格子シミュレーションを用いた
QCDを超えるゲージ理論に対する最近の研究
arXiv:1212.1353[hep-lat] and work in progress

Etsuko Itou (KEK)2013/03/07@ HPCI戦略分野５全体シンポジウム(秋葉原)
2013年3月6日水曜日

Numerical simulation was carried outon
NEC SX-8 and SR16000 @ YITP, Kyoto
NEC SX-8 @ RCNP, Osaka
SR11000 and BlueGene/L
SR16000 and BlueGene/Q @ KEK
100 GPUs @ Osaka and Taiwan
also use JLDG
2013年3月6日水曜日

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
Walking techni-color
Fourth generation.....
2013年3月6日水曜日

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
Walking techni-color
Fourth generation.....
2013年3月6日水曜日

Google“Extra dimension 2013”2013年3月6日水曜日

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
Fourth generation.....
2013年3月6日水曜日

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3月6日水曜日

Is there a theory whose coupling constant show the behavior?

This part may be realized bymany flavor gauge theory.
2013年3月6日水曜日

SU(3) Nf=12 theory
Two loop analysis

perturbative (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)
S-D eq. with large NcNfcr = 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3月6日水曜日

Y.Kusafuka and H.Terao,
arXiv:1104.3606 [hep-ph]

11Nc ∼ 16.5
2

Asymptotically free
Conformal Window

Non freeNfIs there an IR fixed point in SU(3) Nf=12 theory?
Iwasaki et al, '04 ’13 (phase structure, correlation fn.)
Appelquist, Fleming, Neil '07, '09, '11 (running coupling, mass spectrum)
Deuzeman, Lombardo, Pallante, Miura '09, '11(finite temperature)
A. Hasenfratz '09, '10 (MCRG, phase structure)
DeGrand '11 (mass spectrum)
LatKMI ’12 (mass spectrum)
Fodor et al. '09 , '11(running coupling, phase structure, spectrum)
Jin and Mawhinney '09 (phase structure)

2013年3月6日水曜日

Is there an IR fixed point in SU(3) Nf=12 theory?
Iwasaki et al, '04 ’13 (phase structure, correlation fn.)
Appelquist, Fleming, Neil '07, '09, '11 (running coupling, mass spectrum)
Deuzeman, Lombardo, Pallante, Miura '09, '11(finite temperature)

YES

A. Hasenfratz '09, '10 (MCRG, phase structure)
DeGrand '11 (mass spectrum)
LatKMI ’12 (mass spectrum)

Fodor et al. '09 , '11(running coupling, phase structure, spectrum)
Jin and Mawhinney '09 (phase structure)

2013年3月6日水曜日

Why there arecontroversial results?

2013年3月6日水曜日

Running coupling constantAppelquist et al. (SF scheme)
Phys.Rev.D79:076010,2009

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

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

Relationship between two renormalization schemes
scheme transformation: g1 → g2 = f (g1 )beta fn.

2013年3月6日水曜日∂f (g1 )β(g2 ) =β(g1 )∂g1Existence of fixed point isscheme independent.

Why there arecontroversial results?
The continuum extrapolation should be taken carefully.
- conformal theory is realized in the continuum- Lattice law data suffer from O(a)- O(a) effect is renormalization scheme dependent

In the study on the phase structure, parameter search is notenough?
- tuning the beta value2013年3月6日水曜日

chiral symmetry

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.

- Fodor et. al. comparison between two hypotheseschiral extrapolation using conformal hypothesis does not work.
chiral broken hypothesis works well.
chiral symmetry is weakly broken2�Ψ̄Ψ�
=c+cm+cm
fit fn:012

c0 = 0.00282 ± 0.00021- LSD collab. fit the limited # of Fodor’s data using
conformal hypotheses PRD84(2011)054501- DeGrand considers finite-scaling PRD84 (2011) 116901
conformal hypothesis also works well.
2013年3月6日水曜日

Weakly chiral symmetry broken phase
Chang, Hasenfratz and Schaich:Phys.Rev.D85 (2012) 094509

HYP smearing
two jumps of chiral condensate
and in the intermediate region
shift symmetry is broken.

Deuzeman, Lombardo, da Silva and Pallante:
arXiv: 1209.5720

Naik improvement
next-to-nearest neighbor terms
are no longer irrelevant
and indeed modify the patternobserved without improvement.
2013年3月6日水曜日

conjectured phase diagram for many flavor SU(3) gauge theory
de Forcrand, Kim and Unger:
arXiv:1208.2148

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

cf. for SU(Nc),
Tomboulis arXiv:1211.4842
2013年3月6日水曜日

Our result
arXiv:1212.1353 [hep-lat]

2013年3月6日水曜日

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

In the above phase,

�|Pt |� �= 0

In the bottom phase,

�|Pt |� � 0

2013年3月6日水曜日

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3月6日水曜日

Running coupling

2013年3月6日水曜日

Measure the growth ratio
running coupling constant

growth ratio

σ(u)/u = gR2 (1/sL)/gR2 (1/L)

2gRg ∗2g∗2ln(L0 /L)

2gR(µ = 1/L)

systematic error is accumulated

systematic error is not accumulated

No O(a) renormalization scheme
Twisted Polyakov loop (TPL) schemede Divitiis et. al. NPB422(1994)382
2013年3月6日水曜日

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

statistical error
total error

Around the fixed point, the beta fn.
can be approximated by the linear fn.
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3月6日水曜日

2 loop at g 2∗ = 9.4∗
γg = 0.36

4 loop (MS bar)∗γg = 0.28

Anomalous dimensionーPreliminary resultー

2013年3月6日水曜日

The other important critical exponent around the IRFP

m2loop prediction:∗γg = 0.36m2m1∗γmγg∗ IRFP
2013年3月6日水曜日

g2∗
γm = 0.77

Mass anomalous dimfrom the several studies
2 loop3 loop (MS bar)
4 loop (MS bar)
Schwinger-Dyson(ladder)Appelquist et.al
DeGrandOur result (r=1/3)(r=1/4 w. stat. err.)(preliminary)mass deform
direct measure∗γmVermaseren, Larin and Ritbergen PL B405(1997)327
Ryttov and Shrock PRD83 (2011) 056011
Yamawaki, Bando and Matumoto: PRL 56, 1335 (1986)
PRD84(2011)054501
PRD84(2011)116901

2013年3月6日水曜日

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

Yukawa-type fit fn works well.
cf. 2-dim. scalar CFT on a cylinder1�φ(z)φ(0)� = 2hz
2013年3月6日水曜日

�φ(w)φ(0)� =�2πL

�2h2h(2 sinh[πw/L])

Conclusion
標準模型を超える物理を考える時代。
walking technicolor模型には、非摂動論領域の精密な情報が必要。
conformal理論=QCDの赤外領域と異なる理論- conformal理論の存在の判定、方法論の確立- 格子理論の相構造、 有限体積の場合の性質...
我々の結果
SU(3) Nf=12 massless 理論には固定点が存在する連続極限をとる事、パラメータサーチが重要
固定点の共形場の理論の性質、模型の低エネルギー物理への予言はこれから！
2013年3月6日水曜日