Lecture 22: Apr 12th 2012
Reading Griffiths chapter 7
HW:
This problem reassigned to next week
Calculate the e+e-  +- CM frame matrix element. In the ultra-relativistic limit show
that it obeys the s,t,u crossing relationship with e--  e-Also this problem
Determine the matrix element for Compton scattering e-  e- in the ultra relativistic
limit and show that it obeys the s, t, u crossing relationship with electron positron
annihilation to photons. Note this includes demonstrating that the two cross terms are 0.
1) Unpolarized relativistic electron-muon scattering
The matrix element is:
-ge2 é
m
ù
M=
ëu(3)g u(1)ûéëu(4)g m u(2)ùû
t
in general
éëu(a)g m u(b)ùûéëu(a)g n u(b)ùû = éëu(a)g m u(b)ùûéëu(b)g n u(a)ùû = éëu(a)g m u(b)ùûéëu(b)g n u(a)ùûu(a)
= Tr éëg m ( p/ b + mb ) g n ( p/ a + ma )ùû
*
Thus in our example
M1 =
2
*
ge4 é
m
ùûéëu(3)g n u(1)ùû éëu(4)g m u(2)ùû[ u(4)g n u(2)]*
u(3)
g
u(1)
ë
t2
ge4 é
u(3)g m u(1)ùûéëu(1)g n u(3)ùûéëu(4)g m u(2)ùû[ u(2)g n u(4)]
2 ë
t
g4
= 2e éëg m u(1)ùûéëu(1)g n u(3)ùûu(3)éëg m u(2)ùû[ u(2)g n u(4)] u(4)
t
=
ge4 é m
Tr ëg ( p/ 1 + me ) g n ( p/ 3 + me )ùûTr éëg m ( p/ 2 + mm ) g n ( p/ 4 + mm )ùû
2
4t
4 is from the averaging over initial spins
M1 =
2
Tr éëg m ( p/ 1 + me ) g n ( p/ 3 + me )ùû = Tr éëg m p/ 1g n p/ 3 ùû + meTr éëg m p/ 1g n ùû + meTr éëg mg n p/ 3 ùû + me2Tr éëg mg n ùû
Using trace identities
10) Trace of an odd number of gamma matrices is 0
mn
12) Tr g m g n = 4g
[
]
17) Tr éëg p/ 1g n p/ 3 ùû = 4éë p1m p3n + p3m p1n - gmn ( p1 × p3 )ùû
2
g4
M1 = e2 Tr éëg m ( p/ 1 + me ) g n ( p/ 3 + me )ùûTr éëg m ( p/ 2 + mm ) g n ( p/ 4 + mm )ùû
4t
m
}{
}
M1 =
ge4
Tr éëg m p/ 1g n p/ 3 ùû + me2Tr éëg mg n ùû Tr éëg m p/ 2g n p/ 4 ùû + mm2Tr éëg mgn ùû
2
4t
M1 =
ge4
4 p1m pn3 + p3m p1n + g mn ( me2 - p1 × p3 ) p2m p4 n + p4 m p2n + gmn ( mm2 - p2 × p4 )
2
t
M1 =
ge4
8 p × p2 )( p3 × p4 ) + ( p1 × p4 )( p2 × p3 ) - ( p1 × p3 ) mm2 - ( p2 × p4 ) me2 + 2me2 mm2 ]
2 [( 1
t
2
2
2
{
[
][
Often we evaluate these matrix elements in relativistic limits.
In the ultra relativistic limit the Madelstam variables become:
s = ( p1 + p2 ) = 2p1 × p2,
2
t = ( p1 - p3 ) = -2 p1 × p3,
2
u = ( p1 - p4 ) = -2 p1 × p4,
2
and
M1 =
2
2ge4 2
s + u2 ]
2 [
t
with
q
2
2
q
u = ( p1 - p4 ) = -2 p2 (1+ cosq ) = -4p2 cos2
2
2
2
2
2
s = ( p1 + p2 ) = 4(p + m ) = 4p ,
t = ( p1 - p3 ) = -2 p2 (1- cosq ) = -4 p2 sin 2
2
4
q
1+ cos
s2 + u 2
4
2
M = 2ge4
=
2g
e
t2
4 q
sin
2
2
which is valid if the lepton masses can be taken as the same or negligible.
]
This cross section includes and infinity at theta = 0 which corresponds to infinite range
scattering.
2) Crossing symmetry and e-e+  -+
Starting from e--  e-- you can get to e-e+  -+ by moving the final state electron
electron (p3) to the initial state as a positron (-p2) and by moving the initial state muon
(p2) to become a final antimuon (-p3).
Since you are just switching p3 with –p2 or p2 with –p3 then this is equivalent to
changing s and t
M = 2ge4
2
t 2 + u2
s2
Which is quite interesting considering one of the interactions is a scattering interaction
and the other annihilation! This only works at high energy. At low energy there is not
enough energy to form the muon pair and there is no s channel process. Also notice that
the annihilation interaction is not divergent. This makes sense since you can’t have
annihilation at infinite range. In fact we can calculate the total cross section.
In CM d/d = (1/8)2 |M|2 pi/pf 1/E2
using the exact expression for s, t and u in CM
s = ( p1 + p2 ) = 4( p2 + m 2 ),
2
q
t = ( p1 - p3 ) = -2 p2 (1- cosq ) = -4 p 2 sin2 , and
2
2
2
2
2q
u = ( p1 - p4 ) = -2 p (1+ cosq ) = -4 p cos ,
2
2
which can be integrated!
ds =
a2
a2
ò (1+ cos q)df sinqdq = 4s 2p 3 =
4s
2
8
4pa 2
3s
The cross section is finite and goes down at higher energy, which is what is observed.
The t channel scattering cross section is still divergent at =0. The point =0 corresponds
to the point where b=. That the total cross section is infinite just represents the fact that
a 1/r2 potential still has a small effect no matter how far away you get. However, we are
interested in computing scattering cross section or probabilities that for a particle between
b and b+db to scatter to angle theta. As you get to very large b you are probing very
small scattering angles between ~0 and +d. The cross section and the probability to
go to any of the angles will still be finite for any finite b.
3) Various limits of e--  e-a) Mott scattering. Lab frame, M>>m
ge4
M = 2 8[( p1 × p2 )( p3 × p4 ) + ( p1 × p4 )( p2 × p3 ) - ( p1 × p3 ) mm2 - ( p2 × p4 ) me2 + 2me2 mm2 ]
t
2
p1 = (E,p), p3 = (E,pf), p2 = (m,0), p4 = (m,0)
ge4
M = 2 8[ E 2 mm2 + E 2 mm2 - E 2 mm2 + p 2 mm2 cosq - mm2 me2 + 2me2 mm2 ]
t
ge4
2
M = 2 8[(E 2 mm2 + p 2 mm2 cosq + me2 mm2 ]
t
ge4
2
M = 2 8[(( p 2 + me2 )mm2 + p2 mm2 cosq + me2 mm2 ]
t
ge4
2
M = 2 8[ p 2 mm2 (1+ cosq ) + 2me2 mm2 ]
t
2
M =
2
M =
2
16ge4 mm2 æ 2
2
2 qö
2
2q
ç me + p cos ÷, t = -4 p sin
2
è
t
2ø
2
ge4 mm2 æ 2
2
2 qö
ç me - p cos ÷
q
2ø
p 4 sin 4 è
2
In the un-relativistic limit this reduces to the Rutherford formula.
M =
2
ge4 mm2 me2
p 4 sin 4
q
2
In the ultra-relativistic limit, as derived above, this can be expressed all in terms of s, t
and u as above
M = 2ge4
2
s2 + u 2
t2
4) Physical interpretation of s, t and u terms
For electron/positron scattering via a photon s, t and u channel diagrams have matrix
elements probabilities of the form
t 2 + u2
= 8ge4 (1+ cos2 q ), maximum at 0 and , forward
2
s
backward symmetric, s channel wipes out some angular correlations. Not as peaked as
other distributions. Cross section is finite
s, annihilation only: M = 2ge4
2
4
q
1+ cos
s2 + u 2
4
2 , infinite maximum at 0, forward,
t, scattering only: M = 2ge4
=
2g
e
t2
4 q
sin
2
finite minimum backward. Realistic cross section is finite since far forward scattering
corresponds to infinite impact parameters.
2
4
q
1+ sin
s2 + t 2
4
2 , Doesn’t occur by itself. Contributes an infinite
u: M = 2ge4
=
2g
e
u2
4 q
cos
2
maximum at , backward, finite minimum forward.
2
Interference terms
s2
, tu: symmetric with infinite maximum at both 0 and , forward backward,
tu
u2
, st: infinite maximum forward. Realistic cross section is finite.
st
Then:
e-e+  -+ annihilation only: s channel forward backward symmetric and finite.
e--  e-- scattering only: t channel peaked forward
Moller scattering: e-e-  e-e-, t and u channel forward backward symmetric and peaked
at both sides. Interference term is also symmetric.
é s 2 + u2 2s 2 s 2 + t 2 ù
2
M = 2ge4 ê 2 +
+ 2 ú
tu
u û
ë t
é
4 ù
M = 4g ê1- 2 ú
ë sin q û
2
2
4
e
Bhabha scattering: e-e+  e-e+, t (forward) and s (symmetric) channels + a cross term
that will be peaked forward.