What do you get if you take something  that’s infinitely massive and   combining with something else that’s negative  infinitely massive?

You get a single electron,   at least that’s what it looks like in our most  precise way of describing the quantum world.

Today’s episode is an introduction to one  of the most important problems in physics.

It’s the Hierarchy Problem, and its important  because its solution is going to lead us beyond   our current theories.

The Hierarchy Problem,  put simply, is that we don’t understand why the   mass of the Higgs boson is so small.

From our  understanding of quantum field theory and the   standard model of particle physics, interactions  between the Higgs and the quantum vacuum should   drive its mass up to enormous values.

That  would be catastrophic for our universe,   and so we should be grateful that the  small Higgs mass is somehow protected.

We   just don’t know how it's protected.

But to  fully appreciate why it’s surprising that   we haven’t been able to find the mechanism for  this protection, and why the capital-H hierarchy   problem is such a big deal, we’re going to  start with the small-h hierarchy problem   involving the electron.

The electron hierarchy  problem existed before quantum mechanics,   and was solved by quantum mechanics, once we  understood the theory well enough.

By giving   you a sense of how the electron hierarchy  problem was solved, I think you’ll really   appreciate why the Higgs Hierarchy Problem is so  problematic, and what its solution might mean.

We’re building on our last episode here.

Pretty heavily really, so this is a rare   time I’ll recommend watching that first if  you haven’t already.

In that episode we asked   what an electron looks like if we zoom in all  the way.

We discovered that it was ultimately   impossible to say exactly where the electron’s  mass and charge lived.

These properties appear   distributed between the “true” or bare  electron, and the frantic activity in the   quantum fields around it.

Well, this activity not  only makes it difficult to pin an electron down,   it also makes it difficult to fix the mass  and charge to the sensible small values that   we’ve measured.

Instead, this activity tends  to blow these values up to unnatural levels.

But this issue also exists in classical physics.

We saw last time that the potential energy   in an electron’s classical electrostatic  field has a mass equivalent 20,000 times   larger than the measured mass of the  electron, and that’s assuming that an   electron is as large as it could possibly  be according to our best measurements,   at around 10^-17m.

And if the electron is really  point-like then the mass of the field is infinite.

That can’t be right—and the most obvious thing  we did wrong is to assume that classical physics   applies all the way down to such small distances.

The correct description of the world on such small   scales is quantum mechanics—in particular,  quantum field theory—QFT.

In the case of the   electron interacting with the electromagnetic  field, the relevant version of QFT is quantum   electrodynamics, QED.

We’ll use QFT and QED  for most of today’s discussion.

But even in   the classical description we can get a glimpse  of the solution to the enormous field mass,   and also of the even bigger  problem that follows this solution.

Borrowing terms from QFT, let’s call the “true”  mass of the electron the bare mass, and the energy   in its surrounding field the self energy, and  self-energy mass for its E=mc^2 mass equivalent.

When we measure the electron mass, we never  measure the bare mass—we measure the sum of   the bare and self-energy masses.

We’ll call this  the dressed mass, again from QFT because the bare   electron is dressed in quantum fluctuations.

The  measured 511 eV electron mass is its dressed mass.

This equation equating the dressed to the bare  plus self-energy masses can work perfectly well,   even with a tiny dressed mass of 511 eV and  an enormous or even infinite self-energy   mass.

All we need to do is to give the  electron a bare mass that’s also huge,   and negative and just a hair smaller in  magnitude than the self-energy mass.

In fact,   you could even fix an infinite self-energy mass  with a negative infinite bare mass—infinity minus   infinity-minus-one equals one!

No problem!

This  process of adjusting the internal contributions   towards a measured quantity to eliminate  infinities is called renormalization.

It   was first applied in cases like the electron  before quantum mechanics was even invented,   though has since become a key tool in  quantum field theory in particular.

It sounds a little hokey because it assumes that  some mass, in this case that of the bare electron,   is negative.

Negative masses aren’t supposed to  exist.

However remember that this bare mass is   not something we can ever see or measure, so  let’s pretend this odd assumption is OK for   now.

Besides, we now have a bigger problem than  negative bare masses.

It’s the fact that to get   our tiny measured electron mass, the bare and  self energy masses have to be comparatively   huge numbers that are very very nearly, but  just not quite the same—deviating from each   other by only one part in 20,000 if we take  the self-energy mass for a 10^-17 m electron.

This doesn’t feel natural somehow.

It would  require a type of fine tuning for these   seemingly very different sources of mass.

I mean, it would be one thing for them to   be exactly equal, but to be only almost  equal?

This type of problem is generally   called a naturalness or hierarchy problem.

As I said, we’re going to be able to solve   it for the electron, but not for the Higgs  boson.

That’ll be in an upcoming episode.

For now let’s focus on saving our electron, and we  can do that with quantum mechanics.

Last episode,   when we dove in real close to the electron, we  saw that the classical electric field becomes   the jittery quantum electromagnetic field.

In quantum field theory, those jitters are   represented mathematically as the sum of activity  of countless virtual particles.

The EM field   itself is modeled as being made of virtual photons  that communicate the EM force between charged   particles, and the next most important effect  is the interaction between the EM and electron   fields, which QFT models as the very brief  production of a virtual electron-positron pair.

This extreme heft of the classical EM  field is described in QFT as coming from   the energy in this virtual activity.

But how much  self-energy mass does quantum theory predict?

In   QFT we use Feynman diagrams to represent the  possible virtual processes that can be part   of any interaction.

These diagrams are really  shorthand representations of components of an   interaction.

For any combination of ingoing  and outgoing particles for any interaction,   we can calculate something called the  Lagrangian of the interaction by adding   up all possible scenarios that could have  occurred in that interaction.

This allows you   to calculate things like the probabilities  of various interactions occurring.

We can   also calculate energy flowing through the  various processes within the interaction,   and this energy gives us the mass  equivalence produced by that interaction.

In the case of our free electron, the ingoing and  outgoing particles are just the electron itself,   and so we just consider interactions  that leave the electron unchanged.

But that still includes lots of activity—for  example, emitting and absorbing one or more   virtual photons—and that’s what the basic EM field  is in this model.

But those virtual photons may   engage in all sorts of shenanigans with virtual  electron-positron pairs prior to reabsorption.

In order to calculate the self-energy and  corresponding self-energy mass of the electron,   Feynman tells us we need to add up  not only every possible interaction,   but also every way each of  those interactions can play   out.

For example, every energy and momentum that  the intermediate virtual particles could have had.

For simple cases where the electron emits and  reabsorbs a photon that’s no big deal—the energy   of that photon is strictly limited by the law of  energy conservation.

But for cases where there are   entire loops of activity in the intermediate  virtual space, the possible energies of these   intermediate particles are not constrained by the  energy of the electron.

Quantum mechanics tells us   they could be anything, and quantum field theory  tells us that we must add all possibilities.

The   result of summing all possible energies in all  possible Feynman diagrams is infinite energy, which means contributing infinite  self-energy mass.

Something still seems off.

Because of the Heisenberg uncertainty principle,  high energy corresponds to measuring at short   distances.

So the choice to sum energies up to  arbitrarily large numbers is equivalent to trying   to observe the electron with infinite resolution.

By this logic, a truly point-like electron should   have infinite self-energy—so that’s the same bad  answer we got from classical electromagnetism.

But renormalization can also fix this issue.

In  the classical case we imagined a negative bare   mass to the electron.

In QFT we instead need  to cancel the problematic Feynman diagrams one   by one.

This is done by essentially breaking  these “divergent” diagrams into two parts—a   regular term with its infinite contribution and  a counter-term that cancels that infinity.

That   still sounds hokey, don’t worry—the founders  of this approach, people like Richard Feynman,   Julian Schwinger and Freeman Dyson, also  worried about that.

But the results were   stunningly successful.

Quantum field theory,  and in particular quantum electrodynamics   gave incredibly precise predictions of the  quantum behavior of things like the electron.

But the theory didn’t predict the actual  mass of the electron.

The reason QED works   is that all this “hokey” renormalization  is grounded by the fact that we can measure   the mass of the electron, so all of this  canceling that occurs under the hood is … made concrete by calibrating it to reality.

It's still being called “wweeping infinities  under the rug”, and it feels awkward, but   renormalization was put into much more comfortable  context by Ken Wilson, who pointed out that these   infinities arose only because we’d tried to  push quantum mechanics too far.

Remember when   we did the self-energy calculation with classical  electromagnetism, the self-energy mass got out of   hand when we started to apply Coulomb’s law way  beyond its validity, into the quantum realm.

But   maybe something similar is happening here—maybe  all these convenient counter-terms that cancel our   infinities represent real physics beyond the stuff  directly described by our quantum field theory.

Now we know for sure that quantum mechanics  also has limits to its validity.

Quantum   mechanics doesn’t include gravity, so  we know that it must be what we call   an effective theory—emergent from some deeper  theory—just as classical physics emerges from   the quantum.

We know for sure that any  of our quantum field theories fails at   the Planck scale—lengths smaller than around  10^-35 m, or energies larger than 10^28 eV.

So what if we just stop adding energies when  they reach a certain cutoff where we know our   theory is no longer valid?

The process of doing  that is called regularization.

If we set the   Planck scale as our cutoff, and don’t include  energies above that we do get a finite mass,   but that mass is also at the Planck scale—now  10^22 times larger than the measured electron   mass.

A universe with elementary particles  this massive would have collapsed before   it started expanding.

We can still fix  that crazy mass using renormalization,   but now we’re asking Planck-scale physics to  cancel out the positive self-energy of the   electron all the way down to the electron scale.

That means a near-perfect canceling to one part   in 10^22.

And that is a much worse hierarchy  problem than we figured with classical physics..

In the case of the electron, we actually don’t  have to go all the way to the Planck scale to   find new physics to solve the electron’s hierarchy  problem.

I mentioned last time that when we zoom   in real close to the electron so that the energy  in the EM field approaches the rest mass of the   electron itself, a new quantum process becomes  important.

Virtual positrons are now able to   annihilate with the real electron, causing the  companion virtual electron to take its place   in reality.

We saw how this leads to an apparent  blurring of the location of the electron,   which is annoying.

But it also solves  our hierarchy problem, which is great.

It turns out that this particular interaction  has a negative contribution to the self-energy   of the electron—and this perfectly cancels the  leading positive component of the self-energy,   leaving only second-order contributions.

That remaining un-canceled self-energy of   the electron is much smaller than the canceled  part.

In fact it’s similar to the bare mass,   and scales linearly with the bare mass.

We  don’t need any coincidental fine-tuning to   do this canceling because there’s an in-built  symmetry between the negative and positive   energy components of the field.

You can think  of this symmetry as being the matter-antimatter   symmetry.

It’s the fact that the electron has an  antiparticle that protects it from being blown up   to the Planck scale.

More formally, its mass is  protected by the chiral symmetry of the particle.

So, woohoo!

We saved the electron’s mass  from being “unnatural” by finding some new   physics to make its smallness not a complete  fluke.

Unlike the case of the Higgs boson,   whose unnaturalness is yet to be understood,  and very soon we’ll get back to why this is   such a big deal.

We’ll also come back to  some of the tools we saw—renormalization   and regularization.

These are actually  pretty profound subjects in themselves,   and teach us something about the emergence  of effective theories from deeper layers,   and the connection between the  energetic scales of spacetime.