Effect of dtmu quasi-nucleus structure on energy levels of t(5)

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 $

mesicatomwithadeuteronandX,respectively.Inthefollowing,duetolargeinternuclearseparation(ρ r)in(dtµ)Xee,theshort-rangeinteractionV2(|ρ βr|)ofthetµmesicatomwiththesecondnucleusXisnegligibleandwillbeomitted.

Anaturalzeroth-orderapproximationforthecalculationofthe(dtµ)Xeeenergylevelsistotreatthedtµmesicmoleculeasapointquasi-nucleuswiththedtµmassandtheunitcharge.Thecalculationsoftheenergylevelsinthisapproximationarepresentedin[6,7]fordi erentisotopesXofthehydrogen-likemolecule(dtµ)Xee.Clearly,thetreatmentofdtµasapoint-likeparticleisequivalenttothereplacementoftheexactpotentialV(|ρ+αr|)intheSchr¨odingerequation(1)bythepotentialV(ρ)whichdescribestheBOinteractionbetweenXandthepointparticlelocatedatthedtµcenterofmass.Thus,thee ectofthedtµstructure,whichleadstotheshiftofthezeroth-orderenergylevels,originatesfromtheperturbationpotential

Vp=V(|ρ+αr|) V(ρ).(2)

Inthezeroth-orderapproximationVp=0,thesolutionsofEq.(1)withthetotalangularmomentumLanditsprojectionMarewrittenasaproductofthebisphericalharmonicsofnucleiin(dtµ)Xeewiththeangularmomentuml,andtheradialfunctionofrdescribingLM ,r )describingtheangulardependence,theradialfunctionofρdescribingthemotion(ρYlλtheinternalmotioninamesicmoleculewiththeangularmomentumλ.TheunperturbedenergiesEnlandthecorrespondingsquareintegrableradialfunctionsΦnl(ρ)ofthe(dtµ)Xeevibrationalandrotationalstatessatisfytheequation

1

ρ2 ρ +l(l+1)

2µ1 1 r r2

r2 +V1(r) Eφ(r)=0. (4)

HereE= εvλandφ(r)=φvλ(r)fortheboundstatesandE=k2/2µ1andφ(r)=φkλ(r)forthecontinuumstateswiththewavenumberk.Thefunctionsφvλ(r)aresquareintegrableandthefunctionsφkλ(r)arenormalizedbythecondition

∞ 0r2drφ kλ(r)φqλ(r)=δ(k q).

(5)