Resonances are the cornerstone of experimental high-energy particle physics. A resonance is the primary signature of a short-lived exotic particle, and they are relatively easy to detect and measure. Careful study of a resonance can yield a measurement of a particle's mass and lifetime.
Where Resonances Appear
Particle physics experiments, large or small, involve observing reactions between particles. Resonances are cleanest and easiest to see in simple reactions, and the simplest higher-energy reactions in particle physics come from lepton-antilepton collisions. In this writeup, I will primarily consider electron-positron collisions as they are simple but frequently used in particle physics experiments; the simplicity allows for relatively high precision. The mechanism for finding resonances is much the same for more complicated reactions, such as proton-antiproton annihilation.
For a particular choice of reaction, such as e+e- -> μ+μ-, the probability of occurence is quantified with a number called the cross-section, which is a function of the incoming particle energies. The variation of the cross-section contains vast amounts of information about the mechanism of the reaction.
Now in modern particle theory, a reaction like the above is not a direct conversion:
μ+ μ- ^
\ / / \
\ / |
\ / |
. | increasing time
/ \ |
/ \ |
/ \ |
e+ e-
Rather, the particle interaction is mediated by virtual particles, in this case usually a photon. This reaction can then be written as e+e- -> γ -> μ+μ-, or shown schematically as:
μ+ μ-
\ /
\ /
\ /
.
|
| γ (virtual photon)
|
.
/ \
/ \
/ \
e+ e-
The cross-section for this reaction is relatively constant with energy, and can be calculated from theory to high precision.
However, this mechanism is not the only reaction of the form e+e- -> stuff -> μ+μ-. If the total energy of the two particles is close to the mass of a real particle, such as a J/ψ meson (whose double name is due to its near-simultaneous discovery by two groups each with their own ideas on naming), the e+e- collision can produce a real particle rather than a virtual photon that decays to two muons, i.e.
μ+ μ-
\ /
\ /
\ /
O
|
| J/ψ (real meson)
|
O
/ \
/ \
/ \
e+ e-
The probability of J/ψ production adds to the probability of the photon-mediated reaction, but all that is seen is the initial particles and final particles. So, the cross-section for the reaction
e+e- -> μ+μ- increases in the area where J/ψ production is likely, namely with energy close to the mass of the J/ψ. Thus, a peak appears in the graph of cross-section versus energy at the J/ψ mass; this is the J/ψ resonance. (This is actually how the J/ψ particle was discovered.)
What Resonances Mean
Resonances will appear in virtually any reaction, so long as the available quantum numbers match that of the produced particle. An electron-positron collision will never see the resonance of a (single) charged particle, since the total charge available is zero. Similar constraints apply to other particle characteristics ('quantum numbers') such as spin and parity, and so the appearance or non-appearance of resonances in the cross-sections of different reactions will reveal information about the quantum numbers of the corresponding particle.
The centre of a resonance peak will be found at the rest mass of the corresponding particle. This is the best way to measure the masses of the legions of short-lived, massive particles that exist. Furthermore, due to the energy-time uncertainty principle, δEδt ≥ h/4π, the width of a resonance peak is inversely proportional to the particle's lifetime. Thus, careful plotting of a resonance peak can be used to determing both the mass and lifetime of a particle.
Each distinct resonance appears to be a distinct particle. A particle will have the same mass as its antiparticle, but the two particles will have opposite quantum numbers; aside from this every resonance of the same central energy can be said to arise from the same particle. Thus, we classify every set of measured resonances with the same quantum numbers and energy as a particle and assign it a particle name.
Resonance Classification and Particle Structure
In the early days of particle physics, each particle classified in this way was interpreted as its own fundamental particle. Soon, the bewildering array of new particles caused one particle physicist to remark, "Once, the discoverer of a new fundamental particle would earn a Nobel Prize; now, it seems that it should be punishable by a $10,000 fine."
With Murray Gell-Mann's work on the Eightfold Way and the ensuing development of the quark model, it became clear that the vast majority of observed particles are not fundamental particles, but rather various composites of mostly unobserved particles called quarks. The Standard Model of particle physics contains fewer than two dozen particles, plus their corresponding antiparticles, six of which are quarks. All of the composite particles fall into two classes: baryons, which are composites of three quarks or three antiquarks (an antibaryon), and mesons, which are composed of one quark and one antiquark.
Having chosen a compliment of quark and/or antiquark types (whimsically called 'flavours'), there are multiple ways in which they can be combined. The most basic difference that can be applied is different alignment of spin. Quarks all have a spin of 1/2, which can be set up to point in one of two directions. For a meson, the spins of the two particles can be aligned, in which case the spins add giving the meson a spin of 1, or the spins can be anti-aligned, in which case they subtract and the meson has a spin of 0. Baryons work much the same way: either all three are aligned giving a spin of 3/2, or one is opposite the other two giving spin 1/2.
These different spin states apppear, given the classification scheme above, as different particles, even though they have the same constituents. Aligned spins have higher potential energy; spin-bearing charged particles have magnetic moments that cause them to act somewhat like bar magnets, and particles with the same spin direction effectively have their north poles pointing the same direction. Thus, since energy is mass, the higher-spin mesons and baryons will have higher masses.
The physics of a composite particle is even more complicated than spin alignments. Two classical objects can have any given choice of relative energy; a satellite can be orbited around the Earth at any altitude the launching rocket can put it at. For quantum objects such as quarks, and electrons in atoms, there are discrete steps of relative potential energy. Thus, when building a meson, the two quarks can be placed at a number of different 'separations' each having a different potential energy. These different potential energies lead to different masses, and so the energy levels of the two-quark (or three-quark) system show up as different particles, which depending on the arrangement of the quarks may even have different quantum numbers. Borrowing from atomic physics, the lowest-energy of these states is called the ground state and all of the others are called excited states.
This all brings into question exactly what we mean when we declare two sets of resonances to be 'different particles'. For one, the nomenclature of different spin-states is often held over from before the quark model. For example, the spin-0 combination of an up quark and an anti-down quark is called a π+, while the spin-1 combination of the same quarks is called a ρ+. Similarly, we call a spin-1/2 up-up-down baryon a proton (p+), while the spin-3/2 combination is a Δ+. Nomenclature for excited states reflects more modern knowledge; the first excited state of the pion is called π(1300), the second π(1800). The number in parentheses is the resonance energy in MeV (mega-electronvolts).
In my opinion, the convention most consistent with both history and the quark model is to consider all resonances of the same quark content and spin states to be the same particle, i.e. there is a particle called π+ whose ground state is at 140 MeV, and has excited states at 1300 MeV, 1800 MeV and so on. One might say that the π+ and ρ+ are 'really the same particle' too, and that is valid, but this is entirely a matter of taste.
Thanks to IWhoSawTheFace for inspiration. Particle physics knowledge taken from my undergraduate and graduate particle physics courses, the textbook Introduction to Elementary Particles by David J. Griffiths, and the Particle Data Group website at http://pdg.lbl.gov/ .
(CC)
This writeup is copyright 2005 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.5/ .