$D_s\to \eta ,\eta \prime$ semileptonic decay form factors with disconnected quark loop contributions

We calculate for the first time the form factors of the semileptonic decays of the $D_s$ meson to $\eta$ and ${\eta }^{\prime }$ using lattice techniques. As a by-product of the calculation we obtain the masses and leading distribution amplitudes of the $\eta$ and ${\eta }^{\prime }$ mesons. We use $N_f=2+1$ nonperturbatively improved clover fermions on configurations with a lattice spacing $a\sim 0.075\mathrm{fm}$. We are able to obtain clear signals for relevant matrix elements, using several noise reduction techniques, for both the connected and the disconnected contributions. This includes a new method for reducing the variance of pseudoscalar disconnected two-point functions. At zero momentum transfer, we obtain for the scalar form factors, $|f_0^{D_s\to \eta }|=0.564(11)$ and $|f_0^{D_s\to {\eta }^{\prime }}|=0.437(18)$ at $M_{\pi }\approx 470\mathrm{MeV}$, as well as $|f_0^{D_s\to \eta }|=0.542(13)$ and $|f_0^{D_s\to {\eta }^{\prime }}|=0.404(25)$ at $M_{\pi }\approx 370\mathrm{MeV}$, where the errors are statistical only.

Figure 1

Connected (first term) and disconnected (second term) fermion loop diagrams. We use stochastic methods to calculate the blue dashed fermion lines. The labeling of the four-momenta $p,q,k$ reflects the conventions adopted in Eq. (1).

Figure 2

Energies $E(\mathit{p})$ along with fitted lattice and continuum dispersion relations of the ${\eta }^{\prime }$ (top), $\eta$ (middle) and $D_s$ (bottom) mesons at $M_{\pi }\approx 370\mathrm{MeV}$ (Set A, left) and at the SU(3) flavor symmetric point $M_{\pi }\approx 470\mathrm{MeV}$ (Set S, right). The momentum $\mathit{P}=\mathit{p}\times L/(2\pi )$ is in lattice units. Open circles were excluded from the fits. $E(\mathit{p}=\mathbf{0})$ for $\eta$ (Set A) and ${\eta }^{\prime }$ (Sets A, S) were obtained by using the improved method explained in Sec. 3b. For clarity, the different mass determinations are replotted at ${\mathit{P}}^2=0$ with a slight horizontal shift.

Figure 3

Effective masses of the $\pi$, $\eta$ and ${\eta }^{\prime }$ for (left) $M_{\pi }\approx 370\mathrm{MeV}$ (Set A) and (right) $M_{\pi }\approx 470\mathrm{MeV}$ (Set S). Note that for Set S $\pi =\eta ={\eta }_8$ and ${\eta }^{\prime }={\eta }_1$.

Figure 4

Summary plot of recent lattice determinations of the $\eta$ (open symbols) and ${\eta }^{\prime }$ (solid symbols) masses. Results are shown from the RBC/UKQCD (2010, [25]), UKQCD (2011, [26]), HSC (2011, [27]), and ETMC (2013, [28]) Collaborations. The experimental values are taken from the Particle Data Group [29].

Figure 5

The naive zero-momentum ${\eta }^{\prime }$ two-point function for each topological sector, for Set A (left panel) and Set S (right panel).

Figure 6

The constant part of the naive zero-momentum ${\eta }^{\prime }$ two-point function for each topological sector, for Set A (left panel) and Set S (right panel). The green solid circles were obtained using all configurations. We found $⟨Q^2⟩\approx 7.7$ for Set A and $⟨Q^2⟩\approx 9.1$ for Set S. The $Q=0$ finite volume predictions [blue crosses, Eqs. (17) and (15)] were calculated using ${\chi }_t=⟨Q^2⟩/V_4$ and slopes $\alpha$, determined via Eq. (18). The dashed pink lines are linear fits to the fixed $|Q|$ data.

Figure 7

Effective mass of the ${\eta }^{\prime }$, before and after the improvement, for Set A (left panel) and Set S (right panel). Results are shown for three cases (a) using the naive ${\eta }^{\prime }$ two-point correlators, $C_{2\mathrm{pt}}^{{\eta }^{(\prime )}}$, of Eq. (8) (blue squares), (b) using the naive correlators after removing the constant term (green diamonds) and (c) using the improved two-point functions, removing the constant part Eq. (26) for Set S, and solving the generalized eigenvalue problem for Eq. (27) for Set A (red triangles).

Figure 8

Connected (green squares) and disconnected (blue triangles) contributions to the total three-point functions (red circles) for $D_s\to \eta$ (left panel) and $D_s\to {\eta }^{\prime }$ (right panel) matrix elements for Set A with $t_{\text{sep}}=8a$. The $D_s$ is located at $t=0$ and the ${\eta }^{(\prime )}$ is at $t=8a$. The momenta are in the lattice units: $(\mathit{P},\mathit{Q},\mathit{K})=(\mathit{p},\mathit{q},\mathit{k})\times L/(2\pi )$.

Figure 9

Typical examples of fitting $R(t)$ to Eq. (40), to extract $⟨{\eta }^{(\prime )}(\mathit{k})|S(\mathbf{0})|D_s(\mathit{p})⟩$, for Set A. The lower right plot depicts a fit to the first two terms of Eq. (40), while the others use all three terms. The $D_s$ meson is always located at $t/a=16$, while the $\eta$ or ${\eta }^{\prime }$ is located at $t/a=8$ ($t_{\text{sep}}/a=8$), $t/a=6$ ($t_{\text{sep}}/a=10$) and $t=0$ ($t_{\text{sep}}/a=16$). Data points with open symbols were omitted from the fits. The red bands indicate the values of the matrix elements obtained from the fit. The momenta are in the lattice units: $(\mathit{P},\mathit{Q},\mathit{K})=(\mathit{p},\mathit{q},\mathit{k})\times L/(2\pi )$.

Figure 10

The scalar form factor $f_0(q^2)$ for $D_s\to {\eta }^{(\prime )}\ell {\overline{\nu }}_{\ell }$. The errors are statistical only and the dashed lines indicate the fits to the form factors using the parametrization $f_0(q^2)=f_0(0)/(1-b{q}^2)$. On the left are the results for $M_{\pi }\approx 370\mathrm{MeV}$ (Set A) and on the right for $M_{\pi }\approx 470\mathrm{MeV}$ (Set S).

Figure 11

Left: Extrapolation of the ratio $f_0^{D_s\to {\eta }^{\prime }}(0)/f_0^{D_s\to \eta }(0)$ to the physical pion mass. The inner error bar of the extrapolated ratio is statistical, and the outer one includes systematics (see text for details). Right: Histogram of the ratio Eq. (45), varying the ratio $f_0^{D_s\to {\eta }^{\prime }}(0)/f_0^{D_s\to \eta }(0)$ within its errors as well as the parameters $r$ and $\alpha$ within Eq. (47) for the decays into $\eta$ and ${\eta }^{\prime }$.

Figure 12

The relative errors of the disconnected three-point function $⟨C_{\text{disc}}^{3\mathrm{pt},l}(t,t_{\text{sep}},\mathit{p},\mathit{k})⟩$ with and without low mode averaging for Set S. The sink-source separation is $t_{\text{sep}}=10a$, the $D_s$ meson is located at $t=0$ with lattice momentum $\mathit{P}=(1,0,0)$ and the disconnected light-quark loop is located at $t/a=10$ with momentum $\mathit{K}=(1,0,0)$.