aainstitutetext: Department of Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japanbbinstitutetext: Department of Physics &
Research and Education Center for Natural Sciences, Keio University, Hiyoshi
4-1-1, Yokohama, Kanagawa 223-8521, Japan

# Topological confinement of vortices in two-flavor dense QCD

Yuki Fujimoto b Muneto Nitta<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We find a novel confinement mechanism in the two-flavor dense quark matter
proposed recently, that consists of the 2SC condensates and the $P$-wave
diquark condensates of $d$-quarks. This quark matter exhibiting color
superconductivity as well as superfluidity is classified into two phases;
confined and deconfined phases of vortices. We establish that the criterion of
the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices
exhibiting color non-singlet AB phases are confined by the so-called AB
defects to form color-singlet bound states. In the deconfined phase, the most
stable vortices are non-Abelian Alice strings, which are superfluid vortices
with fractional circulation and non-Abelian color magnetic fluxes therein,
exhibiting color non-singlet AB phases. On the other hand, in the confined
phase, these non-Abelian vortices are confined to either a baryonic or mesonic
bound state in which constituent vortices are connected by AB defects. The
baryonic bound state consists of three non-Abelian Alice strings with
different color magnetic fluxes with the total flux canceled out connected by
a domain wall junction, while the mesonic bound state consists of two non-
Abelian Alice strings with the same color magnetic fluxes connected by a
single domain wall. Interestingly, the latter contains a color magnetic flux
in its core, but this can exist because of color neutrality of its AB phase.

##  1 Introduction

Color confinement is one of the most challenging unsolved problems in modern
high energy physics. Particles belonging to non-singlet representations of
color gauge group SU(3), such as quarks and gluons, cannot exist alone.
Instead, they are confined to form color-singlet composite particles, that is,
hadrons such as baryons and mesons. The dual Meissner picture suggests that
color electric fluxes emanating from quarks are squeezed to flux tubes and
confine these quarks, as a dual to the Meissner effect: magnetic fluxes
emanating from monopoles are squeezed to flux tubes or vortices and confine
these monopoles Nambu:1974zg ; Mandelstam:1974pi .

For QED in 2+1 dimensions, a duality maps particles confined by strings to
vortices confined by domain wall strings Polyakov:1976fu . There are several
models exhibiting a vortex confinement, that is, vortices are linearly
confined by solitons, domain walls, or kinks stretching among (between) them.
Bose-Einstein condensates of multiple components serve as test beds for a
vortex confinement; they allow fractional vortices in the system, which are
confined when intercomponent (Rabi or Josephson) couplings are introduced.
Various facets of this model have been known in details: Although
dimensionality is different from QCD, similarities with quark confinement were
pointed out Son:2001td  and were studied extensively Kasamatsu:2004tvg ;
Cipriani:2013nya ; Tylutki:2016mgy ; Eto:2017rfr ; Eto:2019uhe . Phase diagram
of vortex confinement phase transitions was also determined Kobayashi:2018ezm
. Two-gap superconductors also offer another circumstance where fractional
vortices Babaev:2001hv  are confined by domain lines doi:10.1143/JPSJ.70.2844
; PhysRevLett.88.017002 , and vortex confinement was discussed Goryo_2007 . In
3+1 dimensions, these configurations are vortex strings confined by domain
walls (membranes), but they can be applied to confinement of QCD-like theory
on a compactified circle Shifman:2008ja . Another mechanism responsible for
vortex confinement is given in the so-called modified XY model Carpenter_1989
; Kobayashi:2019sus ; Kobayashi:2019npl  in which the modified gradient term
with half periodicity compared with the canonical gradient term plays an
essential role. More recently, a novel vortex confinement mechanism based on
topology was proposed; when some fields receiving nontrivial Aharonov-Bohm
(AB) phases around a vortex develop vacuum expectation values (VEVs), there
must appear kinks or domain walls attached to the vortex in order to
compensate the AB phases to maintain singlevaluedness of the VEVs
Chatterjee:2019zwx ; Chatterjee:2018znk ; Nitta:2020ggi . We referred such
defects AB defects. We may call this confinement as “topological confinement.”

In this paper, we show that such a topological confinement of vortices occurs
in the ground state of cold QCD matter at high baryon densities, which
exhibits color superconductivity Bailin:1983bm ; Iwasaki:1994ij ;
Alford:2007xm . Among various known color-superconducting phases, the color-
flavor locked (CFL) phase Alford:1998mk  in three-flavor symmetric matter is
realized at extremely high density limit while the 2-flavor superconducting
(2SC) phase Alford:1997zt ; Rapp:1997zu  appears at relatively low density
where strange quark mass cannot be neglected. For the case of three-flavor
symmetric matter, the symmetry breaking patterns and low-energy excitations
such as corresponding Nambu-Goldstone modes have one-to-one correspondence
between both phases; this observation led to the concept of quark-hadron
continuity that posit the continuous connection between quark matter (CFL
color superconductor) and hadronic matter (hyperonic superfluid)
Schafer:1998ef . Quark-hadron continuity with three-flavor symmetry has been
well understood Schafer:1998ef ; Alford:1999pa ; Fukushima:2004bj ;
Hatsuda:2006ps ; Yamamoto:2007ah ; Hatsuda:2008is ; Schmitt:2010pf  and has
been repurposed for the modern way of constructing the equation of state of
neutron stars with quark cores Masuda:2012kf ; Masuda:2012ed ; Kojo:2014rca ;
Baym:2017whm ; Baym:2019iky . For the two-flavor symmetric case, the similar
quark-hadron continuity was thought to be absent. However, it was pointed out
recently that the additional diquark pairing between $d$-quarks
$\langle\hat{d}^{\top}\mathcal{C}\gamma^{i}\nabla^{j}\hat{d}\rangle$ (which we
often denote by $\langle dd\rangle$ for simplicity) can arise in the
${}^{3}P_{2}$ channel besides the conventional 2SC condensate Fujimoto:2019sxg
; Fujimoto:2020cho , and this novel quark phase named 2SC+$\langle dd\rangle$
phase opened the possibility for continuity to the ${}^{3}P_{2}$ neutron
superfluid phase Hoffberg:1970vqj ; Tamagaki:1970ptp ; takatsukaPTP71 ;
takatsukaPTP72 ; richardsonPRD72 ; Sauls:1978lna ; Takatsuka:1992ga ;
Masuda:2015jka ; Mizushima:2016fbn ; Yasui:2018tcr ; Yasui:2019unp ;
Mizushima:2019spl . This paper particularly focuses on this 2SC+$\langle
dd\rangle$ phase for a topological vortex confinement.

Quantum vortices, that is vortices with quantized circulations, are essential
degrees of freedom in superfluids. For instance, when superfluids are rotating
rapidly, quantum vortices are created along the rotation axis to form a vortex
lattice. Such the quantum vortices or color magnetic flux tubes are also
present in color-superconducting quark matter Eto:2013hoa . In the CFL phase,
Abelian superfluid vortices exist as the topologically stable configuration
due to the nontrivial homotopy group $\pi_{1}[\mathrm{U(1)_{B}}]=\mathbb{Z}$
Forbes:2001gj ; Iida:2002ev . However, this is unstable against a decay into a
triad of more stable vortices Nakano:2007dr ; Cipriani:2012hr ; Alford:2016dco
, which are non-Abelian vortices with color magnetic fluxes and fractional
circulation of the Abelian vortices Balachandran:2005ev ; Nakano:2007dr ;
Nakano:2008dc ; Eto:2009kg ; Eto:2013hoa . A single non-Abelian vortex carries
the so-called orientational moduli (collective coordinates) of the complex
projective space ${\mathbb{C}}P^{2}$ as Nambu-Goldstone modes trapped in its
core Nakano:2007dr ; Eto:2009bh ; Eto:2013hoa ; Eto:2009tr  as well as a
triplet of gapless Majorana fermions Yasui:2010yw ; Fujiwara:2011za .
Recently, in the context of quark-hadron continuity, vortices penetrating
through the CFL phase into hyperon matter have been extensively discussed
Alford:2018mqj ; Chatterjee:2018nxe ; Chatterjee:2019tbz ; Cherman:2018jir ;
Hirono:2018fjr ; Hirono:2019oup ; Cherman:2020hbe  (see also Ref.
Fujimoto:2021bes  for the similar attempt in two-flavor setup). Vortex
confinement in the CFL phase has been proposed recently Eto:2021nle  as a non-
Abelian generalization of two-component BECs and two-gap superconductors, but
this is not a topological vortex confinement.

On the other hand, the pure 2SC phase, which can be regarded as more realistic
than the CFL phase in the sense that strange quark mass is not degenerate with
light quark masses, cannot support topologically stable vortices because of
the unbroken $\mathrm{U(1)_{B}}$ symmetry resulting in a trivial first
homotopy group Alford:2010qf . However, the 2SC+$\langle dd\rangle$ phase
admits stable topological vortices that winds around the $\langle dd\rangle$
condensate Fujimoto:2020dsa . This 2SC+$\langle dd\rangle$ phase can further
be subdivided into at least two kinds of phases, _deconfined_ and _confined_
phases depending on the form of the 2SC condensate, i.e., if it is restricted
to real value then the phase falls into the deconfined one, and if it is
complex-valued then the confined phase is realized. In the preceding paper
Fujimoto:2020dsa , we only concentrated on the deconfined phase in which we
assumed that value of the 2SC condensate can be taken to be real. In the
deconfined phase, the most stable vortices are non-Abelian Alice vortices,
which carries a 1/3 fractional circulation in $\rm U(1)_{B}$ and the color-
magnetic fluxes. A single non-Abelian Alice string is accompanied by
orientational moduli (collective coordinates) of the real projective space
${\mathbb{R}}P^{2}$ corresponding to the color flux therein in the presence of
the $\langle dd\rangle$ condensates alone. In these respects, the non-Abelian
Alice strings are similar to the CFL non-Abelian vortices. At the same time
there are features unique to the non-Abelian Alice strings such as topological
obstruction. It means that some unbroken generators in the bulk are not
globally defined around the string, akin to Alice strings Schwarz:1982ec ;
Alford:1990mk ; Alford:1990ur ; Alford:1992yx ; Preskill:1990bm ;
Bucher:1992bd ; Lo:1993hp ; Chatterjee:2017jsi ; Chatterjee:2017hya ;
Chatterjee:2019zwx . Another unique feature is that quarks receive color non-
singlet (generalized) AB phases when they encircle the strings, in contrast to
the CFL case where non-Abelian vortices obtain only color-singlet AB phases.
Here, we have called the phase without the baryon circulation as (pure) AB
phase and the one with the baryon circulation as generalized AB phase. The
former is relevant for heavy quarks while the latter is necessary for light
quarks $u$ and $d$ because they participate in the condensations $\langle
dd\rangle$ with a vortex configuration. As a consequence of the nontrivial AB
phases, the “bulk-soliton moduli locking” occurs in the deconfined phase,
i.e., the 2SC condensates develop VEVs whose color fluxes are enforced to be
aligned along the orientational moduli ${\mathbb{R}}P^{2}$ of the Alice string
because of the singlevaluedness of the VEVs Fujimoto:2020dsa . Due to
superfluidity of the system, two separated vortices repel each other and no
bound states are formed in the deconfined phase. Thus, a single Abelian $\rm
U(1)_{B}$ vortex is unstable against a decay into a triad of non-Abelian Alice
strings with total color canceled out among these three.

In this paper, we show that in the confined phase, these non-Abelian vortices
are confined to either a baryonic or mesonic bound state in which constituent
vortices are connected by AB defects Chatterjee:2019zwx ; Chatterjee:2018znk ;
Nitta:2020ggi  that appear to compensate nontrivial color non-singlet
(generalized) AB phases of the 2SC condensates around vortices. The baryonic
bound state consists of three non-Abelian Alice strings with different color
magnetic fluxes with the total flux canceled out, which are connected by a
domain wall junction. Since the domain walls pull vortices by their tension,
these vortices are combined and result in a single Abelian $\rm U(1)_{B}$
vortex. On the other hand, the mesonic bound state consists of two non-Abelian
Alice strings with the same color magnetic fluxes connected by a single domain
wall. Again, they are pulled by the domain wall tension and result in a
doubly-wound non-Abelian string. Interestingly, the latter still contains a
color magnetic flux in its core. Nevertheless, this is screened at large
distances, since it has only a color-singlet AB phase. Thus, we establish that
color neutrality of the AB phases is the criterion of the confinement of
vortices.

Baryonic-type molecules of vortices can be found in three-component BECs
Eto:2012rc , three-gap superconductors Nitta:2010yf , and the
${\mathbb{Z}}_{3}$ modified XY model Kobayashi:2019npl . Baryons in this paper
look similar to these configurations, but a crucial difference lies in that
baryons in this paper are SU(3)C color singlets.

The outline of the paper is as follows. In Sec. 2, we summarize the
2SC+$\langle dd\rangle$ phase with paying a particular attention to the
confined phase. In Sec. 3, we introduce topological vortices in the $\langle
dd\rangle$ phase: Abelian superfluid vortices, non-Abelian Alice strings, and
doubly-wound non-Abelian strings. In Sec. 3.4, we investigate generalized AB
phases around vortices. In Sec. 4, we show topological vortex confinement
mechanism in the confined phase. In Sec. 5, for the consistency, we discuss
the opposite ordering of symmetry breakings in which the 2SC condensate
$\Phi_{\rm 2SC}$ develops VEVs first and $\Phi_{dd}$ condensate develops VEVs
second. Sec. 6 is devoted to a summary and discussion. In Appendix A, we
introduce a pure color flux tube. In Appendix B, we summarize the derivation
of generalized AB phases.

##  2 Two-flavor dense quark matter

In this section, we set forth our setup in this paper by giving a short
summary of the 2SC+$\langle dd\rangle$ phase proposed in Refs.
Fujimoto:2019sxg ; Fujimoto:2020cho . We also turn to the symmetry breaking
induced by diquark condensation. The large part of analysis has already been
carried out in the preceding work Fujimoto:2020dsa , however, here we
introduce new idea of confined and deconfined phases of the vortices and
classify them according to the value of the 2SC condensate. This notion will
be important in the later discussions.

###  2.1 2SC+$\langle dd\rangle$ phase

Let us define the following diquark operators made out of up and down quark
operators $\hat{u}$ and ${\hat{d}}$:

$\displaystyle(\hat{\Phi}_{\rm
2SC})^{\alpha}\equiv\epsilon^{\alpha\beta\gamma}\hat{u}^{T}_{\beta}\mathcal{C}\gamma^{5}\hat{d}_{\gamma}\,,\quad(\hat{\Phi}_{dd})_{\alpha\beta}^{ij}$
$\displaystyle\equiv\hat{d}^{T}_{\alpha}\mathcal{C}\gamma^{i}\nabla^{j}\hat{d}_{\beta}\,,$
(1)

where $\mathcal{C}$ is the charge conjugation, Greek indices
($\alpha,\beta,\gamma,\ldots$) are color indices of the fundamental
representation, and Latin indices ($i,j,\ldots$) denote spatial coordinates.
In the latter diquark, the matrices $\gamma^{i}$ and spatial derivatives
$\nabla^{j}$ correspond to spin and angular momentum in the ${}^{3}P_{2}$
state, respectively. In the 2SC+$\langle dd\rangle$ phase, these two diquark
operators develop VEVs:

$\displaystyle\Phi_{\rm 2SC}\equiv\langle\hat{\Phi}_{\rm
2SC}\rangle\,,\quad\Phi_{dd}\equiv\langle\hat{\Phi}_{dd}\rangle\,.$ (2)

Here, we make a distinction between condensates and operators, i.e.,
quantities without hats are condensates or VEVs and those with hats are
operators. We assumed unitary gauge fixing for these expressions of diquark
condensates. The conventional 2SC condensate is $\Phi_{\rm 2SC}$, and
$\Phi_{dd}$, which is the ${}^{3}P_{2}$ diquark pairing of $d$-quarks, is the
new feature in the 2SC+$\langle dd\rangle$ phase.

We note in passing that this 2SC+$\langle dd\rangle$ phase was originally
proposed in the context of quark-hadron continuity where neutron ${}^{3}P_{2}$
superfluid and this 2SC+$\langle dd\rangle$ phase are continuously connected
Fujimoto:2019sxg ; Fujimoto:2020cho . The meaning of continuity becomes clear
if one considers the order parameter operator of neutron ${}^{3}P_{2}$
superfluid Hoffberg:1970vqj ; Tamagaki:1970ptp ; Takatsuka:1992ga  given by

$\displaystyle\hat{A}^{ij}$
$\displaystyle=\hat{n}^{T}\mathcal{C}\gamma^{i}\nabla^{j}\hat{n}\,,$ (3)

where $\hat{n}$ is a field operator of neutrons. The expectation value of
$\hat{A}^{ij}$ in the hadronic phase reads

$\displaystyle\langle\hat{A}^{ij}\rangle$
$\displaystyle=\langle\hat{n}^{T}\mathcal{C}\gamma^{i}\nabla^{j}\hat{n}\rangle\,.$
(4)

It is finite as long as the ${}^{3}P_{2}$ superfluidity of the neutron is
realized. After rearranging the valence quark content, the expectation value
of $\hat{A}^{ij}$ in the quark phase under the mean field approximation reads

$\displaystyle\langle\hat{A}^{ij}\rangle\simeq(\Phi_{\rm
2SC})^{\alpha}(\Phi_{\rm 2SC})^{\beta}(\Phi_{dd})_{\alpha\beta}^{ij}\,.$ (5)

It is also finite as long as matter is color superconductor. The neutron
superfluid operator is always non-zero in the hadronic and quark matter so the
local order parameter cannot distinguish these two phases, leading to
continuity.

###  2.2 Symmetry of the 2SC+$\langle dd\rangle$ phase: general consideration

The relevant part of the symmetry of QCD in this work is $G_{\rm
QCD}=\mathrm{SU(3)}_{\rm C}\times\mathrm{U(1)}_{\rm B}$. We will see in
particular that $\mathrm{U(1)}_{\rm B}$ symmetry is broken so that
topologically stable vortices can arise. An element $(U,e^{i\theta_{\rm
B}})\in G_{\rm QCD}$ acts on quark fields $\hat{q}$ as

$\hat{q}\to e^{i\theta_{\rm B}}\hat{q}\,,$ (6)

where $U\in\mathrm{SU(3)_{C}}$. The diquark condensates (2) transform as

$\Phi_{\rm 2SC}\to e^{2i\theta_{\rm B}}U^{*}\Phi_{\rm 2SC}\,,\quad\Phi_{dd}\to
e^{2i\theta_{\rm B}}U\Phi_{dd}U^{T}\,.$ (7)

We assume an appropriate structure for the tensor indices $i,j$ of $\Phi_{dd}$
111It is known in the nematic phase Sauls:1978lna  for which
$\mathrm{diag}(1,s,1-s)$ is implied for the $i,j$ indices, with real parameter
$s$., and hereafter we suppress these indices for simplicity.

We discuss symmetry breaking patterns induced by the diquark condensations.
Now we turn on each condensate sequentially instead of turning them on at the
same time. Namely, the VEV of $\Phi_{dd}$ is firstly developed and then
follows the VEV of $\Phi_{\rm 2SC}$. The opposite ordering is also considered.
It is summarized by the equations

$\displaystyle 1.\quad$ $\displaystyle G_{\rm
QCD}\xrightarrow{\Phi_{dd}}H_{dd}\xrightarrow{\Phi_{\rm
2SC}}K_{\mathrm{2SC}+dd}$ (8) $\displaystyle 2.\quad$ $\displaystyle G_{\rm
QCD}\xrightarrow{\Phi_{\rm 2SC}}\tilde{H}_{\rm
2SC}\xrightarrow{\Phi_{dd}}K_{\mathrm{2SC}+dd}$ (9)

and by schematic diagram in Fig. 1. We begin with the first option (8) and
then the second option (9) is mentioned to ensure the consistency that the
physics does not depend on the ordering of the condensations. As shown below,
in a certain assumption, there are at least two possibilities for
$K_{\mathrm{2SC}+dd}$: $K_{\mathrm{2SC}+dd}^{\rm deconf}$ or
$K_{\mathrm{2SC}+dd}^{\rm conf}$.

Figure 1: Spontaneous symmetry breaking patterns in the most symmetric case
for $\Phi_{dd}$. $\Phi_{\rm 2SC}$ takes a value in ${\mathbb{R}}^{3}$ or
${\mathbb{C}}^{3}$ in the deconfined or confined phase where unbroken symmetry
is $K_{\mathrm{2SC}+dd}^{\rm deconf}$ or $K_{\mathrm{2SC}+dd}^{\rm conf}$,
respectively.

###  2.3 Symmetry breaking by the $\langle dd\rangle$ condensate

Let us explain $G_{\rm QCD}\xrightarrow{\Phi_{dd}}H_{dd}$ in Eq. (8). By
suitable gauge rotation, $(\Phi_{dd})_{\alpha\beta}$ can be taken to be
diagonal without loss of generality:

$\Phi_{dd}=\mathrm{diag}\left[(\Phi_{dd})_{11},(\Phi_{dd})_{22},(\Phi_{dd})_{33}\right]\,.$
(10)

We restrict ourselves to the simplest choice of $\Phi_{dd}$:

$\Phi_{dd}=\Delta_{dd}\boldsymbol{1}_{3}\,,$ (11)

i.e., each diagonal component of $\Phi_{dd}$ are taken to be equal:
$(\Phi_{dd})_{11}=(\Phi_{dd})_{22}=(\Phi_{dd})_{33}=\Delta_{dd}$. The unbroken
subgroup of $G_{\rm QCD}$ that keeps $\Phi_{dd}=\Delta_{dd}\boldsymbol{1}_{3}$
invariant is

$H_{dd}=\mathrm{SO(3)_{C}}\rtimes(\mathbb{Z}_{6})_{\rm C+B}\,.$ (12)

This can be understood in the following sense: An element $(U,e^{i\theta_{\rm
B}})\in G_{\rm QCD}$ acts on $\Phi_{dd}=\Delta_{dd}\boldsymbol{1}_{3}$ as
$\Delta_{dd}\boldsymbol{1}_{3}\to\Delta_{dd}e^{2i\theta_{\rm B}}UU^{T}$, so
that the condition under which the condensate does not change is
$e^{2i\theta_{\rm B}}UU^{T}=\boldsymbol{1}_{3}$. This condition is fulfilled
by setting $U\in\mathrm{SO(3)_{C}}$ and $e^{2i\theta_{\rm B}}=1$. This
accounts for $\mathrm{SO(3)_{C}}$ in $H_{dd}$. The discrete group
$(\mathbb{Z}_{6})_{\rm C+B}$ in $H_{dd}$ is defined by

$(\mathbb{Z}_{6})_{\rm C+B}:(X^{k},\omega^{-2k})\in\mathrm{SU(3)_{C}\times
U(1)_{B}}=G_{\rm QCD}\,,$ (13)

where $k=0,1,2,3,4,5$ and $\omega\equiv e^{i\pi/3}$. Due to the semidirect
product $\rtimes$, $X$ is subject to $\mathrm{SO(3)_{C}}$ transformation and
thus the expression is not unique (see Ref. Fujimoto:2020dsa  for precise
meaning of the semidirect product). The typical expressions of $X$ read

$X\equiv\mathrm{diag}(\omega,\omega,\omega^{-2}),\mathrm{diag}(\omega,\omega^{-2},\omega),\mbox{
or }\mathrm{diag}(\omega^{-2},\omega,\omega),$ (14)

generated by $(T_{8}\propto)\mathrm{diag}(1,1,-2)$, $\mathrm{diag}(1,-2,1)$,
or $\mathrm{diag}(-2,1,1)$, respectively of the broken $\mathrm{SU(3)_{C}}$
symmetry.

The order parameter manifold in this breaking $G_{\rm QCD}\to H_{dd}$ is

$\frac{G_{\rm
QCD}}{H_{dd}}=\frac{\mathrm{SU(3)_{C}}\times\mathrm{U(1)_{B}}}{\mathrm{SO(3)_{C}}\rtimes(\mathbb{Z}_{6})_{\rm
C+B}}\simeq\frac{M_{3}\times S^{1}}{(\mathbb{Z}_{6})_{\rm C+B}}$ (15)

with $M_{3}\equiv\mathrm{SU(3)_{C}}/\mathrm{SO(3)_{C}}$. Because of the
relation $\pi_{1}(G_{\rm QCD}/H_{dd})=\mathbb{Z}$, non-Abelian Alice strings
that appears in the later discussion can exist as topologically stable
configuration. Higher homotopy groups are $\pi_{n}(G_{\rm
QCD}/H_{dd})=\pi_{n}(M_{3})$ for $n>1$ and the latter can be found in Refs.
Auzzi:2006ns ; Auzzi:2008hu : $\pi_{2}(M_{3})={\mathbb{Z}}_{2}$,
$\pi_{3}(M_{3})={\mathbb{Z}}_{4}$ and so on.

###  2.4 Symmetry breaking by the 2SC condensate

Next, we switch on the VEV of $\Phi_{\rm 2SC}$ in the existence of
$\Phi_{dd}$. We explain $H_{dd}\xrightarrow{\Phi_{\rm
2SC}}K_{\mathrm{2SC}+dd}$ in Eq. (8). The $(\mathbb{Z}_{6})_{\rm C+B}$ action
on $\Phi_{\rm 2SC}$, as given in Eq. (7), is

$\displaystyle\Phi_{\rm 2SC}\to\omega^{-2}X^{-1}\Phi_{\rm 2SC}$
$\displaystyle=\left\\{\begin{array}[]{c}\omega^{-2}\,\mathrm{diag}(\omega,\omega,\omega^{-2})^{-1}\Phi_{\rm
2SC}=\mathrm{diag}(-1,-1,1)\Phi_{\rm 2SC}\\\
\omega^{-2}\,\mathrm{diag}(\omega,\omega^{-2},\omega)^{-1}\Phi_{\rm
2SC}=\mathrm{diag}(-1,1,-1)\Phi_{\rm 2SC}\\\
\omega^{-2}\,\mathrm{diag}(\omega^{-2},\omega,\omega)^{-1}\Phi_{\rm
2SC}=\mathrm{diag}(1,-1,-1)\Phi_{\rm 2SC}.\\\ \end{array}\right.$ (19)

Then, we find that the unbroken symmetry depends on whether $\Phi_{\rm
2SC}\in{\mathbb{R}}^{3}$ or $\Phi_{\rm 2SC}\in{\mathbb{C}}^{3}$:

$\displaystyle H_{dd}\to K_{\mathrm{2SC}+dd}$
$\displaystyle=\left\\{\begin{array}[]{c}K_{\mathrm{2SC}+dd}^{\rm
deconf}=\mathrm{SO(2)}_{\rm C}\times(\mathbb{Z}_{6})_{\rm C+B}\mbox{ for
}\Phi_{\rm 2SC}\in{\mathbb{R}}^{3}\\\ K_{\mathrm{2SC}+dd}^{\rm
conf}=(\mathbb{Z}_{3})_{\rm C+B}\mbox{ for }\Phi_{\rm
2SC}\in{\mathbb{C}}^{3}.\end{array}\right.$ (22)

The effects of this second symmetry breaking $H_{dd}\xrightarrow{\Phi_{\rm
2SC}}K_{\mathrm{2SC}+dd}$ on Alice strings are main topic of this and previous
Fujimoto:2020dsa  papers. We focused on the deconfined phase in the case
$\Phi_{\rm 2SC}\in\mathbb{R}^{3}$ in the preceding paper Fujimoto:2020dsa ; in
this paper, we investigate the confined phase corresponding to the other
possibility $\Phi_{\rm 2SC}\in\mathbb{C}^{3}$.

In the case of $\Phi_{\rm 2SC}\in{\mathbb{R}}^{3}$, the direction of
$X\in\mathbb{Z}_{6}$ inside $G_{\rm QCD}$ is not unique but transforms under
SO(3)C. The three typical expressions of $X$ in Eq. (14) keep

$\displaystyle\Phi_{\rm 2SC}^{T}\propto(0,0,1),(0,1,0),(1,0,0),$ (23)

unbroken, respectively. An SO(2)C group remains intact, with a direct product
with $X\in\mathbb{Z}_{6}$.

On the other hand, in the case of $\Phi_{\rm 2SC}\in{\mathbb{C}}^{3}$,
$\Phi_{\rm 2SC}$ cannot be taken to be one component by using the SO(3)C
symmetry, but at least two components should be nonvanishing. Consequently,
only even numbers of the $\mathbb{Z}_{6}$ action in Eq. (19), $\Phi_{\rm
2SC}\to\mathrm{diag}(1,1,1)\Phi_{\rm 2SC}$, remain unbroken, thus forming
$\mathbb{Z}_{3}$.

The order parameter manifolds for these symmetry breakings are

$\displaystyle\frac{H_{dd}}{K_{\mathrm{2SC}+dd}^{\rm
deconf}}=\frac{\mathrm{SO(3)_{C}}\rtimes(\mathbb{Z}_{6})_{\rm
C+B}}{\mathrm{SO(2)}_{\rm C}\times(\mathbb{Z}_{6})_{\rm
C+B}}\simeq\frac{\mathrm{SO(3)_{C}}}{\mathrm{SO(2)}_{\rm C}}\simeq S^{2}\,,$
$\displaystyle\frac{H_{dd}}{K_{\mathrm{2SC}+dd}^{\rm
conf}}=\frac{\mathrm{SO(3)_{C}}\rtimes(\mathbb{Z}_{6})_{\rm
C+B}}{(\mathbb{Z}_{3})_{\rm C+B}}\,.$ (24)

###  2.5 The opposite ordering of symmetry breakings

The opposite ordering of the condensation as in Eq. (9) is discussed; we first
consider the 2SC phase by $\Phi_{\rm 2SC}$ and subsequently consider
$\Phi_{dd}$.

We choose the gauge of the 2SC condensate $\Phi_{\rm 2SC}$ as $(\Phi_{\rm
2SC})^{\alpha}=\Delta_{\rm 2SC}\delta^{\alpha 3}$, and this spontaneously
breaks $G_{\rm QCD}$ into

$\tilde{H}_{\rm 2SC}=\mathrm{SU(2)_{C}\times U(1)_{C+B}}\,,$ (25)

where the $\rm U(1)_{C+B}$ symmetry is given by

$\\{(e^{i\alpha T_{8}},e^{2i\alpha})\in\mathrm{SU(3)_{C}\times
U(1)_{B}}:T_{8}=\mathrm{diag}(1,1,-2)\\}\,.$ (26)

The order parameter manifold of this breaking is

$\frac{G_{\rm QCD}}{\tilde{H}_{\rm
2SC}}=\frac{\mathrm{SU(3)_{C}}}{\mathrm{SU(2)_{C}}}\,,$ (27)

allowing trivial first homotopy group or no stable vortices as known before.
Equivalently, this corresponds to the absence of superfluidity.

Let us switch on the $\Phi_{dd}$ condensate. The $\mathrm{SU(2)_{C}}$ symmetry
in $\tilde{H}_{\rm 2SC}$ diagonalizes upper-left block of $\Phi_{dd}$:

$\displaystyle\Phi_{dd}=\begin{pmatrix}\Delta_{dd}^{\prime}&0&(\Phi_{dd})_{13}\\\
0&\Delta_{dd}^{\prime}&(\Phi_{dd})_{23}\\\
(\Phi_{dd})_{13}&(\Phi_{dd})_{23}&\Delta_{dd}^{\prime\prime}\end{pmatrix}\,.$
(28)

We assume here the diagonal components of the upper-left block to be equal.
Unbroken symmetry is identified by looking at the ${\mathbb{Z}}_{6}$ action in
Eq. (13) with $X=\mathrm{diag}(\omega,\omega,\omega^{-2})$ in Eq. (14):

$\displaystyle\Phi_{dd}$ $\displaystyle\to\omega^{-2}X\Phi_{dd}X^{T}$
$\displaystyle=\begin{pmatrix}\Delta_{dd}^{\prime}&0&\omega^{3}(\Phi_{dd})_{13}\\\
0&\Delta_{dd}^{\prime}&\omega^{3}(\Phi_{dd})_{23}\\\
\omega^{3}(\Phi_{dd})_{13}&\omega^{3}(\Phi_{dd})_{23}&\Delta_{dd}^{\prime\prime}\end{pmatrix}\,.$
(29)

There are two phases, deconfined or confined, depending on if the off-diagonal
entries are absent or not. In the previous paper Fujimoto:2020dsa , we
investigated the case for the vanishing off-diagonal components. The
corresponding unbroken group is $K=K_{\mathrm{2SC}+dd}^{\rm
deconf}\simeq\mathrm{SO(2)_{C}}\times(\mathbb{Z}_{6})_{\rm C+B}$ as the same
as the first possibility of Eq. (22). In this paper, our main focus is on the
case that the off-diagonal components are present corresponding to the case
$\Phi_{\rm 2SC}\in\mathbb{C}^{3}$ in Eq. (8). In this case, applying the
${\mathbb{Z}}_{6}$ action even times in Eq. (29) makes $\Phi_{dd}$ invariant
because of $\omega^{6}=1$, thus confirming $K=K_{\mathrm{2SC}+dd}^{\rm
conf}\simeq(\mathbb{Z}_{3})_{\rm C+B}$ in the second possibility of Eq. (22).

The order parameter manifolds for these breakings are

$\displaystyle\frac{\tilde{H}_{\rm 2SC}}{K_{\mathrm{2SC}+dd}^{\rm
deconf}}=\dfrac{\mathrm{SU(2)_{C}}}{\mathrm{SO(2)_{C}}}\times\dfrac{\rm
U(1)_{C+B}}{(\mathbb{Z}_{6})_{\rm C+B}}\simeq S^{2}\times\dfrac{\rm
U(1)_{C+B}}{(\mathbb{Z}_{6})_{\rm C+B}}\,,$ $\displaystyle\frac{\tilde{H}_{\rm
2SC}}{K_{\mathrm{2SC}+dd}^{\rm conf}}=\mathrm{SU(2)_{C}}\times\dfrac{\rm
U(1)_{C+B}}{(\mathbb{Z}_{3})_{\rm C+B}}\,.$ (30)

In the both cases, the ground state admits topologically stable vortex
configurations according to $\pi_{1}(\tilde{H}_{\rm
2SC}/K_{\mathrm{2SC}+dd}^{\rm deconf})\simeq\mathbb{Z}$ and
$\pi_{1}(\tilde{H}_{\rm 2SC}/K_{\mathrm{2SC}+dd}^{\rm conf})\simeq\mathbb{Z}$.
However, the difference in the discrete groups, ${(\mathbb{Z}_{3})_{\rm C+B}}$
or ${(\mathbb{Z}_{6})_{\rm C+B}}$, in the unbroken subgroups implies that
minimal vortices are 1/6 or 1/3 winding in U(1)B, thus carrying 1/6 or 1/3
circulations, respectively.

Comparing the order parameters for the two ways of the symmetry breakings,
Eqs. (8) and (9), look different at a first glance. However, these two are
gauge equivalent and can be transformed to each other by a gauge
transformation, implying the gauge invariant $\hat{A}^{ij}$ are the same.

###  2.6 The overall symmetry breaking $G\to K$

At this stage, we have the overall order parameter manifolds for the whole
symmetry breakings

$\displaystyle\frac{G}{K_{\mathrm{2SC}+dd}^{\rm deconf}}$
$\displaystyle=\dfrac{\mathrm{SU(3)_{C}}\times{\rm
U(1)_{B}}}{\mathrm{SO(2)_{C}}\times(\mathbb{Z}_{6})_{\rm C+B}}\simeq\frac{{\rm
U(3)}}{{\rm SO(2)}\times{\mathbb{Z}}_{2}}$
$\displaystyle\frac{G}{K_{\mathrm{2SC}+dd}^{\rm conf}}$
$\displaystyle=\dfrac{\mathrm{SU(3)_{C}}\times{\rm
U(1)_{B}}}{(\mathbb{Z}_{3})_{\rm C+B}}\simeq{\rm U(3)}$ (31)

for the deconfined and confined phases, respectively. Eventually, the order
parameter manifold U(3) in Eq. (31) for the confining phase is the same with
that of the CFL phase Eto:2013hoa .

##  3 Topological vortices in $\langle dd\rangle$ phase

In this section, we summarize vortices that appear in the presence of
$\Phi_{dd}$. In Sec. 3.1, we introduce an Abelian superfluid vortex. In Sec.
3.2, we introduce a non-Abelian Alice vortex as the topologically most stable
minimal configuration in the $\langle dd\rangle$ phase. In Sec. 3.3, we also
discuss doubly-wound non-Abelian string, which loses the most properties of
the Alice strings. Generalized AB phases around these strings are summarized
in Sec. 3.4.

###  3.1 Abelian superfluid vortices

The simplest vortex is an Abelian superfluid vortex:

$\displaystyle\Phi_{dd}(r,\varphi)=f_{0}(r)e^{i\varphi}\Delta_{dd}{\bf
1}_{3}\sim e^{i\varphi}\Delta_{dd}{\bf 1}_{3}\,,$ (32)

where $(r,\varphi)$ is the polar coordinates. The boundary condition for the
profile function $f_{0}$ is set as $f_{0}(0)=0$ and $f_{0}(\infty)=1$. The
factor $e^{i\varphi}$ accounts for a unit quantized winding in
$\mathrm{U(1)_{B}}$. We call this a $\rm U(1)_{B}$ superfluid vortex or an
Abelian vortex. See the first line of Tab. 1. Due to the relation
$\pi_{1}[\mathrm{U(1)_{B}}]=\mathbb{Z}$, this string is topologically stable,
however, it is unstable against decay into more stable vortices, i.e., triads
of non-Abelian Alice strings with different color fluxes canceled as a whole.
In the presence of $\Phi_{\rm 2SC}$, this string remains unstable against
decay as discussed in the previous paper Fujimoto:2020dsa . However, as
discussed in this paper, this becomes stable in the confined phase.

###  3.2 Non-Abelian Alice strings

The most stable vortex is what we call as a non-Abelian Alice string. It is
has orientational moduli in the internal gauge space of non-Abelian gauge
group. The holonomy at infinite distance

$U(\varphi)=\mathcal{P}\exp\left(i\int_{0}^{\varphi}\boldsymbol{A}\cdot
d\boldsymbol{\ell}\right)\,$ (33)

generates the condensate winding at spatial infinity

$\Phi_{dd}(\varphi)=e^{i\varphi/3}U(\varphi)\Phi_{dd}(\varphi=0)U^{T}(\varphi)\,,$
(34)

where $\Phi_{dd}(\varphi=0)=\Delta_{dd}\boldsymbol{1}_{3}$. The three
representative configurations can be given by

$\begin{split}\Phi_{dd}(r,\varphi)&=\Delta_{dd}\begin{pmatrix}g(r)&0&0\\\
0&g(r)&0\\\ 0&0&f(r)e^{i\varphi}\end{pmatrix}\,,\\\
U(\varphi)&=e^{i(\varphi/6)\mathrm{diag}(-1,-1,2)}\,,\\\
A_{i}&=-\frac{a(r)}{6g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(-1,-1,2)\,\end{split}$
(35)

for a blue ($b$) color magnetic flux

$\begin{split}\Phi_{dd}(r,\varphi)&=\Delta_{dd}\begin{pmatrix}g(r)&0&0\\\
0&f(r)e^{i\varphi}&0\\\ 0&0&g(r)\end{pmatrix}\,,\\\
U(\varphi)&=e^{i(\varphi/6)\mathrm{diag}(-1,2,-1)}\,,\\\
A_{i}&=-\frac{a(r)}{6g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(-1,2,-1)\,\end{split}$
(36)

for a green ($g$) color magnetic flux , and

$\begin{split}\Phi_{dd}(r,\varphi)&=\Delta_{dd}\begin{pmatrix}f(r)e^{i\varphi}&0&0\\\
0&g(r)&0\\\ 0&0&g(r)\end{pmatrix}\,,\\\
U(\varphi)&=e^{i(\varphi/6)\mathrm{diag}(2,-1,-1)}\,,\\\
A_{i}&=-\frac{a(r)}{6g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(2,-1,-1)\,\end{split}$
(37)

for a red ($r$) color magnetic flux. Here, $f$ and $g$ are the profile
functions with the boundary conditions

$f(0)=g^{\prime}(0)=a(0)=0,\quad f(\infty)=g(\infty)=a(\infty)=1.$ (38)

This carries $1/6$ quantized color-magnetic flux,
$\mathcal{F}=\mathcal{F}_{0}/6$ (see Eq. (67) for the definition of
$\mathcal{F}_{0}$) as well as $1/3$ quantized $\rm U(1)_{B}$ circulation, as
summarized in the second line of Tab. 1. As $\Phi_{dd}$ have to be
singlevalued, $U(2\pi)$ belongs to the little group $H_{dd}$ of the condensate
$\Phi_{dd}(0)$. Thus, this configuration connects two elements of $\rm
SU(3)_{C}$: $U(\varphi=0)={\bf 1}_{3}$ and
$U(\varphi=2\pi)=\mathrm{diag}(\omega^{-1},\omega^{-1},\omega^{2}),(\omega^{-1},\omega^{2},\omega^{-1})$,
or $(\omega^{2},\omega^{-1},\omega^{-1})$ for the configuration in Eq. (35),
(36), or (37), respectively. All of these configurations with different color
fluxes are continuously connected by Nambu-Goldstone (NG) modes associated
with this spontaneous symmetry breaking in the vicinity of the vortex:

$\displaystyle O\in\frac{H_{dd}}{\tilde{K}_{\rm vortex}}=\frac{{\rm
SO(3)}_{\rm C}\rtimes\mathbb{Z}_{6}}{{\rm O(2)}_{\rm
C}\times\mathbb{Z}_{6}}\simeq S^{2}/\mathbb{Z}_{2}\simeq\mathbb{R}P^{2}.$ (39)

In the vortex core, there remains the unbroken gauge symmetry.

One of the characteristic features of the Alice string is the presence of the
so-called topological obstruction Schwarz:1982ec ; Alford:1990mk ;
Alford:1990ur ; Alford:1992yx ; Preskill:1990bm ; Bucher:1992bd ; Lo:1993hp ;
Bolognesi:2015mpa ; Chatterjee:2017jsi ; Chatterjee:2017hya ;
Chatterjee:2019zwx , stating that the unbroken generators $T_{x}$, $T_{y}$ and
$T_{z}$ of $\mathrm{SO(3)_{C}}$ are not globally defined around the string.
These generators receive the transformation

$\displaystyle T_{x,y,z}(\varphi)\equiv U(\varphi)T_{x,y,z}U^{-1}(\varphi)\,,$
(40)

around the string, and then we find

$\displaystyle T_{y,z}(\varphi=2\pi)$ $\displaystyle=-T_{y,z}\neq
T_{y,z}(\varphi=0)\,$ $\displaystyle T_{x}(\varphi=2\pi)$
$\displaystyle=+T_{x}=T_{x}(\varphi=0)$ (41)

for the Alice string with the flux of the color $r$ in Eq. (35),

$\displaystyle T_{z,x}(\varphi=2\pi)$ $\displaystyle=-T_{z,x}\neq
T_{z,x}(\varphi=0)\,$ $\displaystyle T_{y}(\varphi=2\pi)$
$\displaystyle=+T_{y}=T_{y}(\varphi=0)$ (42)

for the one with the flux of the color $g$ in Eq. (36), and

$\displaystyle T_{x,y}(\varphi=2\pi)$ $\displaystyle=-T_{x,y}\neq
T_{x,y}(\varphi=0)\,$ $\displaystyle T_{z}(\varphi=2\pi)$
$\displaystyle=+T_{z}=T_{z}(\varphi=0)$ (43)

for the one with the flux of the color $b$ in Eq. (37). We can recover all the
original $T_{x,y,z}$ by rotating $\varphi=4\pi$:

$\displaystyle T_{x,y,z}(\varphi=4\pi)=T_{x,y,z}(\varphi=0)\,,$ (44)

for any kind of Alice strings.

###  3.3 Doubly-wound non-Abelian strings

Among multiply-wound strings, a particularly important string is a doubly-
wound non-Abelian string, given by

$\displaystyle\Phi_{dd}(\varphi)$
$\displaystyle=e^{2i\varphi/3}U(\varphi)\Phi_{dd}(\varphi=0)U^{T}(\varphi)\,$
$\displaystyle=\Delta_{dd}\begin{pmatrix}g(r)&0&0\\\ 0&g(r)&0\\\
0&0&f(r)e^{2i\varphi}\end{pmatrix}\,,$ $\displaystyle U(\varphi)$
$\displaystyle=e^{i(\varphi/3)\mathrm{diag}(1,1,-2)}$ $\displaystyle A_{i}$
$\displaystyle=-\frac{a(r)}{3g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(1,1,-2)\
$ (45)

for the color flux $b$, and similar for other color fluxes. This carries $1/3$
quantized color magnetic flux, $\mathcal{F}=\mathcal{F}_{0}/3$, and $2/3$
quantized circulation. See the fifth line of Table 1. Since
$U(\varphi=2\pi)=e^{i(2\pi/3)\mathrm{diag}(-1,-1,2)}=e^{-2\pi i/3}{\bf 1}_{3}$
this configuration connects two elements ${\bf 1}_{3}$ and $\omega^{-2}$ of
the center of SU(3)C222It is also worth to point out that this property
connecting two center elements of SU(3)C is shared by non-Abelian strings in
the CFL phase Nakano:2007dr ; Eto:2013hoa ., in contrast to a single non-
Abelian Alice string which does not connect center elements of SU(3)C. As we
will see below, this string loses the most of Alice properties that a single
Alice string possesses; there is no topological obstruction from Eq. (44), and
bears only color singlet AB phases as shown in the next section. Nevertheless,
it has a color magnetic flux and the same moduli space with that in Eq. (39)
of a single Alice string.

In the $\langle dd\rangle$ phase, this string is unstable against a decay into
two non-Abelian Alice strings of the same color because of superfluidity. In
the presence of $\Phi_{\rm 2SC}$, this remains unstable in the deconfined
phase, but it will be stabilized in the confined phase, as discussed in later
sections.

###  3.4 Generalized Aharonov-Bohm phases

Here we summarize AB phases of particles and condensation encircling the above
introduced strings. In the CFL case, the AB phases in electromagnetic sector
around a non-Abelian vortex was calculated Chatterjee:2015lbf , while those in
color SU(3)C sector is ${\mathbb{Z}}_{3}$ Cherman:2018jir . In this section,
we consider AB phases of color SU(3)C symmetry around a vortex, without
switching on the electromagnetism.

First, we place the quark field $\hat{q}$, the gauge field $A_{i}$, and the
2SC diquark operator $\hat{\Phi}_{\rm 2SC}$ in the vortex configuration given
above. When they wind around the vortex, they receive a gauge transformation
according to the holonomy action in Eq. (33), as well as a $\mathrm{U(1)_{B}}$
transformation if it participates in the condensation with a vortex winding.
Thus, after $2\pi$ winding around the vortex, the fields receive a phase from
the Wilson loop and the $\mathrm{U(1)_{B}}$ baryon circulation. The phase
without the baryon circulation is called as (pure) AB phase while the one with
the baryon circulation as generalized AB phase. The former is relevant for
heavy quarks such as strange quarks $s$ that do not participate in the
condensation, while the latter is necessary for light quarks $u$ and $d$,
since they participate in condensations $\Phi_{dd}$ containing a vortex
configuration. In the later sections, a criterion for the existence of the
confinement will be given by this (generalized) AB phase.

At any azimuthal angle $\varphi\neq 0$, the light quark field operator
$\hat{q}$ and the diquark operator $\hat{\Phi}_{\rm 2SC}$ are expressed by a
holonomy action as

$\displaystyle\hat{q}(\varphi)$ $\displaystyle\sim e^{i\theta_{\rm
B}(\varphi)}U(\varphi)\hat{q}(\varphi=0)\,,$ (46)
$\displaystyle\hat{\Phi}_{\rm 2SC}(\varphi)$ $\displaystyle\sim
e^{2i\theta_{\rm B}(\varphi)}U^{-1}(\varphi)\hat{\Phi}_{\rm
2SC}(\varphi=0)\,,$ (47)

respectively, where $U(\varphi)$ is defined as in Eq. (33). As mentioned
above, the light quarks $u$ and $d$ that participate in the condensations, and
$\hat{\Phi}_{\rm 2SC}$ receive an additional contribution from the baryon
number symmetry $\rm U(1)_{B}$ other than the usual AB phase of the color
gauge group, because $\hat{\Phi}_{\rm 2SC}$ itself contains a vortex winding.
The total phase is a generalized AB phase. The generalized AB phase $\Gamma$
in the exponentiated form can be read out from the fields at $\varphi=2\pi$
after going around the vortex, i.e.,
$\hat{q}(0)\to\hat{q}(2\pi)=\Gamma\hat{q}(0)$ and $\hat{\Phi}_{\rm
2SC}(0)\to\hat{\Phi}_{\rm 2SC}(2\pi)=\Gamma\hat{\Phi}_{\rm 2SC}(0)$. The gauge
field $A_{i}$ in the vortex configuration is proportional to the diagonal
matrix, so taking $A_{i}\propto\mathrm{diag}(-1,-1,2)$ for instance for the
color flux $b$, the explicit form of the field is

$\displaystyle\begin{split}\hat{q}(2\pi)&\sim e^{i\theta_{\rm
B}(\varphi)}U(2\pi)\hat{q}(0)\\\ &\sim e^{i\pi
B}e^{2i\pi\mathcal{F}\mathrm{diag}(-1,-1,2)}\begin{pmatrix}\hat{q}_{r}(0)\\\
\hat{q}_{g}(0)\\\ \hat{q}_{b}(0)\end{pmatrix}\,,\\\ \hat{\Phi}_{\rm
2SC}(2\pi)&\sim e^{2i\theta_{\rm B}(\varphi)}U^{-1}(2\pi)\hat{\Phi}_{\rm
2SC}(0)\\\ &\sim e^{2i\pi
B}e^{-2i\pi\mathcal{F}\mathrm{diag}(-1,-1,2)}\begin{pmatrix}\hat{\Phi}_{\rm
2SC}^{r}(0)\\\ \hat{\Phi}_{\rm 2SC}^{g}(0)\\\ \hat{\Phi}_{\rm
2SC}^{b}(0)\end{pmatrix}\end{split}$ (48)

where $B$ and $\mathcal{F}$ are the $\mathrm{U(1)_{B}}$ circulation and the
color-magnetic flux, respectively. We tabulate the values of $B$ and
$\mathcal{F}$ for each kind of vortex in Tab. 1, and the detailed derivations
of generalized AB phases are summarized in Appendix B. When an $s$-quark that
does not participate in the condensations encircles a vortex, it receives only
an AB phase of the color gauge group as mentioned above.

phase | vortex |  | $\mathrm{U(1)_{B}}$
---
circul. $B$
| color
---
magnetic
flux ${\cal F}$
| generalized
---
AB phase for
($u,d$) quarks
|  | AB phase for
---
$s$ quark
|  | generalized
---
AB phase for
$\hat{\Phi}_{\rm 2SC}$
|  | color
---
reps.
(de)conf |  $\mathrm{U(1)_{B}}$ vortex | 1 | 0 | $(-1,-1,-1)$ | $\mathbb{Z}_{2}$ | $(1,1,1)$ | $1$ | $(1,1,1)$ | $1$ | singlet
deconf |  |  $\mathrm{U(1)_{C}}$(d) or
---
pure color flux(d)
0 | 1/2 | $\left(\begin{tabular}[]{ccc}$+1$&$-1$&$-1$\\\ $-1$&$+1$&$-1$\\\ $-1$&$-1$&$+1$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{2}$ | $\left(\begin{tabular}[]{ccc}$+1$&$-1$&$-1$\\\ $-1$&$+1$&$-1$\\\ $-1$&$-1$&$+1$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{2}$ | $\left(\begin{tabular}[]{ccc}$+1$&$-1$&$-1$\\\ $-1$&$+1$&$-1$\\\ $-1$&$-1$&$+1$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{2}$ | non-singlet
conf |  |  $\mathrm{U(1)_{C}}$(c) or
---
pure color flux(c)
0 | 1 | $\left(\begin{tabular}[]{ccc}$+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ \end{tabular}\right)$ | $1$ | $\left(\begin{tabular}[]{ccc}$+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ \end{tabular}\right)$ | $1$ | $\left(\begin{tabular}[]{ccc}$+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ \end{tabular}\right)$ | $1$ | singlet
deconf |  | NA Alice string
---
or $\mathrm{U(1)_{C+B}}$(d)
1/3 | 1/6 | $\left(\begin{tabular}[]{ccc}$-1$&$+1$&$+1$\\\ $+1$&$-1$&$+1$\\\ $+1$&$+1$&$-1$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{2}$ | $\left(\begin{tabular}[]{ccc}$\omega^{2}$&$\omega^{-1}$&$\omega^{-1}$\\\ $\omega^{-1}$&$\omega^{2}$&$\omega^{-1}$\\\ $\omega^{-1}$&$\omega^{-1}$&$\omega^{2}$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{6}$ | $\left(\begin{tabular}[]{ccc}$+1$&$-1$&$-1$\\\ $-1$&$+1$&$-1$\\\ $-1$&$-1$&$+1$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{2}$ | non-singlet
conf |  | doubly-wound
---
NA string
or $\mathrm{U(1)_{C+B}}$(c)
2/3 | 1/3 | $\left(\begin{tabular}[]{ccc}$+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ \end{tabular}\right)$ | $1$ | $\left(\begin{tabular}[]{ccc}$\omega^{4}$&$\omega^{4}$&$\omega^{4}$\\\ $\omega^{4}$&$\omega^{4}$&$\omega^{4}$\\\ $\omega^{4}$&$\omega^{4}$&$\omega^{4}$\\\ \end{tabular}\right)$ | $\mathbb{Z}_{3}$ | $\left(\begin{tabular}[]{ccc}$+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ $+1$&$+1$&$+1$\\\ \end{tabular}\right)$ | 1 | singlet

Table 1: (Generalized) AB phases of light $(u,d)$ quarks, heavy ($s$) quark,
and the 2SC condensate $\Phi_{\rm 2SC}\sim ud$ around various vortices
introduced in this section (a pure color flux is introduced in Appendix A).
For row vectors, their columns represent the colors of the quarks or
$\Phi_{\rm 2SC}$ encircling the vortex. For $3\times 3$ matrices, rows
represent the colors of fluxes of the vortices and columns represent the
colors of the quarks or $\Phi_{\rm 2SC}$. The order $k$ of the $s$-quark AB
phase ${\mathbb{Z}}_{k}$ corresponds to the flux $1/k$ of the vortex.

##  4 Vortex confinement

Now let us turn on the VEV of the 2SC condensate $\Phi_{\rm 2SC}$. In the
previous paper, we considered the deconfined phase in which the 2SC
condensates $\Phi_{\rm 2SC}$ are real-valued and can be taken to be one
component by the SO(3)C gauge symmetry unbroken in the presence of the
$\langle dd\rangle$ condensate. Here, we consider the confined phase where
$\Phi_{\rm 2SC}$ are generically complex-valued. In this case, one cannot take
a gauge in which the 2SC condensate $\Phi_{\rm 2SC}$ develops a VEV only in
one component, unlike the case of the deconfined phase. Instead, the
$\Phi_{\rm 2SC}$ has at least two components as VEVs.

In Sec. 4.1, we introduce AB defects attached to non-Abelian Alice strings. In
Sec. 4.2 we construct a baryonic molecule of three Alice strings connected by
a domain wall junction, while in Sec. 4.3 we construct a mesonic molecule of
two Alice strings connected by a single domain wall. In Sec. 4.4 we give a
comment on a collision of two U(1)B strings decaying into three doubly-wound
non-Abelian strings.

###  4.1 Aharonov-Bohm defects in 2SC condensate $\Phi_{\rm 2SC}$

Here, we consider all three components for generality. Then, when the 2SC
condensate $\Phi_{\rm 2SC}$ encircles a single non-Abelian Alice string in
Eqs. (35), (36) and (37), it receives non-trivial AB phases summarized in Eq.
(87). More explicitly, it is

$\displaystyle\Phi_{\rm 2SC}^{\alpha}=\begin{pmatrix}\Delta_{1}\\\
\Delta_{2}\\\ \Delta_{3}\end{pmatrix}\xrightarrow{\text{holonomy}}$
$\displaystyle\begin{cases}\begin{pmatrix}+\Delta_{1}\\\ -\Delta_{2}\\\
-\Delta_{3}\end{pmatrix}\quad\text{around
}M\sim\begin{pmatrix}e^{i\varphi}&&\\\ &1&\\\ &&1\\\ \end{pmatrix}\quad(r)\\\
\begin{pmatrix}-\Delta_{1}\\\ +\Delta_{2}\\\
-\Delta_{3}\end{pmatrix}\quad\text{around }M\sim\begin{pmatrix}1&&\\\
&e^{i\varphi}&\\\ &&1\end{pmatrix}\quad(g)\\\ \begin{pmatrix}-\Delta_{1}\\\
-\Delta_{2}\\\ +\Delta_{3}\end{pmatrix}\quad\text{around
}M\sim\begin{pmatrix}1&&\\\ &1&\\\
&&e^{i\varphi}\end{pmatrix}\quad(b)\end{cases}.$ (49)

Apparently, the inconsistency arises from non-singlevaluedness around the
string if all $\Delta$’s have VEVs. This could be avoided if only one of
$\Delta$’s has a VEV, corresponding to the deconfined phase leading to the
bulk-soliton moduli locking. It is, however, unavoidable in the confined phase
in which at least two of $\Delta$’s must have VEVs.

To overcome this problem, we insert the following function $h(\varphi)$ in the
2SC condensate $\Phi_{\rm 2SC}$ to maintain the singlevaluedness of $\Phi_{\rm
2SC}$:

$\displaystyle\Phi_{\rm
2SC}^{\alpha}=\begin{cases}\begin{pmatrix}\Delta_{1}h(\varphi)\\\
\Delta_{2}h(\varphi)\\\ \Delta_{3}\end{pmatrix}\,\;\;\mbox{ for }(b)\,,\\\
\begin{pmatrix}\Delta_{1}h(\varphi)\\\ \Delta_{2}\\\
\Delta_{3}h(\varphi)\end{pmatrix}\;\;\mbox{ for }(g)\,,\\\
\begin{pmatrix}\Delta_{1}\\\ \Delta_{2}h(\varphi)\\\
\Delta_{3}h(\varphi)\end{pmatrix}\;\;\mbox{ for }(r)\end{cases}\,$ (50)

for the three cases in Eq. (49). Here $h(\varphi)$ is a kink profile inserted
to compensate the AB phase, satisfying the boundary conditions
$h(\varphi=0)=1$ and $h(\varphi=2\pi)=-1$. The domain walls or solitons
appearing to compensate an AB phase are called as the AB defects
Chatterjee:2019zwx ; Chatterjee:2018znk ; Nitta:2020ggi .

Figure 2: Schematic figure of a vortex baryon. In this case, three Alice
strings with different color magnetic fluxes with the total color flux
canceled out are confined by a domain wall junction denoted by grey lines. The
black dotted loops encircling one of the three Alice strings show the Wilson-
loops that pick up color non-singlet generalized AB phases, while the yellow
loops show the color singlet ones. The large yellow loop encircles all of the
three Alice strings and the small yellow loop encircles none of them but
passes through the three domain walls.  Figure 3: Schematic figure of a vortex
meson. In this case, two Alice strings with the same color fluxes are confined
and connected by a single domain wall. The black dotted loop encircling one of
the two Alice strings shows the Wilson-loops that pick up color non-singlet
generalized AB phase, while the yellow loop encircling the two Alice strings
show the color singlet ones. Although the composite state carries a net color
magnetic flux, it is a color singlet state in terms of the generalized AB
phase.

Alice strings are confined due to the formation of the AB defects. A single
Alice string is attached to a single AB defect extending to infinity. For
finite energy configurations, as we illustrate in Figs. 2 and 3, the AB
defects are attached to the Alice strings so that they are confined. There are
two kinds of confinements: “baryonic type” made of three Alice strings in Fig.
2 and “mesonic type” made of two Alice strings in Fig. 3, which we will
discuss in the following subsections.

###  4.2 Baryonic molecule

Let us discuss a molecule of the baryonic type. The three circles red, green
and blue in Fig. 2 denote the vortices, and the lines attached to them are the
kinks that we have considered above. We denote the kinks which are attached to
the red, green and blue vortices as $w_{r}$, $w_{g}$ and $w_{b}$,
respectively. We can show that these three kinks are put together at the
center with the domain wall junction as follows. From Eq. (50), we can define
the domain wall operators

$\displaystyle w_{r}=\mathrm{diag}(-1,-1,1),\quad$ $\displaystyle
w_{g}=\mathrm{diag}(-1,1,-1),\quad$ $\displaystyle
w_{b}=\mathrm{diag}(1,-1,-1)$ (51)

that act on $\Phi_{\rm 2SC}$: When the $\Phi_{\rm 2SC}$ passes through a
domain wall, say $w_{r}$, it undergoes the transformation $w_{r}\Phi_{\rm
2SC}$. Then, we have a relation

$\displaystyle w_{r}w_{g}w_{b}=1$ (52)

implying that when two domain walls $w_{r}$, $w_{g}$ collide, it becomes
$w_{b}^{-1}=w_{b}$. This implies that the three domain walls meet at a
junction. In other words, along the small yellow loop in Fig. 2, $\Phi_{\rm
2SC}$ passes through the all three domain walls, coming back to the original
configuration due to Eq. (52).

Dynamically, these three Alice strings are pulled by the domain walls because
of their tensions. The fate of this baryonic configuration is nothing but a
U(1)B superfluid vortex, having no color magnetic flux. In fact, color
magnetic fluxes of $r$, $g$, $b$ are canceled out when they are combined
together.

Our confinement argument relies on the generalized AB phases of the 2SC
condensate $\Phi_{\rm 2SC}$ encircling the vortices, as we discussed above.
Since $\Phi_{\rm 2SC}$ is color non-singlet carrying a color index, this leads
us to a conjecture that the criterion of the confinement is that the
(generalized) AB phases picking up the color of vortices should disappear at
the spatial infinity in the confined phase. To check the validity of this
conjecture, let us discuss generalized AB phases of $\Phi_{\rm 2SC}$, light
$u,d$-quarks, and a heavy $s$-quark.

##### Generalized Aharonov-Bohm phases of $\Phi_{\rm 2SC}$

If we look at the outer yellow path in Fig. 2, then its generalized AB phase
is apparently trivial. However, if we go around each colored Alice strings,
then we pick up a non-trivial generalized AB phase. These non-trivial AB
phases are compensated by each kinks $w_{r}$, $w_{b}$, or $w_{g}$. Along the
inner yellow path encircling the junction point, the $\Phi_{\rm 2SC}$ receives
no AB phase but it receives domain wall operations in Eq. (51) when it passes
across a domain wall. However, it comes back to the original form due to the
relation in Eq. (52) at the end of the whole path.

##### Generalized Aharonov-Bohm phases of the light $u,d$-quarks

When a $u$ or $d$-quark encircles each Alice string, it picks up a nontrivial
generalized AB phase only when it encircles an Alice string with the
corresponding color, as can be seen from Eq. (85), or the fourth line of Table
1. For instance, only $u_{r}$ ($d_{r}$) recives $-1$ when it encircles a red
Alice string along the corresponding dotted loop in Fig. 2, but $u_{g}$
($d_{g}$) or $u_{b}$ ($d_{b}$) does not. Thus, the up or down quark can detect
the color of the Alice string at the infinite distance. However, if each quark
encircles all of the three Alice strings along the outer yellow loop in Fig.
2, it always picks up a generalized AB phase $-1$ irrespective of its color:

$\displaystyle\mathrm{diag}(1,1,-1)\times\mathrm{diag}(1,-1,1)\times\mathrm{diag}(-1,1,1)=\mathrm{diag}(-1,-1,-1)$
(53)

which is a color singlet. This phase does not have to be canceled because it
develops no VEV, unlike the case of $\Phi_{\rm 2SC}$. The right hand side
coincides with the generalized AB phase of the $u,d$ quarks around a U(1)B
vortex which is in fact a color singlet, see the first line of Table 1. Thus,
the $u,d$ quarks cannot detect the color of the baryonic molecule. This can be
understood as the confinement.

##### Aharonov-Bohm phases of the heavy $s$-quark

Let us discuss a strange quark $s$ encircling the baryonic molecule or each
Alice string. When it encircles one of Alice strings, say the red Alice
string, $s_{r}$ receives $w^{2}$ while $s_{g}$ and $s_{b}$ receive $w^{-1}$,
as can be seen from Eq. (83) or the fourth line of Table 1. Thus, the strange
quark can detect the color of the Alice string at the infinite distance.
However, when it encircles the baryonic molecule, neither $s_{r}$, $s_{g}$ nor
$s_{b}$ receives any AB phase:

$\displaystyle\mathrm{diag}(\omega^{-1},\omega^{-1},\omega^{2})\times\mathrm{diag}(\omega^{-1},\omega^{2},\omega^{-1})\times\mathrm{diag}(\omega^{2},\omega^{-1},\omega^{-1})=\mathrm{diag}(1,1,1)$
(54)

which is a color singlet. Again, the right hand side coincides with the AB
phases of strange quarks around a U(1)B Abelian vortex, which is a color
singlet, see the first line of Table 1. This is also a consequence of the
confinement.

In summary, we observe that each Alice string has a color non-singlet
generalized AB phase and three Alice strings have a color singlet generalized
AB phase as a whole it implies confinement. The 2SC condensate $\Phi_{\rm
2SC}$ receives color non-singlet generalized AB phases, so the AB defects
should be generated to compensate the AB phases to ensure the singlevaluedness
of $\Phi_{\rm 2SC}$. Thus, each Alice string has to be confined by the AB
defect. A baryonic configuration, made of three Alice strings of different
color magnetic fluxes with the total color canceled out, has no generalized AB
phase, where three Alice strings are connected by the domain wall junction.
The domain wall tension pulls the three Alice strings to combine them,
resulting in the U(1)B Abelian superfluid vortex.

###  4.3 Mesonic molecule

There is also another way of confinement. Two Alice strings with the same
color fluxes are confined together as shown in Fig. 3. In this case, we do not
have to introduce the domain wall junction, but simply two colored vortices
are connected with the kink $w_{r}$. Dynamically, the two constituent Alice
strings in the mesonic molecule are pulled by the domain wall tension, and
they are combined together. The fate of this configuration is a doubly-wound
non-Abelian string introduced in Sec. 3.3, which in fact does not have the
Alice properties.

One may wonder why the mesonic-type molecule having a color magnetic flux is
allowed in the confined phase. This can be understood from the fact that no
colored quarks can pick up an AB phase depending on color when it encircles
the molecule along the outer yellow loop in Fig. 3, as explained below.

##### Generalized Aharonov-Bohm phases of $\Phi_{\rm 2SC}$

If we look at the outer yellow path in Fig. 3, then its generalized AB phase
is apparently trivial. However, if we go around each colored string, then we
pick up a non-trivial generalized AB phase, which must be compensated by a
kink $w_{r}$.

##### Generalized Aharonov-Bohm phases of the light $u,d$-quarks

A up or down quark receives no generalized AB phase when it encircles the
molecule, while it receives a non-trivial generalized AB phase depending on
its color, when it encircles each Alice string, as in Eq. (85) or the fourth
line of Table 1; for instance only $u_{r}$ ($d_{r}$) recives $-1$ when it
encircles a red Alice string, but $u_{g}$ ($d_{g}$) or $u_{b}$ ($d_{b}$) does
not. This implies that the up or down quark can detect the color of the Alice
string at the infinite distance but cannot do the color of the molecule;

$\displaystyle\mathrm{diag}(1,1,-1)\times\mathrm{diag}(1,1,-1)=\mathrm{diag}(1,1,1)$
(55)

The right hand side coincides with the generalized AB phases of $u,d$ quarks
around a doubly-wound non-Abelian string, which is a color singlet, see Eq.
(91) or the fifth line of Table 1. Thus, the all $u,d$ quarks receive no
generalized AB phase irrespective of their color, thereby implying that the
$u,d$ quarks cannot detect the color of the molecule.

##### Aharonov-Bohm phases of the heavy $s$-quark

It is further interesting to see what happens when a strange quark $s$
encircles the molecule or each Alice string. When it encircles one of red
Alice strings, $s_{r}$ receives $w^{2}$ while $s_{g}$ and $s_{b}$ receive
$w^{-1}$, and so the strange quark can detect the color of the Alice string at
the infinite distance. However, when it encircles the molecule, $s_{r}$
receives $w^{4}$ and $s_{g}$ and $s_{b}$ receive $w^{-2}=w^{4}$. Therefore, we
have the relation

$\displaystyle\mathrm{diag}(\omega^{-1},\omega^{-1},\omega^{2})\times\mathrm{diag}(\omega^{-1},\omega^{-1},\omega^{2})=\mathrm{diag}(\omega^{4},\omega^{4},\omega^{4})$
(56)

which is nontrivial but a color singlet. The right hand side coincides with
the AB phases of the $s$-quark around a doubly-wound non-Abelian string, which
is a color singlet, see Eq. (89) or the fifth line of Table 1. Thus, all
strange quarks receive the same AB phase irrespective of their color, thereby
implying that the strange quark cannot detect the color of the molecule.

The notion of confinement should be used for color which can be detected by
the (generalized) AB phases at infinity, but not for colors of fluxes that
vortices have. The mesonic molecule possess colored magnetic flux, but it is
not read out by the generalized AB phases as we showed above, so the mesonic
molecule can be stated as confined.

It is interesting to point out that in this definition, non-Abelian vortices
in the CFL phase are already confined as they are Chatterjee:2018nxe ;
Chatterjee:2019tbz .

###  4.4 Baryons-to-mesons decay

Before closing this section, one comment is in order. In the confined phase, a
U(1)B vortex cannot decay into three non-Abelian Alice strings because of the
AB defects connecting them, as seen in the last subsection. However, if we
prepare two U(1)B vortices at the same position, namely a doubly-wound U(1)B
vortex, it can decay into three doubly-wound non-Abelian strings, as
schematically drawn in Fig. 4. Of course, two U(1)B vortices themselves repel
each other, and so it is not easy to prepare a doubly-wound U(1)B vortex.
Also, once we prepare it, it is an open question which decay channel is more
dominant between the decay into two U(1)B vortices or the decay into three
doubly-wound non-Abelian strings.

Figure 4: Decay of a doubly-wound U(1)B string into three doubly-wound non-
Abelian strings with different color fluxes with the total color flux canceled
out.

##  5 Consistency with the opposite ordering in the symmetry breaking

In this section, we discuss vortices formed in the opposite ordering of the
condensation given in Eq. (9): The 2SC condensate $\Phi_{\rm 2SC}$ develops
first, then the $\langle dd\rangle$ condensate. First, the 2SC condensate in
the vacuum can be taken as $(\Phi_{\rm 2SC})^{\alpha}=\Delta_{\rm
2SC}\delta^{\alpha 3}$ as usual for the 2SC phase. Second, vortices arise for
the $\Phi_{dd}$ condensation due to $\pi_{1}(\tilde{H}_{\rm
2SC}/K_{\mathrm{2SC}+dd})=\mathbb{Z}$ with the order parameter manifolds in
Eq. (30), as already given in Sec. 2.5.

###  5.1 Superfluid vortex

The configuration of a superfluid vortex is

$\displaystyle\begin{split}\Phi_{dd}(\varphi)&=f_{0}(r)e^{i\varphi}\Delta_{dd}{\bf
1}_{3}\,,\\\ \Phi_{\rm 2SC}(\varphi)&=h_{0}(r)e^{i\varphi}(0\ 0\ \Delta_{\rm
2SC})^{T}\,,\end{split}$ (57)

where we set the boundary conditions as

$\displaystyle f_{0}(0)=h_{0}(0)=0,\quad f_{0}(\infty)=h_{0}(\infty)=1\,.$
(58)

This is supported by the breaking of the $\rm U(1)_{B}$ symmetry, and is
exactly a $\mathrm{U(1)_{B}}$ vortex considered in Sec. 3.1. This is nothing
but a baryonic bound state of three Alice strings.

###  5.2 $U(1)_{\rm C+B}$ vortices

Here, we consider the vortex with fractional winding and the color-magnetic
flux. In the presence of the 2SC condensate $\Phi_{\rm 2SC}$, the unbroken
symmetry is $\tilde{H}_{\rm 2SC}=\mathrm{SU(2)_{C}\times U(1)_{C+B}}$ as in
Eq. (25). Then, the $\Phi_{dd}$ condensation can be taken as

$\displaystyle\begin{split}\Phi_{dd}&=\begin{pmatrix}\Delta_{dd}^{\prime}&0&(\Phi_{dd})_{13}\\\
0&\Delta_{dd}^{\prime}&(\Phi_{dd})_{23}\\\
(\Phi_{dd})_{31}&(\Phi_{dd})_{32}&\Delta_{dd}^{\prime\prime}\end{pmatrix}\\\
\end{split}$ (59)

without loss of generality. The $\rm U(1)_{C+B}$ symmetry in Eq. (26), keeping
the 2SC condensate $\Phi_{\rm 2SC}$ invariant, acts on $\Phi_{dd}$ as

$\displaystyle\begin{split}\Phi_{dd}&\to\ e^{2i\alpha}e^{i\alpha
T_{8}}\,\Phi_{dd}\,(e^{i\alpha T_{8}})^{T}\\\
&=\begin{pmatrix}\Delta_{dd}^{\prime}&0&e^{3i\alpha}(\Phi_{dd})_{13}\\\
0&\Delta_{dd}^{\prime}&e^{3i\alpha}(\Phi_{dd})_{23}\\\
e^{3i\alpha}(\Phi_{dd})_{31}&e^{3i\alpha}(\Phi_{dd})_{32}&e^{6i\alpha}\Delta_{dd}^{\prime\prime}\end{pmatrix}\,.\end{split}$
(60)

The minimal vortices depends on whether the off-diagonal blocks
$(\Phi_{dd})_{13}$, $(\Phi_{dd})_{23}$, $(\Phi_{dd})_{31}$, and
$(\Phi_{dd})_{32}$ are present or not. When the off-diagonal components vanish
in the deconfined phase, the condensates on the angular coordinate $\varphi$
can be taken as $\varphi=6\alpha$ as the minimal winding vortex, as discussed
in the previous paper Fujimoto:2020dsa . We call such a vortex as U(1)C+B
vortex (d). This is nothing but the Alice string locked with the 2SC
condensate $\Phi_{\rm 2SC}$ in the deconfined phase.

Here, we restrict ourselves to the confined phase in which the off-diagonal
blocks are present as in Sec. 2.5. In this case, as the minimally winding
vortex, we take the dependence of the condensates on the angular coordinate
$\varphi$ as $\varphi=3\alpha$ in Eq. (60) for singlevaluedness of the off-
diagonal components. We call such a vortex as U(1)C+B vortex (c). We thus have
an ansatz,

$\displaystyle\begin{split}\Phi_{\rm 2SC}^{\alpha}&=(0\ 0\ \Delta_{\rm
2SC})^{T}\,,\\\
\Phi_{dd}(\varphi)&=e^{2i\varphi/3}U(\varphi)\Phi_{dd}(0)U^{T}(\varphi)\\\
&=\Delta_{dd}\begin{pmatrix}g(r)&0&h_{1}(r)e^{i\varphi}\\\
0&g(r)&h_{2}(r)e^{i\varphi}\\\ h_{1}(r)e^{i\varphi}\
&h_{2}(r)e^{i\varphi}&f(r)e^{i2\varphi}\end{pmatrix}\,,\\\
U(\varphi)&=e^{i(\varphi/3)\mathrm{diag}(-1,-1,2)}\\\
A_{i}&=-\frac{a(r)}{3g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(-1,-1,2)\end{split}$
(61)

where $f,g,h_{1},h_{2}$ are profile functions with the boundary conditions

$\displaystyle f(0)=g^{\prime}(0)=h_{1}(0)=h_{2}(0)=0,$ $\displaystyle
f(\infty)=g(\infty)=h_{1}(\infty)=h_{2}(\infty)=1.$ (62)

Here, we have set the condensate at $\varphi=0$ as

$\displaystyle\Phi_{dd}(\varphi=0)=\Phi_{dd}\begin{pmatrix}g(r)&0&h_{1}(r)\\\
0&g(r)&h_{2}(r)\\\ h_{1}(r)&h_{2}(r)&f(r)\end{pmatrix}\,.$ (63)

This carries $1/3$ quantized color-magnetic flux $\mathcal{F}_{0}$ and $2/3$
quantized circulation in $\rm U(1)_{B}$. This is precisely a doubly-wound non-
Abelian string locked with $\Phi_{\rm 2SC}$, which can be also understood as a
mesonic bound state of two Alice strings. An interesting fact is that, in the
confined phase, the $\rm U(1)_{C+B}$ vortex with 1/6 flux cannot solely be
observed but only the $\rm U(1)_{C+B}$ vortex with 1/3 flux is allowed, unlike
the deconfined phase allowing a $\rm U(1)_{C+B}$ vortex with 1/6 flux. This
difference can be understood from the order parameter manifolds in Eq. (30).

For each of the above-mentioned vortices other typical configurations are
given by the ones in Eqs. (35), (36) and (37) by $r$, $g$, and $b$,
respectively. These three configurations can be obtained by the color
rotation, only together with the rotation of the 2SC condensate $\Phi_{\rm
2SC}$ in the color space.

##  6 Summary and discussions

We have proposed a novel confinement mechanism in the two-flavor dense quark
matter that has confined and deconfined phases of vortices. As shown in the
previous paper Fujimoto:2020dsa , the most stable vortices in the deconfined
phase are non-Abelian Alice strings, which are superfluid vortices with non-
Abelian color magnetic fluxes therein, exhibiting color non-singlet AB phases.
When the 2SC condensate $\Phi_{\rm 2SC}$ develops VEVs in the confined phase,
it exhibits nontrivial (generalized) AB phases depending on the color around
the non-Abelian Alice string. In the deconfined phase, this leads to the
moduli locking; the 2SC condensate $\Phi_{\rm 2SC}$ and the vortex moduli
${\mathbb{R}}P^{2}$ are locked Fujimoto:2020dsa . On the other hand, in this
paper, we have concentrated on the confined phase and have shown that vortices
exhibiting color non-singlet AB phases are confined by the so-called AB
defects to form color-singlet bound states. More precisely, it is inevitable
that the Alice string is attached by the AB defect appearing to compensate the
AB phase to maintain the singlevaluedness of the 2SC condensate $\Phi_{\rm
2SC}$. We have shown two possibilities of color singlet states as the fate of
confinement; non-Abelian Alice strings are confined to either a baryonic or
mesonic bound state in which constituent vortices are connected by AB defects.
The baryonic bound state consists of three non-Abelian Alice strings with
different color magnetic fluxes with the total flux canceled out, which are
connected by a domain wall junction, while the mesonic bound state consists of
two non-Abelian Alice strings with the same color magnetic fluxes. Although
the latter contains a color magnetic flux in its core, this is already
confined in the sense that it has only a color-singlet AB phase.

Several discussions are in order. In this paper, we have studied the confined
phase, while the deconfined phase was studied in the previous paper
Fujimoto:2020dsa . In order to determine the phase diagram depending on the
magnetic field, temperature and so on, we need for instance the Ginzburg-
Landau (GL) theory of the two-flavor dense quark matter that remains as future
problem.

We have discussed a novel confinement mechanism of vortices in this paper. On
the other hand, this 2SC+$\langle dd\rangle$ phase also admits magnetic
monopoles. Probably these monopoles are also confined as discussed in the
SU(2) toy model Nitta:2020ggi  for which monopoles are twisted Alice strings
and confined by AB defects. If monopoles are confined, this may show a duality
between quark matter and hadronic matter; monopoles are confined where quarks
are condensed in quark matter while quarks are confined where monopoles are
condensed in hadron matter, analogous to the CFL phase Eto:2011mk .

Thus far, we have assumed that eigenvalues of $\langle dd\rangle$ are
degenerate. In general, however, they do not have to be degenerate. They
should be determined in the ground state, for instance, by the Ginzburg-Landau
theory. For non-degenerate eigenvalues of $\langle dd\rangle$, the symmetry
breaking pattern and possible vortex states are different.

We have neglected the electromagnetism in this paper. In the case of the CFL
phase, the electromagnetic interaction induces the effective potential on the
${\mathbb{C}}P^{2}$ moduli space of a non-Abelian string Vinci:2012mc . A
similar potential may exist on the ${\mathbb{R}}P^{2}$ moduli space of Alice
string in the case of the 2SC+$\langle dd\rangle$ phase as well.

Gapless fermions may exist in non-Abelian Alice strings in the 2SC+$\langle
dd\rangle$ as gapless Majorana fermion modes exist in non-Abelian vortices in
the CFL phase Yasui:2010yw ; Fujiwara:2011za . If they do, these modes may
affect the confinement problem. Also, the gapless Majorana fermions trapped
inside vortices endow a non-Abelian exchange statistics to them, thereby
turning them into non-Abelian anyons Ivanov:2000mjr , as is the case of non-
Abelian vortices in the CFL phase Yasui:2010yh ; Hirono:2012ad . It is
interesting to study whether individual non-Abelian Alice strings as well as
baryon and mesonic bound states are non-Abelian anyons before and after the
vortex confinement, respectively.

Finally, the confinement/deconfinement phase transition may be described in
terms of higher-form symmetries (generalized global symmetries)
Gaiotto:2014kfa , which is an indispensable tool to characterize the so-called
topological order. In the CFL phase, higher form symmetries in the presence of
non-Abelian semi-superfluid vortices were studied in Refs. Cherman:2018jir ;
Hirono:2018fjr ; Hirono:2019oup ; Hidaka:2019jtv ; Cherman:2020hbe , in which
a linking between a Wilson loop and a non-Abelian semi-superfluid vortex are
all color singlet. Thus, the confinement and deconfinement phases of vortices
may be distinguished by the higher form symmetry and associated topological
order.

###### Acknowledgements.

We thank Shigehiro Yasui for a discussion at the early stage of this work.
This work is supported in part by Grant-in-Aid for Scientific Research, JSPS
KAKENHI Grant Numbers 20J10506 (YF) and 18H01217 (MN).

##  Appendix A Pure color flux tubes

Here, we summarize a color-magnetic flux tube generated only by the color
gauge group $\mathrm{SU(3)_{C}}$ without using the baryon symmetry U(1)B,
which is a local vortex. It is, however, unstable to decay into the ground
state because of the trivial first homotopy group
$\pi_{1}[\mathrm{SU(3)_{C}}]=0$, as the same as color flux tubes discussed in
the cases of the 2SC and CFL phases in Refs. Iida:2004if ; Alford:2010qf ,
respectively. It is given by a closed loop in the group manifold
$\mathrm{SU(3)_{C}}$ as

$\displaystyle\begin{split}\Phi_{dd}(\varphi)&=\Delta_{dd}\begin{pmatrix}f(r)e^{-2i\varphi}&0&0\\\
0&f(r)e^{-2i\varphi}&0\\\ 0&0&f(r)e^{4i\varphi}\end{pmatrix}\,,\\\
A_{i}&=-\frac{a(r)}{g}\frac{\epsilon_{ij}x^{j}}{r^{2}}\mathrm{diag}(-1,-1,2),\end{split}$
(64)

where the boundary conditions for the profile functions $f$ and $a$ are

$\displaystyle f(0)=a(0)=0,\quad f(\infty)=a(\infty)=1\,.$ (65)

This has a color-magnetic flux

$\displaystyle\int
d^{2}xF_{12}=\frac{2\pi}{g}\mathrm{diag}(-1,-1,2)=\mathcal{F}_{0}\mathrm{diag}(-1,-1,2)\,,$
(66)

where $F_{12}$ is the color-magnetic field strength tensor and a unit color-
magnetic flux $\mathcal{F}_{0}$ is defined as

$\mathcal{F}_{0}\equiv\frac{2\pi}{g}\,.$ (67)

##  Appendix B Generalized Aharonov-Bohm phases around vortices

Here, we describe how to calculate (generalized) AB phases of light quarks,
2SC condensation, and heavy quarks around an Abelian superfluid string, pure
color flux tube, non-Abelian Alice string, and doubly-wound non-Abelian
string. Although these were already obtained in the previous paper
Fujimoto:2020dsa  except for those of the doubly-wound non-Abelian string, we
summarize them for this paper to be self-contained.

###  B.1 Aharonov-Bohm phase around Abelian superfluid strings

Abelian superfluid vortices have $B=1$ and $\mathcal{F}=0$, and let us
substitute these into Eq. (48). The generalized AB phases $\Gamma$ of light
($u,d$) or heavy ($s$) quarks encircling an Abelian U(1)B vortex can be
summarized, by using short hand notation, as

$\displaystyle\Gamma_{\beta}^{u,d}(\varphi)$
$\displaystyle=\bordermatrix{&r&g&b\cr&e^{+i\varphi/2}&e^{+i\varphi/2}&e^{+i\varphi/2}}\,$
(68) $\displaystyle\Gamma_{\beta}^{s}(\varphi)$
$\displaystyle=\begin{pmatrix}1&1&1\\\ \end{pmatrix}\,,$ (69)

respectively, where the columns ($\beta=r,g,b$) denote the colors of the light
($u,d$) or heavy ($s$) quarks encircling the vortex. Here, $\varphi$ is an
azimuthal angle around the vortex. After the complete encirclement
$\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\beta}^{u,d}(\varphi=2\pi)$
$\displaystyle=\begin{pmatrix}-1&-1&-1\end{pmatrix}\,,$ (70)
$\displaystyle\Gamma_{\beta}^{s}(\varphi=2\pi)$
$\displaystyle=\begin{pmatrix}+1&+1&+1\end{pmatrix}\,.$ (71)

Thus, the light quarks receive generalized AB phases originating from vortex
winding since they participate in the condensation with the vortex, while the
heavy quark receive no phase in the absence of a color flux.

On the other hand, when the 2SC operator $\hat{\Phi}_{\rm 2SC}$ encircles the
vortex, its generalized AB phases are

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi)=\begin{pmatrix}e^{+i\varphi}&e^{+i\varphi}&e^{+i\varphi}\end{pmatrix}\,.$
(72)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi=2\pi)=\begin{pmatrix}+1&+1&+1\end{pmatrix}\,.$ (73)

As expected, the generalized AB phases are all color-singlet since the Abelian
vortex contains no color flux.

###  B.2 Aharonov-Bohm phase around pure color flux tubes

A pure color flux tube introduced in Appendix A is generated by only color
gauge symmetry and thus is unstable by the trivial homotopy group
$\pi_{1}[{\rm SU}(3)_{\rm C}]=0$. Nevertheless we discuss AB phases around
them because of usefulness for comparison with other topologically stable
vortices.

####  B.2.1 Pure color flux tube(d)

In the deconfined phase, the pure color flux tube connects the two center
elements 1 and $\omega^{2}$ of SU(3)C and thus carries a half SU(3)C flux:
$B=0$ and ${\cal F}=1/2$. The asymptotic gauge fields of a color flux with a
color $r,g,b$ are given by $A^{r}_{i}\propto\mathrm{diag}(2,-1,-1)$,
$A^{g}_{i}\propto\mathrm{diag}(-1,-2,-1)$,
$A^{b}_{i}\propto\mathrm{diag}(-1,-1,2)$, respectively. Therefore, the AB
phases of light ($u,d$) or heavy ($s$) quarks encircling flux tubes can be
summarized as

$\displaystyle\Gamma_{\alpha\beta}^{u,d,s}(\varphi)=\begin{pmatrix}e^{+i\varphi}&e^{-i\varphi/2}&e^{-i\varphi/2}\\\
e^{-i\varphi/2}&e^{+i\varphi}&e^{-i\varphi/2}\\\
e^{-i\varphi/2}&e^{-i\varphi/2}&e^{+i\varphi}\\\ \end{pmatrix}$ (74)

where the row ($\alpha=r,g,b$) denotes the color of the flux tubes, and the
column ($\beta=r,g,b$) denotes the colors of the light ($u,d$) or heavy ($s$)
quarks encircling them. After the complete encirclement $\varphi=2\pi$, these
phases become

$\displaystyle\Gamma_{\alpha\beta}^{u,d,s}(\varphi=2\pi)=\begin{pmatrix}+1&-1&-1\\\
-1&+1&-1\\\ -1&-1&+1\\\ \end{pmatrix},$ (75)

which are color non-singlet.

On the other hand, when the 2SC condensate operator $\hat{\Phi}_{\rm 2SC}$
encircles the flux tube, its AB phases are

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi)=\begin{pmatrix}e^{-i\varphi}&e^{+i\varphi/2}&e^{+i\varphi/2}\\\
e^{+i\varphi/2}&e^{-i\varphi}&e^{+i\varphi/2}\\\
e^{+i\varphi/2}&e^{+i\varphi/2}&e^{-i\varphi}\\\ \end{pmatrix}.$ (76)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi=2\pi)=\begin{pmatrix}+1&-1&-1\\\ -1&+1&-1\\\ -1&-1&+1\\\
\end{pmatrix},$ (77)

which are color non-singlet as well.

The AB phase of the $u,d,s$ quarks and the 2SC condensate $\Phi_{\rm 2SC}$ are
different among the colors, so they are color non-singlet.

####  B.2.2 Pure color flux tube(c)

In the confined phase, the pure color flux tube is generated by a closed loop
in SU(3)C and thus carries the unit SU(3)C flux: $B=0$ and ${\cal F}=1$. The
asymptotic gauge fields of a color flux with a color $r,g,b$ are the twice of
those of the color flux in the deconfined phase. Therefore, the AB phases of
the light ($u,d$) or heavy ($s$) quarks encircling flux tubes can be
summarized as

$\displaystyle\Gamma_{\alpha\beta}^{u,d,s}(\varphi)=\begin{pmatrix}e^{+2i\varphi}&e^{-i\varphi}&e^{-i\varphi}\\\
e^{-i\varphi}&e^{+2i\varphi}&e^{-i\varphi}\\\
e^{-i\varphi}&e^{-i\varphi}&e^{+2i\varphi}\\\ \end{pmatrix}$ (78)

where the row ($\alpha=r,g,b$) denotes the color of the flux tubes, and the
column ($\beta=r,g,b$) denotes the color of light ($u,d$) or heavy ($s$)
quarks. After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{u,d,s}(\varphi=2\pi)=\begin{pmatrix}+1&+1&+1\\\
+1&+1&+1\\\ +1&+1&+1\\\ \end{pmatrix},$ (79)

which are all color singlets.

On the other hand, when the 2SC operator $\hat{\Phi}_{\rm 2SC}$ encircles the
flux tube, its AB phases are

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi)=\begin{pmatrix}e^{-2i\varphi}&e^{+i\varphi}&e^{+i\varphi}\\\
e^{+i\varphi}&e^{-2i\varphi}&e^{+i\varphi}\\\
e^{+i\varphi}&e^{+i\varphi}&e^{-2i\varphi}\\\ \end{pmatrix}.$ (80)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi=2\pi)=\begin{pmatrix}+1&+1&+1\\\ +1&+1&+1\\\ +1&+1&+1\\\
\end{pmatrix}$ (81)

which are color singlet as well.

The AB phases of the $u,d,s$ quarks and the 2SC condensate operator
$\hat{\Phi}_{\rm 2SC}$ are all color singlets.

###  B.3 Aharonov-Bohm phase around non-Abelian Alice strings

A non-Abelian Alice string has $B=1/3$ and $\mathcal{F}=1/6$, and we thus
substitute these into Eq. (48), together with the asymptotic gauge fields of a
color flux with a color $r,g,b$, given by
$A^{r}_{i}\propto\mathrm{diag}(2,-1,-1)$,
$A^{g}_{i}\propto\mathrm{diag}(-1,-2,-1)$,
$A^{b}_{i}\propto\mathrm{diag}(-1,-1,2)$, respectively.

Therefore, the pure AB phases of heavy ($s$) quark encircling flux tubes can
be summarized, again by using short hand notation, as

$\displaystyle\Gamma_{\alpha\beta}^{s}(\varphi)=\bordermatrix{&r&g&b\cr
r&e^{+i\varphi/3}&e^{-i\varphi/6}&e^{-i\varphi/6}\cr
g&e^{-i\varphi/6}&e^{+i\varphi/3}&e^{-i\varphi/6}\cr
b&e^{-i\varphi/6}&e^{-i\varphi/6}&e^{+i\varphi/3}}\,,$ (82)

where, as explicitly indicated above, the row ($\alpha=r,g,b$) denotes the
color of the flux tubes, and the column ($\beta=r,g,b$) denotes the colors of
the heavy ($s$) quark encircling them. After the complete encirclement
$\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{s}(\varphi=2\pi)=\begin{pmatrix}\omega^{2}&\omega^{-1}&\omega^{-1}\\\
\omega^{-1}&\omega^{2}&\omega^{-1}\\\
\omega^{-1}&\omega^{-1}&\omega^{2}\end{pmatrix}\,,$ (83)

which are color non-singlet. These form a ${\mathbb{Z}}_{6}$ group, and thus a
set of the strange quarks come back to the original fields after complete
encirclements of six times.

When the light quarks $u,d$ encircle the Alice string, they also receive $\rm
U(1)_{B}$ transformation $e^{+i\varphi/6}$ as well as the AB phase that they
have in common with those of the $s$-quarks. Therefore, generalized AB phases
of the light quarks $u,d$ are given by

$\displaystyle\Gamma_{\alpha\beta}^{u,d}(\varphi)$
$\displaystyle=e^{+i\varphi/6}\Gamma_{\alpha\beta}^{s}(\varphi)$
$\displaystyle=\begin{pmatrix}e^{+i\varphi/2}&1&1\\\ 1&e^{+i\varphi/2}&1\\\
1&1&e^{+i\varphi/2}\\\ \end{pmatrix}\,.$ (84)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{u,d}(\varphi=2\pi)=\begin{pmatrix}-1&+1&+1\\\
+1&-1&+1\\\ +1&+1&-1\\\ \end{pmatrix}$ (85)

which are a color non-singlet as well. We see that only quarks of the same
color with that of the flux receive a nontrivial phase $-1$.

On the other hand, when the 2SC operator $\hat{\Phi}_{\rm 2SC}$ encircles the
Alice string, its generalized AB phases are

$\displaystyle\Gamma_{\alpha\beta}^{\rm 2SC}(\varphi)$
$\displaystyle=e^{+i\varphi/3}\Gamma_{\alpha\beta}^{s}(\varphi)$
$\displaystyle=\begin{pmatrix}1&e^{+i\varphi/2}&e^{+i\varphi/2}\\\
e^{+i\varphi/2}&1&e^{+i\varphi/2}\\\
e^{+i\varphi/2}&e^{+i\varphi/2}&1\end{pmatrix}\,.$ (86)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi=2\pi)=\begin{pmatrix}+1&-1&-1\\\ -1&+1&-1\\\ -1&-1&+1\\\
\end{pmatrix}\,,$ (87)

which are color non-singlet.

The (generalized) AB phases of the $u,d,s$ quarks and the 2SC operator
$\hat{\Phi}_{\rm 2SC}$ are different among the colors, and so they are color
non-singlet. Thus, one can read out the color of the flux from infinite
distance by encircling the quarks or the 2SC condensate around the string at
infinite distance.

This situation is in a sharp contrast to the case of the CFL phase, in which
all (generalized) AB phases around non-Abelian vortices (color flux tubes) are
color singlet Chatterjee:2018nxe ; Chatterjee:2019tbz .

###  B.4 Aharonov-Bohm phase around doubly-wound non-Abelian strings

Substituting $B=2/3$ and $\Phi=1/3$ in Eq. (48), one gets the followings. The
pure AB phases of heavy ($s$) quark encircling flux tubes can be summarized as

$\displaystyle\Gamma_{ab}^{s}(\varphi)=\begin{pmatrix}e^{+2i\varphi/3}&e^{-i\varphi/3}&e^{-i\varphi/3}\\\
e^{-i\varphi/3}&e^{+2i\varphi/3}&e^{-i\varphi/3}\\\
e^{-i\varphi/3}&e^{-i\varphi/3}&e^{+2i\varphi/3}\\\ \end{pmatrix}$ (88)

where the row ($a=r,g,b$) denotes the color of the flux tubes, and the column
($b=r,g,b$) denotes the colors of the heavy ($s$) quark encircling them. After
the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{ab}^{s}(\varphi=2\pi)=\begin{pmatrix}\omega^{4}&\omega^{-2}&\omega^{-2}\\\
\omega^{-2}&\omega^{4}&\omega^{-2}\\\ \omega^{-2}&\omega^{-2}&\omega^{4}\\\
\end{pmatrix}=\omega^{4}\begin{pmatrix}+1&+1&+1\\\ +1&+1&+1\\\ +1&+1&+1\\\
\end{pmatrix}.$ (89)

These are nonzero AB phases and are color singlets, forming a
${\mathbb{Z}}_{3}$ group.

When the light quarks $u,d$ encircle the doubly-wound non-Abelian string, they
also receive a U(1)B transformation. Therefore, generalized AB phases of the
light quarks $u,d$ are

$\displaystyle\Gamma_{ab}^{u,d}(\varphi)$
$\displaystyle=e^{+i\varphi/3}\Gamma_{ab}^{s}(\varphi)$
$\displaystyle=e^{+i\varphi/3}\begin{pmatrix}e^{+2i\varphi/3}&e^{-i\varphi/3}&e^{-i\varphi/3}\\\
e^{-i\varphi/3}&e^{+2i\varphi/3}&e^{-i\varphi/6}\\\
e^{-i\varphi/3}&e^{-i\varphi/3}&e^{+2i\varphi/3}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}e^{+i\varphi}&1&1\\\ 1&e^{+i\varphi}&1\\\
1&1&e^{+i\varphi}\\\ \end{pmatrix}.$ (90)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{ab}^{u,d}(\varphi=2\pi)=\begin{pmatrix}+1&+1&+1\\\
+1&+1&+1\\\ +1&+1&+1\\\ \end{pmatrix},$ (91)

which are color singlet.

On the other hand, when the 2SC operator $\hat{\Phi}_{\rm 2SC}$ encircles the
doubly-wound non-Abelian string, its generalized AB phases are

$\displaystyle\Gamma_{\alpha\beta}^{\rm 2SC}(\varphi)$
$\displaystyle=e^{+2i\varphi/3}\begin{pmatrix}e^{-2i\varphi/3}&e^{+i\varphi/3}&e^{+i\varphi/3}\\\
e^{+i\varphi/3}&e^{-2i\varphi/3}&e^{+i\varphi/3}\\\
e^{+i\varphi/3}&e^{+i\varphi/6}&e^{-2i\varphi/3}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&e^{+i\varphi}&e^{+i\varphi}\\\
e^{+i\varphi}&1&e^{+i\varphi}\\\ e^{+i\varphi}&e^{+i\varphi}&1\\\
\end{pmatrix}$ (92)

After the complete encirclement $\varphi=2\pi$, these phases become

$\displaystyle\Gamma_{\alpha\beta}^{\rm
2SC}(\varphi=2\pi)=\begin{pmatrix}+1&+1&+1\\\ +1&+1&+1\\\ +1&+1&+1\\\
\end{pmatrix}$ (93)

which are color singlet as well.

Thus, we conclude that the (generalized) AB phases of the $u,d,s$ quarks and
the 2SC operator $\hat{\Phi}_{\rm 2SC}$ are all color singlets around the
doubly-wound non-Abelian string.

In summary of this Appendix, the (generalized) AB phases are all color singlet
for the Abelian superfluid string and doubly-wound non-Abelian string, while
these are color non-singlet around the non-Abelian Alice string. This fact
indicates that an Abelian superfluid string and doubly-wound non-Abelian
string can be present in the confined phase as discussed in the main text.

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