1. Study of the conformal fixed point in many flavor QCD on the lattice
Tetsuya Onogi (Osaka U)
Based on arXiv: [hep-lat]
+ some work in progress
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)
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.
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.

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. 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.

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. 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

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
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

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

19. 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.

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
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

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 = )
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= )
**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 total error statistical only
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)
Yamawaki, Bando and Matumoto: PRL 56, 1335 (1986)PR D84(2011)054501arXiv: [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. 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

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
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

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. IR fixed point at 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.) .