# Sensitivity of the YbOH molecule to $\mathcal{P}$,$\mathcal{T}$-odd effects
in the external electric field

Alexander Petrov<EMAIL_ADDRESS>Anna Zakharova  Petersburg Nuclear
Physics Institute named by B.P. Konstantinov of National Research Centre
”Kurchatov Institute”, Gatchina, 1, mkr. Orlova roshcha, 188300, Russia  St.
Petersburg State University, St. Petersburg, 7/9 Universitetskaya nab.,
199034, Russia

(Received: date / Revised version: date)

###### Abstract

Electron electric dipole moment (eEDM) search using lasercoolable triatomics
like YbOH is one of the most sensitive probes for physics beyond the Standard
Model. The eEDM-induced energy shift is proportional to polarization ($P$) of
the molecule. Similarly to diatomics with $\Omega-$doubling structure it was
assumed that for triatomics with $l-$doubling structure, related to the
vibrational angular momentum, $P$ can easily be saturated to almost 100% value
with moderate external electric field. We developed the method for calculation
of properties of triatomic molecules and applied it to calculation of $P$ of
174YbOH in the first excited $v=1$ bending mode. Calculations showed that the
most of the levels reach less than 50% efficiency. We showed that this fact is
related to the Hund’s case $b$ coupling scheme of YbOH. As coupling scheme
(for $\Omega=1/2$ molecules) approaches $a$ (or $c$) case polarization
increases up to 100% value. Results of our calculations should be used for
correct extracting of eEDM value from YbOH experiment and similar calculations
are required for other molecules.

Measuring the electron electric dipole moment (eEDM) is now considered as a
most promising test for existence of physics beyond the Standard model
FUKUYAMA (2012); Pospelov and Ritz (2014); Yamaguchi and Yamanaka (2020,
2021). The current limit on the electron electric dipole moment (the ACME II
experiment), $|d_{e}|<1.1\times 10^{-29}$ e$\cdot$cm (90% confidence), was set
by measuring the spin precession using thorium monoxide (ThO) molecules in the
metastable electronic H${}^{3}\Delta_{1}$ state Andreev et al. (2018). For
successful performing of the such kind of experiments the possibility of
suppressing of the systematic effects is of high importance. Previously it was
shown that due to existence of $\Omega$-doublet levels the experiments for
searching of the $\mathcal{P}$,$\mathcal{T}$-odd effects on the ThO Andreev et
al. (2018); DeMille et al. (2001); Petrov et al. (2014); Vutha et al. (2010);
Petrov (2015, 2017) or the HfF+ Cairncross et al. (2017); Petrov (2018) are
very robust against a number of systematics.

In turn cold polar molecules provide unique opportunities for further progress
in search for effects of symmetry violation Isaev and Berger (2016). In such
molecules the sensitivity of the experiments can be strongly enhanced due to
increased coherence time. Both the possibility of laser cooling and the
existence of the close levels of the opposite parity can be realized with
triatomic molecules such as the RaOH Isaev et al. (2017), the YbOH Kozyryev
and Hutzler (2017) etc. In this case the role of the $\Omega$-doublets used in
the diatomic molecular experiments is taken over by the $l$-doublets of the
excited $v=1$ bending vibrational modes Kozyryev and Hutzler (2017); Pilgram
et al. (2021).

Any eEDM experiment searches for an eEDM induced Stark shift

$\delta E=d_{e}E_{\rm eff}P,$ (1)

where $d_{e}$ is the value of electron electric dipole moment, $E_{\rm eff}$
is effective electric field acting on electron in the molecule, which is
subject of molecular calculations Denis et al. (2019); Prasannaa et al.
(2019); Gaul and Berger (2020); Zakharova and Petrov (2021); Zakharova et al.
(2021), $P$ is the polarization of the molecule by the external electric
field. To extract $d_{e}=\delta E/(E_{\rm eff}P)$ from the measured shift
$\delta E$, one needs to know $E_{\rm eff}P$. Therefore value of $E_{\rm
eff}P$ directly influence $d_{e}$ value extracted from experiments Nataraj et
al. (2011); Porsev et al. (2012).

It is well known that for diatomics due to the existence of $\Omega$-doublet
structure $P$ value approaches unity for small laboratory electric fields
DeMille et al. (2001). The same situation was expected for linear triatomic
molecules Kozyryev and Hutzler (2017). In our letter we show that the
$l-doubling$ structure, is, in general, different from $\Omega-doubling$, and
the $P$ value tends to approach half of the maximum value for molecules with
Hund’s case $b$ coupling scheme and 100% value as coupling scheme approaches
$a$ (or $c$) case.

In the present letter we developed the method for computation of energy levels
and different properties of linear triatomic molecules. We applied it for
calculation of the sensitivity of the 174YbOH molecule to eEDM in the ground
rotational $N=1$ level of first excited $v=1$ bending mode to eEDM in the
external electric field. According to eq. (1), these calculations should be
used to determine the limit on the eEDM.

Together with eEDM one always needs to consider scalar T,P-odd electron-
nuclear interaction, since its influence on the spectrum of molecules is
identical to eEDM. For short, only the influence of eEDM is mentioned in the
present letter.

Figure 1: Orientation of the YbOH relative to the molecular frame. Orientation
of the molecular frame relative to the laboratory one is given by Euler’s
angles (not presented in the figure) $\alpha$, $\beta$, $\gamma=0$.
$R,r,\theta$ are Jacobi coordinates.

Consider the molecular Hamiltonian for the electronic-nuclear motion,
excluding nuclear hyperfine structure, and with fixed OH ligand stretch. In
the Jacobi coordinates (represented in Fig. 1) in laboratory frame it reads as

$\hat{\rm H}_{\rm mol}=-\frac{\hbar^{2}}{2\mu}\frac{\partial^{2}}{\partial
R^{2}}+\frac{\hat{\bf L}^{2}}{2\mu R^{2}}+\frac{\hat{\bf
l}^{2}}{2\mu_{OH}r^{2}}+{\rm\hat{H}}_{\rm el},$ (2)

where $\mu$ is the reduced mass of the $Yb-OH$ system, $\mu_{OH}$ is the
reduced mass of the OH ligand, $\hat{\bf L}$ is the angular momentum of the
rotation of the Yb atom and OH around their center of mass, $\hat{\bf l}$ is
the angular momentum of the rotation of the OH, and ${\rm\bf\hat{H}}_{\rm el}$
is the electronic Hamiltonian. For number of applications it is convenient to
rewrite the Hamiltonian in molecular reference frame defined by Euler angles
$\alpha,\beta,\gamma$ (see e.g. Fig 20 in Ref. Landau and Lifshitz (1977)),
with $\alpha,\beta$ corresponding to azimuthal and polar angles of axis (it
will be $z$ axis of the molecular frame) going through Yb and the center of
mass of OH, and $\gamma=0$. In this case the Hamiltonian (2) is rewritten as

$\hat{\rm H}_{\rm mol}=-\frac{\hbar^{2}}{2\mu}\frac{\partial^{2}}{\partial
R^{2}}+\frac{(\hat{\bf J}-\hat{\bf J}^{e-v})^{2}}{2\mu R^{2}}+\frac{\hat{\bf
l}^{2}}{2\mu_{OH}r^{2}}+{\rm\hat{H}}_{\rm el},$ (3)

where $\hat{\bf J}$ is the total molecular less nuclear spins angular
momentum, $\hat{\bf J}^{e-v}=\hat{\bf J}^{e}+\hat{\bf J}^{v}$, $\hat{\bf
J}^{e}$ \- is the total electronic momentum, $\hat{\bf J}^{v}$ is the
vibrational momentum,
$\hat{J}^{v}_{z}=-i\hbar\frac{\partial}{\partial\varphi}$, $\varphi$ is the
angle between $xz$ plane of the molecular reference frame and the plane of the
molecule (see Fig. 1), $\hat{\bf
l}^{2}=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}$,
and $\theta$ is the angle between OH and $z$ axes. The condition $\theta=0$
corresponds to the linear configuration where O atom is between Yb and H ones.

For the purpose of the present letter, we also include hyperfine interaction
and interaction with the external electric field. Finally, our Hamiltonian
reads as

${\rm\hat{H}}={\rm\hat{H}}_{\rm mol}+{\rm\hat{H}}_{\rm hfs}+{\rm\hat{H}}_{\rm
ext},$ (4)

where

${\rm\bf\hat{H}}_{\rm hfs}={g}_{\rm H}{\bf\rm
I}\cdot\sum_{a}\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)$
(5)

is the hyperfine interaction between electrons and the hydrogen nucleus,
${g}_{\rm H}$ is the g-factor of the hydrogen nucleus, $\bm{\alpha}_{a}$ are
the Dirac matrices for the $a$-th electron, $\bm{r}_{a}$ is its radius-vector
in the coordinate system centered on the H nucleus, index $a$ enumerates (as
in all equations below) electrons of YbOH.

${\rm\bf\hat{H}}_{\rm ext}=-{{\bf D}}\cdot{\bf E}$ (6)

describes the interaction of the molecule with the external electric field,
and D is the dipole moment operator.

Wavefunctions were obtained by numerical diagonalization of the Hamiltonian
(4) over the basis set of the electronic-rotational-vibrational wavefunctions

$\Psi_{\Omega
m\omega}P_{lm}(\theta)\chi(R)\Theta^{J}_{M_{J},\omega}(\alpha,\beta)U^{\rm
H}_{M_{I}}.$ (7)

Here
$\Theta^{J}_{M_{J},\omega}(\alpha,\beta)=\sqrt{(2J+1)/{4\pi}}D^{J}_{M_{J},\omega}(\alpha,\beta,\gamma=0)$
is the rotational wavefunction, $U^{H}_{M_{I}}$ is the hydrogen nuclear spin
wavefunctions and $M_{J}$ is the projection of the molecular (electronic-
rotational-vibrational) angular momentum $\hat{\bf J}$ on the lab axis,
$\omega$ is the projection of the same momentum on $z$ axis of the molecular
frame, $M_{I}=\pm 1/2$ is the projection of the nuclear angular momentum
$I=1/2$ of hydrogen on the same axis, $P_{lm}(\theta)$ is associate Legendre
polynomial,

$\Psi_{\Omega
m\omega}=e^{i(\omega-1/2\sum_{a}{\sigma_{a}})\varphi}\sum_{i}\Psi^{e}_{\Lambda_{i}}\Psi^{s}_{\Sigma_{i}},$
(8)

where $\sigma_{a}$ is electronic spin variables, $\Psi^{e}_{\Lambda_{i}}$ and
$\Psi^{s}_{\Sigma_{i}}$ are orbital and spin wavefunctions respectively,
$\Lambda_{i}$ and $\Sigma_{i}$ are projections of electronic orbital and spin
momenta on $z$ axis, i enumerates all possible nonrelativistic wavefunctions.

Eq. (8) is expansion of the electronic wavefunction in terms of
nonrelativistic functions of linear configuration. For linear configuration
only terms with $\Omega=\Lambda_{i}+\Sigma_{i}$ contribute. $\Psi_{\Omega
m\omega}$ depends on $m$ since $\omega$ is chosen so that $\omega=\Omega+m$.
$\Psi_{\Omega m\omega}$ is eigenfunction of the electronic-vibrational
momentum:

$\hat{J}^{e-v}_{z}\Psi_{\Omega m\omega}=\hbar\omega\Psi_{\Omega m\omega}.$ (9)

To prove eq. (9), we note that nonrelativistic wavefunctions depend on
$\varphi$ as $\Psi^{e}_{\Lambda_{i}}\sim e^{-i\Lambda_{i}\varphi}$ and that
$1/2\sum_{a}{\sigma_{a}}\Psi^{s}_{\Sigma_{i}}=\Sigma_{i}\Psi^{s}_{\Sigma_{i}}$.
For the lowest vibrational levels configuration of the molecule is close to
linear, therefore the main contribution in sum (8) is given by the terms with
$\Omega=\Lambda_{i}+\Sigma_{i}$. Therefore, we also have approximate relations

$\hat{J}^{e}_{z}\Psi_{\Omega m\omega}\approx\hbar\Omega\Psi_{\Omega m\omega},$
(10) $\hat{J}^{v}_{z}\Psi_{\Omega m\omega}\approx\hbar m\Psi_{\Omega
m\omega}.$ (11)

In this calculation functions with $\omega-m=\Omega=\pm 1/2$, $l=0-30$ (as in
Ref. Zakharova et al. (2021)) and $m=0,\pm 1,\pm 2$, $J=1/2,3/2,5/2$ were
included to the basis set (7). Note, that the ground mode $v=0$ corresponds to
$m=0$, the first excited bending mode $v=1$ to $m=\pm 1$, the second excited
bending mode has states with $m=0,\pm 2$ etc. Using the basis set (7) allows
us to employ (with a little modification) the codes developed for the
diatomics calculation Petrov (2011); Petrov et al. (2014); Petrov (2015, 2017,
2018); Petrov et al. (2018), and paves the way for computation of the
g-factors, g-factors difference, different systematics in an external electro-
magnetic field etc. These calculations are important for optimization of
experimental conditions for the eEDM experiment.

In the present work we applied the method for calculation of sensitivity of
174YbOH to $\mathcal{P}$,$\mathcal{T}$-odd effects in the external electric
field. As a test, we exactly reproduced with described method frequencies of
bending and stretching vibrational modes (319 and 550 cm-1 respectively) and
value 26 MHz of $l$-doubling for 174YbOH molecule obtained with the
Hamiltonian (2) in Ref. Zakharova et al. (2021).

For the purposes of the present paper, we found that ignoring the R dependence
leads to an error of about 1% or less for polarization $P$ of 174YbOH.
Therefore, we reduced the Hamiltonian (3) and basis set (7) to

$\hat{\rm H}_{\rm mol}=\frac{(\hat{\bf J}-\hat{\bf J}^{e-v})^{2}}{2\mu
R^{2}}+\frac{\hat{\bf l}^{2}}{2\mu_{OH}r^{2}}+{\rm\hat{H}}_{\rm el},$ (12)

and

$\Psi_{\Omega
m\omega}P_{lm}(\theta)\Theta^{J}_{M_{J},\omega}(\alpha,\beta)U^{\rm
H}_{M_{I}}$ (13)

with $R=3.9$ a.u. respectively. In this approximation we neglect the influence
of the stretching modes but nevertheless take into account the bending ones
with fixed $R$. Provided that the electronic-vibrational matrix elements are
known, the matrix elements of ${\rm\bf\hat{H}}$ between states in the basis
set (13) can be calculated with help of the angular momentum algebra Landau
and Lifshitz (1977) in the same way as for the diatomic molecules Petrov
(2011). Note however, that pure electronic matrix elements, in general, depend
on $\theta$, and the selection rules for $\Omega$ quantum number can be
violated. In turn the selection rules for $\omega$ quantum number are rigorous
and the same as for $\Omega$ quantum number in the diatomics. For example the
matrix element for the perpendicular hyperfine structure constant

$A_{\perp}={g_{\rm H}}\times\\\
\langle\Psi_{\Omega=1/2m\omega}P_{lm}|\sum_{a}\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)_{+}|\Psi_{\Omega=1/2m-1\omega-1}P_{l^{\prime}m-1}\rangle$
(14)

is, in general, nonzero. Here

$\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)_{+}=\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)_{x}+i\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)_{y}$

and similarly for other vectors. Since hyperfine interaction on hydrogen is
small, in the present work we neglected their $\theta$ dependence and used the
values

$A_{\parallel}=\frac{g_{\rm H}}{\Omega}\times\\\ \langle\Psi_{\Omega
m\omega}P_{lm}|\sum_{a}\left(\frac{\bm{\alpha}_{a}\times\bm{r}_{a}}{r_{a}^{3}}\right)_{z}|\Psi_{\Omega
m\omega}P_{l^{\prime}m}\rangle\\\ =6.436~{}\delta_{ll^{\prime}}{~{}\rm MHz},$
(15)

$A_{\perp}={g_{\rm H}}\times\\\
\langle\Psi_{\Omega=1/2m\omega}P_{lm}|\sum_{i}\left(\frac{\bm{\alpha}_{i}\times\bm{r}_{i}}{r_{i}^{3}}\right)_{+}|\Psi_{\Omega=-1/2m\omega-1}P_{l^{\prime}m}\rangle\\\
=3.977~{}\delta_{ll^{\prime}}{~{}\rm MHz}$ (16)

from Ref. Pilgram et al. (2021). Spin-rotational interaction is modeled by
matrix element Pilgram et al. (2021)

$p=\frac{1}{\mu R^{2}}\times\\\
\langle\Psi_{\Omega=1/2m\omega}P_{lm}|J^{e}_{+}|\Psi_{\Omega=-1/2m\omega-1}P_{l^{\prime}m}\rangle\\\
=0.5116~{}\delta_{ll^{\prime}}{~{}\rm cm}^{-1}.$ (17)

The matrix element for the dipole moment operator

$\langle\Psi_{\Omega m\omega}P_{lm}|{\bf D}|\Psi_{\Omega
m\omega}P_{l^{\prime}m}\rangle=0.433~{}\delta_{ll^{\prime}}{~{}\rm a.u.}$ (18)

was taken from Ref. Prasannaa et al. (2019). Adiabatic potential was taken
from Ref. Zakharova et al. (2021).

In an experiment to search the $\mathcal{P}$,$\mathcal{T}$-odd effects the
opposite parity levels of the molecule are mixed in the external electric
field to polarize the molecule. For a completely polarized ($P=\pm 1$) 174YbOH
the $\mathcal{P}$,$\mathcal{T}$-odd interaction energy shift approach the
maximum value given by $\delta E^{\rm max}=E_{\rm eff}d_{e}$.

Absolute value of the finite-electric-field shift (1) is smaller than that of
$\delta E^{max}$. In Fig. 2 the calculated polarization $P$ for the lowest
$N=1$ rotational level of the first excited $v=1$ bending vibrational mode of
the 174YbOH as function of the external electric field is presented. The
function $E_{\rm eff}(R,\theta)$ from Ref. Zakharova et al. (2021) was used
for calculations.

Calculations showed that most of the levels have polarization equal to 50% or
less. In particular the stretched $M_{F}=M_{J}+M_{I}=2$ states smoothly
approaches 50% efficiency value. This is in contradiction with Ref. Kozyryev
and Hutzler (2017) where about 100% efficiency was found.

The reason for the fact that 100% efficiency is not reached is the large
Coriolis interaction ($\frac{1}{2\mu R^{2}}J^{e}_{+}J_{-}$) between levels
$|\Omega=1/2,m=1,M_{J}=3/2,M_{I}=1/2,J=3/2\rangle$ and
$|\Omega=-1/2,m=1,M_{J}=3/2,M_{I}=1/2,J=3/2\rangle$ (combination of above
states gives $N=1$ state). The interaction is of the same order of magnitude
as the separation between close rotational levels. In the Hund’s case (b) used
in the Ref. Kozyryev and Hutzler (2017) this fact can be explained by
expression $\langle mNSMM_{S}|\hat{n}|mNSMM_{S}\rangle=sgn(Mm)/N(N+1)$ for
projection of the vector $\hat{n}$ on laboratory axis. Here $\hat{\bf
N}=\hat{\bf J}-\hat{\bf S}$, $\hat{\bf S}$ is electronic spin operator,
$\hat{n}$ is the unit vector along the molecular axis, $M,M_{S}=\pm 1/2$ are
the projections of the $\hat{\bf N},\hat{\bf S}$ on the laboratory axis. The
factor $1/N(N+1)=1/2$ for $N=1$ ($M=\pm 1$ for Hund’s case ($b$)) gives 50%
suppression for the effect. The factor will be different for different
rotational levels. For $N=2$, for example, it is 1/6 for $M=1$ and 1/3 for
$M=2$. The same situation will be for other Hund’s case $b$ molecules e.g.
YbOCH3, RaOH, etc. In Ref. Petrov et al. (2014) it is noted that $J=2$ excited
rotational state of ThO has the same sensitivity to $P,T$-odd effects as
$J=1$, but is more robust against a number of systematic errors. Using excited
rotational levels of YbOH one should take into account that sensitivity will
be smaller than that for N=1 level.

The Spin-rotation interaction couples $\hat{\bf N}$ and $\hat{\bf S}$ to the
$\hat{\bf J}$ angular momentum. The state
$|m=1N=1SJ=3/2M_{J}\rangle=3/4|\Omega=1/2,m=1,M_{J},J=3/2\rangle+1/4|\Omega=-1/2,m=1,M_{J},J=3/2\rangle$
has $|P|=3/4-1/4=1/2$. This is also seen from the consideration above and
facts that $|mN=1SJ=3/2M_{J}=3/2\rangle=|mN=1SM=1M_{S}=1/2\rangle$ and $P$ is
independent of $M_{J}$. The state
$|m=1N=1SJ=1/2M_{J}\rangle=|\Omega=-1/2,m=1,M_{J},J=1/2\rangle$ has $|P|=1$.
(Formulas for $m=-1$ can be obtained by substitution
$m\rightarrow-m,\Omega\rightarrow-\Omega$). The state $J=1/2$ with definite
quantum number $m$ has maximum polarization $|P|=1$. However quantum number
$m$ implies the mixing of $l-$doublets by some finite electric field. Electric
field also mixes $J=1/2$ and $J=3/2$ levels which leads to decreasing of the
polarization up to $|P|=1/2$ value at the limit. Since at zero electric field
$P=0$, the maximum of the $P$ value as function of electric field with
$1/2<|P|^{max}<1$ is observed. The larger is the spin-rotation interaction and
smaller is the $l-$doubling the larger is $|P|^{max}$.

It is important to note that we considered projection of the vector $\hat{n}$
on laboratory axis since in Hund’s case $b$ quantization axis for the electron
spin is an laboratory axis. For Hund’s case $a$ (when both orbital angular and
spin angular momenta have definite projections on molecular axis)
${}^{2}\Pi_{1/2}$ states the matrix element of Coriolis interaction between
$|\Omega=1/2,m=1,M_{J}=3/2,M_{I}=1/2,J=3/2\rangle$ and
$|\Omega=-1/2,m=1,M_{J}=3/2,M_{I}=1/2,J=3/2\rangle$ is zero and $P=1$ value is
reached. For Hund’s case $c$ when ${}^{2}\Pi_{1/2}$ state has admixture of
other $\Omega=1/2$ states (mostly ${}^{2}\Sigma$) Coriolis interaction is not
zero again, but less than one given by eq. (17). In Fig. 2 panel (b) the
calculated polarization for matrix element

$p=0.15~{}\delta_{ll^{\prime}}{~{}\rm cm}^{-1},$ (19)

corresponding to the structure of $X^{2}\Pi_{1/2}$ state of PbF Petrov et al.
(2013) and can be expected on this level for PbOH molecule, is given. As is
expected, for a smaller value of $p$, larger polarization $P$ is reached.

For diatomic molecules, for a given value of external electric field, a
smaller value of $p$ implies larger $P$ value as well. We stress however, that
for triatomics we discuss here the saturated value of $P$ which is independent
of $p$ in diatomics.

Exact values which takes into account mixing of the levels by the hyperfine
interaction should be obtained in molecular calculations as described in our
method. Solid lines on Figure 2 panel (a) show that four of the six levels
$M_{F}=1$ similarly to $M_{F}=2$ have polarization less than 50%. Two of them
($m=\pm 1,J=1/2$ levels) reach values 84% and -72% for electric fields 109 and
166 V/cm respectively and their absolute values decrease for higher values of
the field. Dashed lines on Figure 2 panel (a) correspond to $M_{J}=1/2$ states
assuming zero nuclear spin (and therefore hyperfine interaction is neglected).
One sees that hyperfine interaction only little change the $P$ value.

For dipole moment (18) increased (decreased) on 10% the extreme values for
polarization remain the same but corresponding values for electric field
shifted to 100 and 150 V/cm (122 and 184 V/cm). Energy levels of the six
levels $M_{F}=1$ and four level $M_{J}=3/2$ (assuming zero nuclear spin) on
Figure 3 are shown. At zero electric field the first four $M_{F}=1$ levels are
$J=3/2$ ones and the next two are $J=1/2$ levels. At electric field strength
of about 150 V/cm the avoided crossing between fourth and fifth levels
$M_{F}=1$ is observed. Due to the large changing of the corresponding wave
functions sharp change in $P$ of these levels is observed.

Finally we considered the polarization of 174YbOH molecule in the first
excited bending mode by the external electric field. We found that values for
polarization are smaller than 100% for realistic electric field strengths and
decreases as rotational quantum number increases. The limit for polarization
is $P=Mm/N(N+1)$ (e.g. $P=1/2$ for N=1, $P=1/6,1/3$ for N=2 etc). The spin-
rotation interaction increase $P$ value forming a maximum at some electric
field. The same situation will be for other Hund’s case $b$ molecules e.g.
YbOCH3, RaOH, etc. Hyperfine interaction can potentially increase $P$ value up
to 100%. For 174YbOH $P=0.84$ is reached for electric field strength 109 V/cm.
Results of our letter should be used for correct extracting of eEDM value from
YbOH experiment and optimal choice of the quantum state and electric field
strength. Our results confirm fundamental importance of polyatomic molecules
as candidates for eEDM measurements.

Figure 2:  (Color online) Polarization $P$ (see eq. (1)) for the lowest $N=1$
rotational level of the first excited the $v=1$ bending vibrational mode of
174YbOH as function of the external electric field (a) Solid lines correspond
to the $M_{F}=M_{J}+M_{I}=1$ levels. Dashed lines correspond to the
$M_{J}=1/2$ levels. (b) $M_{F}=M+M_{I}=2$ levels. Solid lines correspond to
the real matrix element (17), dashed lines correspond to reduced matrix
element (19) Figure 3:  (Color online) Energies of the lowest $N=1$ rotational
level of the first excited $v=1$ bending vibrational mode of 174YbOH as
functions of the external electric field. Solid lines correspond to the
$M_{F}=M_{J}+M_{I}=1$ levels. Dashed lines correspond to the $M_{J}=1/2$
levels. Colors of lines correspond to colors of lines in Fig. 2 panel (a).

The authors thank Nicholas Hutzler, Lasner, Zack and Arian Jadbabaie for
useful discussion. The work is supported by the Russian Science Foundation
grant No. 18-12-00227.

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# Band terminations in density functional theory.

A. V. Afanasjev  Department of Physics and Astronomy, Mississippi State
University, MS 39762, USA

###### Abstract

The analysis of the terminating bands has been performed in the relativistic
mean field framework. It was shown that nuclear magnetism provides an
additional binding to the energies of the specific configuration and this
additional binding increases with spin and has its maximum exactly at the
terminating state. This suggests that the terminating states can be an
interesting probe of the time-odd mean fields provided that other effects can
be reliably isolated. Unfortunately, a reliable isolation of these effects is
not that simple: many terms of the density functional theories contribute into
the energies of the terminating states and the deficiencies in the description
of those terms affect the result. The recent suggestion ZSW.05  that the
relative energies of the terminating states in the $N\neq Z,\ A\sim 44$ mass
region given by $\Delta E$ provide unique and reliable constraints on time-odd
mean fields and the strength of spin-orbit interaction in density functional
theories has been reanalyzed. The current investigation shows that the $\Delta
E$ value is affected also by the relative placement of the states with
different orbital angular momentum ${\it l}$, namely, the placement of the $d$
(${\it l}=2$) and $f$ (${\it l}=3$) states. This indicates the dependence of
the $\Delta E$ value on the properties of the central potential.

###### pacs:

PACS:

##  I Introduction

The density functional theory (DFT) in its non-relativistic BHR.03  and
relativistic VRAL  realizations is a standard tool of modern nuclear structure
studies. However, providing global description of atomic nuclei, it still
suffers from the fact that many channels of effective interaction are not
uniquely defined: this is a reason for a large variety of the DFT
parametrizations, the quality of many of which is poorly known. The spin-orbit
interaction and the time-odd mean fields are of particular interest in this
context, since there are considerable variations for these quantities (see,
for example, Refs. DD.95 ; BRRMG.99 ; AR.00 ). The spin-orbit interaction
plays a crucial role in the definition of the shell structure of nuclei, and,
thus its accurate description is required so that theoretical tools have
predictive power for nuclei beyond known regions. The time-odd mean fields (or
nuclear magnetism (NM) in the language of the relativistic mean field (RMF)
theory KR.90 ) contribute to the single-particle Hamiltonian only in
situations where the intrinsic time-reversal symmetry is broken and Kramers
degeneracy of time-reversal counterparts of the single-particle levels is
removed. The rotating nuclei and odd and odd-odd mass nuclei are typical
examples of such situations, see Refs. BHR.03 ; VRAL  and references quoted
therein.

It was recently suggested in Ref. ZSW.05  that the set of terminating states
in the $N\neq Z,\ A\sim 44$ mass region provides unique and reliable
constraints on time-odd mean fields and the strength of spin-orbit interaction
in Skyrme density functional theory (SDFT), see also Refs. S.07 ; ZS.07 .
Later this procedure (called as ’TS-method’ in this manuscript, where ’TS’
refers to ’terminating states’) has been used in the analysis of terminating
states in this mass region within the RMF theory BWSMG.06  which is one of the
versions of covariant density functional (CDFT) theory VRAL .

The authors of Refs. ZSW.05 ; S.07 ; ZS.07 ; BWSMG.06  claim that the TS-
method is free from the drawbacks of standard methods of defining spin-orbit
interaction based on measuring the single-particle energies of the spin-orbit
partner orbitals in spherical nuclei. As a consequence, it is stated that it
allows to define very accurately both isoscalar and isovector channels of
spin-orbit interaction ZSW.05 ; BWSMG.06 ; the feature which was impossible in
the previously existing methods.

The conclusions obtained within the TS-method are drastically different from
the ones previously obtained in the SDFT and RMF frameworks. For example,
based on the comparison of the calculated and experimental energies of spin-
orbit partner orbitals, it was shown in Ref. BRRMG.99  that the experimental
spin-orbit splittings are better reproduced in the RMF approach than in the
SDFT (see Fig. 2 in Ref. BRRMG.99 ). On the contrary, the results obtained in
Refs. ZSW.05 ; BWSMG.06  within the TS-method show that the SDFT provides
better description of spin-orbit splittings than the RMF: it was suggested in
Ref. ZSW.05  that only 5% reduction in isoscalar ls strength is needed in the
SDFT approach in order to reproduce experimental data. Considering the
conflict of these results and the importance of the spin-orbit interaction in
nuclei it is necessary to understand to which extent the basis of the
suggested TS-method is sound and justified. The goal of the present manuscript
is the study of the properties of terminating bands and their terminating
states in the RMF framework. In particular, the question of whether all DFT
contributions have been correctly accounted in the realization of the TS-
method in Refs. ZSW.05 ; BWSMG.06  is addressed is the current manuscript.

The manuscript is organized as follows. Time-odd mean fields in terminating
bands are studied in Sect. II. The basis of the TS-method, its realization in
self-consistent DFT and in the Nilsson potential are discussed in Sect. III.
Sect. IV analyses the contributions of different DFT terms into the relative
energies of terminating states in the $A\sim 44$ mass region. The discussion
of the energy scale, its connection to the effective mass of the nucleon and
their impact on the relative energies of terminating states is presented in
Sect. V. Finally, Sect. VI summarizes main conclusions.

##  II Time-odd mean fields in terminating bands: test case of 20Ne

Previous DFT investigations of the modifications of the moments of inertia
KR.93 ; CRMF ; DD.95 ; YM.00  and single-particle properties AR.00  in
rotating nuclei caused by the time-odd mean fields (nuclear magnetism) were
restricted to the superdeformed (SD) bands. However, these bands are far away
from the termination and are characterized by a relatively stable deformation.
In order to understand how NM affects the properties of the terminating bands,
the ground state configuration in 20Ne has been studied. This band is a
classical example of band termination CNS . It has the
$\pi(d_{5/2})^{2}_{4}\nu(d_{5/2})^{2}_{4}$ configuration relative to the 16O
core with maximum spin $I_{max}=8^{+}$. The selection of this configuration
has been guided by its simplicity, which allows us to understand the role of
time-odd mean fields in terminating bands in greater details. Although the
terminating bands were observed also in heavier nuclei, it is difficult to
trace them from low spin up to band termination in the self-consistent
approaches VRAL  without special techniques such as used in the cranked
Nilsson-Strutinsky (CNS) approach CNS .

Figure 1: (Color online) Calculated total angular momentum (panel (a)), total
binding energy (panel (b)), kinematic and dynamic moments of inertia (panels
(c) and (d)) as a function of rotational frequency $\Omega_{x}$ in the ground
state band of 20Ne. The results obtained with and without nuclear magnetism
are presented. Note that the band termination takes place at rotational
frequency above which the subsequent increase of rotational frequency does not
modify neither total angular momentum nor total binding energy. This point
also corresponds to $\gamma=60^{\circ}$ (Fig. 3c).

The investigation of NM in 20Ne has been carried out within the framework of
the cranked relativistic mean field (CRMF) theory KR.89 ; CRMF  following the
formalism of Ref. AR.00 , where similar study has been performed for the yrast
SD band in 152Dy. In the CRMF calculations of this manuscript, all fermionic
and bosonic states belonging to the shells up to $N_{F}=12$ and $N_{B}=16$ are
taken into account in the diagonalization of the Dirac equation and the matrix
inversion of the Klein-Gordon equation, respectively. The detailed
investigation indicates that this truncation scheme provides very good
numerical accuracy. The NL1 NL1  parametrization of the RMF Lagrangian is
employed in the study of 20Ne, while the studies of terminating states in the
$A\sim 44$ mass region (Sects. III and IV) are performed mostly with the NL1
and NL3 NL3  parametrizations. The pairing is neglected in calculations.

Fig. 1 shows the total angular momentum, total binding energy $E$ and
kinematic (J(1)) and dynamic (J(2)) moments of inertia of the ground state
configuration in 20Ne as a function of rotational frequency obtained in the
calculations with and without NM (the later will be further denoted as WNM).
The band crossing caused by the interaction of the $r=+i$ signatures of the
[220]1/2 and [211]3/2 orbitals both in the proton and neutron subsystems takes
place at lower frequency in the calculations with NM; this is in line with
previous finding that the NM shifts the band crossing frequencies AFR.02 . The
NM also increases the moments of inertia before band crossing (Fig. 1c,d);
similar effect has been seen before in the SD bands (see Ref. AR.00  and
references therein). It also leads to a faster alignment of angular momentum
with rotational frequency (Fig. 1a); full alignment at $I_{max}=8^{+}$
corresponding to band termination takes place at lower frequency in the
calculations with NM.

In the context of study of terminating states two results are important.
First, at the band termination the NM does not modify neither total angular
momentum (Fig. 1a) nor the expectation values of the single-particle angular
momenta $<j_{x}>_{i}$ (Fig. 2). At lower frequency, the impact of NM on
$<j_{x}>_{i}$ is similar to the one previously studied in the SD band of 152Dy
AR.00 , and, thus, it will not be discussed in detail. However, one should
mention that when analyzing the impact of NM on $<j_{x}>_{i}$, the region of
band crossing and the region close to the band termination have to be excluded
from consideration because considerable differences in the deformations of the
NM and WNM solutions at given frequency distort their comparison. Second, the
NM provides an additional binding to the energies of the specific
configuration and this additional binding increases with spin and has its
maximum exactly at the terminating state (Fig. 1b and 3d)). This suggests that
the terminating states can be an interesting probe of the time-odd mean fields
provided that other effects can be reliably isolated.

When the results of the NM and WNM calculations are compared as a function of
total angular momentum, one can see that the quadrupole deformation
$\beta_{2}$ (Fig. 3a), mass hexadecapole moment $Q_{40}$ (Fig. 3b), and
$\gamma$-deformation (Fig. 3c) are almost the same in both calculations. The
only difference is seen in the total binding energies (Fig. 3d), where the NM
solution is more bound than the WNM solution. These results give a hint why
the cranked models based on the phenomenological potentials like Woods-Saxon
or Nilsson, which do not include time-odd mean fields DD.95 , are so
successful in the description of experimental data. When considered as a
function of spin the deformation properties of the rotating system are only
weakly affected by the time-odd mean fields, and the proper renormalization of
the moments of inertia CNS  takes care of the $E$ versus angular momentum
curve.

Figure 2:  (Color online) The expectation values of single-particle angular
momenta $<j_{x}>_{i}$ of the neutron orbitals occupied at low frequency in the
ground state configuration of 20Ne given along the deformation path of this
configuration. The single-particle orbitals are labeled by means of the
asymptotic quantum numbers $[Nn_{z}\Lambda]\Omega$ (Nilsson quantum numbers)
of their dominant component of the wave function at $\Omega_{x}=0.0$ MeV. The
orbitals with signature $r=+i$ and $r=-i$ are shown in right and left panels,
respectively. The results of the calculations with and without NM are shown by
solid and dashed lines, respectively. The region of the band crossing is
located between the dashed lines.  Figure 3: Calculated quadrupole deformation
$\beta_{2}$ (panel (a)), mass hexadecapole moment $Q_{40}$ (panel (b)),
$\gamma$-deformation (panel (c)), and total binding energy $E$ (panel (d)) as
a function of angular momentum. The results obtained with (NM) and without
(WNM) nuclear magnetism are presented.

##  III The TS-method

The TS-method suggested in Ref. ZSW.05  employs the terminating states in the
$A\sim 44$ mass region with the proton $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and
$f_{7/2}^{n}$ structure, but, in general, according to Ref. ZS.07  can be
employed to terminating states in any mass region. The difference $\Delta
E^{exp}=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$ between the excitation
energies $E(d_{3/2}^{-1}\,f_{7/2}^{n+1})$ and $E(f_{7/2}^{n})$ of the
terminating states with the structure $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and
$f_{7/2}^{n}$ is dominated by the size of the magic gap 20 which is surrounded
by the $d_{3/2}$ and $f_{7/2}$ spherical subshells (see Fig. 4).

###  III.1 The spin-orbit splittings in the TS-method

The principal difference between the standard and the TS-methods of defining
the strength of spin-orbit interaction is schematically shown in Fig. 4. The
standard method requires that both partners of spin-orbit ${\it ls}$ doublet
with $\it j=l\pm\frac{1}{2}$ are observed in experiment since the spin-orbit
splitting $\Delta E^{SO}$ is related to the strength of spin-orbit
interaction. This severely restricts the possibilities to study spin-orbit
interaction since both partners should be located in the vicinity of the Fermi
level to be observed: this condition is very difficult to satisfy for high-$j$
orbitals since they are characterized by large spin-orbit splittings, see, for
example, Refs. BRRMG.99 ; IEMSF.02 . On the contrary, the TS-method employs
the terminating states based on the particle-hole excitations involving the
single-particle states with $\it j=l-\frac{1}{2}$ and $\it
j^{\prime}=l^{\prime}+\frac{1}{2}$ which emerge from different $N$-shells and
are characterized by the energy splitting $\Delta E^{TS}$ (Fig. 4). Since
these states are located in the vicinity of the Fermi level, the TS-method
provides also information on the spin-orbit interaction of high-$j$ orbitals
according to Ref. ZSW.05 .

It is necessary to recognize that both methods of defining the spin-orbit
interaction are not free from important drawbacks. The experimental single-
particle states in spherical nuclei used in the standard method are strongly
affected by the couplings with vibrations in many cases MBBD.85 . On the other
hand, the $\Delta E^{TS}$ value used in the TS-method depends not only on the
spin-orbit splitting but also on how well the positions of the single-particle
states with different orbital momenta $l$ and $l^{\prime}$ (Fig. 4) are
described in the DFT calculations. The later fact has been neglected in Ref.
ZSW.05  using an analogy with the Nilsson potential, the validity of which is
questioned below.

Figure 4:  Schematic comparison of the standard and TS-methods of defining the
strength of spin-orbit interaction. In the left and right panels, the single-
particle spectra without (on left) and with (on right) spin-orbit interaction
are compared.

###  III.2 The TS-method in self-consistent approaches

In the self-consistent calculations, the $\Delta
E^{SC}=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$ quantity is defined as
the difference of the binding energies of the corresponding terminating
states. Without going into the details of specific DFT (nonrelativistic or
relativistic), one can conclude that $\Delta E^{SC}$ depends on

  * •

the energy scale of the single-particle spectra which is related to the
effective mass $m^{*}(k_{F})/m$ of the nucleon at the Fermi surface,

  * •

the spin-orbit interaction,

  * •

the relative placement of states with different angular momentum $l$,

  * •

the time-odd mean fields (see Sects. II and IV.2),

  * •

the polarization effects (both in time-even and time-odd channels) on going
from the $f_{7/2}^{n}$ to the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating state.

For simplicity of discussion, they will occasionally be called as ’ingredients
of $\Delta E^{SC}$’.

$\Delta E^{SC}$ can also be split into the terms which depend on time-even
(TE) and time-odd (TO) mean fields

$\displaystyle\Delta E^{SC}=\Delta E^{SC}_{TE}-\Delta E^{SC}_{TO}$ (1)

The minus sign in front of $\Delta E^{SC}_{TO}$ reflects the fact that NM
always decreases the size of $\Delta E$.

####  III.2.1 Coupling constant dependence of $\Delta E^{SC}$

The ingredients of $\Delta E^{SC}$ depend in a complicated way on different
terms of the DFT with at least one term contributing into each of four first
ingredients of $\Delta E^{SC}$ within the nonrelativistic SDFT. In the RMF
theory, the spin-orbit interaction is defined in a natural way without
additional coupling constant VRAL . The time-odd mean fields related to NM are
defined through the Lorentz covariance VRAL  and also do not require an
additional coupling constant. However, both these terms depend in an indirect
way on the coupling constants of other terms of the RMF Lagrangian.

Considering the uncertainties of the description of the ’ingredients of
$\Delta E^{SC}$’ in the DFT, it is not obvious that simple fit (to
experimental $\Delta E^{exp}$ values) of the coupling constants of the DFT
terms related to a pair of ingredients of $\Delta E^{SC}$ (such as time-odd
mean fields and spin-orbit interaction as in Ref. ZSW.05 ; the later treated
perturbatively) will allow to define these constants in a unique way. This is
especially true considering that physical observables depend on many (or
sometimes all) coupling constants simultaneously within the DFT, and the
effect of varying one or two coupling constants may be either enhanced or
cancelled by a variation of others. Strictly speaking, the quality of such
perturbative fits involving only one or two terms of the DFT is not known
until the results of global fit including all the DFT terms are available. For
example, it was shown in Ref. LBBDM.07  that perturbative studies of tensor
terms allow only very limited conclusions.

####  III.2.2 Polarization effects

Figure 5:  (Color online) (right part) Single-particle energies at spherical
shape in 46Ti obtained in the RMF calculations with the NL1 parametrization
and the Nilsson potential with standard set of parameters BR.85  (columns
indicated as “RMF-spher” and “Nilsson”). In order to facilitate the
comparison, the RMF states are given at the calculated energies, while the
energies of the states obtained in the Nilsson potential are shifted in such a
way that the average energy of the $1f_{7/2}$ and $1d_{3/2}$ states is the
same in both calculations. (left part) The magnitudes $\Delta E$ of the
$f_{7/2}-d_{3/2}$ splittings as extracted from the energies of terminating
states ($\Delta E=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$) are shown by
arrows in columns “exp” (experiment), “CRMF-NM” and “CRMF-WNM” (CRMF
calculations with and without NM). The middle points of arrows are located at
the average energy of the $1f_{7/2}$ and $1d_{3/2}$ RMF states. The
terminating states are deformed in the CRMF calculations. Note that the value
of $\Delta E=5.1$ MeV (not shown in figure) obtained in the cranked Nilsson-
Strutinsky calculations compares favorably with experiment ($\Delta
E^{exp}=5.51$ MeV).

Fig. 5 illustrates the polarization effects present in the CRMF calculations
of terminating states. Since fully stretched states with spin $I_{max}$ are
reasonably well described by a single Slater determinant ZSNSZ.07 , the
comparison with experiment is performed without angular momentum restoration
as in all DFT studies of the terminating $I_{max}$ states in this mass region,
see, for example, Refs. ZSW.05 ; BWSMG.06 . At spherical shape, the
$f_{7/2}-d_{3/2}$ energy gap ($\Delta E^{sph}$) in the single-particle spectra
considerably exceeds $\Delta E^{exp}$. The $\Delta E^{sph}$ value is a very
good approximation to the results of the spherical RMF calculations without NM
in which the gap between these states is defined as the difference of the
binding energies of the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and $f_{7/2}^{n}$
states: the difference between two results for all nuclei under study does not
exceed 40 keV. When deformation polarization effects denoted as $\Delta
E^{def-pol}$ are taken into account (the “CRMF-WNM” column in Fig. 5), this
gap becomes even larger and reaches $\Delta E=7.57$ MeV exceeding by 37% the
experimental value. The inclusion of NM decreases the difference between
experiment and calculations considerably (by 1.14 MeV) (column “CRMF-NM” in
Fig. 5). Note that mass and charge quadrupole and hexadecapole moments change
only by $\sim 10^{-4}\%$ on going from the CRMF-WNM to CRMF-NM solutions.
Thus, the deformation differences between these two solutions are almost non-
existant and can be neglected. Based on the consideration of polarization
effects, the $\Delta E^{SC}$ can be approximated as

$\displaystyle\Delta E^{SC}\approx\Delta E^{sph}+\Delta E^{def-pol}-\Delta
E^{SC}_{TO}$ (2)

Considering that the terminating states of interest are close to spherical
(Fig. 6), this approximation which corresponds to a perturbative treatment of
the deformation polarization effects should be quite reasonable. This
approximation also allows to use the results of spherical RMF calculations in
subsequent analysis of $\Delta E$ (Sect. IV).

Figure 6:  (Color online) Quadrupole deformations of the terminating states
obtained in the CRMF calculations with the NL1 parametrization. For each
nucleus, the deformation polarization energies $\Delta E^{def-pol}$ are shown
between the calculated deformation points. Three nuclei with the largest
$\Delta E^{def-pol}$ values are indicated by arrows.

###  III.3 The Nilsson potential analogy of Ref. ZSW.05

In order to overcome the problems discussed in Sect. III.2.1, the authors of
Ref. ZSW.05  use the analogy with the simple form of the Nilsson potential
N.55  for the $\Delta E^{SC}_{TE}$ part. This form of the Nilsson potential is
similar to the one given below (Eq. (5)), but with the parameters $\kappa$ and
$\mu$ independent on principal quantum number $N$. However, no proof (analogy
is not a proof) is provided whether such a transition from self-consistent DFT
to the Nilsson potential is valid and whether the dependence of the energies
of the single-particle states on quantum numbers $N,j,l,$ and $s$ is the same
in the self-consistent DFT and in the Nilsson potential. Fig. 5 clearly shows
that the later is not a case.

In the simple form of the Nilsson potential, the magnitude of the
$f_{7/2}-d_{3/2}$ splitting related to the magic gap 20 is given by

$\displaystyle\Delta E^{Nil}=\hbar\omega_{0}(1-6\kappa-2\kappa\mu).$ (3)

Thus, it depends on three major factors: (i) the energy scale of the single-
particle potential characterized by $\hbar\omega_{0}$, (ii) the flat-bottom
and surface properties entering through the orbit-orbit term $\sim\mu$, and
(iii) the strength of the spin-orbit term $\kappa$. Then, the authors of Ref.
ZSW.05  using the fact that in light nuclei the Nilsson potential resembles
the pure harmonic oscillator potential, which leads to $\mu\sim 0$, conclude
that the magnitude of the $f_{7/2}-d_{3/2}$ splitting is given by

$\displaystyle\Delta E^{Nil}=\hbar\omega_{0}(1-6\kappa).$ (4)

Thus, in this approximation, the $\Delta E^{Nil}$ splitting is defined only by
the energy scale $\hbar\omega_{0}$ and the strength of the
$\it{ls}$-potential. Arguing that the energy scale is rather well constrained
by the data not only for the Nilsson but also for the self-consistent
approaches, the authors of Ref. ZSW.05  conclude that the $f_{7/2}-d_{3/2}$
splitting is directly related to the strength of the $\it{ls}$-potential. Or
alternatively, this approximation corresponds to the situation when the
$\Delta E^{Nils}$ does not depend on orbital motion of the nucleons.

###  III.4 An alternative form of the Nilsson potential

One question of paramount importance we have to ask is whether the simple form
of the Nilsson potential given in Ref. N.55  and used in Ref. ZSW.05  is
unique and how well it describes the experimental data. It turns out that the
modern versions of the Nilsson potential employ the parameters $\kappa$ and
$\mu$ which are dependent on the main oscillator quantum number $N$ and on
nucleon type (proton or neutron), thus facilitating the study of wide range of
nuclei with the same set of single-particle parameters and with comparable
accuracy BR.85 ; Ragbook ; Rag-priv . Since the Nilsson potential is
phenomenological in nature, this procedure is well justified. The accuracy of
the description of different physical quantities such as, for example,
rotational properties and relative energies of different single-particle
configurations BR.85 ; Ragbook ; CNS ; GBI.86  is considerably improved when
different values of $\kappa$ and $\mu$ are used for different $N$-shells.
Although some variations between the parametrizations exist, this approach is
used in almost all parametrizations of the Nilsson potential developed from
the middle of the 80ties of the last century GBI.86 ; BR.85 ; A150 . The
studies employing this description of the Nilsson potential are abundant and
provide systematic information on the accuracy of the description of physical
observables in different mass regions. Even more sophisticated dependence of
the $\kappa$ and $\mu$ parameters on the principal quantum number $N$ and
orbital angular momentum $l$ is introduced in Ref. S.86  and employed in a
number of studies, see, for example, Refs. PRet.97 ; BZ.95 .

####  III.4.1 Terminating states in the $A\sim 44$ mass region

In the Nilsson potential with $\kappa_{N}$ and $\mu_{N}$ parameters dependent
on the principal oscillator quantum number $N$ BR.85 ; Ragbook

$\displaystyle\hat{H}^{Nil-N-dep}-\frac{3}{2}\hbar\omega_{0}=$ (5)
$\displaystyle=$
$\displaystyle\hbar\omega_{0}\\{N-\kappa_{N}[2{\bm{l}}{\bm{s}}+\mu_{N}(\bm{l}^{2}-<\bm{l}^{2}>_{N})]\\},$

the magnitude of the $f_{7/2}-d_{3/2}$ splitting related to the energy
difference of the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and $f_{7/2}^{n}$ terminating
states is given by

$\displaystyle\Delta E^{Nil-N-dep}=$ $\displaystyle=$
$\displaystyle\hbar\omega_{0}(1-3[\kappa_{2}+\kappa_{3}]-3\kappa_{3}\mu_{3}-\kappa_{2}\mu_{2}).$

The superscript ${}^{\prime}Nil-N-dep^{\prime}$ is used to indicate the
dependence of these expressions on the main oscillator quantum number $N$.

Table 1:  The standard parametrization of the Nilsson potential from Ref. BR.85 ; Ragbook . Only the parameters for the $N=2$ and 3 shells are shown. | Protons | Neutrons
---|---|---
N | $\kappa$ | $\mu$ | $\kappa$ | $\mu$
2 | 0.105 | 0.00 | 0.105 | 0.00
3 | 0.090 | 0.30 | 0.090 | 0.25

The $\kappa_{N}$ and $\mu_{N}$ parameters of the so-called standard
parametrization of the Nilsson potential are shown for the shells of interest
in Table 1. For the protons, the value $\Delta E^{Nil-N-
dep}=0.415\hbar\omega_{0}$ is obtained in the calculations employing the
$\kappa_{N}$ parameters from the Table 1 but assuming the $\mu_{N}=0$ as it
was done in the derivation of Eq. (4). This value can be compared with the
$\Delta E^{Nil-N-dep}=0.334\hbar\omega_{0}$ value obtained with the
$\kappa_{N},\mu_{N}$ parameters from the Table 1. One can see that these two
values differ by approximately 25% and this difference is solely attributed to
the non-zero value of the $\mu_{3}$ parameter. Considering that $\Delta
E^{exp}\sim 5.5$ MeV (see Fig. 7), 25% difference correspond to 1.4 MeV; this
difference definetely cannot be ignored when experimental data is compared
with experiment.

####  III.4.2 Concluding remarks

Even for terminating states in the $A\sim 44$ mass region one cannot ignore
the dependence of the energies of the (N,l) and (N’,l’) states, from which the
${\it j=l-\frac{1}{2}}$ and ${\it j^{\prime}=l^{\prime}+\frac{1}{2}}$ states
used in the TS-method emerge (see Fig. 4), on the orbital angular momentum.
This dependence enters through the $\mu_{N}(\bm{l}^{2}-<\bm{l}^{2}>_{N})]$
term of the Nilsson potential (Eq. (5)). This is contrary to the approximation
made in the derivation of Eq. (4) which has a consequence that the energy
difference $\Delta E^{Nils}$ depends only on the energy scale
$\hbar\omega_{0}$ and the strength of the spin-orbit term $\kappa$. The
dependence of $\Delta E^{Nils}$ on the orbital angular momentum in the case of
terminating states involving single-particle states from higher $N$-shells has
been recognized in Ref. ZS.07 .

##  IV Terminating states in the $A\sim 44$ mass region: what we can learn
from the comparison with experiment

Figure 7:  (Color online) The experimental and calculated magnitudes $\Delta
E$ of the $f_{7/2}-d_{3/2}$ splittings as extracted from the energies of
terminating states ($\Delta E=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$).
The results of the CRMF calculations are shown for the NL1 and NL3
parametrizations of the RMF Lagrangian. The experimental data corrected for
the deformation polarization effects (as obtained in the NL1 parametrization)
is shown by open circles. Note that the experimental and deformation
polarization corrected values of $\Delta E$ coincide in the case of 47V.

Fig. 7 compares the results of calculations with experiment. The same data set
as in Refs. ZSW.05 ; BWSMG.06  is used in this comparison, but it is shown as
a function of $n$, where $n$ stands for the number of the $f_{7/2}$ protons in
the $f_{7/2}^{n}$ terminating state. In addition, this figure compares the
absolute values and not the differences between experimental results and
calculations as it was done in Refs. ZSW.05 ; BWSMG.06 : the differences
normalized to 44Ca are compared in Fig. 8a. Since the $n$ value is the same
for each isotope chain ($n=0$ for the Ca isotopes, $n=1$ for the Cs isotopes,
$n=2$ for the Ti isotopes, and $n=3$ for the V isotopes), the isospin
dependence of $\Delta E$ is clearly visible. Different isotope chains show
different isospin dependencies and they are well reproduced in the
calculations (Fig. 7 and Fig. 8a). On the other hand, the calculations
overestimate the absolute value of $\Delta E$ (Fig. 7), and the difference
between the calculated and experimental $\Delta E$ values show pronounced
$n$-dependence (Fig. 8a).

One source of the discrepancy between theory and experiment is related to the
impact of effective mass of the nucleon on the single-particle spectra (see
Sect. V): in general, it should lead to an overestimate of experimental
$\Delta E$ in the calculations. The other sources of these discrepancies are
analyzed in detail in this Section.

Figure 8:  (Color online) (a) The difference $(\Delta E^{th}-\Delta
E^{exp})_{norm}$ between the calculated and experimental values of $\Delta E$
(based on the results of Fig. 7). This difference is normalized to zero for
44Ca. (b) The calculated difference
$E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n})$ shown as a function
of $n$ for the indicated RMF parametrizations (based on the results of Fig.
9).

###  IV.1 Deformation polarization effects

The deformation polarization effects discussed in Sect. III.2.2 are
characterized by the $\Delta E^{def-pol}$ energies. The sign of $\Delta
E^{def-pol}$ depends on relative deformations of the $f_{7/2}^{n}$ and
$d_{3/2}^{-1}f_{7/2}^{n+1}$ terminating states (Fig. 6). It is positive
(negative) when the $\beta_{2}$-deformation of the $d_{3/2}^{-1}f_{7/2}^{n+1}$
state is larger (smaller) than the one of the $f_{7/2}^{n}$ state. The $\Delta
E^{def-pol}$ values almost do not depend on the parametrization of the RMF
Lagrangian: the difference in their values is below 10 keV if the results of
the NL1 and NL3 parametrizations are compared. If the experimental data are
corrected for these deformation polarization effects, then smooth trend (the
curve ’exp. (def-cor)’ in Fig. 7) as a function of $n$ emerges. The $\Delta E$
value along this curve decreases by $\sim 0.25$ MeV on going from $n=0$ to
$n=3$. Assuming that these effects are reasonably well described in the
calculations, one can conclude that the nucleus-dependent fluctuations in
experimental value of $\Delta E$ (the curve ’exp.’ in Fig. 7) are due to
deformation polarization effects.

###  IV.2 Nuclear magnetism (time-odd mean fields) in the terminating states
of the $A\sim 44$ mass region

Fig. 9 shows the additional bindings $E_{TO}(state)$ to the energies of
terminating states due to NM. This quantity increases with the increase of the
$n$-value for the $f_{7/2}^{n}$ and $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating
states. The increase of $E_{TO}$ with isospin within specific isotope chain is
associated with the increase of the number of the occupied neutron $f_{7/2}$
states and corresponding increase in spin. The increase of the values of
$E_{TO}$ correlates with the increase of the spin of the terminating states:
for example, the $f_{7/2}^{n}$ and $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating
states have $I_{max}=6^{+}$ and $I_{max}=11^{-}$ in 42Ca and
$I_{max}=\frac{31}{2}^{-}$ and $I_{max}=\frac{35}{2}^{+}$ in 47V,
respectively.

The results of the calculations confirm previous conclusion obtained in 20Ne
(Sect. II) that the additional binding due to NM is considerably enhanced in
the terminating states. At no rotation, the additional binding due to NM to
the energies of the single-particle configurations in odd-mass nuclei is in
average around $\sim 100$ keV and seldom reaches 200 keV in the mass region of
interest AA.08 . This is much smaller than the additional binding observed in
the terminating states in which it reaches 4 MeV for $n=3$ (Fig. 9).

Because of their magnitude, the $E_{TO}$ values in terminating states are also
a good measure of how well the time-odd mean fields are defined in the
specific version of DFT. The $E_{TO}$ values obtained with different
frequently used non-linear parametrizations of the RMF Lagrangian such as NL1
NL1 , NL3 NL3 , NLSH NLSH , NLRA1 NLRA1  and NLZ NLZ  are shown in Fig. 9.
With increasing $E_{TO}$ and $n$, the absolute variations in the $E_{TO}$
values calculated with different RMF parametrizations increase. However, they
are still within 15% of the absolute value of $E_{TO}$. This result suggests
that within the non-linear versions of the RMF Lagrangian NM is defined with
similar accuracy.

Figure 9:  (Color online) Additional bindings $E_{TO}(state)$ (in absolute
value) to the energies of terminating states due to NM shown for the
terminating states of interest. The results are shown for the indicated
parametrizations of the RMF Lagrangian as a function of $n$.

This value can be used to estimate the uncertainty in the definition of the
moments of inertia in the CRMF calculations due to the uncertainty in NM.
Dependent on the nuclear system, the NM contribution to the total kinematic
moment of inertia is approximately 10-25% CRMF ; AA.08 . Thus, the uncertainty
of the definition of the absolute value of the total kinematic moments of
inertia due to the uncertainty in the definition of NM is modest being in
range of 1.5-3.75%.

It follows from Fig. 8 that the portion of the $n$-independent part of the
discrepancy between experimental and calculated $\Delta E$ values may be
related to the uncertainties in NM since the difference
$E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n})$ somewhat (within
$\approx 200$ keV) depends on the RMF parametrization. The contribution of NM
into the $n$-dependent part of this discrepancy is discussed in Sect. IV.3.3.

###  IV.3 The dependence of $\Delta E$ on orbital angular momentum and spin-
orbit interaction

####  IV.3.1 The impact of density modifications on the single-particle
properties

It is well known fact that the modifications of the central nucleonic
potential and its surface properties affect the single-particle states with
different angular momentum l in a different way (see Refs. MBBD.85 ; RBRMG.98
and references therein). They also alter the spin-orbit potential and lead to
the changes in the spin-orbit splittings. In order to check how big this
effect is in the nuclei under study, the proton density distributions and
single-particle spectra at spherical shape are compared in Fig. 10 for the
42Ca and 47V nuclei. These nuclei represent the lower and upper mass ends of
the data set under investigation. The configuration of 47V has 2 additional
$f_{7/2}$ neutrons and 3 additional $f_{7/2}$ protons as compared with the
configuration of 42Ca. The filling of these high-$j$ orbitals increases the
density near the surface (Fig. 10a). These modifications of the density change
the central and spin-orbit nucleonic potentials (in a similar fashion as it
was discussed in Refs. TPC.04 ; AF.05 ) leading to the modifications of the
single-particle spectra (Fig. 10b).

Figure 10:  (Color online) (a) Proton density distribution in the 42Ca and 47V
nuclei as obtained for ground state in the spherical RMF calculations with the
NL1 parametrization of the Lagrangian. (b) Corresponding energies of the
single-particle states. The states in 42Ca are shown at the calculated
energies, while all states in 47V are shifted by constant value in such a way
that the average energy of the $1d_{3/2}$ and $1f_{7/2}$ states is the same as
in 42Ca.

On going from 42Ca to 47V, the spin-orbit splitting in the $d_{5/2}-d_{3/2}$
doublet decreases by 0.12 MeV (from 6.72 MeV to 6.60 MeV), while the one in
the $f_{7/2}-f_{5/2}$ doublet increases by 0.57 MeV (from 7.0 MeV to 7.57
MeV). If the modifications in the single-particle spectra would be restricted
only to the spin-orbit splittings and their modifications would evenly be
redistributed between the $j=l+1/2$ and $j=l-1/2$ members of the spin-orbit
doublet, this would decrease the $1f_{7/2}-1d_{3/2}$ splitting by 0.22 MeV.

However, the calculations show that the $f_{7/2}-d_{3/2}$ splitting is
increased by 0.34 MeV (Figs. 10). Assuming that the energy scale does not
change on going from 42Ca to 47V, this can only be explained by the change of
the relative positions of the $d$ (${\it l}=2$) and $f$ (${\it l}=3$) states
from which the $d_{3/2}$ and $f_{7/2}$ states emerge. Unfortunately, there is
no straightforward way in the RMF calculations to get an access to the $(N,l)$
states (in sense of Fig. 4). Thus, in order to illustrate the dependence on l,
the centroid energy (denoted as “centr(state)” in Fig. 10b) and defined as an
average energy of the members of spin-orbit doublet is used. Fig. 10b shows
that the energy gap between the centroids of the $d$ and $f$ spin-orbit
doublets increases by 0.55 MeV on going from 42Ca to 47V. As a consequence of
this increase and the above discussed changes in the spin-orbit splittings,
the $1f_{7/2}-1d_{3/2}$ splitting increases by 0.34 MeV. This value represents
more than half of the increase of $(\Delta E^{th}-\Delta E^{exp})_{norm}$ on
going from 42Ca to 47V (Fig. 8a).

####  IV.3.2 Relative placements of the states with different angular momentum
$l$

The fact that the relative placement of states with different orbital angular
momentum $l$ (especially, of high-$l$ states) is not well reproduced in non-
relativistic and relativistic mean field models is well known, see Refs.
MBBD.85 ; RBRMG.98 ; LBBDM.07 . The origin of this problem is connected with
the surface profile of the mean field and kinetic terms. Microscopic
considerations indicate that the effective mass of the nucleon has a
pronounced surface profile which is insufficiently parametrized in the present
mean field models MBBD.85 . In Refs. ZSW.05 ; BWSMG.06 , this fact has been
ignored and no proof has been provided that the placement of the $d$ and $f$
states, from which the $d_{3/2}$ and $f_{7/2}$ states, emerge is correct.

It turns out that the difference in absolute value of $\Delta E$ obtained in
the NL1 and NL3 parametrizations (Fig. 8) is well explained by the differences
in the relative energies of the $d$ and $f$ states in these parametrizations.
The energy gap between the centroids of the $d$ and $f$ spin-orbit doublets is
larger in the NL3 parametrization as compared with the NL1 one by
approximately 400 keV. If one corrects the NL1 results by this energy gap, one
gets the results indicated as NL1cor in Fig. 7. The NL1cor results are very
close to the NL3 ones, which strongly suggests that the difference between the
NL1 and NL3 results is predominantly due to different relative energies of the
$l=3$ and $l=2$ states in these parametrizations of the RMF Lagrangian.

Ref. BWSMG.06  has attributed the fact that the NL1 and NL3 parametrizations
differ in the description of the absolute $\Delta E$ value (Fig. 7) to the
magnitude of the iso-scalar spin-orbit potential. The current investigation
does not support this interpretation.

These results suggest that instead of readjusting the isoscalar strength of
the spin-orbit interaction as it was done in Ref. ZSW.05 , one can attempt to
readjust the coupling constants of the DFT terms influencing the relative
energies of the $l=2$ and $l=3$ states with the same effect on $\Delta E$.
Indeed, 5% reduction of the isoscalar strength of the spin-orbit interaction
introduced in Ref. ZSW.05  reduces $\Delta E$ by $\sim 350$ keV and this
change in $\Delta E$ almost does not depend on nucleus (Fig. in Ref. SW.08 ).
On the other hand, the CRMF results suggests that the same effect can be
achieved if the relative distance of the $d$ and $f$ states is modified.
Indeed, the $\Delta E$ values obtained in the CRMF calculations decrease by
approximately the same amount on going from the NL3 to NL1 parametrization of
the RMF Lagrangian and this decrease only weakly depends on the nucleus (Fig.
7).

####  IV.3.3 The $n$-dependence of $(\Delta E^{th}-\Delta E^{exp})_{norm}$

The $(\Delta E^{th}-\Delta E^{exp})_{norm}$ quantity shows pronounced
dependence on $n$ (Fig. 8a) and its trend (if normalized to a single nucleus)
almost does not depend on the RMF parametrization. In order to understand
which ingredients of $\Delta E^{SC}$ contribute into this $n$-dependence, the
variations $\delta E_{i}=\Delta E_{i}(nucleus)-\Delta E_{i}(^{47}V)$ of
different ($i$-th) terms contributing to $\Delta E^{SC}$ are studied below.
Contrary to Fig. 8a, 47V is selected as a reference in order to get a picture
less disturbed by large fluctuations of some variations in the vicinity of
42Ca (Fig. 11). The $\delta
E_{TO}=\delta(E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n}))$
variation is obtained in the deformed CRMF calculations (from Fig. 8b), while
other variations shown in Fig. 11 are calculated in spherical RMF
calculations. Thus, I effectively employ the approximation given in Eq. (2)
assuming that the deformation polarization effects are the same both in theory
and experiment. All the results presented here are based on the calculations
with the NL1 parametrization, but it was checked that the NL3 results are
similar.

The $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ variation indicates that
the difference between the calculated and experimental $\Delta E$ values
decreases with decreasing $n$. Note that for a given $n$ it is almost constant
indicating only weak isospin dependence of this variation. The largest changes
as a function of nucleus amongst the calculated variations are seen in the
energy gap between the centroids of the $d$ and $f$ spin-orbit doublets (the
curve denoted as “$\delta E({\it l}-{\rm centroids})$” in Fig. 11). It has the
same trend as $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ as a function of
$n$. For a given $n$, it shows very large dependence on isospin. The second
largest variation is seen in the spin-orbit splitting of the $f_{7/2}-f_{5/2}$
spin-orbit doublet (the curve denoted as “$\delta E_{\it
ls}(f_{7/2}-f_{5/2})/2$” in Fig. 11). The factor 1/2 is used in this variation
since only one half of the total variation of spin-orbit splitting contributes
into the $f_{7/2}-d_{3/2}$ splitting (see Sect. IV.3.1). The $\delta E_{\it
ls}(f_{7/2}-f_{5/2})/2$ variation has the wrong trend as compared with
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$. The $\delta E_{\it
ls}(d_{5/2}-d_{3/2})/2$ variation of the spin-orbit splitting in the
$d_{5/2}-d_{3/2}$ doublet is quite small. It has the correct trend as compared
with $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$.

The $\delta E(f_{7/2}-d_{3/2})$ variation of the $f_{7/2}-d_{3/2}$ splitting
approximately satisfies the relation

$\displaystyle\delta E(f_{7/2}-d_{3/2})=\delta E({\it l}-{\rm centroids})+$
$\displaystyle\delta E_{\it ls}(f_{7/2}-f_{5/2})/2+\delta E_{\it
ls}(d_{5/2}-d_{3/2})/2$ (7)

The isospin dependencies seen in $\delta E({\it l}-{\rm centroids})$ and
$\delta E_{\it ls}(f_{7/2}-f_{5/2})/2$ act is opposite directions, thus,
reducing the isospin dependence of $\delta E(f_{7/2}-d_{3/2})$ as compared
with the one of $\delta E({\it l}-{\rm centroids})$. However, the $\delta
E(f_{7/2}-d_{3/2})$ variation (Sect. IV.3.1) cannot completely account neither
for absolute value nor for isospin dependence (for a given $n$) of the
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ variation.

Only when the $\delta E(f_{7/2}-d_{3/2})$ variation is combined with the
$\delta E_{TO}$ variation due to NM by

$\displaystyle\delta E^{sum}=\delta E(f_{7/2}-d_{3/2})+\delta E_{TO}$ (8)

a better description of the $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$
variation emerges. For a given $n$, the isospin dependence of the
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ is well described by $\delta
E^{sum}$. The absolute value of $\delta((\Delta E^{th}-\Delta
E^{exp})_{norm})$ for the Ti nuclei is well described by $\delta E^{sum}$.
However, for the Ca and Sc nuclei, the difference between these two quantities
reaches $30\%$ of the absolute value of $\delta((\Delta E^{th}-\Delta
E^{exp})_{norm})$. The part of this discrepancy is definitely related to the
limitations of the approximation given by Eq. (2).

Thus, the current study clearly shows that the modifications of the relative
placement of the states with different angular momentum $l$, the spin-orbit
splittings and time-odd mean fields on going from 47V to 42Ca contribute into
the $n$-dependence of the difference between the calculated and experimental
$\Delta E$ values (the $\Delta E^{th}-\Delta E^{exp})_{norm}$ quantity).
Previously, this $n$-dependence of $(\Delta E^{th}-\Delta E^{exp})_{norm}$,
expressed in a different form (Fig. 1 in Ref. BWSMG.06 , has been solely
attributed to the deficiency of the iso-vector term of the spin-orbit
interaction BWSMG.06 , but the current investigation does not support such an
interpretation.

Figure 11:  (Color online) The variations $\delta E_{i}$ of different terms
contributing to the $\delta(\Delta E^{th}-\Delta E^{exp})_{norm})$ variation,
see text for detail.

##  V The energy scale and the effective mass of the nucleon

The terminating states are expected to be of predominantly single-particle
nature CNS ; ZSNSZ.07 : the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating states
are obtained from the $f_{7/2}^{n}$ terminating state by particle-hole (p-h)
excitation from the $d_{3/2}$ state into the $f_{7/2}$ state. The energy of
this p-h excitation depends on the energies of above mentioned states, and,
thus, it is affected by the energy scale of the single-particle spectra which
is related to the effective mass $m^{*}(k_{F})/m$ of the nucleon at the Fermi
surface.

In the RMF theory, the spin-orbit interaction is effectively scaled by the
effective mass of the nucleon (Ref. R.89 ), and that is a reason why
experimental data on spin-orbit splittings are well described in the
calculations RBRMG.98 ; BRRMG.99 . This scaling also explains why the spin-
orbit splittings of the $1p_{3/2}-1p_{1/2}$, $1d_{5/2}-1d_{3/2}$ and
$1f_{7/2}-1f_{5/2}$ spin-orbit partner orbitals are almost the same in the RMF
and the Nilsson potential calculations (Fig. 5). Note, that the Nilsson
potential is characterized by the effective mass $m^{*}(k_{F})/m\sim 1$ which
is typical for experimental density of the quasiparticle states. Only in the
case of the $2p_{3/2}-2p_{1/2}$ doublet, the spin-orbit splitting is smaller
in the RMF calculations.

On the other hand, the energies of the centroids of the spin-orbit doublets
are stretched out in the RMF calculations as compared with the Nilsson
potential: the difference between the RMF and Nilsson centroid energies
increases on going away from the Fermi level (Fig. 5). Thus, the stretching
out of the single-particle spectra due to low effective mass of the nucleon
shows up mostly for orbital motion of particles and affects the relative
placement of the levels with different angular momentum l. The origin of this
problem has been discussed in Sect. IV.3.2.

The effective mass of nucleon at the Fermi surface (Lorentz mass in the
notation of Ref. JM.89  for the case of the RMF theory) is $m^{*}(k_{F})/m\sim
0.65$ in the RMF theory BRRMG.99 , $\sim 0.7$ in the case of the Hartree-Fock
(HF) approach based on the Gogny forces BHR.03 , and varies in the range
$0.6-1.0$ in the HF approach based on the Skyrme forces BHR.03  revealing much
larger flexibility of this type of the DFT with respect of effective mass. As
a consequence of low effective mass, the calculated spectra are less dense
than the experimental ones: the well known fact in non-relativistic and
relativistic models both for spherical MBBD.85 ; LR.06 ; RBRMG.98  and
deformed systems A250 ; BBDH.03 . This study shows that the $\Delta E^{SC}$
quantity differs from the $f_{7/2}-d_{3/2}$ energy gap in the spherical
single-particle spectra only by the effects of time-odd mean fields and
deformation polarization effects (Sect. III.2.2). These facts pose an open
problem on how to compare the experimental data on $\Delta E$ with the results
of the DFT calculations (especially, those with low effective mass) since the
experimental data on $\Delta E^{exp}$ is expected to be characterized by
$m^{*}(k_{F})/m\sim 1$. The implicit assumption used in Refs. ZSW.05 ;
BWSMG.06  that the DFT reproduces the empirical $\Delta E$ values relatively
well, say within $\sim 10\%$ SW.08 , may be too optimistic especially for the
DFT with low effective mass.

##  VI Conclusions

In conclusion, the following results were obtained in the study of band
termination within the DFT framework:

  * •

At band termination, the NM does not modify neither total angular momentum nor
the expectation values of the single-particle angular momenta $<j_{x}>_{i}$ of
the single-particle orbitals. NM provides an additional binding to the
energies of the specific configuration and this additional binding increases
with spin and has its maximum exactly at the terminating state. This suggests
that the terminating states can be an interesting probe of the time-odd mean
fields related to NM provided that other effects can be reliably isolated.

  * •

The realization of the TS-method in Refs. ZSW.05 ; BWSMG.06  is based on the
analogy with simple form of the Nilsson potential which allows to neglect the
deficiences in the relative placement of the states with different angular
momentum $l$. This approximation is not valid for terminating states in the
$A\sim 44$ mass region in modern and most frequently used versions of the
Nilsson potential

  * •

The impact of the relative placement of the states with different angular
momentum $l$ on $\Delta E^{SC}$ is also clearly visible in the RMF
calculations. The difference in absolute $\Delta E^{SC}$ values obtained in
the CRMF calculations with the NL1 and NL3 parametrizations of the Lagrangian
is defined by the different relative energies of the $l=3$ and $l=2$ states in
these parametrizations. The modifications of the relative distance of the
states with different angular momentum $l$ on going from 47V to 42Ca
contribute into the $n$-dependence of the difference between the calculated
and experimental $\Delta E$ values (the $\Delta E^{th}-\Delta E^{exp})_{norm}$
quantity) in addition to the ones due to the spin-orbit interaction and time-
odd mean fields.

The detailed analysis of the TS-method in the RMF framework reveals the
picture which is more complicated than the one suggested in Refs. ZSW.05 ;
BWSMG.06 . The relative placement of the states with different angular
momentum ${\it l}$, defined by the properties of central potential, has to be
taken into account in addition to the DFT terms discussed in these references
when the $\Delta E$ quantity is analyzed. Considering the similarities of the
RMF theory and SDFT, it is very likely that these conclusions are also valid
in the SDFT framework. The current investigation calls for a detailed study of
the impact of the relative placement of the states with different orbital
angular momentum ${\it l}$ on the $\Delta E^{SC}$ quantity in the SDFT
framework.

Existing results for superdeformed bands in 32S MDD.00 ; Pingst-A30-60  and
low-spin states in odd mass nuclei AA.08  point to the time-odd mean fields as
a major point of the difference between SDFT and RMF. For example, the
additional binding due to time-odd mean fields and the energy separation
between different signatures of the SD bands are considerable stronger in SDFT
as compared with RMF MDD.00 . The current study clearly shows that the
correlations induced by time-odd mean fields are large: additional binding due
to NM reaches 4 MeV for $n=3$ (Fig. 9), which is by order of magnitude larger
than those seen before in the RMF calculations at low spin. It also has a
considerable impact on $\Delta E^{SC}$: $\Delta E_{TO}\sim 1.2$ MeV (Fig. 8b).
These results call for a comparative study of time-odd mean fields in the
Skyrme DFT and RMF frameworks. Such study is necessary in order to make a
significant progress towards a better understanding of the role of time-odd
mean fields. The work in this direction is in progress and the results will be
presented in a forthcoming manuscript AA.08 .

##  VII Acknowledgements

The material is based upon work supported by the Department of Energy under
Award Number DE-FG02-07ER41459.

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[1]Sebastian Wuchterl 11footnotetext: For the ATLAS and CMS Collaborations.

# Top quark mass and properties in ATLAS and CMS

###### Abstract

For standard model (SM) processes involving top quarks, such as top quark-
antiquark pair production or single top quark production, the theoretical
predictions depend on fundamental parameters of the SM like the top quark
mass. Using the large data sample collected at the CERN LHC in the second
data-taking period by the ATLAS and CMS experiments, these parameters can be
extracted in experimental measurements with high precision. Furthermore,
properties of the production processes such as quark polarization or
asymmetries can be measured. Exploiting the large luminosity of the data set,
new kinematic regimes and methods are explored, as for events in which the top
quarks have a very large transverse momentum. In these proceedings, recent
measurements of SM parameters and top quark properties are presented.
Individual results are also interpreted in terms of effective field theory
extensions of the SM.

##  1 Introduction

At the CERN LHC, top quarks are produced with a high production rate,
predominantly in top quark-antiquark pair ($\mathrm{t\bar{t}}$) production via
the gluon fusion mechanism. However, given the data set available from the
second data taking period (LHC Run II) (2015-2018) at a center-of-mass-energy
of 13 TeV, also processes like single top t-channel production can be studied
with high precision. The large mass of the top quark, $\mathrm{m_{t}}$, is a
free parameter of the SM and needs to be determined experimentally. Its large
value hints that the top quark plays a special role within the standard model
(SM), especially for the electroweak symmetry breaking. Therefore, the precise
study of processes involving top quarks can shed a light also on beyond the SM
physics. Further, theoretical predictions for top quark processes depend on
$\mathrm{m_{t}}$ or other fundamental SM parameters, allowing for their
extraction from measurements of production cross sections or kinematic
observables. Recent measurements performed by the ATLAS [1] and CMS [2]
Collaborations are presented here in these proceedings.

##  2 Top quark mass measurements

Measurements of $\mathrm{m_{t}}$ can be distinguished depending on the
experimental procedures employed. On one hand, $\mathrm{m_{t}}$ can be
extracted in a well-defined theoretical renormalization scheme, e.g., pole or
MSR [3], by comparing absolute or differential cross section measurements to
theoretical predictions at fixed-order. This approach is usually referred to
as indirect measurement. On the other hand, $\mathrm{m_{t}}$ can be measured
by comparing multi-purpose Monte Carlo (MC) predictions to variables sensitive
to the reconstructed energy of the top quarks. Measurements of this type are
commonly denoted as direct measurements, and the value measured can be
referred to as the top quark MC mass $m_{\mathrm{{t}}}^{\mathrm{{MC}}}$. They
lack a clear theoretical interpretation compared to indirect measurements
because of the modeling of non-perturbative effects in the MC. Hence, an
interpretation uncertainty on the order of 0.5–1 GeV reflecting the usage of
probabilistic MC generators [4, 5] is added.

The most precise direct measurement for $\mathrm{m_{t}}$ to date was performed
recently by the CMS Collaboration using $35.9$ $\mathrm{fb^{-1}}$ of pp
collision data [6]. Events in the $\mathrm{t\bar{t}}$ decay channel with one
lepton are analyzed, and a kinematic fit is performed to reconstruct the top
quarks. Using a profiled likelihood fit in 5 dimensions, the top quark mass is
measured. The resulting value is $\mathrm{m_{t}}=171.77\pm
0.38\,\mathrm{GeV}$, which is in good agreement with previous measurements and
improves the precision by $0.12\,\mathrm{GeV}$.

The pole mass $\mathrm{m_{t}^{pole}}$ was also measured by CMS, exploiting the
mass sensitivity of $\mathrm{t\bar{t}}$ production with at least one
additional jet ($\mathrm{t\bar{t}+jet}$) [7]. The normalized differential
cross section as a function of the $\rho$ observable defined as
$340\,\mathrm{GeV}/m_{\mathrm{t\bar{t}+jet}}$ is measured at the parton level
using a profiled likelihood unfolding approach. Machine learning methods are
used for the reconstruction of $\rho$ and the event classification. From a
comparison to next-to-leading-order (NLO) predictions [8],
$\mathrm{m_{t}^{pole}}$is extracted using different parton distribution
functions. Using ABMP16NLO [9], $\mathrm{m_{t}^{pole}}$ is measured to be
$172.94\pm 1.37\,\mathrm{GeV}$. The best-fit predictions and the measurement
are shown in Fig. 1 (left).

Figure 1: Measured cross section as a function of the $\rho$ observable
compared to theoretical predictions for best-fit mass values (left) [7].
Unfolded cross section for $\mathrm{t\bar{t}}$ production as a function of the
boosted jet mass compared to MC predictions for varying top quark masses
(right) [10].

The connection between $\mathrm{m_{t}^{pole}}$ and
$m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ can be probed in the boosted regime by
measuring the invariant mass of the boosted jets. Here, a measurement by CMS
using a data set corresponding to 138 $\mathrm{fb^{-1}}$ [10] is presented for
the first time. The differential cross section is measured as a function of
the invariant jet mass and is unfolded to particle level. Comparing it to MC
predictions, $m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ is determined to be $172.76\pm
0.81\,\mathrm{GeV}$. The unfolded distribution is shown in Fig. 1 (right)
compared to predictions for different values of $\mathrm{m_{t}}$. With respect
to previous measurements, the result improves by a factor of two. This is
mainly due to dedicated calibrations of the jet mass scale and tuning of the
final state radiation scale in the MC.

Using the same observable and theoretical predictions at next-to-leading-log
precision, the relation between $m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ and
$\mathrm{m_{t}^{pole}}$and $\mathrm{m_{t}^{MSR}}$is studied by the ATLAS
Collaboration [11]. Template fits for the theoretical prediction to the
simulation at particle level are performed while the numerical difference
between $m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ and $\mathrm{m_{t}^{pole}}$ or
$\mathrm{m_{t}^{MSR}}$is determined. The difference between
$m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ and
$\mathrm{m_{t}^{MSR}}(R=1\,\mathrm{GeV})$ is measured to be
$80^{+350}_{-410}\,\mathrm{MeV}$, and $350^{+300}_{-360}\,\mathrm{MeV}$ for
$m_{\mathrm{{t}}}^{\mathrm{{MC}}}$ and $\mathrm{m_{t}^{pole}}$. A scale of
$R=1\,\mathrm{GeV}$ is used due to the numerical similarity to the pole mass
($\mathrm{m_{t}^{MSR}}(R=1\,\mathrm{GeV})\approx\mathrm{m_{t}^{pole}}$).

##  3 Measurements of top quark properties

Given the special features of top quarks, the polarization of top quarks and
antiquarks in t-channel single top production can be investigated, as done by
ATLAS using 139 $\mathrm{fb^{-1}}$ of pp collision data [12]. The polarization
vectors for t and $\mathrm{\bar{t}}$ are determined from a likelihood fit to
variables that are constructed to yield maximum sensitivity. All six
components of the polarization vectors are in good agreement with the SM
expectation at next-to-NLO accuracy. Additionally, distributions of the angles
of the lepton in the top quark rest frame are unfolded to particle level and
effective-field-theory (EFT) couplings affecting the tWb vertex are probed.

Using 138 $\mathrm{fb^{-1}}$ of pp collision data, CMS measured the
$\mathrm{t\bar{t}}$ charge asymmetry $A_{C}$ in boosted top quark pair events
with one lepton in the final state [13]. The measurement is presented here for
the first time. The charge asymmetry is predicted to be zero ($\approx 6.6\%$)
in the SM for the gluon fusion (quark-antiquark) production mechanism. Beyond-
the-SM effects could lead to a non-negligible change of the value of $A_{C}$.
It is the first measurement of $A_{C}$ analyzing pp collision data at
$\sqrt{s}=13\,\mathrm{TeV}$ with highly-boosted $\mathrm{t\bar{t}}$ events for
invariant masses of $m_{\mathrm{{t\bar{t}}}}>750\,\mathrm{GeV}$. The value for
$A_{C}$ is measured in three bins of $m_{\mathrm{{t\bar{t}}}}$ in the fiducial
and full phase space using a likelihood unfolding method. The measured data in
the full phase space compared to theoretical predictions is shown in Fig. 2
(left). A good agreement between measurement and SM expectation is observed.

Figure 2: Charge asymmetry in the full phase space compared to theoretical
predictions(left) [13]. Energy asymmetry for $\mathrm{t\bar{t}+jet}$
production in bins of the scattering angle of the additional jet (right) [14].

Similarly, the ATLAS Collaboration measured the energy asymmetry in
$\mathrm{t\bar{t}+jet}$ production [14], which yields an alternative method to
determine the charge asymmetry. The measurement is performed at particle level
as a function of the scattering angle of the additional jet using a likelihood
unfolding method. Also, here, events with boosted top quark jets are analyzed.
The result is shown in Fig. 2 (right). The energy asymmetry is measured to be
$0.043\pm 0.020$, which is in good agreement with the SM prediction.
Additionally, the result is interpreted in the context of EFT, setting limits
on four-quark operators.

##  4 Summary

In this contribution, recent measurements of the top quark mass and standard
model (SM) properties are presented. Significant progress was made by the
ATLAS and CMS Collaborations using the data collected during LHC Run II at a
center of mass energy of $13\,\mathrm{TeV}$, allowing to analyze phase space
regions with unprecedented precision for example in the boosted regime of top
quark-antiquark pair production ($\mathrm{t\bar{t}}$). Multiple top quark mass
measurements by CMS and an interpretation study by ATLAS are presented,
improving the understanding of the relation between well-defined mass
definitions and direct measurements. Both collaborations also measured SM
properties for top quark production processes, such as the charge and energy
asymmetry in $\mathrm{t\bar{t}}$ or $\mathrm{t\bar{t}}$ production with one
additional jet, respectively. Unfolded results are in good agreement with SM
predictions, and interpretations are performed in terms of effective-field-
theory (EFT) extensions of the SM. Finally, the top quark and antiquark
polarizations are measured by ATLAS in t-channel single top production events,
yielding results that confirm the SM expectation. Also here EFT coupling
limits are set, improving with respect to previous measurements.

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