© *Matt Strassler [October 22, 2012]*

*This is article 3 in the sequence entitled How the Higgs Field Works: with Math. Here is the previous article.*

In the previous article I described how and why the Higgs field has a non-zero equilibrium value. Now I want to describe what the Higgs particle is and how its mass emerges from the equations.

I should remind you that except where I state otherwise, I’m always covering the simplest possible form of the Higgs field and particle — the so-called “Standard Model Higgs” — in this set of articles. More complicated forms are possible; for instance there might be several Higgs fields rather than just one. I’ll perhaps briefly describe that more complicated case in a later article, if time permits; but for now let’s just keep it simple.

I didn’t emphasize it in the last post, but among the (apparently-)elementary fields that we’ve discovered so far in nature, the Higgs field is unique. All of the fields except the Higgs satisfy Class 0 or Class 1 equations of motion. In fact (though this is likely not true of all fields in nature) all of the ones we know about that satisfy Class 1 equations do so only because the Higgs field is non-zero; were the Higgs field zero, they would all satisfy Class 0 equations (as I explained in the first article in this series.) Instead, the Higgs field satisfies what we might call a **Class -1** equation.

For a field Z(x,t) the classes I’ve defined are

- d
^{2}Z/dt^{2}– c^{2}d^{2}Z/dx^{2}= – B^{2}Z [Class 1 — gives particles with mass] - d
^{2}Z/dt^{2}– c^{2}d^{2}Z/dx^{2}= 0 [Class 0 — gives particles with*no*mass] - d
^{2}Z/dt^{2}– c^{2}d^{2}Z/dx^{2}= + B^{2}Z [Class -1 — unstable]

where in these equations I assume that B^{2} > 0.

The relative minus sign between Class 1 and Class -1 is crucial. In both cases, the solutions to the field equation includes Z(x,t)=0 as a particular case, but for Class 1, Z(x,t)=0 is **stable**, which means Z(x,t) can oscillate around zero: there are nicely behaved waves whose quanta have a mass. By contrast, for Class -1, Z(x,t)=0 is **unstable**, which means that Z(x,t), rather than oscillating, will grow to larger and larger values of Z(x,t). In fact, unless the equation is modified, the field’s value will fly off to Z(x,t) = infinity. More precisely, while the solution to a Class 1 equation is a oscillation of Z, as in Figure 1, the solution to a Class -1 equation is exponential growth of Z, as in Figure 2.

For the Higgs, as for any field we would find in nature, the Class -1 equation is modified by terms that limits the exponential growth and prevents the field from going off to infinity. As we saw in the previous article, the Higgs field obeys the equation of motion

- d
^{2}H/dt^{2}– c^{2}d^{2}H/dx^{2}= – b^{2}H (H^{2}– v^{2}) = + (bv)^{2 }H – b^{2}H^{3}

which is Class -1 for H near zero, but has an important H^{3} term. (Here b [times Planck’s constant] is a positive number, and v is the equilibrium value for H.) This equation ensures that if the H field starts at H=0 and moves off its unstable equilibrium to positive H, it will end up oscillating around its stable equilibrium at H=v (see Figure 3).

Over time, these oscillations will die away, due to terms in the equation of motion that I haven’t written down (for brevity), which allow some of the energy in the H field’s oscillations to be transferred to waves in other fields *(these are the same types of nonlinear terms that allow Higgs particles to decay, see this article). *So, over time (Figure 4), the field H will settle down to H=v.

If some physical process then kicks the field away from H=v in some small region of space, the field will exhibit waves, 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 its 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. And the quanta of these waves are Higgs particles. The billion dollar question is: * what is the mass of a Higgs particle?* To figure this out, we need, as always for particles (which are quanta of waves in relativistic fields), to determine the relation between the frequency ν and the wavelength λ of the waves of the corresponding field, and then multiply the result by factors of Planck’s constant h to get a relation between the energy and momenta of a quantum of those waves, which tells us the mass of the quantum (i.e. of the particle).

We proceed as described for the field S(x,t) in the overview article that begins this series. We write a *shifted* version of the Higgs field, expressing it as H(x,t) = v + **h**(x,t), and substitute that back into the equation of motion for the H field. [In this article, I am going to put the field **h**(x,t) in boldface to distinguish it from Planck’s quantum mechanics constant h.] The example of the S field given in overview article had a simple equation of motion, so the shift didn’t change the mass of the S particle. But that’s not true here! The Higgs field’s equation of motion is more complicated, so the **h** equation is quite different from the original H equation:

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

where I used the fact that v is a constant and doesn’t depend on space or time. Next, we remember that the quanta of the Higgs field have small amplitude, so in studying one Higgs particle on its own (which is what we need to do to determine its mass) we can drop all terms proportional to **h**^{2} and **h**^{3} :

- d
^{2}**h**/dt^{2}– c^{2}d^{2}**h**/dx^{2}= – 2 b^{2}v^{2}**h**+ …

where the “+ …” reminds us that we dropped some terms. * Notice this equation for h(x,t) is a Class 1 equation, whereas we started with a Class -1 equation for H(x,t)*; that’s because H(x,t) was

*unstable*around H=0, whereas

**h**(x,t) is

*stable*around

**h**=0, which is where H=v. And so we can read off the mass m

_{h}of the Higgs particle

**h**from the form of this Class 1 equation:

- m
_{h}= √2 (h / 2 π) b v / c²

*[The h in the right-hand side is Planck’s quantum mechanics constant.]* If indeed the Higgs-like particle that’s been recently observed at the Large Hadron Collider [LHC] is a Standard Model Higgs, then we now know, for the first time, what b is *(recall that we’ve known v for years)* and therefore we also finally know the quantity a = b v.

- v = 246 GeV;
- m
_{h}≈ 125 GeV/c²*(***if**the new particle is a Higgs) - b ≈ 0.35 (2 π/ h)
*(***if**the new particle is a**Standard Model**Higgs) - a = bv ≈ 87 GeV (2 π/ h)
*(***if**the new particle is a**Standard Model**Higgs)

where again h is Planck’s quantum mechanics constant. Again, the last three are things we didn’t know until this year’s discovery.

Now, if it turns out that the Standard Model isn’t what’s in nature — if additional fields, beyond just a single field H(x,t), have to be added to the other known fields in order to explain the properties of the newly discovered particle with a mass of 125 GeV/c² — if for instance this particle is just one of *several* types of Higgs particles — then it will take some years for us to unravel this more complex situation at the LHC. There are many possibilities that one can imagine, so it doesn’t make sense for me to explain them all to you here, but I did describe some of them roughly here; and if the data from the LHC ever seems to point us in a particular direction, I’ll certainly give you a detailed explanation at that time.

Professor strassler :

Do you take as granted that it is NOT of particular significance to talk to us about 1st and 2nd crisis in cosmology conferences so that we can see that standard models are in trouble and that a new perspective is urgently needed ?

Can any one ignore 700 high caliber cosmologists and physicists and what they say ? can we ignore cahill , giese and many others ?………..etc…….etc…..

I can only write about a few things at a time. The fact that I am not writing about something does not imply I think it is of no significance.

Another excellent article. Class -1 equations had not even occurred to me, yet now it seems such an elegant explanation as to the behavior of the Higgs field.

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echoing Kudzu: Another excellent article. By the way, making it possible to go back and review previous articles by clicking on underlined phrases is VERY helpful.

I am enjoying these articles but am confused by the last part of this one. You give the formula for the mass of the Higgs particle as mh = √2 (h / 2 π) b v / c², but if h(x,t) has a Class 1 equation of motion (approximately) why isn’t mh=hv/c²? Obviously, this can’t be right otherwise mh would be 246GeV/c², and would have been known as soon as the equilibrium frequency was determined. So there seems to be a contradiction here. Where does the formula mh = √2 (h / 2 π) b v / c² come from? Naively, I put 2b²v²=(2 πv)² and substituted v=mhc²/h in the LHS to arrive at your formula, but this only works if b=√2π since I am using the fact that mh=hv/c². Clearly, I’m missing something.

Richard, the reason you’re confused is that the ν_min (from “Particles are Quanta”) is a greek letter nu, representing a frequency. It is an entirely different quantity from the Higgs value ‘v’ used in this article, and therefore does not cancel out when you (correctly) put 2 b² v² = (2 π v_min)² = [2 π (m_h c² / h)]², and thus m_h² = 2 b² v² / (2 π c² / h)² = 2 (h/2π)² b² v² / c^4.

Colin. Thanks very much for this.

Dear Dr.Strassler

Under the Fig.2, a statement you wrote is that “For the Higgs, as for any field we would find in nature, the Class -1 equation is modified by terms that limits the exponential growth and prevents the field from going off to infinity.”

I want to check that the terms you said that limits the exponential growth is -b^2 H^3?

Would you please tell me what the potential is at the center peak of the so called mexican hat? I realize the trough value is 246 GeV.

Thank ,

David Love