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
arXive:0709.1572
In collaboration with K.Tanaka (Juntendo Univ.) & J.Kodaira (KEK)
Hiroyuki Kawamura (RIKEN)
Jiro Kodaira (1951-2006)
RADCOR2005 in Shonan (Oct. 2005)
Hiroyuki Kawamura (RIKEN)
Transversely polarized Drell-Yan process
 tDY (RHIC, J-PARC, GSI,…)
♠ double spin asymmetry:
 Transversity
d  ixP 
 q( x)  
e
PS  | q (0)   5 q (0,   , 0  ) | PS 
4
— 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 δq.
Hiroyuki Kawamura (RIKEN)
Double spin asymmetry at small QT
 ATT for the “QT-integrated” cross sections at NLO
NLO
TT
A
T d NLO

dQ 2 dyd
d NLO
dQ 2 dyd
Martin, Shäfer, Stratmann,Vogelsang (’99)
RHIC (PP) :
GSI (PP-bar ) : Shimizu, Sterman, Yokoya,Vogelsang (’05)
Barone et al. (’06)
(threshold resummation)
 ATT(QT) at small QT
QT
• A bulk of dileptons is produced.
• Soft gluon corrections are dominant : “universal”
→ Extraction of δq(x) can be simpler.
— resummation of “recoil logs”
dˆ
dQ 2 dQT2

  Sn cn (QT2 ) 
QT2
Q2
n

1
QT2
2

k Q 
d
ln

nk
 2 
k 0
 QT  
2 n 1
— spin asymmetry with soft gluon resummation
Hiroyuki Kawamura (RIKEN)
QT distributions at LO
Kodaira, Shimizu, Tanaka, HK (’06)
 Drell-Yan process with transverse polarization
φ: azimuthal angle of one of the leptons
→ phase space integral with φ dependence
(difficult in D-dimension)
♠ soft/col. singularity appear only at QT=0.
ex.
D-dim.
4-dim.
 QT distribution at LO
Altarelli, Ellis,Greco,Martinelli (’84)
Hiroyuki Kawamura (RIKEN)
NLL resummation for tDY
Kodaira, Shimizu, Tanaka, HK (’06)
q
Resummed part:
q
T C
T C
double Mellin space
S
pdf
b : impact parameter
• coeff. function
• Sudakov factor
LL
NLL
• evolution op.
NLL
Hiroyuki Kawamura (RIKEN)
universal
Grazzini, de Florian (‘00)
b
∟
NLL resummation for tDY
C
 “Minimal prescription” + NP function
 QT distribution at “NLL+LO”
bL Landau pole
Laenen, Kulesza, Vogelsang, Sterman, … (’99 - )
Bozzi, Catani, de Florian, Grazzini (’03 - ’07)
→ “unitarity constraint”
Hiroyuki Kawamura (RIKEN)
QT distributions
Kodaira, Shimizu, Tanaka, HK (’06)
 Input function
1
2
 q( x, 02 )  [q( x, 02 )  q( x, 02 )] at 02  0.4GeV 2 + NLO evolution
(GRV98+GRSV00)
T d
dQ dyd dQT
d
dQ dyd dQT
pol.
unpol.
2
Koike et al. (’96)
Kumano et al. (’96)
Vogelsang (’97)
2
pp collision @ RHIC s = 200 GeV, Q = 5GeV, y=2, φ=0 with gNP=0.5GeV2
Hiroyuki Kawamura (RIKEN)
Double-spin Asymmetries at small QT
Kodaira, Tanaka, HK (’07)
ATT (QT )
ATT (QT )
pp collision @ RHIC
s = 200 GeV, Q=2-20 GeV, y=2,φ=0
small-x, (valence) x (sea)
.
pp collision @ J-PARC
s = 10 GeV, Q = 2-3.5 GeV, y=0,φ=0
large-x, (valence) x (sea)
Hiroyuki Kawamura (RIKEN)
Double-spin Asymmetries at small QT
ATT (QT )
ppbar collision @GSI
s = 10 GeV, Q = 2-6 GeV, y=0,φ=0
large-x, (valence)2
Hiroyuki Kawamura (RIKEN)
Double-spin Asymmetry at small QT
 ratios of each component
pp : s = 200 GeV, Q = 5GeV, y=2, φ=0
NLL+LO: XNLL+Y
NLL: XNLL
LL: XLL
NLL
LO
• ATT (QT )  ATT (QT ) → soft corrections are crucial.
NLL  LO
NLL
(QT )  ATT
(QT ) → dominated by the resumed part.
• ATT
• Flat in the peak region ↔ soft gluon corrections almost cancel. (universal)
NLL
LL
But! Some contributions still remain. ATT (QT )  ATT (QT )
What determines (or what can be obtained from) ATT(QT) ?
Hiroyuki Kawamura (RIKEN)
Saddle point evaluation at NLL
Kodaira, Tanaka, HK (’07)
NLL
NLL
Observation: ATT (QT )  ATT (QT )  ATT (QT  0) → saddle point evaluation
 resummed part at QT=0
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
Hiroyuki Kawamura (RIKEN)
Collins, Soper,Sterman (’85)
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.
→ approaches the exact result in the asymptotic limit
Parisi, Petronzio (’79)
Collins, Soper, Sterman (’85)
• Large corrections in ( … ) cancel in the asymmetry.
• Evolution operator
shifts the pdf scale
—  q  x, Q 2    q  x, b0 2
b
2



SP 
Hiroyuki Kawamura (RIKEN)
Asymptotic formula
In the peak region,
  0 b02

 2 b02

0
0 
e  qi  x1 ,
2    qi  x0 ,
2   ( x1  x2 ) 

bSP 
bSP 
cos(2 ) i

 

NLL
ATT
(QT ) 
2
 0 b02

 0 b02

2 
0
0 
e
q
x
,

q
x
,

(
x

x
2
2
i i  i  1 bSP  i  2 bSP  1 2 ) 


2
i
• pdf scale:
2
NLL
LL
NLO
b02 / bSP
 Q 2  ATT
(QT )  ATT
(QT ), ATT
for pp colisions.

⇒
 q ( x,  )
q ( x,  )
• Simple but still contains the essential dynamics which determine
ATT (QT.)
• Only depends on pdf at a fixed (x,μ)
→ useful for extracting pdf from experimental data.
Caution:
LO
Q
b0 / b
The evolution from Q to b0/bSP is given by the NLL approximation
of NLO evolution operator ↔ LO DGLAP kernel.
“mismatch” between resummation and fixed order
Hiroyuki Kawamura (RIKEN)
0
NLO
Asymptotic formula vs. Numerical results
NLL
 ATT (0) 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.)
Hiroyuki Kawamura (RIKEN)
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).
NLL
NLL
— Numerical study shows ATT (QT )  ATT (QT )  ATT (0) in the peak region.
 The saddle-point evaluation at NLL
NLL
→ a novel asymptotic formula for ATT (0)
— pdf at the fixed scale b0/bSP 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 ALL(QT) for vector boson production at RHIC.
Hiroyuki Kawamura (RIKEN)