Leptoproduction of Heavy Quarks I-- General Formalism and Kinematics of Charged Current and Neutral Current Production Processes

Fredrick I. Olness; M. A. G. Aivazis; Wu-Ki Tung

REVIEW

Leptoproduction of Heavy Quarks I-- General Formalism and Kinematics of Charged Current and Neutral Current Production Processes

M. A. G. Aivazis, Fredrick I. Olness, Wu-Ki Tung

High Energy Physics - Phenomenology · Submitted 21 Dec 1993 · 29 references

Pith's verdict: The framework paper behind general-mass heavy-quark DIS: the helicity factorization and the target-mass correction to the slow-rescaling variable are the real content, and both hold up; the phenomenology figure is thin and the factorization ansatz is acknowledged rather than closed.

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arXiv IDhep-ph/9312318 (version 2)

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CitationFredrick I. Olness, M. A. G. Aivazis, Wu-Ki Tung. "Leptoproduction of Heavy Quarks I-- General Formalism and Kinematics of Charged Current and Neutral Current Production Processes." arXiv:hep-ph/9312318 (1993). https://pith.science/paper/NXBW6V7T

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Editor's note

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This is the formalism paper that sits under the general-mass variable-flavour-number schemes people now use for heavy-quark DIS, and it's worth your time if you touch charm or bottom leptoproduction at all. The concrete deliverables are three: the helicity-basis factorization formula Eq. (21), justified by the observation in Section 4.3 that the target momentum P and the parton momentum k1 define the same polarization vectors because they are collinear; the explicit massive leading-order helicity amplitudes in Table 7; and the corrected scaling variable of Eq. (40), chi ~ x(1+m2^2/Q^2)(1-x^2 M^2/Q^2), which says the slow-rescaling variable itself needs a target-mass fix. That last point is the one that actually changed how fixed-target dimuon data get read, and it's derived cleanly. The helicity route (Eq. 105) and the Dirac-trace route (Eq. 107) are cross-checked against each other, and Tables 3-6 give the full transformation matrices so you can re-derive the map yourself. Good practice, though the equality of 105 and 107 is asserted as 'tedious algebra' rather than shown, so budget an afternoon if you care. On the soft spots: the reader weights the factorization ansatz too heavily. Section 4.1 does say Eq. (17) is an ansatz outside the proven regimes, and that's honest, but for the explicit calculation in Section 5 the initial parton is massless (m1=0), so this is ordinary DIS factorization with a light initial parton. The ansatz only bites for massive initial partons and for the neutral-current case, both deferred to paper II. The stress-test note has this right. The genuinely thin part is Fig. 4: the PDF set is unstated, no uncertainties are attached to the quoted 25 percent, and only W+s -> c is included. That's an illustration, not the central claim, and the analytic results don't depend on it, but it should be labelled as such. Section 7 is candid that the LO quark contribution alone is insufficient for phenomenology because of quark-gluon mixing. Who should read it: anyone building or debugging a massive-DIS calculation, and anyone interpreting fixed-target charm data. It deserves a serious referee.

Referee report

Significance

If the program holds together, this is a useful and influential piece of formalism: it supplies a single bookkeeping scheme in which the two conventional heavy-flavour prescriptions appear as limiting cases, and it makes the mass dependence of the longitudinal structure function explicit (Eq. (27): F0 proportional to m2^2/Q^2), which is a clean, falsifiable statement that the Callan-Gross relation fails at finite quark mass. The paper earns credit for several things a referee should name. The transformation matrices between invariant and helicity bases (Tables 3-6) are given explicitly, so the map is checkable rather than asserted. The leading-order result is derived twice, once via helicity amplitudes (Eq. (105)) and once via the Dirac trace (Eq. (107)), and the two routes are claimed identical; I spot-checked the massless limit of Eq. (31) against Eq. (38) and found agreement, and I verified that Eq. (22) with m1 = 0 reproduces Eqs. (25)-(27) using the triangle function of Eq. (45). The generalized scaling variable of Eq. (40) follows correctly from the target-mass-corrected eta of Eq. (57). The helicity cross-section formula Eq. (14) is also internally consistent with the d-matr

Major objections (3)

[§4.1, Eq. (17); Abstract; §7] The abstract and introduction advertise a formalism 'valid for a wide kinematic range from near threshold to energies much higher than the quark mass', but §4.1 states plainly that the factorization theorem is established in this simple form only in specific asymptotic regimes and that Eq. (17) is being applied 'as an ansatz', justified by matching onto the known limits Lambda << m^2 ~ Q^2 and Lambda < m^2 << Q^2. The intermediate region Q^2 ~ m^2 is precisely the region of phenomenological interest, and matching in two limits does not establish correctness there. The authors are commendably explicit about this in the body, so the issue is not concealment but calibration: the abstract and the conclusions of §7 should carry the same caveat, and §7 should say concretely what would count as a test of the ansatz in the intermediate region (for example, the size of the residual scheme/scale dependence at NLO in paper II, or a comparison against a fixed-order massive calculation in the same region). As written, a reader who takes the abstract at face value would overestimate what has been demonstrated.
[§6, Fig. 4 and the paragraph following Eq. (39)] The headline phenomenological number, that the complete leading-order result differs from the slow-rescaling prescription by up to 25%, is not documented well enough to be reproduced or assessed. The parton distribution set is never stated, no factorization or renormalization scale is given, no uncertainty is attached to the deviation, and only the W+s -> c subprocess is included (the d-quark and antiquark channels are dropped, which is defensible for an illustration but should be stated as a limitation of the number, not only of the plot). In addition, the Fig. 4 caption specifies E_nu = 80 GeV while the plotted axes run over E_nu from 50 to 300 GeV, so it is unclear what is actually being varied. Since the 25% figure is repeated in the conclusions as a motivation for the whole program, the inputs and the definition of 'percent deviation' need to be stated explicitly.
[§6, paragraph following Eq. (39)] The sentence 'The main source of discrepancy arises from the charmed quark mass m2 which is only slightly larger than the target mass M; the latter should not be neglected if effects due to the former are significant' does not match what is being compared. Both Eq. (31) and Eq. (39) already contain m2 through the slow-rescaling variable xi = x(1 + m2^2/Q^2), so the difference displayed in Fig. 4 is attributable to the target-mass dependence (the eta of Eq. (57) and the hyperbolic angle psi), not to m2. The intended argument, that m2 and M are numerically comparable so that retaining one while dropping the other is inconsistent, is a good one, but the text as written attributes the discrepancy to the wrong quantity and will confuse readers trying to understand what Fig. 4 isolates.

Minor objections (7)

[§5, Eq. (22)] The on-shell delta function is written as delta(xi*chi - 1) in Eq. (22), whereas Eq. (70) and Eq. (104) give delta(xi/chi - 1), and Eqs. (25)-(27) also use the ratio form. The product form appears to be a typographical error and should be made uniform.
[Appendix A.5, Eqs. (75)-(78)] The sentence comparing the two routes reads 'Eq. (76) and Eq. (76) are identical to Eq. (78) and Eq. (78)', which is garbled; presumably the intended pairing is (75) with (77) and (76) with (78).
[§2, Eq. (4)] The propagator structure G^{mu nu} = g^{mu nu} - q^mu q^nu/M_B^2 is undefined for the photon, where M_B = 0. The text notes that the q^mu q^nu term can be dropped after contraction with the lepton current, but the photon case should be stated explicitly at the point where G^{mu nu} is introduced.
[Abstract] Typographical: 'energies much higher then the quark mass' should read 'than'.
[§3, Eq. (16) versus Appendix B, Eq. (97)] The same relations are given once in terms of the scaling functions F1, F2, F3 and once in terms of the invariant W1, W2, W3. Both are correct, but the reader has to reconstruct the conversion; a one-line reminder that F3 = (nu/M) W3 at the point of Eq. (16) would help.
[Table 7 and Eq. (104)] The Omega amplitudes are labelled by chirality superscripts (RR, RL, LL) while the surrounding text switches between g_R, g_L and g_V, g_A conventions. A sentence tying the two conventions together at Table 7 would prevent misreading, particularly for the mixed term proportional to m1*m2.
[References] Reference [26], on which the NLO and scheme-matching claims of the program rest, is listed as 'to be published'. This is acceptable for a first paper of a series, but the conclusions should be explicit that the reconciliation of quark-scattering and gluon-fusion mechanisms, which is the practical payoff of the formalism, is deferred.

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The paper

The paper claims

Heavy-quark leptoproduction should be described by one factorization formula that keeps the quark mass and the target mass exactly, rather than by a slow-rescaling patch applied to a massless parton formula.

The reading

Existing calculations of charm and bottom production in lepton-hadron scattering split the world into 'light' and 'heavy' quarks and then use different machinery on each side: a massless parton formula with a slow-rescaling variable substitution on one side, gluon-fusion diagrams on the other. The authors argue this split is artificial, and that a single factorization-based treatment can cover the whole range from threshold to energies far above the quark mass, provided the quark masses and the target mass are both kept exactly. The technical vehicle is the helicity formalism: because the target momentum and the struck parton momentum are collinear, they define the same set of vector-boson polarization vectors, so helicity structure functions factorize with no mixing, unlike the invariant structure functions $W_1, W_2, W_3$, which mix in a mass-dependent way. As a concrete demonstration they compute the leading-order quark structure functions with general masses and obtain a neutrino cross section whose effective scaling variable carries a target-mass factor on top of the familiar charm-mass factor. If they are right, the widely used slow-rescaling prescription is not a small correction away from the truth: it is an inconsistent truncation that can miss the complete leading-order answer by up to 25 percent in regions of phase space that fixed-target experiments actually cover.

What carries the argument

The load-bearing device is the helicity decomposition of the hadronic tensor into three structure functions $F_+, F_0, F_-$ (right-handed, longitudinal, left-handed vector boson), combined with the observation that the polarization vectors built from the target momentum $P$ and from the struck parton momentum $k_1$ coincide because the two are collinear in the parton model. That coincidence is what makes factorization transparent in the helicity basis and opaque in the invariant basis, where $k_1^μ = ζ_P P^μ + ζ_q q^μ$ with mass-dependent coefficients $ζ_P, ζ_q$ that mix $W_1, W_2, W_3$ with $W_4, W_5, W_6$. The second device is the exact on-shell condition for the final heavy quark, $k_2^2 - m_2^2 = 0$, which localizes the convolution at $ξ = χ ≡ [η(Q^2 - m_1^2 + m_2^2) + Δ(-Q^2, m_1^2, m_2^2)]/(2Q^2)$, with $η$ the target-mass-corrected Bjorken variable. In the small-mass limit this reduces to $χ ≃ x(1 + m_2^2/Q^2)(1 - x^2 M^2/Q^2)$, which is the usual slow-rescaling variable multiplied by a target-mass correction the authors say has been overlooked.

Extended reading notes

Core claim

The paper's central claim is that charged-current and neutral-current heavy-quark leptoproduction admit a single description, $W = f_N(ξ,μ) ⊗ ω(ξ,Q,m_1,m_2,M)$, in which the partonic hard amplitude $ω$ is computed with the full quark-mass dependence and the convolution variable $ξ$ is fixed by exact kinematics including the target mass. The authors show that in the helicity basis this statement takes an especially clean form, $F^{N}_{λ}(q,P) = Σ_a f_{aN} ⊗ ω^{a}_{λ}$, because the hadron and parton helicity amplitudes are related without any transformation, whereas the invariant structure functions require the mass-dependent mixing coefficients tabulated in the paper. They then evaluate the leading-order diagram with all masses retained and find $ω_+ = 0$, $ω_- = g_{La}^2 (Q^2 + m_1^2 + m_2^2 - Δ)/Δ · δ(ξ/χ - 1)$, and a longitudinal piece $ω_0$ proportional to $(m_2^2 - m_1^2)^2/Q^2 + m_2^2 + m_1^2$ that does not vanish as the quark masses go to zero only in the massless limit — so the Callan-Gross relation fails at finite quark mass already at leading order. The resulting neutrino cross section, their Eq. (31), is claimed to be the correct leading-order answer with full target-mass dependence, and the conventional slow-rescaling prescription is recovered only as a particular truncation of it.

Load-bearing premise

The paper assumes that the factorization theorem holds in its simple convolution form across the whole intermediate region where the quark mass is neither much smaller than nor comparable to the hard scale, even though the authors state plainly that the theorem is proven only in the two asymptotic limits and that they are treating it as an ansatz.

Editorial extensions

If this is right

The Callan-Gross relation fails at leading order once the produced quark has mass: the longitudinal helicity structure function $F_0$ is non-zero and of order $m_2^2/Q^2$ even for massless initial partons.
The conventional slow-rescaling substitution $x → x(1 + m_2^2/Q^2)$ is an inconsistent truncation, because the correct kinematic variable also carries the target-mass factor $(1 - x^2 M^2/Q^2)$.
For fixed-target neutrino charm production, the complete leading-order cross section deviates from the slow-rescaling prescription by more than 25 percent at small $y$ and at large $x$, for neutrino energies between 50 and 300 GeV.
Because the target mass $M$ is comparable to the charm mass $m_c$ in existing neutrino experiments, retaining one mass effect while dropping the other is not a controlled approximation.
The same framework is claimed to apply unchanged to neutral-current production and to bottom quarks, providing the starting point for the next-to-leading-order analysis promised in the companion paper.

Reading between the lines

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

If the target-mass correction to $χ$ is physically real, reanalyses of dimuon data would shift the extracted strange-quark distribution and any $|V_{cs}|$ determination that depends on it, so the correction is testable against existing fits rather than only against new data.
The gap between the ansatz and a proof could be closed numerically: comparing the complete leading-order formula to a full massive next-to-leading-order calculation in the intermediate region $m^2 \sim Q^2$ would show whether the interpolation the authors assume actually holds there.
The helicity bookkeeping relies only on the collinearity of the polarization reference momenta, which suggests the same no-mixing factorization should carry over to hadroproduction of heavy quarks, where the same collinear structure appears.
A sharper experimental discriminator than the total rate would be the small-$y$ shape of $dσ/dy$ at $E_ν \approx 50$–$80$ GeV, where the threshold region makes the two prescriptions diverge most and where the target-mass and quark-mass corrections add rather than cancel.

Original abstract

Existing calculations of heavy quark production in charged-current and neutral current lepton-hadron scattering are formulated differently because of the artificial distinction of ``light'' and ``heavy'' quarks made in the traditional approach. A proper QCD formalism valid for a wide kinematic range from near threshold to energies much higher then the quark mass should treat these processes in a uniform way. We formulate a unified approach to both types of leptoproduction processes based on the conventional factorization theorem. In this paper, we present the general framework with complete kinematics appropriate for arbitrary masses, emphasizing the simplifications provided by the helicity formalism. We illustrate this approach with an explicit calculation of the leading order contribution to the quark structure functions with general masses. This provides the basis for a complete QCD analysis of charged current and neutral current leptoproduction of charm and bottom quarks to be presented in subsequent papers.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,963 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.55 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.71 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Existing calculations of heavy quark production in charged-current and neutral current lepton-hadron scattering are formulated differently because of the artificial distinction of ``light'' and ``heavy'' quarks made in the traditional approach. A proper QCD formalism valid for a wide kinematic range from near threshold to energies much higher then the quark mass should treat these processes in a uniform way. We formulate a unified approach to both types of leptoproduction processes based on the conventional factorization theorem. In this paper, we present the general framework with complete kinematics appropriate for arbitrary masses, emphasizing the simplifications provided by the helicity formalism. We illustrate this approach with an explicit calculation of the leading order contribution to the quark structure functions with general masses. This provides the basis for a complete QCD analysis of charged current and neutral current leptoproduction of charm and bottom quarks to be presented in subsequent papers.

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	Existing calculations of heavy quark production in charged-current and neutral current lepton-hadron scattering are formulated differently because of the artificial distinction of ``light'' and ``heavy'' quarks made in the traditional approach.	background 0.87	0.06	0.06	asserted 1.00
s1	A proper QCD formalism valid for a wide kinematic range from near threshold to energies much higher then the quark mass should treat these processes in a uniform way.	gap 0.78	0.08	0.16	asserted 0.94
s2	We formulate a unified approach to both types of leptoproduction processes based on the conventional factorization theorem.	contribution 0.74	0.93	0.24	asserted 1.00
s3	In this paper, we present the general framework with complete kinematics appropriate for arbitrary masses, emphasizing the simplifications provided by the helicity formalism.	contribution 0.89	0.92	0.14	asserted 0.99
s4	We illustrate this approach with an explicit calculation of the leading order contribution to the quark structure functions with general masses.	contribution 0.81	0.94	0.11	asserted 1.00
s5	This provides the basis for a complete QCD analysis of charged current and neutral current leptoproduction of charm and bottom quarks to be presented in subsequent papers.	future work 1.00	0.26	0.15	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.75 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

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

levels run

P abstract

not run

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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 general lepton-hadron scattering process: N(P) + ℓ1 → X(PX) + ℓ2 via the exchange of a vector boson, B(q). The lepton momenta are ℓi while the initial and final hadronic momenta are P and PX , respectively. The lepton-boson and quark-boson couplings are specified by the following generic expression for the effective fermion-boson term in the electro-weak Lagrangian: L EW int = −gB h j (ℓ) µ (x) + J (h) µ (x) i V µ B (x) (2) where a summation over B is implied. The gauge coupling constant gB for the vector boson field VB depends on B and their values as prescribed by the Standard…

**Figure 2.** Pictorial representation of the factorization theorem for the hadron structure functions for inclusive deeply inelastic scattering. The process on the left is N(P) + B(q) → X(PX), and the factorized process on the right is N(P) → a(k1) (represented by the parton distribution function, f a N ) with the successive hard scattering interaction a(k1) + B(q) (represented by ω a µν). The vertical lines indicate an inclusive sum over the final states, X(PX). In Eq. (17), the label ‘a’ is summed over all parton species. The convolution integral variable ξ is the momentum fraction carried by…

**Figure 3.** Leading order hard-scattering amplitude for heavy quark production. The leading order diagram that contributes to ωλ is shown in

**Figure 4.** Percent deviation of leading-order cross section between the “slow-rescaling,” Eq. (39), and and complete, Eq. (31), for Eν = 80GeV , mc = 1.5GeV : (a) dσ/dy(ν+s → c) integrated in x over the range x = [0.1, 0.6]; (b) dσ/dx(ν +s → c) integrated in y over the range y = [0.1, 0.8] We now present numerical results comparing cross-sections calculated using the complete leading order formula Eq. (31) with that using the slow-rescaling prescription, Eq. (39). In Fig. 4we compare the y and x dependence for ν + N → µ − + c + X for neutrino energies ranging from 50 GeV to 300 Gev—a…

**Figure 5.** Basic process for inclusive boson B(q) nucleon N(P) scattering: N(P) + B(q) → X(PX), summed over the final state, X(PX) The components of the relevant 4-vectors in the lab frame are: P µ = M, 0, 0, 0 ℓ µ 1 = E1, 0, 0, −E1 ℓ µ 2 = E2, −E2 sin θ, 0, −E2 cos θ q µ = ν, +E2 sin θ, 0, −E1 + E2 cos θ (43) where, as throughout this paper, lepton masses are neglected. The cross section for the deep inelastic scattering process is given by the standard form: dσ = 1 2∆(s, m2 ℓ1 , M2) X spin

**Figure 6.** (a) The standard hadron configuration in {x, z}-space. Note that the hadron momenta are colinear with the z-axis, and the lepton momenta define the x − z plane. (b) This frame is related to the standard lepton configuration (

**Figure 7.** (a) The standard lepton configuration in {x, z}-space. Note that the lepton momenta are colinear with the z-axis, and the hadron momenta define the x− z plane; (b) The same fame seen in {x, t}-space. The hyperbolic cosine can be obtained from the formula: coshψ = 2P · (ℓ1 + ℓ2) ∆[−Q2, P2, P2 X] (62) Evaluating the scalar productions in the laboratory frame, we relate coshψ to the more familiar variables: coshψ = E1 + E2 p Q2 + ν 2 = η 2M2 − Q2 + 2η(s − M2 ) η 2M2 + Q2 −→ M → 0 (2 − y) y (63) In developing the helicity formalism (Appendix B), we encounter the “spin-1 rotation matrix”…

Figures reproduced from arXiv:hep-ph/9312318 by the authors.

References

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[27]

T. P. Cheung and Wu-Ki Tung, Phys. Rev. D3, 733 (1971); P. H. Frampton and Wu-Ki Tung, Phys. Rev. D3, 1114 (1971)

View on Pith 1971
[28]

Fredrick Olness and Wu-Ki Tung, Phys. Rev. D35, 833 (1987)

View on Pith 1987
[29]

John C. Collins, Davison E. Soper, George Sterman, Perturbative QCD, A.H. Mueller, ed., WorldScientiﬁc Publ., 1989. 28

View on Pith 1989

Forward citations

Cited by 36 Pith papers

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

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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.

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nCTEQ26 nuclear parton distribution functions
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N³LO DIS implementation in VFNS with improved massive neutral-current coefficient approximation for PDF fits.
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The Four Polarizations of the $W$ at High Energies
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A systematic NNLO data-theory comparison shows that the main PDF sets generalise to unseen LHC and HERA data about equally well once PDF, alpha_s, and missing higher order uncertainties are included.
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The O(alpha_s^3 n_f T_F^2 C_{A,F}) constant terms of the massive gluonic operator matrix elements A_gq,Q and A_gg,Q are derived at general Mellin N, completing the one-massless-one-massive fermion-line sector needed f...
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How later papers used it

60 later papers cite it

Evidence: some. 53 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 9 of the 97 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 13 papers with this label · 1 by the same authors

By the same authors

Included only the dominant target mass effects in the structure function expressions in the ACOT scheme of this paper.

nCTEQ15 - Global analysis of nuclear parton distributions with uncertainties in the CTEQ framework (2015) (same authors) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

49 arXiv papers cite this in 81 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 · 42 sentences without a confident label

Builds on it 1 paper

The S-ACOT-MPS scheme is based on an amended version of the S-ACOT scheme developed for DIS [ref] , and is applied to proton-proton collisions.

Probing heavy-flavor parton distribution functions at hadron colliders · Introduction and Motivations · 0.89

Uses its method or tool 9 papers

The neutron structure functions $F_i^neutron, A$ are computed correspondingly by assuming the isospin symmetry, i.e. [equation] The effects of the heavy quarks are taken into account by the SACOT-prescription [ref] .

Compatibility of neutrino DIS data and global analyses of parton distribution functions · Computation Framework · 0.99
Since each diagram contains different physics, we can use parity invariant projectors [ref] to extract their individual gauge invariant structure functions; for example,

Testing collinear factorization in a spectator model with mass corrections · DIS in a spectator model · 0.97
We work in the ACOT formalism [ref] to compute the structure functions in the helicity basis.

Target mass corrections in lepton--nucleus DIS: theory and applications to nuclear PDFs · The parton model with quark and hadron masses · 1.00

Compares with it 2 papers

We find in this way that for a specified renormalization and factorization scheme (namely $MS$), the FONLL [ref] and ACOT [ref] constructions give formally identical results, to all orders in perturbation theory.

Charm in Deep-Inelastic Scattering · Introduction · 0.81
We draw further comparison to heavy quark PDFs, where quark masses $(m_𝒬)$ violate scaling (“slow rescaling”) at $𝒪(m_𝒬^2/Q^2)$ but vanish in the DIS limit [ref] .

Weak bosons as partons below 10 TeV partonic center-of-momentum · Introduction · 0.96

Cites it as background 22 papers

[ref], the exact form of the mass-dependent part of these coefficient functions is not uniquely fixed in the case of perturbatively generated charm: this has led to the construction of several different (though equally valid) formulations in the literature, such as ACOT [ref] , S-ACOT [ref] , TR and TR$^'$ [ref] , FONLL [ref] , and the recent formulation of Ref. [ref] .

Charm in Deep-Inelastic Scattering · The 4 flavour scheme · 0.99
There are a number of such schemes available, namely the ACOT scheme [ref] (and its simplified variants S-ACOT [ref] , S-ACOT-$χ$ [ref] , and the more recent m-ACOT [ref] for hadron-hadron collisions), the TR scheme [ref] , and the FONLL scheme [ref] .

Resummation and Matching of $b$-quark Mass Effects in $b\bar{b}H$ Production · Introduction · 0.89
A number of such variable flavor number (VFN) schemes have been proposed for DIS structure functions, including ACOT [ref] S-ACOT [ref] , ACOT-$χ$ [ref] , TR and TR' [ref] , and FONLL [ref] , etc.

Mass corrections to the DGLAP equations · Introduction · 0.97

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.

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