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

1. Study of the conformal fixed point in many flavor QCD on the lattice Based on arXiv:1109.5806 [hep-lat] + some work in progress Tetsuya Onogi (Osaka U) KMI miniworkshop “Conformality in Strong Coupling Gauge Theories at LHC and Lattice”, at Nagoya March19, 2012

2. 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

3. 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

4. 1. Introduction

5. 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.

6. LHC are revealing the mechanism for Electroweak symmetry breaking.

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

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

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

10. 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.

11. 2-loop perturbation gauge theory with flavors Conformal window IR fixed point Confinement, Confinement, Confinement, Ladder Schwinger Dyson Conformal window IR fixed point Lattice ? ? Conformal window IR fixed point Asymptotic free Aymptotic nonfree

12. 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.

13. 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).

14. 2. Renormalization schemes in twisted boundary condition

15. 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.

16. 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.

17. 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 We introduce ‘smell’ degrees of freedom: i=1,..,Ns(=Nc) color smell • For staggered fermion: SU(3) with 12-flavors, SU(2) with 8-flavors

18. Twisted Polyakov-Line (TPL) • Running coupling in TPL scheme (di Vitiis et al.)

19. 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 ) Step scaling function • 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.

20. 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

21. 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.

22. 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.

23. 3. 12-flavor SU(3) gauge theory

24. 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 • 3 x Staggered fermion : 3x4=12-flavors • Twisted boundary condition • Polyakov-line correlators  running coupling scheme • Pseudoscalar correlators  running mass scheme • 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

25. 3-a) Running coupling

26. 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

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

28. 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

29. 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**

30. 3-b) mass anomalous dimension

31. Z factor Step scaling function

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

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

34. 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)

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

36. 4. 8-flavor SU(2) gauge theory(Preliminary)

37. 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 • 2 x Staggered fermion : 2x4=8-flavors • Twisted boundary condition • Polyakov-line correlators  running coupling scheme • Pseudoscalar correlators  in progress • 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

38. 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

39. Continuum extrapolation of step scaling function u denpendence of sigma(u)/u IR fixed point at

40. 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.) .