The History and Lessons of the Hagedorn Model

The History and Lessons of the Hagedorn Model K.A.Bugaev Bogolyubov ITP, Kiev, Ukraine in collaboration with J.B.Ellio", L.W. Phair and L.G.More"o 1

Outline: What T do we see in A+A and elementary particle reactions? Do these T signal PT? A bit of history: Stat. Bootstrap Model; MIT Bag Model... Problems with GCE. MCE:Properties of Hagedorn resonances = Perfect Thermostats and Particle Reservoirs Open Questions and Conclusions 2

This Talk: Transition between Partonic World: Deconfinement, Chiral Symmetry Restoration, Color S-Conductivity... Hadronic World: Pairing (S-Conductive), Shape Transitions, Nuclear Liquid-Gas Transition T c 18.1 MeV ρ c 0.53 ρ 0 p c 0.41 MeV/fm 3 3

Temperatures in A+A Reactions Lattice QCD at 0 baryonic density: transition T = 170 +/- 10 MeV, F.Karsch, Nucl.Phys.Proc.Suppl. 83(2000) Chemical Freeze-out at highest SPS and all RHIC energies T = 170 +/- 10 MeV : G.D.Yen, M.I.Gorenstein, PRC 59 (1999), P.Braun-Munzinger et al PLB 465 (1999) This T shows that hadronic composition of a created matter (including decay of resonances!) does not change while system expands and cools down. Remarkably, T = Const while s grows by 12 times! 4

Early Hadronization Temperature in A+A Collisions Remarkably, at highest SPS and all RHIC energies it is also T = 170 +/- 10 MeV! This T evidences that momentum spectra of some hadrons are frozen since the moment of their formation! Necessary conditions: heavy hadrons, small cross-sections with other hadrons, no low lying resonances with pions! T* (MeV) 400 300 200 100 + K! + p (+) # # " $ + $ % ' + ' % D J /& & 0 1 2 3 4 m (GeV) No fit, just data Lines: T = T + m v2 2 Predictions for ψ and D K.A.B. et al., PL B 523 (2001) 5

Momentum Spectra at CERN SPS! J/" " transverse momentum spectra indicate: T = 170+/-10 MeV Is their hadronization T! An elaborate Blast Wave approximation was used to fit data M.I. Gorenstein, K.A.B., M. Gaździcki, Phys. Rev. Lett. 88 (2002) 1323011 10 0 10!2 Pb + Pb at 158 A GeV 200 180 Pb + Pb at 158 A GeV QGP! J/" " FREEZE!OUT dn/(m T dm T ) (a.u.) 10!4 10!6 10!8!!! + T (MeV) 160 140 120 # EXPANSION # 10!10 10!12 J/" " 0 1 2 3 m T!m (GeV/c 2 ) $ FREEZE!OUT 100 0.0 0.2 0.4 0.6 0.8! v T 6

m T -Spectra at RHIC s NN = 130GeV ± φ, Ω. transverse momentum spectra and emission volume τ H RH 2 show: T = 170+/-5 MeV is their hadronization T! K.A.B., M. Gazdzicki, M.I. Gorenstein, PRC 68 (2003): χ 2 /ndf = 0.46 λ Ω = 1.09 ± 0.06, y max T = 0.74 ± 0.09, τ H R 2 H = 275 ± 70 fm3 /c 10 2 10 0 RHIC 2000 1800 1600 low p T high p T low p T high p T dn/(m T dm T dy) (GeV!2 ) 10!2 10!4 10!6 "!!!/100 T * (MeV) 1400 1200 1000 800 600 400 RHIC SPS 10!8 0 1 2 3 m T!m (GeV) 200 0 "! J/# # 0 1 2 3 4 m (GeV) φ data: STAR, Phys. Rev. C 65 041901(R) (2002) ; Ω ± data: G. van Buren [STAR], talk at QM2002. predictions for J/ψ, ψ 7

Evident Explanation for A+A reactions For 0 baryonic charge the Particle Ratios (chemical freeze-out) are frozen since hadronization of QGP at T = 170+/-10 MeV. For 0 baryonic charge the kinetic freeze out of some hadrons ( φ, J/ψ, ψ mesons, Ω. hyperons) occurs at their hadronization from QGP at same T! Surprisingly, similar values of T are seen in el. particle collisions! 8

Hadronization in Elementary Particle Collisions Stat. Hadronization Model: T = 175+/-15 MeV F.Becattini,A.Ferroni, Acta. Phys. Polon. B 35 (2004) %./012345/6423%7%(8*+, #-" #""!-" %9 : ;!! : ; %;%; %* : *! %;%;!!""!" "!"!!" #!" $! %&!'# %()*+,! 9

In wide Kaon Inverse Slopes in Elementary Particle Collisions s range T = 180 +/-20 MeV, M.Kliemant, B.Lungwitz, M.Gazdzicki, PRC 69 (2004) (Open symbols) About the same T is for pions and nucleons T (MeV) 300 200 + K A+A data is a Step, inverse slopes are modified due to transverse expansion M.Gorenstein,M.Gazdzicki,K.A.B., PLB 567 (2003) 100 1 10 p+p (p): 0 K K S + s NN A+A: NA49 AGS RHIC 2 10 (GeV) 10

Problem in our understanding Most of data from A+A collisions are explained by transition to/from QGP, but: Why T values in El. Particle collisions are nearly the same? Is there QGP formed? Why don t we see it? Do Const T values in El. Particle collisions signal a phase transition? Surprisingly, ALL these temperatures are VERY close to the Hagedorn one!!! Can we relate the hadron mass spectrum and PT? 11

There is gap in our understanding of A+A and h+h reactions! A+A data h+h, e+e data 12

Statistical Bootstrap Model 13

Statistical Bootstrap Model The first evidence for ρ(e) = C e αe density of states was found numerically in 1958 having 15 particles only! Result was not understood until a model was formulated in 1963 Model predicted: entropy S = α E energy G. Fast and R. Hagedorn, Nuovo Cimento 27 (1963) 208 Then: T = 1/α = Const leads to the following density of states: ρ(e) = C e S = C e αe i.e. exponentially growing spectrum! R. Hagedorn, Suppl. Nuovo Cimento 3 (1965) 147 It was heresy and Weisskopf forbade to publish it as CERN preprint! But 1964 data confirmed an exponential form. 14

Exponential Mass Spectrum of Heavy Hadrons It showed the existence of a limiting temperature (Hagedorn temperature) for hadrons! => New physics! The grand canonical pressure for such density of states exists ONLY below p(t ) = T 0 d m ρ(m) N d 3 k m 2 +k 2 (2π) 3 e T T H 170 MeV = T 0 d m m 3 N { } d 3 k (2π) 3 e m T H In 1975 Collins and Perry, PRL34 (1975), interpreted the m2 +k 2 Hagedorn temperature as a Phase Transition temperature to quarks and gluons! T 15

Hagedorn Spectrum Follows from Stat.Bootstrap Model, S.Frautschi, 1971 Hadrons are built from hadrons Veneziano Model, K.Huang,S.Weinberg, 1970 Used in string models M.I.T. Bag Model, J.Kapusta, 1981 Hadrons are quark-gluon bags dn/dm 800 700 600 500 Hadron mass spectrum PDG 2004 400 But experimentally it is not seen... 300 200 100 0! " ", K N # $ % & 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 m (GeV) 16

...Hagedorn Spectrum is seen, but not where it is supposed to be! Consider the integral of the mass spectrum N exp (m) = i g i Θ(m m i ), It is exponential for 1 GeV< m< 2.5 GeV! Above 2.5 GeV??? 17

Hagedron Spectrum and Bag Model Bag Model is a foundation of our phenomenology. It gave a first evidence that transition to Partonic World is a phase transition. Resonances are small bags of QGP. How does it explain Hagedorn spectrum with Const T? Consider a single heavy bag of 0 baryonic charge in vacuum. 0 external pressure fixes the temperature (g is # d.o.f): p = g π2 90 T H 4 B = 0 T H = Then entropy of the bag is S = ε(t H)V T H [ 90 gπ 2B [ ] Mass T H ρ(mass) = exp [S] = exp ] 1 4 [ ] Mass T H 18

Everything looks fine, BUT... Example #1: 1-d Harmonic Oscillator For 1-d Harmonic Oscillator with energy & in contact with Hagedorn resonance (just exponential spectrum for simplicity). Total energy is E. K.A.B.et al, Europhys. Lett. 76 (2006) 402 The microcanonical probability of state & is: P( ε) = ρ( E ε) = exp E ε T Η = exp E T exp ε T Η Η Exponent is Grand canonical! With fixed T! ln P(ε) E slope = 1 T H ε Average value of & is E T ε = T H 1 H exp E T H ( ) 1 For E : ε T H 19

Example #2: An Ideal Vapor coupled to Hagedorn resonance Consider microcanonical partition of N particles of ( ) ( mass m and kin. energy &. The total level density is P( E,ε) = ρ H ( E ε)ρ iv ( ε) = 3 V N mε 2 N E mn ε N! 3 exp 2 N 2π T! [ [] ] Η Exponent is Grand canonical! With fixed T! The most probable energy partition is lnp ε = 3N 2ε 1 T Η = 0 ε N = 3 2 T Η T H is the sole temperature characterizing the system: A Hagedorn-like system is a perfect thermostat! 20

Intermediate Conclusion: For Hagedorn resonances the Grand canonical ensemble with T other than Hagedorn T does not make any sense! Because it is equivalent to bring in contact 2 thermostats with different T 21

Example with Explicit Thermostat: Export/import of heat does not change T! T = T c = 273K or 0 T 273K? S = S 0 + ΔQ T = S 0 + E T 0 ( ) = e S = e S 0 + ρ E Is T_o just a parameter? Then add heat E to system with other T: ( ) = deρ E Z T ( )e E T = T T 0 T 0 T es 0 According to this logic, thermostat can have ANY T <T_o! E T 0 22

Why In Previous Works There Was an Upper Temperature? Because they used canonical and grand canonical ensembles which are NOT equivalent to MCE! Since the Hagedorn resonance is a perfect thermostat, the transform to (grand)canonical ensemble with other T does not make ANY SENSE! Z Can d E ρ 0 e E T H E T = ρ 0 T H T T H T it exists for T < T H, but we know that two thermostats of different temperatures CANNOT BE IN EQUILIBRIUM! 23

Example #3: An Ideal Particle Reservoir If, in addition, particles are generated by the Hagedorn resonance, their concentration is volume independent! ρ Η (E) lnp N = m + ln V V T Η N mt Η 2π 3 2 = 0 N V = mt 3 2 Η m exp 2π T Η ideal vapor ρ iv particle mass = m volume = V particle number = N energy = ε Remarkable result because it mean saturation between gas of particles and Hagedorn thermostat! 24

Important Finding! Volume independent concentration of vapor means: for increasing volume of system gas particles will be evaporated from Hagedorn resonance (till it vanishes); by decreasing volume we will absorb gas particles to Hagedorn resonance! Compare to ordinary water! Literally, it is a liquid (Hagedorn resonance) in equilibrium with its vapor! ρ H (E) 25

The Story so far... Anything in contact with a Hagedorn thermostat acquires the Hagedorn temperature. If particles (e.g. pions) can be created from a Hagedorn thermostat, they will form a saturated vapor at fixed (Hagedorn) temperature. If different particles (i.e. of different masses m) are created, they will be in chemical equilibrium. Because of these properties the radiant Hagedorn resonance should be similar to a compound nucleus (same spectra and branching ratios), but at fixed T. 26

The role of the lower mass cut-off So far we ignored that for light hadrons the [ ] spectrum is not exponential. Also translational d.o.f. of the Hagedorn thermostat were ignored. For a single Hagedorn thermostat ( a = const ): ρ H (m H ) = exp[m H /T H ](m o /m H ) a for m H m o The mass cut-off: m o T H From an analysis by W. Broniowski et. al., hep-ph/0407290 m o < 2 GeV. For a single Hagedorn thermostat: δ ln P δ m H = 1 T H + ( 3 2 a) 1 m H 3(N+1) 2 E kin = 0 T (m H ) 2 E kin 3(N + 1) = T H 1 + ( 3 2 a) T H m H. 27

N-dependence and Kinematic Limit ( For such N the maximum of microcanonical partition exists. Otherwise, for N N kin B m H E ) E [mo T H a] T (m o ) m + 3 2 T (m o ) m H m o. N > NB kin, 1 T = T o (N) 2(E mn m o) 3(N+1). Hagedorn resonance does not have sufficient mass to keep equilibrium T/T H a = 0 a = 3 N B kin (a = 0) N B kin (a = 3) A typical behavior (E = 30m) 0 0 5 10 15 20 25 30 N B 28

N-dependence and Kinematic Limit ( For such N the maximum of microcanonical partition exists. Otherwise, for N N kin B m H E ) E [mo T H a] T (m o ) m + 3 2 T (m o ) m H m o. N > NB kin, T = T o (N) 2(E mn m o) 3(N+1). Hagedorn resonance does not have sufficient mass to keep equilibrium T/T H 1 For a > 1.5 Hagedorn T is not a limiting one! a = 0 a = 3 N B kin (a = 0) N B kin (a = 3) A typical behavior (E = 30m) 0 0 5 10 15 20 25 30 N B 28

Inverse Slopes The microcanonical partition can be cast For m + 2 H +Q2 d N NB kin P = V ρ H (m + 3 H ) Q (2π) 3 e T (m + H ) E [ e [ T ] (m + H ) d 3 ] m p 2 +p N 2 V g N! (2π) 3 e T (m + H ). For N > N kin B one has to replace T (m + H ) T o(n) and m + H m o Inverse slope of momentum distribution is a temperature! Lower mass cut-off does not affect our results much. In N kin B vicinity there may exist 10 20 % effect on T 29

How to observe it? In vacuum a Hagedorn thermostat radiates hadrons. For slow radiation the pressure due to radiation is small ( 2-3 % of Bag pressure). Thus, measuring energy and volume (HBT) for vanishing baryon number, one can find the # of d.o.f. g: ε = g π2 30 T 4 H + B p = g π2 90 T 4 H B 0 ε 1.10 ± 0.26 Gev/fm 3 ρ Η (E) g 23.5 ± 6 There was an attempt by Purdue group to measure energy density in el. particle collisions. See T.Alexopoulos et al, PLB 528 (2002) 30

How to observe it? In vacuum a Hagedorn thermostat radiates hadrons. For slow radiation the pressure due to radiation is small ( 2-3 % of Bag pressure). Thus, measuring energy and volume (HBT) for vanishing baryon number, one can find the # of d.o.f. g: ε = g π2 30 T 4 H + B p = g π2 90 T 4 H B 0 ε 1.10 ± 0.26 Gev/fm 3 g 23.5 ± 6 There was an attempt by Purdue group to measure energy density in el. particle collisions. See T.Alexopoulos et al, PLB 528 (2002) 30

Conclusions Exponential mass spectrum is a very special object. It imparts the Hagedorn temperature to particles in contact with it = perfect thermostat! It is also a perfect particle reservoir! The inverse slope that Hagedorn resonances are imparting is a kinetic temperature. K.A.B. et al, hep-ph/0504011 The presence of the mass cut-off of the Hagedorn spectrum DOES NOT ALTER our conclusions: Hagedorn resonances are PERFECT THERMOSTATS and PARTICLE RESERVOIRS! Our results justify the Statistical Hadronization Model and explain why hadronization T and inverse slopes in el. particle collisions are about 170 MeV. This is phase transition in finite system. No liberation of color d.o.f. is necessary for that! 31

A+A data h+h, e+e data 32

Stability Against Fragmentation For no translational entropy the Hagedorn thermostat (=bag) is indifferent to fragmentation. ρ H (m) indifferent exp m = exp T Η ρ H (m 1 ) ρ H (m 2 ) Translational d.o.f. do not change this result. Present model not only EXPLAINS why Becattini s Stat. Hadronization Model gives T close to Hagedorn T, but it also justifies the validity of his major assumption that i=1 T Η all fireballs originate from a Singe Protofireball! k m i ρ H (m 3 ) ρ H (m 4 ) ρ H (m 5 ) ρ H (m 5 ) ρ H (m 6 ) ρ H (m k ) 33

Flash at Double Phi Decay? Different models show that parameter a in Hagedorn mass spectrum is a=3 or even a>3. In this case at the end of radiation ± m H m o and T 1.1 T H 1.2 T H For pions it is unobservable, but for heavy hadrons it is matter! Thus, heavy hadrons emitted about the end of radiation should have an enhanced probability, compared to early time emission! Best candidate to see Flash (V. Koch) is, probably, double phi decay? For more definite predictions we need better model and better data! 34

Conclusions Exponential mass spectrum is a very special object. It imparts the Hagedorn temperature to particles in contact with it = perfect thermostat! It is also a perfect particle reservoir! Grand canonical treatment should be used with great care! Microcanonical one is the right one. Our results justify the Statistical Hadronization Model and explain why hadronization T and inverse slopes in el. particle collisions are about 170 MeV. This is phase transition in finite system. No liberation of color d.o.f. is necessary for that! 35