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

Equations

Gravity trapping

The global string in D= 6

Equations

rr

eom

(constraint)

The global string in D= 6

The global string in D= 6

F(0) = 0

L(0) = 0

L’(0) = 1 (no deficit angle)

( F(r) = f1 r)

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

Dependence on the Higgs scale

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

-V(0) < c < 0

Super heavy limit: (v M6)

Numerical solutions

Region explored by the Higgs field in the super heavy limit

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

A BPS system

BPS equations

A BPS system

EXAMPLE

No cosmological constant

De Carlos & J.M. ’03

A BPS system

Etotal = Egrav + Ekin + Epot 0

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