#  Infrared fixed point of the SU(3) gauge theory with $N_{f}=10$ flavors

Anna Hasenfratz  Department of Physics, University of Colorado, Boulder, CO
80309, USA  Ethan T. Neil  Department of Physics, University of Colorado,
Boulder, CO 80309, USA  Yigal Shamir  Raymond and Beverly Sackler School of
Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel  Benjamin
Svetitsky  Raymond and Beverly Sackler School of Physics and Astronomy, Tel
Aviv University, 69978 Tel Aviv, Israel  Oliver Witzel  Center for Particle
Physics Siegen, Theoretische Physik 1, Naturwissenschaftlich-Technische
Fakultät, Universität Siegen, 57068 Siegen, Germany

###### Abstract

We use lattice simulations and the continuous renormalization-group method,
based on the gradient flow, to calculate the $\beta$ function and anomalous
dimensions of the SU(3) gauge theory with $N_{f}=10$ flavors of fermions in
the fundamental representation. We employ several improvements to extend the
range of available renormalized couplings, including the addition of heavy
Pauli-Villars bosons to reduce cutoff effects and the combination of a range
of gradient flow transformations. While in the weak coupling regime our result
is consistent with those of earlier studies, our techniques allow us to study
the system at much stronger couplings than previously possible. We find that
the renormalization group $\beta$ function develops a zero, corresponding to
an infrared-stable fixed point, at gradient-flow coupling $g^{2}=15.0(5)$. We
also determine the mass and tensor anomalous dimensions: At the fixed point we
find $\gamma_{m}\simeq 0.6$, suggesting that this system might be deep inside
the conformal window.

††preprint: SI-HEP-2023-14

##  I  Introduction

The SU(3) gauge theory with ten Dirac fermions in the fundamental
representation is the subject of continuing debate. The question is whether
its infrared physics is confining or conformal, as determined by the absence
or presence of an infrared fixed point (IRFP). The system has been studied by
several groups, both with domain wall [1, 2, 3, 4, 5] and staggered fermions
[6, 7, 8]. All these studies have used the finite-volume gradient flow (GF)
scheme with a step-scaling renormalization-group transformation [9, 10, 11].
While the results are in reasonable agreement at weak gauge couplings, they
differ at stronger couplings and reach differing conclusions.

> Figure 1:  The $\beta$ function obtained with four different gradient-flow
> transformations in overlapping regions. The orange and dark green solid
> lines, with errors indicated by the dotted lines, are results from staggered
> and DWF simulations [6, 7, 8, 4, 5, 12]. Black solid, dotted and dashed
> curves correspond to the universal 1- and 2-loop and the gradient flow
> 3-loop perturbative results [13].

Using domain-wall fermions, Chiu [1, 2] first claimed an IRFP at $g^{2}\simeq
7$. His later study [3], however, gave a more cautious assessment. Hasenfratz,
Rebbi, and Witzel [4, 5] observed a step-scaling $\beta$ function that
increases in absolute value up to $g^{2}\simeq 9$, where it appears to turn
towards the abscissa and thus hints at an IRFP at some $g^{2}\gtrsim{11}$;
these simulations were limited by a first-order phase transition blocking
access to the $g^{2}>11$ regime. Staggered-fermion calculations by the LatHC
collaboration [6, 7, 8] studied this system in larger volumes, reaching
couplings up to $g^{2}\simeq 10$. In this range their $\beta$ function
increases steadily in magnitude and remains in $2\sigma$ agreement with the
result reported in Refs. [4, 5]. It does not, however, show any sign of a
developing IRFP. No definitive conclusion on the infrared behavior of the
$N_{f}=10$ model has been reached so far.

All the studies listed above were carried out in a range of renormalized
coupling limited by large cutoff effects. Recently we have proposed adding
heavy Pauli-Villars (PV) bosons to remove ultraviolet fluctuations caused by
the many fermion fields [14]. The masses of the PV bosons are kept at the
cutoff scale. Thus they decouple in the continuum limit but they do generate a
local effective gauge action with well-regularized short-distance properties.
We have applied PV improvement successfully in the SU(3) gauge theory with
$N_{f}=12$ [14] and $N_{f}=8$ [15] staggered fermions in the fundamental
representation, as well as in a multirepresentation SU(4) gauge theory with
Wilson fermions [16]. In all cases we found that the PV improved actions
indeed reduced short-distance fluctuations and allowed investigations at
stronger renormalized couplings.

In Ref. [16] we applied the continuous $\beta$ function (CBF) method [17, 18]
and uncovered an IRFP at strong coupling. Going beyond the use of PV bosons,
we further extended the coupling range by combining the results of a number of
lattice gradient flows that possess a common continuum limit. Here we apply
the techniques used in [16] to the SU(3) gauge theory with $N_{f}=10$
fundamental flavors, simulated using Wilson fermions. The presence of heavy PV
bosons permits study of the system at much stronger renormalized couplings
than previously possible. When the CBF method is extended by varying the GF
transformation, the accessible range of renormalized coupling is yet larger.
We summarize our main result—the $\beta$ function of the theory—in Fig. 1. By
combining overlapping results from different gradient flows, we cover the
range $6.5\lesssim g^{2}\lesssim 23$. At weak coupling our prediction is
consistent with that of LatHC [7, 8] and overlaps with an CBF result [12]
obtained by reanalyzing the Möbius DWF data from Refs. [4, 5] (labeled “DWF
Wil” in the figure). Our $\beta$ function turns around at $g^{2}\simeq 10.0$,
confirming the hint of an IRFP reported in [5]. At stronger coupling, the
$\beta$ function rises steadily and crosses zero at an IRFP at
$g_{\text{FP}}^{2}=15.0(5)$, implying the theory is infrared conformal.

This paper is organized as follows. In Sec. II we briefly describe the
numerical simulations and define the gradient flow transformations. In the
following sections we show how we obtain our results and extrapolate them to
the infinite-volume and continuum limits. For the $\beta$ function see Sec.
III and for anomalous dimensions of fermion bilinears see Sec. IV. With the
exception of the volume extrapolation, the techniques used have been explained
at length in our recent paper [16] and hence our presentation here is brief.
We discuss our results further in Sec. V.

##  II  Simulation and gradient flow

Our data emerge from four-dimensional configurations of the Euclidean gauge
theory, generated by hybrid Monte Carlo simulations of the lattice theory. Our
lattice action couples the gauge field to Wilson-clover fermions after
smearing the links with normalized hypercubic (nHYP) smearing [19, 20]. The
clover coefficient is $c_{SW}=1$ [21, 22] and the smearing parameters
$\alpha_{i}$ are the original set (0.75, 0.6, 0.3). The plaquette gauge action
is supplemented by a term for nHYP dislocation suppression (NDS) [23]. To
further tame gauge field roughness, we add 30 Pauli-Villars (bosonic Dirac)
fields—three for each fermion flavor—with bare mass $am_{PV}=1$ [14]. For each
lattice coupling $\beta_{0}$ we set the hopping parameter
$\kappa\simeq\kappa_{c}$ so that the fermion mass, calculated from the axial
Ward identity, is bounded by $|am_{f}|<5\times 10^{-4}$. The ensembles are
listed in Table 1.

$\beta_{0}$ | $\kappa$ | $L/a=24$ | $L/a=28$
---|---|---|---
6.0 | 0.12742 | 135 | 100
6.2 | 0.12695 | 100 | 100
6.3 | 0.12677 | 150 | 100
6.5 | 0.1265 | 155 | 100
6.7 | 0.1263 | 135 |  090
7.0 | 0.12612 | 160 | 110
Table 1:  List of the ensembles, showing the lattice coupling constant
$\beta_{0}$, the hopping parameter $\kappa$, and the number of configurations
with volume $L^{3}\times(2L)$ for the two volumes.

We extract the $\beta$ function and anomalous dimensions using a continuous RG
transformation based on gradient flow [24, 25, 17, 18, 16, 26, 27, 25, 8]. The
RG length scale is given by $\sqrt{t}$ where $t$ is the GF time [9]. In
continuum language, the GF running coupling at scale $\sqrt{t}$ is defined as

$g^{2}=\frac{{\cal N}}{(1+\delta)}t^{2}\left\langle E(t)\right\rangle\ ,$ (1)

with the energy density $E=\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}$,
calculated from the flowed gauge field strength $G_{\mu\nu}^{a}$. The constant
${\cal N}=128\pi^{2}/(3(N_{c}^{2}-1))$ is a normalization factor to match to
the 1-loop $\overline{\text{MS}}$ result, while $\delta$ is a small finite-
volume correction [11]. The average in Eq. (1) is over the ensemble of
configurations generated at given lattice coupling $\beta_{0}$. The $\beta$
function is simply

$\beta(g^{2})=-t\,\frac{\partial g^{2}}{\partial t}\ .$ (2)

The anomalous dimensions (see below) are determined from the derivative of
flowed correlation functions [24, 28, 16].

In the lattice theory, the GF transformation can be performed with different
flows, each originating in a particular discretization of the gauge action. We
make use of four such lattice flows, corresponding to combinations of
plaquette and rectangle terms with coefficients $c_{p}$ and $c_{r}$. Imposing
the perturbative normalization $c_{p}+8c_{r}=1$, we define the Symanzik flow
($c_{p}=5/3$), the Wilson flow ($c_{p}=1$), and flows called C23 ($c_{p}=2/3$)
and C13 ($c_{p}=1/3$) [16]. The different flows correspond to different
renormalized trajectories (RT) of the RG transformation. Ideally, we would
choose the RG transformation whose RT is closest to the simulation action. We
find that GF with smaller $c_{p}$ values are better for this purpose at
stronger gauge coupling.

Besides this, the energy operator $E(t)$ can be defined by a variety of
discretizations. We calculate the W (Wilson), S (Symanzik), and C (clover)
operators in order to gauge the approach of each to the continuum limit. The S
operator gives the smoothest approach, and so we will focus on results
obtained using this operator.

##  III  The $\beta$ function

Raw data for the flowed coupling $g^{2}$ and its derivative $\beta(g^{2})$
from Wilson and C13 flows, are shown for all ensembles in Fig. 2. One sees an
upward flow with increasing $t$, ever faster for ensembles with larger
coupling $g^{2}$. The C13 flow has the shortest paths as $t$ grows, indicating
that its RT is closest. Moreover, while both flows show that the $\beta$
function approaches the axis for large $g^{2}$, the C13 flow actually shows a
positive $\beta$ function at the strongest coupling, even before taking the
continuum limit. This is the first hint of an IRFP. Finally, the figure allows
comparison of lattice sizes $L^{3}\times(2L)$ for $L/a=24$ and $L/a=28$,
whence it is seen that differences are very small but not zero.

We determine the continuous $\beta$ function via the following four steps,
ultimately carrying out the limits $L\to\infty$ and $a\to 0$ [18, 27].

###  III.1  Infinite volume limit

We take the infinite volume limit, for both $g^{2}$ and $\beta(g^{2})$, at
each bare coupling $\beta_{0}$ and at selected flow times. Simple scaling
arguments imply that the leading volume dependence of the renormalized GF
coupling is $\propto t^{2}/L^{4}$. Higher order corrections can hence be
suppressed by choosing $t/L^{2}$ sufficiently small. In this study we consider
two volumes, $L/a=24$ and 28, and restrict the flow time to $2.8\leq
t/a^{2}\leq 3.8$. With this restriction the finite volume effects are well
controlled in all ensembles. In Fig. 3 we plot $\beta(g^{4})/g^{4}$, derived
from Wilson flow at fixed flow time $t/a^{2}=3.8$, at all six bare coupling
values (see Table 1) and both volumes. The plot also shows the infinite volume
extrapolation, assuming the leading order volume dependence is $1/L^{4}$. The
colored bands show quadratic interpolations of the data, which furnish values
of $\beta(g^{2})$ at intermediate values of $g^{2}$ (see below). In the next
two steps we use only the infinite-volume extrapolations.

Figure 2:  Raw data for the flowed coupling $g^{2}$ and its derivative
$\beta(g^{2})$, calculated from the Symanzik operator. The upper panel shows
results for the Wilson flow; the lower, for C13 flow. Each ensemble is
represented by a group of data points: From right to left: $\beta_{0}=6.0$,
6.2, 6.3, 6.5, 6.7, 7.0. In each group, the ensemble average at $t=2.8$ is
plotted as a circle, from which rises the sequence of averages calculated at
increasing $t$ for $2.8<t/a^{2}\leq 3.8$, rising as $t$ grows. The lines at
each point give a sense for the magnitude and direction of the continuum
extrapolation $a^{2}/t\rightarrow 0$. The colors allow comparison of lattice
size $L=24a$ (red) to $L=28a$ (blue).

> Figure 3:  Comparing the interpolations of $L=24a$ (red) and $L=28a$ (blue),
> derived with Wilson flow, S operator at flow time $t/a^{2}=3.8$ (the largest
> flow time we consider). The green symbols show the extrapolation to infinite
> volume. The colored bands correspond to quadratic interpolations as
> explained in Sec. III.2.

###  III.2 Interpolation

In order to take the the continuum limit of $\beta(g^{2};t/a^{2})$, we need to
determine pairs of ($g^{2}$, $\beta(g^{2})$) at selected flow times for values
of $g^{2}$ between those that emerge from the ensembles. As in the example of
Fig. 3, we interpolate $\beta(g^{2};t/a^{2})$ with a quadratic form according
to $\beta(g^{2};t/a^{2})/g^{4}=c_{0}+c_{1}g^{2}+c_{2}g^{4}$ at a series of
flow times. (See Ref. [16] for details.) We do the same for the other
flows.111The C13 flow shows strong cutoff effects at the weakest bare gauge
coupling $\beta_{0}=7.0$, and hence we do not include that value in the
interpolation. Similar phenomena were observed and discussed in Ref. [16]. As
usual in interpolating data, we will not use the interpolating curves outside
the range of the interpolation when taking the continuum limit. The columns
labelled $g^{2}_{\text{min}}$ and $g^{2}_{\text{max1}}$ in Table 2 list the
minimal and maximal values covered by the interpolating curves.

###  III.3  Continuum limit

After determining interpolating curves as in Fig. 3 for many values of
$t/a^{2}$ in the interval $[2.8,3.8]$, we extrapolate $\beta(g^{2};t/a^{2})$
to $t/a^{2}=\infty$ at fixed $g^{2}$. One such extrapolation is shown in Fig.
4. Repeating this in a range of $g^{2}$ gives the $\beta$ function that is
plotted in Fig. 1. We show curves and error bands for the results of Symanzik,
Wilson, C23, and C13 flows. These must agree in the continuum limit. Figure 4,
in particular, shows the excellent agreement of Wilson and C13 flows at
$g^{2}=15.0$, and the fact that $\beta(g^{2})$ is consistent with zero there.

It can be seen in Fig. 1 that each flow is plotted in a restricted range of
$g^{2}$. This stems from a requirement of internal consistency, as follows.

Figure 4:  Continuum extrapolation $a^{2}/t\to 0$ of the S operator for Wilson
and C13 flows at $g^{2}=15.0$ in the infinite-volume limit. The solid symbols
correspond to $2.8\leq t/a^{2}\leq 3.8$, the range of flow time used in the
extrapolations. The curvature observed at small $t$ (large $a^{2}/t$)
indicates that the flow has not yet reached the renormalized trajectory.

###  III.4  Consistency tests

At any given physical coupling $g^{2}$, the $\beta$ functions based on
different discretizations of the flowed energy density—the S, W, and C
operators—must agree in the continuum limit. For each flow, we use this
requirement on the operators to impose cuts on the range of $g^{2}$ where that
flow can be trusted.

We return to the continuum extrapolation of Fig. 4, but we step back to
consider only a single volume (cf. Fig. 3) where we compare the results
stemming from the three operators, S, W, and C. We show an example of these
continuum extrapolations in Fig. 5.

Figure 5:  Continuum limits $a^{2}/t\to 0$ of data from the C23 flow on the
single volume $24^{3}\times 48$ (cf. Fig. 4). We show the extrapolation of
data from S, W, and C operators. We use the S and W extrapolations to test for
consistency. The curvature observed in the data at large flow times (small
$a^{2}/t$) indicates finite volume effects, absent from the $L\to\infty$ limit
shown in Fig. 4.

For all flows and at every $g^{2}$ we observe that the S operator’s
extrapolation has the smallest slope, meaning the smallest cutoff effects,
with the W operator coming next. The C operator is furthest from the continuum
limit, having the largest slopes. This naturally leads to the choice of the S
operator for our main result, but we require consistency between the S and W
operators. We label the extrapolated values of the S and W data at each value
of $g^{2}$ as $\beta_{\text{S}}$ and $\beta_{\text{W}}$.

We base the criterion for consistency on plots like the one shown in Fig. 6,
which is obtained from a bootstrap analysis and includes correlations among
the different operators. The green band is the $\pm 1\sigma$ error band of the
difference $\beta_{\text{S}}-\beta_{\text{W}}$. The solid red curves represent
$\pm\sqrt{2}\sigma_{\text{S}}$, where $\sigma_{\text{S}}$ is the error in
$\beta_{\text{S}}$, while the dashed red curves are $\pm 2\sigma_{\text{S}}$.
The green band is much narrower than the span between the red curves; this
reflects the strong correlations between the operators. For a consistency test
we require $|\beta_{\text{S}}-\beta_{\text{W}}|\lesssim 2\sigma_{S}$ on $L=24$
and 28 volumes separately and simultaneously. This requirement restricts the
$g^{2}$ values where a given flow can be trusted. Table 2 lists the
corresponding values for each flow. The demand that
$|\beta_{\text{S}}-\beta_{\text{W}}|\lesssim 2\sigma_{S}$ gives us the bounds
for each flow that are listed in columns 2 and 4 in the table and shown in
Fig. 1. A slightly stricter constraint is obtained by requiring
$|\beta_{\text{S}}-\beta_{\text{W}}|\lesssim\sqrt{2}\sigma_{S}$. This lowers
some of the upper bounds, as shown in the table (see column 5) and in Fig. 7.
In this case the Wilson flow no longer reaches the fixed point, but there is
little effect on the C23 and C13 flows, which do.

Figure 6:  Comparing the difference $\beta_{S}-\beta_{W}$ (green) with the standard deviation $\sigma$ of the S operator on volume $24^{3}\times 48$ for the C23 flow. We plot $\pm\sqrt{2}\sigma$ (solid red curves) and $\pm 2\sigma$ (dashed red curves). Together with a similar plot for volume $28^{3}\times 56$, these determine the bounds of validity $g^{2}_{\text{max3}}$ and $g^{2}_{\text{max2}}$ in Table 2.  Flow | $g^{2}_{\text{min}}$ | $g^{2}_{\text{max1}}$ | $g^{2}_{\text{max2}}$ | $g^{2}_{\text{max3}}$
---|---|---|---|---
Sym | 6.5 | 18.0 | 8.5 | 7.5
Wil | 7.0 | 20.0 | 16.0 | 10.0
C23 | 8.0 | 21.0 | 17.0 | 16.0
C13 | 12.0 | 23.0 | 23.0 | 21.0
Table 2:  Ranges of $g^{2}$ in which each flow is included in the final result
for $\beta(g^{2})$. $g^{2}_{\text{min}}$ and $g^{2}_{\text{max1}}$ result from
the interpolations, while $g^{2}_{\text{max2}}$ and $g^{2}_{\text{max3}}$ come
from further demanding consistency between the continuum extrapolations
$\beta_{\text{S}}$ and $\beta_{\text{W}}$ (see Fig. 6). We quote all numbers
with a resolution of 0.5. $g^{2}_{\text{min}}$ and $g^{2}_{\text{max2}}$ give
the ranges reflected in Fig. 1. $g^{2}_{\text{max3}}$, rather than
$g^{2}_{\text{max2}}$, gives the stricter bounds shown in Fig. 7.  Figure 7:
Same as Fig. 1, but with stricter bounds on the domain of validity of each
flow. See Table 2.

##  IV  Anomalous dimensions

The calculation of anomalous dimensions follows that of Ref. [16] closely,
with the addition of an extrapolation to infinite volume as described in Sec.
III.1 above. We calculated the two-point function of each flowed mesonic
density $X^{\prime}$ with its unflowed source $X$,

$\left\langle X(0)\,X^{\prime}(t)\right\rangle\sim t^{-(d+\eta+\gamma)/2}\ .$
(3)

Here $\gamma$ is the desired anomalous dimension of the operator and $\eta/2$
is the anomalous dimension of the the elementary fermion field. To eliminate
$\eta$, we divide $\left\langle X(0)\,X^{\prime}(t)\right\rangle$ by the two-
point function of the conserved vector current. Defining the ratio

$R(t)=\frac{\left\langle X(0)\,X^{\prime}(t)\right\rangle}{\left\langle
V(0)\,V^{\prime}(t)\right\rangle}\ ,$ (4)

we have

$R(t)\sim t^{-\gamma/2}\ ,$ (5)

and hence $\gamma$ can be extracted from the logarithmic derivative,

$\gamma=-2\frac{t}{R}\,\frac{\partial R}{\partial t}\ .$ (6)

We require $\sqrt{8t}\ll x_{4}$, where $x_{4}$ is the separation of $X$ and
$X^{\prime}$ in Euclidean time. This means that $x_{4}$ is kept large compared
to the smearing of the operators by the flow. The extrapolation from $L/a=24$,
28 to $L=\infty$, the interpolation in $g^{2}$ at fixed $t$, and the continuum
extrapolation $t/a^{2}\to\infty$ are as described above and in Ref. [16].

Final results for the mass anomalous dimension and for that of the tensor
density are shown in Fig. 8.

Figure 8:  The anomalous dimension of the mass (scalar) operator,
$\gamma_{m}$, and that of the tensor operator, $\gamma_{T}$, obtained with
Wilson and C13 flows, extrapolated to the continuum limit and to infinite
volume.

In the weak-coupling region, the anomalous dimensions agree with one-loop
perturbation theory,

$\gamma_{m}=\frac{6g^{2}C_{2}}{16\pi^{2}}\
,\quad\gamma_{T}=-\frac{1}{3}\gamma_{m}\ ,$ (7)

where $C_{2}=4/3$ is the quadratic Casimir operator of the fermion
representation. At larger couplings, the magnitudes of both $\gamma_{m}$ and
$\gamma_{T}$ drop below the respective one-loop results. At the IR fixed point
$g^{2}\simeq 15$ we find $\gamma_{m}\simeq 0.6$ and $\gamma_{T}\simeq-0.2$.

The LSD collaboration [29, 30] calculated $\gamma_{m}$ in the 4+6 mass-split
system using hyperscaling relations and reported $\gamma_{m}=0.47(5)$. That
work performed simulations near $g^{2}\simeq 10$. Accounting for the
dependence of $\gamma_{m}$ on $g^{2}$ (cf. Fig. 8), the two values are
consistent.

##  V  Conclusions

We have presented a calculation of the $\beta$ function of the SU(3) gauge
theory with $N_{f}=10$. Our result is represented by a consistent set of
curves that derive from gradient-flow transformations with overlapping regions
of validity, giving overall a smooth graph of the $\beta$ function. Our work
makes contact with existing results at weak coupling and reaches much larger
couplings than previously attainable.

We obtain strong evidence for an infrared-stable fixed point at $g^{2}\simeq
15$, whose location is consistently identified by three different flows. This
places the theory inside the conformal window for SU(3) gauge theories with
fermions in the fundamental representation. We remind the reader that the
Banks-Zaks fixed point[31], based on the two-loop $\beta$ function, places the
sill of the window at $N_{f}=8.05$.

We have similarly calculated the anomalous dimensions $\gamma_{m}$ and
$\gamma_{T}$. As shown above, the mass anomalous dimension has the value
$\gamma_{m}\simeq 0.6$ at the fixed point, in good agreement with the value of
0.615 obtained by Ryttov and Shrock using their scheme-independent series
expansion to fourth order [32, 33]. Likewise, there is good agreement for the
tensor anomalous dimension, for which they obtained the value of $-0.149$ at
third order [34].

It is generally expected [35] that $\gamma_{m}\to 1$ at the sill of the
conformal window. Our result suggests that the $N_{f}=10$ theory is well above
the sill.

###### Acknowledgements.

We thank R. Shrock for correspondence about Refs. [32, 34, 33]. Computations
for this work were carried out on facilities of the USQCD Collaboration, which
are funded by the Office of Science of the U.S. Department of Energy. A.H. and
E.N. acknowledge support by DOE grant DE-SC0010005. The work of B.S. and Y.S.
was supported by the Israel Science Foundation under grant No. 1429/21. Our
simulation code is a derivative of the MILC code [36].

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# C14 Automatic Imaging Telescope Photometry of GJ 1214

Gregory W. Henry Center of Excellence in Information Systems, Tennessee State
University, Nashville, TN 37209 USA  Jacob L. Bean Department of Astronomy &
Astrophysics, University of Chicago, Chicago, IL 60637, USA

###### Abstract

GJ 1214b is the highest signal-to-noise sub-Neptune for atmospheric studies.
Although most previous transmission spectroscopy measurements have revealed a
frustratingly featureless spectrum, JWST observations are expected to give new
insights to this benchmark planet. We have performed photometric monitoring of
GJ 1214 (the host star) to provide context for these observations. We find
that GJ 1214 entered a period of relatively high brightness during 2021 and
2022. This implies that the JWST MIRI/LRS phase curve observation of GJ 1214b
in July 2022 was obtained during an epoch of low activity for the spot-
dominated host star. Like previous works, we are unable to definitively
identify the star’s rotation period. Nevertheless, we confirm that it is
likely $\gtrsim$50 days.

Planet hosting stars (1242) — Stellar rotation (1629)

##  1 Introduction

The transiting exoplanet GJ 1214b was discovered by Charbonneau et al. (2009)
with the MEarth Project array of eight 0.40 m automated telescopes designed to
monitor a large number of nearby M dwarfs for transiting exoplanets. They
found GJ 1214b to have a planetary mass of 6.55 M⊕, a radius of 2.68 R⊕, and
an orbital period of 1.58 days. Originally classified as a super-Earth,
consideration of GJ 1214b in light of the Kepler planet demographics (Fulton
et al., 2017; Van Eylen et al., 2018) suggests it is better thought of as a
sub-Neptune (Bean et al., 2021a). Its low density of 1.9 g/cc implies the
presence of a substantial atmosphere (Rogers & Seager, 2010). There has been
extensive effort to detect this atmosphere using transmission spectroscopy
(e.g., Bean et al., 2010; Croll et al., 2011; Bean et al., 2011; Désert et
al., 2011; Berta et al., 2012; Fraine et al., 2013; Kreidberg et al., 2014;
Kasper et al., 2020; Orell-Miquel et al., 2022; Spake et al., 2022), with the
consensus being that the planet has a featureless spectrum due to high-
altitude aerosols.

The 2009 MEarth photometry found the star to vary in brightness by 2% on a
timescale of several weeks (with a dominant period of 83 days). Charbonneau et
al. (2009) concluded that starspots carried around the star by its rotation
was the most likely explanation. Carter et al. (2011) and Kreidberg et al.
(2014) observed 31 transits between 2009 and 2013 and found that four transits
exhibited brightness anomalies as the planet occulted a starspot, confirming
the presence of dark spots as the cause of the star’s brightness variability.

Subsequent studies have confirmed the low-amplitude stellar variability but
have not been very successful at pinning down the true stellar rotation
period. Berta et al. (2011) analyzed new 2010 MEarth photometry with better
sampling and cadence than the 2009 MEarth discovery observations and found a
best period of 53 days. However, they cautioned that if GJ 1214 has well-
spaced active longitudes, its true rotation period may be a higher multiple of
53 days (e.g., $\approx$100 days).

Narita et al. (2013) monitored GJ 1214 for stellar variability over the
relatively short timespan of 78 days in 2012 with the MITSuMe 0.50 m telescope
in Japan and found a shorter period of 44.3 days. Additional photometric
monitoring in 2012 and 2013 was reported by Nascimbeni et al. (2015) who used
the 1.2 m twin robotic telescopes STELLA (STELLar Activity) located on
Tenerife in the Canary Islands. They found possible periods of 83.0, 69.0, and
79.6 days. Mallonn et al. (2018) continued long-term monitoring of GJ 1214
with STELLA to create light curves from 2012 through 2016 primarily in the
Johnson $BV$ pass bands. Their most significant signal was $125\pm 5$ days for
the 2014-2016 $B+V$ data set, which they claimed overrules previous
suggestions of a significantly shorter stellar rotation period.

Figure 1: Nightly Cousins $R$ band photometry of GJ 1214 from the five
observing seasons 2018 through 2022 (small circles), acquired with the C14
automated imaging telescope (AIT) at Fairborn Observatory. The star is
slightly variable within each observing season over a range of 1-2%. The
seasonal means are plotted as the large filled circles and show that GJ 1214b
has brightened by several percent over the course of our observations (see
Table 1). The time of the JWST phase curve observation of GJ 1214b that was
obtained July 20 – 22, 2022 by Bean et al. (2021b) is indicated by the dashed
line.

##  2 Observations

In an attempt to determine the correct rotation period of GJ 1214, we
conducted our own photometric observations with the Tennessee State University
Celestron 14-inch (C14) automated imaging telescope (AIT) located at Fairborn
Observatory in southern Arizona. We acquired 329 good nightly photometric
observations (excluding occasional transit observations) during the five
observing seasons 2018 through 2022. The observations were made through a
Cousins $R$ filter with an SBIG STL-1001E CCD camera. Each nightly observation
consists of 3–5 consecutive exposures of the GJ 1214 field of view. The
individual frames are co-added and reduced to differential magnitudes in the
sense GJ 1214 minus the mean brightness of 13 constant comparison stars in the
same field. Further details of our observing and data reduction procedures can
be found in Sing et al. (2015).

The nightly observations are plotted as small filled circles in Figure 1.
Since GJ 1214 comes to opposition with the Sun on June 11, much of the
observing season occurs during our annual Summer Shutdown when all telescopes
must be closed from early July to early September due to the “monsoon season”
in southern Arizona. Thus, there are gaps in each year’s light curve when no
observations can collected. The 2018 observing season is an exception since
our initial observations did not take place until after the 2018 Shutdown. The
yearly mean differential observations are also plotted in Figure 1 as the
large filled circles and include the observations on both sides of the Summer
Shutdown. The uncertainties in the seasonal means are roughly the size of the
plot symbols.

The observations are summarized by season in Table 1. The standard deviations
of the individual observations from their respective seasonal means are given
in column 4 and range from 6.44 to 9.96 mmag. The typical precision of a
single nightly observation with the C14 AIT is 2–3 mmag on good nights (e.g.,
Fu et al., 2021), so the standard deviations given in column 4 indicate the
presence of low-level, night-to-night brightness variability in GJ 1214 during
each observing season. The seasonal means given in column 5 cover a range of
68 mmag, showing a general brightening trend in GJ 1214 of several percent
over our five years of observation. Finally, we performed period analyses of
the individual yearly light curves using the procedure described in Wong et
al. (2022). The resulting best periods are given in column 6 and range between
56.4 and 99.6 days. The period listed for 2018 is particularly uncertain due
to the low number of observations. Phase curves of the five observing seasons
are plotted with their five individual periods in Figure 2. Peak-to-peak
amplitudes are given in each panel and range from 10 to 20 mmag. Like
previously published results, we are unable to identify confidently the true
stellar rotation period of GJ 1214.

##  3 Conclusion

We can, however, use our photometric results to predict the starspot coverage
at the time of the JWST MIRI/LRS phase curve observation of GJ 1214b that was
taken in July 2022 (Bean et al., 2021b). Radick et al. (2018) examined the
patterns of brightness variation for 72 Sun-like stars and demonstrated that
the brightness variability in young, active (and therefore convective) stars
is driven by dark spots in the sense that a star is fainter when it is more
active (spotted). The combination of Figure 1 and the bottom panel of Figure 2
show that GJ 1214 was near a long-term as well as a short-term brightness
maximum. In other words, the star was near starspot minimum at the time of the
JWST phase curve observation. Continued photometric monitoring would be
valuable to provide context for the upcoming NIRCam transmission spectroscopy
observations of Greene et al. (2017), which are currently scheduled for July
and August 2023

Figure 2: Cousins $R$ band phase curves of GJ 1214 from 2018 to 2022. Periods for each year are given in the upper left of each panel. Peak-to-peak amplitudes are given in the upper right. Like the results given in the literature, our periods cover a broad range. The arrow in the bottom panel marks the phase of the July 20-22, 2022 JWST observation. Table 1: SUMMARY OF C14 AIT PHOTOMETRY OF GJ 1214 Observing |  | Date Range | Sigma | Seasonal Mean | Period
---|---|---|---|---|---
Season | $N_{obs}$ | (HJD $-$ 2,400,000) | (mag) | (mag) | (days)
(1) | (2) | (3) | (4) | (5) | (6)
2018 | 22 | 58386–58439 | 0.00815 | 1.49759 | $\approx 65::$
2019 | 70 | 58571–58789 | 0.00983 | 1.49599 | $81.6\pm 1.6$
2020 | 61 | 58861–59166 | 0.00934 | 1.45758 | $56.4\pm 0.6$
2021 | 67 | 59219–59497 | 0.00644 | 1.42948 | $57.9\pm 0.6$
2022 | 109 | 59584–59896 | 0.00996 | 1.43765 | $99.6\pm 1.7$

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