Improved superscaling description of electron and charged-current neutrino quasielastic scattering using effective mass dynamics

J.E. Amaro; P.R. Casale; V.L. Martinez-Consentino

REVIEW 3 major objections 6 minor 58 references

Improved superscaling description of electron and charged-current neutrino quasielastic scattering using effective mass dynamics

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A revised superscaling model splits the nuclear response into a longitudinal function fitted to separated data and a transverse function that must carry explicit momentum-transfer dependence, improving low-q electron and neutrino…

desk verdict A useful analytic upgrade of the SuSAM scaling model, with a q-dependent transverse function and better low-q behavior, but the validation is partly in-sample and the fT extraction absorbs model-dependent 2p2h chunks. read the letter →

arxiv 2506.03934 v2 pith:3F5EYW7F submitted 2025-06-04 hep-ph nucl-th

V.L. Martinez-Consentino , P.R. Casale , J.E. Amaro This is my paper

classification hep-phnucl-th

keywords superscalingquasielasticelectronscatteringnuclearresponsefunctionseffectivemasstransversescalingfunctionmomentumtransferdependenceneutrino-nucleustwo-particle-two-holeexcitations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper builds an improved superscaling model of quasielastic lepton–nucleus scattering, SuSAM-v2. Superscaling is the phenomenological idea that the nuclear response factorizes into a single-nucleon cross section and a function of one scaling variable $\psi^*$; the improvement is to replace the single universal function of the previous version with two independent ones, a longitudinal $f_L(\psi^*)$ fitted directly to separated longitudinal response data and a transverse $f_T(\psi^*,q)$ extracted from inclusive cross-section data after subtracting the longitudinal part and two-nucleon (2p2h) contributions. The central finding is that $f_T$ must depend explicitly on the momentum transfer $q$ to describe all kinematics, so it is parametrized as a single Gaussian whose height, width, and peak position are smooth Fermi functions of $q$. With this separation, the model simultaneously describes the inclusive quasielastic electron cross section and the separated longitudinal and transverse responses, and it improves charged-current neutrino predictions relative to SuSAM-v1, especially at low momentum transfers. The practical payoff is that an analytic, $q$-dependent transverse scaling function is cheap and transparent enough for use in neutrino event generators and oscillation analyses.

What carries the argument

The carrying mechanism is the factored form of the response, $R_K = (\text{single-nucleon prefactor}) \times \text{scaling function}$, built on the relativistic Fermi gas with effective mass $M^*$ and the scaling variable $\psi^*$. The new content is two scaling functions: $f_L(\psi^*)$, a sum of two Gaussians fitted to separated longitudinal data, and $f_T(\psi^*,q)$, a single Gaussian per $q$-bin whose parameters $a(q)$, $b(q)$, $c(q)$ are each parametrized by a Fermi function. The factorization turns the nuclear response into a product of known single-nucleon kinematics and a phenomenological function, so separating responses in the neutrino channel reduces to assigning $f_L$ to the CC, CL, and LL responses and $f_T$ to the $T$ and $T'$ responses. That identification — longitudinal response inherits the $L$ scaling function, transverse and axial-transverse interference inherit the $q$-dependent $T$ function — is what carries the improved neutrino prediction.

What would settle it

Re-derive $f_T$ from the same electron dataset after replacing the model calculation of two-nucleon meson-exchange currents with an independent microscopic calculation; if the fitted Gaussian parameters $a(q)$, $b(q)$, $c(q)$ shift enough to change the predicted low-angle neutrino cross sections beyond the experimental error bars, then the extracted transverse function is an artifact of that subtraction.

Watch

Extended reading notes

Core claim

SuSAM-v2 claims that the transverse nuclear response in quasielastic scattering is not governed by the same scaling function as the longitudinal response, and that the transverse scaling function carries an unavoidable dependence on the momentum transfer $q$. The evidence is assembled by fitting $f_L(\psi^*)$ to separated longitudinal response data for $^{12}$C and then using roughly three thousand inclusive electron cross-section points to extract $f_T$ per $q$-bin, subtracting the fixed longitudinal contribution, a microscopic two-nucleon meson-exchange current term, and a short-range correlation contribution, while masking the resonance and deep-inelastic backgrounds with additional Gaussians. The authors report that the resulting model reproduces the separated transverse response — which was not directly fitted — across $q = 300$, $380$, and $570$ MeV/c, and that flux-averaged charged-current neutrino cross sections for T2K and MINERvA are described as well as or better than with SuSAM-v1. Their formulation deliberately keeps $f_T$ analytic, with coefficients $a(q)$, $b(q)$, $c(q)$ written as Fermi functions of $q$, so the model can be implemented without numerical tables.

Load-bearing premise

The transverse scaling function is defined as whatever remains after subtracting the longitudinal response and two-nucleon background contributions from the measured cross sections; if those subtracted models are wrong, the fitted function and every neutrino prediction built on it inherit the error.

Editorial extensions

If this is right

SuSAM-v2 reproduces the separated longitudinal and transverse response functions for $^{12}$C at $q = 300$, $380$, and $570$ MeV/c, where SuSAM-v1 failed, making the transverse description a genuine consequence of the fit rather than a direct fit to $R_T$.
In neutrino scattering, the corrected relative weight of longitudinal and transverse responses lowers the predicted cross section at small muon angles in the T2K bins, bringing it into better agreement with data.
MINERvA comparisons show both model versions describe the high-energy data well, so the practical gain of v2 is concentrated at lower momentum transfer, where the $q$-dependence of $f_T$ matters.
Two-nucleon (2p2h) terms from meson-exchange currents and short-range correlations remain necessary, contributing roughly 20 percent of the T2K cross section and about 30 percent in the MINERvA bins.
Replacing the dipole axial form factor with newer parametrizations changes the predicted quasielastic peak cross section by up to 25 percent, so the response separation alone does not remove axial-vector uncertainty.

Reading between the lines

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

If the $q$-dependence of $f_T$ mostly encodes final-state interactions, the same analytic parametrization may need target-specific adjustments beyond effective-mass scaling; applying the extraction to heavier nuclei from electron data alone and then comparing with their neutrino data would test that.
Because the extraction subtracts a model calculation of two-nucleon processes, the neutrino success of SuSAM-v2 doubles as a test of that two-nucleon model; a future calculation that changes the subtracted strength would renormalize $f_T$ and could move the low-angle neutrino predictions.
The explicit momentum-transfer dependence means traditional zeroth-kind scaling fails in the transverse channel, so neutrino event generators should treat the transverse response as kinematics-dependent rather than universal.
Comparing this analytic $f_T$ with the numerically defined transverse function obtained from full relativistic mean-field calculations would quantify how much of the apparent $q$-dependence is nuclear-physics content versus modeling choice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a worthwhile incremental contribution. It separates fL and fT in the SuSAM effective-mass framework, gives the transverse function a smooth q-dependence via Fermi-parametrized Gaussians, and shows real improvement in low-q electron responses and in T2K comparisons. The analytic form is a practical advantage for event-generator use, and the paper is honest about being a phenomenological update.

The main soft spot is exactly the one you flagged: the fT extraction in Eq. (23) subtracts only the MEC 2p2h term, not the SRC term, and then fits the remainder to inclusive cross sections. So the "transverse prediction" in Fig. 5 is partly a refit, not a clean out-of-sample result. The claim that RT is a genuine prediction is overstated: fT is constrained by the same inclusive cross sections that define RT. The neutrino comparisons are more independent, but they inherit whatever systematic error is baked into fT from the 2p2h subtraction and from the visual rejection of low-q points. The paper gives no uncertainties for a(q), b(q), c(q), which makes it hard to judge how significant the improvement over v1 actually is.

The double-counting concern in the stress-test is plausible. If fT already contains part of the EM SRC strength (because only MEC was subtracted), then adding SRC additively in the neutrino channel can double count. The authors should address this explicitly, either by subtracting both 2p2h pieces in the extraction or by showing that the fitted QE Gaussian does not include the SRC tail. This is the main thing I'd want fixed in revision.

The formalism is standard and the paper is clearly written. The citation pattern is fine; the paper builds directly on the SuSAM line and SuSA-v2.

Who should read it: anyone working on neutrino event generators or oscillation systematics who wants a compact, portable QE response. It's not a breakthrough, but it's a solid engineering improvement. I'd send it to review, with a request that the authors clarify the in-sample status and the SRC subtraction.

Referee Report

3 major / 6 minor

Summary. The paper presents SuSAM-v2, a superscaling model for quasielastic lepton-nucleus scattering. Whereas SuSAM-v1 used a single scaling function fitted to inclusive (e,e') cross sections, v2 first fits a longitudinal scaling function fL*(ψ*) to separated longitudinal response data for 12C (Eq. (22), Table I) and then extracts a transverse scaling function fT*(ψ*,q) from inclusive electron cross sections by subtracting a 2p2h-MEC contribution and the fL-based longitudinal part (Eqs. (23)-(25)). The transverse function is fitted bin-by-bin in q as a single Gaussian, and the resulting parameters a(q), b(q), c(q) are smoothed with Fermi functions (Tables II and III). The model is compared with separated RL and RT responses, inclusive (e,e') spectra, and flux-averaged charged-current neutrino cross sections from T2K and MINERvA, with 2p2h (MEC and SRC) contributions added. The authors claim the model simultaneously describes electron responses and cross sections and improves neutrino predictions relative to SuSAM-v1.

Significance. The proposed analytic, q-dependent transverse scaling function is a potentially useful object: if robustly extracted, it offers a transparent and efficient parametrization for event generators and a natural route to improving low-q neutrino predictions. The paper correctly identifies longitudinal/transverse separation as the key limitation of v1, and the neutrino comparisons in Figs. 7-8 are independent of the electron fit, which makes them a genuine test. The analysis of axial-form-factor sensitivity in Figs. 9-10 is also a strength. However, as presented, the electron-scattering validation is largely in-sample: fT is constructed from the same inclusive data that are later compared in Figs. 5-6, no uncertainties are reported for the fitted parameters, the low-q selection is visual, and the neutrino predictions inherit assumptions about the subtracted 2p2h strength. The paper's significance therefore depends on whether these points can be addressed with out-of-sample checks and sensitivity analyses.

major comments (3)

[Sections III.b and IV.A (Eqs. (23)-(25), Fig. 5)] The claim that the agreement with RT in Fig. 5 constitutes a 'genuine prediction' is not supported by the construction. In Eq. (23) the experimental transverse response is defined by subtracting the 2p2h-MEC contribution and the fL-determined longitudinal part from the inclusive cross section, and Eq. (25) fits fT to these same data. Consequently, the RT panels in Fig. 5 and the inclusive cross sections in Fig. 6 are in-sample consistency checks rather than independent tests. This bears directly on the central claim, stated in the abstract, that the model 'simultaneously describes' inclusive cross sections and both response functions. Please provide an out-of-sample validation, for example by refitting fT after excluding selected q-bins or kinematics and then predicting the excluded data, and report residuals or chi-square values for data not used in the fit.
[Section III.b, Tables II and III] The q-dependence of fT is the main new content of the paper, but the evidence for it is presented without uncertainties. The a(q), b(q), c(q) values in Table II are fitted bin-by-bin from roughly 3000 points, and the paper states that for low-q intervals 'certain data points that clearly do not exhibit scaling behavior were not taken into account' after visual inspection. Without parameter errors, per-bin point counts, and a reproducible selection criterion, the reader cannot judge whether the 18-bin model is overfitting, whether the low-q reduction is statistically significant, or whether the improvement over SuSAM-v1 is meaningful. Please report fit uncertainties (including propagation into the Fermi-function parameters of Table III), the number of points per q-bin, and a quantitative description of the data-selection procedure.
[Section III.b and Section IV.B] There is a potential double-counting or mismatch of 2p2h strength between the electron extraction and the neutrino predictions. Eq. (23) subtracts only the RMF-MEC 2p2h contribution from the inclusive cross section before fT is fitted; the SRC/correlated-pair 2p2h contribution of Refs. [40,41] is not subtracted at this stage. The neutrino cross sections in Figs. 7-8 then add both MEC and SRC 2p2h contributions on top of the fT-based 1p1h responses. If any part of the subtracted or unsubtracted 2p2h strength leaks into the fitted QE Gaussian, the weak predictions inherit that leakage; because the weak MEC and SRC channels enter with different axial and isospin couplings than the electromagnetic channels, the claimed T2K improvement could be an artifact of the subtraction scheme. The manuscript should clarify whether the Gaussian decomposition removes SRC strength from fT, and should test the sensitivity of the neutrino results to the assumed 2p2h subtraction model.

minor comments (6)

[Section III.a, Eq. (22) and Table I] The sentence preceding Eq. (22) says the parameters ai, bi, ci are 'center', 'width', and 'height', respectively, but in Eq. (22) a is the height, b the center, and c the width; the wording should be corrected.
[Section III.b, Eq. (24)] The text says the transverse scaling function is obtained by dividing (RT)exp by the single-nucleon transverse response, but the denominator shown is the prefactor Z r_T^p + N r_T^n of Eq. (19), not the response itself; the wording should be aligned with Eqs. (18)-(19).
[Abstract] fT is described as extracted from 'purely transverse data', whereas Eqs. (23)-(25) show it is extracted from inclusive cross sections after subtracting a 2p2h model and the longitudinal contribution; the wording should be adjusted.
[Fig. 2 caption] The abbreviation 'RFGM*' is not defined in the caption or in the text; please define it explicitly.
[Fig. 1 and Fig. 5 captions] The combined citation 'Ref. [3, 48]' should be split into two separate references or formatted consistently with the journal style.
[Section IV.B] The assumption that the same fT applies to both RT and RT' is stated without comment; given that this is a new ingredient of v2, a brief justification or reference to the corresponding SuSA-v2 treatment would help the reader assess the weak predictions.

Circularity Check

1 steps flagged · score 6.0 of 10

The transverse-response 'prediction' in SuSAM-v2 is a refit of the response manufactured from inclusive data via Eq. (23); the neutrino predictions remain genuinely out-of-sample, so the circularity is partial.

fitted input called prediction [Sec. III.b, Eqs. (23)-(25); Sec. IV.A, Eq. (28) and Fig. 5]
"We begin with inclusive electron scattering data in the quasielastic region and, as a first step, subtract the contribution from MEC in the two-particle emission channel. ... In the second step, we subtract the longitudinal contribution already fixed by f∗L(ψ∗). The remaining strength is assumed to be entirely transverse in nature. (RT)exp = ... This constitutes a genuine prediction of the model, since RT data were not used in the fit of the transverse scaling function f∗T —only the inclusive cross section data were employed."

The transverse scaling function f∗T is not extracted from an independent separated-RT dataset. Equation (23) manufactures an 'experimental RT' by taking the inclusive cross section and subtracting the 2p2h-MEC model and the fL-determined longitudinal response; Eq. (24) divides by the transverse single-nucleon prefactor. The f∗T in Eq. (25) is a Gaussian fit to exactly those (f∗T)exp values, and Eq. (28) reconstructs RT as the prefactor times this same fitted f∗T. Therefore the RT curves shown in Fig. 5 are a smoothed refit of the transverse strength obtained from the inclusive data, not a prediction of a held-out observable.

full rationale

The electron-side central claim is partially circular. fL is fitted to separated RL data and inclusive cross sections, while fT is fitted to inclusive cross-section data after subtracting the fL-based longitudinal response and the authors' 2p2h-MEC model (Eqs. 23-25). The model's RT is then the single-nucleon prefactor times that same fitted fT (Eq. 28), so the RT comparison in Fig. 5 is a consistency check of the Gaussian parametrization rather than a genuine prediction. The paper's own statement that 'RL data were used directly in the extraction' correctly acknowledges the refit nature of that channel, but the same logic applies to the transverse channel through Eq. (23). The neutrino predictions for T2K and MINERvA are genuinely out-of-sample applications of the electron-fitted fL and fT, and the differences from SuSAM-v1 there are not forced by construction, so the circularity is partial rather than total. The self-citations to the prior RMF/MEC calculations are not used as a uniqueness theorem and that prior work has independent empirical content, so no score-8 self-citation chain is present. The dominant problem is the 'genuine prediction' language for RT, which reduces to a refit by the paper's own equations.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The model depends on the superscaling factorization ansatz, on the universality of fL and fT across nuclei, on the subtraction of 2p2h MEC and SRC models from previous papers, and on the identification of fT with weak transverse channels. The many fitted parameters of fL and fT are the price paid for the analytic form; the paper itself acknowledges the phenomenological nature. No new physical entities are introduced.

free parameters (5)

Longitudinal scaling function fL Gaussian parameters = a1=0.67732, b1=0.0, c1=0.497381, a2=0.103387, b2=1.05065, c2=0.302541
Fitted to 12C longitudinal response data at q=300, 380, 570 MeV/c (Eq. 22, Table I). Used directly to compute RL and to subtract the longitudinal part from inclusive cross sections.
Transverse scaling function shape parameters a(q), b(q), c(q) = See Table III: 12 alpha values for the Fermi functions in Eq. (26)
Fitted to transverse scaling data derived from inclusive 12C cross sections in 18 q-bins (Tables II and III). This is the central flexible element of the model.
Effective nucleon mass ratio M* = 0.8 for 12C (from Ref. [34])
Determines the scaling variable psi* and all single-nucleon prefactors. Not re-fit in this paper but inherited from earlier fits by the same group.
Fermi momentum kF for target nuclei = not quoted; inherited from prior SuSAM fits
Appears in the normalization of responses via Eq. (16) and in the scaling variable. The paper does not state the kF values used for C or CH.
2p2h SRC semiempirical coefficients = not given; from Refs. [40,41]
The SRC 2p2h contribution used in neutrino predictions is proportional to 2p2h phase space with coefficients fit to the high-energy tail of the experimental scaling function in prior work.

assumptions (6)

domain assumption Factorization of the nuclear response into a single-nucleon prefactor and a scaling function (Eqs. 17-19) remains valid for the separated longitudinal and transverse channels.
This is the defining ansatz of superscaling models. The paper uses it to define fL and fT from data and to extrapolate to neutrino channels.
domain assumption The longitudinal scaling function extracted from 12C at q = 300, 380, 570 MeV/c is universal enough to apply to all kinematics and to CH in MINERvA.
fL is fitted only on one nucleus and three momentum transfers, yet is used for the CH target and for all q in neutrino predictions.
ad hoc to paper The same transverse scaling function fT(psi*,q) applies to the weak transverse and axial interference responses RT and RT' in neutrino scattering.
This is a new assumption introduced in Sec. IV.B; no weak response data are used to test it.
domain assumption Subtraction of the 2p2h MEC and SRC contributions from prior RMF-based calculations leaves a residual transverse response that is purely 1p1h quasielastic.
Eq. (23) subtracts these models from the inclusive cross section; any error in them is absorbed into fT.
ad hoc to paper Fitting Gaussians for the Delta resonance and DIS backgrounds in each q-bin isolates the quasielastic transverse scaling function.
The paper states this is not based on a microscopic model of pion production or DIS; the parameters are determined by data.
standard math Standard electromagnetic and axial nucleon form factors are taken from the literature and are not part of the fitted model.
F1, F2, and GA (dipole with MA=1.032 GeV) are used as inputs; the paper discusses axial form-factor sensitivity only in Fig. 10.

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Cite this review

Pith. "Pith review of Improved superscaling description of electron and charged-current neutrino quasielastic scattering using effective mass dynamics." pith.science (2026). https://pith.science/paper/3F5EYW7F

@misc{pith2026250603934,
  author       = {Pith},
  title        = {Pith review of: Improved superscaling description of electron and charged-current neutrino quasielastic scattering using effective mass dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3F5EYW7F}},
  note         = {Machine review of arXiv:2506.03934}
}

read the original abstract

We present an improved version of the Superscaling Analysis with Relativistic Effective Mass, denoted as SuSAM-v2. In the original SuSAM model, a universal scaling function was fitted to a selected set of quasielastic electron scattering (e,e') cross section data, using a phenomenological ansatz inspired by the Relativistic Mean Field model of nuclear matter. In this work, we refine the procedure by first fitting a longitudinal scaling function directly to experimental longitudinal response data. Subsequently, a separate transverse scaling function is extracted from purely transverse data, after subtracting the longitudinal contribution already determined. We find that the resulting transverse scaling function must exhibit an explicit dependence on the momentum transfer q in order to reproduce all kinematics consistently. The resulting SuSAM-v2 model simultaneously describes inclusive quasielastic cross sections and both longitudinal and transverse response functions in electron scattering. The model is then applied to neutrino-nucleus scattering, showing an improved prediction compared to the previous SuSAM-v1 version, due to a more accurate treatment of the relative contributions of the longitudinal and transverse weak nuclear responses.

Figures

Figures reproduced from arXiv: 2506.03934 by the authors.

**Figure 2.** FIG. 2. Transverse scaling function data obtained from the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Experimental transverse scaling function data sets [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 6.** Figure 6: FIG. 6. Inclusive electron scattering cross section on [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Neutrino charged-current quasi-elastic cross sect [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Neutrino charged-current cross section from the MIN [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9. Axial form factor of the nucleon as a function of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Impact of different axial form factor parametrizati [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

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