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From my understanding of the standard model, I understand that there are 19 or 20 free parameters that we need to put in by hand as, and I'm guessing here, there is as yet no theoretical basis for calculating them. Examples that come to mind are the masses of the elementary particles, the electric charge on them...etc, all resulting from experimental results.

Two quick questions:

  1. Does anybody have a list of these free parameters?

  2. Does the Higgs boson also give us the freedom to choose a parameter value that allows agreement with experimental results?

Qmechanic
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3 Answers3

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Yes, wikipedia has a table which lists the 19 free parameters that need to be tuned by experiments. These include, as you already said, the masses of the elementary particles including the Higgs Boson, and some other notable ones are:

  • CKM Mixing angles and CP-violation phase.
  • Gauge coupling of the three symmetries (U(1), SU(2), SU(3)).
  • Higgs VEV
PhotonBoom
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  • i thought there were also the masses of particles. John Baez says there are 25 "fundamental constants" in the Standard Model (and 1 more for the Cosmological constant) – robert bristow-johnson Mar 28 '15 at 17:11
  • @No worries, and tbf the table was hidden. – PhotonBoom Mar 28 '15 at 17:11
  • of course, the Baez blog is dated (2011): "The Higgs has not yet been seen..." but i think that the mass of the Higgs boson is still fundamental and not derived from other fundamental constants. – robert bristow-johnson Mar 28 '15 at 17:14
  • Photon, that ("numbers which we cannot derive from theory, we have to extract their value from experiments") is precisely the meaning of fundamental constants. – robert bristow-johnson Mar 28 '15 at 17:16
  • It's just that some sources say 19 and some say around 20, it does seem a strange discrepancy over such an important aspect of the S.M. Maybe I'm reading out of date sources, just the difference stuck in my head. That's it sorted anyway. Thanks –  Mar 28 '15 at 17:17
  • well, count 'em out, irish. that's precisely what Baez was doing. he counts 25 in the Standard Model and one more for the cosmological constant. either in this blog (or somewhere else, might have been in Wikipedia) Baez adds that as we learn more about dark matter, we might observe more parameters that cannot (and the moment) be derived from other parameters. he also adds that the number of fundamental constants (or "free parameters" if you like that semantic) will be reduced as the theory advances and some currently "free" values are explained in terms of others. – robert bristow-johnson Mar 28 '15 at 17:21
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    Well, properly, Newton's constant and the cosmological constant are "free parameters of fundamental physics" and not of the standard model, since the standard model does not deal with gravity. And the particle masses (or at least the combination of yukawa couplings hand the higgs VeV that determine the masses) are most definitely amongst these parameters. – Zo the Relativist Mar 28 '15 at 17:30
  • no, @JerrySchirmer. properly $G$ is just an expression of the units we use to express it. with Planck units, $G=1$ (as well as $c$, $\hbar$, and $\frac{1}{4 \pi \epsilon_0}$), but the $\Lambda$ is not normalized in Planck units, so you can call the cosmological constant a fundamental parameter (and its dimensionless value is about $10^{-122})$. – robert bristow-johnson Mar 28 '15 at 18:12
  • and the other question, @JerrySchirmer, is how are the particle masses determined experimentally and expressed among the 25 fundamental constants in the SM. if they are all relative to $m_e$, then there are 24, instead of 25, but $m_e$ still has to have an independent expression of value and we do that as $\frac{m_e}{m_\text{Planck}}$, which is dimensionless. either that or we need (as a fundamental constant) the gravitational coupling constant: $\frac{G m_e^2}{\hbar c}$,the square of $m_e$ relative to the Planck mass. – robert bristow-johnson Mar 28 '15 at 18:25
  • @robertbristow-johnson: Fine, but you've then fixed a scale for everything else and made THEM dimensionless. If you allow the curvature term in the action, it either comes with a dimensionless parameter (which I abstracted to mean G, but fine), or you fix units to make it unity, and that then fixes units for everything else. It's still semantics whether that counts as a "free parameter" – Zo the Relativist Mar 28 '15 at 18:33
  • yes, we've fixed the scale relative to Planck units and while that might seem to be an arbitrary decision, it is not anthropocentric. i don't think it's particularly arbitrary (except for factors of $\sqrt{4\pi}$, i think they should define and use natural units that normalize $c$, $\hbar$, $4 \pi G$, and $\epsilon_0$). but whether you choose $e$ as the unit charge or $\sqrt{4 \pi \epsilon_0 \hbar c}$ as the unit charge, the ratio of the two charges (totally determined from nature) is $\sqrt{\alpha}$. so the fine-structure constant is really just an expression of the elementary charge. – robert bristow-johnson Mar 28 '15 at 18:39
  • the "free parameters" are dimensionless numerical values in which we have no current viable theory of how those numerical values are determined from other parameters so they can only be determined by experiment. the choice of the 25 are arbitrary in the sense that whether it's $g_\text{SU(2)}$ or $\alpha$ is a choice (they are related to each other), but you need one or the other and, in any case, i think you will count 25 free parameters in the SM as does John Baez. – robert bristow-johnson Mar 28 '15 at 18:45
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    i took a look at comparing the table at Wikipedia to the Baez list. what about the "4 numbers for the Pontecorvo-Maki-Nakagawa-Sakata matrix"? is that in the list of free parameters of the Standard Model? – robert bristow-johnson Mar 28 '15 at 18:59
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    @PhotonicBoom: the table appearing in wikipedia is not correct. Either, you consider the Standard Model as in its original formulation with massless neutrinos and thus the 4 parameters of the PMNS matrix should not be listed (obtaining 19 parameters) or you add massive neutrinos considering both PMNS and the 3 masses leading to 26 parameters. – Paganini Mar 28 '15 at 19:53
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In the standard model, technically neutrinos are assumed to be massless. According to special relativity they would travel at the speed of light, so they would be timeless and could not oscillate. Hence, the 3 neutrino masses and 4 mixing angles are technically not parameters of the standard model. However, in simple extensions to the standard model, these parameters are free as well.

It is important to note that the 19 parameters in Wikipedia's table are not the only possible choice to describe the 19 degrees of freedom. i.e there are multiple choices of parameters that span this space.

Constantin Weisser
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    Afaik the Higgs is part of the SM, wouldn't it mean that the SM has massive neutrinos and their Higgs coupling is part of the free parameters? – peterh May 13 '17 at 15:35
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I would add to that list the masses and mixing angles of the three known neutrinos, which are just as arbitrary as the others within the confines of the standard model.

This adds seven parameters to the 19 listed in the Wikipedia article

https://en.wikipedia.org/wiki/Standard_Model#Construction_of_the_Standard_Model_Lagrangian

MikeV
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