Eikonal fit to $pp$ and $\overline{p}p$ scattering and the edge in the scattering amplitude

We make a detailed eikonal fit to current data on the total and elastic scattering cross sections, the ratios $\rho$ of the real to the imaginary parts of the forward elastic scattering amplitudes, and the logarithmic slopes $B$ of the differential cross sections $d\sigma /dt$ at $t=0$, for proton-proton and antiproton-proton scattering at center-of-mass energies $W$ from 5 GeV to 57 TeV. The fit allows us to investigate the structure of the eikonal amplitudes in detail, including the impact-parameter structure of the energy-independent edge in the scattering amplitude shown to exist by Block et al., [Phys. Rev. D 91, 011501(R) (2015)]. We show that the edge region has an essentially fixed shape with a peak at approximately the “black disk” radius $R_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{tot}}/2\pi }$ of the scattering amplitude, a constant width $t_{\text{edge}}\approx 1\mathrm{fm}$, and migrates to larger impact parameters with increasing energy proportionally to $R_{\mathrm{tot}}$. We comment on possible physical mechanisms which could lead to the edge. We show that the eikonal results for the cross sections and $\rho$ values are described to high accuracy by analytic expressions of the forms used in earlier analyses by Block and Halzen, and extend the result to the elastic-scattering slope parameter $B$. These expressions provide simple extrapolations of the results to much higher energies where the cross sections approach the black disk limit with ${\sigma }_{\text{elas}}$, ${\sigma }_{\text{inel}}\to {\sigma }_{\mathrm{tot}}/2$ and $B\to {\sigma }_{\mathrm{tot}}/8\pi$. Finally, we calculate the survival probabilities for large rapidity gaps in the scattering.

Figure 1

Top panel: Fits to ${\sigma }_{\mathrm{tot},\mathrm{pp}}$ (blue dots and solid line) and ${\sigma }_{\mathrm{tot},\overline{\mathrm{p}}\mathrm{p}}$ (red squares and dashed line). Only data above 5 GeV were used in the final fit, with the cross sections constrained to fit compilations of low-energy data at 4 GeV [9]. Bottom panel: Fits to ${\sigma }_{\text{elas},\mathrm{pp}}$ (blue dots and solid line) and ${\sigma }_{\text{elas},\overline{\mathrm{p}}\mathrm{p}}$ (red squares and dashed line). The fit used only data above 10 GeV.

Figure 2

Top panel: Fits to the ratios $\rho$ of the real to the imaginary parts of the forward scattering amplitudes for $pp$ (blue dots and solid line) and $\overline{p}p$ (red squares and dashed line) scattering. The horizontal dashed line is at $\rho =0$. Bottom panel: Fits to the logarithmic slopes of the elastic differential scattering cross sections $d\sigma /dt$ for $pp$ (blue dots and solid line) and $\overline{p}p$ (red squares and dashed line) scattering.

Figure 3

Top: The differential cross section $d\sigma /dt$ from the E710 experiment [20, 21] at $W=1800\mathrm{GeV}$. Bottom: $d\sigma /dt$ from the TOTEM experiment [22] at $W=7000\mathrm{GeV}$.

Figure 4

Plot of the integrand $b(1-\cos {\chi }_R{e}^{-{\chi }_I})J_0(b\sqrt{-t})$ for the imaginary part of $f(s,t)$ versus the impact parameter $b$ for $W=1\mathrm{TeV}$ and $|t|=0.5{\mathrm{GeV}}^2$ (solid blue curve) and $1{\mathrm{GeV}}^2$ (dashed purple curve).

Figure 5

Top: Comparison of the imaginary parts of the different energy-dependent factors in the eikonal function: ${\sigma }_{qq}$ (solid red curve); ${\sigma }_{qg}$ (dot-dashed blue curve barely visible near zero amplitude); ${\sigma }_{gg}$ (long-dashed purple curve); dotted (black) curve, the odd term. Bottom: Comparison of the cross sections calculated with (solid blue curve) and without (dashed red curve) the inclusion of the gluon-gluon ($gg$) term in the eikonal function. The cross section for pure gluon scattering is shown as the long dashed purple curve.

Figure 6

Plots of the eikonal factors (left-hand column) and those factors multiplied by the geometric factor $b$ in the integrands for the $pp$ total cross section (top row), inelastic cross section (middle row), and elastic cross section (bottom row). In the labels for the ordinate, $\eta =e^{-{\chi }_I}$ and $c_R=\cos {\chi }_R$. The curves in each panel correspond, bottom to top, to energies $W=50\mathrm{GeV}$ (red curve), 500 GeV (brown curve), 5 TeV (blue curve), 50 TeV (purple curve) and 1000 TeV (black curve).

Figure 7

Comparisons of the integrands $b(1-\cos {\chi }_R{e}^{-{\chi }_I})$ for ${\sigma }_{\mathrm{tot}}$ (solid red curve), the “black disk” integrands for disk radius $R_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{tot}}/2\pi }$ (long-dashed black curve), and the edge integrands $b(\cos {\chi }_R-e^{-{\chi }_I})e^{-{\chi }_I}$ (short-dashed blue curves) at energies $W=1,5$, and 50 TeV, left to right. The vertical and horizontal scales give the integrand and $b$ in fm.

Figure 8

Plots of the $pp$ edge width $t_{\text{edge}}$ calculated using the present eikonal fit to the $pp$ and $\overline{p}p$ data (solid blue curve) and the black disk radius $R_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{tot}}/2\pi }$ (dashed red curve) as functions of the center-of-mass energy $W$.

Figure 9

Plots of the normalized edge integrands $b(\cos {\chi }_R-e^{-{\chi }_I})e^{-{\chi }_I}/R_{\mathrm{tot}}$ for, left to right, $W=30\mathrm{GeV}$ (red), 500 GeV (green), 5000 GeV (brown), $5\times {10}^4\mathrm{GeV}$ (blue), ${10}^6\mathrm{GeV}$ (purple), and ${10}^8\mathrm{GeV}$ (black).

Figure 10

The curve gives the upper bound on the inclusive cross section for single-particle diffractive dissociation $\overline{p}+p\to \overline{p}+X$ calculated using ${\sigma }_{\mathrm{SD}}\le ({\sigma }_{\mathrm{tot}}-2{\sigma }_{\text{elas}})/2$ and our eikonal fit to the $pp$ and $\overline{p}p$ cross section data from 5 GeV to 70 TeV. The data shown for ${\sigma }_{\mathrm{SD}}$ are from CDF [31].

Figure 11

The model differential dissociation cross sections calculated as the squares of the amplitudes $f_{\mathrm{SD}}$ in Eq. (28) for $W=546\mathrm{GeV}$ (black dashed curve), 1800 GeV (solid blue curve), and 7000 GeV (dot-dashed red curve). The results illustrate the $q^2$ and $W$ dependence expected for dissociation amplitudes which saturate the edge distribution in impact parameter space, but are not predictions for the actual $M_X$-dependent cross sections.