MSUHEP-17-015, MIT-CTP/4942 Prepared for submission to JHEP
Operator classification for nonlocal quark bilinear on lattice Lattice Parton Physics Project (LP3 ) Collaboration Jiunn-Wei Chen,a Tomomi Ishikawa,b Luchang Jin,c,d Huey-Wen Lin,e,f Yi-Bo Yang,e Jian-Hui Zhangg and Yong Zhaoh
arXiv:1710.01089v1 [hep-lat] 3 Oct 2017
a
Department of Physics, Center for Theoretical Sciences, and Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei, Taiwan 106 b T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China c Physics Department, University of Connecticut, Storrs, Connecticut 06269-3046, USA d RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA e Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA f Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA g Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany h Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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[email protected] Abstract: We address the operator-mixing pattern of a class of nonlocal quark bilinear operators that involve a straight Wilson line in a spatial direction. In recent years, this type of nonlocal operator has received a lot of attention in the context of the lattice QCD calculations needed to extract the Bjorken-x dependence of the parton distribution functions (PDFs). The mixing is complicated by the breaking of symmetries on the lattice. Analyzing the behavior of the operators under the symmetries of the lattice action and the equation of motion, we classify the structure of O(a) relevant operators. Going beyond the symmetry point of view, we also investigate the perturbative structure of the O(a) operators due to lattice discretization effects. Since we need a large hadron momentum in the quasi-PDF approach to reduce higher-twist effects, the O(a)-improvement operator will be necessary to obtain a better determination of PDFs, and the analysis provided in this work gives a basis for the improvement.
Contents 1 Introduction
1
2 Operator classification — local quark bilinear operators 2.1 Discrete symmetries and chiral symmetry 2.2 Local operators at O(a0 ) 2.3 Local operators at O(pa) and O(ma) 2.4 Summary for the local quark bilinears
3 3 4 5 6
3 Operator classification — nonlocal quark bilinear operators 3.1 Nonlocal operators at O(a0 ) 3.2 Nonlocal operators at O(pa) and O(ma) 3.3 Summary for nonlocal quark bilinears 3.4 Perturbative analysis of O(a) for nonlocal quark bilinears
7 8 9 11 13
4 Summary
14
1
Introduction
Controlling systematics in numerical simulations on lattice QCD has great significance to obtain physically meaningful results. Nonperturbative renormalization programs, such as Rome-Southampton method [1], have been widely utilized to reduce perturbative uncertainty in converting lattice results into continuum schemes, such as MS scheme, (matching). Discretization errors due to nonzero lattice spacing a is another major source of uncertainties. To systematically reduce the discretization errors, the Symanzik improvement program was proposed [2] and developed [3]. In order to control all of these, it is essential to investigate properties of relevant operators to our physics target. In the present paper, we address the mixing pattern for a class of nonlocal operator, especially quark bilinears: b b x)ψ(x), OΓ (δz) = ψ(x + 3δz)ΓU 3 (x + 3δz;
(1.1)
where two quark fields (ψ and ψ) are separated in a spatial direction, z or 3, by δz, and b x) to make the operator gauge they are connected by a straight Wilson line U3 (x + 3δz; b denotes a unit vector in z-direction and Γ represents general Dirac gamma invariant. 3 matrices. Its renormalization in the continuum has been discussed since the 1980s [4, 5]. In recent years, the nonlocal operator has gotten a lot of attention in the context of quasi parton distribution functions (quasi-PDFs), proposed in ref. [6] and its variants [7, 8] to calculate the Bjorken-x dependence of the light-cone PDFs using lattice QCD. Since the operator involves a Wilson line, it intrinsically suffers from power-law ultraviolet (UV)
–1–
divergences. The nonperturbative subtraction method of the power divergence for this operator was proposed in refs. [9–11]. Since the Wilson line can be represented by auxiliary fields [4, 5], which are similar to static heavy quarks, the power divergence can be seen as a static quark’s self-energy. The renormalization program for the nonlocal operator (1.1) using the auxiliary field was also proposed in refs. [12, 13]. A proof of the renormalizability of the nonlocal operator has been a longstanding issue, since the intrinsic anisotropy of the operator makes simple power-counting argument inapplicable [14, 15]. Recent attempt to prove the renormalizability of the nonlocal operator can be seen in refs. [12, 16]. Lattice actions do not respect all the symmetries possessed by continuum theories. In lacking some symmetries, the operator-mixing pattern in the renormalization could be different from that in the continuum. The operator mixing that occurs on the lattice for the nonlocal operator (1.1) was first reported in ref. [17] at the one-loop perturbative level, and the following work [13] also supported the same mixing pattern. Their finding was that the nonlocal operator (1.1) can mix with b b x)ψ(x), O{Γ,γ3 } (δz) = ψ(x + 3δz){Γ, γ3 }U3 (x + 3δz;
(1.2)
on the lattice. In this work, we investigate the operator mixing in more detail from a view of action symmetry to validate the statement at the nonperturbative level, which is necessary for the nonperturbative renormalization whose numerical demonstrations are seen in refs. [13, 18, 19]. The action symmetries to be harnessed here are chiral symmetry and several discrete symmetries: parity, time reversal and charge conjugation. The symmetry can restrict the operator mixing, and lack of some symmetries affects its pattern. For example, Wilson-fermion formalism, which has often been used in numerical simulations, explicitly lacks chiral symmetry. The mixing mentioned above is caused by the lack of the chiral symmetry. Besides the operator mixing, we address possible O(a)-improvement operators by extending the symmetry argument. In the quasi-PDF method, the nucleon momentum needs to be large to reduce higher-twist effects, which means the lattice discretization error is presumed to be sizable. To reduce the discretization error, it is effective to implement the O(a)-improvement program, so investigating the possible O(a) operators is beneficial. In the usual local-operator case, it is crucial to preserve chiral symmetry to prohibit O(a) operators. By the symmetry analysis, we find the possibility of having O(a) error cannot be excluded for the nonlocal operator even when chiral symmetry is preserved. It is counterintuitive, but partly understandable in the auxiliary-field picture, because the static heavy-light system can have O(a) discretization error; detailed symmetry analysis of this can be found in refs. [20–22]. The possible O(a) error in the nonlocal operator in the auxiliary-field picture was also mentioned in ref. [13]. The present paper is organized in a pedagogical manner, starting with the symmetry analysis for the well-known local quark-bilinear case (section 2). The analysis is extended to the nonlocal case, where we present possible operator mixings and analyses on the O(a) operators (section 3). We also present a supplemental discussion for the O(a) operators using continuum perturbation theory at one-loop (section 3). A summary is given in section 4.
–2–
2
Operator classification — local quark bilinear operators
The lattice action has several important discrete symmetries: parity P, time reversal T , and charge conjugation C. In addition, some lattice actions retain chiral symmetry. (The definitions of the transformation of fields under the discrete symmetries in Euclidean spaces can be found in several textbooks, such as ref. [23].) Those symmetries are important to investigate system properties such as quantum numbers of states. In this section, we review the operator classification using those symmetries for well-known local quark bilinear operators. This argument will be extended in the next section to the nonlocal bilinear case, which is our primary interest in the present paper. 2.1
Discrete symmetries and chiral symmetry
In this subsection, we summarize the discrete symmetries and chiral symmetry used in the following discussion. Throughout this paper, Euclidean spaces are assumed, where three spatial and Euclidean-time directions are denoted (x, y, z, t) or (1, 2, 3, 4). Gamma matrices are chosen to be Hermitian: 㵆 = γµ , and γ5 = γ1 γ2 γ3 γ4 . Parity In Euclidean space, a parity transformation can be defined with respect to any direction, since there is no distinction between time and space directions. The general parity transformation Pµ is Pµ
ψ(x) −−→ ψ(x)Pµ = γµ ψ(Pµ (x)),
(2.1)
ψ(x) −−→ ψ(x)Pµ = ψ(Pµ (x))γµ ,
(2.2)
Pµ
Pµ
Uν6=µ (x) −−→ Uν6=µ (x)Pµ = Uν6†=µ (Pµ (x) − νˆ), Pµ
Uµ (x) −−→ Uµ (x)Pµ = Uµ (Pµ (x)),
(2.3) (2.4)
where Pµ (x) is the vector x with sign flipped except for the µ-direction. Time reversal In Euclidean space, a time reversal can be, again, generalized in any direction. The general time reversal Tµ is Tµ
ψ(x) −→ ψ(x)Tµ = γµ γ5 ψ(Tµ (x)),
(2.5)
ψ(x) −→ ψ(x)Tµ = ψ(Tµ (x))γ5 γµ ,
(2.6)
Uµ (x) −→ Uµ (x)Tµ = Uµ† (Tµ (x) − µ ˆ),
(2.7)
Uν6=µ (x) −→ Uν6=µ (x)Tµ = Uν6=µ (Tµ (x)),
(2.8)
Tµ
Tµ
Tµ
where Tµ (x) is the vector x with sign flipped in the µ-direction.
–3–
Charge conjugation Charge conjugation C transforms particles into antiparticles; it is C
ψ(x) − → ψ(x)C = C −1 ψ(x)> ,
(2.9)
C
ψ(x) − → ψ(x)C = −ψ(x)> C,
(2.10)
C
Uµ (x) − → Uµ (x)C = Uµ (x)∗ = (Uµ† (x))> ,
(2.11)
where the charge conjugation matrix C obeys the relation Cγµ C −1 = −γµ> ,
Cγ5 C −1 = γ5> .
(2.12)
Chiral rotation Chiral rotation of the fermion fields χ is χ
0
χ
ψ(x) − → ψ 0 (x) = eiαγ5 ψ(x), ψ(x) − → ψ (x) = ψ(x)eiαγ5 ,
(2.13)
where α represents a rotation parameter.1 When the quark mass m is small compared to the QCD scale, chiral symmetry in the action is softly broken. The effect of the nonzero quark mass can be analyzed by introducing a spurious chiral transformation χ0
m −→ e−iαγ5 me−iαγ5 ,
(2.14)
so that the quark mass term is invariant under the extended chiral transformation. 2.2
Local operators at O(a0 )
With the symmetries defined in the last subsection, we investigate transformation properties of a local quark bilinear of the form OΓ = ψ(x)Γψ(x),
(2.15)
Γ ∈ {1, γµ , γ5 , iγµ γ5 , σµν },
(2.16)
with
where σµν = 2i [γµ , γν ]. The local bilinear has a property under the Hermitian conjugate: (OΓ )† = −Oγ4 Γγ4 = −G4 (Γ)OΓ ,
(2.17)
where Gµ (Γ) is defined to satisfy γµ Γγµ = Gµ (Γ)Γ.
(2.18)
Thus, when we consider matrix elements of this bilinear operator with the same in and out states: hp|OΓ |pi, they are pure real or pure imaginary, depending on Γ. Under the parity, time-reversal, and charge-conjugation operations, the local operator transforms as Pµ
OΓ −−→ Oγµ Γγµ ,
Tµ
OΓ −→ Oγ5 γµ Γγµ γ5 ,
–4–
C
OΓ − → O(CΓC −1 )> .
(2.19)
Pρ=µ Pρ6=µ Tρ=µ Tρ6=µ C χ
Γ=1 E E E E E V
γµ E O O E O I
γ5 O O O O E V
iγµ γ5 O E E O E I
σµν O O(ρ=ν) /E(ρ6=ν) O O(ρ=ν) /E(ρ6=ν) O V
Table 1. Transformation properties of the local operator OΓ : even/odd (E/O) under parity (Pρ ), time reversal (Tρ ) and charge conjugation (C), and invariant/variant (I/V) under chiral rotation (χ).
The even/oddness of the each Γ under these transformations is summarized in table 1. Since quark bilinears with different Γ have different transformation properties under Pµ (or Tµ ), they do not mix under renormalization. Charge conjugation, on the other hand, does not have such discrimination power. By imposing the chiral rotation onto the quark bilinears, the operators are separated into two groups, depending on whether they are variant or invariant under the rotation, as also shown in table 1. Some lattice fermions, such as Wilson fermions, do not have the chiral symmetry, but chiral symmetry is not essential to prohibit mixing among local bilinear operators. 2.3
Local operators at O(pa) and O(ma)
→ − ← − To describe the O(pa) operator, we need to include covariant derivatives, D µ and D µ , which are defined on the lattice as i → − 1 h D µ φ(x) = Uµ (x)φ(x + µ ba) − Uµ† (x − µ ba)φ(x − µ ba) , (2.20) 2a i ← − 1 h φ(x) D µ = φ(x + µ ba)Uµ† (x) − φ(x − µ ba)Uµ (x − µ ba) , (2.21) 2a
where φ(x) represents any field with color indices. The kinetic term of the fermion action → − tells us that inserting the covariant derivative associated with gamma matrices, γµ D µ or ← − γµ D µ , into quark bilinears does not change the transformation properties with respect to parity and time reversal. In the local-operator case, Euclidean four-dimensional rotational symmetry is assumed, which leads to the insertion being done in a Lorentz-covariant way → − ← − / and D. / The candidate O(pa) operators which have the same parity and timewith D reversal properties as the O(a0 ) operators can then be constructed from → − / ψ(x), → = ψ(x)Γ D QΓ− D → − / Γψ(x), → = ψ(x) D Q− DΓ
← − / − = ψ(x) DΓψ(x), Q← DΓ ← − / ψ(x). − = ψ(x)Γ D QΓ← D
1
(2.22) (2.23)
Anomaly by the single flavor chiral rotation is not relevant in investigating transformation properties of operators.
–5–
Pρ=µ Pρ6=µ Tρ=µ Tρ6=µ O(pa)(+) C(OΓ/Γ ) O(pa)(−)
C(OΓ/Γ
)
Γ=1 E E E E O
γµ E O O E E
γ5 O O O O O
iγµ γ5 O E E O O
σµν O O(ρ=ν) /E(ρ6=ν) O O(ρ=ν) /E(ρ6=ν) E
E
O
E
E
O
I
V
I
V
I
χ
O(pa)(±)
Table 2. Transformation properties of the local operators at O(pa), OΓ/Γ : even/odd (E/O) under parity (Pρ ), time reversal (Tρ ) and charge conjugation (C), and variant/invariant (V/I) under chiral rotation (χ).
Under charge conjugation, the O(pa) operators defined above transform as C
→ − → − − ← −, QΓ− → −Q← D / DΓ D(CΓC −1 )> /(CΓC −1 )>D
C
− ← − − → − → Q← → −Q(CΓC −1 )>− , (2.24) DΓ/ΓD D / D(CΓC −1 )>
where we observe that the O(pa) operators are swapped; hence, it is convenient to define their combination: O(pa)(±)
OΓ
− ±Q − →, = Q← DΓ ΓD
O(pa)(±)
OΓ
− ± Q− → . = QΓ← D DΓ
(2.25)
Their transformation properties under the parity, time reversal, and charge conjugation are summarized in table 2. Note that the charge conjugation property is essential to select out O(pa)(−) OΓ/Γ to have the same transformations as that of OΓ . O(pa)(±)
Imposing the chiral rotation, OΓ and OΓ/Γ
shows different transformation proper-
ties (variant/invariant) from each other. Thus, the existence of the chiral symmetry kills all the O(pa) operators. The O(ma) operator is O(ma)
OΓ
= mψ(x)Γψ(x),
(2.26)
which has the same parity, time reversal and charge conjugation transformation properties as that of OΓ , but the properties are completely opposite under the spurious chiral rotation. As the result, chiral symmetry also kills the O(ma) operator. 2.4
Summary for the local quark bilinears
The discussion in this section shows there is no possible mixing among local quark bilinears, OΓ = ψ(x)Γψ(x),
(2.27)
where charge-conjugation and chiral-rotation properties of the operators are not essential to prohibit them from mixing with each other. There are possible O(a) operators of the form ← ← − → − − → − O(pa) O(pa) / − ΓD / ψ(x), / − DΓ / OΓ = ψ(x) DΓ OΓ = ψ(x) Γ D ψ(x), (2.28)
–6–
O(ma)
OΓ
= mψ(x)Γψ(x).
(2.29)
Again, there is no mixing between O(a) operators with different Γ by parity and time reversal. Note that chiral symmetry kills all the O(pa) and O(ma) operators above. Many lattice fermions, such as the Wilson fermions, do not respect chiral symmetry, choosing to sacrifice this symmetry to resolve the fermion-doubling problem on the lattice. With such fermions, we suffer from O(pa) discretization error. In the clover-Wilson fermion formalism for the O(a)-improved lattice fermion [24], the tree-level O(a)-improvement is achieved by the fermion field rotation [25, 26] h i − ar → / − (1 − z)m ψ, ψ −→ ψ 0 = 1 − zD h 2 ar ← i − 0 / − (1 − z)m , ψ −→ ψ = ψ 1 − −z D 2
(2.30) (2.31)
where r is a Wilson parameter and z is an arbitrary parameter. The O(a) operators that O(pa) O(ma) appear by the rotations in eqs. (2.30) and (2.31) are only OΓ and OΓ . Another O(pa) operator, OΓ , is expected to appear at higher-order in the coupling expansion. Reduction of O(a) operators with the on-shell condition The two O(pa) operators in eq. (2.28) are mutually related through a total derivative: O(pa)
OΓ
O(pa)
− OΓ
=
X µ
(1 − Gµ (Γ))∂µ Oγµ Γ ,
O(pa)
(2.32)
O(pa)
which means OΓ can be represented by the total derivative operator and OΓ . For onshell matrix elements, on-shell conditions can be imposed to reduce the number of operators. O(pa) By imposing the on-shell condition, i.e., using equation of motions, OΓ in eq. (2.28) O(ma) becomes OΓ , and therefore, the O(a) operators can be replaced by the total-derivative O(ma) operator (2.32) and OΓ . For on-shell matrix elements, the O(a)-improved operators are OΓI = (1 + bΓ ma)OΓ + cΓ a∂µ Oγµ Γ−Γγµ ,
(2.33)
where bΓ and cΓ are O(ma)- and O(pa)-improvement coefficients, respectively.
3
Operator classification — nonlocal quark bilinear operators
The operator-mixing properties of the local quark bilinears and their O(a)-improvement operators discussed in the previous section are well known. In this section, we extend the symmetry arguments used for the local operators to nonlocal ones, which are our main interest in this paper.
–7–
P3 Pl6=3
Γ = 1+/− E E/O
T3 Tl6=3
E/O E
C χ
E/O V
γi+/− O E/O(l=i) O/E(l6=i) E/O O(l=i) E(l6=i) O/E I
γ3+/− E O/E
γ5+/− O O/E
O/E E
O/E O
O/E I
E/O V
iγi γ5+/− E O/E(l=i) E/O(l6=i) O/E E(l=i) O(l6=i) E/O I
iγ3 γ5+/− O E/O E/O O E/O I
σi3+/− O O/E(l=i) E/O(l6=i) O/E O(l=i) E(l6=i) O/E V
ijk σjk+/− E E/O(l=i) O/E(l6=i) E/O E(l=i) O(l6=i) O/E V
Table 3. Transformation properties of the nonlocal operator OΓ± (δz): even/odd (E/O) under parity (Pµ ), time reversal (Tµ ) and charge conjugation (C), and variant/invariant (V/I) under chiral rotation (χ). i, j, k 6= 3.
3.1
Nonlocal operators at O(a0 )
We specify the nonlocal operator to be a quark bilinear in which the two quark fields are separated by δz in the spatial direction z, connected by a Wilson line: OΓ (δz) = ψ(x + δz)ΓU3 (x + δz; x)ψ(x).
(3.1)
Because there is asymmetry on the nonlocal operator between z and the remaining directions, we should take this into account in treating Dirac matrices Γ as Γ ∈ {1, γi , γ3 , γ5 , iγi γ5 , iγ3 γ5 , σi3 , ijk σjk },
(3.2)
where i, j and k represent directions perpendicular to z, that is, i, j, k 6= 3. The Hermitian conjugate of the nonlocal operator clearly involves the sign-flipping of δz, thus, the matrix elements hp|OΓ (δz)|pi generally become complex. Under the parity and time reversal, the nonlocal operator transforms as Pl6=3
OΓ (δz) −−−→ Oγl Γγl (−δz), Tl6=3
OΓ (δz) −−−→ Oγ5 γl Γγl γ5 (δz),
P
3 OΓ (δz) −→ Oγ3 Γγ3 (δz),
(3.3)
3 OΓ (δz) −→ Oγ5 γ3 Γγ3 γ5 (−δz).
(3.4)
T
The transformation involves a sign-flip on δz, so again we define combinations: OΓ± (δz) =
1 {OΓ (δz) ± OΓ (−δz)} . 2
(3.5)
Hermitian conjugate of the nonlocal operator above shows (OΓ± (δz))† = ∓G4 (Γ)OΓ± (δz),
(3.6)
and thus the matrix elements hp|OΓ± (δz)|pi are either pure-real or pure-imaginary, depending on the value of G4 (Γ)(= ±1). The even/oddness of the operators under parity and time-reversal transformations are summarized in table 3.
–8–
Under charge conjugation, the quark bilinears transform as C
OΓ± (δz) − → ±O(CΓC −1 )> ± (δz).
(3.7)
The even/oddness of the operators under charge conjugation are also summarized in table 3. We note that whereas the charge-conjugation property was not essential to prohibit the operators from mixing for the local quark bilinear at O(a0 ), it is crucial to the nonlocal case, because only employing parity and time reversal cannot kill the mixing between γi and σi3 or between γ5 and γ3 γ5 . At this stage, possible mixings between 1 and γ3 , and between iγi γ5 and σij are observed. This mixing is possible, because the nonlocal operators have an “extra hand”, δz that provides room to adjust its symmetry property. In a unified way, the mixing can be written as OΓ± (δz) ←→ (1 + G3 (Γ))Oγ3 Γ∓ (δz),
(3.8)
which is consistent with the mixing pattern in refs. [13, 17]. The chiral rotation properties for the nonlocal operators are shown in table 3. By imposing the chiral symmetry onto the quark bilinears, we can see that the operator mixings between different Γ are entirely killed, which is different from the local-operator case where the mixing between the local operators is prohibited even without chiral symmetry. 3.2
Nonlocal operators at O(pa) and O(ma)
We extend the discussion for O(pa)-improvement local operators to the nonlocal case. Because the nonlocal operator has a specific direction, z, the Euclidean spacetime is naturally decomposed into 3 + 1 dimensions. Thus, the candidates for the O(pa) operators are → − b b x)Γ D / α ψ(x), → (δz) = ψ(x + 3δz)U QΓ− (x + 3δz; (3.9) 3 Dα → − b b x) D / α Γψ(x), → (δz) = ψ(x + 3δz)U Q− (3.10) 3 (x + 3δz; D αΓ ← − b b x)ψ(x), / α U3 (x + 3δz; − (δz) = ψ(x + 3δz)Γ D (3.11) QΓ← Dα ← − b D b x)ψ(x), / α ΓU3 (x + 3δz; − (δz) = ψ(x + 3δz) Q← (3.12) D Γ α
→ − − P → − → → − / 3 = γ3 D 3 , D / ⊥ = µ6=3 γµ D µ , where α ∈ [3, ⊥] and we introduce shorthand notations: D → (δz) and Q ← − (δz) are redundant. and so on. Among them, Q− D 3Γ ΓD 3 → − / and As in the case of the local quark bilinear, inserting the covariant derivatives, D ← − / into the nonlocal bilinear does not change the parity and time-reversal properties of the D, O(a0 ) operator, OΓ (δz). Because parity and time reversal contain a sign-flipping of δz, it is natural to define combinations o 1n → − → − → − → (δz) ± Q − → − → (−δz) , QΓ− (δz) = Q (3.13) D α ±/ D α Γ± ΓD α / D α Γ ΓD α / D α Γ 2 o 1n − ← − ← − ← − (δz) ± Q ← − ← − (−δz) . QΓ← (δz) = Q (3.14) D α ±/ D α Γ± ΓD α / D α Γ ΓD α / D α Γ 2 Under charge conjugation, the above operators transform as → QΓ− D
− →
α ±/ D α Γ±
C
− (δz) − → ∓Q← D
α (CΓC
–9–
← −
−1 )> ±/(CΓC −1 )>D
α±
(δz),
(3.15)
P3 Pl6=3 T3 Tl6=3 D (+)
α C(QΓ±/Γ± )
Dα (−) C(QΓ±/Γ± )
χ
Γ = 1+/− E E/O
γ3+/− E O/E
γ5+/− O O/E
O/E E
O/E O
O/E
γi+/− O E/O(l=i) O/E(l6=i) E/O O(i=l) E(l6=i) E/O
E/O
E/O
O/E
I
V
E/O E
γ3 γ5+/− O E/O
O/E
γi γ5+/− E O/E(l=i) E/O(l6=i) O/E E(l=i) O(l6=i) O/E
O/E
σi3+/− O O/E(l=i) E/O(l6=i) O/E O(l=i) E(l6=i) E/O
ijk σjk+/− E E/O(l=i) O/E(l6=i) E/O E(l=i) O(l6=i) E/O
O/E
E/O
E/O
E/O
O/E
O/E
V
I
V
V
I
I
E/O O
D (±)
α Table 4. Transformation properties of the nonlocal operators at O(pa) QΓ±/Γ (δz): even/odd (E/O) under parity (Pµ ), time reversal (Tµ ) and charge conjugation (C), and variant/invariant (V/I) under chiral rotation (χ). i, j, k 6= 3.
− Q← D
← −
α Γ±/ΓD α ±
C
→ (δz) − → ∓O(CΓC −1 )>− D
− →
α ±/ D α (CΓC
−1 )> ±
(δz),
(3.16)
which indicates we should define their combinations D (+)
α − QΓ±/Γ± (δz) = Q← D
← −
→ (δz) + QΓ− D
− →
(δz),
(3.17)
← −
→ (δz) − QΓ− D
− →
(δz),
(3.18)
α Γ±/ΓD α ±
D (−)
α − (δz) = Q← QΓ±/Γ± D
α Γ±/ΓD α ±
α ±/ D αΓ±
α ±/ D αΓ±
and the charge conjugation is D (+)
C
D (+)
(3.19)
D (−)
C
D (−)
(3.20)
α α QΓ±/Γ± (δz) − → ∓Q(CΓC −1 )> ±/(CΓC −1 )> ± (δz), α α QΓ±/Γ± (δz) − → ±Q(CΓC −1 )> ±/(CΓC −1 )> ± (δz).
Their parity, time-reversal, and charge-conjugation properties are presented in table 4. At this stage, we observe that each O(a0 ) operator, OΓ (δz), can have O(pa) operators with Dα α two Γs, QD Γ (δz) and QΓ0 =γ3 Γ (δz). The chiral-rotation properties of the nonlocal operators are important. In the local O(pa) case, the properties differ between OΓ and OΓ , because there is an extra γµ in the Dα O(pa) bilinear. In the present case, QΓγ3 (δz) has the same rotation properties as OΓ (δz) and survives even if the discrete symmetries and the chiral symmetry are all imposed. While this fact is somewhat counterintuitive, it is partly understandable when we consider using an auxiliary-field method to express the Wilson-line operator. The auxiliary field is similar to a static heavy-quark field [12, 13, 27], where the static heavy-light system has O(pa) discretization errors even if the light quarks respect the chiral symmetry. As in the local-operator case, the O(ma) nonlocal bilinear is simply written as b b QM Γ (δz) = mψ(x + 3δz)ΓU3 (x + 3δz; x)ψ(x).
(3.21)
It has the same transformation properties as OΓ (δz) under parity, time reversal, and charge conjugation, but it shows a difference for the chiral rotation. However, chiral symmetry
– 10 –
D (−)
OΓ
Oγ3αΓ
D (−)
, Oγ αΓ 3
, OγM3 Γ
χB
G3 (Γ) = +1 : χB
D (−)
O γ3 Γ
OΓ α
OΓ
Oγ3αΓ
D (−)
, OΓ α
D (+)
, OΓM
D (+)
, Oγ αΓ 3
χB
G3 (Γ) = −1 :
D (−)
OΓ α
D (−)
, OΓ α
, OΓM
Figure 1. Operator-mixing pattern for the nonlocal quark bilinears up to O(a). Up and down arrows represent operator mixing at O(a0 ), while dashed lines express relations between O(a0 ) and O(a). Blue arrows and lines exist due to lack of chiral symmetry. Red lines depict relations present even when imposing chiral symmetry.
does not prevent the O(ma), because QM γ3 Γ (δz) is also allowed as an O(ma) operator of the OΓ (δz). 3.3
Summary for nonlocal quark bilinears
Here, we summarize the operator-mixing pattern described in this section in a more unified way. The discussion reveals the O(a0 ) operator
can mix with
b b x)ψ(x), OΓ (δz) = ψ(x + 3δz)ΓU 3 (x + 3δz;
(3.22)
OχB Γ (δz) = (1 + G3 (Γ))Oγ3 Γ (δz),
(3.23)
when chiral symmetry is broken, where a subscript χB expresses the effect of chiralsymmetry breaking. Having definitions ← − → − D (±) b b x)ψ(x), / αΓ ± Γ D / α U3 (x + 3δz; QΓ α (δz) = ψ(x + 3δz) D (3.24) ← − → − D (±) b b x)ψ(x), /α ± D / α Γ U3 (x + 3δz; QΓ α (δz) = ψ(x + 3δz) ΓD (3.25) b b QM Γ (δz) = mψ(x + 3δz)ΓU3 (x + 3δz; x)ψ(x),
(3.26)
possible O(a) operators are O(pα a)
(δz) = (1 + G3 (Γ))Qγ3αΓ
O(pα a)
(δz) = (1 + G3 (Γ))Qγ αΓ
OΓ OΓ
O(ma)
OΓ
D (−) D (−) 3
(δz) + (1 − G3 (Γ))Oγ3αΓ
D (+)
(δz),
(3.27)
(δz) + (1 − G3 (Γ))Oγ αΓ
D (+)
(δz),
(3.28)
(δz) = (1 + G3 (Γ))QM γ3 Γ (δz),
3
(3.29)
O(p a)
D (−)
(δz),
(3.30)
O(p a)
D (−)
(δz),
(3.31)
OχB Γα (δz) = QΓ α OχB Γα (δz) = QΓ α
– 11 –
O(ma)
OχB Γ (δz) = QM Γ (δz),
(3.32)
where again a subscript χB expresses the effect of chiral-symmetry breaking. Among them, O(p a) O(p a) OΓ 3 and OχB Γ3 are redundant. Figure 1 sketches the operator mixing at O(a0 ) and the relation with O(pa) and O(ma) operators in a unified manner. Notably, there are O(a) contributions even when chiral fermions are employed, which is quite different from the local-operator case. Reduction of O(a) operators with on-shell condition
⊥ ⊥ ) are mutually related through total The two kinds of O(pa) operators (QD and QD Γ Γ derivative operators: X D (+) D (+) QΓ ⊥ (δz) + QΓ ⊥ (δz) = (1 + Gµ (Γ))∂µ Oγµ Γ (δz), (3.33)
µ6=3
D (−) QΓ ⊥ (δz)
−
D (−) QΓ ⊥ (δz)
=
X
(1 − Gµ (Γ))∂µ Oγµ Γ (δz).
(3.34)
µ6=3
D (±)
Therefore, QΓD (δz) can be removed by introducing total-derivative operators. On QΓ ⊥ we can impose on-shell conditions, and rewrite it as D (+)
QΓ ⊥ D (±)
QΓ 3
D (+)
(δz) = −2QΓ 3
D (−)
QΓ ⊥
(δz),
(δz) = 2QM Γ (δz).
(δz),
(3.35)
(δz) can also be written using total-derivative operators: D (+)
QΓ 3
(δz) = [(1 − G3 (Γ))(∂δz − ∂3 ) + ∂3 ] Oγ3 Γ (δz),
(3.36)
= [(1 + G3 (Γ))(∂δz − ∂3 ) + ∂3 ] Oγ3 Γ (δz).
(3.37)
D (−) QΓ 3 (δz)
We classify the O(a) operators with on-shell conditions into four cases: • G3 (Γ) = +1 with chiral symmetry: [2∂δz − ∂3 ] OΓ (δz),
X
mOγ3 Γ (δz),
• G3 (Γ) = −1 with chiral symmetry: [2∂δz − ∂3 ] OΓ (δz),
X
(3.38)
(1 − Gµ (Γ))∂µ Oγµ γ3 Γ (δz).
(3.39)
µ6=3
• G3 (Γ) = +1 without chiral symmetry: [2∂δz − ∂3 ] OΓ (δz),
(1 + Gµ (Γ))∂µ Oγµ γ3 Γ (δz).
µ6=3
X
mOγ3 Γ (δz),
(1 + Gµ (Γ))∂µ Oγµ γ3 Γ (δz),
µ6=3
[2∂δz − ∂3 ] Oγ3 Γ (δz),
mOΓ (δz),
X
(1 − Gµ (Γ))∂µ Oγµ Γ (δz).
(3.40)
µ6=3
• G3 (Γ) = −1 without chiral symmetry: X [2∂δz − ∂3 ] OΓ (δz), (1 − Gµ (Γ))∂µ Oγµ γ3 Γ (δz), µ6=3
∂3 Oγ3 Γ (δz),
mOΓ (δ),
X
(1 − Gµ (Γ))∂µ Oγµ Γ (δz).
µ6=3
– 12 –
(3.41)
b z) (0 + 3
k + p0 p0
(0) k+p
p k
Figure 2. One of the one-loop Feynman diagrams for the Green function with nonlocal quark bilinear, Λ1−loop (p0 , p, m). Incoming and outgoing momenta are p and p0 , respectively. Γ,δz
3.4
Perturbative analysis of O(a) for nonlocal quark bilinears
In this section, it was shown that O(a) operators for the nonlocal operators cannot be prohibited even if chiral symmetry is preserved. This fact can also be seen in the perturbative analysis. To see if the O(a) discretization error exists even when the chiral symmetry is not broken, we can (for simplicity) investigate it using continuum theory external-momentum and quark-mass expansion.2 We calculate the diagram shown in figure 2, which is a simple vertex-type diagram including the nonlocal operator. Because we only check the Dirac structure, the loop integration is not carried out, and any UV regulator is not introduced. For gauge fixing, we use Feynman gauge. 0 The one-loop amputated Green function of the diagram in figure 2, Λ1−loop Γ,δz (p , p, m), is simply written as 0
0 −ip3 δz Λ1−loop Γ,δz (p , p, m)/e Z δµν δAB 1 1 = Γ+ (−igγµ T A ) Γe−ik3 δz (−igγν T B ) 0 2 / / k i(k + p i(k + p /)+m /) + m k O(1) O(m) = 1 + g 2 GF AΓ,δz Γ + g 2 GF AΓ,δz (1 + G3 (Γ))mγ3 Γ n o O(p ) 0 0 +g 2 GF AΓ,δz3 i (1 + G3 (Γ))(−p /3 γ3 Γ − γ3 Γp /3 ) + (1 − G3 (Γ))(−p /3 γ3 Γ + γ3 Γp /3 ) o n O(p ) 0 0 +g 2 GF AΓ,δz⊥ i (1 + G3 (Γ))(−p /⊥ ) /⊥ γ3 Γ − γ3 Γp /⊥ ) + (1 − G3 (Γ))(−p /⊥ γ3 Γ + γ3 Γp n o O(p ) 0 0 +g 2 GF AΓ,δz⊥ i (1 + G3 (Γ))(−γ3 Γp − p γ Γ) + (1 − G (Γ))(−γ Γ p + p γ Γ) /⊥ /⊥ 3 /⊥ 3 3 3 /⊥
+O(p02 , p2 , p0 p, p0 m, pm, m2 ),
(3.42)
with one-loop coefficients Z cos(k3 δz) H(Γ) O(1) AΓ,δz = (H(Γ) − G3 (Γ)) k 2 + (−H(Γ) + 4G3 (Γ)) k32 , (3.43) 2 3 (k ) 3 Zk sin(k3 δz)k3 O(m) (−H(Γ) + 2G3 (Γ)) , (3.44) AΓ,δz = (k 2 )3 k Z sin(k3 δz)k3 H(Γ) O(p3 ) 2 2 AΓ,δz = (−2H(Γ) + 5G (Γ)) k + 2 (H(Γ) − 4G (Γ)) k , (3.45) 3 3 3 (k 2 )4 6 k 2
An appropriate infrared (IR) regulator in loop integrals is implicitly assumed to make the formal expansion being enabled.
– 13 –
Z
sin(k3 δz)k3 G3 (Γ) 2 2 (H(Γ) − 6G (Γ)) k + 2H(Γ)k , (3.46) 3 3 (k 2 )4 6 k Z sin(k3 δz)k3 1 O(p⊥ ) (−H(Γ) + 3G3 (Γ)) k 2 + H(Γ)k32 , (3.47) AΓ,δz = 2 4 (k ) 3 k P where H(Γ) = 4µ=1 Gµ (Γ). The one-loop perturbative calculation reproduces the O(a) structure in eqs. (3.27), (3.28) and (3.29) in momentum representation. The chiral symmetry breaking effects in eqs. (3.30), (3.31) and (3.32) can be seen when we employ lattice fermions with broken chiral symmetry and apply lattice perturbation theory. In the perturbation theory, the existence of the unphysical operator mixings and O(a) discretization errors in the nonlocal operator are attributed to the phase factor e−ik3 δz caused by the nonlocality. While in the loop integral, integrands with odd power of k vanish, the phase factor always can adjust the power of k3 to be even by the “extra hand”, δz, leading to the surviving contributions. O(p )
AΓ,δz⊥ =
4
Summary
In this work, we have presented the operator mixing pattern for a class of nonlocal quark bilinear (1.1) on the lattice by using action symmetries: parity, time reversal, charge conjugation, and chiral symmetry. Unlike local bilinears, the chiral symmetry is so crucial to prevent the mixing. Also, we have shown the classification of O(a)-improvement operators for the nonlocal bilinear by extending the symmetry discussion. One of the significance is that a part of the O(a) operators cannot be prohibited, by the chiral symmetry, from emerging. An one-loop perturbative calculation with external-momentum and quark-mass expansion also suggests this conclusion. The quasi-PDF approach intrinsically has to involve a large hadron momentum to reduce higher-twist effects. The large momentum is challenging because as the momentum becomes large, signal to noise ratio in lattice simulations gets worse. This difficulty is overcome by introducing the momentum-smearing technique presented in ref. [28], whose effectiveness for the quasi-PDF method has been reported in ref. [29]. While the signal in the simulation could be improved by using the momentum-smearing, we would suffer from the lattice artifact due to the large momentum. The lattice discretization error is expected to be reduced by implementing O(a)-improvement program. Determination of the O(a)-improvement coefficients for the nonlocal quark bilinear using the one-loop lattice perturbation, and possibly nonperturbative approach, is soon to be addressed, and with these improvements, we will provide steady progress toward the first principle calculation of the PDFs.
Acknowledgments JWC is partly supported by the Ministry of Science and Technology, Taiwan, under Grant No. 105-2112-M-002-017-MY3 and the Kenda Foundation. TI is supported by Science and Technology Commission of Shanghai Municipality (Grants No. 16DZ2260200). TI and LCJ are supported by the Department of Energy, Laboratory Directed Research and
– 14 –
Development (LDRD) funding of BNL, under contract DE-EC0012704. The work of HL and YY is supposed by US National Science Foundation under grant PHY 1653405. JHZ is supported by the SFB/TRR-55 grant “Hadron Physics from Lattice QCD”, and a grant from National Science Foundation of China (No. 11405104). YZ is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DE-SC0011090 and within the framework of the TMD Topical Collaboration.
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