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In electron-nucleus collisions at the future Electron-Ion Collider, the one-point energy correlator factorizes in TMD QCD and its nuclear modification factor is suppressed to about 0.25–0.4 at small angles, rising with angle, providing a di

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2026-08-03 15:24 UTC pith:THJFIF2S

load-bearing objection Solid TMD factorization for an e+A energy-correlator observable; the numerical suppression curves are real predictions but rest on a thin nuclear-broadening model, so cite with eyes open. the 3 major comments →

arxiv 2512.16847 v2 pith:THJFIF2S submitted 2025-12-18 hep-ph nucl-th

Exploring nuclear modification using one-point energy correlator at the electron-ion collider

Yu Fu , Zhong-Bo Kang , Jani Penttala , Yiyu Zhou This is my paper
classification hep-ph nucl-th
keywords one-point energy correlatortransverse momentum dependent factorizationcold nuclear mattertransverse momentum broadeningnuclear modification factorelectron-ion collidersemi-inclusive deep inelastic scatteringEEC jet function
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish that the one-point energy correlator (OPEC) — the energy-weighted correlation of a hadron's direction with a fixed reference axis — can be computed in e+p and e+A collisions through transverse-momentum-dependent (TMD) factorization, at both the back-to-back limit and the collinear limit inside a jet. It derives the factorized formulas and uses them to predict the observable at EIC kinematics. The central physical claim is that the nuclear modification factor in gold targets is suppressed to roughly 0.25–0.4 at small angles and grows toward unity at larger angles, because cold nuclear matter broadens the intrinsic transverse momentum of quarks. This gives a new, differential window into transverse momentum broadening, complementing traditional TMD measurements.

Core claim

The paper's central claim is that OPEC in electron-nucleus collisions, in both the back-to-back limit (final hadron nearly opposite the beam) and the collinear limit (hadron nearly along the jet axis), factorizes into a hard function, a TMD quark distribution for the nucleus, an EEC jet function built from TMD fragmentation functions, and — for the jet case — soft functions for global and collinear-soft radiation. Working at NLL resummation accuracy with fitted non-perturbative inputs, it shows that for gold at EIC kinematics the nuclear modification factor R_eA is about 0.33 in the back-to-back limit at small τ and 0.25–0.35 in the in-jet limit at small R_L, increasing toward 1 as the angle

What carries the argument

The central object is the one-point energy correlator (OPEC): the energy-weighted distribution measuring the angle between a detected hadron and a fixed reference direction (the beam or the jet axis). The argument is carried by TMD factorization of the OPEC in its two singular limits, in which the unsubtracted TMD PDFs and EEC jet functions are combined with hard and soft functions. Cold-nuclear-matter effects enter by replacing vacuum PDFs/FFs with nuclear sets and by shifting the non-perturbative Gaussian widths of the TMDs as g_1,A = g_1 + a_N (A^{1/3}-1) and g_1,A^D = g_1^D + b_N (A^{1/3}-1), with fitted a_N and b_N. These shifted widths encode the transverse momentum broadening that pro

Load-bearing premise

The predictions rest on modeling the nuclear TMD broadening solely as a shift of the non-perturbative Gaussian widths of the quark TMD PDF and FF, scaled with A^{1/3} using parameters from an earlier fit; if the true medium effect is not captured by that simple shift, or if the soft functions are themselves medium-modified, the R_eA curves would change.

What would settle it

Measure R_eA for the back-to-back OPEC in e+Au and e+p at EIC kinematics (sqrt(s)=90 GeV, Q^2=20 GeV^2) and for the in-jet OPEC at pT=10 GeV, qT=0.5 GeV, over the angular range 10^{-3} to 0.1. If the observed angular shape of R_eA does not rise monotonically from about 0.25–0.4 toward 1, or if the extracted broadening parameters disagree with those from SIDIS TMD measurements, the Gaussian-width model or the factorized formalism would be falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the factorization is right, the EIC can measure the back-to-back OPEC ratio R_eA and directly extract the τ-dependence that carries the nuclear TMD PDF and FF broadening.
  • The in-jet OPEC gives an angle-by-angle map of final-state broadening: its R_eA angular shape isolates the nTMD FF/EEC jet function, so it can be fit to determine the nuclear width parameter b_N independently.
  • Comparisons between the back-to-back and collinear limits separate initial- and final-state cold-nuclear-matter effects, something a single inclusive TMD measurement cannot do.
  • The predicted suppression of ~0.25–0.4 at small angles, growing to ~1 at large angles, provides a distinctive signature that can be checked with early EIC data within the proposed kinematic reach.
  • The factorization framework generalizes to other energy-correlator observables in e+A (e.g., two-point EEC), giving a consistent TMD-based handle on cold nuclear matter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the A^{1/3} Gaussian-broadening model is correct, the same widths should describe R_eA across nuclei; measuring e+Au, e+Cu, and e+Al at the EIC would test the A-scaling of the suppression.
  • The paper keeps all soft functions in vacuum; if the nuclear medium also modifies the soft (low-energy) radiation, the R_eA curves would acquire an additional medium-dependent component, possibly steepening the angular rise. This is a testable extension.
  • Extending the in-jet OPEC to gluon-initiated jets (e.g., through photon-gluon fusion at smaller x) would probe gluon TMD broadening, which the present quark-focused setup does not access.
  • The OPEC could be combined with standard SIDIS TMD measurements at the same kinematics to cross-check the factorization: if both extractions of the broadening parameters agree, the TMD picture of cold nuclear matter is strongly supported.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies one-point energy correlators (OPEC) in e+p and e+A DIS at EIC kinematics. For the back-to-back limit (tau → 0) it derives the factorized expression in Eq. (2.15) in terms of TMD PDFs, an EEC jet function, and the appropriate soft function; for the collinear in-jet limit it derives Eq. (3.19) for back-to-back electron+jet production, with TMD PDFs, global/collinear-soft functions, and an EEC jet function. The paper then presents e+p and e+Au predictions at sqrt(s)=90 GeV, Q^2=20 GeV^2, pT=10 GeV, and defines nuclear modification factors R_eA. Nuclear effects are introduced by replacing collinear PDFs/FFs with nuclear sets (EPPS16, LIKEn21) and by shifting the non-perturbative Gaussian widths via a_N(A^{1/3}-1) and b_N(A^{1/3}-1) from Ref. [83]. The main numerical result is R_eA ≈ 0.25–0.4 at small tau or R_L, increasing with angle, interpreted as a signal of cold-nuclear-matter transverse momentum broadening. Appendix A checks the RG consistency of the Collins–Soper scale choices.

Significance. If the factorization and the nuclear-model assumptions are accepted, this is a useful extension of the EEC/OPEC program to e+A collisions: OPEC is an infrared-safe, energy-weighted angular probe that is complementary to ordinary q_T-differential TMD measurements, and the paper separates initial- and final-state nuclear effects in a clean way (Figs. 2, 3, 7). The factorization derivation is built on standard TMD/SCET ingredients, and the RG consistency checks in Appendix A are a valuable control. The paper also makes falsifiable predictions for the EIC. However, the quantitative suppression is not yet established: the nuclear effect is modeled solely by two Gaussian width shifts taken from one previous extraction, the soft functions are kept vacuum, and all figures lack uncertainty bands. With a careful model-sensitivity study and honest error treatment, the predictions would be a strong EIC target.

major comments (3)
  1. [§2.2.1, Eq. (2.29); §2.2.2, Eq. (2.38)] All nuclear broadening is encoded by changing g1^q and g1^D by a_N(A^{1/3}-1) and b_N(A^{1/3}-1), with values imported from Ref. [83] and no propagated uncertainties. In the ratios R_eA of Eqs. (2.39) and (3.21) the hard function, perturbative Sudakov factor, and vacuum soft functions largely cancel, so the angular dependence of the central predictions is essentially set by these two shifts and by the Gaussian-in-b functional form. The conclusion that OPEC provides a quantitative probe of TMD broadening therefore needs a sensitivity study (e.g., varying a_N,b_N over their extraction uncertainties, testing non-Gaussian b-dependence, or comparing with an independent Monte-Carlo implementation of broadening). Without such a study the quoted 0.25–0.4 suppression is a model result rather than a robust prediction.
  2. [§3.2, Eqs. (3.9), (3.10), (3.13)] The global soft function, the collinear-soft function, and especially the in-jet soft function S_q are taken as vacuum expressions. Since S_q controls the j_perp distribution that enters the collinear-limit OPEC via Eq. (3.12), any cold-nuclear-matter modification of this function would directly alter the R_L dependence of R_eA in Fig. 7. The factorization theorem alone does not prove that all nuclear effects are contained in the nTMD PDFs and nTMD FFs. An explicit argument (e.g., a power-counting estimate of soft-medium interactions at these kinematics) or a numerical uncertainty estimate is needed; this is load-bearing for the angular-shape claim.
  3. [§2.2.2, Eqs. (2.35)–(2.36) and Table 1] The EEC jet function is obtained by replacing the hadron/z-integral in Eq. (2.34) with the fitted form exp[-N b^alpha(1+r b^beta)]. For gold the parameters in Table 1 are Hessian averages from the LIKEn 2021 sets, but the nuclear broadening parameters a_N and b_N are not assigned uncertainties. This puts the R_eA curves three modeling layers away from direct QCD input: TMD factorization, the empirical fit to the z-integral, and the Gaussian broadening ansatz. Please at least show the fitted functional form against the direct integration over the chosen DSS2021 FFs in the range of b relevant for Figs. 2 and 6, or quote the fit uncertainty as part of the predictions.
minor comments (4)
  1. [§4 vs. §2.3/Fig. 2] The text near Fig. 2 states the back-to-back suppression is about 0.33 at small tau, while the Conclusion says R_eA^{b.t.b.} ≈ 0.4 in the small-tau region. Please harmonize the two numbers.
  2. [§3.3] There is a typo: 'yfuture work' should be 'future work'. Some symbols in the figure captions appear as boxes ('□'), likely a rendering issue; please check the compiled PDF.
  3. [General] All numerical figures are shown without uncertainty bands. At minimum, the non-perturbative parameters (g2, g1^q, g1^D, Q0^2, a_N, b_N, and the Table 1 fit parameters) should be varied within their quoted or published uncertainties so the reader can judge the robustness of the R_eA curves.
  4. [§3.3] The statement that 'we do not expect the initial state effect from the nTMD PDFs to affect the angular distribution inside the jet' follows from the factorized form of Eq. (3.19), where fq(x,b,...) is independent of R_L. Consider saying 'in this factorization' to avoid over-generalizing the statement to other jet observables.

Circularity Check

0 steps flagged

No circular derivation: the OPEC predictions are conditional on imported nuclear-broadening fits, not on the OPEC observables themselves.

full rationale

The factorization chain is non-circular. Eq. (2.5) obtains the back-to-back OPEC from the qT-differential SIDIS cross section by the integration identity sum_h ∫ dz_h z_h δ(τ−q_T^2/Q^2), and Eq. (2.15) follows algebraically from the standard soft-function subtraction in Eqs. (2.16) and (2.30); the EEC jet function is a reweighted z-integral of the TMD FFs, not a quantity assumed to equal the predicted ratio. The collinear-limit formulas, Eqs. (3.16)–(3.19), are rearrangements of the lepton-jet TMD factorization with the same EEC jet function, and the RGE consistency checks in Appendix A only fix scales. The numerical suppression R_eA ~ 0.25–0.4 is not fitted to OPEC data: it is controlled by the Gaussian width shifts in Eqs. (2.29) and (2.38), with a_N=0.016 GeV^2 and b_N=0.0097 GeV^2 taken from Ref. [83], plus the vacuum S_EEC_NP fit from Ref. [86]. These references share authors with this paper, so there is a mild self-citation burden; however, the imported parameters come from external fits to SIDIS/DY and TEEC data, not from the OPEC observables being predicted, and the prediction is model-dependent rather than circular. No load-bearing equation reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The numerical predictions depend on a chain of fitted non-perturbative inputs from prior global analyses (nuclear PDFs, nuclear FFs, TMD Sudakov factors, A^{1/3} broadening parameters). The factorization theorems themselves are standard QCD/TMD results; the numerical results are model forecasts rather than parameter-free derivations.

free parameters (5)
  • a_N = 0.016 GeV^2
    Nuclear broadening shift for TMD PDFs (Eq. 2.29), taken from fit in Ref [83]; drives the nTMD PDF contribution to R_eA.
  • b_N = 0.0097 GeV^2
    Nuclear broadening shift for TMD FFs (Eq. 2.38), from Ref [83]; drives the nTMD FF contribution.
  • EEC NP function {N, alpha, beta, r} = p: alpha=0.42, beta=1.00, N=0.45, r=0.63; Au: alpha=0.56, beta=1.21, N=1.21, r=0.138 (GeV units)
    Fitted parameterization S_NP^EEC(b)=N b^alpha (1+r b^beta) (Eqs. 2.35-2.36, Table 1) representing the hadron/z-integral for the EEC jet function; controls the angular shape.
  • b_max = 1.5 GeV^-1
    Cutoff in b*-prescription (Eq. 2.24); standard non-perturbative regulator.
  • NP Sudakov parameters {g2, g^q_1, g^D_1, Q0^2} = g2=0.84, g^q_1=0.106 GeV^2, g^D_1=0.042 GeV^2, Q0^2=2.4 GeV^2
    Non-perturbative Sudakov parameters from global TMD fits [73,74]; used in the NP evolution factors.
axioms (5)
  • domain assumption TMD factorization for qT-differential SIDIS (Eq. 2.6) holds with unsubtracted TMD PDFs/FFs and soft function Snn^h.
    Foundational input for the back-to-back OPEC formula; cited to Refs [29,38,46-51].
  • domain assumption Collinear-limit OPEC factorizes for back-to-back lepton-jet production with TMD fragmenting jet functions and soft functions S_global, S_cs.
    Section 3.2, Eqs. (3.16)-(3.19); built on prior SCET/TMD results [64,89-91].
  • ad hoc to paper Cold nuclear matter modifies only TMD PDFs and TMD FFs; all soft functions remain vacuum ones.
    Sections 2.2 and 3.2; no medium modification of S^nnh, S_global, or S_cs is modeled.
  • domain assumption EPPS16 nuclear PDFs and LIKEn 2021 nuclear FFs describe the collinear nuclear inputs.
    Section 2.2.1-2.2.2; external global fits are used without uncertainty propagation.
  • ad hoc to paper The functional form N b^alpha (1+r b^beta) with alpha,beta>0 captures the hadron/z-integral in the EEC jet function.
    Eqs. (2.35)-(2.36); fitted to pion FFs, not directly to OPEC data.

pith-pipeline@v1.3.0-alltime-deepseek · 22454 in / 23243 out tokens · 204288 ms · 2026-08-03T15:24:45.272578+00:00 · methodology

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read the original abstract

We study the one-point energy correlator (OPEC) at both the back-to-back and collinear limits in electron-proton and electron-nucleus collisions. We provide the factorization formalism for the two types of OPEC and present phenomenological predictions in the kinematic region relevant for the future Electron-Ion Collider. Focusing on cold nuclear matter effects in electron-nucleus scattering, we demonstrate that the OPEC serves as a powerful probe of the transverse momentum dependent (TMD) physics and in characterizing the medium-induced transverse momentum broadening in cold nuclear matter.

discussion (0)

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