Effect of dtmu quasi-nucleus structure on energy levels of t(6)

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