Tensorial charge assignments in unitary groups

Abstract We present an index-based tensorial formulation for computing eigenvalues of charge operators acting on arbitrary tensor representations of unitary gauge groups. The construction follows directly from the action of Cartan generators on tensor products and the additivity of weights, leading to a compact operator acting on general $$(i_p,i_q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> tensors. This framework provides a practical bookkeeping tool for assigning charges to arbitrary-dimensional multiplets appearing in model building. Explicit applications to $$SU(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $$SU(3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and $$SU(5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> representations are discussed.