QCD

Multiple gluon/parton radiation


  1. e+e-→ hadrons at higher orders
  2. Jets
  3. Large logarithms and splitting functions

    IDENTIFYING LARGE LOGS

    Let us now try to analyse the logarithmic structure of the gluon emission process in some more detail. In order to do so, consider the differential cross section for producing a quark with fractional energy x in the qqg final state. It reads
    Here, clearly the first integral diverges logarithmically for z→ 1, but the second one is finite for all values of z. Anyways, without going into any detail considering the divergent z-integral, it is also obvious that this differential cross section diverges for x→ 1. However, from the previous discussion it is known that the inclusion of the virtual contribution will render the total result finite:
    or, written in a more catchy way
    This allows to define specific distributions, also known as "+"-functions, which in this case boils down to including the virtual pieces into the differential cross section. These function in general are defined as
    Keeping this in mind, it is clear that the differential cross section above can be written as
    where
    and
    according to the definition above. From the consideration so far, it is easy to construct the differential +-function. It reads
    where L denotes the logarithmically divergent piece (the z-integration) and f(L) is the remnant function, depending on how the z integral was regularised.

    SPLITTING FUNCTIONS

    However, here P(x) is the splitting function related to the splitting of a quark into a quark and a gluon. It reads
    Of course, in fact it is also a "+"-distribution. Again, due to this, its integral over the interval from 0 to 1 vanishes identically. However, in the following this property of splitting functions to be in fact distributions will be ignored. Here it suffices to state that the splitting function carries the essential logarithmically enhanced pieces of a radiation process in such a way that subsequent emissions can be described by sequences of splitting functions. Of course, there are more splitting functions. They read
    where
    are the structure constants of QCD.

  4. Resummation/exponentiation
  5. DIS: Deep inelastic scattering and the DGLAP evolution equation
  6. Drell-Yan processes at hadron colliders