Collinear and TMD densities from Parton Branching Method Ola Lelek 1 • , Francesco Hautmann 2 , Hannes Jung 1 , Voica Radescu 3 , Radek Žlebčík 1 1 DESY 2 University of Oxford 3 CERN Resummation, Evolution, Factorization 2017 Madrid
Introduction Use a Parton Branching (PB) method to: ● solve DGLAP evolution equation to obtain collinear Parton Distribution Functions (PDFs) But not only! → further advantages: ● obtain Transverse Momentum Dependent (TMD) PDFs Goal: ● TMD PDF sets for all flavours, all kinetically allowed x, k t , μ 2 The project is NOT just an evolution! ● already available in xFitter for determination of PDFs ● TMDlib and TMDplotter web page for easy acces to TMDs and collinear PDFs and plotting T ool ● comparison with measurements through interface with CASCADE and for example Rivet Details in papers: Phys.Lett. B772 (2017) 446-451 2 arXiv:1708.03279
Motivation Parton Shower (PS) crucial for obtaining predictions for processes ● at high-energy hadron colliders open problem: shower’s transverse momentum kinematics ● One of the approaches to deal with the problem: T ransverse ● Momentum Dependent (TMD) formalism based on TMD form of factorization We want to develop an approach in which transverse momentum kinematics will be treated without any mismatch between matrix element (ME) and PS 3
The formalism 4
The formalism n a d n ! a s s n e o c o i t r u p l o g s n i e h v c i n t a a r r e B t i n o n a t r s a a P h f m o e s l m b o r r e p t n s i i h n T o i t a t e r p r e t n i 5
Parton Branching 6
Parton Branching OR 7
k t dependence PB method: for every branching μ is generated and available. How to connect μ with q t,c of the emitted and k t,a of the propagating parton? k t,a contains the whole history of the evolution. In this method kinematics is treated properly at every 8 branching.
z M choice Partons emitted with a transverse momenta smaller than a certain value ● given by a resolution scale can not be resolved → branchings with z >z M are non- resolvable Normally treated with the plus prescription but integrals in evolution ● equation separately divergent for z → 1 : → solved by a parameter z M : Different choices of z M : • z M - fixed • z M - can change dynamically with the scale (resolution scale different for different scales): Replace q t,c with some minimum q 0 : 9
More details in: arXiv:1708.03279 NLO comparison with semi analytical methods Initial distribution: f b0 (x 0 , μ 0 2 ) - from QCDnum The evolution performed with parton branching method up to a given scale μ 2 . Obtained distribution compared with a pdf calculated at the same scale by semi analytical method (QCDnum) Results for fixed 1 − z M = 10 −5 . Upper plots: collinear pdfs from the parton branching method Lower plots: ratios of the pdfs from a parton branching method and pdfs from QCDnum. Very good agreement with the results coming from semi analytical methods (QCDnum). 10
More details in: arXiv:1708.03279 iTMDs: Cross check for different fixed z M Comparison of the results for different fixed z M values (all independent of branching scale). Upper plot: collinear pdfs from the parton branching method Lower plot: ratios of the pdfs from the parton branching method and pdfs from QCDnum. There is no dependence on z M as long as z M large enough. 11
iTMDs with dynamic z M Previous slides: for a fixed value of z M the results the same as from standard evolution (with the same initial conditions). Here: comparison of the PDF from a standard evolution with the PDF from PB with the angular ordering condition to associate k t and μ, with the dynamic z M , for different resolution scales q 0 (with the same initial conditions) μ' - scale at which the branching happens q 0 - a free parameter describing the resolution scale Angular ordering with dynamic z M differs from a standard evolution, especially if the q 0 large. 12 Here starting distribution from HERApdf 2.0 NLO ●
More details in: arXiv:1708.03279 TMDs: Cross check for different fixed z M For collinear PDFs there was no z M dependence (for fixed z M ). ● What about z M dependence for TMDs? ● Here the results for fixed z M at NLO ● large z M - a lot of soft gluons! q t - ordering: for every z M value we obtain different TMD → not physical behavior, q t - ordering shouldn’t be used For virtuality and angular ordering no z M dependence (suppression of soft gluons because of the(1 − z) term) 13
TMDs with dynamic z M Comparison of the TMDs from PB with the angular ordering condition to associate k t and μ, with the dynamic z M , for different resolution scales q 0 q0 = 0.01 q0 = 0.01 q0 = 0.1 q0 = 0.1 q0 = 0.5 q0 = 0.5 No dependence on q 0 parameter in TMD distributions (because of the (1-z) term in the calculation of k t the large z region suppressed) 14 The same conclusion for virtuality ordering condition to associate k t and μ, with the dynamic z M , for different resolution scales q 0
TMDs with fixed and dynamic z M fixed fixed dynamic dynamic Virtuality ordering condition to associate k t and μ Angular ordering condition to associate k t and μ No difference between fixed and dynamic z M No difference between fixed and dynamic z M (because of the (1-z) term in calculation of k t ) (because of the (1-z) term in calculation of k t ) Difference in the TMDs from angular ordering 15 and virtuality ordering
More details in: arXiv:1708.03279 Example of TMDs from PB From PB method we can obtain TMDs for all flavours at all kinematically allowed x, μ and k t Results for angular ordering to associate k t and μ , fixed z M 16
Procedure of the fit to the F 2 data from HERA 1+2 Goal: TMD PDF sets for all flavours, all x, μ 2 and k t 17 More details in talk of Radek Žlebčík, paper in preparation
Application of our TMDs to DY TMDs as a manifestation of transverse momentum resummation up to large k t LO Drell–Yan for qq → Z : ● k t according to TMD (m DY fixed, x 1 , x 2 change) ● 18
Application of our TMDs to DY TMDs as a manifestation of transverse momentum resummation up to large k t LO Drell–Yan for qq → Z : ● k t according to TMD (m DY fixed, x 1 , x 2 change) ● Virtuality ordering and angular ordering condition to associate μ 2 and k t both for fixed z M . 19
Application of our TMDs to DY TMDs as a manifestation of transverse momentum resummation up to large k t LO Drell–Yan for qq → Z : ● k t according to TMD (m DY fixed, x 1 , x 2 change) ● TMD fitted to HERA data reproduces correctly the shape of Z p t spectrum NO tuning /adjustment of parameters is done, all is coming from PDF fit, no free parameters after fit (in contrast to what is being done in MC tuning) transverse momentum originates directly from parton branching difference between angular ordering and virtuality ordering observed also in physical observable Virtuality ordering and angular ordering condition to associate μ 2 and k t both for fixed z M . 20
Application of our TMDs to DY TMDs as a manifestation of transverse momentum resummation up to large k t LO Drell–Yan for qq → Z : ● k t according to TMD (m DY fixed, x 1 , x 2 change) ● TMD fitted to HERA data reproduces correctly the shape of Z p t spectrum NO tuning /adjustment of parameters is done, all is coming from PDF fit, no free parameters after fit (in contrast to what is being done in MC tuning) transverse momentum originates directly from parton branching difference between angular ordering and virtuality ordering observed also in physical observable Virtuality ordering and angular ordering condition to associate μ 2 and k t both for z M changing with the scale μ 2 . Results the same as for the fixed z M because TMDs the same. 21
Application of our TMDs to DY TMDs as a manifestation of transverse momentum resummation up to large k t LO Drell–Yan for qq → Z : ● k t according to TMD (m DY fixed, x 1 , x 2 change) ● TMD fitted to HERA data reproduces correctly the shape of Z p t spectrum NO tuning /adjustment of parameters is done, all is coming from PDF fit, no free parameters after fit (in contrast to what is being done in MC tuning) transverse momentum originates directly from parton branching difference between angular ordering and virtuality ordering observed also in physical observable free parameters: intrinsic k t (here gauss with width=1.5 GeV), scale in α s , fit to F2 Virtuality ordering and angular ordering condition to (including k t dependence of ME) associate μ 2 and k t both for z M changing with the scale μ 2 . Results the same as for the fixed z M because TMDs the same. 22
Work in progress Check the sensitivity to intrinsic k t Angular ordering to associate scale μ 2 with k t , ● Little dependence on intrinsic kt for LHC data. To Do: Check for the low p T data. 23
Work in progress Scale in α s Up to now the scale in α s in was the scale of a branching μ 2 . But the angular ordering ● suggests to use α s (μ 2 (1-z) 2 ) We compare: α s (μ 2 ) ● α s (μ 2 (1-z) 2 ) ● α s (μ 2 (1-z) 2 z 2 ) ● The scale suggested by angular ordering give a very good description of the Z p T spectrum. 24 Note: no tuning of free parameters here!
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