*The key to absorbing this post, following my last two, is to appreciate that the Standard Model ‘particles’/fields can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields* (see header photo). To carry out a Horava-Witten decomposition, flesh out the bulk-field as

with satisfying

and the action is then given by

After an integration by parts and substituting-in , Eq. 1 reduces to

Now, as in typical Kaluza-Klein compactifications, the bulk field manifests itself to a four-dimensional ‘observer’ as an infinite tower of scalars with masses . After changing variables to

we get

Bessel functions of order

give the standard Bertotti-Robinson-solutions. Hence, we have

with a normalization factor. That the differential operator on the LHS of

is self-adjoint means that the derivative of is continuous at the orbifold fixed points, and that gives us two relations that can solve for and , yielding

Now let

and so we have

The exponential suppression can be understood from Eq. 4 above: the modes are larger near the 3-brane at , therefore we can expect to find the Kaluza-Klein excitations in that region and their masses behave exactly in the same way as masses of fields confined to the brane at . For the massless case, m = 0, there is a mode with constant and and can be obtained from Eq. 4 and Eq. 5 above. For small, we get

Without loss of generality or substance, I will assume is of order unity, yielding us

Four-dimensionally, these induce couplings between the Kaluza-Klein modes and so the exponential factor in

where are Lorentz coordinates on the four-dimensional surfaces of constant , plays an essential role in determining the effective scale of the couplings. If the Planck scale sets the scale of the five-dimensional couplings, the low-lying Kaluza-Klein modes will have TeV-range self-interactions!

Start with the term-action

where is of order unity. Then, mode expansion gives us the self-interactions of the light Kaluza-Klein states

Thus, the effective 4-D coupling constants for the interactions are

By Horava-Witten bulk-term derivation, Eq. 7 and Eq. 8 become, respectively, Eq. 10 and 11:

So, from a 4-D ‘observers-perspective’, we get a interaction with a coupling constant

and for ‘large’ , it reduces to

*And the philosophically deep point* is that in both cases, the scale relevant to four-dimensional physics is not , but

*Thus in the Randall-Sundrum compactification model*, bulk scalars acquire low lying Kaluza-Klein modes with four-dimensional masses of order the weak scale and four-dimensional non-renormalizable interactions that are suppressed by powers of the weak scale even though from a five-dimensional brane-perspective their masses and interactions are characterized by the Planck scale, and thus establishes the points of my last three posts relating brane-based cosmology, the Randall–Sundrum model and Klebanov-Strassler warped space-time geometry.

…

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