The Standard Model, gauge theories and grand unification

The Standard Model, developed since the 1970s, has changed little since about 1975. The main later addition is that neutrinos have masses, which can be incorporated.

Guth shows the familiar chart of its particles (slide 6, from Wikimedia Commons; a version in words follows):

• Spin-½ particles (fermions) fall into three generations. There are six flavours of quark, and each comes in three colours. The colours are perfectly symmetric — absolutely identical — while the flavours all differ, with only approximate symmetries between them. With each family of quarks goes a family of leptons, particles without strong interactions: a neutrino and an electron-like particle.
• Spin-1 particles (gauge bosons): the photon of electromagnetism; the gluon, which plays the photon's role for the strong interactions, which are more complicated; and the Z⁰ and W±, carriers of the weak interactions — the only non-gravitational interactions neutrinos feel. A weak process example is neutron decay, $n \to p + e^- + \bar\nu_e$, where the antineutrino balances electron lepton number. Essentially anything involving neutrinos is a weak interaction.
• The chart Guth shows predates 2012 and leaves out the Higgs boson, the spin-0 particle of the Higgs field, found at CERN in 2012; it completes the Standard Model, and the Higgs field is the subject of Lecture 22.

With its interactions this is a complete model of how elementary particles behave, and it agrees with every reliable experiment since the mid-1970s. That has made particle physics a bit dull, Guth jokes, but it is a magnificent achievement.

Why nobody thinks it is final

Guth knows nobody who regards the Standard Model as a candidate for the ultimate theory:

1. It does not include gravity, and there is no simple way to add it.
2. It has no good candidate for the dark matter.
3. It has about 28 free parameters — numbers that must be measured before it predicts anything. That is only an aesthetic argument, but theorists take it seriously: many parameters mean ugliness, and almost all expect the true theory to have far fewer adjustable knobs, perhaps none.

Why is the strong force short-ranged?

A student asks: the weak force is short-ranged because the W and Z are very heavy, but the strong force is short-ranged too, and the gluon is effectively massless. The answer is confinement. Gluons cannot exist as free particles. In a sense the strong force is long-ranged: pull apart a meson, a quark and an antiquark, and a "string" of gluon force between them pulls with a constant force however far you go. What intervenes is that, stretched far enough, it becomes energetically favourable for a new quark–antiquark pair to appear in the middle. The string breaks into two shorter strings, each with a quark at one end and an antiquark at the other — but a string can never simply end in empty space. At very short distances gluons behave like ordinary particles; at larger distances confinement dominates.

Grand unified theories grow out of the Standard Model through the idea of a gauge theory: a theory in which the fields are not themselves reality. There is a set of transformations that take one field configuration to another describing the same physical situation, so many field configurations correspond to one reality.

The example you know is electromagnetism. Writing the fields in terms of potentials,

for any function $\Lambda(t, \vec x)$ one may replace

and $\vec E$ and $\vec B$ — which contain all the physics — are unchanged. [The slide writes $\phi' = \phi - \partial\Lambda/\partial t$; with $\vec E$ defined as above, and $x^0 = ct$ in the covariant form, the factor $1/c$ is needed.] All the important field theories we know are gauge theories.

Two successive gauge transformations must again be a symmetry, so it matters how they combine. In electromagnetism they simply add:

so the group is Abelian (commutative).

The crucial generalisation is to non-Abelian gauge theories, introduced by Yang and Mills in 1954. Nobody knew what to do with them at first, but they became the Standard Model. Here the gauge parameter is an element of a group, and successive transformations multiply according to that group's rules.

The Standard Model uses three gauge groups — itself an example of its ugliness, and a source of extra parameters:

• SU(3): the $3 \times 3$ matrices that are unitary, $U^\dagger U = 1$, and special, with determinant 1.
• SU(2): the same with 2 in place of 3.
• U(1): the group of phases $e^{i\phi}$ — complex numbers of magnitude 1.

The three act independently, so the full gauge group is the product group SU(3) × SU(2) × U(1): an element is just one element of each, written together.

A grand unified theory embeds these three groups in a single simple group (a term with a precise meaning in group theory, but "simple" in the everyday sense will do here). The first one invented uses SU(5), the $5 \times 5$ unitary matrices of determinant 1. The embedding is easy:

• put an SU(3) matrix in the upper-left $3 \times 3$ block;
• put an SU(2) matrix in the lower-right $2 \times 2$ block;
• for the U(1), give each block an overall phase: multiply the whole upper $3 \times 3$ block by $e^{2i\phi}$ and the whole lower $2 \times 2$ block by $e^{-3i\phi}$.

The phases are chosen so the determinant stays 1: $(e^{2i\phi})^3(e^{-3i\phi})^2 = e^{6i\phi - 6i\phi} = 1$. [The transcript says "the product of the five phases is 0"; it is the sum of their exponents that vanishes, so the product is 1.] Any SU(3) matrix, SU(2) matrix and angle $\phi$ together give an SU(5) matrix, and the three pieces commute with one another, so SU(5) contains SU(3) × SU(2) × U(1) as a subgroup. That is how the simplest grand unified theory works.

Guth ends here — out of time, he withholds the grand-unified-theory problem from the problem set, and the monopole discussion is picked up in the next lecture, which Guth says Scott Hughes will give.

• The magnetic monopole problem exists only if grand unified theories are right.
• The Standard Model: three generations of quarks (six flavours, three colours each) and leptons, plus the photon, gluon, W± and Z⁰. It fits every reliable experiment since the mid-1970s, but lacks gravity and a dark matter candidate, and has about 28 free parameters.
• Confinement explains the short range of the strong force despite massless gluons.
• In a gauge theory many field configurations describe the same physics; electromagnetism is Abelian ($\Lambda_3 = \Lambda_1 + \Lambda_2$), while Yang–Mills theories are non-Abelian.
• The Standard Model gauge group SU(3) × SU(2) × U(1) fits inside SU(5), the first grand unified theory.