# Two-dipole and three-dipole interaction coefficients of GROUP XII elements

Neelam Shukla1, Harpreet Kaur2, Bindiya<EMAIL_ADDRESS>and
Rajesh<EMAIL_ADDRESS>1Department of Physics,
Indian Institute of Technology Roorkee, 247667, Roorkee, India  2Department of
Physics, Guru Nanak Dev University, Amritsar, Punjab-143005, India

###### Abstract

The study of long-range interactions is increasingly becoming essential due to
its various applications in cold atomic physics. These interactions can be
conveniently expressed in terms of dispersion coefficients. In the present
work, theoretical calculations have been carried out to estimate the two-
dipole ($C_{6}$) and three-dipole ($C_{9}$) dispersion coefficients between
the group XII atoms and their ions, viz., Zn, Cd, Hg, Zn+, Cd+, and Hg+. To
obtain these coefficients, the dynamic dipole polarizabilities and reduced
matrix elements required are evaluated using the relativistic methods for both
the considered atoms and ions. Further, using the calculated matrix elements,
the oscillator strengths corresponding to leading transitions and static
dipole polarizabilities are determined. These are compared with the available
values in the literature through which the accuracy of the present
calculations ascertains. However, the oscillator strengths for few transitions
of Zn, Cd, Hg, Zn+, Cd+, and Hg+, are not reported earlier, and we have
provided their values in this work. Thereafter, the dispersion $C_{6}$ and
$C_{9}$ coefficients are calculated for various combinations of the group XII
elements, and most of these coefficients were not reported earlier in the
literature. Furthermore, the combinations for which the other theoretical and
experimental values were available for comparison showed good agreement. We,
therefore, believe that our results reported for $C_{6}$ and $C_{9}$
coefficients for various combinations of group XII elements are reliable.

##  I Introduction

The research on laser-cooled ions and atoms is of great interest due to its
involvement in several aspects of ultra-cold atomic physics. It has been
suggested in several reports that the study of controlled ion-atom cold
collisions can be utilized in future quantum information processing Doerk  _et
al._ (2010); Harter and Denschlag (2014). For this purpose, to gain the
precise information about the long-range interatomic interactions among the
cooled atoms and ions is very important because it plays key role in diverse
investigations, viz., atom-ion hybrid traps, determination of stability of
Bose-Einstein condensates & scattering length, experiments on
photoassociation, analysis of Feshbach resonances, and many other Smith  _et
al._ (2005); Zipkes  _et al._ (2010a); Gacesa  _et al._ (2016); Roberts  _et
al._ (1998); Amiot and Verges (2000); Leo  _et al._ (2000); Leanhardt  _et
al._ (2003); Yuju  _et al._ (2004). The involvement of long-range interactions
in low-temperature collisions Zipkes  _et al._ (2010a); Schmid  _et al._
(2010); Bodo  _et al._ (2008); Zhang  _et al._ (2009a, b); Idziaszek  _et al._
(2009); Gao (2010); Cote (2000); Cote and Dalgarno (2000); Makarov  _et al._
(2003) make it essential to study and it also helps in determining the
collisional frequency shift. Further, the investigation of such interactions
provides fundamental knowledge about charge transfer processes Cote (2000);
Cote and Dalgarno (2000); Zhang  _et al._ (2009a, b); Sayfutyarova  _et al._
(2013), ion-atom bound states Cote  _et al._ (2002), formation of cold
molecular ions, and spin exchange reactions Schneider  _et al._ (2010);
Staanum  _et al._ (2010), which helps us to understand them more clearly.

The long-range interaction potential is generally written in terms of the
inverse of the interatomic distance $R$ as a power series (see Eq. (1)). The
multiplier of the term that has $R^{-6}$ represents the two-body dipole
interaction known as $C_{6}$ dispersion coefficient and contributes the most
in the series, due to which it becomes imperative. Similarly, the term with
$C_{9}$ dispersion coefficient acts as a leading contributor to the non-
additive part of the potential when three-body dipole interaction is
considered. Thus, making these two coefficients essential to explore. Due to
this reason, many theoretical and experimental studies have been reported
considering various ion-atom combinations, and details of which are given in
Ref. Shukla  _et al._ (2020). In addition, different experiments have been
performed to understand the nature of force in-between the ultracold ions and
atoms in hybrid traps Smith  _et al._ (2005); Grier  _et al._ (2009); Zipkes
_et al._ (2010b, c); Hall  _et al._ (2011); Hall and Willitsch (2012);
Sullivan  _et al._ (2012); Ravi  _et al._ (2012); Lee  _et al._ (2013);
Tacconi  _et al._ (2011); McLaughlin  _et al._ (2014).

The elements of group XII (Zn, Cd, Hg) are very important because these are
proposed as potential candidates for optical lattice clock Wang and Ye (2007);
Brickman  _et al._ (2007); Hachisu  _et al._ (2008). Further, elements from
group II (alkaline-earth), group XII, and group XVIII (rare-gas) are
considered very suitable for van der Waals molecules Qiao  _et al._ (2012). In
this regard, the dispersion coefficients of alkaline-earth and rare-gas dimers
have been widely studied, and accurately determined Porsev and Derevianko
(2002); Shukla  _et al._ (2020); Mitroy and Zhang (2007); Tang  _et al._
(2010). However, less attention has been paid towards the dispersion
coefficients of group XII dimers, and only limited and sporadic data are
available for immediate use. Therefore, this lack of data motivates us to
calculate the two-dipole and three-dipole interactions for various
combinations of group XII elements (Zn, Cd, Hg) systematically through the ab-
initio method. The $C_{6}$ coefficients are evaluated for atom-atom
(homonuclear and heteronuclear pairs) and atom-ion dimers. In addition, the
$C_{9}$ coefficients are obtained for atom-atom-atom and atom-atom-ion
combinations. Here, we have taken only the singly ionized ions of group XII
elements, viz., Zn+, Cd+, and Hg+.

The evaluation of the dispersion coefficients for any combination is directly
related to the dynamic dipole polarizabilities of atoms/ions, which we have
obtained with the help of electric-dipole (E1) reduced matrix elements. The E1
reduced matrix elements are calculated by employing relativistic methods for
the considered atoms and ions. Further, to ascertain the accuracy of the
determined reduced matrix elements, we obtained using them the oscillator
strengths for the leading transitions in atoms and ions which are then
compared with the values reported by the National Institute of Standards and
Technology (NIST) database Kramida  _et al._ (2018). For the same purpose, the
static dipole polarizabilities in the ground states of atoms/ions are
determined and compared with the measurements Kompitsas  _et al._ (1994); Li
_et al._ (2018); Schwerdtfeger and Nagle (2019) wherever available. The
outline of the present work is as follows: In Section II, we give a brief
overview of the theoretical methodology employed in the present work. The
results are presented and discussed in Section III. The atomic units (a.u.)
have been used throughout the manuscript.

##  II THEORETICAL CONSIDERATION

###  II.1 Dispersion coefficients

The long-range interaction potential among the three atomic species in their
ground states is expressed by following expression Lee  _et al._ (2013),

$\displaystyle V(\vec{R_{12}},\vec{R_{23}},\vec{R_{31}})$ $\displaystyle=$
$\displaystyle-\frac{C_{6}^{(12)}}{R^{6}_{12}}-\frac{C_{8}^{(12)}}{R^{8}_{12}}-\ldots-\frac{C_{6}^{(23)}}{R^{6}_{23}}-\frac{C_{8}^{(23)}}{R^{8}_{23}}$
$\displaystyle-$
$\displaystyle\ldots-\frac{C_{6}^{(31)}}{R^{6}_{31}}-\frac{C_{8}^{(31)}}{R^{8}_{31}}-\ldots$
$\displaystyle+$
$\displaystyle(1+3\cos\theta_{1}\cos\theta_{2}\cos\theta_{3})\frac{C^{123}_{9}}{R^{3}_{12}R^{3}_{23}R^{3}_{31}}.$

In this equation, the maximum contribution comes from the $C_{n}^{(ij)}$
parameters, which are the dispersion coefficients for two-dipole interactions.
The three-dipole interaction terms $C_{n}^{(ijk)}$, are the third-order
correction to the ground-state energy and are known as the non-additive part
of the potential. Therefore, it has different sign as compared to other
interaction terms. Here, $i,j,k=1,2,3$, are representing the three different
atomic species. $R_{ij}$ is the inter-atomic distance between the two atomic
systems. Further, the two $C_{6}^{(ij)}$ and $C_{9}^{(ijk)}$ coefficients can
be expressed in terms of frequency-dependent dipole polarizability
$\alpha(\iota\omega)$ as follows:

$C_{6}^{(ij)}=\frac{3}{\pi}\int_{0}^{\infty}d\omega\alpha_{i}(\iota\omega)\alpha_{j}(\iota\omega),$
(2)

and

$C_{9}^{(ijk)}=\frac{3}{\pi}\int_{0}^{\infty}d\omega\alpha_{i}(\iota\omega)\alpha_{j}(\iota\omega)\alpha_{k}(\iota\omega).$
(3)

In order to calculate dispersion coefficients between the group XII atoms and
their ions precisely, it is essential to use accurate values of frequency
dependent dipole polarizabilities, whose evaluation method is discussed in the
next section.

###  II.2 Dipole Polarizability

Following Ref. Arora  _et al._ (2012), the required dynamic dipole
polarizability ($\alpha^{v}(\iota\omega)$) for an atomic system in its ground
state $v$ can be written through three contributing terms:

$\alpha^{v}(\iota\omega)=\alpha^{v}_{c}(\iota\omega)+\alpha^{v}_{vc}(\iota\omega)+\alpha^{v}_{val}(\iota\omega),$
(4)

where, $\alpha^{v}_{c}(\iota\omega)$, $\alpha^{v}_{vc}(\iota\omega)$ and
$\alpha^{v}_{val}(\iota\omega)$ denote the contributions from the core,
valence-core and valence correlations, respectively. The core contributions
are evaluated using the Dirac-Fock (DF) method for the closed-shell
configurations Safronova  _et al._ (1998) by considering excitations from core
to all valence orbitals. The valence-core $\alpha^{v}_{vc}(\iota\omega)$
contribution involves the transitions from core to occupied valence shells
which are forbidden by the Pauli exclusion principle. Since for the considered
group XII elements the valence orbital is doubly filled, therefore the
$\alpha^{v}_{vc}(\iota\omega)$ contribution for atoms is twice that of singly
ionized ions.

The dominant contribution to the total polarizability comes from the valence
part, which can be calculated using the sum-over-states approach by the
following expression:

$\displaystyle\alpha^{v}_{val}(\iota\omega)=\frac{2}{3(2J_{v}+1)}$
$\displaystyle\times\sum_{p>N_{c},p\neq
v}^{I}\frac{(E_{p}-E_{v})|\langle\psi_{p}||D||\psi_{v}\rangle|^{2}}{(E_{p}-E_{v})^{2}+\omega^{2}}.$

In this equation, the summation is carried over all the possible intermediate
states $p$ allowed by the electric dipole selection rules, and $N_{c}$ refers
to the core orbitals. Static dipole polarizability can be calculated by
substituting $\omega=0$. This major contribution is again categorized in two
parts, i.e., Main and Tail. The Main part includes the contributions primarily
from the ground-state to low-lying allowed transitions, and in the case of the
Tail part, it is from the ground state to the higher states of the atomic
system. In Eq. II.2, $\langle\psi_{p}||D||\psi_{v}\rangle|$ is the reduced
dipole matrix element and is related to $\langle\psi_{p}|D|\psi_{v}\rangle$
through a Clebsch–Gordan coefficient. For the Main part of the valence
contribution, the required matrix elements are calculated using multi-
configurational Dirac-Fock (MCDF) approximation and all-order (SD) method,
respectively, for atoms and ions. The energies $E$s corresponding to the
different states in these calculations are experimental and taken from the
NIST database Kramida  _et al._ (2018). Further, the Tail part of the valence
contributions is evaluated using the same expression, i.e., Eq. (5). The
matrix elements for the Tail part are calculated using the DF method for the
ions and the required energy obtained through this method. However, this
contribution for the atoms is neglected as these might not contribute
significantly and for the sake of simplicity of the calculation.

###  II.3 Evaluation of matrix element

From the above discussion, we find that we need to evaluate the reduced dipole
matrix elements for the determination of necessary dispersion coefficients
among the group XII atoms and their ions. The desired wave functions for group
XII atoms in their initial and final states are calculated through GRASP2K
code Jönsson  _et al._ (2013) under the MCDF approach. In the MCDF approach,
the atomic state wave function (ASFs) in the initial/final states can be
expressed as a linear combination of various configurational state wave
functions (CSFs), having the same total angular momentum and parity, i.e.,

$|\Psi_{v}\rangle_{\rm MCDF}=\sum_{n=1}^{N}a_{n}|\Phi_{n}\rangle$ (6)

here, $n$ denotes the number of CSFs, $a_{n}$ is the mixing coefficient of the
CSF $|\Phi_{n}\rangle$. First, using Dirac-Coulomb Hamiltonian, the required
expansion coefficients and single-particle orbital radial functions are
determined by utilizing the multi-configuration self-consistent field
calculations. Thereafter, by adding Breit and quantum electrodynamic
corrections, the relativistic configuration interaction calculations are
performed. To get accurate ASFs, we have used a large number of CSFs in the
linear contribution, and in the final stage, we keep only those CSFs that have
the corresponding mixing coefficient greater than 10-3.

Further, due to the suitability for the monovalent systems, we have utilized
the singles and doubles excitation approximation in the all-order (SD) method
Blundell  _et al._ (1991); Safronova and Johnson (2008) to compute their wave
functions. As compared to the MCDF approach, this method gives accurate
results for them. However, for divalent systems, employing the SD method is
rather challenging, and there are no codes available.

The wave functions of the group XII ions in their respective state $v$, having
the closed-core with a valence electron, is represented in the SD method by
the following expression,

$\displaystyle|\Psi_{v}\rangle_{\rm SD}$ $\displaystyle=$
$\displaystyle\left[1+\sum_{ma}\rho_{ma}a^{\dagger}_{m}a_{a}+\frac{1}{2}\sum_{mnab}\rho_{mnab}a_{m}^{\dagger}a_{n}^{\dagger}a_{b}a_{a}\right.$
$\displaystyle\left.+\sum_{m\neq
v}\rho_{mv}a^{\dagger}_{m}a_{v}+\sum_{mna}\rho_{mnva}a_{m}^{\dagger}a_{n}^{\dagger}a_{a}a_{v}\right]|\Phi_{v}\rangle,$

here, $|\Phi_{v}\rangle$ represents the Dirac Hartree Fock (DHF) wave function
of the state $v$. $a_{i}$ and $a^{\dagger}_{i}$ are the annihilation and
creation operators. The indices $\\{m,n\\}$ and $\\{a,b\\}$ denote the virtual
and core orbitals, respectively. $\rho_{ma}$ and $\rho_{mnab}$ are the single
and double core coefficients. $\rho_{mv}$ and $\rho_{mnva}$ are the single and
double valence excitation coefficients. The single-particle orbitals for the
SD method have been constructed using the 70 B-spline functions, including a
cavity of radius $R=220$ a.u.

Further for a particular transition of the concerned ions and atoms, with the
help of calculated wave functions, the E1 reduced matrix elements are
determined. To ascertain the accuracy of the matrix elements evaluated using
the above-described methods, we also calculated the oscillator strengths and
compared these with the values available from the NIST database Kramida  _et
al._ (2018). The following expression is used to evaluate the oscillator
strengths Mitroy and Zhang (2008):

$f_{vp}=-\frac{303.756}{g_{v}\lambda_{vp}}\langle\psi_{p}||D||\psi_{v}\rangle|^{2},$
(8)

here $f_{vp}$ is the oscillator strength of the transition from state $v$ to
excited intermediate states $p$. $g_{v}$ is the statistical weight and
$\lambda_{vp}$ represents the transition wavelength in Angstrom.

##  III Results and Discussion

In the present work, the theoretical calculations have been carried out to
estimate the two-dipole ($C_{6}$) and three-dipole ($C_{9}$) dispersion
coefficients between the group XII atoms and their ions viz., Zn, Cd, Hg, Zn+,
Cd+, and Hg+. As explained earlier, to obtain these coefficients, the dynamic
dipole polarizabilities and reduced matrix elements required are evaluated
using the relativistic methods for both the considered atoms and ions.
Further, using the obtained reduced matrix elements, the oscillator strengths
corresponding to leading transitions and static dipole polarizabilities are
determined. All the calculated results are discussed below in their respective
sub-sections.

###  III.1 Oscillator strengths

Using Eq. 8, the oscillator strengths for the group XII atoms and ions are
calculated. The results for few leading transitions of the considered atoms
and ions are represented in Table 1. As can be seen from the table that there
is a close agreement between the present results and reported experimental
values Kramida  _et al._ (2018) except for Hg+. The exact reason for this
discrepancy for Hg+ is not known to us. While our values for Hg+ matches
completely with the theoretical results reported by Safronova and Johnson
Safronova and Johnson (2004). A thorough survey from the available literature
and database (e.g., NIST) reveals that for the transitions of group XII atoms
and their ions, only a limited number of oscillator strengths are reported.
Also, we have not found any reported values of the oscillator strengths for
any higher transitions of these atoms and ions. However, we believe that our
results, for which the previous values are not available for comparison,
should also be reliable. Thus, we can say that the reduced matrix elements of
the considered atomic and ionic systems which are used to calculate the
oscillator strengths, are also accurate as these are further utilized to
evaluate the polarizabilities.

###  III.2 Static dipole polarizabilities

With the help of determined reduced matrix elements, the static dipole
polarizabilities of the group XII atoms and their ions are calculated. As
mentioned above, that the sum-over-states approach has been utilized to obtain
the polarizability. It should be mentioned here that for the ground state
polarizability, only some initial transitions contribute the most; therefore,
the truncation of the summation introduces only 1.5% error in the calculation
for ions and even less for the atoms.

The calculated values of the static dipole polarizabilities for the considered
atoms and ions are given in Table 2, where these are also compared with the
previously reported theoretical Gould and Bucko (2016); Mitroy  _et al._
(2010); Chattopadhyay  _et al._ (2015) and experimental Kompitsas  _et al._
(1994); Li  _et al._ (2018); Schwerdtfeger and Nagle (2019) results. The
comparison from the theoretical results of the polarizability for Zn+ and Hg+
Mitroy  _et al._ (2010) show very close agreement with the present values. On
the other hand, the values reported by Chattopadhyay et al. Chattopadhyay  _et
al._ (2015) only for atoms and by Gould and Bucko Gould and Bucko (2016) for
all ions as well as atoms show some difference in comparison with our results.
This small difference arises due to the different methods used in the current,
and the previous work Gould and Bucko (2016); Chattopadhyay  _et al._ (2015).
Gould and BuckoGould and Bucko (2016) have utilized the time-dependent density
functional theory with exchange kernels to obtain polarizabilities for atoms
and ions. Whereas Chattopadhyay et al. Chattopadhyay  _et al._ (2015)
determined electric dipole polarizability of Zn, Cd, and Hg, by applying
perturbed relativistic coupled-cluster (PRCC) theory. However, in the present
study, the sum-over-sates approach has been employed, which allows us to use
precise experimental values of energies wherever feasible Kramida  _et al._
(2018). Thus, we believe that our determined values are quite reliable.
Further, on comparing the present results with the available measurements
Kompitsas  _et al._ (1994); Li  _et al._ (2018); Schwerdtfeger and Nagle
(2019), we find that there are slight discrepancies with the relative
difference in the range of 0.02-0.1 for ions, and in the case of atoms
difference varies between 0.08 to 0.1.

###  III.3 $C_{6}$ dispersion coefficient

Further, with the help of calculated dynamic dipole polarizabilities, $C_{6}$
dispersion coefficients are evaluated for the combinations of two atoms as
well as the atom and ion of group XII elements in their respective ground
state. The calculation of $C_{6}$ for homonuclear and heteronuclear dimers of
group XII atoms is especially useful for comparison purposes as there are some
previous results available in the literature. This will ascertain the accuracy
of our present method used to evaluate the dispersion coefficients for other
combinations. The values of $C_{6}$ for homonuclear and heteronuclear dimers
are listed in Table 3 and compared with other theoretical Grycuk  _et al._
(2012); Qiao  _et al._ (2012) and experimental results Goebel and Hohm (1995);
Goebel  _et al._ (1996). On comparison, we find that the values calculated by
Qiao et al. Qiao  _et al._ (2012) for all combinations are over-estimated by
their method as compare to our calculations. On the other hand, Grycuk et al.
Grycuk  _et al._ (2012) have computed the dispersion coefficients only for
homonuclear dimers, and their results match considerably with the present
calculated values. One can also see from the Table 3 that the values of Qiao
et al. Qiao  _et al._ (2012) drastically differ from the results of Grycuk et
al. Grycuk  _et al._ (2012). Further, we observe that our obtained
coefficients for homonuclear combinations exhibit excellent agreement with the
reported experiment results Goebel and Hohm (1995); Goebel  _et al._ (1996)
for Cd-Cd and Hg-Hg dimers, where in the case of Zn-Zn, our result deviates
from the measurement by $\sim$ 10%. The overall good agreement of our results
with the previously published works Grycuk  _et al._ (2012); Qiao  _et al._
(2012); Goebel and Hohm (1995); Goebel  _et al._ (1996) gives us confidence
that the dispersion coefficients ($C_{6}$) obtained by us for different
combinations between group XII atoms and their ions which are summarised in
Table 6 are quite reliable.

In addition, from the Tables 3 and 6, we notice that the relative magnitudes
of the determined $C_{6}$ coefficients follow a specific trend, i.e., (Zn-X)
$<$ (Hg-X) $<$ (Cd-X) (where X refers to the atom/ion of the group XII), that
is probably due to the relativistic effects in the polarizability Seth  _et
al._ (1997) which is different from the expected respective order of elements
(Zn, Cd, Hg) in the group XII.

###  III.4 $C_{9}$ dispersion coefficient

In addition to $C_{6}$, the $C_{9}$ dispersion coefficients are calculated for
the interaction of atom-atom-atom and atom-atom-ion combinations of the group
XII elements, and these are listed in Tables 6 and 6. Only for four
combinations, we found the previously reported values for $C_{9}$, and these
are given in Table 6. However, for the rest of the other combinations, the
$C_{9}$ coefficients which we are reporting here are the new ones. Now on
comparison of our results with the available results, we find that values
estimated by Qiao et al. Qiao  _et al._ (2012) drastically differ with ours,
similar to the $C_{6}$ case (as shown in Table 3). In fact, Qiao et al. Qiao
_et al._ (2012) have used Tang’s Tang (1969) one-term approximation formula to
evaluate their values for both the two- and three-dipole interaction terms,
and thus our calculations seem more reliable than there. We have already seen
from Table 3 that their values for $C_{6}$ were large as compare to our and
the experimental results Goebel and Hohm (1995); Goebel  _et al._ (1996).
Consequently, on the basis of good agreement between the present and
previously reported other theoretical and experimental values Grycuk  _et al._
(2012); Goebel and Hohm (1995); Goebel  _et al._ (1996) (Table 3) for the
$C_{6}$ of homonuclear and heteronuclear dimers, gives us the confidence that
the predicted three-dipole coefficients for various interactions are quite
reliable. Here, for the $C_{9}$ coefficients, we also observed the same trend
as we have seen in the $C_{6}$ that the combinations which involve Cd atom in
Tables 6 and 6, have higher value of $C_{9}$ dispersion coefficients with
respect to the other combinations. The reason for this could be the same as
discussed in Section C.

## Conclusion

In the present work, we have investigated in a detailed manner the two-dipole
($C_{6}$) and three-dipole ($C_{9}$) interaction coefficients of the group XII
elements (Zn, Cd, Hg) among themselves and with their singly ionized ions. For
this purpose, we have calculated the reduced dipole matrix elements of atoms
and ions by employing relativistic methods. Further, the oscillator strengths,
dynamic and static dipole polarizability of these atoms, and ions are
determined. Wherever possible, these results are compared with the other
theoretical calculations, measurements, and the values from the NIST database,
and overall good agreement is found. Thereafter, the calculated dynamic dipole
polarizabilities are utilized to evaluate the $C_{6}$ and $C_{9}$ dispersion
coefficients for various combinations. Most of these coefficients are reported
for the first time in the literature. We hope that our values will be useful
for the researchers for immediate use in various fields such as quantum
information processing, designing better atomic clocks, and understanding the
ion-atom hybrid mechanisms.

## Acknowledgements

The authors, N.S. is thankful to the Ministry of Human Resources and
Development (MHRD), Govt. B. A. is thankfull to the SERB-
TARE(TAR/2020/000189), New Delhi, India for research grant. RS are thankful to
the SERB-DST for the sanction of research grant No: CRG/2020/005597, New
Delhi, Govt. of India.

Table 1: Comparison of our calculated oscillator strengths ($f$) of the few
leading transitions with the previously available values. The numbers in
parentheses represent powers of 10.

Ion | Lower level | Upper level | Term | $f_{\rm present}$ | $f_{\rm previous}$
---|---|---|---|---|---
Zn+ | $4s$ | $4p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.262(0) | 0.246(0)a
Zn+ | $4s$ | $4p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.535(0) | 0.501(0)a
Zn+ | $4s$ | $5p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.001(0) | -
Zn+ | $4s$ | $5p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.001(0) | -
Cd+ | $5s$ | $5p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.252(0) | 0.244(0)a
Cd+ | $5s$ | $5p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.536(0) | 0.521(0)a
Cd+ | $5s$ | $6p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.001(0) | -
Cd+ | $5s$ | $6p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.0005(0) | -
Hg+ | $6s$ | $6p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.214(0) | 0.42(0)a
Hg+ | $6s$ | $6p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.507(0) | 0.98(0)a
Hg+ | $6s$ | $7p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{1/2}$ | 0.047(0) | 0.083(0)a
Hg+ | $6s$ | $7p$ |  ${}^{2}S_{1/2}$$\rightarrow$${}^{2}P_{3/2}$ | 0.023(0) | 0.014(0)a
Atom | Upper level | Lower level | Term | $f_{\rm present}$ | $f_{\rm previous}$
Zn | $4s^{2}$ | $4s4p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.164(-4) | -
Zn | $4s^{2}$ | $4s4p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 1.46(0) | 1.46(0)a
Zn | $4s^{2}$ | $4s5p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.654(-5) | -
Zn | $4s^{2}$ | $4s5p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 0.089(0) | 0.089(0)a
Cd | $5s^{2}$ | $5s5p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.543(-3) | 0.194(-2)a
Cd | $5s^{2}$ | $5s5p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 1.26(0) | 1.2(0)a
Cd | $5s^{2}$ | $5s6p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.289(-3) | -
Cd | $5s^{2}$ | $5s6p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 0.887(0) | -
Hg | $6s^{2}$ | $6s6p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.141(-1) | 0.243(-1)a
Hg | $6s^{2}$ | $6s6p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 0.119(0) | 0.115(0)a
Hg | $6s^{2}$ | $6s7p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{3}P_{1}$ | 0.197(-3) | -
Hg | $6s^{2}$ | $6s7p$ |  ${}^{1}S_{0}$$\rightarrow$${}^{1}P_{1}$ | 0.012 | -

aNISTKramida  _et al._ (2018)

Table 2: Comparison of our calculated static dipole polarizabilities
$(\alpha^{v}(0))$ (in a.u.) for the considered group XII atoms and their
singly ionized ions in the ground states with the available measurements and
other theoretical calculations.

Ion | State | Present Calculations | Other Calculations | Measurements
---|---|---|---|---
Zn+ | 4s | 18.05 | 17.9a,18.8b | 15.4c
Cd+ | 5s | 23.41 | 23.1a | 25.2(6)d
Hg+ | 6s | 19.07 | 17.5a,19.36b | -
Atom | State | Present Calculations | Other Calculations | Measurements
Zn | $4s^{2}$ | 35.33 | 38.4a,38.71e | 38.67f
Cd | $5s^{2}$ | 49.61 | 46.7a,48.2e | 48.2f
Hg | $6s^{2}$ | 28.65 | 33.5a,33.63e | 33.91f

aRef.Gould and Bucko (2016), bRef.Mitroy  _et al._ (2010), cRef.Kompitsas  _et
al._ (1994), dRef.Li  _et al._ (2018), eRef.Chattopadhyay  _et al._
(2015),fRef.Schwerdtfeger and Nagle (2019).

Table 3:  The calculated values of the two-dipole interactions ($C_{6}$)
between the two atoms (heteronuclear and homonuclear) of the group XII in
their respective ground state and comparison with the previously reported
results.

Dimers | Present Calculation | Other Calculations | Measurements
---|---|---|---
Zn-Zn | 225 | 298a,359b | 257.5c
Zn-Cd | 332 | 494.5b | -
Zn-Hg | 229 | 368.7b | -
Cd-Cd | 493 | 487a,686b | 466d
Cd-Hg | 347 | 515.8b | -
Hg-Hg | 258 | 272a,392b | 255c

aRef. Grycuk  _et al._ (2012), bRef. Qiao  _et al._ (2012), cRef. Goebel  _et
al._ (1996), dRef. Goebel and Hohm (1995).

Table 4: The calculated values of the two-dipole interactions ($C_{6}$)
between the atoms and singly ionized ions of the group XII in their respective
ground state.

X | Zn-X | Cd-X | Hg-X
---|---|---|---
Zn+ | 124 | 185 | 132
Cd+ | 172 | 259 | 190
Hg+ | 171 | 262 | 201

Table 5:  The calculated values of the three-dipole interactions ($C_{9}$)
among the three atoms of the group XII in their respective ground states and
comparison with the previously reported results.

X | Zn-Zn-X | Cd-Cd-X | Hg-Hg-X
---|---|---|---
Zn | 5667 | 11650 | 4776
| 8940a | 16239a | -
Cd | 8118 | 16751 | 6971
| 12023a | 22033a | -
Hg | 5124 | 10698 | 4666
| - | - | -

aQiao et al.Qiao  _et al._ (2012)

Table 6:  The calculated values of the three-dipole interactions ($C_{9}$)
among the two atoms and singly ionized ions of the group XII atoms in their
respective ground states.

X | Zn-Zn-X | Cd-Cd-X | Hg-Hg-X
---|---|---|---
Zn+ | 2994 | 6185 | 2588
Cd+ | 3960 | 8227 | 3524
Hg+ | 3592 | 7559 | 3401

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# Axion effects on gamma-ray spectral irregularities with AGN redshift
uncertainty

Hai-Jun Li<EMAIL_ADDRESS>Center for Advanced Quantum Studies,
Department of Physics, Beijing Normal University, Beijing 100875, China  Wei
Chao<EMAIL_ADDRESS>Center for Advanced Quantum Studies, Department of
Physics, Beijing Normal University, Beijing 100875, China

###### Abstract

We investigate the photon-axionlike particle (ALP) oscillation effects on TeV
gamma-ray spectral irregularities from the uncertain redshift active galactic
nuclei (AGN) VER J0521+211. The gamma-ray spectra are measured by Fermi-LAT
and VERITAS with the three flux states in 2013 and 2014. We set the combined
constraints on the ALP parameter ($m_{a},g_{a\gamma}$) space with these states
and test the extragalactic background light (EBL) absorption effect on ALP
constraints with the redshift limit scenarios $z_{0}\sim\mathcal{O}(0.1-0.3)$.
The 99% $\rm C.L.$ photon-ALP combined constraints set by VER J0521+211 are
roughly at $g_{a\gamma}\gtrsim 2.0\times 10^{-11}\rm\,GeV^{-1}$ for $1.0\times
10^{-9}\,{\rm eV}\lesssim m_{a}\lesssim 1.0\times 10^{-7}\,{\rm eV}$. We find
no clear connection between the redshift limit scenarios and the photon-ALP
constraints. Both the underestimated and overestimated redshifts can affect
the constraint results.

††preprint: BNU-22-129

##  I Introduction

Axions Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek (1978) and
axionlike particles (ALPs) Arvanitaki  _et al._ (2010); Svrcek and Witten
(2006) are ultralight pseudo-Nambu-Goldstone bosons (pNGBs), which are
potential dark matter (DM) candidates if nonthermally generated in the early
Universe through the misalignment mechanism Preskill  _et al._ (1983); Abbott
and Sikivie (1983); Dine and Fischler (1983); Sikivie (2010); Marsh (2016);
Chao  _et al._ (2022). See $\rm e.g.$ Refs. Di Luzio  _et al._ (2020); Choi
_et al._ (2021); Galanti and Roncadelli (2022) for recent reviews. The
interaction between ALPs and very high energy (VHE; $\sim\mathcal{O}(100)\,\rm
GeV$) photons in the astrophysical magnetic fields with Lagrangian
$-\frac{1}{4}g_{a\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}$ could lead to
detectable effects, such as a reduced TeV opacity of the Universe Mirizzi  _et
al._ (2007); Simet  _et al._ (2008).

The TeV gamma-rays from extragalactic sources are affected by the
extragalactic background light (EBL) absorption effect through the pair
production process, $\gamma_{\rm TeV}+\gamma_{\rm EBL}\to e^{+}+e^{-}$. In
this case, the photon-ALP interaction provides a natural mechanism to reduce
the EBL absorption and constrain the ALP properties (the ALP mass $m_{a}$ and
the photon-ALP coupling $g_{a\gamma}$) Dominguez  _et al._ (2011); Belikov
_et al._ (2011); Horns  _et al._ (2012); De Angelis  _et al._ (2011); Meyer
_et al._ (2013); Abramowski  _et al._ (2013); Ajello  _et al._ (2016); Guo
_et al._ (2021); Li  _et al._ (2021). The common mechanism is considering the
photon-ALP conversions and back-conversions in the astrophysical magnetic
fields. If there is significant photon-ALP mixing, the Universe would appear
to be more transparent than expected based on the pure EBL absorption. See
$\rm e.g.$ Refs. Reynés  _et al._ (2021); Li  _et al._ (2022); Jacobsen  _et
al._ (2022); Li (2022a); Mastrototaro  _et al._ (2022); Dessert  _et al._
(2022); Nakayama and Yin (2022); Eckner and Calore (2022); Bessho  _et al._
(2022); Bernal  _et al._ (2022) for recent studies on photon-ALP conversions
from the different extragalactic astrophysical sources.

In this work, we focus our attention on the photon-ALP oscillation effects on
TeV gamma-ray spectral irregularities from the uncertain redshift active
galactic nuclei (AGN) VER J0521+211. VER J0521+211 ($\rm
RA=05^{h}21^{m}45^{s}$, $\rm Dec=21^{\circ}12^{\prime}51.4^{\prime\prime}$,
J2000) is classified as the intermediate frequency peaked BL Lac (IBL) object,
which was first observed by the Very Energetic Radiation Imaging Telescope
Array System (VERITAS) in 2009 Archambault  _et al._ (2013). Since the lack of
optical emission features of the IBL object, the redshift of VER J0521+211 is
still unknown. Many studies show the redshift limits of this source with
$0.108\leqslant z_{0}\leqslant 0.34$ Archambault  _et al._ (2013); Shaw  _et
al._ (2013); Paiano  _et al._ (2017). Recently, VERITAS reported the TeV
gamma-ray observations of VER J0521+211 in 2013 and 2014 with the Fermi-LAT
and VERITAS data Adams  _et al._ (2022), suggesting the redshift upper limits
$z_{0}\leqslant 0.31$. Here we use these gamma-ray data and redshift limits of
VER J0521+211 to investigate the photon-ALP oscillation effects on TeV gamma-
ray spectral irregularities and test the EBL absorption effect on ALP
constraints. Since the latest gamma-ray data, the redshift uncertainties, and
the magnetic field parameters of VER J0521+211 are given in Ref. Adams  _et
al._ (2022) together, it may be a choice for us to investigate the photon-ALP
oscillation effects with this source.

This paper is structured as follows. In Sec. II, we describe the VHE gamma-ray
data and the redshift limits of VER J0521+211. In Sec. III, we briefly
introduce the ALP constraint method and the magnetic field parameters setup.
The resulting ALP constraints are shown in Sec. IV. Finally, we conclude in
Sec. V.

##  II Gamma-ray data and redshift limits

In this section, we describe the TeV gamma-ray data and the redshift limits of
VER J0521+211. In Ref. Adams  _et al._ (2022), the gamma-ray spectra of VER
J0521+211 in 2013 and 2014 are performed by the Bayesian block (BB) analysis
Scargle  _et al._ (2013), which are defined as the flux states BB1, BB2, and
BB3, respectively.

  * •

BB1: MJD 56580.0 $-$ MJD 56628.5
Corresponds to the intermediate state.

  * •

BB2: MJD 56628.5 $-$ MJD 56632.5
Corresponds to the high state.

  * •

BB3: MJD 56632.5 $-$ MJD 56689.0
Corresponds to the low state.

The redshift lower limits of VER J0521+211 are not confirmed, which can be
defined as the limit scenarios L1 and L2, respectively.

  * •

L1: $z_{0}\geqslant 0.108$ from Ref. Shaw  _et al._ (2013).
Based on a weak emission feature, which however is not confirmed by Ref.
Archambault  _et al._ (2013).

  * •

L2: $z_{0}\geqslant 0.18$ from Ref. Paiano  _et al._ (2017).
The result is also not confirmed, which therefore is still unknown.

The redshift upper limit of VER J0521+211 can be defined as the limit scenario
H1.

  * •

H1: $z_{0}\leqslant 0.308$ from Ref. Adams  _et al._ (2022).
The results in the literature are around $z_{0}\leqslant 0.31$, here we take a
typical value.

Using these flux states and redshift limits, we could investigate the photon-
ALP oscillation effects on gamma-ray spectral irregularities.

The main effect on VHE photon (with the energy $E$) in the extragalactic space
is the EBL photon (with the energy $\omega$) absorption effect with the factor
$e^{-\tau}$. The corresponding optical depth can be described by Franceschini
_et al._ (2008)

$\displaystyle\tau=c\int_{0}^{z_{0}}\frac{{\rm d}z}{(1+z)H(z)}\int_{E_{\rm
th}}^{\infty}{\rm d}\omega\frac{{\rm d}n(z)}{{\rm
d}\omega}\bar{\sigma}(E,\omega,z)\,,$ (1)

where
$H(z)=H_{0}\sqrt{\left(1+z\right)^{2}\left(1+\Omega_{m}z\right)-z\left(2+z\right)\Omega_{\Lambda}}$
is the Hubble expansion rate, with the source redshift $z_{0}$, the threshold
energy $E_{\rm th}$, the integral pair-production cross section
$\bar{\sigma}(E,\omega,z)$, the EBL proper number density ${\rm d}n(z)/{\rm
d}\omega$, $H_{0}\simeq 67.4\,\rm km\,s^{-1}\,Mpc^{-1}$, $\Omega_{m}\simeq
0.315$, and $\Omega_{\Lambda}\simeq 0.685$ Zyla  _et al._ (2020). In this
work, the spectrum of EBL is taken from the model F-08 Franceschini  _et al._
(2008).

In our analysis, the gamma-ray intrinsic spectrum $\Phi_{\rm int}(E)$ is
selected with the minimum best-fit reduced $\chi^{2}_{\rm null}$ from the four
spectra models as discussed in Ref. Li (2022b). Here we adopt the log-parabola
model, which can be described by

$\displaystyle\Phi_{\rm
int}(E)=N_{0}\left(\frac{E}{E_{0}}\right)^{-\Gamma-b\log\left(\frac{E}{E_{0}}\right)}\,,$
(2)

where $N_{0}$ is the normalization constant, $\Gamma$ is the spectral index,
$E_{0}$ and $b$ are free parameters. Then the $\chi^{2}$ value is given by

$\displaystyle\chi^{2}=\sum_{i=1}^{N}\left(\frac{\Phi_{i}-\psi_{i}}{\delta_{i}}\right)^{2}\,,$
(3)

with the expected spectrum $\Phi_{i}=e^{-\tau}\Phi_{\rm int}(E_{i})$, where
$N$ is the gamma-ray spectral point number, $\psi_{i}$ and $\delta_{i}$ are
the detected flux and its uncertainty, respectively.

Figure 1: The SEDs of the three states of VER J0521+211 for the redshift limit scenarios L1 (top; with $z_{0}=0.108$), L2 (middle; with $z_{0}=0.18$), and H1 (bottom; with $z_{0}=0.308$). The blue triangles, red diamonds, and green circles correspond to the three states BB1 (12 points), BB2 (10 points), and BB3 (11 points), respectively. The solid and dashed lines correspond to the best-fit SEDs with/without ALP, respectively. The experimental points are taken from Fermi-LAT and VERITAS Adams  _et al._ (2022). For comparison, the EBL models used in these three plots are taken as F-08 Franceschini  _et al._ (2008). Note that we do not make the ALP analysis with H1.  Table 1: The best-fit $\chi^{2}$ values under the null/ALP hypotheses of the three states of VER J0521+211 for the scenarios L1, L2, H1, and R1. The combined results are also shown. The values of $\chi^{2}_{\rm min}$ correspond to the minimum best-fit points on the ALP parameter space. The effective d.o.f. for the ALP combined analyses are also listed. State | $\chi^{2}_{\rm null}$ | $\chi^{2}_{\rm null}/{\rm d.o.f.}$ | $\chi^{2}_{\rm min}$ | eff. d.o.f.
---|---|---|---|---
L1-BB1 | 4.89 | 0.61 | 2.43 | …
L1-BB2 | 6.69 | 1.11 | 4.25 | …
L1-BB3 | 14.20 | 2.03 | 8.24 | …
L1-combined | 25.78 | … | 18.82 | 5.58
L2-BB1 | 6.62 | 0.83 | 2.25 | …
L2-BB2 | 3.24 | 0.54 | 2.22 | …
L2-BB3 | 11.47 | 1.64 | 6.76 | …
L2-combined | 21.33 | … | 17.30 | 5.46
H1-BB1 | $None$ | $None$ | … | …
H1-BB2 | 15.67 | 2.61 | … | …
H1-BB3 | 19.64 | 2.81 | … | …
R1-BB1 | 11.92 | 1.49 | 2.50 | …
R1-BB2 | 4.00 | 0.67 | 2.12 | …
R1-BB3 | 12.43 | 1.78 | 7.27 | …
R1-combined | 28.35 | … | 18.04 | 5.41

We first take the three redshift limit scenarios L1
[$z_{0}\sim\mathcal{O}(0.1)$], L2 [$z_{0}\sim\mathcal{O}(0.2)$], and H1
[$z_{0}\sim\mathcal{O}(0.3)$] for comparisons. We show the best-fit gamma-ray
spectral energy distributions (SEDs) of the three states BB1, BB2, and BB3 of
VER J0521+211 with these redshift limit scenarios in Fig. 1. The dashed lines
represent the best-fit SEDs under the null hypothesis. The corresponding best-
fit $\chi^{2}_{\rm null}$ values are listed in Table 1. For the scenario H1,
we note that the value of $\chi^{2}_{\rm null}/{\rm d.o.f.}=5.52$ of BB1 is
obviously larger than that of BB2 and BB3, which may be caused by the small
uncertainty of the observed spectrum. In this case ($z_{0}=0.308$), the other
intrinsic spectra models are also checked, of which the log-parabola is still
the best-fit model. Therefore, the redshift upper limit $z_{0}\leqslant 0.308$
may be an overestimate for BB1. Since the large value of $\chi^{2}_{\rm
null}/{\rm d.o.f.}$ in H1-BB1, we will not set the ALP constraint with the
scenario H1. Adopting the redshift upper limit will lead to excessive EBL
absorption, and this redshift limit is not suitable to make the further
analysis with other flux states (BB2 and BB3).

In order to discuss the redshift uncertainty, here we take the another
redshift value of VER J0521+211, which can be defined as the redshift scenario
R1.

  * •

R1: $z_{0}=0.22$.
This redshift value should be around at the middle of L2 and H1 ($\rm i.e.$,
$z_{0}\simeq 0.24$), and the reduced $\chi^{2}_{\rm null}$ of R1-BB1 should be
small. We test three values of $z_{0}=0.24$, 0.23, and 0.22 with R1-BB1,
showing the values of $\chi^{2}_{\rm null}/{\rm d.o.f.}=2.12$, 1.79, and 1.49,
respectively. Therefore, we take the value $z_{0}=0.22$ as R1.

For the scenario R1, we show the best-fit SEDs under the null hypothesis in
Fig. 2. The best-fit $\chi^{2}_{\rm null}$ values are also listed in Table 1.
Compared with H1, the $\chi^{2}_{\rm null}$ values of R1-BB2 and R1-BB3 are
also dramatically depressed. In the following, we will just discuss the ALP
hypothesis with the scenarios L1, L2, and R1.

Figure 2: Same as Fig. 1 but for the scenario R1 ($z_{0}=0.22$).

##  III ALP setup

In this section, we briefly introduce the ALP constraint method and the
magnetic field parameters setup. The photon-ALP oscillation probability in the
homogeneous magnetic field can be simply described by

$\displaystyle\mathcal{P}_{a\gamma}=\left(\frac{g_{a\gamma}B_{T}}{\Delta_{\rm
osc}}\right)^{2}\sin^{2}\left(\frac{\Delta_{\rm osc}x_{3}}{2}\right)\,,$ (4)

where $g_{a\gamma}$ is the photon-ALP coupling constant, $B_{T}$ is the
transverse magnetic field, $\Delta_{\rm osc}$ is the oscillation wave number,
and $x_{3}$ is the propagation direction of photon-ALP. The general photon-ALP
oscillations in the magnetic field can be found in Ref. Li (2022c). Here we
introduce the parameters associated the photon-ALP beam propagating from the
gamma-ray source region to the Earth, which is composed of (i) the source
region, (ii) the extragalactic space, and (iii) the Milky Way. In this case,
the final photon-ALP-photon oscillation probability
$\mathcal{P}_{\gamma\gamma}$ for the propagation distance $s$ is given by De
Angelis  _et al._ (2011)

$\displaystyle\mathcal{P}_{\gamma\gamma}={\rm
Tr}\left(\left(\rho_{11}+\rho_{22}\right)\mathcal{T}(s)\rho(0)\mathcal{T}^{\dagger}(s)\right)\,,$
(5)

where $\mathcal{T}(s)=\mathcal{T}(s_{3})_{\rm
iii}\times\mathcal{T}(s_{2})_{\rm ii}\times\mathcal{T}(s_{1})_{\rm i}$ is the
whole transfer matrix, $\rho_{11}={\rm diag}(1,0,0)$, $\rho_{22}={\rm
diag}(0,1,0)$, $\rho(0)$ and $\rho(s)$ are initial and final density matrices
of the photon-ALP beam, respectively.

(a) L1-BB1. (b) L1-BB2. (c) L1-BB3. (d) L1-combined. (e) L2-BB1. (f) L2-BB2.
(g) L2-BB3. (h) L2-combined. (i) R1-BB1. (j) R1-BB2. (k) R1-BB3. (l)
R1-combined.

Figure 3: The best-fit $\chi^{2}_{\rm ALP}$ distributions on the ALP parameter
($m_{a},g_{a\gamma}$) space for the redshift limit scenarios L1 (top), L2
(middle), and R1 (bottom). The combined results are also shown. These panels
correspond to the scenarios L1-BB1 (a), L1-BB2 (b), L1-BB3 (c), L1-combined
(d), L2-BB1 (e), L2-BB2 (f), L2-BB3 (g), L2-combined (h), R1-BB1 (i), R1-BB2
(j), R1-BB3 (k), and R1-combined (l), respectively.

For (i) the source region of the BL Lac object VER J0521+211, we consider the
photon-ALP oscillation in the blazar jet magnetic field, which can be
described by the poloidal and toroidal components. As discussed in Ref. Li
_et al._ (2021), here we consider the jet magnetic field with the transverse
magnetic field model $B(r)=B_{0}(r/r_{\rm VHE})^{-1}$ and the electron density
model $n_{\rm el}(r)=n_{0}(r/r_{\rm VHE})^{-2}$, where $r_{\rm VHE}$ is the
distance between the source central black hole and the VHE emission region,
$B_{0}$ and $n_{0}$ are the core magnetic field and electron density at
$r_{\rm VHE}$, respectively. For the jet region $r>1\rm\,kpc$, we take the
magnetic field $B=0$. The Doppler factor $\delta_{\rm D}=E_{L}/E_{j}$ is also
considered, which represents the energy transformation between the laboratory
and co-moving frames, $E_{L}$ and $E_{j}$, respectively. The magnetic field
parameters $B_{0}$, $r_{\rm VHE}$, $n_{0}$, and $\delta_{\rm D}$ for the three
states BB1, BB2, and BB3 of VER J0521+211 are listed in Table 2. We note that
the parameters $B_{0}$ and $r_{\rm VHE}$ in Ref. Adams  _et al._ (2022) are
$1.5\times 10^{-2}\,\rm G$ and $\sim\mathcal{O}(0.9)\,\rm pc$, respectively,
while they are taken as $0.25\times 10^{-2}\,\rm G$ and
$\sim\mathcal{O}(2.5)\,\rm pc$ in Ref. Archambault  _et al._ (2013). We also
note that the latter observations are performed to constrain the ALP in Ref.
Jacobsen  _et al._ (2022) with other parameter values. For self-consistency,
the parameters setup used in this work are taken from Ref. Adams  _et al._
(2022). Additionally, for the host galaxy region of VER J0521+211, the photon-
ALP oscillation in this part can be totally neglected.

For (ii) the extragalactic space, we just consider the EBL absorption effect
on VHE photon due to the pair-production process. Since the magnetic field in
the extragalactic space is very weak with the upper limit
$\sim\mathcal{O}(1)\,\rm nG$ Ade  _et al._ (2016); Pshirkov  _et al._ (2016),
we neglect the photon-ALP oscillation in this region.

Finally, we also take into account the photon-ALP oscillation in (iii) the
Milky Way with the Galactic magnetic field model Jansson and Farrar (2012a,
b), which is composed of the disk and halo components (both parallel to the
plane of the Milky Way), and the so-called “X-field” component (out-of-plane)
at the center of the Milky Way. See also Refs. Adam  _et al._ (2016); Unger
and Farrar (2017) for the latest version of this model.

Table 2: The source jet magnetic field parameters of the three states of VER J0521+211. These values can be directly or indirectly obtained from Ref. Adams  _et al._ (2022). State |  $B_{0}$($10^{-2}\,\rm G$) |  $r_{\rm VHE}$(pc) |  $n_{0}$($10^{3}\,\rm cm^{-3}$) | $\delta_{\rm D}$
---|---|---|---|---
BB1 | 1.5 | 0.85 | 0.88 | 26
BB2 | 1.5 | 0.89 | 0.95 | 26
BB3 | 1.5 | 0.93 | 0.68 | 26

##  IV ALP constraints with the redshift uncertainty

Using Eqs. (2) and (5), the expected gamma-ray spectrum under the ALP
hypothesis can be described by $\Phi_{{\rm
ALP},\,i}=\mathcal{P}_{\gamma\gamma}\Phi_{\rm int}(E_{i})$, where $\Phi_{\rm
int}(E_{i})$ is the gamma-ray intrinsic spectrum. For one ALP parameter
($m_{a},g_{a\gamma}$) set, we can derive the best-fit $\chi^{2}_{\rm ALP}$
from Eq. (3) with the notation $\Phi_{i}\to\Phi_{{\rm ALP},\,i}$. Then we can
derive the best-fit $\chi^{2}_{\rm ALP}$ distributions on the whole ALP
parameter space, which are shown in Fig. 3 with the redshift limit scenarios
L1, L2, and R1. In these panels, the distributions of $\chi^{2}_{\rm ALP}$
correspond to the three states BB1, BB2, and BB3 of VER J0521+211. The minimum
best-fit gamma-ray SEDs under the ALP hypothesis of these three states with
the scenarios L1, L2, and R1 are also shown in Fig. 1 and 2 for comparisons,
respectively. The values of minimum best-fit $\chi^{2}_{\rm min}$ on the ALP
parameter space can be found in Table 1. Compared with the null hypothesis,
the minimum best-fit $\chi^{2}_{\rm min}$ under the ALP hypothesis can be
dramatically depressed.

As considered in Refs. Li  _et al._ (2021, 2022), we also set the combined
constraints on ALP with the multistate analysis. In this case, the two or more
states of the same source are selected to fit with the corresponding magnetic
field setup. In order to obtain the $\chi^{2}_{99\%}$ value at 99% $\rm C.L.$,
400 sets of the gamma-ray spectra observations in the pseudoexperiments by
Gaussian samplings are simulated to derive the test statistic (TS)
distribution, ${\rm TS}={\widehat{\chi}_{\rm null}}^{2}-{\widehat{\chi}_{\rm
ALP}}^{2}$, with the best-fit $\chi^{2}$ of the null and ALP hypotheses in the
Monte Carlo simulations, ${\widehat{\chi}_{\rm null}}^{2}$ and
${\widehat{\chi}_{\rm ALP}}^{2}$, respectively. Here the TS distribution obeys
the non-central $\chi^{2}$ distribution with the effective ${\rm d.o.f.}$ and
the non-centrality $\lambda$. Then we assume this TS distribution is
approximated with the ALP hypothesis and can be used to derive
$\Delta\chi^{2}_{99\%}$. Finally, the value of the 99% $\rm C.L.$ $\chi^{2}$
can be obtained by $\chi^{2}_{99\%}=\chi^{2}_{\rm min}+\Delta\chi^{2}_{99\%}$.
More details about the statistical method can be found in Ref. Li (2022a).

Figure 4: The 99% $\rm C.L.$ photon-ALP combined constraints set by VER
J0521+211. The dashed blue, red, and green contours represent the 99% $\rm
C.L.$ combined results of the scenarios L1-, L2-, and R1-combined,
respectively. The other limits are taken from the package AxionLimits O’HARE
(2020).

Then we show the ALP combined constraint results set by VER J0521+211 in Fig.
4. For comparison, we also show the other latest photon-ALP constraints O’HARE
(2020) in this plot. The dashed blue, red, and green contours represent the
99% $\rm C.L.$ combined results of L1, L2, and R1 with the three states
combined, respectively. The corresponding best-fit $\chi^{2}_{\rm min}$ values
and the effective ${\rm d.o.f.}$ are also listed in Table 1 with
$\lambda=0.01$. Compared with the two redshift limit scenarios L1
($z_{0}\geqslant 0.108$) and L2 ($z_{0}\geqslant 0.18$), we find the more
stringent ALP combined constraint with the underestimated redshift. The 99%
$\rm C.L.$ exclusion region of L2-combined is completely covered by
L1-combined, which shows significant difference in the low mass region
$1.0\times 10^{-9}\,{\rm eV}\lesssim m_{a}\lesssim 1.0\times 10^{-8}\,{\rm
eV}$. This is probably because the null hypothesis SEDs of L1-BB2 and L1-BB3
in low energies ($0.1-1\,\rm GeV$) cannot be fitted well with $z_{0}=0.108$,
see Fig. 1. While compared with the scenarios L2 and R1 ($z_{0}=0.22$), we
find they show similar ALP exclusion regions (the dashed red and green
contours in Fig. 4). The small difference of redshifts (0.18 and 0.22) will
reduce the constraint difference on the ALP parameter space. Therefore, no
clear connection is confirmed between the redshift limit scenarios and the
photon-ALP constraints.

Finally, we give the 99% $\rm C.L.$ ALP combined constraints set by VER
J0521+211, which are roughly at $g_{a\gamma}\gtrsim 2.0\times
10^{-11}\rm\,GeV^{-1}$ for $1.0\times 10^{-9}\,{\rm eV}\lesssim m_{a}\lesssim
1.0\times 10^{-7}\,{\rm eV}$. Additionally, we note that our results are
generally similar to the 99% $\rm C.L.$ limit set by the 2009 TeV observations
Archambault  _et al._ (2013) of VER J0521+211 in Ref. Jacobsen  _et al._
(2022), which is performed with the different data (VERITAS+HAWC) and magnetic
field parameters. Since HAWC can measure the VHE gamma-rays exceeding $\sim
1-100\,\rm TeV$, in this case, the photon-ALP oscillation effects will become
more significant. However, the limited energy resolution of HAWC can also
affect the ALP signal sensitivity in this high energy region.

##  V Conclusion

In this paper, we have presented the effects of photon-ALP oscillation on TeV
gamma-ray spectral irregularities from the AGN VER J0521+211, which is
classified as the IBL object with the uncertain redshift $0.108\leqslant
z_{0}\leqslant 0.34$. The gamma-ray spectra are measured by Fermi-LAT and
VERITAS in 2013 and 2014 with the three flux states (BB1, BB2, and BB3), and
analyzed with the four redshift limit scenarios L1 ($z_{0}\geqslant 0.108$),
L2 ($z_{0}\geqslant 0.18$), H1 ($z_{0}\leqslant 0.308$), and R1
($z_{0}=0.22$). The SEDs of these states under the null and ALP hypotheses are
shown for comparisons. Then we set the combined constraints on the ALP
parameter space with these states and test the effect of EBL absorption on ALP
constraints with these redshift limit scenarios. Since the redshift upper
limit scenario H1 may be an overestimate for BB1, we do not set the ALP
constraint with H1. In addition, we take the another redshift scenario R1 to
discuss the redshift uncertainty.

The 99% $\rm C.L.$ photon-ALP combined constraints set by the scenarios L1,
L2, and R1 on the ALP parameter space are roughly at $g_{a\gamma}\gtrsim
2.0\times 10^{-11}\rm\,GeV^{-1}$ for $1.0\times 10^{-9}\,{\rm eV}\lesssim
m_{a}\lesssim 1.0\times 10^{-7}\,{\rm eV}$. Compared with the results of L1-
and L2-combined, we find the more stringent ALP combined constraint with the
underestimated redshift. While compared with the scenarios L2-and R1-combined,
we find they show similar ALP exclusion regions. Then we have the conclusion
that no clear connection is confirmed between the redshift limit scenarios and
the photon-ALP constraints. Both the underestimated and overestimated redshift
limit scenarios can affect the constraint results. In this work, the
underestimated redshift (L1) shows a stringent ALP exclusion region, while the
overestimated redshift (H1) is not suitable to make the ALP analysis.

## Acknowledgments

The authors would like to thank Qi Feng for sharing the experimental data of
VER J0521+211 and valuable comments, and also thank Peng-Fei Yin for helpful
discussions. This work was supported by the National Natural Science
Foundation (NNSF) of China (Grants No. 11775025 and No. 12175027).

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