Recent Progress on the Determination of Nuclear Parton Distribution Functions

Transcription

1 Recent Progress on the Determination of Nuclear Parton Distribution Functions Shunzo Kumano High Energy Accelerator Research Organization (KEK) Graduate University for Advanced Studies (GUAS) Collaborators: Masanori Hirai (TokyoTech) Takahiro Nagai (GUAS) From RHIC to LHC: Achievements and Opportunities INT, University of Washington, Seattle, U.S.A. November 10, 2006

2 Contents (1) Introduction Motivation Parton Distribution Functions (PDFs) in the Nucleon Outline to determine nuclear PDFs (2) Determination of PDFs in Nuclei Analysis Method and Results Comparison with Other Parameterizations Comments on Future Experiments

3 Motivations (1) Basic interest to understand hadron structure, determination of fundamental constants perturbative & non-perturbative QCD fundamental constants: α s, sin 2 θ W (2) Practical purpose: to describe hadron cross sections precisely. For hadron reactions with Q 2 >1 GeV 2, accurate PDFs (parton distribution functions) are needed. For example, heavy-ion reactions: quark-gluon plasma signature exotic events at large Q 2 : physics of beyond current framework neutrino reactions: nuclear effects in ν + 16 O

4 Example 1: to find any exotic signatures in hadron reactions Difference between theory and experiment Comparison of theoretical calculations with CDF experimental data. Jet transverse energy CDF experiment: PRL, 77 (1996) 138. p + p jet + X s = 1.8 TeV, E T jet = GeV Deviations from data Subquark signature??? Could be explained by modification of G(x) at large x. (importance of accurate parton distributions)

5 JET cross sections JET cross sections are explained by the CTEQ6 PDFs. However, it is impossible to predict accurate jet cross sections at large p T. CTEQ6 CTEQ5 MRS2001

6 Example 2: J/ψ suppression dσ J /ψ (E T ) dσ DY (E T ) suppression of J/ψ cross section at large E T in comparison with DY E T small E T large E T NPDF effects J /ψ DY ~ gluon quark / antiquark If the g A (x) modification is significantly different from the quark one, the suppression could be explained partially by the NPDF effect.

7 Parton Distribution Functions (PDFs) in the Nucleon

8 Papers on unpolarized PDFs in the nucleon (Oct. 2005) CTEQ (uncertainties) D. Stump (J. Pumplin) et al., Phys. Rev. D65 (2001) & (CTEQ6) D. Pumplin et al., JHEP, 07 (2002) 012; JHEP, 0310 (2003) 046; Eur. Phys. J. C40 (2005) 145; Phys.Rev. D69 (2004) GRV (GRV98) M. Glück, E. Reya, and A. Vogt, Eur. Phys. J. C5 (1998) no recent update MRST A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, (MRST2001) Eur. Phys. J. C23 (2002) 73; (MRST2002, MRST2003-conservative) Eur. Phys. J. C28 (2003) 455; (theoretical errors) Eur. Phys. J. C35 (2004) 325; (MRST2004-physical gluon) Phys. Lett. B604 (2004) 61; (MRST2004-QED) Eur. Phys. J. C39 (2005) 155. Alekhin S. I. Alekhin, Phys. Rev. D68 (2003) ZEUS S. Chekanov et al., hep-ph/ H1 C. Targett-Adams, hep-ex/ It is likely that I miss some papers! Recent activities uncertainties of PDFs NNLO analysis QED corrections

9 Parton distribution functions are determined by fitting various experimental data. i electron/muon: µ + p µ + X i neutrino: ν µ + p µ + X i Drell-Yan: i p + p µ + µ + X (1) assume functional form of PDFs at fixed Q 2 ( Q 2 0 ) : e.g. f i (x,q 2 0 ) = A i x α i (1 x)β i (1 + γ x), i where i = u v, d v, u, d, s, g (2) calculate observables at their experimental Q 2 points. (3) then, the parameters A i, α i, β i, γ i are determined so as to minimize χ 2 in comparison with data.

10 Recent unpolarized distributions see CTEQ6, JHEP 0207 (2002) 012; GRV98, Eur. Phys. J. C5 (1998) 461; MRST02, Eur. Phys. J. C28 (2003) 455. xg/5 Q 2 =10 GeV 2 x c x s x u, x d x u V x d V

11 Determination of Parton Distribution Functions in Nuclei

12 Introduction Outline to extract nuclear PDFs (Detailed analysis is explained later.)

13 Status of PDF determinations Unpolarized PDFs in the nucleon Investigated by 3 major groups (CTEQ, GRV, MRST). Well studied from small x to large x in the wide range of Q 2. The details are known. (Recent studies: NNLO, QED, error analysis, Polarized PDFs in the nucleon 3 3 Investigated by several groups (GS, GRSV, LSS, AAC, BB, ). Available data are limited (DIS) at this stage. (recent: HERMES, JLab) New data from RHIC-Spin, COMPASS, J-PARC(?), erhic(?), PDFs in nuclei 3 3 Investigated by only a few groups. Details are not so investigated! Available data are limited (inclusive DIS, Drell-Yan). New data from RHIC, LHC, JLab, J-PARC(?), erhic(?), s s, ) J-PARC = Japan Proton Accelerator Research Complex

14 Situation of data for nuclear PDFs Available data for nuclear PDFs Jlab at large x Neutrino factory: 10~15 years later? Small-x, high-energy electron facility? RHIC, LHC, J-PARC RHIC, LHC RHIC, LHC Table from MRST, hep/ph

15 Current nuclear data are kinematically limited. x = Q2 2 p q Q2 ys Q 2 1 fixed target: min(x) = 2M N E lepton 2E lepton (GeV) if Q 2 1 GeV 2 for E lepton (NMC) = 200 GeV, min(x) = = (from H1 and ZEUS, hep-ex/ ) x = F 2 data for the proton x = F 2 & Drell-Yan data for nuclei x = 0.65 region of nuclear data

16 Electron / Muon Scattering dσ = dσ = 1 4 (k p) 2 m 2 M N 2 M = e u(k ',λ ') γ µ u(k,λ) 2M N s M N 2 (2π ) 4 δ 4 (k + p k ' p X ) M 2 d 3 k ' (2π ) 3 2E ' pol X α 2 d 3 k ' Q 4 Lµν W µν E ' g µν q 2 X e J em ν (0) p,λ N L µν = u(k ',λ ') γ µ u(k,λ) = 2 k µ k ' ν + k ' µ k ν (k k ' m 2 )g µν 1 W µν = (2π ) 4 δ 4 (k + p k ' p X ) p,λ N J em µ (0) X X J em ν (0) p,λ N 4π M N λ N X = W 1 g µν q q µ ν q 2 + W 1 2 p 2 µ p q q M N q 2 µ p ν p q q q 2 ν λ, λ k' k q * u(k ',λ ') γ ν u(k,λ) p X p dσ dω de = α 2 4E 2 sin 4 (θ / 2) θ 2W 1 (ν,q 2 )sin W θ 2 (ν,q2 )cos 2 2 F 1 = M N W 1, F 2 = νw 2 2 In parton model, 2xF 1 = F 2, F 2 = x e q [q i (x) + q i (x)] q

17 Nuclear modification Nuclear modification of F 2 A / F 2 D is well known in electron/muon scattering EMC NMC E139 E665 F A 2 2 (LO) = e i x q i (x) + q i (x) i [ ] A Fermi motion shadowing sea quark x original EMC finding valence quark

18 Sea quark e/µ scattering F N 2 = F p n 2 + F 2 2 = 5 18 xv Drell-Yan (lepton-pair production) p 1 + p 2 µ + µ + X = 5 18 x ( u + u + d + d ) x ( u + u + d + d ) xs if q distributions are flavor symmetric dσ q(x 1 ) q(x 2 ) + q(x 1 ) q(x 2 ) q 2 (q 2 ) q 1 (q 1 ) µ + µ at large x F = x 1 x 2 projectile target dσ q V (x 1 ) q(x 2 ) q(x 2 ) can be obtained if q V (x 1 ) is known.

19 Drell-Yan and Antiquark Distributions p + A µ + µ + X Drell-Yan cross-section ratio is roughly equal to antiquark ratio. pca σ DY pd σ DY q Ca q D The Fermilab E772 Drell-Yan data suggested that nuclear modification of antiquark distributions should be small in the region, x 0.1.

20 Scaling Violation and Gluon Distributions logq q + 2 i (x,q 2 ) = α 1 s dy 2π P qi q x y j (x / y) q + j (y,q 2 ) + P qg (x / y) g(y,q 2 ) j dominant term at small x q + i = q i + q i at small x F 2 ( lnq 2 ) 20 α s 27π xg Q 2 dependence of F 2 is proportional to the gluon distribution. No experimental consensus of Q 2 dependence! G A (x) determination is difficult.

21 Nuclear modification of PDFs valence quark nuclear modification in Ca antiquark gluon?

22 Comments on higher-twist effects: Q 2 1 GeV 2 data are included in the NPDF analysis. i Possibility of higher-twist effects at Q 2 < a few GeV 2. i If Q 2 < a few GeV 2 data are excluded, we have difficulty in determining small-x PDFs (x < 0.03). i Furthremore, accurate SLAC data are also excluded even at medium x. i Higher-twist effects are not so obvious in F 2 A / F 2 D data at Q 2 > 1 GeV 2. Higher-twist effects [ Strikman s talk on Nov. 2; L. Frankfurt, V. Guzey, M. Strikman, Phys. Rev. D71 (2005) ] Data may be removed for excluding higher-twist effects.

23 Higher-twist effects in unpolarized PDFs in the nucleon CTEQ analysis F 2 (x,q 2 ) = F NLO 2 (x,q 2 ) 1 + H (x) Q 2 H (x) = h 0 + h 1 x + h 2 x 2 + h 3 x 3 + h 4 x 4 error estimate?

24 Higher-twist effects in polarized PDFs in the nucleon J. Blümlein and H. Böttcher, Nucl. Phys. B636 (2002) 225. h I (x,q 2 ) = 1 + A Q 2 xα (1 x) β, h II (x,q 2 ) = 1+ 1 Q 2 (A + Bx + cx2 ) Higher-twist effects become conspicuous at small Q 2 ; however, they are within the error bands. The present data with Q 2 > 1 GeV 2 do not contain significant higher -twist contributions.

25 Analysis of Data for Determining Nuclear PDFs

26 References There are only a few papers on the parametrization of nuclear PDFs! Need much more works. (EKRS) K. J. Eskola, V. J. Kolhinen, and P. V. Ruuskanen, Nucl. Phys. B535 (1998) 351; K. J. Eskola, V. J. Kolhinen, and C. A. Salgado, Eur. Phys. J. C9 (1999) 61. χ 2 analysis (HKM, HKN) M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) ; M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) ; research in progress. (DS) D. de Florian and R. Sassot, Phys. Rev. D69 (2004) The HKN analysis is first explained. Then, results are compared with the ones for EKRS and DS.

27 A dependence Ref. I. Sick and D. Day, Phys. Lett. B 274 (1992) ρ volume surface R= r 0 A 1/3 roughly speaking σ A = A σ V + A 2/3 σ S σ A A = σ V + 1 A σ 1/3 S ~ 1 1/3 dependence A

28 Nuclear parton distributions (per nucleon) if there were no modification A u A = Z u p + N u n, A d A = Z d p + N d n Isospin symmetry: u n = d p d, d n = u p u u A = Z u + N d A, d A = Z d + N u A Take into accont the nuclear modification by the factors w i (x,a) u VA (x) = w u V (x,a) Z u V(x) + N d V (x) A d V A (x) = w d V (x,a) Z d V(x) + N u V (x) A q A (x) = w q (x,a) q(x) g A (x) = w g (x,a) g(x)

29 Functional form of w i (x,a) f i A (x) = w i (x,a) f i (x), i = u v, d v, q, g first, assume the A dependence as 1/A 1/3 then, use w i (x,a) = 1 + (1 1/A 1/3 ) a i+b i x+c i x 2 +d i x 3 (1 x ) β i a i, b i, c i, d i, β i : parameters to be determined by χ2 analysis Fermi motion: 1 (1 x ) β i as x 1 if β i > 0 Shadowing: w i (x 0, A) = 1 + (1 1/A 1/3 ) a i < 1 Fine tuning: b i, c i, d i

30 Constraints Nuclear charge Z = A dx [ 2 3 (u A u A ) 1 3 (d A d A ) 1 3 (s A s A )] = A dx ( 2 3 u A V 1 3 d A ) V Baryon number: A = A dx 1 3 (u V A + d V A ) Momentum: A = A dx x (u VA + d V A + 6 q A + g A ) Three parameters can be determined by these conditions.

31 Experimental data: total number=951 ( 1241 including D/p) (1) F 2 A / F 2 D 606 data NMC:He, Li, C, Ca SLAC: He, Be, C, Al, Ca, Fe, Ag, Au EMC: C, Ca, Cu, Sn E665: C, Ca, Xe, Pb BCDMS: N, Fe HERMES: N, Kr (2) F 2 A / F 2 A 293 data NMC: Be / C, Al / C, Ca / C, Fe / C, Sn / C, Pb / C, C / Li, Ca / Li (3) σ DY A / σ DY A 52 data E772: C / D, Ca / D, Fe / D, W / D E866: Fe / Be, W / Be

32 Analysis conditions parton distributions in the nucleon MRST01 (Λ QCD =220 MeV) Q 2 point at which the parametrized PDFs are defined: Q 2 = 1 GeV 2 used experimental data: Q 2 1 GeV 2 total number of data: 951 (1241) 606 (F 2 A /F2 D ) (F 2 A /F2 A' ) + 52 (Drell Yan) subroutine for the χ 2 analysis: CERN Minuit χ 2 = Σ i (R i data Ri calc ) 2 (σ i data ) 2, The error of a distribution F(x) is given by δ F( x) 2 =Δ χ 2 F( x) Η 1 F( x) ξ ij i, j i ξ R = F 2 A A pa F D, F 2 2 F A', σ DY pa', σ i data = (σ sys i ) 2 + (σ stat i ) 2 2 σ DY j H = Hessian ξ i = parameter

33 Comparison with F 2 Ca /F 2 D & σ DY pca / σ DY pd data (R exp -R theo )/R theo at the same Q 2 points R= F 2 Ca /F 2D, σ DY pca / σ DY pd χ 2 / d.o.f. = 1490 / 942

34 Analysis results small nuclei Be/D C/D J NMC J J J J J J JJJ J JJ J }EMC J H F NMC E139 E665 F J } } } } J J J J F F } J J J } H H F J } J J J J JHJ F H } } J } } } } H H } J 0.8 Q 2 = 5 GeV x x

35 large nuclei Au/D Pb/D H E139 }E F E H } H H H H H H H F F F 0.8 Q 2 = 5 GeV 2 H H HH 0.8 F F x x

36 Q 2 dependence Due to the lack of accurate Q 2 dependent data, the determination of g(x) is very difficult.

37 Nuclear corrections for 40 Ca with uncertainties f Ca (x,q 2 ) f N (x,q 2 ) Q 2 = 1 GeV 2 Valence quark Antiquark Gluon PDF uncertainties PDF-library code can be obtained from

38 Other works: EKRS (Eskola,( Kolhinen, Ruuskanen, Salgado) EKR (1998), EKS (1999) analysis i "eye fit" (not χ 2 analysis) i F 2 data, Drell-Yan data (see the table for a detailed comparison) i LO (Leading Order of α s ) i Q 2 0 = 2.25 GeV 2 2 = m c [Data with Q GeV 2 are used.] i deuteron (=nucleon): no nuclear modification i u A / u = d A / d = s A / s R S A, u v A / u v = d v A / d v R v A i Divide x into 3 regions (x p 0.1, x eq 0.4) i R A S = R A v = R A F2 at x > x eq, R A S (x p < x < x eq,q 2 0 ) = const = R A S (x p,q 2 0 ) i R A G (x,q 2 0 ) R A F2 (x,q 2 0 ) at small x i 27 parameters (a few of them are fixed) i baryon-number and momentum conservations i nucleonic PDFs: GRV92-LO, CTEQ4L p = plateau: R S A = const x p x eq x

39 Other works: DS (de( Florian, Sassot) DS (2004) analysis i χ 2 analysis i F 2 data, Drell-Yan data (see the table for a detailed comparison) i LO (Leading Order of α s ), NLO (Nexto-to-Leading Order) i Q 0 2 = 0.4 (0.26) GeV 2 in NLO (LO) [Data with Q 2 > 1.0 GeV 2 are used.] i u A / u = d A / d = s A / s, u v A / u v = d v A / d v i modifications are expressed by convolution integrals (see next transparencies) i 27 parameters i baryon-number, charge, and momentum conservations i nucleonic PDFs: GRV98 i Λ QCD : consistent with the unpolarized PDFs i Q 2 evolution: Mellin-transformation method

40 Comparison

41 Comparison of used data set HKN data table data in EKRS data in DS data not included

42 Results for nuclear PDFs Q 2 = 2.25 GeV 2 gluon Ca/D valence antiquark Pb/D Q 2 = 1 GeV 2

43 Nuclear PDFs by DS Q 2 = 10 GeV 2 Valence- and anti-quark distributions are very similar. Gluon shadowing is fairly small, but it is within the HKN uncertainties. Ca/D Q 2 = 1 GeV 2

44 Summary on nuclear-pdf determination (1) Analysis for the nuclear PDFs and their uncertainties. Valence quark: well determined (except for the small-x region). Antiquark: determined at small x, large uncertainties at medium and large x. Gluon: large uncertainties in the whole-x region. (2) There are variations among EKRS, HKN, and DS, but they are roughly within the uncertainties.

45 Research in progress (1) LO and NLO analyses with uncertainties LO & NLO NPDF codes will be provided with possible estimates on uncertainties. (2) F 2D /F 2 p data In future: Higher-twist effects, Small-Q 2 extrapolation,

46 preliminary preliminary

47 Comments on Future Experimental Studies

48 Valence quark 1 2 [F 3 νn +F 3 ν N ] CC u v +d v Q 2 fixed target: min(x) = 2M N E ν 1 2E ν (GeV) 0.01 if Q 2 1 GeV 2, E ν = GeV valence ν factory JLab

49 Antiquark distributions Fermilab J-PARC RHIC LHC

50 Gluon distributions J-PARC? RHIC LHC

51 Summary (1) Nuclear PDF studies are still premature a lot of opportunities for theoretical and experimental investigations (2) Important applications for various high-energy processes e.g. heavy-ion and neutrino reactions (3) Need future experiments J-PARC, LHC, high-energy electron facility, neutrino factory,

52 The End The End