JHEP01(2019)067
near the endpoint in section 2.5, and provide details on our numerical implementation in
section 2.6.
2.1 TMD fragmentation
The cross section for producing a hadron h with energy fraction z
h
and transverse momen-
tum
~
k
⊥
= ~p
h⊥
/z
h
inside a jet with energy E
J
and radius R is given by
dσ
h
dE
J
d
2
~
k
⊥
dz
h
=
X
i
Z
dx
x
H
i
E
J
x
, µ
G
i→h
(x, E
J
R,
~
k
⊥
, z
h
, µ)
1 + O(R
2
)
. (2.1)
Here the collinear approximation R 1 was exploited to factorize the cross section into
a partonic cross section H, that describes the hard scattering, and the fragmenting jet
function G, that captures the formation of the jet. Specifically, the parton i with energy
E
J
/x produced in the hard scattering emits radiation, resulting in a jet with energy E
J
that contains the hadron h. When the hadron is produced close to the center of the jet,
i.e. k ≡ |
~
k
⊥
| E
J
R, we can furthermore separate the effects of the jet boundary B from
the fragmentation,
G
i→h
(x, E
J
R,
~
k
⊥
, z
h
, µ) =
X
j
Z
dy
y
B
ij
x, E
J
R,
z
h
y
, µ
D
j→h
(
~
k
⊥
, y, µ)
1+O
k
2
E
2
J
R
2
.
(2.2)
This requires the effect of the measurement at angular scales θ ≈ k/E
J
and R to factorize,
which was argued to hold for the Cambridge/Aachen and anti-k
T
with the WTA axis in
ref. [6]. Finally, for k Λ
QCD
, the transverse momentum dependence can be calculated,
D
j→h
(
~
k
⊥
, z
h
, µ) =
X
k
Z
dz
z
C
jk
~
k
⊥
,
z
h
z
, µ
d
k→h
(z, µ)
"
1 + O
Λ
2
QCD
k
2
!#
, (2.3)
where d
k→h
(z, µ) are the standard fragmentation functions [40–42].
We will also consider the case E
J
R ∼ k Λ
QCD
, where the hadron is produced fairly
close to the jet boundary. The corresponding factorization theorem is given by
G
i→h
(x, E
J
R,
~
k
⊥
, z
h
, µ) =
X
k
Z
dz
z
J
ik
x, E
J
R,
~
k
⊥
,
z
h
z
, µ
d
k→h
(z, µ)
"
1 + O
Λ
2
QCD
E
2
J
R
2
!#
.
(2.4)
The coefficients J
ik
were also computed in ref. [6]. When k E
J
R, they factorize as
J
ik
(x, E
J
R,
~
k
⊥
, z, µ) =
X
j
Z
dz
0
z
0
B
ij
x, E
J
R,
z
z
0
, µ
C
jk
(
~
k
⊥
, z
0
, µ)
1 + O
k
2
E
2
J
R
2
,
(2.5)
as required by consistency with eqs. (2.2) and (2.3). By using J
ik
when k ∼ E
J
R, instead
of the factorized form on the right-hand side, we capture the k
2
/(E
2
J
R
2
) corrections that
are crucial in this region.
– 4 –
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