Post on 27-Mar-2015
Soft gluon resummation in transversely polarized
Drell-Yan process at small transverse momentum
NPB777 (’07) 203 and more
― double-spin asymmetries and novel asymptotic formula ―
Kazuhiro Tanaka (Juntendo U)
H. Kawamura (RIKEN)
J. Kodaira (KEK)
T
TT
dd dA
dd d
— Transversity
• Chiral-odd : unmeasurable in inclusive DIS• First global fit (LO): SIDIS at HERMES, COMPASS + Collins function from BELLE Anselmino’s talk• Lattice study by QCDSF/UKQCD
• tDY data can provide an direct access to QCD corrections
Transversely Polarized Drell-Yan (t DY) process
RHIC,J-PARC,GSI
p p X
0
( )
5( ) (0) ( )4
Pig d
i xi
n
i
t t
i
n Adf x e PS nn Se PSll g
ld y y l
p ^^ ^
×ò/ =/=ò
0
( )
( ) (0) ( )P4
ig di x
tn
i i
A n
i
tdf x e PS ne PSnlll
y y lp
×
/= =ò
ò
( ,0, ), (0, , )P P n nm + m -^ ^=0 0;
( )if xd
21 2
21 2
LO
( ) ( )cos(2 )
2 ( ) ( )
i ii
TTi i
i
i
i
e f x f xA
e f x f x
• bulk of dileptons produced at small QT • large perturbative corrections from gluon radiations make QT the relevant scale
NLO double-spin asymmetries
NLO NLO
2NLO
2T
TT
d dA
dQ dyd dQ dyd
Martin, Schafer, Stratmann, Vogelsang(’99)
TQ
2 , ,Q y
21 2
2
2 2
21 2
2< a few @RHIC
( , ) ( , )cos(2 )
2 ( , ) (
%, )
i ii
i ii
i
i
e f x f x
e f x f
Q
Q x
Q
Q
“sea-quark region” is probed at the scale2 210 GeVQ >
%
double-spin asymmetries at a measured
2 2( )TT T
TTT
d ddQ dyd dQ
QdQ d dd Qy
A
21 2
NLO2
1
2
22
2
2
( , ) ( , )cos(2 )
> 2 ( , ) ( , )
i i ii
i i
T T
T ii
TT
T
e f x f x
e
Q
Qf f xA
Q
x Q
TQ Q
tDY at ap p measured TQX
2
2
fix
( )
d
,
e
T
QA B S
A BS
Q
P
x
Q
P
x
L
= +
=
?
, Q Qy yA BS Sx e x e-= = transversity
1 12 2
2 cos(2 ) ,( , , ; )( , ) ( ) ,A B A B
A BA B
T
A B
Tiiii x
i A Bi
xT
x xd dddQ dyd dQ
Hf Q fQx x
fx x x x
D sf
d x m d x mmå ò ò:2( , )TQ QÄG%
KLN cancellation
( )2 2 2ln Ts Q Qa
( )2 2ln Ts Q Qa
T
( )2TQd:
4 2D= - e
( )22
2
1 34
2 2Ts Qæ öp ÷ç ÷- - - +ç ÷ç ÷ç e eè ø
a d:
( )
( )( )2 2
2
2 2
2
ln1 3
2
2
1 3
T T
T
T
s
Q QQ Q
Q
++
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üïïï+ ýïïï÷ç
þç ø
e
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a d
L
:
}
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2
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2
2
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2
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,1
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, ,
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,
1, ,
2 (2 )3
( ) ( )
( ) ( )
) (
cos
A B
AA B
A
B
B A B
T
T
i i x x
T
A B
A B
c
i ii ii
T
d x xdS b Qd b
Q
b
Q
b
x x
b b
N SQ
f C C
Y
fe
ddQ dyd dQ
e× x x
x x x xx x
a= f
ìïï´ íïïî
+
d d
D sf
å ò òò b Q
Next-to-leading logarithmic (NLL) resummation formula
( ) ( )2
2
2
2
2
2
2 2
1
log( , ) ( ) ( )Q
s s
b
QdS b Q A B
m
m
m
ì üæ öï ïï ï÷ç=- a m + a m÷í ýç ÷ç ÷ï ïè øï ïî þò
( )
( )
2 21 67 5
2 18 6 9
3
2
s ss F F G f
ss F
A C C C N
B C
p-
ì üæ ö æ öï ïa aï ï÷ ÷ç ça = + -÷ ÷í ýç ç÷ ÷÷ ççï ï è øè øp pï ïî þa
a =-p
Process-indep.: Kodaira, Trentadue (’82)
Davies, Stirling,Webber (’85)
Collins, Soper, Sterman ( ‘85)
22 1
2
2 2
21
22 3
22 2
2 2
LL
NLL
ln
ln
ln
n ns
n T
n
n
T
nn n
T TT T
Q
Q
Q Q
Q Q
Q
Q Q
-¥
- -
=
æ ö÷ç ÷ç ÷ç ÷çè ø
æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç
ìï wïa íïïî
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å144444424444443
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1 1
2 22
2
2
2
2, , ,
( )( , , ,, ,)
(1 )( , ) (1
(( ) ( )
) 1 ( 8)4
)A BA B
A B A BA B
A A
A A
ii
sT A B
i i x
BT
x
A
s
i
i i F
i
d x xd
x x
Q QQ
Q Q x xY f R
bC
Q
C b e
f Qx x
x x x x
x x
a
px x
æ öa ÷ç ÷= d - + p -ç ÷ç ÷
= d d
ç pè ø
å ò ò
2,
valid up to corrections down by
, , is for
the resummation formula
: exact phase space integrals inMS scheme
0
un
(
first complete N
4 2
regular
iv
L
)
L
A B
A B
ii T
s
T
x xQ Q QR
D
x x·
·
a
· = - e
®
·
( )consistent with LO result in the massive gl
trans. spin case
no chi
is realized for ,
but
for : ( ) factorization formula
gluon distribution
a la CSS & CDG
L
ral-o
ersal
dds
str
u t
O
c ure
T sQ Q·
Ü
L O a: ?
uon scheme by Vogelsang, Weber ('93)
Kawamura,Kodaira,Shimizu,Tanaka, PTP 115 (’06) 667
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T A B A B A A A
A AA A B B B B
A A A A Ae
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A A B A A B B
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x x x x xe C e
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xq x t d x
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y A B A B A BA B
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y A B B A A BB A
B A A B B B B
Q Qx x
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x x x x xe
x x x x
x x x x xe
x x x x
t
xq x t d x
x x x
xq x t d x
x x x
+
+
+ + +
+-
+ + +
= + +
üìï ï ï -ï ï ï - -í ýíï ï ï - -ïï ï îî þùüï- ï ú+ - - ýúï- - úïþû
2 2 2 2
, y yT T
A B
Q Q Q Qx e x e
S S
+ + -+ += =
2
00
( , ) ( , )1. -inte 2grati ( )o Jn: TiT
S b Q S b Qd b dbeb bQbe e¥
× = pò òb Q
1442443L L
All-order resummation IR Landau poleÞ
(1) (2)0 0H ( ) H ( )
2T TbQ bQ+
P
20
1
2 ( )1s Q
Qe b a Contour Contour
deformationdeformation
NP( , ) ( , )2. ( )S b Q S b Q F be e® 14424432 2 3 3
2 31 b b+r L +r L= +L
2NPNP ( ) g bF b e-=
°2
NPNP 4 NP
NP
1( ) ,
4 2
Tk
gT T
gF k e k
g
- p= =
p
Laenen, Sterman, Vogelsang (’00) Kulesza, Sterman, Vogelsang (’02)
Landau pole
cut-off at bmax : Collins,Soper,Sterman (’82)2 2
max1 /
bb
bb
b
intrinsic Tk
Elaboration of resummation formalism
systematic reorganization of the resummed series in the b-space2
0 ( )s Q L 2 2ln 1L Q b 1, 1
f 1/ /or 1 Tb
L
for 1, 0 /L b Q
(0)20
( ) ln(1 )FCh
2
(1) 2130 0
2 20
67 5
18 6 9
31 ln(1 )( ) ln (1 ) ln(1 )
2 1 2
ln(1 )2 1
F F
FG f
C Ch
CC N
(0) (1)2
1( , ) ( ) ( ) ( )
( ) ss
S b Q h h OQ
LL NLL NNLL
2,
0,
,2( ) ( )1/ exp ln(1 )2T q
N iNq
Ni
Pbf Qf
( , ) 1S b Qe
● all-order resummation is accomplished at the ``partonic level’’
Bozzi, Catani, de Florian, Glazzini (’06)
● NNLL corrections are down by αs●
uniform accuracy classified by αs !
° ° ° }
1
2
2
2
, ,2
12
22
,
1
2
( , )
1 1( , ,) ,, ,,
22
3
( ) ( ) ) ( )(
B
A B
A B
B
B
T
A x
T A Bij q q
Ag
i Bi j
i A
c Ax
j
T
x x
dS b Q
b bfb
b
b
d
Q Q x x
d
C C Yf
eN SQ
ddQ dyd dQ
eb Q
=
×
x x
x
x
x x
ì xa ïï= íï xïî
´ +
sf ò
å
ò ò
NLL resummation formula for spin-independent cross section
from Altarelli, Ellis, Greco, Martinelli (’84);
° ° ° ° 2, , , , , ,( ), ( ), ( ), ( )A A A
A A A
T A Bgq qx x x
b b b Q Q x xC C C Yx x x
Collins, Soper, Sterman (’85)
( , )S b Qe is universal !( )2
NPNPNP( , ) ( , ) ( ) ( ) = g bS b Q S b Q F b eb Fe e -®
same contour deformation for the b-integration
same elaborations in terms of 20 ( )s Q L 2 2ln 1L Q b
Gluon distribution ( ,1/ )gf b participates !
Transversity distribution 0
2( )
5,( ) ( ( )P0)4
ig dtn A tx
j j j
nid
f x e PS n PSn Se^ ^^
×
= /ò
/ò llld m y y lg
pμ-dep.: NLO DGLAP evolution
Hayashigaki, Kanazawa, Koike (’97)
Kumano, Miyama (’97) Vogelsang(’98)
0
0
,
,
( ) : GRV98
( ) : GRSV00
j
j
f x
f x
m
D m
, ,,
( ) ( )( )
2j j
j
f x f xf x
æ öm+D m÷ç ÷d m £ç ÷ç ÷çè øSoffer (’95)
x-dep.: 0 0
0 0, ,,
( ) ( ) at ( ) 0.6 Ge
2Vj j
j
f x f xf x
m +D mm = md ;
Vogelsang et al (’98)
2NP 0.9 GeV 0.5 GeV from unpolarized DY da taTg k
Nonperturbative inputs
Gaussian smearing factor:
2NPNP ( ) = g bF b e-
Dilepton QT spectrum in tDY Kawamura, Kodaira, Shimizu, Tanaka (’06)
pp collision @ RHIC s = 200 GeV, Q = 5GeV, y=2, φ=0
NLL+LO: total (“NLL+Y’’) prediction
2 2( )TT T
TTT
d ddQ dyd dQ
QdQ d dd Qy
A
2T
T
d
dQ dyd dQ
2
T
d
dQ dyd dQ
NLO LL4.0% ( )TT TT TQA A
( )TT TQA
2T
T
d
dQ dyd dQ
2
T
d
dQ dyd dQ
〔 Kawamura, Kodaira, Tanaka (’07) 〕
flat behavior: universality of soft gluon effectsnonuniversal contribution at NLL enhances
NLL+LO NLL( ) ( )T TTTTT Q QA A
NLL ( )TT TQA
NLL+LO ( )TT TQA @ RHIC (y=2, φ=0)
200 GeVS 500 GeVS
"deepersmaller '' sea- ( ) for quark regionTTT QA and/or smaller larger Q S
NLL NLO+LO ( ) is always larger than by about 20-30%. T TT T TQ AA
( )TT TQA
pp collision @ J-PARC s = 10 GeV, y=0, φ=0
NLL+LO ( )TT TQA
2 GeVQ
flat behavir due to universal soft gluon effects
enhancement mechanism from nonuniversal NLL effects
NLO LL12.8% ( )TT TT TQA A
larger asymmetries than the RHIC case moderate-x region
Novel asymptotoic formulaNLL+LO NLL NLL( ) ( ) ( 0)T TTT TT TTTQ Q Q A A A
NLL
2NP
1 1
2
2
2 2
22
1
2,
( , ) ,1 1
, ,
2 (2
( ) ( )( ) ( )
)3
cos
A B
A B
A B
A B
A B
T
T
i iA
x x
Bi ii b
i
c
i
T
xd xdS b Q gdb b
bb b
N SQ
e f fC C
ddQ dyd dQ
eb Q× x x
xx x x
- x x
a= f
ì üï ïï ïï ï´ í ýï ïï ïïd
ïîd
þ
D sf
å ò òò
(1)( ) ( )2 2
21 2
0
2 21/ 11 2
4 ( )( , (
(, )/)
)SP SP
ih
s SP ispi spie
Qe f x fb bx
Q- z l + l
æ öp ÷ç ÷ç ÷ç ÷¢¢ ÷ç b a z lèd
ødå
0TQ
(0) (1)2
1( , ) ( ) ( )
( )s
S b Q h hQ
20
(0)( )NP
2 2 20
( )( )
( ) ( )s Q
s s
ghe
Q Q Q
( ) 0SP 2
02 ( )1SP
s QSPb e
Q
Saddle-point formula at the NLL level ( ) (NNLL for ) 0TsO O Q
“degree 0 approximation’’ suggested by CSS (’85)
20 ( )s Q L
° °
NLL
2NP
1
2
, ,2 2
2
2
1
2
1,
2
( , ) 1 1( , ) ,,
2
( ) ( ) (
2
3
)B
AB
A BB
BT
A
i A
x
Bij q q g
c
i A
A
j
x
bji
T
x xdS b Q gd b bfb b
b
N
C
SQ
dfe C
ddQ dyd dQ
eb Q
=
× x
x x x
- x x
a=
ì üï ïxï ïï ï´ í ýï ïxï ïï ïî þ
sf
åòò ò
0TQ
(1)( ) ( )2
222 1
2
0
21/ 1( , )1 2
( ,( )
/ )4 ( )
SP SPs
h
s SPi i
ip spie
Qe f x f
Qbxb- z l + l
æ öp ÷ç ÷ç ÷ç ÷¢¢ ÷ç b a z lè øå
“very large prefactor” is universal !
gluon distribution decouples ! ( ) (NNLL for ) 0TsO O Q
extension of LL-level saddle point formula by Parisi, Petronzio (’79) to NLL level
20 ( )s Q L
NLL+LO NLL( 0) ( )T TT TTT Q Q A A2 2
2
21 2
21 2
2
1( , )cos(2
( , )
( ,
/ 1/
1/ 1/) )2 (
)
,
SP SP
SP
i i ii
i i i SPi
e f x f x
e
b b
xbf x f b
Most of soft gluon resummation effets cancel out LOreminiscent of TTA
unconventional scale associated with saddle point, instead of 2
NP22 (dependence on 1 G , is weak)1 V/ eSP Qb gQ
2Q
NNLL corrections would grow at small-x region probed by RHIC.
Useful for direct comparison with experiment
Clearly demonstrates the enhancement mechanism observed in numerical study
a part of NLL-level contribution survives
pp col in liY tD sion p p X
ˆ q qd d
NLOTTA Barone, Cafarella, Coriano, Guzzi, Ratcliffe (‘05)
HOTTA
Shimizu, Sterman, Yokoya, Vogelsang (’05)
directly probe the valence-quark transversity at GSI
For QT unobserved:
Large asymmetries ~ 20-40%
QCD corrections are rather small 10%
We consider the QCD corrections for QT observed case
“threshold resummation’’ at the NNL level
( )TT TQA@ GSI s = 14.5 GeV, y=0, φ=0
NLL+LO ( )TT TQA
4 GeVQ
NLO LL24% ( )TTT TTA Q A
larger asymmetries than RHIC and J-PARC case
collpp ision
flat behavir due to universal soft gluon effects“valence region”
LO is rather robustTTAenhancement mechanism is not significant
NLL+LO NLL( 0) ( )T TT TTT Q Q A A
,
,
2 22
21 2
2
1
2
2 1/ 1/
1/
( , ) ( , )
( , ) ( 1/,
(2 )
2 )
cos iSP SP
SP
qi i i
SPq
iq
i i
q
i
e f x f x
e
b b
b bf x f x
unconventional scale 2 221 GeV1/ SP Qb
NNLL corrections are small
Useful for direct comparison with experiment
does not cause siginificant
modification in the valence region probed by GSI
Summary:
the first complete formula for
well-defined, finite cross section and asymmetrie
MS scheme
all-order r
NLL+LO transverse spin
esummation recoil
channel
logsof due
s
to soft gluon radiat
( )
T TT QA
·
·
( ) ( ) ( )2 1 2 2 2 2 2 2 2 3 2 2ln , ln ln
universal structur
for
ˆ
( ) with satisfying Soffer
transvers inequa
0
ity
i
,
l
o
ity,
e
n
( )
n n n n
T T T
n n
s s s
T
TT T
T
Q Q Q Q Q Q
S f C
Q
C
Q
f
f
d Y
x
eA
- - -a a a
D s Ä Ä Ä sÄ Ä +d
m·
d
d
®
:
flat in the small region
nonuniversal contributions:
universal soft gluo
( ) at
RHIC & J-PAR
C
( ) at
n effects
GSI T
T
TT TT
TTT TT
T
A
AQ
Q
QA
A
A
>
·
:2 2
2 2
21 2
NLL+LO NLL2
1 2
22 2
( , ) ( , )0
( , ) ( , )
1 GeV
useful to extr
cos(2 )( ) ( )
2
soft gluon effe
1
c
/ 1/
1/ 1/
1ts at NL
a
L can be absorbed entirely into
ct tran v
/
s
i i ii
i i ii
SP SP
SP SP
SP
TT TTT
e f x f xb b
bQ Q
e f x f
Qb
x bA
» =f
=d då
å: =
ersity directly from experimental data
Transversely Polarized DY at small QT in QCD