Progress in the calculation of purely leptonic D_{(s)} -- decay constants from lattice QCD
PProgress in the calculation of purely leptonic D ( s ) –decay constants from lattice QCD Justus Tobias Tsang ∗ Higgs Centre for Theoretical PhysicsSchool of Physics & Astronomy, University of EdinburghEH9 3JZ, Edinburgh, United KingdomE-mail: [email protected]
We review recent progress in the calculation of the decay constants f D and f D s from lattice QCD.We focus particularly on simulations with N f = + N f = + + ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b ( s ) purely leptonic decays in lattice QCD Justus Tobias Tsang
1. Introduction
In the Standard Model (SM) the Cabibbo-Kobayashi-Maskawa (CKM) matrix is unitary. Totest the unitarity of the second row of the CKM matrix requires the determination of the matrixelements | V cd | and | V cs | . Neglecting electromagnetic corrections, the SM decay rates of D and D s mesons can be factorised into these CKM matrix elements, the decay constants f D and f D s and known kinematic factors. Experimental measurements of the decay rates combined with non-perturbative calculations of f D ( s ) from lattice QCD calculations hence allow for a determination of | V cd | and | V cs | . The current global averages of (cid:12)(cid:12) V cq (cid:12)(cid:12) f D +( s ) are [1] | V cd | f D + = . ± .
05 MeV and | V cs | f D + s = . ± . . (1.1)On the lattice, the decay constants f D ( s ) are extracted from the Euclidean matrix elements (cid:104) | c γ γ µ q | P ( p ) (cid:105) = i f P p µ P (1.2)for the pseudoscalar mesons P = D , D s and q = d , s , respectively.Lattice QCD calculations differ in many aspects: e.g. in the number of flavours N f in thesea, the lattice gauge and fermion actions, the used lattice spacings and how these are determined,and the simulated pion masses. In all cases an inter- or extrapolation to physical quark masses(chiral inter/extrapolation for the light quark mass) as well as an extrapolation to vanishing latticespacing (continuum limit) is required and only results which carried out these extrapolations willbe considered.In section 2 we will review the recent progress in the determination of f D ( s ) from lattice QCDcalculations. Finally, in section 3 we will conclude and give an outlook of current and ongoingcalculations.
2. Lattice results
Most modern simulations use two degenerate light quarks and a strange quark in the sea sector( N f = + N f = + + N f = + + N f = + + N f = + + ( s ) purely leptonic decays in lattice QCD Justus Tobias Tsang
100 150 200 250 300 350 400 450 500 m π [MeV] a [ f m ] FNAL/MILCRBC/UKQCDJLQCD
100 150 200 250 300 350 400 450 500 m π [MeV] a [ f m ] FNAL/MILCETM
Figure 1:
The N f = + N f = + + D ( s ) meson decay constants. The vertical black dotted line corresponds to physicalpion masses, the black star denotes the physical point, to which simulated data needs to be extrapolated. N f Collaboration f D [ MeV ] f D s [ MeV ] m min π [ MeV ] a [ fm ] a + . ( . ) (cid:0) + . − . (cid:1) . ( . ) (cid:0) + . − . (cid:1) χ QCD [4] 254(2)(4) 300 0.08-0.11 2HPQCD [5] 208.3(1.0)(3.3) 246.0(0.7)(3.5) 245 0.08-0.12 2HPQCD [6] 248.0(2.5) 260 0.045-0.15 5FNAL/MILC [7] 218.9(11.3) 260.1(10.8) 230 0.09-0.15 32 + + . ( . ) (cid:0) + . − . (cid:1) . ( . ) (cid:0) + . − . (cid:1) . ( . ) Table 1:
Summary of results for f D and f D s with N f = + N f = + + m min π , a gives the range of lattice spacings and a the number of distinct lattice spacings used in thecalculation. action setup. They first take the chiral continuum limit of f D s / m D s before extracting f D s in thecontinuum. Similarly, the double ratio ( f D s / f D ) / ( f K / f π ) is extrapolated to the physical pointbefore extracting f D s / f D and finally f D from this. The dominant quoted error is an accumulationof the scale setting, statistics and all errors of inputs needed to eventually extract f D .Even though table 1 lists 6 different results for simulations with N f = +
1, these are basedon only three distinct sets of gauge ensembles. In particular the results of refs [5–7] are based onoverlapping subsets of the FNAL/MILC gauge ensembles using asqtad rooted staggered quarks.In 2010 HPQCD [6] used HISQ valence quarks to obtain a prediction for f D s on these ensembles. No full systematic error budget has been published yet. The second quoted error arises from the scale setting only. ( s ) purely leptonic decays in lattice QCD Justus Tobias Tsang
Their dominant errors arise from the absolute scale setting (0.57%) which is of the same size as thecombined error due to statistics and valence tuning. In 2011, the FNAL/MILC collaborations [7]used the Fermilab action for the charm quark, leading to comparably large heavy quark discretisa-tion errors, which constitutes their dominant contribution to the final error and which is estimatedto be 8 . . f D and f D s , respectively. Finally, in 2012 HPQCD made a predic-tion for f D based on HISQ valence quarks on two lattice spacings [5]. The leading systematic errorhere arises from need of a chiral extrapolation to physical pion masses.Refs [2, 4] use RBC/UKQCD’s gauge ensembles with domain wall fermions. In 2015 the χ QCD collaboration [4] published a prediction for f D s by using Overlap valence quarks. Theyquote three leading contributions to the total error that are of nearly similar size (0.8-1.0%): Theseare the statistical error (which includes discretisation errors and the chiral extrapolation), the quarkmass dependence of the scale setting procedure and the limited reach in the heavy quark mass on thecoarser set of ensembles. A new result by the RBC/UKQCD collaborations appeared in 2017 [2].This result includes three lattice spacings as well as two ensembles with physical pion masses.Domain wall fermions are used for all quarks. Whilst the parameters of the domain wall action forvalence light and strange quarks are kept the same as in the sea, the charm quarks are simulatedwith a slight modification resulting in reduced lattice artefacts [11, 12]. The leading systematicerror is quantified using variations of the global fit, which simultaneously carries out the chiral andcontinuum limit extrapolation and the interpolation to the physical charm quark mass.The JLQCD collaboration [3] recently presented results using stout smeared domain wallfermions in the sea and the valence sector. However, to date, only the systematic error due tothe scale setting has been discussed.The RQCD and ALPHA collaborations have recently presented a status update of a new cal-culation [13] with N f = + a = . , .
085 fm.RBC/UKQCD also presented results with a choice of domain wall parameters that allows for fur-ther reach in the heavy quark mass [14].To significantly improve lattice QCD predictions for f D ( s ) beyond the percent level, electro-magnetic corrections and isospin breaking will have to be considered. FNAL/MILC [8] presenteda first estimate of the isospin breaking effect on the decay constant f D . By considering valence lightquark masses m l = ( m u + m d ) / m l = m d they found f D + − f D = . ( ) stat (cid:0) + − (cid:1) sys MeV.
3. Conclusions and outlook
A number of different lattice calculations with N f = + N f = + + f D and f D s which have started to reach percent precision. Used fermionactions include asqtad, HISQ, twisted mass and domain wall fermions in the sea. In the valencesector there is even more variation additionally including overlap, Fermilab and Osterwalder-Seilerfermions. However, the number of distinct underlying gauge configurations is still limited, as arecalculations with N f = + + ( s ) purely leptonic decays in lattice QCD Justus Tobias Tsang
Currently, the uncertainty on the extraction of the corresponding CKM matrix elements is lim-ited by the experimental precision of 2.3% and 1.6%, respectively (cf (1.1)), so more experimentaldata is needed to constrain | V cd | and | V cs | more tightly. Acknowledgments
JTT is supported by STFC, grant ST/L000458/1.
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