Quark number holonomy and confinement-deconfinement transition
aa r X i v : . [ h e p - ph ] J un YITP-16-13
Quark number holonomy and confinement-deconfinement transition
Kouji Kashiwa ∗ and Akira Ohnishi † Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We propose a new quantity which describes the confinement-deconfinement transition based ontopological properties of QCD. The quantity which we call the quark number holonomy is definedas the integral of the quark number susceptibility along the closed loop of θ where θ is the dimen-sionless imaginary chemical potential. Expected behavior of the quark number holonomy at finitetemperature is discussed and its asymptotic behaviors are shown. PACS numbers: 11.30.Rd, 21.65.Qr, 25.75.Nq
I. INTRODUCTION
Understanding the confinement-deconfinement transi-tion in quantum chromodynamics (QCD) is one of theinteresting and important subjects in nuclear and ele-mentary particle physics. In the heavy quark mass limit,spontaneous breaking of the center ( Z N c ) symmetry isdirectly related to the confinement-deconfinement tran-sition, where N c is the number of color. Then, the holon-omy which is the gauge invariant integral along the closedEuclidean temporal coordinate loop becomes an exactorder-parameter of the confinement-deconfinement tran-sition. It is so called the Polyakov-loop. On the otherhand, we cannot find any exact order-parameters in thecase with dynamical quarks at present, where the directrelation between Z N c symmetry and the confinement-deconfinement transition is lost. Topological order —
The notion of the topological or-der may be of great help in understanding the natureof the confinement-deconfinement transition. Recently,there is an important progress that the confined and de-confined states at zero temperature ( T = 0) are mathe-matically classified based on the topological order [1] inRef. [2]. Motivated by the progress, it has been suggestedthat the confinement-deconfinement transition can be de-scribed by using the analogy of the topological order andthen the free-energy degeneracy plays a crucial role [3].The idea of the topological order is extended to finitetemperature in terms of the Uhlmann phase [4, 5]. TheUhlmann phase is an extension of the Berry phase tomixed quantum states. The Uhlmann phase can describethe topological order at finite T in the one-dimensionalfermion systems such as the topological insulator and thesuperconductor [5]. The Uhlmann phase is defined byusing the amplitude for the density matrix where ampli-tudes form the Hilbert space. There is the U ( n ) gaugefreedom of the amplitude where n is the dimension of thespace and it is a generalization of the U (1) gauge free-dom of pure quantum states. At finite T , the Uhlmannphase includes information of the density matrix of thestatistical mechanics and is calculated by the contour in- ∗ [email protected] † [email protected] tegral along the crystalline momentum. Unfortunately,the calculation of the Uhlmann phase in QCD seems tobe very difficult or impossible at present, Imaginary chemical potential —
In QCD at finite T ,imaginary chemical potential ( µ I ) is an external param-eter, which shows periodicity; Chemical potential µ ap-pears in the form of fugacity e µ/T = e µ R /T e iµ I /T in thefree-energy, and the two states at µ I = 0 and µ I = 2 πT are physically the same. In addition to this 2 πT periodic-ity, characteristic periodicity appears at finite imaginarychemical potential ( µ I ). It is so called the Roberge-Weiss(RW) periodicity [6]. The RW periodicity has deep re-lations with the free-energy degeneracy and thus it isnatural to expect that some hints to understand theconfinement-deconfinement transition are hidden in theimaginary chemical potential region. Quark number holonomy —
We investigate theconfinement-deconfinement transition by using the imag-inary chemical potential in this paper. We discuss thecontour integral along the closed loop of the imaginarychemical potential, µ I = 0 ∼ πT , or the dimensionlessquark imaginary chemical potential, θ = µ I /T = 0 ∼ π .Particularly, we focus on the behavior of the quark num-ber density at finite µ I and propose a new quantity whichcan describe the confinement-deconfinement transitionbased on it. It is a new quantum order-parameter ofthe confinement-deconfinement transition when dynami-cal quarks are acting in the system. We call it the quarknumber holonomy . The quark number holonomy seemsto be a similar quantity with the Uhlmann phase [4, 5].The quark number holonomy defined in Eq. (3) also in-cludes the information of the density matrix via the quarknumber density and is calculated by the contour integralalong the closed loop of θ . It should be noted that thequark number holonomy can be calculated in the effectivemodels of QCD and lattice QCD simulation as discussedlater. It is the most important reason why we propose thenew quantity for the confinement-deconfinement transi-tion in this paper.This paper is organized as following. In the next sec-tion, we propose new quantity which is so called thequark number holonomy. Some discussions for the quarknumber holonomy are shown in Sec. III. Section IV isdevoted to summary. II. QUARK NUMBER HOLONOMY
In this section, we firstly summarize QCD periodic-ities and special transitions which appear at finite µ I .Secondly, we propose a new quantity which describesthe confinement-deconfinement transition based on QCDproperties at finite µ I . Finally, the infinite T and the in-finite bare quark mass ( m ) limit are discussed. A. QCD periodicities and transitions at finite µ I It is known that the QCD partition function ( Z QCD )has the RW periodicity [6]; Z QCD ( θ ) = Z QCD (cid:16) θ + 2 πkN c (cid:17) , (1)where k is any integer. It should be noted that the RWperiodicity is a model independent and exact propertyof the QCD partition function. In the pure gauge limit,there is the Z N c symmetry, but it is explicitly broken bydynamical quark contributions. The RW periodicity isnothing but the remnant of the Z N c symmetry in the puregauge limit. If we neglect the θ dependence of the gaugefield through quark contributions, the RW periodicity islost and then the partition function only has the trivial 2 π periodicity. This situation also appears in the quenchedapproximation.In addition to the RW periodicity, QCD has specialtransition at θ = (2 k − π/N c which is called the RWtransition. The RW periodicity is realized in a differentway at the RW transition in the confined and deconfinedphases. To discuss the RW transition, the phase of thePolyakov-loop is useful. The Polyakov-loop can be ex-pressed asΦ = 1 N c tr P h exp (cid:16) ig I β A ( τ, ~x ) dτ (cid:17)i = | Φ | e iφ , (2)where β is the inverse temperature ( β = 1 /T ), P isthe path-ordering operator and φ is the Polyakov-loopphase. When θ is continuously changed from 0 to 2 π ,the phase of the Polyakov-loop is smoothly rotated be-low T RW , but it becomes discontinuous above T RW at θ = (2 k − π/N c . Such θ dependence of φ can be foundin Ref. [7] for the Polyakov-loop extended Nambu–Jona-Lasinio (PNJL) model [8] and Ref. [9–11] for lattice QCD.The endpoint of the RW transition is called the RW end-point and its temperature is denoted by T RW . B. Deconfinement transition from RW periodicity
In Ref. [3], the authors proposed the new classificationof the confined and deconfined phases at finite T basedon the RW periodicity. The different realization of theRW periodicity plays a crucial role in the classification: Confined phase:
The origin of the RW periodicity isthe dimensionless baryon chemical potential 3 θ inthe form of exp( ± iθ ). For example, it can be seenfrom the strong coupling limit of QCD with themean-field approximation; see Ref. [12, 13]. Deconfined phase:
The origin is the dimensionlessquark chemical potential and the gauge field in theform of exp[ ± i ( gA /T + θ )] where Z images areimportant [6]. It can be clearly seen in the pertur-bative one-loop effective potential [14, 15].Therefore, in our approach for the investigation of theconfinement-deconfinement transition, we focus on theresponse of the system against θ as an indicator of thenon-trivial free-energy degeneracy. The system does notshow singularities along θ at ( T , µ R ) in the confinedphase. By comparison, there should be some singular-ities along θ at ( T , µ R ) in the deconfined phase. Detailsof singularities are explained in Sec. II C; for example,see Fig. 1. Confinement-deconfinement transition tem-peratures determined by the non-trivial free-energy de-generacy and the Polyakov-loop are matched with eachother in the infinite quark mass limit. In the next subsec-tion, we propose a new quantum order parameter of theconfinement-deconfinement transition based on the RWperiodicity. In the following discussions in this section,we concentrate on the case with µ R = 0. C. Definition of quark number holonomy
The quark number density ( n q ) above T RW should havethe gap at θ = (2 k − π/N c which reflects the θ -oddproperty. The schematic behavior of n q with N c = 3is shown in Fig. 1. The periodic solid and dashed linesrepresents n q at sufficiently high and low T comparingwith T RW , respectively. By using the behavior of n q atfinite T , we can construct the order-parameter;Ψ( T ) = hI π n Im (cid:16) d ˜ n q dθ (cid:12)(cid:12)(cid:12) T (cid:17)o dθ i , (3)where ˜ n q is the normalized quark number density definedas ˜ n q ≡ Cn q here the coefficient C [MeV − ] is introducedto make ˜ n q dimensionless. It becomes non-zero at T ≫ T RW and zero at T ≪ T RW because the information of thegap at θ = (2 k − π/N c is missed when we perform thedifferential calculus and the numerical integration. Wecall Eq. (3) the quark number holonomy . The integrandof Eq. (3) can be expressed asIm (cid:16) d ˜ n q dθ (cid:12)(cid:12)(cid:12) T (cid:17) = − CT V h h ˜ N i − h ˜ N i i(cid:12)(cid:12)(cid:12) T ∝ χ q , (4)where V denotes the three-dimensional volume and theoperator ˜ N is R ( q † q ) d x . In Eq. (4), χ q is nothing butthe quark number susceptibility at finite θ . The expectedbehavior of the quark number holonomy as a function of Dimensionless imaginary chemical potential Q u a r k nu m b e r d e n s it y ππ FIG. 1. The schematic behavior of n q as a function of θ for N c = 3. The periodic solid and dashed lines represent thequark number density at T ≫ T RW and T ≪ T RW , respec-tively. T is shown in Fig. 2. We assume that the RW endpoint isthe second (first) order in the case A (B). The schematicphase diagram in the case B is shown in the inset figureof Fig. 2. When T RW is the first-order, the RW endpointcan have two more first-order lines. In this paper, we callit beard line and the endpoint temperature of the beard Temperature Q u a r k nu m b e r ho l ono m y T Beard T RW Case BCase A Ψ T →∞ FIG. 2. The expected behavior of Ψ as a function of T .In the case A, the RW endpoint is the second order, while itis the triple point in the case B. The actual value of Ψ T →∞ is explained in the text of Sec. II D. The inset figure showsthe schematic phase diagram in the case B with N c = 3 as afunction of θ and T . line is denoted by T Beard . This triple point scenario hasbeen predicted by the lattice QCD simulations [16, 17].This behavior may be induced by the correlation betweenthe chiral and deconfinement dynamics, but details arestill under debate.In the case A where the RW endpoint is the secondorder, the quark number holonomy can be expressed asΨ = ± N c lim ǫ → h Im ˜ n q ( θ = θ ∓ RW ) i , (5)where θ ∓ RW = θ RW ∓ ǫ = π/N c ∓ ǫ with the positiveinfinitesimal value ǫ . Below T RW , n q ( θ = π/N c ) is exactlyzero and thus Ψ = 0, but Ψ becomes non-zero above T RW .The coefficient N c in Eq. (5) reflects the number of thegapped point in the 0 ≤ θ ≤ π region.In the case B where the RW endpoint is the triplepoint, situations become complicated in the T Beard D. Asymptotic behavior Here, we discuss the asymptotic behavior of Ψ. Thequark number holonomy in the T → ∞ limit becomesΨ T →∞ = 2 N c lim ǫ → h Im ˜ n T →∞ q ( θ = θ − RW ) i = 0 , (7)where ˜ n T →∞ q is the normalized quark number density inthe T → ∞ limit. Actual value of ˜ n T →∞ q can be obtainedfrom the perturbative one-loop effective potential [14, 15,18] and the value becomeslim ǫ → Im ˜ n T →∞ q ( θ = θ ∓ RW ) N c N f = ± π h θ RW π − (cid:16) θ RW π (cid:17) i −−−→ N c =3 ± . · · · , (8)where the normalization constant C is set to T − .The quark number holonomy in the m → ∞ limit canbe discussed by using the hopping parameter expansionin the lattice formalism. From the straightforward calcu-lation, the normalized quark number density is obtainedas lim ǫ → h Im ˜ n q ( θ = θ − RW ) i = − CN f N s N t lim ǫ → ∞ X n =1 κ n Im D Tr (cid:16) ∂Q∂µ Q n − (cid:17)E , (9)where the hopping parameter κ is related to m as κ =1 / (2 m + 8), N s N t express the space-time lattice vol-ume, h· · · i means the configuration average and Q isthe covariant derivative part of the lattice action; seeRef. [19] as an example. Since the hopping parameter is κ ∝ /m in the heavy quark mass limit, the quark num-ber holonomy is suppressed by 1 /m . Thus, the quarknumber holonomy finally becomes zero at m = ∞ . How-ever, the confinement-deconfinement transition tempera-ture determined by the quark number holonomy perfectlymatches with the transition temperature determined byΦ at m = ∞ by carefully considering m → ∞ limit.Even in the heavy quark mass region, the RW transi-tion may be smeared by the finite size effect in the lat-tice QCD simulation. Thus, the non-zero quark numberholonomy requires finite size scaling analysis to obtainin a straightforward calculation. An alternative way isto fit lattice QCD data by using an oscillating θ -oddfunctions in the region V , { V : 0 ≤ θ ≤ π/N c } , withneglecting data very close to θ = (2 k − π/N c wheredata are strongly affected by the finite size effects. Thefitting function becomes the 2 π/N c periodic function atsufficiently low T and it does the 2 π periodic functionat sufficiently high T in the 0 ≤ θ ≤ π/N c region. Thisdifference may help us to calculate the quark numberholonomy on the lattice. It should be noted that thistreatment is similar to the observation process of non-zero order parameters with vanishing symmetry breakingexternal fields on the lattice and thus it is not a funda-mental problem. III. DISCUSSIONS Firstly, we discuss the current status of the presentdetermination and the ordinary determination of the de-confinement temperature. Readers may doubt the valid-ity of the present definition of the deconfinement transi-tion temperature since the deconfinement temperature, T D ≡ T RW or T Beard , is substantially higher than the chi-ral pseudo-critical temperature. It was considered thatthe chiral and the deconfinement crossover take placeat similar temperatures from the rapid change of thechiral condensate and the Polyakov-loop on the latticewith 2 + 1 flavors; see for example Ref. [20, 21]. Withthe development of the highly improved quark action,it now seems that the Polyakov-loop grows very gradu-ally [22, 23]. An effective model analysis of recent lat-tice data implies that the deconfinement pseudo-criticaltemperature ( ∼ 215 MeV) is substantially higher thanthe chiral pseudo-critical temperature [24]. By compar- ison, a recent lattice determination of the RW endpointtemperature with physical quark masses implies that thecontinuum extrapolated value of T RW is 208(5) MeV [25].Therefore, higher T D does not invalidate the discussion,but is supported by the recent lattice data via effectivemodel analysis.Secondly, we discuss the difference between the quarknumber holonomy and the dual quark condensate [26–30]. The dual quark condensate is defined asΣ ( n ) = − I π dϕ π e − inϕ σ ( ϕ ) , (10)where ϕ = θ + π specifies the boundary condition for thetemporal direction of quarks, σ ( ϕ ) is the ϕ -dependentchiral condensate and n represents the winding numberalong the temporal direction. Particularly, Σ (1) sharessimilar properties with Φ because Φ is also the windingnumber 1 quantity and thus it can be used as the indica-tor of the confinement-deconfinement transition. In thequenched approximation, the dual quark condensate iswell defined, but there is the uncertainty in the dynam-ical quark case [27, 28]. In the calculation of the dualquark condensate, we need to break the RW periodicitybecause Σ (1) should be zero in all T region if the RW peri-odicity exists. It is usually done by imposing the twistedboundary condition on the Dirac operator, while config-urations are sampled under the anti-periodic boundarycondition. This is not a unique procedure. Therefore,there is the uncertainty in the determination of the dualquark condensate. Also, it is well known that the dualquark condensate is strongly affected by the chiral tran-sition or some other transitions [31–33]. On the otherhand, the quark number holonomy (3) can provide non-zero value above T RW or T Beard without any uncertain-ties. It is the important advantage of the quark numberholonomy.Thirdly, we discuss the quark number holonomy fromthe landscape of the effective potential in the complex Φplane. Below T RW or T Beard , the effective potential atany θ can be described by only one minimum which iscontinuously connected with the θ = 0 solution. Above T RW , the Z N c images appear; for example, the Z N c im-ages are e i π/ and e i π/ for Φ = 1 at sufficiently high T for N c = 3. In the confined phase, the fluctuationis strong and thus the Z N c images are collapsed to oneminimum. On the other hand, the Z N c images can with-stand the fluctuation in the deconfined phase. Thus, thequark number holonomy (3) measures the strength of thefluctuation which collapses the Z N c images to one min-imum. It is related to the non-trivial degeneracy of thefree-energy in the deconfined phase discussed in Ref. [3].Therefore, the quark number holonomy can describe theconfinement-deconfinement transition via the nontrivialfree-energy degeneracy. Present discussion may be re-lated with the Polyakov-loop fluctuations discussed inRef. [34] and thus it is interesting to compare the resultsof the Polyakov-loop fluctuations with the quark numberholonomy.Finally, the sign problem is discussed when we calcu-late the quark number holonomy at finite µ R . At finite µ R , Eq.(3) should be replaced asΨ( T ) → Ψ( T, µ R ) . (11)This means that the θ integration in Eq. (11) shouldbe evaluated with fixed T and also µ R . Therefore, wemust consider the complex chemical potential in the cal-culation of the quark number holonomy, where the signproblem arises. At finite imaginary chemical potential( µ R = 0), we can use the γ hermiticity;det D ( µ ) = det[ γ D ( µ ) γ ] = [det D ( − µ ∗ )] ∗ , (12)where D is the Dirac operator. Therefore, the sign prob-lem does not matter at finite imaginary chemical po-tential, µ ∗ = − µ , when we calculate Ψ( T ). On theother hand, at finite real chemical potential, µ R = 0 and θ = 0, the relation (12) can not help us, but the Lef-schetz thimble path integral method [35–37] does. InRef. [38], it is shown that this method leads the saddle-points which manifests the CK symmetry where C and K express the charge and the complex conjugation oper-ator, respectively. The sign problem can be avoided bythe CK symmetric saddle-points and then the mean-fieldcalculation of QCD effective models such as the PNJLmodel is extremely simplified [39, 40]. Unfortunately, the CK symmetry is not preserved at finite complex chemi-cal potential, µ R = 0 and θ = 0, when we calculateEq. (11). Thus, the calculation becomes complicatedeven in the mean-field calculation of the QCD effectivemodels. In this case, we should perform the matter-of-fact calculation based on the Lefschetz thimble path in-tegral method. Actual challenge of the calculation willbe shown elsewhere. IV. SUMMARY In this paper, we have proposed a new quantity todescribe the confinement-deconfinement transition basedon topological properties of QCD in the imaginary chem-ical potential region. We call it the quark number holon-omy which is defined by the contour integral of thequark number susceptibility along the closed loop of θ .The quark number holonomy seems to be similar to theUhlmann phase which can be used to classify the topo-logical order at finite T in the condensed matter physics. The quark number holonomy can have a non-zero valueabove T RW or T Beard and it becomes zero below thesetemperatures. This behavior is related with the differentrealizations of the free-energy degeneracy above and be-low T RW . From the model independent analysis, we findthat the quark number holonomy is proportional to N in the deconfined phase, while it does not in the confinedphase if we determined the confinement-deconfinementtemperature as the topological phase transition. Also,we have shown the behavior of the quark number holon-omy in the T → ∞ and the m → ∞ limit as a landmarkto help the future lattice QCD simulation.We have discussed the similarity between the quarknumber holonomy and the dual quark condensatewhich is sometimes used to investigate the confinement-deconfinement transition. Calculations of the dual quarkcondensate has the uncertainty when the dynamicalquark is taken into account, but the quark number holon-omy does not have such uncertainty. This is the strongadvantage of the quark number holonomy. Also, we haveexplained how the quark number holonomy can describethe confinement-deconfinement transition from the land-scape of the effective potential. In the confined phase,the fluctuation is strong and then the Z N c images col-lapse to one minimum. On the other hand, the Z N c im-ages withstand against the fluctuation in the deconfinedphase. Therefore, the quark number holonomy can de-scribe the confinement-deconfinement transition via thefree-energy degeneracy. Finally, we have discussed thesign problem when we calculate the quark number holon-omy at finite real chemical potential. In this case, weshould consider the complex chemical potential and thuswe need extremely care of the sign problem. ACKNOWLEDGMENTS The authors thank M. Sato whose suggestions for thequantum order parameter of the topological phase transi-tion at finite temperature motivate us to start this study.K.K. thanks H. Tsukiji for helpful comments. K.K. issupported by Grants-in-Aid for Japan Society for thePromotion of Science (JSPS) fellows No.26-1717. 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