PQCD, Transversity, Soft gluon resummation

1
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. 4 2007
PTP115(2006)667 NPB777 (2007)203 arXive0709.1572

In collaboration with K.Tanaka (Juntendo Univ.)
J.Kodaira (KEK)
2
Jiro Kodaira (1951-2006)
RADCOR2005 in Shonan (Oct. 2005)
3
Transversely polarized Drell-Yan process

• tDY

(RHIC, J-PARC, GSI,)

? double spin asymmetry

• Transversity

twist-2, chiral-odd distribution function

• No gluon contribution (? NS-type evolution)
• Not measured in inclusive DIS

Future DY data can provide an direct access to
dq.
4
Double spin asymmetry at small QT

• ATT for the QT-integrated cross sections at NLO

Martin, Shäfer, Stratmann,Vogelsang (99)
RHIC (PP)
Shimizu, Sterman, Yokoya,Vogelsang (05)
GSI (PP-bar )
(threshold resummation)
Barone et al. (06)

• ATT(QT) at small QT

• A bulk of dileptons is produced.
• Soft gluon corrections are dominant
  universal
• ? Extraction of
  dq(x) can be simpler.

resummation of recoil logs
spin asymmetry with soft gluon resummation

5
QT distributions at LO
Kodaira, Shimizu, Tanaka, HK (06)

• Drell-Yan process with transverse polarization

f azimuthal angle of one of the leptons ?
phase space integral with f dependence
(difficult in D-dimension)
? soft/col. singularity appear only at QT0.
ex.
D-dim.
4-dim.

• QT distribution at LO

Altarelli, Ellis,Greco,Martinelli (84)
6
NLL resummation for tDY
Kodaira, Shimizu, Tanaka, HK (06)
Resummed part
double Mellin space
pdf
b impact parameter

• coeff. function

• Sudakov factor

LL

NLL
universal
NLL

• evolution op.

Grazzini, de Florian (00)
7
NLL resummation for tDY
b
?
C

• Minimal prescription NP function

bL
Landau pole
Laenen, Kulesza, Vogelsang, Sterman, (99 -
) Bozzi, Catani, de Florian, Grazzini (03 - 07)

• QT distribution at NLLLO

? unitarity constraint
8
QT distributions
Kodaira, Shimizu, Tanaka, HK (06)

• Input function

NLO evolution
Koike et al. (96) Kumano et al. (96) Vogelsang
(97)
(GRV98GRSV00)
pol.
unpol.
pp collision _at_ RHIC
?s 200 GeV, Q 5GeV, y2, f0 with
gNP0.5GeV2

9
Double-spin Asymmetries at small QT
Kodaira, Tanaka, HK (07)


.
pp collision _at_ RHIC
pp collision _at_ J-PARC
?s 200 GeV, Q2-20 GeV, y2,f0
?s 10 GeV, Q 2-3.5 GeV, y0,f0
large-x, (valence) x (sea)
small-x, (valence) x (sea)

10
Double-spin Asymmetries at small QT
ppbar collision _at_GSI
?s 10 GeV, Q 2-6 GeV, y0,f0

large-x, (valence)2
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Double-spin Asymmetry at small QT

• ratios of each component

pp ?s 200 GeV, Q 5GeV, y2, f0
NLLLO XNLLY
NLL XNLL
LL XLL

• ? 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.


What determines (or what can be obtained from)
ATT(QT) ?
12
Saddle point evaluation at NLL
Kodaira, Tanaka, HK (07)
Observation
? saddle point evaluation

• resummed part at QT0

LL terms
NLL terms

• Around the saddle-point, the resummation formula
  is organized in terms of
• a single parameter.

ex.
?
up to NNLL corrections
degree-0 approximation
Collins, Soper,Sterman (85)
13
Saddle point evaluation at NLL

• Saddle point

,

• The saddle point is determined by LL terms.
• 
  (up to NNLL)

• Result

• Extends the conventional SP evaluation at LL
  level.

Parisi, Petronzio (79) Collins, Soper, Sterman
(85)
? approaches the exact result in the asymptotic
limit

• Large corrections in ( ) cancel in the
  asymmetry.

• Evolution operator
  shifts the pdf scale

14
Asymptotic formula
In the peak region,

• pdf scale

for pp colisions.
?

• Simple but still contains the essential dynamics
  which determine .

• Only depends on pdf at a fixed (x,µ)
• ? useful for extracting
  pdf from experimental data.

LO
Caution

The evolution from Q to b0/bSP is given by the
NLL approximation of NLO evolution operator ?
LO DGLAP kernel.
NLO
mismatch between resummation and fixed order
15
Asymptotic formula vs. Numerical results

• from the asymptotic formula vs.
  numerical results

(1) SP-I asymptotic formula (NLO pdfs LO
DGLAP for Q ? b0/bSP) (2) SP-II asymptotic
formula (NLO pdfs at b0/bSP) (3) NB
numerical b-integration
pp collision
ppbar collision
SP-I coincides with ATT(QT) 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.)
16
Summary

• ATT(QT) for Drell-Yan dilepton production at
  small QT

Soft gluon corrections are crucial. ? QT
resummation at NLL NLL contribution enhances
the asymmetry (for pp collisions). Numerical
study shows
in the peak region.

• The saddle-point evaluation at NLL
• ? a novel asymptotic
  formula for
• pdf at the fixed
  scale b0/bSP at a fixed x.

• Can be useful to extract dq from the
  experimental data.
• The analysis is general and applicable to other
  asymmetries,
• such as ALL(QT) for vector boson production
  at RHIC.