References - Bogoliubov Laboratory of Theoretical Physics - JINR

where
wP = − m
2M 2 −p1 cos ω + pT cos ϕ sin ω
ν p0 + m
�
× 1+ 1
�
p0 −
m
−p1 − (−p1 cos ω + pT cos ϕ sin ω)cosω
sin 2 ��
,
ω
wS = m
�
1+
2Mν
−p1 cos ω + pT cos ϕ sin ω 1
p0 + m m
�
× −p1 cos ω + pT cos ϕ sin ω − −p1 − (−p1 cos ω + pT cos ϕ sin ω)cosω
sin2 ω
Let us remark, that using the notation defined in [3] we can identify
(7)
(8)
��
cos ω .
− cos ω = SL, sin ω = ST , pT sin ω cos ϕ = pT ST , (9)
which appear in definition of the TMDs [2]:
1
2 tr � γ + γ5φ q (x, pT ) � = SLg q
1 (x, pT )+ pT ST
M g⊥q
1T (x, pT ).
The expressions (7),(8) can be reordered in terms of powers of cos ϕ:
(10)
wP =
cos ω
−
2M 2 � 2 pT cos
ν m + p0
2 ϕ +(pT tan ω
+
(11)
pT
� �
p1
p1 (tan ω − cot ω) cos ϕ − p1
m + p0
m + p0
��
+1 ,
wS = 1
� 2 pT cos
2Mν m + p0
2 ϕ − p1pT
�
cot ω
cos ϕ + m .
m + p0
Now, in analogy with Eq.(46) in [7] we define (note that Pq/qS = −M/ cos ω):
(12)
which implies
g q
k =
�
w1 = Mν · wS + M 2 ν
cos ω · wP , w2 = − M 2 ν
cos ω · wP , (13)
�
p0 +
�
p1 dp1d
ΔG (p0) wkδ − x
M 2pT , k =1, 2. (14)
p0
After inserting from Eqs. (11),(12) definition (13) implies
w1 = 1
� �
m + p1 1+
2
p1
� �
− pT tan ω 1+
m + p0
p1
� �
cos ϕ , (15)
m + p0
w2 = 1
� 2 pT cos
2 m + p0
2 ϕ +(pT tan ω (16)
+ pT
� � ��
p1
p1 (tan ω − cot ω) cos ϕ − p1 +1 ,
m + p0
m + p0
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