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Localization of gravity on Higgs vortices
with B. de Carlos
Jesús M. Moreno
IFT Madrid
Hanoi, August 7th
hep-th/0405144
• Topological defects & extra dimensions
• The Higgs global string in D=6
• Numerical solutions
Weak and strong gravity limits
• A BPS system
• Conclusions
Planning
d=5 domain wall
d=6 vortex
d=7 monopole, d=8 instanton
the internal space of a topological defect living in a higher dimensional space-time
Rubakov & Shaposhnikov ´83
Akama ´83
Visser ´85
Our D=4 world:
Topological defects & extra dimensions
Solitons in string theory (D-branes): ideal candidates for localizing gauge and matter fields
Polchinski ´95
REVIVAL:
• Gravity localized in a 3-brane DW in D=5
• Graviton´s 0-mode reproduces Newton’s gravity on the brane
• Corrections from the bulk under control
• Need bulk < 0 to balance positive tension on the brane
Randall and Sundrum ´99
Topological defects & extra dimensions
Gravitational field in D=4
domain walls: regular, non static gravitational field(or non-static DW in a static Minkowski space-time)
Vilenkin ´83
Ipser & Sikivie ‘84
strings: singular metric outside the core of the defectCohen & Kaplan’88
Gregory ‘96
monopoles: static, well defined metric
V 0
Static DW, regular strings … (e.g. SUGRA models)
Barriola & Vilenkin’89
Cvetic et al. 93 ….
(non singular when we add time-dependence)
V 0
Topological defects
Compact transverse space (trapped magnetic flux, N vortices)
Sundrum ’99,
Chodos and Poppitz ’00
Local string/vortex
Non-compact transverse space: local string (Abelian Higgs model)
Gherghetta & Shaposhnikov ´00
Gherghetta , Meyer & Shaposhnikov ´01
Cohen & Kaplan ‘99
previous work: Wetterich’85
Gibbons & Wiltshire ‘87
Global string
Plain generalization to D=6 still singular
However, introducing 0 cures the singularity. Analytic arguments show that, in this case, there should be a non-singular solution Gregory ’00
Gregory & Santos ‘02
The string in D= 6
Matter lagrangian:
Global U(1) symmetry
Let us analyze this system in D=6 space-time
The global string in D= 6
The action for the D=6 system is given by
Metric: preserving covariance in D=4 compatible with the symmetries
coordinates of the transverse space
M(r), L(r) warp factors
and we parametrize
The global string in D= 6
The global string in D= 6
QUESTION:
Is it possible to match BOTH regions having a
regular solution that confines gravity?
ANSWER:
YES! but for every value of v there is a unique
value of that provides such solution
Numerical method
Initial guess ( 5 x N variables)
RELAXATION
ODE finite-difference equations (mesh of points)
Iteration Improvement
The global string in D= 6
Boundary conditions
F(0) = 0
L(0) = 0
m(0) = 0
F’(0) = 0
L’(0) = 0
In general, there will be an angle deficit
L’(0) = 1 c
The global string in D= 6
Numerical solutions
Scalar-field profile
M6 V
V
V (no dependence)
V6
Coincides with thecalculated value
Numerical solutions
Cigar-like space-time metric
Asymptotically AdS5 x S1
Olasagasti & Vilenkin´00
De Carlos & J.M. ‘03
Numerical solutions
Uniqueness of the solution: phase space
Gregory ’00
Gregory & Santos ‘02
In the asymptotic region (far from the Higgs core)
autonomous
dynamical system
Numerical solutions
Flowing towards difficult because is next to a repellor (AdS6)
Only one trajectory, corresponding
to c , ends up in which can
be matched to a regular solution near the core
4 fixed points
Numerical solutions
Plot + fit for small v values
We find a good fit
Gregory´s estimate (v M6)
Numerical difficulties to explore the small v region
Numerical solutions
Is it possible to generate a large hierarchy between M6 and the D=4 Planck mass ?
From the numerical solutions : the hierarchy
is a few orders of magnitue (e.g. 1000 for v = 0.7)
(increases for smaller v values)Gregory ’00
Problem: fine tuning stability under radiative corrections
A BPS system
Solving second order diff. eq. can be very hard and does not give analytical insight
Is it possible to define a subsystem of first order (BPS-like) differential eqs. within the second order one?
Carroll, Hellerman &Trodden ‘99
Conclusions
We have analyzed the Higgs global string in a D=6
space time with a negative bulk c
trapping gravity solutions
For every value of v there is a unique value of c that
that provides a regular solution.
The critical cosmological constant is bounded by
-V(0) < c < 0
It is difficult to get a hierarchy between M6 and MPlanck
Fine tuning, stability