Advances in Mathematical Physics Volume 2009, Article ID 978903,52pages doi:10.1155/2009/978903

*Research Article*

**Spectral Theory for a Mathematical Model of the** **Weak Interaction—Part I: The Decay of the**

**Intermediate Vector Bosons** *W*

^{±}

**J.-M. Barbaroux**

^{1, 2}**and J.-C. Guillot**

^{3}*1**Centre de Physique Th´eorique, Centre National de la Recherche Scientique (CNRS), Luminy Case 907,*
*13288 Marseille Cedex 9, France*

*2**D´epartement de Math´ematiques, Universit´e du Sud Toulon-Var, 83957 La Garde Cedex, France*

*3**Centre de Math´ematiques Appliqu´ees, ´Ecole Polytechnique, UMR-CNRS 7641,*
*91128 Palaiseau Cedex, France*

Correspondence should be addressed to J.-M. Barbaroux,barbarou@univ-tln.fr Received 28 April 2009; Accepted 13 August 2009

Recommended by Valentin Zagrebnov

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.

Copyrightq2009 J.-M. Barbaroux and J.-C. Guillot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

In this article, we consider a mathematical model of the weak interaction as patterned accord-
ing to the Standard Model in Quantum Field Theorysee1,2. We choose the example of
the weak decay of the intermediate vector bosons*W*^{±}into the full family of leptons.

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 from3,4and are based on5,6. Some of the results of this article have been announced in7.

For other applications of the conjugate operator method see8–19.

For related results about models in Quantum Field Theory see20,21in the case of the Quantum Electrodynamics and22in the case of the weak interaction.

The paper is organized as follows. InSection 2, we give a precise definition of the model we consider. In Section 3, we state our main results and in the following sections, together with the appendix, detailed proofs of the results are given.

**2. The Model**

The weak decay of the intermediate bosons*W*^{}and*W*^{−}involves the full family of leptons
together with the bosons themselves, according to the Standard Model see 1, formula
4.139and2.

The full family of leptons involves the electron*e*^{−}and the positron*e*^{}, together with the
associated neutrino*ν** _{e}*and antineutrino

*ν*

*, the muons*

_{e}*μ*

^{−}and

*μ*

^{}together with the associated neutrino

*ν*

*and antineutrino*

_{μ}*ν*

_{μ}*,*and the tau leptons

*τ*

^{−}and

*τ*

^{}together with the associated neutrino

*ν*

*τ*and antineutrino

*ν*

*τ*.

It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity−1/2 and antineutrinos are right handed, that is, antineutrinos have helicity1/2.

In what follows, the mathematical model for the weak decay of the vector bosons
*W*^{} and*W*^{−}that we propose is based on the Standard Model, but we adopt a slightly more
general point of view because we suppose that neutrinos and antineutrinos are both massless
particles with helicity±1/2. We recover the physical situation as a particular case. We could
also consider a model with massive neutrinos and antineutrinos built upon the Standard
Model with neutrino mixing23.

Let us sketch how we define a mathematical model for the weak decay of the vector
bosons*W*^{±}into the full family of leptons.

The energy of the free leptons and bosons is a self-adjoint operator in the correspond- ing Fock spacesee 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 Schr ¨odinger representation the interaction is given bysee1, page 159, equation4.139and2, page 308, equation21.3.20

*I*

d^{3}*xΨ**e*xγ* ^{α}*
1−

*γ*

_{5}

Ψ*ν**e*xW*α*x

d^{3}*xΨ**ν**e*xγ* ^{α}*
1−

*γ*

_{5}

Ψ*e*xW*α*x^{∗}*,* 2.1

where*γ** ^{α}*,

*α*0,1,2,3 and

*γ*5are the Dirac matrices andΨ·xandΨ·xare the Dirac fields for

*e*

_{−},

*e*

_{},

*ν*

_{e}*,*and

*ν*

*.*

_{e}We have
Ψ*e**x*

1 2π

_{3/2}

*s ±1/2*

d^{3}*p*

*b*_{e,}

*p, su*
*p, s*

√*p*0 e^{ip·x}*b*^{∗}_{e,−}

*p, sv*
*p, s*

√*p*0 e^{−ip·x} *,*
Ψ*e*x Ψ*e*x^{†}*γ*^{0}*.*

2.2

Here*p*_{0} |p|^{2}*m*^{2}_{e}^{1/2}where*m*_{e}*>*0 is the mass of the electron, and*up, s*and*vp, s*are
the normalized solutions to the Dirac equationsee1, Appendix.

The operators*b** _{e,}*p, sand

*b*

^{∗}

*p, s resp.,*

_{e,}*b*

*p, sand*

_{e,−}*b*

^{∗}

*p, sare the annihilation and creation operators for the electronsresp., the positronssatisfying the anticommutation relationssee below.*

_{e,−}Similarly we define Ψ*ν**e*x and Ψ*ν**e*x by substituting the operators *c**ν**e**,±*p, s and
*c*^{∗}_{ν}_{e}* _{,±}*p, s for

*b*

*p, s and*

_{e,±}*b*

^{∗}

*p, s with*

_{e,±}*p*

_{0}|p|. The operators

*c*

_{ν}

_{e}*p, s and*

_{,}*c*

_{ν}^{∗}

_{e}*p, s resp.,*

_{,}*c*

_{ν}

_{e}*p, sand*

_{,−}*c*

^{∗}

_{ν}

_{e}*p, sare the annihilation and creation operators for the neutrinos associated with the electronsresp., the antineutrinos.*

_{,−}For the*W**α*fields we havesee24, Section 5.3

*W** _{α}*x
1

2π

_{3/2}

*λ −1,0,1*

d^{3}*k*
2k0

* _{α}*k, λa

_{}k, λe

^{ik·x}^{∗}

*k, λa*

_{α}^{∗}

_{−}k, λe

^{−ik·x}

*.* 2.3

Here*k*0 |k|^{2}*m*^{2}_{W}^{1/2}where*m**W* *>*0 is the mass of the bosons*W*^{±}.*W*^{}is the antiparticule
of*W*^{−}. The operators*a*_{}k, λand*a*^{∗}_{}k, λ resp.,*a*_{−}k, λand*a*^{∗}_{−}k, λare the annihilation
and creation operators for the bosons*W*^{−}resp.,*W*^{}satisfying the canonical commutation
relations. The vectors*α*k, λare the polarizations of the massive spin 1 bosons*W*^{±}see24,
Section 5.2.

The interaction2.1is a formal operator and, in order to get a well-defined operator
in the Fock space, one way is to adapt what Glimm and Jaﬀe have done in the case of the
Yukawa Hamiltoniansee25. For that sake, we have to introduce a spatial cutoﬀ*gx*such
that*g* ∈*L*^{1}R^{3}, together with momentum cutoﬀs*χp*and*ρk*for the Dirac fields and the
*W**μ*fields, respectively.

Thus when one develops the interaction *I* with respect to products of creation and
annihilation operators, one gets a finite sum of terms associated with kernels of the form

*χ*
*p*_{1}

*χ*
*p*_{2}

*ρkg*

*p*_{1}*p*_{2}−*k*

*,* 2.4

where*g*is the Fourier transform of*g. These kernels are square integrable.*

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 24, Section 4.1 that we quote: “The state- vector will be taken to be symmetric under interchange of any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any two fermions with each other, in all cases, whether the particles are of the same species or not.”

Thus, as it follows from24, Section 4.2, fermionic creation and annihilation operators of diﬀerent species of leptons will always anticommute.

Concerning our notations, from now on, ∈ {1,2,3}denotes each species of leptons.

1 denotes the electron*e*^{−}the positron*e*^{}and the neutrinos*ν**e*,*ν**e*. 2 denotes the muons
*μ*^{−},*μ*^{} and the neutrinos*ν** _{μ}*and

*ν*

*, and 3 denotes the tau-leptons and the neutrinos*

_{μ}*ν*

*and*

_{τ}*ν*

*.*

_{τ}Let*ξ*1 p1*, s*1be the quantum variables of a massive lepton, where *p*1 ∈ R^{3} and
*s*_{1} ∈ {−1/2,1/2}is the spin polarization of particles and antiparticles. Let*ξ*_{2} p2*, s*_{2}be the
quantum variables of a massless lepton where*p*_{2}∈R^{3}and*s*_{2}∈ {−1/2,1/2}is the helicity of
particles and antiparticles, and, finally, let*ξ*3 k, λbe the quantum variables of the spin 1
bosons*W*^{}and*W*^{−}where*k* ∈R^{3}and*λ* ∈ {−1,0,1}is the polarization of the vector bosons
see24, Section 5. We setΣ1 R^{3}× {−1/2,1/2}for the leptons andΣ2 R^{3}× {−1,0,1}for
the bosons. Thus*L*^{2}Σ1is the Hilbert space of each lepton and*L*^{2}Σ2is the Hilbert space of
each boson. The scalar product in*L*^{2}Σ*j*,*j* 1,2 is defined by

*f, g*

Σ*j*

*fξgξdξ, j* 1,2. 2.5

Here

Σ1

dξ

*s 1/2,−1/2*

dp,

Σ2

dξ

*λ 0,1,−1*

dk,

*p, k*∈R^{3}

*.* 2.6

The Hilbert space for the weak decay of the vector bosons*W*^{} and*W*^{−} is the Fock
space for leptons and bosons that we now describe.

LetSbe any separable Hilbert space. Let_{n}

*a*Sresp.,_{n}

*s*Sdenote the antisymmetric
resp., symmetric*nth tensor power of*S. The fermionicresp., bosonicFock space overS,
denoted byF*a*S resp.,F*s*S, is the direct sum

F*a*S ^{∞}

*n 0*

*n*
*a* S

resp., F*s*S ^{∞}

*n 0*

*n*

*s* S *,* 2.7

where_{0}

*a*S _{0}

*s*S ≡C. The stateΩ 1,0,0, . . . ,0, . . .denotes the vacuum state inF*a*S
and inF*s*S.

For every*,*F is the fermionic Fock space for the corresponding species of leptons
including the massive particle and antiparticle together with the associated neutrino and
antineutrino, that is,

F

4

F*a*

*L*^{2}Σ1

1,2,3. 2.8

We have

F

*q*≥0,q≥0,r≥0,r≥0

F^{q}_{}^{}^{,q}^{}^{,r}^{}^{,r}^{}^{} 2.9

with

F^{q}_{}^{}^{,q}^{}^{,r}^{}^{,r}^{}^{}
_{q}_{}

*a*

*L*^{2}Σ1 ⊗

⎛

⎝^{q}^{}

*a*

*L*^{2}Σ1

⎞

⎠⊗
_{r}

*a*

*L*^{2}Σ1 ⊗
_{r}

*a*

*L*^{2}Σ1 *.* 2.10

Here*q* resp.,*q** _{}* is the number of massive fermionic particleresp., antiparticlesand

*r*

resp.,*r** _{}*is the number of neutrinosresp., antineutrinos. The vectorΩ is the associated
vacuum state. The fermionic Fock space denoted byF

*L*for the leptons is then

F*L*

3
* 1*

F*,* 2.11

andΩ*L* _{3}

* 1*Ωis the vacuum state.

The bosonic Fock space for the vector bosons*W*^{}and*W*^{−}, denoted byF*W*, is then
F*W* F*s*

*L*^{2}Σ2

⊗F*s*

*L*^{2}Σ2
F*s*

*L*^{2}Σ2⊕*L*^{2}Σ2

*.* 2.12

We have

F*W*

*t≥0,t≥0*

F^{t,t}_{W}*,* 2.13

whereF^{t,t}_{W}_{t}

*s**L*^{2}Σ2⊗_{t}

*s**L*^{2}Σ2. Here*t*resp.,*t*is the number of bosons*W*^{−}resp.,
*W*^{}. The vectorΩ*W*is the corresponding vacuum.

The Fock space for the weak decay of the vector bosons*W*^{}and*W*^{−}, denoted byF, is
thus

F F*L*⊗F*W**,* 2.14

andΩ Ω*L*⊗Ω*W* is the vacuum state.

For every ∈ {1,2,3} let D denote the set of smooth vectors*ψ** _{}* ∈ F for which

*ψ*

_{}^{q}

^{}

^{,q}

^{}

^{,r}

^{}

^{,r}

^{}^{}has a compact support and

*ψ*

^{q}

_{}

^{}

^{,q}

^{}

^{,r}

^{}

^{,r}

^{}^{}0 for all but finitely manyq

*, q*

_{}*, r*

_{}*, r*

*. Let*

_{}D*L* 3

* 1*D*.* 2.15

Here

is the algebraic tensor product.

LetD*W*denote the set of smooth vectors*φ*∈F*W*for which*φ*^{t,t}has a compact support
and*φ*^{t,t} 0 for all but finitely manyt, t.

Let

D D*L*⊗D *W**.* 2.16

The setDis dense inF.

Let*A** _{}*be a self-adjoint operator inFsuch thatD is a core for

*A*

*. Its extension toF*

_{}*L*

is, by definition, the closure inF*L*of the operator*A*_{1}⊗1_{2}⊗1_{3}with domainD*L*when 1, of
the operator 11⊗*A*2⊗13 with domainD*L*when 2, and of the operator 11⊗12⊗*A*3with
domainD*L*when 3. Here 1is the operator identity onF.

The extension of*A** _{}* toF

*L*is a self-adjoint operator for whichD

*L*is a core and it can be extended toF. The extension of

*A*toFis, by definition, the closure inFof the operator

*A*⊗1

*W*with domainD, where

*A*is the extension of

*A*toF

*L*. The extension of

*A*toFis a self-adjoint operator for whichDis a core.

Let*B* be a self-adjoint operator inF*W* for which D*W* is a core. The extension of the
self-adjoint operator*A** _{}*⊗

*B*is, by definition, the closure inFof the operator

*A*

_{1}⊗1

_{2}⊗1

_{3}⊗

*B*with domainDwhen 1, of the operator 11⊗

*A*2⊗13⊗

*B*with domainDwhen 2, and of the operator 11⊗12⊗

*A*

_{3}⊗

*B*with domainDwhen 3. The extension of

*A*

*⊗*

_{}*B*toFis a self-adjoint operator for whichDis a core.

We now define the creation and annihilation operators.

For each 1,2,3, *b** _{,}*ξ1 resp.,

*b*

^{∗}

*ξ1 is the fermionic annihilation resp., fermionic creation operator for the corresponding species of massive particle when and for the corresponding species of massive antiparticle when −. The operators*

_{,}*b*

*,*ξ1 and

*b*

_{,}^{∗}ξ1are defined as usuallysee, e.g.,20,26; see also the detailed definitions in27.

Similarly, for each 1,2,3,*c** _{,}*ξ2 resp.,

*c*

^{∗}

*ξ2is the fermionic annihilationresp., fermionic creationoperator for the corresponding species of neutrino when and for the corresponding species of antineutrino when −. The operators*

_{,}*c*

*ξ2and*

_{,}*c*

^{∗}

*ξ2are defined in a standard way, but with the additional requirements that for any*

_{,}*,*

^{},and

^{}, the operators

*b*

^{}*ξ1and*

_{,}*c*

^{}

_{}*,*

^{}ξ2anticommutes, wherestands either for a∗or for no symbol see the detailed definitions in27.

The operator*a** _{}*ξ3 resp.,

*a*

^{∗}

*ξ3is the bosonic annihilationresp., bosonic creation operator for the boson*

_{}*W*

^{−}when and for the boson

*W*

^{}when −see, e.g.,20,26, or27. Note that

*a*

*ξ3commutes with*

^{}*b*

_{,}*ξ1and*

^{}*c*

^{}

_{}*,*

^{}ξ2.

The following canonical anticommutation and commutation relations hold:

*b** _{,}*ξ1, b

^{∗}

_{}*,*

^{}

*ξ*^{}_{1}

*δ*_{}^{}*δ*_{}^{}*δ*
*ξ*_{1}−*ξ*_{1}^{}

*,*
*c** _{,}*ξ2, c

^{∗}

_{}*,*

^{}

*ξ*_{2}^{}

*δ*_{}^{}*δ*_{}^{}*δ*
*ξ*_{2}−*ξ*^{}_{2}

*,*
*a*ξ3, a^{∗}_{}

*ξ*^{}_{3}

*δ*^{}*δ*
*ξ*_{3}−*ξ*^{}_{3}

*,*
*b*_{,}*ξ*_{1}*, b*_{}*,*^{}

*ξ*^{}_{1}

*c*_{,}*ξ*_{2}*, c*_{}*,*^{}
*ξ*_{2}^{}

0,
*a** _{}*ξ3, a

^{}

*ξ*^{}_{3}
0,
{b*,*ξ1, c^{}*,*^{}ξ2}

*b** _{,}*ξ1, c

^{∗}

_{}*,*

^{}ξ2 0, b

*,*ξ1, a

^{}ξ3

*b**,*ξ1, a^{∗}* _{}*ξ3

*c**,*ξ2, a^{}ξ3

*c**,*ξ2, a^{∗}* _{}* ξ3
0,

2.17

where we used the notation*δξ**j*−*ξ*^{}_{j}*δ*_{λλ}*δk*−*k*^{}.

We recall that the following operators, with*ϕ*∈*L*^{2}Σ1,
*b*_{,}

*ϕ*

Σ1

*b*_{,}*ξϕξdξ,* *c*_{,}

*ϕ*

Σ1

*c*_{,}*ξϕξdξ,*

*b*_{,}^{∗}

*ϕ*

Σ1

*b*^{∗}* _{,}*ξϕξdξ,

*c*

^{∗}

_{,}*ϕ*

Σ1

*c*^{∗}* _{,}*ξϕξdξ

2.18

are bounded operators inFsuch that
*b*^{}_{,}

*ϕ* *c*^{}_{,}

*ϕ* *ϕ*_{L}_{2}*,* 2.19

where*b** ^{}*resp.,

*c*

*is*

^{}*b*resp.,

*c*or

*b*

^{∗}resp.,

*c*

^{∗}.

The operators *b*_{,}* ^{}* ϕand

*c*

^{}*ϕsatisfy similar anticommutaion relations see, e.g., 28.*

_{,}The free Hamiltonian*H*0is given by
*H*_{0} *H*_{0}^{1}*H*_{0}^{2}*H*_{0}^{3}

3
* 1*

* ±*

*w*^{1}* _{}* ξ1b

^{∗}

*ξ1b*

_{,}*,*ξ1dξ1

^{3}

* 1*

* ±*

*w*^{2}* _{}* ξ2c

_{,}^{∗}ξ2c

*,*ξ2dξ2

* ±*

*w*^{3}ξ3a^{∗}* _{}*ξ3aξ3dξ3

*,*

2.20

where

*w*^{1}* _{}* ξ1

*p*

_{1}

^{2}

*m*

^{2}

_{}_{1/2}

*,* with 0*< m*_{1} *< m*_{2}*< m*_{3}*,*
*w*^{2}* _{}* ξ2

*p*

_{2}

*,*

*w*^{3}ξ3

|k|^{2}*m*^{2}_{W}_{1/2}
*,*

2.21

where*m**W* is the mass of the bosons*W*^{}and*W*^{−}such that*m**W* *> m*3.

The spectrum of*H*_{0}is0,∞and 0 is a simple eigenvalue withΩas eigenvector. The
set of thresholds of*H*_{0}, denoted by*T*, is given by

*T*

*pm*_{1}*qm*_{2}*rm*_{3}*sm** _{W}*;

*p, q, r, s*

∈N^{4}*, pqrs*≥1

*,* 2.22

and each sett,∞,*t*∈*T, is a branch of absolutely continuous spectrum forH*0.
The interaction, denoted by*H** _{I}*, is given by

*H**I*

2
*α 1*

*H*_{I}^{α}*,* 2.23

where

*H*_{I}^{1} ^{3}

* 1*

* /*^{}

*G*^{1}* _{,,}*ξ1

*, ξ*2

*, ξ*3b

_{,}^{∗}ξ1c

_{,}^{∗}ξ2aξ3dξ1dξ2dξ3

^{3}

* 1*

* /*^{}

*G*^{1}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}a^{∗}* _{}*ξ3c

*,*

^{}ξ2b

*,*ξ1dξ1dξ

_{2}dξ

_{3}

*,*

*H*_{I}^{2} ^{3}

* 1*

* /*^{}

*G*^{2}* _{,,}*ξ1

*, ξ*2

*, ξ*3b

^{∗}

*ξ1c*

_{,}^{∗}

*ξ2a*

_{,}^{∗}

*ξ3dξ1dξ2dξ3*

_{}^{3}

* 1*

* /*^{}

*G*^{2}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}aξ3c*,*^{}ξ2b*,*ξ1dξ1dξ_{2}dξ_{3}*.*

2.24

The kernels*G*^{2}_{,,}_{}·,·,·,*α* 1,2, are supposed to be functions.

The total Hamiltonian is then

*H* *H*0*gH**I**, g >*0, 2.25

where*g*is a coupling constant.

The operator*H*_{I}^{1}describes the decay of the bosons*W*^{}and*W*^{−}into leptons. Because
of*H*_{I}^{2} the bare vacuum will not be an eigenvector of the total Hamiltonian for every*g >*0
as we expect from the physics.

Every kernel*G** _{,,}*ξ1

*, ξ*

_{2}

*, ξ*

_{3}, computed in theoretical physics, contains a

*δ-distribution*because of the conservation of the momentumsee1and24, Section 4.4. In what follows, we approximate the singular kernels by square integrable functions.

Thus, from now on, the kernels*G*^{α}* _{,,}*are supposed to satisfy the following hypothesis.

*Hypothesis 2.1. Forα* 1,2, 1,2,3,*, *^{} ±, we assume

*G*^{α}* _{,,}*ξ1

*, ξ*2

*, ξ*3∈

*L*

^{2}Σ1×Σ1×Σ2. 2.26

*Remark 2.2. A similar model can be written down for the weak decay of pionsπ*

^{−}and

*π*

^{}see 1, Section 6.2.

*Remark 2.3. The total Hamiltonian is more general than the one involved in the theory of weak*
interactions because, in the Standard Model, neutrinos have helicity−1/2 and antineutrinos
have helicity 1/2.

In the physical case, the Fock space, denoted byF^{}, is isomorphic toF^{}* _{L}*⊗F

*W*, with

F^{}_{L}^{3}

* 1*

F^{}_{}*,*

F^{}_{}

_{2}

*a*

*L*^{2}Σ1 ⊗
_{2}

*a*

*L*^{2}
R^{3} *.*

2.27

The free Hamiltonian, now denoted by*H*_{0}^{}, is then given by

*H*_{0}^{} ^{3}

* 1*

* ±*

*w*_{}^{1}*ξ*_{1}*b*^{∗}_{,}*ξ*_{1}*b*_{,}*ξ*_{1}dξ1^{3}

* 1*

* ±*

R^{3}

*p*_{2}*c*^{∗}_{,}*p*_{2}

*c*_{,}*p*_{2}

dp_{2}

* ±*

*w*^{3}ξ3a^{∗}* _{}*ξ3aξ3dξ3

*,*

2.28

and the interaction, now denoted by *H*_{I}^{}, is the one obtained from *H**I* by supposing that
*G*^{α}ξ1*,*p2*, s*2, ξ3 0 if*s*2 *1/2. The total Hamiltonian, denoted by* *H*^{}, is then given
by*H*^{} *H*_{0}^{} *gH*_{I}^{}. The results obtained in this paper for*H* hold true for*H*^{}with obvious
modifications.

Under Hypothesis 2.1 a well-defined operator on D corresponds to the formal
interaction*H** _{I}* as it follows.

The formal operator

*G*^{1}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}b^{∗}* _{,}*ξ1c

^{∗}

*ξ2aξ3dξ1dξ*

_{,}_{2}dξ

_{3}2.29

is defined as a quadratic form onD⊗D*W*×D⊗D*W*as
*c**,*^{}ξ2b*,*ξ1ψ, G^{1}_{,,}*a*ξ3φ

dξ_{1}dξ_{2}dξ_{3}*,* 2.30

where*ψ,φ*∈D⊗D*W*.

By mimicking the proof of29, Theorem X.44, we get a closed operator, denoted by
*H*_{I,,,}^{1} , associated with the quadratic form such that it is the unique operator inF⊗F*W* such
thatD⊗D*W* ⊂ DH_{I,,,}^{1} is a core for*H*_{I,,,}^{1} and

*H*_{I,,,}^{1}

*G*^{1}* _{,,}*ξ1

*, ξ*

_{2}

*, ξ*

_{3}b

^{∗}

*ξ1c*

_{,}

_{,}^{∗}ξ2aξ3dξ1dξ

_{2}dξ

_{3}2.31

as quadratic forms onD⊗D*W*×D⊗D*W*.

Similarly for the operatorH_{I,,,}^{1} ^{∗}, we have the equality as quadratic forms
*H*_{I,,,}^{1}

_{∗}

*G*^{1}* _{,,}*ξ1

*, ξ*2

*, ξ*3a

^{∗}

*ξ3c*

_{}*,*

^{}ξ2b

*,*ξ1dξ1dξ2dξ3

*.*2.32

Again, there exists two closed operators*H*_{I,,,}^{2} _{} andH_{I,,,}^{2} ^{∗} such that D ⊗D*W* ⊂
DH_{I,,,}^{2} ,D ⊗D*W* ⊂ DH_{I,,,}^{2} ^{∗}, andD ⊗D*W* is a core for*H*_{I,,,}^{2} _{} and H_{I,,,}^{2} ^{∗}and

such that

*H*_{I,,,}^{2} _{}

*G*^{2}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}b_{,}^{∗} ξ1c_{,}^{∗} ξ2a^{∗}* _{}*ξ3dξ1dξ

_{2}dξ

_{3}

*,*

*H*

_{I,,,}^{2}

_{∗}

*G*^{2}* _{,,}*ξ1

*, ξ*2

*, ξ*3aξ3c

*,*

^{}ξ2b

*,*ξ1dξ1dξ2dξ3

2.33

as quadratic forms onD⊗D*W*×D⊗D*W*.

We will still denote by*H*_{I,,,}^{α} andH_{I,,,}^{α} ^{∗}α 1,2their extensions toF. The setD
is then a core for*H*_{I,,,}^{α} andH_{I,,,}^{α} ^{∗}.

Thus

*H* *H*0*g*

*α 1,2*

3
* 1*

* /*^{}

*H*_{I,,,}^{α}
*H*_{I,,,}^{2}

_{∗}

2.34

is a symmetric operator defined onD.

We now want to prove that*H*is essentially self-adjoint onDby showing that*H*_{I,,,}^{α}

andH_{I,,,}^{α} ^{∗}are relatively*H*0-bounded.

Once again, as above, for almost every*ξ*_{3} ∈ Σ2, there exists closed operators inF*L*,
denoted by*B*^{α}_{,,}_{}ξ3andB^{α}* _{,,}*ξ3

^{∗}such that

*B*_{,,}^{1} ξ3 −

*G*^{1}* _{,,}*ξ1

*, ξ*2

*, ξ*3b

*,*ξ1c

*,*

^{}ξ2dξ1dξ2

*,*

*B*

^{1}

_{,,}_{}ξ3

_{∗}

*G*^{1}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}b^{∗}* _{,}*ξ1c

^{∗}

*ξ2dξ1dξ*

_{,}_{2}

*,*

*B*

_{,,}^{2}ξ3

*G*^{2}* _{,,}*ξ1

*, ξ*2

*, ξ*3b

^{∗}

*ξ1c*

_{,}^{∗}

*ξ2dξ1dξ2*

_{,}*,*

*B*

^{2}

_{,,}_{}ξ3

_{∗}

−

*G*^{2}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}b*,*ξ1c*,*^{}ξ2dξ1dξ_{2}

2.35

as quadratic forms onD×D.

We have that D ⊂ DB^{α}* _{,,}*ξ3 resp.,D ⊂ DB

_{,,}^{α}ξ3

^{∗}is a core for

*B*

_{,,}^{α}ξ3 resp., forB

^{α}

*ξ3*

_{,,}^{∗}. We still denote by

*B*

^{α}

*ξ3andB*

_{,,}

_{,,}^{α}ξ3

^{∗}their extensions toF

*L*.

It then follows that the operator*H** _{I}* with domainDis symmetric and can be written in
the following form:

*H**I*

*α 1,2*

3
* 1*

* /*^{}

*H*_{I,,,}^{α}
*H*_{I,,,}^{α}

_{∗}

*α 1,2*

3
* 1*

* /*^{}

*B*^{α}* _{,,}*ξ3⊗

*a*

^{∗}

*ξ3dξ3*

_{}

*α 1,2*

3
* 1*

* /*^{}

*B*^{α}* _{,,}*ξ3

_{∗}

⊗*a*ξ3dξ3*.*

2.36

Let*N*denote the operator number of massive leptonsinF, that is,

*N*_{}

*b*^{∗}* _{,}*ξ1b

*,*ξ1dξ1

*.*2.37

The operator*N* is a positive self-adjoint operator inF. We still denote by*N* its extension
toF*L*. The setD*L*is a core for*N*.

We then have the following.

**Proposition 2.4. For almost every***ξ*_{3} ∈ Σ2*,*DB^{α}* _{,,}*ξ3,DB

^{α}

*ξ3*

_{,,}^{∗}⊃ DN

^{1/2}

*, and for Φ∈ DN*

_{}^{1/2}

*⊂F*

_{}*L*

*one has*

B_{,,}^{α} ξ3Φ

F*L*≤G^{α}* _{,,}*·,·, ξ3

*L*^{2}Σ1×Σ1

N^{1/2}* _{}* Φ

F*L**,*

*B*^{α}* _{,,}*ξ3

_{∗}Φ

F*L*≤G^{α}* _{,,}*·,·, ξ3

_{L}_{2}

Σ1×Σ1

N^{1/2}* _{}* Φ

F*L**.*

2.38

*Proof. The estimates*2.38are examples of*N** _{τ}*estimatessee25. The proof is quite similar
to the proof of20, Proposition 3.7. Details can be found in27but are omitted here.

Let

*H*_{0,}^{3}

*w*^{3}*ξ*_{3}*a*^{∗}_{}*ξ*_{3}*a*_{}*ξ*_{3}dξ3*.* 2.39

Then*H*_{0,}^{3}is a self-adjoint operator inF*W*, andD*W* is a core for*H*_{0,}^{3}.
We get the following.

**Proposition 2.5. One has***B*^{α}_{,,}_{}ξ3_{∗}

⊗*a** _{}*ξ3dξ3Ψ

^{2}

≤

⎛

⎜⎝

Σ1×Σ1×Σ2

G^{α}* _{,,}*ξ1

*, ξ*2

*, ξ*3

^{2}

*w*^{3}ξ3 dξ_{1}dξ_{2}dξ_{3}

⎞

⎟⎠

*N** _{}*1

^{1/2}⊗

*H*_{0,}^{3}1/2

Ψ
^{2}*,*

2.40

*B*^{α}* _{,,}*ξ3⊗

*a*

^{∗}

*ξ3dξ3Ψ*

_{}^{2}

≤

⎛

⎜⎝

Σ1×Σ1×Σ2

*G*^{α}_{,,}_{}*ξ*_{1}*, ξ*_{2}*, ξ*_{3}^{2}

*w*^{3}ξ3 dξ_{1}dξ_{2}dξ_{3}

⎞

⎟⎠

N1^{1/2}⊗

*H*_{0,}^{3}_{1/2}
Ψ

^{2}

Σ1×Σ1×Σ2

G^{α}* _{,,}*ξ1

*, ξ*

_{2}

*, ξ*

_{3}

^{2}dξ

_{1}dξ

_{2}dξ

_{3}

*η*N1

^{1/2}⊗1Ψ

^{2}1 4ηΨ

^{2}

2.41

*for every*Ψ∈ DH0*and everyη >0.*

*Proof. Suppose that*Ψ∈ DN_{}^{1/2}⊗DH_{0,}^{3}^{1/2}. Let
Ψξ3 *w*^{3}ξ3^{1/2}

N1^{1/2}⊗*a*ξ3

Φ. 2.42

We have

*B*^{α}* _{,,}*ξ3

_{∗}

⊗*a*ξ3dξ3Ψ

Σ2

1

*w*^{3}ξ3_{1/2}

*B*_{,,}^{α} ξ3_{∗}

N1^{−1/2}⊗1

Ψξ3dξ3*.*
2.43
Therefore, forΨ∈ DN^{1/2}* _{}* ⊗DH

_{0,}

^{3}

^{1/2},2.40follows fromProposition 2.4.

We now have

*B*_{,,}^{α} ξ3⊗*a*^{∗}* _{}*ξ3Ψdξ3

^{2}
F

*B*_{,,}^{α} _{}ξ3⊗*a*_{}*ξ*_{3}^{}

Ψ, B^{α}_{,,}

*ξ*^{}_{3}

⊗*a** _{}*ξ3Ψ

dξ_{3}dξ^{}_{3}

*B*_{,,}^{}^{α}^{} _{}ξ3⊗1

Ψ^{2}dξ_{3}*,*
2.44

Σ2×Σ2

*B*^{α}* _{,,}*ξ3⊗

*a*

*ξ*

^{}

_{3}

Ψ, B_{,,}^{α}

*ξ*_{3}^{}

⊗*a*ξ3Ψ
dξ_{3}dξ^{}_{3}

Σ2×Σ2

1
*w*^{3}ξ3^{1/2}*w*^{3}

*ξ*_{3}^{}_{1/2}

×

*B*^{α}* _{,,}*ξ3N1

^{−1/2}⊗1 Ψ

*ξ*_{3}^{}
*,*

*B*_{,,}^{α}

*ξ*_{3}^{}

N1^{−1/2}⊗1

Ψξ3
dξ_{3}dξ^{}_{3}

≤

Σ2

1
*w*^{3}ξ3^{1/2}

*B*^{α}_{,,}_{}*ξ*_{3}*N** _{}*1

^{−1/2}

F*L*Ψ*ξ*_{3}dξ3
2

≤

⎛

⎝

Σ1×Σ1×Σ2

*G*^{α}ξ1*, ξ*2*, ξ*3^{2}

*w*^{3}ξ3 dξ_{1}dξ_{2}dξ_{3}

⎞

⎠

N1^{1/2}⊗

*H*_{0,}^{3}_{1/2}
Ψ

^{2}*.*

2.45 Furthermore

Σ2

*B*_{,,}^{}^{α}^{} _{}ξ3⊗1
Ψ^{2}dξ_{3}

Σ2

*B*^{α}* _{,,}*ξ3N1

^{−1/2}⊗1

N1^{1/2}⊗1
Ψ^{2}dξ_{3}

≤

Σ1×Σ1×Σ2

G^{α}* _{,,}*ξ1

*, ξ*2

*, ξ*3

^{2}dξ1dξ2dξ3

*ηN*1Ψ

^{2}1 4ηΨ

^{2}

2.46

for every*η >*0.

By2.40,2.45, and2.46, we finally get2.41for everyΨ ∈ DN_{}^{1/2}⊗DH_{0,}^{3}. It
then follows that2.40and2.41are verified for everyΨ∈ DH0.

We now prove that*H*is a self-adjoint operator inFfor*g*suﬃciently small.

**Theorem 2.6. Let**g_{1}*>0 be such that*
3g_{1}^{2}

*m**W*

1

*m*^{2}_{1} 1

*α 1,2*

3
* 1*

* /*^{}

G^{α}_{,,}^{2}_{L}_{2}

Σ1×Σ1×Σ2*<*1. 2.47

*Then for everyg* *satisfyingg* ≤*g*1*,His a self-adjoint operator in*F*with domain*DH DH0,
*and*D*is a core forH.*

*Proof. Let*Ψbe inD. We have

H*I*Ψ^{2}≤12

*α 1,2*

3
* 1*

* /*^{}

"

*B*^{α}* _{,,}*ξ3

_{∗}

⊗*a** _{}*ξ3Ψdξ3

^{2}

*B*_{,,}^{α} ξ3

⊗*a*^{∗}* _{}*ξ3Ψdξ3

^{2}

#
*.*
2.48

Note that

H_{0,}^{3}Ψ≤H_{0}^{3}Ψ≤ *H*0Ψ,
NΨ ≤ 1

*m** _{}*H0,Ψ ≤ 1

*m*_{1}H0,Ψ ≤ 1

*m*_{1}H0Ψ,

2.49

where
*H*0,

*w*^{1}* _{}* ξ1b

_{,}^{∗}ξ1b

*,*ξ1dξ1

*w*^{2}* _{}* ξ2c

^{∗}

*ξ2c*

_{,}*,*ξ2dξ2

*.*2.50

We further note that
N1^{1/2}⊗

*H*_{0,}^{3}1/2

Ψ
^{2}≤ 1

2 1

*m*^{2}_{1} 1 H0Ψ^{2} *β*

2m^{2}_{1}H0Ψ^{2}
1

2 1 8β

Ψ^{2} 2.51

for*β >*0, and

*ηN*1⊗1Ψ^{2} 1

4ηΨ^{2}≤ *η*

*m*^{2}_{1}H0Ψ^{2} *ηβ*

*m*^{2}_{1}H0Ψ^{2}*η*

1 1 4β

Ψ^{2} 1
4ηΨ^{2}*.*

2.52

Combining2.48with2.40,2.41,2.51, and2.52we get for*η >*0,*β >*0

H*I*Ψ^{2} ≤6

*α 1,2*

3
* 1*

* /*^{}

G^{α}_{,,}^{2}

× 1

2m*W*

1

*m*^{2}_{1} 1 H0Ψ^{2} *β*

2m_{W}*m*^{2}_{1}H0Ψ^{2} 1
2m*W*

1 1

4β

Ψ^{2}
12

*α 1,2*

3
* 1*

* /*^{}

*G*^{α}_{,,}^{2} *η*
*m*^{2}_{1}

1*β*

H0Ψ^{2}

*η*

1 1 4β

1

4η

Ψ^{2} *,*
2.53

by noting

Σ1×Σ1×Σ2

|G*,,*^{}ξ1*, ξ*_{2}*, ξ*_{3}|^{2}

*w*^{3}ξ3 dξ_{1}dξ_{2}dξ_{3}≤ 1

*m** _{W}*G

^{α}

_{,,}^{2}

*.*2.54 By2.53the theorem follows from the Kato-Rellich theorem.

**3. Main Results**

In the sequel, we will make the following additional assumptions on the kernels*G*^{α}_{,,}_{}.
*Hypothesis 3.1.* iFor*α* 1,2, 1,2,3,*, *^{} ±,

Σ1×Σ1×Σ2

*G*^{α}* _{,,}*ξ1

*, ξ*

_{2}

*, ξ*

_{3}

^{2}

*p*_{2}^{2} dξ1dξ2dξ3*<*∞. 3.1

iiThere exists*C >*0 such that for*α* 1,2, 1,2,3,*, *^{} ±,

Σ1×{|p2|≤σ}×Σ2

*G*^{α}_{,,}_{}ξ1*, ξ*_{2}*, ξ*_{3}^{2}dξ_{1}dξ_{2}dξ_{3}

1/2

≤*Cσ*^{2}*.* 3.2

iiiFor*α* 1,2, 1,2,3,*, *^{} ±, and*i, j* 1,2,3

iii.a

Σ1×Σ1×Σ2

$*p*_{2}· ∇*p*2

*G*^{α}_{,,}_{}%

ξ1*, ξ*_{2}*, ξ*_{3}^{2}dξ_{1}dξ_{2}dξ_{3} *<*∞,
iii.b

Σ1×Σ1×Σ2

*p*_{2,i}^{2} *p*_{2,j}^{2}

*∂*^{2}*G*^{α}_{,,}

*∂p*2,i*∂p*2,jξ1*, ξ*2*, ξ*3

2

dξ1dξ2dξ3*<*∞.

3.3