The quark jet function for $$k_T$$-like variables in NNLO QCD

The precise description of jet processes requires observables capable of efficiently capturing the dynamics of the energy flow in hadronic final states. We consider a class of transverse-momentum like resolution variables that smoothly describe the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> to <i>n</i> jet transition in multi-jet processes. We discuss a general method for the computation of the corresponding quark jet function at next-to-next-to-leading order in perturbative QCD. Rapidity divergences are regulated by using a time-like auxiliary vector. We present explicit results for a variant of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi> <mml:mn>23</mml:mn></mml:msub> </mml:math> in the <i>E</i>-scheme and in the WTA scheme.