Spreading of wave packets for neutrino oscillations in vacuum
aa r X i v : . [ h e p - ph ] J a n Spreading of wave packets for neutrino oscillations in vacuum
Y. F. P´erez ∗ Departamento de F´ısica Matem´atica, Instituto de F´ısica,Universidade de S˜ao Paulo, S˜ao Paulo, Brasil
C. J. Quimbay †‡ Departamento de F´ısica, Universidad Nacional de Colombia.Ciudad Universitaria, Bogot´a D.C., Colombia. (Dated: October 30, 2018)
Abstract
The effects originated in dispersion with time on spreading of wave packets for the time-integrated two-flavor neutrinooscillation probabilities in vacuum are studied in the context of a field theory treatment. The neutrino flavor states are writtenas superpositions of neutrino mass eigenstates which are described by localized wave packets. This study is performed for thelimit of nearly degenerate masses and considering an expansion of the energy until third order in the momentum. We obtainthat the time-integrated neutrino oscillation probabilities are suppressed by a factor 1 /L for the transversal and longitudinaldispersion regimes, where L is the distance between the neutrino source and the detector. † Associate researcher of Centro Internacional de F´ısica, Bogot´a D.C., Colombia. ∗ Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Knowing the nature of neutrino fields is an open problem in particle physics [1–3]. This problem might be solved,experimentally, establishing if neutrinos are: (i) Majorana fermions; (ii) Dirac fermions. For the first case, neutrinoand anti-neutrino are the same particle being described by two-component spinorial fields called Majorana fields [1–3].For the second case, neutrinos and anti-neutrinos are described by four-component spinorial fields called Dirac fields[1–3]. However, the theoretical description of neutrino oscillations leads to the same results independently if neutrinosare Majorana or Dirac fermions. This argument was established many years ago using the plane wave formalism ofquantum mechanics [4]. Moreover, the validity of this argument in the context of a quantum field theory treatmentcan also been proved using the plane wave formalism [5].The standard neutrino oscillation probabilities using the plane wave formalism have been obtained in the contextof several quantum mechanics treatments (for instance, see [6–11]). On the other hand, in the context of differentquantum field theory treatments, neutrino oscillations in vacuum have been extensively studied describing neutrinosby Dirac fields [12–24]. In particular, the effects of the spreading of the wave packets on neutrino oscillation prob-abilities have been widely investigated for the case of considering an expansion of the energy until second order inthe momentum [12–22]. For this case, the standard neutrino oscillation probabilities using the wave packet formalismare written in terms of the oscillation and coherence lengths [12–22]. Moreover, some aspects of the time effects onspreading of wave packets for neutrino oscillation probabilities have been also studied [15, 18]. Nevertheless, for thecase of considering an expansion of the energy until third order in the momentum, the study of the effects originatedin dispersion with time on spreading of wave packets for neutrino oscillation probabilities have not been studied untilnow.The main goal of this work is to study the effects originated by dispersion in time on spreading of wave packets forthe time-integrated two-flavor neutrino oscillations in vacuum. To do it, we perform an expansion of the energy untilthird order and we consider the limit of nearly degenerate neutrino masses. The time-integrated two-flavor neutrinooscillation probabilities are calculated in the context of a wave packet extension of the quantum field theory treatmentthat we previously developed for the case in which neutrinos were considered as Majorana fermions and neutrino masseigenstates were described by plane waves [5]. In the present treatment, the neutrino flavor states are considered assuperpositions of neutrino mass eigenstates described by localized wave packets. By methodological reasons, theeffects of the spreading of the wave packets are studied for two cases: (i) Considering the expansion of the energy untilsecond order in the momentum that leads to the standard time-integrated neutrino oscillation probabilities [12–22];(ii) considering the expansion of the energy until third order in the momentum that, leads to a suppression of thetime-integrated neutrino oscillation probabilities by a factor 1 /L for transversal and longitudinal dispersion regimes,where L is the distance between the neutrino source and the detector. This suppression factor is a new result in thecontext of neutrino oscillation probabilities and is in agreement with the one obtained by Naumov, whom has recentlydemonstrated for a theory of wave packets that the integral over time of both the flux and probability densities areproportional to a factor 1 /L , considering the energy expanded until third order in the momentum [25].The content of this work has been organized as follows: In section two, considering neutrinos as Majorana fermions,we show how is possible to obtain the standard two-flavor neutrino oscillation probabilities in the context of a quantumfield theory treatment, for which the mass eigenstate are described by plane waves; in section three, we extend theplane wave treatment presented in the previous section for the case in which the mass eigenstates are described bylocalized wave packets; in section four, we study the effects of spreading of wave packets by performing an expansionof the energy until second order in the momentum and we obtain the standard time-integrated neutrino oscillationprobabilities; in section five, we study the effects originated in dispersion with time on spreading of wave packets forthe time-integrated neutrino oscillation probabilities by performing an expansion of the energy until third order inthe momentum; finally in section six we present some conclusions. II. NEUTRINO OSCILLATION PROBABILITIES USING PLANE WAVES
The standard two-flavor neutrino oscillation probabilities were obtained in the context of a treatment developedin the canonical formalism of quantum field theory for the case in which neutrinos were described as Majoranafermions and neutrino mass eigenstates were described by plane waves [5]. In this treatment, the flavor neutrinoswere considered as superpositions of mass eigenstates with specific momenta. For the case of the relativistic limit( L ≃ T ) and after including a normalization constant, the standard plane wave expressions for the neutrino oscillationprobabilities [5] are obtained P P Wν e ( L ) = 1 − sin [2 θ ] sin (cid:20) ∆ m E L (cid:21) , (1)2 P Wν µ ( L ) = sin [2 θ ] sin (cid:20) ∆ m E L (cid:21) , (2)where ∆ m ≡ m − m , E is the energy of the neutrino, L is the distance between the neutrino source and thedetector and sin [2 θ ] is given by sin [2 θ ] = 4Λ L (1 + Λ L ) , (3)with θ representing the mixing angle between the two mass eigenstates in the vacuum. In the last expression Λ L isΛ L = m ν µL − m ν eL + R L m ν eL ν µL , (4)where R L is defined by means of R L = ( m ν eL − m ν µL ) + 4 m ν eL ν µL , (5)and the parameters m ν eL , m ν µL and m ν eL ν µL are related with the masses m and m of the neutrino fields ν and ν (with definite masses) through the following relations m = 12 ( m ν eL + m ν µL − R L ) , (6) m = 12 ( m ν eL + m ν µL + R L ) e − iα L , (7)being α L the Majorana complex phase [5].The expressions (1) and (2) are obtained starting from the oscillation probabilities defined by P P Wν e ( L ) = (cid:12)(cid:12) h | ˆ ν e L ( x ) (cid:12)(cid:12) ν P We L ( x ) (cid:11)(cid:12)(cid:12) , (8) P P Wν µ ( L ) = (cid:12)(cid:12) h | ˆ ν µ L ( x ) (cid:12)(cid:12) ν P We L ( x ) (cid:11)(cid:12)(cid:12) , (9)in such a way that in the space-time production point ( x ) the initial left-handed neutrino flavor state (electronneutrino flavor) (cid:12)(cid:12) ν P We L ( x ) (cid:11) is defined by the following superposition of the mass eigenstates (cid:12)(cid:12) ν P W ( x ) (cid:11) and (cid:12)(cid:12) ν P W ( x ) (cid:11)(cid:12)(cid:12) ν P We L ( x ) (cid:11) = X h = ± Λ L p L (cid:12)(cid:12) ν P W ( x ) (cid:11) + e − iα L p L (cid:12)(cid:12) ν P W ( x ) (cid:11) , (10)where a sum over helicities is taken in the superposition. The mass eigenstates (cid:12)(cid:12) ν P W ( x ) (cid:11) and (cid:12)(cid:12) ν P W ( x ) (cid:11) involvedin (10) are obtained using plane waves from the vacuum state | i as (cid:12)(cid:12) ν P Wa ( x ) (cid:11) = Ae ip a x ˆ a † a ( ~p a , h ) | i (11)where A is a normalization constant, ˆ a † a is the creation operator of a neutrino of defined mass, x is the space-timepoint where this neutrino is created, p a = ( E a , ~p a ) is the four-momentum of the mass eigenstates and a = 1 ,
2. Wehave assumed that each mass eigenstate involved in (10) has associate a specific four-momentum.In the oscillation probabilities (8) and (9), the flavor neutrino field operator ˆ ν α is defined as a superposition of fieldoperators of neutrinos with defined mass ˆ ν a by means of the expressionˆ ν α ( x ) = X a U L αa ˆ ν a ( x ) , (12)where α = e L , µ L and U L is an unitarian rotation matrix given by [5] U L = 1 p L (cid:18) Λ L e − iα L − L e − iα L (cid:19) . (13)The field operators of neutrinos with defined mass ˆ ν a involved in (12) are defined as [5]ˆ ν a ( x ) = Z d p (2 π ) / (2 E a ) / X h = ± hp E a − h | ~p | ˆ a a ( ~p, h ) χ h ( ~p ) e − ip · x − h p E a + h | ~p | ˆ a † a ( ~p, h ) χ − h ( ~p ) e ip · x i , (14)where E a = | ~p | + m a is the energy of the neutrino field with defined mass and χ h ( ~p ) are Majorana spinors withhelicity eigenvalues ±
1. 3
II. NEUTRINO OSCILLATIONS USING THE WAVE PACKETS FORMALISM
In this section, we will extend the plane wave treatment for neutrino oscillations that we have have presentedbriefly in section two, for the case in which neutrino mass eigenstates are described by localized wave packets. In thistreatment, where neutrinos are considered as Majorana fermions, we do not focus on the study of the details of theinteraction processes in which neutrinos are produced and detected. Here, on the other hand, it is assumed that wavepackets describing mass eigenstates are localized and the coefficients of their superpositions are given by the elementsof the unitarian rotation matrix U L given by (13). The matrix U L establishes a relationship between the flavor andmass eigenstates bases of neutrino fields in vacuum.There are different reasons for understand why the description of mass eigenstates using wave packets is mostappropriate to study the neutrino oscillations with respect to the description from plane waves [1–3]. Some of thesereasons are that the neutrino source and the detector are localized and there exists a spread for the neutrino momentum[1]. Additionally, we have to keep in mind that plane waves localized in some point x are in an obvious contradictionwith the Heisenberg uncertainty principle. Here, we take into account these reasons when we describe the neutrinomass eigenstates in terms of superpositions of localized wave packets. To do the last, we first consider that in a point x ≡ x µ = ( t , ~r ) is created a left-handed electron neutrino which is described by the following superposition ofneutrino mass eigenstates (cid:12)(cid:12) ν W P ( x ) (cid:11) and (cid:12)(cid:12) ν W P ( x ) (cid:11)(cid:12)(cid:12) ν W Pe L ( x ) (cid:11) = X h = ± Λ L p L (cid:12)(cid:12) ν W P ( x ) (cid:11) + e − iα L p L (cid:12)(cid:12) ν W P ( x ) (cid:11) . (15)In contrast to the expression (11), now the mass eigenstates involved in (15) are written in term of localized wavepackets in the form (cid:12)(cid:12) ν W Pa ( x ) (cid:11) = A Z d p (2 π ) / e − i ( E a t − ~p · ~r ) ψ a ( ~p, h ~p a i ) ˆ a † a ( ~p, h ) | i , (16)where A is a normalization constant, ψ a ( ~p, h ~p a i ) is a probability density function which depends on the momentum ~p a and the average momentum h ~p a i , and a = 1 ,
2. In general, ψ a ( ~p, h ~p a i ) may take any form, but it is usuallyapproximated by a Gaussian distribution assuming that is peaked around the average momentum [8, 12–24]. In thiswork, we define the probability density function as [3] ψ a ( ~p, h ~p a i ) ≈ (2 π ) − / [Det Γ] / exp (cid:20) −
14 ( ~p − h ~p a i ) k Γ kj ( ~p − h ~p a i ) j (cid:21) , (17)in such a way that ψ a = ψ a ( ~p, h ~p a i ) satisfies the conditions [3] ∂ ln ψ a ∂p (cid:12)(cid:12)(cid:12)(cid:12) ~p = h ~p a i = 0 , (18) ∂ ln ψ a ∂p k ∂p j (cid:12)(cid:12)(cid:12)(cid:12) ~p = h ~p a i = −
12 Γ kj (19)In the probability density function given by (17), we have taken the convention of summation over the Latin repeatedindex k and j . Additionally, we have assumed that the dispersion over the mass eigenstates (cid:12)(cid:12) ν W P ( x ) (cid:11) and (cid:12)(cid:12) ν W P ( x ) (cid:11) is the same, because these eigenstates are created simultaneously by the same weak production process. This fact isthe reason that justifies why the matrix Γ kj is identic for both mass eigenstates. Moreover, it is important to note thatthis matrix is symmetric and the eigenvalues of its inverse Γ − kj are the squares of the widths in the momentum space [3].The oscillation probabilities between two flavor neutrinos using wave packets are P W Pν e (( T, L ) = (cid:12)(cid:12) h | ˆ ν e L ( x ) (cid:12)(cid:12) ν W Pe L ( x ) (cid:11)(cid:12)(cid:12) , (20) P W Pν µ ( T, L ) = (cid:12)(cid:12) h | ˆ ν µ L ( x ) (cid:12)(cid:12) ν W Pe L ( x ) (cid:11)(cid:12)(cid:12) , (21)where (cid:12)(cid:12) ν W Pe L ( x ) (cid:11) given by (15) is a superposition of mass eigenstate wave packets (cid:12)(cid:12) ν W P ( x ) (cid:11) and (cid:12)(cid:12) ν W P ( x ) (cid:11) ) in thecreation point. The expressions (20) and (21) describe, respectively, the probabilities of finding an electron neutrino( ν e ) and a muon neutrino ( ν µ ) at a distance L in a time T of the creation point x . In the calculation of the transition4robabilities (20) and (21), the relativistic dispersion relation is approximated by means of an expansion around theaverage momentum of the wave packets h ~p a i [16] E a ( ~p ) ≈ ¯ E a + ~v a · ( ~p − h ~p a i ) + 12 ( ~p − h ~p a i ) k Ω kj ( ~p − h ~p a i ) j + · · · , (22)where [16] ¯ E a ≡ E ( h ~p a i ) = p h ~p a i + m a , (23) v ka = ∂E ( ~p a ) ∂p k (cid:12)(cid:12)(cid:12)(cid:12) ~p = h ~p a i = h ~p a i k ¯ E a , (24)Ω akj = ∂ E ( ~p a ) ∂p k ∂p j (cid:12)(cid:12)(cid:12)(cid:12) ~p = h ~p a i = 1¯ E a (cid:0) δ kj − v ak v aj (cid:1) . (25)Additionally, it is possible to write that [3] s E a ( ~p ) ± h | ~p | E a ( ~p ) ≈ s ¯ E a ± h |h ~p a i| E a , (26) χ h ( ~p ) ≈ χ h ( h ~p a i ) . (27)The highest power of the momentum ( ~p −h ~p a i ) in the expansion of the energy given by (22) determines the two differentcases which we will studied below: (i) If the highest power is taken until second order, the effects of the spreadingof the mass eigenstates wave packets lead to the standard time-integrated neutrino oscillation probabilities obtainedusing the wave packet formalism [12–22]; (ii) If the highest power is taken until third order, the effects originated indispersion with time on the spreading of wave packets for the two-flavor neutrino oscillation probabilities are observedas a factor that suppress the standard time-integrated neutrino oscillation probabilities. IV. EXPANSION OF THE ENERGY UNTIL SECOND ORDER IN THE MOMENTUM
In this section, we will obtain the standard time-integrated neutrino oscillation probabilities using the wave packetformalism. To do it, we expand the energy given by (22) up to second order in the power series of ( ~p − h ~p a i ) [16]. Thisfact is justified taking into account that the width of the wave packets is very narrow around the average momentum.For this case, the matrix Γ kj can be diagonalized by means of an orthogonal transformation, i. e. a rotation [3, 16].Given that the expansion of the energy does not change this rotation, without lost of generality we can take a referenceframe where the matrix is diagonal Γ kj = 1 σ p δ kj , (28)with σ p representing the width of the wave packets in the momentum space. We assume that the width has the samevalue for each of the dimensions of the momentum space, due to the wave packets are taken as isotropic. Additionally,we define the width of the wave packets in the coordinate space σ r through the uncertainty relation σ r σ p = 12 . (29)If the energy given by (22) is substituted in (20) and (21), keeping up to the second order in the power series of( ~p − h ~p a i ), we obtain that the neutrino oscillation probabilities are written as P SW Pν e ( T, L ) = πσ r ) / ) (cid:8) Λ exp[ − λ φ S ( T )]+ exp[ − λ φ S ( T )] + Λ ℵ exp[ − λ φ S ( T )] (cid:9) , (30) P SW Pν µ ( T, L ) = πσ r ) / ) (cid:8) Λ exp[ − λ φ S ( T )]+Λ exp[ − λ φ S ( T )] − Λ ℵ exp[ − λ φ S ( T )] (cid:9) , (31)5here λ = λ = 1 / σ r , λ = 1 / σ r , T = t − t , L = | ~r − ~r | and the functions in the arguments of the exponentialsare given by φ S ( T ) =( L − v T ) , (32) φ S ( T ) =( L − v T ) , (33) φ S ( T ) = ( L − v T ) + ( L − v T ) − i σ r ( ¯ E − ¯ E ) T + i σ r (¯ p − ¯ p ) L, (34)with v a = | ~v a | . The quantity ℵ that appears in the oscillation probabilities (30) and (31) is written as ℵ = 1( ¯ E ¯ E ) / X h q ( ¯ E − h |h ~p i| )( ¯ E − h |h ~p i| ) . (35)Now, we take into account the fact that in the atmospheric and reactor neutrino oscillation experiments it is onlypossible to measure the distance between the neutrino source and the detector L , while the neutrino propagation time T is unknown [8, 18, 22]. However, in the case of the accelerator neutrino experiments (for instance K2K, MINOS,OPERA) it is possible to measure the neutrino propagation time T [22]. By this reason, if we focus only on the case ofatmospheric and reactor neutrino oscillation experiments, then it is necessary the elimination of the time dependencepresents in (30) and (31). This last can be performed, if we take the average on the time of the expressions (30) and(37) in the following form P SW Pν e ( L ) = Z P SW Pν e ( T, L ) dT, (36) P SW Pν µ ( L ) = Z P SW Pν µ ( T, L ) dT, (37)Time integrations can be performed using both Gaussian integration and the Laplace approximation method. Afterthe time integrations are performed, we obtain from (36) and (37) the following time-integrated neutrino oscillationprobabilities P SW Pν e ( L ) = 1(1 + Λ ) ( Λ v + 1 v + Λ Ξ (cid:18) v + v (cid:19) / exp (cid:2) if S − f S (cid:3)) , (38) P SW Pν µ ( L ) = 1(1 + Λ ) ( Λ v + Λ v − Λ Ξ (cid:18) v + v (cid:19) / exp (cid:2) if S − f S (cid:3)) , (39)where f S = ( ¯ E − ¯ E ) v + v v + v L − (¯ p − ¯ p ) L, (40) f S = ( v − v ) v + v L σ r + ( ¯ E − ¯ E ) v + v σ r . (41)We have explicitly proved that if the average on the time in the expressions (36) and (37) is performed using Gaussianintegration, the results are the same as those obtained using the Laplace approximation method. In both cases, wehave obtained the oscillation probabilities given by (38) and (39). The functional form of the oscillation probabilities(38) and (39) is in agreement with the one previously obtained by Giunti, Kim and Lee. These authors used aquantum mechanics treatment in which flavor neutrinos were described by a superposition of mass eigenstates wavepackets [8].In order to obtain from (38) and (39) expressions for the oscillation probabilities in the relativistic limit, thefollowing relativistic approximations are used [13, 14, 17]¯ E a ≃ ¯ E + ξ m a E , (42)¯ p a ≃ ¯ E + (1 − ξ ) m a E , (43) v a ≃ − m a E , (44)6here ξ is a dimensionless coefficient, typically of order unity, that depends of the neutrino production process and¯ E is the neutrino energy determined by the kinematics of the production process for a massless neutrino. After therelativistic limit is taken, we obtain from (38) and (39) the standard time-integrated neutrino oscillation probabilities P SW Pν e ( L ) = 1 −
12 sin [2 θ ] ( − exp " i π LL osc − (cid:18) LL coh (cid:19) − π ξ (cid:18) σ r L osc (cid:19) , (45) P SW Pν µ ( L ) = 12 sin [2 θ ] ( − exp " i π LL osc − (cid:18) LL coh (cid:19) − π ξ (cid:18) σ r L osc (cid:19) , (46)where L osc is the oscillation length and L coh is the coherence length given by L osc = 4 π ¯ E ∆ m , (47) L coh = 4 √ E ∆ m σ r , (48)in agreement with the corresponding lengths very well known in the literature [8, 12–22]. To write the expressions(45) and (46), we have used the definition of sin [2 θ ] in terms of the parameter Λ given by (3), where θ is themixing angle in the vacuum between the two mass eigenstates. Specifically, the probability (45) represents the survivalprobability that an electron neutrino ( ν e ) be detected at a distance L in a time T of the creation point x = (0 , ~r ),where by simplicity t = 0. On the other hand, the probability (46) represents the probability of oscillation from anelectron neutrino ( ν e ) created by the source at point x to a muon neutrino ( ν µ ) measured by the detector at a distance L in time T . The dependence of the oscillation probability (46) respect to the mixing angle θ L is in agreement withthe reported by Bernardini and De Leo. These authors studied the effects of positive and negative energy componentsof mass eigenstate wave packets on the two-flavor neutrino oscillation probabilities [23, 24].The dependence of the time-integrated oscillation probabilities (45) and (46) respect to L osc and L coh is in agreementwith the reported in the literature [8, 12–22]. The first term in the argument of the exponentials in (45) and (46) isthe standard oscillation phase proportional to the propagation distance L . The second term in the argument of theexponentials is called the damping factor [8]. This term implies a quadratical decrease of the oscillation probabilitieswith the distance L and determines how far the oscillations take place [13]. For L ≫ L coh , the interference of theneutrino mass eigenstates is suppressed and the oscillations due to L osc disappear [13]. This behavior can be partiallyoriginated in the progressive separation of mass eigenstates wave packets propagating in space [18]. The third termin the argument of the exponential is called the localization factor [13] and this term implies that σ r < L osc √ π = √ E ∆ m , (49)meaning that the neutrino production process, which is characterized by the width of the wave packets σ r , is localizedin a region much smaller that the oscillation length L osc [13]. Thus, this factor does not depends on the distance L .As can be observed from (47) and (48), the oscillation length and the coherence length are related by L coh = √ π σ r ¯ E L osc , (50)showing that the coherence length is much larger than the oscillation length [22]. The maximum number of oscillationscan be obtained from these lengths as [8] N osc = L coh L osc = √ π σ r ¯ E. (51)Because in the neutrino oscillation experiments one has L ≃ L osc , then the term exp (cid:20) − (cid:16) LL coh (cid:17) (cid:21) is nearly equalto one [22]. Additionally, it is easy to show that | v − v | L coh ≃ | ∆ m | E ∼ σ r , then L osc ≫ σ r and thus the termexp (cid:20) − π ξ (cid:16) σ r L osc (cid:17) (cid:21) is also nearly equal to one [22]. In this form, the time-integrated oscillation probabilities (45)and (46) can be written as [22] P SW Pν e ( L ) = 1 −
12 sin [2 θ ] (cid:26) − cos (cid:20) π LL osc (cid:21)(cid:27) , (52)7 SW Pν µ ( L ) = 12 sin [2 θ ] (cid:26) − cos (cid:20) π LL osc (cid:21)(cid:27) , (53)that reduce to the standard neutrino oscillation probabilities (1) and (2) obtained using the plane wave formalism. V. EXPANSION OF THE ENERGY UNTIL THIRD ORDER IN THE MOMENTUM
In this section, we will study the effects originated in dispersion with time on spreading of wave packets for thetime-integrated two-flavor neutrino oscillation probabilities by expanding the energy given by (22) up to third orderin the momentum ( ~p − h ~p a i ). For this case, we take a reference frame where the matrix Γ kj is diagonal and identicalto (28). Substituting the energy given by (22) in (20) and (21), keeping up to the third order in the power series of( ~p − h ~p a i ), we obtain that the neutrino oscillation probabilities are written as P DW Pν e ( T, L ) = πσ r ) / ) n Λ g D ( T ) exp[ − λ φ D ( T )]+ g D ( T ) exp[ − λ φ D ( T )] + Λ ℵ g D ( T ) exp[ − λ φ D ( T )] o , (54) P DW Pν µ ( T, L ) = πσ r ) / ) n Λ g D ( T ) exp[ − λ φ D ( T )]+ Λ g D ( T ) exp[ − λ φ D ( T )] − Λ ℵ g D ( T ) exp[ − λ φ D ( T )] o , (55)with ℵ given by (35), λ = λ = 1 / σ r , λ = 1 / σ r , and the functions in the arguments of the exponentials are φ D ( T ) = ( L − v T ) T ( T L ) , (56) φ D ( T ) = ( L − v T ) T ( T L ) , (57) φ D ( T ) = ( L − v T ) − iTT L + ( L − v T ) − iTT L − i σ r ( ¯ E − ¯ E ) T + i σ r (¯ p − ¯ p ) L, (58)while the functions g T ( T ), g T ( T ) and g T ( T ) are g D ( T ) = (cid:18) T ( T T ) (cid:19) (cid:18) T ( T L ) (cid:19) / , (59) g D ( T ) = (cid:18) T ( T T ) (cid:19) (cid:18) T ( T L ) (cid:19) / , (60) g D ( T ) = (cid:18) − i TT T (cid:19) (cid:18) i TT T (cid:19) (cid:18) − i TT L (cid:19) / (cid:18) i TT L (cid:19) / , (61)where we have defined the longitudinal dispersion time T La as T La = 2 ¯ E σ r /m a , with a = 1 ,
2, while the transversaldispersion time T T has been defined as T T = 2 ¯ Eσ r . The longitudinal dispersion time in neutrino oscillations wasinitially defined in the context of a quantum mechanics treatment [8]. This time was posteriorly considered in thecontext of a quantum field theory treatment of neutrino oscillations [15, 18]. Most recently, the transversal andlongitudinal times that we have defined here were considered in the context of a theory of wave packets in which theenergy that appears in the wave packets is expanded until third order in the momentum [25].Because the longitudinal and transversal times are related as T La = ¯ E m a T L , we observe that T La ≫ T T and these twovery separated times can be used to define three dispersion regimes: (i) The minimum dispersion regime is defined fortimes T that satisfy T < T T ; (ii) the transversal dispersion regime for T T < T < T L ; (iii) the longitudinal dispersionregime for T > T L . These three dispersion regimes were equivalently considered previously by using the distancebetween the neutrino source and the detector L as the quantity to define these regimes [18].8e observe in (54) and (55) the existence of two different longitudinal dispersion times T L and T L . For simplicity,we will work in the limit in which the masses are nearly degenerate m = m = ¯ m . For this limit, it is possible toconsider T L = T L and to work with only one longitudinal dispersion time defined by T L = 2 ¯ E σ r / ¯ m , with ¯ m themass in the degenerate limit [18]. For the nearly degenerate limit, the functions given by the expressions (59), (60)and (61) are written as g D ( T ) = g D ( T ) = g D ( T ) = (cid:18) T ( T T ) (cid:19) (cid:18) T ( T L ) (cid:19) / , (62)In the next, we will study for the three dispersion regimes previously defined the effects originated in dispersionwith time on the spreading of wave packets for the two-flavor neutrino oscillation probabilities. A. Spreading in the minimum dispersion regime
In the minimum dispersion regime
T < T T and for the limit of nearly degenerate masses, the oscillation probabilities(54) and (55) can be written as P MDW Pν e ( T, L ) = πσ r ) / ) (cid:8) Λ exp[ − λ φ S ( T )]+ exp[ − λ φ S ( T )] + Λ ℵ exp[ − λ φ M ( T )] (cid:9) , (63) P MDW Pν µ ( T, L ) = πσ r ) / ) (cid:8) Λ exp[ − λ φ S ( T )]+Λ exp[ − λ φ S ( T )] − Λ ℵ exp[ − λ φ M ( T )] (cid:9) , (64)with the functions φ S ( T ) and φ S ( T ) given by (32) and (33) respectively, and the function φ M ( T ) is φ M ( T ) = ( L − v ) + ( L − v T ) + i [( v − v ) T − L ( v − v ) T ] T /T L − i σ r ( ¯ E − ¯ E ) T + i σ r (¯ p − ¯ p ) L. (65)where we have neglected the terms with powers higher than O ( T /T T ) and O ( T /T L ). Now we focus our attention onthe elimination of the time dependence that is present in the neutrino oscillation probabilities (63) and (64). To doit, we take the average on the time of the expressions (63) and (64). With the integration on the time we obtain P MDW Pν e ( L ) = 1(2 πσ r ) / ) (cid:8) Λ I M + I M + Λ ℵ I M (cid:9) , (66) P MDW Pν µ ( L ) = 1(2 πσ r ) / ) (cid:8) Λ I + Λ I − Λ ℵ I (cid:9) , (67)where the integrals on the time for the minimum dispersion regime I M , I M and I M are I M = Z exp[ − λ φ S ( T )] dT, (68) I M = Z exp[ − λ φ S ( T )] dT, (69) I M = Z exp[ − λ φ M ( T )] dT. (70)(71)The integrals I M and I M can be performed using both Gaussian integration and the Laplace approximation method,while the integral I M can be only performed using the Laplace approximation method. After the time integrationsare performed, we obtain from (66) and (67) the following time-integrated neutrino oscillation probabilities P MDW Pν e ( L ) = 1(1 + Λ ) ( Λ v + 1 v + Λ Ξ (cid:18) v + v (cid:19) / f T exp (cid:2) if T − f T (cid:3)) , (72)9 MDW Pν µ ( L ) = 1(1 + Λ ) ( Λ v + Λ v − Ξ (cid:18) v + v (cid:19) / f T exp (cid:2) if T − f T (cid:3)) , (73)with the functions f T , f T and f T given by f T = ( ¯ E − ¯ E ) v + v v + v L (1 − h T ) − (¯ p − ¯ p ) L, (74) f T = ( v − v ) v + v L σ r (1 + h T ) + ( ¯ E − ¯ E ) v + v σ r (1 − h T ) , (75) f T = (cid:18) v − v )( ¯ E − ¯ E ) ¯ S ( v + v ) (cid:19) − / , (76)where h T = −
12 ( ¯ E − ¯ E )( v − v )( v + v )( v + v ) (cid:2) ( v + v ) + 4 v v (cid:3) ¯ S, (77) h T = −
12 ( ¯ E − ¯ E )( v + v )( v − v )( v + v ) , (78) h T = −
52 ( ¯ E − ¯ E )( v − v )( v + v ) ¯ S, (79)In the relativistic limit, using the approximations (42), (43) and (44), we obtain from (72) and (73) that the time-integrated neutrino oscillation probabilities for the minimum dispersion regime are P MDW Pν e ( L ) = 1 − sin [2 θ ] (cid:26) − − a ) / exp (cid:20) i π LL ′ osc − (cid:16) LL ′ coh (cid:17) − π ξ (cid:16) σ r (1 − a ) L osc (cid:17) (cid:21)(cid:27) , (80) P MDW Pν µ ( L ) = sin [2 θ ] (cid:26) − − a ) / exp (cid:20) i π LL ′ osc − (cid:16) LL ′ coh (cid:17) − π ξ (cid:16) σ r (1 − a ) L osc (cid:17) (cid:21)(cid:27) , (81)where L ′ osc and L ′ coh are written as L ′ osc = L osc a , (82) L ′ coh = L coh (1 + a ) / , (83)and a = 18 ξ (∆ m ) ¯ m ¯ E = 4 ξ σ r L coh T T T L , (84) a = 5 ξ ¯ m ¯ E = 5 ξ T T T L , (85) a = 516 ξ (∆ m ) ¯ m ¯ E = 10 ξ σ r L coh T T T L , (86) a = 34 ξ (∆ m ) ¯ m ¯ E = 24 ξ σ r L coh T T T L . (87)We can observe that the time-integrated oscillation probabilities (80) and (81) have the same functional form thatthe oscillation probabilities (45) and (46) obtained considering the expansion of energy until second order in the10omentum, but now the exponential is multiplied by a factor that includes a . But new, due to the effects originatedin dispersion with time on the spreading of the wave packets, the expressions (82) and (83) show respectively somechanges of the oscillation length (47) and of the coherence length (48). We observe how the oscillation length (82)and the coherence length (83) are respectively a little smaller than the ones obtained for the case in which the energyis expanded until second order in the momentum (47) and (48). Due to the effects originated in dispersion with time,now the maximum number of oscillations is N ′ osc = L ′ coh L ′ osc = (1 + a )(1 + a ) / N osc , (88)which implies that it is smaller than the one obtained for the case in which the energy is expanded until secondorder in the momentum (see the expression (51)). However, the quantities a i , with i = 1 , , ,
4, are very small, so L ′ osc ≃ L osc , L ′ coh ≃ L coh , N ′ osc ≃ N osc , a ≃ a ≃
0. In this way, the time-integrated oscillation probabilities(80) and (81) can reduce to (45) and (46). Thus, for the minimum dispersion regime we find that the effects originatedin dispersion with time on the spreading of the wave packets for the time-integrated neutrino oscillation probabilitiescan be neglected.
B. Spreading in the transversal dispersion regime
In the transversal dispersion regime T T < T < T L and for the limit of nearly degenerate masses, the oscillationprobabilities (54) and (55) can be written as P T DW Pν e ( T, L ) = πσ r ) / ) (cid:8) Λ F T ( T ) exp[ − λ φ S ( T )]+ F T ( T ) exp[ − λ φ S ( T )] + Λ ℵ F T ( T ) exp[ − λ φ M ( T )] (cid:9) , (89) P T DW Pν µ ( T, L ) = πσ r ) / ) (cid:8) Λ F T ( T ) exp[ − λ φ S ( T )]+Λ F T ( T ) exp[ − λ φ S ( T )] − Λ ℵ F T ( T ) exp[ − λ φ M ( T )] (cid:9) , (90)where the function on the time F T ( T ) is given by F T ( T ) = ( T T ) T . In the expressions (89) and (90), we haveneglected the terms with powers higher than O ( T /T L ) and the functions φ S ( T ) and φ S ( T ) are given by (32) and (33)respectively, while the function φ M ( T ) is given by (65). The three integrals that appear in the expressions (89) and(90) can be only performed using the Laplace approximation method. After the time integrations are performed, weobtain from (89) and (90) the following time-integrated neutrino oscillation probabilities P T DW Pν e ( L ) = ( T T ) L P MDW Pν e ( L ) = ( T T ) L P SW Pν e ( L ) , (91) P T DW Pν µ ( L ) = ( T T ) L P MDW Pν µ ( L ) = ( T T ) L P SW Pν µ ( L ) , (92)where the standard time-integrated neutrino oscillation probabilities P SW Pν e ( L ) and P SW Pν µ ( L ) are given respectivelyby (45) and (46), and the transversal dispersion time by T T = 2 ¯ Eσ r . We observe for this case that the standardtime-integrated neutrino oscillation probabilities P SW Pν e ( L ) and P SW Pν µ ( L ) are suppressed by a factor ( T T ) /L . Thisresult is in agreement with the showed by Naumov [25], whom has recently obtained that the integral over time of boththe flux and probability densities are asymptotically proportional to the factor 1 /L , when he considered a theory ofwave packets in which the energy that appears in the wave packets is expanded until third order in the momentum.This author has demonstrated that the origin of the factor 1 /L for quantum objects is their dispersion with time[25]. C. Spreading in the longitudinal dispersion regime
In the longitudinal dispersion regime
T > T L and for the limit of nearly degenerate masses, the oscillation proba-bilities (54) and (55) can be written as P LDW Pν e ( T, L ) = πσ r ) / ) (cid:8) Λ F L ( T ) exp[ − λ φ D ( T )]+ F L ( T ) exp[ − λ φ D ( T )] + Λ ℵ F L ( T ) exp[ − λ φ D ( T )] (cid:9) , (93)11 LDW Pν µ ( T, L ) = πσ r ) / ) (cid:8) Λ F L ( T ) exp[ − λ φ D ( T )]+Λ F L ( T ) exp[ − λ φ D ( T )] − Λ ℵ F L ( T ) exp[ − λ φ D ( T )] (cid:9) , (94)where the function on the time F L ( T ) is given by F L ( T ) = ( T T ) T L T . In the expressions (93) and (94), the functions φ D ( T ), φ D ( T ) and φ D ( T ) are given by (56), (57) and (58) respectively. For this case, also the three integrals thatappear in the expressions (93) and (94) can be only performed using the Laplace approximation method. Afterthe time integrations are performed, we obtain from (93) and (94) the following time-integrated neutrino oscillationprobabilities P LDW Pν e ( L ) = ( T T ) L P MDW Pν e ( L ) = ( T T ) L P SW Pν e ( L ) , (95) P LDW Pν µ ( L ) = ( T T ) L P MDW Pν µ ( L ) = ( T T ) L P SW Pν µ ( L ) , (96)where the standard time-integrated neutrino oscillation probabilities P SW Pν e ( L ) and P SW Pν µ ( L ) are given respectivelyby (45) and (46). We observe also for this case that the standard time-integrated neutrino oscillation probabilities P SW Pν e ( L ) and P SW Pν µ ( L ) are suppressed by a factor ( T T ) /L . VI. CONCLUSIONS
We have studied the effects originated in dispersion with time on spreading of wave packets for the time-integratedtwo-flavor neutrino oscillation probabilities in vacuum. We have calculated the time-integrated two-flavor neutrinooscillation probabilities in the context of a wave packet extension of the quantum field theory treatment that wepreviously developed for the case in which neutrino mass eigenstates were described by plane waves. In the treatmentthat we have presented here, neutrino flavor states have been considered as superpositions of neutrino mass eigenstatesdescribed by localized wave packets.By methodological reasons, we have initially studied the effects of the spreading of the wave packets by consideringthe expansion of the energy until second order in the momentum that leads to the standard time-integrated neutrinooscillation probabilities. After this, we have studied the effects originated by dispersion in time on spreading of wavepackets for the time-integrated two-flavor neutrino oscillations by considering the expansion of the energy until thirdorder in the momentum. We have observed that the standard time-integrated neutrino oscillation probabilities aresuppressed by a factor 1 /L for the transversal and longitudinal dispersion regimes, where L is the distance betweenthe neutrino source and the detector. The existence of this kind of suppression for the standard time-integratedneutrino oscillation probabilities might be proved in reactor neutrino oscillation experiments with beams very narrowin time or experiments at short enough distances [25]. Acknowledgments
Y. F. P´erez thanks for the financial support from the brazilian supporting agencies Funda¸c˜ao de Amparo `a Pesquisado Estado de S˜ao Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). C. J.Quimbay thanks DIB for the financial support received through the research project ”Propiedades electromagn´eticasy de oscilaci´on de neutrinos de Majorana y de Dirac”. C. J. Quimbay would like thank to Maurizio De Sanctis forhelp in the manuscript preparation. [1] Ch. W. Kim and A. Pevsner,
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