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.
Dielectric_constant
Properties of microwave dielectricsIn microwave technique, a large variety of materials are used, includingdielectrics. The measurement of the dielectric properties of microwave materials isnot only helpful in understanding the structural information and studying themicrowave characteristics but also designing microwave devices. The aim of thisexperiment is to understand the property of resonant cavities, and to learn theprinciple and method for measuring the complex dielectric constant of materials.1.PrincipleAccording to the theory of electromagnetic field, a dielectric in alternatingelectric field will be rotational polarized, and it relaxes during the polarization. So itsdielectric constant is a complex number, which can be expressed as follows:"'εεεj r -=where ε’ and ε” are the real part and imagine part of the complex dielectric constant.Due to relaxation of the dielectric in alternating electric field, the electricdisplacement vector in the dielectric has a phase hysteresis of an angel δ in respectiveto that outside the dielectric. The phase angle δ can be calculated from the followingequation:.'"εεδ=tg tgδ is called the dissipation factor or tangent-of-dissipation-angle, because the energydissipation in dielectrics is proportional to tgδ.Selecting a TE 10n rectangle resonant cavity with n an old number, generally takento be 3, and make a small hole at the center (x=a /2, z=l /2), where the electric field ismaximal, and magnetic field is minimal. The dielectric to be measured is processed tobe a long and thin rod, and inserted into thecavity through the small hole.As the dielectric rod is very small, thedistribution of the electromagnetic field in thewhole cavity is kept almost unchanged exceptnear the dielectric rod. Therefore the dielectricrod can be regarded as a micro disturbance tothe field distribution of the cavity. Accordingto the theory of micro disturbance of resonant cavity, one obtains [1]:ss V V f f f 000211'-+=εss V V Q Q 001141"⎪⎪⎭⎫ ⎝⎛-=εwhere V 0 and V s are the volumeof the cavity and that of thedielectric rod, respectively. f oand Q 0 are the resonantfrequency and quality factor ofthe cavity without the dielectric,and f s and Q s are that with thedielectric. The quality factor fora resonant cavity can becalculatedfromfollowing equation:hf f Q ∆=0where Δf h is the half-power frequency width of the resonant curve (Fig.1). Usingreflecting cavity as the sample cavity, then the complex dielectric constant canbeFig.1 The sample cavityFig.2 The resonant curve of a reflection cavitycalculated by measuring the resonant frequency f0 and half-power frequency width Δf h.2. Apparatus and InstrumentationThe experimental apparatus for the measurement of the complex dielectric constant is shown in Fig.3. The klystron working at sawtooth-wave-modulation mode,Fig.3 The experimental setup for dielectric constant measurementoutput a frequency-modulated microwave. The isolator is a single direction device. Microwave power is only transmitted in the direction of the arrow, and waves reflected back towards the klystron are attenuated. The function of the attenuator is to adjust the microwave power. The frequency meter is a cavity, which absorbs microwave in a narrow band. If the band absorbed by the meter, is within the frequency range generated by the klystron, a dip will appear in the oscilloscope displayer of microwave power vs. frequency. The circulator is a kind of power divider. It divides the input power into two parts. One part is transmitted in the sample cavity, and the other in the isolator. The microwave power through the isolator is detected by a crystal detector, and the voltage pulse proportional to the power is then observed in the oscilloscope.3. Contents of the experiment3.1 Observations of the vibration modulus of the klystron with oscilloscope(1)Make the sample cavity non-resonant, through adjusting the frequency ofmicrowave from the klystron by mechanical tuning;(2)Adjust the reflector voltage, and observe the vibration modulus of the klystron onthe oscilloscope.3.2 Observations of the resonant curves of the reflecting cavity(1)Make the klystron to work at the best vibration modulus by adjusting the reflectorvoltage;(2)Make the sample cavity to resonate, by changing the microwave frequencythrough mechanical tuning;(3)Observe the resonant curve.3.3 Measurements of the dependences of frequency on time(1)Measure the f ~t curve;(2)Calculate the frequency scale coefficient K at the vicinity of f 0, which isdefined from the equation:.tf K δδ=3.4 Measurements of the complex dielectric constants εr(1)Measure the resonant frequency f 0 and half-power frequency width Δf h0before the insertion of the sample;(2)Measure the resonant frequency f s and half-power frequency width Δf hs afterthe insertion of the sample;(3)Measure the sample volume V s and cavity volume V 0 in mm 3;(4)Calculate the complex dielectric constant.4. Problems(1) What are the characteristic parameters of a resonant cavity?(2) Which parameter of the resonant cavity determines mainly the real part of the complex dielectric constant?References1. 林木欣，能予莹，高长连，朱文钧，刘战存，冯显灿，等，近代物理实验教程，科学出版社，北京，20002．赫崇骏，韩永宁，袁乃昌，何建国，微波电路，国防科技大学出版社，长沙，1999Vocabularycrystal detector 晶体检波器circulator 环行器dissipation factor损耗因子electric displacement vector 电位移矢量energy dissipation能量损耗klystron 速调管mechanical tuning 机械调谐microwave 微波microwave dielectric 微波介质microammeter微安计non-resonant失谐half-power frequency width 半功率频宽half-power frequency 半功率频率hysterisis 滞后oscilloscope 示波器power divider 功率分配器quality factor 品质因素resonant cavity 谐振腔resonant frequency 谐振频率reflecting cavity 反射式腔reflector voltage 反射极电压resonant curve 谐振曲线sawtooth wave锯齿波sample cavity 样品腔tangent of dissipation angle损耗角正切vibration modulus 振荡模。
Principles of Plasma Discharges and Materials Processing9
CHAPTER8MOLECULAR COLLISIONS8.1INTRODUCTIONBasic concepts of gas-phase collisions were introduced in Chapter3,where we described only those processes needed to model the simplest noble gas discharges: electron–atom ionization,excitation,and elastic scattering;and ion–atom elastic scattering and resonant charge transfer.In this chapter we introduce other collisional processes that are central to the description of chemically reactive discharges.These include the dissociation of molecules,the generation and destruction of negative ions,and gas-phase chemical reactions.Whereas the cross sections have been measured reasonably well for the noble gases,with measurements in reasonable agreement with theory,this is not the case for collisions in molecular gases.Hundreds of potentially significant collisional reactions must be examined in simple diatomic gas discharges such as oxygen.For feedstocks such as CF4/O2,SiH4/O2,etc.,the complexity can be overwhelming.Furthermore,even when the significant processes have been identified,most of the cross sections have been neither measured nor calculated. Hence,one must often rely on estimates based on semiempirical or semiclassical methods,or on measurements made on molecules analogous to those of interest. As might be expected,data are most readily available for simple diatomic and polyatomic gases.Principles of Plasma Discharges and Materials Processing,by M.A.Lieberman and A.J.Lichtenberg. ISBN0-471-72001-1Copyright#2005John Wiley&Sons,Inc.235236MOLECULAR COLLISIONS8.2MOLECULAR STRUCTUREThe energy levels for the electronic states of a single atom were described in Chapter3.The energy levels of molecules are more complicated for two reasons. First,molecules have additional vibrational and rotational degrees of freedom due to the motions of their nuclei,with corresponding quantized energies E v and E J. Second,the energy E e of each electronic state depends on the instantaneous con-figuration of the nuclei.For a diatomic molecule,E e depends on a single coordinate R,the spacing between the two nuclei.Since the nuclear motions are slow compared to the electronic motions,the electronic state can be determined for anyfixed spacing.We can therefore represent each quantized electronic level for a frozen set of nuclear positions as a graph of E e versus R,as shown in Figure8.1.For a mole-cule to be stable,the ground(minimum energy)electronic state must have a minimum at some value R1corresponding to the mean intermolecular separation (curve1).In this case,energy must be supplied in order to separate the atoms (R!1).An excited electronic state can either have a minimum( R2for curve2) or not(curve3).Note that R2and R1do not generally coincide.As for atoms, excited states may be short lived(unstable to electric dipole radiation)or may be metastable.Various electronic levels may tend to the same energy in the unbound (R!1)limit. Array FIGURE8.1.Potential energy curves for the electronic states of a diatomic molecule.For diatomic molecules,the electronic states are specifiedfirst by the component (in units of hÀ)L of the total orbital angular momentum along the internuclear axis, with the symbols S,P,D,and F corresponding to L¼0,+1,+2,and+3,in analogy with atomic nomenclature.All but the S states are doubly degenerate in L.For S states,þandÀsuperscripts are often used to denote whether the wave function is symmetric or antisymmetric with respect to reflection at any plane through the internuclear axis.The total electron spin angular momentum S (in units of hÀ)is also specified,with the multiplicity2Sþ1written as a prefixed superscript,as for atomic states.Finally,for homonuclear molecules(H2,N2,O2, etc.)the subscripts g or u are written to denote whether the wave function is sym-metric or antisymmetric with respect to interchange of the nuclei.In this notation, the ground states of H2and N2are both singlets,1Sþg,and that of O2is a triplet,3SÀg .For polyatomic molecules,the electronic energy levels depend on more thanone nuclear coordinate,so Figure8.1must be generalized.Furthermore,since there is generally no axis of symmetry,the states cannot be characterized by the quantum number L,and other naming conventions are used.Such states are often specified empirically through characterization of measured optical emission spectra.Typical spacings of low-lying electronic energy levels range from a few to tens of volts,as for atoms.Vibrational and Rotational MotionsUnfreezing the nuclear vibrational and rotational motions leads to additional quan-tized structure on smaller energy scales,as illustrated in Figure8.2.The simplest (harmonic oscillator)model for the vibration of diatomic molecules leads to equally spaced quantized,nondegenerate energy levelse E v¼hÀv vib vþ1 2(8:2:1)where v¼0,1,2,...is the vibrational quantum number and v vib is the linearized vibration frequency.Fitting a quadratic functione E v¼12k vib(RÀ R)2(8:2:2)near the minimum of a stable energy level curve such as those shown in Figure8.1, we can estimatev vib%k vibm Rmol1=2(8:2:3)where k vib is the“spring constant”and m Rmol is the reduced mass of the AB molecule.The spacing hÀv vib between vibrational energy levels for a low-lying8.2MOLECULAR STRUCTURE237stable electronic state is typically a few tenths of a volt.Hence for molecules in equi-librium at room temperature (0.026V),only the v ¼0level is significantly popula-ted.However,collisional processes can excite strongly nonequilibrium vibrational energy levels.We indicate by the short horizontal line segments in Figure 8.1a few of the vibrational energy levels for the stable electronic states.The length of each segment gives the range of classically allowed vibrational motions.Note that even the ground state (v ¼0)has a finite width D R 1as shown,because from(8.2.1),the v ¼0state has a nonzero vibrational energy 1h Àv vib .The actual separ-ation D R about Rfor the ground state has a Gaussian distribution,and tends toward a distribution peaked at the classical turning points for the vibrational motion as v !1.The vibrational motion becomes anharmonic and the level spa-cings tend to zero as the unbound vibrational energy is approached (E v !D E 1).FIGURE 8.2.Vibrational and rotational levels of two electronic states A and B of a molecule;the three double arrows indicate examples of transitions in the pure rotation spectrum,the rotation–vibration spectrum,and the electronic spectrum (after Herzberg,1971).238MOLECULAR COLLISIONSFor E v.D E1,the vibrational states form a continuum,corresponding to unbound classical motion of the nuclei(breakup of the molecule).For a polyatomic molecule there are many degrees of freedom for vibrational motion,leading to a very compli-cated structure for the vibrational levels.The simplest(dumbbell)model for the rotation of diatomic molecules leads to the nonuniform quantized energy levelse E J¼hÀ22I molJ(Jþ1)(8:2:4)where I mol¼m Rmol R2is the moment of inertia and J¼0,1,2,...is the rotational quantum number.The levels are degenerate,with2Jþ1states for the J th level. The spacing between rotational levels increases with J(see Figure8.2).The spacing between the lowest(J¼0to J¼1)levels typically corresponds to an energy of0.001–0.01V;hence,many low-lying levels are populated in thermal equilibrium at room temperature.Optical EmissionAn excited molecular state can decay to a lower energy state by emission of a photon or by breakup of the molecule.As shown in Figure8.2,the radiation can be emitted by a transition between electronic levels,between vibrational levels of the same electronic state,or between rotational levels of the same electronic and vibrational state;the radiation typically lies within the optical,infrared,or microwave frequency range,respectively.Electric dipole radiation is the strongest mechanism for photon emission,having typical transition times of t rad 10À9s,as obtained in (3.4.13).The selection rules for electric dipole radiation areDL¼0,+1(8:2:5a)D S¼0(8:2:5b) In addition,for transitions between S states the only allowed transitions areSþÀ!Sþand SÀÀ!SÀ(8:2:6) and for homonuclear molecules,the only allowed transitions aregÀ!u and uÀ!g(8:2:7) Hence homonuclear diatomic molecules do not have a pure vibrational or rotational spectrum.Radiative transitions between electronic levels having many different vibrational and rotational initial andfinal states give rise to a structure of emission and absorption bands within which a set of closely spaced frequencies appear.These give rise to characteristic molecular emission and absorption bands when observed8.2MOLECULAR STRUCTURE239using low-resolution optical spectrometers.As for atoms,metastable molecular states having no electric dipole transitions to lower levels also exist.These have life-times much exceeding10À6s;they can give rise to weak optical band structures due to magnetic dipole or electric quadrupole radiation.Electric dipole radiation between vibrational levels of the same electronic state is permitted for molecules having permanent dipole moments.In the harmonic oscillator approximation,the selection rule is D v¼+1;weaker transitions D v¼+2,+3,...are permitted for anharmonic vibrational motion.The preceding description of molecular structure applies to molecules having arbi-trary electronic charge.This includes neutral molecules AB,positive molecular ions ABþ,AB2þ,etc.and negative molecular ions ABÀ.The potential energy curves for the various electronic states,regardless of molecular charge,are commonly plotted on the same diagram.Figures8.3and8.4give these for some important electronic statesof HÀ2,H2,and Hþ2,and of OÀ2,O2,and Oþ2,respectively.Examples of both attractive(having a potential energy minimum)and repulsive(having no minimum)states can be seen.The vibrational levels are labeled with the quantum number v for the attrac-tive levels.The ground states of both Hþ2and Oþ2are attractive;hence these molecular ions are stable against autodissociation(ABþ!AþBþor AþþB).Similarly,the ground states of H2and O2are attractive and lie below those of Hþ2and Oþ2;hence they are stable against autodissociation and autoionization(AB!ABþþe).For some molecules,for example,diatomic argon,the ABþion is stable but the AB neutral is not stable.For all molecules,the AB ground state lies below the ABþground state and is stable against autoionization.Excited states can be attractive or repulsive.A few of the attractive states may be metastable;some examples are the 3P u state of H2and the1D g,1Sþgand3D u states of O2.Negative IonsRecall from Section7.2that many neutral atoms have a positive electron affinity E aff;that is,the reactionAþeÀ!AÀis exothermic with energy E aff(in volts).If E aff is negative,then AÀis unstable to autodetachment,AÀ!Aþe.A similar phenomenon is found for negative molecular ions.A stable ABÀion exists if its ground(lowest energy)state has a potential minimum that lies below the ground state of AB.This is generally true only for strongly electronegative gases having large electron affinities,such as O2 (E aff%1:463V for O atoms)and the halogens(E aff.3V for the atoms).For example,Figure8.4shows that the2P g ground state of OÀ2is stable,with E aff% 0:43V for O2.For weakly electronegative or for electropositive gases,the minimum of the ground state of ABÀgenerally lies above the ground state of AB,and ABÀis unstable to autodetachment.An example is hydrogen,which is weakly electronegative(E aff%0:754V for H atoms).Figure8.3shows that the2Sþu ground state of HÀ2is unstable,although the HÀion itself is stable.In an elec-tropositive gas such as N2(E aff.0),both NÀ2and NÀare unstable. 240MOLECULAR COLLISIONS8.3ELECTRON COLLISIONS WITH MOLECULESThe interaction time for the collision of a typical (1–10V)electron with a molecule is short,t c 2a 0=v e 10À16–10À15s,compared to the typical time for a molecule to vibrate,t vib 10À14–10À13s.Hence for electron collisional excitation of a mole-cule to an excited electronic state,the new vibrational (and rotational)state canbeFIGURE 8.3.Potential energy curves for H À2,H 2,and H þ2.(From Jeffery I.Steinfeld,Molecules and Radiation:An Introduction to Modern Molecular Spectroscopy ,2d ed.#MIT Press,1985.)8.3ELECTRON COLLISIONS WITH MOLECULES 241FIGURE 8.4.Potential energy curves for O À2,O 2,and O þ2.(From Jeffery I.Steinfeld,Molecules and Radiation:An Introduction to Modern Molecular Spectroscopy ,2d ed.#MIT Press,1985.)242MOLECULAR COLLISIONS8.3ELECTRON COLLISIONS WITH MOLECULES243 determined by freezing the nuclear motions during the collision.This is known as the Franck–Condon principle and is illustrated in Figure8.1by the vertical line a,showing the collisional excitation atfixed R to a high quantum number bound vibrational state and by the vertical line b,showing excitation atfixed R to a vibra-tionally unbound state,in which breakup of the molecule is energetically permitted. Since the typical transition time for electric dipole radiation(t rad 10À9–10À8s)is long compared to the dissociation( vibrational)time t diss,excitation to an excited state will generally lead to dissociation when it is energetically permitted.Finally, we note that the time between collisions t c)t rad in typical low-pressure processing discharges.Summarizing the ordering of timescales for electron–molecule collisions,we havet at t c(t vib t diss(t rad(t cDissociationElectron impact dissociation,eþABÀ!AþBþeof feedstock gases plays a central role in the chemistry of low-pressure reactive discharges.The variety of possible dissociation processes is illustrated in Figure8.5.In collisions a or a0,the v¼0ground state of AB is excited to a repulsive state of AB.The required threshold energy E thr is E a for collision a and E a0for Array FIGURE8.5.Illustrating the variety of dissociation processes for electron collisions with molecules.collision a0,and it leads to an energy after dissociation lying between E aÀE diss and E a0ÀE diss that is shared among the dissociation products(here,A and B). Typically,E aÀE diss few volts;consequently,hot neutral fragments are typically generated by dissociation processes.If these hot fragments hit the substrate surface, they can profoundly affect the process chemistry.In collision b,the ground state AB is excited to an attractive state of AB at an energy E b that exceeds the binding energy E diss of the AB molecule,resulting in dissociation of AB with frag-ment energy E bÀE diss.In collision b0,the excitation energy E b0¼E diss,and the fragments have low energies;hence this process creates fragments having energies ranging from essentially thermal energies up to E bÀE diss few volts.In collision c,the AB atom is excited to the bound excited state ABÃ(labeled5),which sub-sequently radiates to the unbound AB state(labeled3),which then dissociates.The threshold energy required is large,and the fragments are hot.Collision c can also lead to dissociation of an excited state by a radiationless transfer from state5to state4near the point where the two states cross:ABÃðboundÞÀ!ABÃðunboundÞÀ!AþBÃThe fragments can be both hot and in excited states.We discuss such radiationless electronic transitions in the next section.This phenomenon is known as predisso-ciation.Finally,a collision(not labeled in thefigure)to state4can lead to dis-sociation of ABÃ,again resulting in hot excited fragments.The process of electron impact excitation of a molecule is similar to that of an atom,and,consequently,the cross sections have a similar form.A simple classical estimate of the dissociation cross section for a level having excitation energy U1can be found by requiring that an incident electron having energy W transfer an energy W L lying between U1and U2to a valence electron.Here,U2is the energy of the next higher level.Then integrating the differential cross section d s[given in(3.4.20)and repeated here],d s¼pe24021Wd W LW2L(3:4:20)over W L,we obtains diss¼0W,U1pe24pe021W1U1À1WU1,W,U2pe24021W1U1À1U2W.U28>>>>>><>>>>>>:(8:3:1)244MOLECULAR COLLISIONSLetting U2ÀU1(U1and introducing voltage units W¼e E,U1¼e E1and U2¼e E2,we haves diss¼0E,E1s0EÀE11E1,E,E2s0E2ÀE1EE.E28>>>><>>>>:(8:3:2)wheres0¼pe4pe0E12(8:3:3)We see that the dissociation cross section rises linearly from the threshold energy E thr%E1to a maximum value s0(E2ÀE1)=E thr at E2and then falls off as1=E. Actually,E1and E2can depend on the nuclear separation R.In this case,(8.3.2) should be averaged over the range of R s corresponding to the ground-state vibrational energy,leading to a broadened dependence of the average cross section on energy E.The maximum cross section is typically of order10À15cm2. Typical rate constants for a single dissociation process with E thr&T e have an Arrhenius formK diss/K diss0expÀE thr T e(8:3:4)where K diss0 10À7cm3=s.However,in some cases E thr.T e.For excitation to an attractive state,an appropriate average over the fraction of the ground-state vibration that leads to dissociation must be taken.Dissociative IonizationIn addition to normal ionization,eþABÀ!ABþþ2eelectron–molecule collisions can lead to dissociative ionizationeþABÀ!AþBþþ2eThese processes,common for polyatomic molecules,are illustrated in Figure8.6.In collision a having threshold energy E iz,the molecular ion ABþis formed.Collisionsb andc occur at higher threshold energies E diz and result in dissociative ionization,8.3ELECTRON COLLISIONS WITH MOLECULES245leading to the formation of fast,positively charged ions and neutrals.These cross sections have a similar form to the Thompson ionization cross section for atoms.Dissociative RecombinationThe electron collision,e þAB þÀ!A þB Ãillustrated as d and d 0in Figure 8.6,destroys an electron–ion pair and leads to the production of fast excited neutral fragments.Since the electron is captured,it is not available to carry away a part of the reaction energy.Consequently,the collision cross section has a resonant character,falling to very low values for E ,E d and E .E d 0.However,a large number of excited states A Ãand B Ãhaving increasing principal quantum numbers n and energies can be among the reaction products.Consequently,the rate constants can be large,of order 10À7–10À6cm 3=s.Dissocia-tive recombination to the ground states of A and B cannot occur because the potential energy curve for AB þis always greater than the potential energycurveFIGURE 8.6.Illustration of dissociative ionization and dissociative recombination for electron collisions with molecules.246MOLECULAR COLLISIONSfor the repulsive state of AB.Two-body recombination for atomic ions or for mol-ecular ions that do not subsequently dissociate can only occur with emission of a photon:eþAþÀ!Aþh n:As shown in Section9.2,the rate constants are typically three tofive orders of magnitude lower than for dissociative recombination.Example of HydrogenThe example of H2illustrates some of the inelastic electron collision phenomena we have discussed.In order of increasing electron impact energy,at a threshold energy of 8:8V,there is excitation to the repulsive3Sþu state followed by dissociation into two fast H fragments carrying 2:2V/atom.At11.5V,the1Sþu bound state is excited,with subsequent electric dipole radiation in the ultraviolet region to the1Sþg ground state.At11.8V,there is excitation to the3Sþg bound state,followedby electric dipole radiation to the3Sþu repulsive state,followed by dissociation with 2:2V/atom.At12.6V,the1P u bound state is excited,with UV emission tothe ground state.At15.4V,the2Sþg ground state of Hþ2is excited,leading to the pro-duction of Hþ2ions.At28V,excitation of the repulsive2Sþu state of Hþ2leads to thedissociative ionization of H2,with 5V each for the H and Hþfragments.Dissociative Electron AttachmentThe processes,eþABÀ!AþBÀproduce negative ion fragments as well as neutrals.They are important in discharges containing atoms having positive electron affinities,not only because of the pro-duction of negative ions,but because the threshold energy for production of negative ion fragments is usually lower than for pure dissociation processes.A variety of pro-cesses are possible,as shown in Figure8.7.Since the impacting electron is captured and is not available to carry excess collision energy away,dissociative attachment is a resonant process that is important only within a narrow energy range.The maximum cross sections are generally much smaller than the hard-sphere cross section of the molecule.Attachment generally proceeds by collisional excitation from the ground AB state to a repulsive ABÀstate,which subsequently either auto-detaches or dissociates.The attachment cross section is determined by the balance between these processes.For most molecules,the dissociation energy E diss of AB is greater than the electron affinity E affB of B,leading to the potential energy curves shown in Figure8.7a.In this case,the cross section is large only for impact energies lying between a minimum value E thr,for collision a,and a maximum value E0thr for8.3ELECTRON COLLISIONS WITH MOLECULES247FIGURE 8.7.Illustration of a variety of electron attachment processes for electron collisions with molecules:(a )capture into a repulsive state;(b )capture into an attractive state;(c )capture of slow electrons into a repulsive state;(d )polar dissociation.248MOLECULAR COLLISIONScollision a 0.The fragments are hot,having energies lying between minimum and maximum values E min ¼E thr þE affB ÀE diss and E max ¼E 0thr þE af fB ÀE diss .Since the AB Àstate lies above the AB state for R ,R x ,autodetachment can occur as the mol-ecules begin to separate:AB À!AB þe.Hence the cross section for production of negative ions can be much smaller than that for excitation of the AB Àrepulsive state.As a crude estimate,for the same energy,the autodetachment rate is ffiffiffiffiffiffiffiffiffiffiffiffiffiM R =m p 100times the dissociation rate of the repulsive AB Àmolecule,where M R is the reduced mass.Hence only one out of 100excitations lead to dissociative attachment.Excitation to the AB Àbound state can also lead to dissociative attachment,as shown in Figure 8.7b .Here the cross section is significant only for E thr ,E ,E 0thr ,but the fragments can have low energies,with a minimum energy of zero and a maximum energy of E 0thr þE affB ÀE diss .Collision b,e þAB À!AB ÀÃdoes not lead to production of AB Àions because energy and momentum are not gen-erally conserved when two bodies collide elastically to form one body (see Problem3.12).Hence the excited AB ÀÃion separates,AB ÀÃÀ!e þABunless vibrational radiation or collision with a third body carries off the excess energy.These processes are both slow in low-pressure discharges (see Section 9.2).At high pressures (say,atmospheric),three-body attachment to form AB Àcan be very important.For a few molecules,such as some halogens,the electron affinity of the atom exceeds the dissociation energy of the neutral molecule,leading to the potential energy curves shown in Figure 8.7c .In this case the range of electron impact ener-gies E for excitation of the AB Àrepulsive state includes E ¼0.Consequently,there is no threshold energy,and very slow electrons can produce dissociative attachment,resulting in hot neutral and negative ion fragments.The range of R s over which auto-detachment can occur is small;hence the maximum cross sections for dissociative attachment can be as high as 10À16cm 2.A simple classical estimate of electron capture can be made using the differential scattering cross section for energy loss (3.4.20),in a manner similar to that done for dissociation.For electron capture to an energy level E 1that is unstable to autode-tachment,and with the additional constraint for capture that the incident electron energy lie within E 1and E 2¼E 1þD E ,where D E is a small energy difference characteristic of the dissociative attachment timescale,we obtain,in place of (8.3.2),s att¼0E ,E 1s 0E ÀE 1E 1E 1,E ,E 20E .E 28>><>>:(8:3:5)8.3ELECTRON COLLISIONS WITH MOLECULES 249wheres 0%p m M R 1=2e 4pe 0E 1 2(8:3:6)The factor of (m =M R )1=2roughly gives the fraction of excited states that do not auto-detach.We see that the dissociative attachment cross section rises linearly at E 1to a maximum value s 0D E =E 1and then falls abruptly to zero.As for dissociation,E 1can depend strongly on the nuclear separation R ,and (8.3.5)must be averaged over the range of E 1s corresponding to the ground state vibrational motion;e.g.,from E thr to E 0thr in Figure 8.7a .Because generally D E (E 0thr ÀE thr ,we can write (8.3.5)in the forms att %p m M R 1=2e 4pe 0 2(D E )22E 1d (E ÀE 1)(8:3:7)where d is the Dirac delta ing (8.3.7),the average over the vibrational motion can be performed,leading to a cross section that is strongly peaked lying between E thr and E 0thr .We leave the details of the calculation to a problem.Polar DissociationThe process,e þAB À!A þþB Àþeproduces negative ions without electron capture.As shown in Figure 8.7d ,the process proceeds by excitation of a polar state A þand B Àof AB Ãthat has a separ-ated atom limit of A þand B À.Hence at large R ,this state lies above the A þB ground state by the difference between the ionization potential of A and the electron affinity of B.The polar state is weakly bound at large R by the Coulomb attraction force,but is repulsive at small R .The maximum cross section and the dependence of the cross section on electron impact energy are similar to that of pure dissociation.The threshold energy E thr for polar dissociation is generally large.The measured cross section for negative ion production by electron impact in O 2is shown in Figure 8.8.The sharp peak at 6.5V is due to dissociative attachment.The variation of the cross section with energy is typical of a resonant capture process.The maximum cross section of 10À18cm 2is quite low because autode-tachment from the repulsive O À2state is strong,inhibiting dissociative attachment.The second gradual maximum near 35V is due to polar dissociation;the variation of the cross section with energy is typical of a nonresonant process.250MOLECULAR COLLISIONS。
Measurement of the absolute branching ratios for semileptonic K+- decays with the KLOE dete
a r X i v :0712.3841v 3 [h e p -e x ] 28 F eb 2008Preprint typeset in JHEP style -HYPER VERSIONThe KLOE collaboration:F.Ambrosino,d A.Antonelli,a M.Antonelli,a F.Archilli,a C.Bacci,g P.Beltrame,b G.Bencivenni,a S.Bertolucci,a C.Bini,f C.Bloise,a S.Bocchetta,g F.Bossi,a P.Branchini,g R.Caloi,f P.Campana,a G.Capon,a T.Capussela,a F.Ceradini,g S.Chi,a G.Chiefari,d P.Ciambrone,a E.De Lucia,a A.De Santis,f P.De Simone,a G.De Zorzi,f A.Denig,b A.Di Domenico,f C.Di Donato,d B.Di Micco,g A.Doria,d M.Dreucci,a G.Felici,a A.Ferrari,a M.L.Ferrer,a S.Fiore,f C.Forti,a P.Franzini,f C.Gatti,a P.Gauzzi,f S.Giovannella,a E.Gorini,c E.Graziani,g W.Kluge,b V.Kulikov,j cava,f nfranchi,a J.Lee-Franzini,a,h D.Leone,b M.Martini,a P.Massarotti,d W.Mei,a S.Meola,d S.Miscetti,a M.Moulson,a S.M¨u ller,a F.Murtas,a M.Napolitano,d F.Nguyen,g M.Palutan,a E.Pasqualucci,f A.Passeri,g V.Patera,a,e F.Perfetto,d M.Primavera,c P.Santangelo,a G.Saracino,d B.Sciascia,a A.Sciubba,a,e A.Sibidanov,a T.Spadaro,a M.Testa,f L.Tortora,g P.Valente,f G.Venanzoni,a R.Versaci,a G.Xu,a,i a Laboratori Nazionali di Frascati dell’INFN,Frascati,Italy b Institut f¨u r Experimentelle Kernphysik,Universit¨a t Karlsruhe,Germany c Dipartimento di Fisica dell’Universit`a e Sezione INFN,Lecce,Italy d Dipartimento di Scienze Fisiche dell’Universit`a “Federico II”e Sezione INFN,Napoli,Italy e Dipartimento di Energetica dell’Universit`a “La Sapienza”,Roma,Italy f Dipartimento di Fisica dell’Universit`a “La Sapienza”e Sezione INFN,Roma,Italy g Dipartimento di Fisica dell’Universit`a “Roma Tre”e Sezione INFN,Roma,Italy h Physics Department,State University of New York at Stony Brook,USA i Institute of High Energy Physics of Academia Sinica,Beijing,China j Institute for Theoretical and Experimental Physics,Moscow,RussiaAbstract:Using a sample of over 600million φ→K +K −decays collected at the DAΦNE e +e −collider,we have measured with the KLOE detector the absolute branching ratios for the charged kaon semileptonic decays,K ±→π0e ±ν(γ)and K ±→π0µ±ν(γ).The results,BR(K e 3)=0.04965±0.00038stat ±0.00037syst and BR(K µ3)=0.03233±0.00029stat ±0.00026syst ,are inclusive of radiation.Accounting for correlations,we derive the ratio Γ(K µ3)/Γ(K e 3)=0.6511±ing the semileptonic form factors measured in the same experiment,we obtain |V us f +(0)|=0.2141±0.0013.Keywords:e+e −experiments.1.IntroductionAt present,the determinations of V us and V ud provide the most precise verification of the unitarity of the CKM matrix.The relation1−|V ud|2−|V us|2−|V ub|2=0can be tested with an absolute accuracy of few parts per mil using|V ud|as measured in nuclear beta decays and|V us|as derived from semileptonic kaon decays.Since it was already known in 1983that|V ub|2<4×10−5[1]and today|V ub|2is∼1.5×10−5[2],|V ub|2will be ignored in the following.All experimental inputs to V us—branching ratios(BRs),lifetimes,and form factors—can be measured with the KLOE ing tagging techniques,we have already measured the complete set of inputs for K L decays[3,4,5,6],BR(K e3)for the K S[7],and the absolute BRs for K±→µ±ν[8]and K±→π±π0π0[9]decays.Here, we report on the measurement of the absolute BRs for the decays K±→π0e±ν(γ)(K e3) and K±→π0µ±ν(γ)(Kµ3).Our measurements,which make use of a tagging technique, are fully inclusive offinal-state radiation.2.Experimental setupThe data were collected with KLOE detector at DAΦNE,the Frascatiφfactory.DAΦNE is an e+e−collider which operates at a center of mass energy of∼1020MeV,the mass of the φmeson.Positron and electron beams of equal energy collide at an angle of(π−25mrad), producingφmesons with a small momentum in the horizontal plane,pφ∼13MeV.φmesons decay∼49%of the time into nearly collinear K+K−pairs;the detection of aK∓meson(the tagging kaon)therefore signals the presence of a K±(the tagged kaon) independently of its decay mode.This technique is called K±tagging in the following.The results presented here are based on an integrated luminosity of about410pb−1delivered by DAΦNE in2001-02,corresponding to∼6×108K+K−pairs produced.The KLOE detector consists of a large cylindrical drift chamber surrounded by a lead scintillating-fiber electromagnetic calorimeter.A superconducting coil around the calorimeter provides a0.52Tfield.The drift chamber(DC)[10]is4m in diameter and 3.3m long.The momentum resolution for tracks at large polar angles isσp⊥/p⊥≈0.4%. The vertex between two intersecting tracks is reconstructed with a spatial resolution of ∼3mm.The calorimeter(EMC)[11]is divided into a barrel and two endcaps.It is segmented in depth intofive layers and covers98%of the solid angle.Energy deposits nearby in time and space are grouped into calorimeter clusters.The energy and time resolutions areσE/E=5.7%/ E(GeV)⊕100ps,respectively. The trigger[12]uses only calorimeter information.Two energy deposits above threshold (E>50MeV for the barrel and E>150MeV for endcaps)are required.Recognition and rejection of cosmic-ray events is also performed at the trigger level.Events with two energy deposits above a30MeV threshold in the outermost calorimeter plane are rejected.To reject residual cosmic rays and machine background events,we use an offline soft-warefilter(FilFo)that exploits calorimeter information before tracks are reconstructed. As an example,thefilter tests the hypothesis that time difference between pair of clusters be compatible with the time offlight of a muon crossing the detector.The response of the detector to the decays of interest and the various background sources were studied by using the KLOE Monte Carlo(MC)simulation program[13].Changes in machine param-eters and background conditions are simulated on a run-by-run basis in order to properly track the frequent changes in machine operation.The MC sample ofφ→K+K−decays used for the present analysis corresponds to an integrated luminosity of about480pb−1; the sample for the otherφ−mesonfinal states is equivalent in statistics to∼90pb−1of integrated luminosity.3.Method of measurementThe use of a tagging technique allows the measurement of absolute branching ratios.Re-construction of one of the two-body decays K∓→µ∓ν(Kµ2)and K∓→π∓π0(Kπ2)in an event signals the presence of a K±;this provides a clean,counted sample of K±decaysfrom which to select signal events(K±e3or K±µ3decays).Let N Kℓ3be the number of events identified as K e3or Kµ3in a given tagged sample,and N tag the total number of tagged K±events in the sample.The branching ratio of each signal decay,K e3or Kµ3,can bedetermined as:BR(Kℓ3)=N Kℓ3ǫFV is corrected for losses of K±from nuclear interactions in the material traversed by kaons before entering the DC.This material includes the beam pipe(50µm of Be and 500µm of AlBe-met,an alloy of40%of Be and60%of Al)and the inner DC wall(750µm of C and200µm of Al).The probability of interaction in the KLOE setup is negligible (∼10−5)for K+,while it is∼3.4%for K−,as estimated by MC.Therefore,this correction is necessary only for samples tagged by K+decays.The quantityαTB,which we refer to as the tag bias in the following,accounts for the slight dependence of the reconstruction and identification efficiency for the tagging Kµ2(orandαTB,are evaluated from Kπ2)decay on the decay mode of the tagged kaon.BothǫKℓ3MC,with corrections evaluated from data and MC control samples.4.Tag selectionIn theφcenter of mass,the two kaons are produced back-to-back with momentum p∼k 127MeV.Since theφhas a transverse momentum of13MeV,in the laboratory frame ranges between120MeV and133MeV.The K±decay lengthλ±is∼95cm.Before pkentering the DC,kaons have to pass through the beam pipe and through the DC inner wall,is about100MeV, and lose about5MeV of energy.As a result,for kaons in the DC pkand the decay length is reduced to about75cm.Two-body decays are observed as vertices in the drift chamber between an incoming track(the kaon)and an outgoing track of the same charge.Kaons are identified as tracks<130MeV whose point of closest approach to the e+e−collision with momentum70<pkpoint(IP)lies inside a cylinder10cm in radius and20cm in length along the z-axis.1 The kaon decay vertex must be reconstructed within afiducial volume(FV)defined as a cylinder of radius40<r xy<150cm and length|z|<130cm,centered on the IP and coaxial with the beams.A kaon has a probabilityǫFV≃0.56of decaying in the FV as determined by MC.The combined reconstruction efficiency for the kaon and secondary tracks connected with a vertex(which we refer to as the decay chain in the following)is about0.6as estimated by MC.The momentum of the secondary track is computed in the kaon rest frame using the muon and pion mass hypotheses,p∗µand p∗π,respectively.The resulting momentum dis-tributions are shown in Fig.1.Kµ2events are identified as having231<p∗µ<241MeV (the shaded area of Fig.1,left),while Kπ2event candidates are identified as having 201<p∗π<209MeV(as in Fig.1,right).The tails in the distributions are due to resolution effects,and some residual semileptonic contamination on the left of the Kπ2peak.The secondary track is extrapolated to the calorimeter surface and associated to a calorimeter cluster,if possible.The association is based on the distance between the impact point of the track on the calorimeter and the nearest cluster;a cut is made on the component of this distance in the plane orthogonal to the direction of incidence of the track.The efficiency and the acceptance for the extrapolation,together with the efficiency for the association, is about0.7as estimated by MC.6p*µ (MeV)x 106p*π (MeV)Figure1:Momentum distribution of the secondary track in the kaon rest frame using the muon (left)and pion(right)mass.The shaded peaks correspond to events selected as Kµ2(left)and Kπ2 (right)decays.To reduce the tag bias,we require the tagging decay to satisfy the calorimeter trigger by itself.Kµ2decays can independently generate a trigger when the muon incident on the calorimeter traverses two nearby trigger sectors.This happens in about30%of events with identified Kµ2decays.For the remaining events identified as containing a Kµ2decay (referred to as the Kµ2,0sample in the following)the tag bias correction is large(1−αTB∼0.10).We use these events only as a control sample.For Kπ2events,the calorimeter trigger can be satisfied by the two photon clusters from theπ0.To identify Kπ2events,we require theπ0to be reconstructed as follows.For each cluster with E>50MeV not associated to any track,the kaon decay time t Kγ,i is calculated using the cluster time t i cl and the distance L i between the K±decay vertex and the cluster position:t Kγ,i=t i cl−L i/c.This time should have the same value for two photons from the sameπ0decay,so we require the presence of two clusters for which|t Kγ,1−t Kγ,2|<3σt(see Sect.2).Using the energies and the positions of the two clusters,theγγinvariant mass is calculated and a3σcut(σ∼18MeV) about the nominal value of theπ0mass is used to identify theπ0from a Kπ2decay.The calorimeter trigger is satisfied if the two identified photonsfire two different trigger sectors. The combined probability for a Kπ2decay to be identified and to independently satisfy the trigger is about0.25as determined by MC.The overall efficiency for the identification of the tagging kaon ranges between∼4.4% and∼5.7%depending on the sample.In the data set analyzed,about60million tagging decays were identified and divided into the four independent tag samples listed in Table1. MC studies show that the contamination due toφdecays other than K+K−is negligible.N tag213198047220354218742326904949,(5.1)ǫtag(Kℓ3)whereǫtag(i)is the tagging efficiency given that the tagged kaon evolves to afinal state i. In the sum,i indexes all possible outcomes i occurring with probability f(i)for the signal kaon,including not only all decay modes,but also possibly nuclear interactions with the beam pipe or inner DC wall.If the efficiencyǫtag(i)were the same for all i,αTB would be equal to unity.As noted in Sect.4,one of the main sources of tag bias is the dependence of the trigger efficiency on the decay mode of the tagged kaon;the requirement that the tagging kaon independently satisfy the trigger makesǫtrg=1,decreasing the tag bias.αTB refers to the tag bias from other sources,and can be estimated only by using the MC. The values ofαTB for each combination of tag and signal decay mode are listed in Table2. The values ofαTB range from about0.97to1.04,depending on the tag sample used,andK e30.9694(1)(5) 1.0137(3)(5)0.9884(1)(5) 1.0328(2)(3)Kµ30.9756(1)(5) 1.0210(4)(5)0.9963(1)(5) 1.0371(2)(3)efficiency using data and MC samples of K±→π0X events as described in Sect.7.The average correction factor applied to the MC efficiency is∼0.87.The secondary track is extrapolated to the calorimeter and geometrically associated to a cluster.The efficiency for the association is more than99%for K e3events and about 91%for Kµ3events.Both have been estimated using MC.The correction due to data-MC discrepancies is0.4%for K e3(measured using a sample of K L→πeνevents,as described in[5]),and2%for Kµ3.The latter value is obtained by combining the corrections measured using samples of K L→πµν(see[6])and K±µ2events.Thefiducial volume efficiency,ǫFV,is about56%,as in the tag selection,and,for K−’s,is corrected for nuclear interactions.This correction has been checked using data,<1GeV.Since because of lack of knowledge of K±-nuclear interaction cross sections for pkthe geometrical efficiency for the detection of K±decays depends on the K±lifetime τ,so do the values of the BRs:BR(τ)/BR(τ(0))=1−0.0364ns−1 τ−τ(0) ,where τ(0)=12.385±0.024ns,the current world average value[2].To reject the abundant two body decays,we require p∗π<192MeV for signal events. Only poorly reconstructed Kµ2and Kπ2decays and Kπ2events with an earlyπ±→µ±νdecay survive this cut.The procedure used to identify theπ0associated to the decay vertex is similar to that used in the selection of Kπ2decays,the only difference being that for signal events we require E>20MeV for each cluster.The efficiency forπ0identification (including EMC acceptance and cluster efficiency)is about0.57as estimated by MC.The single-photon detection efficiencies for data and MC are evaluated as a function of photon energy using K±π2events;their ratio is used to correct the MC efficiency.The average correction factor is∼0.98.Afterπ0selection,the sample is composed mainly of semileptonic decays,with residual contamination from Kπ2and K±→π±π0π0(K3π)decays.To reject Kπ2events in which theπ±decays toµ±νbefore entering the DC,we evaluate the lepton momentum using the mµmass hypothesis(p∗πµ)in the center of mass of theπ±.Theπ±momentum is defined−pπ0.By requiring p∗πµ>60MeV,we as the missing momentum at the decay vertex,pkreject about95%ofπ±→µ±νdecays while retaining about83%of K e3and78%of Kµ3 events,as estimated by MC.The contamination from K3πevents is reduced by requiring E miss−p miss,calculated using the m e mass hypothesis,to be less than90MeV.After the above cuts,the contamination from non-Kℓ3events is about2.1%in each tag sample,and consists of∼1.4%Kπ2decays and∼0.7%K3πdecays;for K−,a contamination of∼0.3% from nuclear interactions is also present.in The overall efficiencies for reconstruction and identification of Kℓ3events(ǫKℓ3 Eq.3.1),including data-MC corrections,are listed by decay and tag type in Table3.The statistical errors account for both the MC statistics and the statistics of the control sam-ples used to estimate the data-MC efficiency corrections.The uncertainties from control sample statistics represent the largest contributions(about1%)to the total errors on the BR measurements.In particular,for K e3the dominant uncertainty is from the tracking correction,while for Kµ3,the uncertainties from the tracking and muon cluster corrections are at the same level.Further details are given in Sect.7.To isolate K e3and Kµ3decays,the lep-K+µ20.0957(11)(6)0.0815(13)(4)K+π20.0989(16)(6)0.0848(19)(4)K−µ20.0983(12)(6)0.0841(9)(4)K−π20.1008(18)(6)0.0867(20)(4)2lept t lept−t Kπ02−1 ,where frame.The m2lept distribution is shown in Fig.2events.The K e3and Kµ3signals are evident. There is not visible in the distribution and is shown in the to the broad background distribution.For K−’s,The small peak at m2lept equal zero is due to the a background cluster.25005000750010000mlept2 (MeV2)1010101010leptFigure2:Left:MC m2leptdistribution for K+µ2-tagged semileptonic and background events;the ∼2%residual background is not visible and is shown in the inset.Right:m2lept distribution in logarithmic scale for Kℓ3candidate events identified in the K+µ2-tagged sample for data(triangles) and MC afterfit(shaded histograms).The numbers of signal events are obtained from afit to the m2lept data spectrum with a linear combination of the MC distributions for K e3decays,Kµ3decays,and background.N Ke3101733(411)34109(243)108125(430)33887(243) N Kµ355919(339)18999(200)59730(358)18923(205)Source Fractional statistical error(10−2)Total statistical0.80.9Event counting0.20.1Tag bias0.30.3ǫSel0.60.5Stability0.20.5ǫFV0.20.2Table5:Summary of contributions to the uncertainties for BR measurements. parameterized as a function of the kaon polar angle(θK),the decay vertex position(ρvtx), and the lepton momentum(p lab).The average data/MC correction is r dc∼0.87and is mainly due to data-MC differences in the reconstruction efficiency for the kaon track. The large energy loss of the K±in the DC gas2is underestimated in the MC,which results in a higher efficiency for kaon track reconstruction.To check the reliability of the correction applied,the events in each tag sample have been divided into equally populated and statistically independent subsamples withρvtx less than and greater than80cm,and with|θK−90◦|less than and greater than13◦.Wefind that the correction is larger (1−r dc∼0.20)for the samples withρvtx<80cm or|θK−90◦|>13◦than it is for the complementary samples withρvtx>80cm or|θK−90◦|<13◦(1−r dc∼0.05).The branching ratios measured using the full sample and in the two subsamples for each decay and tag type coincide within the errors.The systematic error has been taken to be half of the difference between the BRs measured for the two subsamples inρvtx.The fractional uncertainty in the BR measurements from r dc is about0.54%for K e3and about0.44%for Kµ3,respectively.The correction for the single lepton-cluster association efficiency is1−r∼−0.004lept∼0.02for muons.Both lepton-cluster and decay-chain corrections for electrons and1−rleptstrongly depend on p lab.We have checked the stability of the BR measurements when p lab is additionally required to be greater than50,70,and90MeV.We obtain a systematic error of about0.2%for BR(Kµ3),and a negligible error for BR(K e3).The rπ0correction takes into account differences between data and MC in the cluster reconstruction efficiency for low-energy photons.The single-photon detection efficiencies are evaluated from control samples of K±π2events,which are selected using DC information only.A photon fromπ0decay is identified by requiring that its energy and time offlightbe consistent with K±π2kinematics.This provides a good estimate of the momentum of the second photon.The efficiency is obtained as the probability for the second photon to be found in a cone with opening angle cosω=0.7about the expected direction.Each photon in the MC is weighted with the data/MC ratio of single-photon detection efficiencies evaluated as a function of photon energy.We have studied the effects on the correction rπ0 when the value of the opening-angle cutωis varied between cosω=0.6and cosω=0.9 and when the cut on the miminum energy for photon clusters is varied between10and 40MeV.We obtain a contribution to the uncertainties on the BRs of about0.2%from photon-cluster systematics.In order to evaluate the systematic error associated with thefit procedure,we have performed various studies using the Kµ2,0control sample(see Sect.4).First,we use MC distributions in m2lept for K e3and Kµ3taken from the Kµ2,0sample without applying the background-rejection cuts,which can in principle modify the shape of the distributions. We perform an additional check using K e3and Kµ3fit shapes obtained directly from data. Electron and muon cluster can be distinguished by exploiting the EMC granularity.Cuts on the profile in depth of the energy deposited in the lepton cluster allow the selection of K e3(energy mainly deposited in thefirst EMC plane)or Kµ3(muons behave like minimum ionizing particles in thefirst plane while they deposit a sizeable fraction of their kinetic energy from the third plane onward)events.This allows to obtain K e3and Kµ3fit shapes directly from data.We have tested the stability of the results when using these shapes. Finally,we have checked that the results are stable against changes in the histogram bin-ning andfit range.From these studies,we estimate the fractional systematic uncertainty associated with thefit procedure to range from0.1%to0.4%,depending on the decay mode and tag type.ǫFV has been computed using the MC.For K−decays,ǫFV is corrected for losses due to nuclear interactions.In this case,a contribution to the systematic error is evaluated from the difference between the corrections measured for MC and data.Actually,a suitable selection of K±→π0X events provides a sample containing K±interacting on the beam pipe and on the inner DC wall,and therefore allows comparison of the effects of nuclear interaction in data and MC.We obtain a fractional contribution of0.37%for K+µ2tagged events and0.69%for K+π2tagged events.The tag biasαTB includes the effect of the FilFo and cosmic-ray veto(CV)filters. The FilFo correction has been measured for each tag sample separately.It is about0.1% for Kπ2-tagged events and about1.5%for Kµ2-tagged events.The systematic error has been conservatively taken to be equal to the correction itself.For the CV,the measured correction ranges from0.04%to0.09%,depending on the tag sample;we assign a systematic error equal to half the value of the correction itself.Finally,since K−losses to nuclear interactions contribute to the value ofαTB,we assign an additional fractional error of ∼0.1%for the K+µ2-and K+π2-tagged samples.Last,we use the MC to check the stability of the results with respect to variations of each of the cuts used to increase the purity of the K±ℓ3samples.Moving the p∗πµcut from50to70MeV changes the K e3efficiency from∼0.89to∼0.77and the Kµ3efficiency from∼0.87to∼0.70,while inducing variations of∼0.1%and∼0.4%in the resulting K e3and Kµ3BRs,respectively.We have performed similar studies for the p∗πand E miss−p miss cuts,giving a total contribution to the fractional systematic error of∼0.17%for K e3and ∼0.49%for Kµ3.8.ResultsThe four determinations of the K e3and Kµ3branching ratios are listed in Table6.The fractional uncertainties range from1.5%to2.1%for BR(K e3)and from1.5%to2.7%for BR(Kµ3).Averaging the results for each charge state,we obtain:BR(Kµ3)0.03217(63)0.03223(87)0.03233(49)0.03275(86)corresponding to a correlation coefficient between the errors on BR(K e3)and BR(Kµ3)of 62.7%.With this correlation taken into account,we evaluate the ratio Rµe=Γ(Kµ3)/Γ(K e3) from our results for BR(K e3)and BR(Kµ3).We obtain Rµe=0.6511±0.0064.This value has a fractional error of about1.0%and is in1.5σagreement with the theoretical prediction,R SM=0.6646(61)[15].µeUsing the PDG value for the K±lifetime[2]and the KLOE values for the semileptonic form factors[6],we obtain|V us f+(0)|=0.2148±0.0013from BR(K e3)and|V us f+(0)|=0.2129±0.0015from BR(Kµ3).The average is|V us f+(0)|=0.2141±0.0013,including the correlations between the BR measurements and the use of the same lifetime value for both ing f+(0)=0.961(8) from[16],|V us|is0.2223(23).With|V ud|=0.97418(26)[17],wefind1−|V ud|2−|V us|2= 0.0016(11),so that thefirst-row test of the unitarity of the CKM matrix is satisfied at the level of1.4σ.AcknowledgmentsWe thank the DAΦNE team for their efforts in maintaining low background running conditions and their collaboration during all data-taking.We want to thank our technical staff:G.F.Fortugno and F.Sborzacchi for their dedicated work to ensure an efficient operation of the KLOE Computing Center;M.Anelli for his continuous support to the gas system and the safety of the detector;A.Balla,M.Gatta,G.Corradi,andG.Papalino for the maintenance of the electronics;M.Santoni,G.Paoluzzi,andR.Rosellini for the general support to the detector;C.Piscitelli for his help during major maintenance periods.This work was supported in part by EURODAPHNE,contract FMRX-CT98-0169;by the German Federal Ministry of Education and Research(BMBF) contract06-KA-957;by the German Research Foundation(DFG),’Emmy Noether Programme’,contracts DE839/1-4;by INTAS,contracts96-624,99-37;and by the EU Integrated Infrastructure Initiative HadronPhysics Project under contract numberRII3-CT-2004-506078.References[1]CUSB Collaboration,C.Klopfenstein et al.,Phys.Lett.B1301984444.[2]Particle Data Group,W.-M.Yao et al.,Journal of Physics,G33,1(2006).[3]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B632200643.[4]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B626200515.[5]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B6362006166.[6]F.Ambrosino et al.,[KLOE Collaboration],JHEP0712:105,2007.[7]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B6362006173.[8]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B632200676.[9]F.Ambrosino et al.,[KLOE Collaboration],Phys.Lett.B5972004139.[10]M.Adinolfiet al.,[KLOE Collaboration],Nucl.Instrum.Meth A488200251.[11]M.Adinolfiet al.,[KLOE Collaboration],Nucl.Instrum.Meth A4822002364.[12]M.Adinolfiet al.,[KLOE Collaboration],Nucl.Instrum.Meth A4922002134.[13]F.Ambrosino et al.,[KLOE Collaboration],Nucl.Instrum.Meth A5342004403.[14]C.Gatti,Eur.Phys.J.C,45,417(2006).[15]FlaviaNet Working Group on Kaon Decays,contributed to4th International Workshop onthe CKM Unitarity Triangle,Nagoya,Japan12-16Dec2006,hep-ex/0703013.[16]H.Leutwyler,M.Roos,Z.Phys.C,25198491.[17]I.S.Towner,J.C.Hardy,arXiv:0710.3181[nucl-th](2007).。
Pseudoscalar Decay Constants $ f_{D} $ and $ f_{D_s} $ in Lattice QCD with Exact Chiral Sym
National Taiwan University, Taipei 106, Taiwan
Abstract
We determine the masses and decay constants of pseudoscalar mesons D, Ds, and K in quenched lattice QCD with exact chiral symmetry. For 100 gauge configurations generated with single-plaquette action at β = 6.1 on the 203 × 40 lattice, we compute point-to-point quark propagators for 30 quark masses in the range 0.03 ≤ mqa ≤ 0.80, and measure the time-correlation functions of pseudoscalar and vector mesons. The inverse lattice spacing a−1 is determined with the experimental input of fπ, while the strange quark bare mass msa = 0.08, and the charm quark bare mass mca = 0.80 are fixed such that the masses of the corresponding vector mesons are in good agreement with φ(1020) and J/ψ(3097) respectively. Our results of pseudoscalar-meson decay constants are fK = 152(6)(10) MeV, fD = 235(8)(14) MeV, and fDs = 266(10)(18) MeV.
CURRENTS AND THEIR COUPLINGS IN THE WEAK SECTOR OF THE STANDARD MODEL
(4)
which is effective four-fermion interaction with the Fermi constant given √the familiar 2 2 h ℓ by GF / 2 = g /(8mW ). Here Jµ and Jµ are known as the hadron and lepton currents, where
(2)
1 where t3L (i) is the weak isospin of fermion i ( + 1 2 for ui and νi ; − 2 for di and ℓi ) and qi is the charge of ψi in units of e. The three terms in the interaction Lagrangian of Eq. 1 represent the chargedcurrent weak interaction, the electromagnetic interaction and the neutral-current weak interaction respectively. Note that the Lorentz structure involves only vectors and axial vectors; there is no compelling experimental evidence for scalars, pseudoscalars or tensor constructions. The minimal Standard Model as described by Eq. 1 successfully3 explains W and Z decays, neutrino-hadron scattering, neutrino-electron scattering and parity-violating electron-hadron neutral-current experiments, providing that radiative corrections to order α, the fine-structure constant, are applied. In this chapter we will focus our discussion on the charged-current weak interaction in semi-leptonic decays involving quarks and leptons from the first family. As an example, consider the decay d → ue− ν e , for which the T -matrix is2
The decay constant of the first excited pion from lattice QCD
is
∂µAµ = mqπ
(2)
where π is the interpolating operator for pion states (pseudo-scalar density) and mq is the quark mass. Equation 2 is an operator relation, hence is true between any states. This allows us to write:
decay width of between 200 to 600 MeV. The predominant decay mode is to πππ (this includes ρπ). There is readable discussion about the experimental
2
Group Volkov and Weiss [4]
∂µAµ =
m2πn fπn πn
(5)
n
where πn is the interpolating operator for the n-th excited light 0−+ meson. The PDG [13] quotes the mass of the π(1300) as 1300 ± 100 MeV with a
Elias et al. [5] Maltman and Kambor [7]
Andrianov et al. [6] Kataev et al. [19, 20]
fπ′ MeV 0.68
4.2 ± 2.4 3.11 ± 0.65 0.52 − 2.26
4.3
Table 1: Summary of the values of the π′ decay constant determined from models and sum rules. Our normalisation convention is fπ = 131 MeV.
Leptonic and Semileptonic Charm Decays from CLEO-c
a r X i v :h e p -e x /0702035v 1 23 F eb 2007Leptonic and Semileptonic Charm Decays from CLEO-cS.Stone ∗Physics Department of Syracuse UniversitySyracuse NY,13244,USAI describe CLEO-c purely leptonic decay results leading to f D +=(222.6±16.7+2.8−3.4)MeV,f D +s =(280.1±11.6±6.0)MeV,and f D +s/f D +=1.26±0.11±0.03.Form-factor measurements in Cabibbo favored and suppressed pseudoscalar decays are presented.Some comparisons are made with theoretical predictions.I.INTRODUCTIONThreshold production of D 08πf 2D +(s )m 2ℓ+M D +(s )(1)×1−m 2ℓ+/M 2D +(s )2|V cq |2,Γ(D +s →φπ+)=0.173±0.023±0.035.(2)BaBar recently performed an improved analysis basedon these techniques [10].They used 230fb −1of con-tinuum data.To reduce the background and systematic errors they fully reconstruct a D 0,D +or D ∗meson in the event with the γand µ+candidate.Their data are shown in Fig.2(b).They findΓ(D +s →µ+ν)E n t r i e s /0.01G e V /c FIG.2:The ∆M distributions for µ+candidates after the e +subtraction from (a)CLEO and (b)BaBar.The solid curves are fits to signal plus background.Both of these results,however,need to assume a valuefor B (D +s→φπ+)[11],in order to extract the decay constant.Because of interferences among the final state K +K −π+particles,the rate for φπ+depends on exper-imental cuts [12],and thus has an inherent,sizable,sys-tematic error.(Other experiments also normalize with respect to this or other less well known modes.)CLEO-c eliminates this uncertainty by making abso-lute measurements.We tag a D −s decay and search forthree separate decay modes of the D +s :(1)µ+νand τ+ν,where (2)τ+→π+ν[13].For the first two analyses we require the detection of the γfrom the D ∗s →γD s decay,irrespective if the D ∗s is the parent of the tag or the leptonic decay.In either case,for real D ∗s D s events,the missing mass squared recoiling againstthe photon and the D −s tag should peak at M D +sand is given byMM ∗2=(E CM −E D −E γ)2−(−→p CM −−→p D −−→p γ)2,where E CM (−→p CM )is the center of mass energy (momen-tum),E D (−→p D )and E γ(−→p γ)are the energy of the fully reconstructed D −s tag,and the additional photon.In performing this calculation we use a kinematic fit thatconstrains the decay products of the D −s to M D +s and conserves overall momentum and energy.The MM ∗2from the D −s tag sample data is shown in Fig.3.There are 11880±399±511signal events in the interval 3.978>MM ∗2>3.776GeV 2.MM* (GeV )22FIG.3:The MM*2distribution from events with a photon inaddition to the D −s tag.The curve is a fit to the Crystal Ball function and a 5th order Chebychev background function.Candidate D +s →µ+νevents are searched for by se-lecting events with only a single extra track with opposite sign of charge to the tag;we also require that there not be an extra neutral energy cluster in excess of 300MeV.Since here we are searching for events where there is a single missing neutrino,the missing mass squared,MM 2,evaluated by taking into account the seen µ+,D −s ,and the γshould peak at zero,and is given byMM 2=(E CM −E D −E γ−E µ)2(4)−(−→p CM −−→p D −−→p γ−−→p µ)2,where E µ(−→p µ)is the energy (momentum)of the candi-date muon track.We also make use of a set of kinematical constraints and fit the MM 2for each γcandidate to two hypothesesone of which is that the D −s tag is the daughter of a D ∗−sand the other that the D ∗+s decays into γD +s ,with the D +s subsequently decaying into µ+ν.The kinematical constraints are the total momentumand energy,the energy of the either the D ∗s or the D s ,the appropriate D ∗s −D s mass difference and the invariant mass of the D s tag decay products.This gives us a total of 7constraints.The missing neutrino four-vector needs to be determined,so we are left with a three-constraint fit.We perform a standard iterative fit minimizing χ2.As we do not want to be subject to systematic uncertain-ties that depend on understanding the absolute scale of the errors,we do not make a χ2cut,but simply choose the photon and the decay sequence in each event with the minimum χ2.We consider three mutually exclusive cases:(i)the track deposits <300MeV in the calorimeter,charac-teristic of a non-interacting π+or a µ+;(ii)the track deposits >300MeV in the calorimeter,characteristic of an interacting π+;(iii)the track satisfies our e +selection criteria.The MM 2distributions are shown in Fig.4.The separation between µ+and π+is not unique.Case (i)contains 99%of the µ+but also 60%of the π+,while case (ii)includes 1%of the µ+and 40%of the π+[5].There is a clear peak in Fig.4(i),due to D +s →µ+ν.Fur-thermore,the events in the region between µ+νpeak and 0.20GeV 2are dominantly due to the τ+ν,τ+→π+FIG.4:The MM2distributions from data using D−s tags and one additional opposite-sign charged track and no extra energetic showers(see text).TABLE I:Measured D+s Branching FractionsFinal State B(%)Γ(D+s→µ+ν)=9.9±1.9,(7)Events/5MeVE (GeV)0.5 1.0 1.5extraCCFIG.5:The extra calorimeter energy from data(points),com-pared with the Monte Carlo simulated estimates of semilep-tonic decays in general(dotted),the K0e+νmode specifically(shaded),as a sub-set of the semileptonics,and the expecta-tion from signal(dashed).The peak near150MeV is due totheγfrom D∗s→γD s decay.(The sum is also shown(line).)The arrow indicates the selected signal region below0.4GeV.consistent with the predicted value biningour branching ratios determinations and usingτD+s=0.49ps and|V cs|=0.9737,wefindf D s=(280.1±11.6±6.0)MeV,and(8)fD+s/f D+=1.26±0.11±0.03.These preliminary results are consistent with most re-cent theoretical models.As examples,unquenched lattice [15]predicts1.24±0.01±0.07,while one quenched lat-tice calculation[16]gives1.13±0.03±0.05,with other groups having similar predictions[20].III.SEMILEPTONIC DECAYSOne of the best ways to measure magnitudes of CKM elements is to use semileptonic decays since they are far simpler to understand than hadronic decays and the de-cay width is∼|V cq|2.On the other hand,measurements using other techniques have obtained useful values for V cs and V cd[17],and thus semileptonic D decay mea-surements are a good laboratory for testing theories of QCD.For a D meson decaying into a single hadron(h), the decay rate can be written exactly in terms of the four-momentum transfer defined as:q2=(pµD−pµh)2=m2D+m2h−2E h m D.(9)For decays to pseudoscalar mesons and“virtually massless”leptons,the decay width is given by:dΓ(D→P e+ν)24π3 f+(q2) ,(10)where p P is the three-momentum of P in the D rest frame,and f+(q2)is a“form-factor,”whose normaliza-tion must be calculated theoretically,although its shape can be measured.The shape measurements can distinguish between form-factor parameterizations.In general,f+(q2)=f+(0)m2pole )+1q′2−q2,which incorporates the possibility of a virtual of a nearbypole(first term)with fractional strengthαp.The integralterm can be expressed in terms of an infinite series[18].Typically it takes only a few terms to describe the data.An analytical parametrizationf+(q2)=f+(0)B(%) 3.58(5)(5)0.309(12)(6)CLEO-c TagB(%) 3.56(3)(11)0.301(11)(10)CLEO-c NoTagB(%) 3.58(18)0.360(60)PDG04|f+(0)|0.761(10)(7)0.660(28)(11)CLEO-c Tag|f+(0)|0.749(5)(10)0.636(17)(13)CLEO-c NoTagm pole(GeV) 1.96(3)(1) 1.95(4)(2)CLEO-c Tagm pole(GeV) 1.97(2)(1) 1.89(3)(1)CLEO-c NoTagα0.22(5)(2)0.17(10)(5)CLEO-c Tagα0.21(4)(3)0.32(7)(3)CLEO-c TagE v e n t s / ( 0.00833333 )FIG.9:U distributions for (a)D +and (b)D 0decays into ρe +νcandidates.The dashed curve shows the signal,the dotted curves show various backgrounds and the solid curve the sum.TABLE III:Rare semileptonic decay branching fractions DecayB ×10−4B s Oscilla-tions,”[hep-ex/0609040];see also V.Abazov et al.(D0),[hep-ex/0603029].[3]C.Davies et al.,Phys.Rev.Lett.92,022001(2004).[4]M.Artuso et al.(CLEO),Phys.Rev.Lett.95,251801(2005);[5]G.Bonvicini,et al.(CLEO)Phys.Rev.D70,112004(2004).[6]D.Silverman and H.Yao,Phys.Rev.D38,214(1988).[7]A.G.Akeroyd,Prog.Theor.Phys.111,295(2004);A.G.Akeroyd and C.H.Chen [hep-ph/0701078].[8]J.Hewett,[hep-ph/9505246];W.-S.Hou,Phys.Rev.D48,2342(1993).[9]M.Chadha et al.(CLEO),Phys.Rev.D58,032002(1998).[10]B.Aubert et al.(BaBar)[hep-ex/0607094].[11]W.-M.Yao et al.(PDG),Journal of Physics G 33,1(2006)[12]S.Stone,[hep-ph/0605134].[13]S.Stone,[hep-ex/0610026].[14]P.Rubin et al.(CLEO),Phys.Rev.D73,112005(2006).[15]C.Aubin et al.,Phys.Rev.Lett.95,122002(2005).[16]T.W.Chiu et al.,Phys.Lett.B624,31(2005).[17]M.Artuso,“Status and future perspectives on V cs and V cd ,Expermimental,”presented at 4th Int.Workshop on the CKM Unitarity Triangle,Dec.,2006,Nagoya,Japan.[18]T.Becher and R.J.Hill,Phys.Lett.B633,61(2006);R.J.Hill,[hep-ph/0606023].[19]D.Becirevic and A.B.Kaidalov,Phys.Lett.B478,417(2000).[20]See references to other theoretical predictions in [4].[21]C.Aubin et al.,Phys.Rev.Lett.,95,122002(2006).[22]Y.Gao et al.(CLEO),presented at XXXIII Int.Conf.on High Energy Physics,Moscow,Russia,July,2006.[23]B.Grinstein and D.Pirjol,Phys.Rev.D70,114005(2004)[hep-ph/0404250].[24]M.R.Shepherd et al.(CLEO),Phys.Rev.D74,052001(2006).[25]J.M.Link et al.(FOCUS),Phys.Lett.B 633,183(2006).[26]N.E.Adam et al.(CLEO),Phys.Rev.Lett.97,251801(2006).。
Heavy-heavy form factors and generalized factorization
a r X i v :h e p -p h /9810271v 2 27 O c t 1998ROME1-1222/98Heavy-heavy form factors and generalized factorizationM.Ciuchini 1,R.Contino 2,E.Franco 2,G.Martinelli 21Dipartimento di Fisica,Universit`a di Roma Tre and INFN Sezione di Roma III,Via della Vasca Navale 84,I-00146Rome,Italy.2Dipartimento di Fisica,Universit`a “La Sapienza”and INFN,Sezione di Roma,P.le A.Moro 2,I-00185Rome,Italy.Abstract We reanalyze B →Dπand B →KJ/ψdata to extract a set of parameters which give the relevant hadronic matrix elements in terms of factorized amplitudes.Various sources of theoretical uncertainties are studied,in particular those depending on the model adopted for the form factors.We find that the fit to the B →Dπbranching ratios substantially depends on the model describing the Isgur-Wise function and on the value of its slope.This dependence can be reduced by substituting the BR (B →Dπ)with suitable ratios of non-leptonic to differential semileptonic BR s.In this way,we obtain a model-independent determination of these ing these results,the B →D form factors at q 2=M 2πcan be extracted from a fit of the BR (B →Dπ).The comparison between the form factors obtained in this way and the corresponding measurements in semileptonic decays can be used as a test of (generalized)factorizationfree from the uncertainties due to heavy-heavy form factor modeling.Finally,we present predictions for yet-unmeasured Dπand DK branching ratios and extract f D sand f D ∗s from B →DD s decays.We find f D s =270±45MeV and f D ∗s =260±40MeV,in good agreement with recent measurements and lattice calculations.ROME1-1222/98Oct.19981IntroductionA problem of utmost importance inB phenomenology is the computation of the hadronic amplitudes:in recent years it has been realized that the full determination of the unitarity triangle from B decays can hardly be carried out without an accurate knowledge of these quantities[1,2].Unfortunately,the computation of hadronic am-plitudes requires an understanding of low-energy strong interactions which is missing at present.Even a non-perturbative approach based onfirst principles,such as lattice QCD,fails in computing decay amplitudes involving two or more hadrons in thefinal state[3].In the absence of rigorous methods,some simplifying approaches have been de-veloped.The most popular one consists in the factorization of matrix elements of four-fermion operators in terms of local products of two currents.In this approach, the original matrix element is computed as product of the matrix elements of the two currents.Attempts to give theoretical soundness to this procedure in the framework of the1/N expansion and of the Large Energy Effective Theory(LEET)can be found in refs.[4,5].Unfortunately,there are many problems in both these approaches and their applicability to exclusive decays is questionable.Independently of any theoretical prejudice,there is a priori no reason for this approximation to be accurate in the case of B decays.Indeed,none of the expansions developed so far was able to compute correc-tions to the lowest order results and estimate the size of the errors.On the other hand, the importance of controlling the theoretical uncertainties calls for some phenomeno-logical approach to test predictions obtained using factorized amplitudes.To this end, a popular method consists in reducing the Wick contractions of matrix elements to few topologies using Fierz transformations and color rearrangement.Then,the remaining amplitudes are factorized and expressed in terms of the appropriate decay constants and/or form factors.In this procedure,some phenomenological parameters are intro-duced in order to account for possible deviations from factorization[1,6,7].These factorization parameters,denoted as FP in the following,are meant to befitted to experimental data.In this paper,we introduce a parameterization of the hadronic matrix elements that extends the one of ref.[1]and allows the computation of the hadronic amplitudes rele-vant to Cabibbo-allowed non-leptonic B decays in terms of factorized matrix elements and of three real FP.Wefind that,in thefit of the BR(B→Dπ)and BR(B→KJ/Ψ), there is a strong interplay between the values of the FP and the model used for the heavy-heavy form factors,more specifically on the Isgur-Wise(IW)function and its slopeρ21.This implies that factorization tests are obscured by our ignorance on the values of the form factors in the kinematical region relevant in non-leptonic decays (q2≪q2max).The model dependence is drastically reduced by using,in thefit,suitable ratios of semileptonic and non-leptonic BR s(to be introduced below)instead of the BR(B→Dπ)alone.In this way,we are able to extract(almost)model independent FP.With these FP at hand,we use then the BR(B→Dπ)to determine,within a given model,the value ofρ2which may be compared,as a test of factorization,to the one measured in semileptonic decays.The value ofρ2extracted from thefit depends,Form Factor NRSX0.56[0.49–0.63]0.68[0.61–0.73]0.59[0.54–0.64]0.59[0.53–0.64]0.59[0.53–0.64]Channel NRSX[BR×105][BR×105]14[1–58]<12B d→π0¯D∗013[2–35]6[1–26]<39B d→ρ0¯D∗014[2–38]B d→K+D−23[16–32]22[14–37]–B d→K∗+D−53[38–75]67[43–110]–B+→K+¯D035[18–45]36[13–54]–B+→K∗+¯D069[42–88]87[46–126]–B d→K0¯D0 1.0[0.2–2.7]1.5[0.2–6.2]–B d→K∗0¯D00.7[0.1–1.8]1.7[0.2–7.2]–Table2:Predictions of yet-unmeasured branching ratios.2Factorization,FP and form-factor modelsIn this section,we present our parameterization of the hadronic amplitudes and discuss its relation with other popular choices.We also introduce two different models for the form factors used in our phenomenological analysis.Consider the matrix element of some composite operator appearing in the∆B=1 weak Hamiltonian,between the B meson and twofinal pseudoscalar or vector mesons. In general this operator can be written as the product of two currents.If one of the currents has the correct quantum numbers to create one of thefinal state mesons from the vacuum,then the matrix element can be factorized.The physical idea is the following:the quark pair produced by this current acts as a color dipole,weakly interacting with the surrounding colorfield.If the transferred energy is large,the quark pair has no time to interact before hadronizing far from the interaction point[12].As an example,we discuss the factorization of the amplitudes entering the decay B d→D−π+.In this case,the two relevant matrix elements(αandβare color indices)D−π+|Q1|B d = D−π+|¯bαγµ(1−γ5)cβ¯uβγµ(1−γ5)dα|B d ,D−π+|Q2|B d = D−π+|¯bαγµ(1−γ5)cα¯uβγµ(1−γ5)dβ|B d ,(2) can be Wick-contracted according to two different topologies,that are usually denoted as connected(CE)and disconnected(DE)emissions,respectively.Color indices can be rearranged using the algebraic relation,1δαβδρσ=where N is the number of colors,δis the Kronecker symbol and the t a are the SU(N) color matrices in the fundamental representation,normalized as tr(t a t b)=δab/ing this relation,oneobtainsCE=1N +ǫ8N+ǫ8really exists)at which exact factorization holds,is meaningless.Nevertheless,the FP can be precisely extracted from data,once the renormalization scale and the scheme arefixed.Their values will depend,of course,on these choices.We will use the NDR-q2 +f0(q2)qµ(M2B−M2)εµναβǫ∗νpαpβB V(q2),(8)M B+MV (p,ǫ)|ˆAµ|B (p B ) =2MA 0(q 2)q µǫ∗·q q 2−A 2(q 2)q 2(M 2B −M 2) ,where ˆVµand ˆA µare the vector and axial currents respectively.For heavy finalmesons,the form factors canbe connectedto the HQET functionsξi (y ),see e.g.ref.[16],using the following formulasP (v )|ˆV µ|B (v B ) =M B M iξV (y )εµναβǫ∗νv αv βB ,(9)V (v,ǫ)|ˆAµ|B (v B ) = 2M B M .(10)In the heavy quark limit,the functions ξi (y )are all related to the IW function ξ(y )ξ+=ξV =ξA 1=ξA 3=ξ(y ),ξ−=ξA 2=0,(11)with the normalization of ξfixed by the heavy quark symmetry,ξ(1)=1.The ξi (y )can be written as ξi (y )= αi +αs (¯m ):the first model uses the heavy-heavy form factors defined in eq.(12),taking the βi (y )from ref.[17]and neglecting the O (1/m )corrections,i.e.γi (y )=0.For the IW function,the simplest form is assumed,namelyξ(y )=1−ρ2(y −1).(13)For the heavy-light form factors,LINSR uses those computed with light-coneQCD sum rules [18].•NRSXy +1exp −A y −13The computation of the 1/m corrections is model dependent,relying on the evaluation of a set of hadronic matrix elements of higher dimensional operators in the HQET.where A=2ρ2−1.Concerning the heavy-light form factors,NRSX improves the old WSB model[20]by implementing,for q2∼q2max,the expected heavy quark scaling laws,see ref.[19]for details.We end this section by spending a word of comment about the parameterρ2entering eqs.(13)and(14).In both models,ρ2is defined as the slope of the IW function at the zero-recoil point(i.e.y≡v B·v=1),namelyρ2≡−ddy =G2Fy2−1(y+1)2× 1+4y(1−r)2 |V cb|2F(y)2,(16)where r≡M D∗/M B and F(y)is an effective semileptonic form factor.The latter is a calculable function of theξi(y)s.To make contact with experiments,one defines theslopeˆρ2≡−1dyF(y) y=1,(17)which can be extracted from the measurement of the semileptonic differential rate.The relation between the experimental slopeˆρ2and the theoretical parameterρ2 depends on the model used for the calculation of eq.(16).We foundLINSRρ2=ˆρ2−0.13,NRSXρ2=ˆρ2+0.084,(18) to be compared with the results of ref.[21],ρ2=ˆρ2−(0.14±0.02)+O(1/m)and ρ2=ˆρ2±0.2respectively.In the next section,we will study the dependence of our results on the physical slopeˆρ2,rather thanρ2.3Fitting Cabibbo-allowed decay modes This section contains the results of our phenomenological analysis of Cabibbo-allowed B decays,focused on the rˆo le of heavy-heavy form factors.We proceed as follows:•we show that the bestfit to the BR(B→Dπ)and BR(B→KJ/Ψ)is obtained for different values ofˆρ2,depending on the model used for computing heavy-heavy form factors;•we use the ratios Rπ(B→Dπ)introduced in ref.[24],see below,and show that,by using them instead of the BR(B→Dπ),it is possible tofit the FP (almost)independently ofˆρ2.This method gives our best determination of the FP,free from theoretical uncertainties coming from the assumptions made for the heavy-heavy form factors;Channel300±40276±21790±140670±330B d→D+s D−B d→D+s D∗−B+→D+s¯D0B+→D+s¯D∗0B d→D∗+s D−B d→D∗+s D∗−B+→D∗+s¯D0B+→D∗+s¯D∗089±12Type II135±18∝|a2|299±10147±27⋆B+→π+¯D0⋆B+→π+¯D∗0⋆B+→ρ+¯D0⋆B+→ρ+¯D∗0dq2BR(B→D∗l+ν)(GeV−2)dm2π(0.237±0.026)×10−2(0.35±0.06)×10−2 m2ρ(0.250±0.030)×10−2(0.33±0.06)×10−2m2Ds (0.483±0.033)×10−2–m2D∗s(0.507±0.035)×10−2–Table3:Experimental branching ratios[8,22,23]of decay modes to be used in thefit of the parametersα,ξandδξ.The classification of non-leptonic channels according to their dependence on a1and a2is also shown.The channels marked with⋆have been used for the fit of the ratios defined in eq.(20).•using the(ˆρ2-independent)FP,we perform afit to the BR(B→Dπ)in order to extract a preferred range for the value ofˆρ2;the results are model dependent,in agreement with ourfirstfinding;•we show that the different ranges ofˆρ2actually correspond to the same values of the relevant form factors at q2=M2π∼ing the HQET,the latters can be determined by factorization applied to B→Dπdecays and may be compared with direct measurements from semileptonic decays.The relevant decay modes which we use in thefits are listed in tab.3.It is well known that,in the factorization approximation,only two combinations of DE and CE appear in the amplitudes of these decays.This feature is taken into account by the parameters a1and a2introduced by BSW[6].The relation between a1and a2and our parameters is given bya1=α C2+ξe iδξC1 ,a2=α C1+ξe iδξC2 (19)where C1and C2are the Wilson coefficients of the operators defined in eq.(2)4.In tab.3,the decay modes are organized according to the standard classification in three classes.Amplitudes of Type I,II and III decay modes are proportional to|a1|,|a2| and|x1a1+x2a2|respectively,where x i are generic,process-dependent coefficients.Given the structure of the amplitudes,we have tofit decay modes of all the three classes in order to fully determine the FP.Note that the three classes have a different dependence on the form-factors.While Type-II decays always involve a heavy-light transition,heavy-heavy form factors enter Type-I modes only.The latter is a general feature,since Type-I transitions are always driven by charged currents and are therefore proportional to a1.In general Type-III modes involve transitions of both sorts.In our parameterization,Type-I modes essentiallyfixα,while both Type II and III are needed to constrainξandδξ.Contrary to the common wisdom,the assumptions made on the momentum de-pendence of the heavy-heavy form factors introduce large uncertainties in the deter-mination of the FP from thisfit.This is more easily shownfitting only the effective number of colour as done in the old literature5.In our parametrization this corre-sponds to assumeα=1andδξ=0and tofitξonly.The result of thisfit is shown infig.1,where minimum values ofχ2/dof from thefit ofξto the BR(B→Dπ)and to the BR(B→KJ/Ψ)are plotted as function ofˆρ2.It is apparent that the best fit is obtained for quite different values ofˆρ2,and corresponds to different values ofξ, depending on the model used to compute the heavy-heavy form factors.Consequently, in general,the FPfitted using BR s atfixedˆρ2,as usually done in the literature,suffer from a large theoretical error,which was previously hidden in the choice of a specific model whenfitting the data.The second important remark is that a comparison of the value ofˆρ2,the“physical”slope measured in semileptonic decays,with that extracted from non-leptonic decays is not a good test of factorization since,in the latter case, the result is model dependent.χ2dLINSRˆρ2=0.90χ2/dof1.36Type I1.021.02ξ0.45BR(B→KJ/ψ)0.000.00Rπ(B→Dπ)1.401.42α1.04+0.440.44δξ0.00ˆρ2=1.25ˆρ2=1.45Rπ(B→Dπ)0.390.39α1.01+0.380.38δξ0.00χ2/dof0.70Type I+III1.041.03ξ0.38BR(B→KJ/ψ)0.000.00Table4:Results of thefit using Dπsemileptonic ratios and BR(B→KJ/Ψ)for three different values of the slopeˆρ2and the two form-factor models,LINSR and NRSX,described in the text.For each model,twofits have been performed,the difference being the inclusion of Type-III Dπmodes,which introduce someˆρ2dependence.appreciably the results6.In tab.4we show thefitted values of the FP for severalchoices ofˆρ2,for both NRSX and LINSR.As mentioned above,the results turn out tobe,within a given model,independent ofˆρ2.Havingfitted the FP in aˆρ2-independent way,we now use the BR(B→Dπ)to extract from the data a preferred range ofˆρ2.Notice thatδξis not a critical parameter,since the results of thefits are not very sensitive to its value,and that the values ofαandˆρ2are trivially correlated,because the amplitudes only depend on the productofαwith the effective form factors at q2∼0.Therefore we choose to perform atwo-parameterfit ofξandˆρ2using the BR(B→Dπ),atfixed values ofδξandα,asextracted from the previousfit of tab.4.In this way,we can study the correlationsin the(ξ–ˆρ2)plane and check the consistency of the determination ofξusing differentfitting procedures.Figure2shows the contour plots ofχ2/dof in the(ξ,ˆρ2)plane for NRSX and LINSR.Thefitted value ofξis consistent with tab.4and the preferredˆρ2is largerusing NRSX than LINSR.Moreover,the LINSR BR s are steeper functions ofˆρ2,consistently withfig.1.This observation justifies the choice of the set of values ofˆρ2used in tab.4.It is not surprising that thefit to the BR(B→Dπ)gives values ofˆρ2which are1.81.61.4ˆρ21.210.80.20.40.60.81ξFigure2:Contour plots ofχ2/dof in the plane(ξ,ˆρ2),obtained from a two-parameterfit to the non-leptonic B→Dπdecays.The other parameters(αandδξ)have beenfixed by the fit of tab.4in aˆρ2-independent way.The solid curves refer to LINSR,the dashed ones to NRSX.The different contours correspond toχ2/dof=(1.5,2,3)×χ2min/dof.model dependent.Thefit onlyfixes the values of the relevant heavy-heavy form factorsf i=f+,A0,...at q2=M2π∼07.These form factors can be expressed in terms oftheξi s at q2∼0which,in turn,are related to the IW functionξ(y)by heavy quarksymmetry,see eq.(12).The relation between thefitted form factors at q2=M2πand the values ofξi(1),which arefixed by the HQET,depends on the functional formadopted for the IW functionξ(y)and on the value ofρ2.Thus different values ofˆρ2areobtained byfitting the data with different models.In particular,wefind that the maindifference between NRSX and LINSR relies on the choice of the IW function,eqs.(13)and(14),rather than in the inclusion of1/m corrections.Plotting the results of thefitin the planes(ξ,f i),through eqs.(8)–(12),we obtain almost the same determinationof the f i(M2π)with both NRSX and LINSR,as shown infigs.3and4.Although wehave considered only two models in the present study,we believe that this result isquite general.We stress again that constraints on the heavy-heavy form factors can only be ob-tained by combining the results of two independentfits:thefirst whichfixes the FPusing the ratios Rπ,that are essentially independent on the model used to calculatef +(M 2π)00.20.40.60.8100.20.40.60.81ξf +(M 2π)00.20.40.60.8100.20.40.60.81ξV (M 2π)00.20.40.60.8100.20.40.60.81ξV (M 2π)00.20.40.60.8100.20.40.60.81ξFigure 3:Determination of various B →Dπform factors at q 2=M 2πas functions of ξ.These plots are obtained from fig.2using the relations connecting heavy-heavy form factors to the Isgur-Wise function,eqs.(8)–(12).LINSR (NRSX)form factors are shown in the left (right)column.The different contours correspond to χ2/dof =(1.5,2,3)×χ2min /dof .the form factors,and the second which fixes the form factors using the BR (B →Dπ)at fixed values of the FP.The comparison of the heavy-heavy form factors directly measured in semileptonicdecays at q 2=M 2π∼0with the results in figs.3and 4,and table 1,is a real test of generalized factorization in B →Dπdecays,independently on the choice of the IWfunction and of the value of ˆρ2.This checks the assumptions we made for computinghadronic matrix elements as described in the previous sections.Since we use theHQET relations coming from eq.(12),we are left with only one independent formfactor,namely the IW function.Therefore the only form factor directly measured atq 2=0,f +(0),already allows a test of our approach.Its value,f +(0)=0.66±0.06±0.04[8],showed as a band in the upper plots of fig.3,agrees well with the resultof the fit.The extraction of the other form factors at q 2=0from the CLEO datais under way [25]and would allow a check of the HQET relations among B →DA 0(M 2π)00.20.40.60.8100.20.40.60.81ξA 0(M 2π)00.20.40.60.8100.20.40.60.81ξA 1(M 2π)00.20.40.60.8100.20.40.60.81ξA 1(M 2π)00.20.40.60.8100.20.40.60.81ξA 2(M 2π)00.20.40.60.8100.20.40.60.81ξA 2(M 2π)00.20.40.60.8100.20.40.60.81ξFigure 4:The same plots as in fig.3for A 0,A 1and A 2.form factors near the maximum recoil point.Notice that,at least in principle,the fitting procedure described in this section could be used to extract independently the values of all the form factors at q2=M2π,just including them among the FP.In this case,the comparison of each form factor with the measurements from semileptonic decays would be a test of generalized factorization,independent of the HQET relations eq.(12).Unfortunately the present accuracy of the data does not allow a separate determination of the different form factors.From the discussion above,we conclude that the HQET-inspired parameterization of the heavy-heavy form factors in terms of their value at q2max and of the slopeˆρ2, which is commonly adopted by experimental collaborations and successfully applied to semileptonic decays,is not the most appropriate choice for the factorization analysis of non-leptonic decays.With the results for the FP given in(21),we can predict BR s of yet-unmeasured decay channels,having one D and one light meson in thefinal state.We list our predictions in tab.2,where the ranges in square brackets give an estimate of the theoretical uncertainties.They were found by allowing values ofχ2/dof up to three times larger than the minimum.Flavor SU(3)symmetry justifies the use of parameters obtained from B→Dπand B→KJ/Ψdecays to the decays listed in rge flavour effects are unlikely,since the factorized amplitudes already account for some SU(3)breaking.Finally,we summarize the result of ourfits of the FP by quoting their best values and ranges of variation,obtained by allowing values ofχ2/dof up to three times larger than the minimum.The comparison between the two different models,NRSX and LINSR,gives us an estimate of the theoretical uncertainties due to the form-factor model dependence.As discussed before,these FP parameters are those obtained using the coefficients functions computed at the NLO in theNRSX1.04[0.9–1.2]0.44[0.2–0.5]0.0[-1.5–1.5]|a1| 1.05[1.0–1.2]|a2|0.25[0.0–0.4]0.7[0.7–2.1]dof0.91[0.8–1.0]introduced in the previous section,and define the non-leptonic ratios[7]R P Ds (B→D(∗)D+s)=BR(B→D(∗)D+s)BR(B→D(∗)ρ+).(22)Up to color-suppressed terms,the factorized amplitudes of the decay modes consid-ered here are proportional to f Ds /fπor to f D∗s/fρ.Whereas the ratios R M of eq.(20)are defined in such a way that the main form-factor dependence drops out,in the non-leptonic ratios of eqs.(22),the form factors appearing in the numerator and de-nominator are evaluated at different q2and do not cancel out.In this case,however, it is the dependence on the FP that tends to cancel,as long as penguin contributions are neglected.The non-leptonic ratios above are exactly independent ofξandδξonly if charged B+decays are not considered,as done in ref.[7].In our case,we prefer to double the number of channels in thefit,by including charged B+decays,at the cost of introducing a small dependence on FP and on the D(∗)decay constants,both appearing in color suppressed terms in the decay amplitudes.We take f D=200MeV and f D∗=220MeV[26].In general,all DD s modes suffer from a further theoretical error.This uncertainty originates from using the same FP,obtained from thefit of sec.3,in the calculation of the relevant BR s entering the non-leptonic ratios.Since in DD s decays,the emitted meson is heavy,one may expect,according to the LEET approach,larger violations to the factorization limit.In other words,in this case the FP may significantly differ from thosefixed by the Dπand KJ/Ψmodes.This is a further source of theoretical uncertainty,which we are not able to estimate at present.Using the the semileptonic ratios R Ds (B→D(∗)D+s)and R D∗s(B→D(∗)D∗+s)andthe non-leptonic ratios of eqs.(22),the form factors determined from thefit to theBR(B→Dπ)and the FP from(21),we have extracted f Ds and f D∗s.Results arecollected in tab.5,where we have separately shown the uncertainties coming from the experimental errors on the BR s and from the errors on the FP.As before,in order to estimate this source of theoretical uncertainty,we present results obtained using both LINSR and NRSX.From tab.5,we quotef Ds =270±35MeV,f D∗s=260±30MeV,(23)where the errors indicatively account for all the sources of uncertainty.The value obtained for f Ds is in reasonable agreement with the data[9],f Ds=250±30MeV,and with the lattice results f Ds =218+20−14MeV(quenched),235+22+17−15−9MeV(unquenched)[10],although within large experimental and theoretical uncertainties.Our prediction for f D∗s is in good agreement with the quenched lattice determination,f D∗s =240±20MeV[11].Comparing the NRSX results of tab.5with the analysis of ref.[7],onefinds differ-ences of the order of10–15%.Besides our inclusion of the charged decay modes,this difference arises because we take into account contributions from penguin operators, which were neglected in ref.[7].These contributions amount up to20%in some chan-nels,in particular to those used to determine f Ds.For this reason,it is worth testingLINSRMeV semileptonic nonleptonic 304±42±47253±24±35277±36±43250±31±16of Cabibbo-allowed non-leptonic B decays in terms of factorized matrix elements and three real parametersα,ξandδξ.We have shown the connection of our parameteri-zation with the generalized factorization of ref.[7].In order tofix these parameters, we have reanalized B→Dπand B→KJ/Ψdata.We have found that thefit to B→Dπdecays substantially depends on the model describing the Isgur-Wise functionξand on the value of its slope.This dependence has been drastically reduced byfitting the ratios in eq.(20).We have shown that,once that the FP arefixed in this way,a bestfit to the non-leptonic BR(B→Dπ)determines the values of heavy-heavy form factors at q2=M2π∼0.This provides a costraint on the HQET models which are currently used for the heavy-heavy form factors.We have shown that,in general,different models require different values ofˆρ2to reproduce thefitted values of the form factors.Consequently,a meaningful test of factorization is only provided by the comparison of the values of the form factors extracted from non-leptonic decays with those directly measured,at small values of q2,in semileptonic decays.The only form factor directly measured at q2=0,f+(0)=0.66±0.06±0.04[8], is in good agreement with ourfinding,suggesting that the generalized factorization works well in the case of B→Dπdecays.Our best determination of the FP can be found in(21),where an estimate of the theoretical uncertainties is also ing these FP,we have also presented a set of predictions for the BR s of yet-unmeasured B decays,including Dπ,Dρ,DK(∗)modes, see tab.2.Finally,using non-leptonic ratios of eqs.(20)and(22),we have extracted the charmed meson decay constants from the BR(B→DD s),findingf Ds =270±45MeV,f D∗s=260±40MeV,(24)where errors indicatively account for all sources of uncertainty present in thefit,in-cluding charming penguins.AcknowledgementsWe warmly thank L.Lellouch for very useful discussions.M.Artuso kindly informed us about the current status of CLEO analyses of the B→D(∗)form factors.We acknowledge helpful comments on the experimental results by K.-C.Yang.M.C.,E.F. and G.M.thank the CERN TH Division for the hospitality during the completion of this work.We acknowledge the partial support of the MURST.References[1]M.Ciuchini,E.Franco,G.Martinelli,and L.Silvestrini,Nucl.Phys.B501(1997)271;M.Ciuchini et al.,Nucl.Phys.B512(1998)3.[2]J.-M.G´e rard and J.Weyers,UCL-IPT-97-18[hep-ph/9711469];M.Neubert,Phys.Lett.B424(1998)152;A.F.Falk,A.L.Kagan,Y.Nir,and A.A.Petrov,Phys.Rev.D57(1998)4290;。
关于 ``Banach spaces failing the almost isometric universal extension property'' 的一个注记
Journal of Mathematical Research&ExpositionAug.,2008,Vol.28,No.3,pp.613–616DOI:10.3770/j.issn:1000-341X.2008.03.022A Note of Paper“Banach Spaces Failing the AlmostIsometric Universal Extension Property”ZHAN Hua Ying1,2(1.College of Science,Tianjin University of Technology,Tianjin300384,China;2.School of Mathematics,Nankai University,Tianjin300017,China)(E-mail:zhanhuaying@)Abstract The definition of property A with constantαwas introduced by D.M.Speegle,who proved that every infinite dimensional separable uniformly smooth Banach space has propertyA with constantα∈[0,1).In this paper,we give a sufficient condition for a Banach space tohave property A with constantα∈[0,1),and some remarks on Speegle’s paper[1].Keywords property A with constantα;modulus of convexity;λ-EP;λ-UEP.Document code AMR(2000)Subject Classification46B20Chinese Library Classification O177.2The definition of property A with constantα∈[0,1)was introduced by Speegle in[1].Definition1We say a Banach space E has property A with constantαif there is a normalized weak*null sequence{e∗n},anα∈[0,1),and a normalized sequence{e n}in E such that(1) e∗n(e n)→1and(2)S(e n,α) −S(e m,α)=∅for all integers n and m,where S(e,α)={e∗∈B(E∗):e∗(e)>α}.In[1,Proposition4],Speegle showed us that every infinite dimensional separable uniformly smooth Banach space has property A with constantα∈[0,1).Observing the proof of Proposi-tion4more carefully,we can get the following theorem to give a somewhat weaker condition for an infinite dimensional Banach space to have property A with constantα∈[0,1).In this paper,X,E will stand for separable Banach spaces with infinite dimensions. Theorem1Let X be an infinite-dimensional separable Banach space.If for any sequence{y n} in S(X)we have inf nδ(y n,1/4)>0,wherex+y nδ(y n,1/4)=inf{1−Received date:2006-06-22;Accepted date:2007-01-17Foundation item:the National Natural Science Foundation of China(No.10571090);the Research Foundation for the Doctoral Program of Higher Education(No.20060055010);the Research Foundation of Tianjin Municipal Education Commission(No.20060402).614ZHAN H Yis dense in B(X).Let y∗1be in S(X∗),and let y1be in S(X)such that y∗1(y1)≥1−1/2. Let S1={x∗∈S(X∗):|x∗(y1)|<1/22,x∗(x1)<1/2}.Since X is infinite dimensional,by the Josefson-Nissenzweig’s theorem,there is a normalized sequence which is w*-null.Hence we know that S1=∅.So,let y∗2∈S1and y2∈S(X)such that y∗2(y2)≥1−1/22.Let S2={x∗∈S(X∗):|x∗(y i)|<3/23,x∗(x i)<1/3,1≤i≤2}.Similarly,S2=∅.So,let y∗3∈S2 and y3∈S(X)such that y∗3(y3)≥1−1/23.Continuing in this fashion,we gety∗n+1∈S n={x∗∈S(X∗):|x∗(y i)|<2n−1n+1,1≤i≤n},where y i∈S(X)satisfying y∗i(y i)≥1−1/2i.Obviously,the sequence{y∗n(y n)}is convergent to1.Since{x n}is dense in B(X),we can show that the sequence{y∗n}is convergent toθin the weak*topology.In fact,givenx∈B(X),ε>0,there is x m0∈{x n}such that x−x m<ε.For any large enough n withn>max{m0,1/ε},we have that y∗n(x m)<1/n for y∗n∈S n−1.Hence,|y∗n(x)|≤|y∗n(x)−y∗n(x m0)|+|y∗n(x m)|≤ x−x m+1/n≤2ε.Letα=1−inf nδ(y n,1/4).From the condition of X,we have thatα∈[0,1).So,it remains to show thatS(y i,α) −S(y j,α)=∅for all i,j∈N.Obviously,it holds for i=j.For i>j≥1,y i+y j ≥|y∗i(y i+y j)|>1−12i=12.We claim that x−y i <1/4and x+y j <1/4.Conversely,if x−y i >1/4,then we haveα=1−infn δ(y n,1/4)≥1−δ(y i,1/4)≥x+y i2>1+α2=1+3αA note of paper“Banach spaces failing the almost isometric universal extension property”615We will say that a separable space E has theλ-universal extension property(λ-UEP)if(E,X) has theλ-EP where E imbeds as a(closed)subspace of a separable space X.In[1],Speegle showed us that every Banach space satisfying property A with constantα∈[0,1)fails the(1+ε)-UEP for any0<ε<1−α.1+αProof For any weak*null sequence{e∗n}in S(E∗),define a mapΦ:E→c0byΦ(x)=(e∗n(x))n. Obviously,Φis well defined and linear.Moreover, Φ ≤1.Suppose,on the contrary,that (E,X)has the(1+ε)-into-c0extension property for some X containing E isometrically and some 0<ε<1−α616ZHAN H Y(c0,C b(K′))fails the(1+ε)-EP.In fact,for e∗n,the n th coefficient function of the basis{e n} in B(l1),letΦn be any extension of e∗n defined on C b(K′).ThenΦn|C(K)△=µn is an extension of e∗n defined on C(K).If Φn ≤(1+ε),then µn ≤(1+ε).By the proof of[1,Theorem 3],µn(f)is bounded away from0,where f is the constant one function in C(K).Hence,Φn(f) is also bounded away from0,if f is considered as an element of C b(K′).That is,the sequence {Φn}cannot converge toθin the weak*topology.By[1,Proposition2],(c0,C b(K′))fails the (1+ε)-EP.The Claim is proved.2 From the proof of Claim2above,we can get the following corollary immediately.Corollary1If(E,X)fails theλ-EP and Z is a superspace of X,then(E,Z)fails theλ-EP either.Acknowledgement The author would like to thank Professor Ding Guanggui for his guidance, and thank the referees for their valuable comments and suggestions.References[1]SPEEGLE D M.Banach spaces failing the almost isometric universal extension property[J].Proc.Amer.Math.Soc.,1998,126(12):3633–3637.。
LATTICE CALCULATIONS OF DECAY CONSTANTS a
1
computations within the static approximation 5;6;7 obtained very large answers: fB 300 MeV. I believe it is now generally accepted that the main problem here was contamination by excited states. The static approach is inherently very noisy, and without sophisticated techniques 8 that allow one to extract the signal at short times, one may easily be fooled by \false plateaus" into believing that excited state contamination is under control. In addition, it is now known that the result for fB at the physical B meson is 20% lower than at in nite b quark mass, and that the extrapolation to the continuum also tends to lower fB .
LATTICE CALCULATIONS OF DECAY CONSTANTS a
Department of Physics, Washington University, St. Louis, MO 63130, USA
Lattice attempts to compute the leptonic decay constants of heavy-light pseudoscalar mesons are described. I give a short historical overview of such attempts and then discuss some current calculations. I focus on three of the most important sources of systematic error: the extrapolation to the continuum, the chiral extrapolation in light quark mass, and the e ects of quenching. I brie y discuss the \bag parameters" B and Bs , and then conclude with my expectations of the precision in decay constants and bag parameters that will be possible in the next few years.
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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+