^{1,2}

^{2}

^{3}

^{1}

^{2}

^{3}

We consider a Hamiltonian with cutoffs describing the weak decay of spin 1 massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove the absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.

In this article, we consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory (see [

The mathematical framework involves fermionic Fock spaces for the leptons and bosonic Fock spaces for the vector bosons. The interaction is described in terms of annihilation and creation operators together with kernels which are square integrable with respect to momenta. The total Hamiltonian, which is the sum of the free energy of the particles and antiparticles and of the interaction, is a self-adjoint operator in the Fock space for the leptons and the vector bosons and it has an unique ground state in the Fock space for a sufficiently small coupling constant.

The weak interaction is one of the four fundamental interactions known up to now. But the weak interaction is the only one which does not generate bound states. As it is well known, it is not the case for the strong, electromagnetic, and gravitational interactions. Thus we are expecting that the spectrum of the Hamiltonian associated with every model of weak decays is absolutely continuous above the energy of the ground state, and this article is a first step towards a proof of such a statement. Moreover a scattering theory has to be established for every such Hamiltonian.

In this paper we establish a Mourre estimate and a limiting absorption principle for any spectral interval above the energy of the ground state and below the mass of the electron for a small coupling constant.

Our study of the spectral analysis of the total Hamiltonian is based on the conjugate operator method with a self-adjoint conjugate operator. The methods used in this article are taken largely from [

For other applications of the conjugate operator method see [

For related results about models in Quantum Field Theory see [

The paper is organized as follows. In Section

The weak decay of the intermediate bosons

The full family of leptons involves the electron

It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity

In what follows, the mathematical model for the weak decay of the vector bosons

Let us sketch how we define a mathematical model for the weak decay of the vector bosons

The energy of the free leptons and bosons is a self-adjoint operator in the corresponding Fock space (see below), and the main problem is associated with the interaction between the bosons and the leptons. Let us consider only the interaction between the bosons and the electrons, the positrons, and the corresponding neutrinos and antineutrinos. Other cases are strictly similar. In the Schrödinger representation the interaction is given by (see [

We have

The operators

Similarly we define

For the

The interaction (

Thus when one develops the interaction

In what follows, we consider a model involving terms of the above form but with more general square integrable kernels.

We follow the convention described in [

Concerning our notations, from now on,

Let

The Hilbert space for the weak decay of the vector bosons

Let

For every

The bosonic Fock space for the vector bosons

The Fock space for the weak decay of the vector bosons

For every

Let

Let

Let

The extension of

Let

We now define the creation and annihilation operators.

For each

Similarly, for each

The operator

The following canonical anticommutation and commutation relations hold:

We recall that the following operators, with

The operators

The free Hamiltonian

The spectrum of

The interaction, denoted by

The total Hamiltonian is then

The operator

Every kernel

Thus, from now on, the kernels

For

A similar model can be written down for the weak decay of pions

The total Hamiltonian is more general than the one involved in the theory of weak interactions because, in the Standard Model, neutrinos have helicity

In the physical case, the Fock space, denoted by

Under Hypothesis

The formal operator

By mimicking the proof of [

Similarly for the operator

Again, there exists two closed operators

We will still denote by

Thus

We now want to prove that

Once again, as above, for almost every

We have that

It then follows that the operator

We then have the following.

For almost every

The estimates (

Let

We get the following.

One has

Suppose that

We now have

By (

We now prove that

Let

Let

In the sequel, we will make the following additional assumptions on the kernels

(i) For

(ii) There exists

(iii) For

(iv) There exists

Hypothesis

Hypothesis

Our first result is devoted to the existence of a ground state for

Suppose that the kernels

According to Theorem

Let

Let now

We will denote again by

We also set

We then have the following.

Suppose that the kernels

The spectrum of

Limiting absorption principle.

For every

Pointwise decay in time.

Suppose

The proof of Theorem

Let

For

The operator

Let

Set

Define

We get

We now set, for

Let

We set

Let us define the sequence

Let

Suppose that the kernels

The proof of Proposition

We now introduce the positive commutator estimates and the regularity property of

The operator

Let

Note that we also have

The operators

The operators

By (

Let

We have

The operators

Let

Let

Let

This yields, for any

Suppose that the kernels

For every

We only prove (i), since (ii) and (iii) can be proved similarly. By (

Let

We set

According to [

Recall that

Thus, one of our main results is the following one.

Suppose that the kernels

It follows from Theorem

Furthermore, by Proposition

The following proposition allows us to compute

Suppose that the kernels

for all

and

Part

By (

By (

Combining (

We now introduce the Mourre inequality.

Let

We have, for

We set, for

Let

Furthermore, it follows from the virial theorem (see [

We then have the following.

Suppose that the kernels

Let

Suppose that the kernels

We now prove Theorem

The estimate (

Let

Let us estimate

By (

Statements about

It then follows from (

We first prove Proposition

We have

Let

We have

We have, as sesquilinear forms and with respect to (

By mimicking the proofs of Propositions

Combining (

We have

By (

This, together with (

Proposition

The proof of Theorem

Assume that the kernels

Let

By (

Suppose that the kernels

Let

By mimicking the proof of (

Combining Lemma

Using (

We now prove Theorem

It follows from Proposition

This yields

Multiplying both sides of (

We set

Suppose that the kernels

We use the representation

We get

By Proposition

The proof of Theorem

Suppose that the kernels

We have, for every

Let us now prove that

Combining Hypothesis

Let

Similarly, by mimicking the proof of (

We now prove Theorem

In view of [

Let us prove (

To this end, let

We note that

We also have

Therefore, combining Proposition

In a similar way it follows from Propositions

By (

In this appendix, we will prove Proposition

Let, for

We have

Let us denote by

Let for

Combining (

It follows from [

We have

Let

Combining (

We have, for every

In view of (

For

Recall that, for

By (

Thus, by (