INCLINED DIMENSIONS
Daniel Hernández
The Abdus Salam International Center for Theoretical Physics
D.H, M. Sher; 1101.5695
Phys. Let. B.
FUN WITH PHYSICS
LORENTZ VIOLATION WARPED EXTRA DIMENSIONS
SLANTED BRANE MODEL
(and some nice graduate-level physics)
WHAT LORENTZ VIOLATION?
Inhomogeneous, anisotropic, space-time
Particle physics doesn’t respect Lorentz symmetry
LORENTZ AS A GAUGE SYMMETRY IS VIOLATED
For instance
NOTHING IS VIOLATED REALLY
MOST IMPORTANTLY, WHY ISN’T IS HOPELESS??
Spontaneous LV is a generic prediction of Quantum Gravity:
Some stringy models, Loop QG. Also connected with CPT Non-invariance.
Great improvement in experimental techniques over the last 10-15 years making it possible to test LV effects even if suppressed by the Planck scale!
WHY (true) LORENTZ VIOLATION?
THE SM EXTENSION
Idea: Parametrize Lorentz Violation through effective operators
EFECTIVE FIELD THEORY
………………
Changed vev of the Higgs
Colladay, Kostelecky
BOUNDS ON LORENTZ VIOLATION
S. Liberati, L. Maccione; 0906.0681
Lots of papers, lots of ways, sometimes very ingenious to put bounds
• Photon time of flight
• Vacuum Birrefringence
• Threshold reactions (Very spectacular effects like photon decay)
• …….
I won’t speak about True Lorentz Violation…
However, I will argue that the SM Extension is also useful to describe the field theory of non maximally symmetric spaces
TOTALLY UNRELATED A PRIORI: RANDALL-SUNDRUM
A VERY STUPID LAGRANGIAN
Free field in a perfectly Minkowski BG
Explicitly:
So, what is the PHYSICAL mass of this scalar field??
Wave function normalization
A VERY STUPID LAGRANGIAN (cont.)
What about a lambda(phi)^4 term???
NO CHANGE
Think now of the Higgs and the possibilities?? Mmmmmm
A VERY STUPID LAGRANGIAN (cont.)
Higgs mass depends on the scale factor???
FRW metric
WHY IT IS STUPIDAlways consider the FULL Lagrangian
These two need be the same!
So, nothing wrong with changing variables, but beware of interpretation.Specially when metric perturbations are important.
STUPID IS INTERESTING (rarely)
Nonetheless, if we could have such a Higgs, it would be very cool, right?
A time evolving Higgs!
Or thinking general. A vev for the Higgs that depends on the metric!
WHAT ONE WOULD NEED
Write the full Lagrangian…
Absorb the conformal factor in the fields of the matter Lagrangian
Hope that the remaining Lagrangian «accepts» the new metric
RANDALL-SUNDRUM’S IDEA
WORKS LIKE MAGIC AS WESHALL SEE
Lisa Randall performing her spell
RANDALL-SUNDRUM’S IDEA(explicitly)
Start in 5D Anti de Sitter
We know this fulfills the 5D Einstein equations With negative cosmological constant
RANDALL-SUNDRUM’S IDEA(explicitly)
Put 4D branes, restrict the domain
Higgs on the brane Induced by
RANDALL-SUNDRUM’S IDEA(explicitly)
Normalize Higgs field
Watch the magic
The metrics here are not simply Anti de Sitter and Minkowski but perturbed Anti de Sitter and Minkowski
KEY POINT: HIERARCHY ACHIEVED BETWEEN PLANCK MASS AND EW SCALE
PUTTING MATTER
T. Gherghetta, A. Pomarol; hep-ph/0003129
The procedure is simple and well known
Kaluza Klein decomposition
PUTTING FERMIONS
Substitute in the 5D Lagrangian, integrate for the extradimensional coordinate
T. Gherghetta, A. Pomarol; hep-ph/0003129
The Flavour puzzle encoded in the Yukawas is transferred to the c_i
‘So, what if we tilt the branes?’
‘That would immediately give Lorentz violation’
THE TILTED METRIC
Hubble parameter
Non intersecting branes
THE TILTED METRIC
Is it a solution to the Einstein eqs??
Same as in parallel brane scenario at this order
And flat visible brane!
Induces a vev for the Z boson and modifiesthe Higgs vevOrder a^2. Not measurable.
HIGGS NORMALIZATION
PLANCK MASS
If one wants to absorb the conformal factor in the metric one needs to follow the procedure outlined for the RS metric!!
Won’t explore this further!
VARIABLE HIGGS MASS
The main effect is that it alters fermion masses. For fermionson the branes, the Yukawa couplings are not altered by the tiltand the masses are simply scaled by the warp factor
Search for cosmic variance of the masses of elementary particles
TRIPLE ALPHA PROCESS
Lifetime of order 10^-16 secs
Resonance of 12C state enhances reaction rate!
Energy of the resonance dependent on the nucleon mass
CMB BOUNDS
Temperature of last scattering Binding energy of hydrogen
BEST BOUND OF ALL: AMMONIA
Ammonia clouds in the path of light coming from distant quasars
Inversion transition very sensitive to the proton to electron mass ratio
Comparing with the rotational spectra
FERMIONS IN THE BULK
The above is not valid in a direct manner, essentially because one needs to solve again for the wavefunction. Remember the Kaluza-Klein procedure.
However, very crude estimates point that the previous bounds are valid (maybe even ) improved for fermions in the bulk
GENERAL SOLUTION
Consider the Einstein equation
The ansatz
Gives a diagonal Einstein tensor (to no surprise)
GENERAL SOLUTION
Second order in a
Two more equations appearing from the Einstein eqs
GENERAL SOLUTION
General solution for k
This corresponds to Randall Sundrum
This is the full analytical metric for which the tilted metric is the first order approximation
For example:
CONCLUSIONS• An scenario with tilted branes is plausible, at least at the classical
level.
• Two types of modifications in the Lagrangian. One can be interpreted in the context the Lorentz violating Standard Model extension. The other is a typical sidereal classical field which yields masses for the SM fields that vary with space
• Moreover, generalizing this, a new background solution for the Einstein equations has been found that reduces both to the RS solution and to the tilted metric in the proper limits.
• Bounds on the «tilt parameter» are very stringent as expected from any Lorentz violating parameter. However, since it comes from the metric this parameter is stable.