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Quantum Field Theory of Neutrino Oscillations

Abstract

The theory of neutrino oscillations in the framework of the quantum field perturbative theory with relativistic wave packets as asymptotically free in- and out-states is expounded. A covariant wave packet formalism is developed. This formalism is used to calculate the probability of the interaction of wave packets scattered off each other with a nonzero impact parameter. A geometric suppression of the probability of interaction of wave packets for noncollinear collisions is calculated. Feynman rules for the scattering of wave packets are formulated, and a diagram of a sufficiently general form with macroscopically spaced vertices (a “source” and a “detector”) is calculated. Charged leptons (\(\ell _{\alpha }^{ \pm }\) in the source and \(\ell _{\beta }^{ \mp }\) in the detector) are produced in the space-time regions around these vertices. A neutrino is regarded as a virtual particle (propagator) connecting the macrodiagram vertices. An appropriate method of macroscopic averaging is developed and used to derive a formula for the number of events corresponding to the macroscopic Feynman diagram. The standard quantum-mechanical probability of flavor transitions is generalized by considering the longitudinal dispersion of an effective neutrino wave packet and finite time intervals of activity of the “source” and the “detector”. A number of novel and potentially observable effects in neutrino oscillations is predicted.

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Notes

  1. 1.

    Pauli’s famous letter was published in [2] with comments and insightful historical remarks. The preliminary report of Fermi was published in 1933 [3].

  2. 2.

    A WP is called “effective,” since it incorporates the processes of neutrino production and detection.

  3. 3.

    The packet evolution under the influence of external fields is incorporated into this representation, since any interaction in the \(S\)-matrix QFT formalism should be regarded as a local interaction of real or virtual fields.

  4. 4.

    A more complex WP model was examined in [100].

  5. 5.

    This also follows from simple dimensional considerations: since 3-vector \(\overline {\mathbf{P}} \) depends on just a single vector quantity \({\mathbf{p}}\), \(\left\langle {\mathbf{P}} \right\rangle \propto {\mathbf{p}}\).

  6. 6.

    This is similar in certain respects to the following well-known relativistic effect: the mass of gas in a vessel increases in the process of its (uniform) heating. The vessel acquires no additional momentum, but the internal energy and, consequently, the mass of gas increase.

  7. 7.

    The RGP model was proposed in [57]. It was thoroughly analyzed mathematically and generalized in [101]. Intriguing applications of the RGP (apart from neutrino physics) were discussed in [102]. Another class of models (covariant asymmetric WP, AWP) was proposed in [100]. It was demonstrated that an AWP is never the same as an RGP, although it features all the properties needed for a covariant description of localized “particle-like” quantum states. It was also demonstrated that the WP considered below is the only relativistic “memoryless” packet (i.e., a packet independent of 4-momenta of “parent” states (particles) involved in the reactions or decays producing the WP; see Section 4.1). We follow the logic of [57], which is sufficient for our purpose.

  8. 8.

    The linear dependence of function \(f(t)\) on parameter \(\lambda \) does not preclude one from applying the theorem.

  9. 9.

    It is clear that nonspreading WPs are easier to associate with (quasi) stable particles (i.e., localized objects) and to use as asymptotically free states of in- and out-fields in the \(S\)-matrix QFT formalism instead of common plane waves, which occupy the entire spacetime and are thus unable to serve as an adequate mathematical model of particles.

  10. 10.

     This assumption is verified a posteriori by relation (171).

  11. 11.

     In the physical (rather than in the formally mathematical) sense.

  12. 12.

     It is worth noting here that the states of gamma quanta emerging in processes involving (anti) neutrinos (real or virtual) may be considered in the plane-wave limit, although a rigorous substantiation of this statement is outside the bounds of the present study. See [115, 116] for a thorough discussion of the theory of photon WPs.

  13. 13.

     And, in a more general case, by sets of parameters \({{\sigma }_{a}}\), \({{\sigma }_{b}}\).

  14. 14.

     To avoid unnecessary complications, we discount the possibility that the external one-packet states contain gauge or Higgs bosons.

  15. 15.

     Phase factors \({{e}^{{ + iqx}}}\) and \({{e}^{{ - iqy}}}\) in (192) may be interpreted as WPs of outgoing and incoming neutrinos.

  16. 16.

     Generally speaking, four-momentum \(q\) in (196) is arbitrary and simultaneous vanishing of momenta \({{{\mathbf{p}}}_{\varkappa }}\) implies only that the most probable value of \(\left| {\mathbf{q}} \right|\) is small. The order of smallness of \(\left| {\mathbf{q}} \right|\) is governed by the smallness of ratios \({{{{\sigma }_{\varkappa }}} \mathord{\left/ {\vphantom {{{{\sigma }_{\varkappa }}} {{{m}_{\varkappa }}}}} \right. \kern-0em} {{{m}_{\varkappa }}}}\). We will discuss this in more detail later.

  17. 17.

     “Fortunately”, since otherwise there would be no asymptotically free WPs, and it would be impossible to use the perturbation theory.

  18. 18.

     It bears remembering that the diameter of this ellipsoid perpendicular to the vector of the most probable WP momentum \({\mathbf{p}}\) (and, consequently, the diameter of the classical world line) is \({{d}_{ \star }} \propto {1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-0em} \sigma }\), while the diameter parallel to \({\mathbf{p}}\) is suppressed by factor \({{\Gamma }_{{\mathbf{p}}}}\) emerging as a result of Lorentz contraction.

  19. 19.

     The physical inadequacy of the notions of point-like particles (maximally localized states) and plane waves (states with definite momenta) in the QFT perturbation theory then becomes clear: the former are unable to interact (the amplitude turns to zero if a point-like particle is present in any substate \({{I}_{{s,d}}}\) or \({{F}_{{s,d}}}\)), and the latter interact at arbitrarily large distances.

  20. 20.

     It follows from geometric considerations that \(0 < \left| {{\mathbf{b}}_{a}^{ \star }} \right| < {{d_{a}^{ \star }} \mathord{\left/ {\vphantom {{d_{a}^{ \star }} 2}} \right. \kern-0em} 2}\) (\(d_{a}^{ \star }\) is the effective transverse size of a packet) for such events. Therefore, \(\sigma _{a}^{2}{{\left| {{\mathbf{b}}_{a}^{ \star }} \right|}^{2}} < 0.605\) and \(\exp\left( { - \sigma _{a}^{2}{{{\left| {{\mathbf{b}}_{a}^{ \star }} \right|}}^{2}}} \right)0.546\).

  21. 21.

     This suppression is in no way related to the Pauli Exclusion Principle: if any two packets (not necessarily those characterizing the states of identical fermions) have equal velocities and are sufficiently separated at a certain point in time, their classical world tubes do not intersect. It should also be added that parameters \({{\sigma }_{\varkappa }}\) and \({{\sigma }_{{\varkappa {\kern 1pt} {\text{'}}}}}\) for packets of identical particles should not be regarded as fundamental characteristics of the corresponding quantum field and are not necessarily equal.

  22. 22.

     More specifically, the distance from \({\mathbf{X}}_{{s,d}}^{ * }\) to the classical trajectories of centers of packets \(\varkappa \) and \(\varkappa {\kern 1pt} {\text{'}}\) in the center-of-inertia frame is much shorter than the distance between these centers.

  23. 23.

     Through to the end of the present section, \(\pi \) and \(\mu \) denote particle types rather than Lorentz indices.

  24. 24.

     Comparing (225) with (213), we also find the relation between impact parameters               \({{\left| {{\mathbf{b}}_{{\pi \mu }}^{ * }} \right|}^{2}} = \left( {1 + \frac{{\sigma _{\pi }^{2}}}{{\sigma _{\mu }^{2}}}} \right){{\left| {{\mathbf{b}}_{\pi }^{{(\pi )}}} \right|}^{2}} + \left( {1 + \frac{{\sigma _{\mu }^{2}}}{{\sigma _{\pi }^{2}}}} \right){{\left| {{\mathbf{b}}_{\mu }^{{(\mu )}}} \right|}^{2}},\) which implies, among other things, that \(\left| {{\mathbf{b}}_{\pi }^{{(\pi )}}} \right| = \left| {{\mathbf{b}}_{\mu }^{{(\mu )}}} \right| = 0\) at \(\left| {{\mathbf{b}}_{{\pi \mu }}^{ * }} \right| = 0.\)

  25. 25.

     We leave open the question of the possibility of experimental measurement of impact parameter \(\left| {{\mathbf{b}}_{{\pi \mu }}^{ * }} \right|\), which could help verify the above theoretical constructs.

  26. 26.

     Note that the examples presented in Figs. 8 and 10 (Section 7.2) do not belong to this class, although the technique of calculation of the corresponding amplitudes does not differ much from the one developed below.

  27. 27.

     It was proven in [123] that the asymptotics of integral (243) under somewhat more stringent conditions imposed on function \(\Phi ({\mathbf{q}})\) takes the form    \(\begin{gathered} J({\mathbf{L}},\kappa ) = \frac{{{{e}^{{i\kappa L}}}}}{{4\pi L}} \\ \times \,\,{{\left. {\left\{ {1 - \frac{i}{L}\left[ {\left( {{\mathbf{l}}{{\nabla }_{{\mathbf{q}}}}} \right) + \frac{\kappa }{2}{{{\left( {{\mathbf{l}} \times {{\nabla }_{{\mathbf{q}}}}} \right)}}^{2}}} \right] + \mathcal{O}\left( {\frac{1}{{{{L}^{2}}}}} \right)} \right\}\Phi ({\mathbf{q}})} \right|}_{{{\mathbf{q}} = - \kappa {\mathbf{l}}}}}. \\ \end{gathered} \)     (242) If \(\Phi ({\mathbf{q}}) \in S({{\mathbb{R}}^{3}})\) (which is true in almost all practically relevant cases), integral (243) is an asymptotic series in inverse powers of \(L\), and \({{\left| {J({\mathbf{L}},\kappa )} \right|}^{2}}\) is an asymptotic series in inverse powers of \({{L}^{2}}\). The properties of these series were studied in detail in [123–125], and several physical applications of these results were discussed in [126–129].

  28. 28.

     Note that the term \( \propto {{m}_{j}}\) in (245) does not contribute to amplitude (241) due to matrix factors \({{O}^{{\mu {\kern 1pt} '}}}\) and \({{O}^{{\nu {\kern 1pt} '}}}\).

  29. 29.

     A more accurate modification was discussed, e.g., in [130, 131] (see also the references therein).

  30. 30.

     It is assumed that the minimum is taken over the entire set (defined by the specific experimental conditions) of the most probable momenta \({{{\mathbf{p}}}_{\varkappa }}\) of external packets in the source and the detector.

  31. 31.

     Here, \(\nu \) is not a Lorentzian index.

  32. 32.

     It bears reminding that quadratic forms \(\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\) and \(\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\) are positive for arbitrary 4-vector \(q {\text{≢} }0\).

  33. 33.

     Under additional conditions \(\left| {{{{({{p}_{\nu }} + {{p}_{\alpha }})}}^{2}}} \right| \ll m_{W}^{2}\) and \(\left| {{{{({{p}_{\nu }} - {{p}_{\beta }})}}^{2}}} \right| \ll m_{W}^{2}\), the \(W\)-boson propagator may be written approximately as \({{ - i{{g}_{{\mu \nu }}}} \mathord{\left/ {\vphantom {{ - i{{g}_{{\mu \nu }}}} {m_{W}^{2}}}} \right. \kern-0em} {m_{W}^{2}}}\), which corresponds to the four-fermion theory of weak interaction. Using well-known SM identity \({{{{g}^{2}}} \mathord{\left/ {\vphantom {{{{g}^{2}}} 8}} \right. \kern-0em} 8} = {{{{G}_{F}}m_{W}^{2}} \mathord{\left/ {\vphantom {{{{G}_{F}}m_{W}^{2}} {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }}\), one may then rewrite matrix elements (281) in the following form:                         \(\begin{gathered} {{M}_{s}} \approx - i\frac{{{{G}_{F}}}}{{\sqrt 2 }}{{{\bar {u}}}_{ - }}({{{\mathbf{p}}}_{\nu }})\mathcal{J}_{s}^{\mu }{{O}_{\mu }}v({{{\mathbf{p}}}_{\alpha }}), \\ M_{d}^{ * } \approx - i\frac{{{{G}_{F}}}}{{\sqrt 2 }}\bar {u}({{{\mathbf{p}}}_{\beta }})\mathcal{J}_{d}^{{ * \mu }}{{O}_{\mu }}{{u}_{ - }}({{{\mathbf{p}}}_{\nu }}). \\ \end{gathered} \) However, this slightly restrictive simplification (inapplicable at ultrahigh energies) is not required and will not be used in subsequent analysis.

  34. 34.

     Function \({{\tilde {\mathfrak{D}}_{j}^{2}} \mathord{\left/ {\vphantom {{\tilde {\mathfrak{D}}_{j}^{2}} {E_{\nu }^{2}}}} \right. \kern-0em} {E_{\nu }^{2}}}\) is invariant to within corrections \(\mathcal{O}({{\mathfrak{r}}_{j}})\).

  35. 35.

     In particular, taking into account modern cosmological constraints on the sum of neutrino masses (see, e.g., [132]).

  36. 36.

     The accuracy of this relation is the same as that of formula (282) for the amplitude: it is the accuracy of the saddle-point method used in the derivation.

  37. 37.

     Naturally, this assumption is optional; the “placement of square brackets” in (298) is dictated by the conditions of a specific experiment. For example, in the hypothetical experiment with “labeled” neutrinos, the momenta of all secondary particles both in the source and in the detector should be measurable.

  38. 38.

     For simplicity, we consider an “ideal” experiment with a detection efficiency of 1, although real efficiency and acceptance values, trigger conditions, and event selection criteria are easy to introduce into the formalism.

  39. 39.

     In more correct terms, integral (308) is the number of neutrinos produced in unit time within volume \(d{\mathbf{x}}\), which is centered at point \({\mathbf{x}} \in \mathcal{S}\), propagating within solid angle \(d{{\Omega }_{\nu }}\) in direction \({\mathbf{l}} = {{({\mathbf{y}} - {\mathbf{x}})} \mathord{\left/ {\vphantom {{({\mathbf{y}} - {\mathbf{x}})} {\left| {{\mathbf{y}} - {\mathbf{x}}} \right|}}} \right. \kern-0em} {\left| {{\mathbf{y}} - {\mathbf{x}}} \right|}}\) and crossing a unit area that is centered at point \({\mathbf{y}} \in \mathcal{D}\) and perpendicular to \({\mathbf{l}}\).

  40. 40.

     Naturally, this complication should not be regarded as a drawback of our formalism, since quantum-mechanical formulas are simply inapplicable in this case.

  41. 41.

     The available estimates of the upper bound of \(\sum\nolimits_i {{{m}_{i}}} \), which were obtained by analyzing cosmological data within the \(\Lambda {\text{CDM}} + \sum {{{m}_{\nu }}} \) model and its extensions, span (roughly) the interval from 0.07 to 0.30 eV.

  42. 42.

     Note that all these reservations bear no relation to the applicability domain of our formalism; they just outline the region of applicability of the simplified model used for qualitative analysis of the simplest corollaries of the theory. Models that are more realistic simply require calculations that are more tedious.

  43. 43.

     For example, the suppression does not exceed \({{10}^{{ - 4}}}\) (\({{S}_{0}}(t,t{\kern 1pt} ') > 0.999\)) if \(t > t{\kern 1pt} '\,\, + 2.3273\) (i.e., \({{\tau }_{s}} > {{\tau }_{d}} + {{2.3273} \mathord{\left/ {\vphantom {{2.3273} \mathfrak{D}}} \right. \kern-0em} \mathfrak{D}}\)).

  44. 44.

     This condition is definitely satisfied in experiments with solar, atmospheric, and geophysical (anti) neutrinos, but may be violated in accelerator experiments.

  45. 45.

     This should be a compromise value in the sense that is needs to exceed \({{\tau }_{s}}\) considerably to ensure reliable signal detection but needs not be too large so as to prevent the signal from being lost in the background.

  46. 46.

     For simplicity, we limit ourselves to scalar particles.

  47. 47.

     Compared to the spatial dispersions of functions \({{\psi }_{{s,d}}}(x)\).

  48. 48.

     This is why the “proofs” of Lorentz invariance of the standard formula for the oscillation probability, which one may find in literature, are erroneous.

  49. 49.

     We have prepared some “work pieces” for such potentially interesting problems (and with educational purposes); see Sections 8.2.2 and Appendix D.

  50. 50.

     This result is also reproduced in the quantum-field theory and is the real reason (as opposed to the meaningless “averaging over sources” mentioned in many papers) why astrophysical neutrinos do not oscillate, with an important reservation that the interactions of neutrinos with matter in the astrophysical source are neglected.

  51. 51.

     Only one experiment of this kind, where our model (in its maximally simplified form) was used in the data analysis on vanishing of reactor \({{\bar {\nu }}_{e}}\), has been performed to date [88].

  52. 52.

     The left-hand sides of inequalities (A.6a) are the principal minors of the first and the second orders. It is implied that summation over repeated indices is not performed in (A.6a) and (A.6b).

  53. 53.

     Naturally, the formulas given in this section hold true for any two-particle decay (e.g., decays of relativistic ions via the capture of an orbit electron (such as \(^{{140}}{\text{P}}{{{\text{r}}}^{{57 + }}} \to {{\,}^{{140}}}{\kern 1pt} {\text{C}}{{{\text{e}}}^{{57 + }}}\nu_{*}\)) in the thought experiment on detection of low-energy neutrinos from the EC decay. With certain reservations, the formulas are also applicable to the consecutive processes of emission and resonance absorption (induced by the electron capture) of a Mössbauer antineutrino (e.g., \(^{3}{\text{H}}{{ \to }^{3}}{\text{He}} + \bar {\nu }_{*}\), \(\bar {\nu }_{*} + {{\,}^{3}}{\text{He}} \to {{\,}^{3}}{\text{H}}\)).

  54. 54.

     Subscript \( \star \) denotes the intrinsic reference frame of a meson. Throughout the present section, indices \(a\) and \(\ell \) denote the corresponding particles and should not be confused with Lorentz indices.

  55. 55.

     Wave packets form anew in each (inelastic or elastic) interaction of a particle with the medium and external fields.

  56. 56.

     It should be noted that \(d_{j}^{ \bot }\) is by no means the size of a neutrino packet at rest, since all estimates were obtained in the ultrarelativistic approximation and remain true only in those reference frames where a neutrino is ultrarelativistic. The same important reservation applies to the Lorentz invariance of neutrino wave function \(\Psi _{y}^{j}({{{\mathbf{p}}}_{j}},x)\). The explicit form of the effective wave packet of a nonrelativistic neutrino requires separate consideration, which is beyond the bounds of the present study.

  57. 57.

     However, it should be kept in mind that the applicability conditions of the mathematical averaging procedure are rather restrictive and are by no means guaranteed to be fulfilled in all neutrino experiments. For example, they are definitely violated in accelerator experiments with a baseline shorter than the effective transverse sizes of neutrino packets.

  58. 58.

     We should recall that \(\left| {{\mathbf{u}}_{\ell }^{ \star }} \right|\) in (A.7) is always nonzero.

  59. 59.

     It is worth noting that, in contrast to \({{\mathfrak{F}}_{d}}\), function \({{\mathfrak{n}}_{d}}\) is not a relativistic invariant, although it is expressed (in the laboratory frame) in terms of two invariants.

  60. 60.

     This case is exactly the opposite of the one discussed as an example in Section A.3.2.

  61. 61.

     We should recall that the formulas for dispersion are modified greatly at \({{E}_{\nu }} \sim {{m}_{j}}\); see Section 8.2.2.

  62. 62.

     All formulas are written in the PW\(_{0}\) limit, which implies the exact energy-momentum conservation in reaction \(2 \to 2\) plus \({{m}_{j}} = 0,\forall j\). It is also assumed (unless stated otherwise) that all parameters \({{\sigma }_{\varkappa }}\) are nonzero.

  63. 63.

     It appropriate at this point to note that a more general analysis covering the nonrelativistic case is of potential interest in the context of studies into the possibility of detecting relic neutrinos and for accelerator and astrophysical experiments in the search for hypothetical superheavy neutrinos and sterile neutrinos of keV-scale masses.

  64. 64.

     Naturally, these variations should not violate the conditions of applicability of the SRGP model.

  65. 65.

     In the case of strong hierarchy of scales of momenta dispersions of external packets, but not in the general case.

  66. 66.

     In this study, quantities \({{A}_{{\mu \nu }}}\) and \({{B}_{\mu }}\) are the components of a tensor and a 4-vector, respectively, but this is not significant for subsequent analysis. In addition, (B.4) is independent of the dimensionality and signature of spacetime.

  67. 67.

     Lorentz indices and normal ordering symbols are omitted for brevity in this section (when it can be done without ambiguity).

  68. 68.

     In the general case, this tensor may depend parametrically on the momenta and spins of the initial and final single-particle Fock states of hadrons.

  69. 69.

     Note that \(\left| {\alpha {\kern 1pt} '} \right| \leqslant {\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-0em} 4}\) and \(\left| \beta \right| \leqslant {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}\); the real value of the cubic root is taken in all cases.

  70. 70.

     Note that \({{v}_{{jj}}} = {{v}_{j}},\)\(\tilde {\mathfrak{D}}_{{jj}}^{2} = {{{{\mathfrak{D}}^{2}}} \mathord{\left/ {\vphantom {{{{\mathfrak{D}}^{2}}} {\left( {1 + r_{j}^{2}} \right)}}} \right. \kern-0em} {\left( {1 + r_{j}^{2}} \right)}}\), and \({{{{{\left| {{{{\tilde {\mathfrak{D}}}}_{j}}} \right|}}^{2}}} \mathord{\left/ {\vphantom {{{{{\left| {{{{\tilde {\mathfrak{D}}}}_{j}}} \right|}}^{2}}} {{{{\tilde {\mathfrak{D}}}}_{{jj}}}}}} \right. \kern-0em} {{{{\tilde {\mathfrak{D}}}}_{{jj}}}}} = \mathfrak{D}\).

  71. 71.

     See [142, 143] for details.

  72. 72.

     A more accurate analysis should cover the spatial distributions of colliding and/or decaying particles in the source and the effects of interaction of virtual neutrinos with the source medium (namely, coherent scattering of neutrinos off electrons in the medium, or the MSW effect [144, 145]), which are crucial for astrophysical applications such as experiments with solar neutrinos.

  73. 73.

     In practice, (G.3a) and (G.3b) work fine at \(\left| z \right| \lesssim 1\) and \(1 \lesssim \left| z \right| \lesssim 4.5\), respectively.

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ACKNOWLEDGMENTS

We would like to thank E.K. Akhmedov, V.A. Bednyakov, S.M. Bilenky, Z.G. Berezhiani, F. Vissani, M.I. Vysotsky, A.Z. Gazizov, M.O. Gonchar, D.S. Gorbunov, C. Giunti, I.P. Ivanov, I.D. Kakorin, A.E. Kaloshin, D.I. Kazakov, S.E. Korenblit, K.S. Kuzmin, V.A. Li, W. Potzel, V.A. Rubakov, D.V. Taichenachev, O.V. Teryaev, K.A. Treskov, A.I. Frank, A.S. Sheshukov, M.I. Shirokov, and D.S. Shkirmanov for stimulating discussions.

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Correspondence to D. V. Naumov or V. A. Naumov.

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Translated by D. Safin

Appendices

APPENDIX A

PROPERTIES OF OVERLAP TENSORS

A.1. General Formulas for \(\Re _{{s,d}}^{{\mu \nu }}\) and \(\tilde {\Re }_{{s,d}}^{{\mu \nu }}\)

Let us examine the general properties of overlap tensors

$$\Re _{{s,d}}^{{\mu \nu }} = \sum\limits_\varkappa {T_{\varkappa }^{{\mu \nu }}} = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}} \left( {u_{\varkappa }^{\mu }u_{\varkappa }^{\nu } - {{g}^{{\mu \nu }}}} \right).$$

It is useful in this regard to write matrices \({{\Re }_{{s,d}}} = \left\| {\Re _{{s,d}}^{{\mu \nu }}} \right\|\) in the explicit form:

$${{\Re }_{{s,d}}} = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}} \left( {\begin{array}{*{20}{c}} {\Gamma _{\varkappa }^{2} - 1}&{ - {{\Gamma }_{\varkappa }}{{u}_{{\varkappa 1}}}}&{ - {{\Gamma }_{\varkappa }}{{u}_{{\varkappa - }}}}&{ - {{\Gamma }_{\varkappa }}{{u}_{{\varkappa 3}}}} \\ { - {{\Gamma }_{\varkappa }}{{u}_{{x1}}}}&{1 + u_{{\varkappa 1}}^{2}}&{{{u}_{{\varkappa 1}}}{{u}_{{\varkappa 2}}}}&{{{u}_{{\varkappa 1}}}{{u}_{{\varkappa 3}}}} \\ { - {{\Gamma }_{\varkappa }}{{u}_{{x2}}}}&{{{u}_{{\varkappa 2}}}{{u}_{{\varkappa 1}}}}&{1 + u_{{\varkappa 2}}^{2}}&{{{u}_{{\varkappa 2}}}{{u}_{{\varkappa 3}}}} \\ { - {{\Gamma }_{\varkappa }}{{u}_{{x3}}}}&{{{u}_{{\varkappa 3}}}{{u}_{{\varkappa 1}}}}&{{{u}_{{\varkappa 3}}}{{u}_{{\varkappa 2}}}}&{1 + u_{{\varkappa 3}}^{2}} \end{array}} \right)$$

or, equivalently,

$$\begin{gathered} {{\Re }_{{s,d}}} = \sum\limits_\varkappa {{{{({{\sigma }_{\varkappa }}{{\Gamma }_{\varkappa }})}}^{2}}} \\ \times \,\,\left( {\begin{array}{*{20}{c}} {{\mathbf{v}}_{\varkappa }^{2}}&{ - {{v}_{{\varkappa 1}}}}&{ - {{v}_{{\varkappa 2}}}}&{ - {{v}_{{\varkappa 3}}}} \\ { - {{v}_{{\varkappa 1}}}}&{1 - v_{{\varkappa 2}}^{2} - v_{{\varkappa 3}}^{2}}&{{{v}_{{\varkappa 1}}}{{v}_{{\varkappa 2}}}}&{{{v}_{{\varkappa 1}}}{{v}_{{\varkappa 3}}}} \\ { - {{v}_{{\varkappa 2}}}}&{{{v}_{{\varkappa 2}}}{{v}_{{\varkappa 1}}}}&{1 - v_{{\varkappa 3}}^{2} - v_{{\varkappa 1}}^{2}}&{{{v}_{{\varkappa 2}}}{{v}_{{\varkappa 3}}}} \\ { - {{v}_{{\varkappa 3}}}}&{{{v}_{{\varkappa 3}}}{{v}_{{\varkappa 1}}}}&{{{v}_{{\varkappa 3}}}{{v}_{{\varkappa 2}}}}&{1 - v_{{\varkappa 1}}^{2} - v_{{\varkappa 2}}^{2}} \end{array}} \right). \\ \end{gathered} $$

Here and elsewhere, \({{u}_{{\varkappa i}}}\) and \({{v}_{{\varkappa i}}}\) (\(i = 1,2,3\)) are the components of vectors \({{{\mathbf{u}}}_{\varkappa }} = {{{{{\mathbf{p}}}_{\varkappa }}} \mathord{\left/ {\vphantom {{{{{\mathbf{p}}}_{\varkappa }}} {{{m}_{\varkappa }}}}} \right. \kern-0em} {{{m}_{\varkappa }}}}\) and \({{{\mathbf{v}}}_{\varkappa }}\), respectively (\({{u}_{{\varkappa i}}} = {{v}_{{\varkappa i}}}\)). As above, index \(\varkappa \) designates packets from all four sets of initial (\({{I}_{{s,d}}}\)) and final (\({{F}_{{s,d}}}\)) states. It is evident that \(\left| {{{T}_{\varkappa }}} \right| = {{\left| {{{T}_{\varkappa }}} \right|}_{{{{{\mathbf{v}}}_{\varkappa }} = 0}}} = 0\), but \(\left| {{{\Re }_{{s,d}}}} \right| > 0\). This follows from the positivity of quadratic forms \(\Re _{{s,d}}^{{\mu \nu }}{{x}_{\mu }}{{x}_{\nu }}\) under the assumption that \({{\sigma }_{\varkappa }} > 0\) for at least two packets \(\varkappa \). Note also that \({\text{Tr}}{{T}_{\varkappa }} = T_{{_{{}}\mu }}^{\mu } = - 3\sigma _{\varkappa }^{2}\) and, consequently, \({\text{Tr}}{{\Re }_{{s,d}}} = - 3\sum\nolimits_\varkappa {\sigma _{\varkappa }^{2}} .\) The positivity of all principal minors is another important property of the determinant \(\left| {{{\Re }_{{s,d}}}} \right|\). It then follows that the spatial parts of matrices \({{\Re }_{{s,d}}}\) (i.e., matrices \(\left\| {\Re _{{s,d}}^{{ij}}} \right\|\) (\(i,j = 1,2,3\))) are also positively defined; therefore, quadratic forms \(\Re _{{s,d}}^{{ij}}{{x}_{i}}{{x}_{j}}\) are positive.

In what follows, the following definitions are used:

$$\begin{gathered} {{\omega }_{i}} = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}} \left( {1 + u_{{\varkappa i}}^{2}} \right),\,\,\,\,\omega = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}} {\mathbf{u}}_{\varkappa }^{2}, \\ {{\upsilon }_{i}} = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}{{\Gamma }_{\varkappa }}{{u}_{{\varkappa i}}}} ,\,\,\,\,{{w}_{i}} = \sum\limits_\varkappa {\sigma _{\varkappa }^{2}} {{u}_{{\varkappa j}}}{{u}_{{\varkappa k}}}. \\ \end{gathered} $$
((A.1))

Here and elsewhere, indices \(s\) and \(d\) are omitted for simplicity. Spatial indices \(i,j,k\) assume the values of 1, 2, 3; it is assumed, unless stated otherwise, that \(i \ne j \ne k\), and the \((i,j,k)\) triad in each formula containing all three indices is a cyclic permutation of (1, 2, 3). The determinants of \({{\Re }_{s}}\) and \({{\Re }_{d}}\) may be written in this notational convention as

$$\begin{gathered} \left| {{{\Re }_{{s,d}}}} \right| = \omega \prod\limits_i {{{\omega }_{i}}} + 2\omega \prod\limits_i {{{w}_{i}}} \\ + \,\,\sum\limits_i {{{\upsilon }_{i}}{{w}_{i}}} \left( {{{\upsilon }_{i}}{{w}_{i}} - {{\upsilon }_{j}}{{w}_{j}} - {{\upsilon }_{k}}{{w}_{k}}} \right) \\ + \,\,\sum\limits_i {\left[ {{{w}_{i}}{{\omega }_{i}}\left( {2{{\upsilon }_{j}}{{\upsilon }_{k}} - \omega {{w}_{i}}} \right) - \upsilon _{i}^{2}{{\omega }_{j}}{{\omega }_{k}}} \right]} . \\ \end{gathered} $$
((A.2))

A.2. Inverse Overlap Tensors \(\tilde {\Re }_{{s,d}}^{{\mu \nu }}\)

The matrices inverse to \({{\Re }_{{s,d}}}\) are defined in terms of algebraic cofactors \(\mathfrak{A}_{{s,d}}^{{\mu \nu }}\) of the matrix elements of \({{\Re }_{{s,d}}}\):

$$\Re _{{s,d}}^{{ - 1}} = {{\left| {{{\Re }_{{s,d}}}} \right|}^{{ - 1}}}\left\| {\mathfrak{A}_{{s,d}}^{{\mu \nu }}} \right\|,$$
((A.3))
$$\begin{gathered} \mathfrak{A}_{{s,d}}^{{00}} = \prod\limits_i {{{\omega }_{i}}} - \sum\limits_i {w_{i}^{2}{{\omega }_{i}}} + 2\prod\limits_i {{{w}_{i}}} , \\ \mathfrak{A}_{{s,d}}^{{0i}} = {{\upsilon }_{i}}{{\omega }_{j}}{{\omega }_{k}} - {{\upsilon }_{j}}{{w}_{k}}{{\omega }_{k}} - {{\upsilon }_{k}}{{w}_{j}}{{\omega }_{j}} \\ + \,\,{{w}_{i}}\left( {{{\upsilon }_{j}}{{w}_{j}} + {{\upsilon }_{k}}{{w}_{k}} - {{\upsilon }_{i}}{{w}_{i}}} \right), \\ \mathfrak{A}_{{s,d}}^{{ii}} = \omega \left( {{{\omega }_{j}}{{\omega }_{k}} - w_{i}^{2}} \right) \\ + \,\,2{{w}_{i}}{{\upsilon }_{j}}{{\upsilon }_{k}} - \upsilon _{j}^{2}{{\omega }_{k}} - \upsilon _{k}^{2}{{\omega }_{j}}, \\ \mathfrak{A}_{{s,d}}^{{jk}} = \left( {{{\upsilon }_{j}}{{\upsilon }_{k}} - \omega {{w}_{i}}} \right){{\omega }_{i}} \\ + \,\,{{\upsilon }_{i}}\left( {{{\upsilon }_{i}}{{w}_{i}} - {{\upsilon }_{j}}{{w}_{j}} - {{\upsilon }_{k}}{{w}_{k}}} \right) + \omega {{w}_{j}}{{w}_{k}}. \\ \end{gathered} $$
((A.4))

The matrix elements of \({{\tilde {\Re }}_{{s,d}}}\) are then given by

$$\tilde {\Re }_{{s,d}}^{{\mu \nu }} = {{\left( {\Re _{{s,d}}^{{ - 1}}} \right)}_{{\mu \nu }}} = {{\left| {{{\Re }_{{s,d}}}} \right|}^{{ - 1}}}\tilde {\mathfrak{A}}_{{s,d}}^{{\mu \nu }},$$

where

$$\begin{gathered} \tilde {\mathfrak{A}}_{{s,d}}^{{00}} = \mathfrak{A}_{{s,d}}^{{00}},\,\,\,\,\tilde {\mathfrak{A}}_{{s,d}}^{{0i}} = \tilde {\mathfrak{A}}_{{s,d}}^{{i0}} = - \mathfrak{A}_{{s,d}}^{{0i}}, \\ \tilde {\mathfrak{A}}_{{s,d}}^{{ij}} = \tilde {\mathfrak{A}}_{{s,d}}^{{ji}} = \mathfrak{A}_{{s,d}}^{{ij}}. \\ \end{gathered} $$

Explicit formulas for the components of inverse overlap tensors in terms of the most probable 4-velocities were obtained in [123]:

$$\tilde {\Re }_{{s,d}}^{{\mu \nu }} = \frac{1}{{\left| {{{\Re }_{{s,d}}}} \right|}}\sum\limits_{a,b,c \in \mathcal{S},\mathcal{D}} {\sigma _{a}^{2}\sigma _{b}^{2}\sigma _{c}^{2}\Im _{{s,d}}^{{abc\mu \nu }}} .$$
((A.5))

The multiindex expressions (symmetric in Lorentz indices) appearing in (A.5) take the form

$$\begin{gathered} \Im _{{s,d}}^{{abc00}} = \left[ {{{\Gamma }_{a}}{{\Gamma }_{b}} - \frac{1}{3}({{u}_{a}}{{u}_{b}})} \right]({{u}_{b}}{{u}_{c}})({{u}_{c}}{{u}_{a}}) \\ + \,\,\frac{1}{2}\Gamma _{a}^{2}\left[ {1 - {{{({{u}_{b}}{{u}_{c}})}}^{2}}} \right] + \frac{1}{3}, \\ \end{gathered} $$
$$\begin{gathered} \Im _{{s,d}}^{{abc0i}} = \frac{1}{2}{{\Gamma }_{c}}\left[ {{{{({{u}_{a}}{{u}_{b}})}}^{2}}{{u}_{{ci}}} - {{u}_{{ci}}} - 2({{u}_{a}}{{u}_{b}})({{u}_{b}}{{u}_{c}}){{u}_{{ai}}}} \right. \\ \left. { + \,\,2({{u}_{a}}{{u}_{c}}){{u}_{{ai}}}} \right] - {{\Gamma }_{b}}({{u}_{a}}{{u}_{b}}){{u}_{{ai}}}, \\ \end{gathered} $$
$$\begin{gathered} \Im _{{s,d}}^{{abcij}} = {{\Gamma }_{a}}{{\Gamma }_{b}}\left[ {({{u}_{c}}{{u}_{a}})({{u}_{c}}{{u}_{b}}) - ({{u}_{a}}{{u}_{b}})} \right]{{\delta }_{{ij}}} + \frac{1}{2}\left( {\Gamma _{c}^{2} - 1} \right) \\ \times \,\,\left[ {1 - {{{({{u}_{a}}{{u}_{b}})}}^{2}}} \right]{{\delta }_{{ij}}} \\ + \,\,\left\{ {{{\Gamma }_{c}}\left[ {{{\Gamma }_{a}}({{u}_{b}}{{u}_{c}}) + {{\Gamma }_{b}}({{u}_{c}}{{u}_{a}}) - {{\Gamma }_{c}}{{{({{u}_{a}}{{u}_{b}})}}^{{}}}} \right]} \right. \\ \left. { - \,\,{{\Gamma }_{a}}{{\Gamma }_{b}} + ({{u}_{a}}{{u}_{b}})} \right\}{{u}_{{ai}}}{{u}_{{bj}}} + {{\Gamma }_{b}}\left[ {{{\Gamma }_{b}} - {{\Gamma }_{a}}({{u}_{a}}{{u}_{b}})} \right]{{u}_{{ci}}}{{u}_{{cj}}}; \\ \end{gathered} $$
$$\begin{gathered} \left| {{{\Re }_{{s,d}}}} \right| = \sum\limits_{a,b,c,d \in \mathcal{S},\mathcal{D}} {\sigma _{a}^{2}\sigma _{b}^{2}\sigma _{c}^{2}\sigma _{d}^{2}} \{ {{\Gamma }_{a}}{{\Gamma }_{b}}({{u}_{c}}{{u}_{a}}) \\ \times \,\,\left[ {({{u}_{c}}{{u}_{d}})({{u}_{d}}{{u}_{b}}) - ({{u}_{b}}{{u}_{c}})} \right] + \frac{1}{3}\left[ {({{u}_{a}}{{u}_{b}})({{u}_{b}}{{u}_{c}})({{u}_{c}}{{u}_{a}}) - 1} \right] \\ - \,\,\frac{1}{2}{{\Gamma }_{a}}{{\Gamma }_{b}}({{u}_{a}}{{u}_{b}})\left[ {{{{({{u}_{c}}{{u}_{d}})}}^{2}} - 1} \right] + \frac{1}{6}\Gamma _{d}^{2} \\ \left. { \times \,\,\left[ {3{{{({{u}_{b}}{{u}_{c}})}}^{2}} - 2({{u}_{a}}{{u}_{b}})({{u}_{b}}{{u}_{c}})({{u}_{c}}{{u}_{a}}) - 1} \right]} \right\}. \\ \end{gathered} $$

This result was obtained without assuming energy-momentum conservation.

The positive definiteness of symmetric matrices \({{\Re }_{{s,d}}}\) (and, consequently, \({{\tilde {\Re }}_{{s,d}}}\)) leads to a number of strict inequalities; in particularFootnote 52,

$$\begin{gathered} \tilde {\Re }_{{s,d}}^{{\mu \mu }} > 0,\,\,\,\,\tilde {\Re }_{{s,d}}^{{00}}\tilde {\Re }_{{s,d}}^{{ii}} - {{\left( {\tilde {\Re }_{{s,d}}^{{0i}}} \right)}^{2}} > 0, \\ \tilde {\Re }_{{s,d}}^{{jj}}\tilde {\Re }_{{s,d}}^{{kk}} - {{\left( {\tilde {\Re }_{{s,d}}^{{jk}}} \right)}^{2}} > 0. \\ \end{gathered} $$
((A.6a))

Similar inequalities are valid for the cofactors \(\mathfrak{A}_{{s,d}}^{{\mu \nu }}\) and \(\tilde {\mathfrak{A}}_{{s,d}}^{{\mu \nu }}\) (since \(\left| {{{{\tilde {\Re }}}_{{s,d}}}} \right| > 0\)) and for the elements of matrix \(\left\| {{{R}^{{\mu \nu }}}} \right\| = \left\| {\tilde {\Re }_{s}^{{\mu \nu }} + \tilde {\Re }_{d}^{{\mu \nu }}} \right\|\) (without summation over repeated indices):

$${{R}_{{\mu \mu }}} > 0,\,\,\,\,{{R}_{{00}}}{{R}_{{ii}}} - R_{{0i}}^{2} > 0,\,\,\,\,{{R}_{{jj}}}{{R}_{{kk}}} - R_{{jk}}^{2} > 0.$$
((A.6b))

Inequalities (A.6b) have important corollaries such as the positivity of functions \(\mathcal{R}\) and \(\mathfrak{m} - {{\mathfrak{n}}_{0}} - \mathfrak{n}_{0}^{2}\). Indeed, the following is true in the reference frame with axis \(z\) directed along unit vector \({\mathbf{l}}\):

$$\begin{gathered} \mathcal{R} = {{R}_{{33}}},\,\,\,\,\mathfrak{m} - {{\mathfrak{n}}_{0}} - \mathfrak{n}_{0}^{2} = \frac{{{{R}_{{00}}}\mathcal{R} - {{{({\mathbf{Rl}})}}^{2}}}}{{{{R}^{2}}}} \\ = \frac{{{{R}_{{00}}}{{R}_{{33}}} - R_{{03}}^{2}}}{{{{{\left( {{{R}_{{00}}} - 2{{R}_{{03}}} + {{R}_{{33}}}} \right)}}^{2}}}}. \\ \end{gathered} $$

Since these quantities are rotationally invariant, it follows from (A.6b) that \(\mathcal{R} > 0\) and \(\mathfrak{m} - {{\mathfrak{n}}_{0}} - \mathfrak{n}_{0}^{2} > 0\). It is evident from the latter inequality that quantity (285) is positive. This results in the suppression of probability (297).

Functions \(n\) and \(\bar {m}\) are constructed from the components of 4-vector \(Y = {{Y}_{s}} + {{Y}_{d}}\), where

$$Y_{s}^{\mu } = \tilde {\Re }_{s}^{{\mu \nu }}{{q}_{{s\nu }}}\,\,\,\,{\text{and}}\,\,\,\,Y_{d}^{\mu } = - \tilde {\Re }_{d}^{{\mu \nu }}{{q}_{{d\nu }}},$$

and tensor components \({{R}^{{0i}}} = \tilde {\Re }_{s}^{{0i}} + \tilde {\Re }_{d}^{{0i}}\). Note that scalar product \(Yl = ({{Y}_{s}} + {{Y}_{d}})l = {{E}_{\nu }}R\) and zeroth component \({{Y}^{0}} = Y_{s}^{0} + Y_{d}^{0}\) are the only quantities actually needed to calculate function \(n\). It is also sufficient to determine them in the PW\(_{0}\) limit, where calculations are simplified considerably. In what follows, we use symbol \(\left[\kern-0.15em\left[ f \right]\kern-0.15em\right]\) to denote that function \(f\) is calculated in the PW\(_{0}\) limit. In these terms,

$$\begin{gathered} {{Y}_{{s,d}}}l \to {{\left. {\tilde {\Re }_{{s,d}}^{{\mu \nu }}{{q}_{\mu }}{{l}_{\nu }}} \right|}_{{q = {{E}_{\nu }}l}}} = E_{\nu }^{{ - 1}}\left[\kern-0.15em\left[ {\tilde {\Re }_{{s,d}}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] \\ {\text{and}}\,\,\,\,Y_{{s,d}}^{0} \to \left[\kern-0.15em\left[ {\tilde {\Re }_{{s,d}}^{{0\mu }}{{q}_{\mu }}} \right]\kern-0.15em\right]. \\ \end{gathered} $$

Let us examine several simple types of processes in the source and detector, which are relevant to real-life neutrino experiments, to clarify these general formulas.

A.3. Two-Particle Decay in the Source

Let us study the simplest process: lepton decay \(a \to \ell \nu {\text{*}}\) in the source (\({{a}_{{\ell 2}}}\)). Here, \(a\) is a charged meson (\({{\pi }^{ \pm }}\), \({{K}^{ \pm }}\), \(D_{s}^{ \pm }, \ldots \)), \(\ell \) is a charged lepton (\({{e}^{ \pm }},{{\mu }^{ \pm }},{{\tau }^{ \pm }}\)), and \(\nu {\text{*}}\) is a virtual neutrino or antineutrino. Since such decays are the primary sources of high-energy accelerator, atmospheric, and astrophysical neutrinos and antineutrinos, we examine this example in sufficient detailFootnote 53.

A.3.1. Formulas for arbitrary momenta. In the case under consideration, the determinant of matrix \({{\Re }_{s}}\) is easy to calculate using formula (A.2) written in the intrinsic frame of the meson packetFootnote 54:

$$\left| {{{\Re }_{s}}} \right| = \sigma _{a}^{2}\sigma _{\ell }^{2}\sigma _{2}^{4}{{\left| {{\mathbf{u}}_{\ell }^{ \star }} \right|}^{2}}.$$
((A.7))

Here, \(\sigma _{2}^{2} = \sigma _{a}^{2} + \sigma _{\ell }^{2}\) and \({{{\mathbf{u}}}_{\ell }} = {{{{{\mathbf{p}}}_{\ell }}} \mathord{\left/ {\vphantom {{{{{\mathbf{p}}}_{\ell }}} {{{m}_{\ell }}}}} \right. \kern-0em} {{{m}_{\ell }}}} = {{\Gamma }_{\ell }}{{{\mathbf{v}}}_{\ell }}\). Since determinant \(\left| {{{\Re }_{s}}} \right|\) is Lorentz-invariant, it is transformed to the laboratory frame by substitution

$$\left| {{\mathbf{u}}_{\ell }^{ \star }} \right| = \frac{1}{{{{m}_{a}}{{m}_{\ell }}}}\sqrt {{{{\left( {{{E}_{\ell }}{{{\mathbf{p}}}_{a}} - {{E}_{a}}{{{\mathbf{p}}}_{\ell }}} \right)}}^{2}} - {{{\left| {{{{\mathbf{p}}}_{a}} \times {{{\mathbf{p}}}_{\ell }}} \right|}}^{2}}} = ({{u}_{a}}{{u}_{\ell }}){{{\text{V}}}_{{a\ell }}},$$

where

$${{{\text{V}}}_{{a\ell }}} = \frac{{\sqrt {{{{\left( {{{{\mathbf{v}}}_{a}} - {{{\mathbf{v}}}_{\ell }}} \right)}}^{2}} - {{{\left| {{{{\mathbf{v}}}_{a}} \times {{{\mathbf{v}}}_{\ell }}} \right|}}^{2}}} }}{{1 - {{{\mathbf{v}}}_{a}}{{{\mathbf{v}}}_{\ell }}}}$$

is the invariant relative velocity of a meson and a lepton in the laboratory frame. Note that the kinematic variables corresponding to different particles are independent in this formula because they are not constrained by the energy-momentum conservation law. In the particular case with a virtual neutrino on the mass shell and the neutrino mass and the spreads of lepton and meson momenta being negligible, one may use the common relations of the two-particle decay kinematics, which state that

$$\left| {{\mathbf{v}}_{\ell }^{ \star }} \right| = {{{\text{V}}}_{{a\ell }}} = \frac{{m_{a}^{2} - m_{\ell }^{2}}}{{m_{a}^{2} + m_{\ell }^{2}}}\,\,\,\,{\text{and}}\,\,\,\,\left| {{\mathbf{u}}_{\ell }^{ \star }} \right| = \frac{{m_{a}^{2} - m_{\ell }^{2}}}{{2{{m}_{a}}{{m}_{\ell }}}}.$$

A somewhat more complex calculation involving general formulas (A.4) allows one to determine the explicit form of cofactors \(\mathfrak{A}_{s}^{{\mu \nu }}\):

$$\begin{gathered} \mathfrak{A}_{s}^{{00}} = \sigma _{2}^{2}\left[ {\sigma _{2}^{2}\left( {\sigma _{a}^{2}\Gamma _{a}^{2} + \sigma _{\ell }^{2}\Gamma _{\ell }^{2}} \right) + \sigma _{a}^{2}\sigma _{\ell }^{2}{{{\left| {{{{\mathbf{u}}}_{a}} \times {{{\mathbf{u}}}_{\ell }}} \right|}}^{2}}} \right], \\ \mathfrak{A}_{s}^{{0i}} = \sigma _{2}^{2}\left\{ {\sigma _{a}^{2}} \right.\left[ {\sigma _{a}^{2}{{\Gamma }_{a}} + \sigma _{\ell }^{2}{{\Gamma }_{\ell }}({{u}_{a}}{{u}_{\ell }})} \right]{{u}_{{ai}}} \\ \left. { + \,\,\sigma _{\ell }^{2}\left[ {\sigma _{\ell }^{2}{{\Gamma }_{\ell }} + \sigma _{a}^{2}{{\Gamma }_{a}}({{u}_{a}}{{u}_{\ell }})} \right]{{u}_{{\ell i}}}} \right\}, \\ \mathfrak{A}_{s}^{{ii}} = \sigma _{2}^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{a}^{2}{{u}_{{ai}}} + \sigma _{\ell }^{2}({{u}_{a}}{{u}_{\ell }}){{u}_{{\ell i}}}} \right]} \right.{{u}_{{ai}}} \\ + \,\,\sigma _{\ell }^{2}\left[ {\sigma _{\ell }^{2}{{u}_{{\ell i}}} + \sigma _{a}^{2}({{u}_{a}}{{u}_{\ell }}){{u}_{{ai}}}} \right]{{u}_{{\ell i}}} \\ \left. { + \,\,\sigma _{a}^{2}\sigma _{\ell }^{2}\left[ {{{{\left( {{{\Gamma }_{\ell }}{{{\mathbf{u}}}_{a}} - {{\Gamma }_{a}}{{{\mathbf{u}}}_{\ell }}} \right)}}^{2}} - {{{\left| {{{{\mathbf{u}}}_{a}} \times {{{\mathbf{u}}}_{\ell }}} \right|}}^{2}}} \right]} \right\}, \\ \mathfrak{A}_{s}^{{jk}} = \sigma _{2}^{2}\left[ {\sigma _{a}^{4}{{u}_{{aj}}}{{u}_{{ak}}} + \sigma _{\ell }^{4}{{u}_{{\ell j}}}{{u}_{{\ell k}}} + \sigma _{a}^{2}\sigma _{\ell }^{2}({{u}_{a}}{{u}_{\ell }})} \right. \\ \times \,\,\left. {\left( {{{u}_{{aj}}}{{u}_{{\ell k}}} + {{u}_{{\ell j}}}{{u}_{{ak}}}} \right)} \right],\,\,j \ne k. \\ \end{gathered} $$
((A.8))

An explicitly Lorentz-invariant formula for the components of the inverse tensor may be derived from (A.8) or (A.5):

$$\tilde {\Re }_{s}^{{\mu \nu }} = \frac{{\sigma _{a}^{4}u_{a}^{\mu }u_{a}^{\nu } + \sigma _{\ell }^{4}u_{\ell }^{\mu }u_{\ell }^{\nu } - \sigma _{a}^{2}\sigma _{\ell }^{2}\left\{ {{{g}^{{\mu \nu }}}\left[ {{{{({{u}_{a}}{{u}_{\ell }})}}^{2}} - 1} \right] - ({{u}_{a}}{{u}_{\ell }})(u_{a}^{\mu }u_{\ell }^{\nu } + u_{\ell }^{\mu }u_{a}^{\nu })} \right\}}}{{\sigma _{a}^{2}\sigma _{\ell }^{2}\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right)\left[ {{{{({{u}_{a}}{{u}_{\ell }})}}^{2}} - 1} \right]}}.$$

It follows that

$$\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }} = \frac{{\sigma _{a}^{4}{{{\left( {{{u}_{a}}q} \right)}}^{2}} + \sigma _{\ell }^{4}{{{\left( {{{u}_{\ell }}q} \right)}}^{2}} + 2\sigma _{a}^{2}\sigma _{\ell }^{2}\left( {{{u}_{a}}{{u}_{\ell }}} \right)\left( {{{u}_{a}}q} \right)\left( {{{u}_{\ell }}q} \right)}}{{\sigma _{a}^{2}\sigma _{\ell }^{2}\sigma _{2}^{2}{{{\left| {{\mathbf{u}}_{\ell }^{ \star }} \right|}}^{2}}}} - \frac{{{{q}^{2}}}}{{\sigma _{2}^{2}}}.$$
((A.9))

Let us now find the explicit form of 4-vector \({{Y}_{s}}\) with its components defined as

$$\begin{gathered} Y_{s}^{0} = \tilde {\Re }_{s}^{{0\nu }}{{q}_{{s\nu }}} = \frac{1}{{\left| {{{\Re }_{s}}} \right|}}\left[ {{{\mathfrak{A}}^{{00}}}\left( {{{E}_{a}} - {{E}_{\ell }}} \right) - {{\mathfrak{A}}^{{0i}}}{{{\left( {{{p}_{a}} - {{p}_{\ell }}} \right)}}_{i}}} \right], \\ Y_{s}^{i} = \tilde {\Re }_{s}^{{i\nu }}{{q}_{{s\nu }}} = \frac{1}{{\left| {{{\Re }_{s}}} \right|}}\left[ {{{\mathfrak{A}}^{{ij}}}{{{\left( {{{p}_{a}} - {{p}_{\ell }}} \right)}}_{j}} - {{\mathfrak{A}}^{{i0}}}\left( {{{E}_{a}} - {{E}_{\ell }}} \right)} \right]. \\ \end{gathered} $$

With elementary transformations, the following is derived from (A.8):

$$Y_{s}^{\mu } = \frac{1}{{{{{\left| {{\mathbf{u}}_{\ell }^{ \star }} \right|}}^{2}}}}\left\{ {\left[ {\frac{{({{p}_{a}}{{p}_{\ell }})}}{{m_{a}^{2}}} - 1} \right]\frac{{p_{a}^{\mu }}}{{\sigma _{\ell }^{2}}} - \left[ {\frac{{({{p}_{a}}{{p}_{\ell }})}}{{m_{\ell }^{2}}} - 1} \right]\frac{{p_{\ell }^{\mu }}}{{\sigma _{a}^{2}}}} \right\}.$$
((A.10))

Useful identities:

$$\begin{gathered} \left( {{{u}_{a}}q} \right)\left( {{{u}_{\ell }}q} \right) = {{\Gamma }_{a}}{{\Gamma }_{\ell }}q_{0}^{2} \\ - \,\,\left[ {{{\Gamma }_{a}}\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right) + {{\Gamma }_{\ell }}\left( {{{{\mathbf{u}}}_{a}}{\mathbf{q}}} \right)} \right]{{q}_{0}} + \left( {{{{\mathbf{u}}}_{a}}{\mathbf{q}}} \right)\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right), \\ \left( {{{{\mathbf{u}}}_{a}}{\mathbf{q}}} \right)\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right) \\ = \sum\limits_i {\left[ {\left( {{{u}_{{ai}}}{{q}_{i}}} \right)\left( {{{u}_{{\ell i}}}{{q}_{i}}} \right) + \left( {{{u}_{{aj}}}{{q}_{j}}} \right)\left( {{{u}_{{\ell k}}}{{q}_{k}}} \right) + \left( {{{u}_{{ak}}}{{q}_{k}}} \right)\left( {{{u}_{{\ell j}}}{{q}_{j}}} \right)} \right]} . \\ \end{gathered} $$

A.3.2. PW0limit. Since kinematic constraints were not applied in the derivation of formulas in the previous section, they hold true for an arbitrary 4-vector \(q\). Let us now introduce the exact conservation laws (i.e., pass to the PW\(_{0}\) limit, which implies in this special case that \(q = {{p}_{a}} - {{p}_{\ell }} = {{p}_{\nu }} = {{E}_{\nu }}l\) and \(p_{\nu }^{2} = 0\)):

$${{u}_{a}}{{u}_{\ell }} = \frac{{E_{\ell }^{ \star }}}{{{{m}_{\ell }}}},\,\,\,\,{{u}_{a}}q = E_{\nu }^{ \star },\,\,\,\,{{u}_{\ell }}q = \frac{{{{m}_{a}}E_{\nu }^{ \star }}}{{{{m}_{\ell }}}},$$

where

$$E_{\ell }^{ \star } = \frac{{m_{a}^{2} + m_{\ell }^{2}}}{{2{{m}_{a}}}}\,\,\,\,{\text{and}}\,\,\,\,E_{\nu }^{ \star } = \frac{{m_{a}^{2} - m_{\ell }^{2}}}{{2{{m}_{a}}}}$$

are the energies of a lepton and a neutrino in the intrinsic reference frame of a meson. Inserting these relations into (A.9), we find the PW\(_{0}\) limit for quadratic form \(\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\):

$$\left[\kern-0.15em\left[ {\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] = \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} \gg 1.$$
((A.11))

Estimation of the parameters of an effective neutrino packet. In order to illustrate the result, we consider a special (although fairly realistic) case when the contribution from the reaction in the detector to complete function \(\mathfrak{D}\) may be neglected (i.e., it is assumed that parameters \({{\sigma }_{\varkappa }}\) for all \( \in D\) are sufficiently large compared to \({{\sigma }_{a}}\) and \({{\sigma }_{\ell }}\)). This assumption is valid in experiments with accelerator and atmospheric muon neutrinos and antineutrinos. The interactions of mesons and muons in the source (decay channel, atmosphere) with each other and the medium are normally negligible in such experiments; therefore, the values of \({{\sigma }_{a}}\) and \({{\sigma }_{\mu }}\) are very small (close to the limiting ones in the SRGP model). The values of parameters \({{\sigma }_{\varkappa }}\) for long-lived in- and out-particles \(\varkappa \) (nuclei, hadrons, leptons) in a typical detector are defined by their mean free paths in the detector medium rather than by their decay widthsFootnote 55; thus, \({{\sigma }_{\varkappa }} \gg {{\sigma }_{{a,\mu }}}\). At the very least, the \({{\sigma }_{\varkappa }} \gg {{\sigma }_{\mu }}\) condition is satisfied for short-lived particles and resonances that are produced in the detector and decay prior to interacting with the medium.

Thus, using (A.11), we find

$${{\mathfrak{D}}^{2}} \approx E_{\nu }^{2}{{\left[ {2\left( {\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right)} \right]}^{{ - 1}}} \ll E_{\nu }^{2},$$
((A.12))

and, consequently,

$$\sigma _{j}^{2} \approx \frac{{m_{j}^{2}}}{2}{{\left( {\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right)}^{{ - 1}}} \ll m_{j}^{2}.$$
((A.13))

It is evident that the effective wave packet of a virtual neutrino with a given mass is defined completely in this simplest case by the masses and momentum spreads of packets \(a\) and \(\ell \), and the values of \({{\sigma }_{j}}\) for all three known neutrinos are exceptionally small at any \({{\sigma }_{a}}\) and \({{\sigma }_{\ell }}\) allowed by the SRGP approximation. In addition, since all the other (known) elementary particles with a nonzero mass are many orders of magnitude lower than the three known neutrinos, we conclude that \(\sigma _{j}^{2} \lll \sigma _{{a,\ell }}^{2}\). Although these estimates were obtained without including the contributions of the reaction in the detector, they explain qualitatively the relevance of the standard quantum-mechanical assumption that light neutrinos have definite momenta in spite of the fact that they are produced in processes involving particles with relatively large momentum spreads. It follows from (A.12) that \({{\sigma }_{j}} = 0\) at \({{m}_{j}} = 0\); i.e., massless neutrinos may be regarded as plane waves. With obvious reservations, this remarkable fact may be used in the analysis of processes similar to those depicted in Fig. 8, where light massive neutrinos act as external wave packets.

The conditions of applicability of the SRGP approximation for unstable particles

$${{\left( {{{{{\sigma }_{\varkappa }}} \mathord{\left/ {\vphantom {{{{\sigma }_{\varkappa }}} {\sigma _{\varkappa }^{{max}}}}} \right. \kern-0em} {\sigma _{\varkappa }^{{max}}}}} \right)}^{4}} \ll 1,\,\,\,\,\sigma _{\varkappa }^{{max}} = \sqrt {{{m}_{\varkappa }}{{\Gamma }_{\varkappa }}} $$

(\({{\Gamma }_{\varkappa }} = {1 \mathord{\left/ {\vphantom {1 {{{\tau }_{\varkappa }}}}} \right. \kern-0em} {{{\tau }_{\varkappa }}}}\) is the total decay width of particle \(\varkappa \)) yield the following important constraint:

$$\sigma _{j}^{2} \ll \frac{{m_{j}^{2}}}{2}{{\left( {\frac{{{{m}_{\ell }}}}{{{{\Gamma }_{\ell }}}} + \frac{{{{m}_{a}}}}{{{{\Gamma }_{a}}}}} \right)}^{{ - 1}}}.$$

Thus, two-particle decays of any mesons with a muon in the final state (\({{\pi }_{{\mu 2}}}\), \({{K}_{{\mu 2}}}\), etc.) are constrained from the above:

$$\frac{{\sigma _{j}^{2}}}{{m_{j}^{2}}} \ll \frac{{{{\Gamma }_{\mu }}}}{{2{{m}_{\mu }}}} \approx 1.4 \times {{10}^{{ - 18}}};$$

therefore, \({{\sigma _{j}^{{max}}} \mathord{\left/ {\vphantom {{\sigma _{j}^{{max}}} {\sigma _{\mu }^{{max}}}}} \right. \kern-0em} {\sigma _{\mu }^{{max}}}} \approx {{{{m}_{j}}} \mathord{\left/ {\vphantom {{{{m}_{j}}} {{{m}_{\mu }}}}} \right. \kern-0em} {{{m}_{\mu }}}} \lll 1\). This yields the following lower bounds for the effective sizes of a neutrino packet transverse and longitudinal with respect to the neutrino momentumFootnote 56:

$$\begin{gathered} d_{j}^{ \bot } \gg 2.6\left( {\frac{{0.1\,\,{\text{eV}}}}{{{{m}_{j}}}}} \right)\,\,{\text{km}} \\ {\text{and}}\,\,\,\,d_{j}^{\parallel } = \frac{{d_{j}^{ \bot }}}{{{{\Gamma }_{j}}}} \gg 2.6 \times {{10}^{{ - 5}}}\left( {\frac{{1\,\,{\text{GeV}}}}{{{{E}_{\nu }}}}} \right)\left( {\frac{{0.1\,\,{\text{eV}}}}{{{{m}_{j}}}}} \right)\,\,{\text{cm}}. \\ \end{gathered} $$

The constraints on the characteristics of neutrino packets emerging in \({{a}_{{\tau 2}}}\) decays depend on the type of the decaying particle. For example, \({{\sigma _{j}^{2}} \mathord{\left/ {\vphantom {{\sigma _{j}^{2}} {m_{j}^{2}}}} \right. \kern-0em} {m_{j}^{2}}} \ll 2.2 \times {{10}^{{ - 13}}}\) in the decay of a \({{D}_{s}}\) meson. At \({{m}_{j}} = 0.1\) eV, this corresponds to a limiting effective transverse size of a neutrino packet of “just” 6.6 m.

Effective sizes \(d_{j}^{ \bot }\) and \(d_{j}^{\parallel }\) define (in the order of magnitude) the allowed transverse and longitudinal quantum deviations of the center of the neutrino wave packet from “classical trajectory” \({{{\mathbf{\bar {L}}}}_{j}} = {{{\mathbf{v}}}_{j}}T\). Evidently, transverse deviations \(\delta {\mathbf{L}}_{j}^{ \bot } \sim {{d_{j}^{ \bot }} \mathord{\left/ {\vphantom {{d_{j}^{ \bot }} 2}} \right. \kern-0em} 2}\) may be huge and exceed in magnitude the size of current neutrino detectors and the natural accelerator neutrino beam divergence (all over distances of several hundred or thousand kilometers from the source). This should come as no surprise if one remembers that the standard quantum-mechanical description of a massive neutrino as a state with definite momentum implies (as a direct consequence of the uncertainty principle) that its transverse and longitudinal “sizes” are infinite. When data from common experiments on neutrino oscillations are interpreted, this description does not yield nonphysical results because neither the transverse size of a neutrino packet nor transverse quantum fluctuations are found in the expression for the count rate of neutrino events averaged properly over spatial coordinates \({{{\mathbf{x}}}_{\varkappa }}\) of external packets. This averaging also eliminates automatically the dependence on time components \(x_{\varkappa }^{0}\); only the dependence of the count rate on external (“instrumental”) space-time variables set by the experimental conditionsFootnote 57 remains. Large transverse sizes of a neutrino packet may manifest themselves in specialized neutrino experiments, where both the source–detector distance and the time interval between the neutrino production and its interaction are monitored. Such experiments will be discussed in a separate study.

The effects of noncollinearity of momentum transfers in the source and detector are contained in functions \(\mathfrak{n}\) and . The properties of these functions are discussed below.

Contributions to functions \(\mathfrak{n}\) and \(\mathfrak{m}\). Using general formula (A.10), we find 4-vector Ys in the PW0 approximation:

$$\begin{gathered} \left[\kern-0.15em\left[ {{{Y}_{s}}} \right]\kern-0.15em\right] = \frac{1}{{{{m}_{a}}E_{\nu }^{ \star }}}\left[ {\left( {\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right){{p}_{a}} - \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}{{p}_{\nu }}} \right] \\ = \frac{1}{{{{m}_{a}}E_{\nu }^{ \star }}}\left[ {\left( {\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right){{p}_{\ell }} - \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}{{p}_{\nu }}} \right]. \\ \end{gathered} $$

Scalar products needed to calculate the a𝓁2 contributions to function \(n\) take the form

$$\left[\kern-0.15em\left[ {{{Y}_{s}}l} \right]\kern-0.15em\right] = \left[\kern-0.15em\left[ {\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right]\frac{1}{{{{E}_{\nu }}}} = \left( {\tfrac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \tfrac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right)\tfrac{1}{{{{E}_{\nu }}}}$$
$$\begin{gathered} \left[\kern-0.15em\left[ {{{Y}_{s}}{\mathbf{l}}} \right]\kern-0.15em\right] = \left[\kern-0.15em\left[ {Y_{s}^{0} - {{Y}_{s}}l} \right]\kern-0.15em\right] = \tfrac{{{{\Gamma }_{a}}}}{{E_{\nu }^{ \star }}} \\ \times \,\,\left[ {\left( {\tfrac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \tfrac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}} \right)\left( {1 - \tfrac{{{{m}_{a}}E_{\nu }^{ \star }}}{{{{E}_{a}}{{E}_{\nu }}}}} \right) - \tfrac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}\tfrac{{{{E}_{\nu }}}}{{{{E}_{a}}}}} \right]. \\ \end{gathered} $$

This yields the following:

$$\begin{gathered} \left[\kern-0.15em\left[ {\tfrac{{{{{\mathbf{Y}}}_{s}}{\mathbf{l}}}}{{{{Y}_{s}}l}}} \right]\kern-0.15em\right] = \left[ {{{\Gamma }_{a}} - \left( {\tfrac{{m_{a}^{2}\sigma _{\ell }^{2}}}{{m_{a}^{2}\sigma _{\ell }^{2} + m_{\ell }^{2}\sigma _{a}^{2}}}} \right)\tfrac{{{{E}_{\nu }}}}{{{{m}_{a}}}}} \right] \\ \times \,\,\tfrac{{{{E}_{\nu }}}}{{E_{\nu }^{ \star }}} - 1 \equiv {{\mathfrak{n}}_{s}}\left( {{{E}_{a}},{{E}_{\nu }}} \right). \\ \end{gathered} $$

Function \({{\mathfrak{n}}_{s}}\) may serve as an estimate for complete function \(\mathfrak{n}\) if, just as with function \(\mathfrak{D}\), we neglect the contribution to \(\mathfrak{n}\) from the detector part of the diagram (in the present case, from \(Y_{d}^{0}\) and \({{Y}_{d}}l\)) assuming that parameters \({{\sigma }_{\varkappa }}\) (\(\varkappa \in D\)) are sufficiently large compared to \({{\sigma }_{a}}\) and \({{\sigma }_{\ell }}\). Since function \({{\mathfrak{n}}_{s}}\) depends linearly on Ea at a fixed value of Eν, the following inequality holds true:

$${{\mathfrak{n}}_{s}} \geqslant {{\mathfrak{n}}_{s}}\left( {E_{a}^{{min}},{{E}_{\nu }}} \right),$$

where

$$E_{a}^{{min}} = \frac{{{{m}_{a}}}}{2}\left( {\frac{{{{E}_{\nu }}}}{{E_{\nu }^{ \star }}} + \frac{{E_{\nu }^{ \star }}}{{{{E}_{\nu }}}}} \right)$$

is the minimum energy of particle \(a\) needed to produce a massless neutrino with energy \({{E}_{\nu }}\) in an \({{a}_{{\ell 2}}}\) decay. Thus, the absolute minimum of function \({{\mathfrak{n}}_{s}}\) is negative:

$$\begin{gathered} \mathfrak{n}_{s}^{{min}} = {{\mathfrak{n}}_{s}}({{m}_{a}},E_{\nu }^{ \star }) = - \frac{{\left( {m_{a}^{2} - m_{\ell }^{2}} \right)\sigma _{\ell }^{2}}}{{2\left( {m_{a}^{2}\sigma _{\ell }^{2} + m_{\ell }^{2}\sigma _{a}^{2}} \right)}}, \\ \left| {\mathfrak{n}_{s}^{{min}}} \right| < \frac{1}{2}\left( {1 - \frac{{m_{\ell }^{2}}}{{m_{a}^{2}}}} \right). \\ \end{gathered} $$

Function \({{n}_{s}}\) increases with neutrino energy and may be arbitrarily large at \({{E}_{\nu }} \gg E_{\nu }^{ \star }\):

$$\begin{gathered} {{\mathfrak{n}}_{s}} \geqslant {{\mathfrak{n}}_{s}}\left( {E_{a}^{{min}},{{E}_{\nu }}} \right) = \frac{1}{2}\left( {1 - 2\left| {\mathfrak{n}_{s}^{{min}}} \right|} \right) \\ \times \,\,{{\left( {\frac{{{{E}_{\nu }}}}{{E_{\nu }^{ \star }}}} \right)}^{2}}\left[ {1 + \mathcal{O}\left( {\frac{{E_{\nu }^{ \star }}}{{{{E}_{\nu }}}}} \right)} \right]. \\ \end{gathered} $$

As is well known, the neutrino energy distribution in an \({{a}_{{\ell 2}}}\) decay is uniform (i.e., independent of \({{E}_{\nu }}\)) within the following kinematic bounds:

$$E_{\nu }^{ \star }{{\Gamma }_{a}}(1 - \left| {{{{\mathbf{v}}}_{a}}} \right|) \leqslant {{E}_{\nu }} \leqslant E_{\nu }^{ \star }{{\Gamma }_{a}}(1 + \left| {{{{\mathbf{v}}}_{a}}} \right|).$$

It follows that the mean energy of a decay neutrino is \({{\bar {E}}_{\nu }} = {{\Gamma }_{a}}E_{\nu }^{ \star }\). Therefore, the following is true to within \(\mathcal{O}(\Gamma _{a}^{{ - 2}})\) at high energies of decaying mesons, \({{\Gamma }_{a}} \gg 1\):

$$\begin{gathered} {{\mathfrak{n}}_{s}}\left( {{{E}_{a}},{{{\bar {E}}}_{\nu }}} \right) \approx \Gamma _{a}^{2}\left( {1 - \left| {\mathfrak{n}_{s}^{{min}}} \right|} \right) \\ = \left( {1 - \left| {\mathfrak{n}_{s}^{{min}}} \right|} \right){{\left( {\frac{{{{{\bar {E}}}_{\nu }}}}{{E_{\nu }^{ \star }}}} \right)}^{2}}. \\ \end{gathered} $$

Consequently,

$${{\left. {{{\mathfrak{n}}_{s}}{{r}_{j}}} \right|}_{{{{E}_{\nu }} = {{{\bar {E}}}_{\nu }}}}} \approx \frac{1}{2}\left( {1 - \left| {\mathfrak{n}_{s}^{{min}}} \right|} \right){{\left( {\frac{{{{m}_{j}}}}{{E_{\nu }^{ \star }}}} \right)}^{2}} \ll 1.$$

Under the same assumptions, retaining only the leading terms in \({{\Gamma }_{a}}\) and \({{{{E}_{\nu }}} \mathord{\left/ {\vphantom {{{{E}_{\nu }}} {E_{\nu }^{ \star }}}} \right. \kern-0em} {E_{\nu }^{ \star }}}\), one may estimate the \({{a}_{{\ell 2}}}\)-decay contribution to function \(\mathfrak{m}\):

$$\begin{gathered} {{\mathfrak{m}}_{s}} \approx \Gamma _{a}^{2}\left\{ {1 + \frac{{\sigma _{\ell }^{4}m_{a}^{4}}}{{{{{\left( {m_{\ell }^{2}\sigma _{a}^{2} + m_{a}^{2}\sigma _{\ell }^{2}} \right)}}^{2}}}}\left[ {1 + \frac{{2\sigma _{a}^{2}E_{\nu }^{ \star }}}{{{{m}_{a}}\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right)}}} \right]} \right. \\ \left. { \times \,\,{{{\left( {\frac{{{{E}_{\nu }}}}{{{{E}_{a}}}}} \right)}}^{2}}} \right\}{{\left( {\frac{{{{E}_{\nu }}}}{{E_{\nu }^{ \star }}}} \right)}^{2}}. \\ \end{gathered} $$

It can be seen that \({{\mathfrak{m}}_{s}} \gg {{\mathfrak{n}}_{s}}\); however, inequalities (284) assumed in Section 8.2.1 remain valid at \({{E}_{\nu }} \sim {{\bar {E}}_{\nu }}\).

A.4. Quasi-elastic Scattering in the Detector

Let us consider the quasi-elastic scattering of a virtual neutrino \(\nu_{*}a \to b\ell \), where target particle \(a\) may be an electron, a nucleon, or a nucleus and \(\ell \) is a charged lepton, as the simplest (and the most important) example of a reaction at the detector vertex. Since the velocities of target particles in the laboratory frame are very low (thermal) in a typical neutrino experiment, we assume that the laboratory frame coincides with the intrinsic frame of a wave packet describing the state of particle \(a\). If necessary, all formulas may be rewritten in any other reference frame, since we are concerned just with vectors and tensors, and the laws of their transformations are well known.

A.4.1. Formulas for arbitrary momenta. The determinant of matrix \({{\Re }_{d}}\) in the intrinsic reference frame of packet \(a\) takes the form

$$\begin{gathered} \left| {{{\Re }_{d}}} \right| = \sigma _{3}^{2}\left\{ {\left( {\sigma _{3}^{2} + \sigma _{a}^{2}} \right)\sigma _{b}^{2}\sigma _{\ell }^{2}{{{\left( {{{u}_{b}}{{u}_{\ell }}} \right)}}^{2}}{\text{V}}_{{b\ell }}^{2}} \right. \\ + \,\,2\sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}({{u}_{b}}{{u}_{\ell }})({{{\mathbf{u}}}_{b}}{{{\mathbf{u}}}_{\ell }}) \\ \left. { + \,\,\sigma _{a}^{2}\left[ {\sigma _{b}^{2}(\sigma _{a}^{2} + \sigma _{b}^{2}){\mathbf{u}}_{b}^{2} + \sigma _{\ell }^{2}(\sigma _{a}^{2} + \sigma _{\ell }^{2}){\mathbf{u}}_{\ell }^{2}} \right]} \right\}. \\ \end{gathered} $$
((A.14))

Here, \({{{\text{V}}}_{{b\ell }}}\) is the relative velocity of particles \(b\) and \(\ell \), and \(\sigma _{3}^{2} \equiv \sigma _{a}^{2} + \sigma _{b}^{2} + \sigma _{\ell }^{2}\). One important implication of this formula is that determinant \(\left| {{{\Re }_{d}}} \right|\) remains nonnegative even if one (but only one) of the particles (\(a\)\(b\), or \(\ell \)) is described by a plane wave. If, e.g., one neglects the terms proportional to \(\sigma _{\ell }^{2}\), formula (A.14) takes the form coinciding with that of (A.7):

$$\left| {{{\Re }_{d}}} \right| \approx \sigma _{a}^{2}\sigma _{b}^{2}{{\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right)}^{2}}{{\left| {{{{\mathbf{u}}}_{b}}} \right|}^{2}}.$$
((A.15))

This important feature provides an opportunity to simplify the analysis of multipacket in- and out-states by neglecting the contributions of packets with very large spatial sizes (characterized by very small values of parameters \({{\sigma }_{\varkappa }}\)). However, it should be kept in mind that approximate formula (A.15) is applicable only if \(\left| {{{{\mathbf{u}}}_{b}}} \right| \ne 0\)Footnote 58. The same is true in the general case; i.e., when omitting the contributions of packets with very small values of \({{\sigma }_{\varkappa }}\), the phase-space regions within which determinants \(\left| {{{\Re }_{s}}} \right|\) and \(\left| {{{\Re }_{d}}} \right|\) calculated in this approximation vanish, should be cut-off. Such regions are typically located near the kinematic boundaries of the phase space and do not contribute to experimentally measureable characteristics.

According to (A.4), cofactors \(\mathfrak{A}_{d}^{{\mu \nu }}\) are written as follows:

$$\begin{gathered} \mathfrak{A}_{d}^{{00}} = \sigma _{3}^{2}\left[ {\sigma _{3}^{2}\left( {\sigma _{a}^{2} + \sigma _{b}^{2}\Gamma _{b}^{2} + \sigma _{\ell }^{2}\Gamma _{\ell }^{2}} \right)} \right. \\ \left. { + \,\,\sigma _{b}^{2}\sigma _{\ell }^{2}{{{\left| {{{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }}} \right|}}^{2}}} \right], \\ \mathfrak{A}_{d}^{{0i}} = \sigma _{3}^{2}\left\{ {\sigma _{b}^{2}\left[ {\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right){{\Gamma }_{b}} + \sigma _{\ell }^{2}({{u}_{b}}{{u}_{\ell }})} \right]{{u}_{{bi}}}} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}\left[ {\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right){{\Gamma }_{\ell }} + \sigma _{b}^{2}({{u}_{b}}{{u}_{\ell }})} \right]{{u}_{{\ell i}}}} \right\}, \\ \mathfrak{A}_{d}^{{ii}} = \sigma _{b}^{2}\left( {\sigma _{3}^{2}\sigma _{b}^{2} - \sigma _{a}^{2}\sigma _{\ell }^{2}{\mathbf{u}}_{\ell }^{2}} \right)u_{{bi}}^{2} \\ + \,\,\sigma _{\ell }^{2}\left( {\sigma _{3}^{2}\sigma _{\ell }^{2} - \sigma _{a}^{2}\sigma _{b}^{2}{\mathbf{u}}_{b}^{2}} \right)u_{{\ell i}}^{2} + 2\sigma _{b}^{2}\sigma _{\ell }^{2} \\ \times \,\,\left[ {\sigma _{a}^{2}{{\Gamma }_{b}}{{\Gamma }_{\ell }} + \left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)({{u}_{b}}{{u}_{\ell }})} \right]{{u}_{{bi}}}{{u}_{{\ell i}}} + \sigma _{3}^{{ - 2}}\left| {{{\Re }_{d}}} \right|, \\ \mathfrak{A}_{d}^{{jk}} = \sigma _{b}^{2}\left( {\sigma _{3}^{2}\sigma _{b}^{2} - \sigma _{a}^{2}\sigma _{\ell }^{2}{\mathbf{u}}_{\ell }^{2}} \right){{u}_{{bj}}}{{u}_{{bk}}} \\ + \,\,\sigma _{\ell }^{2}\left( {\sigma _{3}^{2}\sigma _{\ell }^{2} - \sigma _{a}^{2}\sigma _{b}^{2}{\mathbf{u}}_{b}^{2}} \right){{u}_{{\ell j}}}{{u}_{{\ell k}}} + \sigma _{b}^{2}\sigma _{\ell }^{2} \\ \times \,\,\left[ {\sigma _{a}^{2}{{\Gamma }_{b}}{{\Gamma }_{\ell }} + \left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)({{u}_{b}}{{u}_{\ell }})} \right]\left( {{{u}_{{bj}}}{{u}_{{\ell k}}} + {{u}_{{bk}}}{{u}_{{\ell j}}}} \right), \\ j \ne k. \\ \end{gathered} $$
((A.16))

The following is then obtained for an arbitrary 4‑vector \(q\):

$$\begin{gathered} \left| {{{\Re }_{d}}} \right|\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }} = \mathfrak{A}_{d}^{{00}}q_{0}^{2} + \sum\limits_i {\left( { - 2\mathfrak{A}_{d}^{{0i}}{{q}_{0}} + \mathfrak{A}_{d}^{{ii}}{{q}_{i}}} \right)} {{q}_{i}} \\ + \,\,2\sum\limits_{j < k} {\mathfrak{A}_{d}^{{jk}}{{q}_{j}}{{q}_{k}}} \\ = \sigma _{3}^{2}\left[ {\sigma _{3}^{2}\left( {\sigma _{a}^{2} + \sigma _{b}^{2}\Gamma _{b}^{2} + \sigma _{\ell }^{2}\Gamma _{\ell }^{2}} \right) + \sigma _{b}^{2}\sigma _{\ell }^{2}{{{\left| {{{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }}} \right|}}^{2}}} \right]q_{0}^{2} \\ - \,\,2\sigma _{3}^{2}\sigma _{b}^{2}\left[ {\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right){{\Gamma }_{b}} + \sigma _{\ell }^{2}{{\Gamma }_{\ell }}({{u}_{b}}{{u}_{\ell }})} \right]\left( {{{{\mathbf{u}}}_{b}}{\mathbf{q}}} \right){{q}_{0}} \\ - \,\,2\sigma _{3}^{2}\sigma _{\ell }^{2}\left[ {\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right){{\Gamma }_{\ell }} + \sigma _{b}^{2}{{\Gamma }_{b}}({{u}_{b}}{{u}_{\ell }})} \right]\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right){{q}_{0}} \\ + \,\,\sigma _{3}^{2}\left[ {\sigma _{b}^{4}{{{\left( {{{{\mathbf{u}}}_{b}}{\mathbf{q}}} \right)}}^{2}} + \sigma _{\ell }^{4}{{{\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right)}}^{2}} + 2\sigma _{b}^{2}\sigma _{\ell }^{2}({{u}_{b}}{{u}_{\ell }})\left( {{{{\mathbf{u}}}_{b}}{\mathbf{q}}} \right)\left( {{{{\mathbf{u}}}_{\ell }}{\mathbf{q}}} \right)} \right] \\ + \,\,\sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}\left\{ {\mathop {\left[ {({{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }}){\mathbf{q}}} \right]}\nolimits^2 - {{{({{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }})}}^{2}}{{{\mathbf{q}}}^{2}}} \right\} + \sigma _{3}^{{ - 2}}\left| {{{\Re }_{d}}} \right|{{{\mathbf{q}}}^{2}}. \\ \end{gathered} $$

Using (A.16), we determine the components of 4‑momentum \({{Y}_{d}}\):

$$\begin{gathered} Y_{d}^{0} = \frac{{\sigma _{3}^{2}}}{{\left| {{{\Re }_{d}}} \right|}}\left( {c_{a}^{0}{{m}_{a}} - c_{b}^{0}{{E}_{b}} - c_{\ell }^{0}{{E}_{\ell }}} \right), \\ {{{\mathbf{Y}}}_{d}} = \frac{{\sigma _{3}^{2}}}{{\left| {{{\Re }_{d}}} \right|}}\left( {{{c}_{b}}{{{\mathbf{u}}}_{b}} + {{c}_{\ell }}{{{\mathbf{u}}}_{\ell }}} \right). \\ \end{gathered} $$

The coefficient functions found here are given by

$$\begin{gathered} c_{a}^{0} = \sigma _{a}^{2}\left( {\sigma _{3}^{2} + \sigma _{b}^{2}\Gamma _{b}^{2} + \sigma _{\ell }^{2}\Gamma _{\ell }^{2}} \right) + \sigma _{b}^{2}\sigma _{\ell }^{2} \\ \times \,\,\left[ {{{{\left( {{{\Gamma }_{b}}{{\Gamma }_{\ell }} - 1} \right)}}^{2}} - {{{\left( {{{{\mathbf{u}}}_{b}}{{{\mathbf{u}}}_{\ell }}} \right)}}^{2}}} \right] + {{\left( {\sigma _{b}^{2}{{\Gamma }_{b}} + \sigma _{\ell }^{2}{{\Gamma }_{\ell }}} \right)}^{2}}, \\ c_{b}^{0} = \sigma _{3}^{2}\left[ {\sigma _{a}^{2} + \sigma _{b}^{2} + \sigma _{\ell }^{2}\Gamma _{\ell }^{2}\left( {1 - {{{\mathbf{v}}}_{b}}{{{\mathbf{v}}}_{\ell }}} \right)} \right], \\ c_{\ell }^{0} = \sigma _{3}^{2}\left[ {\sigma _{a}^{2} + \sigma _{\ell }^{2} + \sigma _{b}^{2}\Gamma _{b}^{2}\left( {1 - {{{\mathbf{v}}}_{b}}{{{\mathbf{v}}}_{\ell }}} \right)} \right], \\ {{c}_{b}} = \sigma _{b}^{2}\left\{ {\left[ {{{m}_{a}}\sigma _{\ell }^{2}{{\Gamma }_{\ell }} - {{m}_{\ell }}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)} \right]} \right.({{u}_{b}}{{u}_{\ell }}) + {{m}_{a}}\sigma _{b}^{2}{{\Gamma }_{b}} \\ \left. { - \,\,{{m}_{b}}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)} \right\} \\ + \,\,\sigma _{a}^{2}\left\{ {{{m}_{b}}\sigma _{\ell }^{2}{\mathbf{u}}_{\ell }^{2} + \sigma _{b}^{2}\left[ {{{\Gamma }_{b}}\left( {{{m}_{a}} - {{m}_{\ell }}{{\Gamma }_{\ell }}} \right) - {{m}_{b}}} \right]} \right\}, \\ {{c}_{\ell }} = \sigma _{\ell }^{2}\left\{ {\left[ {{{m}_{a}}\sigma _{b}^{2}{{\Gamma }_{b}} - {{m}_{b}}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)} \right]} \right. \\ \times \,\,({{u}_{b}}{{u}_{\ell }}) + {{m}_{a}}\sigma _{\ell }^{2}{{\Gamma }_{\ell }}\left. { - {{m}_{\ell }}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)} \right\} \\ + \,\,\sigma _{a}^{2}\left\{ {{{m}_{\ell }}\sigma _{b}^{2}{\mathbf{u}}_{b}^{2} + \sigma _{\ell }^{2}\left[ {{{\Gamma }_{\ell }}\left( {{{m}_{a}} - {{m}_{b}}{{\Gamma }_{b}}} \right) - {{m}_{\ell }}} \right]} \right\}. \\ \end{gathered} $$

As it should be, the above expressions are symmetric with respect to index interchange \(b \leftrightarrow \ell \), and their explicitly noncovariant form is attributable to the use of a special reference frame.

A.4.2. PW0limit. The kinematics of reaction \(2 \to 2\) in the PW\(_{0}\) limit allows one to write quantities \(\left| {{{\Re }_{d}}} \right|\), \(\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\), and \({{{\mathbf{Y}}}_{d}}{\mathbf{l}}\) in terms of two arbitrary independent invariant variables; we used the standard pair of variables

$$\begin{gathered} s = {{({{p}_{a}} + {{p}_{\nu }})}^{2}} = {{m}_{a}}\left( {2{{E}_{\nu }} + {{m}_{a}}} \right) \\ {\text{and}}\,\,\,\,{{Q}^{2}} = - {{({{p}_{\nu }} - {{p}_{\ell }})}^{2}}. \\ \end{gathered} $$

In order to rewrite the expressions from the previous section in terms of these variables, we use the following exact kinematic relations:

$$\begin{gathered} {{E}_{b}} = \frac{1}{{{{m}_{a}}}}\left( {E_{b}^{ * }E_{a}^{ * } - E_{\nu }^{ * }P_{\ell }^{ * }\cos\theta_{*}} \right), \\ {{{\mathbf{p}}}_{b}}{{{\mathbf{p}}}_{\nu }} = \frac{{{{E}_{\nu }}}}{{{{m}_{a}}}}\left( {E_{b}^{ * }E_{\nu }^{ * } - E_{a}^{ * }P_{\ell }^{ * }\cos\theta_{*}} \right), \\ {{E}_{\ell }} = \frac{1}{{{{m}_{a}}}}\left( {E_{\ell }^{ * }E_{a}^{ * } + E_{\nu }^{ * }P_{\ell }^{ * }\cos\theta_{*}} \right), \\ {{{\mathbf{p}}}_{\ell }}{{{\mathbf{p}}}_{\nu }} = \frac{{{{E}_{\nu }}}}{{{{m}_{a}}}}\left( {E_{\ell }^{ * }E_{\nu }^{ * } + E_{a}^{ * }P_{\ell }^{ * }\cos\theta_{*}} \right), \\ \left| {{{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }}} \right| = \frac{{{{E}_{\nu }}P_{\ell }^{ * }\sin\theta_{*}}}{{{{m}_{b}}{{m}_{\ell }}}},\,\,\,\,\left( {{{{\mathbf{u}}}_{b}} \times {{{\mathbf{u}}}_{\ell }}} \right){{{\mathbf{p}}}_{\nu }} = 0, \\ {{u}_{b}}{{u}_{\ell }} = \frac{{s - m_{b}^{2} - m_{\ell }^{2}}}{{2{{m}_{b}}{{m}_{\ell }}}},\,\,\,\,{{{\text{V}}}_{{b\ell }}} = \frac{{2\sqrt s P_{\ell }^{ * }}}{{s - m_{b}^{2} - m_{\ell }^{2}}}, \\ \end{gathered} $$

where \({{E}_{\nu }} = \left| {{{{\mathbf{p}}}_{\nu }}} \right| = {{\left( {s - m_{a}^{2}} \right)} \mathord{\left/ {\vphantom {{\left( {s - m_{a}^{2}} \right)} {2{{m}_{a}}}}} \right. \kern-0em} {2{{m}_{a}}}}\) and \({{{\mathbf{p}}}_{\nu }} = {{E}_{\nu }}{\mathbf{l}}\) are the energy and the momentum of a massless neutrino in the laboratory frame;

$$\begin{gathered} E_{\nu }^{ * } = \frac{{s - m_{a}^{2}}}{{2\sqrt s }},\,\,\,\,E_{\ell }^{ * } = \frac{{s + m_{\ell }^{2} - m_{b}^{2}}}{{2\sqrt s }}, \\ E_{a}^{ * } = \frac{{s + m_{a}^{2}}}{{2\sqrt s }}\,\,\,\,{\text{and}}\,\,\,\,E_{b}^{ * } = \frac{{s - m_{\ell }^{2} + m_{b}^{2}}}{{2\sqrt s }}, \\ \end{gathered} $$

are the energies of particles \(\nu {\text{*}}\), \(\ell \), \(a\), and \(b\) in the center-of-mass system of colliding particles \(\nu {\text{*}}\) and \(a\), which is determined by

$$E_{\nu }^{ * } + E_{a}^{ * } = E_{b}^{ * } + E_{\ell }^{ * },\,\,\,\,{\mathbf{p}}_{\nu }^{ * } + {\mathbf{p}}_{a}^{ * } = {\mathbf{p}}_{\ell }^{ * } + {\mathbf{p}}_{b}^{ * } = 0;$$

and \(P_{\ell }^{ * } = \left| {{\mathbf{p}}_{\ell }^{ * }} \right|\). Lepton scattering angle \(\theta_{*}\) in the center-of-mass system is related to \({{Q}^{2}}\):

$${{Q}^{2}} = 2E_{\nu }^{ * }\left( {E_{\ell }^{ * } - P_{\ell }^{ * }\cos\theta_{*}} \right) - m_{\ell }^{2}.$$

The kinematically allowed region of the phase space is defined by the following inequalities:

$$s \geqslant {{s}_{{{\text{th}}}}} = max\left[ {m_{a}^{2},{{{\left( {{{m}_{b}} + {{m}_{\ell }}} \right)}}^{2}}} \right],$$
((A.17))
$$Q_{ - }^{2} \leqslant {{Q}^{2}} \leqslant Q_{ + }^{2},\,\,\,\,Q_{ \pm }^{2} = 2E_{\nu }^{ * }\left( {E_{\ell }^{ * } \pm P_{\ell }^{ * }} \right) - m_{\ell }^{2}.$$
((A.18))

Performing elementary (although rather cumbersome) algebraic transformations, we find

$$\begin{gathered} \left[\kern-0.15em\left[ {\left| {{{\Re }_{d}}} \right|} \right]\kern-0.15em\right] = \frac{{\sigma _{3}^{2}}}{{4m_{a}^{2}m_{b}^{2}m_{\ell }^{2}}}\sum\limits_{k,l = 0}^2 {{{A}_{{kl}}}{{s}^{k}}} {{Q}^{{2l}}}, \\ \left[\kern-0.15em\left[ {\left| {{{\Re }_{d}}} \right|\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] = \frac{{\sigma _{3}^{2}}}{{4m_{a}^{2}m_{b}^{2}m_{\ell }^{2}}}\sum\limits_{k,l = 0}^2 {{{B}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} , \\ \left[\kern-0.15em\left[ {\left| {{{\Re }_{d}}} \right|{{{\mathbf{Y}}}_{d}}{\mathbf{q}}} \right]\kern-0.15em\right] = \frac{{\sigma _{3}^{2}}}{{8m_{a}^{2}m_{b}^{2}m_{\ell }^{2}}}\sum\limits_{k,l = 0}^3 {{{C}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} . \\ \end{gathered} $$

Nonzero coefficients \({{A}_{{kl}}}\), \({{B}_{{kl}}}\) (\(0 \leqslant k,l \leqslant 2\)), and \({{C}_{{kl}}}\) (\(0 \leqslant k,l \leqslant 3\)) have the form

$$\begin{gathered} {{A}_{{00}}} = \sigma _{b}^{2}m_{a}^{2}m_{\ell }^{2}\left[ {\sigma _{a}^{2}\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right)\left( {m_{a}^{2} - 2m_{b}^{2}} \right)} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)\left( {m_{\ell }^{2} - 2m_{b}^{2}} \right) - 3\sigma _{a}^{2}\sigma _{\ell }^{2}m_{b}^{2}} \right] \\ + \,\,\sigma _{3}^{2}m_{b}^{2}\left[ {\sigma _{b}^{2}m_{b}^{2}\left( {\sigma _{a}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{a}^{2}} \right)} \right. \\ \left. { + \,\,\sigma _{a}^{2}\sigma _{\ell }^{2}\left( {m_{b}^{4} - 4m_{a}^{2}m_{\ell }^{2}} \right)} \right], \\ {{A}_{{01}}} = \sigma _{a}^{2}\left\{ {\sigma _{b}^{2}} \right.\left[ {2\sigma _{a}^{2}m_{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)\left( {m_{a}^{2} + 2m_{b}^{2}} \right)} \right] \\ \left. { + \,\,2\left[ {\sigma _{b}^{4}m_{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right) + \sigma _{\ell }^{2}m_{b}^{4}\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right)} \right]} \right\}, \\ {{A}_{{02}}} = \sigma _{3}^{2}\sigma _{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right), \\ \end{gathered} $$
$$\begin{gathered} {{A}_{{10}}} = - \sigma _{\ell }^{2}\left\{ {\sigma _{b}^{2}\left[ {\sigma _{a}^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)\left( {2m_{b}^{2} + m_{\ell }^{2}} \right)} \right.} \right. \\ \left. { + \,\,2\sigma _{\ell }^{2}m_{a}^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right] \\ \left. { + \,\,2\left[ {\sigma _{b}^{4}m_{a}^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right) + \sigma _{a}^{2}m_{b}^{4}\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right)} \right]} \right\}, \\ {{A}_{{11}}} = - \sigma _{a}^{2}\sigma _{\ell }^{2}\left[ {\sigma _{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right) + 2\sigma _{3}^{2}m_{b}^{2}} \right], \\ {{A}_{{12}}} = - \sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}, \\ {{A}_{{20}}} = \sigma _{3}^{2}\sigma _{\ell }^{2}\left( {\sigma _{a}^{2}m_{b}^{2} + \sigma _{b}^{2}m_{a}^{2}} \right), \\ {{A}_{{21}}} = \sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}; \\ \end{gathered} $$
$$\begin{gathered} {{B}_{{00}}} = m_{a}^{2}m_{\ell }^{2}\left\{ {\sigma _{b}^{2}\left( {m_{a}^{2} + m_{\ell }^{2}} \right)\left[ {\sigma _{a}^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)} \right.} \right. \\ + \,\,\sigma _{b}^{2}\left( {m_{a}^{2} + m_{\ell }^{2}} \right) + \left. {\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right] \\ \left. { + \,\,m_{b}^{2}\left( {\sigma _{a}^{4}m_{a}^{2} + \sigma _{\ell }^{4}m_{\ell }^{2} + \sigma _{a}^{2}\sigma _{\ell }^{2}m_{b}^{2}} \right)} \right\}, \\ {{B}_{{01}}} = m_{a}^{2}\left\{ {2\sigma _{\ell }^{4}m_{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right. \\ \times \,\,\left[ {\sigma _{b}^{2}\left( {m_{a}^{2} + 2m_{\ell }^{2}} \right) + \sigma _{a}^{2}m_{b}^{2}} \right] + \sigma _{b}^{2}m_{\ell }^{2} \\ \left. { \times \,\,\left[ {\sigma _{a}^{2}\left( {2m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right) + 2\sigma _{b}^{2}\left( {m_{\ell }^{2} + m_{a}^{2}} \right)} \right]} \right\}, \\ {{B}_{{02}}} = \sigma _{3}^{2}m_{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right), \\ \end{gathered} $$
$$\begin{gathered} {{B}_{{10}}} = - m_{\ell }^{2}\left\{ {\sigma _{\ell }^{2}\left[ {\sigma _{b}^{2}m_{a}^{2}\left( {m_{a}^{2} + m_{b}^{2} + 2m_{\ell }^{2}} \right)} \right.} \right. \\ \left. { + \,\,\sigma _{a}^{2}m_{b}^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)} \right] + \sigma _{a}^{2}\sigma _{b}^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)\left( {m_{\ell }^{2} + 2m_{a}^{2}} \right) \\ \left. { + \,\,2m_{a}^{2}\left[ {\sigma _{a}^{4}m_{b}^{2} + \sigma _{b}^{4}\left( {m_{\ell }^{2} + m_{a}^{2}} \right)} \right]} \right\}, \\ {{B}_{{11}}} = - \left[ {2\sigma _{3}^{2}\sigma _{b}^{2}m_{a}^{2}m_{\ell }^{2} + \left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)} \right. \\ \times \,\,\left. {\left( {\sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}} \right)} \right], \\ {{B}_{{12}}} = - \left( {\sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}} \right), \\ {{B}_{{20}}} = \sigma _{3}^{2}m_{\ell }^{2}\left( {\sigma _{a}^{2}m_{b}^{2} + \sigma _{b}^{2}m_{a}^{2}} \right), \\ {{B}_{{21}}} = \sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}; \\ \end{gathered} $$
$$\begin{gathered} {{C}_{{00}}} = m_{a}^{2}m_{\ell }^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{b}^{2}\left( {m_{a}^{4} + m_{a}^{2}m_{\ell }^{2} - m_{a}^{2}m_{b}^{2} + m_{\ell }^{2}m_{b}^{2}} \right)} \right.} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}m_{b}^{2}\left( {m_{b}^{2} - 2m_{a}^{2}} \right)} \right] + \left( {\sigma _{b}^{2}m_{a}^{2} + \sigma _{\ell }^{2}m_{b}^{2} + \sigma _{b}^{2}m_{\ell }^{2}} \right) \\ \left. { \times \,\,\left[ {\sigma _{b}^{2}m_{a}^{2} + \left( {2m_{\ell }^{2} - m_{b}^{2}} \right)\left( {\sigma _{\ell }^{2} + \sigma _{b}^{2}} \right)} \right]} \right\}, \\ {{C}_{{01}}} = m_{a}^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{b}^{2}} \right.} \right.\left( {2m_{a}^{2}m_{\ell }^{2} + m_{b}^{2}m_{\ell }^{2} + m_{\ell }^{4}} \right) \\ \left. { + \,\,\sigma _{\ell }^{2}m_{b}^{2}\left( {m_{\ell }^{2} + m_{b}^{2}} \right)} \right] + \sigma _{\ell }^{4}m_{b}^{2}\left( {3m_{\ell }^{2} - m_{b}^{2}} \right) \\ + \,\,\sigma _{b}^{4}m_{\ell }^{2}\left( {2m_{a}^{2} - m_{b}^{2} + 3m_{\ell }^{2}} \right) \\ + \,\,\sigma _{b}^{2}\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)\left. {\left( {m_{a}^{2} - m_{b}^{2} + 3m_{\ell }^{2}} \right)} \right\}, \\ {{C}_{{02}}} = m_{a}^{2}\sigma _{3}^{2}\left( {m_{b}^{2}\sigma _{\ell }^{2} + \sigma _{b}^{2}m_{\ell }^{2}} \right), \\ {{C}_{{10}}} = - m_{\ell }^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{b}^{2}\left( {2m_{a}^{4} - 2m_{a}^{2}m_{b}^{2} + m_{a}^{2}m_{\ell }^{2} + m_{b}^{2}m_{\ell }^{2}} \right)} \right.} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}m_{b}^{2}\left( {m_{b}^{2} - 3m_{a}^{2}} \right)} \right] + \sigma _{b}^{4}\left[ {2m_{a}^{2}\left( {m_{a}^{2} - m_{b}^{2}} \right)} \right. \\ \left. { + \,\,m_{\ell }^{2}\left( {3m_{a}^{2} - m_{b}^{2}} \right)} \right] - \sigma _{\ell }^{4}m_{b}^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right) + \sigma _{b}^{2}\sigma _{\ell }^{2} \\ \left. { \times \,\,\left[ {m_{a}^{2}\left( {2m_{a}^{2} + 4m_{\ell }^{2} - 3m_{b}^{2}} \right) - m_{b}^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right]} \right\}, \\ \end{gathered} $$
$$\begin{gathered} {{C}_{{11}}} = - \sigma _{a}^{2}\left[ {\sigma _{b}^{2}m_{\ell }^{2}\left( {3m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)} \right. \\ \left. { + \,\,\sigma _{\ell }^{2}m_{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)} \right] - \sigma _{b}^{4}m_{\ell }^{2}\left( {m_{a}^{2} - m_{b}^{2} - m_{\ell }^{2}} \right) \\ + \,\,\sigma _{\ell }^{4}m_{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right) + \sigma _{b}^{2}\sigma _{\ell }^{2} \\ \left[ {{{{\left( {m_{b}^{2} + m_{\ell }^{2}} \right)}}^{2}} - m_{a}^{2}\left( {m_{a}^{2} - m_{b}^{2} + 3m_{\ell }^{2}} \right)} \right], \\ {{C}_{{12}}} = - \sigma _{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right) \\ - \,\,\sigma _{b}^{2}\sigma _{\ell }^{2}\left( {m_{a}^{2} - m_{b}^{2} - m_{\ell }^{2}} \right) + \sigma _{b}^{4}m_{\ell }^{2} + \sigma _{\ell }^{4}m_{b}^{2}, \\ \end{gathered} $$
$$\begin{gathered} {{C}_{{20}}} = m_{\ell }^{2}\left\{ {\sigma _{a}^{2}} \right.\left[ {\sigma _{b}^{2}\left( {m_{a}^{2} - m_{b}^{2}} \right) - \sigma _{\ell }^{2}m_{b}^{2}} \right] \\ + \,\,\sigma _{b}^{4}\left( {m_{a}^{2} - m_{b}^{2}} \right) - \sigma _{\ell }^{2}m_{b}^{2}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right) \\ \left. { + \,\,2\sigma _{b}^{2}\left( {\sigma _{\ell }^{2}m_{a}^{2} - \sigma _{\ell }^{2}m_{b}^{2}} \right)} \right\}, \\ {{C}_{{21}}} = \sigma _{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right) - 2\sigma _{b}^{2}\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right) \\ - \,\,\sigma _{b}^{4}m_{\ell }^{2} - \sigma _{\ell }^{4}m_{b}^{2}, \\ {{C}_{{22}}} = - \sigma _{b}^{2}\sigma _{\ell }^{2}, \\ {{C}_{{31}}} = \sigma _{b}^{2}\sigma _{\ell }^{2}. \\ \end{gathered} $$

Thus, quadratic form \(\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\) and scalar product \({{{\mathbf{Y}}}_{d}}{\mathbf{q}}\) are rational functions of variables \(s\) and \({{Q}^{2}}\):

$$\left[\kern-0.15em\left[ {\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] = \frac{{\sum\limits_{k,l} {{{B}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}{{\sum\limits_{k,l} {{{A}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }} \equiv {{\mathfrak{F}}_{d}}(s,{{Q}^{2}}),$$
$$\left[\kern-0.15em\left[ {{{{\mathbf{Y}}}_{d}}{\mathbf{q}}} \right]\kern-0.15em\right] = \frac{1}{2}\frac{{\sum\limits_{k,l} {{{C}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}{{\sum\limits_{k,l} {{{A}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }} \equiv {{\mathfrak{n}}_{d}}(s,{{Q}^{2}}){{\mathfrak{F}}_{d}}(s,{{Q}^{2}}).$$

The following function was introduced hereFootnote 59:

$${{\mathfrak{n}}_{d}} = \frac{{\left[\kern-0.15em\left[ {{{{\mathbf{Y}}}_{d}}{\mathbf{l}}} \right]\kern-0.15em\right]}}{{\left[\kern-0.15em\left[ {{{Y}_{d}}l} \right]\kern-0.15em\right]}} = \frac{1}{2}\frac{{\sum\limits_{k,l} {{{C}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}{{\sum\limits_{k,l} {{{B}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}.$$

It is also expedient to introduce function \({{\mathfrak{D}}_{d}}\) defined in the following way:

$$\frac{{\mathfrak{D}_{d}^{2}}}{{E_{\nu }^{2}}} = \frac{1}{{2{{\mathfrak{F}}_{d}}}} = \frac{1}{2}\frac{{\sum\limits_{k,l} {{{A}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}{{\sum\limits_{k,l} {{{B}_{{kl}}}{{s}^{k}}{{Q}^{{2l}}}} }}.$$

Although neither \({{\mathfrak{D}}_{d}}\) nor \({{\mathfrak{n}}_{d}}\) have a clear physical meaning, they help illustrate the behavior of functions \(\mathfrak{D}\) and \(\mathfrak{n}\), which are of interest to us, in the special case when the contributions to \(\mathfrak{D}\) and \(\mathfrak{n}\) from the reaction in the source can be neglectedFootnote 60. This case is defined by the following conditions:

$$\left[\kern-0.15em\left[ {\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] \gg \left[\kern-0.15em\left[ {\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right]\,\,\,\,{\text{and}}\,\,\,\,\left[\kern-0.15em\left[ {\left| {{{{\mathbf{Y}}}_{d}}{\mathbf{l}}} \right|} \right]\kern-0.15em\right] \gg \left[\kern-0.15em\left[ {\left| {{{{\mathbf{Y}}}_{s}}{\mathbf{l}}} \right|} \right]\kern-0.15em\right].$$

In the simplest particular case with \({{{{\sigma }_{a}}} \mathord{\left/ {\vphantom {{{{\sigma }_{a}}} {{{m}_{a}}}}} \right. \kern-0em} {{{m}_{a}}}} = {{{{\sigma }_{b}}} \mathord{\left/ {\vphantom {{{{\sigma }_{b}}} {{{m}_{b}}}}} \right. \kern-0em} {{{m}_{b}}}}\) = \({{{{\sigma }_{\ell }}} \mathord{\left/ {\vphantom {{{{\sigma }_{\ell }}} {{{m}_{\ell }}}}} \right. \kern-0em} {{{m}_{\ell }}}} = \lambda = {\text{const}}\) (these relations characterize the scaling of effective sizes of packets), one can demonstrate that functions \({{\lambda }^{2}}{{\mathfrak{F}}_{d}}\) (and, consequently, \({{{{\mathfrak{D}}_{d}}} \mathord{\left/ {\vphantom {{{{\mathfrak{D}}_{d}}} \lambda }} \right. \kern-0em} \lambda }\)) and \({{\mathfrak{n}}_{d}}\) are independent of parameter \(\lambda \) and are defined by the kinematics only. The domains of functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\) are bounded by kinematic conditions (A.17) and (A.18), and the shape variations of surfaces shown in different panels are attributable primarily to the differences in reaction thresholds (A.17) (essentially, the differences in masses of final leptons). Therefore, these variations are smoothed out at sufficiently high energies (i.e., \(s \gg max({{s}_{{{\text{th}}}}})\)). The fact that function \({{\mathfrak{F}}_{d}}\) turns to zero at \({{E}_{\nu }} \to 0\) for thresholdless reaction \(\nu n \to p{{e}^{ - }}\) is irrelevant to our analysis limited to ultrarelativistic neutrinosFootnote 61.

In the general case, the behavior of functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\) is much more complex. Naturally, this assumption, which was adopted for purely illustrative purposes, is completely arbitrary and unrealistic. In a more realistic case, \({{{{\sigma }_{\varkappa }}} \mathord{\left/ {\vphantom {{{{\sigma }_{\varkappa }}} {{{m}_{\varkappa }}}}} \right. \kern-0em} {{{m}_{\varkappa }}}} \lll 1\), even function \(\log ({{\mathfrak{F}}_{d}})\) varies strongly within its domain, and the features of its behavior are hard to reproduce in a two-dimensional plot. Let us consider the most important limiting cases, asymptotics, and inequalities to gain a better understanding of the properties of functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\).

A.4.3. Low-energy limits of functions\({{\mathfrak{F}}_{d}}\)and \({{\mathfrak{n}}_{d}}\). The limits of functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\) at the kinematic threshold of quasi-elastic reaction \(\nu + b \to b + \ell \) in the detector are as followsFootnote 62:

$$\begin{gathered} {{\mathfrak{F}}_{d}}({{s}_{{{\text{th}}}}},Q_{{{\text{th}}}}^{2}) = \frac{{{{{\left( {{{m}_{b}} + {{m}_{\ell }}} \right)}}^{2}}}}{{\sigma _{b}^{2} + \sigma _{\ell }^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}}, \\ {{\mathfrak{n}}_{d}}({{s}_{{{\text{th}}}}},Q_{{{\text{th}}}}^{2}) = \frac{{\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)\left[ {m_{a}^{2} - {{{\left( {{{m}_{b}} + {{m}_{\ell }}} \right)}}^{2}}} \right]}}{{2\left[ {\sigma _{a}^{2}{{{\left( {{{m}_{b}} + {{m}_{\ell }}} \right)}}^{2}} + \sigma _{b}^{2}m_{a}^{2} + \sigma _{\ell }^{2}m_{a}^{2}} \right]}}. \\ \end{gathered} $$

It is assumed here that \({{m}_{a}} < {{m}_{b}} + {{m}_{\ell }}\). The threshold values of \(s\) and \({{Q}^{2}}\) are

$${{s}_{{{\text{th}}}}} = {{({{m}_{b}} + {{m}_{\ell }})}^{2}}\,\,\,\,{\text{and}}\,\,\,\,Q_{{{\text{th}}}}^{2} = {{m}_{\ell }}\left( {{{m}_{b}} - \frac{{m_{a}^{2}}}{{{{m}_{b}} + {{m}_{\ell }}}}} \right).$$

The following is then obtained for the thresholdless reaction (\({{m}_{a}} > {{m}_{b}} + {{m}_{\ell }}\), \({{s}_{{{\text{th}}}}} = m_{a}^{2},\)\(Q_{{{\text{th}}}}^{2} = - m_{\ell }^{2}\)):

$$\begin{gathered} {{\mathfrak{F}}_{d}}({{s}_{{{\text{th}}}}},Q_{{{\text{th}}}}^{2}) = 0,\,\,\,\,{{\mathfrak{n}}_{d}}({{s}_{{{\text{th}}}}},Q_{{{\text{th}}}}^{2}) \\ = 1 - \frac{{\sigma _{3}^{2}\left[ {2\sigma _{a}^{2}m_{b}^{2} + \sigma _{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} - m_{\ell }^{2}} \right)} \right]}}{{2\left[ {\sigma _{a}^{2}\sigma _{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} - m_{\ell }^{2}} \right) + \sigma _{a}^{4}m_{b}^{2} + \sigma _{b}^{4}m_{a}^{2}} \right]}}. \\ \end{gathered} $$

Thus, function \({{\mathfrak{F}}_{d}}\) may assume a value of exactly zero only in a thresholdless reaction (e.g., \(\nu n \to pe\)) at \({{E}_{\nu }} = 0\). Naturally, this formal limit goes well beyond the bounds of ultrarelativistic approximation \(E_{\nu }^{2} \gg max(m_{j}^{2})\), which was used in derivation of the formulas for functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\), and is of no practical importance, because ultrarelativistric neutrino and antineutrino beamsFootnote 63 only are used in all current neutrino experiments. It is of interest to note that the limit of \({{\mathfrak{F}}_{d}}\) at \({{E}_{\nu }} = max({{m}_{j}}) \equiv {{m}_{\nu }}\) and \({{Q}^{2}} = - m_{\ell }^{2}\) (for a thresholdless reaction), which is given by

$$\frac{{4m_{a}^{2}m_{\ell }^{2}m_{\nu }^{2}}}{{\sigma _{3}^{2}\sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}}}\left[ {\frac{{\sigma _{a}^{2}\sigma _{b}^{2}\left( {m_{a}^{2} + m_{b}^{2} - m_{\ell }^{2}} \right) + \sigma _{b}^{4}m_{a}^{2} + \sigma _{a}^{4}m_{b}^{2}}}{{{{{\left( {m_{a}^{2} - m_{b}^{2}} \right)}}^{2}} - 2m_{\ell }^{2}{{{\left( {m_{a}^{2} + m_{b}^{2}} \right)}}^{2}} + m_{\ell }^{4}}}} \right]\,{{\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)}^{{ - 1}}},$$

may still be large in magnitude if at least two out of three parameters \({{\sigma }_{a}}\), \({{\sigma }_{b}}\), and \({{\sigma }_{\ell }}\) are small compared to \({{m}_{\nu }}\).

A.4.4. High-energy asymptotics of functions\({{\mathfrak{F}}_{d}}\)and \({{\mathfrak{n}}_{d}}\). Under the assumption that \({{\sigma }_{{a,b,\ell }}} \ne 0\) and \({{Q}^{2}} < \infty \), the asymptotic behavior of functions \({{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\) and \({{{{\mathfrak{n}}_{d}}(s,{{Q}^{2}})} \mathord{\left/ {\vphantom {{{{\mathfrak{n}}_{d}}(s,{{Q}^{2}})} s}} \right. \kern-0em} s}\) at high energies is independent of \(s\):

$$\begin{gathered} {{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\mathop \sim \limits_{s \to \infty } \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} - \,\,{{\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}} \right)}^{2}}{{\left( {\frac{{{{Q}^{2}}}}{{\sigma _{3}^{2}}} + \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}} \right)}^{{ - 1}}}, \\ {{n}_{d}}(s,{{Q}^{2}})\mathop \sim \limits_{s \to \infty } \frac{s}{{2\sigma _{a}^{2}}}\,{{\left[ {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} + \frac{{\sigma _{3}^{2}m_{\ell }^{2}}}{{\sigma _{\ell }^{2}{{Q}^{2}}}}\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}} \right)} \right]}^{{ - 1}}}. \\ \end{gathered} $$

These asymptotics satisfy the following inequalities:

$$\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} < {{\mathfrak{F}}_{d}}(s,{{Q}^{2}}) < \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}},\,\,\,\,\frac{{\sigma _{b}^{2}\sigma _{l}^{2}\left( {m_{b}^{2} - m_{a}^{2}} \right)}}{{2\sigma _{3}^{2}\left( {\sigma _{a}^{2}m_{b}^{2} + \sigma _{b}^{2}m_{a}^{2}} \right)}} < {{\mathfrak{n}}_{d}}(s,{{Q}^{2}}) < \frac{s}{{2m_{a}^{2}}}\,{{\left[ {1 + \frac{{\sigma _{a}^{2}}}{{m_{a}^{2}}}\left( {\frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)} \right]}^{{ - 1}}},$$

(\(s \to \infty \), \({{Q}^{2}} < \infty \)), and their limiting values at kinematic boundaries are

$$\begin{gathered} \mathop {\lim}\limits_{s \to \infty } {{\mathfrak{F}}_{d}}(s,Q_{ - }^{2}) = \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}},\,\,\,\,\mathop {\lim}\limits_{s \to \infty } {{\mathfrak{n}}_{d}}(s,Q_{ - }^{2}) = \frac{{\sigma _{b}^{2}\left( {m_{a}^{2} - m_{b}^{2}} \right) - \sigma _{\ell }^{2}m_{b}^{2}}}{{2\left( {\sigma _{a}^{2}m_{b}^{2} + \sigma _{b}^{2}m_{a}^{2}} \right)}}, \\ \mathop {\lim}\limits_{s \to \infty } {{\mathfrak{F}}_{d}}(s,Q_{ + }^{2}) = \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}},\,\,\,\,\,\mathop {\lim}\limits_{s \to \infty } {{\mathfrak{n}}_{d}}(s,Q_{ + }^{2}) = \frac{{\sigma _{\ell }^{2}\left( {m_{a}^{2} - m_{\ell }^{2}} \right) - \sigma _{b}^{2}m_{\ell }^{2}}}{{2\left( {\sigma _{a}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{a}^{2}} \right)}}. \\ \end{gathered} $$

It is evident that these quantities are symmetric with respect to index interchange \(b \leftrightarrow \ell \). The \({{\mathfrak{n}}_{d}}(s,Q_{ \pm }^{2})\) threshold values vanish at specific relations between parameters \({{\sigma }_{\varkappa }}\) and masses. The next (\({{ \sim 1} \mathord{\left/ {\vphantom {{ \sim 1} s}} \right. \kern-0em} s}\)) corrections need to be taken into account in these exotic cases.

In the particular case when target particle \(a\) is a nucleon, it follows from dynamic considerations that, at high neutrino energies, the mean scattering angle in the center-of-mass system, \(\left\langle {\theta_{*}} \right\rangle \), is equal in order of magnitude to the inverse Lorentz factor of a lepton \(\Gamma _{\ell }^{ * } = {{E_{\ell }^{ * }} \mathord{\left/ {\vphantom {{E_{\ell }^{ * }} {{{m}_{\ell }}}}} \right. \kern-0em} {{{m}_{\ell }}}}\). Therefore, \(\left\langle {{{Q}^{2}}} \right\rangle \sim m_{\ell }^{2}\) and \(\left\langle \theta \right\rangle \sim {{{{m}_{\ell }}} \mathord{\left/ {\vphantom {{{{m}_{\ell }}} {\sqrt s }}} \right. \kern-0em} {\sqrt s }}\), where \(\theta \) is the scattering angle of a lepton in the laboratory frame (coinciding with the intrinsic frame of a nucleon). It can be demonstrated that the corresponding asymptotics of functions \({{\mathfrak{F}}_{d}}\) and \({{\mathfrak{n}}_{d}}\) take the form

$$\begin{gathered} {{\mathfrak{F}}_{d}}\left( {s,m_{\ell }^{2}} \right)\mathop \sim \limits_{s \to \infty } \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}\,\left[ {1 + \frac{{\sigma _{\ell }^{2}}}{{\sigma _{3}^{2}}}\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}} \right){{{\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{3}^{2}}}} \right)}}^{{ - 1}}}} \right] < 2\frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}, \\ {{\mathfrak{n}}_{d}}\left( {s,m_{\ell }^{2}} \right)\mathop \sim \limits_{s \to \infty } \frac{{\sigma _{b}^{2}\sigma _{\ell }^{2}s}}{{2\left[ {\left( {\sigma _{b}^{2}m_{a}^{2} + \sigma _{a}^{2}m_{b}^{2}} \right)\left( {\sigma _{a}^{2} + \sigma _{b}^{2} + 2\sigma _{\ell }^{2}} \right) + \sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2}} \right]}} < \frac{{{{E}_{\nu }}}}{{2{{m}_{a}}}}. \\ \end{gathered} $$

Since only a narrow region of angles close to \(\theta = \left\langle \theta \right\rangle \) produces a significant contribution to the count rate for quasi-elastic events at high energies, one may conclude that effective asymptotic value \({{\mathfrak{F}}_{d}}\) is almost constant and is defined primarily by momentum dispersion \({{\sigma }_{\ell }}\) of the lepton WP. Arbitrary64 variations of parameters \({{\sigma }_{a}}\) and \({{\sigma }_{b}}\) may change the asymptotics only within a factor of two.

The asymptotic behavior of \({{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\) changes drastically if one (and only one) of the \({{\sigma }_{\varkappa }}\) parameters turns to zero. If \({{\sigma }_{a}} = 0\) or \({{\sigma }_{b}} = 0\), the asymptotics is independent of \(s\):

$${{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\xrightarrow[{s \to \infty }]{}\left\{ \begin{gathered} \frac{{{{Q}^{2}}}}{{\sigma _{b}^{2} + \sigma _{\ell }^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}},\,\,\,\,{\text{at}}\,\,\,\,{{\sigma }_{a}} = 0, \hfill \\ \frac{{{{Q}^{2}}}}{{\sigma _{a}^{2} + \sigma _{\ell }^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}},\,\,\,\,{\text{at}}\,\,\,\,{{\sigma }_{b}} = 0, \hfill \\ \end{gathered} \right.$$

and if \({{\sigma }_{\ell }} = 0\), increases quadratically with \(s\):

$${{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\mathop \sim \limits_{s \to \infty } \frac{{\left( {\tfrac{{{{Q}^{2}}}}{{\sigma _{a}^{2} + \sigma _{b}^{2}}} + \tfrac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \tfrac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}} \right){{s}^{2}}}}{{{{{\left( {{{Q}^{2}} + m_{a}^{2} + m_{b}^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{b}^{2}}},\,\,\,\,{\text{at}}\,\,\,\,{{\sigma }_{\ell }} = 0.$$

Other variables. Certain properties of function \({{\mathfrak{F}}_{d}}\) become more evident if it is rewritten in terms of variables \({{E}_{\nu }}\) and \(\theta_{*}\). Let us consider the asymptotic expansion of \({{\mathfrak{F}}_{d}}\) at \({{E}_{\nu }} \to \infty \) and a fixed value of \(\theta_{*}\). If the values of \(\sin\theta_{*}\) are not too small, it can be written as

$$\begin{gathered} {{\mathfrak{F}}_{d}}({{E}_{\nu }},\theta *) = \frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}} - \frac{{{{a}_{1}}\sigma _{3}^{2}}}{{2{{m}_{a}}{{E}_{\nu }}si{{n}^{2}}\theta_{*}}}\,\left\{ {\left[ {\left( {\frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)\cos\theta_{*}} \right.} \right. + \left( {\frac{{m_{b}^{2}\sigma _{\ell }^{2} - m_{\ell }^{2}\sigma _{b}^{2}}}{{m_{\ell }^{2}\sigma _{b}^{2} + m_{b}^{2}\sigma _{\ell }^{2}}}} \right) \\ \times \,\,{{\left. {\left( {\frac{{2m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)} \right]}^{{ - 2}}} + {{\left( {\frac{{4{{m}_{a}}{{m}_{b}}{{m}_{\ell }}}}{{{{\sigma }_{a}}{{\sigma }_{b}}{{\sigma }_{\ell }}}}} \right)}^{2}}\,{{\left( {\frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)}^{2}}\left. {\left( {\frac{{m_{a}^{2}}}{{\sigma _{a}^{2}}} + \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}} + \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}} \right)} \right\} + \,\,\frac{1}{{si{{n}^{4}}\theta_{*}}}\mathcal{O}\left( {\frac{{m_{a}^{2}}}{{E_{\nu }^{2}}}} \right). \\ \end{gathered} $$

At \(\sin\theta * = 0\), one Footnote 64finds

$${{\mathfrak{F}}_{d}}({{E}_{\nu }},0) = \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}\left[ {1 - \frac{{m_{a}^{2} - m_{b}^{2}}}{{{{m}_{a}}{{E}_{\nu }}}} + \mathcal{O}\left( {\frac{{m_{a}^{2}}}{{E_{\nu }^{2}}}} \right)} \right],\,\,\,\,\,{{\mathfrak{F}}_{d}}({{E}_{\nu }},\pi ) = \frac{{m_{b}^{2}}}{{\sigma _{b}^{2}}}\left[ {1 - \frac{{m_{a}^{2} - m_{\ell }^{2}}}{{{{m}_{a}}{{E}_{\nu }}}} + \mathcal{O}\left( {\frac{{m_{a}^{2}}}{{E_{\nu }^{2}}}} \right)} \right].$$

As was already noted, \(\left\langle {\theta_{*}} \right\rangle \sim \Gamma _{\ell }^{ * } = {{E_{\ell }^{ * }} \mathord{\left/ {\vphantom {{E_{\ell }^{ * }} {{{m}_{\ell }}}}} \right. \kern-0em} {{{m}_{\ell }}}}\) at high energies. It can be demonstrated that the corresponding asymptotic expansion takes the form

$$\begin{gathered} {{\mathfrak{F}}_{d}}({{E}_{\nu }},\theta * = {1 \mathord{\left/ {\vphantom {1 {\Gamma _{\ell }^{ * }}}} \right. \kern-0em} {\Gamma _{\ell }^{ * }}}) \\ = \frac{{m_{\ell }^{2}}}{{\sigma _{\ell }^{2}}}\left[ {{{b}_{0}} - \frac{{{{b}_{1}}\sigma _{3}^{2}}}{{{{m}_{a}}{{{({{b}_{0}} - 1)}}^{2}}{{E}_{\nu }}}} + \mathcal{O}\left( {\frac{{m_{a}^{2}}}{{E_{\nu }^{2}}}} \right)} \right], \\ \end{gathered} $$

where

$$\begin{gathered} {{b}_{0}} = 1 + \frac{{\sigma _{\ell }^{2}\left( {m_{b}^{2}\sigma _{a}^{2} + m_{a}^{2}\sigma _{b}^{2}} \right)}}{{\sigma _{3}^{2}\left( {m_{b}^{2}\sigma _{a}^{2} + m_{a}^{2}\sigma _{b}^{2}} \right) + m_{\ell }^{2}\sigma _{a}^{2}\sigma _{b}^{2}}}, \\ {{b}_{1}} = 8\sigma _{\ell }^{2}{{\left( {m_{b}^{2}\sigma _{a}^{2} + m_{a}^{2}\sigma _{b}^{2}} \right)}^{3}}\left[ {m_{b}^{2}\left( {m_{a}^{2} - m_{b}^{2} + \frac{{m_{\ell }^{2}}}{3}} \right)} \right.\,\sigma _{a}^{2} + m_{a}^{2}\left. {\left( {m_{a}^{2} - m_{b}^{2} + \frac{{4m_{\ell }^{2}}}{3}} \right)\sigma _{b}^{2}} \right] \\ + \,\,2\sigma _{\ell }^{2}\left( {m_{b}^{2}\sigma _{a}^{2} + m_{a}^{2}\sigma _{b}^{2}} \right)\,\left[ {m_{b}^{2}\sigma _{a}^{4} + \left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)\sigma _{a}^{2}\sigma _{b}^{2} + m_{a}^{2}\sigma _{b}^{4}} \right] \\ \times \,\,\left[ {m_{b}^{2}\left( {m_{a}^{2} - m_{b}^{2} - m_{\ell }^{2}} \right)\sigma _{a}^{2} + m_{a}^{2}\left( {m_{a}^{2} - m_{b}^{2} + m_{\ell }^{2}} \right)\sigma _{b}^{2}} \right]. \\ \end{gathered} $$

It is evident that the effective asymptotic \({{\mathfrak{F}}_{d}}({{E}_{\nu }},\theta_ {*})\) value is almost constant and, since \(1 < {{b}_{0}} < 2\), is defined primarily by the momentum dispersion of the lepton packet.

The case of strong hierarchy. Function \({{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\) is simplified considerably in the case of a strong hierarchy of parameters \({{\sigma }_{a}}\), \({{\sigma }_{b}}\), and \({{\sigma }_{\ell }}.\) Calculating the corresponding sequential limits, we obtain the following:

$${{\mathfrak{F}}_{d}}(s,{{Q}^{2}}) \approx \left\{ \begin{gathered} \frac{{{{{\left( {{{Q}^{2}} + m_{\ell }^{2}} \right)}}^{2}}}}{{{{{\left( {s - {{Q}^{2}} - m_{b}^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{a}}}}{{{{\sigma }_{a}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{\ell }} \gg {{\sigma }_{a}} \gg {{\sigma }_{b}}, \hfill \\ \frac{{{{{\left( {{{Q}^{2}} + m_{\ell }^{2}} \right)}}^{2}}}}{{{{{\left( {s - m_{b}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{b}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{a}}}}{{{{\sigma }_{a}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{\ell }} \gg {{\sigma }_{b}} \gg {{\sigma }_{a}}, \hfill \\ \frac{{{{{\left( {s - m_{a}^{2}} \right)}}^{2}}}}{{{{{\left( {{{Q}^{2}} + m_{a}^{2} + m_{b}^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{b}^{2}}}{{\left( {\frac{{{{m}_{b}}}}{{{{\sigma }_{b}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{a}} \gg {{\sigma }_{b}} \gg {{\sigma }_{\ell }}, \hfill \\ \frac{{{{{\left( {s - m_{a}^{2}} \right)}}^{2}}}}{{{{{\left( {s - {{Q}^{2}} - m_{b}^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{\ell }}}}{{{{\sigma }_{\ell }}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{a}} \gg {{\sigma }_{\ell }} \gg {{\sigma }_{b}}, \hfill \\ \frac{{{{{\left( {s - {{Q}^{2}} - m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}}}}{{{{{\left( {s - m_{b}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{b}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{\ell }}}}{{{{\sigma }_{\ell }}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{b}} \gg {{\sigma }_{\ell }} \gg {{\sigma }_{a}}, \hfill \\ \frac{{{{{\left( {s - {{Q}^{2}} - m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}}}}{{{{{\left( {{{Q}^{2}} + m_{a}^{2} + m_{b}^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{b}^{2}}}{{\left( {\frac{{{{m}_{a}}}}{{{{\sigma }_{a}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,\,{{\sigma }_{b}} \gg {{\sigma }_{a}} \gg {{\sigma }_{\ell }}. \hfill \\ \end{gathered} \right.$$

It can be seen thatFootnote 65 the shape and the value of function \({{\mathfrak{F}}_{d}}(s,{{Q}^{2}})\) are not affected by the smallest and the largest of parameters σ. This nontrivial property may be generalized to the case of processes with an arbitrary number of particles in the final state. In the strong hierarchy scenario, the only significant parameter is the second-largest dispersion. This is useful in the analysis of multiparticle processes (with more than two external packets), since one may then treat packets with (comparatively) very small \({{\sigma }_{\varkappa }}\) as plane waves. In particular, the calculation of radiative corrections with plane-wave photons in the external legs of Feynman diagrams is simplified significantly, since the standard QFT calculation methods become available. We should recall that loop electroweak corrections do not introduce additional computational complications associated with the WP formalism, since all of them are formally included in the corresponding matrix elements and are in no way related to the characteristics of external in- and out-states. Under the convention adopted in the main text, the external legs of diagrams do not feature gauge bosons.

A.5. Three-Particle Decay in the Source

The general formulas characterizing three-particle decay \(a \to b + \ell + \nu {\text{*}}\) agree formally with the ones for the \(2 \to 2\) scattering (if they are considered in the intrinsic reference frame of particle \(a\)). The primary difference is of kinematic origin. Therefore, we discuss this case in brief. Just as in the case of the \(2 \to 2\) scattering, functions \(\left| {{{\Re }_{s}}} \right|\) and \(\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\) may be written in terms of two independent invariant variables. One may use, e.g., any pair of invariants

$$\begin{gathered} {{s}_{1}} = {{\left( {{{p}_{b}} + {{p}_{\ell }}} \right)}^{2}} = {{\left( {{{p}_{a}} - {{p}_{\nu }}} \right)}^{2}}, \\ {{s}_{2}} = {{\left( {{{p}_{\nu }} + {{p}_{\ell }}} \right)}^{2}} = {{\left( {{{p}_{a}} - {{p}_{b}}} \right)}^{2}}, \\ {{s}_{3}} = {{\left( {{{p}_{\nu }} + {{p}_{b}}} \right)}^{2}} = {{\left( {{{p}_{a}} - {{p}_{\ell }}} \right)}^{2}}, \\ \end{gathered} $$

which are related by identity \({{s}_{1}} + {{s}_{2}} + {{s}_{3}}\) = \(m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}\), as these variables. The physical domain for them is defined by conditions

$$\begin{gathered} {{\left( {{{m}_{b}} + {{m}_{\ell }}} \right)}^{2}} \leqslant {{s}_{1}} \leqslant m_{a}^{2},\,\,\,\,m_{\ell }^{2} \leqslant {{s}_{2}} \leqslant {{\left( {{{m}_{a}} - {{m}_{\ell }}} \right)}^{2}}, \\ m_{b}^{2} \leqslant {{s}_{3}} \leqslant {{\left( {{{m}_{a}} - {{m}_{b}}} \right)}^{2}}. \\ \end{gathered} $$

For definiteness, we use the \(({{s}_{1}},{{s}_{2}})\) pair. The domain of definition for this pair is the Dalitz diagram

$$s_{1}^{ - } \leqslant {{s}_{1}} \leqslant s_{1}^{ + },\,\,\,\,m_{\ell }^{2} \leqslant {{s}_{2}} \leqslant {{({{m}_{a}} - {{m}_{\ell }})}^{2}},$$

where

$$s_{1}^{ \pm } = m_{b}^{2} + m_{\ell }^{2} - \frac{{\left( {{{s}_{2}} + m_{b}^{2}} \right)\left( {{{s}_{2}} - m_{a}^{2} + m_{\ell }^{2}} \right) \mp \left( {{{s}_{2}} - m_{b}^{2}} \right)\sqrt {{{{\left( {{{s}_{2}} - m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{\ell }^{2}} }}{{2{{s}_{2}}}}.$$

Using the results obtained in the previous section, we find

$$\begin{gathered} \left[\kern-0.15em\left[ {\left| {{{\Re }_{s}}} \right|} \right]\kern-0.15em\right] = \frac{{\sigma _{3}^{2}}}{{4m_{a}^{2}m_{b}^{2}m_{\ell }^{2}}}\sum\limits_{k,l = 0}^2 {A_{{kl}}^{'}s_{1}^{k}s_{2}^{l}} , \\ \left[\kern-0.15em\left[ {\left| {{{\Re }_{s}}} \right|\tilde {\Re }_{d}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] = \frac{{\sigma _{3}^{2}}}{{4m_{a}^{2}m_{b}^{2}m_{\ell }^{2}}}\sum\limits_{k,l = 0}^2 {B_{{kl}}^{'}s_{1}^{k}s_{2}^{l}} . \\ \end{gathered} $$

Therefore, quadratic form \(\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}\) is a rational function of variables \({{s}_{1}}\) and \({{s}_{2}}\),

$$\left[\kern-0.15em\left[ {\tilde {\Re }_{s}^{{\mu \nu }}{{q}_{\mu }}{{q}_{\nu }}} \right]\kern-0.15em\right] = \frac{{\sum\limits_{k,l} {B_{{kl}}^{'}s_{1}^{k}s_{2}^{l}} }}{{\sum\limits_{k,l} {A_{{kl}}^{'}s_{1}^{k}s_{2}^{l}} }} \equiv {{\mathfrak{F}}_{s}}({{s}_{1}},{{s}_{2}}).$$

Nonzero coefficients \(A_{{kl}}^{'}\) and \(B_{{kl}}^{'}\) (\(0 \leqslant k,l \leqslant 2\)) take the form

$$\begin{gathered} A_{{00}}^{'} = \sigma _{\ell }^{2}\left[ {\sigma _{a}^{2}m_{b}^{2}\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right){{{\left( {m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}}_{{_{{_{{}}}}}}} \right. + \,\,\left. {\sigma _{b}^{2}m_{a}^{2}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right){{{\left( {m_{b}^{2} - m_{\ell }^{2}} \right)}}^{2}}} \right] + \sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} \\ \times \,\,\left\{ {\sigma _{\ell }^{2}\left[ {m_{\ell }^{2}} \right.} \right.\left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right) - \left. {7m_{a}^{2}m_{b}^{2}} \right]\left. { + \,\,\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right)\left( {m_{\ell }^{4} - 4m_{a}^{2}m_{b}^{2}} \right)} \right\}, \\ A_{{01}}^{'} = - \sigma _{a}^{2}\left\{ {\sigma _{b}^{2}\left[ {\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right.} \right.\left( {m_{a}^{2} + 2m_{\ell }^{2}} \right) + \,\,2m_{\ell }^{4}\left. {\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right)} \right] + 2\sigma _{\ell }^{2}m_{b}^{2}\left( {m_{\ell }^{2} + m_{a}^{2}} \right)\left. {\left( {\sigma _{a}^{2} + \sigma _{\ell }^{2}} \right)} \right\}, \\ A_{{02}}^{'} = \sigma _{3}^{2}\sigma _{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right), \\ \end{gathered} $$
$$\begin{gathered} A_{{10}}^{'} = - \sigma _{b}^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{\ell }^{2}\left( {m_{\ell }^{2} + m_{a}^{2}} \right)\left( {2m_{\ell }^{2} + m_{b}^{2}} \right)} \right.} \right. + 2m_{\ell }^{4}\left. {\left( {\sigma _{a}^{2} + \sigma _{b}^{2}} \right)} \right] + 2\sigma _{\ell }^{2}m_{a}^{2}\left( {\sigma _{b}^{2} + \sigma _{\ell }^{2}} \right)\left. {\left( {m_{\ell }^{2} + m_{b}^{2}} \right)} \right\}, \\ A_{{11}}^{'} = \sigma _{a}^{2}\sigma _{b}^{2}\left[ {\sigma _{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right) + 2\sigma _{3}^{2}m_{\ell }^{2}} \right],\,\,\,\,A_{{12}}^{'} = - \sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}, \\ A_{{20}}^{'} = \sigma _{3}^{2}\sigma _{b}^{2}\left( {\sigma _{a}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{a}^{2}} \right),\,\,\,\,\,A_{{21}}^{'} = - \sigma _{a}^{2}\sigma _{b}^{2}\sigma _{\ell }^{2}; \\ \end{gathered} $$
$$\begin{gathered} B_{{00}}^{'} = m_{a}^{2}m_{b}^{2}\left\{ {\sigma _{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)\left[ {\sigma _{a}^{2}\left( {m_{a}^{2} + m_{\ell }^{2}} \right)} \right.} \right.\left. { + \,\,\sigma _{b}^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right) + \sigma _{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right)} \right] \\ \left. { + \,\,\,m_{\ell }^{2}\left( {\sigma _{a}^{4}m_{a}^{2} + \sigma _{b}^{4}m_{b}^{2} + \sigma _{a}^{2}\sigma _{b}^{2}m_{b}^{2}} \right)} \right\}, \\ B_{{01}}^{'} = - m_{a}^{2}\left\{ {\sigma _{a}^{2}\left[ {\sigma _{b}^{2}m_{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)} \right.} \right. + \,\,\sigma _{\ell }^{2}m_{b}^{2}\,\left. {\left( {2m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)} \right] + 2m_{b}^{2}\left[ {\sigma _{b}^{4}m_{\ell }^{2} + \sigma _{\ell }^{4}\left( {m_{a}^{2} + m_{b}^{2}} \right)} \right] \\ \left. { + \,\,\sigma _{b}^{2}\sigma _{\ell }^{2}\left( {m_{b}^{2} + m_{\ell }^{2}} \right)\left( {m_{a}^{2} + 2m_{b}^{2}} \right)} \right\},\,\,\,\,\,\,B_{{02}}^{'} = \sigma _{3}^{2}m_{a}^{2}\left( {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{b}^{2}} \right), \\ B_{{10}}^{'} = - m_{b}^{2}\left\{ {2\sigma _{a}^{4}m_{a}^{2}m_{\ell }^{2} + \sigma _{a}^{2}\left( {m_{a}^{2} + m_{\ell }^{2}} \right)} \right.\,\left[ {\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}\left( {2m_{a}^{2} + m_{b}^{2}} \right)} \right] + \sigma _{\ell }^{2}m_{a}^{2} \\ \times \,\,\left. {\left[ {2\sigma _{\ell }^{2}\left( {m_{a}^{2} + m_{b}^{2}} \right) + \sigma _{b}^{2}\left( {m_{a}^{2} + 2m_{b}^{2} + m_{\ell }^{2}} \right)} \right]} \right\}, \\ B_{{11}}^{'} = 2{{\sigma }_{3}}\sigma _{\ell }^{2}m_{a}^{2}m_{b}^{2} + \left( {m_{a}^{2} + m_{b}^{2} + m_{\ell }^{2}} \right)\left( {\sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}} \right), \\ \end{gathered} $$
$$\begin{gathered} B_{{12}}^{'} = - \left( {\sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}} \right), \\ B_{{20}}^{'} = \sigma _{3}^{2}m_{b}^{2}\left( {\sigma _{a}^{2}m_{\ell }^{2} + \sigma _{\ell }^{2}m_{a}^{2}} \right), \\ B_{{21}}^{'} = \sigma _{a}^{2}\sigma _{b}^{2}m_{\ell }^{2} + \sigma _{b}^{2}\sigma _{\ell }^{2}m_{a}^{2} + \sigma _{\ell }^{2}\sigma _{a}^{2}m_{b}^{2}. \\ \end{gathered} $$

We will not study the limiting cases and asymptotics in detail, because they may \({\text{be}}\) obtained from the formulas for quasi-elastic scattering in the detector. As an example, we consider the case of a strong hierarchy of parameters \({{\sigma }_{a}}\), \({{\sigma }_{b}}\), and \({{\sigma }_{\ell }}\), where, as in the case of the \(2 \to 2\) scattering, function \({{\mathfrak{F}}_{s}}({{s}_{1}},{{s}_{2}})\) is especially simple:

$${{\mathfrak{F}}_{s}}({{s}_{1}},{{s}_{2}}) \approx \left\{ \begin{gathered} \frac{{{{{\left( {{{s}_{1}} + {{s}_{2}} - m_{a}^{2} - m_{b}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{2}} - m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{a}}}}{{{{\sigma }_{a}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{\ell }} \gg {{\sigma }_{a}} \gg {{\sigma }_{b}}, \hfill \\ \frac{{{{{\left( {{{s}_{1}} + {{s}_{2}} - m_{a}^{2} - m_{b}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{1}} - m_{b}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{b}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{b}}}}{{{{\sigma }_{b}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{\ell }} \gg {{\sigma }_{b}} \gg {{\sigma }_{a}}, \hfill \\ \frac{{{{{\left( {{{s}_{1}} - m_{a}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{1}} + {{s}_{2}} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{b}^{2}}}{{\left( {\frac{{{{m}_{b}}}}{{{{\sigma }_{b}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{a}} \gg {{\sigma }_{b}} \gg {{\sigma }_{\ell }}, \hfill \\ \frac{{{{{\left( {{{s}_{1}} - m_{a}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{2}} - m_{a}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{\ell }}}}{{{{\sigma }_{\ell }}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{a}} \gg {{\sigma }_{\ell }} \gg {{\sigma }_{b}}, \hfill \\ \frac{{{{{\left( {{{s}_{2}} - m_{b}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{1}} - m_{b}^{2} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{b}^{2}m_{\ell }^{2}}}{{\left( {\frac{{{{m}_{\ell }}}}{{{{\sigma }_{\ell }}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{b}} \gg {{\sigma }_{\ell }} \gg {{\sigma }_{a}}, \hfill \\ \frac{{{{{\left( {{{s}_{2}} - m_{b}^{2}} \right)}}^{2}}}}{{{{{\left( {{{s}_{1}} + {{s}_{2}} - m_{\ell }^{2}} \right)}}^{2}} - 4m_{a}^{2}m_{b}^{2}}}{{\left( {\frac{{{{m}_{a}}}}{{{{\sigma }_{a}}}}} \right)}^{2}},\,\,\,\,{\text{for}}\,\,\,{{\sigma }_{b}} \gg {{\sigma }_{a}} \gg {{\sigma }_{\ell }}. \hfill \\ \end{gathered} \right.$$

It can be seen that, under strong hierarchy, the shape and the value of function \({{\mathfrak{F}}_{s}}({{s}_{1}},{{s}_{2}})\) are not affected by the smallest and the largest of parameters.

APPENDIX B

MULTIDIMENSIONAL GAUSSIAN QUADRATURES

Integrals

$$\mathcal{G}(A,B) = \int {dx} \exp\left( { - {{A}_{{\mu \nu }}}{{x}^{\mu }}{{x}^{\nu }} + {{B}_{\mu }}{{x}^{\mu }}} \right),$$
((B.1))

where \(A = \left\| {{{A}_{{\mu \nu }}}} \right\|\) is a symmetric positively defined matrix and \({{B}_{\mu }}\) are arbitrary complex constantsFootnote 66, are used often in the main text. Although such integrals are well known (see, e.g., [138]), we detail here a simple calculation of (B.1), since the published papers suffer from a certain confusion (or, to put it better, lack of agreement) regarding the definition of the matrix inverse to \(A\) in the Minkowski space. Symmetric matrix \(A\) can always be diagonalized with orthogonal transformation \(O = \left\| {{{O}_{{\mu \nu }}}} \right\|\) (see, e.g., [139]):

$${{A}_{{\mu \nu }}} = \sum\limits_\alpha {{{a}_{\alpha }}} {{O}_{{\mu \alpha }}}{{O}_{{\nu \alpha }}},\,\,\,\,\sum\limits_\alpha {{{O}_{{\mu \alpha }}}} {{O}_{{\nu \alpha }}} = {{\delta }_{{\mu \nu }}},$$
((B.2))

where \({{a}_{\alpha }}\) are (positive) eigenvalues of matrix \(A\). Therefore, the quadratic form in the exponent of the integrand in (B.1) may be rewritten as

$$\begin{gathered} - \sum\limits_\alpha {{{a}_{\alpha }}} \left( {{{O}_{{\mu \alpha }}}{{x}^{\mu }}} \right)\left( {{{O}_{{\nu \alpha }}}{{x}^{\nu }}} \right) + {{B}_{\mu }}{{x}^{\mu }} \\ = \sum\limits_\alpha {\left( { - {{a}_{\alpha }}y_{\alpha }^{2} + \sum\limits_\mu {{{B}_{\mu }}} {{O}_{{\mu \alpha }}}{{y}_{\alpha }}} \right)} , \\ \end{gathered} $$
((B.3))

where \({{y}_{\alpha }} = {{O}_{{\mu \alpha }}}{{x}^{\mu }}\) and, consequently, \({{x}^{\mu }} = \sum\nolimits_\alpha {{{O}_{{\mu \alpha }}}} {{y}_{\alpha }}\). The Jacobian of this transform is \(\left| O \right| = 1\), so \(dx = dy\). Inserting (B.3) into (B.1), we reduce the integral to a product of standard Gaussian quadratures:

$$\mathcal{G}(A,B) = \prod\limits_\alpha {\sqrt {\frac{\pi }{{{{a}_{\alpha }}}}} } \exp\left[ {\frac{1}{{4{{a}_{\alpha }}}}{{{\left( {\sum\limits_\mu {{{B}_{\mu }}{{O}_{{\mu \alpha }}}} } \right)}}^{2}}} \right].$$
((B.4))

According to (B.2),

$$\sum\limits_\alpha {a_{\alpha }^{{ - 1}}} {{O}_{{\mu \alpha }}}{{O}_{{\nu \alpha }}} = {{\left( {{{A}^{{ - 1}}}} \right)}_{{\mu \nu }}}\mathop = \limits^{{\text{def}}} {{\tilde {A}}^{{\mu \nu }}}\,\,\,\,{\text{and}}\,\,\,\,\prod\limits_\alpha {{{a}_{\alpha }}} = \left| A \right|.$$

The following is then obtained for the 3-dimensional Euclidean space:

$$\mathcal{G}(A,B) = \sqrt {\frac{{{{\pi }^{3}}}}{{\left| A \right|}}} \exp\left( {\frac{1}{4}{{{\tilde {A}}}_{{kn}}}{{B}_{k}}{{B}_{n}}} \right),\,\,\,\,\tilde {A} = {{A}^{{ - 1}}}.$$
((B.5))

In the 4-dimensional Minkowski space,

$$\mathcal{G}(A,B) = \frac{{{{\pi }^{2}}}}{{\sqrt {\left| A \right|} }}\exp\left( {\frac{1}{4}{{{\tilde {A}}}^{{\mu \nu }}}{{B}_{\mu }}{{B}_{\nu }}} \right),\,\,\,\,\tilde {A} = g{{A}^{{ - 1}}}g.$$
((B.6))

Note that \(\tilde {A} \ne {{A}^{{ - 1}}}\) in the latter case, because \({{\tilde {A}}_{{0k}}} = {{\tilde {A}}_{{k0}}} = - {{({{A}^{{ - 1}}})}_{{0k}}}\), \(k = 1,2,3.\) It is evident that the eigenvalues of matrix \(\tilde {A}\) are \({1 \mathord{\left/ {\vphantom {1 {{{a}_{\alpha }}}}} \right. \kern-0em} {{{a}_{\alpha }}}} > 0\); therefore, it is positively defined. Naturally, \(\left| {\tilde {A}} \right| = {1 \mathord{\left/ {\vphantom {1 {\left| A \right|}}} \right. \kern-0em} {\left| A \right|}}.\) In the case of most importance to us (with quantities \({{A}^{{\mu \nu }}}\) and \({{B}^{\mu }}\) constituting a tensor and a 4-vector, respectively), \({{\tilde {A}}^{{\mu \nu }}}{{B}_{\mu }}{{B}_{\nu }}\) is a Lorentz scalar, since integral (B.1) is also a Lorentz scalar.

APPENDIX C

FACTORIZATION OF HADRON BLOCKS

Let us demonstrate that, under certain reasonable assumptions formulated below, hadronic matrix element (233) may be reduced to form (235).

To this end, we express the matrix element at the right-hand side of (234), which includes elementary quark currents \({{j}_{q}}\) and \(j_{q}^{\dag }\), in terms of hadronic currents. Using the definition of a chronological product of local operators, we writeFootnote 67

$$\begin{gathered} T\left[ {{{j}_{q}}(x)j_{q}^{\dag }(y){{\mathbb{S}}_{h}}} \right] \\ = \theta ({{x}_{0}} - {{y}_{0}})A(x,y) + \theta ({{y}_{0}} - {{x}_{0}})B(x,y), \\ \end{gathered} $$
((C.1))

where

$$A(x,y) = \mathbb{U}(\infty ,{{x}_{0}}){{j}_{q}}(x)\mathbb{U}({{x}_{0}},{{y}_{0}})j_{q}^{\dag }(y)\mathbb{U}({{y}_{0}}, - \infty ),$$
((C.2a))
$$B(y,x) = \mathbb{U}(\infty ,{{y}_{0}})j_{q}^{\dag }(y)\mathbb{U}({{y}_{0}},{{x}_{0}}){{j}_{q}}(x)\mathbb{U}({{x}_{0}}, - \infty ),$$
((C.2b))

and

$$\mathbb{U}({{\tau }_{2}},{{\tau }_{1}}) = \exp\left[ {i\int\limits_{{{\tau }_{1}}}^{{{\tau }_{2}}} {d{{x}_{0}}} \int {d{\mathbf{x}}} {{\mathcal{L}}_{{\text{h}}}}(x)} \right]$$
((C.3))

is the evolution operator for the hadronic part of the Lagrangian. Utilizing the well-known properties of this operator

$$\begin{gathered} \mathbb{U}({{\tau }_{2}},{{\tau }_{1}}) = \mathbb{U}({{\tau }_{2}},\tau )\mathbb{U}(\tau ,{{\tau }_{1}}),\,\,\,\,\mathbb{U}({{\tau }_{2}},{{\tau }_{1}})\mathbb{U}({{\tau }_{1}},{{\tau }_{2}}) = 1, \\ \mathbb{U}(\infty , - \infty ) = {{S}_{h}},\,\,\,\,\mathbb{U}( \pm \infty , \pm \infty ) = 1, \\ \end{gathered} $$

we rewrite expression (C.2a) in the following way:

$$\begin{gathered} A(x,y) = \mathbb{U}(\infty ,\tau )\mathbb{U}(\tau ,{{x}_{0}}){{j}_{q}}(x)\mathbb{U}({{x}_{0}},\tau )\mathbb{U}(\tau ,{{x}_{0}}) \\ \times \,\,\mathbb{U}({{x}_{0}},{{y}_{0}})j_{q}^{\dag }(y)\mathbb{U}({{y}_{0}},\tau )\mathbb{U}(\tau ,{{y}_{0}})\mathbb{U}({{y}_{0}}, - \infty ) \\ = \mathbb{U}(\infty ,\tau )\mathbb{U}(\tau ,{{x}_{0}}){{j}_{q}}(x)\mathbb{U}({{x}_{0}},\tau ) \\ \times \,\,\mathbb{U}(\tau ,{{y}_{0}})j_{q}^{\dag }(y)\mathbb{U}({{y}_{0}},\tau )\mathbb{U}(\tau , - \infty ). \\ \end{gathered} $$
((C.4))

Here, \(\tau \) is an arbitrary parameter. Let us now define the hadronic current operator (in the Heisenberg representation) as

$$J(x) = \mathop {\lim}\limits_{\tau \to - \infty } \mathbb{U}(\tau ,{{x}_{0}}){{j}_{q}}(x)\mathbb{U}({{x}_{0}},\tau ).$$
((C.5))

Equation (C.4) may then be rewritten as

$$A(x,y) = {{\mathbb{S}}_{h}}J(x){{J}^{\dag }}(y).$$
((C.6a))

In a similar fashion, the following is derived from (C.2b):

$$B(y,x) = {{\mathbb{S}}_{h}}{{J}^{\dag }}(y)J(x).$$
((C.6b))

Inserting (C.6) into (C.1), we find

$$T\left[ {{{j}_{q}}(x)j_{q}^{\dag }(y){{\mathbb{S}}_{h}}} \right] = {{\mathbb{S}}_{h}}T\left[ {J(x){{J}^{\dag }}(y)} \right].$$
((C.7))

As is known, single-particle hadronic states do not change under the effect of a hadronic \(S\)-matrix. Since our states \(\left| {\left\{ {{{k}_{b}}} \right\}} \right\rangle \) are direct products of single-particle hadronic states, it may be assumed that \(\mathbb{S}_{h}^{\dag }\left| {\left\{ {{{k}_{b}}} \right\}} \right\rangle = \left| {\left\{ {{{k}_{b}}} \right\}} \right\rangle \) to within an insignificant phase factor. Therefore, the following expression derived from (C.7) is true to within this factor:

$$\begin{gathered} \left\langle {\left\{ {{{k}_{b}}} \right\}} \right|T\left[ {{{j}_{q}}(x)j_{q}^{\dag }(y){{\mathbb{S}}_{h}}} \right]\left| {\left\{ {{{k}_{a}}} \right\}} \right\rangle \\ = \left\langle {\left\{ {{{k}_{b}}} \right\}} \right|\left| {T\left[ {J(x){{J}^{\dag }}(y)} \right]} \right|\left| {\left\{ {{{k}_{a}}} \right\}} \right\rangle \equiv {{\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle }_{T}}. \\ \end{gathered} $$
((C.8))

Let us now discuss matrix element

$$\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle = \left\langle {\left\{ {{{k}_{b}}} \right\}} \right|J(x){{J}^{\dag }}(y)\left| {\left\{ {{{k}_{a}}} \right\}} \right\rangle .$$

Owing to the translational invariance, current \(J(x)\) satisfies equation

$$i{{\partial }_{\mu }}J(x) = [J(x),{{P}^{\mu }}],$$
((C.9))

where \({{P}^{\mu }}\) is the complete 4-momentum operator corresponding to the hadronic part of the Lagrangian (see, e.g., [140]). Using (C.9), one obtains

$$\begin{gathered} i\left( {\frac{\partial }{{\partial {{x}_{\mu }}}} + \frac{\partial }{{\partial {{y}_{\mu }}}}} \right)\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle = \left\langle {[J(x){{J}^{\dag }}(y),{{P}^{\mu }}]} \right\rangle \\ = {{\left( {K - K{\kern 1pt} '} \right)}^{\mu }}\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle , \\ \end{gathered} $$
((C.10))

where

$$\begin{gathered} K = {{K}_{s}} + {{K}_{d}} = \sum\limits_{a \in {{I}_{s}}} {{{k}_{a}}} + \sum\limits_{a \in {{I}_{d}}} {{{k}_{a}}} , \\ K{\kern 1pt} ' = {{K}_{s}} + {{K}_{d}} = \sum\limits_{b \in F_{s}^{'}} {{{k}_{b}}} + \sum\limits_{b \in F_{d}^{'}} {{{k}_{b}}} . \\ \end{gathered} $$

The formal solution of differential equations (C.10) takes the form

$$\begin{gathered} \left\langle {{{J}_{\mu }}(x)J_{\nu }^{\dag }(y)} \right\rangle = {{e}^{{i\left[ {\left( {K_{s}^{'} - {{K}_{s}}} \right)\left( {x - {{x}_{s}}} \right) + \left( {K_{d}^{'} - {{K}_{d}}} \right)\left( {y - {{x}_{d}}} \right)} \right]}}} \\ \times \,\,\left\langle {{{J}_{\mu }}({{x}_{s}})J_{\nu }^{\dag }({{x}_{d}})} \right\rangle + {{C}_{{\mu \nu }}}(x - y), \\ \end{gathered} $$
((C.11))

where xs and xd are arbitrary space-time 4-vectors, and \({{C}_{{\mu \nu }}}(x)\) are the components of a certain tensorFootnote 68 such that \({{C}_{{\mu \nu }}}({{x}_{s}} - {{x}_{d}}) = 0.\)

A similar result may also be obtained for matrix element (C.8). Indeed, rewriting (C.8) in the explicit form

$$\begin{gathered} {{\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle }_{T}} = \theta ({{x}_{0}} - {{y}_{0}})\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle \\ + \,\,\theta ({{y}_{0}} - {{x}_{0}})\left\langle {{{J}^{\dag }}(y)J(x)} \right\rangle , \\ \end{gathered} $$

and taking relation \(\left( {{\partial \mathord{\left/ {\vphantom {\partial {\partial {{x}_{\mu }}}}} \right. \kern-0em} {\partial {{x}_{\mu }}}} + {\partial \mathord{\left/ {\vphantom {\partial {\partial {{y}_{\mu }}}}} \right. \kern-0em} {\partial {{y}_{\mu }}}}} \right)\theta ({{x}_{0}} - {{y}_{0}}) = 0\) and (C.10) into account, we obtain the following equations:

$$\begin{gathered} i\left( {\frac{\partial }{{\partial {{x}_{\mu }}}} + \frac{\partial }{{\partial {{y}_{\mu }}}}} \right){{\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle }_{T}} \\ = {{\left( {K - K{\kern 1pt} '} \right)}^{\mu }}{{\left\langle {J(x){{J}^{\dag }}(y)} \right\rangle }_{T}}. \\ \end{gathered} $$
((C.12))

Their solution coincides with (C.11), where \(\left\langle \cdots \right\rangle \) is substituted by \({{\left\langle \cdots \right\rangle }_{T}}\). It is convenient for our purpose to choose the impact points for in- and out-packets in the source and detector as 4-vectors \({{x}_{s}}\) and \({{x}_{d}}\), respectively. Since points \({{X}_{s}}\) and \({{X}_{d}}\) are macroscopically separated, currents \(J({{X}_{s}})\) and \({{J}^{\dag }}({{X}_{d}})\) are mutually commuting: \([{{J}_{\mu }}({{X}_{s}}),J_{\nu }^{\dag }({{X}_{d}})] = 0.\) As a result, matrix element \({{\left\langle {{{J}_{\mu }}({{X}_{s}})J_{\nu }^{\dag }({{X}_{d}})} \right\rangle }_{T}}\) is factorized into two cofactors that are associated with the source and detector vertices and depend on the corresponding variables only. Thus, having introduced 4-vectors (\(c\)-number hadronic currents)

$$\begin{gathered} {{\mathcal{J}}_{s}}\left( {{{X}_{s}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right) = {{e}^{{i\left( {{{K}_{s}} - {{K}_{s}}} \right){{X}_{s}}}}}\left\langle {\left\{ {{{k}_{b}}} \right\}} \right|J({{X}_{s}})\left| {\left\{ {{{k}_{a}}} \right\}} \right\rangle , \\ a \in {{I}_{s}},\,\,\,\,b \in F_{s}^{'}, \\ \mathcal{J}_{d}^{ * }\left( {{{X}_{d}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right) = {{e}^{{i\left( {{{K}_{d}} - {{K}_{d}}} \right){{X}_{d}}}}}\left\langle {\left\{ {{{k}_{b}}} \right\}} \right|{{J}^{\dag }}({{X}_{d}})\left| {\left\{ {{{k}_{a}}} \right\}} \right\rangle , \\ a \in {{I}_{d}},\,\,\,b \in {{F}_{d}}, \\ \end{gathered} $$

one obtains

$$\begin{gathered} {{\left\langle {{{J}_{\mu }}(x)J_{\nu }^{\dag }(y)} \right\rangle }_{T}} \\ = \exp\left\{ {i\left[ {\left( {{{K}_{s}} - {{K}_{s}}} \right)x + \left( {{{K}_{d}} - {{K}_{d}}} \right)y} \right]} \right\} \\ \times \,\,J_{s}^{\mu }\left( {{{X}_{s}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right)J_{d}^{{\nu * }}\left( {{{X}_{d}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right) + {{C}_{{\mu \nu }}}(x - y). \\ \end{gathered} $$
((C.13))

Tensor components \({{C}_{{\mu \nu }}}(x - y)\), which satisfy conditions

$${{C}_{{\mu \nu }}}({{X}_{s}} - {{X}_{d}}) = 0,$$
((C.14))

may feature a singularity (no stronger than a \(\delta \) function or its derivatives) only at point \(x = y\). At the same time, by virtue of condition (C.14), they should vanish after integration over \(x\) and \(y\) performed over sufficiently small space-time volumes surrounding the impact points. Functions \({{C}_{{\mu \nu }}}(x - y)\) enter the amplitude only via integral

$$\begin{gathered} \propto \int {dxdy{{e}^{{iq(x - y)}}}} {{C}_{{\mu \nu }}}(x - y) \\ \times \,\,\psi _{\alpha }^{ * }({{{\mathbf{p}}}_{\alpha }},{{x}_{\alpha }} - x)\psi _{\beta }^{ * }({{{\mathbf{p}}}_{\beta }},{{x}_{\beta }} - y), \\ \end{gathered} $$
((C.15))

where \(\psi _{\alpha }^{ * }\) and \(\psi _{\beta }^{ * }\) are \(\psi \) functions describing the WPs of final leptons \(\ell _{\alpha }^{ + }\) and \(\ell _{\beta }^{ - }\) with momenta \({{{\mathbf{p}}}_{\alpha }}\) and \({{{\mathbf{p}}}_{\beta }}\), respectively. The contribution of product \(\psi _{\alpha }^{ * }\psi _{\beta }^{ * }\) to the integrand in (C.15) is significant only within the classical world tubes located near the corresponding impact points. Therefore, the integrand in (C.15) is not negligible only if they have a considerable overlap region (in the limiting case, if the axes of lepton tubes coincide). However, this configuration is strongly suppressed by virtue of the approximate energy-momentum conservation. This is evident from the analysis of the simplest case with \({{C}_{{\mu \nu }}}(x - y) \propto \delta (x - y).\) Integral (C.15) is then proportional to the smeared \(\delta \) function \(\tilde {\delta }({{p}_{\alpha }} + {{p}_{\beta }})\) and is negligible, since \(p_{\alpha }^{0} + p_{\beta }^{0} \geqslant {{m}_{\alpha }} + {{m}_{\beta }}.\) These qualitative arguments enable one to ignore nonphysical long-range correlations induced by term \({{C}_{{\mu \nu }}}(x - y)\) in (C.13). By virtue of (C.8), one can rewrite the right-hand side of (233) as

$$\begin{gathered} \int {\left[ {\prod\limits_a {\frac{{d{{{\mathbf{k}}}_{a}}{{\phi }_{a}}({{{\mathbf{k}}}_{a}},{{{\mathbf{p}}}_{a}})}}{{{{{(2\pi )}}^{3}}2{{E}_{{{{{\mathbf{k}}}_{a}}}}}}}} {{e}^{{i\left[ {\left( {{{k}_{a}} - {{p}_{a}}} \right){{x}_{a}} - {{k}_{a}}x} \right]}}}} \right]} \\ \times \,\,\int {\left[ {\prod\limits_b {\frac{{d{{{\mathbf{k}}}_{b}}\phi _{b}^{ * }({{{\mathbf{k}}}_{b}},{{{\mathbf{p}}}_{b}})}}{{{{{(2\pi )}}^{3}}2{{E}_{{{{{\mathbf{k}}}_{b}}}}}}}} {{e}^{{ - i\left[ {\left( {{{k}_{b}} - {{p}_{b}}} \right){{x}_{b}} - {{k}_{b}}y} \right]}}}} \right]} \\ \times \,\,\mathcal{J}_{s}^{\mu }\left( {{{X}_{s}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right)\mathcal{J}_{d}^{{\nu * }}\left( {{{X}_{d}};\left\{ {{{k}_{a}},{{k}_{b}}} \right\}} \right). \\ \end{gathered} $$

Taking the properties of the form factor in \({{\phi }_{a}}({{{\mathbf{k}}}_{a}},{{{\mathbf{p}}}_{a}})\) and \({{\phi }_{b}}({{{\mathbf{k}}}_{b}},{{{\mathbf{p}}}_{b}})\) into account, one can substitute variables \({{{\mathbf{k}}}_{a}}\) and \({{{\mathbf{k}}}_{b}}\) in functions \({{\mathcal{J}}_{s}}\) and \({{\mathcal{J}}_{d}}\) by the corresponding external momenta \({{{\mathbf{p}}}_{a}}\) and \({{{\mathbf{p}}}_{b}}\). Having introduced this (last) approximation, we arrive at (235), thus completing the proof.

APPENDIX D

STATIONARY POINT FOR AN ARBITRARY CONFIGURATION OF MOMENTA OF EXTERNAL WAVE PACKETS

Let us detail the algorithm for solving Eq. (249) in the general case (i.e., for an arbitrary configuration of external momenta). The general solution is of interest both in the methodological context and for purposes of data processing in neutrino experiments at intermediate (subrelativistic) energies. Although the proposed algorithm is rather cumbersome, it is easy to implement in the form of a computer program and is thus convenient primarily for numerical analysis.

Form (267) of Eq. (249), where the velocity of a virtual neutrino is the unknown quantity, is more convenient to work with. Raising both parts of (267) to the second power, we arrive at algebraic equation of the fourth order

$${{v}^{4}} + {{c}_{3}}{{v}^{3}} + {{c}_{2}}{{v}^{2}} + {{c}_{1}}v + {{c}_{0}} = 0,$$
((D.1))

with coefficients

$$\begin{gathered} {{c}_{0}} = \frac{{{{{({\mathbf{Rl}})}}^{2}} - {{{(\eta {\mathbf{l}})}}^{2}}}}{{{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}}}, \\ {{c}_{1}} = - 2\frac{{R({\mathbf{Rl}}) + 2{{{({\mathbf{Rl}})}}^{2}} - {{\eta }_{0}}(\eta {\mathbf{l}})}}{{{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}}}, \\ {{c}_{2}} = \frac{{{{R}^{2}} + 6{{{({\mathbf{Rl}})}}^{2}} + 4R({\mathbf{Rl}}) + {{{(\eta {\mathbf{l}})}}^{2}} - \eta _{0}^{2}}}{{{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}}}, \\ {{c}_{3}} = - 2\frac{{R({\mathbf{Rl}}) + 2{{{({\mathbf{Rl}})}}^{2}} + {{\eta }_{0}}(\eta {\mathbf{l}})}}{{{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}}}. \\ \end{gathered} $$

Here, \({{\eta }_{\mu }} = {{{{Y}_{\mu }}} \mathord{\left/ {\vphantom {{{{Y}_{\mu }}} {{{m}_{j}}}}} \right. \kern-0em} {{{m}_{j}}}}\); it is assumed from now on that \({{m}_{j}} > 0\) (the case of a massless neutrino is trivial), and index “\(j\)” numbering neutrinos is omitted to simplify the formulas, which are already cumbersome enough. The other designations are the same as in the main text. Equation (D.1) can be solved using the Descartes–Euler method (see, e.g., [141]). Following the corresponding algorithm, we write Eq. (D.1) in the "incomplete” form

$${{\left( {v + \frac{{{{c}_{3}}}}{4}} \right)}^{4}} + {{\tilde {c}}_{2}}{{\left( {v + \frac{{{{c}_{3}}}}{4}} \right)}^{2}} + {{\tilde {c}}_{1}}\left( {v + \frac{{{{c}_{3}}}}{4}} \right) + {{\tilde {c}}_{0}} = 0.$$
((D.2))

The solutions of this equation are constructed from the roots of cubic equation

$${{z}^{3}} + {{a}_{2}}{{z}^{2}} + {{a}_{1}}z + {{a}_{0}} = 0,$$
((D.3))

where

$${{a}_{0}} = - \frac{{\tilde {c}_{1}^{2}}}{{64}},\,\,\,\,{{a}_{1}} = \frac{{\tilde {c}_{2}^{2} - 4{{{\tilde {c}}}_{0}}}}{{16}},\,\,\,\,{{a}_{2}} = \frac{{{{{\tilde {c}}}_{2}}}}{2}.$$

Equation (D.3) may also be reduced to the incomplete (Cardano) form:

$${{\left( {z + \frac{{{{a}_{2}}}}{3}} \right)}^{3}} + \mathfrak{p}\left( {z + \frac{{{{a}_{2}}}}{3}} \right) + \mathfrak{q} = 0.$$

Here,

$$\begin{gathered} \mathfrak{p} = {{a}_{1}} - \frac{{a_{2}^{2}}}{3} = - \frac{{{{{\left[ {{{R}^{2}} + 4R({\mathbf{Rl}}) - \eta _{0}^{2} + {{{(\eta {\mathbf{l}})}}^{2}}} \right]}}^{2}}}}{{48{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{2}}}}, \\ \mathfrak{q} = {{a}_{0}} - \frac{{{{a}_{1}}{{a}_{2}}}}{3} + 2{{\left( {\frac{{{{a}_{2}}}}{3}} \right)}^{3}} = - \frac{A}{{864{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{3}}}}, \\ \end{gathered} $$
$$A = {{A}_{0}} + {{A}_{1}}({\mathbf{Rl}}) + {{A}_{2}}{{({\mathbf{Rl}})}^{2}} + {{A}_{3}}{{({\mathbf{Rl}})}^{3}},$$
$$\begin{gathered} {{A}_{0}} = {{R}^{6}} - 3\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]{{R}^{4}} \\ + \,\,3\left[ {\eta _{0}^{4} + 16\eta _{0}^{2}{{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}} + {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{4}}} \right]{{R}^{2}} - {{\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]}^{3}}, \\ {{A}_{1}} = 12R\left\{ {{{R}^{4}} - 2\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]{{R}^{2}}} \right. \\ \left. { + \,\,\left[ {\eta _{0}^{2} - 7{{\eta }_{0}}({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}}) + {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]{{{(\eta l)}}^{2}}} \right\}, \\ {{A}_{2}} = 48{{R}^{2}}\left[ {{{R}^{2}} - \eta _{0}^{2} + {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right] + 54{{(\eta l)}^{4}}, \\ {{A}_{3}} = 64{{R}^{3}}. \\ \end{gathered} $$

The number of real roots is defined by the sign of function

$$\mathfrak{V} = \frac{{{{\mathfrak{q}}^{2}}}}{4} + \frac{{{{\mathfrak{p}}^{3}}}}{{27}} = \frac{{\mathop {\left[ {{{\eta }_{0}}(\eta {\mathbf{l}})R - {{{(\eta l)}}^{2}}({\mathbf{Rl}})} \right]}\nolimits^2 B}}{{27\,648{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{6}}}},$$

which coincides with the sign of polynomial \(B = {{B}_{0}} + {{B}_{1}}({\mathbf{Rl}})\) + \({{B}_{2}}{{({\mathbf{Rl}})}^{2}} + {{B}_{3}}{{({\mathbf{Rl}})}^{3}}\) with coefficients

$$\begin{gathered} {{B}_{0}} = {{R}^{6}} - 3\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]{{R}^{4}} \\ + \,\,3\left[ {\eta _{0}^{4} + 7\eta _{0}^{2}{{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}} + {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{4}}} \right]{{R}^{2}} - {{\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]}^{3}}, \\ {{B}_{1}} = 6R\left\{ {2{{R}^{4}} - 4\left[ {\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right]} \right.{{R}^{2}} \\ + \,\,\left. {\left[ {2{{\eta }_{0}} - ({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})} \right]\left[ {{{\eta }_{0}} - 2({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})} \right]{{{(\eta l)}}^{2}}} \right\}, \\ {{B}_{2}} = 48{{R}^{2}}\left[ {{{R}^{2}} - \eta _{0}^{2} + {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right] + 27{{(\eta l)}^{4}}, \\ {{B}_{3}} = 64{{R}^{3}}. \\ \end{gathered} $$

Three different real roots are found at \(B < 0\); at \(B > 0\), one real root and two mutually conjugate complex roots are present; at \(B = 0\), two (or all three) real roots may coincide. The validity of the following useful identity may be proven:

$$A = B + 27{{\left[ {{{\eta }_{0}}(\eta {\mathbf{l}})R - {{{(\eta l)}}^{2}}({\mathbf{Rl}})} \right]}^{2}}.$$
((D.4))

Del Ferro–Tartaglia–Cardano solution in radicals. The roots of incomplete cubic equation (D.3) are

$$\begin{gathered} {{z}_{0}} = a + ({{A}_{ + }} + {{A}_{ - }}), \\ {{z}_{ \pm }} = a - \frac{1}{2}({{A}_{ + }} + {{A}_{ - }}) \pm i\frac{{\sqrt 3 }}{2}({{A}_{ + }} - {{A}_{ - }}), \\ \end{gathered} $$

where

$$\begin{gathered} a = \frac{{{{a}_{2}}}}{3} = - \frac{{{{C}_{0}} + {{C}_{1}}({\mathbf{Rl}}) + {{C}_{2}}{{{({\mathbf{Rl}})}}^{2}} + {{C}_{3}}{{{({\mathbf{Rl}})}}^{3}}}}{{12{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{2}}}}, \\ A_{ \pm }^{3} = - \frac{q}{2} \pm \sqrt {\mathfrak{V} } = \frac{{\tfrac{A}{{18}} \pm {{i}^{\delta }}\left| {{{\eta }_{0}}(\eta {\mathbf{l}})R - {{{(\eta l)}}^{2}}({\mathbf{Rl}})} \right|\sqrt {\tfrac{{\left| B \right|}}{3}} }}{{96{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{3}}}}; \\ \end{gathered} $$
$$\begin{gathered} {{C}_{0}} = - \eta _{0}^{2}\left[ {2{{R}^{2}} - 2\eta _{0}^{2} - {{{({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})}}^{2}}} \right], \\ {{C}_{1}} = - 2{{\eta }_{0}}\left[ {4{{\eta }_{0}} - 3({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})} \right]R, \\ {{C}_{2}} = {{R}^{2}} - 2(\eta l)\left[ {5{{\eta }_{0}} - ({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})} \right], \\ {{C}_{3}} = 4R; \\ \end{gathered} $$

\(\delta = 0\) at \(B \geqslant 0\) and \(\delta = 1\) at \(B < 0\). The expression for \({{A}_{ \pm }}\) is simplified if identity (D.4) is taken into account:

$${{A}_{ \pm }} = \frac{{\mathop {\left[ {3\sqrt 3 \left| {{{\eta }_{0}}({\boldsymbol{\mathbf{\eta}}} {\mathbf{l}})R - {{{(\eta l)}}^{2}}({\mathbf{Rl}})} \right| \pm {{i}^{\delta }}\sqrt {\left| B \right|} } \right]}\nolimits^{2/3} }}{{12\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}.$$

Solution in the Vieta’s trigonometric representation. For completeness, we also present the more compact trigonometric solution (Vieta’s representation), which may turn out to be more convenient for numerical calculations. If nothing else, it is useful for monitoring the accuracy of calculations by means of comparison with the canonical solution. The explicit form of the trigonometric solution depends on the sign of function \(B\).

B < 0. As was already noted, Eq. (D.3) has three real roots in this case (sometimes called the irreducible case):

$$\begin{gathered} {{z}_{0}} = a + {{\zeta }_{0}}\cos\frac{\alpha }{3}, \\ {{z}_{ \pm }} = a - {{\zeta }_{0}}\cos\left( {\frac{{\alpha \pm \pi }}{3}} \right), \\ \end{gathered} $$

where

$$\begin{gathered} {{\zeta }_{0}} = \frac{{\left| {{{R}^{2}} + 4R({\mathbf{Rl}}) - \eta _{0}^{2} + {{{(\eta {\mathbf{l}})}}^{2}}} \right|}}{{6\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}, \\ \cos\alpha = - \frac{A}{{\left| {{{R}^{2}} + 4R({\mathbf{Rl}}) - \eta _{0}^{2} + {{{(\eta {\mathbf{l}})}}^{2}}} \right|}}. \\ \end{gathered} $$

B ≥ 0.Equation (D.3) has one real and two complex roots. Let us introduce the following notationFootnote 69:

$$\begin{gathered} tan\alpha {\kern 1pt} ' = \sqrt[3]{{tan\frac{\beta }{2}}},\,\,\,\,\sin\beta = - \frac{4}{{\cos\alpha }} \\ = \frac{4}{A}{{\left| {{{R}^{2}} + 4R({\mathbf{Rl}}) - \eta _{0}^{2} + {{{(\eta {\mathbf{l}})}}^{2}}} \right|}^{3}}. \\ \end{gathered} $$

Then, the roots are

$$\begin{gathered} {{z}_{0}} = a - {{\zeta }_{0}}{\text{cosec}}2\alpha {\kern 1pt} ', \\ {{z}_{ \pm }} = a + \frac{1}{2}{{\zeta }_{0}}\left( {{\text{cosec}}2\alpha {\kern 1pt} '\,\, \pm i\sqrt 3 cot2\alpha {\kern 1pt} '} \right). \\ \end{gathered} $$

Roots of Eq. (D.1). The roots of incomplete equation of the fourth order (D.2) are given by combinations

$${{\Xi }_{n}} = \pm \sqrt {{{z}_{ - }}} \pm \sqrt {{{z}_{0}}} \pm \sqrt {{{z}_{ + }}} ,$$

where four out of the possible eight combinations of signs are chosen so that condition

$$\begin{gathered} - \sqrt {{{z}_{ - }}} \sqrt {{{z}_{0}}} \sqrt {{{z}_{ + }}} = \frac{{{{{\tilde {c}}}_{1}}}}{8} \\ = \frac{{{{D}_{0}} + {{D}_{1}}({\mathbf{Rl}}) + {{D}_{2}}{{{({\mathbf{Rl}})}}^{2}} + {{D}_{3}}{{{({\mathbf{Rl}})}}^{3}} + {{D}_{4}}{{{({\mathbf{Rl}})}}^{4}}}}{{8{{{\left[ {{{{({\mathbf{Rl}})}}^{2}} + \eta _{0}^{2}} \right]}}^{3}}}} \\ \end{gathered} $$

is satisfied. Here,

$$\begin{gathered} {{D}_{0}} = \eta _{0}^{3}(\eta {\mathbf{l}})\left( {{{R}^{2}} + \eta _{0}^{2}} \right), \\ {{D}_{1}} = \eta _{0}^{2}\left[ {{{R}^{2}} - 3\eta _{0}^{2} + 4{{\eta }_{0}}(\eta {\mathbf{l}}) - 2{{{(\eta {\mathbf{l}})}}^{2}}} \right]R, \\ {{D}_{2}} = {{\eta }_{0}}\left\{ {2\left[ {3{{\eta }_{0}} - (\eta {\mathbf{l}})} \right]{{R}^{2}} - (\eta l)} \right. \\ \times \,\,\left. {\left[ {6\eta _{0}^{2} - 3{{\eta }_{0}}(\eta {\mathbf{l}}) + {{{(\eta {\mathbf{l}})}}^{2}}} \right]} \right\}, \\ {{D}_{3}} = \left[ {9\eta _{0}^{2} - 8{{\eta }_{0}}(\eta {\mathbf{l}}) + {{{(\eta {\mathbf{l}})}}^{2}}} \right]R, \\ {{D}_{4}} = 2{{(\eta l)}^{2}}. \\ \end{gathered} $$

All four roots of Eq. (D.1) can then be derived using formula

$${{v}_{n}} = {{\Xi }_{n}} - {{{{c}_{3}}} \mathord{\left/ {\vphantom {{{{c}_{3}}} 4}} \right. \kern-0em} 4}\,\,\,\,(n = 1,2,3,4).$$

The real nonnegative root of interest to us, which corresponds to the stationary point, should satisfy the condition of positivity of the second derivative (251). The solutions found in the main text for two opposite limiting cases (\(1 - v \ll 1\) and \(v \sim 1\)) may serve as additional criteria of uniqueness of the solution of the general form based on the algorithm described here, since they should “join” the correct numerical solution smoothly under the corresponding variations of momenta of the external WPs and discrete parameters defining the effective velocity of a virtual neutrino.

APPENDIX E

SPREADING OF A NEUTRINO WAVE PACKET AT EXTREMELY LONG DISTANCES

Let us consider the generalization of some results from the main text to the case with spreading of an effective neutrino packet at astronomical distances. This generalization can apply in the processing of data from neutrino telescopes (Baikal GVD, IceCube, KM3Net ARCA, etc.), various experiments on radio detection of ultrahigh-energy neutrinos, and future orbital experiments that are aimed, among other things, at measuring the flavor composition of neutrino and antineutrino fluxes from remote astrophysical sources.

The integration over \({{x}^{0}}\) in (302) yields the following expression for the decoherence factor:

$$\begin{gathered} {{S}_{{ij}}} = \frac{{\sqrt \pi }}{{4{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}\exp\left( {{{\Phi }_{{ij}}}} \right)\int\limits_{y_{1}^{0}}^{y_{2}^{0}} {d{{y}^{0}}} \\ \times \,\,\left\{ {{\text{erf}}\left[ {2{{{\tilde {\mathfrak{D}}}}_{{ij}}}\left( {{{y}^{0}} - x_{1}^{0} - \frac{L}{{{{v}_{{ij}}}}}} \right) + \frac{{i\Delta {{E}_{{ij}}}}}{{4{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right]} \right. \\ \left. { - \,\,{\text{erf}}\left[ {2{{{\tilde {\mathfrak{D}}}}_{{ij}}}\left( {{{y}^{0}} - x_{2}^{0} - \frac{L}{{{{v}_{{ij}}}}}} \right) + \frac{{i\Delta {{E}_{{ij}}}}}{{4{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right]} \right\}, \\ \end{gathered} $$
((E.1))
$$\begin{gathered} {{\Phi }_{{ij}}} = - {{\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right)}^{2}}{{\left( {\frac{{\tilde {\mathfrak{D}}_{i}^{*}{{{\tilde {\mathfrak{D}}}}_{j}}}}{{{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right)}^{2}}{{L}^{2}} \\ - \,\,{{\left( {\frac{{\Delta {{E}_{{ij}}}}}{{4{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right)}^{2}} + i\left( {\frac{{\Delta {{E}_{{ij}}}}}{{{{v}_{{ij}}}}} - \Delta {{P}_{{ij}}}} \right)L. \\ \end{gathered} $$
((E.2))

Here,

$$\begin{gathered} \Delta {{P}_{{ij}}} = {{P}_{i}} - {{P}_{j}},\,\,\,\,\Delta {{E}_{{ij}}} = {{E}_{i}} - {{E}_{j}}, \\ \frac{1}{{{{v}_{{ij}}}}} = \frac{1}{{2\tilde {\mathfrak{D}}_{{ij}}^{2}}}\left[ {\frac{{{{{\left( {\tilde {\mathfrak{D}}_{i}^{*}} \right)}}^{2}}}}{{{{v}_{i}}}} + \frac{{\tilde {\mathfrak{D}}_{j}^{2}}}{{{{v}_{j}}}}} \right], \\ \tilde {\mathfrak{D}}_{{ij}}^{2} = \frac{1}{2}\left[ {{{{\left( {\tilde {\mathfrak{D}}_{i}^{*}} \right)}}^{2}} + \mathfrak{D}_{j}^{2}} \right], \\ \end{gathered} $$

and the other notations are the same as in the main text. Using the primitive of error function (306) introduced in the main text and integrating over \({{y}^{0}}\) in (E.1), we find the generalization of formula (303):

$$\begin{gathered} {{S}_{{ij}}} = \frac{{\sqrt \pi }}{{8\tilde {\mathfrak{D}}_{{ij}}^{2}}}\exp\left( {{{\Phi }_{{ij}}}} \right)\sum\limits_{l,l' = 1}^2 {{{{( - 1)}}^{{l + l' + 1}}}} \\ \times \,\,{\text{Ierf}}\left[ {2{{{\tilde {\mathfrak{D}}}}_{{ij}}}\left( {x_{l}^{0} - y_{{l'}}^{0} + \frac{L}{{{{v}_{{ij}}}}}} \right) - i\frac{{\Delta {{E}_{{ij}}}}}{{4{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right]. \\ \end{gathered} $$
((E.3))

Although this expression is much more complex than (303), there are no difficulties in analyzing the decoherence effects numerically. In the present section, we examine only the key properties of complex-valued phase \({{\Phi }_{{ij}}}\). To this end, we need to isolate real and imaginary parts of the phase and determine the length scales governing its behavior.

Using the definitions introduced above and dropping evidently small corrections, we find70

$$\begin{gathered} {{v}_{{ij}}} = \frac{{\left( {{{\tau }_{i}} - {{\tau }_{j}}} \right)\left( {\tfrac{{{{\tau }_{i}}}}{{{{v}_{j}}}} - \tfrac{{{{\tau }_{j}}}}{{{{v}_{i}}}}} \right) + 2\left( {\tfrac{1}{{{{v}_{i}}}} + \tfrac{1}{{{{v}_{j}}}}} \right) - i\left( {{{\tau }_{i}} + {{\tau }_{j}}} \right)\left( {\tfrac{1}{{{{v}_{i}}}} - \tfrac{1}{{{{v}_{j}}}}} \right)}}{{{{{\left( {\tfrac{1}{{{{v}_{i}}}} + \tfrac{1}{{{{v}_{j}}}}} \right)}}^{2}} + {{{\left( {\tfrac{{{{\tau }_{i}}}}{{{{v}_{j}}}} - \tfrac{{{{\tau }_{j}}}}{{{{v}_{i}}}}} \right)}}^{2}}}}, \\ \tilde {\mathfrak{D}}_{{ij}}^{2} = \frac{{{{D}^{2}}\left[ {2 + \tau _{i}^{2} + \tau _{j}^{2} + i\left( {1 - {{\tau }_{i}}{{\tau }_{j}}} \right)\left( {{{\tau }_{i}} - {{\tau }_{j}}} \right)} \right]}}{{2\left( {1 + \tau _{i}^{2}} \right)\left( {1 + \tau _{j}^{2}} \right)}},\,\,\,\,{{\left( {\frac{{\tilde {\mathfrak{D}}_{i}^{*}{{{\tilde {\mathfrak{D}}}}_{j}}}}{{{{{\tilde {\mathfrak{D}}}}_{{ij}}}}}} \right)}^{2}} = \frac{{{{\mathfrak{D}}^{2}}}}{{1 - i{{\tau }_{{ij}}}}}. \\ \end{gathered} $$

Inserting these expressions into (E.2), one obtains

$$\begin{gathered} {{\Phi }_{{ij}}} = - \frac{1}{{1 + \mathfrak{r}_{{ij}}^{2}}}\mathop {\left[ {\frac{{\Delta {{E}_{{ij}}}}}{{4{{\mathfrak{D}}_{{ij}}}}} + \frac{{{{\mathfrak{r}}_{i}} + {{\mathfrak{r}}_{j}}}}{2}\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right){{\mathfrak{D}}_{{ij}}}L} \right]}\nolimits^2 \\ - \,\,{{\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right)}^{2}}\mathfrak{D}_{{ij}}^{2}{{L}^{2}} + i\left\{ {\left[ {\frac{1}{2}\left( {\frac{1}{{{{v}_{i}}}} + \frac{1}{{{{v}_{j}}}}} \right)} \right.} \right. \\ \left. {\left. { - \,\,\frac{{{{\mathfrak{r}}_{{ij}}}\left( {{{\mathfrak{r}}_{i}} + {{\mathfrak{r}}_{j}}} \right)}}{{4\left( {1 + \mathfrak{r}_{{ij}}^{2}} \right)}}\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right)} \right]\Delta {{E}_{{ij}}} - \Delta {{P}_{{ij}}}} \right\}L \\ + \,\,\frac{{i{{\mathfrak{r}}_{{ij}}}}}{{1 + {{r}_{{ij}}}}}\left[ {\left( {1 - {{\mathfrak{r}}_{i}}{{\mathfrak{r}}_{j}}} \right){{{\left( {\frac{{\Delta {{E}_{{ij}}}}}{{4\mathfrak{D}}}} \right)}}^{2}} - {{{\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right)}}^{2}}{{\mathfrak{D}}^{2}}{{L}^{2}}} \right], \\ \end{gathered} $$
((E.4))

where

$$\begin{gathered} {{\mathfrak{D}}_{{ij}}} = \mathfrak{D}{{\left( {1 + \frac{{\mathfrak{r}_{i}^{2} + \mathfrak{r}_{j}^{2}}}{2}} \right)}^{{ - 1/2}}}, \\ {{\mathfrak{r}}_{{ij}}} = \frac{{{{\mathfrak{r}}_{i}} - {{\mathfrak{r}}_{j}}}}{2} \approx \frac{{2\Delta m_{{ij}}^{2}{{\mathfrak{D}}^{2}}L}}{{E_{\nu }^{3}}}. \\ \end{gathered} $$
((E.5))

It follows from the analysis of (E.4) that the real part of phase \({{\Phi }_{{ij}}}\) becomes independent of \(L\) within two (strongly) spatially separated regions defined by the following conditions:

$$L \ll \mathfrak{L} = \frac{{\mathfrak{n}{{E}_{\nu }}}}{{4{{\mathfrak{D}}^{2}}}},$$

and

$$L \gg {{\mathfrak{L}}_{{ij}}} = \left\{ \begin{gathered} \frac{{{{E}_{\nu }}}}{{2(m_{i}^{2} + m_{j}^{2})}}{{\left( {\frac{{{{E}_{\nu }}}}{\mathfrak{D}}} \right)}^{2}} \hfill \\ {\text{if}}\,\,\,\,{{m}_{i}} \gg {{m}_{j}}\,\,\,\,{\text{or}}\,\,\,\,{{m}_{i}} \gg {{m}_{j}}, \hfill \\ \frac{{{{E}_{\nu }}}}{{2\left| {m_{i}^{2} - m_{j}^{2}} \right|}}{{\left( {\frac{{{{E}_{\nu }}}}{\mathfrak{D}}} \right)}^{2}} \hfill \\ {\text{if}}\,\,\,\,\left| {{{m}_{i}} - {{m}_{j}}} \right| \ll {{m}_{i}} + {{m}_{j}}. \hfill \\ \end{gathered} \right.$$

TheFootnote 70 corresponding asymptotic values of \({\text{Re}}{{\Phi }_{{ij}}}\) calculated to an accuracy of \(\mathcal{O}({{r}_{{i,j}}})\) take the form

$$\begin{gathered} {{\left. {{\text{Re}}{{\Phi }_{{ij}}}} \right|}_{{L \ll \mathfrak{L}}}} \approx {\text{Re}}\Phi _{{ij}}^{0} \\ = - {{\left( {\frac{{\pi \mathfrak{n}}}{{2\mathfrak{D}{{L}_{{ij}}}}}} \right)}^{2}}\left[ {1 + \frac{\mathfrak{m}}{\mathfrak{n}}\left( {{{r}_{i}} + {{r}_{j}}} \right)} \right], \\ \end{gathered} $$
((E.6a))
$$\begin{gathered} {{\left. {{\text{Re}}{{\Phi }_{{ij}}}} \right|}_{{L \gg {{\mathfrak{L}}_{{ij}}}}}} \approx {\text{Re}}\Phi _{{ij}}^{\infty } \\ = - {{\left( {\frac{{{{E}_{\nu }}}}{{4\mathfrak{D}}}} \right)}^{2}}\left[ {1 - 4(\mathfrak{n} + 1)\left( {{{r}_{i}} + {{r}_{j}}} \right)} \right]. \\ \end{gathered} $$
((E.6b))

Here and elsewhere, the common definitions for oscillation lengths and differences between the neutrino masses squared are used:

$${{L}_{{ij}}} = \frac{{4\pi {{E}_{\nu }}}}{{\Delta m_{{ij}}^{2}}},\,\,\,\,\Delta m_{{ij}}^{2} = m_{i}^{2} - m_{j}^{2}\,\,\,(i \ne j).$$

It can be seen that the asymptotics at large \(L\), \({\text{Re}}\Phi _{{ij}}^{\infty }\), is almost independent of neutrino masses and factor \(\mathfrak{n}\); both \({\text{Re}}\Phi _{{ij}}^{0}\) and \({\text{Re}}\Phi _{{ij}}^{\infty }\) become arbitrarily large if \(\mathfrak{D}\) tends to zero. In addition, obvious relations

$$\begin{gathered} {{\mathfrak{L}}_{{ij}}} = \frac{\mathfrak{L}}{{\mathfrak{n}\left| {{{r}_{i}} \pm {{r}_{j}}} \right|}}\,\,\,\,{\text{and}}\,\,\,\,{\text{Re}}\Phi _{{ij}}^{0} \\ = \mathfrak{n}({{r}_{i}} - {{r}_{j}}){\text{Re}}\Phi _{{ij}}^{\infty }\left[ {1 + \mathcal{O}\left( {{{r}_{{i,j}}}} \right)} \right] \\ \end{gathered} $$

lead to model-independent inequalities

$$\mathfrak{L} \ll {{\mathfrak{L}}_{{ij}}}\,\,\,\,{\text{and}}\,\,\,\,\left| {{\text{Re}}\Phi _{{ij}}^{0}} \right| \ll \left| {{\text{Re}}\Phi _{{ij}}^{\infty }} \right|.$$

Therefore, as is seen from (E.4), a wide interval of distances \(\mathfrak{L} \ll L \ll {{\mathfrak{L}}_{{ij}}}\), where \(\left| {{\text{Re}}{{\Phi }_{{ij}}}} \right|\) increases quadratically with \(L\), exists for any pair of neutrinos \({{\nu }_{i}},{{\nu }_{j}}\):

$${\text{Re}}{{\Phi }_{{ij}}} \approx - {{\left( {\frac{1}{{{{v}_{i}}}} - \frac{1}{{{{v}_{j}}}}} \right)}^{2}}{{\mathfrak{D}}^{2}}{{L}^{2}} \approx - {{({{r}_{i}} - {{r}_{j}})}^{2}}{{\mathfrak{D}}^{2}}{{L}^{2}}.$$

All these features of behavior of the real part of phase are illustrated by Fig. 25, which presents function \(\log \left| {{\text{Re}}{{\Phi }_{{ij}}}} \right|\) within a wide range of \(L\) values, which vary from terrestrial (\( \gtrsim 100\) km) to cosmological (\( \sim {{10}^{4}}\) Mpc) distances, at several arbitrary values of dimensionless ratio \({\mathfrak{D} \mathord{\left/ {\vphantom {\mathfrak{D} {{{E}_{\nu }}}}} \right. \kern-0em} {{{E}_{\nu }}}}\). The curves in this figure were calculated for \({{m}_{i}} = 0.1\) eV, \({{m}_{j}} = 0.01\) eV, and Eν = 10 GeV. In order to estimate the value of important factor \(\mathfrak{n}\), we assumed, for illustrative purposes, that it is saturated by the contribution from pion decays (the typical source of accelerator, atmospheric, and astrophysical neutrinos) and that the contribution of reactions in the detector is negligible. It was also assumed that the pion WP in the momentum space is much wider than the muon packet; i.e., \({{\sigma }_{\mu }} \ll {{\sigma }_{\pi }}\). It is then easy to demonstrate that factor \(n\) has the following approximate form at ultrarelativistic energies, \({{\Gamma }_{\pi }} \gg 1\):

$$\mathfrak{n} \approx {{({{{{E}_{\nu }}} \mathord{\left/ {\vphantom {{{{E}_{\nu }}} {E_{\nu }^{ \star }}}} \right. \kern-0em} {E_{\nu }^{ \star }}})}^{2}} \approx {{({{{{E}_{\nu }}} \mathord{\left/ {\vphantom {{{{E}_{\nu }}} {29.8}}} \right. \kern-0em} {29.8}}\,\,{\text{MeV}})}^{2}} \approx 1.1 \times {{10}^{5}}.$$
Fig. 25.
figure25

Variation of function \(\log \left| {{\text{Re}}{{\Phi }_{{ij}}}} \right|\) with \(L\) (in kiloparsecs) calculated at \({{m}_{i}} = 0.1\) eV, \({{m}_{j}} = 0.01\) eV, and \({{E}_{\nu }} = 10\) GeV for seven values of ratio \({\mathfrak{D} \mathord{\left/ {\vphantom {\mathfrak{D} {{{E}_{\nu }}}}} \right. \kern-0em} {{{E}_{\nu }}}}\) (indicated next to the curves). Function \(n\) was estimated under the assumption that the dominant contribution to it is produced by the \({{\pi }_{{\mu 2}}}\) decay in the source.

The results of a similar analysis for the imaginary part of phase (E.4) reveal that this part increases linearly (although with different coefficients) with \(L\) in regions \(L \ll {{\mathfrak{L}}_{{ij}}}\) and \(L \gg {{\mathfrak{L}}_{{ij}}}\). In particular,

$${{\left. {{\text{Im}}{{\Phi }_{{ij}}}} \right|}_{{L \ll {{\mathfrak{L}}_{{ij}}}}}} \approx {\text{Im}}\Phi _{{ij}}^{{{\text{st}}}} = \frac{{2\pi L}}{{{{L}_{{ij}}}}}\left[ {1 + n\mathfrak{n}({{r}_{i}} + {{r}_{j}})} \right],$$
((E.7a))
$$\begin{gathered} {{\left. {{\text{Im}}{{\Phi }_{{ij}}}} \right|}_{{L \gg {{\mathfrak{L}}_{{ij}}}}}} \\ \approx {\text{Im}}\Phi _{{ij}}^{\infty } = \frac{{3\pi L}}{{2{{L}_{{ij}}}}}\left[ {1 + \frac{1}{3}(\mathfrak{n} + 1)({{r}_{i}} + {{r}_{j}})} \right]. \\ \end{gathered} $$
((E.7b))

These two regimes are separated by a relatively narrow transition region near \(L = {{\mathfrak{L}}_{{ij}}}\). Nothing unexpected occurs in the \(L \lesssim \mathfrak{L}\) region. This behavior of \({\text{Im}}{{\Phi }_{{ij}}}\) is illustrated by Fig. 26, which shows the \({{{\text{Im}}{{\Phi }_{{ij}}}} \mathord{\left/ {\vphantom {{{\text{Im}}{{\Phi }_{{ij}}}} {{\text{Im}}\Phi _{{ij}}^{{{\text{st}}}}}}} \right. \kern-0em} {{\text{Im}}\Phi _{{ij}}^{{{\text{st}}}}}}\) ratio calculated with the same approximations that were used for the real part of the phase. The unconventional behavior of function \({\text{Im}}{{\Phi }_{{ij}}}\) at \(L \gtrsim {{\mathfrak{L}}_{{ij}}}\) is of a purely academic interest, since it is rendered unmeasurable by an enormous suppression factor

$$ \propto {{\left( {1 + \mathfrak{r}_{{ij}}^{2}} \right)}^{{ - 1/4}}}\exp\left[ {{{ - E_{\nu }^{2}} \mathord{\left/ {\vphantom {{ - E_{\nu }^{2}} {(16{{\mathfrak{D}}^{2}})}}} \right. \kern-0em} {(16{{\mathfrak{D}}^{2}})}}} \right],$$

which eliminates interference terms (\(i \ne j\)) found in the expression for the event count rate. Thus, the oscillatory behavior of the count rate is encountered only at \(L \ll min({{\mathfrak{L}}_{{ij}}})\). This is exactly the region that was examined thoroughly in the main text.

Fig. 26.
figure26

Ratio \({{{\text{I}}{{{\text{m}}}_{{ij}}}} \mathord{\left/ {\vphantom {{{\text{I}}{{{\text{m}}}_{{ij}}}} {{\text{Im}}\Phi _{{ij}}^{{{\text{st}}}}}}} \right. \kern-0em} {{\text{Im}}\Phi _{{ij}}^{{{\text{st}}}}}}\) as a function of \(L\) calculated with the same parameters as those used for the real part of the phase (see the caption of Fig. 25) for three values of ratio \({\mathfrak{D} \mathord{\left/ {\vphantom {\mathfrak{D} {{{E}_{\nu }}}}} \right. \kern-0em} {{{E}_{\nu }}}}\) (indicated next to the curves).

APPENDIX F

SPATIAL AVERAGING

In realistic scenarios, the sizes of the source and detector are not always negligible compared to the distance between them. If this is the case, one needs to perform accurate spatial averaging of the count rate over the effective volumes of the source and detector. The plausible spatial inhomogeneities of the neutrino beam and the detector medium need to be taken into account. In the present section, we limit ourselves to the simple (but methodologically interesting) case when such inhomogeneities are negligible. This implies that density functions \({{\bar {f}}_{a}}({{{\mathbf{p}}}_{a}},{{s}_{a}};{\mathbf{x}})\) are independent of \({\mathbf{x}}\) within the source and detector volumes and turn to zero outside the bounds of these volumes. Within these approximations, we are interested only in \(L\)-dependent factor \({{{{e}^{{{{\Phi }_{{ij}}}(L)}}}} \mathord{\left/ {\vphantom {{{{e}^{{{{\Phi }_{{ij}}}(L)}}}} {{{L}^{2}}}}} \right. \kern-0em} {{{L}^{2}}}}\) (with \({{\Phi }_{{ij}}}(L) = i{{\varphi }_{{ij}}}(L) - {{\mathcal{A}}_{{ij}}}{{(L)}^{2}}\)) in the complete expression for the count rate; it is worth reminding that \(\mathcal{B}_{{ij}}^{2}\) and \({{\Theta }_{{ij}}}\) do not depend on \(L\).

The following simplification is also adopted below: the linear size of the detector in the direction of the neutrino beam is assumed to be negligible compared to the size of the source in the same direction, which, in turn, is small compared to the average source–detector distance. The origin of coordinates is located within the detector (point \({{O}_{d}}\) in Fig. 27), and axis \(z\) is codirectional with unit vector \( - {\mathbf{l}}\) (i.e., directed toward a certain inner point \({{O}_{s}}\) of the source). Let \({\mathbf{x}} = L{\mathbf{l}}\) and \(L = {{O}_{s}}{{O}_{d}}\); the sought-for integral over the spatial volume of the source may then be written as follows:

$${{\mathcal{J}}_{{ij}}} \equiv \int\limits_{{{{\text{V}}}_{s}}} {\frac{{d{\mathbf{x}}}}{{{{L}^{2}}}}} {{e}^{{{{\Phi }_{{ij}}}(L)}}} = \int\limits_{{{\Omega }_{s}}} {d\Omega } \int\limits_{L_{\Omega }^{N}}^{L_{\Omega }^{F}} {dL{{e}^{{{{\Phi }_{{ij}}}(L)}}}} .$$
((F.1))
Fig. 27.
figure27

Schematic illustration of spatial averaging.

Here, \({{{\text{V}}}_{s}}\) is the working volume of the source, \({{\Omega }_{s}}\) is the solid angle under which this volume is seen from point \({{O}_{d}}\), and \(L_{\Omega }^{N}\) and \(L_{\Omega }^{F}\) are the distances from \({{O}_{d}}\) to the near and far boundaries of the source (for simplicity, it is assumed to be convex) along unit vector \(\Omega = \left( {\sin\phi \sin\theta ,\cos\varphi \sin\theta ,cos\theta } \right)\). It is convenient to introduce the following measure of distance between the source and detector:

$$\bar {L} = \frac{1}{{2{{\Omega }_{s}}}}\int\limits_{{{\Omega }_{s}}} {d\Omega } \left( {L_{\Omega }^{F} + L_{\Omega }^{N}} \right).$$

To avoid any misunderstanding, we note that, depending on the angular resolution of the detector and features of the experimental setup, solid angle \({{\Omega }_{s}}\) may turn out to be smaller than the full solid angle under which the source is seen. Figure 27 illustrates this possibility schematically, while equality (F.1) holds true in the general case. Note also that the smallness of angle \({{\Omega }_{s}}\) is not equivalent to the smallness of the source itself; a well-collimated meson beam from an accelerator (neutrino factory) in a long decay channel is an important counterexample. Elementary integration over \(L\) yields

$$\begin{gathered} {{\mathcal{J}}_{{ij}}} = \frac{{{{E}_{\nu }}{{L}_{{ij}}}}}{{4\sqrt \pi D}}\int\limits_{{{\Omega }_{s}}} {d\Omega } \\ \times \,\,\left[ {{\text{erf}}\left( {\frac{{2\pi DL_{\Omega }^{F}}}{{{{E}_{\nu }}{{L}_{{ij}}}}} - \frac{{i{{E}_{\nu }}}}{{2D}}} \right) - {\text{erf}}\left( {\frac{{2\pi DL_{\Omega }^{N}}}{{{{E}_{\nu }}{{L}_{{ij}}}}} - \frac{{i{{E}_{\nu }}}}{{2D}}} \right)} \right] \\ \times \,\,\exp\left[ { - \frac{{2E_{\nu }^{2} + {{{(\Delta {{E}_{{ij}}})}}^{2}}}}{{8{{D}^{2}}}}} \right]\,\,\,(i \ne j), \\ \end{gathered} $$
((F.2a))

and

$${{\mathcal{J}}_{{jj}}} = \int\limits_{{{{\text{V}}}_{s}}} {\frac{{d{\mathbf{x}}}}{{{{L}^{2}}}}} = \int\limits_{{{\Omega }_{s}}} {d{\boldsymbol{\mathbf{\Omega }}}} \left( {L_{\Omega }^{F} - L_{\Omega }^{N}} \right).$$
((F.2b))

These general formulas may be used in processing the data of short-baseline neutrino experiments, where the distance from the source (e.g., a decay channel) to the detector is comparable to the longitudinal size of the source.

In an ideal experiment with

$$\begin{gathered} {{r}_{N}} = \mathop {max}\limits_{\Omega \in {{\Omega }_{s}}} \left( {\bar {L} - L_{\Omega }^{N}} \right) \ll \bar {L} \\ {\text{and}}\,\,\,\,{{r}_{F}} = \mathop {max}\limits_{\Omega \in {{\Omega }_{s}}} \left( {L_{\Omega }^{F} - \bar {L}} \right) \ll \bar {L}, \\ \end{gathered} $$
((F.3))

one can use the following expansion of the error function:

$$\begin{gathered} {\text{erf}}(z + \delta ) \approx {\text{erf}}(z) \\ + \,\,\frac{{2\delta }}{{\sqrt \pi }}{{e}^{{ - {{z}^{2}}}}}\left[ {1 - z\delta + \frac{2}{3}(2{{z}^{2}} - 1){{\delta }^{2}} + \ldots } \right]. \\ \end{gathered} $$
((F.4))

The contributions of order \(\mathcal{O}({{\delta }^{2}})\) and \(\mathcal{O}({{z}^{2}}{{\delta }^{2}})\) in this expansion may be neglected under the assumption that \(\left| \delta \right| \ll 1\) and \(\left| {z\delta } \right| \ll 1\). In the case under consideration, the first condition implies that

$$\frac{{2\pi D{{r}_{{N,F}}}}}{{{{E}_{\nu }}{{L}_{{ij}}}}} \ll 1,$$
((F.5))

while the second one is not necessary due to the approximate cancellation of terms of second order. Using (F.4), we indeed find that

$$\begin{gathered} {{\mathcal{J}}_{{ij}}} \approx \int\limits_{{{\Omega }_{s}}} {d{\boldsymbol{\mathbf{\Omega }}} } \left( {L_{\Omega }^{F} - L_{\Omega }^{N}} \right){{e}^{{{{\Phi }_{{ij}}}(\bar {L})}}} \\ \times \,\,\left\{ {1 - {{\Delta }_{\Omega }}\left[ {\frac{{2i\pi \bar {L}}}{{{{L}_{{ij}}}}} - \mathop {\left( {\frac{{2\pi D\bar {L}}}{{{{E}_{\nu }}{{L}_{{ij}}}}}} \right)}\nolimits^2 } \right]} \right\}, \\ \end{gathered} $$

where

$${{\Delta }_{\Omega }} = 2\left( {1 - \frac{{L_{\Omega }^{N} + L_{\Omega }^{F}}}{{2\overline L }}} \right) = \frac{{L_{\Omega }^{F} - \bar {L}}}{{\overline L }} - \frac{{\bar {L} - L_{\Omega }^{N}}}{{\overline L }}.$$

Evidently, \(\left| {{{\Delta }_{\Omega }}} \right| \ll 1\). Assuming that

$$\mathop {max}\limits_{\Omega \in {{\Omega }_{s}}} {{\Delta }_{\Omega }}\mathop {\left[ {{{{\left( {\frac{{2\pi D\bar {L}}}{{{{E}_{\nu }}{{L}_{{ij}}}}}} \right)}}^{4}} + {{{\left( {\frac{{2\pi \bar {L}}}{{{{L}_{{ij}}}}}} \right)}}^{2}}} \right]}\nolimits^{1/2} \ll 1,$$
((F.6))

we arrive at the following result (valid for any \(i\) and \(j\)):

$${{\mathcal{J}}_{{ij}}} \approx {{e}^{{{{\Phi }_{{ij}}}(\bar {L})}}}\int\limits_{{{\Omega }_{s}}} {d{\boldsymbol{\mathbf{\Omega }}} } \left( {L_{\Omega }^{F} - L_{\Omega }^{N}} \right) \approx {{{\text{V}}}_{s}}\frac{{{{e}^{{{{\Phi }_{{ij}}}(\bar {L})}}}}}{{{{{\bar {L}}}^{2}}}},$$
((F.7))

which could be deduced from the mean-value theorem. Our result, however, is supplemented by fairly nontrivial sufficient conditions (F.3), (F.5), and (F.6), which are impossible to obtain from the mean-value theorem alone. Volume \({{{\text{V}}}_{s}}\) in (F.7) is estimated (with the same accuracy) as

$$\begin{gathered} {{{\text{V}}}_{s}} = \int\limits_{{{{\text{V}}}_{s}}} {d{\mathbf{x}}} = \frac{1}{3}\int\limits_{{{\Omega }_{s}}} {d{\boldsymbol{\mathbf{\Omega }}} } \left[ {{{{\left( {L_{\Omega }^{F}} \right)}}^{3}} - {{{\left( {L_{\Omega }^{N}} \right)}}^{3}}} \right] \\ \approx {{{\bar {L}}}^{2}}\int\limits_{{{\Omega }_{s}}} {d{\boldsymbol{\mathbf{\Omega }}} } (L_{\Omega }^{F} - L_{\Omega }^{N}). \\ \end{gathered} $$

If we now assume that working (reference) detector volume \({{{\text{V}}}_{d}}\) is sufficiently small compared to \({{{\text{V}}}_{s}}\) (this condition is satisfied in a typical neutrino experiment) and that the detector geometry is not too fancy, the integration over \({\mathbf{y}}\) becomes trivial and yields

$$\int\limits_{{{{\text{V}}}_{d}}} {d{\mathbf{y}}} \int\limits_{{{{\text{V}}}_{s}}} {d{\mathbf{x}}} \frac{{{{e}^{{{{\Phi }_{{ij}}}(L)}}}}}{{{{L}^{2}}}} \approx {{{\text{V}}}_{s}}{{{\text{V}}}_{d}}\frac{{{{e}^{{{{\Phi }_{{ij}}}(\bar {L})}}}}}{{{{{\bar {L}}}^{2}}}},$$
((F.8))

where \(\bar {L}\) still denotes the conventional measure of distance between the source and detector.

In order to illustrate the importance and nontrivial nature of conditions (F.3), (F.5), and (F.6), we consider the simplest case of a spheroidal source with radius \(r\). Its angular size \({{\theta }_{s}}\) does not exceed the angular resolution of the detector. It follows from simple geometric considerations that \({{\Delta }_{\Omega }} = 2(1 - \cos\theta )\) and, naturally, \({{r}_{N}} = {{r}_{F}} = r\). Therefore,

$$\mathop {max}\limits_{\Omega \in {{\Omega }_{s}}} {{\Delta }_{\Omega }} = {{\Delta }_{{{{\Omega }_{s}}}}} = 2\left( {1 - \cos{{\theta }_{s}}} \right) \approx \theta _{s}^{2} \approx {{\left( {{r \mathord{\left/ {\vphantom {r {\bar {L}}}} \right. \kern-0em} {\bar {L}}}} \right)}^{2}}.$$
((F.9))

To simplify the scenario further, we assume that \(2\pi \bar {L} \gg \left| {{{L}_{{ij}}}} \right|\) (this inequality is always fulfilled for solar and astrophysical neutrinos detected on the Earth) and \(2\pi \bar {L} \lesssim \left| {{{L}_{{ij}}}} \right|{{{{E}_{\nu }}} \mathord{\left/ {\vphantom {{{{E}_{\nu }}} D}} \right. \kern-0em} D}\) (this is not always true for remote astrophysical sources, but is valid for the Sun). Condition (F.5) is then satisfied automatically, while condition (F.6) takes the form

$$\frac{{2\pi {{r}^{2}}}}{{\bar {L}\left| {{{L}_{{ij}}}} \right|}} \ll 1.$$
((F.10))

This inequality is definitely not fulfilled for the Sun and the currently adopted value of \(\Delta m_{{12}}^{2}\). The regions of efficient neutrino production in the solar core are relatively thin concentric layers with typical radii varying from \(0.1{{R}_{ \odot }}\) for 8B, 7Be, and CNO neutrinos to \(0.3{{R}_{ \odot }}\) for \(pp\), \(pep\), and \(hep\) neutrinosFootnote 71 (\({{R}_{ \odot }}\) is the solar radius). The left-hand side of (F.10) may then be estimated as

$$\frac{{2\pi {{r}^{2}}}}{{\bar {L}\left| {{{L}_{{12}}}} \right|}} \approx 26{{\left( {\frac{r}{{0.2{{R}_{ \odot }}}}} \right)}^{2}}\left( {\frac{{\left| {\Delta m_{{12}}^{2}} \right|}}{{8 \times {{{10}}^{{ - 5}}}\,\,{\text{e}}{{{\text{V}}}^{2}}}}} \right)\left( {\frac{{1\,\,{\text{MeV}}}}{{{{E}_{\nu }}}}} \right),$$

which demonstrates (and this is no news) that approximation (F.8) is definitely not applicable in the analysis of oscillations of solar neutrinos.

Summarizing the results, we note the following: although the above analysis was simplified in many respectsFootnote 72, it demonstrates that the finite size of the source needs to be taken into account to process data correctly in both short-baseline oscillation experiments and very-long-baseline experiments (including the ones with \(L\) on the order of an astronomical unit, which is the case with experiments involving solar neutrinos).

APPENDIX G

COMPLEX ERROR FUNCTION AND ASSOCIATED FORMULAS

The error function and the complementary error function of complex argument have already been studied extensively (see, e.g., [146150] and references therein). Let us list here some results that were used in Section 11 to analyze the decoherence function. For convenience, we first write the following well-known formulas [103]:

$${\text{erf}}(z) = \frac{2}{{\sqrt \pi }}\sum\limits_{n = 0}^\infty {\frac{{{{{( - 1)}}^{n}}{{z}^{{2n + 1}}}}}{{(2n + 1)n!}}} = \frac{2}{{\sqrt \pi }}{{e}^{{ - {{z}^{2}}}}}\sum\limits_{n = 0}^\infty {\frac{{{{2}^{n}}{{z}^{{2n + 1}}}}}{{(2n + 1)!!}}} ,$$
((G.1))
$$\begin{gathered} {\text{erfc}}(z) \sim \frac{{{{e}^{{ - {{z}^{2}}}}}}}{{\sqrt \pi z}}\left[ {1 + \sum\limits_{n = 1}^\infty {\frac{{{{{( - 1)}}^{n}}(2n - 1)!!}}{{{{{(2{{z}^{2}})}}^{n}}}}} } \right] \\ \left( {z \to \infty ,\left| {{\text{arg}}z} \right| < \frac{{3\pi }}{4}} \right). \\ \end{gathered} $$
((G.2))

The following expansions for function \({\text{Ierf}}(z)\), which was introduced in the main text, are derived from (306) and (G.1):

$${\text{Ierf}}(z) = \frac{1}{{\sqrt \pi }}\left[ {1 + {{z}^{2}} - \frac{{{{z}^{4}}}}{6} + \frac{{{{z}^{6}}}}{{30}} - \frac{{{{z}^{8}}}}{{168}} + \mathcal{O}({{z}^{{10}}})} \right],$$
((G.3a))
$$ = \frac{{{{e}^{{ - {{z}^{2}}}}}}}{{\sqrt \pi }}\left[ {1 + 2{{z}^{2}} + \frac{{4{{z}^{4}}}}{3} + \frac{{8{{z}^{6}}}}{{15}} + \frac{{16{{z}^{8}}}}{{105}} + \mathcal{O}({{z}^{{10}}})} \right].$$
((G.3b))

These expansions are useful for small and intermediate values of \(\left| z \right|\)Footnote 73, respectively. In order to determine the asymptotics of functions \({\text{erfc}}(z)\) and \({\text{Ierf}}(z)\) at large \(\left| z \right|\), one needs to use (G.2) for \({\text{erfc}}( - z)\) and then apply the following rule:

$${\text{erfc}}(z) = 2 - {\text{erfc}}( - z).$$

The result derived from (306) and (G.2) is

$$\begin{gathered} {\text{Ierf}}(z) \sim \pm z + \frac{{{{e}^{{ - {{z}^{2}}}}}}}{{2\sqrt \pi {{z}^{2}}}} \\ \times \,\,\left[ {1 - \frac{3}{{2{{z}^{2}}}} + \frac{{15}}{{4{{z}^{4}}}} - \frac{{105}}{{8{{z}^{6}}}} + \mathcal{O}\left( {\frac{1}{{{{z}^{8}}}}} \right)} \right]\,\,\,\,\left( {z \to \infty } \right), \\ \end{gathered} $$
((G.4))

where the upper (lower) sign should be taken for \(\left| {{\text{arg}}z} \right| < {{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right. \kern-0em} 4}\) (\(\left| {{\text{arg}}z} \right| > {{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right. \kern-0em} 4}\)).

The formulas given below are helpful in high-accuracy numerical calculations of the error function. They are based on the following integral representation of the complementary error function (see, e.g., [146149]):

$${\text{erfc}}(z) = \frac{{2z}}{\pi }\int\limits_0^\infty {\frac{{dt{{e}^{{ - ({{t}^{2}} + {{z}^{2}})}}}}}{{{{t}^{2}} + {{z}^{2}}}}} .$$

From this it follows that (cf. the result in [147])

$$\begin{gathered} {\text{Re}}\left[ {{\text{erfc}}(a + ib)} \right] = + \frac{r}{\pi }\exp\left[ { - {{r}^{2}}\cos(2\omega )} \right] \\ \times \,\,\left[ {{{r}^{2}}\cos({{w}_{ + }}){{\mathcal{I}}_{0}}(a,b) + \cos({{w}_{ - }}){{\mathcal{I}}_{2}}(a,b)} \right], \\ {\text{Im}}\left[ {{\text{erfc}}(a + ib)} \right] = - \frac{r}{\pi }\exp\left[ { - {{r}^{2}}\cos(2\omega )} \right] \\ \times \,\,\left[ {{{r}^{2}}\sin({{w}_{ + }}){{\mathcal{I}}_{0}}(a,b) + \sin({{w}_{ - }}){{\mathcal{I}}_{2}}(a,b)} \right], \\ \end{gathered} $$

where

$$\begin{gathered} {{\mathcal{I}}_{n}}(a,b) = \int\limits_{ - \infty }^\infty {\frac{{dt{{t}^{n}}{{e}^{{ - {{t}^{2}}}}}}}{{{{{({{t}^{2}} + {{a}^{2}} - {{b}^{2}})}}^{2}} + 4{{a}^{2}}{{b}^{2}}}}} \\ = \int\limits_{ - \infty }^\infty {\frac{{dt{{t}^{n}}{{e}^{{ - {{t}^{2}}}}}}}{{{{{\left[ {{{t}^{2}} + {{r}^{2}}\cos(2\omega )} \right]}}^{2}} + {{{\left[ {{{r}^{2}}\sin(2\omega )} \right]}}^{2}}}}} , \\ {{w}_{ \pm }} = 2ab + \omega ,\,\,\,\,r = \sqrt {{{a}^{2}} + {{b}^{2}}} , \\ \cos\omega = {a \mathord{\left/ {\vphantom {a r}} \right. \kern-0em} r},\sin\omega = {b \mathord{\left/ {\vphantom {b r}} \right. \kern-0em} r}. \\ \end{gathered} $$

All the quantities found here are real. Note that the integrands in integrals \({{\mathcal{I}}_{n}}(a,b)\) are positive, nonsingular (with the exception of the trivial case of \(a = b = 0\)), and decrease rapidly at large \(\left| t \right|\). These properties are helpful in numerical integration based on the standard quadrature formulas.

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Naumov, D.V., Naumov, V.A. Quantum Field Theory of Neutrino Oscillations. Phys. Part. Nuclei 51, 1–106 (2020). https://doi.org/10.1134/S1063779620010050

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