1

Necessity is the mother of
invention in lattice QCD
Nonperturbative Analysis of Quantum Field Theory
and its Applications, September 22nd, 2022

Tetsuya Onogi (Osaka Univ.)Foreword2

I would like to thank all the organizers of this workshop. In
particular, I would like to express my gratitude to Dr. Fukaya
who initiated and took labor of all the practical work at Osaka
University for the preparation of this workshop.
I would also like to thank the speakers and participants of thisworkshop.
In this talk, I would like to introduce some of the research I
have been involved in, including the stories and lessons. I
would be happy if it could give some hint to young people.

3

Flavor physics and test of SM

4

Lattice QCD has been, and will continue to be, crucial
method to predict hadron matrix element from 1st principles.
At the early stage, we could not foresee the development
which we enjoy today.
Computer power is one of the main driving force.
However, the particle physics phenomenology and thetheoretical study of lattice have also been strong driving
forces which gave birth to many interesting ideas.

5The situation of lattice QCD
in early 1990’s
Lattice errors
Ø Perturbative error: not under good control yetØ Discretization error: big issue
Ø quenched approximation: full QCD with staggered has

started6Perturbative errorØ Failure of bare lattice perturbation theory<latexit sha1_base64="PwCNgeH/jBQAm5zB9hPfmVLpZgE=">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</latexit>

Plaquette VEVCritical mass shift

mc =

1
2c

4

Short distance quantity cannot be reproduced by bare lattice perturbation
è Serious problem in evaluating weak matrix elements

Improved perturbation theory by Lepage-Mackenzie
use of ‘renormalized‘ coupling for PT
7

Precise alpha_s determination by Fermilab
A. El-Khadra et al. PRL 69 (1992) 729
ØInput : Charmonium 1P-1S splitting
Just discovered at Fermilab experiment
Short distance physics: latticesystematic error is well controlleda = (a M )MC / M exp<latexit sha1_base64="NLIIqXQHc7SuJFZm3We/ySYNwMA=">AAACkHichVHLLgRBFD3ae7wGG4lNx4QQyagRQSTiuRCJxGuQGCbVrdDRr3TXTDCZH7CzsrAiEREfYEts/ICFTxBLEhsLd3paBMGtVNWpU/fcOlWluabhS8YeSpTSsvKKyqrqSE1tXX1DtLFpyXcyni6SumM63orGfWEatkhKQ5pixfUEtzRTLGs7E4X95azwfMOxF+WeK9YsvmUbm4bOJVHpaDdXh9VOnpoUpuTqTNd6LuVZ6sxEvueDKjJi182nozEWZ0GoP0EiBDGEMetEz5HCBhzoyMCCgA1J2ASHT20VCTC4xK0hR5xHyAj2BfKIkDZDWYIyOLE7NG7RajVkbVoXavqBWqdTTOoeKVW0s3t2wZ7ZHbtkj+zt11q5oEbByx7NWlEr3HTDQcvC678qi2aJ7U/Vn54lNjEYeDXIuxswhVvoRX12/+h5YWi+PdfBTtkT+T9hD+yWbmBnX/SzOTF/jAh9QOL7c/8ES73xRH+8d64vNjoefkUVWtGGTnrvAYxiCrNI0rmHuMI1bpQmZVAZUcaKqUpJqGnGl1Cm3wF35pjB</latexit>Ø

Determination of alpha_s by improved
perturbation theory<latexit sha1_base64="xQJrXtj4TKRN+sIpXCnv7eJJSOg=">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</latexit>gMS (⇡/a) =1
TrhUP iMC + 0.0252
g08

Our work with KEK group sea quark effect
ICHEP 1992 in Dallas:
Okawa-san “Why not work on this with full QCD at KEK?”
Personal interest: Alpha_s ßà SUSY GUT
c.f. Discovery of Z at SLC, LEP (1989)
Nf=2 staggered fermion full QCD configuration by KEK
Alpha_s from 1P-1S charmonium splitting

Sea quark effect is observed9Later story
Ø Night at Dallas (Lattice 1993, Dallas)

Luscher/Weisz

Lepage/Mackenzie“Lattice theorists should provide
result for the credibility of lattice
QCD calculation.”“Lattice theorist should provide more
timely and useful and appealing
results to stimulate experimentalist.”

2-loop Error is unknown

1-loop Boosted perturbation theory
Error ~ 1 x alpha^22-loop calculation by Luscher/Weisz
Nonperturbative renormalization by
Alpha collaboration

•
•

•
•

Numerical perturbation theory
High-Precision QCD (HPQCD)

c.f. RI-MOM scheme
10

(1992-)1993-1995 (Fermilab) On leave from Hiroshima U

P. Mackenzie, A. Kronfeld,
A. El-Khadra(Fermilab->Illinois),
G. Hockney, J. Simone, B. Gough11

Determination of the quark mass
B. Gough, T.O., et al.

light quark mass
8

O(a)-improved Wilson quark
improved perturbation theory

gives better scaling behavior
for the renormalized quark mass

µ = 2 GeV
}ml(µ) (MeV)••

Wilson64O(a)-improved Wilson
2
0
0

0.4

0.8

1.2

a (GeV-1)<latexit sha1_base64="x8pf2sGXEzlCzca/u14+4oJsPKA=">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</latexit>mMS (2GeV) = 3.6(6)MeV

12

Impact of Lepage-Mackenzie
Success of Pertubation theory
Ø Matrix element of various operators are more reliable(Discrepancies for Wilson and staggered quarks)
Ø 1-loop O(a)-improvement of lattice actions

Wilson fermion à O(a)-improved Wilson fermion
Ø better control of the continuum limit
Ø lattice effective theory (static, NRQCD) became
reliable13

1995-2001 (Back to Hiroshima )Particle Physics Group

Hadron Physics GroupO. Miyamura, S. Hioki
T. Takaishi,
T. Muta, J. Kodaira, T. Morozumi,
K.-I. Ishikawa, N. Yamada, N. Tsusui,
J. Harada, …

H. Matsufuru, T. Umeda, ..

ØNRQCD:B -> pi l nu Form factor
fB, BB mixingØNucleon Decay
14

B physics
Ø Discovery of top quark (CDF, D0: 1993)

top quark is heavy à large FCNC
Ø Discovery of FCNC at CLEOII

B-> K* gamma (1993), b-> s gamma(1995)
Ø Upcoming B factories

Belle (KEK): 1999-2010
Babar(SLAC): 1999-2008

è Test of SM (CKM matrix) was needed.
15

b-quark on the lattice m > a<latexit sha1_base64="00uVrp7n4YiE6NSRrTSulqFlDbk=">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</latexit>b~4 GeV1> ⇤QCD~2GeV

Lattice was too coarse to simulate b-quark.
Effective theory (NRQCD)
was useful.

<latexit sha1_base64="n4h/u09JiMoPwLQ6JvbBVT7wWmk=">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</latexit>

Lheavy ="D2· gB D · gE gE · D
Q iD0 +
+
+
2mQ
2mQ
8m2Q
#
2 2
· (iD ⇥ gE gE ⇥ iD) (D )+++ ··· Q
8m2Q
8m3Q
†

Foldy-Wouthuysen-Tani(FWT)"transformation· (±)

Each term is renormalized
è perturbative matching of
QCD and NRQCD is crucial.

<latexit sha1_base64="T9Dwc9RX4YURuKM3q+W4FeAFo+Y=">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</latexit>

h =

1

2mQ

(2)

⌃ · gB
+
+
8m2Q
8m2Q

i 4 · gE
+ ···
4m2Q

#✓

Q
0

◆

,

16

B physics with NRQCD at HiroshimaØHashimoto-san (Hiroshima-> KEK) was working on
B physics using static quark (Hashimoto-Saeki,1991) and NRQCD (Hashimoto 1994) and Nonrelativistic (Fermilab) interpretation of Wilson quark.

ØHashimoto-san and Matsufuru-san (Hiroshima)
worked on Isgur-Wise function Bà D l nu with
NRQCD in moving frame (Hashimoto-Matsufuru
1995).

Ø

Yamada-san (M2), Ishikawa-san (D1) and I with
Hashimoto-san, Matsufuru-san started working on
flavor physics with NRQCD.

17

Goal: Stability of fB under higher 1/m correction
Other processes ?

ØSimulation code and computers
Matsufuru-san’s code, Paragon workstation

ØPerturbative matching
hard work ( 1/m^2 correction, operator improvement)
Ishikawa-san , Yamada-sanØ

Analysis (Matrix elements of various operators)
Yamada-san, Ishikawa-san

18

f_B from NRQCD, Nf=0
K.I. Ishikawa et al PRD 56 (1997) 7028
Hiroshima Group

K.I. Ishikawa et al, PRD 61 (2000) 074501
JLQCD collab.

Wilson quark tree-level matching

3 lattice spacings a=0.18fm -0.09fm
Operator improvement<latexit sha1_base64="XVm1JH4kRxQcSjzpx5Pn4SnHrp0=">AAADbHicjVHNahRBEK7Z0RjXn2yiByEIQ5Ysu4QsNUskIgiJCnoR8uMmgXQy9HR6d4fMHzO9C3GYF/DqwYN6UBARH8OLL+AhjyBCLhG8eLBmdiDoxmgNPf3VV/VVVXfboevECvFQK+nnzo9dGL9YvnT5ytWJyuTURhz0IyHbInCDaMvmsXQdX7aVo1y5FUaSe7YrN+39+1l8cyCj2An8J+oglDse7/pOxxFcEWVNamPL1oJxt2bUzTmDcTfscYNFvcBa3k3q2EgbBrN5lLAwdlL2kHsehQkzVjbIarRGZWYjNVgn4iKZN9Ok9djC9JQqHlc9u5OwbuamTOwF6oR8IF1FJOXnfU5t0zqzzf81yGfJ8q1KFZuYmzEKzAJUobCVoPIeGOxBAAL64IEEHxRhFzjE9G2DCQghcTuQEBcRcvK4hBTKpO1TlqQMTuw+/bvkbResT35WM87Vgrq4tCJSGjCLX/ADHuNn/Ihf8edfayV5jWyWA9rtoVaG1sSzG+s//qnyaFfQO1GdObOCDtzOZ3Vo9jBnslOIoX7w9MXx+p212aSGb/Ebzf8GD/ETncAffBfvVuXaSyjTA5h/Xvco2Gg1zVtNXF2oLt0rnmIcpmEG6nTfi7AEj2AF2iA0T3uuvdJel4706/q0fnOYWtIKzTX4zfTaL6Mc5AI=</latexit>

1/M vs 1/M^2 corrections
B(0)

A4 =(1 + ↵⇢A ) ¯
1 ¯
(1)
+ ↵⇢A
2M0

D

1 ¯
·
2M0

0.8

1

(2)

0.24

0.25

fB [GeV]

1/2 latt

+ ↵⇢A

0.26

0.30

(fPMP )

·

0.20
0.15

0.22
0.2
0.18
0.16

0.10

0.0

0.2

0.4

0.6

0.8

1/(aMP)

1/M correction is under control

0.14

0

0.2

0.4
0.6
a [GeV-1]

Scaling behavior can be improved19

B à pi l nu form factor from NRQCD Nf=0
<latexit sha1_base64="+iS+FHgXnhGtKG5zTAm2nCY0VlM=">AAADM3icjVFLixNBEK4ZX2t8bNSL4KUxrCTEDZ1hURGEJV487sPsLmaSoWe2Z7bJvNLTCayz8wf8Ax48KYiIZ3+BF0/ePCzexIuIpxW8eLDmIaKLjx6m66uq76uq7rZjXySK0n1NP3L02PETcydrp06fOTtfP3d+I4mm0uF9J/IjuWWzhPsi5H0llM+3YslZYPt80x7fzvObMy4TEYV31W7MhwHzQuEKhykMWfU3ps9Cz+dmLJpjK0WTtfZMm8l0kpkeCwI2Ss1gmtl7vWZs9VqmLOg1Qm4Rd9RuTkZGi6Bn+txVgxwRkhPbP4qV8jKxSExXMicNrN7IWAxKxsjIUqySkUlJzYtJ4e2oIaI2NqFlk/+QXq1Z9Qbt0GKRw6BbgQZUayWqPwMTtiECB6YQAIcQFGIfGCT4DaALFGKMDSHFmEQkijyHDGqonSKLI4NhdIy7h96giobo5zWTQu1gFx9/iUoCC/QtfU4P6Gv6gn6k3/5YKy1q5LPsorVLLY+t+QcX17/+UxWgVbDzU/XXmRW4cKOYVeDscRHJT+GU+tn9hwfrN9cW0iv0Cf2E8z+m+/QVniCcfXGervK1R5A/QPf36z4MNoxO91rHWF1qLPeqp5iDS3AZmnjf12EZ7sAK9MHRlrR7mqNt6y/1d/p7/UNJ1bVKcwF+Wfrn71bx0U4=</latexit>h⇡(k⇡ )|q̄

µ

+

2



b|B(pB )i = f (q ) (pB + k⇡ )

m2B

µ

2
m2⇡ µ
m2⇡ µ
0 2 mB
q + f (q )
q ,
q2
q2

PRD64 (2001),114505
S.Aoki et al.
JLQCD collab.

3.0

+

2

f (q )

2.0

1.0
0

2

f (q )
0.0

20

15
2

25

30

2

q (GeV )20

B-Bbar mixing from NRQCD, Nf=0
Beta=5.7, 5.9, 6.0, 6.1
O(a)-improved Wilson + NRQCC

1.0

BB(mb)

Renormalization of ΔB=2 operator
S. Hashimoto, Ishikawa, Yamada et al.
PRD 62 (2000), 114502

0.8

0.6

0.4
0.0

0.2

0.4

0.6

0.8

1.0

1.2

-1

a (GeV )Good scaling behavior was observed21

Ginsparg-Wilson fermion

22

How to get physical results
and higher precision?
Test with Quenched QCD is done.
All we need to do : Nf=0 à Nf=2

Ø Problems:chiral extrapolation error
wrong operator mixing
23

Weak matrix elements for flavor physics
Uncertainty in chiral extrapolation
example: decay constantOperator mixing with wrong chirality
Ø Bag parameterfor
mixingWithout chiral symmetry, lattice operators can mix
with operators having wrong chirality(huge corrections due to chiral enhancement)
BK from Wilson fermion
S. Aoki et a (1999)
PRD60:114504
quark mass was far from the chiral regime
chiral extrapolation gives 10-15% error
We must reporduce chiral log from lattice

is uncertain due to huge radiative corrections.
24Overlap/Domain-wall fermion
Study of 2dim QED(Schwinger model) with Fukaya-san.
Acknowledgement: Matsufuru-san helped us providing all thebasic codes of full QED.

Ø Test of chiral behavior, condensation
Ø Physics of theta vacuum25Schwinger model case (fixed topology simulation)
There is indeed a nonzero constant forThis constant gives topologicalsusceptibility consisitent withdirect measurement26

Ginsparg-Wilson fermion (JLQCD)
Ø Ginsparg-Wilson relation
Ginsparg and Wilson, Phys.Rev.D 25(1982) 2649.

Exact chiral symmetry on the lattice (index theorem)
Hasenfratz, Laliena and Niedermayer, Phys.Lett. B427(1998) 125
Luscher, Phys.Lett.B428(1998)342.

Ø Overlap fermion ( explicit construction )27

Pioneering studies on Dynamical Overlap fermion
Ø

Fodor, Katz, Szabo JHEP 08(2004), 003;
Nucl. Phys. B(Proc.Suppl.)140(2005)704; JHEP 01 (2006) 049.

Ø

Cundy, Krieg, Arnold, Frommer, Lippert, Schilling
Nucl.Phys.B(Proc.Suppl.)140(2005)841; arXiv: hep-lat/0502007;
Cundy, Krieg, Lippert PoS(LAT2005)107;

Ø

DeGrand, Shafer Phys. Rev. D71(2005)034507;
Phys.Rev.D72(2005)054503; PoS(LAT2005)140.

Dirac operator becomes singular at the zeros ofConclusion: Large scale simulation is hopeless.
28

Det(Hw) term ( Fukaya, Vranas)
Fukaya et al. hep-lat/0607020
Introduce negative heavy mass wilson fermion as a UV
regulator field, whose mass is exactly the same as thatappears in Dov. Infrared physics is unchanged.

This term should kill the breakdown of locality
topology change, and blow-up of numericalcost simultaneouly.
29

JLQCD GW project
KEK BlueGene (10 racks, 57.3 TFlops)
Ø Started on March 1, 2006
Ø 1rack=1024 nodes=2048CPUJLQCD collaboration:
KEK: S. Hashimoto, T. Kaneko,
H. Matsufuru, J. Noaki, M. Okamoto, N. Yamada
RIKEN: H.Fukaya
YITP: T. Onogi
Tsukuba: S. Aoki, K.Kanaya,A.Ukawa,T.Yoshie
Hiroshima: K.-I.Ishikawa, M.Okawa
Taiwan National U: T.W.Chiu collaborators (TWQCD)
30

Test of chiral behaviro
Chiral log visible

Chiral behavior
consistent with chiral log
is reproduced.

Quark mass dependence of31Topological susceptibility
Ø Measure the topological susceptibility

check thermal equilibrium in topology
l Useful for estimate the finite size effectslDefinitionsl

Giusti, Rossi, Testa

Phys.Lett.B587(2004)157

disconnected looplLuescher, Phys.Lett.B(2004)296
n-point function without div.

l

Asymptotic value Fukaya, T.O. Phys.Rev.D70(2004)054508
32

(1) Ward-Takahashi identity(2) Cluster Property

Q distribution(1)&(2)Topological suscpetibility
can be measured indirectlyfrom asymptopic values of
Pseudoscalar 2-pt ftn

33

Reconstruction of θ vacuum physics
from fixed topology
General formula based on field theory
Aoki, Fukaya, Hashimoto, T.O. , PRD76, 054508 (2007).

ØØØ

Systematic formula in 1/V expansion
Topological susceptibility is the key quantity
CP-odd quantities can also be obtained
ex. Nucleon EDM
34

Nucleon sigma term from lattice QCDWith H. Ohki-san and JLQCD collaboration
S. AokiarXiv:0806.4744[hep-lat] (PRD78:054502,2008)

Topology

T. Onogi

35

Nucleon sigma term ßà Higgs coupling of
nucleonDown-type Higgs Yukawa coupling

Chiral perturbation theory (ChPT)+
scattering exp. Data(connected+ disconnected)

Quark model, Lattice(disconnected)

Trace anomaly formula(disconnected)

Strange quark contribution can be dominant but with large uncertainty.
(Huge enhancement in the strange quark mass region?)Naïve question: Can disconnected contribution really so big?
Topology

T. Onogi

36Basic Methods
nucleon mass spectrum
Feynman - Hellman theorem

Moreover, partial derivatives with respect to the valence and sea quark
masses give contributions from ‘connected’ and ‘disconnected’
diagrams.

Topology

T. Onogi37

Connected and disconnected contributions at
Semi quenchedestimate of y

Smaller than older resultsè Why?

Topology

T. Onogi

38

Additive mass shift and sigma term with
Wilson fermion

Wilson fermion has an additive mass shift through tadpole diagram
which mimics mass operator insertion to the valence quark due tothe lack of chiral symmetry

Topology

Normal disconnected
contribution

Operator mixing induced
connected contribution(long distance effect)(short distance effect)

T. Onogi

39

2009- (Osaka U)
Faculty: K. Higashijima, Y. Hosotani, T. Kubota, K. Hashimoto, S.
Kanemura, T. Nishioka, N. Haba, T. Minoru, K. Oda, S. Yamaguchi, H.
Fukaya, N. Iizuka, K. Yagyu, R. Sato, ..
Students: A. Tomiya, K. Kikuchi, M. Hirotsu, A. Kagimura, N. Kawai, Y.
Matsuki, M. Mori, ..
P.D. : Shintani, H. Ohki, S.W. Kim, ..

40

New topics
ØDomain-wall fermion and anomaly
H. Fukaya, S. Yamaguchi, Y. Matsuki, N. Kawai, …

ØGradient flow
K. Kikuchi, S. Aoki, S. Yokoyama, ..

ØGeneral relativity
S. Aoki, Y. Shuichi, T. Yamaoka, è Aoki-san’s talk41

APS index theorem
Ind(D) =π(iD 3D )1+
32 2

sgn( ): eigenvalue of iD3D

2

<latexit sha1_base64="TsKypMsO0F3ivkZJLV1nxpN2Sk4=">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</latexit>[Atiyah, Patodi, Singer 75]

d 4x µtr(F µ F)“eta invariant”
reg(iD3D ) =x4 = 0Can be reformulated with Domain-wall fermion
H. Fukaya, T.O., S. Yamaguchi PRD96 (2017) no.12, 125005 [arXiv:1710.03379]
H. Fukaya, M. Furuta, S. Matsuo, T. O. S. Yamaguchi, M. Yamashita, [arXiv.1910.01987]
H. Fukaya, N. Kawai, Y. Matsuki, M. Mori, K. Nakayama, T.O. S. Yamaguchi [arXiv:1910.09675]

2022/9/26

42

Flow method
Flow equation for FT in D-dim and induced geometry in (D+1)-dim
Ø Using the UV-finite property, one can define induced metric asgµ⌫ (X) = h@µ (X)@⌫ (X)i<latexit sha1_base64="r1l/qVm2Ffy5K3rpADkKR/ZEhH0=">AAACsXichVFNa9RAGH4aq9atulu9CL2ELpV6WSaloBQKRS+Cl3643YWmpJM4jcNOJiGZLLRL/4B/wIMnBRHx1N/QS/+ASG/1WHpsoZcefJMNVFvUN2Tmmed9n3eemfETJTPD2OGIdWP05q3bY3dq43fv3a83Jh6sZXGeBqIdxCpOuz7PhJJatI00SnSTVPDIV6Lj914U+U5fpJmM9WuznYiNiIdabsmAG6K8xqvQG7hR7up8d6b7xF6wXcV1qITtJjw1kiuvyGYyjHiRv2T172xaarxGk7VYGfZ14FSgiSqW4sYXuHiDGAFyRBDQMIQVODL61uGAISFuAwPiUkKyzAvsokbanKoEVXBiezSGtFqvWE3romdWqgPaRdGfktLGNPvOvrJTdsC+sWN28ddeg7JH4WWbZn+oFYlXf/do9fy/qohmg7eXqn96NtjCs9KrJO9JyRSnCIb6/s7709X5lenBY/aJnZD/j+yQ7dMJdP8s+LwsVj6gRg/gXL3u66Az23LmWo6zPNdcfF69xRgmMYUZuvCnWMRLLKFNG+/hB47w05q1utam5Q9LrZFK8xB/hNX7BbNypy0=</latexit>

S. Aoki, T.O. , K. Kikuchi

Ø For CFT, induced metric gives AdS space!
Massless free scalar : [Aoki-TO-Kikuchi] ‘15
General CFT : [Aoki-Yokoyama] ‘17

è A constructive approach to AdS/CFT correspondence?

2022/9/26

43

Summary
PeopleRenormalization
Lepage-Mackenize
Higher-loop
Nonperturbative Renorm
SF-scheme
RI-MOM

Chiral limit
Chiral sym.
Ginsparg-Wilson
Hasenbusch trick
Domain-Decompositoin

Topology
ChangeGradient flow

Aoki-sanHashimoto-san
Matsufuru-san
Ishikawa-san
Yamada-san
Fukaya-san
Ohki-san…

Fixed toplogy

Precise test of SM44

I was fortunate to encounter very interesting developmentof lattice QCD .
I was also very fortunate to get along with great mentors,
excellent collaborators, wonderful colleagues, everywhere I
go at the right time.
Perhaps it was because we shared the same motivation forthe physics goal.
I was influenced and learned a lot from all of them.

Thank you all.
45