# Particles

Many physics majors who plan to do research in grad school on high energy
physics have little or no knowledge on the subject, having read articles
or books written for people with only a high-school education in physics
(or less). (I have no clue why.) Most such physics falls under the
geometrical headings of "particles" or
"

strings".

## Symmetries

There are various kinds of symmetries in particle physics. In particular,
there are

**local** symmetries, which have independent transformations
at each point in spacetime, and

**global** symmetries, which transform
the same way everywhere. There are also

**spacetime** symmetries, which
act on coordinates, and

**internal** symmetries, which don't.

| global | local |

spacetime |
special relativity: Poincaré (Lorentz, translations) |
general relativity: arbitrary coordinates |

internal |
approximate: flavor (isospin, etc.) |
gauge: color, electroweak |

### Relativity

Special relativity, simply stated (more simply than by rotations,
Lorentz transformations, etc.),
is the symmetry that leaves invariant the
infinitesimal distance element (proper time)

*ds*:

-ds^{2} =
-dt^{2} + dx^{2} + dy^{2} + dz^{2}
as a generalization of the nonrelativistic distance element
given by just the last 3 terms, a generalization of the
Pythagorean theorem. (We use units *c=1*.)
General relativity uses arbitrary coordinates, describing
curved space.

### Isospin

Internal symmetries relate similar types of particles.
For example, isospin relates protons and neutrons, which have
the same spin, and about the same mass, but different charge.
The proton and neutron form a doublet that transform into
each other, in the same way as the 2 spin states of a
spin 1/2 particle, hence the name "iso(topic)spin".
(Nuclei that differ by exchanges of protons and neutrons
form different isotopes of the same "isobar".)
In that sense, they are 2 isospin states of a single
type of particle, the "nucleon".
Both spin (actually rotations) and isospin transformations
are examples of the group "SU(2)".

### Flavor

Things will stay simpler if we work in terms of quarks
instead of hadrons. Then the "up" and "down" quarks
that make up the neutron and proton also form a doublet
of SU(2) isospin (and each one is a doublet of SU(2) spin,
again just like the nucleons), These 2 quarks, like the
2 nucleons, are very close in mass, so this SU(2) symmetry
is very accurate. There is a third quark, the "strange"
quark, that is somewhat more different in mass. These 3
quarks together form a triplet of a bigger group, "SU(3)",
that juggles all 3 among themselves. Thus, SU(3)
transformations include SU(2) as a "subgroup". But there
are even more quarks (at least 6 total), which differ greatly
in mass, giving an "SU(6)" symmetry, which is very poor,
at least as far as relating masses. So basically SU(N)
is just the symmetry that says you have N different "flavors"
of quarks that act pretty much the same way, at least with
respect to the strong interactions, except for their masses.

### Electromagnetism

But the quarks have different charges, because they interact
differently with respect to the electromagnetic (and also
weak) interactions. Electromagnetism is described by a "U(1)"
symmetry, because it can relate a single particle to itself,
by multiplying its wave function by a phase. (SU(N) is the
same as U(N) less a U(1). For the above SU(N)'s, that U(1)
symmetry is associated with "baryon number", which is 1 for
baryons like the nucleon, 1/3 for each of the 3 quarks that
make up a baryon, -1/3 for an antiquark, 0 for a meson made
of a quark and antiquark, etc.)

### Weak interactions

This U(1) symmetry associated with charge is a local symmetry
because the coupling of the spin-1 photon of electromagnetism
couples to its current (a local quantity, the distribution of
charge). Similarly, the weak interactions responsible for
the slower particle decays is described by a local SU(2)
symmetry, mediated by the W

^{+}, W

^{-}, and Z
bosons (a triplet of this SU(2), and also each a triplet of
spin SU(2)) and the Higgs. (Actually the photon and Z are mixtures of the
U(1) particle and 1 of the 3 SU(2) particles, because of the way
the Z and W's get mass.)

### Strong interactions

Finally, the strong interactions are mediated by the spin-1
gluons of an SU(3) symmetry that bind quarks (and themselves)
together. There are thus 3 "colors" of quarks (besides their
6 "flavors"), but we never see the colors because they cancel
in their bound states, the hadrons (baryons, antibaryons, and
mesons). The leptons (electron, muon, tauon, and their neutrinos)
also come in 6 flavors, but no colors, so they interact only
electroweakly, not strongly, and are not confined. (You can take
the electron out of the atom, but you can't take the quark out
of the proton.)

## Dynamics

The most important calculational tool in particle physics is
perturbation theory, because it gives a systematic expansion
of exact results that are often accurate already at lowest
order, and can be improved at higher orders. This expansion
is represented by "Feynman diagrams", pretty pictures that
actually have mathematical significance. These "graphs" take
the place of equations that would be much longer and harder to
express as a 1-dimensional line of text. Each such graph
is associated with a quantum mechanical amplitude describing
a specific scattering process.

### Propagators

The diagram consists of various lines (segments), representing
the paths of free particles (waves). They can be solid or dashed,
straight, wavy, or curly, indicating different types of particles.
They can even be multiple parallel lines, indicating the quarks
that make up a hadron. Each line is assigned a momentum, so the
particle is actually a plane wave; as usual, arbitrary states can
be represented as superpositions. Each line is then associated
with a "propagator" ("Green function", "kernel"), as is always
used when giving the general solution of a free (differential)
wave equation. When working with momentum eigenstates (by
Fourier transformation), propagators are just rational functions
of momenta. When the particles carry spin or internal symmetry,
there are also indices associated with the propagator indicating
the corresponding state.

### Vertices

These line segments begin and end at "vertices". These intersections
describe the collisions of these particles at points, where particles
may also be created or annihilated. The pointlike nature of these
collisions is enforced by momentum conservation at the vertex, and the
association of each vertex with a polynomial (usually just linear)
in the momenta of the colliding particles, as well as numerical factors
associated with spin and internal symmetry, and a coupling constant.

### External lines

Finally, there are also short line segments coming out of some
vertices with the other ends free. Each free end is associated
with the wave function of an initial single-particle state, or
the complex conjugate of a final one. As with the propagators,
each of these lines is assigned a momentum and spin/internal
symmetry numbers specifying that state.

### Loops

Putting together all the factors from propagators, vertices, and
external line factors, we get some function of momenta, the
probability amplitude for that process. Some "internal" momenta
of the propagators are determined by the "external" momenta of
the wave functions (and similarly for spin and internal symmetry
by their conservation or interaction-specified breaking).
If there are closed "loops" in the graph, for each
there is a corresponding undetermined momentum, which must then
be integrated over to get the contribution from all possible
internal states (and the corresponding spin and internal symmetry states
must be summed over).