content_image/356029.jpg content_image/356038.jpg
find the potential seen by the negative muon for r < R . If it were sitting exactly at the center of the nucleus, it would (by symmetry) feel no force. That is, there is no term in the potential linear in r , as the force at the origin is − dV nuc /dr ]$_{r}$ =0 = 0. This suggests that we can take V nuc ( r ) = A + Br $^{2}$. The coefficient B can be fixed from the requirement that the slopes dV nuc /dr and dV Coul /dr match at r = R . The coefficient A in turn is fixed by the requirement of continuity at r = R . Thus a potential reasonably appropriate for such a muonic atom is 

 − αZ/r for r > R. This potential no longer has the 1 /r singularity at the origin, so the BCs for the outgoing integration for the 1 s state are more like those in Eqs. (13) and (14) for l = 0: 

<img src='content_image/356029.jpg'>

<img src='content_image/356038.jpg'>

bearing in mind that we here are making the equations dimensionless by setting m$_{µ}$ = 1 and that the eigenenergy of the bound state is less than m$_{µ}$ . Unlike the pure Coulomb case, here f ( r ) will have curvature near r = 0, where it will vanish. 

Running the numerical integrations with potential V$_{v}$ ( r ) for $^{40}$Ca gives a binding energy B = 1 . 07006, some 60 keV smaller than the pure Coulomb result for this nucleus. Figure 17 displays the difference between the ψ$_{a}$ ( x )’s and ψ$_{b}$ ( x )’s for the case with this nuclear charge distribution and the pure Coulomb case. 

## IV. COMBINING THE TWO POTENTIALS 

We now consider the solution of the radial Dirac equations, Eqs. (7) and (8), when both potentials, the Lorentz scalar V$_{s}$ ( x ) and Lorentz vector V$_{v}$ ( x ), come into play. We discuss two different situations. 

## A. When V$_{v}$ ( x ) Is Coulombic 

As our first case, we take the scalar potential V$_{s}$ ( r ) to be the same as in Eq. (9) and the vector potential as V$_{v}$ ( x ) = − α$_{s}$/x . Here α$_{s}$ is the (strong) gluon-quark coupling constant of quantum chromodynamics (QCD), having a value of O (1) at low energies and becoming small at high energies (asymptotic freedom). 

