Hadron Masses and Factorization (in DIS) · 2018-09-24 · Hadron Masses and Factorization (in DIS) Jefferson Lab/Old Dominion University Ted Rogers Quark-Hadron Duality 2018, James

Hadron Masses and Factorization

(in DIS)

JeffersonLab/OldDominionUniversity

TedRogers

Quark-Hadron Duality 2018, James Madison University, Sept 24 2018 1

2

Hadronic vs. Partonic Degrees of Freedom

•  Q≈1GeV,largex.

•  Approachkinematicalissuesintermsofwhattheyrevealaboutunderlyingdegreesoffreedom.

Two Questions

•  Whatarekinematicaltargetmassapproximations?

•  When/howdotheymatter?

3

4

Target Mass Corrections •  LargenumberofTMCformalisms:

– OPEbased

– Feynmangraphbased

– Highertwist

– Standardfactorization(Aivazis,Olness,Tung)

Brady,Accardi,Hobbs,MelnitchoukPHYSICALREVIEWD84,074008(2011)

Standard setup

5

•  Definitionofacrosssection

33

Pb=

✓Q

xN

p2,xNM2

p2Q

,0T

◆(534)

W (P, q)µ⌫ =

Z1

xBj

d⇠

⇠f(⇠;µ/m,↵s(µ))W (xBj/⇠;µ/Q,↵s(µ))

µ⌫ . (535)

F1 =e2f

2�(x/⇠ � 1)

F2 = e2f�(x/⇠ � 1) (536)

+O

✓m2

Q2

◆(537)

F (1)

1=

g2e2f

32⇡3

⇣1�⇠

k2T

� ⇠(2⇠2�6⇠+3)

Q2(1�⇠)2

⌘

q1� 4⇠k2

TQ2(1�⇠)

(538)

T1T2F(1)

1=

g2e2f

32⇡3

(1� ⇠)

k2T

. (539)

g2e2f

32⇡2µ2✏S✏

Z 1

k2cut

dk2T(k2

T)�✏

(1� ⇠)

k2T

�g2e2

f

32⇡2(1� ⇠)

S✏

✏(540)

g2e2f

32⇡2

Zµ2

k2cut

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2

Zµ2

k2max

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2(1� ⇠) ln

µ2

k2max

(541)

F (1)

1⌦ f (0)

=g2e2

f

32⇡2

Zk2max

0

dk2T

0

@(1� xBj)(1�

q1� k

2T

k2max

)

q1� k

2T

k2max

k2T

�xBj(2x2

Bj� 6xBj + 3)

Q2(1� xBj)2

q1� k

2T

k2max

1

A�g2e2

f

32⇡2(1� xBj) ln

µ2

k2max

(542)

=g2e2

f

32⇡2

"2(1� xBj) ln (2)�

2x2

Bj� 6xBj + 3

2(1� xBj)

#�

g2e2f

32⇡2(1� xBj) ln

µ2

k2max

(543)

d� =|Me,P!N |2

2�(s,m2e,M2)1/2

d3p1

(2⇡)32E1

d3p2

(2⇡)32E2

· · ·⇥ d3pN

(2⇡)32EN

(2⇡)4�(4) P + l �

NX

i=1

pi

!(544)

Acknowledgments

This work was supported by...

33

Pb=

✓Q

xN

p2,xNM2

p2Q

,0T

◆(534)

W (P, q)µ⌫ =

Z1

xBj

d⇠

⇠f(⇠;µ/m,↵s(µ))W (xBj/⇠;µ/Q,↵s(µ))

µ⌫ . (535)

F1 =e2f

2�(x/⇠ � 1)

F2 = e2f�(x/⇠ � 1) (536)

+O

✓m2

Q2

◆(537)

F (1)

1=

g2e2f

32⇡3

⇣1�⇠

k2T

� ⇠(2⇠2�6⇠+3)

Q2(1�⇠)2

⌘

q1� 4⇠k2

TQ2(1�⇠)

(538)

T1T2F(1)

1=

g2e2f

32⇡3

(1� ⇠)

k2T

. (539)

g2e2f

32⇡2µ2✏S✏

Z 1

k2cut

dk2T(k2

T)�✏

(1� ⇠)

k2T

�g2e2

f

32⇡2(1� ⇠)

S✏

✏(540)

g2e2f

32⇡2

Zµ2

k2cut

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2

Zµ2

k2max

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2(1� ⇠) ln

µ2

k2max

(541)

F (1)

1⌦ f (0)

=g2e2

f

32⇡2

Zk2max

0

dk2T

0

@(1� xBj)(1�

q1� k

2T

k2max

)

q1� k

2T

k2max

k2T

�xBj(2x2

Bj� 6xBj + 3)

Q2(1� xBj)2

q1� k

2T

k2max

1

A�g2e2

f

32⇡2(1� xBj) ln

µ2

k2max

(542)

=g2e2

f

32⇡2

"2(1� xBj) ln (2)�

2x2

Bj� 6xBj + 3

2(1� xBj)

#�

g2e2f

32⇡2(1� xBj) ln

µ2

k2max

(543)

d� =|Me,P!N |2

2�(s,m2e,M2)1/2

d3p1

(2⇡)32E1

d3p2

(2⇡)32E2

· · ·⇥ d3pN

(2⇡)32EN

(2⇡)4�(4) P + l �

NX

i=1

pi

!(544)

d�

dx dQ2=

Zd⇠

d�

dx dQ2f(⇠) +O

✓m2

(1� x)Q2

◆(545)

E0 d�

d3l0=

2↵2em

(s�M2)Q4Lµ⌫W

µ⌫(546)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +

(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)

Acknowledgments


33

Pb=

✓Q

xN

p2,xNM2

p2Q

,0T

◆(534)

W (P, q)µ⌫ =

Z1

xBj

d⇠

⇠f(⇠;µ/m,↵s(µ))W (xBj/⇠;µ/Q,↵s(µ))

µ⌫ . (535)

F1 =e2f

2�(x/⇠ � 1)

F2 = e2f�(x/⇠ � 1) (536)

+O

✓m2

Q2

◆(537)

F (1)

1=

g2e2f

32⇡3

⇣1�⇠

k2T

� ⇠(2⇠2�6⇠+3)

Q2(1�⇠)2

⌘

q1� 4⇠k2

TQ2(1�⇠)

(538)

T1T2F(1)

1=

g2e2f

32⇡3

(1� ⇠)

k2T

. (539)

g2e2f

32⇡2µ2✏S✏

Z 1

k2cut

dk2T(k2

T)�✏

(1� ⇠)

k2T

�g2e2

f

32⇡2(1� ⇠)

S✏

✏(540)

g2e2f

32⇡2

Zµ2

k2cut

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2

Zµ2

k2max

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2(1� ⇠) ln

µ2

k2max

(541)

F (1)

1⌦ f (0)

=g2e2

f

32⇡2

Zk2max

0

dk2T

0

@(1� xBj)(1�

q1� k

2T

k2max

)

q1� k

2T

k2max

k2T

�xBj(2x2

Bj� 6xBj + 3)

Q2(1� xBj)2

q1� k

2T

k2max

1

A�g2e2

f

32⇡2(1� xBj) ln

µ2

k2max

(542)

=g2e2

f

32⇡2

"2(1� xBj) ln (2)�

2x2

Bj� 6xBj + 3

2(1� xBj)

#�

g2e2f

32⇡2(1� xBj) ln

µ2

k2max

(543)

d� =|Me,P!N |2

2�(s,m2e,M2)1/2

d3p1

(2⇡)32E1

d3p2

(2⇡)32E2

· · ·⇥ d3pN

(2⇡)32EN

(2⇡)4�(4) P + l �

NX

i=1

pi

!(544)

d�

dx dQ2=

Zd⇠

d�

dx dQ2f(⇠) +O

✓m2

(1� x)Q2

◆(545)

E0 d�

d3l0=

2↵2em

(s�M2)Q4Lµ⌫W

µ⌫(546)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


Acknowledgments


Singlephotonexchange

6

Massless Target Approximation (MTA) •  Exact:•  Theapproximation:

•  UsuallytakenforgrantedatlargeQandsmallx

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

✓M2

Q2

◆(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫(P, q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q,M)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q,M)

P · q (555)

Wµ⌫(P , q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2, 0)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2, 0) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2

)µ⌫Wµ⌫(P, q) F2(xN, Q

2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q

2, 0)µ⌫Wµ⌫(P , q) F2(xBj, Q2, 0) ⌘ P2(xBj, Q

2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

✓M2

Q2

◆(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫(P, q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q,M)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q,M)

P · q (555)

Wµ⌫(P , q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2, 0)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2, 0) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2


2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q


2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

36

k2 = O�m2�

(k + q)2 = O�m2�

k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2�

(581)

2k+q� = Q2+O

�m2�

(582)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(583)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(584)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(585)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(586)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(587)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(588)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(589)

⇠ =k+

P+= xBj +O

x2

BjM2

Q2

!+O

✓m2

Q2

◆(590)

y(x) = 1� 1

x2e�x

2

� .95 ln(1 + x) (591)

2P · q ! 2P · q M2/Q2 ! 0 . (592)

Acknowledgments


7

MTA in Light-Cone Fractions •  Light-coneratios:– NoMTA:– MTA:

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘(548)

P ! P = (Pz, 0, 0, Pz) (549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

Acknowledgments


34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫(P, q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q,M)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q,M)

P · q (555)

Wµ⌫(P , q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2, 0)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2, 0) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2


2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q


2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

Structure Functions

8

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2


2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q


2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

Structure Functions

9

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘(548)

P ! P = (Pz, 0, 0, Pz) (549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

✓M2

Q2

◆(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫(P, q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q,M)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q,M)

P · q (555)

Wµ⌫(P , q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2, 0)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2, 0) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

Acknowledgments


34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2


2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q


2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q

2,M2)µ⌫Wµ⌫ F2(xN, Q

2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)

F1(xBj, Q2) ⌘ P1(xBj, Q

2, 0)µ⌫Wµ⌫(P , q) F2(xBj, Q2) ⌘ P2(xBj, Q

2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q


2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)



2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

MTA

10

36

k2 = O�m2�

(k + q)2 = O�m2�

k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2�

(581)

2k+q� = Q2+O

�m2�

(582)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(583)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(584)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(585)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(586)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(587)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(588)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(589)

⇠ =k+

P+= xBj +O

x2

BjM2

Q2

!+O

✓m2

Q2

◆(590)

y(x) = 1� 1

x2e�x

2

� .95 ln(1 + x) (591)

2P · q ! 2P · q M2/Q2 ! 0 . (592)

Acknowledgments


34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q


2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)



2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q


2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)


2, 0)µ⌫Wµ⌫ F2(xBj, Q2) ⌘ P2(xBj, Q

2, 0)µ⌫Wµ⌫ (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q


2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)


2, 0)µ⌫Wµ⌫ F2(xBj, Q2) ⌘ P2(xBj, Q

2, 0)µ⌫Wµ⌫ (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

•  Powerexpansion

•  m2=partonvirtuality,transversemomentum,mass…

•  Whatabouthadronmasses?

11

Factorization

33

Pb=

✓Q

xN

p2,xNM2

p2Q

,0T

◆(534)

W (P, q)µ⌫ =

Z1

xBj

d⇠

⇠f(⇠;µ/m,↵s(µ))W (xBj/⇠;µ/Q,↵s(µ))

µ⌫ . (535)

F1 =e2f

2�(x/⇠ � 1)

F2 = e2f�(x/⇠ � 1) (536)

+O

✓m2

Q2

◆(537)

F (1)

1=

g2e2f

32⇡3

⇣1�⇠

k2T

� ⇠(2⇠2�6⇠+3)

Q2(1�⇠)2

⌘

q1� 4⇠k2

TQ2(1�⇠)

(538)

T1T2F(1)

1=

g2e2f

32⇡3

(1� ⇠)

k2T

. (539)

g2e2f

32⇡2µ2✏S✏

Z 1

k2cut

dk2T(k2

T)�✏

(1� ⇠)

k2T

�g2e2

f

32⇡2(1� ⇠)

S✏

✏(540)

g2e2f

32⇡2

Zµ2

k2cut

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2

Zµ2

k2max

dk2T

(1� ⇠)

k2T

=g2e2

f

32⇡2

Zk2max

k2cut

dk2T

(1� ⇠)

k2T

+g2e2

f

32⇡2(1� ⇠) ln

µ2

k2max

(541)

F (1)

1⌦ f (0)

=g2e2

f

32⇡2

Zk2max

0

dk2T

0

@(1� xBj)(1�

q1� k

2T

k2max

)

q1� k

2T

k2max

k2T

�xBj(2x2

Bj� 6xBj + 3)

Q2(1� xBj)2

q1� k

2T

k2max

1

A�g2e2

f

32⇡2(1� xBj) ln

µ2

k2max

(542)

=g2e2

f

32⇡2

"2(1� xBj) ln (2)�

2x2

Bj� 6xBj + 3

2(1� xBj)

#�

g2e2f

32⇡2(1� xBj) ln

µ2

k2max

(543)

d� =|Me,P!N |2

2�(s,m2e,M2)1/2

d3p1

(2⇡)32E1

d3p2

(2⇡)32E2

· · ·⇥ d3pN

(2⇡)32EN

(2⇡)4�(4) P + l �

NX

i=1

pi

!(544)

d�

dxBj dQ2=

Zd⇠

d�

dxBj dQ2f(⇠) +O

✓m2

Q2

◆(545)

E0 d�

d3l0=

2↵2em

(s�M2)Q4Lµ⌫W

µ⌫(546)

FornowassumeM2≠O(m2)

12

Factorization and partonic light-cone fractions

k

k+qq

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)

Wµ⌫(P, q) ! Wµ⌫

AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j

Nn,jCn+2jA2j+n . (570)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


13


k

k+qq

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

2k+q� = Q2+O

�m2

�(582)

Acknowledgments


14


k

k+qq

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

2k+q� = Q2+O

�m2

�(582)

Acknowledgments


36

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

2k+q� = Q2+O

�m2

�(582)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(583)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(584)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(585)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(586)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(587)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(588)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(589)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(590)

Acknowledgments


15


k

k+qq

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

2k+q� = Q2+O

�m2

�(582)

Acknowledgments


36

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(581)

2k+q� = Q2+O

�m2

�(582)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(583)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(584)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(585)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(586)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(587)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(588)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(589)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(590)

Acknowledgments


36

k2 = O�m2�

(k + q)2 = O�m2�

k+ = O (Q) . (580)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2�

(581)

2k+q� = Q2+O

�m2�

(582)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(583)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(584)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(585)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(586)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(587)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(588)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(589)

⇠ =k+

P+= xBj +O

x2

BjM2

Q2

!+O

✓m2

Q2

◆(590)

Acknowledgments


16

Factorization Power Series

•  DropO(m2/Q2)?:Necessaryforfactorization.

•  DropO(x2BjM2/Q2)?:Notnecessaryforfactorization.

•  Makeapproximationswithexacttargetmomentum:

•  ThendoMTA:

17

MTA with factorization

IntroduceO(m2/Q2)errors

IntroduceO(xBj2M2/Q2)errors

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)

Wµ⌫ ! Wµ⌫

fact(566)

Wµ⌫

fact! Wµ⌫

fact,TMC(567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

k2T

Q2(576)

|k2| ⌧ x2

BjM2

(577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)

Wµ⌫ ! Wµ⌫

fact(566)

Wµ⌫

fact! Wµ⌫

fact,TMC(567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

k2T

Q2(576)

|k2| ⌧ x2

BjM2

(577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

Aivazis, Olness, Tung (AOT)

•  Normalfactorization,justkeepingexactmass.–  MTA

–  TMC

•  Theonly“pure”kinematicalcorrection.Othersinvolveassumptionsaboutdynamics.

18

Phys.Rev.D50,3085(1994)

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘=

✓P+,

M2

2P+,0T

◆(548)

P ! P = (Pz, 0, 0, Pz) =�P+, 0,0T

�(549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

x2

BjM2

Q2

!(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q)

P · q (555)

Wµ⌫ !✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2) ⌘ P1(xN, Q


2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫ (560)


2, 0)µ⌫Wµ⌫ F2(xBj, Q2) ⌘ P2(xBj, Q

2, 0)µ⌫Wµ⌫ (561)

Wµ⌫=

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

35

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (564)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (565)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (566)

Wµ⌫=

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (567)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (568)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (569)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (570)

Wµ⌫=

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (571)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (572)

⇢2 ⌘ 1 +4x2

BjM2

Q2(573)

Wµ⌫ ! Wµ⌫

fact(574)

Wµ⌫

fact! Wµ⌫

fact,TMC(575)

k2 = 0 (576)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (577)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


What if the target mass is important?

•  Howtotest?– ScalingwithNachtmannratherthanBjorkenvariable?

–  Improveduniversality.ExtendrangeofpQCD?

•  Whydoesitgiveimprovement?Somethingaboutnucleonstructure?

19

SeeN.Satotalk

20

Partonic interpretation of target mass effects

k

k+qq•  Smallscales

•  Exacttargetmassusefulifsuppressionbypartonicscalesisgreaterthantargetmass

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2(574)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2(574)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2(574)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2(574)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

Acknowledgments


21


•  Partonvirtualityvs.hadronmass

k

k+qq

?? ??

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2(574)

Acknowledgments


36

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (579)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (580)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (581)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(582)

x2

BjM2 � (583)

⇠ Ax2

BjM2

Q2+B

k2

Q2(584)

|k2| ⌧ x2

BjM2

(585)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(586)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(587)

k2 = O�m2

�(k + q)2 = O

�m2

�k+ = O (Q) . (588)

2k+q� + 2k�q+ �Q2+ k2 = O

�m2

�(589)

2k+q� = Q2+O

�m2

�(590)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(591)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(592)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(593)

⇠ =k+

P+= xBj +

x2

BjM2

Q2+O

✓m2

Q2

◆(594)

⇠ ⌘ k+

P+= xN +O

✓m2

Q2

◆(595)

22


k k+qq

•  Twoscales?

P

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


fact(P, q) (566)

Wµ⌫

fact(P, q) ! Wµ⌫

fact(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

|k2| ⌧ x2

BjM2

(577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

SeeE.Moffattalk

•  AOT(directfactorization):–  Directpowerexpansioninsmallpartonicmassscales–  Keepexactmomentumexpressions

•  OPE:–  TransformtoMellinmomentspace–  Expandin1/Q(bothtwistandtargetmomentum)–  Truncatetwist

•  IdentifyleadingM/Qpart•  IdentifyremainingseriesofM/Q

–  InvertleadingM/Q(F1(0))–  InvertseriesofM/Q(F1TMC)–  RelateF1(0)andF1TMC

23

Operator product expansion versus AOT

Summary

•  NormalfactorizationderivationnaturallyleadstoxNasscalingvariable/independentvariable.

•  Theseareeasytoretain(AOT).

•  Sensitivitytoatargetmassmightsaysomethingaboutnucleonstructure.– CompareProton,Kaon,Pion,Nucleustargets

24

Backup

25

•  AOT(directfactorization):–  Directpowerexpansioninsmallpartonicmassscales–  Keepexactmomentumexpressions

•  OPE:–  TransformtoMellinmomentspace–  Expandin1/Q(bothtwistandtargetmomentum)–  Truncatetwist

•  IdentifyleadingM/Qpart•  IdentifyremainingseriesofM/Q

–  InvertleadingM/Q(F1(0))–  InvertseriesofM/Q(F1TMC)–  RelateF1(0)andF1TMC

26

Operator product expansion versus AOT

OPE-based

•  Georgi-Politzer(1976)

•  WhatisF1,2(0)?

27

34

Wµ⌫=

✓�gµ⌫ +

qµq⌫

q2

◆F1 +


P =

⇣pM2 + P 2

z, 0, 0, Pz

⌘(548)

P ! P = (Pz, 0, 0, Pz) (549)

� q+

P+= xN ⌘ 2xBj

1 +

q1 +

4x2BjM

2

Q2

(550)

� q+

P+= xBj +O

✓M2

Q2

◆(551)

(k + q)2 = 0 (552)

⇠ ⌘ k+

P+= xN (553)

⇡ xBj +O

✓M2

Q2

◆(554)

Wµ⌫(P, q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1 (xN, Q,M)

+

✓Pµ � P · q

q2qµ◆✓

P ⌫ � P · qq2

q⌫◆

F2 (xN, Q,M)

P · q (555)

Wµ⌫(P , q) =

✓�gµ⌫ +

qµq⌫

q2

◆F1(xBj, Q

2, 0)

+(Pµ � qµP · q/q2)(P ⌫ � q⌫ P · q/q2)

P · qF2(xBj, Q

2, 0) (556)

P1(xN, Q2,M2

)µ⌫ ⌘ �1

2gµ⌫ +

2Q2x2

N

(M2x2

N+Q2)2

PµP ⌫(557)

P2(xN, Q2,M2

)µ⌫ ⌘

12Q4x3

N

�Q2 �M2x2

N

�

(Q2 +M2x2

N)4

PµP ⌫ �

�M2x2

N+Q2

�2

12Q2x2

N

gµ⌫!

(558)

(559)

F1(xN, Q2,M2

) ⌘ P1(xN, Q2,M2


2,M2) ⌘ P2(xN, Q

2,M2)µ⌫Wµ⌫(P, q) (560)

F1(xBj, Q2, 0) ⌘ P1(xBj, Q


2, 0)µ⌫Wµ⌫(P , q) (561)

Wµ⌫(P , q) =

Z1

xBj

d⇠

⇠Wµ⌫

(xBj/⇠, q)f(⇠) +O�m2/Q2

�+O

�M2/Q2

�. (562)

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

28

As exact structure function •  Powerseries:

•  Mellinmoments:

•  Leadingtwist• 

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

Acknowledgments


Figure 2: Two di↵erent functions with identical integer Mellin moments.

6 Comparison with / critique of other TMC methods

There are many other theoretical formalisms that are often described as Target Mass Corrections (TMC), butfrom what I can see these actually involve at least some assumptions or modifications of parton dynamics,often in ways that are not easy for me to assess. I will critique these below.

The most famous TMC method is the Georgi-Politzer (GP) [3] method based on an extension of the operatorproduct expansion (OPE). Note that it predates much of the general work on factorization. I will reviewthe steps around [3, Eqs.(4.9-4.19)]. I will sometimes switch to the notation of [4] because I also want toinclude general higher order Wilson coe�cients. The strategy is to expand the forward Compton scatteringamplitude in an OPE and keep only leading twist, but also to keep power suppressed terms associated withthe target mass that are normally dropped. One applies xBj-moments followed by inverse xBj-moments inorder to relate the structure functions to local operator matrix elements. For example, the F2 structurefunction is expressed by the leading twist part of the OPE:

F2 =2

xBj

1X

l=0

1

x2lBj

1X

j=0

✓M

2

Q2

◆j

Nj,lC2l+2j

A2j+2l+2 (58)

where Nj,l is a combinatorial factor whose detailed form is not important for our discussion, the A2j+2l+2 arematrix elements of local operators, and C

2l+2j are hard scattering coe�cients that can be calculated in an ↵s

expansion . This establishes the desired relationship between the local matrix elements of quark fields andthe integer xBj-moments of F2(xN, Q

2,M):

Z 1

0

xn�2Bj F2 =

1X

j=0

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n , (59)

where Nn,j is another combinatorial factor. The steps are: 1.) take these integer xBj-Mellin transforms, 2.)analytically continue to non-integer n, 3.) identify the local matrix elements with Mellin moments of structurefunctions in an MTA, and 4.) inverse Mellin transform to recover a general expression for F2(xN, Q

2,M

2) interms of the MTA structure functions. The analytically continued inverse transverse is

F2 =1

2⇡i

Zi1

�i1dnx

�1+n

Bj

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n . (60)

8

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


DropHigherTwist

29


•  IntegerMellinmoments:

•  Leadingtwist• 

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

Acknowledgments






F2 =2

xBj

1X

l=0

1

x2lBj

1X

j=0

✓M

2

Q2

◆j

Nj,lC2l+2j

A2j+2l+2 (58)




2,M):

Z 1

0

xn�2Bj F2 =

1X

j=0

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n , (59)


2,M


F2 =1

2⇡i

Zi1

�i1dnx

�1+n

Bj

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n . (60)

8

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


SeriesinαsOPEprovidesinfoaboutintegervalues

30


•  IntegerMellinmoments:

•  Leadingtwist,zeromass

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

Acknowledgments






F2 =2

xBj

1X

l=0

1

x2lBj

1X

j=0

✓M

2

Q2

◆j

Nj,lC2l+2j

A2j+2l+2 (58)




2,M):

Z 1

0

xn�2Bj F2 =

1X

j=0

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n , (59)


2,M


F2 =1

2⇡i

Zi1

�i1dnx

�1+n

Bj

✓M

2

Q2

◆j

Nn,jCn+2j

A2j+n . (60)

8

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


Extendtonon-integervalues

31

As exact structure function •  Invert:

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

Acknowledgments


35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

x2

BjM2

Q2

x2

BjM2

X

Q2

k2

Q2

k2T

Q2

M2

J

Q2(574)

x2

BjM2 � (575)

⇠ Ax2

BjM2

Q2+B

x2

BjM2

Q2

k2T

Q2(576)

x2

BjM2 ⌧ |k2| (577)

q =

✓�xNP

+,Q2

2xNP+,0T

◆(578)

q =

✓�xBjP

+,Q2

2xBjP+,0T

◆(579)

320.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x

0.010

0.050

0.100

0.500

1

5

F(x)Structure Functions

•  FunctionswithequalmomentsuptoN=13

33

As fit to phenomenological structure function

•  Finiteorderhardpart

•  Parametrizationofpdf

•  Fitneededallthewaytox=1

•  Theoreticalleadingtwist≠phenofitnearx=1

35

Wµ⌫(P, q) =

Z1

xN

d⇠

⇠Wµ⌫

(xN/⇠, q)f(⇠) +O�m2/Q2

�. (563)

FTMC

1(xBj, Q

2) =

1 + ⇢

2⇢F (0)

1(xN, Q

2) +

⇢2 � 1

4⇢2

Z1

xN

du

u2F (0)

1(u,Q2

) +(⇢2 � 1)

2

8xBj⇢3

Z1

xN

du

Z1

u

dv

v2F (0)

1(v,Q2

) (564)

⇢2 ⌘ 1 +4x2

BjM2

Q2(565)


AP(P, q) (566)

Wµ⌫

AP(P, q) ! Wµ⌫

AP(P , q) (567)

k2 = 0 (568)

F2 =2

xBj

1X

l=0

1

x2l

Bj

1X

j=0

✓M2

Q2

◆j

Nj,lC2l+2jA2j+2l+2 (569)

F2 =1

2⇡i

Zi1

�i1dnx1�n

Bj

✓M2

Q2

◆j


Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (571)

Z1

0

dy yn�2F (0)

2(y,Q2

) ⌘ CnAn . (572)

F2 =

X

j

C↵

n

s

j/i(x/⇠, Q)⌦ fi/P (⇠;Q) (573)

Acknowledgments


As a parton density

•  Ellis-Furmanski-Petronzio(Intro-1982)

•  Imposek2=0butallowkT>0

34

2

ps = 1.8 TeV / 1.97 TeV (13)

�De↵ = 11mb (14)

ps = 7 TeV (15)

Zd2b�2n(s, b; p

ct) = �inc

2n (s, pct) (16)

�De↵ (17)

⇡ 34 mb (18)

pct = 3.5 GeV (19)

�1 di↵(s, b; pct) = �diff (s, b; pct)� �diff (s, b; pct)1X

n=1

(�1)n�1�2n(s, b; pct) (20)

�1 di↵(s, b; pct) = �diff (s, b; pct) exp {��2(s, b; pct)} (21)


�n di↵(s, b; pct) =1

n!�diff (s, b; pct)

n exp {��2(s, b; pct)} (23)

�diff (s, b; pct) =�1� exp

��diff (s, b; pct)

�exp {��2(s, b; p

ct)} (24)

�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p

ct) + · · · ] (25)

�h(s, b; pct) (26)

d�

dq2T(27)

qT (28)

q2 ⇠ Q2 � ⇤2QCD (29)

qT ⇠ ⇤QCD (30)

⇤QCD ⌧ k1T ⌧ Q (31)

qT ⌧ Q (32)

P (33)

II. DISCUSSION

...................

Acknowledgments


k

4

k =�⇠P+, 0,0t

�(52)

F2(x,Q2) =

X

j

e2jxfj(x) 2xF1(x,Q2) = F2(x,Q

2) (53)

d� =X

i

ZH(⇠)i,DIS ⌦ fi/p(⇠) (54)

d� =X

i

ZH(⇠1, ⇠2)i,DY ⌦ fi/p(⇠1)⌦ fi/p(⇠2) (55)

x =Q2

2P · q(56)

d�

d4qd⌦=

X

i

Zd⇠1

Zd⇠2 fi/p(⇠1) fi/p(⇠2)

d�

d4qd⌦(57)

l (58)

k � l (59)

�((k � l + q)2) (60)

⇡�⇠P+, k�,kt

�(61)

k =�⇠P+, 0,0t

�(62)

Zd4k

Zdk+

Zdk�d2kt (63)

�((k + q)2) (64)

II. DISCUSSION

...................

Acknowledgments


35

As a parton density

•  InloworderYukawatheory

2

ps = 1.8 TeV / 1.97 TeV (13)

�De↵ = 11mb (14)

ps = 7 TeV (15)

Zd2b�2n(s, b; p

ct) = �inc

2n (s, pct) (16)

�De↵ (17)

⇡ 34 mb (18)

pct = 3.5 GeV (19)

�1 di↵(s, b; pct) = �diff (s, b; pct)� �diff (s, b; pct)1X

n=1

(�1)n�1�2n(s, b; pct) (20)



�n di↵(s, b; pct) =1

n!�diff (s, b; pct)

n exp {��2(s, b; pct)} (23)

�diff (s, b; pct) =�1� exp

��diff (s, b; pct)

�exp {��2(s, b; p

ct)} (24)

�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p

ct) + · · · ] (25)

�h(s, b; pct) (26)

d�

dq2T(27)

qT (28)

q2 ⇠ Q2 � ⇤2QCD (29)

qT ⇠ ⇤QCD (30)

⇤QCD ⌧ k1T ⌧ Q (31)

qT ⌧ Q (32)

P (33)

II. DISCUSSION

...................

Acknowledgments


k

5

d� = (73)

=�d�PM + d�(µ)one gluon ,↵s

� �f(⇠) + f(⇠, µ)one gluon ,↵s

�+O(↵2

s) +O(⇤/Q) (74)

d� = (75)

=�d�PM + d�(µ)one gluon ,↵s + · · ·

� �f(⇠) + f(⇠, µ)one gluon ,↵s + · · ·

�+O(⇤/Q) (76)

Wµ⌫DIS =

X

j

Zd⇠

⇠WDIS(Q/µ, ⇠, x)µ⌫j fj(⇠;µ) (77)

Wµ⌫DY =

X

i

Zd⇠1⇠1

Zd⇠2⇠2

WDY(Q/µ, ⇠1, ⇠2, x)µ⌫j fi/p(⇠1;µ) fi/p(⇠2;µ) (78)

v1 = (v01 , vx1 , v

y1 , v

z1) v2 = (v02 , v

x2 , v

y2 , v

z2) (79)

v21 = m21 v22 = m2

2 (80)

v+1 ⌘ v01 + vz1p2

, v�1 ⌘ v01 � vz1p2

, v1,t ⌘ (vx1 , vy1 ) (81)

v1 = (v+1 , v�1 ,v1,t) (82)

v1 · v2 = v+1 v�2 + v�1 v

+2 � v1,t · v2,t (83)

v21 = m21 = 2v+1 v

�1 � v21,t (84)

d ! 4� 2✏ gs ! gsµ✏ (85)

Aµ0 ! Z1/2

3 Aµ ! Z1/22 (86)

µd

dµS = 0 (87)

µd

dµ= µ

@

@µ+ �

@

@g� �mm

@

@m(88)

µd

dµg0(µ, g(µ)) = 0 (89)

d↵s(µ)/4⇡

d lnµ2= � (90)

↵s(µ)/(4⇡) = gs(µ)2/(16⇡2) (91)

hP | |P i (92)

hP | 0(0, w�,0t) 0(0, 0,0t) |P i (93)

hP | 0(0, w�,0t)

�+

2 0(0, 0,0t) |P i (94)

Zdw�

(2⇡)e�i⇠P+w�

hP | 0(0, w�,0t)

�+

2 0(0, 0,0t) |P i (95)

36

0.0 0.2 0.4 0.6 0.8 1.0kT (GeV)

0.2

0.6

1.0

1.4

v(G

eV)

Q = 2 GeV

0.0 0.2 0.4 0.6 0.8 1.0kT (GeV)

0.2

0.6

1.0

1.4

Q = 20 GeV

exact

collinear

FIG. 5: The dependence of the parton virtuality v ⌘p�k2 on kT evaluated at exact (solid

red curves) and approximate collinear (dashed blue curves) kinematics, for xbj = 0.6 at fixed

Q = 2 GeV (left panel) and Q = 20 GeV (right panel), for quark mass mq = 0.3 GeV and

spectator diquark mass ms corresponding to v(kT = 0) = 0.5 GeV (see Table I).

kT for the lower Q. At large kT, the virtuality becomes linear with kT, in accordance with

Eq. (59) in the m/Q ! 0 limit. Even assuming v < 1 GeV for kT < 1 GeV, the exact value

of k2 (and its dependence on kT) impacts the shape of the kT distribution and the quality

of the usual factorization approximations.

C. The role of transverse momentum

The factorization approximations discussed in Sec. IV apply to the limit in which kT/Q ⇠

m/Q ⌧ 1. In QCD, however, there are ultraviolet divergences from the integrals over

transverse momentum in the PDF. The standard way to deal with this is to renormalize the

PDF.

When Q is large, vertex corrections involve O (Q2) o↵-shell propagators, so the appropri-

ate renormalization scale is µ ⇠ Q. By comparison, the kinematics of real gluon emission

restrict “large” transverse momentum to be . O�Qp1� xbj)

�[see Eq. (32)], so that the

corresponding scale is µ ⇠ Qp1� xbj. (In our model calculation, the spectator plays the

role kinematically of a real gluon emission.) If xbj is not too large and Q � m, this mismatch

between real and virtual emissions is not a serious problem because kTmax is at least O (Q)

29

As a parton density

•  v=√-k2•  Blue=inpdf•  Red=inunapproximatedgraph

Moffatetal.,(2017)

•  InloworderYukawatheory

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