Decay Constants and Semileptonic Form Factors of Pseudoscalar Mesons
Measurement of the form-factor ratios for D+ -- K l nu
a r X i v :h e p -e x /9809026v 1 24 S e p 1998KSU-HEP-98-001FNAL Pub-98/289-E September 17,1998Measurement of the form-factor ratios forD +→K⋆0ℓ+νℓ,1Centro Brasileiro de Pesquisas F´ısicas,Rio de Janeiro,Brazil 2University of California,Santa Cruz,California 950643University of Cincinnati,Cincinnati,Ohio 452214CINVESTAV,Mexico5Fermilab,Batavia,Illinois 605106Illinois Institute of Technology,Chicago,Illinois 606167Kansas State University,Manhattan,Kansas 665068University of Massachusetts,Amherst,Massachusetts010039University of Mississippi,University,Mississippi3867710Princeton University,Princeton,New Jersey0854411Universidad Autonoma de Puebla,Mexico12University of South Carolina,Columbia,South Carolina2920813Stanford University,Stanford,California9430514Tel Aviv University,Tel Aviv,Israel15Box1290,Enderby,BC,V0E1V0,Canada16Tufts University,Medford,Massachusetts0215517University of Wisconsin,Madison,Wisconsin5370618Yale University,New Haven,Connecticut06511K⋆0ℓ+νℓare an especially clean way to study these effects because the leptonicand hadronic currents completely factorize in the decay amplitude.All informa-tion about the strong interactions can be parametrized by a few form factors.Also, according to Heavy Quark Effective Theory,the values of form factors for somesemileptonic charm decays can be related to those governing certain b-quark de-cays.In particular,the form factors studied here can be related to those for therare B–meson decays B→K⋆e+e−and B→K⋆γ[1,2]which provide windows for physics beyond the Standard Model.With a vector meson in thefinal state,there are four form factors,V(q2),A1(q2),A2(q2)and A3(q2),which are functions of the Lorentz-invariant momentum transfer squared[3].The differential decay rate for D+→K⋆0→K−π+is a quadratic homogeneous function of the four form factors.Unfortunately,the limited size of current data samples precludes precise measurement of the q2-dependence of the form factors;we thus assume the dependence to be given by the nearest-pole dominance model:F(q2)=F(0)/(1−q2/m2pole)where m pole= m V=2.1GeV/c2for the vector form factor V,and m pole=m A=2.5GeV/c2 for the three axial-vector form factors[4].The third form factor A3(q2),which is unobservable in the limit of vanishing lepton mass,probes the spin-0component of the off-shell W.Additional spin-flip amplitudes,suppressed by an overall factor of m2ℓ/q2when compared with spin no-flip amplitudes,contribute to the differential decay rate.Because A1(q2)appears among the coefficients of every term in the differential decay rate,it is customary to factor out A1(0)and to measure the ratios r V=V(0)/A1(0),r2=A2(0)/A1(0)and r3=A3(0)/A1(0).The values of these ratios can be extracted without any assumption about the total decay rate or the weak mixing matrix element V cs.We report new measurements of the form factor ratios for the muon chan-nel and combine them with slightly revised values of our previously publishedmeasurements of r V and r2[5]for the electron channel.This is thefirst set of measurements in both muon and electron channels from a single experiment.We also report thefirst measurement of r3=A3(0)/A1(0),which is unobservable in the limit of vanishing charged lepton mass.E791is afixed-target charm hadroproduction experiment[6].Charm particles were produced in the collisions of a500GeV/cπ−beam withfive thin targets, one platinum and four diamond.About2×1010events were recorded during the 1991-1992Fermilabfixed-target run.The tracking system consisted of23planes of silicon microstrip detectors,45planes of drift and proportional wire chambers, and two large-aperture dipole magnets.Hadron identification is based on the in-formation from two multicellˇCerenkov counters that provided good discrimination between kaons and pions in the momentum range6−36GeV/c.In this momentum range,the probabability of misidentifying a pion as a kaon depends on momentum but does not exceed5%.We identified muon candidates using a single plane of scintillator strips,oriented horizontally,located behind an equivalent of2.4meters of iron(comprising the calorimeters and one meter of bulk steel shielding).The an-gular acceptance of the scintillator plane was≈±62mrad×±48mrad(horizontally and vertically,respectively),which is somewhat smaller than that of the rest of the spectrometer for tracks which go through both magnets(≈±100mrad×±64mrad). The vertical position of a hit was determined from the strip’s vertical position,and the horizontal position of a hit from timing information.The event selection criteria used for this analysis are the same as for the electronic-mode form factor analysis[5],except for those related to lepton identifi-cation.Events are selected if they contain an acceptable decay vertex determined by the intersection point of three tracks that have been identified as a muon,a kaon,and a pion.The longitudinal separation between this candidate decay vertex and the reconstructed production vertex is required to be at least15times the esti-mated error on the separation.The two hadrons must have opposite charge.If the kaon and the muon have opposite charge,the event is assigned to the“right-sign”sample;if they have the same charge,the event is assigned to the“wrong-sign”sample used to model the background.To reduce the contamination from hadron decays inflight,only muon can-didates with momenta larger than8GeV/c are retained.With this momentum restriction,the efficiency of muon tagging was about85%,and the probability for a hadron to be identified as a muon was about3%.To exclude feedthrough from D+→K−π+π+,we exclude events in which the invariant mass of the three charged particles(with the muon candidate interpreted as a pion)is consistent with the D+mass.For ourfinal selection criteria,we use a binary-decision-tree algorithm(CART[7]),whichfinds linear combinations of parameters that have the highest discrimination power between signal and ing this algorithm,we found a linear combination of four discrimination variables[5]:(a)separation significance of the candidate decay vertex from target material;(b)distance of closest approach of the candidate D momentum vector to the primary vertex,taking into account the maximum kinematically-allowedmiss distance due to the unobserved neutrino;(c)product over candidate D decay tracks of the distance of closest approach of the track to the secondary vertex, divided by the distance of closest approach to the primary vertex,where each dis-tance is measured in units of measurement errors;and(d)significance of separation between the production and decay vertices.Thisfinal selection criterion reduced the number of wrong-sign events by50%,and the number of right-sign events by 25%.Although this does not affect our sensitivity substantially,it does reduce systematic uncertainties associated with the background subtraction.The minimum parent mass M min is defined as the invariant mass of Kπµνwhen the neutrino momentum component along the D+direction offlight is ignored.The distribution of M min should have a Jacobian peak at the D+mass,and we observe such a peak in our data(Fig.1).We retain events with M min in the range1.6 to2.0GeV/c2as indicated by the arrows in thefigure.The distribution of Kπinvariant mass for the retained events is shown in the top right of Fig.1for both right-sign and wrong-sign samples.Candidates with0.85<M Kπ<0.94GeV/c2 were retained,yieldingfinal data samples of3629right-sign and595wrong-sign events.The hadroproduction of charm,the differential decay rate,and the detector response were simulated with a Monte Carlo event generator.A sample of events was generated according to the differential decay rate(Eq.22in Ref.[3]),with the form factor ratios r V=2.00,r2=0.82,and r3=0.00.The same selection criteria were applied to the Monte Carlo events as to real data.Out of25million generated events,95579decays passed all cuts.Figure1(bottom)shows the distribution of M Kπfrom real data after background subtraction(“right-sign”minus“wrong-sign”)overlaid with the corresponding Monte Carlo distribution after all cuts are applied.The agreement between the two distributions suggests that wrong-sign events correctly account for the size of the background.The differential decay rate[3]is expressed in terms of four independent kine-matic variables:the square of the momentum transfer(q2),the polar angleθV in theK⋆0and W+decay planes.The definition we use for the polar angle θℓis related to the definition used in Ref.[3]byθℓ→π−θℓ.Semileptonic decays cannot be fully reconstructed due to the undetected neu-trino.With the available information about the D+direction offlight and the charged daughter particle momenta,the neutrino momentum(and all the decay’s kinematic variables)can be determined up to a two-fold ambiguity if the parent mass is constrained.Monte Carlo studies show that the differential decay rate is more accurately determined if it is calculated with the solution corresponding to the lower laboratory-frame neutrino momentum.To extract the form factor ratios the distribution of the data points in the four-dimensional kinematic variable space isfit to the full expression for the differential decay rate.We use the same unbinned maximum-likelihoodfitting technique as inour D+→K⋆0µ+νµcandidates as the previous method, but uses additional neutrino-momentum solutions.This is true for both the data and for the Monte Carlo sample used in the likelihood function calculation,so the results of thisfit could differ from those of the previousfit.The values of the form factor ratios obtained with the two methods agree well, providing further assurance that selecting the lower neutrino momentum solution in the primary method and correcting for the systematic bias gives the correct result.However,the systematic uncertainties for the primary method(see below) were found to be significantly smaller,mainly because the unbinned maximum-likelihood method is more stable against changes in the size of the phase spacevolume.Therefore,the primary method was chosen for quotingfinal results.We classify systematic uncertainties into three categories:(a)Monte Carlo simulation of detector effects and production mechanism;(b)fitting technique;(c) background subtraction.The estimated contributions of each are given in Table I. The main contributions to category(a)are due to muon identification and data selection criteria.The contributions to category(b)are related to the limited size of the Monte Carlo sample and to corrections for systematic bias.The measurements of the form factor ratios for D+→K⋆0e+νe[5]follow the same analysis procedure except for the charged lepton identification.Both results are listed in Table II.The consistency within errors of the results measured in the electron and muon channels supports the assumption that strong interaction effects,incor-porated in the values of form factor ratios,do not depend on the particular W+ leptonic decay.Based on this assumption,we combine the results measured for the electronic and muonic decay modes.The averaged values of the form factor ratios are r V=1.87±0.08±0.07and r2=0.73±0.06±0.08.The statistical and systematic uncertainties of the average results were determined using the general procedure described in Ref.[9](Eqns.3.40and3.40′).Some of the systematic errors for the two samples have positive correlation coefficients,and some nega-tive.The combination of all systematic errors is ultimately close to that which one would obtain assuming all the errors are uncorrelated.The third form factor ratio r3was not measured in the electronic mode.Table II compares the values of the form factor ratios r V and r2measured by E791in the electron,muon and combined modes with previous experimental results.The size of the data sample and the decay channel are listed for each case. All experimental results are consistent within errors.The comparison between the E791combined values of the form factor ratios r V and r2and previous experimental results is also shown in Fig.3(top).Table III and Fig.3(bottom)compare thefinal E791result with published theoretical predictions.The spread in the theoretical results is significantly larger than the E791experimental errors.To summarize,we have measured the values of the form factor ratios in the decay channel D+→K⋆0e+νe gives r V=1.87±0.08±0.07and r2=0.73±0.06±0.08.We gratefully acknowledge the assistance from Fermilab and other participat-ing institutions.This work was supported by the Brazilian Conselho Nacional de Desenvolvimento Cient´ıfico e Technol´o gico,CONACyT(Mexico),the Israeli Academy of Sciences and Humanities,the U.S.Departament of Energy,the U.S.-Israel Binational Science Foundation,and the U.S.National Science Foundation.REFERENCES[1]N.Isgur and M.B.Wise,Phys.Rev.D42(1990)2388.[2]Z.Ligeti,I.W.Stewart and M.B.Wise,Phys.Lett.B420(1998)359.[3]J.G.K¨o rner and G.A.Schuler,Phys.Lett.B226(1989)185.[4]Particle Data Group,Review of Particle Physics,Phys.Rev.D50(1994)1568.[5]Fermilab E791Collaboration,E.M.Aitala et al.,Phys.Rev.Lett.80(1998)1393.The E791electron result for r V quoted in this paper is0.06higher than the value reported in this reference because we have corrected for inaccuracies in the earlier modeling of the D+transverse momentum.[6]J.A.Appel,Ann.Rev.Nucl.Part.Sci.42(1992)367;D.J.Summers et al.,XXVII Rencontre de Moriond,Les Arcs,France(15-22March1992)417. [7]L.Brieman et al.,Classification and Regression Trees(Chapman and Hall,New York,1984).[8]D.M.Schmidt,R.J.Morrison,and M.S.Witherell,Nucl.Instrum.Methods A328(1993)547.[9]L.Lyons,Statistics for Nuclear and Particle Physicists(Cambridge UniversityPress,Cambridge,1986).[10]Fermilab E687Collaboration,P.L.Frabetti et al.,Phys.Lett.B307(1993)262.[11]Fermilab E653Collaboration,K.Kodama et al.,Phys.Lett.B274(1992)246.[12]Fermilab E691Collaboration,J.C.Anjos et al.,Phys.Rev.Lett.65(1990)2630.[13]D.Scora and N.Isgur,Phys.Rev.D52(1995)2783.We have used the q2-dependence assumed in thefits to our data to extrapolate the theoretical form factors from q2=q2max to q2=0.[14]M.Wirbel,B.Stech,and M.Bauer,Z.Phys.C29(1985)637.[15]T.Altomari and L.Wolfenstein,Phys.Rev.D37(1988)681.[16]F.J.Gilman and R.L.Singleton,Jr.,Phys.Rev.D41(1990)142.[17]B.Stech,Z.Phys.C75(1997)245.[18]C.W.Bernard,Z.X.El-Khadra,and A.Soni,Phys.Rev.D45(1992)869,Phys.Rev.D47(1993)998.[19]V.Lubicz,G.Martinelli,M.S.McCarthy,and C.T.Sachrajda,Phys.Lett.B274(1992)415.[20]A.Abada et al.,Nucl.Phys.B416(1994)675.[21]C.R.Alton et al.,Phys.Lett.B345(1995)513.[22]K.C.Bowler et al.,Phys.Rev.D51(1995)4905.[23]P.Ball,V.M.Braun,and H.G.Dosch,Phys.Rev.D44(1991)3567.[24]T.Bhattacharya and R.Gupta,Nucl.Phys.B(Proc.Suppl.)47(1996)481.TABLESTABLE I.The main contributions to uncertainties on the form factor ratios.Sourceσr2σrVσr3Hadron identification0.010.010.02 Muon identification0.040.060.10 Production mechanism0.010.010.02 Acceptance0.030.020.08 Cut selection0.030.040.09MC volume size0.020.020.12 Number of MC points0.010.010.18 Bias0.010.020.06No.of background events0.040.020.06 Background shape0.040.040.06 E7916000(e+µ)1.87±0.08±0.070.73±0.06±0.08 E7913000(µ)1.84±0.11±0.090.75±0.08±0.09 E7913000(e)1.90±0.11±0.090.71±0.08±0.09 E687[10]900(µ)1.74±0.27±0.280.78±0.18±0.10E653[11]300(µ)2.00+0.34−0.32±0.160.82+0.22−0.23±0.11E691[12]200(e)2.0±0.6±0.30.0±0.5±0.2TABLE parison of E791results with theoretical predictions for the form factor ratios r V and r2.Group r V r2ISGW2[13]2.01.3WSB[14]1.41.3KS[3]1.01.0AW/GS[15,16]2.00.8Stech[17]1.551.06BKS[18]1.99±0.22±0.330.70±0.16±0.17 LMMS[19]1.6±0.20.4±0.4ELC[20]1.3±0.20.6±0.3APE[21]1.6±0.30.7±0.4UKQCD[22]1.4+0.5−0.20.9±0.2BBD[23]2.2±0.21.2±0.2 LANL[24]1.78±0.070.68±0.11FIGURES01002003004005006001.251.51.7522.25M min (GeV/ c 2)E v e n t s / 40 M e V / c21002003004005006007000.70.80.91 1.1K π invariant mass (GeV/ c 2)E v e n t s / 10 M e V / c 2050100150200250300K π invariant mass (GeV/ c 2)E v e n t s / 5 M e V / c2FIG.1.Distributions of minimum parent mass M min and Kπinvariant mass for D +→050100150200cos θl (q 2/q 2max < 0.5)E v e n t s / 0.2050100150200cos θl (q 2/q 2max ≥ 0.5)E v e n t s / 0.250100150200cos θV (q 2/q 2max < 0.5)E v e n t s / 0.2050100150200cos θV (q 2/q 2max ≥ 0.5)E v e n t s / 0.250100150200χ (cos θV < 0)E v e n t s / 0.2π050100150200χ (cos θV ≥ 0)E v e n t s / 0.2πparison of single-variable distributions of background-subtracted data (crosses)with Monte Carlo predictions (dashed histograms)using best-fit values for the form factor ratios.1111.21.41.61.822.22.4rr V11.21.41.61.822.22.4r 2r VFIG.3.Top:Comparison of experimental measurements of form factor ratios r V and r 2for D +→。
Masses and decay constants of B_q mesons in the QCD string approach
a rXiv:h ep-ph/61193v116Oct26Masses and decay constants of B q mesons in the QCD string approach A.M.Badalian ∗and Yu.A.Simonov †Institute of Theoretical and Experimental Physics,Moscow,Russia B.L.G.Bakker ‡Vrije Universiteit,Amsterdam,The Netherlands February 2,2008Abstract The relativistic string Hamiltonian is used to calculate the masses and decay constants of B q mesons:they appear to be expressed through onlythree fundamental values:the string tension σ,αs ,and the quark pole masses.The values f B =186MeV,f B s =222MeV are calculated while f B c depends on the c -quark pole mass used,namely f B c =440(424)MeV for m c =1.40(1.35)GeV.For the 1P states we predict the spin-averaged masses:¯M (B J )=5730MeV and ¯M (B sJ )=5830MeV which are in good agreement with the recent data of the D0and CDF Collaborations,at the same time owningto the string correction being by ∼50MeV smaller than in other calculations.1IntroductionThe decay constants of pseudoscalar (P)mesons f P can be directly mea-sured in P →µνdecays [1]and therefore they can be used as an importantcriterium to compare different theoretical approaches and estimate their ac-curacy.Although during the last decade f P were calculated many times:in potential models[2,3,4],the QCD sum rule method[5],and in lattice QCD [6,7],here we again address the properties of the B,B s,B c mesons for several reasons.First,we use here the relativistic string Hamiltonian(RSH)[8],which is derived from the QCD Lagrangian with the use of thefield correlator method (FCM)[9]and successfully applied to light mesons and heavy quarkonia [10,11].Here we show that the meson Green’s function and decay constants can also be derived with the use of FCM.Second,the remarkable feature of the RSH H R and also the correlator of the currents G(x)is that they are fully determined by a minimal number of fundamental parameters:the string tensionσ,ΛMS=250(5)MeV;(1) and the pole masses taken arem u(d)=0;m s=170(10)MeV;m c=1.40GeV;m b=4.84GeV.(2) Third,recently new data on the masses of B c and the P-wave mesons: B1,B2,and B s2have been reported by the D0and CDF Collaborations [12,13],which give additional information on the B q-meson spectra.Here we calculate the spin-averaged masses of the P-wave states B and B s.We would like to emphasize here that in our relativistic calculations no constituent masses are used.In the meson mass formula an overall(fitting) constant,characteristic for potential models,is absent and the whole scheme appears to be rigid.Nevertheless,we take into account an important nonperturbative(NP) self-energy contribution to the quark mass,∆SE(q)(see below eq.(18)).For the heavy b quark∆SE(b)=0and for the c quark∆SE(c)≃−20MeV[10], which is also small.For any kind of mesons we use a universal static potential with pure scalarconfining term,V0(r)=σr−4r,(3)2where the couplingαB(r)possesses the asymptotic freedom property and saturates at large distances withαcrit(n f=4)=0.52[14].The coupling can be expressed throughαB(q)in momentum space,αB(r)=2qαB(q),(4)whereαB(q)=4πβ20ln t BΛ2B.Here the QCD constantΛB,is expressed as[15]ΛB(n f)=Λ2β0· 319n f (6)and M B(σ,ΛB)=(1.00±0.05)GeV is the so called background mass[14]. For heavy-light mesons withΛ2+m2i+p2In (8)m 1(m 2)is the pole (current)mass of a quark (antiquark).The variable ωi is defined from extremum condition,which is taken either from(1)The exact condition:∂H 0p 2+m 2i .(10)ThenH 0ϕn = p 2+m 22+V 0(r ) ϕn =M n ϕn(11)reduces to the Salpeter equation,which just defines ωi (n )=∂˜ωi =0(the so-called einbein approxima-tion).As shown in [9]the difference between ωi and ˜ωi is <∼5%.For the RSH (7)the spin-averaged massM (nL )=ω12+m 212ωb +E n (µ)−2σηf p 2+m 2i nL ;µ=ω1ωbπωf ;(14)with ηf =0.9for a u (d )quark,ηf ∼=0.7for an s quark,ηf =0.4for a c quark,and ηb =0.Therefore,for a b quark ∆SE (b )=0.The mass formula(12)does not contain any overall constant C .Note that the presence of C violates linear behavior of Regge trajectories.The calculated masses of the low-lying states of B ,B s ,and B c mesons are given in Table 2,as well as their values taken from [2,3,6,7].It is of interest to notice that in our calculations the masses of the P -wave states appear to be by 30-70MeV lower than in [2]due to taking into account a string correction [11].4Table1:Masses of the low-lying B q mesons in the QCD String Approach B5280(5)a5279.0(5)5310252753B1(1P)¯M=5730a5721(8)D05734(5)CDFB s5369a5369.6(24)5390253623B s2¯M=58305839(3)D058802B∗c6330(5)a633826321(20)63Current CorrelatorThe FCM can be also used to define the correlator GΓ(x)of the currents jΓ(x),jΓ(x)=¯ψ1(x)Γψ2(x),(15) for S,P,V,and A channels(here the operatorΓ=t a⊗(1,γ5,γµ,iγµγ5)).The correlator,GΓ(x)≡ jΓ(x)jΓ(0) vac,(16) with the use of spectral decomposition of the currents jΓand the definition, vac|¯ψ1γ0γ5ψ2|P n(k=0) =f P n M n,(A,P)vac|¯ψ1γµψ2|V n(k,ε) =f V n M nεµ,(V)(17) can be presented as[3]GΓ(x)d x= n M n0|YΓe−H0T|0ω1ω2N c YΓ=p2 .(20)3Then from Eqs.(18)and(19)one obtains the following analytical expression for the decay constants(for a given state labelled n):f P(V)n 2=2N c M n|ϕn(0)|2.(21)This very transparent formula contains only well defined factors:ω1andωb, the meson mass M n,andϕn the eigenvector ofˆH0.Then in the P channelf P n 2=6(m1m2+ω1ω2− p2 )Table2:Pseudoscalar constants of B q mesons(in MeV)f B189216(34)186(5)f B s218249(42)222(2)f B swhere the w.f.at the origin,ϕn(0),is a relativistic one.In the nonrelativistic limitωi→m i,ϕn(0)→ϕNR n(0)and one comes to the standard expression:f P n(0) 2→12•In our analytic approach with minimal input of fundamental parame-ters(σ,αs,m i)the calculated decay constants are f B=186MeV,f B s=222MeV,f B s/f B=1.19.•For B c the decay constant is very sensitive to m c(pole):f B c=440MeV(m c=1.40GeV)and f B c=425MeV(m c=1.35GeV)References[1]D.Silverman and H.Yao,Phys.Rev.D38,214(1988).[2]S.Godfrey and N.Isgur,Phys.Rev.D32,189(1985);S.Godfrey,Phys.Rev.D70,054017(2004).[3]D.Ebert,R.N.Faustov,and V.O.Galkin,hep-ph/0602110;Mod.Phys.Lett.A17,803(2002),and references therein.[4]G.Cvetic,C.S.Kim,G.L.Wang,and W.K.Namgung,Phys.Lett.B596,84(2004).[5]M.Jamin,nge,Phys.Rev.D65,056005(2002)and referencestherein.[6]A.Ali Khan et al.,Phys.Rev.D70,114501(2004),ibid.64,054504(2004);C.T.H.Davies et al.,Phys.Rev.Lett.92,022001(2004).[7]A.S.Kronfeld,hep-lat/0607011and references therein;I.F.Allison etal.,Phys Rev.Lett.94172001(2005);A.Gray et al.,Phys.Rev.Lett.95,212001(2005).[8]A.Yu.Dubin,A.B.Kaidalov,and Yu.A.Simonov,Phys.Lett.B323,41(1994);Phys.Atom Nucl.56,1745(1993);E.L.Gubankova and A.Yu.Dubin,Phys.Lett.B334,180(1994).[9]H.G.Dosch and Yu.A.Simonov,Phys.Lett.B205,339(1988);Yu.A.Simonov,Z.Phys.C53,419(1992);Yu.S.Kalashnikova,A.V.Nefediev,and Yu.A.Simonov,Phys.Rev.D64,014037(2001);Yu.A.Simonov,Phys.Atom.Nucl.67,553(2004).[10]A.M.Badalian,A.I.Veselov,and B.L.G.Bakker,Phys.Rev.D70,016007(2004);Phys.Atom.Nucl.67,1367(2004).8[11]A.M.Badalian and B.L.G.Bakker,Phys.Rev.D66,034025(2002);A.M.Badalian,B.L.G.Bakker,and Yu.A.Simonov,Phys.Rev.D66,034025(2002).[12]P.Catastini(for the D0and CDF Collab.),hep-ex/0605051;M.D.Cor-coran,hep-ex/0506061.[13]D.Acosta et al.(CDF Collab.),Phys.Rev.Lett.96,202001(2006);hep-ex/0508022.[14]A.M.Badalian and D.S.Kuzmenko,Phys.Rev.D65,016004(2002);A.M.Badalian and Yu.A.Simonov,Phys.Atom.Nucl.60,636(1997).[15]M.Peter,Phys.Rev.Lett.76,602(1997);Y.Schr¨o der,Phys.Lett.B447,321(1999).[16]Yu.A.Simonov,Phys.Lett.B515,137(2001).[17]Particle Data Group,S.Eidelman,et al.,Phys.Lett.B592,1(2004).[18]A.M.Badalian and Yu.A.Simonov(in preparation).9。
Semileptonic B decay as a test of CKM unitarity
U u ij 0 0 U d αβ
uj dβ
.
weak
(5)
Indeed, using the unitarityty violation in the present interest can be expressed as follows,
(7)
where q µ denotes four-momentum of the dilepton and s = q 2 . Notations with hat on the top means it is normalized with the b−quark mass. In the tree-level approximation, new interactions in Eq. (2) contributes to O9 and O10 . Therefore, involving the continuum and resonances parts into calculation gives
Abstract I point out that B → Xq ℓ+ ℓ− decays (q = s, d) are sensitive probes of possible violation of CKM unitarity. I compute the decay rates and asymmetries in a minimal extension of the Standard Model containing an additional isosinglet charge (-1/3) quark, which leads to a deviation from CKM unitar∗ V ) ∼ O (10−2 ), ity. It is shown that even for small mixing ratios zqb /(Vtq tb the contribution of the tree-level Z −FCNC appearing in the model should change the rates and asymmetries significantly. Especially the CP asymmetry, ACP (B → Xs ℓ+ ℓ− ), can be enhanced to be few percents, while in the standard model the size is less than O(10−3 ). On the other hand, ACP (B → Xd ℓ+ ℓ− ) is not altered so much. Constraints for the mixing ratios 0 −B ¯ 0 mixing are extracted from the experiments of B → Xs γ for q = s and Bd d ¯ for q = d under a natural assumption that the couplings of the tree-level Zf f are almost unity, i.e. zαα ∼ 1.
Twist-3 distribution amplitudes of scalar mesons from QCD sum rules
that the second Gegenbauer coefficients of scalar density twist-3 distribution amplitudes for K0∗ and f0 mesons are quite close to that for a0, which indicates that the SU(3) symmetry breaking effect is tiny here. However, this effect could not be neglected for the forth Gegenbauer coefficients
found
that
the
masses
for
isospin
I=1,
1 2
scalar
mesons
are
around
1.27
∼
1.41
GeV
and
1.44
∼
1.56
GeV respectively, while the mass for isospin state composed of s¯s is 1.62 ∼ 1.73 GeV.
2
moments for the above three scalar mesons in section III. The last section is devoted to our conclusions.
II. FORMULATION
In the valence quark model, there are two twist-3 light-cone distribution amplitudes for scalar mesons which are defined as [6]
On the B_{c} leptonic decay constant
Sameer M. Ikhdair∗
∗ Department
of Electrical and Electronic Engineerinsity, Nicosia, North Cyprus, Mersin-10, Turkey PACS NUMBER(S): 12.39.Pn, 12.39.Jh, 13.25.Jx, 14.40.-n
arXiv:hep-ph/0504107v1 14 Apr 2005
(February 2, 2008)
Typeset using REVTEX
∗ Electronic
address (internet): sameer@.tr
1
Abstract
We give a review and present a comprehensive calculations for the leptonic constant fBc of the low-lying pseudoscalar and vector states of Bc -meson in the framework of static and QCD-motivated nonrelativistic potential models taking into account the one-loop and two-loop QCD corrections in the short distance coefficient that governs the leptonic constant of Bc quarkonium system. Further, we use the scaling relation to predict the leptonic constant of the nS-states of the bc system. Our results are compared with other models to gauge the reliability of the predictions and point out differences.
Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}
form factor.
PACS index : 12.15.-y, 13.20.-v, 13.25.Hw, 14.40.Nd, 14.65.Fy Keywards : Factorization, Non-leptonic Decays, Decay Constant, Penguin Effects
∗ experimentally from leptonic B and Ds decays. For instance, determine fB , fBs fDs and fDs
+ the decay rate for Ds is given by [1]
+ Γ(Ds
m2 G2 2 2 l 1 − m M → ℓ ν ) = F fD D s 2 8π s ℓ MD s
1/2
(4)
.
(5)
In the zero lepton-mass limit, 0 ≤ q 2 ≤ (mB − mD )2 .
2
For the q 2 dependence of the form factors, Wirbel et al. [8] assumed a simple pole formula for both F1 (q 2 ) and F0 (q 2 ) (we designate this scenario ’pole/pole’): q2 F1 (q ) = F1 (0) /(1 − 2 ), mF1
∗ amount to about 11 % for B → DDs and 5 % for B → DDs , which have been mentioned in
Heavy-light decay constants with three dynamical flavors
light quark masses, the chiral extrapolation was
a major source of systematic error. Here, chang-
ing from linear to quadratic chiral fits of decay constants changes the results by <∼2%.
quette following Refs. [3,11], and then define the 1-loop coefficient ζA by ZAtad = 1 + αPs ζA.
(2) Fix the scale from mρ on each set, with valence quarks extrapolated to physical values.
∞/∞
8.40 0.09 417 200
0.031/0.031 7.18 0.09 162 40
0.0124/0.031 7.11 0.09 25 –
tors for 5 light and 5 heavy masses. This gives
good control over both chiral and heavy quark
We present preliminary results for the heavy-light leptonic decay constants in the presence of three light dynamical flavors. We generate dynamical configurations with improved staggered and gauge actions and analyze them for heavy-light physics with tadpole improved clover valence quarks. When the scale is set by mρ, we find an increase of ≈ 23% in fB with three dynamical flavors over the quenched case. Discretization errors appear to be small (<∼3%) in the quenched case but have not yet been measured in the dynamical case.
Charmless Three-body Decays of B Mesons
Abstract
An exploratory study of charmless 3-body decays of B mesons is presented using a simple model based on the framework of the factorization approach. The nonresonant contributions arising from B → P1 P2 transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form e−αNR pB ·(pi +pj ) so that the HMChPT results are recovered in the soft meson limit pi , pj → 0. In addition, we have identified another large source of the nonresonant signal in the matrix elements 0 of scalar densities, e.g. KK |s ¯s|0 , which can be constrained from the decay B → KS KS KS or B − → K − KS KS . The intermediate vector meson contributions to 3-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are: (i) All KKK modes are dominated by the nonresonant background. The predicted branching ratios of K + K − KS (L) , K + K − K − and K − KS KS modes are consistent with the data within errors. (ii) Although the penguin-dominated B 0 → K + K − KS decay is subject to a potentially significant tree pollution, its effective sin 2β is very similar to that of the KS KS KS mode. However, direct CP asymmetry of the former, being of order −4%, is more prominent than the latter. (iii) For 0 B → Kππ decays, we found sizable nonresonant contributions in K − π + π − and K π + π − modes, in agreement with the Belle measurements but larger than the BaBar result. (iv) Time-dependent CP asymmetries in KS π 0 π 0 , a purely CP -even state, and KS π + π − , an admixture of CP -even and CP -odd components, are studied. (v) The π + π − π 0 mode is found to have a rate larger than π + π − π − even though the former involves a π 0 in the final state. They are both dominated by resonant ρ contributions. (vi) We have computed the resonant contributions to 3-body decays and determined the rates for the quasi-two-body decays B → V P and B → SP . The predicted ρπ, f0 (980)K and f0 (980)π rates are in agreement with the data, while the calculated φK, K ∗ π, ρK ∗ (1430)π are in general too small compared to experiment. (vii) Sizable direct CP asymmetry and K0 is found in K + K − K − and K + K − π − modes.
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p)
(4)
=
3 gH gH′ 4 π2
I+(p2, p′2, q2)(p + p′)µ + I−(p2, p′2, q2)qµ
= f+(q2)(p + p′)µ + f−(q2)qµ .
Here, q = p − p′, Si(k) = 1/(mi− k) is the propagator of the quark i with mass mi.
DSF-99-35
Decay Constants and Semileptonic Form Factors of
Pseudoscalar Mesons1
M.A. Ivanova and P. Santorellib
a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
2 Our model
Our starting point is the effective Lagrangian describing the coupling between hadrons and quarks. The
Lint(x) = gH H(x) dx1 dx2ΦH (x; x1, x2)q¯(x1)ΓH λH q(x2)
Now, to give an example of the hadronic part of invariant amplitudes, we will evaluate the form fpears in the semileptonic decays of a pseudoscalar meson into another one, H → H′ℓν. The
The coupling constant gH is given by the derivative of the meson mass operator ΠH by the compositeness condition [8]:
ZH
=
1
−
3gH2 4π2
Π′H (m2H )
=
0.
(2)
It is worth noticing that, due to the absence of confinement, the sum of constituent quark masses
arXiv:hep-ph/9910434v1 21 Oct 1999
1 Introduction
Semileptonic decays of pseudoscalar mesons allow to evaluate the elements of the Cabibbo–Kobayashi– Maskawa (CKM) matrix, which are fundamental parameters of the Standard Model. The decay K → πeν provides the most accurate determination of Vus, the semileptonic decays of D and B mesons, D → K(K∗)lν, B → D(D∗)lν and B → π(ρ)lν, can be used to determine |Vcs|, |Vcb| and |Vub|, respectively. This program can be performed if the non-perturbative QCD effects, which are parameterized by the form factors, are known. Up to now, these form factors cannot be evaluated from first principles, thus models, more or less connected with QCD, are usually considered for this purpose. Here we discuss a relativistic quark model [1], previously used to study the baryon form factors [2].
invariant amplitude can be written as:
A(H(p) → H′(p′) e ν) = GF√Vq1q2 2
ℓ¯γµ(1 − γ5) ν
MHµ H′ (p, p′),
(3)
where GF is the Fermi constant, and, in our model, MHµ H′ (p, p′)
−
k2]
.
(5)
They can be evaluated using the standard Feynman α-representation and the integral Cauchy represen-
This model is based on an effective Lagrangian describing the coupling of mesons with their constituent quarks. The physical processes are described by the one-loop quark diagrams and meson-quark vertices related to the Bethe-Salpeter amplitudes. In principle, the vertex functions and quark propagators should be given by the Bethe-Salpeter and Dyson-Schwinger equations, respectively. This kind of analysis is provided by the Dyson-Schwinger Equation (DSE) studies [3] and an unified description of light and heavy meson observables was carried out in [4, 5]. Here, instead, we use free propagators for constituent quarks and consider a Gaussian vertex function as Bethe-Salpeter confining function. The adjustable parameters, the widths of Bethe-Salpeter amplitudes in momentum space, and the constituent quark masses, are determined from the best fit of available experimental data and some lattice simulations. Our results are in good agreement with experimental data and other approaches. We also reproduce the spin-flavor symmetry relations and scaling for leptonic decay constants and semileptonic form factors in the heavy-quark limit [6].
should be larger than the mass of the corresponding meson otherwise, imaginary parts in physical quan-
tities appear. This allows us to consider low-lying pseudoscalar mesons only.
b Dipartimento di Scienze Fisiche, Universit`a “Federico II” di Napoli, Napoli, Italy
and INFN Sezione di Napoli
Abstract
A relativistic constituent quark model is adopted to give an unified description of the leptonic and semileptonic decays of pseudoscalar mesons (π, K, D, Ds, B, Bs). The calculated leptonic decay constants and form factors are found to be in good agreement with available experimental data and the results of other approaches. Eventually, the model is found to reproduce the scaling behaviours of spin-flavor symmetry in the heavy-quark limit.
Vq1q2 is the corresponding is given by:
Cabibbo–Kobayashi–Maskawa
matrix
element,
MHµ H′ (p, p′)
=
3 gH gH′ 4 π2
d4k 4π2i
φH
(−k2
)φH′
(−k
2)tr
γ5S3( k)γ5S2( k+