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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?

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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
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  • $\begingroup$ 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) $\endgroup$ Mar 28 '15 at 17:11
  • $\begingroup$ @No worries, and tbf the table was hidden. $\endgroup$
    – PhotonBoom
    Mar 28 '15 at 17:11
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    $\begingroup$ 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. $\endgroup$ Mar 28 '15 at 17:30
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    $\begingroup$ 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? $\endgroup$ Mar 28 '15 at 18:59
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    $\begingroup$ @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. $\endgroup$
    – 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.

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    $\begingroup$ 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? $\endgroup$
    – 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

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