Lectures on lepton-nucleus quasielastic scattering with relativistic models

Ra\'ul Gonz\'alez-Jim\'enez

REVIEW

Lectures on lepton-nucleus quasielastic scattering with relativistic models

Raúl González-Jiménez

Nuclear Theory · Submitted 9 Jul 2026 · 32 references

Pith's verdict: Solid pedagogical lecture notes on relativistic QE scattering; derivations are careful and the model hierarchy is shown clearly with data.

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arXiv ID2607.08629 (version 1)

Pith ID4A6XYAHA, Pith's permanent record of this paper and its review

CitationRa\'ul Gonz\'alez-Jim\'enez. "Lectures on lepton-nucleus quasielastic scattering with relativistic models." arXiv:2607.08629 (2026). https://pith.science/paper/4A6XYAHA

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This is a set of lecture notes for graduate students entering neutrino-nucleus physics, covering relativistic quasielastic lepton-nucleus scattering across four nuclear models: global RFG, local RFG, PWIA with spectral functions, and RDWIA. The punchline: it does exactly what it sets out to do, and does it carefully. No new physics, no new methods, and it makes no pretense otherwise. What it does offer is a genuinely useful, step-by-step derivation path from the elastic lepton-nucleon cross section through to the full nuclear QE calculation, with all the intermediate Dirac-delta manipulations, recoil factors, and Jacobians written out explicitly. That is the real value here — the stuff that is standard but that nobody writes down in one place. The comparison figures (Figs. 6, 8, 10, 12, 13) showing the progressive improvement from gRFG through to EDRMF with two-body currents are pedagogically effective and consistent with the published literature. The appendices on conventions, Feynman rules, and analytical nuclear densities are a nice touch for students who want to implement the formulas. The soft spots are minor and mostly inherent to the format. The off-shell ambiguity in the SFA approach (Section VII) is mentioned but not quantified — the notes acknowledge it and point to references, which is fine for a pedagogical treatment but a researcher wanting to use these formulas would need to dig deeper. The impulse approximation is the load-bearing assumption for everything in Sections V–VII, and while it is standard and openly acknowledged, the notes could have said a bit more about where it breaks down quantitatively. The CC1/CC2 equivalence discussion in Appendix D is correct for on-shell nucleons but the breakdown for off-shell cases in the SFA is only briefly flagged. The free parameters (p_F, binding energy shift, spectroscopic factors, temperature) are all taken from prior fits to data with appropriate citations. No invented entities, no misattributions, no computational errors that I can find. The reader's assessment and the stress-test note are both on target. This is for Master's or PhD students who need a single, self-contained reference to get from the Born approximation to working QE cross-section code. It is not a research paper and should not be reviewed as one. It deserves a serious referee to check the derivations for correctness, because students will copy these formulas into their code.

Referee report

Significance

The manuscript serves a clear pedagogical purpose for the neutrino-nucleus interaction community. Its primary strength lies in the meticulous, explicit derivation of the cross-section formulas for each nuclear model from a single, consistent starting point (Eq. 17). By systematically averaging the elementary lepton-nucleon process over different nuclear probability distributions, the notes make the physical impact of Fermi motion, Pauli blocking, binding energy, and final-state interactions transparent. The compilation of inclusive electron scattering data alongside the progressive model improvements (culminating in the EDRMF with spectral functions and 1b+2b currents in Fig. 12) provides an excellent visual summary of the field's current theoretical status. The inclusion of detailed appendices on conventions, Feynman rules, and analytical nuclear density parametrizations makes the notes a self-contained and practical reference for students entering the field.

Major objections (2)

[Section VII, Eq. (75) and Appendix D, Eq. (D3)] The notes correctly point out that the CC1 and CC2 current operators are equivalent only for on-shell nucleons (Gordon identity, Eq. D3). However, in the SFA approach (Section VII), the bound nucleons are explicitly off-shell (E^2 != p^2 + M^2). The manuscript mentions 'off-shell ambiguities' but does not specify which current operator prescription (CC1 or CC2) was actually used for the SFA results shown in Figs. 10 and 13, nor does it quantify the uncertainty introduced by this ambiguity. For a pedagogical treatment, a brief remark on the typical magnitude of this ambiguity in the cross-section predictions would be valuable, as it is a known practical issue when implementing these models.
[Section V.A and Fig. 6] The implementation of binding energy (BE) in the gRFG model is described as a simple 24 MeV shift of the energy transfer distribution. While this is a standard ad-hoc procedure, it is applied differently in the lRFG (Section VI, Fig. 8) and SFA (Section VII) models, where binding is handled via the spectral function or local density. The notes should briefly clarify how the binding energy shift is consistently (or inconsistently) applied across the different models compared in Fig. 13, to ensure that the reader understands whether the progressive improvement in data agreement is driven by the nuclear model itself or by the specific treatment of binding energy.

Minor objections (6)

[Section V, Eq. (46)] The transition from Eq. (45) to Eq. (46) involves averaging the elastic cross section over the initial nucleon 4-momentum. The notation in Eq. (46) could be slightly clarified; the factor K/(2π)^2 δ^4(...) L'_{μν} H^{μν} is described as the elastic cross section, but it retains the 4-dimensional delta function. A brief note reminding the student that this is the differential elastic cross section before the final energy-conservation integration would help avoid confusion.
[Section V, Eq. (51)] In the derivation of the two solutions for p_N, the definitions of A and B are given, but the physical interpretation of why p_N^- is negative (and thus discarded) is only briefly mentioned. A one-sentence explanation of the kinematic configuration that leads to the unphysical solution would improve the pedagogical clarity.
[Figures 6, 8, 10, 12, 13] The axis labels in the figures use 'Ei' and 'thl' for the initial electron energy and scattering angle, respectively. It would be clearer to use the manuscript's standard notation (ε_i and θ_f) in the figure captions or axis labels for consistency with the main text.
[Section IX, Eq. (83)] The lepton tensor for CC neutrino scattering includes the term ±i h ε_{μναβ} K_i^α K_f^β. The text states h = +1 for antineutrino and h = -1 for neutrino. It would be helpful to explicitly state the sign convention for the ± symbol (e.g., upper sign for neutrino, lower for antineutrino) to avoid ambiguity.
[Appendix H, Eq. (H6)] In the derivative d/dr ρ(r) = 0, the equation is written with R_p in the exponential, but the text refers to general n,p. It should be R_{n,p} for consistency with Eq. (H1).
[References] Reference [7] (Alvarez Ruso et al., J. Phys. G, 2025) and [18] (Nikolakopoulos and González-Jiménez, arXiv:2512.22763, 2025) appear to be very recent or forthcoming. Ensure that the final bibliographic details (volume, page number, DOI) are updated upon publication.

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The paper

The paper claims

How to model a lepton hitting a nucleus, one nucleon at a time

The reading

These lecture notes provide a unified, step-by-step derivation of quasielastic lepton-nucleus scattering cross sections within a fully relativistic framework, progressing through four nuclear models of increasing sophistication: the global relativistic Fermi gas (gRFG), the local relativistic Fermi gas (lRFG), the plane-wave impulse approximation with a realistic spectral function (SFA), and the relativistic distorted-wave impulse approximation (RDWIA). The central object is the impulse approximation, which reduces the many-body problem of a lepton interacting with a nucleus to an incoherent sum of one-body processes where the exchanged boson couples to a single bound nucleon while the remaining nucleons act as spectators. Within this framework, the notes show how each model incorporates different nuclear effects — Fermi motion, Pauli blocking, binding energy, off-shell nucleons, final-state interactions, and two-body currents — and demonstrate, through comparison with inclusive electron-carbon scattering data across a wide range of kinematics, that models incorporating realistic spectral functions, energy-dependent mean-field potentials, and two-body currents yield progressively better agreement with data. The notes also extend the formalism to charged-current quasielastic neutrino-nucleus scattering, including the flux-averaging required for realistic neutrino experiments.

What carries the argument

The impulse approximation factorization of the many-body hadronic current into one-body processes, combined with the systematic progression of nuclear descriptions from a constant-density on-shell Fermi gas to off-shell spectral functions with distorted waves, serves as the central mechanism. The comparison with electron scattering data provides the empirical anchor.

Extended reading notes

Core claim

The notes establish that the dominant uncertainties in quasielastic lepton-nucleus scattering predictions at intermediate energies (hundreds of MeV to few GeV) arise not from the elementary lepton-nucleon vertex, which is well-constrained by form factors, but from the nuclear model used to describe the initial bound state and final-state interactions of the struck nucleon. By holding the elementary interaction fixed and varying only the nuclear model, the notes isolate and rank nuclear effects: Fermi motion sets the overall peak position and width, binding energy shifts the peak, Pauli blocking suppresses low-energy-transfer strength, realistic spectral functions redistribute strength into高-

Load-bearing premise

The entire formalism rests on the impulse approximation — the assumption that the exchanged boson couples to a single bound nucleon while all other nucleons act as inert spectators. This factorization is expected to work well when the wavelength of the exchanged boson is shorter than the inter-nucleon separation, but its domain of validity at low momentum transfer or near reaction thresholds is not critically assessed in the notes.

Editorial extensions

If this is right

Neutrino oscillation experiments that rely on quasielastic cross sections to reconstruct neutrino energies should propagate nuclear model uncertainties as a leading systematic, since the spread between gRFG and RDWIA predictions is comparable to or larger than typical experimental precision goals.
The demonstrated improvement from two-body currents in the RDWIA framework suggests that the impulse approximation alone is insufficient for precision neutrino-nucleus modeling, and that beyond-IA effects should be incorporated into event generators used by oscillation experiments.
The flux-averaging formalism for neutrino cross sections, combined with the nuclear model hierarchy, implies that neutrino energy reconstruction from final-state lepton kinematics inherits nuclear-model-dependent biases that cannot be removed by flux tuning alone.
The analytical nuclear density parametrizations and Pauli-blocking functions provided for carbon and oxygen could serve as drop-in components for Monte Carlo event generators targeting these common detector materials.

Reading between the lines

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

If the hierarchy of model improvements holds for electron scattering, it should also hold for neutrino scattering at comparable momentum transfers, since the nuclear physics is identical — only the leptonic current differs. This suggests that electron scattering data on the same target nuclei can serve as a calibration tool for neutrino interaction models, a connection the notes mention but do not
The off-shell ambiguity in the SFA approach — where bound nucleons have E² ≠ p² + M² and different prescriptions for E give different cross sections — implies a fundamental limit on how well one can define the initial state of a bound nucleon within a one-body framework, which may set a floor on achievable precision even with perfect nuclear wavefunctions.
The progression from gRFG to RDWIA with two-body currents mirrors a convergence pattern where each added layer of realism yields diminishing returns; extrapolating this trend, one might expect that the remaining gap to data at the highest energies shown is dominated by non-QE channels (resonance pion production, deep inelastic scattering) rather than further refinements within the QE sector.

Original abstract

In these notes, basic concepts and necessary formulas for the modeling of elastic lepton-nucleon and quasielastic lepton-nucleus scattering using relativistic models are discussed. Certain theoretical developments are presented in meticulous detail, particularly those rarely found in papers or standard textbooks. In order to highlight the discrepancies between models and evaluate the impact of various nuclear effects, theoretical predictions are compared with inclusive electron scattering data. These notes are designed for Master's or PhD students embarking on their research in the field of neutrino-nucleus interactions at intermediate energies (ranging from several hundred MeV to a few GeV). They are intended to serve as a supplemental guide, never as a substitute for traditional textbooks on Scattering Theory and Quantum Field Theory. Currently, these notes are included in the course `Relativistic Quantum Theory: Nuclear Processes', which is integrated into both the Interuniversity Master in Nuclear Physics (https://master.us.es/fisicanuclear/index.php/) and the Erasmus Mundus Joint Master Degree in Nuclear Physics (https://www.emm-nucphys.eu/en), with the University of Seville as one of the partner institutions. Previously, this material was part of the course `Weak Interactions', within the same Master's programs.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,712 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.85 · human participants 0.02 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.72 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

In these notes, basic concepts and necessary formulas for the modeling of elastic lepton-nucleon and quasielastic lepton-nucleus scattering using relativistic models are discussed. Certain theoretical developments are presented in meticulous detail, particularly those rarely found in papers or standard textbooks. In order to highlight the discrepancies between models and evaluate the impact of various nuclear effects, theoretical predictions are compared with inclusive electron scattering data. These notes are designed for Master's or PhD students embarking on their research in the field of neutrino-nucleus interactions at intermediate energies (ranging from several hundred MeV to a few GeV). They are intended to serve as a supplemental guide, never as a substitute for traditional textbooks on Scattering Theory and Quantum Field Theory. Currently, these notes are included in the course `Relativistic Quantum Theory: Nuclear Processes', which is integrated into both the Interuniversity Master in Nuclear Physics (https://master.us.es/fisicanuclear/index.php/) and the Erasmus Mundus Joint Master Degree in Nuclear Physics (https://www.emm-nucphys.eu/en), with the University of Seville as one of the partner institutions. Previously, this material was part of the course `Weak Interactions', within the same Master's programs.

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	In these notes, basic concepts and necessary formulas for the modeling of elastic lepton-nucleon and quasielastic lepton-nucleus scattering using relativistic models are discussed.	contribution 0.81	0.38	0.07	asserted 0.96
s1	Certain theoretical developments are presented in meticulous detail, particularly those rarely found in papers or standard textbooks.	contribution 0.89	0.69	0.03	asserted 1.00
s2	In order to highlight the discrepancies between models and evaluate the impact of various nuclear effects, theoretical predictions are compared with inclusive electron scattering data.	method 0.73	0.74	0.09	asserted 0.98
s3	These notes are designed for Master's or PhD students embarking on their research in the field of neutrino-nucleus interactions at intermediate energies (ranging from several hundred MeV to a few GeV).	background 0.67	0.13	0.02	asserted 1.00
s4	They are intended to serve as a supplemental guide, never as a substitute for traditional textbooks on Scattering Theory and Quantum Field Theory.	limitation 0.94	0.12	0.03	asserted 1.00
s5	Currently, these notes are included in the course `Relativistic Quantum Theory: Nuclear Processes', which is integrated into both the Interuniversity Master in Nuclear Physics (https://master.us.es/fisicanuclear/index.php/) and the Erasmus Mundus Joint Master Degree in Nuclear Physics (https://www.emm-nucphys.eu/en), with the University of Seville as one of the partner institutions.	background 0.95	0.28	0.02	asserted 1.00
s6	Previously, this material was part of the course `Weak Interactions', within the same Master's programs.	background 0.98	0.17	0.02	asserted 1.00

What kind of paper the classifier reports this is: review 0.85 · evidence from proof or derivation 0.47.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 1.** FIG. 1: One-boson exchange diagram for elastic lepton-nucleon scattering.

**Figure 2.** FIG. 2: Elastic electron-proton single differential cross section. The results correspond to three values of

**Figure 3.** FIG. 3: Sketch of a quasielastic lepton-nucleon scattering in a reference frame in which

**Figure 4.** FIG. 4: The Impulse Approximation for QE scattering.

**Figure 5.** FIG. 5: Momentum distributions

**Figure 6.** FIG. 6: The inclusive cross section [

**Figure 8.** FIG. 8: As figure 6 but now for the models: RFG(T

**Figure 9.** FIG. 9: The Rome spectral function for protons in

**Figure 10.** FIG. 10: As figure 6 but now for the models: RFG(T

**Figure 11.** FIG. 11: Pauli blocking function within the lRFG computed with the barionic densities from an RMF

**Figure 12.** FIG. 12: As figure 6 but now for the models: EDRMF, EDRMF with a spectral function (SF), and

**Figure 13.** FIG. 13: As figure 6. Model predictions: RFG(T

Figures reproduced from arXiv:2607.08629 by the authors.

References

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How the literature uses it

What this paper does with the arXiv papers it cites

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

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] [1]

Photon-lepton-lepton vertex 37

View on Pith

[2] [2]

Photon-nucleon-nucleon vertex 37 3.Wboson-neutrino-lepton vertex 37 4.Wboson-nucleon-nucleon vertex 38 D. Internal structure of the nucleon38 E. Kinematics for elastic lepton-nucleon scattering with massive leptons39 F. Jacobian for the missing energy distributions with nuclear recoil40 G. Normalization of the momentum distribution in the lRFG model40 3 H...

View on Pith

[3] [3]

Each vertex:i R d4X ¯L, where ¯Lis the Lagrangian without the fields

View on Pith

[4] [4]

Particle propagating from X to Y with 4-momentumP µ: •W boson: DW (X−Y) = Z d4P (2π)4 −igµν P 2 −M 2 W eiP·(X−Y) . •Photon: Dγ(X−Y) = Z d4P (2π)4 −igµν P 2 eiP·(X−Y)

View on Pith

[5] [5]

External particle with 4-momentumP µ = (E,p): Incoming fermion: Ψ(X), outgoing fermion Ψ(X). 37 Appendix C: Lagrangian densities: Couplings to external fields The Standard Model of Particle Physics gives us the full Lagrangian densityLwhich describes all particles and their interactions. Here, we are interested only in the interaction of fermions with ext...

View on Pith

[6] [6]

Photon-lepton-lepton vertex The coupling of an electron to a photon is given by: Lγee =e ψeγµψe Aµ ,(C1) where ψe is the electron field,A µ the photon field andeis the coupling constant (electron charge). For chargeless leptons (neutrinos), the coupling is zero

View on Pith

[7] [7]

Photon-nucleon-nucleon vertex If nucleons are considered as point-like particles, the expression is the same as for the electron but changing the sign of the coupling LγN N =−e ψpγµψp Aµ ,(C2) notice that there is no coupling to the neutron because the neutron net charge is zero. The internal structure of the nucleons is accounted for by replacing the sim...

View on Pith

[8] [8]

W. Greiner and J. Reinhardt, Quantum Electrodinamics (Springer, 2003)

View on Pith 2003

[9] [9]

F. Halzen and A. D. Martin, Quarks and Leptons: An introductory Course in Modern PArticle Physics (John Wiley & Sons., 1984)

View on Pith 1984

[10] [10]

F. Mandl and G. G. Shaw, Quantum Field Theory (John Wiley & Sons., 2010)

View on Pith 2010

[11] [11]

T. W. Donnelly, J. A. Formaggio, B. R. Holstein, R. G. Milner, and B. Surrow, Foundations of Nuclear and Particle Physics (Cambridge University Press, 2017)

View on Pith 2017

[12] [12]

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964)

View on Pith 1964

[13] [13]

L. Alvarez-Ruso et al., Progress in Particle and Nuclear Physics100, 1 (2018)

View on Pith 2018

[14] [14]

L. Alvarez Ruso et al., Journal of Physics G: Nuclear and Particle Physics (2025)

View on Pith 2025

[15] [15]

Wikipedia contributors, “Fermi–dirac statistics — Wikipedia, the free encyclopedia,” (2024), [Online; accessed 11-February-2025]

View on Pith 2024

[16] [16]

W. M. Alberico, A. Molinari, T. W. Donnelly, E. L. Kronenberg, and J. W. Van Orden, Phys. Rev. C38, 1801 (1988)

View on Pith 1988

[17] [17]

D. Day (Quasielastic Electron Nucleus Scattering Archive),https://discovery.phys.virginia. edu/research/groups/qes-archive/(2021)

View on Pith 2021

[18] [18]

C. Horowitz and B. D. Serot, Nuclear Physics A368, 503 (1981)

View on Pith 1981

[19] [19]

M. Sharma, M. Nagarajan, and P. Ring, Physics Letters B312, 377 (1993)

View on Pith 1993

[20] [20]

T. J. Leitner, Neutrino-Nucleus Interactions In A Coupled-Channel Hadronic Transport Model, Ph.D. thesis, Justus-Liebig-Universit¨ at Giessen (2009)

View on Pith 2009

[21] [21]

O. Benhar, D. Day, and I. Sick, Rev. Mod. Phys.80, 189 (2008)

View on Pith 2008

[22] [22]

J. W. Van Orden and T. W. Donnelly, Phys. Rev. C100, 044620 (2019)

View on Pith 2019

[23] [23]

O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nuclear Physics A579, 493 (1994)

View on Pith 1994

[24] [24]

O. Benhar, N. Farina, H. Nakamura, M. Sakuda, and R. Seki, Phys. Rev. D72, 053005 (2005)

View on Pith 2005

[25] [25]

A. Nikolakopoulos and R. Gonz´ alez-Jim´ enez, (2025), arXiv:2512.22763 [nucl-th]

View on Pith arXiv 2025

[26] [26]

J. Walecka, Annals of Physics83, 491 (1974)

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