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, forbidden usually by conservation law (symmetry).

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

但中子是會衰變的！ β Decay 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 子的衰變

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 因次為 𝑡 −1 ~𝐸~𝑚，與能量同單位。

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即是一個中子每秒衰變的機率! ΓN 即是每秒發生衰變的次數，即粒子數的減少 隨時間以指數遞減

β Decay 衰變的產物的可能動量是連續分布的。 物理只能預測衰變為某一個動量組合的機率。 因此衰變率（機率）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。 未衰變的粒子數在 t > τ 後就很少了。 τ 稱為Life Time 生命期

一個粒子的生命期會隨衰變的機制而差別甚大。 π± 的衰變是透過弱作用，生命期約 10-8s，剛好可以看到軌跡。 π0 的衰變是透過電磁作用，生命期約 10-15s， 若粒子透過強作用衰變，生命期約只有10-23s 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 強作用衰變 散射率對質心能量的分布 這個過程會對以上散射率增加一個共振。 在質心能量等於共振態質量時，會被加強。 Breit-Wigner Resonance

強衰變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

碰撞實驗是以粒子束進行。 粒子束越強，單位面積粒子數越多，反應發生的次數越多！ 強度以通量 L Luminosity (亮度)度量： 單位時間通過單位面積的入射粒子數： Event Rate

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，只由該反應的內在性質。 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 定義 dσ： dσ 是一個與粒子束強弱無關的量，只由該反應的內在性質。 一計算出來，就可以用在所有的同類實驗，不管其技術強弱。 dσ 之於 dN，就如同比熱之於熱容量。 dσ 單位是面積。稱為散射截面 Scattering Cross Section。

Classical Scattering 在古典固定靶散射實驗中，𝑑𝜎真的是截面積 在古典散射中，Impact parameter b 與散射角度 θ 有一對一對應

散射實驗中，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 通過左方此一截面的粒子，將散射進入對應的散射角 𝜃 及𝜃+𝑑𝜃 之間 故測量得到散射角為 𝜃 的散射粒子數等於通過該截面的粒子數：

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。但舊稱仍沿用。 在此定義下，dσ 依舊是此反應的內在性質。

亮度的計算： ∆𝑡 時間內通過面積 A 的粒子數 通量 L 為單位時間通過單位面積的粒子數： 對於固定靶實驗：

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

因此入射及生成的粒子應該看成波包。 Δx 波包粒子的動量波函數有一個分布！ 衰變率必須對波包動量波函數作積分： Δk 但我們對粒子位置並未太精密測量 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一個特定動量，而忽略動量分布！

現在開始計算衰變率及散射截面 先將一定會出現的 Factors 從 Γ 及 𝜎 中提出來。 這一些 Factors 與作用的細節無關， 只和入射粒子及產生粒子的數量與身分有關！ 稱為 Phase Space Factors Γ 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 = ℳ 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 2 ∙ P.S. 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛

以衰變為例：實驗會測量衰變後的產物粒子3-動量 py dpy dpx px 衰變後第 i 個粒子的動量在 之間的衰變率記為 dΓ 衰變率dΓ應與所產生粒子的動量所佔相空間的區間大小成正比 ℳ 2 是機率密度！ But 積分元 𝑑 3 𝑝 不是 Lorentz invariant!

𝑑 3 𝑝 ∙ 1 𝐸 這個組合是 Lorentz invariant. 為簡單起見，考慮沿x方向的羅倫茲變換！ 𝑝 𝑥 ′ =𝛾∙ 𝑝 𝑥 − 𝑣 𝑐 2 ∙𝐸 𝐸=𝛾∙ 𝐸−𝑣∙ 𝑝 𝑥 那麼y,z方向是不變的 𝑑 𝑝 𝑦 𝑑 𝑝 𝑧 =𝑑 𝑝 𝑦 ′ 𝑑 𝑝 𝑧 ′ 𝑑 𝑝 𝑥 →𝑑 𝑝 𝑥 ′ =𝑑 𝑝 𝑥 ∙ 𝑑 𝑝 𝑥 ′ 𝑑 𝑝 𝑥 =𝑑 𝑝 𝑥 ∙𝛾∙ 1− 𝑣 𝑐 2 𝑑𝐸 𝑑 𝑝 𝑥 =𝑑 𝑝 𝑥 ∙𝛾∙ 1−𝑣∙ 𝑝 𝑥 𝐸 =𝑑 𝑝 𝑥 ∙ 𝐸′ 𝐸 𝐸 2 𝑐 2 = 𝑝 𝑥 2 + 𝑚 2 𝑐 2 𝑑𝐸 𝑑 𝑝 𝑥 = 𝑐 2 ∙ 𝑝 𝑥 𝐸 𝑑 𝑝 𝑥 ′ =𝑑 𝑝 𝑥 ∙ 𝐸′ 𝐸 𝑑 𝑝 𝑥 ′ 1 𝐸′ =𝑑 𝑝 𝑥 ∙ 1 𝐸 𝑑 3 𝑝 ′∙ 1 𝐸′ = 𝑑 3 𝑝 ∙ 1 𝐸

另一個較高段的推導方法 Dirac 𝛿 function In an integration, enforce the equation that x = a.

只有在 x = ± a 不為零。

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.

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 𝛿 4 𝑝 𝜇 =𝛿 𝑝 0 𝛿 𝑝 1 𝛿 𝑝 2 𝛿 𝑝 3 All the factors are Lorentz Invariant. But is ℳ 2 Lorentz invariant?

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 𝑀 2 再提出一個 1/ E1 現在我們確定 ℳ 2 is Lorentz invariant. It’s called Feynman Amplitude. Lorentz Invariance 使 ℳ 2 簡單.

Dynamic Factors ℳ由交互作用的細節決定。 Γ 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 = ℳ 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 2 ∙ P.S. 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 Dynamic Factors ℳ由交互作用的細節決定。 Kinematic (Phase) factors P.S.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: 1 1

積分式與角度無關！角度積分可以直接執行！ ℳ 2 已代入動量守恆 𝑝 3 = − 𝑝 2 積分式與角度無關！角度積分可以直接執行！ 最後執行 𝑝 2 積分，𝛿函數強迫能量守恆！ 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

ℳ 2 is Lorentz invariant. 我們從截面再拉出這個一定要出現的羅倫茲不變的 Luminosity。 Feynman Amplitude 𝜎 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 = ℳ 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛 2 ∙ P.S. 𝑝 1 , 𝑝 2 ,⋯ 𝑝 𝑛

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

To evaluate the Lorentz Invariant Feynman Amplitude ℳ Feynman Rules To evaluate the Lorentz Invariant Feynman Amplitude ℳ Components of Feynman Diagrams External Lines Determined by Particle Content (Masses and spins) Internal Lines Every line contains a 4-momentum. Vertex Interactions

Every component corresponds to a specific factor! 1 (For spinless particles) External Lines Its momentum is on-shell. 𝑝 𝑖 2 = 𝑚 𝑖 2 propagator Internal Lines 𝜀→ 0 + Its momentum is not on-shell. Vertex -ig Coupling Constant Momentum Conservation

Draw all diagrams with the appropriate external lines, matching the incoming and outgoing particles, with momenta 𝑝 𝑖 fixed by the experiment. Multiply all the factors! Integrate over all internal momenta 𝑞 𝑖 Take out an overall momentum conservation. That’s it! The result is -i𝓜. It’s so simple. The Lorentz Invariance is explicit at every step.

There are 3 scalar particle with masses mA, mB ,mC A toy ABC model There are 3 scalar particle with masses mA, mB ,mC 𝑝 𝐴 2 = 𝑚 𝐴 2 etc. 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 ℳ=𝑔 But remember momenta need to conserve! It is enforced by the phase factor P.S.! Hence: 𝑚 𝐴 > 𝑚 𝐵 + 𝑚 𝐶 As an example, consider that a particle is pion and B,C particles are photons. Assume that the interaction of 𝜋−𝛾−𝛾 has a coupling constant g. For

Could B decay to A and C? A C The Feynamn Amplitude is still the same! ℳ=𝑔 But remember momenta can not conserve! B The phase factor P.S. is zero! If 𝑚 𝐴 > 𝑚 𝐵 + 𝑚 𝐶 , 𝑚 𝐵 < 𝑚 𝐴 + 𝑚 𝐶 If A decay to B and C, B can not decay to A and C. So B and C are stable.

From interaction vertices, we can immediately see some decay possibilities for real particles.

Of Course, massless photon can’t decay. But could electron emit one photon and one electron? 𝑒→𝑒+𝛾 The Feynman amplitude is non-zero (a constant e). But momentum conservation does not allow! 𝑝 𝑒1 = 𝑝 𝑒2 +𝑝 𝛾 𝑝 𝑒2 + 𝑝 𝛾 2 = 𝑝 𝑒1 2 𝑚 𝑒 2 +2 𝐸 𝑒2 ∙ 𝐸 𝛾 − 2 𝑝 𝑒2 ∙ 𝑝 𝛾 = 𝑚 𝑒 2 2 𝐸 𝑒2 ∙ 𝐸 𝛾 − 2 𝑝 𝑒2 𝐸 𝛾 ∙ cos 𝜃 =0 𝐸 𝑒2 − 𝑝 𝑒2 ∙ cos 𝜃 =0 Impossible since 𝐸 𝑒2 > 𝑝 𝑒2 Electron can not emit one photon and one electron. 𝑒↛𝑒+𝛾

But we know accelerating electron can emit EM wave! Electron can emit two photon and one electron. 𝑒→𝑒+𝛾+𝛾 Momentum conservation now allows!

It’s similar to electron-electron scattering! 𝐴+𝐴→𝐵+𝐵 It takes at least two vertices to draw a diagram with appropriate external lines. The leading order diagram: It’s similar to electron-electron scattering!

One propagator and two coupling constants

Carrying out the momentum integration will enforce momentum conservation at one of the vertex: 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 of a vertex 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 mass 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. In Fourier Transformation, you do integration over spacetime xμ 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.

In your integration over spacetime xμ: 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!

To the leading order, there could be more than one diagrams!

This is very similar to electron-electron scattering.

Another possible Scattering 𝐴+𝐵→𝐴+𝐵 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!

冷靜一點，先假設動量有一個上限 cutoff Λ Regularization 如此才能進行計算！ Cut off the momentum q at Λ 冷靜一點看，這個圖對c粒子的質量是一個修正！

質量的輻射修正 Radiative Correction 對質量修正，使真正量到的質量修正為：

那麼在純量場Φ的真空中，規範粒子 𝐴 𝜇 會由與不為零的純量場的作用， 得到一個不為零的質量項：

Mass Renormalization 𝑚 𝑐 2 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,