Triple Regge exchange mechanisms of four-pion continuum production in the $pp\to pp{\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ reaction

We consider exclusive multiperipheral production of four charged pions in proton-proton collisions at high energies with simultaneous exchange of three Pomerons/Reggeons. The amplitude(s) for the genuine $2\to 6$ process are written in the Regge approach. The calculation is performed with the help of the GenEx Monte Carlo code. Some corrections at low invariant masses in the two-body subsystems are necessary for application of the Regge formalism. We estimate the corresponding cross section and present differential distributions in rapidity, transverse momenta and two- and four-pion invariant masses. The cross section and the distributions depend on the value of the cutoff parameter of a form factor correcting amplitudes for off-shellness of $t$-channel pions. A rather large cross section is found for the whole phase space ($\sigma \sim 1–5\mu \mathrm{b}$, including absorption corrections). Relatively large four-pion invariant masses are populated in the considered diffractive mechanism compared to other mechanisms discussed so far in the context of four-pion production. We investigate whether the triple Regge exchange processes could be identified with the existing LHC detectors. We consider the case of ATLAS and ALICE cuts. The ATLAS (or CMS) has a better chance to identify the process in the region of large invariant masses $M_{4\pi }>10\mathrm{GeV}$. In the case of the ALICE experiment the considered mechanism competes with other mechanisms (production of $\sigma \sigma$, $\rho \rho$ pairs or single resonances) and cannot be unambiguously identified.

Figure 1

A diagram for exclusive diffractive production of ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ continuum with three Pomeron/Reggeon exchanges in proton-proton collisions. Here $\mathbb{R}$ denotes the $f_{2\mathbb{R}}$ or ${\rho }_{\mathbb{R}}$ Reggeon exchanges. The four-momentum transfers squared are shown explicitly.

Figure 2

Distributions in transverse momentum of the four-pion system and for the transverse momenta of individual particles (protons and pions) for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1$, 1.5 GeV.

Figure 3

Four-pion invariant mass distribution for the full phase space for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1$, 1.5 GeV.

Figure 4

Rapidity distribution of protons (external peaks) and pions (internal bumps) for the full phase space for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1$, 1.5 GeV.

Figure 5

Rapidity distribution of pions for $(\mathbb{P}+f_{2\mathbb{R}})\times (\mathbb{P}+f_{2\mathbb{R}})\times (\mathbb{P}+f_{2\mathbb{R}})$ (upper curve), $\mathbb{P}\times \mathbb{P}\times \mathbb{P}$ (middle curve) and $(\mathbb{P}+f_{2\mathbb{R}})\times f_{2\mathbb{R}}\times (\mathbb{P}+f_{2\mathbb{R}})$ (lower curve) exchanges for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1$, 1.5 GeV.

Figure 6

$4\pi$ invariant mass distribution for $(\mathbb{P}+f_{2\mathbb{R}})\times (\mathbb{P}+f_{2\mathbb{R}})\times (\mathbb{P}+f_{2\mathbb{R}})$ (upper curve), $\mathbb{P}\times \mathbb{P}\times \mathbb{P}$ (middle curve) and $(\mathbb{P}+f_{2\mathbb{R}})\times f_{2\mathbb{R}}\times (\mathbb{P}+f_{2\mathbb{R}})$ (lower curve) exchanges for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1$, 1.5 GeV.

Figure 7

Dipion invariant mass distributions for the opposite-sign (left) and for the same-sign (right) pions for two different values of ${\mathrm{\Lambda }}_{\text{off},E}=1\mathrm{GeV}$ (lower curve) and ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ (upper curve).

Figure 8

Integrated cross section for different values of $W_{\text{cut}}$ for a sharp (Heaviside-like) cutoff function. The extra (red) cross represents the cross section for the smooth cutoff function with $W_0=2\mathrm{GeV}$ and $a=0.2\mathrm{GeV}$ as taken from [1]. The upper plot is for ${\mathrm{\Lambda }}_{\text{off},E}=1\mathrm{GeV}$ and the lower one for ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$.

Figure 9

Dipion invariant mass distributions for the opposite-sign (left) and the same-sign (right) pions for different values of $W_{\text{cut}}$ for sharp cutoff function (2.20) and ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$.

Figure 10

Distributions in transverse momentum of the four-pion system and for the transverse momenta of individual particles (protons or pions) with the ATLAS cuts (3.2).

Figure 11

Four-pion invariant mass distribution ($M_{4\pi }$) with the ATLAS cuts (3.2) for ${\mathrm{\Lambda }}_{\text{off},E}=1\mathrm{GeV}$ (lower curve) and ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ (upper curve).

Figure 12

Distribution in rapidity of pions and protons for the ATLAS cuts (3.2).

Figure 13

Dipion invariant mass distribution for the opposite-sign (left) and same-sign (right) pions with the ATLAS cuts (3.2) for different values of ${\mathrm{\Lambda }}_{\text{off},E}$.

Figure 14

Dipion invariant mass distributions for the opposite-sign (left) and the same-sign (right) pions for the ATLAS experimental cuts (3.2) with the extra limitations on $M_{ij}>M_{ij,\text{cut}}$. The upper plots are for ${\mathrm{\Lambda }}_{\text{off},E}=1\mathrm{GeV}$ and the lower ones for ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$. “Smooth” means the cutoff (2.19) with $W_0=2\mathrm{GeV}$ and “Sharp” means the cutoff (2.20) with $W_{\text{cut}}=2\mathrm{GeV}$.

Figure 15

Distributions in transverse momentum of the $4\pi$ system and in transverse momenta of individual pions or protons for the ALICE cuts (3.3) and for the two values of ${\mathrm{\Lambda }}_{\text{off},E}$ specified in the figure legends.

Figure 16

Four-pion invariant mass distribution for the ALICE cuts (3.3) and for ${\mathrm{\Lambda }}_{\text{off},E}=1\mathrm{GeV}$ (lower curve) and ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ (upper curve).

Figure 17

Distribution in rapidity of pions (middle bump) and of protons (external peaks) for the ALICE cuts for two different values of ${\mathrm{\Lambda }}_{\text{off},E}$.

Figure 18

Dipion invariant mass distribution for the opposite-sign (left) and the same-sign (right) pions for the ALICE cuts (3.3).

Figure 19

Proton energy distributions for the full phase space (upper plot) and for the ATLAS cuts (3.2) (lower plot). Here we take ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ as an example.

Figure 20

Distributions in the energy left for pions for the full phase space (upper plot) and for the ATLAS cuts (3.2) (lower plot). Here we take ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$. For the full phase space case a sharp break above 1500 GeV is visible, indicating a numerical accuracy limit.

Figure 21

Distributions in the longitudinal momentum of the pion center of mass for the full phase space (upper plot) and for the ATLAS cuts (3.2) (lower plot). Here we take ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ as an example.

Figure 22

Four-pion invariant mass distribution for the full phase space for two different values of ${\mathrm{\Lambda }}_{\text{off},E}$.

Figure 23

Rapidity difference between the second and third pion for ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ for the full phase space.

Figure 24

Rapidity difference between the second and third pion for ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$ for the ATLAS detector cuts.

Figure 25

Four-pion invariant mass distribution ($M_{4\pi }$) with the ATLAS kinematical cuts (3.2) for $\sqrt{s}=7\mathrm{TeV}$. The results correspond to the Born-level calculations. The dotted line represents the triple Regge exchange mechanism obtained for ${\mathrm{\Lambda }}_{\text{off},E}=1.5\mathrm{GeV}$. The solid line represents the contribution from $\sigma \sigma$ mechanism discussed in [8].