Mixing Transformations and Neutrino Oscillations

in Quantum Field Theory

Massimo Blasone

Blackett Laboratory, Imperial College, Prince Consort Road,

London SW7 2BZ, U.K.

and

Dipartimento di Fisica dell’Universita di Salerno

I-84100 Salerno, Italy

Contents

Introduction 2

1. Mixing transformations in Quantum Field Theory 4

1.1 Fermion mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Boson mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 The current structure for field mixing . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Generalization of mixing transformations . . . . . . . . . . . . . . . . . . . . . . . 16

2. Neutrino Oscillations 18

2.1 The usual picture for neutrino oscillations (Pontecorvo) . . . . . . . . . . . . . . . 18

2.2 Neutrino oscillations in QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Green’s functions for mixed fermions . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 The exact formula for neutrino oscillations . . . . . . . . . . . . . . . . . . 23

2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3. Berry Phase for Oscillating Neutrinos 27

Appendix A: Three Flavor Mixing 32

Appendix B: The oscillation formula for mixed bosons 35

1

Introduction

Neutrino oscillations[1, 2, 3] are definitely one of the most important and fascinating topics in

modern Particle Physics. This is especially true after the recent experimental results[4], which

seems to give a first positive evidence for the occurrence of this phenomenon. If confirmed

indeed, these results would represent the first evidence for physics beyond the Standard Model.

However, there are still many unanswered questions about the physics of oscillating neutrinos,

in particular from a theoretical point of view.

Let us first recall some basics facts about neutrinos. In the Standard Model, these particles

appear among the fundamental constituents, together with the corresponding charged leptons

and the quarks. Since however no neutrino mass has been yet directly measured, they appear

there as massless fermions and are consequently described by (two component) Weyl spinors.

There is however no necessity, a priori, for dealing with massless neutrinos. They could

very well have a mass, sufficiently small to be consistent with the present experimental upper

bounds. The fact that they are electrically neutral, makes then possible for two different mass

types, namely Dirac or Majorana. In the first case, the (massive) neutrino would be described

by a (four component) Dirac spinor, similar to the one describing the electron. In the case of

Majorana neutrino, the spinor has two components only, since neutrino and antineutrino are

identified.

However, dealing with massive neutrinos is not yet sufficient for having ”oscillations”. It is

also necessary that mixing is present, i.e. that the neutrinos belonging to different generations

do have a mixed mass term, in analogy to what happens for quarks (CKM mixing).

Then, as pointed out by Pontecorvo[1], the time evolution of a neutrino mixed state would

lead to flavour oscillations, i.e. to a conversion of a neutrino of one flavour into one of another

flavour. This effect is experimentally testable.

It is clear that if such a scheme is accepted, then it is necessary to understand several things,

first of all how to justify the smallness of the neutrino masses with respect to those of the other

leptons. Also the question of the nature of the neutrino mass (Dirac or Majorana) is an open

one. But perhaps the most intriguing aspect is the one related to the mixing: it is not clear how

it arises and also it is difficult to understand the large mixing angles (in contrast with the ones

for quarks, which are small) necessary to fit the latest experimental data.

Thus there is currently a big deal of work in the direction of a proper understanding of the

generation of neutrino masses and mixings as a result of the breakdown of some grand-unifying

symmetry to the one of the Standard Model [5].

On the other hand, from a more general and theoretical point of view, mixing is extremely

interesting, since it appears to be one of the fundamental properties of Nature and is almost

ubiquitous in Particle Physics.

In particular, from a mathematical point of view, there is a problem in defining properly the

Hilbert space for mixed particles and several attempts have been made in this direction [6, 7].

A solution to this problem was achieved only recently [8, 9, 10] and here I report about these

results. I will review also about recent results on neutrino oscillations [11, 12, 13, 14]: the exact

2

formula for flavour oscillations was found and it has been shown also that a geometrical phase

is present for oscillating neutrinos.

The main point of our analysis [10] consists in the observation that a problem of representa-

tion (i.e. of choice of the proper Hilbert space) is involved when mixed fields are considered. This

is due to the peculiar mathematical structure of Quantum Field Theory (QFT), where many

inequivalent representations (many different Hilbert spaces) are allowed for a given dynamics

(field algebra) [15, 16]. A classical example is the one of theories with spontaneous breakdown

of symmetry.

This situation contrast the one of Quantum Mechanics, where only one Hilbert space is

admitted (von Neumann theorem) due to the finiteness of the number of the degrees of freedom

of the systems under consideration.

By a careful analysis of the mixing transformations in QFT, we have been able to show that

a rich non-perturbative structure is associated to the vacuum for mixed fermions (neutrinos),

which appears to be a condensate of particle-antiparticle pairs. The situation has some analogy

with BCS although there the condensate structure is considerably simpler than in the case of

mixing.

A similar situation occurs for mixing of bosons [8, 17, 18], although with a different condensate

structure. In both cases however, the vacuum for the mixed fields is a coherent state, of the

generalized type introduced by Perelomov [19].

As a fact of phenomenological relevance, we have studied neutrino oscillations in the frame-

work of QFT. We have shown that it is impossible to obtain a consistent result, unless the

proper vacuum is used in the calculation of the Green functions for mixed neutrinos. Then the

(exact) oscillation formula has been calculated [12] and it turned out to have an additional oscil-

lating piece and energy dependent amplitudes, in contrast with the usual (quantum mechanical)

Pontecorvo formula, which is however recovered in the relativistic limit.

We have also shown how the concept of a topological (Berry) phase naturally enters the

physics of neutrino oscillations [13]. This is a very novel feature of neutrino oscillations which

was not pointed out previously: we suggest that it can be of possible phenomenological relevance.

The material here presented is organised in the following way:

In Section 1, the mixing transformations are studied in QFT, both for the fermion and the

boson case. Extension of the results to the case of three flavour mixing is given in Appendix A.

In Section 2, neutrino oscillations are considered and the Green’s function formalism for mixed

fermions is introduced. Then the oscillation formula is obtained. Preliminary results about the

oscillation formula for mixed bosons are reported in Appendix B. Finally, Section 3 is about the

Berry phase for oscillating particles (neutrinos).

3

1. Mixing transformations in Quantum Field Theory

In this Section we study the formal structure of the mixing transformations both for fermion

(Dirac) and boson fields [8, 10, 17]. In §1.1 and 1.2 a study of the generator of mixing trans-

formations for fermions and bosons is presented; the Hilbert space for the mixed fields is then

constructed. In §1.3 a general analysis of the current structure for mixed field is given [18].

Finally, in §1.4 a generalization of the results of §1.1 is reported, based on recent work [20, 14].

1.1 Fermion mixing

Since we have in mind neutrinos, for which we will consider flavour oscillations, we will

specialize in the following discussion to neutrino Dirac fields. However, the scheme has general

validity for any Dirac fields.

Let us consider mixing of two flavour fields (for extension to three flavours see Appendix A)

which we will denote by νe(x), νµ(x). The mixing relations, originally proposed by Pontecorvo,

are[1]

νe(x) = ν1(x) cos θ + ν2(x) sin θ

νµ(x) = −ν1(x) sin θ + ν2(x) cos θ , (1.1)

where νe(x) and νµ(x) are the (Dirac) neutrino fields with definite flavours. ν1(x) and ν2(x) are

the (free) neutrino fields with definite masses m1 and m2, respectively. θ is the mixing angle.

The fields ν1(x) and ν2(x) are expanded as

νi(x) =1√V

∑

k,r

eik·x [ur

k,i(t)αrk,i + vr

−k,i(t)βr†−k,i

], i = 1, 2 . (1.2)

where urk,i(t) = e−iωk,itur

k,i and vrk,i(t) = eiωk,itvr

k,i, with ωk,i =√

k2 + m2i . The αr

k,i and the βrk,i

( r = 1, 2), are the annihilation operators for the vacuum state |0〉1,2 ≡ |0〉1 ⊗ |0〉2: αrk,i|0〉12 =

βrk,i|0〉12 = 0. The anticommutation relations are:

{ναi (x), νβ†

j (y)}t=t′ = δ3(x− y)δαβδij , α, β = 1, .., 4 , (1.3)

4

and

{αrk,i, α

s†q,j} = δkqδrsδij; {βr

k,i, βs†q,j} = δkqδrsδij, i, j = 1, 2 . (1.4)

All other anticommutators are zero. The orthonormality and completeness relations are:

ur†k,iu

sk,i = vr†

k,ivsk,i = δrs , ur†

k,ivs−k,i = vr†

−k,iusk,i = 0 ,

∑r

(urk,iu

r†k,i + vr

−k,ivr†−k,i) = I . (1.5)

In QFT the basic dynamics, i.e. the Lagrangian and the resulting field equations, is given in

terms of Heisenberg (or interacting) fields. The physical observables are expressed in terms of

asymptotic in- (or out-) fields, also called physical or free fields. In the LSZ formalism of QFT

[15, 16], the free fields, say for definitiveness the in-fields, are obtained by the weak limit of the

Heisenberg fields for time t → −∞. The meaning of the weak limit is that the realization of the

basic dynamics in terms of the in-fields is not unique so that the limit for t → −∞ (or t → +∞for the out-fields) is representation dependent.

Typical examples are the ones of spontaneously broken symmetry theories, where the same

set of Heisenberg field equations describes the normal (symmetric) phase as well as the symmetry

broken phase. Since observables are described in terms of asymptotic fields, unitarily inequivalent

representations describe different, i.e. physically inequivalent, phases. It is therefore of crucial

importance, in order to get physically meaningful results, to investigate with much care the

mapping among Heisenberg or interacting fields and free fields, i.e. the dynamical map.

With this warnings, mixing relations such as the relations (1.1) deserve a careful analysis,

since they actually represent a dynamical mapping. It is now our purpose to investigate the

structure of the Fock spaces H1,2 and He,µ relative to ν1(x), ν2(x) and νe(x), νµ(x), respectively.

In particular we want to study the relation among these spaces in the infinite volume limit. We

expect that H1,2 and He,µ become orthogonal in such a limit, since they represent the Hilbert

spaces for free and interacting fields, respectively [15]. In the following, as usual, we will perform

all computations at finite volume V and only at the end we will put V →∞.

Our first step is the study of the generator of eqs.(1.1) and of the underlying group theoretical

structure.

Eqs.(1.1) can be put in the following form [10]:

ναe (x) = G−1

θ (t) να1 (x) Gθ(t)

ναµ (x) = G−1

θ (t) να2 (x) Gθ(t) , (1.6)

where Gθ(t) is given by

Gθ(t) = exp[θ

∫d3x

(ν†1(x)ν2(x)− ν†2(x)ν1(x)

)], (1.7)

and is (at finite volume) an unitary operator: G−1θ (t) = G−θ(t) = G†

θ(t), preserving the canonical

anticommutation relations (1.3).

Eq.(1.7) follows from d2

dθ2 ναe = −να

e , d2

dθ2 ναµ = −να

µ with the initial conditions ναe |θ=0 = να

1 ,ddθ

ναe |θ=0 = να

2 and ναµ |θ=0 = να

2 , ddθ

ναµ |θ=0 = −να

1 .

5

We note the time dependence of the generator G: it represent an important feature and will

be carefully considered in Section 2 where neutrino oscillations are discussed.

We also observe that Gθ is an element of SU(2). Indeed, it can be written as

Gθ(t) = exp[θ(S+(t)− S−(t))] . (1.8)

with

S+(t) ≡∫

d3x ν†1(x)ν2(x) , S−(t) ≡∫

d3x ν†2(x)ν1(x) = S†+(t) , (1.9)

By introducing then

S3 ≡ 1

2

∫d3x

(ν†1(x)ν1(x)− ν†2(x)ν2(x)

), (1.10)

together with the Casimir (proportional to the total charge)

S0 ≡ 1

2

∫d3x

(ν†1(x)ν1(x) + ν†2(x)ν2(x)

), (1.11)

the algebra su(2) is closed:

[S+(t), S−(t)] = 2S3 , [S3, S±(t)] = ±S±(t) , [S0, S3] = [S0, S±(t)] = 0 . (1.12)

It is interesting to look at the momentum expansion of the above generators:

S+(t) ≡ ∑

k

Sk+(t) =

∑

k

∑r,s

(ur†k,1(t)u

sk,2(t) αr†

k,1αsk,2 + (1.13)

+vr†−k,1(t)u

sk,2(t) βr

−k,1αsk,2 + ur†

k,1(t)vs−k,2(t) αr†

k,1βs†−k,2 + vr†

−k,1(t)vs−k,2(t) βr

−k,1βs†−k,2) ,

S−(t) ≡ ∑

k

Sk−(t) =

∑

k

∑r,s

(ur†k,2(t)u

sk,1(t) αr†

k,2αsk,1 + (1.14)

+vr†−k,2(t)u

sk,1(t) βr

−k,2αsk,1 + ur†

k,2(t)vs−k,1(t) αr†

k,2βs†−k,1 + vr†

−k,2(t)vs−k,1(t) βr

−k,2βs†−k,1) ,

S3 ≡∑

k

Sk3 =

1

2

∑

k,r

(αr†

k,1αrk,1 − βr†

−k,1βr−k,1 − αr†

k,2αrk,2 + βr†

−k,2βr−k,2

), (1.15)

S0 ≡∑

k

Sk0 =

1

2

∑

k,r

(αr†

k,1αrk,1 − βr†

−k,1βr−k,1 + αr†

k,2αrk,2 − βr†

−k,2βr−k,2

). (1.16)

We observe that the operatorial structure of eqs.(1.13) and (1.14) is the one of the rotation

generator and of the Bogoliubov generator. These structures will be exploited in the following.

Using these expansions it is easy to show that the su(2) algebra does hold for each momentum

component:

[Sk+(t), Sk

−(t)] = 2Sk3 , [Sk

3 , Sk±(t)] = ±Sk

±(t) , [Sk0 , Sk

3 ] = [Sk0 , Sk

±] = 0 ,

[Sk±(t), Sp

±(t)] = [Sk3 , Sp

±(t)] = [Sk3 , Sp

3 ] = 0 , k 6= p . (1.17)

This means that the original su(2) algebra given in eqs.(1.12) splits into k disjoint suk(2) alge-

bras, given by eqs.(1.17), i.e. we have the group structure⊗

k SUk(2).

6

To establish the relation between H1,2 and He,µ we consider the generic matrix element

1,2〈a|να1 (x)|b〉1,2 (a similar argument holds for να

2 (x)), where |a〉1,2 is the generic element of H1,2.

By using eq. (1.6), we obtain:

1,2〈a|Gθ ναe (x) G−1

θ |b〉1,2 = 1,2〈a|να1 (x)|b〉1,2 . (1.18)

Since the operator field νe is defined on the Hilbert space He,µ, eq.(1.18) shows that G−1θ |a〉1,2

is a vector of He,µ, so G−1θ maps H1,2 to He,µ: G−1

θ : H1,2 7→ He,µ. In particular for the vacuum

|0〉1,2 we have (at finite volume V ):

|0(t)〉e,µ = G−1θ (t) |0〉1,2 . (1.19)

|0(t)〉e,µ is the vacuum for He,µ, which we will refer to as the flavour vacuum. Due to the linearity

of Gθ(t), we can define the flavour annihilators, relative to the fields νe(x) and νµ(x) as1

αrk,e(t) ≡ G−1

θ (t) αrk,1 Gθ(t) ,

αrk,µ(t) ≡ G−1

θ (t) αrk,2 Gθ(t) ,

βrk,e(t) ≡ G−1

θ (t) βrk,1 Gθ(t) , (1.20)

βrk,µ(t) ≡ G−1

θ (t) βrk,2 Gθ(t) .

The flavour fields are then rewritten into the form:

νe(x, t) =1√V

∑

k,r

eik·x [ur

k,1(t)αrk,e(t) + vr

−k,1(t)βr†−k,e(t)

]

νµ(x, t) =1√V

∑

k,r

eik·x [ur

k,2(t)αrk,µ(t) + vr

−k,2(t)βr†−k,µ(t)

](1.21)

i.e. they can be expanded in the same bases as ν1 and ν2, respectively.

We observe that G−1θ = exp[θ(S− − S+)] is just the generator for generalized coherent states

of SU(2) [19]2: the flavour vacuum is therefore an SU(2) (time dependent) coherent state. Let us

now obtain the explicit expression for |0〉e,µ and investigate the infinite volume limit of eq.(1.19).

Using the Gaussian decomposition, G−1θ can be written as [19]

exp[θ(S− − S+)] = exp(−tanθ S+) exp(−2ln cosθ S3) exp(tanθ S−) (1.22)

where 0 ≤ θ < π2. Eq.(1.19) then becomes

|0〉e,µ =∏

k

|0〉ke,µ =∏

k

exp(−tanθ Sk+)exp(−2ln cosθ Sk

3 ) exp(tanθ Sk−)|0〉1,2 . (1.23)

The final expression for |0〉e,µ in terms of Sk± and Sk

3 is [10]:

|0〉e,µ =∏

k

[1 + sin θ cos θ

(Sk− − Sk

+

)+

1

2sin2 θ cos2 θ

((Sk−)2 + (Sk

+)2)

+ (1.24)

− sin2 θSk+Sk

− +1

2sin3 θ cos θ

(Sk−(Sk

+)2 − Sk+(Sk

−)2)

+1

4sin4 θ(Sk

+)2(Sk−)2

]|0〉1,2 .

1The annihilation of the flavour vacuum at each time is expressed as: αrk,e(t)|0(t)〉e,µ = G−1

θ (t)αrk,1|0〉1,2 = 0.

2In the following, for simplicity, we will omit the time dependence of the mixing generator.

7

The state |0〉e,µ is normalized to 1 (see eq.(1.19)).

Let us now compute 1,2〈0|0〉e,µ. We obtain

1,2〈0|0〉e,µ =∏

k

(1− sin2 θ 1,2〈0|Sk

+Sk−|0〉1,2 +

1

4sin4 θ 1,2〈0|(Sk

+)2(Sk−)2|0〉1,2

)

=∏

k

(1− sin2 θ |Vk|2

)2 ≡ ∏

k

Γ(k) = e∑

kln Γ(k). (1.25)

where the function Vk is defined in eq.(1.31) and plotted in Fig.1 . Note that |Vk|2 depends on

|k|, it is always in the interval [0, 1[ and |Vk|2 → 0 when |k| → ∞.

By using the customary continuous limit relation∑

k → V(2π)3

∫d3k, in the infinite volume

limit we obtain

limV→∞ 1,2〈0|0〉e,µ = lim

V→∞e

V(2π)3

∫d3k ln Γ(k)

= 0 (1.26)

since Γ(k) < 1 for any value of k and of the parameters m1 and m2.

Notice that (1.26) shows that the orthogonality between |0〉e,µ and |0〉1,2 is due to the infrared

contributions which are taken in care by the infinite volume limit and therefore high momentum

contributions do not influence the result (for this reason here we do not need to consider the

regularization problem of the UV divergence of the integral of ln Γ(k)). Of course, this orthog-

onality disappears when θ = 0 and/or when m1 = m2 (because in this case Vk = 0 for any k

and no mixing occurs in Pontecorvo theory).

Eq.(1.26) expresses the unitary inequivalence in the infinite volume limit of the flavour and

the mass representations and shows the non-trivial nature of the mixing transformations (1.1).

In other words, the mixing transformations induce a physically non-trivial structure in the

flavour vacuum which indeed turns out to be an SU(2) generalized coherent state. In Section

2 we will see how such a vacuum structure may lead to phenomenological consequences in the

neutrino oscillations, which possibly may be experimentally tested. From eq.(1.26) we also see

that eq.(1.19) is a purely formal expression which only holds at finite volume.

Let us now return to the dynamical map, eqs.(1.20): it can be calculated explicitly, thus

giving the flavour annihilation operators

αrk,e(t) = cos θ αr

k,1 + sin θ∑s

[ur†

k,1(t)usk,2(t) αs

k,2 + ur†k,1(t)v

s−k,2(t) βs†

−k,2

]

αrk,µ(t) = cos θ αr

k,2 − sin θ∑s

[ur†

k,2(t)usk,1(t) αs

k,1 + ur†k,2(t)v

s−k,1(t) βs†

−k,1

](1.27)

βr−k,e(t) = cos θ βr

−k,1 + sin θ∑s

[vs†−k,2(t)v

r−k,1(t) βs

−k,2 + us†k,2(t)v

r−k,1(t) αs†

k,2

]

βr−k,µ(t) = cos θ βr

−k,2 − sin θ∑s

[vs†−k,1(t)v

r−k,2(t) βs

−k,1 + us†k,1(t)v

r−k,2(t) αs†

k,1

]

Without loss of generality, we can choose the reference frame such that k = (0, 0, |k|). In

this case the spins decouple and we have the simpler expressions:

αrk,e(t) = cos θ αr

k,1 + sin θ(U∗

k(t) αrk,2 + εr Vk(t) βr†

−k,2

)

8

0

0.25

0.5

0 20 40 60 80 100

|Vk

|2

k

Figure 1.1: The fermion condensation density |Vk|2 in function of k and for sample values of the

parameters m1 and m2.

Solid line: m1 = 1 , m2 = 100Long-dashed line: m1 = 10 , m2 = 100Short-dashed line: m1 = 10 , m2 = 1000

αrk,µ(t) = cos θ αr

k,2 − sin θ(Uk(t) αr

k,1 − εr Vk(t) βr†−k,1

)(1.28)

βr−k,e(t) = cos θ βr

−k,1 + sin θ(U∗

k(t) βr−k,2 − εr Vk(t) αr†

k,2

)

βr−k,µ(t) = cos θ βr

−k,2 − sin θ(Uk(t) βr

−k,1 + εr Vk(t) αr†k,1

)

where εr = (−1)r and

Uk(t) ≡ ur†k,2(t)u

rk,1(t) = vr†

−k,1(t)vr−k,2(t)

Vk(t) ≡ εr ur†k,1(t)v

r−k,2(t) = −εr ur†

k,2(t)vr−k,1(t) . (1.29)

We have:

Vk = |Vk| ei(ωk,2+ωk,1)t , Uk = |Uk| ei(ωk,2−ωk,1)t (1.30)

|Uk| =(

ωk,1 + m1

2ωk,1

) 12

(ωk,2 + m2

2ωk,2

) 12

(1 +

|k|2(ωk,1 + m1)(ωk,2 + m2)

)

|Vk| =(

ωk,1 + m1

2ωk,1

) 12

(ωk,2 + m2

2ωk,2

) 12

( |k|(ωk,2 + m2)

− |k|(ωk,1 + m1)

)(1.31)

|Uk|2 + |Vk|2 = 1 (1.32)

9

We thus see that, at the level of annihilation operators, the structure of the mixing transfor-

mation is that of a Bogoliubov transformation nested into a rotation. The two transformations

however cannot be disentangled, thus the mixing transformations (1.28) are essentially different

from the usual Bogoliubov transformations.

It is possible to exhibit the full explicit expression of |0〉ke,µ (at time t = 0) in the reference

frame for which k = (0, 0, |k|):

|0〉ke,µ =∏r

[(1− sin2 θ |Vk|2)− εr sin θ cos θ |Vk| (αr†

k,1βr†−k,2 + αr†

k,2βr†−k,1)+ (1.33)

+ εr sin2 θ |Vk| |Uk|(αr†

k,1βr†−k,1 − αr†

k,2βr†−k,2

)+ sin2 θ |Vk|2 αr†

k,1βr†−k,2α

r†k,2β

r†−k,1

]|0〉1,2

We see that the expression of the flavour vacuum |0〉e,µ involves four different particle-antiparticle

”couples”, in contrast with the BCS superconducting ground state, which involves only one kind

of couple and is generated by a Bogoliubov transformation.

The condensation density is given by

e,µ〈0|αr†k,1α

rk,1|0〉e,µ = sin2 θ |Vk|2 (1.34)

with a similar result for αrk,2, βr

k,1 and βrk,2. In the case of mixing of three fields, the condensation

densities are different for particles with different masses (see Appendix A).

|Vk|2 =1

2− |k|2 + m1m2

2ωk,1ωk,2

(1.35)

1.2 Boson mixing

Let us now discuss the case of boson mixing [17, 8]. Consider two charged boson fields φi(x),

i = 1, 2 with different masses and their conjugate momenta πi(x) = ∂0φ†i (x), satisfying the usual

commutation relations with non-zero commutators given by:

[φi(x), πi(y)]t=t′ =[φ†i (x), π†i (y)

]t=t′

= iδ3(x− y)[ak,i, a

†p,i

]=

[bk,i, b

†p,i

]= δ3(k− p) (1.36)

The Fourier expansions for these fields are

φi(x) =∫ d3k

(2π)32

1√2ωi

(ak,i e−ik.x + b†k,i eik.x

)(1.37)

πi(x) = i∫ d3k

(2π)32

√ωi

2

(a†k,i eik.x − bk,i e−ik.x

)(1.38)

with k.x = ωt− k · x. Now we define mixing relations as:

φA(x) = φ1(x) cos θ + φ2(x) sin θ

φB(x) = −φ1(x) sin θ + φ2(x) cos θ (1.39)

10

and h.c. and similar ones for πA, πB. We generically denote the mixed fields with A and B. As

for fermions we put eqs.(1.39) into the form:

φA(x) = G−1θ (t) φ1(x) Gθ(t)

φB(x) = G−1θ (t) φ2(x) Gθ(t) (1.40)

and similar ones for πA, πB, where Gθ(t) is given by

Gθ(t) = exp[−i θ

∫d3x

(π1(x)φ2(x)− π†2(x)φ†1(x)− π2(x)φ1(x) + π†1(x)φ†2(x)

)](1.41)

and is (at finite volume) an unitary operator: G−1θ (t) = G−θ(t) = G†

θ(t). Exactly like in the

fermion case, Gθ(t) can be written as

Gθ(t) = exp[θ(S+ − S−)] . (1.42)

where now

S+ = S†− ≡ −i∫

d3x (π1(x)φ2(x)− π†2(x)φ†1(x)) (1.43)

which together with

S3 ≡ −i

2

∫d3x

(π1(x)φ1(x)− π2(x)φ2(x) + π†2(x)φ†2(x)− π†1(x)φ†1(x)

)(1.44)

S0 =Q

2≡ −i

2

∫d3x

(π1(x)φ1(x)− π†1(x)φ†1(x) + π2(x)φ2(x)− π†2(x)φ†2(x)

)(1.45)

close the su(2) algebra associated to the rotation (1.39): [S+, S−] = 2S3 , [S3, S±] = ±S± ,

[S0, S3] = [S0, S±] = 0.

The expansions in terms of annihilation and creation operators are

S3 =1

2

∫d3k

(a†k,1ak,1 − b†−k,1b−k,1 − a†k,2ak,2 + b†−k,2b−k,2

)(1.46)

S0 =1

2

∫d3k

(a†k,1ak,1 − b†−k,1b−k,1 + a†k,2ak,2 − b†−k,2b−k,2

)(1.47)

S+(t) =∫

d3k(U∗

k(t) a†k,1ak,2 − V ∗k (t) b−k,1ak,2 + Vk(t) a†k,1b

†−k,2 − Uk(t) b−k,1b

†−k,2

)(1.48)

S−(t) =∫

d3k(Uk(t) a†k,2ak,1 − Vk(t) a†k,2b

†−k,1 + V ∗

k (t) b−k,2ak,1 − U∗k(t) b−k,2b

†−k,1

)(1.49)

with Uk ≡ |Uk| ei(ωk,2−ωk,1)t , Vk ≡ |Vk| ei(ωk,1+ωk,2)t and

|Uk| ≡ 1

2

(√ωk,1

ωk,2

+

√ωk,2

ωk,1

), |Vk| ≡ 1

2

(√ωk,1

ωk,2

−√

ωk,2

ωk,1

),

|Uk|2 − |Vk|2 = 1 (1.50)

Then one can put |Uk| ≡ cosh σk , |Vk| ≡ sinh σk with σk = 12ln

(ω1

ω2

).

11

The generator of boson mixing transformations does not leave invariant the vacuum of the

fields φ1,2(x), say |0〉1,2, since it induces an SU(2) (bosonic) coherent state structure resulting in

a new state |0〉A,B:

|0(t)〉A,B = G−1θ (t) |0〉1,2 (1.51)

The annihilation operators for the vacuum |0(t)〉A,B are given by ak,A ≡ G−1θ (t) ak,1 Gθ(t), etc..

We have

ak,A(t) = cos θ ak,1 + sin θ(U∗

k(t) ak,2 + Vk(t) b†−k,2

)

ak,B(t) = cos θ ak,2 − sin θ(Uk(t) ak,1 − Vk(t) b†−k,1

)(1.52)

b−k,A(t) = cos θ b−k,1 + sin θ(U∗

k(t) b−k,2 + Vk(t) a†k,2

)(1.53)

b−k,B(t) = cos θ b−k,2 − sin θ(Uk(t) b−k,1 − Vk(t) a†k,1

)(1.54)

Similar expressions can be obtained for ak,B, bk,A and bk,B. From eq.(1.52) and similar, we

see how the only difference with respect to fermion mixing, is in the (internal) Bogoliubov

transformation, which now, as due for bosons, has coefficients which satisfy hyperbolic relations

(cf.eq.(1.50)).

The condensation density of the vacuum is given by

1,2〈0|a†k,Aak,A|0〉1,2 = sin2 θ |Vk|2 = sin2 θ sinh2

[1

2ln

(ωk,1

ωk,2

)](1.55)

which appears to be very different from the corresponding quantity in the fermion case. We

observe (see Fig.2) that still the main contribution to the condensate comes from the infrared

region, although now it is maximal at zero and, most important, not limited to be less than one.

It is also interesting to see how the above scheme works for neutral fields. As for the charged

field case, let us consider two fields φi(x), i = 1, 2 and their conjugate momenta πi(x) = ∂0φi(x),

with the following non-zero commutators:

[φi(x), πi(y)]t=t′ = iδ3(x− y) ,[ak,i, a

†p,i

]= δ3(k− p) (1.56)

and the expansions

φi(x) =∫ d3k

(2π)32

1√2ωk,i

(ak,i e−ik.x + a†k,i eik.x

)(1.57)

πi(x) =∫ d3k

(2π)32

√ωk,i

2i(−ak,i e−ik.x + a†k,i eik.x

)(1.58)

The mixing generator is still given by Gθ(t) = exp[θ(S+(t)−S−(t))] and the su(2) operators are

now realized as

S+(t) ≡ −i∫

d3x π1(x)φ2(x) , S−(t) ≡ −i∫

d3x π2(x)φ1(x)

S3 ≡ −i

2

∫d3x (π1(x)φ1(x)− π2(x)φ2(x)) , (1.59)

S0 ≡ −i

2

∫d3x (π1(x)φ1(x) + π2(x)φ2(x))

12

0

1

2

3

4

5

0 20 40 60 80 100

|V(k

,m1

,m2)|2

k

Figure 1.2: The boson condensation density |Vk|2 in function of k and for sample values of the

parameters m1 and m2.

Solid line: m1 = 10 , m2 = 100Dashed line: m1 = 10 , m2 = 200

We have, explicitely

S+(t)− S−(t) =∫

d3k(Uk(t) a†k,1ak,2 − V ∗

k (t) a−k,1ak,2 + Vk(t) a†k,2a†−k,1 − U∗

k(t) a†k,2ak,1

)

(1.60)

where the Bogoliubov coefficients Uk and Vk are the same of the ones defined in eq.(1.50).

The structure of the annihilators for the mixed field is the following:

ak,A(t) = cos θ ak,1 + sin θ(Uk(t) ak,2 + Vk(t) a†−k,2

)

ak,B(t) = cos θ ak,2 − sin θ(Uk(t) ak,1 − Vk(t) a†−k,1

)(1.61)

The condensation density is the same as in eq.(1.55).

The study of oscillations of mesons in the above framework is in progress (see Appendix B).

Results similar to the ones here presented have been recently obtained in [21].

13

1.3 The current structure for field mixing

We now study the transformations acting on a doublet of free fields with different masses.

The results of this Section clarify the meaning of the su(2) algebraic structure found before and

will be useful in the discussion of neutrino oscillations.

Fermions

Let us consider the following Lagrangian, describing two free Dirac fields, with masses m1

and m2:

L = Ψm (i 6∂ −Md) Ψm (1.62)

where ΨTm = (ν1, ν2) and Md = diag(m1,m2). We introduce a subscript m, in order to distinguish

the quantities here introduced, which are in terms of fields with definite masses, from the ones

of §2.2.2.

Consider now the transformation:

Ψ′m = eiθ· τ

2 Ψm (1.63)

with τ = τ1, τ2, τ3 being the Pauli matrices.

Since the masses m1 and m2 are different, the Lagrangian is not invariant under the above

transformations. The variation of L is given as

δL = −∂µ jµm =

1

2Ψm [Md, τ ] Ψm (1.64)

jµm,i =

1

2Ψm γµ τi Ψm (1.65)

We thus obtain the following currents:

jµm,1 =

1

2[ν1 γµ ν2 + ν2 γµ ν1] (1.66)

jµm,2 =

i

2[ν1 γµ ν2 − ν2 γµ ν1] (1.67)

jµm,3 =

1

2[ν1 γµ ν1 − ν2 γµ ν2] (1.68)

If we now define the charges as Sm,i ≡∫

d3x j0m,i, i = 1, 2, 3, we naturally recover the result of

§1.1 where we found that an su(2) algebra is associated with the mixing transformations (1.1):

[Sm,i, Sm,j] = i εijk Sm,k.

We note that Casimir operator is proportional to the total (conserved) charge Sm,0 = 12Q.

Also Sm,3 is conserved, due to the fact that the mass matrix Md is diagonal. This implies the

14

conservation of charge separately for ν1 and ν2, which is what we expect for a system of two

non–interacting fields.

Explicitely the transformations induced by the three above generators are

Ψ′m =

cos θ1 i sin θ1

i sin θ1 cos θ1

Ψm (1.69)

Ψ′m =

cos θ2 sin θ2

− sin θ2 cos θ2

Ψm (1.70)

Ψ′m =

eiθ3 0

0 e−iθ3

Ψm (1.71)

with θi ≡ 12αi. Thus the transformation considered in §1.1 is the one induced by Sm,2

Bosons

Let us repeat the above analysis for the case of boson fields. We consider the Lagrangian

L = ∂µΦ†m∂µΦm − Φ†

mMdΦm (1.72)

with ΦTm = (φ1, φ2) being charged scalar fields and Md = diag(m2

1,m22).

We have now

Φ′m = e

i2αiτi Φm with i = 1, 2, 3 . (1.73)

and

δL = − ∂µ jµ =i

2Φ†

m [Md , τ ] Φm (1.74)

jµm =

∂L∂(∂µΦm)

δΦm + δΦ†m

∂L∂(∂µΦ†

m)=

i

2Φ†

m τ↔∂µ Φm (1.75)

We thus obtain the currents:

jµm,1 =

i

2

[(∂µφ†1)φ2 + (∂µφ†2)φ1 − φ†1(∂

µφ2) − φ†2(∂µφ1)

](1.76)

jµm,2 =

1

2

[(∂µφ†1)φ2 − (∂µφ†2)φ1 − φ†1(∂

µφ2) + φ†2(∂µφ1)

](1.77)

jµm,3 =

i

2

[(∂µφ†1)φ1 − (∂µφ†2)φ2 − φ†1(∂

µφ1) + φ†2(∂µφ2)

](1.78)

Again, the corresponding charges Sm,i satisfy the su(2) algebra.

15

1.4 Generalization of mixing transformations

We have seen in §1.1 how the fields νe and νµ can be expanded in the same bases as ν1 and

ν2, respectively, the form of the flavor annihilation operators being given in eqs.(1.28).

It has been recently noticed[20], however, that expanding the flavor fields in the same basis

as the (free) fields with definite masses is actually a special choice, and that a more general

possibility exists.

Let us introduce the notation (σ, j) = (e, 1), (µ, 2) and rewrite eqs.(1.21),(1.21) as:

νσ(x) = G−1θ (t) νj(x) Gθ(t) =

1√V

∑

k,r

[ur

k,jαrk,σ(t) + vr

−k,jβr†−k,σ(t)

]eik·x, (1.79)

The flavor annihilation operators are rewritten as

(αr

k,σ(t)

βr†−k,σ(t)

)= G−1

θ (t)

(αr

k,j(t)

βr†−k,j(t)

)Gθ(t) (1.80)

The fact is that in the expansion eq.(1.79) one could use eigenfunctions with arbitrary masses

µσ and write the flavor fields as [20]:

νσ(x) =1√V

∑

k,r

[ur

k,σαrk,σ(t) + vr

−k,σβr†−k,σ(t)

]eik·x, (1.81)

where uσ and vσ are the helicity eigenfunctions with mass µσ3. We denote by a tilde the

generalized flavor operators introduced in ref.[20] in order to distinguish them from the ones

defined in eq.(1.80). The expansion eq.(1.81) is more general than the one in eq.(1.79) since the

latter corresponds to the particular choice µe ≡ m1, µµ ≡ m2.

The relation between the flavor and the mass operators is now:

(αr

k,σ(t)

βr†−k,σ(t)

)= K−1

θ,µ(t)

(αr

k,j(t)

βr†−k,j(t)

)Kθ,µ(t) , (1.82)

with (σ, j) = (e, 1), (µ, 2) and where Kθ,µ(t) is the generator of the transformation (2.7) and can

be written as

Kθ,µ(t) = Iµ(t) Gθ(t) (1.83)

Iµ(t) =∏

k,r

exp

i

∑

(σ,j)

ξkσ,j

[αr†

k,j(t)βr†−k,j(t) + βr

−k,j(t)αrk,j(t)

] (1.84)

with ξkσ,j ≡ (χσ − χj)/2 and cot χσ = |k|/µσ, cot χj = |k|/mj. For µe ≡ m1, µµ ≡ m2 one has

Iµ(t) = 1.

3The use of such a basis simplifies considerably calculations with respect to the original choice of ref.[10].

16

The explicit matrix form of the flavor operators is[20]:

αrk,e(t)

αrk,µ(t)

βr†−k,e(t)

βr†−k,µ(t)

=

cθ ρke1 sθ ρk

e2 icθ λke1 isθ λk

e2

−sθ ρkµ1 cθ ρk

µ2 −isθ λkµ1 icθ λk

µ2

icθ λke1 isθ λk

e2 cθ ρke1 sθ ρk

e2

−isθ λkµ1 icθ λk

µ2 −sθ ρkµ1 cθ ρk

µ2

αrk,1(t)

αrk,2(t)

βr†−k,1(t)

βr†−k,2(t)

(1.85)

where cθ ≡ cos θ, sθ ≡ sin θ and

ρkabδrs ≡ cos

χa − χb

2δrs = ur†

k,ausk,b = vr†

−k,avs−k,b (1.86)

iλkabδrs ≡ i sin

χa − χb

2δrs = ur†

k,avs−k,b = vr†

−k,ausk,b (1.87)

with a, b = 1, 2, e, µ. Since ρk12 = |Uk| and iλk

12 = εr|Vk|, etc., the operators (1.85) reduce to the

ones in eqs.(1.28) when µe ≡ m1 and µµ ≡ m24.

The generalization of the flavor vacuum, which is annihilated by the flavor operators given

by eq.(1.82), is now written as[20]:

|0(t)〉e,µ ≡ K−1θ,µ(t)|0〉1,2 . (1.88)

For µe ≡ m1 and µµ ≡ m2, this state reduces to the BV flavor vacuum |0(t)〉e,µ above defined.

The relation between the general flavor operators of eq.(1.82) and the ones of eq.(1.80) is

(αr

k,σ(t)

βr†−k,σ(t)

)= J−1

µ (t)

(αr

k,σ(t)

βr†−k,σ(t)

)Jµ(t) , (1.89)

Jµ(t) =∏

k,r

exp

i

∑

(σ,j)

ξkσ,j

[αr†

k,σ(t)βr†−k,σ(t) + βr

−k,σ(t)αrk,σ(t)

] . (1.90)

In conclusion, we see that the Hilbert space for the flavor fields is not unique: an infinite

number of vacua (and consequently infinitely many Hilbert spaces) can be generated by intro-

ducing the arbitrary mass parameters µe, µµ. It is obvious that physical quantities must not

depend on these parameters.

4In performing such an identification, one should take into account that the operators for antiparticles differfor a minus sign, related to the different spinor bases used in the expansions (1.79) and (1.81). Such a signdifference is however irrelevant in what follows.

17

2. Neutrino Oscillations

We now consider an application of the theoretical scheme above developed. We will study

neutrino oscillations and will see that a careful field theoretical treatment of the problem leads

to an exact result which is different from the usual one, obtained in a quantum mechanical

framework.

2.1 The usual picture for neutrino oscillations (Pontecorvo)

In the original Pontecorvo and collaborators treatment[1], the mixing relations (1.1) are

assumed to hold also at the level of states - i.e. the vacuum is taken to be the same for flavour

and mass eigenstates - :

|νe〉 = cos θ |ν1〉 + sin θ |ν2〉|νµ〉 = − sin θ |ν1〉 + cos θ |ν2〉 , (2.1)

where the states |νi〉 , i = 1, 2 are eigenstates of the Hamiltonian: H|νi〉 = ωi|νi〉. Then the

time evolution gives

|νe(t)〉 = e−iHt|νe〉 = e−iω1t cos θ |ν1〉 + e−iω2t sin θ |ν2〉|νµ(t)〉 = e−iHt|νµ〉 = −e−iω1t sin θ |ν1〉 + e−iω2t cos θ |ν2〉 , (2.2)

We thus have at time t, the flavour oscillations

Pνe→νe(t) = |〈νe|νe(t)〉|2

= 1− sin2 2θ sin2(

∆ω

2t)

. (2.3)

The number of electron neutrinos therefore oscillates in time with a frequency given by the

difference in the energies of the mass components ∆ω = ω2 − ω1. This is a flavour oscillation

since we have at the same time:

Pνe→νµ(t) = |〈νµ|νe(t)〉|2

= sin2 2θ sin2(

∆ω

2t)

(2.4)

The conservation of probability reads as

Pνe→νe(t) + Pνe→νµ(t) = 1 (2.5)18

2.2 Neutrino oscillations in QFT

We discuss neutrino oscillations in the framework of QFT[9, 12]. We start by a discussion

of the Green’s functions for mixed particles [12]. We then derive the exact oscillation formula,

exhibiting corrections with respect to the Pontecorvo one, eq.(2.3). We finally discuss the oscil-

lation formula in the context of the generalized formulation of §1.4.

2.2.1 Green’s functions for mixed fermions

In order to discuss flavour oscillations is sufficient to consider the following Lagrangian (omit

spacetime dependence for simplicity)

L = νe (i 6 ∂ −me) νe + νµ (i 6 ∂ −mµ) νµ − meµ (νeνµ + νµνe) . (2.6)

Generalization to a higher number of flavours is straightforward. This Lagrangian can be fully

diagonalized by substituting for the fields the mixing relations

νe(x) = ν1(x) cosθ + ν2(x) sinθ

νµ(x) = −ν1(x) sinθ + ν2(x) cosθ , (2.7)

where θ is the mixing angle and me = m1 cos2θ + m2 sin2θ , mµ = m1 sin2θ + m2 cos2θ , meµ =

(m2 − m1) sinθ cosθ . ν1 and ν2 therefore are non-interacting, free fields, anticommuting with

each other at any space-time point. Their expansions are given in eqs.(1.2).

The fields νe and νµ are thus completely determined through eq.(2.7). We have seen that it

is possible to expand the flavour fields νe and νµ in the same basis as ν1 and ν2,

ναe (x) = G−1

θ (t) να1 (x) Gθ(t) = V − 1

2

∑

k,r

[ur

k,1e−iωk,1tαr

k,e(t) + vr−k,1e

iωk,1tβr†−k,e(t)

]eik·x,

(2.8)

ναµ (x) = G−1

θ (t) να2 (x) Gθ(t) = V − 1

2

∑

k,r

[ur

k,2e−iωk,2tαr

k,µ(t) + vr−k,2e

iωk,2tβr†−k,µ(t)

]eik·x,

(2.9)

by means of the generator (1.7). The flavour annihilation and creation operators are given in

eqs.(1.28).

The bilinear mixed term of eq.(2.6) generates four non-zero two point causal Green’s functions

for the mixed fields νe, νµ. The crucial point is about how to compute these propagators: if one

(naively) uses the vacuum |0〉1,2, one gets an inconsistent result (cf. eq.(2.16)). Let us show this

by defining the propagators as Sαβ

ee (x, y) Sαβµe (x, y)

Sαβeµ (x, y) Sαβ

µµ (x, y)

≡ 1,2〈0|

T

[να

e (x)νβe (y)

]T

[να

µ (x)νβe (y)

]

T[να

e (x)νβµ(y)

]T

[να

µ (x)νβµ(y)

]|0〉1,2 , (2.10)

19

where T denotes time ordering. Use of (2.7) gives See in momentum representation as

See(k0,k) = cos2 θ6k + m1

k2 −m21 + iδ

+ sin2 θ6k + m2

k2 −m22 + iδ

, (2.11)

which is just the weighted sum of the two propagators for the free fields ν1 and ν2. It coincides

with the Feynman propagator obtained by resumming (to all orders) the perturbative series

See = Se

(1 + m2

eµ SµSe + m4eµ SµSeSµSe + ...

)= Se

(1−m2

eµ SµSe

)−1, (2.12)

where the “bare” propagators are defined as Se/µ = ( 6 k − me/µ + iδ)−1. In a similar way, one

computes Seµ and Sµe.

The transition amplitude for an electronic neutrino created by αr†k,e at time t = 0 into the

same particle at time t is given by

Pree(k, t) = iur†

k,1eiωk,1t S>

ee(k, t) γ0urk,1 . (2.13)

Here, S>ee(k, t) denotes the unordered Green’s function (or Wightman function) in mixed (k, t)

representation. The upper script > (or <) is related with the corresponding θ function. The

explicit expression for S>ee(k, t) is

S> αβee (k, t) = −i

∑r

(cos2θ e−iωk,1t ur,α

k,1 ur,βk,1 + sin2θ e−iωk,2t ur,α

k,2 ur,βk,2

). (2.14)

The probability amplitude (2.13) is independent of the spin orientation and given by

Pee(k, t) = cos2θ + sin2θ |Uk|2 e−i(ωk,2−ωk,1)t . (2.15)

For different masses and |k| 6= 0 , |Uk| is always < 1 (see eq.(1.31) and Fig.(1.1)). Notice

that |Uk|2 → 1 in the relativistic limit |k| À √m1m2 : only in this limit the squared modulus

of Pee(k, t) reproduces the Pontecorvo oscillation formula.

Of course, it should be limt→0+ Pee(t) = 1. Instead, one obtains the unacceptable result

Pee(k, 0+) = cos2θ + sin2θ |Uk|2 < 1 . (2.16)

This means that the choice of the state |0〉1,2 in (2.10) and in the computation of the Wightman

function is not the correct one. We thus realize the necessity to work in the correct representation

for the flavour fields, i.e. we have to calculate the Green’s functions on the flavour vacuum |0〉e,µ.

We now show that the correct definition of the Green’s function matrix for the fields νe, νµ

is the one which involves the non-perturbative vacuum |0〉e,µ, i.e.

Gαβ

ee (x, y) Gαβµe (x, y)

Gαβeµ (x, y) Gαβ

µµ(x, y)

≡ e,µ〈0(y0)|

T

[να

e (x)νβe (y)

]T

[να

µ (x)νβe (y)

]

T[να

e (x)νβµ(y)

]T

[να

µ (x)νβµ(y)

]|0(y0)〉e,µ . (2.17)

Notice that here the time argument y0 (or, equally well, x0) of the flavour ground state, is chosen

to be equal on both sides of the expectation value. We observe that transition matrix elements

20

of the type e,µ〈0|αe exp [−iHt] α†e|0〉e,µ, where H is the Hamiltonian, do not represent physical

transition amplitudes: they actually vanish (in the infinite volume limit) due to the unitary

inequivalence of flavour vacua at different times (see below). Therefore the comparison of states

at different times necessitates a parallel transport of these states to a common point of reference.

The definition (2.17) includes this concept of parallel transport, which is a sort of “gauge fixing”:

a rich geometric structure underlying the mixing transformations (2.7) is thus uncovered. This

geometric features include also Berry phase [13] (see Section 3) and a gauge structure associated

to the mixing transformations. Further study along this direction is in progress.

In the case of νe → νe propagation, we now have (for k = (0, 0, |k|)):

Gee(k0,k) = See(k0,k) + 2π i sin2 θ[|Vk|2 (6k + m2) δ(k2 −m2

2) (2.18)

− |Uk||Vk|∑r

(εrur

k,2 vr−k,2 δ(k0 − ω2) + εrvr

−k,2 urk,2 δ(k0 + ω2)

) ],

where we used εr = (−1)r . Comparison with eq.(2.11) shows that the difference between the

full and the perturbative propagators is in the imaginary part.

The Wightman functions for an electron neutrino are iG>αβee (t,x; 0,y) = e,µ〈0|να

e (t,x) νβe (0,y)|0〉e,µ,

and iG>αβµe (t,x; 0,y) = e,µ〈0|να

µ (t,x) νβe (0,y)|0〉e,µ. These are conveniently expressed in terms of

anticommutators at different times as

iG>αβee (k, t) =

∑r

[ur,α

k,1 ur,βk,1

{αr

k,e(t), αr†k,e

}e−iωk,1t + vr,α

−k,1 ur,βk,1

{βr†−k,e(t), α

r†k,e

}eiωk,1t

],

(2.19)

iG>αβµe (k, t) =

∑r

[ur,α

k,2 ur,βk,1

{αr

k,µ(t), αr†k,e

}e−iωk,2t + vr,α

−k,2 ur,βk,1

{βr†−k,µ(t), αr†

k,e

}eiωk,2t

].

(2.20)

Here and in the following αr†k,e stands for αr†

k,e(0). These relations show that the definition of the

transition amplitudes singles out one anticommutator by time :

Pree(k, t) ≡ i ur†

k,1eiωk,1t G>

ee(k, t) γ0urk,1 =

{αr

k,e(t), αr†k,e

}

= cos2θ + sin2θ[|Uk|2e−i(ωk,2−ωk,1)t + |Vk|2ei(ωk,2+ωk,1)t

], (2.21)

Pree(k, t) ≡ i vr†

−k,1e−iωk,1t G>

ee(k, t) γ0urk,1 =

{βr†−k,e(t), α

r†k,e

}

= εr |Uk||Vk| sin2θ[ei(ωk,2−ωk,1)t − e−i(ωk,2+ωk,1)t

], (2.22)

Prµe(k, t) ≡ i ur†

k,2eiωk,2t G>

µe(k, t) γ0urk,1 =

{αr

k,µ(t), αr†k,e

}

= |Uk| cosθ sinθ[1 − ei(ωk,2−ωk,1)t

], (2.23)

Prµe(k, t) ≡ i vr†

−k,2e−iωk,2t G>

µe(k, t) γ0urk,1 =

{βr†−k,µ(t), αr†

k,e

}

= εr |Vk| cosθ sinθ[1 − e−i(ωk,2+ωk,1)t

]. (2.24)

21

All other anticommutators with α†e vanish. Notice that in the perturbative case, there were only

two non-zero amplitudes, i.e. Pee and Pµe. We have here two anomalous contributions to the

transition amplitude, represented by Pree and Pr

µe.

The probability amplitude is now correctly normalized: limt→0+Pee(k, t) = 1, and Pee, Pµe,

Pµe go to zero in the same limit t → 0+ . Moreover,

|Pree(k, t)|2 + |Pr

ee(k, t)|2 +∣∣∣Pr

µe(k, t)∣∣∣2+

∣∣∣Prµe(k, t)

∣∣∣2

= 1 , (2.25)

as the conservation of the total probability requires. We also note that the above transition

probabilities are independent of the spin orientation.

At this point we need to understand how to interpret the above result and in particular how

to extract from it the relevant informations, namely the number of neutrinos of both flavors at

time t > 0.

For notational simplicity, we now drop the momentum and spin indices. The momentum

is taken to be aligned along the quantization axis, k = (0, 0, |k|). It is also understood that

antiparticles carry opposite momentum to that of the particles. At time t = 0 the vacuum state

is |0〉e,µ and the one electronic neutrino state is

|νe〉 ≡ α†e|0〉e,µ =[cos θ α†1 + |U | sin θ α†2 − ε |V | sin θ α†1α

†2β

†1

]|0〉1,2 . (2.26)

We thus see that in this state a multiparticle component is present, disappearing in the relativistic

limit |k| À √m1m2 : in this limit the (quantum-mechanical) Pontecorvo state is recovered.

Eq.(2.26) shows that we are actually dealing with a theory which is intrinsecally a many–particle

one: we cannot define the one–neutrino flavor state as a sum of one–particle mass states only!

The presence of the multiparticle component complicates the understanding of the time

evolution of the flavor state. Let us indeed define the time evoluted of |νe〉 as |νe(t)〉 = e−iHt|νe〉.Notice that the flavour vacuum |0〉e,µ is not eigenstate of the free Hamiltonian H. It “rotates”

under the action of the time evolution generator: one indeed finds limV→∞ e,µ〈0 | 0(t)〉e,µ = 0.

Thus at different times we have unitarily inequivalent flavour vacua (in the limit V →∞): this

expresses the different particle content of these (coherent) states and it is direct consequence of

the fact that flavour states are not mass eigenstates. The flavour content of the time evoluted

electronic neutrino state is found to be

|νe(t)〉 =[η1(t) α†e + η2(t) ᆵ + η3(t) α†eα

†µβ

†e + η4(t) α†eα

†µβ

†µ

]|0〉e,µ , (2.27)

with4∑

i=1|ηi(t)|2 = 1.

However, as already pointed out, we cannot directly compare flavor states at different times,

so we need to find a different approach to the problem.

22

2.2.2 The exact formula for neutrino oscillations

Let us then consider the Lagrangian written in the flavor basis (subscript f denotes here

flavor)

L = Ψf (i 6∂ −M) Ψf (2.28)

where ΨTf = (νe, νµ) and M =

(me meµ

meµ mµ

).

In analogy with what was done in §1.3, consider now the variation of the above Lagrangian

under the following transformation:

Ψ′f = e

i2αiτi Ψf i = 1, 2, 3. (2.29)

We have

δL = −∂µ jµf =

1

2Ψf [M, τ ] Ψf (2.30)

jµf =

1

2Ψf γµ τ Ψf (2.31)

and obtain the currents:

jµf,1 =

1

2[νe γµ νµ + νµ γµ νe] (2.32)

jµf,2 =

i

2[νe γµ νµ − νµ γµ νe] (2.33)

jµf,3 =

1

2[νe γµ νe − νµ γµ νµ] (2.34)

and

jµf,0 =

1

2[νe γµ νe + νµ γµ νµ] (2.35)

Again, the charges J0f,i ≡

∫d3x j0

f,i, i = 1, 2, 3, satisfy the su(2) algebra: [J0f,i, J

0f,j] =

i εijk J0f,k.

The Casimir J0f,0 is proportional to the total charge J0

f,0 = S0 = 12Q. However now, because

of the off–diagonal (mixing) terms in the mass matrix M , J0f,3 is not conserved anymore. This

implies an exchange of charge between νe and νµ, resulting in the phenomenon of neutrino

oscillations.

Let us indeed define the flavor charges as

Qe(t) ≡ J0f,0 + J0

f,3(t) (2.36)

Qµ(t) ≡ J0f,0 − J0

f,3(t) (2.37)

23

where Qe(t) + Qµ(t) = Q.

It is then clear that the oscillation formulas are obtained by taking expectation values of the

above charges on the neutrino state.

In terms of the flavor operators, the flavor charge operators read

Qσ(t) =∑

k,r

(αr†

k,σ(t)αrk,σ(t) − βr†

−k,σ(t)βr−k,σ(t)

), σ = e, µ. (2.38)

We thus have:

e,µ〈0|Qe(t)|0〉e,µ = e,µ〈0|Qµ(t)|0〉e,µ = 0 , (2.39)

Qe(t) ≡ 〈νe|Qe(t)|νe〉 =∣∣∣{αe(t), α

†e

}∣∣∣2

+∣∣∣{β†e(t), α

†e

}∣∣∣2

, (2.40)

Qµ(t) ≡ 〈νe|Qµ(t)|νe〉 =∣∣∣{αµ(t), α†e

}∣∣∣2

+∣∣∣{↵(t), α†e

}∣∣∣2

. (2.41)

Charge conservation is obviously ensured at any time: Qe(t) +Qµ(t) = 1. The oscillation

formula for the flavour charges is then

Qe(k, t) =∣∣∣{αr

k,e(t), αr†k,e

}∣∣∣2

+∣∣∣{βr†−k,e(t), α

r†k,e

}∣∣∣2

(2.42)

= 1− sin2(2θ)[|Uk|2 sin2

(ωk,2 − ωk,1

2t)

+ |Vk|2 sin2(

ωk,2 + ωk,1

2t)]

,

Qµ(k, t) =∣∣∣{αr

k,µ(t), αr†k,e

}∣∣∣2

+∣∣∣{βr†−k,µ(t), αr†

k,e

}∣∣∣2

(2.43)

= sin2(2θ)[|Uk|2 sin2

(ωk,2 − ωk,1

2t)

+ |Vk|2 sin2(

ωk,2 + ωk,1

2t)]

.

This result is exact. There are two differences with respect to the usual formula for neutrino

oscillations: the amplitudes are energy dependent, and there is an additional oscillating term.

For |k| À √m1m2, |Uk|2 → 1 and |Vk|2 → 0 and the traditional oscillation formula is recovered.

Work is in progress for the analysis of possible phenomenological relevance of this result

to the present experiments. The case of oscillations involving three flavors and the coherence

properties of the neutrino state (which we have proved to be a coherent state) are in particular

under consideration.

2.2.3 Discussion

A number of considerations about the oscillation formulas eqs.(2.42),(2.43) is in order at this

point.

First of all, we see from the above discussion, that the above quantities have a sense as

statistical averages, i.e. as mean values. This is because, as we have shown, the structure of the

theory for mixed field is that of a many–body theory, where does not make sense to talk about

single–particle states. This situation has a formal analogy with Thermal Field Theory (i.e. QFT

at finite temperature), where only statistical averages are well defined.

24

This situation contrast with the simple quantum mechanical picture of §2.1, which however is

recovered in the ultra–relativistic limit. There, the approximate Pontecorvo result is recovered.

It is an interesting question, however, to ask if there are features of the QFT approach which

persist in this limit: one of this is certainly the coherence properties of the full neutrino state.

We now show [14] that the above results are consistent with the generalization introduced in

§1.4, i.e. that the exact oscillation probabilities are independent of the arbitrary mass parame-

ters.

It can be indeed explicitely checked, through somewhat long direct calculation that∣∣∣{αr

k,e(t), αr†k,e(0)

}∣∣∣2+

∣∣∣{βr†−k,e(t), α

r†k,e(0)

}∣∣∣2

=∣∣∣{αr

k,e(t), αr†k,e(0)

}∣∣∣2+

∣∣∣{βr†−k,e(t), α

r†k,e(0)

}∣∣∣2

(2.44)

∣∣∣{αr

k,µ(t), αr†k,e(0)

}∣∣∣2

+∣∣∣{βr†−k,µ(t), αr†

k,e(0)}∣∣∣

2=

∣∣∣{αr

k,µ(t), αr†k,e(0)

}∣∣∣2

+∣∣∣{βr†−k,µ(t), αr†

k,e(0)}∣∣∣

2

(2.45)

which ensure the cancellation of the arbitrary mass parameters.

We have seen that the quantities in eqs.(2.42),(2.43) are nothing but the expectation values

(on the electron neutrino state at time t) of the charge operators Qσ. We have also seen that the

operator for the total charge Qe + Qµ is the Casimir operator for the su(2) algebra associated with

the mixing transformations eq.(2.7), and consequently it commutes with the mixing generator

(1.7) (and (1.83)).

However, the important point for the full understanding of the result (2.44),(2.45), is that the

charge operators Qσ are invariant under the action of the Bogoliubov generator eq.(1.90), i.e.

Qσ = Qσ, where Qσ ≡ α†σασ−β†σβσ. Besides the direct computations leading to eqs.(2.44),(2.45),

such an invariance provides a strong and immediate proof of the independence of the oscillation

formula from the µσ parameters. Thus, in some sense, expectation values of the flavor charge

operators are the only meaningful quantities from a physical point of view in the context of

the above theory, all other operators having expectation values depending on the arbitrarity

parameters above introduced.

As a last remark, we notice that the mass parameter µσ can be seen as the “bare” mass of

the corresponding field and therefore it can be given any arbitrary value. Indeed, for θ = 0 the

transformation (1.82) reduces to the transformation generated by Iµ(t) given by eq.(1.84): now

note that this is nothing but a Bogoliubov transformation which, at θ = 0, relates unmixed field

operators, αj and, say, aj(ξσ,j), of masses mj and µσ, respectively. In the language of the LSZ

formalism of QFT[16, 15], the αj refer to physical (free) fields and the aj(ξσ,j) to Heisenberg

(interacting) fields. In the infinite volume limit, the Hilbert spaces where the operators αj

and aj are respectively defined, turn out to be unitarily inequivalent spaces. Moreover, the

transformation parameter ξσ,j acts as a label specifying Hilbert spaces unitarily inequivalent

among themselves (for each (different) value of the µσ mass parameter). The crucial point is

that the physically relevant space is the one associated with the observable physical mass mj,

the other ones being associated with the bare masses µσ. It can be shown[16] that the masses

µσ dynamically acquire a convenient mass shift term such that the asymptotic physical αj-fields

are associated with the physical mass mj and the arbitrariness intrinsic to the bare mass µσ does

not affect the observables.

25

Therefore, in principle any one of the ξ-parameterized Hilbert spaces can be chosen to work

with (in other words, the bare masses can be given any arbitrary value). Since, however, one

is interested in observable quantities, in the LSZ formalism the space one chooses to work with

is the free physical field space (associated to the αj operator fields, in our case). This is the

“particular” choice made originally. In the generalized formalism instead, by means of the

Bogoliubov transformation explicitly given by eq.(1.85) written for θ = 0, one first moves to the

operators aj(ξσ,j), leaving the ξ value unspecified (i.e. for arbitrary mass parameter µσ) and

then one considers the mixing problem. Of course, at the end of the computations observable

quantities should not depend on the arbitrary parameters, as indeed we have proven it happens

to be.

26

3. Berry Phase for Oscillating Neutrinos

In this Section we show how the notion of Berry phase [22] enters the physics of mixing by

considering the example of neutrino oscillations.

Since its discovery [22], the Berry phase has attracted much interest [23] at theoretical as

well as at experimental level. This interest arises because the Berry phase reveals geometrical

features of the systems in which it appears, which go beyond the specific dynamical aspects

and as such contribute to a deeper characterization of the physics involved. The successful

experimental findings in many different quantum systems [23] stimulate further search in this

field.

Aimed by these motivations, we show that the geometric phase naturally appears in the

standard Pontecorvo formulation of neutrino oscillations.

Our result shows that the Berry phase associated to neutrino oscillations is a function of

the mixing angle only. We suggest that such a result has phenomenological relevance: since

geometrical phases are observable, the mixing angle can be (at least in principle) measured

directly, i.e. independently from dynamical parameters as the neutrino masses and energies.

Although in the following we treat the neutrino case, we stress that our result holds in general,

also in the case of mixed bosons (Kaons, η′s, etc..).

Let us first consider the two flavour case[1]:

|νe〉 = cos θ |ν1〉 + sin θ |ν2〉|νµ〉 = − sin θ |ν1〉 + cos θ |ν2〉 . (3.46)

The electron neutrino state at time t is [1]

|νe(t)〉 ≡ e−iHt|νe(0)〉 = e−iω1t(cos θ |ν1〉 + e−i(ω2−ω1)t sin θ |ν2〉

), (3.47)

where H|νi〉 = ωi|νi〉, i = 1, 2. Our conclusions will also hold for the muon neutrino state, with

due changes which will be explicitly shown when necessary.

The state |νe(t)〉, apart from a phase factor, reproduces the initial state |νe(0)〉 after a period

T = 2πω2−ω1

:

|νe(T )〉 = eiφ|νe(0)〉 , φ = − 2πω1

ω2 − ω1

. (3.48)

We now show how such a time evolution does contain a purely geometric part, i.e. the

Berry phase. It is a straightforward calculation to separate the geometric and dynamical phases

27

following the standard procedure [24]:

βe = φ +∫ T

0〈νe(t)| i∂t |νe(t)〉 dt

= − 2πω1

ω2 − ω1

+2π

ω2 − ω1

(ω1 cos2 θ + ω2 sin2 θ) = 2π sin2 θ . (3.49)

We thus see that there is indeed a non-zero geometrical phase β, related to the mixing angle θ,

and that it is independent from the neutrino energies ω1, ω2 and masses m1,m2. In a similar

fashion, we obtain the Berry phase for the muon neutrino state:

βµ = φ +∫ T

0〈νµ(t)| i∂t |νµ(t)〉 dt = 2π cos2 θ . (3.50)

Note that βe + βµ = 2π. We can thus rewrite (3.48) as

|νe(T )〉 = ei2π sin2 θe−iωeeT |νe(0)〉 , (3.51)

where we have used the notation

〈νe(t)| i∂t |νe(t)〉 = 〈νe(t)| H |νe(t)〉 = ω1 cos2 θ + ω2 sin2 θ ≡ ωee . (3.52)

We will also use

〈νµ(t)| i∂t |νµ(t)〉 = 〈νµ(t)| H |νµ(t)〉 = ω1 sin2 θ + ω2 cos2 θ ≡ ωµµ , (3.53)

〈νµ(t)| i∂t |νe(t)〉 = 〈νµ(t)| H |νe(t)〉 =1

2(ω2 − ω1) sin 2θ ≡ ωµe , (3.54)

with ωeµ = ωµe.

In order to better understand the meaning of (3.49)-(3.51), we observe that, as well known,

|νe〉 is not eigenstate of the Hamiltonian, and

〈νe(0)|νe(t)〉 = e−iω1t cos2 θ + e−iω2t sin2 θ . (3.55)

Thus, as an effect of time evolution, the state |νe〉 “rotates” as shown by eq.(3.55). However, at

t = T ,

〈νe(0)|νe(T )〉 = eiφ = eiβee−iωeeT , (3.56)

i.e. |νe(T )〉 differs from |νe(0)〉 by a phase φ, part of which is a geometric “tilt” (the Berry phase)

and the other part is of dynamical origin. In general, for t = T + τ , we have

〈νe(0)|νe(t)〉 = eiφ 〈νe(0)|νe(τ)〉= ei2π sin2 θe−iωeeT

(e−iω1τ cos2 θ + e−iω2τ sin2 θ

). (3.57)

Also notice that 〈νµ(t)|νe(t)〉 = 0 for any t. However,

〈νµ(0)|νe(t)〉 =1

2eiφe−iω1τ sin 2θ

(e−i(ω2−ω1)τ − 1

), for t = T + τ , (3.58)

28

which is zero only at t = T . Eq.(3.58) expresses the fact that |νe(t)〉 “oscillates”, getting a

component of muon flavour, besides getting the Berry phase. At t = T , neutrino states of

different flavour are again each other orthogonal states.

Generalization to n−cycles is also interesting. Eq.(3.49) (and (3.50)) can be rewritten for

the n−cycle case as

β(n)e =

∫ nT

0〈νe(t)| i∂t − ω1 |νe(t)〉 dt = 2π n sin2 θ , (3.59)

and eq.(3.57) becomes

〈νe(0)|νe(t)〉 = einφ 〈νe(0)|νe(τ)〉 , for t = nT + τ . (3.60)

Similarly eq.(3.58) gets the phase einφ instead of eiφ. Eq.(3.59) shows that the Berry phase acts

as a “counter” of neutrino oscillations, adding up 2π sin2 θ to the phase of the (electron) neutrino

state after each complete oscillation.

Eq.(3.59) is interesting especially because it can be rewritten as

β(n)e =

∫ nT

0〈νe(t)| U−1(t) i∂t

(U(t) |νe(t)〉

)dt =

∫ nT

0〈νe(t)| i∂t|νe(t)〉 = 2π n sin2 θ , (3.61)

with U(t) = e−if(t), where f(t) = f(0)− ω1t, and

|νe(t)〉 ≡ U(t)|νe(t)〉 = e−if(0)(cos θ |ν1〉 + e−i(ω2−ω1)t sin θ |ν2〉

). (3.62)

Eq.(3.61) actually provides an alternative way for defining the Berry phase [24], which makes

use of the state |νe(t)〉 given in eq.(3.62). From eq.(3.62) we also see that time evolution only

affects the |ν2〉 component of the state |νe(t)〉, so that we have

i∂t|νe(t)〉 = (ω2 − ω1)e−if(0)e−i(ω2−ω1)t sin θ|ν2〉

= (H − ω1)e−if(0)

(cos θ |ν1〉 + e−i(ω2−ω1)t sin θ |ν2〉

)

= (H − ω1)|νe(t)〉 . (3.63)

We thus understand that eq.(3.61) directly gives us the geometric phase because the quantity

i〈νe(t)| ˙νe(t)〉 dt is the overlap of |νe(t)〉 with its “parallel transported” at t + dt.

Another geometric invariant which can be considered is

s =∫ nT

0ωµe dt = π n sin 2θ . (3.64)

Since ωµe is the energy shift from the level ωee caused by the flavour interaction term in the

Hamiltonian [1], it is easily seen that

ω2µe = ∆E2 ≡ 〈νe(t)|H2|νe(t)〉 − 〈νe(t)|H|νe(t)〉2 , (3.65)

and then we recognize that eq.(3.64) gives the geometric invariant discussed in ref.[25], where

it is defined quite generally as s =∫

∆E(t)dt. It has the advantage to be well defined also for

systems with non-cyclic evolution.

29

We now consider the case of three flavour mixing. Consider again the electron neutrino state

at time t [10] (see Appendix):

|νe(t)〉 = e−iω1t(cos θ12 cos θ13 |ν1〉 + e−i(ω2−ω1)t sin θ12 cos θ13 |ν2〉 +

e−i(ω3−ω1)teiδ sin θ13 |ν3〉)

, (3.66)

where δ is the analogous of the CP violating phase of the CKM matrix. Let us consider the

particular case in which the two frequency differences are proportional: ω3 − ω1 = q(ω2 − ω1),

with q a rational number. In this case the state (3.66) is periodic over a period T = 2πω2−ω1

and

we can use the previous definition of Berry phase:

β = φ +∫ T

0〈νe(t)| H |νe(t)〉 dt = 2π

(sin2 θ12 cos2 θ13 + q sin2 θ13

), (3.67)

which of course reduces to the result (3.49) for θ13 = 0. Eq.(3.67), however, shows that β is not

completely free from dynamical parameters since the appearance in it of the parameter q.

Although because of this, β is not purely geometric, nevertheless it is interesting that it does

not depend on the specific frequencies ωi, i = 1, 2, 3 , but on the ratio of their differences only.

This means that we have now (geometric) classes labelled by q.

It is in our plan to calculate the geometric invariant s for the three flavour neutrino state:

this requires consideration of the projective Hilbert space in the line of ref.[25, 26].

The geometric phase is generally associated with a parametric dependence of the time evo-

lution generator. In such cases, the theory exhibits a gauge-like structure which may become

manifest and characterizing for the physical system, e.g. in the Bohm-Aharonov effect[27].

It is then natural to ask the question about a possible gauge structure in the case considered

here. Let us see how, indeed, a covariant derivative may be here introduced.

Let us consider the evolution of the mass eigenstates

i∂t|νi(t)〉 = H |νi(t)〉 , (3.68)

where i = 1, 2. These equations are invariant under the following (local in time) gauge transfor-

mation

|νi(t)〉 → |νi(t)〉 ≡ U(t)|νi(t)〉 = e−if(t)|νi(t)〉 , (3.69)

provided [H,U(t)] = i∂tU(t), i.e.

U−1(t)HU(t) = H + U−1(t)i∂tU(t) = H + ∂tf(t) . (3.70)

This suggests that, by rewriting (3.68) as

(i∂t − H) |νi(t)〉 = 0 , (3.71)

we can consider Dt ≡ ∂t + iH as the “covariant derivative”:

Dt → D′t = U−1(t)DtU(t) = Dt . (3.72)

30

We have indeed

iDt|νi(t)〉 = iDtU(t)|νi(t)〉 = iU(t)Dt|νi(t)〉 = 0 . (3.73)

which in fact expresses the invariance of eq.(3.68) under (3.69).

We thus see that the time dependent canonical transformation of the Hamiltonian, eq.(3.70)

and eq.(3.69) play the role of a local (in time) gauge transformation. Note that the state |νe(t)〉of eq.(3.62) is a superposition of the states |νi(t)〉.

The role of the “diabatic” force arising from the term U−1(t)i∂tU(t) has been considered in

detail elsewhere [28].

Summarizing, we have shown that there is a Berry phase built in in the neutrino oscillations,

we have explicitly computed it in the cyclic two-flavour case and in a particular case of three

flavour mixing. The result also applies to other (similar) cases of particle oscillations.

We have noticed that a measurement of this Berry phase would give a direct measurement

of the mixing angle independently from the values of the masses.

The above analysis in terms of “tilting” of the state in its time evolution, parallel transport

and covariant derivative also suggests that field mixing may be seen as the result of a curvature

in the state space. The Berry phase appears to be a manifestation of such a curvature.

Finally, we remark that the recognition of the geometric phase associated to mixed states

also suggests to us that a similar geometric phase also occurs in entangled quantum states which

can reveal to be relevant in completely different contexts than particle oscillations, namely in

quantum computation [29].

31

Appendix A: Three Flavor Mixing

We consider the mixing of three fermion fields [10]:

νe(x)

νµ(x)

ντ (x)

= M

ν1(x)

ν2(x)

ν3(x)

(A.1)

where M is the mixing matrix.

Among the various possible parameterizations of the three fields mixing matrix, we choose

to work with the following one:

M =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

(A.1b)

with cij ≡ cos θij, sij ≡ sin θij, since it is the familiar parameterization of CKM matrix [1].

To generate the M matrix, we define

G12(θ12) = exp(θ12L12) , G23(θ23) = exp(θ23L23) , G13(θ13) = exp(θ13L13) (A.2)

where

L12 ≡∫

d3x(ν†1(x)ν2(x)− ν†2(x)ν1(x)

)(A.3a)

L23 ≡∫

d3x(ν†2(x)ν3(x)− ν†3(x)ν2(x)

)(A.3b)

L13 ≡∫

d3x(e−iδ ν†1(x)ν3(x)− eiδ ν†3(x)ν1(x)

)(A.3c)

so that

ναe (x) = G−1

12 G−113 G−1

23 να1 (x) G23G13G12 (A.4a)

ναµ (x) = G−1

12 G−113 G−1

23 να2 (x) G23G13G12 (A.4b)

νατ (x) = G−1

12 G−113 G−1

23 να3 (x) G23G13G12 . (A.4c)

The matrix M is indeed obtained by using the following relations:

[να1 (x), L12] = να

2 (x) , [να1 (x), L23] = 0 , [να

1 (x), L13] = e−iδ να3 (x) (A.5a)

32

[να2 (x), L12] = −να

1 (x) , [να2 (x), L23] = να

3 (x) , [να2 (x), L13] = 0 (A.5b)

[να3 (x), L12] = 0 , [να

3 (x), L23] = −να2 (x) , [να

3 (x), L13] = −eiδ να1 (x) . (A.5c)

Notice that the phase δ is unavoidable for three fields mixing, while it can be incorporated

in the definition of the fields in the case of two flavor mixing.

The vacuum in the flavor representation is:

|0〉eµτ = G−112 G−1

13 G−123 |0〉123 . (A.6)

We do not give here the explicit form of this state, which is very complicated and is a

combination of all possible couples αr†k,iβ

r†−k,j with i, j = 1, 2, 3. Nevertheless, we can obtain

physical informations from the structure of the annihilators αrk,l, βr

k,l (l = e, µ, τ). In the reference

frame k = (0, 0, |k|) we obtain:

αrk,e = c12c13 αr

k,1 + s12c13

(Uk∗

12 αrk,2 + εrV k

12 βr†−k,2

)+ e−iδ s13

(Uk∗

13 αrk,3 + εrV k

13 βr†−k,3

), (A.7a)

αrk,µ =

(c12c23 − eiδ s12s23s13

)αr

k,2 −(s12c23 + eiδ c12s23s13

) (Uk

12 αrk,1 − εrV k

12 βr†−k,1

)+

+ s23c13

(Uk∗

23 αrk,3 + εrV k

23 βr†−k,3

), (A.7b)

αrk,τ = c23c13 αr

k,3 −(c12s23 + eiδ s12c23s13

) (Uk

23 αrk,2 − εrV k

23 βr†−k,2

)+

+(s12s23 − eiδ c12c23s13

) (Uk

13 αrk,1 − εrV k

13 βr†−k,1

), (A.7c)

βr−k,e = c12c13 βr

−k,1 + s12c13

(Uk∗

12 βr−k,2 − εrV k

12 αr†k,2

)+ eiδ s13

(Uk∗

13 βr−k,3 − εrV k

13 αr†k,3

), (A.7d)

βr−k,µ =

(c12c23 − e−iδ s12s23s13

)βr−k,2 −

(s12c23 + e−iδ c12s23s13

) (Uk

12 βr−k,1 + εr V k

12 αr†k,1

)+

+ s23c13

(Uk∗

23 βr−k,3 − εr V k

23 αr†k,3

), (A.7e)

βr−k,τ = c23c13 βr

−k,3 −(c12s23 + e−iδ s12c23s13

) (Uk

23 βr−k,2 + εrV k

23 αr†k,2

)+

+(s12s23 − e−iδ c12c23s13

) (Uk

13 βr−k,1 + εrV k

13 αr†k,1

). (A.7f)

with

V kij = |V k

ij | ei(ωk,j+ωk,i)t , Ukij = |Uk

ij| ei(ωk,j−ωk,i)t (A.8a)

33

|Ukij| =

(ωk,i + mi

2ωk,i

) 12

(ωk,j + mj

2ωk,j

) 12

(1 +

|k|2(ωk,i + mi)(ωk,j + mj)

)(A.8b)

|V kij | =

(ωk,i + mi

2ωk,i

) 12

(ωk,j + mj

2ωk,j

) 12

( |k|(ωk,j + mj)

− |k|(ωk,i + mi)

)(A.8c)

|Ukij|2 + |V k

ij |2 = 1 (A.8d)

where i, j = 1, 2, 3 and j > i.

The following identities hold:

(V k

23Vk∗13 + Uk∗

23 Uk13

)= Uk

12 ,(V k

23Uk∗13 − Uk∗

23 V k13

)= −V k

12 (A.9a)

(Uk

12Uk23 − V k∗

12 V k23

)= Uk

13 ,(Uk

23Vk12 + Uk∗

12 V k23

)= V k

13 (A.9b)

(V k∗

12 V k13 + Uk∗

12 Uk13

)= Uk

23 ,(V k

12Uk13 − Uk

12Vk13

)= −V k

23 . (A.9c)

From eqs.(A.7) we observe that, in contrast with the case of two flavors mixing, the conden-

sation densities are now different for different flavors (see eq.(1.34)):

123〈0|Nk,rαe|0〉123 = 123〈0|Nk,r

βe|0〉123 = s2

12c213 |V k

12|2 + s213 |V k

13|2 , (A.10a)

123〈0|Nk,rαµ|0〉123 = 123〈0|Nk,r

βµ|0〉123 ==

∣∣∣s12c23 + eiδ c12s23s13

∣∣∣2 |V k

12|2 + s223c

213 |V k

23|2 , (A.10b)

123〈0|Nk,rατ|0〉123 = 123〈0|Nk,r

βτ|0〉123 =

∣∣∣c12s23 + eiδ s12c23s13

∣∣∣2 |V k

23|2+∣∣∣s12s23 − eiδ c12c23s13

∣∣∣2 |V k

13|2 .

(A.10c)

34

Appendix B: The oscillation formula for mixed bosons

We present here some preliminary results [18] about the oscillation of mixed bosons, also in

connection with ref.[21] where the study of oscillations for the mixed mesons η, η′ and KL, KS

was performed on the line of ref.[10].

We now calculate the oscillation formula, in the line of what has been done in §2.2.2 for

neutrinos.

Let us define the state of the aA particle at time t as |aA(t)〉 ≡ e−iHta†A|0〉A,B and consider

the following quantities:

〈aA(t)| a†AaA |aA(t)〉 =∣∣∣[aA(t), a†A

]∣∣∣2

+ A,B〈0(t)|a†AaA|0(t)〉A,B (B.1a)

〈aA(t)| b†AbA |aA(t)〉 =∣∣∣[b†A(t), a†A

]∣∣∣2

+ A,B〈0(t)|b†AbA|0(t)〉A,B (B.1b)

Thus the net “A-charge” on the state |aA(t)〉 is

QA(t) ≡ 〈aA(t)| a†AaA − b†AbA |aA(t)〉 =∣∣∣[aA(t), a†A

]∣∣∣2 −

∣∣∣[b†A(t), a†A

]∣∣∣2

(B.2)

Similarly, for the net content of “B-charge” of |aA(t)〉, we obtain

QB(t) ≡ 〈aA(t)| a†BaB − b†BbB |aA(t)〉 =∣∣∣[aB(t), a†A

]∣∣∣2 −

∣∣∣[b†B(t), a†A

]∣∣∣2

(B.3)

The total charge is conserved, as follows from the following equation:

∣∣∣[aA(t), a†A

]∣∣∣2 −

∣∣∣[b†A(t), a†A

]∣∣∣2

+∣∣∣[aB(t), a†A

]∣∣∣2 −

∣∣∣[b†B(t), a†A

]∣∣∣2

= 1 (B.4)

The explicit calculation for the two charges gives

QA(k, t) = 1− sin2(2θ)[|Uk|2 sin2

(ωk,2 − ωk,1

2t)− |Vk|2 sin2

(ωk,2 + ωk,1

2t)]

, (B.5)

QB(k, t) = sin2(2θ)[|Uk|2 sin2

(ωk,2 − ωk,1

2t)− |Vk|2 sin2

(ωk,2 + ωk,1

2t)]

. (B.6)

This formula exhibit a negative sign in front of |Vk|2 in contrast with the fermion case. This

means that there is a non–zero probability of transition into antiparticles: this fact does not

produce any violation of charge conservation, since the conserved charge is the total charge of

the system of two mixed fields.

Furthermore, if we consider that for boson mixing the Bogoliubov coefficients are not limited

to the value 1/2 as happens for fermions (see Section 1), we see that the effects in the case of

meson oscillations can be more relevant than in the neutrino case.

35

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