A Very Light Dilatonwith Patipan Uttayarat
arXiv:1105.2370
Grinstein - UCSD
March 19, 2012
Conformality in Strong Coupling Gauge Theories at LFC and Lattice(SCGT12Mini)

vixra log

vixra log

1!
2!

10-2
3!

10-3
10-4
10-5

Global significance
0.8! in range 110-600 GeV
2.1! in range 110-145 GeV

CMS Preliminarys = 7 TeV
L = 4.6-4.8 fb-1

Combined obs.
Exp. for SM Higgs
Comb. ensemble
H " bb (4.7 fb-1)
H " # # (4.6 fb-1)
H " $ $ (4.8 fb-1)
H " WW (4.6 fb-1)
H " ZZ (4.7 fb-1)

200

300

400 500

SM4 Higgs boson mass (GeV)
CMS Preliminary
68% CL bands = 7 TeV
Figure 9: (Left) The
observed and expected 9
2.0
L = 4.6-4.8 fb-1

rameter µ = σ/σSM4 for the SM4 Higgs boso
mass (Right) The observed and expected 95% C
1.5µ = σ/σSM4 as a function of the SM4 Higgs bos
modes and their combination.

1.0
0.5

4!

10-6
110 115 120 125 130 135 140 145

Higgs boson mass (GeV)

2.5 100

1

0.0

Local p-value

10-1Best fit !/!SM

Local p-value

1

1!

-1

-0.5 102!
-2-1.010110 115 120 125 130 135 140 14
-3

10

3!
Higgs boson mass (GeV)

igure 7: The observed local p-value p0 (left) and best-fitµ̂Global= σ/σasforaSM4obs.
functionp-valueSM~0.5(right)Combined
-4
Exp.
Higgs
10
in range 110-600 GeV
Comb.
ensemblehe SM Higgs boson mass in the range 110–145 GeV. The local p-values for individual chann4!
H " bb (4.7 fb )H " # # (4.6 fb ) loca
nd their combination are obtained with the asymptotic
formula (lines); the combined
10-5 CMS Preliminary
H " $ $ (4.8 fb )s = 7pseudo-datasets
TeV
alue is validated by generating ensembles of background-only(points).
T
H " WW(4.6 fb )H " ZZ (4.7 fb )L = 4.6-4.8 fb-6ashed line shows the expected local p-values p0 (mH )10, shoulda Higgs boson with a mass
110 115 120 125 130 135 140 145
xist. The band in the right plot corresponds to the ±1σ uncertainties
the µ̂mass
values.
SM4 Higgson
boson
(GeV)
-1

-1

-1

-1

-1

-1

Outline
1. Introduction
2. The Model
3. Dilatation
4. Phase Structure
5. Conclusions

What and why a Dilaton?
A Nambu-Goldstone boson of a spontaneously broken scale (dilatation)
symmetry. If scale symmetry is only approximate, we get a pseudo NGB
instead, also called dilaton.
Why would one be interested in a light dilaton?

• It can serve as a scalar analog of the graviton. (Sundrum ʼ03)
• If SM is embedded in a CFT, a dilaton could have properties similar to
the Higgs." (Goldberger, BG, Skiba ʼ08; Vecchi ʼ10; Campbell, Ellis,
Olive ʼ11; Matsuzaki, Yamawaki ʼ12)

• It can serve as a force mediator between dark matter and normal
matter. (Bai, Carena, Lykken ʼ09)

• If the dilaton is sufficiently decoupled, can serve as a dark mattercandidate.(Choi, Hong, Matsuzaki ʼ11)

Scale Invariant Theory

Coupling runs from UV fixed point (origin) to IR fixed point(which it reaches in exponential RG-time).

Intorduction(Approximate)
ScaleInvariant
Invariant
Theory(Approximate) Scale
TheoryA schematic
β functionof a WTC-liketheory.
A schematic
β functionof the theory.
The Thecoupling
constant
thecoupling
flow flows
towardtoward
the “would“wouldfixed-point,be”be”
IR IR
fixed-point,g∗g. ✳.
Close
to thethe flow
slow isandClose
to fixed-point,
the fixed-point,
theis flow
the theory
possesslow andthe approximate
theory possesscale
approxisymmetry.

β(g)

g∗

g

mate scale symmetry.

If the RG-trajectory reaches g∗ , scale invariance becomes exact.

If the RG-trajectory reaches g✳, scale invariancebecomesexact.
However,some degree of freedom in the theory can get a vev. If thishappens close to the fixed-point, scale invariance is spontaneously
However, some order parameter may get a vev.
broken and one expect a dilaton in the spectrum.
If this happens close to the fixed-point, scale invariance isspontaneously broken and one expects a light dilaton in the
A field theory with this behavior is not common!
spectrum.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 20115/2

Intorduction(Approximate)
ScaleInvariant
Invariant
Theory(Approximate) Scale
TheoryA schematic
β functionof a WTC-liketheory.
A schematic
β functionof the theory.
The Thecoupling
constant
thecoupling
flow flows
towardtoward
the “would“wouldfixed-point,be”be”
IR IR
fixed-point,g∗g. ✳.
Close
to thethe flow
slow isandClose
to fixed-point,
the fixed-point,
theis flow
the theory
possesslow andthe approximate
theory possesscale
approxisymmetry.

β(g)

g∗

g

mate scale symmetry.

If the RG-trajectory reaches g∗ , scale invariance becomes exact.

If the RG-trajectory reaches g✳, scale invariancebecomesexact.
However,some degree of freedom in the theory can get a vev. If thishappens close to the fixed-point, scale invariance Whatis spontaneously
does this mean?
However, some order parameter may get a vev.
broken and one expect a dilaton in the spectrum.
If this happens close to the fixed-point, scale invariance isspontaneously broken and one expects a light dilaton in the
A field theory with this behavior is not common!
spectrum.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 20115/2

Intorduction(Approximate)
ScaleInvariant
Invariant
Theory(Approximate) Scale
TheoryA schematic
β functionof a WTC-liketheory.
A schematic
β functionof the theory.
The Thecoupling
constant
thecoupling
flow flows
towardtoward
the “would“wouldfixed-point,be”be”
IR IR
fixed-point,g∗g. ✳.
Close
to thethe flow
slow isandClose
to fixed-point,
the fixed-point,
theis flow
the theory
possesslow andthe approximate
theory possesscale
approxisymmetry.

β(g)

g∗

g

mate scale symmetry.

the statementexact.
If the RG-trajectory reaches g∗ , scale invariance It’s
becomes

If the RG-trajectory reaches g✳, scale invariance (equally obscure to me)< g✳
becomesexact.
However,some degree of freedom in the theory thatcangc geta vev. If thishappens close to the fixed-point, scale invariance Whatis spontaneously
does this mean?
However, some order parameter may get a vev.
broken and one expect a dilaton in the spectrum.
If this happens close to the fixed-point, scale invariance isspontaneously broken and one expects a light dilaton in the
A field theory with this behavior is not common!
spectrum.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 20115/2

Intorduction(Approximate)
ScaleInvariant
Invariant
Theory(Approximate) Scale
TheoryA schematic
β functionof a WTC-liketheory.
A schematic
β functionof the theory.
The Thecoupling
constant
thecoupling
flow flows
towardtoward
the “would“wouldfixed-point,be”be”
IR IR
fixed-point,g∗g. ✳.
Close
to thethe flow
slow isandClose
to fixed-point,
the fixed-point,
theis flow
the theory
possesslow andthe approximate
theory possesscale
approxisymmetry.

β(g)

g∗

g

mate scale symmetry.

the statementexact.
If the RG-trajectory reaches g∗ , scale invariance It’s
becomes

If the RG-trajectory reaches g✳, scale invariance (equally obscure to me)< g✳
becomesexact.
However,some degree of freedom in the theory thatcangc geta vev. If thishappens close to the fixed-point, scale invariance Whatis spontaneously
does this mean?
However, some order parameter may get a vev.
broken and one expect a dilaton in the spectrum.
If this happens close to the fixed-point, scale invariance isspontaneously broken and one expects a light dilaton in the
A field theory with this behavior is not common!
spectrum.
A field
theory with this behavior is
not common!
Patipan Uttayarat (UCSD Physics)
A Very Light Dilaton

May 25, 2011

5/2

(Approximate) Scale Invariant Theory (cont.)
In the WTC framework, the fixed-point is strongly interacting.
As a result (wish?), fermion condensate forms and breaks scale
invariance.
But the strong interacting nature of the model makes it difficult toanalyze analytically.
In particular, the existence of a light dilaton in WTC is not clear and is
the subject of a recent rekindled debate• Yes (Appelquist, Bai ʼ10)
• No (Hashimoto, Yamawaki ʼ10; Vecchi ʼ10)
In a nut-shell: small β vs ηʼ-like (QCD anomaly).
Having a perturbative toy model with the above properties – aninteracting fixed-point and an approximate scale invariance whichis broken dynamically – would help me understand this better.
Can it be done??

Why is it non-straightforward? Try the obvious stuff first:

SSB in CFT?

• Take CFT with moduli space (common in SCFT). Say, fordefiniteness:
N = 4 SUSY SU(N) ➠ flat directions ➠ expand about a point
away from origin.

• EFT = SU(N) → SU(N − k) × SU(k) × U(1).
• But EFT has N = 4 SUSY unbroken, “Dilaton” is exactly
massless together with partners ➠ no mass gap.

• Perturbations: flow into ??? (possibly another CFT,
interacting), fate of “dilaton?”a y, running around the polygon, those of the sameabelian-higgs
with a cp, running aroundthe polygon, model.
liesBetter attempt(?):
type, but Coleman-Weinberg
and those shown
inµνFig. 4,µ witha photon running1
dity of
∗
4
L
=
−
F
F
+
D
φ
D
φ
−
λ|φ|
µν
µ
around the 4polygon. Aside from trivial numericalrbifactors, all are of exactly the same structure as
be reFine
tune
mass to zero, classically scale invariant.
•the graphs considered in Sec. III. After some
As wehigher

we obtain
computation,
physicalpotential develops
a minimum
away from origin:• Effectivestraightforward
ive po-ich thethem

4~ 9 c+1152' 64' Wc

M2

6• Gauge and scale symmetries spontaneously broken.
(4.5)
Thisacquires
like the one discussed in Sec. III,function, mass.
CS • Gauge field
has a minimum away from the origin. Here, howBut
would-be-dilatonacquires mass too: trace anomaly spoils•
ed in
ever, the minimum need not be illusory. In Sec.

symmetry.
lectro-scaling III,the minimum arose from balancing a term ofd meorder
Z against
a term of order )Pin(rp/M); thus,
good.
But now:• So far, so
c field.
for small A, , it inevitably occurred at largeof twonge

outside the expected
1n(y/M),
Mdilaton/Mvector ~ e2/16π2domain of validityof our approximation. Here, even for an arbitrarily small coupling constant, we can obtain aminimum by balancing a term of order A. against
a term of order e ln(y/M). Even though the second term formally arises in a higher order of our
expansion than the first, there is no reason in theworld why A. cannot be of the same order of mag-a y, running around the polygon, those of the sameabelian-higgs
with a cp, running aroundthe polygon, model.
liesBetter attempt(?):
type, but Coleman-Weinberg
and those shown
inµνFig. 4,µ witha photon running1
dity of
∗
4
L
=
−
F
F
+
D
φ
D
φ
−
λ|φ|
µν
µ
around the 4polygon. Aside from trivial numericalrbifactors, all are of exactly the same structure as
be reFine
tune
mass to zero, classically scale invariant.
•the graphs considered in Sec. III. After some
As wehigher

we obtain
computation,
physicalpotential develops
a minimum
away from origin:• Effectivestraightforward
ive po-ich thethem

4~ 9 c+1152' 64' Wc

M2

6• Gauge and scale symmetries spontaneously broken.
(4.5)
Thisacquires
like the one discussed in Sec. III,function, mass.
CS • Gauge field
has a minimum away from the origin. Here, howBut
would-be-dilatonacquires mass too: trace anomaly spoils•
ed in
ever, the minimum need not be illusory. In Sec.

symmetry.
lectro-scaling III,the minimum arose from balancing a term ofd meorder
Z against
a term of order )Pin(rp/M); thus,
good.
But now:• So far, so
c field.
for small A, , it inevitably occurred at largeof twonge

outside the expected
1n(y/M),
Mdilaton/Mvector ~ e2/16π2domain of validityof our approximation. Here, even for an arbitrarily small coupling constant, we can obtain a
Want: arbitrarily
dilaton without
interactions
minimum lighta term ofturning
order A.off
by balancing
againsta term of order e ln(y/M). Even though the second term formallyarises in argue
a higherthat
orderof our(note: Hashimoto
and Yamawaki
in WTC:
expansion than the first, there is no reason in theas Mdilatonworld→ 0 why
thenA. thedecay constant f → ∞ and dilaton decouples).
cannot be of the same order of mag-We construct a model which
Is perturbative (naively)
Has a perturbative IR fixed point (naively)
Spontaneously breaks approximate scale symmetry
Dilaton can be made arbitrarily light (mass gap!)
while still interacting
follow-up by Antipin, Mojaza, Sannino ʼ11some qualitative differences; unresolved

In the rest of the talk I will explain this model.

First the big picture:

Theory parameter space: couplings at fixed µ0

Symmetric Phase

Region where we canreliably find minimumof Veff

critical surface
Mσ = 0dilaton of small mass Mσnear critical surface

IRFP
Broken Phase

Theory parameter space: couplings at fixed µ0

Symmetric Phase

IRFP
Broken Phase�φ� = v �= 0

uniformly,arbitrary
Region where we canreliably find minimumof Veff

critical surface
Mσ = 0dilaton of small mass Mσnear critical surface

The Model

Outline1

Intorduction

2

The Model
Fixed-point Structure
Vacuum Structure
Spectrum

3

Dilatation

4

Phase Structure

5

Conclusion

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 2011

6 / 28

The Model

The Model
SU(N) gauge theory with nψ = nχ fundamental fermions ψ and χ
and two scalar singlets φ1 and φ2 .
nχ
!
"

# 1
1
1
µν
j /
j /
2
L = − Tr F Fµν +ψ̄ i Dψj + χ̄ i Dχj + (∂µ φ1 ) + (∂µ φ2 )2
2
2
2
j=1
"
#
− y1 ψ̄ψ + χ̄χ φ1 − y2 (ψ̄χ + χ̄ψ)φ2
1
1
1
4
4
− λ1 φ1 − λ2 φ2 − λ3 φ21 φ2224
24
4This theory is invariant under discrete Z2 as well as SU(nχ ) symmetry
φ1 , ψ → φ1 , ψand
φ2 , χ → −φ2 , −χPatipan Uttayarat (UCSD Physics)

A Very Light Dilaton

ψ → Uψ
χ → Uχ
May 25, 20117 / 28

Nota bene
Masses set to zero (I am not solving the hierarchy problem).
Precisely as with Coleman and Weinberg.
Set them to zero and use dimensional regularization.

Theory has Landau pole. This is a UV issue.
We study the IR properties of the model.
We can take it to be a cut-off theory.

This is not the theory of everything.
It is a Toy Model that displays some behavior that mimics
some wanted behavior of WTC, may answer some questions
and seems to be interesting in its own right (as a field theory).

The Model

Fixed-point Structure

MS β Functions
For large N with nχ = 11N/4 (1 − δ/11), the leading terms are
2
5
δN
25N
g
∂g
= − g3 +(16π 2 )∂t
3
2 16π 22 ∂y1
= 4y1 y22 + 11N 2 y13 − 3Ng 2 y1
(16π )∂t2 ∂y2(16π )= 3y12 y2 + 11N 2 y23 − 3Ng 2 y2∂t
2 ∂λ1
= 3λ21 + 3λ23 + 44N 2 λ1 y12 − 264N 2 y14(16π )∂t2 ∂λ2(16π )= 3λ22 + 3λ23 + 44N 2 λ2 y22 − 264N 2 y24∂t
2 ∂λ3= λ1 λ3 + λ2 λ3 + 4λ23(16π )∂t+ 22N 2 λ3 y12 + 22N 2 λ3 y22 − 264N 2 y12y22
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 20118 / 28

The Model

Fixed-point Structure

Fixed-point
To get a fixed-point for the gauge coupling, need to balance a 1-loop
against a 2-loop.
This is possible because for large N, δ can be made small by acarefully chosen nχ .
The fixed-point to leading order in 1/N is2
2 2g∗ = 16π

δ
75 N

2
3
g
∗
2
2
y1∗
= y2∗
=
11 N
18 g∗22= 6y1∗
λ1∗ = λ2∗ = λ3∗ =11 N

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 2011

9 / 28

The Model

Vacuum Structure

Effective Potential
At tree-level, !φi " = 0 and all the particle are massless. The theory
flows to the IR fixed-point.
However, quantum effect could drastically change the structure of thevacuum (Coleman, Weinberg ’73)
Vef f (φ)

V0 (φ)φφ

The non-trivial vev gives mass to both fermions and scalars and alters
the RG trajectory.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201110 / 28

The Model

Vacuum Structure

Effective Potential (cont.)
The effective potential in MS is
1
1
1
4
4
Veff = − λ1 φ1 − λ2 φ2 − λ3 φ21 φ22
24
24!
4
"
"
2 4
2
2 4 !
2
11N Mf +Mf + 3
11N Mf −Mf − 3
−
ln
−
−
ln
−
2
2
2
2
(64π )2µ2(64π )
2µ
2
!
"
!
"
2
4
2
4
Ms+ 3
Ms−
Ms− 3
Ms+
ln 2 −+
ln 2 −+
22(64π )µ2(64π )2µ2
Mf ± = y1 φ1 ± y2 φ2 ,

2
2
(λ
+
λ
)φ
+
(λ
+
λ
)φ
1
3
2
3
1
2
2
Ms±
=
4
#
(λ1 − λ3 )2 φ41 + (λ2 − λ3 )2 φ42 − 2(λ1 λ2 − λ1 λ3 − λ2 λ3 − 7λ23 )φ21 φ22±
4
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201111 / 28

The Model

Vacuum Structure

Effective Potential (cont.)
Minimizing the potential analytically is difficult. But easy to identify
some local minima. Focus on minimum which preserves discrete Z2(i.e. !φ2 " = 0).
The potential reduces to
!
"
!
"
2 2
2 2
2
2
λ1 φ 1 3(λ3 φ1 )λ3 φ 1 3
λ1 4 (λ1 φ1 )
ln
−
+
ln
−
Veff = φ1 +
2
2
2
2
24
256π
2µ
2
256π
2µ
2
!
"
2 4 4
2 2
22N y1 φ1
y 1 φ1 3−
ln 2 −264πµ
2The extremum, ∂/∂φ1 Veff (!φ1") = 0, is at
!
"
!
"
2
2
2
2
λ1
λ1 !φ1 "λ3
λ3 !φ1 "
λ1
ln
−1 +
ln
−1
− =
2
2
2
2
6
64π
2µ
64π
2µ
!
"
2 4
2
2
88N y1
y1 !φ1 "
−
ln
−1
2
2
64π
µ
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201112 / 28

The Model

Vacuum Structure

Vacuum Expectation
λ1 can be traded with !φ1 " as a free parameter. For consistency,!φ1 "2λ1
ln µ2 # 1. Stability of the vev is determined from the
16π 2
eigenvalues of second derivative matrix∂2
λ23 − 88N 2 y14
2
V
(!φ
",
0)
=
!φ
"
eff
1
1
∂φ21
32π 2
λ3
∂2
2
V
(!φ
",
0)
=
!φ
"
+ O(1-loop)
1
1
2 eff
∂φ2
2
Evaluate Veff at !φ1" yields
λ23 − 88N 2 y14
4
Veff (!φ1 ") = −!φ"
1512π 2Thus when ε ≡ λ23 − 88N 2 y14 ≥ 0, there is a non-trivial minimum.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201113 / 28

Role of ϕ2
Note that ϕ2 never enters any calculations above.
Moreover, one can get an attractive IR fixed-pointwith just one scalar singlet.
This raises the question: what is the purpose of thesecond singlet?

Role of ϕ2
Note that ϕ2 never enters any calculations above.
Moreover, one can get an attractive IR fixed-pointwith just one scalar singlet.
This raises the question: what is the purpose of thesecond singlet?
• It allows us to introduce more couplings, in particular
the cross-coupling λ3.
• Without the second singlet, the extremum found by
perturbative analysis would have been the maximum.

‣ The scalar potential appears to be unboundedfrom below.

‣ Possible to have non-trivial minimum at higher
scale which is inaccessible to perturbative
analysis.

The Model

Spectrum

Pole mass in Broken Phase
The explicit 1-loop pole masses are
"
#$
!
2 2
2
y1 v
g N
3
ln
−4
Mψ (µ) = Mχ (µ) = y1 v 1 −
2
2
16π 2
µ
"
#
"
#
2
2 2
2
2 2
2
λ1 v
λ1 v
5
3λ3 v
λ3 v
1 2λ1
3λ1 v
2π
2
ln
+
ln
M φ1 =
+
− + √
− −
2
2
2
2
2
64π
2µ
3 3 3
64π
2µ
3 3λ3
%
#"
#
"
2 2
2
2
2 2
22N y1 2 2 λ1 v
λ1 v
y1 v
2 2
+
y1 v −ln 2− 3 y1 v −
2
16π
12
12
µ
# "
#'
& 1 "
λ1
x(1 − x)2 2
2−3
dx y1 v −λ1 v ln 1 − x(1 − x) 2
2
2y1
0
ε 2
λ23 − 88N 2y14 2
v =
v
#
2
2
32π
32π

Since v = $φ1%, it has the same anomalous dimension as φ1 .
Using the anomalous dimension and the β functions, one can verify
that the masses are RG invariant at 1-loop.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201115 / 28

Dilatation

Outline1

Intorduction

2

The Model

3

Dilatation
Dilation Current
Decay Constant and Mass

4

Phase Structure

5

Conclusion

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201115 / 28

Dilatation

Dilation Current

Dilatation Current
The dilatation current, D µ , is constructed from the improved
energy-moementum tensor, Θµν , of Callan, Coleman and Jackiw.
D µ = xν Θµν
Θ

µν

= −F

aµλ

1
1
µ ν
ν µ
aν
Fλ + χ̄i (γ D + γ D )χ + ψ̄i (γ µ D ν + γ ν D µ )ψ2

2

1
+ ∂ φi ∂ φi − g L − κ(∂ µ ∂ ν − g µν ∂ 2 )φ2i
2
µ

ν

µν

κ is the improvement term. It is a total derivative.
The CCJ improved tensor is the one with κ = 1/3.
The improvement term does not change the charges constructedfrom Θµν .
The matrix elements of Θµν are
is finite, thus it doesn’t get
renormalized.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201116 / 28

Dilatation

Dilation Current

Trace Anomaly
The divergence of the dilatation current is the trace of the improved
energy-momentum tensor.
Classically Θµµ vanishes for theory without any dimensional couplings.
Quantum effects make Θµµ non-zero, this is known as trace anomaly.
For the theory under consideration4φΘµµ = γφ1 φ1 ∂ 2 φ1 + (4γφ1 λ1 − βλ1 ) 1 + . . .
24

Terms involving other fields are omitted.
Terms proportional to γφ1 are usually omitted.
They cancel when EOM is applied but can contribute to off-shell
matrix element and Green functions.
Also these terms are needed to make the trace RG-invariant.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201117 / 28

Dilaton: The particle (state)
We may look for the dilaton state, σ , by using the following
generic criteria:• spinless state• couples strongly/linearly to the energy-momentum tensor• lightest such state
Clearly ϕ1 satisfies all of the above.

• It is the only state whose mass start at 1-loop (modulo•

gauge fields)
It is the only state which couples linearly to the energymomentum tensor when expanded about
ϕ1⟩ = v, ⟨ϕ2⟩ = 0
Thus we identify σ with a single particle state created by ϕ1.

Dilatation

Decay Constant and Mass

Decay Constanst
Define the decay constant fσ by" ip·xfσ ! µ νµν 2!0|Θ (x)|σ" =p p −g p e
3µν

where p is the momentum state|σ". The form of the right hand sideofis constrained by conservation of Θµν . The factor 1/3 comes from!0|∂µ D µ |σ" = !0|Θµµ |σ" = −fσ Mσ2 e ip·x
Note that Θµν = −1/3v ∂ µ ∂ ν φ1 + · · · .
Thus to lowest order fσ = v + · · · .
The RG invariant expression is easy to guess−1/2
fσ = vZφ1

where Zφ1 is the wavefunction renormalization factor.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201119 / 28

Dilatation

Decay Constant and Mass

Dilaton Mass
Having determined the decay constant fσ , the mass of the dilaton canbe obtained from the trace anomaly.
To lowest order, the mass is
2
2
2 4
λ
+
λ
−
88N
y1 2
1
3
2
v
Mσ =
2
32π
ε 2
2
=
v
=
M
φ1
32π 2

where λ21 term is dropped for consistency.
RG invarianceinvariant of Mσ can be inferred from Mφ1 .
RG
Given the vev, we can tune ε to make the dilaton light compares
the vev.
bytocomparison

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201120 / 28

Phase Structure

Outline1

Intorduction

2

The Model

3

Dilatation

4

Phase Structure
Broken/Symmetric Phase
Numerical Analysis

5

Conclusion

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201120 / 28

Phase Structure

Broken/Symmetric Phase

Broken Phase
Recall the theory admits a non-trivial minimum provided
λ1 is much smaller than other couplings,
ε ≡ λ23 − 88N 2 y14 ≥ 0.
We want to study symmetry breaking close to the IR fixed-point.
However, near the fixed-point these conditions are not satisfied.
Use RGE to trace back the RG trajectory to large RG time where
perturbative analysis of effective potential yields a non-trivial
minimum.
Alternatively,
Similary
can define the theory at scale µ0 where perturbative analysis
yields a non-trivial minimum. Moreover, if the vev is well below µ0 ,RG flows will get closed to the fixed-point before the massive
particles decouple.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201121 / 28

Theory parameter space: couplings at fixed µ0

IRFP?
Flow towards IRFP

Region where we canreliably find minimumof Veff

Phase Structure

Broken/Symmetric Phase

Symmetric Phase
For a point in parameter space where ε < 0
Veff (!φ1 ") becomes positive and the non-trivial minimum
disappears,
the effective potential seems to be unbounded from below along
φ1 direction for large φ1 .
The second point threatens the validity of the model.
However, at large φ1 perturbative analysis breaks down.
Can extend the range of perturbativity using the improved effective
potential which effectively re-sum large logarithms.
imp
Veff
=

! !
1
−4 0t γφ1 dt ! 4
λ̄1 (t)eφ124Here t = ln φ1 /µ0 . This form is valid as long as λ̄1 (t) is perturbative.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201122 / 28

Phase Structure

Broken/Symmetric Phase

Symmetric Phase (cont.)
For points in parameter space closed to the IR fixed-point, gauge
coupling drives the Yukawa coupling to 0 in the UV.
Thus in the far UV, the β-function for λ1 has a Landau pole.
The effective potential is bounded from below because
λ̄1 (t) > 0 for large φ1 .
The theory need a UV cutoff.
One can view the model as being a low energy effective theoryof some UV completed models.
! Since the cutoff will be many order of magnitude above thescale of symmetry breaking, one can (safely) ignore it.
!

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201123 / 28

Theory parameter space: couplings at fixed μ0

critical surface
Mσ =0
Symmetric Phase

IRFP

Broken Phase
Region where we canreliably find minimumof Veff

Phase Structure

Numerical Analysis

Numerical Value
N = 20, nf = 11/2 N, δ = 0.2,
g (µ0) = 94 g∗ , y1 (µ0 ) = 0.32y1∗, y2 (µ0 ) = 15 y2∗ ,1
λ1 (µ0 ) = 30
λ2∗ , λ2 (µ0 ) = 3λ2∗ , λ3 (µ0 ) = 5.2λ3∗ .
These condition corresponds to ε ! 0.
The vev is at
!φ1 "
# −29
ln
µ0
and the spectrum are
M φ1
M φ2
Mψ,χ
−3
−4
# 8.5 × 10 ,# 7.9 × 10 ,# 9.5 × 10−2 .
vvv
Fractional correction to the effective potential from higher orderterms are approximated to be!
" 2 2 #!
2
!
! Ng∗
y
v
1
! # 0.2.
!
ln! 16π 2µ2 !
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201124 / 28

Phase Structure

Numerical Analysis

Numerical: Couplings Evolution
N = 20, nf = 11/2 N, δ = 0.2,
g (µ0) = 94 g∗ , y1 (µ0 ) = 0.45y1∗, y2 (µ0 ) = 15 y2∗ ,1
λ1 (µ0 ) = 30
λ1∗ , λ2 (µ0 ) = 3λ2∗ , λ3 (µ0 ) = 5.2λ3∗ . These condition
corresponds to ε < 0.
λ3 (t)

g(t)

0.025

0.08

λ2 (t)

0.020
0.06
0.015
0.04
0.010
0.02

!400

!300

!200

!100

y1 (t)y2 (t)
100

!400

!300

!200

100

!100

positive
minimum

Fixed point
Patipan Uttayarat (UCSD Physics)

λ1 (t)

0.005

A Very Light Dilaton

May 25, 201125 / 28

Phase Structure

Numerical Analysis

Numerical: Broken Phasey1 (µ0 ) = 0.32y1∗ . This corresponds to a positive ε.
λ3 (t)
0.030
0.00010.025
0.00008
0.020

λ2 (t)

0.015

0.00006
0.00004

0.010

0.00002

0.005

!400

!300

!200

!100

λ1 (t)
100

!80

!60

!40

!20

The coupling λ1 becomes negative during the flow.
This agrees with our expectation from the improved effective
potential.
Gauge coupling walks. Eventually runs again. Endgame: glueballs.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201126 / 28

Conclusion

Outline1

Intorduction

2

The Model

3

Dilatation

4

Phase Structure

5

Conclusion

Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201126 / 28

Conclusion

Summary
We construct a perturbative model which a non-trivial IR fixed
point.
There are two distinct phases in our model:
Symmetric phase: the flow reaches the fixed point (in infinite
RG time.)! Broken phase: the vev is dynamically generated. this phasesomewhat mimics the behavior of walking technicolor.
!

The broken phase can be used to study the dilaton:!

The mass of the dilaton can be made arbitrary lightsmall comparedto the vev by tuning the parameter ε = λ23 − 88N 2 y14 .

The model can be used to verify results/conjectures in literature
which are obtained via indirect argument.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201128 / 28

Conclusion

Applications
We can use this model to verify various results in the literature. For a
specific example we will verify the dilaton potential in nearly
BG, Skiba)
conformal theory
Goldberger,
Grinstein and Skiba. Taking! of(Goldberger,
L = LCFT + n λn On , GGS arrives, via indirect argument, at thedilaton potential
" # $
%
2
Mσ 4
χ
1
Veff (χ) = 2 χ ln−
+ O(γ 2 ).
4fσ
fσ
4
To compare our model with GGS, we view our model as
L(g ) = L(g∗ ) + (L(g ) − L(g∗ )). In our model the dilaton field isidentify with φ1 and the anomalous dimensions are small.
Our effective potential for φ1 turns out to be exactly the same as
GGS.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201127 / 28

Obviously
this adding
basic setup
appliesto ourmodel,smalland[23,
becausedeviations(“deformations”)
fromthe CFT
by adding
perturbations[23,perturbative
24].
ns”) from
the
CFT
by
small
perturbations
24].it is fully
dimensions
γn or the coeffi
!
model
one
shouldbeable
toverifythevalidityofsomegeneralassertions.
ThebviouslyObviously
this
basic
appliestomodel,
becauseitis
fully
perturbative
this setup
basic
setup
appliestoour
ourmodel,
andandbecauseit is fully
perturbative
L
=
L
+
λ
O
.
(30)
CFT
n is
n fully andup appliesIf to our model, andbecause it
perturbative
grounds
oneanomalous
can show thatdeviations(“deformations”)
fromtothe
CFT
bybeadding
small
perturbations[23,24]. thedeviations
fromconformality
can
small
in
one
of
two
ways,
eithermodel
oneshouldbeable
verifythevalidityofsomegeneralassertions.
Theodel one should be able to verifyn the validity of some general assertions. The
ble toObviously
verifythefrom
validityof somegeneralassertions.
this
basic
setup
to ourandbecauseis fullyeither
perturbative
expectation
value
gives risedimensions
γn orapplies
thecoefficients
ofone
the
OThe
On
generaldeviations
conformality
canbemodel,smallλninof operators
twoitways,
thesmall.
anomalousn areterm model
is afrom
CFT
while
thetosumthatfollows
is anofattempt
to eithercapturetheviations
conformality
can
be
small
in
one
two
ways,
the
anomalousone
shouldbeable
verifythevalidityofsomegeneralassertions.
The
grounds
one
can
show
that
for
|γ
|
!
1
the
effective
potential
for
the
field
χ
whose
dimensionsγorthecoefficients
λoftheoperatorsOaresmall.
On
generaln n either the anomalousnality can be smalln in one of two ways,
Veff (χdeviations
fromconformality
can be small
inoneof twoways, eitherO
the anomalous
mensions
γnexpectationor
coefficientsoftheoperators
arefield
small.
value
givesforrise
todilatonis [26]grounds
onethecan showthat|γnλ| n!the1 theeffective
potential forχ whoseOn generaln the
27 O are small.On are
oefficients
λn γofthe operators
Onsmall.
generaldimensions
On generaln or the coefficients λn ofnthe operators!
"#$
expectation
value
gives
rise
to
the
dilatonis[26]
2
oundsgrounds
one can
show
that
for
|γ
|
!
1
the
effective
forcasethe|λnfieldχiswhoseχM
1potential
n
σ
|
!
1
4
2 The
one
can
show
that
for
|γ
|
!
1
the
effective
potential
for
the
field
χ
whose
n
χ
ln
−
+
O(γ
)
.
(31)more cV(χ)=
hat for |γn | ! 1 thethen
effective effpotential
for
the
field
χ
whose
!
"
#
$
2
2
4f
f1σ
4 2
χ
M
σ
4 is σexpectation
value
givesrise
rise
to
the
dilaton[26]
is added does
pectation
value
gives
to
the
dilatonis[26]−+ O(γ ) .
(31) one obtain a
Veff (χ) = 2 χ ln
4f!σ " # fσ $ 4
rise to the dilaton
The case is|λn[26]| ! 1 is morecumbersome. Only in the case that only one perturbation
2
!χ "
M
1 #
σ 42
2 $
or
χ
ln
−
+
O(γ
) . that only one perturbation
(31)
V
(χ)
=
eff
χ
M
1
The case
|λn"
|!
1#
isone
more
cumbersome.
Only
in
the
case
!
$
2 σ
Veff (χ) =
is
added
does
obtainaparameter-freeeffective
potential
4f
f
4
4
2
σ
−
+ O(γ ) .
(31)
Veff (χ) 1= σ 2 χ 2ln
χ
Mσ2 4
!
"
#
$
added
does one obtaina parameter-freeeffective
potential
4f
fσcase that
4γonly one (31)χ |λn|ln+ σO(γ(χ) =Theiscase2 ) .in the! 1 is more−cumbersome.
Only
M
1
χ
2
2
4fσ
fσ
4Veff(χ) 2= ! σ χ4 " #γ $ − 1 +perturbationO(λ) , for more than one pe
while
Mσ 4fσ2 γeffective
1 4+
χpotential
γ fσ1
4 2
is added does one obtain a parameter-free+case
O(λ that),
Veff (χ)= 2 χ
he case |λn | ! 1 is morecumbersome.
Onlyin−theonly one perturbation
f
γ
4
+
γ
f
4
!
"
#
$
σ
σ
γ
2 casee cumbersome.
Onlytheperturbation
Mσ2
Mσone
1thatχonly
1oneonewhile formoreinthan
occur
has
the less restricted
4 perturbation2
Veff (χ) = 2 χ+ O(λpotential),
Veff (χ) a= 2 χ−
added does
obtaineffective
fperturbation
γ
4
+
γ
f
4
while onefor morethan oneparameter-free
occur
one
has
the
less
restricted
fσ
σ
& !
" #γ
$'
2
%
n a parameter-free effectiveσ potential
Mσ &4! !
1#
χ $'
1
2
"
#
"
$
γ
V
(χ)
=
χ
x
−
+
O(λ
),
γ
2
eff
n
%
2
while for more than one perturbation
theMσM4 fσ2 occur 1one1 has4 χχγnless frestricted
1
+
4
1
2 2
!
#
σ $ 4 xnn
where
V"
==
−σ
+ O(λ
) ,) , the coupling constanγ 2 χ&eff (χ)
2
!
"
#
$'
+
O(λ
V
(χ)
χ
−
γ
eff
f
4
+
γ
f
4
Mσ 4
1
χMσ2 4 σ%21(
1 + 2γχ n fσσ 1
n
f
γ
4
4
2
σ
+
O(λ
)
,
) = 2 χ where
Veffthe(χ) coupling= 2χ −
x
−
+
O(λ
)
,
n
constants
have
been
tradedfor constants xby
aren xconstrained
n = 1.
n that
nγ
f
4
+
γ
f
4
n
σ
σ
fσ γ where4the
+
γ
f
4
n
(couplingσ constants
have been traded for constants xn that are constrained
by
γ
x
=
1.
Any model with a conforn( thann nonehile for
more
perturbation
occur
one
has
the
less
restricted
where
the
coupling
constants
have
been
tradedforconstants
x
that
are
constrainedby n γn xn = 1.
nperturbation
occur
one
has
the& less!fixed
restricted
(
Any
model
with
a conformal
point"g∗ canbewrittenin the fashion of Eq. (30)#$'γby nAnyγn xnmodel= 1. with a2conformaln in the fashion of Eq. (30)
% fixed point1g∗ can beχwritten
M
1
σ
2
& !
" 4#
$'
n

n

n

Conclusion

Applications
We can use this model to verify various results in the literature. For a
specific example we will verify the dilaton potential in nearly
BG, Skiba)
conformal theory
Goldberger,
Grinstein and Skiba. Taking! of(Goldberger,
L = LCFT + n λn On , GGS arrives, via indirect argument, at thedilaton potential
" # $
%
2
Mσ 4
χ
1
Veff (χ) = 2 χ ln−
+ O(γ 2 ).
4fσ
fσ
4
To compare our model with GGS, we view our model as
L(g ) = L(g∗ ) + (L(g ) − L(g∗ )). In our model the dilaton field isidentify with φ1 and the anomalous dimensions are small.
identified
Our effective potential for φ1 turns out to be exactly the same as
GGS.
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 201127 / 28

A schematic β function of the theory.
Dilatonin WTC?
n massis givenby
The coupling flow toward the “wouldcg .
AB say:be”
IRfixed-point,
N
s(α
−
α
)
∗
c
f ∗− Nf 2
2
2
Mσ "
Λ,
Close toΛthe"fixed-point,c the flow is
αslow
c
f
and the theoryNposses
approximate scale symmetry.

β(g)

g∗

g
(32)

is the coupling at the fixed point, Nf is the number of flavors and Λ is
If the RG-trajectory reaches g∗ , scale invariance becomes exact.

equation: Inbreaking
our model thecriticaloccurs
coupling onlyis a criticalsurface,of chiralFirst
symmetrywhich
if thecritical coupling αc is

However,
degreeof freedom
in thethe IRFP is on critical surface,
0=0, somecorrect
but not
interesting
andtheory can get a vev. If thishappens
the fixed-point, scale
invarianceis spontaneously
not whatαisc intended
fixed point,< α∗ , which
in closeturntocorresponds
to thenumberof flavors
broken and one expect a dilaton in the spectrum.

ritical value, Nfc . The middle expression in Eq. (32), relating the mass to

A field theory with this behavior is not common!
Second theequation:( Nc - Ncoupling) / N plays roleε, measuresdistance to
ce between
criticalandof thefixed-point,
does not carry overcritical surface, and equationis qualitatively correct!
Patipan Uttayarat (UCSD Physics)

A Very Light Dilaton

May 25, 20115/

del. In our case, the role of the critical value of the coupling constant αc isa critical surface, ε = 0, separating the symmetric and broken phases. Butof the dilaton is not proportional to the distance between this surface andpoint (however one defines distance): the fixed-point lies on the criticalnd the dilaton mass vanishes everywhere on the surface. The rightmost