1 Threshold Resummation Effects in polarized DY at GSI and J-PARC EPOCHAL Tsukuba, Tsukuba, JAPAN, 4/20-24(2006) Ref. H.Shimizu, G.Sterman, W.Vogelsang,

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1 Threshold Resummation Effects in polarized DY at GSI and J-PARC EPOCHAL Tsukuba, Tsukuba, JAPAN, 4/20-24(2006) Ref. H.Shimizu, G.Sterman, W.Vogelsang, HY, Phys.Rev.D71,114007,2005 (hep-ph/ ) Hiroshi Yokoya (Niigata U)

2 Introduction Drell-Yan process : : parton distribution functions : partonic cross section ← perturbatively calculable

3 J-PARC & GSI-FAIR experiments New experiments have been proposed J-PARC : GSI-FAIR (PAX,ASSIA) : 1.polarization may be available → (transversely) polarized Drell-Yan measurements → (transversely) polarized parton distributions 2.rather lower-energy collisions → QCD corrections must be important

4 Mass Distribution

5 Factorization Theorem Drell-Yan cross section formula

6 Hamberg,van Neerven,Matsuura(’91,’02); Harlander,Kilgore(’02) LO : NLO : NNLO : LO NNLO NLO Status of DY higher order calculations Altarelli,Ellis,Martinelli(’78,’79); Kubar-Andre’,Paige(’79); Harada,Kaneko,Sakai(’79) Drell,Yan (’70)

7 LO : NLO : NNLO : Status of DY higher order calculations

8 K-factor NLO/LO NNLO/LO

9 Large corrections come from the partonic threshold region (z~1) real emission suppressed by the phase space restriction imbalance occurs between real and virtual gluon corrections → only soft gluon can be emitted → soft gluon (eikonal) approximation to treat these logs up to all orders (after the cancellation of IR pole) Threshold logs

10 Threshold resummation Sterman(’87);Catani,Trentadue(’89) General Formula : Sudakov Exponent First, goto Mellin-moment space : threshold log →

11 NNLL : Moch,Vermaseren,Vogt( ’ 04) 3-loop split. func. gives Catani,Mangano,Nason,Trentadue(’96) employ “ Minimal Prescription ” : define the inverse Mellin contour as the left of the Landau pole LL :NLL :NNLL : Threshold resummation collinear improvement & qg mixing: Kramer,Laenen,Spira( ‘ 98); Catani,de Florian,Grazzini( ‘ 02); Kulesza,Sterman,Vogelsang( ’ 02, ’ 04) collinear (non-soft) gluon, off-diagonal AD →

12 Threshold resummation LL :NLL :NNLL :

13 not only the convergence of resummation accuracy (N n LL), but also the convergence of the power expansion of Sudakov exponent to Convergence note : “ Minimal Prescription ” defined so that PT has no factorial growth power corr. should be added later if required phenomenologicaly

14 Double Transverse Spin Asymmetry Model of Transverse PDFs → upper limit of Soffer’s inequality with GRV&GRSV Martin,Schafer, Stratmann,Vogelsang

15 Rapidity Distribution

16 Rapidity Distribution Altarelli,Ellis,Martinelli( ’ 79); Kubar,Le Bellac,Meunier,Plaut( ’ 80) Anastasiou,Dixon,Melnikov,Petriello( ’ 04) LO : NLO : NNLO :

17 Fourier & Mellin transform Then, where Resummation Formula Laenen,Sterman( ’ 92);Sterman,Vogelsng( ’ 01); Mukherjee,Vogelsang( ’ 06)

18 LO : NLO : in threshold limit (z → 1) In general : ( in threshold limit) Threshold Limit mass dist. func.

19 M dependence in the hard part; finally inverse Fourier & Mellin transform : (+ matching with fixed order) Now resummation !

20 Numerical Graphs rapidity dependent resummed K-factors (preliminary)

21 become very large at large y (preliminary) Numerical Graphs

22 Numerical Graphs (Spin asymmetry) (preliminary) (to be reported soon)

23 Summary Threshold resummation up to NNLL-NNLO : K=3~10. really large, but seem to be controllable! QCD corrections to the DY mass distribution and the rapidity distribution at J-PARC and GSI spin asymmetries are rather stable under QCD corrections qg subprocess is important for pp collision (J-PARC)