$pp\to pp{\eta }^{\prime }$ reaction at high energies

We discuss double-diffractive (double-elastic) production of the ${\eta }^{\prime }$-meson in the $pp\to p{\eta }^{\prime }p$ reaction within the formalism of unintegrated gluon distribution functions (UGDF). We estimate also the contribution of ${\gamma }^{*}{\gamma }^{*}\to {\eta }^{\prime }$ fusion. The distributions in the Feynman $x_F$ (or rapidity), transferred four-momenta squared between initial and final protons $(t_1,t_2)$ and azimuthal angle difference between outgoing protons ($\Phi$) are calculated. The deviations from the ${sin}^2(\Phi )$ dependence predicted by one-step vector-vector-pseudoscalar coupling are quantified and discussed. The results are compared with the results of the WA102 collaboration at CERN. Most of the models of UGDF from the literature give a too small cross section as compared to the WA102 data and predict angular distribution in relative azimuthal angle strongly asymmetric with respect to $\pi /2$ in disagreement with the WA102 data. This points to a different mechanism at the WA102 energy. Predictions for RHIC, Tevatron and LHC are given. We find that the normalization, $t_{1,2}$ dependences as well as deviations from ${sin}^2(\Phi )$ of double-diffractive double-elastic cross section are extremely sensitive to the choice of UGDF. Possible implications for UGDFs are discussed.

Figure 1

The sketch of the bare QCD mechanism. The kinematical variables are shown in addition.

Figure 2

The sketch of the photon-photon fusion mechanism. Form factors appearing in different vertices are shown explicitly.

Figure 3

Kinematics of exclusive double-diffractive ${\eta }^{\prime }$-meson production.

Figure 4

Kinematics of exclusive ${\gamma }^{*}{\gamma }^{*}$ fusion mechanism of ${\eta }^{\prime }$-meson production.

Figure 5

Cross sections for different prescriptions for off-diagonal UGDFs. The solid line corresponds to our prescription and the dashed line corresponds to the KMR prescription $k_{1t}=k_{2t}=k_{0t}$. In this calculation we have taken KL distribution. No absorption corrections were included here.

Figure 6

Dependence of the cross section on the lower cut on gluon transverse momenta. Solid line corresponds to no cut case, dashed line to $k_{t,\min }^2=0.2{\mathrm{GeV}}^2$ and the dotted line to $k_{t,\min }^2=0.4{\mathrm{GeV}}^2$. No absorption correctons were included here.

Figure 7

Dependence of the cross section on factorization scales for the Gaussian off-diagonal UGDFs. The solid lines correspond to ${\mu }^2=m_{{\eta }^{\prime }}^2$ and the dashed lines correspond ${\mu }^2=m_{{\eta }^{\prime }}^2+P_{M,t}^2$. Three pairs of lines correspond to different choices of ${\mu }_0^2=m_{{\eta }^{\prime }}^2$ (a), $k_{0t}^2$($+\mathrm{\text{freezing}}$) (c), $Q_0^2$ (b) (from top to bottom). No absorption corrections were included here.

Figure 8

$d\sigma /d{x}_F$ as a function of Feynman $x_F$ for $W=29.1\mathrm{GeV}$ and for different UGDFs. The ${\gamma }^{*}{\gamma }^{*}$ fusion contribution is shown by the dash-dotted (red) line (second from the bottom). The experimental data of the WA102 collaboration [12] are shown for comparison. The dashed line corresponds to the KL distribution, dotted line to the GBW distribution and the dash-dotted to the BFKL distribution. The two solid lines correspond to the Gaussian distribution with (a1) (upper) and (b1) (lower) choices of scales. No absorption corrections were included here.

Figure 9

$d\sigma /d{t}_{1/2}$ as a function of Feynman $t_{1/2}$ for $W=29.1\mathrm{GeV}$ and for different UGDFs. The ${\gamma }^{*}{\gamma }^{*}$ fusion contribution is shown by the dash-dotted (red) steeply falling down line. The experimental data of the WA102 collaboration [12] are shown for comparison. The notation here is the same as in Fig. 8. No absorption corrections were included here.

Figure 10

$d\sigma /d\Phi$ as a function of $\Phi$ for $W=29.1\mathrm{GeV}$ and for different UGDFs. The ${\gamma }^{*}{\gamma }^{*}$ fusion contribution is shown by the dash-dotted (red) symmetric around 90° line. The experimental data of the WA102 collaboration [12] are shown for comparison. The notation here is the same as in Fig. 8. No absorption corrections were included here.

Figure 11

${\sigma }_{\mathrm{tot}}$ as a function of center-of-mass energy for different UGDFs. The ${\gamma }^{*}{\gamma }^{*}$ fusion contribution is shown by the dash-dotted (red) line. The world experimental data are shown for reference. The notation here is the same as in Fig. 8. No absorption corrections were included here.

Figure 12

Two-dimensional distribution in $t_1\times t_2$ for the diffractive QCD mechanism (left panel), calculated with the KL UGDF, and the ${\gamma }^{*}{\gamma }^{*}$ fusion (right panel) at the Tevatron energy $W=1960\mathrm{GeV}$. No absorption corrections were included here.

Figure 13

The cross section for $p\overline{p}\to p{\eta }^{\prime }\overline{p}$ as a function of $t$ ($t_1$ or $t_2$) and relative azimuthal angle $\Phi$ for the Tevatron energy $W=1960\mathrm{GeV}$. In this calculation the KL UGDF was used. No absorption corrections were included here.