∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: NRC “Kurchatov Institute” - Institute for High Energy
Physics, Protvino 142 281, Russia

# Central exclusive diffractive production of two-pions from continuum and
decays of resonances in the Regge-eikonal model

R.A. Ryutin e1,addr1

###### Abstract

Calculations of central exclusive diffractive production (CEDP) of two pions
via continuum and resonance mechanism are presented in the Regge-eikonal
approach. Data from STAR, ISR, CDF and CMS were analysed and compared with
theoretical description. Preliminary extraction of $f_{0}(500)$, $f_{0}(980)$
and $f_{2}(1270)$ couplings to Pomeron, and also $\rho$ meson contribution
from photoproduction mechanism to the CEDP cross-section are considered. We
show possible nuances and problems of calculations and prospects of
investigations at present and future hadron colliders.

###### pacs:

11.55.Jy Regge formalism 12.40.Nn Regge theory, duality, absorptive/optical
models 13.85.Ni Inclusive production with identified hadrons13.85.Lg Total
cross sections

## Introduction

In previous papers myCEDP1 ,myCEDP2  general properties and calculations of
the Central Exclusive Diffractive Production (CEDP) were considered. It was
shown, especially in myCEDP2 , that diffractive patterns (differential cross-
sections) of CEDP play significant role in model verification.

In myCEDPpipiContinuum  we considered the low mass CEDP (LM CEDP) with
production of two pion continuum. Here we expand this analysis and add
contributions from CEDP of $f_{0}(500)$, $f_{0}(980)$, $f_{2}(1270)$ and
exclusive vector meson photoproduction (EVMP) of $\rho_{0}$ meson to this
process, taking into account also the interference between continuum and
resonance mechanisms.

CEDP of two pions is one of the basic “standard candles” for LM CEDP. Why do
we need exact calculations and predictions for this process?

  * •

Di-pion LM CEDP is the tool for the investigation of hadronic 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 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 of calculations of “rescattering”
(unitarity) corrections. In the case of di-pion LM CEDP there are several
kinds of corrections, in the proton-proton, pion-proton and also pion-pion
channels.

  * •

The pion is the most fundamental particle in the strong interactions, and LM
CEDP gives us a powerful tool to go deep inside 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 luminocity runs (of the LHC) we need more
time to get enough statistics.

  * •

As was proposed in myCEDPpipiContinuum ,mySD , 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 properties of the Pomeron-Pomeron to pion-pion exclusive cross-
section.

  * •

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 $M_{\pi\pi}$ 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 usual properties of CEDP: clear signature with two final
protons and two large rapidity gaps (LRG) LRG1 ,LRG2  and the possibility to
use the “missing mass method” MMM .

Processes of the LM CEDP were calculated in some other works CEDPw1 -CEDPw6
which are devoted to most popular models for the LM CEDP of di-mesons, where
authors have considered phenomenological, nonperturbative, perturbative and
mixed approaches in Reggeon-Reggeon collision subprocess. Nuances of some
approaches were analysed in the introduction of myCEDPpipiContinuum .

In this article we consider the case, depicted in Fig. 1, and show how it can
describe the data from ISR ISRdata1 , ISRdata2 , STAR STARdata1 -STARdata5 ,
CDF CDFdata1 ,CDFdata2 , CMS CMSdata1 -CMSdata4  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, wich was considered in details
in myCEDPpipiContinuum . Here we take reggeized full (RF) case from four
approaches of myCEDPpipiContinuum  which is the best in the data description.

Also we discuss some nuances of the calculations, which we should take into
account in further more accurate analysis (elastic amplitudes for virtual
particles, off-shell pion form factor, pion-pion elastic amplitude at low
energies, nonlinearity of the pion trajectory, tensorial couplings and spin
effects).

In the second part we analyse the experimental data on the process at
different energies, extract some couplings of resonances to Pomeron, find the
best approach and make some predictions for LHC experiments.

To avoid complicated expressions in the main text, all the basic formulae are
placed to Appendixes.

Figure 1:  Primary amplitudes (absorptive corrections are not shown) of the
process of double pion LM CEDP $p+p\to p+\pi^{+}+\pi^{-}+p$ in the Regge-
eikonal approach for continuum (a),(b), LM CEDP of $f_{0}$ and $f_{2}$
resonances (c) and EVMP of $\rho_{0}$ meson (d),(e) with subsequent decay to
$\pi^{+}\pi^{-}$. (a),(b): central part of the diagram is the primary
continuum CEDP amplitude, where $T_{\pi^{\pm}p}$ are full elastic pion-proton
amplitudes, and the reggeized off-shell pion propagator depicted as dahsed
zigzag line. This is the most applicable case from four approaches considered
in myCEDPpipiContinuum . (c): central part of the diagram contains Pomeron-
Pomeron-resonance fusion with subsequent decay to pions, propagator is taken
in the Breit-Wigner approximation. (d),(e): central part of the diagram
contains full EVMP amplitude with $\rho_{0}$ production and decay. Off-shell
pion form factor on (a),(b) and other suppression form-factors (in the
Pomeron-Pomeron-f or the pion-pion-($f_{0}$,$f_{2}$,$\rho_{0}$ vertices) ) on
(c),(d),(e) are presented as a black circles. Figure 2:  Full unitarized
amplitude of the process of double pion LM CEDP $p+p\to p+\pi^{+}+\pi^{-}+p$.
Proton-proton rescatterings in the initial and final states are depicted as
$V_{pp}$ and $V^{\prime}_{pp}$-blobes correspondingly, and pion-proton
rescattering corrections are also shown as $S_{\pi p}$-blobes. Sum of primary
amplitudes $M_{0}$ from Fig. 1 are shown as a red rectangle.

##  1 General framework for calculations of LM CEDP

LM CEDP is the first exclusive two to four process which is driven basically
by the Pomeron-Pomeron fusion subprocess. It serves as a clear process for
investigations of resonances like $f_{0}(500)$, $f_{0}(980)$, $f_{2}(1270)$
and others with masses less than $3$ GeV. At the noment, for low central di-
pion masses it is a huge problem to use perturbative approach, that 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 godizovpp , godizovpip ,
which describe all the available experimental data on elastic scattering. For
EVMP we use the similar model godizovmesons .

###  1.1 Components of the framework

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

  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 pion propagator in the amplitude for continuum di-pion production is
replaced by the reggeized one

$\displaystyle{\cal P}_{\pi}(\hat{s},\hat{t})$ $\displaystyle=$
$\displaystyle\left(\mathrm{ctg}\frac{\pi\alpha_{\pi}(\hat{t})}{2}-\mathrm{i}\right)\cdot$
(1) $\displaystyle\cdot$
$\displaystyle\frac{\pi\alpha_{\pi}^{\prime}}{2\phantom{\mbox{\large{I}}^{2}}\\!\\!\\!\\!\mathrm{\Gamma}(1+\alpha_{\pi}(\hat{t}))}\left(\frac{\hat{s}}{s_{0}}\right)^{\alpha_{\pi}(\hat{t})},$

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).

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

We use also full eikonalized expressions for proton-proton, pion-proton and
photon-proton amplitudes, which can be found in the Appendix B.

  2. 2.

After the calculation of the primary LM CEDP amplitudes we have to take into
account all possible corrections in proton-proton and proton-pion elastic
channels due to the unitarization procedure (so called “soft survival
probability” or “rescattering corrections”), which are depicted as $V_{pp}$,
$V^{\prime}_{pp}$ and $S_{\pi p}$ blobes in Fig. 1. For proton-proton and
proton-pion elastic amplitudes we use the model of godizovpp , godizovpip
(see Appendix B). Possible final pion-pion interaction 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 CEDPw1
-CEDPw3 , since they give nonleading contributions in our model due to
smallness of the triple Pomeron vertex. Also we have no possible absorptive
corrections in the pion-pion final elastic channel, since the central mass is
low, and also there is a lack of data on this process to define parameters of
the model. Nevertheless one can consider these corrections, as it was done by
some authors recently pipiespagnol , since they could play significant role
for masses less than 1 GeV.

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

Here we use the model, presented in Appendix B for example. You can use
another one, which is proved to describe well all the available data on
proton-proton, proton-pion elastic processes and exclusive $\rho_{0}$
photoproduction. But it is difficult to find now more than a couple of models
which have more or less predictable power (see godizovmodelsfault  for
detailed discussion). That is why we use the model, which is proved to be good
in data fitting, especially in the kinematical region of our interest.

###  1.2 Continuum di-pion production

Final expression for the amplitude for the continuum di-pion production with
proton-proton and pion-proton “rescattering” corrections (see Fig. 1(a),(b))
can be written as

$\displaystyle M^{U}\left(\\{p\\}\right)=$
$\displaystyle=\int\int\frac{d^{2}\vec{q}}{(2\pi)^{2}}\frac{d^{2}\vec{q}^{\prime}}{(2\pi)^{2}}\frac{d^{2}\vec{q}_{1}}{(2\pi)^{2}}\frac{d^{2}\vec{q}_{2}}{(2\pi)^{2}}V_{pp}(s,q^{2})V_{pp}(s^{\prime},q^{\prime
2})$
$\displaystyle\times\;\left[S_{\pi^{-}p}(\tilde{s}_{14},q_{1}^{2})M_{0}\left(\\{\tilde{p}\\}\right)S_{\pi^{+}p}(\tilde{s}_{23},q_{2}^{2})+(3\leftrightarrow
4)\right]$ (2) $\displaystyle
M_{0}\left(\\{p\\}\right)=\phantom{I^{2^{2^{2}}}}$
$\displaystyle=T^{el}_{\pi^{+}p}(s_{13},t_{1}){\cal
P}_{\pi}(\hat{s},\hat{t})\left[\hat{F}_{\pi}\left(\hat{t}\right)\right]^{2}T^{el}_{\pi^{-}p}(s_{24},t_{2}),$
(3)

where functions are defined in (26)-(30) of Appendix B, and sets of vectors
are

$\displaystyle\\{p\\}\equiv\\{p_{a},p_{b},p_{1},p_{2},p_{3},p_{4}\\}$ (4)
$\displaystyle\\{\tilde{p}\\}\equiv\\{p_{a}-q,p_{b}+q;p_{1}+q^{\prime}+q_{1},$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;p_{2}-q^{\prime}+q_{2},p_{3}-q_{2},p_{4}-q_{1}\\},$
(5)

and

$\displaystyle\tilde{s}_{14}$ $\displaystyle=$
$\displaystyle\left(p_{1}+p_{4}+q^{\prime}\right)^{2},\;\tilde{s}_{23}=\left(p_{2}+p_{3}-q^{\prime}\right)^{2},$
(6) $\displaystyle s_{ij}$ $\displaystyle=$
$\displaystyle\left(p_{i}+p_{j}\right)^{2},\;t_{1,2}=\left(p_{a,b}-p_{1,2}\right)^{2},$
(7) $\displaystyle\hat{s}$ $\displaystyle=$
$\displaystyle\left(p_{3}+p_{4}\right)^{2},\;\hat{t}=\left(p_{a}-p_{1}-p_{3}\right)^{2}$
(8)

Off-shell pion form factor is equal to unity on mass shell
$\hat{t}=m_{\pi}^{2}$ and taken as exponential

$\hat{F}_{\pi}=\mathrm{e}^{(\hat{t}-m_{\pi}^{2})/\Lambda_{\pi}^{2}},$ (9)

where $\Lambda_{\pi}\sim 1$ GeV is taken from the fits to LM CEDP of two pions
at low energies (see next section). In this paper we use only exponential
form, but it is possible to use other parametrizations (see CEDPw1 -CEDPw6 ).
Exponential one shows more appropriate results in the data fitting.

Other functions are defined in Appendix B. Then we can use the expression (17)
to calculate the differential cross-section of the process.

###  1.3 CEDP and EVMP of low mass resonances.

For $f_{0}(500)$, $f_{0}(980)$, $f_{2}(1270)$ and $\rho_{0}$ resonances the
general unitarized amplitude (see Fig. 1(c),(d)) is similar to the expression
(2), where amplitude $M_{0}\left(\\{p\\}\right)$ is replaced by the
corresponding central primary amplitude for the resonance production and
futher decay to $\pi^{+}\pi^{-}$.

For $f$ mesons amplitudes are constructed from proton-Pomeron form-factors,
Pomeron-Pomeron couplings to mesons111Here we take the simple scalar one for
every meson, even for $f_{2}(1270)$ (multiplied by the leading tensor term),
although, as was mentioned in our work myWA102 , this vertex can be rather
complicated and can give nontrivial contribution to the dependence on the
azimuthal angle between final protons. But for our goals in this paper,
namely, investigation of the di-pion mass distributions, it is rather good
approximation., off-shell propagators, off-shell form-factors and decay
vertices.

For the EVMP amplitude of $\rho_{0}$ meson we take the full unitarized photon-
proton amplitude contracted with the photon flux, and also with the off-shell
meson propagator, off-shell form-factors and its decay vertex to pions. Also
we have to sum this amplitude with the symmetric one with
$t_{1}\leftrightarrow t_{2}$.

All the primary amplitudes are presented in the Appendix C.

###  1.4 Nuances of calculations.

In the next section one can see that there are some difficulties and puzzles
in the data fitting, which have also been presented in other works CEDPw4
-CEDPw6 . 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. 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 (2)). For this amplitude in the present paper we use Regge-
eikonal model with the eikonal function in the classical Regge form. And “off-
shell” condition for one of the pions is taken into account by additional
phenomenological form factor $\hat{F}_{\pi}(\hat{t})$. But there are at least
two other possibilities.

The first one was considered in PetrovOffshell . For amplitude with one
particle off-shell the formula

$T^{*}(s,b)=\frac{\delta^{*}(s,b)}{\delta(s,b)}T(s,b)=\frac{\delta^{*}(s,b)}{\delta(s,b)}\frac{\mathrm{e}^{2\mathrm{i}\delta(s,b)}-1}{2\mathrm{i}}$
(10)

was used. In our case

$\displaystyle\delta(s,b)=\delta_{\pi
p}(s,b;m_{\pi}^{2},m_{\pi}^{2},m_{p}^{2},m_{p}^{2}),$
$\displaystyle\delta^{*}(s,b)=\delta_{\pi
p}^{*}(s,b;\hat{t},m_{\pi}^{2},m_{p}^{2},m_{p}^{2})$
$\displaystyle\left.\delta_{\pi p}=\delta^{*}_{\pi p}\right|_{\hat{t}\to
m_{\pi}^{2}}.$ (11)

$\delta_{\pi p}$ is the eikonal function (see (22)). This is similar to the
introduction of the additional form factor, but in a more consistent way,
which takes into account the unitarity condition.

Figure 3:  Pion-proton on-shell and off-shell elastic differential cross-
section (in the model of conserved meson currents presented in Appendix C) for
different pion virtualities $\hat{t}_{\pi}$: $m_{\pi}^{2}$(on-shell),$-0.01$
$GeV^{2}$,$-0.15$ $GeV^{2}$,$-0.4$ $GeV^{2}$ in the covariant approach with
conserved currents.

The second one arises from the covariant reggeization method, which was
considered in the Appendix C of myCEDPpipiContinuum . For the case of
conserved hadronic currents we have definite structure in the Legendre
function, which is transformed in a natural way to the case of the off-shell
amplitude. But in this case off-shell amplitude has a specific behaviour at
low t values (see Fig. 3 and myCEDP2  for details). As was shown in myCEDP2 ,
unitarity corrections can mask this behavior.

Since final pion-proton interactions can give rather large suppression (about
10-20%, as in Fig. 4), in our calculations we use the full amplitude as
depicted in Fig. 2.

Figure 4:  Predictions of the model for the continuum (see Fig. 1(a),(b)) at
$7$ TeV. Curves from up to down correspond to the Born term, the amplitude
with proton-proton rescattering corrections only and the one with all the
corrections (proton-proton and pion-proton). $\Lambda_{\pi}=3$ GeV.

##  2 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 size and shape of the interaction region;

  * •

the distribution on the azimuthal angle between final protons gives quantum
numbers of the produced system (see myCEDP2 ,myWA102  and references therein);

  * •

from $M_{c}$ (here $M_{c}=M_{\pi\pi}$) dependence and its influence on
t-dependence we can make some conclusions about the interaction at different
space-time scales and interrelation between them. Also we can extract
couplings of reggeons to different resonances.

Process $p+p\to p+\pi+\pi+p$ is the first “standard candle”, which we can use
to estimate other CEDP processes. In this section we consider the experimental
data on the process and make an attempt to extract the information on
couplings of different resonances to Pomeron.

a)

b)

Figure 5:  The data on the process $p+p\to p+\pi^{+}+\pi^{-}+p$ at
$\sqrt{s}=200$ GeV (STAR collaboration STARdata1 ,STARdata2 ): (a)
$|\eta_{\pi}|<1$, $|\eta_{\pi\pi}|<2$, $p_{T\pi}>0.15$ GeV,
$0.005<-t_{1,2}<0.03$ GeV2; (b) plus additional cut $M_{c}<1$ GeV. Curves
correspond to $\Lambda_{\pi}=1$ GeV in the off-shell pion form factor (9) and
couplings from (12). Solid curves correspond to the sum of all amplitudes and
dotted ones represent the continuum contribution.

a)

b)

c)

d)

Figure 6:  The new data on the process $p+p\to p+\pi^{+}+\pi^{-}+p$ at
$\sqrt{s}=200$ GeV (STAR collaboration STARdata3 -STARdata4a ):
$|\eta_{\pi}|<0.7$, $p_{T\pi}>0.2$ GeV, $p_{x}>-0.2$ GeV, $0.2$
GeV$<|p_{y}|<0.4$ GeV, $(p_{x}+0.3\,\mathrm{GeV})^{2}+p_{y}^{2}<0.25$ GeV2,
where $p$ denotes the momenta of final protons. Curves correspond to
$\Lambda_{\pi}=1$ GeV in the off-shell pion form factor (9) and couplings from
(12). Solid curves correspond to the sum of all amplitudes and the dotted one
in (a) represents the continuum contribution.

###  2.1 STAR collaboration data versus model cases

In this subsection the data of the STAR collaboration STARdata1 -STARdata5
and model curves for continuum and the sum of all cases of Fig. 1 are
presented. In our approach we have several free parameters: $\Lambda_{\pi}$
(for the continuum) and couplings of resonances to Pomeron $g_{\mathbb{PP}f}$,
which we can extract from the data. All the distributions are depicted here
for $\Lambda_{\pi}=1$ GeV and couplings from (12).

Figure 7:  The new data on the process $p+p\to p+\pi^{+}+\pi^{-}+p$ at
$\sqrt{s}=200$ GeV (STAR collaboration STARdata3 -STARdata4a ):
$|y_{\pi\pi}|<0.4$, $0.05$ GeV$<|t_{1,2}|<0.16$ GeV2, $\phi_{0}<45^{o}$.
Dashed curve corresponds to parameters, fixed from the old data from STAR
depicted in the Fig. 5 and explained in the text: $\Lambda_{\pi}=1$ GeV in the
off-shell pion form factor (9) and couplings from (12). Figure 8:  The new
preliminary data on the process $p+p\to p+\pi^{+}+\pi^{-}+p$ at $\sqrt{s}=510$
GeV (STAR collaboration STARdata5 ): $|\eta_{\pi}|<0.7$, $p_{T\pi}>0.2$ GeV,
$p_{x}>-0.27$ GeV, $0.4$ GeV$<|p_{y}|<0.8$ GeV,
$(p_{x}+0.6\,\mathrm{GeV})^{2}+p_{y}^{2}<1.25$ GeV2, where $p$ denotes the
momenta of final protons. Theoretical solid curve correspond to
$\Lambda_{\pi}=1$ GeV in the off-shell pion form factor (9) and couplings from
(12). Fluctuations are due to complex Monte-Carlo integration process, and
they can be smoothed by increasing the integration accuracy.

We begin our analysis by fitting the data from STAR depicted on the Fig. 5(a).
Then we make predictions for other available data on this process.

First of all we have to note a difference between the data and prediction at
the same energy. In the Fig. 5(b) one can see the azimuthal angle
distribution. Here we could explain the difference, because our Pomeron-
Pomeron-meson couplings are constants, but in reality they depends on the
azimuthal angle. The next case, which is depicted in Figs. 6 is more
interesting. We see that predictions slightly overestimate the data in the
region of $M_{c}\sim 0.6\pm 0.1$ GeV (it depends on the Pomeron coupling to
$f_{0}(500)$ and can be corrected) and underestimate the data in the region of
$M_{c}\sim 0.8\pm 0.1$ GeV as in the Fig.5(a). The data are close to
predictions only in the region of $f_{2}(1270)$ meson. This fact can not be
understood at the moment, since, if we fix parameters using this new STAR data
from Figs. 6, we will obtain overestimation of the data at higher energies,
which looks strange. May be we have to renormalize this part of the STAR data,
or Pomeron-Pomeron-meson couplings have some complex dependance on $t_{1,2}$
and $\phi_{0}$. It should be checked in further investigations.

The data for extended kinematical region at $\sqrt{s}=200$ GeV is depicted in
the Fig. 7 with the theoretical curve, which is slightly underestimate the
data at low masses and overestimate the data at the peak of $f_{2}(1270)$.
Since these data were obtained by some extrapolation, it can be considered
only qualitatively.

In the Fig. 8 we see the similar situation with overestimation at $M_{c}\sim
0.7\pm 0.2$ GeV.

###  2.2 ISR and CDF data versus the model curves

a)

b)

Figure 9:  The data on the process $p+p\to p+\pi^{+}+\pi^{-}+p$ (ISR and
ABCDHW collaborations ISRdata1 ,ISRdata2 ): (a) at $\sqrt{s}=63$ GeV,
$|y_{\pi}|<1$, $\xi_{p}>0.9$; (b) at $\sqrt{s}=62$ GeV, $|y_{\pi}|<1.5$,
$\xi_{p}>0.9$. Curves correspond to $\Lambda_{\pi}=1$ GeV in the off-shell
pion form factor (9) and couplings from (12). Solid curves correspond to the
sum of all amplitudes and dotted ones represent the continuum contribution.
Fluctuations in solid curves are due to complex Monte-Carlo integration
process, and they can be smoothed by increasing the integration accuracy.

a)

b)

Figure 10:  The data on the process $p+\bar{p}\to p+\pi^{+}+\pi^{-}+\bar{p}$
(CDF collaboration CDFdata1 ,CDFdata2 ) at $\sqrt{s}=1.96$ TeV: (a)
$|\eta_{\pi}|<1.3$, $|y_{\pi\pi}|<1$, $p_{T,\pi}>0.4$ GeV; (b)
$|\eta_{\pi}|<1.3$, $|y_{\pi\pi}|<1$, $p_{T,\pi}>0.4$ GeV, $p_{T,\pi\pi}>1$
GeV. Curves correspond to $\Lambda_{\pi}=1$ GeV in the off-shell pion form
factor (9) and couplings from (12). Solid curves correspond to the sum of all
amplitudes and dotted ones represent the continuum contribution. Fluctuations
in solid curves are due to complex Monte-Carlo integration process, and they
can be smoothed by increasing the integration accuracy.

Let us look at the ISR ISRdata1 ,ISRdata2  and CDF CDFdata1 ,CDFdata2  data
with parameter $\Lambda_{\pi}$, which we use to describe the data from STAR
collaboration. Different cases are depicted on Figs. 9-10.

We see strong underestimation of the ISR data (by factor of $4$). For these
low energies we have to take into account possible corrections to pion-proton
amplitudes (contributions from secondary reggeons are strong), since our
approach describe data well only for energies greater than $\sim 3$ GeV. And
in each shoulder ($T_{\pi p}$ amplitude in Fig. 1 (a),(b)) energy can be less
than $3$ GeV.

As to the CDF data in the Fig. 10(a), it is overestimated (by factor $2$) at
the region of $f_{2}(1270)$ meson. And another part of the CDF data in the
Fig. 10(b) is close to the prediction, but prediction underestimates the data
at the region of $f_{2}(1270)$ meson. It also looks strange and has no
explanation at the moment, since this is the data from the same experiment.
This is may be due to interference effects with $\gamma\gamma$ or
$\gamma\mathbb{O}$ fusion in the central production process and corrections to
$\pi\pi$ final interaction.

a)

b)

c)

Figure 11:  The data on the process $p+\bar{p}\to p+\pi^{+}+\pi^{-}+\bar{p}$
(CMS collaboration CMSdata1 ,CMSdata2 ): (a) at $\sqrt{s}=5$ TeV with cuts
$|\eta_{\pi}|<2.4$, $p_{T,\pi}>0.2$ GeV; (b) at $\sqrt{s}=7$ TeV with cuts
$|y_{\pi}|<2$, $p_{T,\pi}>0.2$ GeV; (c) at $\sqrt{s}=13$ TeV with cuts
$|\eta_{\pi}|<2.4$, $p_{T,\pi}>0.2$ GeV. Curves correspond to
$\Lambda_{\pi}=1$ GeV in the off-shell pion form factor (9) and couplings from
(12). Solid curves correspond to the sum of all amplitudes and dotted ones
represent the continuum contribution. Fluctuations in solid curves are due to
complex Monte-Carlo integration process, and they can be smoothed by
increasing the integration accuracy.

###  2.3 CMS data and predictions

In Fig. 11 one can see the recent data from the CMS collaboration and curves
of our model, which correspond to $\Lambda_{\pi}=1$ GeV in the off-shell pion
form factor (9) and couplings (12) from the fit to the STAR data on Fig. 6.
Here our predictions are very close to the data.

## Summary and conclusions

In this paper we have considered the process LM CEDP of di-pions and its
description in the framework of the Regge-eikonal approach. Here we summarize
all the facts and conclusions:

  * •

The result is crucially dependent on the choice of $\Lambda_{\pi}$ in the off-
shell pion form factor, i.e. on $\hat{t}$ (virtuality of the pion) dependence.
In the present approach the best description of the old STAR data STARdata1
,STARdata2  is given by the case with $\Lambda_{\pi}=1$ GeV and couplings:

$\displaystyle g_{\mathbb{PP}f_{0}(500)}$ $\displaystyle=$ $\displaystyle
0.77,$ $\displaystyle g_{\mathbb{PP}f_{0}(980)}$ $\displaystyle=$
$\displaystyle 0.3,$ $\displaystyle g_{\mathbb{PP}f_{2}(1270)}$
$\displaystyle=$ $\displaystyle 1.39.$ (12)

The couplings of Pomeron to $f_{0,2}$ can be compared to the value $0.64$,
which is obtained in GodizovResonances .

  * •

the model shows that contributions from $\rho$ meson are not so significant at
available energies, but we have a dip in the theoretical curves at the region
of $\rho$ production (at CMS energies, for example, it is obvious). This also
should be explained in further investigations.

  * •

Rescattering corrections (“soft survival probability”) in $pp$ and $\pi p$
make a significant contribution to the values and form of distributions.

  * •

If we try to fit the data from STAR STARdata3 -STARdata5 , we find that the
curves with parameters fixed from STARdata1 ,STARdata2  overestimate the data
in the region of $M_{c}\sim 0.6\pm 0.1$ GeV (it depends on the Pomeron
coupling to $f_{0}(500)$ and can be corrected) and underestimate the data in
the region of $M_{c}\sim 0.8\pm 0.1$ GeV as in the Fig.5(a). The data are
close to predictions only in the region of $f_{2}(1270)$ meson. These effects
have no explanation at the moment, and may be due to some complex dependance
of Pomeron-Pomeron-meson couplings on $t_{1,2}$ and $\phi_{0}$, wrong
normalization of the data or missed contributions from some other processes
like low mass difractive dissociation.

  * •

We have strong underestimation of the ISR data (by factor of $4$) ISRdata1
,ISRdata2  and contradictory description of two parts of the CDF data CDFdata1
,CDFdata2  with the same parameters (see Figs.9,10).

  * •

However, the CMS data at all energies are described rather well (see Fig. 11).
Discrepancy is observed only in the region $M_{c}\sim 0.8\pm 0.2$ GeV.

The main open problems regarding model parameters are related to interference
terms (we have to know all cut-off form-factor parameters, couplings of pions
and reggeons to resonances and their dependance on $t_{1,2}$ and $\phi_{0}$).
To fix the model parameters correctly we need the comparison with precise
(exclusive) experimental data (STAR, ATLAS+ALFA, CMS+TOTEM) simultaneously in
several differential observables, e.g. the differential distributions
$d\sigma/dt_{1}$, $d\sigma/d\phi_{0}$, the angular distributions in the pi+pi-
rest system and others. It should be done in further works.

We have to take into accout also effects like the interference with
$\gamma\gamma\to\pi\pi$ and $\gamma\mathbb{O}\to\pi\pi$ 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 $M_{\pi\pi}$,
corrections to $T_{\pi p}(s,t)$ for $\sqrt{s}<3$ GeV. In further works we will
take into account possible modifications of the model for best description of
the data.

This model will be implemented to the Monte-carlo event generator ExDiff
ExDiffmanual . It is possible to calculate LM CEDP for other di-hadron final
states ($p\bar{p}$ for “Odderon” hunting, $K^{+}K^{-}$, $\eta\eta^{\prime}$
and so on), which are also very informative for our understanding of
diffractive mechanisms in strong interactions.

## Acknowledgements

I am grateful to Vladimir Petrov, Anton Godizov and Piotr Lebiedowicz for
useful discussions and help.

## Appendix A. Kinematics of LM CEDP

Figure 12:  Total amplitude of the process of double pion LM CEDP $p+p\to
p+\pi^{+}+\pi^{-}+p$ with detailed kinematics. Proton-proton rescatterings in
the initial and final states are depicted as black blobes, and pion-proton
subamplitudes are also shown as shaded blobes. All momenta are shown. Basic
part of the amplitude, $M_{0}$ (see eq. (3)), without corrections is circled
by a dotted line. Crossed lines are on mass shell. Here
$\Delta_{1\perp}=\Delta_{1}-q-q^{\prime}$,
$\Delta_{2\perp}=\Delta_{2}+q+q^{\prime}$,
$\hat{t}=k^{2}=(\Delta_{1\perp}-q_{1}-p_{3}+q_{2})^{2}$,
$\hat{u}=(\Delta_{1\perp}-q_{1}-p_{4})^{2}$,
$\hat{s}=(p_{3}+p_{4}-q_{1}-q_{2})^{2}$.

The $2\to 4$ process $p(p_{a})+p(p_{b})\to
p(p_{1})+\pi(p_{3})+\pi(p_{4})+p(p_{2})$ can be described as follows (the
notation for any momentum is $k=(k_{0},k_{z};\vec{k})$,
$\vec{k}=(k_{x},k_{y})$):

$\displaystyle p_{a}$ $\displaystyle=$
$\displaystyle\left(\frac{\sqrt{s}}{2},\beta\frac{\sqrt{s}}{2};\vec{0}\right),\;p_{b}=\left(\frac{\sqrt{s}}{2},-\beta\frac{\sqrt{s}}{2};\vec{0}\right),$
$\displaystyle p_{1,2}$ $\displaystyle=$
$\displaystyle\left(E_{1,2},p_{1,2z};\vec{p}_{1,2\perp}\right),E_{1,2}=\sqrt{p_{1,2z}^{2}+\vec{p}_{1,2\perp}^{2}+m_{p}^{2}},\;$
$\displaystyle p_{3,4}$ $\displaystyle=$
$\displaystyle\left(m_{3,4\perp}\mathrm{ch}\;y_{3,4},m_{3,4\perp}\mathrm{sh}\;y_{3,4};\vec{p}_{3,4\perp}\right)=$
$\displaystyle=$
$\displaystyle\left(\sqrt{m_{\pi}^{2}+\vec{p}_{3,4\perp}^{2}\mathrm{ch}^{2}\eta_{3,4}},|\vec{p}_{3,4\perp}|\;\mathrm{sh}\;\eta_{3,4};\vec{p}_{3,4\perp}\right),$
$\displaystyle m_{i\perp}^{2}$ $\displaystyle=$ $\displaystyle
m_{i}^{2}+\vec{p}_{i\perp}^{2},\;m_{1,2}=m_{p},\;m_{3,4}=m_{\pi},$
$\displaystyle\vec{p}_{4\perp}$ $\displaystyle=$
$\displaystyle-\vec{p}_{3\perp}-\vec{p}_{1\perp}-\vec{p}_{2\perp},\;$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{1-\frac{4m_{p}^{2}}{s}},\;s=(p_{a}+p_{b})^{2},\;s^{\prime}=(p_{1}+p_{2})^{2}.$
(13)

Here $y_{i}$ ($\eta_{i}$) are rapidities (pseudorapidities) of final pions.

Phase space of the process in terms of the above variables is the following

$\displaystyle d\mathrm{\Phi}_{2\to 4}$ $\displaystyle=$
$\displaystyle\left(2\pi\right)^{4}\delta^{4}\left(p_{a}+p_{b}-\sum_{i=1}^{4}p_{i}\right)\prod_{i=1}^{4}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}=$
$\displaystyle=$
$\displaystyle\frac{1}{2^{4}(2\pi)^{8}}\prod_{i=1}^{3}p_{i\perp}dp_{i\perp}d\phi_{i}\cdot
dy_{3}dy_{4}\cdot{\cal J};$ $\displaystyle{\cal J}$ $\displaystyle=$
$\displaystyle\frac{dp_{1z}}{E_{1}}\frac{dp_{2z}}{E_{2}}\delta\left(\sqrt{s}-\sum_{i=1}^{4}E_{i}\right)\delta\left(\sum_{i=1}^{4}p_{iz}\right)=$
(14) $\displaystyle=$
$\displaystyle\frac{1}{\left|\tilde{E}_{2}\tilde{p}_{1z}-\tilde{E}_{1}\tilde{p}_{2z}\right|},$

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

$\begin{cases}&A=\sqrt{s}-E_{3}-E_{4}=\sqrt{m_{1\perp}^{2}+p_{1z}^{2}}+\sqrt{m_{2\perp}^{2}+p_{2z}^{2}},\\\
&B=-p_{3z}-p_{4z}=p_{1z}+p_{2z},\end{cases}$ (15)
$\displaystyle\tilde{p}_{1z}$ $\displaystyle=$
$\displaystyle\frac{B}{2}+\frac{1}{2(A^{2}-B^{2})}\left[B\left(m_{1\perp}^{2}-m_{2\perp}^{2}\right)+A\cdot\lambda_{0}^{1/2}\right],$
$\displaystyle\lambda_{0}$ $\displaystyle=$
$\displaystyle\lambda\left(A^{2}-B^{2},m_{1\perp}^{2},m_{2\perp}^{2}\right).$
(16)

Here $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$, and then ${\cal
J}^{-1}=\lambda_{0}^{1/2}/2$.

For the differential cross-section we have

$\displaystyle\frac{d\sigma_{2\to
4}}{\prod_{i=1}^{3\phantom{I}}dp_{i\perp}d\phi_{i}\cdot dy_{3}dy_{4}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\beta
s}\cdot\frac{\prod_{i=1}^{3}p_{i\perp}}{2^{4}(2\pi)^{8}\cdot\frac{1}{2}\lambda_{0}^{1/2}}\left|T\right|^{2}=$
(17) $\displaystyle=$
$\displaystyle\frac{\prod_{i=1}^{3}p_{i\perp}}{2^{12}\pi^{8}\beta
s\lambda_{0}^{1/2}}\left|T\right|^{2}.$

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

$\frac{dy_{i}}{d\eta_{i}}=\frac{p_{i\perp}\mathrm{ch}\eta_{i}}{\sqrt{m_{i}^{2}+p_{i\perp}^{2}\mathrm{ch}^{2}\eta_{i}}}$
(18)

to get 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:

$\displaystyle d\mathrm{\Phi}_{2\to 4}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{4}(2\pi)^{8}}\prod_{i=1}^{2}dt_{i}d\phi_{i}\cdot
M_{c}dM_{c}d\eta_{c}dc^{*}d\phi^{*}\cdot{\cal J}^{\prime};$
$\displaystyle{\cal J}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\beta_{M}}{4\beta_{1}\beta_{2}s}\frac{dy_{c}}{d\eta_{c}},\,\,\,\,\,\beta_{i}\simeq\sqrt{1+\frac{4(m_{p}^{2}-(1-\xi_{i})t_{i})}{\beta^{2}s^{2}(1-\xi_{i})^{2}}};$
$\displaystyle\beta_{M}$ $\displaystyle=$
$\displaystyle\sqrt{1-\frac{4m_{\pi}^{2}}{M_{c}^{2}}},\,\,\frac{dy_{c}}{d\eta_{c}}=\frac{p_{c\perp}\mathrm{ch}\eta_{c}}{\sqrt{M_{c}^{2}+p_{c\perp}^{2}\mathrm{ch}^{2}\eta_{c}}};$
(19)

$\displaystyle\frac{d\sigma_{2\to
4}}{\prod_{i=1}^{2\phantom{I}}dt_{i}d\phi_{i}dM_{c}d\eta_{c}dc^{*}d\phi^{*}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\beta
s}\cdot\frac{M_{c}\beta_{M}\frac{dy_{c}}{d\eta_{c}}}{2^{4}(2\pi)^{8}\cdot
4\beta_{1}\beta_{2}s}\left|T\right|^{2}$ (20) $\displaystyle=$
$\displaystyle\frac{M_{c}\beta_{M}\frac{dy_{c}}{d\eta_{c}}}{2^{15}\pi^{8}\beta\beta_{1}\beta_{2}s^{2}}\left|T\right|^{2},$

where $c^{*}=\cos\theta^{*}$, $\theta^{*}$ and $\phi^{*}$ are polar and
azimuthal angles of the pion momenta in the $\pi^{+}\pi^{-}$ rest frame,
$M_{c}$ is the di-pion mass, $\eta_{c}$ is the di-pion pseudorapidity,
$t_{1}=(p_{a}-p_{1})^{2}$, $t_{2}=(p_{b}-p_{2})^{2}$ and

$\xi_{1,2}\simeq\sqrt{\frac{M_{c}^{2}-t_{1}-t_{2}+2\sqrt{t_{1}t_{2}}\cos(\phi_{1}-\phi_{2})}{s}}\mbox{\bf
e}^{\pm y_{c}}.$

For exact calculations of elastic subprocesses (see Fig. 12) of the type
$a(p_{1})+b(p_{2})\to c(p_{1}-q_{el})+d(p_{2}+q_{el})$:

$\displaystyle q_{el}$ $\displaystyle=$
$\displaystyle\left(q_{0},q_{z};\vec{q}\right),$ $\displaystyle q_{z}$
$\displaystyle=$
$\displaystyle-\frac{b}{2a}\left(1-\sqrt{1-\frac{4ac}{b^{2}}}\right),$
$\displaystyle q_{0}$ $\displaystyle=$
$\displaystyle\frac{A_{0}q_{z}+\vec{p}_{1\perp}\vec{q}+\vec{p}_{2\perp}\vec{q}}{A_{z}},$
$\displaystyle a$ $\displaystyle=$ $\displaystyle
A_{z}^{2}-A_{0}^{2},\;b=-2\left(A_{z}\cdot{\cal
D}+A_{0}\left(\vec{p}_{1\perp}\vec{q}+\vec{p}_{2\perp}\vec{q}\right)\right),$
$\displaystyle c$ $\displaystyle=$ $\displaystyle
2A_{z}B_{z}-\left(\vec{p}_{1\perp}\vec{q}+\vec{p}_{2\perp}\vec{q}\right)^{2}+\vec{q}^{2}A_{z}^{2},$
$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
p_{1z}+p_{2z},\;A_{z}=p_{10}+p_{20},$ $\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle
p_{1z}\cdot\vec{p}_{2\perp}\vec{q}-p_{2z}\cdot\vec{p}_{1\perp}\vec{q},$
$\displaystyle B_{z}$ $\displaystyle=$ $\displaystyle
p_{10}\cdot\vec{p}_{2\perp}\vec{q}-p_{20}\cdot\vec{p}_{1\perp}\vec{q},$
$\displaystyle{\cal D}$ $\displaystyle=$ $\displaystyle
p_{1z}p_{20}-p_{2z}p_{10},$ (21)

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

## Appendix B. Regge-eikonal model for elastic proton-proton and pion-proton
scattering

Here is a short review of formulae for the Regge-eikonal approach godizovpp ,
godizovpip , which we use to estimate rescattering corrections in the proton
proton and pion proton channels.

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

$\displaystyle T_{pp,\pi p}^{el}(s,b)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{-2\Omega_{pp,\pi
p}^{el}(s,b)}-1}{2\mathrm{i}},$ $\displaystyle\Omega_{pp,\pi p}^{el}(s,b)$
$\displaystyle=$ $\displaystyle-\mathrm{i}\,\delta_{pp,\pi p}^{el}(s,b),$
$\displaystyle\delta^{el}_{pp,\pi p}(s,b)$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
s}\int_{0}^{\infty}d(-t)J_{0}(b\sqrt{-t})\delta^{el}_{pp,\pi p}(s,t).$ (22)

$\displaystyle\delta^{el}_{pp}(s,t)\simeq$ $\displaystyle
g_{pp\mathbb{P}}(t)^{2}\left(\mathrm{i}+\tan\frac{\pi(\alpha_{\mathbb{P}}(t)-1)}{2})\right)\pi\alpha^{\prime}_{\mathbb{P}}(t)\left(\frac{s}{2s_{0}}\right)^{\alpha_{\mathbb{P}}(t)},$
$\displaystyle\alpha_{\mathbb{P}}(t)=1+\frac{\alpha_{\mathbb{P}}(0)-1}{1-\frac{t}{\tau_{a}}}\,,\,g_{pp\mathbb{P}}(t)=\frac{g_{pp\mathbb{P}}(0)}{\left(1-a_{g}t\right)^{2}}.$
(23)

$\displaystyle\delta^{el}_{\pi p}(s,t)\simeq$
$\displaystyle\left(\mathrm{i}+\tan\frac{\pi(\alpha_{\mathbb{P}}(t)-1)}{2})\right)\beta_{\mathbb{P}}(t)\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{P}}(t)},$
$\displaystyle+\left(\mathrm{i}+\tan\frac{\pi(\alpha_{f}(t)-1)}{2})\right)\beta_{f}(t)\left(\frac{s}{s_{0}}\right)^{\alpha_{f}(t)},$
(24)

$\displaystyle\alpha_{\mathbb{P}}(t)=1+p_{1}\left[1-p_{2}t\left(\arctan\left(p_{3}-p_{2}t\right)-\frac{\pi}{2}\right)\right],$
$\displaystyle\alpha_{f}(t)=\left(\frac{8}{3\pi}\gamma(\sqrt{-t+c_{f}})\right)^{1/2},$
$\displaystyle\gamma(\mu)=\frac{4\pi}{11-\frac{2}{3}n_{f}}\left(\frac{1}{\ln\frac{\mu^{2}}{\Lambda^{2}}}+\frac{1}{1-\frac{\mu^{2}}{\Lambda^{2}}}\right),$
$\displaystyle\beta_{\mathbb{P}}(t)=B_{\mathbb{P}}\mathrm{e}^{b_{\mathbb{P}}t}(1+d_{1}t+d_{2}t^{2}+d_{3}t^{3}+d_{4}t^{4}),$
$\displaystyle\beta_{f}(t)=B_{f}\mathrm{e}^{b_{f}t}.$ (25)

Parameters can be found in tables 1 and 2.

Table 1:  Parameters for proton-proton elastic scattering amplitude. Parameter | Value
---|---
$\alpha_{\mathbb{P}}(0)-1$ | $0.109$
$\tau_{a}$ |  $0.535$ GeV2
$g_{pp\mathbb{P}}(0)$ |  $13.8$ GeV
$a_{g}$ |  $0.23$ GeV-2
Table 2:  Parameters for pion-proton elastic scattering amplitude. Parameter | Value
---|---
$B_{\mathbb{P}}$ | $26.7$
$b_{\mathbb{P}}$ |  $2.36$ GeV-2
$d_{1}$ |  $0.38$ GeV-2
$d_{2}$ |  $0.3$ GeV-4
$d_{3}$ |  $-0.078$ GeV-6
$d_{4}$ |  $0.04$ GeV-8
$B_{f}$ | $67$
$b_{f}$ |  $1.88$ GeV-2

$\displaystyle V_{pp}(s,q^{2})$ $\displaystyle=$ $\displaystyle\int
d^{2}\vec{b}\,\mathrm{e}^{\mathrm{i}\vec{q}\vec{b}}\sqrt{1+2\mathrm{i}T_{pp}^{el}(s,b)}=$
(26) $\displaystyle=$ $\displaystyle\int
d^{2}\vec{b}\,\mathrm{e}^{\mathrm{i}\vec{q}\vec{b}}\mathrm{e}^{-\Omega_{pp}^{el}(s,b)}=$
$\displaystyle=$
$\displaystyle(2\pi)^{2}\delta^{2}\left(\vec{q}\right)+2\pi\bar{T}_{pp}(s,q^{2}),$
$\displaystyle\bar{T}_{pp}(s,q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}b\,db\,J_{0}\left(b\sqrt{-q^{2}}\right)\left[\mathrm{e}^{-\Omega_{pp}^{el}(s,b)}-1\right]$
(27) $\displaystyle S_{\pi p}(s,q^{2})$ $\displaystyle=$ $\displaystyle\int
d^{2}\vec{b}\,\mathrm{e}^{\mathrm{i}\vec{q}\vec{b}}\left(1+2\mathrm{i}T_{\pi
p}^{el}(s,b)\right)=$ (28) $\displaystyle=$ $\displaystyle\int
d^{2}\vec{b}\,\mathrm{e}^{\mathrm{i}\vec{q}\vec{b}}\mathrm{e}^{-2\Omega_{\pi
p}^{el}(s,b)}=$ $\displaystyle=$
$\displaystyle(2\pi)^{2}\delta^{2}\left(\vec{q}\right)+2\pi\bar{T}_{\pi
p}(s,q^{2}),$ $\displaystyle\bar{T}_{\pi p}(s,q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}b\,db\,J_{0}\left(b\sqrt{-q^{2}}\right)\left[\mathrm{e}^{-2\Omega_{\pi
p}^{el}(s,b)}-1\right]$ (29)

Here we take $S_{\pi p}(s,t)=S_{\pi^{+}p}(s,t)=S_{\pi^{-}p}(s,t)$ and

$T^{el}_{\pi^{+}p}(s,t)=T^{el}_{\pi^{-}p}(s,t)=4\pi s\bar{T}_{\pi p}(s,t)$
(30)

Approach (24) describes the data on pion-proton scattering better even at low
energies, that is why we use it instead of the one presented in godizovpp .

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

## Appendix C. Primary amplitudes for CEDP and ERVM resonance production.

Here is the short review of the formulae, which can be obtained by the use of
Refs. godizovmesons , GodizovResonances  and CEDPLIBbasis -CEDPLIBrho .

Let us introduce the general diffractive factor

$F_{\mathbb{P}}(t,\xi)=g_{pp\mathbb{P}}(t)^{2}\left(\mathrm{i}+\tan\frac{\pi(\alpha_{\mathbb{P}}(t)-1)}{2})\right)\frac{\pi\alpha^{\prime}_{\mathbb{P}}(t)}{\xi^{\alpha_{\mathbb{P}}}(t)}.$
(31)

###  $f_{0}$ production

For $f_{0}$ production we have the following expression

$\displaystyle\\!\\!\\!\\!\\!M_{0}^{pp\to p\\{f_{0}\to\pi^{+}\pi^{-}\\}p}=$
(32) $\displaystyle=$ $\displaystyle-
F_{\mathbb{P}}(t_{1},\xi_{1})F_{\mathbb{P}}(t_{2},\xi_{2})\,g_{\mathbb{P}\mathbb{P}f_{0}}(t_{1},t_{2},M_{c}^{2})\times$
$\displaystyle\times$ $\displaystyle\frac{g_{f_{0}\pi\pi}\left({\cal
F}(M_{c}^{2},m_{f_{0}}^{2})\right)^{2}F_{M}(t_{1})F_{M}(t_{2})}{(M_{c}^{2}-m_{f_{0}}^{2})+B_{f_{0}}(M_{c}^{2},m_{f_{0}}^{2})},$

where ($M_{c}>2m_{\pi}$)

$\displaystyle\\!\\!B_{f_{0}}(M_{c}^{2},m_{f_{0}}^{2})=$
$\displaystyle=\mathrm{i}\;\Gamma_{f_{0}}\left({\cal
F}(M_{c}^{2})\right)^{2}\left[\frac{1-4m_{\pi}^{2}/M_{c}^{2}}{1-4m_{\pi}^{2}/m_{f_{0}}^{2}}\right]^{1/2}$
(33) $\displaystyle\\!\\!\\!\\!{\cal
F}(M_{c}^{2},m_{f}^{2})=F^{\mathbb{P}\mathbb{P}f}(M_{c}^{2},m_{f}^{2})=F^{f\pi\pi}(M_{c}^{2},m_{f}^{2})=$
$\displaystyle\\!\\!\\!\\!=\exp\left(\frac{-(M_{c}^{2}-m_{f}^{2})^{2}}{\Lambda_{f}^{4}}\right),\,\Lambda_{f}\sim
1\,\mathrm{GeV},$ (34)
$\displaystyle\\!\\!\\!\\!F_{M}(t)=1/(1-t/m_{0}^{2}),\,m_{0}^{2}=0.5\,\mathrm{GeV}^{2}$
(35)

${\cal F}(M_{c}^{2},m_{f}^{2})$ and $F_{M}(t)$ are off-shell phenomenological
form-factors introduced in CEDPLIBbasis -CEDPLIBrho  to make more good
description of the data. Here we fix mass and width of $f_{0}(500)$ and
$f_{0}(980)$ mesons as in CEDPLIBf0f2

$\displaystyle m_{f_{0}(500)}$ $\displaystyle=$ $\displaystyle
0.6\,\mathrm{GeV},\,\Gamma_{f_{0}(500)}=0.5\,\mathrm{GeV},$ (36)
$\displaystyle m_{f_{0}(980)}$ $\displaystyle=$ $\displaystyle
0.98\,\mathrm{GeV},\,\Gamma_{f_{0}(980)}=0.07\,\mathrm{GeV},$ (37)

and also their couplings to pions

$g_{f_{0}(500)\pi\pi}=3.37,\,g_{f_{0}(500)\pi\pi}=1.55,$ (38)

assuming also $\Gamma(f_{0}\to\pi\pi)/\Gamma_{f_{0}}=100\%$.

In this work coupling of Pomeron to meson $g_{\mathbb{P}\mathbb{P}f_{0}}$ is
the constant and can be extracted from the experimental data as a free
parameter. In GodizovResonances  this coupling is proposed to be $0.64$ GeV
for all mesons related to the “glueball state”. Here due to simplifications
these couplings are of the same order, but may differ by factor $2\div 3$ (see
the main text).

###  $f_{2}$ production

For $f_{2}(1270)$ production we have to modify the expression (32) due to the
tensor nature of this meson. Finally we have

$\displaystyle\\!\\!\\!\\!\\!M_{0}^{pp\to p\\{f_{2}\to\pi^{+}\pi^{-}\\}p}=$
(39) $\displaystyle=$ $\displaystyle
F_{\mathbb{P}}(t_{1},\xi_{1})F_{\mathbb{P}}(t_{2},\xi_{2})\,g_{\mathbb{P}\mathbb{P}f_{2}}\times$
$\displaystyle\times$ $\displaystyle\frac{(g_{f_{2}\pi\pi}/2)\left({\cal
F}(M_{c}^{2},m_{f_{2}}^{2})\right)^{2}F_{M}(t_{1})F_{M}(t_{2})}{(M_{c}^{2}-m_{f_{2}}^{2})+B_{f_{2}}(M_{c}^{2},m_{f_{2}}^{2})}{\cal
P}_{2},$

where

${\cal
P}_{2}=(\Delta_{1}\Delta_{34})^{2}-\frac{(M_{c}^{2}-4m_{\pi}^{2})\lambda(M_{c}^{2},t_{1},t_{2})}{12M_{c}^{2}}$
(40)

is the additional function which can be obtained by contraction of Pomeron-
Pomeron-meson vertex with pion-pion-meson one, $\Delta_{1}=p_{a}-p_{1}$,
$\Delta_{34}=p_{3}-p_{4}$.

$\displaystyle\\!\\!\\!\\!\\!\\!\\!B_{f_{2}}(M_{c}^{2},m_{f_{2}}^{2})=$ (41)
$\displaystyle=$ $\displaystyle\mathrm{i}\;\Gamma_{f_{2}}\left({\cal
F}(M_{c}^{2},m_{f_{2}}^{2})\right)^{2}\left[\frac{1-4m_{\pi}^{2}/M_{c}^{2}}{1-4m_{\pi}^{2}/m_{f_{0}}^{2}}\right]^{5/2}\frac{M_{c}^{4}}{m_{f_{2}}^{4}}$

Fixed parameters for $f_{2}(1270)$ are

$\displaystyle m_{f_{2}(1270)}$ $\displaystyle=$ $\displaystyle
1.275\,\mathrm{GeV},\,\Gamma_{f_{2}(1270)}=0.1851\,\mathrm{GeV},$
$\displaystyle g_{f_{2}(1270)\pi\pi}$ $\displaystyle=$ $\displaystyle
9.26\,\mathrm{GeV}^{-1},\,\Lambda_{f_{2}}=1\,\mathrm{GeV}.$ (42)

After the data analysis the “scalar” coupling of the $f_{2}(1270)$ to Pomeron
is found to be of the order of $2$. Since the Pomeron does not behave as a
simple scalar, the structure of the coupling is, of course, more complicated.
In the “exact” model like in CEDPLIBbasis  Pomeron can be a coherent sum of
different spins, and have several couplings to $f$ mesons. But in real life we
have to obtain the vertex for any spin $J$ and then to make a continuation to
the complex plane in $J$, like in classical Regge theory. This is the
conceptual question, which was partially discussed in myCEDPpipiContinuum ,
but we postpone it to our futher theoretical works. And in the present work we
use simplified model for the $f_{2}$ production. That is why the extracted
value of the “constant” coupling could differ from the real set of tensor
couplings.

###  $\rho_{0}$ production

For $\rho_{0}$ situation is a little bit complicated, since we have to take
into account vector dominance and also $\rho-\omega$ mixing. After all
contractions the primary amplitude looks as follows

$\displaystyle\\!\\!\\!\\!\\!\\!M_{0}^{pp\to
p\\{\rho_{0}\to\pi^{+}\pi^{-}\\}p}=\mathrm{i}\;T^{el}_{\rho_{0}p}(s_{2},t_{2})\frac{C_{T}^{\rho_{0}}(t_{1}){\cal
P}_{\rho}}{|t_{1}|}\times$ (43) $\displaystyle\times$
$\displaystyle\frac{(g_{\rho_{0}\pi\pi}/2){\cal F}_{\rho}(M_{c}^{2}){\cal
F}_{\rho}(t_{1})}{(M_{c}^{2}-m_{\rho_{0}}^{2})+B_{\rho_{0}}(M_{c}^{2},m_{\rho_{0}}^{2})}+(1\leftrightarrow
2),$ $s_{2}=(p_{3}+p_{4}+p_{2})^{2}=(p_{a}-p_{1}+p_{b})^{2},$
$s_{1}=(p_{3}+p_{4}+p_{1})^{2}=(p_{b}-p_{2}+p_{a})^{2}.$

Amplitude $T^{el}_{\rho_{0}p}(s,t)$ can be obtained by the use of the formulae
similar to (22)-(24) with

$\displaystyle\delta^{el}_{\rho_{0}p}(s,t)$ $\displaystyle\simeq$
$\displaystyle
g_{pp\mathbb{P}}(t)g_{\rho\rho\mathbb{P}}(t)\left(\mathrm{i}+\tan\frac{\pi(\alpha_{\mathbb{P}}(t)-1)}{2})\right)\times$
$\displaystyle\times$
$\displaystyle\pi\alpha^{\prime}_{\mathbb{P}}(t)\left(\frac{s}{2s_{0}}\right)^{\alpha_{\mathbb{P}}(t)},$
$\displaystyle g_{\rho\rho\mathbb{P}}(t)$ $\displaystyle=$ $\displaystyle
g_{\rho\rho\mathbb{P}}(0)=7.07\,\mathrm{GeV}\,\,\,\mbox{(see~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{godizovmesons}{\@@citephrase{(}}{\@@citephrase{)}}})};$
(44)

$\displaystyle C_{T}^{\rho_{0}}(t)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3\Gamma_{\rho\to
e^{+}e^{-}}}{\alpha_{e}m_{\rho}}}\frac{m_{\rho}^{2}}{m_{\rho}^{2}-t},$ (45)
$\displaystyle{\cal F}_{\rho}(p^{2})$ $\displaystyle=$
$\displaystyle(1+p^{2}(p^{2}-m_{\rho}^{2})/\Lambda_{\rho}^{4})^{-n_{\rho}},$
(46) $\displaystyle F_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{1-\kappa\,t/(4m_{p}^{2})}{1-t/(4m_{p}^{2})}(1-t/m_{D}^{2})^{-2},$
(47) $\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\mu_{p}/\mu_{N}=2.7928,\,m_{D}^{2}=0.71\,\mathrm{GeV}^{2},$ (50)
$\displaystyle\\!\\!\\!\\!\\!\Gamma_{\rho\to e^{+}e^{-}}=7.04\cdot
10^{-6}\,\mathrm{GeV},$
$\displaystyle\phantom{\frac{A}{B}}\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
P}_{\rho}\simeq
2\sqrt{4\pi\alpha_{e}}F_{1}(t)\left(\vec{\Delta}_{34}^{*}\vec{p}_{a}^{*}\right),$

${\cal P}_{\rho}$ is the factor that is equal to “scalar proton-proton-photon
vertex” contracted with the vector of $\rho\pi\pi$ vertex, and the scalar
product of transverse vectors
$\left(\vec{\Delta}_{34}^{*}\vec{p}_{a}^{*}\right)$ in the $\rho$ meson rest
frame with

$\Delta_{1}^{*}=\\{(M_{c}^{2}+t_{1}-t_{2})/(2M_{c}),0,0,\lambda^{1/2}/(2M_{c})\\}$

can be expressed in terms of four vectors in the central mass frame of
colliding protons:

$\displaystyle\left(\vec{\Delta}_{34}^{*}\vec{p}_{a}^{*}\right)=-\Delta_{34}p_{a}-\Delta_{34,z}^{*}p_{a,z}^{*},$
$\displaystyle\Delta_{34,z}^{*}=-\Delta_{1}\Delta_{34}\frac{2M_{c}}{\lambda^{1/2}},$
$\displaystyle
p_{a,z}^{*}=\left(\frac{p_{a}p_{c}(M_{c}^{2}+t_{1}-t_{2})}{2M_{c}^{2}}-\frac{t_{1}}{2}\right)\frac{2M_{c}}{\lambda^{1/2}},$
$\displaystyle\lambda\equiv\lambda(M_{c}^{2},t_{1},t_{2}),\,p_{c}=p_{3}+p_{4}.$
(51)

Other functions and parameters are

$\displaystyle\\!\\!\\!\\!\\!\\!\\!B_{\rho_{0}}(M_{c}^{2},m_{\rho_{0}}^{2})=$
(52) $\displaystyle=$ $\displaystyle\mathrm{i}\;\Gamma_{\rho_{0}}\left({\cal
F}_{\rho}(M_{c}^{2})\right)^{2}\left[\frac{1-4m_{\pi}^{2}/M_{c}^{2}}{1-4m_{\pi}^{2}/m_{\rho_{0}}^{2}}\right]^{3/2}\frac{M_{c}^{2}}{m_{\rho_{0}}^{2}}$
$\displaystyle m_{\rho_{0}}$ $\displaystyle=$ $\displaystyle
0.7737\,\mathrm{GeV},\,\Gamma_{\rho_{0}}=0.1462\,\mathrm{GeV},$ $\displaystyle
g_{\rho_{0}\pi\pi}$ $\displaystyle=$ $\displaystyle
11.51\,\mathrm{GeV},\,\Lambda_{\rho}=1\,\mathrm{GeV},\,n_{\rho}=0.5.$ (53)

Here we take for all off-shell propagators of resonances simple Breit-Wigner
form, but we can use more complicated expressions, which can be found, for
example, in CEDPLIBbasis -CEDPLIBrho .

## References

  * (1) R. Ryutin, Exclusive Double Diffractive Events: general framework and prospects, Eur. Phys. J. C. 73, 2443 (2013).
  * (2) R. Ryutin, Visualizations of exclusive central diffraction, Eur. Phys. J. C. 74, 3162 (2014).
  * (3) R. Ryutin, Central exclusive diffractive production of two-pion continuum at hadron colliders, Eur. Phys. J. C. 79, 981 (2019).
  * (4) V.A. Petrov, R.A. Ryutin, Single and double diffractive dissociation and the problem of extraction of the proton–Pomeron cross-section, Int. J. Mod. Phys. A 31, 1650049 (2016).
  * (5) J.D. Bjorken, Rapidity gaps and jets as a new-physics signature in very-high-energy hadron-hadron collisions, Phys. Rev. D 47, 101 (1993).
  * (6) F. Abe et al. (CDF Collaboration), Observation of rapidity gaps in $\bar{p}$ $p$ collisions at 1.8 TeV, Phys. Rev. Lett. 74, 855 (1995).
  * (7) M.G. Albrow, A. Rostovtsev, Searching for the Higgs at hadron colliders using the missing mass method, FERMILAB-PUB-00-173 (2000), [arXiv: 0009336[hep-ph]].
  * (8) L.A. Harland-Lang, V.A. Khoze, M.G. Ryskin, Central exclusive production and the Durham model, Int. J. Mod. Phys. A 29, 1446004 (2014).
  * (9) L.A. Harland-Lang, V.A. Khoze, M.G. Ryskin, W.J. Stirling, Central exclusive production within the Durham model: a review, Int. J. Mod. Phys. A 29, 1430031 (2014).
  * (10) L.A. Harland-Lang, V.A. Khoze, M.G. Ryskin, Modeling exclusive meson pair production at hadron colliders, Eur. Phys. J. C 74, 2848 (2014).
  * (11) P. Lebiedowicz, O. Nachtmann, A. Szczurek, Tensor pomeron, vector odderon and diffractive production of meson and baryon pairs in proton-proton collisions, EPJ Web Conf. 206, 06005 (2019).
  * (12) P. Lebiedowicz, O. Nachtmann, A. Szczurek, Exclusive diffractive production of $\pi^{+}\pi^{-}$ continuum and resonances within tensor pomeron approach, EPJ Web Conf. 130, 05011 (2016).
  * (13) P. Lebiedowicz, O. Nachtmann, A. Szczurek, Central exclusive diffractive production of $K^{+}K^{-}K^{+}K^{-}$ via the intermediate $\phi\phi$ state in proton-proton collisions, Phys. Rev. D 99, 094034 (2019).
  * (14) C. Ewerz, M. Maniatis, O. Nachtmann, A Model for Soft High-Energy Scattering: Tensor Pomeron and Vector Odderon, Ann. of Phys. 342, 31 (2014); [arXiv:1309.3478 [hep-ph]].
  * (15) P. Lebiedowicz, O. Nachtmann, A. Szczurek, Central exclusive diffractive production of $\pi^{+}\pi^{-}$ continuum, scalar and tensor resonances in $pp$ and $p\bar{p}$ scattering within tensor pomeron approach, Phys. Rev. D 93, 054015 (2016); [arXiv:1601.04537 [hep-ph]].
  * (16) P. Lebiedowicz, O. Nachtmann, A. Szczurek, Extracting the pomeron-pomeron- $f_{2}(1270)$ coupling in the $pp\to pp\pi^{+}\pi^{-}$ reaction through angular distributions of the pions, Phys. Rev. D 101, 034008 (2020); [arXiv:1901.07788 [hep-ph]].
  * (17) P. Lebiedowicz, O. Nachtmann and A. Szczurek, $\rho_{0}$ and Drell-Soding contributions to central exclusive production of $\pi^{+}\pi^{-}$ pairs in proton-proton collisions at high energies, Phys. Rev. D 91, 074023 (2015); [arXiv:1412.3677 [hep-ph]].
  * (18) P. Lebiedowicz, A. Szczurek, Revised model of absorption corrections for the $pp\to p\pi^{+}\pi^{-}p$ process, Phys. Rev. D 92, 054001 (2015).
  * (19) R. Waldi, K.R. Schubert and K. Winter, Search for glueballs in a pomeron pomeron scattering experiment, Z. Phys. C 18, 301 (1983).
  * (20) A. Breakstone et al. (ABCDHW Collaboration), The reaction Pomeron-Pomeron $\to\pi^{+}\pi^{-}$ and an unusual production mechanism for the $f_{2}(1270)$, Z. Phys. C 48, 569 (1990).
  * (21) L. Adamczyk, W. Guryn and J. Turnau, Central exclusive production at RHIC, Int. J. Mod. Phys. A 29, 1446010 (2014).
  * (22) R. Sikora, Central Exclusive Production of meson pairs in proton-proton collisions at $\sqrt{s}=200$ GeV in the STAR experiment at RHIC, talk at Low x Meeting, 1-5 September 2015, Sandomierz, Poland.
  * (23) J. Adam et al. (The STAR collaboration), Measurement of the central exclusive production of charged particle pairs in proton-proton collisions at $\sqrt{s}=200$ GeV with the STAR detector at RHIC, JHEP 07, 178 (2020); [ arXiv:2004.11078 [hep-ex]]; https://www.hepdata.net/record/ins1792394
  * (24) W. Guryn, From Elastic Scattering to Central Exclusive Production: Physics with Forward Protons at RHIC, Acta Phys. Pol. B 52, 217 (2021); [arXiv:2104.15041 [nucl-ex]].
  * (25) R. Sikora (for the STAR Collaboration), Central exclusive production of charged particle pairs in proton-proton collisions at $\sqrt{s}=200$ GeV with the STAR detector at RHIC, PoS ICHEP2020,501 (2021); [arXiv:2011.14400 [hep-ex]]
  * (26) T. Truhlar (for the STAR Collaboration), Study of the central exclusive production of $\pi^{+}\pi^{-}$, $K^{+}K^{-}$ and $p\bar{p}$ pairs in proton-proton collisions at $sqrt{s}=510$ GeV with the STAR detector at RHIC, [arXiv:2012.06295 [hep-ex]].
  * (27) T.A. Aaltonen et al. (CDF Collaboration), Measurement of central exclusive pi+ pi- production in p pbar collisions at $\sqrt{s}=0.9$ and $1.96$ TeV at CDF, Phys. Rev. D 91, 091101 (2015).
  * (28) M. Albrow, J. Lewis, M. Zurek, A. Swiech, D. Lontkovskyi, I. Makarenko and J.S. Wilson, the public note called Measurement of Central Exclusive Hadron Pair Production in CDF is available at http://www-cdf.fnal.gov/physics/new/qcd/GXG_14/webpage/ .
  * (29) CMS Collaboration, Measurement of exclusive $\pi^{+}\pi^{-}$ production in proton-proton collisions at $\sqrt{s}=7$ TeV, CMS-PAS-FSQ-12-004.
  * (30) K. Osterberg, Potential of central exclusive production studies in high $\beta^{*}$ runs at the LHC with CMS-TOTEM, Int. J. Mod. Phys. A 29 1446019 (2014).
  * (31) A.M. Sirunyan et al., Study of central exclusive $\pi^{+}\pi^{-}$ production in proton-proton collisions at $\sqrt{s}=5.02$ and $13$ TeV, Eur. Phys. J. C 80, 718 (2020); https://www.hepdata.net/record/ins1784063
  * (32) A.A. Godizov, Effective transverse radius of nucleon in high-energy elastic diffractive scattering, Eur. Phys. J. C 75, 224 (2015).
  * (33) A.A. Godizov, Asymptotic properties of Regge trajectories and elastic pseudoscalar-meson scattering on nucleons at high energies, Yad. Fiz. 71, 1822 (2008).
  * (34) A.A. Godizov, The ground state of the Pomeron and its decays to light mesons and photons, Eur. Phys. J. C 76, 361 (2016); [arXiv:1604.01689 [hep-ph]].
  * (35) J.R. Pelaez, A. Rodas, J. Ruiz De Elvira, Global parameterization of $\pi\pi$ scattering up to 2 GeV, e-Print: arXiv:1907.13162 [hep-ph].
  * (36) A.A. Godizov, Current stage of understanding and description of hadronic elastic diffraction, AIP Conf.Proc. 1523, 145 (2013).
  * (37) V.A. Petrov, R.A. Ryutin, A.E. Sobol and J.-P. Guillaud, Azimuthal angular distributions in EDDE as spin-parity analyser and glueball filter for LHC, JHEP 0506, 007 (2005).
  * (38) V.A. Petrov, High-energy implications of extended unitarity, IFVE-95-96, IHEP-95-96, talk given at Blois Conference: 20-24 Jun 1995, Blois, France
  * (39) A.A. Godizov, High-energy central exclusive production of the lightest vacuum resonance related to the soft Pomeron, Phys. Lett. B 787, 188 (2018).
  * (40) R.A. Ryutin, ExDiff Monte Carlo generator for Exclusive Diffraction. Version 2.0. Physics and manual, [arXiv:1805.08591 [hep-ph]].