# A New Look at the YY CrB Binary System

Somayeh Soomandar  Independent astrophysics researcher, Kerman, Iran  Atila
Poro  Astronomy Department of the Raderon Lab., BC., Burnaby, Canada

###### Abstract

This study presented a new analysis for the TESS-observed W Ursae Majoris (W
UMa) binary star YY Coronea Borealis (YY CrB). The light curve was analyzed by
the PHysics Of Eclipsing BinariEs (PHOEBE) Python version together with the
Markov chain Monte Carlo (MCMC) method. The light curve solutions required a
hot spot and $l_{3}$. New eclipse times from the TESS observations were
extracted, and the O-C curve of primary and secondary minima showed an anti-
correlated manner. In order to study the O-C curve of minima, minima times
between 1991 and 2023 were collected. This investigation reported a new linear
ephemeris and by fitting a quadratic function to the O-C curve of minima,
calculated the orbital period rate of $\mathop{P}\limits^{.}\approx
5.786\times{10^{-8}}\frac{{day}}{{year}}$. Assuming mass conservation, a mass
exchange rate of $\mathop{{M_{2}}}\limits^{.}=2.472\times{10^{-8}}$ calculated
from the more massive component to the less massive one. Then, by using the
light travel time function, the possible third body was determined in the
binary and derived the mass of the third body as $0.498{M_{\odot}}$ with a
period of $\simeq 7351.018$ days. The O-C curve analysis and the quantity of
mass indicate that the presence of a third body is unlikely. This binary is
expected to evolve into a broken-contact phase and is a good case to support
the thermal relaxation oscillation model.

binaries: eclipsing – method: photometric

##  1 Introduction

W UMa-type systems are recognised by their eclipsing light curves with almost
equal minima and a short orbital period. These stars have spectral types
ranging from A to middle K, and the convective atmosphere is the main reason
for chromosphere activity, as well as starspots, which are signs of the
existence of dynamo-generated magnetic activity.

YY CrB (HIP 77598, TIC 29287800) is a W UMa binary system discovered by
Hipparcos ESA 1997. This system has been studied in some works; first
Rucinski, et al. (2000) revealed the spectral type F8V for two
components.Vaňko et al. (2004) found the light curves to be asymmetric and
mentioned the existence of starspots on the components. Gazeas et al. (2005)
analyzed the light curve and derived the geometric and photometric parameters
and concluded that this target is a contact binary with weak magnetic
activity.

Essam et al. (2010) combined photometric and spectroscopic solutions and
calculated the fill-out factor approximately equal to 64 percent and mass
ratio of 0.241. In addition, they studied the changes in the orbital period
using the O-C diagram and concluded that the orbital period is decreasing. Yu,
Xiang, & Xiao (2015) studied the orbital period changes and implied that the
decreased period rate and concluded that the sinusoidal oscillatory can be
interrupted as magnetic activity. Essam et al. (2010) and Yu, Xiang, & Xiao
(2015) on the rate of period decrease demonstrate that the value of reduced
rate is lowering progressively, indicating that this system was going through
an orbital expansion stage of thermal relaxation oscillation (TRO) cycles.
Also, understanding the evolutionary status of this target could prove
invaluable. Using new space-based data, we have re-analyzed the light curve
solution and studied the O-C curve in detail. Moreover, we studied the
possibility of a third body in this interesting system. The structure of the
paper is as follows: Section 2 provides information on TESS observations and a
data reduction process. The light curve solution and the estimation of
absolute parameters are included in Sections 3 and 4 respectively, the orbital
period variation analysis is presented in Section 5, and finally, Section 6
contains the discussion and conclusion.

##  2 Observation and Data Reduction

YY CrB was observed by the TESS during sectors 24 and 51 (April 16, 2020-May
13, 2020, and April 22, 2022-May 18, 2022) on Cameras 1 and 3. There is two-
minute cadence data for sector 24 that are processed by the Science Processing
Operations Center (SPOC) pipeline (Jenkins et al. (2016); Jenkins (2015)).
Photometric photos were downloaded using the Lightkurve package (Lightkurve
Collaboration et al., 2018) that provides the functions to download TESS data
from the public data archive at MAST111https://mast.stsci.edu. For sector 24,
we used the Pre-search Data Conditioning flux of the Simple Aperture
Photometry (PDCSAP). There is no detrended light curve for sector 51.
Therefore, we download the TESS Full Frame Images (FFIs) from the MAST and
used Lightkurve package to extract the SAP light curve with a mask that are
defined by the pixels shown in the left panel of Figure 1. We used create
threshold mask function to produce an aperture mask using a threshold equal to
10. This function identifies the pixels in the target pixel file and shows
that a median flux that is brighter than the threshold times the standard
deviation above the overall median. The right panel of Figure 1 shows the
phased light curve that was produced.

Figure 1: Left panel: TESS target pixel file of YY CrB in sector 51. The
pixels included in the computation of the SAP as red bordered pixels. Right
panel: phased light curve during sector 51.

##  3 Light curve solution

Essam et al. (2010) calculated the optimal parameters by combining
simultaneous radial velocity and light curve solutions. We started with the
initial values of the parameters taken from the solution by the Essam et al.
(2010) study. One-day data from TESS sector 24 were utilized to light curve
solution. The observation of 2-min cadence help to better analysis of the
effect of spots on the components. Photometric analysis of the YY CrB system
was carried out using the PHOEBE 2.4.9 version, TESS filter of the code, and
the MCMC approach (Prša & Zwitter 2005, Prša et al. 2016, Conroy et al. 2020,
Poro et al. 2022).

We selected the TESS passband from the code and chose the contact binary mode
in the PHOEBE based on the light curve’s shape and solutions of previous
studies.

The initial and input parameters were as follows: The mass ratio $q=0.241$ and
the effective temperature of primary component ${T_{1}}=5819$ are (Essam et
al., 2010), the gravity darkening coefficients, ${g_{1}}={g_{2}}=0.32$ and the
albedo coefficients, ${A_{1}}={A_{2}}=0.5$ (Lucy 1967, Ruciński 1969).

The limb-darkening coefficients were employed as free parameters, and the
Castelli & Kurucz (2004) method was used to model the stellar atmosphere. The
parameters searched in MCMC include: the orbital inclination $i$, the mean
temperature of the stars $T_{1,2}$, the mass ratio $q$, the fillout factor
$f$, the bandpass luminosity of the primary star (${L_{1}}$), and the third
light in total light (${l_{3}}$). We applied 46 walkers and 1000 iterations to
each walker in MCMC. According to the asymmetry in the brightness of maxima in
the light curve of the close eclipsing binary, the solution requires the
assumption of a hot spot on the primary component (O’Connell 1951). According
to observational and theoretical light curves in this study, it has not been
possible to provide the solution without considering $l_{3}$. The theoretical
fit on the observational light curve for the YY CrB system is given in Figure
2. The corner plot that MCMC produced is displayed in Figure 3. Also, the
geometrical structure is plotted in Figure 4, which has a lower temperature at
the point of contact between companion stars due to the gravity darkening
(Prša et al. 2016). The calculated parameters together with the values
obtained by Essam et al. (2010) are listed in Table 1.

Figure 2: Light curve solution of the eclipsing binary YY CrB. Observational light curve (blank circle), synthetic light curve (solid red line), and the residuals (blue circle). Figure 3: The corner plots of the light curve solution. Figure 4: The geometrical structure of YY CrB. Table 1: The parameters of the eclipsing binary YY CrB. Parameter | This study | Essam et al. (2010)
---|---|---
$q=M_{2}/M_{1}$ | $0.2498_{\rm-0.0024}^{+0.0031}$ | $0.241\pm 0.002$
$T_{1}$ (K) | $5621_{\rm-3}^{+3}$ | 5819
$T_{2}$ (K) | $5944_{\rm-8}^{+6}$ | $6010\pm 72$
$i$ (deg) | $81.50_{\rm-0.29}^{+0.36}$ | $80.26\pm 0.05$
${\Omega_{1}=\Omega_{2}}$ | $2.295\pm 0.079$ | 2.237
$l_{1}/l_{tot}$ | $0.730_{\rm-0.001}^{+0.001}$ | $0.7508\pm 0.0154$
$l_{2}/l_{tot}$ | $0.264\pm 0.001$ | 0.2492
$l_{3}/l_{tot}$ | $0.006_{\rm-0.001}^{+0.001}$ |
$f$ | $0.363_{\rm-0.031}^{+0.025}$ | 0.64
${r_{1}}_{mean}$ | $0.522\pm 0.018$ | 0.537
${r_{2}}_{mean}$ | $0.287\pm 0.028$ | 0.282
Phase shift | $0.08\pm 0.005$ |
Spot on the star 1: |  |
Colatitude $\theta$ | $99\pm 1$ | 90
Longitude $\lambda$ | $325\pm 1$ | $11.25\pm 1.638$
Angular radii $\gamma$ | $18\pm 1$ | $5.250\pm 0.573$
${T_{star}}/{T_{spot}}$ | $1.04\pm 0.02$ | 0.750
Spot on the star 2: |  |
Colatitude $\theta$ |  | $99.487\pm 3.919$
Longitude $\lambda$ |  | 325
Angular radii $\gamma$ |  | $16.300\pm 1.326$
${T_{star}}/{T_{spot}}$ |  | 1.351

##  4 Absolute parameters

The absolute parameters of the binary system including ${M_{v1,2}}$,
${M_{bol1,2}}$, $L_{1,2}$, $R_{1,2}$, $M_{1,2}$, $log(g)_{1,2}$, and $a$ were
calculated. We used Gaia DR3 parallaxes and the parameters of the light curve
solution in this study. We followed the same method as done by Poro et al.
(2022). First the absolute magnitude ${M_{v}}$ of the system was calculated by
Equation (1).

${M_{v(system)}}=V_{max}-5\log(d)+5-{A_{v}}$ (1)

where the distance of the system from Gaia DR3 ($d_{pc}=90.07\pm 0.1$) was
derived and $V_{max}=8.64\pm 0.08$ comes from the
VSX222https://www.aavso.org/vsx/ database. Extinction coefficient
${A_{v}}=0.015\pm 0.002$ was obtained using the DUST-MAPS package in Python
(Green et al., 2019). Also, Equation (2) can be utilized to determine the
primary and secondary components’ absolute magnitude.

${M_{v1,2}}-{M_{vtot}}=-2.5\log(\frac{{{l_{1,2}}}}{{{l_{tot}}}})$ (2)

The bolometric magnitude ${M_{bol}}$ of each component of the binary was
obtained by Equation (3),

${M_{bol}}={M_{v}}+BC$ (3)

where the effective temperature of the stars is employed to obtain the
bolometric correction for the primary and secondary components retrieved
$BC_{1}=-0.111$ and $BC_{2}=-0.052$ respectively (Flower, 1996). The
bolometric correction is presented as polynomial fits in Equation (4).

$BC=a+b(\log{T_{eff}})+c{(\log{T_{eff}})^{2}}+d{(\log{T_{eff}})^{3}}+e{(\log{T_{eff}})^{4}}$
(4)

Then, the luminosity of two components is determined from Pogson’s relation
(Pogson, 1856),

${M_{bol}}-{M_{bol\odot}}=-2.5\log(\frac{L}{{{L_{\odot}}}})$ (5)

where ${M_{bol\odot}}$is taken as $4.73^{mag}$ from Torres (2010). The radius
of primary and secondary components is calculated by the equation (6),

$R={(\frac{L}{{4\pi\sigma{T^{4}}}})^{1/2}}$ (6)

where the $\sigma$ is the Stephen-Boltzmann constant and $T$ is the
temperatures of each components.

Additionally, with considering ${r_{mean1,2}}$ and $a=\frac{R}{{{r_{mean}}}}$,
we calculated the separation $a$ in average ${a_{1}}$ and ${a_{2}}$. The
resulting parameters and the values obtained by the Essam et al. (2010) study
are listed in Table LABEL:tab2.

Table 2: The absolute parameters of YY CrB. Absolute parameters | This study | Essam et al. (2010)
---|---|---
${M_{bol1}}(mag)$ | $4.083\pm 0.074$ | 3.939
${M_{bol2}}(mag)$ | $5.246\pm 0.072$ | 5.173
${L_{1}}({L_{\odot}})$ | $1.832\pm 0.121$ | 2.580
${L_{2}}({L_{\odot}})$ | $0.628\pm 0.041$ | 0.668
${R_{1}}({R_{\odot}})$ | $1.430\pm 0.049$ | 1.427
${R_{2}}({R_{\odot}})$ | $0.749\pm 0.026$ | 0.757
$a({R_{\odot}})$ | $2.674\pm 0.080$ | 2.64
$M_{1}({M_{\odot}})$ | $1.448\pm 0.131$ | $1.467$
$M_{2}({M_{\odot}})$ | $0.362\pm 0.037$ | $0.357$
$log(g)_{1}(cgs)$ | $4.288\pm 0.008$ | 4.295
$log(g)_{2}(cgs)$ | $4.248\pm 0.012$ | 4.232

##  5 The Orbital Period Changes

To calculate the eclipse times of minima, we used the same method as done by
Soomandar & Abedi (2020). First, split the detrended light curves for
individual eclipses and fit a Lorentzian function to each eclipse by using the
least-squares method. We used the Scipy.curve-fit package in Python to fit the
Lorentzian function to the individual eclipses. We used the
np.sqrt(np.diag(cov)) function to calculate the standard deviation errors on
the parameters. TESS observations yielded a total of 220 primary and secondary
minima, as displayed in the table 3. The new observational eclipse times were
calculated. Then, we performed an analysis of observed (O) minus calculated
(C) eclipse times (Sterken, 2005). We calculated the O-C curve using the
following linear ephemeris (Kreiner 2004, Yu, Xiang, & Xiao 2015):

$Min.I=2452500.1757+0.3765545\times E$ (7)

The O-C curve of primary and secondary minima for sectors 24 and 51 are
plotted in Figure 5. The anti-correlated manner between primary and secondary
minima is obvious which is a confirmation of the presence of spots on the
contact binary components (Tran et al. (2013); Balaji et al. (2015)). We
averaged primary and secondary minima to eliminate the anti-correlated impact
when analyzing orbital period changes (Balaji et al., 2015). The calculated
values are shown in Table 3. 109 YY CrB observational minima times were
recorded in the literature over a 31-year period. The appendix contains a list
of the data gathered with uncertainty. Observational minima times converted to
BJD-TDB333https://astroutils.astronomy.osu.edu/time/hjd2bjd.html. Figure 6’s
left panel depicts the O-C curve of the minima. We presented a new ephemeris
for this target as Equation (8) by fitting a linear function on the O-C curve
of primary.

$Min.I=2458955.8598(\pm 1.1e-4)+0.3765581(\pm 1.1e-7)\times E$ (8)

Figure 5: Left panel: primary and secondary O-C curve of minima for sector 24.
Right panel: primary and secondary minima for sector 51 (primary O-C curve in
black circles and secondary O-C curve in red circles.).

The O-C curve of minima is calculated with the new ephemeris and the resulting
curve shows the same shape as the left panel of Figure 6. we fitted a
quadratic function to the O-C curve in order to investigate the variations in
the orbital period:

${T_{mid}}(E)={T_{0}}+PE+\frac{1}{2}\frac{{dP}}{{dt}}{E^{2}}$ (9)

where mid-eclipse times ${T_{mid}}$ are described by ${T_{0}}$ is the
reference mid-eclipse time, P is the orbital period and, E is the epoch of
eclipses (Patra et al., 2017). And the quadratic fit showed a drop in period,
which corresponded to the quadratic plot in Figure 6’s left panel. We
determined the rate of period decline using the model’s quadratic coefficient
as Equation (10)

$\mathop{P}\limits^{.}=\frac{{2\times-1.124\times{{10}^{-11}}}}{{0.3765545}}=-5.786\times{10^{-8}}\pm
9.965\times{10^{-9}}\frac{{day}}{{year}}$ (10)

Considering ${M_{1}}=1.448{M_{\odot}}$ for the primary and
${M_{2}}=0.362{M_{\odot}}$ for the secondary one calculated in this study and
using Equation (11) and mass conservation, the rate of mass exchange between
primary and secondary components was estimated.

$\frac{{\mathop{P}\limits^{.}}}{P}=-3\frac{{{{\mathop{M}\limits^{.}}_{2}}({M_{1}}-{M_{2}})}}{{{M_{1}}{M_{2}}}}\Rightarrow\mathop{{M_{2}}}\limits^{.}=+2.472\times{10^{-8}}\pm
0.190\times{10^{-8}}{M_{\odot}}y{r^{-1}}$ (11)

The positive sign indicates the direction of mass transfer from the more
massive to the less massive component. The cyclic changes are shown by the
residuals of the quadratic fit. As a result, we investigated the Light Travel
Time Effect (LTTE) as a possible cause of the O-C curve variations. The
following periodogram analysis was performed with the Period 04 software (Lenz
& Breger, 2005) for the residuals of a quadratic fit. The peak of frequencies
in the periodogram analysis of residuals shows a period of 7351.018 days.
Then, we used the least square method to fit the Light Travel Time (LTT)
formula on the O-C curve (Irwin, 1952):

${(O-C)_{LTT}}=A\times\left({\frac{{1-{e^{2}}}}{{1+e\cos\upsilon}}\sin(\upsilon+\omega)+e\sin(\omega)}\right)$
(12)

where $A=\frac{{{a_{12}}\sin i}}{c}$, and ${a_{12}}$ is the semi-major axis of
the relative orbit of the eclipsing system around the center of mass (in Au
unit), $i$ is the inclination of the third-body orbit, $e$ is the eccentricity
of the supposed third body, $\omega$ is the longitude of periastron passage in
the plane of the orbit and, $\upsilon$ is the true anomaly. To fit the LTT
function to the residuals of the O-C curve, we have to convert the epoch to
the true anomaly and Kepler’s formula provides the link between the eccentric
anomaly and the observed eclipse time:

${E_{3}}-e\sin E{}_{3}=\frac{{2\pi}}{{{P_{3}}}}(t-{T_{0}})$ (13)

The equation (13) was calculated using Newton-Raphson’s method for every
eclipse time of minima and considering equation (14) the epochs converted to
the true anomaly.

$\begin{array}[]{l}\tan\frac{\nu}{2}={(\frac{{1+e}}{{1-e}})^{1/2}}\tan\frac{{{E_{3}}}}{2}\\\
t={T_{0}}+epoch\times{P_{binary}}\end{array}$ (14)

where ${P_{b}inary}$, t, ${P_{3}}$, ${E_{3}}$, and ${T_{0}}$ are the period of
binary, the time of observed minima, the period of the third body, the
eccentric anomaly and, the time of periastron passage respectively. By
assuming a coplanar orbit ($i=90$), we determined the lower limit for the mass
of the third body. The calculated parameters of the third body are listed in
Table 4 and the related curve is plotted in the right panel of Figure 6.

Figure 6: Left Panel: The O-C data points of the minima (black circle) and, the polynomial fit (red line). Right Panel: LTTE on the residuals of the polynomial fit; LTTE (black line), the O-C data points after subtracted polynomial fit (red circles), and the residuals of the LTT effect fit (blue circles). Table 3: Extracted the times of minima from TESS observations. All times of minima have been reduced to 2450000. Additionally, the minimum error is equal to 0.0001. Min. | Epoch | O-C | Min. | Epoch | O-C | Min. | Epoch | O-C | Min. | Epoch | O-C
---|---|---|---|---|---|---|---|---|---|---|---
8955.8597 | 17144 | 0.0337 | 8966.5927 | 17172.5 | 0.0348 | 8978.8326 | 17205 | 0.0368 | 9702.0170 | 19125.5 | 0.0482
8956.0504 | 17144.5 | 0.0360 | 8966.7802 | 17173 | 0.0340 | 8979.0187 | 17205.5 | 0.0345 | 9702.2047 | 19126 | 0.0351
8956.2360 | 17145 | 0.0336 | 8966.9693 | 17173.5 | 0.0348 | 8979.2092 | 17206 | 0.0368 | 9702.2047 | 19126 | 0.0476
8956.6126 | 17146 | 0.0334 | 8967.1567 | 17174 | 0.0340 | 8979.3952 | 17206.5 | 0.0344 | 9702.3936 | 19126.5 | 0.0482
8956.8034 | 17146.5 | 0.0359 | 8967.3459 | 17174.5 | 0.0349 | 8979.5859 | 17207 | 0.0369 | 9702.5813 | 19127 | 0.0476
8956.9892 | 17147 | 0.0334 | 8967.5334 | 17175 | 0.0341 | 8979.7718 | 17207.5 | 0.0345 | 9702.7702 | 19127.5 | 0.0482
8957.1799 | 17147.5 | 0.0359 | 8967.7225 | 17175.5 | 0.0349 | 8979.9626 | 17208 | 0.0370 | 9702.9578 | 19128 | 0.0475
8957.3657 | 17151 | 0.0002 | 8967.9099 | 17176 | 0.0340 | 8980.1483 | 17208.5 | 0.0344 | 9703.1468 | 19128.5 | 0.0482
8957.3657 | 17148 | 0.0334 | 8968.0990 | 17176.5 | 0.0349 | 8980.3392 | 17209 | 0.0371 | 9703.3343 | 19129 | 0.0476
8957.5565 | 17148.5 | 0.0359 | 8969.4165 | 17180 | 0.0344 | 8980.5249 | 17209.5 | 0.0344 | 9703.5233 | 19129.5 | 0.0482
8957.7421 | 17149 | 0.0333 | 8969.6050 | 17180.5 | 0.0348 | 8980.9014 | 17210.5 | 0.0345 | 9703.7111 | 19130 | 0.0477
8957.9331 | 17149.5 | 0.0359 | 8969.7931 | 17181 | 0.0345 | 8981.2781 | 17211.5 | 0.0346 | 9703.8998 | 19130.5 | 0.0482
8958.1187 | 17150 | 0.0333 | 8969.9817 | 17181.5 | 0.0348 | 8981.4691 | 17212 | 0.0373 | 9704.0876 | 19131 | 0.0478
8958.3096 | 17150.5 | 0.0359 | 8970.1697 | 17182 | 0.0346 | 9692.9793 | 19101.5 | 0.0478 | 9704.2763 | 19131.5 | 0.0481
8958.4952 | 17151 | 0.0332 | 8970.3582 | 17182.5 | 0.0348 | 9693.1681 | 19102 | 0.0482 | 9704.4642 | 19132 | 0.0478
8958.6861 | 17151.5 | 0.0358 | 8970.5464 | 17183 | 0.0347 | 9693.3556 | 19102.5 | 0.0475 | 9704.6531 | 19132.5 | 0.0484
8958.8717 | 17152 | 0.0332 | 8970.7348 | 17183.5 | 0.0348 | 9693.5445 | 19103 | 0.0481 | 9705.9703 | 19136 | 0.0476
8959.0624 | 17152.5 | 0.0356 | 8970.9231 | 17184 | 0.0348 | 9693.7323 | 19103.5 | 0.0476 | 9706.1590 | 19136.5 | 0.048
8959.2482 | 17153 | 0.0332 | 8971.1114 | 17184.5 | 0.0349 | 9693.9211 | 19104 | 0.0482 | 9706.3469 | 19137 | 0.0478
8959.4390 | 17153.5 | 0.0356 | 8971.2997 | 17185 | 0.0349 | 9694.1092 | 19104.5 | 0.0479 | 9706.5354 | 19137.5 | 0.0479
8959.6249 | 17154 | 0.0333 | 8971.4880 | 17185.5 | 0.0349 | 9694.2977 | 19105 | 0.0482 | 9706.7234 | 19138 | 0.0477
8959.8155 | 17154.5 | 0.0355 | 8971.6763 | 17186 | 0.0349 | 9694.4855 | 19105.5 | 0.0478 | 9706.9121 | 19138.5 | 0.0481
8960.0014 | 17155 | 0.0332 | 8971.8645 | 17186.5 | 0.0348 | 9695.2398 | 19107.5 | 0.0489 | 9707.1000 | 19139 | 0.0477
8960.1919 | 17155.5 | 0.0354 | 8972.0531 | 17187 | 0.0351 | 9695.4272 | 19108 | 0.0481 | 9707.2889 | 19139.5 | 0.0483
8960.3778 | 17156 | 0.0331 | 8972.2411 | 17187.5 | 0.0348 | 9695.6154 | 19108.5 | 0.048 | 9707.4766 | 19140 | 0.0477
8960.5685 | 17156.5 | 0.0354 | 8972.4297 | 17188 | 0.0352 | 9695.8037 | 19109 | 0.0480 | 9708.2306 | 19142 | 0.0486
8960.7544 | 17157 | 0.0331 | 8972.6176 | 17188.5 | 0.0348 | 9695.9922 | 19109.5 | 0.0483 | 9709.5480 | 19145.5 | 0.0481
8960.9450 | 17157.5 | 0.0355 | 8972.8063 | 17189 | 0.0353 | 9696.1804 | 19110 | 0.0482 | 9710.4893 | 19148 | 0.048
8961.1310 | 17158 | 0.0331 | 8972.9941 | 17189.5 | 0.0348 | 9696.3688 | 19110.5 | 0.0483 | 9710.6776 | 19148.5 | 0.0480
8961.3215 | 17158.5 | 0.0354 | 8973.1830 | 17190 | 0.0354 | 9696.5567 | 19111 | 0.0479 | 9710.8660 | 19149 | 0.0481
8961.5077 | 17159 | 0.0333 | 8973.3707 | 17190.5 | 0.0348 | 9697.3101 | 19113 | 0.0482 | 9711.0543 | 19149.5 | 0.0481
8961.6980 | 17159.5 | 0.0354 | 8973.5597 | 17191 | 0.0355 | 9697.4983 | 19113.5 | 0.0481 | 9711.2425 | 19150 | 0.0481
8961.8842 | 17160 | 0.0332 | 8973.7471 | 17191.5 | 0.0347 | 9697.6865 | 19114 | 0.0480 | 9711.4307 | 19150.5 | 0.0481
8962.0746 | 17160.5 | 0.0354 | 8973.9364 | 17192 | 0.0357 | 9697.8749 | 19114.5 | 0.0482 | 9711.6190 | 19151 | 0.048
8962.2608 | 17161 | 0.0331 | 8974.1236 | 17192.5 | 0.0346 | 9697.8749 | 19114.5 | 0.0482 | 9711.8073 | 19151.5 | 0.0481
8962.4511 | 17161.5 | 0.0353 | 8974.3130 | 17193 | 0.0375 | 9698.0631 | 19115 | 0.0481 | 9711.9956 | 19152 | 0.0481
8962.6374 | 17162 | 0.0333 | 8974.5001 | 17193.5 | 0.0345 | 9698.2516 | 19115.5 | 0.0483 | 9712.1838 | 19152.5 | 0.048
8962.8276 | 17162.5 | 0.0353 | 8975.0664 | 17195 | 0.0360 | 9698.4395 | 19116 | 0.0479 | 9712.3722 | 19153 | 0.0482
8963.0140 | 17163 | 0.0334 | 8975.2533 | 17195.5 | 0.0347 | 9698.6281 | 19116.5 | 0.0482 | 9712.5602 | 19153.5 | 0.0479
8963.2041 | 17163.5 | 0.0352 | 8975.4430 | 17196 | 0.0360 | 9698.8160 | 19117 | 0.0478 | 9712.7487 | 19154 | 0.0481
8963.3905 | 17164 | 0.0333 | 8975.6298 | 17196.5 | 0.0346 | 9699.0047 | 19117.5 | 0.0483 | 9712.9368 | 19154.5 | 0.0479
8963.5807 | 17164.5 | 0.0351 | 8975.8196 | 17197 | 0.0361 | 9699.1926 | 19118 | 0.0479 | 9713.1252 | 19155 | 0.0481
8963.7673 | 17165 | 0.0335 | 8976.0063 | 17197.5 | 0.0346 | 9699.3812 | 19118.5 | 0.0483 | 9713.5019 | 19156 | 0.0481
8963.9570 | 17165.5 | 0.035 | 8976.1962 | 17198 | 0.0362 | 9699.5691 | 19119 | 0.0479 | 9713.6898 | 19156.5 | 0.0478
8964.1438 | 17166 | 0.0335 | 8976.3828 | 17198.5 | 0.0346 | 9699.7578 | 19119.5 | 0.0482 | 9713.8784 | 19157 | 0.0481
8964.3335 | 17166.5 | 0.035 | 8976.5729 | 17199 | 0.0363 | 9699.9456 | 19120 | 0.0479 | 9714.0664 | 19157.5 | 0.0478
8964.5204 | 17167 | 0.0336 | 8976.7594 | 17199.5 | 0.0345 | 9700.1344 | 19120.5 | 0.0483 | 9714.2550 | 19158 | 0.0482
8964.7101 | 17167.5 | 0.0349 | 8976.9495 | 17200 | 0.0364 | 9700.3221 | 19121 | 0.0478 | 9714.4429 | 19158.5 | 0.0477
8964.8970 | 17168 | 0.0336 | 8977.1360 | 17200.5 | 0.0345 | 9700.3222 | 19122 | 0.004 | 9714.6315 | 19159 | 0.0481
8965.0865 | 17168.5 | 0.0349 | 8977.3260 | 17201 | 0.0009 | 9700.5108 | 19121.5 | 0.0482 | 9714.8195 | 19159.5 | 0.0478
8965.2736 | 17169 | 0.0337 | 8977.5125 | 17201.5 | 0.0345 | 9700.6986 | 19122 | 0.0477 | 9715.0082 | 19160 | 0.0482
8965.4631 | 17169.5 | 0.0349 | 8977.7028 | 17202 | 0.0364 | 9700.8874 | 19122.5 | 0.0482 | 9715.1961 | 19160.5 | 0.0478
8965.6503 | 17170 | 0.0338 | 8977.8891 | 17202.5 | 0.0345 | 9701.0752 | 19123 | 0.0478 | 9715.1961 | 19160.5 | 0.0478
8965.8397 | 17170.5 | 0.0349 | 8978.0794 | 17203 | 0.0366 | 9701.2639 | 19123.5 | 0.0482 | 9715.3847 | 19161 | 0.0482
8966.0269 | 17171 | 0.0338 | 8978.2656 | 17203.5 | 0.0345 | 9701.4517 | 19124 | 0.0478 | 9715.5727 | 19161.5 | 0.0479
8966.2162 | 17171.5 | 0.0349 | 8978.4560 | 17204 | 0.0367 | 9701.6405 | 19124.5 | 0.0481 | 9715.7612 | 19162 | 0.0482
8966.4035 | 17172 | 0.0339 | 8978.6422 | 17204.5 | 0.0345 | 9701.8282 | 19125 | 0.0476 |  |  |
Table 4: Parameters of the third body. Parameters of third body |  | Value
---|---|---
Eccentricity (e) |  | $0.689\pm 0.005$
The longitude of periastron passage ($\omega$) |  | $57.6\pm 1.8$
Period (days) |  | 7351.018
Amplitude (minutes) |  | $17.28\pm 1.44$
The time of periastron passage ($T_{0}$) |  | 2460201
Projected semi-major axis$\times\sin i$ |  | $2.11\pm 0.17$
Mass Function $(MassFunction({f_{m}}))$ |  | $0.023\pm 0.006$
${M_{3}}\sin i(i=90,{M_{\odot}})$ |  | 0.498
$\sum{{{(O-C)}^{2}}}$ |  | 0.001

Assuming the third body is a main-sequence star, this corresponds to the M1V
spectral type with a brightness of $0.041{L_{\odot}}$
444http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt ,
or $0.016$ of the total luminosity which doesn’t agree with the value $l_{3}$
determined in section 3.

We are not certain that $l_{3}$ produced from the TESS light curve analysis
represents a valid observation of flux from a third body in the system,
despite the computation of the effect of the third body, especially as there
is no exact radial-velocity curve. We estimate that this figure is most likely
due to systematic errors in background flux level readings in TESS photos
and/or an underestimation of photometric aperture contamination by other stars
in the image. So, we investigated the Applegate’s effect as a plausible
explanation for the cyclical fluctuations in the O-C curve. We calculated the
observed relative change of the orbital period throughout one cycle of the
binary using the previously obtained modulation period and the O-C amplitude
computed from the orbit of the third body simulated in the previous section.
$\frac{{\Delta P}}{P}=2\pi\frac{{(O-C)}}{{{P_{\bmod}}}}=1.025\times{10^{-5}}$
(Applegate, 1992). The value of $\frac{{\Delta P}}{P}$suggests that the
Applegate effect can explain the cyclic changes in the O-C curve of minima.

This system shows unequal maxima that is known as the O’Connell effect
(O’Connell, 1951) due to the presence of the hot spots (Wilsey & Beaky, 2009).
So, this difference implies that the hemisphere of a component emits a
different amount of radiation than the other hemisphere. These types of
systems have active chromospheres because of the existence of large spots
(Knote et al., 2022). Starspots can alter the depth of minima and have an
obvious effect on the eclipse light curve (Han, Muirhead, & Swift, 2019).

To investigate the effect of the spots on the light curve over both sectors of
TESS observations, we calculated the difference between the two depths of the
primary and secondary minima. We considered the relative fluxes in phases 0
and 0.5 for every complete individual light curve. The curves that resulted
were displayed in Figure 7. YY CrB has the values of the DepthI - DepthII as
large as about $10\%$ of the variable light amplitude. And this is possible
because of the migration and evolution of spots with time on the surface of
two components that cause cyclic magnetic activity.

Figure 7: Right panel: the difference between the depths of Primary and
Secondary minima over the 24th sector of TESS observations. Left panel: the
difference between the depths of Primary and Secondary minima over the 51st
sector of TESS observations.

##  6 Discussion and conclusion

Based on the estimated mass ratio, fill-out factor and inclination angle, YY
CrB is an over-cantact binary with an increased orbital period. Essam et al.
(2010) calculated the decreased period rate
$1.194\times{10^{-6}}\frac{{day}}{{year}}$. Yu, Xiang, & Xiao (2015)
considered all of the minima time published until 2013 and calculated a
secular period decrease with a rate of
$6.727\times{10^{-7}}\frac{{day}}{{year}}$. In this study, the decreasing
value of period rate $5.786\times{10^{-8}}\frac{{day}}{{year}}$ indicates that
the rate of period changes has been decreased. And, when mass conservation is
taken into account, the mass transfer from the Roche-lobe-filling primary
component to the secondary component is
${\mathop{M}\limits^{.}_{2}}=2.472\times{10^{-8}}{M_{\odot}}y{r^{-1}}$. When
the stated results in Yu, Xiang, & Xiao (2015) are compared to the value of
mass transfer in this study, it is clear that mass transfer has been decreased
and the distance between two components is growing while the value of fill-out
factor is decreasing (compare the values of fill-out factors in Table 1). And
this target may evolve to shallow-contact binary via the thermal relaxation
oscillation (TRO) model (Flannery (1976); Robertson & Eggleton (1977)) and
ultimately reach a broken-contact phase ( Lucy (1976)). The mass ratio of the
components, which is related to the mass transfer, is the crucial parameter in
the evolution of the close binary stars. Table 5 contains a list of contact
binaries with low mass ratios $<0.25$. In order to explain the evolutionary
status of the YY CrB system, we provide the mass-luminosity ($M-L$) diagram
displayed in Figure 8. The Zero-Age Main Sequence (ZAMS) and the Terminal‐Age
Main Sequence (TAMS) are plotted along with the selected contact binaries with
low mass ratios. It is obvious that the more massive primary components are
around the ZAMS line, meaning they are not evolved or little evolved. Also,
the less massive secondary components have evolved away from the main sequence
stars and over-luminosity comparing the stars with the same mass in the main
sequence. In addition, the orbital angular momentum of YY CrB has a value of
$51.585\pm 0.067$. The $logJ_{0}-logM$ diagram shows the position of the
system (Figure 9), and this diagram shows that YY CrB is in a contact binary
systems region.

Table 5: Absolute parameters for low mass ratio contact binaries. System | $q$ | ${M_{1}}({M_{\odot}})$ | ${M_{2}}({M_{\odot}})$ | ${R_{1}}({R_{\odot}})$ | ${R_{2}}({R_{\odot}})$ | ${L_{1}}({L_{\odot}})$ | ${L_{2}}({L_{\odot}})$ | Reference
---|---|---|---|---|---|---|---|---
V429 Cam | 0.206 | 1.36(12) | 0.28(3) | 1.55(3) | 0.78(2) | 3.56(9) | 0.85(2) | Li et al. (2021)
V830 Cep | 0.23 | 0.84(5) | 0.19(1) | 0.91(1) | 0.47(1) | 0.98(1) | 0.29(1) | Li et al. (2021)
FP Boo | 0.096 | 1.614(52) | 0.154(21) | 2.310(25) | 0.774(8) | 11.193(99) | 0.920(13) | Gazeas et al. (2006)
DN Boo | 0.103 | 1.428(39) | 0.148(6) | 1.710(67) | 0.670(110) | 3.750(280) | 0.560(170) | Şenavcı et al. (2008)
FG Hya | 0.112 | 1.444(25) | 0.161(7) | 1.405(9) | 0.591(8) | 2.158(86) | 0.412(17) | Qian & Yang (2005)
CK Boo | 0.0108 | 1.442(14) | 0.154(2) | 1.453(3) | 0.577(10) | 2.74(1) | 0.47(2) | Kalci & Derman (2005)
GR Vir | 0.122 | 1.37(16) | 0.17(6) | 1.42(7) | 0.61(4) | 2.87(28) | 0.48(6) | Qian & Yang (2004)
CSS J234807.2$+$193717 | 0.176 | 1.19(4) | 0.21(3) | 1.36 (2) | 0.66 (1) | 1.45(24) | 0.42(5) | Christopoulou et al. (2022)
J170307 | 0.092 | 1.134(253) | 0.105(24) | 1.204(120) | 0.436(48) | 1.874(572) | 0.271(72) | Liu et al. (2023)
J1641000 | 0.095 | 1.402(287) | 0.133(28) | 1.580(144) | 0.577(58) | 3.912(1.109) | 0.512(142) | Liu et al. (2023)
J223837 | 0.093 | 1.541(306) | 0.144(30) | 1.784(159) | 0.646(64) | 5.463(1.534) | 0.704(192) | Liu et al. (2023)
CSS J222607.8$+$062107 | 0.221 | 1.49(3) | 0.33(13) | 1.51(5) | 0.81(2) | 3.35(65) | 1.15(24) | Sun et al. (2020)
CSS J012559.7$+$203404 | 0.231 | 1.38(3) | 0.32(12) | 1.42(4) | 0.77(2) | 2.46(56) | 0.81(18) | Sun et al. (2020)
CSS J153855.6$+$042903 | 0.187 | 1.44(5) | 0.27(12) | 1.37(4) | 0.66(2) | 2.94(96) | 0.30(10) | Sun et al. (2020)
CSS J141923.2$-$013522 | 0.168 | 1.31(5) | 0.22(11) | 1.23(4) | 0.57(2) | 1.97(68) | 0.33(11) | Sun et al. (2020)
CSS J130111.2$-$132012 | 0.108 | 1.38(3) | 0.15(12) | 1.49(5) | 0.61(2) | 2.49(57) | 0.40(9) | Sun et al. (2020)
CSS J165813.7$+$390911 | 0.183 | 1.09(3) | 0.20(9) | 1.05(3) | 0.49(1) | 0.92(24) | 0.24(6) | Sun et al. (2020)
V870 Ara | 0.082 | 1.546(54) | 0.127(37) | 1.64(6) | 0.63(5) | 2.64(17) | 0.42(6) | Poro et al. (2021)
TYC 6995-813-1 | 0.111 | 1.23(1) | 0.135(1) | 1.46(1) | 0.60(1) | 2.293(4) | 0.58(2) | Wadhwa et al. (2021)
NSVS 13602901 | 0.171 | 1.19(2) | 0.203(10) | 1.69(1) | 0.79(1) | 2.05(4) | 0.58(2) | Wadhwa et al. (2021)
NSVS 5029961 | 0.151 | 1.872(468) | 0.284(72) | 1.573(119) | 0.680(53) | 3.403(14) | 0.610(26) | Zheng et al. (2021)
CSS J022914.4$+$044340 | 0.201 | 1.44(25) | 0.29(5) | 1.26(8) | 0.65(4) | 1.718(191) | 0.416(50) | Liu & Li (2021)
HV Aqr | 0.15 | 1.240(28) | 0.186(17) | 1.456(12) | 0.601(5) | 3.326(213) | 0.638(44) | Gazeas et al. (2021)
ZZ PsA | 0.078 | 1.213(8) | 0.095(1) | 1.422(4) | 0.559(4) | 2.20(4) | 0.63(4) | Wadhwa et al. (2021)
NSVS 1926064 | 0.160 | 1.558(38) | 0.249(6) | 1.605(13) | 0.755(42) | 3.91(28) | 0.641(33) | Kjurkchieva et al. (2020)
Figure 8: $M-L$ diagram of selected contact binaries with low mass ratio. The
primary and secondary components of YY CrB are plotted in red and blue colors,
respectively. Figure 9: The location of YY CrB on the $logJ_{0}-logM$ diagram.
The quadratic line is based on a study by Eker et al. (2006).

According to the study Yu, Xiang, & Xiao (2015) and the periodic changes in
the residuals of the quadratic fit on the O-C curve, the potential of
excitability of the third body was investigated. The light-time function
fitting revealed the existence of a body with the value of $0.498{M_{\odot}}$.
This number equates to 0.016 of total brightness, which differs from the value
of $l_{3}$ determined in section 3. We explored the Applegate effect as a
possible explanation for the fluctuation in the O-C curve because there is no
exact radial-velocity curve.

YY CrB is a contact binary with a mass ratio less than $0.3$, so considering
Hut’s criteria (Hut, 1980) to investigate the stability is necessary. We used
Equation (15) to calculate the ratio of the spin angular momentum to the
orbital angular momentum (Yang & Qian, 2015).

$\frac{{{J_{s}}}}{{{J_{o}}}}=\frac{{q+1}}{q}[{({k_{1}}{r_{1}})^{2}}+{({k_{2}}{r_{2}})^{2}}q]$
(15)

where $r_{1}$ and $r_{2}$ are the relative radii for the primary and secondary
components and ${k^{2}}_{1,2}=0.06$ (Li & Zhang, 2006), are the dimensionless
gyration radii. The calculated value of $\frac{{{J_{s}}}}{{{J_{o}}}}=0.087$,
which is less than the threshold value therefore this system is stable. This
target shows a period increase which is attributed to mass transfer. According
to the existing data and the analyses done in this study, the existence of a
third body is unlikely for this system, and detailed spectroscopic and
photometric observations over a longer length of time are required for the
definitive determination.

## Acknowledgements

This manuscript has made use of data from the TESS mission. Funding for the
TESS mission is provided by the NASA Science Mission Directorate. This
research has made use of the SIMBAD and VIZIER databases, operated at CDS,
Strasbourg, France. The time of minima data from the Variable Star Observers
League in Japan (VSOLJ) websites proved invaluable to the assessment of
potential period changes experienced by this variable star. The authors would
like to thank Marco Brentel for his help. We are grateful to Ehsan Paki from
the BSN project (https://bsnp.info/) for providing Figure 4 of this
manuscript, which also shows the color-temperature scale.

## ORCID iDs

Somayeh Soomandar: 0000-0002-9520-9573
Atila Poro: 0000-0002-0196-9732

##  Appendix A Available Minima Times

The appendix table displays the minima times along with their error in the
first column, the epochs in the third column, the O-C values in the fourth
column, and the references in the final column.

Table 1: Available mid-eclipse times of YY CrB system. ons. All minimum have been reduced to 2450000 Min.($BJD_{TDB}$) | Epoch | O-C | Reference |  Min.($BJD_{TDB}$) | Epoch | O-C | Reference
---|---|---|---|---|---|---|---
955.8695(12) | -4101 | -0.0562 | Pribulla et al. (2001) | 4308.3899(2) | 4802 | -0.0005 | Parimucha et al. (2009)
955.8718(6) | -4101 | -0.0539 | Rucinski, et al. (2000) | 4564.2595 | 5481.5 | 0.0003 | Nagai (2009)
1318.4993 | -3138 | -0.04847 | Essam et al. (2010) | 4604.36215(1) | 5588 | -0.0001 | Yilmaz et al. (2009)
1318.5001 | -3138 | -0.0475 | Essam et al. (2010) | 4605.4917(2) | 5591 | -0.0002 | Yilmaz et al. (2009)
1361.4275(3) | -3024 | -0.0475 | Keskin et al. (2000) | 4628.4657(2) | 5652 | 0.0039 | Parimucha et al. (2009)
1361.4275(2) | -3024 | -0.0474 | Keskin et al. (2000) | 4632.4144(3) | 5662.5 | -0.0012 | Parimucha et al. (2009)
1368.3965(4) | -3005.5 | -0.0447 | Keskin et al. (2000) | 4648.4191(30) | 5706 | 0.0000 | Parimucha et al. (2009)
1368.3966(4) | -3005.5 | -0.0446 | Keskin et al. (2000) | 4648.4201(10) | 5705 | 0.0009 | Hubscher et al. (2009)
1370.4659(6) | -3000 | -0.0463 | Keskin et al. (2000) | 4688.3352(3) | 5811 | 0.0013 | Yilmaz et al. (2009)
1372.3494(3) | -2995 | -0.0457 | Keskin et al. (2000) | 4931.4009(1) | 6456.5 | 0.0011 | Hubscher et al. (2011)
1668.3359 | -2209 | -0.0309 | Essam et al. (2010) | 4931.5883(1) | 6457 | 0.0001 | Hubscher et al. (2011)
1669.4602 | -2206 | -0.0363 | Essam et al. (2010) | 4958.5136(20) | 6528.5 | 0.0018 | Hubscher et al. (2011)
1670.3976 | -2203.5 | -0.0403 | Essam et al. (2010) | 4983.7401(3) | 6595.5 | -0.0009 | Diethelm (2009)
1670.3984 | -2203.5 | -0.0395 | Essam et al. (2010) | 5017.44086(3) | 6685 | -0.0017 | Parimucha et al. (2009)
1674.3548 | -2193 | -0.0369 | Karska & Maciejewski (2003) | 5213.6297(3) | 7206 | 0.0022 | Parimucha et al. (2011)
1692.4299 | -2144 | -0.0365 | Essam et al. (2010) | 5219.6532(1) | 7222 | 0.0009 | Parimucha et al. (2011)
1692.4319 | -2145 | -0.0345 | Essam et al. (2010) | 5261.8279(1) | 7334 | 0.0014 | Dvorak (2011)
1975.6050 | -1392 | -0.0304 | Pribulla, et al. (2003) | 5264.4630(3) | 7341 | 0.0006 | Parimucha et al. (2011)
1975.6061(1) | -1392 | -0.0293 | Pribulla et al. (2001) | 5311.3444(2) | 7465.5 | 0.001 | Parimucha et al. (2011)
1975.6064(1) | -1393 | -0.0290 | Pribulla et al. (2001) | 5311.5317(2) | 7466 | 0.000 | Parimucha et al. (2011)
1975.6108(7) | -1392 | -0.0246 | Pribulla & Vanko (2002) | 5351.4463(1) | 7572 | -0.0002 | Hubscher, et al. (2012)
2029.4398(7) | -1250 | -0.0426 | Pribulla & Vanko (2002) | 5354.4587(2) | 7580 | -0.0002 | Parimucha et al. (2011)
2031.5168(7) | -1244.5 | -0.0369 |  Pribulla & Vanko (2002)) | 5420.3573(3) | 7755 | 0.0137 |  Parimucha et al. (2011))
2045.4589 | -1207.5 | -0.0273 | Essam et al. (2010) | 5652.8828(30) | 8372.5 | 0.0045 | Diethelm (2011)
2060.3320 | -1167 | -0.0281 | Essam et al. (2010) | 5665.4931(3) | 8406 | 0.0088 | Parimucha et al. (2013)
2060.3352 | -1167 | -0.0249 | Essam et al. (2010) | 5705.4093(23) | 8512 | 0.0016 | Hubscher, et al. (2012)
2400.1804(2) | -265.5 | -0.0202 | Pribulla et al. (2002) | 6011.5475(2) | 9325 | 0.0051 | Parimucha et al. (2013)
2400.3660 | -264 | -0.0227 | Karska & Maciejewski (2003) | 55987.6371(2) | 9261.5 | 0.0018 | Parimucha et al. (2013)
2469.4699(4) | -81.5 | -0.017 | Demircan et al. (2003) | 5987.6371(2) | 9261.5 | 0.0018 | Parimucha et al. (2013)
2472.2898(2) | -74 | -0.0209 | Petropoulou et al. (2015) | 5992.5319(2) | 9274.5 | 0.0014 | Parimucha et al. (2013)
2473.4197(2) | -71 | -0.0206 | Petropoulou et al. (2015) | 6005.5237(2) | 9309 | 0.0021 | Parimucha et al. (2013)
2473.4247(4) | -71 | -0.0156 | Demircan et al. (2003) | 6005.5237(2) | 9309 | 0.0021 | Parimucha et al. (2013)
2500.1757 | 0 | 0.000 | Kreiner (2004) | 6149.3679(2) | 9691 | 0.0026 | Parimucha et al. (2013)
2719.3200 | 582 | -0.0104 | Nagai (2004) | 2456199.26169(2) | 9823.5 | 0.0028 | Parimucha et al. (2013)
2764.5082(23) | 702 | -0.0088 | Hubscher et al. (2005) | 6742.4439(14) | 11266 | 0.00513 | Hubscher & Lehmann (2015)
2786.3500(2) | 761 | -0.0071 | Ak & Filiz (2003) | 6749.4115(5) | 11284.5 | 0.0066 | Hubscher & Lehmann (2015)
2793.5038(2) | 779 | -0.0079 | Ak & Filiz (2003) | 6011.5483(2) | 9325 | 0.0018 | Parimucha et al. (2013)
7074.9466(1) | 12149 | 0.010 | Nelson (2016) | 6754.4970(35) | 11298 | 0.0086 | Hubscher & Lehmann (2015)
2793.5042(21) | 779 | -0.0075 | Hubscher et al. (2005) | 7074.9466(1) | 12149 | 0.010 | Nelson (2016)
2814.4011(1) | 834.5 | -0.0093 | Selam et al. (2003) | 7084.1734 | 12173.5 | 0.0114 | Nagai (2016)
3151.0470 | 1728.5 | -0.0032 | Nagai (2005) | 7123.5238(28) | 12278 | -0.011 |  Hubscher (2017))
3458.8835(2) | 2546 | 0.000 | Dvorak (2006) | 6749.6002(8) | 11285 | 0.0068 | Hubscher & Lehmann (2015)
3466.41483(1) | 2566 | 0.000 | Pribulla et al. (2005) | 7513.0723 | 13312.5 | 0.0147 | Nagai (2017)
4201.4509(16) | 4518 | 0.0020 | Hubscher (2007) | 7489.5378(18) | 13250 | 0.015 | Hubscher (2017)
4201.6350(17) | 4518.5 | -0.0022 | Hubscher (2007) | 9024.3943 | 17326 | 0.0353 | Nagai (2021)
4245.5055(9) | 4635 | -0.0003 | Brát, et al. (2007) | 9037.3884 | 17360.5 | 0.0382 | Nagai (2021)
4224.4161(2) | 4579 | -0.0028 | Parimucha et al. (2009) | 9269.5368(10) | 17977 | 0.0408 | Paschke (2021)
4300.4828(1) | 4781 | 0.0000 | Parimucha et al. (2009) | 9328.4664(7) | 18133.5 | 0.0398 | Lienhard (2022)
4500.6209(3) | 5312.5 | -0.0006 | Parimucha et al. (2009) | 9605.99382(1) | 18870.5 | 0.0464 | Nelson & Alton (2022)
4504.5751(8) | 5323 | -0.0002 | Parimucha et al. (2009) | 10064.4578(30) | 20088 | 0.0553 | Paschke (2023)
14513.6131(6) | 5347 | 0.0005 | Parimucha et al. (2009) |  |  |  |
14560.1297 | 5470.5 | 0.0126 | Nagai (2009) |  |  |  |

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# Effects of strangeness on the chiral pseudo-critical line

Mahammad Sabir Ali<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni-752050, India  Deeptak Biswas<EMAIL_ADDRESS>School of
Physical Sciences, National Institute of Science Education and Research, An
OCC of Homi Bhabha National Institute, Jatni-752050, India  Amaresh Jaiswal
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of
Science Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India  Institute of Theoretical Physics, Jagiellonian
University, ul. St. Łojasiewicza 11, 30-348 Krakow, Poland  Hiranmaya Mishra
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of
Science Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India

###### Abstract

Within a 2+1 flavor Nambu–Jona-Lasinio model, we have calculated the curvature
coefficients and checked them against available lattice QCD estimations. With
the observation that the flavor mixing due to the ‘t Hooft determinant term
significantly affects the $\kappa_{2}^{S}$, we explore the effect of $\mu_{S}$
on the $T-\mu_{B}$ cross-over lines. With the novel determination of negative
$\kappa_{2}^{B}$ at large $\mu_{S}$, we advocate the importance of studying
the same in lattice QCD.

##  I Introduction

The phase diagram of the strongly interacting matter necessitates the
determination of the chiral transition line in the high-density and high-
temperature regions. The chiral symmetry is broken in the low-density
(temperature) phase of quantum chromodynamics (QCD), which gets restored as
the temperature and/or density increases. At vanishing baryon density, the
restoration of the chiral symmetry is determined to be a cross-over with a
pseudo-critical temperature $T_{pc}=156.5\pm 1.5\text{MeV}$ [1]. On the other
hand, the transition is expected to be a first order at high density, which is
connected to the cross-over line through a critical endpoint (CEP). Although
the determination of the cross-over line for small values of the baryon
chemical potential ($\mu_{B}$) is quite settled with the recent advancements
of lattice QCD (LQCD) [2, 1] calculations, the extension of the line at finite
$\mu_{B}$ suffers from the infamous sign problem which leads to the
oscillatory behaviour of the Monte Carlo sampling method.

For small chemical potential ($\mu_{X}$), the pseudo-critical line can be
Taylor expanded at the lowest order in $\mu_{X}^{2}$, where one defines the
line with the following ansatz, [2, 3, 1],

$\frac{T_{pc}(\mu_{X})}{T_{pc}(0)}=1-\kappa_{2}^{X}\left(\frac{\mu_{X}}{T_{pc}(0)}\right)^{2}-\kappa_{4}^{X}\left(\frac{\mu_{X}}{T_{pc}(0)}\right)^{4}~{}.$
(1)

Here, $\mu_{X}$ corresponds to chemical potential associated with various
charges like baryon charge $B$, electric charge $Q$, and strangeness $S$. Such
parametrization allows for the comparison of results from different models and
lattice QCD calculations within the same baseline. The curvature coefficients
$\kappa_{2}$ and $\kappa_{4}$ have been examined by the Taylor expansion
method on the lattice [4, 5, 1]. Another standard approach relies on
performing the calculations at imaginary chemical potential, followed by
analytic continuation to the real plane [2, 6, 7]. The above-mentioned results
are in good agreement with each other within the respective variances. Similar
studies have been performed within the perturbative QCD [8] as well as in the
ideal and mean-field hadron resonance gas (HRG) model [9, 10], quark-meson
model [11, 12, 13, 14, 15, 16]. Moreover, the Nambu–Jona-Lasinio (NJL) model
has also been employed in this context[17] considering 2 flavors of light
quark.

The effective models, considering the symmetries of the QCD Lagrangian, enable
one to probe the matter at extreme conditions like high temperature and(or)
density, even in the presence of a magnetic field, to understand the phases of
the QCD matter, and provide a bulk description [18, 19]. The NJL model relies
on chiral symmetry and provides a qualitative description of the QCD matter
considering the pseudo-scalar mesons [20, 21]. Despite the analytical
simplicity and the dependence on the parameter sets of such an effective
model, the estimations made with NJL are quite robust [22, 23]. It acts as a
suitable alternative for benchmark estimation at high-density and low-
temperature regions [24], as there is no restriction over the applicability of
this model at finite density.

Over the past few decades, LQCD and NJL have complemented each other while
broadening our understanding of strong interaction in various scenarios. For
example, magnetic catalysis (MC) was first shown within a NJL framework [25,
26]. Two decades later, lattice QCD not only looked at the MC feature [27, 28,
29, 30, 31], but also observed inverse magnetic catalysis (IMC) around the
cross-over temperature [31]. This results in better versions of NJL models
with nonlocal interactions [32] and external agent dependent interaction
strength [33, 34]. Further, in an NJL-like model, the anomalous breaking of
$U(1)_{A}$ symmetry is addressed by explicitly adding the ’t Hooft determinant
interaction (characterized by coupling $G_{d}$), which also represents the
flavor mixing. Recently, Ref. [35, 36] have explored the effect of $G_{d}$ on
isospin-sensitive observables in 2-flavor NJL model and constrained $G_{d}$
using the same from LQCD. In the context of the 3-flavor NJL, $G_{d}$ is the
most ill-constrained parameter with a large allowed range while reproducing
acceptable values of physical observables [22, 23].

In this letter, for the first time in the $2+1$ flavor case with isospin
symmetry, the effect of the $G_{d}$ has been explored by incorporating a
finite strangeness chemical potential. This provides an opportunity to study
the effect of a large $\mu_{S}$ on the pseudo-critical line and provide novel
estimations. Although the large variation of $\mu_{S}$ ($0-200$) MeV is beyond
the scope of the freeze-out lines in heavy-ion collisions owing to strangeness
neutrality, the present investigation is of particular interest for extending
the NJL model at very high density. We have organized the letter as follows,
in Sec. II we describe the model formalism for a $2+1$ flavor NJL model with
the isospin symmetry. We have presented our results in Sec. III and summarized
our findings in Sec. IV.

##  II Formalism

The $2+1$-flavor NJL model Lagrangian is given by

${\cal
L}_{\text{NJL}}=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}-\hat{m}\right)\psi+{\cal
L}_{\text{S}}+{\cal L}_{\text{D}},$ (2)

where the four- and six-point interaction terms are given by

$\begin{split}{\cal
L}_{\text{S}}&=G_{s}\sum_{a=0}^{8}\left[\left(\bar{\psi}\lambda_{a}\psi\right)^{2}+\left(\bar{\psi}\,i\gamma_{5}\lambda_{a}\psi\right)^{2}\right],\\\
{\cal
L}_{\text{D}}&=G_{d}\left[\det\bar{\psi}_{i}(1-\gamma_{5})\psi_{j}+\det\bar{\psi}_{i}(1+\gamma_{5})\psi_{j}\right].\end{split}$
(3)

Here, $\psi^{\text{T}}=(u,d,s)$ is the quark triplet in flavor space with up,
down and strange quark; and $\hat{m}=\text{diag}(m_{u},m_{d},m_{s})$ is the
current quark mass matrix. In the interaction, the $\lambda$’s are the Gell-
Mann matrices, and in ${\cal L}_{\text{D}}$, the determinant is taken in the
flavor space. ${\cal L}_{\text{S}}$ represents the four-quark interaction,
with the coupling strength $G_{s}$, which is symmetric under $U(3)\times U(3)$
symmetry. On the other hand, ${\cal L}_{\text{D}}$, with coupling strength
$G_{d}$, describes the six-quark interactions, known as the ’t Hooft
determinant. ${\cal L}_{\text{D}}$ is included to break the $U(1)_{A}$
symmetry explicitly as $U(1)_{A}$ is anomalous in quantum theory.

To obtain the free energy, it is standard to introduce auxiliary fields using
Hubbard-Stratonovich transformation [37] to make the Lagrangian quadratic in
fermion fields. Within mean-field approximation, we can have nonzero vacuum
expectation values of these auxiliary fields. In the absence of any other
external agents (like a magnetic field, isospin chemical potential, etc.),
symmetry only allows $\bar{\psi}\psi$ channel to acquire nonzero vacuum
expectation values. And the meanfield Lagrangian becomes

${\cal
L}_{\text{MFA}}=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}-\hat{M}\right)\psi-2G_{s}\sum_{i}\sigma_{i}^{2}+4G_{d}\prod_{i}\sigma_{f},$
(4)

where $\hat{M}$ is the constituent mass matrix, and the constituent masses are
given by [22]

$M_{i}=m_{i}-4G_{s}\sigma_{i}+2G_{d}\epsilon_{ijk}\sigma_{j}\sigma_{k},$ (5)

with $\sigma_{i}=\langle\bar{\psi}_{i}\psi_{i}\rangle$ being the condensate
that works as the order parameter of chiral symmetry breaking. As it is
evident from the above equation, $G_{d}$ mixes different flavors.

It is straightforward to integrate out the fermion degrees of freedom from Eq.
(4) to obtain the free energy. To introduce temperature (T) and chemical
potentials ($\mu_{f}$), it is customary to perform the following
transformations [38]

$p_{0}\rightarrow ip_{4}-\mu_{f},\qquad p_{4}=(2n+1)\pi T.$ (6)

With the above transformation, the integration over $p_{0}$ gets replaced by
the sum over Mastubara frequencies, $n$. Moreover, the free energy is given by
[39, 40]

$\Omega=\Omega_{\text{MF}}+\Omega_{\text{Vac}}+\Omega_{\text{Th}},$ (7)

where

$\displaystyle\Omega_{\text{MF}}$ $\displaystyle=$ $\displaystyle
2G_{s}\sum_{i}\sigma_{i}^{2}+4G_{d}\prod_{i}\sigma_{i},$ (8)
$\displaystyle\Omega_{\text{Vac}}$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{i}\int^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\,\varepsilon_{i}(p),$
(9) $\displaystyle\Omega_{\text{Th}}$ $\displaystyle=$
$\displaystyle-2N_{c}T\sum_{i}\int\frac{d^{3}p}{(2\pi)^{3}}\left[\ln{\left(1+e^{-(\varepsilon_{i}(p)-\mu_{i})/T}\right)}\right.$
(10)
$\displaystyle+\left.\ln{\left(1+e^{-(\varepsilon_{i}(p)+\mu_{i})/T}\right)}\right].$

With, $N_{c}=3$ is the number of colors,
$\varepsilon_{i}(p)=\sqrt{\vec{p}^{2}+M_{i}^{2}}$ is the energy of the $i$-th
flavor quark, and $\Lambda$ is the three-momentum cutoff.

To obtain the ground state, one can minimize the free energy, defined in Eqs.
(7), by solving the following gap equations simultaneously

$\displaystyle\frac{\partial\Omega}{\partial\sigma_{u}}=\frac{\partial\Omega}{\partial\sigma_{d}}=\frac{\partial\Omega}{\partial\sigma_{s}}=0.$
(11)

In this study, we have considered the isospin symmetric case; in other words,
the electric charge and associated chemical potential ($\mu_{Q}$) are ignored,
which implies that $\sigma_{u}=\sigma_{d}=\sigma_{l}$. The quark chemical
potential can be written in terms of baryon and strangeness chemical potential

$\begin{split}\mu_{u}&=\mu_{d}=\frac{1}{3}\,\mu_{B},\\\
\mu_{s}&=\frac{1}{3}\,\mu_{B}-\mu_{S}.\end{split}$ (12)

Finally, for a fixed $\mu_{B}$ and $\mu_{S}$, we define the pseudo-critical
temperature ($T_{pc}$) as the inflection temperature where the curvature of
$\sigma_{l}$ changes sign [15]. In the context of LQCD, $T_{pc}$ is generally
determined from the maximum of the chiral susceptibility [15].

|  $\Lambda$ (MeV) |  $G_{s}\Lambda^{2}$ |  $G_{d}\Lambda^{5}$ |  $m_{l}$(MeV) |  $m_{s}$(MeV)
---|---|---|---|---|---
set I | 631.4 | 1.835 | 9.29 | 5.5 | 135.7
set II | 602.3 | 1.835 | 12.36 | 5.5 | 140.7
Table 1: Parameter sets of NJL model used in the present work. Set I and set
II are from Hatsuda and Kunihiro[22] and Rehberg, Klevansky, and Hufner[23]
respectively

Let us note that there are five parameters in this three-flavor NJL model,
namely the current quark mass for the strange and light quarks ($m_{s}$ and
$m_{l}$), two coupling $G_{s}$ and $G_{d}$ and the three-momentum cutoff
$\Lambda$. After choosing the $m_{l}=5.5$ MeV, consistent with chiral
perturbation theory [41], the remaining four parameters are fixed by fitting
the pion decay constant and the masses of pion, kaon, and $\eta^{\prime}$ [22,
23] to their empirical values. With the parametrization of set I, the mass of
$\eta$ meson is underestimated by $11\%$ while for set II, the same is
underestimated by $6\%$. As may be seen from Tab. 1, the dimensionless
coupling $G_{d}\Lambda^{5}$ defer by $30\%$ between the two sets. The value of
$G_{d}$ even differs by about $70\%$. We intend to prescribe here a means of
constraining $G_{d}$. The parameter sets of the present study are given in
Table. 1. With this set of parameters, the mass of the $\eta$ meson is
underestimated by $6\%$ [17] and the constituent masses in the vacuum turn out
to be $M_{l}=368$ MeV and $M_{s}=549$ MeV. Among the widely used parameter
sets [23, 22] (see Tab. 1), $G_{d}\Lambda^{5}$ or $G_{d}$ differ most and in
this work, we intend to provide a prescribe a way to constrain it more
precisely.

##  III Results

For a given value $\mu_{B}$ and $\mu_{S}$, the pseudo-critical temperature
($T_{pc}$) is defined to be the inflection point of light quark condensate
(the order parameter of chiral symmetry breaking) as a function of
temperature. Before proceeding to investigate the effect of finite $\mu_{S}$
on $T-\mu_{B}$ line, it is essential to check the model estimation against the
available lattice QCD results of the curvature coefficients ($\kappa_{2,4}$).
Considering $\mu_{Q}=0$, we have first investigated the $T-\mu_{B}$
($\mu_{S}=0$) and $T-\mu_{S}$ ($\mu_{B}=0$) plane, and find the $\kappa_{2,4}$
by parameterizing the respective pseudo-critical lines with the ansatz of Eq.
(1) for the range $\mu_{B,S}/T_{pc}(0)\leq 1.0$ with $T_{pc}(0)=171.1$ and
$173.4$ MeV for parameter set I and II, respectively.

We have tabulated our estimations for the curvature coefficients $\kappa_{2}$
and $\kappa_{4}$ in Tab. 2 and Tab. 3, respectively. The lattice results are
taken from the HotQCD collaboration [1] and WB collaboration [2] There is
excellent agreement with LQCD estimations for $\kappa_{2}$ in both the
$T-\mu_{B}$ and $T-\mu_{S}$ line. We want to emphasize that the values of
$\kappa_{2}^{B}$ are similar for both the parameter sets, which infers that
the large difference in $G_{d}$ between two parameter sets does not influence
the $T-\mu_{B}$ phase line. On the contrary, for the $\mu_{B}=0$ plane,
$\kappa_{2}^{S}$ is distinctly different for the two parameter sets. Although
these $\kappa_{2}^{S}$ values match with the lattice estimations within the
variances, the estimation with parameter set II has a better agreement with
the mean value. The difference in the $\kappa_{2}^{S}$ is attributed to
$G_{d}$, which brings the influence of strange quark to the light quarks as
pointed out in Eq. 5. This motivates us to examine the effect of $G_{d}$ on
$\kappa_{2}^{B}$ by exploring the $T-\mu_{B}$ line at various values of
$\mu_{S}$.

|  $\kappa_{2}^{B}$ ($\mu_{S}=0$) |  $\kappa_{2}^{S}$($\mu_{B}=0$) | $\kappa_{2}^{B,n_{S}=0}$
---|---|---|---
NJL, set I | 0.01627 | 0.01345 | 0.01478
NJL, set II | 0.01619 | 0.01719 | 0.01350
Lattice QCD | 0.016(6) [1] | 0.017(5) [1] | 0.012(4) [1]
|  |  | 0.0153(18) [7]
Table 2: Estimations of $\kappa_{2}$ for the two parameter sets. The Lattice
QCD results are taken from Ref. [1, 7] (a)  Figure 1: Phase line in
$T-\mu_{B}$ plane for different $\mu_{S}$, evaluated with parameter set I. For
representative purposes, axes are scaled with respective $T_{pc}(\mu_{B}=0)$.
The continuous lines represent our estimations for various values of
$\mu_{S}$. The data and band are from the HotQCD [1]. The dashed line
corresponds to the strangeness neutral case.

To appreciate the effects arising from the strange sector, we have restricted
this study to smaller values of baryon chemical potential (up to
$\mu_{B}/T_{c}(0)\leq 1.0$) and varied the $\mu_{S}$ from 0 to 200 MeV. We
have restricted the $\mu_{S}$ within half the kaon mass to exclude the
possibility of kaon condensation [42]. Due to the difference in the magnitude
of the $T_{pc}$ between NJL and Lattice studies, we have scaled the results
with their respective $T_{pc}(\mu_{B}=0)$ as shown in Fig. 1. As may be
observed, for finite values of $\mu_{S}$, the $T_{pc}$ initially increases
with $\mu_{B}$ and then decreases. For smaller values of $\mu_{B}$, a finite
$\mu_{S}$ decreases the thermal weight in the strange sector ($\mu_{S}$ comes
with a negative sign in the strange thermal distribution, see Eq. (12)) and
therefore leads to a higher value of $T_{pc}/T_{pc}(0)$ for the same $\mu_{B}$
as shown in Fig. 1. As $\mu_{B}$ increases further, this rise in $T_{pc}$ gets
saturated and eventually starts decreasing.

For the first time, such a prominent increase in the pseudo-critical
temperature ($T_{pc}$) along the $T-\mu_{B}$ line is observed, which arises
due to a finite strangeness chemical potential. This trend was not observed in
earlier studies within LQCD [1, 43] and HRG [10], as most of them were
performed along the $\mu_{S}=0$ line or along the freeze-out line, where the
strangeness neutrality sets up the limit of $\mu_{S}\leq\mu_{B}/3$ [43].

(a)  Figure 2: $\kappa_{2}^{B}$ as a function of $\mu_{S}$ in MeV. The red
(dashed) and green (dashed dot) lines are the central values for the set I and
II, respectively. The blue and cyan bands are associated with $\pm 10\%$
change in $G_{d}$ and $\pm 5\%$ change in $G_{s}$ respectively for parameter
set I.

To quantify the increase in the $T_{pc}$ with $\mu_{B}$ for a given value of
$\mu_{S}$, we have used the ansatz of Eq. (1) to extract the curvature
coefficients. We have presented the variation of $\kappa_{2}^{B}$ with
$\mu_{S}$ in Fig. 2 for both the parameter sets. The curvature coefficient
$\kappa_{2}^{B}$ starts from a positive value for $\mu_{S}=0$ and decreases as
we increase the strangeness chemical potential. We wish to emphasize that with
$\mu_{S}$, $\kappa_{2}^{B}$ decreases from its positive value at $\mu_{S}=0$
and eventually becomes negative at some $\mu_{S}=\mu_{S}^{c}$. This negative
sign of $\kappa_{2}^{B}$ is one of the novel results of the present
investigation. This was earlier not observed in the context of pseudo-critical
line [1, 7]. One important observation is that the $\mu_{S}^{\text{c}}$ are
distinctively different for the two parameter sets. $G_{s}\Lambda^{2}$ being
the same for both the sets, this difference in $\mu_{S}^{c}$ is essentially
due to the variance in $G_{d}$. A large $G_{d}$ provides a stronger influence
of the strange quark sector on the light quarks, resulting in a faster
decrease in $\kappa_{2}^{B}$.

At this juncture, it is instructive to check $\kappa_{2}^{B}$ along the
strangeness neutrality line ($n_{S}=0$). A finite $\mu_{B}$ requires the
strangeness chemical potential $\mu_{S}\neq 0$ to achieve zero net
strangeness. This corresponds to $\mu_{S}=\mu_{B}/3$ as we are considering
$\mu_{Q}=0$ and there is no vector interaction in the present model. We have
found $\kappa_{2}^{B,n_{S}=0}<\kappa_{2}^{B,\mu_{S}=0}$ as listed in Table.2.
The decrease of $\kappa_{2}^{B}$ for the strangeness neutral case is
commensurate with the lattice estimations [1, 43] and in accordance with our
findings of the reduction of $\kappa_{2}^{B}$ with $\mu_{S}$. We would like to
comment here that while the behaviour of the $\kappa_{2}^{B}$ ($\mu_{S}\neq
0$) is similar to the lattice QCD calculations. The lattice estimations of
$n_{S}=0$ correspond to values of $\mu_{S}$ which are not large enough to
constrain the flavor mixing determinant coupling. This necessitates LQCD
simulations at a larger value of $\mu_{S}$.

It would be interesting to check the robustness of this negative
$\kappa_{2}^{B}$ on the parametrization of the NJL model itself. For this
purpose, we have varied $G_{s}$, $G_{d}$ by $5\%$ and $10\%$, respectively and
examined the effect on $\kappa_{2}^{B}$ variation as shown in Fig. 2. As
discussed earlier, a larger value for $G_{d}$ increases the coupling between
the light and strange sector resulting in a faster decrease of
$\kappa_{2}^{B}$. It is needless to say that $\kappa_{2}^{B}$ becomes
independent of $\mu_{S}$ at $G_{d}=0$ as the strange and light quark sector
decouple which is evident in the Lagrangian of the NJL model. On the contrary,
the variation of $G_{s}$ has a weaker effect on the features mentioned above.

|  $\kappa_{4}^{B}$ ($\mu_{S}=0$) |  $\kappa_{4}^{S}$ ($\mu_{B}=0$) | $\kappa_{4}^{B,n_{S}=0}$
---|---|---|---
NJL, set I | 0.00006 | 0.001477 | 0.000081
NJL, set II | 0.00005 | 0.001892 | 0.000742
Lattice QCD | 0.001(7) [1] | 0.004(6) [1] | 0.000(4) [1]
|  |  | 0.00032(67) [7]
Table 3: Values of $\kappa_{4}$ for the pseudo-critical line for different
cases.  (a)  Figure 3: $\kappa_{4}^{B}$ as a function of $\mu_{S}$ in MeV.
Color code is same as Fig.2.

Within LQCD, the numerical value of $\kappa_{4}^{B}$ is consistent with zero
[1, 7], as for the small value of $\mu_{B,S}/T$, the $4^{th}$ order
coefficients of $\mu_{X}/T$ expansion is prone to have a weaker effect on the
$T-\mu_{X}$ line. In the present study, we have found $\kappa_{4}$ to have
good agreement for the case (${\mu_{B}\neq 0,\ \mu_{S}=0}$) and (${\mu_{B}=0,\
\mu_{S}\neq 0}$) as shown in Table 3. It would be essential to investigate the
same for the $T-\mu_{B}$ line at various $\mu_{S}$. For larger values of
$\mu_{S}$, we have found the $\kappa_{4}$ to be finite (as in Fig.3), even
with the different parameter sets, as mentioned earlier. These findings
suggest that even within the small $\mu_{B}$ range, a non-zero $\kappa_{4}$ is
possible by switching on a finite strangeness-chemical potential $\mu_{S}$,
which is relevant in the context of lattice simulations.

##  IV Summary and conclusion

In this letter, we explore the chiral phase boundary of the QCD matter within
a $2+1$ flavor Nambu–Jona-Lasinio model with special emphasis on the effect of
strangeness on the curvature coefficients $\kappa_{2}^{B}$ and
$\kappa_{4}^{B}$. To our knowledge, this is the first such exploration within
a $2+1$ NJL model. We have considered the isospin symmetric case and
$\mu_{Q}=0$. To have better control over the lowest-order coefficients
($\kappa_{2}^{X}$), we have limited the study within the range $\mu_{X}/T\leq
1$. As benchmark, we have first estimated the $\kappa_{2,4}^{X}$ for three
separate cases, 1) $T-\mu_{B}$ plane ($\mu_{S}=0$) i.e $\kappa_{2,4}^{B}$ 2)
$T-\mu_{S}$ plane ($\mu_{B}=0$) i.e $\kappa_{2,4}^{S}$ and 3) along the
strangeness neutrality line $\kappa_{2,4}^{B,n_{S}=0}$. We have used two
standard sets of parametrization of $2+1$ NJL model that differ significantly
regarding the flavor mixing determinant interaction. Although we have an
excellent agreement of $\kappa_{2}^{B}$ with the available LQCD finding for
both the parameter sets, we observed that $\kappa_{2}^{S}$ has a strong
dependence on the flavor mixing and $U(1)_{A}$ breaking ’t Hooft interaction.
Between the two parameter sets used, the set II with a higher value of $G_{d}$
reproduces lattice estimation of $\kappa_{2}^{S}$ better. To explore the
effects of flavor mixing through $G_{d}$, it is instructive to study the
$\mu_{S}$ dependence of the $T-\mu_{B}$ lines, which have been quantified by
estimating $\kappa_{2}^{B}$ as a function of $\mu_{S}$.

It is observed that within the model $\kappa_{2}^{B}$ decreases with $\mu_{S}$
and becomes negative for sufficiently large values of $\mu_{S}$. Further, it
is observed that the value of $\mu_{S}$ where $\kappa_{2}^{B}$ vanishes are
different for two parameter sets. This difference is attributed to the fact
that a larger value of $G_{d}$ strengthens the strange contribution to the
light sector, resulting in a faster decrease. This novel observation provides
a direct measure of the flavor mixing through ‘t Hooft interaction. We expect
that outcomes from LQCD investigations for $\kappa_{2}^{B}$ at large enough
$\mu_{S}$ will assist in better constraining the ‘t Hooft coupling $G_{d}$,
thereby enhancing our understanding of effective models like NJL and QCD.

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