## The Origin of Mass?

Hi everyone!

I’m Jim Pivarski, a guest blogger for the Everything Seminar. I was a graduate student in physics at Cornell, and now I am a physics postdoc at Texas A&M university, working on a particle detector at the Large Hadron Collider (LHC) at CERN. In future posts, I’d like to venture beyond my sub-field, but for now I want to talk about reasons why the LHC is exciting, particularly this business about the “God Particle.” It’s fun to see my research in the news, but I soon figured out that all the newspaper journalists read the same press release, and are under the impression that we’re building this expensive machine to discover the origin of mass. I’m uncomfortable with this because mass, or what people commonly mean when they talk about mass in everyday life, is already understood.

We all have an intuitive understanding of mass as the quantity which gives rise to weight and inertia, and this understanding is turned into explicit definitions in freshman physics: inertial mass resists acceleration under the application of a force according to Newton’s second law $F=ma$ and gravitational mass gives rise to a force between bodies according to Newton’s law of gravitation $Fg=GmM/d^2$. Early in the 20th century, it was suspected that mass is a form of energy, and Einstein discovered that this can be realized as a natural consequence of special relativity: $E=mc^2$ (plus other forms of energy). All very famous equations.

It would be glib for me to say at this point, “That’s it! I told you we understand it!” Einstein identified mass as a form of energy, but didn’t know what gave rise to that energy. Energy is a function of the configuration of a system: a spring has energy because (a) it is compressed against internal electromagnetic forces, (b) it contains heat, random motions of its constituent particles, (c) the table it is sitting on refuses to let it sink into the center of the Earth, (d) it is flying, with the solar system, toward the constellation of Virgo at 1.3 million km/h, and (e) it has mass. To really explain the origin of mass, it is necessary to explain how the arrangement of the spring’s particles gives rise to its mass-energy, in much the same way as “being squished” and “not being in the center of the earth” gives rise to its potential energy. This is delightfully ironic, considering how Aristotle sought to boil down all of nature to just matter and form. He couldn’t have guessed that matter itself is derived from form.

When I say that the origin of mass is understood, I mean nearly all of the matter we encounter in everyday life, the kind that is made of protons, neutrons, and electrons. Protons and neutrons account for 99.95% of my mass, leaving 0.05% to the electrons. The mass of the electron is mysterious, exactly what the LHC seeks to explain. But the protons and neutrons are known to be made of quarks, bound by an incredibly strong force called The Strong Force. Converted into conventional units, quarks attract each other with forces typically greater than 15 tons. The potential and kinetic energy of the quark orbits account for 99% of the mass of protons and neutrons; only the last 1% is due to the mass of the quarks themselves. This is relativistic mass in an extreme case— we are made, almost entirely, out of the attraction of quarks.

It may also be hasty to say that this quark attraction is understood, considering that quantitative calculations of the Strong Force are notoriously difficult to make precise. Early calculations in the 1970’s could only reproduce qualitative features of the hadronic mass spectrum— that is, the masses of all composite particles made of two or three quarks, like protons and neutrons. Until 2000, even the best calculations were only in 10-20% agreement with the data. Recently, however, faster computers and, most importantly, improved algorithms have made a new round of precise calculations attainable, achieving 2-3% precision in some cases. Another breakthrough occured earlier this year, as simulations revealing chiral symmetry breaking were performed in Japan.

Chiral symmetry breaking is a subtle effect of the Strong Force in which the a priori theory, with no masses assigned to the quarks, evolves into an effective theory with massive quarks, potentially explaining that last 1% of the mass of protons and neutrons. “Chiral” refers to the mirror-flip symmetry, that is, replacing a right hand with a left hand. Starting from a formulation of the Strong Force with exact mirror symmetry, the pairs of quarks and antiquarks which spontaneously spring out of the vacuum and immediately anihillate each other tend to associate before doing so, turning the vacuum into a condensate with a preferred chirality. The pure, symmetric theory is unstable; any slight perturbation will push it into an asymmetric state— this is what is meant by symmetry breaking. Nature does it easily, but we need immense computer calculations, simulating all the little quark-antiquark pairs, to see it in the theory.

Interestingly enough, when chiral symmetry is broken, quarks interacting with the ambient environment act as though they have mass. Massless quarks travelling at the speed of light are slowed when they interact with the quark-antiquark condensate and may be equivalently described as massive quarks in a free environment. People do a similar thing with light in table-top experiments, by cleverly arranging the electric and magnetic properties of materials, they can slow a light pulse to a few miles per hour, or stop it altogether. One could (but doesn’t, usually) say that they give the photons a mass. This language is more acceptable in the case of quarks, because the ambient environment is the vacuum itself, with its spontaneously broken chiral symmetry. Thus we can explain the mass of quarks with the same theory. But alas, not enough! It has been known for years that this mechanism can only give the quarks a small fraction of their measured masses. Some other effect must account for the rest, as well as the mass of electrons, which are completely blind to the Strong Force.

## The Origin of that last 1% of Mass

Let’s take stock. 99% of my mass is the mutual attraction of quarks inside of protons and neutrons, the majority of the last 1% is the mass of the quarks themselves, and 0.05% of the total is electrons. There’s a spontaneous symmetry breaking effect built into the Strong Force which can give rise to quark masses, but not enough, and not for electrons. But suppose that we extend the theory by adding a particle that (a) forms a condensate by sponteneous symmetry breaking and (b) interacts with all particles known to have a mass? One might be tempted to call such a particle… The God Particle! But I won’t.

I’m talking about the Higgs boson. Peter Higgs’s theory, in its simplest form, posits the existence of a new particle subject to a potential energy curve like the one shown below. The origin of the potential energy curve is the subject of advanced versions of the theory.

When the field value is zero (expected number of particles in a small volume is zero), the potential energy is higher than when the field value is “v” (246 GeV). The ball illustrates how the plus-minus symmetry would be spontaneously broken, as the field takes a non-zero value to minimize potential energy. We imagine this happened in the early universe: as the temperature cooled, the energy available for excitations diminished, and the symmetry just broke. The Higgs field actually has four real components, not one, so the field must choose from a three-sphere for every point in space, and eventually, one value must dominate everywhere to minimize the kinetic energy of the field. Equivalently, we can say that space is filled with a uniform condensate of Higgs particles, perhaps resembling Bose-Einstein condensates in table-top experiments.

Quarks and electrons interact with the ubiquitous Higgses, giving them an effective mass. The qualitative imagery newspapers usually use to describe this process involves thinking about the Higgs field as molasses, with particles struggling to accelerate through it, thus explaining inertial mass. I’m uncomfortable with this analogy because drag forces are always a function of velocity, and will slow a particle to a stop, relative to the ambient fluid. Electrons, however, stream freely through the Higgs field, at a speed which is slower than the speed of light. What’s actually happening is that the electron-Higgs interaction, with the background Higgs field taken as a given fact, can be expressed as a quadratic potential for electron field. This potential provides an energy cost to exciting the electron field from a zero-electron state to a one-electron state. This energy cost is the mass energy, and Einstein’s relationship between energy and mass takes over from there.

Although the Higgs mechanism provides an explaination for the origin of quark masses, electron masses, and the masses of “heavy electrons,” namely muons and taus, that’s not why it was invented, and that’s not why many (most?) physicists find it interesting. The Higgs mechanism is much more important for electroweak symmetry breaking, a story that goes back to the beginning of particle physics in the 1930’s. Enrico Fermi observed that radioactive decays may be described as a new force, much weaker than the rest, which came to be known as The Weak Force. The Weak Force resembles electromagnetism on a deep level, with the exception of being 100,000 times weaker. This weakness could be naturally explained if Weak’s analogs of the photon, $W^+$, $W^-$, and $Z^0$, have huge masses, around 90 GeV. Heavy photons with all the right properties were discovered at CERN in 1980, vindicating Glashow, Salam, and Weinberg’s electroweak theory, the cornerstone of the Standard Model, but with one small problem. A “local gauge symmetry” is key to the derivation of the electromagnetic theory, and by extension, the electroweak, but massive counterparts to the photon spoil local gauge symmetry! The Higgs mechanism is the only known solution to this problem (“higgsless” models derive Higgs-like particles as composites of more fundamental particles), and without it, we can’t really claim to understand how electroweak symmetry is broken into a strong electromagnetic force with massless photons and a weak Weak Force with massive gauge bosons. And thus, the hunt for the Higgs is on! (No discovery yet.)

Moreover, the Higgs field is a key part of supersymmetry, grand unified theories, and almost everything that high-energy physicists are excited about. Many of these extensions of the Standard Model even provide explainations for the shape of the Higgs potential. Precise measurements of the Higgs’s properties might teach us how the electroweak force unifies with Strong.

But precise measurements will take years of study and accumulated data, and are motivated by the breaking or non-breaking of esoteric symmetries. To say that we’re searching for the origin of mass definitely sounds interesting, but it makes me cringe when it’s the only reason given for building the LHC.

## Unless, of course, you’re talking about Dark Matter…

I realized in the midst of writing this article that “the origin of mass” could take a completely different meaning: not the mass of rocks, trees, and people, but the majority of mass in the universe. An astronomical perspective always slants everything. Astronomically speaking, we are incredibly dense, tightly-packed beings— the majority of the universe being gas— unless you compare us with the center of any astronomical body. It’s also peculiar that we’re made of so much carbon and so little helium. In the same spirit, “the origin of mass” could be better interpreted as “the origin of dark matter,” or, the origin of 25% of the universe as seen by astronomers, pretty much the only mass that mattered in the clustering of galaxies that made the universe what it is today.

Sadly, this is not the place to discuss such things, because if I did, this blog entry would become Very Long. I will say only three things: (1) there are good reasons, not bullet-proof, but good, to believe that the LHC will create dark matter particles and allow us to study their properties in a controlled environment, (2) I happen to think that the prospect of dark matter detection is more exciting than the Higgs, and (3) this is certainly not what the newspapers mean by “the Origin of Mass.” They have a whole different set of metaphors for the dark matter/dark energy side of the universe, many of them invoking Darth Vader.

### 21 Responses to “The Origin of Mass?”

1. nc Says:

The qualitative imagery newspapers usually use to describe this process involves thinking about the Higgs field as molasses, with particles struggling to accelerate through it, thus explaining inertial mass. I’m uncomfortable with this analogy because drag forces are always a function of velocity, and will slow a particle to a stop, relative to the ambient fluid.

Even worse: the analogy of inertial mass increase and length contraction by the ‘snowplow’ effect of a particle moving in the Higgs field which can’t quite get out of the way fast enough as the particle speed is increased. This nonsense is even more disconcerting, because it destroys the reliance on the mathematical beauty of special relativity, supplying a mechanism.

It’s a step backwards towards FitzGerald’s discovery of the contraction in 1889 and Lorentz’s development of it in 1893, whereby bodies get contracted for a physical reason (pressure in the direction of motion). It’s far better to pretend that special relativity is the basis of general relativity, instead of an approximation made by absurdly assuming in general relativity that the mass-energy of the universe is M = 0. Put M = 0 into the Schwartzchild metric and you get the Minkowski metric, thus general relativity is ‘built on special relativity’, by the same brilliant reasoning that ‘animals must be made of manure, since they produce manure.’ A very clever and useful deduction to make.

Moreover, the Higgs field is a key part of supersymmetry, grand unified theories, and almost everything that high-energy physicists are excited about.

Making superpartners conveniently too heavy to be observed is – like making extra spatial dimensions too small to be observed – an act of faith. We have to repect those who work as clowns, because they’re a hard act to follow.

2. jpivarski Says:

Thank you for your interest; I’d just like to comment on the bit about the clowns, which I confess I’m not certain I understood. We would be fools if we simply believed beyond-the-Standard-Model theories without evidence. That’s what makes the LHC so exciting: it’s reckoning time.

In my last experiment, I carefully measured decay constants of the well-known Upsilon particles. It was gratifying to see all the tiny effects fall into place— I still had a lingering doubt that physics somehow wasn’t real. But in this next project, I want to search for something unknown, because it’s more exciting that way. Everyone has to tune their preferred ratio of excitement to probability of wasting their time.

The arguments people make that they are not wasting their time, based on naturalness or quadratic divergences or unitarity, are valuable, because without these quantitative hints we would have no guide at all. But we all know that they are a different thing from evidence.

3. dorigo Says:

Wow, great article Jim. I am linking this site before you say quark.

Cheers,
T.

4. Alejandro Rivero Says:

A minor mistery, at least to me, is why the mass of the other two generations of charged leptons happens to be in the typical values of QCD masses. One of them is less than two times the value of the mass of the proton (and near the conjectured glueball) and the other so near of the pion mass than when it was discovered it was believed to be the pion, and only later asessment of the properties concluded it was a particle nobody had predicted (“who ordered this?”). In principle these values can be arbitrary; I can not quote any serious restriction for the muon mass, while for the tau there is only, from GUT arguments, a relationship with the mass of its generation parner, the botton quark.

5. jpivarski Says:

This is a test:

6. jpivarski Says:

Nope, the inline image didn’t work. 🙂

I was thinking about Alejandro’s minor mystery (which I’ve never heard stated before), and decided to make a plot, which can be found at

http://hepr8.physics.tamu.edu/pivarski/tmp/qcd.svg

It’s a histogram of all unflavored QCD meson masses (everything made of a pair of up and down quarks) known to the PDG. By the way, “GeV” ought to be “MeV.”

Indeed, the pion is a lightweight outlier, and it’s very close to the muon mass. Without chiral symmetry breaking, the pion’s mass would be zero, further singling it out as special.

The third generation of the electron family, tau, is in the middle of the unflavored QCD spectrum. I wouldn’t read anything from the fact that it’s in the middle: the shape of that distribution mostly depends on experimentalists’ ability to identify these as individual particles. Many of these “particles” are broad peaks and interfere strongly with their neighbors— two particles with the same mass can interfere constructively to look like one particle with a strong signal, destructively to look like nothing at all, or, more generally, make a very complicated curve that’s difficult to fit.

You know, if we expected electron (0.5 MeV), muon (105 MeV), and tau (1770 MeV) masses to resemble the down-type quark masses, down (5 MeV), strange (95 MeV), and bottom (4500 MeV), only the last one is rather far off. in fact, the down-type/lepton agreement is better than the down-type/up-type agreement: up (2 MeV), charm (1250 MeV), and top (175000 MeV). Both patterns diverge mostly in the third generation. In the Higgs mechanism, these patterns derive from Higgs coupling strengths, which looks about right to me, especially in supersymmetry, where there’s one Higgs for down-type and another for up-type particles.

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8. doc Says:

Sure sure.
The real origin of the pion mass:

e=1.60217648740e-19,
h=6.6260689633e-34,
m0=9.1093821545e-31,
c=2.99792458e8,
alf=1.0/7.297352537650e-3;

m0=0.51099891*e*1e6/(c*c);
E=m0*c*c;
f0=E/h;

m=(139.57018/2.0)*e*1e6/(c*c);
v = sqrt(-(m0*m0* c*c -c*c*m*m)/(m*m));

u=c/alf;
v2=(u+v)/(1.0+u*v/(c*c));
f1=f0*sqrt(1.0+v2/c)/sqrt(1.0-v2/c);
f2=f0*sqrt(1.0-v2/c)/sqrt(1.0+v2/c);
f3=(f1-f2)/2;

u=-u;
v2=(u+v)/(1.0+u*v/(c*c));
f1=f0*sqrt(1.0+v2/c)/sqrt(1.0-v2/c);
f2=f0*sqrt(1.0-v2/c)/sqrt(1.0+v2/c);
f4=(f1-f2)/2;

f=(f3-f4)/2;
E=f*h/(e*1e6);

printf(“%f Mev\n”,E);
printf(“%f m/m(e)\n”,E/0.51099891);

There is double wavepacket.
Its a resonance with atomic electrons. Accuracy is 99.65% !
f3 and f4 is De Broglie wave, f0 is Compton wave.

F is nothing more than an atom in a relativistic point of view. Its Lorentz-contracted.

9. Name Says:

I would call it the God particle.

10. J.A Says:

I will show you Jesus christ with physics. He is at step 2 on this youtube. JESUS CHRIST PHYSICS (making mass) http://youtu.be/AN1jTwHUC7E?a
Just trust me. You can touch a time.

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17. Nicholas Siefert Says:

Quick question from a non-expert:
You mentioned that 99% of the mass of the proton is due to the potential and kinetic energy of the quarks inside of the proton. Shouldn’t the sum of kinetic and potential energy be a term that is negative? (Or is this only for forces like gravity and E&M that drop off as 1/r2?) If the sum is negative (i.e. confinement), this would imply negative mass rather than positive mass, right?
Could you expand on what you mean by 99% of the mass can be explained by kinetic and potential energy?
Thank you

18. Justin Ingalls Says:

Jim Pivarski
I have long been interested in gravity and I think I have the big picture figured out. At the smallest scale the problem is more difficult.
There is a related question I would like the answer to. It may be physics 101 to you but the answer has escaped me. Will you please send me the answer, my e-mail address is below.
An object of weight 10 moving at velocity 10 has X lb/ft energy. If the weight is doubled to 20 and the velocity remains 10 the energy is 2X. However, if the weight remains 10 and the velocity is doubled to 20 the energy becomes 4X.
Why does the energy increase linearly with the increase in weight but it increases by the square of the velocity?
If you are interested in a layman’s insights concerning gravity gained over many years of thought and some study, I would be interested in a correspondence on that subject.

Justin Ingalls

19. Anonymous Says:

i have not yet get the understanding of “what is Chiral symmetry and how it is breaked” i am making my presentation on “mass and its specification in particle physics”
“Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of the strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory describing these interactions, and is responsible for the bulk of the mass (over 99%) of the nucleons, and thus of all common matter, as it converts very light bound quarks into 100 times heavier constituents of baryons”

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