Is the Standard Model overfitted?

[danielcmk]

TL;DR Summary

Is the Standard Model overfitted? It has so many constants.

Is the Standard Model overfitted? I hear that the standard model is the most accurate model that we have so far showing accuracy of 13 digits or so. However I am wondering if this accuracy comes from adding so many constants until it fits.

First of all the standard model takes in 25 constants, many of them unrelated to each other. When I think of the Standard Model, I see parallels with it and how they used to model orbits before Kepler. Before Kepler, astronomers like Tycho Brache used to model orbits as circles inside circles until it fit but you could have modeled it simpler with 1 ellipse. It almost feels like everytime we find something different from our models, we are adding extra constants(analogous to circles).

Why is the Standard Model any different?

 

[malawi_glenn]

Pre Kepler and Newton, they had no mechanism for gravity. They had no predictions.

In the sm, we can predict things based on the values of its free parameters. We measure A and B and can predict C. Then we measure C and compare with our calculation.

You can read the particle data group booklet https://pdg.lbl.gov/ on how the parameters of the SM are measured.

 

[fresh_42]

Pre Kepler and Newton, they had no mechanism for gravity. They had no predictions.

In the sm, we can predict things based on the values of its free parameters. We measure A and B and can predict C. Then we measure C and compare with our calculation.

You can read the particle data group booklet https://pdg.lbl.gov/ on how the parameters of the SM are measured.

Added to the list, thanks!

 

[malawi_glenn]

Added to the list, thanks!

Well it is not an "easy read" but one can look for instance "Z boson" mass and one can read about experiments and "paramter fits" and so on. Most particle physicsists use pdg when referencing values etc

 

[PeterDonis]

Moderator's note: Thread moved to the high energy physics forum.

 

[PeterDonis]

Is the Standard Model overfitted?

No, because an overfitted model is a model that fits the data in its training set well, but does not predict well data beyond its training set. The SM does predict well data beyond its training set.

I am wondering if this accuracy comes from adding so many constants until it fits.

No, because the constants aren't "added". They aren't just put in arbitrarily. They automatically come out when you apply the general principles on which the Standard Model is based.

 

[Orodruin]

Just to add to the choir. Generally there are many many more predictions of the SM than there are fitted parameters and it generally agrees extremely well. When you hear about predictions that agree with experiments to an accuracy of 1 part per billion or more those are values predicted from fitting other measurements to fix the constants.

To add on top of that, there are several measurements of effects that are direct predictions of the SM, such as the universality of coupling constants etc.

 

[mfb]

We have thousands of different measurements (with the exact number depending on how you count) and they can all be described with 19 parameters (26 with neutrino mixing). This single chart shows more measurements than we have free parameters already, and it's just one of many (from CMS).

 

[Orodruin]

26 with neutrino mixing

Do we really need to open that box in this thread?

 

[mfb]

Just wanted to mention why you can find different numbers for the parameter counts.

 

[ohwilleke]

It is also worth noting that while there are a couple of dozen experimentally determined parameters, not all parts of the theory use the same parameters to the same degree.

All But A Few Of The SM Experimentally Measured Parameters Involve W Boson Interactions And/Or Higgs Boson Couplings (Except For The Unknown Case Of Neutrino Masses)

For example, ten of the experimentally measured parameters are all confined entirely to weak force interactions with Standard Model fermions (whose weak force charges are fixed by theory at simple ratios of small whole numbers that are confirmed experimentally): the four parameters of the CKM matrix, the four parameters of the PMNS matrix, the weak force coupling constant, and the W boson mass are all necessary to describe W boson interactions. (Only the weak force coupling constant and the Z boson mass are needed to describe Z boson interactions.)

In electroweak unification, which is part of the SM, four SM experimentally measured parameters: the electromagnetic coupling constants (a.k.a. the fine structure constant), the weak force coupling constant, the W boson mass and the Z boson mass are functionally related to each other in a manner that has only three degrees of freedom, so one can legitimately choose a parameterization of the SM in which the Z boson mass is a derived rather than a free experimentally measured parameter.

All electromagnetic interactions are governed by the experimentally determined electromagnetic coupling constant and electromagnetic charges which are multiples of 1/3e fixed by theory and confirmed experimentally.

The main experimentally determined parameter in the strong force is the strong force coupling constant.

Also, at least twelve the fundamental SM boson masses (the Higgs boson, the W boson, the Z boson, the six quark masses and the three charged lepton masses) can also be considered property of the Higgs boson as these masses of functions of the coupling of these different kinds of fundamental SM particles with the Higgs field which are functionally related to the coupling of these kinds of fundamental SM particles withy the Higgs boson.

So, really, you can think of the SM parameters of making up 9 W boson interaction parameters, 11 Higgs boson interaction parameters (excluding the Z boson mass as a derived quantity), 1 parameter that is both a W boson and a Higgs boson interaction parameter (the W boson mass), one coupling constant to photons, and one coupling constant to gluons.

The Neutrino Masses

Of course, there are also the three neutrino masses which are SM free experimentally measured parameters. But we don't have a consensus SM answer about where those come from unlike all of the masses of the other SM fundamental particles.

There are dozens of papers on arXiv every month speculating on possible answers to the source of the neutrino masses. Still, any HEP theory in which there were three massive neutrinos would need to have three massive neutrino masses either derived or as free experimentally measured parameters. The existence of these parameters one way or another is obviously necessary in that scenario (and we know from observation that these values have to be three distinct values).

Reasons To Be Confident That the SM Is Not Over Fitted

Also, notably, the Higgs interaction is firmly on the electroweak unification side of the SM, in contrast to the strong force side which really has only one experimentally measured parameter that governs all particularly strong force aspects of particle interactions (along with the strong force color charges of quarks and gluons which are theoretically fixed integer values confirmed by experiment).

Put that way, most of the SM doesn't seem over fitted.

Strong force interactions and EM interactions are very elegant and governed predominantly by a single coupling constant each. Throw in a massless graviton in a quantum gravity SM extension and that still isn't very over fitted. Both the strong force and gravity are incredibly elaborate and complex, but that is not because of their large number of parameters. Instead, it is because the carrier bosons of these forces can interact with each other.

The SM as a whole is complex because a single process can (and indeed at some level of high loop precision almost always does) involve all three of the SM forces.

The theory holds together in finding consistent values of these SM parameters in many different contexts which would not have to be consistent in a non-SM theory.

Even with these two dozen or so parameters, consistency of the fundamental particle masses and coupling constants between very different way that they can present experimentally argues for the soundness of the model framework.

For example, the W boson mass that is important in hadron decays is also the same figure that is correct for charged lepton decays and neutrino interactions.

You can determine quark masses, CKM matrix parameters, and coupling constants in all manner of different interactions that wouldn't have to be consistent in some model other than the SM or a minor modification of it, and you get measured values that are consistent in all of them.

For example, most of the quark masses and CKM parameter values are derived from multiple kinds of hadron property measurements yet they are consistent.

The fundamental Higgs field interaction sourced masses in the SM determined independently, and their Higgs boson decay branching fractions, which have a common source in the SM, are consistent in experimental measurements of Higgs boson decays to within experimental uncertainties.

Another confidence inspiring point is that the SM predicts that the numerical values of the experimentally measured parameters of the SM run with energy scale in a fully derived manner. To the full extent that it has been possible to do so, this has been confirmed experimentally, for example, in the cases of the fine structure constant and the strong force coupling constant. Again, outside the framework of the SM this set of observations would be hard to understand.

Prospects For Improvement

Finally, the experimentally determined parameters of the SM are in the current state of science not derived quantities and we don't have a theory to explain that from a smaller number of fundamental physical constants.

There are theories out there, although this sub-forum of the Physics Forums isn't the place to discuss them, that propose possible ways to derive some of these parameters from other parameters that are experimentally measured in the SM.

But these parameters aren't precisely measured enough in many cases (such as the quark masses and neutrino masses) to discriminate convincingly between different theories that could explain them as derived quantities, even if someone already has come up with the right theory to explain their values.

Still, I have yet to meet a physicist or person knowledgeable about physics who really believes deep down that the observed patterns of relationships between these experimentally physical constants don't have a deeper source involving far fewer independent physical quantities, even though we haven't figured it out yet.

There are clear patterns in these two dozen physical constants, only a few of which like the relationship between the electromagnetic coupling constant, the weak force coupling constant, the W boson mass and the Z boson mass have been ascertained yet.

One can think of these parameters a bit like the atomic element masses in the periodic table before we discovered that these masses were a function of proton number, neutron number, and nuclear binding energy arising from the residual strong force mediated by mesons and described by a particular multi-term formula with one component for each kind of carrier meson.

Someday we may be able to explain the relationships between the SM experimentally measured parameters that mostly are necessary to explain in the electroweak part of the SM (1) why W boson interactions work the way that they do, and (2) why SM fundamental particles have the masses (derived from interactions with the Higgs field except possibly in the case of neutrinos) that they do.

But the gap in the SM of not having a way to explain the source of these couple of dozen experimentally measured parameters that mostly relate to W boson and Higgs boson/Higgs field interactions, while vexing, doesn't suggest that the SM is unsound.

Even if we had formulas to derive most of the SM experimentally measured parameters from a few deeper quantities and we would feel better about having a deeper understanding of the laws of nature, the main practical impact of it would be to allow for a more precise determination of some of the two dozen experimentally measured parameters that we do have by using more precisely measurable parameters to calculate parameter that are harder to determine precisely with direct measurements.

On a day to day doing HEP physics basis, we'd still look up the value of the two dozen or so SM parameters in a chart or reference and go from there when making calculations, just as we do now, just as we continue to use the periodic table even though we no longer think of the atomic element masses as fundamental.

A Note On Terminology

As a terminology footnote, I use the language "SM experimentally measured parameters" to underscore the idea that the SM has other parameters that are taken as fixed integer or simple small whole number ratio values that are not experimentally measured except to confirm the validity of the theory generally that are non-moving parts in SM theory. These theoretically assumed constants include the strong force color charge of the quarks and gluons and the number of kinds of strong force color charges, their total angular momentum J a.k.a. spin of the SM fundamental particles, the electromagnetic charges of the SM fundamental particles, the weak force charges of the SM fundamental particles, the number of SM fermion generations and the permissible types of SM fermions, the absence of right handed neutrinos and left handed anti-neutrinos, and the relationships between SM fundamental particles and their antiparticles.

Likewise, I omit from this discussion some physical constants that are necessary for the SM calculations but not usually considered a part of its list of parameters some of which like Planck's constant and the speed of light, arguably ought to be included in the list, and others of which, like the mathematical constants π and Euler's constant e, probably shouldn't be included in the list.

 

[Orodruin]

Of course, there are also the three neutrino masses which are SM free experimentally measured parameters. But we don't have a consensus SM answer about where those come from unlike all of the masses of the other SM fundamental particles. There are dozens of papers on arXiv every month speculating on possible answers to that question. Still, any HEP theory in which there were three massive neutrinos would need to have three massive neutrino masses either derived or as free experimentally measured parameters. The existence of these parameters one way or another is obviously necessary in that scenario (and we know from observation that these values have to be three distinct values).

It is worth pointing out that without the neutrino mass parameters there is no physical PMNS mixing. Hence, the PMNS mixing parameters should go the same way as neutrino masses. This is exactly the kind of rabbit hole I questioned if there was a need for in a B-level thread.

so one can legitimately choose a parameterization of the SM in which the Z boson mass is a derived rather than a free experimentally measured parameter.

Or, more commonly, the W mass. Which is what the hoolabaloo in April was about. Of course, which you take as derived and which as experimentally measured is entirely conventional.

 

[ohwilleke]

The bottom line on the neutrino properties is that they are a work in progress and have been imperfectly shoehorned into the SM after the rest of it was more or less in place when we discovered that they had to be massive due to neutrino oscillation which was observed.

Functionally and operationally, the PMNS matrix and two neutrino mass differences as well as an order of magnitude limitation on absolute neutrino masses is "good enough for government work", but our deeper understanding is lacking compared to the rest of the SM. It is possible to conceptualize the PMNS matrix as a description of W boson interactions, however (there is a published paper spelling this out in detail that I could find again at some point), which I do for the purpose of providing a feel of why the SM has so many parameters from just a couple of main sources in a clear way. But, of course, pretty much everything at a deeper theoretical level about neutrinos is open for theory development.

Or, more commonly, the W mass. Which is what the hoolabaloo in April was about. Of course, which you take as derived and which as experimentally measured is entirely conventional.

Certainly, the truth is the bigger perspective that there are four SM parameters with three degrees of freedom: the W boson mass, the Z boson mass, the fine structure constant, and the weak force coupling constant which is a function of Fermi's Constant. No one of these four parameters is truly more fundamental than the others. You can get the fourth from any possible set of the other three of the four parameters mathematically.

But it is customary to consider the three coupling constants of the SM to be more fundamental that the SM fundamental particle masses whether or not that is true. So, it is a toss up as to whether you think of the W boson mass or the Z boson mass as fundamental.

The W boson mass is the least precisely measured of the four, so in that regard, it makes sense to, in practice, derive the W boson mass from the other three if you are hoping for theoretically maximal precision in the value of the four parameters combined.

But, if one wants to find a way to concentrate the experimentally measured free parameters of the SM into one sector of the model, as I do above to illustrate how few experimentally measured parameters are present in the other parts of the SM, then treating the Z boson mass as derived and putting all of the free parameters into the description of W boson interactions makes sense.

Also, as a practical matter, W boson interactions are at the heart of all sorts are important experimentally measured derived quantities related to hadron and heavy charged lepton decays in the SM, while Z boson mediated interactions are pretty much a side show by comparison, even though they obviously matter and have to be understood too. Z boson mediated interactions are much less complicated and involve far fewer independent parameters.

 

[malawi_glenn]

the W boson mass and the Z boson mass are functionally related to each other in a manner that has only three degrees of freedom

Even beyond tree-level computations?

 

[Orodruin]

It is possible to conceptualize the PMNS matrix as a description of W boson interactions

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).

I still think this discussion goes way beyond B-level.

Even beyond tree-level computations?

B-level is tree level (or lower order ).

 

[malawi_glenn]

B-level is tree level (or lower order ).

I mean, $m_W = m_Z \cos \theta$ where $\theta$ is the Weinberg angle is the tree-level result.
Afaik it does not hold beyond tree-level.

Yeah, the discussion of loop effects is beyond Beginner level for this thread. But the mass relation was mentioned nevertheless and should at least be adressed with a "yes" or "no" answer.
Just to point out the intricacy of how SM parameters enters computations of observables (as in all QFTs)

 

[PeroK]

Whatever happened to the OP?

 

[ohwilleke]

Even beyond tree-level computations?

There is a functional relationship beyond tree-level computations (it also, e.g., includes the Higgs boson mass at some point). But, it is more complicated.

 

[ohwilleke]

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.

 

[Orodruin]

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.

I think this is pretty clear from most (reasonable) descriptions.

The point remains however that the PMNS is unphysical if neutrino masses are all equal.

 

[ohwilleke]

This is the classic quote about overfitting by the way. There’s a saying in physics, attributed to the famous genius John von Neumann: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.”

4gravitons discusses the overfitting issue in more depth at his blog. https://4gravitons.com/2022/10/14/machine-learning-occams-razor-and-fundamental-physics/