Feynman versus Bakamjian-Thomas in Light Front Dynamics

We compare the Bakamjian-Thomas (BT) formulation of relativistic few-body systems with light front field theories that maintain closer contact with Feynman diagrams. We find that Feynman diagrams distinguish Melosh rotations and other kinematical quantities belonging to various composite subsystem frames that correspond to different loop integrals. The BT formalism knows only the rest frame of the whole composite system, where everything is evaluated. The goal of this article is to point out subtle, but important, differences between the BakamjianThomas (BT) formulation of relativistic few-body systems [1] and light front field theories that maintain closer contact with Feynman diagrams (see, e.g., Ref. [2]). To be specific, we consider the triangle diagram that is a major ingredient of recent electromagnetic and weak baryon form factor evaluations in light front dynamics. We start with the effective Lagrangian for the N-q coupling LN−3q = ∑ {i,j,k} Ψ(i)iτ2γ5Ψ C (j)Ψ(k)ΨN (1) where τ2 is the isospin matrix and the sum is over permutations of {1, 2, 3}. The conjugate quark field is ΨC = CΨ>, where C = iγ2γ0 is the charge conjugation matrix. In the nucleon rest frame, and choosing i = 1, j = 2, and k = 3 the spin coupling of the quarks to the nucleon is given by: χ(s1, s2, s3; sN ) = u1γ5u2 u3uN , (2) where the light-front spinor u1 = u(p1, s1) is u(p, s) = p · γ + m √ 2p+2m γγ ( χs 0 ) (3) where p+ = p0 +p3, p− = p0−p3, p⊥ = (p1, p2), and χs is the two component Pauli spinor. The Dirac spinor of the instant form uD(p, s) = p · γ + m √ 2m(p0 + m) ( χs 0 ) (4) carries the subscript D. The expression Eq. (2) appears in the evaluation of the two-loop Feynman diagram of the J+ = J0 + J3 component of the nucleon electromagnetic current once the integrations ∗permanent address: Dept. of Physics, University of Virginia, Charlottesville, U.S.A.