# Inhomogeneous SU(2) gluon matter under rotation

Yin Jiang

###### Abstract

In this work a rotating SU(2) gluon system have been studied with the dyon
ensemble in dilute limit. By solving the rotation-modified Yang-Mills equation
we have obtained rotational corrections to the so-called dyon solutions with
arbitrary centers to $\mathcal{O}(\omega^{2})$ order and the corresponding
semi-classical potential. The radial position dependent deconfinement
temperature have been obtained by minimizing the semi-classical potential in
both real and imaginary angular velocity cases. Although without the
$\omega$-dependent coupling constant the critical temperature behaves
different from the lattice simulation at each radial position as the rotation
goes faster, its radial dependence is qualitatively the same as the lattice.
That is in the real velocity case the outer layer will deconfine more
difficult while the reverse is true in the imaginary velocity case.

###### keywords:

BPS dyon , inhomogeneity , rotation , color deconfinement

††journal: Physics Letters B

[first]organization=Physics Department, Beihang University, addressline=37
Xueyuan Rd, city=Beijing, postcode=100191, country=China

##  1 Introduction

Rotation-induced phenomena have been extensively examined since measurements
[1] of hyperon polarization at the final stage of Quark Gluon Plasma (QGP)
evolution, which is generated through relativistic heavy ion collisions. Both
global and local polarization measurements on hadrons [2, 3, 4, 5, 6, 7, 8, 9]
have spurred numerous investigations into fermionic systems under rotation,
typically associated with chiral restoration and anomalous transport [10, 11,
12, 13, 14, 15, 16, 17]. The rotating gluon system and deconfinement
transition have recently garnered increased attention due to the gluon’s
larger spin for polarization and the dominant population of soft gluons in
QGP. To explore a pure gluon system, lattice simulation of quantum
chromodynamics(QCD) is the primary choice. However, in the rotating system,
the pure gluon action becomes complex, thereby introducing a novel sign
problem. By employing an imaginary angular velocity, lattice simulations can
work and have yielded intriguing results [18, 19, 20, 22, 23, 21, 24].

The sign problem arising from rotation in lattice QCD renders the use of
effective models in theoretical approaches more significant. In the context of
a static, pure SU(N) gluon system, non-perturbative characteristics such as
the well-known color confinement are typically addressed through semi-
classical approaches, given the prevailing belief that the non-trivial
topology of the principal bundle $e^{iA(x)}$ is paramount in the underlying
theory. This conceptual understanding has prompted pivotal research into
various semi-classical solitons of the non-Abelian Yang-Mills equation. To
formulate an effective model, these solitons are often employed as the
fundamental degrees of freedom, with their associated interactions derived by
integrating out quantum fluctuations about them [25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35]. Consequently, selecting an appropriate model for the static
system and subsequently adapting it to incorporate rotation offers a viable
strategy for investigating the non-perturbative properties of a rotating gluon
system. However, this generalization is actually not so straightforward as it
sounds due to the inherent inhomogeneity of a rotating system.

The inhomogeneous problem in rotating matter have attracted great interests in
different branches of physics, such as the vortex state of rotating cold atom
systems [36, 37, 38, 39] and the glitch behavior of pulsar [40, 41]. For the
many-quark system inhomogeneous chiral condensate has also been studied in
[42] in local potential approximation and in [43] self-consistently with a
modified BdG equation. For gluon system and deconfinement problem, the
mechanism of color confinement is no longer particle pairing, thus a self-
consistent computation of soliton spatial distribution should be considered.
With such kind of effective models and lattice QCD simulation, several great
works have been done by introducing the rotation with the rotwisted boundary
condition [44, 45, 18, 19, 46, 20, 21, 22, 23, 24]. In this work we will start
from the SU(2) Yang-Mills theory and solve dyons under rotation with the usual
periodic boundary condition. In our last work [47] the on-axis dyon solutions
have been obtained. Such solutions can only be used to study the system in a
cell on axis. In order to study the inhomogeneity of the system we will solve
them at arbitrary position because the rotation-modified Yang-Mills equation
depends on the radial coordinates explicitly.

The inhomogeneity problem in rotating matter has been studied across various
domains of physics, such as the vortex state of rotating cold atom systems and
the glitch behavior observed in pulsars. Within the realm of many-quark
systems, the inhomogeneous chiral condensate has been explored using both
local potential approximations and a self-consistent approach incorporating a
modified BdG equation. Regarding the gluon system and the deconfinement
problem, the underlying mechanism is no longer particle pairing, thus a self-
consistent computation to ascertain the soliton spatial distribution is more
suitable. Leveraging such effective models of solitons and lattice QCD
simulations with imaginary velocity, pioneering research has been undertaken
by incorporating rotation through the implementation of rotwisted boundary
conditions. In the present study, we will focus on the SU(2) Yang-Mills theory
to resolve dyons under rotation with the standard periodic boundary
conditions. In our preceding work, strict solutions for dyons on the spinning
axis were successfully derived. However, these solutions are exclusively
applicable for examining the thermodynamics within an on-axis cell. To delve
into the system’s inhomogeneity, we are committed to resolving the rotation-
modified Yang-Mills equation exhibiting an explicit dependence on radial
coordinates. And it will be found that in the real velocity case the outer
layer will deconfine more difficult while the reverse is true in imaginary
velocity case as shown in the schematic picture in Fig. 1.

Figure 1: A schematic illustration of the radial position dependence of the
deconfinement temperature in both real(left) and imaginary(right) angular
velocity cases.

##  2 Strict dyon solutions in static and spinning cases

In this section we briefly review strict BPS dyon solutions in the static case
first and then the spinning case. Using these semi-classical solitons one can
model the confinement-deconfinement transition of a SU(2) gluon system at
finite temperature successfully. There are two types of dyons in the SU(2)
Yang-Mills theory, i.e. M and L dyons and their anti-dyons $\bar{\text{M}}$
and $\bar{\text{L}}$. And in the hedge-hog gauge a single M(+) or
$\bar{\text{M}}$(-), whose center locates at $\vec{c}$, is

$\displaystyle A^{a}_{4}(x;\rho)=\pm n_{a}(\frac{1}{r}-\rho\ \coth(\rho r))$
(2) $\displaystyle A^{a}_{m}(x;\rho)=\epsilon_{amk}n_{k}(\frac{1}{r}-\rho\
\text{csch}(\rho r)).$

where $\vec{r}=\vec{x}-\vec{c}$ and $\vec{n}=\vec{r}/r$. And L and
$\bar{\text{L}}$ dyons are obtained by replacing the parameter $\rho$ with
$\rho_{c}=2\pi T-\rho$. Intuitively this parameter $\rho$ characterizes the
size of a dyon. In stringy gauge all the dyons will be gauge transformed to
have the same asymptotic value at spatial infinity. They are strict solutions
of the Yang-Mills equation in the Euclidean time-space ($x_{4}=it$) which
corresponding to a many-gluon system at finite temperature

$\displaystyle\partial_{i}G_{4i}^{a}=-\epsilon_{abc}G_{4i}^{c}A_{i}^{b}$ (3)
$\displaystyle\partial_{i}G_{ij}^{a}=\epsilon_{abc}(G_{4j}^{b}A_{4}^{c}+G_{jn}^{c}A_{n}^{b}).$

In such a imaginary-time thermal field theory, BPS dyons are good solutions
because they satisfy the periodic boundary condition along the imaginary
temporal direction. And in static case the dyon and anti-dyon solutions are
actually sef-dual and anti-self-dual by considering the Bogomol’nyi inequality
[32]. Dyon solutions are non-trivial because their temporal components
$A^{a}_{4}\rightarrow\mp\rho n_{a}$ can be nonzero at spatial infinity, which
leads to a nonzero trace of the Polyakov loop

$\displaystyle
Tr(L)=Tr(\mathcal{P}e^{i\int_{0}^{\beta}dx_{4}A_{4}(x)})=2\cos(\beta\rho/2),$
(4)

at high temperature(deconfinement) and approaching zero(confinement) smoothly
as the fall of temperature. It means that the parameter $\rho$ can be treated
as an equivalent order parameter of confinement. And the $\rho$’s value is
usually solved self-consistently by minimizing the effective potential of the
system. Even when the $\rho$ is fixed there are still infinite number of dyon
solutions because the Eq. 3 is translational invariant. This means an
arbitrary vector $\vec{c}$ gives a different solution of dyon.

To construct a SU(2) gluon system with the semi-classical degree of freedom,
i.e. dyon, one need to obtain the dyon-dyon and dyon-anti-dyon interaction by
solving the semi-classical solution with larger topological charge. These
solutions can be treated as superposition of multi-dyons or anti-dyons in a
specific way, i.e. the Atiyah–Hitchin–Drinfeld–Manin (AHDM) construction [32].
By separating the effective action of such a solutions into summation of
single dyons and residual contributions, the residual terms can be identified
as the interaction between dyons/anti-dyons [28]. Once a type of non-trivial
configurations $\bar{A}(x;\\{c_{i}\\})$ for the vacuum has been chosen, the
1-loop partition function of the system is calculated as the standard
formalism

$\displaystyle\mathcal{Z}=\int DA_{\mu}\exp(-S[A])$
$\displaystyle=\int\Pi_{i}d\xi_{i}\exp(-S[\bar{A}])Det(-D^{2})$ (5)

where $S[A]=\int d^{4}xG^{2}/4g^{2}$ for a general Yang-Mills field. In the
dilute limit the interaction is negligible and the $e^{-S[\bar{A}]}$ can be
computed by summing one L/$\bar{\text{L}}$ and one M/$\bar{\text{M}}$ dyon’s
action together. Consider the vacuum can be a more general system composed
with more than one pair of L/$\bar{\text{L}}$ and M/$\bar{\text{M}}$, such a
dyon ensemble gives the partition function as

$\displaystyle\mathcal{Z}_{D}=\sum_{N_{M},N_{\bar{M}},N_{L},N_{\bar{L}}}\frac{1}{N_{M}!N_{\bar{M}}!N_{L}!N_{\bar{L}}!}$
$\displaystyle\times\left[\int
d^{3}c_{M}|{\bar{\rho}}|^{3}\exp(-S[A_{M}])\right]^{N_{M}+N_{\bar{M}}}$
$\displaystyle\times\left[\int
d^{3}c_{L}|{\bar{\rho}_{c}}|^{3}\exp(-S[A_{L}])\right]^{N_{L}+N_{\bar{L}}}$
(6)

where $\bar{\rho}$ is the uniform asymptotic value for both L and M dyons in
stringy gauge and $\rho d^{3}c$ serves as the integral measure. As a result
the non-perturbative semi-classical potential in dilute limit is

$\displaystyle U_{np}=-\frac{1}{V}ln(\mathcal{Z}_{D})$
$\displaystyle=-c\left[{|\bar{\rho}|}^{3}(\frac{\Lambda}{\pi
T})^{\frac{22\bar{\rho}}{6\pi T}}+|{\bar{\rho}}_{c}|^{3}(\frac{\Lambda}{\pi
T})^{\frac{22|\bar{\rho}_{c}|}{6\pi T}}\right]$ (7)

where $\bar{\rho}_{c}=2\pi T-\bar{\rho}$. Besides doing the full quantum
weight computation in Eq. 2, the perturbative part can also be obtained by
setting the an uniform background $A_{4}=\bar{\rho}\tau_{3}/2$ and integrating
out fluctuations of plane waves about it. It has been done as [48] and gives
the static perturbative potential as

$\displaystyle U_{p}=\frac{1}{12\pi^{2}T}\bar{\rho}^{2}{\bar{\rho}_{c}}^{2}$
(8)

The order parameter $\bar{\rho}$ value is determined by minimizing the total
effective potential $U_{np}+U_{p}$ at different temperature. With typical
parameters one can solve that $\bar{\rho}=\pi T$ corresponding to confinement
at low temperature and deviates from $\pi T$ which means deconfinement as
temperature increase.

From the very brief review in above one can find that the explicit semi-
classical solution is the key quantity for modelling the deconfinement phase
transition. Once a category of soliton has been obtained the effective
potential can be computed according to Eq. 2 by considering a classical
soliton ensemble. For a rotating SU(2) gluon system with uniform angular
velocity $i\vec{\omega}$, the task is to find a solution like

$\displaystyle A=A(x_{4},\varrho,\phi+\omega x_{4},z),$ (9)

satisfying the usual SU(2) Yang-Mills equation. The centripetal force holding
the rotation of each cell in the system is provided by the fundamental
interaction of gluons. The rotwisted boundary condition [18] along the $x_{4}$
dimension requires

$\displaystyle A(x_{4}+\beta,\varrho,\phi-\beta\omega+\omega(x_{4}+\beta),z)$
$\displaystyle=A(x_{4},\varrho,\phi+\omega x_{4},z).$ (10)

It is widely applied in the lattice formalism because no modification need to
be introduced into the Lagrangian density besides considering a unusual
boundary condition of each gauge field configuration. But it is not quite
convenient for analytical computation. By noticing the time derivative
relation as follows

$\displaystyle\partial_{4}A$ $\displaystyle=$ $\displaystyle
A^{(1,0,0,0)}(x_{4},\varrho,\phi+\omega x_{4},z)$ (11) $\displaystyle+$
$\displaystyle\omega A^{(0,0,1,0)}(x_{4},\varrho,\phi+\omega x_{4},z),$

one can find a equivalent formalism for this problem as follows. By solving a
rotation-modified Yang-Mills equation in Eq. 12 with the usual periodic
boundary condition, the Yang-Mills equation is

$\displaystyle\partial_{i}G_{4i}^{a}=-\epsilon_{abc}G_{4i}^{c}A_{i}^{b}$ (12)
$\displaystyle\partial_{i}G_{ij}^{a}=v_{i}\partial_{i}G_{4j}^{a}+\partial_{j}v_{i}G_{4i}^{a}+\epsilon_{abc}(G_{4j}^{b}A_{4}^{c}+G_{jn}^{c}A_{n}^{b})$

where the field strength tensor is

$\displaystyle
G_{4i}^{a}=\partial_{4}A_{i}^{a}-\partial_{i}A_{4}^{a}+\epsilon_{abc}A_{4}^{b}A_{i}^{c}+\partial_{j}v_{i}A_{j}^{a}-v_{j}\partial_{j}A_{i}^{a}$
$\displaystyle
G_{ij}^{a}=\partial_{i}A_{j}^{a}-\partial_{j}A_{i}^{a}+\epsilon_{abc}A_{i}^{b}A_{j}^{c}$
(13)

one can get a solution satisfying
$A(x_{4},\varrho,\phi,z)=A(x_{4}+\beta,\varrho,\phi,z)$. Then the target
solution in Eq. 9 can been obtained as well by simply replacing the $\phi$
here with $\phi+\omega x_{4}$. Evidently this procedure corresponds to going
into a co-rotating frame in which a curved metrics will produce the extra
terms from the derivative in Eq. 11. In the following we will follow this
approach to take the rotation into account.

In the rotating case with an imaginary angular velocity, one can not expect a
spherical symmetric solution as Eq. 2 because of the cylindrical symmetry of
the system. However, when dyons locate at the rotation axis $c_{1,2}=0$, the
rotation correction is very simple and strict. In our last work [47] it has
been found that such a on-axis solution $A^{D,rot}(x)$ is like

$\displaystyle A^{D,rot}=(A^{D}_{4}(x;\rho_{D})+\delta
A_{4},\leavevmode\nobreak\ \vec{A}^{D}(x;\rho_{D})).$ (14)

where the $\rho_{D}$ is the parameter in the static solution of dyon
$A^{D}(x;\rho)$ and the correction under rotation in the hedge-hog gauge is

$\displaystyle\delta A_{4}=\mp\omega_{a}\frac{\tau^{a}}{2}.$ (15)

where $\mp$ for the M and L dyon and $\pm$ for ${\bar{\text{M}}}$ and
${\bar{\text{L}}}$ respectively and $\tau_{a},a=1,2,3$ are Pauli matrices.
Clearly the asymptotic value of $A_{4}$ has been shifted by $\pm\omega$ for
different kind of dyons, but it can be checked explicitly that the solution
will not change the field strength tensor or dyon action. By requiring the
potential multiple dyon/anti-dyon solution has a uniform asymptotic behavior
$\bar{\rho}$ at spatial infinity, the parameter $\rho_{D}$ in the solution
should be chosen to cancel the $\omega$-dependent terms. And this makes the
action of each constituent dyon becomes

$\displaystyle S_{M,\bar{M}}=\frac{(\bar{\rho}\pm\omega)}{2\pi
T}\frac{8\pi^{2}}{g^{2}}$ $\displaystyle
S_{L,\bar{L}}=\frac{(\bar{\rho}_{c}\pm\omega)}{2\pi T}\frac{8\pi^{2}}{g^{2}}$
(16)

where $\bar{\rho}_{c}=2\pi T-\bar{\rho}$ and the scale of this model $\Lambda$
is related with the running coupling as $(\frac{\Lambda}{\pi
T})^{22/3}=e^{-8\pi^{2}/g^{2}}$. Considering the non-interacting dyon ensemble
and noticing the dyon and anti-dyon actions are now different, the partition
function Eq. 2 should be changed as

$\displaystyle\mathcal{Z}_{rot,D}=\sum_{N_{M},N_{\bar{M}},N_{L},N_{\bar{L}}}\frac{1}{N_{M}!N_{\bar{M}}!N_{L}!N_{\bar{L}}!}$
$\displaystyle\times\left[\int
d^{3}c_{M}sgn(\bar{\rho})(\bar{\rho}+\omega)^{3}\exp(-S[\bar{A}_{M}])\right]^{N_{M}}$
$\displaystyle\times\left[\int
d^{3}c_{\bar{M}}sgn(\bar{\rho})(\bar{\rho}-\omega)^{3}\exp(-S[\bar{A}_{\bar{M}}])\right]^{N_{\bar{M}}}$
$\displaystyle\times\left[\int
d^{3}c_{L}sgn(\bar{\rho}_{c})(\bar{\rho}_{c}+\omega)^{3}\exp(-S[\bar{A}_{L}])\right]^{N_{L}}$
$\displaystyle\times\left[\int
d^{3}c_{\bar{L}}sgn(\bar{\rho}_{c})(\bar{\rho}_{c}-\omega)^{3}\exp(-S[\bar{A}_{\bar{L}}])\right]^{N_{\bar{L}}}$
(17)

The $sgn(\rho)$ here corresponding to the $|\rho|$ in Eq. 2, is from that the
$\rho\tau_{3}/2=diag(\rho/2,-\rho/2)$ can be always replaced with
$diag(sgn(Re(\rho))\rho/2,-sgn(Re(\rho))\rho/2)$ in the action computation by
a global gauge transformation. Completing the summation, the non-perturbative
potential is extracted as

$\displaystyle U_{np}(T,\omega)=-\frac{1}{V}ln\mathcal{Z}_{rot,D}$
$\displaystyle=-\frac{c}{2}[sgn(\bar{\rho})(\bar{\rho}+\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})(\bar{\rho}+\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}+\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})(\bar{\rho}_{c}+\omega)}{6\pi T}}]$
$\displaystyle-\frac{c}{2}[sgn(\bar{\rho})(\bar{\rho}-\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})(\bar{\rho}-\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}-\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})(\bar{\rho}_{c}-\omega)}{6\pi T}}].$ (18)

This difference between caloron $\bar{\rho}$ and anti-caloron $\bar{\rho}_{c}$
contributions is induced by the parity violation in a globally rotating system
and will vanish when $\omega$ goes to zero.

The perturbative potential $U_{p}$ are obtained by integrating out Gaussian
fluctuations on top of a multi-dyon background according to reference [48]. In
such a rotating system eigen states of quantum fluctuations around the semi-
classical gauge field are cylindrical waves involving with Bessel functions.
The momentum integration is another three good quantum numbers, i.e.
transverse momentum $k_{T}=\xi_{n}^{(m)}/R$, longitudinal momentum $k_{z}$ and
angular momentum number $m$ corresponding to the polar radius, longitudinal
coordinate and azimuthal angle, where $R$ is the size of system and
$\xi_{n}^{(m)}$ the m-th zero of the Bessel function $J_{n}(r)$. The
transverse momentum is discretized by considering the finite boundary
condition $J_{n}(k_{T}R)=0$. The whole computation procedure is
straightforward but a little lengthy for this paper. One can refer Ref. [45]
for more details. The result is the same as the reference

$\displaystyle U_{p}(T,\omega)=$
$\displaystyle-\sum\limits_{\begin{subarray}{c}s,m=1\\\
n=-\infty\end{subarray}}^{+\infty}\frac{e^{\frac{isn\omega}{T}}cosh(\frac{is\omega}{T})}{\pi^{2}sR^{3}}\frac{4\xi_{n}^{(m)}cos(s\frac{\bar{\rho}}{T})}{J_{n+1}(\xi_{n}^{(m)})^{2}}$
(19) $\displaystyle\times
J_{n}(\xi_{n}^{(m)}\frac{r}{R})^{2}K_{1}(s\frac{\xi_{n}^{(m)}}{TR}).$

where $r=0$ for the on-axis case. One should beware that in all the above
computation the angular velocity is actually an imaginary one. And the result
for the real velocity case can be easily obtained by replacing the $\omega$
with $i\omega$. Although the rotation correction is strict and simple for on-
axis dyons, it is not enough for us to explore the radial dependence of a
rotating system. To study such a inhomogeneous property one should find
rotation corrections to dyons with arbitrary centers.

##  3 Rotation-correction to off-axis dyons

In this section we solve the off-axis dyon corrections with the imaginary
angular velocity. One will find the whole procedure is straightforward
algebraic computation, so the real velocity results will be obtained by
replacing $\omega$ with $i\omega$ eventually. As analyzed from the aspect of
symmetry in last section, it is natural that for a off-axis dyon the rotation-
induced correction are not so simple and strict as the on-axis case in Eq. 15.
Hence we expand the solution as series of $\omega^{n}$ or more precisely in
powers of velocity multiplications $\vec{v}^{n}\vec{u}^{m}$

$\displaystyle
A=A^{(0)}(\omega^{0})+A^{(1)}(\omega^{1})+A^{(2)}(\omega^{2})+...$ (20)

where $\vec{u}=\vec{\omega}\times\vec{c}$ and solve the Eq. 12 to the
$\mathcal{O}(\omega^{2})$. When the $\mathcal{O}(\omega^{0})$ solution is a
off-axis dyon at $\vec{c}=(y_{1},y_{2},y_{3})$, it is more convenient to
redefine the $\vec{x}-\vec{c}$ as the new $\vec{x}$. Such a translation can
preserve the spherical symmetry of the $\omega^{0}$ solution which is better
for the following analytical derivation. In the new $\vec{x}$ coordinate the
Yang-Mills equation is

$\displaystyle\partial_{i}G_{4i}^{a}=-\epsilon_{abc}G_{4i}^{c}A_{i}^{b}$ (21)
$\displaystyle\partial_{i}G_{ij}^{a}=(v_{i}+u_{i})\partial_{i}G_{4j}^{a}+\partial_{j}v_{i}G_{4i}^{a}+\epsilon_{abc}(G_{4j}^{b}A_{4}^{c}+G_{jn}^{c}A_{n}^{b})$

where the field strength tensor is

$\displaystyle
G_{4i}^{a}=\partial_{4}A_{i}^{a}-\partial_{i}A_{4}^{a}+\epsilon_{abc}A_{4}^{b}A_{i}^{c}+\partial_{j}v_{i}A_{j}^{a}-(v_{j}+u_{j})\partial_{j}A_{i}^{a}$
$\displaystyle
G_{ij}^{a}=\partial_{i}A_{j}^{a}-\partial_{j}A_{i}^{a}+\epsilon_{abc}A_{i}^{b}A_{j}^{c}$
(22)

and $\vec{u}$ is from the shift of coordinates. Noting that that both
$\vec{v}$ and $\vec{u}$ are $\omega^{1}$ order, expanding these equations to
$\omega^{2}$ gives

$\displaystyle\partial_{i}G_{4i}^{(0)a}=-\epsilon_{abc}G_{4i}^{(0)c}A_{i}^{(0)b}$
(23)
$\displaystyle\partial_{i}G_{ij}^{(0)a}=\epsilon_{abc}(G_{4j}^{(0)b}A_{4}^{(0)c}+G_{jn}^{(0)c}A_{n}^{(0)b})$

as the equations in static case and the $\mathcal{O}(\omega^{1})$ and
$\mathcal{O}(\omega^{2})$ equations are

$\displaystyle\partial_{i}G_{4i}^{(1)a}=-\epsilon_{abc}(G_{4i}^{(0)c}A_{i}^{(1)b}+G_{4i}^{(1)c}A_{i}^{(0)b})$
(24)
$\displaystyle\partial_{i}G_{ij}^{(1)a}=(v_{i}+u_{i})\partial_{i}G_{4j}^{(0)a}+\partial_{j}v_{i}G_{4i}^{(0)a}$
$\displaystyle+\epsilon_{abc}(G_{4j}^{(0)b}A_{4}^{(1)c}+G_{4j}^{(1)b}A_{4}^{(0)c}+G_{jn}^{(0)c}A_{n}^{(1)b}+G_{jn}^{(1)c}A_{n}^{(0)b})$

and

$\displaystyle\partial_{i}G_{4i}^{(2)a}=-\epsilon_{abc}(G_{4i}^{(1)c}A_{i}^{(1)b}+G_{4i}^{(0)c}A_{i}^{(2)b}+G_{4i}^{(2)c}A_{i}^{(0)b})$
$\displaystyle\partial_{i}G_{ij}^{(2)a}=(v_{i}+u_{i})\partial_{i}G_{4j}^{(1)a}+\partial_{j}v_{i}G_{4i}^{(1)a}$
$\displaystyle+\epsilon_{abc}(G_{4j}^{(1)b}A_{4}^{(1)c}+G_{4j}^{(0)b}A_{4}^{(2)c}+G_{4j}^{(2)b}A_{4}^{(0)c}$
$\displaystyle+G_{jn}^{(1)c}A_{n}^{(1)b}+G_{jn}^{(0)c}A_{n}^{(2)b}+G_{jn}^{(2)c}A_{n}^{(0)b})$
(25)

where the $\omega$-expansion of the field strength tensor is

$\displaystyle
G_{4i}^{(0)a}=\partial_{4}A_{i}^{(0)a}-\partial_{i}A_{4}^{(0)a}+\epsilon_{abc}A_{4}^{(0)b}A_{i}^{(0)c}$
$\displaystyle
G_{ij}^{(0)a}=\partial_{i}A_{j}^{(0)a}-\partial_{j}A_{i}^{(0)a}+\epsilon_{abc}A_{i}^{(0)b}A_{j}^{(0)c}$
(26)

as the $\mathcal{O}(\omega^{0})$ order and

$\displaystyle
G_{4i}^{(1)a}=\partial_{4}A_{i}^{(1)a}-\partial_{i}A_{4}^{(1)a}+\partial_{j}v_{i}A_{j}^{(0)a}-(v_{j}+u_{j})\partial_{j}A_{i}^{(0)a}$
$\displaystyle+\epsilon_{abc}(A_{4}^{(0)b}A_{i}^{(1)c}+A_{4}^{(1)b}A_{i}^{(0)c})$
(27) $\displaystyle
G_{ij}^{(1)a}=\partial_{i}A_{j}^{(1)a}-\partial_{j}A_{i}^{(1)a}+\epsilon_{abc}(A_{i}^{(1)b}A_{j}^{(0)c}+A_{i}^{(0)b}A_{j}^{(1)c})$

and

$\displaystyle
G_{4i}^{(2)a}=\partial_{4}A_{i}^{(2)a}-\partial_{i}A_{4}^{(2)a}+\partial_{j}v_{i}A_{j}^{(1)a}-(v_{j}+u_{j})\partial_{j}A_{i}^{(1)a}$
$\displaystyle+\epsilon_{abc}(A_{4}^{(1)b}A_{i}^{(1)c}+A_{4}^{(0)b}A_{i}^{(2)c}+A_{4}^{(2)b}A_{i}^{(0)c})$
$\displaystyle
G_{ij}^{(2)a}=\partial_{i}A_{j}^{(2)a}-\partial_{j}A_{i}^{(2)a}$
$\displaystyle+\epsilon_{abc}(A_{i}^{(1)b}A_{j}^{(1)c}+A_{i}^{(2)b}A_{j}^{(0)c}+A_{i}^{(0)b}A_{j}^{(2)c})$
(28)

the $\mathcal{O}(\omega^{1})$ and $\mathcal{O}(\omega^{2})$ terms. Taking the
$\bar{M}$ dyon for example, after substituting its spherical symmetric
solution as the $A^{(0)a}_{\mu}$ into above equations, although there are many
terms in each expression, Eq. 24 and 3 are actually all non-homogeneous linear
equations of $A^{(1)a}_{\mu}$ and $A^{(2)a}_{\mu}$. This means these equations
are solvable in principle with the help of corresponding Green functions.
However, the straightforward computation is too tedious because all the
components of field strength tensor are coupling together. Instead, we can try
to find the tensor structure by noticing non-homogeneous terms. Taking the
$G_{4i}^{a}$ equations for example. Non-homogeneous terms in Eq. 24 are in the
R.H.S of the following equation

$\displaystyle\mathcal{L}^{a}[A_{4}^{(1)},A_{k}^{(1)}]=(-x_{m}\omega_{m}x_{a}-\omega_{a}r^{2}+\epsilon_{aij}u_{i}x_{j})Q^{2}(r)$
(29)

where $\mathcal{L}^{a}$ is the homogeneous part in Eq. 24 and
$r^{2}=x_{m}x_{m}$. Then we can factorize $A_{4}^{(1)}$ according to the non-
homogeneous term as

$\displaystyle
A_{4}^{(1)a}=R_{1}(r)\omega_{a}+R_{2}(r)x_{m}\omega_{m}x_{a}+R_{3}(r)\epsilon_{aij}u_{i}x_{j}$
(30)

The first terms gives $\omega_{a}$ in the on-axis solution. And analyzing the
$A_{i}^{(1)a}$ equation in the same way one can choose

$\displaystyle
A_{i}^{(1)a}=S_{1}(r)u_{i}x_{a}+S_{2}(r)u_{a}x_{i}+(S_{3}(r)x_{i}x_{a}+S_{4}(r)\delta_{ia})u_{m}x_{m}$
(31)

Substituting the two factorizations into Eq. 24 we can obtain the solutions

$\displaystyle A_{4}^{(1)a}=\omega_{a}-\epsilon_{ija}u_{i}x_{j}Q(r)$
$\displaystyle A_{i}^{(1)a}=u_{i}x_{a}P(r)$ (32)

Once the $\mathcal{O}(\omega^{1})$ solutions have been obtained, the Eq. 3 can
be solved with the same procedure. The corresponding non-homogeneous terms
indicate second order corrections can be factorized as

$\displaystyle
A_{4}^{(2)a}={\tilde{R}}_{1}u^{2}x_{a}+{\tilde{R}}_{2}(u_{m}x_{m})^{2}x_{a}+{\tilde{R}}_{3}u_{m}x_{m}u_{a}$
$\displaystyle+{\tilde{R}}_{4}\xi_{a}+{\tilde{R}}_{5}\xi_{m}x_{m}x_{a}$
$\displaystyle
A_{i}^{(2)a}={\tilde{S}}_{1}(\omega_{m}u_{i}x_{m}x_{a}-\omega_{i}u_{m}x_{m}x_{a})+{\tilde{S}}_{2}\epsilon_{amn}u_{m}x_{n}u_{i}$
$\displaystyle+{\tilde{S}}_{3}\epsilon_{imn}u_{m}x_{n}u_{a}+{\tilde{S}}_{4}u_{k}x_{k}\epsilon_{amn}u_{m}x_{n}x_{i}$
$\displaystyle+{\tilde{S}}_{5}u_{k}x_{k}\epsilon_{imn}u_{m}x_{n}x_{a}+{\tilde{S}}_{6}u^{2}\epsilon_{iam}x_{m}$
$\displaystyle+{\tilde{S}}_{7}u_{k}x_{k}\epsilon_{iam}u_{m}+{\tilde{S}}_{8}(u_{k}x_{k})^{2}\epsilon_{iam}x_{m},$
(33)

where $\vec{\xi}=\vec{\omega}\times\vec{u}$. Substituting them into Eq. 3, we
obtain the second order corrections as

$\displaystyle
A_{4}^{(2)a}=(Q(r)-\frac{1}{2}r^{2}Q^{2}(r))u^{2}x_{a}+\frac{1}{2}(-1+r^{2}Q(r))\xi_{a}$
$\displaystyle-\frac{1}{2r^{2}}(P(r)+2Q(r)-r^{2}Q^{2}(r))(u_{m}x_{m})^{2}x_{a}$
$\displaystyle+\frac{1}{2}P(r)u_{m}x_{m}u_{a}+\frac{1}{2}Q(r)\xi_{m}x_{m}x_{a}$
$\displaystyle
A_{i}^{(2)a}=\frac{1}{2}P(r)(\omega_{m}u_{i}x_{m}x_{a}-\omega_{i}u_{m}x_{m}x_{a})+\frac{1}{2r^{2}}\epsilon_{amn}u_{m}x_{n}u_{i}$
$\displaystyle+\frac{1}{2}(-Q(r)+\frac{1}{r^{2}})\epsilon_{imn}u_{m}x_{n}u_{a}-(Q(r)-\frac{3}{2r^{2}})u_{k}x_{k}\epsilon_{iam}u_{m}$
$\displaystyle+\frac{1}{r^{2}}(-\frac{2}{r^{2}}+Q(r))u_{k}x_{k}\epsilon_{amn}u_{m}x_{n}x_{i}$
$\displaystyle-\frac{1}{2}(P(r)-Q(r)-r^{2}P(r)Q(r)+\frac{1}{r^{2}})u^{2}\epsilon_{iam}x_{m}$
$\displaystyle-\frac{1}{2r^{2}}(-P(r)+Q(r)+r^{2}P(r))(u_{k}x_{k})^{2}\epsilon_{iam}x_{m}.$
(34)

Now we can obtain the final solution by shifting $\vec{x}$ back to
$\vec{x}-\vec{c}$. Then substituting the solution into the Yang-Mills action
and complete the integration we obtain the action of the off-axis dyon with
the center at $\vec{c}=(y_{1},y2,0)$ as

$\displaystyle S_{\bar{M}}=\frac{8\pi^{2}}{g^{2}}\frac{\rho_{\bar{M}}}{2\pi
T}(1+\omega^{2}(y_{1}^{2}+y_{2}^{2}))$ (35)

The difference between $\mathcal{O}(\omega^{0})$ solutions of M and
$\bar{\text{M}}$ dyon is the sign of the spatial part, which means the higher
order correction of M dyon is

$\displaystyle A_{4}^{(1)}[\text{M}]=A_{4}^{(1)}[\bar{\text{M}}]$
$\displaystyle A_{i}^{(1)}[\text{M}]=-A_{i}^{(1)}[\bar{\text{M}}]$
$\displaystyle A_{4}^{(2)}[\text{M}]=-A_{4}^{(2)}[\bar{\text{M}}]$
$\displaystyle A_{i}^{(2)}[\text{M}]=A_{i}^{(2)}[\bar{\text{M}}]$ (36)

Because all the above derivation is in the hedgehog gauge, the L and
$\bar{\text{L}}$ dyons can be obtained by replacing the $\rho_{D}$ with $2\pi
T-\rho_{D}$ and applying gauge transformation to the stringy gauge. As the
gauge transformation does not change action, the action of all kinds of dyons
in the imaginary velocity case are

$\displaystyle S_{D}=\frac{8\pi^{2}}{g^{2}}\frac{\rho_{D}}{2\pi
T}(1+\omega^{2}(y_{1}^{2}+y_{2}^{2}))$ (37)

To construct a multi-dyon solution with an uniform asymptotic behavior, the
$\rho_{D}$ are $\bar{\rho}\mp\omega$ and $2\pi T-\bar{\rho}\pm\omega$, where
$\mp$ and $\pm$ corresponding to M, $\bar{\text{M}}$, L and $\bar{\text{L}}$
dyons respectively. As a result, the non-perturbative potential is

$\displaystyle U^{RE}_{np}(T,\omega)=-\frac{1}{V}ln\mathcal{Z}_{rot,D}$
$\displaystyle=-\frac{c}{2}\alpha^{3}(\omega)[sgn(\bar{\rho})(\bar{\rho}+i\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})\alpha(\bar{\rho}+i\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}+i\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})\alpha(\bar{\rho}_{c}+i\omega)}{6\pi T}}]$
$\displaystyle-\frac{c}{2}\alpha^{3}(\omega)[sgn(\bar{\rho})(\bar{\rho}-i\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})\alpha(\bar{\rho}-i\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}-i\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})\alpha(\bar{\rho}_{c}-i\omega)}{6\pi T}}].$
(38)

where $\alpha=1-\omega^{2}(y_{1}^{2}+y_{2}^{2})$ in the real rotation velocity
case. And

$\displaystyle U^{IM}_{np}(T,\omega)=-\frac{1}{V}ln\mathcal{Z}_{rot,D}$
$\displaystyle=-\frac{c}{2}\alpha^{3}(\omega)[sgn(\bar{\rho})(\bar{\rho}+\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})\alpha(\bar{\rho}+\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}+\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})\alpha(\bar{\rho}_{c}+\omega)}{6\pi T}}]$
$\displaystyle-\frac{c}{2}\alpha^{3}(\omega)[sgn(\bar{\rho})(\bar{\rho}-\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho})\alpha(\bar{\rho}-\omega)}{6\pi T}}$
$\displaystyle+sgn(\bar{\rho}_{c})(\bar{\rho}_{c}-\omega)^{3}(\frac{\Lambda}{\pi
T})^{\frac{22sgn(\bar{\rho}_{c})\alpha(\bar{\rho}_{c}-\omega)}{6\pi T}}].$
(39)

where $\alpha=1+\omega^{2}(y_{1}^{2}+y_{2}^{2})$ in the imaginary rotation
velocity case.

In the off-axis case we adopt the perturbative potential $U_{p}$ as Eq. 41 in
last section because we only consider the lowest order correction induced by
the background rotation which is supposed to be a even function of of
$\omega$, i.e. $\mathcal{O}(\omega^{2})$. The results in the real and
imaginary angular velocity case are respectively

$\displaystyle U^{RE}_{p}(T,\omega)=$
$\displaystyle-\sum\limits_{\begin{subarray}{c}s,m=1\\\
n=-\infty\end{subarray}}^{+\infty}\frac{e^{\frac{sn\omega}{T}}cosh(\frac{s\omega}{T})}{\pi^{2}sR^{3}}\frac{4\xi_{n}^{(m)}cos(s\frac{\bar{\rho}}{T})}{J_{n+1}(\xi_{n}^{(m)})^{2}}$
(40) $\displaystyle\times
J_{n}(\xi_{n}^{(m)}\frac{r_{c}}{R})^{2}K_{1}(s\frac{\xi_{n}^{(m)}}{TR}).$

and

$\displaystyle U^{IM}_{p}(T,\omega)=$
$\displaystyle-\sum\limits_{\begin{subarray}{c}s,m=1\\\
n=-\infty\end{subarray}}^{+\infty}\frac{e^{\frac{isn\omega}{T}}cosh(\frac{is\omega}{T})}{\pi^{2}sR^{3}}\frac{4\xi_{n}^{(m)}cos(s\frac{\bar{\rho}}{T})}{J_{n+1}(\xi_{n}^{(m)})^{2}}$
(41) $\displaystyle\times
J_{n}(\xi_{n}^{(m)}\frac{r_{c}}{R})^{2}K_{1}(s\frac{\xi_{n}^{(m)}}{TR}).$

where $r_{c}=\sqrt{y_{1}^{2}+y_{2}^{2}}$ for the off-aixs case. Clearly in
both non-perturbative and perturbative computation we have only kept the
lowest order correction, i.e. $\mathcal{O}(\omega^{2})$ and there is no
$\mathcal{O}(\omega)$ term in the effective potential. It is reasonable
because the $\vec{\omega}$ direction is supposed to be non-relevant to the
thermodynamics of the system. In the real velocity case, the negative sign in
$\alpha$ requires the $|\vec{u}|$ can not be too large. Hence in the next
numerical computations we will keep $r_{c}<0.5R$.

##  4 Radial dependence of deconfinement temperature

By minimize the total potential $U_{np}+U_{p}$ at different radial position
and temperature, one can find the corresponding $\bar{\rho}$ as a function of
temperature with a certain $\omega$ and radial position. The first temperature
at which the $\bar{\rho}$ deviates from $\pi T$ is the critical temperature,
because the $\bar{\rho}=\pi T$ gives $\langle TrL\rangle=2cos(\beta\pi T/2)=0$
and thus confinement. As our last work LABEL:Jiang:2023zzu in order to exhibit
the rotation effect more clearly we choose a relatively small radius $R=2(\pi
T_{0})^{-1}$, $T_{0}=0.25$GeV and $\omega_{max}=0.9R^{-1}$. According to Ref.
[33], $c=2$ and $\Lambda=0.28$GeV in numerical computation. In this work we
have not introduced the $\omega$-dependent coupling constant. The main reason
is more careful work is needed to get a rotation and position(possible)
dependent coupling constant. And this work is in process and more results will
be reported in future.

At different radial position the critical temperature as functions of the real
angular velocity is shown in Fig. 2. Clearly the real rotation will speed up
the deconfinement transition no matter where the cell is as the on-axis case.
In the $r_{c}=0.5R$ case, the critical temperature tend to rise around
$\omega=0.8R^{-1}$. It is because the center velocity $u\sim 0.4$ is not quite
a small number. Considering the nonlinear formalism of semi-classical
potential generates higher power terms of $\omega$, this rising behavior could
be quite unphysical. Hence we can focus on the small $\omega$ range. It can be
found that the deconfinement temperature becomes higher when the radial
distance goes larger. It means the outer part is more difficult to deconfine
in the real velocity case.

Figure 2: The critical deconfinement temperature as function of a real angular
velocity at different radial positions.

In order to compare with the lattice result, we also show the imaginary
velocity results in Fig. 3. This is done by the replacement $\omega\rightarrow
i\omega$. The main difference in the imaginary case is the radial distance
dependent factor $\alpha=1+\omega^{2}r_{c}^{2}$ always positive. The effective
potential can be computed without any technical difficulty at arbitrary
$\omega$. It can be found the deconfinement temperature rises when rotation
becomes faster. This is qualitatively the same as the on-axis case. When
radial distance becomes larger, the critical temperature decreases. This means
the outer part is easier to deconfine in the imaginary velocity case.

Figure 3: The critical deconfinement temperature as function of a imaginary
angular velocity at different radial positions.

According to the recent lattice simulation for a SU(3) gauge field system
[21], the radial dependence of the deconfinement temperature is the same as
results in above for both real and imaginary cases. However the $\omega$
dependence at a certain radial position is contradictory to that in this work.
As our last work LABEL:Jiang:2023zzu indicates the $\omega$-dependent coupling
constant will produce the same qualitative behavior as the lattice results.
And different from the on-axis case, the critical temperatures at different
radial distance are different even $\omega=0$. It is the finite-size effect in
the perturbative part which corresponds to $J_{n}(\xi_{n}^{(m)}r)$ in Eq. 41.

##  5 Conclusions

In this work we have solved the rotation-modified Yang-Mills equation in the
SU(2) case and obtained the rotational corrections to the so-called dyon
solutions with arbitrary centers to $\mathcal{O}(\omega^{2})$ order. Because
of the spherical symmetry breaking in the rotating system, the topological
non-trivial solutions can no longer be classified with the homotopy group of
the mapping from space-time to group space of SU(2), i.e. $T^{3}\rightarrow
S^{3}$. Hence there is no strict and spherical symmetric solutions. With these
approximated solutions the semi-classical potential has been computed in
dilute limit. The radial position dependent deconfinement temperature has been
obtained by minimizing the semi-classical potential in both real and imaginary
angular velocity cases. Although without considering the $\omega$-dependent
coupling constant the critical temperature behaves different from the lattice
simulation at each radial position as the rotation goes faster, its radial
dependence is qualitatively the same as the lattice. By analyzing the non-
perturbative and perturbative potentials separately it is found that the non-
perturbative part plays an more important role in the radial dependence of
critical temperature because of the inhomogeneous factor $\alpha$. Constrained
by the small velocity expansion and the light limit of center velocity in
$\alpha$, the computation should stay in the $\omega$ range, or unphysical
behavior of critical temperature will emerge.

In order to compare with the lattice results quantitatively, this work is
expected to be improved in two aspects. First is color number of the theory.
In SU(3) case there are 3 kinds of dyons, and the asymptotic behavior at
spatial infinity are more complicated even in static case. One should consider
the SU(3) case seriously to clarify the reason of the contradictory
qualitative behaviors of the $\omega$-dependence of critical temperature
between dyon ensemble and lattice simulation when no $\omega$-increasing
coupling introduced. Second one may go beyond dilute system limit and consider
the interaction between dyons and introduce quarks as an even further
generalization. By computation the full quantum weight of dyon pairs(caloron)
one can obtain the perturbative potential and dyon interaction simultaneously.
Such a computation can also improve the result of the perturbative part
because the Eq. 41 is the same as that with assumption of a constant
background color field. Furthermore the lattice simulation [22] indicates the
quarks may provide a contrary contribution to the gluons. This can be checked
with the dyon ensemble as well.

## Acknowledgements

The work of this research is supported by the National Natural Science
Foundation of China, Grant Nos. 12375131(YJ). We thanks Xu-Guang Huang very
much for his invaluable comments.

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