398 C. Alexandrou et al. / Nuclear Physics B 923 (2017) 394–415
4 eliminates the mixing that appears in the unpolarized operator, as the bare matrix element is
a linear combination of the unpolarized quasi-PDF and the twist-3 scalar operator. The two
may be disentangled by the construction of a 2×2 mixing matrix.
We adopt a renormalization scheme which is applicable non-perturbati
vely, that is, the
RI
scheme [33]. We compute vertex functions of the operators under study, between external
quark states, with the setup being in momentum space, and the operator defined as:
O
= ψ(x)P e
ig
z
0
A(ζ )dζ
ψ(x + z ˆμ) , (2)
where = γ
μ
, γ
μ
· γ
5
, σ
μν
(ν = μ). The path ordering of the exponential appearing in the above
expression becomes, on the lattice, a series of path ordered gauge links. The renormalization
functions (Z-factors) depend on the length of the Wilson line and, thus, we perform a separate
calculation for each value of z. Typically, z goes up to half of the spatial e
xtent of the lattice.
The renormalization prescription is along the lines of the program de
veloped for local opera-
tors and the construction of the vertex functions is described in Ref. [37]. The difference between
the renormalization of the local operators and the Wilson-line operators is the linear divergence
that appears in the latter case. However, there is no need to separate this di
vergence from the
multiplicative renormalization and, therefore, the technique described below may successfully
extract both contributions at once.
Helicity and transversity quasi-PDFs
We first provide the methodology for a general operator with a Wilson line in the absence of
any mixing. This is applicable for the helicity and transversity quasi-PDFs, provided that their
Dirac structure is chosen along the Wilson line. The renormalization functions of the Wilson-line
operators, Z
O
, are extracted by imposing the following conditions:
Z
−1
q
Z
O
(z)
1
12
Tr
V(p, z)
V
Born
(p, z)
−1
p
2
=¯μ
2
0
= 1 , (3)
where Z
q
is the renormalization function of the quark field obtained via
Z
q
=
1
12
Tr
(S(p))
−1
S
Born
(p)
p
2
=¯μ
2
0
. (4)
The trace is taken over spin and color indices, and the momentum p entering the vertex function
is set to the RI
renormalization scale ¯μ
0
. In Eq. (3) V(p, z) is the amputated vertex function
of the operator and V
Born
is its tree-level value, i.e. V
Born
(p, z) = iγ
3
γ
5
e
ipz
for the helicity op-
erator. Also, S(p) is the fermion propagator and S
Born
(p) is its tree-level. The RI
scale ¯μ
0
is
chosen such that its z-component is the same as the momentum of the nucleon. Such a choice
serves as a suppression of discretization effects, as different classes of spatial components have
different discretization effects, and scales of the form (n
t
, 3, 3, 3) have small discretization ef-
fects [38]. We test both diagonal (democratic) and parallel momenta to the Wilson line (in the
spatial direction), that is a
¯μ
0
=
2π
L
(P
3
, P
3
, P
3
) and a
¯μ
0
=
2π
L
(0, 0, P
3
), respectively. We will
refer to these choices as “diagonal” and “parallel”. The latter are expected to have larger lattice
artifacts, as the ratio
ˆ
P ≡
ρ
¯μ
4
0
ρ
ρ
¯μ
2
0
ρ
2
(5)
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