Discrete ambiguities in the determination of the CP angles

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arXiv:hep-ph/0010304v2 27 Oct 20001DiscreteambiguitiesinthedeterminationoftheCPangles∗J.Charlesa†andL.OliverbaGrupdeF´ısicaTe`orica&IFAE

UniversitatAut`onomadeBarcelona,E-08193Bellaterra(Barcelona),Spainonleavefrom:CentredePhysiqueTh´eorique(CNRS-UPR7061)Case907,F-13288MarseilleCedex9,France

bLaboratoiredePhysiqueTh´eorique(CNRS-UMR8627)

Universit´edeParis-Sud,Bˆatiment210,F-91405Orsay,France

WereviewbrieflytheproblemofthediscreteambiguitiesinthedeterminationoftheCKMangles,byclassifyingthemintodifferentcategories.Wethenfocusoncos2βandtheextractionofαusingQCDfactorization.

WiththefirstmeasurementsinB0d,¯B0d(t)→J/ΨKSoftheCPviolatingparametersin2βbythecollaborationsBaBar,BELLE[1]andCDF[2],themainquestionthatarisesiswhethertheanglesthatbegintobemeasuredinexperimentsthattestCPviolationmatchtheanglesthatarebeingdeterminedwithincreasingaccuracyinex-perimentsthatmeasurequantitiesthatconserveCP,i.e.thelengthsofthesidesoftheunitaritytriangle[3].ThedeterminationofγfromCP-averagedB→Kπ,ππdecayrates[4]alsoen-tersintothelattercategory.Thisisofgreatim-portancebecauseitisrelatedtotheproblemoftheoriginofCPviolation,whethertheStandardModelisthecorrectanswerorifthereissomeothersourceofitfromphysicsbeyondtheStan-dardModel.Thequestionofthediscreteambigu-ities(multiplesolutions)inthedeterminationoftheCPanglesisdirectlyrelatedtotheprecedingissue.Tobeginwith,letuspointoutthattherearethreetypesofdiscreteambiguities:1.Ambiguitiesfromincompleteexperimentalinformation.2.Ambiguitiesfromthetheory,withexper-

1+|λf|2cos∆Mt−2Imλf

p¯M(f)

p¯Mp¯M

p¯M2(twofolddiscreteambiguity).Apossiblesolutionforthiskindofambiguitiesistolookforprocesseswherethedecayproductsdecayweakly,violatingparity,like

B0d,¯B0d(t)→Λ¯Λ→pπ−¯pπ+(4)Here,themorecomplexLorentzstructure(spin1

p¯Mp¯M

p→e2iθd

󰀋q33.AMBIGUITIESFROMTHETHE-ORY,BUTWITHINCOMPLETEEX-PERIMENTALINFORMATION

Onecaninterpretthedataintermsofthethe-oreticalschemesquotedabove,butassumingthatonlysomequantitiesareavailableexperimentallywithrelativelysmallerrors.Oneinterestingresultinthiscaseisthefol-lowing[10]:If(sin2α,sin2β)areknown(α+β+γ=π)allambiguitiesaresolvedknowingsign(cos2α),sign(cos2β),sign(sinα),sign(sinβ).Whilesign(cos2α),sign(cos2β)caninprinci-plebeobtainedinamodel-independentway,sign(sinα),sign(sinβ)canonlybeobtainedinamodel-dependentway,forwhichsomehadronichypothesesarenecessary.Therehasbeenrecenttheoreticalworkonthefollowingtopicsrelatedtodifferenttypesofdis-creteambiguities:

(i)Ambiguitieslinkedtothemeasurementofsin2β.

(ii)AmbiguitiesintheexchangebetweenCPandFSIphases.

(iii)LookingforαinDalitzplotanalysisBd(t)→π+π−π0[11,13].

(iv)AmbiguitiesinthesearchofαinBd(t)→π+π−.

Wewillconcentratehereonthepoints(i)and(iv).

3.1.Ambiguitieslinkedtothemeasure-mentofsin2¯βFromthemeasurementofsin2βinB0d(¯B0d)(t)→J/ΨKSoneisleftwiththefour-foldambiguity

󰀅β,π2−β󰀇(9)

Tolifttheseambiguitiesoneneedstomeasuresign(cos2β)andsign(sinβ)[10].Toobtainsign(sinβ),onecanconsiderthesumofthecoefficientsofsin∆MtintwoCPasymme-tries[10,13],correspondingtothemodesJ/ΨKS

andD+D−:aψKS+aD+D−∼=2rDDcosδDDcos2βsinβ(10)whererDDandδDDarethemodulusandphaseoftheratioofPenguinoverTreeamplitudesforthemodeD+D−.Thereisnorigoroustheoret-icalschemethatallowstocalculatethesequan-tities.Ifoneassumesfactorization3andknowssign(cos2β),thenonecanhaveanindicationonsign(sinβ).Therehasbeenasignificanttheoreticalprogressconcerningmethodstodeterminethesign(cos2β):

(i)DalitzplotanalysesB0d(¯B0d)(t)→D+D−π0.ThismethodusesthedecaychainB0d(¯B0d)(t)→D+D∗∗−+D∗∗+D−→D(∗)+D(∗)−π0.Thein-terferencesintheDalitzplotallowtoobtainsin2βandcos2β[15].Thereishoweveranirre-ductiblecontaminationfromPenguins,thatbe-haveinthesamewayascurrent-currentopera-tors,∆I=1

2󰀂|¯A1|2+|¯A2|2󰀁󰀍󰀃|f+|2+|f−|2󰀄+D2Re[f+(f−)∗]

−cos∆Mt󰀂R󰀃|f+|2−|f−|2󰀄

−Dsinδ2Im[f+(f−)∗]󰀁

−sin∆Mt󰀍D󰀃sin(2β−δ)|f+|2+sin(2β+δ)|f−|2󰀄+sin2β2Re[f+(f−)∗]+Rcos2β2Im[f+(f−)∗]󰀁󰀏(11)where

f(s)∼Y0J

3NotethattheformalismofRef.[14]doesnotallowto

establishfactorizationforB→D¯Dintheheavyquarklimit.