Infrared fixed point for many flavor
SU(N) gauge theory
Etsuko Itou (Osaka University)
To appear arXiv 1108:xxxx

Renormalization Group Approach from Ultra Cold Atoms to the Hot QGP
2011/08/25@ YITP, Kyoto

Collaborators
 T. Aoyama, M. Kurachi, H. Ohki, T. Yamazaki(Nagoya University) H. Ikeda (Sokendai, KEK)
 C.-J David Lin, K. Ogawa (NCTU)
 H. Matsufuru (KEK) T. Onogi (Osaka University)
 E. Shintani (Riken-BNL)
Numerical simulation was carried out on
NEC SX-8 and SR16000 in YITP, Kyoto
UniversityNEC SX-8 in RCNP, Osaka University
SR11000 and BlueGene in KEK
100 GPU in Taiwan

Introduction

Summary of our work
 We study SU(3) gauge theory coupled with Nf=12

fundamental fermions Measure the running coupling constant in this theory andsearch for an IR fixed point Derive the universal quantities around the fixed point

Introduction
In the case of SU(3) coupled with fundamental fermions・Two-loop running coupling :perturbation (MS bar scheme)
2-loop 3-loop 4-loop(alpha) 0.754
(g^2)
9.42

0.435
5.47

0.470
5.90

T. A. Ryttov and R. Shrock, Phys. Rev. D83, 056011 (2011)

S-D eq. with large Nc
Nf > 11.9, there is IR fixed point
Exact RG
H.Gies and J.Jaeckel, Eur.Phys.J.C46:433-438,2006

Conformal Window

Asymptotic free

Asymptotic
NonfreeHiggs sector in the Standard Model Lagrangian

Problem with a fundamental Higgs boson
No fundamental scalars observed in nature
Hierarchy problem (need fine-tuning to cancel a quadratic divergence)
Triviality problem

No interaction at low energy
Running coupling constant diverges at a finite energy
Cuttoff theory?
Candidates for the origin of Higgs sector
Supersymmetry
Extra dimension
Walking techni-color
Fourth generation.....

Walking techni-color (TC)cutoff theory at, (but it is better UV complete theory)
Introduce a new gauge interaction (ex. SU(2), SU(3),SO(N)....)
Add techni-fermions carrying techni-color charge(adjoint rep, fundamental rep,.....)
highenergy

SM + TC gauge interaction
EW symmetry breaks
Standard Model

lowenergy

Conditions which techni-color gauge interaction should be satisfiedTC coupling should be large at TC scale
is larger than 103 TeVbeta fn. has (nearly) zero point at a large TC couplinghighenergy

Constraints for TC couplingtechni-fermion should be condensed(condensate gives quark mass and seedof EW breaking)lowenergy

Techni-color gauge theory is strongly coupled at least TC scale

Constraints for ETC scale
SM+TC theory is Wilsonian effective action.
There are some interaction terms for quark(lepton)( ) and techni-fermions ( )highenergy

4-quark interaction

2-quark and 2-techni-fermion

QCD coupling does not large, then we canneglect the running effects for 4-quark(lepton) interaction

bound from FCNC (lowenergy) gives a lower limit on ETC scale

UFit Collaboration: JHEP 0803,049 (2008)

Constraints for quark mass term
highenergy

lowenergy

2-quark and 2-techni-fermion

quark mass should be enhanced

large coupling but beta fn~0Let’s find such theory
which has an fixed pointwith large couplingRecent Study of Lattice groups
Program at Lattice 2011
There are two plenary talks, more than18 parallel session talks and several poster session
talks concerning with ``conformal window”

SU(3) gauge theory coupled withfundamental fermion

The other gauge theory
The other representation of fermion

The Lattice study for IR behavior of manyflavor QCD theory

 Study the running coupling constant Measure the universal quantities
 Investigate the phase structure (chiral symmetry)

Running couplingA definition of nonperturbative renormalized coupling
Lattice simulation can calculate the vev of <O>NPExamples of scheme• Schrodinger functional scheme• Wilson loop scheme : Phys.Rev.D80:034507(2009)• Twisted Polyakov Loop scheme :no O(a/L) errorscheme

Nonperturbative definition of renormalized coupling in TPL scheme

In SU(3) lattice gauge theory,
the VEV of operator O depends on
Lattice size (L0)
Lattice spacing (a)bare coupling constant“renormalized coupling”
on latticeaL0

How to take the continuum limit

To take the continuum limit, we have to set the scale “ ”.
It corresponds to tuning
to keep a certain input physical parameter constant.

inputoutput

IR cutoff is movingStep scaling method
fixed ``a” (=beta)change L0 -> sL0(continuum limit)

aL0asL0sasL0fixed ``a” (=beta)change sL0 -> s2L0
sas2L0

Step scaling method
renormalized coupling

1.Choose a value of renormalized
coupling constant at energy scale()
2.Tuning the beta (barecoupling) for small lattice size

Lattice size

Step scaling method
renormalized coupling

1.Choose a value of renormalized
coupling constant at energy scale()
2.Tuning the beta (barecoupling) for small lattice size

3.Carry out the simulation forthe large lattice size

Lattice size

Step scaling method
renormalized coupling

1.Choose a value of renormalized
coupling constant at energy scale()
2.Tuning the beta (barecoupling) for small lattice size

3.Carry out the simulation forthe large lattice size
4.Take the continuum limit(energy scale)Lattice size

Step scaling method

Tune the beta at small lattice size

Step scaling method

Take the continuum limit(energy scale)Step scaling method
We obtain the scaling of therunning coupling.

Status of SU(3) Nf=12 theory

Running coupling constant on Lattice
Nf=12

Appelquist’ group (SF scheme) ,Phys.Rev.D79:076010,2009
The running coupling shows a flat region
The value of fixed point coupling g^2~5.
It consists with 3-loop lattice perturbative analysis.
Fodor’s group: (potential scheme)
arXiv:0911:2463 [hep-lat], talk at Lattice2010
The renormalized coupling is larger than 1,2,3loop perturbartive result.
The coupling at low energy region is growing.
(No fixed point.)

Continuum extrapolation in SF scheme
constant extrapolation?

s=2 step scaling
L=6 -> L=12
L=8 -> L=16
L=10 -> L=20

Our result

Continuum Extrapolation(TPL scheme)

Really good scaling behavior.
Each data at the finite lattice grows up.
However, the continuum
extrapolation shows the existence offixed point!!

s=1.5 step scaling
L=6 -> L=9
L=8 -> L=12
L=10 -> L=15
L=12 -> L=18

Running coupling constant in TPL scheme

Each line denotes 100
Jackknife sample

Exponent of the beta function
Around the fixed point, the beta function
can be approximated by linear function:

Our result:

SF scheme (Appelquist)

2 loopMass anomalous dimension-preliminary results-

New definition of the anomalous dimension using psuedo-scalar correlator
Measure the psuedo-scalar correlator
L (lattice size:t direction)
Z factor

at fixed t

scaling function

Here, we fix t/L for eachlattice size.
Choice of t/L corresponds
scheme.

anomalous dimension at the fixed point

PCAC relation in QCDOur result (Mass step scaling fn. ~ Zm factor)¥gamma*~0.29
4-loop ¥gamma*~0.253
S-D eq. ¥gamma*~1

Preliminary

T. A. Ryttov and R. Shrock,Phys. Rev. D83, 056011 (2011)

Comments:We introduce m=10-5 tomeasure the correlator.
We checked the data does notchange if we use m=10-7.

The other works

Study IR behavior of Masses for composite operators andchiral condensate
They measured lattice data include a fermion mass .Then, they extrapolated to massless
limit using small mass expansion.

 Fodor group Appelquist group(arXiv:1104.3124)(arXiv:1106.2148)

Chiral symmetry broken
hypothesis works wellrather than conformal
hypothesis!

Conformal hypothesis also
works well!

 X-Y. Jin (Columbia U group) KMI group

Chiral symmetry is broken in IR
simulationMeasure the psedoscalar mass

Mass deformation
Assuming only the mass op. is relevant in the IR.
Then, if we take a massless limit in the low energy region,
we can get a scaling behavior around IR fixed point.

Raw Data (arXiv:1104.3124, Fodor et al.)The simulation input parameters are lattice size and fermion mass.(beta=2.2 =enoughlow energy)
Appelquist’s group used only largest volume data for each m value .
Lattice size
24^3×4832^3×6440^3×8048^3×96
nearly cont.lim.

Fermion mass
0.035
0.020
....
0.015
0.010massless lim.

Measured masses“Pion”“Pion” decay constant“axial vector”“scalar”….
Chiral condensateConformal hypothesis with mass deformation
The running mass

(if there is a fixed point)
We assume the running mass satisfies at some IR scale M
Mass of physical state X is set by the scale M.
There is a correction term in a small-m expansion:
The chiral condensate can be written by
The leading term should be added in small-m expansion:

Conformal and broken examples
SU(2) Nf=2 ADOJOINT fermions(widely believed to be conformal in IR.)
Gamma* is free parameter for each fit.
Plot the χ^2 as a function of gamma*.
The best fit value: gamma*=0.17(5).
This value is consistent with a direct measurement.
Bursa et al. Phys.Rev.D81,014505 (2010)

SU(3) Nf=2 fundamental fermions(known to be in broken phase,but use the conformal hypothesis fit.)
No clear minimum for any channel.

In the case of SU(3) Nf=12
They fit Fodor’s data at largest volume for each m value.
The small volume data may be suffered from a finite volume effect.

X=pseudoscalar statevector state“nucleon” state“pion” decay constant

The minimum χ^2 for each mass occur at a similar value of gamma*.
Except for chiral condensate…
B_c and C_c is strongly correlated and A_c is dominated.
(Need more simulation data.)

Fitting the data universal anomalous dim.
They global fit the data using universal anomalous dimension.

X=pseudoscalar statevector state“nucleon” state“pion” decay constant

The data can be fitted by a function using conformal hypothesis.
Χ^2/dof~2.508 (dof=53).
The mass anomalous dimension at the fixed point: gamma*= 0.403(13)Conclusion of Appelquist’s papers
Fodor’s data can be described with similar fit quality between
chiral broken hypothesis (x^2/dof~0.81)
withconformal hypothesis (x^2/dof~2.5).
In these hypotheses, m should be small to control massless extrapolation.
Further simulations at additional m will help to distinguish the two scenario.

The other question comes from thisplot in the case of Nf=12.
Why does the minimum of eachchannel appear different gamma?

Mass deformation with additional relevant op.
If there isirrelevant op. around Gaussian FPbutrelevant op. around Non-trivial FP
We have to know how to get theblue line (massless limit)

Related works:
Kusafuka and TeraoarXiv:1104.3606

Summary and Discussion
Is there IR fixed point in SU(3) Nf=12 theory?

Appelquist group:YES (SF scheme g^2*~5, constant continuum extrapolation)
Fodor group: NO (Potential scheme, before taking continuum limit)
A. Hasenfratz:YES (MonteCalro RG, bare step scaling)
Our group:YES! (TPL scheme g^2*~2.4, a^2 quadratic extrapolation)

Chiral condensate

Fodor group: chiral symmetry is broken (huge lattice size and low beta region)
Columbia U group : chiral symmetry is broken
Appelquist group : the same data can be fitted by conformal hypothesis

Anomalous dimension for psuedo-scalar operator

Appelquist group : gamma*= 0.403(13) using conformal hypothesis with mass deformation
KMI group: gamma*~0.47-0.58
Our group: gamma*~0.29 by directly measurement

Discussion
The other theories

SU(3) Nf=10 fundamental fermion by N.Yamada’s group using Wilson fermion
SU(2) Nf=2 adjoint fermion by del Debbio’s group
SU(2) Nf=8 fundamental fermion by H.Ohki (Our group)....
Lattice data will give phenomenological and theoretical information aroundnontrivial fixed point in near future.

BACK UP SLIDESIntroduction
To explain the origin of electro-weak symmetry breaking, we introduce an additional
strong gauge interaction, and the composite state will be a seed of SSB.
There is a scenario based on a conformal theory in IR region.
To suppress FCNC process
To enhance quark mass
Asymptotic free (UV complete)IR fixed point (or walking behavior)
Large anomalous dimension forcomposite state