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2.7 Mass dependence of the binding energy

Let us once more consider the reduced Hamiltonian ^ H of Eq. (2.3) and denote by E(μ) one of the eigenenergies and by ϕ(μ) the corresponding wave function. Obviously

dE- dμ < 0 .
(2.40)

More precisely, from the Feynmann Hellmann theorem [33]

dE- -1-⟨ 2 ⟩ 1- d μ = − μ2 ϕ (μ ) | p | ϕ(μ ) ≡ − μT (μ) .
(2.41)

We also note that H^ depends linearly on μ−1. We can thus use the general theorem that if a Hamitonian depends linearly upon a parameter λ, its ground state energy is concave in λ

d2E0- H = A + λB =⇒ dλ2 ≤ 0 .
(2.42)

This follows from second-order perturbation theory or from the variational principle [33]. Here, once the constituent masses are added, one gets inequalities between the ground-state masses of any given angular momentum l

-- M0,l (m1, m1 ) + M0,l (m2, m2 ) ≤ 2M0,l (m1, m2 ) , or (QQ )l + (qq)l ≤ 2(Qq-)l .
(2.43)

For instance, neglecting spin effects,

- - - φ (ss ) + J∕Ψ (cc) < 2D ∗s(cs),
(2.44)

in agreement with experiment [5] (1.02 + 3.10 < 2 × 2.11 GeV). Similarly, one can derive

- - bs + cd < cs + bd.
(2.45)

The convexity inequality (2.43) cannot be written for radial excitations: one should instead consider the sum of the n first levels.

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