Rigorous inequalities in non-relativistic quantum mechanics with applications to quarkonium spectroscopy

Abstract

We show that for a concave (convex) potential ( l 2 > l 1 > 0) |R 1 S (0)| 2≷2l 1(l 1(l 1+1)〉 1 r 3 〈 1l 1 ≷2l 2(l 2+1)〉 1 r 3 〈 1l 2 , ∂ ∂m ( 1 m 〉 1 r 3 〈 1l)≷0 , 〉 1 r ν 〈 1 S > 〉 1 r ν 〈 2 S , 〉r ν〈 1 S < 〉r ν〈 2 S . If, in addition, (d 2/d r 2)( rV) > 0, then ( m M ) 1 3 < El 2(M) − El 1(M) El 2(m) − El 1(m) ⩽ M m , (M > m) . Further, if the confining potential V c( r) generates spin forces by an admixture of scalar and vector gluon exchanges, then it is shown that the vector fraction must be less than 0.4. In this case a number of rigorous bounds are derived some of which are M J ψ − M η c > 55 MeV, 3344 < M( 1P c 1) ⩽ 3526.1 MeV , Mϒ − Mη b > 23.6 MeV, Mη′ b − Mη b > 550 MeV , Mψ − Mη′ c > 18 MeV . The possibility of V c( r) generating spin forces by scalar gluon exchange alone is also considered in some detail and it is shown that in this case M( 3P b 2 − M( 3P b 2) < 0.8(M ( 3P b 2)− M( 3P b 0)) , M( 3D c 2) − 2 5 M( 3D c 3) − 3 5 M( 3D c 1) < 12.7 MeV , M( 3P b 1) − M( 3P b 0) < 9 8 (M ϒ − Mη b) .