Higgs-confinement phase transitions with fundamental representation matter
Aleksey Cherman, Theodore Jacobson, Srimoyee Sen, Laurence G. Yaffe
HHiggs-confinement phase transitionswith fundamental representation matter
Aleksey Cherman, Theodore Jacobson, Srimoyee Sen, Laurence G. Yaffe School of Physics and Astronomy, University of Minnesota, Minneapolis MN 55455, USA Department of Physics and Astronomy, Iowa State University, Ames IA 50011, USA Department of Physics, University of Washington, Seattle WA 98195-1560, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We discuss the conditions under which Higgs and confining regimes ingauge theories with fundamental representation matter fields can be sharply distin-guished. It is widely believed that these regimes are smoothly connected unless theyare distinguished by the realization of global symmetries. However, we show thatwhen a U (1) global symmetry is spontaneously broken in both the confining and Higgsregimes, the two phases can be separated by a phase boundary. The phase transitionbetween the two regimes may be detected by a novel topological vortex order param-eter. We first illustrate these ideas by explicit calculations in gauge theories in threespacetime dimensions. Then we show how our analysis generalizes to four dimensions,where it implies that nuclear matter and quark matter are sharply distinct phases ofQCD with an approximate SU (3) flavor symmetry. a r X i v : . [ h e p - t h ] J u l ontents O Ω O Ω in the Higgs regime 143.3 O Ω in the U (1) G -broken confining regime 223.4 Higgs-confinement phase transition 243.5 Explicit breaking of flavor permutation symmetry 26 U (1) × U (1) gauge theory and topological order 51 In gauge theories with fundamental representation matter fields, one can often dialparameters in a manner which smoothly interpolates between a Higgs regime and aconfining regime without undergoing any change in the realization of global symme-tries [1, 2]. In the Higgs regime gauge fields become massive via the usual Higgs phe-nomenon, while in the confining regime gauge fields also become gapped (or acquirea finite correlation length) due to the non-perturbative physics of confinement, with– 1 –n approximately linear potential appearing between heavy fundamental test chargesover a finite range of length scales which is limited by the lightest meson mass. In thispaper we examine situations in which the Higgs and confining regimes of such theoriescan be sharply distinguished.This is, of course, an old and much-studied issue. In specific examples, when bothregimes have identical realizations of global symmetries, it has been shown that confin-ing and Higgs regimes can be smoothly connected with no intervening phase transitions[1, 2]. These examples, summarized as the “Fradkin-Shenker theorem,” have inspireda widely held expectation that there can be no useful gauge-invariant order parameterdistinguishing Higgs and confining phases in any gauge theory with fundamental rep-resentation matter fields. But there are physically interesting situations in which theFradkin-Shenker theorem does not apply. We are interested in systems where no localorder parameter can distinguish Higgs and confining regimes and yet the conventionalwisdom just described is incorrect. We will analyse model theories, motivated by thephysics of dense QCD, where Higgs and confining regimes cannot be distinguished bythe realization of global symmetries and yet these are sharply distinct phases necessarilyseparated by a quantum phase transition in the parameter space of the theory.We will consider a class of gauge theories with two key features. The first is thatthey have fundamental representation scalar fields which are charged under a U (1) global symmetry. Second, this U (1) global symmetry is spontaneously broken in both the Higgs and confining regimes of interest. In this class of gauge theories, we argue thatone can define a natural non-local order parameter which does distinguish the Higgs andconfinement regimes. This order parameter is essentially the phase of the expectationvalue of the holonomy (Wilson loop) of the gauge field around U (1) global vortices;its precise definition is discussed below. We will find that this vortex holonomy phaseacts like a topological observable; it is constant within each regime but has differingquantized values in the two regimes. We present a general argument — verifying it byexplicit calculation where possible — that implies that non-analyticity in our vortexholonomy observable signals a genuine phase transition separating the U (1) -brokenHiggs and U (1) -broken confining regimes.The Higgs-confinement transition we discuss in this paper does not map cleanlyonto the classification of topological orders which is much discussed in modern con- By a useful order parameter we mean an expectation value of a physical observable whose non-analytic change also indicates non-analytic behavior in thermodynamic observables and correlationfunctions of local operators. For a rather different take on these issues, see Refs. [3–6]. This statement assumes a certain global flavor symmetry. In the absence of such a symmetry,the phase of the vortex holonomy is constant in the U (1) -broken confining regime and changes non-analytically at the onset of the Higgs regime. – 2 –ensed matter physics [7–10]. The basic reason is that the topological order classifica-tion is designed for gapped phases of matter, while here we focus on gapless phases.Some generalizations of topological order to gapless systems have been considered in thecondensed matter literature, see e.g. Refs. [11–13], but these examples differ in essen-tial ways from the class of models we consider here. Our arguments also do not cleanlymap onto the related idea of classifying phases based on realizations of higher-formglobal symmetries [14–20], because the models we consider do not have any obvioushigher-form symmetries. But there is no reason to think that existing classificationideas can detect all possible phase transitions. We argue that our vortex order param-eter provides a new and useful way to detect certain phase transitions which are notamenable to standard methods.Let us pause to explain in a bit more detail why the Fradkin-Shenker theorem doesnot apply to theories of the sort we consider. The Fradkin-Shenker theorem presupposesthat Higgs fields are uncharged under any global symmetry. This assumption may seeminnocuous. After all, if Higgs fields are charged under a global symmetry, it is temptingto think that this global symmetry will be spontaneously broken when the Higgs fieldsdevelop an expectation value, implying a phase transition associated with a change insymmetry realization and detectable with a local order parameter. In other words, atypical case lying within the Landau paradigm of phase transitions.But such a connection between Higgs-confinement transitions and a change inglobal symmetry realization is model dependent. In the theories we consider in thispaper, as well as in dense QCD, these two phenomena are unrelated. Our scalar fieldswill carry a global U (1) charge, but crucially, the realization of all global symmetrieswill be the same in the confining and Higgs regimes of interest. Consequently, theFradkin-Shenker theorem does not apply to these models and yet the confining andHiggs regimes are not distinguishable within the Landau classification of phases. Nev-ertheless, we will see that they are distinct.The basic ideas motivating this paper were introduced by three of us in an earlierstudy of cold dense QCD matter [21]. We return to this motivation at the end of thispaper in Sec. 4, where we generalize our analysis to cover non-Abelian gauge theories infour spacetime dimensions and explain why it provides compelling evidence against theSchäfer-Wilczek conjecture of quark-hadron continuity in dense QCD [22]. The bulk ofour discussion is focused on a simpler set of model theories which will prove useful torefine our understanding of Higgs-confinement phase transitions.We begin, in Sec. 2, by introducing a simple Abelian gauge theory in three space-time dimensions in which Higgs and confinement physics can be studied very explicitly.In Sec. 3 we introduce our vortex order parameter and use it to infer the existence ofa Higgs-confinement phase transition. Sec. 4 discusses the application of our ideas to– 3 –our-dimensional gauge theories such as QCD, while Sec. 5 contains some concludingremarks. Finally, in Appendices A–C we collect some technical results on vortices,discuss embedding our Abelian model within a non-Abelian theory, and consider theconsequences of gauging of our U (1) global symmetry to produce a U (1) × U (1) gaugetheory. We consider compact U (1) gauge theory in three Euclidean spacetime dimensions. Let A µ denote the (real) gauge field. Our analysis assumes SO (3) Euclidean rotation sym-metry, together with a parity (or time-reversal) symmetry. Parity symmetry precludesa Chern-Simons term, so the gauge part of the action is just a photon kinetic term, S γ = (cid:90) d x e F µν F µν . (2.1)The statement that the gauge group is compact (in this continuum description) amountsto saying that the Abelian description (2.1) is valid below some scale Λ UV , and that theUV completion of the theory above this scale allows finite action monopole-instantonfield configurations whose total magnetic flux is quantized [23]. Specifically, we demandthat the flux through any 2-sphere is an integer, (cid:90) S F = 2 πk , k ∈ Z , (2.2)where F ≡ F µν dx µ ∧ dx ν is the 2-form field strength. Condition (2.2) implies chargequantization and removes the freedom to perform arbitrary field rescalings of the form A → A (cid:48) ≡ ( q (cid:48) /q ) A . As shown by Polyakov, the presence of monopole-instantons,regardless of how dilute, leads to confinement on sufficiently large distance scales [23]. We choose the matter sector of our model to be comprised of two oppositely-chargedscalar fields, φ + and φ − , plus one neutral scalar φ . We assign unit gauge charges q = ± to the charged fields, making them analogous to fundamental representation matterfields in a non-Abelian gauge theory. We require the theory to have a single zero-form The fact that our charged matter fields have minimal charges of ± is an essential difference froma similar model studied by Sachdev and Park [11] in a condensed matter context, see also [13]. Themodel of Ref. [11] has a U (1) global symmetry and fields with charges − and +2 under an emergent U (1) gauge symmetry. The existence of non-minimally charged matter fields allowed Sachdev andPark to use topological order ideas to delineate distinct phases. That approach does not work in ourmodel. – 4 –lobal U (1) symmetry under which the fields φ ± both have charge assignments of − while φ has a charge assignment of +2 . These charge assignments, summarized here: φ + φ − φ U (1) gauge +1 − U (1) global − − (2.3)are chosen in a manner which will allow independent control of the Higgsing of the U (1) gauge symmetry (or lack thereof) and the realization of the U (1) global symmetry byadjusting suitable mass parameters. This is the essential structure needed to examinethe issues motivating this paper in the context of a model Abelian theory.The complete action of our model consists of the gauge action (2.1), standard scalarkinetic terms, plus a scalar potential containing interactions consistent with the abovesymmetries, S = (cid:90) d x (cid:20) e F µν + | D µ φ + | + | D µ φ − | + m c (cid:0) | φ + | + | φ − | (cid:1) + | ∂ µ φ | + m | φ | − (cid:15) (cid:0) φ + φ − φ + h . c . (cid:1) + λ c (cid:0) | φ + | + | φ − | (cid:1) + λ | φ | + g c (cid:0) | φ + | + | φ − | (cid:1) + g | φ | + · · · + V m ( σ ) (cid:21) . (2.4)The mass dimensions of the various couplings are [ e ] = [ λ c ] = [ λ ] = 1 , [ (cid:15) ] = 3 / , and [ g c ] = [ g ] = 0 . The ellipsis ( · · · ) represents possible further scalar self-interactions,consistent with the imposed symmetries, arising via renormalization. The term V m ( σ ) describes the effects of monopole-instantons, and is given explicitly below.The cubic term (cid:15) φ + φ − φ ensures that the model has a single U (1) global symmetry,not multiple independent phase rotation symmetries. From here onward, we will denotethe U (1) global symmetry by U (1) G . The simplest local order parameter for the U (1) G symmetry is just the neutral field expectation value (cid:104) φ (cid:105) . This order parameter has acharge assignment (2.3) of +2 under the U (1) G symmetry; there are no gauge invariantlocal order parameters with odd U (1) G charge assignments.In addition to the U (1) gauge redundancy and the U (1) G global symmetry, thismodel has two internal Z discrete symmetries. One is a conventional (particle ↔ antiparticle) charge conjugation symmetry, ( Z ) C : φ ± → φ ∗± , φ → φ ∗ , A µ → − A µ . (2.5) A zero-form global symmetry is just an ordinary global symmetry which acts on local operators. – 5 –he other is a charged field permutation symmetry, ( Z ) F : φ + ↔ φ − , A µ → − A µ . (2.6)A conserved current j µ mag ≡ (cid:15) µνλ F νλ associated with a U (1) magnetic global sym-metry is also present if monopole-instanton effects are neglected. But for our compact Abelian theory this symmetry is not present. The functional integral representationof the theory includes a sum over finite-action magnetic monopole-instanton configura-tions with all integer values of total magnetic charge. These induce corrections to theeffective potential (below the scale Λ UV ) of the form [23] V m ( σ ) = − µ e − S I cos( σ ) . (2.7)Here S I is the minimal action of a monopole-instanton, and σ is the dual photon field,related to the original gauge field by the Abelian duality relation F µν = ie π (cid:15) µνλ ∂ λ σ . (2.8)With this normalization the dual photon field is a periodic scalar, σ ≡ σ + 2 π , withthe Maxwell action becoming the kinetic term (cid:0) e π (cid:1) ( ∂σ ) . The parameter µ UV isa short-distance scale associated with the inverse core size of monopole-instantons.The U (1) magnetic transformations act as arbitrary shifts on the dual photon field, σ → σ + c . Such shifts are clearly not a symmetry, except for integer multiples of π . Consequently, the U (1) G phase rotation symmetry is the only continuous globalsymmetry in our model.In summary, the faithfully-acting internal global symmetry group of our model is G internal = [ U (1) G (cid:111) ( Z ) C ] × ( Z ) F Z . (2.9)The quotient by Z ⊂ U (1) G : φ ± → − φ ± is necessary because it also lies in the gaugegroup U (1) .When the charged scalar mass squared, m c , is sufficiently negative this theory hasa Higgs regime in which the charged scalar fields are “condensed.” In this regime gaugefield fluctuations are suppressed since the photon acquires a mass term, (cid:0) |(cid:104) φ + (cid:105)| + |(cid:104) φ − (cid:105)| (cid:1) A µ A µ ≡ m A e A µ A µ , (2.10) Expression (2.7) relies on a dilute gas approximation, valid when the instanton action is large, S I (cid:29) . The duality relation (2.8) appears when one imposes the Bianchi identity for F µν by adding aLagrange multiplier term i (cid:82) d x σ π (cid:15) µνλ ∂ µ F νλ to the Euclidean action. Relation (2.8) is the resultingequation of motion for F µν , and integrating out F µν gives the Abelian dual representation of Maxwelltheory. – 6 –to lowest order in unitary gauge). Monopole-instanton–antimonopole-instanton pairsbecome bound by flux tubes with a positive action per unit length T mag . In contrast, for sufficiently positive m c our model should be regarded as a confininggauge theory. Recall that in the context of QCD, the confining regime is character-ized by a static test quark–antiquark potential which rises linearly with separation, V q ¯ q ∼ σr , for separations large compared to the strong scale, r (cid:29) Λ − . But such alinear potential is only present for separations where the confining string cannot break,which requires that σr < m q , with m q the mass of dynamical quarks. So confinementis only a sharply-defined criterion in the heavy quark limit, m q (cid:29) σ/ Λ QCD = O (Λ QCD ) .Nevertheless, it is conventional to speak of QCD as a confining theory even with lightquarks, as this is a qualitatively useful picture of the relevant dynamics. This sum-mary applies verbatim to our compact U (1)
3D gauge theory with massive unit-chargematter, with Λ QCD replaced by an appropriate non-perturbative scale which depends(exponentially) on the monopole-instanton action S I [23].Finally, we note that in the absence of monopole-instanton effects, oppositelycharged static test particles in 3D Abelian gauge theory would experience logarith-mic Coulomb interactions which grow without bound with increasing separation. Sucha phase could be termed “confined,” but for our purposes this terminology is not help-ful. We find it more appropriate to reserve the term “confinement” for situations wherethe potential between test charges is linear over a significant range of distance scales.With this terminology, 3D compact U (1) gauge theory with finite-action monopole-instantons and very heavy charged matter is confining, while the non-compact versionof the theory, which does not have a regime with a linear potential between test charges,is not confining. Our 3D Abelian model is designed to mimic many features of real 4D QCD at non-zerodensity. Explicitly,1. Both theories contain fundamental representation matter fields and are confiningin the sense described above. Of course, the gauge groups are completely different: SU ( N ) versus U (1) . Our charged scalar fields may be viewed as analogs of the electron pair condensate in a Ginsburg-Landau treatment of superconductivity, in which case m A is the Meissner mass whose inverse givesthe penetration length of magnetic fields. On sufficiently long length scales when the flux tube length L (cid:38) S I /T mag , these magnetic fluxtubes can break due to production of monopole-instanton–antimonopole-instanton pairs. This is com-pletely analogous to the situation in the confining regime, discussed next, where electric flux tubesexist over a limited range of scales controlled by the mass of fundamental dynamical charges. – 7 –. QCD with massive quarks of equal mass has a vector-like U ( N f ) / Z N internalglobal symmetry. The quotient arises because Z N transformations are part of the SU ( N ) gauge symmetry. In our model, the corresponding global symmetry is [( Z ) F × U (1) G ] / Z . The ( Z ) F × U (1) G symmetry is analogous to U ( N f ) , whilethe discrete quotient arises for the same reason as in QCD.3. The scalar fields in the 3D Abelian model may be regarded as playing the roleof color anti-fundamental diquark operators which acquire non-zero vacuum ex-pectation values in high density QCD, see Ref. [24] for a review. The symmetrygroup U (1) G is analogous to quark number U (1) ⊂ U ( N f ) , while U (1) G / Z isanalogous to baryon number U (1) B . Note one distinction in the transformationproperties of the scalar fields in our Abelian model and the diquark condensatesin QCD; the former have charge under our U (1) G group whereas the latterhave charge under quark number. The ( Z ) F permutation symmetry of our3D Abelian model is analogous to the Z N f ⊂ U ( N f ) cyclic flavor permutationsymmetry of 4D QCD.4. Since the charged scalars φ ± are analogous to anti-fundamental diquarks in three-color QCD, φ † + φ †− is akin to a dibaryon. This means that φ can also be interpretedas a dibaryon interpolating operator, and the condensation of φ in our model isdirectly analogous to the dibaryon condensation which occurs in dense QCD.5. In QCD, the Vafa-Witten theorem [25] implies that phases with spontaneouslybroken U (1) B symmetry can only appear at non-zero baryon density, while in ourAbelian model U (1) G -broken phases can appear at zero density. This differencereflects the fact that QCD contains only fermionic matter fields, while our Abelianmodel has fundamental scalar fields. We begin analyzing the phase structure of the model (2.4) using the Landau paradigmbased on realizations of symmetries with local order parameters. We will consider thephase diagram as a function of the charged and neutral scalar masses, m c and m .We focus on the regime where quartic and sextic scalar self-couplings are positive, thecubic, quartic and gauge couplings are comparable, (cid:15)/e , | λ c | /e and | λ | /e are all O (1) , and the dimensionless sextic couplings are small, g c , g (cid:28) . The simplest phasediagram consistent with our analysis is sketched in Fig. 1.– 8 – rivial gapped phase(Polyakov mechanism) U (1) G -brokenconfining phaseHiggs phase U (1) G -broken NESENWSW m /e m c /e −∞ + ∞−∞ + ∞ Figure 1 . A sketch of the simplest consistent phase diagram of our model as a function ofthe charged and neutral scalar mass parameters m c and m . The four corners correspond toweakly-coupled regimes in parameter space; curves in the interior of the figure represent phasetransitions. These phase transition curves are robust: they cannot be evaded by varying anyparameters of the model which are consistent with its symmetries. Interpreting Fig. 1 as if it were a map, let us refer to the four weakly-coupledcorners of parameter space by their compass directions:NW : {− m c (cid:29) e , m (cid:29) e } , NE : { m c (cid:29) e , m (cid:29) e } , (2.11a)SW : {− m c (cid:29) e , − m (cid:29) e } , SE : { m c (cid:29) e , − m (cid:29) e } , (2.11b)each of which we discuss in turn. In this section we explain the origin of the phasetransition curve (orange) separating the NE region from the W side of Fig. 1, as well asthe (blue) curve separating the NE and SE regions. The bulk of the paper is dedicatedto understanding the origin of the phase transition curve (green) separating the SEregion from the W side of Fig. 1.First, consider region NE where m c , m (cid:29) e . In this regime our model has aunique gapped vacuum state and no broken symmetry. To see this, one may integrateout all the matter fields and observe that the resulting tree-level effective action is S eff = (cid:90) d x (cid:20) e F µν + V m ( σ ) (cid:21) . (2.12)The monopole potential V m ( σ ) has a unique minimum for the dual photon σ and inducesa non-zero photon mass m γ = 4 π ( µ /e ) e − S I . Hence, the vacuum is gapped andunique. Both the continuous U (1) G and the discrete ( Z ) C and ( Z ) F global symmetriesare unbroken, and hence region NE may be termed “confining and unbroken.”– 9 –ow consider the entire E side where m c (cid:29) e while the neutral mass m isarbitrary. Then one may integrate out the charged fields and the effective actionbecomes S eff = (cid:90) d x (cid:20) e F µν + V m ( σ ) + | ∂ µ φ | + m | φ | + λ | φ | + g | φ | + · · · (cid:21) . (2.13)This is a 3D XY model plus a decoupled compact U (1) gauge theory. The photon isstill gapped by the Polyakov mechanism. If we take m (cid:29) | λ | , then we come backto the discussion of the previous paragraph. If we take − m (cid:29) | λ | , then φ developsa non-vanishing expectation value, the U (1) G / Z symmetry is spontaneously broken,and there is a single massless Nambu-Goldstone boson. So region SE is “confining and U (1) G symmetry broken.” The discrete ( Z ) F symmetry is unbroken in this region, as isa redefined ( Z ) C symmetry which combines the basic ( Z ) C transformation (2.5) with a U (1) G transformation that compensates for the arbitrary phase of the condensate (cid:104) φ (cid:105) .This symmetry-broken regime must be separated from the symmetry-unbroken regimeby a phase transition depending on the value of m /λ . If we take our quartic andsextic couplings to be positive, this is just the well-known XY model phase transition,which is second order in three spacetime dimensions.Next, consider what happens on the W side where − m c (cid:29) e while m is arbitrary.In this case the charged scalar fields φ ± will acquire non-zero expectation values (usinggauge-variant language), with v c ≡ |(cid:104) φ ± (cid:105)| = O ( | m c λ − / c | ) . This has several effects. First, since these fields transform non-trivially under the U (1) G symmetry, this global symmetry is spontaneously broken leading to a masslessNambu-Goldstone excitation. Second, the U (1) gauge field becomes Higgsed, as dis-cussed above, with the photon acquiring a mass m A . Writing φ ± = ( v c + H ± / √ e − iχ ,up to an arbitrary U (1) gauge transformation, the resulting effective action has theform S = (cid:90) d x (cid:20) e F µν + m A A µ A µ + | ∂ µ φ | + m | φ | + λ | φ | + 2 v c ( ∂ µ χ ) − (cid:15)v c Re ( e − iχ φ ) + (cid:88) i = ± (cid:104) ( ∂ µ H i ) + m H H i (cid:105) + · · · (cid:35) , (2.14)where χ is the U (1) G Nambu-Goldstone boson and H ± are real Higgs modes withmass m H . This regime, extending inward from the W boundary of the phase diagram, In this and subsequent parametric estimates, we neglect the cubic coupling and sextic couplings.For the sextic couplings this is justified by our assumption that they are small. We have dropped (cid:15) -dependence purely for simplicity: taking it into account is straightforward but results in much morecumbersome expressions. – 10 –ay be termed “Higgsed and U (1) G symmetry broken.” The discrete symmetries re-main unbroken in the same manner as in region SE. Regardless of the sign of m ,the neutral scalar φ acquires a non-zero vacuum expectation value whose phase, χ ,is set by the phase of the Higgs condensate. As m is varied from large positive tolarge negative values, the magnitude |(cid:104) φ (cid:105)| varies from a small O ( (cid:15)v c m − ) value toa large O ( m λ − / ) value, while always remaining non-zero. Throughout this Higgsregime monopole-instanton–antimonopole-instanton pairs become linearly confined bymagnetic flux tubes as noted earlier.The fact that the U (1) G symmetry is spontaneously broken in this Higgs regimemeans that the entire W region of parameter space with − m c (cid:29) e must be separatedby a phase transition from the trivially gapped region NE where m c and m are largeand positive. But the pattern of global symmetry breaking throughout the W side Higgsregime of − m c (cid:29) e is identical to that in region SE where m c (cid:29) e and − m (cid:29) λ .This raises the central question in this paper: Are the Higgs and confining U (1) G -breaking regimes smoothly connected, or arethey distinct phases? As summarized in the introduction and sketched in Fig. 1, we will find that the Higgsand confining U (1) G -breaking regimes must be distinct phases, separated by at leastone phase transition, even though there are no distinguishing local order parameters.Before leaving this section, we pause to consider the nature of the (cid:15) → limit.Sending (cid:15) → is a non-generic limit of the model, as an additional global symmetrywhich purely phase rotates the charged fields, φ ± → e iα φ ± , is present when (cid:15) = 0 ;we denote this symmetry as U (1) extra . The (cid:15) = 0 theory has four distinct phasesdistinguished by realizations of the U (1) G and U (1) extra symmetries. There is a phasewhere only the U (1) extra symmetry is spontaneously broken, with one Nambu-Goldstoneboson. This phase is not present at non-zero (cid:15) . At (cid:15) = 0 , the U (1) G -broken Higgsphase in Fig. 1 becomes a phase with two spontaneously broken continuous globalsymmetries, U (1) G and U (1) extra , and has two Nambu-Goldstone bosons. This is adistinct symmetry realization from the U (1) G -broken confining regime with only asingle Nambu-Goldstone boson implying, by the usual Landau paradigm reasoning, atleast one intervening separating phase transition.When (cid:15) is non-zero but very small compared to all other scales, there is a paramet-rically light pseudo-Nambu-Goldstone boson with a mass m pNGB ∝ √ (cid:15) in the Higgsregime. Determining whether the U (1) G -broken Higgs and confining regimes remaindistinct for non-zero values of (cid:15) is the goal of our next section in which we examine thelong-distance behavior of holonomies around vortices. In this analysis, it will be impor-– 11 –ant that the holonomy contour radius be large compared to microscopic length scales —which include the Compton wavelength of the pseudo-Goldstone boson, m − . Thereis non-uniformity between the large distance limit of the holonomy and the (cid:15) → limit,and consequently the physics of interest must be studied directly in the theory with (cid:15) (cid:54) = 0 . O Ω Consider the portion of the phase diagram in which the U (1) G symmetry is spon-taneously broken. Then the field φ has a non-vanishing expectation value and thespectrum contains a Nambu-Goldstone boson. The Goldstone manifold has a non-trivial first homotopy group, π ( U (1) G ) = Z . This implies that there are stable globalvortex excitations, which are particle-like excitations in two spatial dimensions. Aswith vortices in superfluid films, these vortex excitations have logarithmic long rangeinteractions. A single vortex, in infinite space, has a logarithmically divergent longdistance contribution to its self-energy. Nevertheless, vortices are important collectiveexcitations and, in any sufficiently large volume, a non-zero spatial density of vorticesand antivortices will be present due to quantum and/or thermal fluctuations. Froma spacetime perspective, vortex/antivortex world lines, as they appear and annihilate,form a collection of closed loops.Vortex excitations may be labeled by an integer winding number w indicating thenumber of times the phase of (cid:104) φ (cid:105) wraps the unit circle as one encircles a vortex. Moreexplicitly, one may write the winding number as a contour integral of the gradient ofthe phase, w = 12 π (cid:73) C dx µ u µ , (3.1)with u µ ≡ − i∂ µ ( (cid:104) φ (cid:105) / |(cid:104) φ (cid:105)| ) . (3.2)Consider the gauge field holonomy, Ω ≡ e i (cid:72) C A , evaluated on some large circularcontour C surrounding a vortex of non-zero winding number k , illustrated in Fig. 2,which we denote by (cid:104) Ω( C ) (cid:105) k . Let r denote the radius of the contour C encirclingthe vortex. We are interested in the phase of the holonomy, but as the size of thecontour C grows, short distance quantum fluctuations will cause the magnitude of theexpectation (cid:104) Ω( C ) (cid:105) k to decrease (with at least exponential perimeter-law decrease). To Using the language of a superfluid, u µ is the superfluid flow velocity, and the winding number w is the quantized circulation around a vortex. – 12 – Figure 2 . A contour C (red dashed curve) which links a vortex world-line (solid black curve).Of interest is the gauge field holonomy Ω ≡ e i (cid:72) C A for contours C far from the vortex core. compensate, we consider the large distance limit of a ratio of the holonomy expectationvalues which do, or do not, encircle a vortex of minimal non-zero winding number, O Ω ≡ lim r →∞ (cid:104) Ω( C ) (cid:105) (cid:104) Ω( C ) (cid:105) . (3.3)Here, the numerator should be understood as an expectation value defined by a con-strained functional integral in which there is a prescribed vortex loop of characteristicsize r and winding number 1 linked with the holonomy loop of size r , with both sizes,and the minimal separation between the two loops, scaling together as r increases. Thedenominator is the ordinary unconstrained vacuum expectation value.The quantity O Ω measures the phase acquired by a particle with unit gauge chargewhen it encircles a minimal global vortex. Or equivalently, it is the phase acquired bya minimal global vortex when it is dragged around a particle with unit gauge charge.Our analysis below will demonstrate that O Ω cannot be a real-analytic function ofthe charged scalar mass parameter m c /e . We will also show that non-analyticities inthe topological order parameter O Ω are associated with genuine thermodynamic phasetransitions.A quick sketch of the argument is as follows. Since the vacuum is invariant underthe ( Z ) C charge conjugation symmetry, the denominator of O Ω must be real and atsufficiently weak coupling is easily seen to be positive. In the constrained expectation One may equally well appeal to reflection symmetry, as this reverses the orientation of a reflectionsymmetric contour like a circle, and hence maps the holonomy on a circular contour to its complexconjugate. This alternative will be relevant for our later discussion in Sec. 4 of dense QCD and relatedmodels with non-zero chemical potential, where charge conjugation symmetry is explicitly broken bythe chemical potential but the ground state remains invariant under reflections. – 13 –alue in the numerator of O Ω , the ( Z ) C symmetry is explicitly broken by the unit-circulation condition that enters the definition of (cid:104) Ω( C ) (cid:105) . But the unit-circulationcondition does not break ( Z ) F permutation symmetry (2.6), which also flips the signof the gauge field. Therefore the numerator of O Ω must be invariant under ( Z ) F ,and hence real. We will see below that it is negative deep in the Higgs regime, but ispositive deep in the U (1) G -broken confining regime. In the large- r limit defining ourvortex observable O Ω , the magnitudes of the holonomy expectations in numerator anddenominator will be identical. Hence, our vortex observable O Ω obeys O Ω = (cid:40) − , U (1) G -broken Higgs regime; +1 , U (1) G -broken confining regime, (3.4)and therefore cannot be analytic as a function of m c /e .In the remainder of this section we support the above claims. We study the prop-erties of vortices in the Higgs and confining U (1) G -broken regimes in Secs. 3.2 and 3.3,respectively. Then in Sec. 3.4 we show that non-analyticities in our topological orderparameter are associated with genuine thermodynamic phase transitions. Finally, inSec. 3.5 we extend the treatment and consider the effects of perturbations which ex-plicitly break the ( Z ) F symmetry. We find that O Ω remains a non-analytic function ofthe charged scalar mass parameter(s) even in the presence of such perturbations. Thisshows that the phase transition line separating the Higgs and confining U (1) G -brokenregimes is robust against sufficiently small ( Z ) F -breaking perturbations. O Ω in the Higgs regime We first consider O Ω deep in the Higgs regime, − m c (cid:29) e and, to begin, neglect quan-tum fluctuations altogether. So the holonomy expectation values in the definition (3.3)of O Ω just require evaluation of the holonomy in the appropriate energy-minimizingclassical field configurations.As always, the holonomy Ω( C ) is the exponential of the line integral (cid:72) C A (times i ) which, in our Abelian theory, is just the magnetic flux passing through a surfacespanning the curve C . For the ordinary vacuum expectation value in the denominatorof O Ω , vacuum field configurations have everywhere vanishing magnetic field and hence (cid:104) Ω( C ) (cid:105) = 1 .For the constrained expectation value in the numerator, one needs to understandthe form of the minimal vortex solution(s). Choose coordinates such that the vortex The ( Z ) F symmetry cannot be spontaneously broken due to the presence of a vortex becausethe vortex worldvolume is one-dimensional, and discrete symmetries cannot break spontaneously inone spacetime dimension. (The exception to this statement involving mixed ’t Hooft anomalies [26] isirrelevant in our case.) – 14 –ies at the origin of space and let { r, θ } denote 2D polar coordinates. For a vortexconfiguration with winding number k , the phase of the neutral scalar φ must wrap k times around the unit circle as one encircles the origin.There exist classical solutions which preserve rotation invariance, and we presumethat these rotationally invariant solutions capture the relevant global energy minima.Such field configurations may be written in the explicit form φ + ( r, θ ) = v c f + ( r ) e iν + θ , φ ( r, θ ) = v f ( r ) e ikθ , (3.5a) φ − ( r, θ ) = v c f − ( r ) e iν − θ , A θ ( r ) = Φ h ( r )2 πr . (3.5b)Here v and v c are the magnitudes of the vacuum expectation values of φ and φ ± ,determined by minimizing the potential terms in the action. The angular wavenumbers ν + , ν − , and k must be integers to have single valued configurations and k , by definition,is the winding number of the vortex configuration. For non-zero values of k and ν ± theradial functions f ( r ) and f ± ( r ) interpolate between 0 at the origin and 1 at infinity.Similarly, to minimize energy the gauge field must approach a pure gauge form at largedistance, implying that h ( r ) may also be taken to interpolate between 0 and 1 as r goesfrom the origin to infinity. The associated magnetic field is B ( r ) = ( rA θ ( r )) (cid:48) r = Φ h (cid:48) ( r )2 πr . (3.6)The gauge field in ansatz (3.5) is written in a form which makes the coefficient Φ equalto the total magnetic flux, Φ B ≡ (cid:90) d x B = 2 π (cid:90) ∞ r dr B ( r ) = Φ (cid:90) ∞ dr h (cid:48) ( r ) = Φ . (3.7)To avoid having an energy which diverges linearly with volume (relative to thevacuum), the phases of φ , φ + and φ − must be correlated in a fashion which minimizesthe cubic term in the action. Below we will suppose that the coefficient of the cubicterm (cid:15) > , but essentially the same formulas would result if (cid:15) < . (The singularpoint (cid:15) = 0 must be handled separately, see the discussion at the end of Sec. 2.3.)Minimizing the cubic term in the action forces the product φ φ + φ − to be real andpositive, implying that ν + = n − k , ν − = − n , (3.8)for some integer n .After imposing condition (3.8), there remains a logarithmic dependence on thespatial volume caused by the scalar kinetic terms which, due to the angular phase– 15 –ariation of the scalar fields, generate energy densities falling as /r . Explicitly, thislong-distance energy density is E ( r ) = v c r (cid:34)(cid:18) n − k − Φ2 π (cid:19) + (cid:18) − n + Φ2 π (cid:19) (cid:35) + v k r + O ( r − ) . (3.9)Minimizing this IR energy density, for given values of k and n , determines the magneticflux Φ , leading to Φ B = Φ = (2 n − k ) π , (3.10)and an IR energy density E ( r ) = ( v c + v ) k /r + O ( r − ) .The explicit form of the radial functions is determined by minimizing the remainingIR finite contributions to the energy. These consist of the magnetic field energy andshort distance corrections to the scalar field kinetic and potential terms, all of whichare concentrated in the vortex core region. Semi-explicitly, E = 2 π (cid:90) r dr (cid:34) h (cid:48) ( r ) e r (2 n − k ) + v c f + ( r ) r [(2 n − k )(1 − h ( r )) − k ] + v k f ( r ) r + v c f − ( r ) r [(2 n − k )(1 − h ( r )) + k ] + v c (cid:2) f (cid:48) + ( r ) + f (cid:48)− ( r ) (cid:3) + v f (cid:48) ( r ) + (potential terms) (cid:35) . (3.11)Minimizing this energy leads to straightforward but unsightly ordinary differentialequations which determine the precise form of the radial profile functions, see Ap-pendix A. Qualitatively, the gauge field radial function h ( r ) approaches its asymptoticvalue of one exponentially fast on the length scale min ( m − A , (cid:101) m − ) , where m A = 2 ev c and (cid:101) m ≡ λ c v c + 2 (cid:15)v . The scalar field profile functions f ( r ) and f ± ( r ) approachtheir asymptotic large r values with /r corrections on the length scales set by thecorresponding masses m and m c .For a given non-zero winding number k , the above procedure generates an infinitesequence of vortex solutions distinguished by the value of n , or more physically by thequantized value of the magnetic flux (3.10) carried in the vortex core. The minimalenergy vortex, for a given winding number, is the one which minimizes this flux. Foreven winding numbers, this is n = k/ and vanishing magnetic flux. In such solutions,the phases of the two charged scalar fields are identical with ν ± = − k/ .For odd winding number k there are two degenerate solutions with n = ( k ± / and magnetic flux Φ = ± π . In these solutions, the charged scalar fields have differingphase windings with ν + = − ( k ∓ / and ν − = − ( k ± / . For minimal | k | = 1 – 16 – igure 3 . There are four distinct minimal energy vortex solutions, with winding number k = ± and magnetic flux Φ = ± π . The ( Z ) F and ( Z ) C discrete symmetries relate thesevortices as shown. vortices, one of the charged scalars has a constant phase with no winding, while theother charged scalar has a phase opposite that of φ .The gauge field holonomy surrounding a vortex, far from its core, is simply ± depending on whether the magnetic flux is an even or odd multiple of π and this, inturn, merely depends on whether the vortex winding number k is even or odd, (cid:104) Ω( C ) (cid:105) k = e i Φ = ( − k . (3.12)The net result is that there are four different minimal energy vortex solutions,illustrated in Fig. 3, having ( k, Φ) = (1 , π ) , (1 , − π ) , ( − , π ) , and ( − , − π ) . As indicatedin the figure, the ( Z ) F symmetry interchanges vortices with identical winding numberand opposite values of magnetic flux, while the ( Z ) C symmetry interchanges vorticeswith opposite values of both winding number and magnetic flux. Therefore, all thesevortices have identical energies. For our purposes, the key result is that the longdistance holonomy is the same for all minimal vortices, namely (cid:104) Ω( C ) (cid:105) k = ± = − .Consequently, we find O Ω = − at tree-level . (3.13)We now consider the effects of quantum fluctuations on this result. Using standardeffective field theory (EFT) reasoning, as one integrates out fluctuations below the UVscale Λ UV , the action (2.4) will receive scale-dependent corrections which (a) renormal-ize the coefficients of operators appearing in the action (2.4), and (b) induce additionaloperators of increasing dimension consistent with the symmetries of the theory. But– 17 –he result (3.13) follows directly from the leading long-distance form (3.9) of the en-ergy density whose minimum fixes the vortex magnetic flux equal to ± π for minimalwinding vortices. Because this /r energy density leads to a total energy which islogarithmically sensitive to the spatial volume, short distance IR-finite contributionsto the energy cannot affect the flux quantization condition (3.10) in the limit of largespatial volume. Only those corrections which modify this /r long distance energydensity have the potential to change the quantization condition.In effective field theory one expands the Wilsonian quantum effective action inpowers of fields and derivatives. Any term in the EFT action with more than twoderivatives will produce a contribution to the energy density which falls faster than /r when evaluated on a vortex configuration, and hence cannot contribute to the O (1 /r ) long distance energy density (3.9). Similarly, terms with less than two derivativesalso do not contribute to the O (1 /r ) long distance energy density (3.9). Hence theonly fluctuation-induced terms that might affect the long distance vortex holonomy arethose with precisely two derivatives acting on the charged scalar fields. Consequently,the portion of the effective action that controls holonomy expectation values aroundvortices can be written in the form S eff , U (1) holonomy = (cid:90) d x (cid:110) f ( φ , φ + , φ − ) (cid:0) | D µ φ + | + | D µ φ − | (cid:1) + f ( φ , φ + , φ − ) [ φ ( D µ φ + )( D µ φ − ) + h.c. ] (cid:111) , (3.14)with coefficient functions f , f depending on the fields φ , φ ± (but not their derivatives)in U (1) G and ( Z ) F invariant combinations. In typical EFT applications one wouldexpand these coefficient functions in a power series (or transseries ) but we have noneed to do so. The f term represents wavefunction renormalizations which simplymodify the overall normalizations in the energy density (3.9), and have no effect on theflux quantization condition (3.10). The f term produces a /r contribution to theenergy density of the form f [ φ ( D µ φ + )( D µ φ − ) + h.c. ] ∝ v v c r (cid:18) n − k − Φ2 π (cid:19) (cid:18) − n + Φ2 π (cid:19) , (3.15)but the extremum of this term (with respect to Φ ) lies at exactly the same point, Φ = (2 n − k ) π , identified in the tree-level analysis. In other words, fluctuation-inducedcorrections to the effective actions have no effect on the flux quantization condition(3.10). For a concrete example of the use of transseries expansions in EFT see, e.g., the review [27]discussing the Euler-Heisenberg effective action for quantum electrodynamics. – 18 –his shows that the minimal vortex expectation value (cid:104) Ω( C ) (cid:105) at large distanceremains real and negative to all orders in perturbation theory, provided that the fluc-tuations are not so large that they completely destroy the Higgs phase. The sizeof quantum fluctuations in this model is controlled by the dimensionless parameter e /m A = O ( eλ / c / | m c | ) = O ( e / | m c | ) , where we have assumed λ / c ∼ (cid:15) / ∼ e and g , g c (cid:28) for simplicity, and hence this conclusion about a negative value of (cid:104) Ω( C ) (cid:105) holds exactly whenever m c /e is sufficiently negative to put the theory into the Higgsphase.As discussed earlier, quantum fluctuations do suppress the magnitude of holonomyexpectation values leading to perimeter law exponential decay. By construction, thissize dependence cancels in our ratio O Ω = (cid:104) Ω( C ) (cid:105) / (cid:104) Ω( C ) (cid:105) . Unbroken ( Z ) F symmetry(or ( Z ) C , or reflection symmetry) in the vacuum state guarantees that the ordinaryexpectation value (cid:104) Ω( C ) (cid:105) in the denominator is real. It is easy to check that it is positiveat tree level, and sufficiently small quantum fluctuations cannot make it negative. So O Ω is determined by the phase of the vortex state holonomy expectation value in thenumerator. The net result from this argument is that within the Higgs phase,Higgs phase: O Ω = − , (3.16)holds precisely. The next subsection gives useful alternative perspectives on the sameconclusion. In the preceding section we analyzed the physics of vortices using effective field theoryin the bulk -dimensional spacetime. This analysis showed that the minimal energyvortices carry quantized magnetic flux ± π , and the phase of the holonomy aroundvortices is quantized, leading to result (3.16). We now reconsider the same physicalquestions from the perspective of an effective field theory defined on the vortex world-line. This will lead to a discussion of vortex junctions, their interpretation as magneticmonopoles, a connection between vortex flux quantization and Dirac charge quantiza-tion, and finally to distinct logically independent arguments for the result (3.16).The (0+1) dimensional effective field theory describing fluctuations of a vortexworldline includes two gapless modes arising from the translational moduli representingthe spatial position of the vortex. The vortex effective field theory must include anadditional real scalar field which may be chosen to equal the magnetic flux Φ carriedby a vortex configuration. This field will serve as a coordinate along field configurationpaths which interpolate between distinct vortex solutions. The field Φ appears in the– 19 –D worldline EFT in the form S vortex EFT = (cid:90) dt (cid:2) c K ( ∂ t Φ) + c V V (Φ) (cid:3) + · · · . (3.17)Here t is a coordinate running along the vortex worldline, Φ is dimensionless, c K and c V are low-energy constants with dimensions of inverse energy and energy, respectively,and the ellipsis represents terms with additional derivatives or couplings to other fieldson the worldline.The potential V (Φ) in expression (3.17) obeys two important constraints. First,since ( Z ) F symmetry acts on Φ by Φ → − Φ , V (Φ) is an even function. Second, Diraccharge quantization in the underlying bulk quantum field theory further constrains thepossible minima of V (Φ) . To see this, suppose that V (Φ) has a minimum at Φ = Φ min (cid:54) =0 . Since V (Φ) is an even function, it must also have a distinct minimum at Φ = − Φ min .For generic values of the microscopic parameters, the potential V is finite for all finitevalues of Φ . This means that there exists a solution to the equation of motion for Φ inwhich Φ interpolates between − Φ min and Φ min as the worldline coordinate t runs from −∞ to + ∞ . This tunneling event has finite action. What is its interpretation in bulkspacetime? It has unit U (1) G circulation at all times, but also possesses a “junction”at some finite time where the magnetic flux changes sign. The magnetic flux througha -sphere surrounding the junction is Φ min − ( − Φ min ) = 2Φ min . Comparing this tothe Dirac charge quantization condition in (2.2) implies that Φ min ∈ π Z . This meansthat unbroken ( Z ) F symmetry and Dirac charge quantization together imply vortexflux quantization.In the preceding section, we saw that in the Higgs phase minimal-energy unit-circulation vortices carry magnetic flux ± π at tree-level. The vortex flux quantizationargument in the paragraph above implies that quantum corrections cannot changethis result, again leading to result (3.16). We also learn that a junction between twominimal-energy unit-circulation vortices with flux π and − π can be interpreted as amagnetic monopole carrying the minimal π flux consistent with Dirac charge quanti-zation, as illustrated in Fig. 4. This is the Higgs phase version of a single monopole-instanton, discussed earlier, when the φ condensate has unit winding.As noted earlier near the end of Sec. 2, Higgs phase monopole–antimonopole pairsare connected by magnetic flux tubes (which can break at sufficiently large separation Since the tunneling event can be interpreted as a monopole-instanton, the value of its actiondepends on the UV completion of our compact Abelian gauge theory. Appendix B describes anexplicit SU (2) gauge theory which reduces to our U (1) gauge theory at long distances, and where S I ∼ m W /e with m W the W -boson mass. If ( Z ) F symmetry is explicitly broken, Dirac charge quantization alone leads to the conclusionthat any two distinct minima Φ , Φ of V (Φ) must satisfy Φ − Φ ∈ π Z . – 20 – igure 4 . A junction between the two minimal energy unit-winding vortex worldlines is amagnetic monopole with flux π . due to monopole–antimonopole pair creation). This is true in the absence of any vor-tices carrying unit U (1) G winding. But in the presence of a unit circulation vortex,a monopole–antimonopole pair can bind to the vortex, with the monopole and anti-monopole then free to separate arbitrarily along the vortex worldline. This is illus-trated in Fig. 5. To see this, note that for fixed separation L between monopole and an-timonopole, the action will be lowered if the monopole and antimonopole move onto thevortex line, provided they are oriented such that adding the monopole–antimonopoleflux tube to the vortex magnetic flux has the effect of merely flipping the sign of vor-tex magnetic flux on a portion of its worldline. This eliminates the cost in action ofthe length L flux tube initially connecting the monopole and antimonopole. As notedabove, the ( Z ) F symmetry guarantees that the vortex action per unit length is inde-pendent of the sign of the magnetic flux. Once the monopole and antimonopole arebound to the vortex worldline, there is no longer any cost in action (neglecting ex-ponentially falling short distance effects) to separate the monopole and antimonopolearbitrarily. In summary, the monopole–antimonopole string tension vanishes on thevortex, and magnetic monopoles are deconfined on minimal Higgs phase vortices. One can also regard the monopole–antimonopole pair as an instanton–antiinstanton Deconfinement of magnetic monopoles on both local and semilocal vortices with and withoutsupersymmetry has been extensively studied previously. In our model the vortices are global but themonopole deconfinement mechanism described here is essentially identical to previous discussions in,for example, Refs. [28–36]. Provided monopoles and antimonopoles alternate along the vortex worldline. There is a directparallel between this phenomenon and charge deconfinement in 2D Abelian gauge theories at θ = π ,see for example Refs. [37–43]. – 21 – igure 5 . Monopole–antimonopole pairs with minimal magnetic flux π are confined inbulk spacetime, but such pairs are attracted to the worldline of a minimal global vortex wherethey become deconfined. pair in the worldline EFT (3.17). We now argue that this perspective leads to yetanother derivation of the result (3.16). The existence of degenerate global minima withflux ± Φ min means that the ( Z ) F symmetry is spontaneously broken on the worldlineto all orders in perturbation theory. But non-perturbatively, the finite-action worldlineinstantons connecting these minima will proliferate and restore the ( Z ) F symmetry.As is familiar from double-well quantum mechanics, the unique minimal energy vortexstate will be a symmetric linear combination of Φ min and − Φ min configurations.From our previous two arguments we know that Φ min = π , so that both of thesevortex configurations have the same − long distance holonomy, and none of this non-perturbative physics has any effect on the validity of the result (3.16) regarding Higgsphase vortices. But suppose that we did not already know that Φ min = π . The existenceof finite-action tunneling events connecting the two Φ minima directly implies that theminimal energy vortex state with a given winding number is unique and invariant under ( Z ) F . Unbroken ( Z ) F symmetry in turn implies that the holonomy expectation valuein the minimal vortex state is purely real. Therefore, on symmetry grounds alone, ourobservable O Ω is quantized to be either +1 or − . Our analysis in the weakly coupledregime serves to establish that in the Higgs phase the value is − , and we again arriveat result (3.16). O Ω in the U (1) G -broken confining regime We now turn to a consideration of holonomies around vortices in the U (1) G -brokenconfining phase. Once again, it is useful to consider the appropriate effective field The deconfinement of magnetic monopoles on unit-circulation vortices corresponds to the fact thatthe separation of an instanton–antiinstanton pair is a quasi-zero mode. – 22 –heory deep in this regime, near the SE corner of the phase diagram of Fig. 1.Suppose that m c (cid:29) e . Given the scale separation, it is useful to to integrate outthe charged fields. The resulting effective action retains the gauge field and neutralscalar φ and has the form S eff = (cid:90) d x (cid:20) e F µν + V m ( σ ) + | ∂ µ φ | + V ( | φ | ) + am c | φ | F µν + · · · (cid:21) , (3.18)where the ellipsis denotes higher dimension terms involving additional powers of fieldsand derivatives. The dimension five term shown explicitly, with coefficient a , is thelowest dimension operator coupling the gauge and neutral scalar fields. This termdescribes “Raleigh scattering” processes in which photons scatter off fluctuations in themagnitude of φ . Within this EFT, the ( Z ) F symmetry simply flips the sign of thegauge field and hence forbids all terms involving odd powers of the gauge field strength.When m is sufficiently negative so that the U (1) G symmetry is spontaneouslybroken and φ condenses, the leading effect of the | φ | F coupling is merely to shiftthe value of the gauge coupling by an amount depending on the condensate v ≡ (cid:104) φ (cid:105) , e → e (cid:48) ≡ e + 4 a | v | m c . (3.19)This is a small shift of relative size O ( e /m c ) within the domain of validity of thiseffective description. The ( Z ) F symmetry (or parity) guarantees that the neutral scalarcondensate cannot source the gauge field strength, so the magnetic field B ≡ (cid:15) ij F ij ( i, j = 1 , ) must have vanishing expectation value.Within this U (1) G broken phase, there are vortex configurations in which the con-densate (cid:104) φ (cid:105) has a phase which winds around the vortex, while its magnitude decreasesin the vortex core, vanishing at the vortex center. As far as the gauge field is concerned,one sees from the effective action (3.18) that the only effect this has is to modulatethe gauge coupling, effectively undoing the shift (3.19) in the vortex core. But suchcoupling renormalizations, or dielectric effects, do not change the fact that the effectiveaction is an even function of magnetic field which is minimized at B = 0 . In otherwords, even in the presence of vortices, the neutral scalar field does not source a mag-netic field. And consequently, both the vacuum state and minimal energy vortex statesare invariant under the ( Z ) F symmetry.Once again, invariance of the both the vacuum and vortex states under the ( Z ) F symmetry implies that holonomy expectation values in both states are real, and henceour observable O Ω must be either +1 or − . The Abelian gauge field holonomy is, ofcourse, nothing but the exponential of the magnetic flux, Ω( C ) = e i (cid:72) C A = e i (cid:82) S B = e i Φ B (with contour C the boundary of disk S ). The above EFT discussion shows that deep in– 23 –he confining U (1) G -broken phase the influence of a vortex on the magnetic field is tinyand hence (cid:104) Ω( C ) (cid:105) is positive, implying that O Ω = +1 . And once again, by analyticity,this result must hold throughout the confining U (1) G -broken phase. In summary, U (1) G -broken confining phase: O Ω = +1 , (3.20)is an exact result within this phase. We have seen that O Ω has constant magnitude but changes sign between the Higgs andconfining, U (1) G -broken regimes; it cannot be a real-analytic function of m c . Hence,there must be at least one phase transition as a function of m c . A single phase transitionwould be associated with an abrupt jump of O Ω from − to at some critical value of m c . If instead O Ω equals − for charged mass-squared below some value, m c < ( m c ) A ,equals +1 above a different value ( m c ) B < m c , and continuously interpolates from − to +1 in the intervening interval ( m c ) A < m c < ( m c ) B , this would indicate thepresence of two phase transitions bounding an intermediate phase in which the ( Z ) F symmetry is spontaneously broken. (This follows since, as discussed above, unbroken ( Z ) F symmetry implies that O Ω must equal ± .)In much of parameter space, phase transitions in our model occur at strong couplingand are not amenable to analytic treatment. But the theory becomes weakly coupledwhen the masses | m c | and | m | are sufficiently large. Specifically, we will assume thatthe dimensionful couplings | λ c | , | λ | and e are all small relative to the masses | m c | and | m | , the cubic coupling obeys (cid:15) (cid:28) min ( | m c | / , | m | / ) , and the sextic couplings aresmall, g c , g (cid:28) . If a first order transition lies within this region, then simple analyticarguments suffice to identify and locate the transition.A first-order transition involving a complex scalar φ with U (1) symmetry requiresmultiple local minima in the effective potential viewed as a function of | φ | . In fourdimensions, a renormalizable scalar potential is quartic and, as a function of | φ | , has atmost a single local minimum. So to find a first-order phase transition in a weakly cou-pled four-dimensional U (1) invariant scalar theory one must either be abnormally sen-sitive to higher order non-renormalizable terms (and thus probing cutoff-scale physics),or else reliant on a one-loop or higher order calculation producing non-analytic termslike | φ | log | φ | . This is illustrated by the classic Coleman and Weinberg analysis [44].But in three spacetime dimensions, renormalizable scalar potentials are sextic, and U (1) invariant sextic potentials can easily have multiple local minima. Consequently, a tree-level analysis can suffice to demonstrate the existence of a first-order phase transition,in a renormalizable theory, without any need to consider higher-order corrections.– 24 – igure 6 . Contour plots of the tree-level scalar effective potential at three different values of m c in the vicinity of the first-order Higgs-confinement phase transition. We have used gaugeand global symmetries to choose the phases of the scalar fields such that the potential can beinterpreted as a function of φ c ≡ φ + and φ , with φ − = | φ + | . We have set m = − e , (cid:15) = 40 e , λ c = λ = − e , and g c = g = 0 . . Decreasing values of the scalar potential arecolored with darker colors, and global minima are marked with red dots. Note that the globalminimum is degenerate when m c ≈ e , and the location of the global minimum jumpsas m c crosses this value, from a point where the charged fields are condensed to one wherethey are not condensed. This shows the presence of a strong first-order Higgs-confinementphase transition, with the U (1) G global symmetry spontaneously broken on both sides of thetransition. Let us see how this works in our model. Consider the region where m , λ c , and λ are all negative. For simplicity, let us also suppose that e (cid:28) | λ c | , | λ | , and (cid:15) (cid:28) e (cid:28) min ( | m c | / , | m | / ) . In Fig. 6 we show contour plots of the scalar potential as afunction of φ c ≡ φ + and φ , with φ − = | φ + | , as m c /e is varied. The figure shows thatthe potential has multiple local minima with relative ordering that changes as m e /e is varied with all other parameters held fixed. With the parameter choices given in thecaption of Fig. 6, the figure shows the existence of a strong first-order phase transitionbetween U (1) G -broken confining and U (1) G -broken Higgs states in the regime where m /e is large and negative and m c /e is large and positive. Correspondingly, thechange in the derivative of the energy density with respect to the charged scalar masssquared in units of e , e − ∆( ∂ E /∂m c ) , is large across the transition. For the parametervalues used in Fig. 6 one finds e − ∆( ∂ E /∂m c ) = 2∆ φ c /e ≈ (cid:29) . This behavioris generic. The effective masses (i.e., curvatures of the potential) at the minima arecomparable to the input mass parameters, so there are no near-critical fluctuations andthe phase transition is reliably established at weak coupling.Finally, the analysis of the previous subsections shows that our vortex holonomy or-– 25 –er parameter O Ω changes sign across this phase transition, confirming that the abruptchange in this “topological” order parameter is associated with a genuine thermody-namic phase transition.As one moves into the interior of the ( m , m c ) phase diagram, out of the weakly-coupled periphery, we certainly expect this direct correlation between a jump in ourvortex order parameter and a thermodynamic phase transition to persist. But onemay contemplate whether this association could cease to apply at some point in theinterior of the U (1) G spontaneously broken domain. In general, a line of first orderphase transitions which is not associated with any change in symmetry realization canhave a critical endpoint (as seen in the phase diagram of water). Could our model havesuch a critical endpoint, beyond which the first order transition becomes a smoothcross-over as probed by any local observable? If so, there would necessarily remainsome continuation of the phase transition line across which our topological observable O Ω continues to flip sign, but all local observables remain smooth. Is this possible?The answer is no. The magnetic flux carried by vortices can change in steps of π due to alternating monopole-instanton fluctuations appearing along the vortex world-line, but such processes do not affect the sign of the holonomy around a vortex. Atthe transition between the Higgs and confining phases the magnetic flux carried byminimal-winding vortices changes by π (modulo π ) . Such a sudden change in thevortex magnetic flux will surely imply non-analyticity in the IR-finite core energy ofa vortex, or equivalently the vortex fugacity. Whenever the U (1) G symmetry is spon-taneously broken, the equilibrium state of the system will contain a non-zero densityof vortices and antivortices due to quantum fluctuations. If the minimal vortex energyis non-analytic this will in turn induce non-analyticity in the true ground state energydensity. (This argument ceases to apply only when the vortex density reaches the pointwhere vortices condense, thereby restoring the U (1) G symmetry.) In other words, ourvortex holonomy observable functions as a useful order parameter, identifying thermo-dynamically distinct gapless phases.This concludes our argument for the necessity of at least one phase transition curveseparating the SE and W regions of Fig. 1. We now generalize our model to include operators which break the ( Z ) F symmetryexplicitly. The simplest such term is just a mass perturbation giving the two chargedfields φ + and φ − distinct masses m + and m − . Let m ≡
12 ( m + m − ) , ∆ ≡ e ( m − m − ) , (3.21)– 26 –enote the average mass squared and a measure of their difference, respectively. Wewill examine the dependence of physics on m /e with ∆ > held fixed.If ∆ is sufficiently large then there are two seemingly different regimes where noglobal symmetries are spontaneously broken: one where no scalar fields are condensed,and another where only φ − is condensed. The latter regime is not a distinct phaseas condensation of the charged field φ − , by itself, does not imply a non-vanishingexpectation value of any physical order parameter. In fact, these two regimes aresmoothly connected to each other and are trivial in the sense that they have a massgap and a vacuum state which is invariant under all global symmetries.The more interesting regimes of the model are those with spontaneously broken U (1) G symmetry. The cubic term in the action (cid:15)φ φ + φ − + h.c. ensures that there is noregime where φ and only one of the two charged fields are condensed. Hence we onlyneed to consider two regimes with spontaneously broken U (1) G symmetry: one whereall scalar fields are condensed, another where only the neutral scalar φ is condensed. Consider the Higgs regime where − m (cid:29) e and all scalars are condensed. Thetree-level long-distance energy density that determines the holonomy around a U (1) G vortex of winding number k is given by an obvious generalization of Eq. (3.9), E ( r ) = v r (cid:18) n − k − Φ2 π (cid:19) + v − r (cid:18) − n + Φ2 π (cid:19) + v k r + O ( r − ) . (3.22)Due to the explicit breaking of ( Z ) F , the magnitudes of the charged scalar expectationvalues v + and v − are no longer equal; let us denote their average by v avg . For givenvalues of k and n , minimizing the above energy density yields Φ = (cid:18) n − k v v (cid:19) π, (3.23)and E = (cid:16) v v − v + v (cid:17) k /r + O ( r − ) . Due to the explicit breaking of ( Z ) F , thereare no longer two degenerate minimal-winding vortices at tree-level. Suppose v −
In the two decades since Shäfer and Wilczek hypothesized quark-hadron continuity inflavor symmetric QCD, based on compatible symmetry realizations and other necessarybut not sufficient correspondences, their conjecture has reached the status of a highlyplausible folk theorem. The expectation of quark-hadron continuity has been used asthe starting point for a large number of further conjectures and developments, see e.g.,Refs. [63, 69–83].Recently, however, three of the present authors argued that a change in particle-vortex statistics between the Higgs regime (quark matter) and the confined regime(nuclear matter) should be interpreted as compelling evidence for invalidity of theSchäfer-Wilczek conjecture [21]. We showed that color holonomies around minimalcirculation U (1) B vortices have non-trivial phases of ± π/ in high density quark mat-ter, noted that these holonomies should have vanishing phases in the nuclear matterregime, and used this sharp change in the physics of topological excitations to arguethat the nuclear matter and quark matter regimes of dense QCD will be separated bya phase transition.Subsequent work by other authors [74, 88] offered some objections to the argumentsin our Ref. [21]. Let us address these objections, starting with Ref. [74] by Hirono and This may sound similar to the Fradkin-Shenker theorem [2] but, as discussed in the introduction,the Fradkin-Shenker theorem does not apply in situations where the Higgs field is charged under globalsymmetries, while the Schäfer-Wilczek conjecture concerns precisely such situations. For other examinations of vortices in dense quark matter, see also Refs. [32, 33, 84–87]. – 35 –anizaki. Changes in particle-vortex statistics are a commonly used diagnostic forphase transitions in gapped phases of matter, see e.g., Refs. [89, 90]. In gapped phases,changes in particle-vortex statistics are connected to changes in intrinsic topologicalorder, which in turn can be related to changes in the realization of higher-form globalsymmetries [17]. Reference [74] tacitly assumed that these statements also hold in gap-less systems, and misinterpreted our work [21] as proposing that the zero temperaturehigh density phase of QCD is topologically ordered. Reference [74] then argued thatthis is not the case by discussing the realization of a putative low-energy “emergent”higher-form symmetry in a gauge-fixed version of N c = 3 Yang-Mills theory coupled tofundamental Higgs scalar fields. Besides relying on a non-manifestly gauge invariantapproximate description to suggest some higher form symmetry, this discussion missedthe central points of Ref. [21] for two reasons. First, Ref. [21] already explicitly em-phasized that the CFL phase of QCD is not topologically ordered according to thestandard definition of that term, so arguing that the CFL phase does not have topo-logical order in no way contradicts the analysis of Ref. [21]. Second, while Ref. [74]agreed with us that in the flavor-symmetric limit, CFL quark matter features non-trivial color holonomies around U (1) B vortices, it did not address the key question ofhow this could be consistent with the expected behavior of color holonomies in thenuclear matter regime. Without addressing this crucial question, one cannot concludethat quark-hadron continuity remains a viable scenario in QCD.Reference [88] by Alford, Baym, Fukushima, Hatsuda and Tachibana accepted themain result of Ref. [21], namely that in the flavor-symmetric limit color holonomiesaround vortices take sharply different values in the nuclear matter and quark matterphases. These authors then considered (straight) minimal-circulation vortices in asetting where the density varies along the direction of a vortex. Ref. [88] invokedthe center-vortex model of confinement [51] to argue that it is not necessary for a“boojum” (i.e., a junction or special defect at points where vortices pass through theinterface between distinct superfluid phases) to form on the interface between quarkand hadron regimes. Along the way, Ref. [88] assumed that such a boojum shouldinvolve three quark-phase vortices joining together at the interface (perhaps inspiredby similar conjectures in Refs. [84, 86]), and argued that this is not necessary becausecolor flux is screened in the nuclear matter regime. But there is no reason for a boojumat the interface between quark matter and nuclear matter to necessarily involve multiplevortices joining together. Instead, given the behavior of the holonomies, it is entirely For example, one can consider a gedanken situation involving a rotating bucket of density stratifiedquark/nuclear matter when the quantized superfluid circulation equals unity on every cross-sectionof the bucket. There must then be a single minimal circulation vortex threading both phases andcrossing the interface between them. – 36 –onsistent for the interface to be a genuine boundary between distinct thermodynamicphases, with minimal-energy boojums involving just one minimal circulation vortex oneither side.In our view, the key limitations of our work in Ref. [21] were that we could notexplicitly compute expectation values of color holonomies in the superfluid nuclearmatter regime and demonstrate that they have trivial phases, nor could we give aproof that a change in the behavior of gauge field holonomies around vortices mustbe associated with a bulk thermodynamic phase transition. (Although we did givephysical arguments for this which we believe are convincing.)In the preceding sections of the present paper, we have analyzed a 3D model whichwas deliberately constructed to be analogous to dense QCD, and to which Schäfer andWilczek’s continuity conjecture applies and predicts that no phase transition separatesthe U (1) G -broken Higgs and confining regimes. This allowed us to examine both of theseearlier limitations in the context of this instructive model, and find that continuity does not hold. The Higgs and confining U (1) G -broken regimes of the 3D theory are distinctphases of matter characterized by a novel order parameter. In earlier sections we focused on our 3D Abelian model because this provided thesimplest setting in which to examine the issue of Higgs-confinement continuity withinsuperfluid (or spontaneously broken U (1) ) phases, with good theoretical control inboth regimes. It is, of course, of interest to understand how the relevant physics mightchange when one turns to 4D gauge theories which are more QCD-like.To that end, we now consider an SU (3) gauge theory coupled to three antifunda-mental representation scalar fields, as well as an additional gauge-neutral complex scalarfield φ . We will build a model with SU (3) flavor symmetry, and write the chargedscalar fields as a × matrix Φ which transforms in the bifundamental representationof SU (3) flavor × SU (3) gauge , Φ → F Φ C † , F ∈ SU (3) flavor , C ∈ SU (3) gauge . (4.1)We also assume the theory has a U (1) global symmetry, which acts as U (1) G : Φ → e iα/ Φ , φ → e iα φ , (4.2)and assume that there exist (or could exist) heavy ‘baryon’ test particles with unitcharge under the U (1) G global symmetry. Since U (1) G phase rotations which lie within Z coincide with the action of SU (3) gauge transformations, the faithfully acting U (1) global symmetry is U (1) G / Z . – 37 –he action defining this model is given by S = (cid:90) d x (cid:20) g tr F µν + tr ( D µ Φ) † D µ Φ + | ∂ µ φ | + m tr Φ † Φ + m | φ | + λ | φ | + λ Φ tr (Φ † Φ) + (cid:15) ( φ † det Φ + h.c ) + · · · (cid:21) . (4.3)As usual, D µ Φ = ∂ µ Φ + i Φ A µ is the covariant derivative in the antifundamental rep-resentation, and the ellipsis denotes possible further scalar self-interactions which areinvariant under the chosen symmetries. The field strength F µν ≡ F aµν t a , with Hermitian SU (3) generators satisfying tr t a t b = δ ab .This 4D model is very similar to the scalar part of the effective field theory thatdescribes high-density three-color QCD in the CFL quark matter regime [24], with U (1) G / Z playing the role of U (1) B in QCD. The matrix-valued scalar Φ representsthree color-antifundamental diquark fields, so that det Φ has the quantum numbers offlavor-singlet dibaryons, which are condensed in both the CFL phase and the SU (3) -symmetric nuclear matter phases. Due to the (cid:15) coupling between the gauge-neutralscalar φ and det Φ , one can think of φ † as a (dynamical) source for flavor-singletdibaryons. Explicitly introducing the neutral scalar φ allows the model (4.3) to de-scribe both the Higgs regime and a regime where dibaryons are light, but the gaugeand charged scalar fields can be integrated out.Of course, the effective action for dense QCD in the CFL regime is rotation-invariant but not Lorentz invariant, and also includes heavy fermionic excitations, incontrast to the purely bosonic Lorentz-invariant theory defined by Eq. (4.3). Thesedifferences are not relevant to our discussion, and we expect the phase structure ofthe model (4.3) to mimic the phase structure of QCD with approximate SU (3) flavorsymmetry.Consider the Higgs regime of the model (4.3) where (in gauge-fixed language) Φ has an expectation value of color-flavor locked form, (cid:104) Φ (cid:105) = v Φ , and there is a residualunbroken SU (3) global symmetry acting as Φ → U Φ U † , with U ∈ SU (3) . The U (1) G global symmetry is spontaneously broken implying, as always, the existence of vortextopological excitations. To describe a straight “superfluid” vortex, using cylindricalcoordinates with r = 0 at the center of the vortex, one may fix a gauge in which thevortex configuration has Φ diagonal and A µ taking values in the Cartan subalgebra, φ ( r, θ ) = v f ( r ) e ikθ , (4.4a) Φ( r, θ ) = v Φ diag (cid:0) f ( r ) e i ( n + k ) θ , f ( r ) e i ( m − n ) θ , f ( r ) e − imθ (cid:1) , (4.4b) A θ ( r ) = a h ( r )2 πr t + b h ( r )2 πr t . (4.4c)– 38 –ere k, m, n ∈ Z , with k the vortex winding number, t ≡ √ diag (1 , , − and t ≡ diag (1 , − , are the usual diagonal SU (3) generator matrices, and the radialprofile functions { f i } and { h i } approach as r → ∞ . Minimizing the long-distanceenergy density of the vortex configuration determines the gauge field asymptotics. Onefinds, a = − π √ ( k + 3 m ) , b = − π ( k + 2 n − m ) . (4.5)The minimal energy vortex with unit circulation ( k = 1 ) corresponds to n = m = 0 (with physically equivalent forms related by Weyl reflections), in which case a = − π √ , b = − π , (4.6)and Φ( r, θ ) = v Φ diag (cid:0) f ( r ) e iθ , g ( r ) , g ( r ) (cid:1) , A θ ( r ) = h ( r )3 r diag ( − , , . (4.7)Here, we have set f ( r ) = f ( r ) , f ( r ) = f ( r ) = g ( r ) , and h ( r ) = h ( r ) = h ( r ) .The minimal-energy vortex configuration (4.7) preserves an SU (2) × U (1) symmetry(cf., Ref. [91]). Hence, these minimal energy unit-winding vortices have zero modesassociated with the moduli space SU (3) SU (2) × U (1) = CP . (4.8)Consequently, the worldsheet effective field theory for a vortex contains a CP non-linear sigma model [33, 35, 92]. But the CP model in two spacetime dimensions (withvanishing topological angle θ ) has a mass gap and a unique ground state angle, seee.g., Refs. [93, 94]. So, despite the appearance of the classical configuration (4.4), the SU (3) global symmetry is unbroken both in the vacuum and in the presence of vortices.Now consider the behavior of our vortex holonomy order parameter in this theory.The gauge field holonomy is now a path-ordered exponential around some contour C , Ω( C ) ≡ P ( e i (cid:82) C A ) , and defines an SU (3) group element. The natural non-Abelianversion of our vortex order parameter involves gauge invariant traces of holonomies, O Ω ≡ lim r →∞ (cid:104) tr Ω( C ) (cid:105) (cid:104) tr Ω( C ) (cid:105) , (4.9)where in the numerator the circular contour C encircles a minimal vortex in the samedirection as the circulation of the U (1) G current. Both expectations in the ratio Once again, the numerator is defined by a constrained functional integral with a prescribed vortexworld-sheet, with the size of that world-sheet and the minimal separation between the vortex world-sheet and the holonomy contour C scaling together as the contour radius r increases. – 39 –4.9) have perimeter-law dependence on the size of the contour C arising from quan-tum fluctuations on scales small compared to r , but this geometric factor cancels byconstruction in the ratio. Unbroken charge conjugation symmetry implies that the de-nominator is real, and it must be positive throughout any phase connected to a weaklycoupled regime. So as in our earlier Abelian model, the behavior of O Ω is determinedby the phase of the vortex expectation value in the numerator.A trivial calculation (identical to that in Ref. [21]) shows that at tree-level, farfrom the vortex, (cid:104) tr Ω( C ) (cid:105) tree1 = e πi/ . (4.10)demonstrating that O Ω = e πi/ at tree-level. An effective field theory argument, anal-ogous to that given in section 3.2 (see also Appendix B), shows that this result isunchanged when quantum fluctuations are taken into account, as long as they are notso large as to restore the spontaneously broken U (1) G symmetry. To see this, considerthe form of the effective action generated by integrating out fluctuations on scales smallcompared to r . Only terms in the effective action with two derivatives acting on thecharged scalar field Φ can contribute to the O (1 /r ) holonomy-dependent part of theenergy density, and hence affect the gauge field asymptotics (4.6) which determines theexpectation value of holonomies far from the vortex core. Consequently, the portion ofthe effective action which controls the holonomy expectation value far from a vortexmay be written in the form S eff , SU (3) holonomy = (cid:90) d x (cid:110) Tr (cid:2) f (Φ , φ )( D µ Φ) † f (Φ , φ )( D µ Φ) (cid:3) + Tr (cid:2) f (Φ , φ )Φ † ( D µ Φ) (cid:3) Tr (cid:2) f (Φ , φ )( D µ Φ) † Φ (cid:3) + (cid:104) (cid:15) ABC (cid:15)
IJK φ † f (Φ , φ ) IA ( D µ Φ) JB ( D µ Φ) KC + (h.c.) (cid:105) (cid:111) , (4.11)where A, B, C are color indices and
I, J, K denote flavor. The five coefficient functions { f i } depend on the fields φ and Φ , but not on their derivatives, only in combinationswhich are are invariant under U (1) G . The functions f , f and f are a color adjointand flavor singlet (like Φ † Φ ), f is color singlet and flavor adjoint (like ΦΦ † ), while f is antifundamental in color and fundamental in flavor (like Φ itself). Plugging in theconfiguration (4.4), one can easily verify that the second line of (4.11) is independentof a, b while the terms in both first and third lines of (4.11) have extrema, with respectto the asymptotic gauge field coefficients a and b , at the same location (4.6) regardlessof the form of the functions { f i } . Therefore small quantum corrections do not perturbthe gauge field asymptotics far from a vortex, and hence cannot shift the phase of the– 40 –ortex holonomy expectation (cid:104) tr Ω( C ) (cid:105) away from π/ . Hence, we learn that U (1) G -broken Higgs phase: O Ω = e πi/ , (4.12)holds exactly throughout the phase connected to the weakly coupled Higgs regime.Alternatively, when m (cid:38) Λ , with Λ the strong dynamics scale of the theory,we can recycle the arguments of Sec. 3.5 to understand the behavior of O Ω . In thisregime, due to the presence of heavy dynamical charged excitations, the expectationvalues of large fundamental representation Wilson loops are (exponentially) dominatedby a perimeter-law contribution. Physically, a Wilson loop describes a process wherea fundamental representation test particle and antiparticle are inserted at some point,separated and then recombined as they traverse the contour C . The perimeter lawbehavior arises from configurations in which dynamical fundamental representationexcitations of mass m Φ are pair-created and dress the test charge and anticharge tocreate two bound gauge-neutral “mesons.” These mesons have physical size of order (cid:96) meson ∼ min (Λ − , ( α s m Φ ) − ) , and experience no long range interactions. Once theWilson loop size exceeds the string breaking scale ∼ m Φ / Λ , pair creation of dynamicalcharges of mass m Φ and the associated meson formation becomes the dominant processcontributing to fundamental Wilson loop expectation values.The perimeter law contribution to large fundamental representation Wilson loopexpectation values arises from fluctuations of the gauge-charged fields within distancesof order of (cid:96) meson from any point on the contour C . The amplitude for such screeningfluctuations, and consequent meson formation, must be completely insensitive to thepresence of a vortex very far away at the center of the loop. This means that theholonomy expectations in the numerator and denominator of the vortex observable(4.9) will be identical (up to exponentially small corrections vanishing as r → ∞ ),leading to the conclusion that U (1) G -broken confining phase: O Ω = 1 . (4.13)Once again, the differing results (4.12) and (4.13), each strictly constant within theirrespective domains, implies that O Ω cannot be a real-analytic function of m . Adaptingthe arguments in Sec. 3.4 regarding the impact of abrupt changes in the properties ofvortex loops on the ground state energy, we see that O Ω functions as an order parameterthat distinguishes the U (1) G -broken Higgs and U (1) G -broken confining phases of thisfour-dimensional SU (3) gauge theory with SU (3) flavor symmetry. Further evidence that changes in our non-local order parameter signal genuine phase transitionsin non-Abelian gauge theories may be gained by considering other calculable examples. One such case – 41 –inally, if the SU (3) flavor symmetry of this theory is explicitly broken by a smallperturbation, a simple generalization of the analysis leading to the gauge field asymp-totics (4.7) implies that the phase of O Ω will now deviate slightly from π/ . But inthe U (1) G -broken confined phase, O Ω remains exactly due to the confinement andstring breaking effects discussed above. This implies that the U (1) G -broken Higgs and U (1) G -broken confining regimes of our 4D SU (3) scalar theory (4.3) must remain sepa-rated by a quantum phase transition even when the SU (3) flavor symmetry is explicitlybroken. Most importantly, essentially the same argument applies to dense QCD.Before leaving this section, we note that one may consider our original 3D model(2.4), or the 4D non-Abelian generalization (4.3), with the addition of a non-zero chemi-cal potential for the U (1) G symmetry. Such a chemical potential explicitly breaks chargeconjugation symmetry, just like the baryon chemical potential in dense QCD. In ourearlier discussion we used unbroken charge conjugation symmetry to conclude that theground state expectation value of the holonomy must be real. But, as noted in footnote10, for a reflection-symmetric holonomy contour (such as a circle), reflection symme-try is an equally good substitute. Consequently, all of our arguments demonstratingthat the phase of the holonomy encircling a vortex at large distance serves as an orderparameter distinguishing “confining” and “Higgs” superfluid phases go through withoutmodification in the presence of a non-zero chemical potential.In summary, we have shown that consideration of our new order parameter impliesthat there is a phase transition between nuclear matter and quark matter in denseQCD near the SU (3) flavor limit. This means that the confining nuclear matter regimeof QCD (at least with approximate SU (3) flavor symmetry) has a sharp definition asa phase of QCD where the expectation values of color holonomies around superfluidvortices are positive, while quark matter — a Higgs regime — can be defined as thephase of QCD where these holonomy expectation values become complex. Given thenotorious difficulties in giving a sharp definition for confining and Higgs regimes in is described in Appendix B. A different example which is closer to the model discussed in this sectionconsists of a version of the theory (4.3) in three spacetime dimensions, with gauge group SU (2) andtwo flavors of SU (2) antifundamental scalar fields, with a global flavor symmetry containing an SU (2) factor. Generalizing the analysis in Sec. 3.4 to this non-Abelian model, we have checked that thereis a set of parameters (essentially identical to the ones in Sec. 3.4) for which the phase transitionbetween the U (1) G -broken confining and Higgs regimes is strongly first-order as a function of themass of the antifundamental scalars. The fact that the transition is strongly first-order allows theexistence of the phase transition to be reliably established despite the fact that the gauge sector isstrongly coupled within the U (1) G -broken confining phase. It is easy to check in this example that ourvortex observable O Ω jumps from +1 to − across the transition, and serves as an order parameterdistinguishing distinct phases, even when the transition is no longer strongly first-order. Finally, it iseasy to check that these statements generalize to N = N f > gauge theories. – 42 –auge theories with fundamental representation matter (see Ref. [51] for a review),this is a satisfying result in the theory of strong interactions. Our results are alsoencouraging for observational searches for evidence of quark matter cores in neutronstars, see e.g. Refs. [54–68], because our results imply that hadronic matter and quarkmatter must be separated by a phase transition as a function of density. We have explored the phase structure of gauge theories with fundamental representa-tion matter fields and a U (1) global symmetry. Motivated by the physics of dense QCD,we considered both Higgs and confining portions of the phase diagram in which the U (1) global symmetry is spontaneously broken, and hence the theory is gapless due tothe presence of a Nambu-Goldstone boson. These two regimes cannot be distinguishedby conventional local order parameters probing global symmetry realizations, nor dothey naturally fit into more modern classification schemes based on topological orderand related concepts. Nevertheless, using a novel vortex order parameter introducedin Sec. 3, we found that U (1) -broken confining and Higgs regimes are sharply distinctphases of matter necessarily separated by at least one phase transition in parameterspace, as illustrated in Fig. 1. In Secs. 2 and 3 (and Appendix B) we examined in-structive parity-invariant Abelian (and non-Abelian) gauge theories in three spacetimedimensions illustrating this physics. Then in Sec. 4 we considered related theories witha U (1) global symmetry in four spacetime dimensions, and explained how our consid-erations serve to rule out the Schäfer-Wilczek conjecture of quark-hadron continuity incold dense QCD.Why are these results interesting? First, we have added to the toolkit of techniquesfor diagnosing phase transitions in gauge theories, and shown that it predicts previ-ously unexpected phase transitions in theories with fundamental representation matterfields. Second, our analysis implies a phase transition between quark matter and nu-clear matter in dense QCD near the SU (3) flavor limit, with possible implications forobservable properties of neutron stars. Third, our analysis provides a sharp distinctionbetween a confined nuclear matter regime of QCD and dense quark matter. In otherwords, it provides sharp answers to some basic questions about strong dynamics: • “What is the confined phase of QCD?” Our work shows that this question has asharp answer when the U (1) B baryon number symmetry is spontaneously broken.The confined phase of QCD with spontaneously broken U (1) B symmetry can bedefined as the phase of QCD where the expectation values of color holonomiesaround minimal-circulation superfluid vortices are positive.– 43 – “What is cold quark matter?” Our analysis shows that cold quark matter can bedefined as the phase of QCD where the expectation values of color holonomiesaround minimal-circulation superfluid vortices have non-vanishing phases.Our results raise a number of other interesting questions that we hope can be addressedin future work. These include: • What is the nature of the point in Fig. 1 where the three different phase transitioncurves intersect? • What can be said in general about the order of the phase transition(s) separating U (1) -broken confining and Higgs phases in the theories we have considered? Asdiscussed in Sec. 3.4, for some ranges of parameters there is a single first orderphase transition. Is this always the case, or is there a range of parameters wherethe transition becomes second-order? How does the answer depend on the space-time dimension? These issues are of more than just theoretical interest, becausethe properties of the nuclear to quark matter phase transition(s) in dense QCDcan have observational impacts for the physics of neutron stars. • Relatedly, when the transition is first order what is the physics on an interfaceseparating coexisting phases? This is also directly connected to potential neutronstar phenomenology. • What happens to the phase structure of the class of theories we have considered,in both three and four spacetime dimensions, at non-zero temperature? • How should the modern classification of the phases of matter be generalized whenconsidering transitions between gapless regimes? Is there a natural embeddingof the constructions in this paper into some more general framework? In Ap-pendix C we gauge the U (1) G symmetry of our 3D Abelian model and showthat the resulting gapped theory (which flows to TQFTs at long distances) has aphase transition analogous to the Higgs-confinement phase transition studied inthe body in the paper. But we also argue that, by itself, this cannot be used toinfer the existence of a phase transition in the original model with a global U (1) G symmetry. • Can our construction be generalized to gauge theories where the U (1) G globalsymmetry is explicitly broken to a discrete subgroup Z k ? Such theories wouldcontain domain walls, and the behavior of gauge field holonomies around domainwall junctions could be used to identify phase transitions.– 44 – Are there condensed matter systems which realize the physics of U (1) -brokenHiggs-confinement phase transitions? Acknowledgments
We are especially grateful to Fiona Burnell for extensive discussions and collabora-tion at the initial stages of this project. We are also grateful to F. Benini, S. Ben-venuti, K.S. Damle, L. Fidkowski, D. Harlow, Z. Komargodski, S. Minwalla, E. Pop-pitz, Y. Tanizaki and M. Ünsal for helpful discussions and suggestions during the longgestation of this paper. AC acknowledges support from the University of Minnesota.TJ is supported by a UMN CSE Fellowship. SS acknowledges the support of IowaState University startup funds. LY acknowledges support from the U.S. Departmentof Energy grant DE-SC0011637.
A Higgs phase vortex profile functions
Recall that the vortex configuration in our 3D model has the form given by Eqs. (3.5)and (3.8), repeated here: φ = v f ( r ) e ikθ , φ + = v c f + ( r ) e i ( n − k ) θ , φ − = v c f − ( r ) e − inθ , A θ = Φ h ( r )2 πr , (A.1)with the radial profile functions h , f ± , and f all approaching 1 as r → ∞ . Theequation of motion for the gauge field profile h ( r ) is Φ2 π (cid:18) d hdr − r dhdr (cid:19) = − e v c (cid:20)(cid:18) n − k − Φ h π (cid:19) f + (cid:18) n − Φ h π (cid:19) f − (cid:21) , (A.2)while the scalar field profile functions obey (cid:34) d f + dr + 1 r df + dr − f + r (cid:18) n − k − Φ h π (cid:19) (cid:35) = −| m c | f + + 2 λ c v c f − (cid:15)v f − f , (A.3a) (cid:34) d f − dr + 1 r df − dr − f − r (cid:18) n − Φ h π (cid:19) (cid:35) = −| m c | f − + 2 λ c v c f − − (cid:15)v f + f , (A.3b) (cid:20) d f dr + 1 r df dr − f r ( k ) (cid:21) = −| m | f + 2 λ v f − (cid:15) v c v f + f − . (A.3c)As discussed in section 3.2 (cf. Eq. (3.10)) the minimal energy solution has Φ = (2 n − k ) π . Inserting this value and examining the resulting large r asymptotic behavior of the– 45 –rofile functions, one finds that h ( r ) equals 1 up to exponentially falling corrections.This will be demonstrated below. Neglecting such exponentially small terms, the scalarprofile functions satisfy (cid:20) d f ± dr + 1 r df ± dr − k f ± r (cid:21) = −| m c | f ± + 2 λ c v c f ± − (cid:15)v f ∓ f , (A.4a) (cid:20) d f dr + 1 r df dr − k f r (cid:21) = −| m | f + 2 λ v f − (cid:15) v c v f + f − . (A.4b)Demanding that the scalar profile functions { f i } approach 1 as r → ∞ and requiringthat the resulting right-hand sides of Eq. (A.4) vanish determines the condensate mag-nitudes v and v c . One finds v c = ( | m c | + (cid:15)v ) / (2 λ c ) with v the positive solution ofthe cubic equation λ λ c v − (2 λ c | m | + (cid:15) ) v − (cid:15) | m c | = 0 . (A.5)One may then verify that the resulting scalar profile functions have the asymptoticforms f ± ( r ) = 1 − (cid:96) c r + O ( r − ) , f ( r ) = 1 − (cid:96) r + O ( r − ) , (A.6)with (cid:96) c = k v c λ v − | m | + 4 (cid:15)v λ c (6 λ v − | m | ) − (cid:15) , (cid:96) = k v (cid:15) + 8 λ c v λ c (6 λ v − | m | ) − (cid:15) . (A.7)The large r power-law tails in scalar field profile functions are characteristic featuresof global vortices. But since we are in a Higgs phase, the gauge field should haveexponential fall-off. To verify this, we parallel the treatment of Ref. [33] and rewritethe coupled equations in terms of sums and differences of the charged field profiles. Let h = 1 + H , f + = 1 + F + G , (A.8a) f = 1 + F , f − = 1 + F − G , (A.8b)and then linearize the field equations in the deviations F , F , G and H . The leadingbehavior of F and F can be read off from Eq. (A.6). The linearized equations for G and H do not involve F or F and read (cid:20) d dr + 1 r ddr − (cid:101) m − k r (cid:21) G = k (2 n − k ) r H , (A.9a) (2 n − k ) (cid:20) d dr − r ddr − m A (cid:21) H = k m A G , (A.9b)– 46 –here m A ≡ e v c and (cid:101) m ≡ λ c v c + 2 (cid:15)v . Taking, for simplicity, k = n = 1 , one maycheck that the two independent homogeneous solutions have the asymptotic forms G I ( r ) ∼ m A ( m A r ) − / e − m A r , H I ( r ) ∼ ( (cid:101) m − m A ) ( m A r ) / e − m A r , (A.10a) G II ( r ) ∼ ( (cid:101) m − m A ) ( (cid:101) mr ) − / e − (cid:101) mr , H II ( r ) ∼ m A ( (cid:101) mr ) − / e − (cid:101) mr . (A.10b)The most general solution is a linear combination of solutions I and II. Depending onwhether m A or (cid:101) m is smaller, either solution I or solution II dominates at large distance.In either case, the gauge field profile function approaches its asymptotic value far fromthe vortex exponentially fast. B Embedding in a non-Abelian gauge theory
Consider a parity-invariant SU (2) gauge theory containing one real adjoint represen-tation scalar field ζ , one fundamental representation scalar Φ , and one SU (2) -singletcomplex scalar field φ . We take the action of the theory to be S = (cid:90) d x (cid:104) g tr F µν + | D µ Φ | + tr ( D µ ζ ) + | ∂ µ φ | + m | Φ | + m | φ | + λ Φ | Φ | + λ | φ | + λ ζ tr (cid:0) ζ − v ζ (cid:1) + ε (cid:0) φ Φ T iσ ζ Φ + h.c. (cid:1) + · · · (cid:105) , (B.1)where the covariant derivatives D µ Φ ≡ ∂ µ Φ − i A µ Φ and D µ ζ ≡ ∂ µ ζ − i [ A µ , ζ ] , thegauge field A µ ≡ A aµ t a , and the SU (2) generators obey tr t a t b = δ ab . The couplings ε , λ Φ , λ , and λ ζ are assumed real and positive, and the ellipsis stands for further scalarpotential terms consistent with the symmetries imposed below.In addition to parity (and gauge and Euclidean invariance), we assume the theoryhas a U (1) G global symmetry acting as U (1) G : Φ → e − iα Φ , φ → e iα φ . (B.2)The Z subgroup generated by the α = π sign flip is part of the SU (2) gauge symmetry,so the faithfully-acting global symmetry is U (1) G / Z . We also assume the existence ofa discrete global symmetry we will call ( Z ) F , acting as ( Z ) F : ζ Φ A µ → − U ζ U † − iU Φ U A µ U † , (B.3)where U ≡ ( ii ) ∈ SU (2) . This transformation leaves the action (B.1) invariant andacts as a Z symmetry on gauge invariant observables.– 47 –e consider this model when v ζ (cid:29) g , leading to Higgsing of the SU (2) gaugegroup down to a U (1) Cartan subgroup. Choosing, for convenience, a gauge where ζ = v ζ σ / , one sees that the “color” components A and A become massive while A µ remains massless. Writing Φ ≡ (cid:0) φ + φ − (cid:1) , the component fields φ ± transform with charge ± / under the unbroken U (1) gauge group. To write the resulting low energy theory,below the scale m W ≡ gv ζ , in the most convenient form let e ≡ g/ and A µ ≡ A µ .This makes A µ an Abelian gauge field with coupling e interacting with fields φ ± havingcharges ± . The final term in the action (B.1) with coefficient ε becomes ε (cid:0) φ Φ T iσ ζ Φ + h.c. (cid:1) = ε φ ( φ + , φ − ) (cid:18) − v ζ / − v ζ / (cid:19) (cid:18) φ + φ − (cid:19) + h.c. = − εv ζ ( φ φ + φ − + h.c. ) , (B.4)after setting ζ to its expectation value. If one now identifies (cid:15) ≡ εv ζ and m c ≡ m ,then the resulting low-energy description of this non-Abelian model, now involving thefields φ ± , φ , and A µ , precisely coincides with our original Abelian model (2.4).Now consider the behavior of the non-Abelian model (B.1) as the mass parameters m and m are varied. To begin, suppose that m (cid:29) g v ζ , so that the fundamentalscalar field Φ is not condensed. If m is sufficiently negative, the neutral scalar φ willcondense and the U (1) G symmetry will be spontaneously broken; otherwise U (1) G willbe unbroken. In either case, the SU (2) adjoint Higgs mechanism leads to the existenceof stable finite-action monopole-instantons [95, 96] whose stability is guaranteed by π ( SU (2) /U (1)) = Z . The Abelian magnetic flux (defined on scales large compared to m − W ) through a spacetime surface M has an SU (2) gauge-invariant definition Φ B ( M ) ≡ | v ζ | (cid:90) M tr ( ζ F ) , (B.5)where F is the two-form field strength. We have normalized the flux Φ B (not to beconfused with the field Φ ) so that when written in terms of the 2-form field strength F of the Abelian gauge field A µ , the flux has the conventional form Φ B = (cid:82) M F . Notethat Φ B is odd under the ( Z ) F global symmetry. The minimal magnetic monopole-instantons in SU (2) gauge theory have Φ B ( S ) = ± π , where S is a spacetime two-sphere surrounding the center of the monopole-instanton. If the center of the monopoleis at r = 0 , then as r → ∞ , then at large distance from the monopole the SU (2) gaugefield and adjoint scalar approaches the asymptotic forms ( A µ ) a → (cid:15) aµν ˆ r ν r , ζ a → v ζ ˆ r a , (B.6)– 48 –n “hedgehog” gauge, with ˆ r µ a radial unit vector. The action of a monopole-instantonhas the form S I = 4 πv ζ g f (cid:18) λ ζ g (cid:19) , (B.7)where the dimensionless and monotonically increasing function f varies between f (0) =1 and f ( ∞ ) = 1 . [97–99]. The associated monopole operator, characterizing theeffect of a monopole on long distance physics, has the form e iσ e − S I , with σ the mag-netic dual of the Abelian field strength F . As discussed in section 2, these monopole-instantons generate a potential of the form e − S I cos( σ ) for the dual photon of A µ , leadingto a mass gap for the low energy Abelian gauge field and confinement of heavy testcharges.Alternatively, if − m (cid:29) g then the fundamental representation scalar Φ willcondense, leading to complete Higgsing of the SU (2) gauge symmetry, along withspontaneous breaking of the global U (1) G symmetry, regardless of the value of m .Monopole-instantons are now confined by magnetic flux tubes, and have negligible ef-fect on long distance physics. All components of the SU (2) gauge field acquire massvia the Higgs mechanism.The question remains: are the confining (via Polyakov mechanism) and fully Hig-gsed regimes, both with spontaneous U (1) G -breaking, smoothly connected? All of theanalysis of section 3 generalizes in a straightforward fashion to this non-Abelian model,and shows that the answer is no. To see this, one may consider the natural general-ization of our previous vortex holonomy observable (3.3) which replaces the Abeliangauge field holonomy with the trace of the non-Abelian holonomy, O SU (2)Ω ≡ lim r →∞ (cid:104) tr Ω( C ) (cid:105) (cid:104) tr Ω( C ) (cid:105) . (B.8)Just as in our original Abelian model, the holonomy expectation values in numeratorand denominator will have perimeter law decay of their magnitudes, but this size de-pendence cancels in the ratio by construction. The denominator is guaranteed to bepositive in weakly coupled regimes because charge conjugation (or reflection) symmetryrequires it to be real, it is positive at tree level, and hence small quantum corrections Alternatively, one might consider writing the Abelian holonomy as the exponential of the magneticflux through a surface spanning the holonomy contour, and then insert the definition (B.5) of theAbelian flux in terms of the underlying non-Abelian field strength. However, this generalization isundesirable as it converts the original line operator into a surface operator, for which one can no longerargue that the magnitude of the expectation value, in the large r limit, must be independent of thepresence of a vortex piercing the surface. With this generalization, the ratio of vortex and ordinaryexpectation values of the surface operator need not be a pure phase. – 49 –annot turn it negative. So the ratio of expectations is determined by the phase of thevortex expectation value in the numerator.One can easily uplift the entirety of the analysis in section 3 to this non-Abeliansetting. In the Higgs phase, a unit-winding vortex configuration has the form Φ( r, θ ) = v Φ (cid:18) f + ( r ) e i ( n − θ f − ( r ) e − inθ (cid:19) , φ ( r, θ ) = v f ( r ) e iθ , A θ ( r ) = a h ( r )2 πr σ . (B.9)The resulting long-distance energy density, generalizing Eq. (3.9), is E ( r ) = v r (cid:20)(cid:16) n − − a π (cid:17) + (cid:16) − n + a π (cid:17) (cid:21) + v k r + O ( r − ) . (B.10)The minimum lies at a = 2 π (2 n − , leading to the tree-level result, (cid:104) tr Ω( C ) (cid:105) = − , (B.11)and a phase of (cid:104) tr Ω( C ) (cid:105) , at long distance, equal to π .To see that the phase of (cid:104) tr Ω( C ) (cid:105) must remain at π even when quantum fluc-tuations are taken into account, one can adapt the effective field theory argument atthe end of Sec. 3.2. Integrating out fluctuations generates corrections to the tree-leveleffective action. The only terms in the effective action that can affect expectationvalue of holonomies along contours far from the vortex core are those with exactly twoderivatives acting on Φ , because only such operators can affect the O (1 /r ) holonomy-dependent part of the energy density. Given the symmetries of our SU (2) model, allsuch terms may be written in the form S eff , SU (2) holonomy = (cid:90) d x (cid:26) D µ Φ † f ( ζ, Φ , φ ) D µ Φ+ (cid:2) φ ( D µ Φ) T iσ ζf ( ζ, Φ , φ ) D µ Φ + h.c. (cid:3) (cid:27) . (B.12)The functions f and f , depending on the indicated fields but not their derivatives,must be constructed so as to be invariant under the U (1) G symmetry, transform in theadjoint representation of the SU (2) group, and be conjugated by U under the ( Z ) F symmetry (B.3). Just as in Sec. 3.2, one may verify that both terms, in the presence ofa unit-winding vortex, have an extremum at the value a = 2 π (2 n − for the asymptoticcoefficient of the gauge field. Therefore, small quantum corrections cannot shift thephase of the holonomy and we learn that:Higgs phase: O SU (2)Ω = − . (B.13)– 50 –n the other hand, in the U (1) G -broken confining regime when m (cid:29) g , one mayreapply the arguments of Sec. 3.3 to show that: U (1) G -broken confining phase: O SU (2)Ω = 1 . (B.14)So, as claimed, O SU (2)Ω serves as an order parameter that distinguishes the U (1) G -brokenconfining and Higgs phases in this SU (2) gauge theory. C U (1) × U (1) gauge theory and topological order Gauging the global U (1) G symmetry of our model (2.4), by adding a second dynami-cal gauge field minimally coupled to the conserved current associated with the U (1) G symmetry, converts the model into a U (1) × U (1) gauge theory. This process has theeffect of converting the massless Nambu-Goldstone boson associated with spontaneousbreaking of the global U (1) G symmetry into the longitudinal component of a massivegauge field, thereby producing a mass gap in the U (1) × U (1) gauge theory. Super-fluid systems (gapless due to global symmetry breaking) and superconducting systems(gapped due to the Meissner effect) are related in precisely this manner.The U (1) × U (1) gauge theory produced by gauging the U (1) G symmetry of ourmodel (2.4) no longer has any continuous global symmetries and is expected to havea non-vanishing mass gap at generic points within its parameter space. This makes iteasier to analyze than the gapless models considered in the body of the paper. Thelong-distance physics of the U (1) × U (1) gauge theory can be described by topologicalquantum field theories (TQFTs) at generic points in parameter space. The phasediagram of the U (1) × U (1) gauge theory produced by gauging the U (1) G symmetryof our model (2.4) turns out to be very similar to the phase structure of our originalmodel (2.4).Let X and Y denote the gauge fields associated each of the U (1) factors of thegauge group, which we henceforth denote as U (1) X × U (1) Y . Let F X and F Y denotethe corresponding field strengths, and e X and e Y the gauge couplings of the two differentgauge fields. The charge assignments of the scalar fields are: φ + φ − φ U (1) X +1 − U (1) Y − − (C.1) We thank Z. Komargodski for urging us to pursue the calculations described in this appendix. – 51 –e assume the standard magnetic flux quantization condition holds for both F X and F Y , (cid:90) S F X = 2 πk X , (cid:90) S F Y = 2 πk Y , k X , k Y ∈ Z , (C.2)and assume that finite action monopole-instantons preclude the existence of any mag-netic U (1) global symmetries. We also assume that the fundamental representationWilson loop operators Ω X = e i (cid:82) C X and Ω Y = e i (cid:82) C Y are genuine line operators in thesense of Ref. [16].The action is a simple extension of the original model (2.4), S = (cid:90) d x (cid:20) e X F X + 14 e Y F Y + | Dφ + | + | Dφ − | + | Dφ | − (cid:15) ( φ + φ − φ + h.c. )+ m | φ + | + m − | φ − | + m | φ | + λ + | φ + | + λ − | φ − | + λ | φ | + · · · + V m ( σ X ) + V m ( σ Y ) (cid:21) . (C.3)The V m ( σ X ) and V m ( σ Y ) terms describe the effects of monopole-instantons for the X and Y gauge fields, respectively. The cubic (cid:15) term ensures that there is no global U (1) symmetry despite the presence of three charged scalar fields and only two gauge bosons.Our model enjoys Euclidean (or Lorentz) invariance, including reflection and time-reversal symmetry. We do not assume any discrete flavor symmetry permuting thedifferent scalar fields, nor any symmetry interchanging the two U (1) subgroups. Giventhe charge assignments (C.1), the Z transformation ( − , − ∈ U (1) X × U (1) Y actstrivially on all three scalar fields φ + , φ − and φ . Consequently, the theory has a 1-formsymmetry, which we denote by ( Z ) (1) XY , which acts on topologically non-trivial Wilsonloops as: ( Z ) (1) XY : Ω X → − Ω X , Ω Y → − Ω Y . (C.4)Now consider the resulting phase diagram. When all three scalars have large posi-tive masses, they can be integrated out resulting in a pure U (1) X × U (1) Y gauge theory,which has a mass gap and a unique vacuum thanks to the Polyakov mechanism. If onlyone of the fields φ ± is condensed, then there is again a mass gap and a unique vacuumthanks to a combination of the Higgs and Polyakov mechanisms. Other portions of thephase diagram can be mapped out by considering: (i) the regime where φ is condensedbut the φ ± fields are not condensed, and (ii) the regime where all three scalar fields, A nearly identical model to (C.3) with a Z flavor permutation symmetry was studied in Ref. [100]. – 52 – ± and φ , are condensed. So long as the cubic coupling (cid:15) (cid:54) = 0 , there is no separateregime where the fields φ + and φ − are condensed, but φ is not. Nor is there a regimewhere, e.g., φ and φ + are condensed but φ − is not. φ condensed phase Suppose that the φ ± fields have large positive masses, so that they may be integratedout. The U (1) X gauge field will be gapped, as usual, thanks to the Polyakov mechanism.Condensation of φ causes Higgsing of the U (1) Y gauge field, showing that this regimeis (generically) gapped. Let us call the regime where only the φ field is condensed theY regime. Despite the fact that it is gapped, the Y regime is not become completelytrivial in the deep infrared because the emergent gauge group at long distances is Z ,and the resultant physics is described by a non-trivial topological quantum field theory(TQFT). Before discussing the TQFT description, let us consider the physics of the systemthrough a more direct approach. After integrating out φ + and φ − , the resulting effectiveaction is S eff = (cid:90) d x (cid:20) e X F X + 14 e Y F Y + | Dφ | + V ( | φ | )+ c X m | φ | F X + c Y m | φ | F Y + b X m S µν F µνX + b Y m S µν F µνY + · · · (cid:21) , (C.5)where m = min( m + , m − ) , D µ ≡ ∂ µ − iY µ , and S µν ≡ i [( D µ φ )( D ν φ † ) − ( D ν φ )( D µ φ † )] .A minimal vortex configuration has the usual form, written as φ ( r, θ ) = v f ( r ) e iθ , Y θ ( r ) = Φ Y h ( r ) / (2 πr ) , (C.6)where v is the φ vacuum expectation value and the radial functions f and h inter-polate between 0 and 1 as r goes from 0 to ∞ . The asymptotic gauge field coefficient Φ Y is determined by minimizing the long-distance energy density, E ( r ) = v r (cid:18) − Φ Y π (cid:19) + O ( r − ) , (C.7)which must vanish to prevent a logarithmic IR divergence in the vortex energy, implyingthat Φ Y = π . One can prove that small quantum corrections cannot shift Φ Y awayfrom this value by using the effective field theory analysis in 3.2. Hence, if C is a large See, e.g., Refs [90, 101]. Reference [17] explains that the TQFT associated with a Z gauge theorycan be viewed as an effective field theory describing a spontaneously broken Z Z symmetry acts by flipping the sign of Ω Y . – 53 –ircular contour centered on the vortex, then the holonomy Ω Y ( C ) has a phase of π .More physically, this means that a test particle with unit charge under U (1) Y picks upa phase of π when it moves around a unit-circulation vortex.Next let us consider the behavior of X holonomies. Consider a test particle withunit charge under U (1) X and zero charge under U (1) Y . What is the phase acquiredby such a test particle when it encircles the φ vortex? The answer is not immediatelyobvious when the coupling b X is non-zero. (A non-zero value for b X may appear in theabsence of any flavor permutation symmetry which also flips the sign of F X .) In thepresence of a winding- k vortex, the antisymmetric tensor S µν is non-vanishing with S rθ = f ( r ) f (cid:48) ( r ) r (cid:20) k − Φ Y π h ( r ) (cid:21) . (C.8)When k = 1 , we know that Φ Y = π . When b X (cid:54) = 0 , the source (C.8) corresponds to anazimuthal J X current encircling the vortex which, in turn, generates a magnetic field B X = (cid:15) ij F ijX localized on the vortex core. Just as in Sec. 3.5, due to confinement, theextent to which this matters depends on size of the loop with which one probes thesystem. Consider a spatial disk D with a φ vortex at its center and the Ω X holonomycalculated along the boundary of D . When the radius of D is small compared to thestring breaking scale, the magnetic flux through D is non-zero, and O Ω X = e i Φ with Φ ∝ b X . But for holonomies on large contours (with radius r (cid:29) L br ) , string breakingeffects remove the sensitivity to b X , and we find O Ω X = 1 . So the expectation valuesof X holonomies that encircle φ vortices on contours C = ∂D are positive in the limitof large contour radius.The information about holonomies around vortices is encoded into the TQFT de-scription of the infrared limit of the system. The action for the topological field theorydescribing the Y regime, which we denote as TQFT Y , can be written using the K -matrix formalism [89, 102–107]. In the regime we are considering, φ vortex excitationscost finite energy and can be viewed as one type of probe excitation, while test particleswith unit charge under U (1) Y are another. Let J Y denote the conserved U (1) Y current,and J V the topologically conserved vortex number current. These currents couple totwo different one-form gauge fields, a i , with i = 1 , , each obeying (cid:82) M da i ∈ π Z forany closed 2-surface M . Physically, we can identify a µ = Y µ , while a µ arises in thederivation of the TQFT description as a Lagrange multiplier that enforces the condi-tion that almost everywhere (cid:15) µνρ ∂ ν ∂ ρ φ = 0 while allowing (cid:72) dx µ ∂ µ φ ∈ π Z . Then adescription of this Z TQFT is provided by the action S TQFT Y = (cid:90) d x (cid:20) i π ( K Y ) ij (cid:15) µνρ a iµ ∂ ν a jρ + a µ J µY + a µ J µV (cid:21) , (C.9)– 54 –here the K -matrix and its inverse (times π ) are given by K Y = (cid:18) (cid:19) , πK − Y = (cid:18) ππ (cid:19) . (C.10)The matrix element (2 πK − Y ) ij gives the phase that an excitation of type i picks upunder braiding around one of type j . This TQFT has | det K Y | g = 4 g ground states oncompact spatial manifolds of genus g . Note that the X gauge field does not appear inthe TQFT description at all. In this way the TQFT (C.9) is implicitly consistent withthe above result that all X holonomies have trivial phases in the long-distance limit. φ , φ + , φ − condensed phase Now consider the regime where all three scalar fields φ , φ + , and φ − are condensed,and the X and Y gauge fields are both Higgsed. We will call this the XY regime. Theinfrared physics of the XY regime be described by a topological field theory which wedenote as TQFT XY .Before discussing the TQFT description, it is again useful to explore the physics ofvortices directly. Consider the weakly coupled corner of the parameter space of the XYregime, and suppose that there is a vortex where φ + winds by πn + and φ − winds by πn − on contours encircling the vortex core. Such a field configuration has the form: φ + ( r, θ ) = v + f + ( r ) e in + θ , X θ ( r ) = Φ X g ( r ) / (2 πr ) , (C.11a) φ − ( r, θ ) = v − f − ( r ) e in − θ , Y θ ( r ) = Φ Y h ( r ) / (2 πr ) , (C.11b) φ ( r, θ ) = v f ( r ) e − i ( n + + n − ) θ . (C.11c)The radial functions f , g and h approach as r → ∞ . The long-distance energy densityof this vortex configuration is E ( r ) = 1 r (cid:20) v (cid:16) n + − Φ X − Φ Y π (cid:17) + v − (cid:16) n − + Φ X +Φ Y π (cid:17) + v (cid:16) n + + n − + Φ Y π (cid:17) (cid:21) + O ( r − ) (C.12)and, for given values of n + and n − , E ( r ) is minimized when the X and Y magneticfluxes Φ X , Φ Y are Φ X = ( n + − n − ) π , Φ Y = − ( n + + n − ) π . (C.13)Since the winding numbers n ± are integers, these two fluxes are identical modulo π .The vortices with minimal winding and minimal magnetic flux correspond to: n + n − Φ X Φ Y V + π − πV − − π − π (C.14)– 55 –he result (C.14) shows that test particles with unit charge under U (1) Y pick up aphase of − π when encircling either V + or V − , while test particles with unit chargeunder U (1) X pick up a phase of ± π when encircling a V ± vortex.The discussion in the body of this paper implies that the XY and Y regimes cannotbe smoothly connected. Both phases contain vortices with flux Φ Y = π mod π , but inthe XY regime any vortices that have Φ Y = π mod π also have Φ X = π mod π . Bycomparison, vortices in the Y regime only carry Y flux. The X flux carried by vorticeschanges non-analytically as we go from one regime to the other. We can repeat thelogic in Sec. 3.4 to argue that such changes are associated with non-analyticities inthermodynamic observables, showing that the Y and XY regimes are distinct phasesof matter. The phase transition between the Y and XY regimes is the parallel of theconfinement to Higgs phase transition discussed in the main part of this paper.We now consider the TQFT description of the long-distance physics of the XYregime. Given the charge assignments (C.1), the unbroken part of the gauge group inthe XY regime is generated by the Z transformation ( − , − ∈ U (1) X × U (1) Y , andthe 1-form symmetry ( Z ) (1) XY , acting as shown in Eq. (C.4), is spontaneously broken.We can derive the appropriate TQFT describing this regime directly from the originalmodel (C.3) in the weakly coupled corner of parameter space of the XY regime. Byvirtue of being topological, the resulting effective action will furnish a valid descriptionof the physics even away from the weak coupling limit. We follow a procedure similar tothat in Sec. 3.3 of Ref. [90]. In the weak coupling and long distance limits, we can freezethe moduli of the scalar fields to their vacuum expectation values because all physicalfluctuation modes around the expectation values are gapped and can be integrated out.Let us denote the phases of the three scalar fields by ϕ , ϕ + , and ϕ − . Since we areinterested in the low energy form of the effective action, we note that minimizing thecubic (cid:15) term in model (C.3) implies that ϕ + ϕ + + ϕ − = 0 . With all this taken intoaccount, the relevant part of the effective action becomes just L Stückelberg = v ( ∂ µ ϕ + − X µ + Y µ ) + v − ( ∂ µ ϕ − + X µ + Y µ ) + v ( ∂ µ ϕ + + ∂ µ ϕ − +2 Y µ ) , (C.15)where v , v + , and v − are the magnitudes of the expectation values of φ , φ + , and φ − ,respectively. We have dropped the Maxwell terms because we are interested in lengthscales which are large compared to /e X and /e Y . The long distance TQFT is obtainedby dualizing ϕ + and ϕ − and taking the low energy limit v , v + , v − → ∞ . To this end,we introduce one-form Lagrange multiplier fields a + and a − satisfying (cid:82) M da ± ∈ π Z for any closed 2-manifold M . Then the Lagrangian, L dual = L Stückelberg + i π (cid:15) µνα a + µ ∂ ν ∂ α ϕ + + i π (cid:15) µνα a − µ ∂ ν ∂ α ϕ − , (C.16)– 56 –nforces (cid:15) µνρ ∂ ν ∂ ρ ϕ ± = 0 almost everywhere while allowing (cid:72) dx µ ∂ µ ϕ ± ∈ π Z . Usingthe resulting equations of motion one finds ∂ µ ϕ ± = ± X µ − Y µ − i π ( v ∓ + v ) (cid:15) µνρ ∂ ν a ± ρ − v (cid:15) µνρ ∂ ν a ∓ ρ v v + v v − + v v − , (C.17)and L dual = i π (cid:15) µνρ (cid:2) ( X µ − Y µ ) ∂ ν a + ρ − ( X µ + Y µ ) ∂ ν a − ρ (cid:3) + O ( v − i ) . (C.18)This Lagrangian (when the v i → ∞ ) describes a topological field theory, TQFT XY .To write the result in a more useful form, denote the set of gauge fields by { a iµ } = { X µ , Y µ , a + µ , a − µ } . We also introduce a set of currents { J iµ } , which are respectivelythe U (1) X and U (1) Y Noether currents and the topological vortex number currentsassociated with φ + and φ − vortices. Then the action for TQFT XY can be written as S TQFT XY = (cid:90) d x (cid:20) i π ( K XY ) ij (cid:15) µνρ a iµ ∂ ν a jρ + a iµ ( J µ ) i (cid:21) , (C.19)with K -matrix (and its inverse) K XY = −
10 0 − − − − − , πK − XY = π − π − π − ππ − π − π − π . (C.20)This TQFT has | det K XY | g = 4 g ground states on spatial manifolds of genus g inaccordance with expectations from the spontaneously broken ( Z ) (1) XY symmetry.It is possible to find an interesting relation between TQFT Y and TQFT XY . Sup-pose we add a new spectator field χ which has charge +1 under a new gauge field Z µ ,obeying standard flux quantization conditions, and assume that the microscopic theoryadmits finite action Z monopole events. We further suppose that χ is condensed in theY regime, and has a large positive mass squared in the XY regime. Then the K -matrixof TQFT XY remains unchanged, but the K -matrix of the Y regime is enlarged andbecomes ˜ K Y = . (C.21)– 57 –he action of the Y regime is now (cid:82) d x (cid:104) i π (cid:15) µνρ a iµ ( ˜ K Y ) ij ∂ ν a jρ + a iµ J µi (cid:105) , with the setof gauge fields { a iµ } = { Y µ , a µ , Z µ , a χµ } . Here a µ and a χµ are auxiliary gauge fields thatcouple to the φ and χ vortex currents, respectively. Then one can verify the congruencerelation G T K XY G = ˜ K Y where G ∈ GL (4 , Z ) is the matrix G = − − − . (C.22)This shows that the extended set of gauge fields in the Y regime are related to thosein the XY regime by the change of basis a Y = G a XY , or explicitly Ya Za χ → − X − a + − X − Ya + + a − . (C.23)We emphasize that the existence of the relation (C.23) does not contradict our assertionabove that the Y and XY regimes cannot be smoothly connected and must be separatedby a phase boundary. Phase transitions and un-gauging limits
Having just seen the U (1) × U (1) gauge theory has distinct gapped phases which arenecessarily separated by phase transitions, in a completely parallel fashion with what weinferred by direct calculations in the original U (1) gauge theory with gapless phases, itis natural to ask whether our direct study in the original U (1) gauge theory was reallynecessary. In other words, can one presume that distinct phases present after oneweakly gauges a continuous global symmetry survive the limit of sending the couplingof the artificially introduced gauge field back to zero? Alternatively, if two regimescan be smoothly connected in a given theory with a dynamical gauge field, does thisnecessarily remain true in the un-gauging limit? We argue that the answer to both ofthese questions is no.For systems with discrete symmetry groups, it has been established that phasetransitions detected by changes in particle-vortex statistics in gauged models implyphase transitions in the parallel un-gauged models, see e.g., Refs. [108–110]. It may be For an analogous example, recall that the high and low temperature regimes of the 2D Isingmodel on a square lattice are related by Kramers-Wannier duality. Nevertheless, the high and lowtemperature regimes are distinct phases separated by a phase transition at the self-dual point. – 58 –empting to assume that the same should be true with continuous symmetries, and insome simple examples this parallel between phases in gauged and un-gauged modelsdoes hold. For instance, the phase structure of a theory of a single parity-invariantcomplex scalar field in three spacetime dimensions with a U (1) global symmetry doesnot change when the U (1) symmetry is gauged (provided there are no monopoles),thanks to particle-vortex duality [111, 112]. Naively one might take this example aspart of a general pattern, and guess that phase transitions in the U (1) X × U (1) Y gaugetheory necessarily imply phase transitions in U (1) X gauge theory obtained via the“un-gauging” limit e Y → .But it is not correct to presume, in general, that the phase structure of a theorywith a continuous global symmetry must be identical to that of the gauged versionof the theory. It is quite possible that as the gauge coupling is sent to zero, a phaseboundary appears between two regimes which were smoothly connected in the gaugedmodel. Similarly, non-analyticities present in thermodynamic functions of the gaugedmodel may disappear when the gauge coupling is sent to zero. It is easy to find examples illustrating the above scenarios. First, consider againa compact U (1) gauge theory with a single complex scalar with charge +1 in threespacetime dimensions. Unlike the discussion above, suppose we specify a UV completionof the theory that does admit finite-action monopole-instanton events with minimalmagnetic flux. Then the Higgs and confining regimes of the gauge theory are smoothlyconnected. However, if we un-gauge the U (1) symmetry, we are left with the XY modelin 3D, which has U (1) global symmetry. The U (1) symmetry broken and unbrokenregimes of the XY theory are the limits of the Higgs and confining regimes of the gaugetheory. But these regimes are separated by a phase boundary in the 3D XY model.It is also possible for phase boundaries of a gauge theory to disappear in the un-gauging limit. To see an example of this consider four free massless Dirac fermions in4D spacetime. Such a system has a global symmetry that includes SU (4) L × SU (4) R . Ifwe gauge the vector-like SU (2) V subgroup of this symmetry, introducing a gauge cou-pling g , we obtain two-color two-flavor massless QCD. It has a SU (2) L × SU (2) R globalsymmetry, and an SU (2) A -breaking phase transition as a function of temperature.The critical temperature has a non-perturbative dependence on g due to dimensionaltransmutation. The phase transition exists for any non-zero value of the SU (2) gaugecoupling g , but disappears at g = 0 where the theory becomes free. Another exampleis given by N c N f free massless Dirac fermions in 4D spacetime when one gauges an This subtlety does not arise in the case of gauged discrete symmetries studied in Refs. [108–110], because gauging discrete symmetries introduces neither local degrees of freedom nor continuouscoupling constants. – 59 – U ( N c ) subgroup of the vector-like global symmetry, yielding massless QCD with N f massless quark flavors. In the large N c limit with N f /N c and g N c fixed, the model isknown to go through at least two quantum phase transitions as a function of N f /N c when g N c is fixed a non-zero, see e.g. Refs. [113, 114]. But there are no such phasetransitions at g = 0 .Similar concerns apply to the continuous Abelian gauge theories we focused onin this paper. In the U (1) X × U (1) Y gauge theory (C.3), one can infer the existenceof phase transitions from the behavior of gauge field holonomies whose values becomequantized on large distance scales. Holonomy quantization only holds on distance scaleslarge compared to all relevant length scales. If one is interested in the behavior of thesystem with generic values of its physical parameters, then this is not a problem. Butthe limit of vanishing gauge coupling, e Y → , is a highly non-generic limit. 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