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Properties of Baryonic, Electric and Strangeness Chemical Potential and Some of Their Consequences in Relatiistic Heavy Ion Collisions

Aram Z. Mekjian

Rutgers University, Department of Physics and Astronomy, Piscataway, NJ. 08854 & California Institute of Technology, Kellogg Radiation Lab 106-38, Pasadena, Ca 91125

Abstract Analytic expressions are given for the baryonic, electric and strangeness chemical potentials which explicitly show the importance of various terms. Simple scaling relations connecting these chemical potentials are found. Applications to ratios and to fluctuations and related thermal properties such as the isothermal

compressibilityκT are illustrated. A possible divergence ofκT is discussed.

PACS numbers: 25.75-q,25.75.Gz, 25.75.Nq

1.Introduction The behavior of hadronic matter at moderate to high temperatureT and density ρ is being studied by heavy ion collisions. Moderate energy collisions, such as those done at National Superconducting Cyclotron at MSU and at GANIL, focus on the liquid/gas phase transition[1]. Much higher energy collisions[2] at CERN or BNL RHIC investigate the very hot and dense regions of phase space which probe a -gluon phase. This phase is then followed by an expansion to lower ρ andT where the colored and anti-quarks form isolated colorless objects which are the well known whose properties are tabulated in ref[3]. Experiments at the proposed FAIR [4] will study the phase diagram of hadronic matter at aT and ρ which fills the gap between medium energy and high energy collisions. Statistical models [1,5-7] of such collisions assume an equilibrium is produced and its predictions are compared with experiment. Properties of the observed particles depend on chemical potentials associated with conservation laws such as number B and electric chargeQ conservation. At higher energies, when strange particles are also produced, strangeness conservation is also present in the strong interactions. This paper focuses on the behavior and properties of the three chemical potentials µ B , µQ , µS . Simple analytic expressions are developed for µ B , µQ , µS which explicitly show the importance of various quantities that appear. These expressions are then used to study particle ratios, particle asymmetries such as the baryon/antibaryon asymmetry, fluctuations and thermal properties of hadronic matter. Results based on a Hagedorn resonance gas model are developed. A possible divergence of the isothermal compressibility is discussed. The critical exponent associated with this divergence is related to properties of the vanishing of chemical potentials withT and the mass spectrum of excited states. 2. Properties of the Phase 2.1 General Results The statistical model of relativistic heavy ion collisions [5-7] assumes that a frozen equilibrium is established in a volumeV and at a temperatureT and proceeds to explore its consequences. The particle yields < Ni > can be written in a simplified notation as

bi qi (−si ) < Ni >= ai y z (1)

3 2 2 in the non-degenerate limit. The ai = g s (i)(VT / 2π )(mi / T ) (K 2 (mi / T ) ). Small mass differences between different states of the same particle will be ignored in ai .The

gs (i) is the spin degeneracy of particlei , which has mass mi , baryon numberbi , charge

qi and strangeness si . The x ≡ exp[µB /T], y ≡ exp[µQ / T ] andz ≡ exp[−µS / T ] are determined by the constraints on B , chargeQ and strangeness S that read:

B = = ΣbI < Ni > , Q = Σqi < Ni > , S = Σsi < Ni > . Because total B,Q, S are each

conserved so are the Y = B + S = Σ(bi + si ) < Ni > and the third component of I Z . The I Z is connected to the hyperchargeY = B + S through the Gell-Mann

Nishijima expression[8]: 2I Z = 2Q − Y = Σ(=2qi − bi − si ) < Ni > Z − N . The Z − N is determined by the initial proton/neutron asymmetry in the target and projectile. In a

heavy ion collision the net strangeness is zero, butµs ≠ 0 . The negative strangeness is from such as Λ0 , Σ+ ,Σ0 , Σ− and anti- K − , K 0 , while the positive strangeness is basically in kaons K + , K 0 . The simple describes the considered here (strange and non- strange and ) in terms ofu, d, and s quarks andu, d , and s anti-quarks. Since baryons are restricted to three quarks and mesons to quark, anti-quark pairs, the

quantum numbers of the hadrons are restricted tobi = ±1, 0 , qi = ±2,±1,0 , and

si = ±3,±2,±1,0 . Contributions of excited states of a given particle can be added to the

lowest members contribution so that ai → Ai . For instance, the number of ∆ like 2 particles < ∆(1232) > + < ∆(1600) >+ … = A∆ xz(y + y +1+1/ y) where A∆ = a(∆(1232) ) * + a(∆(1600) )… . The Aπ will contain π , ρ,... and AN hasp, n, N ,.... Particles are

bi qi (−si ) grouped with J = {N, ∆, Λ, Σ, Ξ, Ω,π , K}and < N J >= AJ x y z . An exponentially increasing Hagedorn density of states can be included as discussed below in sect. 2.3.

2.2 Coupled equations for x, y, z or the chemical potentials µ B , µQ , µS .

The Gell-Mann/ Nishijima expression for 2I Z = 2Q − Y as a constraint condition gives:

2 −1 −1 2 −1 2 I Z = AN x(y −1) + A∆ x(3y + y −1− 3y ) + AΣ xz(2y + 0y − 2y ) + AΞ xz (−y +1) −1 −1 + Aπ (2y + 0y − 2y ) + AK {z(−y +1) − (1/ z)(− y +1)} − 2IW ,Z (B) (2)

The 2 I Z (B) , which is the anti-baryon contribution, is obtained by taking reciprocals of x, y and z . Every numerical coefficient in front of y , when divided by 2, is just the isospin I Z component of each charged state of particle type J .

For S = 0 , 2I Z = Z − N and for symmetric N = Z systems I Z = 0. Eq.(2) explicitly shows that when I Z =0 then y = 1. At y = 1, the µQ = 0. For an asymmetric system

N ≠ Z and thus y ≠ 1. Usually / µQ / < Z and

(N − Z) / B << 1. A lowest order expansion around y = 1gives µQ ~)(Z − N with

µ 2I = 2Q − Y = Z − N Q = Z (3) 1 1 1 1 T (A +10A )(x + ) + 4A (xz + ) + A (xz 2 + ) + 4A + A (z + ) N ∆ x Σ xz Ξ xz 2 π K z from eq.(2). Three terms in the denominator of eq.(3) are important. The large factor of

10 in 10A∆ makes ∆ central in µQ / T . The abundance of coming from the collision results in a large contribution through the 4Aπ factor. The initial protons in the target and projectile are contained in the AN x factor. Of much less importance are the contributions in µQ .The evaluation of x, z for I Z = 0 where y = 1 is given next. As a first approximation these values also apply when I Z ≠ 0 since / µQ / is usually << µS , µB .

The B = N B − N B constraint equation reads: 1 1 1 z3 N = x {A (y +1) + A (y 2 + y +1+ ) + A z + A z(y +1+ ) + A z 2 ( +1) + A } B N ∆ y Λ Σ y Ξ y Ω y 2 3 → (y = 1) → x{2AN + 4A∆ )+ AΛ z +3AΣ z + 2AΞ z + AΩ z } ≡ xC(z) (4)

2 3 The coefficient {(2AN + 4A∆ ) + AΛ z +3AΣ z + 2AΞ z + AΩ z } ≡ C(z) is a polynomial in z .

The N B is obtained from N B by taking the reciprocal of x, y and z so that N B =C(1/ z) / x .

Given that xonly appears in N B and 1/ x in N B , a quadratic equation in x arises from the B constraint equation which reads B = C(z)x − C(1/ z) / x or C(z)x2 − Bx − C(1/ z) = 0 .

2 Thus x = ( B + B + 4C(z)C(1/ z) )/ 2C(z) . Since each AJ ~,V x = f (B /V ,T ) . The result of eq.(4) can be substituted into: − S = N S− − N S+ =0 to find z . The N S− is :

2 3 N S− = x{AΛ z + AΣ z(y +1+1/ y) + 2AΞ z (1/ y +1) + 3AΩ z / y }+ AK z(1/ y +1) 2 3 → (y = 1) → x{AΛ z + 3AΣ z + 4AΞ z + 3AΩ z }+ 2AK z (5)

The N S+ is obtained by taking the reciprocal of x, y and z . The solutions to the resulting polynomial equation for z simplify in certain limits which are now discussed.

2.2 Simplified solutions in symmetric N=Z systems at y=1 and µS / T <<1 or z ≈ 1.

WhenµS / T << 1, z = 1− µS / T , and µS / T is given by

2 µS / T = sinh(µB /T)H S,1 /1 (2AK ) + cosh(µB /T)H S,2 (6)

j j j j The H S , j =)1 (1AΛ ) +1 (3AΣ ) + 2 (2AΞ ) + 3 (1AΩ are strangeness moments of the distribution. If µB /T <<1, cosh(µB / T) →1, sinh(µB / T) → µB / T , and

2 µS / T = (µB / T)H S,1 /(1 (2AK ) + H S,2 ) ≡ CSB µB / T (7)

The slope of µS versus µB involves the magnitude of the strangeness in hyperons to the strangeness fluctuation including kaons. When x ≈ 1, z ≈ 1, y ≈ 1, the < Ni >≈ ai . A factor similar toCSB appears in the discussion of a B / S correlation in ref.[9]. For large µB /T ,

{sinh(µB /T),cosh(µB / T) } → exp(µB / T) / 2 . Substituted this result into Eq.(6) gives a non-linear relation between µS / T and µB /T. The µB /T satisfies:

2 B µ B 2AK H S,0 + (H S,0 H S,2 − (H S,1) ) cosh(µB /T) = sinh( )( 2+AN + 4A∆ ) (8) 2 T 2AK + H S,2 cosh(µB /T)

At low T , where x >>1, and z is small, the main contribution comes from S = ±1 strange particles ( K, Λ, Σ) . Without Ξ, Ω terms, the relation between µS / T, µB / T is now

1(AΛ + (3AΣ )) tanh(µS / T ) = sinh(µB / T ) 2 2 (9) 1 (2AK ) + cosh(µ B / T )[1 (AΛ + (3AΣ ))]

With µB /T >>1,cosh(µB /T) andsinh(µB /T) → exp( µB /T) /2. The hypercharge equationY = (2AN + 4A∆ )2sinh(µB / T) + 2AK 2sinh(µS / T) gives µB /T in closed form. 2.3 Role of the Hagedorn mass spectrum, vanishing of chemical potentials and B, B asymmetries. −τ The Hagedorn density of excited states is ρ = Dτ m exp(β hm) , where Dτ is a constant and βh =1/T0 . TheT0 is the limitingT and the exponentτ is a parameter. This density will affect AJ which are sums over the lowest plus all exited states of J . For a Hagedorn τ −5/ 2 ∞ −x (τ −3/ 2) density the AJ ~ V (y / mJ 0 ) (∫ y (dxe / x )) with y = (T0 − T )mJ 0 / T0T . The mJ 0 is (5/ 2−τ ) the lowest mass of particles of type J . For τ <5/2, AJ → ∞ as 1/ (T0 − T ) . An infinite AJ will result in µ B , µS , µQ → 0 and the associated fugacities x, y, z → 1. As a simplified example, a system with one chemical potential µB has:

N − N 2π y  e−x  ( B B )( )3/ 2 = 2D ( )τ −5/ 2 ∫∞ (dxe )sinh(µ / T) (10) τ  y τ −3/ 2  B V T mJ 0  x 

(5/ 2−τ ) Forτ <5/2, µB will go to zero as (T0 − T ) , whenT → T0 , the Hagedorn limitingT . ∞ −x (τ −3/ 2) The behavior of I y,τ ≡ ∫ y (dxe / x )) → Γ(5 / 2 −τ ) withτ for y → 0 is essential for determining the properties of AJ and µB . For τ >5/2, µB will go to a constant that depends τ −5/ 2 ∞ −x (τ −3/ 2) onτ . For τ =5/2+η, (y) ∫ y dxe / x = 1/η as y → 0 . When 5/2<τ <7/2, the 1/ 2 1/ 2 constant1/η is approached with ∞ slope: y I y,3 = 2(1− (πy) ) . For τ > 7/2, the slope is τ −5/ 2 5/ 2−τ finite. The (NB + N B )~/V y I y,τ cosh(µB /T) which diverges as 1/ (T0 − T ) for − 2 τ < 5 / 2 and forτ → 5 / 2 this quantity → − ln(m0 (T0 − T ) / T0 ) . The asymmetry

Asy(BB ) = (N B + N B ) /(N B − N B ) = coth(µB / T) = (x +1/ x) /(x −1/ x) , with N B − N B = B fixed. An ∞ density presents a problem if particles are not point like. The model is thus limited to ρ andT where composite baryons and mesons don’t overlap. At some 3 value ρ ~1hadron/ fm andT = TQg < T0 , a transition to a quark-gluon phase occurs which truncates behaviors based on this particular model. A discussion of this feature can be found in ref[10] where the Qg phase is treated in a statistical model of Qg bags. 1/ 2 In [6], data are fit to a statistical model with µ B = 80.85(T0 − T ) in MeV andT0 = 167 . 1/ 2 A (T0 − T ) dependence occurs forτ = 2.. This form for µB will be used. Then:

1/ 2 x = exp(µB / T) = exp [ (80.85 / T )(T0 − T ) ] (11)

1/ 2 Because of the large cost factor 80.85T0 / T = 1045 / T ~ mN /,T largeAsy(BB) occur only ifT ≈ T0 . AtT = .99T0 , Asy(BB) ≈ 2 and atT = .9T0 , Asy(BB) =1.06 . When

T >> µB , Asy(BB) → T / µ B .

Given x or µB , the µS and z follow from results noted above while eq.(3) gives µQ and y .

Simple scaling laws relating µS and µQ to µB are developed in sect.2.5. 2.4 Particle ratios and fluctuations Results from the previous section can be used to study particle ratios prior to resonance decays. The final distributions of particles seen experimentally are changed by decays. A few examples are now presented which will illustrate the importance of various factors. + − + − When y = 1, K / K = exp(2 µS / T ). The K / K ratio can also be used to discuss the role of dropping masses [11] which leads to an enhancement of this ratio. The simplest result for this ratio is at low T when µB / T >> 1 and when Ξ, Ω are neglected. Then

+ 2 2 K 1 µB 2mΛ K2 (mΛ / T) + 6mΣ K2 (mΣ / T) − = exp(2 µS / T ) = 2 ≈ 1+ exp( ) 2 K z T 2mK K2 (mK / T ) 3/ 2 3/ 2 µB − mΛ + mK mΛ mΣ − (mΣ − mΛ ) ≈ 1+ exp( ) 3/ 2 {1+ 3 3/ 2 exp( )} (12) T mK mΛ T

+ − The exponential factor µB − mΛ + mK plays a significant role in K / K . For N > Z + − y = exp(µQ / T ) < 1and the K / K = exp(2 µS / T )exp( 2 µQ /T ) is reduced. When µS + − − and µB are connected by eq.(7), the asymmetry (K − K ) / K = 2µS / T =

2CSB µB /T . The CSB is the baryon strangeness correlation coefficient involving the strangeness in hyperons to the strangeness fluctuation in kaons plus hyperons-see eq.(3). − + The π / π ratio = exp(−2µQ / T ) . A good approximation (~1 % error at T=120Mev) is:

− π  N − Z  (2AN + 4A∆ )(x +1/ x) + ≈ 1+ 2  (13) π  B  (AN +10A∆ )(x +1/ x) + 4Aπ

The (π − − π + )/ π + charge asymmetry ratio prior to resonance decays ~(N − Z) / B . This ratio also involves theT dependent amplitudes: AN , A∆ , Aπ , with the numerator containing only the baryon amplitude AN ,A∆ while the denominator also includes the amplitude Aπ . This remark is similar to that for kaons which involvedCSB . Anti-baryons appear as1/ x terms and are small untilT ≈ T0 . At a low enoughT where AN x dominates, theπ − / π + → 1+ 4(N − Z) / B which is a limiting value forπ − / π + . The fluctuation inΦ isδΦ 2 = < Φ 2 > − < Φ >2 . In the grand canonical ensembleδΦ 2 =

2 2 bJ qJ −sJ T (∂ < Φ > / ∂µφ ) = Σφi < Ni > = Σφi ai x y z whereφi is the quantum number associated with particlei . The fluctuations in strangeness, whereΦ = S and µφ = µS , were already shown to be important in the relation between µS and µB as given in eqs.(6,7). The fluctuations in baryon number (where Φ = B and µφ = µ B ) is simply N B + N B since bi = ±1only. In a Hagedorn model this fluctuation can diverge. Such divergences are related to singularities in the baryonic compressibilityκT ,B = − (1/V )(∂V / ∂PB )T . The

(∂PB / ∂V )T = (N B − N B )(∂µB / ∂V )T / T from PBV = (N B + N B )T and (V / T ) (∂µB / ∂V )T

= − (N B − N B ) /(N B + N B ) from B conservation giving:

2 2 < B > (< B > − < B > ) N B + N B V ∂µ B TκT ,B = = = Asy(BB) = −1/( ( )T ) (14) V < B > N B − N B T ∂V

5/ 2−τ From sec.(2.3) , (N B + N B ) diverges as 1/(T0 − T ) for τ < 5 / 2 , while B is fixed at

< B > = (N B − N B ) . A critical exponentγ is associated with the divergence of κT , with γ κT ~1/(T0 − T ) . Consequently, the critical exponentγ and the prefactor exponentτ are connected byγ =5/ 2 −τ . In turn,γ is connected to the exponent in the vanishing 5/ 2−τ behavior of µB ~ (T0 − T ) . The compressibility of a pure gas of massless pions with µ = 0 , including degeneracy terms, is [12] κT ,π =)(ς (2) / ς (3))V /(< Nπ > T which is somewhat larger than an ideal gas resultκT = 1/ P = V /(< N > T) because of the statistical attraction of bosons which increases the compressibility. The zeta functions that appear in this expression for κT arise from degeneracy corrections. The ideal 2 gasκT follows from eq.(14) when δB =< B > , which is the poissonian limit. Then the rhs in the first equality in eq.(14) is 1 and κT ,B =V /(< B > T). By constrast, theκT of a van der Waals gas becomes infinite at a critical point in a liquid gas phase transition which is responsible for the phenomena of critical opalescence. In such a phase transition the density fluctuations become large since the system doesn’t know whether to be in a gas phase or a liquid phase.

Properties of µB are also reflected in the baryonic heat capacityCV ,B at constantV [13]:

2 CV ,B µB ∂ µB 2 µB ∂µB µB 2 = − [coth ( )]⋅T 2 + [csch ( )]⋅T ( − ) (15) N B + N B T (∂T) T ∂T T

Here, the derivatives of µB are with respect toT at constantV . The singularities of

CV ,B are discussed in ref.[13]. The importance of the difference of charge fluctuations in the hadron phase and in the quark gluon phase were discussed in ref.[14,15]. 2.5 Scaling laws

Using eq.(11) for µ B , values of µS and µQ follow from the constraint equations. When

Ξ,Ω contributions are neglected, then eq.(12) is a simple connection of µS to µ B which can be rewritten astanh(µS / T) = f /(1+ f ) with f = (1/ 2) exp(µB / T )R(T ) . TheR(T ) is the ratio of mass terms that appear in eq.(12) with =(aΛ + 3aΣ ) / 2aK . ForT < 100MeV a scaling relation µS / µB ≈ .19 ≈ 1/ 5is found. The behavior of µS / µB ≈ .2 − (.02T /100) is a good approximation for 0 < T < 160 without Ξ,Ω contributions. The Ξ,Ω terms become significant atT ~ 120 MeV . The scaling relation µS / µB ≈ 1/ 5 is still a good approximation with Ξ,Ω terms. Very near T0 , eq.(7) givesµS / µB ≈ . 20 =1/5 and contains not only Ξ,Ω , but also anti-hyperons. Including excited states of hyperons * enhances µS while K ’s decrease µS since (aΛ + 3aΣ ) / aK → (AΛ + 3AΣ ) / 2AK in f .

The µQ follows from eq.(3) and the behavior of µQ is dominated by the N, ∆,π contributions. A scaling law µQ = − ((N − Z) / B)µB /10 also applies. Thus:

1/ 2 1/ 2 µ B ≈ 80.85(T0 − T ) MeV , µS ≈ (15.36 −16.17)MeV (T0 − T ) , 1/ 2 µQ ≈ − ((N − Z) / B)µB /10 =)− ((N − Z) / B 8.085 MeV (T0 − T) (17)

The coefficient 15.36 is when µS / µB ≈ .19 while 16.17 is with 1/5=0.2. Once

µB ,andµS µQ determined, the< Ni /V > then follow from eq.(1). 3.Summary and Conclusions

Properties of the three chemical potentials µ B , µQ , µS were studied using an approach that lead to analytic expressions for them. The Gell-Mann/Nishijima expression for the third component of isospin 2I Z = 2Q − Y, where the hyperchargeY = B + S , was used to obtain the electric chemical potential µQ .The µQ was then shown to depend on the initial proton/neutron asymmetry (Z − N) /(Z + N). The abundance of initial , the large production of charged pions and ∆ ’s with several charged states plays an essential role in determining the value of µQ , besides the asymmetry factor(N − Z) /(N + B ) . The µQ was also shown to scale with µB as µQ ≈ − ((N − Z) / B)(µB /10). Simple connections between the strangeness µS and baryon µB were developed. A main element in this connection was moments of the distribution of strangeness. Specifically, the ratio of strangeness carried in hyperons to the strangeness fluctuation in hyperons plus kaons determined the slope of µS with µB at highT. ForT ≈ T0 µS / µB ≈ 1/ 5 where the1/5 reflects this strangeness ratio. At lowT , where Ξ, Ω contribution can be neglected, a non-linear equation between µS and µB was also obtained. Moreover, solutions to this non-linear equation also show a scaling relation is present that readsµS / µB ≈ 1/ 6valid forT < 140MeV. The role of the Hagedorn mass spectrum in the temperature dependence of chemical potentials was investigated. The prefactor exponent, labeledτ , plays an essential role in the temperature dependence of chemical potentials determining whether they vanish as the critical HagedornT0 is approached. A large baryon/anti-baryon asymmetry, as measured by (NB + N B ) /(N B − N B ) , exists only ifτ ≤ 5 / 2 andT ≈ 0.98T0 . The result thatT has to be within a few % ofT0 has its origin in the high cost associated with mN /.T

The expressions developed for µ B , µQ , µS were used to study particle ratios, such as the charge asymmetry (π − − π + )/ π + , the kaon strangeness ratio K + / K − , prior to adjustments from resonant decays. Terms important in these ratios and asymmetries were + − studied. For example, µB − mΛ + mK is important in K / K . Fluctuations were also investigated and connected to associated thermal quantities such as the relation of the isothermal compressibility to particle number fluctuations. A relation5 / 2 −τ = γ between the prefactor exponentτ and the critical exponentγ , which determines the divergence of the isothermal compressibility, was noted. This work was supported in part by the DOE grant number DE-FG02-96ER-40987. References [1] C.B.Das, S.Das Gupta, W.G.Lynch, A.Z.Mekjian, M.B.Tsang,, Phys. Repts 406,1 (2005) [2] Proceedings of Quark Matter 2004. J. Phys.G, Nucl.&Part. Phys. 30,1 (2005) [3] Review of , Eur. Phys. J. C15 1 (2000) [4] P.Senger, J.Phys. G30, 1087 (2004) [5] P.Braun-Munzinger, K.Redlich and J.Stachel, in Quark Gluon Plasma 3 edited by R.C.Hwa (World Scientific, Singapore, 2004) [6] F.Becattini, M.Gazdzicki, A.Keranen, J.Manninen and R.Stock, Phys. Rev. C69, 24905 (2004) [7] A.Z.Mekjian, Phys. Rev. C17, 1051 (1978), Nucl. Phys. A312, 491 (1978) [8] Quarks and Leptons, F. Halzen and A.D.Martin (John Wiley&Sons, Inc. NY (1984) [9] V.Koch, A.Majumder and J.Randrup, Phys.Rev.Lett. 95, 182301 (2005) [10] M.Gorenstein, M.Gazdzicki and W.Greiner, arXiv:nucl-th/0505050 [11] G.E.Brown and M.Rho, Phys. Repts. 398, 301 (2004) G.E.Brown, B.A.Gelmann and M.Rho, nucl-th/0505037 [12] St. Mrowczynski, Phys. Lett. B430, 9 (1998) [13] A.Mekjian. Phys. Rev. C73, 014901 (2006) [14] S.Jeon and V.Koch, PRL 85, 2076 (2000) [15] M.Asakawa, U.Heinz and B.Muller, Phys.Rev.Lett. 85, 2072 (2000)