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 gqgq
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= Pp)
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 2T 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 2T 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 2T 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 2T 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 An
• 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 2T 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 Ga = +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 h1q(x,pT) in an unpolarized hadron
• The function h1q(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 h1g(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 *qq
Ds g *N X g1N q ( x) Dsˆg *q q
SIDIS
sg *N hX f1N q ( x) sˆg *qq D1qN ( 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 zh1Kh kT pT
2-particle inclusive hadron-hadron scattering
qT z11K1 z21K2 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 zh1Kh 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 *qq 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 2T 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 2T 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,pp1)
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: qqqq 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: qqqq 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 ]qqq CG[U ( D)]sˆ [ D]
sˆ qqqq sˆ [ D]
[ D]
[ D]
(gluonic pole cross section)
etc.
dsˆ qqqq
• for SIDIS:
for DY:
sˆ [q] q sˆ q q
dsˆ(q)qqq
sˆ[q]q sˆqq
y
29
Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]
APPLICATIONS
examples: qgqg 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 qgqg)
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 qgqg)
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 qgqg)
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 qgqg)
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 [( ) ]22 2
[ ]
G
2
2
2( Ncc 1)
[( ) ]
c 1
2( N
1) [( )( 2() N]c 1) 2( NN c N1)
N 1Nc 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