Pion-pion cross section from proton-proton collisions at the LHC

The zero-degree calorimeters (ZDC) installed in the ALICE, ATLAS and CMS experiments at the LHC make possible simultaneous detection of forward-backward leading neutrons, $pp\to nXn$. Such data with sufficiently high statistics could be a source of information about the pion-pion total cross section at high energies, provided that the absorption corrections, which are expected to be strong, are well understood. Otherwise, making a plausible assumption about the magnitude of the pion-pion cross section, one can consider such measurements as a way to study the absorption effects, which is the main focus of the present paper. These effects introduced at the amplitude level are found to be different for the pion fluxes, which either conserve or flip the nucleon helicity. The pion fluxes from both colliding protons are essentially reduced by absorption; moreover, there is a common absorption suppression factor, which breaks down the factorized form of the cross section. We also evaluate the feed-down corrections related to the initial/final state inelastic processes possessing a rapidity gap, and found them to be small in the kinematic range under consideration. The contribution of other isovector Reggeons, spin-flip natural parity $\rho$ and $a_2$ and spin-nonflip unnatural parity $a_1$, are also evaluated and found to be rather small.

Figure 1

Graphical representation for double neutron production with large $z$ in $pp\to nXn$.

Figure 2

The forward flux of pions $f_{{\pi }^{+}/p}^{(0)}(z,q)$ at $q=0$, calculated in the Born approximation with Eq. (16) and including absorption, Eq. (29), plotted by dotted and dashed curves respectively, and by a solid curve after adding the feed-down corrections.

Figure 3

The $q$-integrated pion flux, calculated in the Born approximation, vs neutron fractional momentum $z$. Correspondingly the pion flux carries fraction $1-z$. The fluxes of pions flipping or nonflipping the nucleon helicity are depicted by dashed curves, while the solid curve presents the full pion flux.

Figure 4

The integrated flux of two pions at $z_{\min }=0.5$ calculated in the Born approximation, Eq. (8), and with absorption corrections, Eq. (52), depicted by dotted and dashed curves respectively. The solid curves, including also the feed-down corrections, are calculated with $z_{\min }=0.5–0.9$ (from right to left) as is marked on the plot.

Figure 5

Production of a color octet-octet dipole in $\pi \text{−}\pi$ color-exchange collision in reaction ${\pi }^{+}+p\to X+n$.

Figure 6

Partial survival amplitudes including (${\tilde{S}}_{4q}$) and excluding ($S_{4q}$) interactions with the radiated gluons, as function of $b$, at collision energy $\sqrt{s}=7\mathrm{TeV}$ and $z_1=z_2=0.7$.

Figure 7

The $q$-integrated pion flux $F_{{\pi }^{+}/p}(z)$ calculated with Eqs. (31)–(33), which include absorption. The dotted and dashed curves show the spin parts and the full flux respectively. The full flux enhanced by the feed-down corrections is depicted by solid curve.

Figure 8

The same as in Fig. 7, but integrated over $q$ up to $q_{\max }=0.1–0.4\mathrm{GeV}$, as is marked on the plot. The effects of absorption and the feed-down corrections are included.

Figure 9

Graphical illustration for the double-step correction to neutron production.

Figure 10

Correction to the eikonal approximation due to nucleon excitations in the intermediate state between the pion-exchange and absorption amplitudes.

Figure 11

Graphical representation for the absorption corrected cross section of reaction $pp\to nXn$. Dashed lines show the pions; the double line shows the Pomeron exchange. The flows of transverse momenta, $\overrightarrow{k}$, ${\overrightarrow{k}}^{\prime }$, ${\overrightarrow{q}}_1$ and ${\overrightarrow{q}}_2$, are depicted by red lines.

Figure 12

The triple-Regge graph presenting the cross section of the process $a+b\to c+X$. The Reggeons $i$, $j$ and $k$ can be either the Pomeron $\mathbb{P}$ or the leading Reggeon $\mathbb{R}$. The vertical dashed line shows the unitarity cut.

Figure 13

The $q$-integrated spin-flip meson flux $F_{M/p}^{(s)}(z)$, including pion (upper dashed) and $\rho$ and $a_2$ mesons (bottom dashed) contributions. The solid curve shows the full flux.

Figure 14

The $q$-integrated spin nonflip meson flux $F_{M/p}^{(0)}(z)$, including the pion (upper dashed) and the combined contribution of ${\tilde{a}}_1$ exchange and its interference with the pion (bottom dashed). The solid curve shows the full meson flux conserving helicity.