The two-point functions of kaon can be decomposed as, CKK(P,ts)=∑n=0Nstate-1|⟨Ω|KS|n;P⟩|2(e-Ents+e-En(aLt-ts)),=∑n=0Nstate-1|Zn|22En(e-Ents+e-En(aLt-ts)),(4)with En representing the energy levels and An=Zn/2En being the kaon overlap amplitude ⟨Ω|KS|n;P⟩. By truncating at Nstate=1, one can define the effective mass of kaon using adjacent time slices as shown in Fig. 1. As one can see, the effective mass reach the plateaus around ts≳8a, in which the plateau of Pz=0 around 497 MeV is the physical kaon masses. The plateaus of nonzero momentum all agree with the solid lines evaluated from the dispersion relation E0(Pz)=Pz2+mK2, suggesting the energy spectrum is dominated by the kaon ground state after ts∼8a.

We then perform the N-state fit for the two-point functions of ts in range [tmin,32a]. The results are shown in Fig. 2. The E0 from one-state fit (upper panel) show similar behavior like Fig. 1 but more stable because of combining multiple ts. It is clear that the ground state energy E0 follows the dispersion relation E0(Pz)=Pz2+mK2 well as also observed in Ref. [71]. To further extract the first excited state E1, we use two-state fit and fix E0 with the dispersion relation. The results are shown in the lower panel of Fig. 2. Since the SS correlators are tuned to have better overlap with ground state, the signal of E1 is much worse. Notably, the E1 determined from two-state fit reaches the plateau around tmin∼4a, suggesting the energy spectrum can be well described by the two-state model from there. We will use the En and An in this section for the extraction of bare matrix elements of quasi-DA.

The bare quasi-DA matrix elements has spectral decomposition, CDAγ5γ3(z;P,ts)=∑n=0Nstate-1⟨Ω|Oγ5γ3(z)|n;P⟩⟨n;P|KS|Ω⟩×(e-Ents+e-En(aLt-ts)),=∑n=0Nstate-1Zn2En⟨Ω|Oγ5γ3(z)|n;P⟩×(e-Ents+e-En(aLt-ts)),(5)where both the Zn and En are the same as the two-point functions defined in Eq. (4) and analyzed in Sec. II A. Taking the advantage of correlations between CKK and CDA, we construct the ratio, Rγ5γ3(ts)=CDAγ5γ3(z;P,ts)CKK(P,ts).(6)In Fig. 3 we show the ratios Rγ5γ3(ts) for momentum nz={1,9} at z={1a,2a,4a} as a function of ts. At large ts, plateaus can be clearly observed. In the ts→∞ limit, the quasi-DA matrix elements of kaon ground state can be derived from Rγ5γ3(ts)→⟨Ω|Oγ5γ3(z)|0;P⟩/Z0. As has been observed in Sec. II A that a two-state spectral decomposition can approximate the ts dependence of two-point function beyond tmin=4a, we apply two-state fit to extract the ground state matrix elements, using the ratios Rγ5γ3(ts) started from ts≳4a. The fit results are shown as the bands in Fig. 3, which nicely describe the data. The extracted bare matrix elements ⟨Ω|Oγ5γ3(z)|0;P⟩ for nz∈[0,9] are summarized in Fig. 4.

Considering the Lorentz covariance, the matrix elements can be parametrized as [72], ⟨Ω|Oγ5γμ(z)|0;P⟩=PμH˜(z2,z·P)+zμmK2k˜(z2,z·P).(7)The 2nd term of the right hand side proportional to zμ contributes to the matrix elements of Oγ5γ3 at finite spacial separation z, while it is zero for the case of using Oγ5γ0. In the infinite-momentum limit, the first term H˜(z2,z·Pz) is approaching the light-cone DAs, thus is identified as the actual matrix element H˜(z,Pz)≡H˜(z2,z·Pz) in our notation, while the k˜(z2,z·P) term is suppressed to zero as a power correction. It is worth to mention that, PzH˜(z,Pz) is zero when Pz=0 for the case of Oγ5γ3.

When z=0, the kaon decay constant can be extracted from the matrix elements by H˜(0,Pz)=fK/ZA with ZA=0.969(1) the finite renormalization constant for the axial current operator that has been determined in Ref. [25]. The fK we got are summarized in Fig. 5 as a function of momenta Pz=2πnz/(Nsa). As one can observe, the momentum dependence is mild, suggesting the goodness of our extraction. Our determination at nz=1 and 2 are 153.3(4) and 152.7(4) MeV respectively, which are close to the FLAG average for 2+1 flavor QCD fK=155.7(7)  MeV [73]. The small deviation could come from the lattice artifacts as only one ensemble is used in this calculation. When z≠0, considering the parity and time-reversal symmetry (see appendix in Ref. [72]), the zmK2k˜(z2,z·P) term can only contribute to the imaginary part of matrix elements and leave the real part unaffected. As shown in Fig. 4, the real parts of matrix elements at Pz=0 are zero while the imaginary parts are nonzero as expected. In our calculation, to reduce the twist-3 contamination from the 2nd term in Eq. (7), we ignore the Pz dependence in k˜(z2,z·P), and define the subtracted matrix elements as HB(z,Pz)=1Pμ(⟨Ω|Oγ5γ3(z)|0;P⟩-⟨Ω|Oγ5γ3(z)|0;0⟩).(8)

The multiplicative renormalizablility of quasi-DA [38–40] allows us to define a UV-finite quantity by taking the double ratio of matrix elements at two different momenta [45,46], M(z,P1,P2)=lima→0HB(z,P2,a)/HB(0,P2,a)HB(z,P1,a)/HB(0,P1,a)=HR(z,P2,μ)HR(z,P1,μ),(9)where we have chosen the normalization HB(0,P,a)=HR(0,P,μ)=1. The above ratio is renormalization group invariant, thus is independent of the renormalization scale μ. The ratio can be reformulated through a short-distance factorization at z≪ΛQCD-1 of HR(z,P2,μ) with OPE [25] HR(z,Pz)=∑n=0∞(-izPz2)nn!∑m=0nCnm(z,μ)⟨ξm⟩(μ)+O(z2ΛQCD2),(10)where ⟨ξm⟩(μ)=∫01dx(2x-1)mϕ(x,μ) are the nonperturbative moments of the light-cone DA, and Cnm(z,μ) are the perturbative matching coefficients. Then the ratio M(z,P1,P2) is a function of the moments and known Wilson coefficients when the higher twist correction O(z2ΛQCD2)≪1, M(z,P1,P2)≈∑n=0∞∑m=0n(-izP22)nn!Cnm(z,μ)⟨ξm⟩(μ)∑n∞∑m=0n(-izP12)nn!Cnm(z,μ)⟨ξm⟩(μ),(11)allowing us to extract the moments from these ratios. Given the better signal-to-noise ratio in the real part of our data, we first extract even moments of ξ. Truncating Eq. (11) at m, n≤4 and the perturbation series at NLO [25,74], we have: Cnmγi=δnm+αsCF2π(32L+7200-512L+11124312L-37120-215L+14-1930L+536815L-24736),(12)where L=lnz2μ2e2γE4. The triangular matrix C is a result of the nonmultiplicative renormalization group evolution of the DA moments [75], ddlnμ2(1⟨ξ2⟩(μ)⟨ξ4⟩(μ))=γnm(1⟨ξ2⟩(μ)⟨ξ4⟩(μ))=-αs(μ)CF2π(000-51225120-215-19309130)·(1⟨ξ2⟩(μ)⟨ξ4⟩(μ))+O(αs2).(13)The off-diagonal part of γnm has been calculated up to 3-loop order using conformal symmetry [76]. We fit the moments {⟨ξ2⟩(μ),⟨ξ4⟩(μ)} with both NLO and the RG-resummed (RGR) Wilson coefficients. In the latter case, we fit the moments at an initial scale ⟨ξ2(μi=2e-γEz-1)⟩, where the log terms in Cnm(z,μi) vanish, then evolve to μ=2  GeV by solving Eq. (13). The scale variation is examined by choosing μi=2ce-γEz-1 with c={2,1,1/2} to estimate the uncertainties from higher-order perturbation theory. The fitted ratio and the extracted moments are shown in Fig. 6. We show the fitted moments with both statistical and scale variation error bars. Note that the scale variation becomes large when z increases, because the scale z-1/2∼ΛQCD becomes nonperturbative. The second moment of kaon DA is slightly lower than the moment of pion DA, indicating the kaon DA to be narrower, similar to what has been observed in previous x-dependent calculations [31,32,34].

In principle, the odd moments can also be extracted from the imaginary part of the matrix elements. In our data, we found that the imaginary part are nonvanishing for smaller momenta data, indicating a nonzero first moment of kaon DA. However, for large momenta, the imaginary parts decreases with momentum, and for the largest momentum Im[H(z,nz=9)] are consistent with zero at short distance. Thus in principle we cannot describe all data with a universal first moment. Since we focus on the large momentum expansion approach in this work, we show the fit results for the ratio of imaginary part at the largest momentum nz=9 to the real part of nz=1 in the range of z∈[a,zmax] at different zmax. This results in a first Mellin moment of kaon ⟨ξ⟩K(μ)≈0.0013(60), corresponding to the first Gegenbauer coefficient a1=0.002(10), which is consistent with zero, as shown in Fig. 7. The suppression of the kaon skewness at large momentum may suggest that such skewness originating from the mass difference between light and strange quark is no longer important in the infinite momentum limit, where both partons act just as massless particles. However, considering that our subtracted imaginary part is still potentially contaminated by the Pz dependence in zμk(z2,z·P) term in Eq. (7), and since previous lattice calculations of local twist-2 operator suggest a nonzero kaon first moment [16,17,19,77], the suppressed imaginary part at large momentum in our data may just be a lattice artifact, and needs further investigation.

The bare matrix elements of meson quasi-DAs contain UV divergences. Especially, besides logarithmic divergences, the spatial Wilson line W(0,z) in the operator has a linearly divergent self energy in the lattice regulator a-1 [37–40]. It can be renormalized multiplicatively as [38–40] HR(z,Pz)=HB(z,Pz,a)/ZR(z,a)(14)with a renormalization constant ZR(z,a)∼e-δm(a)|z| containing the same linear divergence in the mass counterterm δm(a)∼a-1. In this work, we used the hybrid renormalization scheme [47] that can be perturbatively matched to the MS¯ scheme in all regions of z. The renormalization constant in the hybrid scheme is defined as Zhybrid(z,a)={ZR(z,a)|z|≤zsZR(zs,a)e-δm(a)(|z|-zs)|z|>zs(15)At short distance, since the correlator of γzγ5 vanishes at Pz=0, we cannot use the ratio scheme ZR(z,a)=HB(z,P=0,a). Instead, we determine it from the self-renormalization approach [35,48], which uses the perturbatively calculated Wilson coefficient ZR(z,a)=e-δm(a)|z|C0(z,μ)(16)for |z|≤zs. And δm(a) is obtained by requiring that the matrix elements after removing the linear divergence eδm(a)|z|HB(z,Pz,a) agree with OPE reconstruction of HR(z,Pz,μ) in Eq. (10) using the moments fitted from the RG-invariant ratio as inputs, up to a constant conversion factor that converts the lattice scheme to MS¯ [50], eδm(a)|z|HB(z,Pz,a)=HR(z,Pz,μ)eIlat(a-1)-I(μ),(17)where I(μ)=∫γβdα|α=αs(μ) are the RG-evolution factors that cancels the UV-regulator dependence μ in the matrix elements. To ensure the linear power accuracy of the factorization, we applied the LRR and RGR-improved Wilson coefficient [50]. CmnLRR(z,z-1)=Cmn(z,z-1)+δnmN⁢[-αs(1+c1)+4πβ0∫0,PV∞due-4πuαs(μ)β0⁢1(1-2u)1+b(1+c1(1-2u)+⋯)],(18)where b=β1/2β02 and c1=(β12-β0β2)/(4bβ04) are from higher orders in the QCD beta function, N(nf=3)=0.575 is the overall strength of the linear renormalon estimated from the quark pole mass correction [78,79]. Taking the moments we fitted from the RG-invariant ratios and tune the value of δm, we find the matrix elements eδm|z|HB(z,Pz,O) agree well with OPE reconstruction Eq. (10) when δm≈0.6  GeV for kaon and δm≈0.57  GeV for pion, as shown in Fig. 8. The consistency with OPE reconstruction suggests that the short-distance data can be well described by perturbation theory. This justifies our choice of using C0(z,μ) calculated from perturbation theory in Eq. (16) to renormalize the short distance matrix elements. Then we apply it to the hybrid scheme with zs=3a in Fig. 9. We find the largest three momenta converge to a universal shape, and the imaginary part for kaon is nonvanishing but close to zero, indicating the existence of small skewness.

The LaMET factorization is naturally defined in momentum space, which requires the x-dependence of quasi-DA to obtain the x-dependent light-cone DA. The extraction of x-dependence requires the Fourier transform of the above renormalized correlation functions to the momentum space. Thus in principle we need to know the distribution to arbitrarily large zPz. On the other hand, the noise increases exponentially at large z, limiting our calculations to a certain range of z values. An exact Fourier transformation is thus impossible. However, the contribution of the unknown long-tail distribution turns out to be bounded by physical constraints. For example, the Euclidean correlation functions must have a finite correlation length λ0∝Pz in the coordinate space, which results in an exponential decay exp{-λ/λ0} of the correlation functions at large λ=zPz. We are thus able to make an extrapolation of the long-tail distribution with these constraints to reduce uncertainty from it. One example of the long-tail modeling is [47], H(λ)⟶λ→∞(c1e-iλ/2(-iλ)d1+c2eiλ/2(iλ)d2)e-λ/λ0,(19)where the parametrization inside the round brackets is motivated from the Regge behavior [80] of the light-cone distribution near the endpoint regions. For symmetric pion DA, c1=c2 and d1=d2. For kaon DA, since the imaginary part is too small to be extrapolated, we model only the real part, and directly discrete Fourier transform the imaginary part truncated at the point when it starts to be consistent with zero. The model dependence of the extrapolation is estimated from three different fits of the long tail, by turning off the exponential decay e-λ/λ0 (labeled with “λ0→∞”) in Eq. (19), or by extrapolating from a different λcut=zcutPz, either from the trough of wave or from a larger λcut where the decaying mode starts to dominate.

The results are shown in Fig. 10. The long-tail extrapolation will eventually introduce a small systematic uncertainties to the light-cone DA near x=0.5 in our case. Compared to the pion quasi-DA, the more precise long-range correlation in kaon quasi-DA helps reduce such a systematic uncertainty.

We can then extract the light-cone DA from the quasi-DA through an inverse matching: ϕ(x,μ)=∫-∞∞dyC-1(x,y,μ,Pz)ϕ˜(y,Pz)+O(ΛQCD2x2Pz2,ΛQCD2(1-x)2Pz2),(20)where the power corrections is improved to quadratic order in ΛQCD2/Pz2 with LRR, the perturbative matching kernel C(x,y,μ,Pz) has been calculated at NLO [53]. Several improvements on the NLO matching kernel have been proposed to increase the accuracy of the perturbative matching in LaMET, including the LRR [35,50] that eliminates the linear power correction, and the RGR [55] and threshold resummation [56] that resums the small-momentum logarithms.

Our other recent work has derived the formalism for DA to resum all the small-momentum logarithms in the threshold limit |x-y|→0 [36], where the matching kernel can be factorized into the Sudakov factor H(xPz,x¯Pz,μ) and jet function J(|x-y|Pz,μ) [56], C(x,y,μ,Pz)→x→yH(xPz,x¯Pz,μ)⊗J(|x-y|Pz,μ)+O((y-x)0).(21)In the threshold limit, the Sudakov factor H(xPz,x¯Pz,μ)=F[C-sgn(z)(xPz,μ)Csgn(z)(x¯Pz,μ)](x-y)(22)incorporates the hard-collinear gluon modes connected to the external quark (antiquark) lines with momentum xPz (x¯Pz), and the jet function J(|x-y|Pz,μ)=F[J˜(z,μ)](x-y) absorbs the soft gluon modes with momentum |x-y|Pz [36,56]. They each follows individual RG equations ∂lnC±(pz,μ)∂lnμ=12Γcusp(αs)(ln4pz2μ2±iπ)+γc(αs),∂lnJ˜(z,μ)∂lnμ=Γcusp(αs)lnz2μ2e2γE4-γJ(αs),(23)where Γcusp,γc,γJ are the corresponding anomalous dimensions [36,55,81].

Solving these RG equations, we can obtain the resummed threshold components in the matching kernel. JHTR(μi,μh1,μh2,μ)=JTR(μi,μ)⊗HTR(μh1,μh2,μ),(24)where μi is the initial scale of jet function, and μh1,μh2 are the initial scale of the quark (antiquark) Sudakov factor, when solving the RG equations. Here the convolution order of H and J does not affect the threshold limit because they are multiplicative in coordinate space. So we average them and consider the difference between the two choices as a systematic error in our final results. The initial scales of the resummation are suggested to be [36], μi=2  min[x,x¯]Pz,μh1=2xPz,μh2=2x¯Pz.(25)

The full matching kernel is then resummed by first subtracting the threshold components JHNLO at fixed order, then add back the resummed threshold terms JHTR, CTR(μ)=JHTR(μi,μh1,μh2,μ)⊗JHNLO-1(μ)⊗CNLO(μ),(26)and the resummed inverse matching kernel can be obtained as CTR-1(μ)=CNLO-1(μ)⊗JHNLO(μ)⊗JHTR-1(μi,μh1,μh2,μ).(27)

Setting μi=μ or μh1=μh2=μ, we can examine the effects of resumming the Sudakov factors or the jet function separately, as shown in Fig. 11. We find that the resummation of Sudakov factor and jet function have opposite effects on the matching. To estimate the uncertainty from higher-order perturbation theory, we vary the initial scale of resummation by a factor of {2-1,2}. When the contribution from large logarithmic terms at higher orders of perturbation theory are significant, the resummed results will become sensitive to the choice of initial scales, indicating that the perturbation theory does not work or converge, so the matched DA is no longer reliable. We show the scale variation for the hard scale μh and semihard scale μi in Fig. 12. The scale dependence are smaller near x=0.5, but when approaching the endpoints, it becomes very large, indicating that the perturbation theory breaks down. We find that the scale variation for kaon DA from Pz=2.3  GeV data is still controllable at x≈0.2. For pion with lower momentum Pz=1.8  GeV, the uncertainty grows much faster, thus the reliable range shrinks. Calculating the same observable with larger hadron momentum helps extend the range of reliability for our calculation.

We consider the systematic error from the choice of zs∈[2a,3a] in the hybrid scheme, the model-dependence in the long-tail extrapolation, and the scale variation c∈{2-1,1,2} in the hard and semihard scales with the replacement μi,h→cμi,h. For each of these choices, we obtain ϕi(x) with a statistical error band. The systematic error band is obtained by requiring the total error band to cover all ϕi(x) results. Figure 13 shows the relative uncertainties as a function of x. Truncating at 10% overall uncertainty, we find that the reliable range of our calculation is roughly x∈[0.25,0.75] for pion, and x∈[0.2,0.8] for kaon.

Combining these results, we show the final results of pion DA and kaon DA and their comparison in Fig. 14. Note that we only work on one single lattice spacing with mixed-action fermions. Thus the error band is not including any estimation of lattice artifacts, such as the discretization effects, finite volume effects, and mixed-action effects. According to previous lattice calculations, the discretization effects are indeed important [32,34], and the finite volume effect could be small [82] since the boosted hadrons have relatively short wavelength. These effects also need to be carefully examined, in combination with our theoretical improvements, to present a reliable lattice benchmark for the meson DAs.

With the LaMET method, we have calculated the pion DA for a range of x∈[0.25,0.75] for pion and x∈[0.2,0.8] for kaon. Beyond this range, the perturbation theory no longer converges, and the power corrections in Eq. (20) are increasingly important. Both effects prevent us from extracting the distribution closer to the endpoint with current hadron momentum Pz∼2  GeV. Although the x-dependence is not directly calculable in this region, it has been proposed to constrain the endpoint regions with the global information of DA, such as the lower moments or short distance correlations, extracted from SDF [35,58]. In coordinate space, the perturbation theory still works at z≪ΛQCD-1, and the power corrections O(znΛQCDn) are also suppressed at short range. Thus the short distance correlations, along with the already determined mid-x distribution, can be utilized to better constrain the distribution in the endpoint regions, thus allowing us to roughly estimate the higher moments of DA.

Here we use a simple power-law model for the distribution near the endpoint, ϕ(x→0,1)∝Axm(1-x)n. Then we get a full ϕ(x,μ,m,n) in [0, 1] depending on the unknown parameters m and n. For the pion we set m=n since its DA is symmetric around x=0.5. By Fourier transforming ϕ(x,μ,m,n) to coordinate space, and implementing the RG-improved matching using SDF, we obtain the quasi-DA correlations H(z,m,n). H(z,m,n)=∫dνZ(ν,z2,μ,zPz)×∫Pzdzei(12-x)νzPzϕ(x,μ,m,n),(28)where Z(ν,z2,μ,zPz) is the matching kernel for coordinate space correlations [25,35,74]. Then the parameters m, n can be obtained by fitting the lattice matrix elements HR(z) to H(z,m,n). The parameters of pion DA are fitted using the real part of the lattice matrix elements because we have enforced the isospin symmetry, while the kaon DA model are constrained by both real and imaginary parts of the lattice matrix elements.

After determining the parameters m and n, we can estimate the higher moments of DA. The Mellin moments ⟨ξi⟩ and Gegenbauer moments ai are calculated by ⟨ξi⟩=∫01dxϕ(x)(2x-1)i,(29)ai=∫01dxϕ(x)Ci3/2[2x-1]4(i+3/2)3(i+1)(i+2),(30)where Ci3/2[2x-1] are the Gegenbauer polynomials. The results are summarized in Table I. The second moment of our kaon DA is consistent with the local twist-2 operators by RQCD collaboration [19] at the physical limit, but the second moment of our pion DA is much larger than their calculation. Our moment extraction highly depends on the short-distance correlations, which are more sensitive to discretization effects, thus the errors presented in the table are missing an important source of lattice artifacts. The first moment of out kaon DA is almost vanishing, very different from the RQCD calculation. It could come from the kinetic contamination of the imaginary part in Eq. (7) not being fully subtracted in our calculation due to the remaining Pz-dependence, making our estimation of the kaon skewness less reliable. If we first ignore the lattice artifacts and just compare different theoretical approaches, in a previous work on the moment analysis of the same pion DA data [25] with NLO matching, it is suggested that the power-law parameter m lies between 0.2 and 0.32 with different analysis choices, consistent with our estimate m=0.31(6) in this calculation. Our second moment and fourth moment are also in 2σ agreement with their analysis of ⟨ξ2⟩=0.287(6)(6) and ⟨ξ4⟩=0.14(3)(3). Note that the fitted second Mellin moments ⟨ξ2⟩ for the pion and kaon data in this approach agree with those fitted from RG-invariant ratios in Fig. 6. Such consistency in the lowest moments provide us more confidence about our estimate of higher moments on this ensemble. From both moments and the x-dependence, the pion DA is flatter than kaon, showing the non-negligible quark mass dependence of the meson DAs. The kaon is only slightly skewed as m and n are very close, and the first Mellin moment is consistent with zero.

Higher moments estimated from this approach can be used as inputs to the pion and kaon phenomenology. An example is to calculate the pion transition form factor Fπγγ*(Q2) at large Q2, which can be factorized as, F(Q2)=2fπ6Q2∫01ϕ(x,μ)T(x,μ,Q2),(31)where the hard kernel T(x,μ,Q2) has been calculated up to 2-loop order [83]. The convolution has also been formulated in terms of Gagenbauer coefficients with the hard coefficients cn provided up to 2-loop [83], Q2F(Q2)=2fπ∑n=0,2,…an(μ)cn(μ,Q).(32)We truncate the above equation at n=6 and show the result from our estimate of moments in Fig. 15. The scale variation is examined by choosing μ∈{Q/2,Q,2Q}, with the corresponding an(μ) evolved from 2 GeV to μ using their anomalous dimensions up to 3 loops [76]. Our band is in good agreement with data from Belle [10] from Q2>4  GeV2.

Note that our estimation of higher moments are model-dependent. For our pion momentum Pz≈1.8  GeV, LaMET only provides reliable prediction in a range of x∈[0.25,0.75]. In principle, this range is not reaching close enough to the endpoint for us to model the DA as a simple power-law function. Moreover, since the lattice spacing is not fine enough, we only have three data points of spatial correlation function at z≤3a that stays within the perturbative region, thus a complicated model could hardly be determined by this approach. To provide a more reliable estimate for the overall shape and higher moments of DA, we should include more calculations on finer lattice spacings and with higher hadron momenta.