Central exclusive diffractive p $\bar{p}$ production in the Regge-eikonal model in the “scalar” proton approximation

Abstract

Central exclusive diffractive production (CEDP) of proton–anti-proton pairs is calculated in the Regge-eikonal approach taking into account continuum and possible $f_0(2100)$ resonance. We use the simple model with the “scalar” proton. Data from ISR and STAR are analysed and compared with theoretical descriptions. Some predictions for the LHC at 13 TeV are also presented. We briefly discuss possible nuances, problems and prospects for investigations of this process at present and future hadron colliders.

1 Introduction

As is often mentioned in papers devoted to the process $p+p\rightarrow p+X+p$, low-mass central exclusive diffractive production (LM CEDP) of resonances and di-hadron continua offers many advantages for the study of hadronic diffraction:

• LM CEDP is a tool for the investigation of hadronic resonances (like $f_2$ or $f_0$) and their decays to hadrons. We can extract different couplings of these resonances to reggeons (pomeron, odderon, etc.) to understand their nature (structure and the interaction mechanisms).

• We can use LM CEDP to fix the procedure for calculation of “rescattering” (unitarity) corrections. For example, in the case of p $\bar{p}$ production, we have corrections in the initial proton–proton and final proton–proton and proton–anti-proton channels.

• Basic hadrons (pion, proton) are the most fundamental particles in the strong interaction, and LM CEDP provides us with a powerful tool to probe deep inside their properties, especially to investigate the form factor and scattering amplitudes for the off-shell (“virtual”) hadron.

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

• As was proposed in [1, 2], it is possible to extract some reggeon–hadron cross-sections. In the LM CEDP of the p $\bar{p}$ we can analyse properties of the pomeron–pomeron to p $\bar{p}$ exclusive cross-section.

• Diffractive patterns of CEDP processes are very sensitive to different approaches (subamplitudes, form factors, unitarization, reggeization procedures), especially differential cross-sections in t and $\phi _{pp}$ (azimuthal angle between final protons), and also $M_{p \bar{p}}$ dependence. That is why these processes are used to verify different models of diffraction [3, 4].

• In particular in the case of LM CEDP of p $\bar{p}$, we have additional possibilities to investigate spin effects, helicity amplitudes and baryon trajectories, to search for the odderon and extract its coupling to the proton.

• All the above items are additional advantages provided by the LM CEDP, which has the typical properties of CEDP: clear signature with two final protons and two large rapidity gaps (LRG) [5, 6] and the possibility to use the “missing mass method” [7].

Processes of the LM CEDP of di-hadrons have been calculated in various works [8,9,10,11,12,13,14,15,16,17,18,19] which are devoted to most popular models. Authors have considered phenomenological, nonperturbative, perturbative and mixed approaches in Reggeon–Reggeon collision subprocesses. The nuances of some approaches were analysed in the introduction of [1].

In our recent works [1, 20] we considered the LM CEDP with production of two pions. Here we consider another possible process, namely, LM CEDP of the p $\bar{p}$ system via resonance and continuum mechanisms. This process was also considered in [19] in the tensor pomeron model.

In this article we consider the case depicted in Fig. 1 and compare the calculations against the data from ISR [21,22,23] and STAR [24,25,26,27]. We also make predictions for the LHC.

In the first part of the present work we introduce the framework for calculations of the di-baryon LM CEDP (kinematics, amplitudes, differential cross-sections) in the Regge-eikonal approach, which was considered in detail in [1, 20]. Here we consider the proton (anti-proton) as a “scalar”, and so we cannot calculate specific spin effects (as was done in [19]). As one can see from Figs. 2 and 3 of [19], cross-sections of spin-1/2 and “zero-spin” protons may differ by several times of magnitude. Our goal at this stage is to make preliminary estimations of some general distributions. Extension of the model to particles with any spin will be discussed in further theoretical works. In this article, calculations are almost the same as in our previous works [1, 20], with some modifications of the amplitudes.

In the second part, we analyse the experimental data on the process at different energies and compare it with our predictions.

To avoid complicated expressions in the main text, all the basic formulae are provided in the appendices.

The purpose of this work is to give experimentalists some hint of what we can expect in the LM CEDP of $p\bar{p}$: possible magnitudes of the couplings and cross-sections (from continuum and resonances), the difference between di-pion and other di-hadron production, how large the spin effects could be, contributions of secondary reggeons or odderon and so on.

Fig. 1

Amplitudes of the process of LM CEDP $p+p\rightarrow p+p\bar{p}+p$ in the Regge-eikonal approach for continuum (a, b), LM CEDP of $f_0$ resonances (c) with subsequent decay to $p\bar{p}$. a, b Central part of the diagram is the continuum CEDP amplitude, where $T_{pp}$, $T_{p\bar{p}}$ are full elastic proton–proton(anti-proton) amplitudes, and the proton propagator is depicted as a dashed zigzag line. c Central part of the diagram contains pomeron–pomeron-resonance fusion with subsequent decay to $p\bar{p}$; the propagator is taken in the Breit–Wigner approximation. Off-shell proton form factor on a, b and other suppression form factors (in the pomeron–pomeron-f or the $p\bar{p}f_0$) on c are presented as black circles. Full unitarized amplitude (d) contains proton–proton rescatterings in the initial and final states, which are depicted as $V_{pp}$ and $V'_{pp}$-blobs, respectively, and proton–proton(anti-proton) rescattering corrections, which are also shown as $S_{pp,p\bar{p}}$-blobs

2 General framework for calculations of LM CEDP

LM CEDP is the first exclusive two-to-four process which is basically driven by the pomeron–pomeron fusion subprocess. It serves as a clear process for investigation of resonances like $f_0$, $f_2$ and others with masses less than 5 GeV. At the moment, for low central masses, the use of the perturbative approach is a huge problem; therefore, we apply the Regge-eikonal method for all the calculations. For proton–proton and proton–anti-proton elastic amplitudes, we use the model in [28, 29], which describes all available experimental data on elastic scattering.

2.1 Components of the framework

The LM CEDP process can be calculated in the following scheme (see Fig. 1):

• 1. We calculate the primary amplitudes of the processes, which are depicted as central parts of diagrams in Fig. 1. Here we consider the case where the bare off-shell proton propagator in the amplitude for continuum $p\bar{p}$ production is taken in its simple form (without reggeization)

  $$\begin{aligned} \mathcal{P}_{p}(\hat{t})=1/(\hat{t}-m_p^2), \end{aligned}$$

  (1)

  where $\hat{t}$ is the square of the momentum transfer between a pomeron and a proton in the pomeron–pomeron fusion process (see Appendix A for details). In the general case, which will be considered in future investigations, we have to do possible reggeization of the spinor proton with the proton trajectory taken, for example, from [30]:

  $$\begin{aligned} \alpha _{p}(\hat{t})=-0.4+0.9\hat{t}+0.125\hat{t}^2. \end{aligned}$$

  As was noted at the beginning, here we consider the proton (anti-proton) as a “spin-0 particle”. Reggeization of the virtual proton propagator is not obvious, since the effect of this is expected to be small, and moreover it is not even clear that we are in the relevant kinematic region ($|\hat{t}|\ll \hat{s}=M_{p\bar{p}}^2$) to include such corrections for central production. This was also verified in the calculations presented in this paper. For example, we can use the replacement

  $$\begin{aligned} \frac{1}{\hat{t}-m_{p}^2}\rightarrow \frac{\textrm{e}^{\alpha _{p}(\hat{t})|\Delta Y|}}{\hat{t}-m_{p}^2}, \end{aligned}$$

  (2)

  as was done in [8,9,10]. This expression gives correct “reggeized” behaviour in the relevant kinematic region, and the usual “bare” proton propagator behaviour for small difference between rapidities of the final central proton (anti-proton). As to the authors of [11,12,13], we could use the phenomenological expression for the virtual proton propagator like (see (3.25), (3.26) of [13] for the pion propagator)

  $$\begin{aligned}{} & {} \frac{1}{\hat{t}-m_{p}^2} F(\Delta Y) + (1-F(\Delta Y)) \mathcal{P}_{p}(\hat{s},\hat{t}),\nonumber \\ {}{} & {} F(\Delta Y)=\textrm{e}^{-c_y \Delta Y},\, \Delta Y=y_{p}-y_{\bar{p}}, \end{aligned}$$

  (3)

  to take into account possible non-Regge behaviour for $\hat{t}\sim \hat{s}/2$, i.e. for small rapidity separation $\Delta Y$ between the final proton and anti-proton. The Regge model really does not work in this area, or it needs to be modified (as was done, for example, in [11,12,13,14,15,16,17,18], with empirical formulae or additional assumptions). We also use full eikonalized expressions for proton–proton and proton–anti-proton amplitudes, which can be found in Appendix B.

• 2. After the calculation of the primary LM CEDP amplitudes, we have to take into account all possible corrections in proton–proton and proton–anti-proton elastic channels due to the unitarization procedure (so-called soft survival probability or rescattering corrections), which are depicted as $V_{pp}$, $V'_{pp}$ and $S_{pp,p\bar{p}}$ blobs in Fig. 1. For proton–proton and proton–anti-proton elastic amplitudes we use the model of [28, 29] (see Appendix B). The possible final interaction between hadrons of the central system is not shown in Fig. 1, since we neglect it in the present calculations.

In this article we do not consider so-called enhanced corrections [8,9,10], since they give nonleading contributions in our model due to the smallness of the triple pomeron vertex. Additionally, we have no possible absorptive corrections inside the $p\bar{p}$ central system, because the central mass is low, and there is also a lack of data on this process to define parameters of the model.

Exact kinematics of the two-to-four process for our case is outlined in Appendix A.

Here we use the model presented in Appendix B as an example. One could use other models which are proved to effectively describe all the available data on proton–proton and proton–anti-proton elastic processes.

2.2 Continuum $p\bar{p}$ production

The final expression for the amplitude for the continuum $p\bar{p}$ production with initial proton-proton and final proton–(anti-)proton “rescattering” corrections (see Fig. 1a, b) can be written as

$$\begin{aligned}{} & {} M^U\left( \{ p \}\right) \nonumber \\{} & {} \quad =\int \int \frac{d^2\textbf{q}}{(2\pi )^2}\frac{d^2\textbf{q}^{\prime }}{(2\pi )^2} \frac{d^2\textbf{q}_1}{(2\pi )^2}\frac{d^2\textbf{q}_2}{(2\pi )^2} V_{pp}(s,q^2) V_{pp}(s^{\prime },q^{\prime 2})\nonumber \\{} & {} \qquad \times \; \left[ S_{p\bar{p}}(\tilde{s}_{14},q_1^2) M_0\left( \{ \tilde{p}\} \right) S_{pp}(\tilde{s}_{23},q_2^2)+ (3\leftrightarrow 4) \right] \end{aligned}$$

(4)

$$\begin{aligned}{} & {} M_0\left( \{ p \}\right) =T^{el}_{pp}(s_{13},t_1) \mathcal{P}_{p}(\hat{t}) \left[ \hat{F}_{p}\left( \hat{t}\right) \right] ^2 T^{el}_{p\bar{p}}(s_{24},t_2), \end{aligned}$$

(5)

where functions are defined in (30)–(31) of Appendix B, and sets of vectors are

$$\begin{aligned}{} & {} \{ p \}\equiv \{ p_a,p_b,p_1,p_2,p_3,p_4\} \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \{ \tilde{p}\}\equiv \{ p_a-q,p_b+q; p_1+q^{\prime }+q_1,\nonumber \\ {}{} & {} \;\;\;\;\;\;\;\;\;\;\;\; p_2-q^{\prime }+q_2,p_3-q_2,p_4-q_1 \} , \end{aligned}$$

(7)

and

$$\begin{aligned} \tilde{s}_{14}= & {} \left( p_1+p_4+q^{\prime } \right) ^2,\; \tilde{s}_{23}=\left( p_2+p_3-q^{\prime } \right) ^2, \end{aligned}$$

(8)

$$\begin{aligned} s_{ij}= & {} \left( p_i+p_j \right) ^2,\; t_{1,2}=\left( p_{a,b}-p_{1,2} \right) ^2, \end{aligned}$$

(9)

$$\begin{aligned} \hat{s}= & {} \left( p_3+p_4 \right) ^2,\; \hat{t}=\left( p_a-p_1-p_3 \right) ^2 \end{aligned}$$

(10)

The off-shell proton form factor is equal to unity on mass shell $\hat{t}=m_p^2$ and taken as exponential

$$\begin{aligned} \hat{F}_{p}=\textrm{e}^{(\hat{t}-m_p^2)/\Lambda _p^2}, \end{aligned}$$

(11)

where $\Lambda _p\sim 1$ GeV is taken from the fits to the LM CEDP of $p\bar{p}$ at low energies (see next section). In this paper we use only the exponential form, but it is possible to use other parametrizations (see [19]).

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

2.3 CEDP of low mass resonances

Here we consider, for example, only one $f_0$ resonance, say, $f_0(2100)$, just to see the final picture and possible changes in the diffractive patterns. The general unitarized amplitude (see Fig. 1c) is similar to the expression (4), where the amplitude $M_0\left( \{ p \}\right)$ is replaced by the corresponding central primary amplitude for the resonance production and further decay to $p\bar{p}$.

For the $f_0(2100)$ meson, the amplitude is constructed from the proton–pomeron form factor, pomeron–pomeron coupling to the meson,¹ the off-shell propagator, the off-shell form factor and the decay vertex.

The primary amplitude is given in Appendix C.

2.4 Nuances of calculations

In the next section one can see that there are some difficulties in the data fitting. In this subsection let us discuss some nuances of calculations which could change the situation.

We have to pay special attention to amplitudes where one or more external particles are off their mass shell. An example of such an amplitude is the proton–(anti-)proton $T_{p p}$ ($T_{p\bar{p}}$), which is 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 baryons is taken into account by the additional phenomenological form factor $\hat{F}_{p}(\hat{t})$. However, there are at least two other possibilities.

The first one was considered in [31]. For an amplitude with one off-shell particle, the formula

$$\begin{aligned} T^*(s,b)=\frac{\delta ^*(s,b)}{\delta (s,b)}T(s,b)=\frac{\delta ^*(s,b)}{\delta (s,b)} \frac{\textrm{e}^{2\textrm{i}\delta (s,b)}-1}{2\textrm{i}} \end{aligned}$$

(12)

was used. In our case,

$$\begin{aligned}{} & {} \delta (s,b) = \delta _{p p,p\bar{p}}(s,b; m_p^2,m_p^2,m_p^2,m_p^2),\nonumber \\{} & {} \delta ^*(s,b) = \delta _{p p,p\bar{p}}^*(s,b; \hat{t},m_p^2,m_p^2,m_p^2)\nonumber \\{} & {} \left. \delta _{p p,p\bar{p}}=\delta ^*_{p p,p\bar{p}}\right| _{\hat{t}\rightarrow m_p^2}. \end{aligned}$$

(13)

$\delta _{p p,p\bar{p}}$ is the eikonal function (see (28)). This is similar to the introduction of the additional form factor, but in a more consistent way, which takes into account the unitarity condition.

The second one arises from the covariant reggeization method, which was considered in Appendix C of [1]. For the case of conserved hadronic currents, we have a definite structure in the Legendre function, which is transformed in a natural way to the case of the off-shell amplitude. In this case, however, off-shell amplitude has a specific behaviour at low t values (see [4] for details). As was shown in [4], unitarity corrections can mask this behaviour. Also, in this case we have to take into account the spinor nature of the proton and modify the covariant reggeization approach presented in  [1].

3 Data from hadron colliders versus results of calculations

Our basic task is to extract the fundamental information on the interaction of hadrons from different cross-sections (“diffractive patterns”):

• From t-distributions we can obtain the size and shape of the interaction region.

• The distribution on the azimuthal angle between final protons gives the quantum numbers of the produced system (see [4, 32] and references therein).

• From $M_c$ (here $M_c=M_{p\bar{p}}$) dependence and its influence on t-dependence, we can make some conclusions about the interaction at different space–time scales and the interrelation between them. We can also extract couplings of reggeons to different resonances.

The process $p+p\rightarrow p+p\bar{p}+p$ is one of the basic “standard candles” which we can use to estimate other CEDP processes. In this section we consider the available experimental data on the process and attempt to extract the information on couplings and form factors.

Fig. 2

The new data on the process $p+p\rightarrow p+p\bar{p}+p$ at $\sqrt{s}=200$ GeV (STAR collaboration [24,25,26,27]): $|\eta _{p,\bar{p}}|<0.7$, $p^{p}_{T}(p^{\bar{p}}_T)>0.4$ GeV, $\min (p^{p}_{T},p^{\bar{p}}_T)<1.1$ GeV, $p_x>-0.2$ GeV, 0.2 GeV$<|p_y|<0.4$ GeV, $(p_x+0.3\,\textrm{GeV})^2+p_y^2<0.25$ GeV$^2$, where p denotes the momenta of final forward protons. Curves correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. Curves correspond to the sum of all amplitudes. The thickness of the curves shows the errors of numerical Monte-Carlo calculations. Additional interpolation was used between calculated points for smoothing. In figure a the continuum contribution is also presented to see the impact of the resonance, which is not so strong for the given values of couplings. $\phi _0$ in b is the azimuthal angle between the final forward protons

3.1 STAR data versus the model distributions

In this subsection, the data from the STAR collaboration [24,25,26,27] and model curves for all the cases of Fig. 1 are presented. In our approach, we have two free parameters, namely, $\Lambda _p$ (for the continuum) and the coupling of $f_0(2100)$ to pomeron $g_{\mathbb{P}\mathbb{P}f}$, which we can extract from the data or fix from some model assumptions. All the distributions are depicted for $\Lambda _p=1.12$ GeV. Here we take only $f_0(2100)$ resonance, with pomeron–pomeron-$f_0(2100)$ coupling equal to 0.64 (this value is inspired by the paper on possible “glueball” states [33]), and the value 3.0 is taken for the $f_0(2100)$-p-$\bar{p}$ just to have their product $\sim 1$, since we do not exactly know these constants from the existing data.

As one can see from Fig. 2, we can describe the data rather well. From these data we fix the parameter $\Lambda _p = 1.12$ GeV, and then we can make some predictions for other energies. A significant difference is seen only in the region of small central masses and small $|t_1+t_2|$. One can see from [1] that $M_c$ distributions (values and shape) are very sensitive to the value of $\Lambda _p$, while in other distributions only the values change while the shape remains almost unchanged.

In Fig. 3 we see predictions for the STAR data at $\sqrt{s}=510$ GeV. For the normalized cross-section we see a good description of the distribution shape. Unfortunately, from these data we cannot make any conclusions on the absolute value of the cross-section versus predictions, since for now there are no official publications on the integrated luminosity in this case. Therefore, a more correct comparison is postponed for further publications.

Fig. 3

a The new data (normalized cross-section) on the process $p+p\rightarrow p+p\bar{p}+p$ at $\sqrt{s}=510$ GeV (STAR collaboration [27]): $|\eta _{p,\bar{p}}|<0.7$, $p^{p}_{T}(p^{\bar{p}}_T)>0.4$ GeV, $\min (p^{p}_{T},p^{\bar{p}}_T)<1.1$ GeV, $p_x>-0.27$ GeV, 0.4 GeV$<|p_y|<0.8$ GeV, $(p_x+0.6\,\textrm{GeV})^2+p_y^2<1.25$ GeV$^2$, where p denotes the momenta of final forward protons. Curves correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. The thick solid curve corresponds to the sum of all amplitudes; the thickness of the curves shows the errors of numerical calculations

Fig. 4

The data on the process $p+p\rightarrow p+p\bar{p}+p$ (WA102 collaboration [21]): at $\sqrt{s}=29.1$ GeV, $|p^c_x|<14$ GeV,$|p^c_y|<0.16$ GeV,$|p^c_z|<0.08$ GeV, $\xi _{1,2\,p}>0.8$. Theoretical curves (multiplied by 740 to compare the shapes) correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. The solid curve corresponds to the sum of all amplitudes; the thickness of the curve shows the errors of numerical calculations. The theoretical curve was normalized in exactly the same way as was done in the experimental data

Fig. 5

The data (black circles) on the process $p+p\rightarrow p+p\bar{p}+p$ (ABCDHW collaboration [22]) at $\sqrt{s}=62$ GeV, $\xi _{1,2\,p}>0.9$, $|y_{p,\bar{p}}|<1.5$, $|t_{1,2}|>0.08$ GeV$^2$. Theoretical curves (multiplied by 200 to compare the shapes) correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. The solid curve corresponds to the sum of all amplitudes; the thickness of the curve shows the errors of numerical calculations. The theoretical curve was normalized in exactly the same way as was done in the experimental data. We should compare with black points (not with the histogram)

Fig. 6

The data on the process $p+p\rightarrow p+p\bar{p}+p$ (AFS collaboration [23]) at $\sqrt{s}=63$ GeV, $\xi _{1,2\,p}>0.95$, $|y_{p,\bar{p}}|<1$, 0.01 GeV$^2<|t_{1,2}|<0.06$ GeV$^2$. The theoretical curves correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. The solid curve corresponds to the sum of all amplitudes; the thickness of the curve shows the errors of numerical calculations. $\sigma _{\textrm{exp}}=2.5\pm 1.25$ nb was calculated from $\textrm{d}\sigma /\textrm{d}t_1\textrm{d}t_2\sim 1\pm 0.5\;\upmu$b GeV$^{-4}$ (see text). The theoretical curve was normalized in exactly the same way as was done in the experimental data

3.2 Low-energy data versus model cases

If we take a look at the low-energy data, we can discover several significant contradictions similar to those found in the production of two pions [20]. This is obvious if we use the data of the WA102 collaboration [21] at $\sqrt{s}=29.1$ GeV (see Fig. 4). The figure shows that the shape of the predicted distribution is close to the experimental one, but the discrepancy by a factor of 720 (integrated cross-section is about 260 pb, and in [21] we have 186 nb), which indicates that the approach used in this energy range fails, and we must take into account other mechanisms for the CEDP of proton–antiproton pairs. It is possible that resonant production plays a key role here, as was pointed out earlier [19], especially at low masses of the central system, and also contributions to the continuum from the resonances below threshold. Also, at low energies, spin effects play a significant role, although their contribution will most likely yield an increase of no more than $\sim 2\div 3$ times (see, for estimations, Figs. 2, 3 of [19]).

The situation is almost the same when we try to compare predictions with the data from the ABCDHW collaboration [22] at $\sqrt{s}=62$ GeV (see Fig. 5). The quality of the data is not so good, errors are large. The experimental integrated cross-section in this case is 0.8±0.17 $\upmu$b, and predictions give only 0.002 $\upmu$b, i.e. about 400 times lower. In this case $|t_{1,2}| > 0.08$ GeV$^2$, and this cut removes the region in which the model works well. We see a similar discrepancy in [19], when one takes into account rescattering corrections.

A more interesting situation arises when we take the data from the AFS collaboration [23]. From this paper we take differential cross-sections

$$\begin{aligned} \left. \textrm{d}\sigma /\textrm{d}t_1\textrm{d}t_2 \right| _{t_{1,2}=-0.035\;\textrm{GeV}^2} = 1.0\pm 0.5 \;\upmu \textrm{b}\;\textrm{GeV}^{-4}. \end{aligned}$$

(14)

If we use a simple form $A\times \textrm{e}^{B(t_1+t_2)}$ for the differential cross-section in a very wide range of B from 3 GeV$^{-2}$ up to 12 GeV$^{-2}$, we can find for the integrated cross-section

$$\begin{aligned} \sigma _{\textrm{exp}} \simeq 2.5\pm 1.25\;\textrm{nb}. \end{aligned}$$

(15)

Theoretical calculations give

$$\begin{aligned}{} & {} \left. \textrm{d}\sigma /\textrm{d}t_1\textrm{d}t_2 \right| _{t_{1,2}=-0.035\;\textrm{GeV}^2} = 0.3 \;\upmu \textrm{b}\;\textrm{GeV}^{-4}, \end{aligned}$$

(16)

$$\begin{aligned}{} & {} \sigma _{th}= 0.75\;\textrm{nb}. \end{aligned}$$

(17)

Also, in Fig. 6 we can see that the curve has no huge discrepancy with the experimental points as in the two previous cases. The discrepancy is of the same order and even less than it was in [20] for the di-pion production at the ISR, and may be compensated by mechanisms proposed above (spin effects, reggeon contributions, resonances below the threshold). Preliminary calculations with the reggeon contribution for di-pion resonance production at low energies show that this is possible.

It is important to note here that we have some contradiction between the dataset [22] at $\sqrt{s}=62$ GeV (ABCDHW collaboration) and [23] at $\sqrt{s}=63$ GeV (AFS collaboration). Let us suppose that the cross-section of [22] has the form of the sum of two exponents

$$\begin{aligned} \textrm{d}\sigma /\textrm{d}t_1\textrm{d}t_2 = A_1\times \textrm{e}^{B_1(t_1+t_2)}+A_2\times \textrm{e}^{B_2(t_1+t_2)}, \end{aligned}$$

where the second exponent plays a more important role for $|t|>0.08$ GeV$^2$. We know from the experimental paper [22] that for $|t|>0.08$ GeV$^2$, the average |t| value is 0.3 GeV$^2$ (which gives us the approximate value of the slope parameter $\sim 4.5$ GeV$^{-2}$), and the integrated cross-section is $0.8\upmu$b, so we can express $A_{1,2}$ in terms of $B_{1,2}$. And for [23], we take the simple exponent as was done in the above calculations. Then we can calculate the integrated cross-section in the full kinematic region for both datasets. The result for [23] is $\sim 10{-}50$ nb when $B\in [4,10]$ GeV$^{-2}$, and for [22] we have $\sim {1}.{6}\div {2}.{4}\upmu$b for $B_1\in [5,16]$ GeV$^{-2}$ and $B_2\in [3,4.5]$ GeV$^{-2}$. Other kinematic cuts are almost the same. And the observed difference by two orders of magnitude looks rather strange, because the energy is almost the same. If we follow similar steps for the di-pion production, the situation is improved (all the cross-sections are of the same order of magnitude). Thus, this contradiction in two datasets is observed only for p $\bar{p}$ production. To my mind, the dataset [23] at $\sqrt{s}=63$ GeV (AFS collaboration) is more adequate, since it shows the same properties (ratios of integrated cross-sections at different energies) as we can observe in di-pion production.

We can conclude that the model has failed to describe the low-energy data if we fix parameters from the STAR data. For these low energies we have to take into account other mechanisms for the CEDP of $p\bar{p}$. Resonant production (especially the contribution of resonances below the threshold) may play a key role. Additionally, these mechanisms may include possible corrections to proton–proton(anti-proton) amplitudes at low energies, since our approach describes data well only for energies greater than $\sim 10$ GeV. And in each $T_{pp,p\bar{p}}$ amplitude in Fig. 1a, b, the energy can be even less than 5 GeV. In the present calculations, just to preliminarily and qualitatively check the effect of secondary reggeons at very low energy and to improve the situation, we use simple exponential parametrization (Born approximation with secondary reggeons) for $T_{pp,p\bar{p}}$ elastic amplitude to cover the energy value down to the threshold $\sim 2$ GeV.

Preliminary calculations, which include reggeon–pomeron and reggeon–reggeon contributions to the amplitude of resonance production (Fig. 1c), show that we can fit the model parameters for the STAR energy in such a way that for ISR energies, the theoretical prediction will be higher than the present one by a factor of 2, and the distribution in $M_c$ will be slightly shifted to larger values of central masses. Also, the effect of more pronounced peaks of resonances may be observed at low energy (as partially confirmed by Fig. 6).

3.3 Predictions for LHC

Fig. 7

Predictions for CMS energies on the process $p+\bar{p}\rightarrow p+p\bar{p}+\bar{p}$ are shown for $\sqrt{s}=13$ TeV with cuts $|\eta _{p,\bar{p}}|<2.4$, $p_{T,p,\bar{p}}>0.2$ GeV. Theoretical curves correspond to $\Lambda _{p}=1.12$ GeV in the off-shell proton form factor (11), and pomeron–pomeron-$f_0(2100)$ coupling is equal to 0.64. The solid curve corresponds to the sum of all amplitudes; the thickness of the curve shows the errors of numerical calculations

In Fig. 7 one can see the prediction for the LHC at 13 TeV for the process that corresponds to the parameter $\Lambda _p=1.12$ GeV, which better fits the STAR data on Fig. 2. The integrated cross-section for these kinematic cuts is about 35 nb, which is about three orders of magnitude smaller than the cross-section of di-pion CEDP. The number of $\pi ^+\pi ^-$ CEDP events is of the order $10^4$ for the LHC at the integrated luminosity $\sim 200\div 500\;\upmu$b$^{-1}$, which is enough to obtain precise distributions in the central mass. That is why for the CEDP of $p\bar{p}$ we should have at least $\sim 50\div 100$ nb$^{-1}$ integrated luminosity for our investigations.

4 Summary and conclusions

To conclude this article, we can summarize the above analysis in a few statements:

• 1. The result is crucially dependent on the choice of $\Lambda _p$ in the off-shell proton form factor, i.e. on $\hat{t}$ (virtuality of the proton) dependence. The $M_c$ distribution is more sensitive to this parameter. This dependence is more significant than in the CEDP of di-pions. In the present approach we take $\Lambda _p=1.12$ GeV and couplings

  $$\begin{aligned} g_{\mathbb{P}\mathbb{P}f_0(2100)}= & {} 0.64,\nonumber \\ g_{p\bar{p}f_0(2100)}= & {} 3.0, \end{aligned}$$

  (18)

  The coupling of the pomeron to $f_0(2100)$ is taken as in [33] just for tests.

• 2. If we try to fit the data from STAR [24,25,26,27], we can fix the parameter $\Lambda _p=1.12$ GeV, for which the description is quite good. The $f_0(2100)$ resonance contribution for the given values of couplings is rather small.

• 3. For the ISR energies, the situation is quite contradictory:

  • at $\sqrt{s}=29$ GeV we observe a difference of almost three orders of magnitude (although the shape of the theoretical and experimental distributions is the same);

  • at $\sqrt{s}=62$ GeV, when $|t_{1,2}|>0.08\; \textrm{GeV}^2$, the difference is already of the order of 200;

  • at $\sqrt{s}=63$ GeV, for $0.01\; \textrm{GeV}^2<|t_{1,2}|<0.06\;\textrm{GeV}^2$, the predictions turn out to be only about $2\div 3$ times smaller.

  We also have unexplained contradictions between the two datasets at the same energy. This has to be explained somehow. We can assume that

  • additional resonant production (below and above the threshold) plays a key role, but now we cannot make any definite inferences about its value;

  • spin effects may be rather large (several times);

  • contributions from reggeon–pomeron and reggeon–reggeon resonance production may also be large (factor of 2) at low energies (for large values of couplings);

  • contributions from other processes ($\gamma \gamma \rightarrow p\bar{p}$, $\gamma \mathbb {O}\rightarrow p\bar{p}$, single and double dissociation) must also be taken into account.

  • There are also effects related to the irrelevance and possible modifications of the Regge approach (for the virtual proton exchange) in this kinematic region, corrections to $T_{pp,p\bar{p}}(s,t)$ for $\sqrt{s}<5$ GeV, and corrections to proton–anti-proton scattering at low $M_{p\bar{p}}$;

• 4. Based on the predictions for $\sqrt{s}=13$ TeV, we can say that in order to study this process at the LHC, we need minimum integrated luminosity of the order of $\sim 50\div 100$ nb$^{-1}$.

In further works we will take into account possible modifications of the model (proton–anti-proton low-energy cross-section, additional off-shell effects in subamplitudes, spin effects, contributions from dissociative processes, contributions of other resonances below and above the threshold, contributions of reggeons to the resonance amplitude at low energies) for the best description of the data. This model will be implemented in the new version of the Monte Carlo event generator ExDiff [34]. It is possible to calculate LM CEDP for other di-hadron final states ($\phi \phi$, $K^+K^-$, $\eta \eta ^{\prime }$ etc.) which are also very informative for our understanding of diffractive mechanisms in strong interactions.

Appendices

Appendix A: Kinematics of LM CEDP

The $2\rightarrow 4$ process $p(p_a)+p(p_b)\rightarrow p(p_1)+p(p_3)+\bar{p}(p_4)+p(p_2)$ can be described as follows (the notation for any momentum is $k=(k_0,k_z;\textbf{k})$, $\textbf{k}=(k_x,k_y)$):

$$\begin{aligned}{} & {} p_a=\left( \frac{\sqrt{s}}{2}, \beta \frac{\sqrt{s}}{2}; \textbf{0} \right) ,\; p_b=\left( \frac{\sqrt{s}}{2}, -\beta \frac{\sqrt{s}}{2}; \textbf{0} \right) ,\nonumber \\ {}{} & {} p_{1,2}= \left( E_{1,2},p_{1,2z}; \textbf{p}_{1,2\perp } \right) , E_{1,2}=\sqrt{p_{1,2z}^2+\textbf{p}_{1,2\perp }^2+m_p^2},\;\nonumber \\{} & {} p_{3,4}= \left( m_{3,4\perp }\textrm{ch}\;y_{3,4}, m_{3,4\perp }\textrm{sh}\;y_{3,4}; \textbf{p}_{3,4\perp } \right) \nonumber \\ {}{} & {} \qquad \quad =\left( \sqrt{m_{p}^2+\textbf{p}_{3,4\perp }^2\textrm{ch}^2\eta _{3,4}}, |\textbf{p}_{3,4\perp }|\;\textrm{sh}\;\eta _{3,4}; \textbf{p}_{3,4\perp } \right) , \nonumber \\ {}{} & {} m_{i\perp }^2=m_i^2+\textbf{p}_{i\perp }^2,\; m_{1,2}=m_p,\; m_{3,4}=m_{h}, \nonumber \\{} & {} \textbf{p}_{4\perp }=-\textbf{p}_{3\perp }-\textbf{p}_{1\perp }-\textbf{p}_{2\perp },\; \nonumber \\ {}{} & {} \beta =\!\sqrt{1-\frac{4m_p^2}{s}},\; s=(p_a\!+\!p_b)^2,\; s^{\prime }\!=\!(p_1+p_2)^2. \end{aligned}$$

(19)

Here, $y_i$ ($\eta _i$) are rapidities (pseudorapidities) of final hadrons.

The phase space of the process in terms of the above variables is as follows:

$$\begin{aligned}{} & {} \textrm{d}\mathrm {\Phi }_{2\rightarrow 4} = \left( 2\pi \right) ^4\delta ^4\left( p_a+p_b-\sum _{i=1}^4p_i\right) \prod _{i=1}^4 \frac{d^3p_i}{(2\pi )^32E_i} \nonumber \\ {}{} & {} \qquad \quad = \frac{1}{2^4(2\pi )^8} \prod _{i=1}^3 p_{i\perp }\textrm{d}p_{i\perp }\textrm{d}\phi _i\cdot \textrm{d}y_3\textrm{d}y_4 \cdot \mathcal{J}; \nonumber \\ {}{} & {} \mathcal{J}= \frac{\textrm{d}p_{1z}}{E_1} \frac{\textrm{d}p_{2z}}{E_2} \delta \left( \sqrt{s}-\sum _{i=1}^4 E_i\right) \delta \left( \sum _{i=1}^4 p_{iz}\right) \nonumber \\ {}{} & {} \qquad \quad =\frac{1}{\left| \tilde{E}_2\tilde{p}_{1z}-\tilde{E}_1\tilde{p}_{2z}\right| }, \end{aligned}$$

(20)

where $p_{i\perp }=\left| \textbf{p}_{i}\right|$, $\tilde{p}_{1,2z}$ are appropriate roots of the system

$$\begin{aligned}{} & {} {\left\{ \begin{array}{ll} &{}A=\sqrt{s}-E_3-E_4=\sqrt{m_{1\perp }^2+p_{1z}^2}+\sqrt{m_{2\perp }^2+p_{2z}^2},\\ &{}B=-p_{3z}-p_{4z}=p_{1z}+p_{2z}, \end{array}\right. } \end{aligned}$$

(21)

$$\begin{aligned}{} & {} \tilde{p}_{1z}= \frac{B}{2}+\frac{1}{2(A^2-B^2)}\left[ B\left( m_{1\perp }^2-m_{2\perp }^2\right) + A\cdot \lambda _0^{1/2} \right] ,\nonumber \\{} & {} \lambda _0=\lambda \left( A^2-B^2,m_{1\perp }^2,m_{2\perp }^2\right) . \end{aligned}$$

(22)

Here $\lambda (x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz$, and then $\mathcal{J}^{-1}=\lambda _0^{1/2}/2$.

For the differential cross-section we have

$$\begin{aligned} \frac{\textrm{d}\sigma _{2\rightarrow 4}}{\prod _{i=1}^{3} \textrm{d}p_{i\perp }\textrm{d}\phi _i\cdot \textrm{d}y_3\textrm{d}y_4 }= & {} \frac{1}{2\beta s}\cdot \frac{\prod _{i=1}^3 p_{i\perp }}{2^4(2\pi )^8\cdot \frac{1}{2}\lambda _0^{1/2}} \left| T\right| ^2 \nonumber \\= & {} \frac{\prod _{i=1}^3 p_{i\perp }}{2^{12}\pi ^8\beta s \lambda _0^{1/2}}\left| T \right| ^2. \end{aligned}$$

(23)

Pseudorapidity is a more convenient experimental variable, and we can use the transform

$$\begin{aligned} \frac{\textrm{d}y_i}{\textrm{d}\eta _i}=\frac{p_{i\perp }\textrm{ch}\eta _i}{\sqrt{m_i^2+p_{i\perp }^2\textrm{ch}^2\eta _i}} \end{aligned}$$

(24)

to obtain the differential cross-section in pseudorapidities.

In some cases it is convenient to use other variables for the integration of the cross-section and calculation of distributions on central mass. For these cases we have

$$\begin{aligned}{} & {} \textrm{d}\mathrm {\Phi }_{2\rightarrow 4} = \frac{1}{2^4(2\pi )^8} \prod _{i=1}^2 \textrm{d}t_i \textrm{d}\phi _i\cdot M_c dM_c \textrm{d}\eta _c dc^* \textrm{d}\phi ^*\cdot \mathcal{J}^{\prime }; \nonumber \\ {}{} & {} \mathcal{J}^{\prime }= \frac{\beta _M}{4\beta _1\beta _2 s} \frac{\textrm{d}y_c}{\textrm{d}\eta _c},\, \,\,\,\, \beta _i \simeq \sqrt{1+\frac{4(m_p^2-(1-\xi _i)t_i)}{\beta ^2s^2(1-\xi _i)^2}}; \nonumber \\ {}{} & {} \beta _M = \sqrt{1-\frac{4m_{p}^2}{M_c^2}},\, \, \frac{\textrm{d}y_c}{\textrm{d}\eta _c}=\frac{p_{c\perp }\textrm{ch}\eta _c}{\sqrt{M_c^2+p_{c\perp }^2\textrm{ch}^2\eta _c}}; \end{aligned}$$

(25)

$$\begin{aligned}{} & {} \frac{\textrm{d}\sigma _{2\rightarrow 4}}{\prod _{i=1}^{2} \textrm{d}t_{i}\textrm{d}\phi _i dM_c \textrm{d}\eta _c dc^* \textrm{d}\phi ^*}= \frac{1}{2\beta s}\cdot \frac{M_c \beta _M \frac{\textrm{d}y_c}{\textrm{d}\eta _c}}{2^4(2\pi )^8\cdot 4\beta _1\beta _2 s} \left| T\right| ^2 \nonumber \\{} & {} \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \quad = \frac{M_c \beta _M \frac{\textrm{d}y_c}{\textrm{d}\eta _c}}{2^{15}\pi ^8\beta \beta _1\beta _2 s^2 }\left| T \right| ^2, \end{aligned}$$

(26)

where $c^*=\cos \theta ^*$, $\theta ^*$ and $\phi ^*$ are polar and azimuthal angles of the central hadron momenta in the $h\bar{h}$ rest frame, $M_c$ is the di-hadron mass, $\eta _c$ is the di-hadron pseudorapidity, $t_1=(p_a-p_1)^2$, $t_2=(p_b-p_2)^2$ and

$$\begin{aligned} \xi _{1,2}\simeq \sqrt{\frac{M_c^2-t_1-t_2+2\sqrt{t_1t_2}\cos (\phi _1-\phi _2)}{s}} \text{ e}^{\pm y_c}. \end{aligned}$$

Fig. 8

Total amplitude of the process of di-hadron LM CEDP $p+p\rightarrow p+h\bar{h}+p$ with detailed kinematics. Proton–proton rescatterings in the initial and final states are depicted as black blobs, and hadron-proton subamplitudes are also shown as shaded blobs. All momenta are shown. The basic part of the amplitude, $M_0$ (see Eq. (5)), without corrections is circled by a dotted line. Crossed lines are on mass shell. Here, $\Delta _{1\perp }=\Delta _1-q-q^{\prime }$, $\Delta _{2\perp }=\Delta _2+q+q^{\prime }$, $\hat{t}=k^2=(\Delta _{1\perp }-q_1-p_3+q_2)^2$, $\hat{u}=(\Delta _{1\perp }-q_1-p_4)^2$, $\hat{s}=(p_3+p_4-q_1-q_2)^2$

For exact calculations of elastic subprocesses (see Fig. 8) of the type

$a(p_1)+b(p_2)\rightarrow c(p_1-q_{el})+d(p_2+q_{el})$:

$$\begin{aligned}{} & {} q_{el}=\left( q_0,q_z; \textbf{q}\right) ,\nonumber \\{} & {} q_z=-\frac{b}{2a}\left( 1-\sqrt{1-\frac{4ac}{b^2}}\right) ,\nonumber \\{} & {} q_0=\frac{A_0q_z+\textbf{p}_{1\perp }\textbf{q}+\textbf{p}_{2\perp }\textbf{q}}{A_z},\nonumber \\{} & {} a=A_z^2-A_0^2,\; b=-2 \left( A_z\cdot \mathcal{D}+A_0 \left( \textbf{p}_{1\perp }\textbf{q}+\textbf{p}_{2\perp }\textbf{q}\right) \right) ,\nonumber \\{} & {} c=2 A_z B_z-\left( \textbf{p}_{1\perp }\textbf{q}+\textbf{p}_{2\perp }\textbf{q}\right) ^2+\textbf{q}^2A_z^2,\nonumber \\{} & {} A_0=p_{1z}+p_{2z},\; A_z=p_{10}+p_{20}, \nonumber \\{} & {} B_0=p_{1z}\cdot \textbf{p}_{2\perp }\textbf{q}-p_{2z}\cdot \textbf{p}_{1\perp }\textbf{q},\nonumber \\{} & {} B_z=p_{10}\cdot \textbf{p}_{2\perp }\textbf{q}-p_{20}\cdot \textbf{p}_{1\perp }\textbf{q},\nonumber \\{} & {} \mathcal{D}=p_{1z}p_{20}-p_{2z}p_{10}, \end{aligned}$$

(27)

and $q_{el}^2\simeq -\textbf{q}^2$.

Appendix B: Regge-eikonal model for elastic proton-proton and proton–anti-proton scattering

The following is a short review of formulae for the Regge-eikonal approach [28, 29] which we use to estimate rescattering corrections in the proton–proton and pion–proton channels.

Amplitudes of elastic proton–proton and proton–anti-proton scattering are expressed in terms of eikonal functions

$$\begin{aligned}{} & {} T_{pp,p\bar{p}}^{el}(s,b)=\frac{\textrm{e}^{-2\Omega _{pp,p\bar{p}}^{el}(s,b)}-1}{2\textrm{i}},\nonumber \\{} & {} \Omega _{pp, p\bar{p}}^{el}(s,b) = -\textrm{i}\,\delta _{pp,p\bar{p}}^{el}(s,b),\nonumber \\{} & {} \delta ^{el}_{pp,p\bar{p}}(s,b)=\frac{1}{16\pi s}\int _0^\infty d(-t) J_0(b\sqrt{-t}) \delta ^{el}_{pp,p\bar{p}}(s,t). ,\nonumber \\ \end{aligned}$$

(28)

$$\begin{aligned}{} & {} \delta ^{el}_{pp,p\bar{p}}(s,t) \simeq g_{pp\mathbb {P}}(t)^2 \left( \textrm{i}+\tan \frac{\pi (\alpha _{\mathbb {P}}(t)-1)}{2} \right) \nonumber \\ {}{} & {} \qquad \quad \qquad \quad \times \pi \alpha ^{\prime }_{\mathbb {P}}(t) \left( \frac{s}{2s_0}\right) ^{\alpha _{\mathbb {P}}(t)},\nonumber \\{} & {} \alpha _{\mathbb {P}}(t)=1+\frac{\alpha _{\mathbb {P}}(0)-1}{1-\frac{t}{\tau _a}},\, g_{pp\mathbb {P}}(t)=\frac{g_{pp\mathbb {P}}(0)}{\left( 1-a_g t\right) ^2}. \end{aligned}$$

(29)

Here we use only the pomeron contribution for the sake of simplicity, and so we cannot analyse possible differences between pp and $p\bar{p}$ elastic scattering due to contributions of other reggeons (Table 1).

Table 1 Parameters for proton–proton(anti-proton) elastic scattering amplitude

$$\begin{aligned} V_{pp,p\bar{p}}(s,q^2)= & {} \int d^2\textbf{b} \,\textrm{e}^{\textrm{i}\textbf{q}\textbf{b}} \sqrt{1+2\textrm{i}T_{pp,p\bar{p}}^{el}(s,b)}\nonumber \\ {}= & {} \int d^2\textbf{b}\, \textrm{e}^{\textrm{i}\textbf{q}\textbf{b}} \textrm{e}^{-\Omega _{pp,p\bar{p}}^{el}(s,b)}\nonumber \\ {}= & {} (2\pi )^2\delta ^2\left( \textbf{q}\right) + 2\pi \bar{T}_{pp,p\bar{p}}(s,q^2), \end{aligned}$$

(30)

$$\begin{aligned} \bar{T}_{pp,p\bar{p}}(s,q^2)= & {} \int _0^\infty b\,db\, J_0\left( b\sqrt{-q^2}\right) \left[ \textrm{e}^{-\Omega _{pp,p\bar{p}}^{el}(s,b)}-1 \right] \nonumber \\ S_{pp,p\bar{p}}(s,q^2)= & {} \int d^2\textbf{b} \,\textrm{e}^{\textrm{i}\textbf{q}\textbf{b}} \left( 1+2\textrm{i}T_{pp,p\bar{p}}^{el}(s,b) \right) \nonumber \\= & {} \int d^2\textbf{b}\, \textrm{e}^{\textrm{i}\textbf{q}\textbf{b}} \textrm{e}^{-2\Omega _{pp,p\bar{p}}^{el}(s,b)}\nonumber \\= & {} (2\pi )^2\delta ^2\left( \textbf{q}\right) + 2\pi \bar{T}_{pp,p\bar{p}}(s,q^2), \end{aligned}$$

(31)

Functions $\bar{T}_{pp,p\bar{p}}$ are convenient for numerical calculations, since its oscillations are not so strong.

Appendix C: Primary amplitude for CEDP f resonance production

The following is a short review of the formulae which can be obtained from Refs. [19] and [33].

Let us introduce the general diffractive factor

$$\begin{aligned} F_{\mathbb {P}}(t,\xi ) = g_{pp\mathbb {P}}(t)\left( \textrm{i}+\tan \frac{\pi (\alpha _{\mathbb {P}}(t)-1)}{2}) \right) \frac{\pi \alpha ^{\prime }_{\mathbb {P}}(t)}{\xi ^{\alpha _{\mathbb {P}}}(t)}. \end{aligned}$$

(32)

For $f_0$ production we have the following expression:

$$\begin{aligned}{} & {} M_0^{pp\rightarrow p \{f_0\rightarrow p\bar{p} \} p} \nonumber \\{} & {} \quad = -F_{\mathbb {P}}(t_1,\xi _1) F_{\mathbb {P}}(t_2,\xi _2) \, g_{\mathbb {P}\mathbb {P}f_0}(t_1,t_2,M_c^2) \nonumber \\{} & {} \qquad \times \frac{g_{f_0 p\bar{p}} \left( \mathcal{F}(M_c^2,m_{f_0}^2)\right) ^2F_M(t_1)F_M(t_2)}{(M_c^2-m_{f_0}^2)+B_{f_0}(M_c^2,m_{f_0}^2)}, \end{aligned}$$

(33)

where ($M_c > 2m_p$)

$$\begin{aligned}{} & {} \!\!B_{f_0}(M_c^2,m_{f_0}^2) = \textrm{i}\; \Gamma _{f_0} \left( \mathcal{F}(M_c^2)\right) ^2\left[ \frac{1-4m_p^2/M_c^2}{1-4m_p^2/m_{f_0}^2}\right] ^{1/2} \nonumber \\\end{aligned}$$

(34)

$$\begin{aligned}{} & {} \!\!\!\! \mathcal{F}(M_c^2,m_f^2) = F^{\mathbb {P}\mathbb {P}f}(M_c^2,m_f^2)=F^{f p\bar{p}}(M_c^2,m_f^2)\nonumber \\ {}{} & {} \qquad \quad \qquad \quad =\exp \left( \frac{-(M_c^2-m_{f}^2)^2}{\Lambda _{f}^4}\right) ,\, \Lambda _{f}\sim 1\,\textrm{GeV}, \nonumber \\\end{aligned}$$

(35)

$$\begin{aligned}{} & {} F_M(t) = 1/(1-t/m_0^2),\, m_0^2 = 0.5\,\textrm{GeV}^2 \end{aligned}$$

(36)

$\mathcal{F}(M_c^2,m_f^2)$ and $F_M(t)$ are off-shell phenomenological form factors introduced in [14,15,16,17] to obtain a better description of the data. Here we fix the mass and width of the $f_0(2100)$ meson as in [35]

$$\begin{aligned} m_{f_0(2100)}= & {} 2.086\,\textrm{GeV},\, \Gamma _{f_0(2100)} = 0.284\,\textrm{GeV}. \end{aligned}$$

(37)

In this work, couplings $g_{f_0 p\bar{p}}$ and $g_{\mathbb {P}\mathbb {P}f_0}$ are constants and can be extracted from the experimental data. Thus, finally, we can use the product of these two couplings as a free parameter, since both are unknown. To estimate $g_{\mathbb {P}\mathbb {P}f_0}$ we can use assumptions from [33], where it is taken to be 0.64 GeV for any “glueball-like” resonance.

Here we take the simple Breit–Wigner form for all off-shell propagators of resonances, but we can use more complicated expressions which can be found, for example, in [14,15,16,17].