Distribution Amplitudes of $K^{*}$ and $\varphi$ at the 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 $\varphi$ using large momentum effective theory. We use the clover fermion action on three ensembles with $2+1+1$ flavors of highly improved staggered quarks action, generated by the MIMD Lattice Computation Collaboration, at physical pion mass and $\{0.06,0.09,0.12\}\mathrm{fm}$ lattice spacings and choose three different hadron momenta $P_z=\{1.29,1.72,2.15\}\mathrm{GeV}$. The resulting lattice matrix elements are nonperturbatively renormalized in a recently proposed hybrid scheme. 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 ab initio theory inputs for analyzing pertinent exclusive processes.

Figure 1

The two-point correlation function for the transversely polarized $K^{*}$ in coordinate space. We make a phase rotation by multiplying a factor $e^{iz{P}_z/2}$ with $P_z=2.15\mathrm{GeV}$.

Figure 2

Quasi-DA and LCDA extracted from it for the transversely polarized $K^{*}$ using data at $a=0.09\mathrm{fm}$, $P_z=2.15\mathrm{GeV}$.

Figure 3

The continuum limit of the LCDA for the transversely polarized $K^{*}$, extrapolated from three different lattice spacings.

Figure 4

LCDAs for the longitudinally polarized $K^{*}$ (upper panel) and transversely polarized $K^{*}$ (lower panel). The results are extrapolated to the continuous limit ($a\to 0$) and the infinite momentum limit ($P_z\to \infty$). Regions with $x<0.1$, $x>0.9$ are shaded, as systematic errors in these regions are difficult to estimate.

Figure 5

Similar to Fig. 4 but for the $\varphi$ vector meson.