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Unifying the Mechanisms for SSAs in Hard Process

Unifying the Mechanisms for SSAs in Hard Process. Feng Yuan RIKEN/BNL Research Center Brookhaven National Laboratory. Outline. Naïve parton model fails for large SSAs Two mechanisms: Sivers and twist-3 Unifying these two Summary. Naïve Parton Model Fails to Explain Large SSAs.

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Unifying the Mechanisms for SSAs in Hard Process

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  1. Unifying the Mechanisms for SSAs in Hard Process Feng Yuan RIKEN/BNL Research Center Brookhaven National Laboratory DIS2006, KEK

  2. Outline • Naïve parton model fails for large SSAs • Two mechanisms: Sivers and twist-3 • Unifying these two • Summary DIS2006, KEK

  3. Naïve Parton Model Fails to Explain Large SSAs • If the underlying scattering mechanism is hard, the naïve parton model generates a very small SSA: (G. Kane et al, PRL41, 1978) • It is in general suppressed by αSmq/Q • We have to go beyond the naïve parton model to understand the large SSAs observed in hadronic reactions DIS2006, KEK

  4. Two Mechanisms in QCD • Transverse Momentum Dependent Parton Distributions • Sivers function, Sivers 90 • Collins function, Collins 93 • Gauge invariant definition of the TMDs: Brodsky, Hwang, Schmidt 02; Collins 02 ; Belitsky, Ji, Yuan 02; Boer, Mulders, Pijman, 03 • The QCD factorization: Ji, Ma, Yuan, 04; Collins, Metz, 04 • Twist-three Correlations (collinear factorization) • Efremov-Teryaev, 82, 84 • Qiu-Sterman, 91,98 DIS2006, KEK

  5. Previous Attempts to Connect these Two • Relations at the matrix elements level TF(x,x)=s d2k |k|2qT(x,k) • Boer, Mulders, Pijman 03 • Attempts to connect these two in some processes, no definite conclusion: • Ma, Wang, 03 • Bacchetta, 05 Sivers Qiu-Sterman DIS2006, KEK

  6. Unifying the Two Mechanisms (P? dependence of SSAs) • At low P?, the non-perturbative TMD Sivers function will be responsible for its SSA • When P?» Q, purely twist-3 contributions • For intermediate P?, QCD¿ P?¿ Q, we should see the transition between these two • An important issue, at P?¿ Q, these two should merge, showing consistence of the theory (Ji, Qiu, Vogelsang, Yuan, hep-ph/0602239, 4023: Drell-Yan hep-ph/0604128: SIDIS) DIS2006, KEK

  7. A General Diagram in Twist-3 Fragmentation function: \hat q(x’) Twist-3 quark-gluon Correlation: TF(x1,x2) Collinear Factorization: Qiu,Sterman, 91 DIS2006, KEK

  8. The TMD Factorization DIS2006, KEK

  9. Factorization guidelines Reduced diagrams for different regions of the gluon momentum: along P direction, P’, and soft Collins-Soper 81 DIS2006, KEK

  10. Detailed Calculations:Soft and Hard Poles • Soft: xg=0 • Hard: xg= 0 DIS2006, KEK

  11. Cross sections • Summing all diagrams, • For P?» Q, many other terms contribute, see Eguchi, Koike, Tanaka, hep-ph/0604003 DIS2006, KEK

  12. Ph?¿Q limit • Keeping the leading order of q?/Q, • Hard-pole contributions play very important role to get the above results, which should be reproduced by the Sivers function at the same kinematical limit, from the TMD factorization DIS2006, KEK

  13. TMD Factorization • When q?¿ Q, a TMD factorization holds, • When q?ÀQCD, all distributions and soft factor can be calculated from pQCD, by radiating a hard gluon DIS2006, KEK

  14. Fragmentation function at p?ÀQCD See, e.g., Ji, Ma, Yuan, 04 DIS2006, KEK

  15. Soft Facotor DIS2006, KEK

  16. Sivers Function from twist-3: soft poles DIS2006, KEK

  17. Hard Poles for Sivers Function DIS2006, KEK

  18. Sivers Function at Large k? • 1/k?4 follows a power counting • Drell-Yan Sivers function has opposite sign • Plugging this into the factorization formula, we indeed reproduce the polarized cross section calculated from twist-3 correlation DIS2006, KEK

  19. Concluding Remarks • Two mechanisms generating the large SSAs are unified for a physical process and observable • This unification will impose a rigorous constraint for phenomenological studies of SSAs • Extension to other processes, and other observables are also interested to follow • Pt scan of SSAs will show the transition from perturbative region to nonperturbative region DIS2006, KEK

  20. Transition from Perturbative region to Nonperturbative region? • Compare different region of P? Nonperturbative TMD Perturbative region DIS2006, KEK

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