Time dependence and the effective potential in
Higgs inflation
Damien P. George
DAMTP & The Cavendish,
The Herchel Smith Fund,
Cambridge, UK
University of Sussex — 25th February 2013
Overview
Higgs inflation paradigm.
Quantum corrections and the calculation of the effective potential:
expanding the action,
split 2-point interactions,
in-in formalism,
computing the tadpole diagrams,
putting it all together.
Work in progress:
full SM,
running of the RGEs.
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Higgs inflation paradigm
Inflation:
solves flatness and horizon
(and monopole) problems,
provides seeds for structure
formation,
is driven by an inflaton field
rolling slowly down its
potential.
The standard model Higgs field can be play the role of the inflaton
Need a single additional term: ξ|H|2 R.
Bezrukov & Shaposhnikov PLB 659 703 (2008).
Such a term is expected to be generated, current bounds: ξ . 2.6 × 1015
Atkins & Calmet PRL 110 051301 (2013).
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Higgs inflation — main results
Z
S=
2
√
d4 x −g . . . + ξH † HR + . . . − λ H † H − v 2 /2 + . . . .
Go to Einstein frame and canonical kinetic term. In large field regime
expand the potential in δ ≡ 1/(ξφ2 ) 1. Slow roll parameters:
= 4δ 2 /3,
η = −4(δ − δ 2 )/3,
Inflation ends for ≈ 1, giving number of e-folds N ≈
With N ≈ 60, we find δ∗ ∼ 6/8N ≈ 1/80.
6
8δ∗ .
Cobe normalisation gives
(V /)∗ = (0.027)4
⇒
λ/ξ 2 = 4 × 10−10
⇒
√
ξ = 5 × 104 λ
Spectral index and tensor-to-scalar ratio are compatible with data:
8
16
ns = 1+ 2η −6 ≈ 1− δ∗ − δ∗2 ≈ 0.967,
3
3
r = 16 = 8δ∗2 = 0.0033.
Running of λ and stability of Higgs vacuum with mh = 126GeV??
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Quantum corrections
To test BSM physics using cosmological data must have a precise
understanding of the scalar field’s dynamics.
→ need to go beyond classical and include dominant quantum effects
One-loop effective action for a scalar field in an expanding universe.
Birrell & Davies, 1982; Candelas & Raine, 1975; Ringwald, 1987a, 1987b.
Describes the backreaction of the quantum fluctuations of the scalar field
on the time-dependent background field, calculated systematically in a
loop expansion.
Here: extend these results by including a coupling to a gauge field.
The toy model is Abelian Higgs with a U(1) gauge field in FLRW
background. Calculation found in DG, Mooij & Postma JCAP 11 043 (2012).
Generalises the Coleman-Weinberg potential to time-dependent
background fields in a curved space-time; Coleman & Weinberg 1973.
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The toy model
A U(1) gauge field Aµ and charge scalar Φ in a curved background.
ds2 = a2 (τ ) dτ 2 − d~x2 ,
1
Φ(xµ ) = √ φ(τ ) + h + iθ .
2
The action in Rξ gauge:
Z
Stot =
√
d4 x −g(L + LGF + LFP ),
with
1
L = − g µα g νβ Fµν Fαβ + g µν Dµ Φ(Dν Φ)† − V (Φ),
4
1
LGF = − G2 ,
G = g µν ∇µ Aν − ξg(φ + h)θ,
2ξ
δG
LFP = η̄g
η.
(α is a U(1) gauge transformation)
δα
Quantum fields h, θ, Aµ , η in a time-dependent background a(τ ), φ(τ ).
Background metric is fixed, backreaction assumed negligible.
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Expanding the action
Expand the action up to 3rd order in the quantum fields.
The one-point vertex is
Z
S (1) = d4 x −λ̂h ĥ ,
where
i
h
λ̂h = ∂τ2 − (H0 + H2 ) φ̂ + V̂φ̂ = a3 φ̈ + 3H φ̇ + Vφ .
Working in conformal frame with H = a0 /a, φ̂ = aφ, V̂ = a4 V (φ̂), etc.
Note A does not need a hat.
Easier to work in conformal time, as the resulting action has a form
similar to the Minkowski action, and all the machinery developed for this
can be used.
Mooij & Postma 2011; Heitmann, 1996.
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Quadratic expansion
S
(2)
Z
=
(
4
d x
1
1
2
2
µ ν
µν
− Aµ −(∂ + m̂(µ) )η + 1 −
∂ ∂ Aν
2
ξ
− A0 (m̂2 )i0 Ai − m̂2Aθ A0 θ̂
)
1
1
− ĥ(∂ 2 + m̂2h )ĥ − θ̂(∂ 2 + m̂2θ )θ̂ − η̄ˆ(∂ 2 + m̂2η )η̂ .
2
2
2-point interactions are:
m̂2(µ) = g 2 φ̂2 +
2
H0 − 2H2 δµ0 ,
ξ
m̂2h = V̂hh − (H0 + H2 ),
m̂2θ = V̂θθ + ξg 2 φ̂2 − (H0 + H2 ),
m̂2η = ξg 2 φ̂2 − (H0 + H2 ).
Off-diagonal 2-point terms are:
m̂2Aθ = 2g(∂τ − H)φ̂,
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2
(m̂2 )i0 = H∂ i .
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Three point vertices
The 3-point interaction vertices are
Z
h 1
1
1
(3)
S = d4 x − λ̂hhh ĥ3 − λ̂hθθ ĥθ̂2 − λ̂hAA ĥÂ2
2
2
2
i
− λ̂hηη ĥη̄ˆη̂ − ĥλ̂hAθ A0 θ̂ ,
where
λ̂hhh = ∂φ̂ m̂2h = V̂φhh ,
λ̂hθθ = ∂φ̂ m̂2θ = V̂φθθ + 2ξg 2 φ̂,
λ̂hAA = ∂φ̂ m̂2A = −2g 2 φ̂,
λ̂hηη = ∂φ̂ m̂2η = 2ξg 2 φ̂,
λ̂hAθ = 2g(−∂τ − H).
Expressed in terms of 2-point interactions; formally, ∂φ̂ H = 0.
Integration by parts used to get λhAθ acting on A0 θ̂. Also discarded a ∂i term.
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Time dependence of interactions
The background is time-dependent: a(τ ), φ(τ ).
→ the “masses” (really 2-point interactions) are also time-dependent.
Split these interaction into time-independent and time-dependent parts:
ˆ 2 + δ m̂2 (τ ),
m̂2 (τ ) = m̄
with the split defined by
δ m̂2 (0) = 0.
ˆ 2 contributes to the free Lagrangian. We call it the mass and it
m̄
determines the propagator.
δ m̂2 (τ ) is treated as a proper 2-point interaction in Feynman
diagrams.
The loop expansion is independent of the split of the 2-point terms into a free and
interacting part.
Also demand off-diagonal 2-point interactions vanish at τ = 0.
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In-in formalism
Since we are interested in expectation values of the background field and
their evolution with time (not scattering amplitudes) we use the in-in
formalism (closed-time-path (CTP) or Schwinger-Keldysh formalism).
Schwinger, 1961; Keldysh, 1964; Jordan, 1986; Bakshi & Mahanthappa 1963a, 1963b;
Calzetta & Hu, 1987; Weinberg, 2005.
Expectation values are computed using an action
−
S = S[φ+
i ] − S[φi ]
−
with boundary condition φ+
i (t) = φi (t).
All fields, propagators and vertices are labelled by ± superscripts.
The propagator D±± (x − x0 ) connects vertices λ± (x) and λ± (x0 ).
The action of the minus-fields is defined with an overall minus sign,
so
2 −
+
mαβ = − m2αβ ,
[λhαβ ]− = − [λhαβ ]+ .
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In-in formalism — propagators
We construct the propagators from the free, time-independent part of
the quadratic action. For example, for h the propagators are defined as
the solutions of
ˆ 2h
∂x2 + m̄
0
−(∂x2
0
ˆ 2h )
+ m̄
Dh++ (x − y)
Dh−+ (x − y)
Dh+− (x − y)
Dh−− (x − y)
= −iδ(x − y)I2 .
This defines D++ as the usual Feynman propagator, D−− as the
anti-Feynman propagator, and D−+ and D+− as Wightman functions.
Dh−+ (xa − xb ) = Dh+− (xb − xa ) = Dh,ab ,
Dh++ (xa − xb ) = Dh,ab Θab + Dh,ba Θba ,
Dh−− (xa − xb ) = Dh,ab Θba + Dh,ba Θab ,
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The one-loop equation of motion
Vanishing of the tadpole → equation of motion for φ(t).
Include 1-loop tadpoles → quantum corrected equation of motion.
hh+ (τ, ~x)i = 0.
Vanishing of the h−
component gives the
same result.
From first principles, expand out the path integral:
0 = hh+ (x)i
Z
+
+
−
−
= Dψα+ Dψβ− h+ (x)ei(S0 [ψα ]+Sint [ψα ])−i(S0 [ψβ ]+Sint [ψβ ])
Z
Z
+
]−iS0 [ψβ− ]
iS0 [ψα
− +
+
−
+
4
= Dψα Dψβ h (x)e
1 + i d y(Lint −Lint ) + . . .
Z
= −i d4 y Dh++ (x − y) − Dh+− (x − y) A(y).
Equation of motion: A(y) = 0.
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The one-loop equation of motion
For divergent terms, need to go to 3rd order in Lint .
0 = Acl + A1 + A2 + A3 + finite
++
= λ̂h (x) + Sαβ λ̂hαβ (x)Dαβ
(0)
Z
± ±+ 0
+±
− iSαβγδ λ̂hαβ (x) d4 x0 Dαγ
(x − x0 ) δ m̂2γδ (x0 ) Dδβ
(x − x)
Z
Z
n
± ±± 0
+±
− Sαβγδρσ λ̂hαβ (x) d4 x0 d4 x00 Dαγ
(x − x0 ) δ m̂2γδ (x0 ) Dδρ
(x − x00 )
o
± ±+ 00
× δ m̂2ρσ (x00 ) Dσβ
(x − x)
+ finite.
Sum over: field-type, Lorentz indices, possibilities for ±.
Sαβ... are appropriate symmetry factors.
ˆ 2α in propagators Dα .
Masses m̄
2
δ m̂αβ (τ ) and λ̂hαβ (τ ): 2- and 3-point interaction vertices.
Concerned only with the UV divergent contributions → consider
only up to three vertices.
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Tadpole diagrams
In terms of diagrams: all 1PI tadpole graphs with one external h+ leg.
Acl + A1 + A2 + A3 =
λ̂h
h+
+
h+
±+
Dαρ
+
+ h
λ̂hαβ
++
Dαβ
±+
Dαρ
δ m̂2ρσ
δ m̂2ρσ
±+
Dσβ
λ̂hαβ
+
h+
λ̂hαβ
±±
Dσκ
±+
Dβτ
δ m̂2κτ
Computing these diagrams gives the quantum corrected equation of
motion at the 1-loop level.
Classical part: Acl = λ̂h (x).
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First order
First order diagrams as in Minkowski
[Mooij & Postma, JCAP 1109, 006 (2011)]
Four diagrams contribute:
h, θ, η, Aµ in the loop.
h+
λ̂hαβ
.
++
Dαβ
Each diagram has the same structure:
1
A1,α = ∂φ m̂2α Dα++ (0)
2
Z Λ̂
ˆ 2α
1
1 1 m̄
2 1
2
= ∂φ m̂α 2
k dk
−
+ ...
2
4π 0
k 2 k3
1 2
1
2
2
2
ˆ ) + finite .
ˆ ln(Λ̂/m̄
∂φ m̂α Λ̂ − m̄
=
16π 2
2 α
The variable k is the comoving momentum with k < Λ̂ a comoving cutoff.
++
++
Gauge loop is expressed as scalar propagators −η µν Dµν
(0) = 3DA
(0) + ξDξ++ (0).
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First order diagrams
For the log-divergent part, sum of all first-order diagrams is
Dh++
"
P
A1,α =
λ̂hhh
ˆ 2h
∂φ m̂2h m̄
Dη++
Dθ++
+
λ̂hθθ
ˆ 2θ
∂φ m̂2θ m̄
+
λ̂hηη
ˆ 2η
−2∂φ m̂2η m̄
++
Dµν
+
λ̂hAA
#
×
−1
32π 2
log Λ2
ˆ 2A (3 + ξ 2 )
∂φ m̂2A m̄
Note that ∂φ m̂2α is time-dependent and evaluated at τ
For example, ∂φ m̂2A = −2g 2 φ̂(τ ).
Thus, A1 is a function of τ .
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Second order
±+
Dαρ
h+
λ̂hαβ
δ m̂2ρσ
±+
Dσβ
At second order in Lint the loop diagrams with one 2-point insertion
contribute. We split them into three parts:
AMink
contains all scalar loops, and the gauge boson loop where
2
only the diagonal part of m̂2(µ) is inserted. Also a mixed θA0 -loop.
This part is analogous to the equivalent Minkowski calculation.
Amass
contains the gauge boson loop with a δ m̂20 .
2
Amix
contains the gauge boson loop with a (δ m̂2 )0i .
2
These last two diagrams are both absent in Minkowski.
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Calculating AMink
2
The scalar Higgs loop with one 2-point insertion gives
Z
i
Mink
2
4
2
A2,h = − ∂φ m̂h (τ ) d xb δ m̂h (τb ) Dh++ (xa −xb )Dh++ (xb −xa )
2
−
i
= − ∂φ m̂2h (τ )
2
Z
4
d
Dh+− (xa −xb )Dh−+ (xb −xa )
xb δ m̂2h (τb )Θab
2
Dh,ab
−
2
Dh,ba
.
Everything expressed in terms of Wightman functions. Fourier transform,
perform the d3 xb integral, integrate by parts to extract the UV divergent
piece:
Z
d3 k
Mink
2
2
A2,h = −∂φ m̂h (τ )δ m̂h (τ )
64π 3~k 3
1
ˆ )2 + finite.
= −∂φ m̂2h (τ )δ m̂2h (τ )
ln(Λ̂/m̄
32π 2
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Calculating AMink
2
The Goldstone boson loop AMink
and ghost loop AMink
are similar to
2,η
2,θ
Mink
A2,h . Ghost has overall factor (−2).
Gauge boson loop follows the same steps but with a non-trivial Lorentz
structure:
Z
i
Mink
2
4
2
µν ρσ
++
++
A2,A = − ∂φ m̂A d xb δ m̂A (τb )η η Dµρ
(xa − xb )Dσν
(xb − xa )
2
+−
−+
− Dµρ (xa − xb )Dσν (xb − xa )
Z 3
Z τ
d k CIJ sin [(ω̄I + ω̄J )(τ − τb )]
2
2
= −∂φ m̂A
dτb δ m̂A (τb )
3
(2π)
(2ω̄I )(2ω̄J )
0
2
(3 + ξ )
ˆ )2 + finite.
ln(Λ̂/m̄
= −∂φ m̂2A δ m̂2A (τ )
32π 2
CIJ encodes the structure arising from Wightman functions in Fourier space.
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Calculating AMink
2
Finally, the mixed θA0 -loop gives
Z
h
i
++
++
+−
−+
Mink
A2,Aθ = −iλ̂hθA (τ ) d4 xb δ m̂2Aθ (τb ) D00,ab
Dθ,ba
− D00,ab
Dθ,ba
Z τ
Z
d3 k CI sin[(ω̄I + ω̄θ )(τ − τb )]
= −2λ̂hθA (τ )
dτb δ m̂2Aθ (τb )
(2π)3
(2ω̄I )(2ω̄θ )
0
(3 + ξ)
ˆ )2 + finite.
= 2λ̂hθA (τ )δ m̂2Aθ (τ )
ln(Λ̂/m̄
128π 2
The 3-point vertex is λ̂hθA (τ ) = 2g(−∂τ − H(τ )).
Adding all the Minkowski pieces together gives
−1 X
(3 + ξ)
ˆ )2 +
ˆ )2 .
AMink
=
Sα ∂φ m̂2α δ m̂2α ln(Λ̂/m̄
λ̂hAθ δ m̂2Aθ ln(Λ̂/m̄
2
2
32π α
64π 2
Symmetry factors Sα = {1, 1, −2, 3, 1} for {h, θ, η, A, ξ}; m̂2ξ = ξ m̂2A .
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Calculating Amass
2
Next, the Lorentz violating mass m20 gives
Z
h
i
i
++
++
+−
−+
mass
2
A2
= − ∂φ m̂A (τ ) d4 xb δ m̂20 (τb )η µν Dµ0,ab
D0ν,ba
− Dµ0,ab
D0ν,ba
2
Z τ
Z
d3 k CIJ sin [(ω̄I + ω̄J )(τ − τb )]
= −∂φ m̂2A (τ )
dτb δ m̂20 (τb )
(2π)3
(2ω̄I )(2ω̄J )
0
2
(3 + ξ )
ˆ )2 + finite.
= −∂φ m̂2A (τ )δ m̂20 (τ )
ln(Λ̂/m̄
4 × 32π 2
This diagram is not present in Minkowski.
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Calculating Amix
2
The off-diagonal interaction (δ m̂2 )0i contains a spatial derivative and
brings down a factor of the momentum.
Z
++
++
+−
−+
2
4
2 0i
µν
Amix
=
i∂
m̂
(τ
)
d
x
(δ
m̂
)
(τ
)η
D
D
−
D
D
φ A
b
b
2
µ0,ab iν,ba
µ0,ab iν,ba
Z τ
Z 3
2
d k 2CIJ cos [(ω̄I + ω̄J )(τ − τb )]
= − ∂φ m̂2A (τ )
dτb H(τb )
ξ
(2π)3
(2ω̄I )(2ω̄J )
0
Z
3
0
2
2CIJ H (τ )
d k
+ finite
= − ∂φ m̂2A (τ )
3
ξ
(2π) (2ω̄I )(2ω̄J )(ω̄I + ω̄J )2
3H0 (τ )(1 − ξ)2
ˆ )2 + finite.
ln(Λ̂/m̄
= ∂φ m̂2A (τ )
64π 2 ξ
We have a cosine instead of a sine → integrate by parts twice to isolate
the leading term in the UV limit. Obtain a result proportional to H0 .
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Summary of second order tadpoles
Dhα+
Dηα+
Dθα+
α+
Dµν
"
P
+
A2,α =
Dhα+
+
Dηα+
Dθα+
∂φ m̂2h δ m̂2h
α+
Dρσ
−2∂φ m̂2η δ m̂2η
∂φ m̂2θ δ m̂2θ
α+
D0µ
+
α+
D0µ
∂φ m̂2A δ m̂2A (3 + ξ 2 )
Dθα+
#
+
+
α+
D0ν
2
∂φ m̂2A δ m̂20 3+ξ
4
×
+
α+
∂(xb )i Diν
0
2
−∂φ m̂2A 3H (ξ−1)
2ξ
−1
32π 2
log Λ2
α+
D00
−λ̂hAθ δ m̂2Aθ 3+ξ
2
These Feynman diagrams are in (conformal) coordinate space.
All time-dependent quantities (λ̂Aθ , m̂2α , δ m̂2α and H) are evaluated at τ .
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Third order
Not done yet!
All 3rd order diagrams with 2-point insertions are UV finite, except for
one.
Power counting:
d4 k in 4d,
1/k 2 per propagator,
k for a derivative from (m̂2 )0i = 2ξ H∂ i .
In 4d, 3 propagators and 2 derivatives → logarithmically divergent.
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The third order diagram
±+
Dαρ
z1 , ±
λ̂hAA
x, +
h
(m̂2 )0i
±±
Dσκ
y, +
z2 , ±
±+
Dβτ
(m̂2 )0j
Z
Z
1
2
4
ab bc
Dσκ Dτcaν .
A3 = ∂φ m̂A (τ ) d xb d4 xc (δ m̂2 )0i (τb )(δ m̂2 )0j (τc )η µν Dµρ
2
Sum over {ρ, σ, κ, τ } from {0, i, j}. Sum over {a, b, c} from ±.
End result is
A3 = ∂φ m̂2A (τ )H2 (τ )
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−6(1 + ξ)
ˆ )2 + finite.
ln(Λ̂/m̄
64π 2 ξ
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Adding it all up
We have computed all quadratically and logarithmically divergent
contributions to the one-loop equation of motion.
The first and second order combined ÂMink = ÂMink
+ ÂMink
is
1
2
Â
Mink
1 X
(3 + ξ)
1 2
2
2
2
ˆ
ˆ )2 .
=
λ̂hAθ m̂2Aθ ln(Λ̂/m̄
Sα ∂φ̂ m̂α Λ̂ − m̂α ln(Λ̂/m̄) +
16π 2 α
2
64π 2
This is independent of how the 2-point interaction is split, since the 1st
ˆ 2α + δ m̂2α .
and 2nd order pieces combine to give m̂2α = m̄
For A0 mass insertions we have the 2nd order piece
Âmass = −∂φ̂ m̂2A δ m̂20
3 + ξ2
ˆ )2 .
ln(Λ̂/m̄
128π 2
For the mixed piece we have contributions from 2nd and 3rd order
Âmix = ∂φ̂ m̂2A
D.P. George
6H2 (1 + ξ)
3H0 (1 − ξ)2
−
ξ
ξ
1
ˆ )2 .
ln(Λ̂/m̄
64π 2
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Effective action
Have the 1-loop equation of motion Â1-loop . Corresponding Lagrangian is
!0
δ L̂1-loop
δ L̂1-loop
1-loop
−
Â
=
.
δ φ̂0
δ φ̂
The action is then simply
1-loop
Γ
Z
=
d3 x dτ L̂1-loop .
All terms polynomial in φ̂ are easily integrated to find the Lagrangian.
Only one is not polynomial; in ÂMink there is:
λ̂hAθ m̂2Aθ = 4g 2 −φ̂00 + H0 φ̂ + H2 φ̂ .
This comes from a Lagrangian
1
− m̂4Aθ = −2g 2 φ̂02 − 2Hφ̂φ̂0 + H2 φ̂2 .
2
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Gauge invariance
Going to coordinate time and taking off the hats, the total 1-loop
effective action is
(
Z
√
−1
d3 xdt −g Vhh + Vθθ + 3m2A Λ2
Γ1-loop =
2
16π
2 2
− Vhh − Ḣ − 2H 2 + Vθθ − Ḣ − 2H 2 + 3m4A
+
2ξVθθ m2A
2 2
− (6 + 2ξ)g φ̇ +
6m2A
Ḣ + 2H
2
ln(Λ/m̄)2
)
.
4
Result is still gauge variant. Use the Nielsen identities
∂Veff ∂φ ∂Veff
+
= 0.
∂ξ
∂ξ ∂φ
Veff is only gauge invariant when the background field is in a minimum of
the potential, i.e. the background field satisfies its equation of motion.
Going on-shell enables us to rewrite the φ̇2 term and eliminate ξ.
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Time dependence and the effective potential in Higgs inflation
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Self-contained summary of the results
FLRW metric background ds2 = dt2 − a2 (t)d~x2 .
Abelian Higgs model with U(1) gauge symmetry
Z
Stot =
√
1
d4 x −g − g µα g νβ Fµν Fαβ + g µν Dµ Φ(Dν Φ)† − V (Φ) .
4
Expand the Higgs around a time-dependent background
1
Φ(xµ ) = √ φ(t) + h(t, ~x) + iθ(t, ~x) .
2
UV divergent contributions at one loop are
−1
Γ1−loop =
16π 2
Z
"
√
d3 xdt −g (Ṽhh + Ṽθθ + 3m2A )Λ2
−
2
Ṽhh
+
2
Ṽθθ
+
3m4A
−
6Ṽθθ m2A
ln(Λ/m̄)2
4
#
,
where the time-dependent “shifted scalar mass” is
Ṽαα ≡ Vαα − Ḣ − 2H 2 .
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Time dependence and the effective potential in Higgs inflation
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Additional scalars and fermions in the loops
If the Higgs field couples to additional scalars χα and/or fermion fields
ψβ , we get an additional contribution
"
Z
2
X
√
1
2
2 ln(Λ/m̄)
1−loop
3
Ṽαα Λ − Ṽαα
δΓ
=−
d xdt −g
16π 2
4
χα
#
2
X
2 2
4
2 ln(Λ/m̄)
.
−
mβ Λ − mβ − Ṽθθ mβ
4
ψβ
The sum is over all bosonic and fermion real degrees of freedom, where a
Weyl (Dirac) fermion counts as 2 (4) degrees of freedom.
The shifted scalar mass is as before Ṽαα ≡ Vαα − Ḣ − 2H 2 .
For fermions, assume a Yukawa interaction mψ ∝ φ.
D.P. George
Time dependence and the effective potential in Higgs inflation
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Comparison with literature
The results agree with the expressions in the literature in the appropriate
limits:
Minkowski case (H = Ḣ = 0, and thus Ṽαα = Vαα ) it matches
Mooij & Postma 2011.
In the de Sitter limit Ḣ = 0, and for a time-independent Higgs field
(Vθθ = 0 by Goldstone’s theorem), it agrees with Garbrecht 2007.
Taking both the Minkowski limit and a static background field we
retrieve the classic CW potential, Coleman & Weinberg 1973.
D.P. George
Time dependence and the effective potential in Higgs inflation
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Next steps
We only calculate the UV divergent terms, as these will generically give
the dominant contribution. Using a renormalisation prescription, these
terms (and wavefunction renormalisation) suffice to derive the RGEs and
find the RG improved action.
To apply our results to Higgs inflation we need to extend them:
1
Include back reaction from gravity.
2
Go from U(1) toy model to SM gauge group.
3
Consider non-minimal coupling to gravity.
4
To relate parameters to low energy observables need RG flow.
D.P. George
Time dependence and the effective potential in Higgs inflation
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Non-minimal coupling to gravity
Include non-minimal coupling to gravity, ξ|Φ|2 R.
Transform to the Einstein frame, then our 1-loop results can be applied.
For the SM, not so straight forward: Higgs H has 4 degrees of freedom,
kinetic term is non-minimal in Einstein frame:
1
6ξ 2
Le
1 δij
√
⊃ γij ∂φi ∂φj =
+ 4 φi φj ∂φi ∂φj .
−g
2
2 Ω2
Ω
Field-space metric γij cannot be diagonalised everywhere. Instead,
diagonalise it at each point in field-space, giving a spectrum of 4 scalars
with masses a function of φBG .
Can then apply our 1-loop results.
D.P. George
Time dependence and the effective potential in Higgs inflation
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Running of the RGEs
To connect low energy observables (at LHC) with high energy ones
(inflation and CMB) need to run the couplings from MZ to Minfl .
Jordan versus Einstein frame:
Jordan has gravity fluctuations which should be important (can we
ignore them?). RGEs:
3λ(ξ + 1/6)
3λm2
, βξ =
,
8π 2
8π 2
m2 (ξ + 1/6)
m4
βκ =
, βΛ =
2
8π
32π 2
βλ =
9λ2
,
8π 2
β m2 =
Einstein has non-minimal kinetic structure (diagonalise at each point
in field-space?) and non-renormalisable terms.
ξ will run, so is reintroduced in Einstein frame. Not such a problem.
D.P. George
Time dependence and the effective potential in Higgs inflation
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Conclusions
Higgs inflation is simple and promising, although slight tension with
mh = 126GeV.
To constrain BSM models using cosmological data need quantum
corrections to scalar potential, and running of the parameters.
We gave the effective potential with time-dependent FLRW background
and time-dependent Higgs vev, with a gauge field.
Work in progress: go to full SM with non-minimal coupling, and
determine RGEs.
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Time dependence and the effective potential in Higgs inflation
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Selected references
S. R. Coleman, E. J. Weinberg, “Radiative Corrections as the Origin of
Spontaneous Symmetry Breaking”, Phys. Rev. D7 (1973) 1888-1910.
Bezrukov & Shaposhnikov, “The Standard Model Higgs boson as the
inflaton”, PLB 659 703 (2008).
K. Heitmann, “ Thermalisierung bei kosmologischen Phasenübergängen:
Eichfelder”, Master’s Thesis, 1996.
K. Heitmann, “Non-equilibirum dynamics of symmetry breaking and gauge
fields in quantum field theory”, PhD Thesis, 2000.
S. Weinberg, “Quantum contributions to cosmological correlations”, Phys.
Rev. D72 (2005) 043514. [hep-th/0506236].
S. Mooij and M. Postma, JCAP 1109, 006 (2011) [arXiv:1104.4897].
D. P. George, S. Mooij and M. Postma, “Effective action for the Abelian
Higgs model in FLRW”, JCAP 1211 (2012) 043 [arXiv:1207.6963].
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