Expectation values of flavor-neutrino numbers with respect to neutrino-source hadron states --Neutrino oscillations and decay probabilities--

On the basis of quantum field theory, we consider a unified description of various processes accompanied by neutrinos, namely weak decays and oscillation processes. The structures of the expectation values of flavor-neutrino numbers with respect to neutrino-source hadron state are investigated. Due to the smallness of neutrino masses, we naturally obtain the old (i.e. pre-mixing) formulas of decay probabilities. Together, it is shown that the oscillation formulas, similar to the usual ones, are applied irrespectively of the details of neutrino-producing processes. The derived oscillation formulas are the same in form as the usually used ones except for the oscillation length.


Introduction
In the preceding short papers [1], it has been pointed out that, in the framework of quantum field theory, the expectation values of the flavor-neutrino numbers at a time x 0 , with respect to the state |Ψ(x 0 ) > generated as a neutrino-source state at a time x 0 I (< x 0 ), are possible to give a unified approach to the neutrino oscillation and decay probabilities of neutrino-source hadrons. In the papers [1], some relations among the quantities corresponding to the decay probabilities have been given. While there was little description on the neutrino oscillation, somewhat complicated oscillation behaviors, different from the usual ones, were suggested.
The main purpose of the present report is to examine the structure of the expectation values of the flavor-neutrino numbers in question and to make clear the conditions for deriving the oscillation formulas together with the decay probabilities. The smallness of neutrino masses in comparison with energies of Mev-or higher-order leads to two oscillation parts with quite different features; the first part is related to the gross energy conservation and the second part causes the neutrino oscillation. By adding the dynamical part of neutrino-producing interaction as the third factor, the expectaion values are shown to be expressed, due to the smallness of neutrino masses, as products of these three factors. The derived neutrino-oscillation formulas are the same in form as the usual ones but have different oscillation length.
The favourable feature of the present expectation-value approach is, as noted in [1], the point that , in order to derive the decay probabilities of neutrino-source particles, we are unnecessary to bother about the problem how to define the one flavor-neutrino state [3,4]. We will give some remarks on this state, which leads to the same relations as those in the expectation value approach under the smallness condition of neutrino masses.
We first summarize the basic requirements of the field-theoretical approach adopted in [1]. We examine the structures of the expectation values and point out that, basing on the smallness of neutrino masses, we obtain the unified description of the neutrino oscillations and the decay probabilities of neutrino-source particles.

Basic Formulas
In quantum field theory [5], the expectation value of a physical observable F (x) at a space-time point x = ( x, x 0 ) with respect to a state |Ψ(x 0 ) is expressed, in the interaction representation, as S(x 0 , x 0 First we summarize definitions of quantities and relations which are used in order to perform considerations along the present purpose. Total Lagrangian L(x) = L 0 (x) + L int (x) related to neutrinos at low-energy (<< m W ) is taken to be For simplicity, we consider only the charged-current weak interaction; thus the source function J F (x) in (4) does not include any neutrino field. Here,ν F (x) represents a set of flavor-neutrino fields ν ρ (x), ρ = e, µ, τ ; this set is related to a set of mass-eigenfields ν j (x), j = 1, 2, 3, by the The matrix Z 1 2 , in anlogy with the renormalization constants, is used in accordance with the field theory of particle mixture [6]. The concrete explanation in the neutrino case is given in [4].
Concretely L int (x) is written as where is the hadronic charged curent. (We use the same notations of γ b 's and other relevant quatities as those employed in [4]. ) We examine the expectation values of the flavor-neutrino and charged-lepton numbers in the lowest order of the weak interaction. The concrete forms of these number operators in the interaction representation are In terms of the momentum-helicity creation-and annihilation-operators, ν M -field is expanded as Here, (k · x) = k x − ω j (k)x 0 with ω j (k) = k 2 + m 2 j ; r represents the helicity (r =↑, ↓); (i/ k + m j )u j (k, r) = 0, (−i/ k + m j )v j (k, r) = 0; α j , β j and their Hermitian conjugates satisfy {α j (k, r), In the same way, we define the number operators of the charged leptons, N ℓρ (x 0 ), and use the expansion of ℓ ρ (x)-field written as where (q · x) = q x − E ρ (q)x 0 with E ρ (q) = q 2 + m 2 ρ ; a e (q, r) and b e (q, r) are the annihilation operators for e − and e + , respectively.
The expectation values now investigated are where |A ± (x 0 I ) is one π ± -or K ± -state which plays a role of a neutrino source. Note that is a quantity in Heisenberg representation, which is taken so as to coincide with the interaction representation at a time x 0 I . For convenience, we use such notations as 3 Concrete forms of the expectation values

Case of N ρ expectation values
There are two kinds of the lowest order (i.e. G 2 F order) contributions; in the case of A + = π + or K + , The dominant contribution, corresponding to the diagram in Fig.1, is included in (16), as seen from the following explanation. In evaluation of (16), it is necessary for us to treat A + (p, We make the vacuum approximation which is expressed by employing the A ± -decay constant f A defined by ρj ν j (x), we obtain from (16), (18) and (19), Performing all spacial integrations, we see R.H.S. of (20) includes the part R(jik, σq, p) := s,r,r ′v which is obtained by setting k ′ = k. This part is rewritten as Then we obtain By employing another set of integration parameters

Case of N ℓσ expectation values
In the same way as the N ρ -case, the main contribution to − N ℓσ , A + (p); x 0 , x 0 I comes from the contribution of Fig.2. We obtain The correspondance between (26) and (23) is seen clearly. Note that the reason why R(jjk, σq, p) appears in (26) comes from the relations In the same way as (23) and (25), we obtain

dacay probability
First tentatively we define the amplitude where (Hereafter we use the notation in R.H.S of (32) instead of − N ℓσ , A + (p), x 0 − x 0 I = T ). The concrete form of (32) is given by (29). When T is so large that we may use we can define P (A + (p) →l σ (q) + ν j (k)), which may be interpreted as the decay probability per unit time (for energetically allowed ℓ σ ) , to be for an energetically allowed σ case.
Here we have to give a remark on the physical meaning of P A + (p) →l σ (q) + ν j (k) . Under the condition m A > m σ + m j (> 0), the concrete calculation of R.H.S. of (34) leads to (See Appendix.) Taking into account the experimental smallness of m ′ j s (e.g. m j /m e ≤ 10 −5 for β-decay neutrino), we obtain from (36) is the same as the expression of the dacay probability, on the basis of which the unversal V − A charged-current interaction (with Cabibo angle [7]) has been recognized. This situation is seen to hold also when the neutrino mixing exists. It is worth noting that, when we define the amplitude (30), we presuppose implicitly T is large enough so that the mass eigenstates |ν j 's are distinguished from each other. A related remark on the definition of one flavor-neutrino state will be given in Section 5.
3.4 Remark on behavior of n ℓσ , A + (p); T when T is not so large In Subsection 3.3, by utilizing the delta-function approximation (33), we have derived the dacay rates (36) as a kind of Golden Rule [8]. Possible deviations from this rule have been investigated by Ishikawa and Tobita [8]. In this connection as well as with aim of examining structures of N ρ , A + (p); T in the next section, we give a remark on the proper range of the approximation (33) in the following.
We apply the relation (33) to the case of the expectation value (29) with p = 0 for simplicity; due to (34) and (38), we obtain As one of ways to see the difference between (39)and (40), we examine the ratio of them; By using (38) and (A.4) in Appendix, we obtain R.H.S of (41) = 2/π As seen from (A.7), it is necessary for us to investigate the deviation of R A + ( p = 0) →l σ ; T /T from 1, since such a deviation gives us an information on the proper range of the delta-function approximation (33). For convenience, R.H.S. of (42) is expressed by employing parameters ( in the m A -unit; A = π ± or K ± ) a σ = mσ mA , b = | k| mA = k mA and also T mAc 2 = T c · mAc = L λA ; then, we obtain where It is easily confirmed that, when the delta-function approximation corresponding to (33) is applied for L >> λ A , R.H.S of (44) goes to a 2 σ (1 − a 2 σ ) 2 π/2. (See (A·8) in Appendix.) Certainly the deviation of R.H.S of (43) from L gives a measure of the departure from Golden formula.
The characteristic length appearing in (44) is the Compton wave length λ A = c m 2 A c 2 ; for π ± and K ± , Thus, the sin-term includes L/λ A , which is larger than ∼ 10 15 for a macroscopic-scale L ≥ 10 0 m. Therefore, we may expect the delta-function approximation to hold well and obtain R(A + ( p = 0),l σ ; L)/L to be nearly equal to 1 for a macroscopic-scale L. Through concrete numerical calculations, we can confirm this expectation. The results of numerical calculations are shown in Fig.3. We see Thus we can say that the delta-function approximation (33) holds well not only in the macroscopic range of L but also in shorter range.  In this section, we consider the way how to apply the delta-function approximation to N ρ , A + (p); T to derive the oscillation formula, similar to the usually used one.
With the use of the approximate forms (50), (52) and (53), the approximate expression of (47) is given by where B 0 ρ,σ (k, q, p; L) = R 0 (k, σq, p)S 0 σ (k, q, p; L) In (55), there are two kinds of L-oscillation terms, which have characteristic defferences. S 0 σ given by (50) leads to the gross energy conversation for a macroscopic-scale L = T c ( Cf. (33)), while the exponential factor, as seen from (52), gives rise to the neutrino oscillation, which is similar to but somewhat different from the ordinary oscillation formulas from the viewpoint of the oscillation length. R 0 (k, σq, p) is the dynamical part which (a) involves information on the neutrino-preparation mechanism and (b) has no (j, i)-dependence by neglecting the neutrino mass diferences in comparison with masses of the relevant hadrons. In Subsection 4.3, the structure of N ρ , A + (p); T with some external constraint on the related momenta is investigated.

Remark on
Although there is no experiment at present corresponding to this ( p = 0) case, we give a remark on a characteristic feature of N ρ , A + ( p = 0); T . Employing relation (A·3) given in Appendix, we obtain from (54) and (55) We consider the case when we can employ the delta-function approximation of S 0 σ with L in the non-microscopic range as explained in Subsection 3.4, that is, Calculations, similar to those in Appendix, lead to Here, k(A, σ) (written as k 0 for simplicity in Appendix) is the solution of The last part of (59) also includes the expression of the 2-body dacay probability (38) or (A·7). We may drop the electron-mode contribution due to (m e /m µ ) 2 ≤ 10 −5 . Then we obtain from (59) R.H.S of (61) corresponds to the oscillation formula used usually [2]; the former coincides with the latter when L in(61) is replaced by 2L.

4.3
N ρ , A + (p); L with conditional leptons It will be meaningful for us to examine the structure of (54) and (55) in high-momentum case, | p| >> m A . With this aim, we perform the momentum integration under a certain additional condition, in which q and k are limited to be nearly parallel to p. This reflects the experimental situation in such as T2K [9], where charged and neutral leptons are produced through π ± (and K ± ) decays and monitored.
In the following, we give an evaluation of (54) under the parallel condition with the energy conservation, i.e.
(62) leads to the solution forω(k) = k which reduces to (60) for p → 0. Under the same conditions (62), we obtain from (A·2) Note that, from | q| = | p| − | k| with a fixed | p|, we obtain Thus, (54) is written as Here, F A (k T , k) plays a role for realizing the parallel condition. We may take e.g.
By using (38), R.H.S. of (66) is rewritten as Because of P0(A + ( p=0)→e + +ν(mass=0) , we drop the first term in the curly bracket; we obtain the oscillation formula with the same structure as R.H.S. of (61), i.e. .

Case of 3-body decays of neutrino-source particle
It seems important for us to examine the features of the expectation value of N ρ under the situation of ν-source in reactor experiments. In this connection, we give a remark on the N ρ (x 0 )-expectation value in the case of 3-body decays of A ± -particle, as shown in Fig.5. For simplicity, A + and B 0 are taken to be scalar particles, and j had b (x) in L int (x) (7) is taken to be, alalogously to the electromagnetic current, where ϕ A (x) and ϕ B (x) are simply assumed to be complex scalar fields. We use similar notations as before; e.g. Along the same way as that in Subsection 3.1, we obtain where ×R(jik, σq, BQ, AP ) · S ABσ,ji (k, q, Q, P ; T ), (72) R(jik, σq, AQ, BP ) = R(jik, σq, p) p=P +Q , S ABσ,ji (k, q, Q, P ; T ) = sin(T ∆E ABσj /2) · sin(T ∆E ABσi /2) L.H.S of (73) comes from the fermion-loop in Fig.5. Comparing this R with the fermion-loop contribution of Fig.1, i.e. (21), we are easy to obtain (73).

Additional remarks and conclusions
1) The motivation of the present expectation-value approach [1] was to evade the trouble concerned with the definition of one-particle flavor-neutrino state [3,4]. Here we give a remark on this subject.
The definition of ρ ij (k) is given below (26). For k-values which effectively contribute to the kintegral ofB ρ,σ (k, q, p), ρ ji (k) can be set nearly equal to 1. Thus, in so far as ρi can be regarded as the one-particle state of a flavor-neutrino.
2) We examined the lowest-order contribution to N ρ , A + (p); T , corresponding to Fig.1. Contributions from other diagrams are relatively small due to suppressing factors such as v + j ( k, k 0 , s) u i (− k, k 0 , s) = i sin (χ j − χ i )/2 with tan χ j = m j /| k|. As noted in Section 4, the important feature of N ρ , A + (p); L , is the existence of two kinds of oscillation factors with qualitatively different behaviors. The one, as given by (50), is nearly (j, i)-independent due to ω j (k)/E σ (q) ∼ = | k|/E σ (q); the other is approximately given by (52) due to ω j (k) − ω i (k) ∼ = ∆m 2 ji /(2k). These two characteristic features are obtained due to the smallness of m j 's in comparison with the relevant energies with the magnitude ∼ = 1Mev or larger.
As noted at the end of Subsection 4.1, the oscillation formula (54) with (55) has the oscillation part which has the oscillation length, different from the usual one.
3) As seen from (80) as well as (86), when a certain condition is added to Q, we obtain a simple model calculation corresponding to the neutrino produced in a reactor.
It seems to be important that N ρ , A + (P ); T and N ρ , A + (P ) → B 0 ; T have the forms of (54) (with (55)) and (80) (with (81)), respectively, derived as the results from the special smallness of neutrino mass. (55) and (81) consist commonly of 3 parts; (i) the part which reflects the structure of neutrino producing interaction, (ii) the part leading to the gross energy conservation, and (iii) the neutrino-oscillation part. These characteristic features lead to the field-theoretical understanding why the quantum-mechanical oscillation formula can be applied irrespectively of the dynamical details of neutrino productions, although the part (iii) in the present approach is different from the usual formula in the point of the oscillation length.