Hiroyuki Kawamura (RIKEN) Soft-gluon resummation in Drell-Yan dilepton (and vector boson) production at small transverse momentum: spin asymmetries and a novel asymptotic formula Hiroyuki Kawamura (RIKEN) RADCOR2007 in Florence Oct In collaboration with K.Tanaka (Juntendo Univ.) & J.Kodaira (KEK) PTP115(2006)667 NPB777 (2007)203 arXive:

Hiroyuki Kawamura (RIKEN) Jiro Kodaira ( ) RADCOR2005 in Shonan (Oct. 2005)

Hiroyuki Kawamura (RIKEN)  tDY  Transversity — twist-2, chiral-odd distribution function (RHIC, J-PARC, GSI,…) Transversely polarized Drell-Yan process ♠ double spin asymmetry: Future DY data can provide an direct access to δq. No gluon contribution (→ NS-type evolution) Not measured in inclusive DIS

Hiroyuki Kawamura (RIKEN) A bulk of dileptons is produced. Soft gluon corrections are dominant : “universal” → Extraction of δq(x) can be simpler.  A TT for the “Q T -integrated” cross sections at NLO Double spin asymmetry at small Q T Martin, Shäfer, Stratmann,Vogelsang (’99)  A TT (Q T ) at small Q T Barone et al. (’06) — resummation of “recoil logs” Shimizu, Sterman, Yokoya,Vogelsang (’05) — spin asymmetry with soft gluon resummation RHIC (PP) : GSI (PP-bar ) : (threshold resummation)

Hiroyuki Kawamura (RIKEN) Q T distributions at LO  Drell-Yan process with transverse polarization φ: azimuthal angle of one of the leptons → phase space integral with φ dependence (difficult in D-dimension) Kodaira, Shimizu, Tanaka, HK (’06) ♠ soft/col. singularity appear only at Q T =0. ex. D-dim.4-dim.  Q T distribution at LO Altarelli, Ellis,Greco,Martinelli (’84)

Hiroyuki Kawamura (RIKEN) NLL resummation for tDY Sudakov factor Kodaira, Shimizu, Tanaka, HK (’06) coeff. function b : impact parameter Resummed part: double Mellin space evolution op. LL NLL universal pdf Grazzini, de Florian (‘00)

Hiroyuki Kawamura (RIKEN) Laenen, Kulesza, Vogelsang, Sterman, … (’99 - ) Bozzi, Catani, de Florian, Grazzini (’03 - ’07) C b bLbL Landau pole  “Minimal prescription” + NP function ∟ NLL resummation for tDY  Q T distribution at “NLL+LO” → “unitarity constraint”

Hiroyuki Kawamura (RIKEN) Q T distributions Kodaira, Shimizu, Tanaka, HK (’06) pol.unpol. pp RHIC  s = 200 GeV, Q = 5GeV, y=2, φ=0 with g NP =0.5GeV 2 + NLO evolution (GRV98+GRSV00) Koike et al. (’96) Kumano et al. (’96) Vogelsang (’97)  Input function

Hiroyuki Kawamura (RIKEN) Double-spin Asymmetries at small Q T pp RHIC  s = 200 GeV, Q=2-20 GeV, y=2,φ=0 pp J-PARC  s = 10 GeV, Q = GeV, y=0,φ=0. Kodaira, Tanaka, HK (’07) large-x, (valence) x (sea) small-x, (valence) x (sea)

Hiroyuki Kawamura (RIKEN) ppbar Double-spin Asymmetries at small Q T  s = 10 GeV, Q = 2-6 GeV, y=0,φ=0 large-x, (valence) 2

Hiroyuki Kawamura (RIKEN) Double-spin Asymmetry at small Q T What determines (or what can be obtained from) A TT (Q T ) ? pp :  s = 200 GeV, Q = 5GeV, y=2, φ=0 NLL+LO: X NLL +Y NLL: X NLL LL: X LL  ratios of each component → soft corrections are crucial. → dominated by the resumed part. Flat in the peak region ↔ soft gluon corrections almost cancel. (universal) But! Some contributions still remain.

Hiroyuki Kawamura (RIKEN) LL terms NLL terms Kodaira, Tanaka, HK (’07) Saddle point evaluation at NLL Observation:→ saddle point evaluation  resummed part at Q T =0 Around the saddle-point, the resummation formula is organized in terms of a single parameter. up to NNLL corrections → ex. “degree-0” approximation Collins, Soper,Sterman (’85)

Hiroyuki Kawamura (RIKEN),  Saddle point  Result Extends the conventional SP evaluation at LL level. Large corrections in ( … ) cancel in the asymmetry. Parisi, Petronzio (’79) Collins, Soper, Sterman (’85) Evolution operator shifts the pdf scale — The saddle point is determined by LL terms. (up to NNLL) → approaches the exact result in the asymptotic limit Saddle point evaluation at NLL

Hiroyuki Kawamura (RIKEN) Asymptotic formula pdf scale : Simple but still contains the essential dynamics which determine. The evolution from Q to b 0 /b SP is given by the NLL approximation of NLO evolution operator ↔ LO DGLAP kernel. Only depends on pdf at a fixed (x,μ) → useful for extracting pdf from experimental data. In the peak region, NLO LO Caution: “mismatch” between resummation and fixed order for pp colisions. ⇒

Hiroyuki Kawamura (RIKEN) Asymptotic formula vs. Numerical results (1) SP-I : asymptotic formula (NLO pdfs + LO DGLAP for Q → b 0 /b SP ) (2) SP-II : asymptotic formula (NLO pdfs at b 0 /b SP ) (3) NB : numerical b-integration pp collision ppbar collision  from the asymptotic formula vs. numerical results — “SP-I” coincides with A TT (Q T ) quite well in all cases. — For the J-PARC & GSI kinematics, “SP-II” also works well. (The difference between LO & NLO kernel is small at large-x.)

Hiroyuki Kawamura (RIKEN) — Soft gluon corrections are crucial. → Q T resummation at NLL — NLL contribution enhances the asymmetry (for pp collisions). — Numerical study shows in the peak region. Summary  A TT (Q T ) for Drell-Yan dilepton production at small Q T  The saddle-point evaluation at NLL → a novel asymptotic formula for — pdf at the fixed scale b 0 /b SP at a fixed x. Can be useful to extract δq from the experimental data. The analysis is general and applicable to other asymmetries, such as A LL (Q T ) for vector boson production at RHIC.