Self-consistent determination of proton and nuclear PDFs at the Electron Ion Collider

Abstract

We quantify the impact of unpolarized lepton-proton and lepton-nucleus inclusive deepinelastic scattering (DIS) cross section measurements from the future Electron-Ion Collider (EIC) on the proton and nuclear parton distribution functions (PDFs). To this purpose we include neutraland charged-current DIS pseudodata in a self-consistent set of proton and nuclear global PDF determinations based on the NNPDF methodology. We demonstrate that the EIC measurements will reduce the uncertainty of the light quark PDFs of the proton at large values of the momentum fraction x, and, more significantly, of the quark and gluon PDFs of heavy nuclei, especially at small and large x. We illustrate the implications of the improved precision of nuclear PDFs for the interaction of ultra-high energy cosmic neutrinos with matter. Introduction – The construction of an Electron-Ion Collider (EIC) [1, 2] has been recently approved by the United States Department of Energy at Brookhaven National Laboratory, and could record the first scattering events as early as 2030. By colliding (polarized) electron or positron beams with proton or ion beams for a range of center-of-mass energies, the EIC will perform key measurements to investigate quantum chromodynamics (QCD) at the intensity frontier. These measurements will be fundamental to understand how partons are distributed in position and momentum spaces within a proton, how the proton spin originates from the spin and the dynamics of partons, how the nuclear medium modifies partonic interactions, and whether gluons saturate within heavy nuclei. In this paper we focus on one important class of EIC measurements, namely inclusive cross sections for unpolarized lepton-proton and lepton-nucleus deep-inelastic scattering (DIS). In particular we study how such data could improve the determination of the unpolarized proton and nuclear parton distribution functions (PDFs) [3] by incorporating suitable pseudodata in a self-consistent set of PDF fits based on the NNPDF methodology (see Ref. [4] and references therein for a comprehensive description). The unique ability of an EIC to measure inclusive DIS cross sections consistently for the proton and a wide range of nuclei will be exploited also to update the proton PDFs used as a boundary condition in the nuclear PDF fit. This feature distinguishes our analysis from previous studies [5, 6], and may be extended to a simultaneous determination of proton and nuclear PDFs in the future. The results presented in this work integrate those contained in Sects. 7.1.1 and 7.3.3 of the upcoming EIC Yellow Report [7]. They systematically account for the impact of projected inclusive DIS measurements at an EIC on the unpolarized proton PDFs for the first time (for projected semi-inclusive DIS measurements see Ref. [8]), and supersede a previous NNPDF analysis of the impact of EIC measurements on nuclear PDFs [6]. Similar studies for polarized PDFs have been performed elsewhere [9–12], including in the NNPDF framework [13]. 1 ar X iv :2 10 2. 00 01 8v 1 [ he pph ] 2 9 Ja n 20 21 The structure of this paper is as follows. We first describe how EIC pseudodata are generated. We then study how they affect the proton and nuclear PDFs once they are fitted. Lastly, we illustrate how an updated determination of nuclear PDFs can affect QCD at the cosmic frontier, in particular predictions for the interactions of highly-energetic neutrinos with matter as they propagate through Earth towards large-volume detectors. Pseudodata generation – In this analysis we use the same pseudodata as in the EIC Yellow Report [7], see in particular Sect. 8.1. In the case of lepton-proton DIS, they consist of several sets of data points corresponding to either the neutral-current (NC) or the charged-current (CC) DIS reduced cross sections, σNC and σCC, respectively. See, e.g., Eqs. (7) and (10) in Ref. [14] for their definition. Both electron and positron beams are considered, for various forecast energies of the lepton and proton beams. In the case of lepton-nucleus DIS, the pseudodata correspond only to NC DIS cross sections, see, e.g., the discussion in Sect. 2.1 of Ref. [6] for their definition. Both electron and positron beams are considered in conjunction with a deuteron beam; only an electron beam is instead considered for other ions, namely 4He, 12C, 40Ca, 64Cu, and 197Au. A momentum transfer Q2 > 1 GeV2, a squared invariant mass of the system W 2 > 10 GeV2 and a fractional energy of the virtual particle exchanged in the process 0.01 ≤ y ≤ 0.95 are assumed in all of the above cases. The pseudodata distribution is assumed to be multi-Gaussian, as in the case of real data. It is therefore uniquely identified by a vector of mean values μ and a covariance matrix Σ, for which the following assumptions are made. The mean values correspond to the theoretical expectations t of the DIS cross sections obtained with a true underlying set of PDFs, and smeared by normal random numbers r sampled from the covariance matrix such that μ = t + rΣ. Specifically we use a recent variant [15] of the NNPDF3.1 determination [16], and the nNNPDF2.0 determination [17], for proton and nuclear PDFs, respectively. The covariance matrix is made up of three components, which correspond to a statistical uncertainty, an additive uncorrelated systematic uncertainty, and a multiplicative correlated systematic uncertainty. The statistical uncertainty is determined by assuming an integrated luminosity L of 100 fb−1 for electron-proton NC and CC DIS, and of 10 fb−1 in all other cases. The systematic uncertainties are instead determined with the djangoh event generator [18], which contains the Monte Carlo program heracles [19] interfaced to lepto [20]. These pieces of software collectively allow for an account of one-loop electroweak radiative corrections and radiative scattering. The Lund string fragmentation model, as implemented in pythia/jetset (see, e.g., Ref. [21] and references therein) is used to obtain the complete hadronic final state. The non-perturbative proton and nuclear PDF input is made available to djangoh by means of numeric tables corresponding to the relevant NC and CC DIS structure functions, which were generated with apfel [22] in the format of lhapdf [23] grids. The optimal binning of the pseudodata is determined accordingly. Two different scenarios, called optimistic and pessimistic henceforth, are considered for systematic uncertainties. The complete set of pseudodata considered in this work is summarized in Table 1. For each pseudodata set, we indicate the corresponding DIS process, the number of data points ndat before (after) applying kinematic cuts (see below), the energy of the lepton and of the proton or ion beams E` and Ep, the center-of-mass energy √ s, the luminosity L, and the relative uncorrelated and correlated systematic uncertainties (in percentage) σu and σc. The optimistic and pessimistic scenarios differ for the size of the projected systematic uncertainties and for the number of data points generated. The kinematic coverage of the EIC pseudodata in the (x,Q2) plane is displayed in Fig. 1 for the optimistic scenario. Pseudodata for lepton-proton and lepton-deuteron are separated from pseudodata for electron-ion collisions via different panels. The approximate coverage of currently available inclusive DIS measurements is shown as a shaded area. Dashed lines correspond to the kinematic cuts used in the PDF fits described below. From Fig. 1, we already can appreciate the relevance of the EIC for the determination of nuclear PDFs. In this case, the EIC measurements extend the kinematic reach of DIS by more than one order of magnitude in both x and Q2. In the case of proton PDFs, instead, the EIC measurements mostly overlap with those already available, in particular from HERA, except for a slightly larger extension at very high x and Q2.