Postgraduate Course: Scattering Amplitudes I: Feynman Integrals (PGPH12001)
Course Outline
| School | School of Physics and Astronomy |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 12 (Postgraduate) |
Availability | Available to all students |
| SCQF Credits | 15 |
ECTS Credits | 7.5 |
| Summary | The course aims to provide an introduction to Feynman integrals. This includes understanding the analytic structure and singularities of Feynman integrals, and their connection to physical principles such as unitarity and causality, as well as methods to compute such integrals. This includes the introduction of parametric representations, integration-by-parts and other identities, differential equation methods, and asymptotic expansions. The course also introduces the special functions to which Feynman integrals evaluate, as iterated integrals with multiple polylogarithms as the primary example. |
| Course description |
This first course on scattering amplitudes provides an introduction to Feynman integrals. It is designed primarily for first year PhD students in either Physics or Mathematics, who have taken (or are currently taking) Quantum Field Theory (QFT) courses.
QFT provides a remarkably successful description of Nature at the fundamental level. It is an elegant mathematical language describing relativistic quantum systems. Scattering amplitudes are the bridge connecting QFT to observable quantities describing scattering of elementary particles. Indeed, it is through scattering that physicists learn about elementary interactions. The computation of scattering amplitudes requires to evaluate Feynman integrals - and in many cases, computing these integrals represents the most difficult step.
The course highlights the connection between the mathematical structure of Feynman integrals and fundamental physics principles, such as locality, unitarity and causality of the interaction.¿It provides hands-on experience in computing Feynman integrals using modern approaches, hence also acquaints students with active research topics on Feynman integrals, which connect to modern topics in algebraic geometry and number theory and are thus of interest from both the mathematics and physics perspectives.
More details are provided in the syllabus and learning outcome sections.
The course is based on two complementary methods of delivery: Ten 2-hour lectures given on a blackboard in which new concepts will be introduced (one 2-hour meeting per week) and in parallel 2-hour exercise classes in which specific problems will be explicitly solved (again one such session per week). The exercises will be given as part of the lecture and solved in the exercise class. Calculations in exercises will be mostly pen-and-paper work, and will occasionally be aided by computer algebra tools such as Mathematica and Maple.
A selected subset of exercises will be defined as hand-in work and will be used for assessment (formative as well as summative). The assessment will be based on the best 3 out of at least 4 hand ins.
Syllabus/Lecture List:
Week 1: Feynman graphs, Feynman rules and the Feynman prescription
Week 2: Parametric representations of Feynman integrals
Week 3: Landau singularities
Week 4: Unitarity, cuts and discontinuities of Feynman integrals
Week 5: Dimensional-shift and integration-by-parts relations between Feynman integrals
Week 6: Computing Feynman Integrals by Differential Equations
Week 7: Differential equations in epsilon form and iterated integrals
Week 8: Multiple Polylogarithms and the Coaction
Weeks 9-10: Asymptotic expansions of Feynman integrals - the method of regions in momentum space and in parameter space
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Entry Requirements (not applicable to Visiting Students)
| Pre-requisites |
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Co-requisites | |
| Prohibited Combinations | |
Other requirements | None |
Information for Visiting Students
| Pre-requisites | Co-requisites Quantum Field Theory (PHYS11065) or equivalent |
Course Delivery Information
| Not being delivered |
Learning Outcomes
On completion of this course, the student will be able to:
- Set up and perform the calculation of basic Feynman integrals in momentum space, in parameter space, or using differential equations
- Perform Landau analysis to identify potential singularities of Feynman integrals, compute unitarity cuts and discontinuities of Feynman integrals, and understand the relation between these operations
- Derive dimensional-shift and integration-by-part relations between Feynman integrals and use them for reducing to a set of master integrals, and to deriving differential equations
- Work with iterated integrals such as multiple polylogarithms and their coaction
- Perform asymptotic expansions using the Method of Regions
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Reading List
| A recommended book on Feynman integrals, which we will use for certain topics, is the book by Stefan Weinzierl, entitled "Feynman Integrals - A Comprehensive Treatment for Students and Researchers". The book is available online via Springer or via the arXiv. |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | Quantum Field Theory,Scattering Amplitudes,Feynman Integrals,Unitarity,Causality,Landau Singul |
Contacts
| Course organiser | Prof Einan Gardi
Tel: (0131 6)50 6469
Email: Einan.Gardi@ed.ac.uk |
Course secretary | Ms Joyce Ternenge
Tel:
Email: joyce.ternenge@ed.ac.uk |
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