matching quasi generalized parton distributions in the ri/mom … · 2019-02-04 · mit-ctp/5074,...
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MIT-CTP/5074, INT-PUB-19-004
Matching Quasi Generalized Parton Distributions in the RI/MOM scheme
Yu-Sheng Liu,1, ∗ Wei Wang,2, † Ji Xu,2, ‡ Qi-An Zhang,3, 4, § Jian-Hui Zhang,5, 6, ¶ Shuai Zhao,2, ∗∗ and Yong Zhao7, ††
1Tsung-Dao Lee Institute, Shanghai Jiao-Tong University, Shanghai 200240, China2SKLPPC, School of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai 200240, China
3Institute of High Energy Physics, Chinese Academy of Science, Beijing 100049, China4School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
5Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany6Center of Advanced Quantum Studies, Department of Physics,
Beijing Normal University, Beijing 100875, China7Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Within the framework of large momentum effective theory (LaMET), generalized parton distri-butions (GPDs) can be extracted from lattice calculations of quasi-GPDs through a perturbativematching relation, up to power corrections that are suppressed by the hadron momentum. In thispaper, we focus on isovector quark GPDs, including the unpolarized, longitudinally and transverselypolarized cases, and present the one-loop matching that connects the quasi-GPDs renormalized in aregularization-independent momentum subtraction (RI/MOM) scheme to the GPDs in MS scheme.We find that the matching coefficient is independent of the momentum transfer squared. As a conse-quence, the matching for the quasi-GPD with zero skewness is the same as that for the quasi-PDF.Our results provide a crucial input for the determination of quark GPDs from lattice QCD usingLaMET.
I. INTRODUCTION
Understanding the internal structure of nucleons hasbeen an important goal of hadron physics. For manydecades, our knowledge on the structure of nucleons hasbeen mainly relying on experimental measurements oftheir form factors (FFs) and parton distribution func-tions (PDFs). The FFs describe the spatial distribu-tion of charge and current within the nucleon and canbe probed in elastic lepton-nucleon scattering, while thePDFs characterize the longitudinal momentum distribu-tion of quarks and gluons in the nucleon and can be mea-sured in deep-inelastic scattering processes.
The proposal of generalized parton distributions(GPDs) (for a review, see e.g. [1–3]) provides a novelopportunity to characterize the partonic structure of nu-cleons. As a generalization of the PDFs to off-forwardkinematics, the GPDs contain a wealth of new informa-tion on nucleon structure. They naturally encompassthe FFs, PDFs as well as the distribution amplitudes(DAs), and offer a description of the correlations be-tween the transverse position and longitudinal momen-tum of quarks and gluons inside the nucleon, therebygiving access to quark and gluon orbital angular momen-tum contributions to the nucleon spin. Experimentally,the GPDs can be accessed through hard exclusive pro-cesses like deeply virtual Compton scattering or meson
∗ [email protected]† [email protected]‡ [email protected]§ [email protected]¶ [email protected]∗∗ [email protected]†† [email protected]
production. Much effort has been devoted to measur-ing such processes at completed and ongoing experiments(HERA [4–9], COMPASS [10], JLab [11–14]), and will becontinued at planned future facilities such as EIC [15, 16]and EicC [17]. Given the complicated kinematic depen-dence of GPDs, extracting them from the accumulatedexperimental data is in general rather difficult, and oneusually needs to resort to certain models that allow for anextrapolation to kinematic regions that are not accessibledirectly [18].
On the other hand, lattice effort of studying GPDshas been mainly focused on the computation of their mo-ments [19–25]. The full distribution can be reconstructedin principle if all their moments are known. However, thenumber of moments that are calculable on lattice is verylimited, owing to power divergent mixing between dif-ferent moments operators and increasing stochastic noisefor high moments operators.
In the past few years, a new theoretical frameworkhas been developed to circumvent the above difficulties,which is now known as the large momentum effective the-ory (LaMET) [26, 27]. According to LaMET, the GPDscan be extracted from lattice QCD calculations of ap-propriately constructed static-operator matrix elements,which are named the quasi-GPDs. The quasi-GPDs areusually hadron-momentum dependent but time indepen-dent, and thus can be readily computed on the lattice.After being renormalized nonperturbatively in an appro-priate scheme, the renormalized quasi-GPDs can then bematched onto the usual GPDs through a factorizationformula accurate up to power corrections that are sup-pressed by the hadron momentum [28, 29].
Since LaMET was proposed, a lot of progress has beenachieved both with respect to the theoretical understand-ing of the formalism [28–71] and the direct calculationof PDFs from lattice QCD [33, 38, 45, 46, 48, 72–82].
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The prospects of extracting transverse momentum de-pendent (TMD) PDFs from lattice with LaMET hasbeen investigated in Refs. [83–87]. In particular, a mul-tiplicative renormalization of both the quark [40, 43, 44]and gluon [69, 70] quasi-PDFs has been established incoordinate space. This allows for a nonperturbativerenormalization in the regularization-independent mo-mentum subtraction (RI/MOM) scheme [88]. For theisovector quark quasi-PDFs, this has been carried out inRefs. [45, 55, 77, 80] (see also [42, 46, 76]). The relevanthard matching kernel in the same scheme has also beencomputed up to one loop [55, 65, 82]. Despite limitedvolumes and relatively coarse lattice spacings, the state-of-the-art nucleon isovector quark PDFs determined fromlattice data at the physical point have shown a reason-able agreement [76, 77, 80] with phenomenological resultsextracted from the experimental data [89–93]. Of course,a careful study of theoretical uncertainties and lattice ar-tifacts is still needed to fully establish the reliability ofthe results.
As for the GPDs, there have been studies on theperturbative matching of the isovector quark quasi-GPDs [28, 29], which are free from contributions of dis-connected diagrams and mixing with gluon quasi-GPDs.Such studies were performed in a transverse momentumcutoff scheme and therefore not well-suited for the lat-tice implementation. In this paper, we reconsider theone-loop matching for isovector quark quasi-GPDs in theRI/MOM scheme. The results can be used to match thequasi-GPDs calculated in lattice QCD and renormalizedin the RI/MOM scheme onto the GPDs in MS scheme.
The rest of the paper is organized as follows: In Sec. II,we establish our definitions and conventions. In Sec. III,we present a rigorous derivation of the factorization for-mula for the isovector quark quasi-GPD based on op-erator product expansion (OPE). Section IV and V aredevoted to the RI/MOM renormalization and matchingprocedure, respectively. We also explain how to obtainthe matching coefficients of DAs from the one-loop resultsof GPDs in Sec. V. Our summary is given in Sec. VI.
II. DEFINITIONS AND CONVENTIONS
The parent function for the quark GPDs, which we callparent-GPD for simplicity, is defined from the Fourier
transform of the off-forward matrix element of a light-cone correlator,
F (Γ, x, ξ, t, µ)
=
∫dζ−
4πe−ixζ
−P+〈P ′′, S′′|O(Γ, ζ−)|P ′, S′〉 , (1)
where x ∈ [−1, 1], the light-cone coordinates ζ± = (ζt ±ζz)/√
2 with ζµ = (ζt, ζx, ζy, ζz), and the hadron state|P ′, S′〉 (|P ′′, S′′〉) is denoted by its momentum and spin.The parent-GPD is defined in the MS scheme and µ isthe renormalization scale. The kinematic variables aredefined as
∆ ≡ P ′′ − P ′, t ≡ ∆2, ξ ≡ −P′′+ − P ′+
P ′′+ + P ′+= − ∆+
2P+,
(2)
where without loss of generality we choose a particularLorentz frame so that the average momentum
Pµ ≡ P ′′µ + P ′µ
2= (P t, 0, 0, P z) , (3)
and only consider the case with 0 < ξ < 1.
The light-cone correlator is given by the gauge-invariant nonlocal quark bilinear
O(Γ, ζ−) = ψ
(ζ−
2
)ΓλaW+
(ζ−
2,−ζ
−
2
)ψ
(−ζ−
2
),
(4)
where Γ = γ+, γ+γ5, and iσ+⊥ = γ⊥γ+ correspond tothe unpolarized, helicity, and transversity parent-GPDs,respectively. λ is a Gell-Mann matrix in flavor space, e.g.,λ3 corresponds to flavor isovector (u − d) distribution.The lightlike Wilson line is
W+(ζ−2 , ζ−1 ) = P exp
[−igs
∫ ζ−2
ζ−1
A+(η−)dη−
]. (5)
The GPDs are defined as form factors of the parent-GPD (we follow the convention of Ref. [2]),
F (Γ, x, ξ, t, µ) =1
2P+u(P ′′, S′′)
{H(Γ, x, ξ, t, µ)Γ + E(Γ, x, ξ, t, µ)
[ /∆, Γ]
4M
+H ′(Γ, x, ξ, t, µ)P [+∆⊥]
M2+ E′(Γ, x, ξ, t, µ)
γ[+P⊥]
M
}u(P ′, S′) , (6)
where [ /∆, Γ] = 2iσ+µ∆µ, 2γ5∆+, and 2(γ+∆⊥− γ⊥∆+) for Γ = γ+, γ+γ5, and iσ+⊥, respectively; M is the
3
hadron mass; H, E, H ′, and E′ are the GPDs. Notethat H ′ and E′ are nonzero only for transversity GPD.
To calculate the quark GPDs within LaMET, we con-sider a quark quasi-parent-GPD defined from an equal-time correlator:1
F (Γ, x, ξ, t, P z, µ)
=
∫dz
4πeixzP
z 2P z
N〈P ′′, S′′|O(Γ, z)|P ′, S′〉 , (7)
where µ is the renormalization scale in a particularscheme, and N is a normalization factor that dependson the choice of Γ. For example, N = 2P z for Γ = γz.
The nonlocal quark bilinear
O(Γ, z) = ψ(z
2
)ΓλaWz
(z2,−z
2
)ψ(−z
2
)(8)
is along the z direction with a spacelike Wilson line
Wz(z2, z1) = P exp
[igs
∫ z2
z1
Az(z′)dz′]. (9)
The kinematic variables are similar to those in Eq. (2)except that the “quasi” skewness parameter
ξ = −P′′z − P ′z
P ′′z + P ′z= − ∆z
2P z= ξ +O
(M2
P 2z
), (10)
which is equal to ξ up to power corrections. From now
on we will replace ξ with ξ by assuming that the powercorrections are small.
The quasi-GPDs are defined as form factors of thequasi-parent-GPD,
F (Γ, x, ξ, t, P z, µ) =1
Nu(P ′′, S′′)
{H(Γ, x, ξ, t, P z, µ)Γ + E(Γ, x, ξ, t, P z, µ)
[ /∆,Γ]
4M
+ H ′(Γ, x, ξ, t, µ)P [z∆⊥]
M2+ E′(Γ, x, ξ, t, µ)
γ[zP⊥]
M
}u(P ′, S′) , (11)
where H, E, H ′, and E′ are the quasi-GPDs with sup-
port x ∈ (−∞,∞). Again, H ′ and E′ are nonzero onlyfor transversity quasi-GPD. In order to minimize oper-ator mixing on lattice, we choose Γ = γt, γzγ5, and
iσz⊥ for the unpolarized, helicity, and transversity quasi-GPDs [42, 47], respectively, which all correspond to thesame normalization factor N = 2P t.
According to LaMET [26, 27], the quasi-GPDs andGPDs are related through a factorization formula. Forexample,
H(Γ, x, ξ, t, P z, µ) =
∫ 1
−1
dy
|y| CΓ
(x
y,ξ
y,µ
µ,µ
yP z
)H(Γ, y, ξ, t, µ) +O
(M2
P 2z
,t
P 2z
,Λ2
QCD
x2P 2z
), (12)
where M2/P 2z and t/P 2
z are kinematic power corrections;Λ2
QCD/(x2P 2
z ) is the higher-twist correction. Since the
choice of Γ corresponds to a unique Γ, we suppress thelabel Γ in the matching coefficient CΓ. Similar factoriza-
tion formulas also exist for H ′, E, and E′. Equation (12)with its explicit form will be rigorously derived in thenext section.
1 We remind the reader that the tilde notation in GPD communityis usually referring to helicity GPDs. In this work, we use tilde
notation to specify quasi-GPDs.
4
III. OPERATOR PRODUCT EXPANSION AND THE FACTORIZATION FORMULA
In this section, we derive the explicit form of the factorization formula for the quasi-GPDs using the OPE of
the nonlocal quark bilinear O(Γ, z). The same method has been used for the “lattice cross section” [94] and quasi-PDF [58], which are both forward matrix elements of a nonlocal gauge-invariant operator. In the case of nonsinglet
quasi-PDF, O(γz, z, µ) (e.g., in the MS scheme) can be expanded in terms of local gauge-invariant operators in the|z| → 0 limit [58],
O(γz, z, µ) =
∞∑n=0
[Cn(µ2z2)
(−iz)nn!
eµ1 · · · eµnOµ0µ1···µn(µ) + higher-twist terms
], (13)
where eµ = (0, 0, 0, 1), µ0 = z, Cn = 1 + O(αs) is the Wilson coefficient, and Oµ0µ1···µn(µ) is the only allowedrenormalized traceless symmetric twist-2 quark operators at leading power in the OPE,
Oµ0µ1...µn(µ) =Zn+1(ε, µ)[ψγ(µ0i
←→D µ1 · · · i←→D µn)ψ − trace
], (14)
where←→D = (
−→D −←−D)/2. Here Zijn+1(ε, µ) are multiplicative MS renormalization factors and (µ0 · · ·µn) stands for the
symmetrization of these Lorentz indices. Similar technique can be applied to gluon and singlet quark quasi-GPDs byincluding the corresponding twist-2 operators on the right-hand side of Eq. (13) as well as the mixing between quarksand gluons. Such an extension has been done for the quasi-PDF in Ref. [95].
The multiplicative renormalization shown in Eq. (14) is valid for the forward case only, as it is known that inthe off-forward case, Oµ0µ1...µn(µ) can mix with other twist-2 operators with overall derivatives i∂µ according to therenormalization group equation [96]
µ2 d
dµ2Oµ0µ1...µn(µ) =
[n/2]∑m=0
Γnm
[i∂(µ1 · · · i∂µ2m ψγµ0i
←→D µ2m+1 · · · i←→D µn)ψ − trace
], (15)
where the anomalous dimension Γ is an upper triangle matrix. In off-forward matrix elements, the overall derivativei∂µ contributes a factor of the momentum transfer ∆µ. As a result, the OPE in Eq. (13) cannot maintain its formunder evolution in µ, so one has to choose the operator bases to be the eigenvectors of Eq. (15) so that each of themis multiplicatively renormalizable.
At leading logarithmic (LL) accuracy, Eq. (15) is diagonalized by the conformal operators [96, 97],
nµ0nµ1 · · ·nµnOµ0µ1...µn(µ) = (in · ∂)nψ/n C3/2n
(n · −→D − n · ←−Dn · −→∂ + n · ←−∂
)ψ − traces , (16)
where nµ is an arbitrary four vector, and C3/2n (η) is the Gegenbauer polynomial. Beyond LL, the conformal operators
start mixing with each other, but Eq. (15) can still be diagonalized with the “renormalization group improved”conformal operators [96, 98]
nµ0nµ1 · · ·nµnO′µ0µ1...µn(µ) =
n∑m=0
Bnm(µ)
[(in · ∂)nψ/n C3/2
m
(n · −→D − n · ←−Dn · −→∂ + n · ←−∂
)ψ − traces
], (17)
where Bnn = 1.As a result, the nonlocal operator O(γz, z, µ) should be generally expanded in terms of these improved conformal
operators with modified kinematic factors.For µ0 = µ1 = · · · = µn = +, the off-forward matrix element of the conformal operator Oµ0µ1...µn(µ) is given by
〈P ′|On+1︷ ︸︸ ︷
+ + . . .+(µ)|P 〉 =〈P ′|(i∂+)nψγ+C3/2n
(−→D+ −←−D+
−→∂ + +
←−∂ +
)ψ|P 〉
=(−∆+)n(2P+)
∫ 1
−1
dy C3/2n
(y
ξ
)F (γ+, y, ξ, t, µ)
=(2P+)n+1ξn∫ 1
−1
dy C3/2n
(y
ξ
)F (γ+, y, ξ, t, µ) , (18)
which is also known as the Gegenbauer moments.
5
Using Lorentz covariance, we have for µ0 = µ1 = · · · = µn = z,
〈P ′|On+1︷ ︸︸ ︷
zz . . . z(µ)|P 〉 =(2P z)n+1ξn∫ 1
−1
dy C3/2n
(y
ξ
)F (γ+, y, ξ, t, µ) +O
(M2
P 2z
,t
P 2z
), (19)
where M2, t � P 2z , and we have used ∆ · P = 0. The power corrections originate from the subtracted traces in the
kinematic part of the matrix element, and their exact form will be derived in the future.
Based on Eq. (19), we have the leading-twist approximation of the off-forward matrix element of O(γz, z, µ),
〈P ′|O(γz, z, µ)|P 〉 =2P z∞∑n=0
Cn(µ2z2)Fn(−zP z)n∑
m=0
Bnm(µ) ξn∫ 1
−1
dy C3/2m
(y
ξ
)F (γ+, y, ξ, t, µ)
+O(M2
P 2z
,t
P 2z
, z2Λ2QCD
), (20)
where Fn(−zP z) are partial wave polynomials whose explicit forms are known in the conformal OPE of current-current correlators for the hadronic light-cone distribution amplitudes [99]. The higher-twist terms contribute toO(z2Λ2
QCD).
The polynomiality of C3/2n allows us to define for m ≤ n,
ξnC3/2m
(y
ξ
)= yn
(ξ
y
)nC3/2m
(y
ξ
)≡ ynC ′m
(ξ
y
), (21)
where C ′m is also a polynomial that satisfies
C ′m(x) = xnC3/2m
(1
x
). (22)
If we define the matching coefficients as
Cγz
(x
ξ,y
ξ,µ
ξP z
)=
∫d(ξzP z)
2πeixξ ξP
zz∞∑n=0
Cn
(µ2
(ξP z)2(ξzP z)2
)Fn(−ξzP z)
n∑m=0
Bnm(µ)
∫ 1
−1
dy C3/2m
(y
ξ
),
Cγz
(x
y,ξ
y,µ
yP z
)=
∫d(yzP z)
2πeixy yP
zz∞∑n=0
Cn
(µ2
(yP z)2(yzP z)2
)Fn(−yzP z)
n∑m=0
Bnm(µ)
∫ 1
−1
dy C ′m
(ξ
y
), (23)
then we can Fourier transform Eq. (20) from z to xP z to obtain the quasi-GPD and its factorization formula,
F (γz, x, ξ, t, P z, µ) =
∫ 1
−1
dy
|ξ| Cγz(x
ξ,y
ξ,µ
ξP z
)F (γ+, y, ξ, t, µ) +O
(M2
P 2z
,t
P 2z
,Λ2
QCD
x2P 2z
), (24)
=
∫ 1
−1
dy
|y|Cγz(x
y,ξ
y,µ
yP z
)F (γ+, y, ξ, t, µ) +O
(M2
P 2z
,t
P 2z
,Λ2
QCD
x2P 2z
), (25)
where the second form in Eq. (25) is postulated in Refs. [28, 29]. Since xP z is the Fourier conjugate to z, the higher-twist contribution of O(z2Λ2
QCD) in Eq. (20) should be of O(Λ2QCD/(x
2P 2z )) in momentum space with an enhancement
at small x. Such enhancement at small x, as well as a 1/(1−x) factor, was also found to exist in the power correctionsfrom renormalon ambiguities in the OPE of quasi-PDFs [71]. Based on Eqs. (24) and (25), we can infer that the
matching coefficients for the quasi-GPDs H, H ′, E, and E′ must be the same.For the helicity and transversity quasi-GPDs, γz in Eq. (13) is replaced by γzγ5 and iσz⊥ respectively, and the local
twist-two operators Oµ0µ1···µn are also replaced accordingly. This will change the kinematic factors in Eqs. (18)–(20),as their tensor structure involves the spin vector of the external state, but it does not affect the form of OPE inEq. (20), nor that of the factorization formulas in Eqs. (24) and (25).
The two matching coefficients in Eqs. (24) and (25) are related to each other by
Cγz
(x
y,ξ
y,µ
yP z
)=
∣∣∣∣yξ∣∣∣∣ Cγz (xξ , yξ , µ
ξP z
). (26)
6
The factorization formulas are similar to the evolution equations for the GPD [100, 101]. Notably, at zero skewnessξ = 0, we have
F (γz, x, 0, t, P z, µ) =
∫ 1
−1
dy
|y|Cγz(x
y, 0,
µ
yP z
)F (γ+, y, 0, t, µ) +O
(M2
P 2z
,t
P 2z
,Λ2
QCD
x2P 2z
), (27)
where the matching kernel Cγz (x/y, 0, µ/(yPz)) is exactly the same matching coefficient for the MS quasi-PDF [58],
even when t 6= 0. Moreover, in the forward limit ξ → 0 and t → 0, Eq. (27) is exactly the factorization formula forthe MS quasi-PDF [58].
On the other hand, in the limit ξ → 1 and t→ 0, we obtain the factorization formula for the quasi-DA,
F (γz, x, 1, t = 0, P z, µ) =
∫ 1
−1
dy Cγz(x, y,
µ
P z
)F (γ+, y, 1, t = 0, µ) +O
(M2
P 2z
,Λ2
QCD
x2P 2z
), (28)
whose explicit form has been postulated in Refs. [38, 59, 102].The same procedure described above also applies to the Γ = γt case. This finishes our derivation of the factorization
formula for the isovector quark quasi-GPD, which will enable us to identify the matching coefficients from the one-loopcalculation in Sec. V.
IV. RENORMALIZATION
Following the strategy in Ref. [102], the UV divergence
of the quasi-GPD only depends on the operator O(Γ, z),not on the external states. We can choose the same renor-malization factor as the one for the quasi-PDF [55, 65].For each value of z, the RI/MOM renormalization fac-tor Z is calculated nonperturbatively on lattice by im-posing the condition that the quantum corrections ofthe correlator in an off-shell quark state vanish at scales{µ} = {p2 = −µ2
R, pz = pzR} [42, 55]
Z(Γ, z, a, µR, pzR) =
〈p|O(Γ, z, a)|p〉〈p|O(Γ, z, a)|p〉tree
∣∣∣∣∣{µ}
(29)
where O(Γ, z, a) is the discretized version of O(Γ, z) onlattice in Eq. (8) with spacing a; the bare matrix element
〈p|O(Γ, z, a)|p〉 is obtained from the amputated Green’s
function Λ(Γ, z, a, p) of O(Γ, z, a), which is calculated onlattice, with a projection operator P for the Dirac matrix
〈p|O(Γ, z, a)|p〉 = Tr [Λ(Γ, z, a, p)P] . (30)
In a systematic calculation of GPD, we start with thebare matrix element of the nonlocal quark bilinear on
lattice
h(Γ, z, ξ, t, P z, a) =1
N〈P ′′, S′′|O(Γ, z, a)|P ′, S′〉 . (31)
After performing RI/MOM renormalization and takingthe continuum limit, the renormalized matrix element is
hR(Γ, z, ξ, t, P z, µR, pzR)
= lima→0
Z−1(Γ, z, a, µR, pzR)h(Γ, z, ξ, t, P z, a) , (32)
which is to be Fourier transformed into the x-space
F (Γ, x, ξ, t, P z, µR, pzR)
= P z∫
dz
2πeixzP
z
hR(Γ, z, ξ, t, P z, µR, pzR) . (33)
Next, we need to disentangle the terms with different
kinematic dependencies to extract quasi-GPDs from F .Finally, we match quasi-GPDs in the RI/MOM schemeto GPDs in MS scheme according to Eq. (12). Note thatthe continuum limit has been taken after the RI/MOMrenormalization, we can therefore calculate the matchingcoefficient in the continuum as the result is regularizationindependent. For simplicity, we choose dimensional reg-ularization in our calculation. The one-loop result willbe presented in the next section.
V. ONE-LOOP MATCHING COEFFICIENT
When the hadron momentum P z is much greater than M and ΛQCD, the RI/MOM quasi-GPD can be matched
onto the MS GPD through the factorization formula [55, 58]
H(Γ, x, ξ, t, P z, µR, pzR) =
∫ 1
−1
dy
|y| CΓ
(x
y,ξ
y, r,
yP z
µ,yP z
pzR
)H(Γ, y, ξ, t, µ) +O
(M2
P 2z
,t
P 2z
,Λ2
QCD
x2P 2z
), (34)
where r = µ2R/(p
zR)2. Here we have chosen the ex- plicit form of factorization in Eq. (25). To obtain the
7
matching coefficient, we calculate their on-shell masslessquark matrix element in perturbation theory by replac-ing the hadron states in Eqs. (1) and (7) with the quarkstates carrying momentum p + ∆/2 and p − ∆/2 withpµ = (pt, 0, 0, pz).
At tree level, the GPDs and quasi-GPDs are
H(0)(Γ, x, ξ, t) = H(0)(Γ, x, ξ, t, pz) = δ(1− x) , (35)
H ′(0) = H ′(0) = E(0) = E(0) = E′(0) = E′(0) = 0 . (36)
At one-loop order, H(1) and H(1) are nonzero and notequal, so their next-to-leading order (NLO) matching
kernel is nontrivial; since H ′(1) = H ′(1), a two-loop cal-culation is needed to extract the NLO matching kernel;
E(1), E(1), E′(1), and E′(1) vanish for massless quarks,which agrees with the GPD quark-target model calcu-
lation [103]. For massive quarks, E(1) = E(1) 6= 0 and
E′(1) = E′(1) 6= 0 according to Refs. [28, 29], so the
NLO matching kernel for E(1) and E′(1) can only be ex-tracted from the two-loop matrix elements in massivequark states. This can be a cross check of the factoriza-tion formulas in Eqs. (24) and (25), which, however, isbeyond the scope of this work.
In order to combine the “real” and “virtual” contribu-tions (defined in Ref. [55]) in a compact form at one-looplevel, we introduce a plus function defined as∫ ∞
−∞dx[h(x)]+g(x) =
∫ ∞−∞
dxh(x)[g(x)− g(1)
](37)
with two arbitrary functions h(x) and g(x) which couldbe piecewise.h(x) can have a single pole at x = 1, whereasg(x) is regular at x = 1. By taking the limit pt → pz, weobtain the matching kernel for the gauge-invariant barequasi-GPD and MS GPD in a quark,
C(1)B
(Γ, x, ξ,
pz
µ,µ
µ′
)= H
(1)B (Γ, x, ξ, t, pz, µ′, ε)−H(1)(Γ, x, ξ, t, µ, ε) , (38)
where the subscript B denotes “bare” and the ultraviolet (UV) divergence is regulated by dimensional regularization
(D = 4 − 2εUV); the infrared (IR) divergences in H(1)B and H(1) are regulated by t and dimensional regularization
(D = 4 − 2εIR), and canceled out in C(1)B ; there is still UV divergence remaining due to the virtual contribution for
transversity GPD. The results are
C(1)B
(Γ, x, ξ,
pz
µ,µ
µ′
)= f1
(Γ, x, ξ,
pz
µ
)+
+ δΓ,iσz⊥δ(1− x)αsCF
4π
[− 1
εUV+ ln
(µ2
µ′2
)](39)
where δa,b is the Kronecker delta,
f1
(Γ, x, ξ,
pz
µ
)=αsCF
2π
G1(Γ, x, ξ) x < −ξG2(Γ, x, ξ, pz/µ) |x| < ξG3(Γ, x, ξ, pz/µ) ξ < x < 1−G1(Γ, x, ξ) x > 1
, (40)
and
G1(γt, x, ξ) = G1(γzγ5, x, ξ) = −[
1
x− 1− x
2ξ+
1 + x
2(1 + ξ)
]lnx− 1
x+ ξ+ (ξ → −ξ) , (41)
G1(iσz⊥, x, ξ) = − x+ ξ
(x− 1)(1 + ξ)lnx− 1
x+ ξ+ (ξ → −ξ) , (42)
G2
(γt, x, ξ, pz/µ
)=
(x+ ξ)(1− x+ 2ξ)
2(1− x)ξ(1 + ξ)
[ln
4(1− x)2(x+ ξ)(pz)2
(ξ − x)µ2− 1
]+
x+ ξ2
ξ(1− ξ2)lnξ − x1− x , (43)
G2 (γzγ5, x, ξ, pz/µ) = G2
(γt, x, ξ, pz/µ
)+
x+ ξ
ξ(1 + ξ), (44)
G2
(iσz⊥, x, ξ, pz/µ
)=
x+ ξ
(1− x)(1 + ξ)
[ln
4(1− x)2(x+ ξ)(pz)2
(ξ − x)µ2− 1
]+
2ξ
1− ξ2lnξ − x1− x , (45)
G3
(γt, x, ξ, pz/µ
)=
1 + x2 − 2ξ2
(1− x)(1− ξ2)
[ln
4√x2 − ξ2(1− x)(pz)2
µ2− 1
]+
x+ ξ2
2ξ(1− ξ2)lnx+ ξ
x− ξ , (46)
G3 (γzγ5, x, ξ, pz/µ) = G3
(γt, x, ξ, pz/µ
)+ 2
1− x1− ξ2
, (47)
G3
(iσz⊥, x, ξ, pz/µ
)=
2(x− ξ2)
(1− x)(1− ξ2)
[ln
4√x2 − ξ2(1− x)(pz)2
µ2− 1
]+
ξ
1− ξ2lnx+ ξ
x− ξ . (48)
8
Some technical details of the calculation are provided inthe Appendix. The above calculation has been carriedout in momentum space. In principle, the same resultcan be obtained from calculations in coordinate spaceand then taking a Fourier transform. For examples in thecase of meson DA and nucleon PDF, see Refs. [71, 104].However, as noticed in [58, 71], the step of taking Fouriertransform is highly nontrivial.
We observe that the bare matching coefficients for Γ =γt, γzγ5, and iσz⊥ reduce to that for the quasi-PDFs [58,
82] when ξ = 0 even if t 6= 0. One can also obtainthe bare matching coefficients of DAs [102] by crossingthe external state with the following replacement ξ →1/(2y − 1), x/ξ → 2x − 1, and the external momentumpz to pz/2 [28].
Next we need the counterterm of the quasi-GPD inRI/MOM scheme. As we argued in Sec. IV, we canuse the renormalization factor for the quasi-PDF torenormalize the quasi-GPD, which leads to the one-loopRI/MOM counterterm [55, 65]
C(1)CT
(Γ, x, r,
pz
pzR,µRµ′
)=
[∣∣∣∣ pzpzR∣∣∣∣ f2
(Γ,pz
pzR(x− 1) + 1, r
)]+
+ δΓ,iσz⊥δ(1− x)αsCF
4π
[− 1
εUV+ ln
(µ2R
µ′2
)], (49)
where r = µ2R/(p
zR)2; f2(Γ, x, r) is the real part of the off-shell quark matrix element of the quasi-PDF calculated at
the subtraction point {µ}; the last term which contains δΓ,iσz⊥δ(1 − x) is the conversion factor between RI/MOM
and MS schemes for the local operator O(Γ, 0). We choose Landau gauge, which is convenient for lattice simulation,and project out the coefficient of Γ (also known as the minimal projection according to [65]) to obtain f2. The resultsfor different spin structures are [65, 82],
f2(γt, x, r) =αsCF
2π
−3r2+13rx−8x2−10rx2+8x3
2(r−1)(x−1)(r−4x+4x2) + −3r+8x−rx−4x2
2(r−1)3/2(x−1)tan−1
√r−1
2x−1 x > 1−3r+7x−4x2
2(r−1)(1−x) + 3r−8x+rx+4x2
2(r−1)3/2(1−x)tan−1
√r − 1 0 < x < 1
−−3r2+13rx−8x2−10rx2+8x3
2(r−1)(x−1)(r−4x+4x2) − −3r+8x−rx−4x2
2(r−1)3/2(x−1)tan−1
√r−1
2x−1 x < 0
, (50)
f2(γzγ5, x, r) =αsCF
2π
3r−(1−2x)2
2(r−1)(1−x) −4x2(2−3r+2x+4rx−12x2+8x3)
(r−1)(r−4x+4x2)2 + 2−3r+2x2
(r−1)3/2(x−1)tan−1
√r−1
2x−1x > 1
1−3r+4x2
2(r−1)(1−x) + −2+3r−2x2
(r−1)3/2(1−x)tan−1
√r − 1 0 < x < 1
− 3r−(1−2x)2
2(r−1)(1−x) + 4x2(2−3r+2x+4rx−12x2+8x3)(r−1)(r−4x+4x2)2 − 2−3r+2x2
(r−1)3/2(x−1)tan−1
√r−1
2x−1x < 0
, (51)
f2(iσz⊥, x, r) =αsCF
2π
3
2(1−x) + r−2x(r−1)(r−4x+4x2) + −r+2x−rx
(r−1)3/2(x−1)tan−1
√r−1
2x−1 x > 11−3r+2x
2(r−1)(1−x) + r−2x+rx(r−1)3/2(1−x)
tan−1√r − 1 0 < x < 1
− 32(1−x) − r−2x
(r−1)(r−4x+4x2) − −r+2x−rx(r−1)3/2(x−1)
tan−1√r−1
2x−1 x < 0
. (52)
Finally, combining Eqs. (39) and (49), we obtain the one-loop matching coefficient CΓ,
CΓ
(x, ξ, r,
pz
µ,pz
pzR
)= δ(1− x) + C
(1)B
(Γ, x, ξ,
pz
µ,µ
µ′
)− C(1)
CT
(Γ, x, r,
pz
pzR,µRµ′
)+O(α2
s) ; (53)
then making the replacements x→ x/y, ξ → ξ/y, and pz → yP z [55, 58], we obtain CΓ in Eq. (34),
CΓ
(x
y,ξ
y, r,
yP z
µ,yP z
pzR
)=δ
(1− x
y
)+
[f1
(Γ,x
y,ξ
y,yP z
µ
)−∣∣∣∣yP zpzR
∣∣∣∣ f2
(Γ,yP z
pzR
(x
y− 1
)+ 1, r
)]+
+ δΓ,iσz⊥δ
(1− x
y
)αsCF
4πln
(µ2
µ2R
)+O(α2
s) . (54)
VI. SUMMARY
Within the framework of LaMET, we have derived theone-loop matching coefficients that connect the isovec-tor quark quasi-GPDs renormalized in the RI/MOMscheme to GPDs in the MS scheme. The calculationwas performed for the unpolarized, longitudinally and
transversely polarized cases defined with Γ = γt, γzγ5,and iσz⊥, respectively. We also presented a rigorousderivation of the factorization formula for isovector quarkquasi-GPDs based on OPE. The matching coefficientturns out to be independent of the momentum trans-fer squared t. As a result, for quasi-GPDs with zeroskewness the matching coefficient is the same as that for
9
p− kp− ∆
2 p + ∆2
k − ∆2 k + ∆
2
FIG. 1. One-loop vertex diagram for the quark quasi-GPD.
the quasi-PDF. Our results will be used to extract theisovector quark GPDs from lattice calculations of the cor-responding quasi-GPDs. This work can also be extendedto gluon and singlet quark quasi-GPDs.
ACKNOWLEDGMENTS
We thank Vladimir M. Braun, Yizhuang Liu, Xiang-dong Ji, and Yi-Bo Yang for enlightening discussions. Y.-S. L. is supported by Science and Technology Commissionof Shanghai Municipality (Grant No. 16DZ2260200) andNational Natural Science Foundation of China (GrantNo. 11655002). W. W., J. X., and S. Z. are supported inpart by National Natural Science Foundation of Chinaunder Grant No. 11575110, 11655002, 11735010, byNatural Science Foundation of Shanghai under Grants
No. 15DZ2272100 and No. 15ZR1423100, Shanghai KeyLaboratory for Particle Physics and Cosmology, and byMOE Key Laboratory for Particle Physics, Astrophysicsand Cosmology. Q.-A. Z. is supported by NationalNatural Science Foundation of China under Grants No.11621131001 and 11521505. J.-H. Z. is supported bythe SFB/TRR-55 grant “Hadron Physics from LatticeQCD,” and a grant from National Science Foundationof China (Grant No. 11405104). Y. Z. is supported bythe U.S. Department of Energy, Office of Science, Of-fice of Nuclear Physics, from de-sc0011090 and withinthe framework of the TMD Topical Collaboration. Y. Z.is also partially supported by the Institute for NuclearTheory at University of Washington during the programINT-18-3 “Probing Nucleons and Nuclei in High EnergyCollisions.”
APPENDIX
In this Appendix, we present some technical details in calculating the following dimensionless integral that arisesfrom the vertex diagram in Fig. 1:
INn (x, ξ) =
∫ 1
0
dy1
∫ 1−y1
0
dy2N(y1, y2){
y1y2t′ + [x− 1 + (1− ξ)y1 + (1 + ξ)y2]2}n+ε , (55)
where N is a function of Feynman parameters y1 and y2 and n is the power of the denominator of the integrand;t′ = −t/(pz)2. In unphysical regions (x < −ξ and x > 1), the integral has no 1/ε pole so that it can be easilycalculated by setting ε = 0. However, this is not the case in the Efremov-Radyushkin-Brodsky-Lepage (ERBL),|x| < ξ, and Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP), ξ < x < 1, regions where there are IR divergences.
As an example, we evaluate the integral with N = 1 and n = 3/2. After integration over y2, the remaining integranddenoted as F (y1, ε) contains hypergeometric functions 2F1. We identify the divergent part of F (y1, ε) as A(ε)/y1+ε
1
in the limit of ε→ 0, and then separate it out from the integral,
I13/2(x, ξ) =
∫ 1
0
dy1
[F (y1, ε)−
A(ε)
y1+ε1
] ∣∣∣∣ε→0
+
∫ 1
0
dy1A(ε)
y1+ε1
, (56)
where the first term is convergent so that we can set ε = 0 before the integration. We suppress x, ξ, and t′ dependencesof F and A for simplicity.
10
y2
y1
(0, 1−x1+ξ )
(1−x1−ξ , 0)
0 1
1
y2
y1
(0, 1−x1+ξ )
(1−x1−ξ , 0)
0 1
1
FIG. 2. Integration in ERBL (left) and DGLAP (right) regions: The singularities are denoted by cross.
In Fig. 2, the singularities are shown in integration regions. We use Pfaff transformation to extract the divergentpart
2F1(a, b; c; z) = (1− z)−a2F1(a, c− b; c; z
z − 1) . (57)
We obtain
A(ε) =2
t′(x− 1)
−1 + 2ε+ ε ln t′(1−x)4(1+ε) |x| < ξ
−1 + 2ε+ ε ln t′(1−x)4(1+ε) − y1+ε
1 (1− 2ε− 2ε ln(1− ξ))∣∣∣y1 − 1−x
1−ξ
∣∣∣−1−2ε
ξ < x < 1. (58)
Finally, we have
I13/2(x, ξ) =
12(1−x)(x2−ξ2) x < −ξ− 2t′(1−x)
[1ε − 2 + ln 4(ξ−x)(1+ξ)2
t′(1−x)2(x+ξ)
]|x| < ξ
− 4t′(1−x)
[1ε − 2 + ln 4(1−ξ2)
t′(1−x)2
]ξ < x < 1
− 12(1−x)(x2−ξ2) x > 1
. (59)
More generally, when calculating the vertex diagram in Fig.(1), we encounter integrals similar to Eq.(55) withnumerator of the integrand replaced by polynomials of y1 and y2. After integrating out y2, we obtain Appell hyper-geometric function F1. In this case, to separate the divergent part, we need Euler transformation
F1(α;β, β′; γ;x, y) = (1− x)−β(1− y)−β′F1(γ − α;β, β′; γ;
x
x− 1,
y
y − 1) . (60)
In the following, we list integrals used in our calculation:
1. In unphysical region x < −ξ, there is no divergence.
I11/2 =
1− x1− ξ2
ln
∣∣∣∣1− xx− ξ
∣∣∣∣+x+ ξ
2ξ(1 + ξ)ln
∣∣∣∣x− ξx+ ξ
∣∣∣∣ , (61)
Iy13/2 = − 1
4ξ(1− ξ)(x− ξ) +1
2(1 + ξ)(1− ξ)2ln
∣∣∣∣1− xx− ξ
∣∣∣∣+1
8ξ2(1 + ξ)ln
∣∣∣∣x− ξx+ ξ
∣∣∣∣ , (62)
Iy23/2 = Iy13/2
∣∣∣ξ→−ξ
, (63)
Iy213/2 =
x− 3xξ + 2ξ2
4ξ2(x− ξ)(1− ξ)2+
1− x(1 + ξ)(1− ξ)3
ln
∣∣∣∣1− xx− ξ
∣∣∣∣+ξ + x
8ξ3(1 + ξ)ln
∣∣∣∣x− ξx+ ξ
∣∣∣∣ , (64)
Iy223/2 = I
y213/2
∣∣∣ξ→−ξ
, (65)
Iy1y23/2 = − 1
4ξ2(1− ξ2)+
1− x2(1− ξ2)2
ln
∣∣∣∣1− xx− ξ
∣∣∣∣− x+ 2xξ + ξ2
8ξ3(1 + ξ)2ln
∣∣∣∣x− ξx+ ξ
∣∣∣∣ . (66)
11
2. In ERBL region |x| < ξ, Iy23/2 and Iy223/2 are divergent.
I11/2 =
x+ ξ
2ξ(1 + ξ)ln
16ξ2
t′+
1− x1− ξ2
ln1 + ξ
2ξ, (67)
Iy13/2 = − 2
t′(1− ξ) ln(ξ − x)(1 + ξ)
2ξ(1− x), (68)
Iy23/2 = − 2
t′(1 + ξ)
[1
ε− 2 + ln
8ξ(1 + ξ)
t′(1− x)(x+ ξ)
], (69)
Iy213/2 = − x+ ξ
t′ξ(1− ξ) +2(1− x)
t′(1− ξ)2ln
2ξ(1− x)
(1 + ξ)(ξ − x), (70)
Iy223/2 = − (1− ξ)(x+ ξ)
t′ξ(1 + ξ)2− 2(1− x)
t′(1 + ξ)2
[1
ε− 2 + ln
8ξ(1 + ξ)
t′(1− x)(x+ ξ)
], (71)
Iy1y23/2 =x+ ξ
t′ξ(1 + ξ). (72)
3. In DGLAP region ξ < x < 1, Iy13/2, Iy23/2, Iy213/2, and I
y223/2 are divergent.
I11/2 =
x− ξ2
2ξ(1− ξ2)lnx+ ξ
x− ξ +1− x1− ξ2
ln4(1− ξ2)
√x2 − ξ2
t′(1− x), (73)
Iy13/2 =1− x
2(1− ξ)I13/2 , (74)
Iy23/2 =1− x
2(1 + ξ)I13/2 , (75)
Iy213/2 =
(1− x)2
2(1− ξ)2I13/2 −
2(1− x)
t′(1− ξ)2, (76)
Iy223/2 =
(1− x)2
2(1 + ξ)2I13/2 −
2(1− x)
t′(1 + ξ)2, (77)
Iy1y23/2 =2(1− x)
t′(1− ξ2). (78)
4. In unphysical region x > 1, the integrals are the same as functions in another unphysical region but with an
overall minus sign, IP (y1,y2)n (x > 1) = −IP (y1,y2)
n (x < −ξ). See Eq. (59) for an example.
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