Ioffe-time distributions instead of parton momentum distributions in description of deep inelastic scattering

L. Mankiewicz; P. G\'ornicki; V. Braun

REVIEW

Ioffe-time distributions instead of parton momentum distributions in description of deep inelastic scattering

V. Braun, P. Górnicki, L. Mankiewicz

High Energy Physics - Phenomenology · Submitted 19 Oct 1994 · 23 references

Pith's verdict: The framework is borrowed, but the sum rule evaluation is new and the u-quark agreement is a real check — the z ≤ 4 claim just runs past the authors' own convergence bound.

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arXiv IDhep-ph/9410318 (version 2)

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CitationL. Mankiewicz, P. G\'ornicki, V. Braun. "Ioffe-time distributions instead of parton momentum distributions in description of deep inelastic scattering." arXiv:hep-ph/9410318 (1994). https://pith.science/paper/BIGRCWFV

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Editor's note

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The thing worth your time here is not the framework — that's Collins-Soper and Balitsky-Braun, and the z-distribution idea goes back to Pestieau-Roy-Terazawa in 1970, all cited. What's new is the actual sum rule evaluation: explicit coefficient functions for valence quarks in Eqs. (46)-(50), and a comparison to GRV. That comparison is the real content. The condensates, Borel parameter, and continuum threshold are standard ITEP/nucleon values, not tuned to the curve, and the normalization is fixed by the Ward identity to the valence quark numbers. So the u-quark curve in Fig. 8 tracking GRV out to z ~ 4 is a nontrivial check, not a fit. The methodological argument — coordinate space separates short and long longitudinal distances and gives smooth coefficient functions instead of delta-function-laden ones — is correct and well made, and the contrast with Ioffe-Belyaev's momentum-space sum rule is the sharpest part of the paper. Soft spots, in order of how much they matter. The z ≤ 4 claim runs past the authors' own convergence criterion, which they state as z < 3; they justify the extension with 'in practice,' which is not an argument. The terms dominating z ~ 3-4 are the dimension-6/8 contact pieces, and the gluonic bilocal power corrections are dropped without any numerical estimate (stated plainly in the Appendix). A number, or a figure isolating their size, would settle it; there isn't one. Second, the small-z agreement is partly guaranteed: the sum rule is normalized to U_val(0)=2 by construction, so the informative region is exactly the contested z ~ 3-4. Third, the d-quark sector fails and the authors say so, with a plausible diagnosis (the cos z term from the single-quark diagram, i.e. delta(1-u) in momentum space) and a nonlocal-condensate patch that helps but doesn't fix it. That's honest. None of this breaks the central point, which is methodological and survives: coordinate-space distributions are a smoother target for nonperturbative methods, and this paper shows the sum rule machinery produces something sensible for u. It's preliminary by the authors' own description, no radiative corrections, but the direction is right and the idea has legs beyond sum rules — lattice being the obvious one. Who should read it: anyone doing nonperturbative parton structure, lattice people thinking about coordinate-space quantities, and anyone who wants to see what a sum rule looks like when the OPE isn't poisoned by singular functions. Recommendation: worth a serious referee. The calculation is checkable in principle, the inputs are stated, and the methodological claim is the kind of thing that should be argued about in review rather than desk-rejected. Ask for the numerical size of the dropped gluonic BPCs and for the z < 3 versus z ≤ 4 gap to be closed.

Referee report

Significance

The reformulation is genuinely useful and has been influential: it makes contact with the OPE directly, avoids the singular-function ambiguities of momentum-space sum rules, and yields coordinate-space evolution equations that are local in the separation. The demonstration that a standard sum-rule calculation with standard ITEP condensate values and standard Borel parameters can reach z ~ 3-4 without tuning to the target is a nontrivial check, and the comparison is made against an independent published fit rather than a fitted curve. The honest reporting of the d-quark failure, and the explicit statement of the reliability criterion, are strengths. If the u-quark agreement survives a quantitative error analysis, the paper provides a template for lattice and instanton-model calculations of Ioffe-time distributions.

Major objections (3)

[Sect. 3, Fig. 8; Abstract] The headline claim is agreement of U_val(z) with the GRV fit up to z <= 4, but the paper's own convergence criterion is stated as z < 3 ('It turns out that for valence quark distributions this criterion is satisfied for z < 3'), and the extension to z ~ 4 is justified only by 'in practice.' Since the z = 3-4 window is exactly where the claim is most striking, a quantitative error estimate there is needed: vary the condensates, s0, and t within their stated ranges and show the spread of the sum-rule curve at z = 3-4, or state explicitly that the agreement in that window is qualitative. Relatedly, the abstract (abstract:s2) quotes 'light-cone distances of order ~1 fm,' while Sect. 3 quotes 'nearly 2 fm' for the same z <= 4; the two statements should be reconciled, since z = 4 corresponds to delta ~ 1.7 fm for M = 0.94 GeV.
[Appendix, after Eq. (A.5); Sect. 3] The calculation retains contact-term bilocal power corrections but explicitly drops BPCs containing explicit gluon fields ('we do not take into account additional BPC's from the expansion of the term with an extra gluon in (A.5)'; Sect. 3: 'we ... neglect contributions of correlation functions involving explicitly the gluonic fields'). The paper offers only a parametric argument that these enter at higher order in coupling and/or dimension; no numerical estimate or figure isolates their size. Because the retained contact terms are what produce the large-z tail that matches GRV, an unquantified gluonic BPC could plausibly shift the curve by an amount comparable to the sum-rule-GRV gap at z ~ 3-4. At minimum, an order-of-magnitude estimate of the leading neglected term, or a figure showing its size relative to the retained dimension-6 and dimension-8 terms, should be provided.
[Fig. 9; Sect. 3] Fig. 9 demonstrates stability only with respect to the Borel parameter t (1.0 versus 1.5 GeV^2). The d-quark discussion shows that the result is highly sensitive to the values of <qbar q> and m0^2: with <qbar q> = -(240 MeV)^3 and m0^2 = 0.8 GeV^2 the d-quark 'prediction' moves into agreement up to z ~ 1. The reader therefore cannot judge whether the u-quark agreement is similarly input-sensitive. A scan of the u-quark curve over the standard ranges of <qbar q>, <(alpha_s/pi)G^2>, m0^2, and s0 would materially strengthen the central claim and should be added.

Minor objections (7)

[Eq. (2) and following sentence] For alpha = -0.5, Eq. (2) gives Q(z) ~ z^{alpha+1} = z^{0.5}, which grows with z, whereas the text immediately below states that 'the valence quark distributions decrease at large z as ~ 1/sqrt(z).' The exponent in Eq. (2) appears to be missing a minus sign (the Fourier transform of u^alpha near u = 0 behaves as z^{-(alpha+1)}). Since Eq. (2) is used to extend the sum-rule curve to large z, the corrected form should be given.
[Eqs. (27) and (29)] Eq. (27) contains gamma(-nu-1) while Eq. (29) contains gamma(-nu+1); solving Eq. (27) with alpha_s(mu^2) = 4 pi / (b ln(mu^2/Lambda^2)) gives the exponent gamma(-nu-1)/b, so the argument in Eq. (29) appears to be a typographical error. Eq. (29) also contains a stray closing parenthesis in 'gamma(-nu+1))/b'. These should be checked, since Sect. 2 is presented as establishing the self-consistency of the coordinate-space formalism.
[Eqs. (37), (50)] The normalization U_val(0) = 2 is imposed by construction through the Ward identity and Eq. (50), so the agreement at z = 0 carries no dynamical information; only the shape at z > 0 is a test. This is implicit in the text but should be stated explicitly so that the small-z agreement in Fig. 8 is not over-read.
[Sect. 3, Figs. 8 and 10] The 'experimental' curves are leading-order GRV parametrizations at two scales; the word 'experimental' should be softened to 'phenomenological' or 'fitted,' and the two thick dashed lines in Figs. 8 and 10 should be labelled individually so the reader can tell which scale is which.
[Sect. 3, radius claim] The statement that 'the radius of the valence u-quark distribution obtained from the sum rule in (45) is close to the measured electromagnetic radius of the proton' is given without a number; the extracted radius and its uncertainty should be quoted.
[Eqs. (60)-(67), Figs. 11-12] The nonlocal-condensate model introduces two parameters, M_D and a, constrained only by Eq. (67); the choice M_D = 0.7-1.0 GeV should be justified or at least its allowed range stated, since Fig. 12 shows the result depends on it.
[General] There are numerous typographical errors: 'Aa anticipated' (Sect. 1), 'illlustrate' (Sect. 1), 'critisize' (Sect. 3), 'explicitely' (Sect. 2), 'dependendence' (Sect. 3), 'presu mably' (Sect. 3), 'thenumber' (Sect. 3), 'of of' (Sect. 3), and the header date 'October 9, 2018,' which is clearly a compilation artifact.

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The paper

The paper claims

Parton distributions are better computed in coordinate space than in momentum space, and a QCD sum rule shows it for the proton's up quark.

The reading

The paper argues that the usual momentum-fraction parton distributions are the wrong object to hand to a nonperturbative method: a fixed value of Bjorken $x$ mixes physics from short and long longitudinal distances, and that mixing shows up as singular, delta-function-like terms in the operator product expansion. The authors propose instead the Ioffe-time distribution, the Fourier transform of the momentum distribution with respect to the light-cone distance travelled by the struck quark, which is a smooth function given directly by a proton matrix element of a nonlocal quark string operator on the light cone. They derive the evolution equations for these distributions directly in coordinate space, where the scale derivative at separation $z$ involves the distribution only at smaller separations, and then compute the valence distributions with QCD sum rules. The calculation reproduces the fitted up-quark distribution out to Ioffe time $z$ of about 4 and fails for the down quark, which the authors trace to the particular structure of the expansion in that channel rather than to the coordinate-space formulation itself.

What carries the argument

The load-bearing object is the Ioffe-time distribution itself, $Q_{ig!rm val}(z,ig!mu^2) = ig!int_0^1 du, ig!cos(uz), q_V(u,ig!mu^2)$, identified with a proton matrix element of a nonlocal light-cone operator carrying a path-ordered exponential. Two pieces of machinery make it usable. First, the coordinate-space evolution equation $ig!mu^2 ig!partial_{ig!mu^2} Q_{ig!rm val}(z;ig!mu^2) = -ig!frac{ig!alpha_s}{2ig!pi}C_F ig!int_0^1 du, K(u), Q_{ig!rm val}(uz;ig!mu^2)$, which shows that the scale dependence at separation $z$ is fixed entirely by the distribution at smaller separations, so the short-distance expansion is visible in the equation itself, unlike the case for fragmentation functions. Second, the sum rule: a double Borel transform in $p_1^2$ and $p_2^2$ with continuum subtraction, plus a Ward identity relating the three-point correlator at $ig!Delta = 0$ to the derivative of the two-point correlator, which enforces the normalization $ig!int du, q_V(u) = N_u = 2$ and $N_d = 1$. The same Ward identity converts the most important bilocal power corrections into contact terms, that is, into ordinary local condensate contributions.

Extended reading notes

Core claim

The central claim is that coordinate-space (Ioffe-time) parton distributions are the natural target for nonperturbative QCD, and that a standard QCD sum rule can actually deliver them for the proton's valence quarks. Using a three-point correlator of two nucleon interpolating currents with the nonlocal vector operator $P e^{ig!int A}$-dressed $ig!bar{Psi}(ig!Delta)ig!slashed{n}[ig!Delta;0]ig!Psi(0)$, the authors obtain sum rules whose coefficient functions are smooth functions of $z$, in contrast to the momentum-space version where the same terms become $ig!delta(u)$, $ig!delta(1-u)$ and derivatives thereof. Keeping perturbative, dimension-4, dimension-6 and dimension-8 contributions, the up-quark Ioffe-time distribution agrees with the Glück–Reya–Vogt fit up to $z ig!simeq 4$, quoted as light-cone distances of order 1 to 2 fm in the proton rest frame; the abstract says about 1 fm while the text says nearly 2 fm, reflecting a factor-of-two convention between light-cone and physical longitudinal distance that the paper states explicitly. The down-quark distribution comes out much shorter-ranged than the fit. The authors attribute this to a $ig!cos z$ term from the diagram in which one quark carries the whole proton momentum, to the absence of the dimension-6 bilocal correction in that channel, and to strong cancellation between dimension-6 and dimension-8 condensate contributions that makes the result sensitive to the normalization point.

Load-bearing premise

That truncating the operator product expansion at dimension 8, and dropping the bilocal power corrections that contain explicit gluon fields (a neglect the authors state in the Appendix), leaves the up-quark prediction numerically intact at $z$ near 4.

Editorial extensions

If this is right

If coordinate-space distributions are the right interface, lattice QCD and instanton-vacuum models can supply parton distributions at moderate $z$ without the singular-term problem that afflicts momentum-space sum rules.
The coordinate-space evolution equations give a practical alternative to the usual momentum-space GLAP equations, with the large-$z$ tail being the part that moves most between 4 and 20 GeV$^2$.
Matching the calculated small-$z$ region onto the Regge asymptotic form $ig!sim z^{ig!alpha+1}$ at $z ig!sim 5$ to 8 would reconstruct the full Ioffe-time distribution and hence, by Fourier transform, the momentum-space distribution.
The same construction applies to the polarized gluon distribution, where knowledge of $ig!Delta G(z)$ up to its maximum at $z ig!sim 6$ would fix the total gluon polarization to roughly 50 percent accuracy.
For the down quark, the failure points at the need for a different interpolating current, radiative corrections, or a large bilocal correction behaving as $z^2$ in coordinate space.

Reading between the lines

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

The argument is really about conditioning: a Fourier transform turns a delta-function-dominated expansion into a smooth one, so the same operator product expansion is far better behaved in $z$ than in $u$. This could be quantified by comparing the convergence of the two expansions term by term, which the paper does not do.
If the finite correlation length of the vacuum is what pushes nonperturbative strength away from $u ig!to 1$, then momentum-space sum rule calculations truncated at local operators will systematically over-weight the endpoint region, a bias that could be checked against existing fits.
The paper's own 'matching window' question, whether a region exists where both the small-$z$ expansion and the Regge asymptotics apply, is empirically answerable with present fits; the figures suggest $z ig!sim 5$ to 8, which is already where the sum rule is failing, so the window may be narrow or absent in the down-quark channel.
A useful independent test would be to compute the same Ioffe-time distributions with a different nonperturbative method and see whether the up/down asymmetry of agreement persists; if it does, it is a property of the interpolating current rather than of the coordinate-space formulation.

Original abstract

We argue that parton distributions in coordinate space provide a more natural object for nonperturbative methods compared to the usual momentum distributions in which the physics of different longitudinal distances is being mixed. To illustrate the advantages of the coordinate space formulation, we calculate the coordinate space distributions for valence quarks in the proton using the QCD sum rule approach. A remarkable agreement is found between the calculated and the experimentally measured u-quark distribution up to light-cone distances $\Delta^- = \Delta^0 - \Delta^3$ of order $\sim 1$ fm in the proton rest frame. The calculation for valence d quarks gives much worse results; the reasons for this discrepancy are discussed.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.95 · human participants 0.02 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.80 · mentions code or data availability 0.02.

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

We argue that parton distributions in coordinate space provide a more natural object for nonperturbative methods compared to the usual momentum distributions in which the physics of different longitudinal distances is being mixed. To illustrate the advantages of the coordinate space formulation, we calculate the coordinate space distributions for valence quarks in the proton using the QCD sum rule approach. A remarkable agreement is found between the calculated and the experimentally measured u-quark distribution up to light-cone distances $\Delta^- = \Delta^0 - \Delta^3$ of order $\sim 1$ fm in the proton rest frame. The calculation for valence d quarks gives much worse results; the reasons for this discrepancy are discussed.

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 argue that parton distributions in coordinate space provide a more natural object for nonperturbative methods compared to the usual momentum distributions in which the physics of different longitudinal distances is being mixed.	contribution 0.47	0.26	0.67	asserted 0.60
s1	To illustrate the advantages of the coordinate space formulation, we calculate the coordinate space distributions for valence quarks in the proton using the QCD sum rule approach.	method 0.51	0.95	0.07	asserted 0.97
s2	A remarkable agreement is found between the calculated and the experimentally measured u-quark distribution up to light-cone distances $\Delta^- = \Delta^0 - \Delta^3$ of order $\sim 1$ fm in the proton rest frame.	result 1.00	0.91	0.37	asserted 1.00
s3	The calculation for valence d quarks gives much worse results; the reasons for this discrepancy are discussed.	limitation 0.73	0.90	0.05	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.71 · evidence from simulation or computation 0.93.

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not examined

full text
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relations to other papers

levels run

P abstract

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

Figures

**Figure 1.** Ioffe time distributions for partons at the scale µ 2 = 4 GeV2 . The solid line denotes Gl¨uck, Reya, Vogt (GRV) [3] set of parametrizations, the dashed line denotes CTEQ [4] parametrization. Uval(z) and Dval(z) are the valence quarks distributions, QU (z) and QD(z) are the C-even up and down quark distributions, and G(z) is the gluon distribution

**Figure 2.** Model of the Ioffe time valence-type distribution corresp

**Figure 3.** Model of the Ioffe time distributions for polarized gluon den

**Figure 4.** Evolution of valence Uval(z, Q2 ) and Dval(z, Q2 ), and gluon G(z, Q2 ) Ioffe time distributions. GRV [3] parametrisation has been used. The solid line corresponds to Q2 = 4 GeV2 , the dashed line to Q2 = 20 GeV2 , respectively. 29

**Figure 5.** Evolution of up and down quark C-even QU (z, Q2 ) and QD(z, Q2 ) Ioffe time distributions. GRV [3] parametrisation has been used. The solid line corresponds to Q2 = 4 GeV2 , the dashed line to Q2 = 20 GeV2 , respectively. 30

**Figure 6.** Typical diagrams contributing to the OPE of the correlato

**Figure 7.** Generic form of a bilocal correction in the OPE of the corre

**Figure 8.** QCD sum rules calculation of the valence u-quark Ioffe time d

**Figure 9.** Stability of the sum rule for Uval(z) against variation of the Borel parameter t. The upper curve corresponds to t = 1 GeV2 , the lower to t = 1.5 GeV2 . 34

**Figure 10.** The sum rule prediction for Dval(z), shown as the thick solid line, is rather far from the leading-order GRV [18] parametrisation, and is much more short-range. As 15

**Figure 11.** The nonlocal condensate contribution to the d-quark sum rule, see Eq.(65). Lines marked as (a) and (b) correspond to the diquark mass parameters MD = 0.7 GeV and 1.0 GeV, respectively. Borel parameter t = 1 GeV2

**Figure 12.** QCD sum rules calculation of the valence d-quark Ioffe time

**Figure 13.** Graphical illustration of the Ward identity, Eq.(A.5). Dashed lines denote pathordered exponentials. 23

Figures reproduced from arXiv:hep-ph/9410318 by the authors.

References

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Power corrections and renormalons in parton quasi-distributions
hep-ph 2018-09 accept novelty 6.0 of 10

Renormalon analysis predicts the x-dependence of leading power corrections to lattice parton quasi- and pseudo-distributions: a 1/(x^2(1-x)) enhancement for quasi-PDFs and a (1-x) suppression for pseudo-PDFs.
Pion distribution amplitude from Euclidean correlation functions: Exploring universality and higher-twist effects
hep-lat 2018-07 conditional novelty 6.0 of 10

A global fit to three Euclidean current-correlation channels yields the pion distribution amplitude's second Gegenbauer moment and a first lattice estimate of the twist-4 parameter δ²_π ≈ 0.20-0.24 GeV², close to QCD ...
Parton distribution functions from reduced Ioffe-time distributions
hep-ph 2018-01 conditional novelty 6.0 of 10

The reduced Ioffe-time distribution must be converted to light-front PDFs through a dedicated factorization formula, not identified directly with the rest-frame-normalized correlator.
Exclusive processes in position space and the pion distribution amplitude
hep-ph 2007-09 conditional novelty 6.0 of 10

The two-electromagnetic-current correlator between vacuum and pion is computed to NNLO plus twist-4 in position space, enabling a lattice extraction of the pion distribution amplitude that avoids operator renormalization.
A simple non-parametric reconstruction of parton distributions from limited Fourier information
hep-lat 2024-12 conditional novelty 5.0 of 10

A fixed-hyperparameter Gaussian-process prior recovers parton distributions from limited Ioffe-time data and yields more realistic small-x uncertainties than simple parametric fits.
Parton Distribution Functions from Ioffe Time Pseudodistributions from Lattice Calculations: Approaching the Physical Point
hep-lat 2020-04 conditional novelty 5.0 of 10

A lattice QCD calculation using Ioffe time pseudo-distributions yields the nucleon valence PDF extrapolated to the physical pion mass, in broad but not precise agreement with global fits.
Generalized quasi parton distributions in a diquark spectator model
hep-ph 2018-08 accept novelty 5.0 of 10

In a scalar diquark model, quasi-GPDs converge to standard GPDs as hadron momentum grows, but the large-x region and small-skewness ERBL region need multi-GeV momenta before they agree.
Moments of Ioffe time parton distribution functions from non-local matrix elements
hep-lat 2018-07 conditional novelty 5.0 of 10

Moments of parton distribution functions can be read off from the small-Ioffe-time expansion of a UV-finite ratio of non-local lattice matrix elements, and the first two moments so obtained roughly match a direct twis...
Parton distributions and lattice QCD calculations: a community white paper
hep-ph 2017-11 conditional novelty 5.0 of 10

Lattice-QCD moments of proton parton distributions agree with global fits at roughly 1 to 2 sigma for polarized quantities but differ by up to 3 sigma for the unpolarized gluon, and only polarized lattice inputs would...
Pion distribution amplitude from Euclidean correlation functions
hep-lat 2017-09 conditional novelty 5.0 of 10

Lattice data at pion momenta up to 1.88 GeV reproduce the momentum-separation dependence predicted by QCD factorization for the pion distribution amplitude, demonstrating feasibility of the position-space method.
Neural-network analysis of Parton Distribution Functions from Ioffe-time pseudodistributions
hep-ph 2020-10 conditional novelty 4.0 of 10

Reduced Ioffe-time pseudodistributions from five lattice QCD ensembles yield the nonsinglet distributions V3 and T3 at 1.6 GeV with chi-squared per point near 1.15, once finite-volume, lattice-spacing, pion-mass and a...
Large-Momentum Effective Theory
hep-ph 2020-04 accept novelty 3.0 of 10

LaMET lets lattice QCD compute x-dependent parton distributions, GPDs, and TMDs from proton matrix elements at finite boost, matched to light-front quantities through effective field theory.
The x-dependence of hadronic parton distributions: A review on the progress of lattice QCD
hep-lat 2020-10 accept novelty 2.0 of 10

Lattice QCD has reached the point of producing x-dependent PDFs, first GPDs and exploratory TMD PDFs with quantified systematics, but reliable hadron boosts near or above 2 GeV remain out of reach.
Mapping Parton Distributions of Hadrons with Lattice QCD
hep-lat 2025-06 accept

A review of lattice QCD methods and results for x-dependent parton distribution functions and generalized parton distributions of hadrons.

How the literature uses it

47 arXiv papers cite this in 73 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 · 33 sentences without a confident label

Uses its method or tool 1 paper

For the non-local condensate we adopt the model proposed in [ref] :

Light-Cone Distribution Amplitudes of the Ground State Bottom Baryons in HQET · QCD Sum Rules · 1.00

Cites it as background 33 papers

Then taking the time component of Eq. [ref] we get [equation] The correlators defined in Eqs. [ref] and [ref] are known in the literature as (bare) Ioffe-time distribution (ITD) and pseudodistribution (pseudo-ITD) respectively [ref] .

Neural-network analysis of Parton Distribution Functions from Ioffe-time pseudodistributions · Introduction · 0.83
As a function of Lorentz invariant variables, $z^2$ and $≡zp_z$ (known as the Ioffe time [ref] or the quasi light-front distance [ref] ), $𝔐$ can be expressed in terms of invariant amplitudes, $ℳ_sp^(+)$ and $ℳ_pp$ [ref] ,

Gluon helicity distribution in the nucleon from lattice QCD and machine learning · Introduction · 0.85
All invariant amplitudes are functions of the invariant interval $z^2$ and the Lorentz invariant $(p ·z) ≡- ν$ [ref] called the Ioffe time due to its relation, up to a sign and normalization of the target's mass, with the original variable from DIS cross section analyses [ref] .

Towards the determination of the gluon helicity distribution in the nucleon from lattice quantum chromodynamics · Theoretical Background of Polarized Gluon pseudo-Distribution · 0.98

Discussion

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