Spectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Computational particle physics: lattice QCD(approaching the continuum limit with physical quark masses)

Z. Fodor
University of Wuppertal, Forschungszentrum Juelich, Eotvos University Budapest

Nara, December 14, 2012
(talk for non-experts)

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Outline1

Spectrum2

Quark masses

3

QCD transition

4

Tc & EoS

5

NN interaction

6

High densities

7

CPU-resourcesZ. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

QCD: need for a systematic non-perturbative method
in some cases (g-2): perturbative approach is good; in other cases: badpressure at high temperatures converges at T=10300 MeV

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Lattice field theory
systematic non-perturbative approach (numerical solution):quantum fields on the latticequantum theory: path integral formulation with S=Ekin -Epot
quantum mechanics: for all possible paths add exp(iS)
quantum fields: for all possible field configurations add exp(iS)
Euclidean space-time (t=iτ ): exp(-S) sum of Boltzmann factors
we do not have infinitely large computers ⇒ two restrictions
a. put it on a space-time grid (proper approach: asymptotic freedom)
formally: four-dimensional statistical systemb. finite size of the system (can be also controlled)⇒ stochastic approach, with reasonable spacing/size: solvableZ. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

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Tc & EoS

NN interaction

High densities

CPU-resources

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yx

fine lattice to resolve thestructure of the proton (<∼0.1 fm)
few fm size is needed
50-100 points in ‘xyz/t’ directionsa⇒a/2 means 100-200×CPU
Z. Fodormathematically
109 dimensional integrals
advanced techniques,good balance andorder of Pflops are needed
Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Lattice Lagrangian: gauge fields
νµPlaquetteaΨ (x)Uµ (x+eµ)Pµνa F aµν + ψ̄(D γ µ + m)ψ
L = − 41 Fµνµ

anti-commuting ψ(x) quark fields live on the sites
gluon fields, Aaµ (x) are used as links and plaquettes
RyU(x, y ) = exp (igs x dx 0 µ Aaµ (x 0 )λa /2)
Pµν (n) = Uµ (n)Uν (n + eµ )Uµ† (n + eν )Uν† (n)S = Sg + Sf consists of the pure gluonic and the fermionic parts
PSg = 6/gs2 · n,µ,ν [1 − Re(Pµν (n))]
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Lattice Lagrangian: fermionic fields
quark differencing scheme:ψ̄(x)γ µ ∂µ ψ(x) → ψ̄n γ µ (ψn+eµ − ψn−eµ )ψ̄(x)γ µ Dµ ψ(x) → ψ̄n γ µ Uµ (n)ψn+eµ + ...
fermionic part as a bilinear expression: Sf = ψ̄n Mnm ψm
we need 2 light quarks (u,d) and the strange quark: nf = 2 + 1(complication: doubling with 16 fermions ⇒ staggered/Wilson)
Euclidean partition function is given by the Boltzmann weights
Z YZ Y−Sg −SfZ=[dUµ (x)][d ψ̄n ][dψn ]e=[dUµ (n)]e−Sg det(M[U])n,µn,µusing two fermionic fields instead of one: det2 (M[U])Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Importance sampling
Z=Z Y[dUµ (n)]e−Sg det(M[U])n,µwe do not take into account all possible gauge configurationeach of them is generated with a probability ∝ its weightimportance sampling, Metropolis algorithm:(all other algorithms are based on importance sampling)P(U → U 0 ) = min 1, exp(−∆Sg ) det(M[U 0 ])/ det(M[U])gauge part: trace of 3×3 matrices (easy, without M: quenched)fermionic part: determinant of 106 × 106 sparse matrices (hard)more efficient ways than direct evaluation (Mx=a), but still hard
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Hadron spectroscopy in lattice QCD
Determine the transition amplitude between:
having a “particle” at time 0 and the same “particle” at time t⇒ Euclidean correlation function of a composite operator O:
C(t) = h0|O(t)O† (0)|0i
insert a complete set of eigenvectors |ii
P
P= i h0|eHt O(0) e−Ht |iihi|O† (0)|0i = i |h0|O† (0)|ii|2 e−(Ei −E0 )t ,where |ii: eigenvectors of the Hamiltonian with eigenvalue Ei .
O(t) = eHt O(0) e−Ht .

andt larget large⇒⇒lightest states (created by O) dominate: C(t) ∝ e−M·t
exponential fits or mass plateaus Mt =log[C(t)/C(t+1)]
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Quenched results
QCD is 40 years old ⇒ properties of hadrons (Rosenfeld table)
non-perturbative lattice formulation (Wilson) immediately appeared
needed 20 years even for quenched result of the spectrum (cheap)
instead of det(M) of a 106 ×106 matrix trace of 3×3 matrices
always at the frontiers of computer technology:
GF11: IBM "to verify quantum chromodynamics" (10 Gflops, ’92)
CP-PACS Japanese purpose made machine (Hitachi 614 Gflops, ’96)
1.8
1.6m (GeV)

ΩΞ

1.4

Σ
1.2

Λ

φN

1.0

Σ*
∆

K*

0.8
0.6

Ξ*

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GF11 infinite volume K−input
CP−PACS K−input
CP−PACS φ−input

0.4

the ≈10% discrepancy was believed to be a quenching effectZ. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resourcesDifficulties of full dynamical calculations
though the quenched result can be qualitatively correct
uncontrolled systematics ⇒ full “dynamical” studies
by two-three orders of magnitude more expensive (balance)
present day machines offer several Pflops
no revolution but evolution in the algorithmic developments
Berlin Wall ’01: it is extremely difficult to reach small quark masses:

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Scale setting and masses in lattice QCD
in meteorology, aircraft industry etc. grid spacing is set by handin lattice QCD we use g,mud and ms in the Lagrangian (’a’ not)measure e.g. the vacuum mass of a hadron in lattice units: MΩ asince we know that MΩ =1672 MeV we obtain ’a’aM

masses are obtained by correlated fits (choice of fitting ranges)
illustration: mass plateaus at the smallest Mπ ≈190 MeV (noisiest)
0.9
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12t/a

volumes and masses for unstable particles: avoided level crossing
decay phenomena included: in finite V shifts of the energy levels
Fodor
Computationalparticle
physics:
lattice
QCD (approach⇒ decay width (coupling) & Z.masses
ofthe heavyandlightstates

Spectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Final result for the hadron spectrum S. Durr et al., Science 322 1224 2008
2000

Budapest-Marseille-Wuppertal collaboration

M[MeV]

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ON
experiment

KwidthinputpQCD

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Light hadron spectrum summary: A. Kronfeld 1209.3468
2400© 2012 Andreas Kronfeld/Fermi Natl Accelerator Lab.

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results with various actions and fermion formulations(!) are the same
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

The mass is not the sum of the constituents’ massusually the mass of “some ordinary thing” is justthe sum of the mass of its constituents (upto tiny corrections)origin of the mass of the visible universe: dramatically different
proton is made up of massless gluons and almost massless quarks

mass of a quark is ≈5 MeV, that of a proton (hadron) is ≈1000 MeV
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

The calculation that we have been dreaming of doing

Nf = 2 + 1 all the way down to Mπ <∼135 MeV to allow smallinterpolation to physical mass point
Large L>∼5 − 6 fm to have sub percent finite V errors
At least three a ≤ 0.1 fm for controlled continuum limit
A reliable determination of the scale with a well measured
physical observable
Full nonperturbative renormalization
Complete analysis of systematic uncertainties

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Where do we stand in lattice QCD?
all simulations with Nf 2,2+1 and 2+1+1 and Mπ < 400 MeV
very first physical mass point simulations: PACS-CS Collaboration
ETMC '09 (2)ETMC '10 (2+1+1)
MILC '10
MILC '12
QCDSF '10 (2)QCDSF-UKQCD '10
BMWc '10
BMWc'08
PACS-CS '09
RBC/UKQCD '10
JLQCD/TWQCD '09
HSC '08
BGR '10
CLS '10 (2)Mπ[MeV]

600

400

200

000.05a[fm]0.1

Z. Fodor

0.15
Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Continuum extrapolation of renormalized masses
Renormalized quark masses interpolated in Mπ2 & MK2 to physical point using:SU(2) χPTor low-order polynomial anszätze
with cuts on pion mass Mπ < 340, 380 MeV
Example of continuum extrapolations:for ms /mud the continuum extrapolation is flat
120

Z. Fodor

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Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Outline: what has lattice done for the phase diagram.
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Outline: what has lattice done for the phase diagram

cross-over: Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, K. Szabo, Nature 443 (2006) 675
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Outline: what has lattice done for the phase diagram

cross-over, absolute scale (Tc )Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Outline: what has lattice done for the phase diagram

cross-over, Tc , EoS
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resourcesOutline: what has lattice done for the phase diagram

cross-over, Tc , EoS, curvature of phase line, µ>0 EoS, fluct.
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Finite size scaling in the quenched theory
look at the susceptibility of the Polyakov-line
first order transition (Binder) =⇒ peak width ∝ 1/V, peak height ∝ V

finite size scaling shows: the transition is of first order
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrumQuark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Approaching the continuum limuit

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Approaching the continuum limuit

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Approaching the continuum limuit

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Approaching the continuum limuit

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Approaching the continuum limuit

Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

The nature of the QCD transition: analytic• finite size scaling analysis with continuum extrapolated T 4 /m2 ∆χthe result is consistent with an approximately constant behavior
for a factor of 5 difference within the volume range
chance probability for 1/V is 10−19 for O(4) is 7 · 10−13
continuum result with physical quark masses in staggered QCD:
the QCD transition is a cross-over
Wuppertal-Budapest Collaboration: Y.Aoki, G.Endrodi, Z.Fodor, S.D.Katz, K.K.Szabo, Nature, 443 (2006) 675
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

The nature of the QCD transition
Y.Aoki, G.Endrodi, Z.Fodor, S.D.Katz, K.K.Szabo, Nature, 443 (2006) 675

analytic transition (cross-over) ⇒ it has no unique Tc :examples: melting of butter (not ice) & water-steam transition

labove the critical point cp and dρ/dT give different Tc s.
QCD: chiral & quark number susceptibilities or Polyakov loop
they result in different Tc values ⇒ physical difference
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Wuppertal-Budapest & hotQCD: staggered Tc & eos
Wuppertal-Budapest: Tc PLB 643 ’06 46; JHEP 906 ’09 88; 1009 ’10 73 eos JHEP 601 ’06 89; 1011 ’10 77
hotQCD: Tc Phys. Rev. D85 (2012) 054503 eos Phys. Rev. D77 (2008) 014511; D80 (2009) 014504
Tc from chiral condensate
1equation of state (eos)
7Δl,sHotQCD results:

r1 scale

0.8

6p4 Nt=8asqtad Nt=8HISQ Nt=8HISQ Nt=1250.4
0.20asqtad: Nτ=8Nτ=12HISQ/tree: Nτ=6Nτ=8Nτ=12stout, cont.
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140(ε-3p)/T40.6432

Wuppertal-Budapest
results:1T [MeV]160

stoutcontinuum

0180

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Wuppertal-Budapest: Tc =157(4) MeV; hotQCD: Tc =154(9) MeV
Wuppertal-Budapest: eos-peak ≈4; hotQCD: ≈70,50,40,25% higher
theoretically more solid: Wilson (WHOT-QCD); best: overlap (JLQCD)
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Project group A02: HAL-BMW cooperation
NN potential & phase-shift for Mπ =300 & 135 MeV (2HEX action)
ETMC '09 (2)ETMC '10 (2+1+1)
MILC '10
MILC '12
QCDSF '10 (2)QCDSF-UKQCD '10
BMWc '10
BMWc'08
PACS-CS '09
RBC/UKQCD '10
JLQCD/TWQCD '09
HSC '08
BGR '10
CLS '10 (2)Mπ[MeV]

600

400

200

000.05a[fm]0.1

Z. Fodor

0.15

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Project group A03: large baryonic densities
Z=Z Y[dUµ (n)]e−Sg det(M[U])n,µwe do not take into account all possible gauge configurationeach of them is generated with a probability ∝ its weight
importance sampling, Metropolis algorithm:(all other algorithms are based on importance sampling)P(U → U 0 ) = min 1, exp(−∆Sg ) det(M[U 0 ])/ det(M[U])for µ 6= 0 the determinant is complex ⇒ sign problem
no importance sampling is possible
reduction of M: staggered (Fodor-Katz) & Wilson (Nagata-Nakamura)
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densitiesCPU-resources

Continuum prediction for the curvature: full result
G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo, JHEP 1104 2011 001

dashed line: freeze-out curve from experiments
lower solid line: Tc from the chiral condensate
upper solid line: Tc from the strange susceptibility
bands (red and blue) indicate the widths of the transition lines
the widths remain in this order approximately the same
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Project group A04: message of computational physics
M’s eigenvalues close to 0: CPU demanding (large condition number)
1

Number of BG/P racks
2448

72

350

Performance/TFlop/s

300
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196K# Parallel tasks

295K

go down to physical pion masses ⇒ algorithmically safeBlue Gene shows perfect strong scaling from 1 midplane to 72 racks
our sustained performance is as high as 37% of the peaksingle K20 GPU card performance upto 570 Gflops (staggered)(other major source: cell based special purpose machine QPACE)
Z. Fodor

Computational particle physics: lattice QCD(approachSpectrum

Quark masses

QCD transition

Tc & EoS

NN interaction

High densities

CPU-resources

Summary: physical quark masses and continuum limit
1Spectrum2

Quark masses

3

QCD transition

4

Tc & EoS

5

NN interaction

6

High densities

7

CPU-resources

Z. Fodor

Computational particle physics: lattice QCD(approach