# Investigating Hadronic Interactions at Ultra-High Energies
with the Pierre Auger Observatory

Isabel Goos<EMAIL_ADDRESS>on behalf of the Pierre Auger Collaboration
222http://www.auger.org/archive/authors_2022_03.html Particles Group,
Astroparticle and Cosmology Laboratory (APC),
10 Rue Alice Domon et Léonie Duquet, 75013 Paris, France

###### Abstract

The development of an extensive air shower depends not only on the nature of
the primary ultra-high-energy cosmic ray but also on the properties of the
hadronic interactions. For energies above those achievable in human-made
accelerators, hadronic interactions are only accessible through the studies of
extensive air showers, which can be measured at the Pierre Auger Observatory.
With its hybrid detector design, the Pierre Auger Observatory measures both
the longitudinal development of showers in the atmosphere and the lateral
distribution of particles that arrive at the ground. This way, observables
that are sensitive to hadronic interactions at ultra-high energies can be
obtained. While the hadronic interaction cross-section can be assessed from
the longitudinal profiles, the number of muons and their fluctuations measured
with the ground detectors are linked to other physical properties. In addition
to these direct studies, we discuss here how measurements of the atmospheric
depth of the maximum of air-shower profiles and the characteristics of the
muon signal at the ground can be used to test the self-consistency of the
post-LHC hadronic models.

##  1 Introduction

The Pierre Auger Observatory is primarily designed to detect cosmic rays with
energies above around $10^{18}\>$\mathrm{e}\mathrm{V}$$ via the extensive air
showers they generate in the earth’s atmosphere. It employs a hybrid technique
to measure the properties of the primary cosmic ray [1]. On the one hand, a
surface detector (SD) consisting of 1600 water-Cherenkov detectors, arranged
on an isometric triangular grid with 1500 m spacing, covers an area of 3000
$\mathrm{k}\mathrm{m}^{2}$. This array samples the lateral distribution of the
particles of extensive air showers that arrive at the ground. The signal and
timing information of triggered stations are used to estimate the primary
energy and reconstruct the shower geometry. On the other hand, a fluorescence
detector (FD), comprising 27 telescopes and spread between four sites,
overlooks the surface detector. These telescopes observe the longitudinal
development of extensive air showers through the fluorescence light emitted by
de-exciting atmospheric nitrogen molecules. They provide a calorimetric
measurement of the primary energy and a complementary description of the
shower geometry. Furthermore, they measure the depth $X_{\mathrm{max}}$ (in
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$) in the atmosphere where an air
shower attains its maximum number of particles. In addition, several
instrumental upgrades, an endeavor collectively called AugerPrime, are being
deployed in order to disentangle the number of muons $N_{\mu}$ [2]. As will be
discussed here, this complementary measurement is important to improve the
understanding of extensive air showers.

Measuring the energy spectrum of ultra-high energy cosmic rays is of prime
importance for understanding the origin and mechanisms of cosmic ray
acceleration and propagation. Another open question regarding the energy
spectrum is about the origin of the flux suppression observed at the highest
energies. Cosmic rays can reach energies up to around $10^{20}$
$\mathrm{e}\mathrm{V}$, which corresponds to $\sqrt{s}\sim 200$
$\mathrm{T}\mathrm{e}\mathrm{V}$ in the center of mass frame. Considering that
the highest available energy in a human-made accelerator is of $\sqrt{s}=13$
$\mathrm{T}\mathrm{e}\mathrm{V}$ at the LHC, this means that cosmic rays open
the unique possibility of studying hadronic interactions at ultra-high
energies. Current high-energy hadronic interaction models used to simulate
extensive air showers rely on extrapolation of physical quantities towards
cosmic ray energies and also phase-space regions inaccessible in man-made
experiments. These models are used to compare the observables obtained from
simulations with those measured at Auger and to understand the relationship
they have with physical parameters.

Figure 1: Energy evolution of the mean of $X_{\mathrm{max}}$ (left) and its
standard deviation (right) as measured at the Pierre Auger Observatory
compared to proton and iron shower simulations. Figures from [6].

Up to now, none of the available high-energy hadronic interaction models has
been able to describe all observables of extensive air showers consistently.
Any observed inconsistency hints at a deficiency in the models. Thus, looking
for inconsistencies helps understanding where improvement in the models is
needed. It is still possible to study the relationship between certain
observables and physical parameters, which provides a way of performing direct
measurements using cosmic ray observations. It is worth noting that air shower
observables are not necessarily independent of one another, which means that
performing these studies, considering simultaneously many observables, might
render a more complete picture.

##  2 Search for inconsistencies

In this section, we present a selection of studies performed to investigate
the consistency between measured observables and their values obtained from
simulations done using different high-energy interaction models. The
generators used in the works reviewed here are EPOS-LHC [3], Sibyll2.3 [4] and
QGSJetII-04 [5].

###  2.1 Depth of maximum development

One of the most robust mass sensitive observables is the depth of the shower
maximum $X_{\mathrm{max}}$. This sensitivity relies on the fact that extensive
air showers generated by lighter primaries develop deeper into the atmosphere
than those initiated by heavier primaries. The latest results [6] for the
energy evolution of the first two moments of $X_{\mathrm{max}}$, superimposing
FD and SD measurements, are shown in figure 1. Their comparison with the
extreme scenarios of proton and iron showers, simulated using different high-
energy interaction models, reveals not only that the models generate values of
$X_{\mathrm{max}}$ consistent with those measured. It also reveals that the
average mass of cosmic rays evolves towards a lighter composition between
$10^{17.2}$ and $10^{18.3}$ $\mathrm{e}\mathrm{V}$, which is linked to their
origin [7]. At higher energies, the trend is reversed, and the average mass
increases with energy.

However, when comparing the energy evolution of the mean of $X_{\mathrm{max}}$
with that of its standard deviation, a certain tension can be observed for
QGSJetII-04. For energies close to $10^{18.2}$ $\mathrm{e}\mathrm{V}$ and
according to QGSJetII-04, the mean of $X_{\mathrm{max}}$ corresponds to that
of a pure proton flux. At the same time, the standard deviation of
$X_{\mathrm{max}}$ is lower than what it would be for a pure proton scenario.

###  2.2 Number of muons

Figure 2: Energy evolution of the average number of muons (left) and its
relative fluctuation (right) measured at the Pierre Auger Observatory for
inclined showers. The results are compared to proton and iron shower
simulations. Figures from [8, 9].

The number of muons is also a mass sensitive observable because the higher the
mass of the primary cosmic ray, the more muons are produced. One way to obtain
the number of muons from extensive air showers is by considering only inclined
showers (exceeding 62∘). For these, the electromagnetic component is absorbed
in the atmosphere, and the particles measured at the ground are only muons or
their decay products. The average number of muons and its relative
fluctuation, measured this way at the Pierre Auger Observatory [8], are shown
as a function of energy in figure 2. More specifically, $R_{\mu}$ is the
integrated number of muons at the ground divided by a reference value given by
$\langle N_{\mu}\rangle$ for simulated showers at $10^{19}$
$\mathrm{e}\mathrm{V}$. The obtained values are compared to the extreme proton
and iron scenarios from simulations. This time, it is evident that the mean
number of muons is not well reproduced by simulations, while the measurement
of the relative fluctuations falls within the range that is expected from
current high-energy hadronic interaction models. Simulations, performed using
any of these models, yield a number of muons which is low compared to
measurements. This is the so-called muon deficit problem.

In order to perform a comparison between measured and simulated values of the
average number of muons and its relative fluctuation simultaneously, a
particular energy value needs to be fixed. The result obtained for a primary
energy of $10^{19}$ $\mathrm{e}\mathrm{V}$ is shown in figure 3, left. Here,
not only the extreme scenarios of proton and iron initiated showers were used
to construct the region of possible values covered by simulations. Sets of
helium and nitrogen initiated showers and of all possible combinations of
these four different primaries were considered as well (represented by colored
contours). None of the predictions from the hadronic interaction models,
considering a mass composition mixture derived from $X_{\mathrm{max}}$
measurements (represented by stars), is consistent with the measurement
(represented by the black dot). The increases in the average number of muons,
necessary to reconcile the simulated values with the measurement, vary from
26$\%$ to 43$\%$, depending on the high-energy interaction model used in the
simulations.

The comparison between measured and simulated values of the mean logarithmic
muon content and the average maximum shower depth are shown in figure 3,
right. Also here it is evident that, even though the simulated mean value of
$X_{\mathrm{max}}$ is consistent with data, the mean logarithmic muon content
is not well reproduced. The correction(s) needed in the high-energy hadronic
interaction models to reproduce the muon content correctly might very well
change the models in a way that the mean $X_{\mathrm{max}}$ values are also
altered. From figure 3 (right) it becomes clear that this would lead to a
change in the interpretation of the mass composition.

When considering the measured cosmic ray flux, which implies having a mixed
composition, an anti-correlation between the number of muons and
$X_{\mathrm{max}}$ can be observed. This derives from the fact that showers
from heavier primaries develop quicker in the atmosphere, rendering thus
smaller $X_{\mathrm{max}}$-values. At the same time, this means a higher
number of muons. This strong interplay between the number of muons and
$X_{\mathrm{max}}$ motivates the study presented in the next section.

Figure 3: Left: Data measured at the Pierre Auger Observatory (black dot)
compared to the average number of muons and its relative fluctuation obtained
from simulations. The simulated expected values are obtained for any mixture
of the four primaries p, He, N and Fe and are represented by the colored
contours. The star symbols indicate the values that correspond to the mass
composition mixture derived from $X_{\mathrm{max}}$ measurements. Right:
Average logarithmic muon content as a function of the average maximum shower
depth. Figures from [8, 9].

###  2.3 Adjustments to the predictions of $X_{\mathrm{max}}$ and the hadronic
signal at the ground

Considering observables one by one only indicated that increasing the muon
content without the necessity of changing the values of $X_{\mathrm{max}}$ was
enough to obtain a consistent picture. However, performing ad-hoc adjustments
to the predictions from hadronic interaction models in order to improve their
consistency with observed two-dimensional distributions proves different [10].
In this work, two-dimensional distributions of the depth of shower maximum
$X_{\mathrm{max}}$ and the signal at the ground were considered. In order to
improve the consistency, a global shift $\Delta X_{\mathrm{max}}$ of the
predicted shower maximum and a re-scaling $R_{\mathrm{Had}}$ of the simulated
hadronic component at the ground (which includes muons and electromagnetic
halos from their decays) are allowed. The latter depends on the zenith angle.

Figure 4: Left: Two-dimensional distributions of
$X_{\mathrm{max}}^{\mathrm{Ref}}$ and $S^{\mathrm{Ref}}(1000)$ measured at the
Pierre Auger Observatory in the energy range $10^{18.5}-10^{19.0}$
$\mathrm{e}\mathrm{V}$. Data are divided into five zenith angle bins. The
distribution for the bin in the middle is shown here. Right: Correlation
between the re-scaling parameter of the hadronic signal for the minimal zenith
angle bin and the shift parameter of the depth of maximum development.
Different high-energy hadronic interaction models are considered. Figures from
[10].

The method is based on a maximum likelihood fit of the two-dimensional
distributions of $X_{\mathrm{max}}$ and the signal at the ground at a distance
of 1000 m from the core, denoted by $S(1000)$. This signal is measured by the
water-Cherenkov stations of the SD at the Pierre Auger Observatory. The
attenuation of this signal depends on the angle of incidence of the shower and
is accounted for in this approach by the factor $f_{SD}(\theta)$. Since this
attenuation is model-dependent, the data are divided into five zenith angle
bins (0∘, 33∘, 39∘, 45∘, 51∘, 60∘). The distribution for the bin in the middle
is shown in figure 4, left (Ref indicates that the observables are given in
reference to a fixed energy $E^{\mathrm{Ref}}=10^{18.7}$
$\mathrm{e}\mathrm{V}$). These distributions are fitted with sums of
templates. The sum goes over the four individual primary species considered
(p, He, N, Fe) and is weighted by their relative fractions. Since these sum up
to 1, the fractions serve as three free parameters. The templates are the
product of a Generalized Gumbel function and a Gaussian function, which
describe the $X_{\mathrm{max}}^{\mathrm{Ref}}$ and $S^{\mathrm{Ref}}(1000)$
distributions, respectively. The ad-hoc adjustments $\Delta X_{\mathrm{max}}$
and $f_{SD}$ are applied to the simulated values
$\widehat{X_{\mathrm{max}}^{\mathrm{Ref}}}$ and
$\widehat{S^{\mathrm{Ref}}}(1000)$ to obtain the corrected values

$\displaystyle X_{\mathrm{max}}^{\mathrm{Ref}}$ $\displaystyle\equiv$
$\displaystyle\widehat{X_{\mathrm{max}}^{\mathrm{Ref}}}+\Delta
X_{\mathrm{max}},$ $\displaystyle S^{\mathrm{Ref}}(1000)$
$\displaystyle\equiv$ $\displaystyle\widehat{S^{\mathrm{Ref}}}(1000)\cdot
f_{SD}(\theta).$

This adds $\Delta X_{\mathrm{max}}$, $R_{\mathrm{Had}}(\theta_{\mathrm{min}})$
and $R_{\mathrm{Had}}(\theta_{\mathrm{max}})$ to the set of free parameters.
$f_{SD}(\theta)$ is a function of the latter two, which are the re-scaling
parameters at the two extreme zenith angle bins.

Indeed, for all three high-energy interaction models, a prediction of a deeper
$X_{\mathrm{max}}$ is preferred, with $\Delta X_{\mathrm{max}}$ ranging from
22 $\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ to 48
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$, depending on the model. This
increase in the Monte Carlo prediction of $X_{\mathrm{max}}$ results in an
increase of the signal at the ground due to the electromagnetic component,
which alleviates the muon deficit problem present in the predictions.
Increases of between 15$\%$ and 24$\%$ are needed at low zenith angles,
depending on the high-energy hadronic interaction model used. For high zenith
angles, the necessary increases vary from 14$\%$ to 17$\%$. These results are
summarized in figure 4, right. As can be intuited from figure 3, right, and is
presented in detail in [10], the corrections on $X_{\mathrm{max}}$ to higher
values imply a heavier mass composition compared to inferences made using the
unaltered hadronic interaction models. This work shows the importance of
considering a multivariate approach in order to not only obtain a better
understanding, but also because this might have an impact on further
conclusions.

##  3 Direct approaches

It is possible to perform direct studies with current air shower models.
Typically, this involves two steps. First, using air shower simulations, a
relationship between one or more observables and one or more physical
properties must be established. This renders a conversion function that can be
used in a second step to deduce the values of the physical parameters
corresponding to the measured values of the observables.

###  3.1 Proton-air cross-section

Figure 5: Left: $X_{\mathrm{max}}$-distribution in one of the studied energy
intervals as measured at the Pierre Auger Observatory. The red line is the
result of the unbinned log-likelihood fit performed to obtain the parameter of
interest $\Lambda_{\eta}$, which describes the tail of the distribution.
Right: Conversion function between $\Lambda_{\eta}$ and the proton-air cross-
section for the same energy interval and for different high-energy hadronic
interaction models. Figures from [11].

The depth of maximum development $X_{\mathrm{max}}$ is strongly correlated
with the depth where the first interaction occurs, $X_{0}$. The distribution
of the latter, for a given energy, is given by the cross-section of proton-air
interactions at that energy. Hence, it is reasonable to look in the
$X_{\mathrm{max}}$-distribution for a parameter related to the cross-section
[11]. The observable, that turns out to be related to the cross-section, is a
parameter $\Lambda_{\eta}$, which describes the exponential shape of the tail
of the $X_{\mathrm{max}}$-distribution via
$\mathrm{d}N/\mathrm{d}X_{\mathrm{max}}\propto\exp{(-X_{\mathrm{max}}/\Lambda_{\eta})}$.
The $X_{\mathrm{max}}$-distribution, measured at the Pierre Auger Observatory
for one of the studied energy intervals, is shown in figure 5, left, together
with the unbinned log-likelihood fit. The observable $\Lambda_{\eta}$ was
measured at the Pierre Auger Observatory using an $X_{\mathrm{max}}$
distribution which included a fraction $\eta\%$ of the most deeply penetrating
air showers. This is done to enhance the proton fraction in the data set used.

In order to make use of the measurement, a conversion function between
$\Lambda_{\eta}$ and the proton-air cross-section needs to be constructed.
This is done by changing cross-sections in the simulations empirically. This
amounts to multiplying all hadronic cross-sections by an energy-dependent
factor, which is equal to 1 for energy values for which the models used in
simulations agree with LHC data. Different modifications lead to a different
value of the cross-section in the energy interval considered and ultimately,
after a full simulation, to a different value of $X_{\mathrm{max}}$. The
conversion functions, obtained this way for the different hadronic interaction
models, are shown in figure 5, right. The cross-section obtained from
projecting the measurement from Auger varies from 497.7 $\mathrm{m}\mathrm{b}$
to 523.7 $\mathrm{m}\mathrm{b}$, depending on the high-energy hadronic
interaction model considered.

###  3.2 Constraining the high-energy tail of the neutral pion spectrum

It is also possible to use the measured number of muons to perform a direct
study on some physical property [12]. A lower number of muons can be
attributed to an energy imbalance towards neutral pions. Since, because of
isospin symmetry, the fraction of neutral pions among all pions is determined,
this imbalance has to happen in the form of the energy distribution among
particles. Thus, very energetic neutral pions must play an important role in
obtaining a lower number of muons. Consequently, it is reasonable to search
for a physical parameter in the high end of the $\pi^{0}$ energy spectrum and
an observable among the lower values of $N_{\mu}$. The distribution of the
fraction $x_{\mathrm{L}}$ of the energy carried by the most energetic neutral
pion of the first interaction is shown in figure 6, left (gray distribution).
The distribution of the number of muons is in the inset plot. The showers
considered here are proton simulations with a primary energy of
$E_{0}=10^{19}$ $\mathrm{e}\mathrm{V}$. The lower values of $\ln{(N_{\mu})}$
can be described through a shape parameter $\Lambda_{\mu}$ by means of
expression $B\exp{(\ln{(N_{\mu})}/\Lambda_{\mu})}$. Similarly, the tail of the
$x_{\mathrm{L}}$-distribution can be fitted by the exponential function
$C\exp{(x_{\mathrm{L}}/\Lambda_{\pi})}$.

It turns out that there is a relation between the physical parameter
$\Lambda_{\pi}$ and the observable $\Lambda_{\mu}$ (see figure 6, right). In
this work, the conversion function is obtained by introducing a small
perturbation in the $x_{\mathrm{L}}$\- and $\ln{(N_{\mu})}$-distributions.
This is achieved by re-sampling the original distributions (gray
distributions). An example of modified distributions is also given in figure
6, left (yellow distributions). As can be seen, a small perturbation in the
$\ln{(N_{\mu})}$-distribution induces a small change in the
$x_{\mathrm{L}}$-distribution. This results in a slightly different conversion
curve for each high-energy hadronic interaction model considered. A precise
determination of $\Lambda_{\mu}$ is still necessary, which will be possible
once the number of events is large enough. In this pursuit, upgrades, such as
AugerPrime, will play a crucial role.

Figure 6: Left: Distribution of the fraction of energy carried by the most
energetic $\pi^{0}$ in the first interaction (in the laboratory frame). The
simulations are performed with proton primaries of energy $E_{0}=10^{19}$
$\mathrm{e}\mathrm{V}$. The gray distribution corresponds to the complete set
of simulations, while the yellow one corresponds to one particular re-
sampling. The respective distribution of the number of muons is given in the
inset plot. Right: Conversion function between the observable $\Lambda_{\mu}$
and the parameter $\Lambda_{\pi}$, which describe the low values and the tail
of the corresponding distributions, respectively. The filled circles
correspond to the nominal values from each hadronic interaction model. Figures
from [12].

##  4 Summary and outlook

With the Pierre Auger Observatory, interesting extensive air shower
observables can be measured. On the one hand, the distribution of the depth of
maximum development $X_{\mathrm{max}}$ and, on the other hand, the muon
content were compared to the corresponding distributions obtained from
simulations. Here, different available high-energy hadronic interaction models
were used. It was observed that the number of muons is not well reproduced. In
addition, two-dimensional distributions of $X_{\mathrm{max}}$ and the signal
at the ground at a distance of 1000 m from the core were considered. This
study revealed that, in fact, also the simulated $X_{\mathrm{max}}$-values
deserve further investigation. Under the conditions considered in [10], the
corrections performed on $X_{\mathrm{max}}$ alleviate the muon deficit present
in simulations.

It is also possible to use air shower simulations to perform direct studies.
There are different techniques to introduce changes in some model parameters
in order to investigate their effect on air shower observables. This way, it
was observed that the steepness of the tail of the
$X_{\mathrm{max}}$-distribution is related to the proton-air cross-section.
This allowed the measurement of this cross-section at energies beyond those
achievable at the LHC. Similarly, once enough events are recorded at the
Pierre Auger Observatory, it will be possible to use measurements of the
number of muons to obtain constraints on the high-energy tail of the neutral
pion spectrum.

It is worth emphasizing that performing a bi-variate comparison between
simulations and measurements (see section 2.3) revealed inconsistencies not
evident from one-dimensional studies. With this in mind, it makes sense to
think that also multivariate direct approaches might give access to physical
properties not accessible with only one observable.

## Acknowledgments

I thank Tanguy Pierog for the fruitful discussions and the organizers of the
Rencontres de Moriond for inviting me to contribute with this talk. I am
supported by LabEx UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001).

## References

  * [1] The Pierre Auger Collaboration, Nucl. Instrum. Methods A 798, 172 (2015).
  * [2] The Pierre Auger Collaboration, arXiv preprint arXiv:1604.03637 (2016).
  * [3] Pierog, T., Karpenko, I., Katzy, J. M., Yatsenko, E., and Werner, K., Phys. Rev. C 92, 034906 (2015).
  * [4] Fletcher, R., Gaisser, T., Lipari, P., and Stanev, T., Phys. Rev. D 50, 5710 (1994).
  * [5] Ostapchenko, S., Phys. Rev. D 83, 014018 (2011).
  * [6] The Pierre Auger Collaboration, EPJ Web of Conferences 208, 08001 (2019).
  * [7] Alves Batista, R. and others, Frontiers in Astronomy and Space Sciences 6, 23 (2019).
  * [8] The Pierre Auger Collaboration, Phys. Rev. Lett. 126, 152002 (2021).
  * [9] Supplemental material of [8]: http://link.aps.org/supplemental/10.1103/PhysRevLett.126.152002.
  * [10] The Pierre Auger Collaboration, 37th International Cosmic Ray Conference (ICRC 2021), 310 (2021).
  * [11] The Pierre Auger Collaboration, 34th International Cosmic Ray Conference (ICRC 2015), 401 (2015).
  * [12] Cazon, L., Conceição, R., Martins, M. A., and Riehn, F., Phys. Rev. D 103, 022001 (2021).


# Energy evolution of the large-$t$ elastic scattering and its correlation
with multiparticle production

S.M. Troshin
Institute for High Energy Physics,
Protvino, Moscow Region, 142280, Russia

It is emphasized that the collective dynamics associated with color
confinement is dominating over a point-like mechanism related to a scattering
of the proton constituents at the currently available values of the momentum
transferred in proton elastic scattering at the LHC. Deep–elastic scattering
and its role in the dissimilation of the absorptive and reflective asymptotic
scattering mechanisms are discussed with emphasis on the experimental
signatures associated with the multiparticle production processes.

## Introduction

Studies of elastic hadron scattering where initial particles are keeping their
identity can lead to a new knowledge on the nonperturbative dynamics of
hadronic interactions, mechanism of confinement and asymptotic regime of
strong interactions.

Concerning relation to the color confinement phenomena, it should be noted
that according to the superselection rules (SSR) colored quarks and gluons
live in the coherent Hilbert subspaces, and those are different from the
physical Hilbert subspace populated by the white hadrons. No self-adjont
operator (related to the observable quantity) describing transition between
colored and bleached Hilbert subspaces can exist. It means that the color
degrees of freedom can never be observed. It is the result of SSR for the
color degrees of freedom which is combined with the non-abelian nature of QCD
[1]. But it is not a proof of confinement yet — according to it, color should
be confined inside hadron. There is no known dynamical mechanism providing
this nowadays. Indeed, what is known is that such mechanism should be based on
the collective dynamics of quarks and gluons and, as it was demonstrated in
[2], the unitarity might be a consequence of the confinement.

We discuss here possible manifestations of the collective effects in hadron
elastic scattering keeping in mind the connection of these effects with
phenomena of color confinement.

##  1 Coherence in the elastic scattering

In this Section the large-$t$ elastic scattering discussed. It should be noted
that in the region of the transferred momenta beyond the second maximum in the
differential cross-sections, additional dips and bumps are absent. This smooth
decrease can be considered as a manifestation of the composite hadron
structure, then a power-like dependence can be used as a relevant function
reproducing the experimental data behavior. On the other hand, the exponential
function can also be applied. Those dependencies are based on the different
dynamical mechanisms, namely, power–like behavior corresponds to the composite
scattering dynamics where coherence is absent and point–like constituents
being independent, while the exponential form should be associated with
coherent collective interactions.

The power-like parametrization $d\sigma/dt\sim|t|^{-7.8}$ has been applied for
the description of the differential cross-section in the region between 1.5
(GeV)2 and 2.0 (GeV)2 in the paper [3]. This dependence is depicted on the
Fig. 1 (red line).

Figure 1: Dependence of the large-$t$ elastic scattering differential cross-
section.

At the LHC energy $\sqrt{s}=7$ TeV the power-like dependence allows to fit
data in the rather narrow region of the transferred momenta. At the same time
the Orear dependence of the form [4]

$d\sigma/dt\sim\exp(-c_{\mbox{o}}\sqrt{-t})$ (1)

can describe the experimental data better with $c_{\mbox{o}}\simeq 12$
(GeV)-1, cf. Fig. 1 (solid line). The slope parameter is about twice as much
bigger compared to the value of $c_{\mbox{o}}$ at the CERN ISR and at lower
energies [5]. It is evident that the exponential dependence on $\sqrt{-t}$
describes experimental data in the wider region of $-t$-values and use of the
power-like dependence for the data analysis seems to be premature and
misleading.

Different various dynamical mechanisms can provide the Orear dependence and
all of them are associted with the non-perturbative dynamics of white hadron
interactions. For the first time such dependence has been obtained in the
multiperipheral model [6], it has also been interpreted as a result of
scattering into a classically prohibited region in [7] and as one originating
from the contribution of the branching point in the complex angular momentum
plane in [8]111I am indebted to S.S. Gershtein and L.N. Lipatov for bringing
the papers [7] and [8], respectively, to my attention.. The presence of poles
in the complex impact parameter plane which can result [9] from the rational
form of the scattering amplitude unitarization leads to such dependence of the
scattering amplitude too. For the case of pure imaginary scattering amplitude
the poles in the impact parameter plane provide the additional oscillating
factors in front of the Orear exponent in the amplitude. Such oscillations are
common for the picture of diffractive scattering. The absence of the
oscillations at lower energies in the region of large $-t$ can be explained by
the significant role of the phase but this explanation could stop working at
the LHC energies.

Alternatively, the smooth dependence of the differential cross–section
observed at lower energies can be associated with the presence of the
essential double helicity-flip amplitude contribution. It has been shown that
the double helicity–flip amplitudes $F_{2}$ and $F_{4}$ are important at large
values of $-t$ and compensates oscillations of the helicity non-flip
amplitudes [10].

If the spin effects can be neglected at the LHC energies, i.e. any helicity-
flip amplitudes would not survive at such high energies, it would result in
appearance of the oscillations at higher $-t$-values. Thus, a possible
appearance of the above oscillations in the differential cross-section at
higher values of $-t$ can be interpreted in this case as an observation of the
$s$-channel helicity conservation in $pp$-scattering at the LHC energies.

##  2 Asymptotics: reflective vs black disk

The existing experimental accelerator and cosmic rays data set for the total,
elastic and total inelastic cross–sections cannot lead to the definite
conclusion on the possible asymptotic hadron scattering mechanism. Therefore
one should try to search for the independent experimental manifestations of
the possible asymptotic mechanism. In this connection it is instrumental to
consider a deep–elastic scattering. The notion of deep–elastic scattering
introduced in the paper [11] uses an analogy with the deep-inelastic
scattering and refers to the elastic scattering with the large transferred
momenta $-t>4$ (GeV/c)2.

With the elastic scattering amplitude being a purely imaginary function,
($f\to if$), the function $S(s,b)$ becomes real ($S=1+2if$) and can be
interpreted as a survival amplitude of the prompt elastic channel. The
relevant expressions for the survival amplitude $S(s,b)$ are the following

$S(s,b)=\pm\sqrt{1-4h_{inel}(s,b)},$ (2)

i.e. the probability of absorptive (destructive) collisions is
$1-S^{2}(s,b)=4h_{inel}(s,b)$ ($h_{inel}(s,b)\leq 1/4$). Simultaneous
vanishing of elastic and inelastic scattering amplitudes at $b\to\infty$
should always take place and therefore only one root in Eq. (2) (with plus
sign) being usually taken into account, while another one (with minus sign) is
omitted as a rule. This is a well known shadow approach to elastic scattering.
This is only valid in the case when $h_{inel}(s,b)$ is a monotonically
decreasing function of the impact parameter and reaches its maximum value at
$b=0$. Thus, the inelastic overlap function has a central impact parameter
profile and approaches its maximum value $1/4$, i.e. $h_{inel}(s,b=0)\to 1/4$
at $s\to\infty$. The survival amplitude described above, vanishes in the high
energy limit in central hadron collisions, $S(s,b=0)\to 0$. However, the self-
damping of inelastic channels at very high energies would lead to a peripheral
dependence on the impact parameter of the inelastic overlap function
$h_{inel}(s,b)$, it is vanishing at $b=0$ in the high energy limit
$s\to\infty$. In this limit the inelastic overlap function $h_{inel}(s,b)$
reaches its maximum value at nonzero values of impact parameter $b=R(s)$ [12].
This conclusion results from the unitarity saturation by the elastic amplitude
when $f(s,b)\to 1$ at $s\to\infty$ and $b=0$. This saturation can be realized
in the framework of the rational form of unitarization (cf. e.g. [12]). Thus,
we should take $S(s,b)=-\sqrt{1-4h_{inel}(s,b)}$ when $1/2<f(s,b)<1$. The
scattering dynamics starts to be reflective in the region where very high
energies combined with small and moderate values of $b$ (it means that hard
core appears) and approaches asymptotically to the completely reflecting limit
($S=-1$) at $b=0$ and $s\to\infty$ since $h_{inel}(s,b=0)\to 0$. The
probability of reflective scattering at $b<R(s)$ is to be determined then by
the magnitude of $S^{2}(s,b)$.

Thus, the deep–elastic scattering (DES) is associated with reflective
scattering at very high energies where the colliding hadrons do not suffer
from absorption anymore. The DES dominates over multiparticle production (at
small impact parameter values $h_{inel}(s,b=0)\to 0$). This ensures favorable
conditions for the experimental measurements, since the peripheral profile of
$h_{inel}(s,b)$, associated with reflective scattering, suppresses the
probability of the inelastic collisions in the region of small impact
parameters. The main contribution to the mean multiplicity is due to the
peripheral region of $b\sim R(s)$. DES in this case is correlated with
inelastic events of low cross–sections, i.e. it has a small background due to
production and high experimental visibility. The reflective mechanism
associated with the complete unitarity saturation will asymptotically decouple
from particle production asymptotically and at finite energies it corresponds
to observation of the DES with decreasing correlations with particle
production. Contrary, the saturation of the black disk limit implies strong
correlation of DES with multiparticle production processes [13].

## Acknowledgement

I am pleased to express the gratitude to IHEP, Protvino and Organizers of
Diffraction 2012 in Puerto del Carmen, Lanzarote, for the support of my
participation to the Conference. The talk is based in part on the papers
prepared in collaboration with N.E. Tyurin. I am grateful to him for the many
useful comments and discussions. I would also like to thank L.L. Jenkovszky,
U. Maor, V.A. Petrov and N.P. Zotov for the interesting remarks and
discussions.

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