# The Standard Model

## Discussion

### introduction

The standard model is the name given in the 1970s to a theory of fundamental particles and how they interact. It incorporated all that was known about subatomic particles at the time and predicted the existence of additional particles as well.

There are seventeen named particles in the standard model, organized into the chart shown below. The last particles discovered were the W and Z bosons in 1983, the top quark in 1995, the tau neutrino in 2000, and the Higgs boson in 2012.

family | particle | predicted/ discovered |
spin number |
charge (e) |
color | *mass* (MeV/c ^{2}) |
|||
---|---|---|---|---|---|---|---|---|---|

f e r m i o n s |
q u a r k s |
u | up quark | 1964 | 1968 | ½ | +⅔+ | r, g, b | 2.16 |

d | down quark | 1964 | 1968 | ½ | −⅓− | r, g, b | 4.67 | ||

c | charm quark | 1970 | 1974 | ½ | +⅔+ | r, g, b | 1,270 | ||

s | strange quark | 1964 | 1968 | ½ | −⅓− | r, g, b | 93.4 | ||

t | top quark | 1973 | 1995 | ½ | +⅔+ | r, g, b | 172,690 | ||

b | bottom quark | 1973 | 1977 | ½ | −⅓− | r, g, b | 4,180 | ||

l e p t o n s |
e | electron | 1874 | 1897 | ½ | −1− | none | 0.510 998 950 | |

μ | muon | 0000 | 1936 | ½ | −1− | none | 105.658 375 5 | ||

τ | tau | 0000 | 1975 | ½ | −1− | none | 1,776.86 | ||

ν_{e} |
electron neutrino | 1930 | 1956 | ½ | 0 | none | < 0.000 000 8 | ||

ν_{μ} |
muon neutrino | s1940s | 1962 | ½ | 0 | none | ? | ||

ν_{τ} |
tau neutrino | s1970s | 2000 | ½ | 0 | none | ? | ||

† | p | proton | 1815 | 1917 | ½ | +1+ | none | 938.272 088 2 | |

n | neutron | 1920 | 1932 | ½ | 0 | none | 939.565 420 5 | ||

b o s o n s |
‡ | H | Higgs boson | 1964 | 2012 | 0 | 0 | none | 125,250 |

v e c t o r |
g | gluon | 1962 | 1978 | 1 | 0 | 8 colors | 0 | |

γ | photon | 0000 | 1899 | 1 | 0 | none | 0 | ||

W | W boson | 1968 | 1983 | 1 | ±1± | none | 80,377 | ||

Z | Z boson | 1968 | 1983 | 1 | 0 | none | 91,187.6 |

### particle families

Fundamental particles are either the building blocks of matter, called fermions, or the mediators of interactions, called bosons. There are twelve named fermions and five named bosons in the standard model.

Fermions obey a statistical rule described by Enrico Fermi (1901–1954) of Italy, Paul Dirac (1902–1984) of England, and Wolfgang Pauli (1900–1958) of Austria called the exclusion principle. Simply stated, fermions cannot occupy the same place at the same time. (More formally, no two fermions may be described by the same quantum numbers.) Leptons and quarks are fermions, but so are things made from them like protons, neutrons, atoms, molecules, people, and walls. This agrees with our macroscopic observations of matter in everyday life. People cannot walk through walls unless the wall gets out of their way.

Bosons, in contrast, are have no problem occupying the same place at the same time. (More formally, two or more bosons may be described by the same quantum numbers.) The statistical rules that bosons obey were first described by Satyendra Bose (1894–1974) of India and Albert Einstein (1879–1955) of Germany. Gluons, photons, and the W, Z and Higgs are all bosons. As the particles that make up light and other forms of electromagnetic radiation, photons are the bosons we have the most direct experience with. In our everyday experience, we never see beams of light crash into one another. Photons are like phantoms. They pass through one another with no effect.

Elementary particles have an intrinsic spin angular momentum *S*. The adjective intrinsic means innate or essential to the thing itself. Elementary particles don't have spin because someone is spinning them. They just spin — or rather, they just have a measurable quantity with the same units as angular momentum. In current physics, elementary particles are featureless — like a mathematical point. In order for something to be perceived as spinning, the thing spinning would need something like a "front" and a "back". Featureless, point particles don't have anything like that. Particle physics is best described with mathematics. Spin is a convenient label for a measurable quality and not a description of reality.

Every elementary particle has associated with it a spin quantum number *s* (often called the spin number or just the spin), where *s* is any whole number multiple of a half. Fermions have half integral spin quantum numbers (½, 1½, 2½, etc.) and bosons have integral spin quantum numbers (0, 1, 2, etc.). No spin numbers are possible in between these. Spin is a quantized quantity.

The elementary fermions have a spin of ½. Particles made from combinations of fermions will have an overall spin that's a combination of the individual spins. A baryon composed three quarks will combine to an overall spin of ½ or 1½, since those are the only possible, non-negative combinations of ½ ± ½ ± ½. That shows that all baryons (like protons and neutrons, for example) are also fermions. Likewise, a meson composed of a quark and an antiquark will combine to an overall spin of 0 or 1 since those are the only possible, non-negative combinations of ½ ± ½. That shows that all all mesons (like the pion of the residual strong interaction, for example) are also bosons.

The force carying bosons of the standard model (gluons, photons, and the W and Z) have spin 1 since they go with vector fields. The Higgs boson corresponds to a scalar field so it has spin 0. If the particle of the gravitational field is ever discovered, it would be called a graviton and would have a spin 2 since it corresponds to a tensor field. A tensor is a mathematical object that's more complex than a vector, which is in turn more complex than a scalar. See the trend? A scalar field with *no direction* gets a particle with spin *0*. A vector field with *a direction* gets a particle with spin *1*. A tensor field that stretches and squeezes space along *two directions* gets a particle with spin *2*.

All fundamental and composite particles have a spin quantum number *s* (lowercase). This is associated with a spin angular momentum *S* (uppercase). The SI unit of angular momentum is the kilogram meter squared per second [kg m^{2}/s] or, equivalently, the joule second [J s], which is much too large for elementary particles. Instead ℏ (h bar), also known as the reduced Planck constant (ℏ = *h*/2π), is used. For reasons that are beyond the scope of this book, the spin quantum number *s* (which is just a number) and the spin angular momentum *S* (which is a number with a unit) are not numerically the same. Instead, they are related by a non-obvious equation.

S = |
⎡ ⎣ |
s |
⎛ ⎝ |
s |
+ 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ | |

For particles with spin quantum number of 0, the solution is a sensible spin angular momentum of 0 ℏ.

S(0) = |
⎡ ⎣ |
0 | ⎛ ⎝ |
0 | + 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ = 0 ℏ | |

For higher spin quantum numbers the spin angular momentum increases, but beyond that there's not much else that can be simply said.

S(½) = |
⎡ ⎣ |
^{1}_{2} |
⎛ ⎝ |
^{1}_{2} |
+ 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ = | ^{√3}_{2} |
ℏ | |

S(1) = |
⎡ ⎣ |
1 | ⎛ ⎝ |
1 | + 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ = | √2 | ℏ | |

S(1½) = |
⎡ ⎣ |
^{3}_{2} |
⎛ ⎝ |
^{3}_{2} |
+ 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ = | ^{√15}_{2} |
ℏ | |

S(2) = |
⎡ ⎣ |
2 | ⎛ ⎝ |
2 | + 1 | ⎞ ⎠ |
⎤^{½}⎦ |
ℏ = | √6 | ℏ | |

Fermions are divided into two groups of six: those that *must* bind together are called quarks and those that *can* exist independently are called leptons.

The word "quark" originally appeared in a single line of the the novel *Finnegans Wake* written by the Irish author James Joyce (1882–1941). The protagonist of the book is a publican named Humphrey Chimpden Earwicker who dreams that he is serving beer to a drunken seagull (no joke). Instead of asking for "three quarts for Master Mark" the inebriated bird says "three quarks for Muster Mark". Since the pre-standard model theory was complete with only three quarks, the name made some sense. The full standard model today needs six quarks. That hasn't made the word any less fun to say. Quark! The six flavors of quark are up, down, strange, charm, top, and bottom. The names of the flavors are essentially meaningless.

Quarks are known to bind into triplets and doublets. The triplets are called baryons, a term derived from the Greek word βαρύς (*varys*) meaning "heavy". The doublets are called mesons, a term derived from the Greek word μέσος (*mesos*) meaning "medium". Collectively baryons (the heavy triplets), mesons (the middleweight doublets), and quarks (the fundamental particles) are known as hadrons, from the Greek word αδρός (*adros*) meaning thick, robust, massive, or large. This name alludes to the ability of the point-like quarks to bind together and form particles that are "thick" in a certain sense.

The other six fermions are called leptons, a name derived from the Greek word λεπτός (*leptos*) meaning thin, delicate, lightweight, or small. These particles don't need to bind to each other, which keeps them "thin" in a certain sense. Originally leptons were considered the "light" particles and hadrons the "heavy" particles, but the discovery of the tau lepton in 1975 broke that rule. The tau (the heaviest lepton) is almost twice as massive as a proton (the lightest hadron).

Baryons found in the nucleus (the proton and neutron) are called nucleons. The Latin word for kernel is *nucleus*. Nucleons are found in the metaphorical "kernel" of the atom. Baryons that contain at least one strange quark but no charm, bottom, or top quarks are called hyperons. The Greek word for beyond is υπέρ (*yper*), which morphed into the English prefix *hyper*-. Hyperons are particles that are "way out" in a certain sense.

The neutrinos are an important subgroup within the leptons. They come in three flavors named for their partner leptons. The electron, muon, and tau are matched with the electron neutrino, muon neutrino, and tau neutrino. Neutrinos have very little mass (even for leptons) and interact so weakly with the rest of the particles that they are exceptionally difficult to detect. The name is a play on words. The Italian word for neutron (*neutrone*) sounds like the word neutral (*neutro*) with an augmentative suffix (-*one*) tacked on the end. That is, it sounds something like "big neutral" to Italian ears. Replace the augmentative suffix -*one* with the diminutive suffix -*ino* and you have a "little neutral", which is a good description of what a neutrino is — a diminutive neutral particle. Aaaaaw, so small and neutral.

Fermions belong to one of three known generations from ordinary (I), to exotic (II), to very exotic (III). (These are the adjectives I selected to describe the generations.) Generation I particles *can* combine to form hadrons with effectively infinite life spans (stable atoms made of electrons, protons, and neutrons for example). Generation II particles always form unstable hadrons. The longest lived hadron containing a generation II quark is the lambda particle (made of an up, down, and strange quark). It has a mean lifetime of less than a billionth of a second, which is considered long-lived for an unstable hadron. Generation III particles are divided in their behavior. The bottom quark isn't much stranger than a strange quark, but the top quark is so short-lived that it doesn't exist long enough to do anything. It falls apart before the world even knows it exists. Top quarks are only known from their decay products.

### particle interactions

Three of the four fundamental fources of nature are included in the standard model of particle physics — electromagnetism, the strong force, and the weak force. (Gravity is not included in the standard model.) Each force acts between particles because of some property of that particle — charge for electromagnetism, color for the strong force, and flavor for the weak force. The bosons associated with each force are called gauge bosons — the photon for electromagnetism, gluons for the strong force, and the W and Z bosons for the weak force.

Charge is the property of matter that gives rise to electric and magnetic phenomena (known collectively as electromagnetism). Charge is quantized, which means it can only exist in discrete amounts with restricted values — multiples and fractions of the elementary charge (e = 1.6 × 10^{−19} C). Particles that exist independently (the electron, muon, and tau) carry multiples of the elementary charge (−1 e), while quarks carry fractions of the elementary charge (+⅔ e or −⅓ e). Quarks always bind together in groups whose total charge is an integral multiple of the elementary charge, which is why no one has ever directly measured a fractional charge. In addition, since opposite charges attract, electrons tend to bind to protons to form atoms that are neutral overall. We don't normally notice the electrical nature of matter because of this.

Charged particles interact by the exchange of photons — the carrier of the electromagnetic force. Whenever one electron repels another or any electron orbits a nucleus, a photon is responsible. Photons are massless, uncharged, and have an unlimited range. The mathematical model used to describe the interaction of *charged particles* through the exchange of *photons* is known as quantum electrodynamics (QED).

Quarks stick to other quarks because they possess a characteristic known as color (or color charge). Quarks come in one of three colors: red, green, and blue. Don't let the words mislead you. Quarks are much too small to to be visible and thus could never have a perceptual property like color. The names were chosen because of a convenient analogy. The colors of quarks in the standard model combine like the colors of light in human vision.

Red light plus green light plus blue light appears to us humans as "colorless" white light. A baryon is a triplet of one red, one green, and one blue quark. Put them together and you get a color neutral particle. A color plus its opposite color also gives white light. Red light plus cyan light looks the same to humans as white light, for example. A meson is a doublet of one colored quark and one anticolored antiquark. Put them together and you get a color neutral particle.

There's something about color that makes it want to hide itself from anything bigger than a nucleus. Quarks can't stand being apart from one another. They just have to join up and always do so in a way that hides their color from the outside world. One color is never favored over another when quarks get together. Matter is color neutral down to the very small scale.

Colored particles are bound together by the appropriately named gluons. Gluons are also colored, but in a more complicated way than the quarks are. Six of the eight gluons have two colors, one has four, and another has six. Gluons glue quarks together, but they also stick to themselves. One consequence of this is that they they can't reach out and do much beyond the nucleus.

The mathematical model used to describe the interaction of *colored particles* through the exchange of *gluons* is known as quantum chromodynamics (QCD). The whole sticky mess is called the strong force or the strong interaction since it results in forces in the nucleus that are stronger than the electromagnetic force. Without the strong force, every nucleus would blow itself to smithereens.

There are twelve named elementary fermions. The difference between them is one of flavor. The word "flavor" is used here to mean "type" and it applies only to fermions. Don't let the word mislead you. Subatomic particles are much too small to have any characteristics that could be directly observed by human senses.

Flavored particles interact weakly through the exchange of W or Z bosons — the carriers of the weak force (also known as intermediate vector bosons). When a neutron decays into a proton, a W^{−} boson is responsible. The mathematical model used to describe the interaction of *flavored particles* through the exchange of *W and Z bosons* is sometimes known as quantum flavordynamics (QFD), but this is a term that is not used by working particle physicists. At higher energies, the weak and electromagnetic forces begin to look more and more alike. The mathematical model that describes these interactions together is known as electroweak theory (EWT). This is the conventional name for the theory of the weak force.

### mass and gravity

All fermions are thought to have a nonzero mass. Particles in generation I are less massive than those in generation II, which are less massive than those in generation III. Within the generations, quarks are more massive than leptons and neutrinos are less massive than the other leptons. Bosons are divided when it comes to mass. Gluons and photons are massless. The W, Z, and Higgs bosons are massive.

Mass is energy. A moving particle is more massive than a stationary particle because it has kinetic energy. Logically then, a stationary particle should have no mass. If we could stop a photon (which we can't) it would weigh nothing. Our logic appears to be working. If we could stop an electron (which we can) it would weigh something. Our logic is broken. Why do some particles weigh something at rest and others weigh nothing?

Mass is energy and energy comes in two types: kinetic energy (the energy of motion) and potential energy (the energy of arrangement). The kinetic energy contribution to mass is minor. Most of the mass around us comes from some sort of potential energy. For example, a proton is made of two up quarks and a down quark. The masses of these three quarks do not add up to the mass of a proton.

m_{p} |
≠ | 2m +1_{u}m_{d} |

938.272 MeV/c^{2} |
≠ | 2(2.3 MeV/c^{2})^{2}) |

938.272 MeV/c^{2} |
≠ | 9.4 MeV/c^{2} |

The masses of the parts are only 1% of the mass of the whole. The remaining 99% comes from the potential energy of the strong force holding the proton together. The particles that mediate the strong force are gluons. The interaction energy of these massleess particles is what gives the proton most of its mass.

So why do quarks have mass but the gluons don't? Or as the question was historically stated, why do the W and Z bosons have mass but the photon doesn't? Maybe there's another kind of potential energy. Maybe there's another interaction out there — an interaction that some particles feel and others don't. If there's an interaction, there's a particle — a particle that gives mass to other particles when they're just sitting around doing nothing. The interaction that gives mass to elementary particles was proposed in 1964 by scientists in three independent locations.

- François Englert and Robert Brout at
*L'Université Libre de Bruxelles*in Belgium - Peter Higgs at the University of Edinburgh in Scotland
- Gerald Guralnik, Carl Hagen, and Tom Kibble at Imperial College, London

It should be called the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism, but it isn't. For whatever reason, the interaction that gives mass to elementary particles is called the Higgs mechanism and the particle that mediates the interaction is called the Higgs boson, the Higgs particle, or (rarely) the higgson.

All of space is assumed to be filled with a Higgs field — a background sea of virtual Higgs bosons that pop in and out of existence. The quarks, leptons, and W and Z bosons moving around through space interact with this field, which is why these particles have mass. The photons and gluons do not interact with the Higgs field, which is why these particles do not have mass. Even the Higgs boson itself interacts with the Higgs field. It gives itself mass! The Higgs boson is different from the other bosons (gluons, photons, W and Z bosons) in that the Higgs mechanism doesn't result in anything resembling a force (like the strong, electromagnetic, and weak forces). The Higgs field is a scalar field and the Higgs boson is a particle with spin zero.

Gravity is the force between objects due to their mass. The mathematical model that would describe gravity on the particle level is sometimes called quantum geometrodynamics (QGD), but is more often referred to as quantum gravitation. The standard model of particle physics does not include gravity (nor could it ever) and there currently is no quantum theory of gravitation. If there was, it would have to include a force carying particle. The proposed name for this particle is the graviton. General relativity describes gravitational waves as a tensor disturbance that propogates — one that shears space-time along two alternating perpendicular directions. This two dimensional behavior leads theoretical physicists to believe that the graviton would have spin two.

It is hoped that gravity will be taken care of in a theory beyond the standard model. In an extreme case of overconfidence, some theorists propose that such a theory would be a theory of everything. Given the history of science (and of life in general), anything that claims to be the ultimate representation of reality (scientific, economic, cultural, or religious) is certainly doomed to be superseded by something bigger and better — or at the very least, something less wrong.

### names, names, names

The theme of this topic seems to be "names, names, names".

group | statistics | eponymous root | |
---|---|---|---|

fermions | fermi- dirac |
Enrico Fermi (1901–1954) Italy |
Paul Dirac (1902–1984) England |

bosons | bose- einstein |
Satyendra Bose (1894–1974) India |
Albert Einstein (1879–1955) Germany |

classical *particles* |
maxwell- boltzmann |
James Clerk Maxwell (1831–1879) Scotland |
Ludwig Boltzmann (1844–1906) Austria |

higgs boson |
higgs mechanism |
Peter Higgs (1929–2024) England |

group | greek root | meaning | |
---|---|---|---|

hadrons | αδρός | (adros) |
thick |

leptons | λεπτός | (leptos) |
thin |

baryons | βαρύς | (varys) |
heavy |

mesons | μέσος | (mesos) |
medium |

hyperons | υπέρ | (yper) |
beyond |

group | latin root | meaning |
---|---|---|

nucleons | nucleus |
kernel |

group | source | explanation |
---|---|---|

neutrinos | Enrico Fermi (1901–1954) Italy |
Italian diminutive form of neutron (neutrone). Neutrino could be translated as the "little neutral" to contrast it with neutrone, which is the "big neutral". |

quarks | Murray Gell-Mann (1929–2019) United States |
An arbitrary utterance later associated with a passage in Finnegans Wake — a novel by Irish modernist author James Joyce. Meant to sound like a drunken seagull ordering "quarts" of beer. |

### group theory

For those who like fancy math, the standard model is described using group theory notation as…

SU(3) × SU(2) × U(1)

where the gauge group of strong interactions is…

SU(3)

and the gauge group of electroweak interactions is…

SU(2) × U(1)

Notes…

- SU(3)
- 3rd order special unitary group
- the set of all 3 × 3 unitary matrices with unit determinant

- SU(2)
- 2nd order special unitary group
- the set of all 2 × 2 unitary matrices with unit determinant
- isomorphic to the group of quaternions of absolute value 1, {
*x*∈ ℍ: |*x*| =1} - diffeomorphic to a hypersphere (3-sphere)
- homomorphic to the rotation group SO(3), the set of all rotations about the origin in ordinary three dimensional euclidean space

- U(1)
- 1st order unitary group
- the set of all 1 × 1 unitary matrices
- isomorphic to the circle group, the multiplicative group of complex numbers with absolute value 1,
**T**= {*x*∈ ℂ: |*x*| =1} - isomorphic to SO(2), the second order special orthogonal group

### lagrangian

What is this? The standard model Lagrangian. What does it say? I'll let you know when I figure it out.

ℒ = −¼*F*_{μν}*F*^{μν} + *i*ψ̄*D̸*ψ*h*.*c*. + ψ* _{i}y_{ij}*ψ

_{j}φ

*h*.

*c*. + |

*D*

_{μ}φ|

^{2}−

*V*(φ)