Transcript Template

TITLE
WHEPP X
January 2008
Non-collinearity
in high energy scattering processes
Piet Mulders
1
[email protected]
OUTLINE
Outline
• Introduction: partons in high energy scattering processes
• (Non-)collinearity: collinear and non-collinear parton correlators
– OPE, twist
– Gauge invariance
– Distribution functions (collinear, TMD)
• Observables
– Azimuthal asymmetries
•
•
•
•
– Time reversal odd phenomena/single spin asymmetries
Gauge links
– Resumming multi-gluon interactions: Initial/final states
– Color flow dependence
Applications
Universality: an example gqgq
Conclusions
2
INTRODUCTION
QCD & Standard Model
•
•
•
QCD framework (including electroweak theory)
_ provides the_ machinery
to calculate cross sections, e.g. g*q  q, qq  g*, g*  qq, qq  qq,
qg  qg, etc.
E.g.
qg  qg
Calculations work for plane waves
0  i( s ) ( ) p, s  ui ( p, s) e ip.
3
INTRODUCTION
Confinement in QCD
•
Confinement limits us to hadrons as ‘quark sources’ or ‘targets’ (with PX= Pp)
X  i( s ) ( ) P e ip.
X  i( s ) ( ) A ( ) P e i ( p  p1 ). ip1 .
•
•
•
These involve nucleon states
At high energies interference terms between different hadrons disappear as
1/P1.P2
Thus, the theoretical description/calculation involves for hard processes, a
forward matrix element of the form
d 3 PX
ij ( p, P)  
 P | j (0) | X  X | i (0) | P   ( P  PX  p)
3
(2 ) 2EX
1
4
i p.

(
p
,
P
)

d

e
 P | j (0)  i ( ) | P 
quark ij
4 
(2 )
momentum
4
INTRODUCTION
Correlators in high-energy processes
•
•
•
•
Look at parton momentum p
Parton belonging to a particular hadron P: p.P ~ M2
For all other momenta K: p.K ~ P.K ~ s ~ Q2
Introduce a generic vector n ~ satisfying P.n = 1, then we have
n ~ 1/Q, e.g. n = K/(P.K)
p  x P  pT  s n
~Q
~M
~ M2/Q
x  p.n ~ 1
s  p.P  xM 2 ~ M 2
• Up to corrections of order M2/Q2 one can perform the s-integration
d ( .P)d 2T i p.
†
ij ( x, pT )   d ( p.P) ij ( p, P) 
e
P

j (0) i ( ) P
3
(2 )
5
 .n  0
INTRODUCTION
(calculation of) cross section in DIS
Full calculation
+
LEADING
(in M/Q)
+
+
+…
6
INTRODUCTION
(calculation of) cross section in SIDIS
Full calculation
+
LEADING
(in M/Q)
+
+
+…
7
INTRODUCTION
Leading partonic structure of hadrons
Need PH.Ph ~ s (large) to
get separation of soft and
hard parts
hard process
p
H
Allows
k
Ph
PH
 ds… =  d(p.P)…
h
fragmentation
correlator
distribution
correlator
Ph
D(z, kT)
(x, pT)
PH
Ph
PH
8
INTRODUCTION
Partonic correlators
The cross section can be expressed in hard squared QCD-amplitudes
and distribution and fragmentation functions entering in forward matrix
elements of nonlocal combinations of quark and gluon field operators
(f   or G). These are the (hopefully universal) objects we are after,
useful in parametrizations and modelling.
Distribution functions
p   xP   pT 
d ( .P)d 2T i p.
†
( x, pT )  
e
P
f
(0)f ( ) P
3
(2 )
Fragmentation functions
p.P  xM
2
n
P.n
 .n  0
1
2
k .K  z M h 
1 

k  K  kT 
n
z
K .n

d ( .K )d 2T i k .
†
D( z, kT )  
e
0
f
(0)
K
,
X
K
,
X
f
( ) 0
3
(2 )
 .n  0
9
(NON-)COLLINEARITY
(non-)collinearity of parton correlators
The cross section can be expressed in hard squared QCD-amplitudes
and distribution and fragmentation functions entering in forward matrix
elements of nonlocal combinations of quark and gluon field operators
(f   or G). These are the (hopefully universal) objects we are after,
useful in parametrizations and modelling.
Distribution functions
p   xP   pT 
d ( .P)d 2T i p.
†
( x, pT )  
e
P
f
(0)f ( ) P
3
(2 )
p.P  xM
2
n
P.n
 .n  0
lightfront:  = 0
TMD
d ( .P) i p.
( x)   d pT ( x, pT )  
e
P f † (0)f ( ) P
(2 )
2
collinear
   dx  ( x)  P f † (0) f (0) P
 .n T 0
lightcone
10
local
(NON-)COLLINEARITY
Spin and twist expansion
Spin n:
• Local matrix elements in 
1
n
~
(P
…P
– traces)
Operators can be classified via their
Twist t:
canonical dimensions and spin (OPE)
dimension  spin
• Nonlocal matrix elements in (x)
Parametrized in terms of (collinear) distribution functions f…(x)
that involve operators of different spin but with one specific twist t
that determines the power of (M/Q)t-2 in observables (cross
sections and asymmetries).
Moments give local operators.
M ( n )  dx x n 1 f ( x)

• Nonlocal matrix elements in (x,pT)
Parametrized in terms of TMD distribution functions f…(x,pT2) that
involve operators of different spin and different twist. The lowest
twist determines the operational twist t of the TMD functions and
determines the power of (M/Q)t-2 in observables.
Transverse moments give
n
2
collinear functions.
 p 
(n)
2
f
2
T
( x)   d pT 
f
(
x
,
p
T)
2 
11
 2M 
(NON-)COLLINEARITY
Gauge invariance for quark correlators
• Presence of gauge link needed for color gauge invariance

[C ]
U[0,
]


 P exp  ig  ds  A 
 0

A  A P  AT  An
• The gauge link arises from all ‘leading’ m.e.’s as  A...A 
d ( .P) i p.
[n]
 ij ( x; n)  
e
P  j (0)U[0,
 ] i ( ) P  .n T  0
(2 )
d ( .P)d 2T i p.
q
[C ]
ij ( x, pT ; n, C )  
e
P

(0)
U
j
[0, ] i ( ) P
3
(2 )
q
 .n  0
• Transverse pieces arise from ATa  Ga = +Aa + …
• Basic gauge links:
[U-] = []
TIME
RERVERSAL
[U+] = []
12
Distribution
(NON-)COLLINEARITY
including the gauge link (in SIDIS)
A+
One needs also AT
G+a =  +ATa
ATa()= ATa(∞) +  d G+a
Belitsky, Ji, Yuan 2002
Boer, M, Pijlman 2003
From (0)AT()() m.e.
(NON-)COLLINEARITY
Gauge invariance for gluon correlators
2
d
(

.
P
)
d
T i p.
[C ]
na
[ C ']
n
a
(
x
,
p
;
C
,
C
')

e
P
U
F
(0
)
U
F
( ) P
g
T
[ ,0]
[0, ]
 (2 )3
 .n  0
• Using 3x3 matrix representation for U, one finds in TMD gluon
correlator appearance of two links, possibly with different paths.
• Note that standard field displacement involves C = C’
Fa ( )  U[[C,] ] Fa ( )U[[C,] ]
• Basic gauge links
g[,]
g[,]
g[,]
g[,]
14
(NON-)COLLINEARITY
Collinear parametrizations
• Gauge invariant correlators  distribution functions
• Collinear quark correlators (leading part, no n-dependence)
 ( x) 
q
(f
q
1
( x)  S L g ( x) g 5  h ( x) g 5 ST )
q
1
q
1
P
2
S  SL
P
 ST
M
• i.e. massless fermions with momentum distribution f1q(x) = q(x),
chiral distribution g1q(x) = Dq(x) and transverse spin polarization
h1q(x) = q(x) in a spin ½ hadron
• Collinear gluon correlators (leading part)
1

g
 g
 g ( x) 

g
f
(
x
)

i
S

(
T
1
L T g1 ( x ) )
2x

• i.e. massless gauge bosons with momentum distribution f1g(x) = g(x)
and polarized distribution g1g(x) = Dg(x)
15
(NON-)COLLINEARITY
TMD parametrizations
• Gauge invariant correlators  distribution functions
• TMD quark correlators (leading part, unpolarized)
pT  P
 q
2
q
2
 ( x, pT )   f1 ( x, pT )  i h1 ( x, pT )

M  2

q
TMD correlators
q[U] and g[U,U’]
do depend on
gauge links!
• as massless fermions with momentum distribution f1q(x,pT) and
transverse spin polarization h1q(x,pT) in an unpolarized hadron
• The function h1q(x,pT) is T-odd!
• TMD gluon correlators (leading part, unpolarized)
 v

1


p
p

g
1

g
2
T T
2 T
 
(
x
,
p
)


g
f
(
x
,
p
)



g
T
T
1
T
2
2x 
M


 g
2
 h 1 ( x, pT ) 


• as massless gauge bosons with momentum distribution f1g(x,pT) and 16
linear polarization h1g(x,pT) in an unpolarized hadron
The quark distributions
(in pictures)
unpolarized quark
distribution
need pT
helicity or chirality
distribution
need pT
transverse spin distr.
or transversity
need pT
need pT
(NON-)COLLINEARITY
OBSERVABLES
Results for deep inelastic processes
DIS
sg *N X  f1N q ( x) sˆg *qq
Ds g *N  X  g1N q ( x)  Dsˆg *q q
SIDIS
sg *N hX  f1N q ( x) sˆg *qq  D1qN ( z)
18
OBSERVABLES
Probing intrinsic transverse momenta
• In a hard process one probes quarks and gluons
p  xP  pT
• Momenta fixed by kinematics (external momenta)
k  z 1P  kT
DIS
SIDIS
x  xB  Q2 / 2P.q
z  zh  P.Kh / P.q
• Also possible for transverse momenta
SIDIS
f2  f1
K1
qT  q  xB P  zh1Kh  kT  pT
2-particle inclusive hadron-hadron scattering
qT  z11K1  z21K2  x1P1  x2 P2  p1T  p2T  k1T  k2T
• Sensitivity for transverse momenta requires  3 momenta
SIDIS: g* + H  h + X
f K
2
pp-scattering
DY: H1 + H2  g* + X
e+e-: g*  h1 + h2 + X
…and knowledge of
hard process(es)!
hadronproduction: H1 + H2  h1 + h2 + X
 h+X
(?)
19
OBSERVABLES
Time reversal as discriminator
W (q; P, S , Ph , Sh )  W (q; P, S , Ph , Sh )
symmetry structure
W* (q; P, S, Ph , Sh )  W (q; P, S, Ph , Sh )
hermiticity
W (q; P, S, Ph , Sh )  W (q; P, S , Ph , Sh )
parity
W* (q; P, S, Ph , Sh )  W (q; P, S , Ph , Sh )
time reversal
W (q; P, S , Ph , Sh )  W (q; P, S , Ph , Sh )
combined
• If time reversal can be used to restrict observable one has only even
spin asymmetries
• If time reversal symmetry cannot be used as a constraint (SIDIS,
DY, pp, …) one can nevertheless connect T-even and T-odd
phenomena (since T holds at level of QCD).
• In hard part T is valid up to order as2
20
OBSERVABLES
Results for deep inelastic processes
qT  q  xB P  zh1Kh  kT  pT
Sivers asymmetry
qT
M
sin(fh  fS )s g *N   X  h 1q ( x)  sˆg *q q  H1(1) q ( z )
Collins asymmetry
qT
M
sin(fh  fS ) s g *N   X  f1T(1) q ( x)  sˆg *qq  D1q ( z )
Function as appearing in
parametrization of [+]
21
GAUGE LINKS
Generic hard processes
• Matrix elements involving
parton 1 and additional
gluon(s) A+ = A.n appear
at same (leading) order in
‘twist’ expansion and
produce link [U](1)
• insertions of gluons
collinear with parton 1 are
possible at many places
• this leads for correlator
(1) to gauge links running
to lightcone ± infinity
Link structure
for fields in
correlator 1
• SIDIS  [+](1)
• DY  [](1)
C. Bomhof, P.J. Mulders and F. Pijlman,
PLB 596 (2004) 277 [hep-ph/0406099];
EPJ C 47 (2006) 147 [hep-ph/0601171]
22
GAUGE LINKS
Integrating [±](x,pT)  [±](x)
d ( .P)d 2T i p.
†
n
T
n
 ( x, pT )  
e
P

(0)
U
U
U
[0,  ] [0T ,T ] [ , ] ( ) P
3
(2 )
[]
collinear
correlator
d ( .P)

 ( x)  
e
[]
(2 )
i p.
n
P  † (0)U[0,
 ] ( ) P
 .n T 0
[]
[]
23
 .n  0
GAUGE LINKS
Integrating [±](x,pT)  a[±](x)
transverse
moments
a []

a [  ] ( x)   d 2 pT pTa [  ] ( x, pT )
d ( .P)d 2T i p.
†
n
a
T
n
( x)   d pT 
e
P

(0)
U
i

U
U
[0,  ] T
[0T ,T ] [  , ] ( ) P
3
(2 )
2
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
Gluonic
pole m.e.
T-odd
24
 .n  0
GAUGE LINKS
A2

2 hard processes: qq
• E.g. qq-scattering as hard
subprocess
• The correlator (x,pT) enters
for each contributing term in
squared amplitude with
specific link

qq
U□ = U+U†
[Tr(U□)U+] = [(□)+]
Traced loop
[U□U+] = [□+]
loop
25
GAUGE LINKS
Gluonic poles
• Thus:
[U](x) = (x)
[U]a(x)
=
~
a(x)
+ CG[U] Ga(x,x)
• Universal gluonic pole m.e. (T-odd for distributions)
• G(x) contains the weighted T-odd functions h1(1)(x) [BoerMulders] and (for transversely polarized hadrons) the function
f1T(1)(x) [Sivers]
~
• (x) contains the T-even functions h1L(1)(x) and g1T(1)(x)
• For SIDIS/DY links: CG[±] = ±1
• In other hard processes one encounters different factors:
CG[□+] = 3, CG[(□)+] = Nc
Efremov and Teryaev 1982; Qiu and Sterman 1991
26
Boer, Mulders, Pijlman, NPB 667 (2003) 201
C. Bomhof, P.J. Mulders and F. Pijlman, EPJ C 47 (2006) 147
APPLICATIONS
examples: qqqq in pp
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]
27
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005) 034030; hep-ph/0505268
APPLICATIONS
examples: qqqq in pp
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
28
Bacchetta, Bomhof, D’Alesio,Bomhof, M, Murgia, PRL2007, hep-ph/0703153
APPLICATIONS
Gluonic pole cross sections
• In order to absorb the factors CG[U], one can define specific hard
cross sections for gluonic poles (which will appear with the functions
in transverse moments)
• for pp:
sˆ[ q ]qqq  CG[U ( D)]sˆ [ D]
sˆ qqqq  sˆ [ D]

[ D]

[ D]
(gluonic pole cross section)
etc.
dsˆ qqqq
• for SIDIS:
for DY:
sˆ [q] q   sˆ q q
dsˆ(q)qqq
sˆ[q]q    sˆqq 
y
29
Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]
APPLICATIONS
examples: qgqg in pp
weigted
Transverse momentum dependent
q 
D1
2
Nc
2
Nc
1
[( )  ]


1
2
Nc
[]
1

  
Nc
2
1
2
Nc
1
 G
Only one factor, but
more DY-like than
SIDIS
D2
D3
D4
Note: also
.
etc
Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019
Boer, M, Pisano, hep-ph/0712.0777
30
UNIVERSALITY
Universality (examples qgqg)
weighted
Transverse momentum dependent
D1
q 
2
Nc
2
Nc  1
[( )  ]


1
2
Nc  1
[]

  
2
Nc
2
1
Nc  1
 G
D2
D3
D4
D5
31
UNIVERSALITY
Universality (examples qgqg)
weighted
Transverse momentum dependent
q 
2
Nc
2
Nc  1
[( )  ]


1
2
Nc  1
[]

  
2
Nc
2
1
Nc  1
 G
32
UNIVERSALITY
Universality (examples qgqg)
weighted
Transverse momentum dependent
q 
q 
q 
2
q

NNc2
c
2
2( N c  1)
2
2( N c  1)
4
Nc
2
( Nc
 1)

[(
[( )(
)( † )  ]

2
[( )(
†

†
)]
2
2


Nc
2( N c  1)
q 
[( )  ]
Nc  1
Nc
) N]c2
2
2
Nc
1
[( )  ]

2
2

[(


[]
) ]
2
( Nc
2
Nc
2
Nc
[( )  ]
 1)
1

2
[(
)(

[]

2
Nc  1
1
1
  
[  ]   
  
 G
2 N2 1
c 1
2( N c N1)
2

1
†

)]
N c c 1
1
2
Nc

[ ]
1

1
2
Nc
1
[ ]
  

2
Nc
1
2
Nc  1
1
2
Nc  1
Nc
2
1
2
Nc
1
 G
 G
 G
    G
33
UNIVERSALITY
Universality (examples qgqg)
weighted
Transverse momentum dependent
It is also possible to group
the TMD
2
2
Nc
Nc  1
1
[(
)

]
[

]
functions in a smart
into
q way
 two!
 2 
   2  G
2
N

1
N

1
Nc  1
c
c
(nontrivial for nine diagrams/four
color-flow possibilities)
2
2
N 2
Nc 1
1 1
[( )( † ) N]c2
[( )  ]
[]
[( )( )  ]
q  N2Nc22c 

† 2  [( ) ]22 2 
[  ]   
 G
2
2
2( Ncc  1)
[( )  ]
c 1
2( N
 1) [( )( 2() N]c  1) 2( NN c N1)
N 1Nc 1 1 [  ]
q
 
†
c
2
2( N c
 G 
q 
 1)


2
Nc
[( )(
42
N
Ncc  1 [( )(

2
2
( N c  1)
†
†
) ]
c
2
2( N c
 1)


c
2
Nc
1

  
1
2
Nc  1
 G
1
 2 2 [ ]
N
)]
 N2c c 12 [( ) ] 
( N c  1)
2
But still no factorization!
Nc
q  2 [(
Nc  1
)(
†
)]
1
2
Nc

[ ]
1

1
2
Nc
1
[  ]
  

Nc
2
1
2
Nc
1
 G
    G
34
UNIVERSALITY
‘Residual’ TMDs
• We find that we can work with basic TMD functions [±](x,pT) + ‘junk’
• The ‘junk’ constitutes process-dependent residual TMDs

[( )(
†
)]
[]
( x, pT )  
 [(

)(
†
)]
( x, pT )  [  ] ( x, pT ) 

 [(
)( † )  ]
( x , pT )
[ ] ( x, pT )  2[ ] ( x, pT )  [ ] ( x, pT )   [ ] ( x, pT )
no definite
T-behavior
definite T-behavior
• The residuals satisfies (x) = 0 and G(x,x) = 0, i.e. cancelling kT
contributions; moreover they most likely disappear for large kT
35
Bomhof, Mulders, Vogelsang, Yuan, NPB, hep-ph/0709.1390
CONCLUSIONS
Conclusions
• Beyond collinearity many interesting phenomena appear
• For integrated and weighted functions factorization is possible
(collinear quark, gluon and gluonic pole m.e.)
• Accounted for by using gluonic pole cross sections (new
gauge-invariant combinations of squared hard amplitudes)
• For TMD distribution functions the breaking of universality
can be made explicit and be attributed to specific matrix
elements
• Many applications in hard processes. Including fragmentation
(e.g. polarized Lambda’s within jets) even at LHC
References:
Qiu, Vogelsang, Yuan, hep-ph/0704.1153
Collins, Qiu, hep-ph/0705.2141
Qiu, Vogelsang, Yuan, hep-ph/0706.1196
Meissner, Metz, Goeke, hep-ph/0703176
Collins, Rogers, Stasto, hep-ph/0708.2833
Bomhof, Mulders, hep-ph/0709.1390
Boer, Bomhof, Hwang, Mulders, hep-ph/0709.1087
36