Energy-energy correlators in Deep Inelastic Scattering
EEnergy-energy correlators in Deep Inelastic Scattering
Hai Tao Li,
1, 2, 3, ∗ Yiannis Makris, † and Ivan Vitev ‡ HEP Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM, 87545, USA INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy
The energy-energy correlator (EEC) is an event shape observable which probes the angular corre-lations of energy depositions in detectors at high energy collider facilities. It has been investigatedextensively in the context of precision QCD. In this work, we introduce a novel definition of EECadapted to the Breit frame in deep-inelastic scattering (DIS). In the back-to-back limit, the observ-able we propose is sensitive to the universal transverse momentum dependent (TMD) parton dis-tribution functions and fragmentation functions, and it can be studied within the traditional TMDfactorization formalism. We further show that the new observable is insensitive to experimentalpseudorapidity cuts, often imposed in the Laboratory frame due to detector acceptance limitations.In this work the singular distributions for the new observable are obtained in soft collinear effectivetheory up to O ( α s ) and are verified by the full QCD calculations up to O ( α s ). The resummationin the singular limit is performed up to next-to-next-to-next-to-leading logarithmic accuracy. Afterincorporating non-perturbative effects, we present a comparison of our predictions to PYTHIA 8simulations. I. INTRODUCTION
Event shape observables (such as thrust, C-parameter,etc.) are measures of the energy flow, multiple particlecorrelations, and the radiative patterns in high energycollisions. They have been extensively investigated atvarious colliders and, over the past several decades, haveplayed a central role in our understanding of the pertur-bative and non-perturbative dynamics of Quantum Chro-modynamics (QCD).The energy-energy correlator (EEC) is such an observ-able, which was originally introduced in the context of e + e − collisions as an alternative to the thrust family ofevent shapes. EEC is defined as follows [1],EEC = (cid:88) a,b (cid:90) dσ ee → a + b + X σ w ab δ (cos χ ab − cos χ ) , (1)where the sum runs over all the hadron pairs ( a, b ) andthe cross section is weighted by w ab ≡ E a E b /s , which isthe product of the energies of a and b normalized by thecenter-of-mass energy of the system. EEC measures theenergy correlations as a function of the opening angle χ ab between particles a and b . Significant theoretical efforthas been devoted to this observable, including fixed-ordercalculations [2–8], and QCD factorization and resumma-tion in the back-to-back ( χ → π ) and collinear ( χ → LL (cid:48) accuracy after ob-taining the O ( α s ) singular distributions. ∗ [email protected] † [email protected] ‡ [email protected] At hadronic colliders, an adaptation of EEC knownas transverse-energy-energy correlation (TEEC) consid-ers only the momenta in the transverse plane in order toconstruct the corresponding observable [15]. TEEC wascalculated including next-to-leading order (NLO) QCDcorrections in ref. [16], and NNLL resummation in thedijet limit was accomplished in ref. [17].For observables like EEC and TEEC soft radiationcontributes only through recoil to the energetic collinearparticles, since direct contributions from soft emissionsare suppressed by the energy weighting factor. Thus, weshould anticipate to have smaller non-perturbative cor-rections compared to other event shape variables. More-over, owing to the high perturbative accuracy achievedboth in resummed and fixed order calculations [9, 14, 17,18], complemented by high precision measurements [19–25], EEC and TEEC observables offer the opportunity forprecision studies in QCD. In particular, EEC and TEEChave been used for precise extractions of the strong cou-pling constant. For a recent review see section 9 ofref. [26].Despite its important applications to precision QCDand the connection of the EEC observable to TMDs, lit-tle has been done in the context of Deep Inelastic Scatter-ing (DIS). With the future Electron-Ion-Collider (EIC)[27] on the horizon, progress in this direction is urgentlyneeded. The first study of TEEC in DIS was performedin [18], where the observable was defined as the correla-tion between final state hadrons and the scattered leptonin the transverse plane of the Laboratory frame. Further-more, in ref. [28] the original definition of TEEC and itsasymmetry were also considered in a fixed-order study inthe Breit frame.In this work we introduce a new definition of EECin the Breit frame, which is the natural frame forthe study of Transverse Momentum Dependent (TMD)physics [29]. In this frame, the target hadron moves along a r X i v : . [ h e p - ph ] F e b ˆ z and the virtual photon moves in the opposite direction.The Born-level process is described by the lepton-partonscattering e + q i → e + q f , where the outgoing quark q f back-scatters in the direction opposite to the proton.Hadronization of the struck quark will form a collimatedspray of radiation close to the − ˆ z direction. In contrast,initial state radiation and beam remnants are moving inthe opposite direction close to the proton’s direction ofmotion. It is this feature of the Breit frame, which leadsto the clean separation of target and current fragmenta-tion that we utilize to construct the novel EEC observablein DIS.We denote the new event shape variable EEC DIS toavoid confusion with the conventional observable. Ourdefinition reads,EEC
DIS = (cid:88) a (cid:90) dσ ep → e + a + X σ z a δ (cos θ ap − cos θ ) , (2)where z a ≡ P · p a P · ( (cid:80) i p i ) , (3)and p µa and P µ are the momenta of the hadron a and theincoming proton respectively. The angle θ ap is the polarangle of hadron a , which is measured w.r.t. the incomingproton. Note that the asymmetric weight function, z a ,is Lorentz invariant and is suppressed for soft radiationand radiation close to the beam direction. Furthermore,this definition of EEC DIS naturally separates the contri-bution to the cos θ spectrum from: i) wide angle softradiation, ii) initial state radiation and beam remnants,and iii) radiation from the hadronization of the struckquark. This unique feature makes the new observable inthe back-to-back limit ( θ → π ) insensitive to experimen-tal cuts on the particle pseudorapidity (in the Labora-tory frame) usually imposed due to detector acceptancelimitations in the backward and forward regions, makingthe comparison of theory and experiment in this regioneven more accurate. This definition of EEC is sphericallyinvariant, however, we discuss in the appendix A a pos-sible definition that is fully Lorentz invariant and can bemeasured directly in any frame.For π − θ ∼ Note that what we propose is different from the discussion in [28],where a fixed order QCD calculation for TEEC (as defined forhadronic colliders) was performed in the Breit frame DIS forthe dijet configuration. This observable exhibits very differentcharacteristics and is suppressed by α s compared to what wepropose here. DIS factorization. This ensures the universality of TMDparton distribution functions (TMD PDFs) appearing inEEC
DIS when we compare to other observables, such assemi-inclusive DIS and jet-TMDs [35–38] in the Breitframe. Moreover, through an operator product expan-sion (OPE) the TMD beam or TMD fragmentation func-tion can be matched to the collinear PDF and collinearfragmentation function, respectively. As discussed inref. [9], after summing over all the final state hadrons, thecollinear fragmentation functions can be removed usingmomentum sum rules and, therefore, the only hadronicmatrix elements that enter the factorization in the back-to-back limit are the TMD PDFs. As a result, EEC
DIS provides a novel approach to TMD physics, avoiding thedependence on fragmentation functions. In this work we construct the resummed cross sectionup to N LL accuracy and merge onto the NLO ( O ( α s ))fixed order QCD result. To achieve the resummed re-sult, we use the four-loop cusp and the three-loop hard,soft, and jet anomalous dimensions. The fixed ordersingular results for the various elements of the factor-ization formula we take from the literature. The TMDPDFs have been calculated up to three loops [39–47],and the soft function at the same accuracy can be foundin refs. [9, 48]. The jet function is the second Mellinmoment of the TMD fragmentation functions with arealso available up to N LO [42–44, 46, 49, 50]. The fixedorder QCD result is obtained numerically using
NLO-Jet ++ [51]. We also perform a consistency check of ourapproach by comparing the LO and NLO singular distribu-tions, obtained by the factorization formula, to the full QCDprediction in the back-to-back limit.The paper is organized as follows. In Section II we intro-duce the relevant modes of EEC
DIS in SCET and the factor-ized formula in the back-to-back limits. In addition, we dis-cuss the hadronization corrections. In Section III we presentour predictions up to N LL+NLO and compare our resultswith PYTHIA simulations. Section IV shows how for theEEC observables we can select a subset of hadrons in finalstate, to be used for flavor tagging in TMD PDFs and TMDFFs. We conclude in Section V. We propose a Lorentz invari-ant definition of EEC in DIS in the Appendix.
II. MODES AND FACTORIZATIONA. Notation and DIS kinematics in Breit frame
In this paper, we work explicitly in the Breit frame, wherea clean geometrical separation of target and current fragmen-tation exists. To avoid contributions from the resolved pho-ton events we will consider only scattering with large photonvirtuality, i.e. Q = (cid:112) − q (cid:29) q µ is the four- As we will discuss later, hadronization corrections enter throughthe so-called EEC jet function. We demonstrate a method forestimating these effects using extractions of TMD fragmentationfunctions (FFs). p ¯ n ? = p n ? p s ?
1. Then, theconstraint in eq. (8) requires that soft radiation and particlesclose to the direction of the target hadron are described byparametrically smaller values of the momentum fraction, i.e., z i (cid:28)
1. Therefore, in the back-to-back limit the contribu-tion to EEC
DIS is dominated by the collinear fragments ofthe struck quark. We refer to the corresponding modes as “¯ n -collinear”. However, from conservation of momentum we havethat “ n -collinear” as well as global soft radiation will con-tribute through recoil, as we demonstrate in the following sub-section. Using the light-cone basis where p µ = ( p + , p − , p ⊥ ),the scaling of the relevant modes is: n -collinear : p µn ∼ Q ( λ , , λ ) , ¯ n -collinear : p µ ¯ n ∼ Q (1 , λ , λ ) , soft : p µs ∼ Q ( λ, λ, λ ) . (9) From the above scaling of momenta we have that the weightfactor, z i , for the three relevant modes is z ¯ n (cid:46) , z n ∼ λ , z s ∼ λ , (10)while beam remnants with momenta almost parallel to thedirection of the proton have a vanishing scaling parameter, z b.r. ∼ B. Factorization
As already mentioned above, the n -collinear and soft modeswill enter the factorization theorem through recoil effects. Todemonstrate that, consider the contribution to the observableof a hadron, a , from the ¯ n sector. As shown in Fig. 1, thehadron- a is labeled by its scaling factor z a and its transversemomentum w.r.t. the fragmenting parton k a ⊥ . This parti-cle will contribute to the measurement at θ = θ ap , which wecan express in terms of q ⊥ ≡ p a ⊥ /z a – the re-scaled trans-verse momenta with respect to the proton/beam direction ofmotion, π − θ ap θ a (cid:39) | q ⊥ | Q (cid:39) Q (cid:12)(cid:12)(cid:12) k a ⊥ z a + p n ⊥ − p s ⊥ (cid:12)(cid:12)(cid:12) . (11)It is, therefore, clear now that the EEC is a reweighted TMDdistribution summed over all hadrons flavours. Followingref. [9], we introduce a dimensionless variable in order to sim-plify the notation, τ = 1 + cos θ . (12)Expanding in the back-to-back limit, θ → π , and for a singlehadron of flavour a we have dσ a dxdQ dzdτ = (cid:90) d q ⊥ dσ a dxdQ dzd q ⊥ δ (cid:16) τ − | q ⊥ | Q (cid:17) × (cid:104) O ( τ ) (cid:105) , where we used the shorthand notation dσ a ≡ dσ ( ep → e + a + X ) and we have dropped the subscript a from the variable z .The TMD cross section can be written as usual, in termsTMD parton distribution functions and fragmentation func-tions: dσ a dxdQ dzd q ⊥ = H ij ( Q, x, y ; µ ) (cid:90) d b (2 π ) exp( − i b · q ⊥ ) × B j/P ( x, b ; µ, ν ) D i/a ( z, b ; µ, ν ) S ( b ; µ, ν ) , (13) where b is the Fourier conjugate of q ⊥ and b = | b | . B j/P isthe TMD beam function, S is the soft function, and D i/a isthe fragmentation function for parton i to hadron a .Integrating against the weighing factor and summing overall possible hadrons, we have, (cid:88) a (cid:90) dz z dσ a dxdQ dzdτ = H ij ( Q, x, y ; µ ) × (cid:90) d q ⊥ δ (cid:16) τ − | q ⊥ | Q (cid:17) (cid:90) d b (2 π ) exp( − i b · q ⊥ ) × B j/P ( x, b, µ, ν ) S ( b, µ, ν ) J i ( b, µ, ν ) , (14)where J q (¯ q ) is the EEC (anti-)quark jet function defined asthe weighed sum of TMD fragmentation functions, J i ( b ; µ, ν ) ≡ (cid:88) a (cid:90) dz zD i/a ( z, b ; µ, ν ) . (15)Note that this is the same jet function that appears in the con-ventional EEC observable for electron-positron annihilation.Through an OPE, the TMDFFs can be expressed in terms ofa convolution of short distance matching coefficients, I ij , andthe collinear fragmentation functions, d i/h , D OPE i/a ( z, b ; µ, ν ) = (cid:88) j (cid:90) z duu I ij (cid:16) bz/u , z/u, µ, ν (cid:17) × d j/a ( u, µ ) (cid:104) O (cid:16) Λ Q , Λ b (cid:17)(cid:105) , (16)where we have used the superscript OPE to denote that thisis the leading contribution in the expansion and is considereda good approximation of the true TMDs in the perturbativeregime where q T (cid:29) Λ QCD . Therefore, for the EEC jet func-tion in the same approximation we have, J OPE i ( b , µ, ν ) = (cid:88) j (cid:90) dw w I ij (cid:16) bw , w ; µ, ν (cid:17) . (17)To obtain this, we used the fragmentation function sum rule, (cid:88) a (cid:90) dz z d i/a ( z ; µ ) = 1 , (18)which gives the normalization of the fragmentation functionsas number density. Renormalization group evolution and resummation
The various elements of the factorized expression haverenormalization scale, µ , dependence. The resummed cross-section, where all large logarithms are summed up to a partic-ular logarithmic accuracy, is obtained by evaluating the ele-ments of factorization at their canonical scales and then using Note that in the literature there are multiple conventions for thematching coefficients. Here we use the ones from ref. [9], howeverother equivalent choices can be made. the renormalization group (RG) equations to evolve them upto a common scale. The RG equation satisfied by each of therelevant functions is, dd ln µ G ( µ ) = γ G ( µ ) G ( µ ) , (19)where G can be any of the functions that appear in the factor-ization theorem in eq. (14). The solution of the RG equationin Laplace space can be written as a product of the evolutionkernel U G and the function G , evaluated at some initial scale µ , G ( µ ) = G ( µ ) × exp (cid:16) (cid:90) µµ d ln µ (cid:48) γ G ( µ (cid:48) ) (cid:17) = G ( µ ) × U G ( µ, µ ) . (20)For each of the functions that appear in the factorization the-orem, a different choice of the initial scale µ is appropriate.In this work the initial scales of the evolution are chosen suchthat they minimize large logarithms in the fixed order expan-sion of the corresponding functions, µ H = Q , µ S = µ B = µ J = µ b ≡ e − γ E b . (21)In our scheme the final scales on the evolution are chosen suchthat they minimize logarithms in the beam and jet functions, µ = µ b . In addition to the renormalization evolution, the soft,beam, and jet functions also satisfy rapidity evolution. Here,we consider the evolution of the soft function alone, S ( b, µ, ν ) = S ( b, µ, ν S ) × exp (cid:16) (cid:90) νν S d ln ν (cid:48) γ R ( µ, µ b ) (cid:17) , (22)and choose the common rapidity scale, ν such that it mini-mizes the rapidity logs in the beam and jet function: ν = ν B = ν J = Q . (23)The anomalous dimension γ R is the soft function rapidityanomalous dimension. Then, the RG evolved cross sectionreads dσdxdQ dτ = H ij ( Q, x, y ; µ H ) (cid:90) d q ⊥ δ (cid:16) τ − | q ⊥ | Q (cid:17)(cid:90) d b (2 π ) exp( − i b · q ⊥ ) B j/P ( x, b, µ, ν ) S ( b, µ S , ν S ) × J i ( b, µ, ν ) R ( b, µ, ν ) . (24)Here, R is the combined evolution kernel for both renormal-ization evolution and rapidity evolution, R ( b, µ, ν ) = exp (cid:16) (cid:90) µµ H dµ (cid:48) µ (cid:48) γ H ( µ (cid:48) )+ (cid:90) νν S dν (cid:48) ν (cid:48) γ R ( µ, µ b ) (cid:17) , (25)where γ H and γ S are the anomalous dimensions of the hardand soft functions.Further details on evolution equations and anomalous di-mensions can be found in ref. [18] and references therein. C. Hadronization and non-perturbative corrections
The result in eq.(15) contains the same jet function found inthe conventional EEC observable in refs. [9, 43] and can be ex-pressed in terms of the perturbatively calculable matching co-efficients after OPE, as done in eq.(17). However, hadroniza-tion corrections are expected to be important, particularly atthe relatively small values of Q anticipated at the EIC. Thefact that EEC DIS factorization involves the universal back-to-back TMD soft function allows us to consider hadronizationand non-perturbative corrections in a universal framework ap-plicable to conventional TMD observables.There are two sources of non-perturbative corrections forTMDs: a) the corrections to the rapidity anomalous dimen-sion; and b) the contribution to the TMD matrix elements.For the soft rapidity anomalous dimension the implementa-tion of the non-perturbative model is done as in conventionalTMD observables at the level of evolution, R ( b, µ, ν ) → R ( b, µ, ν ) × exp (cid:16) g K ( b ) ln νν S (cid:17) , (26)where g K ( b ) is the model function for the non-perturbativecomponent of the rapidity anomalous dimension. Note thatsince the EEC jet function satisfies the same rapidity andrenormalization group evolution as TMDFFs, this also im-plies the same hadronization model for the rapidity anoma-lous dimension. For the hadronization model of the EEC jetfunction we can assume a generic multiplicative ansatz. Thefinal result reads, √ SJ i ( b, µ , ν ) = √ S pert. J OPE i ( b, µ , ν ) j i ( b ) , (27)where S pert. is the perturbative expression for the soft func-tion and j i ( b ) is the multiplicative ansatz for hadronizationeffects in the EEC jet function. The scales µ and ν arearbitrary scales in the soft and jet functions. Then one canextract the model function j i ( b ) by fitting to experimentaldata from e + e − or ep colliders.Alternatively, we can relate the model function j i ( b ) to thestandard TMDFFs as follows, j i ( b ) = (cid:104) J OPE i ( b, µ , ν ) (cid:105) − (cid:88) a (cid:90) dudy ( uy ) I ij (cid:16) bu , u ; µ , ν (cid:17) d i/a ( y, µ ) D NP i/a ( uy, b ) , (28)where the non-perturbative model function D NP i/a ( z, b ) is de-fined in the context of TMDFFs, √ SD i/a ( z, b ; µ , ν ) = √ S pert. D OPE i/a ( z, b ; µ , ν ) D NP i/a ( z, b ) . (29)Therefore, given a specific model for TMD fragmentation wecan explicitly evaluate the model function j i ( b ). Relativelyrecent extractions of the TMDFFs can be found in refs. [52–54]. The non-perturbative model for the TMDPDF can beimplemented the same way as is done for in the case of TMDmeasurements in SIDIS and Drell-Yan processes, √ SB i/P ( x, b ; µ , ν ) = √ S pert. B OPE i/P ( x, b ; µ , ν ) f NP i/P ( x, b ) . (30)In this work we will be assuming a simplified model that isindependent of the Bjorken x : f NP i/P ( x, b ) (cid:12)(cid:12)(cid:12) simplified = f NP i/P ( b ) . (31) Thus, combining all elements at the level of the cross sectionwe can collect all non-perturbative contributions in a singlefunction, F NP i/P ( b ), dσdxdQ dτ = H ij ( Q, x, y ; µ H ) (cid:90) d q ⊥ δ (cid:16) τ − | q ⊥ | Q (cid:17)(cid:90) d b (2 π ) exp( − i b · q ⊥ ) B OPE j/P ( x, b, µ, ν ) S pert. ( b, µ S , ν S ) × J OPE i ( b, µ, ν ) R ( b, µ, ν ) F NP i/P ( b ) , (32)where F NP i/P ( b ) = f NP i/P ( b ) j NP i ( b ) exp (cid:16) g K ( b ) ln νν S (cid:17) , (33)The non-perturbative corrections in this work are imple-mented following the model and parameters in refs. [52, 55], F NP i/P ( b ) ∼ j i ( b ) × exp (cid:104) − . b − .
42 ln (cid:16) b b (cid:17) ln νν S (cid:105) , (34)where the last term in the exponent corresponds to thenon-perturbative rapidity evolution model for g K ( b ) = − .
42 ln(1 + b /b ), with b max = 1 . − . For the EECjet function model j i ( b ) we will consider a LO approximationbased on eq. (28), j i ( b ) (cid:12)(cid:12)(cid:12) LO = (cid:88) a (cid:90) dy y d i/a ( y, D NP i/a ( y, b ) . (35)Note that for the trivial choice D NP i/a = 1 (i.e., no non-perturbative effects in the TMDFF) we have from momen-tum sum rules that also j i ( b ) = 1. For this analysis we usethe DSS collinear fragmentation functions [56, 57] and for theTMDFF model we have, D NP i/a ( y, b ) = exp (cid:16) − . b y (cid:17) . (36)Performing the sum over a ∈ { π + / − / , p/n, K + / − / } andthe integrating over the momentum fraction y in eq. (35) wefind, j i ( b ) = exp (cid:16) − . b − . b (cid:17) , (37)for i ∈ { u, d, s, ¯ u, ¯ d, ¯ s } with very small flavour dependencewhich in this study we will ignore for simplicity.In this section we discussed how the non-perturbative cor-rections are implemented in our work. Our approach is phe-nomenologically motivated and it aligns with what is tradi-tionally used for TMD extractions from semi-inclusive DIS.Nonetheless, there are other ways to take into account non-perturbative effects in EEC. A detailed field theoretic treat-ment of non-perturbative corrections to EEC in e + e − waspresented in ref. [58]. However, the approach discussed inref. [58] is not directly applicable to the observable we pro-pose due to the asymmetric nature of the measurement. Thequestion of power corrections and non-perturbative effects ina field theoretic manner is a rather interesting and importantsubject, but beyond the scope of this paper. - - - - FIG. 2. TEEC
Lab (left) and EEC
DIS (right) distributions from PYTHIA 8 with different rapidity cuts in the lab frame. ForTEEC
Lab we consider cut on the scattered lepton transverse momentum p eT >
20 GeV and for EEC
DIS we consider cut on thephoton virtuality
Q >
20 GeV. For both case we consider cut on y = 2 P · q/s : 0 . < y < . - - - - - - - - - - FIG. 3. Fixed-order ln τ distributions in the τ → III. NUMERICS AND COMPARISON WITHMONTE-CARLO SIMULATIONS
In this section we present numerical results for the EEC
DIS distributions at the future Electron-Ion Collider (EIC) [27].In particular, we consider the following beam energies: 18GeV electrons on 275 GeV protons, which corresponds tothe center-of-mass energy √ s ≈
141 GeV. For all the cal-culations we select events with
Q >
20 GeV and use thePDF4LHC15 nnlo mc PDF sets [59] with the associatedstrong coupling provided by
Lhapdf 6 [60]. - - - - - FIG. 4. Resummed ln τ distributions for EEC. The dark red,dark blue, and dark green bands correspond to NLL, NNLL,and NNNLL distribution. The dot-dashed lines are the LOnon-singular distributions (dark red) and the absolute valueof NLO non-singular ones (dark green).We present the TEEC Lab [18] and EEC
DIS distributionspredicted by
Pythia | η | < . | η | < .
5, and | η | < . | η | < .
5. Because TEEC
Lab measuresthe correlation between hadrons and the final state lepton - - - - - FIG. 5. Resummed ln τ distributions without (solid) and withnon-perturbative factor. The color scheme is the same as inFig. 4.in the lab frame, pseudorapidity cuts have an impact on thefull cos φ range, as shown in left panel of Fig. 2. EEC DIS is defined as the correlation between the final state hadronsand incoming proton in the Breit frame, and the pseudora-pidity cuts only remove particles in the forward region wherethe weighted cross section is small. In the backward region(cos θ → −
1) the EEC
DIS distribution is independent on thepseudorapidity cuts.Fig. 3 shows the fixed order ln τ distributions in QCD com-pared against the singular distributions derived from the fixedorder expansion of the factorized cross section in eq. (14).The factorization and renormalization scales are set to Q .The non-trivial contribution to EEC DIS starts from two-jetproduction in DIS, which is considered as LO . The singu-lar distributions are provided up to NNLO and shown withsolid lines; their τ → NLOJet ++ [51, 63].The non-singular contributions are defined as dσd ln τ (cid:12)(cid:12)(cid:12) non-sing. ≡ dσd ln τ (cid:12)(cid:12)(cid:12) QCD − dσd ln τ (cid:12)(cid:12)(cid:12) sing. , (38)and are represented by the dash-dotted lines. The non-singular results correspond to the power corrections to thefactorized cross section which are of O ( τ ) or higher and van-ish in the limit τ →
0. This, equivalently, implies that thesingular behavior of the full QCD calculation is reproducedby the singular distributions in the same limit. The purposeof this comparison is twofold. First, it is a consistency checkof our approach and our calculations, and, second, it is an in-dicator for the size of power corrections as a function of τ . Inparticular, we find that large power corrections are observedfor τ → τ → ν and µ respec-tively. As discussed in section II B, we chose to evolve the hardfunction to the common soft and jet renormalization scales: µ = µ S = µ B = µ J . For the rapidity evolution we evolve thesoft function up to the common jet and beam rapidity scaleby making the choice ν = ν B = ν J . In our scale choice, inorder to avoid the Landau pole, we adopt the conventionalscheme, ν S = µ S = µ B = µ J : b → b ∗ = b (cid:112) b /b , (39)where b max is fixed as discuss above eq. (34). The scale uncer-tainties are evaluated by varying µ , µ H , ν and ν H by a factorof two independently. Fig. 4 shows resummed distributionsover ln τ in the infrared region. We observe large correctionsfrom NLL to N LL and a good perturbative convergence fromN LL to N LL. Furthermore we find that the scale uncertain-ties are significantly reduced for the N LL compared to thelower accuracy distributions. For comparison, in Fig. 4 wealso show the LO and NLO non-singular distributions. Wefind that the power corrections become numerically relevantfor ln τ (cid:38) − Lab distributions,i.e. Fig. 3 of ref. [18], the peaks of EEC
DIS distributions areat larger τ values, which corresponds to a larger energy scale.This suggests that the EEC DIS observable has a more stableand reliable perturbative behavior at the bulk of the distri-bution. Also, compared to TEEC
Lab , we expect smaller non-perturbative corrections since, in principle, the hadronizationeffects will suppress the cross section in the region q T ∼ DIS , non-perturbative corrections shift the cross sectionsto larger τ .The final EEC DIS distributions are presented in Fig. 6,where we matched the NLL and N LL resummed distribu-tions to the QCD NLO ones in the τ → Pythia 8 simulations runwithout a hadronization modeling. The right panel presentsthe equivalent distributions including non-perturbative effectsand hadronization. For − . < cos θ the distributions aredescribed by the fixed order results and for cos θ < − . − . < cos θ < − . dσd cos θ (cid:12)(cid:12)(cid:12) − . < cos θ< − . =(1 − f ) dσd cos θ (cid:12)(cid:12)(cid:12) QCD + f (cid:18) dσd cos θ (cid:12)(cid:12)(cid:12) non-sing. + dσd cos θ (cid:12)(cid:12)(cid:12) Res. (cid:19) , (40)where f = 12 (cid:104) cos (cid:18) cos θ + 0 . . π (cid:19) + 1 (cid:105) . (41)A similar matching procedure and detailed discussion aboutmatching can be found in ref. [64]. The difference betweenNLL+NLO and N LL+NLO in the region cos θ > − . θ , where the cross section is dominated by thefixed order result, the theoretical uncertainties are estimated - - - - FIG. 6. Comparison of cos θ distributions between the SCET predictions and PYTHIA simulations. The dark red and darkgreen bands are the NLL+NLO and N LL+NLO. The gray bands are from PYTHIA 8 simulations with the default settingsfor uncertainty bands.by varying the renormalization and factorization scales ( µ r and µ f respectively) by a factor of 2 and 1/2: µ r = µ f = κQ with κ = (0 . , , Pythia distributions (gray band) in theleft and right panels of Fig. 6 we see that hadronization effects,as implemented in
Pythia 8 , enhance the distribution forlarge cos θ . In the collinear limit, θ →
0, the distributionfrom hadronic-level
Pythia increases in comparison to thepartonic-level and in comparison with our NLO predictions.Note, however, that this region is sensitive to the rapiditycutoffs and thus comparison to experiment requires a differentformalism to incorporate these effects. Such a formalism willhave to resum logarithmic enchantments from the rapiditycutoff, see for example refs. [65–67].
IV. CORRELATIONS WITH SUBSETS OFHADRONS
In this section we discuss an EEC-like observable, but thistime considering only a subset of hadrons. There are variousreasons why one might want to introduce such a modificationand at the end of this section we give two possibilities. In thediscussion that follows, we denote the subset of hadrons with S . Taking the τ -differential and z -weighed cross section butonly summing over a subset of all hadrons we have , O S = (cid:88) a ∈ S (cid:90) dz z dσ a dxdQ dzdτ . (42)In the back-to-back limit the factorization theorem formula-tion for this observable follows exactly the same steps as forEEC DIS and the final result is given by eq. (14) with the re-placement of the EEC jet function with the appropriate subset For EEC
DIS , there is no collinear singularity in the forward re-gion. In this section we focus on the back-to-back limit. jet function , G i → S : J i ( b, µ, ν ) → G i → S ( b, µ, ν ) = (cid:88) a ∈ S (cid:90) dz zD i/a ( z, b ; µ, ν ) . (43)Beyond the dependence on the initiating parton i , the subsetjet function (SJF) it also depends on the subset of hadrons S and, thus, is a fundamentally non-perturbative function (sim-ilar to fragmentation functions and PDFs). Nonetheless, sim-ilarly to the EEC jet function we can employ the OPE andat leading order match the TMD FF onto the collinear FFs.Doing so we get, G OPE i → S ( b , µ, ν ) = (cid:88) j F j → S ( µ ) (cid:90) dw w I ij (cid:16) b w , w ; µ, ν (cid:17) , (44)where F j → S is interpreted as the average fraction of hadronsincluded in S fragmenting from j , compared to all fragmentsof the parton j , F j → S ( µ ) = (cid:88) a ∈ S (cid:90) dz z d i → a ( z, µ ) . (45)From the fragmentation function sum rules we also have thatif S = all hadrons then F j → S = 1 and we retrieve the EECjet function. In-contrast to the EEC jet function, the SJF ineq. (44) cannot be expressed only in terms of the perturba-tively calculable matching coefficients since it depends explic-itly on the subset S . However, in the OPE limit this depen-dence is rather simple and enter through the fraction F j → S .Beyond this limit, further non-perturbative corrections canbe implemented through a model in a way similar to the onefor the EEC jet function in eq.(46): √ S G i → S ( b , µ, ν ) = √ S pert R ( b , µ, ν ) G OPE i → S ( b , µ , ν ) g i → S ( b ) , (46)where g i → S ( b ) = (cid:104) G OPE i → S ( b , µ , ν ) (cid:105) − (cid:88) a ∈ S (cid:90) dz zD i/a ( z, b ; µ , ν ) , (47) is the non-perturbative ansatz for the SJF to be determinedfrom the flavor sensitive TMD FFs or directly measured fromexperiment.The subset of hadrons S can be chosen accordingly in or-der to accommodate the experimental and phenomenologicalneeds. Two cases we find particularly interesting: • S = C : where C is the subset of all charged parti-cles. This case is important since often experimen-tal measurements consider only charged particles forwhich the energy resolution is much better, yieldingsmaller experimental uncertainties. To apply our for-malism in the back-to-back limit one needs the fraction F i → C which can be obtained from extraction of chargehadron fragmentation functions [56]. Considering onlycharged particles will most definitely have an effect inthe weighed cross section, however, is unclear how muchthis will affect the EEC DIS since it is a normalized ob-servable. • S = h : where h is an identified hadron. This case isparticularly interesting since it allows us to probe theflavor of the incoming parton by appropriately choos-ing the final state hadron h . This is similar to whathas been done in TMD observables [68]. Obviously, thenon-perturbative number F i → h can be obtain by inte-grating the single hadron collinear FF, D i → h .In this work we do not explore any numerical implementationsof this modified EEC-like observable but we foresee those tobe relevant for future studies when comparing to experimentaldata. V. CONCLUSION
In this paper, we introduced a new definition of energy-energy-correlator (EEC) suitable for Breit frame studies ofthe DIS process, where the opening angle θ between finalstate particles and the incoming proton is measured and thecross section is weighted by the four-momentum product offinal state particles and incoming proton. As a consequenceof this definition, the contributions that arise from wide anglesoft particles and initial-state radiation close to the beam di-rection, are suppressed. The usual transverse momentum fac-torization can be applied in the back-to-back limit ( θ → π ).In this limit, the novel EEC observable in DIS is insensitiveto pseudorapidity cuts, usually imposed in the forward andbackward regions due to detector acceptance limitations.We obtained the singular distributions for EEC in DIS upto NNLO from the factorized formula and compared themagainst the full fixed-order QCD calculations up to NLO.The purpose of this comparison is twofold, first the numer-ical agreement in the θ → π limit serves as a validation ofour factorization formalism. Second, the point of deviationof the two distributions indicates the region where power cor-rections become relevant. Predictions up to N LL+NLO werepresented.Non-perturbative and hadronization effects for the EECobservable were investigated by considering non-perturbative form factors extracted from the semi-inclusive hadron produc-tion in DIS. Incorporating these non-perturbative models, wealso presented the comparison of our predictions to
Pythia simulations.Last but not least, we introduced a generalization ofEEC in DIS where a subset of hadrons can be chosen. Themodified observable can be used to tag the initial state andfinal state flavor, or for comparison against experimentalmeasurements where only charged hadrons are considered.To conclude, we propose EEC in DIS as a way to checkthe universality of TMD factorization and study TMDPDFs and FFs with high precision, both experimentally andtheoretically. We remark that EEC can be expanded forthe study of spin effects in a polarized target hadron andtherefore, constitutes a useful tool for the study TMD physicsand nuclear matter effects in electron-ion collisions [69, 70]at the future EIC.
ACKNOWLEDGMENTS
We would like to thank A. Bacchetta, R. Boughezal and F.Petriello for the comments on the manuscript. H.T.L. is sup-ported by the DOE contract DE-AC02-06CH11357 and theNSF grant NSF-1740142. Y.M. is supported by the EuropeanUnion’s Horizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska-Curie grant agreement No.754496 - FELLINI. I.V. is supported by the U.S. Departmentof Energy under Contract No. DE-AC52-06NA25396 and bythe LDRD program at LANL.
Appendix A: Lorentz invariant definition of EEC inDIS
One can also write down a Lorentz invariant definition ofthe EEC in DIS. We denote this definition of the observablewith EEC LI and it is given as follows,EEC LI = (cid:88) a (cid:90) dσ (cid:96) + h → (cid:96) + a + X σ z a δ (tanh ¯ η a − tanh ¯ η ) , (A1)where ¯ η a = 2 (cid:114) q · p a x B P · p a Breit −−−→ frame p a ⊥ p + a . (A2)For particles that satisfy the ¯ n -collinear scaling as describedin eq. (9) we have ¯ η ¯ n (cid:39) π − θ ¯ n . Therefore, in the back-to-backlimit (¯ η → DIS the new ob-servable can be expressed in terms of the conventional TMDfactorization. A advantage of using the above definition isthat, in contrast to the one in eq. (2), it can be applied inthe Laboratory frame and, thus, reduce the propagation ofexperimental uncertainties associated with the Lorentz trans-formation to the Breit frame.[1] C. L. Basham, L. S. Brown, S. D. Ellis and S. T. Love,
Energy correlations in electron-positron annihilation: Testing quantum chromodynamics , Phys. Rev. Lett. (Dec, 1978) 1585–1588.[2] A. V. Belitsky, S. Hohenegger, G. P. Korchemsky,E. Sokatchev and A. Zhiboedov, From correlationfunctions to event shapes , Nucl. Phys.
B884 (2014)305–343, [ ].[3] A. V. Belitsky, S. Hohenegger, G. P. Korchemsky,E. Sokatchev and A. Zhiboedov,
Event shapes in N = 4 super-Yang-Mills theory , Nucl. Phys.
B884 (2014)206–256, [ ].[4] A. V. Belitsky, S. Hohenegger, G. P. Korchemsky,E. Sokatchev and A. Zhiboedov,
Energy-EnergyCorrelations in N=4 Supersymmetric Yang-MillsTheory , Phys. Rev. Lett. (2014) 071601,[ ].[5] L. J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yangand H. X. Zhu,
Analytical Computation ofEnergy-Energy Correlation at Next-to-Leading Order inQCD , Phys. Rev. Lett. (2018) 102001,[ ].[6] J. Gao, V. Shtabovenko and T.-Z. Yang,
Energy-energycorrelation in hadronic Higgs decays: analytic resultsand phenomenology at NLO , .[7] M.-X. Luo, V. Shtabovenko, T.-Z. Yang and H. X. Zhu, Analytic Next-To-Leading Order Calculation ofEnergy-Energy Correlation in Gluon-Initiated HiggsDecays , JHEP (2019) 037, [ ].[8] J. M. Henn, E. Sokatchev, K. Yan and A. Zhiboedov, Energy-energy correlation in N =4 super Yang-Millstheory at next-to-next-to-leading order , Phys. Rev.
D100 (2019) 036010, [ ].[9] I. Moult and H. X. Zhu,
Simplicity from Recoil: TheThree-Loop Soft Function and Factorization for theEnergy-Energy Correlation , JHEP (2018) 160,[ ].[10] L. J. Dixon, I. Moult and H. X. Zhu, Collinear limit ofthe energy-energy correlator , Phys. Rev.
D100 (2019)014009, [ ].[11] M. Kologlu, P. Kravchuk, D. Simmons-Duffin andA. Zhiboedov,
The light-ray OPE and conformalcolliders , .[12] G. P. Korchemsky, Energy correlations in the end-pointregion , JHEP (2020) 008, [ ].[13] H. Chen, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, AnalyticContinuation and Reciprocity Relation for CollinearSplitting in QCD , .[14] M. A. Ebert, B. Mistlberger and G. Vita, TheEnergy-Energy Correlation in the back-to-back limit atN LO and N LL (cid:48) , .[15] A. Ali, E. Pietarinen and W. Stirling, TransverseEnergy-energy Correlations: A Test of PerturbativeQCD for the Proton - Anti-proton Collider , Phys. Lett.B (1984) 447–454.[16] A. Ali, F. Barreiro, J. Llorente and W. Wang,
Transverse Energy-Energy Correlations inNext-to-Leading Order in α s at the LHC , Phys. Rev. D (2012) 114017, [ ].[17] A. Gao, H. T. Li, I. Moult and H. X. Zhu, PrecisionQCD Event Shapes at Hadron Colliders: TheTransverse Energy-Energy Correlator in theBack-to-Back Limit , Phys. Rev. Lett. (2019)062001, [ ].[18] H. T. Li, I. Vitev and Y. J. Zhu,
Transverse-Energy-Energy Correlations in DeepInelastic Scattering , . [19] OPAL collaboration, M. Akrawy et al.,
A Measurementof energy correlations and a determination of alpha-s(M2 (Z0)) in e+ e- annihilations at s**(1/2) =91-GeV , Phys. Lett. B (1990) 159–169.[20]
ALEPH collaboration, D. Decamp et al.,
Measurementof alpha-s from the structure of particle clustersproduced in hadronic Z decays , Phys. Lett. B (1991) 479–491.[21] L3 collaboration, B. Adeva et al., Determination ofalpha-s from energy-energy correlations measured on theZ0 resonance. , Phys. Lett. B (1991) 469–478.[22]
SLD collaboration, K. Abe et al.,
Measurement ofalpha-s from energy-energy correlations at the Z0resonance , Phys. Rev. D (1994) 5580–5590,[ hep-ex/9405006 ].[23] ATLAS collaboration, G. Aad et al.,
Measurement oftransverse energy-energy correlations in multi-jet eventsin pp collisions at √ s = 7 TeV using the ATLASdetector and determination of the strong couplingconstant α s ( m Z ), Phys. Lett. B (2015) 427–447,[ ].[24]
ATLAS collaboration, M. Aaboud et al.,
Determinationof the strong coupling constant α s from transverseenergy–energy correlations in multijet events at √ s = 8 TeV using the ATLAS detector , Eur. Phys. J. C (2017) 872, [ ].[25] ATLAS collaboration,
Determination of the strongcoupling constant and test of asymptotic freedom fromTransverse Energy-Energy Correlations in multijetevents at √ s = 13 TeV with the ATLAS detector , .[26] P. D. Group, P. A. Zyla, R. M. Barnett, J. Beringer,O. Dahl, D. A. Dwyer et al.,
Review of Particle Physics , Progress of Theoretical and Experimental Physics (08, 2020) , [ https://academic.oup.com/ptep/article-pdf/2020/8/083C01/34673722/ptaa104.pdf ].[27] E. C. Aschenauer et al., eRHIC Design Study: AnElectron-Ion Collider at BNL , .[28] A. Ali, G. Li, W. Wang and Z.-P. Xing, Transverseenergy–energy correlations of jets in the electron–protondeep inelastic scattering at HERA , Eur. Phys. J. C (2020) 1096, [ ].[29] J. Collins, Foundations of perturbative QCD , vol. 32.Cambridge University Press, 11, 2013.[30] C. W. Bauer, D. Pirjol and I. W. Stewart,
Soft collinearfactorization in effective field theory , Phys. Rev. D (2002) 054022, [ hep-ph/0109045 ].[31] C. W. Bauer and I. W. Stewart, Invariant operators incollinear effective theory , Phys. Lett. B (2001)134–142, [ hep-ph/0107001 ].[32] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart,
An Effective field theory for collinear and soft gluons:Heavy to light decays , Phys. Rev. D (2001) 114020,[ hep-ph/0011336 ].[33] C. W. Bauer, S. Fleming and M. E. Luke, SummingSudakov logarithms in B → X ( sγ ) in effective fieldtheory , Phys. Rev. D (2000) 014006,[ hep-ph/0005275 ].[34] M. Beneke, A. P. Chapovsky, M. Diehl andT. Feldmann, Soft collinear effective theory and heavy tolight currents beyond leading power , Nucl. Phys.
B643 (2002) 431–476, [ hep-ph/0206152 ].[35] D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn andL. Zoppi,
Transverse momentum dependent distributionswith jets , Phys. Rev. Lett. (2018) 162001, [ ].[36] D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn andL. Zoppi, Transverse momentum dependent distributionsin e + e − and semi-inclusive deep-inelastic scatteringusing jets , JHEP (2019) 031, [ ].[37] D. Gutierrez-Reyes, Y. Makris, V. Vaidya, I. Scimemiand L. Zoppi, Probing Transverse-MomentumDistributions With Groomed Jets , JHEP (2019) 161,[ ].[38] M. Arratia, Y. Makris, D. Neill, F. Ringer and N. Sato, Asymmetric jet clustering in deep-inelastic scattering , .[39] T. Gehrmann, T. Lubbert and L. L. Yang, Transverseparton distribution functions at next-to-next-to-leadingorder: the quark-to-quark case , Phys. Rev. Lett. (2012) 242003, [ ].[40] T. Gehrmann, T. Luebbert and L. L. Yang,
Calculationof the transverse parton distribution functions atnext-to-next-to-leading order , JHEP (2014) 155,[ ].[41] T. Lubbert, J. Oredsson and M. Stahlhofen, Rapidityrenormalized TMD soft and beam functions at twoloops , JHEP (2016) 168, [ ].[42] M. G. Echevarria, I. Scimemi and A. Vladimirov, Unpolarized Transverse Momentum Dependent PartonDistribution and Fragmentation Functions atnext-to-next-to-leading order , JHEP (2016) 004,[ ].[43] M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yangand H. X. Zhu, Transverse Parton Distribution andFragmentation Functions at NNLO: the Quark Case , JHEP (2019) 083, [ ].[44] M.-X. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, Transverse Parton Distribution and FragmentationFunctions at NNLO: the Gluon Case , JHEP (2020)040, [ ].[45] M.-x. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, QuarkTransverse Parton Distribution at theNext-to-Next-to-Next-to-Leading Order , .[46] M.-x. Luo, T.-Z. Yang, H. X. Zhu and Y. J. Zhu, Unpolarized Quark and Gluon TMD PDFs and FFs atN LO , .[47] M. A. Ebert, B. Mistlberger and G. Vita, Transversemomentum dependent PDFs at N LO , JHEP (2020)146, [ ].[48] Y. Li and H. X. Zhu, Bootstrapping Rapidity AnomalousDimensions for Transverse-Momentum Resummation , Phys. Rev. Lett. (2017) 022004, [ ].[49] M. G. Echevarria, I. Scimemi and A. Vladimirov,
Transverse momentum dependent fragmentationfunction at next-to–next-to–leading order , Phys. Rev.
D93 (2016) 011502, [ ].[50] M. A. Ebert, B. Mistlberger and G. Vita,
TMDFragmentation Functions at N LO , .[51] Z. Nagy and Z. Trocsanyi, Multijet cross-sections indeep inelastic scattering at next-to-leading order , Phys.Rev. Lett. (2001) 082001, [ hep-ph/0104315 ].[52] P. Sun, J. Isaacson, C. P. Yuan and F. Yuan, Nonperturbative functions for SIDIS and Drell–Yanprocesses , Int. J. Mod. Phys.
A33 (2018) 1841006,[ ].[53] A. Bacchetta, F. Delcarro, C. Pisano, M. Radici andA. Signori,
Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelasticscattering, Drell-Yan and Z-boson production , JHEP (2017) 081, [ ].[54] I. Scimemi and A. Vladimirov, Non-perturbativestructure of semi-inclusive deep-inelastic and Drell-Yanscattering at small transverse momentum , JHEP (2020) 137, [ ].[55] A. Prokudin, P. Sun and F. Yuan, Scheme dependenceand transverse momentum distribution interpretation ofCollins–Soper–Sterman resummation , Phys. Lett.
B750 (2015) 533–538, [ ].[56] D. de Florian, R. Sassot and M. Stratmann,
Globalanalysis of fragmentation functions for protons andcharged hadrons , Phys. Rev. D (2007) 074033,[ ].[57] D. de Florian, R. Sassot and M. Stratmann, Globalanalysis of fragmentation functions for pions and kaonsand their uncertainties , Phys. Rev. D (2007) 114010,[ hep-ph/0703242 ].[58] Y. L. Dokshitzer, G. Marchesini and B. Webber, Nonperturbative effects in the energy energy correlation , JHEP (1999) 012, [ hep-ph/9905339 ].[59] J. Butterworth et al., PDF4LHC recommendations forLHC Run II , J. Phys.
G43 (2016) 023001,[ ].[60] A. Buckley, J. Ferrando, S. Lloyd, K. Nordstr¨om,B. Page, M. R¨ufenacht et al.,
LHAPDF6: partondensity access in the LHC precision era , Eur. Phys. J.
C75 (2015) 132, [ ].[61] T. Sjostrand, S. Mrenna and P. Z. Skands,
A BriefIntroduction to PYTHIA 8.1 , Comput. Phys. Commun. (2008) 852–867, [ ].[62] T. Sj¨ostrand, S. Ask, J. R. Christiansen, R. Corke,N. Desai, P. Ilten et al.,
An Introduction to PYTHIA8.2 , Comput. Phys. Commun. (2015) 159–177,[ ].[63] Z. Nagy and Z. Trocsanyi,
Three-jet event-shapes inlepton-proton scattering at next-to-leading orderaccuracy , Phys. Lett.
B634 (2006) 498–503,[ hep-ph/0511328 ].[64] T. Becher and M. Hager,
Event-Based TransverseMomentum Resummation , Eur. Phys. J. C (2019)665, [ ].[65] A. Hornig, D. Kang, Y. Makris and T. Mehen, Transverse Vetoes with Rapidity Cutoff in SCET , JHEP (2017) 043, [ ].[66] D. Kang, Y. Makris and T. Mehen, From UnderlyingEvent Sensitive To Insensitive: Factorization andResummation , JHEP (2018) 055, [ ].[67] J. K. Michel, P. Pietrulewicz and F. J. Tackmann, JetVeto Resummation with Jet Rapidity Cuts , JHEP (2019) 142, [ ].[68] A. Signori, A. Bacchetta, M. Radici and G. Schnell, Investigations into the flavor dependence of partonictransverse momentum , JHEP (2013) 194,[ ].[69] H. T. Li, Z. L. Liu and I. Vitev, Heavy mesontomography of cold nuclear matter at the electron-ioncollider , .[70] H. T. Li and I. Vitev, Nuclear matter effects on jetproduction at electron-ion colliders ,2010.05912