Orodruin said:
My point was that any such description is unphysical without the neutrino masses, just as the CKM would be unphysical if quark masses (either up or down type) were identical. The idea of the PMNS describing the W interactions among lepton mass eigenstates presupposes a definition of what those states are.
Generally, I think trying to ”decouple” Ws and Zs is a bit artificial as they both arise from electroweak theory. I’d much rather categorise into stuff owing to the electroweak gauge theory itself and stuff arising from the Higgs Yukawa couplings in the quark case (and whatever mass mechanism exists in the neutrino sector).
The paper I was thinking about that articulates the point I was making in #13 is Gustavo F. S. Alves, Enrico Bertuzzo, Gabriel M. Salla, "
An on-shell perspective on neutrino oscillations and non-standard interactions" arXiv (March 30, 2021) (accepted for publication in the journal Physics Review D). This paper states that it is possible to "obtai[n] the PMNS matrix [which governs neutrino oscillation] without having to ever talk about mass diagonalization and mismatches between flavor and mass basis." It also demonstrates how this is done in the context of W boson interactions strictly analogous to the quark flavor changing W boson interactions that the CKM matrix helps to describe.
It is useful to take that perspective, even though it isn't the only way to interpret it, because it underscores how many of the parameters of the SM are concentrated in one corner of the theory, while lots of other parts of the SM are far less experimentally measured parameter driven.
A quick refresher on the Standard Model and the physical constants within it, at B-level.
Scope
The Standard Model purports to supply essentially all of the laws of Nature, and effectively incorporates special relativity (which basically governs how things moving at or close to the speed of light act) but doesn't include gravity which is explained with general relativity.
The Standard Model is as close as physics get to a "perfect" and "complete" explanation of everything but gravity and phenomena ascribed to dark matter and dark energy. No experiments have been found to definitively violate the Standard Model yet, although there are some tensions between the vast number of experiments that have tried to test it and the theoretical predictions of the Standard Model.
Three Forces
In the Standard Model, there are three kinds of forces, called "gauge forces" that connect on particle to one another through a kind of particle called "bosons". These are the electromagnetic force, the weak force, and the strong force. The electromagnetic force and the weak force are related to each other in non-obvious ways that are explained in "electroweak unification" which is part of the Standard Model.
The whole theory is basically about different kinds of particles interacting with each other through photons, W and Z bosons, and gluons, according to rules that apply to each possible combination of them.
Renormalization
All of the experimentally measured parameters of the Standard Model have specific values based upon the amount of momentum transferred between particles in an interaction commonly described as Q with Q squared often used in practical calculations and physics paper tables.
Some of these parameters get bigger at higher energies, some of these parameters get smaller at higher energies, and one of them (the strong force coupling constant) gets strong up to a peak energy scale and then gets weaker as the energy scale grows beyond that point. The change in the values of these parameters depending upon the energy scale is called the "running" of the constant in question "with energy scale". If someone is being picky, when the tell you the value of a physical constant in the Standard Model, they will also tell you the energy scale at which that experimentally measured physical constant is being measured.
The formula for converting the value of an experimentally measured physical constant in the SM at one energy scale to the value it has at another energy scale is entirely determined from the formulas of the SM, although the formula is very long and complex to determine from first principles without just looking it up (which is what physicists usually do).
This feature of Standard Model physical parameters is a real world physical consequence of a calculation tool used in essentially all Standard Model calculations called renormalization.
The Higgs Mechanism and Neutrino Masses
The Standard Model also has a Higgs boson which is associated with a Higgs field. In the Standard Model, quarks, electrons, muons, tau leptons, W bosons, Z bosons, their antiparticles, and the Higgs boson all get their rest masses (i.e. mass disregarding any kinetic energy or boson frequency energy) from interactions with the Higgs field. The strength of the interaction of the Higgs field with most kinds of fundamental Standard Model particles is called the "Yukawa" of that particle.
The strength of the Higgs field in a vacuum, called the Higgs vacuum expectation value is important in determining these masses as well and is a derived physical constant in the Standard Model based upon the W boson mass the weak force coupling constant.
Photons and gluons don't interact with the Higgs field.
The following Standard Model particles have distinct masses that are known to arise from their interactions with the Higgs field from lightest to heaviest (with names of particles including their antiparticles that have the same mass): The electron, the up quark, the down quark, the strange quark, the muon, the charm quark, the tau lepton, the bottom quark, the W boson, the Z boson, the Higgs boson, and the top quark. The rest masses are associated with twelve Standard Model experimentally measured constants that represent eleven independent degrees of freedom.
There are also in principle three neutrino masses whose source is unknown. We don't really know where neutrino masses come from and there is more than one possible theory to explain that. But it is possible to experimentally measure properties of neutrinos related to their masses (although this is very difficult to do).
The Electromagnetic Force
The photon delivers the electromagnetic force from particles with electromagnetic charge to other particles with electromagnetic charge. Quarks, W bosons (discussed below), electrons, muons (a heavy electron), tau leptons (an even heavier electron than the muon), and their antiparticles have electromagnetic charge.
The electromagnetic force is the only long range force in the Standard Model. Also, essentially all of chemistry is derived from the electromagnetic properties of atoms.
Electromagnetism in the Standard Model has a structure closely related to a mathematical structure known as a U(1) group so when you see U(1) in physics, this is usually shorthand description of electromagnetism.
The strength of the electromagnetic interaction is governed by an experimentally measured constant called the electromagnetic coupling constant a.k.a. the fine structure constant.
The Weak Force
The W+, W- and Z boson govern the weak force that explains how quarks, electrons and their more massive cousins the muon and tau lepton collective called "charged leptons", and neutrinos change from one kind of particle into another kind of particle.
The weak force Z boson also gives rise to a very weak photon-like interaction that only operates at short ranges when particles exchange a Z boson.
All particles with non-zero rest mass interact via the weak force. The most common way we encounter the weak force in every day life is in nuclear beta decay. This is a very short range force because W and Z bosons decay very quickly.
W and Z bosons are very heavy compared to most other particles in the SM and are the fastest decaying particles in the universe.
The weak force in the Standard Model has a structure closely related to a mathematical structure known as an SU(2) group so when you see SU(2) in physics, this is usually shorthand description of the weak force.
The strength of the weak force is governed by the weak force coupling constant, although physicists often use a physical constant that is a function of the weak force coupling constant determined in experiments that is called Fermi's constant, rather than the bare theoretical concept in the Standard Model of th weak force coupling constant itself for many purposes.
The likelihood that a particular quark will turn into a different kind of quark when it interacts with a W boson is described by a three by three matrix of experimentally measured constants called the CKM matrix that can be described completely with four parameters that are independent degrees of freedom, although there is no one unique way to choose these parameters and two main approaches of an infinite number of ways of choosing those four parameters that are possible in principle.
The PMNS matrix is a similar three by three matrix that provides information that governs the likelihood of particular kinds of neutrino oscillations occurring. Like the CKM matrix, this nine experimentally measured physical constants can be summarized with four parameters. And, while there are also an infinite number of ways to parameterize the PMNS matrix, only one way of doing so is commonly used.
The Strong Force
The strong force is delivered by particles with zero mass called gluons from particles that have what is called color charge to other particles that have color charge. Quarks and gluons have color charge but other fundamental particles do not have color charge. The most well- know particles made out of quarks bound by gluons are protons and neutrons. But there are more than a hundred kinds of other particles made out of quarks bound by gluons called hadrons (a term that also includes protons and neutrons) that last only a tiny fraction of a second before they decay. Quarks and gluons other than top quarks are never found outside hadrons except at extremely high temperatures (a property called "confinement"), while top quarks decay almost instantly upon coming into existence before they can form hadrons. Due to confinement, the strong force mostly explains the inner workings of hadrons and usually works at distance scales on the order of a femtometer.
The forces that hold protons and neutrons together in an atomic nucleus work a lot like the strong force (sometimes collectively called the nuclear binding force, or nuclear force, or the residual strong force) but aren't fundamental. Instead, this force (which isn't as strong as the pure strong force) is carried by short lived hadrons called mesons (especially mesons called pions and called kaons) that are made up of quarks bound by gluons from one proton or neutron to another.
In practice, rather than calculating the nuclear force from first principles with the Standard Model, nuclear physicists use a formula for this force which has a different term of each kind of meson that transmits this force that is inspired by the Standard Model but is not rigorously derived from the Standard Model.
The strength of the strong force is governed by the strong force coupling constant.
The strong force in the Standard Model has a structure closely related to a mathematical structure known as an SU(3) group so when you see SU(3) in physics, this is usually shorthand description of the strong force.
Thus the Standard Model as a whole is described as a U(1)*SU(2)*SU(3) theory.
A basic Level Explanation of "tree level" v. "loop effects"
B-level is tree level (or lower order
).
To assist B-level readers, lots of calculations in the Standard Model involve adding up the sum of a great many formulas to get a numerical answer.
Usually these calculations involve the likelihood that a photon, W boson, Z boson, or gluon will cause a particular kind of interaction between two other Standard Model particles.
The first formula term in one of these calculations of the sum of terms that have to be added up to provide a probability of something happening, which is often (but not always) a fairly decent approximate description of what actually happens, is called the "tree-level" or "leading order" (a.k.a. "LO") term. The next term is called the "next to leading order" or "NLO", then you have the NNLO term for next to next to leading order, etc.
The terms of these formulas after the leading order term (at least as far out as they are conventionally calculated, there is a technical exception to this that doesn't come up in practical calculations) are smaller and smaller adjustments to the tree-level calculation, that are harder and harder to calculate at each step. The combined adjustments from the subsequent terms are also called the "loop effects" that modify the "tree-level" calculation.
A crude basic outline form the calculation of what you want to know in a lot of Standard Model calculations looks like:
Answer calculated from theory = L.O. + N.L.O + N.N.L.O + . . .
Loop effects = N.L.O + N.N.L.O + . . .
So you can also say:
Answer calculated from theory = L.O. + Loop effects.
Calculations of electromagnetism and weak force related quantities can usually be done to pretty great precision in a fairly modest and manageable terms, each of which involves advanced calculus calculations, because later terms quickly get much smaller than the previous terms in the series. In a calculation of moderate difficulty with these forces, this is something that one to three scientists might do with computers in a matter of hours or a few days.
In calculations of the strong force, the basic process is the same, but (1) the calculations at each step get much more difficult much more quickly than the do for the electromagnetic and weak forces, and (2) for example, the strong force N.L.O. correction is much larger relative to the L.O. term, and the N.N.L.O correction is much larger relative to the N.L.O. correct, than in the comparable electromagnetic and weak force calculation. So, it takes much more work to get much less accuracy in strong force calculations than it does to do the same thing for the other SM forces. In a calculation of moderate difficulty with the strong force, this is something that three to twelve scientists might do with many supercomputers working together at different locations in a matter of weeks or months.
It wouldn't be unusual for a tree level calculation involving the weak force to be "good enough for government work" and a basic understanding, even though a really precise answer would require consideration of some loop effects.
Tree level calculations are sometimes good enough for a good rough estimate for electromagnetism calculations, but this is less universally true.
Tree level calculations are rarely good enough for anything more than a crude understanding of what is going on in strong force calculations.
