Nuclear parton distributions at next to leading order

D. de Florian; R. Sassot

REVIEW

Nuclear parton distributions at next to leading order

D. de Florian, R. Sassot

High Energy Physics - Phenomenology · Submitted 18 Nov 2003 · 21 references

Pith's verdict: First full NLO extraction of nuclear parton densities, with a real improvement in fit quality and a genuine external check — but the LO-vs-NLO difference is asserted, not quantified.

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arXiv IDhep-ph/0311227 (version 1)

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CitationD. de Florian, R. Sassot. "Nuclear parton distributions at next to leading order." arXiv:hep-ph/0311227 (2003). https://pith.science/paper/MCDMLZKT

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Editor's note

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Worth your time if you use nuclear PDFs at all. The headline is real: this is the first full NLO global extraction of nuclear parton densities, and the convolution ansatz (their own earlier idea, ref. [9]) does the job with nine parameters per nucleus instead of the sprawling multiplicative correction factors. The fit is 300.15 for 393 points against roughly 630 quoted for EKS98 on the same data, and the whole parameter set is in Table II. That is a large, checkable improvement, and the Mellin-space convolution is what makes NLO tractable here at all. They also do the thing I wish more fits did: Figure 10 compares to PHENIX neutral-pion d-Au data that was not in the fit, and it agrees within errors. That is genuine independent support, not a self-check dressed up as one. The authors are upfront that the gluon is poorly constrained by DIS — fair, and I won't hold it against them. The soft spot is the second half of the abstract claim [abstract:s1]. Figure 8 shows LO and NLO ratios differing by amounts comparable to the nuclear effect, and the practical advice — don't pair LO nuclear factors with NLO proton densities — is right. But the comparison is not clean: the LO and NLO fits use different baselines (LO vs NLO GRV98), different initial scales (0.26 vs 0.4 GeV²) and different Λ_QCD (132 vs 167 MeV), so K_i mixes order with boundary conditions. Both fits land at comparable χ² with different parameters, so some of the difference could be re-tuning. And there are no error bands anywhere, so 'substantial' is qualitative. The stress-test note lands on this; it doesn't sink the operational advice, but it does mean the LO-vs-NLO gap shouldn't be quoted as a systematic without a caveat. Two smaller things: the isospin-invariant three-weight-function assumption (Eq. 10) is a real simplification for neutron-rich nuclei, and the grids are available on request rather than shipped. Neither is fatal. This is a solid, useful paper from people who know the subject. Send it out; the referees should push on uncertainty quantification and on making the LO/NLO comparison controlled, not on the central result.

Referee report

Significance

The result is significant: it is the first NLO extraction of nuclear parton densities, and the convolution parameterization is economical (nine parameters per nucleus) and numerically convenient because it permits Mellin-space evolution. The tabulated parameters and the explicit LO/NLO comparison make the analysis reproducible and directly useful to the RHIC and LHC communities. Two features deserve particular credit. First, the alternative nDSg set quantifies the sensitivity of the predictions to gluon shadowing rather than merely asserting it. Second, the comparison with PHENIX neutral-pion RdAu in Fig. 10 is a genuine external and falsifiable test that was not used in the fit and that agrees within experimental errors. The practical warning that LO nuclear correction factors should not be combined with NLO proton densities is important for current practice. The concern that the fit is circular because it is compared with the data it was fitted to does not land: that is standard global-fit practice, and the PHENIX comparison supplies the independent check.

Major objections (5)

[Section IV, Fig. 8, and abstract:s1] The second load-bearing claim of the abstract, that there are 'substantial differences in the scale dependence of nuclear effects compared to leading order analyses', is supported only qualitatively. Figure 8 shows K_i = R_i^NLO / R_i^LO without any uncertainty band, and the largest deviations sit precisely in the region the authors themselves describe as poorly constrained: low x_N, Q^2 below about 10 GeV^2, heavy nuclei, sea and gluon. The paper concedes that 'there is still a large uncertainty' on the gluon density, so the reader cannot judge whether the plotted differences exceed the fit uncertainty. Please provide at least a parameter-variation estimate of the uncertainty on R_i (a Hessian or Lagrange-multiplier band would be preferable) and overlay it on K_i, or state explicitly that the differences are not quantified and soften 'substantial' accordingly.
[Section III, paragraph beginning 'In order to compute the structure function'] The LO and NLO fits differ not only in the order of perturbation theory but also in the baseline densities (LO versus NLO GRV98), the initial scale (Q0^2 = 0.26 versus 0.4 GeV^2) and the QCD scale (Lambda_QCD^(5) = 132 versus 167 MeV). The ratio K_i in Eq. (18) therefore mixes the order effect with baseline and boundary-condition differences, and the attribution to 'order' is not isolated. Some of this is unavoidable, since one cannot consistently use NLO densities in an LO fit, but the paper should either perform a disentangling test (for example, recomputing R_i with a common baseline where possible) or state the caveat explicitly where the claim is made.
[Table II] Several A-dependence parameters differ drastically between the LO and NLO fits: beta_v has gamma = -0.0474 at NLO but 19.925 at LO; epsilon_v has delta = 0.0814 at NLO but 0.9510 at LO; epsilon'_v has delta = 0.9421 at NLO but 0.3657 at LO; and beta_s has delta = 1.1204 at NLO but 0.4284 at LO. Differences of this size in the same parameter at the two orders indicate that the fit is poorly constrained along those directions, which bears directly on the reliability of the A-extrapolations used in Fig. 6 and in the d-Au predictions of Section IV. Please discuss the parameter correlations and degeneracies, or demonstrate that the A-dependence of the ratios is stable under variations of these parameters.
[Section III, paragraph beginning 'It is worth mentioning'] The claim of a 'significant improvement in the agreement with data compared to previous extractions' (abstract:s1) rests on the comparison with the roughly 630 chi-squared attributed to EKS98 on the same data set. This is a strong statement about another collaboration's fit, and the manuscript does not document how it was computed: the cuts, the baseline densities, the Q^2 range and the treatment of correlated systematics are not specified. Please add enough detail (a table or an appendix entry) for the reader to verify the comparison. In addition, the improvement is between two LO fits (316 versus 630), so it is attributable to the convolution parameterization rather than to NLO; the abstract should say so explicitly, since 'significant improvement' could otherwise be misread as an NLO effect.
[Eq. (10) and Section IV, Fig. 10] The fit assumes isospin-invariant nuclear effects, with a single W_v multiplying both u_v and d_v and a single W_s multiplying both anti-u and anti-d. The authors acknowledge this as 'a first approximation', but the consequence for the paper's own predictions is not estimated. The Drell-Yan data that constrain the sea come from nuclei with Z/A well below 0.5, and the d-Au predictions of Fig. 10 involve a neutron-rich Au target, so an isovector component could bias the extracted sea and the predicted RdAu. Please provide a quantitative estimate of the bias, for example by allowing separate up and down sea weights in a variant fit, or at least discuss the expected size of the effect.

Minor objections (9)

[Table I and Section III] The data-point counts listed in Table I sum to 424, whereas the text quotes 420 points and the quoted degrees of freedom (393) correspond to 420 minus 27 parameters. Please reconcile the table with the text and with the quoted chi-squared per degree of freedom.
[Section III and Table II] The text quotes a total chi-squared of 300.15 for the NLO fit, while Table II lists 299.91/393. Please make the two numbers consistent.
[Section III, discussion of Fig. 1] The statement that the LO prediction is 'almost indistinguishable' from the NLO one in Fig. 1 appears to sit uneasily with the substantial LO/NLO differences shown in Fig. 8. Please clarify that the DIS structure-function ratios are insensitive to the sea and gluon differences displayed in Fig. 8, so that the two statements are not read as contradictory.
[Abstract and Section IV] The abstract refers to differences in the 'scale dependence' of nuclear effects, but Fig. 8 shows K_i as a function of x_N at several fixed Q^2 values. The scale dependence proper, dR_i/dln Q^2, is not displayed. Please either show the logarithmic derivative or reword the abstract to describe what is actually plotted.
[Section III, paragraph beginning 'In order to compute the structure function'] Nuclear corrections in the deuteron are neglected throughout. Please state the expected size of this approximation, since the reference ratios are all formed with deuterium and a few-percent bias would propagate into every extracted ratio.
[Section III, paragraph beginning 'Similar results are found'] The assertion that similar results are obtained with nucleon densities other than GRV98 is not supported by any figure or number. Please quantify the statement, for example by quoting the change in chi-squared or the change in R_i for one alternative set.
[Section III, comparison with HKM] The paper notes that the low-x_N behaviour of the valence densities is not constrained by the data, yet the comparison with HKM emphasizes precisely this region. Please state explicitly that the difference there is not data-driven, so that the comparison is not over-read.
[Section III, Drell-Yan treatment] The order at which the Drell-Yan cross section is computed and whether a K-factor or explicit NLO correction is applied are not stated. Please specify this, since it affects the comparison with the E772 data.
[Section III, final paragraph] The nuclear ratios are said to be available as FORTRAN grids 'upon request'. Given that these grids are the paper's main practical output, please deposit them publicly or state the interpolation accuracy of the grids.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A next-to-leading-order global fit to nuclear deep-inelastic and Drell-Yan data, built on a convolution parameterization, describes 420 data points far better than earlier leading-order extractions, and shows that the nuclear correction fac

The reading

The paper asks whether nuclear parton densities — the momentum distributions of quarks and gluons inside a nucleon bound in a nucleus — can be extracted at next-to-leading-order accuracy in QCD, the standard already routine for free protons. The authors answer yes, and the enabling move is to write each nuclear density as the free-proton density convolved with a simple weight function, rather than as the free density multiplied by an ad hoc ratio. With three such weight functions (valence, sea, gluon) and nine parameters per nucleus they fit 420 nuclear deep-inelastic and Drell-Yan data points, reaching $300.15$ units of $χ^2$ for 393 degrees of freedom, against roughly 630 for the widely used leading-order set on the same data. The practical payoff is a warning: the nuclear correction factors extracted at leading order and at next-to-leading order differ by amounts comparable to the nuclear effects themselves at low $Q^2$, so mixing leading-order nuclear factors with next-to-leading-order proton densities is not safe.

What carries the argument

The convolution ansatz $f_i^A(x_N,Q_0^2)=int_{x_N}^{A} rac{dy}{y}, W_i(y,A,Z), f_i(x_N/y,Q_0^2)$, where $W_i$ is an effective density of nucleons carrying a fraction $y/A$ of the nucleus longitudinal momentum. It buys three things: the full kinematic range $0<x_N<A$ instead of truncating at $x_N=1$; closed-form Mellin moments $hat{W}_i(N,A,Z)$ so that evolution and next-to-leading-order coefficient functions are multiplications in moment space; and a physical reading of nuclear effects as a rebalancing of momentum fractions between partons. Non-singlet combinations make $hat{W}_v$ scale independent, while sea and gluon weights mix and must be fixed at an initial scale. Momentum and charge conservation pin two of the parameters, leaving nine per nucleus, with a smooth mass-number dependence $ε_i=γ_i+λ_i A^{δ_i}$.

Extended reading notes

Core claim

Two claims carry the paper. First, the convolution parameterization makes a full next-to-leading-order global fit feasible at all: because the weight functions have closed-form Mellin moments, evolution and the hard coefficient functions reduce to ordinary multiplication in moment space, which the earlier multiplicative-ratio parameterizations with many parameters could not support. The resulting fit describes the NMC and SLAC-E139 structure-function ratios and the E772 Drell-Yan rates far better than the leading-order EKS98 set did on the same points, with $χ^2/mathrm{d.o.f.} = 299.91/393$. Second, and more consequential for practice, the leading-order and next-to-leading-order fits, although of comparable quality, give nuclear correction ratios $R_i(x_N,Q^2,A)$ that differ substantially — most in sea quarks and gluons at small $x_N$ in heavy nuclei — with the discrepancy shrinking as $Q^2$ grows. The authors use this to argue that leading-order nuclear factors should not be bolted onto next-to-leading-order proton densities, a common shortcut.

Load-bearing premise

The fit assumes nuclear modification is isospin-invariant and flavour-blind, so one weight function multiplies both up and down valence quarks and another multiplies both anti-up and anti-down sea quarks, making the extracted weights an average that could bias predictions for neutron-rich nuclei.

Editorial extensions

If this is right

Nuclear correction ratios from this fit can be multiplied onto any modern next-to-leading-order proton density set to build nuclear densities, provided both come from the same order in QCD.
Predictions for neutral-pion production in deuteron-gold collisions at $sqrt{s_{NN}}=200$ GeV agree with the PHENIX data within experimental errors except at the lowest $p_T$, and although the leading-order and next-to-leading-order cross sections differ by roughly 50 percent, the ratio to proton-proton largely cancels.
A variant set with stronger gluon shadowing ($R_g^{Au}=0.75$ at $x_N=0.001$, $Q^2=5$ GeV$^2$) costs about 250 units of $χ^2$, bounding how much small-$x$ gluon shadowing the present data allow.
The authors argue that a much stronger reduction of $R_{dAu}$ than their prediction at $p_T$ near 2 GeV would require physics beyond perturbative QCD.
Gluon densities remain the least constrained ingredient: among the data used, only the $Q^2$ dependence of $F_2^{Sn}/F_2^{C}$ is sensitive to them.

Reading between the lines

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

The paper's own side-by-side comparison of leading-order and next-to-leading-order ratios suggests a broader audit: any phenomenological study that imports a nuclear correction factor should state the order at which it was extracted, and the mismatch size shown here at low $Q^2$ is a rough template for how large that error can be.
Because the weight functions are convolutions rather than pointwise ratios, the framework extends naturally to nuclei where data are absent, so long as the mass-number dependence of the nine parameters is trusted; the risk is that this extrapolation does more work in the lead-lead and deuteron-gold predictions than the fit itself constrains.
The flavour-blind sea weight is the natural place for future data to bite: neutrino-nucleus scattering and Drell-Yan on neutron-rich targets would test whether an isovector piece is hiding inside the fitted average.
The bounded gluon shadowing of the variant set implies that sharper small-$x$ gluon constraints will have to come from future electron-ion colliders or forward hadron measurements rather than from the existing deep-inelastic ratios.

Original abstract

We perform a next to leading order QCD global analysis of nuclear deep inelastic scattering and Drell-Yan data using the convolution approach to parameterize nuclear parton densities. We find both a significant improvement in the agreement with data compared to previous extractions, and substantial differences in the scale dependence of nuclear effects compared to leading order analyses.

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We perform a next to leading order QCD global analysis of nuclear deep inelastic scattering and Drell-Yan data using the convolution approach to parameterize nuclear parton densities. We find both a significant improvement in the agreement with data compared to previous extractions, and substantial differences in the scale dependence of nuclear effects compared to leading order analyses.

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s1	We find both a significant improvement in the agreement with data compared to previous extractions, and substantial differences in the scale dependence of nuclear effects compared to leading order analyses.	result 1.00	0.96	0.79	asserted 1.00

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References

Works cited (21)

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Notice that this is not the case for the ratios R deﬁned in x−space as in Eq.(2)

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How the literature uses it

73 arXiv papers cite this in 97 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 · 51 sentences without a confident label

Contests it 1 paper

However, such an equation has not been found so far, therefore NLO analyses [ref] are not reliable.

Unitarity bound for gluon shadowing · Introduction · contest 0.88

Uses its method or tool 4 papers

Nuclear (isospin and shadowing) corrections of the CTEQ6.5M PDFs [ref] are introduced using the NLO nDSg parametrization [ref] .

Identified hadron spectra in Pb-Pb at sqrt(s) = 5.5 TeV: hydrodynamics+pQCD predictions · Introduction · 0.90
We use several parameterizations of the nuclear modifications in the parton densities to probe the possible range of gluon shadowing effects: EKS98 [ref] , nDSg [ref] , HKN [ref] , EPS08 [ref] and EPS09 [ref] .

Cold Nuclear Matter Effects on J/psi and Upsilon Production at the LHC · Shadowing parameterizations · 0.84
We apply the nuclear corrections $R_f$, defined as $f^A(x,Q^2)=R_f(x,Q^2,A) f(x,Q^2)$, separately for each parton flavour $f$ using the results of a NLO fit by de Florian and Sassot (nDS) [ref] , using the GRV98 [ref] free proton PDFs as input; see Fig. [ref].

Parton distributions for the LHC · Neutrino structure functions · 0.95

Compares with it 7 papers

Figure [ref] (right) shows the [0-10 %] case, where the data points are compared to several model predictions for the nuclear PDF (EPS09 [ref] , nDS [ref] and HKN07 [ref] ).

W, Z and photon production in CMS · Isolated Photons · 0.93
Figure [ref] shows a comparison of the nuclear effects in the average valence quark, average sea quark and gluon distributions at our initial scale $Q_0^2=1.69$ GeV$^2$, as obtained for a Lead nucleus in the LO DGLAP analyses here (EPS08), in HKN07 [ref] , in nDS [ref] , and in our previous works EKPS [ref] and EKS98 [ref] .

An improved global analysis of nuclear parton distribution functions including RHIC data · Discussion · 1.00
The minimal amount of gluon shadowing is nearly negligible, similar to nDS [ref] and even leaves room for some antishadowing in light ions.

Cold Nuclear Matter Effects on J/psi and Upsilon Production at the LHC · Shadowing parameterizations · 0.96

Cites it as support 1 paper

The earlier studies on this topic [ref] have already demonstrated that this non-trivial extension is successful, which strongly supports the collinear-factorization approach in describing the hard nuclear collisions.

EPS09 - a New Generation of NLO and LO Nuclear Parton Distribution Functions · Introduction · 0.99

Cites it as background 29 papers

The fit was done only by eye. * HKM [ref] & HKN [ref] which are LO QCD analyses as well, were the first ones to exploit the automated $χ^2$-minimization procedure and to estimate nPDF uncertainties. * nDS [ref] brought the global analysis of nPDFs to the next-to-leading order level in pQCD.

A global DGLAP analysis of nuclear PDFs · Introduction · 0.90
Since the first of the nPDF sets, EKS98 [ref] , the procedure has been improved by performing the analysis at next-to-leading order (NLO) [ref] and by making uncertainty estimates [ref] in analogy with the free proton case.

An improved global analysis of nuclear parton distribution functions including RHIC data · Introduction · 0.82
Historically, the inclusive pion production at RHIC was the first direct gluon constraint in the fits on nuclear PDFs [ref] .

EPPS21: A global QCD analysis of nuclear PDFs · Inclusive pion production · 0.97

Discussion

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