02 June 2010

The Case For Three Generations of Fermions

LuboŇ° Motl at his blog, The Reference Frame lays out the empirical constraints that support a model of particle physics with precisely three generations of up quarks, down quarks, electrons and neutrinos.

Three generations of each of these types of particles have been discovered empirically, and that data rule out a fourth generation of neutrinos at a twelve standard deviation confidence level, and exclude any top quark with a mass below 335 GeV at a two standard deviation confidence interval.

He notes that even if there is some sort of phenomena generating data at a roughly 450 GeV mass, that it would be a poor fit for a fourth generation quark, and argues that even if there are bona fide new charge-parity (CP) violations, that a fourth generation of fermion (i.e. quarks, electrons and neutrinos in their various generations) is unlikely to be the cause.

How Strong Is The Case For Three Generations of Fermions?

The neutrino evidence is particularly convincing. Neutrinos have by far the lowest mass of the fermions, so our ability to detect them is less limited by the amount of energy that can be pumped into existing particle accelerators. Also, method that has been used to establish neutrino mass is based on oscillations between different generations of neutrinos.

In other words, we determine neutrino mass by looking at the entire system of all neutrinos at once, rather than looking for them one at a time. The presence of a fourth generation of neutrinos would be at odds with a mathematical model that fits the data very well with three generations of neutrinos and unrelated measurements from astronomy that put caps on the mass of particular neutrino generations.

The failure of research so far to detect a fourth generation of electron, which would be the lowest mass fourth generation fermion other than a neutrino, also tends to corroborate the neutrino evidence. This is notable because the kind of evidence used to detect a new generation of quark or electron is quite different from that used to detect new neutrinos.

Almost everyone in theoretical physics assumes that there are the same number of generations for each of the fermions. Even those who don't make assumptions that broad assume that at least electrons and neutrinos should have the same number of generations.

It also doesn't hurt the case for precisely three generations of fermions that the Lie group mathematics (which basically provides a framework into which most already discovered fundamental particles can be neatly fit into mathematically ordained boxes with particular relationships to each other) that has been a seemingly more than coincidentally good fit for the Standard Model is suggestive of the idea that there are a fixed, rather than an infinite, number of fermion generations.

The Standard Model needs some TLC.

The problem is that the cracks genuinely are starting to show in the Standard Model, and all of the leading contenders to solve these problems in the current standard model have their own problems.

The Standard Model Is Inconsistent With General Relativity

To start with the most fundamental issue, the Standard Model is not theoretically consistent with General Relativity, which explains gravity and the nature of time and space. Fortunately, we rarely need to apply either theory in the circumstances where they overlap. But, the flaws are patent.


There are at least two key discrepancies between the Standard Model and the empirical evidence.

First, the Standard Model does not explain the CP symmetry violations to the extent that we observe them, particularly in particles that are made in part from strange quarks. Only left-handed fermions and right-handed antifermions interact with the W boson, which is responsible for the weak force, and the neutrino likewise appears to be chiral, serving as its own antiparticle in its left handed and right handed versions. The strong force and electromagnetic force, in contrast, do not display CP symmetry violations. Incidentally, a CP symmetry violation is equivalent to a time symmetry violation.

Second, the Standard Model predicts the neutrinos have no mass, when we know that they have a slight mass that follows a pattern generally similar to the other three fermions that come in three generations.

Predictions That Have Been Made By Standard Model Extension Aren't Coming True

Another problem looming for the Standard Model is that most of its extensions predict phenomena that are failing to appear.

The Standard Model relies on a Higgs boson, Higgs field, and a score of other empirically derived constants describing interactions with the Higgs field to determine the mass of the fundamental particles. Yet, new experiments are again and again ruling out the existence of Higgs bosons in various mass ranges, providing not even plausible hints that a Higgs boson might be out there. And, the range of masses where it would still be possible to find a Higgs boson consistent with theories that rely upon it to impart mass to fundamental particles is increasingly small.

The Higgs boson isn't the only way to give fundamental particles mass. But, the theories that are the most natural mathematical extensions of the Standard Model predict phenomena like magnetic monopoles and relatively quick proton decay in circumstances that empirical evidence increasingly rules out. Other theories, like supersymmetry, predict the existence of gobs of new particles that there is not the slightest experimental evidence to support, with spin statistics never observed in any particle.

Mathematical progress in string theory has put it in the awkward position of establishing that almost all string theories are essentially just different ways of expressing the same underlying theory, which calls from phenomena like large numbers of dimensions, multiverses, or branes in which our world is embedded, for which there is again, no empirical evidence.

Dark Matter, Dark Energy, and Dark Flow

In addition to failing to produce predicted phenomena, the Standard Model has failed to provide an explanation for observed phenomena known as dark matter, dark energy and dark flow.

The Higgs field could explain dark energy, but we have to find a Higgs boson for that to work.

We need another high mass, neutrino-like particle or beyond the standard model gravitational physics to explain the phenomena we describe as dark matter. None of the currently know particles fit the bill, and if we have only three generations of particles that are aligned in the currently conventional grid, there is no place to put a new one, other than the hypothetical Higgs boson and graviton. And the graviton is expected to be a spin-2 particle with no mass that more or less precisely reproduces general relativity with little or no new physics, so it wouldn't help explain dark matter.

And, dark flow (an apparent bias in the acceleration of the universe from the big bang) is also unexplained.

In short, if the Large Hadron Collider rules out a Higgs boson in the mass range where the theory says that it belongs, the Standard Model is in much deeper trouble than it already is now. This could happen within a few years.

Loop Quantum Gravity Intuitions

Gravity, time symmetry in time dimension and particle mass, the areas where the gaps in quantum mechanics are starting to become clear, are the quintessential questions that General Relativity is designed to answer. My intuition is that most of what we don't know is of a piece. Solve one of the questions and you will solve them all. To that end, loop quantum gravity is starting to look increasingly attractive, not so much for what it does, as for what it does not do.

Emergently, But Not Fundamentally, Four Dimensional and Local

Loop quantum gravity (LQG) is emergently four dimensional, but is not, however, strictly local. This means that if you connect enough nodes of space-time together according to LQG rules, that it starts to behave like a four dimensional space, but the rules don't follow the strict continuity definitions that mathematicians impose in Abstract Analysis. A particular bit of space-time could be adjacent to both several other bits that seem to be next door, and to another bit that is a billion miles distant. These remote connections would be basically random, so they couldn't be used for a "warp drive" but might conceivably be used to transmit information across galaxies almost instantaneously. At the quantum scale, "next door" is ill defined in any manner other than connected to, or not connected to, within n many hops.

A not purely local space-time is something that entanglement phenomena are indicating is a match to the reality of ordinary standard model quantum physics.

LQG is local only emergently and only at the superquantum level, although some of the leading versions of the theory do insist on causality, even though the rate at which time passes varies in different places for different observers. A lack of strict locality may also help explain CP violations in the time dimension. When time is not absolute, it makes far more sense that time symmetries are not always observed. Since, CP violations are equivalent to time symmetry violations, the absence of time symmetry may help explain the time asymmetries of weak force decay. This is appropriate because weak force, which is the force that fuels the decay of nuclear materials with radioactive half-lives is intimately connected with our scientific understanding of time. The metric definition of the second is directly calibrated to a weak force driven form of radioactivity. Thus, LQG does not have the extra dimensions and metaphysical superstructures that string theory requires, and its time dimension related oddities are just where they should be.

Discrete and Background Independent

LQG is background independent, and the fact that it views time and space itself as discrete rather than continuous at approximately the Planck scale prevents singularities from arising. A lack of singularities rules out some of the more absurd macroscopic wormhole ideas, turns the question of what happened before the Big Bang into a sensical question, and fits well with current knowledge of black holes.

A universe where there is more out there than the universe spawned by dark flow would also provide a straightfoward explanation for the newly observed phenomena of dark flow, which is that something outside out universe is exerting a weak gravitational pull on it from the outside.

Mathematically Stable

A space-time made out of discrete chunks also solves a major mathematical problem, which is that a quantum field theory that replicates the results of general relativity at the scales where it has been established to be accurate is not renormalizable.

The equations that describe the strong force, the weak force and electromagnetism are solved with an infinite series approximation. But, one can safely use just the first n many terms of the series and lop off the rest and get a reasonably accurate answer because each new term of the series is increasingly unimportant.

The equivalent infinite series to describe gravity don't converge that fast. You can't just lop off every term after the first few and get a result that is reasonably accurate. Put another way, gravity is mathematically chaotic, i.e. it is sensitive to fine details of the numbers that go into the formula. But, if time-space is finite, then, at a theoretical level at least, the formula is finite rather than infinite, and every finite series is renormalizable. Thus, a discrete space-time makes equations in loop quantum gravity that would be fundamentally unstable at a mathematical level in an infinitely smooth universe stable mathematically, at least in theory.

LQG has other mathematical implications as well. Today, while we have quantum mechanical equations (quantum chromodynamics or QCD for short) that describe the way the strong force that holds atomic nuclei together works, in practice, the calculations are done not with these continuous space functions but with a discete mathematics calculation, generally viewed as an approximation called the "lattice method." If LQG is correct, however, lattice QCD may actually be more physical than the supposedlly "exact" equations used today.

A third mathematical implication is more mundane, but a joy to graduate engineering and physics students. Right now, theoretical physicists need an immense range of estoteric mathematical knowledge, because they need to know the graduate math necessary to consider any conceivably possible physics theory. But, if a theoretically and empirically sound quantum mechanics and quantum gravity are discovered, then this will no longer be necessary. Instead, physics and engineering students who need to understand quantum physics will only need to know the mathematics actually necessary to apply the theories that end up working.

In practice, this means that prospective quantum physicists may be able to learn the math that they need in a couple of college course, instead of a couple of years of college courses. An ambitious student with the mathematics background of a typical senior in physics or engineering could probably master it in a single summer.

Indeed, since a quantum gravity theory will be both more accurate than Einstein's theory of general relativity, and possibly conceptually and computationally simpler as well, it may even be possible to thin down the mathematics currently taught to support the physics of general relativity.

This sounds like mere residual college student laziness, but a key implication is that far more people will be able to master enough math to learn the fundamental laws of physics. It will never be so simple that every high school graduate will know it, but the percentage of college graduates who master this subject in its fully accurate mathematical form could increase tenfold or a hundredfold.

Doesn't Require New Particles Or Fields

LQG dispenses with the need for a Higgs boson and Higgs field, and provides a natural way to integrate dark energy into quantum gravity, while providing an equation to calculate gravitation effects that is at least as simple as the equation in existing non-quantum mechanical general relativity. LQG does not require the new particles that are necessary to make supersymmetry or technicolor theories work.

No results of LQG to date either contradict quantum physics or general relativity, but the possibility of some new physics in ares where general relativity has not been experimentally tested is not ruled out. Indeed, a discrete structure of space-time may provide a direct, simple and fundamental explanation for physical constants like Plank's constant and the fine structure constant.

It will still be necessary to determine what as yet undiscovered particle, or nuance of the quantum mechanical gravitational equation, or combination of the two, give rise to the phenomena observed and described as dark matter. But, discovering a single rather dull heavy particle (cold dark matter, probably in the form of a weakly interactive massive particle a.k.a. WIMP), or a slight twist in the way the gravity equation plays out, seems like a far less of a stretch than a theory that calls for a host of new particles, or needs a new particle for any of the theory to hold together.

An improved understanding of how the fundamental particles arise may also point us more directly toward dark matter candidates (the empty boxes in the theoretical scheme).

May Provide Deeper Understanding Of E=mc^2 and Residual String Theory

LQG may illuminate the fundamental process behind the equivalence of matter and energy. Matter itself in some version of LQG is an emergent property of space-time; basically matter is a tightly packed bunch of the space-time fabric in this theory. The loops of loop quantum gravity may end up being the very same loops of string theory, which may remain viable more or less intact to explain the subject matter that the Standard Model covers today in essentially the same topographical way. And, the underlying structure of LQG may appropriately motivate how those theories need to be adjuated to fit the experimental evidence.

If we understand how space-time turns into matter at a deeper level, it may become possible to calculate fundamental particle masses from first principles, something that would only have to be done once for each type of particle and would probably fall into a clear pattern once a few examples were mastered.


Despite the current mess, I'm more hopeful than at any time in recent history that we are on the brink of putting together a quantum gravity theory that will ultimately lead to the physicists holy grail of a "theory of everything."

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