Flavor states of mixed neutrinos

Flavor states of mixed neutrinos M. Blasone∗ , A. Capolupo† , C. R. Ji∗∗ and G. Vitiello∗

arXiv:0709.1384v1 [hep-ph] 10 Sep 2007

∗

Istituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Salerno and DMI, Università di Salerno, Fisciano (SA) - 84084 Italy † Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT UK ∗∗ Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA

Abstract. By resorting to previous results on flavor mixing in Quantum Field Theory, we show how to consistently define flavor states of mixed neutrinos as eigenstates of the flavor charge operators. Keywords: Neutrino mixing, weak interactions PACS: 14.60.Pq; 11.15.Tk, 12.15.Ff

The issue of a proper definition of flavor states for mixed neutrinos has been object of discussion in recent years [1]-[15]: although the usual Pontecorvo states [16, 17] represent a valid tool for describing the main physical features of neutrino oscillations, it has been clear since some time that conceptual problems arise in connection with a proper definition of flavor states. In fact, it was even stated [17] that it is impossible to construct such states and a formalism has been developed with the aim to avoid their use in the calculation of oscillation probabilities [11]. The root of such difficulties has been found by studying flavor mixing at level of quantum fields. It has then emerged [1] that the vacuum for neutrino fields with definite masses turns out to be unitarily inequivalent to the vacuum for the flavor neutrino fields. The condensate structure of the flavor vacuum leads to a modification of flavor oscillation formulas [2, 6, 13, 10], exhibiting new features with respect to the usual quantum mechanical ones [11, 18]. Further developments include Lorentz invariance violation [19] and neutrino mixing contribution to the dark energy of the Universe [20]. In the Standard Model, flavor neutrinos are produced in charged current weak interaction processes, like W + → e+ + νe . At tree level, flavor charge is strictly conserved in the vertex; the presence of mixing allows for the possibility of violation of lepton number via loop corrections. Such corrections are however extremely small and practically unobservable. Therefore one can assume the tree level as a good approximation of the real processes in which neutrinos are created. This is what is done in practice: neutrinos are identified by the observation of the corresponding charged leptons, assuming flavor conservation in production/detection vertices. It is therefore essential to have a clear understanding of the neutrino flavor charges and states in the presence of mixing. One could nevertheless insist in declaring that flavor states are not to be used because everything can be written in terms of neutrino fields with definite masses and consequently solely neutrino mass eigenstates are being used. This is, however, like “to sweep the dust under the carpet”, since in this way one operates a selection in favor of the Hilbert space of the mass eigenstates. Such a choice has no mathematical basis, because of the existence of a separate Hilbert space (the one for the flavor eigenstates). It is

also lacking of physical motivations: indeed, in practice, it adopts the Pontecorvo states which are not eigenstates of the flavor charges, in contrast with the observed conservation of lepton number in the neutrino production/detection vertices. We therefore devote the present report to: a) properly define flavor charges and states and b) to estimate how much lepton charge is violated by Pontecorvo states. Let us start by considering the charges for flavor (mixed) neutrinos [7, 15]. We discuss here the case of mixing between two generations. Extension to three generations and beyond can also be done [9, 10]. For our purposes, the relevant Lagrangian density terms are L

= L0 + Lint ,

(1)

where L0 is the free lepton Lagrangian:    

νe
e µ µ + (e, ¯ µ¯ ) iγµ ∂ − Ml , L0 = ν¯ e , ν¯ µ iγµ ∂ − Mν νµ µ

(2)

including the neutrino non-diagonal mass matrix Mν and the mass matrix of charged leptons Ml :     mνe mνe µ me 0 Mν = ; Ml = . (3) mνe µ mν µ 0 mµ Lint is the charged current weak interaction Lagrangian: i g h Lint = √ Wµ+ (x) ν e (x) γ µ (1 − γ 5 ) e(x) +Wµ+ (x) ν µ (x) γ µ (1 − γ 5 ) µ (x) + h.c. . (4) 2 2 Of course, L0 can be diagonalized in terms of neutrino fields ν1 , ν2 , with definite masses m1 , m2 . Here, however, we are interested in the construction of flavor charges and therefore we consider the above Lagrangian in terms of flavor fields. To this end, we observe that L is invariant under the global phase transformations e(x) → eiα e(x) ,

νe (x) → eiα νe (x) ,

(5)

µ (x) → eiα µ (x) ,

νµ (x) → eiα νµ (x) .

(6)

together with

These are generated by Qe (t) =

Z

d x e (x)e(x) ,

Qµ (t) =

Z

d 3 x µ † (x)µ (x) ,

3

†

Qνe (t) =

Z

Qνµ (t) =

d 3 x νe† (x)νe (x) , Z

d 3 x νµ† (x)νµ (x) ,

(7) (8)

respectively. The invariance of the Lagrangian is then expressed by [Qtot l , L ] = 0,

(9)

which guarantees the conservation of the total lepton number. Here, Qtot l is the total Noether (flavor) charge: tot tot Qtot l = Qνe (t) + Qνµ (t) + Qe (t) + Qµ (t) = Qe (t) + Qµ (t) ,

(10)

Qtot e (t) = Qνe (t) + Qe (t) ,

(11)

Qtot µ (t) = Qνµ (t) + Qµ (t) .

Note that the presence of the mixed neutrino mass term, i.e. the non-diagonal mass matrix Mν , prevents the invariance of L under the separate phase transformations (5) and (6). Consequently, the flavor charges of Eq.(11) are time dependent. However, family lepton numbers are still good quantum numbers if the neutrino production/detection vertex can be localized within a region much smaller than the region where flavor oscillations take place. This is what happens in practice, since tipically the spatial extension of the neutrino source/detector is much smaller than the neutrino oscillation length. We thus look for the flavor neutrino states as eigenstates of the neutrino flavor charges Qνe and Qνµ . These operators may be expressed in terms of the (conserved) charges for the neutrinos with definite masses Qν1 and Qν2 in the following way [15]: Z

h i † † Qνe (t) = cos θ Qν1 + sin θ Qν2 + sin θ cos θ d x ν1 (x)ν2 (x) + ν2 (x)ν1 (x) , (12) Z i h Qνµ (t) = sin2 θ Qν1 + cos2 θ Qν2 − sin θ cos θ d 3 x ν1† (x)ν2 (x) + ν2† (x)ν1 (x) .(13) 2

2

3

Notice that the last term in the above expressions forbids the construction of eigenstates of the Qνσ (t), σ = e, µ , in the Hilbert space H1,2 for the fields with definite masses. Indeed the vacuum |0ie,µ for the mixed field operators νσ turns out to be orthogonal to the vacuum state |0i1,2 for the fields with definite masses ν j , with j = 1, 2 [1]. One can also show that the above neutrino flavor charge operators are diagonal in the ladder operators ασ , βσ , for the neutrino flavor fields: Z   r† r† 3 r r σ = e, µ (14) :: Qνσ (t) :: = ∑ d k αk,σ (t)αk,σ (t) − β−k,σ (t)β−k,σ (t) , r

where :: ... :: denotes normal ordering with respect to |0ie,µ . This makes straightforward the definition (at reference time t = 0) of flavor neutrino and antineutrino states as: r† |ν rk,σ i ≡ αk, σ |0ie,µ

;

r† |ν¯ rk,σ i ≡ βk, σ |0ie,µ ,

σ = e, µ .

(15)

which are thus by construction eigenstates of the operators in Eq.(14) at t = 0. Let us now turn to the quantum mechanical (Pontecorvo) flavor neutrino states [16]: |ν rk,e iP = cos θ |ν rk,1 i + sin θ |ν rk,2 i ,

|ν rk,µ iP = − sin θ |ν rk,1 i + cos θ |ν rk,2i .

(16) (17)

They are clearly not eigenstates of the flavor charges [12] as can be explicitly seen by using Eqs.(12) and (13). In order to estimate how much the flavor charge is violated in the usual quantum mechanical states, one can take the expectation values of the flavor charges on the above Pontecorvo states. Considering for example an electron neutrino state, we obtain: r P hν k,e | : Qνe

: |ν rk,eiP = cos4 θ + sin4 θ + 2|Uk | sin2 θ cos2 θ < 1,

(18)

r P hν k,e | : Qνµ

: |ν rk,eiP = 2 (1 − |Uk|) sin2 θ cos2 θ > 0,

(19)

for any θ 6= 0, m1 6= m2 , k 6= 0 and where : ... : denotes normal ordering with respect to the vacuum state |0i1,2 . In the relativistic limit, |Uk| → 1 [1, 20] and the Pontecorvo states are a good approximation for the exact charge eigenstates Eq.(15). We thank Arttu Rajantie and the organizers of PASCOS-07 for giving us the opportunity to present this work at the PASCOS conference. We also acknowledge INFN for financial support.

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