References - Bogoliubov Laboratory of Theoretical Physics - JINR

w1 + w2 = 1
� 2 pT cos
2 m + p0
2 ϕ − p1pT
�
cot ω
cos ϕ + m . (17)
m + p0
Apparently, the terms proportional to cos ϕ disappear in the integrals (14) and the remaining
terms give structure functions g1,g2 defined by Eqs. (15),(16) in [6].
Now, the integration over p1 and further procedure can be done in a similar way as
for unpolarized distribution. First, to simplify calculation, we assume m → 0. For w1 we
get
g q
�
1 (x) =1
2
The δ−function is modified as
where
�
ΔGq(p0) 1+ p1
�
�
p0 +
�
p1 dp1d
(p1 − pT tan ω cos ϕ) δ − x
p0
M 2pT . (18)
p0
˜p1 = Mx
2
�
p0 +
�
p1
δ − x
M
dp1 = δ (p1 − ˜p1) dp1
, (19)
x/˜p0
� � � �
pT
2
1 − , ˜p0 =
Mx
Mx
� � � �
pT
2
1+ . (20)
2 Mx
Modified δ−function allows to simplify the integral
g q
1(x) = 1
�
ΔGq(˜p0)(M (2x − ξ) − 2pT tan ω cos ϕ)
2
d2pT ξ
where
Now we define
Δq(x, pT )= 1
2 ΔGq
, (21)
� � � �
pT
2
ξ = x 1+ . (22)
Mx
� �
Mξ
2
(M (2x − ξ) − 2pT tan ω cos ϕ) 1
. (23)
ξ
According to Eq. (40) in [6] we have
� �
Mξ 2
ΔGq =
2 πM3ξ 2
�
3g q
� 1
g
1(ξ)+2
ξ
q
1(y) d
dy − ξ
y dξ gq
�
1(ξ) . (24)
After inserting to Eq. (23) one gets:
1
Δq(x, pT ) =
πM2ξ 3
�
3g q
1 (ξ)+2
� 1
g
ξ
q
1(y) d
dy − ξ
y dξ gq 1 (ξ)
�
(25)
�
× 2x − ξ − 2 pT
�
tan ω cos ϕ .
M
This relation allows us to calculate the distribution Δq(x, pT ) from a known input on
g q
1 (x). Further, it can be shown, that using the notation defined in Eqs. (9),(10), our
result reads
− cos ω · Δq(x, pT )=SLg q
1(x, pT )+ pT ST
M g⊥q
1T (x, pT ), (26)
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