PACS10 project in lattice QCD
Not PACS-X project

Takeshi Yamazaki
University of Tsukuba
Center for Computational Sciences

10th symposium on Discovery, Fusion, Creation of New Knowledge by MultidisciplinaryComputational Sciences@ University of Tsukuba, October 15-16 2018

PACS10 project in lattice QCD
Not PACS-X project

Takeshi Yamazaki
University of Tsukuba
Center for Computational Sciences

10th symposium on Discovery, Fusion, Creation of New Knowledge by MultidisciplinaryComputational Sciences@ University of Tsukuba, October 15-16 2018

PACS10 project
PACS = PACS Collaboration
Tsukuba
Riken-CCS
HiroshimaTohoku

N. Ishizuka, J. Kakazu, Y. Kuramashi, N. Namekawa,T. Taniguchi, N. Ukita, T. Yamazaki, T. Yoshié
Y. Nakamura, E. Shintani
K.-I. Ishikawa
N. Tsukamoto, S. Sasaki

10 = spatial extent of 10 fm

Purpose of PACS10 project
Removing all main uncertainties in lattice QCD
using three ensembles at physical mπ on (10 fm)3 volume1

PACS10 project
PACS = PACS Collaboration
Tsukuba
Riken-CCS
HiroshimaTohokuN. Ishizuka, J. Kakazu, Y. Kuramashi, N. Namekawa,T. Taniguchi, N. Ukita, T. Yamazaki, T. Yoshié
Y. Nakamura, E. Shintani
K.-I. Ishikawa
N. Tsukamoto, S. Sasaki

10 = spatial extent of 10 fm

Purpose of PACS10 project
Removing all main uncertainties in lattice QCD
using three ensembles at physical mπ on (10 fm)3 volume1-a

PACS10 project
PACS = PACS Collaboration
Tsukuba
Riken-CCS
HiroshimaTohokuN. Ishizuka, J. Kakazu, Y. Kuramashi, N. Namekawa,T. Taniguchi, N. Ukita, T. Yamazaki, T. Yoshié
Y. Nakamura, E. Shintani
K.-I. Ishikawa
N. Tsukamoto, S. Sasaki

10 = spatial extent of 10 fm
Purpose of PACS10 project
Removing all main systematic uncertainties in lattice QCD
using three ensembles at physical mπ on (10 fm)3 volume1-b

Outline• PACS10 project– Lattice QCD– pre-PACS10 project– PACS10 project– PACS10 configuration• Result of PACS10 project– Light meson form factors– Nucleon form factors– Kaon semileptonic decay– Muon anomalous magnetic moment• Summary2Lattice QCD
QCD =Quantum Chromodynamics
theory of strong interaction forquarks and gluons2≃2.2 MeV/c2≃1.28 GeV/c2≃4.7 MeV/c2≃96 MeV/c2≃173.1 GeV/c2≃105.66 MeV/cdynamical u, d, s quark eﬀect
in low energy physics
Nf = 2 + 1 QCD(mu = md ≪ ms)

figure from wikipedia

non-perturbative calculation method is necessary for QCD.

Lattice QCD: non-perturbative calculation using supercomputer3

Lattice QCD
QCD =Quantum Chromodynamics
theory of strong interaction forquarks and gluons2≃2.2 MeV/c2≃1.28 GeV/c2≃4.7 MeV/c2≃96 MeV/c2≃173.1 GeV/c2≃105.66 MeV/cdynamical u, d, s quark eﬀect
in low energy physics
Nf = 2 + 1 QCD (mu = md < ms)figure from wikipedia

non-perturbative calculation method is necessary for QCD.

Lattice QCD: non-perturbative calculation using supercomputer

3-aLattice QCD
QCD =Quantum Chromodynamics
theory of strong interaction forquarks and gluons2≃2.2 MeV/c2≃1.28 GeV/c2≃4.7 MeV/c2≃96 MeV/c2≃173.1 GeV/c2≃105.66 MeV/cdynamical u, d, s quark eﬀect
in low energy physics
Nf = 2+1 QCD (mu = md ≪ ms)
what we want to calculate

figure from wikipedia

non-perturbative calculation method is necessary for QCD.

Lattice QCD: non-perturbative calculation using supercomputer

3-bLattice QCD
QCD =Quantum Chromodynamics
theory of strong interaction forquarks and gluons2≃2.2 MeV/c2≃1.28 GeV/c2≃4.7 MeV/c2≃96 MeV/c2≃173.1 GeV/c2≃105.66 MeV/cdynamical u, d, s quark eﬀect
in low energy physics
Nf = 2+1 QCD (mu = md ≪ ms)
what we want to calculate

figure from wikipedia

non-perturbative calculation method is necessary for QCD.

Lattice QCD: non-perturbative calculation using supercomputer

3-cLattice QCD non-perturbative calculation using supercomputeraLUµ(x)ψ(x + µ̂)ψ(x)finite lattice spacing a (regularization)
momentum cut-oﬀ : |p| ≤ π/a
4d finite spacetime = L3 × Tquarks ψ(x) : on site
gluons Uµ(x): link between sites

Expectation value of observable O(ψ, ψ, U )
N√
1 !
−1
⟨O(ψ, ψ, U )⟩ =O(D , Ui) + O(1/ N )N i=1
quark propagator D−1 [U, m ], quark mass mq

Monte Carlo simulationgauge configuration Ui generated with proper probability
each Ui generation needs O(104) D −1[U, mq ]D[U, mq ] = O(109 ) × O(109 ) matrix when L = T = 102
mq → small ⇒ computational time → long

Large computational resource necessary4qLattice QCD
Simulation parameters: mπ (mu,d,s), L3 × T , amπ : pion mass (pion = lightest composite particle of quarks in QCD)

Lattice QCD→QCD need mπ = mphys, L>π∼ 6/mπ , a → 0−(2∼9)

unphysical mπ (mphys= 0.135 GeV): cost ∝ mππ(depend on solver algorithm of D −1 [U, mq ])traditionally extrapolate to mphys
with several data at larger mππfinite L: cost ∝ L3T

finite L eﬀect is negligible in L >∼ 6/mπfinite a: cost ∝ a−(2∼6) (depend on Monte Carlo algorithm)
need several data in small a for a → 0

main systematic uncertainty in lattice QCD

5

pre-PACS10 project
History of lattice QCD Collaboration in University of Tsukuba
Collaboration
Main resource
Nfmπ [GeV]
L [fm]
a [fm]
a→0year

CP-PACS
CP-PACS02> 0.3> 0.53.0
2.5> 0.05 > 0.1
⃝⃝
∼2002

CP-PACS+JLQCD
PACS-CS
PACS
Earth Simulator
PACS-CS, T2KK†
2+12+12+1> 0.6> 0.156
0.146
2.02.98.1> 0.070.09
0.08⃝∼2006∼2009∼2015
Nf = 0no dynamical quark eﬀect
Nf = 2dynamical quark eﬀect for u, d
Nf = 2 + 1 dynamical quark eﬀect for u, d, smphys= 0.135 GeVπ† MEXT SPIRE Field 5 Subject 1

pioneering study in each period

6PACS10 project since 2016
ProjectResource
Nfmπ [GeV]
L [fm]
L3 · Ta [fm]a→0
OFP node
status
Name

1284
0.08
256done
PACS10-1

PACS10OFP
2+1
0.135
1016040.06⃝
512almost done

PACS10-2

2564
0.04

pre-PACS10K
2+1
0.146
8.1
964
0.08

2048
running
PACS10-3——
K confPurposeRemoving all main systematic uncertainties in lattice QCD
using three ensembles at physical mπ on (10 fm)4 volume7

Results of PACS10 projectprecise determination of physical quantity from lattice QCD
I. quantitatively understand property of hadrons (mass, size, etc.)
reproduce experimental values with high accuracy
hadron = composite particles of quarks, e.g. mesons (π, K, · · · ) and nucleon (proton, neutron)

1. Light meson electromagnetic form factors
2. Nucleon form factors

II. search for new physics beyond the standard model
discrepancy between theoretical calculation and experiment
3. Kaon semileptonic decay
4. Muon anomalous magnetic moment

Results from PACS10-1 and PACS10-2 configurations
All results are preliminary.
8Results of PACS10 projectprecise determination of physical quantity from lattice QCD
I. quantitatively understand property of hadrons (mass, size, etc.)
reproduce experimental values with high accuracy
hadron = composite particles of quarks, e.g. mesons (π, K, · · · ) and nucleon (proton, neutron)

1. Light meson electromagnetic form factors
2. Nucleon form factors

Finite volume eﬀect PACS10-1 and (5.4 fm)3 : [arXiv:1807.06237]

II. search for new physics beyond the standard model
discrepancy between theoretical calculation and experiment
3. Kaon semileptonic decay
4. Muon anomalous magnetic moment

Results from PACS10-1 and PACS10-2 configurations
All results are preliminary.
8-a1. Light meson form factors by J. Kakazu
1Electromagnetic form factor F (Q2) = 1 − Q2⟨r 2⟩ + · · ·62⟨r ⟩ mean square charge radius, Q2 momentum transfer
F (Q2) ̸= 1 → hadron has internal structure.
pionkaon1

1experiment
PACS10-1 conf
K conf0.98
0.96

experiment
PACS10-1 conf

0.98

0.94

2FK(Q )

2Fπ(Q )

0.96

0.92
0.9
0.88

0.94

0.92

Preliminary

Preliminary

0.9
0.86
00.02

0.04
20.06

0.08

00.02

2Q [GeV ]advantage of PACS10: smaller q 2 due to large L
pion: agree with the experimentkaon: agree with experiment within 1.5σ0.04
20.06

0.08

2Q [GeV ][experiment from PDG18]

9

1. Light meson form factors by J. Kakazu
1Electromagnetic form factor F (Q2) = 1 − Q2⟨r 2⟩ + · · ·62⟨r ⟩ mean square charge radius, Q2 momentum transfer
F (Q2) ̸= 1 → hadron has internal structure.
comparison of ⟨rπ2⟩

2

0.46
0.44
0.42
0.4
0.38

Preliminary

22<rπ >(fm )

0.48

0.4

experiment
JLQCD Nf=2+1
PACS10-1 Nf=2+10.35

20.50.45<rK>(fm )0.52

experiment
ETMC Nf=2+1+1
JLQCD Nf=2+1
HPQCD Nf=2+1+1Mainz Nf=2PACS-CS Nf=2+1
ETMC Nf=2
RBC/UKQCD Nf=2+1
PACS Nf=2+1 K conf
PACS Nf=2+1 PACS10-1

2⟩
comparison of ⟨rK

0.3

0.25

Preliminary[experiment from PDG18]

pion: agree with the experiment and other lattice resultskaon: agree with experiment and closer to another lattice result
will investigate systematic errors
9-a

! Charge radius of the nucleon

GE (Q ) = 1Proton-sizehr iE + p + . . . puzzle

22. Nucleon
formfactors
Qp22

Nucleon: proton p and neutron nProton size puzzle62! ⇠ 7 discrepancy between µH and H / e-p2 scatteringq
Proton charge radius： GE (q ) = 1⇥r2 ⇤E + O(q 4 ) 2
6μ hydrogen
Lamb shift

ep scattering

Hydrongen Lamb shift[Arrington,arXiv:1506.00873][Arrington,arXiv:1506.00873]
0 the proton charge radius from muonic hydrogen qFIG. 1: Extractions ofmeasurements[1,2], hydrogen
kµµ2µ2p|V(q)|p⇥=u(p)F(q)+i⇥F(q)u(p)
12SlidefromS.
Sasakiin QCDdownunder2017`k✓spectroscopy[3], electron
scattering measurements at Mainz [6, 7], and a2M
global analysis of earlierworld
q average shown is compared to the CODATA-2010 evaluation [3]. Figure courtesy ofdata [4]. The direct
q22µ G (q 2 )2 ⟩ diﬀers inp0 µ-proton and e-proton
(p+p)
qE
⟨r
experiments.
Randolf EPohl. p4M 2 GM (q )= u(p )G (q 2 ) u(p)+i µM2M2M
1 4M 2q2 < 0
While none of these appear likely to resolve to the discrepancy with muonic hydrogenmagneticwarranted.
Form Factors
q2N

Several experiments proposed to resolve this puzzle

measurements, some issues
remain which deserve more detailed examination.
⇥p, s⇥ = ū(p , s )µF1 (q 2 ) + i⇥ µGE (Q2 ) = F1 (Q2 )22F2 (Q2 )2qF2 (q 2 ) u(p, s)
2mII.

GENERAL ISSUES IN THE EXTRACTION OF THE RADII

103. Kaon semileptonic decayStandard model predicts∆CKM = |Vud|2 + |Vus|2 + |Vub|2 − 1 = 0

Vij : Cabbibo-Kobayashi-Maskawa matrix element

if ∆CKM ̸= 0, signal of new physics|Vud| = 0.97420(21)|Vub| = O(10−3)

nuclear β decay [PoS(CKM2016):028]
B meson decay

Determination of |Vus| using lattice QCD
Experimental result of K → πlν decay (l: lepton, ν: neutrino)|Vus|f+(0) = 0.21654(41) [PoS(CKM2016):032]

Kl3 form factor f+(0) from lattice QCD calculation → |Vus|13

3. Kaon semileptonic decay (PACS10-1) by J. Kakazu
Kl3 form factors: f+(q 2) and f0(q 2), f+(0) = f0(0)f+(q 2) and f0(q 2)
1.1

0.23

|Vus|

2f+(q )
20.228f0(q )
f+(0)=f0(0)

1.05

0.226
10.2240.95

Preliminary
0.9

-0.1

-0.05
20.2220comparison of |Vus|
ETM Nf=2+1+1
FNAL/MILC Nf=2+1+1
JLQCD Nf=2+1RBC/UKQCD Nf=2+1
FNAL/MILC Nf=2+1
ETM Nf=2∆CKM=0PDG semileptonic
PDG leptonic
PACS Nf=2+1 semileptonic
PACS Nf=2+1 leptonic

Preliminary

0.052q [GeV ]Our |Vus| agrees with ∆CKM = 0

and another determination of |Vus| (leptonic decay K → lν).
will investigate systematic errors for final result

14

4. Muon anomalous magnetic moment
57. Muon anomalous magnetic moment 5

figure and values from PDG18–301 ± 65

SM predictions

HLMNT 2011
–263 ± 49

DHMZ 2011–289 ± 49

DHMZ 2017

Measurement–268 ± 43

BNL-E821 (world average)
0 ± 63-700-600-500-400-300

aµ – aµexp

ExperimentaexpµBNL-E821 2004

JN 2009-200-100=116592091(63)×10−11aSM= 116591823(43)×10−11µ←− aexp − aSM = 268(76)×10−11µµ

SM: Standard Model

3.5σ diﬀerence = New physics?
Need to check more precisely

0× 10–1111∼ error in ∼2020
24Figure 57.2: Compilation of recent published results for aµ (in units of 10−11 ),subtracted by the central value of the experimental average (57.3). The shaded band
indicates the size of the experimental uncertainty. The SM predictions are taken
from: JN 2009 [4], HLMNT 2011 [23], DHMZ 2011 [19], DHMZ 2017 [18], Notethat the quoted errors in the figure do not include the uncertainty on the subtracted
experimental value. To obtain for each theory calculation a result equivalent toEq. (57.14), the errors from theory and experiment must be added in quadrature.

FNAL(US) and J-PARC(Japan):

TheoryMain! uncertainty from leading order Hadronic Vacuum Polarization
where F (x) =2z(1 − z) /[(1 − z) + x z] dz. For values of ε ∼ 1–2 · 10
hvp originally motivated by cosmology,andcan −11m ∼ 10–100 MeV, the dark photon, awhich was =6931(34) × 10by phenomenologyµprovide a viable solution to the muon g − 2 discrepancy. However, recent experimental
constraints disfavor such a scenario [41] under the assumption that the dark photon
desired
calculation
for new physics searchdecays primarily into charged
lepton pairs.Lattice
Direct searches forQCD
the dark photon
continue
102

22−3V

to be well motivated [42]; but with guidance coming from phenomena outside the muon
anomalous magnetic moment discrepancy.
References:
1.
2.
3.
4.

A. Czarnecki and W.J. Marciano, Phys. Rev. D64, 013014 (2001).
M. Davier and W.J. Marciano, Ann. Rev. Nucl. and Part. Sci. 54, 115 (2004).
J. Miller, E. de Rafael, and B. Lee Roberts, Rept. Prog. Phys. 70, 795 (2007).
F. Jegerlehner and A. Nyﬀeler, Phys. Reports 477, 1 (2009).

15

Summary
PACS10 project
Removing all main uncertainties in lattice QCD
mπ = 0.135 GeV, L = 10 fm, a = 0.8, 0.6, 0.4 for a → 0

Encouraging results obtained from PACS10 configuration– charge radius for π, K, N
– new physics search from |Vus| and aµFuture works– finish PACS10-3 generation– precise physical quantity for stable hadrons
mass, decay constant, charge radius, · · ·[arXiv:1807.06237]– multi-hadron systems
scattering, strong and weak decays, nucleus, · · ·17