Shadowing in inelastic nucleon-nucleon cross section?
Kari J. Eskola Ilkka Helenius Mikko Kuha Hannu Paukkunen
hep-ph nucl-th

















kari.eskola@jyu.fi

ilkka.m.helenius@jyu.fi

mikko.a.kuha@student.jyu.fi

hannu.t.paukkunen@jyu.fi

University of Jyvaskyla, Department of Physics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland

Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland


             



Experimental results of inclusive hard-process cross sections in heavy-ion collisions conventionally lean on a normalization computed from Glauber models where the inelastic nucleon-nucleon cross section σ_ nn^ inel – a crucial input parameter – is simply taken from proton-proton measurements. In this letter, using the computed electro-weak boson production cross sections in lead-lead collisions as a benchmark, we determine σ_ nn^ inel from the recent ATLAS data. We find a significantly suppressed σ_ nn^ inel relative to what is usually assumed, show the consequences for the centrality dependence of the cross sections, and address the phenomenon in an eikonal minijet model with nuclear shadowing.














                             

                              
Shadowing in inelastic nucleon-nucleon cross section?
    Hannu Paukkunen
    December 30, 2023
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§ INTRODUCTION



In a high-energy nucleus-nucleus collision the produced particle multiplicity correlates strongly with the collision geometry: the more central the collision, typically the higher the multiplicity. Experimentally, the centrality classification is obtained by organizing the events according to their multiplicity (or transverse energy) into bins of equal fraction, say 10%, of all events. Conventionally, 0–10% (90–100%) centrality refers to the events of highest (lowest) multiplicities, and 0–100% to all events, minimum bias. 


Inclusive hard processes in turn are rarer processes of a large momentum scale whose cross sections in nucleus-nucleus collisions are traditionally obtained by converting the measured per-event yields N_h^c/N_evt^c in a centrality class c into hard nucleon-nucleon cross sections σ_h^c through

    σ_h^c = 
    σ_nn^inel/⟨ N_ bin⟩_cN_h^c/N_evt^c ,

where ⟨ N_ bin⟩_c is the mean number of independent inelastically interacting nucleon-nucleon pairs, binary collisions, in the centrality class c, and σ_nn^inel is the inelastic nucleon-nucleon cross section. The model-dependent quantity ⟨ N_ bin⟩_c here is obtained from the Monte-Carlo (MC) Glauber model <cit.>. The nuclear modification ratio R_AA^h,c for the hard process, in the centrality class c, is then obtained by dividing σ_h^c by the corresponding minimum-bias cross-section in proton-proton collisions.

This method of constructing hard cross sections is a routine procedure in the heavy-ion measurements at RHIC and at the LHC and it has been used e.g. to construct nuclear modification ratios for jets <cit.> and charged particles <cit.> which, in turn, are widely used in theoretical studies of jet quenching <cit.> and partonic energy loss <cit.>. In the same way, Eq. (<ref>) forms the basis for measuring centrality-dependent cross sections of direct photon <cit.> and electro-weak (EW) boson <cit.> production, which can be used to study e.g. nuclear effects in parton distribution functions (PDFs) <cit.>. 

The basic inputs of the Glauber model are the nuclear geometry and σ_nn^inel <cit.>. In the MC Glauber model the positions of the nucleons are sampled event by event according to the nuclear density profile, usually the Woods-Saxon distribution <cit.>. The probability for an interaction between two nucleons depends on their mutual distance and σ_nn^inel. As a result, cross-section measurements through Eq. (<ref>) depend on σ_nn^inel in a non-trivial way. An established procedure is to take the value σ_nn^inel and its energy dependence from proton-proton measurements. However, at high-enough energies the particle production becomes sensitive to QCD dynamics at small momentum fractions x where some suppression is expected due to gluon shadowing <cit.> or saturation phenomena <cit.>. Such effects become more pronounced in heavy nuclei and towards lower scales so one could argue that in collisions involving heavy ions the value of σ_nn^inel should also be reduced relative to what is measured in proton-proton collisions. Through Eq. (<ref>), this would then change the obtained hard cross sections and nuclear modification ratios, and thereby affect all the subsequent analyses that take these measured cross sections as an input. In this way, the value of σ_nn^inel could be critical and have far-reaching consequences e.g. for the precision studies of jet quenching and other related phenomena. Thus, an alternative benchmark for σ_nn^inel is called for. 

As proposed in Ref. <cit.>, the Glauber model and its inputs could be tested by studying the production of well known ”standard candles”, such as EW bosons, in Pb+Pb collisions at the LHC, but so far this has been limited by the precision of the LHC Run-I measurements <cit.>. Thanks to the increased luminosity and collision energy of Run II, the recent W^±- and Z-boson measurements by ATLAS <cit.> have pushed the precision to a few-percent level enabling now a more precise Glauber model calibration. In the present letter, we use these ATLAS data to study the possible nuclear suppression of σ_nn^inel in Pb+Pb collisions. Since the ALICE measurement <cit.> is less precise and has no reference p+p data we leave it out from the analysis. The idea is to first nail down the EW-boson cross sections by using a next-to-next-to-leading order (NNLO) perturbative QCD (pQCD) with state-of-the-art PDFs for protons and nuclei. Using the theory prediction on the left-hand-side of Eq. (<ref>), we can then determine σ_nn^inel within the same MC Glauber implementation as in the experimental analyses. We find that the data favor a significant suppression in σ_nn^inel. We show that this is compatible with predictions from an eikonal minijet model with nuclear shadowing. We also demonstrate that the unexpected enhancement seen by ATLAS in the ratios R_PbPb^W^±,Z towards peripheral collisions disappears with the found smaller value of σ_nn^inel. 






§ NUCLEAR SUPPRESSION IN Σ_NN^INEL


The observables we exploit in this work to extract σ_nn^inel are the rapidity-dependent nuclear modification ratios for W^± and Z boson production in different centrality classes. Experimentally these are defined as

    R_PbPb^exp(y) = 1/⟨ T_AA⟩1/N_evtdN^W^±, Z_PbPb/d y/dσ^W^±, Z_pp/dy,

where the per-event yield is normalized into nucleon-nucleon cross section by diving with the mean nuclear overlap ⟨ T_AA⟩ = ⟨ N_ bin⟩_c / σ_ nn^ inel obtained from a MC Glauber model calculation. For minimum-bias collisions the same quantity can be calculated directly as a ratio between the cross sections in Pb+Pb and p+p collisions, 

    R_PbPb^theor(y) = 1/(208)^2dσ^W^±, Z_PbPb/d y/dσ^W^±, Z_pp/dy.

We have calculated the cross sections in Eq. (<ref>) at NNLO with the mcfm code (version 8.3) <cit.>. For the protons we use the recent NNPDF3.1 PDFs <cit.> which provide an excellent agreement to ATLAS data for W^± and Z boson production in p+p collisions at √(s)=5.02 TeV <cit.>. The nuclear modifications for the PDFs are obtained from the EPPS16 NLO analysis <cit.> which includes Run-I data for W^± and Z production in p+Pb collisions at the LHC <cit.> and provide an excellent description of the more recent Run-II data <cit.>. The available NNLO nuclear PDFs <cit.> do not include any constraints beyond deeply inelastic scattering, so the applied PDFs provide currently the most accurately constrained setup for the considered observables. The factorization and renormalization scales are fixed to the respective EW boson masses.

The ratios R_PbPb^theor and R_PbPb^exp are compared in the upper panel of Fig. <ref>. For W^±, R_PbPb^exp is formed by diving the normalized yield in Pb+Pb from Ref. <cit.> with the corresponding cross section in p+p from Ref. <cit.> adding the uncertainties in quadrature. The plotted experimental uncertainties do not include the uncertainty in ⟨ T_AA⟩. The theoretical uncertainties derive from the EPPS16 error sets and correspond to the 68% confidence level. Note that the W^± measurement is for 0–80% centrality instead of full 0–100%. However, for rare processes like the EW bosons the contribution from the 80–100% region is negligible so the comparison with the minimum-bias calculations is justified. It is evident that with σ_nn^inel = 70   mb both the W^± and the Z data tend to lie above the calculated result, which we will interpret as an evidence of nuclear suppression in σ_nn^inel as explained below.


By equating Eqs. (<ref>) and (<ref>) we can convert each data point to ⟨ T_AA⟩. The outcome is shown in the upper panel of Fig. <ref>. The obtained values tend to be higher than the nominal ⟨ T_AA⟩ = 5.605  mb^-1 (0–100%) and ⟨ T_AA⟩ = 6.993  mb^-1 (0–80%) which assume σ_nn^inel = 70   mb, see Table <ref>. The fact that the preferred values of ⟨ T_AA⟩ are independent of the rapidity strongly suggests that the original mismatch in R_PbPb is a normalization issue – the nuclear PDFs predict the rapidity dependence correctly. 



Since each ⟨ T_AA⟩ maps to σ_nn^inel through MC Glauber, we can also directly convert R_PbPb^exp to σ_nn^inel. Here, we have used TGlauberMC (version 2.4) <cit.> which is the same MC Glauber implementation as in the considered ATLAS analyses. The centrality classification is done with a two-component model, including negative binomial fluctuations

[While modifying σ_nn^inel, the parameters of the two-component model should be adjusted to maintain a good description of the measured multiplicity or transverse energy distribution. However, as the change in σ_nn^inel can be accurately compensated purely by increasing the mean of the negative binomial distribution not affecting the resulting ⟨ T_AA⟩, the presented results would remain unmodified.]

, similar to the ALICE prescription <cit.> with parameters from Ref. <cit.>. The obtained values of ⟨ T_AA⟩ are in an excellent agreement with the ATLAS values in Refs. <cit.> in all centrality classes when using the nominal, unsuppressed, value σ_nn^inel = 70 mb. The values of σ_nn^inel extracted from each data point are shown in Fig. <ref>. It is obvious that the data prefer a value of σ_nn^inel which is less than the σ_pp^inel = 70  mb obtained from p+p data. 



To quantify the optimal σ_nn^inel we fit its value by requiring a match between R_PbPb^exp and R_PbPb^theor treating the EPPS16 uncertainties as Gaussian correlated errors. In practice we define a χ^2 function by

    χ^2     =    ∑_i[
    𝒩_i R_i^exp- R_i^theor + ∑_k f_k β_i^k/𝒩_iδ_i^exp]^2 + T ∑_k f_k^2
    
    𝒩_i      =    ⟨ T_AA^i(σ_pp^inel)⟩/⟨ T_AA^i(σ_nn^inel)⟩

where i runs over the data points and k=1,…,20 over the number error-set pairs in EPPS16. The factors 𝒩_i with σ_pp^inel = 70 mb account for the shifted normalizations when σ_nn^inel changes. Also the data uncertainties δ_i^exp are scaled by this factor to avoid D'Agostini bias <cit.>. The tolerance T=1.645^2 in the penalty term takes into account scaling the 90% confidence limit uncertainties of EPPS16 into 68% and
β_i^k ≡[ R_i^theor(S_k^+) - R_i^theor(S_k^-)]/2,
where S_k^+ and S_k^- are the positive and negative variations, respectively, of EPPS16 error sets. The χ^2 is minimized with respect to σ_nn^inel and f_k (1+20 parameters). We find 

    σ_nn^inel = 41.5^+16.2_-12.0 mb ,
 where the uncertainties follow from the Δχ^2 = 1 criterion. The resulting values for ⟨ T_AA⟩ and σ_nn^inel are compared to the data-extracted values in Fig. <ref>, and the re-normalized data for R_PbPb are compared with theoretical predictions in the lower panel of Fig. <ref>. It is worth stressing that different final states prefer a very similar, suppressed value of σ_nn^inel and that a very good agreement in R_PbPb is found when normalizing with ⟨ T_AA⟩ calculated using the suppressed cross section in the MC Glauber calculation.








§ CENTRALITY DEPENDENCE


Even the quite significant suppression in σ_nn^inel leads to rather modest modifications in ⟨ T_AA⟩ for central and (close-to) minimum-bias collisions. The impact, however, grows towards more peripheral centrality classes, see Table <ref>. To illustrate this, Fig. <ref> compares the centrality dependent R_PbPb^exp  before and after rescaling the data by ⟨ T_AA(σ_pp^inel)⟩/⟨ T_AA(σ_nn^inel)⟩ using the fitted σ_nn^inel. The left-hand panels show the original ATLAS data including the quoted ⟨ T_AA⟩ uncertainties, and in the right-hand panels the data have been rescaled and the uncertainties follow from the σ_nn^inel fit. The striking effect is that the mysterious rise towards more peripheral collisions in the original data becomes compatible with a negligible centrality dependence, the central values indicating perhaps a mildly decreasing trend towards peripheral bins. As discussed e.g. in the ATLAS publications <cit.>, such a suppression could be expected from selection and geometrical biases associated with the MC Glauber modeling <cit.>. Also other effects such as possible centrality dependence of σ_nn^inel and the neutron-skin effect <cit.> may become relevant to explain the data behaviour in the far periphery. 






§ MINIJETS WITH SHADOWING


To study the plausibility of the obtained suppression in σ_nn^inel, we calculate its value in an eikonal model for minijet production with nuclear shadowing. The model is based on a similar setup as in Ref. <cit.> but in the eikonal function we include only the contribution from the hard minijet cross section σ_jet(√(s_nn), p_0, [Q]), calculated at leading order in pQCD. The transverse-momentum cutoff p_0 (which depends on √(s_nn), scale choice Q and the proton thickness) and the width of the assumed Gaussian proton thickness function we fix so that the model reproduces σ_pp^inel=70 mb matching the COMPETE analysis <cit.> at √(s) = 5.02 GeV. The free proton PDFs are here CT14lo <cit.>, and we take the nuclear PDF modifications from the EPPS16 <cit.> and nCTEQ15 <cit.> analyses. The results for σ_nn^inel, obtained with p_0 and proton thickness function width fixed to the the p+p case, are shown in Fig. <ref>. The error bars are again from the nuclear PDFs scaled to the 68% confidence level. As expected at the few-GeV scales, the predicted σ_nn^inel depends strongly on the factorization/renormalization scale Q, but within the uncertainties the nuclear suppression obtained from the fits to the ATLAS W^± and Z data seems compatible with the eikonal model predictions with both nuclear PDFs.








§ SUMMARY


In the canonical approach the normalization for the measured per-event yields in nuclear collisions is obtained from the Glauber model taking the value of σ_nn^inel from proton-proton measurements. Contrary to this, our strategy was to compare the state-of-the-art pQCD calculations with the measured W^± and Z boson R_PbPb and thereby unfold the value for σ_nn^inel at √(s_nn) = 5.02 TeV. We find that the recent high-precision ATLAS data from Run II prefer the value σ_nn^inel = 41.5^+16.2_-12.0 mb , which is significantly lower than σ_pp^inel = 70± 5 mb. Such a suppression is in line with the expectations from an eikonal minijet model including nuclear shadowing. Remarkably, when using the fitted value for σ_nn^inel, the unexpected enhancements of R_PbPb in peripheral collisions disappear and the results become compatible with no centrality dependence. A possible hint of a slight decreasing trend toward peripheral collisions is observed which would be qualitatively in line with possible selection and geometrical biases. Our results thus suggest that the standard paradigm of using σ_pp^inel as an input to Glauber modeling potentially leads to a misinterpretation of the experimental data.


We thank Mirta Dumancic for discussion concerning the ATLAS measurements. We acknowledge the Academy of Finland, projects 297058 (K. J. E.) and 308301 (H. P. and I. H.), and the Väisälä Foundation (M. K.) for financial support. The Finnish IT Center for Science (CSC) is acknowledged for the computing time through the project jyy2580.



apsrev4-1



