Kaon Distribution Amplitude from Lattice QCD and the Flavor SU(3) Symmetry

We 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. 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.


Introduction
Meson distribution amplitudes (DAs) φ M are important universal quantities appearing in many factorization theorems which allow for the description of exclusive processes at large momentum transfers Q 2 Λ 2 QCD . Some well-known examples of such processes, which are relevant to measuring fundamental parameters of the Standard Model, include B → πlν, ηlν giving the CKM matrix element |V ub |, B → Dπ used for tagging, and B → ππ, Kπ, KK, πη, . . . which are important channels for measuring CP violation (see e.g. [1]). Among those processes, the large difference between the strength of direct CP violation for B ± → π 0 K ± and B 0 → π ∓ K ± [2], and for D 0 → K + K − and D 0 → π + π − [3] 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.
In the chiral limit where m q → 0 with q = u, d, s, SU(3) symmetry predicts φ π = φ K = φ η = φ 0 . Away from the chiral limit, we work in the isospin limit (m u = m d =m) (for simplicity), use the MS scheme, and normalize the DAs such that dx φ M (x) = 1 with meson index M = π, K, η. The leading SU(3) breaking from chiral symmetry takes the form φ M (x, µ) = φ 0 (x, µ) + The functions φ 0 , E P M and F P M are independent of light-quark masses, f P 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 µ χ is the dimensional regularization parameter in chiral perturbation theory (ChPT). The µ χ dependence in F P M and ln(m 2 P /µ 2 χ ) cancel such that φ M is µ χ independent. In Ref. [4], it was proven using ChPT that E P π (x, µ) = E P K (x, µ) = E P η (x, µ) = 0 (1.2) for all P . Hence, at O(m q ), the DAs in Eq. 1.1 are analytic in m 2 P , where we have used m 2 P ∝ m q + O(m 2 q ). Ref. [4] has also shown that and hence, (1.5) 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) [5][6][7]. The LaMET method is based on the observation that, while in the rest frame of the hadron, parton physics corresponds to lightcone correlations, the same physics can be obtained through time-independent spatial correlations (now known as quasi-distributions) in the infinite-momentum frame. For finite but large momenta feasible in lattice simulations, LaMET can be used to relate Euclidean quasidistributions to physical ones through a factorization theorem which involves a matching and power corrections that are suppressed by the hadron momentum [6]. In the past few years, there have been many studies on the one-loop matching kernel for the leading-twist PDFs [8][9][10][11], generalized parton distributions (GPDs) [12,13] and meson DAs [12], as well as on the power corrections [14][15][16][17]. The renormalization property of quasi-distributions was also investigated [18][19][20][21][22][23][24][25][26] with the multiplicative renormalizability established to allloop orders. The LaMET approach has been applied to compute the nucleon unpolarized, helicity and transversity PDFs [14-16, 23, 27, 28], as well as the pion DA [29]. A first lattice PDF calculation at physical pion mass has recently become available [28]. The O(a)-improved operators associated with large hadron momentum were worked out in Ref. [30].
Motivated by LaMET, it was proposed that one can extract the PDFs from the "lattice cross sections" [31,32], and the quasi-PDF is one of them. More recently, it was suggested that one can study instead an Ioffe-time or pseudo distribution [33] which is related to the quasi-distribution through a simple Fourier transform. While this method shows some interesting renormalization feature [34], it is essentially equivalent to the LaMET approach [7,35,36] 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. [37][38][39][40][41]. 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. [29], where the results were improved by a Wilson line renormalization that removes power divergences. The plan of the present paper is to extend the study in Ref. [29] to the K meson and its SU(3) partners, the π and η mesons. A further improvement was made by implementing the momentum-smearing technique proposed recently [42] 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. 1.3 and 1.4.
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 φ K 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.

Meson DAs from LaMET
As was explained in Ref. [29], in the framework of LaMET, the meson DA can be extracted from the quasi-DÃ where µ is a renormalization scale of φ M in the MS scheme, n µ = (1, 0, 0, −1)/ √ 2 is a lightlike vector, Γ is a straight Wilson line that makes the quark bilinear operator gauge invariant, λ a = λ 3 , (λ 4 ± iλ 5 )/2, λ 8 for M = π, K ± , and η, respectively. In theφ M computation, both the quark bilinear and the meson momentum P z are along the z direction. µ R denotes the renormalization scale ofφ M in a given scheme. After removing the power corrections, φ M andφ M are the same in the infrared. Their difference in the ultraviolet can be compensated by the matching kernel Z φ , which can be computed perturbatively [12]: The matching kernel Z φ has the form φ (x , y). The expression for Z (1) φ (x, y) can be found in Ref. [29]. Eq. 2.4 tells us that φ M andφ M differ only at loop level, thus we can write (ignoring the power corrections for the moment) with an error of O α 2 s [31]. 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 mesonmass corrections have been computed to all orders in m 2 M /P 2 z [29], while the higher-twist corrections were removed by a simple fitting with a polynomial form in 1/P 2 z . In this work, we will follow the same procedure but leave out the higher-twist corrections, because we have observed non-monotonic behavior in P z in our lattice data. This implies that the polynomial fit might be too naive to account for the higher-twist effects.

Accessing the Meson DA Matrix Element on the Lattice
We start from the calculation of the correlator, where the sink operator at timeslice τ is the Fourier transform of the quasi-DA, and the quark fields ψ S in the source operator at timeslice 0 have been momentum smeared [42], where U (x, y) is the gauge link that makes ψ S (x) gauge covariant, σ is the smearing radius. Following Ref. [42], the momentum smearing parameter k is determined by optimizing the signal ofC(z, P z , τ ). We found that k = ±0.73 P z for the quark(antiquark) is suitable for our calculations with P z = (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 inC Eq. 2.6. Then, assuming the complete set of states is saturated by the ground state of energy E 0 and an effective excited state of energy E 1 at large t, we haveC What we need for the quasi-DA calculation is the normalizedh defined as which satisfies h M (0, P z ) = 1. Therefore, even if we do not separateh M from the zindependent source matrix element Z src , the determination of h M is not affected.

Accessing the η Distribution Amplitude
For π and K,C in Eq. 2.6 receives contributions from connected diagrams only. For η, in addition to connected diagram contributions,C also receives contributions from disconnected diagrams. However, the disconnected diagram is O((m s −m) 2 ) suppressed because there are two fermion loops; each of which is suppressed by one power of (m s −m) in the diagram. Therefore, it seems that if we just work at O(m q ), we can safely neglect the disconnected diagram of η. However, by dropping the disconnected diagrams, the u(d) and s quark contributions inC yield different values of ground-state energy E 0 ; that is, = 0. Then, when τ > 1/|∆E 0 |,C is dominated by the quark contribution of lower E 0 . However, when the hadron momentum P z is large, such that E 0 P z + m 2 qq /2P z , where mq q is the mass of theqq state without the disconnected diagram, as long as the plateaus for the mass determination appear within τ < 2P z /|m 2 ss − m 2 uu |,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 P z .
There is another complication for η. That is, the operator associated with λ 8 creates the η 8 meson. But the physical η is a linear combination of η 8 and η 0 , the SU(3) singlet. Fortunately, the mixing angle θ is small (θ ≈ −15 • ) [43]. Therefore, in Eq. 2.6, when we insert the physical η state between the two operators at time slices τ and 0, the η 0 contribution is suppressed by a mixing factor sin θ ≈ 0.08 times a factor of (m s −m) coming from the overlap of η 0 with the λ 8 type operator. Hence, numerically, the η 0 contribution can be counted as O((m s −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 < 2P z /|m 2 ss − m 2 uu |, then the connected diagram contribution ofC of Eq. 2.6 with λ a = λ 8 yields contributions from u, d, s quarks in the ratio 1 : 1 : 4. ThisC can be determined from Eq. 2.8 with the matrix element of Eq. 2.10 associated with the η 8 DA. This implies that for the largest P z we use (8π/L), the plateaus should be reached within τ < 1/|∆E| 20a, which is clearly satisfied.
In the following sections, we first present the unphysical η s results (with connecteddiagram contributions only) for different P z values. Then, based on the above discussion, we approximate the η 8 DA for P z = 8π/L with (φ π + 2φ ηs )/3 computed using connected diagrams only, and use the result to check the SU(3) relation Eq. 1.4.

Lattice results
In 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 24 3 × 64 lattice with 2+1+1 flavors (degenerate up and down, strange, and charm degrees of freedom) of highly improved staggered quarks (HISQ) [44] generated by the MILC Collaboration [45]. The pion mass on this ensemble is 310 MeV, and the lattice spacing a ≈ 0.12 fm. In this work, hypercubic (HYP) smearing [46] 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.
The bare matrix elements h M defined in Eq. 2.10 for kaon, pion and η s (with connected diagrams only) are shown in Fig. 1 for P z = 4π/L, 6π/L, and 8π/L. The dispersion relation, E 2 0 (P z ) = m 2 +P 2 z withP z = 2 sin(P z /2), is satisfied up to two standard deviations within statistical uncertainties for all the P z 's used in this work, ruling out sizable systematics due to discretization. The Fourier transform of the DA asymptotic form, φ(x) = 6x(1 − x), is also shown in Fig. 1. If the asymptotic form is correct, it suggests one needs to push for even larger zP z to catch all the short distance features; for the present calculation, this translates into a meson DA with less precision in the regions near x = 0 and x = 1. However, the asymptotic DA is defined in another renormalization scheme, so this is not a direct comparison with h M . We do observe similar oscillating behaviour in our data. When we increase P z , the secondary peaks become more pronounced. But the difference between P z = 6π/L and 8π/L is small already. Nevertheless, we plan to repeat this work with larger boost momentum (to extend the zP z reach) and reduce the lattice spacing by at least a factor of 2 in the future.

Improved Distribution Amplitude
With the DA matrix elements h M , we can then Fourier transform according to Eq. 2.2 to study the meson DAs. To cancel the power divergence in h M arising from the Wilsonline self-energy diagrams we introduce a counterterm δm, as suggested in Refs. [20,29], such that the matching kernel Z φ only has logarithmic divergence but no power-divergent contributions. Thus, the "improved" meson quasi-DA [29] is We then apply the matching kernel Z φ 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 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 where the c −1 term is the Coulomb potential that dominates at short distance and the c 1 term is the confinement linear potential. c 0 is of mass dimension one, so we can break it into a divergent piece and a finite one in the continuum limit: c 0 = c 0,1 /a + c 0,2 . Then where the 2 compensates for the potential using a quark-antiquark pair. Fig. 2 shows the effective potential V (r) at lattice spacings a = 0.06, 0.09, 0.12 fm for M π = 130 MeV and a = 0.12 fm for M π = 310 MeV. The M π dependence for a = 0.12-fm ensembles is almost undetectable. A fit of the potential with four parameters c −1 , c 0,1 , c 0,2 and c 1 , V (r ≥ 5a) has a very good χ 2 /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 obtainφ imp M (x, P z ) for each meson using Eq. 3.1. Next, we apply the one-loop matching kernel Z (1) φ (see Eq. A.6 of Ref. [29]), which is essential in LaMET to obtain lightcone quantities from the quasi-distribution with an error of O α 2 s [31] as discussed earlier in Sec. 2.1. We then apply the mass corrections [29] to φ imp,match M to get the final DAs where c = m 2 M /4P 2 z and f + = √ 1 + 4c + 1. The remaining higher-twist effect is of O(Λ 2 QCD /P 2 z ), which is small at our largest 2 momenta used in this work.

Kaon Distribution Amplitude
Let us now consider the first results for the kaon DA from lattice QCD. The left-hand side of Fig. 3 shows a comparison ofφ imp K (x) (improved quasi-DA shown in blue),φ imp,match (after matching quasi-DA to lightcone DA shown in green) and φ K (x) (DA with meson mass correction added, shown in red) from our largest meson momentum. The distribution after applying the one-loop matchingφ imp,match K (x) changes quite significantly from the quasidistributions. Further treatment with the meson mass correction yields φ K (x) which is very close toφ imp,match K (x). This is expected with the large momentum used here. The right-hand side of Fig. 3 shows the momentum dependence of φ K (x). Note that the higher-twist correction is not extrapolated away as in our previous work due to the non-monotonic behavior in P z . However, we expect its effect to be small at the largest two momenta used in this work, since their difference is small. φ K − is skewed towards large x since its valence s quark is heavier than its valenceū quark. However, the distribution outside the region x ∈ [0, 1] is still quite sizable (though shrinking when P z 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/P z that are not accounted for. Given the large one-loop matching correction seen in the left-hand side of Fig. 3, it will be important to investigate the twoloop matching contributions in the future to check their size. In addition, the truncation of zP z 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 φ K − result (labeled "Lat LaMET") with P z = 8π/L with a few selected results in the literature in Fig. 4: the result from fitting a parametrization to the lowest few moments calculated in lattice QCD [47,48] with pion mass ranging 330-670 MeV (labeled "Lat Mom"), Dyson-Schwinger equation calculations [49] ("DSE-1" and 2), and a calculation with a light-front constituent quark model [50] ("LFCQM"). We observe a broader distribution than the one from LFCQM, without making the assumption on the distribution form of x α (1 − x) β . Our φ K − 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 P z to reduce the Fourier-transformation truncation effects.

SU(3) Symmetry in Meson Distribution Amplitudes
In this work, we also update our previous study [29] of the pion DA and make the first study of the (connected diagrams only) η s case. Fig. 5 shows both DAs obtained after the oneloop matching and mass corrections (but not higher-twist corrections at O(Λ 2 QCD /P 2 z )). Larger boost momentum (with specifically tuned momentum-smearing parameters) and higher statistics are used in this work for φ π . 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 obverse that both φ π and φ ηs 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. Figure 6 shows a comparison of our current results for φ π with earlier results in literature. In the left panel, we show our φ π result along with a result using Dyson-Schwinger equation (DSE) [51], truncated Gegenbauer expansion fit to the Belle data for the γγ * → π 0 form factor (Belle) [52], and from parametrizations to lattice-QCD lowest-moment calculations [53] to extract the pion DA. For the fit to the Belle data, the Gegenbauer-polynomial expansion up to the eighth moment given in Ref. [52] was used at scale 2 GeV. For the with A and B determined from the normalization condition and the lattice calculations of the second moment ("Lat Mom 2") [53]. 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 P z = 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 zP z . This needs to be further investigated before we can draw a definite conclusion on the shape of φ π . In the right panel, we also compare our result on φ π with the calculation using Euclidean current correlators in Ref. [41], where the lattice data was presented for the scalar-pseudoscalar current correlator. In order to make a direct comparison, we have used our result to convolute with the coefficient function up to O(α s ) in Ref. [41], 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. [41] 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.
Comparison of φ π 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") [53], a calculation from the DSE analysis (DSE) [51], one from the LFCQM (LFCQM) [50], a fit to the Belle data (Belle) [52], and the asymptotic form 6x(1 − x) (Asymp). In the right panel, we have converted our result on φ π to the prediction for the scalarpseudoscalar current correlator (blue curve), and compared with the lattice data for the same correlator in Ref. [41] (dark, gray and white circles, which correspond to | P | = 1.08, 1.53, 1.88 GeV, respectively, µ is the renormalization scale).
Since φ π and φ ηs are quite close to each other, φ η is similar to the distribution shown in Fig. 5. We are interested in verifying the following SU(3) symmetry breaking relations:

Conclusion and Outlook
We 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 φ K − − φ K + ). We also present the first results on η s DA (and an indirect determination of η), as well as an improved determination of the pion DAs. Similar to the kaon case, there are non-vanishing 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 [4], and clearly observe δ SU(3),1 > δ SU(3),2 for x ∈ [0, 1] except when x is close to 1/2 where δ SU(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. 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 zP z 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 oneloop 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.  Figure 8. The improved matrix elements H M (z, P z ) defied in Eq. 4.1 shown at three different values of P z . Since the real (imaginary) part of H M (z, P z ) 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 η s , respectively. The imaginary part for kaon is not vanishing, which suggests skewness in the kaon DA.