativistic. We take this into account within the light-cone (LC) wave function formalism and, indeed, found large differences between calculations based on the LC and nonrelativistic (NR) wave functions. We demonstrate that the consistent LC treatment of the $\mathrm{\Delta }-\mathrm{\Delta }$ system plays the key role in the separation of the signal and background. Within the LC approach, the characteristic shape of the momentum distribution of the $\mathrm{\Delta }-\mathrm{\Delta }$ bound system predicted by the meson-exchange model is well visible on the background of usual annihilations at beam momenta between 10 and 15 GeV/$c$.

Figure 1

Impulse approximation graph showing the production of a pion pair in antiproton annihilation on one of the $\mathrm{\Delta }$'s of a $\mathrm{\Delta }-\mathrm{\Delta }$ configuration in the deuteron.

Figure 2

The wave functions of the $\mathrm{\Delta }-\mathrm{\Delta }$ system in (a) momentum and (b) configuration space.

Figure 3

Momentum distribution of the struck $\mathrm{\Delta }$ in the $\mathrm{\Delta }-\mathrm{\Delta }$ c.m. frame. Solid line: CCF model, Eq. (45). Long-dashed line: large-distance asymptotic solution, Eq. (48), obtained by setting $m_{\mathrm{\Delta }}=m_{{\mathrm{\Delta }}_R}=1.232$ GeV. Both lines are multiplied by an extra factor of 1/2, which is the isospin fraction of the ${\mathrm{\Delta }}^{++}-{\mathrm{\Delta }}^{-}$ component. The ordinary deuteron c.m. momentum distribution multiplied by a factor of ${10}^{-6}$ is shown by the short-dashed line.

Figure 4

The background processes due to inelastic CEX of the neutral pion on the proton.

Figure 5

The background process initiated by antiproton annihilation on the neutron into three pions.

Figure 6

Differential cross section of the reaction $\overline{p}d\to {\pi }^{-}{\pi }^{-}{\mathrm{\Delta }}^{++}$ at $p_{\mathrm{lab}}=10$ GeV/$c$ as a function of the Mandelstam $t$, which is defined as the four-momentum transfer squared between one of the outgoing pions (first pion) and the antiproton [Eq. (28)]. The solid, dashed, dotted, and dot-dashed lines correspond to the LC signal, NR signal, CEX background, and $3\pi$ background, respectively. Different panels display the results for the different values of the momentum $p_{\mathrm{\Delta }}$ and the polar angle ${\mathrm{\Theta }}_{\mathrm{\Delta }}$ of the $\mathrm{\Delta }$ isobar and of the relative azimuthal angle ${\varphi }_{\pi \mathrm{\Delta }}={\varphi }_{\pi }-{\varphi }_{\mathrm{\Delta }}$ between first pion and $\mathrm{\Delta }$. All quantities refer to the deuteron rest frame. The residual ${\mathrm{\Delta }}^{++}$ is assumed to be on the mass shell.

Figure 7

Differential cross section of the reaction $\overline{p}d\to {\pi }^{-}{\pi }^{-}{\mathrm{\Delta }}^{++}$ at $p_{\mathrm{lab}}=10$ GeV/$c$ as a function of the LC momentum fraction $\alpha$ of the residual $\mathrm{\Delta }$ [Eq. (31)]. The $\mathrm{\Delta }$ transverse momentum is fixed to zero. The LC momentum fraction of the first pion (a) $\beta =0.5$, (b) $\beta =1$, and (c) $\beta =1.5$. Notations for lines are the same as in Fig. 6.

Figure 8

Same as in Fig. 7 but for $\mathrm{\Delta }$ transverse momentum of 0.4 GeV/$c$. The polar angle between the first pion and the residual $\mathrm{\Delta }$ is ${\varphi }_{\pi \mathrm{\Delta }}={180}^{\circ }$. The range of $\alpha$ within which the signal calculations are shown is restricted by the kinematic region where the struck $\mathrm{\Delta }$ is timelike.

Figure 9

Differential cross section of the reaction $\overline{p}d\to {\pi }^{-}{\pi }^{-}{\mathrm{\Delta }}^{++}$ vs $\alpha$ of the residual $\mathrm{\Delta }$ for (a) $p_{\mathrm{lab}}=10$ GeV/$c$, (b) 12 GeV/$c$, and (c) 15 GeV/$c$. Calculations are done with $p_{\mathrm{\Delta }t}=0,\beta =1.5$. The dot-dot-dashed (green) line shows the LC signal calculated with the large-distance asymptotic form of the $\mathrm{\Delta }-\mathrm{\Delta }$ wave function, Eq. (48). The notations for other lines are the same as in Fig. 6.

Figure 10

The three-pion annihilation background cross section as a function of $\alpha$ of the residual $\mathrm{\Delta }$ for $p_{\mathrm{lab}}=15$ GeV/$c$ at the kinematic condition $p_{\mathrm{\Delta }t}=0, \beta =1.5$. The curves show the calculations with different choice of the artificial pion width as indicated.

Figure 11

Histograms: probability distributions of the residual ${\mathrm{\Delta }}^{++}$ in $\alpha$ at $p_{\mathrm{lab}}=15$ GeV/$c$ in the kinematics with $p_{\mathrm{\Delta }}^2=m_{\mathrm{\Delta }}^2$; $p_{\mathrm{\Delta }t}<0.1$ GeV/$c$; $\beta =1.4-1.6$; $p_1^z,p_2^z>1$ GeV/$c$. Smooth lines: the analytical results of Fig. 9 multiplied by constant factors for appropriate normalization.

Figure 12

Lowest-order contribution to the absorption amplitude of a photon on the deuteron. $q, p_d$ and $p_d^{\prime }$ are the four-momenta of the photon, initial, and final deuteron, respectively. $p_1, p_2$ and $p_1^{\prime }$ are the four-momenta of the intermediate $\mathrm{\Delta }$'s. Time axis is from left to right.

Figure 13

Feynman graphs included in the calculation of the ${\overline{N}}_2{N}_1\to {\pi }_2{\pi }_1$ amplitude. (a) and (b) contain the exchange of a nucleon. (c) and (d) contain the exchange of a $\mathrm{\Delta }$ isobar. The four-momenta of the exchange particle are denoted as $q$ and $q^{\prime }$, respectively, in the $t$-channel (a), (c) and $u$-channel (b), (d) graphs, where $t=q^2, u=q^{\prime 2}$.

Figure 14

Differential cross section of the $\overline{p}p\to {\pi }^{-}{\pi }^{+}$ process at $p_{\mathrm{lab}}=5$ GeV/$c$. Solid line: full calculation. Dashed, dotted, and dash-dotted lines display the separate contributions of neutron, ${\mathrm{\Delta }}^0$ and ${\mathrm{\Delta }}^{++}$ exchange, respectively. Experimental data are from Ref. [29].

Figure 15

Solid line: differential cross section of the $\overline{p}p\to {\pi }^{-}{\pi }^{+}$ process calculated from Eq. (C12) at ${\mathrm{\Theta }}_{\mathrm{c}.\mathrm{m}.}={90}^{\circ }$ as a function of invariant $s$. Dashed line: large $s$ asymptote $\propto s^{-8}$.

Figure 16

Same as Fig. 13, but for the ${\overline{N}}_2{\mathrm{\Delta }}_1\to {\pi }_2{\pi }_1$ amplitude.

Figure 17

Differential cross section of the $\overline{p}{\mathrm{\Delta }}^{-}\to {\pi }^{-}{\pi }^{-}$ process at $p_{\mathrm{lab}}=5$ GeV/$c$. Solid line: full calculation. Dashed and dotted lines display the separate contributions of neutron and ${\mathrm{\Delta }}^0$ exchange, respectively.

Figure 18

Differential cross section of the $\overline{p}{\mathrm{\Delta }}^{-}\to {\pi }^{-}{\pi }^{-}$ process at ${\mathrm{\Theta }}_{\mathrm{c}.\mathrm{m}.}={90}^{\circ }$ (solid line) and ${60}^{\circ }$ (dotted line). The large-$s$ asymptotic behavior $\propto s^{-8}$ is shown for both cases by the dashed lines.

Figure 19

Feynman graph of the inelastic pion CEX amplitude $\pi N\to {\pi }^{\prime }\mathrm{\Delta }$ due to $t$-channel $\rho$-meson exchange.

Figure 20

Differential cross section of the ${\pi }^{+}p\to {\pi }^0{\mathrm{\Delta }}^{++}$ CEX process. Different curves represent calculations with different parameters of the $\rho$ meson Regge trajectory as indicated. Experimental data at 3.6 GeV/$c$, 5.45 GeV/$c$, and 16 GeV/$c$ are from Refs. [38, 39], and [40] respectively.

Figure 21

Fit to the experimental data of Ref. [41] on the $\overline{p}n\to {\pi }^{-}{\pi }^{-}{\pi }^{+}$ cross section by the exponential function given in Eq. (C30).