Introduction
Lattice methods
Results
Conclusions and outlook
Minimal Walking Technicolor on the Lattice
Francis Bursa
Sussex University, 24th January 2011
arXiv:0910.4535, 1010.5909, 1011.0607, 1011.0864, . . .
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Outline
Introduction
Lattice methods
Results
Running coupling
Running mass
Spectrum
MCRG
Conclusions and outlook
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Technicolor
What is the mechanism of Electroweak Symmetry Breaking?
I
The Higgs boson?
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Technicolor
What is the mechanism of Electroweak Symmetry Breaking?
I
The Higgs boson?
I
Strong dynamics at the EW scale?
Technicolor:
I
Techniquarks charged under a new gauge symmetry, which
becomes strong at EW scale.
I
Chiral symmetry of techniquarks breaks spontaneously.
I
hψψi breaks EW symmetry.
Gives masses to W and Z bosons, but not to SM fermions. Need
Extended Technicolor.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Extended technicolor
New gauge bosons at a scale METC couple SM fermions to
techniquarks. Give mass to SM fermions:
m∼
Francis Bursa
hψψi
2
METC
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Extended technicolor
New gauge bosons at a scale METC couple SM fermions to
techniquarks. Give mass to SM fermions:
m∼
hψψi
2
METC
I
Require METC ∼ 10 TeV to get right SM fermion masses.
I
But to avoid FCNC, need METC > 100 TeV!
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Walking technicolor
m∼
hψψiETC
2
METC
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Walking technicolor
m∼
hψψiETC
2
METC
But hψψi is determined at TC scale, and runs to ETC scale:
Z METC
dµ
hψψiETC = hψψiTC exp
γ(µ)
µ
ΛTC
Can running be enhanced?
I Need γ(µ) large over a large range of scales.
γ > 1.0 [Chivukula & Simmons]
I Unlike QCD, where γ(µ) falls rapidly above ΛQCD .
I Need “walking”: coupling runs slowly above ΛTC so γ(µ) can
be large.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Walking technicolor
Conformal window:
If β(g ) = 0 for g 6= 0 there will be an infrared fixed point (IRFP).
Expect this to happen for sufficiently large Nf .
For slightly lower Nf , β(g ) will be small – expect walking.
The existence of an IRFP, and the value of the associated critical
exponents, is scheme-independent. However, walking, and the
associated running of hψψi are scheme-dependent. Can be
changed by changing scheme.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Candidate theories
Phase diagram of possible theories (Sannino):
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Minimal walking technicolor
Simplest model is minimal walking technicolor.
I
SU(2) technicolor gauge group
I
Two techniquarks in adjoint representation
I
Additional weak doublet to cancel Witten anomaly
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Minimal walking technicolor
Simplest model is minimal walking technicolor.
I
SU(2) technicolor gauge group
I
Two techniquarks in adjoint representation
I
Additional weak doublet to cancel Witten anomaly
Is this model phenomenologically viable?
I
Does chiral symmetry breaking occur?
I
Is γ large?
I
S-parameter
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Putting it on the lattice
Want to use lattice QCD experience as far as possible.
Differences:
I
SU(2) instead of SU(3). Easy to change (usually).
I
Nf = 2 instead of Nf = 2.
I
Fermions in adjoint representation instead of fundamental.
More difficult.
Don’t know answers!
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Adjoint fermions
To change representation of fermions, need copy of gauge field in
rep. R.
E.g. Wilson-Dirac operator:
X
y
D(x, y )ψ(y ) = −
1 X
( [(1 − γµ )UµR (x)ψ(x + µ) +
2a µ
(1 + γµ )UµR (x)† ψ(x − µ)] − (8 + 2am)ψ(x))
Uµ (x) and UµR (x) are related by:
Uµ (x) = exp(iAaµ (x)Tfa ) , UµR (x) = exp(iAaµ (x)TRa )
Need to alter HMC to include
Codes: Chroma and HiRep.
∂UµR (x)
∂Uµ (x) .
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Observables
What can the lattice tell us?
I
Phase: conformal or chiral symmetry breaking
I
Running coupling
I
Running mass, γ
I
Spectrum
I
S-parameter
All with controlled systematic errors.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Observables
What can the lattice tell us?
I
Phase: conformal or chiral symmetry breaking
I
Running coupling
I
Running mass, γ
I
Spectrum
I
S-parameter
All with controlled systematic errors.
Warning: lattice and finite box break conformal symmetry. Fermions may break conformal symmetry or be doubled.
Running of coupling and mass are scheme-dependent except at fixed points.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Phase
How to determine phase?
I
Directly measure hψψi.
I
Running coupling has fixed point if conformal.
I
Spectrum in chiral limit.
I
Eigenvalues of Dirac operator
Best to combine more than one approach.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Running coupling
Running coupling can be measured in various schemes. Existence
of a fixed point is scheme-independent. Walking is not scheme
independent.
We use Schrödinger Functional scheme. Other groups using
schemes based on Polyakov loops / Wilson loops.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Schrödinger Functional
Box of size aL with boundary conditions at top and bottom.
I
Boundary conditions impose a chromoelectric field on the
system.
I
Coupling defined through response of the system to this field:
D E−1
g 2 = k ∂S
. To leading order, g 2 = g02 .
∂η
Use step-scaling: compare g 2 on boxes of size L and 4L/3. Do this
at a range of couplings to get β-function over a range of scales.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
SF Coupling
I
Action: Wilson gauge action with unimproved Wilson quarks.
I
Advantage of Schrödinger Functional boundary conditions:
can work at zero quark mass, so no need for chiral
extrapolations.
I
Lattice sizes 6,8,12,16.
I
Choose bare couplings to cover range 0.5 < g 2 < 3.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
SF results
Running of coupling is slow:
5.0
L/a=16
L/a=12
L/a=10 2/3
L/a=9
L/a=8
L/a=6
4.0
g
2
3.0
2.0
1.0
0.0
2.0
3.0
4.0
Francis Bursa
5.0
β
6.0
7.0
8.0
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Continuum limit
Fit as function of β, then extrapolate to a/L = 0 at constant
g 2 (β, L) to obtain continuum limit of step-scaling function.
u=1.0
u=2.0
u=3.0
3.5
Σ(4/3,u,a/L)
3.0
2.5
2.0
1.5
1.0
0.00
0.05
Francis Bursa
0.10
a/L
0.15
0.20
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Errors
Overall fitting procedure quite complicated: several stages, choice
of interpolation functions etc. Use several stages of bootstrapping
to estimate statistical and systematic errors.
Validate by generating fake data and fitting it. Expect true value
to lie within error 68% of the time.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Continuum running coupling
Systematic errors are large. Consistent with fixed point, but not
1.10
conclusive.
1-loop
2-loop
1.08
σ(u)/u
1.06
1.04
1.02
1.00
0.98
0.0
1.0
u
2.0
3.0
Consistent with evidence for fixed point at 2.0 < g 2 < 3.2 from
Hietanen et al. [arXiv: 0904.0864]. Also consistent with walking.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Running mass
We measure the running of the mass by calculating pseudoscalar
renormalisation constant ZP .
ZP (L) =
p
3f1 /fP (L/2)
As for g 2 (β, L), calculate on lattices of size L and 4L/3 and
extrapolate to continuum limit.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
ZP results
Running of ZP is faster than coupling:
1.00
0.90
0.80
0.70
L/a=6
L/a=8
L/a=9
L/a=10 2/3
L/a=12
L/a=16
ZP
0.60
0.50
0.40
0.30
0.20
0.10
2.0
4.0
6.0
Francis Bursa
8.0
β
10.0
12.0
14.0
16.0
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Continuum running mass
Analyse as for running coupling.
1.04
1-loop
Statistical Error
1.02
1.00
σP(4/3,u)
0.98
0.96
0.94
0.92
0.90
0.88
0.86
0.84
0
0.5
1
1.5
u
2
2.5
3
3.5
Errors not consistent with 1, so observe running.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Anomalous dimension
P (u,s)|
Convert to anomalous dimension: γ = − ln|σln|s|
0.5
^γ(u)
0.4
0.3
0.2
0.1
1-loop
Statistical Error
0.0
0.0
0.5
1.0
Francis Bursa
1.5
u
2.0
2.5
3.0
3.5
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Anomalous dimension
Implications of running coupling and mass measurements:
I
γ < 1.0 in this region. Not sufficient for technicolor.
I
If indeed fixed point in 2.0 < g 2 < 3.2, this model probably
ruled out. 0.05 < γ < 0.56 in this range.
I
But don’t want fixed point anyway since then there is no
chiral symmetry breaking.
I
If it is walking in this range, γ still too small.
I
Better if it is walking at stronger coupling. But difficult to
measure there.
Can we determine phase by other methods?
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Measuring masses
Measuring spectrum of low-lying states can give useful information
on phase of the theory.
Behaviour in chiral limit:
I
QCD-like: mPS goes to zero, but other masses have finite
limit.
I
Conformal: all masses go to zero as mq
1/(1+γ)
.
Have to have big enough box size L for this to work.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Pseudoscalar mass
Using high statistics and QCD smearing techniques, can get
accurate results at low masses.
1.6
3
16x8
3
24x12
3
32x16
3
64x24
1.4
amPS
1.2
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
0
0
0.1
0.2
Francis Bursa
0
am
0.02
0.3
0.04
0.06
0.4
0.08
0.5
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Scaling of pseudoscalar mass
2 /m appears to vanish in chiral limit, unlike QCD:
mPS
q
6
3
16x8
3
24x12
3
32x16
3
64x24
5
2
mPS /m
4
3
2
1
0
0
0.1
0.2
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am
0.3
0.4
0.5
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Vector mass
mV also appears to vanish:
3
amV
1.5
16x8
3
24x12
3
32x16
3
64x24
1
0.4
0.3
0.2
0.5
0.1
0
0
0
0.1
0.2
Francis Bursa
am
0
0.02
0.3
0.04
0.06
0.4
0.08
0.5
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Scaling of vector mass
mV /mPS does not diverge:
1.08
1.1
3
16x8
3
24x12
3
32x16
3
64x24
mV/mPS
1.06
1.05
1
0.95
1.04
0
0.02
0.04
0.06
0.08
1.02
1
0
0.1
0.2
Francis Bursa
am
0.3
0.4
0.5
Minimal Walking Technicolor on the Lattice
Running coupling
Running mass
Spectrum
MCRG
Introduction
Lattice methods
Results
Conclusions and outlook
Chiral condensate from GMOR relation
Can also use Gell-Mann–Oakes–Renner relation to estimate chiral
condensate:
0.04
0.03
0.02
0.4
3
2
a (mPSFPS) /am
0.5
0.01
0.3
0
0
0.02
0.04
0.06
0.08
0.2
3
16x8
3
24x12
3
32x16
3
64x24
0.1
0
0
0.1
0.2
Francis Bursa
am
0.3
0.4
0.5
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Glueball masses
Glueball masses also vanish in chiral limit:
3.0
aM
2.0
PS meson mass
++
0 glueball mass
++
2 glueball mass
aσ
1/2
(σ = string tension)
values at a m = ∞
2.5
1.5
1.0
0.5
0.0
0.2
0.4
Francis Bursa
0.6
am
0.8
1.0
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Spectrum implications
Spectrum consistent with conformality, very different from QCD.
1/(1+γ)
Expect masses ∝ mq
low value of γ.
. Difficult to fit, but consistent with a
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Spectrum implications
Spectrum consistent with conformality, very different from QCD.
1/(1+γ)
Expect masses ∝ mq
low value of γ.
. Difficult to fit, but consistent with a
Finite-size scaling: expect scaling law LM ∝ F (L1/(1+γ) mq ).
Fits give γ < 0.5.
Consistent with low values from direct measurements of running of
mass.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Monte Carlo Renormalisation Group
The Monte Carlo Renormalistaion Group is another method for
estimating the running of the coupling and the mass.
Uses repeated blocking transformations to identify values of
coupling or mass that occur at scales differing by a factor of 2.
Specific method for coupling:
I
Carry out simulations at bare coupling β and block n times
I
Carry out simulations at bare coupling β 0 and block n − 1
times.
I
Measure observables on both blocked lattices and tuneβ 0 so
that they match.
I
Then step scaling function is ∆β = β − β 0 .
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Running coupling
Running mass
Spectrum
MCRG
Anomalous dimension
Again, anomalous dimension is easier than running of coupling.
Fit gives γ = 0.49(13). Consistent with direct measurement,
γ = 1.0 ruled out.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Conclusions
The lattice can provide non-perturbative answers about technicolor
models, with systematic errors under control.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Conclusions
The lattice can provide non-perturbative answers about technicolor
models, with systematic errors under control.
I
Consistent picture for Minimal Walking Technicolor, from a
range of observables.
I
MWT is conformal or near-conformal.
I
Anomalous dimension is small.
Francis Bursa
Minimal Walking Technicolor on the Lattice
Introduction
Lattice methods
Results
Conclusions and outlook
Outlook
Outlook for the future:
I
Other models: SU(2) fundamental, SU(3) sextet, SU(3)
fundamental. . .
I
Other observables: S-parameter, chiral condensate.
I
Anomalous dimension is a relatively easy way to tell if a model
is worth further investigation.
I
Improved simulations to reduce O(a) errors.
I
4-fermi interactions may be important; expect them anyway
from ETC sector.
Francis Bursa
Minimal Walking Technicolor on the Lattice