Critical temperature of deconfinement in a constrained space using a bag model at vanishing baryon density
aa r X i v : . [ nu c l - t h ] O c t Critical temperature of deconfinement in a constrained space using a bag model atvanishing baryon density
Zonghou Han, Baoyi Chen, and Yunpeng Liu Department of Applied Physics, Tianjin University, Tianjin 300350, China
The geometry of fireballs in relativistic heavy ion collisions is approximated by a static box, whichis infinite in two directions while finite in the other direction. The critical temperature of deconfine-ment phase transition is calculated explicitly in the MIT bag model at vanishing baryon density. Itis found that the critical temperature shifts to a value higher than that in an unconstrained space.
PACS numbers: 25.75.Nq, 12.38.Mh, 64.60.an, 12.39.Ba
I. INTRODUCTION
Phase transition is one of the most important topics in statistical physics. Besides the well-known liquid-gas phasetransition of nuclear matter [1–8], in nuclear physics it is widely believed that quark matter becomes deconfinedat extremely high temperature and/or density. In theory, the lattice quantum chromodynamics (QCD) predicted acrossover at low baryon density [9, 10], while other models predicted a first order phase transition at high baryondensity [11–13]. In experiments, the quark-gluon plasmas (QGPs) are expected to be found in two kinds of systemsat quite different scales, neutron stars [14] and relativistic heavy ion collisions [15–23]. The former is far larger thana usual real object in condensed matter, while the latter is comparable to that of a nucleus. The order of magnitudeof the transition temperature can be estimated in a very rough but simple way that the radius of a nucleon is about1 fm, therefore the typical temperature to break it up is the inverse of the radius 1 / (1 fm) ≈ . L = 2 ct , where c is the speed oflight and t is the time after the collision. In the first few fm/ c , the length L is obviously smaller than the transversesize of the fireball. Thus the finite size effect is mainly due to the finite length L . Meanwhile the baryon densityin high energy nuclear collisions is also small. In this paper, we focus on the L dependence on the phase transitiontemperature T c in a simple bag model at zero baryon density phenomenologically.The MIT bag model in an unconstrained space is reviewed shortly in Sec. II. It is generalized to a constrained spacein Sec. III to obtain the shifted critical temperature T c . The results are summarized in Sec. IV. II. BAG MODEL IN UNCONSTRAINED SPACE
In the MIT bag model [24, 34, 35], the system experiences a first-order phase transition at critical temperature T c ,which is determined by the mechanical equilibrium condition p QGP = p HG , (1)where p QGP and p HG represent the pressures of QGP and hadron gas, respectively. The pressure of QGP is p QGP = p QGP − B, where p QGP is the pressure of an ideal parton gas, and B is the bag constant counting for the non-perturbtiveinteraction, while the hadron gas is calculated as the ideal gas for simplicity.In relativistic heavy ion collisions, the scale in the beam direction is usually much smaller than that in the transversedirection at early stage of the fireball. As an approximation, we consider a static box, which is constrained in onedirection (taken as the beam direction) with its thickness L and it is not constrained in the other two directions (asthe transverse directions).In such a system, the pressure is not homogeneous, and can not be compared between two phases directly. Thuswe first translate Eq. (1) into the partition function Z in a grand canonical ensemble. Note that for an unconstrainedsystem, the pressures is p = TV ln Z , where T and V are the temperature and volume of the system, respectively. ThenEq. (1) becomes TV ln Z QGP − B = TV ln Z HG , (2)or in a dimensionless form β ln Z QGP V − β B = β ln Z HG V (3)with ln Z QGP = X i ∈{ partons } ln Z i , ln Z HG = X i ∈{ hadrons } ln Z i ,β = 1 T .
In the above, the partition function of one particle i isln Z i = ± Z d ε D i ( ε ) ln (cid:0) ± e − βε (cid:1) , (4)where D i ( ε ) is the density of single particle states at energy ε . The upper and lower signs hereafter are for fermionsand bosons, respectively. Because in high energy collisions the baryon density is small, we have taken the chemicalpotential as zero. In unconstrained space, the density of states reads D i ( ε ) = g i Z d x d p (2 π ) δ (cid:18) ε − q p + m i (cid:19) = g i V εp i π , (5)with p i = p ε − m i in the last term, and g i and m i are the degeneracy of inner degree of freedom and the mass ofparticle i , respectively. Substituting Eq. (5) into Eq. (4) yields β ln Z i V = g i I ± ( ¯ m i )2 π , (6)with I ± ( ¯ m ) = Z + ∞ xf ± ( x, ¯ m )d x. and f ± ( x, ¯ m ) = ± x ln (cid:16) ± e −√ x + ¯ m (cid:17) , with ¯ m i = βm i . In the following, we will omit the subscript ± for simplicity unless it is necessary to be written out.We take the constituents of QGP as massless gluon, massless u and d quarks, and s quark with its mass m s = 150MeV, and the constitute of hadron gas as all hadrons in particle listing [36] below 2 GeV. We also take the bagconstant as B = (236 MeV) , then a critical temperature of T c = 165 MeV is obtained. III. BAG MODEL IN A SPACE CONSTRAINED IN ONE DIRECTION
Now we consider a box, which is infinitely large in two directions, but with a finite length L in the other directionalong the z -axis. We assume that Eq. (3) still holds with the same bag constant B . Note that the group velocity ofthe fireball is strictly constrained by the speed of light, so that no wave function can exceed the lightcone. That is, forparticle i , its wave function ψ i ( x, y, z, t ) vanishes at z = ± ct = ± L , at given time t . This corresponds to a Dirichletboundary condition on the wave functions of the particles. Therefore we work out the calculation mainly under sucha boundary condition. The difference between the results under the Dirichlet boundary condition and the periodicboundary condition will be shortly discussed at the end of this section.For a particle constrained in z direction under the Dirichlet boundary condition, the density of states is D i ( ε ) = g i Z d x T d p T (2 π ) ∞ X n z =1 δ (cid:18) ε − q p T + p z + m i (cid:19) (7)with p z = (cid:0) n z πL (cid:1) . Working the integral out, one can find D i ( ε ) = g i V ε πL (cid:20) p i Lπ (cid:21) . (8)The square brackets here stand for the floor function. With the help of Poisson summation formula, the correspondingresult to Eq. (6) is β ln Z i V = g i I ( ¯ m i )2 π (cid:18) R ( Λ, ¯ m i ) − Λ R ( ¯ m i ) (cid:19) , (9)with R ( Λ, ¯ m ) = 2 I ( ¯ m ) + ∞ X k =1 Z + ∞ d xf ( x, ¯ m ) sin( kΛx ) kΛ ,R ( ¯ m ) = πI ( ¯ m ) Z + ∞ f ( x, ¯ m )d x,Λ = 2 LT.
Details of the calculation can be found in the appedix. The first term 1 in the brackets on the right-hand side ofEq. (9) is exactly the result in unconstrained space; the second term counts the difference between summation andintegral; the third term is from the missing zero-mode ( p z = 0 mode) under the Dirichlet boundary condition. Onecan calculate directly to find R = q π ¯ m K / ( ¯ m ) K ( ¯ m ) = q π ¯ m (cid:16) −
78 ¯ m +15 (cid:17) in a large ¯ m limit and R ± = ∓ ζ (3) ζ (4) π at¯ m = 0. If we replace the definition of f by a classical distribution f cl ( x, ¯ m ) = xe − β √ x + ¯ m , then we have R = π ,which will be used to replace R ± in an estimation later in this section. The values of R and those of R as a functionof ¯ m are shown in Figs. 1 and 2, respectively. They keeps positive and decreases with ¯ m monotonically. As shown inthe figures, the correction from the R term is always larger than that from the R term in the whole range of ¯ m at Λ >
1, and the R term is even negligible at Λ >
3. As a result, the correction is always negative as
Λ >
1. It canalso be seen that the correction is stronger for light particles, and therefore is more strong to the QGP phase thanthe hadron phase. Then the shift of critical temperature can be understood.As an example, we compare the grand potential density ω at L = 4 fm with that at L = + ∞ of both the QGPphase and the hadron gas phase, where the grand potential density is ω ( T, L ) = − T ln Z QGP V + B, for QGP , − T ln Z HG V , for Hadron gas . To be dimensionless, we further scale the ω by a constant ω ( T c , + ∞ ) to define r ( T, L ) = ω ( T, L ) ω ( T c , + ∞ ) . (10)The values of r ( T, L ) at L = + ∞ and L = 4 fm as a function of T is shown in Fig. 3. Because the system in anequilibrium state always minimizes the grand potential, and the constant in denominator on the right hand side ofEq. (10) is negative, the larger r is preferred by nature, and the phase transition happens at the cross point (in red)of the two states of matters. At L = 4 fm, both the curves of the QGP and the hadron gas become smaller due to thecorrection of the R term, and the shift for the QGP is larger due to the small mass of the partons. Therefore thecritical temperature shifts to a higher value. As a result, the critical temperature shifts from 165 MeV to 179 MeV.Repeating this process, one finds the critical temperature as a function of the length L of the system, as shown inFig. 4. Since there is no qualitative difference between different L s, the critical temperature T c shifts to higher values
0 5 10 150.00.51.01.5 m R Boson F e r m i on FIG. 1: Values of R as a function of ¯ m . =1 Boson Λ =1 Fermion Λ =2 Boson Λ =2 Fermion Λ =3 Boson Λ =3 Fermion Λ
0 5 10 150.00.20.40.60.8 m R (cid:9)(cid:9) (cid:9)(cid:9) (cid:9)(cid:9) FIG. 2: (Color online) Values of R as a function of ¯ m . The solid, long dashed, dot-dashed curves are for Λ = 1, 2, and 3,respectively. At each Λ , the upper thicker curve (in blue) is for bosons, while the lower thinner one (in red) is for fermions. in the whole range of L than at L = + ∞ . It can be seen that even at L = 8 fm that is the scale of the radius of anucleus, the shift of T c is still above 5 MeV. It should be noticed that the results at small L ∼ T c for L > s quark in the QGP phase. 2) In the range of L > T ≥
165 MeV, it can be verified directly that
Λ >
3. According to previous discussion, the correction is dominatedby the R term, and we can safely neglect the R term in Eq. (9). 3) Because the qualitative behavior of bosons and
150 160 170 180 190 200 2100246 , QGP ∞ L=+ , HG ∞ L=+L=4 fm, QGPL=4 fm, HG
T (MeV)r (cid:9)(cid:9) (cid:9)
FIG. 3: (Color online) The values of r as a function T at L = + ∞ (maroon) and L = 4 fm (black). Phase transition happensat the red cross point. L (fm) ( M e V ) c T (cid:9) FIG. 4: Critical temperature T c as a function of box length L . fermions are similar, we take a classical limit to neglect the differences between them, that is R = π . With theseassumptions, one obtains the grand potential density of QGP as follows: ω ( T, L ) = − gT π V (cid:16) − π Λ (cid:17) + B (11)with g = P i ∈{ partons } g i . Note that the slope of ω is the entropy density up to a minus sign, it is reasonable to assumethat the entropy in the QGP phase is sizeably larger than that in the hadron gas in such a first order phase transitionmodel. Therefore we further make a rough approximation 4) to neglect the change of ω with respect to T for thehadron gas. Besides, 5) for the same reason as in 1), the correction to hadron gas due to finite L is also neglected.These two assumptions require that ω ( T c , L ) is a constant. Substituting Eq. (11) into this condition yields T c ( L ) = T c (+ ∞ ) (cid:18) − π LT c ( L ) (cid:19) − ≈ T c (+ ∞ ) + π L . (12)Taking L = 4 fm, we have ∆ T c = T c ( L ) − T c (+ ∞ ) ≈ π × = 10 MeV, which is qualitatively consistent with ourprevious calculation. It can be seen from Fig. 3, that the main deviation comes from assumptions 4) and 5).Note that the main correction is from the R term, which is the zero-mode contribution that vanishes under theDirichlet boundary condition. If a periodic boundary condition is used instead, i.e., p z = πn z L with n z ∈ Z in Eq. (7),then Eq. (9) can be replaced by ln Z i = g i V I ( ¯ m i )2 π β (cid:20) R (cid:18) Λ , ¯ m i (cid:19)(cid:21) . The critical temperature is only slightly smaller than that in unconstrained space (because of the R term). Forexample, at L = 4 fm, it gives T c = 164 . . L = + ∞ . The T c shifts to the opposite direction because the sign in front of R is opposite to the sign in front of R in Eq. (9). IV. SUMMARY
We have calculated the shift of the critical temperature T c in a constrained space in the bag model as a simplifiedmodel for the fireball in relativistic heavy ion collisions. When constrained in one direction, the amplitude of thegrand potential density (which is the pressure in an unconstrained space) of ideal gas becomes smaller at the sametemperature due to the Dirichlet boundary condition, especially for light particles. As a result, this effect is strongerfor parton gas. As a result the new balance between QGP and hadron gas can only be established at a higher criticaltemperature T c than that in an unconstrained space. A rough estimation of the shift of the critical temperature∆ T c = π L is also given for relatively large L . Acknowlegements
This work is supported by NSFC under grant numbers 11547043 and 11705125, and by the “Qinggu” project ofTianjin University under grant number 1701.
Appendix A: Derivation of Eq. (9)
By substituting Eq. (8) into Eq. (4), we haveln Z i = ± Z d εD i ( ε ) ln (cid:0) ± e − βε (cid:1) = ± g i S π Z d ε ε (cid:20) p i Lπ (cid:21) ln (cid:0) ± e − βε (cid:1) = ± g i S π Z d p p (cid:20) pLπ (cid:21) ln (cid:16) ± e − β √ p + m i (cid:17) = ± g i S πβ Z d x x (cid:20) xLπβ (cid:21) ln (cid:16) ± e − √ x + ¯ m i (cid:17) = g i Vπβ Λ Z + ∞ d x f ( x, ¯ m i ) (cid:20) Λx π (cid:21) . (A1)We omit the variable ¯ m i in f for simplicity in the following. Define F ( x ) = Z x −∞ f ( ξ )d ξ. (A2)Then the following equations can be verified directly F ( − x ) = F ( x ) , (A3) F (+ ∞ ) = 0 , (A4) F ( x ) = − Z + ∞ x f ( ξ )d ξ. (A5)Therefore, integrating by part, we have Z + ∞ d x f ( x ) (cid:20) Λx π (cid:21) = Z + ∞ d F ( x ) (cid:20) Λx π (cid:21) , = − Z + ∞ F ( x )d (cid:20) Λx π (cid:21) , = − + ∞ X n =1 F (cid:18) πnΛ (cid:19) , = − " + ∞ X n = −∞ F (cid:18) πnΛ (cid:19) − F (0) . (A6)In the above, we have taken advantage of Eqs. (A3) and (A4). Substituting Eq. (A6) into Eq. (A1), and applyingthe Poisson summation formula to the summation, we haveln Z i = − g i V πβ Λ " + ∞ X k = −∞ Z + ∞−∞ F (cid:18) πxΛ (cid:19) e i πkx d x − F (0) = − g i V π β " + ∞ X k = −∞ Z + ∞−∞ F ( y ) e ikΛy d y − πΛ F (0) = − g i V π β "Z + ∞−∞ F ( y )d y + + ∞ X k =1 + −∞ X k = − ! Z + ∞−∞ F ( y ) e ikΛy d y − πΛ F (0) = g i V π β "Z + ∞ xf ( x )d x + 2 + ∞ X k =1 Z + ∞ f ( y ) sin( kΛy ) kΛ d y − πΛ Z + ∞ f ( x )d x = g i V I π β (cid:18) R − Λ R (cid:19) . This is Eq. (9). On the last but one line, we have inserted Eq. (A5). [1] J. Pochodzalla et al. Probing the nuclear liquid - gas phase transition.
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