Higgs BosonThere was excitement at the European Physics Society conference in Grenoble, France last week as physicists gathered to review evidence for the detection of the Higgs particle. With the latest data, the LHC is finally returning on its investment while the Tevatron at Fermilab is bravely battling on before its imminent closure in September. But why is the Higgs particle so important?

The Standard Model of particle physics is the most successful and accurate theory ever created. Physicists such as Anthony Zee have lauded it as mankind’s greatest achievement, the pinnacle of human thought (Quantum Field Theory in a Nutshell, p. 455). Yet it is not widely understood that the Standard Model predicts that all particles will be massless, and so travelling at the speed of light.

You will have noticed that this appears not to be the case. Between 1960 and 1970 a number of physicists worked on the problem of how to introduce mass into the equations of the Standard Model without the whole edifice exploding into infinities. It turns out that there are very, very few ways to do this, and the simplest is to propose that a mysterious field – the Higgs field – pervades the entire universe. Particles which we know to have mass, like the electron and the quark-composites we call protons and neutrons, couple to the Higgs field: that coupling then produces a kind of resistance to acceleration which we interpret as mass.

Want to feel the Higgs field? Wave your arms around and feel how heavy they are. The protons, neutrons and electrons in your arm are interacting with the Higgs field which is all around you, and that’s why your arms feel heavy.

The Higgs field, like all quantum fields, is itself quantized so there will be an associated particle (recall, for example, that the quantum of the electromagnetic field is the photon). The resulting Higgs particle, or Higgs boson as it is called, is the one they’re all looking for. The early signs are encouraging with evidence of some events occurring at the LHC and Tevatron particle accelerators which are consistent with Higgs bosons being produced in collisions and then decaying. But we won’t know for sure until the evidence mounts up, perhaps as early as the end of the year.

And then you’ll finally know for sure why you’re heavy: but I’m afraid there’s a way to go before Higgs engineering can solve that problem for you!

The Higgs Roadmap in Ten Difficult Steps

It’s notoriously difficult to explain the Higgs particle to non-physicists: what it does and why finding it is so important. So for the aspiring physicists out there, here is what you need to know to understand the Higgs, in ten difficult steps.

1. Ordinary Quantum Mechanics is based on the Schrödinger equation. It’s studied in physics undergraduate classes and describes particles moving at non-relativistic speeds. This happens to match the bound states we see in the world around us: the stuff of solids, liquids and gases; the stuff which gives us chemistry and life.

In Quantum Mechanics, matter particles such as electrons, protons and neutrons are represented by waves: standing waves if the particle is in a bound state (such as an electron ‘orbiting’ a nucleus); a wave-packet if the particle is propagating freely, like electrons in a cathode ray tube. Force-mediating fields such as the electromagnetic field (which we sometimes refer to as quantized into photons) occur in the Schrödinger equation as classical Maxwellian fields.

2. At high-energies, due to E = mc2, particles can decay into other particles or collide to create new ones. Ordinary Quantum Mechanics no longer works and we have to move to Quantum Field Theory (QFT). In QFT we solve different equation (the Dirac equation for spin ½ fermions* like the electron or quark; the Proca equation for spin 1 bosons* like the photon or gluon). These equations are interpreted as fields which exist everywhere in the universe. Particles are then seen as “excitations” of these fields.

3. In fact we don’t really start with the respective field equation: we start with an expression called the field Lagrangian, which expresses the difference between the kinetic and potential energy of the quantum field at each spacetime point. There is a procedure in the Calculus of Variations via the Euler-Lagrange equation which lets us derive the explicit field equation from the Lagrangian. It turns out, though, that the Lagrangian is a more fundamental concept.

4. Perhaps the key difference between quantum physics and the classical world is that each quantum entity (electron, proton, etc – modeled as a wave) has both amplitude and phase. The amplitude, when squared, becomes the probability that the particle has some quantum attribute such as position here, or spin pointing in that direction, or momentum of that amount; the phase however makes no difference to what is observed. The phase does make a difference, however, when two waves overlap because it’s the mechanism by which interference takes place. Sometimes the probability of seeing a particle just here vanishes because of destructive interference, even though classically that location should be a hive of activity.

5. This means that if you have a quantum system (e.g. even the whole universe) and you change the phase of your model by the same amount everywhere, there is no physical difference. Specifically the Lagrangian is invariant under a global phase change. This seems difficult to reconcile with the localism of physics, where changes can’t propagate faster than light. Is there a way to make the field Lagrangian invariant over phase changes which can be different at different spacetime locations?

It turns out that this is possible, but at the cost of augmenting the Lagrangian with a compensating potential field. If you start with the Lagrangian corresponding to a spin ½ fermion, such as the electron, and then demand that it be local phase invariant, you have to add an additional potential function to the Lagrangian, which turns out to be the electromagnetic field, the mediator for the electromagnetic force between charged particles.

6. Similar procedures generate the weak force felt by all elementary matter particles (fermions) and the strong (color) force felt by quarks. They are all mathematical consequences of imposing local phase invariance (often called local gauge invariance) on the particle’s field Lagrangian expressing appropriate internal symmetries: for the electromagnetic force the symmetry is U(1) rotation in wavefunction configuration space; for the weak force the symmetry is SU(2) rotation in weak isospin space; for the strong force, SU(3) rotation in color space.

7. Now we have a problem: the mathematics demands that the force carriers, the field quanta, should be massless. This is no problem for the photon of the electromagnetic force or the gluons of the strong (color) force because they are massless; but the weak force carriers, the W+, W and Z0 particles are massive: more than 80 times heavier than a proton. Such masses cannot be added by hand or the underlying equations fail to converge any more (more technically, the theories can no longer be renormalized). The solution is going to be the Higgs field.

8. Let’s talk about the Higgs field energy. Most fields have an energy function like the parabola y = x2 (the field value at any spacetime point is on the x axis while the field energy is on the y axis): the minimum energy occurs at the origin where the field value is zero.

The Higgs field is not like that: close to the origin its energy function is more like a ‘W’, where the energy zeros occur either side of the origin. As the field energy tries to slide down to zero one side or the other, the field value has to make a choice as to which arm of the ‘W’ to choose. This choice is called spontaneous symmetry-breaking.

9. So the Higgs field is postulated to fill the universe with a non-zero value, which unusually represents its lowest energy state. The weak force bosons couple to the non-zero Higgs field; the effect of this coupling is to create a kind of resistance to acceleration which we interpret as mass. Similar mathematical consideration for fermion mass terms in the Standard Model Lagrangian also require that the masses of electrons and quarks start out at zero, and only acquire their masses through their coupling to the Higgs field. More precisely, the coupling to the Higgs field adds an extra interaction term to the Lagrangian, a term which gets transformed into a mass term in the resulting field equation.

10. The Higgs field, like all quantum fields, has excitations and these are Higgs bosons. Because the coupling between the Higgs boson and fermions such as electrons and quarks depends on their masses (pretty much by definition), the low mass electrons or proton-constituents used in particle accelerators don’t produce many. Those Higgs particles that are produced decay to bottom quarks or weak force bosons which then further transform to jets of detectable particles. Unfortunately so do many other kinds of decay processes: finding the Higgs is a needle in a haystack problem.

You will appreciate that this is dramatically over-simplified. To gain a clearer conceptual view, start with Bruce Schumm’s excellent Deep Down Things (John Hopkins University Press, 2004). For an introductory course on all these topics (you should already have studied Quantum Mechanics) David Griffiths’ Introduction to Elementary Particles (Wiley-VCH, 2008) will keep you occupied for many months!

* A fermion is a matter particle defined by having half-integer spin; elementary fermions such as electrons and quarks have spin of ½. A boson is a particle with integer spin (which can be a composite matter particle or a force-carrying particle). However, all elementary force-carrying particles are bosons; the bosons for the electromagnetic, weak and strong forces are all spin 1.