tmds in quantum chromodynamics · 2011. 3. 30. · sidis photon proton hadron fragmentation...
TRANSCRIPT
TMDs in quantum chromodynamics
Alessandro Bacchetta
Wednesday, 26 May 2010
WARNING: the following slides were integrated with blackboard notes and are not completely self-consistent
Wednesday, 26 May 2010
Essential ideas on factorization
Wednesday, 26 May 2010
SIDIS
photon
proton
hadron
Factorization
Wednesday, 26 May 2010
SIDIS
photon
proton
hadronfragmentation
distribution
hard scattering
Factorization
Wednesday, 26 May 2010
SIDIS
photon
proton
hadronfragmentation
distribution
hard scattering
F (x, z,Q2) = x∑
a,b
∫ 1
x
dx
x
∫ 1
z
dz
zfa
(x
x, µ2
F
)Db
(z
z, µ2
F
)Hab
(x, z, ln
µ2F
Q2
)
Factorization
Wednesday, 26 May 2010
SIDIS
photon
proton
hadronfragmentation
distribution
hard scattering
KEY RESULT OF QCD
F (x, z,Q2) = x∑
a,b
∫ 1
x
dx
x
∫ 1
z
dz
zfa
(x
x, µ2
F
)Db
(z
z, µ2
F
)Hab
(x, z, ln
µ2F
Q2
)
Factorization
Wednesday, 26 May 2010
QCD without factorizationis almost useless
Wednesday, 26 May 2010
Universality
Wednesday, 26 May 2010
Universality
Drell--Yan
Wednesday, 26 May 2010
Universality
SIDIS
Drell--Yan
Wednesday, 26 May 2010
Universality
SIDISe–e+ to pions
Drell--Yan
Wednesday, 26 May 2010
Universality
SIDISe–e+ to pions
Drell--Yan
Wednesday, 26 May 2010
Universality
SIDISe–e+ to pions
Drell--Yan
Wednesday, 26 May 2010
Universality
SIDISe–e+ to pions
Drell--Yan
KEY RESULT OF QCD
Wednesday, 26 May 2010
see also lecture of Marco Radici on Thursday
Two words on structure functions
Wednesday, 26 May 2010
Inclusive DIS
y
z
xlepton plane
l′l ST !S
!(l) + N(P )→ !(l′) + X
Wednesday, 26 May 2010
Semi-inclusive DIS
!(l) + N(P )→ !(l′) + h(Ph) + X,
y
z
x
hadron plane
lepton plane
l′l ST
Ph
Ph⊥!h
!S
Wednesday, 26 May 2010
Collinear factorization concepts: “tree level”
Wednesday, 26 May 2010
Dominant light-cone components
Wednesday, 26 May 2010
Dominant light-cone components
P
Ph
q
pk
z
t
n− n+
Wednesday, 26 May 2010
Inclusive DIS
P
q
F (x, Q2)
Wednesday, 26 May 2010
Inclusive DIS
P
q
F (x, Q2)
Wednesday, 26 May 2010
Tree level factorization
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x
)Ha
(x)
+O(
M2
Q2
)
Wednesday, 26 May 2010
Making use of gauge invariance
(a) (b)
(d)(c)
Wednesday, 26 May 2010
Making use of gauge invariance
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Making use of gauge invariance
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Making use of gauge invariance
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Birth of the gauge link
k − l k
−k
k − l
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
Wednesday, 26 May 2010
Birth of the gauge link
2MWµν(q, P, S) ≈∑
q
e2q
12Tr
[Φ(xB , S) γ µγ+γν
].
k − l k
−k
k − l
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
compare with:
Wednesday, 26 May 2010
Birth of the gauge link
2MWµν(q, P, S) ≈∑
q
e2q
12Tr
[Φ(xB , S) γ µγ+γν
].
ξ−
ξT
Φ(a)(x, S) ∼⟨P, S ψ(0) (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ) P, S
⟩
k − l k
−k
k − l
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
compare with:
k − l k
−k
k − l
Wednesday, 26 May 2010
Back to familiar analogy
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
Wednesday, 26 May 2010
Back to familiar analogy
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Back to familiar analogy
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Back to familiar analogy
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
(a) (b)
(d)(c)
Light-cone gauge
Wednesday, 26 May 2010
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
☞
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
☞
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
U(ξ1, ξ2)→ eiα(ξ1) U(ξ1, ξ2) e−iα(ξ2).
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
☞
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
U(ξ1, ξ2)→ eiα(ξ1) U(ξ1, ξ2) e−iα(ξ2).
U[a,b] = P exp[−ig
∫ b
adηµAµ(η)
]
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
☞
Wednesday, 26 May 2010
ψ(ξ)→ eiα(ξ) ψ(ξ)
U(ξ1, ξ2)→ eiα(ξ1) U(ξ1, ξ2) e−iα(ξ2).
U[a,b] = P exp[−ig
∫ b
adηµAµ(η)
]
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)U[0,ξ] ψi(ξ) P, S
⟩
Φij(p, P, S) =1
(2π)4
∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S
⟩
not invariant under
☞
(1 + + + +...)
Wednesday, 26 May 2010
Shape of the gauge link
Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S
⟩
Wednesday, 26 May 2010
Shape of the gauge link
ξ−
ξT
Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S
⟩
Wednesday, 26 May 2010
Shape of the gauge link
ξ−
ξT
ξ−
ξT
Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S
⟩
Wednesday, 26 May 2010
Gauge link
Wednesday, 26 May 2010
Gauge link
Wednesday, 26 May 2010
Gauge link
Wednesday, 26 May 2010
Gauge link
Wednesday, 26 May 2010
Gauge link
Wednesday, 26 May 2010
Collinear factorization: “one-loop level”
Wednesday, 26 May 2010
Literature
1. Fields, “Applications of perturbative QCD”
2. Handbook of Perturbative QCD, CTEQ, http://www.phys.psu.edu/~cteq/
3. Collins, Soper, Sterman (1988), hep-ph/0409313
Wednesday, 26 May 2010
One-loop level
These diagrams have all sorts of divergences:
(v1) (v2) (v3)
(r1)
(r3)
(r3)(r2)
(r4)
Wednesday, 26 May 2010
One-loop level
These diagrams have all sorts of divergences:★ultraviolet
(v1) (v2) (v3)
(r1)
(r3)
(r3)(r2)
(r4)
Wednesday, 26 May 2010
One-loop level
These diagrams have all sorts of divergences:★ultraviolet★collinear (if gluon and quark mass → 0)
(v1) (v2) (v3)
(r1)
(r3)
(r3)(r2)
(r4)
Wednesday, 26 May 2010
One-loop level
These diagrams have all sorts of divergences:★ultraviolet★collinear (if gluon and quark mass → 0)★soft (if gluon mass → 0)
(v1) (v2) (v3)
(r1)
(r3)
(r3)(r2)
(r4)
Wednesday, 26 May 2010
A general rule
If you integrate over everything (total cross section), all divergences disappear
The more you integrate, the more you cancel divergences. For instance, the total cross section is free of any divergence (infrared safe)
Wednesday, 26 May 2010
Cancellations in inclusive DIS
All soft divergences disappear in inclusive DIS, thanks to cancellations between real and virtual diagrams (Kinoshita-Lee-Navenberg theorem)
Wednesday, 26 May 2010
Factorization scale
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x, µ2
F
)Ha
(x, ln
µ2F
Q2
)
Wednesday, 26 May 2010
Factorization scale
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x, µ2
F
)Ha
(x, ln
µ2F
Q2
)
µF
Wednesday, 26 May 2010
Factorization scale
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x, µ2
F
)Ha
(x, ln
µ2F
Q2
)
µF
The factorization scale determines how much we put in the PDF and how much in the hard scattering
∫ µF
λd2lT dl+
1l2T
Wednesday, 26 May 2010
Factorization scale
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x, µ2
F
)Ha
(x, ln
µ2F
Q2
)
µF
The factorization scale determines how much we put in the PDF and how much in the hard scattering
∫ µF
λd2lT dl+
1l2T
Wednesday, 26 May 2010
Factorization theorem
P
q
F (x, Q2) = x∑
a
∫ 1
x
dx
xfa
(x
x, µ2
F
)Ha
(x, ln
µ2F
Q2
)
Wednesday, 26 May 2010
Evolution equations
Wednesday, 26 May 2010
Evolution equations
1. The factorization scale μF is put in “by hand” to separate perturbative from nonperturbative
Wednesday, 26 May 2010
Evolution equations
1. The factorization scale μF is put in “by hand” to separate perturbative from nonperturbative
2. The final result for the structure function cannot depend on μF
Wednesday, 26 May 2010
Evolution equations
1. The factorization scale μF is put in “by hand” to separate perturbative from nonperturbative
2. The final result for the structure function cannot depend on μF
3. The dependence of the PDFs on μF can be computed (DGLAP evolution equations) if μF >> ΛQCD
Wednesday, 26 May 2010
Evolution equations
1. The factorization scale μF is put in “by hand” to separate perturbative from nonperturbative
2. The final result for the structure function cannot depend on μF
3. The dependence of the PDFs on μF can be computed (DGLAP evolution equations) if μF >> ΛQCD
4.The PDFs at a low scale are nonperturbative and have to be extracted from the experiments
Wednesday, 26 May 2010
Transverse-momentum-integrated SIDIS
P
h
q
F (x, z,Q2) = x∑
a,b
∫ 1
x
dx
x
∫ 1
z
dz
zfa
(x
x, µ2
F
)Db
(z
z, µ2
F
)Hab
(x, z, ln
µ2F
Q2
)
analogous to theorems for Drell-Yan or e+e− annihilation, see previous references
Wednesday, 26 May 2010
Integrated SIDIS: tree level
P
h
q
Hab(0)UU,T (x) = e2
b δab δ(1− x)δ(1− z)
Ha(0)UU,L(x) = 0
F (x, z,Q2) = x∑
a,b
∫ 1
x
dx
x
∫ 1
z
dz
zfa
(x
x, µ2
F
)Db
(z
z, µ2
F
)Hab
(x, z, ln
µ2F
Q2
)
Wednesday, 26 May 2010
SIDIS at high transverse momentum
P
h
q
F (x, z,Q2) =1
Q2z2x
∑
a,b
∫ 1
x
dx
x
∫ 1
z
dz
zδ( P 2
h⊥Q2z2
− (1− x)(1− z)xz
)
× fa(x
x, µ2
F
)Db
(z
z, µ2
F
)H ′
ab
(x, z, ln
µ2F
Q2
)
Starts at order αs
Wednesday, 26 May 2010
Factorization theorems
Wednesday, 26 May 2010
Factorization theorems
•Inclusive DISup to twist 4
Wednesday, 26 May 2010
Factorization theorems
P
q•Inclusive DISup to twist 4
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3 P
h
q
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3 P
h
q
✔
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3 P
h
q
✔✔
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3
• SIDIS at low transverse mom.
P
h
q
✔✔
Wednesday, 26 May 2010
Factorization theorems
P
q
✔•Inclusive DISup to twist 4
• Integrated SIDISup to twist 3
• SIDIS at high transverse mom.up to twist 3
• SIDIS at low transverse mom.
P
h
q
✔✔
?Wednesday, 26 May 2010
Important messages
Wednesday, 26 May 2010
Important messages
1. Factorization theorems are the only rigorous way to define what are the objects we call “parton distribution functions”
Wednesday, 26 May 2010
Important messages
1. Factorization theorems are the only rigorous way to define what are the objects we call “parton distribution functions”
2. The intuitive idea, based on parton model and handbag diagram, of PDFs being probability densities is slightly modified by the factorization theorems.
Wednesday, 26 May 2010
Important messages
1. Factorization theorems are the only rigorous way to define what are the objects we call “parton distribution functions”
2. The intuitive idea, based on parton model and handbag diagram, of PDFs being probability densities is slightly modified by the factorization theorems.
3. What is important is that the PDFs are nonperturbative objects, they describe the partonic structure of the nucleon, they can be extracted from experiments
Wednesday, 26 May 2010
TMD factorization:“tree level”
Wednesday, 26 May 2010
P
h
q
Wednesday, 26 May 2010
P
h
q
Wednesday, 26 May 2010
P
h
q
Wednesday, 26 May 2010
P
h
q
Wednesday, 26 May 2010
Wednesday, 26 May 2010
First step to provefactorization
Wednesday, 26 May 2010
First step to provefactorization (at leading
twist)
Wednesday, 26 May 2010
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0
Inclusive DIS
Wednesday, 26 May 2010
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣ η+ = ξ+ = 0ηT = ξT
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0
Inclusive DIS
Semi-inclusive DIS
Wednesday, 26 May 2010
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣ η+ = ξ+ = 0ηT = ξT
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0
☞
Inclusive DIS
Semi-inclusive DIS
Wednesday, 26 May 2010
Shape of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
Wednesday, 26 May 2010
Shape of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
SIDIS
Wednesday, 26 May 2010
Shape of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
SIDIS
The “staple” gauge link
Wednesday, 26 May 2010
Shape of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
ξ−
ξT
pT integrationSIDIS
The “staple” gauge link
Wednesday, 26 May 2010
Final/initial state interactions
P
k − Pk − l − P
k − l
q
q − k
k − l − q
l
P
k − Pk − l − P
q
l
k − l
p − l
k
SIDIS Drell-Yan
Wednesday, 26 May 2010
Gauge link in Drell-Yan
Collins, PLB 536 (02)
k − l k
−k
k − l
2MW (a)µν ∼
∫d4l
∫d4η
(2π)4eil·(η−ξ)〈P, S|ψ(0)γµγ+γα
k/− l/
(k − l)2 + iεγνgAα(η)ψ(ξ)|P, S〉
Wednesday, 26 May 2010
Gauge link in Drell-Yan
Collins, PLB 536 (02)
ik/− l/ + m
(k − l)2 −m2 + iε≈ i
−(−k)−γ+
2l+(−k)− + iε≈ i
2γ+
−l+−iε
k − l k
−k
k − l
2MW (a)µν ∼
∫d4l
∫d4η
(2π)4eil·(η−ξ)〈P, S|ψ(0)γµγ+γα
k/− l/
(k − l)2 + iεγνgAα(η)ψ(ξ)|P, S〉
Wednesday, 26 May 2010
Gauge link in Drell-Yan
Collins, PLB 536 (02)
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
−∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣∣η+=0; ηT =ξT
ik/− l/ + m
(k − l)2 −m2 + iε≈ i
−(−k)−γ+
2l+(−k)− + iε≈ i
2γ+
−l+−iε
k − l k
−k
k − l
2MW (a)µν ∼
∫d4l
∫d4η
(2π)4eil·(η−ξ)〈P, S|ψ(0)γµγ+γα
k/− l/
(k − l)2 + iεγνgAα(η)ψ(ξ)|P, S〉
Wednesday, 26 May 2010
Gauge link in Drell-Yan
Collins, PLB 536 (02)
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
−∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣∣η+=0; ηT =ξT
ik/− l/ + m
(k − l)2 −m2 + iε≈ i
−(−k)−γ+
2l+(−k)− + iε≈ i
2γ+
−l+−iε
k − l k
−k
k − l
2MW (a)µν ∼
∫d4l
∫d4η
(2π)4eil·(η−ξ)〈P, S|ψ(0)γµγ+γα
k/− l/
(k − l)2 + iεγνgAα(η)ψ(ξ)|P, S〉
☞
Wednesday, 26 May 2010
Gauge link in Drell-Yan
Collins, PLB 536 (02)
2MW (a)µν ∼ 〈P, S|ψ(0) γµγ+ γν (−ig)
∫ ξ−
−∞−dη− A+(η) ψ(ξ)|P, S〉
∣∣∣∣∣η+=0; ηT =ξT
ik/− l/ + m
(k − l)2 −m2 + iε≈ i
−(−k)−γ+
2l+(−k)− + iε≈ i
2γ+
−l+−iε
k − l k
−k
k − l
2MW (a)µν ∼
∫d4l
∫d4η
(2π)4eil·(η−ξ)〈P, S|ψ(0)γµγ+γα
k/− l/
(k − l)2 + iεγνgAα(η)ψ(ξ)|P, S〉
☞
☞Wednesday, 26 May 2010
Shapes of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
Wednesday, 26 May 2010
Shapes of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
SIDIS
Wednesday, 26 May 2010
Shapes of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
Drell-Yan
ξ−
ξT
SIDIS
Collins, PLB 536 (02)
Wednesday, 26 May 2010
Shapes of gauge links
Φij(x, pT ) =∫
dξ−d2ξT
8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉
∣∣∣∣ξ+=0
ξ−
ξT
Drell-Yan
ξ−
ξT
ξ−
ξT
pT integration
SIDIS
Collins, PLB 536 (02)
Wednesday, 26 May 2010
Gauge links are not always identical
Wednesday, 26 May 2010
Collins, PLB 536 (02)Bomhof, Mulders, Pijlman, PLB 596 (04) A.B., Bomhof, Mulders, Pijlman, PRD 72 (05)Collins, Qiu, PRD 75 (07)Vogelsang, Yuan, PRD76 (07)
GeneralizedFactorization (factorizationwithout universality)
Gauge links are not always identical
Wednesday, 26 May 2010
pp to hadrons ?Wednesday, 26 May 2010
Rogers, Mulders, arXiv:1001.2977
Wednesday, 26 May 2010
pp to hadronsNo TMD factorization!
Rogers, Mulders, arXiv:1001.2977
Wednesday, 26 May 2010
TMD factorization:“one-loop level”
Wednesday, 26 May 2010
TMD factorization: relevant literature
1. Collins, Soper, NPB 193 (81)
2. Collins, Soper, Sterman, NPB 250 (85)
3. Collins, Acta Phys. Polon. B34 (03)
4. Ji, Ma, Yuan, PRD 71 (05)
5. Collins, Rogers, Stasto, PRD 77 (08)
6. Collins, arXiv:0808.2665 [hep-ph]
7. Cherednikov, Stefanis
Wednesday, 26 May 2010
No cancellations
The problem is that soft divergences do not cancel anymore and a new class of divergences (light-cone or rapidity divergences) appear
Wednesday, 26 May 2010
Factorizing soft divergences
(v1) (v2) (v3)
μ ultraviolet cutoffm quark massλ gluon mass
δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2
αsCF
4π
(− ln
µ2
λ2+ 3 ln
m2
λ2− 4
)]
Wednesday, 26 May 2010
Factorizing soft divergences
(v1) (v2) (v3)
μ ultraviolet cutoffm quark massλ gluon mass
δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2
αsCF
4π
(− ln
µ2
λ2+ 3 ln
m2
λ2− 4
)]
Soft divergence
Wednesday, 26 May 2010
Factorizing soft divergences
(v1) (v2) (v3)
μ ultraviolet cutoffm quark massλ gluon mass
δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2
αsCF
4π
(− ln
µ2
λ2+ 3 ln
m2
λ2− 4
)]
Soft divergence
Collinear divergence
Wednesday, 26 May 2010
Factorizing soft divergences
μ ultraviolet cutoffm quark massλ gluon mass
δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2
αsCF
4π
(− ln
µ2
λ2+ 3 ln
m2
λ2− 4
)]
Soft divergence
Collinear divergence
= ⊗ ⊗⊗
Wednesday, 26 May 2010
Factorizing soft divergences
= ⊗ ⊗⊗
Wednesday, 26 May 2010
Factorizing soft divergences
= ⊗ ⊗⊗
Hard FF PDF Soft factor
Wednesday, 26 May 2010
Factorizing soft divergences
= ⊗ ⊗⊗
= ⊗ ⊗ ⊗
Hard FF PDF Soft factor
Wednesday, 26 May 2010
Factorizing soft divergences
= ⊗ ⊗⊗
= ⊗ ⊗ ⊗
Hard FF PDF Soft factor
Light-cone divergences appear
Wednesday, 26 May 2010
Factorizing soft divergences
= ⊗ ⊗⊗
= ⊗ ⊗ ⊗
Hard FF PDF Soft factor
the light-cone regulators determine what goes in the FF, PDFs, and SF
Light-cone divergences appear
Wednesday, 26 May 2010
parton density. Thus ϕ contains all the infrared sensitive and nonperturbative parts
of the observable.
k
q
p
+
q
k
p
Figure 1: DIS with quark-induced hard scattering.
We work in the γ∗ + hadron reference frame, and we use light-front coordinatesvµ = (v+, v−,vT ) with v± = (v0 ± v3)/
√2. The hadron and photon momenta are
P µ = (P+, m2/2P+, 0T ) and qµ = (−xP+, Q2/2xP+, 0T ). Then we parameterize the
gluon momentum k as
kµ =
(
αxP+, βQ2
2xP+, |kT | φ
)
, (3.1)
where φ is a unit transverse vector at azimuthal angle φ.
For our calculation, the external partons are on-shell, and the incoming quark phas zero transverse momentum, so that Σ can be written as follows:
Σ[ϕ] =∫
∞
0dα
∫
∞
0dβ
∫ 2π
0
dφ
2πϕ(x, Q2, α, β, φ) J(x, α, β)M(α, β). (3.2)
Here, J is the Jacobian factor
J(x, α, β) =1
16π2
1
1 + α − βΘ
(
1 − x
x− α
)
Θ(
1 −x
1 − xα − β
)
, (3.3)
and M is the next-to-leading-order matrix element for γ∗q obtained by contractingthe photon Lorentz indices with the projector corresponding to the structure function
F2 [16]
M = 4 e2q g2
s CF M(α, β) , M(α, β) = (1− β)2 1 + (1 + α − β)2
α β (1 + α − β)+ 2 + 6
(1 − β)2
1 + α − β.
(3.4)
The physical region for α, β is the interior of the triangle in Fig. 2.Standard arguments [17] determine the infrared sensitive regions contributing
to the leading power behavior of Σ[ϕ], which are located on Fig. 2 as follows: Theregion in which the gluon is collinear to the initial state is a neighborhood of the axis
4
parton density. Thus ϕ contains all the infrared sensitive and nonperturbative parts
of the observable.
k
q
p
+
q
k
p
Figure 1: DIS with quark-induced hard scattering.
We work in the γ∗ + hadron reference frame, and we use light-front coordinatesvµ = (v+, v−,vT ) with v± = (v0 ± v3)/
√2. The hadron and photon momenta are
P µ = (P+, m2/2P+, 0T ) and qµ = (−xP+, Q2/2xP+, 0T ). Then we parameterize the
gluon momentum k as
kµ =
(
αxP+, βQ2
2xP+, |kT | φ
)
, (3.1)
where φ is a unit transverse vector at azimuthal angle φ.
For our calculation, the external partons are on-shell, and the incoming quark phas zero transverse momentum, so that Σ can be written as follows:
Σ[ϕ] =∫
∞
0dα
∫
∞
0dβ
∫ 2π
0
dφ
2πϕ(x, Q2, α, β, φ) J(x, α, β)M(α, β). (3.2)
Here, J is the Jacobian factor
J(x, α, β) =1
16π2
1
1 + α − βΘ
(
1 − x
x− α
)
Θ(
1 −x
1 − xα − β
)
, (3.3)
and M is the next-to-leading-order matrix element for γ∗q obtained by contractingthe photon Lorentz indices with the projector corresponding to the structure function
F2 [16]
M = 4 e2q g2
s CF M(α, β) , M(α, β) = (1− β)2 1 + (1 + α − β)2
α β (1 + α − β)+ 2 + 6
(1 − β)2
1 + α − β.
(3.4)
The physical region for α, β is the interior of the triangle in Fig. 2.Standard arguments [17] determine the infrared sensitive regions contributing
to the leading power behavior of Σ[ϕ], which are located on Fig. 2 as follows: Theregion in which the gluon is collinear to the initial state is a neighborhood of the axis
4
parton density. Thus ϕ contains all the infrared sensitive and nonperturbative parts
of the observable.
k
q
p
+
q
k
p
Figure 1: DIS with quark-induced hard scattering.
We work in the γ∗ + hadron reference frame, and we use light-front coordinatesvµ = (v+, v−,vT ) with v± = (v0 ± v3)/
√2. The hadron and photon momenta are
P µ = (P+, m2/2P+, 0T ) and qµ = (−xP+, Q2/2xP+, 0T ). Then we parameterize the
gluon momentum k as
kµ =
(
αxP+, βQ2
2xP+, |kT | φ
)
, (3.1)
where φ is a unit transverse vector at azimuthal angle φ.
For our calculation, the external partons are on-shell, and the incoming quark phas zero transverse momentum, so that Σ can be written as follows:
Σ[ϕ] =∫
∞
0dα
∫
∞
0dβ
∫ 2π
0
dφ
2πϕ(x, Q2, α, β, φ) J(x, α, β)M(α, β). (3.2)
Here, J is the Jacobian factor
J(x, α, β) =1
16π2
1
1 + α − βΘ
(
1 − x
x− α
)
Θ(
1 −x
1 − xα − β
)
, (3.3)
and M is the next-to-leading-order matrix element for γ∗q obtained by contractingthe photon Lorentz indices with the projector corresponding to the structure function
F2 [16]
M = 4 e2q g2
s CF M(α, β) , M(α, β) = (1− β)2 1 + (1 + α − β)2
α β (1 + α − β)+ 2 + 6
(1 − β)2
1 + α − β.
(3.4)
The physical region for α, β is the interior of the triangle in Fig. 2.Standard arguments [17] determine the infrared sensitive regions contributing
to the leading power behavior of Σ[ϕ], which are located on Fig. 2 as follows: Theregion in which the gluon is collinear to the initial state is a neighborhood of the axis
4!
"
(1-x)/x
1
Figure 2: The phase space of Eq. (3.2) in the α,β plane.
β = 0, the region in which the gluon is collinear to the final state is a neighborhoodof the axis α = 0, and the soft region is a neighborhood of the origin α = 0, β = 0.
The truly hard region lies away from the α = 0 and β = 0 axes.To obtain a decomposition for Σ of the type of Eq. (2.3), we now employ the
technique of Ref. [14]. This generalizes the R-operation techniques of renormal-ization. (See Ref. [18] for a related approach.) To ensure that the procedure is
gauge-invariant, each of the terms in the right hand side of Eq. (2.3) is constructedfrom matrix elements involving Wilson line operators,
VI(n) = P exp(
ig∫ 0
−∞
dy n · A(y n))
, VF (n) = P exp(
ig∫ +∞
0dy n · A(y n)
)
,
(3.5)with suitable directions n for the lines. Evolution equations in n enable one to
connect the results corresponding to different directions [12, 13, 14]. We define light-like vectors p = (1, 0, 0T ), p′ = (0, 1, 0T ). We will also use non-lightlike vectorsu = (u+, u−, 0T ), u′ = (u′+, u′−, 0T ), all of whose components are positive. It is
convenient to define η = (2x2P+2/Q2)u−/u+, and η′ = (Q2/2x2P+2)u′+/u′−.As in [14], we start with the smallest region, the soft region α, β → 0, and
determine the corresponding contribution to the matrix element (3.4):
MS(α, β) =2
αβ−
2
(α + η′β) β−
2
α (β + ηα). (3.6)
Observe that the first term in the right hand side of this formula is just obtained
by taking the soft approximation to Eq. (3.4). It can be thought of as the one-loop contribution to the square of a vacuum–to–gluon matrix element of a product
of eikonal Wilson lines taken along lightlike directions p, p′ [14]. This first termreproduces the behavior of the matrix element M when α and β simultaneouslyapproach zero. But there are also logarithms in its integral associated with the
collinear regions where α/β or β/α go to zero. The subtractions provided by the othertwo terms conveniently cancel these regions. They can be derived from operators
analogous to those for the first term, except for replacing one of the lightlike eikonallines by a line along a non-lightlike direction. In particular, the second term subtracts
5
Collins, Hautmann, hep-ph/0009286
Wednesday, 26 May 2010
TMD factorizationCollins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 71 (05)
q
P
h
FUU,T (x, z, P 2h⊥, Q2) = C′
[f1D1
]
= H(Q2, µ2, ζ, ζh)∫
d2pT d2kT d2lT δ(2)(pT − kT + lT − P h⊥/z
)
x∑
a
e2a fa
1 (x, p2T , µ2, ζ) Da
1(z, k2T , µ2, ζh)U(l2T , µ2, ζζh)
Wednesday, 26 May 2010
TMD factorizationCollins, Soper, NPB 193 (81)Ji, Ma, Yuan, PRD 71 (05)
TMD PDF TMD FF Soft factorHard part
q
P
h
FUU,T (x, z, P 2h⊥, Q2) = C′
[f1D1
]
= H(Q2, µ2, ζ, ζh)∫
d2pT d2kT d2lT δ(2)(pT − kT + lT − P h⊥/z
)
x∑
a
e2a fa
1 (x, p2T , µ2, ζ) Da
1(z, k2T , µ2, ζh)U(l2T , µ2, ζζh)
Wednesday, 26 May 2010
Light-cone divergences problems
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
ξ−
ξTpT integration
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
ξ−
ξT
ξ−
ξT
ξ+
pT integration
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
ξ−
ξT
ξ−
ξT
ξ+
fq1 (x, p2
T , ζ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)Uζ
[0,ξ]γ+ψq(ξ)|P 〉
∣∣∣∣ξ+=0
pT integration
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
ξ−
ξT
ξ−
ξT
ξ+
fq1 (x, p2
T , ζ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)Uζ
[0,ξ]γ+ψq(ξ)|P 〉
∣∣∣∣ξ+=0
pT integration
?
Wednesday, 26 May 2010
Light-cone divergences problems
ξ−
ξT
fq1 (x, p2
T ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)U[0,ξ]γ
+ψq(ξ)|P 〉∣∣∣∣ξ+=0
ξ−
ξT
ξ−
ξT
ξ+
fq1 (x, p2
T , ζ) =∫
dξ−d2ξT
16π3eip·ξ〈P |ψq(0)Uζ
[0,ξ]γ+ψq(ξ)|P 〉
∣∣∣∣ξ+=0
pT integration
pT integration ?Cherednikov, Stefanis
?
Wednesday, 26 May 2010
TMD factorization
1. TMD factorization at the one-loop level has been proven in the work of Ji, Yuan, and Ma, extending the earlier work of Collins, Soper, Sterman, etc.
2. Factorization should work for SIDIS, Drell-Yan, and e+e− annihilation
3. The extension to all order is probably just a conjecture
4. Some subtleties have been pointed out by Collins, but I am not aware of any statement that says that the work of Ji, Yuan, and Ma is wrong
Wednesday, 26 May 2010
TMD evolution
Wednesday, 26 May 2010
TMD evolution
1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.
Wednesday, 26 May 2010
TMD evolution
1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.
2.The final result for the structure function cannot depend on the regulators
Wednesday, 26 May 2010
TMD evolution
1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.
2.The final result for the structure function cannot depend on the regulators
3.The dependence on the light-cone regulators can be computed (Collins-Soper evolution equations) in the region where the transverse momentum is >> ΛQCD
Wednesday, 26 May 2010
TMD evolution
1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.
2.The final result for the structure function cannot depend on the regulators
3.The dependence on the light-cone regulators can be computed (Collins-Soper evolution equations) in the region where the transverse momentum is >> ΛQCD
4.The component of the TMDs at small transverse momentum is nonperturbative and has to be extracted from the experiments
Wednesday, 26 May 2010
TMD evolution
1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.
2.The final result for the structure function cannot depend on the regulators
3.The dependence on the light-cone regulators can be computed (Collins-Soper evolution equations) in the region where the transverse momentum is >> ΛQCD
4.The component of the TMDs at small transverse momentum is nonperturbative and has to be extracted from the experiments
5.Everything is done in b space
Wednesday, 26 May 2010