I have academic training a session or
series which is gonna have four sessions
the first one starting today which will
be on a tier introduction and then we
will have some experimental such as
chroming and again at the end some
grandview on the theory and maybe beyond
the standard model type of discussions
so today we have the pleasure of having
Pilar Hernandez who will talk to wasn't
giving in to the introduction
she's from Valencia as you see here but
she's also actually at CERN right now
and as a leader of this so-called new
theory neutrino efforts or some effort
that was build up here as well as there
isn't a P group effort that started last
year this is the theory counterpart and
of course we work closely together with
them and I think you're gonna be here
still til the middle of this year till
summer or so so any discussions on this
thing can of course be followed up later
okay without much I do I would say okay
thank you so as I say as I were just
said this is the first of four talks I
will be given the first and the fourth
talk and the plan of this lecture today
is a following I will start with some
introduction highlighting the landmark
said by neutrinos in the construction of
the standard model then I will Suren
oops
I will explain how neutrinos have forces
to actually step out of the standard
model in accommodating their tiny masses
and then I will describe in the last
part of the talk
the phenomenon of neutrino oscillations
from which we have learned most of what
we know about neutrino masses so
neutrinos are the most elusive particles
in this colourful puzzle which is the
standard model and nevertheless they
have been
he in the discovery not only of the weak
interactions but also in establishing
the two most intriguing features of this
model which are the which we don't
really understand in depth which had a
three-fold repetition or family
structure and the chiral nature of the
weak interactions so let me go back to
the beginning as you know the neutrino
appeared in the history of physics as a
dark fountain particle with the
discovery of radioactivity decay in
which nuclei undergoes a transition
emitting apparently just an electron
which by energy conservation should have
an energy fixed by the rest mass
difference of the mother and daughter
nuclei however when this spectrum was
measured a continuum curve was found and
in a disk you pot you value and it took
many years actually great physicists
like Wolfgang Pauli to come up with what
he called a desperate remedy to save
energy conservation in his famous
radioactive letter he proposed that
another invisible particle was being
emitted together with the electron and
therefore share the available energy now
after some trials they this particle was
finally baptized with an Italian name
neutrino and this was in honor of Enrico
Fermi who was among the first species to
actually take seriously this Paulus
hypothesis and actually build a theory
out of it which is the famous theory of
beta decay now beta decay goes by
through a for failure interaction of
this form which would also imply that
neutrinos should interact with matter to
some at some level now it's interesting
to recall that nature Journal did not
publish this article on the basis that
it contains speculations too remote from
reality to be of interest to the reader
those were the good ol times where
papers had to be of interest to readers
and bitter amperes actually estimated
from this Fermi theory what would be the
neutrino cross-section and they found
this stream Li tiny number declaring
that there was no practically possible
way of detecting a neutrino in fact it's
quite challenging to stop a neutrino the
length of a neutrino in water would be
in the order of thousand light years so
not surprisingly Polly was quite a shame
of what he had done his postulate this
theory might be of the sort that he
hated not even wrong however reveal in
Polly's dark matter was she asked the
question of time ingenuity of many
physicists as you will hear in this in
these lectures particularly on the
experimental side and in fact you can
beat any small cross-section by just I
statistics so you need a large detector
with a lie detector of about thousand
kilograms and a neutral high intense
flux of about ten to eleven neutrinos
per centimeter square per second you
might get a few events per day and such
very intense neutrino sources already
had been invented there was the nuclear
reactors which typical reactor would
produce about 10 to the 20 neutrinos per
second isotropically so if you open your
hand at 100 meters from a reactor you
will be receiving about 10 to the 11
neutrinos per second and this is
precisely how the neutrino was first
discovered in the famous experiment by
rynason Cohen who used the reactor at
Savannah River to discover the anti
neutrino by noticing that an anti
neutrino could give a real really golden
signal that we will discuss tomorrow bye
bye Stefania now this was such a great
idea that modern versions of this Ramos
and Cohen experiment are still making
discoveries today
go under the name of Shah double showed
I have a Reno they are all modern
versions of this very famous experiment
now as soon as neutrinos were first
discovered it was soon realized that
they came in flavors the moon had been
discovered quite early in in cosmic rays
but it took some time to understand what
this beast wars and in particular it was
a heavy version of the electron and no
the nuclear age and the pion and it was
then realized that this process would
take place and produce a phantom
particle yeah just like beta decay now
it was thought that this neutrino could
be different to that in beta decay and
in fact this is what another of the
famous experiments in this field did
Letterman swattin steinberger they
created the first what we call today
accelerator neutrino beam by mimicking
what happens in cosmic rays they hit a
proton beam on a target producing lots
of Pi ons that decay to neutrinos and
they put a detector heavily shielded
from where the signal of this phantom
particle from Payan decay would be the
appearance from time to time of immune
and not an electron okay so they clearly
established family identity as a basic
property of the standard model again
this was such a great idea that modern
versions of Letterman's Watson Steinberg
experiments are what we call oscillator
neutrino experiments they are still
making discoveries today today as you
will hear in following talks and they go
under names such as Minos operated ok
nova etc now since all these i mean in
all these beautiful experiments of
course people search for kinematical
effects of the neutrino mass and the way
in which a neutrino mass finite neutrino
mass could affect this the spectrum of
beta decay of the electron in beta decay
would be by modifying slightly the
endpoint the spectrum we
now they available maximum available
energy for the electron which would not
be the Q value but the Q value - the
rest mass of the neutrino now looking
for such effect has one I mean people
have experiments like mice and toys have
been able to put to set their strongest
on strength on a neutrino mass in that
appears in combination with an electron
poorer limits come from similar
processes involving moons and Taos so
this is why in the standard model
neutrinos were first assumed to be
massless okay now with the salt puzzle
of the standard model summarized in this
table neutrinos fit nicely as the upper
components of the lepton doublet and
they are connected as I said at the
beginning to two of the most striking
features of this table the first of
these features is the left-right
asymmetry which is no where more tear
that done with neutrinos that only have
no counterpart on the right so what does
this left-right asymmetry means so
particles on the left are left-handed
projectors chiral states which in the
relativistic limit is the same thing as
saying that they are left-handed olicity
particles while the particles on the
right right-handed chirality states
which are positively olicity particles
and now the amazing feature of the
standard model is that these two
particle to left and right particles
carry different charges and they're as
u2 and u1 but beyond that all particles
in the standard model except neutrinos
have counterparts on the left and on the
right and this is only neutrinos that
only appear on the left what this means
in more formal terms is that they are by
fermions so they are described by a
field
Spiner field which with two components
that describes a massless particle with
negative velocity and antiparticle with
positive velocity and this is the
minimum this is the minimum dimensional
representation for a spin 1/2
relativistic particle which however
violates in a stream wave parity and
charge conjugation instead on the other
fermions in the standard model fall in
Dirac representations with which are
four components we know representations
which represent the spin 1/2 particles
but also can accommodate parity ok oops
now the second striking feature of the
of the standard model table ya show is
the family replication and of course as
you know very well neutrinos have been
essential in establishing that the
standard model has three families this
is because neutrinos are massless
assumed to be massless in the standard
model and therefore the set ball so can
decay to any pair of neutrinos no matter
how many families there are and in fact
measure in the visible width of the set
zero one can which is proportional to
this number of families one can actually
pin down the not this number to be three
with a great precision here is the
legacy one of the most important legacy
plots from from CERN in the lab area
with this invisible width measurement
for the set zero so yes this tunnel so I
think I hope I have convinced you that
neutrinos have been instrumental in
constructing the standard model but also
they have already taught us the way out
of the standard model and this has been
thanks to the fact that in spite of
their weak interactions we are
surrounded by very intense source of
neutrinos that we have detected
after the next so if you open your hand
you will be receiving about 10 to the 12
neutrinos per second from the Sun the
atmosphere at the very different energy
range is also a source of neutrinos you
will be receiving about 20 per second
the earth natural radioactivity is an
interest source of neutrinos about 10 to
the 9 per second
we have also detected neutrinos coming
from very far away and the famous
supernova 87 produced a flux of
neutrinos that is comparable to that of
the Sun however they started supernova
was 10 to the 8 times farther away from
Earth and our Sun which is quite amazing
this the Big Bang as we know left behind
this relic cosmic micron those photons
well there is a similar background of
neutrinos however this happening is a
quite a lot of them quite a lot of them
out there
unfortunately the energies of these
neutrinos are very very small and it's
very very challenging to detect them and
hasn't been achieved yet we have even
heard from neutrinos from a still
unknown sources so the IceCube detector
has intercepted a few midst of Peavey
energies which come from somewhere
outside the atmosphere and a crescent
the origin of these very high-energy
events is a mystery so using as I said
this many many of these natural sources
and others that have been made by in the
laboratory at least 2 decades of
revolutionary neutrino experiments have
demonstrated that neutrinos are not
quite standard as we thought because
they have a tiny mass and this implies
that we have to extend in one way or
another the standard model and you will
be hearing much more about all these
beautiful experiments
in the following lectures by Stefania so
let's see how we can extend the standard
model to have massive neutrinos for that
let me remind you what is the mass for
the act fermion so direct a mass for
DIAC fermions is simply a term in the
Lagrangian of this form which if we
write it down in terms of left and right
components we see that it links left and
right so the mass can be seen as this
the strength of the transition from left
to right okay now this we cannot do in a
standard model as it stands because they
are known right hundred partners for the
neutrino however it was my Arana that
first realized that actually you can do
the same even if you have at your
disposal just left-handed estate this is
what we call a Meyer on a fermion and it
amounts to adding to the Lagrangian a
term of this form website see is the
charge code you gated field which is a
transformation that essentially takes
you from particles 20 particles in a
very simplistic way so the mass in this
context can be seen as I call s again as
a strength of the transition from left
to right but now right is not at
independent field like for the drug
fermion but it's the antiparticle
of the original field okay so a
propagating mayorana neutrino is
therefore said often to be its own
antiparticle okay you can go from being
a particle state to an antiparticle
state yes by having a mass okay now we
can not do none of these this is not
allowed by the sacred principle of gauge
invariance in the standard model and it
cannot be done for any of the fermions
in the standard model we cannot give
fermion
deokman's because as we have seen left
and right states have different charges
and the ratio to cross u 1 gauge group
and similarly for a Mirena fermion that
changes a particle to its antiparticle
obviously no charge whatsoever whether
he says you to cross one nor any other
global charge can be conserved by this
interaction it's spontaneous symmetry
breaking however can help us here and as
we know very well it allows us to give
direct masses to all the fermions in the
standard model and in the case of
neutrinos it allows us to do either of
these to take any of these two ways and
get give neutrinos a mass so we can give
you twinners a mass by simply extending
the standard model with right-handed
neutrinos so that we can write a yokubou
coupling like we do for the remaining
fermions now this you have a coupling
connects left and right via hips which
matches the charges so that this is
gauge invariant and upon spontaneous
symmetry breaking it takes expectation
value V and therefore the mass of the
fermion is proportional to the you cava
coupling in front of this operator times
the birth of the Higgs however if we
don't want to add right-handed neutrinos
to the standard model we can still do
that we can write down a term that upon
spontaneous symmetry breaking would give
us a Meyer Animas and we can do this
negating Berean way the term however is
more complicated it requires to lepton
doublets and to Higgs fields to make it
gauge invariant and therefore when this
Higgs field takes an expectation value
we know that the it I mean it really
this operator turns into the neutrino
Majorana mass however is proportional to
a second power of the birth of the Higgs
in contracts with previous situation
this is also proportional to this
coupling here alpha which by dimensions
must have dimensions of energy to the
minus 1 so we can write it as in
a dimensional coupling over some high
energy scale that we don't know and
could be related or not to the
electroweak scale now what is the
meaning of this scale the meaning of
this scale is that we know this cannot
be the whole story must much like Fermi
theory of the weak interaction could not
be the whole story at some scale lambda
we expect that new particles will show
up that will induce this interaction at
low energies now the fact that this
neutrino mass is inversely proportional
to a new scale is what is often referred
to as a seesaw mechanism the like the
higher we make this new scale the lower
naturally the neutrino mass will be and
this is of course very interesting
because we see a pattern in which
neutrino masses even if they are mass if
their masses are really much smaller
than the remaining fermions so this
could give us a natural way of
understanding this gap
now whether neutrinos are Dirac or
Marana
their effects the most important
implication of neutrino masses which is
lepton mixing is very similar to both
both cases essentially what I just said
is that neutrino masses in the case of
Dirac in the case of the Dirac option
the thermal masses would be proportional
to the birth of the Higgs with some
Yukawa coupling that is in principle a
generic matrix in a family space so it
can link families okay
now this means that if we write down the
mass matrix for neutrinos and the and
the leptons and the charged leptons we
find that these two matrices are genetic
matrix complex matrices that can be
taken to the diagonal form so this is
what we call going to the massagin basis
by performing some unitary rotations on
the left
on the right which are independent now
if we go to that basis we move we change
basis so that now this masses are
diagonal then we find that all these
unitary rotations disappear from
everywhere except from the charge
current interaction which we're in
general unitary matrix remains which is
of course the complete analogous to the
ckm matrix in the quarks sector and in
this deduct case the analogy is
basically exact this matrix will depend
on three angles and one city violating
face and you will see tomorrow how much
we know about this matrix now if we are
in the Miranda case the situation is
quite similar we have Miami euonymus
instead of right-handed neutrinos now
but no matter what this matrix is a
completely generic matrix in family
space it can be taken to a diagonal form
by a unitary transformation of this form
and again a pmns matrix appears in the
weak in the charge current interaction
the only difference between the Majorana
and the dark case is that now we cannot
really really fine or reabsorb as some
physical as many as the CP faces that we
could in the case of Iraq so the three
physical CP faces in general remain in
this matrix okay
and this will come up later so neutrino
mixing when we produce chart are any
data a nutrient a circuit and
interaction we produce it always in in
connection with an electron immunity
okay so this is what we call a flavor
again state okay and this is not the
same as the Messiaen state they are
related by this unitary matrix which is
usually parametrize in
Euler type fashion in this way
we're these extra faces so this would be
up to here this would be same
parametrization of the ckm and there is
some extra faces that can be
parameterize in this fashion now clearly
if we want to distinguish these two
possibilities the rocker Madonna which
have very important implications for our
understanding of the standard model as
we will see in the last lecture we would
like to to actually distinguish these
two possibilities very one of the most
important questions in neutrino physics
now in principle the deciding whether
you have a Madonna or deduct would give
very clear Sperry mental signatures you
might imagine you create a neutrino beam
from PI plus DK now if neutrinos had the
duck when they interact in the detector
they will only produce new minuses
however if they are Marana they can
produce me minuses as well but then from
time to time quite rarely they can
produce her own sign moon now this is of
course a very clear signal the problem
is that this processes in which you get
the wrong sign are extremely rare and
this is because it requires a carrier
chirality transition which is
proportional to the mass of the neutrino
and therefore the rate for the right
sign with respect to the rate of the
wrong sign is suppressed with a ratio
mass for neutrino over energy squared
and this is completely generic okay so
the first the best hope you have is to
have a small energy as possible and the
best hope we think at the moment to
discover
this Majorana nature of neutrinos is
true neutrinoless double beta decay
now some isotopes cannot undergo a
single beta decay but they can undergo a
double beta decay so the usual process
would go through the production of two
neutrinos in the final state now this
process has been measured for several
isotopes in the range of 10 to the 18 -
10 to 21 years lifetime
however if neutrinos are Moroni the
neutrino producing one of these the case
can be absorbed by the other because it
can turn it into its antiparticle and
then you can have the process but
without neutrinos in the final stage
this is what we call neutrinoless double
beta decay and as I said before the
process compared to the standard one
would be suppressed with a neutrino mass
over energy square in this case however
you gain a lot in phase space from
comparing these two processes so the
situation is as optimal as it gets
okay now this process has been searched
for in several isotopes and here's a
summary table essentially the lifetime
can be computed as a sum phase space
factor which is something calculable a
nuclear matrix element that is very very
hard to to compute and then what we are
interested in which is this neutrino
mass and a combination of neutrino
masses and mixings now the top runners
right now in the field our experiments
Gurda that uses germanium and EXO and
Camel intended uses Senan in particular
columns n has been has done an enormous
effort to and they have or they also had
a bit of luck to get this very very good
limit of this combination which
depending on the nuclear matrix elements
that are assumed would be between 0.06
electron volt and point 1 6 electron
volt
and I will come back to that in the last
in the last talk so let me now turn to
the last topic I want to discuss in this
lecture which is the phenomenon of
neutrino oscillations now as we have
said the fact that neutrinos are massive
implies generically that whatever we
produce in combination with an electron
muon and tau in a charge current
interaction is a combination of the mass
again States this makes any neutrino
experiment that produces a neutrino via
charge current interaction and the dead
seed we attract are an interaction and
in the philometor in flavor space
because neutrinos are so weakly
interacting that the nice counterpart of
this difficulty is that they can keep
coherence over very long distances so if
you produce that we know in flavor alpha
you are producing really a coherent
combination of these Messiaen States
which are the eigenstates of the
Hamiltonian in vacuum and therefore they
evolved picking up phases different
phases so at some distance away which
would usually call the baseline L if
there is some nonzero probability that
the state can be seen as a different
flavor okay this is the probability of
oscillation the general form master
formula for the oscillation probability
from a neutrino alpha to a neutrino beta
where you can see it is a superposition
of sinusoidal oscillatory functions with
wavelengths that are fixed by the
neutrino mass where differences and
amplitudes that depend on different
combinations of the elements of the
mixing matrix now there are many ways to
derive this oscillation probability
master formula in order of difficulty
one could consider neutrinos as plane
waves and just do simple quantum
mechanics this is the derivation you
essentially see
you know most test books you can do a
bit more work and consider neutrinos as
wave packets and again do it in quantum
mechanics you can even go to quantum
field theory and not talk at all about
whether neutrinos are in this state or
another they are just simply
intermediate States now of course the
result of these different derivations is
fortunately the same difference between
these derivations is whether they make
more or less explicit the two basic
ingredients that you need to have
neutrino oscillations and therefore they
are more or less susceptible to the
usual quantum paradoxes so the two basic
ingredients to get neutrino oscillations
is that you need some uncertainty in
momentum both at production and
detection this means that you must not
you must have a situation where you
cannot kinematically distinguish
neutrinos then the different messiaen
States because if that's the case then
they cannot be coherent li produced ok
the second requirement is that of course
these neutrinos are evolving in time at
some point it is clear that they will
lose coherence ok so again when
coherence is lost there cannot be
oscillations anymore so I will just go
very quickly through the first
derivation which is the standard
textbook derivation and I will say a few
words about the second one to show you
how these two ingredients become
explicit in this second derivation so so
as I said you have an experiment in
which you have a source of neutrinos
that produce some flavor alpha and this
flavor alpha as we just said is a
combination through the pmns matrix of
the math-science states that here we
assume to have
some common momentum P now this states
are eigenstates of the of the three
Hamiltonian with eigenvalues given by
the energies which of course this
particles are on shell so they satisfy
the relativistic relation and of course
the Hamiltonian is what determines the
time evolution of the state each of the
stages picks up a face which is e to the
minus the energy of that state times the
delta T now we want to compute what is
the probability that this new state is
in a different flavor beta so we write a
state new beta which is as the same as
this one but with alpha substituted by
beta and the probability is nothing but
the scalar product of these two states
squared now if you do that and then you
do a relativistic approximation so this
neutrinos are route relativistic their
masses are much smaller than their
energies and therefore you can expand
this energy difference to leading order
in a mass of these two states and you
can also assume they move at the
velocity of light so that you can
exchange time by the distance because
this is how we usually do this
oscillation experiments and there you
get the master formula there is so
obviously some well-founded PD system to
this derivation which often is the
source of all these quantum paradoxes
that people have discussed since a long
time now why did I chose to choose the
same momentum for all the mass alien
States why did I choose plane waves if I
talking about neutrinos that are
produced here undetected there and
therefore they are localized they cannot
be strictly speaking plane waves why do
I have to do this an aesthetical t2l
conversion so it turns out that all
these questions can be solved by
slightly more sophisticated analysis in
which you consider neutrinos as wave
packets
okay so I will just sketch the way to do
this but I will not go through all the
algebra of course you can check the
transparencies if you are interested so
here again we have a neutrino flavor
alpha that has been produced as a
sometimes IDO and at some position x0
and this is a superposition of
math/science States which are no longer
plane waves because to ensure this
localization in space-time you need it
to be some kind of wave packet so you
can have a wave packet for each of these
mass state massive states which in
general will be characterized for
example if you have some Gaussian wave
packet by some average momentum of the
wave packet and some momentum
uncertainty which is of course related
to in an inversely proportional way to
the space with of of this wave packet
now the this state at the detector the
state that is detected is again a wave
packet now the detection time and
position is different if T and L and
therefore we essentially write exactly
the same thing with a wave packet and
now in principle might be different
because it depends on the wave which we
will detect these neutrinos now as I say
I will spare you with with the algebra
but if you assume Gaussian wave packets
you can go quite far in having
analytical expressions and finally what
you do is so essentially in order to
compute the amplitude for this process
you just take again the scalar product
the probability will be the square of
that of that amplitude and then you
integrate over time because time the
time difference between production of
the texture is not a measured quantity
okay now if you do that you end up with
again something looking like the master
formula but with two important
differences which are these two
modulation factors okay
one of them ensures that if the bass
line is too long compared to this
quantity which I call a coherence then
the oscillations will be essentially
financially suppressed now what is this
el coherence well this has a very nice I
mean of course you can do the algebra
you get some expression but the physical
interpretation of this quantity is
simply that is the minimum distance at
which the wave packets have traveled
such that their positions are separated
away by one Sigma okay therefore at that
point the hiddens can no longer hold and
oscillations are suppressed so this is
the coherence requirement and there is
another modulation factor which says
that if you have too small so if your
uncertainty in momentum is too good okay
then at some point you will be able to
see to detect the kinematical effects of
the mass of the different messiaen
states okay in this case when this
happens this factor again certain
exponential suppression factor that will
kill any oscillation term so this is a
much nicer result unfortunately in all
this process we have lost the
normalization so I no longer have an
equal sign here
because it's essentially impossible with
this extremely difficult to to get right
the normalization in this way okay so
the way to handle that is of course you
know unitarity must hold so the
oscillation probability from an achene
oh alpha to beta should add up to 1 if I
see if you sum over all final flavours
and this is how you impose them that the
result actually should be this one
without any pre factor now you can do
better than that and not require this
you
eyetality this for that you need to go
to quantum field theory but I will not
cover this because of lack of time so I
hope I I try to pass the message that
this is a robust formula has been tested
in many different ways and I think seen
any deviation for such from such
situation would be extremely difficult
and very dependent on the specific
condition so very hard to go beyond that
so what is the so when we talk about
oscillation probabilities we might have
either different flavor in the initial
and final state this is what we call an
appearance probability or is the same
and in this case we call it a
disappearance or survival probability
one another way to write this formula by
using a unit ality of this matrix is
written here and I just want to show you
this because in this way of writing it
you see that there is a term in the
oscillation probability which it depends
on imaginary part of some combination of
mixing matrices of the mixing matrix
elements okay this means that there is a
sensitivity in the oscillation
probability to complex entries in the
matrix which are basically the CP faces
and in fact if we had computed the
oscillation probability for anti
neutrinos instead of neutrinos we would
have found that this term changes sign
so this means that neutrino oscillations
can be sensitive to CP violation and a
way to enhance this is by looking at the
compare comparing the oscillation
probabilities for neutrinos and
antineutrinos
as you will see much more tomorrow now
of course you have you probably you have
not seen this formula very much if you
are not working in neutrino physics what
formula you have seen most probable we
is the one corresponding to two
neutrinos in which case
is much simpler because there is only
one complex while sorry one unmixing
angle and one del x square this is the
appearance probability and one minus
that is the disappearance probability as
a function of the baseline this is how
the probability looks like is a periodic
function with a period which is
correlated with password differences and
an amplitude that is correlated with a
mixing angle okay now normally neutrino
oscillation experiments do not change or
very few of them have a change in the
baseline because you have to separate
the source from the detector and this is
difficult certainly not in laboratory
experiments so usually however it is
quite usual that you treat of
experiments in both some continuous
spectrum in energy and therefore one can
compute the oscillation probability as a
function of energy and the same formula
gives this curve where the first this
first maximum again gives you
information the position in energy of
this maximum gives you information on
del x square and again the amplitude is
related to the mixing angle now it's
clear that if you want to do an
experiment that tries to pin down both
the mass square difference and the
mixing angle you need to tune your
experiment with an energy neutrino
energy and baseline such that this
condition holds because if you go to
much larger values this means that the
oscillation this oscillation face is
suppressed so if you expand on it you
find that you can no longer separate the
two parameters theta and del x squared
you have essentially a degeneracy if you
go to the other limit to the first
oscillation regime essentially what
happens is that the oscillations are too
fast and you cannot low lorem will
resolve them so what you measure is some
kind of average and the average of this
oscillation probability is just 1/2 of
sine squared 2 theta which means that
you lose completely or sensitivity to
Delta M square now this fastest
and regime is actually exactly
equivalent to propagation without
coherence okay so what this means is
that the oscillation formula has inbuilt
that it has to get to the right limit
when coherence is lost okay so this fast
oscillation regime is equivalent to loss
of coherence
okay so as you will hear tomorrow two
distinct frequencies have been precisely
measured one of them which correspond to
the smaller Delta M Square has been
measured used in reactor neutrinos
shooting at hundred kilometer distances
while the other one the other so this
would be a del x weight of about 10 to
the minus 4 electron volt square I a
much larger one del x square of about
2.5 10 to the minus 3 electron volt has
been measured both using accelerator
neutrinos shooting at distances of a
thousand kilometers for energies in the
GeV and also reactor neutrinos which are
maybe energies shooting much shorter
than in this case now before going to
how you would interpret these three
neutrino oscillations in this master
formula let me turn to an important
topic which is that many neutrino
experiments what I just mentioned refer
to neutrinos propagating in vacuum okay
however many neutrino experiments
involve neutrinos propagating in matter
for example accelerator neutrinos
usually have neutrinos crossing the
earth crust
atmospheric neutrinos come can cross the
earth in many in many ways depending on
definite angle solar neutrinos are
produced close to the center of the Sun
they have to exceed the Sun so they
propagate in a very dense environment of
the Sun and it turns out that neutrinos
propagate in a matter which is form of
electrons
protons and neutrons let's an index or
of refraction
you took Oh hidden forward scattering
that can strongly affect the oscillation
probability and the most important
effect there is coming from this diagram
the charge pattern diagram because the
neutral current we just shift all the
flavors in the same way so it will not
affect oscillations at least in this
standard scenario however the chest
currents select the electron the
neutrino electron of the others and the
most important quantity for the size of
this effect is the number density of the
electrons in the matter that neutrinos
are crossing now what you can show that
actually the fact that neutrinos cross
matter modify the dispersion relation
the relation between energy and momentum
is no longer the mass as we used to
derive all the oscillation formula but
now if it gets a term coming from this
matter potential which actually has a
different sign for neutrinos and
antineutrinos and this is because matter
is of course not
CP invariant as it's made of matter and
not antimatter now what is amounts to is
that in fact the oscillation
probabilities that I wrote can still be
applied to this case however now the
mixing angles and masses are effective
ones they come from the diagonalization
of this combination and not just the
bare mass matrix and so the in general
they will be dependent not only on the
vacuum parameters but also on this
matter potentials and the neutrino
energy sitting here so for the case of
two families this is relatively easy to
work out and what you find is that the
mixing angle for neutrinos crossing some
matter density with a constant density
given by this number density of
electrons the there is a condition this
well if this holds this condition holds
which involves some combination of the
masses and mixings and this density of
electrons and the
then this term here disappears it can
only of course hold for one of the signs
but for one of the signs if it holds
then it doesn't fall for the other but
for the one that holds this becomes zero
and then you can see that the mixing
angle these two terms cancel and you
just get one no matter what about you
mixing angle is okay now this is of
course understood in terms of the famous
MSW resonance so imagine you would have
that your mixing angle would be exactly
zero okay so then the math-science
states in matter as a function of the
matter electron number density will look
like this for either neutrinos or anti
neutrinos okay so you would have a
complete global shift of both massagin
States by the matter density but then
the electrons because of the charge
current coherent scattering will get an
extra contribution and then at some
point these levels might cross okay now
if you have a small mixing angle no
matter how small of course we know that
physically this never happens there is
no level crossings but instead there is
a resonance okay so the heavier state
that for zero matter density is mostly
the moon for a small mixing angle in
vacuum when you go to very large matter
densities is mostly an electron because
this is this state here okay and the
other way around
okay now the position of this closed
closest point is precisely the resonant
condition as I already mentioned it will
hold for neutrinos antineutrinos but not
for both and it will depend on the sign
of this combination whether it is for
neutrinos and antineutrinos
and as you will see this is one of the
handles we have to actually pin down one
of the unknowns in neutrino physics
presently which is the hierarchy in
which the
massagin states are ordered now this MSW
effect is however very essential in
understanding or interpreted solar
neutrinos solar neutrinos don't move in
constant matter the number density of
electrons in the Sun is very highest at
the center of the Sun and then it
decreases exponentially as we go towards
the the radius I mean we exit the Sun so
now if you have this kind of situation
then you can compute the instantaneous
mass eigenstates as a function of radius
okay
so of course as a function of radius if
we are in the center of the Sun we will
have a very large density we are here
and then as we decrease this density as
we exit the Sun the instantaneous mass
eigenstates are changing now if this
variation is slow enough and it is in
the Sun for the matter parameters we
know then it is a good approximation to
the so called adiabatic approximation is
a very good one which says that if a
state is in a massagin state at some
given radius then it will remain it will
let see the Sun in the same higher state
it will not jump okay so the probability
that a neutrino electron producing the
Sun will remain an 18 electron it's
easily then computed from the
probability that the neutrino that in
the center of the Sun is in a neutrino
electron is in a given message instead
and then probably times the probability
that this message has stayed in the when
you exit the Sun is added to inner
electron and some incoherently over all
the states so there is no solutions here
okay
there is no coherence no solutions here
however what happens here is the
following
now as I say imagine that you have a you
start with in the center of the Sun with
a density which is above resonance so
you are
somewhere here now if you produce a
neutrino electron this neutrino electron
is mostly the heaviest guy okay
but if if the diabetic approximation is
good and you are there then you will
exceed that Sun beam mostly immune okay
so the oscillation probability the
survival probability for an 18 electron
in these conditions is basically the you
can actually compute it and you find
that this is actually the vacuum angles
were sign of the vacuum angle squared so
the smaller the angle the most clean
distance the electron neon transition is
now if the density in the center of the
Sun is however below resonance then a
neutrino electron is mostly the lightest
state and it will remain so as as you
exit the Sun now what is the electron
density are we above or below resonance
well this is a question that depends on
the energy of the neutrino it's not just
depending on the the density of the Sun
but it also depends on the energy of the
neutrino because of this condition you
can see there is an energy here and now
the solar neutrinos have a spectrum as
you will see tomorrow that encompasses
several orders of magnitude in energy
and it turns out that if you you have
neutrinos coming from the Sun that are
energetic enough to be above resonance
and some that are below resonance okay
so for those experiments that they take
the higher neutrinos you would expect to
find a survival probability which is
basically this the sine squared theta of
vacuum well if you are below resonance
the oscillation probability is the one
in vacuum very approximately okay so you
don't have information on del x square
by looking only here or here but of
course you have information on the del x
square by knowing where this step
happens okay so this is how many solar
neutrinos get a information on del x
square so here is what you have measure
over many years by this solar neutrino
experiments sister survival probability
as a function of energy and as you can
see there are experiments that are
measuring the higher energy range and
therefore from them we can essentially
read from this curve what is the vacuum
angle relevant for these oscillations in
the Sun okay so in the last five minutes
I will just all what I've talked about
here essentially are I've talked about
neutrinos of two families oscillating
okay
however we know as I already mentioned
that two different frequencies have been
measured in data using solar and
Atmospheric spirimals and many others as
you will see tomorrow and therefore we
cannot longer talk about two families
because we need at least two distinct
mass differences okay so we can assign
in an arbitrary way the largest del x
square to the difference of the two or
three states and the smaller one to the
one two states and this is completely
arbitrary of course is just a labeling
of the states now the mixing of course
involves many mixing angles not just one
okay now it turns out that in spite of
this complexity you can see that
actually the the solar and atmospheric
oscillations the couple has two by two
mixing phenomena quite accurately
provided and because there is a
hierarchy in the del x squares which is
larger than ten and one of the angles in
this matrix turns out to be small so
let's see well how this happens imagine
you have an e over L an experiment with
a new barrel that Tunes to the larger
del x square okay and this del x square
is much larger than the other one so you
can essentially neglect the oscillation
terms that go with this frequency with
this longer frequency and and remain
with the ones that have the
significant oscillation frequency and
this is what you find in that case okay
so of course the all transitions can
take place okay with the same dependence
or since oil dependence on the baseline
but with different combinations of
mixing angles in in front okay
now imagine that besides that we set one
of the angles to zero then what happens
is that essentially we turn into a two
familiar scenario two of the
probabilities are 0 because of this
angle being 0 and there is only one that
survives and has exactly the same form
as if there were only two families ok so
this allows us to interpret whatever we
measure at leading order in this
atmospheric range as coming from these
two parameters ok now what happens if
you now tune your experiment yuri over L
to be at the longer the longer
wavelength or the smaller del x squared
then what happens with the largest
oscillation is that it gets averaged out
okay so if you do that approximation and
average out all the fast frequencies
then you find that the for example the
survival for probability for neutrino
electrons has been spawn now it only
depends of course on the del x squared
that you are not integrating out and
again if you take theta 1 3 this angle
to 0 you recover exactly the 2 by 2
mixing formula again ok so if you do
experiments in the solar range at living
order this is well described by a
two-family oscillation where you
interpret the 2 parameters that you
measure their order to identify the 2
parameters that you measure there with
these two in the full three neutrino
mixing scenario now we know now that we
can we have enough precision as you will
see tomorrow in neutrino experiments to
actually look at this ooh bleeding
effects which implies that
nutrino mixing and oscillations are
really a three family phenomenon and you
can no longer do approximations such as
these to interpret neutrino oscillation
data and you will hear much more
tomorrow and perfect timing thank you so
is there any question
to comment about wave packet in more of
the practical cases solar reactors cases
you have beams of Medina's forgiveness
of neediness
which a Nazi statistical and quantum
mechanical uncertainties so wave packets
becomes completely unnecessary and moral
and even if you work continue to work on
wave packet language
some kind of mind field you can make
many mistakes working wave packet
approximate wave packet language and
just I don't know anything with
statistical uncertainty from the quantum
mechanical uncertainty waste party to
say to me honey yeah I mean my comment
on the wave packet was not of a
practical nature in all ex neutrino
experiments that you can think of
certainly we are extremely far away from
from being sensitive to any of these
effects that show up in the wave packet
approximation okay so it's not a
question of practical relevance for
experiments it is a question of
principle okay
that if you know the usual approximation
that you see in most textbooks which is
this plane wave approximation often
leads to one of paradoxes simply because
you are not formulated the problem
properly and the wave packet provides
that formulation which explicitly makes
explicit when oscillations should
disappear however I agree with you there
is no relevance of this wave packet
treatment for real experiments
well yeah good question I mean of course
when when I when I was I mean when I do
I say this look at this sign I mean this
huh doesn't have to be exact or in any
way okay it's an order of magnitude
statement you want to be close to this
being satisfied if you want to really
sensitive to the frequency that is
driven by this Delta N squared now of
course if you would have a monochromatic
neutrino beam that would have many
advantages I mean especially if you
would be able to tune the energy as you
please because this will I mean this
will help you in many ways
because usually what we measure in
neutrino experiments is though is not a
neutrino energy but the lepton energy
and reconstructing a neutrino energy is
a very difficult task so if you would
know the energy of the neutrino like
would be very very much simple simple in
experiments but that doesn't mean that I
mean this is this should be understood
as an order of magnitude estimate okay
yeah I mean okay there I mean most of
the beams that the neutrino beams that
have been used in experiments are all of
this you know conventional the so-called
conventional neutrino beams which come
from PI in the case and it's very
difficult to know the flux and
everything from first principles okay
now there are other alternatives to this
where you would maybe not get up a
monochromatic 'el monochromatic neutrino
beam but at least to know much better
what your flux is okay and these are
this neutrino factory where you get
neutrinos from neon decay all these beta
beams where you get neutrinos from
radioactive accelerated isotopes now
there is also I think a proposal for a
monochromatic neutrino beam using some
beta decay but I don't know exactly what
the status is there it's probably quite
hard to get intensities you would need I
would say but I mean ideas like that
exist yes
no more questions okay so you will hear
much more tomorrow about these beautiful
experiments and see you on Friday
Thanks
[Applause]
[Music]