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Nuclear Structure Functions at a Future Electron-Ion Collider

E. C. Aschenauer, S. Fazio, M. A. C. Lamont, H. Paukkunen, Pia Zurita

arXiv:1708.05654v1nucl-exhep-exnucl-th

TL;DR

Theoretically robust LHC observables are limited to high Q2, making reliable constraints in the low-x, low-Q2 domain particularly challenging. The study assesses projected EIC measurements using an EPPS16-based nPDF analysis while accounting for parametrization-bias concerns in newly probed kinematics. At Q2 = 1.69 GeV2, the gluon distribution can be determined well down to x ∼ 10^-2, while small-x uncertainties shrink across the domain at higher Q2.

  • Problem

    Theoretically robust LHC observables are limited to high Q2, making reliable constraints in the low-x, low-Q2 domain particularly challenging.

    Source: §I · p. 2

  • Method

    The study assesses projected EIC measurements using an EPPS16-based nPDF analysis while accounting for parametrization-bias concerns in newly probed kinematics.

    Source: §I · p. 2, B. nPDF analysis · p. 9

  • Results

    At Q2 = 1.69 GeV2, the gluon distribution can be determined well down to x ∼ 10^-2, while small-x uncertainties shrink across the domain at higher Q2.

    Source: §VI · p. 16

  • Takeaways & Limitations

    At low Q2, FL measurements could reach x ∼ 7 × 10^-4, where saturation-model predictions differ distinctly from collinear-factorization predictions.

    Source: D. The longitudinal structure function · p. 9

    The Hessian uncertainty method becomes unstable for large uncertainties and complex parameter correlations, so extending the analysis to all parton flavors requires a different methodology.

    Source: B. nPDF analysis · p. 11

Abstract

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The quantitative knowledge of heavy nuclei's partonic structure is currently limited to rather large values of momentum fraction $x$ -- robust experimental constraints below $x \sim 10^{-2}$ at low resolution scale $Q^2$ are particularly scarce. This is in sharp contrast to the free proton's structure which has been probed in deep inelastic scattering (DIS) measurements down to $x \sim 10^{-5}$ at perturbative resolution scales. The construction of an Electron-Ion Collider (EIC) with a possibility to operate with a wide variety of nuclei, will allow one to explore the low-$x$ region in much greater detail. In the present paper we simulate the extraction of the nuclear structure functions from measurements of inclusive and charm reduced cross sections at an EIC. The potential constraints are studied by analyzing simulated data directly in a next-to-leading order global fit of nuclear parton distribution functions based on the recent EPPS16 analysis. A special emphasis is placed on studying the impact an EIC would have on extracting the nuclear gluon PDF, the partonic component most prone to non-linear effects at low $Q^2$. In comparison to the current knowledge, we find that the gluon PDF can be measured at an EIC with significantly reduced uncertainties.

I. INTRODUCTION

Nuclear PDFs remain less constrained than proton PDFs, particularly at low x and low Q2, despite their importance to nuclear physics and searches for nonlinear evolution. The proposed EIC’s broad kinematic reach and flexible beam configurations are intended to improve those constraints.

I. INTRODUCTION

  • Accurate nPDFs matter for interpreting high-energy nuclear collisions and for investigating the transition from linear to nonlinear evolution at low x and low Q2.In the saturation regime, recombination of low-x gluons becomes increasingly important.Source: §I · p. 1
  • Existing fixed-target nuclear DIS data do not reach much below x = 10^-2 in the perturbative region, leaving nPDFs substantially less constrained than proton PDFs.HERA proton measurements reach almost x = 10^-5 at Q2 ≳ 1 GeV2.Source: §I · pp. 1–2
  • LHC observables with robust theoretical constraints generally probe high Q2, making reliable constraints especially challenging in the low-x, low-Q2 region.Low-transverse-momentum p+A observables such as open charm can probe small-x gluons, but have significant theoretical uncertainties.Source: §I · p. 2
  • The paper studies whether EIC measurements can improve nPDF precision, focusing on the EIC’s potential to extend lepton-nucleus measurements to smaller x.The analysis uses EPPS16 as its starting point and notes the risk of parametrization bias when measurements enter previously unconstrained kinematics.Source: §I · p. 2

II. THE ELECTRON-ION COLLIDER PROJECT

  • The EIC proposals aim for a high-luminosity collider with flexible ion species and per-nucleon center-of-mass energies of 20–90 GeV for large nuclei.The proposed kinematic coverage is intended to constrain nuclear PDFs.Source: §II · p. 2
  • The study notes that, at an assumed 10 fb−1 integrated luminosity, structure-function measurements are mostly limited by systematic uncertainties, making detector control of those effects important.The detector requirements include broad coverage and capabilities for lepton identification, hadron identification, and heavy-quark vertex reconstruction.Source: §II · pp. 2–3

III. REDUCED CROSS SECTION AND LONGITUDINAL STRUCTURE FUNCTION

The paper describes inclusive and charm reduced cross sections as complementary ways to access nuclear structure. It explains how their sensitivities and kinematic requirements relate to quark and gluon distributions.

III. REDUCED CROSS SECTION AND LONGITUDINAL STRUCTURE FUNCTION

  • Inclusive DIS is a hard lepton–nucleon interaction in which the nucleon breaks up, while charm production proceeds through photon-gluon fusion.Figure 2 depicts the inclusive process and the charm-production mechanism.Source: §III · pp. 3–4, Figure 2 · p. 3
  • At small x, the reduced cross section is approximately expressed using F2 and FL; F2 is sensitive to quarks, whereas FL has a larger direct gluon contribution.Older fixed-target measurements largely disregarded FL, leaving its information and direct access to nuclear gluons unavailable.Source: §III · p. 3
  • Charm reduced cross sections provide complementary information on nuclear gluons, and an EIC could measure the previously unmeasured charm contribution to FL.Heavy quarks are mainly produced through photon-gluon fusion.Source: §III · p. 4
  • Heavy-quark cross sections require a general-mass variable-flavor-number scheme to retain mass effects at low Q2 and a well-defined high-Q2 limit.The scheme has inherent version and mass-definition dependencies that precise EIC heavy-flavor measurements could benchmark.Source: §III · p. 4
  • FL extraction requires reduced-cross-section measurements at several center-of-mass energies at fixed x and Q2, with sufficient range in Y+ for a precise fit.The lowest x values are unavailable for FL measurements because they can only be reached at the highest center-of-mass energy.Source: §III · p. 4

A. Inclusive reduced cross sections

Simulated inclusive reduced-cross-section measurements at the EIC are expected to have smaller uncertainties than current nPDF predictions, particularly at small x and low Q2. The study models multiple beam energies and includes statistical and systematic uncertainties.

A. Inclusive reduced cross sections

  • The simulation covers three center-of-mass energies—31.6, 44.7, and 89.4 GeV—with a combined integrated luminosity of 10 fb−1.It assumes 1.6% bin-by-bin systematic uncertainty and an additional 1.4% overall luminosity uncertainty.Source: A. Inclusive reduced cross sections · p. 4
  • The simulated e−+Au central values are adjusted to CT14NLO proton PDFs supplemented by EPPS16 nuclear modifications.Statistical and bin-by-bin systematic uncertainties are combined in quadrature; the overall luminosity uncertainty is not shown.Source: A. Inclusive reduced cross sections · pp. 4–5
  • At small x, especially at small Q2, expected EIC inclusive cross-section uncertainties are much smaller than EPPS16 and CT14NLO prediction uncertainties.The advantage is concentrated at small x, while older fixed-target data already constrain nPDFs stringently at larger x.Source: A. Inclusive reduced cross sections · p. 5

B. Reduced cross section in charm production

The study estimates EIC charm reduced-cross-section measurements using kaon tagging and finds projected experimental uncertainties smaller than EPPS16 theory uncertainties, including at large x. The selection has about 30% efficiency and low estimated background contamination.

B. Reduced cross section in charm production

C. QED Corrections

The study evaluates QED radiative corrections to inclusive and charm reduced cross sections at EIC kinematics. Their size and sign depend on inelasticity, Q2, and the hadronic final state, with cuts on hadronic information offering a way to reduce them.

C. QED Corrections

  • Most simulated real radiative photons have energies below 1 GeV and are emitted at very rear angles in the electron-going direction.Source: C. QED Corrections · p. 7, Figure 7 · p. 7
  • Radiative corrections grow with inelasticity y, while increasing Q2 reduces the maximum radiated-photon energy and tends to reduce corrections.Source: C. QED Corrections · p. 8
  • For inclusive σr, corrections become strongly negative at small y and large Q2 as restricted photon-emission phase space lets virtual corrections dominate.Source: C. QED Corrections · p. 8
  • The larger hadronic invariant mass in charm production narrows photon-emission phase space, reducing corrections at high y and high Q2 relative to inclusive σr.Source: C. QED Corrections · p. 8
  • Cuts on hadronic invariant mass or E − pz can reduce radiative corrections, especially at largest y and small x, but may not suffice at larger x.Source: C. QED Corrections · p. 8

D. The longitudinal structure function

The paper studies EIC extraction of inclusive and charm longitudinal structure functions through Rosenbluth separation. It finds that measurements can be precise in several bins, reaching x ∼ 7 × 10^-4 at low Q2, where saturation-model predictions for FL differ from collinear-factorization predictions.

D. The longitudinal structure function

A. Pseudodata for cross-section ratios

The study constructs cross-section-ratio pseudodata using NLO predictions and modeled uncertainties. The samples cover carbon and gold at multiple collision energies, without modeled correlations between energies or nuclei.

A. Pseudodata for cross-section ratios

B. nPDF analysis

The analysis tests how EIC inclusive and charm data could constrain nuclear PDFs using a more flexible small-x gluon parametrization. The simulated data reduce several PDF uncertainties, with the size and flavor reach depending on the observable and fit assumptions.

B. nPDF analysis

  • EPPS16’s small-x gluon form permits only a monotonic rise or fall toward x→0, so the analysis replaces it with a more flexible form to limit parametrization bias.The flexible form is used for gluons only in this work.Source: B. nPDF analysis · p. 10, Figure 11 · p. 10
  • Extending the flexible parametrization to all parton flavors is beyond this work’s scope because Hessian uncertainty estimation becomes unstable with large uncertainties and complex correlations.The authors identify Monte Carlo techniques as an alternative for such extensions.Source: B. nPDF analysis · p. 11
  • The baseline otherwise follows EPPS16, but uses a flexible small-x gluon form and Hessian uncertainties calculated with a linearized prescription and fixed Δχ²=50.The fixed tolerance approximately corresponds to EPPS16’s 90% confidence level.Source: B. nPDF analysis · pp. 10–11
  • High-energy EIC fits reduce uncertainties further than low-energy fits, with wider kinematic reach and approximately twice as many data points.Both the improved reach and the larger number of high-energy data points contribute to the additional reductions.Source: B. nPDF analysis · p. 11
  • Inclusive and charm cross sections do not significantly constrain strange quarks; charged-current charm production or semi-inclusive kaon production are proposed alternatives.The stated alternatives are suggested ways to constrain (anti-)strange quarks at an EIC.Source: B. nPDF analysis · p. 11
  • At small x, inclusive and charm EIC pseudodata reduce up- and down-quark uncertainties by up to a factor of two; charm data strongly constrain mid- and large-x gluons.The comparisons include the baseline and fits with inclusive data alone or inclusive plus charm data.Source: B. nPDF analysis · p. 11, Figure 12 · p. 12, Figure 13 · p. 13
  • At Q²=10 GeV², EIC data suppress small-x light-sea uncertainty by up to a factor of four, while valence-fit differences remain modest and small-x bands are parametrization-biased.The light-sea improvement is less sizable at Q²=1.69 GeV²; small-x bands there are artificially small because the original functional forms are stiff.Source: Figure 14 · p. 14, B. nPDF analysis · p. 15
  • Inclusive cross sections have reduced flavor-separation sensitivity because u- and d-quark contributions can anticorrelate and cancel, including for valence quarks at large x.For 197Au, the small-x flavor-difference term is suppressed by about 0.1 relative to the flavor-average term; selected neutral- and charged-current differences could partly cancel the average terms.Source: B. nPDF analysis · pp. 11, 14, 15

C. Impact on theoretical predictions

EIC pseudodata reduce uncertainties in inclusive and charm cross-section predictions, with the strongest gains at small x for inclusive data and large x for charm data. These constraints complement LHC measurements, which provide limited information at low Q^2.

C. Impact on theoretical predictions

  • At the smallest x, adding high-energy EIC pseudodata reduces inclusive cross-section uncertainty by another factor of two after low-energy data already reduce it.Figure 15 compares baseline, low-energy-only, and low- plus high-energy fits.Source: C. Impact on theoretical predictions · p. 15, Figure 15 · p. 16
  • At large x, EIC pseudodata have diminishing impact as constraints from older fixed-target data begin to dominate.This trend is reported for the inclusive cross-section predictions.Source: C. Impact on theoretical predictions · p. 15
  • Charm data shrink cross-section uncertainties further, especially at large x, after inclusive EIC data have already reduced them.Charm measurements reach x ∼0.3, and their parent gluons have higher x than the measured charm quarks; valence-quark contributions to charm cross sections start only at next-to-next-to-leading order.Source: C. Impact on theoretical predictions · p. 15
  • EIC pseudodata reduce uncertainties in LHC and RHIC observable predictions, complementing those measurements in tests of nPDF universality.Typical LHC p+Pb observables probe much higher Q^2, leaving their constraints on low-Q^2 observables scarce.Source: C. Impact on theoretical predictions · pp. 15–16

VI. SUMMARY

The EPPS16-based study finds that projected EIC inclusive and charm measurements can substantially reduce nPDF uncertainties, particularly for gluons. The gluon constraints extend across small x as Q^2 rises, while the analysis has more limited flexibility for quark flavors.

VI. SUMMARY

  • The study analyzes projected inclusive and charm EIC cross sections in an EPPS16-based global fit, using an extended small-x gluon parametrization.The analysis focuses especially on determining gluon densities.Source: §VI · p. 16
  • EIC pseudodata substantially reduce nPDF uncertainties, particularly for the currently weakly constrained gluon distribution.At Q^2 = 1.69 GeV^2, gluons can be determined well down to x ∼10^-2; at higher Q^2, small-x uncertainties shrink across the small-x domain.Source: §VI · p. 16
  • Charm-tagged cross sections substantially decrease gluon uncertainties at large x.Source: §VI · p. 16, Figure 16 · p. 17
  • Sea-quark uncertainties also decrease, especially above the parametrization scale, but the method cannot flexibly parametrize all quark flavors simultaneously.Source: §VI · p. 17
  • EIC precision and broad x and Q^2 coverage can support stringent tests of nPDF universality, alongside complementary LHC and RHIC results.Source: §VI · p. 17
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