# Effective Field Theories

## Module PH2122

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2020

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions | |||||
---|---|---|---|---|---|

WS 2022/3 | SS 2020 | WS 2018/9 | SS 2018 | WS 2016/7 | SS 2011 |

### Basic Information

PH2122 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

- Specific catalogue of special courses for nuclear, particle, and astrophysics
- Complementary catalogue of special courses for condensed matter physics
- Complementary catalogue of special courses for Biophysics
- Complementary catalogue of special courses for Applied and Engineering Physics
- Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workload | Contact hours | Credits (ECTS) |
---|---|---|

300 h | 90 h | 10 CP |

Responsible coordinator of the module PH2122 in the version of SS 2020 was Antonio Vairo.

### Content, Learning Outcome and Preconditions

#### Content

The module will provide an introduction to effective field theories (EFTs) and renormalization techniques with applications ranging from high energy to atomic physics. The following topics will be covered:

- Principles of EFTs
- Scales and systems in nature
- What is an EFT and how to construct it
- Example: the Euler-Heisenberg Lagrangian
- Example: the Fermi theory of weak interactions at tree level
- Example: the Rayleigh scattering
- Relevant, irrelevant and marginal operators
- Quantum loops of irrelevant operators
- Mass-dependent vs mass-independent regularization schemes
- Dimensional regularization
- Quantum loops of marginal operators
- Example: β function and running coupling constant in QED and QCD
- Decoupling theorem
- Example: the one and two loop matching of the QCD strong-coupling constant in MSbar
- Renormalization group equations in QFTs and EFTs
- Anomalous dimensions
- Mixing
- Example: ΔS = 2 transition amplitude in the Fermi theory of weak interactions

- Heavy quark effective theory
- Heavy-light meson spectrum
- Heavy-quark spin-flavour symmetry
- Static Lagrangian
- Spectroscopy implications
- Heavy meson decay constants
- Transition form factors: Isgur-Wise functions
- Example: B → D transitions and calculation of dΓ(B → D e ν)/dq²
- Renormalization of composite operators
- Example: heavy-light currents and heavy-heavy currents
- Heavy meson decay constants at LL and NLO
- The 1/m expansion of the HQET Lagrangian
- Reparameterization invariance
- Chromomagnetic coupling and hyperfine splitting at LL
- Decoupling in the HQET
- B → D e ν and Luke's theorem

- Applications to atomic physics
- Bound states in QED: physical picture, scales, degrees of freedom
- NRQED: Lagrangian, power counting, matching
- Four-fermion operators
- Example: matching of dimension six four-fermion operators and the positronium decay width
- pNRQED: Lagrangian, power counting, matching
- Example: the hydrogen atom and the Lamb shift
- Example: the Rayleigh scattering in pNRQED

#### Learning Outcome

After successful completion of the module the students are able to

- build effective field theories,
- compute Wilson coefficients and renormalize them,
- solve renormalization group equations
- compute observables in principle at any order in the expansion parameter(s).

#### Preconditions

Quantum Mechanics 1 + 2 (PH0007 +PH1002) and some basic knowledge of Quantum Field Theory and the Standard Model.

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

Type | SWS | Title | Lecturer(s) | Dates | Links |
---|---|---|---|---|---|

VO | 4 | Effective Field Theories | Vairo, A. |
Tue, 14:00–16:00, PH 3343 Thu, 10:00–12:00, PH 3343 |

#### Learning and Teaching Methods

In the thematically structured lecture the learning content is presented. With cross references to many different fields spanning from high energy physics, physics of beyond the Standard Model, low energy physics and condensed matter many different applications and a large number of physical examples are presented. In practice we will work out one loop calculations for non-relativistic effective field theories in QED and QCD. Lectures will be given on a blackboard. Slides presentations may also be used. Additional material will be contained on the web page of the lectures and the blog of the lectures with links to research papers, appropriate chapters of books and further discussion of important questions.

In scientific discussions the students are involved to stimulate their analytic-physics intellectual power.

In the exercise the learning content is deepened and exercised using problem examples and calculations. Thus the students are able to explain and apply the learned physics knowledge independently. Many physical examples will be presented and worked out in all details. Exercises are suggested and will be discussed as requested. Discussions and feedbacks during the lectures are strongly encouraged.

#### Media

Blackboard, slides, examples, exercise sheets

#### Literature

- Principles of EFTs
- Books:
- A. Dobado, A. Gomez-Nicola, A.L. Maroto & J.R. Pelaez:
*Effective Lagrangians for the Standard Model*, Springer Verlag, (1997) - S. Weinberg:
*The Quantum Theory of Fields Vol. II*, Cambridge University Press, (1996), Chapter 19 - Review papers and lecture notes:
- A.V. Manohar:
*Effective field theories*, Schladming, 1996, Perturbative and nonperturbative aspects of quantum field theory* 311-362, e-Print: hep-ph/9606222 - D.B. Kaplan:
*Effective field theories*, 7th Summer School in Nuclear Physics Symmetries, Seattle, e-Print: nucl-th/9506035 - H. Georgi:
*Effective field theory*, Ann.Rev.Nucl.Part.Sci.43:209-252,1993

B.R. Holstein:*Effective effective interactions*, Eur.Phys.J.A18:227-230,2003 - A founding paper:
- S. Weinberg,
*Phenomenological Lagrangians*, Physica A96:327,1979 - EFTs courses:
- School on
*Flavor Physics*, Centro de Ciencias de Benasque (2008) - Effective field theory course (2013) at MIT by Iain Stewart (with emphasis on SCET)

- Heavy quark effective theory
- Books:
- A.V. Manohar & M.B. Wise:
*Heavy quark physics*, Cambridge University Press, (2000) - Review papers and lecture notes:
- M. Neubert:
*Heavy-quark symmetry*, Phys.Rept.245:259-396,1994 - B. Grinstein:
*An introduction to heavy mesons*, 6th Mexican School of Particles and Fields, Villahermosa, e-Print: hep-ph/9508227 - T. Mannel:
*Heavy-quark effective field theory*, Rept.Prog.Phys.60:1113-1172,1997 - Related papers:
- G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, K. Hornbostel:
*Improved nonrelativistic QCD for heavy quark physics*, Phys.Rev.D46:4052-4067,1992, e-Print: hep-lat/9205007 - A.V. Manohar:
*The HQET/NRQCD Lagrangian to order α/m^3*, Phys.Rev.D56:230-237,1997, e-Print: hep-ph/9701294

- Applications to atomic physics
- Related papers:
- W.E. Caswell, G.P. Lepage:
*Effective Lagrangians for bound state problems in QED, QCD, and other field theories*, Phys.Lett.B167:437,1986 - A. Pineda, J. Soto:
*The Lamb shift in dimensional regularization*, Phys.Lett.B420:391-396,1998, e-Print: hep-ph/9711292 - A. Pineda, J. Soto:
*Potential NRQED: the positronium case*, Phys.Rev.D59:016005,1999, e-Print: hep-ph/9805424 - B.R. Holstein:
*Blue skies and effective interactions*, American Journal of Physics 67:422,1999

### Module Exam

#### Description of exams and course work

There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and sample calculations.

In accordance with §12 (8) APSO the exam can be done as an oral exam. In this case the time duration is 25 minutes.

For example an assignment in the exam might be:

- Typical questions will include: how to construct an EFT, how to match it, how to write an RG equation,
- how to write an RG equation. Specific physical examples presented during the lecture (HQET,NRQED,pNRQED, etc.) may be also discussed.
- How to calculate a matching in the effective field theory at tree level and at one loop.
- How to use the effective field theory to calculate physical observers, masses, decays and scattering processes.
- How to do a perturbative calculation inside the effective field theory
- How to use the effective field theory to estimate the scale of new physics.
- How to use the effective field theory to obtain the parametric dependence of a cross section.

In the exam no learning aids are permitted.

Participation in the exercises is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to the exam period directly following the lecture period (not to the exam repetition) and subject to the condition that the students solve 70% of the exercises.

#### Exam Repetition

The exam may be repeated at the end of the semester.