Determination of collinear and TMD photon densities using the Parton Branching method
DDESY 21-0142 Feb 2021
Determination of collinear and TMD photon densities using theParton Branching method
H. Jung , S. Taheri Monfared , and T. Wening Deutsches Elektronen-Synchrotron, D-22607 Hamburg, II. Institut f¨ur Theoretische Physik, Universit¨at Hamburgthe date of receipt and acceptance should be inserted later
Abstract.
We present the first determination of transverse momentum dependent (TMD) photon densitieswith the Parton Branching method. The photon distribution is generated perturbatively without intrinsicphoton component. The input parameters for quarks and gluons are determined from fits to precisionmeasurements of deep inelastic scattering cross sections at HERA. The TMD densities are used to predictthe mass and transverse momentum spectra of very high mass lepton pairs from both Drell-Yan productionand Photon-Initiated lepton processes at the LHC.
PACS.
XX.XX.XX No PACS code given
The Parton Branching (PB) evolution method [1,2] hasbeen applied to evolve both collinear and transverse mo-mentum dependent parton distributions (TMDs) from asmall to large scale using the DGLAP evolution equation.An important feature of this method is that it gives asolution which is fully exclusive, and therefore allowingfor a determination of the TMD parton density. The PBmethod applies the unitarity formulation of the QCD evo-lution equation and has shown to be valid for leading-order(LO), next-to-LO (NLO) and next-to-NLO (NNLO).Given that α s ∼ α over a wide range of scales, it be-comes necessary to include also the corresponding elec-troweak (EW) corrections in the evolution, which can ex-ceed the few percent level and become quantitatively veryimportant for an accurate prediction [3,4]. So far, QEDcorrections have been taken into account for observablesinvolving collinear parton distribution functions (PDFs)[5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Such analyses per-formed up to NNLO in QCD and LO in QED show thatthe photon PDF contribution is not negligible and needs tobe carefully studied for precise predictions at the LHC andeven more for higher energies as the HE-LHC and FCC-hh, where a particularly important aim involves eventswith leptons in the final state. The first significant changein the evolution of parton distributions with QED cor-rections is the appearance of the photon density. In thiscontext, it is necessary and timely to consider the QED contribution to the PB evolution and to extract the firstphoton TMD. Recent phenomenological studies of contri-butions from photon-initiated (PI) channels to lepton pairproduction based on the structure function calculation ofthe underlying process in proton-proton collisions are dis-cussed in [19,20].In this report the determination of parton densitieswith QED corrections obtained using the PB method ispresented together with applications of the obtained pho-ton TMDs to high mass lepton pair production at LHCenergies. In the PB approach the complete evolution of the partondensity including the full information of the kinematic ofthe evolution process is calculated. Soft gluon emissionand transverse momentum recoils are expressed by intro-ducing the soft gluon resolution scale z M . The evolutionwithout resolvable branching from µ to µ is treated viaSudakov form factors ∆ a ( µ , µ ) = exp (cid:32) − (cid:88) b (cid:90) µ µ dq q (cid:90) z M dz z P ( R ) ba ( α s , z ) (cid:33) . (1)Here, z M separates resolvable and non-resolvable branch-ings, z is the longitudinal momentum fraction, α s is the a r X i v : . [ h e p - ph ] F e b H. Jung et al.: Determination of collinear and TMD photon densities using the Parton Branching method strong coupling and P ( R ) ba represents the real emission split-ting functions from flavour a to b . In this approach theTMD evolution equations are written as A a ( x, k t , µ ) = ∆ a ( µ ) A a ( x, k t , µ ) + (cid:88) b (cid:90) dq (cid:48) q (cid:48) dφ π ∆ a ( µ ) ∆ a ( q (cid:48) ) Θ ( µ − q (cid:48) ) Θ ( q (cid:48) − µ ) × (cid:90) z M x dzz P ( R ) ab ( α s , z ) A b (cid:16) xz , k (cid:48) t , q (cid:48) (cid:17) , (2)where A a ( x, k t , µ ) is the TMD distribution of flavour a , carrying the longitudinal momentum fraction x of thehadron’s momentum and transverse momentum k t at theevolution scale µ . The transverse momentum is given by k (cid:48) t = | k + (1 − z ) q (cid:48) | , where q (cid:48) is the rescaled transversemomentum vector of the emitted parton and φ is the az-imuthal angel between q (cid:48) and k . In the application of Eq.2 we consider the scale at which α s is evaluated not nec-essarily equal to the evolution scale. We apply the choiceenforced by an angular ordering of the emissions thereforeensuring quantum coherence of softly radiated partons.The first set of TMDs evolved with NLO QCD DGLAPsplitting functions determined from a fit to HERA I+IIprecision measurements [21] have been described in Ref.[22].Here, we concentrate on QED corrections. The LO-QED kernels are [5] P qq = e q z [1 − z ] + + 32 e q δ (1 − z ) ,P qγ = N e q ( z + (1 − z ) ) ,P γq = e q − z ) z ,P γγ = − N (cid:88) q e q δ (1 − z ) , (3)where the sum (cid:80) q only goes over all active flavours N andwe neglect leptonic contributions to P γγ . The resolvablekernel P ( R ) γγ vanishes, as P γγ only contains a part propor-tional to the Dirac distribution. Since the LO-QED split-ting kernels depend on the electric charge of the quark, theevolution for up-type and down-type quarks is different.The momentum sum rule holds for the LO QED split-ting kernels. A full standard model evolution equations,including a Sudakov form factor is discussed in [3,4].The QED evolution is performed using the PB method,assuming the photon is generated dynamically only fromphoton radiation off the quarks (available in the extendedversion of uPDFevolv [23]). Several groups have determined collinear photon PDFs:The MRST [6] group used a parametrization for the pho-ton PDF based on radiation off of “primordial” up and down quarks. In CT14 [13] a similar phenomenologicalmodel was adopted. The NNPDF group [7] and xFitterDevelopers’ Team [11] treated the photon PDF on thesame footing as the quark and gluon PDFs. Within theirapproach, the photon PDF is parametrized at the startingscale.MMHT [12] provided the photon PDF separated intoelastic (the photon component generated by coherent ra-diation from the proton as a whole) and inelastic contri-butions (the photon component generated from quarks),while the elastic component is less significant at higher Q and negligible below x ∼ . . < Q < and 4 . − 65 at NLO in QCD, for α s ( M Z ) = 0 . χ /dof = 1 . - - - - - x - - - - 10 110 ) m x f ( x , = 10 GeV m photon, from 0.1 up to 1000 GeV t PB-TMDNLOQED-set2-HERAI+II, kCT14qed_proton T M D p l o tt e r . . = 10 GeV m photon, - - - - - x - - - - 10 110 ) m x f ( x , = 10000 GeV m photon, from 0.1 up to 100000 GeV t PB-TMDNLOQED-set2-HERAI+II, kCT14qed_proton T M D p l o tt e r . . = 10000 GeV m photon, Fig. 1. The photon PDF at Q = 10 GeV and Q = 10 GeVplotted versus x . CT14qed-proton is also shown for compari-son.. Jung et al.: Determination of collinear and TMD photon densities using the Parton Branching method 3 We have performed a benchmark test as in Ref. [15]by taking the same parametrization and initial scale inthe FFN scheme with only four active quarks. With thesame assumption of γ ( x, Q ) = 0, an excellent agreementis achieved for all flavors, photon and gluon PDFs.In Fig. 1 we compare the collinear PB photon PDFwith CT14qed at the scale of Q = 10 GeV and Q = 10 GeV. At large scale and small x curves are very similar. TMD parton densities can be obtained within the PBmethod. The procedure for the determination of the TMDdistributions of quarks and gluons is the same as describedin Ref. [1,2,22].Fig. 2 shows the gluon and scaled photon TMD for µ = 10 GeV and µ = 100 GeV. The shape of distributionat low k t ( k t < k t , espe-cially at large scale ( µ = 100 GeV), there is a differencecoming from perturbative gluon-gluon splitting which hasno correspondence in the photon case (this effect appearsin the second term in eq. (2)). - 10 1 10 [GeV] t k - - - - - 10 110 ) m , t x A ( x , k = 10 GeV m PB-TMDNLOQED-set2-HERAI+II, x = 0.01, gluon 1000 · photon T M D p l o tt e r . . = 10 GeV m PB-TMDNLOQED-set2-HERAI+II, x = 0.01, - 10 1 10 [GeV] t k - - - - - 10 1 ) m , t x A ( x , k = 100 GeV m PB-TMDNLOQED-set2-HERAI+II, x = 0.01, gluon 100 · photon T M D p l o tt e r . . = 100 GeV m PB-TMDNLOQED-set2-HERAI+II, x = 0.01, Fig. 2. Transverse Momentum Dependent photon and gluondensities at x = 0 . 01 as a function of k t for different scales µ = 10 GeV and µ = 100 GeV. The CMS experiment [24] has measured the productionof pairs of muons over a wide range of the dilepton invari-ant mass. Dilepton production in hadron-hadron collisionsprovides a unique tool for improving our understanding ofhadronic structure and in particular for testing parton dis-tributions. The contributions from PI lepton production( γγ → l + l − , with l = e, µ ) in hadron-hadron collisions aresizable at high invariant mass [16,25,26,27,28].In Fig. 3 we show the measured dilepton mass spec-trum and compare it with the prediction of collinear NLOPB-QED (Set 2). The spectrum is rather well describedwith the NLO PB-QED prediction. MadGraph5 aMC@NLO [29] is used to calculatethe dilepton and PI lepton production at NLO and LO.The contribution from PI process obtained with the pho-ton PDF is scaled with the factor of 100 for better vis-ibility. The fraction of PI contribution in dilepton massspectrum is generally less than 1%. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb DataMCatNLO PB-NLO-Set (scale)PI PBNLOx − − − − − − − CMS, TeV, DY, full phase-space m( µµ ) [GeV] d σ / d m [ p b / G e V ] Fig. 3. Dilepton high mass distribution production comparedto predictions at QCD+QED using PB-TMDs in the full phasespace. The scaled PI contribution is also shown. In Fig. 4 we present predictions from NLO PB-TMD-QED and NLO matrix elements for Drell-Yan (DY) trans-verse momentum spectra for different lepton pair mass re-gions performed with the Cascade H. Jung et al.: Determination of collinear and TMD photon densities using the Parton Branching method MCatNLO PB-NLO-Set (scale)PI PBNLOx < M µ + µ − < 800 GeV − − − − CMS, TeV, DY, full phase-space p T ( µµ ) [GeV] d σ / dp T [ p b / G e V ] MCatNLO PB-NLO-Set (scale)PI PBNLOx < M µ + µ − < − − − CMS, TeV, DY, full phase-space p T ( µµ ) [GeV] d σ / dp T [ p b / G e V ] MCatNLO PB-NLO-Set (scale)PI PBNLOx < M µ + µ − < − − − CMS, TeV, DY, full phase-space p T ( µµ ) [GeV] d σ / dp T [ p b / G e V ] MCatNLO PB-NLO-Set (scale)PI PBNLOx < M µ + µ − < − − − CMS, TeV, DY, full phase-space p T ( µµ ) [GeV] d σ / dp T [ p b / G e V ] Fig. 4. Standard DY and PI transverse momentum spectrabased on collinear and TMD PB-QED (Set2) at different highmass regions. We determined collinear and TMD photon densities withthe PB method and investigated the mass and transversemomentum spectra of DY lepton-pair production at veryhigh DY masses by matching PB-QED (Set 2) distribu-tions to NLO calculations via MC@NLO. We observed agood description of the dilepton mass measurements in therange 15 to 3000 GeV at √ s = 13 TeV. We also provideda new perspective on the contribution of PI lepton processin the transverse momentum spectrum of very high masslepton pairs.Extracting the photon PB TMD lays the ground workneeded to generate collinear PDFs and TMDs for theheavy gauge bosons Z , W by implementing the EW sec-tor. Acknowledgments. We thank F. Hautmann for various discussions and com-ments on the manuscript. STM thanks the HumboldtFoundation for the Georg Forster research fellowship andgratefully acknowledges support from IPM. References 1. F. Hautmann, H. Jung, A. Lelek, V. Radescu andR. Zlebcik, Phys. Lett. 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