RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-5. Hadron Physics (Theory)

Renormalization in the quasi parton distribution functions†
T. Ishikawa∗1,∗2q(x̃, μ, Pz ) =dδz −iδzx̃Pze4πδz×Pz |ψ(δz)γ exp −ig3dz A3 (z ) ψ(0)|Pz  (1)0for the quasi-quark distribution, where the hadronstate has a large momentum in z-direction Pz . Because of the time-independence, the quantity in Eq. (1)is calculable on the Euclidean lattice. The quasidistributions are matched with usual light-cone distributions through the large momentum eﬀective theory(LaMET).2) After the quasi-distribution approach wasintroduced, several lattice QCD calculations of quasiquark distributions have been carried out. These studies are still exploratory, and thus there are many uncertainties. Among the uncertainties, we address power
divergences and the matching between the continuum
and lattice in this article.
Unlike the original light-cone PDFs, quasi-distributions
are known to have power divergences that originate
from a Wilson line in the operator deﬁnition. The
power divergences must be taken into account in thematching procedure. The renormalization of the nonlocal quark bilinears of the hadronic matrix element in
the r.h.s. of Eq. (1), called O(δz), is assumed to be3)
O(δz) = Zeδm|δz| Oren (δz),(2)

where a superscript “ren” indicates that the operator isrenormalized, Z contains logarithmic divergences, andδm denotes the mass renormalization of a test particle moving along the straight line connecting the two†∗1
∗2

Condensed from the article in arXiv:1609.02018 [hep-lat]
RIKEN Nishina CenterT. D. Lee Institute, Shanghai Jiao Tong University- 124 -unsmear

HYP smear

w/o pw-div subt
w pw-div subt

w/o pw-div subt
w pw-div subt

150
ctadpole(δz)

In the study of high energy scattering processes, suchas deep inelastic scattering (DIS) and the Drell-Yan
process, one of the key concepts is “QCD collinear
factorization,” used to separate the perturbative andnonperturbative parts. The scattering cross sections
are written as a convolution of the perturbative hardpart and nonperturbative parton distribution functions (PDFs), which absorb all collinear divergencesof the partonic scattering. PDFs are universal functions and thus are used to predict the cross sections ofvarious hadronic scattering processes.
Direct calculation of the PDF by lattice QCD could
give us information complementary to global QCD
analyses. However, calculation of the light-cone distributions on the Euclidean lattice is diﬃcult because ofits time-dependence. Ji recently proposed the quasiPDF approach to solve this problem.1) The quasiPDFs are deﬁned with ﬁelds correlated completely
along the spatial direction; for example,

10010500-1020-50δz50
10 -10-50δz510Fig. 1. Tadpole diagram contribution of the matching factor to the one-loop coeﬃcient.

quark ﬁelds. Moreover, δm in the exponential factor
takes care of all the power divergence, which is valid
nonperturbatively. However, the multiplicative renormalizability of the logarithmic divergence is proven
only up to the two-loop level.4) While the all-order
proof of the multiplicative renormalizability is important to give a solid validation to the renormalization in
the quasi-distribution approach,5) we assume the formin Eq. (2) for now. Knowing the renormalization inEq. (2), the power divergence subtraction can be performed by deﬁning a modiﬁed non-local operator for
Eq. (1):6)
Osubt (δz) = e−δm|δz| O(δz),(3)

where the superscript “subt” indicates that this isa power divergence subtracted operator. The mass
renormalization δm now needs to be ﬁxed by imposinga renormalization condition. For this ﬁxing, we canuse static potential V (R) and set V (R0 ) = V0 as thecondition, leading to 2δm = V0 − V (R0 ).
By removing the power divergence, we can perform
continuum-lattice matching. We demonstrate the oneloop matching using the lattice perturbation theory
with a naive fermion and plaquette gluon. The oneloop matching factor is expressed as
Zcont−latt (δz) = 1 +g2
CF c(δz).
(4π)2(4)In the continuum, we introduce a UV cutoﬀ scale andset to the lattice cutoﬀ μ = a−1 . As an example,
we show the tadpole diagram contribution in Fig. 1with and without Wilson line smearing. The smearing
helps to reduce the diﬀerence between the continuum
and lattice caused by the power divergence before thesubtraction. Further study on the matching using thenonperturbative method is underway.
References
1) X. Ji, Phys. Rev. Lett. 110, 262002 (2013).
2) X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014).
3) H. Dorn, Fortsch. Phys. 34, 11 (1986).
4) X. Ji et al., Phys. Rev. D 92, 034006 (2015).
5) T. Ishikawa et al., in preparation.
6) T. Ishikawa et al., arXiv:1609.02018 [hep-lat].