Prepared for Physics of Elementary Particles and Atomic Nuclei. Theory

# Possible studies at the first stage of the NICA collider operation with
polarized and unpolarized proton and deuteron beams

###### Abstract

Nuclotron based Ion Collider fAcility (NICA) project is in progress at the
Joint Institute for Nuclear Research and will start experiments with heavy
ions. In the context of the NICA Hadronic Physics programme double polarized
$pp$-, $dd$\- and $pd$\- collisions even at lower energies of
$\sqrt{s_{NN}}=3.4-10$ GeV, which will be accessible already at the initial
stage of experiments,are essential tools for precise understanding the spin
dependence of the nucleon-nucleon strong interactions, in both elastic and
deep-inelastic regimes. A special interest is interaction in few baryon
systems at double strangeness, charm and beauty thresholds.For instance,
polarized large-angle elastic $pp$ and $pn$ scattering near the charm
threshold allows one to get an access to properties of possible exotic
multiquark states and their relation to the states recently observed at
LHCb.Large angle scattering of protons and deuterons on the deuteron contains
unique information on the short-range structure of the deuteron, its non-
nucleonic degrees of freedom and also on color transparency phenomenon.
Furthermore, double polarized proton-deuteron scattering offer a possibility
to test the Standard Model through the search for time-invariance (or CP-
invariance under CPT symmetry) violation and parity-violation in single-
polarized scattering. This paper contains suggestions for experiments with
usage of the Spin Physics Detector (SPD) and discusses perspectives of the
first stage of the SPD Programme. This includes experiments with non-polarized
beams too as well as collisions like 12С-12С and 40Сa-40Ca.

###### Abstract

The spin-dependent Glauber theory is applied to calculate spin observables of
$pd$ elastic scattering at $3$-$50$ GeV/c using $pp$ amplitudes available in
the literature and parametrized within the Regge formalism. The calculated
vector $A_{y}^{p}$, $A_{y}^{d}$ and tensor $A_{xx}$, $A_{yy}$ analyzing powers
and the spin-correlation coefficients $C_{y,y}$, $C_{x,x}$, $C_{yy,y}$,
$C_{xx,y}$ can be measured at SPD NICA and, thus, will provide a test of the
used $pN$ amplitudes. Quasi-elastic scattering $pd\to\\{pp\\}_{s}n$ with
formation of spin-singlet $pp(^{1}S_{0})$ pair at zero scattering angle is of
special interest. The $dd$ elastic scattering is briefly outlined. The double
polarized $pp$ and $pn$ elastic scattering at large c.m.s. scattering angle
$\theta_{cm}=90^{\circ}$ is considered in the threshold of the charm
production.

###### Abstract

Motivation is outlined for a precise study of high-energy diffractive
scattering of protons at $|t|\lesssim 1$ GeV2 in the experiment SPD. Small
oscillations in the $t$-dependence of the differential cross section at low
and medium $t$, observed in earlier experiments at Protvino, ISR, Fermilab and
now also at LHC, are probably related with the proton’s structure at impact
parameters exceeding the size of the proton’s quark core and thus indicate
involvement of meson periphery of the nucleon to diffractive scattering. The
experiment SPD can provide new precise data on small-angle elastic
$pp$-scattering for exploring this phenomenon.

###### Abstract

The spin effects in the elastic proton-proton scattering are analysed at NICA
energies. It is shown the importance the investigation of the region of the
diffraction minimum in the differential cross sections. Some possible
estimation of spin effects are given for the different NICA energies in the
framework of the new high energy generelazed structure (HEGS) model.

###### Abstract

Physics of single-spin processes for the SPD NICA project is proposed. This
includes transverse single-spin asymmetry ($A_{N}$) and hyperon polarization
($P_{N}$) measurements in various types of collisions, including p+p, d+d, C+C
and Ca+Ca. The polarized $p$\- and $d$-beams in the NICA collider can be used
to study $A_{N}$ for more than several dozen reactions at different energies
in the $3.4<\sqrt{s}<27$ GeV range. A number of interesting phenomena have
been predicted, such as the oscillation for $A_{N}(x_{\rm{F}})$ and
$P_{N}(x_{\rm{F}})$, the resonance dependence on the energy $\sqrt{s}$ for
$A_{N}$ and $P_{N}$, and the threshold dependence of $A_{N}$ on the c.m.
production angle for some reactions. The role of quark composition of
particles involved in the reaction is discussed.

###### Abstract

In the context of NICA-SPD project, the motivation of the study of vector
meson, charm production (hidden) $p+p\to p+p+V$, $V=\rho$,$\phi$, $J/\Psi$ and
open $N+N\to\Lambda_{C}(\Sigma_{C})+\bar{D}+N$ is recalled. Backward vector
meson production, that should be background free in a collider, can possibly
be measured and be used also as an alternative method of producing neutron
beams. Simple estimations of cross sections are presented on the basis of
existing literature. When possible, model independent statements on
polarization effects are highlighted.

###### Abstract

We argue that reaction $p^{2}H\to ppn$ at large momentum transfer to one of
the nucleons of the deuteron provides a sensitive probe of space time
evolution of hard $pN$ scattering . The same process in a different kinematics
allow to study short range correlations in the deuteron. Use of the polarized
deuteron beams would provide a unique opportunity to separate S- and D-wave
contributions to the high momentum component of the deuteron wave function. A
possibility to look for nonnucleonic components of the short range
correlations is also outlined.

###### Abstract

Differential cross sections of various binary reactions with the lightest
nuclei at large fixed scattering angles are in qualitative agreement with the
$s$\- power-law dependence dictated by the constituent counting rules. We
propose to measure differential cross section and deuteron analyzing powers of
the $dp$\- elastic scattering at the SPD NICA to search for transition region
from meson-baryon to quark-gluon degrees of freedom in the deuteron structure.

###### Abstract

In this experimental proposal, we consider a possibility of registering light
dibaryons at the NICA SPD facility, experimental and theoretical indications
of the existence of which were previously obtained at JINR. The main attention
is paid to the description of the observable effects, as well as the
requirements for measuring instruments necessary for their successful
detection.

###### Abstract

Based on our recent study of the lightest neutral hypernuclei with strangeness
$-1$ and $-2$, we propose to look for the neutral hypernucleus [4][ΛΛ]n in
deuteron-deuteron collisions which can be accessed by SPD NICA in the future.
Some advantages and opportunities for hypernuclei and exotic hadrons in the
double $K^{+}$ production channels at NICA are addressed.

###### Abstract

Experiments are proposed directed on solution of three main problems of
physics of soft $pp$ interactions: understanding/description of baryon spectra
in $pp$ collisions, evolution of $<P^{2}_{T}>$ – $x_{F}$ correlations with
energy growth and two-particle $P_{T}$ correlations.

###### Abstract

Over three decades there has been no comprehensive understanding of the
mechanism of soft photons (energy smaller 50 MeV) formation. Experimental data
indicate an excess of their yield in hadron and nuclear interactions in
comparison with calculations performed theoretically. In JINR, in connection
with the building of a new accelerator complex NICA, it has become possible to
carry out such studies in pp, pA and AA interactions at energies up to 25
$A\,$GeV. We prepared the extensive physical program for soft photons that
covers the wide region of investigations in high energy physics. To carry out
this program our group develops the conception of an electromagnetic
calorimeter on type ‘‘shashlik’’ based on gadolinium-gallium garnet (GaGG)
crystals, which have significantly lower the threshold for registration of
photons. The first tests of electromagnetic calorimeters manufactured at JINR
on the basis of the GaGG and a composite of tungsten and copper confirm that
choice.

###### Abstract

The space-time picture of hadron formation in high energy processes with
nuclear targets is still poorly known. It is suggested to test different
models of hadron formation by using collisions of heavy ions. Results of
microscopic transport calculations of proton and charged pion rapidity and
transverse momentum distributions in C+C and Ca+Ca collisions at
$\sqrt{s}_{NN}=11$ GeV are presented.

###### Abstract

It is proposed to use the Drell-Yan process with pair production of $\tau$
leptons to measure the parameters of polarized parton distribution functions
of a proton at the NICA collider in the SPD experiment. To determine the
polarization of tau leptons, we propose to use decays of $\tau$ leptons into a
single charged pi-meson and neutrino. To parameterize the polarization state
of $\tau$ leptons, it is proposed to use the energy of single $\pi$-mesons.

###### Abstract

Firm interpretation of the recent results from the AMS-02 and PAMELA
spectrometers, regarding the antiproton yield in $p$-$p$ and $p$-$d$
collisions, has been hindered by uncertainties in production cross section,
angular, and momentum spectra of the produced antiprotons. The proposed
measurements of antiproton yield at the planned SPD experiment at the NICA
collider could significantly contribute to enhancing the situation in favor of
the search for dark matter WIMPs.

###### Abstract

We present new ideas on tests of fundamental symmetries in polarization
experiments at the NICA facility. Specifically, we explore the possibilities
of high precision tests of the Standard Model by parity violation and searches
of Beyond the Standard Model semistrong breaking of time reversal invariance
in double polarized proton-deuteron scattering, taking advantage of high
intensity beams of polarized protons and deuterons available at NICA. In both
cases, we propose to use the new technique of polarized beam with precessing
horizontal polarizations, and polarized deuterons are the favored choice. The
external target in the extracted beam is optional for the parity violation
experiment, which requires furnishing Nuclotron and/or new Booster with very
modest new instrumentation. One should not overlook this potential for
substantial broadening of horizons of spin physics at the NICA facility.

1 NRC “Kurchatov Institute” - IHEP, Protvino 142281, Moscow region, Russia 2
Skobeltsyn Institute of Nuclear Physics, MSU, Moscow, 119991 Russia 3 P.N.
Lebedev Physical Institute,Leninsky prospect 53, 119991 Moscow, Russia 4
Astronomy Department, Faculty of Science, Cairo University, Giza, Egypt, 12613
5 Veksler and Baldin Laboratory of High Energy Physics, Joint Institute for
Nuclear Research,Dubna, Moscow region, 141980 Russia 6 Joint Institute for
Nuclear Researches, DLNP, Dubna, Moscow reg. 141980 Russia 7 Petersburg
Nuclear Physics Institute NRC KI, Gatchina, Russia 8 St. Petersburg
Polytechnic University, St. Peterburg, Russia 9 Sukhoi State Technical
University of Gomel, Prospect Octiabria, 48, 246746 Gomel, Belarus 10 Budker
Institute of Nuclear Physics of SB RAS, 630090 Novosibirsk, Russia 11
Novosibirsk State University, 630090 Novosibirsk, Russia 12 Novosibirsk State
Technical University,630092 Novosibirsk, Russia 13 Laboratory of Information
Technologies, Joint Institute for Nuclear Research, Dubna, Moscow region,
141980 Russia 14 Joint Institute for Nuclear Researches, BLTP, Dubna, Moscow
reg. 141980 Russia 15 Institut für Theoretische Physik, Justus-Liebig-
Universität, 35392 Giessen, Germany 16 L.D. Landau Institute for Theoretical
Physics, 142432 Chernogolovka, Russia 17 Université de Lyon, Institut de
Physique des 2 Infinis de Lyon, UCBL–IN2P3-CNRS, 4, rue Enrico Fermi,
Villeurbanne, France 18 St. Petersburg State University, St. Peterburg, Russia
19 Pensilvania State University, 104 Davey Laboratory University Park PA 16802
USA 20 DPhN, IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette Cedex,
France 21 Dubna State University, Dubna, Moscow reg. 141980 Russia 22
Department of Physics, M.V. Lomonosov State University, Moscow, 119991 Russia
23 Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, P.R. China 24
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
P.R. China 25 University of Chinese Academy of Sciences, Beijing 100049, P.R.
China 26 Moscow Institute of Physics and Technology (National Research
University), 141701 Dolgoprudny, Russia

###### Contents

  1. 1 The SPD setup and experimental conditions 111This section is written by A.V. Guskov (E-mail<EMAIL_ADDRESS>and A.D. Kovalenko<EMAIL_ADDRESS>  2. 2 Elastic $pN$, $pd$ and $dd$ scattering 333This section is written by Yu.N. Uzikov<EMAIL_ADDRESS>    1. 2.1 Spin amplitudes of $pN$ elastic scattering
    2. 2.2 Polarized $pd$ elastic diffraction scattering within the Glauber model
    3. 2.3  Quasielastic pd-scattering $p+d\to\\{pp\\}(^{1}S_{0})+n$
    4. 2.4  Elastic $dd$ scattering
    5. 2.5 Double polarized large angle $pN$ elastic scattering
    6. 2.6 Summary
  3. 3 Studying periphery of the nucleon in diffractive $pp$ scattering 666This section is written by V.A. Baskov, O.D. Dalkarov, A.I. L’vov (E-mail<EMAIL_ADDRESS>and V.V. Polyanskiy
  4. 4 Hadron structure and spin effects in elastic hadron scattering at NICA energies 888This section is written by O.V. Selyugin; E-mail<EMAIL_ADDRESS>    1. 4.1 HEGS model and spin effects in the dip region of momentum transfer
    2. 4.2 Conclusions
  5. 5 Single-spin physics 101010This section is written by V. Abramov; E-mail<EMAIL_ADDRESS>    1. 5.1 Model of chromomagnetic polarization of quarks
    2. 5.2 Single-spin hadron asymmetry
    3. 5.3 Transverse polarization of hyperons
  6. 6 Vector light and charmed meson production 121212 This section is written by E. Tomasi-Gustafsson; E-mail<EMAIL_ADDRESS>    1. 6.1 Charm production
    2. 6.2 Open charm production
    3. 6.3 Backward meson production
    4. 6.4 Conclusions
  7. 7 Exclusive hard processes with deuteron at NICA151515This section is written by M. Strikman; E-mail<EMAIL_ADDRESS>    1. 7.1 Probing dynamics of nucleon - nucleon interaction in proton - deuteron quasielastic scattering
    2. 7.2 Probing microscopic deuteron structure
  8. 8 Scaling behaviour of exclusive reactions with lightest nuclei and spin observables 171717This section is written by V.P. Ladygin (E-mail<EMAIL_ADDRESS>and Yu.N. Uzikov
  9. 9 Multiquark correlations and exotic hadron state production 191919This section is written by V.T. Kim (kim_vt@pnpi.nrcki.ru), A.A. Shavrin<EMAIL_ADDRESS>and A.V. Zelenov<EMAIL_ADDRESS>    1. 9.1 Multiquark correlations and exotic state production at SPD NICA
    2. 9.2 Multiquark correlations: fluctons in nuclei
    3. 9.3 Few-quark correlations: Diquarks
    4. 9.4 Multiparton scattering
    5. 9.5 Multiquark exotic state production
    6. 9.6 Summary
  10. 10 Study of inelastic d-d and p-d interactions for observation of neutron-proton system under strong compression 212121 This section is written by B.F. Kostenko, E-mail<EMAIL_ADDRESS>    1. 10.1 Introduction
    2. 10.2 Search for new dibaryons at the NICA SPD facility
  11. 11 Proposal for the study of lightest neutral hypernuclei with strangeness $-1$ and $-2$ 242424This section is written by J.-M. Richard, Q. Wang and Q. Zhao<EMAIL_ADDRESS>    1. 11.1 Binding conditions for 3 and 4-body systems with strangeness $-1$ and $-2$
    2. 11.2 Production mechanism for $\isotope[4][\Lambda\Lambda]{n}$ and advantages of double $K^{+}$ productions
    3. 11.3 Summary
  12. 12 Problems of soft $pp$ interactions 262626This section is written by A. Galoyan and V. Uzhinsky.
  13. 13 Puzzles of soft photons pp, pA and AA interactions 282828This section is written by E. Kokoulina<EMAIL_ADDRESS>and V.A. Nikitin (E-mail: nikitin@jinr.ru).
    1. 13.1 The scientific program of SP study
    2. 13.2 The preparation to experimental SP study
  14. 14 Hadron formation effects in heavy ion collisions 303030This section is written by A. B. Larionov; E-mail<EMAIL_ADDRESS>    1. 14.1 The model
    2. 14.2 Numerical results
    3. 14.3 Summary and conclusions
  15. 15  Measurement of characteristics of the processes of pair production of polarized tau leptons in the SPD experiment. 343434 This section is written by A. Aleshko, E. Boos (E-mail: boos@theory.sinp.msu.ru), V. Bunichev (E-mail<EMAIL_ADDRESS>  16. 16 On Measuring Antiproton-Production Cross Sections for Dark Matter Search 363636This section is written by R. El-Kholy; E-mail<EMAIL_ADDRESS>    1. 16.1 Antiproton Production Cross Sections
    2. 16.2 NICA SPD Contribution
    3. 16.3 Summary
  17. 17 Tests of fundamental discrete symmetries at NICA facility: addendum to the spin physics programme 383838 This section is presented by I.A. Koop, A.I. Milstein, N.N. Nikolaev (E-mail: nikolaev@itp.ac.ru), A.S. Popov, S.G. Salnikov, P.Yu. Shatunov,Yu.M. Shatunov.
    1. 17.1 Precessing spin asymmetries in the total $pd$ cross section
    2. 17.2 PV asymmetry: expectations from Standard Model
    3. 17.3 The experimental strategies
    4. 17.4 Summary and outlook

## Tests of QCD basics in the transition region

The Standard Model (SM) of fundamental interactions formulated five decades
ago as a local gauge invariant theory based on the $SU(2)_{L}\times
U(1)_{Y}\times SU(3)_{c}$ spontaneously broken symmetry, was perfectly
confirmed by experiments in electroweak sector. The only part of this model,
Quantum Chromodynamics (QCD), connected with the colored $SU(3)_{c}$ symmetry
and considered as a basis of strong interactions between quarks and gluons is
still under experimental verification.

At low energies, below the GeV region the strong interaction is described in
terms of baryons exchanging mesons in accordance with the chiral effective
field theory, which is based on spontaneously broken chiral symmetry of the
QCD Lagrangian [1]. Recent progress in our understanding of properties of the
light nuclei and nuclear reactions achieved within this approach is outlined
in Refs. [2, 3]. At much higher energies and high transferred 4-momenta,
perturbative Quantum Chromodynamics (pQCD) characterizes the strong force in
terms of quark and gluons carrying color charge, and obeying to parton
distribution functions (PDF) of hadrons and nuclei. Although these two
pictures are well determined in their respective energy scales, the transition
between them is not well identified. Whereas the goal of the Many Purposes
Detector (MPD) NICA project is to search for phase transition of the baryon
matter at high temperature and high density into the quark gluon plasma in
heavy-ions collision, and on this way to study properties of the early
Universe, the main aim of the Spin Physics Detector (SPD) project [4] at its
first stage with lower energies is quite different and, in particular, is just
connected with a search for the transition region from hadron to quark-gluon
degrees of freedom in theoretical describing of collisions of free nucleons or
lightest nuclei. QCD predicts that hadrons produced in exclusive processes at
sufficiently high 4-momentum transfer will experience diminished final
(initial) state interactions. This QCD prediction named as color transparency
(CT) [5], [6] may help to identify the transition between these two
alternative descriptions of strong forces after the onset of CT will be
observed. Another signal for the transition region in structure of the
lightest nuclei is related to onset of the predicted by pQCD dimensional
scaling in reactions with these nuclei. A clear indication for transition to
quark degrees of freedom in strong interactions would give a formation of
multiquark states, like dibaryon resonances observed in sector of light quarks
[7]. Production of heavy quarks in few-nucleon systems can be related to
formation of exotic type of resonances, as ‘‘octoquarks’ $uuds\bar{s}uud$,
$uudc\bar{c}uud$ [8] and the behaviour of double spin correlation $A_{NN}$ of
$pp$ elastic scattering measured near the charm threshold at large angles [9]
supports this assumption. On the other hand, it is important to understand how
this observation is related to recently observed at LHCb pentaquark states
$uudc\bar{c}$ [10]. The SPD NICA has all possibilities for study these and
other issues of QCD. Furthermore, polarization phenomena provide an unique
possibility to search for physics beyond the SM by making test of fundamental
discrete symmetries of the SM related to the space (P), time (T) and charge
(C) inversion. One of these options is connected with double polarized proton-
deuteron scattering providing a search for T-invariance (or CP-invariance
under CPT-symmetry) violation.

Experiments with unpolarized colliding beams are also of importance in study
of reactions at heavy quark thresholds and in search for both color
transparency and scaling onset or multiquark (dibaryon) states.

##  1 The SPD setup and experimental conditions 111This section is written by
A.V. Guskov (E-mail<EMAIL_ADDRESS>and A.D. Kovalenko
<EMAIL_ADDRESS>
The SPD experimental setup is being designed as a universal $4\pi$ detector
with advanced tracking and particle identification capabilities based on
modern technologies that can operate with polarized proton and deuteron beams
at a collision energy up to 27 GeV and a luminosity up to $10^{32}$ cm-2 s-1
(proton collisions). Details of the SPD experimental setup are described in
its Conceptual Design Report [4]. The silicon vertex detector will provide
resolution for the vertex position on the level of below 100 $\mu$m needed for
reconstruction of primary and secondary vertices. The straw tube-based
tracking system placed within a solenoidal magnetic field of up to 1 T at the
detector axis should provide the transverse momentum resolution
$\sigma_{p_{T}}/p_{T}\approx 2\%$ for a particle momentum of 1 GeV/$c$. The
time-of-flight system with a time resolution of about 60 ps will provide
$3\sigma$ $\pi/K$ and $K/p$ separation of up to about 1.2 GeV/$c$ and 2.2
GeV/$c$, respectively. Possible use of the aerogel-based Cherenkov detector
could extend this range. Detection of photons will be provided by the sampling
electromagnetic calorimeter with the energy resolution $\sim 5\%/\sqrt{E}$. To
minimize multiple scattering and photon conversion effects for photons, the
detector material will be kept to a minimum throughout the internal part of
the detector. The muon (range) system is planned for muon identification. It
can also act as a rough hadron calorimeter. The pair of beam-beam counters and
zero-degree calorimeters will be responsible for the local polarimetry and
luminosity control. To minimize possible systematic effects, SPD will be
equipped with a triggerless DAQ system.

It is assumed that up to 30% of the collider running time will be devoted to
polarized deuteron and proton experiments from the beginning of the collider
commissioning. Thus, some polarized $pp$-, $dd$\- and even $pd$\- collisions
at energy range of $\sqrt{s_{NN}}=3.4\div 10$ GeV, could be possible already
at the initial stage of the collider operation. The most accessible is
polarized deuteron beam from the Nuclotron in the energy range of $1\div 4$
GeV/u. Average luminosity of $dd$ \- collisions is estimated to $8\times
10^{27}\div 2.5\times 10^{31}\,cm^{-2}s^{-1}$. Stable direction of the
polarization vector is vertical. A single and double polarized collisions are
possible. Transverse polarization of deuteron beam can be obtained at the
specific energy point $\sim 5.6$ GeV corresponding to the spin integer
resonance. The adequate intensity of polarized proton beam from the Nuclotron
($\geq 10^{10}$ part./pulse) will be reached after commissioning of the new
light ion injector LILAC scheduled to 2025-2026 and the spin control system
have been designed for the collider. The existing proton injected chain put
limit to the beam intensity due to very low output linac energy (5 MeV). Thus,
only experiments on the beam storage and acceleration are planning for the
commissioning phase. Realization of pd - mode is more complicated because
HILAC and LILAC both injection chains should be involved in the process.
Moreover, only single polarized collision mode is available, namely:
unpolarized deuteron with polarized proton. The peak luminosity in symmetric
dp – mode, corresponding to equal momentum of the colliding particles per
nucleon, can reach of $2\times 10^{31}\,cm^{-2}s^{-1}$ at stored intensity of
$6\times 10^{11}$ particles per each collider ring. Light ion collision
studies at the SPD are possible also. The luminosity level can be scaled from
that was specified for gold-gold collisions: $1\times 10^{27}\,cm^{-2}s^{-1}$
at $\sqrt{s_{NN}}=11$ GeV.

##  2 Elastic $pN$, $pd$ and $dd$ scattering 333This section is written by
Yu.N. Uzikov<EMAIL_ADDRESS>
PACS: 25.40.Cm, 13.75.Cs, 13.88.+e

###  2.1 Spin amplitudes of $pN$ elastic scattering

Nucleon-nucleon elastic scattering contains fundamental information on the
dynamics of the $NN$ interaction and constitutes a basic process in physics of
atomic nuclei and hadrons. A systematic reconstruction of spin amplitudes of
$pp$ and $pn$ elastic scattering from $pN$ scattering data is provided by the
SAID partial-wave analysis [11] and covers laboratory energies up to $3$ GeV
($p_{lab}\approx 3.8$ GeV/c) for $pp$ and $1.2$ GeV ($p_{lab}\approx 1.9$
GeV/c) for $pn$ scattering. At higher energies there is only incomplete
experimental information on $pp$ scattering, whereas data for the $pn$ system
are very scarce. In the literature there are several models and corresponding
parametrizations for $pN$ amplitudes. Some of them are obtained in the eikonal
approach for the lab momentum $6$ GeV/c [12] and for LHC energies [13] and
recently in [14] (see Sect. 4). At moderate transferred momenta $-t$ and large
invariant mass $s$ the Regge model is expected to be valid to describe elastic
$pN$ scattering. In literature there are some parametrizations for $pN$
amplitudes, obtained within the Regge phenomenology for values of $s$ above
$6$ GeV2 ($p_{lab}\geq 2.2$ GeV/c) [15] and for $p_{lab}=3$-$50$ GeV/c
(corresponding to $2.77<\sqrt{s}<10$ GeV) [16].

Assuming Lorentz-invariance and parity conservation, the elastic $NN$
scattering is described by eight independent helicity amplitudes $\phi_{i}$
($i=1,\dots 8$) determined in [17, 18]. Under time-reversal invariance, one
has ($\phi_{5}=\phi_{8}$, $\phi_{6}=\phi_{7}$) six independent amplitudes, and
for identical nucleons $pp$ and $nn$ the number of independent helicity
amplitudes is equal to five ($\phi_{5}=-\phi_{6}$, $\phi_{7}=-\phi_{8}$). Full
information about the spin dependent $pN$ amplitudes can be obtained, in
principle, from a complete polarization experiment, which, however, requires
to measure twelve (ten) independent observables at a given collision energy
for $pn$ ($pp$ or $nn$) and, thus, constitutes a too complicated experimental
task. Another possible way to check existing parametrizations in addition to
direct measurement of spin observables of $pN$ elastic scattering is to study
spin effects in proton-deuteron ($pd$) and neutron-deuteron ($nd$) elastic and
quasi-elastic scattering. The polarized $pd$-elastic scattering is discussed
below using the Glauber diffraction theory.

At large $-t$ corresponding to large scattering angles in the c.m.s. pN system
($\theta_{cm}\approx 90^{\circ}$), where the Regge model cannot be applied,
very interesting features were observed in the double spin asymmetry $A_{NN}$
in the elastic pp scattering at laboratory momenta $p_{lab}=5-10$ GeV/c.
Commonly accepted explanation of those features is absent in literature. In
section 2.5 we give a short review of existing models based on usage of the
pQCD amplitudes and non-perturbative exotic multiquark resonances
contribution.

###  2.2 Polarized $pd$ elastic diffraction scattering within the Glauber
model

As was noted above, a possible way to check existing parametrizations of $pN$
elastic amplitudes is to study spin effects in proton-deuteron ($pd$) and
deuteron-deuteron ($dd$) elastic and quasi-elastic scattering. At high
energies and small four-momentum transfer $t$, $pd$ scattering can be
described by the Glauber diffraction theory of multistep scattering, which
involves as input on-shell $pN$ elastic scattering amplitudes. Applications of
this theory with spin-dependent effects included [19] indicate a good
agreement with the $pd$ scattering data at energies about $1$ GeV if the SAID
data on $pN$ scattering amplitudes are used as input of the calculations [20,
21, 22].

Figure 1: Analyzing power for $pp$ elastic scattering as a function of the
four-momentum transfer $-t$ at $4.8$ GeV/c (left) and $45$ GeV/c (right). The
results of calculations [23] based on the Regge model parameterizations from
[16] are shown by the solid line (see details in Ref. [23]). Left: Data are
taken from Refs. [24] (filled squares: $4.4$ GeV/c; open squares: $5.15$
GeV/c), and [25] (circles). Right: Data are taken from Refs. [26] (squares)
and [27] (circles).

fig-uz01

The spin-dependent Glauber theory [19, 20] is applied recently [23] to
calculate spin observables of $pd$ elastic scattering at $3$-$50$ GeV/c
utilizing the $pp$ elastic scattering amplitudes $f_{pp}$ established and
parametrized in Ref. [16] within the Regge formalism. The Regge approach
allows one to construct $pn$ (and $\bar{p}N$) amplitudes together with the
$pp$ amplitudes. This feature allows one to perform a test of broad set of
$pN$ amplitudes and applicability of the Regge model itself to $pN$ elastic
scattering. However, in view of the scarce experimental information about the
spin-dependent $pn$ amplitudes and taking into account that the spin-
independent parts of the $pp$ and $pn$ amplitudes at high energies are
approximately the same, it was assumed in [23] as a first approximation, that
$f_{pn}=f_{pp}$. The amplitudes of $pN$ elastic scattering are written as [19]

$\displaystyle M_{N}({\bf p},{\bf
q};{\mbox{\boldmath$\sigma$}},{{\mbox{\boldmath$\sigma$}}}_{N})=A_{N}+C_{N}{\mbox{\boldmath$\sigma$}}\hat{n}+C_{N}^{\prime}{\mbox{\boldmath$\sigma$}}_{N}\hat{n}+B_{N}({\mbox{\boldmath$\sigma$}}\hat{\bf
k})({\mbox{\boldmath$\sigma$}}_{N}\hat{\bf k})+$ (1)
$\displaystyle+(G_{N}+H_{N})({\mbox{\boldmath$\sigma$}}\hat{\bf
q})({\mbox{\boldmath$\sigma$}}_{N}\hat{\bf
q})+(G_{N}-H_{N})({\mbox{\boldmath$\sigma$}}\hat{\bf
n})({\mbox{\boldmath$\sigma$}}_{N}\hat{\bf n}),$

where the complex numbers $A_{N}$, $C_{N}$, $C_{N}^{\prime}$, $B_{N}$,
$G_{N}$, $H_{N}$ were fixed from the amplitudes of the SAID analysis [11] and
parametrized by a sum of Gaussians. For the double scattering term in $pd$
scattering the unit vectors $\hat{\bf k}$, $\hat{\bf q}$, $\hat{\bf n}$ are
defined separately for each individual $NN$ collision. Numerical values for
the parameters of the Gaussians are obtained by fitting to the helicity
amplitudes from Ref. [16]. Those for $p_{lab}=45$ GeV/c are given in Ref.
[23]. The differential cross section of $pp$ elastic scattering and the vector
analyzing power $A_{y}$ are reproduced with these parameterizations on the
same level of accuracy as in Ref. [16], in the interval of transferred four
momentum $-t<1.5$ (GeV/c)2. An example of calculations of $A_{y}$ at
$p_{lab}=4.8$ GeV/c and 45 GeV/c is shown in Fig. LABEL:fig-uz01.

The spin observables $A_{y}$, $A_{ij}$, and $C_{ij,k}$ considered in the work
[23] are defined in the notation of Ref. [28] as following

$\displaystyle
A_{y}^{d}=TrMS_{y}M^{+}/TrMM^{+},A_{y}^{p}=TrM\sigma_{y}M^{+}/TrMM^{+}$ (2)
$\displaystyle A_{yy}=TrM{\cal P}_{yy}M^{+}/TrMM^{+},A_{xx}=TrM{\cal
P}_{yy}M^{+}/TrMM^{+}$ $\displaystyle
C_{y,y}=TrM{S}_{y}\sigma_{y}M^{+}/TrMM^{+},C_{x,x}=TrM{S}_{y}\sigma_{y}M^{+}/TrMM^{+},$
$\displaystyle C_{xx,y}=TrM{\cal
P}_{xx}\sigma_{y}M^{+}/TrMM^{+},C_{yy,y}=TrM{\cal
P}_{yy}\sigma_{y}M^{+}/TrMM^{+},$

where ${\cal P}_{ij}=\frac{3}{2}(S_{i}S_{j}+S_{j}S_{i})-2\delta_{ij}$ and
$S_{j}$ ($j=x,y,z$) are Cartesian components of the spin operator for the
system with $S=1$, the transition operator $M$ depends on the momentum of the
initial ($\bf p$) and final (${\bf p}^{\prime}$) proton and contains the Pauli
spin matrices ${{\mbox{\boldmath$\sigma$}}}=(\sigma_{x},\ \sigma_{y},\
\sigma_{z})$. We use the Madison reference frame with the axis OZ$||$${\bf
p}$, OY$||$$[{\bf p}\times{\bf p}^{\prime}]$ and OX choosen in such a way to
provide a right-handed coordinate system.

Figure 2: Results for spin-dependent $pd$ observables. Predictions from Ref.
[23] for $p_{lab}=4.8$ GeV/c are shown by dashed lines while those at $45$
GeV/c correspond to the solid lines.

fig-uz02

The unpolarized differential cross section, vector ($A_{y}^{p}$, $A_{y}^{d}$)
and tensor ($A_{xx}$, $A_{yy}$) analyzing powers and some spin correlation
parameters ($C_{x,x}$, $C_{y,y}$, $C_{xx,y}$, $C_{yy,y}$) 555 We use here
notations of Ref. [28] of $pd$ elastic scattering were calculated at
$p_{l}=4.85$ GeV/c and 45 GeV/c at $0<-t<2$ GeV2 using $pN$ amplitudes from
[16]. The results obtained for $A_{y}^{p}$, $A_{y}^{d}$, $C_{xx,y}$ and
$C_{yy,y}$ are shown in Fig. LABEL:fig-uz02. As shown in Ref. [23] available
data on $pd$-elastic differential cross section in forward hemisphere are well
described by this model. Most sensitive to the spin-dependent $pN$ amplitudes
are vector analyzing powers $A_{y}$ and spin correlation parameters $C_{x,x}$
and $C_{y,y}$. So, even measurement of the ratio $A_{y}^{d}/A_{y}^{p}$ at low
$t$ gives valuable information on the transverse spin-spin term in NN-
amplitudes [29]. In contrast, the tenzor analyzing powers $A_{xx}$ and
$A_{xx}$ are very weakly sensitive to those amplitudes and weakly changed with
increasing energy. The calculated in [23] polarization observables can be
measured at SPD NICA that will provide a test of the used $pN$ amplitudes. The
corresponding differential cross section is rather large in the considered
region $p_{lab}=3-50$ GeV/c and $|t|=0-2$ GeV2 being $d\sigma/dt>0.1$ mb/GeV2.
Expected counting rate $N$ at $p_{lab}=50$ GeV/c (${q_{pp}^{cm}}=5$ GeV/c) for
the luminosity $L=5\times 10^{30}cm^{-2}s^{-1}$ and for the solid angle
$\Delta\Omega=0.03$ is $N\geq 10^{2}s^{-1}$.

The $pN$ helicity amplitudes $\phi_{5}$ and $\phi_{1}+\phi_{3}$, which can be
tested in the above described procedure are necessary in search of time-
reversal invariance effects in double-polarized $pd$ scattering [30, 31]. Data
of the spin-correlation parameters of $pp$ elastic scattering being analyzed
in the framework of the eikonal model [13] will allow one to obtain space
structure of the spin-dependent hadron forces [32].

###  2.3  Quasielastic pd-scattering $p+d\to\\{pp\\}(^{1}S_{0})+n$

Spin structure of the amplitude of the reaction of quasielastic $pd$
scattering with formation of the $pp$ pair at small excitation energy $\leq 3$
MeV

$p+d\to\\{pp\\}(^{1}S_{0})+n$ (3)

is of special interest. In this reaction the final $pp$ pair is in the
${}^{1}S_{0}$ state of the internal motion, therefore the number of of
independent transition matrix elements is diminished to six instead of twelve
for the elastic $pd$\- scattering. Since the angular momentum of the
$pp(^{1}S_{0})$ pair is zero, in collinear kinematics the transition matrix
element of this reaction is completely described by two independent amplitudes
${\cal A}$ and ${\cal B}$ as following

${\cal F}={\cal A}({\bf e}\cdot{\bf k})({{\mbox{\boldmath$\sigma$}}}\cdot{\bf
k})+{\cal B}{\bf e}\cdot{{\mbox{\boldmath$\sigma$}}},$ (4)

where ${\bf k}$ is unit vector directed along the beam, ${\bf e}$ is the
deuteron polarization vector and $\sigma$ is the Pauli matrix. The modules of
these amplitudes and cosine of the relative phase $\varphi_{AB}$ can be
determined by measurement of unpolarized cross section of the reaction
$d\sigma_{0}$ and tenzor analyzing powers $T_{20}=A_{zz}/\sqrt{2}$ and
$A_{yy}$. In order to measure the sine of the relative phase $\varphi_{AB}$
one has to measure only the sign of the spin-correlation coefficient
$C_{xz,y}$.

Within the approximation of the $pn$\- single scattering the theoretical
analysis of this reaction becomes more simple. In this case the ${\cal A}$ and
${\cal B}$ amplitudes of the reaction (3) are expressed via the spin
amplitudes of the charge exchange reaction

$p+n\to n+p.$ (5)

The transition matrix element of reaction (5) at zero scattering angle can be
written as

$f_{12}^{collin}=\alpha+\beta({{\mbox{\boldmath$\sigma$}}}_{1}\cdot{{\mbox{\boldmath$\sigma$}}}_{2})+(\varepsilon-\beta)({{\mbox{\boldmath$\sigma$}}}_{1}\cdot{\bf
k})({{\mbox{\boldmath$\sigma$}}}_{2}\cdot{\bf k}),$ (6)

where ${{\mbox{\boldmath$\sigma$}}}_{1}$ (${{\mbox{\boldmath$\sigma$}}}_{2}$)
the Pauli matrix acting on the spin state of the first (second) nucleon.

We can show that measurement of $d\sigma_{0}$ and $T_{20}$ provides the
modules of $|\varepsilon|$ and $|\beta|$ whereas the cosine of the relative
phase ( or $Re\varepsilon\beta^{*}$) is determined by the spin correlation
parameters $C_{x,x}=C_{y,y}$. In order to measure the sine of this phase
($Im\beta\varepsilon^{*}$) one has to measure the sign of
$C_{xz,y}(=-C_{yz,x})$. Therefore, measurement of $d\sigma_{0}$, $T_{20}$,
$C_{y,y}$ and the sign of $C_{xz,y}$ at zero scattering angle completely
determines the spin amplitudes $\varepsilon$ and $\beta$.

###  2.4  Elastic $dd$ scattering

Spin observables of the $dd$\- elastic scattering in forward hemisphere also
can be used to test spin-dependent amplitudes of $pN$ elastic scattering since
the Glauber model can be used for description of these observables.
Unpolarized differential cross section of the $dd$\- elastic scattering in
forward hemisphere measured at energies $\sqrt{s}=53-63$ GeV [33] was well
described by the modified Glauber theory including Gribov inelastic
corrections. At lower energies corresponding to the SPD NICA region, one may
expect that inelastic corrections are not important, that can be checked by
direct calculation of unpolarized cross section and subsequent comparison with
the data. In this calculations the above considered spin dependent amplitudes
of the $pd$ elastic scattering [23] can be used as input for the Glauber
calculations of the $dd$ scattering.

At large scattering angles $\theta_{cm}\sim 90^{\circ}$ the $pd\to pd$ and
$dd\to dd$ processes are sensitive to the short-range (six-quark) structure of
the deuteron. Therefore, measurement of any observables of these processes at
large $\theta_{cm}$ will be important to search for non-nucleonic degrees of
freedom of the deuteron.

###  2.5 Double polarized large angle $pN$ elastic scattering

The $pp$ and $pn$ elastic scattering at high energy $\sqrt{s}=5-7$ GeV and
large transferred momentum $-t=5-10$ GeV2 is powered by short-range properties
of NN-interaction corresponding to small separation between nucleons
$r_{NN}\sim\hbar/\sqrt{-t}\leq 0.1$ fm. There are three following aspects of
QCD dynamics in these processes. (i) First, the differential cross section
$d\sigma^{pp}/dt({s},\theta_{cm})$ at fixed angle $\theta_{cm}\sim 90^{\circ}$
on the whole follows to the pQCD constituent counting rules
$d\sigma^{pp}/dt({s},\theta_{cm})\sim s^{-10}$ [34, 35, 36, 37]. However, a
clear deviation from this prediction in form of oscillations with increasing
energy is observed in the region $s=10\div 40$ GeV2 [34, 35, 36, 37]. The
irregularity in the energy dependence is on the level of $\sim 50\%$ in the
region, where magnitude of the elastic pp- cross section falls down by 8
orders of magnitude. (ii) Second, anomalous polarization asymmetries were
observed in hard pN-scattering at $p_{lab}=11.75$ GeV/c [38, 39, 9]. Elastic
$pp$-cross section with spins of protons parallel and normal to the scattering
plane is almost four time larger than the cross section with antiparallel
spins. The challenge is that in order to generate such large polarization
effect, one needs to have large contribution from double spin-flip helicity
amplitude $\phi_{2}$ or negligible contribution from helicity conserving
$\phi_{1}$ amplitude. However, in pQCD, in contrast, $\phi_{2}$ is the most
suppressed and the $\phi_{1}$ is largest [40]. Predicted within the pQCD
(quark-interchange model) double spin asymmetry $A_{NN}$ does not depend on
energy [41], [42], whereas the measured asymmetry demonstrates ’’oscillating‘‘
energy dependence. (iii) The third QCD aspect of hard NN scattering is related
to the Color Transparency phenomenon (CT), that is a reduction of the
absorption in the nuclear medium of hard produced hadrons, both mesons and
baryons[5], [6]. Being in point like configurations, which are dictated by
mechanism of high momentum transfer, the initial and final hadrons
participating in hard process have small color dipole momenta and, therefore,
small interaction cross section with nuclear medium. These expectations
resulted in huge theoretical and experimental activities in 90’s. While the CT
effect is observed for the hard production of the $q\bar{q}$ systems, the
similar effect for $qqq$ is elusive. The data [43, 44] on the reaction $p+A\to
pp+X$ on the 12C and 27Al show again an ’’oscillatory‘‘ effect, i.e. the
transparency increases with increasing momentum up to $p_{lb}=9$ GeV/c, and
then decreases below the Glauber calculation predictions at 14 GeV/c. An
attempt to connect all three above aspects together into one approach was
undertaken in Ref. [40]. However, recent measurement of the cross section of
the reaction 12C(e,ep)X at $Q^{2}=8-14\,(GeV/c)^{2}$ [45] shows no CT effect
and this fact raises new questions to the analysis made in [40]. On the other
hand, according to [8], the observed large variations in spin correlations of
$pp$-elastic scattering are consistent with formation in the s-channel of
‘‘octoquark’’resonances $uuds\bar{s}uud$ and $uudc\bar{c}uud$ near the
strangeness and charm production thresholds, respectively. The variations with
increasing energy are explained as a result of interference of the pQCD
background amplitude with nonperturbative resonant amplitudes. Furthermore,
the model [8] provides a description of the oscillations in the unpolarized
differential $pp$\- elastic cross section.One should mentioned, however, that
another explanation of the oscillation effect in the
$d\sigma^{pp}/dt({s},\theta_{cm})$ was suggested in Ref. [46].

The considered questions about new types of charm-based resonances [47] became
especially interesting after observation enhancement effects in the decay
$\Lambda_{b}^{0}\to J/\Psi pK^{-}$ interpreted as pentaquark states
$uudc\bar{c}$ [10] (see also Ref. [47]). More insight into this issue can be
get from the data on large angle $pn$ elastic scattering. Different spin-
isospin structure of the transition matrix elements for the near threshold
$J/\Psi$ production in $pn$ and $pp$ collisions [48] means that spin
observables in $pn$\- elastic scattering can give a valuable independent
information on the considered dynamics. Data on these observables are almost
absent in the considered energy region. A task to get such data in the energy
interval $\sqrt{s_{NN}}\approxeq 3\div 5$ GeV from the ${\vec{p}}{\vec{d}}\to
pnp$ and ${\vec{d}}{\vec{d}}\to pnpn$ reactions is accessible for the SPD NCA.

###  2.6 Summary

To conclusion, nucleon-nucleon elastic scattering is a basic process in the
physics of atomic nuclei and the interaction of hadrons with nuclei. Existing
models and corresponding parametrizations of $pp$ amplitudes in the region of
small transferred momenta can be effectively tested by a measurement of spin
observables for $pd$ and $dd$ elastic scattering and a subsequent comparison
of the results with corresponding Glauber calculations. The spin observables
of $pd$ elastic scattering studied and evaluated in [23] are found to be not
too small and, thus, could be measured at the future SPD NICA facility. As
extension of this study, the quasi-elastic processes with formation of the
spin-singlet final $NN$-pair at small excitation energy $<3$ MeV in the
${}^{1}S_{0}$ state of internal motion, $pd\to n\\{pp\\}_{s}$ and $pd\to
p\\{pn\\}_{s}$ can be also investigated.

##  3 Studying periphery of the nucleon in diffractive $pp$ scattering 666This
section is written by V.A. Baskov, O.D. Dalkarov, A.I. L’vov (E-mail:
<EMAIL_ADDRESS>and V.V. Polyanskiy

1\. Scattering of high-energy hadrons at low $t$ is usually described by a
simple phenomenological dependence $d\sigma/dt=Ae^{Bt}$ (not applicable in the
Coulomb region, $|t|\lesssim 0.01$ GeV2, and at $|t|\gtrsim 0.4$ GeV2). In the
impact parameter representation, such a dependence corresponds to a gaussian
profile function $\Gamma(b)\sim\exp(-b^{2}/2B)$ with the average transverse
size $\langle b^{2}\rangle^{1/2}=B^{1/2}\sim 0.6~{}$fm when $B\sim 10$ GeV-2.
This size corresponds well to the quark core size of the nucleon, $r_{q}\sim
0.4-0.5$ fm, where the bulk of the nucleon mass (and the energy and momentum)
is concentrated.

On the other hand, part of the nucleon components is clearly located at larger
distances, pion cloud being the most evident example. The first evidence of
the pion cloud effect in the diffractive scattering, including rapid variation
of the effective slope $B$ at $|t|\sim 0.1$ GeV${}^{2}\approx 4m_{\pi}^{2}$,
have been found in ISR measurements (a comprehensive review of the ISR data
can be found in [49]).

First explanations of this, presumably pion cloud effect, were provided by
Anselm and Gribov [50] (see also [51, 52]). Soon then a dedicated experiment
was conducted in Protvino [53] in order to test the ISR results. However,
beyond confirming findings of ISR, one more oscillation in the differential
cross section at $|t|\sim 0.5$ GeV2 was found. Being located at higher $t$, it
might be related with somewhat heavier mesons around the proton (but not as
heavy as vector mesons that are too heavy).

Actually, S.P. Denisov et al. recently suggested to continue exploring $pp$
elastic scattering in that kinematical region at Protvino [54], and the
current proposal of doing similar experiment at SPD was directly motivated by
the Denisov’s ideas.

2\. It is essential that the Protvino experiment is not the only work
indicating an oscillation at $|t|\sim 0.5~{}\rm GeV^{2}$ in the fine structure
of the $pp$ diffraction cone. In Fig. 3 the most precise data of three
experiments — from Protvino [53] (at the proton beam momentum $p=60$ GeV$/c$),
ISR [49] (at the total energy $\sqrt{s}=52.8$ GeV), and Fermilab [55] (at
$p=200$ GeV$/c$), see also a comprehensive compilation and parameterization of
world data in [56] — are compared to exponential form $F(t)=Ae^{Bt+Ct^{2}}$
beyond the Coulomb region of tiny $|t|$ and the region of $|t|\lesssim 0.1$
GeV2 where effects of the pion cloud contribute. One has to notice that ISR
and FNAL data do not fully cover the region of $t$ with suspected oscillations
and do not have sufficient accuracy there. Therefore, further experimental
studies in that region are well justified.

Figure 3: Deviations of the $pp$ differential cross section from smooth
dependences $F(t)=Ae^{Bt+Ct^{2}}$. Data are from Protvino, ISR and FNAL, see
in the text. Solid line is a polynomial smoothing of shown ratios. Strips show
statistical errors in the polynomial.

In principle. information on the smoothed ratios $R(t)=(d\sigma/dt)/F(t)$
could be used in order to estimate the $pp$ scattering amplitude $f(s,t)$ and
to find then, through a Fourier-Bessel transformation, a profile function
$\Gamma(b)$ of the impact parameter $b$ [57]. A peak in $f(s,t)$ at $|t|\sim
0.5~{}\rm GeV^{2}$ corresponds to a peak in the profile function $\Gamma(b)$
at large distances $b\sim 7.0/\sqrt{|t|}\sim 2$ fm (here 7.0 is the second
maximum of the Bessel function $J_{0}(x)$), However, a straightforward
calculation of $\Gamma(b)$ in this way does not give reliable results in the
region where $\Gamma(b)$ becomes very small and sensitive to assumed phase of
the amplitude used, its spin structure, behavior at higher $|t|$, etc.
Actually, more sophisticated and indirect approaches are to be used in order
to analyze data on the described oscillation — see, for example, [58, 59, 60].

3\. In order to cover the region of interest, $|t|\sim 0.1-0.8$ GeV2, the
experimental setup must detect protons (in coincidence) scattered at angles
$\theta\sim 3-10^{\circ}$ what needs detectors placed at distances $R\sim
4-15$ cm from the beam. Accuracy of determination of the momentum transfer
squared $t$ in individual events of elastic $pp$-scattering must be better
than $\Delta t\sim 0.01{-}0.02$ GeV2, and this can be achieved with planned
tracker endcap detectors and with angular spread of colliding protons
determined by beam emittance and beta-function at IP.

Additional measurements of $d\sigma/dt$ and/or polarization observables at
higher $t$ are also desirable; they do not require high accuracy in
determination of $t$ [61].

Vertex detector, tracker system and software for track reconstruction in SPD
are sufficient for identification and recording $pp$ elastic events at
energies $\sqrt{s}\lesssim 15$ GeV. For higher energies and smaller angles,
when scattered protons fly very close to the beam pipe, installing fast
detectors close to the pipe, with the time resolution $\Delta T\lesssim 50$
ps, for determination of hitting times of forward-flying protons (perhaps,
using the so-called PID system) would make it possible to study the discussed
anomaly at the highest SPD energies.

##  4 Hadron structure and spin effects in elastic hadron scattering at NICA
energies 888This section is written by O.V. Selyugin; E-mail:
<EMAIL_ADDRESS>

PACS: 13.40.Gp, 14.20.Dh, 12.38.Lg

One of the most important tasks of modern physics is the research into the
basic properties of hadron interaction. The dynamics of strong interactions
finds its most complete representation in elastic scattering. It is just this
process that allows the verification of the results obtained from the main
principles of quantum field theory: the concept of the scattering amplitude as
a unified analytic function of its kinematic variables connecting different
reaction channels were introduced in the dispersion theory by N.N.
Bogoliubov[62]. Now many questions of hadron interactions are connected with
modern problems of astrophysics such as unitarity and the optical theorem
[63], and problems of baryon-antibaryon symmetry and CP-invariance violation
[30] The main domain of elastic scattering is small angles. Only in this
region of interactions can we measure the basic properties that define the
hadron structure. Their values are connected, on the one hand, with the large-
scale structure of hadrons and, on the other hand, with the first principles
that lead to the theorems on the behavior of scattering amplitudes at
asymptotic energies [64, 65].

Modern studies of elastic scattering of high energy protons lead to several
unexpected results reviewed, e.g., in [66, 67]. Spin amplitudes of the elastic
$NN$ scattering constitute a spin picture of the nucleon. Without knowledge of
the spin $NN$-amplitudes it is not possible to understand spin observable of
nucleon scattering off nuclei. In the modern picture, the structure of hadrons
is determined by Generalized Distribution functions (GPDs), which include the
corresponding parton distributions (PDFs). The sum rule [68] allow to obtain
the elastic form factor (electromagnetic and gravitomagnetic) through the
first and second integration moments of GPDs. It leads to remarkable
properties of GPDs - some corresponding to inelastic and elastic scattering of
hadrons. Now some different models examining the nonperturbative instanton
contribution lead to sufficiently large spin effects at superhigh energies
[69, 70] The research of such spin effects will be a crucial stone for
different models and will help us to understand the interaction and structure
of particles, especially at large distances. There are large programs of
researching spin effects at different accelerators. Especially, we should like
to note the programs at NICA, where the polarization of both the collider
beams will be constructed. So it is very important to obtain reliable
predictions for the spin asymmetries at these energies. In this paper, we
extend the model predictions to spin asymmetries in the NICA energy domain.

The NICA SPD detector bounded a very small momentum transfer. If in the first
steps the angles start from $16$ mrad, then the minimum momentum transfer,
which can be measured is more than $-0.01$ GeV2. Hence it is needed to exclude
the Coulomb-nuclear interference region, where the real part of the spin-non-
flip amplitude can be determined, We should move our researches on the region
of the diffraction minimum, where the imaginary part of the spin-non-flip
amplitude changes its sign. Note that in some models the absence of the second
diffraction minimum is explained by the contribution in the differential cross
section of the spin-flip amplitude [71] The interference of the hadronic and
electromagnetic amplitudes may give an important contribution not only at very
small transfer momenta but also in the range of the diffraction minimum [72].
However, for that one should know the phase of the interference of the
Columbic and hadronic amplitude at sufficiently large transfer momenta too.

Using the existing model of nucleon elastic scattering at high energies
$\sqrt{s}>9$ GeV - 14 TeV [73, 58], which involves minimum of free parameters,
we are going to develop its extended version aimed to describe all available
data on cross sections and spin-correlation parameters at lower energies down
to the SPD NICA region. The model will be based on the usage of known
information on generalized parton distributions in the nucleon, electro-
magnetic and gravitomagnetic form factors of the nucleon taking into account
analyticity and unitarity requirements and providing compatibility with the
high energy limit, where the pomeron exchange dominates.

###  4.1 HEGS model and spin effects in the dip region of momentum transfer

The differential cross sections of nucleon-nucleon elastic scattering can be
written as a sum of different helicity amplitudes:

$\displaystyle\frac{d\sigma}{dt}=\frac{2\pi}{s^{2}}(|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}+|\Phi_{4}|^{2}+4|\Phi_{5}|^{2}.$
(7)

$\displaystyle
A_{N}\frac{d\sigma}{dt}=-\frac{4\pi}{s^{2}}[Im(\Phi_{1}(s,t)+\Phi_{2}(s,t)+\Phi_{3}(s,t)-\Phi_{4})(s,t)\Phi^{*}_{5}(s,t)]$
(8)

and

$\displaystyle
A_{NN}\frac{d\sigma}{dt}=\frac{4\pi}{s^{2}}[Re(\Phi_{1}(s,t)\Phi^{*}_{2}(s,t)-\Phi_{3}(s,t)\Phi^{*}_{4})(s,t)+|\Phi_{5}(s,t)|^{2}]$
(9)

The HEGS model [73, 58] takes into account all five spiral electromagnetic
amplitudes. The electromagnetic amplitude can be calculated in the framework
of QED. In the high energy approximation, it can be obtained [74] for the
spin-non-flip amplitudes:

$\displaystyle F^{em}_{1}(t)=\alpha f_{1}^{2}(t)\frac{s-2m^{2}}{t};\ \ \
F^{em}_{3}(t)=F^{em}_{1};$ (10)

and for the spin-flip amplitudes: with the electromagnetic and hadronic
interactions included, every amplitude $\phi_{i}(s,t)$ can be described as

$\displaystyle\phi_{i}(s,t)=F^{em}_{i}\exp{(i\alpha\varphi(s,t))}+F^{h}_{i}(s,t),$
(11)

where $\varphi(s,t)=\varphi_{C}(t)-\varphi_{Ch}(s,t)$, and $\varphi_{C}(t)$
will be calculated in the second Born approximation in order to allow the
evaluation of the Coulomb-hadron interference term $\varphi_{Ch}(s,t)$. The
quantity $\varphi(s,t)$ has been calculated at large momentum transfer
including the region of the diffaction minimum [72, 75, 76] and references
therein.

Let us define the hadronic spin-non-flip amplitudes as

$\displaystyle F^{h}_{\rm nf}(s,t)$ $\displaystyle=$
$\displaystyle\left[\Phi_{1}(s,t)+\Phi_{3}(s,t)\right]/2;$ (12)

The model is based on the idea that at high energies a hadron interaction in
the non-perturbative regime is determined by the reggenized-gluon exchange.
The cross-even part of this amplitude can have two non-perturbative parts,
possible standard pomeron - $(P_{2np})$ and cross-even part of the 3-non-
perturbative gluons ($P_{3np}$). The interaction of these two objects is
proportional to two different form factors of the hadron. This is the main
assumption of the model. The second important assumption is that we choose the
slope of the second term four times smaller than the slope of the first term,
by analogy with the two pomeron cuts. Both terms have the same intercept.

The form factors are determined by the Generalized parton distributions of the
hadron (GPDs). The first form factor corresponding to the first momentum of
GPDs is the standard electromagnetic form factor - $G(t)$. The second form
factor is determined by the second momentum of GPDs -$A(t)$. The parameters
and $t$-dependence of GPDs are determined by the standard parton distribution
functions, so by experimental data on deep inelastic scattering and by
experimental data for the electromagnetic form factors (see [13]). The
calculations of the form factors were carried out in [77]. The final elastic
hadron scattering amplitude is obtained after unitarization of the Born term.
At large $t$ our model calculations are extended up to $-t=15$ GeV2. We added
a small contribution of the energy independent part of the spin flip amplitude
in the form similar to the proposed in [78] and analyzed in [14].

$\displaystyle F_{sf}(s,t)\ =h_{sf}q^{3}F_{1}^{2}(t)e^{-B_{sf}q^{2}}.$ (13)

The energy dependent part of the spin-flip amplitude is related to the main
amplitude but with an additional kinematic factor and the main slope taken
twice more, conformity with the paper [79, 80]. The form factors incoming in
the spin-flip amplitude are determined by the GPD functions $H(s,t,x)$ and
$E(s,t,x)$, which include the corresponding PDF distributions. The model is
very simple from the viewpoint of the number of fitting parameters and
functions. There are no any artificial functions or any cuts which bound the
separate parts of the amplitude by some region of momentum transfer.

Now we shell restrict our discussion to the analysis of $A_{N}$ as there are
some experimental data in the region of NICA energies. In the standard
pictures the spin-flip and double spin-flip amplitudes correspond to the spin-
orbit $(LS)$ and spin-spin $(SS)$ coupling terms. The contribution to $A_{N}$
from the hadron double spin-flip amplitudes already at $p_{L}=6\ $GeV/c is of
the second order compared to the contribution from the spin-flip amplitude. So
with the usual high energy approximation for the helicity amplitudes at small
transfer momenta we suppose that $\Phi_{1}=\Phi_{3}$ and we can neglect the
contributions of the hadron parts of $\Phi_{2}-\Phi_{4}$. Note that if
$\Phi_{1},\Phi_{3},\Phi_{5}$ have the same phases, their interference
contribution to $A_{N}$ will be zero, though the size of the hadron spin-flip
amplitude can be large. Hence, if this phase has different $s$ and $t$
dependencies, the contribution from the hadron spin-flip amplitude to $A_{N}$
can be zero at $s_{i},\ t_{i}$ and non-zero at other $s_{j},\ t_{j}$. e
experimental data ($\sum\chi^{2}/n_{dof}=1.24$).

Figure 4:  The model calculation of the diffraction minimum in $d\sigma/dt$ of
$pp$ scattering [left] at $\sqrt{s}=30.4$ GeV; [right] for $pp$ and $p\bar{p}$
at $\sqrt{s}=52.8$ GeV [81] scattering.  Figure 5:  The model calculation of
the diffraction minimum in $d\sigma/dt$ of $pp$ at
$\sqrt{s}=13.4;16.8;30.4;44.7;$ GeV; (lines, respectively, long dash; solid;
thin-solid, and short - dash ); and experimental data [81], respectively, -
the triangle down, the circles (solid), triangle up, and circles )

Now let us examine the form of the differential cross section in the region of
the momentum transfer where the diffractive properties of elastic scattering
appear most strongly - it is the region of the diffraction dip. The form and
the energy dependence of the diffraction minimum are very sensitive to
different parts of the scattering amplitude. The change of the sign of the
imaginary part of the scattering amplitude determines the position of the
minimum and its movement with changing energy. The contributions of the real
part of the spin-non-flip scattering amplitude and the square of the spin-flip
amplitude determine the size and the energy dependence of the dip. Hence, this
depends heavily on the odderon contribution. The spin-flip amplitude gives the
contribution to the differential cross sections additively. So the measurement
of the form and energy dependence of the diffraction minimum with high
precision is an important task for future experiments.

The HEGS model reproduces $d\sigma/dt$ at very small and large $t$ and
provides a qualitative description of the dip region at $-t\approx 1.6$ GeV2,
for $\sqrt{s}=10$ GeV and $-t\approx 0.45$ GeV2 for $\sqrt{s}=13$ TeV. Note
that it gives a good description for the proton-proton and proton-antiproton
elastic scattering or $\sqrt{s}=53$ GeV and for $\sqrt{s}=62.1$ GeV (Fig.2a).

The dependence of the position of the diffraction minimum on $t$ is determined
in most part by the growth of the total cross sections and the slope of the
imaginary part of the scattering amplitude. Figures 2b and 3 show this a
dependence obtained in the HEGS model at different energies.

Figure 6: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=4.9\ $GeV (the experimental data [82]), and b) at $\sqrt{s}=6.8\
$GeV (points - the existence experimental data [83] ).

Figure 7: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=9.2\ $GeV, and (the experimental data [27]), and b) at
$\sqrt{s}=13.7\ $GeV (points - the experimental data [84]).

In Fig.3, the description of the diffraction minimum in our model is shown for
NICA energies. The HEGS model reproduces sufficiently well the energy
dependence and the form of the diffraction dip. In this energy region the
diffraction minimum reaches the sharpest dip at $\sqrt{s}=30$ GeV near the
final NICA energy. Note that at this energy the value of $\rho(s,t=0)$ also
changes its sign in the proton-proton scattering.

The calculated analyzing power at $p_{L}=6\ GeV/c$ is shown in Fig.4a. One can
see that a good description of experimental data on the analyzing power can be
reached only with one hadron-spin flip amplitude.

The experimental data at $p_{L}=11.75\ $GeV/c seriously differ from those at
$p_{L}=6\ $GeV/c but our calculations reproduce $A_{N}$ sufficiently well
(Fig.4b ). It is shown that our energy dependence of the spin-flip amplitudes
was chosen correctly and we may hope that further we will obtain correct
values of the analyzing power and other spin correlation parameters.

From Fig.4 we can see that in the region $|t|\approx 0.2\div 1\ $ GeV2 the
contributions from the hadron spin-flip amplitudes are most important. At
last, Fig.5a shows our calculations at $p_{L}=200\ GeV/c$.

At this energy, the contributions of the phenomenological energy independent
part of the spin-flip amplitude is compared with the energy dependent part.
The spin effect is sufficiently large and has a specifical form, which is
determined by the form of the differential cross section in the diffraction
dip domain.

Figure 8: The analyzing power $A_{N}$ of pp - scattering calculated: a) at
$\sqrt{s}=19.4\ $GeV (the experimental data [85]), and b) at $\sqrt{s}=23.4\
$GeV (points - the experimental data [84])

###  4.2 Conclusions

The Generelized parton distributions (GPDs) make it possible to better
understand the thin hadron structure and to obtain the hadron structure in the
space frame (impact parameter representations). It is tightly connected with
the hadron elastic hadron form factors. The research into the form and energy
dependence of the diffraction minimum of the differential cross sections of
elastic hadron-hadron scattering at different energies will give valuable
information about the structure of the hadron scattering amplitude and hence
the hadron structure and the dynamics of strong interactions. The diffraction
minimum corresponds to the change of the sign of the imaginary part of the
spin-non-flip hadronic scattering amplitude and is created under a strong
impact of the unitarization procedure. Its dip depends on the contributions of
the real part of the spin-non-flip amplitude and the whole contribution of the
spin-flip scattering amplitude. In the framework of HEGS model, we show a deep
connection between elastic and inealastic cross sections, which are tightly
connected with the hadron structure at small and large distances.

The HEGS model reproduces well the form and the energy dependence of the
diffraction dip of the proton-proton and proton antiproton elastic scattering
[86]. The predictions of the model in most part reproduce the form of the
differential cross section at $\sqrt{s}=13$ TeV. It means that the energy
dependence of the scattering amplitude determined in the HEGS model and
unitarization procedure in the form of the standard eikonal representation
satisfies the experimental data in the huge energy region (from $\sqrt{s}=9$
GeV up to $\sqrt{s}=13$ TeV. It should be noted that the real part of the
scattering amplitude, on which the form and energy dependence of the
diffraction dip heavily depend, is determined in the framework of the HEGS
model only by the complex $\bar{s}$, and hence it is tightly connected with
the imaginary part of the scattering amplitude and satisfies the analyticity
and the dispersion relations. Quantitatively, for different thin structures of
the scattering amplitude, a wider analysis is needed. This concerns the fixed
intercept taken from the deep inelastic processes and the fixed Regge slope
$\alpha^{\prime}$, as well as the form of the spin-flip amplitude. Such an
analysis requires a wider range of experimental data, including the
polarization data of $A_{N}(s,t)$, $A_{NN}(s,t)$, $A_{LL}(s,t)$,
$A_{SL}(s,t)$. The obtained information about the sizes and energy dependence
of the spin-flip and double-flip amplitudes will make it possible to better
understand the results of famous experiments carried out by A. Krish at the
ZGS to obtain the spin-dependent differential cross sections [87, 88] and the
spin correlation parameter $A_{NN}$ and at the AGS [89] to obtain the spin
correlation parameter $A_{N}$ showing the significant spin effects at large
momentum transfer.

##  5 Single-spin physics 101010This section is written by V. Abramov; E-mail:
<EMAIL_ADDRESS>
PACS: 12.38.$-$t; 12.38.Qk; 13.60.$-$r; 13.75.$-$n; 13.85.$-$t; 13.85.Ni;
13.87.Fh;

### Introduction

All previous experience in the development of spin physics testifies to its
fundamental importance for understanding the laws of the micro-world,
including for the construction of the theory of strong interactions. It should
be noted that large values of transverse single-spin asymmetries ($A_{N}$) and
hyperon polarizations ($P_{N}$) in a wide energy range have not yet received
an unambiguous and convincing explanation within the framework of the theory
of strong interactions - quantum chromodynamics (QCD), which is one of the
components of the Standard Model. The experimental data accumulated to date
point to a very interesting phenomenology in the field of transverse single-
spin phenomena, including the nontrivial dependence of the spin observables
$A_{N}$ and $P_{N}$ on the collision energy ($\sqrt{s}$), the Feynman variable
($x_{\rm{F}}$), the transverse momentum ($p_{T}$), the atomic weights of the
colliding particles ($A_{1}$ and $A_{2}$), the multiplicity of charged
particles ($N_{ch}$) in the event and the centrality of collisions. It is
equally important to measure $A_{N}$ and $P_{N}$ for as many reactions as
possible in order to understand how spin effects depend on the quark
composition and other quantum characteristics of the particles involved in the
reaction. Data on dozens of reactions have been accumulated, but the accuracy
of measurements and the limited kinematic region in most experiments do not
yet allow unambiguous conclusions to be drawn about the origin of polarization
phenomena and even about their dependence on various variables. The purpose of
this proposal is to significantly expand the amount of polarization data
available for analysis and to improve their accuracy. This will help advance
the creation of adequate models of polarization phenomena and their
discrimination when compared with the entire dataset.

Planned measurements at the SPD facility in the energy range for a pair of
colliding nucleons from 3.4 to 27 GeV in the reaction c.m. frame are very
important for the systematic and detailed study of polarization phenomena and
the study of their dependence on various variables. Analysis of the available
data within the framework of the chromomagnetic polarization of quarks (CPQ)
model [90] shows that the unambiguous determination of the model parameters is
possible only if there are measurements with several (three or more) values
for each of the listed above variables. It should be noted that the maximum
energy of the accelerator in Dubna is high enough to register particles with
large transverse momenta in the range $p_{T}$ = 1 - 4 GeV/c, for which the
polarization effects are significant and the quark degrees of freedom are
already manifested. The identification of particles in this energy range is
much easier than at large accelerators, and this is an important condition for
the systematic study of polarization phenomena in a large number of reactions.

The conditions for making measurements at the SPD facility at the first stage
of the NICA collider can be found in [91]. Maximum energy in the c.m. of two
colliding nucleons will be 27 GeV for p+p collisions and 14 GeV for d+d, C+C
and Ca+Ca collisions. Vector polarization will be 50% for protons and 75% for
deuterons.

Table 1: Inclusive reactions for which the spin-spin asymmetry $A_{N}$ was
measured.
$\rm N^{\underline{o}}$ | Reaction | $\rm N^{\underline{o}}$ | Reaction | $\rm N^{\underline{o}}$ | Reaction
---|---|---|---|---|---
1 | $p^{\uparrow}p(A)\rightarrow\pi^{+}X$ | 10 | $p^{\uparrow}p(A)\rightarrow J/\psi X$ | 19 | $\bar{p}d^{\uparrow}\rightarrow\pi^{0}X$
2 | $p^{\uparrow}p(A)\rightarrow\pi^{-}X$ | 11 | $p^{\uparrow}p(A)\rightarrow\eta X$ | 20 | $\pi^{+}p^{\uparrow}\rightarrow\pi^{+}X$
3 | $p^{\uparrow}p\rightarrow\pi^{0}X$ | 12 | $d^{\uparrow}p(A)\rightarrow\pi^{+}X$ | 21 | $\pi^{-}p^{\uparrow}\rightarrow\pi^{-}X$
4 | $p^{\uparrow}p(A)\rightarrow K^{+}X$ | 13 | $d^{\uparrow}p(A)\rightarrow\pi^{-}X$ | 22 | $\pi^{-}p^{\uparrow}\rightarrow\pi^{0}X$
5 | $p^{\uparrow}p(A)\rightarrow K^{-}X$ | 14 | $p^{\uparrow}p\rightarrow\Lambda X$ | 23 | $\pi^{-}d^{\uparrow}\rightarrow\pi^{0}X$
6 | $p^{\uparrow}p\rightarrow K^{0}_{S}X$ | 15 | $\bar{p}^{\uparrow}p\rightarrow\pi^{+}X$ | 24 | $K^{-}d^{\uparrow}\rightarrow\pi^{0}X$
7 | $p^{\uparrow}p(A)\rightarrow nX$ | 16 | $\bar{p}^{\uparrow}p\rightarrow\pi^{-}X$ | 25 | $K^{-}p^{\uparrow}\rightarrow\pi^{0}X$
8 | $p^{\uparrow}p(A)\rightarrow pX$ | 17 | $\bar{p}^{\uparrow}p\rightarrow\pi^{0}X$ | 26 | $\pi^{-}p^{\uparrow}\rightarrow\eta X$
9 | $p^{\uparrow}p(A)\rightarrow\bar{p}X$ | 18 | $\bar{p}^{\uparrow}p\rightarrow\eta X$ | 27 | $\bar{p}p^{\uparrow}\rightarrow\pi^{0}X$

Table 1 presents 27 inclusive reactions for which there are already data on
the single-spin asymmetry of hadrons [90, 92]. The first 14 reactions from the
Table 1 can potentially be studied at the NICA collider using the SPD
facility. A list of other possible 28 reactions is shown in Table 2 and
includes various particles and resonances. The initial state can be any with a
polarized beam: $p^{\uparrow}p$, $p^{\uparrow}d$, $d^{\uparrow}p$,
$d^{\uparrow}d$. Their detailed study will reveal the dependence of $A_{N}$ on
kinematic and other variables, including the quark composition of the
particles involved, their spin, isospin, and atomic weight.

Table 2: Inclusive reactions to be studied at the SPD for which $A_{N}$ has
not yet been measured. The reaction is $p^{\uparrow}p\rightarrow h+X$. Final
decay mode of the detected particle $h$ is indicated only.
$\rm N^{\underline{o}}$ | Decay mode | $\rm N^{\underline{o}}$ | Decay mode | $\rm N^{\underline{o}}$ | Decay mode
---|---|---|---|---|---
1 | $K^{0}_{L}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ | 10 | $\phi\rightarrow K^{+}K^{-}$ | 19 | $\bar{\Xi}^{0}\rightarrow\bar{\Lambda}\pi^{0}$
2 | $\eta^{\prime}\rightarrow\pi+\pi-\eta$ | 11 | $\rho^{0}(770)\rightarrow\pi^{+}\pi^{-}$ | 20 | $\Sigma^{0}\rightarrow\Lambda\gamma$
3 | $a_{0}(980)\rightarrow\eta\pi^{0}$ | 12 | $\rho^{+}(770)\rightarrow\pi^{+}\pi^{0}$ | 21 | $\bar{\Sigma}^{0}\rightarrow\bar{\Lambda}\gamma$
4 | $K^{0*}(892)\rightarrow K^{+}\pi^{-}$ | 13 | $\rho^{-}(770)\rightarrow\pi^{-}\pi^{0}$ | 22 | $\Delta^{++}\rightarrow p\pi^{+}$
5 | $K^{0*}(892)\rightarrow K^{-}\pi^{+}$ | 14 | $\rho^{0}(770)\rightarrow\mu^{+}\mu^{-}$ | 23 | $\Delta^{+}\rightarrow p\pi^{0}$
6 | $K^{+*}(892)\rightarrow K^{+}\pi^{0}$ | 15 | $\bar{\Lambda}\rightarrow\bar{p}\pi^{+}$ | 24 | $\Delta^{0}\rightarrow p\pi^{-}$
7 | $K^{-*}(892)\rightarrow K^{-}\pi^{0}$ | 16 | $\Xi^{-}\rightarrow\Lambda\pi^{-}$ | 25 | $\Delta^{-}\rightarrow n\pi^{-}$
8 | $\omega(782)\rightarrow\pi^{+}\pi^{-}\pi^{0}$ | 17 | $\Xi^{0}\rightarrow\Lambda\pi^{0}$ | 26 | $\bar{\Delta}^{--}\rightarrow\bar{p}\pi^{-}$
9 | $\omega(782)\rightarrow\gamma\pi^{0}$ | 18 | $\bar{\Xi}^{+}\rightarrow\bar{\Lambda}\pi^{+}$ | 27 | $\bar{\Delta}^{0}\rightarrow\bar{p}\pi^{+}$

Data on the transverse polarization of hyperons and antihyperons are no less
interesting. The list of reactions available to date, for which their
polarization $P_{N}$ was measured, is presented in Table 3 and includes 32
reactions [90, 93]. The first 14 reactions can potentially be studied at the
SPD setup. This list can be supplemented with reactions such as
$pp\rightarrow\Sigma^{0\uparrow}(1385)X$,
$pp\rightarrow\bar{\Sigma}^{0\uparrow}X$,
$pp\rightarrow\Lambda^{\uparrow}(1405)X$,
$pp\rightarrow\Lambda^{\uparrow}(1520)X$. The initial state can be any with a
polarized or unpolarized beam: $p^{\uparrow}p$, $p^{\uparrow}d$,
$d^{\uparrow}p$, $d^{\uparrow}d$ and $AA$.

Table 3: Inclusive reactions for which the polarization ($P_{N}$) of hyperons
was measured.
$\rm N^{\underline{o}}$ | Reaction | $\rm N^{\underline{o}}$ | Reaction | $\rm N^{\underline{o}}$ | Reaction
---|---|---|---|---|---
1 | $pp(A)\rightarrow\Lambda^{\uparrow}X$ | 12 | $A_{1}A_{2}\rightarrow\Lambda^{\uparrow}X$ | 23 | $\pi^{-}A\rightarrow\Xi^{-\uparrow}X$
2 | $pp(A)\rightarrow\Xi^{-\uparrow}X$ | 13 | $A_{1}A_{2}\rightarrow\Lambda^{\uparrow(G)}X$ | 24 | $\pi^{-}A\rightarrow\bar{\Xi}^{+\uparrow}X$
3 | $pp(A)\rightarrow\Xi^{0\uparrow}X$ | 14 | $A_{1}A_{2}\rightarrow\bar{\Lambda}^{\uparrow(G)}X$ | 25 | $\pi^{-}p\rightarrow\Lambda^{\uparrow}X$
4 | $pp(A)\rightarrow\Sigma^{+\uparrow}X$ | 15 | $\Sigma^{-}A\rightarrow\Sigma^{+\uparrow}X$ | 26 | $\pi^{-}p\rightarrow\bar{\Lambda}^{\uparrow}X$
5 | $pp(A)\rightarrow\Sigma^{0\uparrow}X$ | 16 | $\Sigma^{-}A\rightarrow\Xi^{-\uparrow}X$ | 27 | $\pi^{+}p\rightarrow\Lambda^{\uparrow}X$
6 | $pp(A)\rightarrow\Sigma^{-\uparrow}X$ | 17 | $\Sigma^{-}A\rightarrow\Lambda^{\uparrow}X$ | 28 | $K^{-}A\rightarrow\Xi^{-\uparrow}X$
7 | $pp(A)\rightarrow\Omega^{-\uparrow}X$ | 18 | $\Sigma^{-}A\rightarrow\bar{\Lambda}^{\uparrow}X$ | 29 | $\bar{p}A\rightarrow\bar{\Lambda}^{\uparrow}X$
8 | $pp(A)\rightarrow\bar{\Lambda}^{\uparrow}X$ | 19 | $K^{-}p\rightarrow\Lambda^{\uparrow}X$ | 30 | $e^{+}e^{-}\rightarrow\Lambda^{\uparrow}X$
9 | $pp(A)\rightarrow\bar{\Xi}^{+\uparrow}X$ | 20 | $K^{-}p\rightarrow\bar{\Lambda}^{\uparrow}X$ | 31 | $\nu_{\mu}A\rightarrow\Lambda^{\uparrow}X$
10 | $pp(A)\rightarrow\bar{\Xi}^{0\uparrow}X$ | 21 | $K^{+}p\rightarrow\Lambda^{\uparrow}X$ | 32 | $e^{+}A\rightarrow\bar{\Lambda}^{\uparrow}X$
11 | $pp(A)\rightarrow\bar{\Sigma}^{-\uparrow}X$ | 22 | $K^{+}p\rightarrow\bar{\Lambda}^{\uparrow}X$ | 33 | -

It is important to note that for hyperons it is possible to simultaneously
measure both the transverse polarization $P_{N}$ and the single-spin asymmetry
$A_{N}$. Comparing $A_{N}$ and $P_{N}$ for a specific reaction with the
predictions of various models will bring us closer to revealing the mechanism
of the origin of polarization phenomena at high energies and will shed light
on the physics of strong interactions in the confinement region.

A systematic study of polarization data assumes the presence of a model that
describes, within a single mechanism, a large number of reactions depending on
the variables listed above. An example of such a model is the model of
chromomagnetic polarization of quarks (CPQ) [90].

References to most of publications devoted to polarization experiment data can
be found in [90, 92, 93] and other in papers listed in the cited literature.
The following sections describe in more detail the model of chromomagnetic
polarization of quarks and consider examples of existing data and calculations
of $A_{N}$ and $P_{N}$ for various reactions that can potentially be studied
using the SPD setup at the NICA collider in Dubna.

###  5.1 Model of chromomagnetic polarization of quarks

The phenomenological model of chromomagnetic polarization of quark is based on
the following basic assumptions [90]:

1) As a result of collisions of hadrons, a pair of quarks with a large
transferred transverse momentum $p_{T}$ is scattered. Further, the scattered
(test) quark with large $p_{T}$ moves in the effective chromomagnetic field
$\rm\bf{B}^{\it{a}}$ and experiences the action of the Stern-Gerlach force
proportional to the product of the field gradient components and the
corresponding components of the quark chromomagnetic moment. The direction of
the Stern-Gerlach force and the additional transverse momentum received by the
test quark in the effective chromomagnetic field depend on the projections of
the quark spin onto the quantization axis. Subsequently, the polarized quark
from the incident polarized proton recombines with other quarks to form the
observed hadron. The angular distribution of such hadrons has an azimuthal
dependence, i.e., a single-spin asymmetry arises. If unpolarized hadrons
collide, then the action of the Stern-Gerlach force imparts an additional
transverse momentum directed to the left or to the right, depending on the
direction of the projection of the quark spin up or down, when the quark
moves, for example, to the left. Thus, when scattering to the left, a quark
has predominantly one polarization sign, and when scattering to the right, the
opposite. The hyperons formed from these quarks acquire transverse
polarization relative to the scattering plane.

2) The effective chromomagnetic field $\rm\bf{B}^{\it{a}}$ is created by
spectator quarks, that is, all quarks that will not be included in the
recorded hadron. Spectator quarks are moving in the c.m. in the direction of
the colliding hadrons and create for a short time a circular transverse
chromomagnetic field. The sign of the circular chromomagnetic field to the
left and to the right of the collision axis is opposite, but the field
gradient does not change its direction, which ensures a nonzero polarization
effect due to the action of the Stern-Gerlach force. The predominant direction
of polarization of quarks in a chromomagnetic field arises, hence the name of
the model.

3) When taking into account the interaction of a test quark with the field
created by a moving spectator quark, it is necessary to take into account the
color factor for the corresponding pair of quarks (spectator and test quarks).
An analysis of the data showed that the quark-antiquark pair interacts
predominantly in the color-singlet state with the color factor $C_{F}$ = 4/3,
and the quark-quark or antiquark-antiquark pair interacts in the color-triplet
state with $C_{F}$ = 2/3. For a hydrogen-like potential, the wave function of
two quarks or a quark and an antiquark at zero coordinate is proportional to
$|\psi(0)|\propto(C_{F}\alpha_{S})^{3/2}$ [94], which leads to the ratio of
contributions from qq and $q$$\bar{q}$ interactions to an effective field of
the order

$\lambda\approx-|\psi_{qq}(0)|^{2}/|\psi_{q\bar{q}}(0)|^{2}=-1/8=-0.125.$ (14)

The minus sign in (14) takes into account the opposite sign of the field
created by a moving spectator quark and a moving spectator antiquark.
Experimentally, the value of the global parameter, obtained as a result of the
global fit of the polarization data, turned out to be $\lambda=-0.1363\pm
0.0003$.

If the spectator quark is a product of target fragmentation and moves in the
c.m. in the opposite direction, then its contribution to the effective field
will be additionally suppressed by the factor $-\tau$, where $\tau=0.0267\pm
0.0012$ is another important global parameter of the CPQ model. This
suppression of the contribution of quarks from the target is due to the fact
that the chromomagnetic field they create is in a different region of space-
time and, therefore, has almost no effect on the test quarks moving forward.

4) The presence of an effective chromomagnetic field should lead to the
precession of the test quark spin when it moves in the field. Analysis of the
data showed that the effective field length and the corresponding precession
angle are proportional to $x_{A}=(x_{R}+x_{\rm{F}})/2$ and
$x_{B}=(x_{R}-x_{\rm{F}})/2$ in the fragmentation region of the incident
particle A and target B, respectively. As a result, this leads to oscillations
of the dependence of $A_{N}$ and $P_{N}$ on the kinematic variables $x_{A}$
and $x_{B}$, and hence on $x_{\rm{F}}$ and $p_{T}$. These oscillations are the
main feature of the CPQ model and should manifest themselves in the case of
strong fields, when the precession angles reach values of the order of $\pi$
or more.

Figure 9:  The mechanism of origin of single-spin polarization phenomena.

The mechanism of origin of single-spin polarization phenomena is shown
schematically in fig. 9. The interaction of colliding particles $A$ and $B$ is
considered in the c.m. of pair of colliding nucleons.

Observables $A_{N}$ and $P_{N}$ are described by equations (15) and (16):

$P_{N}=C(\sqrt{s})F(p_{T},A)[G(\phi_{A})-\sigma G(\phi_{B})],$ (15)
$G(\phi)=(1-\cos\phi)/\phi+\epsilon\cdot\phi,$ (16)

where function (16) takes into account the action of the Stern-Gerlach forces
and the precession of the quark spin, and where $\epsilon=-0.00497\pm 0.00009$
is the global parameter of the CPQ model, $\sigma$ is the local parameter. The
integral angles of precession of the quark spin are

$\phi_{A}=\omega^{0}_{A}y_{A},\hskip 42.67912pt\phi_{B}=\omega^{0}_{B}y_{B},$
(17)

in the fragmentation region of colliding particles $A$ and $B$, respectively.
The oscillation frequency $\omega^{0}_{A(B)}$ is described by the equation

$\omega^{0}_{A(B)}=g_{s}\alpha_{s}\nu_{A(B)}m_{r}(g^{a}_{Q}-2)/M_{Q},$ (18)

where $\alpha_{s}=g_{s}^{2}/(4\pi)$ is the current strong interaction
constant, $g_{s}$ is the color charge, $M_{Q}$ is the mass of the constituent
quark $Q$, $g^{a}_{Q}$ is the the Lande gyromagnetic color factor of a quark,
$m_{r}=0.2942\pm 0.0072$ GeV is the global parameter that can be considered as
the ratio of the maximum longitudinal extension of the chromomagnetic field to
the square of its radius.

The total contribution of spectator quarks (with weights $\lambda$ and
$-\tau$) to $\nu_{A(B)}$ in the fragmentation region of colliding particles
$A$ and $B$, respectively, is calculated using quark diagrams and the quark
counting rules [90].

Quark diagrams for the reactions $p^{\uparrow}$+$p$$\rightarrow$$\pi^{+}$+$X$
and $p^{\uparrow}$+$p(A)$$\rightarrow$$p$+$X$ are shown in fig. 10a and 10b,
respectively. When nucleus is the target, as in the case of fig. 10b, the
number of target spectator quarks is equal to $3A_{\rm{eff}}\propto A^{1/3}$,
where $A$ is an atomic weight, since all target quarks hit by the incident
proton contribute to the spectator quarks [90]. Below we assume
$A_{\rm{eff}}=A=1$.

|
---|---
(a) | (b)
Figure 10: Quark flux diagrams for the reaction
$p^{\uparrow}$+$p$$\rightarrow$$\pi^{+}$+$X$ (a) and
$p^{\uparrow}$+$p(A)$$\rightarrow$$p$+$X$ (b).

In the approximation of moderate energies ($\sqrt{s}<70$ GeV), we obtain
$\nu_{A}$ for the reaction $p^{\uparrow}$+$p$$\rightarrow$$\pi^{+}$+$X$:

$\nu_{A}=\nu_{B}=3\lambda-3\tau\lambda A_{\rm eff}=-0.398,$ (19)

and for the reaction $p^{\uparrow}$+$p(A)$$\rightarrow$$p$+$X$:

$\nu_{A}=\nu_{B}=2+2\lambda-3\tau\lambda A_{\rm eff}=1.738.$ (20)

To calculate $\nu_{A}$ we have to add up all the contributions ($\nu$) of the
spectator quarks shown to the right of the quark diagram. The $\nu_{A}$ value
for the reaction $p^{\uparrow}$+$p$$\rightarrow$$\pi^{+}$+$X$ is much less
than 1 in absolute value. Consequently, the oscillation frequency
$\omega^{0}_{A(B)}$ is also low, and the $A_{N}(x_{\rm{F}})$ dependence is
close to linear. For the reaction $p^{\uparrow}$+$p(A)$$\rightarrow$$p$+$X$,
the value of $\nu_{A}$ is significantly greater than unity in absolute value,
and for it, as we will see below, a nonmonotonic oscillating dependence
$A_{N}(x_{\rm{F}})$ is indeed observed.

Kinematic variables

$y_{A}=x_{A}-(E_{0}/\sqrt{s}+f_{0})[1+\cos\theta_{cm}]+a_{0}[1-\cos\theta_{cm}],$
(21)
$y_{B}=x_{B}-(E_{0}/\sqrt{s}+f_{0})[1-\cos\theta_{cm}]+a_{0}[1+\cos\theta_{cm}],$
(22)

are expressed in terms of the scaling variables $x_{A}$ and $x_{B}$, the
reaction energy $\sqrt{s}$, the emission angle $\theta_{cm}$ in c.m. and three
local parameters $E_{0}$, $a_{0}$ and $f_{0}$. Function

$C(\sqrt{s})=v_{0}/[(1-E_{R}/\sqrt{s})^{2}+\delta_{R}^{2}]^{1/2},$ (23)

takes into account the dependence of the rate of precession of the spin of a
quark on its energy $E_{Q}$ in c.m. and the effect of attraction ($E_{R}>0$)
or repulsion ($E_{R}<0$) between the test quark and spectator quarks. The
$E_{R}$ sign is determined by the factor $-g_{S}\nu_{A}$, where $g_{S}$ is the
color charge of the test quark (positive for a quark and negative for an
antiquark). An example of a reaction with $E_{R}>0$ is
$p+p\rightarrow\bar{\Lambda}+X$, and reaction with $E_{R}<0$ is
$p+p\rightarrow\Lambda+X$. The global data fit confirms the $E_{R}$ sign rule
for most of 85 investigated reactions (96,5%).

The coefficient $v_{0}$ determines the value of $A_{N}$ and $P_{N}$ and is
calculated as follows:

$v_{0}=-D_{r}g^{a}_{Q}P_{Q}/2(g^{a}_{Q}-2),$ (24)

where $D_{r}$ is a local dimensionless parameter of order 0.8, which is the
ratio of the spectrum slope in $p_{T}$ to the transverse radius of the
effective field, $P_{Q}$ is the polarization of the $Q$ quark in a polarized
proton (+1 for $u$-quark and -1 for $d$-quark), $g^{a}_{Q}$ is the Lande
gyromagnetic factor for the $Q$-type quark, which is a global parameter. The
$A_{N}$ or $P_{N}$ sign for most reactions at small $\phi_{A}$ is the product
of three factors: $-g_{S}\nu_{A}P_{Q}$. When calculating the polarization of
hyperons, we set $P_{Q}=1$.

The color form factor $F(p_{T},A)$ suppresses $A_{N}$ and $P_{N}$ at low
$p_{T}$, when the colored quarks inside the hadron are not visible due to the
uncertainty relation:

$F(p_{T},A)={1-\exp[-(p_{T}/p^{0}_{T})^{2.5}]}(1-\alpha_{A}\ln{A}),$ (25)

where $p^{0}_{T}$ is a local parameter, and the other parameter $\alpha_{A}$
is zero for most of reactions.

The dependence of a number of parameters on the atomic masses $A_{1}$ and
$A_{2}$ turned out to be universal for most of the reactions in Tables 1 and 3
[90, 95]. Further development of the CPQ model is reflected in papers [95, 96,
97, 98, 99, 100, 101, 102, 103, 104].

###  5.2 Single-spin hadron asymmetry

|
---|---
(a) | (b)
Figure 11: $A_{N}(x_{\rm{F}})$ for the reaction
$p^{\uparrow}+p(A)\rightarrow\pi^{+}+X$ (a) and $p^{\uparrow}+p(A)\rightarrow
p+X$ [102] (b) [102].

The most numerous experiments to measure the single-spin asymmetry were
carried out for the reactions of the production of charged and neutral
$\pi$-mesons in $p^{\uparrow}p$ and $p^{\uparrow}A$ collisions. These data are
included in the general database of polarization phenomena, which contains
3608 experimental points for 85 different inclusive reactions, in which the
polarization of one of the particles is known or measured in the initial or
final state [90, 102]. A global fit was performed for the entire dataset using
the CPQ model.

Data on $A_{N}$ for the reaction $p^{\uparrow}+p(A)\rightarrow\pi^{+}+X$ at
different energies are shown in fig. 11a from [102], where they are compared
with the results of calculations using the CPQ model. As seen from fig. 11a,
$A_{N}(x_{\rm{F}})$ dependence for the reactions
$p^{\uparrow}+p(A)\rightarrow\pi^{+}+X$ at moderately high energies
$\sqrt{s}<70$ GeV, is almost linear, which agrees with the predictions of the
CPQ model. This is due to the insignificant value of the parameter
$\nu_{A}=\nu_{B}=3\lambda-3\tau\lambda=-0.398$, which follows from the quark
diagram shown in fig. 10a. The positive sign of $A_{N}(x_{\rm{F}})$ for the
reaction $p^{\uparrow}+p(A)\rightarrow\pi^{+}+X$ is explained by the dominant
contribution of the positively polarized test $u$-quark from a polarized
proton.

A very unexpected and interesting feature of the reaction
$p^{\uparrow}+p(A)\rightarrow\pi^{-}+X$ turned out to be the threshold
dependence of $A_{N}(y_{A})$ on the angle of production $\theta_{cm}$ in c.m.
In fig. 12a from [96] is shown the dependence of the quantity
$(1-E_{R}/\sqrt{s})A_{N}$ on $y_{A}$, where $E_{R}=4.98\pm 0.29$ GeV. It
turned out that this quantity is described by the universal function of
$y_{A}$ if $\theta_{cm}<74^{o}$, and is equal to zero if $\theta_{cm}>74^{o}$.
In fig. 12a, two clearly distinct branches are visible, into which the
experimental points are grouped.

Within the framework of the CPQ model, the threshold effect for $A_{N}(y_{A})$
can be qualitatively explained by the greater mass of the constituent
$d$-quark as compared to the mass of the $u$-quark.

|
---|---
(a) | (b)
Figure 12: Dependence of the value $(1-E_{R}/\sqrt{s})A_{N}$ on $y_{A}$, where
$E_{R}=4.98\pm 0.29$ GeV for the reaction
$p^{\uparrow}p(A)\rightarrow\pi^{-}X$ (a) and $E_{R}=1.92\pm 0.30$ GeV, for
the reaction $p^{\uparrow}p(A)\rightarrow\pi^{+}X$ (b) [96].

In fig. 12b from [96] is shown the dependence of the quantity
$(1-E_{R}/\sqrt{s})A_{N}$ on $y_{A}$ for the reaction
$p^{\uparrow}+p(A)\rightarrow\pi^{+}+X$, where $E_{R}=1.92\pm 0.30$ GeV. Most
of the light test $u$-quarks flying into the front hemisphere will be from a
polarized proton, which means that the asymmetry $A_{N}>0$ for $\pi^{+}$
mesons [96]. All data in fig. 12b are located on the same branch, for a wide
range of energies $\sqrt{s}$ and production angles in c.m.

The positive value $E_{R}=4.98\pm 0.29$ GeV for the reaction
$p^{\uparrow}+p(A)\rightarrow\pi^{-}+X$, found in the framework of the CPQ
model, is a manifestation of the effect of "attraction" of test quarks and
spectator quarks. According to formula (23), $A_{N}$ reaches its maximum value
at energy $\sqrt{s}\approx E_{R}$ [90, 96]. Investigation of the effect of
"attraction" of test quarks for various reactions is one of the objectives of
this proposal and involves scanning in energy $\sqrt{s}$ near $E_{R}$. This
phenomenon is observed not only for single-spin asymmetry, but also for the
polarization of hyperons, in those reactions for which $E_{R}$ is positive and
amounts to several GeV [90].

Finding of scaling (independence of $(1-E_{R}/\sqrt{s})A_{N}$ from energy
$\sqrt{s}$) in the variable $y_{A}$ was one of the stages in the process of
creating the CPQ model [90, 92, 96]. Investigation of the scaling for
polarization observables $A_{N}$ and $P_{N}$ is of independent interest and
can be one of the tasks for the SPD setup. In the framework of the CPQ model,
scaling in polarization phenomena is due to the occurrence of processes at the
quark level, in the limit of high energies and large transverse momentum [90,
92, 93, 96].

Data and calculations of $A_{N}(x_{\rm{F}})$ for the reaction
$p^{\uparrow}+p(A)\rightarrow p+X$, taken from [102], are shown in fig. 11b.
The data of the FODS-2 experiment [105], measured in a wide range in the
variable $x_{\rm{F}}$, at an energy of $\sqrt{s}=8.77$ GeV (solid squares in
fig. 11b and curve (3)), demonstrate a nonmonotonic oscillatory dependence of
$A_{N}(x_{\rm{F}})$. This is a consequence of the large value of the parameter
$\nu_{A}$ and the significant precession angle of the quark spin in the
chromomagnetic field. The quantity
$\nu_{A}=\nu_{B}=[2+2\lambda-3\tau\lambda]=1.738$ is large enough, which
follows from the quark diagram shown in fig. 10b. In the energy region of the
NICA collider, a negative asymmetry $A_{N}(x_{\rm{F}})$ of about 10% is
expected near $x_{\rm{F}}=0.2$ (fig. 11b, curves (3) for $\sqrt{s}=8.77$ GeV).

Another new and interesting direction in the study of polarization phenomena
is associated with the dependence of $A_{N}$ and $P_{N}$ on the multiplicity
of charged particles ($N_{ch}$) in an event. The first results in this region
were obtained for reactions $p^{\uparrow}+p\rightarrow\pi^{\pm}+X$ in the
BRAHMS experiment at an energy $\sqrt{s}=200$ GeV [106]. The single-spin
asymmetry $A_{N}$ increases in absolute value, if we select events with
$N_{ch}$ above the average, and decreases, if we select events with $N_{ch}$
below the average. These data, together with the calculations, are disscussed
in [100]. In the CPQ model, events with a multiplicity above the mean
correspond to quark diagrams with additional quark-antiquark pairs compared to
the minimum required number. This effect, which can manifest itself for both
$A_{N}$ and $P_{N}$, can be studied at the SPD facility.

###  5.3 Transverse polarization of hyperons

Hyperons have the remarkable property that their decay in the weak interaction
makes it possible to determine the transverse polarization to the scattering
plane ($P_{N}$) - the only one possible in strong interactions, due to the
conservation of parity in them. Therefore, the polarization of hyperons can be
studied in collisions of practically any particles. In the case of the first
phase of the SPD NICA project, we are interested in collisions $pp$, $pd$,
$dd$, C+C and Ca+Ca. The available data are discussed in detail in [93].

Quark diagrams for the production of $\Xi^{-}$ hyperons in pp-collisions can
be found in [104]. The effective number of spectator quarks for the reaction
$p+p\rightarrow\Xi^{-\uparrow}+X$ is
$\nu_{A}=\nu_{B}=2+2\lambda-3\tau\lambda\approx 1.7383$. Similar calculations
for the reaction $p+p\rightarrow\Lambda^{\uparrow}+X$ give
$\nu_{A}=\nu_{B}=1+\lambda-3\tau\lambda\approx 0.8746$. Therefore, a
nonmonotonic dependence $P_{N}(x_{\rm{F}})$ can be expected in the case of the
reaction $p+p\rightarrow\Xi^{-\uparrow}+X$.

|
---|---
(a) | (b)
Figure 13: $P_{N}(x_{\rm{F}})$ data and the CPQ model calculations for the
reaction $p+p(A)\rightarrow\Lambda^{\uparrow}+X$ (a) and
$p+p(A)\rightarrow\Xi^{-\uparrow}+X$ (b), taken from [104].

The $P_{N}(x_{\rm{F}})$ data for the reaction
$p+p(A)\rightarrow\Lambda^{\uparrow}+X$ are shown in fig. 13a, and the data
for the reaction $p+p(A)\rightarrow\Xi^{-\uparrow}+X$ are shown in fig. 13b,
together with the CPQ model predictions [104].

As seen from fig. 13b, the $P_{N}(x_{\rm{F}})$ dependence for cascade hyperons
is nonlinear function and $P_{N}(x_{\rm{F}})$ reaches its maximum absolute
value at $x_{\rm{F}}$ in the range 0.5 - 0.6, in agreement with the
calculations by the CPQ model. For the reaction
$p+p(A)\rightarrow\Lambda^{\uparrow}+X$, a close to linear dependence is
observed, since the parameter $\nu_{A}=\nu_{B}\approx 0.8746$ in this case is
approximately two times smaller. The maximum of the absolute value of
polarization for the reaction $p+p(A)\rightarrow\Lambda^{\uparrow}+X$ is
approximately twice that for $p+p(A)\rightarrow\Xi^{-\uparrow}+X$, and
continues to increase with increasing $x_{\rm{F}}$ up to 0.75, in agreement
with the calculations in the framework of the CPQ model.

Detailed calculations of $P_{N}(x_{\rm{F}})$ for reactions
$p+A\rightarrow\Xi^{-}+X$ and $p+A\rightarrow\Xi^{0}+X$ can be found in [104],
which also covers the energy range, available at the NICA collider.

The highest oscillation frequency $P_{N}(x_{\rm{F}})$ is expected, according
to calculations by the CPQ model, in the reactions of antibaryon production in
baryon collisions. This is due to the large number of spectator quarks from
projectile (there are 6 of them, see fig. 14a) accompanying the production of
three antiquarks, which make up an antibaryon. There is a very limited set of
data on the polarization $P_{N}(x_{\rm{F}})$ of antihyperons produced in
nucleon-nucleon collisions. In fig. 14a is shown quark diagram for the
reaction $p+A\rightarrow\bar{\Xi}^{+}+X$. The weighted number of spectator
quarks for both reactions is $\nu_{A}=\nu_{B}=6-3\tau A_{eff}\approx 5.92$.
This leads to a high oscillation frequency $P_{N}(x_{\rm{F}})$ according to
(18), so that in the range $0<x_{\rm{F}}<1$, several complete cycles can be
observed.

|
---|---
(a) | (b)
Figure 14: Quark flow diagram (a) and $P_{N}(x_{\rm{F}})$ data [107] (b) for
the reaction $p+\rm{Be}\rightarrow\bar{\Xi}^{+}+X$, taken from [104].

In fig. 14b are shown the data for the reaction
$p+A\rightarrow\bar{\Xi}^{+}+X$ [107]. There are also shown there the
calculations of $P_{N}(x_{\rm{F}})$ according to the CPQ model [104]. Although
the available data agree with the calculations of $P_{N}(x_{\rm{F}})$ using
the CPQ model, the number of experimental points are clearly insufficient to
prove the phenomenon of $P_{N}(x_{\rm{F}})$ oscillations. New additional data
are required in the range $0<x_{\rm{F}}<1$ to observe several cycles of
$P_{N}(x_{\rm{F}})$ oscillations. Examples of $P_{N}(x_{\rm{F}})$ calculations
for the reactions $p+A\rightarrow\bar{\Xi}^{+}+X$ and
$p+A\rightarrow\bar{\Xi}^{0}+X$ can be found in [104].

The effect of "attraction" in the polarization of antihyperons should manifest
itself most clearly in the reaction $p+A\rightarrow\bar{\Lambda}+X$ [101]. The
dependence of $P_{N}$ on the energy $\sqrt{s}$ of the resonance type is
expected, with a maximum at $\sqrt{s}=E_{R}=6.98$ GeV. This behavior
$P_{N}(\sqrt{s})$ is based on a single non-zero $P_{N}$ report for reaction
$p+A\rightarrow\bar{\Lambda}+X$, observed in experiment E766 at
$\sqrt{s}=7.31$ GeV [108]. It is very important to repeat such measurements
that are within the energy range available at the NICA collider in p+p, d+d,
C+C and Ca+Ca collisions. The width of the "resonant" peak is small, since the
precession of only one test $\bar{s}$-quark is important in this case [101].
In case of the reaction $p+A\rightarrow\bar{\Xi}^{+}+X$, there are two
$\bar{s}$-quarks and one $\bar{d}$-quark with different precession
frequencies, which broadens the "resonant" peak.

Investigation of the dependences $P_{N}(\sqrt{s})$ of the "resonant" type and
$P_{N}(x_{\rm{F}})$ of the "oscillating" type for the reaction
$p+A\rightarrow\bar{\Lambda}+X$ is a very interesting problem affecting many
aspects of strong interactions, such as color forces between quarks,
precession of quark spin in a chromomagnetic field, quark counting rules for
spectator quarks creating the field, anomalous chromomagnetic moment of
quarks, the role of constituent (dressed) quarks in hadron interaction and
formation and quark confinement phenomenon.

An example of possible studies of $P_{N}$ in collisions of ions can be found
in [98]. It is shown that the higher is the atomic weight of the ions, the
higher is the frequency of the oscillations, since the effective
chromomagnetic field is increased by the quarks, coming from colliding ions.

The only available data for the $A+A\rightarrow\Lambda X$ reaction in heavy
ion collisions, where $P_{N}$ was measured, were used as input to the CPQ
model. The data were obtained in a fixed-target experiment, where $\Lambda$
was produced in Au+Au collisions at c.m. energy $\sqrt{s}=4.86$ GeV [109].

Already at the first stage of the SPD NICA project, it is possible to start
studing the transverse polarization of hyperons and antihyperons in ion
collisions. We also note the possibility of simultaneous measurement of the
so-called global polarization with respect to the reaction plane. In this case
the rotation of hadron or quark matter after collision of two nuclei leads to
the hyperon polarization with respect to the reaction plane, determined by an
impact parameter.

To conclusion, the study of single-spin polarization phenomena in the SPD NICA
project makes it possible to reveal the regularities in the behavior of the
single-spin asymmetry of hadrons and the transverse polarization of hyperons
and antihyperons. Such studies are possible due to the $4\pi$ geometry of the
SPD facility, a developed identification system, a fairly wide range of
available energies, the presence of beams of polarized protons and deuterons,
as well as ion beams. Among the most interesting tasks on this topic are the
following:

1) Measurement of $A_{N}$ and $P_{N}$ at several energies $\sqrt{s}$ in a wide
range for $x_{F}$ and $p_{T}$, in order to separate the dependences on these
three kinematic variables. The form of these dependences reflects the
mechanism of the origin of polarization phenomena. These measurements should
be carried out for as many reactions as possible, which is important for
studying the dependence of $A_{N}$ and $P_{N}$ on the type of particles
participating in the reaction. In general, this study will significantly
expand the database available for theoretical analysis and discrimination of
theoretical models.

2) Investigation of the scaling phenomenon for $A_{N}$ and $P_{N}$ and
corrections to it, reflecting the peculiarities of the mechanism of the origin
of polarization phenomena.

3)Investigation of the threshold phenomena for $A_{N}$, including the
measurement of the threshold angle of hadron production in the c.m. on which
$A_{N}$ becomes null.

4) Investigation of the phenomenon of $A_{N}$ and $P_{N}$ oscillations and the
relationship between the oscillation frequency and the number of spectator
quarks and the type of hadrons participating in the reaction. Particularly
interesting in this respect are antihyperons and cascade hyperons, as well as
secondary protons and neutrons, for which the oscillation frequency reaches a
significant value, which facilitates its measurement. High oscillation
frequency is expected also in heavy ion collisions.

5) Investigation of the phenomenon of "resonance" dependence of $A_{N}$ and
$P_{N}$ on energy $\sqrt{s}$. Disclosure of the mechanism of this phenomenon.

6) Study of the dependence of $A_{N}$ and $P_{N}$ on the atomic weights of the
particles involved in collisions. This will allow not only to link the data
obtained with different nuclei, but also to use the nuclei as tools for
investigating the mechanism underlying the polarization phenomena. Research
using ion collisions will provide a new insight into the phenomena previously
studied in hadron-hadron collisions. Until now there is only one experiment in
which the transverse polarization of a hyperon was measured in heavy ion
collisions. Global polarization with respect to the reaction plane can be
measured in addition to the $P_{N}$, which is measured with respect to the
production plane.

7) Additional possibilities for studying the mechanism of polarization
phenomena are provided by the use of such variables as the multiplicity of
charged particles in an event, as well as the centrality of collisions and the
impact parameter in the case of collisions of nuclei.

The data obtained in the proposed studies will significantly expand the
general world database on polarization measurements and become the basis for
their systematic theoretical analysis, within the framework of a unified
approach. One of the models that makes it possible to carry out a systematic
global analysis of polarization data is the model of chromomagnetic
polarization of quarks, which makes it possible to analyze various reactions
in a wide range of kinematic and other variables that determine the
experimental conditions. Global analysis of the entire dataset is suggested.

##  6 Vector light and charmed meson production 121212 This section is written
by E. Tomasi-Gustafsson; E-mail<EMAIL_ADDRESS>
PACS: 13.85.-t; 14.40.-n; 14.40.Lb

### Introduction

Among the wide possibilities opened by the future availability of the beams
from the NICA collider and the operation of the large acceptance SPD detector,
we focus here on two issues: charm production (hidden and open) and backward
vector meson production. The study of such channels will take full advantage
of the possibility of accelerating polarized $p$, $d$ beams (as well as
heavier ions) in a kinematical region where data are scarce on cross sections
and polarization effects are mostly unmeasured. New, precise data will be
extremely useful for the understanding of the mechanism of charm creation and
of hadronic matter dynamics in the non-perturbative region of QCD. In general,
threshold meson production channels in $NN$-collisions, $p+p\to
p+p+\omega(\phi)$, $p+p\to\Lambda(\Sigma^{0})+K^{+}+p$, and $p+p\to
p+p+\eta(\eta^{\prime})$, give deeper insight in the reaction mechanisms as it
is shown by the experimental programs at different proton accelerators as
SATURNE and COSY.

In this respect, $J/\psi$ production has a specific interest: the production
and the propagation of charm in ion-ion collisions have been considered as one
of the most promising probe of quark-gluon plasma (QGP) [110], but in order to
state the evidence of a clear signal, it is necessary to analyze in detail all
possible mechanisms for $J/\psi$ production in ion-ion collisions, and also
all other processes which are responsible for the dissociation of the produced
$J/\psi$ meson. The study of charmonium (hidden strangeness) and $D$ $(D^{*})$
mesons (open charm) are equally important.

###  6.1 Charm production

The elementary $pp$ cross section are collected and illustrated in Ref. [111].

In the energy region that can be investigated with the NICA-SPD facility,
$3.4\leq\sqrt{s}$[GeV]$\leq 27$ [112] for $pp$ collisions, the total $pp$
cross section is relatively constant around 40 mb, whereas the elastic cross
section decreases due to the opening of different inelastic channels as the
energy increases 141414Fundaments of elastic $pp$ scattering up to LHC
energies have been recently reviewed in Ref. [113] and references therein..
The order of magnitude of the inelastic cross section can therefore be
sizable, reaching 30 mb at the highest energies. Among these inelastic cross
sections, the channels $p+p\to p+p+J/\Psi$ and $p+p\to
p+\Lambda_{C}(\Sigma_{C})+D$ open around $\sqrt{s_{Thr}}\sim 5$ GeV and they
are expected to grow up to several $\mu b$ in the considered energy range.

The production mechanisms for charmonium (hidden strangeness) and $D$
$(D^{*})$ mesons (open charm) in nucleon-nucleon collision are not yet
understood. The question is how charm quarks - that are not preexisting in the
nucleon as valence quarks - are formed and how they hadronize. To interpret
the production and the propagation of charm in heavy ion collision as a probe
of quark-gluon plasma (QGP), it is necessary to have a solid theoretical
background based on the understanding of elementary processes.

Experimental data and theoretical studies of $J/\psi$ production in different
processes and of its decays exist: for a review, see [114] and for a most
recent data collection [115]. As a result of high statistics and high
resolution experiments a large information on the properties of the $J/\psi$
meson have been collected, on the production processes and on its numerous
decays. From a theoretical point of view, the interpretation of the data, in
particular in confinement regime, is very controversial. As an example, the
$c-$quark mass is too large, if compared to predictions from chiral symmetry,
but for theories based on expansion of heavy quark mass (Heavy Quark Effective
Theory), this mass is too small [116].

In the threshold region, the final particles are produced in $S$-state and the
spin structure of the matrix element is largely simplified. Simple
considerations indicate that this region is quite wide: the effective proton
size, which is responsible for charm creation, has to be small, $r_{c}\simeq
1/m_{c}\simeq 0.13$ fm, where $m_{c}$ is the $c$-quark mass, selecting small
impact parameters [117]. The $S$-wave picture can therefore be applied for
$q\leq m_{c}$, where $q$ is the norm of the $J/\psi-$ three-momentum in the
reaction center of mass (CMS). The momenta of the produced particles are
small, but the mechanisms for the production of charmed quarks must involve
large scales. In Ref. [48], the near-threshold $J/\psi-$ production in
nucleon-nucleon collisions was analyzed in the framework of a general model
independent formalism, which can be applied to any reaction $N+N\to
N+N+V^{0}$, where $V^{0}=\omega$, $\phi$, or $J/\psi$. Such reactions show
large isotopic effects: a large difference for $pp$\- and $pn$-collisions,
which is due to the different spin structure of the corresponding matrix
elements.

In Ref. [48] an estimation of $J/\Psi$ production was suggested from the
comparison of the cross sections for the $\phi$ and $J/\psi$ production in
$pp$ collisions. The same approach, namely $\pi$ exchange in $N+N\to
N+N+V^{0}$ and $\rho$ exchange for the sub process $\pi+N\to N+V^{0}$, with
$V^{0}=\phi$ or $J/\psi$ was considered. For the same value of the energy
excess, $Q=\sqrt{s}-2m-m_{V}$, taking into account the different phase space
volumes, coupling constants for the decay $V\to\pi\rho$, monopole-like
phenomenological form factor for the vertex $\pi^{*}\rho^{*}V$, with virtual
$\pi$ and $\rho$, one finds the following simple parametrization for the cross
section, holding in the near threshold region only:

$\sigma[nb]=0.097(Q[\mbox{GeV}])^{2}.$ (26)

In Ref. [118] a parametrization of exponential form

$\sigma[nb]=ae^{-bM_{J/\Psi}/\sqrt{(}s)};$ (27)

was suggested. The values $a$= 1000 [nb], and $b$=16.7 GeV reproduce well the
experimental data over threshold.

The threshold for this reaction is $E_{th}$=12.24 GeV which corresponds to
$\sqrt{s}=2m+m_{J/\psi}\simeq$ 4.97 GeV. In Fig. 15 the data for $p+p\to
J/\psi+p+p$ (red circles) and $p+A\to J/\psi+X$ (blue squares) are plotted
from the recollections in Refs. [114] (filled symbols) and [115] (open
symbols). Different symbols differentiate $J/\psi$ production in $pp$ or
(extrapolated from) $pA$ collisions. The data, mostly collected at CERN, are
reconstructed from the measurement using models and/or assumptions, and the
compiled total cross section for $J/\Psi$ production may differ up to a factor
of two. For example, the original reference for the measurement from Protvino
at $\sqrt{s}=11.5GeV$ [119] gives $\sigma(pp\to(J/Psi\to\mu+\mu^{-})+X)=9.5\pm
2.5$ nb, whereas the same experimental point is referenced as $\sigma=11\pm 3$
nb, in Ref. [114] and $\sigma=20\pm 5.2$ nb, in Ref. [115]. The cross section
from Ref. [48] is also plotted in Fig. 15 (solid line).

Taking the value of luminosity ${\cal L}=10^{30}$ cm-2 s-1, one expects 3
counts/hour for such a process with a cross section of the order of 1 nb. This
number is not corrected for the detector efficiency and reconstruction with
identification, for example, in a missing mass. The reconstruction of $J/\Psi$
through its decay into a lepton pair, that is the preferred mode, requires two
additional orders of magnitude as the branching ratio is $(\simeq 5.9\pm
0.5)10^{-2}$.

Note also that in the framework of the considered model, one can find a large
isotopic effect, due to the different spin structure of the matrix element at
threshold:

$\displaystyle\frac{\sigma(np\to npJ/\psi)}{\sigma(pp\to ppJ/\psi)}=5,$

which would require a correction of the experimental data on $pA$ reaction,
where equal $np$ and $pp$ cross sections are usually assumed for the
extraction of the elementary cross section in $pp$ collisions.

Figure 15: Experimental data on $J\psi$ production in $pp$ (red circles) and
$pA$ (blue squares) reactions, from the recollections in Refs. [114] (filled
symbols) and [115] (open symbols). The solid line is the calculation from Ref.
[48]. The plot is drawn from the $J/\psi$ production threshold (black line).
The green filled region represents the range that can be investigated with
NICA-SPD.

From Ref. [48] one also learns that only one polarization observable, the
$J/\psi$-polarization, is identical for $pp$ and $pn$ collisions: the $J/\psi$
meson is transversely polarized - even in collisions of unpolarized nucleons.
The experimental determination of the ratio of the total cross sections for
$np$ and $pp$ collisions gives important information for the identification of
the reaction mechanism.

The possibility of presence of intrinsic charm as a higher order component of
the development of the Fock expansion for the proton state has been discussed
in Ref. [120]. Near threshold, all partons must transfer their energy to the
charm quarks, within a time $t\sim 1/m_{c}$, thus selecting short range
correlations between the valence quarks. Most interesting is the deuteron
case, where all six quarks must be involved coherently, giving access to the
hidden color part of the deuteron wave function.

###  6.2 Open charm production

Open charm production, $N+N\to N+\bar{D}+\Lambda_{C}(\Sigma_{C})$ gives
information on scattering lengths, effective radius, hadronic form factors,
and coupling constants and is also related to the dynamics of charm creation
in $NN$, $NA$, $AA*$ collisions. Some predictions can be done from an analogy
with strangeness production, relying on the equivalence of SU(3) and SU(4)
symmetries, that is however, not totally reliable. Existing information and
estimation indicate that near threshold cross section can be of the order of
microbarns. The threshold cross section, normalized at the lowest existing
value is plotted in Fig. 16, where the insert highlights the threshold region.
A dedicated simulation should be done to evaluate the counting rates, as the
charmed particles should be reconstructed from the most suitable decay
channels.

Figure 16: Total charm production in $pp$ and $pA$ collisions. Data are from
Ref. [121]. The line is a threshold parametrization (see text).

The spin and isospin structure of the matrix element for the reactions
$N+N\to\Lambda_{C}(\Sigma_{C})+\bar{D}+N$ was derived for open charm in Ref.
[122]. Detailed estimation of cross sections and the expressions of the
polarization observables can be found there.

The charm production near threshold cross section follows the behaviour:

$\sigma[\mu b]=0.03(Q[\mbox{GeV}])^{2}$ (28)

that can be useful for simulation purposes. It is plotted in Fig. 16 over a
collection of data from Ref. [121] reanalyzed from several experiments on
charm production in $pp$ and $pA$ collisions at different facilities.

We stress that these are difficult measurements, with low counting rates, but
that even setting upper limits will be important, as no data at all are
present in the threshold region.

###  6.3 Backward meson production

Larger counting rates are expected for light meson productions, since cross
sections are of the order of mb. The $\rho^{0}$ meson production in elementary
collisions and on nuclei has been discussed for example in Ref. [123] and
references therein. The $\rho^{0}$ inclusive cross section has been measured
at different accelerators since the 70s, mostly at CERN [124], and more
recently by the HADES collaboration [125]. In Ref. [126], the inclusive cross
section for $\rho$ production in $pp$ collision is calculated in frame of a
generalized vector meson dominance model, and the existing data up to
$\sqrt{s}=65$ GeV are fairly reproduced and compared to other models. In Ref.
[127] the following parametrization was suggested

$\sigma(pp\to\rho^{0}X)=(0.38\pm 0.02)\ln^{2}s-(2.1\pm 0.4).$ (29)

This parametrization is shown together with the data for the inclusive cross
section of $p+p\to\rho+X$ are in Fig. 17.

Figure 17:  $\rho$ production in $pp$ and $pA$ collisions. The red circles
(black squares) are for inclusive (exclusive) $\rho$ production in different
experiments. The line is the parametrization from Ref. [127]). The shaded
region represents the SPD energy range.

One can see that it is of the order of the mb from the near threshold region,
and therefore measurable at SPD already in the first phase of the experiment.

In Ref. [128] a specific kinematics, the backward light meson production in
$pp$ or $pA$ collisions, was discussed in similarity with the ’quasi real
electron method’, where a hard photon is produced on the collision of
electrons on any target [129]. Two important characteristics have been proved
for the electron case: -the collinear emission probability has a logarithmic
enhancement - the cross section can be factorized in a term related to the
probability of the meson emission with a given energy at a given angle from
the beam particle, and a term related to the interaction of the beam remnant
after emission on the target.

Figure 18: Feynman diagram for collinear hard photon emission in $eT$
reactions (T stands for any target). The hadron equivalent is obtained by
replacing the photon by a $\rho$-meson and the electron by a proton.

The cross sections for the reactions of interest are:

$\displaystyle d\sigma^{pT\to h_{+}X}(s,x)$ $\displaystyle=$
$\displaystyle\sigma^{nT\to X}(\bar{x}s)dW_{h_{+}}(x),$ $\displaystyle
d\sigma^{pT\to h_{0}X}(s,x)$ $\displaystyle=$ $\displaystyle\sigma^{pT\to
X}(\bar{x}s)dW_{h_{0}}(x),$ (30)

where $h$ is a hadron. The quantity $dW_{\rho}(x)$ can be inferred using the
QED result:

$\displaystyle\frac{dW_{\rho^{i}}(x)}{dx}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4\pi^{2}}\frac{1}{x}\sqrt{1-\frac{m_{\rho}^{2}}{x^{2}E^{2}}}$
$\displaystyle\left[\left(1-x+\frac{1}{2}x^{2}\right)L-(1-x)\right],$
$\displaystyle
1>x=\frac{E_{\rho}}{E}>\frac{m_{\rho}}{E},L=\ln\left(1+\frac{E^{2}\theta_{0}^{2}}{M^{2}}\right),\rho^{i}=\rho^{+},\rho^{-},\rho^{0},$

where $M$, $m_{\rho}$, $E$, and $E_{\rho}$ are the masses and the energies of
the initial proton and the emitted $\rho$-meson in the Laboratory system.

The integrated quantities $W_{h}$, $h={\rho,\pi}$ can, in general, exceed
unity, violating unitarity. To restore unitarity, we have to take into account
virtual corrections: the vertex for the emission of a single pion (charged or
neutral) from the proton has to include ’radiative corrections’, which account
for emission and absorption of any number of virtual pions. For this aim we
use the known expression for the probability of emission of $n$ "soft" photons
in processes of charged particles hard interaction, $i.e.,$ the Poisson
formula for emission of $n$ soft photons $W^{n}=(a^{n}/n!)e^{-a}$ (where $a$
is the probability of emission of a single soft photon) [130].

The probability of emission of ’soft’ neutral pions follows a Poisson
distribution, which is not the case for the emission of charged pions.
Fortunately, in our case, it is sufficient to consider the emission of one
charged pion at lowest order (the process of one charged pion emission) plus
any number of real and virtual pions with total charge zero. In such a
configuration, this vertex has the form of the product of the Born probability
of emission of a single pion multiplied by the Poisson-like factor:

$P_{\pi,\rho}=e^{-W_{\pi,\rho}},$ (32)

which takes into account virtual corrections.

The final result is obtained using the replacement:

$\sigma(s)\to\sigma(s)\times{\cal R_{\pi}},~{}{\cal
R_{\pi}}=P_{\pi}\sum_{k=0}^{k=n}\frac{W^{k}_{\pi}}{k!},$ (33)

where ${\cal R}_{\pi}$ is the renormalization factor in order to account for
the emission of $n$ real soft neutral pions escaping the detection.

Concerning the production of two charged pions, accompanied by a final state
$X$, we can write:

$d\sigma^{p\bar{p}\to\rho^{0}X}=2\frac{dW_{\rho}(x)}{dx}\sigma^{p\bar{p}\to
X}(\bar{x}s)\times P_{\rho},~{}$ (34)

where the factor of two takes into account two kinematical situations,
corresponding to the emission along each of the initial particles and
$P_{\rho}$ is the survival factor (32) which takes into account virtual
radiative corrections. The cross section is shown in Fig. 19 as a function of
the $\rho$ energy fraction, for two values of the incident energy and of the
emission angle.

Figure 19: Cross section $d\sigma(p,\bar{p}\to\rho^{0}X)$ as function of the
$\rho$ energy fraction for two values of the incident energy and of the $\rho$
emission angle: $E=10$ GeV and $\theta_{0}=10^{\circ}$ (black, solid line),
$E=10$ GeV and $\theta_{0}=20^{\circ}$ (red, dashed line), $E=20$ GeV and
$\theta_{0}=10^{\circ}$ (green, dotted line), $E=20$ GeV and
$\theta_{0}=20^{\circ}$ (blue, dash-dotted line).

The $x$ dependence shows a characteristic peak at $x=x_{max}$ that has the
same nature as for the QED process $e^{+}+e^{-}\to\mu^{+}+\mu^{-}+\gamma$. As
explained in Ref. [131], it is a threshold effect, corresponding to the
creation of a muon pair, where $x_{max}=1-4M_{\mu}^{2}/s$, $M_{\mu}$ is the
muon mass.

The prediction of the model for backward $\rho$-meson production in $pp$
collisions is shown in Fig. 20, as a black solid thick line. The red dashed
line is the renormalization factor, from Eq. (33), integrated over $x$. The
total $pp$ cross section is the black dash-dotted line, of the order of 40 mb,
and it is quite flat in all the considered energy region. The blue line is the
parametrization from Ref. [127]) of the inclusive $\rho$ cross section. The
available data are also shown, as different symbols and colors for inclusive
measurements and as black squares for exclusive $\rho$ production. Backward
production can be of the order of several mb, therefore accessible at NICA-SPD
also with the initial lower luminosity.

An original application is the possibility of creating neutron beams by
tagging the incident proton beam with a negative meson emitted backwards.
Charge exchange reaction takes place, and the beam remnant is a neutron
impinging on the target beam.

Figure 20: Cross section for $\rho$-meson production in $pp$ collisions:
inclusive (different symbols and colors from different experiments) and
exclusive data from $pp\to pp\rho$ (black squares). The present calculation is
shown as a black line. The red dashed line is the renormalization factor from
Eq. (33). The black dash-dotted line is the total $pp$ cross section.The first
red point is the inclusive measurement from Ref. [125]. The blue line is the
parametrization from Ref. [127].

###  6.4 Conclusions

The understanding of charm production (open or hidden) should unify the
different steps: parton-level hard process with production of $c\overline{c}$
pairs, after hadronization of $c\overline{c}$ into $J/\psi$ or into charmed
hadrons (mesons and baryons) including the final state interaction of the
produced charmed hadrons with other particles. The relatively large
transferred momenta involved in most processes of $J/\psi$ production in
hadron-hadron collisions allow to treat the first step in framework of
perturbative QCD. But the applicability of QCD is not so straightforward for
the description of the $c$-quark hadronization. In this respect, precise data
collected in the NICA-SPD energy range will bring important information,
especially if covering a wide range above threshold. Light meson as $\rho$
meson production is definitely easier to be measured. Collecting precise,
systematic data should help to refine the models and of great interest also
for the collision on heavy targets. Backward kinematics could constitute an
original contribution to the field, offering an alternative possibility to
produce neutron beams.

##  7 Exclusive hard processes with deuteron at NICA151515This section is
written by M. Strikman; E-mail<EMAIL_ADDRESS>
PACS number(s) 13.75.Cs, 25.10.+s, 25.40Ep

Our understanding of the dynamics of NN interactions at the energy range of
$\sqrt{s}\sim 5\div 20\,GeV$ is still rather limited. In particular, it is not
clear yet where transition occurs from nonperturbative to perturbative
dynamics in few body processes with a large momentum transfer ($-t$). This
includes even the most basic process of the large $-t$ elastic nucleon -
nucleon scattering at large $-t$. Among the puzzles are large spin effects in
large angle scattering of polarized protons[38] and a complicated energy
dependence of the nuclear transparency in large angle scattering of incident
protons off the protons embedded in nuclei [44]. Also the recent observations
of two nucleon short range / high momentum correlations in nuclei mostly in
electron - nucleus scattering (see review in [132, 133]) require confirmation
and testing universality of the SRCs using other projectiles - protons,
photons, etc.

Questions involved in studies of the short-range / high momentum nuclear
structure and understanding microscopic nucleon structure and dynamics of
large momentum transfer processes are delicately intertwined: understanding of
hard dynamics of two body processes is also necessary for precision studies of
the short range nuclear structure.

Several strategies are possible to address these questions. Here we will
concentrate on reactions with the deuteron since the nonrelativistic deuteron
wave function is well known and hence the measurements could be matched to
detailed theoretical calculations. Also, the use of the deuteron allows to
choose special kinematic domains where $p^{2}H$ scattering is sensitive to the
short-range nuclear correlations. The collider kinematics presents a number of
advantages as all particles in the reactions in question have large momenta
and hence can be easily detected.

###  7.1 Probing dynamics of nucleon - nucleon interaction in proton -
deuteron quasielastic scattering

The simplest reaction which would be interesting to study is the process
$p^{2}H\to ppn$ where one of the nucleons has small transverse momentum and
two others have approximately back to back large transverse momenta[134, 135].

In the impulse approximation this process corresponds to elastic scattering of
the projectile proton off a quasifree nucleon of the target. There exist
however kinematical conditions where the dominant contributions are due to
soft rescatterings of the initial and final nucleons, which accompany the hard
$pp(pn)$ reaction. The eikonal approximation, which accounts for relativistic
kinematics as dictated by the Feynman diagrams, reveals the important role
played by the initial and final state interactions in the angular and momentum
dependences of the differential cross section in well defined kinematics. The
condition for the applicability of the generalized eikonal approximation [136]
is that the c.m. scattering angle and invariant mass of the two nucleon system
are large enough so that $-t,-u\geq\mbox{2 GeV}^{2}$.

It was suggested in [5, 6] that nucleons in the elementary reaction interact
in small size configurations with a small cross section - so called color
transparency phenomenon. This effect is suppressed by the space - time