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 $
IncorrespondencewiththeVesmanmechanism,(dtµ)Xeecontainsadtµmesicmoleculeintheweaklyboundstatewiththebindingenergyε11(v=1,λ=1).Otherdtµstates,whosebindingenergiessigni cantlyexceedallthecharacteristicenergiesoftheproblemunderconsideration,willnotbetakenintoaccountinthecalculationoftheenergyshifts.
B.Perturbationtheory
Thee ectofthedtµstructureissmallduetosmallnessofdtµmesicmoleculeincom-parisonwithacharacteristiclengthofdtµmotioninthemolecularpotentialV(ρ).Inotherwords,theperturbationVpissmallincomparisonwithV(ρ)andcanbeexpandedinpowersofthesmallparameterαr.Correspondingly,thedimensionlessparameteroftheperturba-tiontheoryistheratiooftheaveragedistancebetweenthedeuteronandthedtµcenterofmassα r totheaverageamplitudeofvibrations ρ a inthemolecularpotentialneartheequilibriuminternucleardistancea.
Oneshouldnotethatthelowest-ordertermoftheexpansionVp,whichisproportionaltoαr,doesnotcontributetotheenergyshiftsinthe rst-orderPT;therefore,theenergyshiftoforder(αr)2mustbeobtaineduptothesecond-orderPT.Besides,Vpcouplestherotationalstateswithl=L±1whilethestatewithl=Lremainsuncoupled.Astheseparationoftherotationallevelsiscomparativelysmall,thelevelcouplingcannotbeapriorineglectedandrequiresexplicittreatment.Thus,theenergyshiftswillbedeterminedinthesecond-orderdegeneratePTbysolvingasecularequation
det[Vn+Wn+En E]=0(6)
whereVnandWnarethematriceswiththematrixelementsofthe rst-andsecond-order
nnPTVllandW,respectively,thematrixelementsofEnare(Enl+ε11)δll1,andEisthell11
levelenergy.
The rst-orderPTmatrixelementsare
nVll1= ,r )YlLM ,r )d3rd3ρVp|φ11(r)|2Φnl(ρ)Φnl1(ρ)YlLM(ρ(ρ111 (7)
andthesecond-orderPTmatrixelementsincludeasumandanintegraloverintermediatestatesdescribingsimultaneousexcitationsoftheexoticmoleculewiththequantumnumbersνand andadtµmesicmoleculewiththecontinuum-statewavenumberkandtheangular