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Computing Hadron PropertiesC.B. Lang :
MPI Munich, 17.10.2006
Computing Hadron Properties
C. B. LangInst. F. Physik / FB Theoretische Physik
Karl-Franzens-Universität Graz
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
1. Concepts
2. Methods
3. The fermion adventure
4. Applications: Hadron masses
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Concepts of Quantumchromodynamics
QCD is the quantum field theory of quarks and gluons, defined by a Lagrangian (action)
This theory can be solved from first principles, withminimal number of input parameters (bare quark massesand a scale fixing parameter)Hadron properties should be computable from QCD
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
High energies, hard scattering:perturbative expansion
Low energies, hadron structure: non-perturbative methods
Mass, decay constants,matrix elements, distribution functions,form factors, finite temperature and density, …..
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Quantization
(Euclidean space-time and )
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Fundamental challenge
Non-perturbative resultsare hard to get.
Rigorous proof of confinement brings1,000.000 $ prize
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
QCD on a space-time grid: Lattice QCD
Gauge field
QuarksAnti-quarks
Ken Wilson
Lattice QCD is an approximation – not a modelIt opens a way to compute the path integral
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Let‘s compute
…functional integral is approximated bycomputer (Monte Carlo) sampling of fieldconfigurations……..
Mike Creutz
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Computational requirements
20 x 20 x 20 x 40 grid has- 11 M complex numbers for the gauge variables- 4 M complex numbers for quark variables
The Dirac operator beomes a huge matrix- e.g. 4M x 4M
Algorithms:MC simulationMatrix inversion by CGR type methods
Competitive projects need >1 TeraFlopYear
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
„Berlin Wall“
C. Urbach, LAT06
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Custom computersPC Clusters„Self-built“ computers(apeNEXT, QCD-on-a-Chip)
BlueGeneL at Jülich (46 TFl, 8K nodes)
Altix at LRZ (26 TFl, 4K nodes )
QCDOC
QCDOC at BNL(20 TFl, 14 K nodes)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
1. Concepts
2. Methods
3. The fermion adventure
4. Applications: Hadrons
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Simulate gauge fields nonperturbatively
The vacuum is not perturbativeGluons are represented by SU(3) matrices – like a 4D system of bosonicspinsMonte Carlo importance sampling of gauge configurationsMeasure all observables (propagators, matrix elements) in this gauge fieldbackground:„quenched approximation“
(Gluonic vaccuum fluctuations, movie © Leinweber et al.)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Work with valence quarks
Sea quarksValence quarks
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Work with valence quarks
„quenched approximation“Valence quarks
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Measure masses
Compute the correlationfunction for an operator withcorrect quantum numbers, e.g.
GMOR
Mπ=280 MeV
BGR, Nucl.Phys. B677 (2004)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Fix the scale
Simulate at some values of bare lattice couplings (g, mf)
All lattice quantities are dimensionless combinations(like e.g. mlatt= a mphys)
Trade one measured quantity with one experimental value to get the lattice spacing a
E.g. Sommer parameter = distance r0, where string tension givesmeasured Regge slope (r0
2 F(r0)=1.65)E.g. compute the (dimensionless lattice) pion decay constant(a fπ) and use fπ = 93 MeV Typical values used are a=0.05…0.25 fm
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo: Simulate dynamical quarks
$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Fermions(Grassmann)
Bosons„pseudo-fermions“
„Hybrid Monte Carlo“ = Molecular dynamics + MC step
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
HowTo remove the constraints?
Technical constraints
Conceptual issuesExcited states (and decays)?Chemical potential?Chiral theories (Standard model)?Supersymmetry?
Lattice spacinga 0
„continuum limit“Renormalization group, improvement (e.g. Symanzik)
Volume sizeL ∞
„thermodynamic limit“Finite size corrections, Chiral Perturbation Theory
Quark massm 0
…or at least small enough, i.e. mπ 0 ,„Chiral limit“, ChPT
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Typical parameters for today‘s simulations
a=0.2 - 0.07 fm
Lattice size 123x24…323x64Spatial volume (1.5-2.5 fm)3
mπ around 350-800 MeV
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Some topics (LAT06)
Algorithms for dynamical overlap fermionsLight dynamical fermions on the lattice: toward the chiral regime of QCDRooted staggered fermions: good, bad, or ugly?Status report on ILDG activitiesThe one-flavor quark condensate and related problemsThe rational hybrid Monte Carlo algorithmTwisted mass QCD for weak matrix elements
Algorithms
Heavy flavor physics from lattice QCDProgress in Kaon Physics on the LatticePushing Lattice to the Edge: Flavor PhysicsRecent lattice QCD results on nucleon structureSearch for Gluonic Excitations in Light Unconventional MesonsTowards a quantitative understanding of the Delta I=1/2 ruleTwisted mass QCD for weak matrix elements
Hadron properties
Color superconductivity in ultra-dense quark matterHot QCD and warm dark matterLattice QCD at finite densityQCD phase diagram: an overviewRecent progress in finite temperature lattice QCDStrong Correlations in Hot QCDThe sign problem in the epsilon-regime of QCD
Extreme environments
Advances and applications of lattice supersymmetryD-branes and coherent topological charge structure in QCDLattice Field Theory Methods in Modern BiophysicsThe search for the fundamental QCD string in anti de Sitter spaceQCD physics beyond the standard model and LHC searches
„Transgressing boundaries“
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
1. Concepts
2. Methods
3. The fermion adventure
4. Applications: Hadrons
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
A problem for lattice QCD: chiral symmetry
Continuum QCD: Chiral symmetry SU(2)R x SU(2)L is spontaneouslybroken: SU(2) multiplets + Goldstone pions
The theory should allow for explicit chiral symmetry, such that it canbe broken spontaneously!
Local term
Lattice QCD:
This „Ginsparg-Wilson condition“ is violated bysimple Dirac operators (simple fermion actions)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
What fermions are on the market?
StaggeredWilson-improvedTwisted mass
Domain WallFixed PointChirally Improved
Overlap
Non-GW-type
GW-type
GW-exact
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Non-GW actions: „Clash of discretisations“
Staggered (improved: Asqtad)
Wilson (improved: clover)Twisted mass
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Non-GW actions: Staggered type
Sugar (LAT06)
Staggered (improved: Asqtad, MILC collaboration)
Quark degrees distributed in hypercubeRemnant chiral symmetrySimple to implementHarder to interpret (taste splitting)4 fermions:
- 4th root trick: (det D)1/4 ???
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Non-GW actions: Wilson type
m m ?
lGW lWilsonWilson (improved: clover)Simple to implementDoubler modesProblem with small quarkmasses: Spurious low-lyingeigenmodes of the Dirac operator
Twisted mass (© Frezotti,Grassi, Sint, Weisz)
Wilson + „twist“ i mg5t3
No spurious modesBreaks parity and flavor
DelDebbio et al. JHEP 0602(2006)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
GW-type fermions…fewer or no spurious modes…can reach smaller pion and quark masses
Domain Wall5th „dummy“ dimension (© Kaplan, Furman, Shamir)
Fixed pointRG transformation, parametrisation (© Hasenfratz, Niedermeier)
CI (chirally improved)Systematic expansion, parameterisation (© Gattringer, Hip, L.)
Overlap„exact“ but very expensive (© Neuberger)
1+γ5 sign(γ5 DW) m
lGW
+ . . .+ += +
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
1. Concepts
2. Methods
3. The fermion adventure
4. Applications: Hadrons
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Fundamental constants
2004 MILC/HPQCD/UKQCD/Fermilab „Gold plated“(?) results
2006 values range
CP-PACS, JLQCD- mud= 3.5 (4) MeV- ms = 92 (9) MeV- Fπ = 141(9) MeV- fK = 161(9) MeV
Davies et al. PRL92 (04)022001
MILC- mu = 2.0 (3) MeV, md =4.6 (5) MeV- ms = 90 (9) MeV- fπ = 129 (3) MeV - fK = 155 (3) MeV
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Ground state masses
CP-PACS (2002, Wilson, quenched, extrapolated)
MILC(2004, staggered, dynamical, extrapolated)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
How to identify excited states?
Propagator = sum of exponential decay terms:
ground state (large t)
excited states (smaller t)
Previous attempts: biased estimators (Bayesian analysis), maximum entropy,...
Significant improvement: Variational analysis
Mathur et al. (05), Lee et al (03),Sasaki et al. (05), Juge et al. (06),Zanotti et al. (03), Melnichouk et al.(03)
Burch et al. (03-06), Basak et al. (05, 06)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Variational analysis (Michael, Lüscher/Wolff)
Use several interpolators
Compute all cross-correlations
Solve the generalized eigenvalue problem:
The eigenvectors are „fingerprints“ of the state
Allows to separate ghost contributions (cf. Burch et al.)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Analysis of excited hadron masses
Interpolation fields:
E.g. for nucleon
Allow for different quark sourceshapes
Analyse cross-correlations
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Exciting nucleons
Roper Level crossing (from + - + - to + + - -)?
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN
[G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
negative parity
N(1535)
N(1650)
N(2090)
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN
[G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(3), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
positive parity
N(938)
N(1440)
N(1710)
Burch et al., 2006
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Exciting nucleons
Roper Level crossing (from + - + - to + + - -)?
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN
[G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
negative parity
N(1535)
N(1650)
N(2090)
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN
[G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(3), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
positive parity
N(938)
N(1440)
N(1710)
Burch et al., 2006
?
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Chiral PT: Expansion around Mπ = 0
Chiral dynamics:Goldstone pion
- Effective theory for vanishing pion masses
- Quark masses as perturbation- Low energy theorems like PCAC are
exact
Systematic expansion in Mπ/Lch , E/Lch :- Lagrangian with phenomenological
parameters
(Weinberg, Leutwyler, …):
Meissner (LAT05)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Chiral perturbation theory for excited states?
… new scale parameters(non-vanishing in the chiral limit) … new assumptions…
Roperbehaves similar to theNucleon ?
Bernard, Hemmert, Meissner (2004)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Baryons (chiral extrapolations)
1.0
1.5
2.0
2.5
MB [
GeV
]
N Σ Ξ Λ ∆++ Ω− −
negative parity
1.0
1.5
2.0
2.5
MB [
GeV
]
N Σ Ξ Λ ∆++ Ω− −
positive parity
Burch et al., 2006
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Baryons (chiral extrapolations)
1.0
1.5
2.0
2.5
MB [
GeV
]
N Σ Ξ Λ ∆++ Ω− −
negative parity
Predictions:W 1st excited state, pos.parity: 2300(70) MeVW ground state, neg.parity: 1970(90) MeVX ground state, neg.parity: 1780(90) MeVX 1st excited stated, neg.parity: 1780(110) MeV
1.0
1.5
2.0
2.5
MB [
GeV
]
N Σ Ξ Λ ∆++ Ω− −
positive parity
Burch et al., PRD 74 (2006)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
What to expect for the next years?
Dynamical quarks („full QCD“, 2+1 flavors)
2008: reasonable dynamical results for mp around 200 MeV 2010: reliable dynamical results for mp around 200 MeV 2012: reasonable dynamical results for mp around 140 MeV 2014: reliable dynamical results for mp around 140 MeV
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
The future
Prediction is very difficult, especially about the future(US: Yogi Berra)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
The future
Prediction is very difficult, especially about the future(EU: Niels Bohr)
C.B. Lang : Computing Hadron Properties
Concepts Methods The fermion adventure Applications: Hadron masses
Thank you !
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