Fine and hyperfine interaction on the light cone

The formalism for the spin interactions in the front form (light-cone) is re-phrased in terms of an instant form formalism. It is shown how to unitarily transform the Brodsky-Lepage spinors to BjørkenDrell spinors and to re-phrase the so called spinor matrix in terms of the interactions one is familiar with from atomic and Dirac theory. — One retrieves the (relativistic) kinetic correction, the hyperfine and the Darwin term which acts even when wave function is spherically symmetric. One also retrieves angular momentum dependent terms like the spin-orbit interaction in a relativistically correct way; and one obtains additional terms which thus far have not been reported particularly various L-dependent terms. Since the approach includes the full retardation, one gets additional, thus far unknown terms. The differ from atomic and Dirac theory, since there only that part of the vector potential is usually included which is generated by the atomic nucleus. Quite on purpose, the paper is kept formal. — PACS. 11.10.Ef – 12.38.Aw – 12.38.Lg – 12.39.-x 1 The light-cone integral equation This paper number 3 in a row of 3 [1,2] on the bound state problem in gauge theory [3] deals with the technical question of how to formulate the fine and hyperfine interaction in the one-body integral ‘master’ equation which has been previously derived [2,3]. I therefore jump immediately to Eq.(16) of [2], Mψh1h2(x,k⊥) = [ m1 + k 2 ⊥ x + m2 + k 2 ⊥ 1− x ] ψh1h2(x,k⊥) − 1 4π2 ∑ h 1 ,h 2 ∫ dxdk ⊥ ψh1h2(x ,k ⊥) √ x(1− x)x′(1− x′) αc(Q) Q2 R(Q) × [u(k1, h1)γu(k 1, h′1)] [v(k 2, h′2)γμv(k2, h2)] . (1) Here, M is the eigenvalue of the invariant-mass squared. The associated eigenfunction ψh1h2(x,k⊥) is the probability amplitude 〈x,k⊥, h1; 1−x,−k⊥, h2|Ψqq̄〉 for finding the quark with momentum fraction x, transversal momentum k⊥ and helicity h1, and correspondingly the anti-quark. Their (effective) masses are denoted by m1 and m2, and u(k1, h1) and v(k2, h2) are their Dirac spinors in Lepage Brodsky convention, as given in [3]. The (effective) coupling function αc(Q) = 4 3α(Q) is also given in [3]. The kernel is governed by the mean four-momentum transfer, Q = 12 ( Qq +Q 2 q̄ ) , where Qq = −(k1 − k 1) and Qq̄ = −(k2 − k 2) , (2) are the Feynman four-momentum transfers of quark and anti-quark, respectively. The regulator function R(Q), finally, removes the ultraviolet singularities and regulates the interaction. Note that the equation is fully relativistic and covariant. It coincides literally with Eq.(4.101) of [3]. Krautgärtner et al [4] and Trittmann et al [5] have shown how to solve such an equation numerically with high precision. But since the numerical effort is considerable, it is reasonable to work first with simpler models. The aim of the present work is to derive such ones. The aspects of regularization and renormalization have been emphasized in [1,2], resulting in an explicit construction of the regulator function R(Q). The case was worked out within the so called Singlet-Triplet model. Here, I address to go beyond that, particularly to derive a model for the spin-orbit interaction, which had been suppressed on purpose in [2]. 2 Transforming the integral equation The light-cone integral equation (1) has the unpleasant aspect that the integration variables have a completely different support, 0 < x < 1 , −∞ < k⊥ < +∞ . Therefore, practically in all of the numerical work particularly in [4] and [5], the variable transform x(kz) = E1 + kz E1 + E2 , (3) with E1,2 = E1,2(k) ≡ √ m 2 1,2 + k 2 z + k 2 ⊥ , has been used to transform to integration variables −∞ < kz < +∞ , −∞ < k⊥ < +∞ , 2 Hans-Christian Pauli: Fine and hyperfine interaction on the light cone with the same support. While kz varies from −∞ to +∞, the x(kz) varies from 0 to 1. The particles are then described by their front form four-momenta k 1 = kz + E1(k) , k + 2 = − kz + E2(k) , k⊥1 = k⊥ , k⊥2 = − k⊥ , k 1 = − kz + E1(k) , k 2 = kz + E2(k) . Or, they are described by the instant form four-momenta k 1 = E1(k) , k 0 2 = E2(k) , k1 = k , k2 = − k , with k ≡ (k⊥, kz). Such a switching between front form and instant form parameterization is possible, since the four-vectors of the constituents refer to free particles. The free invariant mass of the two particles, M free = m1 + k 2 ⊥ x + m2 + k 2 ⊥ 1− x = (E1(k) + E2(k)) 2 , (4) looks like in the rest frame of the instant form (P = 0). For vanishing k it is (m1 +m2) . The discrepancy can be calculated exactly as [6] (E1 + E2) 2 − (m1 +m2) = (E1 + E2 −m1 −m2) (E1 + E2 +m1 +m2) ,