Lattice study of conformality in twelve-flavor QCD
Hiroshi Ohki
for LatKMI collaboration
Kobayashi-Maskawa Institute for the Origin of Particles and the UniverseNagoya University@SCGT14mini, March, 5-7

LatKMI collaboration

Y. AokiT. Aoyama
M. Kurachi
E. Bennett
T. Maskawa
K. Miura
K.I. Nagai
H. O.
T. YAmazaki
K .Yamawaki

K. Hasebe

A. Shibata

E. Rinaldi

Introduction

Walking and conformal behavior -> non-perturbative dynamics

Many flavor QCD: benchmark test of walking dynamics: Number of flavor(µ): running gauge coupling

Asymptotic non-free

Conformal window

Walking technicolor
QCD-like•Understanding of the conformal dynamics is important (e.g. critical phenomena)•Walking technicolor (WTC) could be realized just below conformal window.
•What the value of the anomalous dimensions γ? (γ : critical exponent )•Rich hadron structures may be observed in LHC.

LatKMI-Nagoya project (since 2011)
Systematic study of flavor dependence in Large Nf QCD
using single setup of the lattice simulation
Our goals:• Understand the flavor dependence of the theory• Find the conformal window• Find the walking regime and investigate the anomalous dimension!

Status (lattice):
T. Yamazaki (poster)
Nf=16: likely conformal
Nf=12: controversial
This talk
Nf=8: controversial, our study suggests walking behavior?
Nf=4: chiral broken and enhancement of chiral condensate!

M. Kurachi (poster)Observables:
talk by K.-i. Nagai (next)
pseudoscalar, vector meson -> chiral behavior
Glueball (O++) and/or flavor-singlet scalar
Is this lighter compared with others? If so, Good candidate of “Higgs” (techni-dilaton).
talk by T. Yamazaki
E. Rinaldi for gluonic observables (poster)Our work••use of improved staggered action
Highly improved staggered quark action [HISQ]
use MILC version of HISQ action
use tree level Symanzik gauge actionno (ma)2 improvement (no interest to heavy quarks)= HISQ/tree

Simulation setup• SU(3), Nf=12 flavor

simulation parameters
two bare gauge couplings (β) & four volumes & various fermion masses• β=6/g2=3.7, 3.8, and 4.0• V=L3xT: L/T=3/4; L=18, 24, 30, 36• 0.03≦mf≦0.2 for β=3.7, 0.04≦mf≦0.2 for β=4.0
Statistics ~ 2000 trajectory•Measurement of meson spectrumin particular pseudoscalar (“NG-pion”) mass (Mπ), decay constant (Fπ)
vector meson mass (Mρ)

Machine: φ @ KMI, CX400 @ Kyushu Univ.

Nf=12 Result [LatKMI, PRD86 (2012) 054506]
andSome updates

Preliminary

Fπ and Mπ

Nf=121

0.2
L=18
L=24
L=30
L=360.8

0.15

L=18
L=24
L=30
L=36F/

M/

0.6
0.1

0.4
0.05

0.2
000.05

0.1
mf

0.15

0.2

000.050.1
mf0.150.2

Nf=12 theory:
Conformal phase v.s. Chiral broken phase
From the fermion mass (mf) dependence of the hadron mass,
we study the phase structure of the theory.

Conformal hypothesis: critical phenomena near the fixed point
hyper-scaling, γ : mass anomalous dimension at the fixed pointMH  mf1/(1+γ)•Fπ  mf1/(1+γ) + … (for small mf)• Fπ/Mπ → constant (mf→0)Mρ/Mπ → constantmf
RG flow in mass-deformed
conformal field theory(CFT)βcβNf=12 theory:
Conformal phase v.s. Chiral broken phase
From the fermion mass (mf) dependence of the hadron mass,
we study the phase structure of the theory.

Conformal hypothesis: critical phenomena near the fixed point
hyper-scaling, γ : mass anomalous dimension at the fixed pointMH  mf1/(1+γ)•Fπ  mf1/(1+γ) + … (for small mf)•
 Fπ/Mπ → constant (mf→0)Mρ/Mπ → constant
mf hypothesis: π is NG-boson.
Chiral symmetry breaking
Chiral perturabation theory (ChPT) works.
RG flow in mass-deformed
2  mf (PCAC relation)M
conformalfield•π theory(CFT)Fπ=F+c Mπ2 + …
(for small mf)• Fπ/Mπ → ∞ (mf → 0) ββc

Nf=12A primary analysis, Fπ/Mπ vs Mπ

0.22

LatKMI
L=18
L=24
L=30
L=24
L=30
L=36
L=36

0.21

F//M/

β=3.7
0.2

β=40.19

0.18

0.2

0.4

0.6

0.8

1M/

In both of β=3.7 and 4.0, both ratios at L=30 and L=36 seem to be flat in the small
mass region, but small volume data (L≦24) shows large finite volume effect.
This behavior is contrast to the result in ordinary QCD system

Mρ/Mπ vs Mπ

Nf=121.3Flat region

L=18
L=24
L=30
L=24
L=30
L=36
L=36Ml/M/

1.25

β=3.7

1.2

β=4
1.15

1.1
00.2

0.4
M/

0.6

0.8

Ratio is almost flat in small mass region (wider than Fπ/Mπ)-> consistent with hyper scaling
Volume dependence is smaller than Fπ/Mπ.
In the large mass region, large mass effects show up.
Mρ/Mπ should be 1, as mf -> infinity.

Conformal hypothesis in infinite volume & finite volume••

Universal behavior for all hadron masses (hyper-scaling)
Mass dependence is determined by scaling dimension (mass-deformed CFT.)MH

1/(1+ )mf,F⇥

1/(1+ )

mf

(inﬁnite'volume'result)

Our interest : the same low-energy physics with the one obtained
in infinite volume limit
But all the numerical simulations can be done only in finite size system (L).
we use Finite size scaling hypothesis-> Finite size hyper-scaling for hadron mass in L^4 theory[DeGrand et al. ; Del debbio et. al., ’09 ]Note: In order to avoid dominant finite volume effect andto connect with infinite volume limit result,
we focus on the region of L >> ξ (correlation length), (LMπ >>1).

Finite size hyper-scaling••

Universal behavior for all hadron masses
From RG argument the scaling variable x is determined as a combination of massand size•

The universal description for hadron masses are given by the following forms as,

Ref [DeGrand et al. ; Del debbio et. al., ’09 ]c.f. Finite Size Scaling (FSS) of 2nd order phase transition

Test of Finite size hyper-scaling

We test the finite hyper-scaling for our data at L=18, 24, 30, 36.
The scaling function f(x) is unknown in general,
But if the theory is inside the conformal window,
the data should be described by one scaling parameter x.

Data alignment at a certain γ
303018^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

25203018^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

25
2020

1515151010

10= 0.1500

2468101202468101264

4= 0.1

14686810

12101210

1218^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

32= 0.4

1042422205

3300618^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

518^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

5= 0.7

5xxx
good alignment! How'to'quan4fy'this'situa4on?

60= 0.45018^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

2502468

10= 0.71
12002468!To quantify the alignment and obtain the optimal γ
We define a function P(γ) to quantify how much the data “align” as a function of x.

1P( ) =
N|y j
L j KLf (KL )(xj )|2|⇥y j |2
[LatKMI, PRD86 (2012) 054506]

y = LM

182430|y j

f (KL ) (xj )|16

!To quantify the alignment and obtain the optimal γ
We define a function P(γ) to quantify how much the data “align” as a function of x.

1P( ) =N|y j
L j KLf (KL )(xj )|2|⇥y j |2
[LatKMI, PRD86 (2012) 054506]

Optimaly = value
LM of γ for alignment will minimize P(γ).1824
our analysis: three observables of yp=LMp for p=π, ρ; yF=LF30 π .

A scaling function f(x) is unknown,→ f(xj) is obtained by interpolation (spline) with linear ansatz (quadratic for a|y j f (KL ) (xj )|systematic error).
If ξj is away from f(xi) by δ ξj as average → P=1.

!
!
16

P(γ) analysis• P(γ) has minimum at a certain value of γ,
from which we evaluate the optimal value of γ.
• At minimum, P(γ) is close to 1.

Results for data for L=18, 24, 30 at β=3.7
L > ξ is satisfied in our analysis.

(LM > 8.5 for our simulation parameter region)

!Result of gamma (data L=18,24,30)[LatKMI, PRD86 (2012) 054506]

M/ (`=3.7)
M/ (`=4.0)

2012 ResultF/ (`=3.7)
F/ (`=4.0)
Ml (`=3.7)
Ml (`=4.0)

0.3

0.4

a0.5

0.6

0.7

• The error -> both statistical & systematic errors<- estimation by changing x range of the analysis•Remember: Fπ data seems to be out of scaling region
due to finite mass & volume corrections. Flat range is smaller than Mρ/Mπ.

!Result of gamma (data L=24,30,36 with lighter mass region)

M/ (`=3.7)
M/ (`=4.0)

2013-14 Update

F/ (`=3.7)
F/ (`=4.0)
Ml (`=3.7)
Ml (`=4.0)

0.30.4

a0.50.60.7• γ(Mπ) is stable against the change of the mass (x) and β .
• smaller mass with larger volume (18,24,30 ->24,30,36)→closer value to γ(Mπ)
The universal scaling is obtained for both values of β =3.7 & 4.0
γ=0.4-0.5.

Further corrections
tothe hyperscaling■Possible corrections to the finite size hyper scaling

We consider simultaneous fit for the three quantities of!
!

with finite mass (volume) correction.
We consider following possibilities by adding different massdependence as
1/(1+ )

= c0 + c1 Lmf

1/(1+ )

= c0 + c1 Lmf· · · (no correction)+ c2 Lmf

1/(1+ )= (c0 + c1 Lmf)(1 + c2 m )α. ω … unknown exponente.g.
1. ladder Schwinger-Dyson eq. analysis:
2. lattice (am)^2 artifact :!
!

[LatKMI PRD85(2012)074502]

3. exponent of the gauge coupling[c.f. A. Cheng, et al. ’14]

=y0 /(1 + )y0= -0.36 (2-loop perturbation theory)We demonstrate simultaneous fit for three observables ofMπ, Fπ, Mρ using following functions.
10 fit parameters(ω is fixed to some specific value)= LF
618^3 x 24
24^3 x 32
30^3 x 40
36^3 x 48

5We consider following fit
region I … LFπ > 2(LMπ >8)

I

4
3
2
10024681012

Result for “region I”•The data with empty symbols are not used in the fit
306L=18
L=24
L=30
L=36
no correction
t=0.4

25

4
LF/LM/20

L=18L=24
L=30
L=36
no correction
t=0.415102

5000.05

0.1
mf

0.2

0.15

Fit result with L=18, 24, 30, 36
ω [ﬁxed]γχ^2/dof

0.457(1)15

0.40.398(5)

2.6

0.8

0.425(2)2.00(no correction)

000.05

0.1
mf

0.15

L=24, 30, 36
ω [ﬁxed]γχ^2/dof0.459(2)120.4

0.406(5)2.4

0.8

0.430(4)2.00(no correction)

In various trials of this analysis: γ=0.2ー0.450.2

Short summary in Nf=12•

β=3.7-4.0: Mπ, Fπ, Mρ show conformal hyper scaling•

Fπ : large mass corrections in our whole mass parameters, likely too
heavy mf to be neglect.
→ Approaching small mass region, we obtain hyper-scaling behavior.

•The hyper-scaling is realized in larger volume region together withsmaller mass region.

!
•We consider possible corrections to the finite size hyper scaling, tounderstand both the outsides of the scaling region.

→ The large fermion mass region can be described by such a correction.
The value of γ could be smaller as γ ~ 0.2-0.45.

!
•ChPT expansion is not valid, expansion parameter is much larger than 1.
(Not yet exclude chiral broken scenario (very small Fπ))β dependence (UV cutoff) effect•β dependence is important to study the lattice phase structure (existenceof bulk transition, asymptotic free or non-free)
and to obtain the continuum limit physics!
•In the conformal phase, we demonstrate some scaling matching
analyses,!
1. Matching of the dimension-less ratio
2. Matching of hyper scaling curves for L M

!scale (β) dependence

1.3
L=24
L=30
L=36

1.25

Ml/M/

Why is there difference in theratio between β＝3.7 and 4.0?
Note: This ratio is dimension-lessquantity.

β=3.7
β=41.2

1.15

1.1

0

0.2

0.4

0.8

0.6
M/

To study more about β dependence,
we use the hyper scaling relation in infinite volume limit for simplicity.
continuum theory
1/(1+ )

M = c mf

1/(1+ )

M = c mf

M /M+ ···
+ ···c+ ···
ca possible discretization (cutoff) effectc = c + a2 c̃
c = c + a2 c̃M /Mcc

1 + a2c̃cc̃
c+ ···The discretization error appears in the overall factor.
This can make the difference of the ratio.

!How to match the scale
1. Matching the factor of the ratio (which come from the disc. effects) byintroducing a factor R to multiply Mρ/Mπ for β=4.
1.3

1.3

Ml/M/

1.25

1.2

M
M

Rfor β=4

L=24
L=30
L=36

1.25

1.2

1.15

1.15

1.1

M
MMρ/Mπ

L=24
L=30
L=360

0.2

0.4

0.6

1.1

0.8

0M/

0.2

0.4

0.6
Mπ

0.8

11.22. Further tuning for the remaining difference which may appear at the tailby introducing the horizontal factor r as r Mπ.

aM

raM

for β=4

!How to match the scale

R=1.03, r=1.00
1.3
L=24
L=30
L=36/Mπ
RMMρ/M

1.25

1.2

1.15

1.1

0

0.2

0.4

0.6
Mπ

raM

0.8

11.2

!How to match the scale

R=1.03, r=1.10
1.3
L=24
L=30
L=36/Mπ
RMMρ/M

1.25

1.2

1.15

1.1

00.2

0.4

0.6
Mπ

raM

0.8

11.2

!How to match the scale

R=1.03, r=1.20
1.3
L=24L=30
L=36/Mπ
RMMρ/M

1.25

1.2

1.15

1.1

00.2

0.4

0.6
Mπ

raM

0.8

11.2

!How to match the scale

R=1.03, r=1.30
1.3
L=24
L=30
L=36/Mπ
RMMρ/M

1.25

1.2

1.15

1.1

0

0.2

0.4

0.6
Mπ

raM

0.8

11.2

!How to match the scale

R=1.03, r=1.40
1.3
L=24
L=30
L=36/Mπ
RMMρ/M

1.25

1.2

1.15

1.1

00.2

0.4

0.6
Mπ

raM

0.8

11.2

!How to match the scale

R=1.03, r=1.50
1.3
L=24
L=30
L=36/Mπ
RMMρ/M

1.251.2

1.15

1.1

0

0.2

0.4

0.6
Mπ

raM

0.8

11.2

!The scale matching
The value of r~ 1.2- 1.3 shows a consistency between β=3.7 and 4.0
for a quantity of the ratio Mρ/Mπ

a1=ra2

1.2

1.3

This result is consistent with being in theasymptotically free region for our β’s.

!Comparison between different beta’s using
We assume that the two scales!

have the following relation!

andwhere b is a factor.

!Comparison between different beta30
24^3 x 32
30^3 x 40
36^3 x 482520
151050024681012Fit results for combined data of beta=3.7 and 4.0

This results suggest that the data for both beta areconsistent with the finite volume scaling
and asymptotically free.

ChPT analysis

!Fit result on π mass (β=3.7 to see near the chiral limit)
We analyze the largest volume data only.
mf=0.03-0.05
mf=0.035-0.06

The fit results

0.2

ﬁt range

M/2[0.03-0.05]

0.1[0.035-0.06]

M 2 = c0 + c1 mf + c2 m2f

000.020.04

Fit results for Mπ•mf0.060.08

c0χ2/dof

dof

[0]

2.4

2-0.023(7)

0.16

1

[0]

5.6

2-0.02(1)0.16

1LM = 8.71 (mf = 0.030)
0.1

LM = 9.79 (mf = 0.035)

Polynomial fit is reasonable for small fermion mass range.
For the smallest mass range, Mπ goes to zero or negative.

!Fit result on Fπ (β=3.7)
0.2
mf=0.03-0.05
mf=0.035-0.06

The fit results

F/0.150.1
0.05

F000.02= c0 + c1 mf + c2 m2f
0.04mf

Fit results for Fπ•0.06

0.08

ﬁt rangec0χ2/dof

dof[0.03-0.05]-0.003(7)1.1

1[0.035-0.06]

0.012(5)

0.01

10.1

Polynomial fit is reasonable for small fermion mass range.
For the smallest mass range, Mπ goes to zero or negative.
Fπ in the chiral limit is tiny non-zero or consistent with zero.

Note on ChPT fit in many flavor QCD!
!
•Natural chiral expansion parameter is!
!
!

[M. Soldate and R. Sundrum, Nucl.Phys.B340,1 (1990)],[R. S. Chivukula, M. J. Dugan and M. Golden, Phys. Rev. D47,2930 (1993)]The parameter  should be less than 1 to be consistent with ChPT expansion.
 ~ 3.5 at the lightest mass point and >30 using F in the chiral limit.

!
->It is difficult to tell real chiral behavior. e.g. Fπ in the chiral limit, if it exists.

Summary•Large Nf SU(3) gauge theory is being investigated in LatKMI project.
•We focus on the Nf=12 case.

!

[LatKMI, PRD 2012 and some update].
•Finite size hyper scaling is observed for the π (“NG-boson”) mass,decay constant and rho meson mass.
•Nf=12 is consistent with conformal gauge theory.
•The resulting universal γ ~0.4-0.5 (without correction), 0.2-0.4(withcorrection), (not favored as Walking Technicolor)•ChPT expansion is not valid, expansion parameter is much larger than 1.
(Not yet exclude chiral broken scenario (very small Fπ))!

How about other # of fermions??
-> e.g. 8 flavor case, talk by K.-i. Nagai (next)

END
Thank you