BNL-52613 Formal Report

Proceedings of RIKEN BNL Research Center Workshop Volume 28

EQUILIBRIUM & NON-EQUILIBRIUM

ASPECTS OF HOT, DENSE QCD

July 17-30,200O

Organizing Committee:

D. Boyanovsky, H.J. de Vega, L.D. McLerran, R.D. Pisarski

RIKEN BNL Research Center Building 51 OA, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Other RIKEN BNL Research Center Proceedings Volumes:

Volume 30 - RBRC Scientific Review Committee Meeting - BNL-52603

Volume 29 - Future Transversity Measurements - BNL-52612 ’

Volume 27 - Predictions and Uncertainties for RHIC Spin Physics & Event Genera- tor for RHIC Spin Physics III -Towards Precision Spin Physics at RHIC- BNL-52596

Volume 26 - Circum-Pan-Pacific RIKEN Symposium on High Energy Spin Physics - BNL-52588

Volume 25 - RHIC Spin - BNL-52581

Volume 24 - Physics Society of Japan Biannual Meeting Symposium on QCD Physics at RIKEN BNL Research Center - BNL-52578

Volume 23 - Coulomb and Pion-Asymmetry Polarimetry and Hadronic Spin Dependence at RHIC Energies - BNL-52589

Volume 22 - OSCAR II: Predictions for RmC - BNL-52591

Volume 21 - RBRC Scientific Review Committee Meeting - BNL-52568

Volume 20 - Gauge-Invariant Variables in Gauge Theories - BNL-52590

Volume 19 - Numerical Algorithms at Non-Zero Chemical Potential - BNL-52573

Volume 18 - Event Generator for RHIC Spin Physics - BNL-52571

Volume 17 - Hard Parton Physics in High-Energy Nuclear Collisions - BNL-52574

Volume 16 - RIKEN Winter School - Structure of Hadrons -Introduction to QCD Hard Processes- BNL-52569

Volume 15 - QCD Phase Transitions - BNL-52561

Volume 14 - Quantum Fields In and Out of Equilibrium - BNL-52560

Volume 13 - Physics of the 1 Teraflop RIKEN-BNL-Columbia QCD Project First Anniversary Celebration - BNL-66299

Volume 12 - Quarkonium Production in Relativistic Nuclear Collisions - BNL-52559

Volume 11 - Event Generator for RHIC Spin Physics - BNL-66116

Volume 10 - Physics of Polarimetry at RHIC - BNL-65926

Volume 9 - High Density Matter in AGS, SPS and RHIC Collisions - BNL-65762

Volume 8 - Fermion Frontiers in Vector Lattice Gauge Theories - BNL-65634

Volume 7 - RHIC Spin Physics - BNL-65615

Volume 6 - Quarks and Gluons in the Nucleon - BNL-65234

Volume 5 - Color Superconductivity, Instantons and Parity (Non?)-Conservation at High Baryon Density - BNL-65105

Volume 4 - Inauguration Ceremony, September 22 and Non-Equilibrium Many Body Dynamics - BNL- 64912

Volume 3 - Hadron Spin-Flip at RHIC Energies - BNL-64724

Volume 2 - Perturbative QCD as a Probe of Hadron Structure - BNL-64723

Volume 1 - Open Standards for Cascade Models for RHIC - BNL-64722

Preface to the Series

The RIKEN BNL Research Center (RBRC) was established in April 1997 at Brookhaven National Laboratory. It is funded by the “Rikagaku Kenkysho” @KEN, The Institute of Physical and Chemical Research) of Japan. The Center is dedicated to the study of strong interactions, including spin physics, lattice QCD and RHIC physics through the nurturing of a new generation of young physicists.

During the fast year, the Center had only a Theory Group. In the second year, an Experimental Group was also established at the Center. At present, there are seven Fellows and nine post dots in these two groups. During the third year, we started a new Tenure Track Strong Interaction Theory RHIC Physics Fellow Program, with six positions in the academic year 1999-2000; this program will increase to include eleven theorists in the next academic year, and, in the year after, also be extended to experimental physics. In addition, the Center has an active workshop program on strong interaction physics, about ten workshops a year, with each workshop focussed on a specific physics problem. Each workshop speaker is encouraged to select a few of the most important transparencies from his or her presentation, accompanied by a page of explanation. This material is collected at the end of the workshop by the organizer to form proceedings, which can therefore be available within a short time.

The construction of a 0.6 teraflop parallel processor, which was begun at the Center on February 19, 1998, was completed on August 28, 1998.

T. D. Lee September 29,200O

*Work performed under the auspices of U.S.D.O.E. Contract No. DE-AC02-98CH10886.

CONTENTS

Preface to the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*............*........*..................................~................ i Motivation & Purpose of the Workshop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~................ 1

**********************xxx*******************

Transport Coefficients in Hot Gauge Theories

L. Yaffe . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*......................................................... 3

QCD Thermodynamics: from Lattice to Quasi Particles

E. Bra&en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*......................................... 11

Low-Momentum Correlations from Real-Time Theory

P. Daniele wicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Calculating the Viscosity (the hard way)

R. Kobes . . . . . . . . . . . . . . . ..*..................................................................... 23

Non-Interacting Non-Relativistic Bose Gas

P. Arnold . . . . . ..*............................................................................... 29

A Space-Time View of Thermal Field Theory

H. A. Weldon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~........................... 35

Resonant Decay of Parity Odd Bubbles in Hot Hadronic Matter

R. Baier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.......... 41

The Quantum Field Limit of Classical Field Dynamics

B. Miiller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*.......................... 51

The Renormalization-Group Method Applied to Transport Equations

T. Kunihiro . . . . . . . . . . . . . . . . . . . . . . . . . . ..*..............................*......................... 57

Thermodynamics of Hot QCD; the Effective Theory Approach

K. Kajan tie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Non-Equilibrium Quantum Plasmas, Quantum Kinetics and the

Dynamical Renormalization Group

H. J. de Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.................. 65

Domain Walls and Cosmology at the QCD Scale

A. Zhitnitsky . . . ..*............................................................................ . 69 .

QCD at Finite Density of Isospin: from pion to qq condensation

M. Stephanov . . . . . . . . . . . . . . ..*......................................................*.......... 75

Properties of Gluons in Color Superconductors

D. H. Rischke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Quark Stars from Perturbative QCD

E, S. Fraga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.................. 87

The Electrical Conductivity in a QED Plasma

E. Mottola . . . . . . ..*......................... *...........*..................*........~.........* 99

HTLlHDL Thermodynamics of the Quark-Gluon Plasma

A. Rebhan . . . . . . . . . . . . . . . . . . . . . . . . ..*......................................................... 107

Dynamics of the Chiral Phase Transition at Finite Chemical Potential

F. Cooper . . . . . . . . . . . . ..‘...................................................................... 113

Hard Thermal Loops on the Lattice . . D. BGdeker . . . . . . . . . . . . . . . . . . . . . . . . . ..*........................................................ 123

Yang-Mills in (2+1), Magnetic Mass, Covariance, etc.

V. P, Nair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.._......................................... 131

Solving the Problem of Initial Conditions in Heavy Ion Collisions

R. Venugopalan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.......... 137

Reorganizing Finite Temperature Field Theory

M. Strickland . . . . . . . . . . . . . . . . . . . . . . . ..“........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

QCD at Theta - Pi

M. Tyfgat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.......... 153

Decrease of the Non-Perturbative Cutoff in QGP

E. Shuryak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..a.................. 159

Brief Closing Remarks

L. McLerran . . . . . . . . . . . . . . . . . . . . . . ..*..*........................*................... . . . . . . . . . . . 165

*********xx*********************************

List of Registered Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Agenda of Workshop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

******************************************

Motivation & Purpose of the Workshop

The Relativistic Heavy Ion Collider (RHIC) at Brookhaven, beginning operation this year, and the Large Hadron Collider (LHC) at CERN, beginning operation -2005, will provide an unprecedented range of energies and luminosities that will allow us to probe the Gluon-Quark plasma.

At RHIC and LHC, at central rapidity typical estimates of energy densities and temperatures are e * l-10 GeV/fm3 and TO * 300 - 900 MeV. Such energies are well above current estimates for the GQ plasma. Initially, this hot, dense plasma is far from local thermal equilibrium, making the theoretical study of transport phenomena, kinetic and chemical equilibration in dense and hot plasmas, and related issues a matter of fundamental importance.

During the last few years a consistent framework to study collective effects in the Gluon-Quark plasma, and a microscopic description of transport in terms of the hard thermal (and dense) loops resummation program has emerged. This approach has the potential of providing a microscopic formulation of transport, in the regime of temperatures and densities to be achieved at RHIC and LHC. A parallel development over the last few years has provided a consistent formulation of non-equilibrium quantum field theory that provides a real-time description of phenomena out of equilibrium. Novel techniques including non-perturbative approaches and the dynamical renormalization group techniques lead to new insights into transport and relaxation. A deeper understanding of collective.excitations and transport phenomena in the GQ plasma could lead to recognize novel potential experimental signatures. New insights into small-c physics reveals a striking similarity between small-c and hard thermal loops, and novel real-time numerical simulations have recently studied the parton distributions and their thermalizations in the initial stages of a heavy ion collision.

Recently new exciting theoretical developments in understanding cold dense plasmas revealed the possibility of novel phases of dense quark matter in which color superconductivity can occur and novel features of the possible phase diagram of QCD in the temperature-chemical potential plane had been studied. AGS at Brookhaven can study a region of large chemical potential and low temperatures which can potentially probe these new phases of QCD. Furthermore, event-by-event analysis of fluctuations had been proposed as definite experimental signatures of these new phases. These phases are not only interesting from the perspective of the GQ plasma, but also could exist at the cores of neutron stars and could have observational implications in pulsar glitches.

Clearly, a deep understanding of equilibrium and non-equilibrium phenomena described from first principles from quantum field theory is a necessary step for recognizing experimental signatures in the new generation of ultrarelativistic heavy-ion colliders.

Moreover, the theme of transport phenomena on extremely short time and spatial scales is truly interdisciplinary: in cosmology a deeper understanding of inflationary dynamics and reheating has been benefiting from the application of techniques brought to bear on the problem of GQ plasma, and the QCD phase transitions in the early universe can have an imprint in the CMB. In condensed matter physics, recent advances in femtosecond spectroscopy is beginning to probe kinetics and relaxation phenomena on unprecedented small scales and the field can benefit from developments in equilibrium and non-equilibrium techniques from the GQ plasma program.

1

4

5

6

i.-

CJ

i

10

11

12

13

1 .

2 .

3 .

14

15 .

- ___ __- ..-. . . ---

- .-. __.- -. __._. _ - ___. -_._-_-__-._-_... .--

_._. __...___ - -..-._. _.-- . .._ --

-. . .

16

LOW-MOMENTUM CORRELATIONS

FROM REAL-TIME THEORY

Pawel Danielewicz and David Brown .

NSCL, Michigan State INT, Seattle

l Inclusive A-ptcle yields from real-time theory

- Nonrelativistic theory 4-*

- A-ptcle sources (bv Y ; >

0 .::. z -; ~

l Application c

- pp measurements I kk e

- Simultaneous pp, nn and np measurements

- Three-body problem for emission ou c

- Numerical results

- Discussion: sudden vs adiabatitb limits

e Conclusions

In the A-ptcle case:

dNv

s

b -= dv

dtl dxl .dtA dx/+dt; dxa J&-I* (x1 il .XA TV)

where A-ptcle source-function is

(+)AC;(xl tl.xA tA; x; t’l.& t;)

XL-)(x&tl.xA tA) = (-5?)A lim t’-kxY s dx; .dx>

xGJxl tl.xA tA; X’l .Xa, t’) eXp [-i&,t’] (ai-)(X;.Xk)

PlQOPA GATE? J%CYb’A@O5 4PYkC-rf//rlC

and the advanced A-particle Green’s function

(+)A GJxl tl.xA t& X’l.+ t’)

= c

X

cGI(A)

““r”x e(t’ - b(l)) ’ o(ta(A-1) - b(A))

Finally, when effects of symmetrization of A ptcles with the remainder of the system a& insignificant, then

where

P*C SEMf- P 7L L(

x~(xl ‘).~(XA--l ‘)j(XA ‘))A-irred

J, - w-LLE sou*cc

+lx s dx; dxi v(xj xk; x; xk) xc-)(x,.x;.x;.xA, j<k

Simultaneous pp, nn & np

Measurements C’HIC Collaboration. Phys. L&t.. B 317(1993)505: R. G1lct.t.i. PhD ‘97 40Ar + lg7Au at E/A = 30 MeV/nucltvn: symbols - data

6

4

C(s) 2

0 0 20 40 60 80 100

1.5 ,-a’-

PP

I -., J..- --L .__ ,_ - 20 40 60 80 100

T.‘- .- ~ --.--7’-

--2--.-I -A - 0 20 40 60 80 100

q (MeV/c)

Structures in nn and pp, but not in rip!

SIMPLE EXPLAN.ATION

As particles move out, proton, gains momentum VC

APC a 2,

relative to neutron. For Au, change in the relative momentum &J = Ap/2 - 35 MeV/c (w N 0.2c), is large compared to the range in Q where the correlation is enhanced:

0 5 10 15 20 25 30 35 .10

4 WV/c)

,

THREE-BODY PROBLEM np wavef1lllc.t ic)!l in the vicinity of the emitting

PROTOnf -ME kTRON source: IN ffRA~Ti0~

a F’: 0;

(-

L(

%@ = - Jr,1 + 2777

>

a + ‘I~&-, - l-2) (1) I

+C,(rl,t)Q,+ C,(r,./)+

where \ wltcg&-#urLEuJ ?oTL*,:f*’

c; = vc.+ VN - iw,y,

and at-1 -.+ ei(P(rl+r2)/2+A-Ef) 4Ly)(r1 _ r2)

as r1,2 -+ cm, t + 00 e mm HAVE A UfSouAn PffAK A7 INFl~~n-

Stationary eq:

i-tliillclll~ioili\l (:(I. assuming splic~rkal s)*nllll(~t rJ’*

Angultir momentum expansion?: Intrinsic and pair CM momenta couple. + For every total LM coup1e.d eqs in ~-1. ~2. k1. i!2 and m.1. No transpa.rency. Boundary conditions??

3

NSCL-MSU 1G

ClassiCal method?? Changing velocity, barrier . . .

Anything else?

Possibility: \L

@(r. R.) = ,yp(rl) xn(rz) +(r, R) c-‘~’

where the 1-ptcle wavefunctions x satisfy the 1-ptcle eqs

v2 Ex=-~x+C-X

Then eq for intrinsic wavefunction dependent on CM position: 3

$ VR (l(% XI, )(.n)) vR+q

= -$Pq-

x

Slow variation of intrinsic wavdimction with CM poSition + largest variatibn of l-ptcle wveFLa*ction along the &@f CM mom&um

p starts slower than n towards Au. Should be slowed down more, i.e. q should increase on approach to the nucleus.

Relative wavefunction at the distance R 7: 25 f?~ from Au:

5oo.o t’ 200.0

100.0

50.0

‘20.0

10.0

5.0

1.0 =-/ L

0.5 l’,,,,‘,,,,‘,,,,‘,,,‘I,,“I,,‘,I, -6 -4 -2 0 2 4 6

p AWAY FROP1 Acr 2 (fm) p ?OdhROS AL.

2 Rf LAllV& h p

Posrrrorv

Closer to the nucleus: 5oo.o m---T--“” 200.0

t

I'P

q=lOMc ,

t -- s=o

sv/c E,=25MeV i 100.0

N- 50.0

3 7 20.0

4 b Ie 10.0

22 5.0

2.0

Changes in the correlation function: 3.5 I ’ ’ ’ I I I ’ ’ ’ I I ! I I I ’ ’ ’ ’ I ’ ’ ’ ’ ’ 4 I ’ * ’ I I ’ ’

3.0

2.5

2.0 cd

1.5

1.0

0.5 1

0.0 J’~‘~“~‘ll’~‘l’t”ll”‘l”““l”l’~,i 25 50 75 100 125 150 175 200

R b-4

SUDDEN vs ADIABATIC LIMITS

* Naive expectation (BDS) ’ Full adjustment of the intrinsic wavefunction TV 0

external changes - the adiabatic limit.

- + Intrinsic wavefunction actually changes VGV-)” \kk

Conditions for sudden and adiabatic limits formulated normally for discrete states.

Wave spends at short distances the time:

+$!!I! yv 4-3 q dE AL-

. ‘I F-/ cc/

where 6 - phase shift. Wavefunction can adjust itself if:

T << At

where At - time for significant changes in the exterior. If

r >> At

no adjustment can be expected.

5-

In the case at hand 6 M -ai, q, where a;, = -23.7 fm. Then

TZ:- 2 $&, mN

Q - 2($n)2 mN

for relevant q N l/ Iu& 1. Time for significant changes given by

APP -2qz FAt

w i.e.

2R

At - V(R) a;,

The wavefunction can adjust if

R >> R, = ad?f&&i-i = 190 f/71. -- -- t

2 =?3

1. Many-body theory allows to formulate clo;lnlv the problem of A-particle emission in a complicated heavy-ion reaction.

2. A-particle sources can be defined in terms of expectation values.

3. Relative np wavefunction undergoes only modest changes in the field of heavy nuclei.

4. The lack of a structure in the correlation function measured by CHIC is not explained.

5. The fate of a 2-ptcle scattering wavefunction generally depends on &i/&S.

23

24

0 R= P D rl - I2

D fi = 11 - D D 21

0 \a - D 1% c D at - D 22ro 25

H .

r, 0

+

A 1 G-7

F S

T U

i = -k-

27

"' - J, t

.i“. . . . . .

,, .” .‘: .,:.: . . .T

;., ,, .,__: .‘ . . .,,,,, . . ...>.

.’ :....‘. : : .. .) ‘.

: : -,.:. : $

..,:

” :. : . .+. :,

), : ..

.’ : :.

. .

,,::

. . .

._. .i a.,

.L/ :,

' R \ esw e

31

.

a7 6.

32

(* 100 7,)

( + 40%)

33

34

H.A. Weldon

Department of Physics, West Virginia University

Space-time is preferable to momentum space for certain aspects

of quantum field theory such as lattice gauge theory and the

short distance operator product expansion. Here the asymp-

totic behavior of the photon propagator DpV(t, r”) in the region

r -+ 00 and t -3 00 with fixed ratio r/t < 1 is used to under-

stand the behavior of the electron self-energy near the mass-shell

in various temperature regimes. (1) At zero temperature the

free photon propagator falls like l/t2 and this causes the elec-

tron self-energy to have a branch point at the mass-shell. (2)

At low temperature, 0 < T << m,, the free photon propaga-

tor falls exponentially at large time and this causes the electron

self-energy to have a simple pole at the mass-shell. (3) At high

temperature, m, << T, the resummed hard-thermal-loop prop-

agator for the photon falls like T/r even in the deep time-like

region and this cause the electron to have a logarithmically di-

vergent damping rate.

35

2.

Electron self-energy to one loop

c(p) = ie2 J ‘y4 o”“(K) @(P + K)rv (2 > Electron propagator

y(P+K)+m S(PfK)= (p+K)2-m2+ic

For soft K approximate as

S(P+ K) = y-P-+-m

P2-m2+2P.K-+-ic

Use exponential form

1

-I

O3 dt P2-m2+2P. K+ic =-’ o 2p. eit(P2--m2+2P.K+i~)/2po

Behavior near mass shell controlled by photon propagator

Dpu(t, i7t) in space-time as t --+ 00.

36

Define l-I(P) = &Tr((y-P+m)C(P)).

Near mass shell po = E the ferrnion propagator is

S’(P) 73 Numerator

PO - E - H(P)

where the self-energy is

e2 O” II(P) = 3 J & e’t(PO-EE+i~) P,pu~PU(t) 5-t)

0

. 1. Self-energy at zero temperature

Use zero-temperature photon propagator. In Feynman gauge

P"(t) F) = -igP"

47~“(t~ - r2)

The electron self-energy near mass-shell depends on P” (t, 8) :

n(p) = i:22 Jm $eit(loo-E+ie) + :;I (PO - E) 14Po - E) t0

where to is an UV regulator.

The denominator of the electron propagator becomes

PO - E - n(P) -+po-E++(po-E)ln(po-E) 7r

Q/T is infrared anomalous dimension.

This result is well-known.

37

4 .

2. Self-energv at low temperature (0 < T << m,)

Puzzle: Exact momentum-space calculation at one-loop shows

that for nonzero temperature, 0 < n-T << m,, the electron self-

energy does not contain a term (po - E) ln(po - E).

The singularity is a simple pole. (HAW, Phys. Rev. D 59)

Space-time explanation: In Feynman gauge the free, thermal

propagator is

iT lY(t, F) = gpu-

[

1 1 4rr $rT(tSr) - 1 - eznT(tT) - 1 1

-gpu& [6(t + r) - d(t - r)]

(See HAW, hep-ph/0007072 for this and results in other gauges.)

The fermion self-energy is

ie2m2T ITcp) = gnvE2

J [ “’

eit(po -E+ic)

to t e 2~Tt(l+~) _ 1 - eZZZC~JII~~ 1 1 This integral converges at po = E and so do all its derivatives

with respect to PO. The self-energy analytic at po = E.

The electron propagator has a simple pole for any small, but

nonzero, temperature.

38

3. Self-energy at high temperature (nT > m)

For ,hard fermions can neglect resummed vertex functions.

Resummed fermion propagator introduces thermal m = eT/1/8.

Need HTL photon propagator *W”(t, 3 in space-time.

TTJPe Deep space-like Deep time-like

Free massless T/r (T/r) exp(-2rT[t - r])

HTL */Do0 pole rnglr l/2 3/2 mLl It HTL *Do0 cut exp small Tlb-+t3>

HTL *DiJ’ pole Wmgr3) l/2 312 mcl It HTL *@j cut T/r (a) T/r ca)

(*I A magnetic mass changes both these to T/(mgagr3)

The cut contribution to *Dij in the deep space-like limit is

r > t :* Vijcut(t, f) -+ I!&jU + gi$j + 4TE2r3 (-@j + 3j+jj.

9

In the deep time-like limit the leading behavior is the same:

where ?? = r/t is fixed.

39

6 ,

The fact that the space-like behavior is T/r is trivial to ver-

ify in the Matsubara formalism. The fact that the time-like

behavior is also T/r is very non-trivial. (The free thermal prop-

agator falls exponentially in the deep time-like region.)

Fermion self-energy is

-ie2Tv qlJoJ3 = 4r J O" g eit(po-E+ie)

t0 t

This diverges at po = E like ln(po - E).

Logarithmically divergent damping rate in abelian gauge

theories shown by Pisarski, Phys. Rev. D 47 (1993).

Fourier transform

-ie2Tv e--iEt m;Dq = 471. t

Agrees with Blaizot-Iancu, Phys. Rev. D 55 & 56 (1997)

Boyanovsky, de Vega, et al, Phys. Rev. D 58 (1998); 60 (1999)

For nonabelian, magnetic mass changes asymptotic behavior

of *TV@, 8) from T/r to l/t”. Then II(P) is analytic and

damping rate is finite.

40

Resonant decay ofparity odd bubbles

in hot hadronic matter

Daria Ahrensmeier, Rudolf Baier, Marcus Dirt’&

Faculty of Physics, University of Bielefeld

PLB 484 (2000) 58 (hep-ph/OO 05 051) ’

WITTEN - DH~CCHIA - VENEZIANO (79/80)

c f eff = : [tr(~&Ja’lu+) + tr(MU + MU+)

a -- NC (

8 * - %tr(lnU - 2

In U+) )I in the following: NC = 3, “real world” 19 = 0

meson fields U= exp(iqS/f,)

Ah 7r+ J2K+

with g5= fiqql+

7r” + $$

&r- -7rO + z l/5K0

JZK- JZI-CO --@3

mass matrix Mij.

with EL9 = 4

4

IQ2

topological susceptibility

= Pf6ij = - m:, = - 2ma - irag

= (P? +P; +&I3 N - 0.171 GeV2

a = 2Nf XYM-/fjf

cz 0.726 GeV2

42

Temperatanre dependence of singlet effective potential

Kff 3 =P2f2 ( > - cos 3 + 2

$33 >

WI/f)

i / v/f

with q G 71 and f E I/- Wf7r

T=O

all-J2 l=- WP2)sp

T = Tsp

(frj/fjsp = 4.493

0 < ah2 < WP2jSP KHARZEEV, PISARSKI, TYTGAT 98

parity odd metastable states

T = Td

a/p2 = 0

43

Application to heavy ion collision:

little BANG

phase transition

+-l-f-

T=O

T = Td

T&, < T < Td

field trapped in false vacuum

CP-odd bubble in hadronic phase

T = TSp ==: 0.86 Td

decay of bubble:

field starts rolling down

energy transfer to fluctuations

* particle production

see also: EC. BUCKI~EI-. T. F'IJCTLEBER.G and A. ZHITIVITSIO

(l~~~-~l~~/00oCi057) ;

~infla.t.ion. llC(‘l. axion prod~lctior~ )

44

, -..-

Evolution eqleations

decompose field into expectation value and fluctuations:

(e.g. D. BOYANOVSKY, H. DE VEGA et al., A. LINDE et al.)

rl(K t> = cp(t> + x(6 t> with (~(a, t)) = 0

Hartree-type approximation x3 + 3(x2)x and x2 + (x2)

dimensionless time r z pt

initial conditions: 7 cp(o) 5 y = 4.493 and f = d(O) 0

expectation value of fluctuations

dimensionless momentum K E k/p

initial conditions: xIG (0) = Jk, & (0) = -i,,/y

45

Characteristics of the evolution equations

coupled non-linear evolution equations:

6 zero mode transfers energy to mode functions

via time dependent frequency W&(T)

+ particle production

l mode functions react back on zero mode

via fluctuations (“I back react ion”)

+ damping

46

Solving the evolution equations II

Numerical solution of system including back reaction

zero mode Vml

back reaction

p(O) N 412 MeV; &I’&,) C-Y 676 A&V

f&J) ” f(O)

47

4t> V- mw3]

1

0.8

0.6

0.4

0.2

0

Particle number density

0 2 4 6 8 10 12

tFm1

PGP) E 676 MeV

P(O) N 412 MeV

inclusion of back reaction destroys parametric resonance

48

Momentum spectrum

of produced particles at tend % 10 fm

for p(T = 0) N 412 MeV

non-thermal

0 0.2 0.4 0.6 0.8 1

does not fit Bose-Einstein distribution

m = 0, T = 1.16 GeV

m = 0, T =,130 MeV

49

Summary

We investigated the decay of metastable states with

broken CP-symmetry which are expected to form in

hot hadronic matter.

l amplification of the low momentum modes

by parametric resonance

l back reaction must be taken intb account G=

amplification of 7’ by parametric resonance

suppressed

0 for tF5 10fm

- particle density n(t) z 0.7 f me3

- correlation length < FZ 2.4 fm

- soft, non-thermal momentum spectrum

with k,,, z 190 MeV

50

The Quantum Field Limit Of Classical Field Dynamics

Berndt Miiller Duke University

BNL - RIKEN, July 20,200O

51

flk ~“(4) = Jdr a(z) .

53

55

56

The Renormalization-group Method Applied to Transport Equations 1

TEIJI KUNIHIRO

Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto, 60643502, Japan E-mail: kunihiro@yukawa.kyoto-u.ac.jp

The renormalization group (RG) method is applied to deduce a slow dynamics from trans- port equations, i.e., Boltzmann equation and Langevin equations. We deduce a hydro- dynamic equation from the Boltzmann equation and also Fokker-Planck equations from Kramers equations which correspond to Langevin equations with multiplicative noise.

Following [I], we start with derivation of an exact RG equation for asymptotic analysis of evolution equations, which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method con- structs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set to = t is naturally understood where to is the arbitrary initial time. We show that the integral constants in the unperturbed solution constitutes natural co- ordinates of the invariant manifold.

We show that the RG equation determines the slow motion of the would-be integral constants in the unperturbed solution on the invariant manifold. We emphasize that the underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto[2] a decade ago. We indicate that the reduction procedure of evolution equations has a good corre- spondence with the renormalization procedure in quantum field theory.’

We work out two simplest examples from ordinary differential equations; the damped harmonic oscillator and the Rayleigh oscillator which has a limit cycle. Then we examine a generic non-linear equation in which the linear operator in the unperturbed equation has degenerate zero eigenvalues, to show how a reduction of dynamics is described in the RG method: We explicitly show how the invariant or attractive manifold and the reduced dynamics on it are naturally deduced perturbatively in the RG method.

As good examples of this generic system, we examine the hydrodynamic limit of the Boltzmann equation and the reduction of the Kramers equation to Fokker-Planck equa- tion. In the latter case, we discuss on the way how the It&Storatanovich dilemma is resolved when the Langevin equation with multiplicative noise is extended to Newton equation with an inertia.

IThe original title was “The Renormalization-group Method Applied to Kinetic Equations”. However, I feel now that the changed title reflects more the content of the talk.

57

References

[I] S.-I. Ei, K. F u 11 and T. Kunihiro, Ann. Phys. 280, 236 (2000). j” See also, T. Kunihiro, Prog. Theor. Phys. 94(1995)503; (E) ibid., 95(1996)835; Jpn. J. Ind. Appl. Math. 14 (1997), 51. T. Kunihiro, Prog. Theor. Phys. 97(1997),179. T. Kunihiro, Phys. Rev. D57 (1998), R2035; Prog. Theor. Phys. Suppl. 131 (1998), 459; patt-soU979003.

[2] Y. Kuramoto, Prog. Theor. Phys. Suppl. 99, 244 (1989).

58

.

N

’

61

62

63

1.l e

I-’ t

t

\ en=0

eo=9

\

.

.

.- . . . . . . .

I

100

64

Non-equilibrium quantum plasmas, quantum kinetics and the dynamical renormalization group.

H. J. de Vega, LPTHE, Paris, France.

We study the real time nonequilibrium dynamics in hot quantum plasmas both in QED and scalar theories. We derive the relaxation of scalar, gauge and fermion fields implementing the dynamical renormalization group and using the hard thermal loop (HTL) approximation. We derive quantum kinetic equations from a quantum field theory diagrammatic perturbative expansion improved by a resummation via the dynamical renormalization group. This method resums secular terms in real time and leads directly to the quantum kinetic equation. We compute photon production taken into account transient process at one loop (order eA2). Such contribution grows only logarithmic with time instead of linearly as the two loop contribution does. However, this new one-loop photon production contribution gives comparable effects at the time scales to be observed at RHIC.

65

C’

I .

5 t-J

68

Domain Walls and Cosmology at the QCD scale.

Ariel Zhitnitsky

Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1.31, Canada

Abstract

QCD was shown to have a nontrivial vacuum structure due to the topology of the 8 E 8 -I- 2ma parameter. Specifically, the transitions 6’ --f 9+ 2nn are homotopically distinct-the homotopy classes are de- scribed by the winding number n. These transitions can be described exclusively in terms of QCD degrees of freedom (which are identified with the q’ N (A + 4 d meson) without involving the dynamical ax- ) ion field. As a result of this nontrivial transition, quasi-stable QCD domain walls will appear. I construct and discuss the physics of these QCD domain walls. I argue that QCD domain walls, though classi- cally stable, are unstable on the quantum level due to the tunneling. Therefore, this object is irrelevant for study at the present epoch. The lifetime is short enough that no cosmic QCD domain walls remain to- day, however, short-lived QCD domain walls might play an important role in evolution of early universe and may be detectable in energetic collisions such as those at RHIC. We also argue that the interactions with baryons make domain walls ferromagnetic. This feature may play an important role in physics shortly after the QCD phase transition in early Universe. In this case the domain walls formed at the QCD phase transition can serve as the magnetic seeds of the primordial galactic magnetic fields.

69

3.

72

73

74

--

75

76

1

--__

78

79

_” -... _ &,fi , .’ ‘j .‘(, . -

.

81

_’ ,._

,.

., _. . . : 1

:, ::. .’ 0 . . . . ,j. : i_:

. .

P. ‘.-

i ., ,. ‘,

.: .

.,,. ;Q: ,. ._.. 5 ‘.

4-7 8

;.

_.

.r

i.

I y 0.2862

82

i

83

.. Im I&r -(.p,, p) in 24avor CSC

8 00 . k k 1 0 1 2 3 4 0.1 2 3 4 5

pd@ pd2+

gluon momentumJp = 4@/ I 8 I ’ 1 ’ , 1 I ’ I ’ ‘I ’ I A

.

1. 2 3 4. 5 p&3

85

86

BNL - July/2000

Quark Stars from

Perturbative QCD

Eduardo S. Fraga

Nuclear Theory - BNL

Work done in collaboration with JGrgen Schaffner-Bielich (RIKEN-B&L)

87

88

- 1975178: Physical description (basics):

Many papers on quantum-hadron phase transition, high-density regime and neu- tron stars (Baym & Chin, Kislinger & Morley, Baluni, Freedman & McLerran, . ..) N eu ron stars, strange stars, hybrid stars, . . . too many possibilities. How to t

decide?

- 1978: Freedman and McLerran develop the first systematic quark star phenomenology

by using high-density perturbative &CD. Astronomical observables:

- 1984: Witten proposes the idea of stable stran,ge matter, ie, quark matter rather than

nuclear matter might be the ground state of &CD at finite baryon number (Followed by studies by Farhi and Jaffe) -+ Quark matter in neutron star cores --+ Strange stars!

- 1986: Baensel, Zdunik and Schaeffer / Alcoclc, Farhi and Olinto discuss in detail the

phenomenology of self-bound strange stars

. . . . Hybrid stars, different families of neutron stars, strange stars, etc. (see, eg, Glen-

denning, Compact Stars)

M: total mass R: total radius I: moment of inertia

Example (Bombaci, astro-ph/0002524):

\ z? 1.5

3 1.

/ 1.0 /

/ /

/

0.5 ,/

2.5 / /

I

/ I

/ /

2.0

I /

0.0 0 5 10 15 20

R (km)

How to. c%Jculate &hem - TOV equations: 2. Usual approsch to quark stars and strange stars

Einstein’s equations + static + spherically symmetric stars:

dp GM(+W z=- ?.2

I[

1 _ 2GMb9 r

dM - = 47rr2+) ; dr

M(R) = M

\D 0

Given M(0) = 0, E(O) = E,, and the equation of state (EoS) p = p(e), one can integrate the TOV equations from the origin until the pressure p(r) becomes zero at T = R.

Given p = p(e), M(Q) defines a family of stars. Extrema in Aa signal gravita- tional instability (theorem, see Weinberg) --+ maximum mass. Then:

Physical picture:

- Strange matter described by a Fermi gas of u, d and s quarks, and electrons, where the region the quarks live in is characterized by a constant energy density B.

- Star temperature << typical chemical potentials ---+ T = 0

- Chemical equilibrium:

d--+u+e-+q

u+e--+d+v,

s-+-u-l-e-+q

u+e-*s-l-v,

s+u-sd+u

Then: different types of stars (neutron, strange, . ..) give different EoS + differ- ent astronomical output.

Remark: The usual approaches to quark stars rely on the MIT bag model and provide results that depend strongly on th.e bag phameter B. It would be nicer to obtain results which depend on quantities like CI’, or p(o+%) instead.

- Overall charge neutrality:

2 1 1 -nu - -nd - -n3 - ne = 0 3 3 3

Then: There is only one independent chemical potential.

C!?Sli ni=-alli

CC&: thermodynamic potential (free gas + eventual O(cr,) corrections, (Y, = const)

v) Y

For the simple case m, = md = m, = 0, 01, = 0, and pe = 0 (n,/n, M 0), the EoS is:

p(e) = $5 - 4B)

Remark: ior intermediate values of m,, the correction is less than 4%. The EoS is dominated essentially by B !

Global properties of strange stars (from TOV equations): .

p=-B-C CT& ; e = B + C (sli + pini) ; B1J4 = 145 MeV i i

M 0.0258

nMz = (ypB’,2 = 2Kola,

0.095

e y,= 19.2B M 2 x 1015 g/cm3 M 860

e,,,.f = 4B x 4 x lOI g/cm3 M 2~0

EO = Enudea1 M 2.5 x 1014 g/cm3

ENS - su,.f - %olidFe !,-a 7.8 g/Cm3

Remarks:

- Results depend crucially on B.

- Very different patterns for NS and SS. However, for M M 1.4A&,lar most of the properties are almost the same.

- Surface details are complicated and will not be discussed.

d I

FG l.-Density (p) vs. radius (r) lor strange stars of mass (4 0.53 MO. (b) : 4 Me,(c) 1.95 M,. and kf) 1.99 M,.

._

0’ 1 1

10” (0’” 1O’6

R (km) - Charge neutrality + chemical equilibrium:

FIG. 3.-Total mass(M) vs. radius(R) for stable strange stars

3. Quark stars from perturbative QCD

Physical picture:

- Star is made only of a gas of u, d and s quarks.

- Strong interaction in this dense quark-gluon plasma is taken into account per- turbatively upto,O(c$‘) ; a, = g2/4n.

- cx, is allowed to run according to the renormalization group equation.

- No bag constant is introduced.

- Star temperature << typical chemical potentials ---+ T = 0

Then: Use high-density zero-temperature perturbative QCD. The typical densi- ties one can find inside quark stars allow for a sensible use of perturbation theory [Pisarski & Rischke (2000); cold, dense quark matter].

Warning: It is not our aim to provide a realistic a.nd accurate description of the phenomenology related to quarlc stars by using such a crude model. We intend to highlight the essential difference between the bag equation of state approach (strong dependence on B) and the perturbative QCD approach (dependence on cr,, /3(crs), etc.) -’

Rema.rk: Nevertheless, it is natural to expect that the results obtained from bag models should correspond to some limit in this more fundamental approach.

The thermodynamic potential:

The thermodynamic potential of a plasma of massless quarks and gluons with NC colors and Nf flavors upto O(a$), in the MOM subtraction scheme and using the Landau gauge, is given by [Freedman & McLerran (77)]

i&f& = c i-p {l-(T?$y(~)+(!zg)2(~)x

x jr:--$(21n(z).--D) --8(P)]}

\9 P do) = -$N,/(12n2) (ideal gas 1 flavor massless ?. z I 1 1 T = 0)

C = -2.25ON, + 0.409Nf - 3.697 - 4.24 f 0.12

N c

p”=cpf ; Ng = N,” - 1 i

defined. ; PO : Euclidean subtraction point at which the charge is

Translation from MOM to m [Celmaster & Sivers (81); Raczka & Raczka (SS)]

A = -6.623 + 0.854Nf

Then: Up to this order in the expression of s2, the translation between schemes corresponds to a shift in the constant of the 2nd order term of @‘O”.

Final form for 0 in the MS scheme:

n(p) = Jg {l-2(:) -(~)z~+N~ln~]}

GE NfInNf+;(Nf -l)ln4-$c+N,D)

c=c+4.A ; D=3.249

The coupling cr,(ps) runs according to

a(p0) = /30 In (iY/*&=-,J [ ’ 2P1 ln [In (cl”/&)] +

-2 PO ln (P”l+)

1” + /?a4 ln2 ;:“l/*$&j ((In [ln (P22/hb)] - i$’ f t$$ - p)]

PO = 11 - 2Nf/3 ; ,Br = 51- 19Nf/3

& = 2857 - 5033Nf/9 + 325Nf2/27 ; Am = 364.834MeV

Choice of 16s:

Since the only physical scale of the system is given by the Fermi momentum of the interacting massless quarks, we choose p. = ,u,.

Another “physically motivated” choice: ~0 = PB = 311 -+ leads to results that reproduce the features obtained by using the bag model approach.

From the knowledge of W(U):

Then: TOV equations --+ astrophysical features that follow can be written in terms of crsr @(IX,), . . .

In fact, one could even define a “bag function”:

4. Discussion of results

Equation of state:

1.5

-1 C-Y E e > Q) u x 0.5

0 4

60 = 2.5 x 1Or” g/cm3

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Mass-radius relation:

0.5

0.4

L 0.3 L? 8

;I - 0.2 E

\o 4 0.1

0 0

’ I? (km) *

In the usual bag model approach:

3

5. C.onclusion and future work

Quark stars from perturbative QCD have the following features:

- They are smaller (I? M 3 km), less massive (M x 0.5Mso~a,), and denser (% M 80~s) than bag model quark stars and neutron stars. --+ Very different phenomenology!

- Phenomenological results depend on QI, and ~(cY~), not on B.

- They represent a new class of stars (new branch in p = p(e)).

Related problems:

- More complicated and realistic models.

- Relation to MACHOS (Massive Compact Halo Objects) ? [Best fit mass M 0.5Mso~arl

- Relation to color superconductivity, CFL phase, etc ?

98

The Electrical Conductivity in a QED Plasma --Luis Bettencourt and Emil Mottola, Los Alamos

A calculation of the electrical conductivity of a high temperature plasma of $e*{\pm}$ and photons in QED in the real time formalism is described. The methods employed in this calculation are applicable to nonequilibrium processes in other field theories, and offer an interesting prototype of extracting the hydrodynamic limit from a microscopic quantum theory.

Linear response about thermal equilibrium is the simplest such real time process and the abelian QED case is the simplest gauge theory, where weak coupling methods should be applicable. Nevertheless a non-trivial resummation of .perturbation theory is required to extract the weak coupling result for the DC conductivity, due the extreme sensitivity of the relevant long wavelength, large time hydrodynamic limit to infrared effects in the microscopic theory. ’

We propose a specific truncationof the infinite Schwinger-Dyson hierarchy of correlation functions in QED to just the electron/positron and photon two-point functions, together with the 3-point vertex function, which satisfies the relevant Ward identities. This set of S-D equations is the next non-trivial order beyond the leading order in large N which includes Landau damping and Hard Thermal Loops. The S-D equations are renormalizable and include both hard and soft scattering processes as well as the fermion damping rate in a self-consistent way. The Boltzmann equation with two-to-two scattering processes in the collision kernel can be derived from our S-D equations in a certain limit, but our approach is to keep careful track of the errors made in passing to the Boltzmann limit from the microscopic point of view. The method of solution of the S-D equations in real time in the high temperature QED plasma is sketched and the final results will appear shortly.

99

r

+ 104

SJ -= SP

o=

A -\ w

,, 7 = f 2, : . 1s ..___.._. % = : . isA ,’ . -1 , \’ . * /3 = G’:z

105 :- __ . . . _ _ _... ., -- - e--L. -

(Approximately self-consistent)TM HTL/HDL thermodynamics of the

quark-gluon plasma

Anton Rebhan

Institut fiir Theoretische Physik, Technische Universitat Wien

Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria

Abstract

Conventional thermal perturbation theory yields a series expan- sion in powers and logarithms of the coupling g with poor convergence properties, caused in particular by contributions associated with col- lective effects such as screening. Usually only the effects of static Debye screening are utilized in resumming perturbation theory. A more complete description should involve the full hard thermal/dense loop (HTL/HDL) propagators. It is shown that in a self-consistent <P-derivable two-loop approximation of the thermodynamic potential one can derive simple effectively one-loop expressions for the entropy and the quark densities which are formally UV finite. Gauge indepen- dent and UV finite results are obtained by restricting the propagators to those of HTL/HDL perturbation theory up to order g3. The results for entropy and density need not be truncated into a polynomial in g (with poor convergence properties), but can retain numerically im- portant HTL/HDL effects that are formally of higher order in g. The resummation in entropy and density exhibits an interesting reorgan- isation of LO and NLO contributions: The LO interaction terms are caused entirely by the so-called asymptotic thermal masses at hard momentum scales, as given by the light-cone values of HTL/HDL self- energies. The plasmon effect - g3, which usually comes from soft momentum scales only, is found to arise mostly (to 75 %) from NLO corrections to the hard asymptotic masses, calculable from HTL/HDL perturbation theory. Numerically, including the NLO corrections to the asymptotic masses through an approximately self-consistent gap equation achieves remarkable agreement with lattice data at T > 2T, while being perturbatively equivalent up to and including order g3.

107

108

. . :cFs 5 IL

= --

i-2 d& s

IDynamics of the chiral phase transition at finite --

chemical potential

Talk by Fred Cooper at Brookhaven, July 2000

Alan CHODOS*

Department of Physics, Yale University, New Haven, CT 06520-8120

Fred COOPER+

Los Alamos National Laboratory, Los Alamos, NM 87545

Wenjin MAO

Department of Physics, Boston College, Chestnut Hill, MA 02167

Anupam SINGH

Los Alamos National Laboratory, Los Alamos, NM 87545

*Elettronic address : chodosQhepvms .physics . yale . edu

+Electronic address: cooperQschuinger . lanl . gov

113

Here we want to study the dyamics of what happens when we

start at high temperatures and traverse the 3 possible QCD phase

transitions- first and second order chiral tr$&ion and first order

superconducting transition.

l Obtain a Model with phase structure of QCD

0 Determine Phase Structure as a function of p and T

l Use model to study dynamics of QCD-like phase transitions

l look for signatures for RHIC

. HERE WE IGNORE THE SUPERCONDUCTING PHASE CI = MODEL:

The COMPLETE MODEL with chiral and superconducting

phase:

L = p)i $p + ~92w(“)~(i)]~~(3)7jl(i)]

+ 2G2(~(a)y,~(j))(~(“)ys~li)) _ p$t(i)$(i), (1).

The flavor indices, summed on from 1 to N, have been explicitly

indicated. The first term is the Gross-Neveu interaction, whereas

the second such term, which differs in the arrangement of its flavor

indices, induces the pairing force to leading order in h. In the final

term, p is the chemical potential. The FULL phase structure is in

the Fig 1

a2=1/4n

FIG. 1. Phase Structure at 6 = &

NEXT we restrict ourselves to the Chiral Sector G=O! Lagrangian

is

c = -&&yl”d,l.p - $2 pii+, (2)

which is invariant under the discrete chiral group: @i + ysqi. In

leading order in large N the effective action is

Seff = j d2x -iqfi o-2 pigi - i&LQi - - 2g2 I

+ hbS+l [a], (3)

where S-l(x, y)[o] = [~‘“a~ + g]S(x - y).

The equilibrium properties of the GN model a,t finite temperature

and chemical potential been known for a long time -L. Jacobs, Phys.

Rev. D 18 (1974) 3956. U. Wolff, Phys. Lett. 33 157, 303 (1985).-

and is summarized by Fig. 2.

” for 0u=0.66,T=0.8 " for mu:O.66,T=O.Z

FIG. 2. Phase structure at finite temperature and chemical potential p, The

phase below the line has (&J) # 0

The phase structure is determined from the renormalized effective

potential

T/eff(02, T, p) = g [In ZY - l] 1723

- j sooo g [In (1 + e--IO@+)) + ln(1 + e-B(E+P))] (4 Here mf is the physical mass of the fermion in the vacuum sector,

The critical temperature at small chemical potential is given by

T, = !?lleU[l _ 7p2c(3)]e 7-i 4yrn;er

The tricritical point occurs at

- = ,608, 2 = .318. PC

rnf mf We have chosen to renormalize the effective potential so its value

at T = 0 in the false vacuum 0 = 0 is zero. In the true vacuum

cr = mf the energy density has the value

-0.25 I

1.2 1.4 " for mu.0.66.T.O.15 Y for mu=O.66,T=O.05

FIG. 3. Evolution of V,,f as a function of T. This is for a first order transition.

" far cu=O.1,T=O.0 " for mu.O.l,T=O.5S

FIG. 4. Evolution of V,,, as a function of T. This is for a second order transition.

1 iz/m2, = -4n.

EXPANSION OF THE PLA~MATHROUGHAPHASETRAN-

SITION -

Following a heavy ion collision, the ensuing plasma expands and

cools traversing the chiral phase .transition. In hydrodynamic sim-

ulations of these collisions, a reasonable approximation is’ 40 treat

the expansion as a 1+1 dimensional boost invariant expansion (B-

jorken,Landau,Cooper et.al) along the beam (x) axis. . In this

approximation, the fluid velocity scale as

. . v = z/t

CL g In terms of the variables fluid rapidity

,=;I, t+z ( ) t-z q- = (t2 - z2y2

physical variables such as 0, E, p become independent of q.

Although the effective mass G and E is a function solely of 7, two

point correlation functions depend on fluid rapidity (q) differences

as well.

We shall use the metric convention (-+). In our approxima-

tion, the dynamics are described by the Dirac equation with self-

consiste&ly determined mass term. Resealing the fermion field,

and introducing conformal time u via

7 = mfe”

we obtain

where 5(u

[r”a, + r3a, + qu)] q2-g = 0 ) (5)

>= Pi- = zeU. Further letting g2 = X/2N,we have the

gap equation

c = -iX 2N’ ([

q&yOqi I> = -ii ([tit> TO@]) > (6) * and we have assumed here that we have N identical Qi = $J.

These equations are to be solved subject to initial conditions at

r = 70. It is sufficient to describe the initial state of the charged

fermion field by the initial particle and anti particle number densities,

which we will assume to be Fermi-Dirac distributions described by

,QO and To.

Expanding the fermion fields Cp in terms of Fourier modes at fixed

conformal time u,

C@(x) = J ~[b(k)@(u)eix~v + d+(-k)f$~k(u)e-ik~II], (7)

the 4: then obey

[ 70-g + iy3k, + 6(u)] q&u) = 0. (8) *

The superscript f refers to positive- or negative-energy solutions.

Introducing

,(9) -where the momentum independent spinors x& are chosen to be the

ortho+omal fl eigenstates of iy’, we obtain

d2 -- du2 (10)

where 61 = Ici+S2(~). We parametrize the positive-energy solutions

fz in a WKB manner:

!& obeys the real equation

fik + &- 2c?1, -

(ii + c$($ + 3&)

4ng =q(u)-q.

The lowest order WKB has

Using the mode decompositon and the definitions’ (bt(ic-)b(q)) =

2nS(k - q)N+(q) and (dj(k)d(q)) = 2nd(k: - a)N-(cJ), we obtain

for the gap equation

s= x j $I- N+(k) - N+))Rk(u)

where &k(U) = 1 - 2k: If$(u)j2. and

(13)

This equation is solved simultaneously with eq.. (10).

We choose our initial state to be in local thermal and chemical so

that

N&(/c, p, T) = [e(wk(o)Tp)/T + 11-l

where &(O) = E = /%G7@j = 3J9 Since we start oul sim-

ulation in the unbroken mode, C(O) =” 0. We choose the initial

70 = $ and measure the proper time in these units. We use adia-

batic initial conditions on the mode functions f, i.e. fk(0) = --$j$=,

f;(O) = -4$&(O) and N,$ = [&(O) + C(O)]-‘.

We have studied three separate starting points on the phase di-

agram of Fig. 1 in our numerical simulations. We determined the

energy density a.nd the pressure from the expectation value of the

energy momentum tensor.

02 ($ = (;Iy(,‘Qo - ;v(po~v)~ - gpy&. (14

In the 77, r coordinate system TPV is diagonal which allows us to read

off the comoving pressure and energy density. After renormalization

we obtain

(N+ + N-1 [= + 4%(w; - 02)lfk12 + 2(/c, - g)]] , (15)

Pr2= LA 2 [(I - N+ - N-) 4 (5 + fik)(i?2 - w;)lfk12

+ 2{kF - 2k, - cr2 1 #q-jx$’

(16)

In the massless phase, one finds that the exact equation of state is p =

E. To compare our field theory calculation with a local equilibrium

hydrodynamical model we assume

The conservation law of energy and momentum

Tap = 0 ;P ’

combined with scaling law w = z/t and p = E yields

From Eq. 15 and Eq. 16 we can also determine p(& 2’) and E(P, T).

Assuming T/To = TO/~ and ~/PO = 7-e/7- we find that the local

equilbrium expressions for E and p evolve identically to the numer-

ically determined field theory evolution before the phase transition.

Wth the same assumptions we find the distributions for N& plotted

against k, are independent of r. This also agrees with the exact

evolution before the phase transition.

We want to understand is how the paricle number distributions

evolve in time. In relativistic quantum mechanics, particle number is

not conserved. However in a mean field approximation on can define

an interpolating number operator which at late times becomes the

outstate number operator. By fitting the interpolating number den-

sities for both ferrnions and’antifermions to Fermi-Dirac distributions

we extract the-best value of 1-1 and T for that value of the proper

time. To define the interpolating number operator we use a set of or-

thonormal mode functions yk which are the adiabatic approximation

to the exact mode functions.

with

-iypk, + 5 uk = /w ??I -‘-‘--k = /$$!-$kq .%I-’

, The creation and annihilation operators then become time depen-

dent and the expansion of the quantum field becomes

@(xc> = / $[a(k, u)y,i!(u) + ct(k, u)yi(u)]eil”vv.

This is an alternative expansion to that found in eq.(7), and the two

sets of creation and annihilation operators are related by a Bogoli-

ubov transformation

d&u)= ‘X&@(k) + /3,$!!(k)

c+(k, u)= -,&(U)b(k) + o@+(k). - (18)

To ensure that at u = 0 the two number operators match, one choos-

es adiabatic initial conditions: y = 4, so that C&(O) =, l;&(O) = 0.

The interpolating number operators for “quarks” and ‘Lanitquarks”

are defined by

fi+(k, u) = (d(k, ?+(k, U)}; N-(k, U) = (c+(k, U)c(k, u)j.

With A, 7 2Rk u we have explicitly

(19)

N*(k, u) = N*(k) + [l - N+(k) - N-(k)](Pk(u)(2. (20)

We have solved the simultaneous equations Eq. 10 and Eq. 12

. numerically. Comparing N*((Ic, u) with an equilibrium paramatriza-

tion we have determined T(k, U) and p(k, u) as a function of k as in

E Aarts. When these quantities are independent of k, = hr this de-

fines a time evolving temperature and chemical potential. We found

that ‘2’ and ,u are independent of k except at high momentum before

the chiral phase transition.

From Fig. 5 we see that for both the 1st and 2nd order transitions,

O(T) shows a sharp transition during evolution from the unbroken

mode to the broken symmetry mode. Before the phase transition the

temperature falls consistent with the equation of state p = E. For the

2nd order transition, the chemical potential follows the temperature

and falls as $. After the phase transition, there is now a mass scale

rnf which leads to oscillations of B. For the 1st order transition

the chemical potential falls faster than 5. If instead we chose initial

FIG. 5. Evolution of T,p and D as a function of U. Top figure is for 1st order

transition. Bottom figure is for 2nd order phase transition

conditions so that the trajectory passed through’the tricritical point,

one finds a fall off for, p intermediate between the two above cases.

The order of the transition has a more noticable effect on the

spectrum of particles and antiparticles. As we have discussed, if the

system evolves in local thermal equilbrium with 0 = 0, then when

Nf(k eta, u) is plotted vs. k, = kT it should be independent of u.

Thus any change in this spectra is an indication of the system going

out of equilibrium. We expect and find that because of the latent.

heat released during a first order transition that the distortion of the

spectra is greatest in that case. (see Fig. 6). Going through a second

order phase transition has a smaller effect in distorting the Fermi-

Dirac distribution wheras passing throught the tricritical regime one

finds results (not shown here) just intermediate between these two

’

cases.

In local equilibrium with g = 0, E = p oc re2 . Simulation-

s,shown in Fig. 6 agree with this before the phase transition occurs,

FIG. 6. Evolution of A$ as a function of u. Top figure is for a first order

transition, bottom figure for a second order transition.The momentum displayed

. is k, = kr

is 0

After the phase transition we find that the energy density oscillates

around the true broken symmetry values discussed earlier, namely

EO = -1/4~ These oscillations would be damped if we went beyond

mean field theory and include hard scatterings between the fermions

-See Recent work of Berges and Cooper on web..

I. “PION” COkRELATION FUNCTION

We can define an effective (neutral) pion field via

where c is a constant. Using our mode expansion we find that

= / dk,k,P - N+(k) - N-(k)~-&f/c~2 (22) Because the integrand in eq. 22 is odd, the expectation value is zero

(otherwise there would be spontaneous breakdown of parity).

,a IG ”

FIG. 7. Evolution of the pressure and energy density as a function of u Top figure

is for lst order transition, bottom figure is for second order phase transition

For the equal time correlation function in lowest order in large-N,

we obtain the usual Fermion self energy loop, with the interpolating

number density for fermions and anti-fermions participating in the

loop evolving with the proper time. Apart from an overall constant

one can write the connected correlation function in the form:

-WI - 7’; 4 f (djh +o’, +h = -$T+‘S(q - 7’; +y3S(q’ - q; 7-)],

(23)

where at equal times, the prop&gator is just

Here a, /3 take on the values { 1,2) and are the spinor indices. Using

the mode expansion we find that the equal time propagator can be

written as,

Evaluating the trace we find that we obtain

dk dq *-L 2

T~D(~ - Y,I’; T) = 1 ,Ge i(k-dkd)D(~, Q; 7) (26)

where

D(k, q)= ([l - 2iV+(k)][l- 2N+(q)l+ [I - 2N-(k)1[1 - 2N-(q)l)Fl(h 4)

+([1 - 2N+(k)][l - 2N-(cl)] + [I.- 2~+(4)1P - 2WWWk 4) (2

and

F2@, q; 7)” Ifiil”l.&l”([k - &An + (f&t + C)(f& + 6)12 +[A,(fi, + 6) + A,(& + if)12). (29)

Correlation function: m i 0.8, p= 0.3 at u = 2 8000

6000

4000

2000

‘. 50 100 150 200 250 Correlation function: m = 0.8, T= 0.3 at u = 2.04

6000

5000

4000 I

3000

2000

1000

L 50 100 150 200 250 Correlation function: m I 0.8, Tr 0.1 at u E 2.00

5000

Correlation function: m = 0.8. T i 0.3

50000 R 40000

30000 /q-

\

M-= y

20000 !

10000 : !

50 100 150 200 250 FIG. 1. Evolution of the Correlation function as a function

of time. This is for a first order transition. of time. This is for a second order transition.

Correlation function: m = 0.5, T= 0.5 at u = 2 10000

Correlation function: m = 0.5, T: 0.5 at u = 2.04

Correlation function: m = 0.5, T: 0.5 at u = 2.08 6000

5000

4000

2000

1000

L 50 100 150 200 250 Correlation function: m = 0.5, T = 0.5 at u _ 4 35000,

30000 n

25000

20000

15000

1OOOQ

5000 \

50 100 150 200 250 FIG. 2. Evolution of the Correlation function as a Cunction

122

Hard thermal loops on the lattice

D.B6deker, in collaboration with G.D. Moore and K. Rummukainen

We have investigated an out-of-equilibrium problem in a hot non-Abelian gauge theory. The rate for baryon number dissipation y in the electroweak theory is much larger than the thermalization of the other degrees of freedom (except, of course, hydrodynamic modes). Therefore, the system is in a quasi- equilibrium state when observed on time scales of order y-i. Due to the chiral anomaly the change of baryon number is proportional to the change of the Chern-Simons number of the gauge fields. A fluctuation-dissipation theorem relates y to the diffusion rate of the Chern-Simons number I’cs.

The rate I’os is obtained from an unequal time correlation function in thermal equilibrium. It is non-perturbative since it is mainly determined by soft gauge fields with momenta of order of the magnetic scale g2T. Therefore it has to be evaluated on a lattice. It is not possible to perform lattice simu- lation of a quantum fields theory in real time. However, the non-perturbative gauge fields have long wavelengths and therefore behave classically. Thus one can use an effective classical theory to determine Ios.

To obtain such an effective theory the hard k - 2’ modes, which are not classical, must be integrated out. But even then one has to deal with the problem that the resulting effective theory does not have a continuum limit. One can also integrate out the k N gT modes. In this way one obtains an effective theory which does have a continuum limit and which has recently been used to compute I’ os. The drawback is that it is only valid at leading and next-to-leading order in log( l/g).

Here we have used an effective theory, the hard thermal loop effective theory, which keeps the IG N gT physics in the game. It contains the classical gauge fields and, in addition, the fields W(rc, 5) representing the color charge due to hard particles with velocity v’ (v” = 1). We have expanded W(Z, 3) in spherical harmonics Wlm(z) with respect to v’ and we kept only components with 1 < I,,. This gives a field theory in (3+1) dimensions which was put on a lattice. We have measured I’os for various values of the lattice spacing a, of Lmx and the Debye mass mn. While analytic estimates indicate that one has to use rather large values of I,, in order to reach the Z,, + co limit, we found to our surprise that the I,, dependence is very weak for even 2,,. already the result for I,, = 2 is basically identical to the I,, -+ 03 limit.

The u-dependence could be reduced by adding an u-dependent counter-term to razt. Our results agree remarkably well with two independent calculations, one which uses particle degrees of freedom to represent the hard modes, and one which uses only gauge field degrees of freedom. Our result is about a factor 4-5 larger than the leading log result, while the agreement appears to be better when the next-to-leading log corrections are included in the latter.

123

*’ msu Its

124

125

126

127

t

128

60

50

40 7c’

30

20

10

Chem-Simons diffusion

I

A&=8

. PO = 8.7 n p, = 12.7 0 particles * no HTL d.o.f

9 I cs = ?d

129

0.40 0.50

tx s 9 t

p rob (QUA L, Lot QCD

130

4

WJY @+f) ~IIA/ENSKWS ? 4

e KIM

4, M’A~NISTIC MASS: MASS GAP OF Y%WI

x MdBNET/C MASS OF

jh& AT rib.

I-IAMLL TONUIN ANALYSIS

+ SOME EXACT RESUb7-S

t; CI -+ MASS Gap, VACUUM ~JAVEFIJNCTION,

STRRNG TENSloN.

REcAPIT\JLATI~N OF RESULTS (~6s + ~YEAR>

2% 'f?AaE- rNvARtANT vAR,‘AHLES

2. GAIJ&E- INVAA~ANT MEASURE ) INNER ?RO~VCT.

3, =ti IN T‘HESE VAR/A0L. ES

4. PRO ? A GAro R NAss cot4PARE WVH 5. %c@J , STRING TENS\ON kATT\cE

‘\WN/VfRsALW OF 5”

8, NASS GAP ~MPROVE~” PERr~RdArto# ~#tSoRy >

N

132

d r-3

c”o

133

b

-+ e2-+ ea & %I?@)

Pd IN r QF blSCRE PAW)‘. ?

(NOT ewt)

WITH COMPARABLE

MASS TERt’# = ma F

@EOUIREb t ZIESIRABG PROPER Ttfs

2. SHouL& AGREE WITH F ABoYP

3. sHour.b HAVE u. HOLO Mot? PHlc ” ZNVARIANCB

E A= - 3,M M-’ w b-d &J, 3 SAME A

Solving the problem of initial conditions in heavy ion collisions

Raju Venugopalan

Physics Department and RBRC, Brookhaven National Laboratory, Upton, NY 11979, USA

October 27, 2000

The initial conditions for particle production at central rapidities in heavy ion colli- sions are determined in a classical effective field theory of &CD. The space-time evolution of the produced glue is determined by solving Yang-Mills equations for two singular light cone sources. Boost invariance is assumed-thereby allowing for a dimensional reduction of the theory to a 2+1-dimensional-theory. The energy and number distributions of gluons at ,late times are determined and shown to depend simply on the saturation scale QS of gluons in the nuclear wavefunction. With initial conditions motivated by the lattice sim- ulations, a Landau equation for parton transport is solved to study the equilibration of t.he system achieved by 2 to 2 processes. The equilibration time, the initial temperature and the chemical potential are determined as a function of QS.

137

t h

r\ F E

ii P

.

+

0.1 r

t

0 3 6 9 12 15

g2P-

Figure 1: 0-/(g2/.t)3 as a function of g2/1T for g2pL = 5.66 (diamonds), 35.36 (pluses) and 296.98 (squares). Both axes are in dimensionless units. Note that ET = 0 at T = 0 for all g2pL. The lines are exponential fits Q + ,L? e-r7 including all points beyond the peak.

141

142

At Equilibrium

100

600

5 2.75 3

u, (GeV)

FilllltI~llillllilllli~~lliIlolilI 1 1.25 1.5 1.75 2 ,,2.25 2.. A .

l-i. i i i I

1 l.LJ 1.3

143

144

Reorganizing Finite Temperature Field Theory

Michael Strickland Physics Department, University of Washington, Seattle, WA 98195-1560, USA

The heavy-ion collision experiments at RHIC and LHC, will allow us to study the properties of high- temperature phase of QCD for the first time. There are many methods that can be used to calculate the properties of the quark-gluon plasma. On the non- perturbative end, there are now reliable lattice results for equilibrium (Euclidean) properties. In the other extreme, the weak-coupling limit, standard perturba- tion theory techniques can be applied to both static and dynamical quantities. The pressure of a gas of quarks and gluons, for instance, is known up to O(g5). [I] Unfortunately, in order for the series to converge, the temperature has to be on the order of lo5 GeV. At temperatures expected in RHIC and LHC heavy- ion collisions (0.2-l GeV) the weak-coupling expansion seems to be of little use.

1.04, I 1 I

1.02 .

0.92 - q g5 Q

0.9 0 1 2 3 4

c@nT)

Figure 1. Weak-coupling expansion to orders g2, g3, g4, and g5 for the pressure normalized to that of an ideal gas.

Here I will concentrate on scalar d4 since the poor convergence of the weak-coupling expansion also shows up there. In Part II, I will discuss the extension of the method to gauge theories. [2] The weak-coupling expansion of the pressure is known up to order g5. [3] In Fig. 1, we show the successive perturbative approximations to ?/?i&a) as a function of g(2rrT). Each partial sum is shown as an error band obtained by varying the renormalization scale /I from 7rT to 4nT. The shaded bands can be considered as a lower bound on the theoretical error. The difference between successive approximations should also be considered which leads us to conclude that the error grows quickly for g > 1.5.

Clearly perturbation theory is not. converging in this regime. This is not suprising since the “soft” scale (p - gT) and the “hard” scale (p - T) are of the same order. In this regime we have to more carefully take into account the interaction of the soft and hard degrees of freedom. One possible way to do this is to change the expansion point for the loop expansion. Instead of expanding around a massless ground state we expand around a gas of massive quasiparticles which incorporate the Ysoft” scale physics. In the following section I will discuss a technique called screened perturbation theory that can be used to do this in a systemaeic manner.

I. FORMALISM

Within screened perturbation theory the scalar lagrangian density is written as

1 &p~ = -Eo + i8P@P4 - ~(TTL~ - YTX~)~~ - &g2+4 -t AL: + A.&T , (1)

where & is a vacuum energy density parameter and we have added and subtracted mass terms. If we set Es = 0 and rn: = mz, we recover the original lagrangian. Screened perturbation theory is defined by taking m3 to be of order go and rn: to be of order g2, expanding systematically in powers of g2, and setting rnf = m” at the end of the calculation. This defines a reorganization of perturbation theory with

,!z free (2)

L-h = -$g244 + im:$2 + AL + A&T . (3)

At each order in screened perturbation theory, the effects of the m2 term in (2) are included to all orders. Rowevcr. when. we set rn: = m2, the dependehce on m is systematically subtracted out at higher orders in perturbation theory by the rn: term in (3). -At nonzero temperature, screened perturbation theory does not generate any infrared

145

divergences, because the mass parameter m2 .’ m the free part of the lagrangian provides an infrared cutoff. The resulting perturbative expansion is therefore a power series in g2 and rn: = m2 whose coefficients depend on the mass parameter m.

In order to complete the calculation we must specify the way in which this mass determined. In the data shown here I will identify m with the tadpole mass. For a more complete discussion of the various mass prescriptions and results obtained with each see Ref. [4].

(a) xTTP<-~ET (b) im.r p 5 2m.

0.9 0 1 2 3 4 0 1 2 3 4

g(2z-V WXT)

FIG. 1. One-, two-. and three-loop results for the pressure of Xd4 theory as a function of g(20) for two different values of the renormalization scale /.K (a) rrT < p < 4xT (b) l/Zm < /I < 2m.

In ‘Figure 1, the one-, two-, and three-loop renormalized pressures are shown as a function of g(2nT) The bands in Figure la and lb were obtained varying the renormalization scale by a factor of two about the values of 2nT and rn. respectively. In both cases, the convergence of the successive approximations is much better than the conventional perturbative calculation with the two- and three-loop results lying within the minimal error bars of the conventional 0(g5) calculation. [4] The convergence for ,u cc m seems to be better than with 1-1 oc T, but at this point it is impossible to copclusively settle this issue. A reasonable estimate would take the appropriate scale to be somewhere between these two scales.

In ‘order to make contact with RHIC and LHC we need to extend these methods to &CD. For gauge theories you can’t simply add and subtract a momentum-indepedent mass because this would violate gauge invariance. One possibility is to use instead the non-local functions related to the gluon self-energy and vertex functions obtained in the high-temperature limit of &CD. [5]

ACKNOWLEDGEMENTS

This work was done in collaboration with J.O. Andersen and E. Braaten. I am grateful to the organizers of the “Eyui- librium and non-equilibrium aspects of hot, dense &CD” workshop. This work was supported by the U.S. Department, of Energy Division of High Energy Physics grant DE-FG03-97-ER41014.

[1] P. .4rnold and C. Zhai, Phys. Rev. D50, 7603 (1994); Phys. Rev. D51, 1906 (1995); B. Kastening and C. Zhai, Phys. Rev. D52, 7232 (1995); E. Braaten and A. Nieto, Phys. Rev. Lett. 76, 1417 (1996); Phys. Rev. D53, 3421 (1996).

[2] M. Strickland, “Reorganizing Finite Temperature Field Theory, Part II. Gauge Theory”, Proceedings of the BNL In and Out of Thermal Equilibrium Workshop, [FILL].

[3] P. Arnold and C. Zhai, Phys. Rev. D50, 7603 (1994); R.R. P arwani and H. Singh, Phys. Rev. D51, 4518 (1995); E. Braaten and A. Nieto, Phys. Rev. D51, 6990 (1995).

[4] J.O. Andersen, E. Braaten and M. Strickland, “Screened Perturbation Theory to Three Loops”, hep-ph/0007159 (2000); Phys. Rev. D 62, 45004 (2000).

[5] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83, 2139 (1999); Phys. Rev. D61, 014017 (2000); D61. 074016 (2000).

146

HTL and Screened Perturbation Theory

Michael Strickland

Quantum Fields “Equilibrium and non-equilibrium aspects of hot, dense QCD”

Brookhaven National Labs, Thursday July 27, 2000

Failure of perturbative QCD

l The weak coupling expansion of the QCD free energy, F, has been calculated

to order CY~‘~. lY2

l The successive terms contributing to F can strictly only form a decreasing

series if 01 .z l/20 which corresponds to T N lo5 GeV.

2.0

1.5

1.0

0.5

0.0

+ i ,.’ 1

+ - Lattice Results i 2 - pQCD Order g2 3 - pQCD Order g” 4 - pQCD Order g4 5 - pQCD Order g5

Figure 1. Perturbative QCD Free Energy vs Temperature. (XT 5 p4 2 47rT)

Lattice results from G. Boyd et al, 95/96.

147

Screened perturbation theory

0 Within screened perturbation theory, a mass, which can be treated as a vari-

ational parameter, is added to the Lagrangian and the loop expansion is re-

computed.778

LSPT =

We can split this into free and interaction parts

The additional counterterms that are required have the form

with

AE, = ZE (m2 - mTj2

Am2 = (Z+Zm - l)m2

= (Z+Zm - l>mf

e The previous work performed a two-loop calculcation for N = B, and a three-

loop calculation only in the large-N limit. We have recently finished the full

N = 1 three-loop calculation of the thermodynamic functions.”

’ F. Karsch, A. Patkhs, and P. Petreczky, 97. 8 S. Chiku and T. Hatsuda, 98. ’ ABS, hep-ph/0002048,

148

(1)

(2)

(3)

(4)

(5)

(a) ‘IETIPLI ~TCT (b) h m,l CL -2 2m,

1

0.98

5 e 0.96

& 0.94

0.9 0 1 2 3 4 0 1 2 3 4

g(2n-U g (2n-U

0.8 t . 0.8

'- 0.6

E

'm 0.6

1 1

'iij 0.98 a, 0

z

0.96

Ti 0.98

0 G r/3

0.96

0.94 0

g(2fO g(2n-U

Figure 2. SPT results for the pressure, screening mass, and entropy. 149

\ . I

E 1 fCJ One Loop

Hard-thermal-loop perturbation theory

Hard-thermal-loop (HTL) perturbation theory is a reorganization of the perturbative

series ‘for QCD.1°

The HTL improvement term is

c 3 HTL = --

2 mfj - ?%i>n-

Transverse propagator

( g’T”

w2 - k2 -+ w2 - k2 - rr,(w,k)

ITT (w, kJ

Longitudinal propagator

> 3 2w2

w2 - k2 = -m - 2 g it2

[

l- 2(& log ; ?I ;

II~(w,lc) = 3rnz

k2 + /k241~(W,k)

[ w+k

$%w-k-l 1 Weak coupling limit (High T)

47r ,lirs,rnz = -aa,T2

s 3

(6)

(7)

(8)

(9)

(10)

150

Conclusions and Outlook

* We have completed the three-loop calculation of the free energy of a

massive scalar field. We were also able to calculate the three-loop free

energy (pressure), two-loop screening mass, and three-loop entropy for

massless scalar fields using screened perturbation theory.

l For QCD, we have completed the first step in systematically including

some of the effects of gluonic and fermionic quasi-particles, screening,

and Landau damping in the free energy of a quark-gluon plasma.

l All divergences can be dealt with in a systematic way and quantities are

properly renormalized.

l The htlQCD results agree with the lattice results considering that order

01, and higher order corrections will be generated at the two-loop level.

l We are in the process of computing the htlQCD two-loop contribution.

l Once the thermodynamics have been described, this method can be used

to calculate real-time processes within a systematic framework.

l Some immediate phenomenological applications include:

(1) heavy quark production in QGP

(2) photon/dilepton emission from QGP

151

152

We study some aspects of the vacuum energy of pure Yang-Mills and QCD at 8 - 7~ We show that there are domain walls, separating phases in which CP is spontaneously broken, compute or estimate their tension and exhibit the phase diagram of QCD with two light fla- vors as function of the quark mass difference and the angle 8.

Reference: M. Tytgat, Phys.Rev. D61, 114009

(2000), hep-ph/9909532

153

. Why O’- 7~ ?

In Nature, 8 - 0 if CP is conserved by Strong Interactions. A priori, 8 = r is also CP con- serving, because

7r c+p -7r = Z- mod 2~

however, CP can be spontaneously broken! This is Dashen’s phenomenon.

This is forbidden at 0 = 0 by the Vafa-Witten

theorem.

154

Puzzles at large NC

Consider pure Yang-Mills with large number of colors, NC. The vacuum energy as function of 6 takes the form

This is because there are O(Nz) degrees of freedom, and of the ‘t Hooft resealing of the coupling,

with ‘7 fixed as NC + 00.

This seems incompatible with the fact that 0 + 8+2x should describes the same physics.

To leading order in NC, Witten’s solution is

WI - - $ m;n(O + k2~)~

where x is the topological susceptibility of pure Yang-Mills.

155

Simple construction of Witten% solution

‘Consider QCD with one light flavor. At large I\lc and low energy, the effective Iagrangian is

vcr7,e) = -mm/ cos rl f + $7 + e12 7r

where no is the quark mass and IX is the quark condensate.

For fixed 8 and if rr~> >> xi there are many I:ocaI minima at (q/&J z k27~ Integrating out the 7 field gives

v(e) ==: z 2 ryy@ + b7/fd2 which reproduces Witten’s solution.

4-t 8 = r, there are two degenerate vacua,

(l-r> e-- (q), and CP is spontaneously broken. There is a Domain Wall, with tension 0 - NC. For 8 + 0, there is a unique groundstate. The other vacua become stable in the limit NC + 00. The false vacuum decay rate is

Shifman; Gabadadze; M.T.

156

QCD with two light flavors

At 0 - x and for three light flavors, CP is spontaneously broken if the quark mass satisfy

mumd >'lmd - muI

ms

Naively, as ms + 00, CP is never sponta- neously broken if there are only two flavors !?

But we expect a first order phase transition at 8 - -2-r. A second order transition would imply that the meson excitations get massless, but chiral symmetry is always broken for mg $ O?

Smilga

If one keeps track, of the singlet mode, there is no puzzle. The expression valid for two light flavors is

mu-md 2 m,

<- mhmd m;

M.T.

157

This can be summarized by the following phase

ktructure diagram for QCD with two light fla- vors: *

CM.S.S-Ol’er

2nd order

1st order

2nd order

3-c

0

2n

Where x = ??z&&. For large or small mass differences, the line of first order transitions

at 8 = r become second order at two end- points.

158

.

. . .

159

--

‘U - --a . If% iQ

\

163

164

I. . .

. . .^

fa

; (s

167

:’ ,: .

. . ..,

. .

.f.‘.

, :’

168

cl s

/A =O

Mu=+? d = 0

169

z 0

Equilibrium & Non-Equilibrium Aspects of Hot, Dense QCD July 17* - 3oth, 2000

A RIKJZN BNL Research Center & BNL Nuclear Theory Workshop

LIST OF REGISTERED PARTICIPANTS

NAME

Peter Arnold

Rudolf Baier

Dietrich Bodeker

Daniel Boyanovsky

Eric Braaten

Gregory Carter

Fred Cooper

Pawel Danielewicz

Hector J. de Vega

Glen Doki

Valeri Emiel’yauov

Eduardo S. Fraga

Kazunori Itakura

Sergio E. Joras

AFFILIATION AND ADDRESS E-MAIL ADDRESS University of Virginia parnold@virainia.edu Department of Physics P.O. Box 400714 Charlottesville, VA 22904-4714 Fakultaet Physik Baier@phvsik.uni-bielefeld.de Universitaet Bielefeld Postfach 10 0131 D-33501 Bielefeld Germany Niels Bohr Institute bodeker@nbi.dk Blegdamsveg 17 2 100 Copenhagen Denmark University of Pittsburgh boyan+@oitt.edu Physics Department Pit&urgh,hA 15260 Ohio State University braaten@oacific.mps.ohio-state.edu Columbus, Ohio Department of Physics SUNY Stony Brook Stony Brook, N.Y. 11794-3800 Los Alamos National Lab MS B-285 Los Alamos, NM 87545 NSCL/Cyclotron Lab Michigan State University LPTHE, Univ. Paris VI Pierre et Marie Curie Lab de Physique Tour 16, ler etage 4, Place Jussieu F-75252 Paris CEDEX 05 France Duke University Dept. of Physics Durham, NC 277080305 Physics Department Moscow Engineering Phys. Inst. Kashirskoe shosse 3 1 115409 Moscow Russia Nuclear Theory Group Physics Department, Bldg 5 10 Brookhaven National Laboratory Upton, N.Y. 11973-5000 RIKEN BNL Research Center Physics Department, Bldg 5 1 OA Brookhaven National Laboratory Upton, N.Y. 11973-5000 Department of Physics Brown University Box 1843 Providence, RI 029 12

Carter@tonic.phvsics.sunysb.edu

cooper@schwinger.lanl.aov

danielewicz@nscl.msu.edu

devega@lpthe.iussieu.fr

gdoki@,phv.duke.edu

vemel@ref.rhic.bnl.gov

fraga@,cuark.phv.bnl.zov

itakura@bnl.gov

joras@het.brown.edu

list of participants 171 07/30/2000

Equilibrium & Non-Equilibrium Aspects of Hot, Dense QCD July 17th - 30*, 2000

A RIKEN BNL Research Center & BNL Nuclear Theory Workshop

LIST OF REGISTERED PARTICIPANTS

NAME Keijo Kajantie

Randy ,Kobes

Raymond Krisst

Teiji Kunihiro

Larry Mclerran

Agnes Mocsy

Emil Mottola

Bemdt Mueller

V. P. Nair (Parameswaran) Ken Oyama

Robert D. Pisarski

Krishna Rajagopal

Anton (Tony) Rebhan

Dirk Rischke

Julien Serreau

AFFILIATION AND ADDRESS University of Helinski Finland University of Winnipeg Physics Department Winnipeg, MB - R3B2E9 Canada Physics Department Yale University P.O. Box 208120 New Haven, CT 06520 Yukawa Institute for Fundamental Physics Kyoto University Nuclear Theory Group Phyqics Department, Bldg. 5 10 Brookhaven National Laboratory Upton, N.Y. 11973-5000 School of Physics & Astronomy University of Minnesota 116 Church St S.E. Minneapolis, MN 55455 Los Alamos National Laboratory Theoretical Div. T-8 MS B285 Los Alamos, NM 87545 Duke University Dept. of Physics Durham, NC 27708-0305 City College of New York

E-MAIL ADDRESS kaiantie@ocu.heIsinki.fi

randv@,theorvx5uwinniue%ca

krisrim@att.net

kunihiro@,yukawa.kyoto-u.ac.jp

mclerran@bnl.aov

amocsv@ohysics.spa.umn.edu

emil@lanl.gov

muller@ohv.duke.edu

vpn@,fabbro.rockefeller.edu

ovama@bnl.sov

pisarski@bnl.aov

krishna@,itp.mit.edu

rebhana@,tph.tuwien.ac.at

rischke@,bnl.gov

julien.serreau(th.u-osud.fr

CNS Physics Department, Bldg 5 1 OC Brookhaven National Laboratory Upton, N.Y. 11973-5000 RIKEN BNL Research Center & Nuclear Theory, Bldg. 5 10 Brookhaven National Laboratory MIT 6-308A 77 Massachusetts Ave. Cambridge, MA 02139 Inst. f. Theoret. Phys. Techn. Univ. Vienna Wiedner Hauptstr. 8- 101136 A- 1040 Vienna, Austria RIKEN BNL Research Center Bldg 5 10, Physics Dept. Upton, N.Y. 11973-5000 Laboratoire de Physique Theorique Universite Paris Sud-Batiment 2 10 9 1405 ORSAY

list of participants 172 07/30/2000

Equilibrium & Non-Equilibrium Aspects of Hot, Dense QCD July 17’h - 3oth, 2&M

A RIKEN BNL Research Center & BNL Nuclear Theory Workshop

LIST OF REGISTERED PARTICIPANTS

NAME Edward Shury&

Misha Stephanov

Michael Strickland

Michel Tytgat Raju Venugopalan

H. Arthur Weldon

AFFILIATION AND ADDRESS SUNY Stony Brook Stony Brook, N.Y. Illinois University, Chicago RIKEN BNL Research Center University of Washington Physics Department Se&e, W&hington 98 195-1560 CERN / BNL Brookhaven National Laboratory Nuclear Theory Group Building 5 10, Physics Upton, N.Y. -11973-5000 Department of Physics West Virginia University Morgantown, WV 26506-63 15 University of Washington Department of Physics Seattle, Washington 98195-1560 Physics Department British Columbia University 6224 Agricultrual Road Vancouver, BC VGTlZI

E-MAIL ADDRESS shurvak@dau.ohvsics.sunysb.edu

misha@uic.edu

mike@ohvs.washinston.edu

mtvtgat@bnl.gov raiu@bnl.sov

hweldon@,wvu.edu

Laurence Yaffe

Ariel Zhitnitsky

Igv@newton.phvs.washinzton.edu

arz@physics.ubc.ca

list of participants 173 07/30/2000

EQUILlBRlUM & NON-EQUILIBRIUM ASPECTS OF HOT, DENSE QCD July 17’h -July 30th, iOO0

A RIKEN BNL Research Center & BNL Nuclear Theory Workshop AGENDA

All lecturers are allotted one hour plus halfhour for discussions.

Monday 17 July

Morning Small Seminar Room, Physics, Bldg. 510 1” floor

lo:oo - 10:55 Registration & Welcome Reception (Coffee & Break$ast Treats will be provided)

11:OO - 12:30 Laurence Yaffe Non-Perturbative Dynamics of Hot Non- (University of Washington) Abelian Gauge Fields

Discussion

12:30 - LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, Is’ floor

14:15 - 15:45 Eric Eraaten Thermodynamics of Hot QCD (Ohio State University)

Discussion

16:30 - l&O0 Pawel Dauielewicz Low-Momentzrm Correlation from Real- (Michigan State University) Time Theory

Discussion

Tuesday 18 July

Morning Room 2-l 60, Physics Bldg. 510, 2Rdjloor

11:OO - 12:30 Krishna Rajagopal Traversing the QCD Phase Transition: (MIT) Quenching, Slowing or Bubbling Out of

Equilibrium Discussion

12:30 LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, l”f1oor

14:15 - 15:45 Randy Kobes Calculating Viscosity (University of Winnipeg)

Discussion

16:iO - 18:OO Peter Arnold Hot Scalar Theories in Large N: Bose- (University of Virginia) Einstein Condensation

Discussion

agenda 174 07J3Ol2000

Wednesday 19 July

Mokninp Berkner Hall Auditorium (Cafeteria Building)

ll:oo - 11:30 Coffee

. 11:30 - 12:30 SPECIAL Announcement “First Physics Results porn PHOBOSG2WIC” Wit Busza, PHOBOS Collaboration

12:30 - LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, lSfJloor

14:1,5 - 15:45 H. Arthur Weldon A Space Time View of Thermal Field Theory (W. Virginia University)

Discussion

16:30 - 18:00 Rudolf Baier (Univerky of Bielefeld)

Resonant Decay of Parity Odd Bubbles in Hot Hadronic Matter

Discussion

Thursday 20 July

Mokng Room .2-l 60, Physics Bldg. 510, 2”dfloor

1 l:qO - 12:30 Berndt Mueller The Quantum Limit of Classical Field (Duke University) Dynamics

Discussion

12:30 - LUNCH

Afternoon Room 2-l 60, Physics Bldg. 510, 2ndjloor

14:15 - 15:45 Teij i Kunihiro (Kyoto University)

The Renormalization Group Method Applied to Kinetic Equations

Discussion

ageida 175 07/30/2000

Thursday 20 July (cant)

16:30 - l&00** Keijo Kajantie (University of Helsinki)

Thermodynamics of Hot QCD; the Effective Theory Approach

Discussion

** k6:30 talk & discussion to be held in the Small Seminar Room, Physics, lstjloor

18:00 - 21:00 WORKSHOP BBQ at Brookhaven Center, Bldg 30 - Patio

Friday 21 July

Morning, Small Seminar Room, Physics Bldg. 510, Is’ floor

1l:OO - 12:30 Hector de Vega (Paris VI)

Discussion

Non-Equilibrium Quantum Plasmas, Quantum Kinetics and the Dynamical Renormalization Group

12:30 - LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, 1” floor

14:15 - 15:45 Ariel Zhitnitsky Early Universe at the QCD Scale (British Columbia University)

Discussion

agenda 176 07/30/2000

Monday 24 July

Morning Small Seminar Room, Physics Bldg. 510, lstjloor

11:bO - 12:30 Misha Stephanov QCD at Finite Density of Isospin (Illinois University & RIKEN BNL)

,Discussion

12:30 - LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, l”jloor

14:15 - 15:45 Dirk Rischke Properties of Gluons in Color (RIKEN BNL Theory) Superconductors

Discussion

16:30 - 18:00 Eduaxdo Fraga (BW

Quark Stars from Perturbative QCD

Discussion

Tuesday 25 July

Morning Room 2-l 60, Physics Bldg. 510, 2”‘floor

11:bO - 12:30 Emil Mottola (LAW

The Electrical Conductivity of a QED Plasma

Discussion

12:30 - LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, lstjloor

14:15 - 15:45 Anton Rebhan Hard-thermal/dense-loop Thermodynamics (University of Vienna) of the Quark-gluon Plasma

Discussion

16:30 - 18:OO Fred Cooper CL@=)

Simulations of the Chiral Phase Transition on Both Sides of the Tricritical Point

Discussion

agenda 177 07/3Ol2000

Wednesday 26 July

Morning Small Seminar Room, Physics Bldg. 510, I”‘jloor

11:OO - 12:30 Dietrich Bodeker Hard Thermal Loops on the Lattice (Heidelberg University)

Discussion

LUNCH

Afternoon Small Seminar Room, Physics Bldg. 510, lS’Jloor

15:30 - 17:oo V;P. Nair Covariance & Magnetic Mass (City College of New York)

Discussion

Thursday 27 July

Morning Room 2-160, Physics Bldg. 510, 2”djloor

11:OO - 12:30 Raju Venugopalan (BNL Theory)

Discussion

Solving the Problem of Initial Conditions in HI Collisions: from the Nuclear Wave Function to a Gluon Plasma

12:30 - LUNCH

Afternoon Room 2-l 60, Physics Bldg. 510, Tdfloor

14:15 - 15:45 Michael Strickland (Ohio State University)

HTL Peturbation Theory

Discussion

16:30 - 18:00** Michel Tytgat (CERN / BNL)

QCD at Theta -Pi

Discussion

**16:30 talk and discussion to be held in the Small Seminar Room, Physics Bldg 510, lstjloor

agenda 178 07/30/2000

Friday 28 July

Mkning Small Seminar Room, Physics Bldg. 510, 1”‘floor

11:OO - 12:30 Ed Shuryak Decrease of the Non-peturbative Cutoff in (Stony Brook University) QCP at RHIC

Discussion

12: 30 - Lunch

Afternoon Small Seminar Room, Physics Bldg. 510, Is’ floor.

14:15 - 15:45 Larry MC Lerran (BNL Theory)

CLOSING LECTURE

Discussion

agenda 179 07/30/2000

For information please contact:

Ms. Tammy Heinz or Ms. Pamela Esposito RIKEN BNL Research Center Building 510A, Brookhaven National Laboratory Upton, NY 11973-5000, USA Phone: (631)344-5864 (631) 344-3097 Fax: (631)344-2562 (631) 344-4067 E-Mail: theinz@bnl.gov pesposit@bnl.gov Homepage: http://quark.phy.bnl.gov/www/riken.html

http://penguin.phy.bnl.gov/www/riken.htm.l

(3 a RIKEN BNL RESEARCH CENTER

kWlLlBRlUM’& NON-EQUILIBRIUM I ASPECTS OF HOT, DENSE QCD

Speakers:. F? Arnold ’ F. Cooper K. Kajantie E. Mottola A. Rebhan p.Ttrx-kland

.

R. Baier D. Bijdeker E. Braaten P Danielewicz H. de Vega E. Fraga R. Kobes T. Kunihiro L. McLerran B. Miiller V.P. Nair K. Rajagopal D. Rischke E. Shut-yak M. Stephanov M. T@gat R. Venugopalan H.A. Weldon A. Zhitnitsky

Organizers: D. Boyanovsky, H.J. de Vega, L.D. McLerran, R.D. Pisarski