## Drell-Yan at the Tevatron and the LHC

#### 1.

Calculate analytically the tree-level decay rate of the W boson to leptons. The formula for the decay rate is given by

where denotes the matrix element describing the decay, is the mass of the decaying particle and is the two-particle phase space measure.

You may also have a look at the following Mathematica notebook.

#### 2.

The partonic cross-section near the resonance is described by the Breit-Wigner formula:

where , and denote the partial and total decay rates of the W (See Exercise 1.), and denotes the partonic center of mass energy.\ In the limit where , we can use the narrow width approximation for the cross-section. Use

to derive the expression of the cross-section in the narrow width approximation.

Fold the partonic cross-section with PDF's to obtain the full cross-section for Drell-Yan production at Tevatron,

w here and denote the PDF's of the and quarks inside the proton. For this exercise we choose You may also have a look at the following Mathematica notebook.

#### 3.

Use Madgraph/MadEvent to generate at the Tevatron and the LHC. Compare the cross sections and indentify the qualititative differences.

#### 4.

Consider the rapidity asymmetry for production at the Tevatron. defined as:

Give an estimate of such asymmetry and show that it is proportional to the slope of evaluated at . Plot the rapidity distributions of the the charged leptons coming from decays at the Tevatron.

#### 5.

Is it possible to define an asymmetry at the LHC too?