A formalism for the systematic treatment of rapidity logarithms in Quantum Field Theory

TLDR: In this article, the authors present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group".

Abstract: Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the relevant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. Such observables include: transverse momentum distributions at p T much less then the high energy scattering scale, jet broadening, exclusive hadroproduction and decay, as well as the Sudakov form factor. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a “rapidity renormalization group”. That is, a Collin-Soper like equation is realized as a renormalization group equation, but has a more universal applicability to observables beyond the traditional transverse momentum dependent parton distribution functions (TMDPDFs) and the Sudakov form factor. This formalism has the feature that it allows one to track the (non-standard) scheme dependence which is inherent in any scenario where one performs a resummation of rapidity divergences. We present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form factor. The formalism is then used to study observables of current interest. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. Our formalism allows one to define gauge invariant TMDPDFs which are independent of both the hard scattering amplitude and the soft function, i.e. they are universal. We present details of the factorization and re- summation of the jet broadening cross section including a renormalization in p⊥ space. We furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.

Figures

Figure 14.

Figure 1. The mass-shell hyperbolae showing the distinction between the different sectors [5]. The separation between soft and collinear modes is arbitrary and leads to rapidity divergences. The soft sector has two distinct rapidity (UV) divergences that must cancel with rapidity (IR) divergences arising from the collinear sector.

Figure 11. (a) W -boson and its decay is not shown. buW vertex is represented by •. u quark represented with the thick line is the hard parton with off-shellness mBΛQCD. Hard interaction takes place at vertex A and external on-shell partons are at vertex B and C. (b) Corresponding Sudakov form factor demonstrating hard interaction at vertex A. On-shell n-collinear parton is at vertex B and on-shell n̄-collinear parton is at vertex C.

Figure 6. A contribution to the production cross section with central jets.

Figure 12. Modes and Frames: relationship between symmetric, lab and super-boosted frame is shown along with the modes in each frame with their location on the hyperbola.

Figure 16. The correspondence between the full theory diagram and the diagrams in the effective theory. The first column on the right is the soft function contribution while the next two columns correspond to the collinear functions in the n and n̄ directions respectively. Not shown are mirror diagrams in which n↔ n̄. The blob corresponds to possible hard pieces or soft non-hadronic (e.g. the Higgs) final states. The darkened lines are eikonalized. The incoming lines in the collinear functions are either quarks or gluons.

Frequently asked questions

Q1. What have the authors contributed in "A formalism for the systematic treatment of rapidity logarithms in quantum field theory" ?

In this paper the authors will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a “ rapidity renormalization group ”. The authors present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form factor. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. The authors present details of the factorization and resummation of the jet broadening cross section including a renormalization in p⊥ space. The authors furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.