Feynman  rules  for  Scalar  Electrodynamics  

LSZ  for  scalars:    

LSZ  for  vectors:  

(c.f.6.1  Schwartz)  Time  ordered  

Feynman  rules  for  Scalar  Electrodynamics  

LSZ  formula  c.f.  scalar  case  ch  10  Srednicki  

δ j ≡

1i

δδ J (x j )

Feynman  rule  example:  vertex  rule  

0 T φ x1( )φ x2( )φ x2( )( ) 0 = δ1δ 2δ3Z(J ) |J=0 ,

δ j ≡

1i

δδ J x j( )

0 T φ x1( )φ x2( )φ x2( )( ) 0 = δ1δ 2δ3Z(J ) |J=0

= δ1δ 2δ3

ig3!

1i

⎛⎝⎜

⎞⎠⎟

3i2

⎛⎝⎜

⎞⎠⎟

3

23 d 4 y1,2,3,a∫ Π i=1,2,3J ( yi )Δ yi − ya( ) at  O(g)  

= ig

3!1i

⎛⎝⎜

⎞⎠⎟

3i2

⎛⎝⎜

⎞⎠⎟

3

23 1i

⎛⎝⎜

⎞⎠⎟

3

3! d 4 yaΠ iΔ(xi − ya )∫

Insert  in  LSZ  formula:  

f i = −g i( )3

d 4x1,2,3d4 yae

i(k1x1+k2x2−k3x3 )∫ Π i −∂i2+ m2( )Δ(xi − ya )

= ig 2π( )4δ 4(k1 + k2 − k3) ≡ 2π( )4

δ 4(k1 + k2 − k3)iT

iT = ig Feynman  rule  for  vertex  

−( )

−∂i

2+ m2( )Δ(xi − ya ) = δ 4(xi − ya )

Feynman  rules  

× No  scaIering  

δ1δ2δ3δ4

δ1 removes a source and labelsthe propagator end-point x1

(photon :−∂i2Δµν xi − y( ) = gµνδ 4 xi − y( ) in Lorentz gauge)−∂i

2 +m2( )Δ xi − y( ) = δ 4 xi − y( )

Feynman  rules    

iT is given by the sum of all diagrams

ie k + k '( )µ −2ie2gµν −iλ

NB  Combinatoric  factors  

Feynman  rules  for  Scalar  Electrodynamics  

Feynman  rules  for  Scalar  Electrodynamics  

ie k + k '( )µ −2ie2gµν −iλ

ie k + k '( )µ −2ie2gµν −iλ

Feynman  rules  for  Scalar  Electrodynamics  

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

ελiµ*(k), ελi

µ (k) for incoming and outgoing photons respectively

ApplicaNon  of  the  Feynman  rules:      I.  Tree  level  

e−e− → e−e−

ξ dependence vanishes (gauge invariance)...here just through k µJµ = 0

(Moller  scaIering)  

⇒

α = e2

4π, fine structure constant

⇒

α = e2

4π, fine structure constant

3 32 42 4 2

4 6

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

C DC D A B

A B C D

d p d pVd p p p p VE E V E E

πσ δπ

= + − −Av

M

Exercise

+

Ward  idenNty  and  gauge  invariance    

Gauge  invariance:  

c.f.  Ward  idenNty   if  matrix  element  for  on-­‐shell  photon  is     εµ Mµ

Ward  IdenNty  more  general,  applies  even  if  photon  non-­‐physical  

e.g.  Consider  

Assuming  only  electron  on-­‐shell    

−

+

Ward  idenNty   ε3µ

* → p3µ ?

≠ 0

−

Ward  idenNty   ε3µ

* → p3µ ?

≠ 0

−

≠ 0

−