Infrared conformal gauge theory and composite Higgs
Kari Rummukainen
University of Helsinki and Helsinki Institute of Physics
Work done in collaboration with:
Ari Hietanen, Tuomas Karavirta, Viljami Leino, Anne-Mari Mykkänen,
Jarno Rantaharju, Teemu Rantalaiho, Kimmo Tuominen
University of Sussex 2/2015
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
1 / 34
Introduction: strongly coupled BSM physics?
The Higgs particle has been found!
the Standard Model is in excellent
shape
I Higgs field is centrally important:
drives the EW symmetry breaking!
I Scalar → theoretical problems:
⇒ naturalness, vacuum stability, unitarity
bound . . .
I
There is still room (and need!) for BSM
physics.
Most BSM models aim to ameliorate
the problems in the SM by e.g.
I
I
I
Pairing bosons with fermions (SUSY)
Cutoff (extra dimensions)
No scalars: composite Higgs,
Technicolor and related models
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
2 / 34
Motivation
Infrared conformality: gauge theories with with large enough (but not too
large!) number of fermions generically feature an infrared fixed point.
phenomenology: may enable light composite Higgs
theoretical curiosity: strongly coupled conformal phase, sQGP, “unparticles”
Lot of recent activity both on and off the lattice (non-perturbative → lattice)
On LATTICE 2013 conference, 38 contributions on this topic!
Slow running of the coupling g 2
→ Lattice studies technically very difficult
Here we mostly discuss SU(2) gauge with different fermion contents
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
3 / 34
Introduction: Conformal Window
Consider 2-loop perturbative
β-function
SU(3) with Nf massless fermions
0.4
β(g ) = µ
g3
g5
dg
= −β0
−
β
1
dµ
16π 2
(16π 2 )2
0.2
Small Nf : β0 > 0, β1 > 0
running coupling, confinement
and χSB (QCD-like)
Medium Nf : β0 > 0, β1 < 0
IR fixed point, no χSB
β = µ dg/dµ
Generically 3 different behaviours:
18
14
13
0
12
-0.2
0
3
10
[Banks,Zaks]
Large Nf : β0 < 0
Asymptotic freedom lost
-0.4
0
K. Rummukainen (Helsinki)
2
4
6
8
10
2
g
Conformal window: range of Nf
where IRFP exists
Conformal gauge
Sussex 2015
4 / 34
Walking coupling
What happens when we approach the lower edge of the conformal window
from below?
Competition between IR conformal behaviour and non-perturbative
confinement/χSB
The β-function may get close to zero at finite coupling
⇒ The coupling evolves slowly, walks.
β
g2
IR fixed point
walking
g2
g*2
QCD−like
walking
IR fixed
point
QCD−like
µ
Typically at strong coupling: perturbation theory not applicable, lattice
simulations needed
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
5 / 34
Technicolor
Technigauge + massless techniquarks Q
Techniquarks have both technicolor and EW charge (exactly like quarks in
the SM)
Chiral symmetry breaking in technicolor −→ Electroweak symmetry breaking
Scale: ΛTC ∼ fTC ∼ ΛEW
After chiral symmetry breaking looks like SM:
⇒
decay constant fTC
scalar Q̄Q σ-meson
Goldstone pseudoscalars, “pions”
exotic technihadrons (observable!)
↔
↔
↔
Higgs expectation value v
Higgs particle
W,Z -longitudinal modes
Describes well the W , Z +Higgs sector (depending on the model, may have
too many Goldstone bosons)
Does not explain fermion masses (Yukawa). For that, we need additional
structure → Extended technicolor
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
6 / 34
Extended technicolor
In addition to the “pure” technicolor, introduce a new higher-energy
interaction coupling Standard Model fermions q (quarks, leptons) and
techniquarks (Q): extended technicolor (ETC)
Several options, e.g. massive gauge boson, METC :
METC
q,Q
q,Q
[Eichten,Lane,Holdom,Appelquist,Sannino,Luty. . . ]
1
Q̄Q q̄q −→ SM fermion mass mq ∝ 2 hQ̄QiETC
2
METC
METC
1
q̄qq̄q −→ extra FCNC’s (harmful!)
2
METC
1
Q̄Q Q̄Q −→ explicit χSB in the techniquark sector
2
METC
1
hQ̄QiETC : condensate evaluated at the ETC scale
hQ̄QiEW : condensate at TC∼EW) scale
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
7 / 34
Extended technicolor
I) In order to avoid unwanted FCNC’s need to push
ΛETC ≈ METC >
∼1000 × ΛEW (ΛTC ≈ ΛEW )
II) For EWSB we must have hQ̄QiEW ∝ Λ3EW
2
(top quark!)
III) On the other hand, hQ̄QiETC ∝ mq METC
Using RG evolution
"Z
METC
hQ̄QiETC = hQ̄QiEW exp
ΛEW
#
γ(g 2 )
dµ
µ
where γ(g 2 ) is the mass anomalous dimension.
In weakly coupled theory γ is small, and hQ̄Qi is ∼ constant.
Thus, it is not possible to satisfy the constraints I), II), II) in a QCD-like
theory, where the coupling is large only on a narrow energy range above χSB.
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
8 / 34
Walking coupling
If the coupling walks, i.e. if g 2 ≈ g∗2 (constant) over the range from TC to
γ(g∗2 )
ΛETC
hQ̄QiTC (condensate
ETC, then we can solve hQ̄QiETC ≈
ΛTC
enhancement)
Inserting II) and III) we obtain
γ(g∗2 ) ≈ 1 − 2
g2
g*2
β
QCD−like
IR fixed point
walking
walking
g2
IR fixed
point
QCD−like
Λ EW
Λ ETC
µ
Walking → Higgs naturally light, “dilatonic”
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
9 / 34
Conformal window in SU(N) gauge
fundamental
[Appelquist, Lane, Mahanta, Cohen,
Georgi, Yamawaki, Schrock, Sannino,
Tuominen, Dietrich]
2-index antisymmetric
2-index symmetric
adjoint
Upper edge of band: asymptotic freedom lost
Lower edge of band: ladder approximation
Walking can be found near the lower edge of the conformal window: large
coupling, non-perturbative - lattice simulations needed!
In higher reps it is easier to satisfy EW constraints [Sannino,Tuominen,Dietrich] →
recent interest
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
10 / 34
Existence of the IRFP essentially non-perturbative
Example: Perturbative β-function of SU(2) gauge with Nf = 6 fundamental rep
fermions
2
3-loop
0
4-loop
β
-2
2-loop
-4
1-loop
-6
-8
0
2
4
6
2
g /4π
8
10
12
[4-loop MS: Ritbergen, Vermaseren, Larin]
Results from lattice: existence of IRFP inconclusive
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
11 / 34
“Walking” at Nf <
∼6
Interestingly, the fixed point vanishes from 4-loop MS beta function if Nf is
slightly lowered from 6:
2
0
4-loop, Nf = 6
4-loop, Nf = 5.945
β
-2
-4
-6
-8
0
1
2
3
4
5
2
g /4π
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
12 / 34
Goals:
Take SU(N) gauge theory with Nf fermions in some representation.
Locate the lower edge of the conformal window
Measure β(g 2 )-function
Measure γ(g 2 )
Can we find IRFP? Or walking? (how to define it?)
How IRFP appears? More fixed points?
The most interesting case is a theory which
I
I
I
is walking or
is just within conformal window (easy to deform into walking)
has large anomalous exponent γ near FP
F
AdS-QFT: Indications that γ = 1 at the lower edge of the conformal window
[Järvinen et al.]
I
Can it have “light” scalar (Higgs)? Walking should help!
“Hadron” spectrum, chiral symmetry breaking pattern
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
13 / 34
Models studied
Red: conformal
Blue: χSB
Black: unclear
SU(3) + Nf = 8–16 fundamental rep:
I
Nf = 8:
Appelquist et al; Deuzeman et al; Fodor et al; Jin et al; Aoki et al; Schaich et al;
Gelzer et al
I
I
I
Nf = 9: Fodor et al
Nf = 10: Hayakawa et al; Appelquist et al
Nf = 12: Hasenfratz; Appelquist et al; Deuzeman et al; Xin and Mawhinney; Fodor et al;
Okawa et al; Aoki et al; Cheng et al; Itou; Lin et al; Gelzer et al
I
Nf = 16:
Damgaard et al; Heller; Hasenfratz; Fodor et al; Deuzemann et al
SU(2) + fundamental rep fermions:
I
I
I
I
Nf
Nf
Nf
Nf
= 4: Karavirta et al
= 6: Del Debbio et al; Karavirta et al; Appelquist et al;
= 8: Iwasaki et al; Lin et al
= 10: Karavirta et al
K. Rummukainen (Helsinki)
Conformal gauge
Tomii et al; Voronov et al
Sussex 2015
14 / 34
Models studied
Red: conformal
Blue: χSB
Black: unclear
SU(2) + Nf = 1 adjoint rep:
Athenodorou et al
SU(2) + Nf = 2 adjoint rep:
Catterall et al; Bursa et al; Hietanen et al; Rantaharju et al; De
Grand et al; Del Debbio et al; August and Maas; Arthur et al
SU(3) + Nf = 2 2-index symmetric rep:
SU(3) + Nf = 2 adjoint rep:
DeGrand et al; Sinclair and Kogut; Fodor et al
DeGrand et al
SU(4) + Nf = 2 2-index symmetric rep:
DeGrand et al
SU(4) + Nf = 2 2-index antisymmetric rep:
SO(4) + Nf = 2 vector rep:
K. Rummukainen (Helsinki)
DeGrand et al
Hietanen et al
Conformal gauge
Sussex 2015
15 / 34
Classifying conformal / χ SB ?
Measure β-function directly
I
I
I
Schrödinger functional
MCRG
Gradient flow methods
Measure technihadron and glueball masses and string tension as functions of
the techniquark mass mQ :
I
I
I
Non-zero mQ takes us away from the (possible) IRFP
1/(1+γ)
Conformal: M ∝ mQ
, incl. string tension
1/2
χSB: Mπ ∝ mQ , others remain massive
Dirac operator eigenvalue distribution: scales with γ
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
16 / 34
Evolution of the coupling: Schrödinger
functional
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
17 / 34
Evolution of the coupling
Schrödinger functional: Generate a background chromoelectric field using
non-trivial fixed boundary conditions, parametrised by a twist angle η
At the classical level, we have
dSclass.
A
= 2
dη
g
t=T
where A(η) is a known constant.
At the quantum level, we define the coupling
through
1
g2
=
=
E
1 dS
A dη
const. × h(boundary plaq.)i
special fixed
boundaries
t=0
Evaluates g 2 directly at scale µ = 1/L, the lattice size
Can use mQ = 0
Has been used very succesfully in QCD by the Alpha collaboration
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
18 / 34
Evolution of the coupling: QCD-like
Test with Nf = 2 fundamental
representation, QCD-like test case:
fundamental rep.
βL=2.2
12
L/a grows, k ∼ a/L decreases,
g 2 (L) increases: asymptotic
freedom, OK!
βL=2.5
βL=2.8
βL=3.1
βL=3.5
9
Large βL → small lattice spacing
→ small volume
g
2
6
Continuous line: coupling
evaluated from the 2-loop
β-function (integration constant
fixed to measurement at
L/a = 16)
Not a continuum limit, but shows
consistency
K. Rummukainen (Helsinki)
3
0
0
Conformal gauge
5
10
15
20
L/a
Sussex 2015
19 / 34
Evolution of the coupling: QCD-like
Test with Nf = 2 fundamental
representation, QCD-like test case:
fundamental rep.
12
L/a grows, k ∼ a/L decreases,
g 2 (L) increases: asymptotic
freedom, OK!
βL=2.5
βL=2.8
9
Large βL → small lattice spacing
→ small volume
Continuous line: coupling
evaluated from the 2-loop
β-function (integration constant
fixed to measurement at
L/a = 16)
Not a continuum limit, but shows
consistency
K. Rummukainen (Helsinki)
βL=2.2
g
βL=3.1
βL=3.5
2
6
3
0
Conformal gauge
1
10
2
10
3
10
L/L0
Sussex 2015
19 / 34
Evolution of the coupling: SU(2) + Nf = 2 adjoint
7
With two adjoint representation
fermions:
6
At small g 2 (L): increases with L
(asymptotic freedom)
At large g 2 (L): decreases as L
increases
⇒ β-function positive here!
As L/a → ∞, apparently
g 2 (L) → g∗2 ≈ 3.
⇒ IRFP
Here: using “HEX” improved
Wilson-Clover action
4
2
Agrees with 2-loop
β-function
g
I
5
3
A
β=1
β=1.05
β=1.1
β=1.2
β=1.3
β=1.5
β=2
β=4
β=5
β=6
β=8
2
1
A
0
0
4
A
8
A
A
12
A
A
16
20
24
28
L/a
[Karavirta, Rantaharju, Rantalaiho, KR, Tuominen, to be
published]
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
20 / 34
Evolution of the coupling: SU(2) + Nf = 2 adjoint
0.69
β=6
With two adjoint representation
fermions:
g
2
0.68
At small g 2 (L): increases with L
(asymptotic freedom)
Agrees with 2-loop
β-function
As L/a → ∞, apparently
g 2 (L) → g∗2 ≈ 3.
⇒ IRFP
Here: using “HEX” improved
Wilson-Clover action
β=8
0.51
2
At large g 2 (L): decreases as L
increases
⇒ β-function positive here!
0.66
0.515
g
I
0.67
0.505
0.5
0.495
5
10
15
20
L/a
[Karavirta, Rantaharju, Rantalaiho, KR, Tuominen, to be
published]
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
20 / 34
Step scaling function
Step scaling: coupling when the lattice size is (e.g.) doubled
Σ(u, L/a) = g 2 (g02 , 2L/a)u=g 2 (g02 ,L/a)
Continuum limit:
σ(u) = lim Σ(u, L/a)
a/L→0
Step scaling is related to β-function:
Z
σ(u)
√
−2 ln 2 =
u
dx
√
xβ( x)
Close to the fixed point:
β(g ) ≈
g
2 ln 2
σ(g 2 )
1−
g2
1-loop analysis indicates that finite lattice spacing effects large (∼ 50% at
L/a = 10) ⇒ improvement! [Alpha; Karavirta et al.]
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
21 / 34
Schrödinger functional in SU(2)+Nf = 2 adjoint fermions
[Rantaharju et al; preliminary]
Step scaling with volume pairs L/a = 6-12, 8-16, 10-20
Raw step scaling
Interpolation + continuum extrap.
1.05
L/a=6
L/a=8
L/a=10
2-loop
1.02
1.02
0.99
2
2
σ(g )/g
2
σ(g ,2)/g
2
0.96
0.96
0.9
L/a=Continuum
L/a=10
2-loop
0.93
0.84
0
4
2)
(g
8
0
1
2
3
4
5
2
g
Questionable continuum extrapolation? Lattice artifacts on small volumes?
2
Largest volume: IRFP at gSF
= 2 – 2.5. Compatible with previous results [Hietanen
et al; Del Debbio et al; DeGrand et al].
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
22 / 34
Modern alternative: Gradient flow coupling
Is currently being applied to QCD, more exotic theories...
Gradient flow coupling in SU(2) + 8 Fundamental flavours
1.1
v=8
v=10
v=12
2-loop
2
σ(g ,2)/g
2
1.05
1
0.95
0
1
2
3
4
2
g
5
6
7
8
[Rantaharju et al, LATTICE 2014]
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
23 / 34
Mass spectrum and
anomalous exponent γ
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
24 / 34
Example: mass spectrum of SU(2)+Nf = 2 adjoint
fermions
[Del Debbio et al 09]
1/(1+γ)
As “quark” mass mQ → 0, masses vanish with M ∝ mQ
very different from QCD-like
More recent measurements, taking into account Finite volume scaling:
γ ≈ 0.371 at estimated IRFP value of coupling [Del Debbio et al, Lattice 2013]
Problem: scalar (“Higgs”) mass very difficult to measure technically!
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
25 / 34
SU(3) Nf = 12 spectrum
Fπ : non-zero intercept as mQ → 0? Looks QCD-like (χSB)
F/ linear fit
0.1
F/ = F + c1 m
0.08
0.06
F= 0.00784 ± 0.00050
c1= 1.600 ± 0.021
F/
r2/dof= 0.52
0.04
r2 sum = 3.10
0.02
m fit range: 0.01 ï 0.035
0
0
0.01
0.02
m
0.03
0.04
[Fodor, Holland, Kuti, Nogradi, Schroeder, 2011]
Expect IRFP, but the results do not look like it?
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
26 / 34
Mass spectrum measurement
1/(1+γ)
If the massless mQ = 0 theory has an IRFP, excitation masses M ∝ mQ
Excitation size ∝ M −1 , diverges as mq → 0
[Del Debbio and Zwicky]
All excited states in a given channel become massless → excitation spectrum
∼ continuous, “unparticles”.
→ great care needed in mass spectrum measurment – not yet fully under control.
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
27 / 34
Other measurements of γ
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
28 / 34
Dirac operator eigenvalues
Eigenvalue density of the lattice (D † D + m2 ):
[DeGrand; Del Debbio and Zwicky; Patella]
hQ̄Qi ∝ mη ⇔ ρ(λ) ∝ λη
⇒ Mode number
ν(Ω) = C + (Ω2 − m2 )2/(1+γ)
In SU(2) + Nf = 2 adjoint rep fermions, γ = 0.37(2):
1e-01
3
64x24 am0=-1.15
-4
a [ν(Ω) - ν0]
1e-02
1e-03
1e-04
1e-05
1e-06
0.01
0.1
2
1
2 1/2
[ (aΩ) - (am) ]
K. Rummukainen (Helsinki)
Conformal gauge
[Patella]
Sussex 2015
29 / 34
Determining γ(g 2 ) using Schrödinger functional
The mass scaling exponent γ can be determined with Schrödinger functional
methods
in SU(2) + Nf = 2 adjoint fermions:
0.5
Continuum
L=12
1-loop
0.4
*
2
γ (g )
0.3
0.2
0.1
0
0
1
2
3
4
5
2
g
∗2
∗2
At IRFP gSF
≈ 2.5 – 3.5, γ(gSF
) ≈ 0.2 – 0.3
∗
Smaller
than
the one from the eigenvalue
≈ 0.37
K.
Rummukainen
(Helsinki)
Conformal gauge spectrum γ
[Patella]
Sussex 2015
30 / 34
What do the results imply?
Lattice analysis is significantly more difficult than in QCD-like theories:
Slow evolution → small signal
Slow evolution → strong bare coupling
⇒ Conflicting results, unknown systematics
I
I
Improvement is needed: better actions, better methodology, understanding of
limitations
Schrödinger functional is running out of steam at (L/a)4 ≈ 244 : Noisy signal,
huge statistics required (several ×105 trajectories per point)
Gradient flow is a promising new tool
I
I
Can reach larger volumes than with SF - potentially less lattice artifacts
Preliminary studies in this context by Nogradi et al; Fritzsch and Ramos;
Rantaharju; Cheng et al; Hasenfratz et al
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
31 / 34
Conclusions
Status of the field: not yet quite conclusive results. No full consensus yet of
the “best practices” – getting there
Interesting new tools
Mapping out the theory space: IRFP, χSB
Walking has not been unambiguously observed (except in toy models)
Not yet clear quantitative phenomenology: Higgs and exotica masses,
branching ratios . . .
Theories are considered in isolation: coupling to EW?
I
I
Has an effect on the physics
Axial gauge coupling: we do not even know how to do it!
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
32 / 34
Walking in 2d O(3)
2-d O(3) model with topological charge
Z
1
S=
d 2 x ∂i ua ∂i ua + iθQ
2g 2
[de Forcrand, Pepe, Wiese]
with |u| = 1
Z
Q=
d 2 xij abc Ua ∂i Ub ∂j Uc
asymptotically free
mass gap
has a IR fixed point at θ = π! (integrable model)
Adjusting θ the degree of “walking” can be changed
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
33 / 34
Walking in 2-d O(3)
0
-0.5
β(α)
-1
-1.5
IRFP
θ=0
θ=π/2
θ=3π/4
θ=0.92 π
θ=π
-2
-2.5
-3
0
0.5
1
1.5
2
2.5
3
3.5
α
K. Rummukainen (Helsinki)
Conformal gauge
Sussex 2015
34 / 34