localization of gravity on higgs vortices with b. de carlos jesús m. moreno ift madrid hanoi,...
TRANSCRIPT
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