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 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 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?