Light pseudoscalar mesons play a fundamental role in quantum chromodynamics (QCD) as they are the (pseudo) Nambu-Goldstone bosons associated with dynamical chiral symmetry breaking [1,2], an important nonperturbative phenomena in the standard model. Their internal structure and its impact on experimental measurements have been actively investigated for many years.

The leading-twist pion and kaon light-cone distribution amplitudes (DAs) are among the simplest physical quantities characterizing such internal structure, and provide a probability amplitude interpretation on how the longitudinal momentum of the pion or kaon is distributed among quarks in its leading Fock state [3]. They are critical inputs for the description of hard exclusive reactions, such as the B meson weak decays [4,5] that provide useful information on CP violation and the Cabibbo-Kobayashi-Maskawa matrix, and play a crucial role for probes of new physics [6]; they are also important for the study of the pion elastic form factors [7], the pion-photon transition form factor [8–10], and of hard exclusive meson production that may give access to nucleon generalized parton distributions [11,12].

At the asymptotically large renormalization scale, it is well-known that these DAs follow a simple form, ϕ(x)=6x(1-x) [3], as other components are suppressed logarithmically through anomalous dimensions of higher-spin operators. However, their shapes at hadronic scales are a nonperturbative QCD problem. A QCD sum rule calculation for the pion DA has stimulated much theoretical debate and many experimental measurements [13–16]. In the past few decades, various nonperturbative models and phenomenological analyses have been proposed to understand this interesting physical quantity; see, for example, Refs. [17–19]. Clearly, a first-principle calculation from lattice QCD will shed more light on this issue.

There have been many lattice studies on the pion and kaon DAs using the traditional moments approach [20–26]. The proposal of large-momentum effective theory (LaMET) [27–29] allows one to access the entire x dependence of the DAs from first-principle lattice calculations, instead of only the first few moments (for other proposals with applications to the DAs, see Refs. [30–33]). Using LaMET, several calculations of the x dependence of meson DAs have been carried out [34–37]. However, a recent analysis [38] showed that the nonperturbative renormalization of the quasi-light-front (quasi-LF) correlation in LaMET could be highly nontrivial, especially when off-shell quark matrix elements are used. In such a case, even after renormalization there may still be residual linear divergences rendering the continuum extrapolation problematic. To resolve this issue, a self-renormalization strategy [39] has been proposed, where one fits the divergence structure to a quasi-LF correlation and uses it for renormalization. The present Letter provides the first full implementation of this strategy, and shows that it indeed gives promising results.

Let us begin with the following definition of the leading-twist light-cone DA of a pseudoscalar meson: ∫dξ-2πeixp+ξ-⟨0|ψ¯1(0)nγ5U(0,ξ-)ψ2(ξ-)|M(P)⟩=ifM(p·n)ϕM(x),(1)where U(0,ξ-)=Pexp[igs∫ξ-0ds n·A(sn)] is the path-ordered gauge link defined along the minus light-cone direction n (n2=0). To extract this quantity, we calculate the following quasi-LF correlation on the lattice with momentum P→ along the z direction [34]: C2m(z,P→,t)=∫d3ye-iP→·y→⟨0|OΓ1(z;y→,t)ψ¯2(0,0)Γ2ψ1(0,0)|0⟩,where OΓ1(z;y→,t)≡ψ¯1(y→,t)Γ1U(y→,y→−zz^)ψ2(y→−zz^,t) is the quasi-LF operator with z^ being the unit vector in the z direction, U(x→,x→−z→) is the spatial Wilson line connecting lattice sites x→ and x→-z→, ψ2Γ2ψ1 is the interpolating field of the meson m, and Γ1,2 are chosen as Γ1=γzγ5, Γ2=γ5 for the pseudoscalar meson. The ground-state matrix elements can be extracted from the following two-state fit formula: C2m(z,P→,t)C2m(z=0,P→,t)=HmB(z)(1+cm(z)e-ΔEt)(1+cm(0)e-ΔEt),(2)where HmB(z) is the normalized ground-state matrix element, cm and ΔE are free parameters accounting for (one or more) excited state contamination, which are exponentially suppressed in the large time limit. Based on the comparison of one- and two-state fits (see Supplemental Material [40]), we use the one-state fit results in the analysis below with tmin=0.72, 0.54, 0.42 fm (for Pz=1.29, 1.72, 2.15 GeV), which is large enough to eliminate the excited states contamination.

In this Letter, the simulation is done using the clover fermion action on three ensembles with 2+1+1 flavors of highly improved staggered quarks generated by the MILC Collaboration [41,42], at physical pion mass with three lattice spacings: 0.057, 0.088, and 0.121 fm. Hypercubic smeared fat links [43] are used in both the fermion action and the quasi-LF operators in C2m to improve the signal-to-noise ratio. The rest of the simulation setup is summarized in Table I. In addition, we use momentum smeared 2-2-2 grid sources, repeat the calculation at several time slices, and average the forward and backward correlation functions to improve statistics. In total, we have 570  (configurations)×8  (grid source)×8  (source time slices)×2  (forward or backward), 730×8×6×2, and 970×8×4×2 measurements at three ensembles with a=0.057, 0.088, and 0.121 fm, respectively.

The bare quasi-LF correlation calculated above contains both linear and logarithmic ultraviolet (UV) divergences that have to be removed by renormalization. On the lattice, the numerical subtraction of linear divergences is extremely delicate. In particular, such divergences may not be fully removed if the RI/IMOM renormalization scheme is used [38]. We also try the RI/IMOM scheme [44] with different momentum transfer Q2 at the current operator, and found that the linear divergence can be sensitive to Q2 in certain cases [40]. Thus, simple modifications on the momentum setup of the RI/IMOM scheme cannot solve the problem.

Here, we adopt the self-renormalization proposed in Ref. [39], which amounts to fitting the bare quasi-LF correlation and subtracting the relevant UV divergences. To be more precise, one fits the bare quasi-LF correlation at given hadron momentum and multiple lattice spacings with a perturbative-QCD-dictated parametrization that contains a linear divergence, a logarithmic divergence, and discretization effects. After removing all the UV divergences and discretization effects, one is left with the renormalized quasi-LF correlation encoding the intrinsic nonperturbative physics.

As suggested in Ref. [39], the UV divergences in the quasi-LF correlator can be determined by using, e.g., the pion parton distribution function (PDF) matrix elements M(z)≡⟨π|Oγt|π⟩ in the rest frame at multiple lattice spacings, and fitting the bare data MB to the following form [39]: MB(z,a)=Zself(z,a)MR(z),(3)with the renormalization factor parametrized as [39] Zself(z,a)≡exp{kzaln[aΛQCD]+m0z+f(z)a+3CFb0ln{ln[1/(aΛQCD)]ln[μ/ΛQCD]}+ln[1+dln(aΛQCD)]},(4)where the first term in the curly bracket is the linear divergence, m0 denotes a finite mass contribution arising from renormalon ambiguity, etc., and f(z)a accounts for the discretization effects. (The O(a) correction here arises from the mixed action effect in using clover valence fermions on highly improved staggered quark sea ones.) The last two terms come from the resummation of leading and subleading logarithmic divergences, which only affect the overall normalization at different lattice spacings. To partially account for higher-order perturbative effects as well as remaining lattice artifacts, we also treat d and ΛQCD as fitting parameters [39]. The renormalized matrix element is required to be equal to the continuum perturbative MS¯ result at short distances (chosen to be z∈zs=[0.06,0.18]  fm as defined in [45]), MR(z)|z∈zs=MMS¯,1-loop(z)≡1+αsMS¯CF4π[3lnz2μ24e-2γE+5]+O(αs2,MS¯),(5)which helps the determination of m0 and d. In the calculation we use the MS¯ renormalization scale μ=2  GeV and ΛMS¯=0.24  GeV.

In the present case, we follow the same strategy as above, except that the renormalized matrix element in the MS¯ scheme, HmMS¯(z)=HmB(z,a)/Z˜self(z,a),(6)is now required to be matched to the continuum perturbative MS¯ result of the normalized quasi-DA matrix element at short distances in the rest frame, which reads at one loop HmMS¯,1-loop(z)≡1+αsMS¯CF4π[3lnz2μ24e-2γE+7].(7)Z˜self turns out to be the same as Zself except for the value of d.

In Fig. 1, we show a comparison between the self-renormalized quasi-LF correlations Re[e(izPz/2)HmMS¯(z)] (after linear O(a) continuum extrapolation and phase rotation) with the perturbative one-loop result HmMS¯,1-loop(z). As can be seen from the figure, all quasi-LF correlations agree well with the perturbative result for small z, indicating a mild Pz dependence in that region.

It is worth pointing out that the self-renormalization strategy above does not apply at very small z due to finite lattice spacing artifacts in the data. In the ratio scheme [46], some degree of cancellation happens in the bare correlations between large momentum states and nonperturbative lattice renormalization factors. However, in the present case, the agreement of the self-renormalized LF correlation with the perturbative result extends down to z∼0.06  fm, which is our smallest lattice spacing. Thus, we only need to supplement it with the renormalized quasi-LF correlation at z=0, which is normalized to 1. In this way, we obtain the fully renormalized quasi-LF correlation. To facilitate the subsequent matching procedure, we define a modified renormalized correlation by further dividing out the perturbative factor HmMS¯,1-loop(z) so that the ratio scheme matching applies: HmR(z)=HmB(z,a)Z˜self(z,a)·HmMS¯,1-loop(z)=HmMS¯(z)HmMS¯,1-loop(z).(8)Note that this is equivalent to using the hybrid renormalized quasi-LF correlation and matching, as the perturbative difference in the quasi-LF correlation is exactly compensated by that in the matching.

From Fig. 1, we can see that the uncertainty of the renormalized quasi-LF correlation grows rapidly at large distance. A brute-force truncation of the correlation introduces unphysical oscillations [36] in momentum space after Fourier transformation. To resolve this issue, we adopt a physics-based extrapolation form [45] at large quasi-LF distance (λ=zPz): HmR(z,Pz)=[c1(iλ)a+e-iλc2(-iλ)b]e-λ/λ0,(9)where the algebraic terms in the square bracket account for a power law behavior of the DAs in the endpoint region and the exponential term comes from the expectation that at finite momentum (P→) the correlation function has a finite correlation length (denoted as λ0) that becomes infinite when the momentum goes to infinity. In this Letter, the Lorentz boost factor γ for the pion at the physical point is very large {9.21,12.29,15.36}, and thus the correlation length is very large. We therefore drop the e-λ/λ0 factor, and directly perform a polynomial extrapolation, as suggested in [45]. The details of this extrapolation can be found in the Supplemental Material [40].

We perform a phase rotation eizPz/2 to the renormalized quasi-LF correlation, so that the imaginary part directly reflects the flavor asymmetry between the strange and up or down quarks. As an example, in Fig. 2, the imaginary part of eizPz/2HmR(z) for the pion (upper panel) and kaon (lower panel) at different lattice spacings with Pz=2.15  GeV. It reflects the SU(3) flavor breaking effects between the valence quarks in the light meson. For the pion it is consistent with zero within errors as expected, since we have used degenerate valence u/d quark masses in the ensembles. While in the case of kaon there is a nonvanishing imaginary part, such an imaginary part increases slightly with Pz, as observed in previous DA studies using LaMET [36,45], and a comparison of the results at different momenta can be found in the Supplemental Material [40].

The factorization can be done either in momentum space [47,48] or in coordinate space. Here, we choose the latter, which results in HmR(z,λ,μR)=∫01dxdy θ(1-x-y)×C(x,y,z2,μR,μ)hmR(x,y,λ,μ)+O(ΛQCD2z2,M2z2),(10)where we take renormalization scale and factorization scale to be the same and set μ=μR=2  GeV in this Letter. hmR is the LF correlation related to the light-cone DA through the following Fourier transformation: hmR(x,y,λ,μ)=∫01du eiu(x-1)λ-i(1-u)yλϕ(u,μ).(11)The perturbative matching kernel C up to the next-to-leading order is given in the Supplemental Material [40].

The impact of the perturbative matching is illustrated in Fig. 3, where a Fourier transformation to momentum space has been performed. As can be seen from the figure, the matching broadens the quasi-DA in the physical region. Outside the physical region (x<0 or x>1), there still exists a nonvanishing tail, indicating potential effects of higher-order matching and higher-twist contributions. Nevertheless, in the unphysical region, the results are consistent with zero within ∼2 standard deviations.

With the results for Pz=1.29, 1.72, 2.15 GeV above, we can perform an extrapolation to Pz→∞ using the following functional form: ϕ(x,Pz)=ϕ(x,Pz→∞)+c2(x)Pz2+O(1Pz4).(12)The final results of the π, K DAs are given in Fig. 4, where systematic uncertainties from renormalization scale μ dependence, large λ extrapolation, and continuum and infinite momentum extrapolation have been taken into account. While the systematic uncertainty from continuum extrapolation dominates for the kaon, the statistical (larger than for the kaon due to the lighter quark mass) and continuum extrapolation uncertainties are comparable and dominate for the pion. Therefore, we expect that the kaon DA uncertainty can get significantly suppressed when data at smaller lattice spacings are available that enable us to do a more reliable continuum extrapolation; whereas for the pion, besides data at smaller lattice spacings, we also need increasing statistics to reduce the statistical uncertainty [40]. In the endpoint region, which cannot be reliably predicted by LaMET, we adopt a phenomenological xa(1-x)b extrapolation (taken as 0<x<0.1 and 0.9<x<1). The unreliable region is expected to shrink if a larger Pz can be reached in future calculations. For comparison, we also plot the asymptotic form 6x(1-x) and results from QCD sum rules [49], Dyson-Schwinger equations (DSE) [50], and reconstructed from moments calculations (OPE) [26]. As can be seen from the figure, both π and K DAs deviate significantly from the asymptotic form, but are close to the results from DSE and OPE calculations. The shape of π DA is much broader than the asymptotic form, manifesting the impact of dynamical chiral symmetry breaking at such a low scale. A similar behavior has also been observed in a recent analysis using QCD sum rules with nonlocal condensates [51].

We present a state-of-the-art lattice calculation of π and K DAs using LaMET. The renormalization is done in the hybrid scheme with self-renormalization proposed recently. Based on the results at physical light and strange quark masses with three lattice spacings and momenta, we perform an extrapolation to the continuum and infinite momentum limit. The final results exhibit a significant deviation from the asymptotic form, while they are close to the DSE and OPE results, especially in the middle x region where our method is reliable. However, there are still some significant differences in the endpoint regions. This could be due to missing higher-power or high-order corrections in LaMET that can be improved in future calculations, or due to effects of higher moments ignored in the OPE and DSE calculations. A more accurate determination of the endpoint behavior of the DAs would be an important step toward a better understanding of quantities like the pion-photon transition form factor.