Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 49 (2016)

Perturbative matching for quasi-PDFs between continuum and lattice

Parton distribution functions (PDFs) play an important role in understanding the structure of nucleons.
In the search for the nucleon structure through experiment, the PDF is important; however it is currently
estimated using the model assumption. Its direct lattice QCD calculation is impossible because it basicallyinvolves light-cone dynamics. In principle, the calculation can be made by operator product expansion;
this method, however, is quite diﬃcult because in thehigher moments, the signal-to-noise ratios worsen. Recently, a strategy has been proposed, in which a computable quantity on the lattice (quasi-PDF) can be related to the (normal) PDF by perturbative matching.1)
The relation between the normal-PDF q(x, µ) and thequasi-PDF q�(x, P3 , Λ) can be written as()∫ 1dyx µ ΛZ, ,q(y, µ) + · · · , (1)q�(x, Λ, P3 ) =y P3 P3−1 |y|where x is the momentum fraction of the parent hadron
being carried by a quark, µ is the renormalization scale,and Λ is the UV cutoﬀ scale. The quasi-PDF is presented by∫dδz −ix̃P3 δzq�(x̃, Λ, P3 ) =e
⟨N (P3 )|Oδz |N (P3 )⟩, (2)2πwhere the matrix element (ME) is deﬁned as the transition amplitude between the in- and out-nucleon stateswith momentum P3 in the z-direction; the non-local
operator elongated in the z-direction is described as∫Oδz =ψ(x + 3̂δz)γ3 U3 (x + 3̂δz; x)ψ(x),(3)xwith a Wilson line operator U3 (x + 3̂δz; x). The essential part in Eq. (2) is the nonperturbative ME, which
we call quasi-PDF ME and write Mδz (P3 ). This ME
can be calculated using the lattice QCD simulation because it is not time-dependent. We need a matching toconvert the ME obtained from the lattice calculation
to the continuum counterpart, which has been omitted in the current lattice calculations and is a main
purpose of this report.
To achieve matching between lattice and continuum,
we have two choices: matching “in momentum space”
or “in coordinate space”. The ﬁrst one is similar to thecontinuum matching in Eq. (1). In this approach, δz
is ﬁrst integrated out; thus, the z-component of the incoming and outgoing momentum at the vertex is ﬁxed
to be x̃P3 . However, we take the second approach,which is simple and more easily controllable in the actual lattice QCD simulation. The matching is achievedby a δz dependent matching factor:∗1RIKEN Nishina Center- 121 -C(δz)

T. Ishikawa∗11050-5-10-50δz

510Fig. 1. One-loop coeﬃcient of the matching factor.

[Mδz (P3 )]cont= Z(δz) [Mδz (P3 )]

latt.

(4)

As the matching factor for operators does not dependon the external states, we can set P3 = 0 for the calculation of the amplitude.
The transition amplitude involves power divergences, which arises from the Wilson line operator inboth continuum and lattice. The power divergencemust be subtracted nonperturbatively so that the theory is well-deﬁned. There are several ways to performthe subtraction and we choose to use a static QQ̄ potential that shares the same power divergence. The
Wilson line operator with a contour C, WC , can berenormalized as
WC = eδmℓ(C) WCren ,(5)where the superscript “ren” indicates that the operator is renormalized, ℓ(C) denotes the length along thecontour C, and δm denotes the mass renormalizationof a test particle moving along the contour C.2) By
choosing the renormalization condition for the staticQQ̄ potential at a distance R0 so that V ren (R0 ) = V0 ,the mass renormalization can be obtained as3) δm =(V0 − V (R0 ))/2. Using the renormalization condition
above, we can deﬁne power divergence free ME asMSδz (P3 ) = e

V (R0 )−V0|δz|
2Mδz (P3 ).

(6)

We only show the matching factor between continuum and lattice here using a naive fermion for simplicity, while it is not realistic for the actual simulation. We calculate it at one-loop order with HYP2
link smearing of the Wilson line and with mean-ﬁeldimprovement. Figure 1 shows the one-loop coeﬃcientof the matching factor
Z(δz) = 1 +g2 4C(δz) + O(g 4 ).
(4π)2 3(7)

Using this, we can obtain the ME in continuum andthe quasi-PDF can be calculated. We are currently
planning to perform the actual simulation with thismatching strategy.
References
1) X. Ji, Phys. Rev. Lett. 110, 262002 (2013).
2) H. Dorn, Fortsch. Phys. 34, 11 (1986).
3) B. U. Musch et al., Phys. Rev. D 83, 094507 (2011).