## Conservation Laws of Particle Interactions

Definitions:

Lepton number : +1 for the particles in the green boxes (see the diagram above), namely, electron, muon, tau, electron type neutrino, muon type neutrino, tau type neutrino, -1 for their antiparticles, 0 for others.

Baryon number = 1/3 (number of quarks  –  number of antiquarks)

Baryons  (3-quark formations like proton and neutron) have a baryon number of +1. Mesons (quark-antiquark formations) have a Baryon number of 0. Antibaryons (antiproton for example) have a baryon number of -1.

Electron family number : +1 for electron and its neutrino, -1 for antielectron and its neutrino, 0 for all others.

Muon family number : +1 for muon and its neutrino, -1 for antimuon and its neutrino, 0 for all others.

Tau family number : +1 for tau and its neutrino, -1 for antitau and its neutrino, 0 for all others.

3 generations of fermions: (see the diagram above)
(physicists confuse us by referring to generations as families/flavors also)

Color charge: quarks and the entities that glue them together called gluons have an internal degree of freedom known as “color”. This degree of freedom is 3-valued (labeled as “red”, “green”, “blue”). Since there are antiquarks there are also “antired”, “antigreen”, “antiblue” degrees of freedom. Physicist use the term “color charge” but “charge” is a misnomer. For elementary particles a “charge” (any kind of charge) is a property of the particle. Electron’s electric charge is -1. This does not change. The “color” of a quark, on the other hand, changes quickly.  “Red” u-quark, for example, can be “blue” or “green” after a fraction of a second.

Any particle composed of quarks and gluons is “colorless” at all times. This is known as “color confinement“. As quarks exchange gluons their color change but the composite particle remains “colorless” at all times.

More information about color charge, color confinement and the answer to what “colorless” means can be found here and here (Ethan Siegel articles are excellent).

Isospin, hypercharge: these quantities became obsolete after the establishment of the 3 generations of quarks and gluons as the explanatory factors of nuclear reactions.

Conserved quantities in particle interactions

Definition of “interaction” : decay process or a point-like interaction where two particles get very close to each other. It can be an elastic scattering where two particles simply scatter from each other or an inelastic scattering (very energetic interactions) where particle transformations take place.

In the interactions of elementary particles the following quantities are conserved (the total before and after the interaction is the same).

• Baryon number
• Lepton number
• Electron family number
• Muon family number
• Tau family number
• Electric charge
• Energy
• Angular momentum
• 4-momentum

These are empirical facts. It is important to point out that the conservation laws are not derived from theory. These laws are based on observations repeated millions of times over the past decades. Yes, it is true that physicists have gained some understanding about the relationship between the conserved quantities and the symmetries observed in nature. For example the symmetries of the Maxwell equations imply the conservation of charge but Maxwell equations themselves were the generalization and quantification of the empirical observations of electromagnetic phenomena in the first place. Emmy Noether demonstrated that every symmetry observed in nature – mathematically expressed as a continuous transformation – has a corresponding conservation law. This is a GREAT INSIGHT , of course. But, again, we are not deriving the conservation law from theory, the theory itself is built on top of observations.

We may discover the unknown symmetries of a given theory or we may hypothesize various symmetries theoretically, of course. Hypothesis building is part of science too with the understanding that those symmetries and associated conservation laws have to be experimentally demonstrated.

Measuring the properties of elementary particles involve repeated measurements. Demonstrations of symmetries and conservation laws are much more difficult. Demonstration requires the test of time, experiments repeated over many decades, in very different experimental conditions. Why the “test of time”? Because with improved technology the sensitivity of the experimental equipment increases. We may then discover that some of the symmetries (and their associated conservation laws) are not exact. This means that in special conditions there may be violations – with very small probability – of the symmetry. We also know that some of the symmetries of nature existed in the early universe (when the energy density was very high) but they were broken in later stages when the energy density decreased as the universe expanded. We should distinguish “near” and “broken” symmetries from the “exact” symmetries.

The conserved quantities mentioned above are conserved “exactly” at the interaction point as far as we can test with our current technology up to the energy density scales we can reach.

In the literature you may see confusing statements declaring the conservation of lepton number not being exact or the conservation of baryon number not being exact. Those claims are based on proposed BSM (beyond Standard Model) theories but none of those BSM theories have experimental support yet.

You may also see statements saying that lepton (electron, muon, tau) family number is not conserved based on the experimental observations that neutrinos can switch their families. Remember that this does not happen at the interaction point. Neutrinos may switch their family (this is known as neutrino oscillation) over long distances not at the interaction point.

Electron family number

Electron is the most stable particle in the universe. Electron never decays.  But electron can be annihilated in electron-positron head-on collisions. Electron and electron type neutrino can be transformed into each other as well. Conservation of the electron family number is strictly observed. In the case of electron-positron annihilation the net total electron family number before the interaction is zero, after the interaction it is zero as well.

Muon family number

Muon decays in a fraction of a second. The decay products are an electron plus muon-type-neutrino and an electron-type-antineutrino. The conservation rule applies: the total muon family number before and after interaction is the same.

Tau family number

Tau is very unstable. Tau never decays into muon or electron without neutrinos showing up. The total tau family number before and after interaction will be the same.

Neutrino-less charged lepton flavor violation is not allowed

Example: The decay process where a muon decays into 2 electrons plus a positron without any muon-type-neutrinos showing up has never been observed.

The term “lepton flavor violation” refers to the violation of the “electron family number”, or the violation of the “muon family number” or the violation of the “tau family number”. This type of violation never happens.

• The total electron family number stays the same through an interaction
• The total muon family number stays the same through an interaction
• The total tau family number stays the same through an interaction

Quark family numbers are not conserved

Consider the 3 generations (flavors or families) of quarks. We could assign quark family numbers similar to lepton (electron, muon, tau) family numbers but we know that quark family number is not conserved in the weak decays in the CKM matrix.

Conservation of total 4-momentum

We have to consider not just the natural decays and low energy scatterings but also the high energy interactions where relativistic kinematics has to be considered. If you look above you will see that I have not listed “total momentum” as a conserved quantity. The correct statement is that the “total 4-momentum” is conserved.

Remember the relativistic formula for the total energy of an individual particle

$E=\sqrt{p^2c^2 + {m_0}^2c^4}$

where $m_0$ is the “invariant mass”, $c$ is the speed of light, $p$ is the momentum of the particle.

Written in a different way

$E^2/c^2 - p^2 = {m_0}^2c^2$

Since $m_0$ is the invariant mass (a constant) and $c$ is a constant the quantity $E^2/c^2 - p^2$ will stay constant. This applies to all particles. Therefore the sum total of the $E^2/c^2 - p^2$ will stay constant in particle interactions or decays.

The quantity $E^2/c^2 - p^2$ is the square of a 4-vector known as the 4-momentum $p^\mu$

$p^\mu = (E/c, - \overrightarrow{p})$

where $\overrightarrow{p}$ is the momentum 3-vector (vector in 3 spatial dimensions). In particle interactions the total 4-momentum before and after the interaction is the same. There is a double conservation here.

• The total sum of the squares of the 4-momenta is conserved.
• The total 4-vector sum of the 4-momenta is conserved as well.

Conservation of angular momentum

“In any particle-particle or particle decay interaction the total spin before and total spin after can differ only by integer changes”  [1]. That’s because all interactions are mediated through force carrying bosons (photon, gluon, W and Z) which carry 1 unit of spin. For example, consider the neutron decay:

neutron –> proton + electron + antineutrino

Neutron, proton, electron, antineutrino are spin=1/2 particles. We see that the total spin before the interaction is 1/2, after the interaction is 3/2, the difference is 1, so the rule mentioned above is not broken.

The confusing part is the conservation of angular momentum. “Spin” is part of angular momentum. The rule mentioned above gives the impression that the angular momentum is not conserved. The answer is simple but many students do not get this. When we say spin=1/2 particle we are referring to the absolute value of the spin angular momentum. When we talk about the conservation of angular momentum we are talking about the conservation of a vector not a scalar.

For example, in the neutron decay the electron and the antineutrino helicities will oppose each other. Angular momentum vectors of the interaction (decay) products will be such that their vector sum will be equal to the vector sum of the angular vectors before the interaction (decay).

Conservation of energy

This is a subtle subject! In the theory of gravitation known as the General Relativity the total energy of the universe is not well defined. If the empirical observation that the expansion of the universe is accelerating is true then we have to conclude that at the cosmic scale energy is not conserved. Take a look my previous post titled “Recent discussions on dark energy“.

Leaving the unknown of “cosmic scale” aside, we know for certain that energy is conserved in particle interactions and decays. This has been observed to be true in billions of particle interactions and decays in the past decades.

Conservation of lepton number

Sum of the lepton numbers is conserved in an interaction. The counter example  neutrino-less double beta decay has never been observed.

Point-like self-coupling (cloning) is forbidden

Point-like self-coupling refers to self-cloning of a particle and interaction with its clone. Only the Higgs particle can do this (not experimentally verified yet).

• Charged fermions, an electron for example, cannot be cloned because that would violate the law of the conservation of charge.
• Neutral fermions cannot be cloned either. A neutrino cannot be cloned because the total spin before cloning is 1/2, after cloning 1. The difference is 1/2. This is not allowed. Remember, the total spin before and total spin after interaction can differ only by integer changes.
• Bosons cannot be cloned because of the law of “no-cloning of quantum mechanical state”. See below.

No cloning of a quantum mechanical state

We cannot create an identical copy of a quantum state without destroying the orginal state. You must read this article by William K. Wootters and Wojciech H. Zurek: “The No-Cloning Theorem.

Therefore, we cannot clone a photon without destroying it. In other words we cannot have 2 identical copies of the original photon. The caveat is that a photon can be split into 2 identical entangled photons but the new photons will have a different frequency (different state) from the original photon.

The law of “no-cloning of a quantum mechanical state” covers the “no-cloning of fermions” as well. In the case of charged fermions we can invoke the conservation of charge as an additional explanation.

Quotes

[1] Tommaso Dorigo

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