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Standard Model

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The Standard Model of particle physics is a theory that accounts for all known elementary particles and their interactions, with the exception of gravity. It emerged in the 1960s and 1970s as a result of several discoveries, notably of quarks, the quantum-chromodynamics (QCD) theory of the strong interactions, and electroweak unification. Since then, there have been nearly forty years of Kuhnian normal science with it — it has been very successful in agreeing with experiments as particle accelerators have gone up to the teraelectronvolt (TeV) energy range.

Contents (Low Energy)[edit]

"Low energy" here means energy scales less than the electroweak-unification energy scale of about 250 gigaelectronvolts (GeV). Below that energy scale, the Higgs boson has a nonzero vacuum or ground-state field value, and that nonzero value makes it always present to every particle that it interacts with. That is what gives the other massive Standard-Model particles their masses.

Particles[edit]

The Standard Model contains a variety of elementary particles.

These particles have masses and several quantum numbers: spin, electric charge, and QCD multiplicity. The masses here are all given in GeV for easy comparison. Also here is whether a particle is its own antiparticle (X for yes, blank for no). The graviton is the hypothetical quantum of the gravitational field, and it is presented here for comparison.

Particle Spin E Charge QCD Mult Self Anti Mass(es)
Higgs Boson (H) 0 0 1 X 126
Photon (γ) 1 0 1 X 0
W+ 1 1 1 80.4
W- 1 -1 1 80.4
Z 1 0 1 X 91.2
Gluon (g) 1 0 8 X 0
Up-type Quarks 1/2 2/3 3 up: 0.0023, charm: 1.275, top: 173.07
Down-type Quarks 1/2 -1/3 3 down: 0.0048, strange: 0.095, bottom: 4.18
Neutrinos 1/2 0 1  ? nonzero, but likely less than about 10^(-10)
Charged Leptons 1/2 -1 1 electron: 0.000511, muon: 0.1057, tau: 1.777
Up-type Antiquarks 1/2 -2/3 3* up: 0.0023, charm: 1.275, top: 173.07
Down-type Antiquarks 1/2 1/3 3* down: 0.0048, strange: 0.095, bottom: 4.18
Antineutrinos 1/2 0 1  ? nonzero, but likely less than about 10^(-10)
Charged Antileptons 1/2 1 1 electron: 0.000511, muon: 0.1057, tau: 1.777
Graviton (hypothetical) 2 0 1 X 0

A particle's antiparticle has the same mass and spin, but opposite charge and conjugate QCD multiplicity. Thus, the antiparticle of the W+ is the W- and vice versa. The QCD multiplicity is a bit complicated. The colorless multiplicity 1 and the gluon multiplicity 8 have themselves as conjugates, but the quark multiplicity 3 has a separate multiplicity 3* (or 3 bar) for antiquarks.

Neutrinos may be their own antiparticles if they are "Majorana neutrinos". Flipping the spin of a Majorana neutrino makes it a Majorana antineutrino and vice versa. However, neutrinos would not be their own antiparticles if they were "Dirac neutrinos", analogs of the charged elementary fermions.

Interactions[edit]

The best-known Standard-Model interaction is the electromagnetic interaction. It is a "gauge interaction", and it preserves a "gauge symmetry". For EM, that symmetry involves modifying elementary-particle fields by multiplying them by factors of exp(i*a*Q) for charge Q and a the same for all particles. Using conservation of charge, the particle fields' equations of motion reduce to their original ones. Since a is constant here, this is a global symmetry. Make it a function of space-time, and the symmetry becomes local. But it produces extra terms involving a in the equations of motion. But one can get rid of them by adding EM interactions, complete with its own terms involving a that cancel the other such terms. Thus, EM gauge symmetry is local and not just global.

Quantum chromodynamics is constructed in a similar fashion. Quarks come in three QCD states that have been named "colors": red, green, blue. Antiquarks have anticolors or cyan, magenta, yellow. QCD gauge symmetries are mixings of color states. To make these mixings local, one needs an analog of the electromagnetic field: the gluon field. Thus, the gluon has eight possible states: color-anticolor states with a colorless combination subtracted out. Since these mixings can interact with each other, gluons can interact with each other as well as with (anti)quarks.

The W and the Z are similar to these particles. The W does up-type <-> down-type for quarks and neutrinos <-> charged leptons for leptons. The Z does not do such "flavor" changes, being like a photon but with different interaction strengths.

Gauge-symmetry operations form continuous abstract-algebra groups called "Lie groups", after mathematician Sophus Lie ("Lee"), not deliberate falsehoods. They are generated by "Lie algebras", mathematical structures that are often much easier to work with than the groups themselves. Electromagnetism has symmetry group U(1) and QCD symmetry group SU(3).

General relativity may be interpreted as a gauge theory of gravity, where the gauge transformations are spacetime coordinate transformations.

The Higgs boson interacts with all the massive particles of the Standard Model, including itself. It is not a gauge interaction, since it has no associated symmetry.

Contents (Electroweak Unification)[edit]

The Standard Model has electroweak symmetry, though it is broken down to electromagnetic symmetry at low energies. This symmetry breaking is caused by the Higgs boson getting a nonzero field value. This always-present field then induces masses in most of the Standard-Model particles. Only the photon and the gluon are unaffected by it, and they stay massless.

Particles[edit]

Their quantum numbers are somewhat different from the low-energy case. The elementary fermions here have chirality or handedness (left or right), a "weak hypercharge" value, and a "weak isospin" multiplicity. The WHC value is closely related to electric charge, while the WIS multiplicity is a shrunken version of the QCD multiplicity (only two "colors": up and down). Masses are not included because all these particles but the Higgs boson are massless.

Particle Spin Hand WHC Val WIS Mult QCD Mult
Higgs 0 -1/2 2 1
Higgs Conj. 0 1/2 2 1
WHC boson (B) 1 0 1 1
WIS boson (W) 1 0 3 1
Gluon (g) 1 0 1 8
Left Quark (Q) 1/2 L 1/6 2 3
Right Up (U) 1/2 R 2/3 1 3
Right Down (D) 1/2 R -1/3 1 3
Left Lepton (L) 1/2 L -1/2 2 1
Right Neutrino (N) 1/2 R 0 1 1
Right Electron (E) 1/2 R -1 1 1
Left Antiquark (Q*) 1/2 R -1/6 2 3*
Right Anti-Up (U*) 1/2 L -2/3 1 3*
Right Anti-Down (D*) 1/2 L 1/3 1 3*
Left Antilepton (L*) 1/2 R 1/2 2 1
Right Antineutrino (N*) 1/2 L 0 1 1
Right Antielectron (E*) 1/2 L 1 1 1
Graviton (hypothetical) 2 0 1 1

The right-handed neutrino and the hypothetical graviton are both included for completeness. The WHC reverses for antiparticles, just like the electric charge, but the WIS has no conjugate multiplets as QCD does.

The Higgs conjugate is listed as a separate particle for convenience, even though its field is a flipped complex-conjugate version of the "canonical" Higgs particle's field.

Interactions[edit]

As noted earlier, the unbroken Standard Model has two new quantum numbers, weak isospin and weak hypercharge. WIS is carried by a field that is much like a gluon field, but that is defined only over two "colors". Its name comes from a mathematical analogy with quantum-mechanical angular momentum. WHC is carried by a field that is much like the electromagnetic field, complete with different possible values of its "charge". Their symmetry groups are SU(2) for the WIS "W" field and U(1) for the WHC "B" field. Thus making the unbroken Standard Model's symmetry

SU(3) * SU(2) * U(1)

for QCD, WIS, and WHC. Electroweak symmetry breaking makes it SU(3) * U(1) for QCD and EM.

The symmetry-breaking process breaks the weak-isospin field into three parts: W+, W-, and W0. The W+ and W- become the familiar low-energy W's, and the W0 and the weak-hypercharge field B mix to make the low-energy electromagnetic (photon, γ) and the Z fields. Likewise, the left-handed quark and lepton weak-isospin doublets split into left-handed up-type quarks, left-handed down-type quarks, left-handed neutrinos, and left-handed charged leptons. The Higgs boson connects these left-handed parts to their right-handed counterparts to make the low-energy elementary fermions, masses and all. The electric charges of the resulting particles are given by

(electric charge) = (projected weak isospin) + (weak hypercharge)

For the quarks, it is

  • Left-handed up-type: +1/2 + 1/6 = 2/3
  • Left-handed down-type: -1/2 + 1/6 = -1/3
  • Right-handed up-type: 0 + 2/3 = 2/3
  • Right-handed down-type: 0 - 1/3 = -1/3

Incompleteness[edit]

The Standard Model is evidently a very complicated theory, something that has provoked the construction of grand unified theories (GUT's). It is also incomplete, not including several observed phenomena and having some theoretical problems.

Observed non-Standard-Model effects:

If the latter is produced by the Higgs mechanism, then this means some very curiously small Higgs-interaction strengths. An alternative is the "seesaw model", featuring a "Majorana mass" for the right-handed neutrinos and a "Dirac mass" that connects the left-handed and right-handed parts in the fashion of the charged elementary fermions. With plausible Higgs-induced Dirac masses, the Majorana masses must be close to the GUT mass scale to make the observed neutrino masses.

Theoretical problems:

  • The "hierarchy problem", where the Higgs boson gets a very large mass by quantum-mechanical effects.
  • The "strong CP problem", where QCD may give electric dipole moments to protons and neutrons.
  • The number of parameters: 19
  • Interrelationships. Though the Standard Model's multiplet structure is very complicated, it has certain regularities that suggest some GUT.
  • Gauge interaction-strength convergence at high energies, from quantum-mechanical effects. This convergence also suggests some GUT.

The first problem is solved by an additional symmetry called "supersymmetry". It extends spacetime symmetry to include interrelationships between particles with different spins. For the Higgs boson, it introduces a spin-1/2 "higgsino" whose interactions cancel out that extra mass. Supersymmetry can also improve the high-energy convergence of gauge interaction strengths.

External links[edit]