Centrality-dependent modification of hadron and jet production in electron-nucleus collisions

Centrality-dependent measurements of hadron and jet cross section attenuation in deep inelastic scattering on nuclei can shed new light on the physics of final-state interactions in the nuclear matter, including the path-length dependence of the in-medium parton shower formation and evolution. Recent simulation studies have demonstrated the feasibility of experimental centrality determination in $e$A reactions at the electron-ion collider via neutron detection in the zero-degree calorimeter. Motivated by these results, we present the first theoretical calculation of the production rate modification for hadrons and jets in central and peripheral $e$Pb collisions. We find that the variation in the suppression of inclusive jet cross section as a function of centrality is less than a factor of two. In more differential measurements, such as the distribution of hadrons versus the hadronization fraction $z_h$, the difference can be enhanced up to an order of magnitude.


I. INTRODUCTION
Reactions with nuclei have been an integral part of the study of quantum chromodynamics (QCD) for more than 40 years [1].
Medium-induced radiative corrections have attracted a lot of attention as a natural mechanism of cross section modification in cold nuclear matter.Specifically, they have been applied to interpret [17][18][19][20][21][22] Drell-Yan and J/ψ suppression at large Feynman-x in minimum bias pA [23,24], and jet modification in central pA at very high energies [25,26].Furthermore, in the framework of different theoretical formalisms, including parton energy loss, in-medium evolution, a hybrid approach and renormalization group analysis [27][28][29][30], bremsstrahlung from final-state interactions was shown to lead to hadron cross section attenuation in semi-inclusive deep inelastic scattering (SIDIS) on nuclei.The overwhelming majority of these calculations have focused on HERMES collaboration measurements on helium (He), neon (Ne), krypton (Kr) and xenon (Xe) [31][32][33], but early EMC collaboration results [34,35] show the same type of nuclear modification using carbon (C), copper (Cu) and tin (Sn) as targets.
Final-state radiative corrections are not the only possible explanation of HERMES and EMC results.Models on early hadron formation and absorption in nuclear matter have been developed [36,37] and the possibility of universal fragmentation function modification has also been suggested [38,39].It was found that light hadron measurements at HERMES do not have sufficient discriminating power to uniquely validate or exclude theoretical models [40,41].
The electron-ion collider (EIC) will provide flexible center-of-mass energies and the opportunity to access final states that have not been studied thus far in SIDIS on nuclei.Recently, significant progress has been made in extending the theory of light and heavy hadron suppression [29,30], and jet and jet substructure modification [42][43][44] in eA at the EIC.All of these studies have been limited to minimum bias collisions.Centrality-dependent measurements can provide new insights into the physics of final-state interactions in nuclear matter and centrality class determination has been shown to be feasible via neutron tagging [45,46].To this end, we present theoretical results on hadron and jet modification in central and peripheral electron-lead (ePb) collisions at the future facility.
The rest of this paper is organized as follows: in Sec.II we briefly review the theoretical formalism for hadron and jet production on protons and nuclei.Discussion of centrality determination in SIDIS and phenomenological results in central and peripheral ePb collisions are contained in Sec.III.We present our conclusions in Sec.IV.

II. THEORETICAL FORMALISM
Recent developments in perturbative QCD have allowed us to place the calculation of semi-inclusive hadron and jet production on the same footing.Using the formalism of jet functions [47,48], the collinear differential hadron and jet cross sections can be written arXiv:2303.14201v1[hep-ph] 24 Mar 2023 in a similar factorized form Here, f i/N is the parton distribution function (PDF) of parton i carrying a fraction x of in nucleon N momentum.We denote by σi→f is the lepton-parton scattering cross section producing a final-state parton f .The processes that we study as a function of p T receive contributions from electron scattering at small angles, where the lepton becomes a source of quasi-real photons.The corresponding γq → q(g), γq → g(q) and γg → q(q) processes contribute to the cross section at order α 2 em α s and the Weizsäcker-Williams (WW) distribution of quasi-real photons is given by a perturbative distribution function f γ/ ren (y, µ) [49,50] with s, t, u the lepton-parton Mandelstam variables.The analytical results for σi→f , σγi→f and f γ/ ren (y, µ) up to O(α 2 em α s ) are taken from Ref. [51].D h/f is the standard fragmentation function (FF) from parton f to hadron h, taking a momentum fraction z.J f is the semi-inclusive jet function (SiJF) initiated by parton f .When the jet radius R is small, logarithms of the type ln R can be resummed by evolving the jet function from the jet scale p T R to the factorization scale µ.
In eA reactions initial-state effects parametrized via nPDFs can alter hadron and jet cross sections.Our main focus in this paper is the centrality dependence of final-state medium-induced radiative corrections and we consider observables that minimize or eliminate FIG.1: Illustration of the concept of centrality in electron-nucleus collisions.The struck quark and the jet initiated by it will see nuclear matter of different mean interaction length d .
the cross section modification due to nPDFs.Parton branching in nuclear matter is described by in-medium splitting kernels dN med ji /dzd 2 k ⊥ for the i → j + k channel.We use the results derived in the framework of soft-collinear effective theory with Glauber gluon interaction (SCET G ) [52,53] and verified using a lightcone wavefunction formalism [54,55].
We calculate numerically the real part of the branching processes, for averaged interaction length d corresponding to different centrality classes, as illustrated in Fig. 1.
The corresponding virtual corrections P med,vir ji are obtained using flavor and momentum sum rules [56,57].Final-state in-medium radiation leads to additional scaling violations [58] in the fragmentation functions and we implement them in medium-modified DGLAP evolution equations [22,28,29,59] We solve these equations numerically using Hoppet [60].
The SiJFs used to calculate the semi-inclusive jet cross sections also receive medium-induced radiative corrections.We implement them at next-to-leading order as shown in Refs.[42,43,61,62].The results for quark and gluon initiated jets of transverse momentum p T and radius parameter R read where we have denoted dN med ji /dzd 2 k ⊥ ≡ f med i→jk (z, k ⊥ ) for brevity.In Eq. ( 6) In the equations above all, singularities when z → 1 are regularized by the plus-distribution function that has the standard definition.

III. CENTRALITY DEPENDENT NUCLEAR MODIFICATION
To study the centrality dependent nuclear modification, we are motivated by recent simulations of constraints on nuclear geometry in eA reactions using the Monte Carlo event generator BeAGLE [46].The idea behind this more differential approach is to measure the energy deposited in the zero-degree calorimeter [45] at the EIC and correlate it to collision centrality, and the effective path length d .A subset of effects, such as shadowing or assumed initial particle formation time, were studied and found to not significantly affect the energy distribution in the ZDC.The correlation between the centrality classes and the energy deposition remains robust when such effects are taken into account in simulation.
With this in mind, the average interaction length of a parton in a Pb nucleus as a function of centrality obtained in BeAGLE is given in Table I.In the top 0-1% central events d is twice as large as the one in minimum bias collisions.In the most peripheral 90-100% events d is almost twice as small as the minimum bias one.In this paper, we pick two more representative examples of centrality selection -a central -0-10% class and a peripheral 80-100 % class.Next, we calculate grids of in-medium splitting functions [52][53][54][55] while constraining nuclear geometry to yield the enhancement or reduction of the average interaction length relative to the minimum bias one as given in Table I.
With the numerically evaluated splitting functions at hand and the theoretical framework described in Sec.II we now turn to phenomenology.In our calculations for the baseline ep collisions we use CT14nlo PDF sets [63] with the strong coupling constant provided by Lhapdf6 [64].For the case of semi-inclusive hadron production, fragmentation functions into light pions are taken directly from the HKNS parameterization in Ref. [65].
Heavy quark fragmentation into D-and B-mesons at the scale µ = 2m Q is evaluated perturbatively using heavy quark effective theory (HQET) [66,67] and evolved to a higher scale.When we consider reactions with nuclei, such as the ePb case of interest, we use the nCTEQ15FullNuc PDF sets [6].Consistent with Ref. [29] and more recently Ref. [30], we fix the gluon transport coefficient in cold nuclear matter k 2 ⊥ /λ g = 0.12 GeV 2 /fm and k 2 ⊥ /λ q = 0.053 GeV 2 /fm.
We first consider jets reconstructed with a radius parameter R and define the centrality dependent nuclear modification in electron-nucleus collisions through the ratio Here, the nuclear thickness function at impact parameter b is and ∆ b = 2πbdb is the differential area around the impact parameter b such that b ∆ b T A (b) = A. In other words, R eA (R) is the per nucleon cross section modification for the relevant impact parameters corresponding to the centrality class.Earlier work on hadron and jet production in minimum bias eA collisions has already provided useful guidance on how to study final-state interactions [29,42,43].In particular, they can be separated from initial-state nuclear PDFs [6,7] by taking the ratio of nuclear modification for a small radius jet to the modification for a large radius jet R eA (R)/R eA (R = 1).This strategy works very well, eliminating initial-state effects to less than a few % [42,43].
In Fig. 2 we show the double modification ratio R eA (R)/R eA (R = 1) for three different choices R = 0.3 (red band), 0.5 (blue band), and 0.8 (green band).The bands correspond to varying the cold nuclear matter transport parameters by a factor of two relative to the nominal values quoted above.The idea behind normalizing this observable to the R eA for a large radius jet is that final-state effect for R = 1 will be minimal.Even though the medium induced parton shower is broader than the vacuum one, most of it will be contained in a unit radius.Conversely, by choosing smaller radii an increasingly larger fraction of the shower energy will be redistributed outside of the jet cone, leading to cross section suppression.We also choose the forward proton/nucleus going direction 2 < η < 4 since the jet energy in this kinematic region is the smallest in the rest frame of the nucleus, leading to larger final-state effects.
The top row of panels shows 10 GeV (e) × GeV (Pb) collision and the bottom row of panels is for 18 GeV (e) × 275 GeV (Pb) collisions.On the left we show the 0-10% centrality selection and the 80-100% centrality class is on the right.
Our calculations show that the nuclear modification is the largest at relatively small transverse momenta.At the same time, it depends on the steepness of the p T spectra and the effects become larger again close to the kinematic edges of phase space, as seen in the upper panels of Fig. 2. For large radius jets the relative modification R eA (R = 0.8)/R eA (R = 1) is small, ≤ 5%.On the other hand, for small radius jets R eA (R = 0.8)/R eA (R = 1) in central ePb reactions can show more than a factor of two suppression.At higher center of mass energies the modification is smaller, as expected, and decreases monotonically with p T .By comparing the left and right panels of Fig. 2 we see clearly that final-state effects depend on the thickness of nuclear matter.
Another method to directly investigate the interaction length dependence of final-state cold nuclear matter effects using jet production is to compare the cross sections in peripheral and central collisions.Thus, we define the ratio as where the initial-state effects are reduced and most of the contribution is from final-state interactions.As discussed above, the medium induced energy loss is smaller and the per-nucleon cross section is larger for peripheral collisions.Thus, the ratio defined in Eq. ( 12) is expected to be larger than one.Figure 3 displays our predictions for 10 GeV (e) × 100 GeV (Pb) and 18 GeV (e) × 275 GeV (Pb) collisions in the forward rapidity region for various jet radii.The R = 0.3 case is shown in the insets since, as expected, the ratio is much larger than in other cases and is very sensitive to the thickness of the nuclear matter in kinematic regions where the jet p T distribution is steeper in particular when the collision energy is small.For 10 GeV (e) × 100 GeV (Pb) collisions the ratio can be around 1.1 for R = 0.5 and R = 0.8 in the small jet p T region.It shows only about a few percent deviation from one for R = 1.The ratio in the large p T region is enhanced since jets are produced close to the edges of phase space.For 18 GeV (e) × 275 GeV (Pb) collisions, the ratio decreases with increasing jet p T and is smaller than 1.1 for most of the cases.The R dependence indicates that the energy loss for larger radii is smaller which is consistent with Fig. 2. In summary, the centrality class-dependent modification in matter is clearly observed in Fig. 3. Next, we discuss the cross-section modification for hadron production at the EIC, including π + and the heavy D 0 and B 0 mesons.As shown in [29], the following double ratio as a function of momentum fraction z is a suitable observable for cold nuclear matter tomography at the EIC Here, we use the shorthand notation N h (p T , η, z) ≡ dσ h /dηdp T dz for the distribution of hadrons versus the hadronization fraction z and N inc (p T , η) ≡ dσ J /dηdp T for the distribution of large radius jets.In practice we integrate over suitably chosen rapidity and transverse momentum bins before taking the ratio.The idea behind normalizing by N inc (p T , η) is to once again minimize initial-state effects and emphasize physics of final-state interactions in nuclear matter.For eA collisions we can further define per-nucleon cross sections by dividing out the 1/∆ b T A (b) geometric factor.Our results for R h ePb (z) are shown in Fig. 5.We consider electron-proton/nucleus collisions with energy 5 GeV (e) × 40 GeV (A), and the transverse momenta of final-state hadrons are fixed in the range 2 GeV to 3 GeV.Consequently, the momentum fraction z distribution corresponds to the variation of ν constrained by the kinematics of the scattered electron in experiment.The left column of panels is for the 0 − 10% central events, and the right column is for 80 − 100% peripheral events.For each hadron species, the red, blue and green bands correspond to the predictions in rapidity regions −2 < η < 0, 0 < η < 2 and 2 < η < 4, respectively.Just as in the case of jet production, the bands reflect the variation of the nuclear matter transport parameter by a factor of two.Top to bottom rows show the differential π + , D 0 and B 0 modification.Because lower energy partons receive larger medium corrections induced by the final-state interactions in the nucleus, the medium modification is more significant in the forward rapidity region 2 < η < 4. In this region the energy of the final-state parton is lower in the nuclear rest frame in comparison, for example, to backward rapidity.It is instructive to observe that for light hadrons at large z the differential cross section suppression can reach a factor of two even in peripheral collisions.In central events the energy loss effect can lead to more than an order of magnitude reduction.For heavy flavor, just as in minimum bias reactions [29], R h ePb (z) shows transition from suppression at large z to enhancement at small z because of the non-monotonic behavior of the heavy quark fragmentation function into heavy mesons [66,67].In central reactions nuclear effects are noticeably larger.
To compare the cross section modification in central and peripheral collisions for differential hadron distributions quantitatively, we define and note that the baseline ep cross sections will drop out.As we expect, central collisions result in more significant medium corrections than peripheral ones, as shown in Fig. 4. The steep fragmentation distribution when z → 1 enhances the differences for light pions to an order of magnitude.As we go forward in rapidity the enhancement in Peripheral/Central(h) extends to smaller z.For D 0 mesons this enhancement can also be very significant when z → 1 but at intermediate fragmentation fractions the double ratio can dip below unity -a consequence of the transition from suppression to enhancement in R h eA (z).The qualitative behavior is is similar for B 0 mesons.

IV. CONCLUSIONS
We presented theoretical predictions for the nuclear modification of semi-inclusive hadron and jet production in ePb collisions at the EIC as a function of centrality.We took advantage of recent simulations that were able to demonstrate robust correlation between centrality classes in eA and energy deposition in the zero-degree calorimeter, and to determine the mean interaction length seen by partons.We constructed observables that minimize initial-state nPDF effects and are sensitive to the inelastic final-state interactions of the struck parton in the nucleus.Future measurements of these observables at the EIC can provide essential information on the path length dependence of parton shower formation and hadronization in cold nuclear matter.
Our theoretical results indicate that the dependence of in-medium shower formation and energy loss on the transport properties and size of the nuclear medium can be easily identified and studied at the EIC.The exact sensitivity, however, depends on the choice of observables.
We found that for inclusive jets of small radius at moderate center-of-mass energies and at forward rapidities the per-nucleon cross sections variation between 0-10% and 80-100% collision can reach a factor of two.Because of the high integrated luminosity that EIC is expected to deliver [3], such peripheral-to-central differences will be easily measurable, but they are smaller than the differences in the mean interaction length d seen by the jet.The reason for this is that even for R = 0.3 only a fraction of the medium-induced shower is redistributed outside of the jet cone.eA in peripheral to central collisions.The electron and proton/nucleus beam energies, pT and η ranges are the same as in Fig. 4. We show π + (red), D 0 (blue) and B 0 (green).From top to bottom panels cover backward to forward rapidities.be performed at even lower center-of-mass energies.Our theoretical calculations showed that the per-nucleon differential particle distributions versus the fragmentation fraction z h depend much more significantly on centrality.For light pions at large z h the peripheral-to-central ratio can reach a factor of 10, exceeding the ratio of effective interaction lengths for these centrality classes.Furthermore, the nuclear modification due to final-state interactions and its centrality variation are strong enough to be detected near mid rapidity and even at backward rapidity.The nuclear cross section modification also depends on the hadron flavor and has a predicted non-monotonic behavior for D-and B-mesons.We conclude by pointing out that in the future it will be important to explore the centrality dependence of other more differential jet observables such as jet substructure.

FIG. 2 :
FIG. 2: Relative modifications of the inclusive jet cross section ReA(R)/ReA(R = 1) for three radius choices R = 0.3, 0.5, 0.8 in the rapidity interval 2 < η < 4. The upper panels are for 10 × 100 GeV ePb collisions and the bottom panels are for 18 × 275 GeV ePb collisions.Central reactions are on the left and peripheral reactions are on the right.

FIG. 5 :
FIG. 5: The ratio of R heA in peripheral to central collisions.The electron and proton/nucleus beam energies, pT and η ranges are the same as in Fig.4.We show π + (red), D 0 (blue) and B 0 (green).From top to bottom panels cover backward to forward rapidities.

TABLE I :
Selected centrality classes in ePb collisions at the EIC, the corresponding effective length of cold nuclear matter seen by the scattered parton, and the ratio relative to the one in minimum bias (0 -100 %) collisions.