Walking dynamics
from lattice gauge theory
C.-J. David Lin
National Chiao-Tung University
and
National Centre for Theoretical Sciences, Taiwan
NTHU
31/12/09
Work done in collaboration with
• Erek Bilgici (University of Graz)
• Antonino Flachi (University of Kyoto)
• Etsuko Itou (Kogakuin University)
• Masafumi Kurachi (Tohoku University)
• Hideo Matsufuru (KEK)
• Hiroshi Ohki (University of Kyoto)
• Eigo Shintani (University of Kyoto)
• Tetsuya Onogi (Osaka University)
• Takeshi Yamazaki (University of Tsukuba)
Based on: PRD80, 034507 (2009) and arXiv:0910.4196.
Outline
• Phenomenological motivation
• The step-scaling method
• The finite-volume Wilson loop scheme
→ Definition and pure-YM
• The twisted Polyakov loop scheme
→ The Nf = 12 case
• Conclusion and outlook
Fundamental scalar Higgs in the SM
• Hierarchy problem
→ m2H = 2λv 2 ∼ Λ2UV .
• The theory is trivial
→ Higgs self-coupling: λ(µ) =
λ(ΛUV )
1+(24/16π )λ(ΛUV )log(ΛUV /µ)
2
→ The coupling vanishes for all µ when the cut-off Λ → ∞.
• A solution: strong interactions involving fermions
2
2
→ ΛEW ∼ ΛUV e−gc /gUV .
Motivation
Extended technicolour model
• Standard-model fermion masses
→ mf =
C(µ)
hψ̄ψi
Λ2ET C
via dim-6 operators
C(µ)
ψ̄ψ f¯f .
Λ2ET C
→ ΛET C ∼ TeV via estimating hψ̄ψi using MW and slow running of ψ̄ψ.
• FCNC processes
→
1 ¯ ¯
ffff.
Λ2ET C
→ ΛET C ∼ 102 ∼ 103 TeV from constraints imposed by K 0 −K̄ 0 mixing.
• A solution: fast running of ψ̄ψ (large anomalous dimension).
Motivation
Slow and fast running of the condensate
• Slow running (“TeV-QCD”)
→ hψ̄ψiETC ≈ hψ̄ψiEW [1 + α log(ΛETC /ΛEW )] ∼ Λ3EW .
→ ΛETC ∼ ΛEW (ΛEW /mf )1/2
• Fast running (IR fixed point with γ ∗ = 1)
→ hψ̄ψiETC ≈ hψ̄ψiEW (ΛETC /ΛEW ) ∼ Λ2EW ΛETC .
→ ΛETC ∼ ΛEW (ΛEW /mf )
Motivation
Walking technicolour
ΑHΜL
LTC
LETC
Μ
• Generates large anomalous dimension for ψ̄ψ to solve the FCNC problem.
• Modifies the relevant spectral function and the OPE to elude the Sparamter criticism a’la Peskin and Takeuchi.
• ΛETC /ΛTC ∼ 102 ∼ 103 .
→ Compared to the typical lattice size L/a ∼ 30 in each direction.
Motivation
IR-conformal/Walking β function
Β
Β
3
0.5
2
0.5
1
0.5
1.0
1.5
2.0
2.5
g
-0.5
-1
-1.0
-2
• Need a scale to have a gap.
• How do we look for a walking theory?
1.0
1.5
2.0
2.5
g
Motivation
Technicolour in the twenty-first century
• Except for “TeV-QCD”, technicolour has not been ruled out.
• Serious lattice calculations to support/kill technicolour
– Hadron spectrum.
L. Del Debbio, A. Patella and C. Pica
S. Catterall and F. Sannino
A. Hietanen et al.
– Pheses of candidate walking theories.
S. Catterall at al.
T. DeGrand, B. Svetitsky and Y. Shamir
– Spectrum of the Dirac operator.
Z. Fodor et al.
– The step-scaling method for the coupling constant.
T. Appelquist, G. Fleming and E. Neil
This work
The step-scaling method
The idea
ḡ 2 (L0 ) ppppppp
ppppppp
pppp
ppppppp
pppp
ppppppp
pp pp pp pp pp pp pp
- pp pp pp pp pp pp pp
pp pp pp pp pp pp pp pp pp pp pp
pp pp pp pp
ppppppp
ḡ 2 (8L0 )
pp pp pp pp pp pp pp
p p p p p p p
p p p p - p p p p p p p
p p p p
p p p p p p p
p p p p
p p p p p p p
p p p p p p p
p p p p
• Extrapolate to the continuum limit at every step.
• Vary the scale by changing the dimensionful lattice size L0.
The step-scaling method
Implementation: An example
4
1. Prepare lattices of size Lˆ0 :
Lˆ0 = L0/a = 6, 8, 10, 12,
and tune the lattice spacing (via varying the bare coupling).
→ The renormalised coupling g(L0) is the same on these lattices.
2. Double the lattice size to be
Lˆ0 = L0/a = 12, 16, 20, 24,
and calculate g(2L0, a/L0).
→ Extrapolate to the continuum limit to get g(2L0).
3. Go back to the lattices in 1., and tune the lattice spacing.
→ The renormalised coupling equals g(2L0) on all these lattices.
→ Repeat 2. to obtain g(4L0).
The step-scaling method
Results for the Schödinger Functional scheme
T.Appelquist, G.Fleming and E.Neil, PRD79, 076010 (2009).
• SU(3) fundamental fermions, Nf = 12.
• Scheme dependence?
• Controversy from the study of the Dirac operator eigenvalue spectrum?
A new non-perturbative scheme
General considreations
ALO = k g02
ANP(µ) = k g 2(µ)
Lattice
ANP(L, a) = k(a) g 2(L, a)
The finite-volume Wilson-loop scheme
The idea
L0
T
T0
a
R
• Set up T0 = L0 and T = R.
• When a → 0, with fixed r̂ ≡ (R + a/2)/L0, there is only one scale.
→ Varying r̂ corresponds to changing scheme.
The finite-volume Wilson-loop scheme
More details
• The Creutz Ratio (CR):
−R2
∂2
LO
lnhW (R, T ; L0 , T0 )i|T =R,T0 =L0 −→ kg02 .
∂R∂T
• On the lattice (R̂ = R/a and L̂0 = L/a):
W (R̂ + 1, T̂ + 1; L̂0 )W (R̂, T̂ ; L̂0 )
.
−(R̂ + 1/2) χ(R̂ + 1/2, T̂ + 1/2; L̂0 ) = −(R̂ + 1/2) ln
W (R̂ + 1, T̂ ; L̂0)W (R̂, T̂ + 1; L̂0 )
• The factor k can be calculated analytically (periodic BC)

!2 2πn1 R 
i L
X sin πnL0T
2
0
e
4
0
2 ∂


+ zero − mode contrib.
k = −R
2
4
2
∂R∂T
(2π)
nµ 6=0
n0
n0 + ~
n
T =R
Numerical test of the FV Wilson-loop scheme
Simulation parameters in pure-YM
CR
1.4
L̂0 = 18
L̂0 = 16
1.2
L̂0 = 14
L̂0 = 12
1
L̂0 = 10
0.8
0.6
0.4
0.2
6
6.5
7
β = 6/g 2(a)
7.5
8
8.5
• Take CR = 0.2871 as the starting point (step 0, scale L̃0).
• Take the step size s = 1.5.
Numerical test of the FV Wilson-loop scheme
Continuum extrapolation
CR
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.0005
0.001
0.0015
(a/L0)
0.002
2
0.0025
0.003
Numerical test of the FV Wilson-loop scheme
Result for pure-YM
2
gW
14
12
10
8
6
4
2
0
2 7
3
2 6
3
2 5
3
2 4
3
2 3
3
2 2
3
2
3
1
L̃0/L0
Twisted Polyakov Loop scheme
Twisted Boundary Condition
• Uµ (x + ν̂L) = Ων Uµ (x)Ω†ν , ν = 1, 2,
where the twist matrices Ων satisfy
Ω1 Ω2 = e2iπ/3Ω2Ω1 , Ωµ Ω†µ = 1, (Ωµ )3 = 1, Tr(Ωµ ) = 0.
’t Hooft, 1979.
• If ψ(x + ν̂L) = Ων ψ(x) ⇒ ψ(x + ν̂L + ρ̂L) = ΩρΩν ψ(x) 6= Ων Ωρψ(x)
• The ”smell” d.o.f. for fermions, Ns = Nc :
†
b
ψαa (x + ν̂L) = eiπ/3Ωab
ν ψβ (x)(Ων )βα .
G.Parisi, 1983.
Twisted Polyakov Loop scheme
The scheme
G.M.de Divitiis et al., 1994
• Polyakov loops in the twisted direction:
P1(y, z, t) = TrhΠj U1(j, y, z, t)Ω1e2iyπ/3Li (gauge and translation invariant).
• The coupling constant
∗ (0,0,0)i
h
y,zP
(y,z,L/2)P
1
2 (L) = 1 P
1
gTP
k h x,yP3(x,y,L/2)P3∗(0,0,0)i ,
P
where k =
1
24π 2
(−1)n
n=−∞ n2 +(1/3)2
P∞
2 →
• Special feature: At L → ∞, gTP
1
k
∼ 0.031847
∼ 32 if there is no IRFP.
Twisted Polyakov Loop scheme
Result for pure-YM
Twisted Polyakov Loop scheme
Nf = 12 simulation with staggered fermion
• Nf = Nt × Ns = 4 × 3 = 12.
• s = 1 ⇒ L = 4, 6, 8, 10;
s = 1.5 ⇒ L = 6, 9, 12, 15.
Twisted Polyakov Loop scheme
Nf = 12 simulation at low−β
0.11
L/a=4
L/a=6
L/a=8
L/a=10
L/a=12
0.1
0.09
0.08
0.07
0.06
0.05
4
4.5
5
5.5
6
6.5
Very different from the SF-scheme.....
7
Twisted Polyakov Loop scheme
Compare with the SF scheme
T.Appelquist, G.Fleming and E.Neil, PRD79, 076010 (2009).
Twisted Polyakov Loop scheme
Step-scaling behaviour
2 (1.5L)
gTP
3
data
2-loop
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
2 (L)
gTP
Twisted Polyakov Loop scheme
Obtaining the β function
1. g 2(1.5L) = g 2(L) + s0 g 4(L) + s1 g 6(L) + s2 g 8(L) + s3 g 10(L),
where we use perturbative results for s0 and s1, then fit s2 and s3.
2. Then the value of the β function at each step is
∂g 2(1.5L)
g(L)
β(g(1.5L)) = β(g(L)) g(1.5L) ∂g2(L) .
!
Twisted Polyakov Loop scheme
Result for β function
β(g)
0
TPL beta fn.
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.045
0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5
gTP
Concluding remarks and outlook
• We have designed a new non-perturbative Wilson Loop scheme for calculating the running coupling constant.
• We have demonstrated it is valid through numerical tests in pure-YM.
• Using the Twisted Polyakov Loop scheme, we have seen evidence (preliminary!) that Nf = 12 QCD contains a non-trivial IR fixed point.
• Other studies under way.
It is a new lattice research avenue.