We published a paper based on my Bachelor’s thesis – submitted in July 2011 at Humboldt University. You can read it on arXiv or PRD.

**What did we do?**

We analysed the __efficiency__ of new __analytical expressions__ for __tree-level__ __amplitudes__ in __massless__ __QCD__. To give an idea what this means, you have to read the last sentence backwards (almost word for word):

**QCD**

QCD stands for Quantum Chromo Dynamics and is a quantum field theory to describe the fundamental particles of matter, namely quarks and their interaction through gluons. Protons and Neutrons (in the nucleus of every atom) are built from quarks. Understanding QCD is crucial to understand the behavior of matter on high energies, for example in high energy experiment at the Large Hadron Collider (LHC) at CERN.**Massless**

Quarks are not massless (otherwise we would not feel gravity). However, on high energies the mass is negligible compared to the high-energy in their momentum: a bullet shot with a canon will still fall down, but as approximation for a few meters we can assume that it follows almost a straight line. In the same way, we can neglect mass if a particle is accelerated to a very high speed and momentum.**Amplitudes**

For many experiments in high-energy physics we want to know how likely it is that accelerated particles scatter and produce new particles. This can be answered by calculating a probability distribution which depends on the absolute value of a complex quantity called scattering amplitude. Roughly speaking, a (scattering) amplitude is the function that allows us to calculate the probability of a scattering event to happen and it depends on the properties of the particles involved.**Tree-level**

Calculating scattering amplitudes is difficult: Even with computer there is no way to calculate scattering amplitudes to arbitrary precision. Instead we have to use a perturbation series of very complicated expressions. Feynman found a way to represent these complicated expressions by simple diagrams called Feynman diagrams and the perturbation series**Analytical exprressions**

Calculation of such amplitudes is mainly done through a numerical recursion that repeats certain steps a number of times to get a numerical result. In contrast to this, an analytical expression is one formula describing how you get a numerical value by just applying certain operations (like addition, multiplication, squareroots etc.) and is mathematically speaking a function with explicit form. In our case, we have such analytical expressions for the tree-level amplitudes, even though they are very long and complicated.**Efficiency**

In order to predict certain outcomes of a scattering experiment one needs to predict probabilities by calculating the amplitudes for different possible processes. Because even on tree-level (leading order) one finds a huge number of possible processes one has to evaluate thousands and millions of amplitudes which makes efficiency very important: the less time one evaluation of an amplitude takes the more realistic is it to make certain predictions. Our whole analysis has the goal to compare the efficiency of just evaluating our analytical expressions with one commonly used numerical procedures (called Berends-Giele) that use a recursion to approximate the value. Furthermore, we compared the precision of the different approaches.

**What did we get?**

As expected before, our implementation of the analytical formulae (the main work of my bachelor thesis) is faster for the first two orders (MHV and NMHV). The third order (NNMHV) that I implemented, as well, becomes slower than the Berends-Giele approach as the number of involved particles increases. I did not implement higher orders because it was clear that higher orders will be slower than Berends-Giele. Finally, the accuracy of the analytical formulae was a bit higher than using the Berends-Giele scheme.