Lectures on Feynman Integrals

26 Sept 2026, 13:00 → 1 Jul 2027, 16:00 Asia/Shanghai

1502 (SEU-YC)

Baining Peng (东南大学), LINGYU XIAO (东南大学), Zhengwen Liu (Southeast University), jia Zhang (东南大学)

 As high-energy colliders and gravitational-wave detectors continue to advance, increasingly precise experiments demand ever more accurate theoretical predictions. Multi-loop calculations in quantum field theory are becoming essential for precision tests and the search for new physics, with the evaluation of the vast number of Feynman integrals arising from multi-loop diagrams at their core.

 

This seminar is designed to provide graduate and senior undergraduate students with a systematic introduction to Feynman integrals and their calculation methods. Through lectures and discussions, participants will develop an understanding of the theoretical foundations of Feynman integrals and learn how to reduce multi-loop integrals, with applications to particle physics and gravitational wave physics.

Topics will include the basics of Feynman integrals, integration-by-parts (IBP) identities and reduction to master integrals, the differential-equation method, parameterizations, the method of regions for asymptotic expansions, and an introduction to polylogarithms and related special functions.

 

Prerequisites:A strong passion for science, together with a deep curiosity and enthusiasm for exploring frontier problems in physics. You should have a basic background in calculus and algebra, as well as some programming experience. Prior knowledge of quantum field theory is not required, but a basic familiarity with the subject will be highly beneficial.

Instructors: Zhengwen Liu, Baining Peng, Lingyu Xiao, Jia Zhang

Website: indico.global/e/feyn-int-lect2026 (all schedules and materials will posted and updated here)

Time: Saturday 13:00--16:00 (Start Date: 26 Sep 2026)

Location: 东南大学四牌楼校区 (具体教室将在日程处更新)

Additional Information

• Formal registration is not required, but for planning and logistical purposes, we kindly ask interested individuals to send your basic information to seuqft@yeah.net.
• These seminars are voluntarily organized by the QFT Group at SEU. It carries no academic credit (non-credit bearing).
• The format will be discussion-based, with active participation strongly encouraged.

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References

[1] V. A. Smirnov,  Analytic Tools for Feynman Integrals, Springer 2012 

[2]S. Weinzierl,  Feynman Integrals: A Comprehensive Treatment for Students and Researchers, Springer 2022

 

[3]J. M. Henn, J. C. Plefka, and S. Zoia, Scattering Amplitudes in Gauge Theories, Springer 2014

 

[4]C. Duhr,  Mathematical aspects of scattering amplitudes, TASI Lecture Notes 2014, arXiv:1411.7538

 

[5]C. Duhr and F. Dulat,  PolyLogTools --- polylogs for the masses, JHEP 08 (2019) 135, arXiv:1904.07279

Saturday 26 September

1 Lecture 1: Introduction to gravitational-wave science

We begin our journey with a broad overview of the science of Gravitational Waves (GWs). In this lecture, we will address fundamental questions such as, "What are gravitational waves?" and "What astrophysical events in the Universe can produce them?" We will also review the key discoveries of the past decade, following the first direct detection of GWs. This first lecture begins with a brief overview of the limitations of Newtonian gravity and its successor, Einstein's General Theory of Relativity. We then introduce the astrophysical sources of gravitational waves and trace the key historical developments leading to their discovery.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

2 Lecture 2: Introduction to gravitational-wave science (II)

In this second lecture, we introduce the basic principles of gravitational wave detection, the characteristic waveforms produced by compact binary systems, and their important physical properties. We conclude the lecture with a discussion on the future prospects of Gravitational-Wave Astronomy and its potential to revolutionize our understanding of the cosmos.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 3 October

3 Lecture 3: Basic review on special and general relativity

In the first lecture this Saturday, we will review the elementary tools of Special and General Relativity that are essential for introducing Linearized Gravity. We will begin by covering the foundational principles of both theories, then reviewing basic definitions of their geometric descriptions, before moving on to the basics of these theories' symmetry groups.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

4 Lecture 4: Introduction to the linearized theory of gravity

In this second lecture, we will study the Linearized Theory of Gravity, starting from the expansion of the Einstein field equations. As we will see, the symmetry group of General Relativity plays a crucial role. In particular, by exploring its properties, we will arrive at a wave equation that gives rise to the notion of Gravitational Waves. This lecture covers the textbook's §1.1.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 10 October

5 Lecture 5: The transverse-traceless gauge

In this lecture, we will study the wave equation in a vacuum. We will show how a suitable gauge choice allows us to write the wave solutions explicitly in terms of only two physical modes. As we will see, this is accomplished by adopting the so-called TT (transverse-traceless) gauge. This lecture covers the textbook's §1.2.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

6 Lecture 6: Geodesic equation and geodesic deviation

This lecture will prepare us for next week's topic—the interaction of gravitational waves with test masses. To that end, we will study the geodesic equation and the equation of geodesic deviation in detail, in particular, going through the derivation of the latter starting from the former. These two equations are important to understand how one can detect the passage of a GW. This lecture covers the textbook's §1.3.1.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 17 October

7 Lecture 7: Interaction of GWs with test masses - The TT frame

In this lecture, we will cover how GWs interact with test masses, focusing on the specific frame of the TT gauge. In this part, we will learn how to understand physically the coordinate system related to the TT gauge. Today's lectures cover the textbook's §1.3.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

8 Lecture 8: Interaction of GWs with test masses - The detector frame

In this lecture, we will cover how GWs interact with test masses, focusing on another relevant frame, called the detector frame. Similarly to the previous lecture, we will understand in physical terms what does this reference frame corresponds to. With this, we finish the textbook's §1.3.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 24 October

9 Lecture 9: Separation of scales on generic curved background

Today's pair of lectures will be devoted to the study and derivation of the energy-momentum tensor of GWs based on the geometrical approach. In particular, in this first part, corresponding to the ninth lecture, we will introduce the concept of separation of scales, which will be important to understand how, in general, GWs can be meaninfully defined on curved backgrounds.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

10 Lecture 10: The energy-momentum tensor of GWs

For the second part, lecture number ten, we will separate the Einstein's equation into its low- and a high-frequency parts. As we will see, this separation is grounded on the separation of scales introduced in the previous lecture, each having a deep physical meaning. In particular, we will be able to explicitly derive the energy-momentum tensor of GWs from the low-frequency part.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 31 October

11 Lecture 11: The energy flux of GWs

Having derived the energy-momentum tensor carried by the GWs in the previous lecture, today we will obtain important relations out of it. In particular, we will derive the energy flux, showing that it can be written in terms of the two physical modes of the GW, i.e., the plus and cross polarizations. Also in this lecture, we will discuss to what extent an expansion around flat spacetime makes sense.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

12 Lecture 12: A brief introduction to the Mathematica package xAct

During today's second lecture, we will introduce xAct, a Mathematica package for efficient tensor calculation. Our focus on this brief introduction will be the derivation of important formulae used along the first chapter of our textbook. In particular, used in the derivation of the energy-momentum tensor of GWs, we will derive the expression for the Ricci tensor up to second order in the perturbation.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 7 November

13 Lecture 13: A brief review of the classical field theory

In the first lecture, I would like to give a brief review of the classical field theory, including the derivation of Euler-Lagrange equation of fields, symmetry transformation and Noether's theorem in a general form. After that we will discuss a particular symmetry transformation, the translation under which we will get the expression of the energy-momentum tensor as the conservative current.

Speaker: Hongbin Wang (Southeast university)

14 Lecture 14: The energy-momentum tensor

In this lecture, I will derive the expression of energy-momentum tensor of the classical electromagnetic field and the gravitational field, from the field-theoretical point of view. We will talk about the electromagnetic field as a good example to understand the basic concepts and techniques in field theory, and then derive the expression of energy-momentum tensor which should be consistent with the one derived in chapter 1. Finally we will discuss some difference between the electromagnetic field and the gravitational field.

Speaker: Hongbin Wang (Southeast university)

Saturday 14 November

15 Lecture 15: Angular momentum for gravitational waves

In this lecture, we derived the angular momentum expression for gravitational waves from the Noether's theorem, by the symmetry of the action under rotations in three dimensions.

Speaker: Hongbin Wang (Southeast university)

16 Lecture 16: Why spin-2

In this lecture, we talked about why the gravitational field is the spin-2 fields. We examine different cases, from spin-0, spin-1 and spin more than 3. We found all of these cases except spin-2 are ruled out by some physical prediction contradictory to the facts.

Speaker: Hongbin Wang (Southeast university)

Saturday 21 November

17 Lecture 17: The Pauli-Fierz action

In this lecture, we derived the Pauli-Fierz action, which is gauge-invariant and leads to the correct equation for linear order of gravitional field.

Speaker: Hongbin Wang (Southeast university)

18 Lecture 18: The propagator

From the action leading to the linear equation of motion, we derived the propagator of gravitons in De Donder gauge. From the propagator we derived the expression of the gravitional interaction between two static point particles, which showed the consistency with Newton's gravity.

Speaker: Hongbin Wang (Southeast university)

Saturday 28 November

19 Lecture 19: Higher-order terms

In this lecture, we explained the nonlinearity of gravity, and how construct higher-order terms in the action. From the dimensions, we saw how to determine the higher-order interacting terms in the action.

Speaker: Hongbin Wang (Southeast university)

20 Lecture 20: GR as the effective theory

In the low-energy case, we saw we need only consider the leading terms in the expansion, from which point of view we explained the general relativity as a totally adequate low-energy field theory.

Speaker: Hongbin Wang (Southeast university)

Saturday 5 December

21 Lecture 21: The theory of massive gravitons

In this lecture, we considered the theory of massive gravitons. From some basic observations, we could determine the range of mass of gravitons.

Speaker: Hongbin Wang (Southeast university)

22 Lecture 22: The problems in theory of massive gravitons

By analogy with the massive electromagnetic field, we considered the massive gravitational field, deriving the propagators of massive gravitons. From some contradiction, we haven't established a "correct" theory of massive gravitons.

Speaker: Hongbin Wang (Southeast university)

Saturday 20 February

23 Lecture 23: Warm up and review

In this lecture, we will first warm up by reviewing some key ingredients about general relativity and some contents in previous chapters.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

24 Lecture 24: Gravitational waves from weak field source

Then we will derive gravitational waves from weak-field sources, furthermore focus on the non-relativistic sources. We will also discuss a commonly used technique/formula, the quadruple formula. Through a careful study of two examples, we will clarify the relation between two different ways, the direct integration and the quadruple formula, of calculating the leading order of gravitational waves. More importantly, we will get a clear physical picture of generation of the gravitational waves.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

Saturday 27 February

25 Lecture 25: Binary system bound by scalar field

In this lecture, we will explicitly work out two examples where gravitational waves are generated from circular motions. In the first example, we will consider two point masses interacting through a scalar field. Through this example, we will see how the different parts we have learned so far come into play just at the right place to form a complete physical picture.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

26 Lecture 26: Gravitationally-bound binary system

In the second example, we will consider two point masses interacting through gravity. Through the knowledge we have learned from the first example, we can see that we have to go beyond the linear order to correctly calculate the gravitational waves generated.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

Saturday 20 March

27 Lecture 27: Quadrupole radiation: radiated power, radiated angular momentum and radiation reaction

In this lecture, we finish our discussion of leading-order GW by computing the radiated power, radiated angular momentum, and discussing its backreaction on the binary system.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

28 Lecture 28: Mass octupole and current quadrupole

In this lecture, we discuss the subleading order of GW, generated by the mass octupole and current quadrupole of the source. This serves as a warm-up for the upcoming discussion on systematic multipole expansion next week.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

Saturday 27 March

29 Lecture 29: Poisson's equation and group theory

In this lecture, we will use the Poisson's equation as a trigger to introduce two different ways to write down the general solution to the equation. To really understand the structure of the solutions and the relation between the two different forms, we review the group theory. We focus on concepts that we will use later.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

30 Lecture 30: Multipole expansion: Symmetric-trace-free form and tensor spherical harmonics form

In this lecture, we will see how we utilize those concepts we have learned in group theory to write down the general solutions to the relativistic wave equation for scalar/vector/tensor in different ways (more precisely, in the STF form and tensor spherical harmonics form). We also use Schur's lemma to give a elegant proof of the vanishment of the interference terms in the radiated power.

Speaker: Yuchen Du (Shing-Tung Yau Center, Southeast University)

Saturday 3 April

31 Lecture 31: GWs Emitted by a Compact Binary System in Circular Motion

In this lecture, we will apply our foundational theory to an important astrophysical phenomenon: the inspiral of compact binaries. Starting from the quadrupole formula, we will derive generic expressions for the two polarizations of GWs emitted by a compact binary in a circular orbit, evaluated along an arbitrary direction of propagation. Using these results, we will then calculate the total radiated power.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

32 Lecture 32: Back-Reaction and Quasi-Circular Motion

Building on our derivations of wave polarizations and radiated power, we will examine how GW emission back-reacts on the system's orbital dynamics. We will explore how this energy loss naturally drives the binary into an adiabatic regime of quasi-circular motion. Consequently, we will derive the equations governing the time evolution of both the orbital frequency and the corresponding GW frequency. The lecture concludes with a discussion of the typical frequencies and coalescence timescales for actual astrophysical sources.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 17 April

33 Lecture 33: GW Cycles, Orbital Separation, and the Plunge

After briefly reviewing the adiabatic evolution of quasi-circular motion, we will calculate key quantities for assessing detector sensitivity, primarily the number of GW cycles within a detector's bandwidth. We will then derive the evolution of the binary's orbital separation before exploring the transition from the inspiral phase to the plunge, where the quasi-circular approximation is no longer valid.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

34 Lecture 34: Quasi-Circular GW Waveforms and the "Chirp" Signal

In the second half of the lecture, we will examine the detailed waveform produced during the quasi-circular regime. By accounting for the effects of GW emission, we will derive an expression that demonstrates how the signal's amplitude and frequency increase as coalescence approaches, producing the characteristic gravitational wave "chirp."

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Saturday 24 April

35 Lecture 35: Elliptic Keplerian Orbits in Newtonian Gravity

In the first part of our final lecture, we will review elliptic Keplerian orbits within the framework of Newtonian gravity. We will derive the orbit's equation from two fundamental constants of motion: energy and angular momentum. Next, we will demonstrate how the parametric solutions for the radial distance and polar angle as functions of time naturally lead to Kepler's third law. Finally, applying the quadrupole formula to these derived orbital dynamics, we will compute the total radiated power.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

36 Lecture 36: Orbital Evolution Under Back-Reaction

We begin the second half of the lecture by calculating the angular momentum lost to gravitational wave emission. Combining this result with our previous expression for total radiated power, we will derive the equations governing the time evolution of the semi-major axis and the eccentricity, effectively quantifying the orbital back-reaction. By examining the decay of the eccentricity over time, we will conclude with a discussion on the phenomenon of orbital circularization.

Speaker: Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)