It depends. For example, the free parameters of the standard model lack any theoretical justification or explanation but are extracted through the standard regularization and renormalization schemes. One crucial problem, both from an operational perspective and a theoretical one, is incredible degree of fine-tuning to the necessary quantum corrections required for the divergences when the parameters are supposed to be independent of one another. Fine-tuning of this sort in mean field theory and many-body physics (e.g., condensed matter physics, statistical field theory, even fluid dynamics!) is not an issue because the cutoffs and regularizations are not supposed to be fundamental but are calculated from known physical forces or constraints that we are ignoring or summing over or otherwise "subtracting" and replacing in order to make problems tractable. In the standard model, the delicate corrections and cancellations required cannot be so justified. Even from an EFT perspective, that so many supposedly independent parameters should require such precision in order for the most fundamental physical theory in existence to make any sense at all is a problem. But whether one views it as a problem that should be solved by a more fundamental theory that explains it or a change in how we interpret both the "bare" terms and the renormalization schemes used (or both!) is a matter of debate.
This is without getting into the difficulties (again, operational as well as theoretical) of trying to explain the difficulties faced the delicate fine-tuning processes when one tries to seek answers from models where the parameters can take on different values or be explained by underlying parameters which do or in which we examine the interrelationships by altering known parameters and so forth.
Things are worse when it comes to issues of the cosmological constant because even if the standard model requires so many unexplained, unjustified parameters when it is supposed to be as fundamental as it gets, at least we are able to obtain predictions (or in the case of QCD and other similar or smaller scale processes, finite) results. This is not true of gravitation, which is a non-renormalizable force and thus there isn't even an answer as to how we might go about trying explain the fine-tuning of the cosmological constant and other such parameters, still less why it is that again the precision required (in this case not just operationally but also for the universe to exist) is so fine.
By the way, the multiverse concept is not used as a justification. Nor is it a metaphysical speculation. It arises from the mathematics of physics.
Almost nothing about the above is correct. Firstly, it only "arises" from the mathematics as one class of solutions to resolve metaphysical or aesthetic issues of a particular sort. Second, most of the relevant mathematics comes from as yet unformulated speculations about what would be required in particular BSM theories should they ever be formulated (still less tested).
"Despite the growing popularity of the multiverse proposal, it must be admitted that many physicists remain deeply uncomfortable with it. The reason is clear: the idea is highly speculative and, from both a cosmological and a particle physics perspective, the reality of a multiverse is currently untestable...For these reasons, some physicists do not regard these ideas as coming under the purvey of science at all.
Since our confidence in them is based on faith and aesthetic considerations (for example mathematical beauty) rather than experimental data, they regard them
as having more in common with religion than science."
from the editor's introduction to Carr, B. (Ed.). (2007).
Universe or multiverse?. Cambridge University Press.
Thirdly, it is indeed true that a central justification for multiverse cosmologies is a sort of Copernican-like principle in which even something like the cosmological constants and other values that enter into cosmological models in BSM physics (rather than into HEP equations or particle physics
per se) can be explained as simply inevitable given the vast numbers of existing universes. This is a decades old explanation that has been "strengthened" in a certain sense over the past ~30 years or so in particular.
But even (perhaps especially) its propenents stress repeatedly how it serves to explain the particular precision and fine-tuning of cosmological and standard model paramers, e.g.:
"If our universe is just one among very many in an enormous multiverse, then observed universes will be those that contain certain complex structures necessary for observers. Such arguments from environmental selection can potentially solve the cosmological constant problem, and yield a statistical prediction for the dark energy in observed universes. In this paper, we consider the extent to which nuclear stability and electroweak symmetry breaking provide evidence for environmental selection.
Many physicists, however, are reluctant to countenance any form of anthropic argument...Why make the extraordinary leap of postulating an extra-horizon multiverse, which has the smell of a secular form of God? In short, many believe that appeals to the environment are an escape from true science and that, in the absence of data confirming the conventional symmetry approach, it would be better to change fields than to succumb to the philosophy of anthropics.
The case of the cosmological constant demolishes these arguments. Traditional methods have not given any satisfactory understanding for why the cosmological constant is small. In contrast, the environmental argument not only explains why it must be small, but makes a statistical prediction for a nonzero value..to make an observer. Of course, the prediction does require a multiverse..."
Hall, L. J., & Nomura, Y. (2008). Evidence for the Multiverse in the Standard Model and Beyond.
Physical Review D,
78(3), 035001.
Like I said, when examined the improbabilities are not there.
I'm going to go out on a limb and guess that your familiarity of the literature and debate here is limited to popular media, and further that you are not familiar with the relevant distinctions between so-called fine-tuning arguments of the type theists make and the more general issues of naturalness (including so-called technical naturalness but also the more general principles initiated by Dirac) and fine-tuning either in particle physics or cosmology.