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.