A few days ago, I wrote about a measurement of the "width" of the Higgs boson, meaning the width of its mass distribution. It is a bit of a surprise that particles even have a mass distribution; surely a particle should have a particular mass, not a distribution of different masses? The clue is in the words. But, as I described in that piece, quantum mechanics, in the shape of Heisenberg's uncertainty, says no.

There is a particular mass associated with particles, it is just that they do not always have to have that mass. If they have the right mass, they are "real", or "on mass-shell". If not, they are called "virtual" or "off mass-shell". Particles which are off mass-shell are unstable, living a very short time before decay. Hence, if your budget estimate is described this way, it is probably not a compliment.



A piece of jargon like this has a precise meaning amongst experts, and like all jargon it can speed up and improve communication in a technical area, while being unintelligible to outsiders. Often such jargon is based on more common, everyday words, which a field of expertise appropriates and makes precise and specific. It amuses me when it goes the other way, when a piece of quantum-field-theoretical terminology is used to describe something more everyday such as a budget. Or colleague's lifestyle: "She's continually flying back and forth between CERN and Fermilab - totally off mass-shell." It's a bit of an in-joke, a badge of membership which doesn't much help communication and can also exclude outsiders, but the way the human brain continually finds analogies between even the most abstruse things is charming.

These off-mass-shell particles occur in the middle of Feynman diagrams and the associated calculations of how particles interact, and they do lead to observable effects, so despite their name, they are important. The Einstein and Pythagoras connections are to do with why we use the curious term "mass shell".

To get this, think of the equation for a circle. The thing that defines a circle is that fact that all the points on it are the same distance from a single special point - the centre of the circle. If you want to define a circle in the usual (x,y) Cartesian coordinate system, you'd do it like this:

A circle, in a Cartesian coordinate system Photograph: /Wikimedia Photograph: Wikimedia

That circle has a radius of 2, and for every point on the circle, the square of the value of x plus the square of the value of y is equal to the square of the radius. That comes from Pythagoras' theorem. Imagine drawing a line from the centre to the circumference, then imagine getting there by going along in the x (horizontal) direction some way, then in the y (vertical) direction some distance, until you hit the circumference. The x and y paths will make two sides of a right-angle triangle, the radius of the circle (r) will make the third side, the hypoteneuse, and so as Pythagoras tells us "the square on the hypoteneuse is equal to the sum of the squares of the other two sides", or x² + y² = r². The x and the y make a vector - a line with a direction and a length. If you rotate the vector around the centre through 360°, without changing its length, r, you draw the circle.

This can be made three dimensional simply by adding a third coordinate, call it z. Now we rotate same vector, with the same length r, in three dimensions and we draw a sphere. Every point on the sphere has coordinates satisfying x² + y² + z² = r².

This is where it starts to make sense to think of a shell. That sphere is a super-thin shell, like a bubble, with the point in the centre.

Then comes Einstein. There are four dimensions. In relativity the fourth dimension is time. It is not quite the same as the other three, the properties of space-time are such that it picks up a minus sign in the Pythagoras-like equation. And physics means that to get from from time to distance, we have to multiply by a speed (which is distance divided by time). So it is like this:



x² + y² + z² - c²t² = r²,



where c is the speed of light. We have rotated a space-time vector of a fixed length around, and made some four-dimensional surface, which won't be a sphere anymore, but which it still makes sense to call a "shell".



Finally, to get to the "mass" part into "mass-shell", we look at the energy and momentum vector of a particle. These form another space-time vector, with x, y and z components, call them X, Y and Z. The energy E forms the time component, with a factor of 1/c to get the units right this time. So the size of this "four-vector", R, is given by



X² + Y² + Z² - E²/c² = R².

Because X, Y and Z are momenta now, not distances, R is not just a length, and it is worth asking, what is the physical meaning of the length of this energy-momentum four-vector? The answer may not come as a surprise if you remember Einstein's most famous equation (which by happy coincidence featured on these pages yesterday as I was writing this piece). It relates energy and mass. Let's say the particle we are looking at is not moving - it has no momentum, so X, Y and Z are all zero. We end up with



- E²/c² = R².

or, rearranging it,

E² = -R²c²

But we know E = mc², so E² = m²c⁴. Which means that, to make both equations work, -R² must equal m²c². Or, the length of this four-vector - which doesn't change no matter how much you move it around in space time to describe the weird four-dimensional "shell" - is just a constant multiplied by the mass (often called the "invariant mass", precisely because it does not vary when you rotate it around in space-time).



So the thing to remember is that the mass of a particle is the "radius" of a shell in four-dimensional space-time. Anywhere on that shell, a particle will have different total energy and different speed, but the mass, the invariant mass, will be the same.

Virtual quantum particles, in the middle of Feynman diagrams, don't have to have that mass. They can live (briefly) "off mass-shell".

The Higgs discovery relied on measuring events in which off-mass-shell Z bosons feature as the Higgs decays to four leptons (electrons or muons). The Higgs width result I wrote about last week relied on four-lepton events in which off-mass-shell Higgs bosons play a role. And also last week, ATLAS made a nice measurement of another bump in that distribution - the Z to four lepton cross section - which can only happen via off-mass-shell photons or Z bosons.

So there we are, back at the LHC, courtesy of Pythagoras, Einstein and Feynman. I realise that was a bit more maths than usually features in the Guardian, I think this blog may have gone briefly off mass-shell itself there. Still, I hope it was entertaining.

Jon Butterworth’s book, Smashing Physics, is out on 22 May. You can order it now!