Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter and Analysis of Moment of Inertia
KKEK-TH-2290, J-PARC-TH-0236
Deconfining Phase Boundary of Rapidly Rotating Hot and Dense Matter andAnalysis of Moment of Inertia
Yuki Fujimoto a , Kenji Fukushima a , Yoshimasa Hidaka b,c,d a Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan b Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801 Japan c Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan d RIKEN iTHEMS, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Abstract
We discuss the effect of rapid rotation on the phase diagram of hadronic matter. The energy dispersion relation is shiftedby an effective chemical potential induced by rotation. This suggests that rotation should lower the critical temperatureof chiral restoration, but it is still controversial how the deconfinement temperature should change as a function ofangular velocity. We adopt the hadron resonance gas model as an approach free from fitting parameters. We identify thedeconfinement from the thermodynamic behavior and find that rotation decreases the deconfinement temperature. Wealso discuss the spatial inhomogeneity of the pressure and give a semi-quantitative estimate of the moment of inertia.
Keywords: hadronic matter, quarks and gluons, phase transition, deconfinement, rotation
1. Introduction
Rotation effects are ubiquitous in various systems andhave been one of the central topics in nuclear physics. Aprominent example is the successful classification of nu-clear spectra by the collective rotational modes. Someheavy nuclei spontaneously break rotational symmetry bydeformation and rotate to restore the broken symmetryleading to the rotational band. A classic review is foundin Ref. [1]; see also Ref. [2] for a modernized picture.The present work addresses a nuclear system with morerapid rotation and higher temperature created in non-central relativistic heavy-ion collisions (see Refs. [3–5] forrecent reviews). In the experiment a nonzero value of Λand ¯Λ polarization has been confirmed and the observedpolarization is translated to an angular velocity or vortic-ity ω of created matter as large as ω (cid:39) (9 ± × s − [6].Some theoretical studies imply even higher values for vor-ticity [7–9]. Although expected ω values are extraordinar-ily large, the corresponding energy scale is ω ∼ T , thebaryon chemical potential µ , and the magnetic field B , thevorticity ω plays a role as a relevant parameter to charac-terize the properties of hot and dense matter in heavy-ioncollision phenomenology. Now, a lot of theoretical effortsare aimed to establish a full dynamical description of thepolarization in terms of hydrodynamics or kinetic theory(see Refs. [4, 5] and references therein). Email addresses: [email protected] (Yuki Fujimoto), [email protected] (Kenji Fukushima), [email protected] (Yoshimasa Hidaka)
Apart from phenomenology, the rotation has been alsointerested from the wider point of view of quantum fieldtheory or quantum chromodynamics (QCD). A well-knownexample is an anomalous transport phenomenon; namely,the chiral vortical effect [10–13]. Another interesting pos-sibility is an exotic ground state such as the charged pioncondensation emerging from a combination of rotation andmagnetic field [14, 15]. Then, it is a natural anticipationthat ω should be a useful probe to investigate the QCDphase diagram. In theory ω could be chosen to be com-parable to a typical QCD scale, and it would be a quiteinspiring question how the QCD phase diagram evolveswith increasing ω . In fact, the chiral phase transition hasbeen already examined extensively in the literature [16–22]. It is a more or less accepted consensus that the ro-tation effect suppresses the chiral condensate just like thefinite density effect, so that the chiral critical temperaturedrops with increasing ω .While most of the works have been concentrated onthe chiral aspects, the deconfinement transition in QCD isrecently being focused [23–25]. One of the latest lattice-QCD calculations, which builds upon the formulation inRef. [26], claims that the deconfinement temperature, T c ,increases with growing ω by measuring the Polyakov loopon the lattice in rotating frames [23]. A holographic QCDapproach, by contrast, suggests the opposite behavior, i.e., T c decreases with growing ω [24], which is in accordancewith the behavior of the chiral critical temperature. Thereis also an alternative proposal of a mixed inhomogeneousphase supporting spatially separated confinement and de-confinement sectors [25]. Preprint submitted to Elsevier January 25, 2021 a r X i v : . [ h e p - ph ] J a n n the present work we perform a simple yet robustanalysis based on the hadron resonance gas (HRG) modelto estimate thermodynamic quantities in a rotating frames.This model has no free parameter adjustable by hand andthe input variables are all fixed by experimentally observedparticle spectra. The virtue of the HRG model lies in itsunambiguousness of the minimal model definition. TheHRG model (or the thermal model fit) has manifestedeminent successes in reproducing the particle abundancesin heavy-ion collision experiments [27]. Furthermore, theHRG model has been found to be consistent with ther-modynamic properties measured in lattice-QCD simula-tions up to ∼ T c or even for higher T once interactions areincluded [28, 29]. The ideal (i.e., non-interacting) HRGmodel prevails as long as we stay below T c , but for T > T c the thermodynamic quantities predicted from the idealHRG model blow up. The breakdown of such a hadronicmodel based on the Hagedorn picture [30] should be iden-tified as the deconfinement transition [31]. We will discussthis characterization later (see Ref. [32] for more details).To this end we formulate how to calculate the pressurein the rotating frame. With global rotation the pressure isinhomogeneous to be balanced with the centrifugal force,from which we can infer the distribution of the angular mo-mentum and also the moment of inertia of hot and densematter. Our numerical results agree with empirical depen-dence on the radial distance from the rotation axis, andwe make a consistency check in a semi-quantitative way.This paper is organized as follows. In Sec. 2 we willbriefly review the field theoretical treatment of rotationand the energy dispersion relation gapped by the causal-ity bound. In Sec. 3 we will set forth our strategy todescribe deconfinement within the HRG model based onthe Hagedorn picture. In Sec. 4 we will give an explicitexpression for the pressure in the rotating frame and spellout calculational procedures. The pressure has explicitdependence on the radial distance from the rotation axisand we closely discuss the physical interpretation in Sec. 5.Section 6 constitutes our central results in this paper andwe will show a 3D phase boundary surface as a function ofthe baryon chemical potential µ and the angular velocity ω . In Sec. 7 we will revisit the r dependence to make aconsistency check between the physical interpretation andthe numerical results. Section 8 is devoted to the summaryand outlooks.
2. Causality bound in the rotating frame
The most straightforward approach to treat rotatingsystems is to describe physics in a rotating frame by trans-forming non-rotating coordinates, ¯ x µ , into x µ rotating withthe angular velocity ω . We take the rotation axis along the z direction, so that local quantities in the rotating frameare given as functions of x = ¯ x cos ωt + ¯ y sin ωt , y = − ¯ x sin ωt + ¯ y cos ωt . (1) We can read the metric as g µν = η ab ∂ ¯ x a ∂x µ ∂ ¯ x b ∂x ν = − ( x + y ) ω yω − xω yω − − xω − − . (2)Here, η ab represents the Minkowskian metric: η = diag(1 , − , − , − η ab = e µa e νb g µν ,where e t = e x = e y = e z = 1 e x = yω, e y = − xω , (3)and the other components are zero. The explicit calcula-tions using the metric and the vierbein lead to a shift inthe Hamiltonian as ˆ H → ˆ H − J · ω (4)in the rotating frame, where J is the total angular momen-tum; namely, J = L + S , with the orbital part L and thespin part S . Accordingly the energy dispersion relationfor spin- S particles should be shifted as ε → ε − ( (cid:96) + s ) ω . (5)Here, s = − S, − S + 1 , . . . S − , S and (cid:96) denotes the quan-tum number corresponding to the z component of the or-bital angular momentum, i.e., L z .The energy shift above is analogous to the chemical po-tential for finite density systems: ( (cid:96) + s ) ω can be regardedas an effective chemical potential. Then, one might thinkthat, for ( (cid:96) + s ) ω > ε , a Bose-Einstein condensate shouldform for bosons, or, a Fermi surface should appear forfermions. This is, however, unphysical because Eq. (1)is merely a coordinate transformation and the vacuumphysics must not change, while the rotation together withexternal effects such as the temperature, the electromag-netic fields, etc could make physical differences.In fact, ( (cid:96) + s ) ω > ε is prohibited by the causalitycondition. In the cylindrical coordinates ( r, ϕ, z ), we canperform the Bessel-Fourier expansion to define the modeswith corresponding momenta, ( k r , (cid:96), k z ), where (cid:96) is noth-ing but the orbital angular momentum. We can then im-pose a boundary condition at r = R , such that the wave-functions are normalized within r ≤ R . The causalityrequires, R ω ≤ . (6)We note that the right hand side in the above condition isthe speed of light, c = 1, in the natural unit. The bound-ary condition at r = R makes the momenta discretized We can find explicit calculations for S = 0 and S = 1 / − J · ω generallyholds for S = 1 , / k r = ξ (cid:96),n /R with ξ (cid:96),n being the n th zero of the Besselfunction: J (cid:96) ( ξ (cid:96),n ) = 0 [12, 18, 34]. It is important to notethat J (cid:96) ≥ ( ξ ) has a zero at ξ = 0 but such a zero modeis identically vanishing due to the boundary condition at r = R , and there is no zero mode contribution to physi-cal quantities. Strictly speaking, thus, ξ (cid:96),n indicates the n th zero excluding a trivial zero at ξ = 0. In this waywe can conclude that the energy is always gapped at leastby ξ (cid:96), /R for any particles. We combine this gap and thecondition (6) to confirm, ε ≥ ξ (cid:96), R ≥ ξ (cid:96), ω . (7)It is known that ξ (cid:96),n ’s satisfy the following inequality: ξ (cid:96), > (cid:96) + 1 . (cid:96) / + 0 . (cid:96) − / for (cid:96) ≥ ξ , =2 . ε > ( (cid:96) + s ) ω for suffi-ciently large (cid:96) even when s is large. Our present calcula-tions, as we explain in details later, contain hadrons up to S = 2 and ε > ( (cid:96) + s ) ω holds for | s | ≤ S and all (cid:96) .
3. Deconfinement transition in the Hagedorn pic-ture
The HRG model analysis is thoroughly hadronic, butwe can still discuss the deconfinement transition as follows.Historically speaking, the Hagedorn limiting temperaturewas first recognized within the framework of hadronic boot-strap model [30]. Later, then, Cabibbo and Parisi realizedthat the limiting temperature should be given a correctphysical interpretation as the transition temperature tomore fundamental degrees of freedom than hadrons [31].Let us suppose that the hadron mass spectrum risesexponentially, i.e., ρ ( m ) = e m/T H , (8)where T H is not a physical temperature but just a slopeparameter to characterize the mass spectrum. Then, theintegration weighted with the Boltzmann factor, e − m/T ,gives us the partition function as Z = (cid:90) dm ρ ( m ) e − m/T . (9)For simplicity we omit the phase space volume (that wouldgive a polynomial factor) and focus on the exponential be-havior only. In other words the integration measure of dm is implicitly defined in a consistent way. Now, it is obviousthat the integration diverges for T > T H , and Hagedornconsidered that T H should be the limiting temperature:any physical systems of hadrons cannot be heated above T H . This conjecture should be revised once internal struc-tures of hadrons are taken into account. The existence of T H should be correctly interpreted as a breakdown pointof such a simple hadronic description and the physical sys-tems should be better characterized by quarks and gluonsat T > T H . In the HRG model, the hadron mass spectrum is takenfrom the experimental data, and interestingly, ρ ( m ) showsexponential growth up to m ∼ T ∼ T H ),though they do not diverge strictly. Therefore, we canphysically identify the deconfinement crossover point fromthe blowup behavior of thermodynamic quantities in theHRG model. It is straightforward to extend the abovementioned picture to a finite density case by replacing theBoltzmann factor with e − ( m − µ ) /T for baryons that also ex-hibit exponential spectra ∼ e m/T B (see Ref. [32]). We willexplain our working criterion for deconfinement in laterdiscussions.
4. Rotating hadron resonance gas model
The HRG model has been well established and forour purpose to investigate rotating systems we need torewrite the formulas in terms of the cylindrical coordi-nates, ( k r , (cid:96), k z ). The pressure in the HRG model hascontributions from both mesons ( m ) and baryons ( b ) upto an ultraviolet mass scale, Λ: p ( T, µ, ω ; Λ) = (cid:88) m ; M i ≤ Λ p m + (cid:88) b ; M b ≤ Λ p b , (10)The mesonic and the baryonic pressures are given by p m = p − i = m , p b = p + i = b , (11)where the generalized pressure functions are p ± i = ± T π ∞ (cid:88) (cid:96) = −∞ (cid:90) dk r (cid:90) dk z (cid:96) +2 S i (cid:88) ν = (cid:96) J ν ( k r r ) × log { ± exp[ − ( ε (cid:96),i − µ i ) /T ] } . (12)The energy spectrum is ε (cid:96),i = (cid:112) k r + k z + m i − ( (cid:96) + S i ) ω with S i and m i being the spin and the mass of the particle i . We note that the radial integration is with respect to k r in the above form; that is, dk r = 2 k r dk r . The aboveexpression needs some more explanations. The rotationeffect shifts the energy dispersion relation by the crankingterm, i.e., − J · ω , which varies as ( (cid:96) + s i ) ω from s i = − S i to s i = + S i . We reorganize the sum over s i and (cid:96) so that theenergy shift can be the same, − ( (cid:96) + S i ) ω , to simplify theexpression. Then, the spin sum is translated to the sumwith respect to ν with the square of the Bessel function J ν ( k r r ) as in Eq. (12). The Bessel function arises fromthe weight in the Bessel-Fourier expansion. The simplestnontrivial example is the spin-1/2 calculation (see Ref. [18,35] for more details). After the appropriate redefinitionof (cid:96) in such a way that the total angular momentum is3 = (cid:96) + 1 /
2, one particle solutions of the Dirac equationread: u + = e − iεt + ik z z √ ε + m ( ε + m ) J (cid:96) ( k r r ) e i(cid:96)ϕ k z J (cid:96) ( k r r ) e i(cid:96)ϕ ik r J (cid:96) +1 ( k r r ) e i ( (cid:96) +1) ϕ . (13)The other solution, u − , can be expressed similarly (theexplicit expression is found in Ref. [18]). From these solu-tions the fermionic propagator can be constructed and itstrace involves J (cid:96) ( k r r ) + J (cid:96) +1 ( k r r ), that is nothing but thesum we see in Eq. (12) for S i = 1 / ω → p ± i → ± g i T π (cid:90) ∞ k dk log (cid:40) ± exp (cid:34) − (cid:112) k + m i − µ i T (cid:35)(cid:41) , (14)where g i = 2 S i + 1 is the spin degeneracy factor andthis expression is certainly convergent. The dispersionrelation involves an exponentially growing factor, e (cid:96)ω/T ,but J ν ≥ (cid:96) ( k r r ) has stronger exponential suppression andEq. (12) is finite.There is, however, one subtlety in Eq. (12). As dis-cussed in Sec. 2, we can avoid unphysical condensates fromthe causality bound, but it is time consuming to take thediscrete sum of k r . Here, instead, we shall employ an ap-proximate and minimal prescription to evade unphysicalcondensates. As long as ω is not significantly larger thanΛ QCD , the discretization in high momentum regions is ex-pected to be a minor effect, and the leading discretizationeffect in the low momentum regions is the mass gap. Wecan thus introduce an infrared cutoff for the k r integration,Λ IR (cid:96) , defined by Λ IR (cid:96) = ξ (cid:96), ω , (15)where, as we already noted, an obvious zero at ξ = 0 isexcluded. The k r integration in Eq. (12) is then replacedas (cid:90) dk r → (cid:90) (Λ IR (cid:96) ) dk r . (16)We will elucidate technical procedures in more details inSec. 6.
5. Radial dependence
We note that our main formula (12) depends on theradial coordinate r through J ν ( k r r ). There are twofoldintuitive origins for this r dependence. One is possible r dependence from the boundary effect at R ∼ /ω . Theboundary effect exists even for non-rotating matter. Weare interested in not surface singularities (as discussed inRef. [35] for example) but bulk properties, and so we cantake as small r as possible for numerical implementation. Another origin is that the centrifugal force should be sup-ported by the r dependent part of the pressure.Let us consider the r dependence from the latter origin.From the analogy to the relation between the baryon num-ber density and the pressure: n = ∂p/∂µ , we can expressthe angular momentum density as (cid:104) j (cid:105) ( r ) = ∂p ( r ) ∂ω . (17)When ω is small in the linear regime, the angular momen-tum is related to the moment of inertia in the infinitesimalvolume dV as (cid:104) j (cid:105) ( r ) dV (cid:39) dI ( r ) ω . (18)For homogeneous matter with mass density ρ , we can eas-ily find the moment of inertia as dI ( r ) = ρr dV . If thebaryon chemical potential is vanishing, ρ should be char-acterized by the temperature T , i.e., ρ = σT . We canroughly approximate σ from the enthalpy density; namely, σ = 2 νπ /
45 with the thermal degrees of freedom ν . Then,we can approximate: p ( r ) = p (0) + ∆ p ( r ) , ∆ p ( r ) (cid:39) σ T r ω . (19)Because σ may differ for confined hadronic matter anddeconfined matter of quarks and gluons, the deconfinementpoint could be in principle dependent on r . Indeed inthe cylinder with a boundary, the possibility of spatiallyseparated regions of confinement and deconfinement waspointed out [25].In the present work, to avoid ambiguous interpretation,we shall take rω (cid:28) r dependence: we fix r = 0 .
01 GeV − throughout this work.If we take the strict limit of r → (cid:96) and theintegration with respect to k r are harmless), all the termsinvolving J ν (cid:54) =0 (0) = 0 should vanish. Then, only termswith ν = 0 survive, which are allowed for (cid:96) = − S i to (cid:96) = 0, corresponding to the energy shifts from − S i ω to+ S i ω . Since we redefined (cid:96) to simplify Eq. (12), it is abit nontrivial to see, but the surviving terms are differentspin states with zero orbital angular momentum. This isvery natural: at r = 0 the orbital angular momentum isidentically zero and the rotation couples to the spin only.
6. Numerical results
In our HRG model treatment we have adopted the par-ticle data group list of particles contained in the package ofTHERMUS-V3.0 [36] and incorporated the data into ourown numerical codes. To reduce the numerical cost, we im-pose an ultraviolet mass cutoff as Λ = 1 . . f (1270), a (1320), K ∗ (1430), and f (1430)with S = 2. The effect of Λ on the chemical freezeoutcurve has been examined in Ref. [37], and they have found4 .10 0.12 0.14 0.16 0.18 0.20 T [GeV] p / T without mass cutoffwith mass cutoff m <1.5 GeV 0.10 0.12 0.14 0.16 0.18 0.20 T [GeV] / T without mass cutoffwith mass cutoff m <1.5 GeV 0.10 0.12 0.14 0.16 0.18 0.20 T [GeV] s / T without mass cutoffwith mass cutoff m <1.5 GeV Figure 1: Thermodynamic quantities, the pressure (left), the energy density (middle), and the entropy (right), calculated in the HRG modelwith and without imposing the mass cutoff m <
Λ with Λ = 1 . that the changes of the chemical freezeout curve are assmall as around 10 MeV.We quantitatively study the effect of Λ. In Fig. 1 weplot the thermodynamic quantities with and without thecutoff from Eq. (14) in the standard non-rotating HRGmodel. The left panel shows the pressure p , the middleshows the energy density ε , and the right shows the en-tropy density s as functions of T . To check the validityof our simplification with Λ, we shall compare the criticaltemperature T c read out from a thermodynamic criterion.The critical temperature without Λ is known from thelattice-QCD simulation as T c = 154 MeV [38]. We canfind the corresponding critical p/T , ε/T , and s/T at T c from the crossing points of the orange dashed curvesand the dotted vertical lines. Then, we can estimate theΛ modified T c from the crossing points of the blue solidcurves and the dotted horizontal lines in Fig. 1. The shiftsin T c read out from p/T , ε/T , and s/T are 3 . . . T c are less than 10 MeV.In conclusion, our simplification by Λ = 1 . T c and also at the quantitative level the possible error is ∼ ω as well.Now let us discuss the deconfinement phase boundariesat finite µ and ω . For this purpose we should make thethermodynamic quantities not only with T (as in Fig. 1)but with some proper combination of T , µ , and ω . Weemploy the normalization given by the Stefan-Boltzmannlimit of a rotating quark-gluon gas: p SB ≡ ( N − p g + N c N f ( p q + p ¯q ) , (20)where the number of colors and flavors are N c = 3, N f = 2,respectively. The gluon pressure reads: p g = − T π ∞ (cid:88) (cid:96) = −∞ (cid:90) Λ IR (cid:96) dk r (cid:90) dk z (cid:2) J (cid:96) ( k r r ) + J (cid:96) +2 ( k r r ) (cid:3) × log (cid:40) − exp (cid:34) − (cid:112) k r + k z − ( (cid:96) + 1) ωT (cid:35)(cid:41) . (21)Here, we note that the possible angular momenta are only j = (cid:96) − j = (cid:96) + 1 and there is no contribution from [ G e V ] [ G e V ] T c [GeV] Figure 2: Deconfinement transition surface as a function of thebaryon chemical potential µ and the angular velocity ω . s z = 0 because gluons are massless gauge bosons. Thisis why J (cid:96) ( k r r ) + J (cid:96) +2 ( k r r ) appears above. The quarkpressure reads more straightforwardly: p q = − T π ∞ (cid:88) (cid:96) = −∞ (cid:90) Λ IR (cid:96) dk r (cid:90) dk z (cid:2) J (cid:96) ( k r r ) + J (cid:96) +1 ( k r r ) (cid:3) × log (cid:40) (cid:34) − (cid:112) k r + k z +( (cid:96) + ) ω − µN c T (cid:35)(cid:41) (22)and the anti-quark pressure, p ¯q , takes almost the sameexpression with µ → − µ .Here our criterion for the deconfinement transition isprescribed, in a way similar to Ref. [39], as pp SB ( T c , µ, ω ) = γ . (23)Here, γ is a constant, which is chosen to reproduce T c ( µ = ω = 0) = 154 MeV in accordance with the lattice-QCDresults [38]. This condition fixes γ = 0 .
18 in our calcula-tion. Now we can numerically solve Eq. (23) to identify T c = T c ( µ, ω ) as plotted in Fig. 2.Now it is evident that T c is a decreasing function withincreasing ω just like the behavior along the µ direction.We cannot directly study the chiral properties within theHRG model, but it is conceivable that the deconfinement5 .00 0.03 0.06 0.09 0.12 0.15 0.18 r [GeV ] p
1e 6 = 0.1GeV= 0.2GeV= 0.3GeV
Figure 3: ∆ p as a function of r for three different values of ω . T c and the chiral restoration temperature are linked evenat finite ω . We can also notice that the effect of ω makes T c drop faster than that of µ . We understand this from the ω induced effective chemical potential which is proportionalto (cid:96) + S i . Because (cid:96) becomes arbitrarily large, the systemcan be more sensitive to the effective chemical potentialthan the baryon chemical potential. From our parameterfree analyses we make a conclusion that the deconfiningtransition temperature is lowered by the rotation effect.
7. Revisiting the radial dependence
It would be an interesting problem to make systematicinvestigations of the r and ω dependence in the pressure.The main focus of the present work is the survey of thephase diagram, so we will not go into systematic discus-sions here. Still, it would be instructive to verify our phys-ical interpretation of the r and ω dependence in Eq. (19)from the numerical calculation.We fix the temperature, T = 0 .
15 GeV, and change r for three different values of ω = 0 .
1, 0 .
2, 0 . r is [0 . , .
17] GeV − . Our numerical calcula-tions lead to the r dependence as shown in Fig. 3. Wehave checked that each curve on Fig. 3 is well fitted by aquadratic function ∝ r as expected from Eq. (19). Fromthis quadratic r dependence we can numerically estimate σ defined in Eq. (19). For ω = 0 . p/r (cid:39) . × − GeV . Thecorresponding value of σ is σ (cid:39) .
21, from which we caninfer, ν ( ω = 0 . (cid:39) . (24)For different ω the results are slightly changed, but of thesame order. This value of ν is comparable to the thermaldegrees of freedom of light mesons, i.e., pions and Kaons.We have a full expression of Eq. (12) and we do not haveto rely on an Ansatz like Eq. (19). In this sense the abovementioned estimate of ν should be understood as a consis-tency check. It would be a very intriguing question to seethe spatial distribution of the angular momentum density, (cid:104) j (cid:105) ( r ), as well as the moment of inertia, dI ( r ), directlyfrom Eq. (12). We will report a thorough analysis in aseparate publication and stop our discussions at the levelof the consistency check in the present paper.
8. Summary
We studied the effect of rotation on the deconfinementtransition from hadronic to quark matter. We devised thehadron resonance gas (HRG) model in a rotating frameand formulated a practical scheme for the pressure calcu-lation that is dependent on the radial distance r from therotation axis. Adopting a working criterion for deconfine-ment in the view of the Hagedorn picture, we found thatincreasing the angular velocity ω lowers the deconfinementtransition temperature, which is similar to the effect ofbaryon chemical potential. We then drew the 3D phasediagram of rotating hot and dense matter in Fig. 2. Ourphysics discussions include not only the phase diagram butalso the physical interpretation of the spatial dependenceof the pressure. The numerical results are consistent withthe physical interpretation in terms of the moment of in-ertia.There are many interesting directions for future ex-tensions. In the context of the QCD phase diagram re-search it would provide us with an inspiring insight tostudy whether the deconfinement and the chiral restora-tion transitions should be locked or unlocked by rotationaleffects. Also, a more comprehensive analysis involving themagnetic field on top of rotation would be desirable forphenomenological applications. Since we formulated thepressure as a function of r and ω , it have paved a clearpath for the microscopic computation of the angular mo-mentum and the moment of inertia of rotating hot anddense matter. Such quantities should be valuable for phe-nomenological modeling. We are making progress alongthese lines using perturbative QCD as well as the HRGmodel. Acknowledgments
This work was supported by Japan Society for the Pro-motion of Science (JSPS) KAKENHI Grant Nos. 18H01211(KF,YH), 19K21874 (KF), 17H06462 (YH), and 20J10506(YF).
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