Study of the conformal fixed point in many flavor QCD on the

lattice

Tetsuya Onogi (Osaka U) Based on arXiv:1109.5806 [hep-lat] + some work in progress

KMI miniworkshop“Conformality in Strong Coupling Gauge Theories at LHC and Lattice”, at Nagoya March19, 2012

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Collaborators:

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KMI, Nagoya: T. Aoyama, M. Kurachi, H. Ohki, T. Yamazaki

KEK: E. Itou, H. Ikeda, H. Matsufuru

National Chiao-Tung U.: C.-J.D. Lin, K. Ogawa Riken-BNL: E. Shintani

Outline

1. Introduction

2. Renormalization schemes in twisted boundary condition

3. 12-flavor SU(3) gauge theory a) running coupling b) mass anomalous dimension (preliminary)

4. 8-flavor SU(2) gauge theory (preliminary)

5. Summary3

1. Introduction

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Listening to previous talks, I found that my slides (pages 6-12) must be skipped, because you have seen similar slides too many times.

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Before starting my talk, ……

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LHC are revealing the mechanism for Electroweak symmetry breaking.

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Allowed range of Higgs mass

Results of the Higgs search at LHC

Light SUSY Higgs ?, 　　　 or Heavy strong coupling Higgs?

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Recent results of the Higgs search at LHC

Light SUSY Higgs ?, 　　　 or Heavy strong coupling Higgs?

Moriond 2012

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Strong coupling Higgs has large self-interactions.The RG-flow hits the Landau pole at not so high energy.

Therefore, it should be replaced by a more fundamental theory at 1-10TeV scale.

We need to understand strong dynamics from UV complete theory such as gauge theory.

Higgs triviality bound (PDG)

Conformal dynamics from QCD?• Large Nf flavor QCD has an Infrared Fixed Point (IRFP) (Caswell-Banks-Zaks) in perturbation theory.

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Does this IR fixed point exist beyond perturbation theory? Lattice Studies are needed.

　　　　 gauge theory with flavors

Aymptotic nonfreeAsymptotic free

?

Conformal windowIR fixed point

2-loop perturbation

Ladder Schwinger Dyson

Confinement,

Confinement,

Lattice

Confinement, Conformal windowIR fixed point

Conformal windowIR fixed point

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?

Previous Lattice Studies in Nf=12 QCD

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Appelquist, Fleming et al. (SF scheme)Phys. Rev. D79:076010, 2009

Kuti et al. (potential scheme)PoS LAT2009:055, 2009

IRFP! No IRFP…..

The existence of the IRFP should not depend on the scheme.The situation is still controversial.

on the running coupling

Other approaches MCRG: Hasenfratz Spectrum: Pallante et al., LatKMI collab. Finite temperature: Pallente et al., Kuti et al.

Goal of this work

• We give a lattice study of the running coupling constant (and mass anomalous dimension) in QCD with many flavor in fundamental representation. SU(3), nf=12 SU(2), nf=8

• We take continuum limit using schemes in twisted boundary condition, which are free from discretization errors of O(a).

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2. Renormalization schemes in twisted boundary condition

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Definition of the running coupling scheme

• We study the conformal fixed using renormalization scheme in finite volume.

• RG-flow is probed by “step-scaling”, which is the change under the change of the volume.

• In order to avoid the (perturbative) infra-red divergence in finite volume, we need to kill both the gluonic and fermionic zero-modes by some boundary condition.

example: Dirichlet boundary condition ( SF scheme ) Our choice Twisted boundary condition.

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Discretization error in Dirichlet boundary condition. There exist O(a) counter terms in the action in 3-dim

Dirichlet boundary, which are not prohibited by the symmetry and can be the source of O(a) errors.

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Better to avoid 3-dim boundary Twisted boundary condition.

• Twisted boundary condition in SU(Nc) gauge theory (‘t Hooft NPB153:131)

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kills the zero-modes in finite volume

• Boundary condition for fermions, Parisi 1983, unpublished

color smellWe introduce ‘smell’ degrees of freedom: i=1,..,Ns(=Nc)

•For staggered fermion:

SU(3) with 12-flavors, SU(2) with 8-flavors

• Twisted Polyakov-Line (TPL)

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• Running coupling in TPL scheme (di Vitiis et al.)

Step scaling function

• Using the renormalized coupling defined in finite box of L^4, The renormalization group evolution can be obtained by studying the volume dependence. (renorm. scale )

• If holds for some u*, we can verify the existence of the IR fixed point.

• Change the volume by factor s as L s L ( s=1.5, or 2, ….) and consider the step scaling function: the RG for finite change of scale.

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• mass and pseudoscalar operator are related by PCAC relation.

• The renormalization factor Zm is the inverse of Zp

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Definition of the renormalization scheme for mass operator

A new scheme for

• Compute the 2-point Pseudoscalar correlator

• Then impose the renormalization condition

• The renormalization factor Zp is defined as

at fixed t = rL.

• We choose r(=t/L) =1/3 as the optimal choice.

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Step scaling

Step scaling function for pseudoscalar operator P can be defined by the ratio

To take the continuum limit we use the renormalized gauge coupling as input.

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3. 12-flavor SU(3) gauge theory

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Lattice Setup

3 x Staggered fermion : 3x4=12-flavors Twisted boundary condition Polyakov-line correlators running coupling scheme Pseudoscalar correlators running mass scheme

Wilson Plaquette gauge actionStaggered fermion action (exact partial chiral symmetry)Box size L/a=6,8,10,12,16,20Bare coupling :# of trajectories: Hybrid Monte Carlo algorithm

Simulations were carried out onNEC SX-8, SR16000 at YITP, Kyoto UNEC SX-8 at RCNP, Osaka USR11000 and BlueGene/L at KEK100 GPUs in XinChu University

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3-a) Running coupling

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Raw dataWe fit beta dependence of the data with the function

We take s=1.5 for the step size

Data for L/a=9,15,18 are obtained by linear interpolation (a/L)^2

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Continuum limitAt each step, we make a linear extrapolation in (a/L)^2 with 3 points or 4 points.

Input value

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Fixed point

Running coupling

( c.f. )

Anomalous dimension

Our Result: There exists a fixed point at

Nf=12 QCD is in the interacting Coulomb phase!

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Is there IR fixed point?

Appelquist, Flemming et al. : YES (SF scheme g^2*~5, gamma_g =0.10-0.16)

Fodor, Kuti et al. : NO (Potential scheme)

Hasenfratz : YES (Monte Carlo RG, bare step scaling)

Our group : YES (TPL scheme g^2*~2.5, gamma_g=0.28-0.79)

**The critical exponent is not consistent with each other** 29

3-b) mass anomalous dimension

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Z factor

Step scaling function

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Continuum extrapolation (linear in (a/L)^2)

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Preliminary Results

total errorstatistical only

At the fixed point, the mass anomalous dimension is given as

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Mass anomalous dimension from various groupsVermaseren, Larin and Ritbergen PL B405(1997)327Ryttov and Shrock PRD83 (2011) 056011Yamawaki, Bando and Matumoto: PRL 56, 1335 (1986)PR D84(2011)054501arXiv:1109.1237[hep-lat]

2 loop

3 loop (MS bar)

4 loop (MS bar)

Schwinger-Dyson

Appelquist et.al

de Grand

Our result (r=1/3)

Our result (r=1/4)

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IRFP

Mass dependence for fixed lattice spacing may probe the anomalous dimension in mass deformed theorybut not that in the conformal theory itself.

4. 8-flavor SU(2) gauge theory (Preliminary)

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Lattice Setup

2 x Staggered fermion : 2x4=8-flavors Twisted boundary condition Polyakov-line correlators running coupling scheme Pseudoscalar correlators in progress

Wilson Plaquette gauge actionStaggered fermion action (exact partial chiral symmetry)Box size L/a=6,8,10,12,16,18Bare coupling :# of trajectories: Hybrid Monte Carlo algorithm

Simulations were carried out onNEC SX-8, SR16000 at YITP, Kyoto UNEC SX-8 at RCNP, Osaka USR11000 and BlueGene/L at KEK

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Raw dataWe fit beta dependence of the data with the function

We take s=1.5 for the step size

Data for L/a=9,15 are obtained by polynomial interpolation (a/L)^2

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Continuum extrapolation of step scaling function

u denpendence of sigma(u)/u

IR fixed point at

5. Summary• We study Nf=12 SU(3) and Nf=8 SU(2) gauge theories

with TPL scheme using lattice.• We find that there are Infrared (IR) fixed points.• We also obtained preliminary results on the anomalous

dimension for the running coupling and the mass for SU(3).

Future prospects• Further studies are need to control the systematic

errors for the anomalous dimensions. (Larger volume.)• Our method can be applied to other theories (e.g.: adjoint rep.) .

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