On flavor violation for massive and mixed neutrinos
aa r X i v : . [ h e p - ph ] D ec On flavor violation for massive and mixed neutrinos
M. Blasone a , A. Capolupo a , C.R. Ji b and G. Vitiello aa Dipartimento di Matematica e Informatica, Universit`a di Salerno and Istituto Nazionale di FisicaNucleare, Gruppo Collegato di Salerno, 84100 Salerno, Italy, b Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA
We discuss flavor charges and states for interacting mixed neutrinos in QFT. We show that the Pontecorvostates are not eigenstates of the flavor charges. This implies that their use in describing the flavor neutrinosproduces a violation of lepton charge conservation in the production/detection vertices. The flavor states definedas eigenstates of the flavor charges give the correct representation of mixed neutrinos in charged current weakinteraction processes.
In this report we analyze the definition of theflavor charges in the canonical formalism for in-teracting (Dirac) neutrinos with mixing. On thisbasis, we study the flavor states for mixed neu-trinos in the QFT formalism [1]-[7] and in thePontecorvo formalism [8]-[11]. We show thatPontecorvo mixed states are not eigenstates ofthe neutrino flavor charges and we estimate howmuch the leptonic charge is violated on thesestates.A realistic description of flavor neutrinos startsby taking into account the (charged current) weakinteraction processes in which they are created,together with their charged lepton counterparts.In the Standard Model, flavor is strictly con-served in the production and detection vertices ofsuch interactions. The flavor violations are dueonly to loop corrections and are thus expectedto be extremely small [12]. Therefore, we definethe flavor neutrino states as eigenstates of flavorcharges. This is obtained in a QFT treatmentwhere the flavor charges are defined in the usualway from the symmetry properties of the neutrinoLagrangian.Here we consider the decay process W + → e + + ν e and we study the case where the neu-trino mixing is present. We consider for simplic-ity the case of two generations. After spontaneoussymmetry breaking, the relevant terms of the La-grangian density for charged current weak inter- action are L = L + L int , where the free leptonLagrangian L L = (¯ ν e , ¯ ν µ ) ( iγ µ ∂ µ − M ν ) (cid:18) ν e ν µ (cid:19) + (¯ e, ¯ µ ) ( iγ µ ∂ µ − M l ) (cid:18) eµ (cid:19) , (1)includes the neutrino non-diagonal mass matrix M ν and the mass matrix of charged leptons M l : M ν = (cid:18) m ν e m ν eµ m ν eµ m ν µ (cid:19) , M l = (cid:18) m e m µ (cid:19) . L int is the interaction Lagrangian given by [13] L int = g √ h W + µ ( x ) ν e ( x ) γ µ (1 − γ ) e ( x )+ W + µ ( x ) ν µ ( x ) γ µ (1 − γ ) µ ( x ) + h.c. i . L is invariant under the global phase transfor-mations: e ( x ) → e iα e ( x ) , ν e ( x ) → e iα ν e ( x ) , (2)together with µ ( x ) → e iα µ ( x ) , ν µ ( x ) → e iα ν µ ( x ) . (3)These are generated by the charges Q e ( t ), Q ν e ( t ), Q µ ( t ) and Q ν µ ( t ) respectively, where Q ν e ( t ) = Z d x ν † e ( x ) ν e ( x ) , (4) Q ν µ ( t ) = Z d x ν † µ ( x ) ν µ ( x ) . (5)1Similar expressions hold for Q e , Q µ . The in-variance of the Lagrangian is then expressed by[ Q totl , L ( x )] = 0 , which guarantees the conser-vation of total lepton number. Here, Q totl is thetotal Noether (flavor) charge: Q totl = Q ν e ( t ) + Q ν µ ( t ) + Q e ( t ) + Q µ ( t ) (6)Note that the presence of the mixed neutrinomass term, i.e. the non-diagonal mass matrix M ν ,prevents the invariance of the Lagrangian L un-der the separate phase transformations (2) and(3). Consequently, Q ν e and Q ν µ are time depen-dent. However, family lepton numbers are stillgood quantum numbers if the neutrino produc-tion/detection vertex can be localized within aregion much smaller than the region where fla-vor oscillations take place. This is what happensin practice, since typically the spatial extensionof the neutrino source/detector is much smallerthan the neutrino oscillation length.We now proceed to define the flavor states aseigenstates of the flavor charges Q ν e and Q ν µ .Till now, our considerations have been essentiallyclassical. In order to define the eigenstates of theabove charges, we quantize the fields with defi-nite masses as usual. Then, the normal orderedcharge operators for free neutrinos ν , ν are: Q ν i : ≡ Z d x : ν † i ( x ) ν i ( x ) : (7)= X r Z d k (cid:16) α r † k ,i α r k ,i − β r †− k ,i β r − k ,i (cid:17) , where i = 1 , .. : denotes normal orderingwith respect to the vacuum | i , . The neutrinostates with definite masses defined as | ν r k ,i i = α r † k ,i | i , , i = 1 , , (8)are then eigenstates of Q ν and Q ν , which canbe identified with the lepton charges of neutrinosin the absence of mixing.The situation is more delicate when the mix-ing is present. In such a case, the flavor neutrinostates have to be defined as the eigenstates of theflavor charges Q ν σ ( t ) (at a given time). The re-lation between the flavor charges in the presenceof mixing and those in the absence of mixing is given by Q ν e ( t ) = cos θ Q ν + sin θ Q ν (9)+ sin θ cos θ Z d x h ν † ( x ) ν ( x ) + ν † ( x ) ν ( x ) i , and similarly for Q ν µ ( t ). Notice that the presenceof the last term in Eq.(9) forbids the construc-tion of eigenstates of the Q ν σ ( t ), σ = e, µ , in theHilbert space for free fields H , .The normal ordered flavor charge operators formixed neutrinos are then written as:: Q ν σ ( t ) :: ≡ Z d x :: ν † σ ( x ) ν σ ( x ) :: (10)= X r Z d k (cid:16) α r † k ,σ ( t ) α r k ,σ ( t ) − β r †− k ,σ ( t ) β r − k ,σ ( t ) (cid:17) , where σ = e, µ , and :: ... :: denotes normal order-ing with respect to the flavor vacuum | i e,µ , thevacuum for the Hilbert spaces of interacting fields H e,µ . At finite volume it is given by: | t ) i e,µ = G − θ ( t ) | i , , where G θ ( t ) is the mixing genera-tor. Indeed, the mixing transformations can bewritten as ν ασ ( x ) = G − θ ( t ) ν αi ( x ) G θ ( t ), where( σ, i ) = ( e, , ( µ, ν σ ( x ) at each timeare expressed as: α r k ,σ ( t ) = G − θ ( t ) α r k ,i ( t ) G θ ( t ), β r k ,σ ( t ) = G − θ ( t ) β r k ,i ( t ) G θ ( t ), where α r k ,i and β r k ,i , i = 1 , , r = 1 , | i , .Eq.(10) shows that the flavor charges are diago-nal in the flavor annihilation/creation operators.Note that : : Q ν σ ( t ) : : = G − θ ( t ) : Q ν j : G θ ( t ) , with ( σ, j ) = ( e, , ( µ, , and:: Q ν :: = :: Q ν e ( t ) :: + :: Q ν µ ( t ) ::= : Q ν : + : Q ν : = : Q ν : . (11)The flavor states are defined as eigenstates of theflavor charges Q ν σ at a reference time t = 0: | ν r k ,σ i ≡ α r † k ,σ (0) | i e,µ , σ = e, µ. (12)Let us turn now to the Pontecorvo states [8]-[11] | ν r k ,e i P = cos θ | ν r k , i + sin θ | ν r k , i , (13) | ν r k ,µ i P = − sin θ | ν r k , i + cos θ | ν r k , i . (14)They are clearly not eigenstates of the flavorcharges [7] as can be seen from Eqs.(9) and(10). In order to estimate how much the leptoncharge is violated in the usual quantum mechan-ical states, we consider the expectation values ofthe flavor charges on the Pontecorvo states. Weobtain, for the electron neutrino charge, P h ν r k ,e | :: Q ν e (0) :: | ν r k ,e i P = cos θ + sin θ + 2 | U k | sin θ cos θ + X r Z d k , (15)where | U k | ≡ u r † k , u r k , = v r †− k , v r − k , and , h | :: Q ν e (0) :: | i , = X r Z d k . (16)The infinities in Eqs.(15) and (16) may be re-moved by considering the expectation values of: Q ν σ ( t ) :, i.e. the normal ordered flavor chargeswith respect to the mass vacuum | i , . Then, , h | : Q ν e (0) : | i , = 0 . (17)However, [7] P h ν r k ,e | : Q ν e (0) : | ν r k ,e i P = cos θ + sin θ +2 | U k | sin θ cos θ < , (18) P h ν r k ,e | : Q ν µ (0) : | ν r k ,e i P = 2 sin θ cos θ ×× (1 − | U k | ) > , (19)for any θ = 0 , m = m , k = 0 . In conclusion, the correct flavor states describ-ing the neutrino oscillations must be those definedin Eq.(12).We also note that similar results about flavorviolation for massive neutrinos have been recentlydiscussed in Ref.[14], although from a differentpoint of view.
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