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

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

Kanji Fujii and Norihito Toyota        1)Department of Physics, Faculty of Science, Hokkaido University,
Sapporo 060-0810, Japan
Hokkaido Information University, Ebetsu, Nisinopporo 59-2, Japan
fujii@particle.sci.hokudai.ac.jp, toyota@do-johodai.ac.jp
###### Abstract

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.

## 1 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 , with respect to the state generated as a neutrino-source state at a time , 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.

## 2 Basic Formulas

In quantum field theory[5], the expectation value of a physical observable at a space-time point with respect to a state is expressed, in the interaction representation, as

 ⟨Ψ(x0)|F(x)|Ψ(x0)⟩=⟨Ψ(x0I)|S−1(x0,x0I)F(x)S(x0,x0I)|Ψ(x0I)⟩, (1) S(x0,x0I)=1+∑m=1(−i)m∫x0x0Id4y1∫y01x0Id4y2⋯∫y0m−1x0Id4ymHint(y1)⋯Hint(ym). (2)

First we summarize definitions of quantities and relations which are used in order to perform considerations along the present purpose.

Total Lagrangian related to neutrinos at low-energy () is taken to be

 L0 (3) Lint =−(¯νF(x)JF(x)+¯JFνF(x))=−Hint(x). (4)

For simplicity, we consider only the charged-current weak interaction; thus the source function in (4) does not include any neutrino field. Here, represents a set of flavor-neutrino fields ; this set is related to a set of mass-eigenfields , by the unitary transformation where

 νF(x)=⎛⎜⎝νe(x)νμ(x)ντ(x),⎞⎟⎠,νM(x)=⎛⎜⎝ν1(x)ν2(x)ν3(x)⎞⎟⎠, (5)
 diag(Mdiag)=(m1,m2,m3),Z12†Z12=I,Z12=[Z12ρj]. (6)

The matrix , 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 is written as

where ; is the hadronic charged curent. (We use the same notations of ’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

 Nρ(x0) =i∫d3x:j4ρ(→x,x0):withjaρ(x)=−i¯νρ(x)γaνρ(x), (8) Nℓρ(x0) =i∫d3x:j4ℓρ(→x,x0):withjaℓρ(x)=−i¯ℓρ(x)γaℓρ(x). (9)

In terms of the momentum-helicity creation- and annihilation-operators, -field is expanded as

 νj(x)=∑→k,r1√V[αj(k,r)uj(k,r)ei(k⋅x)+β†j(k,r)vj(k,r)e−i(k⋅x)]. (10)

Here, with ; represents the helicity ; , ; and their Hermitian conjugates satisfy , In the same way, we define the number operators of the charged leptons, , and use the expansion of -field written as

 ℓρ(x)=∑→q,r1√V[aρ(q,r)uρ(q,r)ei(q⋅x)+b†ρ(q,r)vρ(q,r)e−i(q⋅x)], (11)

where with and are the annihilation operators for and , respectively.

The expectation values now investigated are

 ⟨A±(x0I)|S−1(x0,x0I)Nρ(x0)S(x0,x0I)|A±x0I)⟩, (12)

where is one - or -state which plays a role of a neutrino source. Note that

 NHρ(x0):=S−1(x0,x0I)Nρ(x0)S(x0,x0I) (13)

is a quantity in Heisenberg representation, which is taken so as to coincide with the interaction representation at a time .

For convenience, we use such notations as

 ⟨Nρ,A±;x0,x0I⟩ :=⟨A±(x0I)|NHρ(x0)|A±(x0I)⟩, (14) ⟨Nℓρ,A±;x0,x0I⟩ :=⟨A±(x0I)|NHℓρ(x0)|A±(x0I)⟩. (15)

## 3 Concrete forms of the expectation values

### 3.1 Case of Nρ expectation values

There are two kinds of the lowest order ( order) contributions; in the case of or ,

 ⟨Nρ,A+;x0,x0I⟩I := ⟨A+(x0I)|∫x0x0Id4y∫x0x0Id4zHint(z)Nρ(x0)Hint(y)|A+(x0I)⟩, (16) ⟨Nρ,A+;x0,x0I⟩II := ⟨A+(x0I)|i2∫x0x0Id4y∫y0x0Id4z[Hint(z)Hint(y)Nρ(x0) (17) + Nρ(x0)Hint(y)Hint(z)]|A+(x0I)⟩.

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 We make the vacuum approximation

which is expressed by employing the -decay constant defined by

Using , we obtain from (16), (18) and (19),

 ⟨Nρ,A+(p);x0,x0I⟩vac=[GFfA√2]2∫x0x0Idz0∫x0x0Idy0∫d→z∫d→y∫d→x12EA(p)Vei(p⋅(y−z)) (20)

Performing all spacial integrations, we see R.H.S. of (20) includes the part

 (21)

which is obtained by setting . This part is rewritten as

 R(jik,σq,p);=1ωj(k)ωi(k)Eσ(q)[(→k2+ωj(k)ωi(k)+mjmi){(p⋅p)Eσ(q)−2(q⋅p)EA(p)} +(ωj(k)+ωi(k)){−(p⋅p)→k→q+2(q⋅p)→k→p}]. (22)

Then we obtain

 ⟨Nρ,A+(p);x0,x0I⟩vac=[GFfA√2]2∫x0x0Idz0∫x0x0Idy0∑→q∑→kδ(→p,→k+→q)12EA(p)V ×∑σ∑j,iZ12σjZ12∗ρjZ12ρiZ12∗σiR(jik,σq,p)ei{x0(ωj(k)−ωi(k))+z0(EA−Eσ−ωj)−y0(EA−Eσ−ωi)}. (23)

By employing another set of integration parameters

 ty=y0−x0+x0I2,tz=z0−x0+x0I2, (24)

with their ranges for , , we rewrite the ()-integration () part in (20) as

 ∫x0x0Idz0∫x0x0Idy0expi{x0(ωj(k)−ωi(k))+z0(EA−Eσ−ωj)−y0(EA−Eσ−ωi)} =expi{(x0−x0+x0I2)(ωj−ωi)}⋅∫T/2−T/2dtz∫T/2−T/2dtyei{tz(EA−Eσ−ωj)−ty(EA−Eσ−ωi)} (25)

### 3.2 Case of Nℓσ expectation values

In the same way as the -case, the main contribution to comes from the contribution of Fig.2. We obtain

 −⟨Nℓσ,A+(p);x0,x0I⟩:=⟨A+(p,x0I)∣∣−NHℓσ(x0)∣∣A+(p,x0I)⟩≅[GFfA√2]212EA(p)V ×∫x0x0Idz0∫x0x0Idy0∑→q∑→kδ(→p,→k+→q)∑jZ12σjZ12∗σjR(jjk,σq,p)ei(z0−y0)(EA(p)−Eσ(q)−ωj(k)). (26)

The correspondance between (26) and (23) is seen clearly. Note that the reason why appears in (26) comes from the relations , .(See [4].) The concrete form of , from (22), is given by

 R(jjk,σq,p) = 2ωj(k)Eσ(q)[m2A(k(j)⋅q)+2(q⋅p)(k(j)⋅p)]; (27) (k(j)⋅q) = →k→q−ωj(k)Eσ(q). (28)

In the same way as (23) and (25), we obtain

 −⟨Nℓσ,A+(p);x0−x0I=T⟩≅[GFfA√2]212EA(p)V∑→q∑→kδ(→p,→k+→q) ×∑jZ12σjZ12∗σjR(jjk,σq,p)[sin(T(EA(p)−Eσ(q)−ωj(k))/2)(EA(p)−Eσ(q)−ωj)/2]2. (29)

### 3.3 Relation of ⟨Nℓσ,A+(p);x0−x0I=T⟩ to dacay probability

First tentatively we define the amplitude

 A(A+(p)→¯ℓσ(q,s)+νj(k,r);x0−x0I=T) :=⟨¯ℓσ(q,s)+νj(k,r);x0∣∣−i∫d→z∫x0x0Idz0Hint(z)∣∣A+(p);x0I⟩, (30)

where is the charged-current interaction with . Then, from

 A(A+(p)→¯ℓσ(q,s)+νj(k,r);x0−x0I=T)=[GFfA√2]1√2EA(p)Vδ(→p,→k+→q)∫T/2−T/2dy0Z12∗σj (31)

one can easily confirm by remembering (26)

 ∑j∑→q,s∑→k,r∣∣A(A+(p)→¯ℓσ(q,s)+νj(k,r);T)∣∣2=⟨¯nℓσ,A+(p),x0−x0I=T⟩. (32)

(Hereafter we use the notation in R.H.S of (32) instead of The concrete form of (32) is given by (29). When is so large that we may use

 [sin(T(EA(p)−Eσ(q)−ωj(k))/2)(EA(p)−Eσ(q)−ωj(k))/2]2≅2πδ(EA(p)−Eσ(q)−ωj(k))T, (33)

we can define , which may be interpreted as the decay probability per unit time (for energetically allowed , to be

 P(A+(p)→¯ℓσ(q)+νj(k)):=∑→q,s∑→k,r[∣∣A(P(A+(p)→¯ℓσ(q)+νj(k));T∣∣2/T]T→large =[GFfA√2]2|Z1/2σj|2∫d→q(2π)3⋅2πδ(EA(p)−Eσ(q)−ωj(k))2EA(p)R(jjk,σq,p)∣∣→p=→q+→k. (34)

From (32), we obtain the relation

 [⟨¯nℓσ,A+(p);x0−x0I=T⟩/T]T→large=∑jP(A+(p)→¯ℓσ(q)+νj(k)) (35)

for an energetically allowed case.

Here we have to give a remark on the physical meaning of . Under the condition , the concrete calculation of R.H.S. of (34) leads to

 P(A+(→p=0)→¯ℓσ+νj)=(GFfA)28π|Z1/2σj|2mAm2σ(1+m2jm2σ−(m2σ−m2j)2m2Am2σ) × ⎷{1−(mσ+mj)2m2A}{1−(mσ−mj)2m2A}. (36)

(See Appendix.) Taking into account the experimental smallness of (e.g. for -decay neutrino), we obtain from (36)

 P(A+(→p=0)→¯ℓσ+νj)≅(GFfA)28π|Z1/2σj|2mAm2σ(1−m2σm2A)2; (37)

thus

 ∑jP(A+(→p=0)→¯ℓσ+νj) ≅ (GFfA)28πmAm2σ(1−m2σm2A)2 (38) := P0(A+(→p=0)→¯ℓσ+ν(mass=0)).

is the same as the expression of the dacay probability, on the basis of which the unversal 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 is large enough so that the mass eigenstates ’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 ⟨¯nℓσ,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 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 for simplicity; due to (34) and (38), we obtain

 ⟨¯nℓσ,(A+(→p=0);T⟩=∑j((GFfA)√2)2|Z1/2σj|2∫d→q(2π)312EA(p)R(jjk,σq,p) (39) ×[sin(T(EA(p)−Eσ(q)−ωj(k))/2)(EA(p)−Eσ(q)−ωj)/2]2∣∣→p=→q+→k=0 deltaapprox.(???)−−−−−−−−−−−→T∑jP(A+(→p=0)→¯ℓσ(q)+νj(−→q)) ≅T⋅P0(A+(→p=0)→¯ℓσ+ν(mass=0)). (40)

As one of ways to see the difference between (39)and (40), we examine the ratio of them;

 (41)

By using (38) and (A.4) in Appendix, we obtain

 R.H.Sof(???)=2/πm2σ(1−m2σm2A)2∫∞0dkk2(Eσ(k)−k)Eσ(k)[sin(T(EA(p)−Eσ(k)−k)/2)(EA(p)−Eσ(k)−k)/2]2. (42)

As seen from (A.7), it is necessary for us to investigate the deviation of 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 -unit; or )

, and also

then, we obtain

 R(A+(→p=0)→¯ℓσ;T)=L⋅2/πa2σ(1−a2σ)2I(A+(→p=0),σ;L), (43)

where

 I(A+(→p=0),σ;L)=λAL∫∞0db⋅b2(1−b√a2σ+b2)[sin(L2λA(1−√a2σ+b2−b))(1−√a2σ+b2−b)/2]2. (44)

It is easily confirmed that, when the delta-function approximation corresponding to (33) is applied for , R.H.S of (44) goes to . (See (A8) in Appendix.) Certainly the deviation of R.H.S of (43) from gives a measure of the departure from Golden formula.

The characteristic length appearing in (44) is the Compton wave length ; for and ,

 λπ≅197.3MeV⋅10−15m139.6MeV,λK≅197.3MeV⋅10−15m493.7MeV. (45)

Thus, the sin-term includes , which is larger than for a macroscopic-scale m. Therefore, we may expect the delta-function approximation to hold well and obtain to be nearly equal to 1 for a macroscopic-scale . Through concrete numerical calculations, we can confirm this expectation.

The results of numerical calculations are shown in Fig.3. We see

 (46)

Thus we can say that the delta-function approximation (33) holds well not only in the macroscopic range of but also in shorter range.