Revised model of absorption corrections for the $pp\to pp{\pi }^{+}{\pi }^{-}$ process

We include new additional absorption corrections into the double Pomeron/Reggeon exchange (nonresonant) model for $pp\to pp{\pi }^{+}{\pi }^{-}$ or $p\overline{p}\to p\overline{p}{\pi }^{+}{\pi }^{-}$ processes. They are related to the $\pi N$ nonperturbative interaction in the final state of the reaction. We present predictions of cross sections for RHIC, Tevatron, and LHC experiments. The new absorption corrections lead to a further decrease of the cross section by about a factor of 2. The role of the absorption corrections is quantified for several differential distributions. They change the shape of some distributions ($d\sigma /dt$, $d\sigma /d{p}_{t,p}$, $d\sigma /d{\varphi }_{pp}$) but leave almost unchanged shape of other distributions ($d\sigma /d{M}_{\pi \pi }$, $d\sigma /d{y}_{\pi }$, $d\sigma /d{p}_{t,\pi }$, $d\sigma /d{\varphi }_{\pi \pi }$). The effect may have an important impact on the interpretation of the recent STAR and CDF data as well as the forthcoming data of the ALICE, ATLAS/ALFA, and CMS/TOTEM collaborations.

Figure 1

(a) Born amplitudes for the $pp\to pp{\pi }^{+}{\pi }^{-}$ process. (b) Absorptive correction amplitudes due to the $pp$ interaction.

Figure 2

New absorptive correction amplitudes for the $pp\to pp{\pi }^{+}{\pi }^{-}$ process due to the $\pi p$ interaction in the final state included in the present analysis.

Figure 3

Differential distributions $d\sigma /dt$ for ${\pi }^{+}p$ (left panel) and ${\pi }^{-}p$ (right panel) elastic scattering at incident beam momenta $P_{\text{lab}}=50\mathrm{GeV}$ ($\sqrt{s}\simeq 9.7\mathrm{GeV}$) [34, 35, 36] and $P_{\text{lab}}=200\mathrm{GeV}$ ($\sqrt{s}\simeq 19.4\mathrm{GeV}$) [34, 37, 38]. The dashed lines show results obtained with formula (2.10), while the solid lines are obtained via the replacement (2.11), where $B_0=6.5{\mathrm{GeV}}^{-2}$ and ${\mu }^2=0.6{\mathrm{GeV}}^2$.

Figure 4

Two-pion invariant mass distribution at ISR energies with the ISR kinematical cuts indicated in the legend. The ISR data [28, 30] are shown for comparison. The blue dashed lines represent the results obtained for the monopole form factors [(2.13), ${\mathrm{\Lambda }}_{\text{off},M}=1.2$ and 1.6 GeV], while the black solid lines are for the exponential form [(2.12), ${\mathrm{\Lambda }}_{\text{off},E}=1.6\mathrm{GeV}$]. The bottom panels represent the $|t|$ distributions without (dotted lines) and with (dashed lines) the $pp$ absorption corrections, and with all ($pp$ and $\pi N$) absorption corrections included (solid lines). Results for the exponential (left panel) and for the stretched exponential (right panel) parametrizations of the $\pi p$ subsystem are shown in addition.

Figure 5

Two-pion invariant mass distribution at $\sqrt{s}=200\mathrm{GeV}$ with the STAR kinematical cuts specified in the legend. The dotted line in the left panel corresponds to the Born calculation, the long-dashed and solid lines to calculations with the absorption effects due to the $pp$ and the $\pi p$ rescattering, respectively. In the right panel, the blue dashed lines represent the results with all absorption effects and obtained for the monopole form factors (2.13) for different choices of the cutoff parameter ${\mathrm{\Lambda }}_{\text{off},M}=0.8,1.2,1.6\mathrm{GeV}$ (from bottom to top). The red dotted line represents result for ${\mathrm{\Lambda }}_{\text{off},M}=1.2\mathrm{GeV}$ multiplied by the extra suppression factor $f(M_{\pi \pi })$, as explained in the text. The black solid lines are for the exponential form (2.12), and ${\mathrm{\Lambda }}_{\text{off},E}=1.0$ and 1.6 GeV. The STAR preliminary data [5] are shown for comparison.

Figure 6

The distributions in the pseudorapidity of the produced ${\pi }^{+}{\pi }^{-}$ system (${\eta }_{\pi \pi }$) and in the azimuthal angle between the outgoing pions (${\varphi }_{\pi \pi }$) and between the outgoing protons (${\varphi }_{pp}$) at $\sqrt{s}=200\mathrm{GeV}$ in the range of $0.5<M_{\pi \pi }<1\mathrm{GeV}$. In the right bottom panel, we show the results without the absorption effects (the dotted line), with the $pp$ rescattering (the long-dashed line), and with the additional $\pi p$ rescattering (the solid line). The STAR preliminary data [5] are shown for comparison.

Figure 7

Two-pion invariant mass distribution at $\sqrt{s}=1.96\mathrm{TeV}$, with the CDF kinematical cuts specified in the legend. The meaning of the lines in the left panel is the same as in Fig. 5 (the right panel). In the right panel we show the results with an additional cut on transverse momentum of the pion pair $p_{t,\pi \pi }>1\mathrm{GeV}$ and for two different $t$ dependences of the $\pi p$-subsystem amplitudes. The thick upper lines represent results for the replacement (2.11), while the thin lower lines show the results with formula (2.10). The CDF data [6, 7] are shown with only statistical errors; systematic uncertainties are approximately 10% at all masses.

Figure 8

Mean value of $p_{t,\pi }$ (left panel) and $p_{t,\pi \pi }$ (right panel) as a function of two-pion invariant mass at $\sqrt{s}=1.96\mathrm{TeV}$ calculated with the CDF kinematical cuts specified in the legend.

Figure 9

Mean value of the first even Legendre polynomials $P_L(\cos {\theta }_{{\pi }^{+}}^{r.f.})$ as a function of the two-pion invariant mass, with the CDF kinematical cuts specified in the legend. The results correspond to two types of off-shell pion form factors: the exponential one (2.12) and the monopole one (2.13) without (thin lines) and with (thick lines) absorption effects. The red dotted line represents the result for the monopole type of form factor and with an extra suppression factor $f(M_{\pi \pi })$. The CDF preliminary data [6] are shown for comparison.

Figure 10

Two-pion invariant mass distribution at $\sqrt{s}=7\mathrm{TeV}$ with the ALICE kinematical cuts specified in the legend. In the left panel we show results with a cutoff parameter ${\mathrm{\Lambda }}_{\text{off},E}=1.6\mathrm{GeV}$ [see (2.12)], without the absorption effects (the dotted line), with the $pp$ rescattering (the long-dashed line), and with the additional $\pi p$ rescattering (the solid line). The meaning of the lines in the right panel is the same as in Fig. 5.

Figure 11

Differential cross sections $d\sigma /d{p}_{t,\pi }$, $d\sigma /d{p}_{t,\pi \pi }$, and $d\sigma /d{\varphi }_{\pi \pi }$ at $\sqrt{s}=7\mathrm{TeV}$ with the ALICE kinematical cuts specified in the legend. In the calculations we have used different values of the cutoff parameter ${\mathrm{\Lambda }}_{\text{off}}$.

Figure 12

Mean value of the $p_{t,\pi }$ (left panel) and $p_{t,\pi \pi }$ (right panel) as a function of two-pion invariant mass at $\sqrt{s}=7\mathrm{TeV}$, with the ALICE kinematical cuts specified in the legend. In the calculation we have used the cutoff parameter ${\mathrm{\Lambda }}_{\text{off},E}=1.6\mathrm{GeV}$.

Figure 13

Two-pion invariant mass distribution at $\sqrt{s}=8\mathrm{TeV}$ with the CMS kinematical cuts specified in the legend.

Figure 14

The distributions in proton transverse momentum (top panels), in azimuthal angle between the outgoing protons (middle panels), and in proton four-momentum transfer $t_1$ (bottom panels) at $\sqrt{s}=8\mathrm{TeV}$ with the CMS kinematical cuts specified in the legend. In the left panels we show the distributions without and with the absorption corrections. In the calculation, results of which are shown on the right panels, we have used two forms for the off-shell pion form factors and different cutoff parameters ${\mathrm{\Lambda }}_{\text{off}}$.

Figure 15

Two-dimensional distributions in $p_{t,p}$ and ${\varphi }_{pp}$ at $\sqrt{s}=8\mathrm{TeV}$ with the CMS kinematical cuts specified in the legend. We show the distributions without and with the absorption corrections. In this calculation we have used the cutoff parameter ${\mathrm{\Lambda }}_{\text{off},E}=1.6\mathrm{GeV}$.