An appealing interpretation of a parton density is that it is a number density of partons in a target hadron. As we saw in Sec. 6.7, a parton density in a simple theory is an expectation value of a light-front number operator, integrated over transverse momentum. A similar interpretation applies to fragmentation functions: Sec. 12.4.

As explained in Secs. 6.8 and 12.4, it is equally natural to define unintegrated, or transverse-momentum-dependent (TMD), parton densities and fragmentation functions, simply by omitting the integral over transverse momentum. In a sense, the TMD functions are more fundamental and present more information on non-perturbative phenomena than do the ordinary integrated functions. Therefore it is useful to find situations where TMD functions are needed.

In this chapter, I treat two characteristic cases. One is two-particle-inclusive e+e- annihilation when the detected hadrons are close to back-to-back. This process needs TMD fragmentation functions. Then I will extend this work to semi-inclusive DIS (SIDIS) with a detected hadron of low transverse momentum. In SIDIS, TMD parton densities are needed as well as fragmentation functions. A further extension to the Drell-Yan process at low transverse momentum will be covered in Sec. 14.5.

There are substantial complications in QCD. Although the discussion about light-front quantization and the associated definitions of number densities gives a general motivation, it does not work correctly in QCD (or any other gauge theory). The actual definitions are whatever is appropriate to consistently obtain a valid factorization theorem.