Spin Physics and Transverse Structure

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Spin Physics and Transverse Structure
Piet Mulders
mulders@few.vu.nl
1
ABSTRACT
Spin Physics and Transverse Structure
Piet Mulders (Nikhef Theory Group/VU University Amsterdam)
Spin is a welcome complication in the study of partonic structure that has led to new
insights, even if experimentally not all dust has settled, in particular on quark flavor
dependence and gluon spin. At the same time it opened new questions on angular
momentum and effects of transverse structure, in particular the role of the transverse
momenta of partons. This provides again many theoretical and experimental
challenges and hurdles. But it may also provide new tools in high-energy scattering
experiments linking polarization and final state angular dependence.
2
Parton distribution (PDF) and fragmentation (PFF) functions
Parton densities (PDFs) and decay functions (PFFs) are natural way of dealing
with quarks/gluons in high energy processes (several PDF databases)
PDFs can be embedded in a field theoretical framework via Operator Product
Expansion (OPE), connecting Mellin moments of PDFs with particular QCD
matrix elements of operators (spin and twist expansion)
Hard process introduces necessary directionality with light-like directions,
P – M2 n and n = P’/P.P’ (satisfying P.n = 1)
The lightcone momentum fraction x of the parton momentum p = x P is linked
to xB = Q2/2P.q (DIS) or x1 = q.P2/P1.P2 and x2 = q.P1/P1.P2 (DY)
Polarization, MS = SL P + MST - M2 SL n (longitudinal or transverse) is a
welcome complication, experimentally challenging but new insights emerged
Consideration of partonic transverse momenta, p = x P + pT + (p2 pT2) n opens
new avenues to Transverse Momentum Dependent (TMD) PDFs, in short TMDs
with experimental and theoretical challenges
The measurement is mostly not for free, but requires dedicated final states
(jets/flavor), polarized targets or polarimetry e.g. through decay orientation of
final states (r or L)
3
Link to matrix elements  hadron correlators
Structure of PDFs and PFFs as correlators built from nonlocal field configurations,
extending on OPE and useful for interpretation, positivity bounds, modelling, …
ui ( p,s)u j ( p, s) Þ Fij ( p | p) ~ å P y j (0) | X >< X | yi (0) P d ( p - P + PX )
X
yi (x )
=
y j (0)
dx i p.x
ò 2p e P y j (0) yi (x ) P
parametrized using symmetries (C, P, T)
|X><X| gateway to models: e.g. diquarks
WG6/12 Mao, Lu, Ma
AUT predictions (diquark model)
ui (k,s)u j (k,s) Þ Dij ( p | p) ~ å 0 yi (0) | K h X >< K h X | y j (0) 0 d (k - K h - K X )
X
=
ò
dx i k .x
e
0 yi (x ) ah+ ah y j (0) 0
2p
no T-invariance!
Collins & Soper, NP B 194 (1982) 445
4
Link to matrix elements  hadron correlators
Gluonic matrix elements
e m ( p, l )e n * ( p, l ) Þ F g ab ( p | p) ~ å P Aa (0) | X >< X | Ab (x ) P d ( p - P + PX )
X
=
F g ab ( p)
Also needed are multi-parton correlators
a
F A;ij ( p - p1 , p1 | p) =
or better:
ò
ò
dx i p.x
e
P F na (0)F nb (x ) P
2p
FA(p-p1,p)
d 4x d 4h i ( p- p1 ).x +ip1.h
a
e
P
y
(0)A
(h ) yi (x ) P
j
8
(2p )
a
F D;ij ( p - p1 , p1 | p) =
a
F F ;ij ( p - p1 , p1 | p) =
(color gauge invariance)
ò
d 4x d 4h i ( p- p1 ).x +ip1.h
a
e
P
y
(0)D
(h ) yi (x ) P
j
8
(2p )
ò
d 4x d 4h i ( p- p1 ).x +ip1.h
na
e
P
y
(0)F
(h ) yi (x ) P
j
8
(2p )
5
(Un)integrated correlators
F(x, pT , p.P) =
F(x, pT ;n) =
ò
d 4x i p.x
e
P y (0) y (x ) P
4
(2p )
ò
d(x .P)d 2xT
(2p )
3
e
i p.x
P y (0) y (x ) P
unintegrated
TMD (light-front)
x .n=x + =0
p- = p.P integration makes time-ordering automatic.
The soft part is simply sliced at a light-front instance
F(x) =
ò
d(x .P) i p.x
e
P y (0) y (x ) P
(2p )
x .n=xT =0 or x 2 =0
Is already equivalent to a point-like interaction
F = P y (0) y (x ) P
x =0
collinear (light-cone)
local
Local operators with calculable anomalous dimension
6
TMDs and color gauge invariance
Gauge invariance in a non-local situation requires a gauge link U(0,x)
y (0)y (x ) = å
n
1 m1 m N
x ...x y (0)¶m ... ¶m y (0)
1
N
n!
y (0)U (0,x ) y (x ) = å
n
x


U (0, x )  P exp  -ig  ds  A 
 0

1 m1 m N
x ...x y (0) Dm ... Dm y (0)
1
N
n!
Introduces path dependence for F(x,pT)
F[U ] (x, pT ) Þ F(x)
x
T
x
0
x.P
7
Parametrization of F(x)  collinear PDFs
Quarks in nucleon:
p=xP
compare
u( p,s)u( p,s) = p/ + m
Fq (x) µ x f1q (x) P/ + .... = f1q (x) p/ + ....
unpolarized
quarks
8
Parametrization of F(x)  collinear PDFs
compare
Quarks in nucleon:
u( p,s)u( p,s) = p/ + m
Fq (x) µ f1q (x) p/ + ....
unpolarized
quarks
Operator connection through Mellin moments
x N -1 F(x) =
x = p+ = p.n
ò
=
d(x .P) i p.x
[n]
e
P y (0)(¶xn ) N -1U [0,
y (x ) P
x]
(2p )
ò
x .n=xT =0
d(x .P) i p.x
[n]
n N -1
e
P y (0)U [0,
(D
) y (x ) P
x]
x
(2p )
x .n=xT =0
F( N ) = ò dx x N-1 F(x) = P y (0)(Dn ) N-1 y (0) P
Anomalous dimensions can be Mellin transformed into the splitting functions
that govern the QCD evolution.
9
Collinear PDFs without polarization
Quarks in nucleon:
p=xP
compare
u( p,s)u( p,s) = p/ + m
Fq (x) µ x f1q (x) P/ + ....
unpolarized
quarks
10
Collinear PDFs with polarization
æP
ö
+ Mn ÷ + ST
Quarks in polarized nucleon: S = S L ç
èM
ø
0 £ S L2 - ST2 £1
Fq (x) µ x f1q (x) P/ + S L xg1q (x) P/ g 5 + xh1q (x)S/ T P/ g 5 + ...
unpolarized
quarks
chiral quarks in
L-polarized N
T-polarized quarks
in T-polarized N
compare
u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ )
spin  spin
DIS/SIDIS programme to obtain
• Flavor PDFs: q(x) = f1q(x)
• Polarized PDFs: Dq(x) = g1Lq(x)
dq(x) = h1Tq(x)
WG6/70
JAM global QCD analysis
WG6/142 Drachenberg (STAR)
Transversity in polarized pp
WG6/150 Gunarathne (STAR)
Longitudinal spin asymm. in W prod.
11
TMDs with polarization
æP
ö
+ Mn ÷ + ST
Quarks in polarized nucleon: S = S L ç
èM
ø
0 £ S L2 - ST2 £1
q
Fq (x, pT ) µ xf1q (x, pT2 ) P/ + S L xg1L
(x, pT2 ) P/ g 5 + xh1Tq (x, pT2 )S/ T P/ g 5 + ...
unpolarized
quarks
… but also
chiral quarks in
L-polarized N
T-polarized quarks
in T-polarized N
compare
u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ )
( pT × ST ) q
F (x, pT ) µ ... +
xg1T (x, pT2 ) P/ g 5 + ...
M
q
spin  spin
chiral quarks
in T-polarized N
(worm-gear)
12
TMDs with polarization
æP
ö
+ Mn ÷ + ST
Quarks in polarized nucleon: S = S L ç
èM
ø
0 £ S L2 - ST2 £1
q
Fq (x, pT ) µ xf1q (x, pT2 ) P/ + S L xg1L
(x, pT2 ) P/ g 5 + xh1Tq (x, pT2 )S/ T P/ g 5 + ...
unpolarized
quarks
chiral quarks in
L-polarized N
T-polarized quarks
in T-polarized N
compare
u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ )
… but also
( pT × ST ) ^q
2 p
F (x, pT ) µ ... +
xh1T (x, pT ) / T P/ g 5 + ...
M
M
q
spin  spin
T-polarized quarks
in T-polarized N
(pretzelocity)
WG6/103 Prokudin, Lefky
Extraction pretzelosity distribution
13
TMDs with polarization
… and T-odd functions
p/ T
( pT ´ ST ) ^q
F (x, pT ) µ ... + ih (x, p ) P/ + i
xf1T (x, pT2 ) P/ + ...
M
M
^q
1
q
T-polarized quarks
in unpolarized N
(Boer-Mulders)
spin  orbit
2
T
unpolarized quarks in
T-polarized N (Sivers)
compare
u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ )
spin-orbit correlations are T-odd correlations. Because of T-conservation they
show up in T-odd observables, such as single spin asymmetries, e.g. left-right
asymmetry in p p­ ® p X
14
TMD structures for quark PDFs
QUARKS
U
p/
f1T^
p/ g ag 5
h1^
f1
L
T
p/ g 5
g1L
h1L^
g1T
h1T , h1T^
15
gluon TMD’s without polarization
Also for gluons there are new features in TMD’s
circularly polarized
gluons in L-pol. N
g
F g mn (x, pT ) µ - gTmn xf1g (x, pT2 ) + iS LeTmn xg1L
(x, pT2 )
m ö
æ p m pn
p
mn
^g
2
T T
T
ç
÷
+
g
xh
(x,
p
) + ...
unpolarized gluons
T
1
T
ç M2
2÷
2M ø
è
in unpol. N quarks
compare
F g mn ( p)
e m ( p, l )e n * ( p, l ) = -gTmn +...
Collinear situation:
Unpolarized gluon PDFs: g(x) = f1g(x)
linearly polarized
gluons in unpol. N
(Rodrigues, M)
WG6/75 Li (STAR)
Jet and di-jet in pol. pp
Polarized gluon PDF: Dg(x) = g1Lg(x)
WG6/63 Guragain (PHENIX)
ALL results, impact on Dg
Some of the TMDs also show up in
resummation. An example are the linearly
polarized gluons (Catani, Grazzini, 2011)
WG6/316 Pisano
Linear polarization Higgs + jet prod
16
TMD structures for quark and gluon PDFs
QUARKS
U
p/
GLUONS
U
g1L
h1L^
f1T^
g1T
h1T , h1T^
-gTab
eTab
pTab
h1^g
f1g
L
T
p/ g ag 5
h1^
f1
L
T
p/ g 5
f1T^g
g1Lg
h1L^g
g1Tg
h1Tg , h1T^g
17
TMD structures for quark and gluon PFFs
QUARKS
U
p/
GLUONS
U
G1L
H1L^
D1T^
G1T
H1T , H1T^
-gTab
eTab
pTab
H1^g
D1g
L
T
p/ g ag 5
H1^
D1
L
T
p/ g 5
D1T^g
G1Lg
H1L^g
G1T
H1Tg , H1T^g
18
Requirements and possibilities to measure QT
Parton collinear momentum (lightcone fractions)  scaling variables
Parton transverse momenta appear in noncollinear phenomena (azimuthal
asymmetries)
Jet fragments: k = Kh/z + kT or kT = k – Kh/z = kjet – Kh/z
SIDIS:
q + p = k but q + x PH ≠ Kh/z
Hadrons:
p1 + p2 = k1 + k2 but x1 P1 + x2 P2 ≠ k1 + k2
Two ‘separated’ hadrons involved in a hard interaction (with scale Q), e.g.
SIDIS-like (g*H  h X or g*H  jet X), DY-like (H1H2  g* X or H1H2  dijet X
or H1H2  Higgs X), annihilation (e+e-  h1h2 X or e+e-  dijet X)
Number of pT’s in parametrization  rank of TMDs  azimuthal sine or
cosine mf asymmetry  Need for dedicated facilities at low and high energies
19
Rich phenomenology
WG6/151 Seder (COMPASS)
Charged kaon multiplicities in SIDIS
WG6/75 Li (STAR)
Jet and di-jet in pol. pp
WG6/161 Vossen
FFs at Belle
WG6/136 Diefenthaler (SeaQuest)
Polarized DY measurements
WG6/135 Sbrizzai (COMPASS)
Transverse spin asymmetries SIDIS
WG6/153 Barish (PHENIX)
Transverse Single-spin asymm.
WG6/84 Anulli
Collins Asymmetry with BaBar
WG6/139 Allada
Recent SIDIS results at JLab
WG6/22 Lyu
Collins Asymmetry at BESIII
WG6/318 Liyanage
Recent polarized DIS results at
JLab
20
TMDs and factorization
Collinear PDFs involve operators
of a given twist
x ‘measurable’
TMDs involve operators of all twist
pT in a convolution (scale and rapidity
cutoffs)
ds ~ F(x1; m )F(x2 ; m )H(Q, m )
ds ~
ò dp
1T
dp2T d ( p1T + p2T - qT ).....
..... F(x1 , p1T ; V 1 , m )F(x2 , p2T ; V 2 , m )
Impact parameter representation
including also large bT
ds ~
ò db
T
exp(iqT bT )....
...F(x1 ,bT ; V 1 , m )F(x2 ,bT ; V 2 , m )
+ .........
At small bT, large kT collinear physics
(Collins-Soper-Sterman)
WG6/33 Collins
TMD factorization and evolution
J.C. Collins, Foundations of
Perturbative QCD, Cambridge Univ.
WG6/312 Kang
TMD evolution and global analysis
21
Non-universality because of process dependent gauge links
F
q[ C ]
ij
d (x .P)d 2xT i p.x
[C ]
( x, pT ; n)  
e
P

(0)
U
j
[0,x ] i (x ) P
3
(2 )
x .n  0
TMD
path dependent gauge link
 Gauge links associated with dimension zero (not suppressed!) collinear An = A+
gluons, leading for TMD correlators to process-dependence:
SIDIS
DY
… A+ …
(resummation)
… A+ …
F[-]
F[+]
Time reversal
Belitsky, Ji, Yuan, 2003; Boer, M, Pijlman, 2003
22
Non-universality because of process dependent gauge links
[C ,C ']
Fab
(x, pT ;n) =
g
ò
d(x .P)d 2xT
(2p )3
[C ']
nb
ei p.x P U [[Cx ,0]] F na (0)U[0,
F
(x ) P
x]
x .n=0
 The TMD gluon correlators contain two links, which can have different paths.
Note that standard field displacement involves C = C’
F  (x )  U[[C,x] ] F  (x )U[[xC,] ]
 Basic (simplest) gauge links for gluon TMD correlators:
Fg[+,+]
Fg[-,-]
Fg[+,-]
Fg[-,+]
gg  H
in gg  QQ
Bomhof, M, Pijlman, 2006; Dominguez, Xiao, Yuan, 2011
23
he t ransverse moment
um of more[Ut han
one hadron
such as e.g. in t he
pT · SisT involved,
[U ]
]
[U ]
2
2
for(x,
of
TMDs
(x,be
pT impossible
) = SL g1L
pT ) −just a single
g1T T(x,
pT for
).
(12)
DY case above, Consequences
itg1smay
t oparametrizations
have
MD
a
given
hadron
M
5,6
because color get s ent angled .
linkinclude
dependence
r quarks,
t hese
not only
t helink
functcan
ions
survive
pT -intofegrat
ion,
T heGauge
correlat
ors
including
a gauge
bet hat
paramet
rizedupon
in t erms
T MD
q
7,8
2 h q (x) = δq(x), which are t he well-known collinear
(x)
=
q(x),
g
(x)
=
∆
q(x)
and
PDFs
depending
on x and pT , 1
1
quark
TMDs quark and nucleon spin) but also moment um-spin denn-spinLeading
densit ies
(involving
pT S T
⊥q
2
[U ion
]
[U
]
T
[U ]
es suchΦas
f
p
)
(unpolarized
a t ransversely
[U ] t he Sivers funct
21T (x, ⊥
2
(x, pT ; n) = f 1 (x, pT ) − f 1TT (x, pT )
+quarks
g1s (x, in
pT )γ
5
M
larized nucleon) and spin-spin-moment um densit ies such as g1T (x, p2T ) (longit uγ5 p
p P
/
/ T nucleon).
nally polarized quarks
polarized
[U ]
⊥ [U ]
⊥ [U ]
2in a t/ransversely
2 /T
+ h1T (x, pT ) γ5 ST + h1s (x, pT )
+ i h1 (x, pT )
, (11)
M naming convent ion
M in2 Ref. 9, is
T he paramet rizat ion for gluons, following t he
en by
wit h t he
spin vect
or TMDs
paramet rized as Sµ = SL P µ + STµ + M 2 SL n µ and short hand
Leading
gluon
[U ]
⊥ [U ]
pT S T
not at ions for g1s and h1s ,
g[U
]
⊥ g[U ]
µ
ν
µ
ν
2x Γ µ ν [U ] (x,pT ) = − gT f 1 (x,p2T ) + gT T
f 1T (x,p2T )
pT · ST M[U ]
[U ]
[U ] µ
2
2
ν )−
2
g
(x,
p
)
=
S
g
(x,
p
g
(x,
p
).
(12)
T
L
1L pT pTT
µ ν1s g[U ]
µ ν MpT 1T ⊥ g[UT ]
2
+ i T g1s (x,pT ) +
− gT
h1
(x,pT )
2
2
M
2M
For quarks, t hesepTinclude
tνhat
survive
{ µ ν } not only t he functpions
}
S T { µ ν }upon pT -int egrat ion,
T {µ
q
q
pT ⊥ g[U
Swhich
] hq (x) = δq(x),
g[U ]
2collinear
T
T
T + Tare t p
T well-known
and
he
1 (x) = q(x), −g1 (x) = ∆ q(x)
1
h
(x,p
)
−
h
(x,p
(13)
T
T ).
1s
1T
2
2M
4M
spin-spin densit ies (involving quark and nucleon spin) but also moment um-spin den⊥q
sit ies such as t he Sivers funct ion f 1T
(x, p2T ) (unpolarized quarks in a t ransversely
polarized nucleon) and spin-spin-moment um densit ies such as g1T (x, p2T ) (longit u24
Transverse moments  operator structure of TMD PDFs
Operator analysis for TMD functions: in analogy to Mellin moments consider
transverse moments involving gluonic operators
pTa F[±] (x, pT ;n) =
ò
d(x .P)d 2xT
(2p )3
ei p.x P y (0)U [0,±¥]iDTaU[±¥,x ]y (x ) P
calculable
T-even
Fa¶ (x) = FaD (x) - FaA (x)
FaD (x) =
D
FaG (x) = p FnFa (x,0 | x)
T-odd
T-even (gauge-invariant derivative)
FaA (x) = PV ò
ò dx Fa (x - x , x | x)
1
x .n=0
1
dx1 na
F F (x - x1 , x1 | x)
x1
1
T-odd (soft-gluon or gluonic pole, ETQS m.e.)
Efremov, Teryaev; Qiu, Sterman;
Brodsky, Hwang, Schmidt; Boer, Teryaev; Bomhof, Pijlman, M
25
TMD and collinear twist 3
There are many more links between TMD factorized results, pQCD
resummation results and collinear twist 3 results:
WG6/57 Pitonyak et al.
Transverse SSA in pp  /g X
WG6/120 Koike
Collinear twist 3 for spin asymm.
WG6/309 Metz
Twist-3 spin observables
WG6/108 Dai Lingyun
Sivers asymmetry in SIDIS
WG6/104 Granados
Left-right asymm. in chiral dynamics
WG6/315 Burkardt
Transverse Force on Quarks in DIS
26
Operator classification of TMDs according to rank
factor
TMD PDF RANK
0
1
F(x, pT2 )
[U ]
CG,c
[U ]
CGG,c
[U ]
CGGG,c
Buffing, Mukherjee, M
1
2
3
F¶ (x, pT2 )
F¶¶ (x, pT2 )
F¶¶¶ (x, pT2 )
FG,c (x, pT2 )
F{G¶},c (x, pT2 )
F{G¶¶},c (x, pT2 )
FGG,c (x, pT2 )
F{GG¶},c (x, pT2 )
FGGG,c (x, pT2 )
27
Distribution versus fragmentation functions
Operators:
Operators:
F[U ] ( p | p) ~ P | y (0)U[0,x ] y (x ) | P
D(k | k)
~ å 0 | y (x ) | K h X
out state
K h X | y (0) | 0
X
DaG (x) = p D nFa ( Z1 ,0 | Z1 ) = 0
T-even
T-odd (gluonic pole)
FaG (x) = p FnFa (x,0 | x) ¹ 0
Collins, Metz; Meissner, Metz;
Gamberg, Mukherjee, M
T-even operator combination,
but still T-odd functions!
28
Operator classification of TMDs
factor
TMD PDF RANK
0
1
F(x, pT2 )
[U ]
CG,c
1
2
F¶ (x, pT2 )
F¶¶ (x, pT2 )
F¶¶¶ (x, pT2 )
FG,c (x, pT2 )
F{G¶},c (x, pT2 )
F{G¶¶},c (x, pT2 )
FGG,c (x, pT2 )
F{GG¶},c (x, pT2 )
[U ]
CGG,c
FGGG,c (x, pT2 )
[U ]
CGGG,c
factor
TMD PFF RANK
0
1
3
D(z -1,kT2 )
Buffing, Mukherjee, M
1
D¶ (z -1 ,kT2 )
2
D¶¶ (z -1 ,kT2 )
3
D¶¶¶ (z -1,kT2 )
29
Operator classification of quark TMDs (unpolarized nucleon)
factor
QUARK TMD PDF RANK UNPOLARIZED HADRON
0
1
2
3
f1
1
CG[U ]
h1^
[U ]
CGG,c
Example: quarks in an unpolarized target are described by just 2 TMD structures;
in general rank is limited to to 2(Shadron+sparton)
T-even
Gauge link dependence:
(coupled to process!)
T-odd
[B-M function]
h1^[U ] (x, pT2 ) = CG[u]h1^ (x, pT2 )
30
Operator classification of quark TMDs (polarized nucleon)
factor
QUARK TMD PDFs RANK SPIN ½ HADRON
0
1
f1 , g1L , h1T
CG[U ]
[U ]
CGG,c
1
2
g1T , h1L^
h1T^( A)
h1^ , f1T^
h1T^( B1) , h1T^( B2)
Three pretzelocities:
Process dependence also for (T-even) pretzelocity,
[U ]
[U ]
h1T^[U ] = h1T^(1)( A) +CGG,1
h1T^( B1) +CGG,2
h1T^(1)( B2)
Buffing, Mukherjee, M
3
A : y ¶¶y = Trc é붶yy ùû
B1: Trc éëGGyy ùû
B2 : Trc éëGG ùû Trc éëyy ùû
31
Operator classification of TMDs (including trace terms)
factor
GLUON TMD PDF RANK UNPOLARIZED HADRON
0
1
f1g
1
2
3
h1g^( A)
h1g^( Bc)
[U ]
CGG,c
Note also process dependence of (T-even) linearly polarized gluons
[U ] ^( Bc)
h1g^[U ] = h1^( A) + åCGG,c
h1
c
Buffing, Mukherjee, M
32
Operator classification of TMDs (including trace terms)
factor
TMD PDF RANK
0
1
F(x, pT2 )
[U ]
CG,c
[U ]
CGG,c
1
2
3
F¶ (x, pT2 )
F¶¶ (x, pT2 )
F¶¶¶ (x, pT2 )
FG,c (x, pT2 )
F{G¶},c (x, pT2 )
F{G¶¶},c (x, pT2 )
FGG,c (x, pT2 )
F{GG¶},c (x, pT2 )
FG.G,c (x, pT2 )
[U ]
CGGG,c
FGGG,c (x, pT2 )
Trace terms affect pT width (note FG.G(x) = 0)
Boer, Buffing, M, arXiv:1503.03760
33
Operator classification of TMDs (including trace terms)
factor
GLUON TMD PDF RANK UNPOLARIZED HADRON
0
1
2
1
f1g
h1g^( A)
[U ]
CGG,c
d f1g ( Bc)
h1g^( Bc)
3
Note process dependence, not only of linearly polarized gluons but also
affecting the pT width of unpolarized gluons
[U ] ^( Bc)
h1g^[U ] = h1^( A) +CGG,c
h1
[U ]
f1g[U ] = f1g +CGG,c
d f1g ( Bc)
Boer, Buffing, M,
M arXiv:1503.03760
with df1g(Bc)(x) = 0
34
Operator classification of TMDs (including trace terms)
factor
QUARK TMD PDFs RANK SPIN ½ HADRON
0
1
CG[U ]
[U ]
CGG,c
f1 , g1L , h1T
1
2
g1T , h1L^
3
h1T^( A)
h1^ , f1T^
d f1, d g1, dh1
h1T^( B1) , h1T^( B2)
Process dependence in pT width of TMDs is due to
gluonic pole operator (e.g. Wilson loops)
[U ]
f1[U ] = f1 +CGG,c
d f1( Bc)
with df1(Bc)(x) = 0
Boer, Buffing, M, arXiv:1503.03760
35
Entanglement in processes with two initial state hadrons
Resummation of collinear gluons coupling onto external lines contribute to
gauge links
…. leading to entangled situation (Rogers, M), breaking universality
y (x 2 )
y (02 )
[ - ,0 1 ]
[x 1 , - ]
[  ,x 1 ][  ,x 2 ]
Gauge knots (Buffing, M)
[0 1 ,  ][02 ,  ]
[x 2 , - ]
[ - ,02 ]
y (x1 )
y (01 )
36
t ransverse separat ions involving collinear and t ransverse
into account
gauge links,
which
are also
leading
gluons.
In hadrons
t he
t he color
remains
ent angled
as or
ilCorrelators
description
of process
hard (e.g.
process
(e.g.
DY)
Twoininitial
state
DY)
cont
rat ed tin
1 and
given
lust
ratribut
ed inions,
Fig. is
2. illust
Bypassing
heFig.
det ails
of get
t ingby
gauge
links in t he first place, we not e t hat
at measured qT t he
∗
dσDY
∼ cont
Tr c ribut
Φ(xe1 ,tpo1Tt he
)Γ gauge
Φ(x 2 , p
2T )Γappear in
ingredient
s t hat
links
different part s of t he
1 diagram and∗ cannot be t rivially ab
=
Φ(xof1 , tphe
Φ(xcorrelat
1T )Γ
2 , p2T )Γ,
sorbed in t he definit
ion
T
MD
ors, nor can
Nc
t hey be incorporat ed by a simple redefinit ion of t he corre
Complications
transverse
momentum
ofher
twot han
lat or.
T herefore, tifhethe
name
gauge connect
ion rat
initial state hadrons is involved, resulting for DY at
gauge link is used at t his point . T he result is
measured QT in
dσDY = Tr c U−† [p2 ]Φ(x 1 , p1T )U− [p2 ]Γ ∗
× U−† [p1 ]Φ(x 2 , p2T )U− [p1 ]Γ
(2)
†
1 [− ]
∗ [− ]
=
Φ (x 1 , p1T )Γ Φ (x 2 , p2T )Γ,
Nc
This leads
color
just
ascross
for twist-3
suppressing
allto
part
s offactors
t he (part
ial)
sect ion t hat are
squared
in
collinear
DY
not of direct import ance for our purpose, e.g. t he phase
1
(xors.
, x ,qT ) =argument
f (x , sp of) Ä
f1 (xWilson
, p2T ) lines we have
spacesfact
t he
DY
1 2 As
2
N c 1 1 1T
used a not at ion wit h t he moment a p1 and p2 in square
1
^ from which
^
bracket s, merely- t 1o indicat
e
orj )t he
h
(x
,
p
)
Ä
h
(x2 correlat
, p2T )cos(2
1
1
1T
1
2
N c -1 cont ribut ions in t he form o
gauge connect ionsNreceive
c
gluon emissions. Also, in Eq. 2 t he dagger indicat es t he
Buffing, M, PRL 112 (2014), 092002
37
Attempts to provide a useful database have started
TMDlib project
http:/ / tmdlib.hepforge.org
4
38
Conclusions and outlook
TMDs extend collinear PDFs (including polarization 3 for quarks and
2 for gluons) to novel TMD PDF and PFF functions
Although operator structure includes ultimately the same operators
as collinear approach, it is a physically relevant
combination/resummation of higher twist operators that governs
transverse structure (definite rank linked to azimuthal structure)
Knowledge of operators for transverse structure is important for
interpretation, relations to twist 3 and lattice calculations.
Transverse structure for PDFs (in contrast to PFFs) requires careful
study of process dependence linked to color flow in hard process
Like spin, the transverse structure does offer valuable tools, but you
need to know how to use them!
39
Some other (related) contributions
WG6/21 Lowdon (ArXiv:1408.3233)
Spin operator decompositions
WG6/58 Courtoy, Bacchetta, Radici
Collinear Dihadron FFs
WG6/102 Prokudin et al.
Tensor charge from Collins asymm.
WG2/87 Yuan
TMDs at small x
WG6/304 Kasemets
Polarization in DPS
WG6/315 Burkardt
Transverse Force on Quarks in DIS
WG2/112 Tarasov
Rapidity evolution of gluon TMD
WG2/87 Yuan
TMDs at small-x
WG2/326 Stasto
Matching collinear and small-x fact.
WG4/88 Yuan
Soft gluon resumm for dijets
WG7/297 Zhihong
SoLID-SIDIS: future of T-spin, TMD
WG1/325 Kasemets
TMD (un)polarized gluons in H prod
40
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