== 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 %$\begin{cal}M\end{cal}$% denotes the matrix element describing the decay, %$m$% is the mass of the decaying particle and %$\rm{d}\Phi_2$% is the two-particle phase space measure.

You may also have a look at the following Mathematica [http://cp3wks05.fynu.ucl.ac.be/twiki/bin/viewfile/Physics/WAsymm?rev=1;filename=WDecay.nb notebook].

==== 2. ====

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



where %$\Gamma_{\ell\nu}$%, %$\Gamma_{u\bar d}$% and %$\Gamma$% denote the partial and total decay rates of the W (See Exercise 1.), and %$\hat s$% denotes the partonic center of mass energy.\ In the limit where %$m_W\gg \Gamma$%, 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 %$u(x)$% and %$d(x)$% denote the PDF's of the %$u$% and %$d$% quarks inside the proton. For this exercise we choose %$ u(x)=6(1-x)^2,\qquad d(x)=3(1-x)^2. $%

You may also have a look at the following Mathematica [http://cp3wks05.fynu.ucl.ac.be/twiki/bin/viewfile/Physics/WAsymm?rev=1;filename=DYNarrowWidth.nb notebook].

==== 3. ====

Use Madgraph/MadEvent to generate %$pp \to W^\pm \to e^\pm \nu_e$% at the Tevatron and the LHC. Compare the cross sections and indentify the qualititative differences.

==== 4. ====

Consider the rapidity asymmetry %$A_W(y)$% for %$W^\pm$% production at the Tevatron. defined as:



Give an estimate of such asymmetry and show that it is proportional to the slope of %$d(x)/u(x)$% evaluated at %$x=M_W/\sqrt{s}$%. Plot the rapidity distributions of the the charged leptons coming from %$W^\pm$% decays at the Tevatron.

==== 5. ====

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