The Relativistic Quantum Field Calculator for Elementary Particle Interactions

1. The Relativistic Quantum Field Calculator for Elementary Particle Interactions Brandon Green , Dr. Edward Deveney (Mentor) Department of Physics, Bridgewater State College, MA 02325 Abstract Generating Feynman Diagrams QED Interaction Table Before we developed the generator, we constructed an algorithm to figure out how to generalize all possible Feynman diagrams. We need to be able to distinguish each interaction as well as feed our generator the proper components to construct a list of Feynman diagrams. I used a set of logic here to distinguish each particle interaction based on the user’s particle selections. We have built a relativistic quantum field calculator to compute QED particle interactions. This requires original code capable of generating Feynman diagrams. To build the code, we constructed an algorithm through a detailed understanding of electron-muon scattering, which included meticulous documentation of Casamir's trick as well as the completeness relation. After the algorithm was built, we used LabView to assemble our generator for tree-level diagrams as well as higher order diagrams. In addition to the generator, we developed an incoming QED interaction table to use in accordance to a set of reaction filters. Once proven with tree-level calculations, higher order events will be computed. Early computational studies have been done with this generator. We started with a simple system for our tree-level diagrams, 6-components and 3 elements. Systematically listing, we saw a definite pattern in every new sequence. Fig.2: Generalization of electron-muon scattering into an ABCCAB sequence. The following relationship denotes the number of states, where p = # of components, [N] = # of elements, and S = # of states What is Relativistic Quantum Field Theory? From here, we can now build up every combination of p-component states. Let n{X1, X2, ..Xc} be a column denoted by n amount of X’s, where n is any integer and X is any component. The final result will be a column of states expressed by Fig.7: LEFT: Front Panel of LabView, showing the Mott scattering reaction for the two incoming particles. RIGHT: Back Panel of LabView, shows the logic involved for each reaction based on user input. Our algorithm helped us develop the following program in LabView to generate any possible Feynman diagram. Casimir’s Trick and The Completeness Relation Extracting the amplitude is important to analyze particle interactions, but we are interested in < M>2. Casimir’s Trick allows us to extract < M>2 without dealing with the spinors involved. Understanding this step in extreme detail is crucial to our code. Starting with the solution for < M>2 for electron-muon scattering, We need to define a general form, G, to help reduce |M|2 Note the completeness relation, Summing over b spins on G, we can see that Fig.4: Back Panel of LabView, code written for our generator FIG. 1: Brief summary of the history and development of Relativistic Quantum Field Theory (RQFT). Reducing and summing over a spins, The relativistic theory in Feynman’s formulation is primarily suited to explain decays and scattering reactions. Feynman diagrams are pictorial representations of elementary particle interactions that permit finding amplitudes, or M. Fig.3: Front Panel of LabView interface showing the combinations for 3[3] Rewriting into explicit matrix multiplication and rearranging, Why Develop The Calculator? Next is the filtering process to select all valid diagrams for a desired interaction. Applying the completeness relation, Filtering Valid Feynman Diagrams • Feynman diagrams, although a useful tool, have a complex mathematical framework behind them. • Extracting M is a complicated process that requires not only the Feynman Rules, but close attention to detail. • Our computer program generates all possible Feynman Diagrams based on user input, and proceeds with the calculation of < M>2. All of the generated values for {p[N]} need to be filtered somehow in order to pick the Feynman diagrams were interested in. Which gives us the final result, This is huge because it effectively gives a direct method of calculating < M>2 The filter section, shown in Fig.5, was designed to accept all incoming permutations from our generator. We developed a selector capable of picking the designated filter to choose the interaction the user wishes to calculate. We designed a test filter first, with the combination of CCCCCC. Conclusions Research Goals • We have successfully created a working generator that creates all valid Feynman Diagrams up to any order. • We have produced filters, selectable by the user, to properly filter the correct diagrams designated by user input. • A QED interaction table has been constructed, and tested to properly denote the particle interaction based on user input of incoming particles. • Casimir’s Trick and the completeness relation have been documented in extreme detail for further code processing. • To work through a develop a working generator of Feynman Diagrams of any order. • Construct a set of filters to pick all valid diagrams pertaining to particle interactions. • Develop an interaction table for all elementary particles. • Implement a system of values for all fundamental particles. • Apply Feynman Rules to construct < M>2. • Use Casmir’s Trick and the completeness relation to reduce < M>2to a number. Fig.5: Back Panel of LabView interface showing code for the filter selector. Acknowledgements Fermi Golden Rule Although our filter chooses the correct diagram(s), we had to discard all other permutations not used. This section, shown in Fig. 6, labels all permutations that do not pass the filter test as -1, which are then discarded. Only the valid Feynman Diagram(s) remain! Thank you to the Adrian Tinsley Program coordinators at Bridgewater State College, Dr. Edward Deveney, my peers, and anyone else who has helped me to perform this project. Fermi’s Golden Rule for any particle interaction, determines the transition rate for a reaction using the extracted amplitude and the phase space information. Fig.6: Back Panel of LabView interface showing code for selection of valid diagrams. The amplitude contains all dynamical information, while the phase space is reserved for kinematics. Calculating M is difficult and requires Feynman diagrams. Contact: b2green@bridgew.edu