Conformal window
on the lattice
伊藤悦子 (KEK)

PTEP (2013) 083B01 and arXiv:1307.6645[hep-lat]「離散的手法による場と時空のダイナミクス」@KEK 2013/9/27
2013年9月27日金曜日

Summary of our work
• Study SU(3) gauge theory coupled to 12 masslessfundamental fermions• Measured the running coupling constant in Twisted
Polyakov loop scheme and found an IR ﬁxed point• Derived the universal quantities around the ﬁxed point• critical exponent of the beta fn.
• mass anomalous dimension
2013年9月27日金曜日

Introduction
Conformal ﬁxed point is the most important object in QFT
Scalar field theory
2d Gaussian ﬁxed point
unitary discrete series
minimal model (IR ﬁxed point of Landau-Ginzburg model)
3d Gaussian ﬁxed point
Wilson-Fisher ﬁxed point
non-linear sigma model
scaling dim. for the primary ﬁeld
4d Gaussian only...
central charge
̀̀c function”
Gauge theory
4d Gaussian ﬁxed point
nontrivial ﬁxed point?

2013年9月27日金曜日

H.Kawai and Y.Kikukawa: Phys.Rev.D83:074502,2011
S.Kamata and H.Suzuki: Nucl.Phys.B854:552-574,2012

Introduction
Higgs sector in the Standard Model Lagrangian

Problem with a fundamental Higgs boson
Hierarchy problem (need fine-tuning to cancel a quadratic divergence)
Triviality problem

No interaction at low energy
Running coupling constant diverges at a finite energy
Cuttoff theory?
Candidates for the origin of Higgs sector
Supersymmetry
Extra dimension
Introduce an additional gauge interaction
Walking techni-colorand fermions.....

2013年9月27日金曜日

Walking Technicolor
To suppress FCNC process To enhance quark mass
Asymptotic free (UV complete)
IR fixed point (or walking behavior)
Large anomalous dimension forcomposite state

2013年9月27日金曜日

Is there a theory whose coupling constantshow the behavior?

This part may be realized bymany ﬂavor gauge theory.
2013年9月27日金曜日

SU(3) Nf gauge theoryβ(α) = −bα −cα2Two loop analysis

3Perturbative (MS bar scheme)
2-loop 3-loop 4-loop(alpha) 0.75
0.44 0.47(g^2) 9.4
5.5
5.9
T.A.Ryttov and R.Shrock,Phys.Rev.D83,056011 (2011)
20th order in Wilson loop scheme is also doneby Horsley et.al.
Phys.Rev. D86 (2012) 054502
S-D eq. with large Nc

Nfcr = 11.9
Exact RG

Nfcr = 10.0+1.6−0.7

H.Gies and J.Jaeckel,
Eur.Phys.J. G46:433-438,2006

Exact RG (+ 4 fermi interaction)

Nfcr = 11.58
Phase structure based on two loop038
Nc ∼ 8.8
13chiral sym.broken
conﬁnement
2013年9月27日金曜日

Asymptotically free

Conformal Window(chiral symmetric)

Y.Kusafuka and H.Terao,
Phys.Rev. D84 (2011) 125006

11Nc ∼ 16.5
2

Asymptotic non-freeNfIs there an IR ﬁxed point in SU(3) Nf=12 theory?
Ishikawa, Iwasaki, Nakayama, Yoshie (phase structure, correlation fn.)

Appelquist, Fleming, Neil, M.Lin, Schaich (running coupling, mass spectrum)

Deuzeman, Lombardo, Pallante, Miura, da Silva (ﬁnite temperature)

Cheng, A. Hasenfratz, Petropoulos, Schaich (MCRG, phase structure, Dirac eigenmodes)

DeGrand (mass spectrum)

LatKMI (mass spectrum)

D.Lin, Ogawa, Ohki, Shintani (running coupling)

Fodor, Holland, Kuti, Nogradi, Schroeder, (running coupling, phase structure, spectrum)
Jin and Mawhinney (phase structure)

2013年9月27日金曜日

Why are these studies
contradictory?

2013年9月27日金曜日

相互作用をする赤外固定点の見つけ方(1) Step scaling 法によるrenormalized couplingの測定
Luescher, Weisz and Wolﬀ, NPB 359 (1991) 221(2) low betaでのchiral symmetry(3) 質量変形した理論のhyperscaling
mass spectrum
Miransky, PRD59(1999)105003
Luty, JHEP 0904(2009)050
Del Debbio and Zwicky, PRD82(2010)014502(4) Dirac固有モードのhyperscaling (Volume-scaling)
Patella,PRD86(2012)025006
Cheng, Hasenfratz, Petropulos and Schaich, JHEP1307(2013)061(5) メソン的演算子の相関関数の形
Ishikawa, Iwasaki, Nakayama and Yoshie, PRD87(2013)071503

2013年9月27日金曜日

(1) Step scaling method- measuring the running coupling constant Luescher, Weisz and Wolﬀ, NPB 359 (1991) 221- tune beta to reproduce the input renormalized coupling- measure the g^2 at the larger lattice with the tuned beta
- take the continuum limit

2gRσ(u)2σ(u) ≡ gR
(1/sL)
2u ≡ gR(1/L)

input u0(1/16)2 (1/12)2

We can apply this method to any
renormalization schemes on the lattice.
2013年9月27日金曜日(1/8)2 (1/6)2(a/L)

2

2013年9月27日金曜日

Step scaling method
Running coupling constant(Nf=12)
Appelquist et al. (SF scheme)
Phys.Rev.D79:076010,2009

Fodor et al. (potential scheme)
PoS LAT2009:055,2009, talk at Lattice2010

Taking constant extrapolation in (a/L)

Plot: Slide of K.Holland’s talk at Lattice2010

The continuum extrapolation was not considered.
(O(a) eﬀects depends on the renormalization scheme)
2013年9月27日金曜日

Several renormalization schemes and universality
scheme transformation

P

g1 → g2 = f (g1 )57

f (g1 ) is an analytic fn. ofbeta fn.

γ‫ٮ‬P‫ٮ‬,5)3Jγ

Jg1∂f (g1 )β(g2 ) =β(g1 )∂g1The existence of the ﬁxed point isscheme independent.

Note that the renormalized coupling constant at the FP depends on the scheme.
∗∗∗
g2 = f (g1 )(�= g1 )∗∗The critical exponents ( γg , γm ) around the ﬁxed point are scheme independent.

2013年9月27日金曜日

(2),(3) Chiral symmetryとMass deformed theory
Z.Fodor et al.: Phys.Lett.B703:348-358,2011.

measured mass spectrum and chiral condensate at beta=2.2
in several lattice sizes and fermion bare masses.

Comparison between two hypotheses
5Appelquist et. al:PRD84(2011)054501! Ψ̄Ψ" − c 1 m − c 2 m 2 plottedï3x 101∗1+γm! Ψ̄Ψ" = c 0 + c 1 · m + c 2 · m (fitted)
c0= 0.00282 ! 0.00021
c = 13.251 ! 0.023
1c = ï4.21 ! 0.56
243M X ∼ mq

2

2

1

r2/dof= 0.81

DeGrand:
PRD84 (2011) 116901

m fit range: 0.01 ï 0.035
000.010.02m

0.030.04

Also Jin and Mawhinneyʼs work∗
11+γm
= F (mq L)
MX L

Also LatKMIʼs work

Both chiral broken hypothesis and conformal hypothesis work well.
It might be hard to show the existence of IRFP from the ﬁt quality.
2013年9月27日金曜日+ mq

Mass deformed conformal theory in the continuum limit
If only the mass op. is the relevant op. around the nontrivial ﬁxed point,a scaling of the mass of the composite op. is described by the mass anomalousdim.
1∗
1+γmMiransky scaling MX ∼ mq
P

57

Remarks(1) Around ﬁxed point(Tuning the values of g and ma is needed.
We need two independent observables)‫ٮ‬P(2) Continuum extrapolationγ‫ٮ‬,5)3Jγ2loop prediction:∗γg = 0.36
2013年9月27日金曜日∗γm= 0.77

JFit fn on the lattice

MX = CX m1/(1+γ ∗ )+DX m(If gamma is small, then the ﬁt becomes diﬃcult.
If gamma is large, then the other operator might
become relevant or marginal.)(3) Finite volume eﬀectPhase structure on the lattice
Cheng, Hasenfratz and Schaich:
PRD85 (2012) 094509

HYP smearing

In the intermediate region the shift symmetry is broken.

Deuzeman, Lombardo, da Silva and Pallante:
PLB720(2013)358

Naik improvement
next-to-nearest neighbor terms are no longer irrelevant
and indeed modify the pattern observed without improvement.
Conjectured phase diagram     de Forcrand, Kim and Unger:JHEP 1302(2013)051

In the strong coupling limit, the chiral symmetry is broken Nf<52.

               

2013年9月27日金曜日

    In the studies on the phase structure,
a careful parameter search is important foreach lattice setup.

(4) Volume scaling for the Dirac eigenmodes
In the massless theory, 1/L gives IR cutoﬀ. Instead of the mass, there is a scalingof (1/L). In the chiral limit, a single observable suﬃce to ﬁnd the IR conformality.
Del Debbio and Zwicky, PRD82(2010)014502
Patella,PRD86(2012)025006
de Forcrand, Kim and Unger, JHEP 1302(2013)051

Dirac eigenvalues D(x, y)ψλ = iλψλν(λ) = V� λ−λeigenvalue density in IR
conformal theory

At tree level: λ = 4ρ(ω)dω ∝ V λ241+γm (λ)

�sin2k̂a∼ ( )22LCheng, Hasenfratz, Petropulos and Schaich, JHEP1307(2013)061

IR limit
λ→0L→∞∗γm (λ) → γm

Nf=8 still questionable.
walking?
IRFP with the otherrelevant op.?

Studying the volume dependence is important.
2013年9月27日金曜日

(5) Correlation fn. of nearly conformal theory
Ishikawa, Iwasaki, Nakayama and Yoshie: arXiv:1301.4785
two point fn. of a meson state

conformal theory (massless, continuum)

conﬁned theory(nearly) conformal theory

To extract the mass spectrum, we have to use the Yukawa-type ﬁt fn.

2013年9月27日金曜日

Our result
PTEP (2013) 083B01

2013年9月27日金曜日

Simulation detailHybrid Monte Carlo algorithm
Wilson gauge action+ naive staggered fermion
beta=4.0--100 on (L/a)^4 lattice
L/a=6,8,10,12,16,20
Twisted boundary condition for x,y directionsµ = x, y, z, t

Link variable†Uµ (x + ν̂L/a) = Ων Uµ (x)ΩνFermion†aiπ/3 ab bψα (x + ν̂L/a) = eΩν ψβ (Ων )βα

ν = x, yΩν is twist matrices (3x3 complex matrix)†3i2π/3Ων Ων = I, (Ων ) = I, Tr[Ων ] = 0, Ωx Ωy = eΩy Ωx
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Phase diagramin the lattice setup

2013年9月27日金曜日

In our simulation set up,
there is a bulk phase transition in small mass region.

(L/a) = 44

2013年9月27日金曜日

4In our simulation set up,
there is a bulk phase transition in small mass region.

In the above phase,

�|Pt |� �= 0

In the bottom phase,

�|Pt |� � 0(L/a) = 44

2013年9月27日金曜日

4Phase diagram for SU(3) Nf=12 naive staggered fermion
with the twisted boundary condition.

2013年9月27日金曜日

Phase diagram for SU(3) Nf=12 naive staggered fermion
with the twisted boundary condition.

Our simulation
to search for the IRFP

We also see that the chiral symmetry is preserved in this region.
2013年9月27日金曜日

Running coupling

2013年9月27日金曜日

Measuring the growth ratio
Observe the growth ratio of renormalized coupling constant
to see the precise running behavior.
running coupling constant

growth ratio

σ(u)/u = gR2 (1/sL)/gR2 (1/L)

2gRg ∗2gln(L0 /L)

systematic error is accumulated
2013年9月27日金曜日∗22gR(µ = 1/L)

systematic error is not accumulated

Twisted Polyakov loop (TPL) scheme
Examples of renormalization scheme
Schroedinger functional scheme
Wilson loop scheme
Twisted Polyakov Loop scheme
Wilson flow scheme....

}Definition of Twisted Polyakov loop (TPL) schemeon the latticeno O(a/L) error schemede Divitiis, Frezotti, Gaugnelli and Petronzio,
NPB422(1994)382�†
1 � y,z Px (y, z, L/2a)Px (0, 0, 0) �2
�gTPL = lim†�a → 0 klatt �P(x,y,L/2a)P(0,0,0)zx,y zklatt is determined by the tree level value to satisfy
�22�gTPL=g0tree
2013年9月27日金曜日

Raw data in TPL scheme
2-3 % statistical error.
# of Trj is 64,400- 1,892,800.
Fitting fn. for beta interpolationN�
6Cj (L/a)2gT P L (β, L/a) = +β j=1 β j+1

s=1.5 step scaling
L/a=6 -> L/a=9
L/a=8 -> L/a=12
L/a=10 -> L/a=15
L/a=12 -> L/a=18
For L/a = 9, 15 and 18,
we estimate values of g2 for a given betaby the linear interpolation in (a/L)2.

2013年9月27日金曜日

Growth ratio of TPL coupling(global ﬁt analysis)
TPL coupling shows the ﬁxed pointaround∗2gTPL ∼ 2.7
This is the ﬁrst zero point of thebeta function from theasymptotically free region,
it must be IR ﬁxed point.

Unfortunately, the growth ratio with errorbar does not cross over the unity line.

2013年9月27日金曜日

Local ﬁt analysis
Focus on the low beta region (u>2.0)
Add the data (more than 30 data points)

2013年9月27日金曜日

Growth ratio of TPL coupling(local ﬁt analysis)
0∗2gTPL= 2.69 ± 0.14(stat.)−0.16 (syst.)

statistical error
total errorAround the ﬁxed point, the beta fn.
can be approximated by the linearfn.
β(g 2 ) ∼ γg∗ (g 2∗ − g 2 )Our result

0+0.35∗γg = 0.57−0.31 (stat.)−0.16 (syst.)

SF scheme∗
γg = 0.13 ± 0.03
2013年9月27日金曜日

2 loop at g 2∗ = 9.4∗
γg = 0.36

4 loop (MS bar)∗γg = 0.28

Continuum extrapolation
s=1.5 step scaling
L/a=6 -> L/a=9
L/a=8 -> L/a=12
L/a=10 -> L/a=15
L/a=12 -> L/a=18

2 loop predictionin this region isσ(u = 2.69) ∼ 2.78

The systematic error is small in the strong coupling region in this scheme.
(Fit range dependence and “s” (step scaling parameter) dependence are also small.)2013年9月27日金曜日

Is there an IR ﬁxed point in SU(3) Nf=12 theory?
Ishikawa, Iwasaki, Nakayama, Yoshie (phase structure, correlation fn.)

Appelquist, Fleming, Neil, M.Lin, Schaich (running coupling, mass spectrum)

Deuzeman, Lombardo, Pallante, Miura, da Silva (ﬁnite temperature)

Cheng, A. Hasenfratz, Petropoulos, Schaich (MCRG, phase structure, Dirac eigenmodes)

DeGrand (mass spectrum)

LatKMI (mass spectrum)

D.Lin, Ogawa, Ohki, Shintani (running coupling)

Fodor, Holland, Kuti, Nogradi, Schroeder, (running coupling, phase structure, spectrum)
Jin and Mawhinney (phase structure)

2013年9月27日金曜日

Is there an IR ﬁxed point in SU(3) Nf=12 theory?
Ishikawa, Iwasaki, Nakayama, Yoshie (phase structure, correlation fn.)

Appelquist, Fleming, Neil, M.Lin, Schaich (running coupling, mass spectrum)

Deuzeman, Lombardo, Pallante, Miura, da Silva (ﬁnite temperature)

YES

Cheng, A. Hasenfratz, Petropoulos, Schaich (MCRG, phase structure, Dirac eigenmodes)

DeGrand (mass spectrum)

LatKMI (mass spectrum)

D.Lin, Ogawa, Ohki, Shintani (running coupling)

Fodor, Holland, Kuti, Nogradi, Schroeder, (running coupling, phase structure, spectrum)
Jin and Mawhinney (phase structure)

2013年9月27日金曜日

相互作用をする赤外固定点の見つけ方(1) Step scaling 法によるrenormalized couplingの測定(2) low betaでのchiral symmetry
bulk phaseがあるのでこれだけでは固定点があるかどうかわからない(3) 質量変形した理論のhyperscaling
mass spectrum
ﬁtのクォリティでは固定点があるかどうかわからない(4) Dirac固有モードのhyperscaling (Volume-scaling)
質量以外のrelevantな演算子がある場合は？

(5) メソン的演算子の相関関数の形
ﬁtのクォリティでは固定点があるかどうかわからない？
2013年9月27日金曜日

Anomalous dimension
arXiv:1307.6645[hep-lat]

2013年9月27日金曜日

異常次元の求め方(1) ステップスケーリング法

Luescher, Weisz and Wolﬀ, NPB 359 (1991) 221
ALPHA collaboration, NPB 544 (1999) 669

γm = −γP(2) 質量変形した理論に対するhyperscaling法
複合状態のmass spectrum
Miransky, PRD59(1999)105003
Luty, JHEP 0904(2009)050
Del Debbio and Zwicky, PRD82(2010)014502(3) ゼロ質量におけるDirac固有モードのhyperscaling法(Volume-scaling)ν(λ) ∼ λ4∗
1+γmPatella,PRD86(2012)025006
Cheng, Hasenfratz, Petropulos and Schaich, JHEP1307(2013)061(4) メソン的演算子の相関関数の形から求める方法
Ishikawa, Iwasaki, Nakayama and Yoshie, PRD87(2013)071503

�φ(x)φ(0)� ∼2013年9月27日金曜日c∗ )2(d−γm|x|

Step scaling method- measuring the wave function renormalization constant ALPHA collaboration NPB 544 (1999) 669

g (β, L/a)2r

PCAC relation in QCD
L/a = 16
L/a = 12input

L/a = 8
L/a = 6βZp(β, L/a)

L/a = 6
L/a = 8
L/a = 12
L/a = 16β

2013年9月27日金曜日−1Zm∂ µ (AR )µ = 2mPR= 2(Zm · m0 )(ZP · P0 )Mass scaling function

�ZP (go , a/sL) ��2
σP (g , s) = lim
a→0 ZP (g0 , a/L) � 2
g =const

We measure Z factor of pseudo-scalar op.
at the IRFP.
A new definition of Z factor�
CPtree (t)ZP (g0 ) ≡CP (t)at fixed t固定点を実現する格子セットアップ
各格子サイズでのbetaの値
L/a u=2.475 u=2.686 u=2.823

statistical error
total error

2013年9月27日金曜日

65.378

4.913

4.600

85.7965.4145.181

105.998

5.653

5.450

12

6.121

5.786

5.588

16

6.241

5.909

5.709

20

6.296

5.944

5.733

Nf=12赤外固定点での異常次元に関する現状
E.I., arXiv:1307.6645[hep-lat]mass deformation
Dirac eigenmodestep scaling
Fodor’s data: PLB703 (2011) 348-358
Fit(I): PRD84(2011) 054501
Fit(II): PRD84 (2011) 116901
LatKMI: PRD86 (2012) 054506
Cheng et.al : JHEP1307 (2013) 061
Ours: PTEP (2013)083B01
arXiv: 1307.6645
2013年9月27日金曜日∗γm= 0.044+0.062−0.040Nf=12赤外固定点での異常次元に関する現状
E.I., arXiv:1307.6645[hep-lat]mass deformation
Dirac eigenmodestep scaling

格子間隔を有限に止めて、無限体積極限をとる。
データを更新して、 有限体積効果とbetaの臨界指数の効果を入れたら我々の値に近づいた。＠Lattice2013
2013年9月27日金曜日

phenomenological prediction?!

Assume that ΛT C ∼ 1TeV, ΛET C ∼ 1000TeV

∗
γm=1

∗
γm= 0.5∗γm= 0.25∗
γm= 0.15
2013年9月27日金曜日

Mq ∼1GeV

Mq ∼30MeV

Mq ∼5MeV

Mq ∼2MeV

残った課題 :step scaling法
pseudo-scalarのZ因子の連続極限の様子
s=1.5 step scaling
L/a=6 -> L/a=9(離散化誤差大きすぎる)
L/a=8 -> L/a=12
L/a=10 -> L/a=15
L/a=12 -> L/a=18

3つのデータ点を用いた２パラメータフィットによる外挿では系統誤差の見積もりが不十分
もう一つ大きい格子サイズデータが必要
2013年9月27日金曜日

残った課題 :mass deformed法
P57- critical betaへのtuning- massless 近傍へのtuning
1
∗
1+γm

γ‫ٮ‬
P,5)3γ‫ٮ‬JM X ∼ mqJ

beta=4.0 HISQ action
LatKMI
arXiv:1305.6006
2013年9月27日金曜日

他の理論の場合2013年9月27日金曜日

mass deformed theoryによるhyperscaling
SU(2) Nf=2 adjoint fermions

横軸: PCAC mass
Patella et.al., PoS Lattice2010:068,20102013年9月27日金曜日

Mass anomalous dim.
SU(2) Nf=2 adjoint fermion(minimal walking technicolor)
2 loop3 loop (MS bar)
4 loop (MS bar)

step scaling

PRD81 (2010) 014505
PRD83 (2011) 074507
PRD82 (2010) 014510

mass deformation

PRD82 (2010) 014509
PRD85 (2012) 097503

Dirac eigenmode

PRD86 (2012) 025006
0

2013年9月27日金曜日

0.2

0.4

am*

0.6

0.8

running coupling constantSU(2) Nf=2 adjoint fermions
Del Debbio et. al., PRD81 (2010) 014505

連続極限の系統誤差はコントロール
できていないけど…

beta=2.25
2013年9月27日金曜日

Conclusion and Discussion
The IRFP exists in SU(3) Nf=12 massless theory.
Continuum extrapolation and parameter search are important.
(1) The phenomenological model construction by using the mass
anomalous dimension from the lattice simulation.
Nf=12 model must be killed by lattice results. (Also minimal walking technicolor: Del Debbio et.al.)(2) Spectrum for several modes, who is the lightest state?
pseudoscalar? dilaton?
(3) Study on universal quantities(anomalous dimension, “central charge” in 4-dim)(4) We have to understand the property of conformal ﬁeld theory on theﬁnite lattice volume. (functional form of correlation fn.)
Lattice precise data can give phenomenological and theoretical information
around nontrivial fixed point.
2013年9月27日金曜日