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Finetuned Cancellations and Improbable Theories

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Abstract

It is argued that the \(x-y\) cancellation model (XYCM) is a good proxy for discussions of finetuned cancellations in physical theories. XYCM is then analyzed from a statistical perspective, where it is argued that a finetuned point in the parameter space is not abnormal, with any such point being just as probable as any other point. However, landing inside a standardly defined finetuned region (i.e., the full parameter space of finetuned points) has a much lower probability than landing outside the region, and that probability is invariant under assumed ranges of parameters. This proposition requires asserting also that the finetuned target region is a priori established. Therefore, it is surmised that a highly finetuned theory (i.e., remaining parameter space is finetuned) is generally expected to be highly improbable. An actionable implication of this moderate naturalness position is that the search for a non-finetuned explanation to supplant an apparently finetuned theory is likely to be a valid pursuit, but not guaranteed to be. A statistical characterization of this moderate position is presented, as well as those of the extreme pro-naturalness and anti-naturalness positions.

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Notes

  1. It is straightforward to generalize the probability discussions to come to models with more random variables cancelling. Nevertheless, the two-variable XYCM captures much of the essence of any multi-variable model.

  2. An attack on this viewpoint (e.g., concern for what observables, or functions of observables, should be used to compute finetunings) and a defense against such attacks can be found in [14].

  3. If we had instead chosen x, y to be flat over the interval \(\xi \le x,y\le 1\) instead of \(0\le x,y\le 1\) the probability of landing in the FT region would increase modestly to \(P_{\Delta _\mathrm{FT}}(\xi )=\frac{2}{\mathrm{FT}+1}\left( \frac{1+\xi }{1-\xi }\right) \) when \(\mathrm{FT}\cdot (1-\xi )\gg 1\).

  4. The importance of \(3\sigma \) (99.73% probability of falling within it) is highlighted in the Intergovernmental Panel on Climate Change (IPCC) Reports which give \(>99\%\) likelihood the English phrase “virtually certain” [22]. Thus, it is “virtually certain” that a sampled point should lie within \(3\sigma \) of the mean.

  5. By “\(n\, \sigma \)” one means that the measurement is more than n standard deviations away from expected measurement within the standard theory under consideration. For very large n, the probability of the measurement fluctuation so far away from the true value is extremely rare, signifying a breakdown of the standard theory (i.e., breakdown of understanding of what physics is at play to give the results obtained).

  6. All the discussion in this article points more to the utility of declaring an unambiguous threshold to decide if a theory is “unnatural” (or improbable), since a theory on the other side of that threshold may also not be probable.

  7. To name two specific examples, the extreme anti-naturalness position would imply that requiring an extreme splitting of the doublet and triplet in minimal grand unified theories [23] should not concern anyone, nor should we be concerned that the cosmological constant appears finetunely small compared to ordinary quantum field theory expectations [24].

  8. The rescaled variables of the rXYCM are \(\hat{x},\hat{y},\hat{z}\) to not be mistaken with XYCM variables x, y, z.

  9. More technically, the probability density integral over \(\Delta _X\) must yield a tiny probability for falling within \(\Delta _X\).

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Acknowledgements

This work was JDW supported in part by the DOE under Grant No. DE-SC0007859. I wish to thank G. Giudice, S. Martin, A. Pierce, N. Steinberg and Y. Zhao for helpful conversations on these issues.

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  1. Leinweber Center for Theoretical Physics, Physics Department, University of Michigan, Ann Arbor, MI, 48109-1040, USA

    James D. Wells

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Wells, J.D. Finetuned Cancellations and Improbable Theories. Found Phys 49, 428–443 (2019). https://doi.org/10.1007/s10701-019-00254-2

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