The low-energy spectra of 12 and 16 flavored QCD
on the lattice	

Hiroshi Ohki
for LatKMI collaboration
KMI, Nagoya University

March, 17, 2012	

LatKMI collaboration members	

Introduction

Oringin of the electroweak symmetry breaking	• Technicolor (dynamical symmetry breaking model,alternative to Higgs mechanism in SM )	

Electroweak symmetry breaking-> techni-fermion Q condensation (scale up of QCD)
(c.f. Chiral symmetry breaking in QCD)	

a new strong interaction	
Ex.	

Walking (conformal) behavior -> non-perturbative dynamics

Large flavor QCD: benchmark test of walking dynamics

Asymptotic non-free	Conformal window	

Walking technicolor	
QCD-like	•Walking technicolor could be realized just below conformal window.
•What the value of the anomalous dimensions?
•Rich hadron structures may be observed.
5Possible signals of the conformal behavior (on lattice)•Measurement of (hadron) mass spectrums & decay constant with finite small bare mass-> Hyper scaling [Miransky ’96]

Introducing the bare mass (explicit breaking of conformality)
If the theory is in the conformal window,IRFP

1.

m->0, L->∞, CFT is realized. (no mass gap)2.

m≠0, this theory has mass gap at low energy (M) and give rise to asymptotic state(bound state).

(1)	
IRFP

(2)	
IRFPM(m)6Rich hadronic structures
QCD•
•
•

Pion (pNG-boson)rho meson ()Scalartechnicolor•••

Techni-pion
Techni-rho
Techni-dilation, (techni-) glueballcould be lighter than other mesons
due to approximate scale inv.
(These could be observed in LHC)

Conformal•

Universal behavior for above all bound state masses (hyper scaling)
e.g.	

•

Finite volume formula for hyper scaling (Del Debbio et.al.)7

SU(3) with fundamental fermions
Systematic study of flavor dependence using single setup of the lattice simulation

Our goals:•Understand the flavor dependence of the theory•
Find the conformal window•
Find the walking regime and investigate the anomalous dimension
Status (lattice):•
Nf=16: likely conformal
This talk	•Nf=12: controversial•
Nf=8: studies suggests no conformal behavior or walking behavior?
•
Nf=4: chiral broken and enhancement of chiral condensate
Next talk	

Lattice gauge theory + numerical simulation : powerful tool

Our work••use of improved staggered action
Highly improved staggered quark action [HISQ]• to get nearly continuum results from non-zero lattice spacing• to reduce flavor violation for good SU(N) chiral symmetry• bound to Nf=4 n
We use MILC version of HISQ action
use tree level Symanzik gauge actionno (ma)2 improvement (no interest to heavy quarks)= HISQ/tree•

Measurement of meson spectrumin particular Goldstone pion mass and decay constant varying volume•

The results shown are preliminary	910simulation setup• Parameters search•

SU(3), Nf=12 flavor

Volume (= L^3 x T)
• L =18, T=24
• L =24, T=32
• L =30, T=40
Bare coupling constant (• beta=3.7, 4L:T =3:4)

bare quark mass
• Mf=0.04, 0.05, 0.08, 0.1, 0.12, 0.16, 0.2

11

ChPT v.s. conformal hypothesis
CHPT hypothesis• In usual QCD chiral symmetry is spontaneously broken.
Low energy effective theory (ChPT) is described by the pion (NG-boson).
leading order
Including higher order(PCAC relation)

Pion decay constant(c0=0 -> chiral restoration)
Conformal hypothesis• Universal behavior for all bound state masses.

Lattice results Nf=12

Raw data of Spectra (beta=3.7)fπ	mπ	mq	mq	

• analysis•••

Taking infinite volume limit
CHPT fit (Polynomial fit )Study in the chiral limit

14Infinite volume limit (beta=3.7)

ChPT-like finite volume scalingmπ	fπ	

• results••

reasonably fitted
The value at infinite volume is consistent with one at the largest L(=30)

15Fit result on pion mass (beta=3.7)

Fit results• Negative constantC0= -0.0345(33)• If we fix c0- > zeroChi^2/dof=3.22-> 1716

Chiral condensation (beta=3.7)

GMOR relation
Fit results• Negative constantC0= -0.0109 (29)or consistent with zero17Results at beta=418

short summary of CHPT fit•••

ChPT fit of Mpi^2 and (fpi mpi)^2/mf does not work well.
• Non-zero constant (negative) for pion mass is obtained• Chiral condensation in massless limit is negative or zero consistent
Our results do not favor ChPT behavior
Two analyses for beta=3.7 & 4 give similar results.

•We need to investigate smaller masses and larger volumes.

-> Next we also try to fit with hyper scaling hypothesis

Test of Finite size hyper scaling
Conformal hypothesis• Universal behavior for all hadron masses.
•Scaling variable x

We test the finite hyper-scaling for our data at L=18, 24, 30.
If the theory is inside the conformal window,
the data should be described by one scaling parameter x.

γ = 0.0	x 	γ = 0.1	x 	γ = 0.2	x 	γ = 0.3	x 	γ = 0.4	x 	γ = 0.5	x 	γ = 0.6	x 	γ = 0.7	x 	γ = 0.8	x 	γ = 0.0	x 	γ = 0.1	x 	γ = 0.2	x 	γ = 0.3	x 	γ = 0.4	x 	γ = 0.5	x 	γ = 0.6	x 	γ = 0.7	x 	γ = 0.8	x 	Fit analysis
Near the optimal value of γ, the data become to align,
where linearity is observed.	

• L-> infinity, the (infinite volume) hyper-scaling relation is obtained.
• 3 fit parameters : c0, c1, γ39Results (beta=3.7)•The data with empty symbols are not used in the fit40ρ/π ratio

Beta=3.7	
Beta=4	

• The data near the chiral limit -> almost constant.
(c.f. it becomes blow-up in QCD.)•This ratio is useful as the definition of constant physics•a(β=3.7) > a(β=4).
=> Our results suggests asymptotically free region for beta=3.7-4.
41

Lattice results Nf=16

Nf=16••This theory should be in the deep conformal window
Two loop perturbation suggests
the existence of the IRFP in very weakly couling region.

Bare coupling constantbeta=3.15, 3.5, 5Volume (= L^3 x T)
• L =8, T=24
• L =12, T=24
• L =16, T=24
• L=24, T=32

• Test of the finite size hyperscaling for pion mass43

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.15)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=3.5)

Pion mass (beta=5)Pion mass (beta=5)Pion mass (beta=5)Pion mass (beta=5)Pion mass (beta=5)•The preliminary result of gamma

beta	

3.15	

3.5	5	

γ	0.42	

0.35	

0.25	

63

Summary•Large flavor SU(3) gauge theory is being investigated.
•We focus on the Nf=12,16 case.
•We also study other models with fermions #=4,8(talk by Nagai)•We measure the pion (NG-boson) mass anddecay constant.
•Finite size hyper scaling is observed.
•Nf=12 &16 favor conformal gauge theory64

Future prospect• Questions
In Nf=12
difference of γ between several observables<- due to mass and volume corrections? or walking?
In Nf=16
Large γ compared with perturbation theory
Beta dependence of γ->need to take continuum limit orvery small IRFP (large beta simulations )• Study of other spectra (ongoing)scalarglueball mass

END
Thank you