Distribution Amplitudes of $K^*$ and $\phi$ at Physical Pion Mass from Lattice QCD

We present the first lattice QCD calculation of the distribution amplitudes of longitudinally and transversely polarized vector mesons $K^*$ and $\phi$ using large momentum effective theory. We use the clover fermion action on three ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ) action, generated by MILC collaboration, at physical pion mass and \{0.06, 0.09, 0.12\} fm lattice spacings, and choose three different hadron momenta $P_z=\{1.29, 1.72, 2.15\}$ GeV. The resulting lattice matrix elements are nonperturbatively renormalized in a hybrid scheme proposed recently. An extrapolation to the continuum and infinite momentum limit is carried out. We find that while the longitudinal distribution amplitudes tend to be close to the asymptotic form, the transverse ones deviate rather significantly from the asymptotic form. Our final results provide crucial {\it ab initio} theory inputs for analyzing pertinent exclusive processes.

Introduction.-Searching for new physics beyond the standard model (SM) is a primary goal of particle physics nowadays. A unique possibility of doing so is to investigate flavor-changing neutral current processes which are highly suppressed in the SM. Some prominent examples of such processes include B → K * + − and B s → φ + − decays. Recent experimental analyses by Belle and LHCb collaborations [1][2][3][4][5] have revealed notable tensions between the SM predictions of such processes and data, and attracted quite considerable theoretical interests (see Refs. [6][7][8] and many references therein). Various new physics interpretations have been proposed to resolve such tensions, but to firmly establish their existence requires an accurate and reliable theoretical understanding of the dynamics of weak decays.
In the low recoil region (high q 2 ), the B → K * and B s → φ form factors can be directly calculated on the lattice (see for instance Refs. [9,10]), however these decays at large recoil are also of experimental interests, and for instance the P 5 anomaly has attracted many theoretical and experimental attentions [11,12]. In the latter kinematics region, decay amplitudes are split into shortdistance hard kernels and long-distance universal inputs. The universal inputs that enter include the light-cone distribution amplitudes (LCDAs) of the vector mesons K * , φ which, to the leading-twist accuracy, specify the longitudinal momentum distribution amongst the valence quark and antiquark in the meson. While the hard scattering kernel is perturbatively calculable, the LCDAs can only be extracted from nonperturbative methods or from fits to relevant data. A reliable knowledge of LCDAs is essential in making predictions on physical observables, and in particular the transition form factors at large recoil can be typically affected by O(10%) by the non-asymptotic terms of LCDAs in light-cone sum rules approach [13,14]. To date most of the available analyses have made use of estimates based on QCD sum rules [15] or Dyson-Schwinger equation [16], but a first-principle description of LCDAs for the vector (K * , φ) meson is still missing.
Lattice QCD provides an ideal ab initio tool to access nonperturbative quantities in strong interaction. Though some lowest moments of the ρ LCDA have been studied in Ref. [17], a direct calculation of the entire distribution has not been feasible until the proposal of large momentum effective theory (LaMET) [18,19] recently. This is realized by simulating on the lattice appropriately chosen equal-time correlations, and then convert them to the lat- ter through a perturbative matching. Since LaMET was proposed, a lot of progress has been achieved in calculating various parton distribution functions [20,21] (and many references therein) as well as distribution amplitudes for light pseudoscalar mesons [22][23][24]. Other variants have also been explored in Refs. [25][26][27].
In this Letter, we present the first lattice calculation of LCDAs for vector mesons K * , φ in LaMET with the clover fermion action, on three ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ) action [28], generated by MILC collaboration [29], at physical pion mass and 0.06, 0.09 and 0.12 fm lattice spacings. To improve the signal-to-noise ratio of simulation, we take the smearing transformation of hyperubi(HYP) fat link [30], the other simulation setup is given in Table  I. A hybrid renormalization scheme [31] is used to renormalize bare quantities, after which an extrapolation is taken to the continuum limit as well as to the infinite momentum limit based on data at three hadron momenta, P z = {1.29, 1.72, 2.15} GeV. It should be noticed that, a momentum boost close to or larger than the inverse lattice spacing may introduce uncontrolled discretization effects, while the dispersion relation is satisfied with the O(a 2 ) corrections as shown in the supplemental material [32]. In the calculation we neglect the strong decays of K * , φ due to their narrow decay widths. The finite width corrections should be solved with a proper finitevolume analysis, which is beyond the scope of this work. Our final results indicate that, while the longitudinal LCDAs are close to the asymptotic form, the transverse ones deviate considerably from the asymptotic form.
LCDAs from LaMET.-The leading-twist LCDAs for longitudinally and transversely polarized vector mesons, Φ V,L and Φ V,T , are defined as follows [34]: where U (0, ξ − ) = P exp ig s 0 ξ− dsn + · A(sn + ) is the gauge-link defined along the minus lightcone direction, is the polarization vector of the vector meson, and n + is the unit vector along the plus lightcone direction. f V and f T V are the decay constants defined by the local vec-tor and tensor current, respectively. Here for K * , ψ 1 denotes the strange quark field and ψ 2 is the light u/d quark. For the φ meson, both ψ 1,2 are strange quark fields. According to LaMET, the above LCDAs can be obtained by first calculating the following bare equal-time correlations on the lattice where the Lorentz indices in the second line are chosen as {ν, ρ} = z, y, and the gauge-link U (0, zẑ) is along the z direction. The quantities H V,{L,T } (z) can be renormalized nonperturbatively in an appropriate scheme [31,[35][36][37][38][39][40]. Here we choose the hybrid scheme [31] proposed recently which has the advantage that the renormalization factor does not introduce extra nonperturbative effects at large z which distort the IR property of the bare correlations. This scheme works as follows: At |z| ≤ z S where z S is within the region where the leading-twist approximation is valid, we can choose the RI/MOM scheme [38] to avoid certain discretization effects (alternative choices include, e.g., the ratio [27] scheme), while for |z| > z S one applies the gauge-link mass subtraction scheme where the superscript R denotes the renormalized quantity,μ denotes the intrinsic scale dependence of the gauge-link including both UV and IR. We have chosen Z(z, a) as the RI/MOM renormalization factor computed from . (5) The Z hybrid denotes the endpoint renormalization constant which can be determined by imposing a continuity condition at z = z S , The mass counterterm δm(μ) can be extracted from the RI/MOM renormalization factor [31]. The z S are chosen as 0.24f m and 0.36f m within perturbative region, and their difference is treated as a systematic error. By Fourier transforming H R V,{L,T } to momentum space, we then obtain the quasi-DAs where the continuum limit has been taken. It can be factorized into the LCDAs through the factorization theorem [41]: where the matching kernel C V,{L,T } was derived first in the transverse momentum cutoff scheme in Ref. [42] and then in the RI/MOM scheme in Ref. [43]. The µ and µ R reflect the generic renormalization scale dependence of LCDAs and quasi-DAs. The matching formula and more details of the hybrid scheme can be found in the supplemental material [32]. Numerical setup.-On the lattice, one directly calculates the two-point correlation function defined as: where the longitudinal polarization case (m = L) has Γ 1 = γ t , and Γ 2 = γ z , and the transverse polarization case (m = T ) has Γ 1 = σ zy , and Γ 2 = γ x /γ y . Then the quasi-DAs can be extracted from the following parameterization: where c m (z) and ∆E are free parameters accounting for the excited state contaminations, and H b V,m (z) is the bare matrix elements for quasi-DA. When t is large enough, the excited state contaminations parameterized by c m (z) and ∆E are suppressed exponentially, and the ratio defined in Eq. (10) approaches the ground state matrix element H b V,m (z). Based on the comparison between the joint two-state fit and constant fit shown in the supplemental material [32], we choose to use the constant fit in the range of t ≥ 0.54 fm to provide a conservative error estimate in the following calculation.
The numerical simulation is based on three ensembles with 2+1+1 flavors of HISQ [28] at physical pion mass with 0.06, 0.09 and 0.12 fm lattice spacings. The momentum smeared grid source [44] with the source positions (x 0 + j x L/2, y 0 + j y L/2, z 0 + j z L/2) are used in the calculation, where (x 0 , y 0 , z 0 ) is a random position and j x,y,z = 0/1. It allows us to obtain the even momenta in unit of 2π/L with ∼ 8 times of the statistics. We also repeat the calculation at 8, 6, 4 time slices and fold the data in the normal and reversed time directions, which is equivalent to having 570 × 8 × 8 × 2, 730 × 8 × 6 × 2 and 970 × 8 × 4 × 2 measurements at three ensembles at a =0.06, 0.09 and 0.12 fm, respectively. We have further reversed theẑ direction in Eq. (9) to double the statistics. Based on the numerical results, we confirmed that the dispersion relation can be satisfied for all the cases up to the O(a 2 p 4 ) correction, and the continuum extrapolation in the coordinate space or momentum space provides consistent results [32].
Results.-After renormalization in the hybrid scheme, we perform a phase rotation e izPz/2 to the renormalized correlation, so that the imaginary part directly reflects the flavor asymmetry between the strange and up/down quarks. Taking the transversely-polarized K * as an example, we show in Fig. 1 the real (upper panel) and imaginary part (lower panel) of the renormalized quasi-DA matrix elements e izPz/2 H K * ,T (z) with the momentum P z = 2π/L × 10 = 2.15 GeV. As shown in the upper panel, the matrix elements at different lattice spacings are consistent with each other, indicating that linear divergences arising from the gauge-link have been canceled up to the current numerical uncertainty. In the lower panel, we find a positive imaginary part at all the lattice spacings, which corresponds to a non-zero asymmetry with the peak at x < 1/2. This is consistent with expectations that lighter quarks carry less momentum of the parent meson.
As one can see from Fig. 1, the uncertainty of lattice data grows rapidly with the spatial separation of the nonlocal operator. Thus, to have a reasonable control of uncertainties in the final result we need to truncate the correlation at certain point. The missing long-range information can be supplemented by a physics-based extrapolation proposed in Ref. [31], which removes unphysical oscillations in a naive truncated Fourier transform with the price of altering the endpoint distribution (at x ∼ 0 or 1), which cannot be reliably predicted by LaMET anyway due to increasingly important higher-twist contributions. Following Ref. [31], we adopt the following extrapolation form: where the exponential term accounts for the finite correlation length for a hadron at finite momentum, and the two algebraic terms account for a power law behavior of the momentum distribution at x close to 0 and 1, respectively. λ = zP z , and the parameters c 1,2 , a, b, λ 0 are determined by a fitting to the lattice data in the region where it exhibits an exponential decay behavior.
To account for systematics from such an extrapolation, we have done two different extrapolations, one including the exponential term and the other not, and taken their difference as an estimate of systematics. This can be attributed as a source of the uncertainty from the inverse problem, and a more systematic strategy to handle the inverse problem of the Fourier transform is available in Ref. [45]. The detailed comparison of two extrapolations can be found in the supplemental material [32]. After renormalization and extrapolation, we can Fourier transform to momentum space and apply the corresponding matching. In Fig. 2, we show as an example the comparison of the quasi-DA and extracted LCDA for the transversely polarized K * . The results correspond to the case with P z = 2.15 GeV, and a = 0.09 fm. One notices that there is a non-vanishing tail for quasi-DA (yellow curve) in the unphysical region (x > 1 or x < 0), but it becomes much better for the LCDA (blue curve) after the perturbative matching is applied. We have performed a simple extrapolation to the continuum limit using the results at three different lattice spacings and the following formula with O(a) correction being due to the mixed action effect from the clover valence fermion on HISQ sea. As an example, we show the extrapolated results for the transversely polarized K * in Fig. 3 for three different momenta, P z = {1.29, 1.72, 2.15} GeV. From this figure, one can see that the asymmetry slightly increases with P z . Defining the asymmetry as c asy = 1/2 0 dxφ(x)/ 1 1/2 dxφ(x), we find c asy is 1.090(15), 1.176(07), 1.227(08) for the three momenta. Since the strange quark is heavier than the up/down quark, a slight preference of x < 1/2 to x > 1/2 is expectable. It suggests that a large P z extrapolation is essential to suppress the power corrections and reproduce this correct preference behavior. Such a behavior is also observed in the study of Kaon LCDAs in Ref. [24].
After matching from Quasi-DA to LCDAs with µ = 2 GeV, our final results for LCDAs of the K * and φ are given in Fig. 4 and 5, respectively, where the upper and lower panels correspond to the longitudinal and transverse polarization cases. In these figures, we have made a P z → ∞ extrapolation using the following simple form: tions [15] and the Dyson-Schwinger equations(DSE) results [16] in Fig. 5. Our results indicate that while the longitudinal LCDAs tend to be close to the asymptotic form, the transverse LCDAs have relatively large deviations from the asymptotic form. These behaviors might have important implications on the study of semileptonic B → K * + − decay towards the search for new physics, and can be explored in the future.
Summary.-We have presented the first lattice QCD calculation of LCDAs of longitudinally and transversely polarized vector mesons K * , φ using LaMET. We did not consider the ρ meson due to its large width which will introduce sizable systematic errors. The continuum and infinite momentum limits are taken based on calculations at physical light and strange quark mass with three lattice spacings and momenta. Our final results are then compared to the asymptotic form and QCD sum rule results. While the longitudinal LCDAs tend to be close to the asymptotic form, the transverse ones have relatively large deviations from the asymptotic form. Our final results provide crucial ab initio theory inputs for analyzing pertinent exclusive processes.