Physics is all about the dynamics,
on and off the lattice

Shoji Hashimoto (KEK, SOKENDAI)

Hiroshima, 1990~
Nov 1991, Hiroshima• Tetsuya joined the particle theory lab of Hiroshima
University, as the youngest faculty member.
• The head of the lab was Taizo Muta, who had an
intention to study non-perturbative dynamics ofthe EW symmetry breaking: techni-color, topcondensation, NJL model, …
• Tetsuya was willing to learn new subjects (for him)with us students. In fact, it was an ideal learning
experiences… Memories with (always) lost mugs.

Lattice gauge theory
Volume 266, number 1,2• Muta gave me a theme: simulations of
Lattice Gauge Theory (LGT). For me, LGT
was a kind of toy to play with using
numerical simulations. I learned how toprogram.
• Tetsuya was again willing to study together,and gave me a homework: plaquette →cube. I didn’t even know what the model
was representing. Memory remains with
Tetsuya’s herpes.

PHY

2. Compact lattice gauge theory of the antisymmetr
PHYSICS LETTERS B
tensor fieldPhysics Letters B 266 ( 1991 ) 107-111
North-Holland

Gauge theory ofWe
antisymmetrictensorformulatethefields
lattice gauge theory of the ansymmetric tensor field. The partition function is giv
by the following path integral:
Received 8 April 1991

T e t s u y a O n o g i a n d Shoji H a s h i m o t oDepartment of Physics, Hiroshima University, 1-1-89 Senda-Machi, Nakaku, Hiroshima 730, Japan

Z= ~ ~ d0(p)exp(-Sg),

The compact lattice gauge theory of the second-rank antisymmetric tensor field in four dimensions is studied. We calculate thehypersurface tension by Monte Carlo simulation on a 104 lattice, It is found that the tension does not go to zero for a finite valueof the coupling constant. This suggests that the theory has the confinement phase only. This result is consistent with the result
obtained from the dilute gas approximation of the instanton. Some phenomenological implications on the string model are discussed also.

(

where

I. Introduction
It is well known that the compactifications on the
Calabi-Yau manifolds in superstring theories lead toan enormous number of four-dimensional models
which possess N = 1 space-time supersymmetry. Unfortunately, due to the lack of understanding the nonperturbative aspects o f string theory we have not yet
succeeded in making low energy predictions fromthose models. So far one is only able to treat low energy effective field theories to do some phenomenological analyses. Although such analyses might givenew predictions for a certain class o f models, theywould not clarify the novel model-independent features which are low energy remnants peculiar to string
theory.

The tree-level mass scale o f the coupling of the superstring axion is the Planck scale. If the tree-level lagrangian for the superstring axion remains to be correct down to the low energy scale, there will be aserious contradiction to the cosmological constraints.
Since the model independent axion comes from thesecond-rank antisymmetric tensor gauge field, thegauge invariance guarantees the axion to be masslessat least in the perturbative sense. One would thenimagine that some non-perturbative effect may
change the low energy behavior of the antisymmetrictensor field drastically. In fact this possibility is thepoint we consider in this paper. In 1982, Orland [ 4]studied compact antisymmetric tensor gauge theory
using the instanton approximation and found that thesystem is in a confinement phase for any value of the

Sg= ~ p[l -cos(o~ O(p) ) ] .

(Here c and p denote unit cubes and plaquettes. T
lattice action has the invariance under the followi
gauge transformation:

O..( i )--,Ou.( i) +A~(i+/~)Strong coupling constant
VOLUME 74, NUMBERPH YS ICAL REVIEW

1

0.20

VOLUME 74, NUMBER
VOLUME 74, NUMBER

0.20
PH YS ICAL REVIEW

1

S. Aoki, M. Fukugita,* S. Hashimoto,

N. Ishizuka,'

H. Mino,

0. 14

0. 12

I:101

M. Okawa,

T. Onogi,

~

0. 18

[11]

12

~/a (GeV)

10
f. I12](a) 1/a„—(p)(b

f. [13]

14

~

~

I

I

s

I

1

~

e

a

I

s

i

~

I

a

~

~

I

~

I

0

~

~

0.1

4

6

10 1. Comparison of nFIG.
f. [13]of charmonium

0 1A\/
0 2~

f. I12]0. 18
employing the 15-1P mass splitting
estimated
for two-flavor full QCD.
0. 16

14

12

0. 12

that sea quark effects of a magnitude
expected from renormalizationgroup
0. 10 constant measured in current full QCD
considerations are clearly visible in the strong coupling(~) 2
6 modified 8 minimal 10
simulations.
Building on this result an estimate of nMs(Mz)(MS 4denotes thesubtraction scheme) is made employing the charmonium IS 1P ma-ss splitting calculated~/aon (GeV)
full QCD
0.
20
configurations generated with two flavors of dynamical Kogut-Susskind quarks to fix the scale.

it
A distinctive feature of QCD that differentiatesfrom phenomenologicalquark models of hadrons is the

10

8

0
0.1

0. 14

We demonstrate

12.38.Gc

6

2 JANUARY 1995

0. 16

and A. Ukawa''Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan'Yukavva Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan
0. 14
Department of Physics, Hiroshima University, Higashi Hir-oshima 724, Japan'Faculty of Engineering, Yamanashi University, Kofu 400, Japan'National Laboratory for High Energy Physics
0.
12 (KEK), Ibaraki 305, Japan(Received 25 July 1994)

PACS numbers:

4

0.
1020

2 JANUARY 1995

of Sea Quark Effects in the Strong Coupling Constant in Lattice QCD

'

[11]

2

LETTERS(a

I:101

0. 10 LETTERSPH YS ICAL REVIEW

1

0.18

Manifestation

10

0.18

• Tetsuya joined a group of LQCD, later called
JLQCD. The first subject was a determination of
αs. It was the first attempt to do it including theeffect of dynamical fermion.

LE

8x/a (GeV)(Xf)~ (GeV)

10

12

10

14

—(m/a) estimated via (1) for two-

flavor full QCD (filled symbols) and quenched QCD (open—(~)
scaleWi~fixed by the p meson mass.
with1ruthe
symbols) (b)αs runs differently, indeed!

statesare extrapolated

linearly in the sea quark mass to VEq = 0.

0.1

FIG. 2. Ev

QCD. Band(&f)
n(p, ).
(a) Fukugitaal. (Nf —0
5.6) [12] and—heavy quarks on the lattice• Tetsuya left us for a visit at Fermilab in 1993-94 (?).
His study there was about applications of LQCD
to pheno.
• My Ph.D thesis was also about heavy quark
effective theory (or called NRQCD) on the lattice.
• A lot of “chat” on UNIX between US and Japan
about perturbative matching calculation. Throughsuch discussions, I learned that “lattice” is indeed akind of (lovely!) QFT.

physical momenta of the three points shown. We find thatin
the continuum extrapolation in a2 is also mild, even though(3) Aa potentially significant discretization effect due to theeheavy quark mass of the form ðamQ Þ2 is expected. This isfapartly because the renormalization factor discussed in theCprevious section absorbs the bulk of the discretization
effects. The global fit forms of Eqs. (20) and (21) assume
C. discretization effect PHYSICAL
DIFFERENTIAL DECAY RATE OF B→ ! l , SEMILEPTONIC DECAY
that. .theapplies as REVIEW
an overallfactor
D 64
114505(4) A
22
ð1 þ Ca2 ðΛQCD aÞ þ CðamQ Þ2 ðamQ Þ Þ, independent of lightfand light quark masses so that the extrapolation in v •k ! quark masses and energies v · pπ ¼ Eπ . This choice isatoward the soft pion limit becomes more stable, which is still justified because the dependence on each such parameter is(5) A
beyond the scope of this paper.
small as we saw above. In principle this allows the global fit
fto discriminate between the ðamQ Þ2 and ðΛQCD aÞ2 effects;
ð1
G. Pole dominancein practice, both terms in our fits return coefficientsinIn the soft pion limit, the heavy meson effective Lagrang- consistent with zero.
S
The
final
results
for
f
ðv·p
Þþf
ðv·p
Þ
and
f
ðv
·
p
Þ
1
π
2
π
2
π
ian predicts the B * pole dominance "2.11#, that is,(6) A
at the physical quark masses and in the continuum limit are
e
shown
in
Fig.
11
as
a
function
of
v
·
p
¼
E
.
The
bands
!
f B * m B * v •k !
ππb. "6.15# represent the one standard deviation regions with only thelim f 2 " v •k ! # !g2f!
v •k ! "$ Bv •k ! →0Cstatistical uncertainties included. The region that our latticedata cover is from 0.225 GeV to 0.975 GeV. The results
C
%M
#M
is
much
smallerSincethehyperfinesplitting$
B
B*
B
PHYSICAL REVIEW D, VOLUME 64, 114505
outside of this region are obtained from the fit functions in(7) A
than the momentum transfer v •k ! we measure, we can apEqs. (20) and (21). In the soft pion limit, the form factoroDifferential decay rate of B\ ! l " semileptonic
decay withlattice nonrelativistic
proximateits functional
form by a QCD
constant in our data reFIG. 9. Same as Fig. 7, but with estimated systematic errors.
f 2 ðv · pπ Þ rapidly goes to zero as a resultofthe pole
term
20
years
after
fi
gion.
Our
data
supporttheconstant
behavior
and
give
1
2
3
3
1,4
1,4
3
1,4
S. Aoki, M. Fukugita, S. Hashimoto, K.-I. Ishikawa, N. Ishizuka, Y. Iwasaki, K. Kanaya, T. Kaneko,
We plot
3
1,4
3
1,4
!
resultofthechiral
limit
and
that
of
the
lightest
+
.
This
gives
am
/2f
)!0.35(18),which
reducestog
Y. Kuramashi,3 M. Okawa,3 T. Onogi,5,6 S. Tominaga,4 g(
N. fTsutsui,A.
Ukawa,N.
Yamada,andT.
YoshieBB!
**
1
represenInstitute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan
(
•k
)"
f
(
•k
)
and
1%
for
f
(
•k
).
10%
for
f
!0.30(16).
This
agreeswiththephenomenological
value
v
v
v
2
1
!
2
!
2
!
Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan3
The total error is estimated by adding these errors in 26.40 G* →D !Japan
decay 0.27"6# &29', and also withextracted
High Energy Accelerator Research Organization(KEK),from
TsukubaD305-0801,
4
Center for Computational Physics, University
ofrecent
Tsukuba, lattice
Tsukuba 305-8571,
Japan g!0.42(4)(8) &30', which isquadrature together with the statistical error. In Fig. 9 the results a
thecalculation
5
Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan*
6
form factors f 1 ( v •k ! )" f 2 ( v •k ! ) and f 2 ( v •k ! ) are plotted lighter gfor Kyoto
the 606-8502,
static heavy
Yukawa Institute for Theoretical Physics, obtained
Kyoto University,
Japan† quark. The agreement suggests!Received 2 July 2001; published
7 November
2001"
with the estimated systematic uncertainties. Numerical re- which isthat the1/M correction
is small for the form factors.
band diWe present a lattice QCD calculation of B→ # l $ semileptonic decay form factors in the small pion recoil
sults
are
listed
in
Table
XIII.
momentum region. The calculation is performed on a quenched 163 !48 lattice at % "5.9 with the nonrelativeffects t
Systematic
errorsistic QCD action including the full 1/M terms. The form factors f 1 ( v •k # ) and f 2 ( v •kH.
)
defined
in
the
heavy
#
2–7 add
quark effective theory for which the heavy quark scaling is manifest are adopted, and we find that the 1/M
VII. COMPARISON WITH OTHER CALCULATIONS
correction to the scaling is small for the B meson. The dependenceof thediscuss
form factors
on the lightsources
quark mass of systematic errors andWe nowpossibleWe aland on the recoil energy is found to be mild, and we use a global fit of the form factors at various quark masses
A. f 1 „ v "k ! … and f 2 „ v "k ! …
their estimates. Since the statistical error, the discretizationof the lis
Colquhounetal(2022)
and recoil energies to obtain model independent results for the physical differential decay rate. We find that the22), theandperturbative
error of O( ( s ), and theB * pole contribution dominates the form factor f # (q 2 )error
for smallof
pionO(arecoil energy,
obtain the differential
El-Khadra et al. calculated the form factors at the b quark both for222
decay rate integrated over the kinematic region q $18 GeV to be ! V ub ! !(1.18%0.37%0.08
chiral extrapolation error are large, we consider only thesemass using a nonrelativistic interpretation of the relativistic have da%0.31) psec&1 , where the first error is statistical, the second is that from perturbative calculation, and the third
dominantsources Weof alsoerrorsneglect other subleading eris the systematic error from the finite lattice spacing andthe chiral extrapolation.
discussand
the systemlattice action &11'. A comparison is made with our results for above, i.
2022atic errors in the soft pion limit for f (q max ) in the presentrors simulation.
such as O„( /(aM )…, O( ( a)), O( ( )/M ),can therSemileptonic!

• After a bunch of works with younger students in
Hiroshima (I was already at KEK), Tetsuya joined
the JLQCD collaboration for a calculation of
B→πlν form factors.

I. INTRODUCTION
Chiral symmetry and topology are tightly related witheach other in the gauge field theory through the quantum
correction. Namely, the axial anomaly appears at the oneloop level, and its integral over space-time leads to the
topological charge of the background gauge field. In principle, one should be able to analyze the implication of thisrelation for physical observables, such as the neutron electric dipole moment, using the lattice gauge theory, which
Developmentof“chiral”
latticefermions(domainprovides a rigorous formulation of the non-Abelian gaugewall, overlap,
opened new
directions.
theoriesetc.,
even 1998~)
in the nonperturbative
regime.
Such study isvery difficult with the Wilson-type Dirac operator, since
the chiral symmetry is explicitly violated on the lattice.
• ConnectiontoχPT,
simplerrenormalization,
The overlap-Dirac operator [1,2]denominator as in (3) is not introduced for this case, but itof jaHW j is bounded from below by a positive (nonzero)is known that the rectangular term has an effect to suppress
constant [5]. This condition is proved to be satisfied if thethedislocations.
gauge field configuration is smooth and each plaquette is
The
extra
fermionterm
SiswrittenasE
close enough to one;X
SE !k1 ' !P#x$D
W #m<0$$ #x$
for all x; $"; #%:(3)"# $x%kx
Xthe plaquette variable at x on the "-#Here, P"# $x%% is%y #x$&DW #m0 $ % i"&5 '3 '%#x$;(4)plane, and k ( ( (x k denotes the norm of the operator. In
the four-dimensional case, the parameter $ ’ 1=20:49 iswheredenotesof extra condition
heavy Wilson
a sufficient(buttwo
notflavors
a necessary)[6]. fermion
This iswitha negative
mass m0 . The
second term is a pseudofercalled
the ‘‘admissibility’’bound.
Suppressor
evenprohibitthe
topologychangemion
term
introduced
in
order
to
cancelunwanted
effects
One can construct a gauge action, which generates
ofgaugethe Wilson
fermion respecting
especially thein theultraviolet
region,configurations
condition(3). For
inwhichleads
to a large
shift theof the! valuestance,Lüscherproposedaction[7] to be used in theoneway:
simulation
discussed$x%=3
later in Sec. VII. Because of an! Pas 1'ReTrP"# y% mass; 'when
1'ReTrP
additional
term
%
i"&
%,
which
is
twistedinthe"# $x%=3<$;53$x%=3%=$
1'$1'ReTrP
"#
SG !
;
P
flavor spaceby '3 , the extraotherwise
Wilson fermion works to
1
suppress the near-zero modes of HW #m0 $ as expected.(4)In
fact,
action SE generates the suppression factor
andthefinally:which has the same continuum
limit
as
the
standard
Wilson!
"2
H#m$
W limit0gauge action does.det
In fact, the$ ! 1 corresponds(5)to2action.
2 Unfortunately, thethe standard Wilson Hgauge
#m
$
%
"
W
0
bound $ ’ 1=20:49 is too tight to produce gauge field
inensembles
the partition
function.
The
twistedmass"
controls
thecorresponding to the lattice spacing aroundrange
of thesuppressedby thenu0.1 fm;for near-zeropractical eigenvalues
purposes, onemust choosemuchmerator.
The eigenvalues
whose question
absoluteis,valueislowerlargervaluesof$.
Aninterestingthen,whether
Formed
the
basis
of
our
next
project:than"arestrongly
suppressed,
while
the
other
higherthe action can keep the good properties for $ significantly
modes
less
affected.
The
limit ofone" expects! 1 corresponds
dynamical
overlap
fermionlarger arethan
1=20:49.
To be
explicit,that (i) the
totopology
the case where
pseudofermion
term is switched off.
changetheduring
the molecular-dynamics-type
When" ! 0,
cancellation
is exact,
and thereofis thenosimulationis thesuppressed,and (ii)
the appearance

New direction: chiral, topology•

and of course, topology.

1
D ! "1 # !5 sgn$aHW %&;
a!

a
;
a! !
1#s

(1)

• Yet, its simulation was impractical, due to the cost
realizes the exact chiral symmetry at finite lattice spacing aand the topology
change.
[3] satisfying theGinsparg-Wilson relation [4]!5 D # D!5 ! aD!5 D:(2)• Tetsuya (then at YITP) together with his student
is constructedfrom theWilson-Dirac
operatoraDW with
HidenoriIt Fukaya,
startedlooking
at theproblemthe Wilson parameter r ! 1; the Hermitian Wilson-Diracfrom theoperator
viewpoint
of!theorist,notalattice
aHW !
$aD
'
1
'
s%
entersas an argument
5Wof the sign function sgn$x%. The parameter s in (1) is a fixedpractitioner.
number in the region jsj < 1.
Since the definition (1) contains a nonsmooth function,
the locality of the Dirac operator could be lost when there
are near-zero eigenvalues of jaHW j. This is consistent with

sio
es
ga
pa
in
lat
th
by
th
is

ex
clu
lim

32

Th

I. INTRODUCTION
In the construction of the lattice chiral fermions, theconventional Wilson-Dirac operator DW still plays a crucial role. In the domain-wall fermion [1–3] the lattice
Dirac operator is nothing but the Wilson-Dirac operator
in four dimensions, but the fermion field has interactions
also with its fifth dimensional neighbors. The overlapDirac operator [4] D contains a matrix sign function ofchiralsymmetryoperator
realizedHWby# #5 DW as
the Exact
Hermitian
Wilson-Diracdislocation). An analytic example
of! such#4$a gauge#3$config
"
#3$
E V
4E V
5#E V$2
hhsii !exact"%given1 mode""[6
ration and its associated
zero
is
in
#2$
3=2
#2$
2
#2$
3
2#E V$
3#E V$
8
4#E V$
Because such a zero mode is localized in space-time, t#3$ V
5E
number of the near-zero modes increases
as
the
latti
"3=2
hhs3 ii ! "
%
O#V
#2$
3=2
2#E
V$
volume V is increased. In other words, the spectral densi
we finally!!"W " ofHW obtain
is nonzero at "W # 0, which is true at a!
#4$ localization#3$
2"
finite value of gauge coupling
[7].
The
pro
1
E
V
5#E
V$
#2$
#4$ #
GQ ! G#!c $ % G #!c $ #2$ 1 "
%
%
G
#2$
2E V
2#E#2$ V$2 4#E
V$3 studi
erty of the near-zero modes
has
recently
been
# #3$
!
#4$
#3$
2"
#5$
E
V
4E
V
5#E
V$
E
V
#1$
extensively [8–10],
and
it
is
found
that
they
are
expone" G #!c $
%
1"
%
#2$
2
#2$
2
#2$
3
2#E V$
3#E V$
8#E#2$ V
4#E V$
tially localized unless one enters the Aoki #n$phase, where t— “No,”Theit’s
merelyfinitevolumeeffect.
above
expansion a
is valid
for any!c as long as
G #!c $=G#!c $ ! O#1$ aflavor-parity
symmetry
spontaneously[11
Dependingon the size of is!c , wecan further expand thebroken
above formula.
O#1$],
we haveof the exponential falloff is of the ord
Since theradius!
"modof lattice spacinga, attheeffectofthenear-zero
2#4$
Finite volume QCD
fixed topologicalcharge
!c
1
E V
#1$
#2$
#2$
#
GQ ! G#0$ % G #0$!c % G #0$ % G #0$ #2$ 1 "%Gdisappears
in thecontinuum
limit,
andisV$a2 latti
Sinya Aoki,
Hidenori
Fukaya, Shoji Hashimoto,Onogi
2 and Tetsuya
2Etherefore
2#E#2$
V
$
# #3$
!
"
#4$
#5$
#3$
E
V
4E
V
E
V
E
artifact.
#2$"
G" G#1$ #0$
#0$!c
1
"
%
#2$
2
#2$
2
#2$
3
#2$
2#E
3#E
8#E
2#E
V$
V$
V$
The effect of the near-zeromodes
appears
as
a
sm
#
$
2
1
Q
c4
1
#2$
#4$
#1$
1 "symmetry" 2#0$ chiral#0$ domain-w%
G#0$! G#0$ % G of
%in
G the
residual breaking
2
2
2"t V
"
"t V 2"t V
8"t V
formulation [12]. Namely, the four-dimensional fermi
mode receives additive mass renormalization mres whthe lattice
inGthedimension
L5 is finiteThe[13,14
formulIf G extent
is CP even,is an fifth
even functionof !, so that#m
$ but theeffectdue to2is finite,
The problem
is
not
just
that
suppre
res
1
Q
c4
even
#2$
is o
" that
GQ ! G#0$ % G #0$ one region1 "of phase space
are separated by correctionsome2
I. INTRODUCTION
2""bysion is only by 1=L5 rather
expectpotentialwall, becausethe2"‘‘kineticof"theas
mo-compensatt Vthant V exp!(cLt V energy’’ 5islecular dynamics system may not be enough to go throughQuantum chromodynamics (QCD) in four space-time
ChPT
it is ev
wall. Thissituation happens for
QCD inthe
four overl
dimensions
allowstopologically
nontrivial gauge
configu-1 the potential
#4$
"3
for
the
extendednonzeromodes[10,15].
For#0$ 2 2dimensions% O#Vbecause$; of the nontrivial topological(28) sectors. In
rations labeled by a winding number%orGa topologicalmass m and th
8" Vcharge Q. The path integral to define the partition functiont the continuum theory the potential barrier is infinite andfermion,theresidual
mass
can
be
made
arbitrarily
small
the gauge
configurations
in different
topological sectorsof QCD
includes an integral
overconfigurations withofflavorsNf
while,
if
G
is
CP
odd,
we
have
cannot be reached by a single stream of the continuous
arbitrary Q. In order to ensure the cluster decomposition
whene! evolution."On
the lattice, andthe potential
barrier is pressedfinitethem
propertyof physical observables,
the weight
among
differprojectingoutthenear-zeromode
treating
iQthe ! (orderc4of inverse lattice spacing
iQ1=a) and the probability
ent topological sectors must be e , which defines

Overlap fermion & topology
1
D # $1 % #5 sgn!aHW "&;
a!

a
:
a! '
1%s

(1)

An •importantdifference
from
the
usual
Wilsonfermionis
It has a discontinuity, which poses athat the mass term is given with a negative value m0 #problem of locality.
(!1 % s"=a of the order of the inverse lattice spacing 1=a.
•Itoccursonthebackground
gauge
field
In the limit of vanishing gauge coupling, the parameter sforbetween
which thechargeQ isthenotsingle
must be(1 topologicaland 1 in orderto obtain
Chiralsymmetrytoflavor well-defined.
massless fermion;
the signfunctionisisrelated
well definedsince there
is a lower
boundtheorem
in the eigenvalue
topology
as index
dictates.spectrum ofjHW•j. Can
In thebe
presenceof whenthe gaugeinteraction,
HW could
avoidedQ doesn’tchange.
develop zero or near-zero eigenvalues, which makes theAny problem?
the matrix sign function singular. The near-zero mode does
not exist for sufficiently smooth background gauge fields
fU$ !x"g satisfying a condition jj1 ( P$% !x"jj < & for all

PHYSICAL REVIEW D 76, 054508 (2007)

1,2

3

4,5

6

1

Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Ibaraki Japan
2
Riken BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
3
Theoretical Physics Laboratory, RIKEN, Wako 351-0198, Japan
4
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
5
School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
6
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan(Received 5 July 2007; published 19 September 2007)
In finite volume the partition function of QCD with a given ! is a sum of different topological sectors
with a weight primarily determined by the topological susceptibility. If a physical observable is evaluatedonly in a fixed topological sector, the result deviates from the true expectation value by an amountproportional to the inverse space-time volume 1=V. Using the saddle point expansion, we derive formulas
to express the correction due to the fixed topological charge in terms of a 1=V expansion. Applying thisformula, we propose a class of methods to determine the topological susceptibility in QCD from various
correlation functions calculated in a fixed topological sector.

DOI: 10.1103/PhysRevD.76.054508

odd i!Q#1$

PACS numbers: 12.38.Gc, 11.15.Ha#3$

it becomes si

A lot of applications
0.005

Dirac eigenvalue spectrum
0.005
0.002

Cη'(t) = − N1 Σ X

0.004
0.001

+ΣX
x

0

πρQ(λ)xJLQCD&TWQCD (2007)

x

0.002

connected
η'

0

0.01

0.02

0.002
+ cosh of the
In order to preserve the exact chiral symmetry,
which is essential for the Adefinition

"

"

(4)

5 singlet pseudo-scalar
10
15density P (x) ≡al suggestion is in [11]), where P (x) is the flavor

f

00

0.01
0.005

0.02
0.01

0.03
λ

0.04

0.05

0.03
λ

0.04

0.05

0.06

0

0.01

0.02

0.03
λ

0.04

0.05

0.

πρQ(λ)

0.004

25

0
0 t
contains"
x #P (x)P (0)$#connected and a disconnected diagram as shown in Fig. 1. If we pick the disconnected piecefrom the Q=0 sector!

nd identify a “topological charge density”, it can be written as ρ1 (x) = mq tr[γ5 (Dc +mq )−1x,x ],!
FIG.
2: Time-correlation function of the flavor singlet η (circles) and its connected (triangle
here Dc is a chirally-symmetric (γ5 Dc + Dc γ5 = 0) nonlocal operator, relating to D(0) by

Dc = [1 − aD(0)/(2m0 )]−1 D(0) [14]. Integrated over the entire lattice volume, ρ1 (x) reduces
down) and disconnected (triangle up) contributions. Data at mq = 0.025 are shown.

0.006

lattice
ChPT LOChPT NLO

0.004
0.001
πρQ(λ)

020

f
f
f =1 ψ̄ (x)γ5 [1 − aD(0)/(2m0 )]ψ (x). The correlator Cη! (t) ≡

4

0.003

0.005
0.002

mq P 0 (0)), that were shown to give the same asymptotic constant as (2) [6] (the origi# Nf

0.06

by solid curves. For comparison, the prediction at the leading order of $ expansion (dashed curves) is also shown.

ith a large negative mass term −m0 . -0.002
In place of the topological charge density ρ(x) (and ρ(0)) in (2), we use mq P 0(x) (and

1

00

0.006
lattice
FIG. 1: Spectral density πρQ (λ) (left) and mode number NQ (λ) (right) of the Dirac operator at mud = 0.015, ms = 0
ChPT LO
Q = 0. The lattice result (given by histogram
compared with the ChPT formula (ChPT(left)
NLO or solid symbols (right)) is 0.005
0.004

ith mass mq . The kernel operator HW (−m0 ) is the conventional Wilson-Dirac operator

0

0.05

6

0.005

mq
0.000
+
m0 −
γ5 sgn [HW (−m0 )]
2
!

0.04

lattice
ChPT LOChPT NLOΣeffmq
D(mq ) = m0 +
2
!

0.01

8 1.5
2

0.005
0.004

0.003

Σeff

C(t)

A

opological charge, we employ the overlap-Dirac operator [12, 13]

0.03
λ

0.005

0.001 density πρQ (λ) (left) and mode number NQ (λ) (right)
2
FIG. 1: Spectralof the Dirac operator at mudlattice= 0.015, ms
ChPTChPTLO
Q = 0. The lattice result (given by histogram (left) or solid symbols (right)) is compared with the
form
ChPT
NLO
by solid curves.
0 For comparison, the prediction at the leading order0 of $ expansion (dashed curves) is also shown

0.004
IG. 1: A schematic diagram for the time-correlation function of the flavor singlet
operator P 0 (x).
disconnected

ach solid line denotes the valence quark propagator.

0.02

0

JLQCD (2009)

0
0.01

0
4

0.003

0

6

10

lattice
ChPT LOChPT NLO

0

x

f

0.003

8 1.5

NQ(λ)

πρQ(λ)

0.004

NQ(λ)

topological susceptibility

10

lattice
ChPT LOChPT NLO

0
0.002

0

0.01

0.02

0.03
λ

0.04

0.05

0.06

FIG. 2:0.001
Same as the left panel of Fig. 1, but calculated on
the $ regime lattice, at mud = 0.002 and ms = 0.080.

0.003

0.003

lattice(chiral limit)
4pt fi5pt fi6pt fi

0.002
0.001

0

0.02

0.04

0.06
mud

even near m=0!

0.08

lattice(chiral limit)
0.002
4pt fit
FIG. 3: Three parameter fit of Σeff to 5pt
thefitNf =6pt fit0.001

me 1 . For the sake of simplicity we represent the vacuum subtracted matrix¯q|N" − V !0|q̄q|0" by !N|q̄q|N" in what follows. (q represents a quark field: up

Nucleon!

), or strange (s).) Note that the sigma term is renormalization group invariant,ormalization factor cancels between the quark mass mq andthequark
scalarStrange
contentoperatorof the nucleon and dark matter searches
R. Young at Lattice 20120

100 200 300 400 500• Noquarks
additive
renormalization
withup and down
contributeto σπN bothasoverlap
valence and sea quarks, theFukugita et al. 1995
fermion.
It
enabledanother
directionofDong et al. 1996
k appears only as a sea quark contribution. As a measure of the strange quark
Lewis et al. 2002
studies: sigma terms, the strange quark
SESAM 1998
e nucleon, thecontent,y parameterin particular.
JLQCD 2008
JLQCD 2010
2!N|s̄s|N"y≡,(2)
QCDSF 2011¯
!N|ūu + dd|N"
Young & Thomas 2009
Toussaint & Freeman 2009
introduced.
characterizing
thehispurelysea quarkcontent of the nucleon,
Martin-Camalich et al. 2010
• Besides
Tetsuya,
together withstudent
HiroshiDürr et al. 2011
Ohki,
initiatedacomputation.
s a clear distinction from the quark model picture of hadrons, this parameter
QCDSF-UKQCD 2011
Freeman & Toussaint 2012ortant role to determine the detection rate of possible neutralino dark matter
Shanahan et al. 2012
• Followed by other groups, then. Many of
JLQCD 2012
ymmetric extensionof theproblematic,Standard Model[1, 2, 3, 4, 5, 6, 7]. Already with thethem were
though.
Ren et al. 2012
t dark matter search experiments one may probe a part of the MSSM model
Engelhardt 2012
0 100 200 300 400 500
pace, and new experiments such as XMASS and SuperCDM will be able toss HMeVL
Ú

Ú

Ú

‡

‡

‡

‡

Ê

Ê

Ê

Ê

Ê

Ê

Ê

Ê

Ê

Ê

FIG. 17:

The disconnected diagram contributing to the renormalization of flavor-singlet scalarsensitivity by 2–3 orders of magnitude. Therefore, a precise calculation of theoperator (cross). At higher orders, the quark loop and the quark line on the bottom may be

R. D. You

ation is recovered in the continuum limit, this term can be expanded in terms ofQµ as∆Jµν (Q) =Strong coupling constant

! "

δµν

m,n=1

!
ρ

#
2(m−1)
2n
Q̂2m
−
Q̂
Q̂
Q̂
Q
µ ν
ρ
µ
ν Fmn (Q̂).

(13)

Fmn denotes the scalar function depends on the index m, n and momentum Q. It$
the condition µ Q̂µ ∆Jµν (Q) = 0 coming from the WT identity for Jµa,cv . In this2
we confirmed that
this
term
is
numericallynegligible
in
the
range
(aQ)
< 1, and• Mostly done (at the time) through the QCD

ts contribution as potential
we discuss later.
(or the Wilson-loop), but one can use any

perturbative quantities, say the Adler function.

IT WITH THE PERTURBATIVE FORMULA•
startedat
Tetsuya’s
office.
(0)Discussion(1)ning Π (Q) = Π (Q) + Π (Q), the Operator Product Expansion (OPE) of VPF,
JJJ

Q) = ΠV (Q) + ΠA (Q), is given by

0.16

ΠV+A, ms=0.08

2

0.14 2

ΠV +A |OPE (Q , αs ) = c + C0 (Q , µ2 , αs )
2
m̄
(Q)
0.12
V +A
2
2
=0.015+ Cm (Q , µ ,mα
q s)
Q2
! 0.1
"mq q̄q#
V
+A
2
0.16
+
C
(Q , αs )q̄q4Q
q=u,d,smq=0.025

0.14"(αs /π)GG#−6+ CGG (Q , αs )
+
O(Q
)
4
Q
0.12
2

mq=0.035

2

eQ.

(14)
mq=0.050(V +A)
0.1
The perturbative expansion of the coefficients CX(X = 0, q̄q and GG) is
0.4
0.7
0.8
0.9
1 0.4
0.7
0.5
0.6
0.5
0.6

Shintaniet(aQ)al. (2010)(aQ)
up to two- to four-loop order in the continuum
renormalization
scheme, i.e. the MS
2depending on the terms.

2

0.8

0.9

0.16

ΠV+A, ms=0.10

1

Not competitive in precision…

I learned a lot of things from our decades-long collaborations.
• LGT, LQCD in particular, is not just a toy to play on computers. In
fact, it includes everything.
• Chiral symmetry is central to χSB, index theorem, topologicalsusceptibility… It works on the lattice! All new.
• Perturbative at short distances, even on the lattice, of course!
→ excited states, and inclusive processes
our (more recent) works

0.0
0.0

1.0

4.0

2.0

3.0

λ(µ = 6 GeV) [GeV]

4.0

5.0

mPV = 3

3.5

Short distances
F(λ;µ= 6 GeV)

3.0

Current correlator

2.5
2.0

O(αs5)

1.5

O(αs3)1.0

O(αs1)O(αs4)O(αs2)

Dirac spectrum

Borel transformation

β = 4.47

0.5

β = 4.35
β = 4.17

0.0
0.0
4.0

1.10

3.5

1.05

3.0

F(λ;µ= 6 GeV)

RV+A

1.15

1.00
0.95
ALEPH
Perturbation
β = 4.17, amq = 0.0070, ams = 0.0300
β = 4.35, amq = 0.0042, ams = 0.0180
β = 4.47, amq = 0.0030, ams = 0.0150

0.90
0.85
0.80

1.0

0

0.1

0.2

0.3

0.4 0.5|x| [fm]

2.0

3.0

λ(µ = 6 GeV) [GeV]

4.0

5.0

mPV → ∞

2.5
2.0

O(αs5)

1.5

O(αs3)1.0

O(αs1)O(αs4)O(αs2)

β = 4.47
0.5

β = 4.35
β = 4.17

0.6

0.7

0.8

0.9

0.0
0.0

1.0

2.0

3.0

λ(µ = 6 GeV) [GeV]

4.0

5.0

FIG. 5. Exponent of the Dirac spectral density F (λ) with the Pauli-Villars mass amPV = 1 (top),

Tomii etmal.
(2017)etFIG.
al.scale(2018)
Ishikawa
al.the(2021)
2 ) in the continuum
3 (middle), and amPV → ∞ (bottom).Nakayama
The renormalization
λ(µ)
is set toofµ ⇧(M=
˜ 6 GeV.
PV =14.
Comparison
limitetwithexperimental

G. 6. Same as Figure 5 but calculated on ensembles with different lattice cutoffs. Pion masses

Mπ ! 300 MeV.

1.15
1.10

The lattice data at β = 4.17 (crosses), 4.35 (triangle) and 4.47
(dots)
are shown. Grayed data
meson
contribution.
points are those with aλ > 0.5. Lines represent results from the perturbative expansion.
VI.

CONCLUSION AND OUTLOOK

Smeared spectrum
Correlation function:
Spectral function:
Smearing:
approx:by?

Bailas, SH, Ishikawa, PTEP 2020, 4, 043B07 (2000); arXiv:2001.11779

narrow
0.020

0.020
N = 12
N = 14
N = 16

0.010

0.010

0.005

0.005

0.000
0.0

0.000
0.0

0.5

1.0

1.5

!

2.0

2.5

N = 12
N = 14
N = 160.015

Ω̄¢(!)

0.015

Ω̄¢(!)

wide

when the smearing kernel is …

3.0

0.5

1.0

1.5

!

2.0

2.5

3.0

Inclusive processes
Diﬀeren'al decay rate:

Gambino, SH (2020)

Structure func'on (or hadronic tensor):

Total decay rate:

kinema'cal (phase-space) factor

Energy integral to be evaluated:=smeared spectrum!

Compton amplitude obtained on the la?ce:

Jµ†

J⌫

B

B

tsrc

t1

t2

tsnk

Approx :K(H)̂ = k0 + k1e

−Ĥ

+ k2e

−2Ĥ

+ ⋯ + kN e

−kN Ĥ

Gambino et al., 2203.11762

X̄ [GeV2 ]3

XV V kXV V ?
XAA?
XAAk

2

4

band from OPE

3
2

1

1
0
0.0

0
0.0

0.2

0.4

0.6

0.8

q2 [GeV2 ]

1.0

1.2

0.2

0.4

0.6

0.8

1.0

1.2

More challenges ahead• QCD seems to be the theory at all distances: short and long. But,
harder to treat in the middle. Can we go without the factorization,
which is the basis of PDF, say?
• Related: Can we approach the light-cone from Euclidean lattice?
• (smeared) Spectral function: more uses, including hadronic B decays.
We need Tetsuya’s insight on QFT, one more time!