11institutetext: 1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow
State University, RU-119991, Moscow, Russia
2 Bogoliubov Laboratory of Theoretical Physics, JINR, RU-141980, Dubna, Russia
a<EMAIL_ADDRESS>
b<EMAIL_ADDRESS>
c<EMAIL_ADDRESS>
# Exploring alignment phenomenon through the transverse momentum disbalance

I.P. Lokhtin 1a1a A.V. Nikolskii 112b2b A.M. Snigirev 112c2c

###### Abstract

The hypothesis of the relation between the observed alignment of spots in the
X-ray films in cosmic ray emulsion experiments and the selection procedure of
the highest-energy particles itself together with the transverse momentum
conservation is tested in the framework of the HYDJET++ model. It is shown
that the high degree of alignment can appear at the reasonable values of
transverse momentum disbalance of selected most energetic particles.

##  1 Introduction

In our previous study Lokhtin:2023tze ; LokhtinYF24  in the framework of
geometrical approach it has been shown that the high degree of alignment can
appear partly due to the selection procedure of the highest-energy particles
itself and the threshold on the energy deposition together with the transverse
momentum conservation. This hopeful result encourages us to investigate more
intently this hypothesis in the framework of some realistic model of hadron
interactions. It is about the observation of strong angular correlations in
the families of hadrons and photons that are products of interaction between
cosmic rays and target nuclei pamir1 ; pamir2 ; pamir3 ; pamir4 ; book . This
phenomenon can potentially indicate the coplanar structure of the observed
events and was named alignment pamir4 . At the qualitative level, alignment
shows a deviation from the straight line of most energetic registered
particles and their clusters in the plane of emulsion film. In addition to
that the strong angular correlations of particles was also observed in
stratospheric strat1 ; strat2  X-ray-emulsion chamber experiments. So far
there is no commonly accepted explanation of this intriguing phenomenon in
spite of numerous attempts to find it (see, for instance, book ; halzen ; man
; mukh ; Lokhtin:2005bb ; Lokhtin:2006xv ; deroeck  and references therein).

The discovery of the ridge effect (long-range azimuthal correlations ), made
in heavy ion collision at the RHIC phobos  and in $pp$-interactions with high
multiplicity cms  at the LHC, stimulated the search of any manifestations
Dremin:2010yd ; Lokhtin ; mukh2  of alignment phenomenon at the LHC
conditions. It is worth noting that the collision energy at the LHC is already
higher than the estimated threshold $\sqrt{s_{\rm
eff}}\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}4$
TeV after which the alignment or some its collider analog should be clearly
seen. However, the determination of any correspondence between these striking
phenomena was unsuccessful because they are observed in the quite different
intervals of the rapidity practically without overlapping and in the different
reference frames. Moreover the subsequent studies allowed finding the
explanation of the ridge effect within established theoretical approaches: for
instance, as an simple interplay between the elliptic and triangular flows
Eyyubova:2014dha . This fact basically agrees with a rather widespread point
of view (questioned in mukh ; mukh3 ) that the alignment phenomenon is not
more than a tail in a distribution caused fluctuations. The main purpose of
the present paper is just to add more argumentations in favour of this opinion
in the spirit of our previous study Lokhtin:2023tze , basing on more realistic
modeling now. In Sect. 2 we remind the main definitions and kinematical
relations in the problem under consideration. Basic characteristics of the
applied HYDJET++ model are sketched in Sect. 3. Section 4 describes the
results of numerical simulation made under conditions close to emulsion
experiments in the framework of HYDJET++ model. A summary can be found in
Sect. 5.

##  2 Basic definitions and kinematical relations

To characterize the azimuthal angular correlations the Pamir Collaboration
defines the alignment of spots in term of the variable introduced by A.
Borisov pamir4 :

$\lambda_{N}~{}=~{}\frac{\sum^{N}_{i\neq j\neq
k}\cos(2\phi_{ijk})}{N(N-1)(N-2)},$ (1)

where $\phi_{ijk}$ is the angle between two straight lines connecting the
$i$spot with the the $j$th and $k$th spots. $N$ is the number of spots used in
an analysis. This parameter $\lambda_{N}$ describes the degree of alignment
better than the possible other parameters of asymmetry like the eccentricity
or the thrust. For instance, $\lambda_{4}$ will be equal to 1 if all four
points lie strictly on the same straight line, but it will be considerably
less than 1 if these points form four vertices of a long rectangle. The
combinatorial factor $(N(N-1)(N-2))$ normalizes the parameter $\lambda_{N}$,
because it takes into account the number of variants of choosing three of $N$
points. For instance, for the case of $N=3$ (triangle) each angle is taken
into account twice by the definition (1), the combinatorial factor
$N(N-1)(N-2)=6$ and thus normalizes $\lambda_{3}$. By way of example,
$\lambda_{3}=-0.5$ in the case of the symmetrical configuration of three
points in a plane (the equilateral triangle). It is important to note that
$\lambda_{N}=1$ independently of the number of points under consideration if
they lie exactly along the straight line. The degree of alignment $P_{N}$ is
understood as the fraction of the events for which $\lambda_{N}>0.8$ book
among the total number of events in which the number of energy centres not
less than $N$.

The needed azimuthal angles are calculated over the positions ${\bf r}_{i}$ of
the particles (spots) in the $(xy)$-plane in the film:

${\bf r}_{i}~{}=~{}\frac{{\bf v}_{ri}}{v_{zi}}~{}h~{}=\frac{{\bf
p}_{Ti}}{\sqrt{p^{2}_{Ti}+m^{2}_{i}}~{}\sinh(\eta_{0}+\eta_{i})}~{}h~{},$ (2)

where $v_{zi}$ and ${\bf v}_{ri}$ are the longitudinal and radial components
of particle velocity respectively. $h$ is the height of several hundred meters
to several kilometers in the atmosphere above chamber where the primary
interaction takes place.

For convenience we parametrize the 4-momentum of each particle $i$ under
consideration with its transverse momentum $p_{Ti}$ (with respect to the
collision axis $z$), pseudorapidity $\eta_{i}$ and azimuthal angle $\phi_{i}$
in the center-of-mass system:

$\displaystyle[\sqrt{p^{2}_{Ti}+m^{2}_{i}}~{}\cosh\eta_{i},~{}~{}~{}p_{Ti}\cos\phi_{i},~{}~{}~{}p_{Ti}\sin\phi_{i},$
$\displaystyle\sqrt{p^{2}_{Ti}+m^{2}_{i}}~{}\sinh\eta_{i}].$ (3)

At the considered interacting energies particles are ultrarelativistic, and,
as is known, in this case the values of rapidity and pseudorapidity are
practically the same. For this reason, in what follows, under rapidity we
understand pseudorapidity. The transformation from the center-of-mass system
to the laboratory frame amounts to the rapidity shift:
$\zeta_{i}=\eta_{0}+\eta_{i}$, where $\eta_{0}$, $\zeta_{i}$ are the
rapidities of the center-of-mass system and the particle $i$ respectively in
the laboratory reference frame.

Because the size of the observation region is about several centimeters, these
distances $r_{i}$ must satisfy the following relations:

$r_{\rm min}~{}<~{}r_{i},$ (4) $r_{i}~{}<~{}r_{\rm max}.$ (5)

The condition (4) speaks about noncoincidence of the points in the film with
the center formed by the particles emitted along the collision axis
(predominantly region of incident-hadron fragmentation). The distance between
particles is determined by the expression:

$d_{ij}~{}=~{}\sqrt{r^{2}_{i}~{}+~{}r^{2}_{j}~{}-~{}2r_{i}r_{j}\cos(\phi_{i}~{}-~{}\phi_{j})}$
(6)

and must satisfy the condition:

$d_{ij}~{}>~{}r_{\rm min}~{},$ (7)

which means the distinguishability of spots $i$ and $j$ from each other.
Otherwise the particles are combined into a cluster until there remain only
particles and/or particle clusters with mutual distances larger than $r_{\rm
min}$. The coordinates of the new cluster formed by two particles are
determined in just the same way as the center-of-mass coordinates of two
bodies in classical mechanics:

${\bf r}_{ij}=({\bf r}_{i}E_{i}+{\bf r}_{j}E_{j})/(E_{i}+E_{j}).$ (8)

Among clusters that satisfy the restrictions (4), (5) and (7) one selects the
$2,...,7$ clusters/particles which are most energetic. After that one
calculates the alignment $\lambda_{N}$ using the definition above and taking
into account the central cluster, i.e. $N-1=2,...,7.$

##  3 HYDJET++ model

An absence of any collider experiment evidences for exotic types of
interactions encouraged us to investigate more intently the influence of the
selection procedure of most energetic particles itself and the threshold on
the energy deposition in appearing of alignment phenomenon. These studies
Lokhtin:2023tze  was based on the geometrical and kinematical considerations
being not influenced by the specific dynamics. Now we would like spreading
this hopeful investigation to some realistic model of hadron interactions. For
this purpose we use the popular and known HYDJET++ model which successfully
describes the large number of physical observables measured in heavy ion
collisions during RHIC and LHC operation. The details on this model may be
found in the HYDJET++ manual HYDJET_manual .

The event generator includes two independent components: the soft, hydro-type
state and the hard state resulting from in-medium multi-parton fragmentation.
The soft component is the “thermal” hadronic state generated on the chemical
and thermal freeze-out hypersurfaces prescribed by the parametrization of
relativistic hydrodynamics with preset freeze-out conditions (the adapted
event generator FAST MC Amelin:2006qe ; Amelin:2007ic ). Particle
multiplicities are calculated within the effective thermal volume using a
statistical model approach. The effective volume absorbs the collective
velocity profile and the form of hypersurface and cancels out in all particle
number ratios. Therefore, the particle number ratios do not depend on the
freeze-out details as long as the local thermodynamic parameters are
independent of spatial coordinates. The concept of the effective volume is
applied to calculate the hadronic composition at both chemical and thermal
freeze-outs. The number of particles in an event is calculated according to
Poisson distribution around its mean value, which is supposed to be
proportional to the number of participating nucleons for a given impact
parameter in a A+A collision. To simulate the elliptic and triangular flow
effects, the hydro-inspired parametrization for the momentum and spatial
anisotropy of soft hadron emission source is implemented HYDJET_manual ;
Wiedemann:1997cr .

In the hard sector the model propagates the hard partons through the expanding
quark gluon plasma and takes into account both collisional loss and gluon
radiation due to parton rescattering. It is based on the PYQUEN partonic
energy loss model Lokhtin:2005px . The number of jets is generated according
to the binomial distribution. Their mean number in an A+A event is calculated
as the product of the number of binary nucleon-nucleon (NN) sub-collisions at
a given impact parameter and the integral cross section of the hard process in
NN collisions with the minimum transverse momentum transfer $p_{T}^{\rm min}$.
The latter is the input parameter of the model. In the HYDJET++ framework,
partons produced in (semi)hard processes with the momentum transfer lower than
$p_{T}^{\rm min}$ are considered as being “thermalized”, so their
hadronization products are included in the soft component of the event
“automatically”.

It should be noted that applying this model we have the natural threshold on
collision energy in appearing of alignment phenomenon since the conception of
quark gluon plasma is valid after some minimal colliding energy.

##  4 Simulation of alignment in HYDJET++ model

As it was mention above the model calculations such as transverse momentum
spectra, pseudorapidity and centrality dependence of inclusive charged
particle multiplicity and $\pi^{\pm}\pi^{\pm}$ correlation radii in central
Pb+Pb collisions Lokhtin:2012re , centrality and momentum dependence of second
and higher-order harmonic coefficients Bravina:2013xla , flow fluctuations
Bravina:2015sda , jet quenching effects jet1 ; jet2 , and angular dihadron
correlations Eyyubova:2014dha  are in fair agreement with the experimental
data. Therefore, to be specific we consider Pb+Pb collisions in the centrality
class $c=0-5\%$ at center-of-mass energy 5.02 TeV per nucleon pair with free
parameters of the model tuned and fixed early. Following the consideration in
the previous section 2 we set the parameters $r_{\rm min}=r_{\rm res}=1$mm,
$r_{\rm max}=15$mm, $h=1$km, which are close to the conditions of emulsion
experiments, with the additional restriction on the energy threshold of
particle registration in the emulsion $E_{i}>E^{\rm thr}=4$TeV. Our
calculations are practically insensitive to this threshold in the wide
interval of its variation. The results for the degree of alignment $P_{N}$ are
the following:

$P_{3}\approx 0.2,~{}P_{4}\approx 0.04,~{}P_{5}\approx 0.008~{}\rm
at~{}\lambda_{N}>0.8.$ (9)

and are close to that obtained early Lokhtin:2023tze ; LokhtinYF24  for
chaotically located spots in the X-ray film. It is surprising at first glance
only. In fact the azimuthal angle distribution of particles in most central
collisions is practically isotropic in the HYDJET++ generator as the same as
the angular distribution of points in the procedure applied in Lokhtin:2023tze
; LokhtinYF24  and the existing angular model correlations (for instance, with
the reaction plane) are not enough to reproduce the correlations needed for
appearing of alignment phenomenon without the special selection procedure.

Figure 1: The degree of alignment $P_{3}$ for the three particles as a
function of the disbalance $\Delta$. (a) – the results for the only soft
particles (“without jets”) at the different values of the height $h$ in the
centrality class collisions $c=0-5\%$; (b) – the same the as (a), but in the
centrality class collisions $c=0-75\%$; (c) – comparing results for the soft
particles and the case with included jet mechanism (“with jets”) at the height
$h=1$km.

Figure 2: The degree of alignment $P_{4}$ for the four particles as a function
of the disbalance $\Delta$. (a) – the results for the only soft particles
(“without jets”) at the different values of the height $h$ in the centrality
class collisions $c=0-5\%$; (b) – the same the as (a), but in the centrality
class collisions $c=0-75\%$; (c) – comparing results for the soft particles
and the case with included jet mechanism (“with jets”) at the height $h=1$km.

Figure 3: The degree of alignment $P_{5}$ for the five particles as a function
of the disbalance $\Delta$. (a) – the results for the only soft particles
(“without jets”) at the different values of the height $h$ in the centrality
class collisions $c=0-5\%$; (b) – the same the as (a), but in the centrality
class collisions $c=0-75\%$; (c) – comparing results for the soft particles
and the case with included jet mechanism (“with jets”) at the height $h=1$km.

Here it is worth noting that the majority of soft particles are generated
independently in the statistical model approach. In this approach the total
momentum-energy together with the particle number vary (fluctuate) from an
event to an event. Only their mean values are meaningful that is the
transverse momentum of all particles is equal to zero not every event, but in
average. To decide this long-standing conceptual problem, the possible
influence of transverse momentum conservation in every event we take into
account in the form of missing transverse momentum:

$|{\bf p}_{T_{1}}+{\bf p}_{T_{2}}+...+{\bf p}_{T_{(N-1)}}|<\Delta,$ (10)

where ${\bf p}_{T_{i}}$ is the transverse momentum of $i$ particle. The
smaller the value $\Delta$ the better the transverse momentum of the selected
highest-energy particles should be balanced and vice versa – the larger
$\Delta$ the greater the disbalance of the transverse momentum is permissible
for the selected particles.

For the two selected most energetic particles the value of their total
transverse momentum $\Delta_{2}$ is equal to

$\displaystyle\Delta_{2}=\sqrt{p^{2}_{T1}+p^{2}_{T2}+2p_{T1}p_{T2}\cos{(\phi_{1}-\phi_{2})}}.$
(11)

In the case of the collinear ($\phi_{1}-\phi_{2}\simeq 0$) and back-to-back
($|\phi_{1}-\phi_{2}|\simeq\pi$) configurations one has the “line up events”
with the large value of the parameter $\lambda_{3}$. The share of such
configurations among all possible ones is just $\sim 20\%$ and can be directly
estimated if the azimuthal angular distribution of particles is known.

With the decreasing of $\Delta$ the condition (10)( $\Delta_{2}<\Delta$)
strengthens the fraction of back-to-back configurations, since they have the
minimal value of $\Delta_{2}\simeq|p_{T1}-p_{T2}|$ among its possible values.
The condition (4) about noncoincidence of the points in the film with the
center means also that there is some minimal transverse momentum $p_{T{\rm
min}}$ of selected highest-energy particles in according with the kinematical
relation (2). This minimal transverse momentum depends on the size ($r_{\rm
min}$) of the region observation, the height ($h$) of primary interaction and
the collision energy (over $\eta_{0}$). In fact it characterises the scale of
missing transverse momentum and the some measure of transverse momentum
resolution. Thus the restriction on the transverse momentum from below means
that at $\Delta<\sqrt{2}p_{T{\rm min}}$ survives back-to-back configurations
only and we have practically the $100\%$-degree of alignment. This peculiarity
in the behavior of $P_{3}$ as a function of transverse momentum disbalance
$\Delta$ is shown in Fig.1. In according with the kinematical relation (2) the
region of high degree of alignment becomes wider with the height decreasing,
that is also seen in Figs.1a–b, because in this case the minimal transverse
momentum becomes larger too. However this widening is not so large since the
inverse proportionality between the transverse momentum and the height in the
numerator of (2) is partially compensated by the energy decreasing in the
denominator.

The results of our modelling with restriction (10) are shown in Fig.2 and
Fig.3 for the four and five of selected highest-energy particles respectively.
The degree of alignment increases noticeably by a factor of the order of 2 for
the $P_{4}$ and a factor of order 5 for the $P_{5}$ in comparison with its
values (9) without restriction (10). The effect of the odd number of
considered particles takes place also and which is observed in our previous
investigation Lokhtin:2023tze ; LokhtinYF24  at the odd number of the points
randomly generated. The reason is similar: the disbalance $\Delta$ for the
three selected particles is of the order of $p_{T{\rm min}}$ or larger, if
their location is close to the same line passing through an origin (center).
The small value of $\Delta$ does not allow such configurations to be generated
and the degree of alignment is close to zero.

To disentangle the influence of different mechanics of particle production we
considered at first only the soft part of the HYDJET++ model (“without jets”
in Figs.1–3), that is the adapted version of event generator FASTMC and which
yields the bulk of particles with the transverse momentum up to $\sim 5$ GeV.
In the interval of the transverse momentum larger than $\sim 5$ GeV the
influence of hard part of the considered model can be noticeable.

Figures1c–3c show the compare of the results with included jet mechanism
(“with jets”) and with only soft particles. The form of curves is not
practically changed, but significant increasing of alignment degree takes
place for the four and five particles analyzed. Moreover the effect of odd
particle number is less pronounce, since the small disbalance is achievable
for the back-to back configurations with the large enough value of $\lambda$,
because the transverse momentum of one jet particle can be easily compensated
by the total transverse momentum of two soft particles having relatively less
characteristic momentum.

To investigate the possible influence of elliptic and triangular flow effects
(included in the event generating mode) on the alignment degree $P_{N}$ we
consider also so-called “minimum bias”, in fact centrality class collisions
$c=0-75\%$, Figs.1b–3b. It is very difficult to assess the effect on alignment
from an anisotropic flow with the jet mechanism turned on, since the
contribution of jets to the full multiplicity depends significantly on
centrality and this introduces additional uncertainty into the simulation
result, so it is reasonable to compare the results for $P_{N}$ only for soft
particles (without jets). The results for $P_{3}$ and $P_{5}$ remain
practically unchanged, but for $P_{4}$ we observe less pronounce effect of the
odd number of considered particles as it shown in Fig.2b. The explanation is
close to the case with jets mechanism: the small disbalance $\Delta$ is
achievable for the back-to back configurations with the large enough value of
$\lambda$, because the transverse momentum of one particle strengthen by
anisotropic flow can be easily compensated by the total transverse momentum of
two soft particles having relatively less characteristic momentum.

Thus our modeling demonstrates clearly that the conservation law of the
transverse momentum in the form of the disbalance of the total transverse
momentum $\Delta$ of highest-energy particles allows one to select
automatically more line up configurations. The application of this procedure
to the HYDJET++ model can be partially justified by the fact that the majority
of soft particles are generated independently as it was mention above and the
transverse momentum conserves only in average, but in any rapidity interval.
Conceptually similar modification of the statistical approach has been
recently done Chernyshov_2023  to take into account the charge conservation in
event-by-event to describe successfully the LHC data on the charge balance
function. The high degree of alignment $P_{3}$ is achievable at the reasonable
values (up to $\sim 3$ GeV) of transverse momentum disbalance of selected most
energetic particles.

It worth noting that the degree of alignment $P_{N}$ is not enough large to
match the central values of experimental measurements for $N\geq 4$. For
comparison the Pamir measurements are $P_{4}^{\rm exp}=0.67\pm 0.33$ and
$P_{5}^{\rm exp}=0.33\pm 0.23$ man , while our “best” values are $P_{4}\approx
0.07$ and $P_{5}\approx 0.04$. Nevertheless these values are in the limits of
the two standard deviations from the experimentally measured values as a
minimum.

##  5 Conclusions

We have analyzed the influence of the selection procedure of most energetic
particles itself in appearing of alignment phenomenon in the framework of
HYDJET++ model. Event-by-event the transverse momentum conversation has been
taken into account in the form of missing transverse momentum since the
majority of particles are independently generated in the statistical model
approach. It has been shown that the high degree of alignment appears at the
reasonable values (up to $\sim 3$ GeV) of transverse momentum disbalance of
selected most energetic particles. Applying this model we have the natural
threshold on collision energy in appearing of alignment phenomenon since the
conception of quark gluon plasma is valid after some minimal colliding energy.
However one should note that the degree of alignment $P_{N}$ in our modeling
is not enough large to match the central values of experimental measurements
for $N\geq 4$ living some room for other explanations and further
investigations.

###### Acknowledgements.

It is pleasure to thank A.S. Chernyshov, A.I. Demianov, S.N. Nedelko for
fruitful discussions. This work is supported by the Russian Science
Foundation, grant 24-22-00011.

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