Title: 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 arXiv1109.5806 hep-lat
some work in progress
KMI miniworkshop Conformality in Strong Coupling
Gauge Theories at LHC and Lattice,
at
Nagoya March19, 2012
2Collaborators
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
3Outline
- 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
41. 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.
6LHC are revealing the mechanism for Electroweak
symmetry breaking.
7Results of the Higgs search at LHC
Allowed range of Higgs mass
Light SUSY Higgs ?, or Heavy strong coupling
Higgs?
8Recent results of the Higgs search at LHC
Moriond 2012
Light SUSY Higgs ?, or Heavy strong coupling
Higgs?
9Strong 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.
10Conformal dynamics from QCD?
- Large Nf flavor QCD has an Infrared Fixed Point
(IRFP) (Caswell-Banks-Zaks) in perturbation
theory. -
-
-
Does this IR fixed point exist beyond
perturbation theory? ? Lattice Studies are
needed.
11- gauge theory with flavors
2-loop perturbation
Conformal window IR fixed point
Ladder Schwinger Dyson
Conformal window IR fixed point
Lattice
?
?
Conformal window IR fixed point
Asymptotic free
Aymptotic nonfree
12Previous Lattice Studies in Nf12 QCD
on the running coupling
Appelquist, Fleming et al. (SF scheme) Phys.
Rev. D79076010, 2009
Kuti et al. (potential scheme) PoS LAT2009055,
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), nf12
- SU(2), nf8
- We take continuum limit using schemes in twisted
boundary condition, which are free from
discretization errors of O(a).
142. Renormalization schemes in twisted boundary
condition
15Definition 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 NPB153131)
kills the zero-modes in finite volume
- Boundary condition for fermions, Parisi 1983,
unpublished
We introduce smell degrees of freedom
i1,..,Ns(Nc)
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 L4, The renormalization group evolution
can be obtained by studying the volume
dependence. (renorm. scale )
20Definition 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
21A 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.
22Step 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.
233. 12-flavor SU(3) gauge theory
24Lattice Setup
- Wilson Plaquette gauge action
- Staggered fermion action (exact partial chiral
symmetry) - Box size L/a6,8,10,12,16,20
- 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 3x412-flavors
- Twisted boundary condition
- Polyakov-line correlators ? running coupling
scheme - Pseudoscalar correlators ? running mass scheme
253-a) Running coupling
26Raw data
We fit beta dependence of the data with the
function
We take s1.5 for the step size
Data for L/a9,15,18 are obtained by linear
interpolation (a/L)2
27Continuum limit
At each step, we make a linear extrapolation in
(a/L)2 with 3 points or 4 points.
Input value
28Our Result There exists a fixed point at
Nf12 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
g25, 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
g22.5, gamma_g0.28-0.79) - The critical exponent is not consistent
with each other
303-b) mass anomalous dimension
31Z factor
Step scaling function
32Continuum extrapolation (linear in (a/L)2)
33Preliminary Results
At the fixed point, the mass anomalous dimension
is given as
total error statistical only
34Mass 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)054501arXiv1109.1237hep-lat
35Mass dependence for fixed lattice spacing may
probe the anomalous dimension in mass deformed
theory but not that in the conformal theory
itself.
364. 8-flavor SU(2) gauge theory
(Preliminary)
37Lattice Setup
- Wilson Plaquette gauge action
- Staggered fermion action (exact partial chiral
symmetry) - Box size L/a6,8,10,12,16,18
- 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 2x48-flavors
- Twisted boundary condition
- Polyakov-line correlators ? running coupling
scheme - Pseudoscalar correlators ? in progress
38Raw data
We fit beta dependence of the data with the
function
We take s1.5 for the step size
Data for L/a9,15 are obtained by polynomial
interpolation (a/L)2
39Continuum extrapolation of step scaling function
u denpendence of sigma(u)/u
405. Summary
- We study Nf12 SU(3) and Nf8 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.) .