Randall-Sundrum Compactification Models and Bulk-Field Actions

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

    \[\Phi \left( {x,\phi } \right) = \sum\limits_n {{\psi _n}} \frac{{{y_n}\left( \phi \right)}}{{\sqrt {{r_c}} }}\]

with {y_n}\left( \phi \right) satisfying

    \[\int\limits_{ - 1\pi }^\pi {d\phi {e^{ - 2\sigma (\phi )}}} {y_n}\left( \phi \right){y_m}\left( \phi \right) = {\delta _{nm}}\]

and the action is then given by

 

eq1

Eq. 1

 

After an integration by parts and substituting-in {e^{ - 2\sigma \left( \phi \right)}}, Eq. 1 reduces to

 

eq2

Eq. 2

 

Now, as in typical Kaluza-Klein compactifications, the bulk field \Phi \left( {x,\phi } \right) manifests itself to a four-dimensional ‘observer’ as an infinite tower of scalars {\psi _n}(x) with masses {m_n}. After changing variables to

    \[\left\{ {\begin{array}{*{20}{c}}{{z_n} = {m_n}{e^{\sigma \left( \phi \right)}}/k}\\{{f_n} = {e^{ - 2\sigma \left( \phi \right)}}{y_n}}\end{array}} \right.\]

we get

 

eq3

Eq. 3

 

Bessel functions of order

    \[v = \sqrt {4 + \frac{{{m^2}}}{{{k^2}}}} \]

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

 

eq4

Eq. 4

 

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

 

Eq. 5

Eq. 5

 

is self-adjoint means that the derivative of {y_n}\left( \phi \right) is continuous at the orbifold fixed points, and that gives us two relations that can solve for {m_n} and {b_{n\nu }}, yielding

    \[{b_{n\nu }} = \frac{{2{J_\nu }\left( {\frac{{{m_n}}}{k}} \right){{J'}_\nu }\left( {\frac{{{m_n}}}{k}} \right)}}{{2{Y_\nu }\left( {\frac{{{m_n}}}{k}} \right){{Y'}_\nu }\left( {\frac{{{m_n}}}{k}} \right)}}\]

Now let

 

Cnd. 1

Cnd. 1

 

and so we have

    \[2{J_\nu }\left( {{x_{n\nu }}} \right) + {x_{n\nu }}{J'_\nu }\left( {{x_{n\nu }}} \right) = 0\]

The exponential suppression can be understood from Eq. 4 above: the modes {y_n}\left( \phi \right) are larger near the 3-brane at \phi = \pi, 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 \phi = \pi. For the massless case, m = 0, there is a mode with {y_1} constant and {x_{12}} = 0 and can be obtained from Eq. 4 and Eq. 5 above. For {x_{1\nu }} small, we get

    \[{x_{1\nu }} \simeq \frac{1}{{\sqrt 2 }}\left( {\frac{m}{k}} \right){e^{k{r_c}\pi }}\]

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

 

Cnd. 2

Cnd. 2

 

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

 

Eq. 6

Eq. 6

 

where {x^\mu } are Lorentz coordinates on the four-dimensional surfaces of constant \phi, 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

 

Eq. 7

Eq. 7

 

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

 

Eq. 8

Eq. 8

 

Thus, the effective 4-D coupling constants for the \psi _n^{2m} interactions are

 

Eq. 9

Eq. 9

 

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

 

Eq. 10

Eq. 10

 

 

Eq. 11

Eq. 11

 

So, from a 4-D ‘observers-perspective’, we get a \psi _n^{2m} interaction with a coupling constant

 

Eq. 12

Eq. 12

 

and for ‘large’ k{r_c}, it reduces to

 

Eq. 13

Eq. 13

 

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

    \[\nu = M{e^{ - k{r_c}\pi }}\]

  • 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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