evolution of nucleon wave packets [137, 138]. However effect of evolution is
very small for the deuteron where typical distances between nucleons in the
rescattering amplitude are is $\leq$ 1.5 fm. Hence the discussed process
allows to measure the wave packet size of a nucleon practically right in the
interaction point.

It was pointed out that the hard dynamics in $pp$ and $pn$ elastic scattering
may be rather different [139]. Hence it would be instructive to compare the
channels where $pp$ and $pn$ are produced with large $p_{t}$.

Experiments with polarized beams would greatly add to this program: a) study
of the dependence of the cross section on the deuteron polarization allows to
improve separation of kinematic domains where impulse approximation, double
and triple scattering dominate, while the studies $\vec{p}\vec{d}\to pNN$
processes will allow both to study spin structure of $pp$ and $pn$ elastic
scattering at large $t$ (the later is practically not known). Also, it would
be possible to find out whether the A. D. Krisch effect [38] (a strong
difference between the cross sections of elastic scattering of protons with
parallel and antiparallel spins) involves collisions of protons in
configurations with sizes depending on the spin orientation.

It would be possible also to study effects of coherence in the channels where
exchange by gluons in t-channel is not possible, for example $pd\to\Delta NN$.
In particular, it would be possible to test the effect of chiral transparency
suggested in [140] \- suppression of the pion field in the nucleons
experiencing large $-t$ scattering.

###  7.2 Probing microscopic deuteron structure

As emphasized in the introduction, the dominant source of the SRC in nuclei
are proton - neutron correlations with the same quantum numbers as the
deuteron and with high momentum tail similar to that in the deuteron. Hence
the deuteron serves as a kind of the hydrogen atom of the SRC physics. Only
after it would be tested experimentally that approximations currently used for
the description of the $p^{2}H$ reaction work well, it would be possible to
perform high precision studies of SRC in heavier nuclei.

It was demonstrated in Ref.[134, 135] that under specific kinematical
conditions (in particular low transverse momenta of a slow nucleons in the
deuteron rest frame)) the effect of initial and final state interactions can
be accounted for by rescaling the cross section calculated within the plane
wave impulse approximation. In this kinematics it would be possible to check
universality of the wave function - in particular its independence on the
momentum transfer in the elementary reaction. Such factorization is expected
to break down at sufficiently large $-t$ and $-u$ where scattering involves
interaction of nucleons in the small size configurations (the color
transparency regime) since the small size configurations are suppressed in
bound nucleons with suppression growing with the nucleon off shellness [138].

Studies of the nonnucleonic configurations in the deuteron as well as
relativistic effects. are of separate interest. One option is a search for
non-nucleonic degrees of freedom like 6 quark, two$\Delta$ isobars via
production reaction $p^{2}H\to\Delta^{++}+p+\Delta^{-}$ with $\Delta^{++}$ and
proton back to back and $\Delta^{-}$ in the deuteron fragmentation region for
the light cone fraction $\alpha_{\Delta}=2p^{-}_{\Delta}/p^{-}_{d}\geq 1$
(slow in the deuteron rest frame; proton momentum is along $z$ direction).

In a long run, when polarized deuteron beams would become available, it would
be possible to to separate contribution of the S- and D- wave to the SRCs and
compare different relativistic models of the deuteron – light -cone vs virtual
nucleon approximations. The difference between two predictions is large
already for the nucleon momenta $\sim\mbox{300 MeV/c}$ in the deuteron rest
frame.

##  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

PACS: 21.45.+v; 24.70.+s; 25.10.+s

The structure of the lightest nuclei at short distances $r_{NN}<0.5$ fm or
high relative momenta ($q>\hslash/r_{NN}\sim$ 0.4 GeV/$c$ constitues a
fundamental problem in nuclear physics. One of the most important questions is
related to search for onset of transiton region from meson-baryon to quark-
gluon picture on nuclei. A definite signature for transition to the valence
quark region is given the constituent counting rules (CCR) [141, 142].
According the dimensional scaling the differential cross section of a binary
reaction at enough high incident energy can be parametized as $d\sigma/dt\sim
s^{-(n-2)}f(t/s)$, where $n$ is the sum of costituent quarks in all
participants, $s$ and $t$ are Mandelstam variables. Many hard processes with
free hadrons are consistent with CCR at energies of several GeV. The CCR
propetrties of the reactions with the lightest nuclei were observed in
photodisintegration of the deuteron $\gamma d\to pn$ at $E_{\gamma}=1-4$ GeV
and ${}^{3}He$ nucleus ${}^{3}He(\gamma,pp)n$, $\gamma^{3}He\to dp$. Earlier
data on the reaction $dd\to^{3}Hp$, $dd\to^{3}Hen$ [143] and $pd\to pd$, as
was show in Ref.[144] also follow CCR behavior $s^{-22}$ and $s^{-16}$,
respectively, at surprising low energies, 0.5 GeV. Recently the CCR behaviour
of the reaction $pd\to pd$ was observed in [145, 146] at higher energies. On
the other hand, the reaction with pion production $pp\to d\pi^{+}$ does not
follow CCR rule demostrating the differential cross section $\sim s^{-9}$
instead of $s^{-12}$. One possible way to explain this is a partial
restortaion of chiral symmetry at enough high excitaion energy [147]. However,
systematic study of these properties of the reactions with lightest nuclei are
absent. So, important to know whether reaction $pn\to d\rho^{0}$ follows the
CCR behavior and at what minimal energy there is the CCR onset. Assuming the
model of the vector meson dominance and taking into account the observed CCR
behavior of the $\gamma d\to pn$ reaction, one may expect the $\sim s^{-12}$
dependence of the cross section of the reaction $pn\to d\rho^{0}$.
Furthermore, possible relation between CCR behavior of the upolarized cross
section and spin observables of the same reaction are practically not known.
The SPD NICA facility provides a good opportunity for this study using
polarized beams in $pp$, $dd$ and $pd$ collisions.

The results on the polarization observables for deuteron induced reactions at
large transverse momenta are controversial. Results on the tensor polarization
$t_{20}$ for the $ed$\- elastic scattering [148] obtained at JLab can be
reproduced quite well at $Q^{2}\leq 1.7$ (GeV/c)2 by the covariant
relativistic model [149] without contribution of non-nucleonic degrees of
freedom. The perturbative QCD (pQCD) predictions [150] are not reliable for
these momentum transfers. On the other hand, while the electromagnetic form
factors for the deuteron in the soft-wall AdS/QCD model [151] are well in
agreement with the experimental data and the form factors display correct
$1/Q^{10}$ power scaling for large $Q^{2}$ which is consistent with the CCR,
the tensor analyzing power $t_{20}$ [148] demonstrates the discrepancy with
the soft-wall AdS/QCD predictions [152].

The cross section of high energy two-body photodisintegration of the deuteron,
$\gamma d\to pn$, at large angles in the cms [153, 154] has shown the scaling
behavior up to 5.5 GeV predicted by CCR [141, 142]. Recent measurements of the
proton polarization [155] at energies up to 2.4 GeV are also consistent with
the pQCD hadron helicity conservation prediction [156], but not the
polarization transfers obtained in the same experiment.

Figure 21: Vector $A_{y}$ (left panel) and tensor $A_{yy}$ (right panel)
analyzing powers in $dp$\- elastic scattering obtained at the fixed angles in
the cms: $60^{\circ}$, $70^{\circ}$, $80^{\circ}$ and $90^{\circ}$.

The dependencies of the vector $A_{y}$ and tensor $A_{yy}$ analyzing powers in
$dp$\- elastic scattering obtained at $60^{\circ}$, $70^{\circ}$, $80^{\circ}$
and $90^{\circ}$ in cms versus transverse momentum $p_{T}$ are shown in Fig.21
in the left and right panels, respectively. The open points are the world data
obtained at RIKEN, Saclay and ANL, while the black ones represent the results
obtained at Nuclotron [157, 158] The analyzing powers demonstrate the sign
changes at $p_{T}\sim$ 650 MeV/$c$ and the asymptotic behavior at large
transverse momenta. Note, that negative sign of $A_{yy}$ is observed also in
deuteron inclusive breakup at large $p_{T}$ [159],[160].

It would be interesting to extend the range of the measurements to larger
$p_{T}$, where the manifestation of non-nucleonic degrees of freedom is
expected. New precise measurements with small statistical and systematic
uncertainties at the energies higher than $\sqrt{s}\geq$3.3 GeV and at
different scattering angles are required to make a conclusion about the
validity of CCR [141, 142] in $dp$ \- elastic scattering. We propose to
measure also different vector and tensor analyzing powers in $dp$\- elastic
scattering at SPD energies.

The tensor $A_{yy}$ and vector $A_{y}$ analyzing power in $dp$\- elastic
scattering obtained at $60^{\circ}$, $70^{\circ}$, $80^{\circ}$ and
$90^{\circ}$ in cms versus transverse momentum $p_{T}$ [157, 158] demonstrates
the negative and positive asymptotics, respectively. Note, that negative sign
of $A_{yy}$ is observed also in deuteron inclusive breakup at large $p_{T}$
[159],[160]. It would be interesting to extend the range of the measurements
to larger $p_{T}$, where the manifestation of non-nucleonic degrees of freedom
is expected. New precise measurements with small statistical and systematic
uncertainties at the energies higher than $\sqrt{s}\geq$3.3 GeV and at
different scattering angles are required to make a conclusion about the
validity of CCR [141, 142] in $dp$ \- elastic scattering. We propose to
measure also different vector and tensor analyzing powers in $dp$\- elastic
scattering at SPD energies.

The measurements of $dp$\- elastic scattering can be performed either with
polarized deuterons and unpolarized protons, or with unpolarized deuterons and
polarized deuterons. The $dp$\- elastic scattering events can be selected
using cuts on the azimuthal and polar scattering angles correlations. The
vector $A_{y}$ and tensor $A_{yy}$ and $A_{xx}$ analyzing powers will be
measured simultaneously in the case of the vertically polarized deuteron beam.
The precision on the tensor $\Delta A_{yy}\sim$0.09 and $\Delta
A_{xx}\sim$0.09 and on the vector $\Delta A_{y}\sim$0.03 analyzing powers can
be achieved for the scattering angle $\sim$90${}^{\circ}\pm$5∘ at
$\sqrt{s}\sim$4.5 GeV ($p_{T}\sim$1.7 GeV/$c$) for 30 days of the beam time at
the luminosity ${\cal L}\approx$10${}^{29}cm^{-2}\cdot s^{-1}$. We assume
$\sim$75% of the beam polarization from the ideal values of polarization for
different spin modes. The counting rate has been estimated using $dp$\-
elastic scattering cross section parameterization from Ref.[146]. The spin
correlations can be obtained in quasi-free $dp$\- elastic scattering using
$dd$\- collisions.

##  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>
Multiquark correlations in the collisions of particles and nuclei at NICA
energies play an important role in understanding of QCD. Processes involving
multiquark degrees of freedom can shed light on various aspects from
multiquark fluctons, diquarks, multiparton scattering to exotic resonance
production and fulfil a broad and rich physics program at SPD experiment.

PACS: 44.25.$+$f; 44.90.$+$c

###  9.1 Multiquark correlations and exotic state production at SPD NICA

Multiquark correlations in the collisions of particles and nuclei play an
important role in understanding of QCD. Multiquark correlation phenomena may
be divided into four classes.

The first one can be related with parton distribution functions (PDFs) of the
colliding nuclei. In the leading twist approximation, in the nuclear PDFs
there is a contribution at large $x>1$, which is related with objects
fluctuations of nuclear matter [161] known as multiquark fluctons [162, 163,
164] (see for a review, e. g., Refs. [165, 166] and references therein) or
few-nucleon short-range correlations (see, e. g., Ref. [167, 138, 168] and
references therein). Fluctons are compact multinucleon states with defrosen
color quark-gluon degrees of freedom, see, e. g., [162, 163, 164, 169, 170,
171, 172, 173, 174, 175, 176, 177, 178, 179, 180].

The second class is connected with higher twist contributions of two- and/or
three- quark correlations in PDFs of hadrons and nuclei.

The third class is dealing with multiparton scattering subprocesses in
hadronic and nuclear collisions. Multiparton scattering occurs when two or
more partons from each colliding objects simultaneously scatter off each
other.

The forth class phenomena can be related with production of exotic multiquark
resonance states, e. g., pentaquark and tetraquark states.

Below one can briefly outline possible studies, which can shed light on the
all mentioned above classes of multiquark phenomena and fulfil a broad and
rich physics program at SPD experiment.

###  9.2 Multiquark correlations: fluctons in nuclei

Fluctuations of nuclear matter were considered in Ref. [161], which were
discussed after discovering backward scattering off nuclei [181, 182], would
form fluctons, compact multi-nucleon states or in other language, short-range
few nucleon correlations.

Fluctons are directly connected with cumulative hadron production in the
nuclear fragmentation region [183, 184]. The flucton approach [179], which is
based on hard QCD-factorisation and EMC-ratio constraints, predicts an extra
nuclear quark sea, which has rather hard momentum distribution: the extra
nuclear sea $x$-slope is equal to the $x$-slope of the valence quarks at
$x>1$. It leads to "superscaling" for cumulative hadron production at $x>1$ in
the nuclear fragmentation region: the $x$-slope of all cumulative hadron
distributions including "sea" ones [179, 185] are the same. The superscaling
phenomenon was experimentally confirmed by ITEP group [186, 187].

Nuclear fluctons consist of the nucleons compressed in distances comparable
with nucleon size, so the flucton with five or six nucleons could be
considered as a cold dense baryon matter since the effective nuclear density
would be high as that in the core of neutron stars [132]. In such a dense
nuclear medium there would be $CP$-violating effects [188].

In high-$p_{T}$ cumulative processes at the central region, other
contributions should be added to the contribution of the nuclear PDFs at
$x>1$, such as the contributions from the PDFs of the other colliding object
and possible intranuclear rescattering effects [189, 190]. So, beyond the
nuclear fragmentation region one should observe deviations from superscaling
for cumulative production.

For many observables there are two popular approaches: fluctons and short-
range nucleon correlations yield similar predictions, e. g., cumulative
particle production, nuclear structure functions in deep inelastic processes
with leptons, etc. However, there is the main difference with flucton and
short-range nucleon correlation approaches: extra sea quark degrees of freedom
atributted to flucton sea in nuclear PDF at $x>1$.

The hard flucton antiquark sea can manifest in massive lepton pair [191, 192,
193] and $J/\psi$ production in cumulative region [194, 195]. SPD can study
$J/\psi$ production process in $pd$ and $dd$ collisions in $x>1$, which should
be highly sensitive for hard flucton antiquark sea of deutrons.

###  9.3 Few-quark correlations: Diquarks

Another type of multiquark correlations is few-quark short-range correlations:
diquark and triquark states in baryons. Diquark states were discussed soon
after suggesting quark model for hadrons, see for a review Refs. [196, 197].
This is an important source [198, 199, 200, 201, 202] of large-$p_{T}$ baryon
production [203, 204, 200]. Being a higher-twist the diquark contribution can
describe the strong scaling violation in deep inelastic scattering off
nucleons and in large-$p_{T}$ baryon production in hard nucleon collisions at
SPD energies [201, 202, 205] (Fig. 22)

Figure 22: $p$/$\pi^{+}$-ratio of inclusive invariant cross sections at
$\theta_{cms}=90^{0}$ in $pp$-collisions confronted with the data from IHEP,
Protvino [204] ($\bullet$) at $\sqrt{s}=11.5$ GeV and from FNAL, Batavia [203]
($\blacksquare$) at $\sqrt{s}=23.4$ GeV. The preliminary predictions from MC
event generator ULYSSES [205] with incorporated diquark subprocesses: the
solid red line corresponds diquark state probablity in proton equals 0.5, the
dash-dotted green line - 1.0. Prediction by the standard PYTHIA8.3 4Cx [206]
without diquark subprocesses is shown by the blue dashed line.

Most of diquark studies were done within inclusive approach. ABCDHW
Collaboration at CERN ISR have found more convincing indications [207, 208] on
diquark manifestations in two-particle correlations. SPD allows to study two-
particle correlations between large-$p_{T}$ baryons (proton, $\Delta^{++}$,
$\Lambda^{0}$, etc. ) and back-to-back hadrons to reveal of quark structure of
proton in more detail.

Therefore, measuring by SPD experiment various two-particle correlations
should reveal more detail diquark structure of proton.

###  9.4 Multiparton scattering

There is an important aspect of hadronic collsions, which was studied since
70s [209, 210, 211, 212, 213, 214, 215, 216]. There are two basic approaches
for description of multiparton scattering, depending on view whether it is
considered as a higher twist effect or not [217, 218, 219, 220, 221]. In the
all approaches the main object is momentum multiparton momentum-dependent
distribution function, which is complicated in addition by the multiparton
distribution in space (impact plane). Therefore, measuring few-particle
correlation at SPD one can study multiparton scattering processes [205], which
are related with 2D- and 3D- PDFs in momentum and coordinate spaces.

(a)

(b)

Figure 23: Two-proton correlation function at $\theta_{cms}=90^{0}$ in
$pp$-collisions at $\sqrt{s}=23.4$ GeV as a function of $p_{T}$ ($\approx
p_{T1}\approx p_{T2}$) for two regions of azimuthal angle difference: (a) for
back-to-back region ($\Delta\varphi=|\varphi_{1}-\varphi_{2}|\approx\pi$) and
(b) out of it ($\Delta\varphi\not\approx\pi$). The preliminary predictions
from MC event generator ULYSSES [222] with incorporated diquark subprocesses
in different tunes (red squares and magenta stars) and PYTHIA8.3 4Cx [206]
(green triangles) without diquark subprocesses.

SPD experiment can study multiparton distribuion in proton via multiparton
scattering, which yield, in particular, certain azimuthal two-particle
correlations. For instance, double-parton scattering should increase two-
particle correlations out of back-to-back scattering region, where single
parton-parton scattering is dominant [179] yielding a ridge structure [223].
Predictions from MC event generators ULYSSES [205] with incorporated diquark
subprocesses and PYTHIA8.3 for two-particle correlation function of two
protons

$R_{12}=\sigma^{in}\frac{\frac{E_{1}E_{2}d^{6}\sigma}{d^{3}p_{1}d^{3}p_{2}}}{\frac{E_{1}d^{3}\sigma}{d^{3}p_{1}}\frac{E_{2}d^{3}\sigma}{d^{3}p_{2}}}$

is shown in Fig. 23 for back-to-back region and outside it at $\sqrt{s}=23.4$
GeV.

Two-particle correlation function for out of back-to-back azimuthal region is
directly connected with double-parton scattering cross section.

Multiparton scattering can be significant for production of multiquark systems
and and nuclei [224, 205].

###  9.5 Multiquark exotic state production

Multiparton scattering provide a unique opportunity to study production of
various multiquark states and light nuclei [224, 205] at SPD energies.

An excitement has been flourished when LEPS [225], DIANA [226] and CLAS [227]
Collaborations observed a resonance state, which was correspondent to quantum
flavour numbers of $\Theta^{+}$-pentaquark, with the narrow width as predicted
in Ref. [228].

However, in a meantime there was a somewhat contradictory situation. Most of
$\Theta^{+}$-pentaquark observation claims were ruled out while the others
[226, 229, 230, 231] were still not, see for a review of the experimental
situation, e. g., Ref. [232, 233]. There were also papers tried to explain why
one experiments could observe the $\Theta^{+}$-pentaquark state while the
others would not, see, e. g., Ref. [234]. A similar situation with the other
multiquark states: H-dihyperon [235, 236], etc. [237, 238].

Figure 24: Production of multiquark systems (pentaquarks,H-dihyperons, etc.)
and deutrons on large angles via double diquark-quark scattering [224, 205] in
nucleon collisions.

Multiquark state production cross section depends from its structure and
production mechanism. For instance, the atomic or molecular structures lead to
different mechanisms of production and to different branching ratios of the
final states.

Production of multiquark systems with possible diquark structure [239, 240,
241, 242, 222] on large angles can be enhanced due to double diquark-quark
scattering [224, 205]. Inclusive and two-particle correlation studies with
production of light nuclei and antinuclei should help establishing two
production mechanism stages: multiparton scattering and fragmentation/fusion
process [224, 205] (Fig. 24).

So far, most of the negative pentaquark and H-dihyperon searches were
performed at small production angles. Therefore, SPD experiment provides an
uniqe opportunity to search such multiquark states at unexplored kinematic
domain.

###  9.6 Summary

To summarise this section:

  * •

SPD with study of the inclusive particle production and few-particle
correlations at different kinematic regions in $pp$-collisions at high
luminosity should reveal such quark correlations as diquark states in proton;

  * •

Multiparton distribution in proton can be studied via multiparton scattering,
which yield certain azimuthal two-particle correlations; In particular,
double-parton scattering should increase two-particle correaltions out of
back-to-back scattering region;

  * •

SPD can carry out investigations of the novel production mechanisms as double
quark-diquark scattering and the other multiparton scattering subprocesses,
which would lead to enhanced production of exotic multiquark resonance states
and light nuclei at large angles;

  * •

$pd$\- and $dd$-collisions at high luminosity can shed light on the difference
between multiquark fluctons and short-range nucleon correlations by measuring
cumulative production of antiprotons and $K^{-}$ mesons. Collision processes
with light nuclei, if available, should enhance sensitivity to the hard
flucton antiquark sea.

Therefore, SPD experiment at NICA has a unique opportunity, from one side, to
test and extend our knowledge of various aspects of multiquark correlations:
from diquark substructure of proton, double-parton scattering, the cold dense
baryon matter, to the exotic multiquark resonance production, and, from the
other side, it has a rather high potential for discovering new exotic
multiquark states and novel mechanisms of production at high energy hadronic
collisions.

##  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>
PACS: 24.85.+p, 13.88.+e, 13.85.Fb

###  10.1 Introduction

At present, there are different theoretical indications that cold nucleonic
matter under strong compression should undergo phase transitions, see, e.g.,
[243]. Experiments at NICA MPD will be devoted to the search for signs of
these transitions, in which the manifestations of the transformations of the
nucleon phase into quark-gluon one, as well as other transitions, is expected
to be seen in collisions of heavy nuclei. Unfortunately, traditional nuclear
physics, as well as astrophysical restrictions on the equation of state for
cold dense matter, do not yet give direct indications of the need to take into
account such processes [244]. A similar situation has arisen in the
description of small nucleon systems, which can be used as the basis for the
microscopic theory of phase transitions in nuclear matter. Thus, it has long
been suggested that various multibaryon states can exist in nature in the form
of a configurational admixture together with ordinary nuclei [245]. At the
same time, models continued to develop, suggesting that nucleons in nuclei can
also retain their individuality even in very exotic high-momentum states
[246]. An attempt to find experimental manifestations of the multiquark
component in the wave function of atomic nuclei was undertaken in the frame of
the flucton model[247, 248]. In parallel, the model of short-range
correlations according to which, to explain the same processes, the nucleon
models, taking into account the high-momentum component of the wave function
of nuclei, were also quite suitable [249].

Experiments on the NICA MPD facility should in the future answer some of the
questions associated with the problem of phase transitions of nucleonic matter
into other forms of nuclear substance at high densities, which can be achieved
in collisions of heavy nuclei accelerated to relativistic velocities. The
theory of these phenomena is extremely difficult, so that the interpretation
of the obtained experimental data may turn out to be highly uncertain. In this
regard, it is of particular interest to study the transitions of nucleonic
matter to other states, which can occur in few-nucleon systems. Drawing an
analogy with the theory of the formation of an electron-ion plasma, it can be
argued that, in this respect, knowledge of the characteristics of few-nucleon
systems under conditions of strong compression can contain information as
indispensable as knowledge of the ionization characteristics of atoms for
building a theory of ordinary plasma.

###  10.2 Search for new dibaryons at the NICA SPD facility

There are a number of theoretical and experimental indications of the
existence of light dibaryon excitations in 2-nucleon systems [250, 251, 252].
An analysis of the experimental data suggests the possible observation of
events, the cross sections of which in a certain kinematic region are
comparable to the cross sections of hard elastic nucleon-deuteron collisions.
For example, in [252] a peak was observed, which is interpreted to date as a
quasi-free knockout by the target deuteron of a nucleon from the incident
deuteron in the reaction d $+$ d $\to$ d $+$ X. However, its large width can
hardly be explained only by the internal motion of nucleons in the deuteron
being destroyed. At the same time, kinematic analysis shows [250] that
scattering processes with the participation of virtual dibaryons preexisting
in the incident deuteron in a virtual form before the collision can also
contribute to the observed peak. In experiments [251], where the effective
masses of two-nucleon systems formed after the emission of several (up to 5)
pions from highly excited neutron-proton systems were measured, there are
indications of the observation of narrow dibaryons, the masses of which can be
described by a simple formula [250] $M_{n}=M_{d}+10.08n$ MeV, where
$M_{d}=1875.6$ MeV is the value of deuteron mass232323Hereinafter, the system
of units is used, in which $c=1$.. It can hardly be interpreted as a mere
coincidence that the broadening of the peak caused by the ‘‘quasi-free’’
nucleon knockout in the experiment [252] can also be explained [250] by
scattering d $+$ d $\to$ d $+$ d ∗ with the participation of the same
dibaryons as in the experiment [251]. Of course, these arguments can only be
considered as indirect evidence in favor of the existence of light dibaryons.
On the other hand, it must be emphasized that the criticism of Troyan’s
experiment presented in [253] also does not look convincing enough. Firstly,
in [251] a larger number of events were studied than in [253]; secondly, in
[253] , events with a poorer ‘‘cooling’’ of a highly excited dibaryon were
used, i.e. the work is based on events with fewer secondary pions emitted by a
highly excited dibaryonic system. And finally, even in spite of all that,
significant variations in the measured cross sections were observed in [253],
which can hardly be explained by simple statistical fluctuations. The above
facts suggest that light dibaryons could have been present in a sufficiently
large number already in some of the earlier experiments, but were not
recognized due to insufficient spectral resolution of the corresponding
experimental facilities. Drawing an analogy with the history of discoveries
associated with the invention of the microscope, we can say that a medium that
can contain a large number of interesting microscopic objects is presumably
known, but a device that allows them to be clearly seen has not yet been
created. Below we shall consider the demands that would have to be advanced to
an experimental setup so that it could be used to obtain unambiguous answers
to all the above questions.

In experiments on colliding beams of deuterons with momenta of the same
modulus, the energy $q_{0}$ transferred in the reaction d + d $\to$ d + d ∗
from one of the deuterons to the dibaryon being produced is

${q_{0}}=\frac{{M_{*}^{2}-M_{d}^{2}}}{{4{E}}},$ (35)

where ${E}=\sqrt{P^{2}+M_{d}^{2}}$ is the deuteron energy in the laboratory
coordinate system (l.c.s.). For the $n-$th excited level of the dibaryon
${M_{*}}={M_{d}}+n\varepsilon$. Here $\varepsilon$ is the distance between the
levels, which we will take equal to 10 MeV [250]. The relation (35) allows one
to estimate the influence of the instability of the beam energy $E$ on the
accuracy of determining the value of $\varepsilon$. For this purpose, let us
bring into correspondence the spread in energy $\Delta E$ of deuterons in
beams and the inaccuracy of setting the level number $0<\delta<1$, which will
describe the deviation of the measured level energy from its exact value,
referred to $\varepsilon$. Substitution of $E\to E+\Delta E$ and $n\to
n+\delta n$ into the relation (35) gives a quadratic equation for $n$, which
determines the highest level for which the uncertainty in the distance between
adjacent levels is does not yet exceed the value of $\delta$:

$\frac{{\Delta
E}}{E}=\frac{{2{M_{d}}+(2n+\delta)\varepsilon}}{{2{M_{d}}n+{n^{2}}\varepsilon}}\delta.$
(36)

For $n\varepsilon\ll{M_{d}}$, this relation is reduced to a directly
proportional relationship between the relative fluctuation of the beam energy
and the relative error in measuring the distance between adjacent levels,
caused by these fluctuations:

$\delta\approx\frac{{\Delta E}}{E}n.$

It turns out to be rather weak (so far, the influence of the beam momentum
spread on the accuracy of measuring the $M_{*}$ dibaryon mass is not taken
into account). For example, for $\Delta E=100$ MeV, the error in determining
the energy of the tenth level of a dibaryon is only 1 MeV for the momentum of
colliding beams at the level of 10 GeV in l.c.s.

It is also useful to have the correspondence between the 4-momentum $q$
transferred to the nascent dibaryon, which can be measured experimentally, and
the relativistic Mandelstam variable $t\equiv q_{0}^{2}-\mathbf{q}^{2}.$ One
can make sure after elementary but cumbersome calculations that for the
experiments with colliding beams the following simple formula holds:

$t=2(E{q_{0}}-P{q_{\parallel}}).$ (37)

Here $q_{\parallel}$ is longitudinal, i.e. directed along the beams′ axis, the
component of the momentum transfer, which is directly responsible for the
excitations. It can be seen from (37) that the processes of dibarion
production from a kinematic point of view are possible at both negative and
positive, as well as zero values of the varibale $t$, which seems to be
extremely unlikely. It is clear that the dynamic theory of the processe should
be based on the relativistic-invariant amplitude using the variable $t$, since
the fact of dibarion production does not depend on the frame in which it is
registered. In the absence of such a theory, the answer to the question of
whether the dibaryons under consideration are produced, for example, ar zero
values of the variable $t$, can only be given by experiment. Since the
relativistic-invariant amplitude also depends on the Mandelstm variable $s$,
the search for the region of localization of dibaryons should also be
accompanied by a variation in the momentum of deuterons in colliding beams.

Traditionally, formula (35) is used to identify resonances, in which $M_{*}$
is the effective mass of decay products of a resonance. An individual
resonance can be seen if an error of its mass determining, which includes its
own width, and also takes into account the measurement errors of the momenta
of its decay products, is significantly less than distances to adjacent
levels. It is clear that, to recognize resonances, one can also use the
relationship (37), in which resonances are seen as separate lines. However, to
formulate the requirements for the experimental setup, the most appropriate is
a method of resonance recognition directly based on the quantities to be
measured. Indeed, resonances, as well as the degree of reliability of their
recognition, can also be seen on graphs of the dependence
$q_{\bot}(q_{\parallel})$ with a direct indication of the measurement errors
of longitudinal $q_{\parallel}$ and perpendicular $q_{\bot}$ components of the
momentum transfer. The exact analytical dependence
${q_{\bot}}({q_{\parallel}})$ for colliding beams with equal momenta has the
form

${q_{\bot}}({q_{\parallel}})=\sqrt{{{\left({\frac{{4{E^{2}}+M_{*}^{2}-M_{d}^{2}}}{{4E}}}\right)}^{2}}-M_{*}^{2}-{{(P-{q_{\parallel}})}^{2}}}.$
(38)

Figure 25: Dependence of the transverse momenta of light dibaryons on their
longitudinal momenta measured in MeV for an experiment with colliding beams
with $P=3$ GeV. The uppermost curve corresponds to the first excited level,
and the lowest one corresponds to the tenth level.

This dependence is shown in Fig. 25 for the supposed theoretical values of the
masses of dibaryons and $P=3$ GeV. In a real experiment, values of the momenta
of decay products of a dibaryon should be measured. For light dibaryons, these
are the momenta $\mathbf{p}_{n}$ and $\mathbf{p}_{p}$ of the secondary neutron
and proton. The transverse component of the momentum transferred to the
dibaryon is equal to the transverse component of the total momentum of the
secondary nucleons $q_{\bot}=p_{n\bot}+p_{p\bot}$, which is known with the
measurement accuracy $\Delta p_{n,p\bot}=\Delta p_{n\bot}+\Delta p_{p\bot}$.
In addition, the nonzero fluctuation of the transverse momentum of the beam
$\Delta P_{\bot}$ makes a contribution to the measurement error of the
transverse momentum transfer. Thus, the total measurement error of the
transverse component of the momentum transfer is calculated by the formula

$\Delta q_{\bot}=\Delta p_{n\bot}+\Delta p_{p\bot}+\Delta P_{\bot}.$

Similarly, taking into account that
$q_{\parallel}=P-(p_{n\parallel}+p_{p\parallel})$, we have

$\Delta q_{\parallel}=\Delta p_{n\parallel}+\Delta p_{p\parallel}+\Delta
P_{\parallel},$

where $\Delta P_{\parallel}$ is a value of the beam momentum fluctuations.

In paper [252], in which the light dibaryons could be observed at $t=$ -0.5
GeV2, interactions of deuterons with momentum $P=$ 8.9 GeV with a stationary
deuteron target were used. The colliding beam experiments at $|P|=$ 3 GeV
correspond to the deuteron momentum $P=$ 11.3 GeV in experiments with the
stationary target, which is close to the value in [252].

Using the formulas (35) and (37), it can be verified that the kinematics of
colliding beams corresponds to similar events at $q_{\parallel}=$ 100 $\pm$ 20
MeV. In its turn, the formula (38) tells us that at $q_{\parallel}=$ 100 MeV
the value of $q_{\bot}$ will be 755.8 MeV for elastic d-d scattering, 743.2MeV
for scattering with excitation of the first level, 730.3 MeV for scattering
with excitation of the second level, … 631.0 MeV – for the tenth level. In
this range, the distances between the levels grow as follows: 12.3, 12.6,
12.9, … 14.9, 15.3 MeV. Thus, it can be seen that the measurement accuracy of
the transverse momentum at a level of 3 MeV is quite enough to register all
the dibaryons described above.

To estimate the required accuracy of measuring the longitudinal component of
the momenta of the secondary particles, the dependence (38) should be
reversed, expressing $P-q_{\parallel}$ in terms of $q_{\bot}$. One can check
that the distances in the horizontal direction between the levels of dibaryons
shown in Fig. 25, in this case only slightly more than 3 MeV at full
longitudinal momentum $P-q_{\parallel}$ of the decay products at a level of
3000 MeV. This dictates very high requirements both for the measurement
accuracy of the longitudinal components of momentum of the secondary particles
and the degree of monochromaticity of the beam – at the level of $10^{-4}$.
Nevertheless, even in this respect, the experiments with colliding beams have
significant advantages in relation to experiments with the stationary target.
In the latter case, a similar experiment would require increasing relative
accuracy by a factor of 7, because of the larger momentum $P$ of the
accelerated beam and also because of a different analytical dependence
$q_{\bot}(q_{\parallel})$.

The requirements for the accuracy of measuring the longitudinal component of
the momentum transfer very fast decrease with decreasing momentum of the
colliding beams. So, if it is equal to 1 GeV, then the required relative
accuracy becomes equal to 10-2. Since the exact range of values of the
Mandelstam variable $s$, within which the production of light dibaryons is
possible, we do not know, then it makes sense to set up an experiment even at
lower than in the experiment [252] values of $s$.

Fig.25 suggests the possibility of searching for any dibaryons, including
heavier ones, the existence of which quark models predict. For this, it is
enough to use the fact of a distinct concentration events near some line
${q_{\bot}}({q_{\parallel}})$, described by formula (38), which should be very
similar to those shown in Fig.25.

##  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>
PACS: 21.80.+a, 21.30.Fe, 21.30.-x, 21.45.-v, 25.45.De

There have been significant progresses in the study of hypernuclei in both
experiment and theory during the past decades. The discovery of a number of
hypernuclei with $S=-1$ and $S=-2$ has greatly enriched our knowledge about
the hyperon-nucleon and hyperon-hyperon interactions [254, 255], and the
search for stable hypernuclei still serves as a direct probe for many
interesting physics with the presence of hyperons in nuclear matter.

So far, there is no evidence for a stable $(n,\Lambda)$ nor $(p,\Lambda)$
bound state for baryon number $A=2$, except for a resonance peak of about 2.06
GeV seen in $d+\pi^{-}$ [256]. The situation becomes more complicated in the
3-body or 4-body systems made of nucleons and hyperons. For $A=3$ in the
sector of isospin $I=1$, $\isotope[3][\Lambda]{He}=(p,p,\Lambda)$ and
$\isotope[3][\Lambda]{H}=(n,p,\Lambda)$ are found bound. The latter has a
3-body binding energy of $E_{3}=-2.45$ MeV, which is just below the threshold
of a deuteron plus an isolated $\Lambda$, i.e. $E_{2}=-2.20\,$ MeV. In
contrast, the existence of $\isotope[3][\Lambda]{n}=(n,n,\Lambda)$ still needs
confirmation [256].

For $A=4$ with $I=0$, the nonstrange tetraneutron, $\isotope[4]{n}=(n,n,n,n)$,
is naively suggested to exist by the stability of [8]He isotope, but so far
has received only controversial experimental indications [257]. Calculations
based on realistic neutron-neutron potentials [258, 259, 260, 261, 262] or
potentials made artificially deeper to produce a dineutron [2]n do not support
its stability due to the Pauli principle. To avoid the Pauli blocking effects
and benefit from the convening role played by the hyperon, it is thus
interesting to consider the stability of the doubly-strange tetra-baryon
system, $T=\isotope[4][\Lambda\Lambda]{n}=(n,n,\Lambda,\Lambda)$ with $S=-2$
and $I=1$. In particular, the ground state should favor the spin-singlet
assignments for both the $nn$ and the $\Lambda\Lambda$ pairs in order to
satisfy the antisymmetrisation.

In the sector of $A=4$ and $S=-2$, namely the ‘‘double $\Lambda$’’
hypernuclei, it includes the states, $\isotope[4][\Lambda\Lambda]{He}$,
$\isotope[4][\Lambda\Lambda]{H}$, and $\isotope[4][\Lambda\Lambda]{n}$. Note
that the ‘‘Nagara’’ event [263], has set a limit on the $\Lambda\Lambda$
effective attraction in $\isotope[6][\Lambda\Lambda]{He}$ with a precise
measurement of its binding energy. This may provide some constraints on
lighter $S=-2$ systems [264, 265, 266], and in particular hypernuclei with
$A=4$.

The $\isotope[4][\Lambda\Lambda]{H}$ was studied in Ref. [267] by Filikhin and
Gal, and it was found to be unbound within the models adopted. However, with a
more sophisticated method for solving the four-body problem, the calculation
by Nemura, Akaishi and Myint [268] found a small amount of binding below the
threshold of dissociation into $\isotope[3][\Lambda]{H}+\Lambda$. As stressed
in Refs. [267, 268, 264], the interaction of $\Lambda\Lambda\leftrightarrow
N\Xi\leftrightarrow\Lambda\Lambda$, which is sizeable in free $\Lambda\Lambda$
scatterings, will be suppressed in a dense nucleus due to the
antisymmetrisation between the nucleons in the core and the nucleon in $N\Xi$.
Such a Pauli suppression effect was invoked to explain the relatively weak
binding of $\isotope[6][\Lambda\Lambda]{He}$. However, since it requires a
$(n\Lambda\Lambda)$ correlation it should not play a significant role in the
limit of weak binding.

It is no doubt that the question of stability of the $A=4$ double-$\Lambda$
hypernuclei would be crucial for our understanding of the role played by
hyperons in nuclear matter. While it is still controversial for model
calculations of such a four-body problem in the regime of weak binding, we
would propose that some general properties arising from the weakly binding
systems involving the 2-body and 3-body bound-state energies may provide a
guidance for a possible stability of [4][ΛΛ]n. Meanwhile, we propose a
sensitive reaction process for the search for [4][ΛΛ]n in deuteron-deuteron
scatterings, i.e. $d+d\to K^{+}+K^{+}+\isotope[4][\Lambda\Lambda]{n}$, which
is accessible at NICA. After all, it would rely on the experimental study to
decide the dedicate dynamics for such an exotic system.

As follows, we will first demonstrate that [4][ΛΛ]n may exist given the
satisfaction of Thomas condition, where [4][ΛΛ]n is likely to be a
‘‘Borromean’’ system. We then propose and discuss its production mechanism in
$d+d\to K^{+}+K^{+}+\isotope[4][\Lambda\Lambda]{n}$. Advantages of using the
double $K^{+}$ production channels to probe exotic hypernuclei and exotic
hadrons are also addressed. A brief summary is given in the end.

###  11.1 Binding conditions for 3 and 4-body systems with strangeness $-1$
and $-2$

The Thomas condition is referred to the relation between the range of nuclear
forces and the ratio of 3-body to 2-body bound-state energies which was
discovered by Thomas in 1935 [269]. Namely, it shows that the ratio of 3-body
to 2-body bound-state energies, $E_{3}/E_{2}$, becomes very large if the range
of the interaction decreases. In such a case one may have a deep 3-body
binding of $E_{3}/E_{2}\to\infty$ for a given (short) range if the coupling
$g$ approaches (from above) the minimal value $g_{2}$ required by the 2-body
binding. Here, the coupling $g$ is defined by the potential energy $g\,\sum
v(r_{ij})$, where $v$ accounts for the attractive parts of the potential, and
$r_{ij}$ denotes the inter-particle separation. Implication of the Thomas
condition is that the minimal coupling $g_{3}$ to bind three particles is
smaller than $g_{2}$. Therefore, it allows a coupling value of
$g_{3}<g<g_{2}$, which will lead to a ‘‘Borromean’’ 3-body binding system.
Namely, the 3-body system become bound but its two-body subsystems are
unbound.

The rigorous boundaries on the allowed domain of coupling constants for the
Borromean system have been studied in the literature [270], and the Borromean
window as a function of the potential shape [271] or of the low-energy
parameters of the pair interactions [272] have been investigated. In Ref.
[273] it was shown that such a Borromean binding for 3-body (such as
$nn\Lambda$) and 4-body (such as $nn\Lambda\Lambda$ systems may exist based on
some general features implemented for the 2-body interactions.

While the detailed discussions on the Thomas conditions can be found in Refs.
[273, 274], we only outline the key points which are relevant to the study of
the $nn\Lambda\Lambda$ system. First, we adopt some simple potentials for the
2-body interaction:

$\displaystyle-g\,\exp(-\mu\,r)\qquad\text{(exponential)}~{},$ (39)
$\displaystyle-g\,\exp(-\mu\,r)/r\quad\text{(Yukawa)}~{},$ (40) $\displaystyle
g\exp[-2\,\mu\,(r-R)]-2\,g\,\exp[-\mu\,(r-R)]\ \text{(Morse)}~{},$ (41)

where for the Morse potential, we use $R=0.6$ for the illustration purpose.

Table 4: Values (in fm) adopted for the scattering length and effective range parameter in the two models. Pair | ESC08 |  | CEFT
---|---|---|---
a | $r_{\text{eff}}$ |  | a | $r_{\text{eff}}$
$nn$ | -16.51 | 2.85 |  | -18.9 | 2.75
$(n\Lambda)_{s=0}$ | -2.7 | 2.97 |  | -2.9 | 2.65
$(n\Lambda)_{s=1}$ | -1.65 | 3.63 |  | -1.51 | 2.64
$\Lambda\Lambda$ | -0.88 | 4.34 |  | -1.54 | 0.31

We then solve the 2-body problem to reproduce the deuteron binding energy, and
$nn$, $n\Lambda$ and $\Lambda\Lambda$ scattering lengths and effective ranges
extracted by the Nijmegen-RIKEN model [254, 275, 254] and the chiral effective
field theory (CEFT) [276, 277]. The values for the scattering lengths and
effective ranges from these two models are listed in Table 4. It shows that
these two models have similar results for the $nn$ and $n\Lambda$
interactions, but they produce quite different values for the $\Lambda\Lambda$
interaction. In particular, the effective range of $\Lambda\Lambda$ is
remarkably small, and for a given scattering length, this eases the occurrence
of Borromean binding involving a pair of $\Lambda$. However, in a more
advanced CEFT study [278], the same group found a larger value for $r_{\rm
eff}$, and this modifies the conclusions for the 3-body and 4-body systems at
the edge of binding. Hopefully, some new experimental results, e.g., from
final-state correlations in heavy-ion collisions [279] would allow for a
better tuning of the models. The main results can be outlined as follows:

  * •

The model reproduces the observed binding of the [2]H and [3][Λ]H systems. For
[3][Λ]H both spin $s=1/2$ and $s=3/2$ are found bound since there is no much
difference between the spin-triplet and the spin-averaged nucleon-hyperon
interactions.

  * •

We find that [3][Λ]H with isospin $I=1$ and spin $s=1/2$ is marginally
unbound. However, it may become bound if some masses increase by about 10%.
Namely, the unequal masses between nucleon and hyperon will bring more
binding.

  * •

We also find that $\isotope[3][\Lambda]{n}$ is marginally unbound. Our results
for $\isotope[3][\Lambda]{n}$ agree with the conclusions of the recent studies
[280, 281, 282].

  * •

The state [4][ΛΛ]H with isospin $I=0$ is found weakly bound (about 3 MeV) in
the Nijmegen-RIKEN model, and slightly more (about 9 MeV) in the CEFT one. The
state [4][ΛΛ]H with isospin $I=1$ and [4][ΛΛ]n deviate from binding by a very
small amount with the Nijmegen-RIKEN parameters, but become bound by about 1
MeV with the CEFT parameters.

The above analysis is based on some general properties arising from few-body
systems. In particular, the satisfaction of the Thomas condition is crucial
for the stability of [4][ΛΛ]n as a Borromean system. However, it should be
noted that the detailed dynamics for the nucleon-hyperon and hyperon-hyperon
interactions will decide whether [4][ΛΛ]n could exist at all. At this moment,
there are still significant discrepancies between some of the most popular
models. For instance, as shown in Table 4, the $\Lambda\Lambda$ scattering
length and effective range determined by the Nijmegen-RIKEN model and CEFT
model turn out to be signficantly different. It indicates that experimental
constraints on the $\Lambda\Lambda$ interaction is desired.

Another point needs to be addressed is that so far the 3-body forces have not
been considered. Whether or not they contain an attractive component would be
crucial for the stability of close-to-binding systems. In case that some spin-
dependence of the 3-body forces can play a significant role, it would keep the
spin $s=1/2$ state of [3][Λ]H bound and move the $s=3/2$ in the continuum. It
is also possible that the 3-body and $n$-body forces with $n>3$ contain a
short-range repulsive component. This is due to the Pauli exclusion of the
constituent quarks when several hyperons (or several hyperons and nucleons)
overlap within a small distance. The repulsive component seems to be necessary
in large systems containing strangeness [283]. While the calculations of
hyperon-nucleon and hyperon-hyperon forces should be pushed to higher order
within theoretical models, experimental search for these strangeness $-2$
hypernuclei would provide crucial constraints on the model parameters.

###  11.2 Production mechanism for $\isotope[4][\Lambda\Lambda]{n}$ and
advantages of double $K^{+}$ productions

We turn to the possible experimental search for
$\isotope[4][\Lambda\Lambda]{n}$ and propose a production mechanism which can
be accessed at SPD NICA. In Ref. [273] we have shown that the deuteron-
deuteron collisions around the energy region above $E_{cm}\simeq 5.2$ GeV is
favored to produce $\isotope[4][\Lambda\Lambda]{n}$ with the total cross
section of about 2.5 nb. Here, based on the same analysis we try to clarify
some key points and make a rough estimate of its production rate at the
kinematics of SPD NICA.

As mentioned earlier, the quantum numbers of the ground state
$\isotope[4][\Lambda\Lambda]{n}$ will favor $J^{P}=0^{+}$, where the neutron
pair and $\Lambda$ pair have spin 0, namely, their spins are anti-parallel,
respectively. Meanwhile, the total isospin is $I=1$. Thus, the total
wavefunction of the ground state is anti-symmetric under the interchange of
the two nucleons or the two $\Lambda$. In principle, one has to construct a
dynamic wavefunction for the $(n,n,\Lambda,\Lambda)$ system, which is a
nontrivial work and strongly model-dependent due to the unknown
$\Lambda\Lambda$ interactions. But for the purpose of making an estimate of
the production rate, we can simply introduce a momentum distribution for the
$n\Lambda$ clusters for the $(n,n,\Lambda,\Lambda)$ system [273].

The most ideal reaction for producing $\isotope[4][\Lambda\Lambda]{n}$ should
be $d+d\to K^{+}+K^{+}+T$ which is an extremely clean process since the
background processes involving the $K^{+}K^{-}$ productions become irrelevant.
It makes the measurement of the missing mass spectrum recoiling against the
$K^{+}K^{+}$ pairs sensitive to the existence of any pole structure in the
$nn\Lambda\Lambda$ system. The transition matrix element can be expressed as

$\displaystyle{\cal M}$ $\displaystyle=$
$\displaystyle\int\psi_{T}^{*}(\boldsymbol{p}_{1}^{\prime},\boldsymbol{p}_{2}^{\prime},\boldsymbol{p}_{3}^{\prime},\boldsymbol{p}_{4}^{\prime};\boldsymbol{P}_{T}^{\prime})\psi^{*}_{K_{1}}(\boldsymbol{P}_{K_{1}})\psi^{*}_{K_{2}}(\boldsymbol{P}_{K_{2}})\hat{{\cal
O}}(\boldsymbol{p}_{1}^{\prime},\boldsymbol{p}_{3}^{\prime},\boldsymbol{p}_{1},\boldsymbol{p}_{3},\boldsymbol{P}_{K_{1}},\boldsymbol{P}_{K_{2}})$
(42) $\displaystyle\times$
$\displaystyle\psi_{d_{1}}(\boldsymbol{p}_{1},\boldsymbol{p}_{2};\boldsymbol{P}_{d_{1}})\psi_{d_{2}}(\boldsymbol{p}_{3},\boldsymbol{p}_{4};\boldsymbol{P}_{d_{2}})\delta(\boldsymbol{P}_{T}^{\prime}+\boldsymbol{P}_{K_{1}}+\boldsymbol{P}_{K_{2}}-\boldsymbol{P}_{d_{1}}-\boldsymbol{P}_{d_{2}})$
$\displaystyle\times$
$\displaystyle\delta(\boldsymbol{p}_{1}+\boldsymbol{p}_{2}-\boldsymbol{P}_{d_{1}})\delta(\boldsymbol{p}_{3}+\boldsymbol{p}_{4}-\boldsymbol{P}_{d_{2}})\delta(\boldsymbol{p}_{1}^{\prime}+\boldsymbol{p}_{2}^{\prime}+\boldsymbol{p}_{3}^{\prime}+\boldsymbol{p}_{4}^{\prime}-\boldsymbol{P}_{T}^{\prime})$
$\displaystyle\times$
$\displaystyle\delta(\boldsymbol{p}_{2}-\boldsymbol{p}_{2}^{\prime})\delta(\boldsymbol{p}_{4}-\boldsymbol{p}_{4}^{\prime})d\boldsymbol{p}_{1}d\boldsymbol{p}_{2}d\boldsymbol{p}_{3}d\boldsymbol{p}_{4}d\boldsymbol{p}_{1}^{\prime}d\boldsymbol{p}_{2}^{\prime}d\boldsymbol{p}_{3}^{\prime}d\boldsymbol{p}_{4}^{\prime}\
,$

where the kinematic variables are defined in Fig. LABEL:fig:reac. Note that
Fig. LABEL:fig:reac illustrates one of the leading transitions favored by the
[4][ΛΛ]n production in the central deuteron-deuteron collisions.

Figure 26: (Color online) Mechanism for $T$ production in $dd$ collisions.

fig:reac

In Fig. LABEL:fig:cross the total cross section is estimated by the leading
transition process of Fig. LABEL:fig:reac. As the leading-order test the
relative internal momenta between the proton and neutron inside the incident
deuterons has been neglected. Namely, the Fermi motion inside the initial
deuterons is neglected. Also, the final-state interactions among the final-
state baryons are overlooked, which means that the neutrons are treated as
spectators and their contributions to the amplitude will be via the
convolution of the final-baryon momentum distributions. The $S_{11}(1535)$
resonance is included in the transition amplitude which is found to be
relatively small, mainly due to the smaller couplings to $\pi N$ and
$K\Lambda$. Since other processes with intermediate $N^{*}$ excitations may
contribute, our estimate including only the Born term and $S_{11}(1535)$
excitations can be regarded as a conservative estimate of the production cross
section for $\isotope[4][\Lambda\Lambda]{n}$.

Figure 27: (Color online) Total cross section for $d+d\to
K^{+}+K^{+}+(n,n,\Lambda,\Lambda)$). From upper to lower the curves stand for
the total cross sections of the full calculations, exclusive process from the
nucleon Born terms, exclusive process from the double $S_{11}(1535)$
excitations, and exclusive process from the one Born transition and one
$S_{11}(1535)$ excitation.

fig:cross

In Fig. LABEL:fig:missingmass the $K^{+}K^{+}$ missing mass spectra for the
recoiled $(n,n,\Lambda,\Lambda)$ at different energies are shown above the
production threshold. The peak position is located at the four baryon
$nn\Lambda\Lambda$ threshold since only a momentum distribution for [4][ΛΛ]n
is considered. However, our estimate is sufficient to demonstrate the behavior
of the correlated system recoiled by the $K^{+}$ pair. For uncorrelated
$K^{+}K^{+}$ events, i.e. the final-state $(n,n,\Lambda,\Lambda)$ are not
bound, there would no peak in the missing mass spectrum.

Figure 28: (Color online) The missing mass spectra for the recoiled
$(n,n,\Lambda,\Lambda)$ at different energies above the production threshold
in $d+d\to K^{+}+K^{+}+(n,n,\Lambda,\Lambda)$

fig:missingmass

The c.m. energy at SPD NICA starts at $E_{cm}=6.7$ GeV with a luminosity of
$L=10^{27}\mbox{cm}^{-2}\mbox{s}^{-1}$. We estimate that the total cross
section at $E_{cm}=6.7$ GeV will drop about one order of magnitude compared
with the peak value of about 2 nb. Thus, the events expected in one-year
runtime are

$N=\sigma_{total}\times L\times t=0.2\mbox{nb}\times
10^{27}\mbox{cm}^{-2}\mbox{s}^{-1}\times 1\ \mbox{year}\simeq 6.3\ ,$ (43)

which is a small event counting. It could be even smaller taking into account
the detection efficiency which generally will reduce the event counting by one
order of magnitude. However, if the luminosity can reach $10^{29}$, which is
an approximate average between the lower limit of
$10^{27}\mbox{cm}^{-2}\mbox{s}^{-1}$ and the upper limit of $2\times
10^{30}\mbox{cm}^{-2}\mbox{s}^{-1}$, the event counting can be significantly
increased:

$N_{m}=\sigma_{total}\times L\times t=0.2\mbox{nb}\times
10^{29}\mbox{cm}^{-2}\mbox{s}^{-1}\times 1\mbox{year}\simeq 630\ ,$ (44)

which is sufficient for establishing the state. For the highest luminosity,
one would expect about 12000 events in one-year runtime. Even though the
detection efficiency will reduce the events, there will be tens to hundreds of
events to count.

It shows that the double $K^{+}$ production channel has great advantages for
the study of hypernuclei and exotic hadrons. Apart from the proposed process,
$d+d\to K^{+}+K^{+}+(n,n,\Lambda,\Lambda)$, it is also interesting to look at
the proton-proton collisions, $p+p\to K^{+}+K^{+}+\Lambda+\Lambda$, where the
missing mass spectrum of $K^{+}K^{+}$ also provides a clean and direct way to
search for the di-baryon $\Lambda\Lambda$, or to study the $\Lambda\Lambda$
interactions.

For the proton-deuteron collisions, the double $K^{+}$ channel is $p+d\to
K^{+}+K^{+}+n+\Lambda+\Lambda$. The recoiled part of the double $K^{+}$ is
$n\Lambda\Lambda$. It is also unknown whether such an exotic system, i.e. $H$
di-baryon, can exist or not. A direct measurement of such a system would
provide rich information about both $\Lambda\Lambda$ and $n\Lambda$
interactions. Nevertheless, notice that the final states have access to the
$nK^{+}$ invariant mass spectrum. The exclusive measurement of this process
can also tell whether the light pentaquark state $\Theta^{+}(1540)$ exists or
not.

###  11.3 Summary

The stability of $\isotope[3][\Lambda]{n}$ and, more likely, of
$T=\isotope[4][\Lambda\Lambda]{n}$ is within the uncertainties of our
knowledge of the baryon-baryon interaction. Although many effects need to be
considered in order to refine the predictions, we propose that direct
experimental evidence would be extremely useful for model constraints and a
better understanding of the hyperon dynamics within nuclear matter. We also
emphasize the advantages of the double $K^{+}$ production channels as a probe
for topical exotic hypernuclei and exotic hadron studies. The SPD NICA
facility would be the ideal place for such an experimental effort in the
future.

##  12 Problems of soft $pp$ interactions 262626This section is written by A.
Galoyan and V. Uzhinsky.

PACS: 24.10.Lx, 13.85.Ni, 14.20.-c

Description of proton spectra in hadron-nucleon interactions is a long
standing problem of high energy physics. By tradition, three reggeon
phenomenology is used for its solution at $x_{F}\rightarrow$ 1 (see Ref.
[284]). Especially, it is assumed that the three pomeron graph (PPP) is
responsible for the high mass diffraction dissociation. The non-vacuum reggeon
– two pomeron’s graph (RPP) is connected with so-called low mass diffraction
dissociation. The three non-vacuum reggeon’s graph (RRR) gives a contribution
at $x_{F}\geq$ 0.8. A.B. Kaidalov and O.I. Piskunova [285] proposed a method
of the spectra description in the central region in the framework of the
Quark-Gluon-String model. Up to now various Monte Carlo models cannot describe
the spectra sufficiently well [286]. All of them are used the LUND
fragmentation scheme [287] for a treatment of decays of quark-diquark strings.
An example of the description is presented in Fig. 1 where Pythia 6.4 [288]
and Geant4 FTF model’s predictions are shown.

The Geant4 FTF model [289] considers RRR graphs. As seen, the Pythia model
gives abnormal humps at $y\sim$ 1.7 and 2.3 at $P_{lab}=$ 158 and 400 GeV/c,
correspondently. Thus, detailed experimental data at low energies, where the
diffractions are dominant processes, are highly desired for a development of
the theoretical models.

The most impressive experimental data on $pp$ interactions were presented in
the last decade by the NA61/SHINE collaboration for $\sqrt{s_{NN}}=$ 6.2, 7.6,
8.8, 12.3 and 17.3 GeV [290]. It is pitty, but the data are not sufficiently
detailed. Before there were data at $P_{lab}=$ 12 and 24 GeV/c [291]. They
were based on low statistics according to a modern point of view.

On the whole, we can say that the FTF model of Geant4 toolkit describes quite
well the multiplicities and kinematical spectra of produced particles in
proton-proton interactions in a wide laboratory energy range from 1 GeV up to
1000 GeV.

Figure 29: Distributions of $\pi^{\pm}$ and protons on rapidity in the center-
of-mass system of $pp$ interactions at $P_{lab}$= 158 and 400 GeV/c. Points
are experimental data [290, 292] without systematical errors. Solid and dashed
curves are FTF and Pythia 6.4 model calculations.

Another problem, recognized by us, was observed in $<$$P^{2}_{T}$$>$ – $x_{F}$
correlations in $pp$ interactions. The NA61/SHINE collaboration did not
provide corresponding experimental data for the correlations, though, they can
be extracted in principle. The NA49 collaboration presented the needed data
[293, 294, 295] only at the page –
http://spshadrons.web.cern.ch/ppdata.html. The data are very different from
analogous data by the LEBC-EHS collaboration [292] at $\sqrt{s_{NN}}=$ 27.5
GeV. The models cannot describe the last data quite well.

Figure 30: $<P^{2}_{T}>$ – $x_{F}$ correlations in $pp$ interactions at
$\sqrt{s_{NN}}$=17.3 GeV. Points are the NA49 data [293, 294, 295]. Solid and
dashed lines are FTF and Pythia 6.4 model calculations.

As seen in Fig. 2, the models describe the general behaviour of the data
except the correlations for protons. Predictions of the models are close to
each other and to the data for $\pi^{+}$, $\pi^{-}$, $K^{+}$ and $K^{-}$
mesons at $\sqrt{s_{NN}}=$ 17.3 GeV. The FTF model overestimates the
correlations for protons at $x_{F}\geq$ 0.2, and does not reproduce the hump
at $x_{F}\sim$ 0\. The Pythia model gives the hump, but it is essential lower
than the experimental data. In other regions of $x_{F}$, the Pythia model does
not describe the shape of the data. At higher energy, the situation becomes
worse for the proton correlation. This shows that both models have a problem
with baryon production in $pp$ interactions.

It is interesting to look at evolution of the correlations with energy. In
Fig. 3 we present $<P^{2}_{T}>$ – $x_{F}$ correlations at various energies. As
seen, there is a smooth evolution, though there is a change of process
contributions with energy in the FTF model. Quark exchange processes and one-
vertex diffractions dominate at low energies. At higher energies, the
diffraction is stay on the same lavel, but an yield of non-diffractive
processes is increased.

Figure 31: $<P^{2}_{T}>$ – $x_{F}$ correlations in $pp$ interactions at
$\sqrt{s_{NN}}$=3.2, 6.3, 8.8 and 17.3 GeV (lines from bottom to top). Points
are the NA49 data [293, 294, 295]. Lines are FTF model calculations.

In order to clarify a nature of the correlations, we propose to study two-
particle $P_{T}$ correlations in soft interactions. The correlation can be
also studied by current experiments at RHIC and the LHC.

As known, two-particle correlations were intensively used in the past at study
of particles and jets in high energy physics (see, for example, [296, 297,
298, 299, 300]). At the present time, there are well developed methods of jet
recognition [301, 302] and analysis of thier properties. More complicated
situation takes place at sufficiently lower energies, especially for nucleus-
nucleus interactions, when hard jets cannot be produced. Though, some methods
of high energies can be adjusted for lower energies. Instead of jets with high
$P_{T}$, one can choose as a trigger a particle of any type in an event. An
associated particle can be choosen in the same manner in the event, where the
trigger paticle is produced.

In general, two-particle $P_{T}$ correlation function can be determined as:

$C(\vec{P}^{tr}_{T},\vec{P}^{as}_{T})=\frac{1}{N_{tr}}\ \frac{d\
N(tr,as)}{d^{2}P^{tr}_{T}\ d^{2}P^{as}_{T}},$ (45)

where $\vec{P}^{tr}_{T}$ is a transverse momentum of a trigger particle.
$\vec{P}^{as}_{T}$ is a transverse momentum of an associated particle.
$N_{tr}$ is a number of the trigger particles. $N(tr,as)$ is a number of pairs
– trigger particles and associated particle having pre-determined values of
$\vec{P}^{tr}_{T}$ and $\vec{P}^{as}_{T}$.

The function $C$ is a function of 4 independent variables. Though, accounting
azimuthal symmetry of interactions of unpolarized particles there must be only
3 independent variables. We propose to use as the variables the module of
transverse momentum of the trigger particle ($|\vec{P}^{tr}_{T}|$), and 2
projections of the vector $\vec{P}^{as}_{T}$ on the direction of the vector
$\vec{P}^{tr}_{T}$, and on the direction perpendicular to $\vec{P}^{tr}_{T}$
(see Fig. 4). In Fig. 4 we choose $\Lambda$ as a trigger particle, and
$K$-meson or $\pi$-meson as associated particles.

Fig. 4

$P^{as}_{T,L}\ =\ \vec{P}^{as}_{T}\cdot\vec{P}^{tr}_{T}/|\vec{P}^{tr}_{T}|,$
(46) $P^{as}_{T,T}\ =\
|\vec{P}^{as}_{T}\otimes\vec{P}^{tr}_{T}|/|\vec{P}^{tr}_{T}|$ (47)

Instead of calculations of scalar and vector products, one can use the
following method: Determine the azimuthal angle of flying of the trigger
particle – $\phi=\arctan{P^{tr}_{T,y}/P^{tr}_{T,x}}$. Make the Euler transform
of the coordinate system, and find new components of $\vec{P}^{as}_{T}$.

$\displaystyle P^{as}_{T,x^{\prime}}=\ P^{as}_{T,x}\cdot\cos{\phi}\ +\
P^{as}_{T,y}\cdot\sin{\phi},$ (48) $\displaystyle
P^{as}_{T,y^{\prime}}=-P^{as}_{T,x}\cdot\sin{\phi}\ +\
P^{as}_{T,y}\cdot\cos{\phi}.$ (49)

It is obvious, that the new components of $\vec{P}^{tr}_{T}$ will be
$P^{tr}_{T,x^{\prime}}=|\vec{P}^{tr}_{T}|$ and $P^{tr}_{T,y^{\prime}}=0$. In
the following, we will omit subscripts "$T$", and apostrophes of $x$ and $y$
for the new components of vectors.

Trigger and associated particles can be chosen in various rapidity/pseudo-
rapidity windows. Types of the particles can or cannot coincide. It is useful
to consider also the correlation functions integrated on one or two variables.

Let us consider a connection of the functions with transverese momentum
generation mechanism (see Fig. 5). Let us take a baryon ($\Lambda$) as a
trigger particle. The transverse momentum of the baryon ($\vec{P}^{\Lambda}$)
is a sum of the transverese momentum of diquark ($\vec{P}_{qq}$) after
previouse fragmentation steps and the transverse momentum of $s$-quark
($P_{s}$) produced from the vacuum: $\vec{P}^{\Lambda}=\vec{P}_{qq}\ +\
\vec{P}_{s}$. The transverse momentum of an associated hadron
$\vec{P}^{as}=-\vec{P}_{s}\ +\vec{P}_{q}$. $\vec{P}_{q}$ is a transverse
momentum of a quark produced at the next fragmentation step from the vacuum.

5.5

Fig. 5

According to the Schwinger’s model [303] implemented in the LUND fragmentation
model [287] distributions of $\vec{P}_{s}$ and $\vec{P}_{q}$ are Gaussian
ones. Assume for simplicity, that distribution of $\vec{P}_{qq}$ has also
Gaussian shape.

In the case, a probability density of the process is given by the expression:

$W=\left[\frac{1}{\pi\ \sigma_{1}}\
e^{-\vec{P}_{qq}^{2}/\sigma_{1}}\right]\left[\frac{1}{\pi\ \sigma_{2}}\
e^{-\vec{P}_{s}^{2}/\sigma_{2}}\right]\left[\frac{1}{\pi\ \sigma_{2}}\
e^{-\vec{P}_{q}^{2}/\sigma_{2}}\right]$

$\propto\exp(-\vec{(}P^{tr})^{2}/\sigma_{1}\ -\ \vec{(}P^{as})^{2}/\sigma_{2}\
+\ \frac{\sigma_{1}\ \sigma_{2}}{2\sigma_{1}\ +\
\sigma_{2}}(\vec{P}^{tr}/\sigma_{1}-\vec{P}^{as}/\sigma_{2})^{2}).$

Choosing $x$-axis direction along the vector $\vec{P}^{tr}$, we have:

$W(\vec{P}^{tr},\vec{P}^{as})\ \propto\
\exp\left(-\frac{\sigma_{1}+\sigma_{2}}{\sigma_{2}(2\sigma_{1}+\sigma_{2})}\left[P^{as}_{x}+\frac{\sigma_{2}}{\sigma_{1}+\sigma_{2}}P^{tr}\right]^{2}-\frac{\sigma_{1}+\sigma_{2}}{\sigma_{2}(2\sigma_{1}+\sigma_{2})}(P^{as}_{y})^{2}\right)$
(50)

Using Exp. 50, one can obtain for average values of projections of vector
$\vec{P}^{as}$ on $x$ and $y$ axes the following expressions:

$<P^{as}_{x}>=-\frac{\sigma_{2}}{\sigma_{1}+\sigma_{2}}P^{tr},$ (51)
$<(P^{as}_{y})^{2}>=\frac{1}{2}\frac{2\sigma_{1}+\sigma_{2}}{\sigma_{1}+\sigma_{2}}\cdot\sigma_{2}.$
(52)

It is useful to consider the following cases:

$1)\ \ \sigma_{1}=0,\ \ \ <(P^{as}_{y})^{2}>=\sigma_{2}/2,\ \ \ \
<P_{as,x}>=-P_{tr}$

$2)\ \ \sigma_{1}=\sigma_{2},\ <(P^{as}_{y})^{2}>=\frac{3}{4}\sigma_{2},\ \ \
<P^{as}_{x}>=-\frac{1}{2}P^{tr}$

$3)\ \ \sigma_{2}\simeq 0,\ \ \ <(P^{as}_{y})^{2}>\simeq 0,\ \ \ \ \ \
<P_{as,x}>\simeq 0$

In the first case, we assume that diquarks have no transverse momentum. The
second case corresponds to the assumptions that average momenta of quarks and
diquarks are equal. The last one shows the obvious results, when transverse
momenta of sea quarks are equal zeros.

Let us test the method on events generated by Pythia 6.4 and Geant4 FTF models
at $\sqrt{s_{NN}}=$ 10 GeV. We chose for the testing $\Lambda$-hyperons as
trigger particles, and considered anti-$\Lambda$ hyperons, $K$-mesons and
$\pi$-mesons as associated particles. We expected that there will be an
essential difference between the model preductions due to the difference in
the baryon production mechanisms. As seen in Fig. 6, it is so. We present in
Fig. 6 $<P^{as}_{x}>$ as functions of triggered $\Lambda$ momentum for various
associated particles.

Figure 34: $<P^{as}_{x}>$ as functions of triggered $\Lambda$ momentum for
associated particles – $\bar{\Lambda}$, $K^{0}$, $K^{+}$, $K^{-}$, $\pi^{+}$
and $\pi^{-}$, figures a, b, c, d, e and f, respectively, in $pp$ interactions
at $\sqrt{s_{NN}}=$ 25 GeV. Open and filled symbols are Pythia and FTF
calculation results, respectively.

As seen, first of all, the correlations are approximately linear functions of
$P^{tr}$, as it was expected. A large deviation between the model results
takes place for $\Lambda$ correlations with $\bar{\Lambda}$, $K^{0}$, $K^{+}$
and $\pi^{-}$ mesons. The models give close predictions for $\Lambda$
correlations with $K^{-}$ and $\pi^{+}$ mesons. In general, the FTF model
predicts strongest correlations between baryons and anti-baryons. Other strong
correlations are between triggered $\pi$ mesons and baryons. The correlations
are weaker in the Pythia model. There is practically no correlations between
$\Lambda$ and $\bar{\Lambda}$ hyperons in the Pythia model.

According to the FTF model, there is an evolution of the correlations from
strong to less strong ones in the energy range $\sqrt{s_{NN}}=$ 3 – 15 GeV. It
is connected with dying out of the quark exchange processes in the FTF model.
At higher energies the correlations become "frozen". A next step in the
function’s evolution can be at copious gluon production.

Correlations of $<(P^{as}_{y})^{2}$ with $P^{tr}$ give a possibility of a
direct check of the Schwinger’s mechanism. The correlations are presented in
Fig. 7.

As seen, the correlations are practically constant. The model predictions are
close to each other for $\pi^{\pm}$ and $K^{\pm}$ mesons. There is an
essential difference between the predictions for $\bar{\Lambda}$ hyperons and
$K^{0}$ mesons.

Figure 35: $<(P^{as}_{y})^{2}>$ as functions of triggered $\Lambda$ momentum
for associated particles – $\bar{\Lambda}$, $K^{0}$, $K^{+}$, $K^{-}$,
$\pi^{+}$ and $\pi^{-}$, figures a, b, c, d, e and f, respectively, in $pp$
interactions at $\sqrt{s_{NN}}=$ 25 GeV. Open and filled symbols are Pythia
and FTF calculation results, respectively.

The correlations evolve in the energy range $\sqrt{s_{NN}}=$ 3 – 15 GeV in the
FTF model. We believe all of the considered correlations can be studied at the
SPD experiment.

##  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).

PACS: 12.39.-x, 12.40.-y

Over more than three decades, a question about the mysterious origin and
properties of soft photons (SP), has been lacking a comprehensive
understanding. A few experimental groups found out an increased yield of
photons in the region of low transverse momentum, $p_{T}$, smaller than 50
MeV/$\it c$ [304, 305, 306, 307, 308, 309, 310, 311, 312]. These photons are
not the decay products of short-living particles (including resonances).

According to QCD, at high energies, $q\overline{q}$, $qg$ and $gg$
interactions lead to the emission of photons called direct photons. In spite
of the fact that the majority of photons in high energy interactions comes
from decays of secondary hadrons (neutral mesons and others), the direct
photons present a unique opportunity to study the soft gluon component of a
nucleon and a stage of hadronisation. Direct photons interact with the
surrounding medium only electromagnetically as opposed to the strong
interaction of hadrons. So SP keep more information about the medium at all
stages of the interaction. This is especially valuable, along with the
information about the secondary hadrons.

The first convincing experiment on the detection SP has been carried out at
the Big European Bubble Chamber (BEBC) [304]. The study of SP production in
$K^{+}p$ interactions at 70 GeV/c beams of $K^{+}$-mesons after the
subtraction of all known hadron decays indicated their excess compared to what
was expected from QED inner bremsstrahlung.

The next series of experiments have been conducted at higher energies [305,
306, 307, 308, 309]. The excess of the SP yield was fourfold or even higher,
eightfold. The last European experiment has been realised at the SPS
accelerator, CERN by the DELPHI Collaboration [310]. The yield of SP as a
function of the neutral pion number turned out to be completely unexpected
[311]. The excess of the inner bremsstrahlung rates over the predicted ones as
a function of the neutral multiplicity in the quark jet turned out to be
seventeen-fold for the largest recorded number of pions $N_{neu}$ = 6. The
muon bremsstrahlung photons in the reaction e+e- $\to$ Z0 $\to\mu^{+}\mu^{-}$
has demonstrated a good agreement of the observed photon rate with predictions
from QED for the muon inner bremsstrahlung [312].

Our BGO calorimeter has been installed at the NIS-GIBS setup at an angle of
16∘ relative to the beam direction, registering the energy release of gamma
quanta. The Monte Carlo simulation has also been carried out in the conditions
of the last assembly and with the beam energy 3.5 $A\,$GeV. Spectra of photon
energy release in deuterium-carbon and lithium-carbon (see [313]) interactions
have been obtained. In the region of energy release below 50 MeV, a noticeable
excess over the Monte-Carlo simulation is observed.

There are a few phenomenological models which have been worked out to describe
the SP spectra [314, 315, 316, 317]. The most attractive among them is the
model of the cold quark-gluon plasma of L. Van Hove [314]. But until now, a
complete comprehensive understanding of the nature of the anomalous yield of
SP has not been achieved.

###  13.1 The scientific program of SP study

In accordance of the gluon dominance model (GDM) design [317, 318]
multiparticle production is described as the convolution of two stages. The
first stage is described in QCD as a quark-gluon cascade. For the second one,
the phenomenological scheme of hadronization is applied. GDM evidences that
the sources of secondary particles are gluons that we call active, and an
abundance of soft gluons can be the sources of SP. They are picked up by newly
born quarks with a subsequent dropping of energy by emission of SP:
$g+q\to\gamma+q$ or $q+\bar{q}\to\gamma$. At that valence quarks are staying
in the leading particles. Our estimations of the emission region of SP (its
linear size) for $pp$ $\to$ hadrons + photons at U-70 in case of a near-
equilibrium state at using the black body radiation exceeds the typical
hadronisation region (1 fm) and reaches a value of about 4–6 fm [318]. We
consider the soft gluon component of protons is relevant for the understanding
of the spin structure of nucleons.

The formation of the pionic (Bose-Einstein) condensate in the region of high
total multiplicity (N $>>$ ${<N>}$, $N=N_{ch}+N_{0}$, ${<N>}$ – the average
multiplicity of charged and neutral pions) can be related with the exceeding
yield of SP [315]. The growth of the scaled variance
$\omega^{0}~{}=~{}({<N_{0}^{2}>}-{<N_{0}>}^{2})/{<N_{0}>}$ will be the signal
of pionic condensate formation [319].

The increasing yield of photons at low energy gives us the opportunity to
calculate two particle correlations of direct photons. Similar results were
obtained by the WA98 Collaboration [320]. The deviation from theoretical
predictions is at the level of a fraction of a percent.

The next item of our scientific program appeared thanks to the RHIC
experiment, namely with using the variable flow or v2. There is an interesting
prediction in [321] about the growth of v2 in the region of small $p_{T}$ of
photons. This dependence can be the evidence of a coherent SP emission. Our
program proposes to test this behaviour.

The scientist from the US Cheuk-Yin Wong [316] develops an intriguing model of
open string QED mesons to explain and describe the $p_{T}$-spectrum of SP.
Wong believes that since $q$ and $\bar{q}$ cannot be isolated, the intrinsic
motion of the $q\bar{q}$ system in its lowest-energy state lies predominantly
in 1+1 dimensions as in the open string with $q$ and $\bar{q}$ at its two
ends. Extrapolating into the $q\bar{q}$ QED sector in which $q$ and $\bar{q}$
interact with the QED interaction, he finds an open string QED meson state at
17.9 $\pm$ 1.5 MeV and QED meson state at 36.4 $\pm$ 3.8 MeV. These predicted
masses of the isoscalar and isovector QED mesons are close to the masses of
the hypothetical X17 [322] and E38 [323] particles observed recently, making
them good candidates for these particles. This hypothesis has generated a
great deal interest [324].

The decay of these particles can show up as an excess of $e^{+}e^{-}$ and
$\gamma\gamma$ pairs in the SP phenomenon [316]. Wang shows that an
astrophysical object consisting of a large assembly of QED mesons such as the
X17 particle with a mass $m_{X}$ = 17 MeV will be an electron-positron and
gamma-ray emitters. If the temperature of such an assembly is low, it can form
a Bose-Einstein condensate. Such assemblies of QED mesons present themselves
as good candidates as $e^{+}e^{-}$ emitters, gamma-ray emitters, or a part of
the primordial cold dark matter.

###  13.2 The preparation to experimental SP study

To carry out our scientific program we have to manufacture a space saver
electromagnetic calorimeter (ECal) with the capacity of low energy gamma
quanta detection . As it is known the production of homogeneous crystalline
ECal’s is expensive. Usually physicists prefer making heterogeneous assemblies
[325]. They are cheaper and at that possess satisfactory properties. Our
activity was aimed at testing of two types of ECal’s. The ‘‘spaghetti’’ type
was the first prototype that we produced and irradiated with photon beams in
Germany [313].

We substituted light scintillator material for a very dense (heavy) crystal
having a high specific light yield [325]. It allowed us to build a device that
is capable of maintaining considerable compactness (space-saving of about
30$\%$). To produce the ‘‘spaghetti’’ type we have chosen a mono-crystal of
gadolinium-gallium garnet, Gd3Al2Ga3O12:Ce (GaGG), as a scintillator and
tungsten+copper composite by way of the absorber. We expect the decay time to
be $\sim$ 90 ns; the light yield to be $\sim$ 45,000–55,000 ph/MeV; an
estimated price of 25–35 $\$$/cm3 of volume, and a good radiation resistance.

GaGG crystal is a fast-acting scintillator, its light yield is 4 times greater
than that of BGO crystals. And we can acquire these crystals from the well-
known domestic firm ‘‘Fomos-Materials’’. The Monte-Carlo simulation has shown
that the necessary energy resolution for SP in the case of the ECal
‘‘spaghetti’’ type turns out lower than when at using the ‘‘shashlik’’ type.
See figures in [313]) for an example of a typical view of the ECal
‘‘shashlik’’ type.

We plan to assemble the ECal ‘‘shashlik’’. It is assumed that it will consist
or 16 or more of GaGG plates (100$\times$100$\times$3)mm3, 15 plates of 2-mm-
absorber (W:Cu composite, 1:19), and its total length will be about 138 mm.
GEANT4 simulation shows that the energy resolution of the ‘‘shashlik’’ is
considerably better than ‘‘spaghetti’’ type. The dimensions of both assemblies
were optimised by the Monte-Carlo simulation for obtaining the required
characteristics. The design of the ‘‘shashlik’’ allows expanding it up to any
length. We also can exclude the W/Cu absorber from the ‘‘shashlik’’ and then
to assemble the homogeneous device for the detection of MeV photons if the
energy resolution of the ECal will not be of a good enough quality.

We would like to thank all participants of SP study activity for a long
years.and also the leadership of our laboratory for significant support.

##  14 Hadron formation effects in heavy ion collisions 303030This section is
written by A. B. Larionov; E-mail<EMAIL_ADDRESS>
PACS: 25.75.-q; 25.75.Bh; 25.75.Dw

Hadrons produced in a hard exclusive particle-nucleon collisions emerge first
in a form of prehadrons having a reduced transverse size $\sim 1/\sqrt{Q^{2}}$
where $Q^{2}$ is a hard scale. For exclusive meson electroproduction $eN\to
e^{\prime}N^{\prime}M$ ($M=\pi,\rho$), the hard scale is the photon virtuality
(cf. Ref. [326]). In the case of large-angle hadronic scattering $hN\to
h^{\prime}N^{\prime}$ with $-t\simeq-u\simeq s/2$, the hard scale is given by
the momentum transfer, i.e. $Q^{2}=-t$, (cf. Ref. [327] and refs. therein).
The prehadrons, called also the point-like configurations (PLCs), interact
with nucleons with reduced strength which is known as the color transparency
(CT) phenomenon, see Ref. [328] for the most recent review of CT.

In inclusive production channels of the $NN$ collisions, there is no fixed
hard scale, even at large $\sqrt{s}$. Thus, formation of PLCs is questionable
here. However, the hard scale will fluctuate event-by-event and one can expect
that in average the transverse size of the outgoing hadrons will be still
reduced. This motivates inclusion of the hadron formation effects in
microscopic transport models for high-energy heavy ion collisions, such as
UrQMD [329], HSD [330], GiBUU [331]. 323232Based on the CT concept, also the
incoming nucleons are transversely ’squeezed’ if they are in the initial state
of a hard collision. This effect is totally neglected in transport models.

Hadron formation studies in the current fragmentation region [332] based on
the Deep Inelastic Scattering (DIS) data from HERMES and EMC concluded the
linear increase of the prehadron-nucleon cross section with time (or length
traveled by a prehadron), in agreement with the quantum diffusion model (QDM)
[137]. In the QDM, the prehadron is converted to the ’normal’ hadron after
passing the distance called formation (or coherence) length

$l_{h}\simeq\frac{2p_{h}}{|M_{h}^{2}-M_{h^{\prime}}^{2}|}\sim
0.4-0.6\mbox{(fm/GeV)}\cdot p_{h}\mbox{(GeV)}~{},$ (53)

where $p_{h}$ is the momentum of hadron $h$, $M_{h}$ is the mass of the hadron
$h$, and $M_{h^{\prime}}$ is roughly the mass of the closest radially excited
state $h^{\prime}$. Thus, based on the QDM one can estimate that particles
with momenta above $\sim 10$ GeV leave the nucleus almost without
interactions. On the other hand, the analyses of E665 data on the low-energy
($E<10$ MeV) neutron production in the 470 GeV muon DIS on Pb target [333,
334] favor a surprisingly low cutoff value of the momentum $\sim 1$ GeV above
which the prehadrons do not interact in the nucleus. This can not be explained
by the QDM or any other existing model of hadron formation and calls for the
new experiments on hadron formation in the target fragmentation region, e.g.
in virtual photon - nucleus collisions at the future Electron-Ion Collider
(EIC) or in ultraperipheral heavy ion collisions at the LHC and RHIC [334].
The physics of target fragmentation is one of perspectives for the
experimental program of the EIC [335].

Another opportunity to study hadron formation effects is offered by heavy ion
collisions at NICA-SPD regime ($\sqrt{s}_{NN}\sim 10$ GeV). Here, the stopped
nucleon in the $NN$ center-of-mass system (c.m.s.) has
$x_{F}=m_{N}/\sqrt{s}_{NN}\sim 0.1$. Therefore, slow nucleons in the $NN$
c.m.s. represent a mixture of the current ($x_{F}\ll 1$) and target
($x_{F}\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\scriptstyle<}}{{\sim}}$}1$)
fragmentation products. This paper addresses the sensitivity of heavy ion
collision observables to the different treatments of hadron formation. The
calculations are performed within the Giessen Boltzmann-Uehling-Uhlenbeck
(GiBUU) model.

In sec. 14.1, the GiBUU model is briefly described with a focus on hadron
formation. Sec. 14.2 contains the results of numerical calculations of the the
rapidity- and $p_{t}$ spectra of protons and pions in C+C and Ca+Ca collisions
at $\sqrt{s}_{NN}=11$ GeV. The summary and estimation of rates at NICA-SPD are
given in sec. 14.3.

###  14.1 The model

The GiBUU model (see detailed description in ref. [331]) solves the system of
kinetic equations for the baryons ($N$, $N^{*}$, $\Delta$, $\Lambda$,
$\Sigma,\ldots$), corresponding antibaryons ($\bar{N}$, $\bar{N}^{*}$,
$\bar{\Delta}$, $\bar{\Lambda}$, $\bar{\Sigma},\ldots$), and mesons ($\pi$,
$K,\ldots$). The kinetic equations are coupled via collision integrals and
mean field potentials. The latter are determined self-consistently, i.e. they
depend on the actual particle distributions in the six-dimensional phase space
of position and momentum. The distribution function is projected onto the set
of point-like test particles. Kinetic equation is then solved if the test
particle positions and momenta propagate in time according to the Hamiltonian
equations of motion between collisions. 333333In the present calculations we
disregard mean field potentials. Thus, the test particles propagate along
straight-line trajectories. The two test particles experience a two-body
collision during the time interval $[t-\Delta t/2;t+\Delta t/2]$ if they
approach their minimal distance $d$ in position space during this time
interval and if $d<\sqrt{\sigma_{\rm tot}/\pi}$. Here, $\Delta t$ is the time
step in the computational frame that is chosen to be the nucleus-nucleus
c.m.s. and $\sigma_{\rm tot}$ is the total interaction cross section of
colliding particles.

The model also includes three-body collision processes $\pi NN\to NN$ and
$\Delta NN\to NNN$ which are simulated by the direct calculation of the three-
body absorption rate. Unstable resonances experience decays with a probability
$P=\Gamma\Delta t$ where $\Gamma$ is the total decay width of a resonance. The
final states of the two- and three-body collisions and resonance decays are
sampled by the Monte-Carlo method according to the partial channel cross
sections, widths and angular distributions. Pauli blocking is taken into
account for the final state nucleons.

For low-energetic elementary binary collisions, the resonance model
complemented with empirical background cross sections (e.g. $s$-wave direct
pion production $NN\to NN\pi)$ is implemented. For such collisions, it is
assumed that all produced particles are formed immediately and interact with
usual hadronic cross sections.

High-energy elementary binary collisions are simulated by the PYTHIA and
FRITIOF models. (The latter is applied for antibaryon-baryon collisions only.)
The corresponding transition values of $\sqrt{s}$ for meson-baryon, baryon-
baryon, and antibaryon-baryon collisions are $2.2,4.0$ and 2.38 GeV. Note that
in the narrow $\sqrt{s}$ region $\pm(0.1-0.2)$ GeV centered at the transition
value, the low- and high-energy events are mixed to ensure a smooth behaviour
of the cross sections between the two energy regimes.

The particles produced in a high-energy binary collision are supposed to be in
a prehadronic state. Their effective interaction cross section $\sigma_{\rm
eff}$ with nucleon becomes time-dependent and deviates from the ordinary
interaction cross section $\sigma_{0}$. We applied the three different models
for the ratio of the effective and ordinary interaction cross sections:
(i) Based on JETSET-production-formation points (see ref. [336], used as
default in GiBUU) favored by analysis of hadron attenuation at HERMES and EMC
[332]:

$\sigma_{\rm eff}(t)/\sigma_{0}=X_{0}+(1-X_{0})\frac{t-t_{\rm prod}}{t_{\rm
form}-t_{\rm prod}}~{},$ (54)

where $X_{0}=r_{\rm lead}a/Q^{2}$, $a=1$ GeV2, $r_{\rm lead}$ is the ratio of
the number of leading quarks to the total number of quarks in the prehadron.
$Q^{2}$ is defined for the hard $2\to 2$ subprocess by the PYTHIA variable
VINT(52). By default, it gives $Q^{2}\sim p_{\perp}^{2}$ where $p_{\perp}$ is
the transverse momentum of the hard scattering (see PYTHIA 6.4 manual [337]
for detail).
(ii) QDM [137]:

$\sigma_{\rm eff}(t)/\sigma_{0}=X_{0}+(1-X_{0})\frac{c(t-t_{\rm
hard})}{l_{h}}~{}.$ (55)

Here we set $X_{0}=0$ for simplicity. $t_{\rm hard}$ is the time moment of the
collision in GiBUU calculation.
(iii) Momentum cutoff:

$\sigma_{\rm eff}/\sigma_{0}=\Theta(p_{\rm cut}-p_{h})~{},~{}~{}~{}p_{\rm
cut}\sim 1-2~{}\mbox{GeV/c}.$ (56)

Some comments are in order with regard to the Lorentz covariance. In
principle, all three above models are explicitly non-covariant. Thus, one
needs to specify the frame in which they are applied. In the case of models
(i) and (ii) the non-covariance, however, should largely cancel-out in the
ratios of time- or length differences. The model (iii) is most problematic: it
is clear that the result will certainly depend on the frame in which the
particle momentum is calculated. Note that originally Eq.(56) has been
suggested for the target nucleus rest frame [334] where DIS is calculated in
GiBUU. In contrast, in the present work, the nucleus-nucleus c.m.s. is used as
the calculational frame. The problem related to the choice of the
calculational frame is beyond the scope of this work and should be addressed
in future studies.

###  14.2 Numerical results

The calculations were performed for the systems C+C and Ca+Ca at
$\sqrt{s}_{NN}=11$ GeV. The maximum impact parameter was set to $8$ fm for C+C
and to $11$ fm for Ca+Ca that corresponds to the minimum bias trigger. The
time evolution of the colliding system was calculated until the maximum time
30 fm/c with the time step $\Delta t=0.2$ fm/c. The total accumulated
statistics is $2\cdot 10^{6}$ ($1\cdot 10^{6}$) nucleus-nucleus collision
events for C+C (Ca+Ca).

Figure 36: Proton and charged pion rapidity distributions in C+C and Ca+Ca at
$\sqrt{s}_{NN}=11$ GeV as indicated. Dot-dot-dashed (magenta) line –
calculation without formation (immediately formed hadrons), Solid (black) line
– default formation (i). Dot-dashed (red) line – QDM model (ii). Dashed (blue)
line – $p_{\rm cut}=2$ GeV/c (iii). Dotted (brown) line – $p_{\rm cut}=1$
GeV/c (iii). Here (i), (ii) and (iii) refer to the models defined in the text,
see Eqs.(54),(55),(56), respectively.

fig:dNdy

Fig. LABEL:fig:dNdy shows the rapidity spectra of protons and pions. Note that
we do not distinguish free protons from those bound in nuclear clusters. This
explains the maxima in the proton rapidity spectra at the target and
projectile rapidities. Calculation without hadron formation gives the largest
production at midrapidity ($y=0$). Hadron formation leads to the reduced FSI
of the outgoing hadrons. Hence, the produced protons lose less longitudinal
momentum which results in the depletion of the proton spectrum at midrapidity.
Since less FSI implies less inelastic production by secondary protons, also
the pion spectrum gets depleted at midrapidity in calculations with hadron
formation. As expected, the depletion is stronger for heavier system and for
calculations with more restrictions on FSI ($p_{\rm cut}=1$ GeV/c).

Figure 37: Proton and charged pion transverse momentum distributions in C+C
and Ca+Ca at $\sqrt{s}_{NN}=11$ GeV as indicated. C.m.s. rapidity cut
$|y|<0.5$ is applied. Line notation is the same as in Fig. LABEL:fig:dNdy.

fig:dNdpt

Fig. LABEL:fig:dNdpt shows the proton and charged pion transverse momentum
spectra at midrapidity. The proton $p_{t}$ spectrum becomes softer due to
hadron formation. Indeed, protons effectively gain transverse momentum in the
rescattering processes while hadron formation suppresses rescattering. Pion
$p_{t}$ spectrum is influenced by hadron formation only very weakly. In the
collisions of heavier nuclei [338] the hadron formation effect becomes more
pronounced, especially for central events.

###  14.3 Summary and conclusions

The purpose of this work was to clarify whether heavy ion collisions can be
used to test various models of hadron formation. To this end, the GiBUU
transport model was used that includes the three optional models of hadron
formation. The first two models, (i) and (ii) [Eqs.(54),(55)] are based on the
CT mechanism and differ only in the details of the hadron formation length.
Both these models were successful in the description of hadron attenuation in
DIS reactions. The third model (iii) [Eq.(56] assumes that the prehadrons with
momenta above 1-2 GeV/c do not interact with nucleons which is in stark
contrast with widely accepted momentum dependence of the hadron formation
length, Eq.(53). This model was originally adopted to describe the low-energy
neutron production induced by the muon DIS on nuclear targets [334].

The GiBUU model calculations were performed for inclusive proton and charged
pion production in C+C and Ca+Ca at $\sqrt{s}_{NN}=11$ GeV. In the C+C system,
the secondary rescattering effects are minimal and the reaction process is
governed by first-chance $NN$ collisions. The Ca+Ca system is more suitable
for the study of FSI effects. Proton and charged pion rapidity distributions
and proton $p_{t}$ distributions are strongly sensitive to the hadron
formation effects. More restrictions on FSI of prehadronic states leads to
less production at midrapidity and to softer proton $p_{t}$ spectra.

In the C+C system the proton and $\pi^{\pm}$ production cross sections in the
c.m.s. rapidity window $|y|<0.5$ are about 0.4 b and 3 b, respectively. In the
Ca+Ca system the corresponding cross sections are about one order of magnitude
higher. Thus, the measurements of the proton and charged pion spectra seems
easily possible at the first stage of NICA-SPD operation ($L=10^{25}$ cm-2
s-1) with the production rate of about 1-10 events per second (without
efficiency and acceptance corrections). The accuracy needed to separate
different models of hadron formation should be better than $\sim 10\%$ for the
production cross sections at midrapidity which is reachable within few hours
of beam time.

##  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>
PACS: 24.10.Lx, 13.85.Ni, 14.20.-c

The probability of the process with a particular parton is described by the
function, which is dependent on the fraction of the total momentum of the
proton carried by that parton and the momentum transfer scale. These functions
are known as parton distribution functions (PDF) and contain information about
the internal structure of corresponding nucleons [339, 340]. Hence, for a
consistent description of processes with polarized nucleons in a wide energy
range and the most complete description of the proton structure a generalized
parton model was developed [341, 342, 343], which consist of 8 specialized
parton distribution functions. These functions are: the distribution of the
parton density in unpolarized nucleon (Density); distribution of the
longitudinal polarization of quarks in the longitudinally polarized nucleon
(Helicity); distribution of transverse polarization of quarks in transversely
polarized nucleon (Transversity); the correlation between the transverse
polarization of a nucleon and the transverse momentum of unpolarized quarks
(Sivers); the correlation between the transverse polarization of a nucleon and
the longitudinal polarization of quarks (Worm-gear-T); the distribution of the
transverse momenta of quarks in unpolarized nucleon (Boer-Mulders); the
correlation between longitudinal polarization of nucleon and the transverse
momenta of quarks (Worm-gear-L); the distribution of the transverse momenta of
quarks in transversely polarized nucleon (Pretzelosity).

At the current level of our knowledge of the nucleon structure, parton
distribution functions cannot be obtained analytically and have to be measured
experimentally. The deep inelastic scattering (DIS,SIDIS) and the Drell-Yan
(DY) processes are typically used for the measurements of PDFs [344, 345, 346,
347, 348, 349, 350, 351, 349, 352, 353, 354, 355, 356].

In the Drell-Yan processes, the annihilation of a quark and antiquark, from
colliding nucleons, through an intermediate photon or Z-boson leads to the
production of a lepton-antilepton pair [357, 349]. The matrix element of the
Drell – Yang process consists of a convolution of two tensors [358, 359, 349].
The first one is responsible for the hadron part with initial partons and
includes a dependence on the polarized parton distribution functions. The
second tensor corresponds to the lepton part of the process. Such a simple
structure and the fact that Drell-Yan process is one of the cleanest hard
hadron-hadron scattering processes makes it a very convenient tool for
studying PDFs.

The conventional way for studying parton distribution functions via Drell-Yan
process is to use processes with the production of electron-positron (and muon
anti-muon) pairs [349, 354, 356], because in this case the mass of leptons can
be neglected and one can consider them as having some definite helicity.
Within this approach angular parameters, which define the spatial orientation
of the momenta of leptons relative to the momenta of colliding nucleons, are
used as parameters describing the polarization state of the lepton pair. Then
one can measure azimuthal asymmetries and extract particular structure
functions from them.

In the case of $\tau$ production, this approach will work only at very high
energies. At lower energies, however, the mass of tau cannot be neglected.
Massive particles do not have definite helicity. Therefore, for $\tau$
leptons, the angular parameters as spin-sensitive variables can be used only
at collision energy much greater than the mass of the $\tau$ lepton and are
not suitable for energies comparable to its mass. Consequently, we need a more
suitable parameterization of the polarization state of $\tau$ lepton.

Our idea to tackle this problem is to utilize the unique decays properties of
$\tau$-lepton. Consider the hadronic decay of $\tau$ to a single charged
$\pi$-meson and neutrino. Due to the weak nature of this decay and the fact
that neutrinos are always left-handed, the energy spectra of $\pi$-meson is
strictly correlated with the polarization state of decaying $\tau$ lepton
[360, 361, 362, 363]. For instance, in the case of mostly right-handed $\tau$
decay, due to the angular momentum conservation law, most of the tau’s
momentum is transferred to the $\pi$-meson. On the contrary, in the case of
mostly left-handed $\tau$, the significant part of its momentum is transferred
to the neutrino. Thereby, the energy of single $\pi$-meson is a convenient
characteristic of the polarization state of its parent $\tau$. So, to
determine the polarization of $\tau$ leptons, we suggest to use hadron decays
of $\tau$ leptons to a single charged $\pi$-meson and neutrino. The diagram of
the corresponding process is shown in the Figure 38. We propose to use the
energy of single $\pi$ meson as variable that parametrize the polarization
state of $\tau$ leptons. A key feature of our method is that we do not
summarize the $\tau$ lepton polarizations, but keep information about the
polarizations of both $\tau$ leptons through the energies of single $\pi$
mesons from $\tau$ decays. Due to the fact that a pair of $\tau$ leptons is
produced at the NICA collider near the reaction threshold, depending on the
parton momenta, the $\tau$ leptons can be produced almost at rest or have a
significant momentum, which is reflected in the decay products of the $\tau$
leptons and makes the process with $\tau$ leptons are very sensitive to the
state of the spins and momenta of the initial partons and nucleons.

Figure 38: Drell-Yan process with the pair-production of tau leptons and their
subsequent decay into single charged $\pi$-meson and neutrino.

One of the closest opportunities for testing this approach is the future NICA
SPD experiment [364]. Since at the NICA collider a pair of $\tau$ is produced
near the reaction threshold, the leptons can be at rest or possess a
significant momentum, which affects the $\tau$ decay products and makes this
process very sensitive to the polarization states and momenta of the initial
hadrons.

To confirm the efficiency of our method, we carried out a detailed numerical
simulation. First, using the CompHEP [365] package, we computed the tree level
cross-section of the Drell-Yan $\tau$ pair production process. Also, we used
the $MadGraph5\\_aMC@NLO$ package [366] to estimate the cross section of the
same process at the next to the leading order in perturbation theory. In both
cases we have used the NNPDF3.1 PDF set [367]. As it has been mentioned, the
important advantage of the NICA facility is its energy range $\sqrt{s}=10-26$
GeV, which lies close to the $\tau$ pair production threshold. However, one of
the problems arising in calculating the cross-sections at such low energies is
the correct choice of the QCD factorization scale. At present, certain
ambiguity exists as to how to choose this scale properly. When dealing with
Drell-Yan process it is often suggested to choose the scale to be of the order
of the invariant mass of the lepton pair $Q$, since it is the most obvious
characteristic scale of the process. However, the important point given in the
aforementioned work is that the choice of the factorization scale $\mu_{F}=Q$
is not always lead to satisfactory results. The problem especially evident at
lower energies, which is the case for NICA. In [368] authors suggest using
scale $\mu_{F}\sim 0.5Q$. Other popular choices of the factorization scale
include the transverse energy $E_{T}$ and transverse mass $m_{T}$ of the
event. The results obtained using different types of scales at energy
$\sqrt{s}=24$ GeV shown in Table 5.

Order | LO
---|---
Scale | cross-section, $\sigma$ pb | MC error, $\Delta\sigma$ pb
$Q$ | 70.0 | 0.3
$Q/2$ | 84.8 | 0.4
$E_{T}$ | 75.7 | 0.4
$m_{T}$ | 70.7 | 0.3
Order | QCD NLO
Scale | cross-section, $\sigma$ pb | MC error, $\Delta\sigma$ pb
$Q$ | 83.5 | 0.7
$Q/2$ | 83.8 | 0.7
$E_{T}$ | 108.5 | 0.6
$m_{T}$ | 105.7 | 0.7
Table 5: Cross-sections of the Drell-Yan tau pair production at energy
$\sqrt{s}=24$ GeV calculated with different choices of factorization scale
$\mu_{F}$.

To assess the approximate birthrate of corresponding events in NICA
conditions, we utilize the known branching ratio of corresponding $\tau$ decay
channel: $Br(\tau^{-}\to\pi^{-}\nu_{\tau})=(10.83\pm 0.06)$ % according to
[369]. Table 6 contains the estimation of the number of events at two
energies, which correspond to the highest proposed luminosities.

$\sqrt{s}=24\text{GeV}(L=1.0\cdot 10^{32}\ \text{cm}^{-2}\text{s}^{-1})$
---
Lower cut on $M_{l^{+}l^{-}}$, GeV | 3.56 | 4
$\sigma_{pp\to\tau^{+}\tau^{-}}\cdot Br_{\tau^{-}\to\pi^{-}\nu_{\tau}}\cdot Br_{\tau^{+}\to\pi^{+}\bar{\nu_{\tau}}}$, pb | 1.00 | 0.71
Approximate number of events per 7000h | 2500 | 1800
$\sqrt{s}=26\text{GeV}(L=1.2\cdot 10^{32}\ \text{cm}^{-2}\text{s}^{-1})$
Lower cut on $M_{l^{+}l^{-}}$, GeV | 3.56 | 4
$\sigma_{pp\to\tau^{+}\tau^{-}}\cdot Br_{\tau^{-}\to\pi^{-}\nu_{\tau}}\cdot Br_{\tau^{+}\to\pi^{+}\bar{\nu_{\tau}}}$, pb | 1.22 | 0.88
Approximate number of events per 7000h | 3100 | 2200
Table 6: Estimation of the number of events
$pp\to\pi^{-}\nu_{\tau}\pi^{+}\bar{\nu_{\tau}}$ per year ($\sim 7000$h) of
data taking, assuming 100% efficiency of the detector.

The next important step is to demonstate the manifestation of the effect
discussed. In order to do so, we have carried out numerical simulation of the
full process $pp\to\pi^{-}\nu_{\tau}\pi^{+}\bar{\nu_{\tau}}$ with polarized
protons and different polarization of initial partons. Events were generated
in CompHEP package. For the simulation of the polarized processes, we have
used polarized PDF set NNPDFpol provided by NNPDF collaboration [370].
Currently, only the longitudinally polarized proton version of the PDF is
available. Generated events were transferred to Pythia 8 package [371] for
parton showering and hadronization. Finally, events were treated by Delphes
package [372] for detector simulation and reconstruction.

Figure 39: Energy spectra of $\pi$-mesons in unpolarized case.

Figure 40: Energy spectra of $\pi$-mesons produced in processes with different
polarization of the initial partons: left picture corresponds to the case of
left-handed quark and right-handed anti-quark, while right one corresponds to
the case of right-handed quark and left-handed anti-quark

The energy spectra of pions produced in the unpolarized case can be seen in
Figure 39. Both $\pi^{+}$ and $\pi^{-}$ mesons manifest a similar behaviour.
The picture becomes interesting in the case of polarized initial states. The
Figure 40 shows the energy spectra of pions produced in the processes with
different polarization of initial partons. The first figure corresponds to the
case of left-handed quark interacting with right-handed anti-quark while the
second one corresponds to the case of right-handed quark interacting with
left-handed anti-quark. As we can see, the energy spectra of the pions are
correlating with the polarization states of quarks in the way described in the
previous section. Moreover, the clear asymmetry is seen between the energy
spectra of $\pi^{+}$ and $\pi^{-}$ mesons, which can also be utilised as a
characteristic of the polarization state of the initial quarks.

The first numerical evaluations carried out show the potential of the new
method. In the next step, we will carry out simulations with different initial
combinations of longitudinally and transversely polarized protons. Next, we
will compile a detailed table of correspondence between the pair distributions
of $\pi$ mesons and various combinations of the initial polarizations of
protons and their constituent quarks.

Following this approach one can determine the polarization of interacting
partons by measuring energies of $\pi$-mesons produced in corresponding decays
of $\tau$-leptons. Considering the fact that pions are rather easy detectable
[373], the proposed approach is potentially very convenient and powerful tool
for studying nuclons PDF via tau production in polarised Drell-Yan process.

##  16 On Measuring Antiproton-Production Cross Sections for Dark Matter
Search 363636This section is written by R. El-Kholy; E-mail:
<EMAIL_ADDRESS>
PACS: 95.35.+d, 95.55.Vj, 96.50.sb

The dark matter (DM) problem is a long-standing puzzle in modern cosmology.
Even though DM is known to make more than 26% of the total energy-matter
content of the Universe[374], we still have no confirmed conclusions about its
identity. Evidence of DM is abundant and diverse, but mainly gravitational.
The effects of DM have been observed in the rotation curves of galaxies, the
mass discrepancy of galaxy clusters, and the lens-less gravitational lensing,
among others[375]. Perhaps most of our knowledge about DM properties was a
product of the _Bullet Cluster_ event, where two galaxy clusters collided with
one another. Astronomical observations and lensing analyses of the event
assured astronomers that DM particles are only weakly interacting[376]. Once
merely a tentative hypothesis, this has become widely accepted. Consequently,
among many DM candidates, Weakly Interacting Massive Particles (WIMPs) became
the most favored[377].

There are several search approaches applied in the case of DM. Each of them
has an underlying paradigm. The main three approaches are collider searches,
direct detection, and indirect detection. Collider searches are based on the
hypothesis that DM particles can be pair-produced in collisions of Standard-
Model (SM) particles, either directly or via a mediator. Direct detection
experiments try to measure the recoil energy of DM particles colliding with SM
nuclei. Indirect detection seeks to prove the hypothesis that DM particles
decay and pair-annihilate producing SM particles as final products[377]. Thus,
astrophysical searches try to accurately measure rare fluxes in Cosmic Rays
(CRs), in order to detect a secondary anomalous signature of DM. To identify
an anomalous signal, it is crucial to first subtract other ordinary
astrophysical sources. Secondary CR fluxes are produced in collisions of
primary CRs on Inter-Stellar Medium (ISM). In particular, secondary
antiprotons are produced in proton-nucleus, nucleus-proton, and nucleus-
nucleus collisions.

During the last two decades, two satellite-borne experiments, namely,
PAMELA[378] and AMS-02[379], have measured the secondary antiproton flux and
antiproton-to-proton ratio with unprecedented accuracy, covering a wide energy
range. The AMS-02 experiment has measured antiproton-to-proton ratio with an
accuracy higher than 95% for kinetic energies from 1 to 450 GeV[380]. However,
we still cannot come to firm conclusions about any exotic signal because the
AMS-02 measurements are still surrounded with several sources of uncertainties
as shown in Fig.41. These sources stem from uncertainties on: (i) the primary
spectra slopes at high energies, (ii) the solar modulation at lower energies,
(iii) the propagation parameters in the galactic environment, and (iv) the
antiproton-production cross sections. The first two can be relatively reduced
through new AMS-02 measurements. The third can be minimized via parameter
variation. However, the last of these sources is the most significant and
ranges between 20% and 50%, depending on the energy[381].

Figure 41: The combined total uncertainty on the predicted secondary
$\bar{p}/p$ ratio superimposed to the PAMELA and the AMS-02 data. Each
envelope accounts for uncertainties enclosed in it; such that the green band
is the total band of all four sources of uncertainty[381].

In addition to the scarcity of cross-section measurements of antiproton-
production, most available datasets date back to before 1980, and did not
account for hyperon decay to antiprotons[382]. Thus raising the need for new
measurements of antiproton-production cross sections. The kinematic phase-
space coverage required to catch up to the AMS-02 measurements has been
outlined[383]. Here, we affirm the capability of the Spin Physics Detector
(SPD), planned at the NICA collider currently under construction at JINR, to
make a sizable contribution to the desired measurements[384, 385].

###  16.1 Antiproton Production Cross Sections

The contribution of each interaction channel of primary CRs with ISM to the
production of secondary antiprotons depends on the abundances of different
nuclei in both CRs and ISM. Based on the abundances outlined in[384], the most
dominant channel of antiproton production would be proton-proton collisions,
followed by proton-helium, and proton-deuteron collisions, respectively. Other
nucleus-nucleus collisions have a negligible contribution. In addition to
different production channels, there are also different production mechanisms.
The dominant mechanism is the direct production. However, antiprotons are also
produced via decay of anti-baryons; namely, antineutrons, and the
$\bar{\Lambda}$ and $\bar{\Sigma}^{-}$ hyperons. The overall antiproton-
production cross section can be expressed as[386]

$\sigma_{\bar{p}}=\sigma_{\bar{p}}^{0}(2+\Delta_{\text{IS}}+2\Delta_{\Lambda}),$
(57)

where $\Delta_{\text{IS}}=\sigma_{\bar{n}}^{0}/\sigma_{\bar{p}}^{0}-1$ is the
isospin enhancement factor of antineutron direct production, and
$\Delta_{\Lambda}=\sigma^{\Lambda}_{\bar{p}}/\sigma_{\bar{p}}^{0}$ is the
hyperon factor, assuming
$\sigma_{\bar{n}}^{\Lambda}=\sigma_{\bar{p}}^{\Lambda}$.

The currently-available data on proton-proton collisions is extremely scarce.
It is graphically summarized in Fig.42, in terms of transverse momentum,
$p_{T}$, and the radial-scaling variable, $x_{R}$, which is given by[382]

$x_{R}=\dfrac{E_{\bar{p}}^{*}}{E_{\bar{p}.\text{max}}^{*}},$ (58)

where $E_{\bar{p}}^{*}$ is the antiproton energy, and
$E_{\bar{p}.\text{max}}^{*}$ is the maximal energy it can acquire, both in the
CM frame. The maximal antiproton energy can be obtained by[382]

$E_{\bar{p}.\text{max}}^{*}=\dfrac{s-8m_{p}^{2}}{2\sqrt{s}}.$ (59)

Moreover, most of these datasets date back to before 1980, and thus do not
include a hyperon-decay feed-down. As for datasets on other production
channels, they are essentially non-existent. The first ever proton-helium
dataset has been released by the LHCb collaboration in 2018[387].

Figure 42: The available data on $d^{3}\sigma_{pp\rightarrow\bar{p}}/dp^{3}$
in the $p_{T}-x_{R}$ space[382].

In addition to measuring the cross sections of direct antiproton production,
it is necessary to accurately estimate the hyperon factor, $\Delta_{\Lambda}$,
as well. The hyperon factor can be expressed as[386]

$\Delta_{\Lambda}=\frac{\bar{\Lambda}}{\bar{p}}\times\text{BF}_{\bar{\Lambda}\to\bar{p}\pi^{+}}+\frac{\bar{\Sigma}^{-}}{\bar{p}}\times\text{BF}_{\bar{\Sigma}^{-}\to\bar{p}\pi^{0}},$
(60)

where $\bar{\Lambda}/\bar{p}$ and ${\bar{\Sigma}}^{-}/\bar{p}$ are the hyperon
to promptly-produced antiproton ratios, and
$\text{BF}_{\bar{\Lambda}\to\bar{p}\pi^{+}}$ $=0.639\pm 0.005$ and
$\text{BF}_{\bar{\Sigma}^{-}\to\bar{p}\pi^{0}}=0.5157\pm 0.0003$ are the
branching fractions of the corresponding decays. While there are some existing
data on the ratio $\bar{\Lambda}/\bar{p}$, there is not any data on the ratio
$\bar{\Sigma}^{-}/\bar{p}$; and it is only estimated based on the symmetry
argument that $\bar{\Sigma}^{-}/\bar{\Lambda}=0.33$[388]. Thus, expression
(60) can be rewritten as

$\Delta_{\Lambda}=(0.81\pm 0.04)\times\bar{\Lambda}/\bar{p}.$ (61)

Even existing data on the ratio $\bar{\Lambda}/\bar{p}$ has a high energy-
dependent uncertainty ranging from 12% to 18%[386].

From all of the above, the need for new measurements, of the antiproton-
production cross sections and hyperon-to-antiproton production ratios, becomes
clear. The phase space that requires coverage in order to bridge the gap
between antiproton data and the AMS-02 measurements has been outlined. Fig.43
shows it in terms of transverse momentum and the radial-scaling variable,
where the areas within the contours need to be covered with uncertainty no
more than 3%, while the areas outside the contours require coverage with
uncertainty no more than 30%[383].

Figure 43: The kinematic range accessible to the SPD[385] superimposed on the
required measurement range to match the AMS-02 accuracy[383].

###  16.2 NICA SPD Contribution

The SPD detector is planned to operate at the NICA collider currently under
construction at JINR[389]. Polarized proton and deuteron beams will be used.
The $pp$ collisions are expected to reach $\sqrt{s}=27$ GeV, and a luminosity
of $5\times 10^{30}$ cm-2s-1 should be achieved[390]. Preliminary MC results
using PYTHIA8[391] show that the antiproton-production cross section in $pp$
collisions multiplied by the average antiproton-multiplicity in this energy
range is on the level of a few millibarns[384], corresponding to a production
rate $>10^{5}$ s-1. Thus, the measurements will not be hindered by statistical
uncertainty. The $4\pi$ angular acceptance of the SPD will also allow access
to a wider kinematic range, in comparison to fixed-target experiments. Fig.43
shows the kinematic range accessible to the SPD based on a preliminary MC
study[385], superimposed on the required measurement range to match the AMS-02
accuracy.

The time-of-flight system of the SPD detector will enable particle
identification. In addition, reconstruction of secondary-vertices will allow
the study of secondary hyperon decays[385]. Thus, the SPD can also contribute
to measurement of the $\bar{\Lambda}/\bar{p}$ ratio, where Fig.44 shows the
energy range accessible to the SPD.

Figure 44: $\bar{\Lambda}/\bar{p}$ ratio in proton-proton collisions as
measured by several experiments[386]. The range of $\sqrt{s}$ accessible at
the NICA SPD is indicated at the lower left part[385].

###  16.3 Summary

The Spin Physics Detector can make a sizable contribution to the search of
physics beyond the Standard Model. The SPD can measure energy and angular
distributions of antiprotons produced in proton-proton and proton-deuteron
collisions both directly and from the decays of $\bar{\Lambda}$ and
$\bar{\Sigma}^{-}$ hyperons in the kinematic range starting from the
threshold, for the astrophysical searches for dark matter. The collider mode
and the geometry of the SPD detector provide a unique possibility to study the
production of antiprotons at high transverse momenta which is unavailable for
fixed-target experiments. The possibility for NICA to operate with beams of
light nuclei, such as ${}^{3}He$ and ${}^{4}He$, could extend this program.

##  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.

PACS: 13.40.Em, 11.30.Er, 29.20.Dh, 29.27.Hj

### Introduction

The SPD project at NICA aims primarily at polarization experiments in the
collider mode. In the broader context, the main virtue of NICA facility is
high current beams of polarized protons and deuterons [392, 393]. We report
here new ideas developed in the framework of the Russian Fund for Basic
Research Grant No. 18-02-40092 MEGA. Basically, we propose addendum to the
previously discussed spin physics programme of the NICA facility. For decades
to come, NICA will be a holder of a unique potential to conduct fixed target
experimental tests of fundamental symmetries within or beyond the Standard
Model (SM), and one should not overlook this outstanding opportunity.

Our knowledge about high energy parity violation (PV) in the pure non-leptonic
sector of SM is as yet scarce. The most accurate result on PV asymmetry in
$pp$ scattering at 45 MeV, $A_{PV}=-(1.5\pm 0.22){\cdot}10^{-7}$, is based on
the data collected for several years [394]. A result of PV experiment at ZGS
wtih 5.1 GeV protons interacting with the water target, $A_{PV}=-(26.5\pm
6.0\pm 3.6){\cdot}10^{-7}$, was also collected for several years [395].
Theorists failed to acomodate the latter, anomalously large, asymmetry in the
SM (see review [396]). Only modest upper bounds were set at Fermilab in $pp$
and $p\bar{p}$ interactions at 200 GeV: $A_{PV}<10^{-5}$ [397].

Of our prime concern is a feasibility of precision measurements of PV
asymmetry at the NICA facility [398, 399, 400]. Our key suggestion is using
beams with polarization idly rotating in the ring plane. Such a polarization
has for the first time been successfully applied in 1986 at the Budker INP,
with participation of members of the present team, to a record precision
comparison of magnetic anomalies of $e^{+}$ and $e^{-}$ [401]. We also mention
the 2002 idea to accelerate oscillating polarization beams in Nuclotron [402].
Subsequently, the JEDI collaboration, including members of the present team,
had extended this approach to studies of fine aspects of spin dynamics of
polarized deuterons stored at the COSY accelerator [403, 404, 405, 406, 407].
The principal point is that time-stamped precessing polarization has matured,
and can be viewed as useful as the static ones. Namely, oscillating spin
asymmetries in cross section generated by the in-plane-precessing
polarization, can readily be isolated by Fourier analysis. We find it
imperative to dwell on the aspects of precessing polarization approach which
are not that familiar to the NICA community.

In this report we mostly focus on the preferred external fixed target PV
experiments at the new Booster or Nuclotron. The already published
collaboration results [398, 399, 400] are mentioned only briefly. Execution of
the PV experiment would require furnishing the Nuclotron or Booster ring with
only very modest extra instrumentation: the radiofrequency spin flipper and
internal target polarimeter to fix the time stamp of the spin phase.
Extraction of the signal of PV violation from the difference of attenuations
of extracted beams of opposite helicity will not require any sophisticated
external detectors – the anticipated non-invasive measurement of the total
charge of the bunch incident on the condensed matter target and of the
transmitted bunch will be performed by a system of Rogowski coils.

Polarized deuterons are favored because the NICA energy range is free of the
deuteron spin resonances. An outstanding record of experimentation with
polarized deuterons at the Nuclotron is noteworthy [408, 409]. Our starting
point was the PV studies, but the project outgrew its original boundary: it
was understood that the precessing polarization deuterons can give an access
to tests of still another fundamental symmetry — the time reversal invariance
[410, 411]. Specifically, one can search for the semistrong T- and CP-
violating, P-conserving and flavor-conserving interaction, suggested by Okun
[412], Prentki & Veltman [413], and Lee & Wolfenstein [414]. An intriguing
open issue is whether this manifestly beyond SM semistrong CP-violation can
resolve the puzzle of the anomalously large baryon asymmetry of the Universe,
where SM fails by many orders in magnitude [415]. On top of that, we mention
also a possibility of spin crisis experiments with oscillating deuteron
polarization at the electron-ion collider eIC [411].

###  17.1 Precessing spin asymmetries in the total $pd$ cross section

We illustrate the polarization effects on the example of total $pd$ cross
section:

$\begin{split}\sigma_{\text{tot}}=&\,\sigma_{0}+\sigma_{\text{TT}}\left[\left({\bf
P}^{\text{d}}\cdot{\bf P}^{\text{p}}\right)-\left({\bf P}^{\text{d}}\cdot{\bf
k}\right)\left({\bf P}^{\text{p}}\cdot{\bf
k}\right)\right]+\sigma_{\text{LL}}\left({\bf P}^{\rm d}\cdot{\bf
k}\right)\left({\bf P}^{\text{p}}\cdot{\bf
k}\right)+\sigma_{\text{T}}Q_{mn}k_{m}k_{n}\\\
&+\sigma_{\text{PV}}^{\text{p}}\left({\bf P}^{\text{p}}\cdot{\bf
k}\right)+\sigma_{\text{PV}}^{\text{d}}\left({\bf P}^{\text{d}}\cdot{\bf
k}\right)+\sigma_{\text{PV}}^{\text{T}}\left({\bf P}^{\text{p}}\cdot{\bf
k}\right)Q_{mn}k_{m}k_{n}\\\ &+\sigma_{\text{TVPV}}\left({\bf
k}\cdot\left[{\bf P}^{\text{d}}\times{\bf
P}^{\text{p}}\right]\right)+\sigma_{\text{TVPC}}k_{m}Q_{mn}\epsilon_{nlr}P_{l}^{\text{p}}k_{r}\,.\end{split}$
(62)

Here ${\bf P}^{\rm d}$ and ${\bf P}^{\rm p}$ are polarizations of the deuteron
and proton, $Q_{mn}$ is the tensor polarization of the deuteron, and ${\bf k}$
is the collision axis (the $z$-axis). The $y$-axis is normal to the ring
plane, $x$-axis is the radial one. In the tensor polarization dependent terms
$Q_{mn}k_{m}k_{n}=Q_{zz}$, and $k_{m}Q_{mn}\epsilon_{nlr}P_{l}^{\rm
p}k_{r}=Q_{xz}P_{y}^{\rm p}-Q_{yz}P_{x}^{\rm p}\,$ is odd under flip of the
proton polarization. The cross sections $\sigma_{0}$, $\sigma_{\text{TT}}$,
$\sigma_{\text{LL}}$, and $\sigma_{\text{T}}$ correspond to the ordinary
P-even and T-even interactions, $\sigma_{\text{PV}}^{\text{p}}$,
$\sigma_{\text{PV}}^{\text{d}}$, and $\sigma_{\text{PV}}^{\text{T}}$ give PV
signals , and $\sigma_{\text{TVPV}}$ is the TVPV component, while
$\sigma_{\text{TVPC}}$ is the null observable for the TVPC semistrong
interaction [20, 30, 31].The last three polarization effects have not been
studied before.

In our approach, the equilibrium vertical spin of the stored beam will be
subjected to parametric resonance driven by the radiofrequency (RF) flipper
(solenoid):

$\displaystyle\vec{P}(n)=P_{y}(0)[$ $\displaystyle\cos(\epsilon
n)\vec{e}_{y}+\sin(\epsilon
n)[\cos(\theta_{s}n)\vec{e}_{x}-\sin(\theta_{s}n)\vec{e}_{z}]\,.$ (63)

Here $n$ is the turn number, $\theta_{s}=2\pi\nu_{s}$, where $\nu_{s}=G\gamma$
is the spin tune, $G$ is the magnetic anomaly of the particle, $\gamma$ is its
relativistic $\gamma$-factor, $\epsilon=2\pi\nu_{R}$ and $\nu_{R}$ is the spin
resonance tune, related to the amplitude of spin rotation in a single pass
through the spin flipper. In this equation, $\cos(\epsilon n)$ and
$\sin(\epsilon n)$ have conspicuous interpretation as envelopes of the
vertical and in-plane idly precessing polarizations, respectively. The flipper
is turned off when $\epsilon n=\pi/2$ is reached and then spin will keep idly
precessing in the storage ring plane.

Oscillating $P_{z}=-P_{y}(0)\sin(\theta_{s}n)$ gives rise to the oscillating
PV signal in total cross section, see Eq. 62. The internal target polarimetry
of oscillating $P_{x}$ is used by JEDI collaboration to monitor the spin
precession frequency [403, 404] and simultaneously to provide the time stamp
for $P_{z}$ . In our proposal this time stamp will be used to trigger the
single-turn dump of beam of the desired $P_{z}$ onto the external target. In
JEDI experiments [405, 406] the spin coherence time of idly precessing
deuterons has been maximized up to 1400 s by fine tuning the sextupole lenses,
as was suggested already in 1987 by the present collaborations members (Koop
and Shatunov [416]).

We mention briefly only major points about flipper driven evolution of the
tensor polarization of deuterons starting with
$Q_{yx}(0)=Q_{yz}(0)=Q_{xz}(0)=0$, and
$Q_{xx}(0)=Q_{zz}(0)=-\frac{1}{2}Q_{yy}(0)$ [411]:

$\begin{split}Q_{yy}(n)&=\phantom{-}\frac{1}{2}Q_{yy}(0)\cdot\left[-1+3\cos^{2}(\epsilon
n)\right]\,,\\\
Q_{xx}(n)&=\phantom{-}\frac{1}{2}Q_{yy}(0)\cdot\left[-1+3\sin^{2}(\epsilon
n)\cdot\cos^{2}(\theta_{s}n)\right]\,,\\\
Q_{zz}(n)&=\phantom{-}\frac{1}{2}Q_{yy}(0)\cdot\left[-1+3\sin^{2}(\epsilon
n)\cdot\sin^{2}(\theta_{s}n)\right]\,,\\\
Q_{yx}(n)&=\phantom{-}\frac{3}{2}Q_{yy}(0)\cdot\sin(\epsilon
n)\cdot\cos(\epsilon n)\cdot\cos(\theta_{s}n)\,,\\\
Q_{yz}(n)&=-\frac{3}{2}Q_{yy}(0)\cdot\sin(\epsilon n)\cdot\cos(\epsilon
n)\cdot\sin(\theta_{s}n)\,,\\\
Q_{xz}(n)&=-\frac{3}{4}Q_{yy}(0)\cdot\sin^{2}(\epsilon
n)\cdot\sin(2\theta_{s}n)\,.\end{split}$ (64)

At the working point $\epsilon n=\pi/2$ we have $Q_{yx}=Q_{yz}=0$,
$Q_{yy}=const$ and $Q_{xx,zz}\propto 1\pm 3\cos 2\theta_{s}n$. There is a
conspicuous difference of Fourier harmonics of the vector and tensor