1
00:00:05,160 --> 00:00:07,040
MARKUS KLUTE: Welcome
back to 8.701.

2
00:00:07,040 --> 00:00:12,030
So now after we introduced
the weak interaction

3
00:00:12,030 --> 00:00:13,980
and the Feynman rules
for weak interaction,

4
00:00:13,980 --> 00:00:17,770
we can now look at decays
of muons, and in this case,

5
00:00:17,770 --> 00:00:19,230
the decay of a pion.

6
00:00:19,230 --> 00:00:21,930
Decay of the pion is
specifically interesting.

7
00:00:21,930 --> 00:00:23,700
And we discussed the
decay of the pion

8
00:00:23,700 --> 00:00:27,660
before when it came to the
discussion of helicity states.

9
00:00:27,660 --> 00:00:31,170
Now, let's look at this again
with the information we have

10
00:00:31,170 --> 00:00:32,790
and what we learned.

11
00:00:32,790 --> 00:00:34,800
Now, if you look
at the pion decay,

12
00:00:34,800 --> 00:00:39,480
the two or three leading
decay modes are given here.

13
00:00:39,480 --> 00:00:42,630
The one is where the pion, in
this case a negatively charged

14
00:00:42,630 --> 00:00:45,510
pion, decays into an
anti-electron neutrino

15
00:00:45,510 --> 00:00:50,880
and an electron, or V or the
W in a muon and an antimuon

16
00:00:50,880 --> 00:00:52,620
neutrino.

17
00:00:52,620 --> 00:00:55,740
If you look at this in the
rest frame of the pion,

18
00:00:55,740 --> 00:00:59,490
we can see that the neutrino
and the lepton, charged lepton,

19
00:00:59,490 --> 00:01:01,470
are produced back-to-back.

20
00:01:01,470 --> 00:01:04,560
Now, the spin of
the pion is 0, which

21
00:01:04,560 --> 00:01:09,060
means that the
opposite-direction outgoing

22
00:01:09,060 --> 00:01:12,210
leptons have to have the
same helicity states.

23
00:01:12,210 --> 00:01:17,160
Since the neutrino is massless,
the antineutrino is massless,

24
00:01:17,160 --> 00:01:20,790
the antineutrino is
produced right-handed.

25
00:01:20,790 --> 00:01:22,110
It is always right-handed.

26
00:01:22,110 --> 00:01:26,160
The chiral state of the
neutrino and the helicity state

27
00:01:26,160 --> 00:01:28,380
of the neutrino are
essentially the same,

28
00:01:28,380 --> 00:01:30,330
because they're massless.

29
00:01:30,330 --> 00:01:33,630
Means the projection
of the spinor

30
00:01:33,630 --> 00:01:35,940
is basically the
same as a projection

31
00:01:35,940 --> 00:01:41,520
of the spin on the
momentum [? direction. ?]

32
00:01:41,520 --> 00:01:42,420
All right.

33
00:01:42,420 --> 00:01:44,070
But the charged
lepton is massive.

34
00:01:44,070 --> 00:01:46,650
If the charged lepton
would be massless,

35
00:01:46,650 --> 00:01:48,210
the decay would not be allowed.

36
00:01:48,210 --> 00:01:51,100
There would not be a
right-handed helicity

37
00:01:51,100 --> 00:01:55,230
state for a charged lepton.

38
00:01:55,230 --> 00:01:58,200
Now, this causes
quite some confusion.

39
00:01:58,200 --> 00:02:01,500
And I've seen, even in this
course, some students being

40
00:02:01,500 --> 00:02:04,140
confused by this.

41
00:02:04,140 --> 00:02:11,020
I can write the, let's say,
right-handed charged lepton

42
00:02:11,020 --> 00:02:15,030
and decompose its
right-handed helicity state.

43
00:02:15,030 --> 00:02:19,700
So this is, let's say, the
right-handed helicity state.

44
00:02:22,650 --> 00:02:26,310
And I can decompose this
through the chiral states,

45
00:02:26,310 --> 00:02:29,430
the right-handed
and the left-handed.

46
00:02:29,430 --> 00:02:31,980
And you have seen in
the previous lectures

47
00:02:31,980 --> 00:02:36,140
that only the left-handed
component participates.

48
00:02:36,140 --> 00:02:38,840
Now, you can also see
from this equation

49
00:02:38,840 --> 00:02:41,840
here that if the momentum
and energy would be the same,

50
00:02:41,840 --> 00:02:45,380
as it is the case for massless
particle, this would be 0,

51
00:02:45,380 --> 00:02:48,110
this would be 1,
this would be 1.

52
00:02:48,110 --> 00:02:51,650
And therefore, this
right-handed helicity state

53
00:02:51,650 --> 00:02:55,370
would be the same
as the chiral state,

54
00:02:55,370 --> 00:02:58,650
and it wouldn't be coupling
to the weak interaction.

55
00:02:58,650 --> 00:03:00,470
Now let's erase
this really quickly,

56
00:03:00,470 --> 00:03:02,854
because you want to
actually look at this decay.

57
00:03:05,760 --> 00:03:08,430
And so now we have all the tools
together-- almost all the tools

58
00:03:08,430 --> 00:03:12,510
together to calculate
this, the decay rates,

59
00:03:12,510 --> 00:03:14,460
or the ratio of decay rates.

60
00:03:14,460 --> 00:03:16,530
And you want to do this
in the pion rest frame,

61
00:03:16,530 --> 00:03:20,500
so the momenta are given here.

62
00:03:20,500 --> 00:03:22,560
See that the pion momentum is 0.

63
00:03:22,560 --> 00:03:25,290
And for momentum, the
energy is equal to the mass

64
00:03:25,290 --> 00:03:27,030
for the charged lepton.

65
00:03:27,030 --> 00:03:28,640
And for the neutrino,
just produce

66
00:03:28,640 --> 00:03:30,940
in an opposite direction.

67
00:03:30,940 --> 00:03:34,700
So neutrino in this case goes
into negative d direction.

68
00:03:34,700 --> 00:03:38,450
Then we can write
the leptonic current,

69
00:03:38,450 --> 00:03:40,520
as we have just seen in
the previous lecture.

70
00:03:40,520 --> 00:03:44,840
You see this 1 minus
gamma 5 term here.

71
00:03:44,840 --> 00:03:45,340
Good.

72
00:03:45,340 --> 00:03:48,915
And I could have just
called this left-handed here

73
00:03:48,915 --> 00:03:52,400
and put this into the
definition of the spinor.

74
00:03:52,400 --> 00:03:54,080
Fine.

75
00:03:54,080 --> 00:03:56,720
When we put a real spinor,
this comes out immediately

76
00:03:56,720 --> 00:03:57,220
[INAUDIBLE].

77
00:03:57,220 --> 00:03:58,100
Immediately.

78
00:03:58,100 --> 00:04:00,620
You have to keep this in mind.

79
00:04:00,620 --> 00:04:03,500
The matrix element, then, is
a little bit more complicated.

80
00:04:03,500 --> 00:04:06,980
And here is an additional-- so
you see the current here again.

81
00:04:06,980 --> 00:04:10,790
You see the
propagator, and I went

82
00:04:10,790 --> 00:04:12,770
into the low-energy
approximation here.

83
00:04:12,770 --> 00:04:16,579
You see that instead of having
a [? q ?] square minus m square,

84
00:04:16,579 --> 00:04:19,700
I'm just keeping the m
square component of this.

85
00:04:19,700 --> 00:04:23,120
And then I have this part here
for the current, for the pion

86
00:04:23,120 --> 00:04:24,890
current here.

87
00:04:24,890 --> 00:04:28,040
And I simply parameterize
my missing understanding

88
00:04:28,040 --> 00:04:31,730
or missing ability to calculate
[? non-prohibitive ?] QCD with

89
00:04:31,730 --> 00:04:32,610
a form factor.

90
00:04:32,610 --> 00:04:35,300
So I introduce this
form factor for a pion.

91
00:04:35,300 --> 00:04:37,580
This is not an important
part of the discussion,

92
00:04:37,580 --> 00:04:40,410
we just keep track of this here.

93
00:04:40,410 --> 00:04:41,100
All right.

94
00:04:41,100 --> 00:04:43,570
Then we can calculate
this matrix element fine.

95
00:04:43,570 --> 00:04:46,920
We then have to be explicit
about the spinors we are using,

96
00:04:46,920 --> 00:04:49,920
and we use the momentum
as defined above.

97
00:04:49,920 --> 00:04:52,090
So this step here I'm
not doing explicitly.

98
00:04:52,090 --> 00:05:02,040
If you want, you can go to
Thomson and read in chapter 11.

99
00:05:02,040 --> 00:05:05,490
He gives quite some
detail on this.

100
00:05:05,490 --> 00:05:06,480
All right.

101
00:05:06,480 --> 00:05:09,660
So moving on, there
is one extra thing.

102
00:05:09,660 --> 00:05:15,840
When we try to calculate the
spin-averaged matrix element,

103
00:05:15,840 --> 00:05:18,750
we find that we don't have to
do any work because there's

104
00:05:18,750 --> 00:05:21,300
a spin [? 0 set, ?] there's
only one state contributing,

105
00:05:21,300 --> 00:05:23,610
so we don't have to do any work.

106
00:05:23,610 --> 00:05:25,947
We just have to square
the matrix element.

107
00:05:25,947 --> 00:05:27,780
We find this as a
solution here, and there's

108
00:05:27,780 --> 00:05:30,420
an additional factor we
haven't introduced yet.

109
00:05:30,420 --> 00:05:32,190
This is the Fermi coupling.

110
00:05:32,190 --> 00:05:38,580
Again, this comes out in the
low-energy approximation.

111
00:05:38,580 --> 00:05:44,970
And G Fermi is simply defined
over the coupling to the W

112
00:05:44,970 --> 00:05:48,470
over the W mass
squared, as shown here.

113
00:05:48,470 --> 00:05:48,970
All right.

114
00:05:48,970 --> 00:05:50,387
Again, this is
just a factor which

115
00:05:50,387 --> 00:05:53,830
is not relevant to the
discussion at this point.

116
00:05:53,830 --> 00:05:57,400
But we can then, using
Fermi's golden rule,

117
00:05:57,400 --> 00:06:04,090
calculate the partial decay
width of the pion decay.

118
00:06:04,090 --> 00:06:04,590
OK?

119
00:06:04,590 --> 00:06:07,680
So we just put in the
matrix element here,

120
00:06:07,680 --> 00:06:10,980
and we replace the
momentum with the energy,

121
00:06:10,980 --> 00:06:14,730
being equal to the
mass of the pion.

122
00:06:14,730 --> 00:06:19,590
And voila-- we get this as an
answer for the partial decay

123
00:06:19,590 --> 00:06:20,560
width.

124
00:06:20,560 --> 00:06:21,460
OK?

125
00:06:21,460 --> 00:06:24,370
If you now want to know some
experimental information,

126
00:06:24,370 --> 00:06:26,490
like the partial decay
width of the pion,

127
00:06:26,490 --> 00:06:29,050
charged pion, to
electrons over the muon,

128
00:06:29,050 --> 00:06:31,570
you want to know what this
factor is, we immediately

129
00:06:31,570 --> 00:06:32,240
can do this.

130
00:06:32,240 --> 00:06:35,290
We don't need to know any
of the details F, G Fermi

131
00:06:35,290 --> 00:06:37,540
as a structure
function of the pion.

132
00:06:37,540 --> 00:06:39,440
All of those factors cancel out.

133
00:06:39,440 --> 00:06:43,030
And what is left here
are the parameters

134
00:06:43,030 --> 00:06:46,000
of the electron mass, the
muon mass, and the pion mass.

135
00:06:46,000 --> 00:06:51,640
And if you just use
values like this,

136
00:06:51,640 --> 00:06:55,510
the mass of the
muon with 205 MeV

137
00:06:55,510 --> 00:07:03,100
and the mass of the pion 240
MeV, you find 10 to the minus 4

138
00:07:03,100 --> 00:07:09,033
as a value for this ratio
of part [? indicators. ?]

139
00:07:09,033 --> 00:07:10,450
And you see where
this comes from.

140
00:07:10,450 --> 00:07:11,867
This basically
comes from the fact

141
00:07:11,867 --> 00:07:15,310
that the electron mass
is much, much smaller

142
00:07:15,310 --> 00:07:18,890
than the mass of the muon.

143
00:07:18,890 --> 00:07:21,590
And factually, you can
expand this by the fact

144
00:07:21,590 --> 00:07:25,190
that a right-handed
helicity state for a muon

145
00:07:25,190 --> 00:07:28,690
can have a much
larger contribution

146
00:07:28,690 --> 00:07:33,220
of the left-handed chiral state
of a muon, while this is not

147
00:07:33,220 --> 00:07:35,560
possible-- or, that
is, the component

148
00:07:35,560 --> 00:07:38,360
is much smaller for
the lighter electron.

149
00:07:38,360 --> 00:07:44,260
And again, only the left-handed
component of the charged lepton

150
00:07:44,260 --> 00:07:47,580
contributes to the
weak interaction.