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Source lifetime dependence of neutrino oscillations - a simple wave-function derivation

When neutrinos are produced from long lived particles the life-time of the source particle affects the flavour conversion probability formula. Due to this source dependence experiments (like LSND) which use muon decay neutrinos are two orders of magnitude more sensitive to lower values of mass square difference compared to other experiments where the sources are pions or kaons.

The standard formula for neutrino oscillations is derived with the assumption [1] that the uncertainity in the initial position of the neutrino is small compared to the distance between the production and the detection sites of the neutrinos . We shall show here that in a covariant treatment, the effective spread of the netutrino wavefunction in space is given by (σ x + cτ ) where σ x is the initial spatial uncertainity of the production point of the neutrinos and τ is the lifetime of the sources (eg. pions, kaons , muons etc) whose decay produce the neutrinos. We find that the effect of long lifetime sources is to exponentially suppress the oscillation term in the conversion probability fromula. The flavor conversion probability as a function of distance , for relativistic neutrinos produced from long lived resonances, turns out to be P (ν e → ν µ ; X) = 1 2 where ∆m 2 is the mass square difference in eV 2 ,L where ∆m 2 is the mass square difference in eV 2 ,X is the detector distance in m (km), τ is the lifetime of the source particle in the lab frame in m (km) and E is the energy in MeV (GeV ). When cτ is comparable to the detector distance spatial oscillations of the conversions probability is not seen. However the neutrino mass squared difference that can be probed is lower for longer lived sources. We find that for the LSND experiment where the neutrinos are produced from muons (cτ µ = 658.65m), when fitted with the covariant oscillation formula (1) gives a bound on ∆m 2 which is two orders of magnitude lower than the corresponding bounds obtained from other similar experiments but where the neutrino sources are pions or kaons (cτ π = 7.8m).
We start with an initial wave function of a general Gaussian form The initial spread of the wavefunction in space around the mean initial position x i is denoted by σ x and the spread in the time direction around the mean initial time of production t i is denoted by σ t . The magnitudes of σ x and σ t depends on how the state is prepared. The time evolved wavefunction can be obtained from the initial wavefunction (2) by the linear superposition principle , which can be written formally as where is the probability amplitude of a particle initially located at (x, t) to be detected at another spacetime point ( for a free particle is given by the expression ( [2][3][4]) then we can use the asymptotic expansion of the Bessel function to obtain from (4) the expression for the propagation amplitude at large time-like separation This expression for the free particle propagator is valid for bosons. For fermions their is a extra factor (i ∂ + m a ) operating on the l.h.s of the expressions (4) and (6). This factor only changes the normalisation and the expressions for the conversion probability is identical for bosons and fermions. The time evolved wave-function can be evaluated by substituting the where the phase factor in the exponential is of the form and the constant coefficients have been clubbed together as the factor N which normalises the wave-function. We perform the integrations in (7) by the method of stationary phases [5]. The integral is approximated by the integrand along the trajectory where the phase is an extremum. The extremum of the phase (8) is given by One can solve for t and x using these two equations and substitute in the integrand of (7) to get the expression for the integral in the stationary phase approximation , which turns out to be where again we have clubbed together all the constants as the normalisation coefficient N ′ . Denoting and v a = (P a /E a ) which can be identifired by distance of propagation, time of propagation and the group velocity of the particle respectively we can write the expression for the time evloved wave-function compactly as The state vector of a mass eigenstate can be in the mass basis |ν a > in the form On the other hand the state of a weak interaction state say ν e is expressed as a linear combination The probability amplitute of a neutrino to be produced at (x i , t i ) as a ν e and to be detetected at (x f , t f ) as another weak eigenstate ν µ is given by and the corresponding probability is given by Restricting ourselves to mixing between two flavour generations, and denoting the mixing matrix elments as < ν a |ν e >= − < ν b |ν µ >= sinθ and < ν a |ν µ >=< ν b |ν e >= cosθ, the expression (14) for the probability reduces to the form This expression expresses the probability of a ν e produced at (x i , t i ) to be converted to a ν µ at some other spactime point (x f , t f ). In actual experiments only the distance X between the source and the detector is known but the arrival time of the neutrinos is not measured.
The expression for the conversion probability as a function of only X is obtained by taking the time average of (15). The normalisation is chosen so that Ψ a (X, T ) represents a single particle wave-function and therefore, The non-trivial contribution to the time averaging comes from the interference term of (15) This integral can be evaluated by the completing the squares, to give We express the energy, momentum and the masses in terms of their averages E = (E a + In terms of these variables (18) reduces to the form where we have retained the terms to the first order in ∆E/E, ∆P/P and ∆m/m. The energy and momenta of the neutrinos are determined by the energy-mommentum conservation laws at the production vertex. Since niether the energy nor the momenta of the remaining outgoing particles are measured, one cannot fix either the energy or the momenta of the different neutrino mass eigenstates in the linear combination state. So one cannot assume either E a = E b or P a = P b . Only the mass shell relation E 2 a = P 2 a + m 2 a for each mass eigenstate ν a ( [2,3]). In terms of the average and difference , the mass shell conditions imply, Using the relation (22) in (21) we see that the interference term reduces to the form we see that the oscillation length is L osc = (4πP/∆m 2 ) for both relativistic as well as nonrelativistic particles. The significant difference in the formula comes from the suppression factor A (21). For intereference term to be non-zero, both the terms of the suppression factor must be small. The second term in (21) is << 1 as long as X << L coh with the coherence length for relativistic neutrinos given by , This expression for the coherence length differs from the expression derived in the standard wave-packet treatments [1] by the presence of the σ t term. In all accelerator neutrino oscillation experiments the criterion X << L coh is satisfied , so the effect of this term is negligible. The first term of (21) plays a more significant role.
For relativistic neutrinos, the suppresion factor (21) reduces to the form, In the accelerator experiments σ x -the spread in the beam of primary particles is of the order of a few cm. The neutrinos are produced from the secondary decays of pions and kaons produced in the primary collision. The uncertainity in time of production of such neutrinos σ t is then given by the lifetime τ of the pions, kaons or muons which are the source of neutrinos for that particular experiment. In such situations σ x << σ t and (σ x + vσ t ) ≃ vτ and the expression (25) reduces to Using these results we see that the expression for the time average of the conversion probabilty, for relativistic neutrinos produced from long lived resonances is given by where ∆m 2 is the mass square difference ,X is the detector distance , τ is the lifetime of the source particle in the lab frame and E is the neutrino energy. In practice one averages over the energy flux n(E) of the neutrinos and the average probability which is fitted with the experimental number to obtain the allowed regions of ∆m 2 and sin 2 2θ is given formally by the expression,  Table I and plotted in Fig 1, Fig 2 and Table I plotted using the covariant oscillation formula is shown in Fig 3. The dependence of the oscillation term on the source lifetime can be understood as follows. When the time uncertainity is large the uncertainity in energy becomes small and the wave-functions are then eigenstates of energy. In the time smearing the overlap of two different energy eigenstates dissapears and therefore the oscillation term vanishes.  (27). τ is the lifetime of the neutrino source in the lab frame, < E ν > is the average ν energysource in the lab frame, < E ν > is the average ν energy, X is the detector distance and P is the experimental value of the conversion probability.

TABLES
LSND (µ) [  The region between the continuous lines is allowed by the LSND µ experiment [6] .The region between the dotted lines is allowed by the LSND π experiment [11] . Region ruled out by E776 [7] is above the dashed-dotted curve and by Karmen [8] is above dashed curve. The region above the top-most dashed curved is ruled out by CCFR [9].