Symmetry Properties of Nonlocal Quark Bilinear Operators on a Lattice

Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators with a straight Wilson line along a spatial direction. We find that mixing among the ${\cal O}(a^0)$ operators can happen if the lattice action breaks chiral symmetry, while mixing among ${\cal O}(a^0)$ and ${\cal O}(a)$ operators can happen even the lattice action preserves chiral symmetry. Our result is relevant for the quasi parton distribution function (PDF) approach which aims at computing the Bjorken-$x$ dependence of hadron PDFs on a Euclidean lattice. The requirement of using large hadron momentum ($p$) in this approach makes the control of ${\cal O}(p a)$ error even more important.


Introduction
Taming systematic uncertainties is critical to obtain meaningful results in lattice QCD. For example, the nonperturbative renormalization of the Rome-Southampton collaboration [1] has been widely used to convert from the the lattice renormalization scheme to other continuum schemes to remove higher order errors from the slowly converging lattice perturbation theory. Another example is using the Symanzik improvement program [2,3] to systematically reduce the discretization errors due to nonzero lattice spacing a. In those programs, understanding the mixing patterns of the operators involved is crucial. A powerful, nonperturbative method to reveal the mixing pattern is to use the symmetries of the problem. Symmetries could protect certain mixings from happening. For those that are not protected by symmetries, mixings could happen under quantum corrections. Although symmetry considerations do not provide a quantitative analysis of the mixing, they do provide a complete mixing pattern among operators in the problem.
In this work, we use symmetries of the theory to analyze the mixing pattern for a class of nonlocal quark bilinear operators defined in Eq. (3.1). Their renormalization in the continuum has been discussed since the 1980s [4,5]. In recent years, there are renewed interests in the renormalization of those operators in the context of quasi parton distribution functions (quasi-PDFs) of hadrons [6] and its variations [7,8] to calculate the Bjorken-x dependence of the hadron PDFs using lattice QCD, for recent progress see Refs. . A special feature of these nonlocal quark bilinears is that the Wilson line connecting the quark fields receives power divergent contributions. A nonperturbative subtraction of the power divergence was proposed in Refs. [33,34,57] by recasting the Wilson line as a heavy quark field in the auxiliary-field approach [4,5] such that the counterterm needed to subtract the power divergence is just the counterterm for heavy quark mass renormalization. The renormalization for the nonlocal quark bilinears in the continuum was proposed in Refs. [19,35,36] and on a lattice in Ref. [27], and in non-perturbative renormalization schemes [14,15].
A lattice theory has less symmetries than its corresponding continuum theory. This implies that there will be more mixing among operators in a lattice theory than in the corresponding continuum theory. For example, a pioneering one-loop lattice perturbation theory calculation using Wilson fermions shows that the breaking of chiral symmetry for the Wilson fermions induces the mixing shown in Eq. (3.8) [26]. In this work, instead of performing explicit computations, we use symmetries to systematically study the mixing patterns among non-local quark bilinears. We study not only the mixing among the O(a 0 ) non-local quark bilinears as Ref. [26] did, but also the mixing between the O(a 0 ) and O(a) operators 1 which cannot be avoided even if chiral symmetry is preserved. 2 This feature is confirmed by the direct computation of an example one-loop diagram.
Our study is particularly relevant for the quasi-PDF 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 the momentum-smearing [13,58]), but at the same time the mixing to O(pa) is still under control. In the following we first review the symmetry analysis of local quark bilinear operators, and then move to the nonlocal ones.

Review for local quark bilinear operators
If the θ term is neglected, the lattice action exhibits important discrete symmetries: the action is invariant under discrete parity (P), time reversal (T ) and charge conjugation (C) transformations (see e.g. Ref. [59]). Chiral symmetry, which is a continuous symmetry, however, might be broken after the fermion fields are discretized. In this section, we review the symmetry properties for a specific set of local quark bilinear operators under these transformations. Then we extend the analysis to non-local quark bilinear operators in the next section. The importance of these analyses is that if two operators transform differently, then symmetries will protect them from mixing with each other under quantum corrections to all orders in the coupling. For operators whose mixing are not protected by symmetries, then in general, they will mix.

P, T , C and Axial Transformations
In this subsection, we summarize the transformations of fields under P, T , C and the axial transformation (the vector transformation in chiral symmetry is conserved in all the 1 It might be more appropriate to call the mixing to lower dimensional operators lattice artifacts while saving the term mixing exclusively for mixing to operators of the same mass dimension. Here we just use the term mixing for both cases. operators that we study). We work in Euclidean spacetime with coordinate (x, y, z, τ ) = (1, 2, 3, 4) throughout this paper. Gamma matrices are chosen to be Hermitian: γ † µ = γ µ , and Since there is no distinction between time and space in Euclidean space, parity transformation, denoted as P µ with µ = 1-4, can be defined with respect to any direction.
where P µ (x) is the vector x with sign flipped except for the µ-direction.
Analogously, time reversal transformation, denoted as T µ , can be generalized in any direction in Euclidean space.
Charge conjugation C transforms particles into antiparticles, and The continuous axial rotation (A) of the fermion fields is where α is the x independent rotation angle for the global transformation. 3 The explicit axial symmetry breaking pattern by the quark mass m can be studied by introducing a spurious transformation so that the quark mass term is invariant under this extended axial transformation.

Local operators at O(a 0 )
Now we study the transformation properties for a class of local quark bilinear operators of the form The Hermitian conjugate is where G µ (Γ), which has a value of either +1 or −1, satisfies Therefore, depending on Γ, the expectation value of O Γ can be purely real or imaginary. Under P, T , and C, the local quark bilinear transforms as  Table 1. Some lattice fermions, such as Wilson fermions, breaks the axial symmetry. But from the above discussion, we see that axial symmetry is not essential in protecting O Γ from mixing to each other. Only P µ or T µ is needed.
Euclidean four-dimensional rotational symmetry dictates that O(pa) operators being con- It can be shown that those operators transform in the same way as O Γ under P µ and T µ . While under C, with − → / D and ← − / D related operators transforming into each other. Therefore, it is convenient to define the combinations Therefore, we conclude that if the lattice theory preserves axial or chiral symmetry, then the O(a 0 ) and O(a) operators (including the O(pa) and O(ma) operators) studied above will not mix.

Nonlocal quark bilinear operators
After reviewing the operator-mixing properties of the local quark bilinears, we now apply the analysis to a specific type of non-local quark bilinears.

Nonlocal operators at O(a 0 )
We are interested in the nonlocal quark bilinear operators with quark fields separated by δz in the z-direction: where a straight Wilson line U 3 is added such that the operators are gauge invariant.
Treating the z-direction differently from the other directions, we write where i, j, k = 3. Under P µ and T µ , The transformation property of O Γ± (δz) are listed in Table 3. We see that P µ , T µ and C symmetries cannot protect the mixing between 1 and γ 3 and between iγ i γ 5 and ijk σ jk at O(a 0 ). This can be summarized as which is consistent with the mixing pattern found using lattice perturbation theory in Refs. [26,27]. However, if the lattice theory preserves axial or chiral symmetry, then none of the O(a 0 ) operators will mix with each other: Table 3. Transformation properties of the O(a 0 ) nonlocal operators O Γ± (δz). i, j, k = 3. Other notations are the same as in Table 1.

Nonlocal operators at O(pa) and O(ma)
Now we extend the discussion for O(pa) local operators to nonlocal ones. We can insert / D at any point on the Wilson line. The symmetry properties will not depend on where / D is inserted.
where 0 ≤ δz ≤ δz. The z-direction is treated differently by writing α ∈ [3, ⊥] and − → / As in the local quark bilinear case, inserting − → / D and ← − / D does not change the transformation properties under P µ and T µ . So these operators transform in the same way as O Γ (δz).
It is useful to define combinations that are even or odd under P µ and T µ : Under C, those operators transform as So we define the combinations Their properties under P µ , T µ and C are listed in Table 4. By comparing with Table 3, we find that P µ , T µ and C symmetries do not protect O Γ (δz) from mixing with Q Dα Γ (δz, δz ) and Q Dα γ 3 Γ (δz, δz ). If the lattice theory preserves axial or chiral symmetry, then the mixing with Q Dα Γ (δz, δz ) is forbidden, but the mixing with Q Dα γ 3 Γ (δz, δz ) is still allowed. This is analogous to the static heavy-light system (Wilson line can be described as a heavy quark propagator in the auxiliary-field approach [19,27,35]) which has O(pa) discretization errors even if the light quarks respect chiral symmetry.
The O(ma) nonlocal bilinear is It has the same transformation properties as O Γ (δz) under P µ , T µ and C. but it shows a difference for the chiral rotation. However, chiral symmetry does not prevent O Γ (δz) from mixing with Q M γ 3 Γ (δz) at O(ma).

A mixing example in perturbative theory
In the previous section, it was shown that P µ , T µ , C, and chiral symmetries cannot protect O(a 0 ) nonlocal quark bilinears from mixing with O(a) ones. This is a distinct feature from local quark bilinears in which O(a 0 ) operators are protected from mixing with O(a) ones. Here we use the diagram shown in Fig. 1 to demonstrate where the effect comes from. For our purpose, we can simplify our calculation by taking the Feynman gauge and the limit of small external momenta and quark masses, and we will work in the continuum limit with Figure 1. One of the one-loop Feynman diagrams for the nonlocal quark bilinear. p and p are incoming and outgoing external momenta, respectively.
appropriate UV and IR regulators imposed implicitly. Then the one-loop amputated Green function in figure 1, Λ 1−loop Γ,δz (p , p, m), yields +O(p 2 , p 2 , p p, p m, pm, m 2 ), (3.22) where the coefficients are and where H(Γ) = 4 µ=1 G µ (Γ). It is easy to see that when δz = 0 (corresponding to a local quark bilinear), the mixing to O(a) operators all vanish, while when δz = 0 (corresponding to a nonlocal quark bilinear) the mixing to O(a) operators are not protected even the theory has P, T , C, and chiral symmetries.

Summary
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 those operators. Below we summarize our findings. This study is particularly relevant for the quasi-PDF 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, but at the same time the mixing to O(pa) is still under control. For future work, in light of the similarity between the Wilson line and the heavy quark propagator, it would be valuable to apply techniques developed for heavy quark effective field theory on the lattice [60,61] and the associated improvement treatment of lattice artifacts [62][63][64][65][66][67].