Collinear and TMD densities from Parton Branching Methods

Transcription

1 Ola Lelek 1 Francesco Hautmann 2 Hannes Jung 1 Voica Radescu 3 Radek Žlebčík 1 1 Deutsches Elektronen-Synchrotron DESY) 2 University of Oford 3 CERN Rencontres de Moriond, QCD and High Energy Interactions La Thuile 1 / 23

2 Table of Contents Motivation Introduction to the method Collinear PDFs from parton branching method Results for TMDs Fit of integrated TMDs for all flavours to HERA DIS data with Fitter Summary 2 / 23

3 Motivation Motivation 3 / 23

4 Motivation Advantages of the parton branching method To be discussed today: a parton branching method in analogy to a Parton Shower PS) but used to solve DGLAP evolution equation. Parton branching solution reproduces eactly semi-analytical results for collinear PDFs. Similar codes eist use similar formalism): eample: evolution code EvolFMC by Cracow group S. Jadach et al., Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations, Comput.Phys.Commun ) / 23

5 Motivation Advantages of the parton branching method To be discussed today: a parton branching method in analogy to a Parton Shower PS) but used to solve DGLAP evolution equation. Parton branching solution reproduces eactly semi-analytical results for collinear PDFs. Similar codes eist use similar formalism): eample: evolution code EvolFMC by Cracow group S. Jadach et al., Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations, Comput.Phys.Commun ) Advantages of the parton branching method A possibility of: studying different orderings Q T -ordering, virtuality ordering, angular ordering) this will be not discussed in detail here), etraction of TMDs structure of the grid suitable for usage in Fitter) this will be discussed). 4 / 23

6 Motivation Why Transverse Momentum Dependent PDFs? Goal: TMD PDFs for all flavours, all, µ 2 and k T What is Transverse Momentum Dependent TMD) Parton Distribution Function PDF)? TMD PDF is a generalization of a concept of the PDF. TMD: depends not only on and µ 2 but also on k T : TMD, µ 2, k T ) 5 / 23

7 Motivation Why Transverse Momentum Dependent PDFs? Goal: TMD PDFs for all flavours, all, µ 2 and k T What is Transverse Momentum Dependent TMD) Parton Distribution Function PDF)? TMD PDF is a generalization of a concept of the PDF. TMD: depends not only on and µ 2 but also on k T : TMD, µ 2, k T ) TDMs are important in studies on: resummation of all orders in the QCD coupling to many observables in high-energy hadronic collisions, nonperturbative information on hadron structure at very low k T,... perturbative region where QCD evolution equations describe processes a proper and consistent simulation of parton showers, multi-scale problems in hadronic collisions, Acta Physica Polonica B, Vol ) 5 / 23

8 Introduction to the method Introduction to the method 6 / 23

9 Introduction to the method DGLAP evolution equation DGLAP evolution equation for momentum weighted parton density f, µ 2 ) = f, µ 2 ) d f a, µ 2 ) d ln µ 2 = 1 dzp ab α s µ 2 ) ) ), z fb b z, µ2 1) a, b- quark 2N f flavours) or gluon, - longitudinal momentum fraction of the proton carried by a parton a, z = i - splitting variable, µ- evolution mass scale i 1 splitting function: P ab α s µ 2 ) ), z = D ab α s µ 2 ) ) δ1 z) + K ab α s µ 2 ) 1 ) + R ab α s µ 2 ) ), z, 2) 1 z) + f )g) + d = 1 0 f ) f 1))g)d R ab αs µ 2 ) ), z has no power divergences 1 z) n for z 1. ) As long as P ab α s µ 2 ), z has this structure, the formalism presented today can be applied LO, NLO, NNLO). 7 / 23

10 Introduction to the method DGLAP evolution equation DGLAP evolution equation for momentum weighted parton density f, µ 2 ) = f, µ 2 ) d f a, µ 2 ) d ln µ 2 = 1 dzp ab α s µ 2 ) ) ), z fb b z, µ2 1) a, b- quark 2N f flavours) or gluon, - longitudinal momentum fraction of the proton carried by a parton a, z = i - splitting variable, µ- evolution mass scale i 1 splitting function: P ab α s µ 2 ) ), z = D ab α s µ 2 ) ) δ1 z) + K ab α s µ 2 ) 1 ) + R ab α s µ 2 ) ), z, 2) 1 z) + f )g) + d = 1 0 f ) f 1))g)d R ab αs µ 2 ) ), z has no power divergences 1 z) n for z 1. ) As long as P ab α s µ 2 ), z has this structure, the formalism presented today can be applied LO, NLO, NNLO). Two potential problems for numerical solution: details in Backup!) presence of the delta function solved by momentum sum rule c ) dzzp ca α s µ 2 ), z = 0, integrals separately divergent for z 1) solved by a parameter z M : 1 z M 7 / 23

11 Introduction to the method Sudakov formalism After some algebraic transformations: d f a, µ 2 ) d ln µ 2 = where P R ab αs µ 2 ), z zm dz P R ab b ) = R ab αs µ 2 ) ), z + K ab α s µ 2 ) ) ), z fb z, µ2 αs µ 2 )) 1 1 z f a, µ 2) c - real part of the splitting function. zm dz 0 zp R ca α s µ 2 ) ), z 3) 8 / 23

12 Introduction to the method Sudakov formalism After some algebraic transformations: d f a, µ 2 ) d ln µ 2 = where P R ab αs µ 2 ), z zm dz P R ab b ) = R ab αs µ 2 ) ), z + K ab α s µ 2 ) ) ), z fb z, µ2 αs µ 2 )) 1 1 z f a, µ 2) c - real part of the splitting function. zm dz 0 zp R ca α s µ 2 ) ), z 3) Define the Sudakov form factor: ln µ aµ 2 2 ) = ep ln µ 2 0 d ln µ 2) b zm dzzp R ba 0 α s µ 2 ), z) 4) insert it in Eq.3) and integrate ln µ fa, µ 2 ) = f a, µ ) aµ2 ) + ln µ 2 0 d ln µ 2 aµ2 ) aµ 2 ) zm dzp R ab b α s µ 2 ) ) ), z fb z, µ 2. 5) 8 / 23

13 Introduction to the method Iterative solution Eample for a= gluon: ln µ fa, µ 2 ) = f a, µ ) aµ2 )+ ln µ 2 0 Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching. d ln µ 2 aµ2 ) aµ 2 ) zm dzp R ab b ) α s µ 2 ), z) fb z, µ2 0 b µ 2 ) +... Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching. OR This problem has an iterative solution which can be easily implemented in the MC code. 9 / 23

14 Introduction to the method Iterative solution Eample for a= gluon: ln µ fa, µ 2 ) = f a, µ ) aµ2 ) + ln µ 2 0 d ln µ 2 aµ2 ) aµ 2 ) zm dzp R ab b ) α s µ 2 ), z) fb z, µ2 0 b µ 2 )+... Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching. Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching. OR OR OR This problem has an iterative solution which can be easily implemented in the MC code. 9 / 23

15 Introduction to the method Iterative solution Eample for a= gluon: ln µ fa, µ 2 ) = f a, µ ) aµ2 ) + ln µ 2 0 d ln µ 2 aµ2 ) aµ 2 ) zm dzp R ab b ) α s µ 2 ), z) fb z, µ2 0 b µ 2 ) +... Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching. Sudakov: probability of evolving from µ 2 0 to µ2 without any resolvable branching.... OR OR OR This problem has an iterative solution which can be easily implemented in the MC code. 9 / 23

16 Collinear PDFs from parton branching method Collinear PDFs from parton branching method / 23

17 Collinear PDFs from parton branching method LO comparison with semi analytical methods Initial distribution: f b0 0, µ 2 0 ) - 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) f) ratio down 2 µ 2 = GeV 2 µ 2 = 00 GeV 2 µ 2 = 0000 GeV f) ratio Upper plots: collinear pdfs from the parton branching method Lower plots: ratios of the pdfs from a parton branching method and pdfs from QCDnum. 5 gluon 2 µ 2 = GeV 2 µ 2 = 00 GeV 2 µ 2 = 0000 GeV Very good agreement with the results coming from semi analytical methods QCDnum) / 23

18 Collinear PDFs from parton branching method NLO comparison with semi analytical methods Initial distribution: f b0 0, µ 2 0 ) - 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) f) ratio down 2 µ 2 = GeV 2 µ 2 = 00 GeV 2 µ 2 = 0000 GeV f) ratio Upper plots: collinear pdfs from the parton branching method Lower plots: ratios of the pdfs from a parton branching method and pdfs from QCDnum. 5 gluon 2 µ 2 = GeV 2 µ 2 = 00 GeV 2 µ 2 = 0000 GeV Very good agreement with the results coming from semi analytical methods QCDnum) / 23

19 Collinear PDFs from parton branching method Cross check for different z M Comparison of the results for different z M values. Upper plot: collinear pdfs from a MC method Lower plot: ratios of the pdfs from a MC method and pdfs from QCDnum. There is no dependence on z M as long as z M large enough. Here results at NLO, at LO the same conclusion. 13 / 23

20 Results for TMDs Results for TMDs 14 / 23

21 Results for TMDs k T dependence Parton branching method: for every branching µ 2 is generated and available. How to connect µ with Q T of the emitted and k T of the propagating parton? Q T - ordering: Q 2 T,n = µ2. virtuality ordering: Q 2 T,n = 1 z)µ2 k T,n = k T,n 1 Q T,n k T contains the whole history of the evolution. In this method k T is treated properly from the beginning of the evolution- no etra reshuffling at the end is required. 15 / 23

22 Results for TMDs TMD PDFs from different k T definition at LO Reminder: for collinear PDFs there was no z M dependence. What about z M dependence for TMDs? Q T - ordering: Q 2 T,n = µ2 virtuality ordering: Q 2 T,n = 1 z)µ2 large z - soft gluons! Q T - ordering: for every z M value we obtain different TMD not physical behaviour, Q T - ordering shouldn t be used Virtuality ordering: no z M dependence because of the 1 z) term soft gluons suppressed) 16 / 23

23 Fit of integrated TMDs for all flavours to HERA DIS data with Fitter Fit of integrated TMDs for all flavours to HERA DIS data with Fitter 17 / 23

24 Fit of integrated TMDs for all flavours to HERA DIS data with Fitter Procedure of the fit to the HERA 1+2 F 2 data Goal: TMD PDF sets for all flavours, all, Q 2 and k T A kernel A b a is determined from the parton branching method from a toy starting distribution: f 0,b = δ1 ). Fitter chooses a starting distribution A 0,b and performs a convolution of the kernel A b a with the starting distribution A 0,b to obtain a parton density fa, µ 2 dk 2 ) T ) = 0 k T 2 d A 0,b ) Ab a, k2 T, µ2 }{{} fa,µ 2,k 2 ) T Obtained parton density f a, µ 2 ) is fitted to the F 2 data and χ 2 is calculated. Data: arxiv: v3, Abramowicz, H. and others. The procedure is repeated with the new starting distributions A 0,b many times to minimize χ 2. A very good χ 2 /ndf 1.2 is obtained for 3.5 < Q 2 < GeV 2 and > ) σ red Theory/Data e + p e + X NC) HERA1+2 Data Q δ uncorrelated δ total = / 23

25 Fit of integrated TMDs for all flavours to HERA DIS data with Fitter TMDs from fits - comparison of LO and NLO TMDs TMDs with eperimental uncertainties from the fit. Comparison of the LO and NLO TMDs from the fit. 19 / 23

26 Fit of integrated TMDs for all flavours to HERA DIS data with Fitter TMDs from the fit at NLO From parton branching method we can obtain TMDs for all flavours At small k T no branching or just a few branchings), the difference in the quark TMDs comes from initial distributions. At large k T many branchings) TMDs for quarks the same. TMDs sets available soon! check on TMDplotter and TMDlib / 23

27 Summary Summary 21 / 23

28 Summary Summary New approach to solve coupled gluon and quark DGLAP evolution equation with a parton branching method was shown. Advantages: it reproduces eactly semi-analytical solution for collinear PDFs results consistent with QCDNum), moreover: etraction of TMD PDFs possible and done, fit to F2 Hera data at LO and NLO was performed within Fitter, TMDs sets for all flavours with uncertainties were obtained from the fit, options to study different orderings for collinear and TMD PDFs available within this framework. Prospects: TMD sets released soon, application in measurements, long term goal: direct usage in PS matched calculation. 22 / 23

29 Summary Thank you! 23 / 23

30 Back up Back up 1 / 8

31 Back up DGLAP evolution equation DGLAP evolution equation DGLAP evolution equation for momentum weighted parton density f, µ 2 ) = f, µ 2 ) 1 d f a, µ 2 ) d ln µ 2 = b a, b- quark 2N f flavours) or gluon, - longitudinal momentum fraction of the proton carried by a parton a, z = i - splitting variable, µ- evolution mass scale and i 1 a structure of a splitting function: dzp ab αsµ 2 ), z ) fb z, µ2) 7) P ab αsµ 2 ), z ) = D ab αsµ 2 ) ) δ1 z) + K ab αsµ 2 ) ) 1 1 z) + + R ab αsµ 2 ), z ), 8) f )g) + d = 1 0 f )g)d 1 0 f 1)g)d D ab αs µ 2 ) ) = δ ab da αs µ 2 ) ), K ab αs µ 2 ) ) = δ ab ka αs µ 2 ) ), R ab αs µ 2 ) ), z contains logarithmic terms in ln1 z) and has no power divergences 1 z) n for z 1. 2 / 8

32 Back up DGLAP evolution equation DGLAP evolution equation DGLAP evolution equation for momentum weighted parton density f, µ 2 ) = f, µ 2 ) 1 d f a, µ 2 ) d ln µ 2 = b a, b- quark 2N f flavours) or gluon, - longitudinal momentum fraction of the proton carried by a parton a, z = i - splitting variable, µ- evolution mass scale and i 1 a structure of a splitting function: dzp ab αsµ 2 ), z ) fb z, µ2) 7) P ab αsµ 2 ), z ) = D ab αsµ 2 ) ) δ1 z) + K ab αsµ 2 ) ) 1 1 z) + + R ab αsµ 2 ), z ), 8) f )g) + d = 1 0 f )g)d 1 0 f 1)g)d D ab αs µ 2 ) ) = δ ab da αs µ 2 ) ), K ab αs µ 2 ) ) = δ ab ka αs µ 2 ) ), R ab αs µ 2 ) ), z contains logarithmic terms in ln1 z) and has no power divergences 1 z) n for z 1. d f a, µ 2 ) d ln µ 2 = b 1 b dz K ab αsµ 2 ) ) 1 1 z) + R ab αsµ 2 ), z )) fb z, µ2) + fb, µ 2 ) 1 dz K ab αsµ 2 ) ) z) D ab αsµ 2 ) ) ) δ1 z) 9) 2 / 8

33 Back up DGLAP evolution equation DGLAP evolution equation DGLAP evolution equation for momentum weighted parton density f, µ 2 ) = f, µ 2 ) 1 d f a, µ 2 ) d ln µ 2 = b a, b- quark 2N f flavours) or gluon, - longitudinal momentum fraction of the proton carried by a parton a, z = i - splitting variable, µ- evolution mass scale and i 1 a structure of a splitting function: dzp ab αsµ 2 ), z ) fb z, µ2) 7) P ab αsµ 2 ), z ) = D ab αsµ 2 ) ) δ1 z) + K ab αsµ 2 ) ) 1 1 z) + + R ab αsµ 2 ), z ), 8) f )g) + d = 1 0 f )g)d 1 0 f 1)g)d D ab αs µ 2 ) ) = δ ab da αs µ 2 ) ), K ab αs µ 2 ) ) = δ ab ka αs µ 2 ) ), R ab αs µ 2 ) ), z contains logarithmic terms in ln1 z) and has no power divergences 1 z) n for z 1. d f a, µ 2 ) d ln µ 2 = b 1 b dz K ab αsµ 2 ) ) 1 1 z) + R ab αsµ 2 ), z )) fb z, µ2) + fb, µ 2 ) 1 dz K ab αsµ 2 ) ) z) D ab αsµ 2 ) ) ) δ1 z) Two potential problems for numerical solution: presence of the delta function, integrals separately divergent for z 1. 2 / 8 9)

34 Back up DGLAP evolution equation Momentum sum rule To get rid of the delta function: We use momentum sum rule c ) dzzp ca α s µ 2 ), z = 0: 3 / 8

35 Back up DGLAP evolution equation Momentum sum rule To get rid of the delta function: We use momentum sum rule c ) dzzp ca α s µ 2 ), z = 0: d f a, µ 2 ) d ln µ 2 = 1 dz K ab αsµ 2 ) ) 1 b 1 z) + R ab αsµ 2 ), z )) fb z, µ2) + fb, µ 2 ) 1 dz K ab αsµ 2 ) ) 1 b 0 1 z) D ab αsµ 2 ) ) ) δ1 z) + f a, µ 2 ) 1 dzzp ca αsµ 2 ), z ) = c 0 = 1 dz K ab αsµ 2 ) ) 1 b 1 z + R ab αsµ 2 ), z )) fb z, µ2) + f a, µ 2 ) 1 dz z K ca αsµ 2 ) ) 1 c 0 1 z + Rca αsµ 2 ), z ) ) ) 3 / 8

36 Back up DGLAP evolution equation Momentum sum rule To get rid of the delta function: We use momentum sum rule c ) dzzp ca α s µ 2 ), z = 0: d f a, µ 2 ) d ln µ 2 = 1 dz K ab αsµ 2 ) ) 1 b 1 z) + R ab αsµ 2 ), z )) fb z, µ2) + fb, µ 2 ) 1 dz K ab αsµ 2 ) ) 1 b 0 1 z) D ab αsµ 2 ) ) ) δ1 z) + f a, µ 2 ) 1 dzzp ca αsµ 2 ), z ) = c 0 = 1 dz K ab αsµ 2 ) ) 1 b 1 z + R ab αsµ 2 ), z )) fb z, µ2) + f a, µ 2 ) 1 dz z K ca αsµ 2 ) ) 1 c 0 1 z + Rca αsµ 2 ), z ) ) ) We got rid of the delta function, both pieces of the equation written in the same way. Virtual and non-resorvable pieces still included. 3 / 8

37 Back up DGLAP evolution equation Divergence for z 1 To avoid divergence when z 1 a cut off must be introduced: 1 dz b K ab αsµ 2 ) ) 1 1 z + R ab αsµ 2 ), z )) fb z, µ2) + f a, µ 2 ) 1 dz z K ca αsµ 2 ) ) 1 c 0 1 z + Rca αsµ 2 ), z ) ) zm dz K ab αsµ 2 ) ) 1 b 1 z + R ab f a, µ 2 ) c zm 0 dz z αsµ 2 ), z )) fb z, µ2) + K ca αsµ 2 ) ) 1 1 z + Rca αsµ 2 ), z ) It can be shown that terms 1 zma skipped in the integral in eq. 11) are of order O1 zma ). ) 11) 4 / 8

38 Back up DGLAP evolution equation Divergence for z 1 To avoid divergence when z 1 a cut off must be introduced: 1 dz b K ab αsµ 2 ) ) 1 1 z + R ab αsµ 2 ), z )) fb z, µ2) + f a, µ 2 ) 1 dz z K ca αsµ 2 ) ) 1 c 0 1 z + Rca αsµ 2 ), z ) ) zm dz K ab αsµ 2 ) ) 1 b 1 z + R ab f a, µ 2 ) c zm 0 dz z αsµ 2 ), z )) fb z, µ2) + K ca αsµ 2 ) ) 1 1 z + Rca αsµ 2 ), z ) It can be shown that terms 1 zma skipped in the integral in eq. 11) are of order O1 zma ). Different choices of z ma : z ma - fied z ma - can change dynamically with ) the scale: angular ordering: z ma = 1 Q0 Q ) or virtuality ordering: z ma = 1 Q0 2 Q In this presentation: results from fied z ma. 4 / 8 ) 11)

39 Back up Sudakov formalism Sudakov form factor d f a, µ 2 ) d ln µ 2 where P R ab = b zm dz P R ab αsµ 2 ), z ) fb z, µ2) f a, µ 2 ) zm c 0 αs µ 2 ) ), z = R ab αs µ 2 ) ), z + K ab αs µ 2 )) 1 - real part of the splitting function. 1 z dz zpca R αsµ 2 ), z ) 12) 5 / 8

40 Back up Sudakov formalism Sudakov form factor d f a, µ 2 ) d ln µ 2 = b zm dz P R ab αsµ 2 ), z ) fb z, µ2) f a, µ 2 ) zm c 0 where P ab R αs µ 2 ) ), z = R ab αs µ 2 ) ), z + K ab αs µ 2 )) 1 - real part of the splitting function. 1 z Defining the Sudakov form factor: aµ 2 ) = ep ln µ 2 ln µ 2 0 d ln µ 2) b zm 0 dz zpca R αsµ 2 ), z ) 12) dzzpba R αsµ 2 ), z )) 13) d f a, µ 2 ) d ln µ 2 = b zm dzpab R αsµ 2 ), z ) fb z, µ2) + f a, µ 2 ) 1 d aµ 2 ) aµ 2 ) d ln µ 2, 14) 5 / 8

41 Back up TMDs from fits at LO TMDs from the fit at LO From parton branching method we can obtain TMDs for all flavours At small k T no branching or just a few branchings), the difference in the quark TMDs comes from initial distributions. At large k T many branchings) TMDs for quarks the same. 6 / 8

42 Back up TMDs from fits at LO integrated TMD from parton branching method and HERA pdf 7 / 8

43 Back up TMDs from fits at LO Role of the limit in k T integration Comparison of int TMDs integrated up to a diffent k T values The integral over k T has to be performed up to a value higher than the evolution scale to obtain collinear PDF which agrees well with the HERA pdf. 8 / 8