Randall-Sundrum Braneworld Scenario, Klebanov-Strassler Geometry and the Standard Model

In this post, the mathematics applies to both, Randall-Sundrum-1and-2 models, hence I will not distinguish between them here. One of the most powerful aspects of M-theory’s braneworld scenarios is that the bosonic and fermionic fields of the Standard Model of physics can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields, after extra dimensional modulus stabilization, and recalling that Randall-Sundrum bulk/brane interactions yield a very deep solution to the EW hierarchy problem. Start with the theory defined by the following action:

    \[S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi - {m^2}{\phi ^2}} \right)\]

with the bulk field given by:

    \[\Phi \left( {x,\phi } \right) = \sum\limits_n {\frac{{{\Upsilon _n}\left( \phi \right)}}{{\sqrt {{\tau _c}} }}} \]

where generally, the bulk action, with worldsheet-uplift, is given by:

    \[\begin{array}{l}{S_B} = - \frac{1}{{2{k^2}}}\int {{d^D}} X\left\{ {\frac{1}{2}} \right.\left[ {{\eta ^{\mu \rho }}} \right.{\eta ^{\nu \sigma }} + {\eta ^{\mu \sigma }}{\eta ^{\nu \rho }}\\ - \frac{2}{{D - 2}}{\eta ^{\mu \nu }}\left. {{\eta ^{\rho \sigma }}} \right]{h_{\mu \nu }}{\partial ^2}{h_{\rho \sigma }}\left. { + \frac{4}{{D - 2}}\bar \Phi {\partial ^2}\bar \Phi } \right\}\end{array}\]

and {\Upsilon _n}\left( \phi \right) satisfying:

    \[\int\limits_{ - \pi }^\pi {d\phi {e^{ - \sigma \left( \phi \right)}}} {\Upsilon _n}\left( \phi \right){\Upsilon _m}\left( \phi \right) = {\delta _{nm}}\]

with a Dirac-Born-Infeld brane interaction term:

    \[{S_{BI}} = - {\tau _\rho }\int {{d^{p + 1}}} \xi \left( {\left( {\frac{{2p - D + 4}}{{D - 2}}} \right)\bar \Phi - \frac{1}{2}{h_{aa}}} \right)\]

which, after integration by parts and upon substituting {e^{ - \sigma \left( \phi \right)}} in our action, we get the Horava-Witten action variant:

    \[\begin{array}{c}S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}d\phi } \left( {{e^{ - 2\sigma \left( \phi \right)}}} \right.\\{\eta _{\mu \nu }}{\partial _\mu }\Phi {\partial _\nu }\Phi + \frac{1}{{r_c^2}}\left( {{e^{ - 4\sigma \left( \phi \right)}}\partial \Phi } \right)\\ - {m^2}{e^{ - 4\sigma \left( \phi \right)}}\left. {{\Phi ^2}} \right)\end{array}\]

Now, the bulk fields manifest themselves to 4-D ‘observers’ as infinite towers of scalars {\psi _n}\left( x \right) with masses {m_n}. After change of variables to:

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

our actions reduce to two interaction terms:

    \[S_{{\mathop{\rm int}} }^G = \int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \frac{\lambda }{{{M^{5m - 5}}}}{\left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi } \right)^m}\]

and:

    \[\begin{array}{l}S_{{\mathop{\rm int}} }^\Upsilon = \int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}} d\phi {e^{ - 4\sigma \left( \phi \right)}}\frac{\lambda }{{{M^{5m - 5}}}} \cdot \\\psi _n^{2m}\left( {\frac{{{{\left( {{\partial _\phi }{\Upsilon _n}} \right)}^2}}}{{r_c^3}}} \right)\end{array}\]

where we have:

    \[z_n^2\frac{{{d^2}{f_n}}}{{dz_n^2}} + {z_n}\frac{{d{f_n}}}{{d{z_n}}} + \left[ {z_n^2 - \left( {4 + \frac{{{m^2}}}{{{k^2}}}} \right)} \right]{f_n} = 0\]

and the Bessel functions of order:

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

yield the standard Bertotti-Robinson-solutions. Hence, we have:

    \[\begin{array}{*{20}{l}}{{\Upsilon _n}\left( \phi \right) = \frac{{{e^{2\sigma \left( \phi \right)}}\phi }}{{{N_n}}}\left[ {{J_\nu }\left( {\frac{{{m_n}}}{k}{e^{\sigma \left( \phi \right)}}} \right)} \right. + }\\{\left. {{b_{n\nu }}{\Upsilon _\nu }\left( {\frac{{{m_n}}}{k}{e^{\sigma \left( \phi \right)}}} \right)} \right]}\end{array}\]

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

    \[ - \frac{1}{{r_c^2}}\frac{d}{{d\phi }}\left( {{e^{ - 4\sigma \left( \phi \right)}}\frac{{d{\Upsilon _n}}}{{d\phi }}} \right) + {m^2}{e^{ - 4\sigma \left( \phi \right)}}{\Upsilon _n} = m_n^2{e^{ - 2\sigma \left( \phi \right)}}{\Upsilon _n}\]

is self-adjoint means that the derivative of {y_n}\left( \phi \right) is continuous at the orbifold fixed points, giving us:

    \[\left\{ {\begin{array}{*{20}{c}}{{N_n} \simeq \frac{{{e^{k{r_c}\pi }}}}{{\sqrt {k{r_c}} }}{A_n}}\\{{A_n} = {J_n}\left( {{x_{n\nu }}} \right)\sqrt {1 + \frac{{4 - {\nu ^2}}}{{x_{n\nu }^2}}} }\end{array}} \right.\]

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

    \[\left\{ {\begin{array}{*{20}{c}}{d{s^2} = {e^{ - 2k{r_c}\left| \phi \right|}}{\eta _{\eta \nu }}}\\{d{x^\mu }d{x^\nu } - r_c^2d{\phi ^2}}\end{array}} \right.\]

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

Now, a Klebanov-Strassler geometry naturally arises by considering string theory compactification on Ad{S_5} \times {X_5} where {X_5} is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is:

    \[\begin{array}{*{20}{c}}{{\mathcal{L}^{KS}}_{\psi \bar \psi {H^0}}\frac{1}{{{M^{3/2}}}}\bar \psi \left[ {i{\gamma ^\mu }} \right.{\sigma ^{\mu \nu }}H_{\mu \nu \lambda }^0\left( {{x^\mu }} \right)}\\{\left. {\frac{{{\chi ^0}(r)}}{{\sqrt {\tau c} }}} \right]\psi }\end{array}\]

which, after integrating over the extra dimensional part, the effective 4-D Lagrangian reduces to:

    \[\begin{array}{*{20}{c}}{\mathcal{L}_{\psi \bar \psi {H^0}}^{KS} = i\bar \psi {\gamma ^\mu }{\sigma ^{\mu \nu }}\left[ {\frac{{{e^{ - 4\pi K/{3_{{g_s}}}M}}}}{{{M_{pl}}}}} \right. \cdot }\\{\left. {\left( {\frac{{{r_{\max }}}}{{{r_0}}}} \right)} \right]H_{\mu \nu \lambda }^0\psi }\end{array}\]

with the fundamental Planck scale M and the 4-D Planck scale {M_{pl}} related as

    \[{M_{pl}} = \frac{{{M^{3/2}}}}{{\sqrt {2R} }}{r_{\max }}{\left( {1 - \frac{{r_0^2}}{{r_{\max }^2}}} \right)^{1/2}}\]

Hence, in light of the Klebanov-Strassler/Randall-Sundrum throat-bulk isomorphism, this defines a background geometry given by:

    \[S = - {M^3}\int {{d^5}} x\sqrt G \left[ {R - \Lambda } \right]\]

    \[ + \]

    \[\int {{d^5}} x\sqrt G \left[ {\frac{1}{2}{G^{MN}}{\partial _M}\Phi {\partial _N}\Phi - V\left( \Phi \right)} \right]\]

    \[ - \]

    \[\int {{d^4}} x\sqrt { - {g_{hid}}} {\lambda _{hid}}\left( \Phi \right)\]

    \[ - \]

    \[\int {{d^4}} x\sqrt { - {g_{vis}}} {\lambda _{vis}}\left( \Phi \right)\]

with {{g_{hid}}} and {{g_{vis}}} the induced metric on the hidden and visible brane-sectors, {G_{MN}} the 5-D metric, with M the 5-D Planck scale, \Lambda the cosmological ‘constant’, \Phi the scalar field and {V\left( \Phi \right)} the corresponding potential.

Working in the A\left( \Phi \right)-warp-factor metric:

    \[d{s^2} = \exp \left[ { - 2A\left( \phi \right)} \right]{\eta _{\mu \nu }}d{x^\mu }d{x^\nu } - r_c^2d{\phi ^2}\]

the corresponding 5-D Einstein and scalar field equations are:

    \[\begin{array}{l}\frac{4}{{r_c^2}}{{A'}^2}\left( \phi \right) - \frac{1}{{r_c^2}}A''\left( \phi \right) = - \frac{{2{k^2}}}{3}V\left( \Phi \right)\\ - \frac{{{k^2}}}{3}\sum {{\lambda _i}} \left( \Phi \right)\delta \left( {\phi - {\phi _i}} \right)\end{array}\]

    \[\frac{1}{{r_c^2}}{{A'}^2}\left( \phi \right) = \frac{{{k^2}}}{{12r_c^2}}{{\Phi '}^2} - \frac{{{k^2}}}{6}V\left( \Phi \right)\]

and

    \[\frac{1}{{r_c^2}}\Phi ''\left( \phi \right) = \frac{4}{{r_c^2}}A'\Phi ' + \frac{{\partial V}}{{\partial \Phi }} + \sum {\frac{{\partial {\lambda _i}}}{{\partial \Phi }}} \delta \left( {\phi - {\phi _i}} \right)\]

with the index over the branes and our boundary-conditions of A\left( \phi \right) and \Phi \left( \phi \right) are given by:

    \[\frac{1}{{{r_c}}}{\left[ {A'\left( \phi \right)} \right]_i} = \frac{{{k^2}}}{3}{\lambda _i}\left( {{\Phi _i}} \right)\]

    \[\frac{1}{{{r_c}}}{\left[ {\Phi '\left( \phi \right)} \right]_i} = {\partial _\Phi }{\lambda _i}\left( {{\Phi _i}} \right)\]

To analytically solve in the backreacted Randall-Sundrum model-type, we use the quadratic/quartic bulk/brane dualized potential:

    \[V\left( \Phi \right) = \frac{1}{2}{\Phi ^2}\left( {{\upsilon ^2} + 4\upsilon k} \right) - \frac{{{k^2}}}{6}{\upsilon ^2}{\Phi ^4}\]

with:

    \[k = \sqrt { - {k^2}\frac{\Lambda }{6}} \]

Now we can derive solutions:

    \[A\left( \phi \right) = k{r_c}\left| \phi \right| + \frac{{{k^2}}}{{12}}\Phi _P^2{e^{ - 2\upsilon {r_c}\left| \phi \right|}}\]

    \[\Phi \left( \phi \right) = {\Phi _P}{e^{ - \upsilon \pi {r_c}}}\]

where {\Phi _P} is the scalar field on the Planck brane. Hence, {\lambda _{hid}} and {\lambda _{vis}} are given by:

    \[{\lambda _{hid}} = \frac{{6k}}{{{k^2}}} - \upsilon \Phi _P^2\]

and

    \[{\lambda _{vis}} - = \frac{{6k}}{{{k^2}}} - \upsilon \Phi _P^2{e^{ - 2\upsilon \pi {r_c}}}\]

We can now address the modulus stability of the braneworld. Substituting {\Phi _P} into:

    \[S = - {M^3}\int {{d^5}} x\sqrt G \left[ {R - \Lambda } \right]\]

    \[ + \]

    \[\int {{d^5}} x\sqrt G \left[ {\frac{1}{2}{G^{MN}}{\partial _M}\Phi {\partial _N}\Phi - V\left( \Phi \right)} \right]\]

    \[ - \]

    \[\int {{d^4}} x\sqrt { - {g_{hid}}} {\lambda _{hid}}\left( \Phi \right)\]

    \[ - \]

    \[\int {{d^4}} x\sqrt { - {g_{vis}}} {\lambda _{vis}}\left( \Phi \right)\]

gives us the 4-D potential for the radion:

    \[{V_{eff}}\left( {{r_c}} \right) = {r_c}\int {d\phi } \exp \left[ { - 4A\left( \phi \right)} \right]\left[ {{\upsilon ^2}\Phi _P^2{e^{ - 2\upsilon {r_c}\phi }}} \right.\]

    \[ + \]

    \[\left( {{\upsilon ^2} + 4\upsilon k} \right)\Phi _P^2\exp \left( { - 2\upsilon r{\phi _c}} \right) - \frac{{{k^2}{\upsilon ^2}}}{3}\left. {\Phi _P^4} \right]\]

    \[ \cdot \]

    \[{e^{ - 4\upsilon {r_c}\phi }} + {e^{\left[ { - 4A\left( 0 \right)} \right]\upsilon \Phi _P^2}}\]

    \[ - \]

    \[{e^{\left[ { - 4A\left( {\pi {r_c}} \right)} \right]\upsilon \Phi _P^2}}{e^{\left( {\pi - 2\upsilon \pi {r_c}} \right)}}\]

One then achieves inter-brane stabilization by minimizing the above potential with respect to the radion:

    \[\begin{array}{l}\frac{{\partial {V_{eff}}}}{{\partial \left( {\pi {r_c}} \right)}} = {e^{\left[ { - 4A\left( {\pi {r_c}} \right)} \right]}}{e^{\left( { - 2\upsilon \pi {r_c}} \right)}} \cdot \\\left[ {4{\upsilon ^2}\Phi _P^2 + 8\upsilon k\Phi _P^2 - {k^2}{\upsilon ^2}} \right.\left. {{e^{\left( { - 2\upsilon \pi {r_c}} \right)}}} \right]\end{array}\]

Hence, for the modulus field {r_c}, the stabilization condition is:

    \[k\pi {r_c} = \frac{{4{k^2}}}{{m_\Phi ^2}}{\rm{In}}\left\{ {\frac{{{k^{{\Phi _P}}}}}{{2\sqrt {1 + \frac{{2k}}{\upsilon }} }}} \right\}\]

Note now, in a backreacted RS model, 

    \[{V_{eff}}\]

has no minima that is consistent with inflationary coupling-running. Thus, a quartic term of the …