NUPHB14531S0550-3213(18)30360-210.1016/j.nuclphysb.2018.12.020The Author(s)Frontiers ArticleFig. 1The correlator C˜ in Eq. (11) receives contributions from connected and disconnected diagrams. Point O is at origin, point A is at coordinate (x→+z,τ) and point B is at (x→,τ). The arrowed solid lines are full quark propagators, while the double lines are gauge links. The gluon propagators and closed fermion loops are not shown.Fig. 1Fig. 2The π, K and ηs energies squared as functions of momentum Pˆz2, where the lines with error bands are fits to the dispersion relation E2(Pz)=m2+c2Pˆz2. The fitted c2 are 1.02(5), 0.98(3), 0.96(2) for π, K and ηs, respectively, which is consistent with the desired value of 1 within 2σ.Fig. 2Fig. 3The quasi-DA matrix elements hM(z,Pz) for M = K− (top), π (middle) and ηs (bottom), respectively, shown as functions of zPz at three different values of Pz. The purple dashed lines are Fourier transforms of the asymptotic DA ϕ(x)=6x(1 − x). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)Fig. 3Fig. 4The potential between a static quark and a static antiquark as function of distance r (in fm) computed using Eq. (17) for a = 0.06 fm (triangles), 0.09 fm (circles), and 0.12 fm (squares) with pion mass around 135 MeV and for heavier pion mass 310 MeV at a = 0.12 fm (inverted triangles). The difference between pion masses is negligible for 310 and 135 MeV at 0.12-fm lattice spacing. The line indicates our attempted fit to the potentials using the form of Eq. (18) with r ≥ 5a to extract the δm counterterm, and the fit describes the majority of the potential well.Fig. 4Fig. 5(Left) Improved kaon quasi-DA, ϕ˜K−imp(x) (blue), kaon DA with one-loop matching applied, ϕK−imp,match (green), and ϕK− (red) with the meson-mass correction, which is a very small effect, further added. The asymptotic DA is shown as the purple dashed line. (Right) DAs for K− after the one-loop matching and mass corrections but not higher twist corrections are shown with each Pz we study in this work.Fig. 5Fig. 6Comparison of ϕK− of this work (“Lat LaMET”) to a few selected works in literature. This includes a parametrized fit to lattice moments (“Lat Mom”) [52,53], the Dyson–Schwinger equation calculation (DSE-1 & -2) [56], and a light-front constituent quark model (LFCQM) [55]. The error in the “Lat LaMET” curve is statistical error only. A broader distribution than the one predicted in LFCQM is clearly preferred; further studies are planned to investigate the nonzero distribution outside x ∈ [0,1] together with the systematics from unphysical pion mass, nonzero lattice spacing and finite volume.Fig. 6Fig. 7DAs of π (left) and ηs (right) as functions of Bjorken-x after the one-loop matching and mass corrections with different Pz. The errors in the lattice calculation are statistical errors only. The asymptotic form ϕ(x)=6x(1 − x) is shown as the purple dashed lines.Fig. 7Fig. 8Comparison of ϕπ from this work (“Lat LaMET”) to previous determinations in literature. In the left panel, this includes the results from parametrized fits to the lattice moments (“Lat Mom 1” and “Lat Mom 2”) [58], a calculation from the DSE analysis (DSE) [56], one from the LFCQM (LFCQM) [55], a fit to the Belle data (Belle) [57], and the asymptotic form 6x(1 − x) (Asymp). The error in the “Lat LaMET” curve is statistical error only. In the right panel, we have converted our result on ϕπ to the prediction for the scalar-pseudoscalar current correlator (blue curve), and compared with the lattice data for the same correlator in Ref. [44] (dark, gray and white circles, which correspond to |P→|=1.08,1.53,1.88 GeV, respectively, μ is the renormalization scale). Systematics from unphysical pion mass, nonzero lattice spacing and finite volume have not been quantified and included in this calculation.Fig. 8Fig. 9Results for flavor SU(3) symmetry breaking: δSU(3),1=(ϕK−−ϕK+)/2 (left) and δSU(3),2=(ϕπ+3ϕη−2ϕK+−2ϕK−)/8 (right) using the corrected distribution of Pz = 8π/L. Our results support the ChPT [7] prediction |δSU(3),1| > |δSU(3),2|.Fig. 9Fig. 10An example demonstrates that the truncated Fourier transform could cause an error that is non-monotonic in Pz. The asymptotic DA 6x(1 − x) is Fourier transformed to the z space, then Fourier transformed back to the x space with the region outside of z = [−L/2,L/2] truncated. The resulting truncated asymptotic DA is non-monotonic in Pz at x = 1/2.Fig. 10Fig. 11The improved matrix elements HM(z,Pz) defined in Eq. (A.1) shown at three different values of Pz. Since the real (imaginary) part of HM(z,Pz) is even (odd) in z in definition, only the z > 0 part is shown. The magenta lines are derived from the asymptotic DA ϕ(x)=6x(1 − x). The plots in the top, middle and bottom rows are for K−, π, and ηs, respectively. The imaginary part for kaon is not vanishing, which suggests skewness in the kaon DA.Fig. 11Kaon distribution amplitude from lattice QCD and the flavor SU(3) symmetryLattice Parton Physics Project (LP3) CollaborationRuiZhangabcJiunn-WeiChend⁎jwc@phys.ntu.edu.twLuchangJinefHuey-WenLinag⁎hwlin@pa.msu.eduAndreasSchäferhPengSunaYi-BoYanga⁎yangyibo@pa.msu.eduJian-HuiZhangh⁎jianhui.zhang@ur.deYongZhaoiaDepartment of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USADepartment of Physics and AstronomyMichigan State UniversityEast LansingMI48824USAbKey Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, ChinaKey Laboratory of Theoretical PhysicsInstitute of Theoretical PhysicsChinese Academy of SciencesBeijing100190ChinacSchool of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, ChinaSchool of Physical SciencesUniversity of Chinese Academy of SciencesNo. 19A Yuquan RoadBeijing100049ChinadDepartment of Physics, Center for Theoretical Physics, and Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei, 106, TaiwanDepartment of PhysicsCenter for Theoretical PhysicsLeung Center for Cosmology and Particle AstrophysicsNational Taiwan UniversityTaipei106TaiwanePhysics Department, University of Connecticut, Storrs, CT 06269-3046, USAPhysics DepartmentUniversity of ConnecticutStorrsCT06269-3046USAfRIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USARIKEN BNL Research CenterBrookhaven National LaboratoryUptonNY11973USAgDepartment of Computational Mathematics, Michigan State University, East Lansing, MI 48824, USADepartment of Computational MathematicsMichigan State UniversityEast LansingMI48824USAhInstitut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyInstitut für Theoretische PhysikUniversität RegensburgRegensburgD-93040GermanyiCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USACenter for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeMA02139USA⁎Corresponding authors.Editor: Hong-Jian HeAbstractWe present the first lattice-QCD calculation of the kaon distribution amplitude using the large-momentum effective theory (LaMET) approach. The momentum-smearing technique has been implemented to improve signals at large meson momenta. We subtract the power divergence due to Wilson line to high precision using multiple lattice spacings. The kaon structure clearly shows an asymmetry of the distribution amplitude around x=1/2, a clear sign of its skewness. Our result also prefers a broader distribution than the asymptotic form. We also study the leading SU(3) flavor symmetry breaking relations for the pion, kaon and eta meson distribution amplitudes, and the results are consistent with the prediction from chiral perturbation theory.1IntroductionMeson distribution amplitudes (DAs) ϕM are important universal quantities appearing in many factorization theorems which allow for the description of exclusive processes at large momentum transfers Q2≫ΛQCD2 [1,2]. Some wellknown examples of such processes, which are relevant to measuring fundamental parameters of the Standard Model, include B→πlν,ηlν giving the CKM matrix element |Vub|, B→Dπ used for tagging, and B→ππ,Kπ,KK¯,πη,… which are important channels for measuring CP violation (see e.g. [3]). Among those processes, the large differences between the strength of direct CP violation for B±→π0K± and B0→π∓K± [4], and for D0→K+K− and D0→π+π− [5] clearly highlight the importance of understanding the flavor SU(3) symmetry breaking among light flavors before attributing the effects to enhancement of higher-order amplitudes or even new physics. A first-principles theory determination of the kaon DA will help to make predictions for the large-momentum transfer kaon form factors that will be measured as part of the upcoming Jefferson Laboratory 12-GeV upgrade program [6].In the chiral limit where mq→0 with q=u,d,s, SU(3) symmetry predicts ϕπ=ϕK=ϕη=ϕ0. Away from the chiral limit, we work in the isospin limit (mu=md=m¯) (for simplicity), use the MS‾ scheme, and normalize the DAs such that ∫dxϕM(x)=1 with meson index M=π,K,η. The leading SU(3) breaking from chiral symmetry takes the form [7](1)ϕM(x,μ)=ϕ0(x,μ)+∑P=π,K,ηmP2(4πfP)2[EMP(x,μ)lnmP2μχ2+FMP(x,μ,μχ)]+O(mP4). The functions ϕ0, EMP and FMP are independent of light-quark masses, fP is the decay constant for meson P, x is the fraction of the meson momentum held by the quark, μ is the factorization scale, and μχ is the dimensional regularization parameter in chiral perturbation theory (ChPT). The μχ dependence in FMP and ln(mP2/μχ2) cancel such that ϕM is μχ independent. In Ref. [7], it was proven within the context of ChPT that(2)EπP(x,μ)=EKP(x,μ)=EηP(x,μ)=0 for all P. Hence, at O(mq), the DAs in Eq. (1) are analytic in mP2, where we have used mP2∝mq+O(mq2). Ref. [7] has also shown that(3)ϕK+(x,μ)−ϕK−(x,μ)=ϕK0(x,μ)−ϕK¯0(x,μ)∝ms−m¯. Furthermore, at O(mq),(4)ϕπ(x,μ)+3ϕη(x,μ)=2[ϕK+(x,μ)+ϕK−(x,μ)]=2[ϕK0(x,μ)+ϕK¯0(x,μ)], and hence,(5)ϕπ(x,μ)+3ϕη(x,μ)−2ϕK+(x,μ)−2ϕK−(x,μ)=O(mq2). It will be interesting to investigate whether the above leading SU(3) breaking relations derived from ChPT emerge from direct computations of meson DAs in lattice QCD.Such direct computations have become possible recently, thanks to the large-momentum effective theory (LaMET) [8–10]. The LaMET method calculates time-independent spatial correlations (called quasi-distributions) on the lattice and takes the infinite-momentum limit to recover the true lightcone distribution. For large momenta feasible in lattice simulations, LaMET can be used to relate Euclidean quasi-distributions to physical ones through a factorization theorem, which involves a matching and power corrections that are suppressed by the hadron momentum [9]. In the past few years, there have been many studies on the one-loop matching kernel for the leading-twist PDFs [11–14], generalized parton distributions (GPDs) [15,16] and meson DAs [15], as well as on the power corrections [17–20]. The renormalization property of quasi-distributions was also investigated [21–29] with the multiplicative renormalizability established to all-loop orders. The LaMET approach has been applied to compute the nucleon unpolarized, helicity and transversity PDFs [17–19,26,30,31], as well as the pion DA [32]. A first lattice PDF calculation at physical pion mass has recently become available [31]. The O(a)-improved operators associated with large hadron momentum were worked out in Ref. [33].Motivated by LaMET, it was proposed that one can extract the PDFs from the “lattice cross sections” [34,35], and the quasi-PDF is one of them. More recently, it was suggested that one can study instead an Ioffe-time or pseudo distribution [36] which is related to the quasi-distribution through a simple Fourier transform. While this method shows some interesting renormalization features [37], it is essentially equivalent to the LaMET approach [10,38,39] and offers no new physics regarding the factorization into PDFs. In addition, there are proposals using current-current correlators to compute PDFs, the pion DA, etc. [40–44]. Different approaches can have different systematics to reach the same goal; therefore, they can be complementary to each other.A first lattice calculation of the leading-twist pion DA using LaMET is done in Ref. [32], where the results were improved by a Wilson line renormalization that removes power divergences. The final result favors a single-hump form for the pion DA, which is broader than its asymptotic form. The plan of the present paper is to extend the study of the π meson in Ref. [32] to the K meson and their SU(3) partner, the η meson. A further improvement was made by implementing the momentum-smearing technique proposed recently [45] to increase the overlap with the ground state of a moving hadron. With our results computed directly from lattice QCD, we will examine the ChPT prediction of the leading flavor SU(3) breaking relations in Eqs. (3) and (4).The rest of the paper is organized as follows: In Sec. 2 we briefly review the procedure for extracting meson DAs from the quasi-DAs defined in LaMET and explain how we access the η DA. In Sec. 3, we show our lattice results. The final result on ϕK− clearly shows its skewness. Alongside, we have the corresponding results for π and η mesons and study the leading SU(3)-breaking relations. The conclusion and outlook are given in Sec. 4.2Methodology2.1Meson DAs from LaMETAs was explained in Ref. [32], in the framework of LaMET, the meson DA(6)ϕM(x,μ)=ifM∫dξ2πei(x−1)ξn⋅P〈M(P)|ψ¯(0)n⋅γγ5Γ(0,ξn)λaψ(ξn)|0〉 can be extracted from the quasi-DA(7)ϕ˜M(x,μR,Pz)=ifM∫dz2πe−i(x−1)Pzz〈M(P)|ψ¯(0)γzγ5Γ(0,z)λaψ(z)|0〉, where μ is a renormalization scale of ϕM in the MS‾ scheme, nμ=(1,0,0,−1)/2 is a lightlike vector, Γ is a straight Wilson line that makes the quark bilinear operator gauge invariant, λa=λ3, (λ4±iλ5)/2, λ8 for M=π, K±, and η, respectively. In the ϕ˜M computation, both the quark bilinear and the meson momentum Pz are along the z direction. μR denotes the renormalization scale of ϕ˜M in a given scheme. ϕM and ϕ˜M are the same in the infrared. After removing the power divergence in ϕ˜M non-perturbatively, the difference between ϕM and ϕ˜M in the ultraviolet can be compensated by the matching kernel Zϕ, which can be computed perturbatively:(8)ϕ˜M(x,μR,Pz)=∫01dyZϕ(x,y,μ,μR,Pz)ϕM(y,μ)+O(ΛQCD2Pz2,mM2Pz2). Currently, the matching kernel is only computed to one-loop with the quasi-DA computed with a transverse-momentum cutoff regulator and the DA computed in the MS‾ scheme [15]. Therefore, the Zϕ factor is accurate up to the leading logarithm but not for the numerical constant. To determine this constant, a lattice perturbation theory calculation using the same lattice action is required.The matching kernel Zϕ has the form(9)Zϕ(x,y)=δ(x−y)+αs2πZ‾ϕ(x,y)+O(αs2)=δ(x−y)+αs2π(Zϕ(1)(x,y)−Cδ(x−y))+O(αs2) with C=∫−∞∞dx′Zϕ(1)(x′,y). The expression for Zϕ(1)(x,y) can be found in Ref. [32]. Eq. (9) tells us that ϕM and ϕ˜M differ only at loop level, thus we can write (ignoring the power corrections for the moment)(10)ϕM(x)≃ϕ˜M(x)−αs2π∫dyZ‾ϕ(x,y)ϕ˜M(y)≃ϕ˜M(x)−αs2π∫−∞∞dy[Zϕ(1)(x,y)ϕ˜M(y)−Zϕ(1)(y,x)ϕ˜M(x)] with an error of O(αs2) [34]. For simplicity, we have also extended the integration range of y to infinity, which will introduce an error at higher order. To account for the power corrections, we need to know higher-twist and meson-mass corrections as well. The meson-mass corrections have been computed to all orders in mM2/Pz2 [32], while the higher-twist corrections were removed by a simple fitting with a polynomial form in 1/Pz2. In this work, we will follow the same procedure but leave out the higher-twist corrections, because we have observed non-monotonic behavior in Pz in our lattice data. This implies that the polynomial fit might be too naive to account for the higher-twist effects.2.2Accessing the meson DA matrix element on the latticeWe start from the calculation of the correlator,(11)C˜(z,Pz,τ)=〈∫d3xeiP→⋅xψ¯(x→,τ)γzγ5Γ(x→,x→+z)λa†ψ(x→+z,τ)ψ¯S(0,0)γ5λaψS(0,0)〉, where the sink operator at timeslice τ is the Fourier transform of the quasi-DA, and the quark fields ψS in the source operator at timeslice 0 have been momentum smeared [45],(12)ψS(x)=∫d3ye−|x−y|22σ2−ik→⋅(x→−y→)U(x,y)ψ(y), where U(x,y) is the gauge link that makes ψS(x) gauge covariant, σ is the smearing radius. Following Ref. [45], the momentum smearing parameter k→ is determined by optimizing the signal of C˜(z,Pz,τ). We found that k→=±0.73Pz→ for the quark (antiquark) is suitable for our calculations with Pz→=(0,0,{4,6,8}π/L). Note that we need to generate the quark and antiquark propagators separately, since the optimal k→'s for them have opposite signs.Following the standard procedure, we insert a complete set of states between the two operators at timeslices t and 0 in C˜ Eq. (11). Then, assuming the complete set of states is saturated by the ground state of energy E0 and an effective excited state of energy E1 at large t, we have(13)C˜(z,Pz,τ)=Zsrch˜M(z,Pz)2E0e−E0τ+B(z,Pz)e−E1τ with the matrix element(14)h˜M(z,Pz)=〈M(P)|ψ¯(0)γzγ5λaΓ(0,z)ψ(z)|0〉. We varied the fitting range but did not see significant differences; the χ2/d.o.f. for most matrix elements remains smaller than or around 1.3. We also extended our analysis by performing a three-state fit for the matrix elements and confirm that three-state fit results are consistent with the two-state fit ones but with larger error bars, as expected.What we need for the quasi-DA calculation is the normalized h˜ defined as(15)hM(zPz,Pz)=h˜M(z,Pz)PzfM, which satisfies hM(0,Pz)=1. Therefore, even if we do not separate h˜M from the z-independent source matrix element Zsrc, the determination of hM is not affected.2.3Accessing the η distribution amplitudeFor π and K, C˜ in Eq. (11) receives contributions from “connected diagrams” (see Fig. 1) only. For η, in addition to connected diagram contributions, C˜ also receives contributions from “disconnected diagrams” (see Fig. 1). However, the disconnected diagram is O((ms−m¯)2) suppressed because there are two fermion loops, each of which is suppressed by one power of (ms−m¯) in the diagram. Therefore, it seems that if we just work at O(mq), we can safely neglect the disconnected diagram of η.However, by dropping the disconnected diagrams, the u(d) and s quark contributions in C˜ yield different values of ground-state energy E0; that is, ΔE0≡E0s−E0u(d)≠0. Then, when τ>1/|ΔE0|, C˜ is dominated by the quark contribution of lower E0. However, when the hadron momentum Pz is large, such that E0≃Pz+mq¯q2/2Pz, where mq¯q is the mass of the q¯q state without the disconnected diagram, as long as the plateaus for the mass determination appear within τ<2Pz/|ms¯s2−mu¯u2|, C˜ remains equally balanced between u(d) and s quark contributions. Therefore, even without including the disconnected diagrams, the error from this ambiguity can be systematically reduced by increasing Pz.There is another complication for η. That is, the operator associated with λ8 creates the η8 meson, but the physical η is a linear combination of η8 and η0, the SU(3) singlet. Fortunately, the mixing angle θ is small (θ≈−15∘) [46]. Therefore, in Eq. (11), when we insert the physical η state between the two operators at time slices τ and 0, the η0 contribution is suppressed by a mixing factor sinθ≈0.08 times a factor of (ms−m¯) coming from the overlap of η0 with the λ8 type operator. Hence, numerically, the η0 contribution can be counted as O((ms−m¯)2) and can be neglected in our calculation.The above discussion leads to the conclusion that if the plateaus for the meson-mass determination appear within τ<2Pz/|ms¯s2−mu¯u2|, then the connected diagram contribution of C˜ of Eq. (11) with λa=λ8 yields contributions from u,d,s quarks in the ratio 1:1:4. This C˜ can be determined from Eq. (13) with the matrix element of Eq. (15) associated with the η8 DA. This implies that for the largest Pz we use (8π/L), the plateaus should be reached within τ<1/|ΔE|≃20a, which is clearly satisfied.In the following sections, we first present the unphysical ηs results (defined as the ss¯ pseudoscalar meson with connected diagrams only) for different Pz values. Then, based on the above discussion, we approximate the η8 DA for Pz=8π/L with (ϕπ+2ϕηs)/3, and use the result to check the SU(3) relation Eq. (4).3Lattice resultsIn this section, we present our lattice setup and the results for the π, K and η DAs. The simulations were performed using clover valence fermions on a 243×64 lattice with 2+1+1 flavors (degenerate up and down, strange, and charm degrees of freedom) of highly improved staggered quarks (HISQ) [47] generated by the MILC Collaboration [48]. The pion mass on this ensemble is 310 MeV, and the lattice spacing a≈0.12 fm. In this work, hypercubic (HYP) smearing [49] is applied to the configurations; the bare quark masses and clover parameters are tuned to recover the lowest pion mass of the staggered quarks in the sea. The results shown in this section were obtained using the correlators calculated from 3 momentum-smearing sources and 4 source locations on each of the 967 configurations.We first check whether the dispersion relation E2(Pz)=m2+c2Pˆz2 for Pˆz=2/asin(Pza/2) is satisfied up to Pz=8π/L (the largest momentum we used) for π, K and, motivated by the discussion in section 2.3, the unphysical ηs. As shown in Fig. 2, this dispersion relation is satisfied within two sigma statistical uncertainties. For Pˆz=1, Pz/Pˆz≃1.05.The bare matrix elements hM defined in Eq. (15) for kaon, pion and ηs are shown in Fig. 3 for Pz=4π/L,6π/L, and 8π/L. The Fourier transform of the DA asymptotic form [50], ϕ(x)=6x(1−x), is also shown in Fig. 3. However, the asymptotic DA is defined in another renormalization scheme, so this is not a direct comparison with hM. We do observe similar oscillating behavior in our data. When we increase Pz, the secondary peaks become more pronounced. But the difference between Pz=6π/L and 8π/L is small already. Nevertheless, we plan to repeat this work with larger boost momentum (to extend the zPz reach) and reduce the lattice spacing by at least a factor of 2 in the future.3.1Improved distribution amplitudeWith the DA matrix elements hM, we can then Fourier transform according to Eq. (7) to study the meson DAs. To cancel the power divergence in hM arising from the Wilson-line self-energy diagrams we introduce a counterterm δm, as suggested in Refs. [23,32], such that the matching kernel Zϕ only has logarithmic divergence but no power-divergent contributions. Thus, the “improved” meson quasi-DA [32] is(16)ϕ˜Mimp(x,Pz)=∫−∞∞dz2πe−i(x−1)zPz+δm|z|PzhM(z,Pz). We then apply the matching kernel Zϕ and mass correction, as discussed in Sec. 2, to obtain the final DA.First, we need to calculate the counterterm δm. The Wilson line can be equivalently described by a quark propagator in the heavy-quark limit and the only dimensionful counterterm in the heavy-quark Lagrangian is the mass counterterm δm. Therefore, δm can be determined by the Wilson loop W(τ,r) with width r and length τ, which has the negative effective action of a static quark–antiquark pair with interquark distance r at temperature 1/τ. The quark–antiquark effective potential is approximated by(17)V(r)=−1alimτ→∞ln〈Tr[W(τ,r)]〉〈Tr[W(τ−a,r)]〉, using a combination of such Wilson loops. The cusp anomalous dimensions from the four sharp corners of the Wilson loop are canceled between the numerator and denominator of the expression, and keeping 1/τ larger than the inverse of the energy gap between the ground state and the first excited state ensures that higher excitations are sufficiently suppressed.When r is larger than the confinement scale but shorter than the string-breaking scale, this can be fit by(18)V(r)=c−1r+c0+c1r, where the c−1 term is the Coulomb potential that dominates at short distance and the c1 term is the confinement linear potential. c0 is of mass dimension one, so we can break it into a divergent piece and a finite one in the continuum limit: c0=c0,1/a+c0,2. Then(19)δm=−c0,12a, where the 2 compensates for the potential using a quark–antiquark pair.Fig. 4 shows the effective potential V(r) at lattice spacings a=0.06,0.09,0.12 fm for Mπ=130 MeV and a=0.12 fm for Mπ=310 MeV. The Mπ dependence for a=0.12-fm ensembles is almost undetectable. A fit of the potential with four parameters c−1,c0,1,c0,2 and c1, V(r≥5a) has a very good χ2/d.o.f.=1.04 (46 degrees of freedom). This fit yields δm=0.154(2)/a, which corresponds to 253(3) MeV at a=0.12 fm.With the thus determined δm, we can now obtain ϕ˜Mimp(x,Pz) for each meson using Eq. (16). Next, we apply the one-loop matching kernel Zϕ(1) (see Eq. A.6 of Ref. [32], with the lattice cut-off scale π/a and the MS‾ renormalization scale μ=2 GeV), which is essential in LaMET to obtain lightcone quantities from the quasi-distribution(20)ϕMimp,match(x,Pz)≃ϕ˜Mimp(x,Pz)−αs2π∫−∞∞dy[Zϕ(1)(x,y)ϕ˜Mimp(y,Pz)−Zϕ(1)(y,x)ϕ˜Mimp(x,Pz)] with an error of O(αs2) [34] as discussed earlier in Sec. 2.1. We then apply the mass corrections [32] to ϕMimp,match to get the final DAs(21)ϕM=1+4c∑n=0∞(4c)nf+2n+1[(1+(−1)n)ϕMimp,match(12−f+2n+1(1−2x)4(4c)n)+(1−(−1)n)ϕMimp,match(12+f+2n+1(1−2x)4(4c)n)], where c=mM2/4Pz2 and f+=1+4c+1. The remaining higher-twist effect is of O(ΛQCD2/Pz2), which is small at our largest 2 momenta used in this work.3.2The renormalon ambiguityThe analogy of HQET is used in two different parts in this work, one is the subtraction of the power divergence using the δm counterterm, where the Wilson line is described using the HQET language, the other is the matching between quasi-DA and DA. Here we discuss whether the renormalon ambiguity will affect each of them.For the first part, the subtraction of power divergence using the δm counterterm, if we were doing the subtraction perturbatively, then the IR renormalon ambiguity could arise and give an O(ΛQCD) uncertainty to δm. However, this subtraction is done non-perturbatively, so the renormalon ambiguity does not enter [51].For the second part, the relation between quasi-DA to DA is similar to that of the quark mass in QCD and the pole mass in HQET, with matching and power corrections involved in both cases. In the quark mass case, a perturbative computation of the matching coefficient has the IR renormalon ambiguity which is completely canceled by the corresponding ambiguity from power corrections. Therefore the IR physics of the full theory remains the same as that of HQET. This is exactly the same as in the DA case. The IR renormalon ambiguity in the matching kernel can be removed systematically by performing the power corrections consistently. In this work, the renormalon ambiguity arising in the kernel is numerically O(ΛQCD2/Pz2), which is of the same order as the twist-4 effect. Here we have neglected this effect because it is parametrically a small effect. But in an improved calculation, the renormalon ambiguity at O(ΛQCD2/Pz2) can also be removed by computing the twist-4 contribution consistently.3.3Kaon distribution amplitudeLet us now consider the first results for the kaon DA from lattice QCD. The left-hand side of Fig. 5 shows a comparison of ϕ˜Kimp(x) (improved quasi-DA shown in blue), ϕ˜K−imp,match(x) (after matching quasi-DA to lightcone DA shown in green) and ϕK−(x) (DA with meson mass correction added, shown in red) from our largest meson momentum. The distribution after applying the one-loop matching ϕ˜K−imp,match(x) changes quite significantly from the quasi-distributions. Further treatment with the meson mass correction yields ϕK−(x) which is very close to ϕ˜K−imp,match(x). This is expected with the large momentum used here.The right-hand side of Fig. 5 shows the momentum dependence of ϕK−(x). Note that the higher-twist correction is not extrapolated away as in our previous work due to the non-monotonic behavior in Pz. However, we expect its effect to be small at the largest two momenta used in this work, since their difference is small. ϕK− is skewed towards large x since its valence s quark is heavier than its valence u¯ quark. However, the distribution outside the region x∈[0,1] is still quite sizable (though shrinking when Pz is increased). Given that the DA for the largest 2 momenta are already quite close to each other, it seems unlikely that the residual effect in the unphysical region is totally due to higher-twist power corrections in 1/Pz that are not accounted for. Given the large one-loop matching correction seen in the left-hand side of Fig. 5, it will be important to investigate the two-loop matching contributions in the future to check their size. In addition, the truncation of zPz in the Fourier transformation can yield nonzero distribution outside x∈[0,1]. This has become more visible in this work than our previous work in the pion case because of the larger momentum reach.Finally, we compare our ϕK− result (labeled “Lat LaMET”) with Pz=8π/L with a few selected results in the literature in Fig. 6: the result from fitting a parametrization to the lowest few moments calculated in lattice QCD [52,53] with pion mass ranging 330–670 MeV (labeled “Lat Mom”), Dyson–Schwinger equation calculations [54] (“DSE-1” and 2), and a calculation with a light-front constituent quark model [55] (“LFCQM”).We stress that systematics from unphysical pion mass, nonzero lattice spacing and finite volume have not been quantified and included in this first calculation. Likewise, the systematics of other approaches are likely not fully taken into account either. Therefore, it is premature to draw firm conclusions based on the comparison shown in Fig. 6. With this in mind, we observe a broader distribution than the one from LFCQM, without making the assumption on the distribution form of xα(1−x)β. Our ϕK− noticeably has smaller peak near x=0.5; this is mainly due to the sizable distribution outside the [0,1] region, since the integral of the kaon DA is normalized to 1. Therefore, the DA has to have a smaller peak to produce the same integral. We plan to study the higher-loop matching as well as go to large Pz to reduce the Fourier-transformation truncation effects.3.4SU(3) symmetry in meson distribution amplitudesIn this work, we also update our previous study [32] of the pion DA and make the first study of the ηs case. Fig. 7 shows both DAs obtained after the one-loop matching and mass corrections (but not higher-twist corrections at O(ΛQCD2/Pz2)). Larger boost momentum (with specifically tuned momentum-smearing parameters) and higher statistics are used in this work for ϕπ. The dominant systematic uncertainty, due to the counterterm δm using a single spacing in the previous study, is significantly improved with the use of 3 lattice-spacing determinations in this work. We also observe that both ϕπ and ϕηs are symmetric with respect to x=1/2 due to charge-conjugation symmetry. As in the kaon case in the previous subsections, there is sizable distribution outside x∈[0,1]. As discussed earlier, we suspect finer lattice spacing and higher-loop matching in future studies may improve these properties.Fig. 8 shows a comparison of our current results for ϕπ with earlier results in literature. Again, systematics from unphysical pion mass, nonzero lattice spacing and finite volume have not been quantified and included in this calculation and one should not draw a confirm conclusion based on this comparison. In the left panel, we show our ϕπ result along with a result using Dyson–Schwinger equation (DSE) [56], truncated Gegenbauer expansion fit to the Belle data for the γγ⁎→π0 form factor (Belle) [57], and from parametrizations to lattice-QCD lowest-moment calculations [58] to extract the pion DA. For the fit to the Belle data, the Gegenbauer-polynomial expansion up to the eighth moment given in Ref. [57] was used at scale 2 GeV. For the fit to the lattice-moment distribution, two different parametrizations are shown here. The first one is a simple truncation of the Gegenbauer-polynomial expansion to the pion DA parametrization to the second order ϕ(x)=6x(1−x)[1+a2C23/2(2x−1)] (“Lat Mom 1”) with the value of a2 taken from Ref. [58]. The second distribution is ϕ(x)=A[x(1−x)]B with A and B determined from the normalization condition and the lattice calculations of the second moment (“Lat Mom 2”) [58]. The two parametrizations using lattice moment calculations yield significantly different pion DA. The difference between them can be viewed as a rough estimate of errors due to the moment truncation. With more lattice moment data the parametrization dependence may improve; however, with individual distributions the systematic error is currently underestimated. Our distribution has a lower peak at x=1/2 mainly due to the nonvanishing contribution outside the [0,1] region, since the integral of the distribution over all regions is normalized to 1 by definition. Given the smallness of the mass corrections and that our curves at Pz=6π/L and 8π/L are very close to each other, we expect the higher-order matching kernel will play an important role in reducing the contribution in the unphysical region. Also higher boosted momentum will help improve the truncation systematics in Fourier transform in zPz. This needs to be further investigated before we can draw a definite conclusion on the shape of ϕπ. In the right panel, we also compare our result on ϕπ with the calculation using Euclidean current correlators in Ref. [44], where the lattice data was presented for the scalar-pseudoscalar current correlator. In order to make a direct comparison, we have convolved our result with the coefficient function up to O(αs) in Ref. [44], and then included the higher-twist contributions obtained there. Our final result is shown as the blue curve. The dark, gray and white circles are the lattice data in Ref. [44] for |P→|=1.08,1.53,1.88 GeV, respectively, μ is the renormalization scale. As can be seen from the plot, both approaches yield consistent results at small P→⋅z→.Finally, we investigate the leading SU(3) flavor breaking effect predicted using ChPT. In this work, we did not calculate ϕη directly; as discussed in Sec. 2.3, at the largest boost momentum Pz=8π/L we can approximate ϕη from ϕπ and ϕηs:(22)ϕη=(ϕπ+2ϕηs)/3. Since ϕπ and ϕηs are quite close to each other, ϕη is similar to the distribution shown in Fig. 7. We are interested in verifying the following SU(3) symmetry breaking relations:(23)δSU(3),1=(ϕK−−ϕK+)/2,(24)δSU(3),2=(ϕπ+3ϕη−2ϕK+−2ϕK−)/8. ChPT [7] predicts the magnitude of δSU(3),1 to be O(mq) while the magnitude of δSU(3),2 is O(mq2); thus, the lattice results should see |δSU(3),1|>|δSU(3),2|.Fig. 9 shows the Bjorken-x dependence of both δSU(3),1 (left) and δSU(3),2 (right) at the largest boost momentum Pz=8π/L. δSU(3),1 shows a clear sign of the skewness in the kaon. |δSU(3),1|>|δSU(3),2| within x∈[0,1] (except when x is close to 1/2 where δSU(3),1=0), so the ChPT prediction is indeed supported by our lattice study. In addition, δSU(3),2 is consistent with zero within the statistical errors at the pion mass of 310 MeV. Future studies at lighter pion mass can check the quark-mass dependence directly.Finally, we discuss a possible cause of the non-monotonic Pz dependence in our extracted DA's after we have corrected for the one loop matching kernel and the all order mass corrections. Here we take the asymptotic DA 6x(1−x), Fourier transform it to the z space, truncate the region outside of z=[−L/2,L/2], then Fourier transform it back to the x space. The resulting truncated asymptotic DA is shown in Fig. 10 where a non-monotonic Pz dependence is seen at x=1/2, similar to other meson DAs we obtained above. This example demonstrates that the truncated Fourier transform could cause an error that is non-monotonic in Pz which needs to be taken into account in the Pz→∞ extrapolation.4Conclusion and outlookWe have presented the first lattice calculation of the kaon distribution amplitude using the large-momentum effective theory (LaMET) approach with a pion mass of 310 MeV. Momentum smearing has been implemented to improve signals up to meson momentum 1.7 GeV. We subtract the power divergence due to Wilson line using the counterterm δm determined to 2% accuracy using multiple lattice spacings—a significant improvement over our previous pion-DA work. We clearly see the skewness of kaon from the asymmetric distribution with respect to x=1/2 (or equivalently the nonvanishing ϕK−−ϕK+).We also present the first results on ηs DA (and an indirect determination of η), as well as an improved determination of the pion DAs. Similar to the kaon case, there are nonvanishing contributions outside the physical region [0,1]. Without eliminating them, we are unable to draw a definite conclusion on the shape of the DAs, since the result in the physical region will be affected by the total normalization. With all 3 meson DAs, we are able to investigate the leading SU(3) flavor symmetry breaking in meson DAs suggested by ChPT [7], and clearly observe |δSU(3),1|>|δSU(3),2| for x∈[0,1] except when x is close to 1/2 where δSU(3),1=0. The quark-mass dependence can be studied in the future using lighter pion masses.Given that some of these exciting results are being studied first time on the lattice, there are possible improvements for future work. Before a solid conclusion can be made, the calculation should use physical light quark masses, smaller lattice spacing, large enough volume, and matching kernels computed using lattice perturbation theory (or RI/MOM renormalization) on the quasi-DA side. We speculate, however, some of the effects will be small: for example, in view of the small difference between ϕπ and the ϕηs despite m¯≪ms, the error from not using physical quark masses might be small for the meson DAs. Also, the correlators hM at large z are already quite small as shown in Fig. 11, therefore, we do not expect the finite volume effect to be large. On the other hand, the simulation should be repeated at different lattice spacings to confirm the linear divergence has been fully removed, and the O(Pza) effect is under control when we increase Pz.With the improved signal due to the usage of the momentum-smearing sources and better determination of Wilson-loop counterterm δm, the distribution outside the x∈[0,1] region remains sizable and not consistent with zero by a few standard deviations. This leads to a few possible directions to achieve more reliable meson DAs (removing the residual DA outside the [0,1] region): Doubling the momentum on finer lattice spacing, say 0.06 fm, can reduce the systematics due to the truncation in zPz in Fourier transform from lattice nonlocal matrix elements. This will also reduce the size of higher-twist contributions, which seems to be more noticeable outside x∈[0,1] than within. In addition, the finite meson-momentum correction using the one-loop matching kernel dominates the sums of all corrections (including the mass correction and the higher-twist estimation). This suggests that moving to higher-loop level for the matching kernel can have sizable contribution. We plan to work out the exact form in a future study.AcknowledgementsWe thank the MILC Collaboration for sharing the lattices used to perform this study. The LQCD calculations were performed using the Chroma software suite [59] and multigrid solver for clover fermions package [60]. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231; facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy, and supported in part by Michigan State University through computational resources provided by the Institute for Cyber-Enabled Research. JC is partly supported by the Ministry of Science and Technology, Taiwan, under Grant No. 105-2112-M-002-017-MY3 and the Kenda Foundation. LJ is supported by the Department of Energy, Laboratory Directed Research and Development (LDRD) funding of BNL, under contract DE-EC0012704. RZ, HL and YY are supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for New-Physics Searches”. AS and JZ are supported by the SFB/TRR-55 grant “Hadron Physics from Lattice QCD”, JZ is also supported by a grant from National Science Foundation of China (No. 11405104). JZ thanks G. Bali and P. Wein for help with the comparison plot in the right panel of Fig. 8. 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.Appendix ASkewness of the kaon distribution amplitudeThere is another way to see the skewness of ϕK− in the lattice data. We can consider the following improved bare matrix elements:(A.1)HM(z,Pz)=ei2zPz+δm|z|hM(z,Pz). Note that the linear divergence has been removed by δm. Using HM(z,Pz) to study the skewness removes any systematics due to the truncation in zPz Fourier transform, the matching or power corrections. We show the results of HM(z,Pz) for kaon (as well as for pion and ηs) in Fig. 11. The real (imaginary) part of the improved meson DA matrix elements HK−(z,Pz) is even (odd) in z; therefore, we only plot the positive-z region. The imaginary parts for Hπ and Hηs are consistent with zero due to charge-conjugation symmetry, which make ϕ˜π,ηsimp(x) and ϕπ,ηs(x) even functions of (x−1/2). On the other hand, the nonvanishing imaginary part of HK−(z,Pz) leads to an asymmetry around x=1/2 and shows the skewness of ϕK− (that is, ϕK−(x)≠ϕK−(1−x)).References[1]M.BenekeG.BuchallaM.NeubertC.T.SachrajdaNucl. Phys. B6062001245arXiv:hep-ph/0104110[2]M.BenekeG.BuchallaM.NeubertC.T.SachrajdaPhys. Rev. 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