fermilab physicists really care about
the mass of the w boson they spent
nearly a decade recording collisions at
the tevotron collider and another decade
analyzing the data this culminated in
the april 7th announcement that this
obscure particle's mass seems to be 0.1
percent heavier than expected so why do
we care
well because understanding why this
particle even has mass was one of the
most important breakthroughs in our
understanding of the subatomic world and
because measuring its precise mass
either doubles down on our current
understanding or reveals a path to even
deeper knowledge
the fermilab discrepancy is a
tantalising hint of the latter
[Music]
the timing of fermilab's discovery is
weirdly convenient over several previous
episodes we've been building towards an
understanding of how the forces of
nature are unified
the most powerful clue driving this is
the weirdness of the weak force in
particular the particles that carry this
force its w and z bosons have a property
that we once thought no force carrying
particle should have
they have mass
the w bosons are especially weak and
they also have electric charge this
allows the weak force to trespass on the
province of electromagnetism suggesting
a connection between the two this
connection hints at a unification of the
forces of nature
the path to that unification leads to
the higgs mechanism which not only
explains the mass of the weak bosons but
teaches us about the nature of massey
itself to get to all of that good stuff
we need a bit of a refresher on the
episodes that led to this
fields and forces and symmetries and all
that similar to how the fabric of a drum
has vibrational modes so does the fabric
of reality every point in space can
wiggle twist and oscillate in different
ways a quantum field just represents one
of these modes
these wiggles are quantized they come in
discrete packets of energy that can move
around and those are the particles of
the field one special type of field is
the gauge field these arise from the
fact that physics often doesn't care
what coordinate system you use the laws
of physics are symmetric under certain
transformations for example physics
works the same no matter where you
decide to center your xyz axes or where
you put the zero point of your angles in
polar coordinates we saw in our episode
on notice theorem that these symmetries
lead to conservation laws in quantum
mechanics this sort of redundant degree
of freedom leads to a gauge field
we've seen an example of this the exact
phase of the quantum wave function from
one pointed space to the next local
phase doesn't affect measurable
quantities only relative phase matters
when we enforce this requirement we find
that we have to add a new quantum field
to the schrodinger equation that lets
the universe counteract these phase
shifts
that gauge field turns out to be
electromagnetism
and oscillations in this field are the
photon our first gauge boson and carrier
of the electromagnetic force so one of
the fundamental forces arises from
asymmetry of nature in this case the
fact that the laws of physics are
invariant under changes of local phase
the set of transformations that can
change local phase are an example of a
symmetry group in this case unitary
group 1 or u1
these transformations can be represented
with just one number in this case
rotation of a phase angle vector of unit
length see this episode for all of the
nitty gritty this is just a refresher in
the next episode we tried the same trick
to explain the weak interaction as
arising from symmetries we saw that we
could invent a pair of totally abstract
degrees of freedom and demand that the
universe be invariant to transformations
of these we call this symmetry group
su-2
for reasons
that requirement gave us a new gauge
field that had three-force carriers that
look awfully like the weak force bosons
so far so good
except that the predicted particles are
massless well the real weak force bosons
are as i mentioned pretty hefty
and that is a huge deal breaker actually
massive bosons break gauge symmetries
according to something called
goldstone's theorem all gauge bosons are
massless
so so far so bad
but we're going to forge on anyway and
hope that this all gets sorted out
another problem with this first pass at
the weak force is that it has absolutely
no connection to electromagnetism is
bosons do have properties that look like
weak iso spin and weak hypercharge but
no electric charge
in our universe these three quantities
are sort of locked together only taking
on certain values relative to each other
to see if we can duplicate that in our
theory we need to combine the u1 and the
su-2 symmetries so they all apply at the
same time we call this combined symmetry
group u1 cross su2 the resulting gauge
field still has bosons that look a bit
like the photon and the three weak force
bosons but the latter are still massless
and the resulting charges are completely
unconnected to each other they're all
free to be whatever they want unlike the
real universe where iso spin and hyper
charge are tightly coupled and their
combination defines electric charge
so is it time to give up on this
symmetry stuff
well there's one more long shot clue
i said that massive bosons break the
gauge symmetry but what if that's okay
we've also talked about the idea of
symmetry breaking before using this
example of a bunch of bar magnets
these magnets have high temperature
which makes them move randomly but as
the system cools down this random
thermal motion gets overpowered by the
magnetic interaction and they all end up
aligning
the equations of magnetism don't start
out with a preferred orientation but in
certain conditions namely cold ones the
system chooses a preferred direction
this is an example of spontaneous
symmetry breaking so if a bunch of
magnets can live in a state that
violates the symmetries of their ruling
equations
maybe the universe can too
here's another analogy consider a ball
rolling back and forth in a valley the
equations describing its motion are
symmetric between left and right because
the valley is symmetric
now imagine it wasn't one valley but two
valleys with a hill in the middle the
system is still symmetric but if the
ball starts at the top of the hill it
will randomly roll down into one of the
valleys
now the current state of the system has
a broken symmetry
even if the symmetry of the landscape
remains let's see if we can break the
symmetries of the universe in a similar
way the equivalent of the simple valley
does exist a quantum field can oscillate
around some zero point value like a ball
rolling back and forth in the valley
the walls of the valley are just the
potential energy the energy stored when
the field moves away from the zero point
trying to pull it back to the center
graphing this the vertical represents
potential energy and the horizontal
represents the field strength particles
of the field are just oscillations of
its field strength across the lowest
point where the field strength is zero
but if there are no particles around and
the field just sits at the lowest point
we call that the vacuum state easy stuff
right
in that case you won't mind seeing
the math
okay don't freak out it's not on the
test
this is a lagrangian something we've
covered before
it's really just the difference between
the kinetic energy and the potential
energy in the field
our plot was of the potential energy
part this particular lagrangian
describes a simple quantum field made of
massive particles which interact with
each other let me talk you through the
hieroglyphics firstly phi is the quantum
field itself it just means there's a
numerical strength of the field
everywhere in space
the potential energy part is the shape
of the field and it's made of various
powers of the field strength
representing ways that the fields
particles can interact for example this
phi to the power of four term says that
the particles of the field interact with
particles of the same field with
strength lambda
this phi squared term represents the
field interacting with itself that
self-interaction is what leads to the
property of mass
gauge fields
shouldn't interact with themselves so
shouldn't have a phi squared term and
that means this probably isn't a gage
field
the gauge field is a thing that comes
from the degrees of freedom within this
field this particular lagrangian has the
simple symmetry that it's the same if
you reflect it around the y-axis
if we complicate things by adding a
second field component phi 1 and phi 2
we get a parabolic bowl
if the current state of the field is at
the bottom of the dip then it has a
single continuous degree of freedom in
that you can rotate this thing and
nothing changes that would be a global
u1 symmetry
repeating our electromagnetism trick
means requiring local u1 invariants we
need the laws of physics to still make
sense if there are rotations from one
point in space to the next
that means adding a new gauge field to
the lagrangian that allows the angle of
this rotational degree of freedom to
vary oscillations in that field would be
a gauge boson
you can think about those oscillations
as the angle twirling around freely and
it doesn't need any minimum energy to
oscillate so the gauge boson is massless
on the other hand the particle of the
original field needs a rest mass energy
to be able to oscillate up and down the
potential walls so while this potential
has its own massive particles it doesn't
give us massive gauge bosons
let's make a slight change we're going
to switch the sign in front of the mass
term then the potential would change to
this just like the two valleys we saw
earlier but now the second fuel
component gives us this 3d shape it's
called a mexican hat potential
this potential shape is the heart
of the higgs field
it's still a symmetric potential it has
the same global u1 symmetry as the
original but if the universe suddenly
transitions from the old potential to
this one
then we have a problem
the field strength would find itself at
the top of the little hill but then
quickly rolled down in a random
direction
the new state of the field would not be
symmetric to the same rotations because
different points around this valley
correspond to different physical states
we say that the symmetry is
spontaneously broken the current state
of the field is a state of broken
symmetry even if the global field shape
keeps its old symmetry this is just like
how the field of magnets can be in a
state of broken symmetry even though the
overall equations of magnetism have not
changed the transition from the central
bump to the valley is an example of
vacuum decay and we saw how this can
lead to the creation of cosmic strings
in a recent episode but cosmic strings
are perhaps the least interesting part
of this process once the field has
reached the base of this new minimum we
have a new stable vacuum state
but this state is very different from
the original for two reasons
the first is that the lowest energy
state the vacuum state isn't where the
field strength itself is zero
so this is called a non-zero vacuum
expectation value
the original massive particles of the
field now oscillate in the radial
direction
but no longer centered on the zero field
value
and this is the higgs boson the second
weird thing there's not just a single
vacuum state there's a ring of valid
states the universe will just have
chosen one state randomly
but the field can also oscillate along
the base of the valley in what we'll
call the theta direction
the resulting particle is called a gold
stone boson and it's massless because
the valley is flat there's no energy
differential
however this is not a gauge boson
because location around the valley
represents a real physical difference in
the field state okay let's bring all of
this together
we want to combine the weak and
electromagnetic interactions so we need
simultaneous local u1 and su-2 symmetry
let's just do u1 because that gives us
the basic picture because the potential
is the same all around the valley it
shouldn't be possible to tell where in
the valley you are
it matters if two adjacent patches of
the universe are in different parts of
the valley the relative difference in
the angle theta matters but exact values
don't
so we're going to demand local u1
invariants and come up with a gauge
field that shakes out the shifts in our
arbitrary choice of the zero point of
theta but now this gauge field finds
itself in a much more complex lagrangian
with this mexican hat potential
weird stuff happens when the gauge field
couples to the particles of that
potential first of all we see that the
goldstone bosons which are just
oscillations in the theta angle
can be absorbed into this u1 gauge field
both are oscillations around the valley
in the lagrangian they get lumped
together into a single field term that
still looks like a gauge field with a
single gauge boson
the gauge field sort of eats
the gold stone bozo but when we do this
the new combined field has a field
square term in the lagrangian which is
weird because those are mass terms
except that neither the original gauge
boson nor the gold stone boson that it
just ate had any mass
so where does this mass come from
ultimately it's because the gauge boson
is now anchored to the higgs field the
goldstone boson that is now part of the
gauge boson is coupled to the higgs
field and that means that the gauge
boson is also so coupled
ultimately this happens because of the
non-zero vacuum state of the higgs field
that little bit of higgsiness everywhere
refuses to cancel out in the lagrangian
giving us our mass term all of this was
a simplified explanation i know
horrifying right imposing the full
electro weak u1 cross su-2 invariants on
the true higgs potential gets you three
goldstone bosons that are eaten by three
of the four electro weak gauge bosons
those gain mass and become our two w
bosons and one zed boson of the weak
interaction the fourth boson manages to
escape unscathed and massless becoming
the photon that we know and love
it flew free of its heavier cousins the
independent mediator of a part of the
old electroweak field what we now
experience is electromagnetism
while the w and z slog on through the
mire of their coupling with the higgs
field
their mass shortened their life spans
and so enormously reduce their range
weakening the force that they mediate so
that's where we are today this is the
higgs mechanism the higgs field also
gives mass to the matter particles the
fermions but that is for another time
but what about this new measurement of
the w boson mass
well mass results from interaction with
the higgs field
but also from other subtle interactions
that a particle can undergo the
predicted mass of the w boson takes into
account all standard model particles
that could have a ghostly presence as
virtual particles in the energy field of
the boson
the fact that fermilab measured a larger
mass than was predicted suggests an
unknown particle or particles flickering
around the w boson the discovery of the
higgs boson 10 years ago verified the
idea that the underlying symmetries of
nature explain and unify some of the
forces of nature
perhaps new particles will lead us to
new clues about yet deeper unifying
symmetries
of space-time hey everyone comments will
return next week but before we go i
wanted to thank those of you who filled
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