Lattice QCD, RHIC, and LHC

1
Lattice QCD, RHIC, and LHC
Masayuki Asakawa
Former Aggie Department of Physics, Osaka
University
2
QCD Phase Diagram
T
QGP (quark-gluon plasma)
CP(critical point)
160-190 MeV
crossover
1st order
order ?
Hadron Phase

• chiral symmetry breaking
• confinement

CSC (color superconductivity)
mB
5-10r0
3
One of the most striking findings _at_RHIC
Possible Formation of Strongly Interacting
Matter
The beginning of the universe is a drop?
The quark is liquid
Asahi Shinbun (Newspaper), April 2005
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Is (Q)GP really close to perfect liquid?
Two lattice calculations (Pure Gauge, Nf 0)
Meyer, 2007
Nakamura and Sakai, 2004

• Both have problems

• assumption for the spectral function
• direct inversion from Euclidean data to
  Minkowski NOT unique

Unless we know T-dependence of viscosities,
unable to discuss validity of hydrodynamical
calculation at RHIC and LHC
LQGP collaboration in progress
5
Strong coupling not necessarily perfect liq.

• Low viscosity Little momentum transport

Long range and many body correlation leads to
large viscosity
What is generally experienced in condensed matter
physics
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Importance of Understanding Hadrons _at_Finite T

• v2M(pTM) v2 B(pTB) 2 3 for pTM pTB
  2 3

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Constituent Quark Number Scaling

• v2M(pTM) v2 B(pTB) 2 3 for pTM pTB
  2 3

• Partons are flowing and Partons recombine to
  make mesons and baryons

Assumption
All hadrons are created at hadronization
simultaneously
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Hadrons above Tc?
Hadrons above Tc

• No a priori reason that no hadrons exist above
  Tc
• QGP looks like strongly interacting system (low
  viscosity...etc.)

• Definition of Spectral Function (SPF)

• Information on
• Dilepton production
• Photon Production
• J/y suppression...etc.

encoded in SPF
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Microscopic Understanding of QGP
Importance of Microscopic Properties of
matter, in addition to Bulk Properties

• In condensed matter physics, common to start
  from one particle states,
• then proceed to two, three, ... particle
  states (correlations)

Spectral Functions

• One Quark
• Two Quarks
• mesons
• color singlet
• octet
• diquarks
• Three Quarks
• baryons
• ......

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Photon and Dilepton production rates

• Photon production rate

• Dilepton production rate

(for massless leptons)
where rT and rL are given by
rmn QCD EM current spectral function
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Lattice calculation of spectral functions

• No calculation yet for finite momentum light
  quark spectral functions

LQGP collaboration, in progress
So far, only pQCD spectral function has been used

• Calculation for zero momentum light quark
  spectral functions by two groups

rT rL _at_ zero momentum
Since the spectral function is a real time
quantity, necessary to use MEM (Maximum Entropy
Method)
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QGP is strongly coupled, but...
DEnterria, LHC workshop _at_CERN 2007
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Lattice calculation of spectral functions

• No calculation yet for finite momentum light
  quark spectral functions

LQGP collaboration, in progress
So far, only pQCD spectral function has been used

• Calculation for zero momentum light quark
  spectral functions by two groups

rT rL _at_ zero momentum
Since the spectral function is a real time
quantity, necessary to use MEM (Maximum Entropy
Method)
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Is Parametrization of SPF at finite T/m Easy?
Sometimes hear statements like
Finite T/m Spectral Functions are not always
given by shift broadening
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A Good Example (for r meson)
and many more examples in many fields
Rapp and Wambach (1999)
Due to D-hole contribution, non-Lorentzian

• Lorentzian Assumption ab initio not justified

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Lattice? But SPF cannot be measured ...

• Whats measured on the Lattice is
• Imaginary Time Correlation Function D(t )

K(t,w) Known Kernel
However,
c2-fitting inconclusive !

• Measured in Imaginary Time
• Measured at a Finite Number of discrete points
• Noisy Data Monte Carlo Method

Direct Inversion ill-posed !
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Similar Difficulties in Many Areas

• Lattice

• Analytic Continuation to Imaginary Time is
  measured
• Measured at a Finite Number of discrete points
• Noisy Data

• X-ray Diffraction Measurement in Crystallography

• Fourier Transformed images are measured
• Measured at a Finite Number of data points
• Noisy Data

• Observational Astronomy

• Smeared Images due to Finite Resolution are
  measured
• Measured by a Finite Number of Pixels
• Noisy Data

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Example of MEM Application

• Lattice

Will be shown shortly

• X-ray Diffraction Measurement in Crystallography

• Observational Astronomy

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MEM

• Maximum Entropy Method

• a method to infer the most statistically
  probable image (such as A(w))
• given data, instead of solving the (ill-posed)
  inversion problem

• Theoretical Basis Bayes Theorem

• In Lattice QCD

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Result of Mock Data Analysis
N( of data points)-b(noise level) dependence
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Stat. and Syst. Error Analyses in MEM
Generally,
The Larger the Number of Data Points and the
Lower the Noise Level
The Closer the Result is to the Original Image
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Lattice Parameters

• Lattice Sizes 323 Nt
• b 7.0, x0 3.5 x as/at 4.0
  (anisotropic)
• at 9.75 10-3 fm Ls 1.25 fm
• Standard Plaquette Action
• Wilson Fermion

• Heatbath Overrelaxation 1
  41000 sweeps between measurements
• Quenched Approximation
• Gauge Unfixed
• p 0 Projection
• Machine CP-PACS

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Spectral Functions above Tc
at T/Tc 1.4
peak structure
spectral function
Lattice Artifact
Asakawa, Nakahara Hatsuda hep-lat/0208059
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Another calculation
log scale!
massless quarks
smaller lattice
dilepton production rate
2 GeV _at_1.5Tc
Karsch et al., 2003
In almost all dilepton calculations from QGP,
pQCD expression has been used
How about for heavy quarks?
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J/y non-dissociation above Tc
Lattice Artifact
J/y (p 0) disappears between 1.62Tc and 1.70Tc
Lattice Artifact
Asakawa and Hatsuda, PRL 2004
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Result for PS channel (hc) at Finite T
hc (p 0) also disappears between 1.62Tc and
1.70Tc
Asakawa and Hatsuda, PRL 2004
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Microscopic Understanding of QGP
Importance of Microscopic Properties of
matter, in addition to Bulk Properties

• In condensed matter physics, common to start
  from one particle states,
• then proceed to two, three, ... particle
  states (correlations)

Spectral Functions

• One Quark
• Two Quarks
• mesons
• color singlet
• octet
• diquarks
• Three Quarks
• baryons
• ......

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Baryon Operators

• Nucleon current

• Euclidean correlation function at zero momentum

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Spectral Functions for Fermionic Operators

• For Meson currents, SPF is odd

Thus, need to and can carry out MEM analysis in
-wmax, wmax
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Analysis Details

• Default Model

At zero momentum,
Espriu, Pascual, Tarrach, 1983

• Relation between lattice and continuum currents

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Stat. and Syst. Error Analyses in MEM
Generally,
The Larger the Number of Data Points and the
Lower the Noise Level
The Closer the Result is to the Original Image
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Below Tc Light Baryon
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Below Tc Charm Baryon
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Above Tc Light Baryon
peak near zero symmetric and equally separated
peaks
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_at_Higher T
only symmetric and equally separated peaks
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Above Tc Charm Baryon
peak near zero symmetric and equally separated
peaks
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_at_Higher T
only symmetric and equally separated peaks
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Statistical Analysis Light Baryon _at_T0
Peaks are statistically significant
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Statistical Analysis Charm Baryon _at_Finite T
Peak near zero is statistically significant
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Origin of Near Zero Structure
Scattering Term
a.k.a. Landau damping

• This term is non-vanishing only for

• For J/y (m1m2), this condition becomes

zero mode
cf. QCD SR (Hatsuda and Lee, 1992)
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Scattering Term (two body case)
(Boson-Fermion case, e.g. Kitazawa et al., 2008)
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Scattering Term (three body case)
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Negative parity a possible interpretation
anti-quark parity -
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Origin of Symmetric Structure

• Wilson Doublers

Mass of Wilson Doublers with r 1 in the
continuum limit

• If quark mass can be neglected

• Masses of baryons with doublers
• Scattering term peaks with quark-doubler,
  doubler-doubler pairs

Approximately equally separated and symmetric in w
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QCD Phase Diagram
Quark-antiquark correlations
T
Diquark correlations
Baryon correlations
CEP(critical end point)
160-190 MeV
crossover
100MeV 1012 K
1st order
order ?

• chiral symmetry breaking
• confinement

mB
5-10r0
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Summary (Baryon Part)

• Baryons disappear just above Tc

• A sharp peak with negative parity near w0 is
• observed in baryonic SPF above Tc
• This can be due to diquark-quark scattering
  term and
• imply the existence of diquark correlation
  above Tc

• Diquarks disappear below meson disapperance
  temperature

• Direct measurement of SPF of one and two quark
  operators
• with MEM is desired

• To understand doubler contribution, calculations
  with finer lattices
• are desired

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Ingredients of MEM

given by Shannon-Jaynes Entropy
For further details, Y. Nakahara, and T. Hatsuda,
and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459
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Error Analysis in MEM (Statistical)

• MEM is based on Bayesian Probability Theory

• In MEM, Errors can be and must be assigned
• This procedure is essential in MEM Analysis

• For example, Error Bars can be put to

Gaussian approximation
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Meson-Baryon Universality
Partons are flowing and Partons recombine to make
mesons and baryons
Evidence of Deconfinement !
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Statistical Analysis below Tc