Counting operators in Effective Field Theories
| Thesis title in Czech: | Generující funkce pro operátory v Efektivních Teoriích Pole |
|---|---|
| Thesis title in English: | Counting operators in Effective Field Theories |
| Key words: | Klasická teorie pole|Efektivní teorie pole|Teorie grup|Lieovy grupy|Reprezentace|Schurovy relace ortogonality|Hilbertova generující funkce |
| English key words: | Classical field theory|Effective field theory|Group theory|Lie groups|Representation theory|Schur character orthogonality relations|Hilbert series |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | Bachelor's thesis |
| Thesis language: | angličtina |
| Department: | Institute of Particle and Nuclear Physics (32-UCJF) |
| Supervisor: | Mgr. Petr Vaško, Ph.D. |
| Author: | Bc. Jonáš Dujava - assigned and confirmed by the Study Dept. |
| Date of registration: | 27.11.2021 |
| Date of assignment: | 27.11.2021 |
| Confirmed by Study dept. on: | 07.01.2022 |
| Date and time of defence: | 07.09.2022 09:00 |
| Date of electronic submission: | 21.07.2022 |
| Date of submission of printed version: | 21.07.2022 |
| Date of proceeded defence: | 07.09.2022 |
| Opponents: | Vasja Susič, Ph.D. |
| Guidelines |
| An Effective Field Theory is the most efficient framework for describing physics at different length scales, by only capturing the relevant degrees of freedom at the given energy/length scale (and neglecting all the rest in a controlled way). A basic characteristic of an operator that can appear in a Lagrangian of an Effective Field Theory is the number of fields and derivatives (and hence a given mass dimension). Additionally, these operators are severely constraint by various symmetries. Before actually attempting to construct all these operators (which is hard in general), the first simpler step is just to enumerate all allowed inequivalent possibilities. This is the topic of the thesis. The student will get familiar with techniques (known as the Hilbert series) for counting operators in effective field theories. The Hilbert series is a generating function in two formal variables whose coefficients provide the number of operators in the effective Lagrangian with a given number of fields and a given number of derivatives (modulo equivalence relations -- integrations by parts and equations of motion). The main idea of the Hilbert series is to construct a group character with respect to all relevant symmetries (space-time, global), decompose into irreducible representations, project onto the singlet (using Schur orthogonality relations) and thus count the invariants, i.e. operators that can appear in the Lagrangian (group theory takes care of the equivalence relations). In summary, the topics of this thesis are classical field theory, group theory (in particular Schur orthogonality relations) and a little bit of complex analysis (residue theorem). The basic goal of this thesis is to reproduce the Hilbert series for a single scalar field. Advanced students might manage to go beyond that and compute the Hilbert series of a non-linear sigma model based on a coset target space G/H. |
| References |
| [1] Henning, Lu, Melia and Murayama, Hilbert series and operator bases with derivatives in effective field theories,Commun. Math. Phys. 347 (2016), [arXiv:1507.07240 [hep-th]]
[2] Henning, Lu, Melia and Murayama, Operator bases, S-matrices, and their partition functions,JHEP 10 (2017) 199, [arXiv:1706.08520 [hep-th]] [3] Graf, Henning, Lu, Melia and Murayama, 2, 12, 117, 1959, 45171, 1170086, ...: a Hilbert series for the QCD chiral Lagrangian, JHEP 01 (2021) 142, [arXiv:2009.01239 [hep-ph]] [4] Lehman and Martin, Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox, Phys. Rev. D 91 (2015) 105014, [arXiv:1503.07537 [hep-ph]] [5] Benvenuti, Feng, Hanany and He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050, [arXiv:hep-th/0608050 [hep-th]] |
| Preliminary scope of work in English |
| All students who are curious how to continue the sequence 2, 12, 117, 1959, 45171, 1170086, ... should pick this thesis. Hint: the next term in the sequence counts the basis of operators in the effective Lagrangian (describing mesons, e.g. pions) of Quantum Chromodynamics with three light quarks at order O(p^14), i.e. operators in the effective Lagrangian with 14 derivatives (those have never been constructed explicitly and are unimportant for phenomenology, however it is cool to know their number anyway). This "pion Lagrangian" is based on the coset space G/H with G=SU(3)xSU(3) and H=SU(3). Advanced students might be able to either reproduce these numbers or obtain their own new results for a different non-linear sigma model (even if, admittedly, this is the most famous one as it describes Nature). Historical note: the Hilbert series is a standard technique developed by mathematicians within the framework of commutative rings; to my knowledge it was first applied in physics around 2006 in [5] for counting special types of operators in supersymmetric Quantum Field Theories and then later around 2015 in the context of Effective Field Theories (which is the topic of this thesis). |