Continuum limit of the quasi-PDF operator using chiral fermion
CContinuum limit of the quasi-PDF operator using chiral fermion
Kuan Zhang,
1, 2
Yuan-Yuan Li, Yi-Kai Huo, Peng Sun, ∗ and Yi-Bo Yang
2, 5, 6, † ( χ QCD Collaboration) University of Chinese Academy of Sciences, School of Physical Sciences, Beijing 100049, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China Nanjing Normal University, Nanjing, Jiangsu, 210023, China Physics Department, Columbia University, New York, NY 10027 School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China (Dated: December 21, 2020)Non-perturbative renormalization using the regularization independent momentum subtraction(RI/MOM) scheme has achieved great success in dealing with local operators under the latticeregularization, while a high accuracy and systematic examination on its validity of the non-localoperator likes the quasi-PDF operator, such as the quark bi-linear operator with Wilson link whichsuffers from the linear divergence, is still absent. In this work, we compute the pion unpolarizedquasi-PDF matrix element in the rest frame at 11 lattice spacings from 0.032 to 0.121 fm, usingthe discretized fermion actions with or without additive chiral symmetry breaking, on 3 kinds ofdynamical gauge ensembles. The result shows that RI/MOM renormalization cancels all the lineardivergence for the chiral fermion at least up to Wilson link length z ∼ Introduction:
The intrinsic properties of nucleon andalso nuclear, is at the center of the non-perturbative QCDstudy. Based on the fundamental theory, the nucleoncollision cross sections can be factorized into perturba-tive calculable short distance kernel and universal nu-cleon parton distribution functions (PDFs) . PDF de-scribes the energy distribution of given parton inside ahigh-momentum nucleon, and it has several extensionsto describe the 3D picture of nucleon, such as transversemomentum dependent PDF, generalized PDF and so on.One can use them to describe the nucleon and nuclearproperties, such as the gluon and sea quark contribu-tions to the nucleon mass, momentum and spin, and alsonuclear medium effects based on the difference betweenthe nucleon and nuclear PDFs.Currently, PDFs are mainly determined phenomeno-logically with the global analysis of the experimentaldata. The first principle prediction of PDF was notpossible with lattice QCD calculations, since the light-cone gauge link in the PDF definition can not be im-plement on a Euclidean lattice. The Large MomentumEffective theory (LaMET) suggests a quasi-PDF oper-ator O Γ ( z ) = ¯ ψ (0)Γ U (0 , z ) ψ ( z ) with spatial gauge link U (0 , z ) = exp( − ig (cid:82) z dz (cid:48) A z ( z (cid:48) )), and connects its renor-malized nucleon matrix element to PDF through the fac- ∗ † [email protected] torization theorem [1–5]. But the bare O Γ ( z ) under lat-tice regularizations looks like O Γ ( z ) = Γ (cid:0) g ( γ log( p a ) − m − za ) + . . . . . . (cid:1) , (1)at 1-loop level [6] with lattice spacing a , and then candeviate from the tree level value exponentially at eitherlarge z or small a when higher order effects are summedover. Thus a non-perturbative renormalization with ac-curate cancellation on the linear divergence is essentialfor the quasi-PDF, to ensure the existence of a finite con-tinuum limit.RI/MOM scheme has been proposed for more than20 years [7], and has been applied to various quark andgluon local operators. Since the theoretical studies showthat the quasi-PDF operator is multiplicative renormal-izable [8–11] in the continuum, there are many studies toapply RI/MOM renormalization to the quasi-PDF oper-ator, starting from the investigations in Refs. [6, 11–14].The recent studies using multiple lattice spacings [15, 16]studied the lattice spacing dependence on the quasi-PDF,however the precision of their data were not enough tojustify whether the dependence is a discretization erroror not.In this work, we study the RI/MOM renormalized pionquasi-PDF matrix element in the rest frame, on 11 lat-tice spacings from 0.032 to 0.121 fm with different quarkand gluon actions. The result shows that the linear diver-gences can be cancelled effectively and the lattice spacingdependence is at 1% level with the chiral fermion. On the a r X i v : . [ h e p - l a t ] D ec other hand, the residual linear divergence is still clearlyvisible using the clover fermion, especially when the lat-tice spacing is much smaller than 0.1 fm. Numerical setup:
When we consider the quasi PDFnucleon matrix element in the moving frame, the gap δm between the ground and first excited states can de-crease significantly with large momentum and then alarge source/sink separation t sep is required to eliminatethe excite state contaminations. At the same time, thesignal to noise ratio also decreases exponentially on t sep ,and then it decreases our ability to identify the latticespacing dependence from the data with large statisticaland/or systematic uncertainties.In order to avoid the defects above, the pion matrixelement in the rest frame can be a better choice. It isknown that the first excited state of pion is higher than1GeV, and the signal to noise ratio is almost independentto t sep . Thus we can set the t sep to be one half of thelattice length T along the temporal direction which canbe larger than 2 fm, and then obtain the ground statematrix element h π = (cid:104) π | O γ t ( z ) | π (cid:105) with high accuracy: R π ( T / , t, z ; a ) ≡ (cid:104) O π ( T / (cid:80) (cid:126)x (cid:0) O Γ ( z ; ( (cid:126)x, t )) + O Γ ( z ; ( (cid:126)x, T − t )) (cid:1) O † π (0) (cid:105)(cid:104) O π ( T / O † π (0) (cid:105) = h π, Γ ( z ) + O ( e − δmt ) + O ( e − δm ( T/ − t ) ) + O ( e − δmT/ ) , (2)where O π is the pion interpolation field. Note that the de-nominator includes both the forward and backward twopoint functions needed by the two terms in the numera-tor, due to the loop around effect in the temporal direc-tion.In the RI/MOM renormalization, we define the nor-malized renormalization constant as [17], Z γ t ( z, µ ) = Tr[ γ t (cid:104) q ( p ) | O γ t ( z ) | q ( p ) (cid:105) ]Tr[ γ t (cid:104) q ( p ) | O γ t (0) | q ( p ) (cid:105) ] | p = − µ ,p z = p t =0 , (3)where | q ( p ) (cid:105) is the off-shell quark state with external mo-mentum p . When the Landau gauge fixed volume( V )source [13] is used in the calculation, the statical uncer-tainty is suppressed by a factor 1 / √ V comparing to thepoint source case. By eliminating the z and t componentsof p , the matrix element is free of the operator mixing andavoids the related systematic uncertainties.Then we can apply Z γ t on the bare pion matrix element h π,γ t ( z ) to obtain the non-perturbative renormalized andnormalized matrix element at given RI/MOM scale µ , h rπ,γ t ( z, µ ) = Z γ t ( z, µ ) h π,γ t ( z ) h π,γ t (0) . (4)The joint effect of the normalizations on both quarkand pion matrix elements is equivalent to the originalRI/MOM definition which uses the conserved vector cur-rent to define the quark self energy. In order to check if the linear divergence is relate tothe fermion action definition, we consider two kinds offermion actions in this work: the clover and overlapactions. It is relatively cheap to calculate the quarkpropagator for the clover action and it is widely usedin the quasi-pdf calculations, and the overlap propagatoris much more expensive but it conserves chiral symmetry.The clover fermion action is defined by the Wilson ac-tion S w q = (cid:88) x,y ¯ ψ ( x ) D w ( m w q ; x, y ) ψ ( y ) ,D w ( m w q ; x, y ) = 1 − γ µ a U ( x, x + ˆ n µ ) δ x +ˆ n µ ,y + 1 + γ µ a U ( x, x − ˆ n µ ) δ x − ˆ n µ ,y − ( 4 a + m w q )) δ x,y , (5)with an additional “clover” term, S clv q = S w q + ac sw (cid:88) x ¯ ψ ( x ) σ µν F µν ( x ) δ x,y ψ ( y ) , (6)where ˆ n µ is the unit vector along the µ direction, m w q − m cri is the multiplicative renormalizable bare quarkmass, and m cri is the bare quark mass which vanishes thepion mass. m cri is O ( α s a ) at the leading order for the Wil-son action and always negative, but it can be suppressedto O ( α s a ) (but usually still negative) with a clover coef-ficient c sw = 1 + O ( α s ). It can be further suppressed byapplying the smearing on the gauge link U and/or tuning c sw manually, while it can not exactly vanish due to therandom gauge fluctuations.To eliminate m cri accurately, one can use the chi-ral fermion which satisfies the Ginsburg-Wilson rela-tion D ov γ + γ D ov = aρ D ov γ D ov [18]. For example,Refs. [19, 20] define the overlap fermion as S ov q = (cid:88) x,y ¯ ψ ( x ) D ov ( x, y ) ψ ( y ) , (7) D ov = ρ (cid:16) D w ( − ρ ) (cid:113) D † w ( − ρ ) D w ( − ρ ) (cid:1)(cid:17) , where − ρ should be lower than m cri , to make D ov to bethe same as the standard Dirac operator in the contin-uum limit. The chiral fermion propagator can be furtherdefined through D ov ,1 D c + m ov q = 1 D ov − ρ D ov + m ov q = 1 − ρ D ov D ov + m ov q (1 − ρ D ov ) , (8)where D c satisfies the relation D c γ + γ D c = 0 and then m q → m cri . Results:
In this work, we use the 2+1+1 flavors (de-generate up and down, strange, and charm degrees offreedom) of highly improved staggered quarks (HISQ) tag 6 /g L T a (fm) c sw m w q a c (cid:48) sw m w (cid:48) q a m ov q a tag 6 /g L T a (fm) m ov q a HISQ12 3.60 24 64 0.1213(9) 1.0509 -0.0695 1.31 0.010 0.015 DW11 2.13 24 64 0.1105(3) 0.015HISQ09 3.78 32 96 0.0882(7) – – – – 0.011 DW08 2.25 32 64 0.0828(3) 0.011HISQ06 4.03 48 144 0.0574(5) 1.0349 -0.0398 1.25 0.0014 0.008 DW06 2.37 32 64 0.0627(3) 0.008HISQ03 4.37 96 288 0.0318(3) 1.0287 -0.0333 1.26 0.0030 –tag 6 /g L T a (fm) c sw m w q a tag 6 /g L T a (fm) c sw m w q a CLS10 3.34 24 48 0.0980(12) 2.06686 -0.3437 CLS08 3.40 32 96 0.0854(10) 1.98625 -0.3468CLS06 3.55 48 128 0.0644(08) 1.82487 -0.3525 CLS04 3.85 64 192 0.0390(06) 1.61281 -0.3478TABLE I. Setup of the ensembles, including the bare coupling constant g , lattice size L × T and lattice spacing a . m w q and m w (cid:48) q are the bare quark masses using the clover fermion action with two clover coefficient c sw and c (cid:48) sw respectively, and m ov q isthe bare quark mass of the overlap fermion. The pion masses in all the cases are in the range of 310-360 MeV. and one-loop Symanzik improved gauge ensembles fromthe MILC Collaboration [21] at four lattice spacings,2+1 flavor domain wall (DW) quarks and Iwasaki gaugeensembles from the RBC/UKQCD collaboration [22] atthree lattice spacings, and also 2+1 flavor clover quarkand Luescher-Weisz (equivalent to Symanzik) gauge en-sembles from CLS collaboration [23]. With 1-step HYPsmearing [24] is applied on the MILC/RBC gauge con-figurations, we use ρ = 1 . /a for the overlap fermion,and two choices of the clover coefficients for the cloveraction: one is the tadpole improved tree level coefficient c sw which is very close to one after the configuration isHYP smeared, and the other one c (cid:48) sw ∼ . m cri q to be around zero. Thus onecan see that the bare overlap quark mass m ov q a is roughlyproportional to the lattice spacing if we require the pionmass to be around 310 MeV, but m w q a are always neg-ative with c sw ∼ c sw to ∼ p = 2 π/L (5 , , , p x,y on theRBC/UKQCD ensembles to make µ = (cid:112) p to be roughlythe same on all the ensembles within 6%. As shown inFig. 1 for the ratio of the renormalization constants, R RI/MOM ( z, µ, µ ) = Z ( z, µ ) /Z ( z, µ (cid:48) ) , (9)with µ (cid:48) = (cid:112) − ( p (cid:48) ) ∼ . p (cid:48) = 2 π/L (3 , , , z =1 fm regardless the actions and lattice spacings,and then the systematic uncertainty from a 6% differencein µ can be smaller than the statistical uncertainty. Onthe other hand, such a good convergence at small lat-tice spacing suggests that the UV divergence due to thediscretized fermion actions can be perfectly cancelled upto the discretization errors. However, small residual zdependence may indicates some nonpertuabative effect FIG. 1. The z -dependent ratio R RI/MOM ( z, µ, µ ) defined inEq. (9) between µ (cid:39) µ (cid:48) (cid:39) z depen-dence is weak but doesn’t vanish. FIG. 2. RI/MOM renormalized pion matrix element using theoverlap fermion on DW (bands) and also HISQ (data points)configurations. All the cases shows good agreement within ∼
1% percent difference.) in Z ( z, µ ). More informations of the linear divergencecoefficients in different cases can be found in the supple-mentary materials [25].Then we apply the renormalization constant atRI/MOM µ =3.0(2) GeV to h π,γ t ( z, a ), starting from theoverlap case. As in Fig. 2, the renormalized pion matrix FIG. 3. Same as Fig 2, the cases using the clover fermionon HISQ configurations at different lattice spacings (datapoints), comparing to the overlap case (purple band). Theleft and right panels show the cases with c sw ∼ a ≤ .
06 fm. element h rπ,γ t ( z, a ) on different ensembles with differentsea quark, gauge actions and lattice spacings are consis-tent with each other within ∼
1% difference and doesn’tshow any obvious lattice spacing dependence. The dataon different ensembles can have 2-3 σ deviation based ontheir tiny statistical uncertainty at 0.5% level; such a de-viation can be the systematic uncertainty coming fromthe mismatch in the pion mass, renormalization scale,finite volume and etc. The z dependence of h π,γ t ( z, a )can be considered as the non-perturbative power cor-rections; and the fit gives λ = − (0 . and λ = (0 . in the range of z ∈ [0 . , .
2] fmwith χ /d.o.f. ∼
1, if we use a simple 1+ (cid:80) i =1 , λ i z i formto describe the data and consider the data on differentensembles as independent samples. Both λ , are at theorder of Λ nQCD and it is consistent with the naive powercounting.In the clover fermion case, the RI/MOM renormalized h rπ,γ t ( z, a ) can have a very strong lattice spacing depen-dence, and such a dependence becomes even stronger atsmaller lattice spacing. As in the left panel of Fig. 3, thechanges in h rπ,γ t ( z, a ) from a = 0 .
057 fm to 0 .
032 fm ismuch larger than that from a =0.121 fm to 0.057 fm, andthe difference becomes exponentially larger with a biggerWilson link length z . Such a behavior can be explainedby a residual linear divergence which is not fully cancelledby Z ( z, µ ). If we change the clover coefficient by ∼ O (0.01), thelattice spacing dependence behavior is still the same asin the right panel of Fig. 3. More figures without HYPsmearing on the Wilson link are provided in the supple-mentary materials [25], while all the conclusions here arebasically unchanged.Since m cri q is very sensitive to whether the gauge config-uration used in the clover fermion action is HYP smeared,and the clover on HISQ setup can suffer from O ( a ) mixedaction effects, we also provide the unitary clover fermioncase using the CLS enemasbles in Fig. 4 without HYP FIG. 4. Same as Fig 3, the cases of the unitary clover valenceon clover sea with CLS ensembles (data points), comparingto to the overlap case (gray band). Note that the Wilson linkis not HYP smeared for both the clover and overlap cases.
FIG. 5. The same as Fig. 2 but with the scalar current andcorresponding RI/MOM renormalization, using the overlapfermion. The z dependence is stronger than the vector currentcase but still O (Λ QCD z ). smearing on either the fermion action or Wilson link.The situation seems to be better except the smallest lat-tice spacing, but the results at given z still increases on1 /a except the case at largest lattice spacing which suf-fers from obvious discretization error.By definition, the operator product expansion of h π ( z, a ) includes the trace terms likes ¯ ψγ t D ψ , whichequals to m ¯ ψγ t ψ with the chiral fermion and would in-troduce the z dependence in Fig. 2; but ¯ ψγ t D ψ using theclover fermion action would have additional linear diver-gence from the Wilson, clover and residual mass terms.Those unphysical terms could bring in additional lineardivergences which may not be caught by the RI/MOMrenormalization.At the end of this section, we present the overlap pionmatrix element with the scalar current instead of the vec-tor one in Fig. 5. One can see that the lattice spacingdependence is much larger than the vector case, and wehave to use the form 1 + (cid:80) i =1 , , λ i z i with higher or-der correction to describe the data at a (cid:39) .
06 fm with λ = − (0 . , λ = (0 . and λ = − (0 . , where the second uncertaintycomes from the difference between the fitting parameterswith MILC and RBC ensembles. Again, all the coeffi-cients are at the order of Λ nQCD . Note that the parame-terizations for the vector and scalar matrix elements arenot the only possibility to describe the data, and it justtargets to a simple way to reconstruct our result for fur-ther studies.We skipped the clover action case with the scalar cur-rent since it is similar to the vector case and the scalarcurrent with the clover fermion is known to be more prob-lematic due to the chiral symmetry breaking effect intro-duced by m cri . Summary:
We have demonstrated that the lin-ear divergence in the quark quasi-PDF proposed inLaMET framework can be properly eliminated using theRI/MOM renormalization, with the chiral fermion in thissimplest case. It paves the way to a systematic studyon PDF through the quasi-PDF approach, and also pro-vides a solid numerical check on the factorization theoremof the quasi-PDF matrix elements. The corrections are O (Λ QCD z ), while the effect using the vector current, ismuch smaller than that using the scalar current.Our results also show clear evidences that there areresidual linear divergences in the RI/MOM renormalizedpion matrix element h Rπ,γ t when the clover action is usedfor the valence quark, on the dynamical configurationswith different quark and gluon actions, and it is irrele-vant to the mixed action effect. The residual linear di-vergence is small when the lattice spacing is around 0.1fm, but becomes very conspicuous at small lattice spac-ing likes 0.04 fm or so. Thus it is essential to check other actions with the relatively good chiral symmetry like thedomain wall fermion, and the ones with partly the chiralsymmetry like the twisted mass fermion. If so, one canmake a solid conclusion on the proper quark action thatcan be used in the quasi-PDF calculation, and answerwhether the residual linear divergence in h Rπ is relatedto the explicit chiral symmetry breaking in the fermionaction or not. ACKNOWLEDGEMENT
We thank the CLS, MILC and RBC/UKQCD col-laborations for providing us their gauge configurations,and Long-Cheng Gui, Xiangdong Ji, Keh-Fei Liu, An-dreas Schaefer, Wei Wang and Jian-Hui Zhang, YongZhao for useful information and discussion. The cal-culations were performed using the Chroma softwaresuite [26] with QUDA [27–29] and GWU-code [30, 31]through HIP programming model [32]. The numeri-cal calculation is supported by Strategic Priority Re-search Program of Chinese Academy of Sciences, GrantNo. XDC01040100, and also HPC Cluster of ITP-CAS,Jiangsu Key Lab for NSLSCS. P. Sun is supported byNatural Science Foundation of China under grant No.11975127, as well as Jiangsu Specially Appointed Profes-sor Program. Y. Yang is supported by Strategic Prior-ity Research Program of Chinese Academy of Sciences,Grant No. XDB34030303 and XDC01040100. [1] X. Ji, Phys. Rev. Lett. , 262002 (2013),arXiv:1305.1539 [hep-ph].[2] X. Ji, Sci. China Phys. Mech. Astron. , 1407 (2014),arXiv:1404.6680 [hep-ph].[3] Y.-Q. Ma and J.-W. Qiu, Phys. Rev. D98 , 074021 (2018),arXiv:1404.6860 [hep-ph].[4] Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. , 022003(2018), arXiv:1709.03018 [hep-ph].[5] T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao,Phys. Rev.
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SUPPLEMENTAL MATERIALSA. fitting the renormalization constant
Since different quark flavors, quark and gauge actions used by MILC and RBC/UKQCD makes their bare couplingto be differ by a factor ∼ Z ( z ; a ) = Z ( a )exp (cid:8) α s ( a ) α s ( a ) (cid:2)(cid:0) m − a + m (cid:1) z + d z log( zz ) (cid:3)(cid:9) , (10)where z =0.3 fm and a =0.1 fm. We also introduce a dummy systematic uncertainty factor δ sys to enlarge theuncertainty of the renormalization constants into (cid:113) ( δZ ) + δ sys Z , as the statistical uncertainty at small z cansmaller than kinds of the systematic uncertainty from discretization, partially quenching, lattice spacing determination,valence/sea quark mass and volume dependence. Sea HISQ+S DW+I CLV+SValence OV CLV CLV’ OV CLV m − δ sys (%) 0.006 0.013 0.024 0.008 0.013TABLE II. The linear divergence parameters we obtained with the parameterization in Eq. (10) using a =0.1 fm and z =0.3fm. “S” means the Symanzik gauge action and “I” is used for the Iwasaki gauge action. Then we fit the renormalization constant in the range z ∈ [0 . , .
2] fm with the same actions but at different latticespacings, and tune the δ sys to ensure χ /d.o.f ∼ ≥ m − are ∼ . δ sys in the unitary clover case is largerthan the other cases, and the major difference is that the HYP smearing is applied on the valence fermion actions inall other cases. B. HYP smearing dependence for the overlap fermion
Fig. 6 compares the RI/MOM renormalized the pion matrix element with the vector current h π,γ t ( z, a ), usingoverlap fermion with and without HYP smearing on the Wilson link. The results just differ by less the 1% whilethe cases without HYP smearing have larger statistical uncertainties. Such a difference can come from systematicuncertainties of the mismatch of the pion mass, mixed action effect and etc. FIG. 6. The RI/MOM renormalized pion matrix element with the vector current, using the overlap fermion on RBC ensembles(left panel) and MILC ensembles (right panel). The data points correspond to the cases with 1-step of the HYP smearing, andthe bands are the cases without HYP smearing on the Wilson link.
FIG. 7. The RI/MOM renormalized pion matrix element with the vector current, using the clover fermion on MILC ensembles(upper panels) and CLS ensembles (lower panel), with (left panels) and without (right panels) HYP smearing. The residuallinear divergences are always exist.
C. HYP smearing dependence for the clover fermion
Fig. 7 compares the cases with and without HYP smearing on the Wilson link, for the RI/MOM renormalized thepion matrix element h π,γ t ( z, a ), using the clover fermion. The upper two panels are the cases using the clover fermionon the MILC ensembles (the gauge link in the clover action is HYP smeared, and such a case has O ( a ) mixed actioneffect), and lower two panels are the cases using the clover fermion on CLS ensembles (the gauge link in the cloveraction is the original one, and it is an unitary case without any mixed action effect). The left two panels apply theHYP smearing on the Wilson link, but the right two panels are not. Note that the overlap results shown in the grayband uses the HYP smearing in left two panels but not in right two panels, to provide a fair comparison even thoughtheir difference are smaller than 1%. The linear divergences are obvious for all the simulation setups using the cloverfermion, and we can see that the behavior of h π,γ t ( z, az, a