# Student Seminar: Frontier Researches in Particle Physics

## The First meeting of the study group

Yu Dai Tsai
Discussion #1: Precision tests for QED
We went back to the topic we discussed before, the order-alpha QED prediction for g-factor, generations of physicist improved both theoretical and experimental accuracy.
A.    Fine-structure constant:
We fit the experimental data to a theoretical expression to contain the alpha as a parameter
1.)    The transition energy in hydrogen and hydrogen-like atoms
2.)    The anomalous magnetic moment of the muon
3.)    The decay rates of singlet and triplet positrunium.
*the error assigned to the alpha depends on both experiment and theory
B.     Hyperfine splitting in the e- muon+ atoms
Discussion #2: the standard deviation (use other computer to check the nthu phys out)
We discuss the definition of the standard deviation and its usage
Discussion #3 Delta function’s properties
We refresh the memory of those property of the Dirac delta function
1.      delta(f(x)) =Sum delta(x-xi)/|f ‘(x) ||x=xi
2.      delta(ax-b)

Discussion #6: The meaning of the perturbation theory (recall) (expansion)
We discussed about how to expand the interaction action and promote it out of the field functional integral (path integral) by present it as derivatives
The idea behind the path integral perturbation is that we only knows how to do the integration of quadratic term on the exponential (gauss integral), for tri-linear and quartic terms we have to use derivation to do it (recall the more familiar situations), or gamma function.
Problems:
1.      The determinant of a operator
2.      The reason for 0<=xi<=1 with boost explanation

## The 5th Discussion of Study Group

By Yu Dai Tsai
Outline
1.      Isospin symmetry
2.      SU(S) symmetry and quark model
3.    Strangness and associated production(V particles)
4.      Lamb shift
6.      reduced mass
7.      Error: Statistical error (standard error)
8.      QED has been tested in most stringent ways.
9.      Field theory correction
Discussion #1: Quark model
We used the method of group theory to understand standard model and predict some properties of hadrons.
Discussion #2: Lamb Shift
The splitting between the j=1/2 2S (l=0) and 2P(l=1) levels of hydrogen
Related to order-alpha Vacuum polarization (A.K.A photon self-ennergy). However, the interaction between electron and the vacuum shift the energy of 2S ½. This measurement stimulated the renormalization theory.
A.    Precision test in QED  (p
B.     Interpretation of Pi_2 ( P.252
Discussion #3: Strangeness
Strangeness S is a quantum number assigned to some specific particles to describe decays involving strong and electromagnetic interactions.
Discussion #4 strangeness conservation
Kaons and hyperons tend to be created easily in particle collisions and decay much slower than our expectation according to their mass and cross section of production.
On the other hand, these collisions seem to produce pairs of particles so that Gell-Mann and Nishijima proposed a conservation quantity, Strangeness, to describe this “associated production”.
Nowadays, we know that lightest strange particles are created in strong interaction. But because they are the light, they can’t decay through strong processes; instead, they have to decay in weak interactions.

References
[1]. Wikipedia: strangeness
[2]. Wikipedia: lamb shift

## The Beautiful structure of Standard model in particle physics

The Student Seminar of Frontier Researches in Particle Physics
Title: The Beautiful structure of Standard model in particle physics
Author: Yu Dai Tsai.
Summary:
Quantum field theory and the model based on it (standard model) is the most successful theory to explain our universe.
Introduction:
We discuss those interesting facets of the theory and modern development of these concepts. The important experiments are included in our discussion.
Discussions:
Our discussions will be introduced below following the time order of each meeting.
Meeting 1
Discussion #1: The Ghost in the diagram
We discussed about how the ghost field came into play in the Feynman diagram calculations. It is originated from the fact that Grassman number can generate the determinant of a matrix. Hence, we can use it as the way to promote the determinant onto an exponential and make it a term in the Lagrangian.
Ghost particle play a rather important role in loop calculations of Yang-Mills theory.
Discussion #2: Feddeev-Popov’s method
This method is to get rid of infinity in gauge integral. Adding gauge-fixing and ghost terms to get rid of the "volume” divergence in group transforming space.
Discussion #3: The detection of the proton radius
This discussion brings us to the work of a Professor back in our department. Professor Lou and international team he cooperates with discovered that the size of proton is not as big as our expectation.
Feedbacks :
(from meeting 3)
It is very interesting to understand how to transform the general Euclidean coordinate to the spherical coordinate not only in 3-D situation but in arbitrarily dimensional world.
Questions and Solutions
(from meeting 3)
Why there is only one 0 ~2 pi angle and while others are 0 ~ pi
Discussion 3.   The modern device to measure the time: atomic clock
The atomic clock use the transitions between atomic levels to measure the time.
By detecting its frequency, We can
Conclusion
1.      The necessity to fix the gauge, and we can fix the gauge by imposing constraints.
2.      Using F-P method we can calculate photon or gluon propagators .

Acknowledgement
I want to thank professor Li for his accompany throughout the meeting.
References

## 高能物理讀書心得

Discussion #1:Spontaneous Symmetry Breaking

首先,我們探討了真空自發性破缺，這是基於鐵磁性物質內的未成對電子所攜帶的自旋，因排列方向相同而有較強的磁場和較低的能量.
自旋的序列性會給出空間一個方向，即空間不再有均勻性質，這就是簡單的真空自發性破缺的概念，而利用這個概念，場論中將量子場的真空期望值平移可以得到類似的結果，其結果是物質可因此有質量的獲取，其特別的地方是真空場必須是多場的，即必須要有兩種不同的場，如此真空自發性破缺才能給出質量和多出來的耦合項，其理論的結果是要解釋質量的來源，

Discussion #2:Goldstone Theory

進一步的探討,物理學家發現如果真空自發性破缺成立，則必有一無質量自旋為零的波色子，這個假設的粒子場相對應的是一個么正算符的連續變換函數，這可以解釋為什麼我們真實空間中光子的質量為零，但規範W和Z波色子卻具有相當大的質量。

## The 4th meeting of the study group

By Yu Dai Tsai
We already build basic skills in QED calculations, now it is time to explore higher-order contributions, known as radiative corrections.
The order-alpha correction to the cross section comes from the interference term between these diagrams and the tree diagrams
There is basically three kinds of 1 loop corrections:
1.      Vertex correction
2.      External leg correction
3.      Vacuum polarization

Discussion #1: The Electron vertex Function: Formal Structure
We want to discuss about the general properties of vertex correction diagrams. Consider the correction to electron scattering that comes from the presence of an additional virtual photon.
Remark:
Although it is difficult to calculate the loop diagram, we can set the constraints to limit the form of vertex term Gamma
There are three constraints for the form of the vertex
1.      The basic requirement of Lorentz invariance,
2.      The discrete symmetries of
3.      The Ward identity

*By using the master formula for S-matrix elements, we can get the amplitude  of electron scattering with the heavy target.
Reference
[1] Peskin and Schroeder “An introduction to Quantum field theory

## The discussions and inspired thoughts of the 3rd group activity

By Yu Dai Tsai

Discussion 1. The idea of the Pauli-Villars regularization:
We discussed about the idea of this regularization formula and how it can preserve the Lorentz covariant.
Discussion 2. n-dimensional spherical coordinate integration
Professor Li taught us how to calculate the n-dimensional spherical coordinate integration.
It is very interesting to understand how to transform the general Euclidean coordinate to the spherical coordinate not only in 3-D situation but in arbitrarily dimensional world.
Question:
Why there is only one 0 ~2 pi angle and while others are 0 ~ pi
Discussion 3.   The modern device to measure the time: atomic clock
The atomic clock use the transitions between atomic levels to measure the time.
By detecting its frequency, We can
Feedback:
Discussion 4. The anomalous magnetic moment by QED
We discussed how to calculate g/2 to a high accuracy to 1+alpha/2pi
by using
1.       Feynman parameter trick
2.       Wick rotation
Discussion 5. Anomalous magnetic moment of the muon
Our member Luo gave us a short introduction of the anomalous magnetic moment of the muon.
You can see directly from Wikipedia that the standard model calculation have to include effects of QED, Electro-weak and Hadronic contribution of the alpha and combine them together.
Also W and Z particle need to be included in the calculations
This experiment actually showed some deviation from the theoretical calculation so there might be some things to improve either experimentally or theoretically.

## The discussions and inspired thoughts of 2nd study group activity

Discussion #1: The Ghost in the diagram

We discussed about how the ghost field came into play in the Feynman diagram calculations. It is originated from the fact that Grassman number can generate the determinant of a matrix. Hence, we can use it as the way to promote the determinant onto an exponential and make it a term in the Lagrangian.
Ghost particle play a rather important role in loop calculations of Yang-Mills theory.
Feedback: I was amazed by the freedom of a theory. To adjusted for practical needs, a seemly pure mathematically trick actually plays such an intuitive role if you regard it in an alternative way.

Discussion #2: Feddeev-Popov’s method
This method is to get rid of infinity in gauge integral. Adding gauge-fixing and ghost terms to get rid of the “volume” divergence in group transforming space.

Discussion #3: The detection of the proton radius
This discussion brings us to the work of a Professor back in our department. Professor Lou and international team he cooperates with discovered that the size of proton is not as big as our expectation.
This is a very interesting discover and went on the cover of the “nature” magazine.
Check this out
Rydberg constant is a number features the energy level of atoms. Now giving the need to modify this number, we will have to change many results and probably the way we measure the time.

*Inspired question #1: Gribov ambiguities
The gauge fixing constraint may not be good enough to intersect with the gauge orbit only one time. So we may need to modify it. Gribov has some discussion and we can have further readings.
Conclusions:
1.       The necessity to fix the gauge, and we can fix the gauge by imposing constraints.
2.       Using F-P method we can calculate photon or gluon propagators .