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Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Higgs ξ-inflation for the 125–126 GeV Higgs
(Based on arXiv:1306.6931)
Kyle Allison
University of Oxford
Seminar, University of Sussex
November 4, 2013
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Outline
¶ Inflation from a particle physics perspective
· Higgs ξ-inflation: Tree level
¸ Radiative corrections
¹ Conclusions
Radiative corrections
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Part I
Inflation from a particle
physics perspective
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
What is inflation?
I
A (supposed) period of accelerating
expansion of the early universe
I
Before the radiation-dominated era,
the scalar potential dominated the
energy density of the universe
,→ space grows exponentially
Proposed to explain
I
. Flatness problem
. Horizon problem
−→
I
Produces a nearly scale-invariant spectrum of density fluctuations
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Achieving inflation
I
Let scalar field φ be the “inflaton”
V (φ)
Inflation
φ(~x , t) = φ(t) + δφ(~x , t)
I
Need a slowly rolling field for inflation
φ̇2 V (φ),
I
φ̈ 3H φ̇
Inflation ends
In terms of the slow roll parameters
00 M2 V 0 2
V
2
≡ Pl
, η ≡ MPl
,
2
V
V
Reheating
φ(t)
inflation occurs so long as
< 1,
I
|η| < 1
⇐⇒
Inflation
After inflation, φ oscillates about the minimum and its kinetic energy
is converted into SM particles ←− Reheating
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Matching observations
I
The amount of inflation described by N = number of times space
expands by a factor of e (e-folds)
Z
tf
N=
ti
I
1
Hdt =
MPl
Z
φi
φf
dφ
√
2
Precise constraints come from measuring primordial density
fluctuations via the cosmic microwave background (CMB)
δφ(x)
δgµν
δTµν
δT
δρ
Quantum fluctuations exist at all scales during inflation and produce
a nearly scale-invariant spectrum
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Matching observations
I
The predicted temperature anisotropies from inflation are
δT 2
16π V (φ` )
field value when scale ` leaves
=
,
φ
=
`
4 (φ )
horizon during inflation
T `
45MPl
`
I
Measurement of δT /T for ` ' 3000 Mpc gives
V (φ∗ )
field value N∗ ' 50–60 e-folds
4
' 5.6 × 10−7 MPl
, φ∗ =
before end of inflation
(φ∗ )
I
Two other parameters are particularly useful for constraining
inflationary models
ns = 1 − 6 + 2η
r = 16
Spectral index, deviation from scale-invariance
Tensor-to-scalar ratio, size of gµν perturbations
As above, ns (φ) and r (φ) are evaluated at φ∗
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Matching observations
I
Many inflationary models have been constructed
(Planck collaboration, 2013)
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Candidates for the inflaton
I
I
Identity of inflaton still unknown; often assumed to be a new
weakly-coupled scalar field, but . . .
. . . we have just discovered our first fundamental scalar field
Can the Higgs boson be the inflaton?
.
.
.
.
.
h4 chaotic inflation (Linde, 1983) ←− experimentally disfavoured 7
Quasi-flat SM potential (Isidori et al., 2008) ←− too few e-folds 7
False vacuum inflation (Masina & Notari, 2012) ←− needs second scalar 7
New Higgs inflation (Germani & Kehagias, 2010) ←− new scale M < MPl ?
Higgs ξ-inflation (Bezrukov & Shaposhnikov, 2008) ←− unitarity issues ?
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Part II
Higgs ξ-inflation: Tree level
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Model definition
I
I
I
Higgs ξ-inflation is based on a non-minimal coupling of the Higgs
doublet to the Ricci scalar
M2
L = − Pl R − ξH † HR + LSM
2
This is the only local, gauge-invariant interaction with dimension ≤ 4
For computing tree-level predictions, use unitary gauge H = √12 h0
,→ Jordan frame action
Z
2 MPl
λ 4
ξh2
4 √
2
SJ = d x −g −
1 + 2 R + (∂µ h) − h
2
4
MPl
Slow-roll calculations require a minimal gravity sector
,→ Conformal transformation
ξh2
gµν → g̃µν = Ω2 gµν ,
Ω2 = 1 + 2
MPl
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Model definition
I
The resulting Einstein frame action is
Z
SE =
d 4x
2
Ω2 = 1 + ξh2 /MPl
2 p
M2
1 Ω2 + 6ξ 2 h2 /MPl
λh4
2
−g̃ − Pl R̃ +
(∂
h)
−
µ
2
2
Ω4
4Ω4
I
Define a new scalar field χ with canonical kinetic term
s
2
Ω2 + 6ξ 2 h2 /MPl
dχ
=
dh
Ω4
I
The Einstein frame action is then
Z
2
p
MPl
1
λ[h(χ)]4
4
2
SE = d x −g̃ −
R̃ + (∂µ χ) − U(χ) , U(χ) =
2
2
4Ω4
√
U(χ) flattens for h(χ) & MPl / ξ ←− Inflationary region
I
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Model predictions
I
Can perform slow-roll calculations
with U(χ)
2 0 2
MPl
4M 4
U
=
' 2 Pl4
2
U
3ξ h
4
00
4MPl
ξh2
2 U
η = MPl
' 2 4 1− 2
U
3ξ h
MPl
I
√
Inflation ends at hend ' 1.07MPl / ξ
when ' 1
I
√
Working backwards, N = 59 e-folds produced at h∗ ' 9.14MPl / ξ
I
The spectral index and tensor-to-scalar ratio for Higgs ξ-inflation are
ns (χ∗ ) ' 0.967,
(Bezrukov & Shaposhnikov, 2008)
r (χ∗ ) ' 0.0031
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Model predictions
I
Can perform slow-roll calculations
with U(χ)
2 0 2
MPl
4M 4
U
=
' 2 Pl4
2
U
3ξ h
4
00
4MPl
ξh2
2 U
η = MPl
' 2 4 1− 2
U
3ξ h
MPl
I
√
Inflation ends at hend ' 1.07MPl / ξ
when ' 1
I
√
Working backwards, N = 59 e-folds produced at h∗ ' 9.14MPl / ξ
I
The spectral index and tensor-to-scalar ratio for Higgs ξ-inflation are
ns (χ∗ ) ' 0.967,
(Planck collaboration, 2013)
r (χ∗ ) ' 0.0031
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Perturbative unitarity breakdown
I
Generating the observed primordial density fluctuations requires
√
U(χ∗ )
4
' 5.6 × 10−7 MPl
=⇒ ξ ' 48000 λ ' 17000
(χ∗ )
I
Such a large value ξ ∼ 104 creates a problem:
√
(1) Perturbative unitarity breaks down at MPl /ξ MPl / ξ
p
2 + ξh2
ξ MPl
ξ 2
2
ξh R −→ 2
ĥ2 ĝ '
ĥ ĝ
for h ' 0
2
2
2
MPl
MPl + ξh + 6ξ h
(2) New physics entering at Λ = MPl /ξ is naively expected to affect the
potential in an uncontrollable way
(3) Self-consistency of Higgs ξ-inflation is questionable
I
Proponents argue that the scale of perturbative unitarity breakdown
depends on h (it is larger during inflation) and so does not spoil the
inflationary predictions
,→ Assumes scale of new physics is background field-dependent
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Perturbative unitarity breakdown
I
For Mh ' 125–126 GeV, λ can run
to very small values near the Planck
scale (where inflation occurs)
I
How does this affect the predictions
for Higgs ξ-inflation? We expect
√
ξ ' 48000 λ 17000
Can this address the perturbative
unitarity breakdown problem?
I
Actually, Mt must be about 2–3σ
below its central value for Higgs
ξ-inflation to be possible
(Degrassi et al., 2012)
. Glass half empty: Model disfavoured at 2–3σ
. Glass half full: Special region within 2–3σ of measurements
I
Need to go beyond the tree level
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Part III
Radiative corrections
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Renormalization group equations
I
The most important higher-loop effect is the running of λ
I
Need the RG equations for Higgs ξ-inflation
2
p
∂LE
dχ
Jordan
πh =
g̃ 0ν ∂˜ν h −−−−→
= −g̃
dh
∂ ḣ
Ω
2√
−g
dχ
dh
Canonical commutation relation gives
[h, πh ] = i~δ 3 (~x − ~y )
where s(h) =
I
2
1+ξh2 /MPl
2
1+(1+6ξ)ξh2 /MPl
=⇒
'
The physical Higgs propagators
are suppressed during inflation,
but not those of the NambuGoldstone bosons
1
1+6ξ
[h, ḣ] = s(h)i~δ 3 (~x − ~y )
for large h
s
t
h
t̄
h
2
ḣ
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Renormalization group equations
I
Two slightly different ways of dealing with the suppressed Higgs
propagators
(1) Insert a factor s for each off-shell Higgs in the RG equations of the
SM (De Simone, Hertzberg & Wilczek, 2009)
(2) View the effect as a suppression of the Higgs coupling to SM fields.
For ξ 1, use the RG equations from the chiral electroweak theory
(Bezrukov & Shaposhnikov, 2009)
I
Have tried to reconcile the differences, but have been unable to
reproduce the results from method (2) using Feynman diagrams
I
For this analysis, use the two-loop RG equations derived using
method (1) with leading three-loop corrections to λ, yt and γ
I
Note the running of ξ is given by
1 βm2
,
βξ = ξ +
6 m2
ξ(MPl /ξ0 ) = ξ0
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Effective potential
I
The SM effective potential must also be modified by the suppressed
Higgs propagators ←− result seems to be frame-dependent
Renormalization prescription II (Jordan frame)
I
Perturb the tree-level SM potential about the background value, compute
the masses of the perturbations, then transform to the Einstein frame
Mh2 = 3sλh2 ,
MG2 = λh2 ,
2
MW
=
g 2 h2
,
4
MZ2 =
(g 2 + g 02 )h2
, ...
4
The higher-loop corrections take the usual Coleman-Weinberg form with
these modified particle masses
4 Mh
λh4
1
Mh2
3
3MG4
MG2
3
+
ln
−
+
ln
−
+
.
.
.
U(χ) =
4Ω4
16π 2 4Ω4
µ2
2
4Ω4
µ2
2
I
Choose the renormalization scale µ = h to minimize the log terms
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Effective potential
I
The Einstein frame prescription is similar but we perform the
conformal transformation before computing the particle masses
Renormalization prescription I (Einstein frame)
I
Transform the tree-level SM potential to the Einstein frame, perturb it
about the background value, and compute the masses of the perturbations


ξh2
2
4
2
1
−
2
λh
3sλh 
g 2 h2
MPl
2
 , MG2 = λh , MW
U0 =
=⇒ Mh2 =
=
,...
2
4
4
4
ξh
4Ω
Ω
Ω
4Ω2
1+ 2
MPl
4
1
Mh2
3
3MG4
MG2
3
λh4
Mh
U(χ) =
+
ln 2 −
+
ln 2 −
+ ...
4Ω4
16π 2 4
µ
2
4
µ
2
I
There is an additional suppression of Mh2 and MG2 in this prescription
I
Choose the renormalization scale µ = h/Ω to minimize the log terms
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Effective potential
I
The additional suppression of Mh2 and MG2 makes little numerical
difference to the effective potential since the contributions of these
masses are already small (λ 1)
I
The most important difference between the Einstein and Jordan
frame renormalization prescriptions is the functional dependence µ(h)
(
√ h 2 2 Prescription I (Einstein frame)
1+ξh /MPl
µ=
h
Prescription II (Jordan frame)
Can have a large impact on the effective potential during inflation!
I
For this analysis, consider both prescriptions and use the two-loop
effective potential with the appropriately modified particle masses
I
Moreover, define the effective Higgs self-coupling λeff (µ) through
U(χ) ≡
λeff (µ)h4
4Ω4
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Two-loop inflationary predictions
Use the RG equations to run the initial values of Mh , Mt , αs , . . .
up to the inflationary scale and use the two-loop effective potential
I
To explore λeff (µ) 1, replace Mt with λmin
eff ≡ min {λeff (µ)}
,→ fine-tuning
0.02
For a fixed Mh , λmin
and
α
,
M in GeV
s
eff
I
Minimum effective Higgs quartic coupling Λmin
eff
I
h
0.01
(1) Choose ξ0 . Adjust Mt to
give the desired λmin
eff
0.00
(2) Determine U/ at N = 59
e-folds before the end of
Α HM L=0.1184
-0.01
Ξ =1000
inflation
(3) Repeat the steps above
-0.02
170
171
172
173
until the correct U/
Top quark mass M in GeV
normalization is obtained
(4) Compute the predictions for ns and r
s
Z
0
t
I
124
125
126
127
The numerical results are presented as a function of λmin
eff
174
175
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
I
I
Mh in GeV
s
-5
-4.5
-4
-3.5
Z
-3
-2.5
-2
min
eff
0.971
0.970
0.969
0.968
0.967
Mh in GeV
124
125
126
127
Αs HMZ L=0.1184
0.966
10-5 10-4.5 10-4 10-3.5 10-3 10-2.5 10-2
Minimum effective Higgs quartic coupling Λmin
eff
Tensor-to-scalar ratio r
I
5000
Conclusions
124
Non-minimal coupling can be as
2000
125
−4.4
126
small as ξ ∼ 400 for λmin
∼
10
eff
1000
127
,→ still too large to address the
500
Α HM L=0.1184
perturbative unitarity problem
200
min
−4.4
10
10
10
10
10
10
10
Need larger ξ for λeff < 10
Minimum effective Higgs quartic coupling Λ
Eventually the effective potential
develops a second minimum that spoils Higgs ξ-inflation
Predictions for ns and r remain within the 1σ region
Spectral index ns
I
Non-minimal coupling Ξ0
Results for prescription I
Radiative corrections
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
Mh in GeV
124
125
126
127
Αs HMZ L=0.1184
0.0000
10-5 10-4.5 10-4 10-3.5 10-3 10-2.5 10-2
Minimum effective Higgs quartic coupling Λmin
eff
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
I
I
Non-minimal coupling Ξ0
Results for prescription II
Radiative corrections
5500
5000
4500
4000
3500
Mh in GeV
Two regions for prescription II:
124
125
large and small λmin
regions
126
eff
127
3000
min
The large λeff region behaves
2500
similarly to prescription I but the
Α HM L=0.1184
2000
10
10
10
10
running of λeff (µ) more important
Minimum effective Higgs quartic coupling Λ
Can only have ξ as small as about
ξ ∼ 2000–4000 before the potential develops a second minimum
Predictions for ns and r show more variation, still within 1σ
s
-3.5
0.98
Tensor-to-scalar ratio r
I
-3
Z
-2.5
-2
min
eff
Spectral index ns
I
Conclusions
0.97
0.96
Mh in GeV
124
125
126
127
0.95
0.94
0.93
0.92
10-3.5
Αs HMZ L=0.1184
10
-3
-2.5
10
Minimum effective Higgs quartic coupling
-2
10
Λmin
eff
0.004
0.003
0.002
Mh in GeV
124
125
126
127
Αs HMZ L=0.1184
0.001
0.000
10-3.5
-3
10
10-2.5
10-2
Minimum effective Higgs quartic coupling Λmin
eff
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Results for prescription II
I
The small λmin
eff region is very different from the other results
0.35
0.35
Tensor-to-scalar
ratio
Tensor-to-scalar
ratio
r r
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
H10-4.10 , 58L
H10-4.15 , 75L
H10-4.10 , 58L
-4.10
H10-4.15H10
, 75L
, 76L
H10-4.10 , 76L
H10-4.05 , 78L
H10-4.05 , 78L -4.0
H10 , 80L
H10-4.0 , 80L
H10-3.95 , 84L
Planck+WP+BAO: LCDM+r
H10-3.95 , 84L
Planck+WP+BAO: LCDM+r
Planck+WP+BAO: LCDM+r+Neff
Planck+WP+BAO: LCDM+r+Neff
Planck+WP+BAO: LCDM+r+YHe
Planck+WP+BAO: LCDM+r+YHe
H10-4.05 , 63L
H10-4.05 , 63L
H10-3.9 , 90L
H10-3.9 , 90L
Mh in GeV
Mh in GeV
124
124
124.5
124.5
H10-3.875 , 96L
H10-3.875 , 96L
0.05
0.05
0.00
0.00
I
0.94
0.94
0.96
0.96
0.98
0.98
Spectral index ns
Spectral index ns
1.00
1.00
1.02
1.02
Allows ξ ∼ 90 at 2–3σ with an observable level of r , but there is still
a perturbative unitarity problem
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Part IV
Conclusions
Radiative corrections
Conclusions
Inflation from a particle physics perspective
Higgs ξ-inflation: Tree level
Radiative corrections
Conclusions
Conclusions
I
Higgs ξ-inflation is one of the few remaining inflationary models that
does not require scalar fields in addition to those in the SM
I
The breakdown of perturbative unitarity at MPl /ξ (below the scale
of inflation) has long been a potential problem for this model
I
We have investigated whether the recently measured Higgs mass, for
which λeff (µ) 1 near the Planck scale, can address this problem
µ
Prescription I
Prescription II
I
h
2
1+ξh2 /MPl
√
h
λmin
eff
ξ
ns
r
−4.6
& 400
0.97
. 0.003
−3.3
& 2000
∼ 90
0.96–0.97
0.97–1.00
. 0.003
0.15–0.25
& 10
& 10
∼ 10−3.9
The perturbative unitarity problem remains but small λmin
eff allows a
new region of Higgs ξ-inflation with observable tensor-to-scalar ratio
Thank you for your attention!