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I know that the search for Higgs would be quite pointless if there was no estimation of its mass.

Namely the perturbative violation of unitatity, gave us an upper bound on its mass.Unitarity Constraints on Heavy Higgs Bosons

I wonder if there are enough parameters in the Standard Model of particle physics, such that it introduces sharp bounds on SUSY parameters that cannot escape experiments?

Is there in general any model independent prediction made by SUSY or it can always be rescued by means of introducing new fields and couplings phenomenologically?

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Since supersymmetry is not realized in our world, any supersymmetric model must include supersymmetry breaking. Similar to how the Higgs mechanism breaks the $SU(2) \times U(1)$ symmetry of the standard model to a $U(1)$ subgroup, so too must there be a variant of the Higgs mechanism that breaks supersymmetry to explain why our vacuum is not manifestly supersymmetric.

Any symmetry breaking mechanism comes with a scale. Above the energy scale of symmetry breaking, the symmetry is approximately restored; below the energy scale of symmetry breaking, the symmetry is broken. This scale will typically be associated with the masses of the supersymmetric partners. Traditional solutions of the hierarchy problem will place the scale of supersymmetry breaking somewhere around where the LHC can probe the new superpartners. However, there is no a priori reason for the SUSY breaking scale to be within reach of the LHC; theoretically it could take any value beyond current experiments (at least up to the Planck scale where one presumably has to incorporate quantum gravity as well). There are models such as split supersymmetry where the SUSY breaking scale could be of order 100 TeV while still being relevant for the hierarchy problem.

So, to summarize:

  • For SUSY in general, the SUSY breaking scale could take on any value up to the Planck scale, so we will essentially never be able to rule out SUSY as a possibility with lab experiments.
  • For SUSY relevant to the hierarchy problem, there are published models where the SUSY scale could be as high as the 100 TeV range. However, at least as far as I know, there is no guarantee that if experiments ruled out SUSY breaking up to about 100 TeV, that theorists would not discover interesting models with a SUSY breaking scale in (say) the 1000 TeV range (and so on). The further we get above the mass of the Higgs, the harder it is to argue that the model solves the hierarchy problem, but there is no sharp line where it becomes impossible to make that connection.
  • There are other SUSY parameters that I did not talk about which are necessary to understand how constraints arise on specific instantiations of SUSY. However, the presence of a SUSY breaking scale is a universal feature of phenomenological SUSY models.
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  • $\begingroup$ About the sharp line: I want to say, the sharp line can be where the "fine tuning" can be avoided by another fine tuning. $\endgroup$
    – Bastam Tajik
    Commented Jun 25, 2023 at 18:17
  • $\begingroup$ @BastamTajik The point is that there is no sharp line defining exactly what counts as "fine-tuned" and what does not. $\endgroup$
    – Andrew
    Commented Jun 26, 2023 at 2:10
  • $\begingroup$ Well one can quantify fine tuning by different means. Exponential sensitivity is one such way. At the very least, one can self-referentially determine it: whatever and as much as you count as fined tuned in case of Higgs. There are several ways to me to formulate the concept. $\endgroup$
    – Bastam Tajik
    Commented Jun 26, 2023 at 6:41
  • $\begingroup$ @BastamTajik "There are several ways to me to formulate the concept." Indeed, that's the problem :) More seriously, yes you can define a metric that has a sharp cutoff, but 10 other physicists can come up with 10 different metrics, and the 10 precise cutoffs won't agree. To be fair I think that if a solution to the hierarchy problem doesn't show up by, say, 100 TeV, then it becomes extremely hard to imagine there is a solution that isn't fine tuned by any reasonable definition, but by definition fine tuning comes from subjective aesthetic concerns, not mathematical consistency. $\endgroup$
    – Andrew
    Commented Jun 26, 2023 at 22:57

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