1
00:00:05,280 --> 00:00:07,860
MARKUS KLUTE: Welcome
back to 8.701.

2
00:00:07,860 --> 00:00:10,440
We have seen in
the previous video

3
00:00:10,440 --> 00:00:13,110
how neutrinos can acquire mass.

4
00:00:13,110 --> 00:00:15,870
When they have mask, ["mass"?]
their weak eigenstate is not

5
00:00:15,870 --> 00:00:17,890
equal to their mass eigenstate.

6
00:00:17,890 --> 00:00:20,760
So we observe the
same mixing as we

7
00:00:20,760 --> 00:00:22,740
have seen in the quark sector.

8
00:00:22,740 --> 00:00:25,260
So let's review
this a little bit.

9
00:00:25,260 --> 00:00:29,730
So just starting from
two neutrino generations,

10
00:00:29,730 --> 00:00:32,189
we can write the
flavoring states.

11
00:00:32,189 --> 00:00:35,190
We are mixing of
mass eigenstate.

12
00:00:35,190 --> 00:00:39,900
If we do this, it's
a simple matrix.

13
00:00:39,900 --> 00:00:44,260
You find that there
is one angle used

14
00:00:44,260 --> 00:00:46,360
for the rotation of
the mass eigenstate

15
00:00:46,360 --> 00:00:49,430
into the flavor eigenstate.

16
00:00:49,430 --> 00:00:49,940
All right.

17
00:00:49,940 --> 00:00:53,570
So we can add some time
to 0, write our neutrino

18
00:00:53,570 --> 00:00:58,100
or our muon neutrino as a
combination of the 1 and 2

19
00:00:58,100 --> 00:00:59,610
mass eigenstates.

20
00:00:59,610 --> 00:01:02,840
If we then have this
neutrino evolve as time,

21
00:01:02,840 --> 00:01:07,990
we see that the relative
contribution of the 1 and 2

22
00:01:07,990 --> 00:01:10,830
mass eigenstate
actually changes.

23
00:01:10,830 --> 00:01:15,350
So if we do that, so obviously
you find some time evolution.

24
00:01:15,350 --> 00:01:20,630
If we then ask ourselves,
what is the probability

25
00:01:20,630 --> 00:01:24,140
that we start from
a muon neutrino

26
00:01:24,140 --> 00:01:27,470
that we actually find in an
interaction electron neutrinos.

27
00:01:27,470 --> 00:01:29,870
Through this mixing
of mass eigenstates,

28
00:01:29,870 --> 00:01:32,750
we can calculate
this probability

29
00:01:32,750 --> 00:01:36,050
just by squaring the amplitudes.

30
00:01:36,050 --> 00:01:39,260
If you do this, you
just use this part here.

31
00:01:39,260 --> 00:01:45,140
We find that there is a
cosine E2 minus E1 term.

32
00:01:45,140 --> 00:01:45,770
All right.

33
00:01:45,770 --> 00:01:46,400
Good.

34
00:01:46,400 --> 00:01:48,680
So let's analyze this
a little bit further.

35
00:01:48,680 --> 00:01:50,690
We know that the masses
need to be small.

36
00:01:50,690 --> 00:01:53,030
So one thing we
can also do here is

37
00:01:53,030 --> 00:01:56,270
do a Taylor expansion
of our energy and then

38
00:01:56,270 --> 00:01:58,430
just revise the term.

39
00:01:58,430 --> 00:02:01,430
If you then analyze
it some more,

40
00:02:01,430 --> 00:02:05,930
you find that the oscillation
probability simply

41
00:02:05,930 --> 00:02:13,190
depends on the mass difference
squared, the length of distance

42
00:02:13,190 --> 00:02:16,400
the neutrino had time
from 0 to oscillate,

43
00:02:16,400 --> 00:02:19,050
and the energy of the neutrino.

44
00:02:19,050 --> 00:02:20,970
So this is fantastic,
because now

45
00:02:20,970 --> 00:02:25,430
by studying the
probability for a neutrino

46
00:02:25,430 --> 00:02:31,280
to change its flavor, we can
infer the mass differences

47
00:02:31,280 --> 00:02:33,110
of two states.

48
00:02:33,110 --> 00:02:34,440
This is fantastic.

49
00:02:34,440 --> 00:02:37,880
I should add here that in
this case, in this formula,

50
00:02:37,880 --> 00:02:45,460
the length is given in
meters, the energy is given--

51
00:02:45,460 --> 00:02:50,506
unit of the energy MeV
and the mass difference

52
00:02:50,506 --> 00:02:57,010
is in eV, otherwise the
equation doesn't make sense.

53
00:02:57,010 --> 00:03:01,870
So again, we have seen, if
you start from two neutrino

54
00:03:01,870 --> 00:03:04,570
kind of model, two
neutrino flavor model,

55
00:03:04,570 --> 00:03:07,990
that the experimental
parameters of interest

56
00:03:07,990 --> 00:03:12,100
here are the length of distance
from the neutrino source

57
00:03:12,100 --> 00:03:15,760
to the detector on the
place where we generate

58
00:03:15,760 --> 00:03:17,890
a specific neutrino
of a specific flavor,

59
00:03:17,890 --> 00:03:21,820
to where we actually observe
the flavor of the neutrino

60
00:03:21,820 --> 00:03:24,130
and the energy of the neutrino.

61
00:03:24,130 --> 00:03:26,710
And then, the appearance
or disappearance

62
00:03:26,710 --> 00:03:28,870
of a muon neutrino,
for example, if we

63
00:03:28,870 --> 00:03:32,290
start from a beam
of muon neutrinos,

64
00:03:32,290 --> 00:03:34,810
is a function of the
length of the source.

65
00:03:34,810 --> 00:03:39,810
And this is shown here for
neutrinos of a specific energy.

66
00:03:39,810 --> 00:03:42,330
So you can observe
or can try to measure

67
00:03:42,330 --> 00:03:44,700
the disappearance
of muon neutrinos

68
00:03:44,700 --> 00:03:48,570
or you can try to find
the appearance of electron

69
00:03:48,570 --> 00:03:51,870
neutrinos in the specific
two-neutrino model.

70
00:03:51,870 --> 00:03:52,740
All right.

71
00:03:52,740 --> 00:03:54,360
So all we find later
is that we want

72
00:03:54,360 --> 00:03:58,650
to look for disappearance
and appearance of neutrinos

73
00:03:58,650 --> 00:04:03,317
of specific flavors in order
to probe mass differences.

74
00:04:05,940 --> 00:04:08,260
Instead of doing this
for two generations,

75
00:04:08,260 --> 00:04:10,600
you already know how to do
this in three generations,

76
00:04:10,600 --> 00:04:13,060
you can find that
the unitary matrix

77
00:04:13,060 --> 00:04:17,529
has three angles,
three rotations and one

78
00:04:17,529 --> 00:04:18,670
complex phase.

79
00:04:18,670 --> 00:04:20,380
And this looks
very much the same

80
00:04:20,380 --> 00:04:22,120
here as in the quark sector.

81
00:04:22,120 --> 00:04:25,720
The big difference is that
the values of those parameters

82
00:04:25,720 --> 00:04:26,980
are quite different.

83
00:04:26,980 --> 00:04:30,400
For the quarks, we have seen
it's dominated by the diagonal.

84
00:04:30,400 --> 00:04:31,990
And then we have
seen, for example,

85
00:04:31,990 --> 00:04:34,120
in the Wolfenstein
parameterization

86
00:04:34,120 --> 00:04:37,540
that we can do an
expansion of the matrix

87
00:04:37,540 --> 00:04:41,110
and see terms which are all of
this lambda, which would all

88
00:04:41,110 --> 00:04:45,620
point to two and number
square and number acute.

89
00:04:45,620 --> 00:04:47,780
Here, on the leptons
sector, the situation

90
00:04:47,780 --> 00:04:49,830
seems to be quite different.

91
00:04:49,830 --> 00:04:52,940
We have a later lecture where
we look at the extra parameters

92
00:04:52,940 --> 00:04:55,340
and the numerical values.

93
00:04:55,340 --> 00:04:57,800
But what you see
here is that it's

94
00:04:57,800 --> 00:05:02,320
more like democracy between
the individual values.

95
00:05:02,320 --> 00:05:05,380
Question is, do we
have sensitivity

96
00:05:05,380 --> 00:05:06,700
to the complex phase?

97
00:05:06,700 --> 00:05:09,250
We can only have
that sensitivity

98
00:05:09,250 --> 00:05:12,250
if the value of this
matrix element is non-zero.

99
00:05:12,250 --> 00:05:14,570
And this has been
observed already.

100
00:05:14,570 --> 00:05:19,410
So that's good news in order to
allow further neutrino studies.

101
00:05:19,410 --> 00:05:22,670
So in general, you can write
the oscillation from one flavor

102
00:05:22,670 --> 00:05:27,920
to another flavor state using
this rotation of matrices

103
00:05:27,920 --> 00:05:31,310
we have seen, and
with that measure

104
00:05:31,310 --> 00:05:34,300
the individual
components of the matrix.