With regard to the ‘dark matter’ conclusions based on the observations of the motion of gas (rather than stars) in galaxies, see also my webpage Galactic Rotation Curves and the Dark Matter Myth.

Thomas

]]>There are also other, field-theoretic, ways of localizing matter on a submanifold, by using a topological defect.

]]>Bee has a nice description of how extra dimensions can make gravity weaker in her post on extra dimensions. As near as I can tell, no one knows of an actual reason for electromagnetism to be confined to 4+1 dimensions while gravity isn’t though, It’s just a way of explaining the discrepancy.

]]>The spherical harmonics are the angular portion of the solution to Laplace’s equation in spherical coordinates where azimuthal symmetry is not present. And there are three types of galaxies.

elliptical galaxy e.g. NGC4881 Three Dimension GM(

Mark wrote:

PK: I am indeed using renormalization in the QFT sense. The hierarchy problem is the following. Write down a classical Lagrangian containing all the fields of the standard model, including the Higgs field, with its mass set at the GeV or TeV scale. Then compute the quantum loops in the system (do renormalization). The Higgs, because it is a scalar particle, picks up quadratic corrections to its mass. If the standard model holds unchanged up to the Planck scale, then the Higgs mass will be renormalized up to that scale. The only way to avoid it is to fine tune the original bare mass you put in the theory to many, many decimal places. This doesn’t really happen for the other particles because if they’re fermions they only experience logarithmic running, which doesn’t require fine tuning to fix. Hope this helps.

This is a good argument if and only if we view a QFT model as a theory that are effective only upto a certain scale. Since the Standard Model is not a theory of everything it is legitimate to expect new physics to kick in at some scale, and then the bare Higgs mass at that scale may require fine tuning.

But if we take the Standard Model purely as a mathematical model with no concern about physics at a higher energy scale, then the cutoff goes to infinity, and the question of fine tuning to many decimal places is as bad or as irrelevant for logarithmic running as for quadratic running. Either this model can have non-zero, non-infinite masses naturally, or not.

If the model can have non-zero, non-infinite masses naturally, then there is no hierarchy problem in a fundamental sense, it is purely an artifact of our limited computational methods. We may have a secondary hierarchy problem, which is how does the model generate mass scales from the fraction of a eV to a TeV, but the finite small (compared to infinity!) Higgs mass is not a problem. In other words, why doesn’t everything weigh the same as the Higgs?

If the model cannot have non-zero, non-infinite masses, what we’re saying is that our model as a physical theory has finite GeV-TeV masses purely because of the intervention of new physics at some high energy scale, and to me, this is even more mysterious than the original hierarchy problem.

Another possibility is that the Standard Model is not well defined as a mathematical object.

The question is – is any of the above correct?

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