An Overview of Transverse Momentum Dependent Factorization and Evolution

I review TMD factorization and evolution theorems, with an emphasis on the treatment by Collins and originating in the Collins-Soper-Sterman (CSS) formalism. I summarize basic results while attempting to trace their development over that past several decades.Comment: 14 pages, 1 figure, Submission to EPJ A topical issue on "3D Structure of the Nucleon

TMD Evolution at Moderate Hard Scales

We summarize some of our recent work on non-perturbative transverse momentum dependent (TMD) evolution, emphasizing aspects that are necessary for dealing with moderately low scale processes like semi-inclusive deep inelastic scattering.Comment: 6 pages, 1 figure, proceedings for QCD Evolution 2015 26-30 May 2015, Jefferson Lab (JLAB), Newport News Virginia, US

A representation of the natural numbers by means of cycle-numbers, with consequences in number theory

In this paper we give rules for creating a number triangle T in a manner analogous to that for producing Pascal's arithmetic triangle; but all of its elements belong to {0, 1}, and cycling of its rows is involved in the creation. The method of construction of any one row of T from its preceding rows will be defined, and that, together with starting and boundary conditions, will suffice to define the whole triangle, by sequential continuation. We shall use this triangle in order to define the so-called cycle-numbers, which can be mapped to the natural numbers. T will be called the 'cyclenumber triangle'. First we shall give some theorems about relationships between the cyclenumbers and the natural numbers, and discuss the cycling of patterns within the triangle's rows and diagonals. We then begin a study of figures (i.e. (0,1)- patterns, found on lines, triangles and squares, etc.) within T. In particular, we shall seek relationships which tell us something about the prime numbers. For our later studies, we turn the triangle onto its side and work with a doubly-infinite matrix C. We shall find that a great deal of cycling of figures occurs within T and C, and we exploit this fact whenever we can. The phenomenon of cycling patterns leads us to muse upon a 'music of the integers', indeed a 'symphony of the integers', being played out on the cycle-number triangle or on C. Like Pythagoras and his 'music of the spheres', we may well be the only persons capable of hearing it!

Connecting Different TMD Factorization Formalisms in QCD

In the original Collins-Soper-Sterman (CSS) presentation of the results of transverse-momentum-dependent (TMD) factorization for the Drell-Yan process, results for perturbative coefficients can be obtained from calculations for collinear factorization. Here we show how to use these results, plus known results for the quark form factor, to obtain coefficients for TMD factorization in more recent formulations, e.g., that due to Collins, and apply them to known results at order $\alpha_s^2$ and $\alpha_s^3$. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the two schemes. We compile the higher-order perturbative inputs needed for the updated CSS scheme by appealing to results obtained in a variety of different formalisms. In addition, we derive the connection between both versions of the CSS formalism and several formalisms based in soft-collinear effective theory (SCET). Our work uses some important new results for factorization for the quark form factor, which we derive.Comment: 30 pages, 2 Figures; Fixed typos including missing term in Eq.(60

Observational signatures of convectively driven waves in massive stars

We demonstrate observational evidence for the occurrence of convectively driven internal gravity waves (IGW) in young massive O-type stars observed with high-precision CoRoT space photometry. This evidence results from a comparison between velocity spectra based on 2D hydrodynamical simulations of IGW in a differentially-rotating massive star and the observed spectra.We also show that the velocity spectra caused by IGW may lead to detectable line-profile variability and explain the occurrence of macroturbulence in the observed line profiles of OB stars. Our findings provide predictions that can readily be tested by including a sample of bright slowly and rapidly rotating OB-type stars in the scientific programme of the K2 mission accompanied by high-precision spectroscopy and their confrontation with multi-dimensional hydrodynamic simulations of IGW for various masses and ages.Comment: 4 pages, 3 figures, accepted for publication in The Astrophysical Journal Letter

The Response of Normal Shocks in Diffusers

The frequency response of a normal shock in a diverging channel is calculated for application to problems of pressure oscillations in ramjet engines. Two limits of a linearized analysis arc discussed: one represents isentropic flow on both sides of a shock wave; the other may be a crude appr'l'I;imation to the influence of flow separation induced hy the wave. Numerical results arc given, and the influences of the shock wave on oscillations in the engine are discus,ed