![Page 1: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/1.jpg)
Based on:
S.M. Aybat, TCR (2011)
TCR, P.J. Mulders (2010)
See also, Foundations of Perturbative QCD, J.C. Collins.(available May 2011)
TMD Factorization,TMD Factorization, Factorization Factorization Breaking and EvolutionBreaking and Evolution
Ted C. RogersVrije Universiteit Amsterdam
DIS 2011 April 13, 2011
![Page 2: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/2.jpg)
2
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Factorization:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! ( see pre-DIS talks of Aybat, Cherednikov, Collins)
��
�!!
![Page 3: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/3.jpg)
3
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Factorization:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
��
�!!
![Page 4: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/4.jpg)
4
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization Theorems:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
Definitions dictated by requirements
for factorization!
![Page 5: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/5.jpg)
5
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization Theorems:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
![Page 6: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/6.jpg)
6
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization Theorems:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
��
�
Watch out for sign flips!
![Page 7: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/7.jpg)
7
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization Theorems:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Evolution and its implementation.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementation of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
��
�!!
Watch out for sign flips!
![Page 8: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/8.jpg)
8
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization and Universality:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Implementation and Evolution.
– Existing fixed-scale fits / no evolution.– Existing “Old fashioned” implementations of Collins-Soper-
Sterman formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
!!
![Page 9: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/9.jpg)
9
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization and Universality:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Implementation and Evolution.
– Existing fixed-scale fits with no evolution.– Existing “Old fashioned” implementations of Collins-Soper-
Sterman formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
!!
![Page 10: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/10.jpg)
10
TMD-Factorization
• TMD Parton model intuition (Drell-Yan):
Wµν =∑
f
∣∣Hf(Q)2∣∣µν
∫d2k1T d
2k2T Ff/P2(x1,k1T )Ff̄/P2(x2,k2T )×
× δ(2)(k1T + k2T − qT )
Generalized Parton Model
Leading order hard part
No evolution
![Page 11: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/11.jpg)
11
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization and Universality:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Implementation and Evolution.
– Existing fixed-scale with no evolution.– Existing “old fashion” implementations of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
Much progress! (see pre-DIS talks)
!!
![Page 12: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/12.jpg)
12
Evolved Cross Section:
• Contrast: Typical appearance of Collins-Soper-Sterman formalism:
∫ 1
x1
dx̂1x̂1C̃f/j(x1/x̂1, b∗;µ
2b , µb, g(µb))fj/P1(x̂1, µb)
∫ 1
x2
dx̂2x̂2C̃f/j(x2/x̂2, b∗;µ
2b , µb, g(µb))fj/P2(x̂2, µb)
exp
[∫ Q2
1/b2
dµ′ 2
µ′ 2
{A(αs(µ′)) ln
Q2
µ′2+ B(αs(µ′))
}]
exp
[−gk(b) ln
Q2
Q20− g1(x1, b)− g2(x2, b)
]
dσ ∼∫d2b e−ib·qT
(1985)
+ Large qT term
![Page 13: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/13.jpg)
13
Status:• Complicated issues involved in defining TMDs.
– Divergences.– Wilson lines / gauge links.– Universality / Non-universality.–
• Related: Factorization and Universality:– Semi-Inclusive deep inelastic scattering.– Drell-Yan.– e+/e- annihilation.– p + p h1 + h2 + X
• Related: Implementation and Evolution.
– Existing fixed-scale with no evolution.– Existing “old fashion” implementations of Collins-Soper-Sterman
formalism.
• How are these related? How to connect to phenomenology?
!!
Much progress! (see pre-DIS talks)
![Page 14: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/14.jpg)
14
What is needed?
• TMD Parton model intuition (Drell-Yan):
• Using newest definitions:
Wµν =∑
f
∣∣Hf(Q)2∣∣µν
∫d2k1T d
2k2T Ff/P2(x1,k1T )Ff̄/P2(x2,k2T )×
× δ(2)(k1T + k2T − qT )
Wµν =∑
f
|Hf (Q;µ)2|µν
×∫d2k1T d
2k2T Ff/P1(x1,k1T ;µ; ζ1)Ff̄/P2(x2,k2T ;µ; ζ2)
× δ(2)(k1T + k2T − qT )+ Y (Q, qT ) +O((Λ/Q)a).
![Page 15: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/15.jpg)
15
What is needed?
• TMD Parton model intuition (Drell-Yan):
• Using newest definitions:
Wµν =∑
f
∣∣Hf(Q)2∣∣µν
∫d2k1T d
2k2T Ff/P2(x1,k1T )Ff̄/P2(x2,k2T )×
× δ(2)(k1T + k2T − qT )
Wµν =∑
f
|Hf (Q;µ)2|µν
×∫d2k1T d
2k2T Ff/P1(x1,k1T ;µ; ζ1)Ff̄/P2(x2,k2T ;µ; ζ2)
× δ(2)(k1T + k2T − qT )+ Y (Q, qT ) +O((Λ/Q)a).Process dependence
in hard part Universal PDFs with evolution
![Page 16: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/16.jpg)
16
TMD PDF, Complete Definition:
Ff/P (x, b;µ; ζF ) =
+∞
−∞
−∞
+∞
ys
ys
−∞
“Unsubtracted”
Implements Subtractions/Cancellations
From Foundations of Perturbative QCD , J.C. Collins,
See also, Collins, TMD 2010 Trento Workshop
![Page 17: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/17.jpg)
17
Our Strategy:
• Use evolution to extrapolate between existing fits, to build unified fits that include evolution.
– PDFs: • Start with DY:
• Modify to match to SIDIS:
• Can supply explicit, evolved TMD PDF fit.
(Landry et al, (2003); Konychev, Nadolsky (2006))
(Schweitzer, Teckentrup, Metz (2010))
(BLNY)
(STM)
(S.M. Aybat, TCR (2011))
(For details, see Aybat talk, pre-DIS meeting.)
![Page 18: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/18.jpg)
18
Evolving TMD PDFs
(Landry et al, (2003))(Schweitzer, Teckentrup, Metz (2010))
(SIDIS) (Drell-Yan)
![Page 19: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/19.jpg)
19
Evolving TMD PDFs
JLabEnergies
TevatronEnergies
![Page 20: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/20.jpg)
20
Evolving TMD PDFs
Gaussian fit good at small kT.
![Page 21: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/21.jpg)
21
Unambiguous Hard Part
• Higher orders follow systematically from definitions:
• Drell-Yan:
–
• SIDIS
–
H = H0
(1 +
CFαsπ
(π2
2− 4))
+O(α2s)
H = H0
(1− 4CFαs
π
)+O(α2s)
Wµν = |Hf (Q;µ/Q)2|µν Ff/P1 ⊗ Ff/P2
|Hf (Q;µ/Q)2|µν =Wµν
Ff/P1 ⊗ Ff/P2
![Page 22: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/22.jpg)
22
Unambiguous Hard Part
• Definition:
• Drell-Yan:
• SIDIS
|Hf (Q; µ/Q)2|µν =
e2f |H 20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4 + π2
2
])+ O(α2s)
|Hf (Q; µ/Q)2|µν =
e2f |H20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4
])+ O(α2s)
(MS)
|Hf (Q;µ/Q)2|µν =Wµν
Ff/P1 ⊗ Ff/P2
![Page 23: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/23.jpg)
23
Unambiguous Hard Part
• Definition:
• Drell-Yan:
• SIDIS
|Hf (Q; µ/Q)2|µν =
e2f |H 20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4 + π2
2
])+ O(α2s)
|Hf (Q; µ/Q)2|µν =
e2f |H20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4
])+ O(α2s)
(MS)
|Hf (Q;µ/Q)2|µν =Wµν
Ff/P1 ⊗ Ff/P2
![Page 24: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/24.jpg)
24
Unambiguous Hard Part
• Definition:
• Drell-Yan:
• SIDIS
|Hf (Q; µ/Q)2|µν =
e2f |H 20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4 + π2
2
])+ O(α2s)
|Hf (Q; µ/Q)2|µν =
e2f |H20 |µν
(1 +
CFαs
π
[3
2ln(Q2/µ2
)− 1
2ln2(Q2/µ2
)− 4
])+ O(α2s)
(MS)
|Hf (Q;µ/Q)2|µν =Wµν
Ff/P1 ⊗ Ff/P2
Space-like photon!
![Page 25: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/25.jpg)
25
Long-Term Goal:
• Repository of new TMD fits with evolution.
• Based on well-understood operator definitions.
– Take Collins definitions.
![Page 26: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/26.jpg)
26
Agenda:
• Improve fits. Combine SIDIS, DY, e+e- in global fit. Extend to higher orders. Gaussian fits.
• Extend to polarization dependent functions (Sivers, Boer-Mulders, etc…).
• TMD gluon distribution.
• Factorization breaking??
• Updates to appear at:https://projects.hepforge.org/tmd/
![Page 27: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/27.jpg)
27
Thanks!
![Page 28: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/28.jpg)
28
Backup Slides
![Page 29: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/29.jpg)
29
Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
Naïve Factorization:
dσ = |H|2 F̃ unsub1 (y1 − (−∞))× F̃ unsub2 (+∞− y2).
![Page 30: Ted C. Rogers · 2011. 4. 12. · Based on: S.M. Aybat, TCR (2011) TCR, P.J. Mulders (2010) See also, Foundations of Perturbative QCD, J.C. Collins. (available May 2011) TMD Factorization,](https://reader035.vdocument.in/reader035/viewer/2022081623/614acb5612c9616cbc69a591/html5/thumbnails/30.jpg)
30
Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
Naïve Factorization:
dσ = |H|2 F̃ unsub1 (y1 − (−∞))× F̃ unsub2 (+∞− y2).
y = 0 y = +∞y = −∞
Soft
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Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 F̃unsub1 (y1 − (−∞))× F̃unsub2 (+∞− y2)
S̃(+∞,−∞).
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
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32
Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 F̃unsub1 (y1 − (−∞))× F̃unsub2 (+∞− y2)
S̃(+∞,−∞).
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
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Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 F̃unsub1 (y1 − (−∞))× F̃unsub2 (+∞− y2)
S̃(+∞,−∞).
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
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Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}
dσ = |H|2 F̃unsub1 (y1 − (−∞))× F̃unsub2 (+∞− y2)
S̃(+∞,−∞).
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Understanding the Definition:• Start with only the hard part factorized:
• Separate soft part:
• Multiply by:
• Rearrange factors:
dσ = |H|2 F̃unsub1 (y1 − (−∞))× F̃unsub2 (+∞− y2)
S̃(+∞,−∞).
dσ = |H|2 Funsub1 (y1 − (−∞))√S̃(+∞,−∞)
× F̃unsub2 (+∞− y2)√S̃(+∞,−∞)
.
√S̃(+∞, ys) S̃(ys,−∞)
√S̃(+∞, ys) S̃(ys,−∞)
dσ = |H|2{
F unsub1 (y1 − (−∞))√
S̃(+∞, ys)S̃(+∞,−∞)S̃(ys,−∞)
}
×{
F̃ unsub2 (+∞− y2)√
S̃(ys,−∞)S̃(+∞,−∞)S̃(+∞, ys)
}Separately Well-defined
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• Collins-Soper Equation:
–
• RG:
–
–
Evolution
∂ ln F̃ (x, bT , µ, ζ)
∂ ln√ζ
= K̃(bT ;µ)
dK̃
d lnµ= −γK(g(µ))
d ln F̃ (x, bT ;µ, ζ)
d lnµ= −γF (g(µ); ζ/µ2)
K̃(bT ;µ) =1
2
∂
∂ynlnS̃(bT ; yn,−∞)S̃(bT ; +∞, yn)
Perturbatively calculable, from definitions
Perturbatively calculable from definition at small b.
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TMD-Factorization
• TMD-factorization with consistent definitions:
• Compare with generalized parton model:
Wµν =∑
f
∣∣Hf(Q)2∣∣µν
∫d2k1T d
2k2T Ff/P2(x1,k1T )Ff̄/P2(x2,k2T )×
× δ(2)(k1T + k2T − qT )
Wµν =∑
f
|Hf (Q;µ)2|µν
×∫d2k1T d
2k2T Ff/P1(x1,k1T ;µ; ζ1)Ff̄/P2(x2,k2T ;µ; ζ2)
× δ(2)(k1T + k2T − qT )+ Y (Q, qT ) +O((Λ/Q)a).
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• After evolution:
Implementing Evolution
b∗(bT ) ≡bT√
1 + b2T /b2max
µb(bT ) ∼ 1/b∗
× exp{gj/H(x, bT ) + gK(bT ) ln
√ζ
Q0
}
× exp{ln
√ζ
µbK̃(b∗;µb) +
∫ µ
µb
dµ′
µ′
[γF (g(µ
′); 1)− ln√ζ
µ′γK(g(µ
′))
]}×
F̃f/H(x, bT , µ, ζ) =∑
j
∫ 1
x
dx̂
x̂C̃f/j(x/x̂, b∗;µb, g(µb))fj/H(x, µb)×
CSS matching procedure
A
B
C
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