Precise energies of rovibrational states of the exotic hydrogen-like molecule $(dt\mu)Xee$ are of importance for $dt\mu$ resonant formation, which is a key process in the muon-catalyzed fusion cycle. The effect of the internal structure and motion of the $
Furthermore,twoelectronsinthehydrogen-likemolecule(dtµ)Xeemovemuchfasterthantheheavyparticlesd,X,andtµ,whichmakesitpossibletousethefamiliarBorn-Oppenheimer(BO)approximation,i.e.,tosolveelectronicproblemwiththe xedchargedparticlesdandXthusobtainingtheBOenergywhichplaysaroleofthee ectivepoten-tialbetweendandX.Theelectronicexcitations,whichrequireaconsiderableamountofenergy[13],arenottakenintoaccountforthelow-energyprocessesunderconsideration.Asaresult,thedescriptionof(dtµ)Xeeisreducedtosolutionofathree-bodyproblemforthreeparticlestµ,dandX.TheinteractionbetweenchargeddandXisdescribedbythewell-knownBOpotentialforthehydrogenmolecule.Inaccordwiththetreatmentofthetµmesicatomasapoint-likeneutralparticle,thepresentcalculationdoesnotexplicitlyusethetµ+de ectivepotential,rathertheresultisexpressedviathelow-energytµ+dscatteringphaseshiftsandcharacteristicsofthetµmesicmoleculeinthelooselyboundexcitedstate.
The(dtµ)Xeestatesareeithertrueboundstatesornarrowresonancesiftheirenergyisbeloworabovethetµ+DXthreshold.Astheenergyshiftsaremainlydeterminedbythecouplingwithclosedchannels,inthepresentcalculationbothresonancesandboundstatesaretreatedonanequalfootingthusneglectingasmallcontributiontotheenergyshiftswhichcomesfromthecouplingwiththeopentµ+DXchannel.
A.Three-bodydescription
Undertheaboveapproximations,theSchr¨odingerequationforthehydrogen-likemolecule(dtµ)Xeereads
1
2µ2 ρ+V1(r)+V2(|ρ βr|)+V(|ρ+αr|) EΨ=0(1)
wheretheJacobicoordinatesrandρarethevectorsfromdtothepoint-likemesicatomtµandfromthesecondnucleusXtothedtµcenterofmass,respectively.Thereducedm1m2m1massesandparametersαandβareµ1=,α=m1+m2+m3
,wherem1,m2andm3arethemassesoftµ,d,andX,respectively.Them1+m2atomicunitsareusedthroughoutthepaperunlessotherisspeci ed.InEq.(1),V(|ρ+αr|)
denotesthewell-knownBOpotentialdescribingtheinteractionbetweenchargeddandXwhiletheshort-rangepotentialsV1(r)andV2(|ρ βr|)describetheinteractionofatµ