Effect of dtmu quasi-nucleus structure on energy levels of t(11)

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 $

Thesecond-orderPTmatrixelementsWlln

1inEq.(6)arecalculatedbysubstituting

UnνD(22)andIλ( )(28)inEq.(12),whichgivestheexpression

W32ω

lln

1= v0ε11 (34)

viatheenergyscalev0andthedimensionlessfactors.Solvingthesecularequation(6),onecansafelyreplace,uptoanaccuracyofthesecond-orderPT,theeigenvalueEintheargumentofJλbytheunperturbedvalueEnl.Thus,thecalculationoftheenergyshiftsisbasicallyaccomplishedbyderivationofEqs.(32-34).

However,itisreasonabletomakefurthersimpli cationof(34)byneglectingthedi erenceoftherotationalenergiesintheargumentofJλ,whichallowsobtaininganexplicitandsu cientlyaccuratedependenceoftheenergyshiftsonthequantumnumbersnandl.Astherotationalenergyismuchsmallerthanthevibrationalquantumω,onereplacestheenergydi erencesEn±1l EintheargumentofJλbythel-independentvaluesEn±ingtheangularintegralsAL n0=±ω1(l1l1λ)(A5,A6)andintroducingthenotationJ±

λ=Jλ(1±ω/ε11)forintegralsindependentofnandloneobtains

Wlln

1=v0 1+αn(2 β βn,l,l1=L(L=0)n)δll1+(αl,l(35)n 1)Bll1,1=L 1

where

αn=1 16ωJ2+)+n(J

0+1

3πε 5(n+1)(J0++7J

1152) (37)

determinetheexplicitdependenceonthevibrationalquantumnumbern.Asaresult,thesumofVll(32)andWlln

11(35)takesasimpleform

V,l=l1=L(L=0)

ll1+Wlln

1=v0 βn β αnl,l,(38)

nδll1 αnBll1,1=L 1

i.e.,theparameterβndeterminestheconstantshiftv0βnofalllevelenergiesEnlwhereasαing(38)and(21)inthesecularequation(6)oneobtains

0(n)=v0(βn αn),(39) ±(nl)=v0βn vr[2(l 1)+1]±