JWC is partly supported by the Ministry of Science and Technology (1052112M002017MY3) and the Kenda Foundation. TI is supported by Science and Technology Commission of Shanghai Municipality (16DZ2260200). TI and LCJ are supported by the Department of Energy, Laboratory Directed Research and Development (LDRD) funding of BNL (DEEC0012704). The work of HL is supported by US National Science Foundation (PHY 1653405). JHZ is supported by the SFB/TRR55 grant “Hadron Physics from Lattice QCD”, and a grant from National Science Foundation of China (11405104). YZ is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DESC0011090 and within the framework of the TMD Topical Collaboration
Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimensionthree operators will always mix with dimensionfour operators, even if chiral symmetry is preserved. Also, the number of dimensionfour operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken
Article funded by SCOAP^{3}
Controlling the systematic uncertainties is critical for obtaining meaningful results in lattice QCD. For example, the nonperturbative renormalization method of the RomeSouthampton collaboration [
In this work, we use the symmetries of lattice QCD to analyze the mixing pattern of a class of nonlocal quark bilinear operators defined in Eq. (27). Their renormalization in the continuum has been discussed since the 1980s [
A lattice theory has fewer symmetries than its corresponding continuum theory. This implies that there will be more mixing among the operators in a lattice theory than in the corresponding continuum theory. For example, a pioneering oneloop lattice perturbation theory calculation using Wilson fermions showed that the breaking of chiral symmetry for the Wilson fermions induces the mixing shown in Eq. (34) [
This is different for the case of local operators, where mixing between the dimension three and four operators is forbidden by symmetries. If a lattice action is
Our study is particularly relevant to the quasiPDF approach, which receives power corrections in inverse powers of hadron momentum. It is important to find the window where hadron momentum is large enough to suppress power corrections (good progress was made using momentum smearing in [
If the
In this subsection, we summarize the transformations of fields under
Since there is no distinction between time and space in the Euclidean space, the parity transformation, denoted
where
Similarly, the time reversal transformation, denoted as
where
Charge conjugation
and
The continuous axial rotation (
where
The anomaly induced by the singleflavor axial rotation is identical for all the operators that we study. Hence, it can be safely neglected in the operator classification.
. The explicit axial symmetry breaking pattern induced by the quark mass
so that the quark mass term is invariant under this extended axial transformation.
We now study the transformation properties of a class of local quark bilinear operators of the form
with
where
where
Therefore, depending on
Under
Properties of the dimensionthree local operator








E  E  O  O  O 

E  O  O  E  O


E  O  O  E  O 

E  E  O  O  O


E  O  E  E  O 

V  I  V  I  V 
Under an axial rotation (with Eq. (14) included),
For dimensionfour, we can further classify the operators into
The Euclidean fourdimensional rotational symmetry dictates that
It can be shown that these operators transform in the same way as
with the operators
which are either even or odd under
Transformation properties of the dimensionfour








E  E  O  O  O 

E  O  O  E  O


E  O  O  E  O 

E  E  O  O  O


O  E  O  O  E 

E  O  E  E  O 

I  V  I  V  I 
We now consider the
which transforms in the same way as
Therefore, we conclude that if the lattice theory preserves axial or chiral symmetry, then the dimensionthree and dimensionfour operators (including both the
Having reviewed the operatormixing properties of the local quark bilinears, we now apply the analysis to a specific type of nonlocal quark bilinears.
We are interested in the nonlocal quark bilinear operators with quark fields separated by
where a straight Wilson line
where
Under
The transformations could change the sign of
whose Hermitian conjugate yields
Thus, the expectation value of
The transformation properties of
Transformation properties of the dimensionthree nonlocal operators











E  O  E  O  E  O  O  E 

E/O  E/O

O/E  O/E  O/E

E/O  O/E

E/O

O/E

E/O

E/O

O/E



E/O  E/O  O/E  O/E  O/E  E/O  O/E  E/O 

E  O

E  O  E

O  O

E

E

O

E

O



E/O  O/E  O/E  E/O  E/O  E/O  O/E  O/E 

V  I  I  V  I  I  V  V 
which is consistent with the mixing pattern found using the lattice perturbation theory in Refs. [
The mixing among dimensionthree operators of different
We now extend the discussion for
where
As in the local quark bilinear case, inserting
Under
Hence, we define the combinations
such that
Their properties under
Transformation properties of the dimensionfour











E  O  E  O  E  O  O  E 

E/O  E/O

O/E  O/E  O/E

E/O  O/E

E/O

O/E

E/O

E/O

O/E



E/O  E/O  O/E  O/E  O/E  E/O  O/E  E/O 

E  O

E  O  E

O  O

E

E

O

E

O



O/E  E/O  E/O  O/E  O/E  O/E  E/O  E/O 

E/O  O/E  O/E  E/O  E/O  E/O  O/E  O/E 

I  V  V  I  V  V  I  I 
The
It has the same transformation properties as
In the previous section, it was shown that
One of the oneloop Feynman diagrams for the nonlocal quark bilinear.
where the coefficients are
and where
We have used the symmetry properties of nonlocal quark bilinear operators under parity, time reversal and chiral or axial transformations to study the possible mixing among these operators. Below, we summarize our findings.
1) If the lattice theory preserves chiral symmetry, then the dimensionthree nonlocal quark bilinear operators
where
2) If the lattice theory breaks chiral symmetry, then the dimensionthree nonlocal quark bilinear
The operator
This study is particularly relevant for the quasiPDF approach, which receives power corrections in inverse powers of hadron momentum. It is important to find a window where hadron momentum is large enough to suppress power corrections, but at the same time the mixing with