Kazuyuki Kanaya

1
An Introduction toField Theoryon the
Latticeab initio calculations of hadrons from
quarks

• Kazuyuki Kanaya
• Inst. of Physics, Univ. of Tsukuba
• kanaya_at_rccp.tsukuba.ac.jp

2003/12 Nagoya
2
Topics in field theories on the lattice
1. Basics

• Formulation of Lattice Field Theories
• Lattice as a non-pert. regularization of UV
  divergences
• Scalar field, gauge fields
• Fermions naive, Wilson, KS, DW, Overlap, ...
• Weak / strong coupling expansions, 1/mass
  expansion
• Scaling and continuum limit
• Improvement
• Heavy quarks
• Numerical methods
• Monte Carlo methods (pseudo) heat bath, HMC,
  PHMC, ...
• Matrix inversion CG, CR, MR, BiCGStab,
  BiCGStab-L, ...
• Techniques random number, error estimation,
  reweighting, MEM, ...
• Machines vectorization, parallelization,
  (dedicated) parallel computers, clusters, ...

3
2. Applications

• Fundamental properties of QCD
• Hadron masses, decay constants gt critical test
  of QCD
• Quark mass, astrong
• Hadronic properties
• Scattering length, phase shift, hadronic ME for
  weak decays, Proton decay, etc.
• Test / parameter-determination of models LEC of
  ChPT, hadron models, ...
• High temperature -- early Universe, RHIC
• Critical phenomena and universality
• Pure gauge system
• With light quarks
• High density -- heavy ion collisions,
  neutron/quark star
• Sign problem, reweighting method, ...
• Gedanken experiments of QCD -- Nc, Nf, ...

4

• In this Lecture
• Start-up for lattice simulations
• Overview of recent topics (if possible)

• Motivations some results
• Basic formulation of LQCD
• Numerical methods
• Fermions on the lattice
• Chiral extrapolation
• Improvement
• Summary and future prospects

5

• Text books
• H.J. Rothe, "Lattice Gauge Theories An
  Introduction"
• (World Sci., Lecture Notes in Physics Vol.43)
• M. Creutz, "Quarks, Gluons and Lattices"
  (Cambridge)
• I. Montvay and G. Münster, "Quantum Fields on a
  Lattice" (Cambridge)
• Lecture notes
• in "YITP School on Lattice Field Theory" (2002)
• http//www2.yukawa.kyoto-u.ac.jp/onogi/school/OHP
  /
• in "Phenomenology and Lattice QCD" (Uehling
  Summer School 1993)
• ed. G. Kilcup and S. Sharpe (World Sci. 1995)
• Lecture on Finite Temperature QCD by A. Ukawa,
  etc.
• K. Kanaya, at YKIS'97, Prog. Theor. Phys. Suppl.
  131 ('98) 73
• R. Gupta, hep-lat/9807028

6
1. Motivations
Quarks constituents of hadrons provide an
economical description of large of hadrons QCD
high energy experiment ? AF in UV cannot
isolate q's from hadrons ? confinement in IR
Fundamental parameters of QCD mq, astrong Direct
measurement not possible by an experiment!

• Calculate hadronic obs. by QCD as functions of
  mq, astrong
• Compare with experiment
• However, hadron masses, etc. incalculable
  analytically.
• non-perturbative calculation required.
• Can hadrons really made of quarks alone?
• Is QCD correct also at low energies?

7
1. Motivations (2)
Conventional QCD Renormalization using
regularizations based on the weak-coupling
expansion.
A non-perturbative definition of QCD required to
calculate hadrons as strongly coupled
quarks! Lattice as a non-perturbative
regularization Define fields on a discrete
space-time lattice.
Natural cut-off of O(p/a). UV divergences in the
continuum limit a gt 0
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1. Motivations test of QCD

• Direct test of QCD light hadron spectrum
  (present status)
• Quenched QCD
• quench disregard pair creation/annihilation of
  sea quarks
• Important properties of QCD (confinement, AF,
  ChSB) retained.
• Major effects at low energies renormalization /
  running (?)
• except for U(1) problem, finite T chiral
  transition, decays, etc.
• Computer time lt 1/100.

3 parameters a(g), mud, ms 3 inputs Mp,
Mr, MK or Mf predict other light hadron masses,
etc.
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1. Motivations test of QCD (2)
The first well-controlled study CP-PACS,
PRL84('00)238, PRD67('03)034503 3 parameters
a(g), mud, ms lt 3 inputs, Mp, Mr, MK or Mf
La 3 fm a-1 2, 2.5, 3, 4 GeV a 0.1-0.05
fm 323x56 - 643x112 lattices

• Experiment approximately
• reproduced by only 3 inputs.
• Deviations of O(10) remain.
• Limitation of quenched approx.

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1. Motivations test of QCD (3)

• Nf 2 full QCD
• dynamical u,d quarks (degenerate), quenched s.
• CP-PACS, PRL85('00)4674 E 90(03)029902,
  PRD65('02)054505 E D67('03)059901
• JLQCD, PRD68('03)054502
• Discrepancy largely removed!
• K- and f-inputs agree.
• Most direct proof of QCD.
• Inclusion of dynamical s quark
• Nf 3 full QCD
• in progress gt Seminar

• ? Experiment
• ? NF 2 QCD
• ? quenched QCD

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1. Motivations quark mass
Light quark masses (present status) Various
definitions because mq is not a direct
observable. "VWI" Mbare Zren, "AWI" ?A /
P gt different values at a gt 0.

• Different definitions converge in the cont. lim.
• Different actions converge in the cont. lim. ?
  universality
• Quenched results depend on the input.
• Input-dependence absent within errors in Nf 2
  fQCD.

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1. Motivations quark mass (2)
Note For u,d quark mass, difference between K-
and f-inputs absent in qQCD and Nf 2 fQCD. It
is negligible also in Nf 3 fQCD.
Current best Nf 2 fQCD
Errors statistical only. ms/mud 26 consistent
with 1-loop ChPT 24.4(1.5) Leutwyler, PL
B378('96)313
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• Fundamental properties of QCD
• Hadron masses, decay constants gt critical test
  of QCD
• Light quark mass, astrong
• Hadronic properties
• Scattering length, phase shift, hadronic ME for
  weak decays, Proton decay, etc.
• Test / parameter-determination of models
• High temperature -- early Universe, RHIC
• Critical phenomena and universality
• Pure gauge system
• With light quarks
• High density -- heavy ion collisions,
  neutron/quark star
• Sign problem, reweighting method, ...
• Gedanken experiments of QCD -- Nc, Nf, ...

14
2. Formulation basics
Euclidian field theory Path-integral
representation of field theory Euclidian
space-time free propagator euclidian field
theory ? classical statistical mechanics VEV,
Green functions ensemble av., correlation
funct's mass 1 / correlation
length Techniques in statistical mechanics
applicable. gt numerical study
15
2. Formulation basics (2)

• Lattice field theory lattice discretization of
  euclidian space-time
• provides an axiomatic definition of QFT.
• Scalar field
• Lattice discretization

site
link
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2. Formulation basics (3)

• Fourier transformation

17
2. Formulation basics (4)

• Dimension-less field
• K "hopping" parameter
• control prob. to hop to neighbors
• Path-integral on the lattice

eS gives on each link.
only even powers of f on each site
(2K)length Short path' dominate when K ltlt 1 (m
large). "hopping parameter expansion"
fx
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2. Formulation basics (5)
Interaction weak perturbations
strong potentials (potential)n for each
site
for each site
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2. Formulation gauge

• Local gauge symmetry in the continuum
• Gauge transformation change the origin of
  (generalized) phase at each x.
• Am(x) intermediate difference of phase origin at
  x and x dxm
• (absorb the mismatch of phase origin at
  these points).
• When 2 points are apart
• "connection"

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2. Formulation gauge (2)

• Lattice QCD finite distance a to neighboring
  points
• Fundamental gauge d.o.f. connection from x to
  xm.
• For small a,
• Connection has orientation
• from xm to x Ux,m

K.G. Wilson, PR D10(74)2445
"link variable"

• Pure gauge theory
• Ux,m must form closed loops to be gauge-inv.
• (junctions eab... with Nc lines also possible)

C
"plaquette"
"Wilson loop"
Smallest Wilson loop "plaquette" simplest
non-trivial object
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2. Formulation gauge (3)

• Wilson gauge action (standard one-plaquette
  action)

when b 6/g2
a gt 0 here is at the level of bare variables
("classical continuum limit"). True continuum
limit ("quantum continuum limit")
DU invariant measure (Harr measure) gauge
transf. U gt Ug , e.g., U(1) S1 0,2p),
U(1) transf. rotation in S1 gt DU ?02p dq
SU(2) S3 in 4d, SU(2) transf. rotation in S3
gt DU ?d4a d(a2-1)
Compact group ?DU lt 8 no divergence from the
gauge volume, i.e. no gauge fixing
required. (But we sometimes do fix the gauge to
reduce noise in the measurement etc.)
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2. Formulation gauge (4)

• Strong coupling b 6/g2 0

area low
Static quark potential
i.e., quark confinement at b 0. Confinement
proven if b 8 is analitically connected, i.e.,
no phase transitions in between.
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2. Formulation gauge (5)

• Weak coupling b 6/g2 8

• Expand in Am using the identification Ux,m
  expigaAm(x).
• Gauge-fixing required.
• Feynman rules similar to conventional QFT's in
  the continuum,
• but more complicated
• propagators, momentum integrations, ..
• more vertices
• Ux,m expigaAm(x) gt
  ...
  ...
• Haar measure DU
• Reguralization through a.

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2. Formulation lattice scale

• No dimensionful quantities on a computer.
• The scale a should be determined through a
  physical interpretation
• i.e., through an identification of a
  dimensionful observable
• with its experimental value in MeV.
• (e.g.) r meson correlation function
• dim.-less corr. length
• (measured on the lattice)
• Aoki et al. (CP-PACS), PR D67(03)034503
• quenched QCD, a 0.1 fm, L 3.3 fm
• Identify x 1/mra with mr 770 MeV

25
2. Formulation lattice scale (2)
Any dimensionful quantities should be equally
valid. Favorite choices mr s (420440
MeV)2 r0 0.5 fm (Sommer scale) 1S-1P HFS for
heavy quarks CP-PACS Nf2 fQCD data (PR
D65) 1/r0 0.88 vs Final results depend
on the choice in approximate calculations!
26
2. Formulation continuum limit
Smaller a ? larger x i.e. a gt 0 ? x gt
8 The continuum limit locates at a 2nd order
phase transition point in the coupling
parameter space. Physics of IR modes (large x)
near a 2nd order point ? theory of critical
phenomena "universality" holds Near the
continuum limit ("scaling region"), Physics
does not depend on the details of the lattice
action. Ambiguities in defining lattice action
does not matter. Recovery of rotation/Lorentz
symmetry.
27
2. Formulation continuum limit (2)
QCD x gt 8 (continuum limit) at b gt 8 (g
0) ? asymptotic freedom lattice perturbation
theory
We already know how observables should scale near
the continuum limit!
CF non-perturbative constant to be determined
on the lattice Accurate extrapolation possible!!
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3. Numerical methods

• Lattice QCD axiomatic definition of QCD, valid
  also at strong couplings
• On finite lattices with a gt 0 and L lt 8,
• system is free from UV and IR divergences.
• Physics is defined in the limits of
• a gt 0
• L gt 8 gt
• Before these limits, the theory is finite and
  well-defined.
• We can apply various techniques beside the
  perturbation theory.
• Axiomatic studies using exact inequalities
• Strong coupling expansion, hopping parameter
  expansion, ...
• Approximate models (mean field, random matrix
  theory, Polyakov loop model, ...)
• Numerical methods

29
3. Methods (2)

• Numerical evaluation of Euclidian path integral
• Difficulties
• Large dimensional multi-integrals.
• e.g., QCD on 324 gt 50,000,000 dimensional
• Integrand is sharply peaked due to eS note S
  µ V
• peak position classical orbit
• width quantum fluctuations
• Conventional mesh method inefficient.
• ß
• Monte Carlo method
• Generate sampling points important sampling
• with the probability µ eS
• Note Euclidian space-time essential for eS to
  be a probability.

30
3. Methods MC

• Monte Carlo method
• Generate a chain of configurations f1, f2,
  ...
• such that Pf Z1 eSf.
• Measure observables as the average over
  configurations.
• "Update" generation of fn1 from fn
• P(f' f) Probability to have f' as the
  successor of f
• We assume "strong ergodicity" or "irreducibility"
  (after finite steps)
• P(f' f) gt 0 for (f',f) "Markov
  process"
• In order to reproduce Pf Z1 eSf,
• P(f' f) must satisfy eSf' Sf P(f'
  f) eSf.
• A sufficient condition
• P(f' f) eSf P(f f') eSf'
  "detailed balance"
• Many possibilities.

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3. Methods Metropolis

• Metropolis algorithm Metropolis (1953)
• P(f' f) min(1, eDS), DS Sf'
  Sf
• ?
• generate a uniform random number r Î 0,1
• accept f' if r lt eDS
• reject f' and accept f otherwise
• Applicable to many systems (only DS required to
  compute).
• For efficiency, we need a way to generate f'
  with small DS .
• Naively, DS is small when f' f.
• e.g., f'(x) f(x) e(x) e O(1/V) or
  single site only.
• But, then, f' and f are strongly correlated
  ("autocorrelation"),
• i.e., long chain required to achieve a
  statistical accuracy.
• More sophisticated way necessary. (MD, combine
  with OR, etc.)

32
3. Methods HB

• (Pseudo) heat-bath algorithm
• P(f' f) Z1 eSf' independent of f
  
• Analytic calculation of such f' is possible
  only in simple models
• Heisenberg model (O(3) s-model)
• SU(2) pure gauge O(4) spin with von Neumann
  trick
• Creutz ('80)
• SU(N) pure gauge using SU(2) subgroups
• Cabibbo-Marinari, Okawa ('82)

33
3. Methods HB (2)

• SU(2)
• Let us suppose to update a particular link U.
• O(4) representation DU µ d4x d(x2 1), x
  (x1, x2, x3, x4)
• The probability distribution for the new link U
  is, then

M SstaplesUUU xm x1m1 x2m2 x3m3 x4m4,
mi functions of M and b.
Polar coordinate for x.
Now the task is to generate z with the
probability
34
3. Methods HB (3)

• von Neumann method to generate a rand. variable
  with distribution
• generate trial x (xmax xmin)r1 xmin
• accept x if y lt f(x), where y fmax r2
• Repeat this many times. Chain of accepted x has
• the desired distribution.
• The method is inefficient if f is sharply
  peaked like

• ß
• generate trial z with the distribution emz
• accept z if r2 lt (1z2)1/2
• Vectorize/parallerize updating independent links
  simultaneously.
• Fixed of trials ("hits") per sweep. (Adopt
  previous config. if accepted nonce.)

35
3. Methods HB (4)

• SU(N) Cabibbo-Marinari-Okawa method
• Cabibbo and Marinari, PL B119(82)387
  Okawa, PRL 49(82)353
• Update all SU(2) subgroups by pseudo-HB.
• e.g., for SU(3)

u Î SU(2), pHB update possible. Repeat for all
SU(2) subgroups.
36
3. Methods random number

• For LQCD, random number with long period is
  required.
• e.g., SU(3) on 324 HB 10,000 sweeps
  324x4x8x3x4x10000 4x1012
• (cf.) usual congruential rand. number period lt
  230 109
• M-series random number (Generalized feedback
  shift register algorithm)
• T.G. Lewis and W.H. Payne, JACM 20(73)456
• period 2p, p 532, 607, etc.
• Vectorization, parallelization easy.
• See lecture note by T. Yoshié at YITP Lattice
  School (2002).

37
3. Methods errors

• Errors in the lattice results
• Statistical errors
• Systematic errors
• Due to a ? 0
• Perform simulations at several a. gt Extrapolate
  to a 0.
• Errors from extrapolation (higher orders in a).
• Due to V lt 8
• Perform simulations at several V. gt Estimate or
  extrapolate.
• Due to mq gt Mud
• Perform simulations at several mq. gt Chiral
  extrapolation.
• Errors from extrapolation (higher orders in mq).
  gt Sec.5
• Estimation case by case for each observables.
• lt Theoretical inputs.

38
3. Methods errors (2)

• Estimation of statistical errors Particle Data
  Group booklet
• Naive estimate
• If F follows a distribution with variance s2,
  (DF)2 s2/N.
• Error propagation for G G(F1, F2, ...) for
  small errors
• V Error matrix (correlation matrix, covariance
  matrix)
• Combined with fittings, complicated/difficult to
  estimate reliably.
• Assumed in these formulae statistical
  independence of each configuration fi
• In reality, configurations are not independent.

39
3. Methods errors (3)
Autocorrelation
tF auto-correlation length lt simulation
parameters, simulation algorithm, F, L, ...
tmax necessary to avoid fluctuations.
CP-PACS, PR D65(02)054505 F Ninv
40
3. Methods errors (4)

• Jackknife method with binning
• naive jackknife B. Efron, SIAM Rev. 21(79)460
  R.G. Miller, Biometrika 61(74)1
• Identical with the naive estimate for simple
  averages.
• Applicable to complicated G's including fittings
  etc.
• Suppress statistical fluctuations in the error
  analysis.
• Correlations among F's automatically included.
• bining Gottlieb et al., NP B263(86)704
• (xxxxx)(xxxxx)(xxxxx) ... (xxxxx) devided into K
  bins.
• b(1) b(2) b(3)
  b(K) bin size b(i) N/K
• Identical with the naive estimate for
  uncorrelated data.
• With full correlation, DF increases as (bin
  size)1/2.

41
3. Methods errors (5)
Bin size gt tF needed for a reliable estimation
of DF. gt bin size dependence should be studied!
Nf2 fQCD 203x48 Plaq.Clover a0.09 fm HMC
algorithm Aoki et al. (JLQCD), PR
D68(03)054502 PS meson mass
qQCD 362x48x6 pseudo-HB OR algorithm Iwasaki et
al. (QCDPAX), PR D46(92)4657 P plaquette, c
plaq. susceptibility