© *Matt Strassler [October 14, 2012]*

*This is article 2 in the sequence entitled How the Higgs Field Works: with Math. Here is the previous article.*

How does it happen that the Higgs field has a non-zero average value in nature, while the other (apparently-)elementary fields of nature that we know about so far do not? *[Very fine point: other fields excepting the lowest-level gravitational field, called the "metric", that helps establish the very existence of space and time.]*

First, it is impossible for fermion fields to have a large constant non-zero average value in nature. This is related to the difference between fermions and bosons; bosons can be non-zero on average, but fermions really can’t. So we can forget about electrons (and their cousins the muons and the taus), about neutrinos, and about the quarks. *[Fine point: Fermions can pair with each other or with anti-fermions to make composite bosons, and those composite bosons can be non-zero on average. In fact this is true of the up and down quarks and their antiquarks, and it is true of electrons in a superconductor. But that's a long story, and not our immediate concern.] *

What about the photon field, the gluon field, the W and the Z field? These are all bosons. In principle these fields could have a constant non-zero value on average throughout the universe. It is experiment, not theory, that says this isn’t the case. A large non-zero value for the electric field would have all sorts of effects that we do not observe, including most significantly an apparent violation of rotational invariance at large distance scales. The electric field is a vector (spin-1) — it points in a particular direction — so if it were non-zero, the direction in which its non-zero value points would be different from the other directions. See Figure 1, lower left.

By contrast the Higgs field is a scalar (spin-0) — it does not point anywhere. Other (non-elementary and non-relativistic) scalar fields include the density field of the air, the pressure field inside the earth, the temperature inside the ocean; at every point in space and in time, the density or pressure or temperature is just a number, whereas the electric field is a number and a pointing direction. So if the Higgs field has a non-zero value, it does not result in a preferred direction; see Figure 1, lower right. More bizarre is the fact that (because it is a *relativistic* field) it produces no preferred frame at all. For the density of the air, there is a preferred frame, because one is either at rest with respect to the air or moving through it. But this isn’t true for the Higgs field; all observers are at rest with respect to the Higgs field. Therefore the success of Einstein’s special relativity in describing all sorts of phenomena is not inconsistent with the presence of a non-zero value for a relativistic scalar field, such as the Higgs. In short the non-zero value of the Higgs field leaves the vacuum behaving much the way it does even when H=0; remarkably, you can only tells it’s there through its effects on the masses of particles (or by doing something dramatic, such as using the Large Hadron Collider to make Higgs particles.)

The simplest way for the Higgs field to end up with a non-zero value throughout the universe would be if has a non-zero equilibrium value H_{0} that appears in its Class 1 equation of motion:

- d
^{2}H/dt^{2}– c^{2}d^{2}H/dx^{2}= -(2 π ν_{min})^{2}(H – H_{0})

(It has to be Class 1, not Class 0, for reasons that we’ll see when we discuss the Higgs particle.) In fact, the situation is a bit more complicated. The correct equation turns out to be

- d
^{2}H/dt^{2}– c^{2}d^{2}H/dx^{2}= a^{2}H – b^{2}H^{3}

where a and b are constants *(whose squares are positive! notice the plus sign in front of a ^{2}H, and compare it with the minus sign in the previous equation)* which we’ll learn something about in a minute. We can rewrite this as

- d
^{2}H/dt^{2}– c^{2}d^{2}H/dx^{2}= – b^{2}H (H^{2}– [a/b]^{2})

Now if H(x,t) is a constant over space and time, then dH/dt=dH/dx=0, and so

- 0 = – b
^{2}H (H^{2}– [a/b]^{2})*(when H(x,t) is constant in x and t)*

which has the solutions (slightly oversimplifying for now)

- H = 0
- H = + a/b
- H = – a/b

In other words, it has *three* equilibria rather than one. *[Fine point: I'm oversimplifying here for the moment, but harmlessly.]*

It isn’t instantly obvious, but the solution at H = 0 is ** unstable**. The situation is analogous to the equation of motion for a ball in a bowl shaped like the one shown in Figure 2 — like the bottom of a wine bottle. This also has three equilibria, one at 0 and ones at ±x

_{0}. But clearly the equilibrium at 0 is unstable, in that even a tiny push will cause the red ball to roll far from x=0, a dramatic change. By contrast the equilibrium at x=x

_{0}is stable, in that any little push will just cause the green ball to oscillate with small amplitude around the point x=x

_{0}, a not very dramatic change. (The same is true for the light green ball at x=-x

_{0}.) In a similar way, although H = 0 is a solution to the Higgs field’s equation, our universe’s history has been complex enough to assure that the Higgs field has been knocked around a bit, and so it can’t possibly be sitting there. Instead, the Higgs field ends up in a solution with a non-zero value, a situation which is stable.

We have known for decades, from a combination of experiment and theory, that the Higgs field’s value (which we traditionally call “v”) is 246 GeV. That tells us something about those two constants a and b: in particular

- a = v b = (246 GeV) b

So that determines a in terms of b, and we can rewrite the Higgs equation of motion as

- d
^{2}H/dt^{2}– c^{2}d^{2}H/dx^{2}= – b^{2}H (H^{2}– v^{2})

**But it doesn’t tell us what b itself is.** We’ll learn more about the quantity b in the next article.

Now, although I’ve set things up so that H could be either v or -v, it doesn’t matter whether the value of the Higgs field is positive or negative (actually there are even more possibilities, see below); the world comes out looking the same, with the same physics, because nothing depends on the overall sign of H. This isn’t instantly obvious, but it’s true; one hint is that wherever you find H in the equations described in my overview of how the Higgs field works, it’s always H^{2} that appears, not H alone — and H^{2} doesn’t depend on whether H = v or H = -v. *[Fine point: In fact, H is a complex field, with a real and an imaginary part, so H can be v times any complex number z with |z|=1; and in fact it's always H*H = |H| ^{2} that appears in the equations, which is independent of z. Even that's not the whole story! but it's good enough for today.]*

If you find a way *(perhaps using the proton-proton collisions at the Large Hadron Collider)* to push or disturb the Higgs field a little bit somehow, it will wiggle back and forth — i.e., waves will develop, of the form

- H = v + A cos[2 π (ν t - x / λ)]

where A is the amplitude of the wave, ν and λ are the frequency and wavelength, and the relationship between ν and λ depends on the precise form of the equation of motion, in particular on b and v. Since the Higgs field is a quantum field, these waves have a quantized amplitude, and quanta of these waves are what we call Higgs particles. We’ll look at the properties of these particles next.

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This is good. I particularly enjoy how you are now introducing mathematics to it whilst keeping the original style of writing which makes this blog so good.

Are there any thoughts in your head of perhaps publishing a version of these blogs as a book when you are reasonably happy with the amount of content? I would very much like that.

I second Jax’s “write a book” comment! You are writing at just the right level for us “advanced physics buffs,” and I think this would fill an under-served niche in the market. Kind of like Penrose’s “Road to Reality” but more easily digestible. And frankly, more direct and to the point. I would buy it, even though I’ve already read all your articles online!!!

thanks — something like this will eventually emerge.

Is H(x,t) constant over time and space? Given that mass is in part derived from H would a “no” answer imply that the mass of fundamental particles could change (admittedly by a very small amount) over time and from one region of the universe to another?

On a related note, do disturbances in the Higgs field cause a fluctuation in the mass of nearby particles? If a Higgs Boson were to go zipping by, would I lose weight? :)

Mike – I think professor Strassler partially answered here:

http://profmattstrassler.com/articles-and-posts/the-higgs-particle/360-2/higgs-faq-1-0/#comment-13959

Even naively, H(x,t) is not

necessarilyconstant over time and space. However, in most places and at most times, it is constant, because it naturally settles down to a constant value if left alone, and it takes an enormous amount of energy per unit volume (relative to ordinary earthly or even stellar processes) to make it non-constant: the easiest way to make it non-constant is to create a ripple (i.e. one or more Higgs particles) in it, and it takes a huge particle accelerator like the Large Hadron Collider to do it.The reason it settles down to a constant value is that non-constancy can be removed through the non-linear terms in the equations that I keep referring to; for instance, if you make ripples in it (i.e. Higgs particles) those Higgs particles will rapidly decay to other particles, leaving the Higgs field again as a constant.

But we are being naive: we should also note that saying it is “constant” is a lie. It is constantly varying due to quantum fluctuations, see http://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/ball-on-a-spring-quantum/ and http://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/5-waves-quantum/. Only its average is constant.

Would it be possible for the Higgs field to have the value x0 in one region of space, and -x0 in another? If so, what would happen at the boundary? If the Higgs field had the opposite sign in a distant region of space, would we be able to tell?

I think you have to take into account it’s a complex number ( so more like a unit circle ). Just treating Ho it as two discrete values +/-xo is a simplification which is leading your thinking slightly astray.

“H can be v times any complex number z with |z|=1;”

This turns out to be a very subtle subject.

There *are* fields in nature (not any of the known elementary ones) that can do this, in principle. An example is the magnetization in a magnet, you can make a chunk of magnetic material where the magnet points in one direction in one part of the chunk and points in another direction in the rest of the chunk. At the boundary between the two — a wall, of sorts, called a “domain wall” — will be stored some energy, because the magnetic field has to transition from one direction to the other, and that transition creates a certain amount of strain.

If there were a field that could potentially take multiple values in the vacuum of empty space, then indeed such a domain wall between different regions could occur. We would indeed potentially observe effects (just gravitational ones, if nothing else, because of the large energy in the domain wall).

However: (a) the Higgs actually can take continuous values, not just the two shown in Figure 2 (that was my point about it being a complex field — and in fact there’s even a richer story than that); so if there are two regions with different values for the Higgs field they will, over the early history of the universe, continuously shift until they take the same value; and (b) there’s an even more subtle issue, which I mention just to alert future experts in the field, that has to do with how the Higgs field relates to the W and Z fields, and means that Figure 2 is impoverished, in that changes in the Higgs field have to be related to what the W and Z fields are doing and can’t be discussed independently of them.

All of this is to say that this particular issue doesn’t arise for the Standard Model Higgs field, or for most Higgs-like fields. It can arise for other fields, however, and the non-observation of domain walls out in our universe puts constraints on the types of fields that might exist in nature and on how the Big Bang took place.

But when the higgs field starts off random it could still settle down to a network of topological defects named “cosmic strings”, depending on theory of course: http://en.wikipedia.org/wiki/Cosmic_string

You say that H is a complex field, but if H = vz, where z = √i, then wouldn’t H^2 have magnitude i? What does that mean? or have I got something wrong?

The magnitude of a complex number |z| has a special meaning – http://www.clarku.edu/~djoyce/complex/abs.html

Once again I am informed of something of which I was unaware. This aspect of absolute value I had not encountered before. Many thanks.

It’s |H|^2 = H H* that appears in the equations; that’s always a real and positive quantity.

sqrt(i) is -1, so sqrt(i)*sqrt(i) = 1. Complex numbers are fun.

Oh dang, got that wrong. sqrt(i) = +/- 1/sqrt(2) * (1+i), and indeed if you square that you get i. Sorry, shouldn’t post late at night. :)

From laymen’s view, a closed system(matter?) radiates energy in quantas. But due to conservation of energy, it retains the energy by absorbing from outside its system(from field)?. Can we call this outside field as Higgs Field?, in which only a non-zero equilibrium could provide the necessary prison(closed system or mass?) for quarks and gluons, which(Gauge boson) may be other than imprisoned gluons, and have constant interaction with Higgs Field(Higgs boson)?.Because zero equilibrium fields like photons could provide energy but not the rest mass?

So, the mass of the presumed Higgs boson is 125.3 GeV, and the value of the Higgs field is 246 GeV. Is there any significance to the mass of the boson being just a hair more than half the value of the field? Or is that just numerology?

Prof. Strassler answered this in comments on a previous post: http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/#comment-13999 . As far as anyone knows, it’s just coincidence.

Although presumably whatever mechanism resolves the hierarchy problem is causing the mass (125.3 GeV) and vev (246 GeV) to be of the same scale.

Hi – very interesting and as you say bizarre

In which part of my following thought process did I go wrong?

Higgs field gives particles like electrons mass

According to Lorenz or Einstein the mass increases with the speed of the particle

The Higgs field is relativistic so has the same effect independent of electron speed – is there not a contradiction?

Thanks

Zbynek

I think Higgs field gives particles their rest mass. The rest mass does not change with speed. The “mass” changing with speed is something else: it is called “relativistic mass”, but from what I understand it is an abandoned concept. Instead of “relativistic mass” it makes more sense to talk about the total energy.

The Higgs field gives particles their ‘rest mass’ or the lightest they can possibly be. Because the Higgs field is relativistic this mass never changes. Moving particles get more energy, which looks like they are getting heavier.

If there was no Higgs field the electron for example would have no ‘rest mass’ and so would move at light speed.

Weak nuclear forces makes “rest mass”, rest mass creats “relativistic local symmetry” or space time. This “space time” creats gravitation. This gravitation creates “geometrical symmetry(asymmetry?)” in which we live and Newton’s laws start to function?

No: weak nuclear forces do not make rest mass (the Higgs field is NOT causing the weak nuclear force, that’s the W and Z fields). And rest mass does not create spacetime.

Gravitation and space-time are part of a single story. Whether there are massive particles sitting inside of spacetime is a separate question.

Moreover, objects with rest mass can be created by combining together objects that don’t have any: you can make a massive black hole by smashing enough massless high-energy photons together. I think you may want to read this post: http://profmattstrassler.com/2012/10/15/why-the-higgs-and-gravity-are-unrelated/

/The Einsteinian law of gravity involves a complicated combination of momentum and energy, in which mass does not explicitly appear./- the curve the matter(mathematical human comprehension) makes in Gravitational wave field is mass?(mathematical human comprehension) or “space time”?

/masses of the proton and neutron are as much due to those energies, and to the energy that is needed to trap the quarks etc. inside the bag, as it is due to the masses of the quarks and antiquarks contained within the bag./-

What is the difference between the “curve in gravitational wave field or space time” created by “restmass” of “photon combination” and “this bag”?. Is that a duality of matter and non matter uncertanity? or both of them are human mathematical comprehension with slight mathematical anomaly?

I understand, the explicity of momentum and energy is the intuition out of mathematical human cognition as “mass”.

The energy that is needed to trap the quarks etc. inside the bag does not come from Higgs Field?. Rest mass does not create spacetime, but the gravitational potential energy curve(lowest energy level of an object) creats spacetime?. But the lowest-level gravitational field needs non-zero(creation of “mass”) for the the very existence of space and time. Known particles acquire “mass” from Higgs Field but not the “rest mass bag” ?. The Einsteinian gravity(universal) of momentum and energy creats spacetime(gravitational curve) at the lowest energy level of objects- means not restmass?. The energy carriers can become non-zero but Fermions not, for the sake of explicity and very existence. The Higgs Field cannot be zero for the sake of “creation” of explicity. So Einsteinian gravity of momentum and energy is not explicit, but sustain the “energy or massive particle bag(explicit)” – both of them create restmass, but only gravity due to its lowest energy level appears as spacetime?

To Dr. Matt Strassler: you said “..objects with rest mass can be created by combining together objects that don’t have any: you can make a massive black hole by smashing enough massless high-energy photons together.” Does a single massive particle(proton, electron etc.) or a photon(massless) “alone” with “enough kinetic energy” spontaneously turn into a black hole without smashing them? It seems it cannot turn into a black hole because kinetic energy depends on a relative motion, so one observer with a low kinetic energy relative to a particle sees no black hole formation, while another observer with a high kinetic energy sees a particle turning into a black hole. This is not possible, correct? Does this mean no matter how much energy you add to a massive or massless particle, it just keeps traveling faster(&heavier) (for a massive one) and bluer(higher frequency, for massless one) and nothing happens even if you accelerate a massive particle beyond “the planck mass” or add energy to a photon equivalent to/beyond “the planck mass”? No gamma ray-range photon can turn into a black hole no matter how extremely high frequency it has? Or something else happens to a single massive/massless particle at very high kinetic energy around/beyond “the planck mass”?

Has anyone given a plausible explanation of why the Higgs vev (vacuum expectation value) has the particular value that is experimentally found?

No. Or more precisely, many plausible explanations have been offered, but none is obviously better than the others, and it is likely that all of them are wrong. None that have experimental implications have seen those implications show up (yet) at the LHC.

I’ve never really understood the wine bottle analogy as applied to the Higgs field. Clearly the source of the gravitational potential energy curve for the wine bottle is a nearby massive object – the Earth. But what is the analogous source for the Higgs potential energy curve? Is there a physical explanation for the Higgs potential energy curve or is it simply that the equations of motion are similar? I have the same question for the inflaton field, but maybe that’s a bit off-topic.

That’s why I use equations of motion and not wine-bottle pictures most of the time. It’s a mathematical analogy, not a physical one. You can understand the equation of motion for the Higgs field as

mathematically analogousto the equation of motion for a ball in a wine bottle. Consequently, you can use a physical picture of how a ball behaves in a wine bottle to gain intuition for how the Higgs field will behave according to this equation.But that doesnotmean that the physical picture of a ball held by gravity in a wine bottle isphysically analogousto a Higgs field held by some kind of force in some kind of constrained place.It’s mainly just the equations of motion that are similar, and some of the resulting behavior (though be cautious about quantum mechanics and about the fact that H(x,t) is a field — there are many things that are are not similar!)

You could extend this slightly by saying that what is really analogous is the potential energy function — that the potential energy of a ball as a function of its position in a wine bottle has a similar mathematical form as the energy of a constant Higgs field as a function of its constant value. but the reasons they have the same form do not have a physical linkage.

Same for the inflaton.

Thanks, that’s very helpful. Is there any (speculative or established) physical or intuitive explanation of why the Higgs potential energy function is what it is? Same question for inflaton field.

Not really. We observe the Higgs field has a non-zero value; we infer that it must have a potential energy function something like what we draw; we have not yet checked experimentally that this is precisely right; however in the Standard Model with no other particles and fields other than gravity, that’s the only possible form.

Is it fair to say that the Higgs potential energy curve is really a just a function of various self interactions of the Higgs field, and we simply have no idea why the interaction constants, a, b, etc., are what they are, just as we have no idea why many other constants of the Standard Model are what they are?

That’s right; we do not know anything about what causes the Higgs potential energy function to take the form that it does. And it’s that function and its parameters that determine BOTH how large the Higgs field is AND the mass of the Higgs particle.

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How was the Higgs vev measured? How precisely is it known?

If I’m reading the previous article properly, a particle’s mass is equal to its Yukawa parameter times vev — but are these Yukawa parameters measurable independently?

[quote=jax] Are there any thoughts in your head of perhaps publishing a version of these blogs as a book when you are reasonably happy with the amount of content?[/quote]

And one for the mathematically challenged, please.

It will probably happen, yes… I am hoping to do so eventually.

Dear Sir,

Shouldn’t line 2 at figure-1 above rather read.. “if the electric field were NON-zero…”??

Thanks! Fixed.

Hi Prof. Strassler, I have some technical conceptual question about the non-zero higgs field vev. Since the higgs field itself isn’t gauge invariant, why does it make physical sense to talk about a “non-zero” higgs field? Is there an instrinsic physical way of talking about this non-zero vev without using non-gauge invariant concept (i.e. would it be better to say gets a vev, or another way of describing the same thing without ever mentioning any vev)?

I meant “would it be better to say |H|^2 gets a vev…”

Tim, in future just send me an email for technical stuff like this; we can have a more elaborate and sophisticated conversation. These pages are really for people who took a little bit of undergraduate physics and never went any further. You should be learning all the technicalities, without any shortcuts, and I’m happy to discuss those with you.

The more correct statement is indeed that the square of the Higgs vev (which is gauge invariant) is v^2. But this is a technical point, as you say. I don’t discuss gauge invariance here because it is both confusing and really unnecessary for the presentation — gauge invariance is introduced only as a technical device to make calculations simpler, as it isn’t a real symmetry, so why confuse a non-expert by adding it in and taking it out again at the end?

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To Matt Strassler: If Higgs field is a elementary scalar field, it does not flow nor have a current in the field? Or does it? If it has a Higgs current, then the Higgs field has a direction and a vector field and is no longer a scalar field? Is this the case or not?

you said “Higgs field is a scalar (spin-0) — it does not point anywhere. Other (non-elementary and non-relativistic) scalar fields include the density field of the air, the pressure field inside the earth, the temperature inside the ocean; at every point in space and in time, the density or pressure or temperature is just a number, whereas the electric field is a number and a pointing direction.”

“temperature” and “pressure” sound like good ordinary examples for scalar field but do they not have a direction? as temperature distribution or temperature points can have a flow (can drift with a chunk of air etc.) or pressure can point to a direction? Are they not vector fields? but not true scalar fields?

A scalar field is a number at each point in space. It’s not a material.

Temperature is a number at each point in space. Now if the temperature is different at one point and another, there will be a change, because one area will cool while another will warm. BUT THAT IS NOT A TEMPERATURE CURRENT. Temperature is not flowing. What is flowing is heat — and heat flow is a field that does have a direction. Or if the air moves, then what is flowing is air and heat — not temperature. The temperature is still, simply, a number at every point.

The density of the air is a number at each point in space. If the density is higher at one point in the room than another, then the air will move to equalize the density. BUT THAT IS NOT A DENSITY CURRENT. It is a wind [a current of air] — and the wind most definitely has a direction.

Similarly, if the Higgs field were different at one place than another, then the Higgs field would adjust in some way (not quite the same way as temperature — the equations that govern them are different). The changing Higgs field would not be a Higgs current… nothing flows at all, in this case. It is simply a changing Higgs field, which, at every point, is a number. The thing which points is a *gradient* in the Higgs field — not the Higgs field itself.

To Matt Strassler: what happens to “..a *gradient* in the Higgs field..”? Does it average out if you wait for a while and eventually have more or less (or exactly?) equal field amplitude in different points? and as this happens, do multiple Higgs waves or disturbances propagate (assuming finite volume)? By the way, do all fields values/amplitudes(photon(electromagnetic), electron, Higgs fields etc.) have “exact” values? without any uncertainty? For example, does electric and magnetic field value/amplitude of a photon can be determined experimentally and theoretically arbitrarily accurately?

about the temperature field: It seems to me “temperature field (of air etc.)” is like an artificial construct, because if you zoom in you see molecules and its building constituents moving with kinetic energy and velocity (which comes with “direction”, and “vector”). and where is “temperature” in small scale? (there is only average temperature? because each molecule has different kinetic energy and there is “no point” with that (average) temperature but “atoms/molecules” at that small scale. true?) so, it seems temperature field is just a mathematical concept/construct which is convenient but fails to describe underlying physical phenomenon. In this case(temperature of air/matter etc.), it seems there is no scalar temperature field in microscopic scale but only “vector field” of molecules/atoms. Is this true?

On the other hand, Higgs field sounds/appears like a (true?) scalar field so far. Does Higgs field stay “scalar” under any conditions/scale? Are there any other examples of scalar fields that remain scalar “at any scale”?

You can’t tell what’s a “True” field when you first learn about it. We have no idea whether it is “true” or not. The same was true with “temperature”. We know now about atoms; but temperature has been around a lot longer than our knowledge that atoms exist.

Yes, gradients in Higgs field’s smoothe out just as gradients in air pressure do.

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