On the Treatment of Neutrino Oscillations Without Resort to Weak Eigenstates

C. Giunti; C.W. Kim; J.A. Lee

REVIEW

On the Treatment of Neutrino Oscillations Without Resort to Weak Eigenstates

C. Giunti, C.W. Kim, J.A. Lee

High Energy Physics - Phenomenology · Submitted 17 May 1993 · 10 references

Pith's verdict: A self-contained QFT derivation that delivers the factorization claim, marred only by a dimensional slip in the coherence-length summary formulas (26)-(27).

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arXiv IDhep-ph/9305276 (version 1)

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CitationC. Giunti, C.W. Kim, J.A. Lee. "On the Treatment of Neutrino Oscillations Without Resort to Weak Eigenstates." arXiv:hep-ph/9305276 (1993). https://pith.science/paper/VPJ3G7V2

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Editor's note

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The thing to know up front: this is the field-theoretic version of the wave-packet story this group has been telling, and it delivers what the abstract promises. They build the amplitude for PI → PF μ+ ν, ν + DI → DF e− with Gaussian wave-packet external particles, deform the q0 contour to pick the neutrino pole, saddle-point the three-momentum, and time-average at fixed baseline. The result, Eq. (20), genuinely does not factor into a common piece times a distance-dependent probability: the prefactors A_a A_b*, the energy-conservation factors exp[-S_a], and the coherence damping all carry mass-eigenstate labels. Then the ultra-relativistic expansion of Eq. (22) makes the mass dependence common to leading order, and out comes Eq. (23), whose first line is the textbook oscillation probability and whose damping term reproduces the coherence length of [7] and the intuition of [1,6,10]. That cross-check is the real content, and it is done properly. Relative to [3] and [7], neither of which computed a production-to-detection cross section with wave-packet external states, time-averaged it, or derived L_coh from the cross section rather than from QM propagation, this is a real step forward. The self-citations are published PRD papers with independent content; no circularity here. Now the soft spots. The one concrete error is in the summary formulas. From their own Eqs. (24) and (25), L_coh/L_osc ~ q0z σ_x/(2π), so Eq. (26) should read L_coh ~ q0z σ_x L_osc and Eq. (27) N_osc ~ q0z σ_x ~ q0z/(2σ_p). As printed, q0z σ_p has dimensions of momentum squared, and the sentence 'σ_p ≪ q0z and N_osc ≫ 1' only connects if σ_p sits in the denominator. It is a typo in a secondary formula, not a flaw in the factorization argument, but Eq. (27) is the paper's quantitative decoherence statement, so it should be corrected. Beyond that: the common-width Gaussian of Eq. (29) is doing a lot of work, and the saddle-point truncations, including dropping F^ab(L), come with no error estimates. The parametric scaling σ_x E²/Δm² is robust; the numerical coefficient is not. The authors acknowledge the single-process scope in Section 1, and I do not read abstract:s3 as a general theorem. Who should read it: anyone working on neutrino decoherence, wave-packet treatments, or the domain of validity of the standard oscillation formula. It is a foundational reference in that literature. Recommendation: send it out. The derivation is self-contained and the central claim survives; a referee should insist on fixing Eqs. (26)-(27) and on at least a sentence about how L_coh changes for unequal widths.

Referee report

Major objections (2)

[§2, Eq. (12)] In passing from the propagator of Eq. (6) to the residue at the neutrino pole, the term proportional to m_a in q/+m_a is dropped without comment. Since the paper's stated motivation is the regime in which some mass eigenstates are not extremely relativistic (Introduction and §1), m_a/E_a is not small there and this term is not suppressed. It should be retained or its omission explicitly justified. This does not threaten the factorization claim, which rests on the L-dependent phase, but it does affect the quantitative cross section and the prefactor A_a.
[§4, Eqs. (26)-(27)] The coherence length and number of oscillations are written as L_coh ~ q_0z σ_p L_osc and N_osc ~ q_0z σ_p. With σ_p the momentum width of Eq. (29) and σ_x σ_p = 1/2 (Eq. 33), q_0z σ_p has dimensions of momentum squared, not length or dimensionless. The correct expressions following from Eqs. (24)-(25) are L_coh ~ q_0z σ_x L_osc and N_osc ~ q_0z σ_x = q_0z/(2σ_p). The sentence 'Since we have assumed that σ_p ≪ q_0z and N_osc ≫ 1' is only consistent with the corrected form. This is a correctable error in a secondary result, but it should be fixed before publication.

Minor objections (7)

[§4, Eq. (23)] The coefficient of the L^2 damping term in the second line should be checked against Eq. (20); with v_a → 1 and v_b → 1 the denominator of the damping exponent becomes 2[Ω_0]_zz, which appears to introduce a factor of order unity relative to the coefficient written in Eq. (23).
[Abstract] The statement that the usual oscillation probability 'can be factored out' should be qualified by the coherence condition L ≪ L_coh, as the text does in §4; even in the relativistic limit the factorization holds only up to the wave-packet damping factor, which reduces to unity only for distances shorter than the coherence length.
[Appendix A, Eq. (29)] The common-width Gaussian ansatz for all external particles is a strong simplification. A comment on how the coherence length of Eq. (24) would change for unequal widths or non-Gaussian profiles would strengthen the paper and clarify the generality of the quantitative coherence criterion.
[§3, Eq. (20)] The neglect of F^ab(L) is justified by its becoming constant for large L, but since it is a≠b dependent it could affect the relative phases between mass eigenstates. A more explicit justification, or a demonstration that the residual a,b dependence is higher order in the relativistic expansion, would be helpful.
[§4, Eq. (28)] The '~' between (m_a^2-m_b^2)/q_0z and 1/L_osc^ab hides a factor of 4π. Please state the numerical factor or replace '~' with '∝' to avoid confusion.
[§1] The paper acknowledges that a general formula cannot be derived and that a specific process is used as an example. A sentence on the robustness of the conclusions to the choice of process, beyond the relativistic limit, would help the reader assess the scope of the result.
[Manuscript] There are some spacing and typographical issues from the arXiv source (for example, 'e xternal', 't he', 'neu trino'); these should be cleaned up in the final version.

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The paper

The paper claims

Computing neutrino oscillations in full quantum field theory, without ever introducing 'weak eigenstates', the authors show that the familiar oscillation probability can be factored out of the measurable cross section only in the extremely

The reading

The paper asks what a neutrino oscillation experiment actually measures if you refuse to introduce 'weak eigenstates' — the flavour superpositions of the textbook treatment — and instead compute the entire production-propagation-detection process in quantum field theory. The external particles are described by localized wave packets, the production and detection interactions sit at a macroscopic separation, and the different mass-eigenstate neutrinos travel between them. The measurable quantity is the cross section averaged over the time separation, which experiments do not record, and the authors compute it. The result is that the standard oscillation probability can be pulled out of that cross section only in the extremely relativistic limit; away from that limit the cross section carries mass-eigenstate-dependent prefactors and coherence factors that do not separate into a common cross section times a phase. The payoff is that the textbook oscillation formula becomes a derived approximation with a stated domain of validity, together with an explicit coherence length beyond which the interference that produces oscillations dies away.

What carries the argument

The carrier of the argument is the wave-packet amplitude of Eq. (15), $A propto sum_a U^*_{mu a} U_{e a} A_a exp[-iE_a T + i q_a cdot L - S_a( q_a) - (1/2)( L - v_a T) v_a^{-1} ( L - v_a T)]$, obtained by deforming the $q_0$ contour to pick up the neutrino pole and then saddle-pointing over $vec q$ and over the angles. The matrix $v_a$, defined in Eq. (16), is built from the velocities of the production and detection particles and is proportional to $1/sigma_p^2$, so it acts as a coherence filter: the amplitude survives only when $| L - v_a T| sim< sigma_x$. Averaging over the unmeasured time $T$ turns that filter into the distance-dependent damping of Eq. (20), and the coherence length of Eq. (21) follows directly from it. The machinery is what lets the authors compare mass eigenstates on unequal footing — each has its own saddle point $q_a$, its own velocity, and its own overlap with the source and detector — which is precisely why the oscillation phase cannot be pulled out in general.

Extended reading notes

Core claim

The paper establishes that neutrino oscillations are not, in general, a property that can be attributed to a propagating flavour state, but rather a feature of a cross section for a specific pair of weak processes. Working with a concrete flavour-changing chain, $P_I + D_I$ producing a muon and a neutrino that is later absorbed to make an electron, the authors keep the full momentum dependence of the external wave packets and evaluate the amplitude by a sequence of saddle-point approximations. The resulting time-averaged cross section, their Eq. (20), is a sum over mass-eigenstate pairs with three ingredients that have no common factor: mass-dependent prefactors $A_a A_b^*$, energy-conservation suppression $e^{-S_a( q_a) - S_b( q_b)}$, and a Gaussian damping $e^{-(L^2/2)(v_{az}-v_{bz})^2/[ (v_{az}^2 [v^{-1}]_{zz} + v_{bz}^2 [v^{-1}]_{zz})]}$ that enforces coherence of the two contributions along the line of sight. Only after expanding $q_{az}$, $E_a$ and $v_{az}$ to leading order in $m_a^2/q_{0z}$ does the cross section factorize as $d(L) = P(L) d_0$, with $P(L)$ given by Eq. (23), whose leading line is exactly the usual oscillation probability. Two consequences are drawn: a coherence length $L^{ab}_{ ext{coh}} sim sigma_x, 2q_{0z}/(m_a^2 - m_b^2)$ beyond which the two mass eigenstates contribute incoherently, and a suppression of the interference unless $|E_a - E_b| sim< sigma_p$, i.e. unless energy is conserved within the wave-packet width at both vertices.

Load-bearing premise

Every external particle in the production and detection processes is given the same Gaussian momentum width $sigma_p$, and the contour deformation, the momentum saddle point and the angular saddle point that drops the residual $F^{ab}(L)$ factor all inherit that ansatz, with no demonstration that the coherence length or the factorization condition survives unequal widths or non-Gaussian profiles.

Editorial extensions

If this is right

In the limit $m_a ll E_a$ the cross section factorizes and the standard quantum-mechanical oscillation probability is recovered exactly, so existing relativistic analyses are not disturbed.
Oscillations are damped beyond a coherence length $L^{ab}_{ ext{coh}} sim sigma_x, 2q_{0z}/|m_a^2 - m_b^2|$; the paper writes the number of oscillations visible before decoherence as $N_{ ext{osc}} sim q_{0z} sigma_p$, independent of the neutrino masses.
If the velocity difference satisfies $|v_a - v_b| T sim> sigma_x$, the two mass-eigenstate wave packets separate in space, no longer overlap, and the experiment sees only a constant flavour-changing probability rather than an oscillating one.
Interference between mass eigenstates is suppressed unless $|E_a - E_b| sim< sigma_p$, so the oscillation signal is controlled by energy-momentum conservation holding at both the production and the detection vertex within the wave-packet width.
Because the prefactors $A_a$ differ between mass eigenstates, a non-relativistic neutrino source would give a flavour-conversion rate that depends on the details of the production and detection reactions, not only on mixing angles and masses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Read as an experimental statement, the coherence length turns 'which mass eigenstate is the neutrino in' from a philosophical question into a knob: a tightly localized source and detector shorten $L_{ ext{coh}}$, while a loose one lengthens it, so the same mixing parameters can look oscillating or decohered depending on apparatus design.
Since the non-factorizable prefactors depend on the production channel, comparing flavour-conversion rates from two different neutrino sources (for example pion decay versus beta decay) at the same baseline would in principle expose the residual mass dependence the standard formula drops.
The suppression condition $|E_a - E_b| sim< sigma_p$ suggests the relevant uncertainty is that of the whole apparatus rather than of the source alone, which connects this treatment to the more phenomenological 'energy-resolution decoherence' often invoked for reactor and atmospheric data.
If the factorization genuinely fails outside the relativistic regime, a strictly correct analysis of such data would fit the full cross section of Eq. (20) rather than a product of flux, cross section and oscillation probability, which is a different statistical model from the one usually assumed.

Original abstract

We discuss neutrino oscillations in the framework of the quantum field theory without introducing the concept of neutrino weak eigenstates. The external particles are described by wave packets and the different mass eigenstate neutrinos propagate between the production and detection interactions, which are macroscopically localized in space-time. The time-averaged cross section, which is the measurable quantity in the usual experimental setting, is calculated. It is shown that only in the extremely relativistic limit the usual quantum mechanical oscillation probability can be factored out of the cross section.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.64 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss neutrino oscillations in the framework of the quantum field theory without introducing the concept of neutrino weak eigenstates. The external particles are described by wave packets and the different mass eigenstate neutrinos propagate between the production and detection interactions, which are macroscopically localized in space-time. The time-averaged cross section, which is the measurable quantity in the usual experimental setting, is calculated. It is shown that only in the extremely relativistic limit the usual quantum mechanical oscillation probability can be factored out of the cross section.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss neutrino oscillations in the framework of the quantum field theory without introducing the concept of neutrino weak eigenstates.	contribution 0.95	0.89	0.10	asserted 0.87
s1	The external particles are described by wave packets and the different mass eigenstate neutrinos propagate between the production and detection interactions, which are macroscopically localized in space-time.	method 0.99	0.52	0.05	asserted 0.99
s2	The time-averaged cross section, which is the measurable quantity in the usual experimental setting, is calculated.	method 0.72	0.90	0.07	asserted 0.96
s3	It is shown that only in the extremely relativistic limit the usual quantum mechanical oscillation probability can be factored out of the cross section.	result 0.97	0.90	0.62	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (10)

[7]

C. Giunti, C.W. Kim and U.W. Lee, Phys. Rev. D 44 (1991) 3635

View on Pith 1991
[3]

C. Giunti, C.W. Kim and U.W. Lee, Phys. Rev. D 45 (1992) 2414

View on Pith 1992
[1]

S.M. Bilenky and B. Pontecorvo, Phys. Rep. 41 (1978) 225

View on Pith 1978
[2]

S.M. Bilenky and S.T. Petcov, Rev. Mod. Phys. 59 (1987) 671; See, for a recent review, C. W. Kim and A. Pevsner, Neutrinos in Physics and Astrophy sics, Con- temporary Concepts in Physics, Vol. 8, Harwood Academic Press, 1 993

View on Pith 1987
[4]

R.G. Winter, Lett. Nuovo Cimento 30 (1981) 101

View on Pith 1981
[5]

See, for example, R.E. Shrock, Phys. Lett. 96B (1980) 159; Phys. Rev. D24 (1981) 1232,1275

View on Pith 1980
[6]

B. Kayser, Phys. Rev. D 24 (1981) 110

View on Pith 1981
[8]

P.H. Frampton and P. Vogel, Phys. Rep. 82 (1982) 339

View on Pith 1982
[9]

F. Boehm and P. Vogel, Physics of Massive Neutrinos , Cambridge University Press, Cambridge, 1987

View on Pith 1987
[10]

S. Nussinov, Phys. Lett. B 63 (1976) 201. 18

View on Pith 1976

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Real and Virtual Propagation in Neutrino Oscillations
hep-ph 2026-06 unverdicted novelty 6.0 of 10

In Gaussian wave-packet QFT, flavor oscillations switch on only after a propagation-time threshold set by the wave-packet energy uncertainty and the intermediate particle's decay width; below it the neutrino is purely...
Oscillations of neutrinos and mesons in quantum field theory
hep-ph 2001-09 accept novelty 6.0 of 10

A single external wave packet framework in quantum field theory reproduces stationary oscillation models as a limit and yields a unified oscillation formula for stable and unstable neutrinos and mesons, including deco...
Quantum field theory treatment of neutrino flavor oscillations in matter
hep-ph 2024-11 conditional novelty 5.0 of 10

The virtual-particle QFT formalism with exact matter propagators for Majorana neutrinos reproduces the standard MSW oscillation probability in uniform matter.
Neutrino oscillations: Quantum mechanics vs. quantum field theory
hep-ph 2010-01 accept novelty 5.0 of 10

The QM neutrino wave packet parameters are derived from QFT, and the correctly normalized oscillation probability is shown to follow from QFT without an ad hoc unitarity imposition.
Paradoxes of neutrino oscillations
hep-ph 2009-05 accept novelty 5.0 of 10

The standard neutrino oscillation formula follows without the 'same energy' or 'same momentum' assumptions whenever production/detection and propagation coherence hold, and a reordered summation convention explains wh...

Show 2 more citing papers

Quantum field theory treatment of the spin-flavor precession of Dirac neutrinos with diagonal magnetic moments
hep-ph 2026-11 conditional novelty 4.0 of 10

For ultrarelativistic Dirac neutrinos with diagonal magnetic moments, an exact-propagator QFT calculation reproduces the standard quantum-mechanical spin-flavor precession probability at leading order in source-detect...
keV Neutrino Model Building
hep-ph 2013-02 accept novelty 2.0 of 10

The review organizes the model-building literature for keV sterile neutrino dark matter, showing which mechanisms can produce the keV scale and which models survive the X-ray, seesaw, and neutrino-oscillation constraints.

How the literature uses it

33 arXiv papers cite this in 49 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 31 sentences without a confident label

Contests it 1 paper

This differs from statements frequently found in the literature, e.g., Refs. [ref] , where it is said that all neutrino mass eigenstates are produced at the moment the source decays, and the different mass eigenstates propagate with different velocities in the lab frame.

Neutrino Oscillation Measurements Computed in Quantum Field Theory · Discussion and Conclusions · contest 0.79

Uses its method or tool 4 papers

Localization is implemented by assigning Gaussian wave packets to all external lines and performing the calculation accordingly [ref] .

Real and Virtual Propagation in Neutrino Oscillations · External Wave Packet Model · 0.85
In this paper we contribute to this discussion by adopting the quantum field theoretical (QFT) approach to neutrino oscillations [ref] .

Decoherence effects in reactor and Gallium neutrino oscillation experiments -- a QFT approach · Introduction · 0.90
The idea is to treat the neutrino production and detection together as a single process [ref] :

Consistent QFT description of non-standard neutrino interactions · Oscillations in QFT · 0.87

Compares with it 1 paper

[ref] their uncertainties are taken into account.

Neutrino oscillations and scattering theory · Summary and discussion · 0.89

Cites it as background 12 papers

From a theoretical perspective, the wave-packet formulation in quantum field theory (WPQFT) provides one of the most fundamental frameworks for describing neutrino oscillations, and the basics of the formalism were established in [ref] .

Real and Virtual Propagation in Neutrino Oscillations · Introduction · 0.91
In the QFT treatment [ref] , one considers neutrino production, propagation and detection as a single process, described by a tree-level Feynman diagram with the neutrino in the intermediate state (see fig.

Neutrino oscillations: Quantum mechanics vs. quantum field theory · Introduction · 0.83
The formulation of such a non-local S-matrix theory is that we wish to put forward in this paper$$[For other attempts along this research direction, see [ref] .].

Towards a Localised S-Matrix Theory · Non-Locality in QFT · 0.85

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [7]

C. Giunti, C.W. Kim and U.W. Lee, Phys. Rev. D 44 (1991) 3635

View on Pith 1991

[2] [3]

C. Giunti, C.W. Kim and U.W. Lee, Phys. Rev. D 45 (1992) 2414

View on Pith 1992

[3] [1]

S.M. Bilenky and B. Pontecorvo, Phys. Rep. 41 (1978) 225

View on Pith 1978

[4] [2]

S.M. Bilenky and S.T. Petcov, Rev. Mod. Phys. 59 (1987) 671; See, for a recent review, C. W. Kim and A. Pevsner, Neutrinos in Physics and Astrophy sics, Con- temporary Concepts in Physics, Vol. 8, Harwood Academic Press, 1 993

View on Pith 1987

[5] [4]

R.G. Winter, Lett. Nuovo Cimento 30 (1981) 101

View on Pith 1981

[6] [5]

See, for example, R.E. Shrock, Phys. Lett. 96B (1980) 159; Phys. Rev. D24 (1981) 1232,1275

View on Pith 1980

[7] [6]

B. Kayser, Phys. Rev. D 24 (1981) 110

View on Pith 1981

[8] [8]

P.H. Frampton and P. Vogel, Phys. Rep. 82 (1982) 339

View on Pith 1982

[9] [9]

F. Boehm and P. Vogel, Physics of Massive Neutrinos , Cambridge University Press, Cambridge, 1987

View on Pith 1987

[10] [10]

S. Nussinov, Phys. Lett. B 63 (1976) 201. 18

View on Pith 1976