Energy and Number Density Created at RHIC What’s in the PHENIX White Paper, and a little bit more Paul Stankus, ORNL PHENIX Focus, Apr 11 06.

Energy and Number Density Created at RHIC

What’s in the PHENIX White Paper, and a little bit more

Paul Stankus, ORNL PHENIX Focus, Apr 11 06

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Energy Density, Take 1

2R ~ 14 fm 2R/ ~ .13 fm

Rest Frame

= E/V = M/V0

~ 0.14 GeV/fm3 =

Boosted Frame

= E/V = M/(V0/) = 02

RHIC = 106

~ 1570 GeV/fm3 (!!)

Just divide energy by volume, in some frame.

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Energy Density, Take 2Examine a box with total momentum zero.

= 0 ~ 3150 GeV/fm3

= ?

Very high, but very short-lived!

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Energy Density, Take 3Count up energy in produced particles/matter.

Define produced as everything at velocities/rapidities intermediate between those of the original incoming nuclei.

Two extremes:

All particles Bjorken

All fluid Landau

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The Bjorken Picture: Pure Particles

Key ideas:

• Thin radiator

• Classical trajectories

• Finite formation time

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Particles in a thin box with random velocities

Release them suddenly, and let them follow

classical trajectories without interactions

Strong position-momentum

correlations!

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J.D. Bjorken, Phys. Rev. D 27 (1983) 140“Highly relativistic nucleus-nucleus

collisions: The central rapidity region”

Key idea: Use the space-momentum correlation to translate between spatial density dN/dz and momentum density dN/dpZ

Thin radiator

The diagram is appropriate for any frame near mid-rapidity, not just the A+A CMS frame specifically.

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x

z

pT = pX= p pZ

= 0 y = 0

E=√m2+pT2mT

x’

z’

pT = pX mT =√m2+pT2

pZ = mT sinh(y) E = √m2+p2 = mT

cosh(y) Z= pZ/E = sinh(y)/cosh(y) y =tanh-1(Z) y

Z for Z<<1

Useful relations for particles in different Lorentz frames

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dZ

Exercise: Count particles in the green box at some time t, add up their energies, and divide by the volume.

Particles in the box iff 0<Z<dZ/t (limit of infinitely thin source)

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Number of particles dN =dN

dβ Z

dZ

t=

dN

dy

dZ

t

Average energy per particle E = mT

Volume of the box V = A ⋅dZ

A ≡ xy cross section area (limit t << R)

R

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(t) =Energy

Volume=

E dN

V=

1

t A

dN(t)

dymT (t) =

1

t A

dET (t)

dy

Valid for material at any rapidity and for any shape in dET(t)/dy! A plateau in dET(t)/dy is not required.

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(t) =1

t A

dET (t)

dyHow low can t go? Two basic limits:

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t >> R 1 γ A +1 γ B( ) Crossing time

t ≥ τ Form Formation time

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(τ Form ) =1

τ Form A

dET (τ Form )

dy≈

1

τ Form A

dETFinal State

dy

Bjorken

For many years this Bjorken formula was used with a nominal Form=1.0 fm/c with no real justification, even when it manifestly violated the crossing time limit for validity!

2R/ = 5.3 fm/c for AGS Au+Au, 1.6 fm/c for SPS Pb+Pb.

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Better formation time estimates

Generic quantum mechanics: a particle can’t be considered formed in a frame faster than hbar/E

Translation: Form 1/mT ~ 1/<mT>

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mT =dET (τ Form ) /dy

dN(τ Form ) /dy≈

dET /dη

dN /dη (Final State)

PHENIX Data: (dET/d)/(dNch/d) ~ 0.85 GeV

Assuming 2/3 of particles are charged, this implies Form ~ 0.35 fm/c

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Some assumptions we’ve used

• Transverse energy density dET/dy only goes down with time.

• The number density of particles does not go down with time (entropy conservation).

• We can estimate, or at least bound, thermalization time from other evidence.

An unanswered question: What are the initially produced particles? (Bj: “quanta”)

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Identifying the intial “quanta”

Multiplicities in Au+Au at RHIC were lower than initial pQCD predictions. Indicates need for “regularization”. Good candidate is CGC.

CGC identifies intial quanta as high-ish pT gluons (~1 GeV), which is consistent with our particle picture.

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The Landau Picture: Pure Fluid

Key ideas:

• Complete, instant thermalization

• Fluid evolves according to ideal relativistic fluid dynamics (1+1)

• Very simple √s dependences for multiplicity and dN/dy (Gaussian)

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Courtesy of P. Steinberg; see nucl-ex/0405022

Multiplicities

Widths

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dE = TdS − PdVBasic

Thermodynamics

Sudden expansion, fluid fills empty space without loss of energy.

dE = 0 PdV > 0 therefore dS > 0

Gradual expansion (equilibrium maintained), fluid loses energy through PdV work.

dE = -PdV therefore dS = 0Isentropic Adiabatic

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