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8

Dimensional regularization (i.e., dim-reg) is a method to regulate divergent integrals. Instead of working in $4$ dimensions where loop integrals are divergent you can work in $4-\epsilon$ dimensions. This trick enables you to pick out the divergent part of the integral, as using a cutoff does. However, it treats all divergences equally so you can't ...


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One notable class of exceptions are what are called "hierarchy problems" in particle physics. For example, if you identify the Planck mass as a fundamental energy scale, you end up with huge dimensionless numbers which don't have an obvious explanation (i.e. ratio of Planck to electroweak scale, etc.). Explaining these large (or small depending on how you ...


5

Here are two examples of where dimensional estimates fail: Divergent expressions I have a laser pointer pointed at a wall directly facing me, 1 meter away. I turn the laser pointer by 90 degrees over the course of 1 second. What is the average speed of the laser spot during the turn? The dimensional analysis argument is 1meter/1second, which would be ...


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The holographic principle tells us that the description of what happens in a volume of space can be encoded on a surface that surrounds it. This is related to the Bekenstein bound that tells us that the amount of information in a volume of space cannot be more than the area of the surrounding surface in units of a quarter of a planck area. This in turn as ...


4

Well, of course you have to pick the quantities in your dimensional analysis right. Example: Use dimensional analysis to estimate the potential energy of a star, hold together only by gravitation. Solution: Newtons gravitational constant $G$ better show up somewhere. This requires us to include something with units $kg^2 / m$. We can get this by inserting ...


4

All we can do precisely is give a probability for some physical quantity to have its observed value. For example (subject to various assumptions!) the probability of the cosmological constant having it's observed value is around 1 in $10^{120}$. Since this is absurdly low we say it's fine tuned. But where you draw the line between fine tuned and not fined ...


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The argument for fine tuning is far broader than just the Higgs. For example it's argued that the existance of any elements heavier than lithium relies on a fine tuned resonance in the carbon 12 nuclues that allows three helium nuclei to stay together long enough to form a carbon nucleus. There are lots of books that explore these ideas. A good start would ...


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It's the same problem because the low scale matches in both definitions; and the high scale matches in both definitions, too. Both problems are the puzzle why the two scales are so much different. First, the low scale. In the Higgs fine-tuning, you define the low scale as the Higgs mass. But the Higgs mass can't be parameterically greater than the Z-boson ...


3

The hierarchy problem is not only about big numbers, such as $M_{pl}/M_{EW}$, per se'. In fact in QCD there is no hierarchy problem associated to the ratio $M_{pl}/\Lambda_{QCD}$. The problem is actually about the quantum numbers of certain operators in a Wilsonian EFT. The point is that we understand the SM as an effective low-energy description of the ...


3

The question is dealt with on pages 302 to 304 of Smolin's book The Life of the Cosmos. The reasoning is based on a proposal by Hans Bethe and Gerald Brown (I don't have the reference to hand) that in a neutron star kaons can become light (by a mechanism analogous to superconductivity), and indeed can become light enough for an electron to decay into a kaon ...


2

The example of unnaturalness you describe is the example of the mexican hat for the higgs mechanism ( if you look at this page up on the left you will see the mexican hat in the PHYSICS logo). As all should know this symmetry is naturally broken at our energy levels, as in this the example, which is correct, that the pencil sits precariously and can break ...


2

The gauge bosons such as the photon and chiral fermions may be massless due to the symmetry – gauge symmetry and chiral symmetry, respectively. We may consistently require the theory to respect these symmetries at the exact level, including all quantum corrections. For the gauge symmetry, it means that $1+1=2$ polarizations of the photon or another particle ...


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It seems plausible that the probability of the universe being just right for human life by pure chance is pretty low, and possibly absurdly low. That means if the universe only had one chance at existing the probability we'd be around to see it is vanishingly small. The various multiverse theories suggest that huge numbers of disconnected universes get ...


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1) The cosmological constant in the context of quantum field theory is a set of calculations and leads to a sum that will look something schematically like this cc = 3-5+22-120+3042-50242+... +O(M^4) where M is some heavy mass scale The problem is we don't know the exact numbers in the sum, we can only calculate a few of them and the best we can do is ...


1

To me, it seems like there are 3 different concepts being discussed: (1) fine-tuning, (2) wanting unitless constants to be of order unity, and (3) wanting theories to have a simple form. The WP link defines "naturalness" as #2, although I don't think that's universally understood. A very old example of #3 would be the history of models of the solar system, ...


1

Almost by definition, everything that can influence us today is part of our universe. So something in another universe (roughly) cannot have any relevance to our universe today. However, in some (very speculative) models, what is now in another universe might have influenced our universe a long time ago, and we might be able to find signatures of that in ...


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One more exception: weak localization. Second class of exceptions (Lagerbaer named first where interaction constant is involved) originates from two characteristic lengths/times/... in the system. [Function of] their ratio may appear in the answer.


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A notable exception would be Debye's model for low temperatures. The molar heat capacity is then approximated as $ C = 234 \ k \ \left( \frac{T}{T_D} \right)^3 $ with k the Boltzmann constant and $T_D$ the Debye temperature. I think the biggest reason why constant are always so small is that if they're not small, they're absorbed in other quantities.


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Another exception: the factors can depend on the number of dimensions. Esp in models with extra dimensions you gather easily some (2 pi)^d that can be a factor 1000 or larger.


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Analogously to the use of SU(2) instantons for describing tunelling between topologically distinct vacuum sectors, I've heard people talk about gravitational instantons as describing a tunneling process connecting "nothing" with an expanding Minkowski signature universe. People talk heuristically of a universe having emerged by tunnelling from "nothing". ...


1

Let us suppose that that the Standard Model is an effective field theory, valid below a scale $\Lambda$, and that its bare parameters are set at the scale $\Lambda$ by a fundamental, UV-complete theory, maybe string theory. The logarithmic corrections to bare fermion masses if $\Lambda\sim M_P$ is a few percent of their masses. The quadratic correction to ...


1

The upper mass limit for neutron stars depends both on general relativity and on the equation of state for dense nuclear matter. General relativity we seem to understand pretty well. Dense nuclear matter we understand less well, so estimates of the neutron star upper mass limit vary by about a factor of two. If Smolin's idea really does provide another, ...



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