# Quark-hadron continuity under rotation: vortex continuity or boojum?

###### Abstract

Quark-hadron continuity was proposed as crossover between hadronic matter and quark matter without a phase transition, based on matching of symmetry and excitations in both phases. In the limit of light strange quark mass, it connects hyperon matter and color-flavor locked (CFL) phase exhibiting color superconductivity. Recently, this conjecture was proposed to be generalized in the presence of superfluid vortices penetrating the both phases Alford:2018mqj (), in which they suggested that one hadronic superfluid vortex in hyperon matter could be connected to one non-Abelian vortex (color magnetic flux tube) in the phase. Here, we argue that their proposal is consistent only at large distances: Instead, we show that three hadronic superfluid vortices must join together to three non-Abelian vortices with different colors with the total color magnetic fluxes canceled out, where the junction was called a colorful boojum. We rigorously prove this both in a macroscopic theory based on the Ginzburg-Landau description, in which symmetry and excitations match including vortex cores, and a microscopic theory based on the Bogoliubov de-Gennes equation, in which the Aharanov-Bohm phase of quarks around vortices match.

## I Introduction

The presence or absence of phase transitions is the most important issue to understand phases of matter. In last few decades, a lot of efforts was paid to understand the phase structure of matter at high density and/or temperature Fukushima:2010bq (). In particular, the region of high density and low temperature is relevant for cores of compact stars such as neutron stars, in which nuclear matter and quark matter are present. Superfluidity of nucleon-nucleon pairing is expected in nuclear matter, and there appears, in high density region, nuclear matter of hyperons, nuclei containing strange quarks, is expected Takatsuka:2000kc () (see Ref. Sedrakian:2018ydt () as a recent review). In higher density, quark matter is expected; in asymptotically high density much higher than strange quark mass, the color-flavor locked () phase is realized, in which three (up, down and strange) quarks participate in a diquark paring, exhibiting color superconductivity as well as superfluidity Alford:1997zt (); Alford:1998mk (), see Refs. Alford:2007xm (); Rajagopal:2000wf () as a review. In addition to superfluid vortices Forbes:2001gj (); Iida:2002ev (), there are non-Abelian vortices or color magnetic flux tubes Balachandran:2005ev (); Nakano:2007dr (); Nakano:2008dc (); Eto:2009kg (); Eto:2009bh (); Eto:2009tr (); Alford:2016dco (), see Ref. Eto:2013hoa () as a review. The former is dynamically split into three of the latter with the total magnetic fluxes canceled out Nakano:2007dr (); Alford:2016dco ().

The quark-hadron continuity conjecture was proposed as crossover between hadronic matter and quark matter, based on the matching of elementary excitations and existing global symmetries in both the matter, in particular hyperon matter and phase Schafer:1998ef (); Alford:1999pa (), as summarized in Table 1. The continuity was further studied in interior of neutron stars Masuda:2012kf (); Masuda:2012ed (); Baym:2017whm (). Since neutron stars are rapidly rotating, there appear superfluid vortices both in nuclear matter and quark matter, thereby it is natural to extend the quark-hadron continuity in the presence of vortices penetrating both the matter Alford:2018mqj (). They defined a continuity of vortices by matching of the Onsager-Feynman circulation of vortices in both the phases, and suggested a continuity to connect one hadronic vortex to one non-Abelian vortex during the hadron- crossover.

In this Letter, with pointing out that their conclusion is consistent only in large distance behavior of vortices but is not compatible with symmetry structures of vortex cores, we reach a conclusion that arguably only possibility that is left is to form a connection of three hadronic vortices in the hyperon matter of condensation with three non-Abelian vortices in the phase with total color magnetic fluxes cancelled out, forming a colorful boojum Cipriani:2012hr (), analogous to a boojum in helium 3 superfluid Mermin (); Volovik (). We prove this both in a macroscopic theory based on the Ginzburg-Landau (GL) description, in which symmetry and excitations match including vortex cores, and a microscopic theory based on the Bogoliubov de-Gennes (BdG) equation, in which the Aharanov-Bohm (AB) phase of quarks around vortices match.

hadronic phase | CFL phase | ||

unbroken symmetry | |||

# of NG bosons | 8 | 8 | |

# of massive vector mesons | 8 | 8 | |

# of quasi fermions | 8 | 8+1 | |

vortex configurations | 3 | , , | |

vortices | circulation of 3 vortices | ||

AB phase of quarks |

## Ii Vortex continuity in macroscopic theory

The concept of continuity is defined by continuation of symmetries and elementary excitations at ground state while going through crossover. Now we would like to discuss the concept of continuity in the presence of general background. For example, the vortices which are present in two different phases should be joined together so that all physical quantities remain smoothly connected and symmetry structure remains the same through the crossover.

On the other hand, in the presence of solitonic objects may sometime break existing unbroken global symmetries present at the ground state. Since the condensate eventually reaches its ground state expectation value (modulo gauge transformations) at large distances, the large distance symmetry structure in general remains the same as the ground state. However, scenario may change inside solitonic objects and the existing bulk symmetry may be broken spontaneously inside. In this case there appear extra Nambu-Goldstone (NG) zero modes inside the solitons, which should be carefully handled during crossover. In other words, to maintain the continuity of solitonic objects along with elementary excitations, one should check the symmetry structure everywhere.

Let us focus our interest on crossover
of the hadronic phase to the phase.
At high densities, one may expect the appearance of strange quarks
as hyperon states in the hadronic side.
In general, the first hyperon expected to appear is , which is the lightest one with an
attractive potential in nuclear matter. Here we consider only flavor symmetric pairing in channel
for our purpose to be fulfilled.^{1}^{1}1We consider the singlet channel as the most attractive one in the limit. We use the paring as an abbreviation of paring for nucleon () and , , baryons deSwart:1963pdg (). In this case we may consider the existence of superfluid vortices since would break
baryon number symmetry and we may express the vortex ansatz as

(1) |

with the distance from the center of the vortex and the angle around the vortex axis. The exact nature of the profile function can be derived from the GL theory of the system and we are not going to discuss this here. Since the condensate is flavor symmetric in this phase, the flavor symmetry would be intact everywhere including the vortex cores. The Onsager-Feynman circulation which is defined as where and are the winding number and chemical potential of the condensate, can be computed for a single vortex to be where is the chemical potential for a single baryon. Here is the superfluid velocity at large distance from the core of the vortex.

In the phase, the order parameter is a matrix with a color index and a flavor index , where . The Ginzburg-Landau formulation of the CFL phase has been derived in Refs. Giannakis:2001wz (); Iida:2000ha (); Iida:2001pg (). Symmetries in the CFL phase are summarized in Appendix. The order parameter for an Abelian superfluid vortex can be written as Forbes:2001gj (); Iida:2002ev ()

(2) |

where is a profile function vanishing at the center of the vortex, , and eventually reaching the ground state value at large distances. is the absolute value of the gap (condensate) at the bulk in the CFL phase. The Onsager-Feynman circulation of Abelian vortices in the CFL phase is found to be , since the chemical potential of di-quark is . So a single vortex cannot connect continuously to a single vortex in the CFL phase. Instead, we may conclude that three vortices would join to form one CFL vortex.

Now let us discuss non-Abelian vortices or color magnetic flux tubes. In this case, the simplest vortex ansatz can be expressed as Balachandran:2005ev (); Eto:2009kg (),

(3) | |||||

with the gauge coupling constant . The profile functions , and can be computed numerically with boundary conditions, Eto:2009kg (). We call this an up-red () vortex since the component has a vortex winding. We also define two other vortices by changing the position of the vortex winding () from to and , which can be called as down-green () and strange-blue () vortices, respectively. At large distances, the order parameter of these three vortices behave as where is the large distance configuration of the gauge field corresponding to the color flux present inside the vortex core. So it is easy to check that at large distances the symmetry remains unbroken. In this case one may derive superfluid velocity at large distances by replacing ordinary derivative to covariant derivative in the expression of the current. The Onsager-Feynman circulation of non-Abelian vortices in the phase is found to be , which coincides with the circulation of a single vortex. Therefore one would expect that a single vortex would be smoothly connected to a single non-Abelian vortex during the crossover Alford:2018mqj (). Below we show that this is consistent only at large distances but contradict at short distance at the vortex core.

First let us consider the symmetry structures in the presence of non-Abelian vortices. According to hadron-quark continuity the unbroken symmetry can be smoothly connected to the unbroken flavor symmetry in the hadron phase. So it seems that as if there would not be any problem also for continuation of a non-Abelian vortex to a single vortex. However, the missing point is that the symmetry present at the bulk is spontaneously broken at the core of a non-Abelian vortex to . This generates NG modes inside the vortex core Nakano:2007dr (); Eto:2009bh (). The low-energy effective theory of the NG modes was obtained along the vortex line Eto:2009bh (); Eto:2009tr (); Chatterjee:2016tml (). This helps us to distinguish two different vortices by flavor quantum numbers. The three kinds of vortices , and , where the color part is chosen in a particular gauge for our own convenience, lie in three points of the moduli space, and they are continuously connected by the flavor symmetry. This can be understood directly from the structure of the order parameters at the center of vortices. We may write the order parameters at the center of vortices for these three cases as

(4) |

with a constant which can be fixed numerically. The flavor symmetry is spontaneously broken by these matrices to three different unbroken subgroups of the . Since the flavor is unbroken in the hadronic vortex, following symmetry principle of continuity we can say that a single vortex cannot smoothly transform into any single non-Abelian vortex.

We need to have a construction where the flavor symmetry is recovered in a vortex core while connecting to the hadronic phase.
In other words, we have to terminate the NG modes.
This is possible only when three different non-Abelian vortices join
to one vortex in whose core the flavor symmetry
is not broken.^{2}^{2}2
The reason in the support of the above proposal is related to the fact
that a vortex is
energetically unstable to break into three non-Abelian vortices Nakano:2007dr (); Alford:2016dco ().
The NG modes of the three different kind of non-Abelian vortices describe fluctuations from the three different points
of the moduli space.
When we join all of them, these NG modes can smoothly
move from one patch to another patch at the junction.
As we already discussed,
one vortex can be connected to three vortices during the hadron- crossover,
and then we reach Fig. 1.
The junction point was called a colorful boojum Cipriani:2012hr (),
analogous to those in helium 3 superfluids Mermin (); Volovik ().

One important notice is that we did not require cancellation of color magnetic fluxes at the junction point. Instead, we only required the termination of the NG modes. The color of a non-Abelian vortex is gauge dependent as emphasized in Ref. Alford:2018mqj (), but the termination of the NG modes implies the cancellation of the color magnetic fluxes in our gauge choice.

## Iii Vortex continuity in microscopic theory

We now prove the same result from a microscopic point of view, by requiring a continuity of quark wave functions in the presence of vortices penetrating the and hadronic phases. More precisely, we achieve a continuity of AB phases of quarks encircling around vortices.

In the hadronic phase, the BdG equation in the vortex, in Eq. (1), can be written as

(5) |

for the baryon ( particle component, hole component) in the Nambu-Gor’kov formalism. Here, is the baryon mass, is a profile function of the vortex. By noting that the BdG equation has the symmetry , , we find that the rotation of a baryon around the vortex with an angle induces the above phase transformations on particle and hole components of the baryon. A complete encircling yields an AB phase . We call this factor as a charge or simply a AB phase of the baryon around the vortex. Let us understand this in the quark level. Since the quarks are confined inside the baryon in a symmetric way, each quark quasi-particles in should have AB phases given by

(6) |

This is because the phase should be independent of flavor and color. Here, we indicated the AB phases of particle components, while the hole components have simply opposite sign. The question is then how the phases of quarks can be connected smoothly from the hadronic phase to the phase.

Before investigating the AB phases of quarks in the phase, we remind us that, as a simpler case of a single-component Dirac fermion () in the BdG equation has an AB phase because of the phase as in for the rotation as shown in Appendix B.

The BdG equation in the presence of a non-Abelian vortex is given by Yasui:2010yw (); Fujiwara:2011za () (see also Ref. Sadzikowski:2002in ())

(7) |

where we have used the notation e.g.,
in the Nambu-Gor’kov representation.^{3}^{3}3The BdG equation was used to find a Majorana fermion zero mode on a non-Abelian vortex in Refs. Yasui:2010yw (); Fujiwara:2011za (), and it was applied to a non-Abelian statistics of exchanging multiple non-Abelian vortices Yasui:2010yh (); Yasui:2011gk (); Hirono:2012ad (); Yasui:2012zb (). The coupling of the Majorana fermion zero modes and the NG modes was obtained in Ref. Chatterjee:2016ykq ().
We define
and

(8) |

where corresponds to the vortex configuration with winding number one, and does not have a winding number. We also obtain the BdG equations for the and vortices, where the quarks couple to the gap profile functions differently. The AB phases , , of quarks around non-Abelian , , vortices are obtained respectively as

(9) |

Now we consider the connection of the vortices in the hadronic and phases. The AB phase of quarks around a vortex in Eq. (6) is apparently different from those of quarks in any single non-Abelian vortex (either of , , ) in the phase:

(10) |

Therefore, one non-Abelian vortex cannot be connected to one vortex without discarding the continuity at the quark level, although such a connection could be consistent only at large distance scale in the GL equation Alford:2018mqj ().

To achieve a smooth connection with the obtained charges in Eq. (9), we consider the case that the quark turns around a bundle of non-Abelian , , vortices simultaneously as shown in Fig. 2. We notice that all the quarks acquire the AB phase irrespective to the flavor and color components. For example, the quark acquires a phase for the path around the vortex, for the and vortices, and hence it acquires in total. The same phase is obtained for the other quarks. The AB phases in the path turning around the , , vortices simultaneously are equal to the sum of the AB phases in the paths turning around each of them: . As a result, the quarks with any flavor and color acquire a common charge, which turns out to be exactly equal to the AB phase in the presence of three vortices:

(11) |

Therefore, the continuity of the AB phases of quarks is allowed only when the bundle of , , vortices is connected to the bundle of three vortices.

We prove that the , , vortices and the three vortices meet at one point in transverse directions. First of all, we notice that the quarks can take an arbitrary path. One may think of a path that does not necessarily encircle all the , , vortices when those vortices are separated in space. However, such a path precludes a continuity of the quark wave functions between the and hadronic phases. Therefore, only the paths turning simultaneously around the , , vortices should be allowed to exist: the , , vortices meet at one point. There, they are connected to the vortex, as shown in Fig. 1. In summary, the continuity of the quark wave function induces that the bundle of the , , vortices and the three vortices are connected via the vortex.

## Iv Summary and Discussion

In this Letter we have discussed a continuity of vortices during the crossover between the hadronic and CFL phases. By using macroscopic (GL) and microscopic (BdG) descriptions, we have proved that three vortices in the hadronic phase must join together and transform to three different non-Abelian () vortices in the phase to maintain a smooth connection [see Eq. 11)]. The colorful boojum is inevitable for quark-hadron continuity.

We have ignored (strange) quark masses and electromagnetic interaction, whose effects on a non-Abelian vortex were investigated in Refs. Eto:2009tr () and Vinci:2012mc (); Hirono:2012ki (); Chatterjee:2015lbf (), respectively. We should take into account them for more realistic situations. One of questions is whether fermion zero modes in a vortex core in the CFL phase Yasui:2010yw (); Fujiwara:2011za () have a continuity to the hadron phase. It is also interesting what a role a confined monopole in the CFL phase Eto:2011mk (); Gorsky:2011hd () plays for quark-hadron duality. It is also interesting to study how vortex lattices Kobayashi:2013axa () are connected during continuity. Finally, it will be important to study impacts of the presence of vortex junctions (boojums) on dynamics of neutron stars.

## Acknowledgment

We would like to thank Motoi Tachibana for discussions. This work is supported by the Ministry of Education, Culture, Sports, Science (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006). C. C. acknowledges support as an International Research Fellow of the Japan Society for the Promotion of Science (JSPS) (Grant No: 16F16322). This work is also supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No. 16H03984 (M. N.), No. 18H01217 (M. N.), No. 17K05435 (S. Y.)), and also by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” No. 15H05855 (M. N.).

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## Appendix A The symmetries of Phase

We summarize symmetry of the CFL phase. The color-flavor-locked phase can be expected when density becomes asymptotically high. The order parameters in phase are defined by the di-quark condensates (close to the critical temperature ) as where are left/right handed quarks carrying fundamental color indices () and fundamental flavour () indices . The chiral symmetry is spontaneously broken at ground state . The order parameter transforms as After subtraction of the redundant discrete symmetries the actual symmetry group is given by . At ground state the full symmetry group is spontaneously broken down to and the oder parameter is defined as where depends on the GL parametersGiannakis:2001wz (); Iida:2000ha (); Iida:2001pg (). The existence of stable vortices can be confirmed by a non-trivial first homotopy group of the order parameter space .

## Appendix B BdG equation for a single component Dirac fermion

We consider a single component (massless) Dirac fermion in the presence of a vortex with a winding number . The explicit form of the BdG equation is expressed as

(12) |

with particle component and hole component in the Nambu-Gor’kov representation. Here, is the chemical potential. The rotation of the quark around the vortex changes to . This is compensated by the phase rotations for and , to maintain the above equation, by changing to . Therefore, the particle (hole) in the presence of the vortex has an AB phase .