# nLab phi^n interaction

Contents

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

In quantum field theory of the scalar field $\Phi$, the canonical local interaction term is a Lagrangian density of the form

$\mathbf{L}_{int} \;=\; \phi^n \, dvol_\Sigma$

(with notation as at A first idea of quantum field theory).

For $g_{sw} \in C^\infty_{cp}(\Sigma)$ any bump function on spacetime, the corresponding adiabatically switched local observable is

\begin{aligned} S_{int} & = \underset{\Sigma}{\int} \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) } } \, dvol_\Sigma(x) \\ & = \underset{\Sigma}{\int} : \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \mathbf{\Phi}(x) } } : \, dvol_\Sigma(X) \end{aligned} \,,

where in the first line we have the integral over a pointwise product (this def.) of $n$ field observables (this def.), which in the second line we write equivalently as a normal ordered product, by the discusssion at Wick algebra (this def.).

The interacting field theory with Lagrangian density that of the free scalar field plus interactions of the form $\phi^k$ as above, up to order $n$, is often called simply “$\Phi^n$-theory”.

## Examples

The mass term of the free scalar field is a $\Phi^2$-interaction.

The Higgs field involves a quadratic and quartic interaction of this form.

The potential for the inflaton field in chaotic cosmic inflation is a $\Phi^2$-interaction.

## References

An introduction to $\Phi^4$ theory could be found in lecture 13 of

• Sourav Chatterjee, Introduction to Quantum Field Theory for Mathematicians, pdf

The weak adiabatic limit for mass-less $\Phi^4$ theory was established in

• P. Blanchard and R. Seneor, Green’s functions for theories with massless particles (in perturbation theory), Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (Numdam)