Walking signals
in Nf=8 QCD on the lattice
Kei-ichi NAGAI
for LatKMI Collaboration
Kobayashi-Maskawa Institutefor the Origin of Particles and the Universe

素粒子宇宙起源研究機構

Kobayashi-Maskaw
for the Origin of Particles aKobayashi-Maskawa Institute for the Origin of Particles
and the Universe (KMI),

宙起源研究機構

Nagoya University

素粒子宇宙起源研究機

SCGT14Mini, 06 March 2014 @ KMI, Nagoya

LatKMI Collaboration:
Y. AokiT. Aoyama
E. Bennett Kobayashi-Maskawa Institute

宙起源研究機構

for the Origin of Particles and the Universe

M. Kurachi

T. Maskawa
K. Miura

素粒子宇宙起源研究機構
K.-i. Nagai
H. OhkiK. Yamawaki

yashi-Maskawa Institute for the Origin of Particles and the Universe

T. Yamazaki

A. Shibata
素粒子宇宙起源研究機構

E. Rinaldi

K. Hasebe

素粒子宇宙

Kobayashi-Maskawa Institute forPlan of the Talk:
1. Introduction
2. Lattice study of Nf=8 QCD{Chiral Perturbation Theory (ChPT)
Finite Size Hyperscaling (FSHS)

3. Summary♠ Nf=8 is a candidate of the walking behavior.
Walking signals in Nf=8 QCD on the lattice
Yasumichi Aoki, Tatsumi Aoyama, Masafumi Kurachi, Toshihide Maskawa, Kei-ichi Nagai, Hiroshi Ohki, Akihiro Shibata,
Koichi Yamawaki, Takeshi Yamazaki,
Published in Phys.Rev. D87 (2013) 094511, e-Print: arXiv:1302.6859 [hep-lat].

+ update

1. Introduction• LQCD with many fermions• BSM (quark mass, FCNC, ..., model building)• Candidate of the walking technicolor(WTC)

conformal window and walking coupling- non-Abelian gauge theory with Nf massless fermions Nf
Asymptotic non-free

NfAF

_(µ)

[Caswell, Banks-Zaks]

Conformal window_(µ)µ[Yamawaki-Bando-Matumoto]NfcritWalking Technicolor_(µ)µQCD-likeµ• Walking Technicolor could be realized just below the conformal window• crucial information: Nfcrit & mass anomalous dimension around Nfcrit

Many ﬂavor QCD
Candidate of walking/conformal
Our investigation in Nf=12 (Ohkiʼs talk)→ consistent with the conformal with γ=0.4--0.5 .
not favored as WTC (model building)Thus, we investigate Nf=8 QCD.
strong coupling dynamics and non-perturbative
Lattice simulation of Nf=8 QCD

One-family
ExtendedTechniColormodelΛ1 uctU1
 d   s   b   D1   u   c   t   U1   d   s   b   D1   u   c   t   U1   d   s   b   D1   e   µ   τ   E1 νe
νµντ
N1



UNTC
 DNTC  UNTC  DNTC  UNTC  DNTC  ENTC NNTC{{{SU (3 + NTC )SU (2 + NTC )Λ2SU (1 + NTC )Λ3

Why Nf=8 QCD?

SU (NTC )8-ﬂavor SU(NTC )technicolor
We consider

NTC = 3

What is the signal of walking?

`(_)

_(µ)
_µ

In Gauge Theories:Spectrum?

Heuristically

SχSB and/or conformal ?

What is the signal of walking?
Scenario of Walking Dynamics ;Schematic picture of the gauge coupling

Spectrum?
SχSB and/or conformal ?

What is the signal of walking?
Scenario of Walking Dynamics ;Case-1: probe mf << mD → SχSB-likeSpectrum?
SχSB and/or conformal ?

What is the signal of walking?
Scenario of Walking Dynamics ;Case-2: probe mf >> mD → conformal-likeSpectrum?
SχSB and/or conformal ?

What is the signal of walking?
Scenario of Walking Dynamics ;Case-1: probe mf << mD → SχSB-like
Case-2: probe mf >> mD → conformal-like
Schematic picture of the gauge coupling

Spectrum?
SχSB and/or conformal ?

2. Lattice Study of Nf=8 caseSimulation for Nf=8 (same setup with Nf=12)
lattice action (Hybrid Monte-Carlo simulation)
・Tree-level Symanzik gauge action
・Highly Improved Staggered Quarks = HISQ
(without tadpole improvement and mass correction in Naik term)★ parameter set�・ β ≡ 6/gVmf

12^3x162

�

= 3.8,18^3x24 24^3x32

V=L3xT , T/L=4/3 ﬁxed.
30^3x4036^3x48 42^3x56

0.01~0.16 0.04~0.1 0.02~0.1 0.02, 0.03,0.04~0.07 0.015,0.02(updated)★ Gauge conﬁgurations for scalar measurement★ Measurements (P+AP method

double size in T-dir.)・Mπ, Fπ, Mρ, chiral condensate
・analysis for MπL > 60.012(new)data in the Nf=8 paper +updated dataPreliminary

HISQ with Nf=8:
eﬀective mass for the lowest mf(=0.015)
on the largest size(L=36)

MπPS=MπSC, MρPV=MρVT→ good ﬂavor symmetry

Mπ2, Fπ vs. mf on various L
3

2

F/ vs mf, L x(4L/3)

3

M/ vs mf, L x(4L/3)

0.2

0.8

0.18
0.16

0.6

0.14
M/F/2

0.12
0.1
0.08

L=12
L=18
L=24
L=30
L=36
new

0.06
0.04
0.02
00

0.02

0.04

0.06

0.08 0.1
mf

0.12

0.14

0.16

0.4
L=12
L=18
L=24
L=30
L=36
new

0.2

0.18

00

0.02

0.04

0.06

0.08 0.1
mf

0.12

0.140.160.18

Size dependence of Mπ and Fπ at β=3.8△=updated

M/ vs L at each mf

F/ vs L at each mf

L^3x(4L/3)
0.9
0.8
0.7
0.6
0.5

0.14
0.12
0.1

F/M/L^3x(4L/3)

mf=0.10
mf=0.08
mf=0.07
mf=0.06
mf=0.05
mf=0.04
mf=0.03
mf=0.020.4

0.08

mf=0.10
mf=0.08
mf=0.07
mf=0.06
mf=0.05
mf=0.04
mf=0.03
mf=0.02

0.06

0.3
0.04

0.2

0.02

0.1
004

812

1620
L24

2832

36

40

0048

12

1620L242832

36

On the larger volume, there is not (or very tiny) size dependence.

40

Fπ/Mπ for Nf=12, 8 and 4 (ﬂat or divergent inχ-limit?)
0.3
L=12
L=18
L=24
L=30
L=36
L=42

0.28

F//M/

0.26
0.24
0.22
Nf=8, `=3.8

0.2
0.18

0

0.1

0.2

0.3

0.4

0.5
M/

0.6

0.7

0.8

0.9

1Mρ/Mπ for Nf=12, 8 and 4 (ﬂat or divergent inχ-limit?)
1.6

4
L=12
L=18
L=24
L=30
L=36
L=42

1.55
1.5

Ml/M/Ml/M/

1.45
1.4
1.35
1.3

3

2.5

Nf=4, `=3.721.25
1.2

B3.7 L12T18
B3.7 L16T24
B3.7 L20T30

3.5

Nf=8, `=3.8
00.1

0.2

0.3

0.4

0.5
M/

0.6

0.7

0.8

0.9

1

1.5

00.1

0.2

0.3
M/

0.4

0.5

ChPT analysis

Walking signals in Nf=8 QCD on the lattice
Yasumichi Aoki, Tatsumi Aoyama, Masafumi Kurachi, Toshihide Maskawa, Kei-ichi Nagai, Hiroshi Ohki, Akihiro Shibata,
Koichi Yamawaki, Takeshi Yamazaki,
Published in Phys.Rev. D87 (2013) 094511, e-Print: arXiv:1302.6859 [hep-lat].

ChPT(SχSB) in 0.015 mf 0.04

In the χ-limit;

Fπ=0.031(1), Mρ/(F/ 2)=7.7(1.5)
condensate〜0.0005

Re-analysis with updated data ⇓Fπ vs mf with quadratic-func. ﬁt : y=C0+C1*mf+C2*mf^2
F/ vs mf, and quadratic fit
0.1
0.09
0.08
0.07

F/

0.06
0.05

mf=0.012-0.03
mf=0.012-0.04
mf=0.015-0.04

0.04
0.03
0.02
0.01
000.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
mf

red : C0=0.0202(13), χ2/dof=0.60 (dof=1)
green: C0=0.0240(5),χ2/dof=5.9 (dof=2)
blue : C0=0.0246(7), χ2/dof=9.4 (dof=1)

ChPT is good in 0.012 mf 0.03.

Mπ2 vs mf with quadratic-func. ﬁt
2

M/ vs mf, and quadratic fit
0.14
mf=0.012-0.03
mf=0.012-0.04
mf=0.015-0.04

0.12

M/2

0.1
0.08
0.06
0.04
0.02
000.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
mf

red : C0=-0.0012(11), χ2/dof=0.79 (dof=1)
green: C0=-0.0025(6),χ2/dof=1.4 (dof=2)
blue : C0=-0.0034(8), χ2/dof=0.24 (dof=1)

ChPT is good in 0.012 mf 0.03.

Mρ vs mf with quadratic-func. ﬁt
Ml vs mf, and quadratic fit

0.5

Ml

0.4
0.3

mf=0.012-0.03
mf=0.012-0.04
mf=0.015-0.04

0.2
0.1
000.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
mf

red : C0=0.121(29), χ2/dof=0.27 (dof=1)
green: C0=0.138(15),χ2/dof=0.36 (dof=2)
blue : C0=0.149(20), χ2/dof=0.025 (dof=1)

Condensate (PBP and Σ) with quadratic-func. ﬁt
PBP=Tr[Prop(x,x)/4]
0.03
0.028
0.026
0.024
0.022
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0-0.002

enlarged

Condensate (PBP) vs mf, and quadratic fit

0.01
0.009

PBPY
mf=0.012-0.03
mf=0.012-0.04
mf=0.015-0.04

0.008
0.007
0.006
cond.

cond.

Condensate (PBP) vs mf, and quadratic fit

0.005
0.004
0.003
0.002

PBP
mf=0.012-0.03
mf=0.012-0.04
mf=0.015-0.04
22(Fmf->0) *(M/ /mf/4)mf->00.001
000.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
mf-0.001-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05mf

PBP
Σ=FπMπ2/mf/4

red : C0=0.00018(5),χ2/dof=0.89
green: C0=0.00024(2), χ2/dof=1.61(Fπ)2mf->0*(Mπ2/mf/4)mf->0 = 0.00021(2)blue : C0=0.00025(2), χ2/dof=3.18

consistent valueSummary-1,old data

• The quadratic ﬁt• Nf=8 is consistent with ChPT(SχSB) in the small mf
region (0.015 mf 0.04).

• In the χ-limit. F=0.031(1), M /(F/ 2)=7.7(1.5)ρ

condensate〜0.0005.

• The expansion parameter χ=O(1) of ChPT in thesmallest mf (self-consistent), in contrast to Nf=12.

•

in the intermediate mf region?

Summary-1,+updated• The quadratic ﬁt• Nf=8 is consistent with ChPT(SχSB) in the small mf
region (0.012 mf 0.03).

• In the χ-limit. F=0.0202(13), M /(F/ 2)=8.5(2.1)
ρcondensate〜0.0002.

• The expansion parameter χ=O(1) of ChPT in thesmallest mf (self-consistent), in contrast to Nf=12.

•

in the intermediate mf region?

Finite size Hyperscaling analysis
in a intermediate mf region
LMH = FH (x), LFH = GF (x)(DeGrand, Del Debbio et al.)1/1+γ (: universal γ)x ≡ Lm

If the coupling is in slow-running, ...

information
at (approximate) IRFP

Finite size Hyperscaling analysis
in a intermediate mf region
LMH = FH (x), LFH = GF (x)(DeGrand, Del Debbio et al.)1/1+γ (: universal γ)x ≡ Lm

If the coupling is in slow-running, ...

information
at (approximate) IRFP

Dig it out!

from the data in this region

The hyperscaling with mass corrections is needed,from the lesson of Schwinger-Dyson analysis (our paper, PRD(2012)).

Comparison with Nf=4
: hyperscaling of Fπ (in SχSB)`=3.7, a = 1.0`=3.7, a = 2.0

3

3

2.5

2.5

2.5

2

2

2

1.5
1

1.5
1

L=12, T=18
L=16, T=24
L=20, T=30

0.50

L f/3L f/L f/`=3.7, a = 0.0012341/(1+a)
Lm561
L=12, T=18
L=16, T=24
L=20, T=300.5

701.5012341/(1+a)
Lm

Fπ：no scaling forComparison with Nf=1256

L=12, T=18
L=16, T=24
L=20, T=30

0.5

7000<γ<212341/(1+a)
Lm567Finite size Hyperscaling analysis
Comformal → Finite size Hyperscaling behavior with universal γ(Critical exponent obtained from the ﬁnite volume setup)
LMH = FH (x), LFH = GF (x)1/1+γx ≡ LmFπLMπLMρL

in the middle region of mf 0.05 and ξπ (=MπL) 8γ(Fπ)〜1.0,γ(Mπ)〜0.6,γ(Mρ)~0.8

diﬀerent from Nf=4 and12

Simultaneous ﬁt of Hyperscaling with mass correctionsin the middle region of mf 0.05 and ξπ (=MπL) 8(Schwinger-Dyson eq. analysis with large mass)ξH(=MHL) vs mf (not X): α=1ﬁxed (example)

In this case, the mass correction works wellwith γ=0.874(25), χ2/dof=0.75, dof=32.

• In various trials of this analysis: γ=0.78ー0.93 〜1Summary-2, Finite-size Hyperscaling analysis• In the region of mf 0.05, hyperscaling is seen. (diﬀerentfrom Nf=4)• non-universal γ for each observable in the naive ﬁnitesize hyperscaling (diﬀerent from Nf=12)• Lesson from the Schwinger-Dyson analysis with the(large) mass.

• Simultaneous ﬁt of hyperscaling with mass correction
gives the universal γ=0.78--0.93 〜1. [requirement forthe successful walking technicolor.]• Nf=8 has the “remnant” of the conformality in themiddle range of mf.

SummarySU(3) gauge theories with 4, 12, 16 and 8 HISQ quarks.
Nf=4 → Kurachiʼs poster
Nf=12 → Ohkiʼs talkNf=16 → Yamazakiʼs poster

Preliminary (data updated: 2013→2014)
Nf=8; possibility of SχSB in the small mass region of oursimulation and the remnant of the conformality in the middle
region of mf with γ~1. (In contrast to Nf=4 and 12 cases.)
Nf=8 → Candidate of Walking dynamics
`(_)

_(µ)
_

Phenomenologically preferable?

µIn Progress:
Simulation on larger volumes at lighter masses(one-volume: mf=0.015 on L=36, mf=0.012 on L=42)
Finite Size Eﬀect
Finite Size Scaling
Lattice spacing dependence (Enhancement) ←many βSpectroscopy (Mglueball, M”scalar”, Mbaryon, Mmeson, Fρ/σ, S-param.,
string tension, etc.)
Mglueball, string tension (→ Rinaldiʼs poster for Nf=12)
M”ﬂavor-singlet light scalar”→ Yamazakiʼs talkThank youBackup

KMI computer, ϕComparison with Nf=4
: hyperscaling of MρL (in SχSB)
1818

16

16

16

14

14

14

12
10

12
10

L=12, T=18
L=16, T=24
L=20, T=30

86L Ml18L Ml

L Ml`=3.7, a = 2.0`=3.7, a = 1.0`=3.7, a = 0.00

12341/(1+a)
Lm

56

10
L=12, T=18
L=16, T=24
L=20, T=308
76120

12341/(1+a)
Lm56

L=12, T=18
L=16, T=24
L=20, T=3087

Mρ：no scaling for 0 < γ < 26012341/(1+a)Lm

567

Simultaneous ﬁt of hyperscaling with mass corrections(same method with Nf=12)
Hyperscaling? in the middle region of mf (mf 0.05 and ξπ (=MπL) 8)
The mass corrections might be needed, as done in Nf=12,
from the lesson in SD analysis.

good χ2/dof, but unclear which α is better.

Tools for analysis(1) Chiral Perturbation Theory (ChPT)
Spontaneous chiral symmetry breaking (SχSB)(2) (Finite size) Hyperscaling(critical phenomena in conformal transition)

LMH = FH (x), LFH = GF (x)1/1+γx ≡ Lm

Inﬁnite volume limit:(: universal γ)(DeGrand, Del Debbio et al.)