$D \rightarrow Kl{\nu}$ semileptonic decay using lattice QCD with HISQ at physical pion masses

The quark flavor sector of the Standard Model is a fertile ground to look for new physics effects through a unitarity test of the Cabbibo-Kobayashi-Maskawa (CKM) matrix. We present a lattice QCD calculation of the scalar and the vector form factors (over a large $q^2$ region including $q^2 = 0$) associated with the $D \rightarrow Kl{\nu}$ semi-leptonic decay. This calculation will then allow us to determine the central CKM matrix element, $V_{cs}$ in the Standard Model, by comparing the lattice QCD results for the form factors and the experimental decay rate. This form factor calculation has been performed on the $N_f =2+1+1$ MILC HISQ ensembles with the physical light quark masses.


Introduction
The flavour changing weak interactions between quarks via emission of W bosons can be parametrised in terms of the Cabbibo-Kobayashi-Maskawa (CKM) unitary matrix in the Standard Model given by [1,2] Precise and independent determination of each of the CKM matrix elements is crucial to test the Standard Model stringently and any deviation from unitarity would signal the existence of physics beyond the Standard Model. The uncertainties in the unitarity checks of the second row and second column of the CKM matrix are dominated by that of |V cs |, the central CKM matrix element. This element is calculated from the studies of the leptonic and semileptonic meson decays involving charged flavour changing current from c to s by combining the experimental decay rate with the vector form factor calculated from lattice QCD [1,2]. The best experimental result to date is achieved by combining the experimental data from BaBar [3], Belle [4], BES [5] and CLEO [6]. However, in the present scenario, the uncertainty in |V cs | is dominated by the lattice uncertainty in the form factors.
Here, we present a calculation of the semileptonic D → Klν decay on the N f = 2 + 1 + 1 lattices generated by MILC using highly improved staggered quark (HISQ) formalism [7], which is an improvement over HPQCD's previous work reported in [8,9]. In contrast to the work done by Fermilab lattice and MILC collaboration in the reference [10], we have calculated both the scalar and vector form factors of this decay over the whole range of kaon momentum instead of only at the maximum kaon momentum. A similar study [11] has been recently done using twisted mass fermions.

Formalism
The matrix element of the D → Klν semileptonic decay via the charged electroweak current gets contribution only from the vector current. The vector current matrix element can be parametrised in terms of the scalar and vector form factors f 0 (q 2 ) and f + (q 2 ), and can be written as - where, q µ = p µ D − p µ K is the exchanged 4-momentum from D with a 4-momentum p µ D to K with a 4-momentum p µ K and carried away by the W boson. M D amd M K are the masses of the D and K mesons respectively. In our setup, we give momentum to the strange quark inside the K meson.
The local vector current using the HISQ formalism is not conserved, and therefore requires a renormalisation factor Z V to obtain the continuum result. We determine this renormalisation by using the partially conserved vector current (PCVC) relation [8] and also the scalar current amplitude The scalar current amplitude is parametrised as Table 1. Sets of MILC configurations used with their β = 10/g 2 [13], w 0 /a for w 0 = 0.1715 (9) fm fixed from f π [14], L s /a, L t /a, number of configurations N con f , number of independent time sources for each configuration t 0 , multiple values of the source-sink separation T for each t 0 , (HISQ) sea quark massesm l (at physical pion mass), m s and m c in lattice units [13]. In this way we can cover the full physical range of q 2 starting from q 2 max where the momentum exchange is maximum i.e. K meson is rest to q 2 = 0 where K gets the maximum possible momentum in the opposite direction to leptons.
The differential decay rate is dominated by the vector channel in the vanishing lepton mass limit [12] and we get Staggered quarks have four tastes running in the correlator loop. To get a non-zero expectation value of the scalar and vector current operator matrix elements, we need to choose correct combinations of operators at the source, sink and current insertion point such that the correlator becomes taste-singlet. For generating the scalar current amplitude, the current carries spin-taste 1 ⊗ 1. We keep the spin-taste at the K-meson annihilation point the same for both scalar and vector currents as we want to use the same strange propagators in both cases. This end has the spin-taste content γ 5 ⊗ γ 5 . To nullify the tastes, the simplest choice of operators for the D meson end is also the Goldstone pseudoscalar operator γ 5 ⊗ γ 5 .
For the local temporal vector current, at the current insertion point we have used the γ t ⊗γ t operator. As mentioned before, the same K meson propagators are used in this case as well, therefore we have a spin-taste operator γ 5 ⊗ γ 5 at this end. Now, to cancel the overall taste the simplest operator choice at the D meson end would be the local non-Goldstone operator γ 5 γ t ⊗ γ 5 γ t which generates a D meson with slightly different mass. To use these three-point correlators we need to make Goldstone and local non-Goldstone two-point D correlators and Goldstone two-point K correlators as well.

Lattice setup
We have used publicly available MILC HISQ N f = 2 + 1 + 1 configurations with three different lattice spacing ∼ 0.09 f m (fine), ∼ 0.12 f m (coarse), ∼ 0.15 f m (very coarse) and the physical values of all of the sea quark masses. The details of these configurations are given in Table 1.
The values of the time sources t 0 have been chosen randomly to reduce autocorrelation and for each configuration multiple values of t 0 , uniformly placed on the lattice, have been used to get better statistics. To increase the statistics further, multiple values of the source-sink separation T have been used for each t 0 value. The valence light quark mass am val l is taken to be the same as the sea light quark mass am sea l whereas the valence strange quark mass am val,tuned s is tuned [15] to give the mass of the η s meson to be 0.6885(22) GeV [15]. We also tuned the valence charm quark mass am val,tuned c to get the mass of the η c meson m η c = 2.9863(27) GeV [15].
The three-point correlation functions on the lattice have been generated using the "sequential technique" shown in Figure 1. In this set up, zero momentum D meson is created at time t 0 + T on lattice, after it propagates to time t on lattice, the current (scalar or local temporal vector) is inserted at time t which changes the flavor c inside the D meson to flavor s to create a K meson and emits a W boson.

Fits and data analysis
The two point heavy-light D and K meson correlators have the following fit form - Here, both mesons in staggered quark formalism have oscillation in the correlators; E n represents the energy of the n−th excited state whereas E no represents the energy of the n-th oscillating state. Similarly, a n and a no respectively give the non-oscillating and oscillating pieces of the amplitude for the n-th state of the meson. We have taken for simplicity t 0 = 0 by always shifting the source time in the correlators to the origin of the lattice. In the two-point correlators apart from ground state, other excited states are also present, but we are only interested in the mass, energy and amplitude of the ground state for this calculation.
The ground state probability of the D/K meson is extracted as Here, χ D and χ K are the interpolating operators for the D and the K mesons respectively; a is the lattice spacing.
The three-point correlators (for both the scalar and vector currents) have oscillations at both ends and can be written as - Here, following a similar notation, "nn", "no", "on" and "oo" represent the non-oscillating/nonoscillating, non-oscillating/oscillating, oscillating/non-oscillating, and oscillating/oscillating states respectively. We use multi-exponential Bayesian fitting methods [16] to simultaneously fit the two-point and three-point correlators for multiple T s with all correlations among errors taken into account to extract the three-point amplitude V nn .
The ground state nonoscillating-nonoscillating amplitude of the three-point function for any current J is For the first sum in equation 5 (and for p K = 0) we have used the priors as (energy in the units of GeV) We have assigned analogous priors for the second sum as well, but with log(Eo D/K (0) ) = log(E D /K (0) + (0.23, 0.12)).
For the three point amplitudes we assign The energy priors for other kaon momenta p K are given following the dispersion relation E 2 =

Results
While extracting meson ground state properties, we have fitted starting from number of exponentials n exp = 2 up to n exp = 7 to get a stable fit with a χ 2 /dof < 1. For the D and K meson properties, we have achieved stable fit results from the 3rd exponential fits and hence, these results are taken as the final results. The behavior of these results with number of exponentials is shown in Figure 2.
We have also tested the taste-effects between the Goldstone and γ 5 γ 0 non-Goldstone D mesons arising from the staggered formalism and as expected their mass difference became zero in the continuum, as shown in Figure 3.
We have tested the relativistic dispersion relation as it is only approximate on the lattice. We check the deviation of the square of the velocity of light from 1 for different kinematics and on all ensembles. Generally, on the lattice we expect to get violations O(α S (pa) 2 ) for the HISQ formalism. Figure 4 shows that in our calculation we do not see   any significant deviation and the relativistic dispersion relation holds within 1 − 2% statistical deviation, which is within our expectation using the HISQ formalism. However the statistical uncertainties increase in the fitted results for the kaon energies with non-zero momenta. The scalar and vector form factors are extracted from the simultaneous fits of all data -including two-point and three-point correlators for all q 2 values on each ensemble. Generally the vector current is noisier and hence the vector form factor f + (q 2 ). The results for the form factors and their q 2 dependence is shown in Figure 5. These results come from an uncorrelated fit and so is only preliminary at this stage.
The results we show here include u/d quarks with physical masses. We plan to extend the study to heavier u/d masses, however, in order to map out the light quark mass dependence. This may also improve our uncertainties somewhat since heavier u/d masses typically give smaller statistical errors. We will then fit our results to a power series expansion in z-space (converting from q 2 to z) following [9]. Taking the coefficients in the z-expansion to depend on lattice spacing and light quark mass allows a smooth connection to the continuum physical point where we can compare to experiment, q 2 -bin by q 2 -bin, to optimise the final uncertainty on V cs .