# Precision Lattice Calculation of D and Ds decay constants

###### Abstract

We present a determination of the decay constants of the and mesons from lattice QCD, each with a total error of about , approximately a factor of three better than previous calculations. We have been able to achieve this through the use of a highly improved discretization of QCD for charm quarks, coupled to gauge configurations generated by the MILC collaboration that include the full effect of sea u, d, and s quarks. We have results for a range of u/d masses down to and three values of the lattice spacing, which allow us to perform accurate continuum and chiral extrapolations. We fix the charm quark mass to give the experimental value of the mass, and then a stringent test of our approach is the fact that we obtain correct (and accurate) values for the mass of the and mesons. We compare and with and , and using experiment determine corresponding CKM elements with good precision.

## I Introduction

Precision calculations in lattice QCD play a crucial role in testing our non-perturbative theoretical tools, by comparing the results of the calculation with precisely measured quantities. In addition accurate calculations of non-perturbative QCD quantities are very important in the extraction of information from analysis of experimental data, for example in the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.

This is most clearly seen in the case of “gold-plated” processes, for example the leptonic decay of , , and mesons. In this process the corresponding meson, with quark content (or ) annihilates weakly into a W boson, with a width given, up to calculated electromagnetic corrections marciano (); pdg06 (), by:

(1) |

is the corresponding element of the CKM matrix, and the decay constant parametrizes the amplitude for W annihilation. By combining a measurement of with an accurate calculation of (1) can be used to determine . If is known from elsewhere we can use (1) to get a value for .

The decay constant is conventionally defined to be a property of the pseudoscalar meson, calculable in QCD without QED effects, and is given by:

(2) |

The calculation of is a hard non-perturbative problem, which at present can only be done fully with lattice QCD. There are very precise experimental measurements for the leptonic decay rates in the case of the and , and new results are appearing for and , which make the calculations a highly non-trivial test of lattice QCD, and ultimately of QCD itself. This tests are important to give us confidence in similar lattice QCD predictions of matrix elements in B systems, for which experimental results are much harder to obtain.

## Ii Improved Staggered Quarks

We use HISQ staggered quarks in the valence sector, whereas the sea quarks are ASQTAD staggered quarks with the fourth root trick Sharpe (); Creutz (); Kronfeld ().

Staggered quark actions suffer the doubling problem: there are four “tastes” (non-physical flavours) of fermions in the spectrum, which couple through taste-changing interactions. These are lattice artifacts of order , involving at leading order the exchange of a gluon of momentum . Although quite large in the original one-link (Kogut-Susskind) staggered action, such interactions are perturbative for typical values of the lattice spacing, and can be corrected systematically a la Symanzik. By judiciously smearing the gauge field we can remove the coupling between quarks and high momentum gluons.

The most widely used improved staggered action is called ASQTAD, and removes all tree-level discretization errors in the action Naik (); Lepage1 (); Orginos ().

The HISQ (highly improved staggered quarks) staggered Dirac operator involves two levels of smearing with an intermediate projection onto . It is designed so that, as well as eliminating all tree-level discretization errors, it further reduces the one-loop taste-changing errors (see HISQ () for a more detailed discussion.) This action has been shown to substantially reduce the errors associated with the taste-changing interactions HISQ (); spectrum1 (); spectrum2 ().

When we put massive quarks on the lattice, the discretization errors grow with the quark mass as powers of . Therefore to obtain small errors we would need . For heavy quarks this would require very small lattice spacings. On the other hand, to keep our lattice big enough to accommodate the light degrees of freedom, we need . The fact that we have two very different scales in the problem makes difficult a direct solution. What we can do instead is to take advantage of the fact that is large, by using an effective field theory (NRQCD, HQET). This program has been very successful for b quarks NRQCD1 (); NRQCD2 (); FNAL ().

The charm quark is in between the light and heavy mass regime. It is quite light for an easy application of NRQCD, but quite large for the usual relativistic quark actions, . However, if we use a very accurate action (HISQ) and fine enough lattices (fine MILC ensembles), it is possible to get results accurate at the few percent level. A non-relativistic analysis HISQ () shows that for HISQ charm quarks the largest remaining source of error is due to the quark’s energy, and can be further suppressed by powers of , where is the typical velocity of the quark in the system of interest, simply by retuning the overall coefficient of a term called Naik term to impose the correct relativistic dispersion relation for low lattice momemtum .

One advantage of the use of a relativistic action is the existence of a partially conserved current, which implies the non-renormalization of the lattice result for . We can extract from the PCAC relation for zero momentum meson P:

(3) |

## Iii Results

Lattice/sea | valence | |

, | , , , | |

0.0194, 0.0484 | 0.0264, 0.066, 0.85, 0.66 | 2.129(11) |

0.0097, 0.0484 | 0.0132, 0.066, 0.85, 0.66 | 2.133(11) |

0.02, 0.05 | 0.0278, 0.0525, 0.648, 0.79 | 2.650(8) |

0.01, 0.05 | 0.01365, 0.0546, 0.66, 0.79 | 2.610(12) |

0.005, 0.05 | 0.0067, 0.0537, 0.65, 0.79 | 2.632(13) |

0.0124, 0.031 | 0.01635, 0.03635, 0.427, 0.885 | 3.711(13) |

0.0062, 0.031 | 0.00705, 0.0366, 0.43, 0.885 | 3.684(12) |

We use gluon field configurations including flavours of sea quarks generated by the MILC collaboration MILC1 (); MILC2 (); MILC3 (). The parameters of the ensembles we have used for both the sea and the valence sectors are in table 1. The lattice results are converted to physical units through the heavy quark potential parameter , as determined by the MILC collaboration (table 1, MILC2 ()). The physical value of is determined from the spectrum calculated in NRQCD with b quarks on the same MILC ensembles NRQCD2 (), with the result fm, GeV.

We use multiple precessing random wall sources, which gives a 3-4-fold reduction in statistical errors with respect to conventional local sources.

The mass of the charm quark is fixed by adjusting the mass of the “goldstone “ to its experimental value. The light (u/d) and strange quark masses are fixed using the experimental values for the masses of and . Our results use masses for the u and d quarks that are substantially larger (by a factor of around three) than the real ones. In order to get physical answers we extrapolate to the correct u/d mass using chiral perturbation theory. Once the masses have been thus fixed, there is no remaining freedom to change any parameters, and in particular the results we obtain for the masses of heavy-light mesons are a stringent test of our method. In figure 1 we show the spectrum of charmonium. We obtain an hyperfine splitting of MeV (experiment 117(1) MeV) We have made no attemp as yet to optimize the calculation of the excited states.

In addition to the chiral extrapolation, we have systematic errors coming from a variety of sources fds (), among them from the finite lattice spacing. Because we have three different lattice spacings and very precise data, we can extrapolate to the continuum limit. This extrapolation is linked to the chiral extrapolation through discretization errors in the light quark action. We therefore perform a simultaneous bayesian fit for both chiral and continuum extrapolations, allowing for expected functional forms in both. We tested the validity of the method by fitting hundreds of fake datasets generated using staggered chiral perturbation theory with random couplings. We fit simultaneously to the masses and the decay constants, that is, we fit , , and simultaneously, and similarly for , , and . We present some of the results in figures 2 and 3.

We get an excellent agreement with experiment for the masses: GeV (experiment GeV), and GeV (experiment GeV). Our calculation also reproduces correctly the difference in binding energies between a heavy-heavy () and a heavy-light ( and ) state: (experiment ). Our charm quark action is the first one to be accurate enough to do this calculation (which also cannot be done, for example, in potential models.)

We also have agreement with experiment for the light-light decay constants fds (). The result for the ratio is very accurate, , and shows tiny discretization effects (figure 4). Combining this ratio with experimental leptonic branching fractions MILC3 (); marciano2 () we get , where the first error is theoretical and the second experimental. This gives the unitarity relation .

Our results for the heavy-light decay constants are 4-5 times more accurate than previous lattice QCD results and existing experimental measurements: MeV, MeV, and a ratio of (see figure 5). For the double ratio , which is estimated to be close to 1 from low order chiral perturbation theory becirevic (), we get a value of .

The experimental leptonic branching rates, together with CKM matrix elements determined from other processes (assuming ) give MeV for decays and MeV for decay from CLEO-c cleocfds () and 283(23) MeV from BaBar babar (), and for 223(17) MeV from CLEO-c for decay cleocfd (). Using our results for and and the experimental values from CLEO-c cleocfds () for decay (since the electromagnetic corrections are well-known in that case) we can directly determine the corresponding CKM elements: and = 4.42(4)(41). The first error is theoretical and the second experimental. The result for improves on the direct determination of 0.96(9) given in the Particle Data Tables pdg06 ().

Our calculation is precise enough that we can see the difference between in the bottom sector and the similar quantity in the charm sector (figure 6). These mass differences are small compared to the absolute masses of the states, and should be the same in the infinitely heavy quark limit. We can see that our calculation correctly reproduces the small difference due to the finite value of the mass of the charm and bottom.

## Iv Conclusions and outlook

We have shown that the use of a highly improved relativistic action on fine enough lattices is capable of delivering very precise results on systems with a charm quark. The high statistical accuracy of our data combined with calculations at several values of the lattice spacing and light quark masses allows us to make a controlled joint chiral and continuum extrapolation.

We can calculate accurately the mass of heavy-light systems, which provide a stringent test of the calculation. We can calculate precise values for the decay constants of pseudoscalar heavy-light mesons (as well as light-light mesons), and especially for the ratio of such decay constants.

The very precise calculation of the masses of heavy-heavy pseudoscalar mesons should make possible a direct lattice determination of the mass of the charm quark. Because we use the same relativistic action through the calculation for both the charm and the light quarks, we can also obtain a very precise value for the ratio , and therefore if is determined through another method use the ratio to get . We are also working (in collaboration with the Karlsruhe group) on a new method for the determination of by combining continuum perturbation results with lattice data.

Another quantity which we plan to calculate in the near future is the leptonic decay width , as well as the semileptonic form factors for , .

###### Acknowledgements.

We are grateful to the MILC collaboration for the use of their configurations and to Quentin Mason and Doug Toussaint for useful discussions. The computing was done on Scotgrid and the QCDOCX cluster. This work was supported by PPARC, the Royal Society, NSF and DoE.## References

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