Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-. I

We consider a Hamiltonian with cutoffs describing the weak decay of spin one 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 absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.


Introduction
In this article, we consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory (see [18,31]). 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 from [4] and [13] and are based on [3] and [25]. Some of the results of this article has been announced in [8].
For related results about models in Quantum Field Theory see [7] and [28] in the case of the Quantum Electrodynamics and [2] in the case of the weak interaction.
The paper is organized as follows. In section 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. Acknowledgments. One of us (J.-C. G) wishes to thank Laurent Amour and Benoît Grébert for helpful discussions. The authors also thank Walter Aschbacher for valuable remarks. The work was done partially while J.M.-B. was visiting the Institute for Mathematical Sciences, National University of Singapore in 2008. The visit was supported by the Institute.

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 [18,Formula (4.139)] and [31]).
The full family of leptons involves the electron e − and the positron e + , together with the associated neutrino ν e and antineutrinoν e , the muons µ − and µ + together with the associated neutrino ν µ and antineutrinoν µ and the tau leptons τ − and τ + together with the associated neutrino ν τ and antineutrinoν τ .
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 mixing [27].
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 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 [18, p159, (4.139)] and [31, p308, (21.3.20)]) (2.1) where γ α , α = 0, 1, 2, 3 and γ 5 are the Dirac matrices and Ψ . (x) and Ψ . (x) are the Dirac fields for e − , e + , ν e andν e .
We have Ψ e (x) = 1 2π Here p 0 = (|p| 2 + m 2 e ) 1 2 where m e > 0 is the mass of the electron and u(p, s) and v(p, s) are the normalized solutions to the Dirac equation (see [18,Appendix]).
The operators b e,+ (p, s) and b * e,+ (p, s) (respectively b e,− (p, s) and b * e,− (p, s)) are the annihilation and creation operators for the electrons (respectively the positrons) satisfying the anticommutation relations (see below).
The interaction (2.1) is 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 Hamiltonian (see [16]). For that sake, we have to introduce a spatial cutoff g(x) such that g ∈ L 1 (R 3 ), together with momentum cutoffs χ(p) and ρ(k) 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 whereĝ 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 [30, 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, wether the particles are of the same species or not". Thus, as it follows from section 4.2 of [30], fermionic creation and annihilation operators of different species of leptons will always anticommute.
Let S be any separable Hilbert space. Let ⊗ n a S (resp. ⊗ n s S) denote the antisymmetric (resp. symmetric) n-th tensor power of S. The fermionic (resp. bosonic) Fock space over S, denoted by F a (S) (resp. F s (S)), is the direct sum where ⊗ 0 a S = ⊗ 0 s S ≡ C. The state Ω = (1, 0, 0, . . . , 0, . . .) denotes the vacuum state in F a (S) and in F s (S).
For every ℓ, F ℓ is the fermionic Fock space for the corresponding species of leptons including the massive particle and antiparticle together with the associated neutrino and antineutrino, i.e., We have ) . Here q ℓ (resp.q ℓ ) is the number of massive particle (resp. antiparticles) and r ℓ (resp.r ℓ ) is the number of neutrinos (resp. antineutrinos). The vector Ω ℓ is the associated vacuum state. The fermionic Fock space denoted by F L for the leptons is then ℓ=1 Ω ℓ is the vacuum state. The bosonic Fock space for the vector bosons W + and W − , denoted by F W , is then We have ). Here t (resp.t) is the number of bosons W − (resp. W + ). The vector Ω W is the corresponding vacuum.
The Fock space for the weak decay of the vector bosons W + and W − , denoted by F, is thus and Ω = Ω L ⊗ Ω W is the vacuum state. For every ℓ ∈ {1, 2, 3} let D ℓ denote the set of smooth vectors ψ ℓ ∈ F ℓ for which ψ (q ℓ ,q ℓ ,r ℓ ,r ℓ ) ℓ has a compact support and ψ (q ℓ ,q ℓ ,r ℓ ,r ℓ ) ℓ = 0 for all but finitely many (q ℓ ,q ℓ , r ℓ ,r ℓ ). Let Here⊗ is the algebraic tensor product.
Let D 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 many (t,t). Let The set D is dense in F. Let A ℓ be a self-adjoint operator in F ℓ such that D ℓ is a core for A ℓ . Its extension to F L is, by definition, the closure in F L of the operator A 1 ⊗ 1 2 ⊗ 1 3 with domain D L when ℓ = 1, of the operator 1 1 ⊗ A 2 ⊗ 1 3 with domain D L when ℓ = 2, and of the operator The extension of A ℓ to F L is a self-adjoint operator for which D L is a core and it can be extended to F. The extension of A ℓ to F is, by definition, the closure in F of the operatorÃ ℓ ⊗ 1 W with domain D, whereÃ ℓ is the extension of A ℓ to F L . The extension of A ℓ to F is a self-adjoint operator for which D is a core.
Let B be a self-adjoint operator in F W for which D W is a core. The extension of the self-adjoint operator A ℓ ⊗ B is, by definition, the closure in F of the operator A 1 ⊗ 1 2 ⊗ 1 3 ⊗ B with domain D when ℓ = 1, of the operator 1 1 ⊗ A 2 ⊗ 1 3 ⊗ B with domain D when ℓ = 2, and of the operator 1 1 ⊗ 1 2 ⊗ A 3 ⊗ B with domain D when ℓ = 3. The extension of A ℓ ⊗ B to F is a self-adjoint operator for which D is a core.
The free Hamiltonian H 0 is given by where m W is the mass of the bosons W + and W − such that m W > m 3 .
The spectrum of H 0 is [0, ∞) and 0 is a simple eigenvalue with Ω as eigenvector. The set of thresholds of H 0 , denoted by T , is given by T = {p m 1 + q m 2 + r m 3 + s m W ; (p, q, r, s) ∈ N 4 and p + q + r + s ≥ 1} , and each set [t, ∞), t ∈ T , is a branch of absolutely continuous spectrum for H 0 .
The operator H the bare vacuum will not be an eigenvector of the total Hamiltonian for every g > 0 as we expect from the physics.
In what follows, we approximate the singular kernels by square integrable functions.
Thus, from now on, the kernels G (α) ℓ,ǫ,ǫ ′ are supposed to satisfy the following hypothesis .
A similar model can be written down for the weak decay of pions π − and π + (see [18, section 6.2]). Remark 2.3. The total Hamiltonian is more general than the one involved in the theory of weak interaction because, in the Standard Model, neutrinos have helicity −1/2 and antineutrinos have helicity 1/2.
In the physical case, the Fock space, denoted by F ′ , is isomorphic to F ′ L ⊗ F W , with ) . The free Hamiltonian, now denoted by H ′ 0 , is then given by and the interaction, now denoted by H ′ I , is the one obtained from H I by supposing that The total Hamiltonian, denoted by H ′ , is then given by 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 By mimicking the proof of [24, Theorem X.44], we get a closed operator, denoted by H (1) I,ℓ,ǫ,ǫ ′ , associated with the quadratic form such that it is the unique operator Again, there exists two closed operators H and (H (2) I,ℓ,ǫ,ǫ ′ ) * and such that We shall still denote H (α) is a symmetric operator defined on D.
We now want to prove that H is essentially self-adjoint on D by showing that H (α) Once again, as above, for almost every ξ 3 ∈ Σ 2 , there exists closed operators in It then follows that the operator H I with domain D is symmetric and can be written in the following form Let N ℓ denote the operator number of massive leptons ℓ in F ℓ , i.e., The operator N ℓ is a positive self-adjoint operator in F ℓ . We still denote by N ℓ its extension to F L . The set D L is a core for N ℓ . We then have The estimates (2.16) and (2.17) are examples of N τ estimates (see [16]). We give a proof for sake of completeness. We only consider B Let (2.18) By the Fubini theorem we have By (2.9), and the Cauchy-Schwarz inequality we get By the definition of b 1,+ (ξ 1 )Φ (Q) and the Cauchy-Schwarz inequality we get for every Ψ ∈ D L . Therefore we get and by (2.19) we finally obtain Since D L is a core for N 1 2 1 and B (1) Then H 0,ǫ is a self-adjoint operator in F W , and D W is a core for H 0,ǫ . We get for every Ψ ∈ D(H 0 ) and every η > 0.
We get as it follows from Proposition 2.4.
We now have We now prove that H is a self-adjoint operator in F for g sufficiently small.
Theorem 2.6. Let g 1 > 0 be such that Then for every g satisfying g ≤ g 1 , H is a self-adjoint operator in F with domain D(H) = D(H 0 ), and D is a core for H.
Proof. Let Ψ be in D. We have We further note that (2.27) for β > 0, and (2.28) Combining ( by noting By (2.29) the theorem follows from the Kato-Rellich theorem.
Our first result is devoted to the existence of a ground state for H together with the location of the spectrum of H and of its absolutely continuous spectrum when g is sufficiently small. According to Theorem 3.3 the ground state energy E = inf σ(H) is a simple eigenvalue of H and our main results are concerned with a careful study of the spectrum of H above the ground state energy. The spectral theory developed in this work is based on the conjugated operator method as described in [23], [3] and [25]. Our choice of the conjugate operator denoted by A is the second quantized dilation generator for the neutrinos.
Let a denote the following operator in L 2 (Σ 1 ) The operator a is essentially self-adjoint on C ∞ 0 (R 3 , C 2 ). Its second quantized version dΓ(a) is a self-adjoint operator in F a (L 2 (Σ 1 )). From the definition (2.4) of the space F ℓ , the following operator in F ℓ is essentially self-adjoint on D L .
Let now A be the following operator in F L Then A is essentially self-adjoint on D L . We shall denote again by A its extension to F. Thus A is essentially self-adjoint on D and we still denote by A its closure.
For every s > 1/2 and ϕ, ψ in F, the limits exist uniformly for λ in any compact subset of (inf σ(H), as t → ∞.
The proof of Theorem 3.4 is based on a positive commutator estimate, called the Mourre estimate and on a regularity property of H with respect to A (see [23], [3] and [25]). According to [13], the main ingredient of the proof are auxiliary operators associated with infrared cutoff Hamiltonians with respect to the momenta of the neutrinos that we now introduce.
We then get Proposition 3.5. Suppose that the kernels G (α) ℓ,ǫ,ǫ ′ satisfy Hypothesis 2.1, Hypothesis 3.1(i) and 3.1(iv). Then there exists 0 <g δ ≤ g (1) δ such that, for g ≤g δ and n ≥ 1, E n is a simple eigenvalue of H n and H n does not have spectrum in ( E n , E n + (1 − 3gD γ )σ n ). The proof of Proposition 3.5 is given in Appendix A. We now introduce the positive commutator estimates and the regularity property of H with respect to A in order to prove Theorem 3.4 The operator A has to be split into two pieces depending on σ. Let η σ (p 2 ) = χ 2σ (p 2 ) , η σ (p 2 ) = χ 2σ (p 2 ) , a σ = η σ (p 2 ) a η σ (p 2 ) , a σ = η σ (p 2 ) a η σ (p 2 ) .
Note that we also have The operators a, a σ and a σ are essentially self-adjoint on C ∞ 0 (R 3 , C 2 ) (see [3, Proposition 4.2.3]). We still denote by a, a σ and a σ their closures. Ifã denotes any of the operator a, a σ and a σ , we have The operators dΓ(a), dΓ(a σ ), dΓ(a σ ) are self-adjoint operators in F a (L 2 (Σ 1 )) and we have dΓ(a) = dΓ(a σ ) + dΓ(a σ ) . By (2.4), the following operators in F ℓ , denoted by A σ ℓ and A σℓ respectively, are essentially self-adjoint on D ℓ .
Let A σ and A σ be the following two operators in F L , The operators A σ and A σ are essentially self-adjoint on D L . Still denoting by A σ and A σ their extensions to F, A σ and A σ are essentially self-adjoint on D and we still denote by A σ and A σ their closures.
We have The operators a, a σ and a σ are associated to the following C ∞ -vector fields in (3.11) Let V(p) be any of these vector fields. We have for some Γ > 0 and we also have where theṽ's are defined by (3.11) and (3.12), and fulfill |p| α d α d|p| αṽ (|p|) bounded for α = 0, 1, 2.
Let U (t), U σ (t) and U σ (t) be the corresponding one-parameter groups of unitary operators in L 2 (Σ 1 ). The operators a, a σ , and a σ are the generators of U (t), U σ (t) and U σ (t) respectively, i.e., ℓ (ξ 2 )) ℓ=1,2,3 and dΓ(w (2) Let V (t) be any of the one-parameter groups U (t), U σ (t) and U σ (t). We set Here ψ t is the flow associated to V (t).
Let N be the smallest integer such that N γ ≥ 1.
Set a n = a σn , a n = a σn , We further note that (3.28) a nχσn (p 2 ) = a n .  (3.29) P n [H n , iA n ]P n = 0 .

Existence of a ground state and location of the absolutely continuous spectrum
We now prove Theorem 3.3. The scheme of the proof is quite well known (see [5], [20]). It follows from Proposition 3.5 that H n has an unique ground state, denoted by φ n , in F n , H n φ n = E n φ n , φ n ∈ D(H n ), φ n = 1, n ≥ 1 .
Therefore H n has an unique normalized ground state in F, given byφ n = φ n ⊗ Ω n , where Ω n is the vacuum state in F n , H nφn = E nφ n ,φ n ∈ D(H n ), φ n = 1, n ≥ 1 .
Since φ n = 1, there exists a subsequence (n k ) k≥1 , converging to ∞ such that (φ n k ) k≥1 converges weakly to a stateφ ∈ F. We have to prove thatφ = 0. By adapting the proof of Theorem 4.1 in [2] (see also [7]), the key point is to estimate c ℓ,ǫ (ξ 2 )Φ n F in order to show that uniformly with respect to n. The estimate (4.1) is a consequence of the so-called "pull-through" formula as it follows.

Proof of the Mourre Inequality
We first prove Proposition 3.9. In view of Proposition 3.8(a) (iii) and (3.22), we have, as sesquilinear forms, ℓ ) be the Fock space for the massive leptons ℓ (respectively the neutrinos and antineutrinos ℓ).
We have We have F (1) is the Fock space for the massive leptons and the bosons W ± , and F (2) is the Fock space for the neutrinos and antineutrinos.
We have, as sesquilinear forms and with respect to (5.2), and where 1 j is the identity operator in F (j) . By mimicking the proofs of Proposition 2.4 and 2.5, we get, for every ψ ∈ D, Noting that |(a η σ )(p 2 )| ≤ C uniformly with respect to σ, it follows from hypothesis 2.1 and 3.1 that there exists a constant C(G) > 0 such that This yields Combining (5.1), (5.3) with (5.4), we obtain 0 n .
By (3.24), (3.26) and (5.6) we get δ . This, together with (5.5), yields for g ≤ g δ . Proposition 3.9 is proved by settingg The proof of Theorem 3.10 is the consequence of the following two lemmata.
We now prove Theorem 3.10.
Proof. It follows from Proposition 3.9 that for n ≥ 1 and g ≤g (1) δ .
This yields for some constantC > 0 and for g ≤ inf(g (2) ,g δ ). Multiplying both sides of (5.18) with E ∆n (H − E) we then get
This yields It is easy to prove that By Proposition 3.8(b)(i) and (6.1) we finally get, for g ≤ g 1 sup 0<|t|≤1 ad At ϕ(H) < ∞ .
Recall that e −ita is an one parameter group of unitary operators in L 2 (Σ 1 ×Σ 1 ×Σ 2 ). Combining Hypothesis 3.1(iii.a) and (iii.b), with (6.8)-(6.12) we finally get This yields Let V (p 2 ) denote any of the two C ∞ -vector fields v σ (p 2 ) and v σ (p 2 ) and letã denote the corresponding a σ and a σ operators. We get Combining the properties of the C ∞ fields v σ (p 2 ) and v σ (p 2 ) together with Hypothesis 2.1 and 3.1 we get, from (6.13) and by mimicking the proof of (6.14), Similarly, by mimicking the proof of (6.15), we easily get, for g ≤ g 1 , This concludes the proof of Proposition 6.2 We now prove Theorem 3.7. Proof of Theorem 3.7. In view of [3, Lemma 6.2.3] (see also [13,Proposition 28]), the proof of Theorem 3.7 will follow from Proposition 6.1 and the following estimates for every ϕ ∈ C ∞ 0 ((−∞, m 1 − δ/2)) and for g ≤ g 1 . Let us prove (6.16). The inequalities (6.17)-(6.19) can be proved similarly. To this end, let φ be an almost analytic extension of ϕ satisfying It follows that We note that We also have (6.21) Therefore, combining Proposition 3.8 (b)(i) and (6.20) we obtain Inequality (6.22) together with (6.21) yields (6.16), and H is locally of class C 2 (A) on (−∞, m 1 − δ/2) for g ≤ g 1 .
In a similar way it follows from Proposition 3.8(b), Proposition 6.1 and Proposition 6.2 that H is locally of class C 2 (A σ ) and C 2 (A σ ) in (−∞, m 1 − δ/2) and that H σ is locally of class C 2 (A σ ) in (−∞, m 1 − δ/2), for g ≤ g 1 . This ends the proof of Theorem 3.7.
Appendix A.
In this appendix, we will prove Proposition 3.5. We apply the method developed in [4] because every infrared cutoff Hamiltonian that one considers has a ground state energy which is a simple eigenvalue.
The operators H n + andH n + are self-adjoint operators in F n and F n+1 respectively. Here 1 n and 1 n+1 n are the identity operators in F n and F n+1 In view of (A.3) and (A.6) it follows from (A.7) that Recall that for n ≥ 0, (A.11) σ n+1 < m 1 .
where g ≤ g (2) δ . This yields and by (A.15), we obtain We now prove Proposition 3.5 by induction in n ∈ N * . Suppose that E n is a simple isolated eigenvalue of H n such that inf σ(H n + ) \ {0} ≥ (1 − 3gD γ )σ n , n ≥ 1 .