Feynman’rules’ Interac/on’term’ ·...

Feynman rules

J = J (x),φ = φ(x)

Interac/on term

Feynman rules

Z(J ,λ)

Expansion in λ :

λJ 4

e.g.

J = J (x),φ = φ(x)

Connected v/s disconnected diagrams

e.g.

Sum of connected diagrams

Feynman diagrams – a convenient way to represent the double series expansion

Expansion in J (Wick): Z(J ,λ) = Z(0,0) 1

s!J sG (s)

s=0

∞∑

Green func/ons

e.g. 2-point function

for λ=0, G = iD(x1 − x2 )

ScaHering amplitudes for Scalar Electrodynamics

LSZ for scalars:

LSZ for vectors:

(Lehmann-‐Symanzik-‐Zimmerman)

Feynman rules for Scalar Electrodynamics

LSZ formula c.f. scalar case ch 10 Srednicki

Time ordered

2 incoming 2 outgoing par/cles

Feynman rules

× No scaHering

W (J ) is sum of all connected diagrams with no tadpoles and at least two sources

δ1δ2δ3δ4

δ1 removes a source and labelsthe propagator end-point x1

ie k + k '( )µ −2ie2gµν −iλ

NB Combinatoric factors


ie k + k '( )µ −2ie2gµν −iλ


−igµν / k2 − iε( ) −i / k2 +m2 − iε( )

ελiµ*(k), ελi

µ (k) for incoming and outgoing photons respectively

Applica/on of the Feynman rules: I. Tree level

Compton scaHering of a π meson

k,µ k ',ν

p 'pp k+ p 'p'p k−

k,µ k ',ν k,µ k ',ν

p 'p

iM fi = (ie)2[ε .(2 p + k) −i( p + k)2 − m2 ε '.(2 p '+ k ')

+ ε .(2 p '− k) −i( p − k ')2 − m2 ε '.(2 p − k ')− 2iε .ε ']

dσ = 1

vA 2EA2EB

M2 1

(2π )2 δ4( pC + pD − pA − pB )

d 3 pC

2EC

d 3 pD

2ED

γπ γπ→

c. f . dσ = 14 k1 CM

sT

2dLIPSn ' (k1 + k2 ) Srednicki (11.22)

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= −ie2ε 'µ T µνεν

Cross section = Transition rate x Number of final states

Initial flux

42 4

32 32

64

1 (2 ) ( )2 (22 ) 2 2C D A B

A B C D

C Dd p p p pE E

dE E

pVV

p d Vπ δπ

σ = + − −Av

M

dσ =

M2

FdQ

3 34 4

3 3(2 ) ( )(2 ) 2 (2 ) 2

C DC D A B

C D

d p d pdQ p p p pE E

π δπ π

= + − −Lorentz Invariant Phase space

2 2 2 1/ 2

2 2

4(( . ) )A B

A B A B

F E E

p p m m

=

= −Av

The cross section

Sum over final polarisa/ons and average over ini/al polarisa/ons

εµ

*

pol∑ (k)εν (k)→−gµν c.f. Peskin & Schroeder P159

k µ (2 p + k)µ ( p + p '+ k)λ

( p + k)2 − m2 +(2 p '− k)µ ( p + p '− k)λ

( p + k)2 − m2 − 2gµλ

⎛

⎝⎜⎞

⎠⎟

(2 p + k).k = ( p + k)2 − m2 − ( p2 − m2 ) (2 p '− k).k = −( p '− k)2 + m2 + p '2− m2( )

p + p '+ k( )λ − p + p '− k( )λ = 2kλ

0 0}

⇒ 0 QED

Ward Iden/ty k µΓµ ...(k,...) = 0

Sum over final polarisa/ons and average over ini/al polarisa/ons

εµ

*

pol∑ (k)εν (k)→−gµν c.f. Peskin & Schroeder P159

12 Pol∑ M

2

p.k = mω , p.k ' = mω '

(using k '.k =ωω '(1− cosθ ))

12 Pol∑ M

2

Lab Frame:

'

12 Pol∑ M

2

+ +

+...

Ward Iden/ty k µΓµ ...(k,...) = 0

δSδφa (x)

= ∂L(x)∂φa (x)

− ∂µ

∂L∂(∂µφa (x))

Formal proof: = 0 along classical path

Noether current

= 0 0 =

jµ

δSδφa (x)

δφa (x) = −∂µ jµ (x)

Under a symmetry φ(x)→φ(x)+δφ(x) that leaves L invariant

Take n-functional derivatives w.r.t. Ja j

(x j )

δSδφa (x)

δφa (x) = −∂µ jµ (x)

Ward Takahashi iden/ty

Contact terms vanish in scaHering amplitude

LSZ formula

0 in scattering amplitude

i.e. k µΓµ ...(k,...) = 0

Fundamental experimental objects

1 1 2( ... )na b b bΓ → Decay width = 1/lifetime

1 2 1 2( ... )na a b b bσ → Cross section


Initial flux

3

31 (2 )Vd pn

i π=Π

a1v 1V V×

1a 1av

# particles passing through unit area in unit time

# target particles per unit volume

2a

(Lab frame)

The transition rate 4 * ( ) ( ) ( ) ...fi f iT d x x V x xφ φ= − +∫

φi, f → f p

± = e(− ,+ )ip.x 1

2 p0V≡ N

Ve(− ,+ )ip.x

A

B

C

D

Transition rate per unit volume 2

fiTfi TVW =

24 4(2 ) ( )A B C DN N N N

fi C D A BVT p p p pπ δ= − + − − fiM

e.g.

φ f ,i = e(− ,+ )ip.x

(2π )4δ 4(kin − kout )⎡⎣ ⎤⎦

2= (2π )4δ 4(kin − kout )× (2π )4δ 4(0)

(2π )4δ 4(0) = d 4x ei0.x =VT∫

The transition rate 4 * ( ) ( ) ( ) ...fi f iT d x x V x xφ φ= − +∫

φi, f → f p

± = e(− ,+ )ip.x 1

2 p0V≡ N

Ve(− ,+ )ip.x

A

B

C

D

Transition rate per unit volume 2

fiTfi TVW =

24 4(2 ) ( )A B C DN N N N

fi C D A BVT p p p pπ δ= − + − − fiM

e.g.

φ f ,i = e(− ,+ )ip.x

244

4

( ) 1 1 1 1(2 )2 2 2 2

C D A Bfi

A B C D

p p p pW

V E E E Eδ

π+ − − ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

M


Initial flux

42 4

32 32

64

1 (2 ) ( )2 (22 ) 2 2C D A B

A B C D

C Dd p p p pE E

dE E

pVV

p d Vπ δπ

σ = + − −Av

M

2

d dQF

σ =M

3 34 4

3 3(2 ) ( )(2 ) 2 (2 ) 2

C DC D A B

C D

d p d pdQ p p p pE E

π δπ π

= + − −Lorentz Invariant Phase space

2 2 2 1/ 2

2 2

4(( . ) )A B

A B A B

F E E

p p m m

=

= −Av

The cross section

The decay rate

212 A

d dQE

Γ = M

1

1

1

334 4

3 3(2 ) ( ... ) ...(2 ) 2 (2 ) 2

n

n

n

BBA B B

B B

d pd pdQ p p p

E Eπ δ

π π= − −

Feynman’rules’ Interac/on’term’ ·...

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