7.2 The Born–Oppenheimer approximation

The Born–Oppenheimer method is very often used in molecular physics and other few-body problems and always turns out to be very efficient and to actually work better than expected. This method seems particularly suited for baryons bearing two units of heavy flavour, since the heavy quarks move much more slowly than the light quark.

Let us consider a (1, 2, 3) = (QQq) baryon, with quark masses m, m and m′. The generalization to m₁≠m₂ is obvious. With the Jacobi coordinates of Eq. (3.54), the Hamiltonian reads

(7.3)

By including the recoil of the charmed core, we greatly improve the accuracy of the calculation, at no expense. For any fixed , one can compute the eigenstates corresponding to the stationary states of the light quark, i.e., 

(7.4)

This corresponds to a one-particle problem in a non-central potential. Astute calculations could make use of elliptic or bi-polar coordinates. In fact, one may simply use an ordinary expansion into partial waves

(7.5)

resulting in coupled equations for the u_l’s.

Consider first the ground state. If one adds the light-quark energy ϵ(ρ) to the direct interaction between the heavy quarks, one gets the simplest (and well-known) form of the Born–Oppenheimer approximation, sometimes referred to as the “extreme adiabatic” [72]

(7.6)

which overestimates the binding, i.e., is antivariational. Indeed, E^EA ≤ E^exact results from the operator inequality [33]

(7.7)

The method is in fact more systematic. The exact wave function Ψ(,) of the (QQq) system can be expanded as

(7.8)

reducing the three-body problem to an infinite set of coupled equations for the ϕ_n(). Keeping only the first term in the above expansion corresponds to a variational approximation, with a trial wave function ϕ₀f₀. This is sometimes called [72] the

uncoupled adiabatic or

variational adiabatic approximation

(7.9)

For improved calculations, one has to keep more terms in Eq. (7.8). An optimal choice consists of grouping together the f_n(,) in a combination which provides the maximal correction. This is reminiscent of the potential harmonics in hyperspherical expansions, as studied in Section 5.4. Here, we remark that f₀(,) is coupled to the higher adiabatics through the kinetic energy. So, we use a normalized ∇f₀(,) to supplement f₀(,), namely

(7.10)

(7.11)

Solving the two coupled equations in f₀ and f_b gives the so-called “coupled adiabatic” approximation [72], which is always extremely close to (above) the exact result.

Detailed numerical studies of the Born–Oppenheimer approximation have been performed in the context of studies of baryons with double charm [7, 73]. The method works quite well for ccq configurations, as expected, but also for the ssq or even qqq cases. In Table 7.1, we display a comparison of the extreme and uncoupled adiabatic approximations with exact results for the mmm′ system with masses m = 1 and m′ = 0.2, 0.5 and 1, bound by the smooth 1∕2 ∑ r_ij^0.1 potential. The quality of the approximation is impressive for both the energy of the first levels and the short-range correlation.

What about the excited states in the Born–Oppenheimer approximation? We consider here the optimal case m′≫ m. This implies that it is much more economical to excite the relative motion of the heavy quarks than the motion of the light quark around them. For instance, in the harmonic oscillator model, there is a ratio exactly [2m∕(2m′ + m)]^1∕2 between the corresponding excitation energies. Then, for the first excited states, the light-quark wave function remains in the lowest adiabatic f₀(,): the binding energy and the qq distribution is obtained from the low-lying excited states of Eq. (7.9) or (7.6).

Higher states may also consist of an excitation of the light quark, corresponding to a dominant ₀f_n component. In general, one hardly avoids mixing between such a configuration and the previous ones, so that accurate estimates require solving coupled equations. Still, the Born–Oppenheimer method provides one of the most efficient accesses to these states.