Phase transitions from Hadronic to Partonic Worlds L. G. Moretto, K. A. Bugaev, J. B. Elliott and L. Phair

Lawrence Berkeley National LaboratoryNuclear Science Division

Phase transitions in the Hadronic world•Pairing (superconductive) Transition

finite size effects: correlation length•Shape transition

all finite size effects, shell effects

•Liquid-vapor (with reservations) van der Waals-like finite size effects due to surface

Tc ≈ 18.1 MeVc ≈ 0.53 0

pc ≈ 0.41 MeV/fm3

Phase transitions in the partonic world•Q. G. P. . . . Finite size effects?

Firm and fleeting features of phase transitions

• Usually very firm:• Change in aggregation state:

• Solid - Liquid - Vapor

• Change in symmetry:• Different crystalline structure

• Appearance of new features:• Superconductivity - Superfluidity

• Evidence for hadronic to partonic (QGP) transition is (so far) not as firm as the above

examples.

• Perhaps there are features so prominent that they have escaped the detailed inspection of

experimental data.

sulfur

Origin of the bag pressure

•To make room for a bubble of volume V an energy E = BV is necessary.•To stabilize the bubble, the internal vapor pressure p(T) must be equal to the external pressure B.

•Notice that the surface energy coefficient in this example is not obviously related to the volume energy coefficient.

10 m

B = 1 atm

The partonic world (Q.G.P.)(a world without surface?)

• The M.I.T. bag model says the pressure of a Q.G.P. bag is constant:

• ; g: # degrees of freedom, constant p = B, constant .

• The enthalpy density is then

•

• which leads to an entropy of

•

• and a bag mass/energy spectrum (level density) of

• .

• This is a Hagedorn spectrum:

•

Partonic vacuum

Hadronic vacuum

€

p =gπ 2

90TΗ

4 = B

€

ε =H

V=E

V+ p =

gπ 2

30TΗ

4 + B

€

S =δQ

T∫ =

dH

T0

H

∫ =H

TΗ

≡m

TΗ

€

m( ) = exp S( )∝ expm

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

H m( )∝m0

m

⎛

⎝ ⎜

⎞

⎠ ⎟x

expm

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

TH = B90

gπ 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1

4

?

mm0

ln

(m

)

Can a “thermostat” have a temperature other than its own?

•

•

• Is T0 just a “parameter”?

•

• According to this, a thermostat, can

have any temperature lower than its

own!€

Z T( ) = dEρ E( )e−E T∫ =T0T

T0 −TeS0

€

E( ) = eS = eS0 +

E

T0

€

S = S0 +ΔQ

T= S0 +

E

T0

T = Tc = 273Kor

0 ≤ T ≤ 273K ?

Thermal equilibriumSystem A System B

No thermostat: any temperature

One thermostat: one temperature

Two thermostats: no temperature

S

E

S

ES

E

S

ES

E

S

E

€

∂S1

∂E1

=∂S2

∂E2

∴ 1

T1

=1

T2

Equilibrium with Hagedorn bags:Example #1: the one dimensional harmonic oscillator

•BEWARE! A linear dependence of S on E spells danger to the unaware!•Beware of the canonical and grand canonical ensembles!•When in doubt, Go MICRO-canonical !

•For a one dimensional harmonic oscillator with energy ε in contact with a Hagedorn bag of energy E:

•The probability P(ε) is:

€

P ε( ) = ρ E −ε( ) = expE −ε

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟= exp

E

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟exp −

ε

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

E ε

ln Pε

€

slope =1

TH•The most probable value of ε:

•

•For E : ε TH

€

ε =TH 1−E TH

exp E TH( ) −1

⎛

⎝ ⎜

⎞

⎠ ⎟

• The total level density:

•

• Most probable energy partition:

•

• TH is the sole temperature characterizing the system:

• A Hagedorn-like system is a perfect thermostat.

• If particles are generated by the Hagedorn bag, their concentration is:

•

• Volume independent! Saturation! Just as for ordinary water, but with

only one possible temperature, TH!

Equilibrium with Hagedorn bags:Example #2: an ideal vapor of N particles of mass m and energy ε

€

P E,ε( ) = ρH E −ε( )ρ iv ε( ) =V N

N!3

2N

⎛

⎝ ⎜

⎞

⎠ ⎟!

mε

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2N

expE − mN −ε

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∂ lnP

∂ε=

3N

2ε−

1

TΗ

= 0⇒ε

N=

3

2TΗ

€

∂ lnP

∂N V

= −m

TΗ

+ lnV

N

mTΗ

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥= 0⇒

N

V=

mTΗ

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2exp −

m

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

(E)

ideal vapor iv• particle mass = m• volume = V• particle number = N• energy = ε

Saturated vapor with free mass

• If we allow the mass of the particle to be free, its most probable value is:

€

∂ lnP E,ε( )∂m

= N3

2m−

1

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟= 0⇒ m =

3

2TΗ

1. Anything in contact with a Hagedorn bag acquires the temperature

TH of the Hagedorn bag.

2. If particles (e.g. s) can be created from a Hagedorn bag, they will

form a saturated vapor at fixed temperature TH.

3. If different particles (i.e. particles of different mass m) are created

they will be in chemical equilibrium.

H(E)

The story so far . . .

(E)

The radiant Hagedorn bag(initial energy E0, initial radius R0)

• At coexistence:• flux in in = flux out out = particles:

•

• Energy flux E:

• €

φparticles =c

4

m

m + 2TH

⎛

⎝ ⎜

⎞

⎠ ⎟

8TH

πm

€

φE = m + 2TH( )φparticles

• With no containing volume the Hagedorn bag radiates itself away.

• Upper limit for lifetime of Hagedorn bag:

•

• Similarities with compound nucleus:• Same spectra and branching ratios

• Differences with compound nucleus:• All quantities calculated at fixed TH

€

∝ E0

mTHR0( )2 exp

m

TH

⎛

⎝ ⎜

⎞

⎠ ⎟

(E)

Stability of the Hagedorn bag against fragmentation

• If no translational or positional entropy, then the Hagedorn bag is indifferent to fragmentation.

H(m)H(mk)

H(m3)H(m2)

H(m4)

H(m5)

H(m5)

H(m1)

H(m6)

€

expm

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟= exp

mi

i=1

k

∑TΗ

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

indifferent

Resonance gas - A gas without pressure

No intrinsic energy and/or entropy penalty for aggregation

How many particles?

1 ≤ N ≤ Nmax

Ideal gas law:

€

p =N

VT

Resonance Gas Cont’d

€

A ⇔ nB

€

cBn

cA

=qBn

qA

=qAn−1

n3n / 2

€

qB =2πmAT

h2n

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

=1

n3 / 2qA

€

limn→∞

cB = limn→∞

cA1/ nqA

n−1

n

n3 / 2= 0

€

%dissoc =1

n5 / 2q

n−1

ncA

1/ n

c0

• The total level density:

•

• Most probable energy partition:

•

• TH is the sole temperature characterizing the system:

• A Hagedorn-like system is a perfect thermostat.

• If particles are generated by the Hagedorn bag, their concentration is:

•

• Volume independent! Saturation! Just as for ordinary water, but with

only one possible temperature, TH!

Equilibrium with Hagedorn bags:Example #2: an ideal vapor of N particles of mass m and energy ε

€

P E,ε( ) = ρH E −ε( )ρ iv ε( ) =V N

N!3

2N

⎛

⎝ ⎜

⎞

⎠ ⎟!

mε

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2N

expE − mN −ε

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∂ lnP

∂ε=

3N

2ε−

1

TΗ

= 0⇒ε

N=

3

2TΗ

€

∂ lnP

∂N V

= −m

TΗ

+ lnV

N

mTΗ

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥= 0⇒

N

V=

mTΗ

2π

⎛

⎝ ⎜

⎞

⎠ ⎟

3

2exp −

m

TΗ

⎛

⎝ ⎜

⎞

⎠ ⎟

(E)

ideal vapor iv• particle mass = m• volume = V• particle number = N• energy = ε

The EoS for the Gas of Hagedorns

For the mass spectrum

Is the most probable concentration of H with mass m

Energy conservation defines the maximal mass of Hagedorns mmax and fixes the number of Hagedorns Ntot and pressure P of the system

Maximal number of Hagedorns

The EoS for the Gas of HagedornsEnergy conservation defines the maximal mass of Hagedorns mmax

And gives EoS as P(E/V) : for M = 15.9 GeV to fit Ntot of all resonances with masses < 1.85 GeV

For plots E = 2 GeV

Is speed of sound square

A bag with a surface?•Remember the leptodermous expansion:

•

•Notice that in most liquids aS ≈ -aV

•However, in the MIT bag there is only a volume term

•

• Should we introduce a surface term? Although we may not know the magnitude of as, we know the sign (+). The consequences of a surface term:

•

•

€

M = E ≅ H = aV A + aSA2 3 + aCA

1 3

€

εV = H = f T( ) + B[ ]V + aSV2 3 ?( )

V

TTc

€

p =1

3f T( ) − B +

2

3aSV

−1 3 ⎛

⎝ ⎜

⎞

⎠ ⎟= 0 at equilibrium

€

T = f −1 3 B +2

3aSV

−1 3 ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

V

0

Cp=

0

V

εV

Stability of a gas of bags

Bags of different size are of different temperature. If the bags can fuse or fission, the lowest temperature solution at constant energy is a single bag. The isothermal solution of many equal bags is clearly unstable.

A gas of bags is always thermodynamically unstable.

A bag decays in vacuum by radiating (e.g. pions). As the bag gets smaller, it becomes HOTTER! Like a mini-black hole.

The decay of a bag with surface

Is the surface energy temperature dependent?

•

•From the bag stability condition:

€

cs = cs0 1−

T

Tc

⎛

⎝ ⎜

⎞

⎠ ⎟

Tc

€

2c0V−2 3

•For V very small T Tc

€

σTH

4 = 3 B +2

3cs

0 1−T

Tc

⎛

⎝ ⎜

⎞

⎠ ⎟V

−2 3 ⎡

⎣ ⎢

⎤

⎦ ⎥

T€

3B

V

TTc

Tindep

Tdep

The perfect “Granulator”

Tc

TH

Conclusion1. Any system with a Hagedorn-like spectrum

is a perfect thermostat.

2. If such a system evaporated non-Hagedorn particles they constitute a saturated vapor in physical and chemical equilibrium.

3. A vapor of Hagedorn bags is indifferent to aggregation/fragmentation P=0.

4. Surface energy further destabilizes the vapor. Smaller bags are at higher temperature.

ConclusionsNuclear dropletsIsing lattices

• Surface is simplest correction for finite size effects (Rayleigh and Clapeyron)

• Complement accounts for finite size scaling of droplet

• For ground state droplets with A0<<Ld, finite size effects due to lattice size are minimal.

• Surface is simplest correction for finite size effects(Rayleigh and Clapeyron)

• Complement accounts for finite size scaling of droplet

• In Coulomb endowed systems, only by looking at transition state and removing Coulomb can one speak of traditional phase transitions

Bulk critical pointextracted whencomplement takeninto account.

Origin of the bag pressure

•To make room for a bubble of volume V an energy E = BV is necessary.•To stabilize the bubble, the internal vapor pressure p(T) must be equal to the external pressure B.

•Notice that the surface energy coefficient in this example is not obviously related to the volume energy coefficient.

10 m

B = 1 atm