Presented at the PHOTON2005 Conference, 31 August–4 September, 2005, Warsaw, Poland.In the last year four new parametrizations of the Hadronic Photon
Structure Function at Next to Leading Order have appeared. In this talk,
I briefly review the main features of the three of them: the FFNSCJK, CJK
and AFG.This work has been partially supported by Junta de Andalucia (contract FQM-330), Ministerio de educación y Ciencia (contract FPA2003-09298-c02-01) and the European Community’s Human Potential Programme (contract HPRN-CT-2002-00311 EURIDICE)

Cornet Sánchez del Águila, Fernando

Acta Physica Polonica B

English

Repositorio Institucional Universidad de Granada

Vol. 37 (2006) ACTA PHYSICA POLONICA B No 3NEW PARAMETRIZATIONS FOR THEPHOTON STRUCTURE FUNCTION∗Fernando CornetDepartamento de Física Teórica y del CosmosandCentro Andaluz de Física de Partículas, Universidad de GranadaCampus de Fuente Nueva, E-18071 Granada, Spain(Received January 30, 2006)In the last year four new parametrizations of the Hadronic PhotonStructure Function at Next to Leading Order have appeared. In this talk,I briefly review the main features of the three of them: the FFNSCJK, CJKand AFG.PACS numbers: 13.60.Hb, 14.70.Bh, 14.65.Dw, 14.65.Fy1. IntroductionThe photon structure function has been recognized as an interestingquantity for QCD since long ago [1, 2] because it was expected that theasymptotic point-like Q2 evolution could be calculated without additionalassumptions. Unfortunately, further studies showed the need for a hadroniccomponent that required extra assumptions at an input scale. However, agood knowledge of the parton content of the photon is still needed and usefulfor many phenomenological applications. A review of the situation in theearly days can be found in these proceedings [3].The main problem found in the study the Photon Structure Functionsome years ago was the lack of experimental data [4, 5]. Indeed, there werevery few data and they covered a very limited region in the plane (x,Q2).The situation has improved very much in the last years with the measure-ments performed by the four LEP experiments. These measurements re-duced the experimental errors in regions of (x,Q2) that were already stud-ied at previous experiments and also covered regions in this plane where∗ Presented at the PHOTON2005 Conference, 31 August–4 September, 2005, Warsaw,Poland.(663)664 F. Cornetthere were no previous measurements. These has prompted the appear-ance in the last year of four new parametrizations for the Photon StructureFunction at LO and NLO. They are, in chronological order: FFNSCJK,CJK [6], AFG05 [7] and SAL [8]. In this talk I will just cover the first threeparametrizations for the fourth one will be covered in the next talk [9].One important difference, certainly not the only one, among the newparametrizations is the way they deal with the heavy quark thresholds.There are three schemes to introduce these thresholds.• The Fixed Flavor Number Scheme (FFNS), where one considers onlythe three light quarks and gluons as partons of the photon for all en-ergy scales. The heavy quarks, c and b, contribute only as externalparticles in the final state produced in hard processes either in a directproduction or through the partonic content of the photon. In the cal-culation of the heavy quark contributions one keeps their mass fixed totheir physical value. This scheme is expected to give a poor descriptionof the Photon Structure Function for energy scales much larger thanthe heavy quark masses, where one would expect the contributions ofthe heavy quarks to be similar to the ones of the light quarks.• The Zero-mass Variable Flavor Number Scheme (ZVFNS). In thisscheme the number of active flavors as partons of the photon increasesin one whenever the energy goes through a heavy quark threshold.From then on, the heavy quark is treated as massless in the evolutionof the parton densities, just in the same way as the light quarks aretreated. This scheme is expected to solve the problem of the FFNSscheme at large energies but, obviously, should have problems at en-ergies near the thresholds, where one cannot neglect the heavy quarkmasses.• The Variable Flavor Number Scheme (VFNS) attempts to solve theproblems of the previous schemes. Here, one considers both contri-butions: the heavy quarks are produced in the final state taking intoaccount their masses but also they are included as massless partonsof the photon. In this way both energy regions are treated properly.Unfortunately, this scheme is not free of problems either. It is clearthat there is a double counting that should be avoided introducingsome subtraction terms.2. The FFNSCJK NLO and CJK NLO parametrizationsIt is clear from the name that the FFNSCJK uses the Fixed FlavorNumber Scheme, while the CJK parametrization is using the VFNS. De-tailed expressions for the Photon Structure Function, F γ2 (x,Q2), for bothNew Parametrizations for the Photon Structure Function 665parametrizations, involving a description of the way the subtraction termsare chosen in the CJK parametrization can be found in Ref. [6]. In addi-tion, the CJK parametrization uses the ACOT(χ) scheme. The idea is toenforce that the heavy quark distribution functions vanish for W = 2mh(and below), where W is the invariant mass of the hadronic final state.This is achieved substituting the x variable by χh = x(1 + 4m2h/Q2) inthe heavy quark densities. In this way χ → 1 and qh(χ,Q2) → 0 forW 2 = (1 − x)Q2/x → 4m2h.Both parametrizations are written as a function of the quark and gluondistribution functions that obey an inhomogeneous DGLAP set of equations.In order to solve these equations we introduce the same input for bothparametrizations at Q20 = 0.765 GeV2, based on Vector Meson Dominance(VMD):fγ(x,Q20) =∑V4παf̂2VfV (x,Q20), (1)with the sum running over all light vector mesons (V ) into which the photoncan fluctuate. The parameters f̂2V can be extracted from the experimentaldata on Γ (V → e+e−) width. In practice we take into account the ρ0meson while the contributions from the other mesons are accounted for viaa parameter κfγ(x,Q20) = κ4παf̂2ρfρ(x,Q20), (2)which is left as a free parameter in the fits. The scale Q0 has been fixed tothis value because it is the one that allows a better fit to the experimentaldata.For the ρ meson we assume the following form for the valence quark andgluon distributionsxvρ(x,Q20) = Nvxα(1 − x)β, (3)xGρ(x,Q20) = ÑGxvρ(x,Q20) = NGxα(1 − x)β ,where Nv, NG, α and β are free parameters. The sea quark distributionis assumed to vanish at this scale. This is similar to what was done inthe GRV parametrization [11], but there the authors fixed the values of theparameters α and β to the ones they had previously obtained for the piondistribution functions. Since there are more data available now, we preferto leave these parameters as free parameters in the fit.We have included in the fit all the available data in year 2004 except theDELPHI LEP2 data because they present three sets of mutually inconsistent666 F. Cornetdata. The total number of points used in the fit is 192 covering a kinematicalrange of 0.001 ≤ x ≤ 0.65 and 1.3 <GeV2 ≤ Q2 ≤ 780 GeV2. The resultsof the fit are shown in Table I. Introducing the DELPHI LEP2 data inthe fit the χ2/DOF increases to 1.50 (TWOGAM), 1.54 (PHOJET) or 1.66(PYTHIA), depending on the Monte Carlo used to analyze the data.TABLE IThe χ2 and parameters of the final fits for 192 data points for FFNSCJK NLO andCJK NLO models with assumed Q20 = 0.765 GeV2. The α, β and κ errors areobtained from Minos requiring ∆χ2 = 1.NLO models χ2 χ2/DOF κ α βFFNSCJK 243.3 1.29 2.288+0.108−0.096 0.502+0.071−0.066 0.690+0.282−0.252CJK 256.8 1.37 2.662+0.108−0.099 0.496+0.063−0.057 1.013+0.284−0.255A comparison of the CJK predictions with the recent L3 data for Q2 =12.4 GeV2 and Q2 = 16.7 GeV2, not included in the fit because they havebeen published after the fit was performed [12,13] is shown in Fig. 1. Datafor similar Q2 from CELLO [14], DELPHI [15], OPAL [16] and TOPAZ [17]are also included. We see that the CJK parametrization provides a gooddescription of the data, even though not all the data sets are fully compatiblewith each other.Fig. 1. Comparison of the CJK NLO prediction with various sets of data, includingthe new L3 data not included in the fit. The kink observed in the curve at large xis the charm quark threshold.3. The AFG NLO parametrizationThe third parametrization I will briefly review here has been performedby Aurenche, Fontannaz and Guillet [7]. It is an update of a previousparametrization obtained by the same authors [18]. This parametrizationNew Parametrizations for the Photon Structure Function 667uses the ZVFNS with Nf = 5, however they keep terms O(m2h/Q2) in the di-rect contribution in order to have a smooth threshold behavior. At Q2 = Q20the structure function is given by:F γ2 (x,Q20)x= Cγ(x) +Nf∑f=1[e2f (qNPf (Q20) + q̄NPf (Q20) − Cfγ,c], (4)where Cγ(x) is the direct contribution and Cfγ,c is given by the “hand-bag”diagram. The non-perturbative input is also based on VMD, identifyingthe form of the parton distributions for the ρ meson with the ones for thepion obtained in Ref. [19], but leaving a normalization factor, Cnp, as a freeconstant:xuγvalence= Cnpα49xuπvalence = Cnpα491B(p2, 1 + p3)xp2(1 − x)p3 ,xuγsea = Cnpα23xuπsea = Cnpα23Cs(1 − x)p8 ,xGγ = Cnp23xgπ = Cnp23Cg(1 − x)p10 , (5)where p2 =0.48, p3 =0.85, p8 =7.5, p10 =1.9, Cs =1.2, Cg =0.447(1+p10) andB(x, y) is the beta function. In summary, there are only two free parameters:the input scale Q0 and the normalization factor Cnp.Fig. 2. Figure from Ref. [7] where it is presented the ∆χ2 = 1 contour in the(Q20, Cnp) plane as well as the individual best fits for each LEP experiment. Thepoint for DELPHI(12.7) is outside the figure, while for DEPHI (3.7) no minimumis found for Q20 < 1.9 GeV2.The values of the two free parameters are obtained performing a fit toall the LEP experimental data. The best fit, with a χ2/DOF = 1.03, gives668 F. CornetQ0 = 0.7 GeV2 (very similar to the one used in the FFNSCJK and CJKparametrizations) and Cnp = 0.78. Aurenche, Fontannaz and Guillet havealso performed independent fits for each one of the four LEP experiments.The result is summarized in Fig. 2, where one can see that the best fitsfrom the DELPHI experiment give very different values of the parameterscompared with the ones obtained from the global fit as well as from eachone of the other three experiments.The authors have also explored two other parametrizations allowing fora harder gluon component modifying the value of p10to p10 = 1.0 or a softergluon component with p10 = 4.0.4. Web pagesInstead of a summary I will finish just referring the interested reader tothe web pages where he can find FORTRAN routines with these parametriza-tions:• FFNSCJK and CJK: http://www.fuw.edu.pl/∼pjank/param.html• AFG05: http://www.lapp.in2p3.fr/lapth/PHOX FAMILY/main.htmlI thank P. Jankowski and M. Krawczyk for the nice atmosphere in thecollaboration and PJ for his help in preparation of this talk. I also thankM. Krawczyk and H. Abramowicz for inviting me to this very nice meeting.This work has been partially supported by Junta de Andalucia (contractFQM-330), Ministerio de educación y Ciencia (contract FPA2003-09298-c02-01) and the European Community’s Human Potential Programme (con-tract HPRN-CT-2002-00311 EURIDICE).REFERENCES[1] T.F. Walsh, P.M. Zerwas, Nucl. Phys. B41, 551 (1972); Phys. Lett. B44, 195(1973).[2] E. Witten, Nucl. Phys. B120, 189 (1977).[3] A.J. Buras, Acta Phys. Pol. B 37, 609 (2006) these Proceedings.[4] M. Krawczyk, M. Staszel, A. Zembrzuski, Phys. Rep. 345, 265 (2001).[5] R. Nisius, Phys. Rep. 332, 165 (2000).[6] F. Cornet, P. Jankowski, M. Krawczyk, Phys. Rev. D70, 093004 (2004)[hep-ph/0404063]; Acta Phys. Pol. B 35, 2213 (2004) [hep-ph/0404244];Nucl. Phys. Proc. Suppl. 126, 28 (2004); hep-ph/0310029; F. Cornet,P. Jankowski, M. Krawczyk, A. Lorca, Phys. Rev. D68, 014010 (2003)[hep-ph/0212160].New Parametrizations for the Photon Structure Function 669[7] P. Aurenche, M. Fontannaz, J.Ph. Guillet, Eur. Phys. J. C44, 395 (2005)[hep-ph/0503259].[8] W. Slominski, H. Abramowicz, A. Levy, hep-ph/0504003.[9] H. Abramowicz, A. Levy, W. Slominski, Acta Phys. Pol. B 37, 671 (2006)these Proceedings.[10] M. Glück, E. Reya, A. Vogt, Phys. Rev. D45, 3986 (1992).[11] Gluck, Reya, Vogt, Phys. Rev. D46, 1973 (1992).[12] P. Achard et al. (L3 Collaboration), Phys. Lett. B622, 249 (2005)[hep-ex/0507042].[13] M.N. Kienzle-Focacci, Acta Phys. Pol. B 37, 677 (2006) these Proceedings.[14] CELLO Collaboration, Phys. Lett. B126, 391 (1983).[15] P. Abreu et al. (DELPHI Collaboration), Z. Phys. C69, 223 (1996); I. Tyapkin(DELPHI Collaboration), Proceedings of the International Conference on theStructure and Interactions of the Photon and 14th International Workshopon Photon–Photon Collisions (Photon 2001), Ascona (Switzerland) 3–7 Sept.2001.[16] G. Abbiendi et al. (OPAL Collaboration), Eur. Phys. J. C18, 15 (2000).[17] K. Muramatsu et al. (TOPAZ Collaboration), Phys. Lett. B332, 477 (1994).[18] P. Aurenche, M. Fontannaz, J.Ph. Guillet, Z. Phys. C64, 621 (1994).[19] P. Aurenche et al., Phys. Lett. B233, 517 (1989).

New parametrizations for the photon structure function

https://digibug.ugr.es/bitstream/10481/32682/1/Cornet_PhotonStructure.pdf

New parametrizations for the photon structure function

Abstract

Similar works

Full text

Available Versions

Repositorio Institucional Universidad de Granada