a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark effective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suffices to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus effectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the phase redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which define the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the effective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coefficient function.possible subtraction terms.Because of theflavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The effective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usfirst present the result for the nonperturbative power corrections.WefindK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we find that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our final result,there is one more effect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the effective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the effective theory[16,18].Onefinds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,wefind stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourfinal result forχ3(y)is not very different from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisfigure one can also deduce how changes in the values of the vacuum condensates would affect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Wefind that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonfield and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesfinancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientific,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisfigure and are both represented by the narrow dots.11。
a r X i v :h e p -e x /0207007v 1 1 J u l 2002BELLEBelle Prerpint 2002-18KEK Preprint 2002-59Study of B →ρπdecays at BelleBelle Collaboration A.Gordon u ,Y.Chao z ,K.Abe h ,K.Abe aq ,N.Abe at ,R.Abe ac ,T.Abe ar ,Byoung Sup Ahn o ,H.Aihara as ,M.Akatsu v ,Y.Asano ay ,T.Aso aw ,V.Aulchenko b ,T.Aushev ℓ,A.M.Bakich an ,Y.Ban ag ,A.Bay r ,I.Bedny b ,P.K.Behera az ,jak m ,A.Bondar b ,A.Bozek aa ,M.Braˇc ko t ,m ,T.E.Browder g ,B.C.K.Casey g ,M.-C.Chang z ,P.Chang z ,B.G.Cheon am ,R.Chistov ℓ,Y.Choi am ,Y.K.Choi am ,M.Danilov ℓ,L.Y.Dong j ,J.Dragic u ,A.Drutskoy ℓ,S.Eidelman b ,V.Eiges ℓ,Y.Enari v ,C.W.Everton u ,F.Fang g ,H.Fujii h ,C.Fukunaga au ,N.Gabyshev h ,A.Garmash b ,h ,T.Gershon h ,B.Golob s ,m ,R.Guo x ,J.Haba h ,T.Hara ae ,Y.Harada ac ,N.C.Hastings u ,H.Hayashii w ,M.Hazumi h ,E.M.Heenan u ,I.Higuchi ar ,T.Higuchi as ,L.Hinz r ,T.Hokuue v ,Y.Hoshi aq ,S.R.Hou z ,W.-S.Hou z ,S.-C.Hsu z ,H.-C.Huang z ,T.Igaki v ,Y.Igarashi h ,T.Iijima v ,K.Inami v ,A.Ishikawa v ,H.Ishino at ,R.Itoh h ,H.Iwasaki h ,Y.Iwasaki h ,H.K.Jang a ℓ,J.H.Kang bc ,J.S.Kang o ,N.Katayama h ,Y.Kawakami v ,N.Kawamura a ,T.Kawasaki ac ,H.Kichimi h ,D.W.Kim am ,Heejong Kim bc ,H.J.Kim bc ,H.O.Kim am ,Hyunwoo Kim o ,S.K.Kim a ℓ,T.H.Kim bc ,K.Kinoshita e ,S.Korpar t ,m ,P.Krokovny b ,R.Kulasiri e ,S.Kumar af ,A.Kuzmin b ,Y.-J.Kwon bc ,nge f ,ai ,G.Leder k ,S.H.Lee a ℓ,J.Li ak ,A.Limosani u ,D.Liventsevℓ,R.-S.Lu z,J.MacNaughton k,G.Majumder ao, F.Mandl k,D.Marlow ah,S.Matsumoto d,T.Matsumoto au,W.Mitaroffk,K.Miyabayashi w,Y.Miyabayashi v,H.Miyake ae,H.Miyata ac,G.R.Moloney u,T.Mori d,T.Nagamine ar,Y.Nagasaka i,T.Nakadaira as,E.Nakano ad, M.Nakao h,J.W.Nam am,Z.Natkaniec aa,K.Neichi aq, S.Nishida p,O.Nitoh av,S.Noguchi w,T.Nozaki h,S.Ogawa ap, T.Ohshima v,T.Okabe v,S.Okuno n,S.L.Olsen g,Y.Onuki ac, W.Ostrowicz aa,H.Ozaki h,P.Pakhlovℓ,H.Palka aa,C.W.Park o,H.Park q,L.S.Peak an,J.-P.Perroud r, M.Peters g,L.E.Piilonen ba,J.L.Rodriguez g,F.J.Ronga r, N.Root b,M.Rozanska aa,K.Rybicki aa,H.Sagawa h,S.Saitoh h,Y.Sakai h,M.Satapathy az,A.Satpathy h,e,O.Schneider r,S.Schrenk e,C.Schwanda h,k,S.Semenovℓ,K.Senyo v,R.Seuster g,M.E.Sevior u,H.Shibuya ap,V.Sidorov b,J.B.Singh af,S.Staniˇc ay,1,M.Stariˇc m,A.Sugi v, A.Sugiyama v,K.Sumisawa h,T.Sumiyoshi au,K.Suzuki h,S.Suzuki bb,S.Y.Suzuki h,T.Takahashi ad,F.Takasaki h, K.Tamai h,N.Tamura ac,J.Tanaka as,M.Tanaka h,G.N.Taylor u,Y.Teramoto ad,S.Tokuda v,S.N.Tovey u,T.Tsuboyama h,T.Tsukamoto h,S.Uehara h,K.Ueno z, Y.Unno c,S.Uno h,hiroda h,G.Varner g,K.E.Varvell an,C.C.Wang z,C.H.Wang y,J.G.Wang ba,M.-Z.Wang z,Y.Watanabe at,E.Won o,B.D.Yabsley ba,Y.Yamada h, A.Yamaguchi ar,Y.Yamashita ab,M.Yamauchi h,H.Yanai ac,P.Yeh z,Y.Yuan j,Y.Yusa ar,J.Zhang ay,Z.P.Zhang ak,Y.Zheng g,and D.ˇZontar aya Aomori University,Aomori,Japanb Budker Institute of Nuclear Physics,Novosibirsk,Russiac Chiba University,Chiba,Japand Chuo University,Tokyo,Japane University of Cincinnati,Cincinnati,OH,USAf University of Frankfurt,Frankfurt,Germanyg University of Hawaii,Honolulu,HI,USAh High Energy Accelerator Research Organization(KEK),Tsukuba,Japani Hiroshima Institute of Technology,Hiroshima,Japanj Institute of High Energy Physics,Chinese Academy of Sciences,Beijing,PRChinak Institute of High Energy Physics,Vienna,Austria ℓInstitute for Theoretical and Experimental Physics,Moscow,Russiam J.Stefan Institute,Ljubljana,Slovenian Kanagawa University,Yokohama,Japano Korea University,Seoul,South Koreap Kyoto University,Kyoto,Japanq Kyungpook National University,Taegu,South Korear Institut de Physique des Hautes´Energies,Universit´e de Lausanne,Lausanne,Switzerlands University of Ljubljana,Ljubljana,Sloveniat University of Maribor,Maribor,Sloveniau University of Melbourne,Victoria,Australiav Nagoya University,Nagoya,Japanw Nara Women’s University,Nara,Japanx National Kaohsiung Normal University,Kaohsiung,Taiwany National Lien-Ho Institute of Technology,Miao Li,Taiwanz National Taiwan University,Taipei,Taiwanaa H.Niewodniczanski Institute of Nuclear Physics,Krakow,Polandab Nihon Dental College,Niigata,Japanac Niigata University,Niigata,Japanad Osaka City University,Osaka,Japanae Osaka University,Osaka,Japanaf Panjab University,Chandigarh,Indiaag Peking University,Beijing,PR Chinaah Princeton University,Princeton,NJ,USAai RIKEN BNL Research Center,Brookhaven,NY,USAaj Saga University,Saga,Japanak University of Science and Technology of China,Hefei,PR ChinaaℓSeoul National University,Seoul,South Koreaam Sungkyunkwan University,Suwon,South Koreaan University of Sydney,Sydney,NSW,Australiaao Tata Institute of Fundamental Research,Bombay,Indiaap Toho University,Funabashi,Japanaq Tohoku Gakuin University,Tagajo,Japanar Tohoku University,Sendai,Japanas University of Tokyo,Tokyo,Japanat Tokyo Institute of Technology,Tokyo,Japanau Tokyo Metropolitan University,Tokyo,Japanav Tokyo University of Agriculture and Technology,Tokyo,Japanaw Toyama National College of Maritime Technology,Toyama,Japanay University of Tsukuba,Tsukuba,Japanaz Utkal University,Bhubaneswer,Indiaba Virginia Polytechnic Institute and State University,Blacksburg,VA,USAbb Yokkaichi University,Yokkaichi,Japanbc Yonsei University,Seoul,South KoreaB events collected with the Belle detector at KEKB.Thebranching fractions B(B+→ρ0π+)=(8.0+2.3+0.7−2.0−0.7)×10−6and B(B0→ρ±π∓)=(20.8+6.0+2.8−6.3−3.1)×10−6are obtained.In addition,a90%confidence level upper limitof B(B0→ρ0π0)<5.3×10−6is reported.Key words:ρπ,branching fractionPACS:13.25.hw,14.40.Nd1on leave from Nova Gorica Polytechnic,Nova Gorica,Sloveniamodes are examined.Here and throughout the text,inclusion of charge con-jugate modes is implied and for the neutral decay,B0→ρ±π∓,the notation implies a sum over both the modes.The data sample used in this analysis was taken by the Belle detector[9]at KEKB[10],an asymmetric storage ring that collides8GeV electrons against3.5GeV positrons.This produces Υ(4S)mesons that decay into B0B pairs.The Belle detector is a general purpose spectrometer based on a1.5T su-perconducting solenoid magnet.Charged particle tracking is achieved with a three-layer double-sided silicon vertex detector(SVD)surrounded by a central drift chamber(CDC)that consists of50layers segmented into6axial and5 stereo super-layers.The CDC covers the polar angle range between17◦and 150◦in the laboratory frame,which corresponds to92%of the full centre of mass(CM)frame solid angle.Together with the SVD,a transverse momen-tum resolution of(σp t/p t)2=(0.0019p t)2+(0.0030)2is achieved,where p t is in GeV/c.Charged hadron identification is provided by a combination of three devices: a system of1188aerogelˇCerenkov counters(ACC)covering the momentum range1–3.5GeV/c,a time-of-flight scintillation counter system(TOF)for track momenta below1.5GeV/c,and dE/dx information from the CDC for particles with very low or high rmation from these three devices is combined to give the likelihood of a particle being a kaon,L K,or pion, Lπ.Kaon-pion separation is then accomplished based on the likelihood ratio Lπ/(Lπ+L K).Particles with a likelihood ratio greater than0.6are identified as pions.The pion identification efficiencies are measured using a high momentum D∗+data sample,where D∗+→D0π+and D0→K−π+.With this pion selection criterion,the typical efficiency for identifying pions in the momentum region0.5GeV/c<p<4GeV/c is(88.5±0.1)%.By comparing the D∗+data sample with a Monte Carlo(MC)sample,the systematic error in the particle identification(PID)is estimated to be1.4%for the mode with three charged tracks and0.9%for the modes with two.Surrounding the charged PID devices is an electromagnetic calorimeter(ECL) consisting of8736CsI(Tl)crystals with a typical cross-section of5.5×5.5cm2 at the front surface and16.2X0in depth.The ECL provides a photon energy resolution of(σE/E)2=0.0132+(0.0007/E)2+(0.008/E1/4)2,where E is in GeV.Electron identification is achieved by using a combination of dE/dx measure-ments in the CDC,the response of the ACC and the position and shape of the electromagnetic shower from the ECL.Further information is obtained from the ratio of the total energy registered in the calorimeter to the particle momentum,E/p lab.Charged tracks are required to come from the interaction point and have transverse momenta above100MeV/c.Tracks consistent with being an elec-tron are rejected and the remaining tracks must satisfy the pion identification requirement.The performance of the charged track reconstruction is studied using high momentumη→γγandη→π+π−π0decays.Based on the relative yields between data and MC,we assign a systematic error of2%to the single track reconstruction efficiency.Neutral pion candidates are detected with the ECL via their decayπ0→γγ. Theπ0mass resolution,which is asymmetric and varies slowly with theπ0 energy,averages toσ=4.9MeV/c2.The neutral pion candidates are selected fromγγpairs by requiring that their invariant mass to be within3σof the nominalπ0mass.To reduce combinatorial background,a selection criteria is applied to the pho-ton energies and theπ0momenta.Photons in the barrel region are required to have energies over50MeV,while a100MeV requirement is made for photons in the end-cap region.Theπ0candidates are required to have a momentum greater than200MeV/c in the laboratory frame.Forπ0s from BE2beam−p2B and the energy difference∆E=E B−E beam.Here, p B and E B are the momentum and energy of a B candidate in the CM frame and E beam is the CM beam energy.An incorrect mass hypothesis for a pion or kaon produces a shift of about46MeV in∆E,providing extra discrimination between these particles.The width of the M bc distributions is primarily due to the beam energy spread and is well modelled with a Gaussian of width 3.3MeV/c2for the modes with a neutral pion and2.7MeV/c2for the mode without.The∆E distribution is found to be asymmetric with a small tail on the lower side for the modes with aπ0.This is due toγinteractions withmaterial in front of the calorimeter and shower leakage out of the calorimeter. The∆E distribution can be well modelled with a Gaussian when no neutral particles are present.Events with5.2GeV/c2<M bc<5.3GeV/c2and|∆E|< 0.3GeV are selected for thefinal analysis.The dominant background comes from continuum e+e−→qB events and jet-like qi,j|p i||p j|P l(cosθij)i,k|p i||p k|,r l=),where L s and L qqD0π+ decays.By comparing the yields in data and MC after the likelihood ratiorequirement,the systematic errors are determined to be4%for the modes with aπ0and6%for the mode without.Thefinal variable used for continuum suppression is theρhelicity angle,θh, defined as the angle between the direction of the decay pion from theρin the ρrest frame and theρin the B rest frame.The requirement of|cosθh|>0.3 is made independently of the likelihood ratio as it is effective in suppressing the background from B decays as well as the qB events is used[14].The largest component of this background is found to come from decays of the type B→Dπ;when the D meson decays via D→π+π−,events can directly reach the signal region while the decay D→K−π+can reach the signal region with the kaon misidentified as a pion.Decays with J/ψandψ(2S) mesons can also populate the signal region if both the daughter leptons are misidentified as pions.These events are excluded by making requirements on the invariant mass of the intermediate particles:|M(π+π−)−M D0|>0.14 GeV/c2,|M(π+π0)−M D+|>0.05GeV/c2,|M(π+π−)−M J/ψ|>0.07GeV/c2 and|M(π+π−)−Mψ(2S)|>0.05GeV/c2.The widest cut is made around the D0mass to account for the mass shift due to misidentifying the kaons in D0 decays as pions.Fig.1shows the∆E and M bc distributions for the three modes analysed after all the selection criteria have been applied.The∆E and M bc plots are shown for events that lie within3σof the nominal M bc and∆E values,respectively. The signal yields are obtained by performing maximum likelihoodfits,each using a single signal function and one or more background functions.The signal functions are obtained from the MC and adjusted based on comparisons of B+→B0are assumed to be equal.The M bc distribution for all modes isfitted with a single Gaussian and an ARGUS background function[15].The normalization of the ARGUS function is left tofloat and shape of the function isfixed from the∆E sideband:−0.25 GeV<∆E<−0.08GeV and5.2GeV/c2<M bc<5.3GeV/c2.For the mode with only charged pions in thefinal state,the∆E distribution isfitted with a single Gaussian for the signal and a linear function withfixed shape for the continuum background.The normalization of the linear function is left to float and the slope isfixed from the M bc sideband,5.2GeV/c2<M bc<5.26GeV/c2,|∆E|<0.3GeV.There are also other rare B decays that are expected to contaminate the∆E distribution.For the mode without aπ0,these modes are of the type B0→h+h−(where h denotes aπor K),B→ρρ(including all combinations of charged and neutralρmesons,where the polarizations of theρmesons are assumed to be longitudinal)and B→Kππ(including the decays B+→ρ0K+,B+→K∗0π+,B+→K∗0(1430)0π+,B+→f0(980)K+ and B+→f0(1370)K+)[16].These background modes are accounted for by using smoothed histograms whose shapes have been determined by combining MC distributions.The three B→ρρmodes are combined into one histogram. The normalization of this component is allowed tofloat in thefit due to the uncertainty in the branching fractions of the B→ρρmodes.Likewise,the B→hh and all the B→Kππmodes are combined to form one hh and one Kππcomponent.The normalizations of these components arefixed to their expected yields,which are calculated using efficiencies determined by MC and branching fractions measured by previous Belle analyses[16,17].The∆Efits for the modes with aπ0in thefinal state have the signal compo-nent modelled by a Crystal Ball function[18]to account for the asymmetry in the∆E distribution.As for the B+→ρ0π+mode,the continuum background is modelled by a linear function withfixed slope.Unlike the B+→ρ0π+mode, a component is included for the background from the b→c transition.The pa-rameterization for rare B decays includes one component for the B→Kππ0 modes(B0→ρ+K−and B0→K∗+π−)[19]and one for all the B→ρρmodes.The normalization of the B→ρρcomponent is left tofloat while the other components from B decays arefixed to their expected yields.Table1summarizes the results of the∆Efits,showing the number of events, signal yields,reconstruction efficiencies,statistical significance and branching fractions or upper limits for eachfit.The statistical significance is defined assystematic error in thefitted signal yield is estimated by independently varying eachfixed parameter in thefit by1σ.Thefinal results are B(B+→ρ0π+)=(8.0+2.3+0.7−2.0−0.7)×10−6and B(B0→ρ±π∓)=(20.8+6.0+2.8−6.3−3.1)×10−6where thefirst error is statistical and the second is systematic.For theρ0π0mode,one standard deviation of the systematic error is added to the statistical limit to obtain a conservative upper limit at90%confidence of5.3×10−6.The possibility of a nonresonant B→πππbackground is also examined.To check for this type of background,the M bc and∆E yields are determined for differentππinvariant mass bins.Byfitting the M bc distribution inππinvariant mass bins with B→ρπand B→πππMC distributions,the nonresonant contribution is found to be below4%.To account for this possible background, errors3.7%and3.2%are added in quadrature to the systematic errors of the ρ+π−andρ0π+modes,respectively.Theππinvariant mass distributions are shown in Fig.2.Two plots are shown for theρ+π−andρ0π+modes,one with events from the M bc sideband superimposed over the events from the signal region(upper)and one with events from signal MC superimposed over events from the signal region with the sideband subtracted(lower).Fig.3 shows the distribution of the helicity variable,cosθh,for the two modes with all selection criteria applied except the helicity condition.Events fromρπdecays are expected to follow a cos2θdistribution while nonresonant and other background decays have an approximately uniform distribution.The helicity plots are obtained byfitting the M bc distribution in eight helicity bins ranging from−1to1.The M bc yield is then plotted against the helicity bin for each mode and the expected MC signal distributions are superimposed.Both the ππmass spectrum and the helicity distributions provide evidence that the signal events are consistent with being fromρπdecays.The results obtained here can be used to calculate the ratio of branching frac-tions R=B(B0→ρ±π∓)/B(B+→ρ0π+),which gives R=2.6±1.0±0.4, where thefirst error is statistical and second is systematic.This is consistent with values obtained by CLEO[20]and BaBar[21,22]as shown in Table2. Theoretical calculations done at tree level assuming the factorization approx-imation for the hadronic matrix elements give R∼6[3].Calculations that include penguin contributions,off-shell B∗excited states or additionalππres-onances[4–8]might yield better agreement with the the measured value of R.In conclusion,statistically significant signals have been observed in the B→ρπmodes using a31.9×106BWe wish to thank the KEKB accelerator group for the excellent operation of the KEKB accelerator.We acknowledge support from the Ministry of Ed-ucation,Culture,Sports,Science,and Technology of Japan and the Japan Society for the Promotion of Science;the Australian Research Council and the Australian Department of Industry,Science and Resources;the National Science Foundation of China under contract No.10175071;the Department of Science and Technology of India;the BK21program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation;the Polish State Committee for Scientific Research under contract No.2P03B17017;the Ministry of Science and Technology of the Russian Federation;the Ministry of Education,Science and Sport of the Republic of Slovenia;the National Science Council and the Ministry of Education of Taiwan;and the U.S.Department of Energy.References[1] A.E.Snyder and H.R.Quinn,Phys.Rev.D48,2139(1993).[2]I.Bediaga,R.E.Blanco,C.G¨o bel,and R.M´e ndez-Galain,Phys.Rev.Lett.81,4067(1998).[3]M.Bauer,B.Stech,and M.Wirbel,Z.Phys.C34,103(1987).[4] A.Deandrea et al.,Phys.Rev.D62,036001(2000).[5]Y.H.Chen,H.Y.Cheng,B.Tseng and K.C.Yang,Phys.Rev.D60,094014(1999).[6] C.D.Lu and M.Z.Yang,Eur.Phys.J C23,275(2002).[7]J.Tandean and S.Gardner,SLAC-PUB-9199;hep-ph/0204147.[8]S.Gardner and Ulf-G.Meißner,Phys.Rev.D65,094004(2002).[9]Belle Collaboration,A.Abashian et al.,Nucl.Instr.and Meth.A479,117(2002).[10]E.Kikutani ed.,KEK Preprint2001-157(2001),to appear in Nucl.Instr.andMeth.A.[11]G.C.Fox and S.Wolfram,Phys.Rev.Lett.41,1581(1978).[12]This modification of the Fox-Wolfram moments wasfirst proposed in a seriesof lectures on continuum suppression at KEK by Dr.R.Enomoto in May and June of1999.For a more detailed description see Belle Collaboration,K.Abe et al.,Phys.Lett.B511,151(2001).[13]CLEO Collaboration,D.M.Asner et al.,Phys.Rev.D53,1039(1996).[14]These MC events are generated with the CLEO group’s QQ program,see/public/CLEO/soft/QQ.The detector response is simulated using GEANT,R.Brun et al.,GEANT 3.21,CERN Report DD/EE/84-1,1984.[15]The ARGUS Collaboration,H.Albrecht et al.,Phys.Lett.B241,278(1990).[16]Belle Collaboration,A.Garmash et al.,Phys.Rev.D65,092005(2002).[17]Belle Collaboration,K.Abe et al.,Phys.Rev.Lett.87,101801(2001).[18]J.E.Gaiser et al.,Phys.Rev.D34,711(1986).[19]Belle Collaboration,K.Abe et al.,BELLE-CONF-0115,submitted as acontribution paper to the2001International Europhysics Conference on High Energy Physics(EPS-HEP2001).[20]CLEO Collaboration,C.P.Jessop et al.,Phys.Rev.Lett.85,2881(2000).[21]Babar Collaboration,B.Aubert et al.,submitted as a contribution paper tothe20th International Symposium on Lepton and Photon Interactions at High Energy(LP01);hep-ex/0107058.[22]BaBar Collaboration,B.Aubert et al.,submitted as a contribution paper tothe XXXth International Conference on High Energy Physics(ICHEP2000);hep-ex/0008058.Table1∆Efit results.Shown for each mode are the number of events in thefit,the signal yield,the reconstruction efficiency,the branching fraction(B)or90%confidence level upper limit(UL)and the statistical significance of thefit.Thefirst error in the branching fraction is statistical,the second is systematic.ρ0π+15424.3+6.9−6.29.68.0+2.3+0.7−2.0−0.74.4σρ+π−30144.6+12.8−13.46.820.8+6.0+2.8−6.3−3.13.7σρ0π0116−4.4±8.58.5<5.3-Experiment B(B0→ρ±π∓)×10−6B(B+→ρ0π+)×10−6RE v e n t s /16 M e VE v e n t s /3 M e V /c2(b) ρ0π+Signal backgrd02.557.51012.51517.52022.55.25.225 5.25 5.2755.3E v e n t s /18 M e VE v e n t s /2 M e V /c2(d) ρ+π-Signal backgrd051015202530355.25.225 5.25 5.2755.3∆E(GeV)E v e n t s /18 M e V(e) ρ0π024681012-0.2-0.10.10.2(GeV/c 2)E v e n t s /2 M e V /c2M bc (f) ρ0πSignal backgrd02468101214165.25.225 5.25 5.2755.3Fig.1.The ∆E (left)and M bc (right)fits to the three B →ρπmodes:ρ0π+,ρ+π−and ρ0π0.The histograms show the data,the solid lines show the total fit and the dashed lines show the continuum component.In (a)the contribution from the B →ρρand B →hh modes is shown by the cross hatched component.In (c)the cross hatched component shows the contribution from the b →c transition and B →ρρmodes.102030405060+0(GeV/c 2)E v e n t s /0.1 G e V /c2M(π+π0)(GeV/c 2)E v e n t s /0.1 G e V /c2(GeV/c 2)E v e n t s /0.1 G e V /c2+-(GeV/c 2)E v e n t s /0.1 G e V /c2M(π+ π-)510152025Fig.2.The M (ππ)distributions for B 0→ρ±π∓(left)and B +→ρ0π+(right)events in the signal region.Plots (a)and (b)show sideband events superimposed;plots (c)and (d)show the sideband subtracted plots with signal MC superimposed.-1-0.500.51M b c y i e l d (E v e n t s )cos θh-1-0.500.51M b c y i e l d (E v e n t s )cos θhFig.3.The ρmeson helicity distributions for B 0→ρ±π∓(a)and B +→ρ0π+(b).Signal MC distributions are shown superimposed.。
编者按：大地涵藏万物，孕育生命，被誉为人类的母亲。
但是，近年来，伴随我国工业化的快速发展，大地不断遭到各种污染的伤害。
仅仅因土壤污染防治不足、环境监管乏力，导致的食品药品安全事件就频频发生，2008年以来，全国已发生百余起重大污染事故。
目前我国大地污染现状严峻，成因十分复杂，形成令人扼腕的“大地之殇”。
《经济参考报》以此为主题，探寻大地污染背后所触及的我国农业、工业、城市化进程中关于生存与发展的一系列深层矛盾与两难抉择，并以“大地之殇”系列报道的形式在“深度”版推出，敬请关注。
大地之殇一·黑土地之悲占全国粮食总产五分之一的东北黑土区是我国最重要的商品粮基地，但一个并不为多数人了解的严峻事实是，支撑粮食产量的黑土层却在过去半个多世纪里减少了50%，并在继续变薄，几百年才形成一厘米的黑土层正以每年近一厘米的速度消失。
照此速度，部分黑土层或将在几十年后消失殆尽，东北这一中国最大粮仓的产能也将遭受无法挽回的损失。
□记者孙彬管建涛连振祥吉哲鹏娄辰李松南京哈尔滨兰州昆明济南重庆报道毒土：GDP至上的恶果当前，我国土壤污染出现了有毒化工和重金属污染由工业向农业转移、由城区向农村转移、由地表向地下转移、由上游向下游转移、由水土污染向食品链转移的趋势，逐步积累的污染正在演变成污染事故的频繁爆发。
日益加剧的污染趋势可能还要持续30年“目前，我国土壤污染呈日趋加剧的态势，防治形势十分严峻。
”多年来，中国土壤学会副理事长、中国农业科学院研究员张维理教授一直关注我国土壤污染问题“我国土壤污染呈现一种十分复杂的特点，呈现新老污染物并存、无机有机污染混合的局面。
”“现在我国土壤污染比各国都要严重，日益加剧的污染趋势可能还要持续30年。
”中国土壤学专家，南京农业大学教授潘根兴告诉《经济参考报》记者，这些污染包括随经济发展日益普遍的重金属污染、以点状为主的化工污染、塑料电子废弃物污染及农业污染等。
国土资源部统计表明，目前全国耕种土地面积的10%以上已受重金属污染。
