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Refactorizing NRQCD short-distance coefficients in exclusive quarkonium production Graduate University of the CAS Deshan Yang Based on Yu Jia & DSY, NPB814, 217 (2009) 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 1 Outline 1. Introduction & Motivations 2. Refactorization of NRQCD coefficients 3. Example 1 – e e b 4. Example 2 – h Υ 5. Summary 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 2 Charmonium production at B factories Exlusive quarknium production at B-factories Belle : [e e J / c ] Bc [ 2] (25.6 2.8 3.4) fb BaBar : [e e J / c ] Bc [ 2] (17.6 2.8 12..51 ) fb Abe et al (Belle Collaboration), PRL 2002; Aubert et al (BaBar Collaboration), PRD 2005; Suprises for theorists [ e e J / c ] (2.3 5.5) fb LONRQCD: Braaten and Lee, PRD 2003; PRD 2005; Liu, He and Chao, PLB 2003; Hagiwara, Kou and Qiao, PLB 2003; 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 3 Battlefield for NRQCD guys NLO QCD Corrections: K~ 2 Zhang, Gao, Chao, PRL 2006; Gong, Wang, PRD 2008; Relativistic Corrections: K~ 2 Bodwin, Kang, Kim, Lee, Yu, AIP Conf. Proceeding 2007; He, Fan, Chao, PRD 2007; Triumph? Confusions? Questions? Perturbation stability Non-relativistic vs. Relativistic 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 4 Salute to those guys Gong, QWG2008 Non-pert. quantities! Kill renormalization scale dependence K ( s / m ) NLO / LO 2 c 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 5 NRQCD factorization For single quarkonium production : NRQCD operator with definite velocity power counting multi-scale problem: Q>>m stability of the perturbation: large log(Q/m) may need the resummation. 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 6 Light-cone framework Ma and Si, PRD 2004;Bonder and Chernyak, PLB 2005;Braguta, arXiv:0811.2640[hep-ph] From Ma and Si, PRD 2004; LCDA: Non-pert. quantities! Lead to end-point singularity! Fit to the expr.! 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 7 Motivations 1 2 3 Key Large logs in NRQCD short-distance coefficients could ruin the perturbation! How to harmonize LC factorization and NRQCD factorization? Understand the mechanisms to heavy quarkonium exclusive production! s mc mc v mc v 2 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 8 Inclusive quarkonium production at hadron collider J.W.Qiu’s talk at this program Absent from electron-positron collider! Less complications! Perturbative region could be furthur fined into two regions! 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 9 Exclusive single quarkonium production 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 10 Refactorization At the leading power of velocity, The hard kernel is the same as the similar process in which the quarkonium is replaced by a flavor singlet light meson. Since , the LCDA of bounded heavy quark and anti-quark can be calculated perturbatively. Ma and Si, PRD 2006; Bell and Feldman 2007; 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 11 Example-1: e e b Amplitude up to leading power of velocity: The short-distance contribution is parameterized as The equivalent computation is to calculate the on-shell heavy quark anti-quark pair with equal momentum and the same quantum number as the quarkonium. At the tree level, 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 12 One-loop level 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 13 Leading regions Hard Region: Collinear region: Anti-collinear region: Potential region: Soft region: Ultra-soft region: 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 14 Form factor NRQCD: Collinear factorization: Hard-kernel: at tree level Light-cone distribution amplitude 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 15 RGE for LCDA Brodsky-Lepage kernel: Resum the leading logrithms where 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 16 Resummation in moment space Both LCDA and Brodsky-Lepage kernel can be expanded in Gegenbauer polynomial In moment space, the LL resummed form-factor is 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 17 Truncated LL approximation In momentum space, solve the RGE recursively, at two-loop level, we have which is equivalent to the expressions in moment space 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 18 Comparisons LL 2-loop LL 1-loop LL 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 19 Example-2: h Amplitude up to the lowest order of velocity: The short-distance contribution is parameterized as The factorization formula for the form-factor involves the twist-2 LCDA for transversely polarized quark pair which corresponding B-L kernel is 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 20 Comparisons 1-loop LL 2-loop LL LL 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 21 Summary For the single heavy quarkonium exclussive production at very large center energy collision, it is possible to derive the refactorization formula for the relevant NRQCD short-distance coefficients; In the two examples we showed, the LL resummation does not lead to large corrections in the accessible ernergy; However, the refactorization property of the NRQCD short-distance coefficients may offer an easier way to obtain the higher order radioactive corrections with price of m/Q suppressed corrections; For the double charmonium production, double logarithm appears with the overall m/Q suppression at one-loop level. However we do not know how to resum such logarithm with the aid of B-L equations. 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 22 2009.9.10 KITPC-EFT in Nuclear Physics & Particle Physics 23