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 $
Thee ectofthedtµ nitesizewaspreviouslyinvestigatedinasimpleapproach[8,9,10]wheretheenergyshiftsforthe(dtµ)deewereobtainedbymultiplyingby1.45theshiftcalculatedfortheatom-likesystem(dtµ)einthesecondorderperturbationtheory(PT).Withintheframeworkofthissimpleapproachitisnotpossibletotakeaccountofthemolecularstructure;inparticular,thecalculatedenergyshiftisindependentoftherovibrationalquantumnumbers.Thee ectofthemolecularstructure,i.e.,thedependenceonangularmomentum,wasexplicitlydemonstratedintheelaboratesix-bodycalculation[8]ofthe(dtµ)deeenergyshiftsinthe rstorderoftheperturbationtheory.Note,however,thatthe rst-andsecond-orderPTcontributionstotheenergyshiftarecomparable.Recently,resonancepositionsinthelow-energytµ+D2scatteringhavebeenobtainedintheelaboratethree-bodycalculation[11,12].Onlyfewresonancestateswiththezerototalangularmomentumhavebeenconsideredinthispaper.
Themainaimofthepresentpaperistocalculatetheenergyshiftswhichariseduetotheinternalstructureandmotionofthedtµmesicmoleculeembeddedinthehydrogen-like(dtµ)Xeemolecule.Thecalculationisreducedtosolutionofathree-bodyproblemforheavyparticlestµ,dandX.Thisapproachisbasedonthehierarchyofscalesandcorrespondingenergiesofconstituentsubsystemsofthe(dtµ)Xeethusreliablytakingintoaccountthespeci cfeaturesofthismolecule.Asaresult,theenergyshiftsareobtainedforanumberofvibrationalandrotationalstatesof(dtµ)deeand(dtµ)teeinthesecond-orderPT.II.METHOD
Thestructureoftheexoticmolecule(dtµ)Xeeischaracterizedbyahierarchyofscalesandcorrespondingenergiesofitsconstituentsubsystems.Inthisrespect,atµmesicatomissmallincomparisonwithitsmeanseparationfromadeuteroninthelooselybounddtµmesicmolecule,whichallowstµtobetreatedasapoint-likeneutralparticleinteractingwithadeuteronbytheshort-rangee ectivepotential.ThereisalsointeractionoftµwiththesecondnucleusX;however,thismightbeneglectedduetolargeseparationbetweentheseparticles.Inturn,thesizeofadtµmesicmoleculeissmallincomparisonwiththeamplitudeofvibrationsin(dtµ)Xee;therefore,itmovesasapoint-likequasi-nucleusneartheequilibriumposition.Forthisreason,thee ectofthedtµstructureisconsideredwithintheframeworkoftheperturbationtheory.