11 June Physics Letters B D. Delepine, J.-M. Gerard, J. Pestieau, J. Weyers

Transcription

1 June 998 hyic Lee B Final ae ineacin hae in B K D Deleine, J-M Gead, J eieau, J Weye decay amliude Iniu de hyique Theique, UniÕeie cahlique de LuÕain, B-348 LuÕain-la-NeuÕe, Belgium Received Mach 998 Edi: V Lhff Abac A imle Regge le mdel f K caeing exlain he lage hae e id beween iin amliude which i beved a he D men ma df I edic df48y8 a he B ma Imlicain f B K decay exenin f he mdel he w-bdy decay channel ae biefly dicued q 998 Elevie Science BV All igh eeved Inducin Wih B-facie fhcming, deailed check f he ecie C-vilain aen ediced by he ad mdel will becme ible weve i i by n mean ivial exac eliable infmain n C-vilain aamee fm vaiu B-decay mde One f he blem i f cue hw eimae hadnic effec uch a final ae ineacin FSI hae Alhugh hee hae ae f n aicula inee by hemelve, hey d lay an iman le f many enial ignal f C vilain in hadnic B-decay The elevan quein cncening hee FSI hae i whehe hey ae ignificanly diffeen fm n Clealy he anwe hi quein deend n he hadnic channel cnideed ee we will fcu u aenin n K channel whee exeimenal daa al exi f B decay wx Thee ae w iin invaian caeing amliude I I3 he quaniy ne wan eimae, a a funcin f enegy, i d d yd namely 3 he diffeence beween he S-wave hae hif in he I 3 d I d 3 amliude A a mae f fac d ha been meaued a he D ma m whee i i fund wx D be aund Naively ne de n exec uch a huge FSI angle a m D becme negligible a m B bu, bviuly, a me quaniaive agumen i called f The main ue f hi lee i ugge a Regge mdel a a geneal aegy f deemining FSI angle wx 3 a exeience wih N KN caeing amliude ngly ugge ha uch a mdel huld wk quie well f K caeing ve an enegy ange which include he D B men mae The dminan Regge exchange cnide in K K caeing ae, eecively, he men he exchange degeneae y f ajecie in he -channel in he u-channel he exchange degeneae K ) yk )) ajecie In he nex ecin we biefly ecall a few eie f hee ajecie hen ceed hw in Secin $9 q 998 Elevie Science BV All igh eeved II: S

2 D Deleine e alhyic Lee B 49 ( 998) 6 7 ha wih all aamee fixed henmenlgically u mdel aumaically accun f he beved d m D, Fm he knwn enegy deendence f Regge ajecie ne hen eadily edic dm B cle degee, namely quie a izeable FSI angle a he B ma, a naıvely execed Thee ae u main eul T cnclude hi ne we fi cmmen n bviu imlicain f u eul f B K decay hen end wih eveal geneal emak n he aameizain f any quai w-bdy decay amliude f he B men ( ) A Regge mdel f K K caeing amliude We ake,,u be he uual Melam vaiable In he -channel, K K caeing amliude ae linea cmbinain f he iin invaian amliude A A 3 In he -channel KK, we have iin invaian amliude A iin A iin, imilaly, in u he u-channel K K, we define A A u 3 The elain beween hee amliude ae given by he cing maice A ' 6 A A 3 A y ' 6 A u 3 43 u y3 3 A 3 In a Regge mdel, -channel amliude a high enegy lage ae aameized a um ve Regge le exchange in he ced channel: nea he fwad diecin mall, -channel exchange dminae while nea he backwad diecin cle y u mall, i i he u-channel exchange which ae elevan The geneic fm, a lage, f a Regge le exchanged in he -channel i given by a yia qe yb ina In Eq, b i he eidue funcin, he ignaue " a a qa X 3 i he linea Regge ajecy wih inece a le a X ; finally i a cale fac uually aken a GeV F a Regge le exchanged in he u-channel, he geneic fm i imila Eq bu wih he vaiable elaced by u The leading ajecy highe inece i he -called men I ha he quanum numbe f he vacuum I,q i exchange decibe diffacive caeing The men alway cnibue elaic caeing decibe quie well he bulk f hadnic diffeenial c-ecin ve a wide enegy ange In he enegy ineval which i f inee u hee, namely 3 GeV QQ35 GeV 4 a vey imle bu excellen henmenlgical aameizain f he men ajecy eidue funcin i given by a 5 b b e b 6 wih 5 GeV y Qb Q3 GeV y 7 bained fm fi wx 4 elaic, K diffeenial c-ecin uing facizain A a eul he men cnibuin A nw ead GeV A ib e b 8 The nex ajecie cnide ae he yf ajecie in he -channel he K ) yk )) ajecie in he u-channel The ajecy ha T,y while he f ajecy ha T, q; imilaly he K ) ajecy ha T, y while he K )) ajecy ha he ie ignaue Becaue f he abence f exic enance n K enance wih I3, he f ajecie a well a he K ) yk )) ne mu be

3 8 D Deleine e alhyic Lee B 49 ( 998) 6 exchange degeneae Secifically hi mean ha he f ajecie cıncide a a ( q 9 f ha hei eidue ae elaed ie bf b ' 6 Similaly, f he K y K ajecie in he SU 3 -limi ) )) a ) u a )) K K u ( qu yb ) u b )) u K K Eq 9, Eq, guaanee ha he nn diffacive imaginay a f A 3 vanihe They ued be called dualiy cnain wx 5 We neglec lwe lying ajecie uch a he X I,y he f I,q in he -channel a well a hei SU 3 ane in he u-channel Wee we include hem hey huld al be aken a exchange degeneae I i cumay wie he eidue funcin f he ajecy a b b 3 G a ' 4 G a ina fg a ina, b fb, 5 in wiing he ajecy cnibuin A 5q A, mall qiex yi 6 Since G a ina i a vey mh funcin f, n ham i dne in uing a mall he aximain An exacly imila eaning give f he f ajecy cnibuin A Af, mall 3 5q ( yqiex yi, 7 a while f he K ) K )) ajecie cnibuin A u ne wie A ) K, mall u b ) K 5qu qiex yi u, 8 A )) K wih, mall u b ) K 5qu y yqiex yi u, 9 3 b ) K 4 b in he SU 3 limi uing eveyhing gehe uing he cing maice given in Eq, u Regge mdel f K caeing i nw cmleely defined by he amliude i b 5q A, mall b e q ' 6 ' 3i yi 5q q e, a ' 5qu A, mall u, b i b 5q A3, mall b e y, ' 6 a 5qu A3, mall u y b 3 S-wave ecaeing hae The emaining ak i nw exac fm Eq, he l aial wave amliude a a 3 Neglecing K mae, we have, u ielevan eal fac ai A dai, 3 y

4 D Deleine e alhyic Lee B 49 ( 998) 6 9 Fm he hyical idea undelying Eq, i i clea ha uide he fwad backwad egin, he inegal in Eq 3 give a negligibly mall cnibuin a We hu wie I I I u I a A da, mall q dua, mall u 4 Wih he exlici exein given in Eq,, he inegal in Eq 4 ae ivial efm Fuheme, he inegaed cnibuin a he u bundaie aund GeV ae cnideably malle han a he bunday f bh inegal in Eq 4 Neglecing hee cnibuin, ne hu bain i b b a q ' 6 b ln 3i ln qi q 5 ' ln q i b b a3 y 6 ' 6 b ln Im ai fm which he an di ae aighf- Re ai wad cmue Ne ha bh an d an d 3 deend n ne ingle henmenlgically deemined aamee namely b x 7 b Fm fi wx 6, K al c ecin in he enegy ange given in Eq 4 again uing facizain, we find b 9" 8 Fm Eq 7, we hu cnclude ha x i cle ne x7"7 9 Simila eul ae bained uing he fi given in Ref wx 7 f a lage enegy ange Wih hee value f x, he ange f he FSI angle a he D ma i calculaed be D 3 D D d m 'd m yd m 85"6 8 3 in ecacula ageemen wih he ecen analyi f wx CLEO daa D d m 96"3 8 3 We e ha bh he analyi f CLEO daa u calculain ae baed n he quai-elaic aximain A he B ma, we edic a izeable angle cle degee, namely d m 7"3 8 3 B Befe cmmening n u edicin f dm B, i may be whwhile in u a few fac abu u calculain f d Ø i i a n-aamee calculain: x i deemined d fm he daa n al c-ecin wx 6 d wx 4; Ø in efming u calculain f d, we have made eveal aximain a we negleced lwe ajecie a well a he inemediae egin in he S-wave jecin inegal Thee aximain ae ceainly und fm a henmenlgical in f view hey becme bee bee a inceae A he D ma we d n believe ha u end eul huld be ued bee han % bu in any cae, ageemen wih he daa emain excellen; Ø he calculain eened hee f K caeing can f cue be eeaed f KK caeing A deailed accun dicuin f hee calculain will be eened elewhee wx 8 ee we imly in u ha he eul f bh calculain ae nce again in excellen ageemen wx wih he daa available a he D ma : d i KK fund be aund 3 d aund 6 Thee eul cnideably enghen u cnfidence in a imle Regge aameizain f hadnic caeing amliude 4 Cncluin The main eul f hi lee ae given by Eq 3 3 can be ummaized a fllw: a Regge mdel f K caeing exlain he lage S-wave ecaeing hae diffeence d beved a he D men ma namely d m D f, edic dm f8 B

5 D Deleine e alhyic Lee B 49 ( 998) 6 Such a izeable FSI angle a he B men ma lead iman imlicain f B K decay wx 9 : i invalidae he Fleiche-Mannel bund wx n he Cabibb-Kbayahi-Makawa angle g im- " lie a enially lage C aymmey, a, in B K " decay: af4 ing % 33 Sng ineacin hadnic hae can be aameized a la Regge f any quai w bdy decay mde f he B men, KK a aleady ) ) menined, bu al, K, K, ec The fac ha u quai-elaic eamen f he caeing amliude f K, KK agee well wih he daa a he D men ma i a ng agumen f neglecing inelaic effec n hadnic hae In view f he eviu cmmen, a geneal aameizain f all w-bdy decay mde f he B men naually ugge ielf The decay amliude can be wien a a um f educed maix elemen ²² B N N M M, I:: W f he effecive weak hamilnian, mulilied by he aiae hadnic FSI hae e d I Thee educed maix elemen ae in geneal cmlex numbe which can be yemaically calculaed in em f ee-level, clu ueed, enguin, exchange annihilain quak diagam Of cue, n iin vilaing caeing hae ae allwed beween hee diagam fuheme, a aleady hwn elewhee wx 9, clae f diagam which wuld naıvely be excluded can id eaea due fac f he ye ye I On he he h, enguin diagam can vide an abive ie imaginay cmnen he educed maw x Bu hee imaginay a ae vey ix elemen mdel-deenden bably quie mall Thee- fe we ugge w x, a a fi aximain imly igne hee quak hae wheneve he hadnic hae ae izeable Thi wa aumed in Ref wx 9 Thi haen be he cae f B K decay Acknwledgemen We ae gaeful Chihe Smih Fank Wuhwein f ueful dicuin cmmen Refeence wx R Gdang e al, CLEO 97-7, CLNS 975, heex97 wx M Bihai e al, CLEO Cllabain, hy Rev Le wx 3 F ealie aem, ee eg Zheng, hy Le B ; JF Dnghue, E Glwich, AA ev, JM Sae, hy Rev Le ; B Blk, I alein, hy Le B ; AN Kamal, CW Lu, Univ f Albea ein Thy-8-97, 997, he-h975396; AF Falk, AL Kagan, Y Ni, AA ev, Jhn kin Univ ein TIAC-978, 997, he-h975 wx 4 R Sebe, hy Rev Le wx 5 See f examle, J Mula, J Weye, G Zweig, Ann Rev Nucl Sc wx 6 V Bage, RJN hilli, Nucl hy B wx 7 A Dnnachie, V Lhff, hy Le B wx 8 J-M Gead, J eieau, J Weye, in eaain wx 9 J-M Gead, J Weye, UCL ein IT-97-8, 997, he-h97469 wx R Fleiche, T Mannel, Univ f Kaluhe ein TT-97-7, 997, he-h97443 wx M Be, D Silveman, A Sni, hy Rev Le wx F anhe in f view, ee M Neube, CERN ein T97-34, 997, he-h974