TMDs in quantum chromodynamics · 2011. 3. 30. · SIDIS photon proton hadron fragmentation distribution hard scattering F (x,z,Q2)=x a,b " 1 x dxˆ xˆ " 1 z dzˆ zˆ f a # x xˆ,µ2

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


Essential ideas on factorization


SIDIS

photon

proton

hadron

Factorization


SIDIS

photon

proton

hadronfragmentation

distribution

hard scattering

Factorization


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


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


QCD without factorizationis almost useless


Universality


Universality

Drell--Yan


Universality

SIDIS

Drell--Yan


Universality

SIDISe–e+ to pions

Drell--Yan


Universality


Drell--Yan


Universality


Drell--Yan


Universality


Drell--Yan

KEY RESULT OF QCD


see also lecture of Marco Radici on Thursday

Two words on structure functions


Inclusive DIS

y

z

xlepton plane

l′l ST !S

!(l) + N(P )→ !(l′) + X


Semi-inclusive DIS

!(l) + N(P )→ !(l′) + h(Ph) + X,

y

z

x

hadron plane

lepton plane

l′l ST

Ph

Ph⊥!h

!S


Collinear factorization concepts: “tree level”


Dominant light-cone components


Dominant light-cone components

P

Ph

q

pk

z

t

n− n+


Inclusive DIS

P

q

F (x, Q2)


Inclusive DIS

P

q

F (x, Q2)


Tree level factorization

P

q

F (x, Q2) = x∑

a

∫ 1

x

dx

xfa

(x

x

)Ha

(x)

+O(

M2

Q2

)


Making use of gauge invariance

(a) (b)

(d)(c)



(a) (b)

(d)(c)

Light-cone gauge



(a) (b)

(d)(c)

Light-cone gauge



(a) (b)

(d)(c)

Light-cone gauge


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〉



2MWµν(q, P, S) ≈∑

q

e2q

12Tr

[Φ(xB , S) γ µγ+γν

].

k − l k

−k

k − l


∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

compare with:



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


∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

compare with:

k − l k

−k

k − l


Back to familiar analogy

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)



(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

Light-cone gauge



(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

Light-cone gauge



(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

(a) (b)

(d)(c)

Light-cone gauge


Φij(p, P, S) =1

(2π)4

∫d4ξ eip·ξ⟨P, S ψj(0)ψi(ξ) P, S

⟩


ψ(ξ)→ eiα(ξ) ψ(ξ)

Φij(p, P, S) =1

(2π)4


⟩

not invariant under



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


⟩

not invariant under



Φij(p, P, S) =1

(2π)4


⟩

Φij(p, P, S) =1

(2π)4


⟩

not invariant under

☞



Φij(p, P, S) =1

(2π)4


⟩

Φij(p, P, S) =1

(2π)4


⟩

not invariant under

☞



U(ξ1, ξ2)→ eiα(ξ1) U(ξ1, ξ2) e−iα(ξ2).

Φij(p, P, S) =1

(2π)4


⟩

Φij(p, P, S) =1

(2π)4


⟩

not invariant under

☞



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


⟩

Φij(p, P, S) =1

(2π)4


⟩

not invariant under

☞



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


⟩

Φij(p, P, S) =1

(2π)4


⟩

not invariant under

☞

(1 + + + +...)


Shape of the gauge link

Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S

⟩



ξ−

ξT

Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S

⟩



ξ−

ξT

ξ−

ξT

Φ(x, S) ∼⟨P, S ψ(0)U[0,∞−] U[∞−,ξ−]ψ(ξ) P, S

⟩


Gauge link


Gauge link


Gauge link


Gauge link


Gauge link


Collinear factorization: “one-loop level”


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


http://www.phys.psu.edu/~cteq/

http://www.phys.psu.edu/~cteq/

One-loop level

These diagrams have all sorts of divergences:

(v1) (v2) (v3)

(r1)

(r3)

(r3)(r2)

(r4)


One-loop level

These diagrams have all sorts of divergences:★ultraviolet

(v1) (v2) (v3)

(r1)

(r3)

(r3)(r2)

(r4)


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)


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)


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)


Cancellations in inclusive DIS

All soft divergences disappear in inclusive DIS, thanks to cancellations between real and virtual diagrams (Kinoshita-Lee-Navenberg theorem)


Factorization scale

P

q

F (x, Q2) = x∑

a

∫ 1

x

dx

xfa

(x

x, µ2

F

)Ha

(x, ln

µ2F

Q2

)


Factorization scale

P

q

F (x, Q2) = x∑

a

∫ 1

x

dx

xfa

(x

x, µ2

F

)Ha

(x, ln

µ2F

Q2

)

µF


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


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


Factorization theorem

P

q

F (x, Q2) = x∑

a

∫ 1

x

dx

xfa

(x

x, µ2

F

)Ha

(x, ln

µ2F

Q2

)


Evolution equations


Evolution equations

1. The factorization scale μF is put in “by hand” to separate perturbative from nonperturbative


Evolution equations


2. The final result for the structure function cannot depend on μF


Evolution equations



3. The dependence of the PDFs on μF can be computed (DGLAP evolution equations) if μF >> ΛQCD


Evolution equations



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


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


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

)


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


Factorization theorems



•Inclusive DISup to twist 4



P

q•Inclusive DISup to twist 4



P

q

✔•Inclusive DISup to twist 4



P

q


• Integrated SIDISup to twist 3



P

q



• SIDIS at high transverse mom.up to twist 3



P

q



• SIDIS at high transverse mom.up to twist 3 P

h

q



P

q




h

q

✔



P

q




h

q

✔✔



P

q




• SIDIS at low transverse mom.

P

h

q

✔✔



P

q




• SIDIS at low transverse mom.

P

h

q

✔✔

?Wednesday, 26 May 2010

Important messages


Important messages

1. Factorization theorems are the only rigorous way to define what are the objects we call “parton distribution functions”


Important messages


2. The intuitive idea, based on parton model and handbag diagram, of PDFs being probability densities is slightly modified by the factorization theorems.


Important messages


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


TMD factorization:“tree level”


P

h

q


P

h

q


P

h

q


P

h

q



First step to provefactorization


First step to provefactorization (at leading

twist)



∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0

Inclusive DIS



∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

∣∣∣∣ η+ = ξ+ = 0ηT = ξT


∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0

Inclusive DIS

Semi-inclusive DIS



∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

∣∣∣∣ η+ = ξ+ = 0ηT = ξT


∫ ξ−

∞−dη− A+(η) ψ(ξ)|P, S〉

∣∣∣∣ η+ = ξ+ = 0ηT = ξT = 0

☞

Inclusive DIS

Semi-inclusive DIS


Shape of gauge links

Φij(x, pT ) =∫

dξ−d2ξT

8π3eip·ξ〈P |ψj(0)U[0,ξ]ψi(ξ)|P 〉

∣∣∣∣ξ+=0



Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

SIDIS



Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

SIDIS

The “staple” gauge link



Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

ξ−

ξT

pT integrationSIDIS

The “staple” gauge link


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


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〉




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


k/− l/






∫ 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


k/− l/






∫ 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


k/− l/


☞





∫ 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


k/− l/


☞

☞Wednesday, 26 May 2010

Shapes of gauge links

Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0



Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

SIDIS



Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

Drell-Yan

ξ−

ξT

SIDIS




Φij(x, pT ) =∫

dξ−d2ξT


∣∣∣∣ξ+=0

ξ−

ξT

Drell-Yan

ξ−

ξT

ξ−

ξT

pT integration

SIDIS



Gauge links are not always identical


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


pp to hadrons ?Wednesday, 26 May 2010

Rogers, Mulders, arXiv:1001.2977


pp to hadronsNo TMD factorization!

Rogers, Mulders, arXiv:1001.2977


TMD factorization:“one-loop level”


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


No cancellations

The problem is that soft divergences do not cancel anymore and a new class of divergences (light-cone or rapidity divergences) appear


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

)]



(v1) (v2) (v3)


δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2

αsCF

4π

(− ln

µ2

λ2+ 3 ln

m2

λ2− 4

)]

Soft divergence



(v1) (v2) (v3)


δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2

αsCF

4π

(− ln

µ2

λ2+ 3 ln

m2

λ2− 4

)]

Soft divergence

Collinear divergence




δ(1− xB)δ(1− zh)δ2(Ph⊥)[1 + 2

αsCF

4π

(− ln

µ2

λ2+ 3 ln

m2

λ2− 4

)]

Soft divergence

Collinear divergence

= ⊗ ⊗⊗



= ⊗ ⊗⊗



= ⊗ ⊗⊗

Hard FF PDF Soft factor



= ⊗ ⊗⊗

= ⊗ ⊗ ⊗




= ⊗ ⊗⊗

= ⊗ ⊗ ⊗


Light-cone divergences appear



= ⊗ ⊗⊗

= ⊗ ⊗ ⊗


the light-cone regulators determine what goes in the FF, PDFs, and SF

Light-cone divergences appear


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


of the observable.

k

q

p

+

q

k

p





gluon momentum k as

kµ =

(

αxP+, βQ2

2xP+, |kT | φ

)

, (3.1)



Σ[ϕ] =∫

∞

0dα

∫

∞

0dβ

∫ 2π

0

dφ

2πϕ(x, Q2, α, β, φ) J(x, α, β)M(α, β). (3.2)


J(x, α, β) =1

16π2

1

1 + α − βΘ

(

1 − x

x− α

)

Θ(

1 −x

1 − xα − β

)

, (3.3)


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)



4


of the observable.

k

q

p

+

q

k

p





gluon momentum k as

kµ =

(

αxP+, βQ2

2xP+, |kT | φ

)

, (3.1)



Σ[ϕ] =∫

∞

0dα

∫

∞

0dβ

∫ 2π

0

dφ

2πϕ(x, Q2, α, β, φ) J(x, α, β)M(α, β). (3.2)


J(x, α, β) =1

16π2

1

1 + α − βΘ

(

1 − x

x− α

)

Θ(

1 −x

1 − xα − β

)

, (3.3)


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)



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


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)


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)


Light-cone divergences problems

fq1 (x, p2

T ) =∫

dξ−d2ξT

16π3eip·ξ〈P |ψq(0)U[0,ξ]γ

+ψq(ξ)|P 〉∣∣∣∣ξ+=0



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+ψq(ξ)|P 〉∣∣∣∣ξ+=0



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+ψq(ξ)|P 〉∣∣∣∣ξ+=0

ξ−

ξTpT integration



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+ψq(ξ)|P 〉∣∣∣∣ξ+=0

ξ−

ξT

ξ−

ξT

ξ+

pT integration



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+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



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+ψq(ξ)|P 〉∣∣∣∣ξ+=0

ξ−

ξT

ξ−

ξT

ξ+

fq1 (x, p2

T , ζ) =∫

dξ−d2ξT


[0,ξ]γ+ψq(ξ)|P 〉

∣∣∣∣ξ+=0

pT integration

?



ξ−

ξT

fq1 (x, p2

T ) =∫

dξ−d2ξT


+ψq(ξ)|P 〉∣∣∣∣ξ+=0

ξ−

ξT

ξ−

ξT

ξ+

fq1 (x, p2

T , ζ) =∫

dξ−d2ξT


[0,ξ]γ+ψq(ξ)|P 〉

∣∣∣∣ξ+=0

pT integration

pT integration ?Cherednikov, Stefanis

?


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


TMD evolution


TMD evolution

1.The light-cone regulators are put in “by hand” to separate what belongs to PDFs, FFs, SF.


TMD evolution


2.The final result for the structure function cannot depend on the regulators


TMD evolution



3.The dependence on the light-cone regulators can be computed (Collins-Soper evolution equations) in the region where the transverse momentum is >> ΛQCD


TMD evolution




4.The component of the TMDs at small transverse momentum is nonperturbative and has to be extracted from the experiments


TMD evolution




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


Documents

TMDs in quantum chromodynamics · 2011. 3. 30. · SIDIS photon proton hadron fragmentation distribution hard scattering F (x,z,Q2)=x a,b " 1 x dxˆ xˆ " 1 z dzˆ zˆ f a # x xˆ,µ2