Open Access Library Journal
Vol.06 No.05(2019), Article ID:92571,32 pages
10.4236/oalib.1105352
Spectral Theory for the Weak Decay of Muons in a Uniform Magnetic Field
Jean-Claude Guillot
CMAP, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau, France
Copyright © 2019 by author(s) and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 26, 2019; Accepted: May 20, 2019; Published: May 23, 2019
ABSTRACT
In this article we consider a mathematical model for the weak decay of muons in a uniform magnetic field according to the Fermi theory of weak interactions with V-A coupling. With this model we associate a Hamiltonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is self-adjoint and has a unique ground state. We specify the essential spectrum and prove the existence of asymptotic fields from which we determine the absolutely continuous spectrum. The coupling constant is supposed sufficiently small.
Subject Areas:
Mathematical Analysis, Particle Physics
Keywords:
Weak Decay of Muons, Fermi’s Theory, Uniform Magnetic Field, Spectral Theory
1. Introduction
In this paper we consider a mathematical model for the weak decay of muons into electrons, neutrinos and antineutrinos in a uniform magnetic field according to the Fermi theory with V-A (Vector-Axial Vector) coupling,
(1.1)
(1.2)
(1.2) is the charge conjugation of (1.1).
This is a part of a program devoted to the study of mathematical models for the weak interactions as patterned according to the Fermi theory and the Standard model in Quantum Field Theory. See [1] .
In this paper we restrict ourselves to the study of the decay of the muon
whose electric charge is the charge of the electron (1.1). The study of the decay of the antiparticle
, whose charge is positive, (1.2) is quite similar and we omit it.
In [2] we have studied the spectral theory of the Hamiltonian associated with the inverse
decay in a uniform magnetic field. We proved the existence and uniqueness of a ground state and we specify the essential spectrum and the spectrum for a small coupling constant and without any low-energy regularization.
In this paper we consider the weak decay of muons into electrons, neutrinos associated with muons and antineutrinos associated with electrons in a uniform magnetic field according to the Fermi theory with V-A coupling. Hence we neglect the small mass of neutrinos and antineutrinos and we define a total Hamiltonian H acting in an appropriate Fock space involving three fermionic massive particles―the electrons, the muons and the antimuons―and two fermionic massless particles―the neutrinos and the antineutrinos associated with the muons and the electrons respectively. In order to obtain a well-defined operator, we approximate the physical kernels of the interaction Hamiltonian by square integrable functions and we introduce high-energy cutoffs. We do not need to impose any low-energy regularization in this work but the coupling constant is supposed sufficiently small.
We give a precise definition of the Hamiltonian as a self-adjoint operator in the appropriate Fock space and by adapting the methods used in [2] we first state that H has a unique ground state and we specify the essential spectrum for sufficiently small values of the coupling constant.
In this paper, our main result is the location of the absolutely continuous spectrum of H. For that we follow the first step of the approach to scattering theory in establishing, for each involved particle, the existence and basic properties of the asymptotic creation and annihilation operators for time t going to
. We then have a natural definition of unitary wave operators with the right intertwining property from which we deduce the absolutely continuous spectrum of H. Scattering theory for models in Quantum Field Theory without any external field has been considered by many authors. See, among others, [3] - [19] and references therein. A part of the techniques used in this paper is adapted from the ones developed in these references. Note that the asymptotic completeness of the wave operators is an open problem in the case of the weak interactions in the background of a uniform magnetic field. See [20] for a study of scattering theory for a mathematical model of the weak interactions without any external field.
In some parts of our presentation we will only give the statement of theorems referring otherwise to some references.
The paper is organized as follows. In the second section we define the regularized self-adjoint Hamiltonian associated to (1.1). In the third section we consider the existence of a unique ground state and we specify the essential spectrum of H. In the fourth section we carefully prove the existence of asymptotic limits, when time t goes to
, of the creation and annihilation operators of each involved particle, we define a unitary wave operator and we prove that it satisfies the right intertwining property with the Hamiltonian and we deduce the absolutely continuous spectrum of H. In Appendices A and B we recall the Dirac quantized fields associated to the muon and the electron in a uniform magnetic external field together with the Dirac quantized free fields associated to the neutrino and the antineutrino.
2. The Hamiltonian
In the Fermi theory the decay of the muon
is described by the following four fermions effective Hamiltonian for the interaction in the Schrödinger representation (see [1] , [21] and [22] ):
(2.1)
here
and
are the Dirac matrices in the standard representation.
and
are the quantized Dirac fields for
and
.
.
is the Fermi coupling constant with
. See [23] .
We recall that
.
and
are massless particles.
2.1. The Free Hamiltonian
Throughout this work notations are introduced in appendices A and B.
Let
(2.2)
Let
(2.3)
Let
(resp.
,, and
) be the Dirac Hamiltonian for the electron (resp. the muon, the antimuon and the neutrino).
The quantization of
, denoted by
and acting on
, is given by
(2.4)
Likewise the quantization of
,, and
, denoted by
, and
respectively, acting on
, and
respectively, is given by
(2.5)
We set
.
is defined on
.
For each Fock space
let
denote the set of vectors
for which each component
is smooth and has a compact support and
for all but finitely many r. Then
is well-defined on the dense subset
and it is essentially self-adjoint on
. The self-adjoint extension will be denoted by the same symbol
with domain
).
The spectrum of
in
is given by
(2.6)
is a simple eigenvalue whose the associated eigenvector is the vacuum in
denoted by
.
is the absolutely continuous spectrum of
.
Likewise the spectra of
, and
in
, and
respectively are given by
(2.7)
, and
are the associated vacua in
, and
respectively and are the associated eigenvectors of
, and
respectively for the eigenvalue
.
The vacuum in
, denoted by
, is then given by
(2.8)
The free Hamiltonian for the model, denoted by
and acting in
, is now given by
(2.9)
is essentially self-adjoint on
.
Here
is the algebraic tensor product.
and
is the eigenvector associated with the simple eigenvalue
of
.
Let
be the set of the thresholds of
:
with
.
Likewise let
be the set of the thresholds of
:
with
.
Then
(2.10)
is the set of the thresholds of
.
Throughout this work any finite tensor product of annihilation or creation operators associated with the involved particles will be denoted for shortness by the usual product of the operators (see e.g. (2.13) and (2.14)).
2.2. The Interaction
Similarly to [2] [24] - [29] in order to get well-defined operators on
, we have to substitute smoother kernels
and
for the δ-distribution associated with (2.1) (conservation of momenta) and for introducing ultraviolet cutoffs.
Let
(2.11)
We get a new operator denoted by
and defined as follows
(2.12)
here
(2.13)
and
(2.14)
describes the decay of the muon and
is responsible for the fact that the bare vacuum will not be an eigenvector of the total Hamiltonian as expected from physics.
We now introduce the following assumptions on the kernels
and
in order to get well-defined Hamiltonians in
.
Hypothesis 2.1
(2.15)
These assumptions will be needed throughout the paper.
By (2.12)-(2.15)
is well defined as a sesquilinear form on
and one can construct a closed operator associated with this form.
The total Hamiltonian is thus
(2.16)
g is the coupling constant that we suppose non-negative for simplicity. The conclusions below are not affected if
.
The self-adjointness of H is established by the next theorem.
Let
(2.17)
For
we have
(2.18)
(2.18) follows from standard estimates of creation and annihilation operators in Fock space (the
estimates, see [30] ). Details can be found in ( [31] , proposition 3.7).
Theorem 2.2 (Self-adjointness). Let
be such that
(2.19)
Then for any g such that
H is self-adjoint in
with domain
. Moreover any core for
is a core for H.
By (2.18) and (2.19) the proof of the self-adjointness of H follows from the Kato-Rellich theorem.
stands for the spectrum and
denotes the essential spectrum. We have
Theorem 2.3 (The essential spectrum and the spectrum) Setting
we have for every
with
.
In order to prove the theorem 2.3 we easily adapt to our case the proof given in [29] (see also [2] , [32] and [33] ). The mathematical model considered in [29] involves also one neutrino and one antineutrino. We omit the details.
3. Existence of a Unique Ground State
In the sequel we shall make some of the following additional assumptions on the kernels
and
.
Hypothesis 3.1 There exists a constant
such that for
1)
2)
3)
4)
We then have
Theorem 3.2 Assume that the kernels
and
satisfy Hypothesis 2.1 and 3.1. Then there exists
such that H has a unique ground state for
.
In order to prove theorem 3.1 it suffices to mimic the proofs given in [2] [25] and [29] . We omit the details.
In [34] fermionic Hamiltonian models are considered without any external field. Without any restriction on the strength of the interaction a self-adjoint Hamiltonian is defined for which the existence of a ground state is proved. Such a result is an open problem in the case of magnetic fermionic models.
4. The Absolutely Continuous Spectrum
As stated in the introduction, in order to specify the absolutely continuous spectrum of H, we follow the first step of the approach to scattering theory in establishing, for each involved particle, the existence and basic properties of the asymptotic creation and annihilation operators for time t going to
. The existence of a ground state is quite fundamental in order to get a Fock subrepresentation of the asymptotic canonical anticommutation relations from which we localize the absolutely continuous spectrum of H.
4.1. Asymptotic Fields
Let
(4.1)
where, for
, and, for
,.
The strong limits of
when the time t goes to
for models in Quantum Field Theory have been considered for fermions and bosons by [14] [15] [16] and [8] [9] [10] [11] [12] and, more recently, by [3] [5] [7] [17] [18] and [19] and references therein.
In the sequel we shall make some of the following additional assumptions on the kernels
and
.
Hypothesis 4.1
1)
2)
Hypothesis 4.2
1)
2)
We then have
Theorem 4.3 Suppose Hypothesis 2.1-Hypothesis 4.2 and
. Let
and
. Then the following asymptotic fields
(4.2)
exist.
Proof. The norms of the
’s are uniformly bounded with respect to t. Hence, in order to prove theorem 4.1 it suffices to prove the existence of the strong limits on
with smooth
.
Strong limits of
and
.
Let
(4.3)
Let
. According to [8] (lemma 1) we have
(4.4)
Moreover we have
(4.5)
where
.
Let us first prove the existence of
.
Let
and
. By (4.4), (4.5) and the strong differentiability of
we get
(4.6)
By using the usual canonical anticommutation relations (CAR) (see (A.4)) we easily get for all
(4.7)
(4.8)
(4.9)
where
.
Similarly we get
(4.10)
with
(4.11)
and
(4.12)
(4.13)
By (4.6) and (4.10), in order to prove the existence of
, we have to estimate
and
for large
.
By (B.5), the
estimates (see [30] and [31] , Proposition 3.7), (A.8), (A.11) and (A.13) we get
(4.14)
and
(4.15)
By (2.18) and (2.19) we have
(4.16)
with
and
Hence we obtain
(4.17)
with
Therefore we have
(4.18)
where
is the mass of the muon.
Hence we get
(4.19)
Moreover we have
(4.20)
where
are the four components of the vectors (A.8) and (A.11)
.
Note that
(4.21)
By (4.20) and (4.21), by a two-fold partial integration with respect to
and by Hypothesis 4.1 one can show that there exits for every j a function, denoted by
, such that
(4.22)
Here
is the characteristic function of the support of
and (A.13) is used.
By (4.6) and (4.19)-(4.22) the strong limits of
on
when t goes to
and for all
exist for every
.
By (4.11)-(4.13) and by mimicking the proof of (4.14) and (4.15) we get
(4.23)
It follows from (4.10) and (4.20)-(4.23) that the strong limits of
exist when t goes to
, for all
and for every
.
We now consider the existence of
Let
and
with
. By (4.4), (4.5) and the strong differentiability of
we get
(4.24)
with
(4.25)
and
(4.26)
Similarly we obtain
(4.27)
It follows from (4.16)-(4.18) that
(4.28)
Hence
(4.29)
Moreover we have
(4.30)
where
are the four components of the vectors (A.8) and (A.11)
for
.
By (4.30), by a two-fold partial integration with respect to
and by Hypothesis 4.1 one can show that there exits for every j a function, denoted by
, such that
(4.31)
here
is the characteristic function of the support of
and (A.13) is used.
Similarly we have
(4.32)
with
(4.33)
and
(4.34)
Similarly we obtain
(4.35)
It follows from (4.29), (4.31) and (4.35) that the strong limits of
exist when t goes to
, for all
and for every
.
Let us now consider the strong limits of
.
We have for all
(4.36)
with
(4.37)
(4.38)
By mimicking the proofs given above we get
(4.39)
and
(4.40)
where
are the four components of the vectors (A.14)-(A.16)
for
.
By (4.40), by a two-fold partial integration with respect to
and by Hypothesis 4.1 one can show that there exists for every j a function, denoted by
, such that
(4.41)
Here
is the characteristic function of the support of
and (A.17) is used.
It follows from (4.36), (4.39)-(4.41) that the strong limits of
exist when t goes to
, for all
and for every
.
We now have for all
(4.42)
with
(4.43)
(4.44)
Similarly to (4.39) we get
(4.45)
It follows from (4.43), (4.45), (4.40) and (4.41) that the strong limits of
exist when t goes to
, for all
and for every
.
Strong limits of
and
.
Let
(4.46)
Let
. According to ( [8] , lemma1) we have
(4.47)
Moreover we have
(4.48)
where
.
Let
and
where
. By (4.4), (4.5) and the strong differentiability of
we get
(4.49)
By using the usual anticommutation relations (CAR) (see (A.4) and (B.4)) we easily get for all
(4.50)
(4.51)
and
(4.52)
By (B.5) we get
(4.53)
and
(4.54)
Moreover we have
(4.55)
where
are the four components of the vector (12)
.
By a two-fold partial integration with respect to
and
and by Hypothesis 4.2 one can show that there exit for every j a function, denoted by
, such that
(4.56)
Here
is the characteristic function of the support of
.
By the
estimates and by (4.18), (A.13), (A.17) and (B.14) it follows from (4.52)-(4.56) that, for every
,
(4.57)
Furthermore we have
(4.58)
with
(4.59)
(4.60)
and
(4.61)
By adapting the proof of (4.53)-(4.57) to (4.58)-(4.61) we obtain
(4.62)
here
is the characteristic function of the support of
.
It follows from (4.49), (4.47), (4.58) and (4.62) that the strong limits of
exist when t goes to
, for all
and for every
.
(4.63)
By using the usual canonical anticommutation relations (CAR) (see (A.4) and (B.4)) we easily get for all
(4.64)
(4.65)
and
(4.66)
By (B.5) we get
(4.67)
where
is the scalar product in
.
And
(4.68)
By adapting the proof of (4.57) to (4.67) and (4.68) one can show that there exists for every j a function, denoted by
, such that
(4.69)
with
Here
is the characteristic function of the support of
.
Similarly we have
(4.70)
with
(4.71)
(4.72)
and
(4.73)
By (B.5) we get
(4.74)
where
is the scalar product in
.
And
(4.75)
By adapting the proof of (4.57) and (4.67) to (4.74) and (4.75) one gets
(4.76)
It follows from (4.63), (4.69), (4.70) and (4.76) that the strong limits of
exist when t goes to
, for all
and for every
.
This concludes the proof of theorem 4.3.
4.2. Existence of a Fock Space Subrepresentation of the Asymptotic CAR
From now on we only consider the case where the time t goes to
. The following proposition is an easy consequence of theorem 4.1.
Proposition 4.4
Suppose Hypothesis 2.1-Hypothesis 4.2 and
. We have
1) Let
and
. The following anticommutation relations hold in the sense of quadratic form.
Here
.
2)
and the following pull trough formulae are satisfied:
3)
Here
is the ground state of H.
Our main result is the following theorem
Theorem 4.5 Suppose Hypothesis 2.1-Hypothesis 4.2 and
. Then we have
Proof. By (2.2) we have, for all sets of integers
in
,
(4.77)
with
(4.78)
Here p is the number of electrons, q (resp.
) is the number of muons (resp. antimuons), r is the number of antineutrinos
and s is the number of neutrinos
.
Let
, and
be tree orthonormal basis of
. Let
and
be two orthonormal basis of
.
Consider the following vectors of
(4.79)
The indices are assumed ordered,
,,, and
.
The set, for
given in
,
is a dense domain in
. The set of vectors of the form (4.79) is an orthonormal basis of
(see [35] , Chapter 10). Hence the vectors obtained in this way for
form an orthonormal basis of
and the set
is a dense domain in
.
On the other hand we now introduce the following vectors of
(4.80)
Let
denote the closed linear hull of vectors of the form (4.80). It follows from proposition 4.4 that the set of vectors of the form (4.80) is an orthonormal basis of
.
The set, for
given in
,
is a dense domain in
.
The asymptotic outgoing Fock pace denoted by
is then defined by
(4.81)
The vectors of the form (4.80) obtained for
form an orthonormal basis of
and the set
is a dense domain in
.
We now introduce the following linear operators, denoted by
, and defined on
by
(4.82)
can be uniquely extended to linear operators from
to
. It then follows from prposition 4.4. that the operators
can be uniquely extended to unitary operators from
to
Let
(4.83)
Hence
is a unitary operator from
to
.
The operators
,,,,,,,, and
defined on
generate a Fock representation of the ACR (see Proposition 4.4 1)).
By proposition 4.4 2) we have
(4.84)
Hence
leaves
invariant and H is both reduced by
and
. Thus
In view of (4.5), (4.48) and (4.84) we get
(4.85)
This yields
(4.86)
Hence the reduction of H to
is unitarily equivalent to
. Thus
. This concludes the proof of theorem 4.5.
Acknowledgements
J.-C.G. acknowledges J.-M Barbaroux, J. Faupin and G. Hachem for helpful discussions.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Appendix
Appendix A. The Dirac Quantized Fields for the Electrons and the Muons
The appendices are based on the section 2 and section 3 of [2] . See also [36] [37] [38] and [39] .
are quantum variables of the electrons, the positrons, the muons and the antimuons in a uniform magnetic field. Here
,,,.
Let
be the quantum variables of a electron and let
be the quantum variables of a muon and of an antimuon.
We set
for the configuration space for the electrons, the muons and the antimuons.
is the Hilbert space associated to each species of fermions.
(A.1)
Let
and
denote the Fock spaces for the electrons and the muons respectively. Remark that
is also the Fock space for the antimuons.
We have
(A.2)
is the antisymmetric n-th tensor power of
.
is the vacuum state in
for
.
We shall use the notations
(A.3)
(resp.
is the annihilation (resp. creation) operator for the electron.
Let
.
(resp.
) are the annihilation (resp. creation) operators for the muon when
and for the antimuon when
.
The operators
,, and
fulfil the usual anticommutation relations (CAR)(see [2] and [40] ). Therefore the following anticommutation relations hold
(A.4)
where
and
or
.
In addition, following the convention described in ( [40] , Section 4.1) and ( [40] , Section 4.2), we assume that the fermionic creation and annihilation operators of different species of particles anticommute (see [24] arXiv for explicit definitions). In our case this property will be verified by the creation and annihilation operators for the electrons, the muons, the antimuons, the antineutrinos associated with the electron and the neutrinos associated with the muons..
Recall that for
, the operators
(A.5)
are bounded operators on
and
respectively satisfying
(A.6)
Set
. The Dirac quantized fields for the electron and the muon, denoted by
, are given by
(A.7)
where
and
. See [41] .
Throughout this work e will be the positive unit of charge taken to be equal to the proton charge.
For
we have
.
For
and
is given by
(A.8)
where
(A.9)
Here
is the Hermite polynomial of order n and we set
(A.10)
For
and
we set
For
and
is given by
(A.11)
,, is given by
(A.12)
Note that
(A.13)
where
is the adjoint in
.
In order to study the spectral theory of our Hamiltonian it is not necessary to know
in (6). We have to know
explicitly.
For
with
we have
(A.14)
For
and
is given by
(A.15)
and for
we set
For
and
is given by
(A.16)
Note that
(A.17)
where
is the adjoint in
.
Appendix B. The Dirac Quantized Fields for
and
We suppose that neutrinos and antineutrinos are massless as in the Standard Model.
The quantum variables of the neutrinos and antineutrinos are the momenta and the helicities.
Let
be the generators of space-translations.
is the helicity operator
where
and
with for
(B.1)
The helicity of the neutrino associated with the muon is
.
is left-handed. The helicity of the antineutrino associated with the electron is
.
is right-handed.
Let
be the quantum variables of the antineutrino
where
is the momentum and
is the helicity. Let
be the quantum variables of the neutrino
where
is the momentum and
is the helicity.
is the Hilbert space of the states of the neutrinos
and of the antineutrinos
. Let
and
denote the Fock spaces for the neutrinos and the antineutrinos respectively.
We have
(B.2)
is the vacuum state in
for
.
In the sequel we shall use the notations
(B.3)
and
) are the annihilation and creation operators for the antineutrino associated with the electron respectively.
and
) are the annihilation and creation operators for the neutrino associated with the muon respectively. The operators
and
, fulfil the usual anticommutation relations (CAR) and they anticommute with
and
according to the convention described in ( [40] , Section 4.1). See [24] arXiv for explicit definitions.
Therefore the following anticommutation relations hold
(B.4)
Recall that, for
, the operators
(B.5)
are bounded operators on
and
respectively satisfying
(B.6)
The Dirac quantized fields for the neutrinos and antineutrinos associated with the electron and the muon respectively are denoted by
and
.
We have
(B.7)
and
(B.8)
where
and
with, for
,
(B.9)
See [5] .
For the purpose of this paper one only needs to know
and
explicitly.
and
are given in [2] .
By ( [2] , (3.6), (3.7), (3.24), (3.32)) and [35] we have
(B.10)
with
(B.11)
and for
we set
Moreover we have
(B.12)
with
(B.13)
and for
we set
Note that
(B.14)
Cite this paper
Guillot, J.-C. (2019) Spectral Theory for the Weak Decay of Muons in a Uniform Magnetic Field. Open Access Library Journal, 6: e5352. https://doi.org/10.4236/oalib.1105352
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