Lattice inputs for the determination of | V cd | and | V cs | from (semi-)leptonic decays
OOctober 17, 2018
Lattice inputs for the determination of | V cd | and | V cs | from(semi-)leptonic decays Jonna Koponen
SUPA, Department of Physics and AstronomyUniversity of Glasgow, Glasgow G12 8QQ, UK
This paper is a review of recent lattice QCD results for D and D s meson leptonic and semileptonic decays. The theory inputs needed forthe determination of V cd and V cs from experimental results are the mesondecay constants (leptonic decays) and the form factors (semileptonic de-cays). In addition one can compare the shape of the form factors fromlattice QCD and experiment, and use the full experimental q range (par-tial decay rates in q bins) to determine the CKM matrix elements.PRESENTED AT The 8th International Workshop on the CKM UnitarityTriangle (CKM 2014), Vienna, Austria, September 8-12,2014 a r X i v : . [ h e p - l a t ] N ov Leptonic and semileptonic decays
In a leptonic decay a meson (here D or D s ) decays to a lepton and its neutrino via avirtual W boson. The decay rate is given byΓ D s → (cid:96)ν = G F π m (cid:96) M D s (cid:32) − m (cid:96) M D s (cid:33) f D s | V cs | , (1)and ( f D s | V cs | ) can thus be cleanly extracted from experiment.On the other hand, consider a semileptonic decay where a D meson decays to akaon (or a pion), a lepton and its neutrino. The partial decay rate for a decay whereboth the initial and final state mesons are pseudoscalars can be written asdΓ D → K d q = G F p π | V cs | | f D → K + ( q ) | . (2)Here p = | (cid:126)p | is the momentum of the K meson in the rest frame of the D , and q isthe four-momentum transfer between the two mesons, q = ( M D − E K ) − p . In thiscase experiment can tell us ( | V cs || f D → K + ( q ) | ) in a given q bin.The meson decay constants f D and f D s and the form factors f + ( q ) for the semilep-tonic decays can be calculated non-perturbatively in lattice QCD from first principles.Combining these experimental and theoretical results allows us to determine the cor-responding elements of the CKM matrix, providing a test of the Standard Model andconstraints for new physics. The aim of this review is to summarize recent latticeQCD results and extract | V cs | and | V cd | . The current status of calculations of the D and D s meson decay constants is shownin Fig. 1, tagged by the names of the lattice groups. The results are from [1, 2, 3, 4,5, 6, 7, 8]. Note that some of the results are still preliminary. Here n f denotes thenumber of flavors used in the calculation: n f = 2 is two light quarks in the sea ( u and d quarks that both have the same mass), n f = 2 + 1 means light and strange quarksand n f = 2 + 1 + 1 has in addition charm quarks in the sea. Different discretizationsof the Dirac equation for quarks are denoted by the tags “HISQ”, “twisted mass”,“Fermilab” and “clover” — these should all agree in the continuum limit.Let us now turn to D meson semileptonic decays: There are two form factorsassociated with a pseudoscalar to pseudoscalar semileptonic decay, a scalar form factor f ( q ) and a vector form factor f + ( q ). The scalar form factor is not accessible inexperiment as it is suppressed in the decay rate by the lepton mass. However, itis quite straightforward to consider a scalar and a vector current on the lattice andcalculate both form factors. 1
80 190 200 210 220 230 240FLAG averagesETMC ’14, twisted massAlpha ’13, cloverHPQCD/UKQCD ’07, HISQFNAL/MILC ’11, FermilabHPQCD ’12, HISQETMC ’13, twisted massFNAL/MILC ’14, HISQMeVn f =2+1n f =2+1+1n f =2n f =2n f =2+1
220 230 240 250 260 270 280FLAG averagesETMC ’14, twisted massAlpha ’13, cloverHPQCD/UKQCD ’07, HISQFNAL/MILC ’11, FermilabHPQCD ’12 & ’10, HISQETMC ’13, twisted massFNAL/MILC ’14, HISQMeVn f =2+1n f =2+1+1n f =2+1n f =2n f =2 prelim. prelim.prelim.prelim. f D f D s Figure 1: Decay constants: f D on the left, f D s on the right. Different discretizations“HISQ”, “twisted mass”, “Fermilab” and “clover” should all agree in the continuumlimit, and the agreement is seen to be good. The best results at the moment, i.e. re-sults with smallest errors and most modern lattice configurations ( n f = 2+1+1, phys-ical pion mass), are from Fermilab Lattice and MILC Collaborations (FNAL/MILC’14): f D = 212 . ± . +1 . − . | syst MeV and f D s = 249 . ± . +1 . − . | syst MeV.For completeness, averages from Flavor Lattice Averaging Group (FLAG) [9] are alsoshown for 2 and 2 + 1 flavors. (GeV )0.60.91.21.5 f ( q ) o r f + ( q ) c D to Kc005 D to Kf D to K f f + c=coarse, f=fine (GeV )0.60.91.21.5 f ( q ) o r f + ( q ) c D to ! c005 D to ! f D to ! c D s to Kc005 D s to Kf D s to K f f + Figure 2: On the left: Scalar and vector form factors of D → K(cid:96)ν semileptonicdecay [10]. Note the kinematic constraint f + (0) = f (0). On the right: Form factorsof D → π(cid:96)ν and D s → K(cid:96)ν semileptonic decays. Note that the shape of the formfactor is insensitive to the mass of the spectator quark: the form factors for the two c → l decays are the same within ∼ B → D(cid:96)ν and B s → D s (cid:96)ν as well [11]. 2 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 q Total 0.850.90.9511.051.11.151.2 P a r ti a l r a t e r a ti o E xp / L a t , V c s CLEOBaBarBelleBaBar+lattice, fit ACLEO+lattice, fit ACLEO+BaBar+Belle total rates +lattice, fit BUnitarity
Bin1 Bin2 Bin3 Bin4 Bin5 Bin6 Bin7 Bin8 Bin9 q [GeV /c ] D ! K ! ! Figure 3: | V cs | extracted from D → K(cid:96)ν decay using all experimental q bins.Here we will only consider lattice results for decays D → K(cid:96)ν and D → π(cid:96)ν .Several lattice groups have calculated the form factors — see Refs. [10, 12, 13, 14, 15].Fig. 2 shows results by HPQCD from different lattice spacings [coarse ( a = 0 .
12 fm)and fine ( a = 0 .
09 fm) lattice] and extrapolation to continuum and physical lightquark mass (more details can be found in [10]). The shapes of the vector form factorsagree well with experiment — see for example Fig. 5 in [10] and Fig. 5 in [14]. | V cs | and | V cd | Now we have the needed input from lattice QCD to determine | V cs | and | V cd | fromleptonic and semileptonic decays. In the case of a semileptonic decay, we can inte-grate the form factor calculated in lattice QCD over the experimental q bins anddetermine the CKM element from each bin: the experimental result divided by thelattice result for a given bin is | V cs | (or | V cd | ) as shown in Fig. 3. This also showsthat the shape of the form factor agrees very well between lattice QCD and experi-ment. One can then do a weighted average fit to these values, including bin to bincorrelations. This is more accurate compared to earlier calculations that extractedCKM elements from experimental knowledge of ( | f + (0) || V cs | ) (or ( | f + (0) || V cd | ) ) anda lattice determination of the form factor at q = 0, since this uses more information.The current status of V cd and V cs from leptonic and semileptonic decays is shownin Fig. 4. The tags are the same as for the decay constants. Leptonic decays tend togive a higher value for V cs than the unitarity value, but note that all data points in3he plot would shift to left or right, if the experimental average changed. All latticeresults agree with each other very well, and the semileptonic determination of V cs andboth leptonic and semileptonic determinations of V cd agree with the assumption ofCKM matrix unitarity. References [1] A. Bazavov et al. (Fermilab Lattice and MILC Collaborations), Phys. Rev. D (2014) 074509, arXiv:1407.3772.[2] P. Dimopoulos et al. , PoS LATTICE (2013) 314, arXiv:1311.3080.[3] H. Na et al. (HPQCD), Phys. Rev. D (2012) 054510, arXiv:1206.4936.[4] C.T.H. Davies et al. (HPQCD), Phys. Rev. D (2010) 114504, arXiv:1008.4018.[5] A. Bazavov et al. (Fermilab Lattice and MILC Collaborations), Phys. Rev. D (2012) 114506, arXiv:1112.3051.[6] E. Follana et al. (HPQCD and UKQCD Collaborations), Phys. Rev. Lett. (2008) 062002, arXiv:0706.1726.[7] J. Heitger, G. M. von Hippel, S. Schaefer and F. Virotta, PoS LATTICE (2013) 475, arXiv:1312.7693.[8] N. Carrasco et al. (ETMC), JHEP (2014) 016, arXiv:1308.1851.[9] FLAG averages: http://itpwiki.unibe.ch/flag [10] J. Koponen et al. (HPQCD Collaboration), arXiv:1305.1462.[11] J. A. Bailey et al. (Fermilab Lattice and MILC Collaborations), Phys. Rev. D (2012) 114502, arXiv:1202.6346; M. Atoui et al. (ETMC), arXiv:1310.5238and PoS LATTICE et al. (HPQCD), Phys. Rev. D (2011) 114505, arXiv:1109.1501.[13] C. Aubin et al. (Fermilab Lattice and MILC and HPQCD Collaborations), Phys.Rev. Lett. (2005) 011601, arXiv:hep-ph/0408306.[14] J. A. Bailey et al. (Fermilab Lattice and MILC Collaborations), PoS LATTICE (2012) 272, arXiv:1211.4964[15] F. Sanfilippo, D. Becirevic, V. Lubicz and S. Simula, PoS LATTICE et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).4 .9 0.95 1 1.05 1.1 neutrino scatteringFLAG averageETMC ’13, twisted massFNAL/MILC ’04, FermilabHPQCD ’11 & ’13, HISQn f =2+1n f =2n f =2+1 f =2+1n f =2+1+1n f =2+1n f =2n f =2 f =2+1n f =2n f =2+1 f =2+1n f =2+1+1n f =2+1n f =2n f =2 LEPTONIC prelim.prelim.prelim.prelim. V cd V cs SEMILEPTONIC prelim.prelim. V cd V cs Figure 4: Summary of the CKM elements. Top row from left to right: | V cd | and | V cs | from leptonic decays. Bottom row from left to right: | V cd | and | V cs | fromsemileptonic decays. Vertical error bands show the unitarity value for reference.The best values using the latest lattice results (most modern lattice configurationswith n f = 2 + 1 + 1 and physical pion mass, and smallest errors) are: V cd (leptonic) =0 . expt (13) lattice and V cs (leptonic) = 1 . expt (6) lattice , taking decay don-stants from [1] (FNAL/MILC ’14 in Fig. 1); V cd (semileptonic) = 0 . expt (10) lattice from [12] and V cs (semileptonic) = 0 . expt (14) lattice from [10]. Experimental aver-ages used here are from [16]. [10] is the first calculation to use all experimental q binsto extract a CKM element from a semileptonic decay. For comparison, averages fromFlavor Lattice Averaging Group (FLAG) [9] for n f = 2 and n f = 2 + 1 are also shownin the plots, as well as the result for | V cd ||