Update: Precision $D_s$ decay constant from full lattice QCD using very fine lattices

We update our previous determination of both the decay constant and the mass of the $D_s$ meson using the highly improved staggered quark formalism. We include additional results at two finer values of the lattice spacing along with improved determinations of the lattice spacing and improved tuning of the charm and strange quark masses. We obtain $m_{D_s}=1.9691(32)\mathrm{GeV}$, in good agreement with experiment, and $f_{D_s}=0.2480(25)\mathrm{GeV}$. Our result for $f_{D_s}$ is $1.6\sigma$ lower than the most recent experimental average determined from the $D_s$ leptonic decay rate and using $V_{cs}$ from Cabibbo-Kobayashi-Maskawa unitarity. Combining our $f_{D_s}$ with the experimental rate we obtain a direct determination of $V_{cs}=1.010(22)$, or alternatively $0.990\genfrac{}{}{0ex}{}{+0.013}{-0.016}$ using a probability distribution for statistical errors for this quantity which vanishes above 1. We also include an accurate prediction of the decay constant of the ${\eta }_c$, $f_{{\eta }_c}=0.3947(24)\mathrm{GeV}$, as a calibration point for other lattice calculations.

Figure 1

Results for the mass of the pseudoscalar meson made of quarks with masses close to that of the charm quark mass for the full set of ensembles from Table 1. The $x$-axis is the lattice bare mass of the quark, which runs with lattice spacing from right to left. Very coarse ensembles are triangles; coarse, crosses; fine, squares; superfine, pluses; ultrafine, bursts. Results for heavier sea $u/d$ quark masses at each lattice spacing are in red, lighter ones are in blue. On the coarse lattices the very heavy sea masses of set 6 are in pink, the lighter strange sea mass of set 7 in grey and the large volume results on set 5 are in green, on top of the result from set 4. Statistical errors are too small to be visible on this plot. The results show that tuning the quark mass to that of charm depends very little on the sea-quark masses or on the volume. The dotted line gives the physical value, with its error, appropriate to lattice QCD, see text.

Figure 2

Results for the square of the mass of the pseudoscalar meson made of quarks with masses close to that of the strange quark mass for the full set of ensembles from Table 1. Errors are statistical errors from the fits to the meson correlators. The $x$-axis is the lattice bare mass of the quark, which runs with lattice spacing from right to left. Very coarse ensembles are triangles; coarse, crosses; fine, squares; superfine, pluses; ultrafine, bursts. Results for heavier sea $u/d$ quark masses at each lattice spacing are in red, lighter ones are in blue. On the coarse lattices the very heavy sea masses of set 6 are in pink, the lighter strange sea mass of set 7 in grey and the large volume results on set 5 are in green, on top of the result from set 4. The results show that tuning the quark mass to that of strange depends very little on the sea-quark masses or on the volume. The dotted line gives the physical value, with its error, appropriate to lattice QCD, see text.

Figure 3

Results for the effective mass of the $D_s$ correlator and the effective mass of the noise in the $D_s$ correlator plotted as a function of lattice time for one correlator on the fine lattices (set 8). The effective mass is obtained from the log of the ratio of the correlator (or its error) at successive times. At large times it becomes the mass of the lowest state in the correlator or its error. The lines compare the results to the expected mass i.e. the $D_s$ mass for the signal and $(m_{{\eta }_s}+m_{{\eta }_c})/2$ for the noise.

Figure 4

Results for the mass of the $D_s$ meson (specifically the difference between that mass and one half of the ${\eta }_c$ mass) as a function of the square of the ${\eta }_s$ meson mass, acting as a proxy for the strange quark mass. Results are for a range of different quark masses around the masses of the $c$ and $s$ quark masses on very coarse set 1 (triangles), coarse set 4 (crosses) and fine set 7 (squares). The lines are fits to the results for each ensemble allowing linear terms in $m_{{\eta }_s}^2$ and $m_{{\eta }_c}$. Here the lines join points for a fixed $c$ quark mass. See Fig. 5 for the equivalent as a function of $m_{{\eta }_c}$.

Figure 5

Results for the mass of the $D_s$ meson (specifically the difference between that mass and one half of the ${\eta }_c$ mass) as a function of the ${\eta }_c$ meson mass, acting as a proxy for the charm quark mass. Results are for a range of different quark masses around the masses of the $c$ and $s$ quark masses on very coarse set 1 (triangles), coarse set 4 (crosses) and fine set 7 (squares). The lines are fits to the results for each ensemble allowing linear terms in $m_{{\eta }_s}^2$ and $m_{{\eta }_c}$. Here the lines join points for a fixed $s$ quark mass. See Fig. 4 for the equivalent as a function of $m_{{\eta }_s}^2$.

Figure 6

Results for the mass of the $D_s$ meson tuned to the correct valence $c$ and $s$ mass on each ensemble from Table 4 as a function of the square of the lattice spacing. The line shows the result of the fit described by Eq. (15), taken at the physical values for the sea-quark masses. The shaded band gives our final result adjusted for electromagnetic effects and with the full error as described in the text. The black burst gives the experimental result.

Figure 7

Results for the mass of the $D_s$ meson tuned to the correct valence $c$ and $s$ mass on each ensemble from Table 4 as a function of the difference between the sea light quark mass and the physical value scaled by the physical strange quark mass (i.e. the parameter $\delta x_l$). The results are clearly separated by their lattice spacing value with very coarse at the top and ultrafine at the bottom. The lines show the result of the fit described by Eq. (15), taken at the value of the sea strange quark mass ($\delta x_s$) and using the lattice spacing value corresponding to the ensemble with smallest $\delta x_l$ in that group. The results on the coarse lattices at $\delta x_l=0.25$ include numbers at two different values of $\delta x_s$ as well as at two different volumes. This gives an idea of the spread in results from these effects. The lowest line is the fit curve in $\delta x_l$ at $a=0$ and $\delta x_s=0$. The shaded band gives our final result adjusted for electromagnetic effects and with the full error as described in the text. The black burst gives the experimental result.

Figure 8

Results for the decay constant of the $D_s$ meson as a function of the square of the ${\eta }_s$ meson mass, acting as a proxy for the strange quark mass. Results are for a range of different quark masses around the masses of the $c$ and $s$ quark masses on very coarse set 1 (triangles), coarse set 4 (crosses) and fine set 7 (squares). The lines are fits to the results for each ensemble allowing linear terms in $m_{{\eta }_s}^2$ and $m_{{\eta }_c}$. Here the lines join points for a fixed $c$ quark mass. See Fig. 9 for the equivalent as a function of $m_{{\eta }_c}$.

Figure 9

Results for the decay constant of the $D_s$ meson as a function of the ${\eta }_c$ meson mass, acting as a proxy for the charm quark mass. Results are for a range of different quark masses around the masses of the $c$ and $s$ quark masses on very coarse set 1 (triangles), coarse set 4 (crosses) and fine set 7 (squares). The lines are fits to the results for each ensemble allowing linear terms in $m_{{\eta }_s}^2$ and $m_{{\eta }_c}$. Here the lines join points for a fixed $s$ quark mass. See Fig. 8 for the equivalent as a function of $m_{{\eta }_s}^2$.

Figure 10

Results for the $D_s$ decay constant tuned to the correct $c$ and $s$ mass on each ensemble as a function of the square of the lattice spacing. The line shows the result of the fit at the physical value for the sea-quark masses, as described in the text. The shaded band gives our final result with the full error bar as described in the text.

Figure 11

Results for the decay constant of the $D_s$ meson tuned to the correct valence $c$ and $s$ mass on each ensemble from Table 4 as a function of the difference between the sea light quark mass and the physical value scaled by the physical strange quark mass (i.e. the parameter $\delta x_l$). The results are clearly separated by their lattice spacing value with very coarse at the top and ultrafine at the bottom. The lines show the result of the fit described by Eq. (19), taken at the value of the sea strange quark mass ($\delta x_s$) and using the lattice spacing value corresponding to the ensemble with smallest $\delta x_l$ in that group. The results on the coarse lattices at $\delta x_l=0.25$ include numbers at two different values of $\delta x_s$ as well as at two different volumes. This gives an idea of the spread in results from these effects. The lowest line is the fit curve in $\delta x_l$ at $a=0$ and $\delta x_s=0$. The shaded red band gives our final result with the full error as described in the text.

Figure 12

Results for the ${\eta }_c$ decay constant as a function of the ${\eta }_c$ mass for the different ensembles in Table 1. As in Fig. 1, very coarse ensembles are triangles; coarse, crosses; fine, squares; superfine, pluses. Errors shown are statistical only. Results for heavier sea $u/d$ quark masses at each lattice spacing are in red, lighter ones are in blue. On the coarse lattices the very heavy sea masses of set 6 are in pink, the lighter strange sea mass of set 7 in grey and the large volume results on set 5 are in green, on top of the result from set 4. The lines are fits to the results for one ensemble at each lattice spacing allowing linear and quadratic terms in $m_{{\eta }_c}$.

Figure 13

Results for the ${\eta }_c$ decay constant tuned to the correct $c$ mass on each ensemble as a function of the square of the lattice spacing. The line shows the result of the fit described in the text. The shaded band gives our final result with the full error bar as described in the text.

Figure 14

Results for the decay constant of the ${\eta }_c$ meson tuned to the correct valence $c$ mass on each ensemble from Table 4 as a function of the difference between the sea light quark mass and the physical value scaled by the physical strange quark mass (i.e. the parameter $\delta x_l$). The results are clearly separated by their lattice spacing value with very coarse at the top and ultrafine at the bottom. The lines show the result of the fit described in the text, taken at the value of the sea strange quark mass ($\delta x_s$) and using the lattice spacing value corresponding to the ensemble with smallest $\delta x_l$ in that group. The results on the coarse lattices at $\delta x_l=0.25$ include numbers at two different values of $\delta x_s$ as well as at two different volumes. This gives an idea of the spread in results from these effects. The lowest line is the fit curve in $\delta x_l$ at $a=0$ and $\delta x_s=0$. The shaded blue band gives our final result with the full error as described in the text.

Figure 15

Summary of full lattice QCD results for the difference of binding energy between charm-strange and charmonium states. $\Delta$ uses pseudoscalar mesons $D_s$ and ${\eta }_c$ and compares the result from this paper to experiment, given by the appropriate red point and shaded band). ${\Delta }^{\prime }$ uses a spin-average of the pseudoscalar and vector states and compares the result from the Fermilab Lattice/MILC Collaborations to experiment. Our result corresponds to the complete error budget given in Table 5 and is corrected for missing electromagnetic effects. The Fermilab Lattice/MILC result includes both errors given in [45] but has not been corrected for missing electromagnetic effects.

Figure 16

Comparison of the result from this paper for the $D_s$ decay constant with those from other lattice QCD calculations that include the effect of sea quarks. The Fermilab Lattice/MILC result is a preliminary one but also includes the effects of $u$, $d$ and $s$ quarks in the sea. The ETMC result includes only $u$ and $d$ quarks in the sea. We show also a recent average of experimental results from the Heavy Flavor Averaging Group [25] and two separate averages over the $\mu \nu$ and $\tau \nu$ channels. Experimental results for $f_{D_s}$ convert the leptonic decay rate to a decay constant using Eq. (21) and using an input value for $V_{cs}$ (see text).

Figure 17

Comparison of our new result for the $D_s$ decay constant with recent experimental results from CLEO [21, 22, 23] and BABAR [26]. These are derived from leptonic decay modes of the $D_s$ in various channels, and using Eq. (21) with an input value for $V_{cs}$ (see text). The CLEO numbers are taken from the compilation in [23], using consistent values for $V_{cs}$, $m_{D_s}$ and ${\tau }_{D_s}$ and so differ slightly from the historical numbers in Fig. 18. We also include the HFAG 2010 world average for experiment [25].

Figure 18

Values for $f_{D_s}$ from experiment and from lattice QCD since 2005, excluding results from conference proceedings. Later results from a given collaboration and process supersede the earlier ones. Experimental results are divided into those from the $\mu \nu$ channel [11, 12, 15, 21, 26] (in red) and those from the $\tau \nu$ channel [11, 14, 21, 22, 23, 24, 26] (in several $\tau$ decay modes, in pink). The HFAG October 2010 world average for experiment [25] is included as a light orange band. Note that the leftmost red point (from BABAR [12]) appears with dashed error bars—the lower value with solid error bars is the result adjusted by HFAG [65], although this number is not now included in the HFAG average. Lattice QCD points are in dark blue for full QCD [3, 7]—the rightmost point is from this paper. The light blue point is from ETMC [49] including only $u$ and $d$ in the sea.

Figure 19

Bounds in the $\tan \beta /\mathrm{\text{charged}}$ Higgs mass plane for a 2-Higgs doublet model of Type II given by our lattice QCD (i.e. standard model) result for $f_{D_s}$ and two different experimental scenarios. The green area is excluded at $3\sigma$ by existing experimental results [25] and the light blue area will be excluded by BESIII results [58] if the central experimental value does not change. The light grey band shows the direct limit from LEP searches [60].

Figure 20

Results for the ratio of the physical masses of the strange quark using the HISQ formalism [35] and the asqtad formalism [31]. The mass for the asqtad formalism in this ratio has the factor of $u_0$ removed. The points are lattice results combining our work and that of the MILC Collaboration. The shaded band represents a fit of the form given in Eq. (a2).