# Janus-faced tomograms and retrieval of quadrature moments for $q$-deformed
states

S. Kannan1 and C. Sudheesh2 1 Department of Quantum Science and Technology,
Research School of Physics, The Australian National University, Canberra, ACT
2601, Australia.
2 Department of Physics, Indian Institute of Space Science and Technology,
Thiruvananthapuram 695 547, India.  $ ^1<EMAIL_ADDRESS>$ ^2
<EMAIL_ADDRESS>
###### Abstract

In this work, we derive the optical tomograms of various $q$-deformed quantum
states. We found that the optical tomograms of the states under consideration
exhibit a fascinating ‘Janus faced’ nature, irrespective of the deformation
parameter $q$. We also derived a general method to extract the quadrature
moments from the optical tomograms of any $q$-deformed states. We also note
that this technique can be used in high-precision experiments to observe
deviations from the standard quantum mechanical behavior.

##  1 Introduction

The quantum state provides all the information of a given system. Usually,
this is as a vector ($|{\psi}\rangle$) for pure states and a density matrix
($\rho$) for mixed states. In an experiment, if one is unaware of whether the
system is pure or mixed, many sets of measurements are required to reconstruct
the density matrix. This is briefly the method of quantum state tomography
[1]. Optical tomography was introduced in quantum optics by reconstructing
Wigner distribution and the density matrix of squeezed states of light [2].
Optical tomography is based on the finding that the rotated quadrature phase
can be expressed in quasiprobability distributions and vice versa [3]. Thus an
optical tomogram of a quantum system contains all the information encoded in
the system’s density matrix. The study of the physical properties of quantum
states and the estimation of errors in the experimentally observed values can
be evaluated using an optical tomogram [4]. In recent literature, optical
tomograms were used to study various nonclassical properties [5, 6], quantum
mechanical evolution of states [7], and also as an indicator of the
nonclassicality in a quantum system [8]. Optical tomograms were also used to
study $q$-deformed coherent states [9] and string coherent states [10].

Deformed quantum mechanics can play a vital role in the search for possible
new physics at a high energy scale. The general feature of such spaces is that
they are non-commutative and have a well-defined mathematical structure from
quantum group symmetries. $q$-deformation in quantum mechanics is widely
studied, ranging from the nonclassical features of $q$-deformed states [11,
12, 13, 14], a study of cosmic microwave background radiation [15], in loop
quantum gravity to construct observables with cosmological constants [16], the
realization of quasibosons [17], etc. In this work, we apply the concept of
optical tomograms to $q$-deformed quantum states. We derive the $q$-deformed
versions of well-known nonclassical states and an expression to calculate
quadrature moments from the optical tomograms of any $q$-deformed states. We
note that this can be used to derive various nonclassical features of deformed
quantum states and can be used in the search for new physics.

The paper is organized as follows: section 2 compares the optical tomogram of
$q$-deformed ‘Janus faced’ partner states. We introduce $q$-deformed squeezed
vacuum and excited states as the eigenstates of $q$-deformed two-photon
annihilation operators. In section 3, the optical tomogram is expressed as an
expectation value of normal ordered power series of $q$-deformed annihilation
and creation operators. We also derive an equation to obtain moments from the
optical tomogram of any $q$-deformed state. Section 4 is devoted to the
conclusion.

##  2 Janus-faced tomograms

It is known that $q$-deformed states, such as the $q$-deformed cat and
squeezed states, are nonclassical. While analyzing some of these $q$-deformed
states, we noticed that some of their optical tomograms reveal a ‘Janus-faced’
nature. We found that this property of quantum states was earlier reported in
non-deformed scenarios [18]. This section analyzes two ‘Janus-faced’ optical
tomogram pairs: (i). $q$-deformed even cat state and $q$-deformed squeezed
vacuum state, and (ii). $q$-deformed odd cat state and $q$-deformed squeezed
excited state.
Let us commence our study by considering a one-dimensional $q$-deformed
oscillator satisfying the algebra

$AA^{\dagger}-q^{2}A^{\dagger}A=1,\textnormal{ }|{q}|<1,$ (1)

where $A$ and $A^{\dagger}$ are the $q$-deformed annihilation and creation
operators [19]. In literature, this kind of deformation is called the math-
type $q$-deformation. Using $q$-deformed integers $[n]_{q}$, the algebra in
Eq. 1 can be defined on the $q$-deformed Fock space such that the following
relations are satisfied:

$A|{n}\rangle_{q}=\sqrt{[n]_{q}}|{n-1}\rangle_{q},\textnormal{
}A^{\dagger}|{n}\rangle_{q}=\sqrt{[n+1]_{q}}|{n+1}\rangle_{q},$ (2)

where the $q$-deformed integer $[n]_{q}$ and $|{n}\rangle_{q}$ is defined as

$[n]_{q}=\frac{1-q^{2n}}{1-q^{2}}\textnormal{,
}|{n}\rangle_{q}=\frac{A^{\dagger n}}{\sqrt{[n]_{q}!}}|{0}_{q}\textnormal{ and
}[n]_{q}!=\prod_{k=1}^{n}[k]_{q}.$ (3)

The algebra described above, which deviates from the non-deformed version, is
an example of non-standard quantum mechanical behavior. In the limit
$q\rightarrow 1$, everything converges to the non-deformed scenario satisfying
the algebra $[a,a^{\dagger}]=1$.

The nonclassical properties of $q$-deformed cat states are well documented in
the literature [11]. In terms of $q$-deformed coherent state
$|{\alpha}\rangle_{q}\,$[20], these are given by

$|{\alpha,\pm}\rangle_{q}=\frac{1}{N_{q}(\alpha,\pm)}\left(|{\alpha}\rangle_{q}\pm|{-\alpha}\rangle_{q}\right),$
(4)

where $|{\alpha,+}\rangle_{q}$ and $|{\alpha,-}\rangle_{q}$ represents the
even and odd $q$-deformed cat states with the normalization constant
$N_{q}(\alpha,\pm)$. The ‘Janus-faced’ $q$-deformed partner states of the even
and odd cat states are the eigenstates of the $q$-deformed version of two-
photon annihilation operators [21], $A^{\dagger-1}A$ and $AA^{\dagger-1}$. For
the non-deformed scenario, the eigenstates of the two-photon annihilation
operators (TAO’s) are analogous to the squeezed vacuum state and the squeezed
first excited state [21]. From Eq. 2, we obtain the action of TAO’s on the
$q$-deformed number state for $n\geq 2$

$A^{\dagger-1}A|{n}\rangle_{q}=\sqrt{\frac{[n]_{q}}{[n-1]_{q}}}|{n-2}\rangle_{q},\textnormal{
}AA^{\dagger-1}|{n}\rangle_{q}=\sqrt{\frac{[n-1]_{q}}{[n]_{q}}}|{n-2}\rangle_{q}.$
(5)

Now let us calculate the eigenstates for these operators. Let
$|\xi\rangle_{q}$ be the right eigenstate of $A^{\dagger-1}A$. In the deformed
Fock basis, $|\xi\rangle_{q}$ can be expanded as

$|\xi\rangle_{q}=\sum_{n=0}^{\infty}C_{n}|n\rangle_{q}.$ (6)

The action of the first TAO on $|\xi\rangle_{q}$ gives

$A^{\dagger-1}A|\xi\rangle_{q}=\sum_{n=0}^{\infty}C_{n+2}\sqrt{\frac{[n+2]_{q}}{[n+1]_{q}}}|n\rangle_{q}=\xi|\xi\rangle_{q}.$
(7)

We can see that the coefficient $C_{n}$ satisfies the recurrence relation

$\xi C_{n}=\sqrt{\frac{[n+2]_{q}}{[n+1]_{q}}}C_{n+2}.$ (8)

For even $n$, we can see that the state

$|\xi\rangle_{q}=N\sum_{n=0}^{\infty}\xi^{n}\sqrt{\frac{[2n-1]_{q}!!}{[2n]_{q}!!}}|2n\rangle_{q},$
(9)

where $N$ is the normalization factor and for $\xi=-e^{i\theta}\tanh(r)$,
$|\xi\rangle_{q}$ becomes the $q$-deformed squeezed vacuum state [22].
Similarly, we can compute the right eigenstate of the second TAO, and for odd
$n$, the eigenstate $|\xi_{1}\rangle_{q}$ is

$|\xi_{1}\rangle_{q}=N_{1}\sum_{n=0}^{\infty}\xi_{1}^{n}\sqrt{\frac{[2n+1]_{q}!!}{[2n]_{q}!!}}|2n+1\rangle_{q}.$
(10)

Here $N_{1}$ is the normalization constant and for
$\xi_{1}=-e^{i\theta}\tanh(r)$, we obtain the $q$-deformed squeezed first
excited state.

Figs. 1 and 2 are the $q$-deformed tomograms (explained in the next section)
for the states defined above. We have taken the parameters $\alpha$, $\xi$,
and $\xi_{1}$ to be real. These tomograms reveal their ‘Janus-faced’ nature.
Interestingly, the ‘Janus-faced’ tomogram pairs appear the same except for a
phase difference of $\pi/2$. These properties exist for all $q$ values. Also,
with an increase in deformation (small $q$ value), the strands in the
tomographic plane become narrower and denser (large value of
$\omega(X_{\theta},\theta)$).

Figure 1: $q$-deformed optical tomogram for $q$-even cat state ((a) and (c))
and $q$-squeezed vacuum state ((b) and (d)) with $|\alpha|^{2}=r=0.5$ for
$q=0.9$ ((a) and (b)) and $q=0.7$ ((c) and (d)).

Figure 2: $q$-deformed optical tomogram for $q$-odd cat state ((a) and (c))
and $q$-squeezed excited state ((b) and (d)) with $|\alpha|^{2}=r=0.5$ for
$q=0.9$ ((a) and (b)) and $q=0.7$ ((c) and (d)).

##  3 Estimation of moments from optical tomogram

Expressing density operator $\rho$ using normal ordered moments $(\langle
a^{\dagger\alpha}a^{\beta}\rangle)$ for non-deformed algebra was derived in
[23, 24]. Here we will derive the $q$-deformed version of the same.

Let us consider the expansion of an operator $F$ as a normal ordered power
series in $A$ and $A^{\dagger}$

$F=\sum_{\alpha,\beta=0}^{\infty}F_{\alpha,\beta}A^{\dagger\alpha}A^{\beta}.$
(11)

Using Eq. 2, it can be easily shown that

$_{q}\langle{m}|F|{n}\rangle_{q}=\sum_{r=0}^{min(\alpha,\beta)}F_{\alpha-r,\beta-r}\left(\frac{[m]_{q}![n]_{q}!}{[r]_{q}!^{2}}\right)^{1/2}.$
(12)

Now, we can guess the expression for $F_{\alpha,\beta}$:

$F_{\alpha,\beta}=\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}{{}_{q}\langle}{\alpha-k}|F|{\beta-k}\rangle_{q}.$
(13)

This can be proved as follows:

$\displaystyle{}{}\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}{{}_{q}\langle}{\alpha-k}|F|{\beta-k}\rangle_{q}}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}$
$\displaystyle=\sum_{k=0}^{min(\alpha,\beta)}\sum_{l=0}^{min(\alpha-k,\beta-k)}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}\left(\frac{[\alpha-k]_{q}![\beta-k]_{q}!}{[l]_{q}!^{2}}\right)^{1/2}F_{\alpha-
k-l,\beta-k-l}.$ (14)

Using the $q$-binomial formula [25], we can express

$\delta_{p,0}=(1-1)^{p}_{q}=\sum_{k=0}^{p}\frac{[p]_{q}!}{[k]_{q}![p-k]_{q}!}(-1)^{k}q^{k(k-1)}.$
(15)

Inserting Eq. 15 into RHS of Eq. 3 along with the change of variable $l=p-k$,
we obtain

$\sum_{p=0}^{min(i,j)}\sum_{k=0}^{p}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}![p-k]_{q}!}F_{\alpha-p,\beta-p}=F_{\alpha,\beta},$
(16)

which concludes our proof. $F$ can also be expressed as

$F=\sum_{m,n=0}^{\infty}|{n}\rangle_{q}{{}_{q}\langle{m}}|Tr\\{|{m}\rangle_{q}{{}_{q}\langle{n}|}F\\}.$
(17)

Now by applying Eq. 13 for $|{m}\rangle_{q}{{}_{q}\langle{n}|}$, we have

$\displaystyle F$
$\displaystyle{}{}=\sum_{m,n=0}^{\infty}\frac{|{n}\rangle_{q}{{}_{q}\langle{m}}|}{\sqrt{[m]_{q}![n]_{q}!}}\sum_{k=0}^{\infty}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}!}Tr\\{A^{\dagger
m+k}A^{n+k}F\\}$ (18)
$\displaystyle=\sum_{\alpha,\beta=0}^{\infty}\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}|{\beta-k}\rangle_{q}{{}_{q}\langle{\alpha-k}}|}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}Tr\\{A^{\dagger\alpha}A^{\beta}F\\}.$

Thus using Eq. 18, we can express the density operator $\rho$ in terms of
moments of $q$-deformed annihilation and creation operator as

$\displaystyle\rho=\sum_{\alpha,\beta=0}^{\infty}\rho(\alpha,\beta){{}_{q}\langle
A^{\dagger\alpha}A^{\beta}\rangle_{q}},\textnormal{ where}$ (19)
$\displaystyle\rho(\alpha,\beta)=\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}|{\beta-k}\rangle_{q}{{}_{q}\langle{\alpha-k}}|}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}.$
(20)

Using the homodyne $q$-deformed quadrature $X_{\theta}\,$[9], we can compute
the optical tomogram $\omega(X_{\theta},\theta)$ for the density matrix $\rho$
as [3]

$\omega(X_{\theta},\theta)={{}_{q}\langle
X_{\theta}}|\rho|X_{\theta}\rangle_{q}.$ (21)

After inserting Eq. 19 into the above equation, we have

$\omega(X_{\theta},\theta)=\sum_{\alpha,\beta=0}^{\infty}{{}_{q}\langle
X_{\theta}}|\rho(\alpha,\beta)|X_{\theta}\rangle_{q}{{}_{q}\langle}A^{\dagger\alpha}A^{\beta}\rangle_{q}.$
(22)

With the quadrature representation of $q$-deformed Fock state, we compute

$\displaystyle{{}_{q}\langle
X_{\theta}}|\rho(\alpha,\beta)|X_{\theta}\rangle_{q}=$
$\displaystyle\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}$
(23) $\displaystyle\times
J_{{\alpha-k}_{q}}(X_{\theta})J_{{\beta-k}_{q}}(X_{\theta})e^{i(\alpha-\beta)\theta}|{\Psi_{0_{q}}(X_{\theta})}|^{2},$

where $J_{n_{q}}(X_{\theta})$ is an $n^{th}$ degree polynomial that satisfies
the three-term recurrence relation [9]

$J_{n+1_{q}}(X_{\theta})=\frac{1}{\sqrt{[n+1]_{q}}}\left[\frac{2X_{\theta}}{\sqrt{1+q^{2}}}J_{n_{q}}(X_{\theta})-\sqrt{[n]_{q}}J_{n-1_{q}}(X_{\theta})\right].$
(24)

Here, $\Psi_{0_{q}}(X_{\theta})$ is the wavefunction for the $q$-deformed
vacuum state. We can show that for finite sums over the polynomial
$J_{n_{q}}$, the following identity is satisfied:

$\sum_{k=0}^{min(\alpha,\beta)}\frac{(-1)^{k}q^{k(k-1)}}{[k]_{q}!\sqrt{[\alpha-k]_{q}![\beta-k]_{q}!}}J_{{\alpha-k}_{q}}J_{{\beta-k}_{q}}=\frac{\sqrt{[\alpha+\beta]_{q}!}}{[\alpha]_{q}![\beta]_{q}!}J_{{\alpha+\beta}_{q}}.$
(25)

Now by inserting Eq. 25 into Eq. 23, Eq. 22 can be expressed as

$\omega(X_{\theta},\theta)=\sum_{\alpha,\beta=0}^{\infty}|{\Psi_{0_{q}}(X_{\theta})}|^{2}e^{i(\alpha-\beta)\theta}\frac{\sqrt{[\alpha+\beta]_{q}!}}{[\alpha]_{q}![\beta]_{q}!}J_{{\alpha+\beta}_{q}}(X_{\theta}){{}_{q}\langle}A^{\dagger\alpha}A^{\beta}\rangle_{q}.$
(26)

As the polynomial $J_{n_{q}}$ satisfies the three-term recurrence relation
(Eq. 24), Favard’s theorem guarantees its orthogonality. Now, by multiplying
Eq. 26 with $J_{{\gamma}_{q}}(X_{\theta})$$\,(\gamma=\alpha+\beta)$,
integration over the variable $X_{\theta}$, one obtains a linear combination
of normally ordered moments
${{}_{q}\langle}A^{\dagger\alpha}A^{\gamma-\alpha}\rangle_{q}$. Using a little
algebra, one can find all the normally ordered moments. For example, let us
elaborate on the calculation for ${{}_{q}\langle}A\rangle_{q}$. Consider two
arbitrary different angles $\theta_{1}$, $\theta_{2}$,
$\theta_{1}-\theta_{2}\neq\pi$. Now, one can obtain
${{}_{q}\langle}A\rangle_{q}$ by performing the following operation on the
optical tomogram $\omega(X_{\theta},\theta)$.

${{}_{q}\langle}A\rangle_{q}=\frac{1}{2i\sin(\theta_{2}-\theta_{1})}\int_{-\infty}^{\infty}dX_{\theta}\big{(}e^{i\theta_{2}}\omega(X_{\theta},\theta_{1})-e^{i\theta_{1}}\omega(X_{\theta},\theta_{2})\big{)}J_{1_{q}}(X_{\theta}).$
(27)

The optical tomogram can be used to calculate the squeezing and higher-order
squeezing for $q$-deformed states. Nonclassical properties like these can be
obtained from instantaneous tomograms for a system evolving in time.

##  4 Conclusion

In this paper, we analyzed the optical tomograms of $q$-deformed versions of
nonclassical states. The optical tomograms of these states are computed for
different deformation parameters. We arranged the states into ‘Janus faced’
partners, and this holds for all deformation values. We also found that with a
decrease in the value of $q$ (increase in deformation), the strands in the
tomographic plane became thin; also, the maximum value of
$\omega(X_{\theta},\theta)$ increased.

We note that the optical tomograms are well suited to study various
nonclassical properties of quantum states. We derived a method to extract the
quadrature moments from the optical tomogram of deformed states, and this is
useful in calculating nonclassical properties such as quadrature squeezing.
Also, this technique can be applied to tabletop experiments where the
deviation from standard quantum mechanical behavior is to be tested.

## Data availability statement

All data supporting this study’s findings are included in the article.

## References

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11institutetext: Konkoly Observatory, Research Center for Astronomy and Earth
Sciences, Eötvös Loránd Research Network
Budapest, 1121 Konkoly Thege ut. 15-17, Hungary
11email<EMAIL_ADDRESS>
# Bright single-mode RR Lyrae stars: matching Gaia EDR3 with pulsation and
evolutionary models

Geza Kovacs and Behrooz Karamiqucham

(Received September 09, 2021; accepted …)

We combine observed metallicity, optical and infrared magnitudes with
evolutionary and pulsation models to derive average luminosities for $156$
single-mode RR Lyrae stars. These luminosities are compared with those
obtained from the Gaia EDR3 parallaxes, and found in excellent agreement with
the high accuracy subsample ($62$ stars, with relative parallax errors less
than $2$%). With the temperature and metallicity scale used, no parallax shift
seems to be necessary when $\alpha$-enhanced evolutionary models are employed.
Some $10$% of the sample shows curious ‘distance keeping’ between the
evolutionary and pulsation models. The cause of this behavior is not clear at
this moment but can be cured by an excessive increase of the reddening.

###### Key Words.:

Stars: fundamental parameters – stars: distances – stars: variables: RR Lyrae
– stars: oscillations – stars: horizontal branch

##  1 Introduction

In a former paper (Kovacs & Karamiqucham, 2021, hereafter K21) we used
pulsational and evolutionary models to close the relations governing the basic
physical parameters of double-mode RR Lyrae (RRd) stars. The so-derived
luminosities of $30$ galactic RRd stars were compared with the direct
estimates from the Gaia Early Data Release 3 (EDR3, Lindegren et al., 2021).
The agreement between these independent sets was satisfactory, considering the
overall low brightness of the available objects and the consequentially low
accuracy of their parallaxes.

One of the issues in the case of RRd stars is the lack of chemical composition
measurements. In K21 we determined the overall metal abundance [M/H] as a by-
product of the matching procedure of the pulsation and the evolutionary
models. Opposite to the RRd stars, single-mode RR Lyrae (RRabc) stars have
very abundant chemical analyses, many of them are based on high-dispersion
spectra and multiple visits by different studies. The work presented in this
paper heavily relies on past and current spectral analyses by fixing [M/H] and
thereby letting us to use RRabc stars in matching the pulsation and
evolutionary models.

The ultimate goal of this work is to check the consistency of the nearly
directly derived Gaia luminosities with the model values by using basic
observables only. Because of the significantly higher accuracy of the input
data and the weaker dependence on the more intricate details of the pulsation
models, we consider this study to be more stringent for the compatibility of
the Gaia and theoretical luminosities than the one presented in K21 (even
though the two studies are consonant within the error limits).

##  2 Method

Our method of deriving the basic physical parameters (mass, luminosity,
temperature – $M$, $L$, $T_{\rm eff}$, respectively) is similar to those used
in our earlier papers (Kovacs & Walker, 1999; Kovacs, 2000, K21). A simple
flowchart of the analysis is shown in Fig. 1. Some additional details are as
follows.

Figure 1: Main steps of the star-by-star procedure to obtain estimates of the
RR Lyrae luminosities. Boxes on the left exhibit the steps to compute the
theoretical luminosity $\rm L_{\rm HB/LNA}$, whereas those on the right show
the steps leading to the empirical luminosity $\rm L_{\rm Gaia}$. We follow
standard notation (i.e., $RP$ for the Gaia red and $W2$ for the WISE W2
magnitudes).

For any given star, we have the following input parameters: period $P$,
intensity-averaged Johnson V and K magnitudes and iron abundance [Fe/H] (see
Sect. 3 for the derivation of these quantities). From these parameters we
calculate $T_{\rm eff}$ – the weak gravity ($\log g$) dependence can be
handled via the pulsation equation as given in K21. We use the $T_{\rm eff}$
zero point of González Hernández & Bonifacio (2009) as detailed in K21.

Once $T_{\rm eff}$ has been estimated, for its fixed value we can compute the
corresponding $(M,L)$ values for the evolutionary and pulsation models. To
avoid the somewhat cumbersome procedure of directly interpolating the model
values to the observed metallicity, we opted to compute the iso-$T_{\rm eff}$
lines for three metallicity values bracketing the observed metallicity. The
following metric is used to measure the distance between the pulsation and
evolution models for any matching metallicity $Z$

$\displaystyle{\cal D}(M,L,Z)=\sqrt{\log^{2}\Biggl{(}{L^{HB}\over
L^{LNA}}\Biggr{)}+\log^{2}\Biggl{(}{M^{HB}\over M^{LNA}}\Biggr{)}}\hskip
5.69054pt,$ (1)

where the superscripts $HB$ and $LNA$ denote, respectively, the horizontal
branch evolution and linear non-adiabatic pulsation models. Once the minima of
${\cal D}(M,L,Z)$ for the three metallicities are found, we use quadratic
interpolation in $\log(Z)$ to derive the closest matching $(M,L)$ values for
the $HB$ and $LNA$ models.

We use the recently updated $HB$ models from the Bag of Stellar Tracks and
Isochrones (BaSTI) project, as described by Hidalgo et al. (2018). The models
have fixed metallicity values spanning the full range of galactic field RR
Lyrae stars (i.e., $-3.0<$ [Fe/H] $<0.1$). All models are without convective
overshooting. We use two sets of models: those with scaled solar-composition
and those with enhanced $\alpha$-elements (i.e., O, Ca, Mg, etc). In the
latter case, the degree of enhancement is [$\alpha$/Fe]$=+0.4$ and all models
have diffusion and mass loss of $\eta=0.3$ (none of the solar-scaled models we
use have either of these). In searching for the best HB/LNA matches, we
consider all evolutionary stages, not only those close to the zero-age HB.

The pulsation models are linear non-adiabatic models, without
$\alpha$-enhancement – the effect of enhancement on the period is in the range
of few times $10^{-3}$ days, too small to have an influence on the derived
stellar parameters. All models have a Hydrogen abundance of $0.76$, very close
to those of the $HB$ models, and matching $Z$ values to the $HB$ metallicity
grid (altogether $18$ discrete metallicities). For any fixed $Z$, there are
$4095$ models on the $(M,L,T_{\rm eff})$ grid given in Table 2 of K21.

Since we use observed [Fe/H], we need to convert this to the corresponding
overall metallicity $Z$ for the enhanced models. We do this conversion by
using the table available at the BaSTI home page. The $+0.4$ dex increase in
the $\alpha$-elements results in a factor of two overall increase in $Z$
throughout the full range of [Fe/H].

##  3 Datasets

Most of the RR Lyrae stars we deal with in this paper are bright, ($8<V<13$)
and therefore, have abundant observational material. Nevertheless, various
observational settings and details in the analyses may introduce differences
in the derived parameters. Consequently, in gathering the essential data we
tried to rely on homogeneous datasets. For [Fe/H] they are apparently not
available, therefore, we rescaled the data from the same source and adjusted
the zero point to some chosen level.

Our basic dataset contains $67$ stars with EDR3 parallax errors less than
$2$%. For assessing error dependence we also use a larger dataset, containing
additional $94$ stars with larger relative parallax errors, up to $10$%. Both
sets are based on the combination of the target lists of Dambis et al. (2013),
Monson et al. (2017) and several recent spectroscopic studies, detailed in
Sect. 3.1. The basic set is complete in respect of the above source lists with
the parallax constraint. The extended set is incomplete, and satisfies mostly
the constraints of “relatively low parallax error”, “easy data access” and
[Fe/H] availability.

###  3.1 Iron abundance

In recent years, several high-dispersion spectroscopic (HDS) works have been
devoted to deep chemical analysis of RR Lyrae stars. Here we use the combined
lists of Magurno et al. (2018) and Crestani et al. (2021), based also on the
works of Clementini et al. (1995), Fernley & Barnes (1996), Lambert et al.
(1996), For et al. (2011), Hansen et al. (2011), Liu et al. (2013), Nemec et
al. (2013), Govea et al. (2014), Pancino et al. (2015), Chadid et al. (2017),
Sneden et al. (2017) and Andrievsky et al. (2018). The individual [Fe/H]
values in these compilations may differ by as much as $\sim 0.4$ dex, due to,
e.g., differences in the reduction methods or the turbulent velocity used.
Therefore, simple averaging is, in general, does not seem to be justified.
Consequently, we followed the procedure below to put all [Fe/H] on the same
scale.

Figure 2: Published iron abundances: Jurcsik & Kovacs (1996) vs high-
dispersion spectroscopic [Fe/H] for the data sources given in the upper left
corner of each panel (C17, A18 and L13 for Chadid et al., 2017; Andrievsky et
al., 2018; Liu et al., 2013, respectively). In the panel at the lower right we
plot the transformed values for all [Fe/H] matching the list of Jurcsik &
Kovacs (1996). The HDS identity line is shown by red continuous line.

To establish a common base of comparison, we used the metallicities compiled
by Jurcsik & Kovacs (1996) (hereafter J96). This list is based on the low-
dispersion spectroscopic (LDS) measurements primarily by Blanco (1992), Layden
(1994) and Suntzeff et al. (1994). We found that the J96 metallicities
correlate well with the values of the individual HDS studies but there are
systematic differences in the zero points (ZPs) and that the HDS values are
lower toward lower [Fe/H]. In particular, for Chadid et al. (2017) (hereafter
C17) we find that

$\displaystyle{\rm[Fe/H]}_{\rm C17}=1.10{\rm[Fe/H]}_{\rm J96}-0.22\hskip
5.69054pt,$ (2)

with an RMS scatter of $0.12$ dex.111Omitting the single outlier SS Leo, we
have $\sigma=0.07$ dex. This excellent agreement (see Fig. 2) between the two
independent sets of abundances led us to transform all HDS and the LDS
datasets of Layden (1994) and Layden et al. (1996) to the scale of C17, and
then, average them out for each target. For the HDS sets we employed only ZP
shifts. Several datasets did not have overlap with C17. Then, we used the
overlaps with those that had overlaps with the set in question and with C17 to
transform the published [Fe/H] for that dataset. Altogether we ended up with
$177$ objects, most of them with HDS metallicities, often with independent
measurements from multiple (up to six) sources per target.

###  3.2 Reddening

One of the sensitive parameters entering in the estimation of the luminosity
is interstellar reddening. Because the commonly used all-sky maps give the
total reddening at a certain direction, the exact amount of the extinction
between the object and the observer remains hidden. In the case of excessive
reddening this might lead to a serious overestimation of the reddening. For
instance, based on the map of Schlafly & Finkbeiner (2011), we got over
$10^{4}$ K for $T_{\rm eff}$ in the case of RZ Cep, AR Per, BN Vul, SW Cru and
BI Cen with $E(B-V)>0.7$ for all of them.

For RRab stars one may use some of the formulae available to estimate their
intrinsic colors (Sturch, 1966; Blanco, 1992; Kovacs & Jurcsik, 1997). We
found this approach unsatisfactory, mainly because of the need of additional
color and light curve information and the relative low accuracy of the derived
reddenings. The ideal approach would be to use spectroscopic $T_{\rm eff}$ and
employ the related intrinsic color index to estimate $E(B-V)$ from the
observed colors. Although HDS [Fe/H] values are often derived from multiple
epoch data, the phase coverage is usually rather limited, and it is out of the
scope of this paper to derive reliable average $T_{\rm eff}$ values from those
sparsely-sampled sets.

We also tested the recent 3D map of Green et al. (2019) derived from 2MASS and
Pan-STARRS 1 colors, aided by the Gaia parallaxes. Except for the targets with
large reddenings, we found no improvement in the correlation between the
HB/LNA and Gaia luminosities. Consequently, we stay with the map of Schlafly &
Finkbeiner (2011) accessible at the NASA/IPAC Infrared Science
Archive.222https://irsa.ipac.caltech.edu/applications/DUST/

###  3.3 Johnson V, K photometry

The intensity-averaged $V$ magnitudes have been gathered from three sources.
In the order of priority, they come from Monson et al. (2017), Dambis et al.
(2013) and from ASAS (Pojmanski, 1997), whenever the given target was not
available in the first two sources.333For the ASAS light curves we fitted
Fourier series of order $4$–$10$ to derive the intensity-averaged $V$
magnitudes – see K21 for further details.

For the intensity-averaged 2MASS $K$ ($Ks$) magnitudes we could use the
recently compiled set of Layden et al. (2019). However, even though it is the
biggest homogenized sample of direct $Ks$ measurements available today, it
misses some Gaia objects with high-precision parallaxes (e.g., CS Eri, EL Aps,
etc). In addition, several $Ks$ values have been derived via template fitting
due to the low number of data points available. As a result, we inclined to
use the unWISE catalog of Schlafly et al. (2019) and sought for a
transformation from the WISE and Gaia simple average magnitudes to approximate
the $Ks$ values of Layden et al. (2019). With the $145$ stars used, we found
that by leaving out eight $2.5\sigma$ outliers, the following formula fits
Layden’s et al. data with an RMS scatter of $0.028$ mag

$\displaystyle Ks$ $\displaystyle=$ $\displaystyle\phantom{(}9.958\pm 0.004$
(3) $\displaystyle+$ $\displaystyle(0.884\pm 0.028)(W2-9.949)$
$\displaystyle+$ $\displaystyle(0.120\pm 0.030)(RP-10.736)\hskip 5.69054pt.$

Here $RP$ denotes the Gaia EDR3 red magnitude. The unWISE $W2$ magnitude is
calculated from the published FW2 fluxes from $W2=22.5-2.5\log_{10}(FW2)$.
Unlike the transformation presented in K21, Eq. 3 is applicable also to RRab
stars, due to the correction introduced by the Gaia red magnitude.

##  4 Luminosities

To investigate the data quality dependence of the level of agreement between
the Gaia and model (HB/LNA) luminosities, we divided the total sample into
three groups. The division is based on the relative parallax errors (see Table
1). Set A (the basic dataset without the five stars with overestimated
reddenings – see Sect 3.2) is the most populated and of the highest accuracy.
Figure 3 shows the correlation between the Gaia and model luminosities
following the division mentioned and completed by a similar plot for the RRd
stars from K21.

Table 1: Gaia and model luminosities

$\alpha$-enhanced models
---
Set | $(r_{\rm min},r_{\rm max})$ | $\langle\Delta\log L\rangle$ | RMS | N
A | $(0.00,0.02)$ | $\phantom{-}0.006\pm 0.004$ | 0.029 | 62
B | $(0.02,0.03)$ | $\phantom{-}0.015\pm 0.006$ | 0.041 | 43
C | $(0.03,0.10)$ | $\phantom{-}0.026\pm 0.009$ | 0.062 | 51
solar-scaled models
A0 | $(0.00,0.02)$ | $-0.012\pm 0.004$ | 0.030 | 62
B0 | $(0.02,0.03)$ | $-0.003\pm 0.006$ | 0.042 | 43
C0 | $(0.03,0.10)$ | $\phantom{-}0.007\pm 0.009$ | 0.063 | 51
solar-scaled models
RRd | $(0.03,0.13)$ | $\phantom{-}0.015\pm 0.016$ | 0.087 | 30

Notes: $(r_{\rm min},r_{\rm max})$: range of the relative parallax errors –
$\langle\Delta\log L\rangle$: average of $\log(L_{\rm Gaia}/L_{\rm model})$ –
$RMS$: standard deviation of the residuals – $N$: number of data points in the
given set.

Figure 3: Gaia EDR3 vs evolutionary and pulsation model luminosities for
single-mode RR Lyrae stars (panels A, B, C). The single-letter labels denote
independent datasets in the order of increasing parallax error (see Table 1).
We used $\alpha$-enhanced evolutionary models. For comparison, we also plotted
the parallax-shifted RRd models of K21. The Gaia identity lines are shown by
red continuous lines.

Let us recall that (except for the RRd stars) no adjustments were made for any
of the parameters used. The dependence on the data quality is clearly visible.
However, the Gaia luminosities seem to be systematically overestimated for
higher luminosities. The difference depends on the quality of the data and it
increases for poorer data. We suspect that most of the effect comes from the
increasing errors of the Gaia luminosities that are, on the average, twice as
large as the HB/LNA errors. This difference leads to a flattening of the $\log
L-\log L$ relation, with the Gaia luminosities plotted on the abscissa (see,
in particular, dataset B in Fig. 3). Then, the finite sample size and non-
uniform sampling lead to further distortion of the ridge, causing apparent
systematic deviations for certain groups of stars.

For set A with accurate parallaxes, the $\alpha$-enhanced models show a
preference against the solar-scaled models with respect to the Gaia
luminosities. For sets B and C this preference diminishes, primarily because
of the higher parallax errors. For set A no parallax shift (with a sufficient
significance) is needed. In K21 we found a shift of $+0.02$ mas to be
appropriate for the RRd stars with low overall parallax accuracy. However, the
significance of this shift is only a few sigma, even if we omit the stars with
apparent outlier status. The $\alpha$-enhanced models perform also better in
terms of the HB/LNA solution (see Sect. 5), lending further support for to use
of these models.

##  5 The distance keepers

Our model luminosities are based on the solutions provided by the
intersections of the iso-$T_{\rm eff}$ HB and LNA lines. Unfortunately, as in
K21, this solution may not exist for some observed parameter sets, even if we
consider the errors associated with the given objects. We refer to these
‘stubborn’ objects as “distance keepers”, or DKs.

Figure 4: Examples on the solution types from the HB and LNA models. Filled
circles show the closest $(M,L)$ pairs of the two types of models from $20$
random realizations with the errors of $V$, $Ks$, [Fe/H], $E(B-V)$ and
parallax, centered around their observed values. Gray and blue coloring are
used for the HB and LNA models, respectively. The closest values at the
observed parameters are shown by yellow squares for the HB and by red
triangles for the LNA models. The odds for being a DK are shown under the
variable names.

As a part of the error estimate of the model luminosities, we calculated the
solutions for each object by using the observed $V$, $Ks$, [Fe/H], $E(B-V)$
and parallax values and for $20$ perturbed sets, obtained by adding
perturbations computed from the observational errors and assuming Gaussian
distribution for these errors. We searched for the closest $(M,L)$ pairs from
the HB and LNA models and derived the metric given by Eq. 1 for each
realization. These simulations yielded not only the errors of the theoretical
luminosities but also informed us if the high ${\cal D}(M,L,Z)$ values could
be explained by observational errors. Figure 4 shows the various scenarii
found in our sample. The two upper panels show examples when solutions were
found for the observed parameters – albeit with slightly different statistical
significance. The lower two panels are examples for the DKs. Indeed, both
stars, although at various degrees, “keep the distance” between the nearest HB
and LNA points in the $(M,L)$ space.

By (somewhat arbitrarily) fixing the maximum ${\cal D}(M,L,Z)$ to $0.01$, we
counted the number of realizations for each object exceeding this limit. The
relative number of these cases to the $20$ realizations ($P_{\rm DK}$) for
each object yielded an approximation of the probability that a given object is
a DK. We found that for the $\alpha$-enhanced models $122$ have lower than
$50$% chance of being a DK (and, $\sim 100$ have $0$%). However, there are
some $15$ stars without HB/LNA solution (i.e., practically none of the
simulated values yield lower than $0.01$ for ${\cal D}(M,L,Z)$).

The analysis of $P_{\rm DK}$ is also useful in the comparison of the overall
performance of $\alpha$-enhanced models with the solar-scaled models. We
checked set A (see Table 1) and found that, on the average, $P_{\rm DK}$ is
smaller for the enhanced models by $0.2$ for those with $P_{\rm DK}>0.5$ for
the solar-scaled models. The number of severe DKs (i.e., those with $P_{\rm
DK}>0.9$) shrank to $3$ from 7 when switching from the solar-scaled to the
enhanced models. Some stars with low ${\cal D}(M,L,Z)$ of $0.01-0.02$ have
completely recovered from the DK status by using enhanced models (e.g., SW
And, AV Peg, U Pic). We consider these figures as additional pieces of
evidence for the likely higher relevance of the $\alpha$-enhanced models.

We made a few tests to investigate the possibility of major parameter errors
causing the DK classification of the more stubborn stars. We introduced large
and separate changes in the period ($\pm 0.01$), $V$ magnitude ($\pm 0.03$),
[Fe/H] ($\pm 0.4$) and in $E(B-V)$ (as much as it was needed to go below the
cutoff for ${\cal D}(M,L,Z)$). We found that even with the above large changes
in $P$, $V$ and [Fe/H], the more stubborn DKs are left in the same status.
However, varying $E(B-V)$ may help. We examined the top $21$ DKs with $P_{\rm
DK}>0.8$. We found solutions for each of them by increasing $E(B-V)$. The
amount of increase varied between $0.01$ and $0.04$ (well above the nominal
errors of their reddenings). Two variables (ST Oph and TV Lib) required even
more excessive increase ($0.07$ and $0.10$, respectively). Together with its
short period and light curve shape, in the case of TV Lib this might indicate
a different variable classification (e.g., evolved high-amplitude $\delta$
Scuti status).

Although the all out increase of $E(B-V)$ might indicate the need for shifting
the zero point of the $T_{\rm eff}$ scale, we found that an increase of
$0.0045$ in $\log T_{\rm eff}$444That is, we use the $T_{\rm eff}$ scale
compatible with the Castelli et al. (1997) models rather than the recent scale
of González Hernández & Bonifacio (2009), based on the infrared flux method.
is insufficient to cure the notorious DKs. The increased temperature leads to
more luminous theoretical models and thereby a difference of $-0.015$ in $\log
L$ (Gaia minus models). Although this can be remedied for the enhanced models
of set A by an overall parallax decrease of $0.02$ mas, the final gain of
these zero point shifts does not seem to be justified in seeking the root
cause of the DK phenomenon.

One may also wonder if a slight change in the Helium abundance (Y) may alter
the DK status. Although a deeper discussion of the effect of Helium is out of
the scope of this study, we note the following. Increasing Y increases the HB
luminosity (e.g., Sweigart & Gross, 1976; Marconi et al., 2018) and with the
pulsation periods remaining basically the same, the DK status exacerbates.
Decreasing Y by $0.02$–$0.05$ may match the evolutionary and pulsational
luminosities, however, the smaller Helium abundance may raise other questions
concerning its currently accepted primordial value555Which is Y$\sim 0.245$,
used also by BaSTI. and the errors involved (e.g., Cooke & Fumagalli, 2018).

##  6 Conclusions

With the aid of evolutionary and linear stellar pulsation models we computed
the luminosities of bright RR Lyrae stars by using basic observed parameters,
such as period, average magnitudes (Johnson $V$, Gaia $RP$, WISE $W2$), iron
abundance and interstellar reddening. These luminosities have been compared
with those derived from the latest release of the Gaia parallaxes. By using
$\alpha$-enhanced evolutionary models, for the bright part of the sample ($62$
stars, with relative parallax errors less than $2$%), we found high level of
agreement between the two sets (including both fundamental and first overtone
pulsators). The $\log L$ values span a range of $\sim 0.5$ with an RMS of
$0.029$. Solar-scaled models for the same sample yield similarly good
correlation between the two (essentially independent) luminosity sets.
However, it seems that the solar-scaled models are brighter by $0.012\pm
0.004$ in $\log L$. For the remaining $94$ stars with larger parallax errors
we observe a tendency for increasing Gaia luminosities (relative to the
theoretical luminosities) both for the $\alpha$-enhanced and solar-scaled
models.

Based on these results, it seems that stars with accurate parallaxes yield
well-matching luminosities with the theoretical values without the need for
parallax shift. This assertion likely holds whenever the ‘theoretical values’
are derived by using:

  * •

$\alpha$-enhanced evolutionary models – BaSTI (Hidalgo et al., 2018) or
similar;

  * •

standard radial pulsation models – e.g., Kovacs (2000);

  * •

$T_{\rm eff}$ zero point, based on the infrared flux method – e.g., González
Hernández & Bonifacio (2009).

Our result is essentially in agreement with the conclusion of Marconi et al.
(2021), based on theoretical period-Wesenheit relations, largely relying on
nonlinear pulsation models and Gaia DR2 parallaxes. In our previous paper on
$30$ double-mode RR Lyrae stars (Kovacs & Karamiqucham, 2021) we landed on a
slightly different conclusion, by accepting an overall shift of $+0.02$ mas
for the EDR3 parallaxes. Because of the considerably lower accuracy of data
used in that study, this shift may seem to be justified, considering the
similar trend observed in the present sample. However, due to the different
method used in our double-mode study (because of the lack of metallicity
measurements for those faint stars), and the size of the sample (together with
the large errors involved) make us cautious to put strong weight on the
conclusion concerning the parallax shift from that study.

Finally, it is important to note that $\sim 10$% of the stars show curious
‘distance keeping’ between the evolutionary and pulsation models (i.e., in a
strict sense they do not yield theoretical solutions – with zero distance in
the $(L,M)$ space, even stretching the errors of the observed parameters). We
found that an object-dependent increase of the reddening (well above the
formal error limits) solves the problem. This, however, is not very easy to
understand, since the reddening maps are believed to be based on the total
dust density, so any increase in the extinction are expected to be related to
the star rather than to the interstellar matter.

###### Acknowledgements.

We thank the referee for the quick and helpful report (in particular,
suggesting the investigation of the effect of Helium abundance). This research
has made use of the VizieR catalogue access tool, CDS, Strasbourg, France
(DOI: 10.26093/cds/vizier). This work has made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of data products from the Wide-field Infrared Survey Explorer, which is a
joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration. This research has made use of
the NASA/IPAC Infrared Science Archive, which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology. This research has made use of the International Variable Star
Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA.
Supports from the National Research, Development and Innovation Office (grants
K 129249 and NN 129075) are acknowledged.

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