effective actions in particle physics and...
TRANSCRIPT
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Thomas Konstandin
in collaboration with
J. Elias-Miro, J.R. Espinosa, M. Garny & T. Riotto
Nikhef, May 19, 2017
Effective actions in particle physics and cosmology
[1205.3392,1406.2652, 1607.08432, 1608.06765]
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Electroweak phase transition
[Kajantie, Laine, Rummukainen, Shaposhnikov '96]
The effective potential is the standard tool to study phase transition at finite temperature.
[S. Weinberg ’73, ’74]
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Higgs properties
[S. Martin '13]
The effective action can also be used for Higgs precision studies.
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Vacuum stability
[Degrassi, Di Vita, Miro, Espinosa, Giudice, Isidori, Strumia '12]
Likewise, the stability of the electroweak vacuum can be analyzed.
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Technical aspects
[Coleman & E. Weinberg ’73][S. Weinberg ’73, ’74][Dolan & Jackiw ’74]
[N.K. Nielsen ’75][Fukuda & Kugo ’76][Aitchison & Fraser ’84][Kobes, Kunstatter & Rebhan ’91][Metaxas & E. Weinberg ’95][Laine ’95]
[Patel & Ramsey-Musolf ’11][Andreassen, Frost, Schwartz ’14][Plascencia, Tamarit ’15][Lalak, Lewicki, Olszewski ’16]
SSB
gauge-dependence
it’s a Higgs!
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Layers of complication
gauge invariance
derivative expansion
perturbation theory
So what? If I do my calculation correctly, everything should be gauge independent.
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Layers of complication
gauge invariance
derivative expansion
perturbation theory
So what? If I do my calculation correctly, everything should be gauge independent.
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Layers of complication
gauge invariance
derivative expansion
perturbation theory
So what? If I do my calculation correctly, everything should be gauge independent.
If the standard model is unstable, at what energy scale do we expect new degrees of freedom?
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Outline
Introduction
Effective action and symmetry breaking
IR problems
Gauge dependence
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Toy model
We expect that the symmetry is broken. But how can one see this? After all by symmetry
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Sources
Idea: Introduce a small source that breakes the symmetry and study how the system reacts:
We see now that W is the generating functional of n-point functions
Symmetry breaking will happen when
But there is no way to study this as long as we do perturbation theory in J! We have to reorganize perturbation theory for that.
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Effective action
The effective action is defined as
and is given by
>1-loop vacuum diagrams in the shifted theory without tadpoles
= 1PI vacuum diagrams = 1-particle-irreducible diagrams
Coleman-Weinberg
[Dolan & Jackiw '74]
EoM:
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The symmetry breaking
Maxwell construction[E Weinberg & Wu '78]
kink!
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Summary
The 1PI effective action is the main tool to study symmetry breaking.
It resums tadpole diagrams and can be calculated perturbatively.
In contrast, the energy functional is not analytic in .
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Outline
Introduction
Effective action and symmetry breaking
IR problems
Gauge dependence
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Motivation : IR problems
The effective potential in the SM is known (partially) up to three loops.
[Martin '13] G = Goldstone mass squaredT, H, W, Z = other masses squaredL = Log
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IR problems
Since the Goldstone mass vanishes in (or close to) the broken phase, this poses a problem.
Solution: Resummation of Goldstone self-energies → CW term with the full propagator
By construction and an expansion in the self-energy will not converge.
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IR problems
Calculating the self-energy and expanding in the self-energy will remove the offending terms in the effective action at 2- and 3-loops.
This is non-trivial, since terms of the form
and
are removed simultaneously. Notice that is not the full Goldstone self-energy
and that the scheme becomes more involved at higher loop orders.
[Elias-Mir, Espinos, TK ’14]
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Solutions
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Summary
A consistent effective action might require to resum a larger set of diagrams. In particular, be aware of
IR problems
non-decoupling termsSee also [Huber, Konstandin, Nardini, Rues '15]
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Outline
Introduction
Effective action and symmetry breaking
IR problems
Gauge dependence
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Gauge dependence is a controversial topic
is gauge parameter
[Patil & Ramsey-Musolf '11]
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Summary
The effective action is gauge-dependent.
Observables are gauge-independent.
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Nielsen identity
The Nielsen identity is based on the following observation
gauge invariance
invariance
BRSTinvariance
Ward-Takahashi identitiese follow from the BRST invariance.
Changes in the gauge fixing parameter can be moved to the source term → Nielsen identities
[Nielsen '75]
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Nielsen identity
With
For example in gauge
Notice that for any solution of the field equations, the effective action is invariant. In particular, the value of the effective potential in the minimum, tunnel amplitues and sphaleron rate are gauge-independent.
Using
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Effective potential
For constant fields the Nielsen identity reads
This means the the function C can be interpreted as
And a change in gauge just stretches the potential
Plot: Ramsey-Musolf, Patil '11
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Consistent effective potential
-dependent resummed masses
The potential is gauge dependent
And we explicity checked the relation
And that the value of the potential in the minimum is gauge-independent.
[Garny & TK '12]
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Nielsen identity of the gradient expansion
In order to study vacuum transitions, one has to consider the effecitve action, eventually expanded in gradients
Expanding also the Nielsen function C
One arrives at the following relation
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Explicit check
Nielsen identity fulfilled at order g3
Gradient expansion breaks down for small VEVs !
[Garny & TK '12]
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Wall tension & sphaleron mass
The wall tension should be gauge-independent since the potential difference is.
This is indeed true up to the order we are calculating
[Garny & TK '12]
We also find a gauge-independent sphaleron mass
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SM vacuum stability and GI scales
Physical question: What is a good indication of the scale where new physics is to expected to make an unstable vacuum stable?
[Espinos, Garny, Riotto, TK ’16]
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Higher dimensional operators
Add to the effective action an operator
[Espinos, Garny, Riotto, TK ’16]
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Higher dimensional operators
Higher-dimensional operators are to a very high a degree gauge-invariant probe of new physics.
[Espinos, Garny, Riotto, TK ’16]
Likewise, the critical temperature is a gauge-invariant measure of the scale of new physics.
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Inflation
Quantum fluctuations during inflation can drive the Higgs field over the potential barrier.
The dynamics of the Higgs field is described by the Fokker-Planck or Langevin equations.
[Espinos, Garny, Riotto, TK ’16]
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Tunneling probability
The probability that the Higgs field is driven by fluctuations over the potential barrier is to high degree gauge-independent.
[Espinos, Garny, Riotto, TK ’16]
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Lessons
The Higgs VEV is gauge-dependent.
The effective action is gauge-dependent. Don't try to find a gauge-independent effective action. If you find one, you probably fixed the gauge.
Observables are gauge-independent. Use Landau gauge or unitary gauge. But be aware of the convergence properties of perturbation theory and the derivative expansion.
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The end
Thank you