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]

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]

Higgs properties

[S. Martin '13]

The effective action can also be used for Higgs precision studies.

Vacuum stability

[Degrassi, Di Vita, Miro, Espinosa, Giudice, Isidori, Strumia '12]

Likewise, the stability of the electroweak vacuum can be analyzed.

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!

Layers of complication

gauge invariance

derivative expansion

perturbation theory

So what? If I do my calculation correctly, everything should be gauge independent.

Layers of complication

gauge invariance

derivative expansion

perturbation theory

So what? If I do my calculation correctly, everything should be gauge independent.

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?

Outline

Introduction

Effective action and symmetry breaking

IR problems

Gauge dependence

Toy model

We expect that the symmetry is broken. But how can one see this? After all by symmetry

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.

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:

The symmetry breaking

Maxwell construction[E Weinberg & Wu '78]

kink!

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 .

Outline

Introduction

Effective action and symmetry breaking

IR problems

Gauge dependence

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

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.

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]

Solutions

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]

Outline

Introduction

Effective action and symmetry breaking

IR problems

Gauge dependence

Gauge dependence is a controversial topic

is gauge parameter

[Patil & Ramsey-Musolf '11]

Summary

The effective action is gauge-dependent.

Observables are gauge-independent.

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]

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

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

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]

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

Explicit check

Nielsen identity fulfilled at order g3

Gradient expansion breaks down for small VEVs !

[Garny & TK '12]

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

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]

Higher dimensional operators

Add to the effective action an operator

[Espinos, Garny, Riotto, TK ’16]

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.

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]

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]

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.

The end

Thank you