© 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
- d2Z/dt2 – c2 d2Z/dx2 = – B2 Z [Class 1 -- gives particles with mass]
- d2Z/dt2 – c2 d2Z/dx2 = 0 [Class 0 -- gives particles with no mass]
- d2Z/dt2 – c2 d2Z/dx2= + B2 Z [Class -1 -- unstable]
where in these equations I assume that B2 > 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
- d2H/dt2 – c2 d2H/dx2 = – b2 H (H2 – v2) = + (bv)2 H – b2 H3
which is Class -1 for H near zero, but has an important H3 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:
- d2h/dt2 – c2 d2h/dx2 = – b2 (v+h) (2 v h + h2)
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 h2 and h3 :
- d2h/dt2 – c2 d2h/dx2 = - 2 b2 v2 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 mh of the Higgs particle h from the form of this Class 1 equation:
- mh = √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;
- mh ≈ 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.