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 $
PTquadrupolecontributiontotheenergyshiftsisinagreementwiththeelaboratesix-bodycalculation[8],whichisagoodargumentforthevalidityofthepresentapproach.Furthermore,itisshownthatforalltheconsideredstatesthee ectofcouplingoftherotationalstateswithl=L 1andl=L+1isbeyondtheaccuracyofthepresentcalculation.Inaddition,theenergyshiftandsplittingisconvenientlyexpressedbysimpleanalyticaldependence(39),(41)ontherotational-vibrationalquantumnumbersnandl.Itisofinteresttocomparethepresentresultwiththedirectnon-perturbativethree-bodycalculationoftµ+D2scattering[11,12]inwhichthee ectofthedtµstructureisexplicitlytakenintoaccount.Thepositionsofnarrowresonancescalculatedinthispapercorrespondtoenergylevelsof(dtµ)deeforL=0(l=1)andn=3,4.Fortwoapproximationsofthee ectivepotentialbetweentµanddusedin[11,12],energyshiftsare,respectively,1meVand4meVforn=3and2meVand5meVforn=4.Thecauseofthenoticeabledi erence(about3meV)isnotclearsincebothpotentialsallowagooddescriptionofthelow-energypropertiesofthetµ+dsystem.Thedependenceonthechoiceofthee ectivepotentialandalimitationonlybyL=0hindersaquantitativecomparisonofthepresentresultsandthoseof[11,12].Qualitatively,theenergyshiftsobtainedinRef.[11,12]exceedthepresentonesand,contrarytoPTconsiderations,thevalueforn=4ishigherthanforn=3.Thisn-dependenceclearlydeservesfurtherinvestigation.
Finally,itshouldbementionedthatthepresentapproach,whichreliablytakesintoac-countthestructureoftheexoticmolecule,ispromisingforwiderapplications,inparticular,fordeterminationofresonancepositionsandformationratesbeyondPTbysolvingthescat-teringproblem.Tillnow,exceptRefs.[11,12],theformationrateshavebeencalculatedonlyinthe rst-orderPT.Inthisrespect,theresultofRef.[26]showsthatthe rst-orderPTdipoleapproximationisquestionableandoneshoulddomorere nedcalculations.Inaddition,itisofinteresttoapplythepresentapproachtotheproblemoftheresonanceformationofmetastabledtµmesicmolecules[27,28]incollisionsofexcitedtµmesicatomswithD2molecules.
APPENDIXA:ANGULARINTEGRALS
Thefollowingangularintegralsarenecessarytocalculatethematrixelements
ALK(lλl1λ1)= LM , , d (ρr)YlLM(ρr)dρrPK(cosθ)Ylλ1λ1 (A1)