The heart of particle physics How do we predict?

The major phenomena we observe in particle physics are Decays and Collisions.

Decay 衰變 Everything which is not forbidden is allowed (a principle of English Law) That which is not explicitly forbidden is guaranteed to occur. Every particle will decay if it get a chance.

Relativity allows particles to decay by transforming mass into energy. Heavier particles usually don’t exist in nature. They decay soon after they are produced. Decay chain will continue until it can decay no more.

The only five stable particles in nature Neutron Proton Electron Photon Neutrino

β Decay 但中子是會衰變的！ 基本粒子應該是不會變的宇宙基本成分。 當初以為中子是電子與質子的束縛態！ 但並未發現中子的激發態。 基本粒子是會消失而形成其他粒子的！

在交互作用的交點，粒子是會消失的。

π 會衰變

Bubble Chamber picture of pions

衰變是一個不確定的機率過程。 我們無法預測單一一顆中子何時衰變，只能預測衰變發生的機率。 單位時間內的衰變機率：衰變率 Γ

如果是處理一大群中子，知道衰變的機率就足夠了： 單一一顆中子的衰變機率即對應一大群中子的衰變分布。 Γ即是一個原子核每秒衰變的機率! 不衰變的機率即不衰變的粒子數 隨時間以指數遞減

β Decay 衰變的產物可以有無數可能的動量。 物理只能預測衰變為某一個動量組合的機率。 衰變率是產物粒子的能量及動量的函數。

甚至衰變產物也有一個以上的可能。 不同的可能稱為 Channel 每一個 Channel 就對應一個衰變率。 總衰變率就是所有衰變率的和。 位衰變的粒子數在 t > τ 後就很少了。 τ 稱為Life Time 生命期

Strongly decaying particle 這樣的粒子即使產生，都無法在實驗室內看到他的痕跡。

這樣的粒子會以共振曲線的形式出現在其衰變產物的的散射分布上 散射率對質心能量的分布 在質心能量等於共振態質量時，會被加強。 Breit-Wigner Resonance

這樣的粒子會以共振曲線出現在其衰變產物的的散射分布上 共振曲線中心即粒子質量。 共振曲線寬度即衰變率。 衰變率 Γ又稱為 Decay Width

Collision

粒子束越強，單位面積粒子數越多，反應發生的次數越多！ Event Rate Luminosity(亮度，即通量） Number of incoming particles per unit area per unit time

電流密度：單位面積的電流，為空間的區域性質。

達到的 Luminosity 是加速器效率的度量

dσ 是一個與粒子束強弱無關的量，只由該反應決定。 粒子束越強，單位面積粒子數越多，反應發生的次數越多！ Event Rate Luminosity(亮度，即通量） dσ 是一個與粒子束強弱無關的量，只由該反應決定。 dσ 是此反應的內在性質。 dσ 一計算出來，就可以用在所有的實驗。 dσ 之於 dN，就如同比熱之於熱容量。 dσ 是一個面積。

Classical Scattering Impact parameter b 與散射角度 θ 有一對一對應

散射角在 θ 與 θ+dθ 間的粒子，其 b 必定在對應的 b 與 b+db之間 通過此一截面的粒子，將散射進入對應的散射角

L 散射截面 Scattering Cross Section 粒子物理的粒子已不再有特定軌跡了。古典的計算已不適用。 但在此定義下，dσ 依舊是此反應的內在性質。

Event recorded with the CMS detector in 2012 at a proton-proton centre-of-mass energy of 8 TeV. The event shows characteristics expected from the decay of the SM Higgs boson to a pair of Z bosons, one of which subsequently decays to a pair of electrons (green lines and green towers) and the other Z decays to a pair of muons (red lines).

Fixed Target實驗的 dσ 與 Colliding Beam實驗一樣。 在沿射向的羅倫茲變換下是不變的！ Fixed Target實驗的 dσ 與 Colliding Beam實驗一樣。

生成的粒子可以觀察到軌跡

因此生成的粒子可以看成波包。 Δx 衰變率與位置關係不大 與動量較為相關。 Δk 假設我們對位置並為太精密測量 Δp 就不會太大 我們可以以 p0 的衰變率來近似！ 討論時會近似使粒子都具有特定動量！

Phase Space Factors 先將一定會出現的 Factors 從衰變率及截面中提出來。 只和入射粒子及產生粒子的數量與身分有關！

But this integral is not Lorentz invariant! Particle decay: 第 i 個粒子的動量在 之間的衰變率 dΓ py dpy dpx px 衰變率dΓ應與產生粒子的動量所佔相空間的區間大小成正比 But this integral is not Lorentz invariant!

We make it more complicated by allowing an indefinite p0 integration and then fixing it by requiring the on-shell condition: But in this form, we can be sure it is Lorentz invariant! We can perform the p0 integration to recover the 3 space form. This is Lorentz invariant.

只有在 x = ± a 不為零。

All the factors are Lorentz Invariant. But is M 2 Lorentz invariant? No matter what, the overall 4-momenta are conserved! 從衰變率再拉出一個執行動量守恆的 δ function All the factors are Lorentz Invariant. But is M 2 Lorentz invariant?

But M 2 is not Lorentz invariant since Γ is not. For a particle, Γ transforms like 1/t1. t1 transforms like E1. 我們可以從M2再提出一個 1/ E1 Now we can be sure M 2 is Lorentz invariant. It’s called Feynman Amplitude. The Lorentz Invariance makes M 2 simple.

Dynamic Factors 由交互作用的細節決定。 Kinematic factors只和入射粒子及產生粒子的數量與身分有關！ We separate the kinematics and dynamics in such an elegant way that the still dynamic part M 2 is Lorentz invariant. M 2

Now we apply this to pion two photon decay. Choose the rest frame of pion: Total Decay Rate:

式子與角度無關！

若生成粒子有質量：見課本推導。 注意M 2是沒有因次的。

M 2 is Lorentz invariant since dσ almost is. Two body scattering: M 2 is Lorentz invariant since dσ almost is. If we consider only Lorentz transformation along the 1-2 colliding axis, the cross section dσ is invariant! But we do want to pull out a Lorentz invariant factor that reflects the inverse luminosity that must appear in cross section:

Lorentz invariant factor that becomes luminosity in rest frame: In the rest frame of particle 2

It’s called Feynman Amplitude. 我們從截面再拉出這個一定要出現的羅倫茲不變的 Luminosity。 M 2 is Lorentz invariant. It’s called Feynman Amplitude. The Lorentz Invariance makes M 2 simple.

Two body scattering in CM

Carry out the p4 integration Enforce the 3 momentum conservation 散射截面的角度分布 注意M 2是沒有因次的。

Two body scattering in CM 如果生成粒子與入射粒子質量相等

Feynman Rules To evaluate the Lorentz Invariant Feynman Amplitude M Components of Feynman Diagrams External Lines Particle Content (Masses and spins) Internal Lines Vertex Interactions

Every component corresponds to a specific factor! 1 (For spinless particles) External Lines Internal Lines propagator Vertex -ig Coupling Constant Momentum Conservation

Draw all diagrams with the appropriate external lines, momenta fixed by the experiment. Multiply all the factors! Integrate over all internal momenta Take out an overall momentum conservation. That’s it! The result is -iM. It’s so simple. The Lorentz Invariance is explicit at every steps of the calculation.

Three scalar particle with masses mA, mB ,mC For a toy ABC model Three scalar particle with masses mA, mB ,mC 1 External Lines Internal Lines Lines for each kind of particle with appropriate masses. C Vertex -ig A B The configuration of the vertex determine the interaction of the model.

Vertices for real Interactions

Particle A decay Take out overall momentum conservation. The Feynman Amplitude is just a constant!

There are other possible diagrams, in fact infinite number of them. Each extra vertex carries an extra factor of –ig, which is small. Consider first the diagrams with the fewest number of vertices. Feynman Rule is a perturbation theory.

Lifetime of A For

Scattering It’s similar to electron-electron scattering!

Take out overall momentum conservation. One propagator and two coupling constants Take out overall momentum conservation.

The overall momentum conservation is always there. The remaining momentum conservations can be enforced by immediately carrying out the integration in the beginning! The result can be written down right away!

After momentum conservation is enforced, the momentum of internal particle c is It does not satisfy momentum mass relation of a particle! The internal particle is not a real particle, it’s virtual.

It does not satisfy momentum mass relation of a particle! The internal particle is not a real particle, it’s virtual. In a sense, the relation is only for particles when we “see” them! Internal lines are by definition “unseen” or “unobserved”. It’s more like: It’s more a propagation of fields than particles!

Field fluctuation propagation can only proceed forward that is along “time“ from past to future. This diagram is actually a Fourier Transformation into momentum space of the spacetime diagram. The vertices can happen at any spacetime location xμ and the “location” it happens need to be integrated over. After all you do not measure where interactions happen and just like double slit interference you need to sum over all possibilities.

is actually the sum of the above two diagrams! For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2. For those amplitude where time 2 is ahead of time 1, propagation is from 2 to 1. B B B B 2 1 1 C 2 C A A A A is actually the sum of the above two diagrams! Feynman diagram in momentum space is much simpler than in space-time!

In this diagram, particles seem to have certain momenta In this diagram, particles seem to have certain momenta. According to uncertainty principle, their position uncertainty is infinity! This diagram is only an approximation of more precise treatment using wave packet! You need to “superpose” similar diagrams with slightly uncertain momentum to produce incoming and outgoing wave packets.

This is very similar to electron-electron scattering.

Scattering To the leading order, there could be more than one diagrams! In the diagrams, the B lines in some vertices are incoming particle. The striking lesson: Lines in a vertices can be either outgoing or incoming, depending on their p0. p0 for an observed particle is always positive. So this vertex diagram is actually 8 diagrams put together! That’s the simplicity of Feynman diagram.

If all the momenta in the diagrams can be determined through momentum conservation, the diagram has no loop and is hence called tree diagram! If there is a loop in the diagram, some internal momentum is not fixed and has to be integrated over! These are called loop diagram.

Oops! Loops are infinite! 將所有的動量守恆事先執行 若有 loop 就會有某些動量無法確定 無法確定的動量必須積分。 Oops! Loops are infinite!

Renormalization Cut off the momentum q at Λ δm中的 ln Λ正好使 Tree 圖與 Loop圖的 ln Λ抵消 結果會是有限的！

Mass Renormalization δm中的 ln Λ正好與 Σ 的 ln Λ抵消 c 的原始質量是無限大，加上無限大的修正，量到的質量是有限的

Charge Renormalization 原始耦合常數是無限大，加上無限大的修正，量到的有限的耦合常數

All the infinities can be cancelled out by a finite numbers of parameter renormalization.

Schrodinger Wave Equation He started with the energy-momentum relation for a particle Expecting them to act on plane waves he made the quantum mechanical replacement: How about a relativistic particle？

The Quantum mechanical replacement can be made in a covariant form The Quantum mechanical replacement can be made in a covariant form. Just remember the plane wave can be written in a covariant form: As a wave equation, it does not work. It doesn’t have a conserved probability density. It has negative energy solutions.

Plane wave solutions for KG Eq. Time dependence can be determined. There are two solutions for each 3 momentum p (one for +E and one for –E ) It has negative energy solutions.

The proper way to interpret KG equation is it is not a Wavefunction Equation but actually a Field equation just like Maxwell’s Equations. Plane wave solutions just corresponds to Plane Waves. It’s natural for plane waves to contain negative frequency components. Expansion of the KG Field by plane： If Φ is a real function, the coefficients are related:

Add a source to the equation: We can solve it by Green Function. G is the solution for a point-like source at x’. By superposition, we can get a solution for source j.

Green Function for KG Equation: By translation invariance, G is only a function of coordinate difference: The Equation becomes algebraic after a Fourier transformation. This is the propagator!

Green function is the effect at x of a source at x’. KG Propagation Green function is the effect at x of a source at x’. That is exactly what is represented in this diagram. The tricky part is actually the boundary condition.

For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2. B B B B 2 1 1 C 2 C A A A A is actually the sum of the above two diagrams! To accomplish this,