Particle-level parton densities and uncertainties for the resolved photon at NLO

Andy Buckley; Frank Krauss; Peter Meinzinger; Vithyaban Anjelo Narendran

REVIEW

Particle-level parton densities and uncertainties for the resolved photon at NLO

Andy Buckley, Frank Krauss, Peter Meinzinger, Vithyaban Anjelo Narendran

High Energy Physics - Phenomenology · Submitted 22 Oct 2026 · 76 references

Pith's verdict: A genuinely new particle-level fitting method with an unusually honest first PDF set; the proof of method is solid, the error bands should be treated with caution.

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arXiv ID2610.31224 (version 1)

Pith ID6Q6VLJ5O, Pith's permanent record of this paper and its review

CitationAndy Buckley, Frank Krauss, Peter Meinzinger, Vithyaban Anjelo Narendran. "Particle-level parton densities and uncertainties for the resolved photon at NLO." arXiv:2610.31224 (2026). https://pith.science/paper/6Q6VLJ5O

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Editor's note

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PLgamma26 is a real step forward: the first resolved-photon PDF fit done at particle level, with hadronisation and MPI inside the fit rather than applied as external corrections, and the first photon PDF set constrained jointly by a broad slate of HERA and LEP jet data. The paper's two best features are its technical completeness (per-histogram selections, parameter ranges, fit statistics, the whole pipeline in the appendices) and its candour. Nearly every weakness a referee would list is already stated in Sections 7.2 and 8, including the MPI/PDF inconsistency [b247].

The proof-of-method claim holds. The machinery is real and reusable: PDF reweighting through SHERPA at NLO+PS, PROFESSOR surrogates, the APFEL++-derived evolution code, RIVET analyses, and a released 101-member LHAPDF set. The improvements over the SaS 1M baseline (ZEUS dijet chi2/bin from ~3.9 to ~0.8, inclusive jets from 8.4 to 1.9) show the fitted PDFs carry real information. The GS96/ACFGP comparisons are honest relative benchmarks; the paper does not oversell them as validation.

Soft spots, in proportion. The weak part is the quantitative error-set claim. The two-step fit tunes MPI with SaS 1M frozen, then fits six PDF parameters to overlapping data; PDF and MPI parameters are genuinely degenerate there, and the 100 replicas fluctuate only the experimental bins. The authors acknowledge the inconsistency but do not quantify how much the PDFs would shift under a PLgamma26-consistent MPI tune, so the 68% bands plausibly understate model uncertainty. Second, fit quality is moderate: total chi2/Ndf of 2.97 excludes the shape-only ZEUS four-jet data, which sit at 19.9 per bin, and the L3/OPAL data remain poorly described. Third, the gluon is effectively unconstrained: AhadG pins at its boundary and the low-x gluon band reflects the parametrisation restriction, not data; again, the authors say this plainly. Minor reproducibility notes: the modified evolution code is documented in a Master's thesis and the generator is a pre-release SHERPA. This is a fit, so in-sample agreement is expected; the error bands are experimental-only by construction.

Bottom line: read it as a proof of method with a usable-for-exploration PDF set, not as a precision catalogue. Anyone working on photon PDFs, EIC photoproduction, or generator-level fitting gets value. It deserves a serious referee; the useful demands are a combined jet plus F2gamma fit, a more flexible gluon parametrisation, and a quantification of the MPI/PDF degeneracy.

Referee report

Significance

The proof-of-method claim is credible and timely: demonstrating that a PDF can be fitted at particle level through a shower+hadronisation+MPI generator, rather than through external hadronisation corrections, opens a practical route to using the many low-scale jet measurements that fixed-order fits cannot exploit. The engineering is a genuine deliverable: the modified APFEL++′ evolution, SHERPA NLO reweighting, new RIVET analyses, and a public 101-member LHAPDF set. The relative comparisons with GS96 and ACFGP under the same tuned generator are an appropriate independent benchmark; that agreement with the fitted data is in-sample is inherent to any fit and is not, by itself, a flaw. The quantitative part of the central claim — an error set for quantitative uncertainty estimation — is not yet commensurate with what is delivered: the replica bands are conditional on a single MPI tune whose degeneracy with the PDFs is acknowledged but unquantified, and the overall fit quality is poor by conventional standards. With the additional quantification or explicit re-scoping proposed below, this would be a valuable contribution to the photon-PDF and MC-tuning literature.

Major objections (3)

[Secs. 5.3, 6.2, 8] The most serious gap concerns the sequential MPI-then-PDF fit. As described in Sec. 5.3, the AMISIC MPI model uses the full photon PDF for secondary scatters and applies only to the resolved (HERA) and double-resolved (LEP) components, precisely the components that carry the photon-PDF information; the MPI parameters of Table 2 are tuned to largely the same data as the PDF parameters of Table 5 (Appendices A and B). The two parameter sets are therefore degenerate in principle: a different MPI tune could be compensated by different fitted PDFs. The replica recipe of Eq. (31) fluctuates only the experimental bin values at one frozen MPI configuration, so the 68% bands do not sample this degeneracy. The authors explicitly acknowledge the mismatch in Sec. 8 ("there is an inconsistency between it and the final use-case") and recommend using SaS 1M for the MPI in applications, but the size of the induced PDF shift is never estimated. I request a quantification: for example, re-tune the MPI with PLγ26 or with an alternative photon PDF such as GS96, re-fit the six PDF parameters, and compare the shifted PDFs with the quoted replica bands; if the shift exceeds the bands, the error-set claim must be re-scoped. If such a scan is not feasible, the abstract and conclusions should state that the error set is conditional on the SaS 1M-based MPI tune. The authors acknowledge this: “As the MPI tunes were performed using the SaS PDF , there is an inconsistency between it and the final use-case.”
[Table 4; Secs. 6.3, 7.2] The quantitative deliverable advertised in the title and abstract is an error set for quantitative uncertainty estimation, but the achieved fit does not yet support that reading. Table 4 gives χ2/Nbins = 2.97 for 659 bins even though Eq. (30) already includes a 5% ad hoc theory-uncertainty term; the ZEUS three-jet data are described at 4.98/bin, OPAL dijets at 4.53/bin, L3 inclusive jets at 6.09/bin, and the ZEUS four-jet data enter only as unit-area shapes because the predictions overshoot the measured normalisation (Sec. 6.3; the shape-only four-jet χ2 is itself 19.89/bin). The replica bands of Eq. (31) propagate only the experimental uncertainties of the fitted bins, so the dominant data-theory discrepancies are not represented in the quoted uncertainties. The paper should either complement the replica bands with a model-uncertainty component (for example a scan over scale, shower, or MPI settings), or explicitly re-scope the abstract and conclusions to describe the bands as bin-wise experimental-uncertainty estimates conditional on the generator setup, and demote the "quantitative uncertainty estimation" wording.
[Table 5; Sec. 7.2; Figs. 3–4] Part of the quoted uncertainty is driven by the parametrisation box rather than by the data. The central fit lies at the edge of the sampling range for APL, Ahad, Bhad and AhadG; the fit was restricted to the sub-box Bhad, BhadG > −2.0 to keep the momentum integral finite; and the Table 5 note states that replicas preferring less physical B values are pinned at that boundary. The authors themselves state that the low-x gluon band "reflects the restricted gluon parametrisation discussed below rather than a constraint from data". Consequently the 68% intervals in Figs. 3–4 are not uniformly interpretable as fit uncertainties. I ask for a passage identifying, per flavour and x range, which regions of the bands are data-driven and which are artefacts of the box or of the fixed exponents Chad, ChadG and rs, and for correspondingly qualified wording in the conclusions. The authors acknowledge this: “The exponent BhadG , which is primarily responsible for the low- x tail, can, at very large negative values, also affect the high-x gluon region that we are sensitive to, but such values make the momentum integral diverge, so we restricted the fit to Bhad,B hadG >− 2.0 (Table 5).”

Minor objections (3)

[Table 2] The header χ2/Ndf is used with different conventions in the two collider rows: the HERA value 1698.2/508 matches 514 bins minus 6 fit parameters, while the LEP value 498.2/129 uses the raw bin count (129 bins, 5 parameters). Please state the convention explicitly in the caption.
[Fig. 2 caption; Sec. 7.2] For the PLγ26 predictions shown in Fig. 2, clarify which photon PDF is used inside the MPI model; Sec. 7.2 and the conclusions imply that SaS 1M remains in the MPI even for the PLγ26 curves, and the caption should state this explicitly, as Fig. 1 does for the tunes.
[Eq. (30)] Report the total χ2 also with the 5% regularisation term removed, or state the sensitivity of the quoted χ2/Ndf values to ε, so that the reader can gauge how much of the quoted goodness of fit is carried by the ad hoc term.

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The paper

The paper claims

The PLγ26 photon PDFs are the first fitted at particle level to jet data from both HERA and LEP, with hadronisation and multi-parton interactions built into the NLO predictions rather than corrected for afterwards.

The reading

The paper presents PLγ26, a next-to-leading-order set of parton distribution functions (PDFs) for the resolved photon — the state in which a photon briefly fluctuates into quarks and gluons. It is the first photon-PDF set fitted to a wide range of jet measurements from several experiments, combining ten analyses from HERA photoproduction and LEP photon–photon collisions. Its distinctive ingredient is that the fit is performed at particle level: predictions come from the SHERPA event generator with NLO matrix-element-plus-shower matching, and hadronisation and multi-parton interactions enter the fit itself instead of being removed from the data as external correction factors. Because the jet data sit at low scales where such non-factorisable effects are large, this lets the fit use far more measured bins than a fixed-order fit could. If the approach is sound, PLγ26 is the most jet-constrained resolved-photon PDF set available, and the first one whose 100-replica ensemble puts a quantitative error bar on that constraint.

What carries the argument

The argument runs on four linked pieces. First, the evolution: photon PDFs obey the inhomogeneous DGLAP equation, whose extra $\gamma\to q\bar q$ source term is implemented in the MS scheme by APFEL++′, a modified version of the APFEL++ evolution library running with four active flavours. Second, the parametrisation: the six-parameter SAL ansatz, specified at the starting scale $Q_0 = 1.3$ GeV, describes the $u$, $d$, $s$ quarks and the gluon, with charm generated radiatively in the evolution. Third, the generator: SHERPA produces particle-level predictions from MC@NLO-matched NLO matrix elements through the parton shower, hadronisation and the AMISIC multi-parton-interaction model, and the MPI parameters are tuned first — separately for HERA and for LEP — with the photon PDF fixed to SaS 1M. Fourth, the optimisation: 3000 randomly sampled PDF parameter points are propagated through a single SHERPA run by on-the-fly NLO reweighting, a polynomial surrogate is built for every observable bin in the manner of the PROFESSOR tuning toolkit, and a least-squares fit to the selected bins yields the central PDFs, with 100 Gaussian-smeared replica datasets providing asymmetric 68% uncertainty bands.

Extended reading notes

Core claim

On its own terms, the paper establishes that particle-level PDF fitting is practical and that it changes the picture of the photon's parton content. The fitted set PLγ26 describes the resolved-enriched ZEUS dijet data within 5–8% where the older GS96 and ACFGP sets deviate by up to 20%, and it markedly improves the description of ZEUS dijet and inclusive-jet photoproduction relative to the fixed SaS 1M PDF used in the preparatory MPI tune. At LEP the improvement is partial: PLγ26 overshoots the OPAL dijet data by 20–30%, where the older sets overshoot by 30–60%. Compared with recent fits to the photon structure function $F_2^\gamma$, PLγ26 has larger up- and strange-quark densities and a smaller gluon; the authors argue that jet data fix the combined quark-plus-gluon effect, so the difference reflects a compensation that only a combined fit to both jet and $F_2^\gamma$ data could fully disentangle.

Load-bearing premise

The fit assumes that MPI and PDF parameters do not trade off against each other: the multi-parton-interaction model is tuned once with the fixed SaS 1M photon PDF and then frozen while the six PDF parameters are fitted to the same data, so if MPI absorbs resolved-photon structure the central PLγ26 set and its replica error bands are both biased — a risk the authors acknowledge in Section 8.

Editorial extensions

If this is right

PLγ26 is the first MS resolved-photon PDF set whose uncertainty comes from jet data, so photoproduction predictions for the Electron–Ion Collider and for ultraperipheral LHC collisions can now carry a quantitative PDF error band.
The pipeline — PDF reweighting inside a single generator run, surrogate interpolation, replica fits — makes particle-level fitting computationally cheap enough to repeat, opening the same treatment to other low-scale observables where fixed-order fits depend on large external non-perturbative corrections.
The separately tuned HERA and LEP MPI configurations, which prefer less MPI activity than the SHERPA defaults and a photon smaller than the proton, are a standalone outcome for underlying-event modelling in photon-induced collisions.
A combined fit of the PLγ26 jet data with the $F_2^\gamma$ measurements used by the VALO 1.0 fit should separate the quark and gluon contributions that jet data only fix jointly, and would test whether the PLγ26 quark–gluon ratio is physical or an artefact of the restricted gluon parametrisation.

Reading between the lines

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

The decisive stability test the paper leaves open is to close the loop by re-tuning MPI with PLγ26 itself and refitting; if the central PDFs then move by more than a replica standard deviation, the published bands understate the uncertainty, and the paper's own recommendation to keep using SaS 1M for MPI in the meantime is a sign the authors regard this as an open issue.
Because non-perturbative modelling now lives inside the fit, the hadronisation and parton-shower models become part of the very definition of PLγ26; a refit with an independent generator would reveal how much of the set's shape is generator-specific, a source of uncertainty the paper currently folds into a flat 5% term.
Read broadly, the paper points toward a future in which PDF sets and generator tunes become a single joint product rather than separate artefacts reconciled afterwards — a direction the authors themselves gesture at in the conclusion.
The narrow low-x gluon band should not be taken as a data constraint, since the paper states it reflects the restricted gluon parametrisation; users of PLγ26 at small x should carry that caveat until $F_2^\gamma$ data or a freer parametrisation weighs in.

Original abstract

We present a new set of parton distribution functions (PDFs), PL$\gamma$26, for the resolved photon at next-to-leading order (NLO), the first to be fitted to a wide range of jet measurements from several experiments. Jet production in photoproduction at HERA and in $\gamma \gamma$ collisions at LEP is sensitive to both the quark and the gluon content of the photon, but the measurements lie at low scales, where non-perturbative corrections are large. We therefore choose a novel approach and fit the PDFs to these measurements at particle level, using NLO predictions from the general-purpose event generator SHERPA, matched to the parton shower and including hadronisation and multi-parton interactions (MPI). As part of the procedure, we tune the MPI model separately for photon--proton and photon--photon collisions. Modelling these non-factorisable effects in the fit, rather than through external correction factors, lets us include many more measurements and data points. We estimate the experimental uncertainties with replicas and compare the fitted PDFs with previous sets.

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, 2026-10-23: 1 request, 5,063 input tokens.

What the abstract itself states, as probabilities: claims first 0.92 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.95 · mentions code or data availability 0.04.

The abstract, sentence by sentence

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

We present a new set of parton distribution functions (PDFs), PL$\gamma$26, for the resolved photon at next-to-leading order (NLO), the first to be fitted to a wide range of jet measurements from several experiments. Jet production in photoproduction at HERA and in $\gamma \gamma$ collisions at LEP is sensitive to both the quark and the gluon content of the photon, but the measurements lie at low scales, where non-perturbative corrections are large. We therefore choose a novel approach and fit the PDFs to these measurements at particle level, using NLO predictions from the general-purpose event generator SHERPA, matched to the parton shower and including hadronisation and multi-parton interactions (MPI). As part of the procedure, we tune the MPI model separately for photon--proton and photon--photon collisions. Modelling these non-factorisable effects in the fit, rather than through external correction factors, lets us include many more measurements and data points. We estimate the experimental uncertainties with replicas and compare the fitted PDFs with previous sets.

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 a new set of parton distribution functions (PDFs), PL$\gamma$26, for the resolved photon at next-to-leading order (NLO), the first to be fitted to a wide range of jet measurements from several experiments.	contribution 1.00	0.96	0.62	asserted 1.00
s1	Jet production in photoproduction at HERA and in $\gamma \gamma$ collisions at LEP is sensitive to both the quark and the gluon content of the photon, but the measurements lie at low scales, where non-perturbative corrections are large.	background 0.99	0.08	0.04	asserted 1.00
s2	We therefore choose a novel approach and fit the PDFs to these measurements at particle level, using NLO predictions from the general-purpose event generator SHERPA, matched to the parton shower and including hadronisation and multi-parton interactions (MPI).	method 0.93	0.96	0.13	asserted 0.99
s3	As part of the procedure, we tune the MPI model separately for photon--proton and photon--photon collisions.	method 0.99	0.96	0.03	asserted 1.00
s4	Modelling these non-factorisable effects in the fit, rather than through external correction factors, lets us include many more measurements and data points.	interpretation 0.53	0.83	0.10	asserted 0.98
s5	We estimate the experimental uncertainties with replicas and compare the fitted PDFs with previous sets.	method 0.96	0.95	0.08	asserted 1.00

What kind of paper the classifier reports this is: dataset or benchmark 0.46 · evidence from analysis of existing data 0.93.

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.** Selected observable distributions from the MPI tunes: data compared with

**Figure 2.** Selected observable distributions with final MPI tunes and PDF fits: H

**Figure 3.** All PLγ26 flavours at different Q 2 values, with the central fit and the 68 % interval of the replicas. The c PDF error band is too small to be visible, as charm-quark content arises predominantly via perturbative γ splitting in the evolution, and is insensitive to the replica variations of parton content. 19

**Figure 4.** PLγ26 (±1σ, i.e. the 68 % interval of the replicas) compared with other photon-PDF sets at Q 2 = 5 GeV2 for x ≥ 0.1, and for x ≥ 10−3 in the low-x gluon plot. The lower panels show the ratio to the central PLγ26. The charm-quark plot does not contain VALO 1.0 as the quark is negative at these Q 2 values due to near-threshold DISγ to MS conversion. 20

Figures reproduced from arXiv:2610.31224 by the authors.

References

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