*In the process of revising a chapter of my book, I found a clip I would like to share here, as it contains an analogy I cooked up and which I find nice enough to be proud of. Well, two analogies, as you'll soon find out; here I am speaking of the cat weighing trouble at the end of the piece - the other is quite trivial.*

The topic is the widely different masses of fermions, the building blocks of our universe, and the trouble in making sense of it and of measuring precisely their values. Comments welcome!

The topic is the widely different masses of fermions, the building blocks of our universe, and the trouble in making sense of it and of measuring precisely their values. Comments welcome!

Mass is a mysterious intrinsic property of matter corpuscles. We are accustomed to confuse the mass of an object with its weight, since the two quantities are proportional to one another as long as we remain in the roughly constant gravitational field of Earth’s surface. But mass is not a force: rather, it is the attribute of a material body specifying how hard it is to change its status of motion. The larger the mass of an object, the stronger the pull one has to apply to set it in motion at a given speed, or to modify its trajectory. It was the large mass of 2nd and 3rd generation fermions what prevented their discovery for the better part of the twentieth century; all that was known in the thirties were the proton, the neutron, the electron and its antiparticle, and the photon. In fact, mass can be created in particle reactions when the kinetic energy released in a collision is converted into new matter states, following the famous formula E=mc^2, where the squared speed of light in vacuum c^2 acts as a proportionality constant between energy E and mass m.

To visualize just how different the masses of fermions are, and how mysterious it is that those incongruous entities all fit in the tidy scheme outlined above, try imagining that we were measuring fermion masses with a yardstick as if they were lengths in space, and we agreed that the length corresponding to the mass of the lightest charged fermion, the electron, fit in the first notch of our scale –say one inch. Under such circumstances, in order to measure distances corresponding to masses like that of the top quark, we would need a yardstick about a kilometer long! The whole point here is that the masses of elementary fermions constitute an unsolved enigma. We have learned how to measure them with good precision (except those of neutrinos, for which so far we only know that a proper yardstick should be able to discern lengths well below the nanometer), but the standard model gives neither any clue on the reason for the enormous range in results we find, nor for their observed pattern. In fact, fermion masses are free parameters of the model, which we are bound to infer only from the experimental measurement in order to complete the picture.

It would be rather silly for us to stick to the dubious analogy of measuring masses with a yardstick. In the following chapters you will instead become familiar with a more standard treatment of mass and energy, two quantities which –with a convention simplifying matters a whole lot –can be sized up with the same units. All works in multiples of the electron-Volt, the energy that is imparted to a particle carrying the electric charge of one electron by moving it “upstream” through a potential difference of one Volt. The electron-Volt (usually written eV) is a quite small amount of energy: its order of magnitude is the energy of atomic transitions between nearby energy levels. It takes little more than a dozen eV to free the electron of a hydrogen atom from its electromagnetic bond with the nucleus, and less than one eV is enough to make the same electron jump from one atomic orbit to another. In this book we will often use a multiple of the eV called GeV, where the G stands for giga-, the ancient Greek prefix for a billion. A proton’s mass is close to a GeV, and so are the masses of many other composite bodies whose study has opened our way to the understanding of sub-nuclear physics.

Now armed with a reasonable measurement unit, let us give a look at the mass hierarchy of the fundamental objects we have encountered so far. Neutrinos are the lightest elementary particles among fermions. They are much lighter than one eV! In fact they are so light that we have so far been unable to measure their mass. Next up is the electron, with a mass of 511 thousand eV –511 keV, where the prefix k signifies a thousand. The muon is still 200 times heavier, at 105 million electron-Volts: 105 MeV, where M stands for a million. The tauon, the heaviest lepton, is some further 17 times heavier at 1.77 GeV –where as noted above G stands for a billion. As for quarks, the lightest is the up quark, which weighs about 3 MeV, and the heaviest is the top quark, whose hefty mass totals 173 GeV. In-between lie the down quark (7 MeV), the strange quark (90 MeV), the charm quark (1.2 GeV), and the bottom quark (4.2 GeV).

The numbers above let you appreciate the existence of an unsolved mystery. We in fact have no clue of the reason of that hierarchy of values, although the discovery of the Higgs boson has at least allowed us to cast the problem in different terms. You should also bear in mind that the ones I gave above are approximate numbers: the masses of all quarks except the top cannot be measured directly, as those particles can only be studied within hadrons, the bound states they form by combining in pairs or triplets. Like a cat that would not be convinced to stay still on a scale, quarks require us to use indirect methods to infer their mass. You can estimate your cat’s mass by subtraction, measuring yourself with and without the cat in your arms; similarly, you can size up quark masses by comparing the masses of hadrons that contain them or others.

Dear Tommaso,

It would take you one week at the most to thoroughly study/analyze The gem 5, 6, 7, 8, 9, and both the mass and inertia would not be the mystery for you any more.

The first 4 texts are about how could had been derived that what we today call “relativity theory”, in the year 1900 already. You can skip that, and start from The gem (5).

Please, do read The gem 5, 6, 7, 8, 9

My intentions are good. I do not want to distract you. I just ask you to read something that I think might help you, might give you some new idea, might make your book better.

Best wishes,

Vera