At The Heart Of The Problem

At the heart of the problem here is the Planck scale or , where GN is the TEV, where . is the Newton’s coupling constant. The essence of this problem can be captured in a single question: why is gravity some 10^38 orders of magnitude weaker than other forces of Nature? Behind this major problem is the question of the mass of these KK series Neutrinos. Their mass, another unknown, sets limits on the size of the extra dimensions. The whole problem is similar to balancing a billiard cue on its point. It is mechanically possible, but, but the amount of fine tuning required for such a delicate balance is too demanding to be of a practical use. And I might add it’s not natural. Again back to the heart of the problem Similarly, making the SM work for all energies up to the Planck scale would require tremendous amount of fine tuning of its parameters to a precision of without such fine tuning and without some solution to the hierarchy problem. But it is this problem that places so many variables towards solving the actual Neutrino mass problem. Models with extra dimensions appearing at or near the TeV scale have become very popular in recent years. In these models, Kaluza-Klein (KK) excitations appear as particles with masses of the scale of the extra dimension. Due to conservation of momentum in the higher dimensions, a symmetry called KK parity can arise which can, in some cases, make the lightest KK particle (LKP) stable However the size of the extra dimension is a variable itself. To provide an observable flux of neutrinos, dark matter particles must be gathered in high concentrations. Deep gravitational wells such as the Sun, Earth or galactic centre are examples of regions where such concentrations may be present. In the following calculation, we will focus on the Sun, as its prospects are the most promising. The rate at which WIMPs are captured in the Sun depends on the nature of the interaction the WIMP undergoes with nucleons in the Sun. For spin-dependent interactions, the capture rate is given by

Where ρlocal is the local dark matter density, σH,SD the spin-dependent WIMP-on-proton (hydrogen) elastic scattering cross-section, the local rms velocity of halo dark matter particles and mDM is our dark matter candidate. The analogous formula for the capture rate from spin-independent (scalar) scattering is

Here, σH,SI is the spin-independent WIMP-on-proton elastic scattering cross-section and σHe,SI the spin-independent WIMP-on-helium elastic scattering cross-section. Typically, σHe,SI 16.0σH,SI. Factors of 2.6 and 0.175 include information on the solar abundances of elements, dynamical factors and form factor suppression. Although these two rates appear to be comparable in magnitude, the spin-dependent and spin-independent cross-sections can differ radically. For example, with KK dark matter, the spin-dependent cross-section is typically 3-4 orders of magnitude larger than the spinindependent cross-section, and solar accretion by spin-dependent scattering dominates. If the capture rates and annihilation cross-sections are sufficiently high, the Sun will reach equilibrium between these processes. For N (number of) WIMPs in the Sun, the rate of change of this number is given by

where C is the capture rate and A the annihilation cross-section times the relative WIMP velocity per volume. C is as defined in (1), whereas A is

with Veff being the effective volume of the core of the Sun determined roughly by matching the core temperature with the gravitational potential energy of a single WIMP at the core radius. This yields

The present WIMP annihilation rate is

where t 4.5 billion years is the age of the solar system. The annihilation rate is maximized when it reaches equilibrium with the capture rate. This occurs when

For the majority of particle physics models which are most often considered (e.g., most supersymmetry or KK models), the WIMP capture and annihilation rates reach or nearly reach equilibrium in the Sun. The rate of neutrinos produced in WIMP annihilations is highly model-dependent as the annihilation fractions to various products can vary largely from model to model. Neutrinos which are produced lose energy as they travel through the Sun It is my prediction that of those that survive and stay on brane that they lose energy to brane lensing. But the range of this brane lensing is an unknown variable also. The probability of a neutrino escaping the Sun without interaction is given by

where Ek is 130 GeV for νμ, 160 GeV for ντ, 200 GeV for and 230 GeV for . Thus we see that neutrinos above a couple of hundred GeV are especially depleted, although those which escape are also more easily detected. But the two issues together rather rule out detecting them on earth no matter the size of the extra dimension. It is this issue which has suggested to me we need either a closer study of the Sun or to seek external observational evidence for brane lensing effects. Now, the 8 meters per second difference in C in system to external system is telling us something about those messing neutrino masses. Get that value out of all the variables and we could solve a whole lot of problems at the same time. We can from the above assume they are less than a couple of hundred GeV because of the known depletion rate there. I’d say the upper limit is 200 GeV. The size of large extra dimensions (R) is fixed by their number (n) and the fundamental Planck scale in the (4+n)-dimensional space-time (MD). By applying Gauss’s law, one finds the gravitational potential would fall of as 1/rn+1 for Currently, the best limit on the size of extra dimensions in the ADD model from gravity measurements comes from the Eot-Washington group and constrains the largest ED size to at the 95% confidence level which corresponds to in the case of two ED. The observation of a handful of neutrinos from the SN1987A explosion by the IMB and Kamiokande detectors allowed them to put a constraint of for N=2 models and for N=3 models. As you see the energy level goes down the more extra dimensions one adds. This going down could be seen as a good thing as far as

our energy requirements go. But somewhere in all this variables is the real solution that our 8 meters per second difference is telling us about. Actually, in our real case we do not want this energy level to drop. As it drops the amount of neutrinos needed increases. Given the Sun does its 8 meters per second effect increase the amount of neutrinos needed and you multiply our problem manifold. One good thing, N=2 case seems to be largely disfavored by the host of the astrophysical and cosmological constraints. So lets assume for a sake of starting point here that N=1 and we will ignore the other possible values of N. One good argument for this is brane lensing takes on a different perspective with multiple brane cases. In some ways it becomes more of an optical effect with no real ruler change at N=2 cases. The best limits come from LEP electroweak precision measurements; the combined limit on the compactification scale of the

dimensions

approaches

This sets some strong limits on our search area for the messing KK series neutrino masses. In fact, it drops us way below the Ek is 130 GeV for νμ, 160 GeV for ντ, 200 GeV range we started with above. Again our problem has been compounded. But it can be worked with in this range. Say we took 6.8 TeV as a base line. We could plug this mass value back into the original Israel condition and figure out how many neutrinos the Sun produces at that value for our given 8 meters per second ruler change. Then we’d have to look for a way to generate at least twice that amount of neutrinos to get any lab based experimental result we could measure. Parameters of the two DØ highest diEM-mass candidates. All energy variables are in GeV.

I have added this in to demonstrate that certain events detected in lab experiments do favor the ED models to date. So the evidence is mounting out there for this whole general idea. This was based upon Searches for Large Extra Dimensions via Virtual Graviton Effects. I could add to this Dimuon (left) and diphoton (right) mass spectrum from the CDF search for Randall-Sundrum gravitons. Points are data; solid line is the total expected background.

Individual 95% CL limits on the RS model parameters M1 and k/MPl from the CDF search in three channels. The shaded area to the left of the corresponding curve is excluded. This gets at least a scale range we need to start with. As you can see one is confronted with a lot of variables here when it comes to my giving any solid suggestion on what we are up against. The problems can be solved. But they require further research. On the other side, as far as problems go, is the drop of range of brane lensing in general. As I have mentioned before we have no solid way to measure differences in C below a certain error range. But, if the difference where we can measure is 8 meters per second beyond the orbit of Pluto then we can assume the maximum range is around that range. We can also assume it drops of slowly somewhere in between. So we have some version of the general 1/R^2 rule going on here. The reason I point this out is it will come into play for not only testing eventually, it also will enter in with craft and field generator design. Actually, I rather hope it has a long range effect. We can scan ahead within the range of our field no matter the multiple of C simply because EM signals will run at the fields altered C value while the craft runs at a slower speed. This applies only within the brane lensed region. External to that region we cannot expect to have scan ability. So the longer the lensed region the better warning we have as far as objects in our path go. Any way it goes we still face some limits upon navigation here and it still may require something like a pulsed run mode to navigate and avoid objects. It also might be suggested something akin to our old shield idea could be required even though brane lensing might offer a solution there.

Now a third area that needs investigation concerns how a field like this in motion will behave. Our Sun produces a similar field. It is not only in motion. It has other bodies within its field in motion and it moves within the whole galaxy field, which itself has its own combined brane lensing effect. The question here concerns drag effects and possible turbulence.