Abstract
Efficient methods for describing non abelian charges in worldline approaches to QFT are useful to simplify calculations and address structural properties, as for example color/kinematics relations. Here we analyze in detail a method for treating arbitrary non abelian charges. We use Grassmann variables to take into account color degrees of freedom, which however are known to produce reducible representations of the color group. Then we couple them to a U(1) gauge field defined on the worldline, together with a Chern-Simons term, to achieve projection on an irreducible representation. Upon gauge fixing there remains a modulus, an angle parametrizing the U(1) Wilson loop, whose dependence is taken into account exactly in the propagator of the Grassmann variables. We test the method in simple examples, the scalar and spin 1/2 contribution to the gluon self energy, and suggest that it might simplify the analysis of more involved amplitudes.
Highlights
Efficient methods for describing non abelian charges in worldline approaches to QFT are useful to simplify calculations and address structural properties, as for example color/kinematics relations
There remains an integration over a modulus associated to the U(1) gauge field, an angle φ, that implements the projection in the amplitudes
In our model we see how that suggestion is explicitly realized. We find it useful to encode the coupling of the Grassmann variables to the modulus φ by using twisted boundary conditions
Summary
Where xμ are the coordinates of the particle, e is the einbein, q is the charge of the particle, and Aμ(x) is the background abelian gauge field. To select the fundamental representation one must project to the sector with occupation number one, so to isolate the wave function φα(x) with an index in the fundamental representation This can be achieved by coupling the variables cα and cα to a U(1) gauge field a(τ ) living on the worldline, with in addition a Chern-Simons term with quantized coupling s. The gauge field a acts as a Lagrange multiplier that imposes a constraint on physical states, and setting n = 1 selects precisely the sector with occupation number one as possible physical states, corresponding to wave functions in the fundamental representation of the color group. Let us analyze this mechanism in detail.
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