Long-range multi-scalar models at three loops
Dario Benedetti, Razvan Gurau, Sabine Harribey, Kenta Suzuki
Abstract
Abstract We compute the three-loop beta functions of long-range multi-scalar models with general quartic interactions. The long-range nature of the models is encoded in a kinetic term with a Laplacian to the power 0 < ζ < 1, rendering the computation of Feynman diagrams much harder than in the usual short-range case ( ζ = 1). As a consequence, previous results stopped at two loops, while seven-loop results are available for short-range models. We push the renormalization group analysis to three loops, in an ϵ = 4 ζ − d expansion at fixed dimension d < 4, extensively using the Mellin–Barnes representation of Feynman amplitudes in the Schwinger parametrization. We then specialize the beta functions to various models with different symmetry groups: O ( N ), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:mo>⋊</mml:mo> <mml:msub> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and O ( N ) × O ( M ). For such models, we compute the fixed points and critical exponents.