Effective actions in particle physics and cosmology

Transcription

1 Effective actions in particle physics and cosmology Thomas Konstandin in collaboration with J. Elias-Miro, J.R. Espinosa, M. Garny & T. Riotto [ , , , ] Nikhef, May 19, 2017

2 Electroweak phase transition The effective potential is the standard tool to study phase transition at finite temperature. [S. Weinberg 73, 74] [Kajantie, Laine, Rummukainen, Shaposhnikov '96]

3 Higgs properties The effective action can also be used for Higgs precision studies. [S. Martin '13]

4 Vacuum stability Likewise, the stability of the electroweak vacuum can be analyzed. [Degrassi, Di Vita, Miro, Espinosa, Giudice, Isidori, Strumia '12]

5 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 gaugedependence it s a Higgs!

6 Layers of complication So what? If I do my calculation correctly, everything should be gauge independent. gauge invariance derivative expansion perturbation theory

7 Layers of complication So what? If I do my calculation correctly, everything should be gauge independent. gauge invariance derivative expansion perturbation theory

8 Layers of complication So what? If I do my calculation correctly, everything should be gauge independent. gauge invariance derivative expansion perturbation theory If the standard model is unstable, at what energy scale do we expect new degrees of freedom?

9 Outline Introduction Effective action and symmetry breaking IR problems Gauge dependence

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

11 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.

12 Effective action The effective action is defined as EoM: Coleman-Weinberg and is given by >1-loop vacuum diagrams in the shifted theory without tadpoles = 1PI vacuum diagrams = 1-particle-irreducible diagrams [Dolan & Jackiw '74]

13 The symmetry breaking kink! Maxwell construction [E Weinberg & Wu '78]

14 Summary The 1PI effective action is the main tool to study symmetry breaking. It resums tadpole diagrams and calculated perturbatively. can be In contrast, the energy functional analytic in. is not

15 Outline Introduction Effective action and symmetry breaking IR problems Gauge dependence

16 Motivation : IR problems The effective potential in the SM is known (partially) up to three loops. [Martin '13] G = Goldstone mass squared T, H, W, Z = other masses squared L = Log

17 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 converge. and an expansion in the self-energy will not

18 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]

19 Solutions

20 Summary A consistent effective action might require to resum a larger set of diagrams. In particular, be aware of IR problems non-decoupling terms See also [Huber, Konstandin, Nardini, Rues '15]

21 Outline Introduction Effective action and symmetry breaking IR problems Gauge dependence

22 Gauge dependence is a controversial topic is gauge parameter [Patil & Ramsey-Musolf '11]

23 Summary The effective action is gauge-dependent. Observables are gaugeindependent.

24 Nielsen identity The Nielsen identity is based on the following observation [Nielsen '75] gauge invariance invariance BRST invariance Ward-Takahashi identitiese follow from the BRST invariance. Changes in the gauge fixing parameter source term Nielsen identities can be moved to the

25 Nielsen identity Using 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.

26 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

27 Consistent effective potential [Garny & TK '12] -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.

28 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

29 Explicit check Nielsen identity fulfilled at order g3 Gradient expansion breaks down for small VEVs! [Garny & TK '12]

30 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 We also find a gaugeindependent sphaleron mass [Garny & TK '12]

31 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]

32 Higher dimensional operators Add to the effective action an operator [Espinos, Garny, Riotto, TK 16]

33 Higher dimensional operators Higher-dimensional operators are to a very high a degree gaugeinvariant probe of new physics. Likewise, the critical temperature is a gauge-invariant measure of the scale of new physics. [Espinos, Garny, Riotto, TK 16]

34 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]

35 Tunneling probability The probability that the Higgs field is driven by fluctuations over the potential barrier is to high degree gaugeindependent. [Espinos, Garny, Riotto, TK 16]

36 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.

37 The end Thank you