Alice meets Boojums in neutron stars: vortices penetrating two-flavor quark-hadron continuity
AAlice meets Boojums in neutron stars:vortices penetrating two-flavor quark-hadron continuity
Yuki Fujimoto ∗ and Muneto Nitta † Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Department of Physics & Research and Education Center for Natural Sciences,Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
Alice and Boojums are both representative characters created by Lewis Carroll. We show thatthey possibly meet in cores of rotating neutron stars. Recent studies of quark-hadron continuitysuggest that neutron superfluid matter can connect smoothly to two-flavor symmetric quark matterat high densities. We study how this can be maintained in the presence of the vortices. In theneutron matter, quantized superfluid vortices arise. In the two-flavor dense quark matter, vorticescarrying color magnetic fluxes together with fractionally quantized superfluid circulations appearas the most stable configuration, and we call these as the non-Abelian Alice strings. We showthat three integer neutron superfluid vortices and three non-Abelian Alice strings of different colormagnetic fluxes with total color flux canceled out are joined at a junction called a Boojum.
I. INTRODUCTION
Neutron stars, particularly pulsars, provide us with aunique opportunity to study states of matter under ex-treme conditions: highest known baryon density in theuniverse, rapid rotation, strong magnetic fields, etc (see,e.g., Refs. [1–3] for reviews). In the present work, weaddress the combined effect of high-density and rapid ro-tation; namely the quantum vortices that appear in theneutron superfluid and the color-superconducting quarkmatter.There are several ongoing observations of neutron starssuch as the ones with two-solar-mass [4–6], probed bygravitational wave detectors [7, 8], and investigated inthe NICER mission [9, 10], etc., and they have led toa flurry of studies about neutron stars in diverse fieldsof research. Amongst such efforts, numerous articles fo-cus on the equation of state (EoS). The recent develop-ment in this direction was constructing a phenomenolog-ical hybrid EoS with a smooth crossover for the hadron-to-quark phase transition [1, 11–14] rather than a first-order phase transition as conventionally done in manyliteratures. The crossover construction enables neutronstars to have a sizable quark core inside consistently withthe constraints put by observations. Such constructionshould not be regarded as an exotic alternative becausethere can indeed be a possibility that a substantial quarkcore inside a heavy neutron star is realized as suggestedby the model-independent analysis [15]. The plausibleground state of such cold dense quark matter is colorsuperconductor [16, 17] with various patterns of the di-quark pairing being known such as color-flavor locked(CFL) phase [18] in three-flavor symmetric matter andtwo-flavor superconducting (2SC) phase [19, 20] in two-flavor symmetric matter. ∗ [email protected] † [email protected] The idea of crossover construction of the EoS can betraced back to the concept of quark-hadron continuity be-tween the CFL phase and the hyperon superfluid phase[21] (see also [22–26]). The both color-superconductingand hadronic superfluid phases share the same symme-try breaking patterns and low-lying excitations, so thesephases can be connected continuously. The recent break-through is an extension to the case with rapid rotationunder which superfluid vortices are created in hadronicand quark matter [27–31]. In the CFL phase, super-fluid vortices appear [32, 33], but each superfluid vor-tex is unstable against a decay into a set of more sta-ble vortices [34, 35]. The most stable vortices are non-Abelian semi-superfluid vortices carrying color magneticfluxes and fractionally quantized 1/3 circulation of theAbelian superfluid vortices [34–38]. Thus, an importantquestion was raised in Ref. [27] how these non-Abelianvortices penetrate into hyperonic matter; It was sug-gested in Ref. [27] that one non-Abelian vortex in theCFL phase should be connected to one superfluid vortexin the hyperon matter. This work stimulated the discus-sion whether there is a discontinuity between the vorticesin two phases [29] or not [30] from the viewpoint of topo-logical order based on a Wilson loop linking a vortex.Aharonov-Bohm (AB) phases of quasi-particle encir-cling vortices provide us with further insight into thisproblem by comparing these in the hyperonic and CFLphases. It was shown in Ref. [28] that three non-Abelianvortices, which carry different color magnetic fluxes withtotal color canceled out, must joint at one point to threeinteger vortices in the hyperonic matter. Such a junctionpoint of vortices is called a colorful
Boojum [39]; origi-nally, the similar structures found in helium superfluidswere named Boojums by Mermin [40], [41] and have beenpredicted to occur in He superfluids [42] in particularat the A-B phase boundary [43, 44], liquid crystals [45],Bose-Einstein condensates [46], and quantum field the-ory [47]. Moreover, in neutron stars, Boojums were sug-gested to explain pulsar glitch phenomena, which is a a r X i v : . [ h e p - ph ] F e b sudden speed-up in rotation [48].Thus far, we have discussed quark-hadron continuitywithin the ideal three-flavor symmetric setup in the limitof strange ( s ) quark mass degenerate with up ( u ) anddown ( d ) quark masses. In the realistic setup, however,the hadronic matter is dominated by neutrons, which arecomposed of two-flavor u and d valence quarks, and the s -quark masses are heavy so that they do not partici-pate in condensation. Neutrons are superfluids, for whichpairing in P channel are responsible [49–52] (see alsoRefs. [53, 54] and references therein for recent studies).By contrast, the conventional 2SC phase does not ex-hibit the property of superfluidity because U(1) B sym-metry stays intact in this phase. It was, however, pro-posed recently in Ref. [55] that the two-flavor color su-perconductor can be superfluids if we take into accountthe P pairing of d -quarks in addition to the 2SC pair-ing, and this novel phase was named 2SC+ (cid:104) dd (cid:105) phase.The consideration of the superfluidity led to the two-flavor counterpart of the quark-hadron continuity, i.e.,the 2SC+ (cid:104) dd (cid:105) phase and P neutron superfluid phaseare continuously connected. The two-flavor continuityis indeed consistent with the crossover construction ofthe EoS and more natural in the sense that we use thehadronic EoS dominated by nucleons in the most of thecases; see Ref. [14] for an explicit crossover constructionof the EoS within the two-flavor setup.Then, a natural question arises that if Boojums arealso present in the two-flavor quark-hadron continuity.The most stable vortices in the two-flavor dense QCDare “ non-Abelian Alice strings” [56], which are the non-Abelian counterpart of the so-called Alice strings [57–60].The vortices support 1/3 fractional windings in U(1) B aswell as the color-magnetic fluxes, and a single AbelianU(1) B vortex is unstable against a decay into triple ofnon-Abelian Alice strings, as in the case of the CFLphase. Quasi-particles winding around a non-AbelianAlice string pick up nontrivial (color non-singlet) ABphases, unlike those of CFL vortices with color singletAB phases.In this work, we show that in the two-flavor quark-hadron continuity picture, three Alice strings with red,blue, green color magnetic fluxes in two-flavor quark mat-ter must join at a junction point to three integer vorticesin P neutron matter, forming a colorful Boojum. II. TWO-FLAVOR DENSE MATTER
We give a brief synopsis of the two-flavor hadronic andquark matter at high density [55].The hadronic phase is a neutron P superfluid, forwhich the order parameter operator is given by [49, 50]ˆ A ij = ˆ n T C γ i ∇ j ˆ n , (1)where ˆ n denotes a neutron field operator, C is the chargeconjugation operator, and indices ( i, j, . . . ) denote spa-tial coordinates. In this paring, the matrices γ i and spatial derivatives ∇ j account for spin and angular mo-mentum contributions in the P pairing, respectively.We further assume that neutrons made out of u - and d -quarks can be described as a quark-diquark systemˆ n = (cid:15) αβγ (ˆ u Tα C γ ˆ d β ) ˆ d γ with the Greek letters ( α, β, . . . )being the color indices.In the quark matter side, which is the 2SC+ (cid:104) dd (cid:105) phase,the expectation value of ˆ A ij can be taken, under themean field approximation, as A ij = (cid:104) ˆ A ij (cid:105) (cid:39) (Φ ) α (Φ ) β (Φ dd ) ijαβ , (2)Φ ≡ (cid:104) (cid:15) αβγ ˆ u Tβ C γ ˆ d γ (cid:105) , Φ dd ≡ (cid:104) ˆ d Tα C γ i ∇ j ˆ d β (cid:105) , (3)where we have suppressed the spatial indices i, j of Φ dd ,and hereafter an appropriate tensor structure is implied.Note that quantities without hat symbols denote conden-sates while those with hat symbols are operators. Thetwo condensates Φ and Φ dd at the mean field level inEq. (2) account for the color superconductivity of thequark matter: Φ is the so-called 2SC condensate,while the novel feature here is represented by Φ dd , whichis the diquark condensate of d -quarks in the P channel. A ij is always non-zero in both phases, so the local orderparameter indeed cannot distinguish these two phases,leading the continuity [55].Let us discuss the symmetry action on the diquarkcondensate, which will be used in the later analysis ofthe vortices. The relevant part here in the symmetry ofQCD is G QCD = SU(3) C × U(1) B . Under the element( U, e iθ B ) ∈ SU(3) C × U(1) B acting on quark fields ˆ q asˆ q → e iθ B U ˆ q as a column vector belonging to the funda-mental representation of SU(3) C with U ∈ SU(3) C , thediquark condensates (3) transform asΦ → e iθ B U ∗ Φ , Φ dd → e iθ B U Φ dd U T . (4) III. VORTICES IN TWO-FLAVOR DENSEMATTER
We classify the vortices that appear in the two-flavorneutron and quark matter following Ref. [56].First, let us discuss P neutron superfluid vortices.In terms of the P order parameter in Eq. (1), a sin-gle integer vortex in the P neutron superfluid can beasymptotically written as [51, 61–63] A ij ( ϕ ) ∼ e iϕ A ij ( ϕ = 0) (5)at large distance. In the weak coupling limit, the P superfluid is in the nematic phase [52] with A ij ( ϕ = 0) =diag( s, s, − s ) with a real parameter s , which is actuallydetermined by the temperature and magnetic field [53,54]. At higher magnetic field, half-quantized vortices arethe most stable [64], but we do not consider such a casefor simplicity.Next, let us discuss vortices in 2SC+ (cid:104) dd (cid:105) phase [56].The simplest vortex is a superfluid vortex of the formΦ dd ( ϕ ) = f ( r ) e iϕ ∆ dd ∼ e iϕ ∆ dd Φ = h ( r ) e iϕ √ (1 , , T (6)with the boundary condition f (0) = h (0) = 0 and f ( ∞ ) = h ( ∞ ) = 1 for the profile function f and h ( r ),and ( r, ϕ ) being the polar coordinates. We also assumeda unitary gauge fixing for the diquark condensates. Thisvortex configuration carries a unit quantized circulationin U(1) B as encoded in the factor e iϕ . We call thisa U(1) B superfluid vortex or an Abelian vortex. Thisstring is created under rotation because of the superflu-idity. Although this string is topologically stable due to π [U(1) B ] = Z , it is unstable against decay into threenon-Abelian Alice strings introduced below. The P or-der parameter given in Eq. (2) behaves as A ij ∼ e iϕ A ij ( ϕ = 0) (7)at large distance. Compared with Eq. (5), this corre-sponds to three integer vortices in the P neutron mat-ter.Next, we present a non-Abelian Alice string as themost stable vortex. The condensate at spatial infinitycan be written asΦ dd ( ϕ ) = e iϕ/ U ( ϕ )Φ dd ( ϕ = 0) U T ( ϕ ) . (8) U ( ϕ ) = P exp (cid:18) ig (cid:90) ϕ A · d (cid:96) (cid:19) . (9)The full vortex ansatz is of the form ofΦ dd ( ϕ ) = ∆ dd diag( f ( r ) e iϕ , g ( r ) , g ( r )) ,U ( ϕ ) = e i ( ϕ/ , − , − ,A i = − a ( r )6 g (cid:15) ij x j r diag(2 , − , − , Φ = (∆ , , T , (10)for the red color magnetic flux ( r ),Φ dd ( ϕ ) = ∆ dd diag( g ( r ) , f ( r ) e iϕ , g ( r )) ,U ( ϕ ) = e i ( ϕ/ − , , − ,A i = − a ( r )6 g (cid:15) ij x j r diag( − , , − , Φ = (0 , ∆ , T , (11)for the green one ( g ), andΦ dd ( ϕ ) = ∆ dd diag( g ( r ) , g ( r ) , f ( r ) e iϕ ) ,U ( ϕ ) = e i ( ϕ/ − , − , ,A i = − a ( r )6 g (cid:15) ij x j r diag( − , − , , Φ = (0 , , ∆ ) T , (12) for the blue one ( b ), with the boundary conditions forthe profiles f (0) = g (cid:48) (0) = a (0) = 0 , f ( ∞ ) = g ( ∞ ) = a ( ∞ ) = 1. These carry 1 / F = F /
6, and 1 / B .For all the three cases, the P order parameter givenin Eq. (2) behaves as A ij ( ϕ ) ∼ e iϕ A ij ( ϕ = 0) (13)at large distance. IV. VORTEX CONTINUITY AND BOOJUMS
As mentioned earlier, the idea of the vortex conti-nuity was originally proposed in Ref. [27]. Their dis-cussion of the vortex continuity was based on quantitycalled the Onsager-Feynman circulation whose definitionis given by C = (cid:72) v · d (cid:96) = 2 πn/µ with n and µ beingthe winding number and chemical potential of the con-densate, respectively. The circulations of vortices in thehadronic and quark phase are turned out to be identi-cal, so, it led to the observation that a single hadronicvortex would be smoothly connected to a single non-Abelian vortex. One can also calculate the circulationfor our case: The circulation of a neutron P vortexis C nn = π/µ B with µ B being the baryon chemical po-tential. The circulation of a non-Abelian Alice string isgiven by C NA = π/ µ q = π/µ B , where µ q = µ B / C nn = C NA . The ex-pressions for the neutron superfluid order parameter inEqs. (5, 13) also coincide with each other. Thus, at aglance one might think that a single non-Abelian Alicestring would be connected to a single integer P vortex.It is, however, not true as shown below.In order to investigate how vortices are connected,we employ the (generalized) AB phases of quarks encir-cling around vortices. When the neutron field ˆ n encirclesaround a single integer vortex given in Eq. (5), it re-cieves a phase factor (generalized AB phase) exp( iϕ/ ϕ , and after the complete encirclement it ob-tains exp( iπ ) = −
1. Then, we next take quarks as probeparticles and calculate the AB phase that the quarks re-cieve. Since we have assumed the neutron operator asˆ n = (cid:15) αβγ (ˆ u Tα C γ ˆ d β ) ˆ d γ , light quarks ˆ q = ˆ u, ˆ d obtain thephase Γ u,dnn ( ϕ ) ≡ exp( iϕ/
6) when they encircle the neu-tron vortex at angle ϕ :ˆ q ( ϕ = 0) → ˆ q ( ϕ ) ∼ Γ u,dnn ( ϕ )ˆ q ( ϕ = 0) , (14)Thus, at the quark level, the generalized AB phase formsa Z group. The heavy quark field ˆ s ves no phase aroundthe vortex, i.e., Γ snn ( ϕ ) ≡
1. We explain our notationthat Γ ψnn is the generalized AB phase around neutron P vortex probed by particle ψ = ˆ u, ˆ d, ˆ s .Let us turn to quark matter and calculate phase factorsof quarks around the Alice string. For any ϕ (cid:54) = 0, thelight quark field (ˆ q = ˆ u, ˆ d ) and heavy quark field (ˆ s ) aregiven by a holonomy action asˆ q ( ϕ ) ∼ e iθ B ( ϕ ) U ( ϕ )ˆ q ( ϕ = 0) , (15)ˆ s ( ϕ ) ∼ U ( ϕ )ˆ s ( ϕ = 0) , (16)respectively, where U ( ϕ ) is defined in Eq. (9). Here, weuse the following shorthand notation for the generalizedAB phase. The field ˆ ψ (= ˆ q, ˆ s, . . . ) of color β (= r, g, b )encircling around the flux tubes of the α (= r, g, b ) colormagnetic flux acquires the AB phase denoted by Γ ψαβ :ˆ ψ β ( ϕ = 0) → ˆ ψ β ( ϕ ) ∼ Γ ψαβ ( ϕ ) ˆ ψ β ( ϕ = 0) , (17)Note that the index β is not contracted in the aboveformula. The pure AB phases of heavy quark encirclingaround flux tubes areΓ sαβ ( ϕ ) = r g br e + iϕ/ e − iϕ/ e − iϕ/ g e − iϕ/ e + iϕ/ e − iϕ/ b e − iϕ/ e − iϕ/ e + iϕ/ , (18)where, as explicitly indicated above, the row ( α = r, g, b )denotes the color of the flux tubes, and the column ( β = r, g, b ) denotes the colors of the heavy ( s ) quark encirclingthem. Thus, the heavy quark field s forms a Z grouparound the Alice string.When the light quarks u, d encircle the Alice string,they also recieve U(1) B transformation e + iϕ/ as well asthe AB phase that they have in common with those ofthe s -quarks. Therefore, the generalized AB phases areΓ u,dαβ ( ϕ ) = e iϕ/ Γ sαβ ( ϕ ) = e iϕ/ e iϕ/
11 1 e iϕ/ . (19)Thus, the light quarks u, d form a Z group around theAlice string.From the above calculations of generalized AB phasesaround the vortices, one can immediately conclude thatone integer vortex in P phase cannot be connected toone Alice string with any color flux. We can check thisby the fact that the AB phases of the light quarks do notmatch between the neutron matter and two-flavor quarkmatter: Γ u,dnn ( ϕ ) (cid:54) = Γ u,dαβ ( ϕ ) for any α, β = r, g, b (20)and the same for heavy quarks:Γ snn ( ϕ ) (cid:54) = Γ sαβ ( ϕ ) for any α, β = r, g, b (21)Only possibility is that a bundle of three integer vor-tices in the P neutron matter can be connected to abundle of three Alice strings with different color fluxes r, g, b . In this case, the (generalized) AB phases for bothphases completely coincide:[Γ u,dnn ( ϕ )] = Γ u,drβ ( ϕ )Γ u,dgβ ( ϕ )Γ u,dbβ ( ϕ ) for β = r, g, b (22) P superfluid vortices non-Abelian Alice strings FIG. 1. A schematic figure of the Boojum. Three P neutronvortices in the hadronic phase are joined to three non-AbelianAlice strings in the color-superconducting phase. We alsoshow the Aharonov-Bohm phase around each vortices. for the light quarks u, d of the color β , and[Γ snn ( ϕ )] = Γ srβ ( ϕ )Γ sgβ ( ϕ )Γ sbβ ( ϕ ) for β = r, g, b (23)for the heavy quark s of the color β . We thus reach thepicture of Boojum illustrated in Fig. 1.So far we have assumed that two-flavor dense QCD isin the so-called deconfined phase in which non-AbelianAlice strings can exist. On the other hand, if it is inthe confined phase, non-Abelian Alice strings must beconfined either to doubly quantized non-Abelian stringsaround which all AB phases are color singlet, or to U(1) B Abelian strings. In the context of quark-hadron conti-nuity in Fig. 1, three non-Abelian Alice strings in thetwo-flavor quark matter is confined to one U(1) B Abelianstring. Thus, three integer vortices in the P neutron su-perfluid are combined to one Abelian string in the two-flavor quark matter.The mismatch in the AB phase tells us that if we try toconnect a single hadronic vortex to a single non-Abelianvortex, it might lead to the discontinuity. The Boojumsare therefore needed to maintain the continuity. Our dis-cussion have been carried out on the level of the oper-ator in this work. There can be a possibility that theAB phase may recieve the non-trivial contributions andthe phase may be dynamically screened if we take theexpectation value under the vacuum, which may leadto the different result, but we believe our present studyalready captures the important feature and the furthernon-perturbative analysis is the beyond the scope of thiswork. V. SUMMARY
In summary, we have found that the Boojums betweenthe non-Abelian Alice strings in the two-flavor quarkmatter (the 2SC+ (cid:104) dd (cid:105) phase) and the P neutron vor-tices in the hadronic matter. As previously suggested inthe three-flavor case, Boojum structures are ubiquitousin quark-hadron continuity. Neutron stars are rapidly ro-tating object so that they are accompanied typically byabout 10 of vortices. The vortices are believed to playa crucial role in pulsar glitches, and it would be an in-teresting and important question how our work influencesuch events.The problem of the quark-hadron continuity is not onlyof the phenomenological relevance, but also tightly re-lated with the fundamental problem of the gauge theory,particularly, the phase structure of gauge theory withfundamental Higgs field. The idea that the Higgs phaseand the confinement phase is indistinguishable, which iscommonly referred to as Fradkin-Shenker theorem [65–67], is a baseline for the idea of quark-hadron continuity, and is still under a debate nowadays [31]. Since the non-perturbative studies of gauge theories are still limited atpresent, we believe that the quark-hadron continuity inthe bulk matter and vortices from the neutron star phe-nomenology should serve an important clue. ACKNOWLEDGMENTS
We thank Shigehiro Yasui for a discussion at the earlystage of this work. This work is supported in partby Grant-in-Aid for Scientific Research, JSPS KAK-ENHI Grant Numbers 20J10506 (Y.F.) and JP18H01217(M.N.). [1] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song,and T. Takatsuka, Rept. Prog. Phys. , 056902 (2018),arXiv:1707.04966 [astro-ph.HE].[2] V. Graber, N. Andersson, and M. Hogg, Int. J. Mod.Phys. D26 , 1730015 (2017).[3] T. Kojo (2020) arXiv:2011.10940 [nucl-th].[4] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, andJ. Hessels, Nature , 1081 (2010), arXiv:1010.5788[astro-ph.HE]; E. Fonseca et al. , Astrophys. J. , 167(2016), arXiv:1603.00545 [astro-ph.HE]; Z. Arzoumanian et al. (NANOGrav), Astrophys. J. Suppl. , 37 (2018),arXiv:1801.01837 [astro-ph.HE].[5] J. Antoniadis et al. , Science , 1233232 (2013).[6] H. T. Cromartie et al. , Nature Astron. , 72 (2019),arXiv:1904.06759 [astro-ph.HE].[7] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 161101 (2017).[8] B. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J.Lett. , L3 (2020), arXiv:2001.01761 [astro-ph.HE].[9] T. E. Riley et al. , Astrophys. J. Lett. , L21 (2019),arXiv:1912.05702 [astro-ph.HE].[10] M. Miller et al. , Astrophys. J. Lett. , L24 (2019),arXiv:1912.05705 [astro-ph.HE].[11] K. Masuda, T. Hatsuda, and T. Takatsuka, Astrophys. J. , 12 (2013), arXiv:1205.3621 [nucl-th]; PTEP ,073D01 (2013), arXiv:1212.6803 [nucl-th].[12] T. Kojo, P. D. Powell, Y. Song, and G. Baym, Phys.Rev. D , 045003 (2015), arXiv:1412.1108 [hep-ph].[13] G. Baym, S. Furusawa, T. Hatsuda, T. Kojo, and H. To-gashi, Astrophys. J. , 42 (2019), arXiv:1903.08963[astro-ph.HE].[14] T. Kojo, D. Hou, J. Okafor, and H. Togashi, (2020),arXiv:2012.01650 [astro-ph.HE].[15] E. Annala, T. Gorda, A. Kurkela, J. N¨attil¨a, andA. Vuorinen, Nature Phys. (2020), 10.1038/s41567-020-0914-9, arXiv:1903.09121 [astro-ph.HE].[16] D. Bailin and A. Love, Phys. Rept. , 325 (1984).[17] M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨afer,Rev. Mod. Phys. , 1455 (2008), arXiv:0709.4635 [hep-ph].[18] M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys.B , 443 (1999), arXiv:hep-ph/9804403.[19] M. G. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett.B , 247 (1998), arXiv:hep-ph/9711395. [20] R. Rapp, T. Sch¨afer, E. V. Shuryak, and M. Velkovsky,Phys. Rev. Lett. , 53 (1998), arXiv:hep-ph/9711396.[21] T. Sch¨afer and F. Wilczek, Phys. Rev. Lett. , 3956(1999), arXiv:hep-ph/9811473.[22] M. G. Alford, J. Berges, and K. Rajagopal, Nucl. Phys. B558 , 219 (1999), arXiv:hep-ph/9903502 [hep-ph].[23] K. Fukushima, Phys. Rev. D , 094014 (2004),arXiv:hep-ph/0403091.[24] T. Hatsuda, M. Tachibana, N. Yamamoto, andG. Baym, Phys. Rev. Lett. , 122001 (2006), arXiv:hep-ph/0605018; N. Yamamoto, M. Tachibana, T. Hat-suda, and G. Baym, Phys. Rev. D76 , 074001 (2007),arXiv:0704.2654 [hep-ph].[25] T. Hatsuda, M. Tachibana, and N. Yamamoto, Phys.Rev.
D78 , 011501 (2008), arXiv:0802.4143 [hep-ph].[26] A. Schmitt, S. Stetina, and M. Tachibana, Phys. Rev. D , 045008 (2011), arXiv:1010.4243 [hep-ph].[27] M. G. Alford, G. Baym, K. Fukushima, T. Hatsuda,and M. Tachibana, Phys. Rev. D , 036004 (2019),arXiv:1803.05115 [hep-ph].[28] C. Chatterjee, M. Nitta, and S. Yasui, Phys. Rev. D , 034001 (2019), arXiv:1806.09291 [hep-ph]; JPS Conf.Proc. , 024030 (2019), arXiv:1902.00156 [hep-ph].[29] A. Cherman, S. Sen, and L. G. Yaffe, Phys. Rev. D ,034015 (2019), arXiv:1808.04827 [hep-th].[30] Y. Hirono and Y. Tanizaki, Phys. Rev. Lett. , 212001(2019), arXiv:1811.10608 [hep-th]; JHEP , 062 (2019),arXiv:1904.08570 [hep-th].[31] A. Cherman, T. Jacobson, S. Sen, and L. G. Yaffe,(2020), arXiv:2007.08539 [hep-th].[32] M. M. Forbes and A. R. Zhitnitsky, Phys. Rev. D ,085009 (2002), arXiv:hep-ph/0109173.[33] K. Iida and G. Baym, Phys. Rev. D , 014015 (2002),arXiv:hep-ph/0204124.[34] E. Nakano, M. Nitta, and T. Matsuura, Phys. Rev. D ,045002 (2008), arXiv:0708.4096 [hep-ph]; Prog. Theor.Phys. Suppl. , 254 (2008), arXiv:0805.4539 [hep-ph].[35] M. G. Alford, S. Mallavarapu, T. Vachaspati, andA. Windisch, Phys. Rev. C , 045801 (2016),arXiv:1601.04656 [nucl-th].[36] A. Balachandran, S. Digal, and T. Matsuura, Phys. Rev.D , 074009 (2006), arXiv:hep-ph/0509276.[37] M. Eto and M. Nitta, Phys. Rev. D , 125007(2009), arXiv:0907.1278 [hep-ph]; M. Eto, E. Nakano, and M. Nitta, Phys. Rev. D , 125011 (2009),arXiv:0908.4470 [hep-ph]; M. Eto, M. Nitta, andN. Yamamoto, Phys. Rev. Lett. , 161601 (2010),arXiv:0912.1352 [hep-ph].[38] M. Eto, Y. Hirono, M. Nitta, and S. Yasui, PTEP ,012D01 (2014), arXiv:1308.1535 [hep-ph].[39] M. Cipriani, W. Vinci, and M. Nitta, Phys. Rev. D ,121704 (2012), arXiv:1208.5704 [hep-ph].[40] N. D. Mermin, Surface singularities and superflow in3He-A, Quantum Fluids and Solids, edited by S. B.Tickey, E. D. Adams, and J.W. Dufty (Plenum, NewYork, 1977) p. 322.[41] Originally the boojum is a particular variety of the fic-tional animal species called snarks created by Lewis Car-roll in his nonsense poem “The Hunting of the Snark.”See [68] for the story how Mermin made “boojum.[42] G. E. Volovik,