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

Collaborators: 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 Introduction Renormalization schemes in twisted boundary condition 12-flavor SU(3) gauge theory a) running coupling b) mass anomalous dimension (preliminary) 4. 8-flavor SU(2) gauge theory (preliminary) 5. Summary

1. Introduction

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

LHC are revealing the mechanism for Electroweak symmetry breaking. As we know the LHC experiment is running to reveal the electroweak symmetry breaking mechanism. Out of possible scenarios, I would like to focus on the strong dynamics scenario. Unfortunately many of them are just effective theories so that the prediction power is low. I would like to know if there is a description from a UV completed theory such as gauge theory.

Results of the Higgs search at LHC Allowed range of Higgs mass Light SUSY Higgs ?, 　　　or Heavy strong coupling Higgs?

Recent results of the Higgs search at LHC Moriond 2012 Light SUSY Higgs ?, 　　　or Heavy strong coupling Higgs?

We need to understand strong dynamics Strong coupling Higgs has large self-interactions. The RG-flow hits the Landau pole at not so high energy. Higgs triviality bound (PDG) Therefore, it should be replaced by a more fundamental theory at 1-10TeV scale. As we know the LHC experiment is running to reveal the electroweak symmetry breaking mechanism. Out of possible scenarios, I would like to focus on the strong dynamics scenario. Unfortunately many of them are just effective theories so that the prediction power is low. I would like to know if there is a description from a UV completed theory such as gauge theory. We need to understand strong dynamics from UV complete theory such as gauge theory.

Conformal dynamics from QCD? Large Nf flavor QCD has an Infrared Fixed Point (IRFP) (Caswell-Banks-Zaks) in perturbation theory. e.g. 2-loop beta function Does this IR fixed point exist beyond perturbation theory?  Lattice Studies are needed.

Lattice gauge theory with flavors ? ? 2-loop perturbation Confinement, Conformal window IR fixed point Ladder Schwinger Dyson Conformal window IR fixed point Confinement, Lattice ? ? Confinement, Conformal window IR fixed point Asymptotic free Aymptotic nonfree

Previous Lattice Studies in Nf=12 QCD on the running coupling 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. 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).

2. Renormalization schemes in twisted boundary condition

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.

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. Better to avoid 3-dim boundary  Twisted boundary condition.

Twisted boundary condition in SU(Nc) gauge theory (‘t Hooft NPB153:131) kills the zero-modes in finite volume Boundary condition for fermions, Parisi 1983, unpublished color smell We 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) 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 ) 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. If holds for some u*, we can verify the existence of the IR fixed point.

Definition of the renormalization scheme for mass operator mass and pseudoscalar operator are related by PCAC relation. The renormalization factor Zm is the inverse of Zp

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.

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.

3. 12-flavor SU(3) gauge theory

Lattice Setup Wilson Plaquette gauge action Staggered fermion action (exact partial chiral symmetry) Box size L/a=6,8,10,12,16,20 Bare coupling : # of trajectories: Hybrid Monte Carlo algorithm Simulations were carried out on NEC SX-8, SR16000 at YITP, Kyoto U NEC SX-8 at RCNP, Osaka U SR11000 and BlueGene/L at KEK 100 GPUs in XinChu University 3 x Staggered fermion : 3x4=12-flavors Twisted boundary condition Polyakov-line correlators  running coupling scheme Pseudoscalar correlators  running mass scheme

3-a) Running coupling

Raw data We 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

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

Our Result: There exists a fixed point at Nf=12 QCD is in the interacting Coulomb phase! Anomalous dimension ( c.f. ) Running coupling Fixed point

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**

3-b) mass anomalous dimension

Z factor Step scaling function

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

Preliminary Results total error statistical only At the fixed point, the mass anomalous dimension is given as total error statistical only

Mass anomalous dimension from various groups Vermaseren, Larin and Ritbergen PL B405(1997)327 Ryttov and Shrock PRD83 (2011) 056011 Yamawaki, 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)

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

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

Lattice Setup Wilson Plaquette gauge action Staggered fermion action (exact partial chiral symmetry) Box size L/a=6,8,10,12,16,18 Bare coupling : # of trajectories: Hybrid Monte Carlo algorithm Simulations were carried out on NEC SX-8, SR16000 at YITP, Kyoto U NEC SX-8 at RCNP, Osaka U SR11000 and BlueGene/L at KEK 2 x Staggered fermion : 2x4=8-flavors Twisted boundary condition Polyakov-line correlators  running coupling scheme Pseudoscalar correlators  in progress

Raw data We 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

IR fixed point at 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.) .