Determinations of quark mixing matrix elements | V cd | and | V cs | from leptonic and semileptonic D Decays
aa r X i v : . [ h e p - e x ] S e p Determinations of quark mixing matrix elements | V cd | and | V cs | from leptonic and semileptonic D Decays
G. Rong , Y. Fang , and H. L. Ma Institute of High Energy Physics, Beijing 100049, People’s Republic of China
29 August, 2014
Abstract
With the recent measurements of purely leptonic D +( s ) decays and semileptonic D decays in conjunction with decay constants f D +( s ) and form factors f π ( K )+ (0) calculated inLQCD, we extract the magnitudes of V cd and V cs to be | V cd | = 0 . ± .
005 and | V cs | =0 . ± . | V cd | and | V cs | are improved by more than 2.0 and 1.5 factors, respectively.With the newly extracted | V cd | and | V cs | together with other CKM matrix elementsgiven in PDG2013, we check the unitarity of the CKM matrix, which are | V ud | + | V cd | + | V td | = 0 . ± . | V us | + | V cs | + | V ts | = 1 . ± .
032 and | V cd | + | V cs | + | V cb | =1 . ± . In the Standard Model (SM) of particle physics, the D +( s ) meson can decay into ℓ + ν ℓ (where ℓ = e , µ , or τ ) via annihilation mediated by a virtual W + boson. The decay rate dependsupon the wave function overlap of the two quarks at the origin, which is parameterized by the D +( s ) decay constant, f D +( s ) . All of the strong interaction effects between the two initial-statequarks are absorbed into f D +( s ) . In the SM, the decay width of D +( s ) → ℓ + ν ℓ is given byΓ( D +( s ) → ℓ + ν ℓ ) = G F f D +( s ) π | V cd ( s ) | m ℓ m D +( s ) − m ℓ m D +( s ) , (1)where G F is the Fermi coupling constant, V cd ( s ) is the c → d ( s ) Cabibbo-Kobayashi-Maskawa(CKM) matrix element [1], m ℓ is the lepton mass, and m D +( s ) is the D +( s ) meson mass.1imilarly, in the SM, neglecting the positron mass, the differential decay rate of D → π ( K ) e + ν e process is given by d Γ dq = X G F π | V cd ( s ) | | ~p π ( K ) | | f π ( K )+ ( q ) | , (2)where ~p π ( K ) is the three-momentum of the π ( K ) meson in the rest frame of the D meson, f π ( K )+ ( q ) represents the hadronic form factor of the hadronic weak current depending onsquare of the four-momenta transfer q , and X is a factor due to isospin, which equals to 1for D → π − e + ν e , D → K − e + ν e and D + → ¯ K e + ν e , and equals to 1 / D + → π e + ν e .The form factor f π ( K )+ ( q ) measures the probability to form the final state π ( K ) meson inthis decay.Recently, the branching fractions for leptonic D + and D + s decays were well measured atthe e + e − experiments near threshold of the D ¯ D production (CLEO-c and BESIII) and near10 . f D + and f D + s were calculated in LQCDat precisions of ∼ .
6% and ∼ . f D +( s ) calculated in LQCD, the magnitudes of CKM quark mixingparameters V cd and V cs can be well extracted. In addition, the precisions of these measuredbranching fractions for D → πe + ν e and D → Ke + ν e decays or measured products of | V cd ( s ) | and f π ( K )+ (0) are at an accuracy level of about 1%, while the LQCD calculations of theseform factors f π + (0) and f K + (0) also reach to about 4 .
4% and 2 . | V cd ( s ) | and f π ( K )+ (0) together with inputs of the form factors calculatedin LQCD, the magnitudes of V cd and V cs can also be well extracted.In this article, we extract | V cd | and | V cs | with these measured branching fractions and/or | V cd ( s ) | f π ( K )+ (0) in conjunction with decay constants f D +( s ) and/or form factors f π ( K )+ (0) cal-culated in LQCD. In determinations of | V cd | and | V cs | , we use G F , masses of D +( s ) meson andleptons, and lifetimes of D +( s ) meson given in PDG2013 [1]. D + decays In 2008, the CLEO-c Collaboration accumulated 460055 ± D − tags by analyzing818 pb − data taken at 3.773 GeV and selecting D − mesons from 6 hadronic decay modesof the D − meson. They observed 149 . ± . D + → µ + ν µ decays inthe system recoiling against these D − tags. They measured the decay branching fraction B ( D + → µ + ν µ ) = (3 . ± . ± . × − [2].In 2014, the BESIII Collaboration measured the branching fraction for D + → µ + ν µ decays by analyzing 2.92 fb − data taken at 3.773 GeV. From 9 hadronic decay modes of2 − meson, the BESIII Collaboration accumulated 1703054 ± D − tags. In this D − tag sample they observed 409 . ± . D + → µ + ν µ decays and measuredbranching fraction B ( D + → µ + ν µ ) = (3 . ± . ± . × − [3].Averaging these two branching fractions, we obtain B ( D + → µ + ν µ ) = (3 . ± . × − , (3)where the error is the combined statistical and systematic errors together. D + s decays In 2009, the CLEO-c Collaboration studied the D + s → ℓ + ν ℓ decays based on 600 pb − data taken at 4.17 GeV. From this data sample, they tagged D − s mesons from 9 hadronicdecay modes. By examining distribution of missing mass-squared of the D − s and γ systemthey accumulated 43859 ± D + s mesons; by analyzing distribution of missing mass-squaredof the D − s γµ + system, they selected D + s → µ + ν µ decay events and measured the branchingfraction B ( D + s → µ + ν µ ) = (0 . ± . ± . B ( D + s → τ + ν τ ) = (5 . ± . ± . τ + → π + ¯ ν τ [4], τ + → e + ν e ¯ ν τ [5] and τ + → ρ + ¯ ν τ decays [6].In 2013, the Belle Collaboration measured the branching fractions for leptonic D + s de-cays. They selected leptonic D + s decays from the e + e − → c ¯ c continuum production, inwhich the D tag K frag X frag D ∗ + s is produced from the quark fragmentation, where D ∗ + s → γD + s , K frag is either K + or K S , and X frag indicates several pions or photons. By reconstructingthe recoil mass of the D tag K frag X frag γ , they observed clear D + s signal in the system recoil-ing against the D tag K frag X frag γ . By fitting the recoil mass spectra of D tag K frag X frag γ , theyaccumulated 94360 ± ± D + s mesons. To search for D + s → µ + ν µ de-cays, they examined the missing mass-squared M ( D tag K frag X frag γµ ) distribution of the D tag K frag X frag γµ system. Fitting the M ( D tag K frag X frag γµ ) distribution yields 492 ± D + s → µ + ν µ decays. With these numbers of events, the Belle Collaborationmeasured the decay branching fraction B ( D + s → µ + ν ) = (0 . ± . ± . ±
83 signal events for D + s → τ + ν τ decayswith τ + → e + ν e ¯ ν τ , τ + → µ + ν µ ¯ ν τ and τ + → π + ¯ ν τ decays, and measured the decay branchingfraction B ( D + s → τ + ν τ ) = (5 . ± . +0 . − . )% [7].In 2010, using the similar technique as the one used by the Belle Collaboration, theBaBar Collaboration made measurements of the branching fractions for leptonic D + s decays.By analyzing 521 fb − data taken at 10.6 GeV, the BaBar Collaboration measured the decaybranching fractions B ( D + s → µ + ν µ ) = (0 . ± . ± . B ( D + s → τ + ν τ ) =(5 . ± . ± . B ( D + s → µ + ν µ ) = (0 . ± . B ( D + s → τ + ν τ ) = (5 . ± . , (5)where the errors are the combined statistical and systematic errors together. D decays In 2008, the CLEO-c Collaboration studied the semileptonic decays of D → π − e + ν e , D → K − e + ν e , D + → π e + ν e and D + → ¯ K e + ν e by analyzing 818 pb − data taken at 3.773GeV. They extracted the products f π + (0) | V cd | = 0 . ± . ± .
001 and f K + (0) | V cs | = 0 . ± . ± .
005 by fitting their measured partial decay rates with form factor parameterizedwith three parameter series expansion [9].Recently, the BESIII Collaboration reported their new preliminary results of D → π − e + ν e and D → K − e + ν e decays obtained by analyzing 2.92 fb − data taken at 3.773 GeV.They obtained f π + (0) | V cd | = 0 . ± . ± . f K + (0) | V cs | = 0 . ± . ± . f K + ( q ) by analyzing 75 fb − data collected at 10.6 GeV and determined f K + (0) = 0 . ± . ± . ± .
007 [11].Multiplying this form factor by | V cs | = 0 . ± . f K + (0) | V cs | = 0 . ± . ± . ± . D → π − e + ν e decay by analyzing 347.2 fb − data collectedat Υ(4 S ) and reported preliminary results at ICHEP2014. They measured f π + (0) | V cd | =0 . ± . ± . ± . f π ( K )+ (0) | V cd ( s ) | measured at the CLEO-c, BESIII and BaBar experi-ments, we obtain f π + (0) | V cd | = 0 . ± .
002 (6)and f K + (0) | V cs | = 0 . ± . , (7)where the errors are the combined statistical and systematic errors together. | V cd | Before 2012, the CKM matrix element | V cd | was usually determined with the ν ¯ ν inter-action or the semileptonic decay of D → πe + ν e . Actually, using the measured branch-ing fraction for D + → µ + ν µ decays in conjunction with the LQCD calculation on D + V cd can also be extracted via the Eq. (1). AtCharm2012, the BESIII Collaboration reported preliminary result on the determination of | V cd | based on their measured branching fraction for D + → µ + ν µ decay, which is | V cd | =0 . ± . ± . f D + calculated in LQCD. The averaged D + decay con-stant calculated in LQCD is f D + = (209 . ± .
3) MeV [14]. Inserting the averaged branchingfraction for D + → µ + ν µ decays as given in Eq. (3) and this averaged f D + into Eq. (1) yields | V cd | D + → µ + ν µ = 0 . ± . ± . , (8)where the first uncertainty is from the measured branching fractions and the second mainlyfrom the uncertainties of f D + and the lifetime of D + meson.Dividing the averaged f π + (0) | V cd | from semileptonic D → πe + ν e decays by the form factor f π + (0) = 0 . ± .
029 calculated in LQCD [15] yields | V cd | D → πe + ν e = 0 . ± . ± . , (9)where the first uncertainty is from the measured f π + (0) | V cd | , and the second uncertainty isfrom f π + (0).Figure 1 shows the comparison of | V cd | determined from purely leptonic D + decay andsemileptonic D decay. Averaging the determined | V cd | D + → µ + ν µ and | V cd | D → πe + ν e yields | V cd | = 0 . ± . . (10)Figure 2 shows the comparison of the newly determined | V cd | and the one given in PDG2013 [1]. | V cs | Using the measured decay branching fractions for D + s → ℓ + ν ℓ together with the D + s mesondecay constant calculated in LQCD, the magnitude of V cs can be extracted via Eq. (1). Weherein use the value of f D + s = (248 . ± .
7) MeV, which is the FLAG average of severaldecay constants calculated in LQCD [14], to extract | V cs | . Inserting the averaged branchingfractions for D + s → ℓ + ν ℓ decays and the f D + s into Eq. (1) yields | V cs | D + s → µ + ν µ = 1 . ± . ± .
013 (11)and | V cs | D + s → τ + ν τ = 1 . ± . ± . , (12)where the first uncertainties are from the measured branching fractions, and the seconduncertainties are mainly from f D + s and the lifetime of D + s meson. Combining the above twovalues, we obtain | V cs | D + s → ℓ + ν ℓ = 1 . ± . ± . , (13)5 .05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 | V cd | CLEO-c (2008) △ BESIII (2014) ◦ Average • CLEO-c (2009) △ BESIII (2014) Preliminary ◦ BaBar (2014) Preliminary ⋄ Average • D + → µ + ν µ D → πe + ν e . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . Figure 1: Comparison of | V cd | determined from leptonic D + and semileptonic D decays. | V cd | PDG2013 △ This work (cid:13) . ± . . ± . Figure 2: Comparison of the newly determined | V cd | from both the leptonic D + and semilep-tonic D decays and the one given in PDG2013.6here the first uncertainty is from the measured branching fractions, the second uncertaintyis mainly from f D + s and the lifetime of D + s meson.Dividing the averaged f K + (0) | V cs | from semileptonic D → Ke + ν e decays by the formfactor f K + (0) = 0 . ± .
019 calculated in LQCD [16] yields | V cs | D → Ke + ν e = 0 . ± . ± . , (14)where the first uncertainty is from the measured f K + (0) | V cs | and the second uncertainty isfrom f K + (0).Figure 3 shows the comparison of | V cs | determined from purely leptonic D + s decays andsemileptonic D decays. Averaging the determined | V cs | D + s → ℓ + ν ℓ and | V cs | D → Ke + ν e yields | V cs | = 0 . ± . . (15)Figure 4 shows the comparison of the newly determined | V cs | and the one given in PDG2013 [1]. Using the newly extracted | V cd | = 0 . ± . | V ud | = 0 . ± . | V td | = (8 . ± . × − [1], the first column unitarity of CKM matrix is checked, whichis | V ud | + | V cd | + | V td | = 0 . ± . . (16)Using the newly extracted | V cs | = 0 . ± . | V us | = 0 . ± . | V ts | = (42 . ± . × − [1], we find | V us | + | V cs | + | V ts | = 1 . ± .
032 (17)for the second column of the CKM matrix. Using these newly extracted | V cd | and | V cs | , andthe PDG value | V cb | = (40 . ± . × − [1], we find | V cd | + | V cs | + | V cb | = 1 . ± .
032 (18)for the second row of the CKM matrix. The unitarity check results for the first column,second column and second row of the CKM matrix are shown in Fig. 5 together with theunitarity checks given in PDG2013 [1]. The newly determined | V cd | and | V cs | give morestringent checks of the CKM matrix unitarity compared to those in PDG2013. Combining the precise measurements of leptonic D +( s ) → µ + ν µ decays and semileptonic D → π ( K ) e + ν e decays at the CLEO-c, Belle, BaBar and BESIII together with the improved7 .7 0.8 0.9 1.0 1.1 1.2 1.3 | V cs | CLEO-c (2009) △ Belle (2013) • BaBar (2010) ⋄ Average • CLEO-c (2009) △ Belle (2013) • BaBar (2010) ⋄ Average • CLEO-c (2009) △ BESIII (2014) Preliminary ◦ BaBar (2007) ⋄ Average • D + s → µ + ν µ D + s → τ + ν τ D → K e + ν e . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . Figure 3: Comparison of | V cs | determined from leptonic D + s decays and semileptonic D → Ke + ν e decays. 8 .96 0.98 1.0 1.02 1.04 | V cs | PDG2013 △ This work (cid:13) . ± . . ± . Figure 4: Comparison of the newly determined | V cs | from both the leptonic D + s and semilep-tonic D decays and the one given in PDG2013. | V ud | + | V cd | + | V td | ◦ ◦ | V us | + | V cs | + | V ts | ⋄ ⋄ | V cd | + | V cs | + | V cb | • • . ± . . ± . . ± . . ± .
005 (PDG2013)1 . ± .
046 (PDG2013) . ± .
047 (PDG2013)
Figure 5: Unitarity checks for the first column, second column and second row of the CKMmatrix. 9 +( s ) decay constants and semileptonic D decay form factors calculated in LQCD, we extractthe magnitudes of V cd and V cs to be | V cd | = 0 . ± .
005 and | V cs | = 0 . ± . | V cd | and | V cs | give more stringent unitaritychecks of the CKM matrix compared to those given in PDG2013. Acknowledgements
This work is supported in part by the Ministry of Science of Technology of China underContracts No. 2009CB825204; National Natural Science Foundation of China (NSFC) underContacts No. 10935007 and No. 11305180.
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