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
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
This work was supported by...
Singlephotonexchange
6
Massless Target Approximation (MTA) • Exact:• Theapproximation:
• UsuallytakenforgrantedatlargeQandsmallx
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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)
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
This work was supported by...
7
MTA in Light-Cone Fractions • Light-coneratios:– NoMTA:– MTA:
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
This work was supported by...
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
)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)
Structure Functions
8
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
)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)
Structure Functions
9
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
This work was supported by...
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
)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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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)
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
This work was supported by...
34
Wµ⌫=
✓�gµ⌫ +
qµq⌫
q2
◆F1 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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⌫ 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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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⌫ 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
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
13
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
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
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
This work was supported by...
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)
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
This work was supported by...
14
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
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
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
This work was supported by...
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)
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
This work was supported by...
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
This work was supported by...
15
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
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
q =
✓�xBjP
+,Q2
2xBjP+,0T
◆(579)
k2 = O�m2
�(k + q)2 = O
�m2
�k+ = O (Q) . (580)
Acknowledgments
This work was supported by...
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)
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
This work was supported by...
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)
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
This work was supported by...
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
This work was supported by...
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
This work was supported by...
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
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
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
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
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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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⌫ 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
Nn,jCn+2jA2j+n . (578)
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)
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(574)
Acknowledgments
This work was supported by...
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(574)
Acknowledgments
This work was supported by...
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(574)
Acknowledgments
This work was supported by...
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(574)
Acknowledgments
This work was supported by...
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)
Acknowledgments
This work was supported by...
21
Partonic interpretation of target mass effects
• 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)
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(574)
Acknowledgments
This work was supported by...
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
Partonic interpretation of target mass effects
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)
Wµ⌫(P, q) ! Wµ⌫
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
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)
|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 +
(Pµ � qµP · q/q2)(P ⌫ � q⌫P · q/q2)P · q F2 (547)
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
)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)
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)
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)
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⌫(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)
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⌫(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)
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)
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)
Acknowledgments
This work was supported by...
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)
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)
Acknowledgments
This work was supported by...
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)
Acknowledgments
This work was supported by...
DropHigherTwist
29
As exact structure function • Powerseries:
• 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)
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)
Acknowledgments
This work was supported by...
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)
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)
Acknowledgments
This work was supported by...
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)
Acknowledgments
This work was supported by...
SeriesinαsOPEprovidesinfoaboutintegervalues
30
As exact structure function • Powerseries:
• 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)
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)
Acknowledgments
This work was supported by...
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)
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)
Acknowledgments
This work was supported by...
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)
Acknowledgments
This work was supported by...
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)
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)
Acknowledgments
This work was supported by...
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)
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⌫(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)
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⌫(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)
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)
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)
Acknowledgments
This work was supported by...
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)
�1 di↵(s, b; pct) = �diff (s, b; pct) exp {��2(s, b; pct)} (22)
�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
This work was supported by...
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
This work was supported by...
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)
�1 di↵(s, b; pct) = �diff (s, b; pct) exp {��2(s, b; pct)} (21)
�1 di↵(s, b; pct) = �diff (s, b; pct) exp {��2(s, b; pct)} (22)
�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
This work was supported by...
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