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Subatomic Physics: Particle Physics Lecture 3: Quantum Electro-Dynamics & Feynman Diagrams 6th November 2009 Antimatter Feynman Diagram and Feynman Rules Quantum description of electromagnetism Virtual Particles Yukawa Potential for QED 1 Schrödinger and Klein Gordon ˆ E = i t ˆ p = i Ψ E ( r,t)= ψ E ( r) exp {iEt/} However for particles near the speed of light E 2 =p 2 c 2 +m 2 c 4 Solutions with a fixed energy, Ep=+(p 2 c 2 +m 2 c 4 ) ! , and three-momentum, p: Ψ( r,t)= N exp {i( p · r E p t)/} Also solutions with a negative energy, En="Ep = "(p 2 c 2 +m 2 c 4 ) ! , and momentum, !p: Negative energy solutions are a direct result of E 2 =p 2 c 2 +m 2 c 4 . We interpret these as anti-particles Quantum mechanics describes momentum and energy in terms of operators: E=p 2 /2m gives time-dependent Schödinger: The solution with a definite energy, E: Ψ ( r,t)= N exp {i( p · r + E p t)/} 2 2m 2 Ψ( r,t)= i t Ψ( r,t) 2 2 t 2 Ψ( r,t)= 2 c 2 2 Ψ( r,t)+ m 2 c 4 Ψ( r,t) Klein-Gordon equation is non- examinable 2

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### Transcript of Subatomic Physics: Particle Physics - School of Physics ...

Subatomic Physics:

Particle PhysicsLecture 3: Quantum Electro-Dynamics & Feynman Diagrams

6th November 2009

Antimatter

Feynman Diagram and Feynman Rules

Quantum description of electromagnetism

Virtual Particles

Yukawa Potential for QED

1

Schrödinger and Klein Gordon

E = i� ∂

∂t�p = −i��∇

ΨE(�r, t) = ψE(�r) exp {−iEt/�}

• However for particles near the speed of light E2=p2c2+m2c4 ⇒

• Solutions with a fixed energy, Ep=+(p2c2+m2c4)!, and three-momentum, p:

Ψ(�r, t) = N exp {i(�p · �r − Ept)/�}• Also solutions with a negative energy, En="Ep = "(p2c2+m2c4)!, and momentum, !p:

• Negative energy solutions are a direct result of E2=p2c2+m2c4.

• We interpret these as anti-particles

• Quantum mechanics describes momentum and energy in terms of operators:

• E=p2/2m gives time-dependent Schödinger:

• The solution with a definite energy, E:

Ψ∗(�r, t) = N∗ exp {i(−�p · �r + Ept)/�}

− �2

2m∇2Ψ(�r, t) = i� ∂

∂tΨ(�r, t)

−�2 ∂2

∂t2Ψ(�r, t) = −�2c2∇2Ψ(�r, t) + m2c4Ψ(�r, t)

Klein-Gordon equation is non-

examinable2

Antimatter

Dirac Interpretation:

• The vacuum is composed of negative energy levels with E<0. Each level is filled with two electrons of opposite spin: the “Dirac Sea”.

• A “hole” in the sea with charge "e and E<0, appears as a state with charge +e and E>0.

• This idea lead Dirac to predict the positron, discovered in 1931.

Nuclear and Particle Physics Franz Muheim 14

AntiparticlesAntiparticles

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+)\$(+))%8%0+'%/)

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%)(.!+&#(+)\$('%5#

! "# \$! "# \$xpEti

xptEi!!

!!

%&&'

&%&&&&&

(exp

)()())((exp

Klein-Gordon equation predicts negative energy solutions.

t

xe+ (x, t)

E>0

e" ("x,"t)E<0

≡Feynman-Stueckelberg Interpretation:

• negative energy particles moving backwards in space and time correspond to...

• positive energy antiparticles moving forward in space and time

Ψe−(−r,−t) ∝ exp−i/��(−E)(−t)− (−p) · (−r)

Ψe+(+r,+t) ∝ exp−i/��(+E)(+t)− (+p) · (+r)

3

Feynman Diagrams• A Feynman diagram is a pictorial representation of a

particular process (decay or scattering) at a particular order in perturbation theory.

• Feynman diagrams can be used to represent and calculate the matrix elements, M, for scattering and decays.

• Feynman diagrams are very useful and powerful tools. We will use them a lot in this course. We use them a lot in our research!

Richard Feynman receiving the 1967 Noble prize in

physics for his invention of this technique.

Conventions• Time flows from left to right (occasionally upwards)

• Fermions are solid lines with arrows• Anti-fermion are solid lines with backward pointing arrows. • Bosons are wavy (or dashed) lines

Use Feynman Rules to calculate M at different orders in perturbation theory.Nuclear and Particle Physics Franz Muheim 4

Feynman DiagramsFeynman Diagrams

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Nuclear and Particle PhysicsFranz Muheim4

Feynman Diagrams Feynman Diagrams

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Nuclear and Particle Physics Franz Muheim 4

Feynman DiagramsFeynman Diagrams

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23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

time

4

Quantum Electrodynamics (QED)

Classical electromagnetism:

• Force between charged particle arise from the electric field

• act instantaneously at a distance

Quantum Picture:

• Force between charged particle described by exchange of photons.

• Strength of interaction is related to charge of particles interacting.

Nuclear and Particle Physics Franz Muheim 2

Quantum ElectrodynamicsQuantum Electrodynamics

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QED is the quantum theory of electromagnetic interactions.

�E =Q r

4π�0r2

e.g. electon-proton scattering ep#eppropagated by the exchange of

photonscharge of electron

charge of proton

photon propagator

M(ep→ ep) ∝ e · 1q2 −m2

γ

· eFeynman rules:

• Vertex term: each photon!charged particle interaction gives a factor of fermion charge, Q.

• Propagator term: each photon gives a factor of where is the photon four-momentum.

• Matrix element is proportional to product of vertex and propagator terms.

q1/(q2 −m2

γ) = 1/q2

5

Basic electromagnetic process:

• Initial state charged fermion (e, µ, \$ or quark + anti-particles)

• Absorption or emission of a photon

• Final state charged fermion

Examples: e"#e"% ; e"%#e"

Mathematically, EM interactions are described by a term for the interaction vertex and a term for the photon propagator

Electromagnetic Vertex

√α

QED Conservation Laws

• Momentum, energy and charge is conserved at each vertex

• Fermion flavour (e, µ, \$, u, d ...) is conserved: e.g. u#u% allowed, c#u% forbidden

Coupling strength

• Matrix element is proportional to the fermion charge:

• Alternatively use the fine structure constant, &

⇒ strength of the coupling at the vertex is

α =e2

4π�0�c=

e2

4π≈ 1

137

M ∝ e

∝√

α

6

The force between two charged particles is propagated by virtual photons.

• A particle is virtual when its four-momentum squared, does not equal its rest mass:

• Allowed due to Heisenberg Uncertainty Principle: can borrow energy to create particle if energy ('E=mc2) repaid within time ('t), where "E"t # (

Virtual Particles

• In QED interactions mass of photon propagator is non-zero.

• Only intermediate photons may be virtual. Final state ones must be real!

Example: electron-positron scattering creating a muon pair: e+e"#µ+µ".

q

p1

p3

p4

p2

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

• Four momentum conservation:

• Momentum transferred by the photon is:

• Squaring,

p1

+ p2

= p3

+ p4

q = (p1

+ p2) = (p

3+ p

4)

q2 = (p1)2 + (p

2)2 + 2p

1· p

2

= 2m2e + 2(E1E2 − �p1 · �p2) > 0q2 �= m2

γ

m2X �= E2

X − �p 2X

7

• QED is formulated from time dependent perturbation theory.

• Perturbation series: break up the problem into a piece we can solve exactly plus a small correction.

• e.g. for e+e"#µ+µ" scattering.

• Many more diagrams have to be considered for a accurate prediction of !(e+e"#µ+µ").

• As & is small the lowest order in the expansion dominates, and the series quickly converges!

• For most of the course, we will only consider lowest order contributions to processes.

Perturbation Theory

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

Lowest Order

|M|2 ∝ α2

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

2nd Order

+ ...|M|2 ∝ α4

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

3rd Order

+ ...

|M|2 ∝ α6

8

• Any test charge will feel the e+e" pairs: true charge of the electron is screened.

• At higher energy (shorter distances) the test charge can see the “bare” charge of the electron.

QED Coupling Constant• Strength of interaction between electron and photon

• However, & �s not really a constant...

• An electron is never alone:

• it emits virtual photons, these can convert to electron positron pairs...

24

Running of

! specifies the strength of the interaction

between an electron and photon.

! BUT isn’t a constant

Consider a free electron: Quantum fluctuations lead to a

‘cloud’ of virtual electron/positron pairs

-e

γ γ

γ γ

γ

-e

-e

-e

-e

+e

+e+e

this is just one of

many (an infinite set)

such diagrams.

! The vacuum acts like a dielectric medium

! The virtual pairs are polarized

! At large distances the bare electron charge is screened.

-

+

+

+

+

+

+

+

Test Charge

At large R test chargesees screened e

- charge

-

-

-

-

-

-

-

+

+

+

+

+

+

+ Test Charge

At small R test chargesees bare e

- charge

-

-

-

-

-

-

Dr M.A. Thomson Lent 2004

24

Running of

! specifies the strength of the interaction

between an electron and photon.

! BUT isn’t a constant

Consider a free electron: Quantum fluctuations lead to a

‘cloud’ of virtual electron/positron pairs

-e

γ γ

γ γ

γ

-e

-e

-e

-e

+e

+e+e

this is just one of

many (an infinite set)

such diagrams.

! The vacuum acts like a dielectric medium

! The virtual pairs are polarized

! At large distances the bare electron charge is screened.

-

+

+

+

+

+

+

+

Test Charge

At large R test chargesees screened e

- charge

-

-

-

-

-

-

-

+

+

+

+

+

+

+ Test Charge

At small R test chargesees bare e

- charge

-

-

-

-

-

-

Dr M.A. Thomson Lent 2004

∝ α =e2

4π�0≈ 1

137

Nuclear and Particle Physics Franz Muheim 12

Running of Running of !!

!"#\$%&'()*"'+,-',)!!"#\$%&"'()*(\$+\$,"#)-.&%\$"/,(/%"\$#.,"/)%

! 0(\$123" /4(%)"(.(,)%4".%"(."(.++(5/4".%,\$4

.-*##/6)"(\$-7"89(.#):%5(*#\$\$(\$+\$,"#)%(

,#\$."/)%(.%5(.%%/'/+."/)%

)*(;/#":.+(\$+\$,"#)%<7)4/"#)%(7./#4

0*122'&'(=.#\$(,'.#&\$ .%5(-.44 )*(\$+\$,"#)%()%+8(;/4/>+\$(

."(;\$#8(4')#"(5/4".%,\$4

! /%,#\$.4\$4(?/"'(?/"' +.#&\$#(-)-\$%":-("#.%4*\$#

@+.44/,.+(+/-/"A1 0(B((((((((((((((! ! 0(C2CDE

F"(4')#"(5/4".%,\$4A1 0(GHB(I\$JK1 ! ! 0(C2C1L

#\$#\$% &&ee& varies as a function of

energy and distance

9

Yukawa PotentialThe quantum and classical descriptions of electromagnetism should agree.

Yukawa developed theory whereby exchange of bosons describes force / potential.

• Klein-Gordon equation:

• Non-time dependent solutions obey:

• Spherically symmetric solutions of this are:

• Interpret this as a potential, V, caused by a particle of mass, m.

• For electromagnetic force, m=0, g=e.

• Potential felt by a charged particle due to the exchange of a photon.

∇2Ψ(�r) =m2c2

�2Ψ(�r)

V (r) = − g2

4πrexp

�− r

R

�with R =

�mc

Ψ(|�r |) = − g2

4πrexp

�−mc

� |�r |�

VEM(r) = − e2

4πr

−�2 ∂2

∂t2Ψ(�r, t) = −�2c2∇2Ψ(�r, t) + m2c4Ψ(�r, t)

10

• Elastic electron-proton scattering: e"p#e"p

• Momentum transferred to photon from e":

• Rutherford scattering: e" Au#e" Au, can neglect recoil of the gold atoms: E=Ei=Ef

QED Scattering Examples

Nuclear and Particle Physics Franz Muheim 8

ElectronElectron--Proton ScatteringProton Scattering

!"#\$%&'()*+*,# !" #\$! !" #%&'()*+*,(\$-.#/*+01!

0)!\$2!3(.'( &0/!)

)*.#/!\$*4\$(!5/!6+*(5\$)#*()

-\$.//'/*0#%.,7&,8'8*/*+3\$4,&\$)6'++!&*(5\$

9".,.!(+0.\$+&'()4!&

:;!(\$(!5/!6+*(5\$.!

12#3*\$4.\$5'60"##*\$%,7</')+*6\$<4 =\$<* =\$<>\$\$\$\$(!5/!6+\$#&,+,(\$&!6,*/

22

2

2

41

qq

ee

qeM

!"##\$

% &% &

'(

)*+

,-#

--#

.-/#-##

2sin4

cos22

2

2

2

2222

0

0

111

1

if

ifife

ififif

EE

ppEEm

ppppppqqq

!!% &% &

iii

fff

pEp

pEp

!

!

,

,

#

#1

1

'(

)*+

,#

22

sin442

2

0"3

Ed

d

Lab

4

22

4

42 16

qq

eM

d

d "!3#\$\$

2

''(

)**+

,'(

)*+

,-'(

)*+

,

'(

)*+

,#

2 2sin

22cos

2sin4

2

2

2

2

42

2 000

"3

pi

f

Lab M

q

E

E

Ed

d

12#3*\$4.\$5'60"##*\$%,7'12#3*\$4.\$5'60"##*\$%,7'

q2 = (pf− p

i)2 = p2

f+ p2

i− 2p

f· p

i

= 2m2e − 2(EfEi − |�pf ||�pi| cos θ)

≈ −4EfEi sin2(θ/2)

σ ∝ |M|2 ∝ e4

q4=

16π2α2

q4

M ∝ e2

q2=

4πα

q2

σ ∝ Z2π2α2

E4 sin4(θ/2)

• Inelastic e"e+#µ+µ"

• Momentum transferred by photon:

• For this situation need full density of states, ), (which we won’t do...)

σ =16πE2

3|M|2 =

4πα2

3s

23

Higher Orders

So far only considered lowest order term in the

perturbation series. Higher order terms also

contribute

Lowest Order:

e-

e+

µ-

µ+γ

Second Order:

e-

e+

µ-

µ+

e-

e+

µ-

µ+

+....

Third Order:

+....

Second order suppressed by relative to first

order. Provided is small, i.e. perturbation issmall, lowest order dominates.

Dr M.A. Thomson Lent 2004

q2 = (pe+

+ pe−

)2 = s

M ∝ e2

s2=

4πα

s2

11

Relativistic quantum mechanics predicts negative energy particles: antiparticles. Two interpretations: • a negative energy particle travelling backwards in time.• a ‘hole’ in a vacuum filled with negative energy states.

Quantum Electro Dynamics (QED) is the quantum mechanical description of the electromagnetic force.Electromagnetic force propagated by virtual photons:

Feynman diagrams can be used to illustrate QED processes. Use Feynman rules to calculate the matrix element, M.

All QED interactions are described by a fermion-fermion-photon vertex:• Strength of the vertex is the charge of the fermion, Qf.

• Fermion flavour and energy-momentum are conserved at vertex.The photon propagator ~ where is the 4-momentum transferred by the photon.M is proportional to product of vertex and propagator terms.

Summary of Lecture 3

1/q2 q

q2 �= m2γ

12