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It's my understanding that nothing in the Standard Model predicts the mass values of the fundamental particles, so I guess that means we don't currently know how to make models of Higgs interactions predict the masses theoretically. My question is more about upcoming experiments when the LHC comes back online. As more data on the details of the mechanism come in, might a more accurate picture of Higgs (and the details the mass acquisition process with it) yield empirical models that specify how much mass a particle gets? Or are the reasons the Standard Model doesn't predict the masses too deep to be resolved by simple extensions of the theory?

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The model – so far compatible with everything you know – isn't just an empirical model and it won't be found in 2015. It was already found in the 1960s, with the QCD completion in the early 1970s, and it's called the Standard Model; it is the most complete realistic example of a quantum field theory (or, using a more specific classification, a gauge theory with matter fields). The LHC has so far agreed with every single prediction of the Standard Model.

The known symmetries etc. are so constraining that the model with the minimal spectrum – the particles that have been already observed – is complete up to 20 free parameters or so. And we know the values of all these parameters at a pretty good accuracy level. Except for one parameter, effectively the Higgs mass, we have known all these values of parameters for decades.

Both the Higgs vev $v$ (how far the Higgs field at each point of space is from the "central value" for which the masses would be zero) and the Higgs mass itself may be calculated from the coefficients of the quartic ($\lambda$) and quadratic ($-\mu^2$) terms in the Higgs potential energy. The W-mass and Z-mass are calculated from the Higgs vev and the gauge couplings (which may also be independently measured from the rate of the weak interactions etc.), essentially by $m_{\rm boson}=g_\text{its force} v$.

And the masses of the fermions are calculated as the product $m_{\rm fermion}=yv$ of the Higgs vev $v$ and the corresponding Yukawa coupling $y$ (how much the Higgs interacts with the given fermion species). The gauge couplings, Yukawa couplings, and $\lambda$ (the fourth-order Higgs' self-interaction) are dimensionless (without units) parameters of the Standard Model that have to be measured; they determine the masses of the particles. One needs a deeper theory to calculate them from first principles (string theory to calculate all of them). The Higgs mass or Higgs vev or Higgs quadratic coefficient is the only dimensionful parameter of the Standard Model (with the units of mass) which determines the natural scale for all masses of particles in the Standard Model. (A mystery why this mass scale is so much lower than the Planck scale, the natural scale of quantum gravity, is known as the hierarchy problem.)

There won't be any "new or better" model found by the LHC if the spectrum of particles remains what it is. The LHC won't find any new values of parameters in the Standard Model; we already know all of them. At most, the accuracy will improve a bit. If the LHC finds new physics in 2015 (or in the 2012 data that haven't been processed yet), we will have to switch to a more comprehensive model that also describes the Beyond the Standard Model physics.

But to explain the values of the parameters from deeper principles, one needs to make theoretical and not experimental advances – advances in very high-energy theoretical physics near the fundamental scale. Grand unified theories may do a part of the job but only string theory has the potential to explain and calculate all the values of the parameters.

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What a very lucid summary! Thanks –  David H Sep 7 '13 at 7:37

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