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UV completion with R2-term
Some like it hot: R2 term heals Higgs inflation,but does not cool it
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov,C. Shepherd
Institute for Nuclear Research, Moscow, Russia
May, 10
The talk is based on arxiv:1807.02392, arxiv:1904.04737
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Standard Model Higgs inflation
A minimal model for the inflationary stage — no new fields added
Action for the Higgs boson in a unitary gauge (H = h/√2)
S =
∫d4x√−g(−M2
p + ξh2
2R +
(∂µh)2
2− λ
4(h2 − v 2)2
)Einstein frame action
S =
∫d4x√−g(−M2
p
2R +
(∂µχ)2
2− λ
4
(h(χ))4
(1 + ξh(χ)2/M2p )2
)dχ
dh=
√1 + ξ(1 + 6ξ)h2/Mp2
1 + ξh2/M2p
h > Mp/√ξ, V (χ) '
λM4p
4ξ2
(1− e−2χ/(
√6Mp)
)2
In order to produce CMB normalization one needs λ/ξ2 ' 4× 10−10, ξ ∼ 104.
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Field-dependent cutoff scale
Until which scale is it a valid description?
In the small field domain (gµν = ηµν + hµν/Mp) the minimalsuppression scale of non-renormalizable operator
Lint =ξh2∂2hµµ
Mp→ Λ =
Mp
ξ
Λ/Mpeff
1
1/ξhMpMp/√ξMp/ξ
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Why UV completion is reqiured?
Connection between the inflationary parameters and lowenergy physics
Description of the preheating and reheating process
Y.Ema, R. Jinno et al., arXiv:1609.05209M. DeCross, D. Kaiser, A. Prabhu, arXiv:1610.08916
Treh ∼ 10−3Mp &Mp
ξ
Reheating can not be described in the Higgs inflation model!
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
UV completion with R2-term
The Starobinsky inflation is safe from the low cutoff scale problem:Λ ∼ Mp.
S0 =
∫d4x√−g
(−M2
p + ξh2
2R +
β
4R2 +
(∂µh)2
2− λ
4(h2 − v2)2
)How does it work?
Introduce a Lagrange multiplier L and an auxiliary scalar R,
S =
∫d4x√−g
(Lh −
M2p + ξh2
2R+
β
4R2 − LR+ LR
)Integrate out the field R: the problematic ξ appears only in thepotential
S =
∫d4x√−g
(Lh + LR − 1
4β(L +
1
2ξh2 +
1
2M2
p )2
)L is a dynamical field connected to the scalar graviton (scalaron)
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Einstein frame action
Hereafter we use Mp = 1/√
6
S =
∫d4x√−g
(− R
12+
1
2e−2φ(∂h)2 +
1
2(∂φ)2−
−1
4e−4φ
(λh4 +
1
36β(e2φ − 1− 6ξh2)2
))Y. Ema, arXiv:1701.07665
Bounds on β:
No strong coupling → ξ2/β . 1 < 4π
CMB normalization can be satisfied if β + ξ2/λ = 2× 109
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Bounds on the scalaron mass
1500 2000 2500 3000 3500 4000 4500
1.×10- 5
5.×10- 5
1.×10- 4
5.×10- 4
0.001
0.005
strong coupling
λ=0.005 λ=0.01
m/M
P
ξ
allo
wed r
ange
allo
wed r
ange
scalaron dominated inflation
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
R2 term: a proper variables
Which field interacts with gauge bosons and fermions?
Lkin =1
2e−2φ(∂h)2 +
1
2(∂φ)2
It is a mixture of φ and h.New variables: h = eΦ tanhH, φ = eΦ/ coshH → Higgs iscanonical
Lkin =1
2cosh2 H(∂χ)2 +
1
2(∂H)2
V =1
4
(λ sinh4 H +
1
36β(1− e−2Φ cosh2 H − 6ξ sinh2 H)2
)Lgauge =
g2h2
4e−2φW+
µ W−µ =
g2
4sinh2 H W+
µ W−µ
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Unitarity breaking scale
Standard model:
mW =1
2gh
R2- Higgs:
mW =g
2sinhH
The growing part of amplitude
A ∼ g2p2
m2W
(4
g2
(dmW (H)
dH
)2
− 1
)∼ p2
6M2p
ΛU =√
6Mp
Higgs inflation:
ΛU =Mp
ξ, H ∼ v ΛU =
Mp√ξ, H ∼ Mp
F. Bezrukov, A. Magnin, M. Shaposhnikov and S. Sibiryakov, arxiv:1008.5157
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Potential: who drives inflation?
V =1
4
(λ sinh4 H +
1
36β(1− e−2χ cosh2 H − 6ξ sinh2 H)2
)
H
Φ
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Tragectory
- 0.015 - 0.010 - 0.005 0.005 0.010 0.015
0.5
1.0
1.5
2.0
Ne=60
- 0.005 0.000 0.005
0.02
0.04
0.06
0.08
0.10
0.12
0.14
H H
ΦΦ
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Reheating: background dynamics
Oscillation timescales:
Scalaron: m2 = 1/(18β)
Oscillations around the minimum: M2min = λΦ/ξ
Fast oscillations in the valley: M2 = 2Φ/(3β)
6.2×106 6.4×106 6.6×106 6.8×106 7.0×106 7.2×106 7.4×106
0.02
0.04
0.06
0.08
6.2×106 6.4×106 6.6×106 6.8×106 7.0×106 7.2×106 7.4×106
- 0.001
0.001
0.002
t
t
HΦ̇
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Which process is responsible for reheating?
Production of weak gauge bosons (longitudinal components)?
6.2×106 6.4×106 6.6×106 6.8×106 7.0×106 7.2×106 7.4×106
5.×10- 7
1.×10- 6
mW2
t
?
?
Minxi He, Ryusuke Jinno, Alexei A. Starobinsky et al., arXiv:1812.10099
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Chaotic behaviour of the background: resonant points ofthe scalaron mass
5.0×10 1.0×10 1.5×10 2.0×10-1.5×10
-1.0×10
-5.0×10
0
min
m2 H/M
2 P
5.0×10 1.0×10 1.5×10 2.0×100.00
0.25
0.50
0.75
1.00
maxH2/6
2M
2 P
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Reheating: two field dynamics
Oscillations after inflation
-0.002 -0.001 0.000 0.001 0.002H/MP
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
/MP
-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003H/MP
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
/MP
-0.004 -0.002 0.000 0.002 0.004H/MP
-0.02
0.00
0.02
0.04
/MP
0.950 0.955 0.960 0.965 0.970-0.010.000.010.020.03
0.950 0.955 0.960 0.965 0.970
tMP/106
-0.002-0.0010.0000.0010.002
H
0.9450.9500.9550.9600.9650.9700.975
0.00
0.02
0.04
0.06
0.9450.9500.9550.9600.9650.9700.975
tMP/106
-0.002-0.0010.0000.0010.002
H
0.94 0.96 0.98 1.00 1.02-0.0250.0000.0250.0500.0750.1000.125
0.94 0.96 0.98 1.00 1.02
tMP/106
-0.004
-0.002
0.000
0.002
0.004
H
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Reheating: tachionic modes
The inflaton background can produce:
’Higgs’ modes, m2H = VHH
weak gauge bosons, m2 = VH/ tanhH − 4V
0.954 0.957 0.960 0.963 0.966
tMP/106
0
2.5×10
5.0×10
7.5×10
[M2 P]
m2H
m2W(k/a mT)
m2L
0.955 0.960 0.965 0.970
tMP/106
0
2.5×10
5.0×10
7.5×10
1.0×10
[M2 P]
m2H
m2W(k/a mT)
m2L
0.95 0.96 0.97 0.98 0.99
tMP/106
0
1×10
2×10
3×10
[M2 P]
m2H
m2W(k/a mT)
m2L
Decay of the modes:
ΓW = 0.8αW 〈mW 〉 ∼ m
ΓH = 0.1y2b 〈mH〉 � m
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Energy balance
0.95 1.00 1.05 1.10
tMP/106
0
2.5×10 ¹²
5.0×10 ¹²
7.5×10 ¹²
1.0×10 ¹¹
[M4 P]
H20/2
20/2
3M2P
2
0.95 1.00 1.05 1.10 1.15 1.20
tMP/106
0
2.0×10 ¹²
4.0×10 ¹²
6.0×10 ¹²
8.0×10 ¹²
1.0×10 ¹¹
1.2×10 ¹¹
[M4 P]
H20/2
20/2
3M2P
2
0.9 1.0 1.1 1.2 1.3
tMP/106
0
5.0×10 ¹²
1.0×10 ¹¹
1.5×10 ¹¹
2.0×10 ¹¹
[M4 P]
H20/2
20/2
3M2P
2
0.954 0.957 0.960 0.963 0.966
tMP/106
0
2.5×10
5.0×10
7.5×10
[M2 P]
m2H
m2W(k/a mT)
m2L
0.955 0.960 0.965 0.970
tMP/106
0
2.5×10
5.0×10
7.5×10
1.0×10
[M2 P]
m2H
m2W(k/a mT)
m2L
0.95 0.96 0.97 0.98 0.99
tMP/106
0
1×10
2×10
3×10
[M2 P]
m2H
m2W(k/a mT)
m2L
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Reheating: production of radial Higgs modes andlongitudinal gauge bosons
0.955 0.960 0.965 0.970
tMP/106
10 12.5
10 10.0
10 7.5
10 5.0
10 2.5
100.0
[M4 P]
H
W
3M2P
2
0.955 0.960 0.965 0.970 0.975
tMP/106
10 14
10 13
10 12
10 11
10 10
[M4 P]
H
W
3M2P
2
0.96 0.97 0.98 0.99 1.00 1.01 1.02
tMP/106
10 15
10 14
10 13
10 12
10 11
10 10
[M4 P]
H
W
3M2P
2
0.0000 0.0025 0.0050 0.0075 0.0100
k/a(ts)/MP
10 30
10 20
10 10
100
[M2 P]
t=965033.0/M_P
nHk2/2 2a3
nwk2/2 2a3
0.0000 0.0025 0.0050 0.0075 0.0100
k/a(ts)/MP
10 24
10 22
10 20
10 18
10 16
[M2 P]
t=964007.0/M_P
nHk2/2 2a3
nwk2/2 2a3
0.0000 0.0025 0.0050 0.0075 0.0100
k/a(ts)/MP
10 28
10 26
10 24
10 22
10 20
[M2 P]
t=979047.0/M_P
nHk2/2 2a3
nwk2/2 2a3
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Results for the reheating temperature
What if we are not in the ’resonant’ point?
Back reaction has to be included.
However, after small number of crossings we anyway fall intothe region where a lot of energy is stored in the Higgs field.
6.5×106 7.0×106 7.5×106 8.0×106 8.5×106
5.×10- 14
1.×10- 13
1.5×10- 13
2.×10- 13
2.5×10- 13
7.0×106 8.0×106 9.0×106 1.0×107 1.1×107 1.2×107
2.×10- 14
4.×10- 14
6.×10- 14
8.×10- 14
1.×10- 13
1.2×10- 13
1.4×10- 13
t t
3 H2 MP2
H2/2.
3 H2 MP2
H2/2.
Instant reheating, Treh ' 1015 GeV Predictions:
Ne = 59 ns = 0.97 r = 0.0034
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Conclusions
Higgs inflation can be UV completed up to the Planck scale
The completion can be achieved by means of introducing onlyone extra term in the lagrangian (R2-term)
At the beginning, inflation is driven by the R2 degree offreedom, then the trajectory turns to a Higgs direction
R2 term can also improve the stability of Higgs potential for acertain range of top quark masses
In this model, the reheating is instant. It lasts much less thanthe Hubble time.
For several special values of the scalaron mass the reheatingcomes after the first zero crossing of the scalaron. For otherpoints, the back reaction is important.
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
UV completion with R2-term
Thanks for your attention!
A.A. Tokareva in collaboration with F. Bezrukov, D. Gorbunov, C. ShepherdSome like it hot: R2 term heals Higgs inflation, but does not cool it
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