LHC as $\pi p$ and $\pi\pi$ Collider

We propose an experiment at the LHC with leading neutron production.The latter can be used to extract from it the total $\pi^+ p$ cross-sections. With two leading neutrons we can get access to the total $\pi^+\pi^+$ cross-sections. In this note we give some estimates and discuss related problems and prospects.


Introduction 2 Kinematics
The diagram of the process p(p 1 ) + p(p 2 ) → n(p n ) + X(p X ) is presented in the Fig. 1a.
In the center-of-mass frame momenta of particles can be represented as follows (arrows denote transverse momenta): Figure 1: Diagrams for the signal and background processes in pp collisions. a) process with a single pion exchange (SπE) p + p → n + X, M is the mass of the system X; b) process with other reggeon exchanges; c) double dissociation process with a pion exchange; d) double dissociation with Pomeron and reggeon exchanges. S represents soft rescattering corrections.
One of the important questions is the definition of the kinematical region of the process, especially in rapidity y (pseudorapidity η) (see Section 5). If we have several secondaries from π + p scattering with momenta and y i ≃ 1 2 ln For negative ξ i we have y i → y i,max , η i → ∞ for k i → 0. It means that we have no pseudorapidity gap for low momenta of produced hadrons even if we have the rapidity one Experimentally, it is difficult situation when we need to cut momenta of secondary particles from below (see Section 5). For example, if y = 6 for pions of energy 30 GeV, then η ≃ 9.
3 Calculation of the cross-section. Absorptive effects.
In addition we should take into account other possible processes with neutron production. We also have to include contributions from ρ, a 2 exchanges (Fig. 1b), and from resonance decays, such as ∆ and N * in processes shown in Figs.1c,d. The calculation of the neutron spectra [27]- [29] shows that the contribution depicted in Fig. 1a dominates, while the contribution corresponding to Fig. 1b is about 20%. Other reggeons give also small contribution due to spin effects [27]. The main background can arise from minimum bias events and Fig. 1d, which was estimated to be about 0.06 · σ(p + p → p + X) [30] at low energies. It has, however, inverse missing mass dependence and is suppressed at intermediate ξ (see Fig. 3). This fact is used to eliminate the background.
Another important suppression factor arises from absorptive corrections. All possible corrections are discussed in Ref. [20]. We estimate only absorption in the initial state, since it gives the main contribution. For this task we use our model with 3 Pomeron trajectories [24]: which are the result of a 20 parameter fit of the total and differential cross-sections in the region 0.01GeV 2 < |t| < 14GeV 2 and 8GeV< √ s < 1800GeV, χ 2 /d.o.f. = 2.74.
Although χ 2 /d.o.f. is rather large, the model gives good predictions for the elastic scattering (especially in the low-t region with χ 2 /d.o.f. ∼ 1). It was also noted in [24], that this approach may be an artefact of the more general one with Regge cuts or nonlinear Pomeron trajectory. Following the procedure described in [22], [23], we can estimate absorptive corrections. Finally we obtain (an effective factorized form of the following expression (16) is only used for convenience, there is no factorization): where functions Φ 0 and Φ s arise from different spin contributions to the amplitude and both are equal to Φ B in the Born approximation. Hereσ i are Pauli matrices andΨ n , Ψ p are neutron and proton spinors. All the above functions can be calculated from the following formulae: the values of parameters can be found in (14) and in Table 1.
We can use any other reliable parametrization. Nevertheless our principal aim is to extract σ π + p (s) from the data. For this task we need a procedure of such extraction. One of the methods [1] is to evaluate the factor in front of σ π + p in (12) at some low t ≃ −0.014, and then divide the data by this factor. For fixed value of t we obtain ξ ≃ 0.125 for q ∼ 0. It is convenient for the estimation. We have to take the ISR data [13] for the cross-section E dσ/d 3 p at x = 1 − ξ = 0.875 and we get an approximate formula for the total π + p cross-section where S(s/s 0 , 0.125, 0) ≃ 0.68. For low energies we obtain quite reasonable predictions for the cross-section (see Table 2 and Fig. 6), which are compatible with real π + p measurements [33], if we take into account reggeon corrections.
In the real situation factor S is the model dependent function presented in the Fig. 7, but at t → 0 it tends to unity and we can obtain more model independent cross-section σ π + p .´  Table 3: Total p + p → n + X cross-sections in the kinematical region 0 < | q| < 0.5 GeV, ξ min = 10 −6 < ξ < ξ max for two parametrizations (29)( (30)). (57) 175(244) 576(820) 921(1320) Differential cross-sections for the process p + p → n + X at √ s = 10 TeV are depicted in Figs. 8,9. The total rate is rather large to make proposed investigations(see Table 3) and total absorptive corrections are about 0.35 for ξ max < 0.15. For low energies the region of applicability of the presented model is usually given by inequalities 0.01 GeV 2 < |t| < 0.5 GeV 2 , 10 −6 < ξ < 0.4, but for higher energies this region may be smaller (like ξ < 0.1), since this corresponds to masses M = 3 TeV at √ s = 10 TeV, and for large masses it may not work.

Double pion exchange
As was said above the double pion exchange inclusive process is a possible source of information on both total and elastic ππ sross-sections. Early attempts to extract ππ crosssection were made with the use of the exclusive cross-section. The results are presented in Fig. 10 [7]. There is some tendency of the early flattening out of the ππ cross-sections. In πp and pp cross-sections this flattening begins at higher energies and precedes the onset of the growth. One could argue about probable early onset of the growth in ππ interactions. Figure 10: Elastic and total cross-sections for π − π + and π − π − scattering from the data on exclusive reactions as a function of the dipion invariant mass (Fig.5 from Ref. [7]).
Since there are no data on this process, we can make only predictions for higher energies. Numerically calculated functions for DπE are presented in Figs. 11,12 and in Table 4 for two parametrizations.

Experimental possibilities
We propose to perform measurements of SπE, Fig.2 (a), and DπE, Fig.2 (b), processes at LHC with the CMS detector [34]. In this chapter we discuss perspectives of such measurements at 10 TeV, c.m. energy of the LHC protons in the first year runs. For the leading neutron detection Zero Degree Calorimeter (ZDC) [25] could be used. ZDCs are placed on the both sides of CMS at the distance 140 m from the interaction point. ZDC consists of electromagnetic and hadronic parts. It is designed for neutron and gamma measurements in the pseudorapidity region |η| > 8.5. This type of detectors is widely used at RICH experiments since 2001 year [35].
To study SπE and DπE processes a generator has been developed in the framework of more general simulation package EDDE [36]. Kinematics of SπE and DπE reactions are defined by ξ n and t n of the leading neutron. Vertex pπ + virt n is generated on the basis of the model described above. PYTHIA 6.420 [37] is used for the π + virt p → X generation in the SπE and π + virt π + virt → X generation in the DπE. Inelastic processes, including single and double diffractive dissociation (SD and DD) and minimum bias events 1 (MB), have been studied as possible background for SπE and DπE. All background processes have been generated with PYTHIA 6.420. Cross sections for signal and background at 10 TeV have the following ratio 2 : DπE : SπE : DD : SD : MB = 0.2 : 2.6 : 9.7 : 14 : 50 mb.
Pseudorapidity distributions for the processes are shown in Fig. 13. All processes have leading neutrons in the acceptance of ZDC, |η| > 8.5. Thus, SD, DD, and MB can imitate SπE and DπE events and SπE can be a strong background for DπE measurement.
for DπE study. Here, N f n (N b n ) is the number of neutron hitting the forward (backward) ZDC, ξ f n (ξ b n ) is the relative energy loss of the forward (backward) leading neutron. I.e., for SπE selection we choose events with neutrons in the forward or backward ZDC and with the absence of neutrons in the opposite one. For DπE, we select events with neutrons in the forward and backward ZDCs. Such selections suppress ∼ 90% of background events for SπE and suppress background for DπE by a factor of 240, see Tables 5 and 6    Nevertheless, the signal/background ratio remains ∼ 1/3 for SπE and ∼ 2/3 for DπE. Figure 14 shows distributions of ξ and t of the leading neutron and M = √ ξs after TSπE selection for SπE, SD, DD and MB events. It is seen that a cut in t (|t| < 0.25 GeV 2 ) should suppress background for SπE very efficiently. Unfortunately, in the present design of ZDC this feature is supported very restrictedly 3 . To search for another selections suppressing the rest of background, we chose SπE events with neutrons in the forward ZDC and looked at the data from other CMS calorimeters: Barrel, Endcap, HF, CASTOR and electromagnetic part of ZDC. It was found essential difference in the number of hits and in the total energy deposit for SD and DD comparing with SπE in the Barrel, Endcap and HF. As an example, we show the number of hits and the total energy deposit in the forward and backward HF for all studied processes, see Figure 15.
where N HF hits is the number of hits in HF, makes SD and DD background negligible, but it has no any influence on MB events. Table 7 shows signal/background ratios with selection (52). In the mass region below 5000 GeV we could expect S/B∼10/6. For MB suppression we have to require additional cut using t of the leading neutron: As shown in the table 7 selection (53) suppresses MB events by a factor of 8.7 and makes S/B ratio ∼ 100/8. Figure 16 presents M distribution for SπE and background with selections (52) and (53). The same S/B ratio for DπE process could be achieved with selections THFhitsDπE : N HFF hits > 4 .and. N HFB hits > 4 and TtmaxDπE :

Discussions and conclusions
In conclusion, our study of SπE and DπE processes and background on the generator level shows that we could expect SπE and DπE observation at LHC with CMS in the first runs as realistic. Some modifications of ZDC is required to measure t of the leading neutron. Using combination of 2 simple cuts, for N hits in HF and t n , one could suppress background by a factor of ∼ 760 for SπE and ∼ 9500 for DπE, saving ∼ 35% of SπE events and ∼ 60% of DπE. Without ZDC modification we could expect observation of As was said the main motivation for this work is the extraction of the total πp and ππ cross-sections from proton-proton scattering measurements. Procedure is quite simple for low values of t, because absorptive factor goes to unity and backgrounds are completely suppressed. If this task is done than we will have additional, more rich, data in the high energy region to check predictions of different models for strong interactions, quark counting rules and so on.
Further task is the exact estimation of backgrounds presented in Figs. 1b,c,d (especially ρ, a 2 exchanges) to be sure that the extraction procedure is correct.
To make some estimations we use several parametrizations for the total cross-sections. All they show similar behaviour. That is why we can use them to make Monte-Carlo simulations to see possible experimental difficulties. Unfortunately, the present data on SπE is not so clear (problems in normalization), so we can make only more or less plausible estimations for high energies (especially for the LHC).
At the Monte-Carlo simulation on the generator level we can see that the main background can arise from minimum bias events (if we integrate cross-sections in variable t). We have found that signal to the minimum bias background ratio for SπE and DπE is about 1.5-2. It can be increased significantly only if we take a cut |t| < 0.2 ÷ 0.3 GeV 2 (which was done, for example, in low-energy experiments). Since that the precision measurement of t n is not possible with the present design of ZDC, extraction of πp and ππ cross-sections sets aside for the future ZDC modifications 4 . Nevertheless, we can look for SπE and DπE events in the fist LHC run for rough estimations of cross-sections.
We have also some perspectives of measurements of the parton distribution functions in pions in hard processes (see Fig. 18). This would be also very interesting, since we can get parton distributions in the pion at smaller x and higher Q 2 . But cross-sections for hard processes are small and this task is also for high luminocity runs. It seems to be fairly realistic, especially taking into account new prospects for high-energy πp interactions.
Investigation of SπE and DπE processes can also provide us with unique measurements of πp and ππ elastic cross-sections.
In spite of all difficulties, proposed measurements are so exciting that it makes all sense to pursue our aim further.