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Quark-hadron continuity was proposed as a crossover between hadronic matter and quark matter without a phase transition, based on the matching of the symmetries and excitations in both phases. In the limit of a light strange-quark mass, it connects hyperon matter and the color-flavor-locked (CFL) phase exhibiting color superconductivity. Recently, it was proposed that this conjecture could be generalized in the presence of superfluid vortices penetrating both phases [arXiv:1803.05115], and it was suggested that one hadronic superfluid vortex in hyperon matter could be connected to one non-Abelian vortex (color magnetic flux tube) in the CFL phase. Here, we argue that their proposal is consistent only at large distances; instead, we show that three hadronic superfluid vortices must combine with three non-Abelian vortices with different colors with the total color magnetic fluxes canceled out, where the junction is called a colorful boojum. We rigorously prove this in both a macroscopic theory based on the Ginzburg-Landau description in which symmetry and excitations match (including vortex cores), and a microscopic theory in which the Aharonov-Bohm phases of quarks around vortices match.

The presence or absence of phase transitions is the most important issue to understand phases of matter. In the last few decades, a lot of effort was made to understand the phase structure of matter at high density and/or temperature

The quark-hadron continuity conjecture was proposed as a 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

The properties in the CFL phase and in the hadron phase.

In this paper—by pointing out that the conclusion in

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

On the other hand, the presence of solitonic objects may break existing unbroken global symmetries in 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 that in the ground state. However, this scenario may change inside solitonic objects and the existing bulk symmetry may be broken spontaneously. In this case, there appear extra Nambu-Goldstone (NG) zero modes inside the solitons, which should be carefully handled during the 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 the crossover between the hadronic phase to the CFL phase. At high densities, one may expect strange quarks to appear as hyperon states on the hadronic side. In general, the first hyperon expected to appear is

We consider the singlet channel as the most attractive one in the

In the

Now let us discuss NA vortices or color magnetic flux tubes. In this case, the simplest vortex ansatz can be expressed as

First let us consider the symmetry structures in the presence of NA vortices. According to hadron-quark continuity the unbroken

We need to have a construction where the

The reasoning behind the above proposal is related to the fact that a

A schematic diagram of connection of three

One important point is that we do not require the cancellation of color magnetic fluxes at the junction point. Instead, we only require the termination of the

We comment that the present treatment of the symmetry breaking at the vortex is based on the mean-field approximation. The quantum fluctuations recover the spontaneously broken symmetry at the vortex core

We now prove the same result from a microscopic point of view, by requiring the continuity of quark wave functions in the presence of vortices penetrating the

Fermions

In the hadronic phase, we consider the vortex

Let us understand this vortex phase factor at the quark level. Since the

Before investigating the vortex phases and AB phases of quarks in the

Now, let us investigate the vortex phases and AB phases for

Let us consider the cases of the

Now we consider the connection of the vortices in the hadronic and CFL phases, by requiring the continuity of quark wave functions, that is, the matching of generalized AB phases (including vortex phases) in both phases. The generalized AB phase

To achieve a smooth connection with the generalized AB phases in Eq.

Alternatively, we consider the case where the quark encircles a

We prove that the three NA vortices and the three

In Table

The groups for the (generalized) AB phases around a

In this paper we have discussed the continuity of vortices during the crossover between the hadronic and CFL phases. By using macroscopic (GL) and microscopic (quark) descriptions, we have proved that three

We have ignored (strange) quark masses and electromagnetic interactions, whose effects on an NA vortex were investigated in Refs.

We would like to thank Motoi Tachibana for discussions. This work is supported by the Ministry of Education, Culture, Sports, Science and Technology (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.).

We summarize the symmetries of the CFL phase. The color-flavor-locked phase can be expected when the density becomes asymptotically high. The order parameters in the CFL phase are defined by the diquark condensates (close to the critical temperature

The BdG equation in the

We consider a single-component (massless) Dirac fermion in the presence of a vortex with winding number 1. The explicit form of the BdG equation is

The BdG equation

The BdG equation was used to find a Majorana fermion zero mode in an NA vortex in Refs.