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Rolf Hagedorn Lecturing 1994 at Divonne: First Transparency Nuclear Matter at Hagedorn Temperature

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Rolf Hagedorn Lecturing 1994 at Divonne: First Transparency Nuclear Matter at Hagedorn Temperature Rolf Hagedorn lecturing 1994 at Divonne: First Transparency Nuclear Matter at Hagedorn Temperature Rolf Hagedorn at CERN 1977 2 Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 3 Thermal Hadron Spectra and Limiting Temperature Multihadron production in pp reactions produced spectra with exponential fall- o® as function of transverse momenta. The exponential (inverse) slope was the same for all particles, TH 160 MeV, and it did not change as the collision energy varied. Rolf Hagedorn (RH)' recognized behavior akin to particle evaporation from boiling elementary matter. In the boiling process a maximum temperature cannot be exceeded even if we provide more and more heating. The spectrum here shown is for the recently (2000) measured NA49 ©(ss¹) in 158 GeV (psNN = 17:2 GeV) pp reac- tions. 95% of ©(ss¹) is directly produced (as opposed to being a decay product). We ¯t a Boltzmann thermal shape allowing for the cuts in rapidity and binning in p . There are 5 data points, ¯t parameters? are: normalization and T . Â2 dof = 1:2 3 as required for high con- ¯dencej ¯t withj a few degrees of freedom. In AA collisions we have greater slopes since the compressed bulk matter flows apart, use Doppler formula J. Letessier 2003 1 + v T T slope ' 1 v r ¡ Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 4 Exponential Hadron Mass Spectrum RH discovered that the exponential growth of the hadronic mass spectrum could lead to an understanding of the limiting hadron temperature T 160 MeV, H ' The solid line is the ¯t: ½(m) c(m2 + m2)a=2 exp(m=T ) ¼ a H with a = 3, ma = 0:66 GeV, TH = 0:158 GeV. Long-dashed¡ line: 1411 states of 1967. Short-dashed line: 4627 states of 1996. Experimental lines include Gaussian smoothing: 2 gm¤ (m m¤) ½(m) = exp ¡2 : p2¼σ ¤ ¡ 2σ ¤ m¤=m ;m ;::: m m X¼ ½ µ ¶ σ = ¡=2, ¡ = (200) MeV is the assumed width of the O resonance, excluding the `stable' pion, a special case. Note the missing resonances at m > 1:4GeV . The newly discovered `pentaquark' resonances needed to ¯ll this gap. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 5 Limiting Hagedorn Temperature A gas of hadrons with exponentially rising mass spectrum: 3=2 cl T 1 a m=T 3=2 m=T ln = cV m e Hm e¡ dm + D(T; M); ZHG 2¼ µ ¶ ZM Cuto® M > m > T is arbitrary, its role is to separate o® D(T; M) < . Because a H 1 of the exponential factor, the ¯rst integral can be divergent for T > TH, and the partition function is singular for T T for a range of a: ! H (a+5=2) (a+7=2) 1 1 ¡ 1 1 ¡ ; for a > 5; ; for a > 7; T ¡ T ¡2 T ¡ T ¡2 8 µ H ¶ 8 µ H ¶ > 1 1 > 1 1 P (T ) > ln ; for a = 5; ² > ln ; for a = 7; ! > T ¡ T ¡2 ! > T ¡ T ¡2 <> µ H ¶ <> µ H ¶ constant; for a < 5; constant; for a < 7: > 2 > 2 > ¡ > ¡ > 7> The energy:> density ² goes to in¯nity for a :>, when T TH. ¸ ¡2 ! Mass spectrum slope TH appears as the limiting Hagedorn temperature beyond which we cannot heat a system which can have an in¯nite energy density. The partition function can be singular even when V < . 1 Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 6 a = ::: 3:5 3 2:5 ¡ ¡ ¡ energy density @ ² = ln (¯; V; ¸) ¡@¯ Z exponentially growing mass spectrum pressure @ P = ln (¯; V; ¸) @V Z exponentially growing mass spectrum Straight lines: P , ² based on actually known mass spectrum. This is for point hadrons, year was 1965. Understanding of quark structure of hadrons implies their ¯nite. This is important here given the exponential growth of mass spectrum which over¯lls a ¯xed volume with (di®erent) particles. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 7 Finite Volume Hadron Gas Model Point hadron gas in free available volume ¢ to have the properties of ¯nite size hadron gas in total mean volume V (RH 1978+) h i ln (T; ¢; ¸) ln (T; V ; ¸) Zpt ´ Z h i Proper particle volume in the rest frame is assumed to be proportional to mass. For a gas of moving hadrons, in gas rest- frame: V = ¢ + E =4 . h i h i B @ E = V ²(¯; ¸) = ln (¯; V ; ¸) = h i h i ¡@¯ Z h i @ = ln (¯; ¢; ¸) = ¢² (¯; ¸) ¡@¯ Zpt pt V = ¢ 1 + ² (¯; ¸)=4 ; h i pt B E ² (¯; ¸) h i ²(¯¡; ¸) = pt ¢ ; V ´ 1 + ² (¯; ¸)=(4 ) h i pt B P (¯; ¸) P = pt : 1 + ² (¯; ¸)=4 pt B The gas of ¯nite size hadrons with exponential mass spectrum has nearly the same properties as a gas of point hadrons with today experimentally observed mass spectrum. That is why `statistical hadronization works'. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 8 Statistical Bootstrap Model (SBM) Rolf Hagedorn constructed a theoretical model Statistical Bootstrap in which the exponentially rising hadron mass spec- trum naturally occurred. STATISTICAL THERMODYNAMICS OF STRONG INTERACTIONS AT HIGH-ENERGIES. Nuovo Cim. Supp. 3 (2): 147 (1965) citation classic ON HADRONIC MASS SPECTRUM Nuovo Cim. A 52 (4): 1336 (1967) HADRONIC MATTER NEAR BOILING POINT Nuovo Cim. A 56 (4): 1027 (1968) 75th Birthday, Divonne 1994 Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 9 Statistical Bootstrap Equation Resonant scattering comprises the essence of strong interaction and implies that there is clustering of hadrons into new hadrons. What happens if we compress? We look at N-particle level density, N N N 1 gV 3 3 σN (E; V ) = ± "i E ± p~i d pi; N! (2¼)3 ¡ i=1 Z i=1 i=1 Y ¡ X ¢ ¡ X ¢ COMPRESS Mix gases with di®erent masses m e.g. 2 2 g ½(m)dm, with "i = m + pi . We sum! over N allowing all microcanoni- cal con¯gurationsR which preservp e the im- posed constraints. We apply the Boot- Natural cluster strap hypothesis: Macroscopic volume V volume V c ( m,b ) σ(E; V ) ½(m) c ¡! In covariant notation we ¯nd the Bootstrap (SBM) equation for ½: N N 1 1 ½(p2) = ± (m2 m2 ) + ±4 p p ½(p2)d4p ; H H 0 ¡ in N! ¡ i H i i N=2 Z i=1 i=2 X ¡ X ¢ Y This is a nonlinear inhomogeneous integral equation. to be ¯xed by experi- H ment, related to proper volume Vc. For min m¼, the lightest hadron, the pion generates higher hadron clusters. Further belo! w we include baryons. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 10 Singularity of the Bootstrap Equation ¯ p 2 4 ¯ p 2 2 4 2 K1(¯min) G e¡ ¢ ½(p ) d p; '(¯) = e¡ ¢ ± (p m )d p = 2¼m ; ´ H H 0 ¡ in H in ¯m Z Z in Laplace Transform of the SBM equation yields G(') = ' + eG(') G(') ¡ or ' = 2G(') eG(') + 1. ¡ For ' ' , G(') ln 2 constant p' ' + : and since ' regular at ' we ! 0 ¼ § £ 0 ¡ ¢ ¢ ¢ 0 have G(¯ ¯0) = ln 2 constant p¯ ¯0. Such a singularity can only exist if ' ¡ £ 3¡m¯0 ½ m¡ e ¯ = 1=T : / 0 H Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 11 Statistical Bootstrap with Mesons and Baryons Each hadron `i' has now baryon number bi and we have the constraint that the baryon number in all partitions has to add up to b. The SBM equation becomes: N N N 2 2 2 1 1 4 2 4 ¿(p ; b) = gb±0(p mb) + ± p pi ±K b bi ¿(pi ; bi) d pi; H H ¡ N! à ¡ ! à ¡ ! H N=2 Z i=1 bi i=1 i=1 X X Xf g X Y The generalized Laplace transform method works, with : ¯ p 1 b 2 2 4 1 b 2 K1(mb¯) '(¯; ¸) e¡ ¢ ¸ gb±0(p mb) d p = 2¼ ¸ gbmb ; ´ H ¡ H mb¯ b= b= Z X¡1 X¡1 ¯ p 1 b 2 4 ©(¯;¸) ©(¯; ¸) e¡ ¢ ¸ ¿(p ; b) d p implies ©(¯; ¸) = '(¯; ¸) + e ©(¯; ¸) 1: ´ H ¡ ¡ b= Z X¡1 The singularity point becomes a line ¯cr(¸cr) derived from: cr m¼ 1 mN ln(4=e) = '(¯ = 1=T ; ¸ = e¹b =Tcr) = 2¼ T 3m K + 4 ¸ + m K : cr cr cr H cr ¼ 1 T cr ¸ N 1 T · µ cr ¶ µ cr ¶ µ cr ¶¸ cr This condition de¯nes a critical curve ¹b = f(Tcr) in the T; ¹b plane. The value cr of is ¯xed by the critical temperature TH = 160 MeV associated with ¹b = 0. BeloHw the Hagedorn temperature we have matter made of baryons and mesons. Approaching this critical curve for ¯nite volume hadrons pressure vanishes and all hadrons combined to one only hadronic cluster. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J. Rafelski page 12 Clustering of Hadrons and New Phase of Matter ? Dilute Onset of Particles with Hadron Quark−gluon gas clustering proper volume matter plasma µ [MeV] 1000 800 Liquid−gas equilibrium 600 Heavy clusters QGP Light clusters 400 Dilute 200 Gas 0 20 60 100 140 T [MeV] Vacuum T0 At the critical curve for ¯nite volume hadron pressure vanishes and all hadrons combine to one only hadronic cluster. This drop of matter is made of hadron constituents. The Hagedorn Temperature has become the condition in which the internal degrees of freedom of hadrons (quarks and gluons) become dynamical. Nuclear Matter at Hagedorn Temperature Hagedorn Memorial CERN, November 28, 2003, J.
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