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 $
BOpotentialisindependentoftheisotopiccomposition,bothµ2ω2andαMareindependentofthemassesofheavyparticlesduetoEq.(20).Theresultofthe tgivesω=321.8meVfor(dtµ)dee(correspondingly,ω=273.1meVfor(dtµ)tee)withafewpercentaccuracyandtheparameterαM=0.7.Fortheseparameters,theenergiesofthelowestvibrationalstatesintheapproximatepotentialareinreasonableagreementwiththeresultsof[6,7].
Therotationalspectracalculatedin[6,7]are ttedtoEq.(21)for1≤l≤10andeach1≤n≤4.Forthelowestvibrationalstaten=1,oneobtainsvr=2.43meVfor(dtµ)deeandvr=1.85meVfor(dtµ)tee.Thesevaluesagreewiththesimpleestimate2µ2vr
determinestheisotopicdependenceofvv≈1/a2thatr.Althoughrslightlydecreasesforthehighervibrationalstates,theabovevalueswillbeusedforn>1,whichleadstoafewpercenterror.
Determinationofthebindingenergyε11ofadtµlooselyboundstatewasasubjectofnumerouselaboratedcalculations.Asaresult,thevalueε11=596meV[1,2,4]isobtainedforthelowesthyper nestatebytakingintoaccountrelativistice ects,hyper nee ects, nitenuclearsize,andvacuumpolarization.TheasymptoticconstantCawasdeterminedinanumberofpapers[9,16,17]byacomparisonoftheasymptoticexpression(24)three-bodywavefunction.Inthefollowing,itisacceptedthevalueCa=0.874/√withthe
(m)m,theparameterv0(31)isindependentofm
isthesameforanyisotopeX1+m.Given2the33,i.e.,it
abovenumericalvaluesoneobtainsv0=1.81meV.
Forthesakeofcompleteness,itisinterestingtoestimatetheenergyscaleforthemolecule(ddµ)Xeetoobyusingthevaluesε11=1975meV,Ca=1.006/√