よく知られている理論の不思議な性質 --大規模数値計算によるQCDの相境界近傍の解析
青木保道

理化学研究所 計算科学研究センター 連続系場の理論研究チーム

素粒子標準模型に登場する素粒子
QCD
最後のピース
2012年発見

電弱理論

素粒子標準模型に登場する素粒子とパラメタ• 結合定数 (QCD, 電弱)• Higgs (vev, mass)• フェルミオン質量• クォーク• レプトン

が
係
関
証
の
検
間
密
タ
精
メ
の
ラ
型
パ
模
の
準
ら
れ
→標
こ

• フェルミオン混合• 小林-益川 行列 (3x3)• レプトン混合

な
盾
矛

？
か
い

素粒子標準模型の動力学•

対称性とその破れ•

ゲージ対称性 :•電弱理論

ヒッグス機構➡ フェルミオン質量の起源•

カイラル対称性 : QCD•

南部 - ゴールドストン➡ ハドロンの質量の起源: 陽子、中性子含む•

目に見える宇宙の99%の質量はこちらから

QCD 標準模型中の強い相互作用の理論• Quantum Chromo Dynamics• カイラル対称性: 右巻きと左巻きのクォークの交換で理論が変わらない• 理論式を見る限り対称性がある。しかし、、、• 自発的に破れている• 真空でカイラル凝縮: クォークと反クォークのペア: “同じ方向”を向いている• ゼロ質量の粒子の存在: 南部ーゴールドストン粒子: パイオン (π)• π: 複合粒子: (ud), (du), (uu-dd) 粒子反粒子 ペア• 核力を説明する粒子

QCDにおける π と σ (シグマ)• 谷に添って動くモード:π• 等高線と垂直に動くモード: σ

ゼロ質量複合粒子 (ud), (du), (uu-dd)
ミステリアスな複合粒子 (uu+dd)• 複合粒子の質量 m2=曲率 : mπ=0, mσ>0
• 実世界では → 傾きが入る(mud>0)

有効ポテンシャル• 対称性が少し破れる• mπ = 135 MeV• mσ = 400〜550 MeV• QCDのスケール: π崩壊定数• fπ = 93 MeV

インプット• simulationで求まる: mπ/fπ, mσ/fπ ,,,• 他の複合粒子(陽子,中性子…)の質量はクォーク質量を決めれば全て決まる

QCDにおける π と σ (シグマ) <= 有効模型•

有効模型による記述: 結構良い ← かなり昔から分かっている••

カイラル対称性の自発的破れで定性的理解
有効模型のパラメタをいくつか決めると••しかし、それでは、標準模型の精密検証はできない•••

それなりに世界を記述できる

π や σ 等の複合粒子を作る元になっている理論の第一原
理計算必要
しかも高精度!

実は、標準模型中のヒッグス部分 → この有効模型とほぼ同じ•

背後にQCDの様な理論がある？

カイラル対称性の回復
•

温度

•

２つの異なる相: やぶれ(低温)←→回復(高温)••相転移の存在クォーク閉じ込め←→非閉じ込めTchiral symmetry
restoredTcchiral symmetry
broken

m

T>Tc

T<Tc

出典: 東大理学部
https://www.s.u-tokyo.ac.jp/ja/press/2004/02.html

カイラル対称性の回復
•

温度

•

２つの異なる相: やぶれ(低温)←→回復(高温)••クォーク閉じ込め←→非閉じ込めクォークの数 Nf

Nf>Nc

Nf<Nc

出典: 東大理学部
https://www.s.u-tokyo.ac.jp/ja/press/2004/02.html

Asymptotic non-freeNfAF

結合強度 vs エネルギーα(µ)

Conformal windowα(µ)

Nfcritµ

Walking Technicolor

α(µ)

µ

QCD-likeµカイラル対称性の回復
•

温度

•

２つの異なる相: やぶれ(低温)←→回復(高温)•クォーク閉じ込め←→非閉じ込め

T>Tc

T<Tc

出典: 東大理学部
https://www.s.u-tokyo.ac.jp/ja/press/2004/02.html•

クォークの数α(µ)µ

カイラル対称性の回復
•

温度

•

２つの異なる相: やぶれ(低温)←→回復(高温)••クォーク閉じ込め←→非閉じ込めクォークの数
•

Nf>Nc

Nf<Nc

出典: 東大理学部
https://www.s.u-tokyo.ac.jp/ja/press/2004/02.html

ヒッグスセクターの背景にある新法則の可能性

α(µ)

µ

理論解析: モンテカルロ法と格子ゲージ理論• 簡単な例: 円周率 π をモンテカルロ法で求める• 一様乱数(x, y) 0<x, y<1 を大量に生成• 円の中に入る確率は面積の比• 円の面積／正方形の面積=π／４• 面積←モンテカルロ法による数値積分• 基礎理論: 格子場のラグランジアンの経路積分

1

Latti

y

• Analysis of Quantum Field T
perturbative calculation.

0(x), Aµ(x), x ⌅ R4
1
x divergences
quantuma⇥(n)• 4次元時空の各点で場の変数の積分を行う• 点の数と評価する関数の複雑さ→膨大な計算

Uµ(n)⇥(n + µ̂)• Analysis of Quantum Field
perturbative calculation.

困難：体積、点の数は多い方がいい(x), Aµ(x), x ⌅ Rquantum divergenc• 理想的にはa• 格子の体積(格子点数) → 無限大• 格子間隔⇥(n)a →T=00 の格子計算は現状⇥(n + µ̂)Uµ(n)• 物理点クォーク質量直上計算• とたんに世界最速スーパーコンピュータでも扱えなくなる• 複数の a から連続極限 a→0
1. Accumulate samples of va• a ≠ 0 では： カイラル対称性はあからさまに破れる: ニールセン-二宮の定理• 工夫が可能になってきた

configuration Uµ(n) on di

2. Then measure physical ob• カイラル対称性の改良／厳密化• 格子間隔 a が比較的大きくても 連続極限 a→0 のエッセンスをとらえる⇧O⌃

Taku Izubuchi, Wako, Mini Workshop on Lattice QC

form factor calculation: one example•

Kaon semi-leptonic decay• K→π+l+ν
tf

t

ti

1. Quark flow diagram
for a 3pt function2with initial and final states Pi and P
2h⇡(pf )|VµFigure(0)|K(p
)i
=
f
(q
)(p
pi )
i
+
f + pi ) + f (q )(pf

where E is the energy, m is the meson’s mass and ✓ is the di↵erence of tha CKM matrix
element
Vus isobtainedfor thetwo valencequarksin the frommeson. By varying the twist angles, arbit
2 = 0. In the quark fl
can be reached.
Herewechoosetheanglessuchthatq
Vus f+ (0) = 0.2165(4)figure 1, we twist the strange (s) and light quarks (q) coupling to the vecto
phases
✓K of
and
the
and pion
respective
average[KL✓⇡e3,inKorder, K±momenta] bykaon
Mouslon2014
L µ3, Kto
S e3give
e3 , and Kto± µ3of twisting angles is discussed further in section 3 and their values are given
The matrix element in (2.1) can be extracted from the time dependence ofof Euclidean two- and three-point correlation functions in lattice QCD. T
function is defined by
2 ⇣form factor calculation: one example
with unphysical ud mass simulation
RBC/ULQCD 2013
unphysically heavy pion

extrapolation

simulation points
Figure 3. Illustration of results for three different fit-models. In each case the fit to the full set of simulation
results is shown.

chiral expansion such effects are compensated by a change in the decay-constant’s contribution at
higher order. Surprisingly, for a value of around f = 100 MeV all the results seem to be reasonably
well described by the NLO-ansatz without any NNLO corrections. We have therefore attempted todetermine the decay constant from a fit to the data of only 1 + f2 (fit A ). The fits were of goodquality (cf. the c 2 /d.o.f.-values in Table 4). The functional form of fit A is shown in Figure 3

form factor calculation: one example
one big step forward: physical ud mass simulation
RBC/ULQCD 2015

RBC/ULQCD 2013

NO model dep.

large model dep.
extrapolationinterpolation to phys. pt.

unphysically heavy pion

Isospin symmetry. The unitary light quarks in our simulations
We approximate the isospin broken theory by interpolating in the vaof ∆M 2 corresponding to the physical point [32]. This still leaves adue to the sea-quark isospin breaking which is diﬃcult to quantify
central value between this fit result andhowever
fit F as theresidual
that
thesemodel-dependence.
eﬀects are small compared to the other compone
considerations our simulation
final result points
is,Techniques
include such eﬀects in future calculations are being
ure 3. Illustration of results for three different fit-models. In each casethe fit to the full settoof simulation
lts is shown.
6.
Illustration
for fit E tolead
all data
for theform
factor renormalised with
These
considerations
to
our
final
result:+ Figure
0K⇡
simulation
pointsf (0) = 0.9670(20) ( )
(7)
(17)

stat 42 model
FSE
cuto↵
+
A0 is assumed
to agreefor ensembles A and C. Note the two sets of error band
al expansion such effects are compensated0.2%
by a change0.4%
in the decay-constant’s
contribution
at
0.07%
0.2% C.
Kπ (6.1)
A
and
one
for
ensembleher order. Surprisingly, for a value of around f =+18
100 MeV all the results seem to be reasonably
f+ (0) = 0.9685(34)stat (14)finite volume .
=0.9670(20)(),
l described by the NLO-ansatz without any NNLO46
corrections. We have therefore attempted toermine the decay constant from a fit to the data of only 1 + f2 (fit A ). The fits wereKπ of good2

Using |Vus |f(0) = 0.2163(5), as determined in [3] from experimen

form factor calculation: one example
this is a best understood quantity → average available
Eur. Phys. J. C (2017) 77:112

Page 39 of 228 112

RBC/ULQCD 2015
interpolation to phys. pt.

NO model dep.

Isospin symmetry. The unitary light quarks in our simulations
We approximate
the isospin broken theory by interpolating in the va[224][225] 2of ∆Mcorresponding to the physical point [32]. This still leaves a
[226]
[227]
[228]
due to the sea-quark isospin breaking which is diﬃcult to quantify
however that these eﬀects are small compared to the other compone
Fig. 7 Comparison of lattice results (squares) for f + (0) and f K ± / f π ±breaking correction (see Sect. 4.3). The black squares and grey bands
Techniquestoincludeeﬀectsfutureof calculations
are inbeing
Flavour
Lattice
AveragingGroupwith various model estimates based on χPT (blue circles). The ratio
indicate suchour estimates.
The insignificancethe colours is explainedf K ± / f π ± is obtained in pure QCD including
the 6.
SU Illustration(2) isospinSect.fit
2 E toleadforall data
for theform
factor renormalised with
These
considerations
to ourfinal
result:
WG to provide average ofFigure
latticeresults
simulation
pointsA0(2019
is assumed
to outagreesoon)for ensembles A and C. Note the two sets of error bandissues: 2010, 2013, 2016,
will beKπ
A and one for ensemble C.
f(0) = 0.9685(34)stat (14)finite volume .
+
so
far
does
not
containany
baryonic
quantities
unknown parameters. The latter can be determined from thenumbers obtained in this way correspond to a positive or an
nucleon ofaxial
chargethenextdependence
the lattice
resultstoonbe
the included
masses of theinquarks.
almost vanishing rather than a negative value for # f . We note
Kπ
Using
|V
|f
(0)
= 0.2163(5), as determined in [3] from experimen
us +Note that any calculation that relies on the χ PT formulaforthat FNAL/MILC 12I [23] have made an attempt at deter-form factor calculation: one example
to flavor physics pheno

Page 39 of 228 112
Page 43 of 228 112

Eur. Phys. J. C (2017) 77:112
Eur. Phys. J. C (2017) 77:112

simulations of SU (2) isospin-breaking effects are therefore
required. To remain on the conservative side we add a 100%
error to the correction based on SU (3) chiral perturbation
theory. For further analyses we add (in quadrature) such anusuduncertainty to the systematic error.
Using the results of Table 15 for N f = 2 + 1 we obtain

V

and V

determined to sub % for Nf=2+1 (+1)
without assuming unitarity

direct, N f = 2 + 1 + 1 : f K ± / f π ± = 1.193(3)Refs. [14,26,27],(64)

direct, N f = 2 + 1 : f K ± / f π ± = 1.192(5)Refs. [28–31],(65)
Unitarity satisfied
to ~1σdirect, N f = 2 : f K ± / f π ± = 1.205(6)(17)
Ref. [32],

[224]
[225]
[226]
[227]
[228]

(66)
2+|Vus|2+|Vub|2=1
unitary
line
with
|V
|
ud
Fig.
8 The
plot comparesthe4.3).
information|Vud |,and|Vusgrey| obtained
breaking
correction(see Sect.
The blackfor
squares
bands
on
the latticewith the withexperimental
resultextracted
nuclearβ
indicate
our estimates.
Thesignificanceofthecolours fromis explained
betterdetermined|V
input
ub| in7 withComparison
lattice results (squares) for f + (0) and f K ± / f π ±for Fig.
QCD
brokenof isospin.
various modelon χPT(blue forcircles).
Theandratio
Itwithis instructive
toestimatesconvert based
the above
resultsf + (0)transitions.
The dotted line indicates the correlation between |Vud | andf K ± / f π ± is obtained in pure QCD including the|V
SUus
(2)|f
isospinSect.
2(0)=0.2165(4)+±±f K / f π into a corresponding range for the CKM matrix ele|Vus | that follows if the CKM-matrix is unitary|Vus |,ausing
the relations(54).with
Considerfirst K decay constants (leptonic decay) on
ments |V
ud | and
andusing
similarrelationπ andthe results for N f = 2+1+1. The range for f + (0) in Eq. (58)
unknownparameters.
The|Vlattercan be determined from thenumbers
in this way
positive or
the outcome
for correspondthe sum of tothea squares
of an
the
fK obtained
a us Numerically,is mapped
into
the interval
us | = 0.2231(9), depicted as V
2 = fordependenceofthelatticeresultsonthemassesofthequarks.
almostvanishingratherthananegativevalue# f . Wewhich
note|
0.980(9),
first
row
of
the
CKM
matrix
reads
|V
u
±
±
horizontal red band in Fig. 8, while the one for f K / f π inVfud
⇡FNAL/MILC
Note
that
any
calculation
that
reliesontheχPTformulaforthat[23]level
haveofmade
attemptdeviations.
at deterdeviatesfrom unity12I
at thetwo anstandard
Eq. (64) is converted into |Vus |/|Vud | = 0.2313(7), shownf 2 is subject to the uncertainties inherent in NLO results: Still,miningof somethe low-energy
constants
it isa combination
fair to say thatat thisoflevel
the Standard
Model

FLAG 2016 coverage I
112 Page 4 of 228

Eur. Phys. J. C (2017) 77:112

Table 1 Summary of the main results of this review, grouped in termsof N f , the number of dynamical quark flavours in lattice simulations.
Quark masses and the quark condensate are given in the MS scheme atrunning scale µ = 2 GeV or as indicated; the other quantities listed are
specified in the quoted sections. For each result we list the referencesthat entered the FLAG average or estimate. From the entries in this col-umn one can also read off the number of results that enter our averages
for each quantity. We emphasize that these numbers only give a very
rough indication of how thoroughly the quantity in question has been
explored on the lattice and recommend to consult the detailed tables
and figures in the relevant section for more significant information andfor explanations on the source of the quoted errors

Quantity

Sects.

Nf = 2+1+1

Refs.

Nf = 2+1

Refs.

Nf = 2

Refs.

m s [MeV]

3.1.3

93.9(1.1)[4,5]

92.0(2.1)[6–10]

101(3)[11,12]

m ud [MeV]

3.1.3

3.70(17)[4]3.373(80)[7–10,13]

3.6(2)[11]m s /m ud

3.1.4

27.30(34)[4,14]

27.43(31)[6–8,10]27.3(9)[11]m u [MeV]

3.1.5

2.36(24)[4]

2.16(9)(7)a2.40(23)[16]m d [MeV]

3.1.5

5.03(26)[4]4.68(14)(7)a4.80(23)[16]

m u /m d

3.1.5

0.470(56)[4]

0.46(2)(2)a0.50(4)[16]m c (3 GeV) [GeV]

3.2

0.996(25)[4,5]0.987(6)[9,17]

1.03(4)[11]m c /m s

3.2.4

11.70(6)[4,5,14]

11.82(16)[17,18]11.74(35)[11,132]m b (m b ) [GeV]

3.3.4

4.190(21)[5,19]

4.164(23)[9]4.256(81)[20,21]f + (0)4.3

0.9704(24)(22)[22]0.9677(27)[23,24]0.9560(57)(62)[25]f K ± / fπ ±4.3

1.193(3)[14,26,27]

1.192(5)[28–31]1.205(6)(17)[32]f π ± [MeV]

4.6

130.2(1.4)[28,29,31]

f K ± [MeV]

4.6

155.6(4)[14,26,27]

155.9(9)[28,29,31]

157.5(2.4)[32]" 1/3 [MeV]

5.2.1

280(8)(15)[33]

274(3)[10,13,34,35]

266(10)[33,36–38]

Fπ /F

5.2.1

1.076(2)(2)[39]

1.064(7)[10,29,34,35,40]

1.073(15)[36–38,41]

ℓ̄3

5.2.2

3.70(7)(26)[39]

2.81(64)[10,29,34,35,40]

3.41(82)[36,37,41]

ℓ̄4

5.2.2

4.67(3)(10)[39]

4.10(45)[10,29,34,35,40]

4.51(26)[36,37,41]

ℓ̄6

5.2.2

15.1(1.2)[37,41]

B̂K

6.1

0.727(22)(12)[46]0.717(18)(16)[42]0.7625(97)[10,43–45]

a This is a FLAG estimate, based on χPT and the isospin averaged up- and down-quark mass m

ud [7–10,13]

FLAG 2016 coverage II
Page 5 of 228 112

Eur. Phys. J. C (2017) 77:112
Table 2 Summary of the main results of this review, grouped in termsof N f , the number of dynamical quark flavours in lattice simulations.
The quantities listed are specified in the quoted sections. For each result
we list the references that entered the FLAG average or estimate. Fromthe entries in this column one can also read off the number of results that
Quantity

Sects.

f D [MeV]

7.1

f Ds [MeV]f Ds / f D
f +Dπ (0)enter our averages for each quantity. We emphasize that these numbers
only give a very rough indication of how thoroughly the quantity inquestion has been explored on the lattice and recommend to consult thedetailed tables and figures in the relevant section for more significant
information and for explanations on the source of the quoted errors

T=0 の素粒子物理で重要な
Nf = 2+1+1

Refs.

Nf = 2+1

Refs.

Nf = 2

Refs.

212.15(1.45)[14,27]209.2(3.3)[47,48]208(7)[20]

基本的な物理量は数%誤差で求まっている

f +D K (0)7.1

248.83(1.27)[14,27]249.8(2.3)[17,48,49]

250(7)[20]

7.1

1.1716(32)[14,27]

1.187(12)[47,48]1.20(2)[20]

7.2

0.666(29)

[50]

7.2

0.747(19)[51]

1%以下のものもある

f B [MeV]

8.1

186(4)[52]

192.0(4.3)[48,53–56]

188(7)[20,57,58]

f Bs [MeV]

8.1

224(5)[52]

228.4(3.7)[48,53–56]

227(7)[20,57,58]

f Bs / f B!
f Bd B̂ Bd [MeV]!
f Bs B̂ Bs [MeV]

8.1

1.205(7)[52]1.201(16)[48,53–55]

1.206(23)[20,57,58]

8.2

219(14)[54,59]216(10)[20]

8.2

270(16)[54,59]262(10)

[20]

B̂ Bd

8.2

1.26(9)[54,59]1.30(6)

[20]

B̂ Bs

8.2

1.32(6)[54,59]1.32(5)

[20]

ξ

8.2

1.239(46)[54,60]1.225(31)[20]

B Bs /B Bd

8.2

1.039(63)[54,60]1.007(21)[20]

他の話題はどうだろう？

Quantity

Sects.

N f = 2 + 1 and N f = 2 + 1 + 1

Refs.

(5)
αMS
(M Z )

9.9

0.1182(12)[5,9,61–63]

[MeV]
$(5)
MS

9.9

211(14)[5,9,61–63]

• Analysis of Quantum Field
perturbative calculation.

困難：体積、点の数は多い方がいい(x), Aµ(x), x ⌅ Rquantum divergenc• 理想的にはa• 格子の体積(格子点数) → 無限大⇥(n)⇥(n + µ̂)Uµ(n)• 格子間隔a → 0• とたんに世界最速スーパーコンピュータでも扱えなくなる

1. Accumulate samples of va• a ≠ 0 では： カイラル対称性はあからさまに破れる: ニールセン-二宮の定理
configuration Uµ(n) on di• 工夫

2. Then measure physical ob• カイラル対称性の改良／厳密化• 格子間隔 a が比較的大きくても 連続極限 a→0 のエッセンスをとらえる⇧O⌃

Taku Izubuchi, Wako, Mini Workshop on Lattice QC

格子作用と対称性
U(1)B

SU(Nf)V

Wilson✓✓staggered✓domain wall✓

overlap✓✓✓

SU(Nf)A

計算コスト
普通

部分的

安い

ほぼ厳密

高い

✓

高すぎ／
非現実的

「改良」& 系の性質により、改善しうる対称性••

Wilsonstaggeredカイラル対称性の回復
•

温度

•

２つの異なる相: やぶれ(低温)←→回復(高温)••クォーク閉じ込め←→非閉じ込めクォークの数
•

Nf>Nc

Nf<Nc

出典: 東大理学部
https://www.s.u-tokyo.ac.jp/ja/press/2004/02.html

ヒッグスセクターの背景にある新法則の可能性

α(µ)

µ

LatKMI collaboration• KMI / Nagoya Univ.

素粒子宇宙起源研究機構

Kob
for the Orig

T.Maskawa, K.Nagai, K.Yamawaki

素粒子宇宙起源• KEK, Kyoto, Swansea, Keio, CPT Marseille, Nara, RBEC, Tsukuba
Determination of |Vub /Vcb |Kobayashi-Maskawa Institute for the Origin
using baryonic decays
5IF -BU,.*$PMMBCPSBUJPO
Y. A, T.Aoyama,
E.Bennett,
M.Kurachi, K.Miura, H.Ohki,
E.Rinaldi, A.Shibata, T.Yamazaki
Stefan Meinelnnj : "PLJj 5 "PZBNB

j , .JVSB

0.5189(3)0.5223(4)0.2

0.5221(4)0.5252(4)0.5250(4)

ξi

ξ4

Mπ

ξiξj
before carrying out fit analyses, which will be discussedinξiξ0
the following sections.
0.1
In Fig. 15, we show the ratios Fπ =Mπ and Mξρ4ξ=M
5
π for
ξiξ5
N f ¼ 8 as a function of Mπ for various lattice volumes.
Upξ5
to some exceptions suffering from finite-volume
effects,
both ratios
monotonically increase as M π decreases. The
0
0
0.02
0.04
0.08
0.1
0.12
0.06
present
resultsareconsistentwith
ourprevious
work
[15]
m
f
• Highly Improved Staggered Quark (HISQ) の利用
C. Hadron mass ratios
The purpose of this subsection is to give an overview of FIG. 11.0.06 The spectra for the NG pion and its taste partners, M πξ
• our理論の対称性hadron spectrum dataSU(Nusing ratiosof the hadron spectra for N f ¼ 8.
f)V のチェック

infrared dynamics. We will investigate various hadronic
channels in more depth in the following sections. Although
we will test only one or two taste partners in each channel,
thef results in the pion sector here lead to an expectation thatthe effects of taste symmetry breaking will be small for themass range we simulate for the N f ¼ 8 theory.

00.5277(4)Iξ

0.5188(3)20.5147(3)

(Mπ

24

2

0.08

0

0

N =8 QCD

FIG. 13N f ¼ 8,and its

2

2

Mπ

ξiξj
ξiξ0

ξ4ξ5
ξiξ5

0.03

ξ

ξ4

0.2
0.02

ξiξ4

ξξ

0.01

ξ4ξi 5j

0.1

ξ4ξ5

0.04

0.06
mf

0.08

0.1

0.12

0

ξξ

ξiξi 5 40

π (63個) の質量がほぼ等しい
FIG. 11. The spectra for the NG pion and its taste partners, M πξfor N f ¼ 8.

• 欲しい性質の間接的チェック
α(µ)

0.3

Nf=4, β=3.7

ξI

Mπ

2

ξi
0.2

ξ4

ξiξj

0.02

0.04

0.06
mf

0.08

0.1ξiξ50.12

ξ50 The taste symmetry breaking ðM 2π − M 2π Þ=M 2π for
FIG. 13.
00.02
0.04 ξ
0.06
N f ¼ 8, where M π and M πξ represent
mf the mass of the NG pionand its taste partner with an index ξ, respectively.

π(15個) 対称性破れている

FIG. 12. The spectra for the NG pion and its taste partners, M πξfor N f ¼ 4.
2

α(µ)
2
2
Mπ )/Μπ

0.02

2
Mπ
ξ

0

ξI

ξ4

ξiξi j

ξ5
0

ξi

2
2
Mπ )/Μπξ40.1

Nf=4, β=3.7

2
(Mπ
ξ

2

ξi

ξI

ξ

ξI

Mπ (Mπ

0.2

0.04
0.3

2

Nf=8

Mπ )/Μπ

2

0.05

1.5

1

FIG. 14N f ¼ 4,and its

ξI

α(µ)

ξi

ξ4

ξiξj
ξiξ4

ξ4ξ5
ξiξ5

014508-15

Nf=8 spectrum — σ: flavor singlet scalar• σ is a candidate of Higgs in a successful walking technicolor theory• observed hierarchy of spectrum (parametrically)• mπ ≃ mσ < mρ (Nf=8)
0.5• unlikely due to “heavy quark”

0.4• also in other (near) conformal th.

• contrast to QCD (physical point)

0.3

MH

• Nf=12, Nf=2 sextet, SU(2) 2 adj..

σ L=42σ L=36
σ L=30σ L=24πρ(PV)σ L=480.2
0.1
0• mπ ≪ mσ < mρ (~Nf=2+1)

-0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

real world
QCD:• eventually mπ < mσ should
be seenupdated from Phys.Rev.D96(2017)• mπ = 135 MeV
mf=0.009 (lightest) is new• but, far from our simulation points• mσ = 400〜550 MeV• this continues to even• lighter
points:m = 770
MeVsee LSD 2016ρmf

何がおもしろいのか？
•

素朴には、複合模型 Higgs の質量は••

ところが、2012年に発見された Higgs 粒子は••

mH ~ 1000 GeV

mH ~ 125 GeV「軽いσ」の存在は 「軽い Higgs」を説明できる可能性 !

格子作用と対称性
U(1)B

SU(Nf)V

Wilson✓✓staggered✓domain wall✓

overlap✓✓

SU(Nf)A

計算コスト
普通

部分的

安い

ほぼ厳密

高い

✓

高すぎ／
非現実的

✓

JLQCDの有限温度計算: カイラルフェルミオン
••domain wall fermion (DW)→ “reweighting” to overlap (OV) [JLQCD]
時間を節約しつつ、最終結果は厳密な対称性を保証•

ただし、有効統計の減少とDWの近似の精度には注意が必要

“chiral fermion”• Domain-wall fermion: Kaplan 92, Furman-Shamir 95• もとの４次元理論→５次元• ５次元方向に 質量の domain wall を作る• エッジモードとして、カイラルフェルミオンが出現する• 物性ではよく知られた現象• 5次元方向の長さが有限の場合、微少なやぶれがある• Overlap fermion• Domain-wall fermion の ５次元方向を 無限大極限• 厳密な格子上のカイラル対称性を実現する a Ginsparg-Wilson fermion• どちらも Wilson Fermion のライブラリを援用して構成

JLQCD collaboration
す
で
y
r
a
n
ポスト京重点課題9: サブ課題A - QCD相転移
i
m
i
l
e
r
p
て
全
は
果
青木慎也
[JLQCD collaboration]
結
の
降
青木保道
以
れ
こ
Cossu, Guido
深谷英則
橋本省二
金児隆志
鈴木渓

GL-DW
GL-OV

gluonic charge on DW
gluonicensemble
charge on OVOV ensemble
index on DWOV-OV OV indexensemble
on OV ensemble

OV-DW

χt(mf) for Nf=2 T=220 MeV
3

32 x12, β=4.3

∞

2e+08

ud quark のみの世界の話です
GL-DW
GL-OV
OV-DW
OV-OV

もし現実世界がこうであったら4

χ [MeV ]1.5e+08

1e+08

宇宙初期に一次相転移
重力波?

5e+07

0
0

axion window が閉じる…
5

10

physical ud

15
mf [MeV]

20

25

30

ms

0

mud

∞

32 x12, β=4.3

2e+08
GL-DW
GL-OV
OV-DW
OV-OV

4

χ [MeV ]1.5e+08

有限温度 Nf=2 QCD

1e+08

5e+07

0
0

∞

T

5

10

15
mf [MeV]

20

25

30

physical pt.

chiral symmetry
restoredmsTcchiral symmetry
broken

1st order line ?

m

0

mud

∞

chiral symmetry=SU(2)• test if there is any 1st order transition line there ?
• investigating the symmetry at m→0✴SU(2)LxR or SU(2)LxR x U(1)A ?

現在でも: Columbia Plot = 大方の人の理解 || 期待∞

二次転移physical pt.

ms

クロスオーバー

0

mud

一次転移

∞

u, d, s クォークの質量: mu<md ≪ ms < ΛQCD ≪ mc< mb ≪ mt
[original Columbia plot: Brown et al 1990]

Nf=2+1相図• 連続極限で分かっていること• Nf=0: 一次転移∞

• 右上隅はよく分かっている

physical pt.

ポスト京を用いて、これを解明していきますms• Nf=2+1 物理点: cross-over• staggered (Wuppertal 2006)• 他の正則化でも反証なし• 厳密なカイラル対称性を持つアプローチでは未踏• こんな図を普通書きますが、大部分はよく分かっていない

0

mud

∞

QCD 有限温度相転移の理論: Nf=2+1 Lattice∞• Nf=2+1 相図が完成すればphysical pt.

• QCD の理解

ms

• 物理点の相転移の存在、次数が分かる。
• 遠回りだが確実な方法

0

mud

∞

• 相境界(μ=0)の μ>0 への伸び方を調べる→(T,μ)臨界終点の研究へつなげる• 大変重要／有用である！

32 x12, β=4.3

2e+08
GL-DW
GL-OV
OV-DW
OV-OV

4

χ [MeV ]1.5e+08

まずは Nf=2

1e+08

5e+07

0
0

5

10

15
mf [MeV]• Nf=2+1 physical pt. から遠い？
• ms ~100 MeV → ∞∞• T=0 では s のあるなしは微細効果• boundary の情報としては有用

ms

physical pt.

• Nf=2• Wilson, staggered: 未確定• 厳密な格子カイラル対称性➡U(1)A 回復を示唆[JLQCD16]➡一次転移の可能性 → χt(m)に飛び?
[Pisarski&Wilczek]

0

mud

∞

20

25

30

一次転移だとどうなるか？
• 0 ≤ mf < mc : 一次転移3

32 x12, β=4.3

2e+08
GL-DW
GL-OV
OV-DW
OV-OV

4

• 一つの可能性として: Nf=3の一次転移領域と繋がる

χ [MeV ]1.5e+08

1e+08

5e+07

• 物理点への影響も考えられる

0
0

∞
ms

0

5

10

15
mf [MeV]∞?

physical pt.

mud

ms

∞

0

mud

∞

20

25

30

Topological charge

トポロジカル電荷 (ゲージ場の幾何学的指数)→! 感受率Correspondence with the index is only approximate.
1µ⌫⇢
lat latq(x) =
Tr✏
F
µ⌫ F⇢
2
32⇡

32x12 beta=4.23 m=0.005 (T=191 MeV)
3

lat
Fµ⌫
=

after WF|gluonic Q|

2
1
0
-1
-2
-3
-3

Q=

S. Hashimoto (KEK)

X

q(x)

x

2

hQ i
t =
V

-2

chargePage
18susceptibility-10|OVindex|

1

2

3

Sep 27, 2016

実際、我々の研究でトポロジカル感受率 χt(m) に一次転移的兆候が見えた。。。

3

32 x12, β=4.3

2e+08
GL-DW
GL-OV
OV-DW
OV-OV

4

χ [MeV ]1.5e+08

1e+08

5e+07

0
0

5

10

15
mf [MeV]

20

25

30

系統誤差?
3

32 x12, β=4.3

今年度のフォーカス
• V=323
1.5e+08 GL-DW➡ ∞ 必要OV-OV
4e+07• a = 0.07 fm➡ 0 必要• a = 0.11 fm と比較• 誤差はa2• T=220 MeV➡ 温度を下げていき
Tc 近傍まで調べたい

3

32 x12, β=4.3

GL-DW
GL-OV
OV-DW
OV-OV

4
4

χ [MeV ]• 熱力学極限: m→の前に

χ [MeV ]

2e+08

2e+07

1e+08

5e+07

a→0
0
0
0
0

5
5

10
10
mf [MeV]15
mf [MeV]

20
15

25

30

Results of χt(m) at T=220 MeV; multiple volume
OV-OV

•

3

32
24
3
32
3
48

8

4

χt [MeV ]1.5×108

1.0×107

5.0×10

0.0
0

5

10

15
m [MeV]

20

25

Statistics in trajectory~30k, 30k, 10k

Results of χt(m) at T=220 MeV; multiple volume
GL-DW

OV-OV

3

24
3
32
3
48

8

3

32
24
3
32
3
48

8

1.5×108

1.0×10χt [MeV ]744χt [MeV ]1.5×10

5.0×10

0.0
0

5

10

15
m [MeV]

20

25

Statistics in trajectory~30k, 30k, 10k

81.0×107

5.0×10

0.0
0

•

•

5

10

15
m [MeV]

20

25

V dependence at m=10 MeV is strange• non-monotonic: cannot take thermodynamic limit• important region, where a phase boundary was suggested w/ 323
Let’s look at the histogram of Qsummary of histogram: T=220 MeV, m=10 MeV

3

24
3
32
3
48

8

4

χt [MeV ]1.5×108

1.0×107

5.0×103

OV index

24 x12, β=4.3, m=0.00375

1000

100

10

1

1
-2

-1

0
Qt

1

2

3

-1

0
Qt

1

gluonic

gluonic

24 x12, β=4.3, m=0.00375

32 x12, β=4.3, m=0.00375

2

3

1000

GL-DW
GL-OV

histogram

100

10

若干
1000

GL-DW
GL-OV

100

10

1
-2

-1

0
Qt

1

2

3

# trajectory: ~30k
sample rate: 100

10

100

-3

-2

積
体
low

20

OV-DW
OV-OV

い
な

許
心

-1

継
て
し
ら

減
を

0
Qt

1

2

中
続
3

gluonic

3

48 x12, β=4.3, m=0.00375

GL-DW
GL-OV

.
t
a
st

1
-3

-2

-1

0
Qt

1

~30k
20

2

3

-3

-2

25

OV index

が
計

統
だ

15
m [MeV]

10

1
-3

low

ま

3

histogram

1000

100

1

-2

3

5

48 x12, β=4.3, m=0.00375

.
t
a
st

10

-3

0

1000

OV-DW
OV-OV

3

histogram

OV index

100

10

-3

3

483x1232 x12, β=4.3, m=0.00375

1000

OV-DW
OV-OV

histogramhistogram

323x12histogram

243x12

0.0

-1

0
Qt

1

~10k
100

2

3

まとめ (未言及含む)••

カイラル対称性が厳密な計算手法でQCD相転移を追跡•

特に Nf=2 QCD では他の手法は信頼に足りない•

先行するJLQCD研究より連続理論に近い計算を推進

高温相 T=220 MeV で χt(m) について特に 体積依存性を追求•••

一次転移と思われた mc 近傍で 体積依存性が不自然

比較的大きいクオーク質量で不思議な体積依存性: U(1)A•相転移を示唆？

•フレーバー一重項スカラーが軽くなる新奇な現象？

継続中。。。

Nf=2+1相図• 連続極限で分かっていること• Nf=0: 一次転移∞

• 右上隅はよく分かっている

physical pt.
ポスト京 & カイラルフェルミオン
ms• N =2+1 物理点: cross-overを用いて、これを解明していきますf• staggered (Wuppertal 2006)• 他の正則化でも反証なし• 厳密なカイラル対称性を持つアプローチでは未踏• こんな図を普通書きますが、大部分はよく分かっていない

0

mud

∞

ご静聴ありがとうございました。