Contents

# Contents

## Idea

In quantum field theory, a given vacuum state is called stable if in a suitable sense it does not evolve into or from any orthogonal state.

In perturbative quantum field theory with specified S-matrix $\mathcal{S}(g S_{int})$ it makes sense to say that a vacuum state is stable if there is vanishing quantum probability for the vacuum state to scatter into a non-vacuum state, or for a non-vacuum state to scatter into the vacuum state. (e.g. Nikolic 01, p. 6)

## Definition

###### Definition

(vacuum stability)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to (this def.), let $\mathcal{S}$ be a corresponding S-matrix scheme according to (this. def.), and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle$ be a local observable, regarded as an adiabatically switched interaction action functional.

We say that the given Hadamard vacuum state (this prop.)

$\langle - \rangle \;\colon\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ] \longrightarrow \mathbb{C}[ [ \hbar, g, j ] ]$

is stable with respect to the interaction $S_{int}$, if for all elements of the Wick algebra

$A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]$

we have

$\left\langle A \mathcal{S}(g S_{int}) \right\rangle \;=\; \left\langle \mathcal{S}(g S_{int}) \right\rangle \, \left\langle A \right\rangle \phantom{AA} \text{and} \phantom{AA} \left\langle \mathcal{S}(g S_{int})^{-1} A \right\rangle \;=\; \frac{1} { \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle A \right\rangle$

## Properties

###### Example

(scattering amplitudes as vacuum expectation values of interacting field observables)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to this def., let $\mathcal{S}$ be a corresponding S-matrix scheme according to this def., and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a local observable regarded as an adiabatically switched interaction-functional, such that the vacuum state is stable with respect to $g S_{int}$ (def. ).

Consider local observables

$\array{ A_{in,1}, \cdots, A_{in , n_{in}}, \\ A_{out,1}, \cdots, A_{out, n_{out}} } \;\;\in\;\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]$

whose spacetime support satisfies the following causal ordering:

$A_{out, i_{out} } {\gt\!\!\!\!\lt} A_{out, j_{out}} \phantom{AAA} A_{out, i_{out} } {\vee\!\!\!\wedge} S_{int} {\vee\!\!\!\wedge} A_{in, i_{in}} \phantom{AAA} A_{in, i_{in} } {\gt\!\!\!\!\lt} A_{in, j_{in}}$

for all $1 \leq i_{out} \lt j_{out} \leq n_{out}$ and $1 \leq i_{in} \lt j_{in} \leq n_{in}$.

Then the vacuum expectation value of the Wick algebra-product of the corresponding interacting field observables (this def.) is

\begin{aligned} & \left\langle {\, \atop \,} (A_{out, 1})_{int} \cdots (A_{out,n_{out}})_{int} \, (A_{in, 1})_{int} \cdots (A_{in,n_{in}})_{int} {\, \atop \,} \right\rangle \\ & = \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \right| \; \mathcal{S}(g S_{int}) \; \left| A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \\ & \coloneqq \frac{1}{ \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \; \mathcal{S}(g S_{int}) \; A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \,. \end{aligned}

These vacuum expectation values are interpreted, in the adiabatic limit where $g_{sw} \to 1$, as scattering amplitudes (see this remark).

For proof see at S-matrix, this prop..

## References

Discussion for QED:

Discussion for Higgs field:

• J.R. Espinosa, G. Giudice, A. Riotto, Cosmological implications of the Higgs mass measurement, JCAP 0805:002, 2008 (arXiv:0710.2484)

• John Ellis, J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto, The Probable Fate of the Standard Model, Phys. Lett. B679:369-375, 2009 (arXiv:0906.0954)

• Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian Giudice, Filippo Sala, Alberto Salvio, Alessandro Strumia, Investigating the near-criticality of the Higgs boson (arXiv:1307.3536)

• Anson Hook, John Kearney, Bibhushan Shakya, Kathryn M. Zurek, Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation, J. High Energ. Phys. (2015) 2015: 61 (arXiv:1404.5953)

• Jose R. Espinosa, Gian F. Giudice, Enrico Morgante, Antonio Riotto, Leonardo Senatore, Alessandro Strumia, Nikolaos Tetradis, The cosmological Higgstory of the vacuum instability (arXiv:1505.04825)

• William E. East, John Kearney, Bibhushan Shakya, Hojin Yoo, Kathryn M. Zurek, Spacetime Dynamics of a Higgs Vacuum Instability During Inflation, Phys. Rev. D 95, 023526 (2017) (arXiv:1607.00381)

• Gordon Kane, Exciting Implications of LHC Higgs Boson Data (arXiv:1802.05199)

Last revised on April 5, 2018 at 04:51:34. See the history of this page for a list of all contributions to it.