How seriously can we take the success of the Standard Model when it has so many input parameters?

Question

The Standard Model of particle physics is immensely successful. However, it has many experimentally fitted input parameters (e.g. the fermion masses, mixing angles, etc). How seriously can we take the success of the Standard Model when it has so many input parameters?

On face value, if a model has many input parameters, it can fit a large chunk of data. Are there qualitative and more importantly, quantitative, predictions of the Standard Model that are independent of these experimentally fitted parameters? Again, I do not doubt the success of the SM but this is a concern I would like to be addressed and be demystified.

Answer 1

It is inaccurate to think that all of the standard model of particle physics was determined through experiment. This is far from true. Most of the time, the theoretical predictions of particle physics were later confirmed experimentally and quite often to a very high accuracy.

For example, theoretical physicists predicted the existence of the $W^\pm$ and $Z$ bosons and their masses, the Higgs boson, the existence of gluons, and many of their properties before these particles were even detected. Pauli postulated the existence of the neutrino to explain energy conservation in beta decay, before the neutrino was observed. The anomalous magnetic moment of the electron, whose value was predicted by Julian Schwinger, agrees with experiment to up to 3 parts in $10^{13}$. Parity violation in the weak interaction, predicted by Lee and Yang, was later confirmed experimentally. The prediction of the positron by Dirac, was detected four years later by Anderson.

The list goes on and the list is huge$^1$. Particle physics is arguably the most successful physics theory because time and time again its predictions were later confirmed by experiment to surprisingly high accuracy (though sometimes our theories needed to be improved to explain some details of experimental data). I may be biased coming from a theoretical particle physics background, but I've always agreed that the Standard Model is the most mathematically beautiful, deep and profound model of all of physics. This is reflected in its almost miraculously accurate predictive power.

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$^1$Some more of the highlights:
1935 Hideki Yukawa proposed the strong force in order to explain interactions between nucleons.
1947 Hans Bethe uses renormalization for the first time.
1960 Yoichiro Nambu proposes SSB (chiral symmetry breaking) in the strong interaction.
1964 Peter Higgs and Francois Englert, propose the Higgs mechanism.
1964 Murray Gell-Mann and George Zweig put forth the basis of the quark model.
1967 Steven Weinberg and Abdus Salam propose the electroweak interaction/unification.

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Answer 2

The Standard Model may have many parameters, but it also talks about many things, each typically only involving a very limited number of parameters. For example, the muon lifetime$^\dagger$ $$\tau_\mu=\frac{6144\pi^3M_W^4}{g^4m_\mu^5}$$depends on only $M_W,\,g,\,m_\mu$ ($g$ is the weak isospin coupling), and the tauon lifetime $\tau_\tau$ satisfies$$\frac{\tau_\tau}{\tau_\mu}=\frac{1}{3(|V_{ud}|^2+|V_{us}|^2)+2}\frac{m_\mu^5}{m_\tau^5},$$which only depends on $|V_{ud}|,\,|V_{us}|,\,m_\mu,\,m_\tau$. So these two predictions need six parameters, which doesn't in itself sound that impressive. But that misses the point here. The full slate of SM predictions uses all the parameters in a variety of subsets, creating a system of far, far more simultaneous equations than we have parameters. If (to take one estimate discussed herein) there are $37$ parameters, it's not like we're fitting a degree-$36$ polynomial $y=p(x)$ by OLS. It's more like requiring the same few coefficients to simultaneously fit many different regression problems.

$^\dagger$ I'm sure someone will object my formula for $\tau_\mu$ has a time on the LHS and an inverse mass on the RHS, is using natural units, and should read$$\tau_\mu=\frac{\hbar}{c^2}\frac{6144\pi^3M_W^4}{g^4m_\mu^5}$$in SI units to expose the use of two more parameters $c,\,\hbar$. Bear in mind, however, these both have exact SI values by definition. Anyway, it doesn't really affect the argument.

Answer 3

The LHC has produced 2,852 publication as of today: September 24, 2021. Let's say each publication has 5 plots. Each plot has 50 points. We'll round that up to 1,000,000 data points, along with a comparison to theory.

What fraction of particle physics data is LHC? 1%? I don't know. Particle physics started in 1908 with Rutherford's $^{197}{\rm Au}(\alpha,\alpha)^{197}{\rm Au}$ experiment. Let's say LHC is 0.1% of all data.

That means 1 Billion data points explained with $N$ free parameters. I remember $N=37$, but maybe it's changed. Wikipedia say $N=19$. I don't know about that.

Either way, the quantity of data over orders of magnitude of energy, involving all known forms of matter, in EM, weak, and strong sectors, explained by so few parameters is extraordinary.

Dark things notwithstanding.

Answer 4

The question is ultimately not the one of physics, but of statistics. It is for a good reason that particle physics remains one of few fields of physics where statistics is still practiced on an advanced level (in many other fields high precision of measurements reduced the need in statistical analysis to calculating standard deviations). In particular, the statistics chapter in PRD is an excellent crash course about statistical analysis. 

How many parameters is many? 

In physics we are used to models where the number of parameters can be counted on our fingers, because we are aiming at understanding the elementary interactions/processes/etc. Describing any real world phenomena necessarily results in combining many elements and using more parameters. The models used in engineering, e.g., to design airplanes, or in government planning contain hundreds or thousands of parameters. The high promise of machine learning is due to the modern computational ability to use models with millions of parameters, often having very obscure meaning (to humans) - but they still work very well, as we see by Facebook tagging photos or growing quality of Google translate.

How much data? 

Whether we have too many parameters depends on how much data we have. The rule-of-thumb is having more data points than we have parameters. However, more principled approaches are built around the likelihood that is the probability of observing data, given our values of parameters: $$P(D|\theta).$$ Model in this context is the means of expressing this relationship between the parameters and the data mathematically.

Now, if our model is any good, the likelihood will be increasing as we increase the amount of the data (the number of the data points) - although this increase is not strictly monotonuous, due to the random effects. If this does not happen, our model is not good - perhaps it is too simplistic, has too few parameters - this is called underfitting.

Comparing models 

Given a wealth of data, the model with more parameters will generally result in higher likelihood - this is where the problem raised in the OP lies. Let me note in passing that we can never prove or disprove a model by itself - rather we compare different models and choose a better one. A model with more parameters may be simply better, because it better aporoximates physical reality. But such a model can result in a higher likelihood simply because we have more parameters to tune - this is what we call overfitting.

Methods have been developed for correcting for the number of parameters when correcting model. One of the most well-known is Akaike information criterion (AIC), where on compares quantities $$AIC=k_M -\log P(D|M),$$

where $k_M$ is the number of parameters un model $M$. The model with the lowest value of AIC is then considered to be the one that achieves the best results with the smallest number of parameters.

Lest this simple criterion appears too intuitive, let me point out that justifying it rigorously requires quite a bit of math. There exist also more elaborate versions, as well as alternative criteria, such as Bayesian information criterion (where $k_M$ is replaced by its logarithm). 

This is how choosing the best model is done in a nutshell. The physics comes in in formulating the logically motivated models to choose from. I suspect that, if we look at the publications in the times when the standard model was formulated, there were quite a few alternative proposals, and even more were probably floated in discussions among the scientists. Yet, the beauty of physics is that it allows to significantly narrow the choice of models - as alternative to machine learning approaches, where all possible models are equal, and the choice is based solely on their compatibility with the data.

Answer 5

If you have $n$ input parameters in a deterministic theory you can perfectly fit at most $n$ data points just by adjusting those parameters. In a probabalistic theory that is more subtle, but there is a similar association. Regardless of how many parameters the standard model needs, it is a lot less than what would be necessary to fit the 1 petabyte of data collected at the LHC per second.

Answer 6

We should take the standard model seriously because it has exquisite predictive power

I feel like focusing on the many hand-input parameters in the Standard Model misses the forest for the trees. There is no requirement for nature's laws to conform to our ideas of what a theory should look like. The ultimate arbiter for the theory is whether it makes testable predictions. Since the Standard Model is so good at that, we should take it seriously.

Answer 7

30 odd parameters is not large compared to 30 thousand, or indeed 30 million. Of course physicists would like to whittle down the number of parameters to just one, or indeed none. However, we can take it seriously because of its experimental success. A more mathematically elegant derivation of the standard model goes via non-commutative geometry. This produces the full model including neutrino mixing in a naturally geometric fashion - even if non-commutative.