# The Meaning of the Higgs

Jul 05 2012 Published by under Good Math, Physics

This isn't exactly my area of expertise, but I've gotten requests by both email and twitter to try to explain yesterday's news about the Higgs' boson.

The questions.

• What is this Higgs' boson thing?
• How did they find it?

• What does the five sigma stuff mean?
• Why do they talk about it as a "Higgs'-like particle"?

So, first things first. What is a Higgs' boson?

When things in the universe interact, they usually don't actually interacts by touching each other directly. They interact through forces and fields. What that means is a bit tricky. I can define it mathematically, but it won't do a bit of good for intuition. But the basic idea is that space itself has some properties. A point in space, even when it's completely empty, it has some properties.

Outside of empty space, we have particles of various types. Those particles interact with each other, and with space itself. Those interactions are what end up producing the universe we see and live in.

Fields are, essentially, a property of space. A field is, at its simplest, a kind of property of space that is defined at every point in space.

When particles interact with fields, they can end up exchanging energy. They do that through a particular kind of particle, called an exchange particle. For example, think about an electromagnetic field. An electron orbits an atomic nucleus, due to forces created by the electromagnetic fields of the electrons and protons. When an electron moves to a lower-energy orbital, it produces a photon; when it absorbs a photon, it can jump to a higher orbital. The photon is the exchange particle for the electromagnetic field. Exchange particles are instances of a kind of particle called a boson.

So.. one of the really big mysteries of physics is: why do some particles have mass, and other particles don't? That is, some particles, like protons, have masses. Others, like photons, don't. Why is that?

It's quite a big mystery. Based on our best model - called the standard model - we can predict all of the basic kinds of particles, and what their masses should be. But we didn't have a clue about why there's mass at all!

So, following the usual pattern in particle physics, we predict that there's a field. Particles moving through that field, if they interact with the field, experience a sort of drag. That drag is mass. So - just like particles like neutrinos aren't affected by electromagnetic fields, some particles like photons won't have mass because they don't interact with the field that produces mass. We call that field the Higgs' field.

(The previous paragraph formerly contained an error. The higgs field produces mass, not gravity. Just a stupid typo; my fingers got ahead of my brain.)

So physicists proposed the existence of the Higgs' field. But how could they test it?

It's a field. Fields have exchange particles. What would the exchange particles of the Higgs' field be? Exchange particles are bosons, so this one is, naturally, called a Higgs' boson. So if the Higgs' field exists, then it will have an exchange particle. If the standard model of physics is right, then we can use it to predict the mass that that boson must have.

So - if we can find a particle whose mass matches what we predict, and it has the correct properties for a mass-field exchange particle, then we can infer that the Higgs' field is real, and is the cause of mass.

How did they find the Higgs' boson?

We have a pretty good idea of what the mass of the Higgs' boson must be. We can describe that mass in terms of a quantity of energy. (See the infamous Einstein equation!) If we can take particles that we can easily see and manipulate, and we can accelerate them up to super-super high speed, and collide them together. If the energy of a collision matches the mass of a particle, it can create that kind of particle. So we slam together, say, two protons at high enough energy, we'll get a Higgs' boson.

But things are never quite that easy. There are a bunch of problems. First, the kind of collision that can produce a Higgs' doesn't always produce one. It can produce a variety of results, depending on the specifics of the collision as well as purely random factors. Second, it produces a lot more than just a Higgs'. We're talking about an extremely complex, extremely high energy collision, with a ton of complex results. And third, the Higgs' boson isn't particularly stable. It doesn't really like to exist. So like many unstable things in particle physics, it decays, producing other particles. And many of those particles are themselves unstable, and decay into other particles. What we can observe is the last products of the collision, several steps back from the Higgs'. But we know what kind of things the Higgs' can decay into, and what they can decay into, etc.

So, we slam these things together a couple of thousand, or a couple of million times. And we look at the results. We look at all of the results of all of the collisions. And we specifically look for a bump: if there's really a specific collision energy level at which Higgs' bosons are produced, then we'll see a bump in the number of Higgs' decay products that are produced by collisions at that energy. And what the announcement yesterday showed is that that's exactly what they saw: a bump in the observations inside the expected range of values of the mass of a Higgs' boson.

What does five sigmas mean?

Whenever we're making observations of a complex phenomenon, there are all sorts of things that can confound our observations. There are measurement errors, calculation errors, random noise, among many other things. So we can't just look at one, or two, or ten data points. We need to look at a lot of data. And when you've got a lot of data, there's always a chance that you'll see what appears to be a pattern in the data, which is really just the product of random noise

For example, there are some people who've won the lottery multiple times. That seems crazy - it's so unlikely to win once! To win multiple times seems crazy. But probabilistically, if you keep observing lotteries, you'll find repeat winners. Or you'll find apparent patterns in the winning numbers, even though they're being drawn randomly.

We don't want to be fooled by statistics. So we create standards. We can compute how unlikely a given pattern would be, if it were occuring do to pure randomness. We can't even absolutely rule out randomness, but for any degree of certainty, we can determine just how unlikely a given observation is to be due to randomness.

We describe that in terms of standard deviations. An observation of a phenomenon has a roughly 68% chance of being measured within one standard deviation (one sigma) of the actual value, or a roughly 32% chance of being observed outside of one sigma. At two sigmas, there's only a roughly 5% chance of being outside. At three sigmas out, you're down to a roughly 0.3% chance of randomly observing an event outside. The odds continue that way.

So, the Higgs' hunters computed probabilities of observing the data that they found if they assumed that there was no Higgs'. The amount of data that they found exceeded 5 sigmas away from what you would expect by random chance if there was no Higgs'. That translates as damned unlikely. The ultimate choice of 5 sigmas is arbitrary, but it's accepted as a threshold for certainty in particle physics. At five sigmas, we realistically rule out random chance.

Why do they keep saying Higgs'-like particle?

Remember up above, I said: "So - if we can find a particle whose mass matches what we predict, and it has the correct properties for a mass-field exchange particle, then we can infer that the Higgs' field is real, and is the cause of mass"? There are two thing we need to show to conclude that we've found the mediator of the Higgs' field. There needs to be a particle with the right mass, and it needs to have the properties of a mass-mediator. What we've got right now is an observation that yes, there is a particle at the mass we'd expect for a Higgs'. But we don't have observations yet of any properties of the particle other than its mass. Assuming the standard model is right, the odds of finding another particle with that mass is damned unlikely, but the standard model could be wrong. It's not likely at this point, but people like to be careful. So at this point, to be precise, we've observed a Higgs'-like particle - a particle that according to all of the observations we've made so far appears to be a Higgs'; but until we observe some properties other than mass, we can't be absolutely certain that it's a Higgs'.

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• John Fringe says:

That seems like a good summary to me, but just two things. I would not imply that SM gives us particles' masses. And of course, it's not only mass what have been measured, but also interaction and decay ratios, which seem appropiate, the spin (0 or 2, most probably 0), just not in a conclusive way.

• dete says:

Still don't get it!

Here's where I get confused:

1 - You (and everyone else) talk about how the Higgs Field causes mass. Then you talk about "the mass of a Higgs Boson". That seems self-referential and therefore meaningless.

2 - Mass is never defined in these types of articles. It seems to me that mass has two interesting properties: Inertia and gravitational pull. The standard model has nothing to say about gravity, I'm fine with that. But why do we talk about "mass" instead of "inertia". Is there something feature of "mass" OTHER than these two phenomena?

3 - The supposed "best" explanation of how the Higgs Field creates mass is the cocktail party analogy, where the Higgs Bosons "drag" certain particles more than others, and those with more drag are the ones with mass. However, this analogy makes me angry because it seems to presuppose that you have an overly simplistic view of "mass": i.e. heavy things are hard to move, and slow down and stop on their own. But that's not how inertia works at all! Heavy things keep on trucking, they don't get "slowed down" by their mass.

I think my biggest problem is that I really want a more sophisticated explanation than "the layman", but I want it to be intuitive, not mathematical... and I'm not sure such an explanation exists...

• MarkCC says:

The higgs field causes mass. The higgs boson is the exchange particle of the higgs field. You've got to keep those two things distinct.

The higgs boson is a particle. Particles that interact with the higgs field have mass. If you think about it that way, there's nothing wrong with having the higgs field transfer particle be a particle that interacts with the higgs field. It's actually pretty sensible, in its way.

But to get to the next step is really hard.

What is mass? To be honest, I can't follow the math. I just don't have the background for the full formal mechanics of it.

The simplest form is to say that it's inertia. That is, a certain amount of force is required to change the motion of a particle. The measure of the underlying property of how hard you have to push something to make it accelerate is that things mass.

To get anywhere beyond that, you need to start talking about symmetry breaking, which I don't understand. The kinds of symmetries that we'd need to talk about to understand the whole concept of symmetry breaking which leads to mass is just well beyond anything I've studied. From my failed attempts to understand it, you need to work with something called spinor math - which is a sort-of second or third-order vector with special rotational properties.

And gravity is a whole different kettle of fish. Roughly speaking - very roughly speaking - gravity is a distortion of space caused by mass. You can have mass without gravity - at least as I understand it, it's a different phenomenon. Obviously, there's got to be some relationship between gravity and the higgs field, because mass affects gravity, but that's a whole mess about which I know even less.

• dete says:

The higgs field causes mass. The higgs boson is the exchange particle of the higgs field. You've got to keep those two things distinct.

Really?

Because, I thought that quantum theory basically said that fields don't exist. That what we think of as fields (and what we can conveniently represent as fields mathematically in aggregate) are actually huge quantities of exchange quanta. So, there is no EM field, there are just lots and lots of photons each carrying a little bit of force, and those photons sort of pop in-and-out of existence to do their job.

Like the current in the river: We can think of there being a current, and we can model a current, but in reality there are just a bazillion individual water molecules, each with their own velocities that average out to the current.

• Um.. quantum field theory.

• dete says:

I can't believe how upset this comment makes me. I'm trying to learn here. I want to understand this stuff, and I come to Mark's blog because he seems to genuinely want to engender understanding to a wider audience. I post my questions because I don't believe my misunderstandings are unique, and the corrections of them may help someone else understand.

Your comment basically says, "No, you're wrong," which would be unhelpful enough, but you decided to be clever and make the point sarcastically, implying that I'm not only wrong, but stupid.

No thank you.

• Ciaran says:

@dete
It's my understanding that fields are the fundamental building blocks of the universe in quantum field theory and particles are ripples in these fields.

I don't know if it's OK to post links to other blogs in the comments here but I found Prof Matt Strassler's explanations to be pretty understandable. The Higgs page at http://profmattstrassler.com/articles-and-posts/the-higgs-particle/360-2/ has some explanation of fields and particles.

• dete, your comment didn't read as a question about your misunderstanding but as a snide statement about your superior understanding, at least to me. I apologize for the misinterpretation.

• MarkCC says:

Oh yeah, and...

Intuitive and detailed but not mathematical is, I believe, impossible.

You can have intuition without math. Or you can have detail without intuition. And I suspect that if you really understand the math, you can derive some intuition from it. But I don't think that it's possible to have detail without the math.

• Entirely non-mathematical? not really possible. Less mathematical and only slightly blurry? I think it's possible, and I'm hoping to work on a short series of less-technical posts at my weblog soon. In fact, the basic Higgs mechanism actually occurs in classical field theory, and all the really hard bits are in the quantization (which is admittedly really mind-breaking stuff).

The basic idea is this: we've got some collection of number-valued functions defined in spacetime -- fields -- that collect together to give a big high-dimensional vector-valued field. Fields -- like everything else -- like to minimize energy (actually they like to extremize action, but work with me here). Normally, just like a marble rolling in the bottom of a salad bowl, the field energy is higher the further away from the zero vector it is. It's also higher the more the field varies from point to point, which has a lot to do with what makes this stuff "fieldy". Incidentally, the salad bowl visual is dealing with a 2-D massless field, which will be important later.

Anyway, the Higgs field defines a potential that's not like the bottom of a salad bowl. It's more like the bottom of a wine bottle with a punt in it: a bump in the middle that makes the zero vector a local maximum for energy. So the marble -- the field -- "rolls down the side" to some point around the circle of potential energy minima. The place where it settles is the "effective" vacuum, as opposed to the "actual" vacuum where the field values vanish at the zero vector.

So, when energies are pretty low the marble will jiggle around near the effective vacuum, and we find it convenient to describe our fields in terms of the effective vacuum plus small perturbations. That is, we want to CHANGE VARIABLES (this is the important bit). And when we plug that change of variables back into our physics that we wrote in terms of the original field, we get new equations! These equations are the equations of the effective fields -- the small perturbations around the effective vacuum. We still have two directions -- two field components -- but one of them now comes with a new term in our equations, which corresponds to that field component "having a mass".

The specific Higgs mechanism everyone's so interested in is about the electroweak field. That is, we start with four massless fields. We mix in a potential field with a "bump" to "break the symmetry" just like before, and three of those four fields gain mass terms when we do the change of variables. The remaining massless field is the photon field, while the others become the weak force-mediating bosons W^+, W^-, and Z^0.

• Andy M says:

The photons emitted when an electron moves to a lower orbital aren't examples of exchange particles are they? They're not virtual, for one thing.

• Most people wouldn't call them exchange particles, since within the scope of the system you describe they're not being "exchanged" with anything. On the other hand, to be observed the photon does have to interact with something else later, so it gets a little fuzzy.

One thing we can do is define a condition that proves a particle is an exchange particle: all non-exchange particles live "on the mass shell". That is, if a particle has mass m and you measure its energy E and momentum p you will find that E^2-p^2=m^2 (I'm using a system of units where c=1). Luckily, special relativity works out so that it doesn't matter how the particle is moving with respect to you; all observers will measure this same identity.

"Virtual" particles are not constrained to live on the mass shell (handwaving about Heisenberg uncertainty goes here), so if you see a particle in your equations that doesn't obey this condition it must be part of an exchange. However, you're never going to observe a particle like this in the real world.

• Verena says:

So the particle we found (and hope to be the Higgs boson) only exists for the fraction of a moment, and we have not even managed detect it itself, just its left-overs... How could we possible go about observing its properties to be confident it's indeed a Higgs?

Is this something that the scientists have a plan for? Will it be shown in our lifetime?

• MarkCC says:

We can observe its properties the same way we observe its existence: by indirect effects.

That is, the particle, during its existence, has some set of properties. Those properties will have an effect on the properties of the particles that those break down into, down through the cascade. So by looking at the breakdown products, we'll be able to figure it out.

My understanding is that there's already quite a lot of supporting data for this being the higgs - but that the amount of data about the necessary properties hasn't yet reached 5 sigmas.

• There's an old joke that says particle physics works as if you were studying cars by crashing them into each other and studying the debris. Taking this metaphor seriously for a moment, if you're looking at the debris and you see a NiMH battery you know one of the cars must have been a hybrid or electric vehicle.

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• ben says:

all force "particles have a corresponding wave function- ie a photon is both a particle and a wave- depending on your observation. this new boson is a "force particle, hence it has a corresponding wave function, similar to an electron partnering with an electromagnetic field, this new boson corresponds to a "higgs or higgs like field. any given position within this field, is a quanta, or potential particle, of the higgs. Since the entire cosmos is saturated with this field, then all of space, is full of these actual potential, virtual particles. matter, as in particles that are fermions, must exist in "space/time, so these particles must also be within this higgs field. All force particles mediate or transmit a "force. its silly to think that the higgs wouldn't decay nearly instantly, as it process of exchanging its "force, interferes with its own wave function, similar to how a photon, as a wave, isn't a particle until it interacts with a mass that is itself made of particles. Also note that this new boson is the heaviest of the known particles, some 120 times the mass of a proton- keep in mind, that it almost doesn't matter if this new boson, is in fact the boson predicted by higgs. It IS a NEW boson. Its real, and inquires much further study. We have a new force particle guys! Someone posted that this isn't yet a sigma 5 , idk how much research he did, but this is indeed, officially a sigma 5. So much of what we know is theory. its all we really have. For until were able to mathematically show that we have discovered the grand theory of everything, then experimentally prove it, all of even everything we think we know, is just a part of something, we know very little about. Our theories is like a flashlight in the darkness. Even the known laws breakdown as they fall under this subset of conditions that were just beginning to explore. The meaning of the higgs- it ushers in a new and exciting era of physics, allowing us to finally close the book on the classic model, and open a new book, that takes us to a whole new level. hmm, i understand that they say we dont yet have room temperature superconductors, i wander should a manufacturing technique come about, that lets us produce massless sheets of copper, what kind of electrical resistance, if any would it have- i bet this holds key into our understanding of Bose-condensates as well, also now we have to change what we know in particle entanglement- sheessh this discovery is going to change everything we know- it holds hands with everything, like the first link in a long chain. Im so proud of all of the people responsible for this discovery

• Man, I can barely make heads or tails of this. On the few things I can make out:

"Boson" and "force particle" are not necessarily identical. Actually, "force" is really sort of a misnomer in the Standard Model, so talking about "force particles" is as well.

The new particle is not the heaviest known particle; it's about 125 GeV and the top quark is about 172 GeV.

Photons do not "decay nearly instantly"; they can be and are generated as particles and transmitted effectively infinitely far as particles without decay.

From there on I'm lost.

• Ethan Bradford says:

Thanks, @MarkCC, for excellent explanations. I am also a physicist who doesn't fully understand particle physics trying to understand this phenomenon.

On mass, your comment about needing to understand spinor math to fully grasp how the Higgs gives mass applies to the anomalously large masses of the W and Z bosons; it doesn't apply to how it gives mass (or rest energy, to avoid circularity in our arguments) to the leptons (e.g. electron, quarks, etc). For them, the rest energy comes directly from the coupling with the scalar field.

The analogy that works for me is taking two massless particles with positive electric charge and bringing them together. Because you had to put energy into this system, it has an effective mass even though none of the input particles have mass. Similarly, bringing a particle with a Higgs coupling into existence in the presence of the omnipresent Higgs field requires energy, and thus gives the particle an effective mass.

On whether we this is in fact the Higgs boson, what we know to five sigmas of significance is that there is a particle of this mass, and that that particle has to have spin 0 or spin 2 because it produces two spin-1 particles (i.e. photons). Therefor, it is (to five sigma) a boson and it's compatible with being the Higgs boson (which has spin 0).

We don't know to five sigma that it is the Higgs boson, but since the standard model points to there being a Higgs boson at about this mass, it would be very surprising to everybody if it turned out not to be. We will distinguish that possibility from other models by comparing how often the new particle decays in various ways; the standard model for the Higgs boson predicts certain proportions which this observed particle is compatible with, and when we get a lot more data, if that continues to match, then we can pretty much rule out any other alternative.

• Actually you don't need spinors at all. And spinors are secretly just physicists' fancy word for vectors in the fundamental representation of either SU(2) or SL(2,C) -- the latter of which is secretly the Lorentz group, and contains the former.

More directly, Ethan: did you do much Lagrangian mechanics or field theory? Just take a Lagrangian of the form (not sure how to use LaTeX here):

where $F_{munu}$ is the electromagnetic Lorentz field, $phi$ is the scalar "Higgs" field, and $D_mu=partial_mu-ieA_mu$ is the "covariant derivative", which is how the Higgs field gets coupled to the electromagnetic potential $A_mu$.

Now notice that the "true vacuum" where the fields are zero is not an energy minimum (the bracketed term on the right is an effective potential). Pick an effective vacuum:

write your fields as

and change variables to write out the Lagrangian in terms of the perturbation fields $a_mu$ and $xi$ fields. Hey, presto, a mass term pops up for the new $a_mu$ field!

So, where does the Higgs mass come into it? See that first term in the brackets in the original Lagrangian? The $m$ there is the mass of the Higgs field, since it's the coefficient of the $phi^*phi$ term.

• leon says:

Thanks for the nice article.

On a slightly off-topic note, can you (or someone else) recommend a good introductory physic textbook? I am not afraid of math but I don't know any higher calculus. Thanks!

• kik says:

Heh, the series of data points in that graph has a bunch of bumps and deviations, but only the bump corresponding to the higgs boson causes any deviation in the "S+B fit" curve. Probably doesn't mean anything but it's curious to look at.

• s. Vik says:

Regarding the explanation:

>>>>>>Particles moving through that field, if they interact with the field, experience a sort of drag. That drag is mass.

I've heard this in the LHC statements too, but it does not make sense as a "drag" would slow down the particle and then it would be stationary and have no mass.

Does a particle get have a constant stream of Higg boson exchanges to give it mass?
Or does it just get one Higgs to give it mass once like an electron elevated to a higher energy level?

Is this Higgs confered mass the rest mass of a particle. Or does it include kinetic energy when it is moving.

Are Higgs exchanges needed to increase the Momentum. If so the Higgs field should be proportional to the acceleration and not "movement as in velocity".
Is this similar to and inductor resisting change in the current??

How does a 125 GeV Higgs give smaller masses to the Quarks and Leptons.

Where does the Higgs fit into SM 4x4 elementary particle table.
Does it really fit there at all????

And why is the square of the Higgs Mass about equal to the Square of the W and Z masses. Is there a square dance between W and Z which produces Higgs in the diagonal :).

Also the Higgs theory has another 20 or so parameters which have all been measured, except for the Higgs mass. Why could it not be calculated from some other parameters in SM. Or is it that independent in the equations.

Just wondering.....

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