Color transparency in ${{\bar{p}}} d \rightarrow \pi ^- \pi ^0 p$ reaction

Abstract

We consider exclusive two-pion production in antiproton-deuteron interactions at the beam momenta around 10 GeV/c in the kinematics with large momentum transfer in the underlying hard process ${{\bar{p}}} n \rightarrow \pi ^- \pi ^0$. The calculations are performed taking into account the antiproton and pion soft rescattering on the spectator proton in the framework of the generalized eikonal approximation. We focus on the color transparency effect that is modeled by introducing the dependence of rescattering amplitudes on the relative position of the struck and spectator nucleons along the momentum of a fast particle. As a consequence of the interplay between the impulse approximation and rescattering amplitudes the nuclear transparency ratio reveals a pretty complicated behaviour as a function of the transverse momentum of the spectator proton and the relative azimuthal angle between the $\pi ^-$-meson and the proton. Color transparency significantly suppresses rescattering amplitudes which leads to substantial modifications of the nuclear transparency ratio moving it closer to the value obtained in the impulse approximation. By performing the Monte-Carlo analysis we determine that this effect can be studied at PANDA with a reasonable statistics.

Appendices

Appendix A: elementary amplitudes

1.1 ${{\bar{p}}} n \rightarrow \pi ^- \pi ^0$

For the ${{\bar{N}}} N \rightarrow \pi \pi$ annihilation amplitude we apply the nucleon and $\Delta$ exchange model in a version described in Appendix C of Ref. [34]. The powers of the vertex form factors are chosen from the condition of the $s \rightarrow \infty ,~t/s=\text{ const }$ asymptotic scaling law [24, 25]

$$\begin{aligned} \frac{d\sigma }{dt} = \frac{f(t/s)}{s^n},~~~n=\sum n_i - 2, \end{aligned}$$

(A1)

with $n_i$ being the number of quarks in each incoming and outgoing hadron. This leads to the $\pi N N$ and $\pi N \Delta$ vertex form factors having powers of 2 and 5/2, respectively. The cutoff parameters $\Lambda _{\pi NN}=2.0$ GeV and $\Lambda _{\pi N\Delta }=1.8$ GeV are chosen to reproduce the shape of the t-dependence of the differential cross section ${{\bar{p}}} p \rightarrow \pi ^- \pi ^+$ at $p_\mathrm{lab}=5$ GeV/c (see Fig. 14 in ref. [34]). The absolute value of that cross section at $\Theta _\mathrm{c.m.}=90^\circ$ is accounted for by multiplying the invariant amplitude by the factor $\sqrt{\Omega }$ with $\Omega =0.008$ motivated by significant absorption in the incoming ${{\bar{N}}} N$ channel. It is clear that such a description of the elementary ${{\bar{N}}} N \rightarrow \pi \pi$ amplitude is pretty simple. However, we believe that it is good enough for our purposes to address reactions at $p_\mathrm{lab} \sim$ 5–15 GeV/c. Figure 10 shows the t-dependence of the ${{\bar{p}}} n \rightarrow \pi ^- \pi ^0$ differential cross section $d\sigma /dt$ at $p_\mathrm{lab}=5$ GeV/c (a) and the s-dependence of the same cross section at $\Theta _\mathrm{c.m.}=90^\circ$ (b). The nucleon exchange contributions dominate at $\Theta _\mathrm{c.m.}=90^{\circ }$ while the $\Delta$ exchanges are important at forward and backward scattering angles. The local minimum at $s=6.77$ GeV$^2$ ($p_\mathrm{lab}=2.5$ GeV/c) is due to destructive interference of n and p exchanges. In the case if only neutron exchange is possible (${{\bar{p}}} p \rightarrow \pi ^- \pi ^+$) or the neutron and proton exchanges interfere constructively (${{\bar{p}}} p \rightarrow \pi ^0 \pi ^0$) the s-dependence is smooth.

Fig. 10

Differential cross section ${{\bar{p}}} n \rightarrow \pi ^- \pi ^0$ as a function of $-t$ at $p_\mathrm{lab}=5$ GeV/c (a) and as a function of s at $\Theta _\mathrm{c.m.}=90^\circ$ (b). In panel (a), the solid, dashed, dotted, dash-dotted and dash-double-dotted lines correspond to full cross section and to the partial contributions of the neutron, proton, $\Delta ^0$ and $\Delta ^+$ exchanges. In panel (b) the solid line shows the calculated full cross section while the dashed line – the power law fit of Eq. (A1) with $n=8$

1.2 ${{\bar{p}}} p \rightarrow {{\bar{p}}} p$

The antiproton-proton elastic scattering amplitude at high energies and forward scattering angles can be conveniently parameterized by the expression

$$\begin{aligned} M_{{{\bar{p}}} p}(t)= 2 i p_\mathrm{lab} m_N \sigma _{{{\bar{p}}} p}^{\mathrm{tot}} (1-i\rho _{{{\bar{p}}} p}) \text{ e }^{B_{{{\bar{p}}} p}t/2}, \end{aligned}$$

(A2)

where the $p_\mathrm{lab}$-dependent parameterizations $\sigma _{{{\bar{p}}} p}^{\mathrm{tot}}$ and $B_{{{\bar{p}}} p}$ are described in ref. [51]. While the total cross section and the slope of momentum transfer dependence are well constrained by experiment, the data on $\rho _{{{\bar{p}}} p}=\text{ Re }M_{{{\bar{p}}} p}(0)/\text{ Im }M_{{{\bar{p}}} p}(0)$ at $p_\mathrm{lab}=5-15$ GeV/c are quite scarce and thus we rely here on the extrapolation of the Regge–Gribov fit [52] towards low beam momenta. Fortunately, the sensitivity of our results to $\rho _{\bar{p} p}$ is quite weak.

1.3 $\pi N \rightarrow \pi N$ elastic

Elastic scattering of charged pions on protons is thoroughly studied and the elastic amplitude can be thus parameterized in a usual way:

$$\begin{aligned} M_{\pi ^\pm p}(t)= 2 i p_\mathrm{lab} m_N \sigma _{\pi ^\pm p}^{\mathrm{tot}} (1-i\rho _{\pi ^\pm p}) \text{ e }^{B_{\pi ^\pm p}t/2}. \end{aligned}$$

(A3)

The $p_\mathrm{lab}$-dependent total $\pi ^\pm p$ cross sections are well described by the CERN-HERA fit [47]. The ratios $\rho _{\pi ^\pm p}=\text{ Re }M_{\pi ^\pm p}(0)/\text{ Im }M_{\pi ^\pm p}(0)$ are taken from the Regge–Gribov fit [52]. The parameterizations of the slope parameters $B_{\pi ^\pm p}$ at $|t|=0.2$ GeV$^2$ are provided in ref. [53]. The amplitude of the $\pi ^0 p$ elastic scattering can be calculated from the isospin relation:

$$\begin{aligned} M_{\pi ^0 p}(t)=\frac{1}{2}(M_{\pi ^- p}(t)+M_{\pi ^+ p}(t))~. \end{aligned}$$

(A4)

Appendix B: the differential cross section in the LC variables

The purpose of this Appendix is to derive Eq. (32). Let us consider the frame where the four-momentum of the ${{\bar{p}}} + d$ system is ${{{\mathcal {P}}}} = p_{{{\bar{p}}}} + p_d = ({{{\mathcal {P}}}}^0,\varvec{0},P)$ with $P \rightarrow +\infty$. In that frame the four-momenta of the pions are $k_i=(\omega _i,\varvec{k}_{it},{\tilde{\alpha }}_i P),~i=1,2$ and the four-momentum of the spectator is $p_s=(E_s,\varvec{p}_{st},{\tilde{\alpha }}_s P)$. The particle energies can be written as

$$\begin{aligned} \omega _i= & {} {\tilde{\alpha }}_i P + \frac{m_{it}^2}{2{\tilde{\alpha }}_i P},~~~m_{it}^2=m_\pi ^2+\varvec{k}_{it}^2, \end{aligned}$$

(B1)

$$\begin{aligned} E_s= & {} {\tilde{\alpha }}_s P + \frac{m_{st}^2}{2{\tilde{\alpha }}_s P},~~~m_{st}^2=m_N^2+\varvec{p}_{st}^2, \end{aligned}$$

(B2)

$$\begin{aligned} {{{\mathcal {P}}}}^0= & {} P + \frac{{{{\mathcal {P}}}}^2}{2P}. \end{aligned}$$

(B3)

After simple algebra the invariant phase space volume element (28) can be rewritten in terms of the LC variables ${\tilde{\alpha }}_i$, $i=1,2,s$ and the transverse momenta as follows:

$$\begin{aligned} d\Phi _3= & {} = 2\delta ({{{\mathcal {P}}}}^2 - \frac{m_{1t}^2}{{\tilde{\alpha }}_1} - \frac{m_{2t}^2}{{\tilde{\alpha }}_2} - \frac{m_{st}^2}{{\tilde{\alpha }}_s}) \nonumber \\&\times \delta ^{(2)}(\varvec{k}_{1t}+\varvec{k}_{2t}+\varvec{p}_{st}) \delta (1-{\tilde{\alpha }}_1-{\tilde{\alpha }}_2-{\tilde{\alpha }}_s)\nonumber \\&\times \frac{d^2k_{1t}d{\tilde{\alpha }}_1}{(2\pi )^32{\tilde{\alpha }}_1} \frac{d^2k_{2t}d{\tilde{\alpha }}_2}{(2\pi )^32{\tilde{\alpha }}_2} \frac{d^2p_{st}d{\tilde{\alpha }}_s}{(2\pi )^32{\tilde{\alpha }}_s}. \end{aligned}$$

(B4)

After successive integrations over $d^2k_{2t}d{\tilde{\alpha }}_2$ and $dk_{1t}$ the phase space volume (B4) becomes:

$$\begin{aligned} d\Phi _3 = \frac{2}{|\partial {{{\mathcal {F}}}}/\partial k_{1t}|} \frac{k_{1t}d\phi _1d{\tilde{\alpha }}_1}{(2\pi )^32{\tilde{\alpha }}_1} \frac{1}{(2\pi )^32{\tilde{\alpha }}_2} \frac{p_{st}dp_{st}d\phi _s d{\tilde{\alpha }}_s}{(2\pi )^32{\tilde{\alpha }}_s},\nonumber \\ \end{aligned}$$

(B5)

where

$$\begin{aligned} {{{\mathcal {F}}}} = {{{\mathcal {P}}}}^2 - \frac{m_{1t}^2}{{\tilde{\alpha }}_1} - \frac{m_{2t}^2}{{\tilde{\alpha }}_2} - \frac{m_{st}^2}{{\tilde{\alpha }}_s} \end{aligned}$$

(B6)

and, therefore,

$$\begin{aligned} \frac{\partial {{{\mathcal {F}}}}}{\partial k_{1t}}= & {} -\frac{\partial }{\partial k_{1t}}\left( \frac{m_{1t}^2}{{\tilde{\alpha }}_1} + \frac{m_{2t}^2}{{\tilde{\alpha }}_2}\right) \nonumber \\= & {} - 2(k_{1t}/{\tilde{\alpha }}_1+(k_{1t}+p_{st}\cos \phi )/{\tilde{\alpha }}_2)~. \end{aligned}$$

(B7)

For the non-polarized particles the differential cross section is invariant with respect to rotations about beam axis. Thus, we can integrate the cross section over $d\phi _s$ keeping the relative azimuthal angle $\phi$ of Eq. (33) fixed. This leads to the following expression for the four-differential cross section:

$$\begin{aligned} {\tilde{\alpha }}_s {\tilde{\alpha }}_1 \frac{d^4\sigma }{d{\tilde{\alpha }}_s\, d{\tilde{\alpha }}_1\, d\phi \, p_{s t} dp_{s t}} = \frac{\overline{|M|^2} k_{1t}}{16(2\pi )^4 p_\mathrm{lab} m_d |\partial {{{\mathcal {F}}}}/\partial k_{1t}| {\tilde{\alpha }}_2}.\nonumber \\ \end{aligned}$$

(B8)

By using the relations between the LC variables

$$\begin{aligned} \left| \frac{d{\tilde{\alpha }}_1}{{\tilde{\alpha }}_1}\right|= & {} \left| \frac{d\beta }{\beta }\right| = \left| \frac{ dk_1^z}{\omega _1}\right| ~, \end{aligned}$$

(B9)

$$\begin{aligned} \left| \frac{d{\tilde{\alpha }}_s}{{\tilde{\alpha }}_s}\right|= & {} \left| \frac{d\alpha _s}{\alpha _s}\right| = \left| \frac{dp_s^z}{E_s}\right| \end{aligned}$$

(B10)

we finally obtain Eq. (32) where $\kappa _t=|\partial {{{\mathcal {F}}}}/\partial k_{1t}| {\tilde{\alpha }}_2$.