Combined threshold and transverse momentum resummation for inclusive observables

We present a combined resummation for the transverse momentum distribution of a colorless final state in perturbative QCD, expressed as a function of transverse momentum p_T and the scaling variable x. Its expression satisfies three requirements: it reduces to standard transverse momentum resummation to any desired logarithmic order in the limit p_T→0 for fixed x, up to power suppressed corrections in p_T; it reduces to threshold resummation to any desired logarithmic order in the limit x→1 for fixed p_T, up to power suppressed correction in 1−x; upon integration over transverse momentum it reproduces the resummation of the total cross cross at any given logarithmic order in the threshold x→1 limit, up to power suppressed correction in 1−x. Its main ingredient, and our main new result, is a modified form of transverse momentum resummation, which leads to threshold resummation upon integration over p_T, and for which we provide a simple closed-form analytic expression in Fourier-Mellin (b,N) space. We give explicit coefficients up to NNLL order for the specific case of Higgs production in gluon fusion in the effective field theory limit. Our result allows for a systematic improvement of the transverse momentum distribution through threshold resummation which holds for all p_T, and elucidates the relation between transverse momentum resummation and threshold resummation at the inclusive level, specifically by providing within perturbative QCD a simple derivation of the main consequence of the so-called collinear anomaly of SCET.