Transcript Document 7406348

DIS2006, Tsukuba
April 20-24, 2006
Universality of single spin
asymmetries in hard processes
Cedran Bomhof and Piet Mulders
[email protected]
Content
Universality of Single Spin Asymmetries (SSA) in hard processes
• Introduction
• SSA and time reversal invariance
• Transverse momentum dependence (TMD)
Through TMD distribution and fragmentation functions to
transverse moments and gluonic poles
•
•
•
•
Electroweak processes (SIDIS, Drell-Yan and annihilation)
Hadron-hadron scattering processes
Gluonic pole cross sections
Conclusions
Introduction: partonic structure of hadrons
For (semi-)inclusive measurements, cross sections in hard scattering
processes factorize into a hard squared amplitude and distribution and
fragmentation functions entering in forward matrix elements of
nonlocal combinations of quark and gluon field operators (f  y or G)
( x)  



p  xP  pT 
d ( .P) i p.
e
P f † (0) f ( ) P
(2 )
p.P  xM
2
n
 .n T  0

P.n
d ( .P)d 2T i p.
†
( x, pT )  
e
P
f
(0)f ( ) P
3
(2 )
TMD
FF
lightcone
 .n  0
lightfront
d ( .P)d 2T i k .
†
( z, kT )  
e
0
f
(0)
P
,
X
P
,
X
f
( ) 0
3
(2 )
 .n  0
Introduction: partonic structure of hadrons
FORWARD
matrix
elements
x section
one hadron
in inclusive
or semiinclusive
scattering
• Quark distribution functions (DF) and
fragmentation functions (FF)
– unpolarized
q(x) = f1q(x) and D(z) = D1(z)
– Polarization/polarimetry
q(x) = g1q(x) and dq(x) = h1q(x)
– Azimuthal asymmetries
g1T(x,pT) and h1L(x,pT)
– Single spin asymmetries
h1(x,pT) and f1T(x,pT); H1(z,kT) and D1T(z,kT)
OFFFORWARD
• Form factors
Amplitude
• Generalized parton distributions
Exclusive
NONLOCAL
lightcone
NONLOCAL
lightfront
LOCAL
NONLOCAL
lightcone
SSA and time reversal invariance
• QCD is invariant under time reversal (T)
• Single spin asymmetries (SSA) are T-odd observables, but they are
not forbidden!
• For distribution functions a simple distinction between T-even and Todd DF’s can be made
– Plane wave states (DF) are T-invariant
– Operator combinations can be classified according to their Tbehavior (T-even or T-odd)
• Single spin asymmetries involve an odd number of (i.e. at least one)
T-odd function(s)
• The hard process at tree-level is T-even; higher order as is required
to get T-odd contributions
Intrinsic transverse momenta
p  x P + pT
k  z-1 K + kT
f2  f1
df K
2
pp-scattering
• In a hard process one probes partons (quarks and gluons)
• Momenta fixed by kinematics (external momenta)
DIS
x = xB = Q2/2P.q
SIDIS z = zh = P.Kh/P.q
• Also possible for transverse momenta
SIDIS qT = kT – pT
= q + xBP – Kh/zh  Kh/zh
2-particle inclusive hadron-hadron scattering
K1
qT = p1T + p2T – k1T – k2T
= K1/z1+ K2/z2 x1P1 x2P2  K1/z1+ K2/z2
• Sensitivity for transverse momenta requires 3 momenta
SIDIS: g* + H  h + X
DY: H1 + H2  g* + X
e+e-: g*  h1 + h2 + X
hadronproduction: H1 + H2  h + X
 h1 + h2 + X
TMD correlation functions
(unpolarized hadrons)
quark
correlator
i [ p , P ] 
1
 ( x, pT )   f1 ( x, pT2 ) P  h 1 ( x, pT2 )  T

2
2M 
In collinear
cross section
(x, pT)
 ( x)   d 2 pT  ( x, pT ) 
1
2
f1 ( x) P
a ( x)   d 2 pT pTa ( x, pT )  14 i h 1 (1) ( x)[ P , g a ]
In azimuthal
asymmetries
Transverse moment
h
 (1)
1
• T-odd
• Transversely
polarized quarks
pT2 
2
( x)   d pT
h
(
x
,
p
1
T)
2
2M
2
Color gauge invariance
• Nonlocal combinations of colored fields must be joined by a gauge link:
y (0)y ( )  y (0)U (0,  )y ( )



U (0,  )  P exp  ig  ds  A 
 0

• Gauge link structure is calculated from collinear A.n gluons exchanged
between soft and hard part
DIS  [U]
SIDIS  [U+] = [+]
DY  [U-] = []
• Link structure for TMD functions
depends on the hard process!
Integrating [±](x,pT)  [±](x)
d ( .P)d 2T i p.
†
n
T
n
 ( x, pT )  
e
P
y
(0)
U
U
U
[0,  ] [0T ,T ] [  , ]y ( ) P
3
(2 )
[]
collinear
correlator
d ( .P)

 ( x)  
e
[]
(2 )
i p.
n
P y † (0)U[0,
 ]y ( ) P
 .n T  0
 .n  0
Integrating [±](x,pT)  a[±](x)
transverse
moment
a [  ] ( x)   d 2 pT pTa [  ] ( x, pT )
d ( .P)d 2T i p.
†
n
a
T
n
 ( x)   d pT 
e
P
y
(0)
U
i

U
U
[0,  ] T
[0T ,T ] [  , ]y ( ) P
3
(2 )
d ( .P) i p.
n
a
a [  ] ( x)  
e [ P y † (0)U[0,
iD
]
T y ( ) P
(2 )

a []
2
n
n
na
n
 P y (0)U[0,
d
(

.
P
)
U
gG
(

)
U
 ] 
[  , ]
[ , ] y ( ) P

a [  ] ( x)  aD ( x)   dx1
i
x1 i
 aD ( x)   dx1 P
i
x1
aG ( x, x  x1 )
G(p,pp1)
aG ( x, x  x1 )   aG ( x, x)
a ( x)
T-even
T-odd
 .n  0
]
LC
Gluonic poles
• Thus
•
•
•
•
•
[±]a(x) = a(x) + CG[±] Ga(x,x)
CG[±] = ±1
with universal functions in gluonic pole m.e. (T-odd for
distributions)
There is only one function h1(1)(x) [Boer-Mulders] and (for
transversely polarized hadrons) only one function f1T(1)(x)
[Sivers] contained in G
These functions appear with a process-dependent sign
Situation for FF is more complicated because there are no T
constraints
What about other hard processes (in particular pp scattering)?
Efremov and Teryaev 1982; Qiu and Sterman 1991
Boer, Mulders, Pijlman, NPB 667 (2003) 201
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277
Other hard processes
•
•
•
qq-scattering as hard
subprocess
insertions of gluons collinear
with parton 1 are possible at
many places
this leads for ‘external’ parton
fields to a gauge link to
lightcone infinity
Link structure
for fields in
correlator 1
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277
Other hard processes
•
•
•
•
qq-scattering as hard
subprocess
insertions of gluons collinear
with parton 1 are possible at
many places
this leads for ‘external’ parton
fields to a gauge link to
lightcone infinity
The correlator (x,pT) enters
for each contributing term in
squared amplitude with specific
link
□)U+]
(x,p
[Tr(U
[U□U+](x,pT)
T)
U□ = U+U†
Gluonic pole cross sections
• Thus
[U]a(x) = a(x) + CG[U] Ga(x,x)
• CG[U±] = ±1
CG[U□ U+] = 3, CG[Tr(U□)U+] = Nc
• with the same uniquely defined functions in gluonic pole m.e. (Todd for distributions)
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268
examples: qqqq
Tr (U )
U
U U
Nc
D1
q 
2
Nc  1
2
Nc
1

[( )  ]

2
2
Nc
[ ]
1

  
D4
2
Nc
1
 G
CG [D1] = CG [D2]
D2
D3
2
Nc  5
q 
2
2 Nc
2
Nc
1

[( )  ]

2
Nc  1
2
Nc
1
[ ]

  
2
Nc  3
2
Nc
1
 G
CG [D3] = CG [D4]
Gluonic pole cross sections
• In order to absorb the factors CG[U], one can define specific hard
cross sections for gluonic poles (to be used with functions in
transverse moments)
• for pp:
ˆ ( qg ) q qq  CG[ D ]ˆ [ D ]
ˆ qq qq  ˆ [ D ]


[D]
[D]
(gluonic pole cross section)
etc.
dˆ qq qq
• for SIDIS:
for DY:
ˆ
( qg )  q
  ˆ
q q
dˆ ( qg ) q  qq
ˆ ( qg ) q    ˆ qq 
• Similarly for gluon processes
y
Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171
examples: qqqq
D1
q 
1
2
Nc
1

[(
†
)]

2
Nc  2
2
Nc
1

[ ]
  
2
Nc  3
2
Nc
1
 G
For Nc:
CG [D1]  1
(color flow as DY)
q 
2
Nc
2
Nc  1

[(
†
)]

1
2
Nc  1

[ ]
  
2
Nc  1
2
Nc  1
 G
Conclusions
• Single spin asymmetries in hard processes can exist
• They are T-odd observables, which can be described in terms of
T-odd distribution and fragmentation functions
• For distribution functions the T-odd functions appear in gluonic
pole matrix elements
• Gluonic pole matrix elements are part of the transverse moments
appearing in azimuthal asymmetries
• Their strength is related to path of color gauge link in TMD DFs
which may differ per term contributing to the hard process
• The gluonic pole contributions can be written as a folding of
universal (soft) DF/FF and gluonic pole cross sections
Belitsky, Ji, Yuan, NPB 656 (2003) 165
Boer, Mulders, Pijlman, NPB 667 (2003) 201
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030
Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171
Eguchi, Koike, Tanaka, hep-ph/0604003
Ji, Qiu, Vogelsang, Yuan, hep-ph/0604023