PHYSICAL REVIEW D 97, 093002 (2018)

Neutrino oscillation processes in a quantum-field-theoretical approach

Vadim O. Egorov1,2 and Igor P. Volobuev1 1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University 119991 Moscow, Russia 2Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia

(Received 3 October 2017; published 2 May 2018)

It is shown that oscillation processes can be consistently described in the framework of using only the plane wave states of the particles. Namely, the oscillating survival probabilities in experiments with neutrino detection by charged-current and neutral-current interactions are calculated in the quantum field–theoretical approach to neutrino oscillations based on a modification of the Feynman propagator in the momentum representation. The approach is most similar to the standard technique. It is found that the oscillating distance-dependent probabilities of detecting an electron in experiments with neutrino detection by charged-current and neutral-current interactions exactly coincide with the corresponding probabilities calculated in the standard approach.

DOI: 10.1103/PhysRevD.97.093002

I. INTRODUCTION It was first discussed in Ref. [4], where it was suggested that the produced neutrino mass eigenstates are virtual is a well-known and experimentally and their motion to the detection point should be described confirmed phenomenon, which is usually understood as the by the Feynman propagators. In this approach, neutrino transition from a neutrino flavor state to another neutrino oscillations occur as a result of interference of the ampli- flavor state depending on the distance traveled [1–3]. tudes of processes due to all three intermediate virtual However, the situation with the theoretical explanation neutrino mass eigenstates. However, the calculations of the of this phenomenon is paradoxical: there is no generally amplitudes in this approach are essentially different from accepted explanation of this phenomenon, which is both the standard calculations in the framework of the Feynman quantum and relativistic, in the framework of quantum field diagram technique in the momentum representation. This is theory, which is a synthesis of quantum mechanics and the due to the standard S-matrix formalism of QFT, which is special theory of relativity. Nowadays, the generally not appropriate for describing processes at finite distances accepted explanation of this phenomenon is the quantum and finite time intervals. To describe the localized particles mechanical description in terms of plane waves based on or nuclei, which produce and detect , one has to the notion of neutrino flavor states. This standard approach use wave packets, which impede the calculation. is not perfect: the neutrino flavor states are superpositions Later, this approach was developed in [5], where an of the neutrino mass eigenstates, and for this reason the important theorem was proved that the virtual particles processes with the flavor states cannot be consistently propagating at macroscopic distances are almost on the described within quantum field theory. The problem is the mass shell. A detailed review of the approach can be found violation of energy-momentum conservation in such proc- in [6,7]. esses, because in local quantum field theory, where the An approach combining the features of the one discussed four-momentum is conserved in any interaction vertex, above and of the standard approach was put forward in [8]. different mass-eigenstate components of the flavor states A crucial point of the approach is that neutrino oscillations must have different momenta as well as different energies. are due to interference of the amplitudes, corresponding to This problem was repeatedly discussed in the literature the processes mediated by the neutrino mass eigenstates. (see, e.g., [4–9]). However, unlike the approach discussed above, the authors A possible solution to the problem with the violation of treat the neutrinos to be detected as real particles and also energy-momentum conservation is to go off the mass shell. make use of wave packets in their reasoning to disentangle the produced neutrino mass eigenstates and to account for energy-momentum conservation. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. In the present paper, we will show that neutrino Further distribution of this work must maintain attribution to oscillation can be consistently described in the framework the author(s) and the published article’s title, journal citation, of quantum field theory using only plane wave states and DOI. Funded by SCOAP3. of the particles. Namely, we will explicitly calculate the

2470-0010=2018=97(9)=093002(9) 093002-1 Published by the American Physical Society VADIM O. EGOROV and IGOR P. VOLOBUEV PHYS. REV. D 97, 093002 (2018) probabilities of the neutrino oscillation processes in experi- associated with nuclei 1, 10 and 2, 20. As it is customary in ments with neutrino detection by charged-current and QFT, we assume that the incoming nuclei 1 and 2 have neutral-current interactions within a modified perturbative definite momenta. Therefore all the three virtual neutrino S-matrix formalism, which enables one to calculate the eigenstates and the outgoing particles and nuclei also amplitudes of the processes passing at finite distances and have definite momenta. In what follows, a 4-momentum finite time intervals. This formalism was put forward in of the virtual neutrino mass eigenstates defined only by the [10]. It is based on the Feynman diagram technique in the energy-momentum conservation in the production vertex coordinate representation [11] supplemented by new modi- will be denoted by pn and the one selected also by the fied rules of passing to the momentum representation, experimental setting will be denoted by p. which will be discussed below in detail. The amplitude in the coordinate representation corre- sponding to diagram (2) can be written out in the standard II. OSCILLATIONS IN EXPERIMENTS WITH way using the Feynman rules formulated in textbook [12]. NEUTRINO DETECTION BY CHARGED- According to the prescriptions of the standard perturbative CURRENT INTERACTION S-matrix theory ([12], Sec. 24), in order to obtain the amplitude in the momentum representation, next we would In the framework of the minimal extension of the have to integrate it with respect to x and y over the (SM) by the right neutrino singlets, we Minkowski space. However, in this case, we would get consider the case, where the neutrinos are produced and the amplitude of the process lasting an infinite amount of detected in the charged-current interaction with nuclei. time and lose the information about the distance between After the diagonalization of the terms sesquilinear in the the production and detection points defined by the exper- neutrino fields, the charged-current interaction Lagrangian imental setting. In order to retain this information, we have of leptons takes the form to integrate with respect to x and y in such a way that the   X3 distance between these points along the direction of the g ¯ μ 5 − L ¼ − pffiffiffi l γ ð1 − γ ÞU ν Wμ þ H:c: ; ð1Þ neutrino propagation remains fixed. Of course, this is at cc 2 2 i ik k i;k¼1 variance with the standard S-matrix formalism. However, we recall that the diagram technique in the coordinate where li denotes the field of the charged lepton of the i-th representation was developed by R. Feynman [11] without generation, νi denotes the field of the neutrino mass reference to S-matrix theory. Thus, the Feynman diagrams eigenstate most strongly coupled to li and Uik stands for in the coordinate representation make sense beyond this the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. theory, and for this reason we can integrate with respect to x Due to this structure of the interaction Lagrangian any and y in any way depending on the physical problem at process involving the production of a neutrino at one point hand. In particular, in the case under consideration, we have and its detection at another point, when treated perturba- to integrate in such a way that the distance between the tively, can be represented in the lowest order by the points x and y along the direction of the propagation of following diagram, neutrino with momentum p⃗is equal to the distance between a source and a detector, which will be denoted by L. This can be achieved by introducing the delta function δð⃗ð⃗− ⃗Þ j⃗j − Þ ð2Þ p y x = p L into the integral. When passing to the momentum representation and integrating with respect to x and y over the Minkowski space, the introduction of this delta function is formally equivalent to replacing which should be summed over all three neutrino mass the standard Feynman propagator of the neutrino mass ν cð − Þ eigenstates. To be specific, we assume that the virtual W eigenstate i in the coordinate representation Si y x by cð − Þδð⃗ð⃗− ⃗Þ j⃗j − Þ are produced and absorbed in interactions with Si y x p y x = p L . However, we emphasize A1 once again that the propagation of the neutrino mass nuclei. Namely, we suppose that a nucleus Z1 X that will be þ eigenstates is still described by the Feynman propagator, called nucleus 1 radiates W and turns into the and the introduced delta function serves only to represent A1 10 nucleus Z1−1X that will be called nucleus , and a nucleus the experimental situation that the neutrino mass eigen- A2 þ states with momentum p⃗are detected at a distance L from Z2 X that will be called nucleus 2 absorbs W boson and turns into the nucleus A2 X that will be called nucleus 20. the source. Z2þ1 The Fourier transform of the expression Scðy − xÞ × Thus, the filled circles stand for the matrix elements of the i δðp⃗ðy⃗− x⃗Þ=jp⃗j − LÞ was called in [10] the distance- charged weak hadron current dependent propagator of the neutrino mass eigenstate νi

ð1Þ A1 ðhÞ A1 ð2Þ A2 ðhÞ A2 in the momentum representation. It will be denoted by μ ¼h j μ j i ρ ¼h j ρ j i c j Z1−1X j Z1 X ;j Z2þ1X j XZ2 X ; ð Þ Si p; L and is defined by the integral:

093002-2 NEUTRINO OSCILLATION PROCESSES IN A QUANTUM … PHYS. REV. D 97, 093002 (2018) Z The advantage of the time-dependent propagator is that Scðp; LÞ¼ dxeipxScðxÞδðp⃗x=⃗ jp⃗j − LÞ: ð3Þ i i there exists the inverse Fourier transformation of this propagator, which allows one to retain the standard rules This integral can be evaluated exactly by the method of ⃗2 2 − 2 of the Feynman diagram technique just by replacing the contour integration [10], and for p >mi p the result is Feynman propagator by this time-dependent propagator. given by For macroscopic time intervals T, i.e., for the particles close  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2−m2 to the mass shell, it looks explicitly like ˆ þ γ⃗⃗ 1 − 1 þ i þ p p p⃗2 mi c 2 2 S ðp; LÞ¼i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m −p i 2 2 2 ˆ þ −i i nT 2 ⃗ þ − pn mi 2 0 p p mi cð Þ¼ pn ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Si pn;T i 0 e : 7 2 2 2 2p −iðjp⃗j− p⃗ þp −m ÞL n × e i : ð4Þ In case all the neutrinos in a beam have the same (In [10], this distance-dependent propagator was defined by momentum p defined by the experimental setting, we δð⃗⃗−j⃗j Þ substituting the dimensionless delta function p x p L can express the time T in terms of the distance L and j⃗j into the integral, which results in an extra factor p in the the neutrino speed jp⃗j=p0 as T ¼ Lp0=jp⃗j, neglect the cð Þ denominator of Si p; L . Below we will see that the present neutrino mass that is small compared to pˆ and get a definition is more natural.) We emphasize that this dis- distance-dependent propagator, tance-dependent propagator makes sense only for 2 2 macroscopic distances L. m −p pˆ −i i L The results of [5] imply that the virtual particles Scðp; LÞ¼i e 2jp⃗j ; ð8Þ i 2p0 propagating at macroscopic distances are almost on the j 2 − 2j ⃗2 ≪ 1 mass shell. This means that p mi =p and we can which coincides with the above defined distance-dependent ð 2 − 2Þ ⃗2 expand the square roots to the first order in p mi =p . propagator of neutrinos (5) in the approximation of small It is clear that this term can be dropped everywhere, except neutrino masses. In what follows, we will use propagators in the exponential, where it is multiplied by a large (7), (8) for describing neutrino oscillation processes. We macroscopic distance L. In this approximation distance- also note that the time-dependent scalar field propagator is dependent propagator (4) takes the simple form adequate for calculating the probabilities of oscillation 2 2 ˆ m −p processes with massive scalar mesons, where we cannot p þ mi −i i L cð Þ¼ 2jp⃗j ð Þ Si p; L i 2j⃗j e : 5 neglect their masses. p Now we will calculate the amplitude in the momentum representation corresponding to diagram (2) in the case, It is worth noting that this distance-dependent fermion 0 0 propagator taken on the mass shell has no pole and does not where the time difference y − x is fixed and equal to T. depend on the distance, which is also true for the exact Since the momentum transfer in both production and propagator in formula (4). detection processes is small, we can use the approximation In fact, we have discussed this distance-dependent of Fermi’s interaction. Then making use of time-dependent propagator in order to explain better the motivations for propagator (7) and keeping the neutrino masses only in the introducing such an object, because it exactly corresponds exponential we can explicitly write out the amplitude in the to the experimental situation in neutrino oscillation proc- momentum representation corresponding to diagram (2) esses. However, this distance-dependent propagator is not summed over all three neutrino mass eigenstates: convenient for calculating amplitudes, because there is no 2− 2 2 X3 m pn −i i T inverse Fourier transformation for the propagator in for- GF 2 2 0 ð2Þ ρ 5 M ¼ −i jU1 j e pn jρ u¯ðkÞγ ð1 − γ Þ mula (3). It turns out that a more convenient and a more 4 0 i pn i¼1 fundamental object is the time-dependent propagator of the μ 5 ð1Þ neutrino mass eigenstates, which can be defined as the × pˆ nγ ð1 − γ ÞvðqÞjμ : ð9Þ cð Þδð 0 − Þ Fourier transform of Si x x T . A similar time- ð1Þ ð2Þ dependent scalar field propagator was introduced in [10]. Here jμ and jρ stand for the matrix elements of the Using the results of the calculations of the time-dependent charged weak hadron current associated with nuclei 1, 10 0 scalar field propagator in this paper, one can easily find that and 2, 2 ; k, pn, and q are the 4-momenta of the electron, the time-dependent fermion field propagator in the momen- the intermediate virtual neutrinos and the positron, respec- tum representation is tively, and we do not write out explicitly the fermion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi polarization indices. pˆ − γ0ðp0 − ðp0Þ2 þ m2 − p2Þþm c n npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin i n i Averaging with respect to the polarizations of the S ðpn;TÞ¼i i 2 ðp0Þ2 þ m2 − p2 incoming nuclei and summing over the polarizations of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin i n ð 0 − ð 0 Þ2þ 2− 2 Þ the outgoing particles and nuclei one gets the expression for i pn pn mi pn T ð Þ × e : 6 the squared amplitude as follows:

093002-3 VADIM O. EGOROV and IGOR P. VOLOBUEV PHYS. REV. D 97, 093002 (2018)

4 4 2 GF ð1Þ μνρσ ð2Þ hjMj i¼ Wμν A Wρσ 2 2 μν μ ν μ ν 0 2 hjM1j i¼4G ð−g ðp qÞþðpnq þ q p Þ ðpnÞ F n n    ð1Þ 3 2 2 μναβ X − þ iε p αqβÞWμν ; ð14Þ 2 2 2 mi mk n 1 − 4 jU1 j jU1 j T ; × i k sin 4 0 i;k¼1 pn i

1 2 μνρσ μ 5 ν 5 Here hjM1j i is the squared amplitude of the decay process A ¼ trðpˆ nγ ð1 − γ Þðqˆ − mÞγ ð1 − γ Þ 64 of nucleus 1 into nucleus 10, positron and a massless ˆ σ 5 ˆ ρ 5 hj j2i × pnγ ð1 − γ Þðk þ mÞγ ð1 − γ ÞÞ fermion and M2 is the squared amplitude of the process of electron production in the collision of the massless (the factor 1=64 is introduced in order to separate the fermion and nucleus 2. numerical coefficient from the Lorentz structure proper), Now we are in a position to calculate the probability of the ð1Þ ð2Þ the tensors Wμν , Wρσ characterizing the interaction of process depicted in diagram (2), when the time difference nuclei 1 and 2 with virtual W bosons are defined as between the points x and y is equal to T. We will do these calculations in accordance with the rules of the standard ð1Þ ð1Þ ð1Þ þ ð2Þ ð2Þ ð2Þ þ Wμν ¼hjμ ðjν Þ i;Wρσ ¼hjρ ðjσ Þ i: ð11Þ perturbative S-matrix theory, although we are aware that the rules of calculating the probabilities of processes passing at Here and below, the angle brackets denote the averaging finite time interval and finite distances may be different from with respect to the polarizations of the incoming those of the standard S-matrix theory. We will discuss this particles and the summation over the polarizations of difference below. To this end we denote the 4-momenta of ð Þ 0 0 ð 0Þ the outgoing particles; i.e., in the previous formula, they the nuclei by PðiÞ ¼ðEðiÞ;P⃗i Þ, Pði Þ ¼ðEði Þ;P⃗i Þ, i ¼ 1,2, denote the averaging with respect to the polarizations of and recall that the amplitude in the momentum representa- nuclei 1,2 and the summation over the polarizations of tion corresponding to diagram (2) contains, along with the nuclei 10, 20. expression in formula (9), the delta function of energy- Since we have dropped the neutrino masses in the momentum conservation. Thus, to calculate the probability time-dependent propagators, we have actually calculated of the process per unit time per unit volume, we have to 0 the amplitude in the approximation of zero neutrino multiply amplitude (13) by ð2πÞ4δðPð1Þ þ Pð2Þ − Pð1 Þ − 0 masses. As we have already noted, for macroscopic time Pð2 Þ − q− kÞ and to integrate it with respect to the momenta intervals T the virtual neutrinos are almost on the mass of the outgoing particles and nuclei. shell and, therefore, the squared momentum of the virtual 2 Since the momentum pn of the virtual neutrinos is defined neutrinos pn is also of the order of the neutrino masses by the energy-momentum conservation in the production 0 squared and can be neglected. In other words, we may vertex, p ¼ Pð1Þ − Pð1 Þ − q, this integration can lead to calculate the squared amplitude in the approximation n 2 variation in the virtual neutrino momentum, which contra- pn ¼ 0. Direct calculations show that, in this approxima- μνρσ dicts the experimental situation in neutrino oscillations, tion, the tensor A factorizes: where the virtual neutrinos propagate in the direction defined by the relative position of a source and a detector. This μνρσ ¼ð− μνð Þþð μ ν þ μ ν Þþ εμναβ Þ A g pnq pnq q pn i pnαqβ means that we have to calculate the differential probability of ρσ ρ σ ρ σ ρσαβ × ð−g ðpnkÞþðpnk þ k pnÞ − iε pnαkβÞ: the process with pn fixed by the experimental setting. Let us denote by p⃗the momentum that is directed from the ð12Þ source to the detector and satisfies the momentum con- ð1Þ ð10Þ servation condition P⃗ − P⃗ − q⃗− p⃗¼ 0 in the produc- Correspondingly, the squared amplitude in formula (10) 0 factorizes as follows: tion vertex and define the four-momentum p ¼ðp ; p⃗Þ, p0 > 0, p2 ¼ 0. The required differential probability of the 1 hj j2i¼hj j2ihj j2i process with pn fixed can be obtained by multiplying M M1 M2 0 2 δð − Þ 4ðpnÞ amplitude (13) by the delta function pn p or, equiv-    3 2 2 alently, by replacing pn by p in the amplitude and multi- X − 0 1 − 4 j j2j j2 2 mi mk plying it by δðPð1Þ − Pð1 Þ − q − pÞ. This is consistent, × U1i U1k sin 4 0 T ; i;k¼1 pn becausewework in the approximation of massless neutrinos. i

093002-4 NEUTRINO OSCILLATION PROCESSES IN A QUANTUM … PHYS. REV. D 97, 093002 (2018)    4 X3 2 − 2 dW GF 2 2 2 mi mk ¼ 1 − 4 jU1 j jU1 j sin T ⃗ 16ð2πÞ8 ð1Þ ð2Þð 0Þ2 i k 4 0 dp E E p i;k¼1 p Z i

It is easy to verify that, due to the factorization of the squared amplitude, this differential probability also factorizes. Now, since the momentum of virtual neutrinos is fixed, we can replace T by Lp0=jp⃗j, as it was explained after formula (7), which gives    1 X3 2 − 2 dW ¼ dW1 1 − 4 j j2j j2 2 mi mk ð Þ ⃗ 2π ⃗ W2 U1i U1k sin 4j⃗j L ; 17 dp dp i;k¼1 p i

Z 3 3 ð10Þ dW1 1 1 1 d q 1 d P 1 2 4 ð1Þ ð10Þ ¼ 0 hjM1j ið2πÞ δðP − P − q − pÞ dp⃗ 2Eð1Þ ð2πÞ3 2p0 ð2πÞ3 2q0 ð2πÞ3 2Eð1 Þ Z 2 1 GF 3 3 ð10Þ μν μ ν μ ν μναβ ¼ d qd P 0 ð−g ðpqÞþðp q þ q p Þþiε pαqβÞ 4ð2πÞ5Eð1Þp0 q0Eð1 Þ ð1Þ ð1Þ ð10Þ ð1Þ ð10Þ × Wμν ðP ;P ÞδðP − P − q − pÞð18Þ is the differential probability of the decay of nucleus 1 into nucleus 10, positron and a massless fermion with momentum p⃗, which coincides with the sum of the differential probabilities of the decay of nucleus 1 into nucleus 10, positron and all the three neutrino mass eigenstates, and

Z 3 3 ð20Þ 1 d k 1 d P 1 2 4 ð2Þ ð20Þ W2 ¼ 0 hjM2j ið2πÞ δðP þ p − P − kÞ 2Eð2Þ2p0 ð2πÞ3 2k0 ð2πÞ3 2Eð2 Þ Z 2 1 GF 3 3 ð20Þ ρσ ρ σ ρ σ ρσαβ ¼ d kd P 0 ð−g ðpkÞþðp k þ k p Þ − iε pαkβÞ 4ð2πÞ2Eð2Þp0 k0Eð2 Þ ð2Þ ð2Þ ð20Þ ð2Þ ð20Þ × Wρσ ðP ;P ÞδðP þ p − P − kÞð19Þ is the probability of the scattering process of a massless The appearance of this extra factor can be explained fermion with momentum p⃗and nucleus 2 resulting in the as follows: in fact, the detector registers not only the production of nucleus 20 and an electron, which coincides neutrinos with momentum p⃗from a pointlike source, but with the sum of the probabilities of the scattering processes also the neutrinos with the momenta, which lie inside a of all the three neutrino mass eigenstates and nucleus 2. The small cone (see Fig. 1) with the axis along the vector p⃗. terms in the square brackets in formula (17) reproduce the This is due to a nonzero size of the detector. Obviously, the standard expression for the oscillating neutrino or electron picture has an approximate circular symmetry about the survival probability. direction of momentum p⃗, which gives the factor 2π after The physical considerations suggest that the differential dW → 0 probability dp⃗ for L should be equal to the product dW1 2π dp⃗ W2; i.e., there is an extra factor in the denominator of formula (17). This means that the standard rules of calculating the process probabilities in perturbative S-matrix theory should be modified so as to include the FIG. 1. Illustration of variance in the direction of the virtual extra factor 2π. neutrino momenta due to a nonzero detector size.

093002-5 VADIM O. EGOROV and IGOR P. VOLOBUEV PHYS. REV. D 97, 093002 (2018) the integration with respect to the azimuthal angle. Thus, which fixes the 4-momentum p of the intermediate neu- the rules of the standard perturbative S-matrix theory trinos. Correspondingly, the final formula for the differ- should be modified in our case so as to include the factor ential probability of the process under consideration 0 2π along with the delta function δðPð1Þ − Pð1 Þ − q − pÞ, looks like

Z 3 3 3 ð10Þ 3 ð20Þ dW 1 d k 1 d q 1 d P 1 d P 1 2 ¼ 0 0 hjMj i dp⃗ 2Eð1Þ2Eð2Þ ð2πÞ3 2k0 ð2πÞ3 2q0 ð2πÞ3 2Eð1 Þ ð2πÞ3 2Eð2 Þ 0 0 0 ×ð2πÞ4δðPð1Þ þ Pð2Þ − Pð1 Þ − Pð2 Þ − q − kÞ2πδðPð1Þ − Pð1 Þ − q − pÞ; ð20Þ which eliminates the contradiction and provides the con- their squared amplitudes arising due to the extremely dW j ¼ dW1 small neutrino masses. However, we believe that these sistent result: dp⃗ ¼0 dp⃗ W2. L rules can be used for calculating the probabilities of any In the approximation of massless neutrinos dW1 coincides dp⃗ processes passing at finite distances and finite time inter- with the neutrino probability flux and W2 coincides with vals. In particular, they can also be used for calculating the the cross section of the scattering process of a massless probabilities of oscillation processes with neutral kaons, fermion on nucleus 2. Thus, we have obtained that the where the differential probability of the processes factorizes probability of detecting an electron is equal to the prob- exactly like in formula (17) due to a simpler structure of the ability of the neutrino production in the source multiplied amplitude of such processes in the case of scalar particles. by the probability of the neutrino interaction in the detector and the standard distance-dependent electron or neutrino survival probability; i.e., we have actually exactly repro- III. OSCILLATIONS IN EXPERIMENTS WITH duced the result of the standard approach to neutrino NEUTRINO DETECTION BY NEUTRAL- oscillations in the framework of QFT without making CURRENT AND CHARGED-CURRENT use of the neutrino flavor states. INTERACTIONS In formula (17) and in the subsequent formulas, the Now we consider the case, where the neutrinos are distance L appears as the distance between two neutrino produced in the charged-current interaction with nuclei and interactions, which take place in the neutrino source and in detected in both neutral-current and charged-current inter- the detector. However, if we consider the same process actions with , as it is done in the Kamiokande passing at a different point of the detector at a distance experiment. The corresponding processes are described by 0 ¼ þ Δ jΔ j ≪ L L L, L L, we get the same differential the following Feynman diagrams: probability up to terms of the order ΔL=L, which can be dropped. If the sizes of the source and the detector are much less than the distance between them, which is always the case in the neutrino oscillation experiments, the ð Þ differential probability of all the neutrino production and 21 detection processes passing within their volumes is the same with a very high accuracy. Therefore, the distance L, just like in the standard approach, can be considered as the distance between the source and the detector, and differ- ential probability (17) can be considered as the differential probability per unit source volume and per unit detector volume. If nuclei 1 in the source have a momentum distribution, the total neutrino probability flux can be dW1 ð22Þ obtained by performing the average of dp⃗ over the momentum distribution of nucleus 1, and the number of events in the detector per unit time can be found by dW integrating the corresponding differential probability dp⃗ and the densities of nuclei 1 and nuclei 2 over the volumes It is clear that in calculating the amplitude of the process the of the neutrino source and detector. contribution of diagram (22) should be taken with all three The rules for calculating the probabilities of neutrino neutrino mass eigenstates, i.e., k ¼ 1, 2, 3. Since only the oscillation processes passing at finite distances and finite final electron is registered experimentally, the probabilities time intervals were suggested by the factorized structure of of the processes with different final neutrino states should

093002-6 NEUTRINO OSCILLATION PROCESSES IN A QUANTUM … PHYS. REV. D 97, 093002 (2018) be summed up to give the probability of registering an and the momentum of the outgoing neutrino is k2. Again electron. we use the approximation of Fermi’s interaction and Now let us denote the particle momenta as follows: the take time-dependent propagator (7) keeping the neutrino momentum of the positron is q, the momentum of the masses only in the exponential. Then the amplitude virtual neutrinos is pn, the momentum of the outgoing corresponding to diagram (21) in the momentum repre- electron is k, the momentum of the incoming electron is k1 sentation looks like

2 2 2 m −p −i i nT ðiÞ GF 2 0 μ 5 ρ 5 M ¼ i U e pn ν¯ ðk2Þγ ð1−γ Þpˆ γ ð1−γ ÞvðqÞjρ nc 4p0 1i i n n   1 − þ 2θ ¯ð Þγ ð1−γ5Þ ð Þþ 2θ ¯ð Þγ ð1þγ5Þ ð Þ ð Þ × 2 sin W u k μ u k1 sin Wu k μ u k1 : 23

Similarly, the sum over k of the amplitudes corresponding to diagram (22) can be written out to be

 2− 2  2 X3 m pn −i i T ðiÞ GF 2 2 0 μ 5 ρ 5 M ¼ −i U jU1 j e pn u¯ðkÞγ ð1 − γ Þpˆ γ ð1 − γ ÞvðqÞjρ cc 4 0 1i k n pn k¼1 5 × ν¯iðk2Þγμð1 − γ Þuðk1Þ: ð24Þ

ðiÞ Next, it is convenient to use the Fierz identity, which transposes the spinors u¯ðkÞ and ν¯iðk2Þ in the amplitude Mcc and makes ðiÞ this amplitude look similar to Mnc, and to introduce the following notations for the time-dependent factors:

2− 2  2− 2  m pn X3 m pn −i i T −i i T 2 0 2 2 0 ¼ pn ¼ j j pn ð Þ Ai U1ie ;Bi U1i U1k e : 25 k¼1

ðiÞ ðiÞ Then the total amplitude of the process with neutrino νi in the final state, which is the sum of the amplitudes Mnc and Mcc , can be represented as follows:

2 ðiÞ GF μ 5 ρ 5 M ¼ i ν¯ ðk2Þγ ð1 − γ Þpˆ γ ð1 − γ ÞvðqÞjρ tot 4p0 i n n    1 þ − þ 2θ ¯ð Þγ ð1 − γ5Þ ð Þþ 2θ ¯ð Þγ ð1 þ γ5Þ ð Þ ð Þ × Bi Ai 2 sin W u k μ u k1 Aisin Wu k μ u k1 : 26

Now, we have to calculate the squared amplitude, averaged with respect to the polarizations of the incoming nucleus and particles and summed over the polarizations of the outgoing nucleus and particles. Similar to the case of the neutrino 2 detection in charged-current interaction, in the approximation pn ¼ 0 the squared amplitude factorizes as follows: 1 hj ðiÞj2i¼hj j2ihj ðiÞj2i ð Þ Mtot M1 M2 0 2 ; 27 4ðpnÞ

hj j2i¼4 2 ð− μνð Þþð μ ν þ μ ν Þþ εμναβ Þ ð1Þ ð Þ M1 GF g pnq pnq q pn i pnαqβ Wμν ; 28    1 2 hj ðiÞj2i¼64 2 þ − þ 2θ ð Þ2 þj j2 4θ ð Þ2 M2 GF Bi Ai 2 sin W pnk1 Ai sin W pnk      1 − 2θ þ − þ 2θ ð Þ 2 ð Þ sin WRe Bi Ai 2 sin W Ai pnk2 m : 29

2 0 Here, hjM1j i is the squared amplitude of the decay process of nucleus 1 into nucleus 1 , positron and a massless fermion, ð1Þ ðiÞ 2 Wμν denoting the corresponding averaged product of the matrix elements of the charged weak hadron current, and hjM2 j i is the squared amplitude of the scattering process of a massless fermion and the incoming electron.

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As we have found in the previous section for the case of approximation of zero neutrino masses. Similarly, in neutrino registration in charged-current interaction, to ðiÞ accordance with formula (32), the probability W2 is the obtain the differential probability of the process, we have probability of the process of scattering of a massless neutral hj ðiÞj2i to multiply the amplitude Mtot by the delta function of fermion and electron with the production of electron and 4 ð1Þ energy-momentum conservation ð2πÞ δðP þ k1 − the neutrino mass eigenstate νi in the final state. Obviously, ð10Þ ð1Þ P − q − k − k2Þ and by the delta function 2πδðP − this probability is the sum of the probabilities of the 0 Pð1 Þ − q − pÞ that selects the momentum of the virtual processes of scattering of electron and all the three neutrino neutrinos, substitute p instead of p in it and to integrate mass eigenstates with the production of electron and the n ν with respect to the momenta of the outgoing particles and neutrino mass eigenstate i in the final state. nucleus. This gives To obtain the probability of finding an electron in the final dWðiÞ ¼1 state we have to sum the probability dp⃗ over i ,2,3. ðiÞ dW dW1 ð Þ Obviously, this reduces to summing over i the squared ¼ i ð Þ ð Þ 2 W2 ; 30 hj i j i dp⃗ dp⃗ amplitude M2 , because only this amplitude depends on i. Z 2 hj j i 0 Since now the virtual neutrinos have fixed momentum p, dW1 ¼ M1 ð2πÞ4δð ð1Þ − ð1 Þ − − Þ ð1Þ P P q p 0 j⃗j dp⃗ 2E we can replace T by Lp = p in all the subsequent 0 formulas. Then the definition of the time-dependent factors d3q d3Pð1 Þ 1 ð Þ A and B in (25) leads to the following expressions for their × 3 0 3 ð10Þ 3 0 ; 31 i i ð2πÞ 2q ð2πÞ 2E ð2πÞ 2p absolute values and products: Z ðiÞ 2 j j2 ¼j j2 ð Þ ðiÞ hjM2 j i 4 Ai U1i ; 33 W ¼ ð2πÞ δðk1 þ p − k − k2Þ 2 2p02k0    1 X3 2 − 2 3 3 2 2 2 2 2 mk ml jB j ¼jU1 j 1 − 4 jU1 j jU1 j sin L ; d k d k2 ð Þ i i k l 4j⃗j × 3 0 3 0 : 32 k;l¼1 p ð2πÞ 2k ð2πÞ 2k2 k

     X3 1 2 X3 2 − 2 hj ðiÞj2i¼64 2 4θ ð Þ2 þ þ 2θ − 8 2θ j j2j j2 2 mk ml ð Þ2 M2 GF sin W pk 2 sin W sin W U1k U1l sin 4j⃗j L pk1 i¼1 k;l¼1 p k

Next, substituting this expression into formula (32) summed over i and evaluating the integrals with the help of the formulas for neutrino-electron scattering kinematics presented in chapter 16 of Ref. [13], we arrive at the following result:        2 2j⃗j2 2j⃗j 1 2j⃗j 2 GFm p 2 p 4 p W2 ¼ 1 − 2sin θ 1 þ þ 4sin θ 1 þ 2π 2jp⃗jþm W 2jp⃗jþm W 3 2jp⃗jþm   3   2jp⃗j X m2 − m2 þ 4 2θ 1 þ 1 − 4 j j2j j2 2 k l ð Þ sin W 2j⃗jþ U1k U1l sin 4j⃗j L : 37 p m k;l¼1 p k

093002-8 NEUTRINO OSCILLATION PROCESSES IN A QUANTUM … PHYS. REV. D 97, 093002 (2018)

IV. CONCLUSION consistently obtained using only the mass eigenstates of these particles. There is a good reason to believe that the In the present paper, we have shown that it is possible to developed approach is equivalent to the approach based on calculate consistently neutrino oscillation processes in a the use of the Feynman propagators and wave packets just quantum field-theoretical approach within the framework of like the Feynman diagram technique using plane wave the SM minimally extended by the right neutrino singlets states of particles is equivalent to the calculations of the using only plane wave states of the particles. To this end amplitudes of scattering processes with the help of wave we have adapted the standard formalism of perturbative packets. However, the use of plane waves, which essen- S-matrix for calculating the amplitudes of the processes tially simplifies the calculation of amplitudes, results in the passing at finite distances and finite time intervals by formalism in its present form being unable to reproduce the modifying the Feynman propagator in the momentum coherence and localization terms, which naturally arise in representation. The developed approach is physically trans- the approaches using wave packets. Thus, the problem of parent and, unlike the quantum mechanical description in describing the coherence and localization properties of terms of plane waves based on the notion of neutrino flavor neutrino oscillations in this approach needs a special states, has the advantage of not violating energy-momentum investigation. conservation. It can be considered as a formalization of the approach based on the use of the Feynman propagators and ACKNOWLEDGMENTS wave packets [4–7], which allows one to simplify the calculations essentially. In its framework, the calculation The authors are grateful to E. Boos, A. Lobanov, and of amplitudes is carried out in the way that is most similar to M. Smolyakov for reading the manuscript and making the one used in the usual perturbative S-matrix formalism. important comments and to L. Slad for useful discussions. The application of this modified formalism to describing Analytical calculations of the amplitudes have been carried the neutrino oscillation processes with neutrino detection out with the help of the COMPHEP and REDUCE pack- by charged-current and neutral-current interactions with ages. The work was supported by Grant No. NSh- electrons showed that the standard results can be easily and 7989.2016.2 of the President of the Russian Federation.

[1] C. Giunti and C. W. Kim, Fundamentals of Neutrino [7] E. K. Akhmedov and J. Kopp, Neutrino oscillations: Quan- Physics and Astrophysics (Oxford University Press, tum mechanics vs. quantum field theory, J. High Energy New York, 2007). Phys. 04 (2010) 008; 10 (2013) 52. [2] S. Bilenky, Introduction to the physics of massive and mixed [8] A. G. Cohen, S. L. Glashow, and Z. Ligeti, Disentangling neutrinos, Lect. Notes Phys. 817, 1 (2010). Neutrino Oscillations, Phys. Lett. B 678, 191 (2009). [3] K. Nakamura and S. T. Petcov (Particle Data Group), [9] A. E. Lobanov, Oscillations of particles in the Standard Neutrino masses, mixing, and oscillations, Chin. Phys. C Model, Teor. Mat. Fiz. 192, 70 (2017); Theor. Math. Phys. 38, 090001 (2014). 1921000 (2017). [4] C. Giunti, C. W. Kim, J. A. Lee, and U. W. Lee, On the [10] I. P. Volobuev, Quantum field-theoretical description of treatment of neutrino oscillations without resort to weak neutrino and neutral kaon oscillations, arXiv:1703.08070. eigenstates, Phys. Rev. D 48, 4310 (1993). [11] R. P. Feynman, Space-time approach to quantum electro- [5] W. Grimus and P. Stockinger, Real oscillations of virtual dynamics, Phys. Rev. 76, 769 (1949). neutrinos, Phys. Rev. D 54, 3414 (1996). [12] N. N.BogoliubovandD. V.Shirkov,IntroductiontotheTheory [6] M. Beuthe, Oscillations of neutrinos and mesons in quan- of Quantized Fields, 3rd edition (Wiley, New York, 1980). tum field theory, Phys. Rep. 375, 105 (2003). [13] L. B. Okun, Leptons and Quarks (North Holland, New York, 1984).

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