The DIS structure functions $F_4$ and $F_5$ at NLO and their impact on $\tau$-neutrino scattering

E. Spezzano; I. Schienbein; M. Klasen; P. Risse; T. Je\v{z}o

REVIEW

The DIS structure functions $F_4$ and $F_5$ at NLO and their impact on $\tau$-neutrino scattering

T. Ježo, M. Klasen, P. Risse, I. Schienbein, E. Spezzano

High Energy Physics - Phenomenology · Submitted 10 Nov 2026 · 12 references

Pith's verdict: F4 and F5 contributions to tau-neutrino DIS: plausible and useful, but the -40% number comes from a proceedings paper with no uncertainty bands and no independent validation.

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arXiv ID2611.13618 (version 1)

Pith IDR2YQNYVF, Pith's permanent record of this paper and its review

CitationE. Spezzano, I. Schienbein, M. Klasen, P. Risse, T. Je\v{z}o. "The DIS structure functions $F_4$ and $F_5$ at NLO and their impact on $\tau$-neutrino scattering." arXiv:2611.13618 (2026). https://pith.science/paper/R2YQNYVF

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Tom,

This is a short proceedings paper from the DIS 2026 conference. The new result is the first numerical quantification of how F4 and F5 affect the charged-current tau-neutrino DIS cross section. The central claim is that these structure functions suppress the cross section by up to -40% near the kinematic boundary at low Q2 and large x, and that they must be included for precision predictions at DUNE, SHiP, and IceCube.

What the paper does well: it uses a standard framework (ACOT scheme, CT18NLO PDFs, APFEL++), the formalism is clearly summarized, and the suppression pattern is physically plausible given the known behavior of heavy-quark contributions near threshold. The companion paper [1] contains the full derivation of the coefficient functions, so the proceedings is not trying to be self-contained in that respect. The figures are clear and the message is straightforward.

Soft spots: this is a proceedings paper, so it is thin. There are no uncertainty bands (scale, PDF, or scheme variation), no validation against an independent calculation, and no code or data artifacts. The -40% number is presented as a definite result, but the calculation depends on the ACOT scheme near the b-quark threshold, where the scheme choice could matter. The authors do not discuss this uncertainty. The paper also does not state any limitations—the classifier found none—which is a minor red flag for a proceedings that is making a phenomenological claim. The reader's weakest assumption (ACOT scheme validity near threshold) is a real concern, but it is not a flaw in the paper's argument; it is a general theoretical uncertainty that applies to any calculation using that scheme. The paper does not claim to resolve it, only to present results within that framework.

Who this is for: neutrino phenomenologists working on tau-neutrino cross sections for DUNE, SHiP, or IceCube. They will want to read the companion paper [1] for the full derivation and more extensive results. This proceedings is a useful preview but not a standalone reference.

Recommendation: this paper deserves a serious referee. The claim is important enough to warrant peer review, and the framework is sound. The referee should ask for uncertainty estimates and a discussion of scheme dependence near threshold. But the paper is not fatally flawed; it is a legitimate conference contribution that reports a plausible new result.

Best, [Your name]

Referee report

Significance

If the results are correct, this paper provides the first numerical assessment of the phenomenological impact of F4 and F5 on tau-neutrino charged-current DIS. The finding that these structure functions suppress the cross section by tens of percent in a kinematic region relevant to future experiments is important and actionable: it implies that standard calculations using only F1, F2, and F3 overestimate event rates in that region. The calculation uses the well-established ACOT variable-flavor-number scheme and public CT18NLO PDFs, and the implementation is in the open-source APFEL++ framework. The paper is concise and clearly written, with informative figures. As a proceedings contribution, it appropriately refers the reader to the companion paper for full derivations and additional results. The main limitation is the absence of quantitative uncertainty estimates, but this does not undermine the qualitative conclusion that F4 and F5 cannot be neglected near the kinematic boundary.

Minor objections (6)

[Section 2, Eq. (1)] The symbols \tilde{\eta} and \eta_W in Eq. (1) are not defined; please define them explicitly and clarify the meaning of the superscript W on the structure functions.
[Figure 1] The captions contain the placeholder "W□" and the y-axis labels in the lower panels appear truncated; please replace with the proper symbol (W^-) and ensure all labels are visible.
[Section 3] The statement that δR is negative "throughout the physical domain" is based on the four y-values shown in Fig. 2; please state how the full domain was checked, or show a representative scan over y.
[Section 3] Please add a brief comment on the sensitivity of the -40% maximum to scale choice, PDF uncertainties, and the ACOT subtraction scheme near the b-quark threshold, even if a full analysis is deferred to the companion paper.
[Summary] In the last paragraph, "The combined contribution ... suppress" should read "... suppresses" to agree in number.
[References] Consider updating reference [12] to include more recent tau-neutrino DIS cross-section calculations that may be relevant to the kinematic region discussed.

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The paper

The paper claims

Including the F4 and F5 structure functions reduces the charged-current tau-neutrino deep-inelastic cross section by up to 40 percent near low Q2 and large x.

The reading

Deep-inelastic scattering is the process where a neutrino or charged lepton hits a quark inside a nucleon and knocks it apart. The scattering is usually summarized by a few structure functions, F1, F2, and F3, which are well measured. Two additional functions, F4 and F5, are usually invisible because their contribution is multiplied by the lepton mass squared. Tau leptons are heavy enough that these functions can matter.

This paper computes F4 and F5 for heavy-quark contributions at next-to-leading order in quantum chromodynamics, using the ACOT scheme to handle heavy quarks with mass, and implements the result in the open-source APFEL++ code with CT18NLO parton distributions. The authors then use the computed functions to estimate the charged-current tau-neutrino scattering cross section with and without F4 and F5. They find that the two functions suppress the cross section throughout the allowed phase space, strongly near the kinematic boundary at low momentum transfer and high momentum fraction, up to minus 40 percent. At high momentum transfer, above about 1,000 GeV squared, the effect drops below one percent.

The paper is a short conference contribution. The detailed derivation and the W-plus and neutral-current results are given in a companion article. The new numerical result here is the size and shape of the suppression, which is presented for several values of the inelasticity y.

Extended reading notes

Core claim

The central claim, stated in the abstract and Section 4, is that the combined contribution of F4 and F5 suppresses the charged-current tau-neutrino deep-inelastic cross section throughout the physical phase space, by up to -40 percent near the kinematic boundary at low Q2 and large x, so these structure functions must be included in precision predictions for DUNE, SHiP, and IceCube. If correct, event rates from standard F1-F3-only calculations are too high by tens of percent in that corner, and F4 and F5 cannot be dropped there.

Load-bearing premise

The numerical suppression is computed using the ACOT scheme at NLO with CT18NLO PDFs, as described in Section 2: heavy-quark masses are retained in coefficient functions and quasi-collinear logarithms are removed by ACOT subtraction terms. The load-bearing premise is that this scheme and subtraction prescription give the correct physics at low Q2 and large x, close to the b-quark threshold, where the suppression is largest. If the ACOT subtraction or PDF treatment mis-specifies the near-threshold heavy-quark region, the -40 percent number changes even though the F4 and F5 formulas are internally consistent.

Original abstract

We present next-to-leading-order calculations of heavy-quark contributions to the deep-inelastic scattering structure functions $F_4$ and $F_5$ within the Aivazis-Collins-Olness-Tung scheme, implemented in the open-source library APFEL++ and evaluated using CT18NLO parton distribution functions. These structure functions are suppressed by lepton mass effects in light-lepton processes, but become important in $\tau$-lepton and neutrino scattering at facilities such as SHiP, IceCube, and DUNE. We furthermore quantify the direct impact of $F_4$ and $F_5$ on the charged-current $\tau$-neutrino cross section, for the first time presented here, showing they can suppress it by up to $-40\%$ near the kinematic boundary at low $Q^2$ and large $x$.

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, 2026-11-11: 1 request, 3,246 input tokens.

What the abstract itself states, as probabilities: claims first 0.83 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.15 · mentions code or data availability 0.47.

The abstract, sentence by sentence

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

We present next-to-leading-order calculations of heavy-quark contributions to the deep-inelastic scattering structure functions $F_4$ and $F_5$ within the Aivazis-Collins-Olness-Tung scheme, implemented in the open-source library APFEL++ and evaluated using CT18NLO parton distribution functions. These structure functions are suppressed by lepton mass effects in light-lepton processes, but become important in $\tau$-lepton and neutrino scattering at facilities such as SHiP, IceCube, and DUNE. We furthermore quantify the direct impact of $F_4$ and $F_5$ on the charged-current $\tau$-neutrino cross section, for the first time presented here, showing they can suppress it by up to $-40\%$ near the kinematic boundary at low $Q^2$ and large $x$.

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 present next-to-leading-order calculations of heavy-quark contributions to the deep-inelastic scattering structure functions $F_4$ and $F_5$ within the Aivazis-Collins-Olness-Tung scheme, implemented in the open-source library APFEL++ and evaluated using CT18NLO parton distribution functions.	contribution 0.65	0.94	0.06	asserted 1.00
s1	These structure functions are suppressed by lepton mass effects in light-lepton processes, but become important in $\tau$-lepton and neutrino scattering at facilities such as SHiP, IceCube, and DUNE.	background 0.99	0.09	0.06	asserted 1.00
s2	We furthermore quantify the direct impact of $F_4$ and $F_5$ on the charged-current $\tau$-neutrino cross section, for the first time presented here, showing they can suppress it by up to $-40\%$ near the kinematic boundary at low $Q^2$ and large $x$.	result 0.98	0.95	0.86	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.81 · evidence from simulation or computation 0.99.

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.** The CC structure functions 𝑥𝐹4 (left) and 𝑥𝐹5 (right) for 𝑊− exchange, computed in the ACOT scheme using CT18NLO PDFs. Absolute predictions at LO (dashed) and NLO (solid) are shown for 𝑄 2 ∈ {2, 10, 100, 1000} GeV2 (yellow, red, blue, green). The lower panels show the NLO/LO ratios (Kfactors) across 𝑥 ∈ [10−3 , 0.7]. As shown in

**Figure 2.** Relative cross-section change 𝛿𝑅 due to 𝐹4 and 𝐹5 in the 𝑊+ CC 𝜏-neutrino cross section, for 𝑦 = 0.3, 0.4, 0.5, 0.6. The grey region is kinematically forbidden; dash-dotted and dotted contours mark −20% and −5% levels. The dashed orange line indicates 𝑄 2 = 𝑚 2 𝑏 . The ratio 𝛿𝑅 is negative throughout the physical domain, confirming a suppression from 𝐹4 and 𝐹5. The suppression reaches up to −40% near the kinematic boundary at low 𝑄 2 and large 𝑥 and gets more pronounced with 𝑦. The −20% contour lies at 𝑄 2 ≲ 10–30 GeV2 for 𝑥 ≳ 0.1–0.3, and the −5% contour extends to 𝑄 2 ≈ 80 Ge

Figures reproduced from arXiv:2611.13618 by the authors.

References

Works cited (12)

[1]

E. Spezzano, T. Je z o, M. Klasen, P. Risse, and I. Schienbein, Heavy-quark contributions to the DIS structure functions F_ 4 and F_ 5 at NLO in the ACOT scheme , JHEP 05 (2026) 012, [ http://arxiv.org/abs/2512.11569 arXiv:2512.11569 ]

View on Pith arXiv 2026
[2]

Particle Data Group Collaboration, S. Navas et al., Review of particle physics , Phys. Rev. D 110 (2024), no. 3 030001

View on Pith 2024
[3]

SHIP Collaboration , The ship experiment at the proposed cern sps beam dump facility , 2022

View on Pith 2022
[4]

IceCube Collaboration, R. Abbasi et al., Observation of Seven Astrophysical Tau Neutrino Candidates with IceCube , Phys. Rev. Lett. 132 (2024), no. 15 151001, [ http://arxiv.org/abs/2403.02516 arXiv:2403.02516 ]

View on Pith arXiv 2024
[5]

B. Abi et al. , Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume II: DUNE Physics , 2020

View on Pith 2020
[6]

T. Stavreva, F. I. Olness, I. Schienbein, T. Jezo, A. Kusina, K. Kovarik, and J. Y. Yu, Heavy Quark Production in the ACOT Scheme at NNLO and N3LO , Phys. Rev. D 85 (2012) 114014, [ http://arxiv.org/abs/1203.0282 arXiv:1203.0282 ]

View on Pith Pith review arXiv 2012
[7]

M. A. G. Aivazis, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 1. General formalism and kinematics of charged current and neutral current production processes , Phys. Rev. D 50 (1994) 3085--3101, [ http://arxiv.org/abs/hep-ph/9312318 hep-ph/9312318 ]

View on Pith Pith review arXiv 1994
[8]

M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies , Phys. Rev. D 50 (1994) 3102--3118, [ http://arxiv.org/abs/hep-ph/9312319 hep-ph/9312319 ]

View on Pith Pith review arXiv 1994
[9]

S. Kretzer and I. Schienbein, Heavy quark initiated contributions to deep inelastic structure functions , Phys. Rev. D 58 (1998) 094035, [ http://arxiv.org/abs/hep-ph/9805233 hep-ph/9805233 ]

View on Pith Pith review arXiv 1998
[10]

V. Bertone, S. Carrazza, and N. P. Hartland, APFELgrid: a high performance tool for parton density determinations , Comput. Phys. Commun. 212 (2017) 205--209, [ http://arxiv.org/abs/1605.02070 arXiv:1605.02070 ]

View on Pith Pith review arXiv 2017
[11]

T.-J. Hou, J. Gao, T. Hobbs, K. Xie, S. Dulat, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, I. Sitiwaldi, D. Stump, and C.-P. Yuan, New cteq global analysis of quantum chromodynamics with high-precision data from the lhc , Physical Review D 103 (Jan., 2021)

View on Pith 2021
[12]

S. Kretzer and M. H. Reno, Tau neutrino deep inelastic charged current interactions , Physical Review D 66 (Dec., 2002)

View on Pith 2002

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]

E. Spezzano, T. Je z o, M. Klasen, P. Risse, and I. Schienbein, Heavy-quark contributions to the DIS structure functions F_ 4 and F_ 5 at NLO in the ACOT scheme , JHEP 05 (2026) 012, [ http://arxiv.org/abs/2512.11569 arXiv:2512.11569 ]

View on Pith arXiv 2026

[2] [2]

Particle Data Group Collaboration, S. Navas et al., Review of particle physics , Phys. Rev. D 110 (2024), no. 3 030001

View on Pith 2024

[3] [3]

SHIP Collaboration , The ship experiment at the proposed cern sps beam dump facility , 2022

View on Pith 2022

[4] [4]

IceCube Collaboration, R. Abbasi et al., Observation of Seven Astrophysical Tau Neutrino Candidates with IceCube , Phys. Rev. Lett. 132 (2024), no. 15 151001, [ http://arxiv.org/abs/2403.02516 arXiv:2403.02516 ]

View on Pith arXiv 2024

[5] [5]

B. Abi et al. , Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume II: DUNE Physics , 2020

View on Pith 2020

[6] [6]

T. Stavreva, F. I. Olness, I. Schienbein, T. Jezo, A. Kusina, K. Kovarik, and J. Y. Yu, Heavy Quark Production in the ACOT Scheme at NNLO and N3LO , Phys. Rev. D 85 (2012) 114014, [ http://arxiv.org/abs/1203.0282 arXiv:1203.0282 ]

View on Pith Pith review arXiv 2012

[7] [7]

M. A. G. Aivazis, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 1. General formalism and kinematics of charged current and neutral current production processes , Phys. Rev. D 50 (1994) 3085--3101, [ http://arxiv.org/abs/hep-ph/9312318 hep-ph/9312318 ]

View on Pith Pith review arXiv 1994

[8] [8]

M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies , Phys. Rev. D 50 (1994) 3102--3118, [ http://arxiv.org/abs/hep-ph/9312319 hep-ph/9312319 ]

View on Pith Pith review arXiv 1994

[9] [9]

S. Kretzer and I. Schienbein, Heavy quark initiated contributions to deep inelastic structure functions , Phys. Rev. D 58 (1998) 094035, [ http://arxiv.org/abs/hep-ph/9805233 hep-ph/9805233 ]

View on Pith Pith review arXiv 1998

[10] [10]

V. Bertone, S. Carrazza, and N. P. Hartland, APFELgrid: a high performance tool for parton density determinations , Comput. Phys. Commun. 212 (2017) 205--209, [ http://arxiv.org/abs/1605.02070 arXiv:1605.02070 ]

View on Pith Pith review arXiv 2017

[11] [11]

T.-J. Hou, J. Gao, T. Hobbs, K. Xie, S. Dulat, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, I. Sitiwaldi, D. Stump, and C.-P. Yuan, New cteq global analysis of quantum chromodynamics with high-precision data from the lhc , Physical Review D 103 (Jan., 2021)

View on Pith 2021

[12] [12]

S. Kretzer and M. H. Reno, Tau neutrino deep inelastic charged current interactions , Physical Review D 66 (Dec., 2002)

View on Pith 2002