Exclusive $pp\to pp{\pi }^0$ reaction at high energies

The amplitudes for the $pp\to pp{\pi }^0$ process applicable at high energy are discussed in detail. Both diffractive bremsstrahlung (Drell-Hiida-Deck-type model), photon-photon and photon-omega exchange mechanisms are included in the calculation. Large cross sections of the order of mb are predicted. The corresponding differential cross sections in rapidities and transverse momenta of outgoing protons and pions as well as relative azimuthal angle between outgoing protons are calculated for RHIC and LHC energies. The hadronic bremsstrahlung contributions dominate at large (forward, backward) pion rapidities. The diffractive nonresonant background contributes at small ${\pi }^{0}p$ invariant mass and could be therefore misinterpreted as the Roper resonance. We predict strong dependence of the slope in $t$ (squared four-momentum transfer between ingoing and outgoing proton) on the mass of the supplementary excited ${\pi }^{0}p$ system. At high energies and midrapidities, the photon-photon contribution dominates over the diffractive components, however, the corresponding cross section is rather small. The photon-odderon and odderon-photon contributions are included in addition and first estimates (upper limits) of their contributions to rapidity and transverse momentum distribution of neutral pions are presented. We suggest a search for the odderon contribution at midrapidity and at $p_{\perp ,{\pi }^0}\sim 0.5\mathrm{GeV}$. Our predictions are ready for verification at RHIC and LHC. The bremsstrahlung mechanisms discussed here contribute also to the $pp\to p(n{\pi }^{+})$ reaction. Both channels give a sizable contribution to the low-mass single diffractive cross section and must be included in extrapolating the measured experimental single diffractive cross section.

Figure 1

Diagrams of the ${\pi }^0$-bremsstrahlung amplitudes driven by the Pomeron exchange in proton-proton collisions: (a) pion exchange, (b) proton exchange, and (c) direct production. The direct-channel $p^{*}$ in (c) is an off-shell proton, not a proton resonance. Some kinematical variables are shown in addition.

Figure 2

Typical absorptive correction diagrams to: (a) pion exchange and (b) proton exchange. The stars attached to protons and ${\pi }^0$ meson denote the fact that they are off-mass shell.

Figure 3

A sketch of the photon-photon (a) and photon-omega meson (b) exchanges induced production of ${\pi }^0$ in proton-proton collisions.

Figure 4

Diagrams with the photon-odderon and odderon-photon exchanges in the $pp\to pp{\pi }^0$ reaction.

Figure 5

The distribution of ${\pi }^0$ in rapidity at $\sqrt{s}=45\mathrm{GeV}$ (ISR), 500 GeV (RHIC), and 14 TeV (LHC). The ${\pi }^0$-bremsstrahlung contribution (black solid lines) and $\omega \gamma$ ($\gamma \omega$) exchanges (violet dashed lines) peaks at forward (backward) region of $y_{{\pi }^0}$, respectively, while $\gamma \gamma$ fusion (blue dotted lines) contributes at midrapidity. In this calculation we have used ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ of the hadronic form factors. No absorption effects are included here.

Figure 6

The distribution of ${\pi }^0$ in rapidity for $\sqrt{s}=0.5$, 14 TeV. In panel (a) we show individual contributions to the Born cross section. A large cancellation between the initial ($p$-exchange) and final state radiation (direct production) can be observed. In panels (b) and (c) the upper solid line corresponds to calculations without absorption effects, the lower solid line with absorption effects. The solid lines are for ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ while the dashed lines are for ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$.

Figure 7

The average value of subsystem energies $⟨W_{13}⟩(y_{{\pi }^0})$ (solid line) and $⟨W_{23}⟩(y_{{\pi }^0})$ (dashed line) at $\sqrt{s}=0.5$, 14 TeV. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$.

Figure 8

The distribution in pseudorapidity of protons in the backward (left panel) and the forward (right panel) hemisphere at $\sqrt{s}=14\mathrm{TeV}$ and for $y_{{\pi }^0}>0$. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 9

The distribution of outgoing protons in transverse momentum for $\sqrt{s}=14\mathrm{TeV}$ and for $y_{{\pi }^0}>0$. As in the previous figure we show results without and with absorption effects. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 10

The distribution of ${\pi }^0$ mesons in transverse momentum for $\sqrt{s}=0.5$, 14 TeV and for $y_{{\pi }^0}>0$. In panel (a) we show individual contributions to the Born cross section. In the other panels as in the previous figures we show theoretical uncertainties (in, e.g., form factors). Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 11

Distribution in the four-momentum transfer squared between initial and final protons at $\sqrt{s}=14\mathrm{TeV}$ and for $y_{{\pi }^0}>0$. In panels (a) and (b) we show individual contributions to the Born cross section. The theoretical uncertainties are shown in panels (c) and (d). Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 12

Distribution in $(t_1,t_2)$ for the ${\pi }^0$-bremsstrahlung contribution at $\sqrt{s}=14\mathrm{TeV}$ (top panels) and $\sqrt{s}=500\mathrm{GeV}$ (bottom panels) and for $y_{{\pi }^0}>0$ without (left panels) and with (right panels) absorption effects. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$.

Figure 13

Energy spectrum of pions at $\sqrt{s}=0.5$, 14 TeV and for $y_{{\pi }^0}>0$. In panel (a) we show individual contributions to the Born cross section. Theoretical uncertainties are shown in panels (b) and (c). Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 14

Distribution in proton-pion invariant mass $M_{13}$ at $\sqrt{s}=0.5$, 14 TeV and for $y_{{\pi }^0}>0$. In panel (a) we show individual contributions to the Born cross section. Theoretical uncertainties are presented in panels (b) and (c). Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 15

The distribution in azimuthal angle between outgoing protons for $\sqrt{s}=0.5$, 14 TeV. In panel (a) we show individual contributions to the Born cross section. Theoretical uncertainties are presented in panels (b) and (c). Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ (solid line) or ${\Lambda }_N=0.6\mathrm{GeV}$ and ${\Lambda }_{\pi }=1\mathrm{GeV}$ (dashed line).

Figure 16

Distribution in $(t_2,M_{13})$ for the ${\pi }^0$-bremsstrahlung contribution at $\sqrt{s}=14\mathrm{TeV}$ and for $y_{{\pi }^0}>0$ without [panel (a)] and with absorption effects in the final state only [panels (b) and (c)] and absorption effects in the initial state only [panel (d)]. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$ [panels (a) and (b)] and ${\Lambda }_N=0.6\mathrm{GeV}$, ${\Lambda }_{\pi }=1\mathrm{GeV}$ [panels (c) and (d)].

Figure 17

Distribution in $(y_{{\pi }^0},p_{t,{\pi }^0})$ at $\sqrt{s}=14\mathrm{TeV}$ for the ${\pi }^0$-bremsstrahlung contribution and $y_{{\pi }^0}>0$ without (left panel) and with (right panel) absorption effects. Here ${\Lambda }_N={\Lambda }_{\pi }=1\mathrm{GeV}$.

Figure 18

Rapidity distribution of neutral pions (left panel) produced in $\gamma \mathcal{O}$-fusion and $\mathcal{O}\gamma$-fusion (black upper solid lines) compared to the $\gamma \gamma$ contribution (blue lowest solid line) for $\sqrt{s}=14\mathrm{TeV}$. Individual contributions of photon-odderon (short dashed line) and odderon-photon (long dashed line) are shown separately. We show predictions for the HERA upper limit (${\sigma }_{\gamma p\to {\pi }^{0}p}=49\mathrm{nb}$) and for the Ewerz-Nachtmann estimate (${\sigma }_{\gamma p\to {\pi }^{0}p}=6\mathrm{nb}$). In the right panel we make similar comparison of contributions of the two mechanisms for transverse momentum distribution of ${\pi }^0$’s in the rapidity region $-2.5<y_{{\pi }^0}<2.5$. For the odderon contributions we have used different values of slope parameters $B=4$, 6, $8{\mathrm{GeV}}^{-2}$.