# Central exclusive diffractive production of two-pion continuum at hadron colliders

## Abstract

Calculations of central exclusive diffractive di-pion continuum production are presented in the Regge–eikonal approach. Data from ISR, STAR, CDF and CMS were analyzed and compared with the theoretical description. We also consider theoretical predictions for LHC, possible nuances and problems of calculations and prospects of investigations at present and future hadron colliders.

## 1 Introduction

In previous papers [1, 2] the general properties and calculations of the Central Exclusive Diffractive Production (CEDP) were considered. It was shown, especially in [2], that diffractive patterns (differential cross-sections) of CEDP play a significant role in model verification.

Di-pion LM CEDP is the basic background process for CEDP of resonances (like \(f_2\) or \(f_0\)), since one of the basic hadronic decay modes for these resonances is the two-pion one.

We can use LM CEDP to fix the procedure of calculations of “rescattering” (unitarity) corrections. In the case of di-pion LM CEDP there are two kinds of corrections, in the proton–proton and the pion–proton subamplitudes. They will be considered in the present work.

The pion is the most fundamental particle in the strong interactions, and LM CEDP gives us a powerful tool to thoroughly investigate its properties, especially to investigate the form factor and scattering amplitudes for the off-shell pion.

LM CEDP has rather large cross-sections. It is very important for an exclusive process, since in the special low luminosity runs (of the LHC) we need more time to get enough statistics.

As was proposed in [3], it is possible to extract some reggeon–hadron cross-sections. In the case of single and double dissociation it was the Pomeron–proton one. Here, in the LM CEDP of the di-pion we can analyze the properties of the Pomeron–Pomeron to pion–pion exclusive cross-section, and also check again the predictions of the covariant reggeization method [3].

Diffractive patterns of this process are very sensitive to different approaches (subamplitudes, form factors, unitarization, reggeization procedure), especially differential cross-sections in

*t*and \(\phi _{pp}\) (azimuthal angle between final protons), and also the \(M_{\pi \pi }\) dependence. That is why this process is used to verify different models of diffraction.All the above items are additional advantages provided by the LM CEDP of two pions, which has the usual properties of CEDP: a clear signature with two final protons and two large rapidity gaps (LRG) [4, 5] and the possibility to use the “missing mass method” [6].

In Refs. [10, 11, 12, 13] the authors do not introduce “enhanced” corrections, but they take into account pion–proton interactions in the final state (see Fig. 2). There still is an issue in this approach, since they mix a partially Regge approach with continuous complex spin and a model with fixed “Pomeron spin” (exactly vector or tensor Pomerons). It may be convenient for calculations and gives results close to reality, but we have no clear physical explanation. On the one hand we have collision of two particles with fixed spin (1 or 2), and on the other hand we use the Regge expression, where the spin is replaced by the complex Regge trajectory.

In this article we consider several cases, depicted in Fig. 3, and we show how they describe the data from the ISR [15, 16], STAR [17, 18], CDF [19, 20], CMS [21, 22] collaborations.

In the first part of the present work we introduce the framework for calculations of double pion LM CEDP (kinematics, amplitudes, differential cross-sections) in the Regge-eikonal approach.

In the second part we analyze the experimental data on the process at different energies, find the best approach and make some predictions for LHC experiments.

## 2 General framework for calculations of LM CEDP

LM CEDP is the first exclusive two to four process which is driven by the Pomeron–Pomeron fusion subprocess. That is why it serves as a basic background for LM CEDP of resonances like \(f_0(980)\), \(f_2(1270)\). At the moment, for low central pion–pion masses (less than \(\sim 3~\hbox {GeV}\)), it is a huge problem to use a perturbative approach, which is why we apply the Regge-eikonal method for all the calculations. For proton–proton and proton–pion elastic amplitudes we use the model of [23, 24], which describes all the available experimental data on elastic scattering.

### 2.1 Components of the framework

- 1.
We calculate the primary amplitude of the process, which is depicted as the central part of diagrams in Fig. 3. Here we consider four cases to show that only one of them gives the best description of the data on this process. The case PB represents the Born term (the letter P means that we use the usual “bare” virtual pion propagator \(1/(\hat{t}-m_{\pi }^2)\), and B means the Born term (28) in each shoulder or pion–proton elastic subamplitudes of the central primary amplitude). The expressions of pion–proton elastic subamplitudes can be found in Appendix B.

The case RB is similar to the previous one, and the letter R (instead of P) means that the bare off-shell pion propagator is replaced by the reggeized one;where \(\hat{s}\) is the di-pion mass squared and \(\hat{t}\) is the square of the momentum transfer between a Pomeron and a pion in the Pomeron–Pomeron fusion process (see Appendix A for details).$$\begin{aligned} {\mathcal {P}}_{\pi }(\hat{s},\hat{t})= & {} \left( \mathrm {ctg}\frac{\pi \alpha _{\pi }(\hat{t})}{2} -\mathrm {i} \right) \cdot \nonumber \\&\cdot \frac{\pi \alpha _{\pi }^{\prime }}{2\varGamma (1+\alpha _{\pi }(\hat{t}))} \left( \frac{\hat{s}}{s_0}\right) ^{\alpha _{\pi }(\hat{t})}, \end{aligned}$$(3)The cases PF (RF) can be obtained from PB (RB) if we replace the Born pion–proton elastic amplitudes to full eikonalized expressions (which is reflected in the replacement of B to F in the notation for these cases), which could be found in Appendix B.

- 2.
After the calculation of the primary LM CEDP amplitude we have to take into account all possible corrections in proton–proton and proton–pion elastic channels due to the unitarization procedure (the so-called “soft survival probability” or “rescattering corrections”), which are depicted as \(V_{pp}\), \(V'_{pp}\) and \(S_{\pi p}\) blobs in Fig. 3. For the proton–proton and proton–pion elastic amplitudes we use the model of [23, 24] (see Appendix B). A possible final pion–pion interaction is not shown in Fig. 3, since we neglect it in the present calculations. RB and PB cases of Fig. 3 are similar to the one in Fig. 2.

Exact kinematics of the two to four process is outlined in Appendix A.

Here we use the model, presented in Appendix B for example. One can use another one, which has been proved to describe well all the available data on proton–proton and proton–pion elastic processes. But it is difficult to find now more than a couple of models which have more or less predictable power (see [26] for a detailed discussion). That is why we use the model, which has been proved to be good in data fitting, especially in the kinematical region of our interest.

Other functions are defined in Appendix B. Then we can use Eq. (23) to calculate the differential cross-section of the process.

### 2.2 Nuances of calculations

In the next section one can see that there are some difficulties in the data fitting, which have also been presented in other work [10, 11, 12, 13]. In this section let us discuss some nuances of calculations, which could change the situation.

We have to pay special attention to the amplitudes, where one or more external particles are off their mass shell. The example of such an amplitude is the pion–proton one \(T_{\pi ^+ p}\) (\(T_{\pi ^- p}\)), which is the part of the CEDP amplitude (see (4)). For this amplitude in the present paper we use the Regge-eikonal model with the eikonal function in the classical Regge form. The “off-shell” condition for one of the pions is taken into account by an additional phenomenological form factor \(\hat{F}_{\pi }(\hat{t})\). But there are at least two other possibilities.

The second one arises from the covariant reggeization method, which is considered in Appendix C. For the case of conserved hadronic currents we have the definite structure in the Legendre function (50), which is transformed in a natural way to the case of the off-shell amplitude. But in this case the off-shell amplitude shows a specific behavior at low t values (see Fig. 4 and [2] for details). As was shown in [2], unitarity corrections can mask this behavior. To check this we need to make all the calculations and fitting of the data for the process \(p+p\rightarrow p+\pi +\pi +p\), but with an amplitude like (50) instead of (33). This will be done in further work on the subject.

In the present calculations we use the linear pion trajectory \(0.7(\hat{t}-m_{\pi }^2)\). The nonlinear case was also verified, and the difference in the final result is not significant.

## 3 Data from hadron colliders versus results of calculations

from t-distributions we can obtain the size and shape of the interaction region;

the distribution on the azimuthal angle between the final protons gives the quantum numbers of the produced system (see [2, 28] and the references therein);

from \(M_c\) (here \(M_c=M_{\pi \pi }\)) dependence and its influence on the t-dependence we can draw some conclusions about the interaction at different space-time scales and the interrelation between them.

### 3.1 STAR collaboration data versus model cases

### 3.2 ISR and CDF data versus RF case of the model

We see an underestimation of the ISR data. For these low energies we have to take into account possible corrections to the pion–proton amplitudes, since our approach describes data well only for energies greater than \(\sim 3~\hbox {GeV}\). In each shoulder (\(T_{\pi p}\) amplitude in Fig. 3 RF) the energy can be less than \(3~\hbox {GeV}\).

### 3.3 CMS data and predictions

In Fig. 13 one can see the recent data from the CMS collaboration and curves of our model. The upper curve, which corresponds to the parameter \(\varLambda _{\pi }\), which better fits the STAR data on \(\phi _{pp}\) distribution (but gives higher values for \(M_{\pi \pi }>1~\hbox {GeV}\) as depicted in Fig. 5b), also describes the data of CMS collaboration well (but overestimates the CDF data, as was shown in Figs. 11, 12). The lower curve underestimates the data from STAR, ISR and CMS, but it is close to the CDF data. Interference with resonances and modifications of the model in the region \(|\hat{t}|\sim \hat{s}/2\sim 1~\hbox {GeV}^2\) can change the picture, especially for low \(M_{\pi \pi }\), which is why we have to take it into account when fitting the data (Fig. 14).

### 3.4 Summary

In our approach the best description is given by the case RF (Fig. 3). That is why effects from rescattering (unitarity) corrections are very important.

The result is crucially dependent on the choice of \(\varLambda _{\pi }\) in the off-shell pion form factor, i.e. on \(\hat{t}\) (virtuality of the pion) dependence.

If we try to fit the data from STAR [17, 18], we find that the best description gives overestimation of the CDF data [19, 20] (especially in the region \(M_{\pi \pi }< 1.5~\hbox {GeV}\)) and underestimation of the ISR data [15, 16]. This is due to effects like the interference with resonance contributions or \(\gamma \gamma \rightarrow \pi \pi \) and \(\gamma \mathbb {O}\rightarrow \pi \pi \) processes, effects related to the irrelevance and possible modifications of the Regge approach (for the virtual pion exchange) in this kinematical region, as was discussed in the introduction, corrections to pion–pion scattering at low \(M_{\pi \pi }\), and corrections to \(T_{\pi p}(s,t)\) for \(\sqrt{s}<3~\hbox {GeV}\).

Predictions for CMS are close to the data, if we use the best fit to the STAR data on \(\phi _{pp}\) distribution (see Fig. 5b). We need also an estimate of the interference with resonance terms to see the full picture and draw final conclusions.

## 4 Pomeron–Pomeron to pion–pion cross-section

Another interesting question, which we can discuss here, concerns the Pomeron–Pomeron cross-section. As was shown in [3], it is possible to extract the Pomeron–proton cross-section from the data on single (SD) and double (DD) dissociation, and this numerical value appears to be of the order of typical hadron–hadron cross-sections. It was done by the use of covariant reggeization method with conserved spin-J meson currents, which helps to solve the old problem of the very small Pomeron–proton cross-section extracted by other authors [30, 31].

Reggeon–hadron and reggeon–reggeon scattering can be considered as a scattering of all possible real mesons lying on the Regge trajectory of hadrons. Conceptually it is similar to hydrogen–hadron or hydrogen–hydrogen scattering, since hydrogen has the spectrum of the states, and each of them has its own probability to scatter on a hadron or another hydrogen atom. A specific feature is that we deal in this case with “off-shell atoms”. There is some misunderstanding concerning the physical nature of reggeons. Actually, the reggeon is quite a general notion to qualitatively describe a generic quantum composite system. Conceptually a hadronic reggeon differs from the familiar hydrogen atom only by constituent content: electron–proton in the latter case and, say, quark–antiquark in the former. Energy levels of both are given by the corresponding Regge trajectories. So the reggeon is a full fledged composite particle in the same sense as an atom. Certainly the specific properties are different due to different binding forces.

As was shown also in [3], absorptive corrections play the crucial role in high energy scattering, and make the extraction procedure rather complicated and model dependent. We should propose some appropriate parametrization for Pomeron–hadron cross-section, then we apply unitarization procedure to obtain real SD or DD cross-sections.

Results of calculations (15)–(18) are shown in Fig. 15. In the case of conserved meson currents we obtain Pomeron–Pomeron to pion–pion cross-section \(10\div 100\) times higher than in the case of non-conserved currents and with more specific and strong dependence on the Pomeron virtuality. In the old work [32, 33, 34, 35, 36] the extracted Pomeron–Pomeron total cross-section was of the order 100 \(\upmu \mathrm{b}\) at \(\sqrt{\hat{s}}<3~\hbox {GeV}\) and almost independent on Pomeron’s virtuality. \(\sigma _{\mathbb {P}\mathbb {P}\rightarrow \pi \pi }\) should be at least less than this number.^{1} Our calculations in the same kinematical region give numbers of the order \(0.1{-}5~\upmu \mathrm{b}\) for non-conserved currents and \(0.3{-}100~\upmu \mathrm{b}\) for the case of conserved currents. There are strong contributions of other processes (especially production of resonances) in this region, which is why we should have \(\sigma _{\mathbb {P}\mathbb {P} \rightarrow \pi \pi }\ll \sigma ^{tot}_{\mathbb {PP}}\), which is obvious in the case of non-conserved currents where we have both extracted numbers to compare.

In Sect. 3 it is shown that RF and, possibly, PF modes (see Fig. 3 for notations) give an appropriate description of the data, i.e. we have to take into account all rescattering corrections (even in the \(T_{\pi p}\) amplitudes). If to use the RB mode, as some other authors do [7, 8, 9, 10, 11, 12, 13],^{2} it is possible to extract the Pomeron–Pomeron cross-section more easily (“almost model independent method”, as was done, for example, for the pion–proton cross-section [37]).

## 5 Conclusions

After calculations of several cases (see Fig. 3) we can see that the RF case is better suited to describe the data on the process \(p+p\rightarrow p+\pi ^++\pi ^- +p\).

When we try to fit the data from STAR collaboration [17, 18] with different values of \(\varLambda _{\pi }\) (different behavior of the virtual pion form factor), we obtain an underestimation of the ISR data [15, 16] and an overestimation of the CDF [19, 20]. Possible explanations are interference terms with resonances, corrections to \(T_{\pi p}(s,t)\) for \(\sqrt{s}<3~\hbox {GeV}\), off-shell pion effects, and some other mechanisms in the Pomeron–Pomeron to pion–pion process at low \(M_{\pi \pi }\).

We have rather good predictions to the CMS data, when we use the fit to the STAR data on \(\phi _{pp}\) distribution depicted on Fig. 5b, but we have to take into account interference with resonances to see the full picture. These main open problems regarding the model parameters are related to interference terms (we have to know all couplings of pions to resonances), which require full spectrum simulation comparisons with data simultaneously in several differential observables. This should be done in further work.

After estimations of the Pomeron–Pomeron to pion–pion cross-section in the framework of covariant reggeization approach we obtain cross-sections which are (at least in the case of non-conserved Pomeron currents) much lower (\(\sim 0.1\div 5\,\upmu \mathrm{b}\)) than the total Pomeron–Pomeron cross-section, estimated by other authors [32, 33, 34, 35, 36] (\(\sim 100{-}300\,\upmu \mathrm{b}\)), which shows strong contributions from other processes (especially from resonance production). For the case of conserved currents we need re-evaluate old results [32, 33, 34, 35, 36] in the same approach.

## Footnotes

- 1.
Let us note that the authors of [32, 33, 34, 35, 36] extracted \(\sigma ^{tot}_{\mathbb {P}\mathbb {P}}\) in the classical approach, which corresponds to the case of non-conserved currents in the present paper, which is why we should compare their result with the solid curve on Fig. 15b, which gives \(\sigma _{\mathbb {P}\mathbb {P}\rightarrow \pi \pi }\sim 0.1\div 5~\upmu \mathrm{b}\) for \(\sqrt{s}<3~\hbox {GeV}\) and \(|\hat{t}|=0.1~\hbox {GeV}^2\) as found in [32, 33, 34, 35, 36]. To compare our results for the case of conserved currents we need to use the same approach also when we extract \(\sigma ^{tot}_{\mathbb {P}\mathbb {P}}\)

- 2.
Of course, they have fitted the data on pion–proton elastic cross-sections, using the Born term only, but in our approach we use the full eikonalized amplitude. That is why their description of LM CEDP (in the RB mode) is also good, but only with their own parameters. In our approach the RB mode is not good in the data description.

## Notes

### Acknowledgements

I am grateful to Vladimir Petrov and Anton Godizov for useful discussions.

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