How can the mass of Higgs give preference to SUSY vs multiverse? According to the documentary Particle Fever, the precise value of the Higgs boson's mass could give more credence to either SUSY or multiverse theories. If the mass had been 115 GeV or below SUSY would have been favored, whereas a mass above 140 GeV would have given preference to multiverse.
Is there a way to understand this connection? How can the mass of a particle give input to the likelihood of a particular physical theory, (and/or in particular the two discussed here}?
I'm especially interested in a graduate-level but qualitative explanation, though any level would be great.
Note: a Physics.SE question discusses Higgs, SUSY, and multiverse, but does not give an explanatory answer for the connection.
 A: I have not seen the film. But this was not "supersymmetry versus multiverse". It was "supersymmetry without multiverse" versus "supersymmetry with multiverse". 
According to quantum field theory, a light Higgs boson (light compared to "grand unification" energies) should still look heavy because of virtual particle effects, unless these effects mostly cancel each other out. This is a feature of traditional supersymmetric models, and 115 GeV was a value coming from that sort of theory. 
However, it can be difficult to make such a model that gets everything right, experimentally. You may have to suppose that the various parameters of the theory assume values that are "just right", e.g. that one parameter is very small, or that two parameters are almost the same - and there will be nothing in your theory which implies this. You will just be "fine-tuning" it, in order to have certain undesirable effects not show up. 
In the past, the need for such parametric fine-tuning might be regarded as reason to reject a theory, if a causal mechanism for the fine-tuning couldn't be found. But "the anthropic principle" or "environmental selection" gives us a potential new reason why physics might look fine-tuned: perhaps other values of the parameters are inconsistent with the existence of life/atoms/etc. There might be a "multiverse" in which the parameters take different values in different places, but life is only possible in those places where the parameters take values which allow, e.g., something like complex stable chemistry to develop. 
140 GeV was a prediction coming from one of those arguments. Here is the paper. But you'll see that this is still a theory with supersymmetry! It's just that it's a supersymmetric model which contains some anthropic fine-tuning too. 
I want to very strongly emphasize that 115 GeV and 140 GeV are in no way the predictions coming from these two approaches - they are just examples. They may have been discussed in the film, because there were some experimental false alarms (in the search for the Higgs boson) at those energies. But we are talking about two types of theory - a supersymmetric theory with untuned parameters, and a supersymmetric theory with parameters tuned by anthropic selection - and if the details are different, the predictions are different. 
Indeed, go to pages 25-26 of the multiverse paper, and you will see no less than four special values of the Higgs boson mass listed, each of which they think might be indicative of anthropic selection within a multiverse. The reason is that they don't have an exact model of how physics works throughout their multiverse - they are just guessing at the principles governing what variations are allowed from place to place. In the paper they favor 140 GeV, but here they are saying that if the truly fundamental physics works in some other way, then maybe one of these other values would be favored. 
They list 128 GeV, which is sort of close to the value that was ultimately found, and say (page 26) "a Higgs mass near 128 GeV would provide strong evidence for the multiverse, although not quite as strong as might occur for a value near 141 GeV". In this regard, one should consider a "secretly famous" paper by Shaposhnikov and Wetterich, which actually did predict the right value - 126 GeV - several years in advance, and which didn't use the multiverse or supersymmetry. Instead, they assumed that quantum gravity has the property of "asymptotic safety" at high energies. This is an unfashionable assumption because it seems to contradict standard ideas about black hole entropy... However, my real point is that the right mass for the Higgs boson can possibly be obtained without the use of anthropic effects. And indeed, there are now some string-theory models in which the right value is produced by a physical cause rather than an anthropic tuning.
A: Mitchell Porter’s answer is very interesting.  However, it is also very divergent from what the movie conveyed and the current state of the SUSY vs. Multiverse debate.  He uncovers an interesting paper “A Finely-Predicted Higgs Boson Mass from a Finely-Tuned Weak Scale” published in 2009 or four years before the release of the movie.  This paper contradicts much of what you will read elsewhere on the topic.  The paper advances that SUSY is either consistent or accommodating of Higgs boson mass up to 141 GeV.  It also advances that SUSY and Multiverse may not be mutually exclusive, but instead Multiverse may serve as an extension of SUSY. 
The paper’s position is in direct contradiction of the one conveyed in the movie and in current debate over the issue.  As it currently stands, it appears well accepted that to confirm SUSY in its various forms the Higgs boson mass should come in at close to 115 GeV.  It is also advanced by some proponents of Multiverse that if it came in at 140 GeV it would be supportive of Multiverse.  Those two outcomes are obviously mutually exclusive.  This is why those theories are deemed competitors.  That much was explicitly stated by the various particle physicist theorists interviewed in the movie, including: David Kaplan and Savas Dimopoulos, supporters of SUNY (who confirmed the 115 threshold), and Nima Arkani-Hamed, supporter of Multiverse (who confirmed the 140 threshold).
The movie was very well received not only by the public but also the scientific community.  Peter Voit, a theoretical physicist, called the movie “fantastically good.”  He expressed some reservations regarding the opinions of Arkani-Hamed mentioned later in this writing. 
When the discovered boson (still not 100% sure it is actually Higgs) comes in at 125 to 126 GeV, it causes great confusion within the particle physicist theorists community.  This is because it essentially confirms both theories are not supported by this finding.  In plain English, those theories are wrong.  They are not supported by the ample data generated by the Large Hadron Collider (LHC).  At the end of the movie, David Kaplan stated that much.  Also, near the end another particle physicist supportive of SUSY conveys his genuine despair on the subject to his colleague Savas Dimopoulos.  The latter attempts to shore up his colleague’s mood by suggesting how exciting it is to finally uncover the truth after all those decades of research.  But, the colleague finds little comfort in finding he has been wrong for decades.
Regarding SUSY, a Higgs boson coming in at 115 GeV would have been consistent with what the Standard Model and SUSY would have predicted (caveat excluding the theory advanced by the mentioned paper).  Given it came in much above at 125 to 126, some of the foundation of the Standard Model and SUSY is questionable.  The underlying math of those theories will need more adjustments.  They already have many of those that question their scientific robustness.  Indeed, the Standard Model has numerous major weaknesses.  It does not explain gravity (as one of the four forces).  It entirely misses out cold dark matter and it overweights dark energy by a very large multiple.  The Standard Model has also 19 numerical constants to make the math work that are considered arbitrary.  Given that SUSY depends on the Standard Model, it relies on a somewhat tenuous foundation.  SUSY has mathematical adjustments of its own that may compound the probability of errors when combined with the 19 somewhat arbitrary constants from the Standard Model.  Scientists have advanced that SUSY theory should be abandoned as the LHC has not come up with any supporting evidence of its veracity despite trying hard to do so since 2010 (reference: Natalie Wolchover, November 29, 2012.  “Supersymmetry Fails Test, Forcing Physics to Seek New Ideas”.  Scientific American).
The Minimal Supersymmetric Standard Model (MSSM) is a recent effort to fully integrate the Standard Model and SUSY.  However, it does introduce 120 new parameters.  Several of those are deemed less than scientific as they can’t readily be tested.  Similarly, the LHC has not generated any supporting evidence for the MSSM.  That’s in good part for the simple reason that the Higgs boson came in too heavy at 125 - 126 GeV. 
Regarding Multiverse, a Higgs boson coming in at 140 GeV would suggests that at this level all hell breaks loose.  Our body of particle physics can’t precisely explain what is going on and needs an entire overhaul.  Peter Woit criticized Arkami-Hamed’s linking of the Higgs boson to Multiverse.  This is because since Multiverse follows a set of unknown and different physical laws this theory is entirely not testable.  In other words, it is a speculation that does not belong to the body of science.  And, a Higgs boson threshold of 140 GeV or any other threshold to supposedly support Multiverse is arbitrary by nature. 
The scientific method is a self-correcting mechanism.  Over time, most everything I have written will be obsolete.  Nevertheless, it will still serve as a fair representation of what the movie was about, what it conveyed, and the state of the SUSY vs. Multiverse debate at such time.         
