# On the nature of jets from a main sequence companion at the onset of common
envelope evolution

Noam Soker Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS>
###### Abstract

I consider a flow structure by which main sequence companions that enter a
common envelope evolution (CEE) with giant stars might launch jets even when
the accreted gas has a sub-Keplerian specific angular momentum. I first show
that after a main sequence star enters the envelope of a giant star the
specific angular momentum of the accreted gas is sub-Keplerian but still
sufficiently large for the accreted gas to avoid two conical-like openings
along the two opposite polar directions. I suggest that the high-pressure zone
that the accreted gas builds around the main sequence equatorial plane
accelerates outflows along these polar opening. Most of the inflowing gas is
deflected to the polar outflows, now turning to two opposite jets. The actual
mass that the main sequence star accretes is only a small fraction, $\approx
0.1$, of the inflowing gas. However, the gravitational energy that this gas
releases powers the inflow-outflow streaming of gas and adds energy to the
common envelope ejection. This flow structure might take place during a
grazing envelope evolution if occurs, during the early CEE, and possibly in
some post-CEE cases. This study increases the parameter space for main
sequence stars to launch jets in shaping some planetary nebulae, in adding
energy to mass removal in CEE, and in powering some intermediate luminosity
optical transients.

(stars:) binaries (including multiple): close; stars: winds, outflows; stars:
jets; planetary nebulae

##  1 Introduction

The morphologies of many bipolar and elliptical planetary nebulae (PNe) with
central binary systems (e.g., Bujarrabal et al. 2018; Wesson et al. 2018;
Jones 2019; Kővári et al. 2019; Miszalski et al. 2019; Orosz et al. 2019;
Jones 2020) suggest that a main sequence companion accretes mass from the
asymptotic giant branch (AGB) progenitor, or in some cases a red giant branch
(RGB) progenitor, of the PN and launches jets that shape the PN (e.g., Morris
1987; Soker 1990; Sahai & Trauger 1998; Rechy-García et al. 2017; Akashi &
Soker 2018; Balick, Frank, & Liu 2019; Derlopa et al. 2019; Estrella-Trujillo
et al. 2019; Tafoya et al. 2019; Balick, Frank, & Liu 2020; Rechy-García et
al. 2020; for an alternative view for some bipolar PNe see, e.g., Baan, Imai,
& Orosz 2021). The companion can launch the jets before the common envelope
evolution (CEE), during, or after the CEE (e.g., Tocknell, De Marco, & Wardle
2014; Guerrero, Suzett Rechy-García, & Ortiz 2020; Soker 2020b). Blackman &
Lucchini (2014) study the kinetic properties of outflows from 19 pre-planetary
nebulae (pre-PNe; taken from Bujarrabal et al. 2001 and Sahai et al. 2008).
They conclude that a main sequence companions to the AGB progenitor of the
pre-PNe should accrete at a very high rate to explain these kinetic
properties. Accretion is likely via a Roche lobe overflow (RLOF) or during a
CEE.

Theoretical studies should reveal the processes that allow main sequence
companions to PN progenitors to launch energetic jets, before, during and/or
after the CEE. This is the goal of this study.

Based on the arguments listed above for PNe, as well as other systems, like
intermediate-luminosity optical transients (ILOTs) and supernova impostors, I
argued in Soker (2020b) that main sequence stars in binary systems can accrete
mass at a high rate from an accretion disk and launch jets, and that such high
mass accretion rates can take place in the CEE phase, in particular at the
onset of the CEE, including a possible grazing envelope evolution (GEE) phase.
In this paper I examine a possible flow structure that allows a main sequence
star to accrete mass and launch jets.

Several CEE studies consider jets from a neutron star (NS) companion (e.g.,
Armitage & Livio 2000; Chevalier 2012; Moreno Méndez et al. 2017; López-Cámara
et al. 2019; Schreier et al. 2019; López-Cámara et al. 2020; Hillel, Schreier,
& Soker 2022; Lopez-Camara et al. 2022; Soker 2022; Schreier, Hillel, & Soker
2023). Other consider jets that a main sequence companion in a CEE launches
(e.g., Shiber & Soker 2018; Shiber et al. 2019; Lopez-Camara et al. 2022; Zou
et al. 2022). It is much more difficult to envision the launching of jets by a
main sequence companion in a CEE than by a NS or a black hole (BH) because the
accretion flow structure onto a main sequence star in a CEE differs from that
onto NS/BH in two significant manners. (1) NSs and BHs have much smaller radii
than a main sequence star and therefore a centrifugally supported disk is
formed around a NS/BH in a CEE according to the above studies (but see
Murguia-Berthier et al. 2017 for difficulties). This is not the case for a
main sequence star accreting mass during a CEE because of its much larger
radius. (2) The accreted mass onto a NS/BH in a CEE loses energy in neutrinos,
and the energy of the accreted mass onto a BH can in addition be accreted by
the BH. The accretion process onto a main sequence star in a CEE has no energy
sinks.

Shiber, Schreier, & Soker (2016) study the removal of accretion energy from
accretion disks by magnetic fields. They argue that this might allow a high
mass accretion rate onto main sequence stars, up to $\approx
10^{-2}M_{\odot}{~{}\rm yr}^{-1}$ for solar type stars, and up to $\approx
1M_{\odot}{~{}\rm yr}^{-1}$ for very massive main sequence stars. In the
present study I consider the case of sub-Keplerian accretion flow that forms
an accretion belt rather than a centrifugally-supported accretion disk
(section 2) and where the energy removal is by the jets even when the role of
magnetic fields is small (section 3). The flow structure that I study might
facilitate the launching of jets by a main sequence star in a CEE, at least
while the main sequence companion spirals-in in the outer zones of the giant
envelope where the flow is not in a steady state (section 4). In section 5 I
discuss the implications of such an accretion process that launches jets.

##  2 Accretion disk to accretion belt transition

I consider a secondary main sequence star of mass $M_{2}$ that enters the
envelope of a giant star, which might be an RGB star, an AGB star, or a red
super giant (RSG). The secondary star accretes mass from the envelope in a
Bondi-Hoyle-Lyttleton (BHL) type accretion flow. In Soker (2004) I derived a
crude condition for the accreted mass to form an accretion disk around the
secondary star. The assumptions there, as I adopt here as well, are as
follows. (1) The accretion process is a BHL type accretion flow. (2) The
secondary-envelope relative velocity is $v_{r}=[GM_{1}(a)/a]^{1/2}$, where $a$
is the orbital separation and $M_{1}(a)$ is the giant mass inside radius
$r=a$. Namely, $v_{r}$ is approximately the Keplerian velocity of the
secondary inside the envelope. (3) The three-dimensional hydrodynamical
simulations of Livio et al. (1986) that were performed in a rectangle box
without neither the circular orbital motion of the secondary star nor the
gravity of the giant hold for the CEE. For a density gradient perpendicular to
the secondary-envelope relative velocity in the form $\rho=\rho_{0}(1+y/H)$
Livio et al. (1986) find that the specific angular momentum of the accreted
gas is

$j_{\rm acc}=\frac{\eta}{4H}\frac{(2GM_{2})^{2}}{v^{3}_{r}},$ (1)

where $\eta\simeq 0.25$. For an envelope profile of

$\rho_{\rm env}(r)\propto r^{-\beta}$ (2)

at $r=a$ the density gradient has $H=a/\beta$.

The condition for the formation of an accretion disk (i.e., supported by the
centrifugal force alone) is $j_{\rm acc}>j_{2}$, where $j_{2}$ is the specific
angular momentum of a Keplerian motion on the companion’s equator. We better
express the radius $R_{\rm d2}$ of the disk that the accreted gas would form
with its average specific angular momentum (the subscript ‘2’ indicates that
the accretion disk is around the secondary star). The condition for the
formation of an accretion disk is then

$R_{\rm d2}=\frac{j^{2}_{\rm
acc}}{GM_{2}}=\eta^{2}\beta^{2}\left[\frac{M_{2}}{M_{1}(a)}\right]^{3}a>R_{2},$
(3)

where $R_{2}$ is the radius of the secondary star.

Consider a main sequence star that enters the envelope of a red giant
(AGB/RGB/RSG). In the very outer envelope the density profile is very steep,
i.e., $\beta\gg 1$, becoming $\beta\simeq 2$ in the main part of the envelope.
In the outer region of the envelope, inner to the steep density profile, there
is even a density inversion. To enter a CEE the main sequence companion cannot
be too massive. If it is, then it brings the giant rotation to a stable
synchronisation with the orbital motion. As well, I do not consider a low mass
secondary star that has a small influence on the envelope. I consider
therefore the range of $0.1M_{1}\lesssim M2\lesssim 0.5M_{1}$. Scaling then
equation (3) gives

$\frac{R_{\rm
d2}}{R_{2}}=0.39\left(\frac{\eta}{0.25}\right)^{2}\left(\frac{\beta}{2}\right)^{2}\left[\frac{M_{2}}{0.25M_{1}(a)}\right]^{3}\left(\frac{a}{100R_{2}}\right).$
(4)

When the main sequence secondary star is outside the envelope and accretes via
a RLOF type accretion or when it just enters the outskirts of the giant
envelope, i.e., at about $a\simeq 0.9R_{1}-R_{1}$ where $\beta\gg 2$, an
accretion disk is formed outside the secondary star as $\beta\gg 1$. As the
companion enters deeper into the envelope, but still in the outer parts, say
$a\simeq 0.5-0.9R_{1}$, then as $a$ decreases so is $\beta$ and the scaling of
equation (4) shows that the accretion disk forms very close to the secondary
surface and then the accreted gas becomes sub-Keplerian. Namely, centrifugal
forces alone cannot support the gas from reaching directly the secondary star
and a thermal pressure builds around the accreting secondary star.

Because centrifugal forces alone cannot hold anymore against gravity, the
accreted gas spreads to the two sides of the equatorial plane and forms an
accretion belt rather than an accretion disk. Namely, centrifugal forces and
pressure gradients together support the accreted gas around the secondary
star. I consider that such accretion belts can launch energetic jets (e.g.,
Schreier & Soker 2016).

The sub-Keplerian accretion will start for higher values of $\beta$ than what
the scaling in equations (3) and (4) give. The reason is that the negative jet
feedback mechanism in CEE implies that the mass accretion rate is below the
BHL accretion rate (e.g., Soker 2016b; Grichener, Cohen, & Soker 2021; Hillel,
Schreier, & Soker 2022; Lopez-Camara et al. 2022). A smaller accretion rate
implies a smaller ratio of the BHL accretion radius to the density scale-
height $H$, and therefore, by equation (1), a smaller value of $j_{\rm acc}$.
Indeed, Lopez-Camara et al. (2022) find in their simulations of CEE of a main
sequence companion inside a RGB stellar envelope that the specific angular
momentum of the accreted mass is not large enough to form an accretion disk.
In section 3 consider a way by which even a sub-Keplerian accretion flow in a
CEE might launch jets.

The flow structure is more complicated than what I have described above
because the flow itself depends on the ability of the flow to lose energy.
MacLeod & Ramirez-Ruiz (2015b) conduct a detailed study of the BHL accretion
flow by 3D simulations and present thorough discussions with references to
earlier studies. They find that when the adiabatic index of the flow is
$\gamma=1.1$ an accretion disk is much more likely to form than for
$\gamma=5/3$. The $\gamma=1.1$ mimics an accretion flow that can lose energy.
Here I attribute this energy loss (valve as Chamandy et al. 2018 term it) to
the jets that carry the energy. However, the simulations of MacLeod & Ramirez-
Ruiz (2015b) take the central accreting body to be a sink, i.e., they impose a
spherical absorbing boundary condition surrounding the central point mass.
This is different from a main sequence star that does not have an absorbing
surface. Therefore, I do not expect simulations with absorbing inner boundary
condition to have the flow structure I present here. I encourage common
envelope evolution with a main sequence accreting mass.

Murguia-Berthier et al. (2017) find in their wind tunnel simulations (rather
than full CEE simulation) that accretion disks form only for lower effective
$\gamma$, namely, when the gas is more compressible. They mention the partial
ionization zones of a common envelope as one where the gas is more
compressible. The present study is relevant to these outer envelope zones.
Their simulations, however, lack two important ingredients that I have here,
namely, a non-absorbing accreting body and a full CEE simulation (as they
simulate a wind tunnel). The mass accretion rates that simulations find
demonstrate the importance of including full CEE simulation. The wind tunnel
formalism that some studies employ lead to mass accretion rate much below the
values according to the BHL accretion flow (e.g., MacLeod & Ramirez-Ruiz
2015a, b; De et al. 2020). On the other hand, full simulations of accretion
from a wind, i.e., simulations that include the orbital motion in a binary
system, derive the BHL accretion rate (Kashi, Michaelis, & Kaminetsky, 2022).

##  3 The sub-Keplerian accretion flow

There are many studies of sub-Keplerian accretion flows (e.g., Jiao & Wu 2011;
Garain & Kim 2023). When the accreting body is a main sequence star (rather
than a NS or a BH) in the conditions that I studied in section 2, such a flow
does not form a thin accretion disk but rather a rotating flow on the surface
of the accreting body which is termed an accretion belt, or a slim disk if it
extends to larger distances from the accreting body. These two papers (Jiao &
Wu 2011; Garain & Kim 2023) as examples, study the inflow of a sub-Keplerian
flow and present the outflow along the two opposite polar directions. The flow
structure I present here is qualitatively similar to their study. I also note
that some observations support the notion that sub-Keplerian disk can launch
jets (e.g., Nandi et al. 2018; Banerjee et al. 2020; Debnath et al. 2020,
2021).

In what follows I use Greek letters to mark coordinates around the center of
the secondary star, $\Delta$ will be the distance from the center of the
secondary star and $\varpi$ the distance from the symmetry (polar) axis with
respect to the center of the secondary star. I keep $R_{d}$ to be the radius
of the disk measured from the center of the secondary star even when $R_{\rm
d2}<R_{2}$, and $R_{2}$ to be the radius of the secondary star.

The situation is as I present in Fig. 1. Magnetic fields do play a role in
launching the polar outflow, i.e., jets (e.g., Schreier & Soker 2016; Shiber,
Schreier, & Soker 2016) but I do not show them in the figure. Note that I
present a laminar flow structure. However, simulations (e.g., Garain & Kim
2023 for sub-Keplerian accretion onto a black hole) show that the flow can be
turbulent.

Figure 1: A schematic drawing of the sub-Keplerian accretion flow onto the
secondary star (yellow region) with the suggested jets drawn in one quarter of
the meridional plane. There is an axial-symmetry around the polar (vertical)
axis and a mirror symmetry about the equatorial (horizontal) plane. Magnetic
fields also play roles in launching the jets but they are not drawn. The
dashed-blue lines depict the sub-Keplerian inflow while the solid-red lines
depict the outflow, i.e., the formation of jets (only one jet is drawn). The
green line is the boundary of the inflowing gas according to equation (5) for
$R_{\rm d2}=0.1R_{2}$.

I consider the case where the radius of the secondary star is much smaller
than the BHL accretion radius. I therefore consider that most of the material
is flowing onto the secondary star with the average specific angular momentum
$j_{\rm acc}$ that corresponds to a Keplerian disk at radius $R_{\rm
d2}<R_{2}$. Namely, a centrifugally-supported disk cannot form. Consider then
a parcel of gas with a specific angular momentum $j_{\rm acc}$ that is
accreted above (or below) the equatorial plane and it is at $\Delta>R_{2}$.
There is a minimum distance $\varpi(\Delta,j_{\rm acc})$ below which the
parcel of gas cannot flow towards the symmetry axis with respect to the
secondary star (vertical black line in Fig. 1). This distance is given by the
balance of the gravitational component perpendicular to the equatorial plane,
$GM_{2}\varpi/\Delta^{3}$ with the centrifugal force $j^{2}_{\rm
acc}/\varpi^{3}$. The distance $\varpi$, neglecting forces due to pressure
gradients perpendicular to the symmetry axis (perpendicular to the angular
momentum direction), is then

$\varpi(D,j_{\rm acc})=\left(\frac{R_{\rm d2}}{\Delta}\right)^{1/4}\Delta,$
(5)

where $R_{\rm d2}$ is given by equations 3 and 4. On the surface of the
secondary star there is an avoidance cap with an angle $\theta_{a}$ from the
symmetry axis that is given by

$\sin\theta_{a}=\frac{\varpi(R_{2},j_{\rm acc})}{R_{2}}=\left(\frac{R_{\rm
d2}}{R_{2}}\right)^{1/4}.$ (6)

In Fig. 1 I draw this angle and $\varpi(\Delta,j_{\rm acc})$ for $R_{\rm
d2}=0.1$.

Equations (5) and (6) teach us that there are wide openings (two opposite
openings) along the polar directions that allow the gas to flow out. These
openings are not completely devoid of gas because a fraction of the accreted
gas does have a low specific angular momentum (as mixing of the inflowing gas
is not complete). However, the density there is low and I assume that the
accelerated outflow continues to flow out.

The high-pressure region (red-dotted area in Fig. 1) accelerates gas towards
the low-density zones near the polar directions. This reduces the pressure
around the mass-accreting secondary star and acts as a valve to allow the high
accretion rate process to continue, even to be super-Eddington (e.g., Chamandy
et al. 2018).

I also note that magnetic fields play roles in accelerating jets by accretion
belts (e.g., Schreier & Soker 2016). The magnetic fields channel rotational
energy and turbulence to kinetic energy of the outflowing jets. In the flow
structure I study here that takes place inside a common envelope the thermal
pressure around the mass-accreting secondary star might dominate the role of
magnetic fields.

In the flow structure I study here there is no energy loss beside the outflow
(jets). There is no neutrino cooling as in the accretion process onto a NS,
there is no energy sink as in the accretion process onto a BH, and the entire
region is optically thick and there is no energy loss in radiation as in
active galactic nuclei. Therefore, the energy is either stored in the
accreting main sequence star or is carried away by the outflow (jets).
Simulations of accretion flows without radiative cooling (e.g., Stone,
Pringle, & Begelman 1999; Mosallanezhad et al. 2021) show that most of the
energy is carried away by the jets and that most of the mass flowing in is
diverted to the polar outflow. Super-Eddington accretion flow also show this
property (e.g., Jiao et al. 2015; Jiao 2023).

The typical outflow velocity is the escape velocity from the accreting object.
This implies that the specific energy of the outflowing gas (the jets) is
about equal to the binding energy of the gas near the surface of the accreting
body, and is about equal to the specific energy of the accreted gas.
Therefore, most of the inflowing gas is ejected back (e.g., Stone, Pringle, &
Begelman 1999). Only a small fraction of the inflowing gas is accreted.

The flow is basically a large circularization flow involving a mass streaming
rate of about the BHL accretion rate $\simeq\dot{M}_{\rm BHL}$ which is driven
by small accreted mass $\dot{M}_{\rm acc}=\epsilon\dot{M}_{\rm
inflow}\simeq\dot{M}_{\rm BHL}$ where $\epsilon\ll 1$. The outflow mass rate
is then $\dot{M}_{\rm jets}=(1-\epsilon)\dot{M}_{\rm inflow}$. Numerical
simulations of accretion flow onto main sequence stars in CEE will eventually
determine the value of $\epsilon$. I will scale with a value of
$\epsilon\simeq 0.05-0.1$ based on calculations of super-Eddington accretion
flows onto BHs (e.g., Jiao et al. 2015).

Since the outflow mass rate in the jets is about equal to the inflow mass
rate, the specific angular momentum of the gas in the jets is about equal to
that of the inflowing gas, $j_{\rm jet}\simeq j_{\rm acc}$. This implies, by
equation (5), that most of the material in the jets flows close to the
boundary along $\varpi(\Delta,j_{\rm acc})$ as is shown in Fig. 1 by the dense
three red lines and as calculations of sub-Keplerian accretion (e.g., Jiao &
Wu 2011) and also simulations of super-Eddington accretion flow (e.g., Asahina
& Ohsuga 2022) show.

##  4 Timescales

During the rapid plunge-phase of the CEE the accretion flow is not in a steady
state because the spiralling-in timescale $a/\dot{a}$ is on about the
dynamical timescale of the giant star (e.g., Ohlmann et al. 2016; Iaconi et
al. 2017b; Law-Smith et al. 2020; Glanz & Perets 2021a, b; González-Bolívar et
al. 2022; Lau et al. 2022a, b; Moreno et al. 2022; Ondratschek et al. 2022;
Trani et al. 2022; Roepke & De Marco 2023 for some recent papers and
references to earlier studies therein). I take the plunge-in timescale as
$\tau_{1}=2\pi R^{3/2}_{1}(GM_{1})^{-1/2}$, i.e., the orbital Keplerian time
of a test particle on the giant surface. I take therefore
$\dot{a}=R_{1}/\tau_{\rm Kep}$ where $R_{1}$ is the unperturbed giant radius
at which the secondary starts the plunge-in phase. I take again the relative
secondary-envelope velocity to be the local Keplerian orbital velocity
$v_{r}$. The BHL mass accretion rate is then

$\dot{M}_{\rm
acc}=4\pi\sqrt{GM_{2}}\rho_{0}R^{\beta}_{0}\left[\frac{M_{2}}{M_{1}(a)}\right]^{3/2}a^{3/2-\beta},$
(7)

where $R_{0}$ is the radius, measured from the center of the primary giant
star, at which the density is $\rho_{0}$ (the location $R_{0}$ is not
important).

Since the mass of the primary $M_{1}(a)$ decreases as the secondary spirals-in
the accretion rate increases faster than $a^{3/2-\beta}$ as $a$ decreases in
the spiralling-in process. This implies that the ram pressure of the inflowing
gas that hits the high pressure zone near the secondary star increases with
time. This is true at a given distance from the secondary, because if the
high-pressure zone expands the inflow velocity and density decrease and so is
the ram pressure of the inflowing gas.

The second time scale is the one during which the accretion flow establishes
itself, $\tau_{f}(R_{\rm acc})\simeq R_{\rm acc}/v_{r}$, where $R_{\rm acc}$
is the accretion radius, i.e., the distance from which mass is accreted.
Taking the accretion radius to be the BHL accretion radius and the relative
velocity the Keplerian velocity at $a$, gives the ratio of the flow time to
the spiralling-in dynamical time

$\frac{\tau_{f}(R_{\rm
acc})}{\tau_{1}}\simeq\frac{1}{\pi}\frac{M_{2}}{M_{1}(a)}\left[\frac{M_{1}}{M_{1}(a)}\right]^{1/2}\left(\frac{a}{R_{1}}\right)^{3/2}\ll
1.$ (8)

Because $\tau_{f}\ll\tau_{1}$ we can consider a BHL-like accretion flow.
However, because the mass inflow rate increases there is no real steady state.
Note also that because the feedback mechanism reduces the mass accretion rate,
the accretion radius is smaller than the BHL radius that I take here.
Therefore, a typical ratio, for $0.1M_{1}\lesssim M2\lesssim 0.5M_{1}$, might
be $\tau_{f}(R_{\rm acc})/\tau_{1}\simeq 0.1-0.5$.

Another relevant timescale is the viscosity timescale of the gas circulating
the boundary of the inflow from the polar conical-like zones (green line in
Fig. 1). In accretion disks this time scale can be $Q\approx 10-100$ times the
orbital period at a given radius in the disk. Consider then a circulating flow
around the inner boundary of the inflow at distance $\varpi(\Delta,j_{\rm
acc})$ from the axis (depicted by the green line in Fig. 1). Consider a parcel
of gas with a specific angular momentum around the symmetry axis $j_{\rm
acc}$, a disk radius $R_{\rm d2}<R_{2}$, and a distance $\varpi$ given by
equation (5). The parcel of gas at the boundary surface $\varpi$ will complete
an orbit on a time of $P(\varpi)=2\pi\Delta^{3/2}/\sqrt{GM_{2}}$. I find the
ratio of the viscous time $\tau_{v}=QP(\varpi)$ to the flow time to be

$\begin{split}\frac{\tau_{v}(\Delta)}{\tau_{f}(R_{\rm acc})}\approx\pi
Q&\left[\frac{M_{1}(a)}{M_{2}}\right]^{3/2}\left(\frac{\Delta}{a}\right)^{3/2}=0.25\\\
\times\left(\frac{Q}{10}\right)&\left[\frac{M_{1}(a)}{4M_{2}}\right]^{3/2}\left(\frac{\Delta}{0.01a}\right)^{3/2}.\end{split}$
(9)

The more relevant time to compare the viscosity timescale at $\Delta$ with is
the flow time from $\Delta$ to the secondary star, which is shorter than
$\tau_{f}(R_{\rm acc})$ that measures the flow time from the accretion radius
to the secondary star. This would give $\tau_{v}(\Delta)>\tau_{f}(\Delta)$. I
conclude that the loss of angular momentum by viscosity does not close the two
opposite conical-like openings along the polar directions (the regions inner
to the green line in Fig. 1).

##  5 Summary and Implications

The morphologies of many PNe with central binary stars force us to consider
main sequence companions to the PN progenitors that launch energetic jets
(Section 1). Not at all jet-launching phases the jets breakout from the
envelope to shape the PN (e.g., Lopez-Camara et al. 2022). The present study
aim at suggesting a flow structure that might take place during a GEE if
occurs, and during the early CEE, i.e., the early phase of the plunge-in stage
of the CEE. At these phases the specific angular momentum of the accreted gas
might be sub-Keplerian with respect to the main sequence secondary star.
Namely, the accreted gas cannot form a centrifugally supported accretion disk.
The flow structure might be as I schematically draw in Fig. 1.

Numerical hydrodynamical, and possibly magnetohydrodynamical, simulations
should confirm whether the flow structure that I propose here does indeed take
place. It is impossible yet to resolve both the entire CEE volume and the
accretion flow around a main sequence companion. Instead, the simulations
should impose appropriate boundary conditions and set for an inflow of sub-
Keplerian gas at the boundaries of the grid. Such simulations should also
conserve angular momentum and have a high resolution to resolve turbulence
that is likely to develop. Both the differential rotation and turbulence will
form a dynamo that amplifies magnetic fields. For that, the following stage
will bee the conduction of high-resolution magnetohydrodynamical simulations.

Although PNe motivate me to introduce this sub-Keplerian flow structure to
launch jets I note the following. Firstly, the jets that shape PNe might be
launched only before the CEE, like in a GEE phase, or after the CEE (e.g.,
Soker 2020b) because during the the CEE the jets are deflected and even
chocked within the envelope (e.g., Lopez-Camara et al. 2022). Nonetheless, the
sub-Keplerian accretion that accelerates jets might be in some cases relevant
to the pre-CEE evolution, in particular the GEE phase if occurs. The sub-
Keplerian accretion might also be relevant to the post-CEE phase when the main
sequence companion accretes mass from a circumbinary disk (e.g., Soker 2020b).
Secondly, even if the jets are chocked within the common envelope they deposit
their energy into the envelope and by that can contribute to the removal of
the common envelope, at least when the companion is in the outer parts of the
envelope. i.e., the early plunge-in phase.

As I noted in section 2 the negative jet feedback mechanism in CEE reduces the
mass accretion rate to be below and even much below the BHL accretion rate
(e.g., Soker 2016b; Grichener, Cohen, & Soker 2021; Hillel, Schreier, & Soker
2022; Lopez-Camara et al. 2022). This reduces also the specific angular
momentum of the accreted gas $j_{\rm acc}$. On the other hand the jets remove
mainly gas from the polar directions. This gas has a lower specific angular
momentum to begin with. The removal of gas from the polar directions increases
somewhat $j_{\rm acc}$ with respect to an accretion without polar mass
removal. Overall, I expect that with the launching of jets the specific
angular momentum of the accreted gas be smaller than what equation (1) gives.
Namely, the effective value of $\eta$ is smaller than $0.25$. This in turn
implies that the condition for a sub-Keplerian accretion flow, $R_{\rm
d2}<R_{2}$, is met for steeper density gradients, namely higher values of
$\beta$, i.e., even for up to $\beta\simeq 10$ (see equation 4), which is the
case in the very outer parts of the giant’s envelope.

Most relevant to the present study are the simulations that Lopez-Camara et
al. (2022) conducted of a main sequence star inside the envelope of a RGB
star. They find that during a GEE and early in the CEE (the early plunge-in
phase) jets might breakout. However, jets cannot break out from the envelope
when the companion is deep inside the envelope. Note that they inject the jets
manually and did not follow the formation of the jets by the accretion flow.
In my study here I suggest a way for such main sequence companions in their
simulations to launch jets as they inserted, even when accretion is sub-
Keplerian

The sub-Keplerian accretion onto a main sequence star that powers jets might
be also relevant to some intermediate luminosity optical transients (ILOTs) if
they are powered by jets (e.g., Soker & Kashi 2012; Soker & Kaplan 2021).111I
follow Kashi & Soker (2016) in using ILOTs (for usage of the term ILOT see
also Berger et al. 2009 and Muthukrishna et al. 2019) to include all subgroups
that are powered by gravitational energy. These subgroups of ILOTs include
intermediate-luminosity (red) transients, red novae, luminous red novae (e.g.
Jencson et al. 2019; Blagorodnova et al. 2021), LBV giant eruptions and SN
Impostors. Another general term is gap transients (e.g., Kasliwal 2011;
Pastorello & Fraser 2019). The accretion belt/disk is formed by either a mass
transfer onto a main sequence star or by tidal disruption of one star onto the
other main sequence star (e.g., Kashi & Soker 2016). It is possible that in
some cases the accreted mass in ILOTs is sub-Keplerian. Since jets are very
efficient in powering ILOTs (e.g., Soker 2020a) the present study increases
the parameter space of binary interaction to power ILOTs with jets.

Overall, the present study suggests a flow structure that increases the
parameter space for main sequence stars to launch jets in shaping some
planetary nebulae, in adding energy to mass removal in CEE, and in powering
some ILOTs.

## Acknowledgments

This research was supported by a grant from the Israel Science Foundation
(769/20) and by the Pazy Research Foundation.

## References

  * Akashi & Soker (2018) Akashi M., Soker N., 2018, MNRAS, 481, 2754. doi:10.1093/mnras/sty2479
  * Armitage & Livio (2000) Armitage, P. J., & Livio, M. 2000, ApJ, 532, 540
  * Asahina & Ohsuga (2022) Asahina, Y. & Ohsuga, K. 2022, ApJ, 929, 93. doi:10.3847/1538-4357/ac5d37
  * Baan, Imai, & Orosz (2021) Baan W. A., Imai H., Orosz G., 2021, RAA, 21, 275. doi:10.1088/1674-4527/21/11/275
  * Balick, Frank, & Liu (2019) Balick B., Frank A., Liu B., 2019, ApJ, 877, 30. doi:10.3847/1538-4357/ab16f5
  * Balick, Frank, & Liu (2020) Balick B., Frank A., Liu B., 2020, ApJ, 889, 13. doi:10.3847/1538-4357/ab5651
  * Banerjee et al. (2020) Banerjee A., Bhattacharjee A., Debnath D., Chakrabarti S. K., 2020, RAA, 20, 208. doi:10.1088/1674-4527/20/12/208
  * Berger et al. (2009) Berger, E., Soderberg, A. M., Chevalier, R. A., et al. 2009, ApJ, 699, 1850
  * Blackman & Lucchini (2014) Blackman E. G., Lucchini S., 2014, MNRAS, 440, L16. doi:10.1093/mnrasl/slu001
  * Blagorodnova et al. (2021) Blagorodnova N., Klencki J., Pejcha O., Vreeswijk P. M., Bond H. E., Burdge K. B., De K., et al., 2021, A&A, 653, A134. doi:10.1051/0004-6361/202140525
  * Bujarrabal et al. (2001) Bujarrabal, V., Castro-Carrizo, A., Alcolea, J., & Sánchez Contreras C. 2001, A&A, 377, 868
  * Bujarrabal et al. (2018) Bujarrabal V., Castro-Carrizo A., Van Winckel H., Alcolea J., Sánchez Contreras C., Santander-García M., Hillen M., 2018, A&A, 614, A58. doi:10.1051/0004-6361/201732422
  * Chamandy et al. (2018) Chamandy L., Frank A., Blackman E. G., Carroll-Nellenback J., Liu B., Tu Y., Nordhaus J., et al., 2018, MNRAS, 480, 1898. doi:10.1093/mnras/sty1950
  * Chevalier (2012) Chevalier, R. A. 2012, ApJ, 752, L2
  * De et al. (2020) De S., MacLeod M., Everson R. W., Antoni A., Mandel I., Ramirez-Ruiz E., 2020, ApJ, 897, 130. doi:10.3847/1538-4357/ab9ac6
  * Debnath et al. (2020) Debnath D., Chatterjee D., Jana A., Chakrabarti S. K., Chatterjee K., 2020, RAA, 20, 175. doi:10.1088/1674-4527/20/11/175
  * Debnath et al. (2021) Debnath D., Chatterjee K., Chatterjee D., Jana A., Chakrabarti S. K., 2021, MNRAS, 504, 4242. doi:10.1093/mnras/stab1169
  * Derlopa et al. (2019) Derlopa S., Akras S., Boumis P., Steffen W., 2019, MNRAS, 484, 3746. doi:10.1093/mnras/stz193
  * Estrella-Trujillo et al. (2019) Estrella-Trujillo D., Hernández-Martínez L., Velázquez P. F., Esquivel A., Raga A. C., 2019, ApJ, 876, 29. doi:10.3847/1538-4357/ab12e1
  * Garain & Kim (2023) Garain S. K., Kim J., 2023, MNRAS, 519, 4550. doi:10.1093/mnras/stac3736
  * Glanz & Perets (2021a) Glanz H., Perets H. B., 2021a, MNRAS, 500, 1921. doi:10.1093/mnras/staa3242
  * Glanz & Perets (2021b) Glanz H., Perets H. B., 2021b, MNRAS, 507, 2659. doi:10.1093/mnras/stab2291
  * González-Bolívar et al. (2022) González-Bolívar M., De Marco O., Lau M. Y. M., Hirai R., Price D. J., 2022, MNRAS, 517, 3181. doi:10.1093/mnras/stac2301
  * Grichener, Cohen, & Soker (2021) Grichener A., Cohen C., Soker N., 2021, ApJ, 922, 61. doi:10.3847/1538-4357/ac23dd
  * Guerrero, Suzett Rechy-García, & Ortiz (2020) Guerrero M. A., Suzett Rechy-García J., Ortiz R., 2020, ApJ, 890, 50. doi:10.3847/1538-4357/ab61fa
  * Hillel, Schreier, & Soker (2022) Hillel S., Schreier R., Soker N., 2022, MNRAS, 514, 3212. doi:10.1093/mnras/stac1341
  * Iaconi et al. (2017b) Iaconi, R., Reichardt, T., Staff, J., De Marco, O., Passy, J.-C., Price, D., Wurster, J., & Herwig, F. 2017b, MNRAS, 464, 4028
  * Jencson et al. (2019) Jencson, J. E., Kasliwal, M. M., Adams, S. M., et al. 2019, ApJ, 886, 40
  * Jiao (2023) Jiao C.-L., 2023, arXiv, arXiv:2301.10378. doi:10.48550/arXiv.2301.10378
  * Jiao et al. (2015) Jiao C.-L., Mineshige S., Takeuchi S., Ohsuga K., 2015, ApJ, 806, 93. doi:10.1088/0004-637X/806/1/93
  * Jiao & Wu (2011) Jiao C.-L., Wu X.-B., 2011, ApJ, 733, 112. doi:10.1088/0004-637X/733/2/112
  * Jones (2019) Jones D., 2019, hsax.conf, 340. doi:10.48550/arXiv.1810.10782
  * Jones (2020) Jones D., 2020, Galax, 8, 28. doi:10.3390/galaxies8020028
  * Kashi, Michaelis, & Kaminetsky (2022) Kashi A., Michaelis A., Kaminetsky Y., 2022, MNRAS, 516, 3193. doi:10.1093/mnras/stac1912
  * Kashi & Soker (2016) Kashi A., Soker N., 2016, RAA, 16, 99. doi:10.1088/1674-4527/16/6/099
  * Kasliwal (2011) Kasliwal, M. M. 2011, Bulletin of the Astronomical Society of India, 39, 375
  * Kővári et al. (2019) Kővári Z., Strassmeier K. G., Oláh K., Kriskovics L., Vida K., Carroll T. A., Granzer T., et al., 2019, A&A, 624, A83. doi:10.1051/0004-6361/201834810
  * Lau et al. (2022a) Lau M. Y. M., Hirai R., González-Bolívar M., Price D. J., De Marco O., Mandel I., 2022a, MNRAS, 512, 5462. doi:10.1093/mnras/stac049
  * Lau et al. (2022b) Lau M. Y. M., Hirai R., Price D. J., Mandel I., 2022b, MNRAS, 516, 4669. doi:10.1093/mnras/stac2490
  * Law-Smith et al. (2020) Law-Smith J. A. P., Everson R. W., Ramirez-Ruiz E., de Mink S. E., van Son L. A. C., Götberg Y., Zellmann S., et al., 2020, arXiv, arXiv:2011.06630
  * Livio et al. (1986) Livio M., Soker N., de Kool M., Savonije G. J., 1986, MNRAS, 222, 235. doi:10.1093/mnras/222.2.235
  * López-Cámara et al. (2019) López-Cámara, D., De Colle, F., & Moreno Méndez, E. 2019, MNRAS, 482, 3646
  * Lopez-Camara et al. (2022) López-Cámara D., De Colle F., Moreno Méndez E., Shiber S., Iaconi R., 2022, MNRAS, 513, 3634. doi:10.1093/mnras/stac932
  * López-Cámara et al. (2020) López-Cámara, D., Moreno Méndez, E., & De Colle, F. 2020, MNRAS, 497, 2057
  * MacLeod & Ramirez-Ruiz (2015a) MacLeod M., Ramirez-Ruiz E., 2015a, ApJL, 798, L19. doi:10.1088/2041-8205/798/1/L19
  * MacLeod & Ramirez-Ruiz (2015b) MacLeod M., Ramirez-Ruiz E., 2015b, ApJ, 803, 41. doi:10.1088/0004-637X/803/1/41
  * Miszalski et al. (2019) Miszalski, B., Manick, R., Rauch, T., Iłkiewicz, K., Van Winckel, H., & Mikołajewska J. 2019, PASA, 36, e042. doi:10.1017/pasa.2019.36
  * Miszalski et al. (2019) Miszalski B., Manick R., Van Winckel H., Mikołajewska J., 2019, MNRAS, 487, 1040. doi:10.1093/mnras/stz1315
  * Moreno et al. (2022) Moreno M. M., Schneider F. R. N., Röpke F. K., Ohlmann S. T., Pakmor R., Podsiadlowski P., Sand C., 2022, A&A, 667, A72. doi:10.1051/0004-6361/202142731
  * Moreno Méndez et al. (2017) Moreno Méndez, E., López-Cámara, D., & De Colle, F. 2017, MNRAS, 470, 2929
  * Morris (1987) Morris M., 1987, PASP, 99, 1115. doi:10.1086/132089
  * Mosallanezhad et al. (2021) Mosallanezhad A., Zeraatgari F. Z., Mei L., Bu D.-F., 2021, ApJ, 909, 140. doi:10.3847/1538-4357/abde49
  * Murguia-Berthier et al. (2017) Murguia-Berthier A., MacLeod M., Ramirez-Ruiz E., Antoni A., Macias P., 2017, ApJ, 845, 173. doi:10.3847/1538-4357/aa8140
  * Muthukrishna et al. (2019) Muthukrishna, D., Narayan, G., Mandel, K. S., Biswas, R., & Hložek, R. 2019, PASP, 131, 118002
  * Nandi et al. (2018) Nandi A., Mandal S., Sreehari H., Radhika D., Das S., Chattopadhyay I., Iyer N., et al., 2018, Ap&SS, 363, 90. doi:10.1007/s10509-018-3314-1
  * Ohlmann et al. (2016) Ohlmann S. T., Röpke F. K., Pakmor R., Springel V., 2016, ApJL, 816, L9. doi:10.3847/2041-8205/816/1/L9
  * Ondratschek et al. (2022) Ondratschek P. A., Röpke F. K., Schneider F. R. N., Fendt C., Sand C., Ohlmann S. T., Pakmor R., et al., 2022, A&A, 660, L8. doi:10.1051/0004-6361/202142478
  * Orosz et al. (2019) Orosz G., Gómez J. F., Imai H., Tafoya D., Torrelles J. M., Burns R. A., Frau P., et al., 2019, MNRAS, 482, L40. doi:10.1093/mnrasl/sly177
  * Pastorello & Fraser (2019) Pastorello, A., & Fraser, M. 2019, Nature Astronomy, 3, 676
  * Rechy-García et al. (2020) Rechy-García, J. S., Guerrero, M. A., Duarte Puertas, S., Chu, Y.-H., Toalá, J. A., & Miranda, L. F., 2020, MNRAS, 492, 1957. doi:10.1093/mnras/stz3326
  * Rechy-García et al. (2017) Rechy-García J. S., Velázquez P. F., Peña M., Raga A. C., 2017, MNRAS, 464, 2318. doi:10.1093/mnras/stw2524
  * Roepke & De Marco (2023) Roepke F. K., De Marco O., 2023, arXiv:2212.07308
  * Sahai et al. (2008) Sahai, R., Claussen, M., Sánchez Contreras, C., Morris, M., & Sarkar G. 2008, ApJ, 680, 483
  * Sahai & Trauger (1998) Sahai R., Trauger J. T., 1998, AJ, 116, 1357. doi:10.1086/300504
  * Schreier et al. (2019) Schreier, R., Hillel, S., & Soker, N. 2019, MNRAS, 490, 4748. doi:10.1093/mnras/stz2914
  * Schreier, Hillel, & Soker (2023) Schreier R., Hillel S., Soker N., 2023, arXiv:2209.13573
  * Schreier & Soker (2016) Schreier R., Soker N., 2016, RAA, 16, 70. doi:10.1088/1674-4527/16/5/070
  * Shiber et al. (2019) Shiber S., Iaconi R., De Marco O., Soker N., 2019, MNRAS, 488, 5615. doi:10.1093/mnras/stz2013
  * Shiber, Schreier, & Soker (2016) Shiber S., Schreier R., Soker N., 2016, RAA, 16, 117. doi:10.1088/1674-4527/16/7/117
  * Shiber & Soker (2018) Shiber, S. & Soker, N. 2018, MNRAS, 477, 2584
  * Soker (1990) Soker N., 1990, AJ, 99, 1869. doi:10.1086/115465
  * Soker (2004) Soker N., 2004, NewA, 9, 399. doi:10.1016/j.newast.2004.01.004
  * Soker (2016b) Soker N., 2016, NewAR, 75, 1. doi:10.1016/j.newar.2016.08.002
  * Soker (2020a) Soker N., 2020a, ApJ, 893, 20. doi:10.3847/1538-4357/ab7dbb
  * Soker (2020b) Soker N., 2020b, Galax, 8, 26. doi:10.3390/galaxies8010026
  * Soker (2022) Soker N., 2022, RAA, 22, 122003. doi:10.1088/1674-4527/ac9782
  * Soker & Kaplan (2021) Soker N., Kaplan N., 2021, RAA, 21, 090. doi:10.1088/1674-4527/21/4/90
  * Soker & Kashi (2012) Soker N., Kashi A., 2012, ApJ, 746, 100. doi:10.1088/0004-637X/746/1/100
  * Stone, Pringle, & Begelman (1999) Stone J. M., Pringle J. E., Begelman M. C., 1999, MNRAS, 310, 1002. doi:10.1046/j.1365-8711.1999.03024.x
  * Tafoya et al. (2019) Tafoya D., Orosz G., Vlemmings W. H. T., Sahai R., Pérez-Sánchez A. F., 2019, A&A, 629, A8. doi:10.1051/0004-6361/201834632
  * Tocknell, De Marco, & Wardle (2014) Tocknell J., De Marco O., Wardle M., 2014, MNRAS, 439, 2014. doi:10.1093/mnras/stu079
  * Trani et al. (2022) Trani A. A., Rieder S., Tanikawa A., Iorio G., Martini R., Karelin G., Glanz H., et al., 2022, PhRvD, 106, 043014. doi:10.1103/PhysRevD.106.043014
  * Wesson et al. (2018) Wesson R., Jones D., García-Rojas J., Boffin H. M. J., Corradi R. L. M., 2018, MNRAS, 480, 4589. doi:10.1093/mnras/sty1871
  * Zou et al. (2022) Zou Y., Chamandy L., Carroll-Nellenback J., Blackman E. G., Frank A., 2022, MNRAS, 514, 3041. doi:10.1093/mnras/stac1529


# Do charged-pions condense in a magnetic field with rotation?

Hao-Lei Chen  Physics Department and Center for Particle Physics and Field
Theory, Fudan University, Shanghai 200433, China  Xu-Guang Huang  Physics
Department and Center for Particle Physics and Field Theory, Fudan University,
Shanghai 200433, China  Key Laboratory of Nuclear Physics and Ion-beam
Application(MOE), Fudan University, Shanghai 200433, China  Kazuya Mameda
Physics Department and Center for Particle Physics and Field Theory, Fudan
University, Shanghai 200433, China  RIKEN Nishina Center for Accelerator-Based
Science, Wako 351-0198, Japan

###### Abstract

We revisit the condensation scenario of charged pions in external magnetic
field and rotation, which was first considered by Y. Liu and I. Zahed. Based
on the Ginzburg-Landau analysis of the Nambu–Jona-Lasinio model, we find that
the charged-pion condensation takes place only when both a strong coupling
constant and negatively large baryon chemical potential are applied. Besides,
our numerical calculation shows that the chiral restoration induced by the
interplay between magnetic field and rotation (i.e., the rotational magnetic
inhibition) interrupts the formation of the charged-pion condensate. This
suggests that the analysis of such condensation requires a careful treatment
of the inner structure of pions, which was not taken into account before. We
also discuss the underlying physical mechanism of our finding and the
indication of charged-rho condensation.

##  I Introduction

In the last few decades, quark matter under a magnetic background has
attracted many attentions in a wide range of physical systems from the early
universe to the neutron stars and heavy ion collision experiments. One of the
active research topics is the anomalous chiral transport phenomena caused by
the quantum anomaly in a magnetic field, for example, the chiral magnetic
effect (CME) Kharzeev  _et al._ (2008); Fukushima  _et al._ (2008) and its
cousins. Their experimental searches are the frontiers of the recent heavy-ion
collision physics Huang (2016); Hattori and Huang (2017); Kharzeev  _et al._
(2016). The magnetic response of quark matter also strongly affects the phase
structure of quantum chromodynamics (QCD). A lot of interesting phenomena have
been discussed in this context, e.g., the magnetic catalysis of chiral
condensate Gusynin  _et al._ (1994, 1996); Miransky and Shovkovy (2015),
inverse magnetic catalysis at finite temperature Bali  _et al._ (2012a, b) and
density Preis  _et al._ (2011), and others Fukushima and Warringa (2008);
Ferrer and de la Incera (2007); Son and Stephanov (2008); Chernodub (2011);
Hidaka and Yamamoto (2013); Sinha  _et al._ (2013); Cao and Huang (2016);
Brauner and Yamamoto (2017); Hattori  _et al._ (2017); Ozaki  _et al._ (2016);
Chen  _et al._ (2017).

Additionally, various properties of rotating relativistic matter has been
actively discussed, motivated by extremely strong fluid vorticity found in
heavy-ion collisions Deng and Huang (2016); Jiang  _et al._ (2016); Wei  _et
al._ (2019) via the spin polarization of spinful particles Liang and Wang
(2005); Adamczyk  _et al._ (2017). Such a vorticity can generate the parity
violating current called the chiral vortical effect (CVE) Vilenkin (1979,
1980); Son and Surowka (2009); Liu  _et al._ (2019), which is analogous to the
CME. It is also revealed that rotation affects even the phase structure of QCD
Chen  _et al._ (2016); Jiang and Liao (2016); Ebihara  _et al._ (2017);
Chernodub and Gongyo (2017a); Huang  _et al._ (2018); Chernodub and Gongyo
(2017b, c); Liu and Zahed (2018a); Wang  _et al._ (2019a); Zhang  _et al._
(2018); Wang  _et al._ (2019b). In a uniformly rotating system at finite
temperature and/or density, angular velocity plays a role as an effective
chemical potential, and thus suppresses the spin-0 pairing of quarks Jiang and
Liao (2016). On the other hand, it is confirmed that such a rotational effect
on thermodynamics is invisible at zero temperature and density Ebihara  _et
al._ (2017); Chernodub and Gongyo (2017a, b). In Refs. Wang  _et al._ (2019a,
c), the authors showed that under a nonuniform rotation the ground state
exhibits a vortex structure under sufficiently rapid rotation.

On top of these, a remarkable property of rotating matter is that combined
with magnetic field, rotation creates more fruitful QCD phase structure. In
Ref. Chen  _et al._ (2016), Fukushima and the present authors found, from the
Nambu–Jona-Lasinio (NJL) model analysis, that the chiral condensate decreases
as magnetic field increasing, and eventually the chiral symmetry is restored.
This novel phenomenon was named the “rotational magnetic inhibition”, as an
analogy to the magnetic inhibition phenomenon in the finite density system.
Furthermore, it is argued in Refs. Liu and Zahed (2018a, b) that the interplay
between magnetic field and rotation can induce a charged-pion condensation.
The essential idea is that such a combined effect leads to an energy splitting
between $\pi^{+}$ and $\pi^{-}$, if pions are taken into account as point-like
particles (see Section II). Namely, an effectively isospin chemical potential
is induced. As a result, a charged-pion Bose-Einstein condensation (BEC) takes
place, similarly to that induced in an isospin imbalanced system Son and
Stephanov (2001); He  _et al._ (2005); Sun  _et al._ (2007); He  _et al._
(2006).

In the above argument of the charged-pion condensate, however, it is unclear
whether the point-particle picture is safely admitted. Indeed, it would be
expected that the inner structure is important, in the following two senses.
First, due to the rotational magnetic inhibition, magnetic field and rotation
would suppress not only the chiral condensate but also the charged-pion
condensate. Second, since both magnetic field and rotation tend to align the
angular momenta of the paired quark and antiquark, the magnetized and rotating
quark system energetically disfavors the condensation of charged-pion, which
is a spin-0 composite state.

The main purpose of this paper is to quantify how the quark dynamics affects
the charged-pion condensation induced in magnetic field and rotation. For this
reason, we perform the Ginzburg-Landau analysis of the $\pi^{\pm}$ fields in
the two-flavor NJL model. We find that within this analysis, the charged-pion
BEC is not observed for a small coupling strength. However, if a negatively
high baryon chemical potential is present, the charged-pion condensation can
take place for a strong coupling constant. We also discuss the underlying
physical meaning of these results.

##  II General Arguments

First let us review the condensation mechanism of the point-like particle
$\pi^{\pm}$ Liu and Zahed (2018a, b). We consider the magnetized and rotating
cylindrical system with a magnetic field $\vec{B}=B\hat{z}$ , angular velocity
$\vec{\Omega}=\Omega\hat{z}$, and the transverse system radius $R$. In this
paper, we always assume $e>0$ and $B>0$ and $\Omega>0$. The dispersion
relations of the pions are given by

$E_{\pi^{\pm}}=\sqrt{eB(2n+1)+p_{z}^{2}+m_{\pi}^{2}}\mp\Omega l,$ (1)

where $n$ ($l$) is the quantum number of the Landau levels (angular momentum).
The above dispersion implies that $\pi^{\pm}$ is affected by an effective
chemical potential $\pm\Omega l$. Therefore the lowest Landau level (LLL)
$\pi^{+}$ forms a Bose condensate when $\mu_{N}=N\Omega$ (with $N$ being the
magnetic flux, and thus the maximum value of $l$) exceeds
$m_{0}=\sqrt{m_{\pi}^{2}+eB}$. Besides, as $\Omega$ increased, a higher Landau
level $\pi^{+}$ may Bose-condense; see Refs. Liu and Zahed (2018a, b) for more
details. We note that the condition $\Omega\leqslant 1/R\ll\sqrt{eB}$ is
implicitly assumed: the first inequality is due to the causality, and the
second one is to make the Landau quantization sensible Chen  _et al._ (2016).

This picture cannot be applicable, however, when pions are regarded as
composite spin-0 particles of quarks and antiquarks (especially, in a large
magnetic field $eB>\Lambda_{\rm QCD}$). Let us consider $\pi^{+}$, which
comprise a $u$ quark and a $\bar{d}$ antiquark with their angular momenta
(both the spin and the orbital one) antiparallel to each other. Since the
magnetic field and rotation would tend to align the angular momenta of $u$ and
$\bar{d}$, the $\pi^{+}$ condensate is suppressed. In addition, if a magnetic
field is strong enough, the chiral restoration also takes place via the
rotational magnetic inhibition Chen  _et al._ (2016) with an effective baryon
chemical potential $\sim\Omega N$ 111The appearance of the baryon density
contains a contribution from chiral anomaly such that $n_{\rm{B}}\sim\Omega
B/(4\pi^{2})$ Hattori and Yin (2016); Ebihara  _et al._ (2017), which persists
even in low-energy effective theory due to anomaly matching. Thus, even if we
disregard the inner quark structure of pions, a finite baryon density still
exists.. It is hence necessary to discuss the charged-pion condensation in the
language of quarks.

In the rest of this article, we will use an NJL model to study the above two
competing effects. The NJL model lacks the confinement and has no scale like
$\Lambda_{\rm QCD}$ to separate the hadronic degrees of freedom from the
quarks. The results from the NJL model may differ from that of QCD, but
usually provide useful insights to QCD physics, especially those related to
symmetry breaking. The $\pi^{+}$ condensate in NJL model is considered as
$\langle\bar{d}i\gamma^{5}u\rangle$, which can smoothly change from the
condensates of loosely correlated pairs to tightly bound bosons upon tuning
the coupling constant or densities, a.k.a., BCS (Bardeen-Cooper-
Schrieffer)-BEC crossover. For a wide range of the coupling constant, we do
not observe qualitatively different results, so in the following we present
our results for one fixed coupling constant.

We note that the charged-pion condensation also triggers an electric
superconductivity, which always obstructs the penetration of the magnetic
field (the Meissner effect 222 For a rotating superconductor, the superfluid
velocity is entangled with the magnetic field, which makes the discussion of
the Meissner effect more involved. But the main implication to the charged-
pion condensation is unchanged.). As a result, there are two possibilities;
the magnetic field is repelled from the charged-pion condensate, and thus it
cannot induce pion condensation at all; or the magnetic field penetrates into
the condensate from, e.g. the magnetic vortices if it is a type-II
superconductor, to turn the system very inhomogeneous. These issues will not
be discussed in the following.

##  III Dirac Equation

We start from the Dirac equation in rotating frame with a magnetic field
$\vec{B}$. The frame is rotating with a constant angular velocity
$\vec{\Omega}=\Omega\hat{z}$ ($\Omega>0$), which can be described by the
metric tensor

$g_{\mu\nu}=\begin{pmatrix}1-r^{2}\Omega^{2}&\Omega y&-\Omega x&0\\\ \Omega
y&-1&0&0\\\ -\Omega x&0&-1&0\\\ 0&0&0&-1\\\ \end{pmatrix}$ (2)

with $r^{2}=x^{2}+y^{2}$. The Dirac equation in such a frame is

$\Bigl{[}i{\gamma}^{\mu}(\partial_{\mu}+iqA_{\mu}+\Gamma_{\mu})-m+\mu\gamma^{0}\Bigr{]}\psi(x)=0\,,$
(3)

where $A_{\mu}$ is the background gauge field in the rotating frame, $q$ the
charge of the Dirac fermion, $\mu$ the fermion chemical potential. The spin
connection $\Gamma_{\mu}$ is defined by

$\begin{split}\Gamma_{\mu}&=-\frac{i}{4}\omega_{\mu ab}\sigma^{ab}\,,\\\
\omega_{\mu
ab}&=g_{\alpha\beta}e^{\alpha}_{a}(\partial_{\mu}e^{\beta}_{b}+\Gamma^{\beta}_{\nu\mu}e^{\nu}_{b})\,,\\\
\sigma^{ab}&=\frac{i}{2}[\gamma^{a},\gamma^{b}]\,.\end{split}$ (4)

The Greek and the Latin indices stand for the curved and local Lorentz
coordinates, respectively. The vierbein field $e^{\mu}_{a}$ is chosen to be

$e^{t}_{0}=e^{x}_{1}=e^{y}_{2}=e^{z}_{3}=1,\quad e^{x}_{0}=y\Omega,\quad
e^{y}_{0}=-x\Omega,$ (5)

and other components are zero.

We now consider the situation that a constant magnetic field is set to be
along the rotating axis. In the non-rotating Minkowski spacetime with the
coordinate $(t^{\prime},x^{\prime},y^{\prime},z^{\prime})$, the magnetic field
is along the $z^{\prime}$-axis, i.e., $\vec{B}=B\hat{z}^{\prime}$. We adopt
the symmetric gauge $A_{a}^{\prime}=(0,By^{\prime}/2,-Bx^{\prime}/2,0)$, then
get the gauge potential in the rotating frame by a coordinate transformation:
$A_{\mu}(x)=(-\frac{1}{2}B\Omega r^{2},\frac{1}{2}By,-\frac{1}{2}Bx,0)$. Thus,
the Dirac equation is written as

$\begin{split}i\partial_{t}\psi&=(\hat{H}_{D}-\Omega\hat{J}_{z}-\mu)\psi\,,\\\
\hat{H}_{D}&=-i\gamma^{0}\gamma^{1}\left(\partial_{x}+iqB{y\over
2}\right)-i\gamma^{0}\gamma^{2}\left(\partial_{y}-iqB{x\over 2}\right)\\\
&\;\;\;\;\,-i\gamma^{0}\gamma^{3}\partial_{z}+m\gamma^{0}\,,\\\
\hat{J}_{z}&=iy\partial_{x}-ix\partial_{y}+\frac{\sigma_{z}}{2}=-i\partial_{\theta}+\frac{\sigma_{z}}{2}\,.\end{split}$
(6)

A uniformly rotating system must be finite. We consider a cylindrical system
with a radius $R$ satisfying the causality constraint $\Omega R<1$, and
suppose that an appropriate boundary condition is imposed. For $qB>0$, the
solutions can be written as

$\psi^{(a)}_{s}=\frac{1}{\sqrt{\pi
R^{2}N_{l,k}^{2}}}\frac{1}{\sqrt{2\varepsilon}}\mathrm{e}^{-i(a\varepsilon-\Omega
j-\mu)t}\Psi^{(a)}_{s}$ (7)

with $\varepsilon=\sqrt{2qB\lambda_{l,k}+p_{z}^{2}+m^{2}}$ and $\lambda_{l,k}$
being the $k$-th eigenvalue of the radial Hamiltonian for angular momentum
quantum number $l$. Also $a=\pm$ stands for positive or negative frequency
solution, $s=\pm$ represents the spin state, and $j=l+1/2$. Note that
$\lambda_{l,k}$ and $N_{l,k}$ depend on what types of boundary condition is
imposed Chen  _et al._ (2017). The eigenfunctions of $\hat{H}_{D}$ with the
eigenvalue $a\varepsilon$ reads 333Note that the present convention is
different from that in Ref. Chen  _et al._ (2017).

$\displaystyle\Psi^{(+)}_{+}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{ip_{z}z}}{\sqrt{\varepsilon+m}}\begin{pmatrix}(\varepsilon+m)\phi_{l,k}\\\
0\\\ p_{z}\phi_{l,k}\\\ i\sqrt{2qB\lambda_{l,k}}\varphi_{l,k}\end{pmatrix},$
(8) $\displaystyle\Psi^{(+)}_{-}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{ip_{z}z}}{\sqrt{\varepsilon+m}}\begin{pmatrix}0\\\
(\varepsilon+m)\varphi_{l,k}\\\ -i\sqrt{2qB\lambda_{l,k}}\phi_{l,k}\\\
-p_{z}\varphi_{l,k}\end{pmatrix},$ (9) $\displaystyle\Psi^{(-)}_{+}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{-ip_{z}z}}{\sqrt{\varepsilon+m}}\begin{pmatrix}-p_{z}\phi_{l,k}\\\
i\sqrt{2qB\lambda_{l,k}}\varphi_{l,k}\\\ -(\varepsilon+m)\phi_{l,k}\\\
0\end{pmatrix},$ (10) $\displaystyle\Psi^{(-)}_{-}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{-ip_{z}z}}{\sqrt{\varepsilon+m}}\begin{pmatrix}-i\sqrt{2qB\lambda_{l,k}}\phi_{l,k}\\\
p_{z}\varphi_{l,k}\\\ 0\\\ -(\varepsilon+m)\varphi_{l,k}\end{pmatrix}.$ (11)

Here we have introduced

$\begin{split}\phi_{l,k}&=\mathrm{e}^{il\theta}\Phi_{l}(\lambda_{l,k},\tfrac{1}{2}qBr^{2})\,,\\\
\varphi_{l,k}&={\rm
sgn}(j)\mathrm{e}^{i(l+1)\theta}\Phi_{l+1}(\lambda_{l,k}-1,\tfrac{1}{2}qBr^{2})\,,\end{split}$
(12)

where

$\begin{split}\Phi_{l}(\lambda,x)=&\frac{1}{|l|!}\Bigg{[}\frac{\Gamma(\lambda+l+1)}{\Gamma(\lambda+1)}\Bigg{]}^{\frac{{\rm
sgn}(j)}{2}}x^{\frac{|l|}{2}}\mathrm{e}^{-x/2}\\\
&\times{}_{1}F_{1}(-\lambda+\frac{|l|-l}{2},|l|+1,x)\end{split}$ (13)

with ${}_{1}F_{1}$ the Kummer’s function of the first kind. If $\lambda$ is
integer, i.e. $R\to\infty$, we have

$\Phi_{l}(\lambda,x)=\biggl{[}\frac{\lambda!}{(\lambda+l)!}\biggr{]}^{\frac{{\rm
sgn}(j)}{2}}x^{\frac{|l|}{2}}\mathrm{e}^{-x/2}L^{|l|}_{\lambda-(|l|-l)/2}(x)$
(14)

with $L^{l}_{n}$ the Laguerre polynomials. We can construct the propagator
from the solutions,

$\begin{split}S(x,x^{\prime})&=i\int\frac{\mathrm{d}p_{0}\mathrm{d}p_{z}}{(2\pi)^{2}}\sum_{l=-\infty}^{\infty}\sum_{k=1}^{\infty}\frac{1}{\pi
R^{2}N^{2}_{lk}}\\\ &\quad\times\frac{\mathrm{e}^{-i(p_{0}-\Omega j-\mu)\Delta
t+ip_{z}\Delta
z}}{p_{0}^{2}-\varepsilon^{2}+i\epsilon}\mathcal{S}(p_{0}),\end{split}$ (15)

with $\Delta x=x-x^{\prime}$ and

$\displaystyle\mathcal{S}(p_{0})=\begin{pmatrix}\mathcal{M}_{+}&\mathcal{N}\\\
-\mathcal{N}&\mathcal{M}_{-}\end{pmatrix},$ (16)
$\displaystyle\mathcal{M}_{\pm}=(\pm
p_{0}+m)\begin{pmatrix}\phi\phi^{*\prime}&0\\\
0&\varphi\varphi^{*\prime}\end{pmatrix},$ (17)
$\displaystyle\mathcal{N}=\begin{pmatrix}-p_{z}\phi\phi^{*\prime}&i\sqrt{2qB\lambda_{l,k}}\phi\varphi^{*\prime}\\\
-i\sqrt{2qB\lambda_{l,k}}\varphi\phi^{*\prime}&p_{z}\varphi\varphi^{*\prime}\end{pmatrix}.$
(18)

Here we have abbreviated $\phi_{l,k}(x)$ by $\phi$,
$\phi^{*}_{l,k}(x^{\prime})$ by $\phi^{*\prime}$, and so on.

The propagator for $qB<0$ is obtained from Eq. (15). The difference between
the propagators for $qB>0$ and $qB<0$ appears only in the sign of angular
momentum; when the sign of $qB$ changes, a charged particle moves to an
opposite direction and its spin is also flipped. For this reason, we take the
complex conjugate of $\mathcal{S}(p_{0})$, interchange $\phi$ and $\varphi$,
and replace $\mathrm{e}^{i\Omega j}$ with $\mathrm{e}^{-i\Omega j}$. Then we
can construct the propagator for $qB<0$, as follows:

$\begin{split}\tilde{S}(x,x^{\prime})&=i\int\frac{\mathrm{d}p_{0}\mathrm{d}p_{z}}{(2\pi)^{2}}\sum_{l=-\infty}^{\infty}\sum_{k=1}^{\infty}\frac{1}{\pi
R^{2}N^{2}_{l,k}}\\\ &\quad\times\frac{\mathrm{e}^{-i(p_{0}+\Omega
j-\mu)\Delta t+ip_{z}\Delta
z}}{p_{0}^{2}-\varepsilon^{2}+i\epsilon}\tilde{\mathcal{S}}(p_{0}),\end{split}$
(19)

with

$\displaystyle\tilde{\mathcal{S}}(p_{0})=\begin{pmatrix}\tilde{\mathcal{M}}_{+}&\tilde{\mathcal{N}}\\\
-\tilde{\mathcal{N}}&\tilde{\mathcal{M}}_{-}\end{pmatrix},$ (20)
$\displaystyle\tilde{\mathcal{M}}_{\pm}=(\pm
p_{0}+m)\begin{pmatrix}\varphi^{*}\varphi^{\prime}&0\\\
0&\phi^{*}\phi^{\prime}\end{pmatrix},$ (21)
$\displaystyle\tilde{\mathcal{N}}=\begin{pmatrix}-p_{z}\varphi^{*}\varphi^{\prime}&-i\sqrt{2|qB|\lambda_{l,k}}\varphi^{*}\phi^{\prime}\\\
i\sqrt{2|qB|\lambda_{l,k}}\phi^{*}\varphi^{\prime}&p_{z}\phi^{*}\phi^{\prime}\end{pmatrix}.$
(22)

The sign difference in front of $\Omega j$ in $\mathcal{S}(p_{0})$ and
$\tilde{\mathcal{S}}(p_{0})$ implies that $\Omega j$ can be regarded as an
effective isospin chemical potential. The same is true for charged scalar
particles, as seen in the dispersion (1) Liu and Zahed (2018a). We should
mention that this effective isospin chemical potential is induced only if both
of magnetic field and rotation are applied. Indeed, when $B=0$, the symmetry
$j\leftrightarrow-j$ is restored, so $\Omega j$ cannot be interpreted as an
isospin chemical potential; and when $\Omega=0$, there is simply no such
effective chemical potential.

##  IV Schwinger Phase

The Dirac propagator in a constant magnetic field is in general gauge
dependent (the results presented in Sec. III is for the symmetric gauge). In
Minkowski spacetime, the gauge dependence is completely incorporated by the
Schwinger phase. This is true also for the rotating frame as we show in the
following, and the generalization to an arbitrary curved spacetime will be
present elsewhere.

The Schwinger phase $\Theta(x,x^{\prime})$ is defined through
$S(x,x^{\prime})=\mathrm{e}^{i\Theta(x,x^{\prime})}S_{\rm inv}(x,x^{\prime})$
so that $\Theta(x,x)=0$ is satisfied and $S_{\rm inv}$ is gauge invariant. The
condition $\Theta(x,x)=0$ means that the Schwinger phase is independent of the
path connecting $x$ and $x^{\prime}$. As a biscalar, $\Theta(x,x^{\prime})$ is
invariant under coordinate transformation. The expression in rotating frame
can thus be obtained from the usual Schwinger phase in Minkowski spacetime.
For a constant $F_{\mu\nu}$ (i.e., $\nabla_{\rho}F_{\mu\nu}=0$ with the
covariant derivative $\nabla_{\rho}$), the result is

$\Theta(x,x^{\prime})=-q\int_{x}^{x^{\prime}}\left[A_{\mu}(z)+\frac{1}{2}F_{\mu\rho}\nabla^{\rho}_{z}s(z,x)\right]\mathrm{d}z^{\mu}\,,$
(23)

where the integral is along an arbitrary path from $x$ to $x^{\prime}$. We
have introduced the Synge’s world function $s(a,b)$, which is half the squared
geodesic distance between $a$ and $b$ Poisson  _et al._ (2011):

$s(a,b)=\frac{1}{2}\int_{0}^{1}g_{\mu\nu}(\xi(\tau))\frac{\mathrm{d}\xi^{\mu}}{\mathrm{d}\tau}\frac{\mathrm{d}\xi^{\nu}}{\mathrm{d}\tau}\mathrm{d}\tau$
(24)

with $\xi^{\mu}(0)=a^{\mu}$ and $\xi^{\mu}(1)=b^{\mu}$. This is reduced to
$s(a,b)=(1/2)\eta_{\mu\nu}(a-b)^{\mu}(a-b)^{\nu}$ in Minkowski spacetime. We
can verify the path independence of $\Phi(x,x^{\prime})$, by showing that the
curl of the integrand in Eq. (23) vanishes after substituting the relation
$\nabla^{a}_{\mu}\nabla^{a}_{\nu}s(a,b)=g_{\mu\nu}(a)$ 444This relation holds
for any spacetime with zero Riemann curvature..

When we choose the integral path as the geodesic from $x$ to $x^{\prime}$, the
second term in Eq. (23) vanishes because $\nabla^{\mu}_{z}s(z,x)$ is tangent
to the geodesic. We are left with a neat expression for the Schwinger phase

$\Theta(x,x^{\prime})=-q\int_{x}^{x^{\prime}}A_{\mu}(z)\mathrm{d}z^{\mu}$ (25)

with geodesic being the integral path. See Appendix A for an explicit
representation of the geodesic in rotating frame.

To end this section, we show how the Schwinger phase can be factored out. For
simplicity, we focus on an unbounded system, where the fermion propagator is
computed with the following replacement Chen  _et al._ (2016):

$\begin{split}&\lambda\in\mathbb{N}\,,\quad\frac{1}{\pi
R^{2}N^{2}_{lk}}\to\frac{|qB|}{2\pi}\,,\quad-\lambda\leq l\leq-\lambda+N\
\end{split}$ (26)

with $N=\lfloor qBS/2\pi\rfloor$. We should note that $N$’s of $u$ and $d$
quarks are different because so are their charges: $q_{u}=2e/3$ and
$q_{d}=-e/3$. Also we consider that magnetic field is strong enough to justify
the LLL approximation. Summing over all $l$’s under the replacement
$N\to\infty$ , then we arrive at

$\begin{split}S_{\rm
LLL}(x_{1},x_{2})&=i\chi_{\perp}\int\frac{\mathrm{d}p_{0}\mathrm{d}p_{z}}{(2\pi)^{2}}\frac{\mathrm{e}^{-i(p_{0}-\Omega/2-\mu)\Delta
t+ip_{z}\Delta z}}{p_{0}^{2}-p_{z}^{2}-m^{2}+i\epsilon}\\\
&\quad\times(p_{0}\gamma^{0}-p_{z}\gamma^{3}+m)(1+i\gamma^{1}\gamma^{2})\,,\end{split}$
(27)
$\displaystyle\displaystyle\chi_{\perp}(x_{1},x_{2})=\exp\biggl{[}\frac{i}{2}qBr_{1}r_{2}\sin\Delta\vartheta-\frac{1}{4}qBC^{2}_{\perp}\biggr{]}\,,$
(28) $\displaystyle
C^{2}_{\perp}=r_{2}^{2}-2r_{1}r_{2}\cos\Delta\vartheta+r_{1}^{2}\,,$ (29)

where we define $\Delta\vartheta=\Delta\theta+\Omega\Delta t$. Here we notice
that the first term in Eq. (28) is the Schwinger phase with the integral path
along the geodesic:

$-q\int^{x_{2}}_{x_{1}}A_{\mu}\mathrm{d}z^{\mu}=\frac{1}{2}qBr_{1}r_{2}\sin\Delta\vartheta\,,$
(30)

where we employ the symmetric gauge. It should be noticed here that the second
term in Eq. (28) is gauge invariant. We note that

$\begin{split}\int^{1}_{0}\sqrt{g_{\mu\nu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\tau}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}\tau}}\mathrm{d}\tau=\sqrt{\Delta
t^{2}-C^{2}_{\perp}-\Delta z^{2}}\,.\end{split}$ (31)

This implies that $C_{\perp}$ is just the perpendicular distance measured in
Minkowski spacetime. For $\Omega=0$, Eq. (27) reproduces the usual LLL Dirac
propagator under a constant magnetic field in Minkowski spacetime.

##  V condensates

In this section, we will use the two-flavor NJL model to analyze the chiral
condensate and charged-pion condensate. For zero current quark mass, the NJL
Lagrangian in rotating frame is

$\begin{split}\mathcal{L}_{\rm
NJL}&=\bar{\psi}iD\psi+\frac{G}{2}\Bigl{[}(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma^{5}\vec{\tau}\psi)^{2}\Bigr{]}\,,\\\
iD&=i\gamma^{\mu}\nabla_{\mu}+\mu_{\rm{B}}\gamma^{0}\,.\end{split}$ (32)

Here $\psi=(u,d)^{T}$ is the two flavor quark field,
$\nabla_{\mu}=\partial_{\mu}+iQA_{\mu}+\Gamma_{\mu}$ the covariant derivative,
$\mu_{\rm{B}}$ (1/3 of) the baryon chemical potential,
$Q={\rm{diag}}(q_{u},q_{d})={\rm{diag}}(2e/3,-e/3)$ the charge matrix in
flavor space, and $\vec{\tau}$ the Pauli matrix. In the mean field
approximation, the one-loop effective action reads

$\begin{split}\Gamma(\sigma,\vec{\pi})&=-\int\mathrm{d}^{4}x\frac{\sigma^{2}+\vec{\pi}^{2}}{2G}-i\,{\rm
Tr}\ln(iD-\sigma-i\gamma^{5}\vec{\pi}\cdot\vec{\tau})\,,\end{split}$ (33)

where we introduce $\sigma=-G\langle\bar{\psi}\psi\rangle$ and
$\vec{\pi}=-G\langle\bar{\psi}i\gamma^{5}\vec{\tau}\psi\rangle$. While the
presence of the magnetic field spoils the $SU_{\rm V}(2)\otimes SU_{\rm A}(2)$
symmetry, its diagonal part remains. This fact allows us to eliminate the
neutral pion condensate by a chiral rotation. Hereafter we will always assume
$\pi^{0}\equiv\pi^{3}=0$.

Now let us employ the imaginary-time formalism. Then the effective action
transforms into the thermodynamic potential

$\displaystyle V_{\rm{eff}}(\sigma,\vec{\pi})$ $\displaystyle=$
$\displaystyle-i\frac{1}{\beta V}\Gamma_{\rm E}(\sigma,\vec{\pi}),$ (34)

with $\Gamma_{E}$ being the one-loop effective action after the Wick rotation
to Euclidean spacetime. For the current setup, it is practically hard to find
the global minimum of $V_{\rm{eff}}(\sigma,\vec{\pi})$, which corresponds to
the ground state. Instead, we will first calculate the minimum of
$V_{\rm{eff}}(\sigma,\vec{0})$, and then analyze its stability against the
charged-pion fluctuation. In other words, we perform the Ginzburg-Landau
expansion up to the second order in the pion fields. This is justified for the
analysis around a second-order or a weak first-order phase transition, which
is enough for our purpose. The thermodynamic potential that we examine is thus
written as

$\begin{split}V_{\rm{eff}}&=V_{\rm{eff}}^{(0)}+V_{\rm{eff}}^{(2)}+\dots\,,\\\
V_{\rm{eff}}^{(0)}&=\frac{1}{\beta V}\int\mathrm{d}^{4}x_{\rm
E}\frac{\sigma^{2}}{2G}-\frac{1}{\beta V}{\rm Tr}\ln(iD-\sigma)\,,\\\
V_{\rm{eff}}^{(2)}&=\frac{1}{\beta V}\int\mathrm{d}^{4}x_{\rm
E}\frac{\vec{\pi}^{2}}{2G}-\frac{1}{2\beta V}{\rm
Tr}[(iD-\sigma)^{-1}\gamma^{5}\vec{\pi}\cdot\vec{\tau}]^{2}\,.\end{split}$
(35)

The linear terms do not appear due to the invariance of
$\Gamma(\sigma,\vec{\pi})$ under the $\mathbb{Z}_{2}$ transformation
$\vec{\pi}\rightarrow-\vec{\pi}$. We note that each term in Eq. (35) is gauge
invariant, as the gauge transformation leads to $A_{\mu}\to
A_{\mu}+\partial_{\mu}\alpha(x)$ and $\pi^{\pm}(x)\rightarrow e^{\pm
ie\alpha(x)}\pi^{\pm}(x)$.

Let us prepare several parameter choices in our NJL model analysis. For the
numerical calculation in this section, we choose the model parameters as

$G=11\Lambda^{-2},\ 17\Lambda^{-2}\,,\quad R=30\Lambda^{-1}\,.$ (36)

Besides, we note that the NJL model is nonrenormalizable. Hereafter, when we
carry out the momentum sum, we shall implicitly introduce the following smooth
cutoff regulator Gorbar  _et al._ (2011); Chen  _et al._ (2016):

$f(k_{f};\Lambda,\delta\Lambda)=\frac{\sinh(\Lambda/\delta\Lambda)}{\cosh(k_{f}/\delta\Lambda)+\cosh(\Lambda/\delta\Lambda)}$
(37)

with $k_{f}=\sqrt{2q_{f}B\lambda+p_{z}^{2}}$. Here $\Lambda$ is the UV cutoff
scale, and $\delta\Lambda$ characterizes the smoothness of the UV cutoff,
which we choose

$\delta\Lambda=0.05\Lambda\,.$ (38)

The parameter choice in Eqs. (36)-(38) is almost the same as that in Ref. Chen
_et al._ (2017) 555The former coupling $G=11\Lambda^{-2}$ in the two-flavor
NJL model is the same as $G=22\Lambda^{-2}$ in the one-flavor case.. In the
following, we focus on the zero density case $\mu_{\rm{B}}=0$ and the
negatively finite density case $\mu_{\rm{B}}<0$. The reason why we adopt
$\mu_{\rm{B}}<0$ will be explained in Section V.2.

###  V.1 Chiral condensate

Figure 1: Chiral condensate $\sigma$ as a function of the angular velocity for
different baryon chemical potentials with $G=11\Lambda^{-2}$ (upper panel) and
$G=17\Lambda^{-2}$ (lower panel). We employ $|q_{d}B|=0.1\Lambda^{2}$ and the
temperature is set to be zero.

Due to the numerical difficulty to calculate the Ginzburg-Landau coefficients,
we focus on an unbounded system, where Eq. (LABEL:eq:Landau) is applicable.
The calculation of the chiral condensate in a finite cylinder with no-flux
boundary condition is represented in Appendix B. We also treat $\sigma$ to be
independent of the spacetime coordinate. Then the chiral condensate $\sigma$
is determined through the gap equation $\partial
V^{(0)}_{\rm{eff}}/\partial\sigma=0$, that is,

$\begin{split}\frac{\sigma}{G}=\frac{\sigma}{S}\int\frac{\mathrm{d}p}{2\pi}\sum_{\lambda,l,a,f}\frac{\alpha_{\lambda}}{\varepsilon}\biggl{[}\frac{1}{2}-n_{F}(\varepsilon_{f}-a\tilde{\mu}_{f})\biggr{]}\,,\end{split}$
(39)

where we denote $n_{F}(x)=1/(\mathrm{e}^{\beta x}+1)$, $\tilde{\mu}_{f}={\rm
sgn}(q_{f})\Omega j+\mu_{\rm{B}}$,
$\varepsilon_{f}=\sqrt{p_{z}^{2}+2q_{f}B\lambda+\sigma^{2}}$, and
$\alpha_{\lambda}=2-\delta_{0\lambda}$. The factor $\alpha_{\lambda}$ comes
from that $\Phi_{l+1}(-1,\tfrac{1}{2}|qB|r^{2})=0$ for arbitrary $r$; in the
unbounded system either the spin-up or down cannot occupy the LLL. We also
note that the first and second terms in the brackets of Eq. (39) correspond to
the vacuum part and the effective finite-density part with chemical potential
$\tilde{\mu}_{f}={\rm sgn}(q_{f})\Omega j+\mu_{\rm{B}}$, respectively. At zero
temperature, these terms (after the summation over $a$) are reduced as

$\sum_{a}\biggl{[}\frac{1}{2}-n_{F}(\varepsilon_{f}-a\tilde{\mu}_{f})\biggr{]}\to\theta(\varepsilon_{f}-|\tilde{\mu}_{f}|)\,.$
(40)

For this reason, we expect that Eq. (39) for large $\Omega$ or $\mu_{\rm{B}}$
has no nontrivial solution. This reflects that the effective Fermi surface
induced by $\tilde{\mu}$ suppresses the low-energy mode excitation, which is
required to form the chiral condensate. At the same time, we readily find that
the scale to character the onset of the rotational effect is

$\mu_{N}=\Omega N\,,$ (41)

which is the rotation-induced effective chemical potential for the maximum
angular momentum Chen  _et al._ (2016).

In Fig. 1, the upper (lower) panel shows the numerical solution for Eq. (39)
at zero temperature with $G=11\Lambda^{-2}$ ($G=17\Lambda^{-2}$). The result
with $\mu_{\rm{B}}=0$ and $G=11\Lambda^{-2}$ (the black line on the upper
panel) is parallel to that obtained in Ref. Chen  _et al._ (2016), expect for
the number of flavors. We confirm from Fig. 2 that the chiral condensate is
more destructed either for larger $\Omega$ or larger $\mu_{\rm{B}}$, as we
have explained below Eq. (40). The destructions of the condensate by rotation
correspond to the rotational magnetic inhibition Chen  _et al._ (2016).

At the end of this subsection, we discuss the difference between the
rotational effects on each flavor. From Eq. (40), we find that a $j$-mode of
$u$ quark receives the finite-density effect with $\tilde{\mu}_{u}=\Omega
j-|\mu_{\rm{B}}|$, while the one of $d$ with $\tilde{\mu}_{d}=-\Omega
j-|\mu_{\rm{B}}|$. Hence, it might seem that the rotational effect on $u$ and
$d$ behaves only as an effective isospin chemical. On top of this, however,
the rotational effect involves the aspect of the baryon chemical potential.
The main reason is because the upper bound of $l$ for each flavor are
different, i.e., $l\lesssim N$ with $N\sim 2eBS/3$ for $u$ and with $N\sim
eBS/3$ for $d$. The rotational effect on $u$ thus differ from that on
$\bar{d}$, which results in the difference between $\langle\bar{d}d\rangle$
and $\langle\bar{u}u\rangle$. This imbalance of the condensates is one of the
reasons why there appears the bumpy structure for $\mu_{\rm{B}}=0$ and
$G=11\Lambda^{-2}$ around $0.4\lesssim\Omega R\lesssim 0.7$, while it is
absent in the single flavor case Chen  _et al._ (2016).

###  V.2 Charged-pion condensate

To calculate $V_{\rm{eff}}^{(2)}$, we make the following ansatz for the two-
pion function:

$\pi^{+}(x^{\prime})\pi^{-}(x)=\exp\biggl{[}ie\int^{x^{\prime}}_{x}A_{\mu}\mathrm{d}z^{\mu}\biggr{]}\tilde{\pi}^{+}(x^{\prime})\tilde{\pi}^{-}(x)\,,$
(42)

where the integral path is along the geodesic between $x$ and $x^{\prime}$,
and $\tilde{\pi}^{+}(x^{\prime})\tilde{\pi}^{-}(x)$ is gauge invariant. In
other words, the gauge dependent part of the correlator is assumed to be
extracted only as the Wilson line. We emphasize that a similar ansatz is also
employed to construct the gauge invariant two-point observables in the context
of the usual electric superconductivity Frohlich  _et al._ (1981); Bricmont
and Frohlich (1983). To proceed, we further suppose $\tilde{\pi}^{+}$ and
$\tilde{\pi}^{-}$ to be constant spatially and temporally. Then the second
order term of the thermodynamic potential is written as

$\begin{split}V_{\rm{eff}}^{(2)}=C^{(2)}\tilde{\pi}^{+}\tilde{\pi}^{-}\,,\end{split}$
(43)

where $C^{(2)}$ is evaluated with the quark propagator. It is important to
note that $C^{(2)}$ is gauge independent since the Schwinger phases in the
quark propagators compensate the Wilson line in Eq. (42) exactly.

In the Ginzburg-Landau analysis, $C^{(2)}<0$ characterises the instability of
the $\sigma$ condensed state against the charged-pion fluctuation. Thus this
is the criterion for the phase transition towards the charged-pion
condensation. We can write down $C^{(2)}$, as follows:

$\displaystyle\displaystyle C^{(2)}=\frac{1}{2G}+\mathcal{C}$
$\displaystyle\displaystyle\mathcal{C}=\frac{q_{u}B|q_{d}B|}{S}\int\frac{\mathrm{d}p_{z}}{2\pi}\sum_{\lambda_{u},l_{u}}\sum_{\lambda_{d},l_{d}}\sum_{a}\frac{g_{u}-g_{d}}{\varepsilon_{u}\varepsilon_{d}(\varepsilon^{2}_{u}-\varepsilon^{2}_{d})}\,,\
\ $ (44)
$\begin{split}g_{f}&=\varepsilon_{f}\biggl{[}2q_{f}B\lambda_{f}I_{1}-\,{\rm
sgn}\Bigl{(}j_{u}j_{d}\Bigr{)}\sqrt{2q_{u}B\lambda_{u}}\sqrt{2|q_{d}B|\lambda_{d}}I_{2}\biggr{]}\\\
&\quad\times\biggl{[}\frac{1}{2}-n_{F}(\varepsilon_{f}-a\tilde{\mu}_{f})\biggr{]}\,,\\\
I_{1}&=\int_{0}^{\infty}\mathrm{d}rr\int_{0}^{\infty}\mathrm{d}r^{\prime}r^{\prime}J_{j_{u}+j_{d}}\Bigl{(}\tfrac{1}{2}eBrr^{\prime}\Bigr{)}\\\
\times&\biggl{[}\Phi^{u}_{\uparrow}(r)\Phi^{d}_{\downarrow}(r)\Phi^{u}_{\uparrow}(r^{\prime})\Phi^{d}_{\downarrow}(r^{\prime})+\Phi^{u}_{\downarrow}(r)\Phi^{d}_{\uparrow}(r)\Phi^{u}_{\downarrow}(r^{\prime})\Phi^{d}_{\uparrow}(r^{\prime})\biggr{]}\,,\\\
I_{2}&=2\int_{0}^{\infty}\mathrm{d}rr\int_{0}^{\infty}\mathrm{d}r^{\prime}r^{\prime}J_{j_{u}+j_{d}}\Bigl{(}\tfrac{1}{2}eBrr^{\prime}\Bigr{)}\\\
&\quad\times\Phi^{u}_{\uparrow}(r)\Phi^{d}_{\downarrow}(r)\Phi^{u}_{\downarrow}(r^{\prime})\Phi^{d}_{\uparrow}(r^{\prime})\,.\end{split}$
(45)

Here we denote $j_{f}=l_{f}+1/2$ as the total angular momentum, and $J_{l}(x)$
represents the first kind of the Bessel function. Also we introduced shorthand
notations:
$\Phi^{f}_{\uparrow}(r)=\Phi_{l_{f}}(\lambda_{f},\tfrac{1}{2}|q_{f}B|r^{2})$
and
$\Phi^{f}_{\downarrow}(r)=\Phi_{l_{f}+1}(\lambda_{f}-1,\tfrac{1}{2}|q_{f}B|r^{2})$.
We note that again the (effective) finite-density effect enters into Eq.
(V.2), as well as the gap equation (39). In this sense, similarly to the
chiral condensate $\sigma$, the low-energy mode accumulation is necessary to
generate the pion condensate; otherwise, we have $C_{2}\simeq(2G)^{-1}>0$, and
thus no pion condensate is generated. In Fig. 2, we present the zero-
temperature numerical results of $C^{(2)}$ with $|q_{d}B|=0.1\Lambda^{-2}$.
The results for $G=11\Lambda^{-2}$ ($G=17\Lambda^{-2}$) are plotted in the
upper (lower) panel. We note that there is a numerical cost to increase the
precisions of $I_{1}$ and $I_{2}$, which are the multi-integration of products
of highly oscillating functions. The nonsmooth behaviors in Fig. 2 is hence
the numerical artifacts that comes from less precisions of such evaluations.

Figure 2: The Ginzburg-Landau coefficient $C^{(2)}$ as a function of the
angular velocity with $G=11\Lambda^{-2}$ (upper panel) and $G=17\Lambda^{-2}$
(lower panel). We employ $|q_{d}B|=0.1\Lambda^{2}$ and the temperature is set
to be zero.

Let us first look at the $\mu_{\rm{B}}=0$ case, which corresponds to the black
lines in Fig. 2. Although the strong magnetic field $eB=0.1\Lambda^{2}$
generates a lot of the zero modes (the Landau quantization), it also increases
a constituent quark mass $\sigma$ (the magnetic catalysis). That is, the low-
energy modes never contribute to $C_{2}$ unless the chiral restoration takes
place due to an effective finite density effect (the rotational magnetic
inhibition). This is nothing but a Silver Blaze phenomenon; the rotational
effect on microscopic quantities at zero temperature becomes visible only if
the effective chemical potential $\mu_{\rm{eff}}\sim\Omega N$ exceeds the
infrared threshold $\sigma(\Omega=0)$. This is the reason why the
$\mu_{\rm{B}}=0$ case in Fig. 2 involves plateau behaviors around small
$\Omega$ regions. Also we have numerically checked that at zero baryon
chemical potential, arbitrary strong $B$ leads to $C^{(2)}>0$ irrelevantly to
the strength of $\Omega$. At the zero density system, hence, the charged-pion
condensate is disfavored Cao and Zhuang (2015); Adhikari (2019). This fact is
consistent with the argument based on the spin-alignment of $u$ and $\bar{d}$
in magnetic field and rotation; two fermions with the parallely aligned spins
cannot form the spin-0 composite states.

On the other hand, for a negatively large $\mu_{\rm{B}}$, the chiral
condensate is washed out, as seen in Fig. 1. In this case, the low-energy
modes can contribute to $C_{2}$. Therefore $C_{2}$ decreases only if $\sigma$
also does, which we find from Figs. 1 and 2. This is plausible if we would
regard $C_{2}$ as the charged-pion mass, which is partially determined by the
chiral condensate Gell-Mann  _et al._ (1968) except for the magnetic-induced
part. We also mention that the plateaus in the lower panel represent a Silver
Blaze phenomena for the effective chemical potential
$\mu_{N}+\mu_{\rm{B}}=\Omega N+\mu_{\rm{B}}$.

Furthermore, when the coupling constant is strong enough, $|\mathcal{C}|$
exceeds $(2G)^{-1}$ and thus the pion condensation can be realized. In Fig. 2,
the relevant case is only the result with $\mu_{\rm{B}}=-0.7\Lambda^{2}$ and
$G=17\Lambda^{-2}$ (the red line in the lower panel). This is understandable
in the sense that condensate cannot be formed in a weakly interacting system
and mimics a BCS to BEC type crossover. We note that a large $\mu_{\rm{B}}$
that overwhelms the rotational effects would always destroy mesonic
condensations. Only if we choose a negative $\mu_{\rm{B}}$ to let $\sigma$
disappear earlier, there will be a window allowing charged-pion condensation.

Here we explain why the negative chemical potential is required for the
charged-pion condensation. As we have mentioned already in Section V.1,
rotation affects each flavor not only as an effective isospin chemical
potential but also the baryon chemical potential. This would implies that we
schematically define the following effective chemical potentials induced by
rotation;
$\mu^{\text{rot}}_{u}=\mu^{\text{rot}}_{\text{isospin}}+\mu^{\text{rot}}_{\text{baryon}}$
and
$\mu^{\text{rot}}_{d}=-\mu^{\text{rot}}_{\text{isospin}}+\mu^{\text{rot}}_{\text{baryon}}$.
Therefore, combined with the negative baryon chemical potential that
compensates $\mu^{\text{rot}}_{\text{baryon}}$, the rotational effect yields
the purely isospin density system, which can yield the charged-pion condensate
Son and Stephanov (2001); He  _et al._ (2005); Sun  _et al._ (2007); He  _et
al._ (2006). Indeed, it can be checked that the critical baryon chemical
potential $|\mu_{\rm{B}}|\sim 0.7\Lambda$ is of the same order as
$\mu_{N}=\Omega N$ Chen  _et al._ (2016).

##  VI Discussion

In this paper, we revisited the possibility of a charged-pion condensation in
a magnetic field with rotation. We first discussed the gauge dependence of the
Dirac propagator in curved spacetime and show that the Schwinger phase is
given by a Wilson line along the geodesic. Then we calculated the second-order
coefficient in charged-pion field in the Ginzburg-Landau expansion of the two
flavor NJL model. We find that, at zero baryon chemical potential, the
charged-pion condensation is unlikely to happen in a wide region of the
parameters. However, turning on a negative baryon chemical potential to
compensate the positive effective baryon density due to the interplay between
magnetic field and rotation, we do find that charged-pion condensation may
happen for certain values of angular velocity and magnetic field.

As a by-product, we also studied the chiral condensate in a finite cylinder
with no-flux boundary condition in Appendix B, as an extension of the analysis
in Ref. Chen  _et al._ (2017) to a rotating system. We find that due to the
centrifugal force, the chiral condensate is suppressed near the boundary, and
there is a nontrivial competition between the rotational magnetic inhibition
and the surface magnetic catalysis.

Some comments are in order: (1) Our Ginzburg-Landau analysis would not be
rigorous for a large $|\mu_{\rm{B}}|$, which could leads to a strong first-
order phase transition. Nevertheless, our numerical studies provide a clearer
underlying picture of the charged-pion condensation in finite $eB$ and
$\Omega$. (2) The charged-pion condensate triggers also an electric
superconductivity, which may repel the magnetic field. This disfavors the
formation of the charged-pion condensate itself, especially in small magnetic
field. For large magnetic field, a vortex lattice may form to accommodate the
magnetic field, if it is a type-II superconductor. It will be interesting to
explore these phenomena. (3) Our analysis suggests that the condensation of
charged spin-one particles (such as $\rho$) may be favored by a rotation with
a magnetic field. This mechanism differs from the ones in pure magnetic field
and in finite isospin chemical potential; The former may suffer from the
constraint due to Vafa-Witten theorem Hidaka and Yamamoto (2013); In the
latter case, the inevitable pion condensate may make $\rho$ harder to condense
Brauner and Huang (2016). We emphasize that the $\rho$ condensation under
rotation is not prohibited by the Vafa-Witten theorem, because the presence of
rotation violates the positivity of the Dirac determinant. We leave this topic
to future works.

## Acknowledgment

We thank Gaoqing Cao, Kenji Fukushima, Lianyi He, and Dirk-Hermann Rischke for
useful discussions. This work is supported by NSFC through Grants No. 11535012
and No. 11675041 (X.-G. H.), and the China Postdoctoral Science Foundation
under Grant No. 2017M621345 (K. M.).

Note Added.— We learnt that Gaoqing Cao and Lianyi He had been working on the
same subject Cao and He (2019). They use a different integral path for the
Schwinger phase and the two-pion function, which results in different
conclusions from ours; see also the Appendix C.

##  Appendix A Geodesic curve in a rotating frame

We derive the geodesic between two points $x_{1}$ and $x_{2}$ in a rotating
frame in this appendix. The geodesic equation is

$\displaystyle\frac{\mathrm{d}^{2}x^{\mu}}{\mathrm{d}\tau^{2}}+\Gamma^{\mu}_{\nu\rho}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}\tau}\frac{\mathrm{d}x^{\rho}}{\mathrm{d}\tau}=0,$
(46)

where $\sigma$ is the affine parameter. For the $t$ and $z$ coordinates, the
solutions are

$\displaystyle t(\tau)=t_{1}+(t_{2}-t_{1})\tau,$ $\displaystyle
z(\tau)=z_{1}+(z_{2}-z_{1})\tau,$ (47)

which is the same as that in the Minkowski coordinate. For the transverse
coordinates,

$\begin{split}\frac{\mathrm{d}^{2}x}{\mathrm{d}\tau^{2}}-\Omega^{2}\Delta
t^{2}x-2\Omega\Delta t\frac{\mathrm{d}y}{\mathrm{d}\tau}&=0,\\\
\frac{\mathrm{d}^{2}y}{\mathrm{d}\tau^{2}}-\Omega^{2}\Delta
t^{2}y-2\Omega\Delta t\frac{\mathrm{d}x}{\mathrm{d}\tau}&=0,\end{split}$ (48)

where $\Delta t=t_{2}-t_{1}$. Introducing $\xi=x+iy$ we get

$\displaystyle\ddot{\xi}-\Omega^{2}\Delta t^{2}\xi+2i\Omega\Delta
t\dot{\xi}=0.$ (49)

The solution is

$\displaystyle\xi=C_{1}\mathrm{e}^{-i(\Omega\Delta
t\tau+\phi_{1})}+C_{2}\mathrm{e}^{-i(\Omega\Delta t\tau+\phi_{2})}\tau,$ (50)

i.e.,

$\begin{split}x(\tau)&=C_{1}\cos(\Omega\Delta
t\tau+\phi_{1})+C_{2}\tau\cos(\Omega\Delta t\tau+\phi_{2}),\\\
y(\tau)&=-C_{1}\sin(\Omega\Delta t\tau+\phi_{1})-C_{2}\tau\sin(\Omega\Delta
t\tau+\phi_{2}),\end{split}$ (51)

where $C_{1}$, $C_{2}$, $\phi_{1}$ and $\phi_{2}$ can be determined by the
condition

$\begin{split}x(0)&=x_{1},\quad x(1)=x_{2},\\\ y(0)&=y_{1},\quad
y(1)=y_{2}.\end{split}$ (52)

Finally, we have

$\begin{split}C_{1}&=r_{1},\;\;\;\phi_{1}=-\theta_{1},\\\
C_{2}^{2}&=r_{2}^{2}-2r_{1}r_{2}\cos(\Delta\theta+\Omega\Delta
t)+r_{1}^{2}\equiv C_{\perp}^{2},\\\ \phi_{2}&=-\Omega\Delta
t-\arctan\frac{r_{1}\sin(\theta_{1}-\Omega\Delta
t)-r_{2}\sin\theta_{2}}{r_{1}\cos(\theta_{1}-\Omega\Delta
t)-r_{2}\cos\theta_{2}}.\end{split}$

##  Appendix B Inhomogenous Chiral Condensate in a Cylinder

Figure 3: Chiral condensate in a finite cylinder as a function of the radial
coordinate $r$ for the choice of $R=30\Lambda^{-1}$ and
$|q_{d}B|=0.1\Lambda^{2}$. The temperature is zero.

In a finite cylinder with transverse radius $R$, the thermodynamic potential
$V_{\rm{eff}}^{(0)}$ becomes the functional of $\sigma(r)$ due to the breaking
of the translational invariance. Accordingly, the gap equation for $\sigma(r)$
reads $\delta V_{\rm{eff}}^{(0)}/\delta\sigma(r)=0$. Now we employ the local
density approximation, where $|\partial_{r}\sigma(r)|$ is supposed to be
negligible compared with $\sigma(r)^{2}$ Jiang and Liao (2016). Then the gap
equation is reduced to

$\begin{split}\frac{\sigma(r)}{G}&=\int\frac{\mathrm{d}p_{z}}{2\pi}\sum_{l,k}\sum_{a=}\sum_{f=u,d}\frac{\sigma(r)}{2\varepsilon_{f}(r)}\\\
&\quad\times\biggl{[}\frac{1}{2}-n_{F}(\varepsilon_{f}(r)-a\tilde{\mu}_{f})\biggr{]}\frac{[\Phi^{f}_{\uparrow}(r)]^{2}+[\Phi^{f}_{\downarrow}(r)]^{2}}{\pi
R^{2}N_{lk}^{2}}\,,\end{split}$ (53)

where
$\varepsilon_{f}(r)=\sqrt{p_{z}^{2}+2|q_{f}B|\lambda_{l,k}+\sigma(r)^{2}}$.
Note that $\lambda_{l,k}$ depends on $q$ when a boundary condition is imposed.
Here we adopt the same boundary condition as Chen  _et al._ (2017)

$\begin{split}&\Phi^{f}_{\uparrow}(R)=0\quad\text{for}\quad l\geq 0\\\
&\Phi^{f}_{\downarrow}(R)=0\quad\text{for}\quad l\leq-1\,.\end{split}$ (54)

This is one of the conditions so that the incoming $u$\- and $d$-quark current
fluxes (i.e., the isospin current flux) vanishes individually at the boundary
$r=R$, or equivalently, so that the hermiticity of the Hamiltonian is
respected. In the numerical calculation we choose the regulator (37) and the
parameters (36), (38) 666 In this section, we employ only the larger coupling
constant $G=17\Lambda^{-2}$..

The result for $\sigma(r)$ is shown in Fig. 3. We also show the case with
$\Omega=0$, which has been thoroughly studied in Ref. Chen  _et al._ (2017).
The surface magnetic catalysis is clearly seen 777The numerical result at the
vicinity of the boundary obviously contradicts the local density
approximation. This strong enhancement of $\sigma$ however comes from a number
of accumulating modes at $r\sim R$. In this sense, this magnetic response is
expected to be qualitatively unchanged even with stronger inhomogeneity of
$\sigma$ (see Ref. Chen  _et al._ (2017) for a detailed discussion).. Once the
rotation is turned on, the rotational suppression first occurs in the region
far away from the rotating axis where the centrifugal force is large. Also the
rotational effect is enhanced toward the center, as $\Omega$ increases. On the
other hand, at the vicinity of the boundary, the chiral condensate is
amplified by the surface magnetic catalysis, but with its effective region
shrinking with increasing $\Omega$; thus we observe a coexistence of and
competition between the rotational magnetic inhibition and surface magnetic
catalysis Chen  _et al._ (2016, 2017).

Figure 4: Chiral condensate $\sigma$ and Ginzburg-Landau coefficient $C^{(2)}$
as a function of the angular velocity for $|q_{d}B|=0.1\Lambda^{2}$ with the
integral path given in Eq. (55). The temperature is zero, and we choose
$G=17\Lambda^{-2}$.

##  Appendix C Another Choice of the Integral Path

In Section. IV we have shown that the integral path in the Schwinger phase
should be the geodesic. For the sake of comparison, we in this appendix choose
another integral path for the Schwinger phase. This also results in a
corresponding change of the integral path in Eq. (42). This path from $x_{1}$
to $x_{2}$ is specified by

$\begin{split}(t_{1},x_{1},y_{1},z_{1})&\to(t_{1},0,0,z_{1})\to(t_{2},0,0,z_{2})\\\
&\to(t_{2},x_{1},y_{1},z_{2})\to(t_{2},x_{2},y_{2},z_{2}),\end{split}$ (55)

where the arrows mean being connected by geodesics. Then the integral is

$iq\int^{x_{2}}_{x_{1}}A_{\mu}\mathrm{d}z^{\mu}=-i\frac{1}{2}qBr_{1}r_{2}\sin\Delta\theta,$
(56)

which has the same expression as the Schwinger phase in the Minkowski
coordinate, except that the coordinates here correspond to rotating frame.
This formula is used in Ref. Cao and He (2019).

The numerical result in this case is shown in Fig. 4. We find that $C^{(2)}$
begins to decrease with increasing $\Omega$, and the charged-pion condensation
can happen without adding a negative baryon chemical potential. However, we
still find that in certain region a negative baryon chemical potential can
catalyze the charged-pion condensation.

## References

  * Kharzeev  _et al._ (2008) D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph] .
  * Fukushima  _et al._ (2008) K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-ph] .
  * Huang (2016) X.-G. Huang, Rept. Prog. Phys. 79, 076302 (2016), arXiv:1509.04073 [nucl-th] .
  * Hattori and Huang (2017) K. Hattori and X.-G. Huang, Nucl. Sci. Tech. 28, 26 (2017), arXiv:1609.00747 [nucl-th] .
  * Kharzeev  _et al._ (2016) D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Prog. Part. Nucl. Phys. 88, 1 (2016), arXiv:1511.04050 [hep-ph] .
  * Gusynin  _et al._ (1994) V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499 (1994), [Erratum: Phys. Rev. Lett.76,1005(1996)], arXiv:hep-ph/9405262 [hep-ph] .
  * Gusynin  _et al._ (1996) V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Nucl. Phys. B462, 249 (1996), arXiv:hep-ph/9509320 [hep-ph] .
  * Miransky and Shovkovy (2015) V. A. Miransky and I. A. Shovkovy, Phys. Rept. 576, 1 (2015), arXiv:1503.00732 [hep-ph] .
  * Bali  _et al._ (2012a) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, JHEP 02, 044 (2012a), arXiv:1111.4956 [hep-lat] .
  * Bali  _et al._ (2012b) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, and A. Schafer, Phys. Rev. D86, 071502 (2012b), arXiv:1206.4205 [hep-lat] .
  * Preis  _et al._ (2011) F. Preis, A. Rebhan, and A. Schmitt, JHEP 03, 033 (2011), arXiv:1012.4785 [hep-th] .
  * Fukushima and Warringa (2008) K. Fukushima and H. J. Warringa, Phys. Rev. Lett. 100, 032007 (2008), arXiv:0707.3785 [hep-ph] .
  * Ferrer and de la Incera (2007) E. J. Ferrer and V. de la Incera, Phys. Rev. D76, 045011 (2007), arXiv:nucl-th/0703034 [NUCL-TH] .
  * Son and Stephanov (2008) D. T. Son and M. A. Stephanov, Phys. Rev. D77, 014021 (2008), arXiv:0710.1084 [hep-ph] .
  * Chernodub (2011) M. N. Chernodub, Phys. Rev. Lett. 106, 142003 (2011), arXiv:1101.0117 [hep-ph] .
  * Hidaka and Yamamoto (2013) Y. Hidaka and A. Yamamoto, Phys. Rev. D87, 094502 (2013), arXiv:1209.0007 [hep-ph] .
  * Sinha  _et al._ (2013) M. Sinha, X.-G. Huang, and A. Sedrakian, Phys. Rev. D88, 025008 (2013), arXiv:1306.3300 [astro-ph.HE] .
  * Cao and Huang (2016) G. Cao and X.-G. Huang, Phys. Lett. B757, 1 (2016), arXiv:1509.06222 [hep-ph] .
  * Brauner and Yamamoto (2017) T. Brauner and N. Yamamoto, JHEP 04, 132 (2017), arXiv:1609.05213 [hep-ph] .
  * Hattori  _et al._ (2017) K. Hattori, K. Itakura, and S. Ozaki, Phys. Lett. B775, 283 (2017), arXiv:1706.04913 [hep-ph] .
  * Ozaki  _et al._ (2016) S. Ozaki, K. Itakura, and Y. Kuramoto, Phys. Rev. D94, 074013 (2016), arXiv:1509.06966 [hep-ph] .
  * Chen  _et al._ (2017) H.-L. Chen, K. Fukushima, X.-G. Huang, and K. Mameda, Phys. Rev. D96, 054032 (2017), arXiv:1707.09130 [hep-ph] .
  * Deng and Huang (2016) W.-T. Deng and X.-G. Huang, Phys. Rev. C93, 064907 (2016), arXiv:1603.06117 [nucl-th] .
  * Jiang  _et al._ (2016) Y. Jiang, Z.-W. Lin, and J. Liao, Phys. Rev. C94, 044910 (2016), [Erratum: Phys. Rev.C95,no.4,049904(2017)], arXiv:1602.06580 [hep-ph] .
  * Wei  _et al._ (2019) D.-X. Wei, W.-T. Deng, and X.-G. Huang, Phys. Rev. C99, 014905 (2019), arXiv:1810.00151 [nucl-th] .
  * Liang and Wang (2005) Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005), [Erratum: Phys. Rev. Lett.96,039901(2006)], arXiv:nucl-th/0410079 [nucl-th] .
  * Adamczyk  _et al._ (2017) L. Adamczyk _et al._ (STAR), Nature 548, 62 (2017), arXiv:1701.06657 [nucl-ex] .
  * Vilenkin (1979) A. Vilenkin, Phys. Rev. D20, 1807 (1979).
  * Vilenkin (1980) A. Vilenkin, Phys. Rev. D21, 2260 (1980).
  * Son and Surowka (2009) D. T. Son and P. Surowka, Phys. Rev. Lett. 103, 191601 (2009), arXiv:0906.5044 [hep-th] .
  * Liu  _et al._ (2019) Y.-C. Liu, L.-L. Gao, K. Mameda, and X.-G. Huang, Phys. Rev. D99, 085014 (2019), arXiv:1812.10127 [hep-th] .
  * Chen  _et al._ (2016) H.-L. Chen, K. Fukushima, X.-G. Huang, and K. Mameda, Phys. Rev. D93, 104052 (2016), arXiv:1512.08974 [hep-ph] .
  * Jiang and Liao (2016) Y. Jiang and J. Liao, Phys. Rev. Lett. 117, 192302 (2016), arXiv:1606.03808 [hep-ph] .
  * Ebihara  _et al._ (2017) S. Ebihara, K. Fukushima, and K. Mameda, Phys. Lett. B764, 94 (2017), arXiv:1608.00336 [hep-ph] .
  * Chernodub and Gongyo (2017a) M. N. Chernodub and S. Gongyo, JHEP 01, 136 (2017a), arXiv:1611.02598 [hep-th] .
  * Huang  _et al._ (2018) X.-G. Huang, K. Nishimura, and N. Yamamoto, JHEP 02, 069 (2018), arXiv:1711.02190 [hep-ph] .
  * Chernodub and Gongyo (2017b) M. N. Chernodub and S. Gongyo, Phys. Rev. D95, 096006 (2017b), arXiv:1702.08266 [hep-th] .
  * Chernodub and Gongyo (2017c) M. N. Chernodub and S. Gongyo, Phys. Rev. D96, 096014 (2017c), arXiv:1706.08448 [hep-th] .
  * Liu and Zahed (2018a) Y. Liu and I. Zahed, Phys. Rev. Lett. 120, 032001 (2018a), arXiv:1711.08354 [hep-ph] .
  * Wang  _et al._ (2019a) L. Wang, Y. Jiang, L. He, and P. Zhuang, Phys. Rev. C100, 034902 (2019a), arXiv:1901.00804 [nucl-th] .
  * Zhang  _et al._ (2018) H. Zhang, D. Hou, and J. Liao, (2018), arXiv:1812.11787 [hep-ph] .
  * Wang  _et al._ (2019b) X. Wang, M. Wei, Z. Li, and M. Huang, Phys. Rev. D99, 016018 (2019b), arXiv:1808.01931 [hep-ph] .
  * Wang  _et al._ (2019c) L. Wang, Y. Jiang, L. He, and P. Zhuang, (2019c), arXiv:1901.04697 [nucl-th] .
  * Liu and Zahed (2018b) Y. Liu and I. Zahed, Phys. Rev. D98, 014017 (2018b), arXiv:1710.02895 [hep-ph] .
  * Son and Stephanov (2001) D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592 (2001), arXiv:hep-ph/0005225 [hep-ph] .
  * He  _et al._ (2005) L.-y. He, M. Jin, and P.-f. Zhuang, Phys. Rev. D71, 116001 (2005), arXiv:hep-ph/0503272 [hep-ph] .
  * Sun  _et al._ (2007) G.-f. Sun, L. He, and P. Zhuang, Phys. Rev. D75, 096004 (2007), arXiv:hep-ph/0703159 [hep-ph] .
  * He  _et al._ (2006) L. He, M. Jin, and P. Zhuang, Phys. Rev. D74, 036005 (2006), arXiv:hep-ph/0604224 [hep-ph] .
  * Hattori and Yin (2016) K. Hattori and Y. Yin, Phys. Rev. Lett. 117, 152002 (2016), arXiv:1607.01513 [hep-th] .
  * Poisson  _et al._ (2011) E. Poisson, A. Pound, and I. Vega, Living Rev. Rel. 14, 7 (2011), arXiv:1102.0529 [gr-qc] .
  * Gorbar  _et al._ (2011) E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D83, 085003 (2011), arXiv:1101.4954 [hep-ph] .
  * Frohlich  _et al._ (1981) J. Frohlich, G. Morchio, and F. Strocchi, Nucl. Phys. B190, 553 (1981).
  * Bricmont and Frohlich (1983) J. Bricmont and J. Frohlich, Phys. Lett. 122B, 73 (1983).
  * Cao and Zhuang (2015) G. Cao and P. Zhuang, Phys. Rev. D92, 105030 (2015), arXiv:1505.05307 [nucl-th] .
  * Adhikari (2019) P. Adhikari, Phys. Lett. B790, 211 (2019), arXiv:1810.03663 [nucl-th] .
  * Gell-Mann  _et al._ (1968) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968).
  * Brauner and Huang (2016) T. Brauner and X.-G. Huang, Phys. Rev. D94, 094003 (2016), arXiv:1610.00426 [hep-ph] .
  * Cao and He (2019) G. Cao and L. He, Phys. Rev. D100, 094015 (2019), arXiv:1910.02728 [nucl-th] .