Fermilab physicists really care about the
mass of the W boson. They spent nearly a decade recording collisions in the Tevatron
collider and another decade analysing the data. This culminated in the April 7 announcement
that this obscure particle’s mass seems to be 0.1% heavier than expected. So why do we care? 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 an even deeper knowledge. The FermiLab
discrepancy is a tantalizing hint of the latter. —- This 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 weird in
that 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 mass itself. To get to all 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, oscillate in different ways. A quantum field just represents one of these
modes. And these wiggles are quantized - they come in discrete packets of energy that can move
around - and those are the particles of a 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 x-y-z axes, or where you put the zero point of
your angles in polar coordinates. We saw in our episode on Noether’s theorem that
these symmetries lead to conservation laws. In quantum mechanics, such a “redundant degree
of freedom” leads to a gauge field. We’ve seen an example of this. The exact phase of the quantum
wavefunction from one point in 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 electromagnetic force. So one of the fundamental forces arises from
a symmetry of nature - in this case the fact that the laws of physics are invariant
under changes in local phase. The set transformations that can change local phase are
an example of a symmetry group - in this case “unitary group 1”, or U(1). These
transformations can be represented with just “one” number in this case rotation of
phase angle vector of “unit” length. See this episode for the nitty gritty of all
of this. This is a refresher, remember. 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 has 3 force carriers that
look awfully like the weak force bosons. So far so good except that the predicted particles
are massless, while the real weak force bosons are, as I mentioned, pretty hefty. That’s 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 gonna
forge on anyway and hope this gets sorted out. Another problem with this first pass at the weak
force is that it has absolutely no connection to electromagnetism. Its bosons do have properties
that look like weak isospin 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 U(1) and SU(2) symmetries so that they apply at the same time. We call this
combined symmetry group U(1)xSU(2). 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 isospin and hypercharge are tightly coupled,
and their combination defines electric charge. Is it time to give up on this symmetry stuff yet? 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 end up all 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 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 valley.
Now the current state of the system has a broken symmetry, even if the
symmetry of the landscape remains. So let’s just see if we can break
the symmetries of the universe in a similar way. The equivalent of the
simple valley exists. A quantum field can oscillate around some “zero-point” value
like a ball rolling back and forth in a valley. The “walls” of the valley are just the
potential energy - the energy stored when the field value 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 this field are just oscillations
of the field strength across the lowest point, where the field strength is zero.
But if there are no particles around then 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. OK, don’t freak out. It’s not on the test. This
is a Lagrangian - something we covered previously. 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 is made of various powers of the field strength that represent ways the field’s particles
can interact. For example, this phi^4 term says the particles of the field interact with particles
of the same field with strength lambda. This phi^2 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^2 term. And that means this isn’t a gauge field. The gauge field is a new 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 compone nt - 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 U(1) symmetry. Repeating
our electromagnetism trick means requiring local U(1) invariance. 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 in the Lagrangian that allows the angle of this rotational degree
of freedom to vary. Call that angle theta. Oscillations in that field would be a gauge
boson. You can think about those oscillations as the theta 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 this potential 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
with this extra field component to give us this 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 U(1) 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 on top of
this little hill, but then quickly roll 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 the perhaps least
interesting thing about 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 is zero. 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. This is the Higgs boson itself. Second weird thing: there’s not just a single
vacuum state - there’s a rin g 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
Goldstone boson and it’s massless because the valley is flat - there’s no energy
differential. However it isn’t a gauge boson because location around that valley represents
a real physical difference in the field state. OK, let’s try to bring all of this together.
We want to combine the weak and electromagnetic interactions, so we need simultaneous local
U(1) and SU(2) symmetry. Let’s just do U(1) 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 this angle theta matters, but exact values don’t. So we’re
going to demand local U(1) invariance and come up with a gauge field that shakes out 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 U(1) gauge
field. Both are oscillations around the valley. In the Lagrangian they get
lumped together into a single field that still looks like a gauge field with a single gauge boson. The gauge field eats the
Goldstone boson. But when we do this, the new combined field has field-squared
terms in the Lagrangian, which is weird because those are mass terms - except
neither the original gauge boson nor the Goldstone boson that it just ate had any
mass. So where does this mass come from? It’s because the gauge boson is now coupled to the
Higgs field. The Goldstone boson that is now part of the gauge boson is coupled to the Higgs field,
which means 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 electroweak U(1)xSU(2) invariance on the true Higgs potential
gets you three Goldstone bosons that are eaten by 3 of the 4 electroweak gauge bosons. Those gain
mass and become the two W and one Z bosons 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 as electromagnetism, while the W and Z slog on through the mire of their coupling
with the Higgs field. Their mass shortened their lifespans, and so enormously reduces 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’s for another time. But what about this new measurement of the W boson mass? Mass
results from interaction with the Higgs field, but also all of the 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 that 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 say thank you to
everyone who filled out our episode ideas survey and if you haven't yet, there's still time. It’s just a short 8 question survey that you can use to share your episode ideas, tell us what topics you’d like us to explore, and in general tell us a little bit about you and what you'd like to see happen in the Space Time universe.
There’s a link in the description.
I felt so lost on this one. I guess all the only slightly beyond what I can maybe understand culminated here into not a chance
I understood all of the words. Just not in those combinations.
This one was a real head scratcher
I feel like I’m no longer the target audience for this show.