For two free leptons the uncoupled equations are first taken as Pauli equations

# For two free leptons the uncoupled equations are

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For two free leptons, the uncoupled equations are first taken as Pauli equations (2.58), including external vector potentials. Using again the com- pact notation π i = π i σ i , they may be written in forms which also comprise KG particles, p 02 i ψ as = K i ψ as , K i = m 2 i + π 2 i . (4.388) The first step is to combine the two equations into a single one which contains only a common time shift in the form p 0 1 + p 0 2 p 0 . With p 02 1 p 02 2 = p 0 ( p 0 1 p 0 2 ), the difference of the equations gives ( p 0 1 p 0 2 ) p 0 ψ as = ( K 1 K 2 ) ψ as . (4.389)
4.11 Dirac Structures of Binary Bound States 197 An eigenvalue K 0 of p 0 will be assumed. As p 0 1 p 0 2 commutes with K 1 and K 2 , a second application of p 0 1 p 0 2 gives ( p 0 1 p 0 2 ) 2 ψ as = ( K 0 ) 2 ( K 1 K 2 ) 2 ψ as . (4.390) Next, consider the sum of the two equations (4.388), ( p 02 1 + p 02 2 K 1 K 2 ) ψ as = 0 . (4.391) With p 0 1 = 1 2 K 0 + 1 2 ( p 0 1 p 0 2 ) etc, this leads to [ K 02 / 2 + ( K 1 K 2 ) 2 / 2 K 02 K 1 K 2 ] ψ as = 0 . (4.392) In terms of the triangle function λ (4.76), ( K 0 ) 2 λ ( K 02 , K 1 , K 2 ) ψ as = 0. In the “constraint Hamiltonian mechanics” which dates back to Dirac, the coupling between the two particles is introduced already in equations (4.388), but practical success has been limited (Crater and Van Alstine 1994). For particles of equal spins, λ may be decomposed symmetrically in the indices 1 and 2: λ = ( K 02 K 1 K 2 ) 2 4 K 1 K 2 . (4.393) With m 2 2 m 2 1 = m + m , λ = ( K 02 m 2 1 m 2 2 ) 2 2 π 2 1 ( K 02 + m + m ) 2 π 2 2 ( K 02 m + m ) 4 m 2 1 m 2 2 + ( π 2 1 π 2 2 ) 2 . (4.394) For A = 0, the transformation (4.291), (4.307) of variables, p 1 = p lab + K E 1 /E, p 2 = p lab + K E 2 /E, E ( K 02 K 2 ) 1 / 2 , (4.395) yields together with p lab ,x = p x , p lab ,y = p y , p lab ,z = γp z , λ = γ 2 [( E 2 m 2 1 m 2 2 ) 2 4 m 2 1 m 2 2 4 E 2 ( p 2 x + p 2 y + p 2 lab ,z 2 )] . (4.396) The factor γ 2 in front of the square bracket corresponds to the separation of one factor γ from (4.308). The square bracket is factorized by means of the Dirac matrices β, γ 5 = β x and 5 β = β y as follows: λ/γ 2 = ( f 0 + f z β + f x β x + f y β y )( f 0 f z β f x β x f y β y ) , (4.397) f 0 = E 2 m 2 1 m 2 2 , f z = 2 m 1 m 2 , f x = 2 E 1 , f y = 0 . (4.398) The second factor of (4.398) is precisely that of the leptonium equation (4.237). More general forms include a rotation by an arbitrary angle ω D about the β -axis in Dirac space (2.103), f x = 2 E 1 cos ω D , f y = 2 E 1 sin ω D , or a rotation (4.270) in the Pauli product space. For constant f 0 , one may also take f y = 2 E 2 sin ω D , as σ 1 i σ 1 j and ( σ 1 i cos ω D + 2 i sin ω D )( σ 1 j cos ω D 2 j sin ω D ) are identical in their symmetric tensor components. (In the quark
198 4 Scattering and Bound States model, f y is not excluded a priori even when f 0 is r -dependent. For small ω D , it would mainly affect the hyperfine structure of quarkonium.) The inclusion of A is trivial in the 16-component equation (3.186) or (3.190), whereas (3.218) is still unclear. The first-order relativistic Zeeman shift δE of the ground state has been calculated by Faustov (1970) and by Grotch and Hegstrom (1971). Besides the factor (1 α 2 Z / 3) from H Zee (2.285) in the Pauli reduction of the GY-equation (4.378), these authors find essen-