Announcements

10/3

• Today: Problems 5.11, 5.12, 5.13• Friday: Problems 6.1, 6.2, 6.4• Monday: Read 6E - 6G

Problem 6.4 says “scalar couplings” ig

Too many formulas

pu mu

pv mv

up um

vp vm

uu p m

vv p m

5 5

25 1

q q

5 5

q q

Tr 4 , Tr 4ab a b abcd a b c d a d c b a c d b

General Procedure• Draw Feynman diagrams• Find intermediate momenta• Write Feynman amplitude for each diagram• Add or subtract them• Simplify if you can• Multiply by complex conjugate• For unpolarized cross-sections:• Sum over final spins• Average over initial spins

• Rewrite as traces• Recall, any number equals its trace• You can then move back to front in a trace

• Use sum over spins rules• Simplify using trace rules, etc.• Finish the problem in the usual way

*

5u qv 5vq u

, , ,pu mu pv mv up um vp vm

5 5 5 5Tru qvvq u u qvvq u

5 5Tr uu qvvq

,uu p m vv p m

Plus or minus?

For each pair of diagrams, should I add or subtract?

Add if they differ only by switching external boson linesSubtract if they differ only by switching external fermion lines

Possible answers: Add, Subtract, or Trick Question

Subtract Trick

AddAdd

Trick

Only add diagrams with identical

particles in initial and final states

A hard computation6.8 Calculate the cross section for scattering. Treat all masses as zero.

1p

2p

3p

4p

3p

4p

1p

2p 3 5 1i u g uM 4 5 2u g u

2 21 3

i

p p M

4 5 1 3 5 2 2 2

1 4

iu g u u g u

p p M

2 23 5 1 4 5 2 4 5 1 3 5 2

2 2

1 3 1 4

ig u u u u ig u u u ui

p p p p

M

1 2 3 4p p p p

2 23 5 1 4 5 2 4 5 1 3 5 2

1 3 1 42 2

ig u u u u ig u u u u

p p p p

Multiply by complex conjugate 2

3 5 1 4 5 2 4 5 1 3 5 2

1 3 1 42

u u u u u u u uigi

p p p p

M

2* 1 5 3 2 5 4 1 5 4 2 5 3

1 3 1 42

u u u u u u u uigi

p p p p

M

3 5 1 4 5 2 1 5 3 2 5 4 3 5 1 4 5 2 1 5 4 2 5 32

41 3 1 42 1 3

4 5 1 3 5 2 1 5 3 2 5 4 4 5 1 3 5 2 1 5 4 2 5 32

1 3 1 4 1 4

4

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

Sum/Average over spins

3 5 1 4 5 2 1 5 3 2 5 4 3 5 1 4 5 2 1 5 4 2 5 32

41 3 1 42 1 3

spins spin 4 5 1 3 5 2 1 5 3 2 5 4 4 5 1 3 5 2 1 5 4 2 5 32

1 3 1 4 1 4

1

4 16 s

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

• Just as we sum over final momenta, we sum over final spins too• Initial spin usually random, so average over it

• Combine them so they each start and end with the same Dirac spinor

3 5 1 1 5 3 4 5 2 2 5 4 3 5 1 1 5 4 4 5 2 2 5 32

41 3 1 42 1 3

spins spin 4 5 1 1 5 3 3 5 2 2 5 4 4 5 1 1 5 4 3 5 2 2 5 32

1 3 1 4 1 4

1.

4 16 s

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

Replace with traces

3 5 1 1 5 3 4 5 2 2 5 4 3 5 1 1 5 4 4 5 2 2 5 32

41 3 1 42 1 3

spins spin 4 5 1 1 5 3 3 5 2 2 5 4 4 5 1 1 5 4 3 5 2 2 5 32

1 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Trs

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

3 3 5 1 1 5 4 4 5 2 2 5 3 3 5 1 1 5 4 4 5 2 2 52

41 3 1 42 1 3

spins spin 4 4 5 1 1 5 3 3 5 2 2 5 4 4 5 1 1 5 3 3 5 2 2 52

1 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Trs

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

Use the sum rules

,s s

uu p m vv p m

3 3 5 1 1 5 4 4 5 2 2 5 3 3 5 1 1 5 4 4 5 2 2 52

41 3 1 42 1 3

spins spin 4 4 5 1 1 5 3 3 5 2 2 5 4 4 5 1 1 5 3 3 5 2 2 52

1 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Trs

u u u u u u u u u u u u u u u u

p p p pp pgi

u u u u u u u u u u u u u u u u

p p p p p p

M

3 5 1 5 4 5 2 5 3 5 1 5 4 5 2 5

24

1 3 1 42 1 3

spins 4 5 1 5 3 5 2 5 4 5 1 5 3 5 2 5

21 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Tr

p p p p p p p p

p p p pp pgi

p p p p p p p p

p p p p p p

M

Announcements

10/5

• Today: Problems 6.1, 6.2, 6.4• Monday: Read 6E - 6G• Wednesday: Problems 6.6, 6.7

6.6 …For an added challenge, let m 0 but keep M =0.

6.7 Write the full Feynman amplitude for (k)(k’) (p)(p’) for pseudoscalar couplings.

6.8 Calculate the cross section for scattering. Treat all masses as zero.

1 2 3 4p p p p

Get rid of the 5’s

25 5 5, 1q q

3 5 1 5 4 5 2 5 3 5 1 5 4 5 2 5

24

1 3 1 42 1 3

spins 4 5 1 5 3 5 2 5 4 5 1 5 3 5 2 5

21 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Tr

p p p p p p p p

p p p pp pgi

p p p p p p p p

p p p p p p

M

3 1 4 2 3 1 4 2

24

1 3 1 42 1 3

spins 4 1 3 2 4 1 3 2

21 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Tr

p p p p p p p p

p p p pp pgi

p p p p p p p p

p p p p p p

M

Simplify and take traces

Tr 4 , Tr 4ab a b abcd a b c d a d c b a c d b

3 1 4 2 3 1 4 2

24

1 3 1 42 1 3

spins 4 1 3 2 4 1 3 2

21 3 1 4 1 4

Tr Tr Tr

1

4 16 Tr Tr Tr

p p p p p p p p

p p p pp pgi

p p p p p p p p

p p p p p p

M

1 3 2 4 1 3 2 4 1 4 2 3 1 2 3 42

1 3 1 42 1 34

spins 1 4 2 3 1 3 2 4 1 2 3 4 1 4 2 32

1 3 1 4 1 4

41

4

4

p p p p p p p p p p p p p p p p

p p p pp pi g

p p p p p p p p p p p p p p p p

p p p p p p

M

• Note middle two terms are identical

Work on dot products

2 1 2 3 4 2 4 1 3 2 3 1 44 2 32 4

spins 1 3 1 4 1 3 1 4

1

4 2

p p p p p p p p p p p pp pp pi g

p p p p p p p p

M

• What are the momenta? • Recall all particles massless

1 2 3 4p p p p

1 3

2 4

,0,0, , sin cos , sin sin , cos

,0,0, , sin cos , sin sin , cos

p E E p E E E E

p E E p E E E E

2 2 21 2 3 4 1 3 2 4 1 4 1 32 , 1 cos , 1 cosp p p p E p p p p E p p p p E

2 22 4

spins

4 1 cos 1 cos11 1

4 2 1 cos 1 cosi g

M

2

4

2

4 2 2cos2

2 1 cosg

2 4

spins

13

4i g M

Finish the problem …2 4

spins

13

4i g M

• Find D:

2 1 1 24

D

E E p

p 28

D

E

2

216 cm

pD i d

E M

2

2spins

1

16 4cm

pD i d

E M

2

2spins

1

16 2 4

ED i d

E M

4

2

3

32

gd

4

2 2

3

128

gd

E

4

2 2

3

128

d g

d E

• Recall we have identical final state particles: 4

2

3

64

g

E

43

16

g

s

A hard computation6.5 Calculate the cross section for scattering.

5 5

2 2

u g i p k m g ui

p k m

M

5 5 5 5

5 5

, ,q q m m

q m q m

p k p k

p p k k

p p k k

p k p k

5 5

2 2

u g i p k m g u

p k m

5 522 2 22

u p k m ui ig

p k p k m

M

5 522 2 22

u k p m uig

p k p k m

2 22 22 2

u p k m u u k p m ui ig ig

M p k M p k

M

25 1

pu mu

u p u m

2 2

2 22 2

ig u ku ig u ku

M p k M p k

Square and sum/average on spins p k p k

22 2

1 1

2 2i ig u ku

M p k M p k

M

* 22 2

1 1

2 2i ig uku

M p k M p k

M

2

2 42 2

1 1

2 2i g uku u ku

M p k M p k

M

24

2

2 2spins spins

1 1 1

2 2 2 2

gi uku u ku

M p k M p k

M

24

2

2 2spins

1 1 1Tr

2 2 2 2

gi p m k p m k

M p k M p k

M

uu p m

u u p m

Do the traces

24

2

2 2spins

1 1 1Tr

2 2 2 2

gi p m k p m k

M p k M p k

M

2Tr Trp m k p m k pkp k m kk

• Only even number of Dirac matrices contribute

Tr 4 4 4 , Tr 4 .abcd a b c d a d c b a c b d ab a b

2 2Tr 4 4 4 4p m k p m k p k p k p k p k p p k k m k 2 2 28 4 4p k p k p p M m M

2

2 4 2 2 22 2

spins

1 1 12 2

2 2 2i g p k p k p p M m M

M p k M p k

M

Announcements

10/8

• Today: Read 6E - 6G• Wednesday: Problems 6.6, 6.7

6.6 …For an added challenge, let m 0 but keep M =0.

6.7 Write the full Feynman amplitude for (k)(k’) (p)(p’) for pseudoscalar couplings.

Write out the momenta explicitly p k p k

• In the cm frame, the initial particles must have equal and opposite momenta p

• But the initial energies will not match• The final particles also have matching momenta p’• The final energies will be:• But energy is conserved:

2 2 2 2,p kE p m E p M

2 2 2 2,p kE p m E p M

2 2 2 2 2 2 2 2p m p M p m p M

• To make this work, p = p’ ,p p k kE E E E

,0,0, , sin cos , sin sin , cos

,0,0, , sin cos , sin sin , cos

p p

k k

p E p p E p p p

k E p k E p p p

Replace all the dot products ,0,0, , ,0,0, , , sin cos , sin sin , cosp k pp E p k E p p E p p p

2

2 4 2 2 22 2

spins

1 1 12 2

2 2 2i g p k p k p p M m M

M p k M p k

M

2 2 2 2, cos , cosp k p k pp k E E p p k E E p p p E p

2

2 42 2 2 2

spins

2 2 2 2 2 2 2

1 1 12

2 2 2 cos 2 2

2 cos cos

p k p k

p k p k p

i gM E E p M E E p

E E p E E p E p M m M

M

2

2

1 1

24 16k p p k

d pi

d E E E E

p kM

How do you average over spins?

When you have a spin ½ particle in the initial state, you sum over spins and divide by two. When you have two spin ½ particles in the initial state, you sum over spins and divide by four. What do you divide by if you have n spins in the initial state?

Does your formula work for n = 0?

1 spin: ,

2 spin: , , ,

3 spin: , , , , , , ,

Questions from the Reading Quiz“ I'm still confused on the whole pseudoscalar vs scalar thing. How do we pick the "scalar theory" of the "pseudoscalar theory". Which one is right?”

Answer: Neither is right, because it’s not a real theory.

2 2

3/22 2 2 22

4 , 48 8s P

g gM m M m

M

Questions from the Reading Quiz“Could we please go over the B coupling on page 101?”

iB

Scalar vs. Pseudoscalar couplings

1 1 2 2 1 1 2 2, , , ,p s p s p s p s

1 1 2 2 1 1 2 2, , , ,p s p s p s p s

1 1 1 5 1.i u u vs i u u M M