# Effect of antiprotons on hydrogen-like ions in external magnetic fields

A. Anikin1,2<EMAIL_ADDRESS>A. Danilov1 D. Glazov1,4 A. Kotov1
D. Solovyev1,4 1 Department of Physics, St. Petersburg State University,
Peterhof, Oulianovskaya 1, 198504, St. Petersburg, Russia
2 D.I. Mendeleev Institute for Metrology, St. Petersburg, 190005, Russia
3 School of Physics and Engineering, ITMO University, Kronverkskiy pr. 49,
197101 St. Petersburg, Russia
4 Petersburg Nuclear Physics Institute named by B.P. Konstantinov of National
Research Centre ’Kurchatov Institut’, St. Petersburg, Gatchina, 188300, Russia

###### Abstract

In the present work, quasi-molecular compounds consisting of one antiproton
($\bar{p}$) and one hydrogen-like ion are investigated:
$\mathrm{He}^{+}-\bar{p}$, $\mathrm{Li}^{2+}-\bar{p}$,
$\mathrm{C}^{5+}-\bar{p}$, $\mathrm{S}^{15+}-\bar{p}$,
$\mathrm{Kr}^{35+}-\bar{p}$, $\mathrm{Ho}^{66+}-\bar{p}$,
$\mathrm{Re}^{74+}-\bar{p}$, $\mathrm{U}^{91+}-\bar{p}$. For the calculations,
the Dirac equation with two-center potential is solved numerically using the
dual-kinetically balanced finite-basis-set method adapted to systems with
axial symmetry (A-DKB). Adiabatic potential curves are constructed for the
ground state of the above quasi-molecular compounds in the framework of the
A-DKB approach. Calculations were also performed for the case of an external
magnetic field (the field is taken into account non-perturbatively). Zeeman
shifts of the quasi-molecular terms are obtained for a homogeneous magnetic
field with a strength of the laboratory order (up to 100 Tesla) directed along
the axis of the molecule.

##  I Introduction

Investigation of hydrogen-like ions (ions with only one electron) is of great
interest in many fields of physics. Throughout the whole periodic table H-like
ions serve as one of a key tools to measure electronic $g$-factor [1, 2, 3, 4,
5, 6], Lamb shift [7, 8, 9, 10] etc. Of particular importance is the
investigation of hyperfine structure of the one-electron highly-charged ions
(HCI). Being one of the most sensitive tools for testing quantum
electrodynamics (QED), such systems represent the basis for studies of the
structure of the nucleus, approaching the sector of strong interactions [11,
12]. In turn, collisions of HCI with both neutral atoms and other ions,
leading to the formation of quasi-molecular systems, provide information for
studying critical QED phenomena of bound states, such as the creation of
electron-positron pairs [13, 14, 15, 16, 17, 18, 19, 20]. Quasi-molecules
consisting of H-like HCIs have been widely studied in literature, see, for
example, [21, 22, 23] and references therein.

The study of antiprotons colliding with atoms, molecules and hydrogen-like
ions is another interesting subject of modern physics [24]. Light antiproton
systems offer an opportunity for the successful study of ionisation processes
[25] or the formation of protonium atoms ($p-\bar{p}$) [26]. Matter-antimatter
compounds can also be used to test fundamental interactions or search for
effects that violate known physical symmetries: such systems are used to
straightforward verification of CPT-invariance by comparing the spectra of
matter and antimatter [27]. The collisions of heavy HCI and atoms with
antiprotons can be also used to study the antiproton capture processes [28]
and such phenomenon as Coulomb glory [29, 30, 31].

Referring to the well-known problem of solving the many-body problem exactly,
the study of quasi-molecules is carried out using numerical methods. One of
the most widely used and effective approaches is the nonrelativistic theory.
Within this theory the one-electron problem is solved with high accuracy,
whereas relativistic and quantum electrodynamics (QED) effects are described
in the framework of nonrelativistic quantum electrodynamics (NRQED) [32, 33,
34]. While providing a powerful tool for calculations in the case of light
systems, NRQED is a poor approximation for heavy element compounds. In this
case, it is more advantageous to solve the one-electron problem for dinuclear
compounds using the Dirac equation with the corresponding two-center Coulomb
potential. As in NRQED, this method uses the Born-Oppenheimer approximation
along with various decompositions of the two-center potential [35, 36, 37].

In this paper, we apply a dual-kinetic-balance (DKB) method based on the
decomposition of the wave function over the finite basis set of B-splines
[38], generalized to the axially symmetric systems (A-DKB) [39]. The DKB
method is widely used in various relativistic and QED calculations for highly
charged ions. Recently, the A-DKB method has been successfully worked out to
describe both heavy [21, 22, 23] and light [40, 41, 42] few-electron quasi-
molecules. The present work is devoted to the study of quasi-molecules
consisting of antiproton and different (light and heavy) hydrogen-like ions
subjected to an external uniform magnetic field directed along the molecular
axis. In this study, we restrict ourselves to fields with strengths relevant
to laboratory conditions. Experimental research on HCI is often carried out
using ion traps. The external magnetic field in these devices is typically a
few Tesla [43], whereas to date, experimentally achievable magnetic field
strengths have reached the extraordinary value of $100$ T (National High
Magnetic Field Laboratory [44, 45]). It should be noted that the Born-
Oppenheimer approximation used in the present work is valid only as a rough
approach in the case of very strong magnetic fields. For the fields of the
order of up to $10^{11}-10^{12}$ T, as in the atmospheres of neutron stars
[46, 47, 48, 49, 50], the contribution of the nuclei motion is comparable to
the Coulomb binding energy and requires special consideration [51].

The paper is organized as follows. A brief description of the method used is
presented in section II. Results of the electronic energy calculations are
given in section III, in which nuclear charge dependence of the adiabatic
potentials is analyzed, as well as dependence of Zeeman shifts on both nuclear
charge and on the inter-nuclear distance. The last section provides a
discussion of the obtained numerical results. Numerical results are collected
in Tables 1-10. The atomic units $\hbar=e=m_{e}=1$ ($\hbar$ is the Planck
constant, $e$ is the electron charge modulus, $m_{e}$ is the mass of electron)
are used throughout the paper. In these units, $c$ (the speed of light) can be
expressed through the fine structure constant $c=1/\alpha$, but for clarity it
is written explicitly.

##  II The dual kinetic balance method for systems with axial symmetry (A-DKB)

In frames of the Born-Oppenheimer approximation the stationary Dirac equation
for the two-center potential is

$\displaystyle\left[c\bm{\alpha}\bm{p}+\beta
c^{2}+V(\bm{r})\right]\Psi_{n}(\bm{r})=E_{n}\Psi_{n}(\bm{r}),$ (1)

where $\bm{r}$ is the electron radius vector, $\bm{p}$ is the momentum
operator and $\bm{\alpha}$, $\beta$ are the Dirac matrices. The two-center
potential is expressed by

$\displaystyle
V(\bm{r})=V_{1}(|\bm{r}-\bm{R}_{1}|)+V_{2}(|\bm{r}-\bm{R}_{2}|)$ (2)
$\displaystyle\equiv-\frac{Z_{1}}{\left|\bm{r}-\bm{R}_{1}\right|}-\frac{Z_{2}}{\left|\bm{r}-\bm{R}_{2}\right|},$

with the Coulomb potentials $V_{1,2}$, generated by nuclei with charges
$Z_{1}$ and $Z_{2}$ at the point $\bm{r}$ and located at a distance of
$R\equiv|\bm{R}|=|\bm{R}_{1}-\bm{R}_{2}|$ from each other. While in Eq. (2)
the potential is written for point-like charges, in the A-DKB calculations
various finite-nucleus charge models are used. In the present work we
nonperturbatevely study effect of the external magnetic field directed along
the molecular axis. This magnetic interaction is accounted for by the operator

$\displaystyle
V_{\mathrm{ext}}=-\frac{e}{2}([\bm{r}\times\bm{\alpha}]\cdot\bm{B}),$ (3)

where $e/2$ is Bohr magneton (in a.u.) and $\bm{B}$ is magnetic field
strength. The coordinate origin is chosen between the two nuclei, the $z$ axis
coincides with the molecule axis, see Fig. 1.

Figure 1: The coordinate system for diatomic one-electron compounds oriented
along the $z$-axis. The bound electron is marked with a filled circle, the
nuclei are marked with circles with the nucleus charges $Z_{1}$ and $Z_{2}$.
The corresponding position vectors from the coordinate origin $O$ are
indicated by $\bm{R}_{1}$ and $\bm{R}_{2}$. The position vector of the bound
electron placed on one of the nuclei is given by $\bm{r}$ directed from $O$.
Since axial symmetry, the azimuth angle $\theta$ is introduced.

Within the relativistic approach, the wave function can be written as

$\displaystyle\psi\left(r,\theta,\varphi\right)=\frac{1}{r}\begin{pmatrix}G_{1}\left(r,\theta\right)e^{i(m_{j}-1/2)\varphi}\\\
G_{2}\left(r,\theta\right)e^{i(m_{j}+1/2)\varphi}\\\
iF_{1}\left(r,\theta\right)e^{i(m_{j}-1/2)\varphi}\\\
iF_{2}\left(r,\theta\right)e^{i(m_{j}+1/2)\varphi}\end{pmatrix},$ (4)

where $m_{j}$ is the projection of the total angular momentum of the electron
onto the nuclear axis, which is conserved in the axially symmetric case. Then,
the components $G_{i}(r,\theta)$ and $F_{i}(r,\theta)$ ($i=1,2$), according to
the A-DKB method [39], are decomposed through $B$-splines as follows:

$\displaystyle\phi\left(r,\theta\right)\cong\sum^{4}_{u=1}\sum^{N_{r}}_{i_{r}=1}\sum^{N_{\theta}}_{i_{\theta}=1}C^{u}_{i_{r}i_{\theta}}\Lambda
B_{i_{r}}(r)Q_{i_{\theta}}(\theta)e_{u}.$ (5)

Here $\left\\{B_{i_{r}}(r)\right\\}^{N_{r}}_{i_{r}=1}$ are the B-splines,
$\left\\{Q_{i_{\theta}}(\theta)\right\\}^{N_{\theta}}_{i_{\theta}=1}$ are
given by the Legendre polynomials of the argument $2\theta/\pi-1$, and
$\left\\{e_{u}\right\\}^{4}_{u=1}$ are the four-component basis vectors. To
avoid the problem of spurious states [52, 53], the matrix $\Lambda$ introduced
in Eq. (5) imposes the dual-kinetic-balance (DKB) conditions on the basis set
[38, 39]. The eigenvalues and eigenfunctions are found numerically by solving
the generalized eigenvalue problem.

Application of the DKB approach for axially symmetric systems is implied for
the extended charge nucleus only. The point-like nucleus case can be accessed
by the extrapolation of the extended-nucleus results in vanishing nuclear
size. The expression above is given for arbitrary basis sets
$\\{B_{i_{r}}(r)\\}_{i_{r}=1}^{N_{r}}$ and
$\\{Q_{i_{\theta}}(r)\\}_{i_{\theta}=1}^{N_{\theta}}$. A particular choice of
one-component basis functions corresponds to $B$-splines of the second order
and Legendre polynomials as
$Q_{i_{\theta}}(\theta)=P_{i_{\theta}-1}\left(\frac{2\theta}{\pi}-1\right)$,
forming the set of one-component $\theta$-dependent basis functions of
polynomial degrees $l=0\dots N_{\theta}-1$. By varying the values $N_{r}$,
$N_{\theta}$, such a sample allows the convergence of numerical calculations
to be monitored.

In the following we consider the ground states of one-electron quasi-molecules
consisting of one hydrogen-like ion and one antiproton:
$\mathrm{He}^{+}-\bar{p}$, $\mathrm{Li}^{2+}-\bar{p}$,
$\mathrm{C}^{5+}-\bar{p}$, $\mathrm{S}^{15+}-\bar{p}$,
$\mathrm{Kr}^{35+}-\bar{p}$, $\mathrm{Ho}^{66+}-\bar{p}$,
$\mathrm{Re}^{74+}-\bar{p}$, $\mathrm{U}^{91+}-\bar{p}$. This choice covers a
wide range of $Z_{1}$, permitting the main features to be identified, and the
corresponding H-like ions have been studied theoretically and experimentally,
see e.g. [43, 54, 55, 56, 57]. The magnetic field strength considered in this
paper is: $1-5$, $10$ and $100$ Tesla.

##  III Results

To study the antiproton impact on hydrogen-like ions, we calculate fully
relativistic electronic energies within the A-DKB approach. The following grid
parameters were used: $N_{r}=395$ and $N_{\theta}=40$, see Eq. (5). For the
H-like ions the nuclei radii were taken from [58] and for the antiproton
$r_{\bar{p}}=r_{p}=0.8410$ fm was used ($r_{p}$ is the proton charge radius
[59]). We used the spherical nucleus model for light ions $\mathrm{He}^{+}$,
$\mathrm{Li}^{2+}$, $\mathrm{C}^{5+}$, and the antiproton, and the Fermi model
for heavy ion nuclei $\mathrm{S}^{15+}$, $\mathrm{Kr}^{35+}$,
$\mathrm{Ho}^{66+}$, $\mathrm{Re}^{74+}$, $\mathrm{U}^{91+}$. To study
compounds with such a broad range of nuclear charges, for every element we
consider a ”chemical” distance, which in atomic units equals to $2/Z_{1}$. For
binuclear quasi-molecules, this distance is often used as a characteristic
scale, and in this paper, for a given $Z_{1}$, the electronic energy is
calculated for inter-nuclear distances within the interval $R\in[0;4/Z_{1}]$
(twenty one points from zero to twice the chemical distance in a.u.).
Recently, the A-DKB method has been used in [40, 41, 42, 21, 22, 23] to study
both light and heavy one-electron binuclear compounds. Accuracy was achieved
at $10^{-8}$ relative magnitude for the grid parameters given above. For the
Zeeman shifts the same level of accuracy was reached in the present work for
the considered systems.

Adiabatic electron energy curves scaled according to the relation
$E_{\mathrm{A-DKB}}\rightarrow E_{\mathrm{A-DKB}}/Z_{1}^{2}$, for inter-
nuclear distances $\tilde{R}\rightarrow R\times Z_{1}$ are represented by the
plots in Fig. 2. The numerical values for each compound can be found in Tables
3-10, given for brevity at the end of the paper.

Figure 2: Adiabatic potential energy curves for the ground state of H-like
ion-antiproton compounds. The inter-nuclear distance is scaled as
$\tilde{R}\rightarrow R\times Z_{1}$, and the energy is depicted according to
$\tilde{E}_{\mathrm{A-DKB}}\rightarrow E_{\mathrm{A-DKB}}/Z_{1}^{2}$. All
values are given in a.u.

Fig. 2 shows that the greater the nuclear charge, the weaker the influence of
the antiproton on the corresponding ion. In particular, an almost threefold
energy change is evident for the quasi-molecule $\mathrm{He}^{+}-\bar{p}$
(upper curve), whereas for the compound $\mathrm{U}^{91+}-\bar{p}$ the
electronic energy variation is about $\approx 10\%$ (lower curve). This
behaviour is expected for the following reasons. When the charge of the
nucleus is large, the electron is exposed to a Coulomb field that is orders of
magnitude (especially for $\mathrm{U}^{91+}$) larger than that of the
antiproton, and hence the bound electron almost does not ”feel” the
antiproton. The situation is quite different for the light elements: the total
field of charges $Z_{1}=2$ and $Z_{2}=-1$ considerably weakens with the
approach of the antiproton to the one-electron ion, which affects the
electronic binding energy. Among this, ionisation occurs for $Z_{1}=1$ and
$Z_{2}=-1$, see e.g. [40] and references therein.

From the graphs in Fig 2, it also follows that for ions with higher nucleus
charge, relativistic corrections combined with a correction for finite nucleus
size have to be taken into account. In particular, the chosen scaling allows
one to clearly see the difference of the obtained electron energies from their
nonrelativistic values. In the limit of the joined atom ($R\rightarrow 0$) or
infinitely distant nuclei ($R\rightarrow\infty$), the nonrelativistic energy
for a point nucleus is $E_{\mathrm{nr}}^{(R=0)}=-(Z_{1}-1)^{2}/2$ and
$E_{\mathrm{nr}}^{(R=\infty)}=-Z_{1}^{2}/2$ in atomic units. The latter gives
$0.5$ a.u. when using the above scaling, whereas the former is approximately
$0.49$ for $\mathrm{U}^{91+}-\bar{p}$ compound.

The main goal of the present work, however, is to study the influence of
antiprotons on H-like ions in an external homogeneous magnetic field. In the
framework of the A-DKB approach, such a combination can be analysed quite
simply for a field directed along the molecule axis. The calculations can be
performed in a non-perturbative way, i.e., by including the interaction of Eq.
(3) in the Hamiltonian of Eq. (1). To this end, the fully relativistic
electron energies have been calculated in magnetic fields with different
values of field strength ($1-5$, $10$, $100$ Tesla). Then the corresponding
Zeeman shifts $\Delta E_{\mathrm{Z}}$ can be retrieved by comparing with
numerical results obtained for the case of absence in the Hamiltonian of
interaction with the external field. The results are combined in Table 1 for
chemical distances, and at the end of the paper Tables 3-10 present the values
for inter-nuclear distances in the range $R\in[0;4/Z]$ a.u. All Zeeman shifts
coincide in sign with the sign of the projection of the electron total angular
momentum onto the molecular axis.

Table 1: Absolute values of Zeeman shifts, $\Delta E_{\mathrm{Z}}$, in atomic
units (a.u.) for compounds of hydrogen-like ions with an antiproton. The ions
considered are listed in the first column, the magnetic field strength values
in Tesla (T) in the second row of the table. All values in the table are
calculated for chemical inter-nuclear distances ($2/Z$ a.u.) and multiplied by
a factor $10^{6}$.

ion |  $|\Delta E_{\mathrm{Z}}|$ $\times 10^{6}$, a.u. (at a chemical distance, 2/Z a.u.)
---|---
$B=1$ T |  $B=2$ T |  $B=3$ T |  $B=4$ T |  $B=5$ T |  $B=10$ T |  $B=100$ T
$\mathrm{He}^{+}$ | $2.13$ | $4.25$ | $6.38$ | $8.51$ | $10.63$ | $21.27$ | $212.69$
$\mathrm{Li}^{2+}$ | $2.13$ | $4.25$ | $6.38$ | $8.51$ | $10.63$ | $21.27$ | $212.68$
$\mathrm{C}^{5+}$ | $2.13$ | $4.25$ | $6.38$ | $8.50$ | $10.63$ | $21.26$ | $212.58$
$\mathrm{S}^{15+}$ | $2.12$ | $4.23$ | $6.35$ | $8.47$ | $10.59$ | $21.17$ | $211.76$
$\mathrm{Kr}^{35+}$ | $2.08$ | $4.15$ | $6.23$ | $8.31$ | $10.39$ | $20.77$ | $207.76$
$\mathrm{Ho}^{66+}$ | $1.95$ | $3.89$ | $5.84$ | $7.79$ | $9.73$ | $19.46$ | $194.65$
$\mathrm{Re}^{74+}$ | $1.90$ | $3.79$ | $5.69$ | $7.59$ | $9.48$ | $18.96$ | $189.65$
$\mathrm{U}^{91+}$ | $1.76$ | $3.52$ | $5.29$ | $7.05$ | $8.81$ | $17.62$ | $176.17$

As follows from Table 1, the Zeeman shifts are linear in the field strength
for all compounds considered. In the case of (relatively weak) fields involved
in this study, perturbation theory can be used to calculate the energy shifts,
in which a linear approximation is sufficient. It can be found that the value
of the Zeeman shift is almost independent of the nuclear charge in the case of
light elements and varies significantly for heavy ions (starting from
$Z_{1}\sim 36$ in Table 1).

From Tables 3-10 one can also analyze a dependence of the Zeeman shifts on
inter-nuclear distance. For light elements, $\Delta E_{Z}$ is practically
independent of the inter-nuclear distance: at $10$, $100$ T the shift varies
only at the level of $10^{-7}-10^{-8}$ a.u. As the nuclear charge increases,
the Zeeman shift begins to depend weakly on the inter-nuclear distance. This
is already apparent for the hydrogen-like Krypton Kr35+. For the compound
$\mathrm{U}^{92+}-\bar{p}$ at a field of $100$ T, the Zeeman shift changes by
$\sim 10^{-6}$ a.u. as the distance changes from $R=0$ fm to $R\approx 2000$
fm.

##  IV Discussion and conclusions

In this paper, within the framework of a completely relativistic method the
effect of antiprotons on various hydrogen-like ions is studied using the
following quasi-molecules as an example: $\mathrm{He}^{+}-\bar{p}$,
$\mathrm{Li}^{2+}-\bar{p}$, $\mathrm{C}^{5+}-\bar{p}$,
$\mathrm{S}^{15+}-\bar{p}$, $\mathrm{Kr}^{35+}-\bar{p}$,
$\mathrm{Ho}^{66+}-\bar{p}$, $\mathrm{Re}^{74+}-\bar{p}$,
$\mathrm{U}^{91+}-\bar{p}$. To obtain the ground state electron energy of
these compounds, we utilized the Dual Kinetic Balance Approach [38], extended
to the Dirac equation for axially symmetric systems in [39]. The A-DKB method
has already proven its effectiveness for both heavy [21, 22, 23] and light
[40, 41, 42] one-electron quasi-molecules, so in this study we focused on the
electron energy of the ground term and its dependence on the nuclear charge.

To analyze how presence of antiproton affects hydrogen-like ions with
different nuclear charges we calculated adiabatic potential energy curves and
plotted the corresponding scaled graphs, see Fig. 2. It was found that, as one
might expect, for elements with higher nuclear charges relativistic effects
and corrections due to the finite nuclear size play an increasingly important
role. At the same time, for higher nuclear charges, due to the large Coulomb
potential, the electron energy is almost independent of the inter-nuclear
distance, in contrast to the light elements.

The study of the antiproton effect on the energy states of a bound electron in
hydrogen-like ions with different nuclear charges has been extended to the
case of the presence of an external magnetic field. For this purpose adiabatic
potential energy curves has been obtained for $1-5$, $10$, $100$ T magnetic
fields. As a consequence the Zeeman shifts were determined, see numerical
results in Tables 1 \- 10.

It has been established that in the presence of an external field, light and
heavy compounds behave quite differently. In particular, the Zeeman shift for
an arbitrary nuclear charge can be estimated with high accuracy as linear
field strength contribution in the considered range of $1-100$ T. However, its
value vary significantly with $Z_{1}$. The effect is manifested in a weak
dependence of the Zeeman shift on the charge of the nucleus. At large $Z_{1}$
($Z_{1}\geqslant 36$), the deviation becomes noticeable and increases with
growing field strength. It should be emphasized that in the nonrelativistic
limit, the Zeeman shift does not depend on $Z_{1}$. Thus, this behavior refers
to relativistic effects in combination with the finite size of the nucleus,
which are taken into account in the A-DKB method.

Moreover, from Tables 3 \- 10 it is clear that in the case of small nuclear
charges, Zeeman shifts are practically independent of the inter-nuclear
distance (up to double the chemical distance $4/Z_{1}$). For
$\mathrm{He}^{+}-\bar{p}$ even at $100$ T the corresponding change is of the
order of $10^{-8}$ a.u. absolute value, i.e. at the level of the numerical
error of the A-DKB method. For large nuclear charges $Z_{1}$, for example in
the compound $\mathrm{U}^{92+}-\bar{p}$, such a slight change in the Zeeman
shift is observed for the field $1$ T, and, hence, at $100$ T it increases to
$10^{-6}$ a.u. However, even such a result can be considered independent of
the inter-nuclear distance with an accuracy limit of $10^{-6}$.

Additionally, the results collected in Tables 3-10 can serve to better
determine the capture energy of antiprotons by hydrogen-like ions. In
particular, the binding energy of the electron initial state
$E_{\mathrm{A-DKB}}$, given in the second column for each specified ion in the
mentioned tables, includes the entire relativistic value taking into account
the effect of finite nucleus size. The latter is more pronounced for heavy
ions. For example, the Dirac energy of the bound electron for hydrogen-like
uranium is $E_{\mathrm{D}}=-4861.1979$ a.u. (the antiproton is infinitely
distant) and with correction for the finite size of the nucleus amounts to
$-4853.8957$ a.u. [60]. The bound electron energy found in the framework of
the A-DKB approach is $E_{\mathrm{A-DKB}}\approx-4830.4778$ a.u. at the inter-
nuclear distance $R=0.043478$ a.u. or $2300.77065$ fm (see the last row in
Table 10), which therefore represents a significant shift of the resonance of
the antiproton capture process [61]. At the same time, the level shift in the
magnetic field, even with a strength of $100$ T, can be considered as a
correction of the next order of smallness: the variation of the energy of the
bound electron state differs by hundreds of eV with approaching antiproton to
the nucleus and the Zeeman shift in the field of $100$ T is, respectively,
meV, remaining practically unchanged.

Finally, the correction for the finite nucleus size should be analysed in
conjunction with the model used to describe it. As was recently observed in
[40, 42], in light binuclear compounds the electronic spectrum is independent
of the nuclear size correction with an accuracy of about $10^{-8}$ for the
relative magnitude. Obviously, this correction becomes significant for heavy
ions. In turn, antiproton compounds should also be sensitive to this effect.
Since the nuclear field changes essentially as the aniti-proton approaches,
the model used to describe the nucleus is also important. To demonstrate this,
the relevant calculations are given in the following Table 2 (for brevity, for
the U${}^{91+}-\bar{p}$ compound only).

Table 2: The binding energy of an electron for the ground term in a quasi-
molecule U${}^{91+}-\bar{p}$. Energy values are given in atomic units (a.u.)
for inter-nuclear distances of $0.0$, $1150.383$ fm ($2/Z_{1}$), and
$2300.771$ fm ($4/Z_{1}$) in the corresponding columns. To determine the
Zeeman shift the electron energies are given in each first and second rows for
the zero field and the field strength $10$ T, respectively. The Zeeman shift,
$\Delta E_{Z}$, multiplied by a factor of $10^{6}$ is indicated in every third
row in a.u. To reveal the energy dependence on the finite nucleus size model,
the radius (in fm) and the model used for comparison are given in the first
column.

Fermi-model |  $R=0.0$ fm |  $R=1150.383$ fm |  $R=2300.771$ fm
---|---|---|---
$r^{-}_{N}=5.8538$ fm | $-4731.62494651$ | $-4809.44455498$ | $-4830.48421214$
$B=10$ T | $-4731.62496422$ | $-4809.44457260$ | $-4830.48422976$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.61$
$r_{N}=5.8571$ fm111This row contains the values given in Table 10. | $-4731.61940826$ | $-4809.43830386$ | $-4830.47781493$
$B=10$ T | $-4731.61942597$ | $-4809.43832148$ | $-4830.47783255$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.61$
$r^{+}_{N}=5.8604$ fm | $-4731.61386859$ | $-4809.43204318$ | $-4830.47141770$
$B=10$ T | $-4731.61388630$ | $-4809.43206080$ | $-4830.47143531$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.62$
sphere-model |  $R=0.0$ fm |  $R=1150.383$ fm |  $R=2300.771$ fm
$r^{-}_{N}=5.8538$ fm | $-4731.61239329$ | $-4809.41349294$ | $-4830.41091172$
$B=10$ T | $-4731.61241100$ | $-4809.41351056$ | $-4830.41092934$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.62$
$r_{N}=5.8571$ fm | $-4731.60685922$ | $-4809.40704558$ | $-4830.40439320$
$B=10$ T | $-4731.60687693$ | $-4809.40706320$ | $-4830.40441081$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.61$
$r^{+}_{N}=5.8604$ fm | $-4731.60132373$ | $-4809.40059737$ | $-4830.39787536$
$B=10$ T | $-4731.60134144$ | $-4809.40061499$ | $-4830.39789297$
$\Delta E_{Z}\times 10^{6}$, a.u. | $17.71$ | $17.62$ | $17.62$

First, the values listed in each column of Table 2 show the independence of
the Zeeman shift from the radius and the nuclear model used. This follows from
the fact that these effects shift the energy term equally for cases with and
without a field. Thus, the Zeeman shift in our work is determined with an
accuracy of $10^{-4}$ relative magnitude or $10^{-8}$ on the absolute scale
for the compound U${}^{91+}-\bar{p}$.

Second, the dependence of the results on the nuclear radius can be observed by
comparing the corresponding rows in each sector of the table. In particular,
one finds that the results vary by about $10^{-7}$ when changing the radius
near the error of its determination [58], i.e. $r_{N}=5.8571(33)$ fm. To
demonstrate the dependence on the nuclear radius at different inter-nuclear
distances for the elements $\mathrm{He}^{+}-\bar{p}$,
$\mathrm{Li}^{2+}-\bar{p}$, $\mathrm{C}^{5+}-\bar{p}$,
$\mathrm{S}^{15+}-\bar{p}$, $\mathrm{Kr}^{35+}-\bar{p}$,
$\mathrm{Ho}^{66+}-\bar{p}$, $\mathrm{Re}^{74+}-\bar{p}$,
$\mathrm{U}^{91+}-\bar{p}$, plots of the relative value of
$\Delta_{r_{N}}=\sqrt{(\Delta_{r^{+}_{N}})^{2}+(\Delta_{r^{-}_{N}})^{2}}$,
where
$\Delta_{r^{\pm}_{N}}=\left|E_{\mathrm{A-DKB}}^{(r^{\pm}_{N})}-E_{\mathrm{A-DKB}}^{(r_{N})}\right|/E_{\mathrm{A-DKB}}^{(r_{N})}$
are shown in Fig. 3. The following notations are introduced for the relative
deviation $\Delta_{r_{N}}$: $r_{N}$, $r^{-}_{N}$, and $r^{+}_{N}$ denote the
nuclear radius, nuclear radius minus error, and nuclear radius plus error,
respectively, see [58]. For the light elements ($Z_{1}\leq 6$) the spherical
model of the nucleus is used, and for the others the Fermi model is employed.

Figure 3: The graph showing the dependence of the relative magnitude
$\Delta_{r_{N}}$ on the inter-nuclear distance, $R$.

It should be noted that the errors in the values of the nuclear radii [58]
differ significantly for different nuclei. For example,
$r_{\mathrm{Re}}=5.3596\pm 0.0172$, whereas $r_{\mathrm{U}}=5.8571\pm 0.0033$,
which leads to a considerably stronger deviation of $\Delta_{r_{N}}$ for
hydrogen-like ions with $Z_{1}=75$ and $Z_{1}=92$. The graph in Fig. 3 shows
that in the case of light elements the dependence on the nucleus size is an
order of magnitude smaller compared to heavy nuclei.

In turn, the dependence on the model of the nucleus was tested for the Fermi
and spherical forms of the charge distribution of the nucleus in case of heavy
ions and for spherical and shell models for light nuclei. Comparing the
corresponding rows in different sectors of Table 2, we find a discrepancy of
numbers at the level of $10^{-5}-10^{-6}$. Since the dependence on the nucleus
model shows a larger scatter of values than the dependence on the charge
radius, it can be concluded that collisions of antiprotons with hydrogen-like
ions can be used to refine the parameters of the used models. The dependence
on the inter-nuclear distance of the finite nuclear size model is shown in
Fig. 4.

Figure 4: The graphs showing the dependence of the relative magnitude
$\Delta_{\mathrm{model}}=\left|E_{\mathrm{A-DKB}}^{(\rm{model}_{1})}-E_{\mathrm{A-DKB}}^{(\rm{model}_{2})}\right|/E_{\mathrm{A-DKB}}^{(\rm{model}_{1})}$
at fixed nuclear radius and zero field on the inter-nuclear distance, $R$.

In particular, the behaviour of the relative magnitude
$\Delta_{\mathrm{model}}=\left|E_{\mathrm{A-DKB}}^{(\rm{model}_{1})}-E_{\mathrm{A-DKB}}^{(\rm{model}_{2})}\right|/E_{\mathrm{A-DKB}}^{(\rm{model}_{1})}$
clearly demonstrates this possibility. The final result for the binding energy
should not depend on the utilised model.

Finally, it can be expected that the results obtained in the present study may
be useful for the research areas discussed recently in [62].

##  V Acknowledgement

This work was supported by RSF grant №23-22-00250.

##  VI Data availability

The data that support the findings of this study are available within the
article. The data that support the findings of this study are available from
the corresponding author upon reasonable request.

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Table 3: The values of the Zeeman shift $\Delta E_{\mathrm{Z}}$ (in a.u.) for the ground state of the compound $\mathrm{He}^{+}-\bar{p}$ at a magnetic field strength of $100$ T are indicated in the third column. The first column shows the values of the inter-nuclear distances (in fermi, fm). The second column shows the energies of electrons in the absence of a magnetic field in atomic units. The values of the Zeeman shift at a different magnetic field strength are not given due to their independence from the inter-nuclear distance. $R$, fm |  $E_{\mathrm{A-DKB}}$, a.u. |  $|\Delta E_{\mathrm{Z}}|\times 10^{6}$, au
---|---|---
$0.0$ | $-0.50000665$ | $212.76$
$5291.77249$ | $-0.51232902$ | $212.76$
$10583.54498$ | $-0.54658328$ | $212.75$
$15875.31747$ | $-0.59948446$ | $212.75$
$21167.08996$ | $-0.66690910$ | $212.74$
$26458.86245$ | $-0.74359023$ | $212.73$
$31750.63494$ | $-0.82415224$ | $212.73$
$37042.40743$ | $-0.90419211$ | $212.73$
$42334.17992$ | $-0.98072434$ | $212.72$
$47625.95241$ | $-1.05206955$ | $212.72$
$52917.72490$ | $-1.11751295$ | $212.72$
$58209.49739$ | $-1.17695850$ | $212.72$
$63501.26988$ | $-1.23066400$ | $212.72$
$68793.04237$ | $-1.27906400$ | $212.72$
$74084.81486$ | $-1.32266143$ | $212.72$
$79376.58735$ | $-1.36196558$ | $212.72$
$84668.35984$ | $-1.39745954$ | $212.72$
$89960.13233$ | $-1.42958499$ | $212.72$
$95251.90482$ | $-1.45873706$ | $212.72$
$100543.67731$ | $-1.48526460$ | $212.72$
$105835.44980$ | $-1.50947315$ | $212.72$
Table 4: The same as in Table 3 for the $\mathrm{Li}^{2+}-\bar{p}$ compound. $R$, fm |  $E_{\mathrm{A-DKB}}$, a.u. |  $|\Delta E_{\mathrm{Z}}|\times 10^{6}$, au
---|---|---
$0.0$ | $-2.00010648$ | $212.71$
$3527.84833$ | $-2.03167805$ | $212.71$
$7055.69665$ | $-2.11360541$ | $212.71$
$10583.54498$ | $-2.22996804$ | $212.71$
$14111.39331$ | $-2.36677964$ | $212.71$
$17639.24163$ | $-2.51210687$ | $212.70$
$21167.08996$ | $-2.65691048$ | $212.70$
$24694.93829$ | $-2.79524842$ | $212.70$
$28222.78661$ | $-2.92381308$ | $212.69$
$31750.63494$ | $-3.04120258$ | $212.69$
$35278.48327$ | $-3.14723436$ | $212.69$
$38806.33159$ | $-3.24242367$ | $212.69$
$42334.17992$ | $-3.32763343$ | $212.69$
$45862.02825$ | $-3.40385842$ | $212.69$
$49389.87657$ | $-3.47210176$ | $212.69$
$52917.72490$ | $-3.53330998$ | $212.69$
$56445.57323$ | $-3.58834337$ | $212.69$
$59973.42155$ | $-3.63796645$ | $212.69$
$63501.26988$ | $-3.68284942$ | $212.69$
$67029.11821$ | $-3.72357495$ | $212.69$
$70556.96653$ | $-3.76064721$ | $212.69$
Table 5: The same as in Table 3 for the $\mathrm{C}^{5+}-\bar{p}$ compound. $R$, fm |  $E_{\mathrm{A-DKB}}$, a.u. |  $|\Delta E_{\mathrm{Z}}|\times 10^{6}$, au
---|---|---
$0.0$ | $-12.50416195$ | $212.63$
$1763.92416$ | $-12.59936726$ | $212.62$
$3527.84833$ | $-12.83172810$ | $212.62$
$5291.77249$ | $-13.14049340$ | $212.61$
$7055.69665$ | $-13.48287107$ | $212.61$
$8819.62082$ | $-13.82992424$ | $212.60$
$10583.54498$ | $-14.16368683$ | $212.60$
$12347.46914$ | $-14.47431391$ | $212.59$
$14111.39331$ | $-14.75748499$ | $212.59$
$15875.31747$ | $-15.01233849$ | $212.59$
$17639.24163$ | $-15.23999327$ | $212.59$
$19403.16580$ | $-15.44257313$ | $212.59$
$21167.08996$ | $-15.62260484$ | $212.59$
$22931.01412$ | $-15.78266972$ | $212.59$
$24694.93829$ | $-15.92521715$ | $212.59$
$26458.86245$ | $-16.05247671$ | $212.58$
$28222.78661$ | $-16.16642752$ | $212.58$
$29986.71078$ | $-16.26879914$ | $212.58$
$31750.63494$ | $-16.36108853$ | $212.58$
$33514.55910$ | $-16.44458386$ | $212.58$
$35278.48327$ | $-16.52039028$ | $212.58$
Table 6: The same as in Table 3 for the $\mathrm{S}^{15+}-\bar{p}$ compound. For this quasi-molecule, the values of the Zeeman shift in the $10$ T and $100$ T fields are indicated in the third and fourth columns, respectively. $R$, fm | $E_{\mathrm{A-DKB}}$, a.u. |  $|\Delta E_{\mathrm{Z}}|\times 10^{6}$, a.u.
---|---|---
$B=10$ T |  $B=100$ T
$0.0$ | $-112.83886661$ | $21.19$ | $211.87$
$661.47156$ | $-113.16215620$ | $21.19$ | $211.86$
$1322.94312$ | $-113.91709032$ | $21.18$ | $211.84$
$1984.41468$ | $-114.88068920$ | $21.18$ | $211.83$
$2645.88624$ | $-115.91503204$ | $21.18$ | $211.81$
$3307.3578$ | $-116.93787284$ | $21.18$ | $211.80$
$3968.82936$ | $-117.90372807$ | $21.18$ | $211.78$
$4630.30092$ | $-118.79068655$ | $21.18$ | $211.77$
$5291.77249$ | $-119.59129326$ | $21.18$ | $211.77$
$5953.24405$ | $-120.30645964$ | $21.18$ | $211.76$
$6614.71561$ | $-120.94155048$ | $21.18$ | $211.76$
$7276.18717$ | $-121.50397157$ | $21.17$ | $211.75$
$7937.65873$ | $-122.00174664$ | $21.17$ | $211.75$
$8599.13029$ | $-122.44272501$ | $21.17$ | $211.75$
$9260.60185$ | $-122.83417789$ | $21.17$ | $211.75$
$9922.07341$ | $-123.18262617$ | $21.17$ | $211.75$
$10583.54498$ | $-123.49380039$ | $21.17$ | $211.75$
$11245.01654$ | $-123.77267191$ | $21.17$ | $211.75$
$11906.48810$ | $-124.02351854$ | $21.17$ | $211.75$
$12567.95966$ | $-124.25000340$ | $21.17$ | $211.75$
$13229.43122$ | $-124.45525546$ | $21.17$ | $211.75$
Table 7: The same as in Table 6 for the $\mathrm{Kr}^{35+}-\bar{p}$ compound. $R$, fm | $E_{\mathrm{ADKB}}$, a.u. | $|\Delta E_{\mathrm{Z}}|\times 10^{6}$
---|---|---
$B=10$ T |  $B=100$ T
$0.0$ | $-622.81876775$ | $20.80$ | $208.02$
$293.98736$ | $-623.71972536$ | $20.80$ | $208.00$
$587.97472$ | $-625.69949444$ | $20.80$ | $207.96$
$881.96208$ | $-628.13546078$ | $20.79$ | $207.91$
$1175.94944$ | $-630.68467991$ | $20.79$ | $207.87$
$1469.93680$ | $-633.15910731$ | $20.78$ | $207.84$
$1763.92416$ | $-635.46389756$ | $20.78$ | $207.81$
$2057.91152$ | $-637.55844470$ | $20.78$ | $207.79$
$2351.89888$ | $-639.43357940$ | $20.77$ | $207.78$
$2645.88624$ | $-641.09748949$ | $20.77$ | $207.76$
$2939.87361$ | $-642.56707902$ | $20.77$ | $207.76$
$3233.86097$ | $-643.86273369$ | $20.75$ | $207.75$
$3527.84833$ | $-645.00528107$ | $20.74$ | $207.74$
$3821.83569$ | $-646.01440491$ | $20.74$ | $207.74$
$4115.82308$ | $-646.90793501$ | $20.74$ | $207.74$
$4409.81041$ | $-647.70161658$ | $20.74$ | $207.74$
$4703.79777$ | $-648.40913245$ | $20.74$ | $207.74$
$4997.78513$ | $-649.04225174$ | $20.73$ | $207.74$
$5291.77249$ | $-649.61103315$ | $20.74$ | $207.74$
$5585.75985$ | $-650.12404223$ | $20.74$ | $207.74$
$5879.74721$ | $-650.58856102$ | $20.74$ | $207.74$
Table 8: The same as in Table 6 for the $\mathrm{Ho}^{66+}-\bar{p}$ compound. For this quasi-molecule, the values of the Zeeman shift in the $3$, $4$, $5$, $10$ and $100$ T fields are indicated in the last five columns, respectively. $R$, fm | $E_{\mathrm{ADKB}}$, a.u. | $|\Delta E_{\mathrm{Z}}|\times 10^{6}$
---|---|---
$B=3$ T |  $B=4$ T |  $B=5$ T |  $B=10$ T |  $B=100$ T
$0.0$ | $-2321.02291453$ | $5.86$ | $7.81$ | $9.76$ | $19.52$ | $195.20$
$157.96336$ | $-2323.82451644$ | $5.85$ | $7.81$ | $9.76$ | $19.51$ | $195.14$
$315.92672$ | $-2329.01021529$ | $5.85$ | $7.80$ | $9.75$ | $19.50$ | $195.04$
$473.89007$ | $-2334.84243831$ | $5.85$ | $7.80$ | $9.75$ | $19.49$ | $194.95$
$631.85343$ | $-2340.60253542$ | $5.85$ | $7.79$ | $9.74$ | $19.49$ | $194.87$
$789.81679$ | $-2345.96445844$ | $5.84$ | $7.79$ | $9.74$ | $19.48$ | $194.80$
$947.78015$ | $-2350.80386659$ | $5.84$ | $7.79$ | $9.74$ | $19.47$ | $194.75$
$1105.74350$ | $-2355.10160294$ | $5.84$ | $7.79$ | $9.74$ | $19.47$ | $194.71$
$1263.70686$ | $-2358.87442259$ | $5.84$ | $7.79$ | $9.73$ | $19.47$ | $194.68$
$1421.67022$ | $-2362.16494487$ | $5.84$ | $7.79$ | $9.73$ | $19.47$ | $194.66$
$1579.63358$ | $-2365.03189436$ | $5.84$ | $7.79$ | $9.73$ | $19.46$ | $194.65$
$1737.59694$ | $-2367.53574608$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.64$
$1895.56029$ | $-2369.72982539$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.63$
$2053.52365$ | $-2371.65927895$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.63$
$2211.48701$ | $-2373.36217338$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.63$
$2369.45037$ | $-2374.87051899$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.62$
$2527.41373$ | $-2376.21115360$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.62$
$2685.37708$ | $-2377.40656969$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.62$
$2843.34044$ | $-2378.47569427$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.62$
$3001.30380$ | $-2379.43460173$ | $5.84$ | $7.78$ | $9.73$ | $19.46$ | $194.62$
$3159.26716$ | $-2380.29712949$ | $5.84$ | $7.79$ | $9.73$ | $19.46$ | $194.62$
Table 9: The same as in Table 8 for the $\mathrm{Re}^{74+}-\bar{p}$ compound. The values of the Zeeman shift in the $2$, $3$, $4$, $5$, $10$ and $100$ T fields are indicated in the last six columns, respectively. $R$, fm | $E_{\mathrm{ADKB}}$, a.u. | $|\Delta E_{\mathrm{Z}}|\times 10^{6}$
---|---|---
$B=2$ T |  $B=3$ T |  $B=4$ T |  $B=5$ T |  $B=10$ T |  $B=100$ T
$0.0$ | $-2972.30805239$ | $3.81$ | $5.71$ | $7.61$ | $9.51$ | $19.03$ | $190.29$
$141.11393$ | $-2976.09312938$ | $3.80$ | $5.71$ | $7.61$ | $9.51$ | $19.02$ | $190.21$
$282.22787$ | $-2982.65931510$ | $3.80$ | $5.70$ | $7.60$ | $9.50$ | $19.01$ | $190.10$
$423.34180$ | $-2989.80282984$ | $3.80$ | $5.70$ | $7.60$ | $9.50$ | $19.00$ | $189.98$
$564.45573$ | $-2996.70635801$ | $3.80$ | $5.70$ | $7.59$ | $9.49$ | $18.99$ | $189.89$
$705.56966$ | $-3003.04369779$ | $3.80$ | $5.69$ | $7.59$ | $9.49$ | $18.98$ | $189.81$
$846.68360$ | $-3008.68956528$ | $3.79$ | $5.69$ | $7.59$ | $9.49$ | $18.97$ | $189.76$
$987.79753$ | $-3013.65734919$ | $3.79$ | $5.69$ | $7.59$ | $9.49$ | $18.97$ | $189.71$
$1128.91146$ | $-3018.00053903$ | $3.79$ | $5.69$ | $7.59$ | $9.48$ | $18.97$ | $189.68$
$1270.02540$ | $-3021.77354569$ | $3.79$ | $5.69$ | $7.59$ | $9.48$ | $18.97$ | $189.66$
$1411.13933$ | $-3025.03933807$ | $3.79$ | $5.69$ | $7.59$ | $9.48$ | $18.96$ | $189.65$
$1552.25326$ | $-3027.86970304$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.64$
$1693.36720$ | $-3030.33588984$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.63$
$1834.48113$ | $-3032.49857997$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.62$
$1975.59506$ | $-3034.40616085$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.62$
$2116.70900$ | $-3036.09723779$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.62$
$2257.82293$ | $-3037.60300317$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.62$
$2398.93686$ | $-3038.94886299$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.62$
$2540.05079$ | $-3040.15561730$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.61$
$2681.16473$ | $-3041.24039834$ | $3.79$ | $5.69$ | $7.59$ | $9.48$ | $18.96$ | $189.62$
$2822.27866$ | $-3042.21745820$ | $3.79$ | $5.69$ | $7.58$ | $9.48$ | $18.96$ | $189.61$
Table 10: The same as in Table 8 for the $\mathrm{U}^{91+}-\bar{p}$ compound. The Zeeman shift is presented at the field strength $1$, $2$, $3$, $4$, $5$, $10$, $100$ T. $R$, fm | $E_{\mathrm{ADKB}}$, a.u. | $|\Delta E_{\mathrm{Z}}|\times 10^{6}$
---|---|---
$B=1$ T |  $B=2$ T |  $B=3$ T |  $B=4$ T |  $B=5$ T |  $B=10$ T |  $B=100$ T
$0.0$ | $-4731.61940826$ | $1.77$ | $3.54$ | $5.31$ | $7.08$ | $8.85$ | $17.71$ | $177.06$
$115.03853$ | $-4739.21337367$ | $1.77$ | $3.54$ | $5.31$ | $7.08$ | $8.85$ | $17.69$ | $176.92$
$230.07706$ | $-4750.31646276$ | $1.77$ | $3.53$ | $5.30$ | $7.07$ | $8.84$ | $17.67$ | $176.74$
$345.11560$ | $-4761.37516295$ | $1.76$ | $3.53$ | $5.30$ | $7.06$ | $8.83$ | $17.66$ | $176.58$
$460.15413$ | $-4771.48208068$ | $1.76$ | $3.53$ | $5.29$ | $7.06$ | $8.82$ | $17.64$ | $176.46$
$575.19266$ | $-4780.37762099$ | $1.76$ | $3.53$ | $5.29$ | $7.05$ | $8.82$ | $17.64$ | $176.36$
$690.23119$ | $-4788.10088334$ | $1.76$ | $3.53$ | $5.29$ | $7.05$ | $8.82$ | $17.63$ | $176.30$
$805.26973$ | $-4794.73586551$ | $1.76$ | $3.52$ | $5.29$ | $7.05$ | $8.81$ | $17.62$ | $176.25$
$920.30826$ | $-4800.37950112$ | $1.76$ | $3.52$ | $5.29$ | $7.05$ | $8.81$ | $17.62$ | $176.21$
$1035.34679$ | $-4805.23045976$ | $1.76$ | $3.52$ | $5.29$ | $7.05$ | $8.81$ | $17.62$ | $176.19$
$1150.38532$ | $-4809.43838298$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.62$ | $176.17$
$1265.42386$ | $-4813.08395527$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.16$
$1380.46239$ | $-4816.22345098$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$1495.50092$ | $-4818.91377728$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$1610.53945$ | $-4821.22305291$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$1725.57799$ | $-4823.22648989$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$1840.61652$ | $-4824.99169849$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$1955.65505$ | $-4826.56937959$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.62$ | $176.15$
$2070.69358$ | $-4827.99460806$ | $1.76$ | $3.52$ | $5.28$ | $7.05$ | $8.81$ | $17.61$ | $176.15$
$2185.73211$ | $-4829.29165902$ | $1.76$ | $3.52$ | $5.28$ | $7.04$ | $8.81$ | $17.61$ | $176.15$
$2300.77065$ | $-4830.47781493$ | $1.76$ | $3.52$ | $5.28$ | $7.04$ | $8.81$ | $17.61$ | $176.15$


# Centrality and cumulative activities in hadron collisions

S.M. Troshin, N.E. Tyurin
NRC “Kurchatov Institute”–IHEP
Protvino, 142281, Russian Federation,
<EMAIL_ADDRESS>
We discuss the notion of cumulative activity for elastic events in hadron
collisions, its energy dependence and connection with the reflective
scattering mode. The interrelation between centrality and cumulative
activities of the elastic and inelastic events has been established for hadron
scattering. Considerations on its application are presented.

##  1 Introduction. The distinctive features of reflective scattering mode

An apparently natural expectation is that hadron collision energy growth must
be accompanied by an increase of the relative weight of the inelastic
interactions because the hadrons are extended objects. However, a
significantly increasing relative contribution of the elastic scattering
events to $pp$–interactions was observed [1].

The elastic scattering matrix element in the impact parameter representation
$S(s,b)$ should obey inequality $|S|\leq 1$. In the case of pure imaginary
elastic scattering amplitude we obtain $S=1-2f$. The reflective scattering
mode corresponds to negative values of the function $S$ [2]. The term
reflective was adopted from optics.

Evolution of central density of a scatterer with the energy and appearance of
the reflective scattering mode can be illustrated as the scattering of hadrons
with hard cores (Fig. 1):

Figure 1: Illustration of hadron scattering in the reflective scattering mode.
It corresponds to an appearance of hadron repulsive cores [3, 4].

This picture assumes that the unitarity saturation takes place in the limit of
$s\to\infty$, i.e the imaginary part of partial elastic scattering amplitude
${\imaginary}f_{l}(s)$ tends to unity.

In the impact parameter representation, the amplitude $f(s,b)$ obey the
unitarity relation:

$\mbox{Im}f(s,b)=h_{el}(s,b)+h_{inel}(s,b).$ (1)

The function $h_{el}(s,b)\equiv|f(s,b)|^{2}$ is a contribution of the elastic
intermediate state while $h_{inel}(s,b)$ is a contribution of the inelastic
intermediate states.

A central profile of the elastic overlap function and peripheral form of the
inelastic overlap function are encoded into the relation between derivatives
(in the case of a pure imaginary scattering amplitude) when the reflective
scattering mode ($S<0$) occures

$\frac{\partial h_{el}}{\partial s}=\left(\frac{1-S}{S}\right)\frac{\partial
h_{inel}}{\partial s}.$ (2)

The qualitative $b$–dependencies of $h_{el}$ and $h_{inel}$ are presented in
Fig. 2 where arrows indicate changes with the energy of the respective
profiles. The evolution in the opposite directions of the elastic and
inelastic overlap funcitions is a characteristic feature of the reflective
scattering. It is to be noted that the both functions change in the same way
in the shadow scattering mode.

Figure 2: Profiles and energy evolution (indicated by arrows) of the elastic
and inelastic overlap functions.

The centrality definition in the case of hadron interactions is [5]:

$c^{b}(s)\equiv\frac{\sigma^{b}_{tot}(s)}{\sigma_{tot}(s)},$ (3)

where

$\sigma^{b}_{tot}(s)=8\pi\int_{0}^{b}\mbox{Im}f(s,b^{\prime})b^{\prime}db^{\prime}.$
(4)

Note, that $\sigma^{b}_{tot}(s)\to\sigma_{tot}(s)$ at $b\to\infty$. In. Eq.
(3) the total cross–section replaces the inelastic one used in the case of
nuclear scattering. The straightforward extension of the nuclear scattering
case to hadron collisions would not lead to correct representation of the
impact parameter.

The definition of centrality reflects the level of perepherality under hadron
collisions. The origin can obviously be traced from the impact parameter
concept but the appeal to centrality and other cumulative collision
characteristics is dictated by the complexity of high energy dynamics.

The cumulative properties of the inelastic events have been discussed in [6].
The introduced quantity $a_{inel}^{b}(s)$ closely related to the inelastic
overlap function is a way to express a degree of peripherality for the
inelastic events.

Here we concentrate on cumulative characteristics of the elastic events and
their relation to cumulative properties of the inelastic events and centrality
related through unitarity.

##  2 Cumulative activity of elastic events in hadron interactions

Definition of cumulative elastic activity in hadron interactions is similar to
definition of the inelastic activity [6]:

$a^{b}_{el}(s)\equiv\frac{\sigma^{b}_{el}(s)}{\sigma_{el}(s)},$ (5)

where

$\sigma^{b}_{el}(s)=2\pi\int_{0}^{b}\sigma_{el}(s,b^{\prime})b^{\prime}db^{\prime}$
(6)

and $\sigma_{el}(s,b)$ is the differential contribution
($\sigma_{el}(s,b)\equiv d\sigma_{el}/d^{2}\mathbf{b}$) to the elastic
cross–section at the impact parameter $\mathbf{b}$ ($b\equiv|\mathbf{b}|$).
The function $a^{b}_{el}(s)$ is positive, it varies from zero to unity and
determines the fraction of elastic events generated in hadron collisions with
the impact parameter values in the interval from zero to $b$.

The dimensionless function $\sigma_{el}(s,b)$ is proportional to the elastic
overlap function $h_{el}(s,b)$:

$\sigma_{el}(s,b)=4h_{el}(s,b).$

Evidently, the differential elastic activity is determined by the elastic
overlap function $h_{el}(s,b)$:

$\frac{\partial a^{b}_{el}(s)}{\partial b}=\frac{8\pi
b}{\sigma_{el}(s)}h_{el}(s,b).$ (7)

The advantage of using the cumulative elastic activity is related to a
probabalistic nature of the impact parameter reconstruction. The combined
studies of centrality and cumulative elastic and inelastic activities in
hadron interactions would be useful for the studies of hadron interaction
dynamics. Relationship between these quantities follows from unitarity:

$c^{b}(s)=a_{el}^{b}(s)\frac{\sigma_{el}(s)}{\sigma_{tot}(s)}+a_{inel}^{b}(s)\frac{\sigma_{inel}(s)}{\sigma_{tot}(s)}$
(8)

Eq. (8) has a form similar to the relation between averaged values of the
impact parameter $\langle b^{2}\rangle_{i}$ $(i=tot,el,inel)$ [7, 8]. However,
the interrelation between the functions $\langle b^{2}\rangle_{i}$ is less
sensitive to the pecularities of the impact parameter dependencies since,
contrary to Eq. (8), it includes quantities integrated over an entire region
of the impact parameter variation.

Eq. (8) is just another expression of the unitarity relation projected onto
the space of the relative event’s numbers. It is to be noted that there are no
similar relations between centrality and events activities for the cases of
nucleus–nucleus or proton–nucleus collisions.

To illustrate and discuss energy dependence of the cumulative elastic activity
we use the unitarization scheme based on representation of the scattering
amplitude $f(s,b)$ in a rational form. This scheme allows to cover the entire
range of the elastic scattering amplitude variation allowed by unitarity [9].

$S$–matrix element $S(s,b)$ of the elastic scattering is written as the Cayley
transform which maps nonnegative real numbers (in the pure imaginary case
$f\to if$) to the interval $[-1,1]$ of $S(s,b)$ variation111For the general
case, this one-to-one transform is a solution (in a rational form) of Eq. (1)
provided that $\imaginary U\geq 0$; it maps upper half– plane of complex
values of $U$ onto a unit circle in the complex plane of $S$ [both $U$ and $S$
$(S=1+2if)$ are complex functions in this case]; the point $S=0$ is a regular
one, it corresponds to finite values of the energy and impact parameter [10].:

$S(s,b)={[1-U(s,b)]}/{[1+U(s,b)]}.$ (9)

The function $U(s,b)$ is considered as an input amplitude and is an object for
application of the unitarization procedure. The choice of this function in the
form

$U(s,b)=g(s)\omega(b),$ (10)

has been described in [6]. In Eq. (10) $g(s)\sim s^{\lambda}$ ($\lambda>0$) at
large values of $s$. Such function $g(s)$ implies saturation of unitarity for
the scattering amplitude $f(s,b)$. The function $\omega(b)\sim\exp({-\mu b})$
has been chosen to meet analytical properties of the scattering amplitude. The
use of Eq. (10) allows then to calculate the cumulative elastic activity with
the following asymptotic dependence at large $s$ and $b$–fixed:

$a^{b}_{el}(s)\propto[1-{C}{s^{-\lambda}}]{\ln^{-2}s},\,C>0.$ (11)

This dependence results from unitarity saturation assumed by Eqs. (9) and
(10).

##  3 Conclusion. Comments on the experimental measurements.

Estimation of the impact parameter for a given $pp$-collision event from the
experimental data has been considered in [5].

The event classification by multiplicity of the final state seems to be
appropriate for the cumulative inelastic activity estimation but it is not a
relevant tool for centrality reconstruction since the elastic channel
contribution is almost neglected in that case because the average number of
produced particles is much greater than two at high energies. However, the
total fraction of the elastic events is about $30$% at the LHC energies [1].

The relevant observable for centrality determination can be a sum of the
transverse energies of the final state particles measured in the detector
equiped with a hadron calorimeter. The ALICE, ATLAS and CMS experiments [11]
are, in principle, feasible for solution of this task. The respective method
was described in [12].

To extract the cumulative activity of elastic events $a^{b}_{el}(s)$ it is
sufficient to restore the functions $c^{b}(s)$ and $a^{b}_{inel}(s)$ from the
respective experimental data on the total transverse energy and multiplicity
hits registered by the detector and apply then Eq. (8).

It should be emphasized that any quantity integrated over entire region of its
$b$–variation (i.e. those taken at $-t=0$) is not sensitive to the
pecularities of $b$– and/or $t$–behaviour and, therefore, cannot provide a
complete information.

The $b$-dependent differential quantities are responsible for a number of
important conclusions such as transition to a peripheral form of the inelastic
overlap function contrary to a persisting central distribution of the elastic
overlap function over the impact parameter. This allows to highlight the
structure of the hadron interaction region [6]. However, the use of
$b$-dependent differential quantities implies reduced statistical significance
under the experimental studies. It can make the final results disputable. In
that respect, the cumulative quantities allow to partly resolve the case. The
respective cumulative activities and centrality extracted from the
experimental data can be used for the elastic scattering amplitude
reconstruction in the impact parameter space. The proposed method is universal
and can be extended to the case of hadron scattering [5].

Unitarization procedure as a way to satisfy unitarity constraints serves also
to mitigation of the assumed strong energy dependence of the input amplitude
under transition to the final scattering amplitude. Strong energy dependence
of the input contribution transforms into the observed effects of soft
interactions dynamics. Unitarization procedure results also in additional
dynamic mechanisms such as generation of the energy growth of the slope
parameter in the elastic scattering differential cross–section.

Saturation of unitarity corresponds to the principle of maximal strength of
strong interactions proposed long ago by Chew and Frautchi [13, 14]. It leads
to energy decrease ($b$ is fixed) of the inelastic activity $a^{b}_{inel}(s)$
to zero and increase of the normalized elastic activity $a^{b}_{el}(s)$ to
unity as well as evolution of hadron interaction region to a black ring with
inner reflective core [6]. Studies of those quantities could also provide an
important information on the asymptotic regime of QCD in soft domain.

The reflective ability of the inner part of the interaction region increases
with the energy [10] along with similar increase of the cumulative activity of
elastic events.

Further searches for experimental manifestations of the reflective mode in the
elastic and inelastic hadron interactions at the available energies is an
important issue. The measurements of centrality combined with measurements of
the cumulative elastic and inelastic activities in $pp$-collisions can be
promising for that purpose.

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