PoS(LATTICE 2015)151 http://pos.sissa.it/ emical potential based on the nt (pseudo) critical temperature ) critical temperature. Expected e condensed matter physics. The v-Bohm phase, then the Roberge-Weiss y, Kyoto 606-8502, Japan y, Kyoto 606-8502, Japan e Commons al License (CC BY-NC-ND 4.0). ∗ [email protected] [email protected] Speaker. We propose new determination of theby confinement-deconfineme using the analogy ofimaginary chemical the potential topological is interpreted order as the discussed Aharono in th endpoint would define the confinement-deconfinementQCD (pseudo phase diagram diagrams are presented at finite complex ch perturbative calculation. ∗ Copyright owned by the author(s) under the terms of the Creativ c
Akira Ohnishi Yukawa Institute for Theoretical Physics, Kyoto Universit E-mail: E-mail: Kouji Kashiwa Yukawa Institute for Theoretical Physics, Kyoto Universit The 33rd International Symposium on Lattice14 Field -18 Theory July 2015 Kobe International Conference Center, Kobe, Japan* Topological feature and phase diagram ofcomplex QCD chemical at potential Attribution-NonCommercial-NoDerivatives 4.0 Internation PoS(LATTICE 2015)151 article , respec- I ormed so µ and Kouji Kashiwa sition plays a iant holonomy R critical tempera- µ nement transition t extend the chemical e deconfinement tran- alculation, there is no 2. Also, we shows the CD simulation has the R nite quark mass regime, μ ) and real quark chemical po- iagram at finite ment transition is very limited. T region in our present determination of R µ Critical point e. 2 O RW T /3 T π /3 π 2 Chiral critical endpoint π . Even if several methods to circumvent the sign problem are used, we can R 1 region. Therefore, several effective model calculations are perf ) symmetry is directly related with the decofinement transition in this case. RW endpoint µ 3 θ > Z T / R µ ) is one of the interesting and important subjects in the nuclear and elementary p Schematic figure of our current expectation of the QCD phase d R µ In the study of the QCD phase diagram, the confinement-deconfinement tran Understanding of QCD phase diagram at finite temperature ( In this talk, we discuss the new determination of the deconfinement (pseudo) the deconfinement (pseudo) critical temperature. not reach the tively. Solid lines represent the first order transition lin far. Figure 1 shows our current expectation of the QCD phase diagram. Figure 1: Topological feature and phase diagram of QCD 1. Introduction physics. If we can obtainunclearness. the However, QCD such first phase principle diagram calculationsign from which problem the at is first-principle finite the c lattice Q ture by using the analogy ofexpected the QCD topological phase order diagrams which at is complex chemical explainedpotential potential in because to Sec. we complex mus from real when we consider finite the Polyakov-loop is usually treated as thesition, approximate and order the parameter QCD of phase th diagra is drown. Although, the Polyakov-loop is no longer the exact order parameter in the fi crucial role. However, our currentAt understanding least of in the the deconfine heavyfor quark the mass imaginary limit, time the circlebecause Polyako-loop becomes the which center the is ( exact the order gauge parameter invar of the deconfi tential ( PoS(LATTICE 2015)151 . is 3. / RW 1-th π T RW in the + T = i D T θ is uniquely Kouji Kashiwa re constructed -th and RW kov-loop is the ate (a), (b) and i flux. Since the T ntial at relations of those ) value as termined by using 1 uld exist. ( , but not below tion (b) and (c) follw hm phase [1]. In this l is similar in the ac- erent behaviors in the . U nfined and deconfined ture QCD was already y the flux insertion via RW minima of the effective d in the condensed mat- , we propose that T erefore, we can find the ed phase, it is inconsistent tion relation is commutable imaginary chemical potential. systems used in the topological hadron degrees of freedom with ations become non-commutable 3 above T turbative one-loop calculation [4, 5] and / . The Polyakov-loop is no longer 3 π is not unique, but Z . From the lattice QCD and effective T D = 3 T Z θ ≤ T RW T at zero temperature. There are introduce three flux can be inserted to the closed loop of the ≤ 3 3 ) Φ in the infinite quark mass limit is unambiguously 1 T T ( D T U flux to spatial closed loops, (b) exchange plane. We call it ) 1 Φ ( is the critical temperature determined by the susceptibility of U Φ . It is induced by the nontrivial appearance of the RW periodicity 1 T - Im Φ changed to the pseudo-critical temperature. The upper bound of the is the deconfinement critical temperature. When the dynamical quarks Φ where which agrees with D T T RW RW T T = 0 in the Re Φ 0 from the non-trivial free energy degeneracy of the effective pote , the topological order cannot be well defined, because thermal states a = T I T = µ = I µ D T At finite For example, this fact can be seen from the comparison between the per In the imaginary-time formalism, the It should be noted that the present definition and the standard definition de In Ref. [3], the author consider the torus 1 tells us the non-trivial degeneracy of the free energy minima. Therefore I µ This degeneracy seems similar to the vacuum degeneracy in zero the exact order parameter and thus the determination of order and the analogy can be found; the response of hidden local minima b potential at pseudo-critical temperature may be determined by the appearance of local the Polyakov-loop and quarks and (c) move a quarkthe along loops. Braid Commutation group, relations of and theoperations the opera with Aharonov-Bohm operation (a). effect If determines quarksbecause are the quarks deconfined, have commutation above the oper fractional charge.if On the quarks other are hand, confined commuta because physicalinteger states charges. are characterized Therefore, by if therewith is the only non-commutability of one the vacuum operations in and the thus deconfin vacuum degeneracy sho model calculations, we can have the relation the strong coupling limit of QCD with the mean field approximation [6, 7] at finite are taken into account, determined. Thus, determined in the lattice QCD and effective models and provides a reasonable in the deconfined phase. This fact suggests that we can distinguish the co phases at by a mixture of(c), pure adiabatically. states Nevertheless, with the the RWconfined Boltzmann periodicity and shows factor, deconfined significantly phases and diff then we cannot oper Actually, we can have degenerated free energy minima at the pseudo-critical temperature of the confinement-deconfinement transition the Polyakov-loop are consistent inexact the order-parameter infinite of quark the massrelation confinement-deconfinement limit transition. where the Th Polya Topological feature and phase diagram of QCD 2. Free energy degeneracy and deconfinement transition imaginary-time direction and then the vector potential is induced by the adiabatic operations: (a) Insert the appearance form of the vectortion, potential the and imaginary the chemical imaginary potential chemical caninterpretation, potentia be we interpreted may as use the the Aharonov-Bo discussionter of physics the [2]. topological order It discusse discussed should in be Ref. noted [3]. that actual applications to zero tempera PoS(LATTICE 2015)151 d ). V (3.1) (3.2) . This ε region. + which is R C µ e can sketch Kouji Kashiwa axis. The RW θ tour method [10, able properties of , te that Eq. (3.2) is diagrams expected ve expansion of the 3 is a infinitesimal real vin method [16, 17]. ) y. 0 limit, we can distin- We call the configura- g the ε ider the Polyakov-loop; T sinio (PNJL) model [8], e summarized in Fig. 2. . chemical potential at the / ritical temperature of the 0 → R ) = ε 0 µ R eral difficulties. One of the ( = µ correction term makes Re order as
where R µ q [9]. There are some proposals ε 2 2 | R R at least in the small θ O ) − µ R µ I d in the dn -dependence of the RW endpoint. R C µ µ + R ε µ , 0 iT µ + R = deconfinement configuration R − C µ µ ) ( . The
q = ε 0 )] I RW + = Tn T R µ ( , we can say that the deconfinement transition µ , T
RW R D correction term is then positive with / ) R T T µ R T 2 ( R µ ( T 4 µ q q µ / as to that with I d ε µ dn Tn − i [ − RW which is labeled as ( ) d I T C d can be expanded up to 2 µ 0) may be connected with the first order phase transition T , R 0 µ R = , = T with µ 0 T R decreases with increasing ( because the = µ V R ε 2 1 V µ
+ RW C T − T as a first step to investigate the q / ) = R R I plane ( -dependence of the deconfinement pseudo-critical temperature define dn µ µ µ ) R , d θ µ R plane. Two of the first order transition lines starting from the RW endpoint , T µ ) T , R ( and it becomes moderate above T µ decreases or increases if we can estimate the third term of r.h.s. of Eq. (3.1 ( , T confinement configuration V RW ( RW T ) T . By comparing Re ε ε − + C higher than that with RW ε T . We here give an model independent argument based on the perturbati ( − C Here, we consider the confinement and deconfinement configurations. The effective potential with small By taking into account our perturbative result and the symmetry argument, w We now discuss the RW T effective potential at finite expected QCD phase diagrams at finite complexfrom chemical potential. the QCD symmetry phase argument andBecause our of present the perturbative RW calculations periodicity, ar the phase structure should be periodic alon by It should be notedpromising that effective models non-perturbative is model the Polyakov-loop approaches extendedbut Nambu–Jona-La have the sev PNJL model also has the model sign problem at finite where transition line on the guish whether positive value. Also, we call the configuration at fact means that the first-order with to circumvent the model sign problem,11, for 12] example the based complex on integral the pathUnfortunately con Lefschetz those approaches thimble cannot [13, be 14, directlypresent 15] used because at and finite we the complex complex cannotQCD. Lange maintain the RW periodicity and some other desir This behavior is consistent withdeconfinement the transition decreasing defined behavior by using of usual the determinationsfor pseudo-c which example, cons see Refs [8, 20, 21]. labeled as is the topological phase transition. 3. QCD phase diagram at complex chemical potential Topological feature and phase diagram of QCD case with the dynamical quark. If we adopt Equation (3.2) is real. The second term in r.h.s. of Eq.tion (3.1) at is pure imaginar boundary on the From the lattice QCD andnegative effective below model calculations [18, 19], we can estima PoS(LATTICE 2015)151 ) R RW the and µ T , R e fol- T µ ( . Kouji Kashiwa plane. Therefore, 0 as shown in the should be related tematically under- plane is connected ) R = [22]. Then, it is not R ere is the possibility ) region and forms an imaginary to the real µ µ θ T , R , me temperature. There d with a strength of the ace. T µ T plane are separated from lytic continuation on the ( ( to construct the canonical ) correlated case plane has less relevance to θ ) the , θ at finite T hemical potential. The left (right) , ( T space, and connects the ( RW ) T θ , R µ , T chiral transition lines ( en the chiral and RW transition surfaces. plane. In the continuation, we may miss some 2 5 µ plane, as shown in the bottom panel of Fig. 2. and the former possibility ) R µ , 0 is used to clarify realized temperatures in experiments space. The RW endpoint may reach T plane. In this case, the first order phase boundary on the ( = ) ) θ R R , µ µ R , µ T at , ( T Chiral transition surface ( RW uncorrelated case T , but does not goes across the chiral transition surface. In this case, line becomes smaller than the chiral critical surface at moderate RW transition surface R µ RW T plane. The RW transition line extends in the finite ) θ , T ( Two possible schematic QCD phase diagrams at finite complex c plane forms a chiral transition surface in the ) R The QCD phase diagram at finite complex chemical potential may be related with th Another possibility is that the first order transition lines on the µ , T lowing subjects. (I) In Ref.partition [23, function. 24], the Then, authors use experimental data information such as an inhomogeneous chiral condensate [27]. The ana unreasonable to expect that the endpoint of the chiral critical line on the RW transition surface in the with the critical point on the we can call this possibility panel represents the correlated (uncorrelated) case betwe have chiral transition natures and are referred to as the the first order phase boundary, which would be the chiral transition, on the first order phase boundary on the through the Lee-Yang zero analysis [25, 26]. In the analysis, ( Figure 2: Topological feature and phase diagram of QCD finally becomes larger than the chiralcorrelation transition between surface.. the chiral It transition is surface deeply and relate the RW transition surf plane and the deconfinement transition represented by the RW endpoint on the with zeros inside the unit circlethat on we can the strictly complex determine quark realized fugacitystand temperatures plane. in the If experiments behavior if so, of we th zeros. canchemical sys (II) potential The is analytic usually continuation performed in on QCD the from the first decreases at small top panel of Fig. 2 oris it the may possibility deviate that from the chiral transition surface at so PoS(LATTICE 2015)151 0. = µ . In the and also the Kouji Kashiwa eavy ion collider . The topological e connected at the ate the topological T nov-Bohm phase in- entanglement and topo- W endpoint is separated r example, see Ref. 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