The heart of particle physics

The heart of particle physics

How do we predict?

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

Decay 衰變

That which is not explicitly forbidden is guaranteed to occur.

Every particle will decay if it get a chance.

Everything which is not forbidden is allowed (a principle of English Law)

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

β Decay epn

但中子是會衰變的！

基本粒子是會消失而形成其他粒子的！

W

W

d

u

在交互作用的交點，粒子是會消失的。

eπ 子的衰變

Bubble Chamber picture of pions

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。衰變是一個不確定的機率過程。

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：

NdtdN

teNN 0 隨時間以指數遞減

Γ 即是一個中子每秒衰變的機率 !

不衰變的機率即不衰變的粒子數

單一一顆中子的衰變機率即對應一大群中子的衰變分布。

β Decay epn

衰變率（機率密度）是產物粒子的能量及動量的函數。

衰變的產物可以有連續分布的可能的動量。物理只能預測衰變為某一個動量組合的機率密度。

epeep EEppp ,,,,

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。

τ 稱為 Life Time 生命期

t

t eNeNN 00

未衰變的粒子數在 t > τ 後就很少了。

π± 的衰變是透過弱作用，生命期約 10-8s ，

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。

20

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

pp

pp 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

4

12

2

mE

強作用衰變 p

這個過程會對以上散射率增加一個共振。

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

4

12

2

mE

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

LdN

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

通量 L Luminosity( 亮度 ) 為單位時間通過單位面積的入射粒子數：

tvAN

通量 L 為單位時間通過單位面積的粒子數：

時間內通過面積 A 的粒子數

vAt

NL

達到的 Luminosity 是加速器效率的度量

LdNd

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！

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 是一個面積。稱為散射截面 Scattering Cross Section 。

dσ 之於 dN ，就如同比熱之於熱容量。

Event Rate LdN 定義 dσ

Classical Scattering

Impact parameter b 與散射角度 θ 有一對一對應

在古典散射真的是截面積

通過左方此一截面的粒子，將散射進入對應的散射角 之間

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

dLdN

dddbbdbbd )(2)(2

故散射角為 的散射粒子數：

dLdN

LdNd 散射截面 Scattering Cross Section

LdNd

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。

L

Bunch Crossing 4 107 Hz

7x1012 eV Beam Energy 1034 cm-2 s-1 Luminosity 2835 Bunches/Beam 1011 Protons/Bunch

7 TeV Proton Proton colliding beams

Proton Collisions 109 Hz

Parton Collisions

New Particle Production 10-5 Hz (Higgs, SUSY, ....)

p pH

µ+

µ-

µ+

µ-

Z

Zp p

e- e

q

q

q

q1

-

g~

~

20~

q~

10~

Selection of 1 event in 10,000,000,000,000

7.5 m (25 ns)

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).

fANNvL BA

dσ 是一個垂直於射向的面積。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。

Δk

Δx

衰變率必須對動量波函數積分：

21

kx

假設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！

2)()( pbpdp

)( 0p

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！

粒子的動量波函數有一個分布！

先將一定會出現的 Factors 從衰變率及截面中提出來。

Phase Space Factors

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！

n

jjjzjy

n

jjx pdMdpdpdpMd

2

32

2

2

But this integral is not Lorentz invariant!

Particle decay: )()(3)(2)(1 321 npnppp

px

py

dpx

dpy

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

衰變後第 i 個粒子的動量在 iii pdpp , 之間的衰變率記為 dΓ

1

axdx

axax if0

Dirac function

In an integration, enforce the equation that x = a.

afxfaxdx

10

222

axdx

0

22

0

22

0

222 221 axdxxaxdxxaxdx

120

1

axcdxx

22 ax 只有在 x = ± a 不為零。 )(2122 axcaxcax

12 1 ca

120

2

axcdxx 12 2 ca

)(2122 axaxa

ax jj j

xxxf

xf )('

1)( 0)( jxf

xk

kx 1

3

3

3

3

222 221

22

1

pdE

pd

cmp

We make it more complicated by allowing an indefinite p0

integration and then fixing it by requiring the on-shell condition:

0222

4

4

22

pcmppd

But in this form, we can be sure it is Lorentz invariant!We can perform the p0 integration to recover the 3 space form.

22220222 cmppcmp

)(2122 axaxa

ax

2220

222

022220

2

1 cmppcmp

pcmpp

This is Lorentz invariant.

n

jjpdMd

2

32

n

jjj

n

j j mcpE 2 2222 2

12

1

n

j

j

j

pdE

Md2

3

32

221

npppp 321442

No matter what, the overall 4-momenta are conserved!

n

j

j

jn

pdE

ppppMd2

3

3

321442

2212

我們從衰變率再拉出這個 Factor使積分是羅倫茲不變

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?

n

j

j

jn

pdE

ppppMd2

3

3

321442

2212

But M 2 is not Lorentz invariant since Γ is not.For a particle, Γ transforms like 1/t1. t1 transforms like E1.

2

2

1cv

t

2

2

2

1cv

mcE

1

1E

n

j

j

jn

pdE

ppppME

d2

3

3

321442

1 22121

Now we can be sure M 2 is Lorentz invariant. It’s called Feynman Amplitude.

我們可以從 M2 再提出一個 1/ E1

The Lorentz Invariance makes M 2 simple.

n

j

j

jn

pdE

ppppME

d2

3

3

321442

1 22121

We separate the kinematics and dynamics in such an elegant way that the still dynamic part M 2 is Lorentz invariant.

M 2

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！

Now we apply this to pion two photon decay. 20

Choose the rest frame of pion:

mE 1 01 p

3

33

332

3

2321

442

1 221

22121

pd

Epd

EpppM

Ed

Total Decay Rate:

)()( afaxxfdx

式子與角度無關！

ddpdppd sin23

若生成粒子有質量：見課本推導。

注意 M 2 是沒有因次的。

Two body scattering: n 321

n

j

j

jn

pdE

ppppMd3

3

3

321442

2212

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:

M 2 is Lorentz invariant since dσ almost is.

LdNd

221

2214 mmpp

In the rest frame of particle 2

1121121221

212

221

221 444 vvEEpmmEmmmmE

E

cvx

2

1

11

Lorentz invariant factor that becomes luminosity in rest frame:

22 mE 02 p

n

j

j

jn

pdE

ppppMd3

3

3

321442

2212

221

2214

1

mmpp

n

j

j

jn

pdE

ppppMmmpp

d3

3

3

321442

221

221

2212

4

1

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

散射截面的角度分布

注意 M 2 是沒有因次的。

Carry out the p4 integrationEnforce the 3 momentum conservation

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 fi pp

2CM

2

EM

dd

Feynman Rules

To evaluate the Lorentz Invariant Feynman Amplitude M

Components of Feynman Diagrams

External Lines

Internal Lines

Vertex

ip

iq

Determined by Particle Content (Masses and spins)

Interactions

Every line contains a 4-momentum.

Every component corresponds to a specific factor!

External Lines

Internal Lines

Vertex

ip

iqpropagator

-ig

1 (For spinless particles)

imqi

jj 22

i

ip442

Coupling Constant

Momentum Conservation

Its momentum is on-shell.𝑝❑2 =𝑚2

Its momentum is not on-shell.

Draw all diagrams with the appropriate external lines, matching the incoming and outgoing particles with momenta fixed by the experiment.

Integrate over all internal momenta iqd 4

421

Take out an overall momentum conservation.

npppp 321442

That’s it! The result is -iM. It’s so simple.

The Lorentz Invariance is explicit at every step.

Multiply all the factors!

A toy ABC model

There are 3 scalar particle with masses mA, mB ,mC

External Lines

Internal Lines

Vertex

ip

iq

-ig

1

imqi

jj 22

321442 kkk

A B

C

1k

2k

3k

Lines for each kind of particle with appropriate masses.

The configuration of the vertex determine the interaction of the model.

etc.

qe,

qe,

Vertices for real Interactions

CBA

Particle A decay

gM

The Feynman Amplitude is just a constant!

321442 pppigiM

Take out overall momentum conservation.

Consider first the diagrams with the fewest number of vertices.

Each extra vertex carries an extra factor of –ig, which is small.

There are other possible diagrams, in fact infinite number of them.

Feynman Rule is a perturbation theory.

Lifetime of A

CBA

gM

20 For cm

g16

2

0 CB mm22mmp A

As an example, consider that a particle is pion and B,C particles are photons.

Assume that the interaction of has a coupling constant g.

It’s similar to electron-electron scattering!

Scattering

It takes at least two vertices to draw a diagram with appropriate external lines.

The leading order diagram:

One propagator

and two coupling constants

2ig

24 ppq

Carrying out the momentum integration will enforce momentum conservation at one of the vertex:

24

444

4

22

ppqdq

Take out overall momentum conservation.

24 pp

The remaining momentum conservations can be enforced by immediately carrying out the integration in the beginning!

The result can be written down right away!

243144

2444 22 ppppppq

The overall momentum conservation is always there.

24

444

4

22

ppqdq

24 pp

After momentum conservation is enforced, the momentum of internal particle c is 24 pp

It does not satisfy momentum mass relation of a particle! 22224 cmpp c

The internal particle is not a real particle, it’s virtual.

24 pp

It does not satisfy momentum mass relation of a particle! 22224 cmpp c

The internal particle is not a real particle, it’s virtual.

24 pp

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!

24 pp

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.

B

A

B

A

For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2.

C1

2B

A

B

A

For those amplitude where time 2 is ahead of time 1, propagation is from 2 to 1.

C

1

2

24 pp is actually the sum of the above two diagrams!Feynman diagram in momentum space is much simpler than in space-time!

24 pp

23 pp

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!

24 pp

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!

31 pp 31 pp

qpp 31

q

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。

Renormalization

24 pp

Cut off the momentum q at Λ

22Physical,

2 mmm cc

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！

ln

imp

i

mpimpmp

i

mpimp

i

c

c

cc

c

c22

22

2222

22

22

1

1

1

Mass Renormalization

ln2mi

22Physical,

2 mmm cc

22cmp

i

2Physical,

2cmpi

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 2Physical,cm

pp p

q

pq p pp p

2222cc mp

imp

i

222222

ccc mpi

mpi

mpi

δm 中的 ln Λ 正好與 Σ 的 ln Λ抵消

Charge Renormalization

lng

Physicalg

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 Physicalg

ggg Physical

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

Schrodinger Wave EquationHe started with the energy-momentum relation for a particle

he made the quantum mechanical replacement:

How about a relativistic particle？

Expecting them to act on plane waves

ipxrpiiEt eee

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.

ipxrpiiEt eee

0222 cmp

0p E

There are two solutions for each 3 momentum p (one for +E and one for –E )

ipxxpitip eaeax 0

)(Plane wave solutions for KG Eq.

ipxipxipx eameapeap 2220

2220 mpp

p

xpiiEtxpiiEt

p

ipxipx ebeaebeax

)(

Time dependence can be determined.

It has negative energy solutions.

Expansion of the KG Field by plane ：

p

xpiiEtxpiiEt

p

ipxipx ebeaebeax

)(

p

ipxipx eaeax

)(

If Φ is a real function, the coefficients are related:

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.

Add a source to the equation:

)(2 xjm

We can solve it by Green Function. ')',()',( 2 xxxxmxxG

G is the solution for a point-like source at x’.

By superposition, we can get a solution for source j.

)'()',(')()( 40 xjxxGxdxx

Green Function for KG Equation: ')',()',( 42 xxxxmxxG

By translation invariance, G is only a function of coordinate difference:

)'()',( xxGxxG

The Equation becomes algebraic after a Fourier transformation.

)(~

2)'( )'(

4

4

pGepdxxG xxip

1)(~22 pGmp

)'(4

44

2)'( xxipepdxx

22

1)(~mp

pG

This is the propagator!

)'(

4

422)'(

4

4

2)(~

2xxipxxip epdpGmpepd

'x x

Green function is the effect at x of a source at x’.That is exactly what is represented in this diagram.

KG Propagation

The tricky part is actually the boundary condition.

B

A

B

A

For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2.

C1

2B

A

B

A

For those amplitude where time 2 is ahead of time 1, propagation is from 2 to 1.

C

1

2

is actually the sum of the above two diagrams!

To accomplish this, 22

1)(~mp

pG

imp

pG

22

1)(~