Study of the Reactions $B to D^ pi pi$ and $B to D^ rho pi$

a r X i v :h e p -p h /9301241v 1 15 J a n 1993December 1992BARI-TH/92-131Study of the reactions B →D ∗ππand B →D ∗ρπPietro Colangelo a ,Fulvia De Fazio a ,Giuseppe Nardulli a,b a Istituto Nazionale di Fisica Nucleare,Sezione di Bari,Italy b Dipartimento di Fisica,Universit´a di Bari,Italy ABSTRACT We evaluate the non leptonic B meson decays B →D ∗ππand B →D ∗ρπ,in the factorization approximation and in the limit of infinitely heavy quarks,assuming the dominance of intermediate positive parity charmed resonances.We find that the branching ratios are of the order 10−3.Positive parity charmed mesons have been recently investigated by a number of authors[1,2,3,4]in the framework of the Heavy Quark Effective Theory (HQET).In particular in [1]it has been shown that the semileptonic decays :B →D ∗∗ℓν(1)can be described in terms of two universal form factors,τ1/2(ω)and τ3/2(ω),where ω=v ·v ′and v µ,v ′µare the B and D ∗∗meson four-velocities respectively.In eq.(1)D ∗∗is one of the four positive parity charmed mesons:2+3/2,1+3/2,1+1/2,0+1/2,that one expects in the infinite heavy quark mass limit;here we employ the notation JPs ℓ,where s ℓ=1/2or 3/2is the total angular momentum of the light degrees of freedom.The form factor τ3/2(ω)describes the decays into the 2+3/2and 1+3/2states,whereas τ1/2(ω)is related to the 1+1/2(ω)and 0+1/2states.Both form factors have been estimated in [2]by using the QCD sum rules approach.The 2+3/2state has been observed experimentally [5]with a mass of 2460MeV :it is denoted by D ∗2(2460).The 1+3/2meson has to be basically identified with D 1(2420)1.As shown in refs.[3,4]they are both narrow (Γ≤20MeV )since their strong decays occur in D -wave,in contrast with the states D 0(the 0+1/2state)and D ′1(the 1+1/2state)that canalso decay by S -wave.On the experimental side,some evidence has been gathered on the semileptonic decays (1)[6,7],but it is not yet conclusive.A different way to study the transition B →D ∗∗is by the non leptonic reactions [8]:B →D ∗ππ(2)B →D ∗ρπ(3)which can occur by the intermediate production of positive parity resonances with subse-quent decay into D ∗π.Examples of these processes are as follows:B −→D ∗∗0π−→D ∗+π−π−(4)B −→D ∗∗0ρ−→D ∗+π−ρ−(5)¯B 0→D ∗0π+π−(6)where in the last case we could have either D ∗0ρ0or D ∗∗+π−as intermediate resonant states.In this letter we wish to study the processes (4-6)in the framework of the factorization approximation.This approach has first been proposed by Feynman [9],and then has been extensively applied by Bauer,Stech and Wirbel (BWS)[10]and by a number of other authors 2.As well known,in this approximation one considers matrix elements of the weak non leptonic effective hamiltonianH NL =G 2V ∗cb V ud :a 1¯c γµ(1−γ5)b ¯dγµ(1−γ5)u +a 2¯dγµ(1−γ5)b ¯c γµ(1−γ5)u :(7)and evaluates them by inserting the hadronic vacuum state between the V −A quark currents appearing in (7);the constants a 1and a 2are given by:a 1=c 1+c 2N c (9)where c 1=1.1and c 2=−0.24are Wilson coefficients evaluated at the b −quark mass scale.As for N c ,experimental data for B →Dπand B →Dρseem to favour the value of the BWS model [10]N =∞(see e.g.[11])instead of the value N c =3;we shall comment on this point later on.A test of factorization has been worked out in ref.[13],where the experimental ratio between the ¯B0→D ∗+π−non leptonic width and semileptonic spectrum d Γ(¯B 0→D ∗+ℓ−¯ν)m B m D ξ(v ·v ′)(v +v ′)µ(10)<D ∗(v ′,ǫ)|(V −A )µ(0)|B (v )>=√m B m D ∗∗τ1/2(v ·v ′)(v −v ′)µ(12)<D ′1(v ′,ǫ)|(V −A )µ(0)|B (v )>=√m B m D ∗∗τ3/2(v ·v ′)[(1−(v ·v ′)2)2ǫ∗µ+(ǫ∗·v )2[−3v µ+(v ·v ′−2)v ′µ]+i (v ·v ′−1)2ǫµναβǫ∗ν(v +v ′)α(v −v ′)β] (14)<D ∗2(v ′,ǫ)|(V −A )µ(0)|B (v )>=√33(v ·v ′+1)ǫ∗µαv α+√2m B m D ∗∗|q 2≃m 2π,m 2ρin our case,we use τ1/2(v ·v ′)=0.20and τ3/2(v ·v ′)=0.19.We have also to consider the matrix elements of the currents between the vacuum and the charmed resonances in the m Q→∞limit:<0|Vµ(0)|D0(p)>=if(+)/√m cǫµ(17) where f(+)depends only logarithmically on the heavy quark mass and has been determined in ref.[2]by QCD sum rules(m c=1.35GeV):f(+)≃0.46GeV3/2.(18) In contrast,the matrix elements between the vacuum and the sℓ=3/2resonances vanish in the m Q→∞limit[2].Finally,we have to consider the weak current matrix elements between B and a light mesonπ,ρ;they can be written as follows:<π(p′)|Vµ|B(p)>=(pµ+p′µ−(m2B−m2π)q2qµF0(q2)(19) <ρ(p′,ǫ∗)|Vµ−Aµ|B(p)>=ǫµνρσǫ∗νpρp′σ2V(q2)(m B+mρ)(p+p′)µ+(ǫ∗·p)qµΓ(B−→D∗∗0(1+3/2)π−)andΓ(B−→D∗∗0(2+3/2)ρ−)Γ(B−→D∗∗0(1+3/2)π−)=4z(1+√(1−√z)2−y]2(21)andΓ(B−→D∗∗0(2+3/2)ρ−)z)2]zω2+5ω−6√m B )2,y=(mπ,mρ2√than10%from the asymptotic(m Q→∞)value reported in Table I,which provides a perhaps optimistic estimate of the theoretical errors introduced by the heavy quark mass limit.If we observe that uncertainties in the phenomenological value of N c(see eqs.(8,9)) would strongly affect only the B−decays into the0+1/2state,which however does not decay into D∗πin the infinite heavy quark limit,we can conclude that our predictions (26)-(28)should provide reasonable estimates of the non leptonic B−decays into charmed multipionfinal states.AcknowledgmentsWe thank S.Stone for having suggested this calculation,and N.Paver for useful dis-cussions.References[1]N.Isgur and M.B.Wise,Phys.Rev.D43(1991)819[2]P.Colangelo,G.Nardulli and N.Paver,Phys.Lett.B293(1992)207[3]A.F.Falk and M.Luke,Phys.Lett.B292(1992)119[4]U.Kilian,J.G.Korner and D.Pirjol,Phys.Lett.B288(1992)360[5]Particle Data Group,Review of Particle Properties,Phys.Rev.D45(1992)[6]CLEO Collab.,R.Fulton et al.,Phys.Rev.D43(1991)651;CLEO Collab.,S.Henderson et al.,Phys.Rev.D45(1992)2212[7]ARGUS Collab.,H.Albrecht et al.,preprint DESY92-146,october1992[8]S.Stone,private communication.[9]R.P.Feynman,”Symmetries in elementary particle physics”,A.Zichichi ed.p.167,Academic Press(N.Y.)1965[10]M.Bauer,B.Stech and M.Wirbel,Z.Phys.C-Particles and Fields34(1987)103[11]M.Tanimoto,K.Goda and K.Senba,Phys.Rev.D42(1990)3741[12]L.L.Chau,H.Y.Cheng,W.K.Sze,B.Tseng and H.Yao,Phys.Rev.D45(1992)3143[13]D.Bortoletto and S.Stone,Phys.Rev.Lett.65(1990)2951[14]N.Isgur and M.Wise,Phys.Lett.B232(1989)113[15]A.F.Falk,Nucl.Phys.B378(1992)79[16]M.B.Wise,Phys.Rev.D45(1992)R2188[17]R.Casalbuoni,A.Deandrea,N.Di Bartolomeo,F.Feruglio,R.Gatto and G.Nardulli,Preprint UGVA-DPT1992/11-790,BARI-TH/92-125(November1992)[18]N.Isgur,D.Scora,B.Grinstein and M.Wise,Phys.Rev.D39(1989)799[19]S.Godfrey and N.Isgur,Phys.Rev.D32(1985)189[20]A.Abada et al.,Nucl.Phys.B376(1992)172;C.R.Allton et al.,Nucl.Phys.B349(1991)1[21]M.E.Luke,Phys.Lett.B252(1990)447[22]P.Colangelo,G.Nardulli,A.A.Ovchinnikov and N.Paver,Phys.Lett.B269(1991)201Table CaptionsTab.I Widths and branching ratios(B.R.)of B−→D∗∗0(J P sℓ)π−.The B−lifetime is τB=1.2×10−12sec.The branching ratios are obtained for a1=1.1,a2=−0.24 and V cb=0.045.Tab.II Widths and branching ratios(B.R.)of B−→D∗∗0(J P sℓ)ρ−.The B−lifetime is τB=1.2×10−12sec.The branching ratios are obtained for a1=1.1,a2=−0.24 and V cb=0.045.Table IJ P sWidth(GeV) B.R.ℓ)2610−40.045)2410−40.045)2610−40.045)2310−30.045Table IIWidth(GeV) B.R. J P sℓ)2110−30.045)2110−30.045)2110−30.045)2410−50.045。