Gravitational Effects of Weak Interactions at TeV Energies
11 Gravitational Effects of Weak Interactions at TeV Energies
C Sivaram and Kenath ArunIndian Institute of Astrophysics, Bangalore
Abstract:
Recently there has been a lot of interest in the search for extra dimensions. If gravity propagates in extra dimensions then gravity would become as strong as other interactions. This could also lead to the production of mini black holes. Here we have discussed how even without considering extra dimensions gravitational effects of weak interactions could show up at TeV energies.There is a lot of interest in the search for extra dimensions at TeV energies, including at the LHC accelerator. If gravity propagates in extra dimensions then n gravity would become as strong as other interactions (such as electroweak) and unification would occur at cml u , corresponding to TeV energies. This could also lead to the production of mini black holes with a radius u l , when we have a strong (unified) gravitational field (or coupling). Such black holes should decay on time scales scl u by Hawking radiation. However, even without considering extra dimensions it appears that gravitational effects of weak interactions, could show up at TeV energies. For a region of radius r, the associated (short range) weak interaction energy is ( F G is the Fermi constant): rGE FW … (1)So the self gravitational energy is: c GrGrc GrGE FFGW … (2)Where G is the Newtonian gravitational constant. Note the r dependence!For this to be a region of strong interaction, we have: rcEE GW … (3)( cg is the strong coupling, ~ ce is the electromagnetic coupling, and so on)This would imply that the gravitational effects of the weak interactions would become strong when (from equations (2) and (3)): cmcGGr F … (4)This corresponds to an energy scale of around 100 TeV!The above scaling suggests (that is r or E ) that at energies ~10 TeV (like in the LHC), with the E dependence of the gravitational-weak effects, the corresponding energies would be ~several MeV. This is testable. (That is the gravitational effects of the highly localised weak interactions at several TeV would amount to energies ~several MeV, causing measurable differences in particle energies, etc.)Again two wave packets of extent given by equation (4) and separated by similar distancewould gravitationally interact with energy: ~ rrrGG F again with similar strength. The interaction energy would also be ~several TeV. However for a mini black hole to form (with the usual four dimensions) the energy required to be squeezed into the region of scale given by equation (4) is several orders more. The energy required for a region of extent r , to form a black hole is: GrcE bh … (6)With equations (2) and (6), we get the required r as:
818 22 cGGr F … (7)This gives cmr for an energy of the particles of TeV ! This could have consequence in astrophysics, especially for high energy cosmic rays which have energies of eV . Independently of extra dimensions, the extended uncertainty principle, if it has a weak fundamental length of w L , could have testable effects at such TeV energies. In particular, equations (2)-(5) would get modified at lengths w L . The generalised uncertainty principle (GUP) has the form: pLpx w … (8)So for w Lp , we just have the second term! This implies that x now increases with p . So equation (2) would be modified as:
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EL cGGEcLc GGpLc GGrc GGE wFwFwFFGW … (9)And x increases with p as: pLx w … (10)If w L corresponds to TeVcm , equation (9) would suggest a sharp drop of the self gravitational energy with interaction energy at energies TeV , in contrast to the earlier increase with energy. This could reveal the existence of a fundamental weak length scale and a corresponding modified uncertainty principle at these energies as a consequence. These are all testable effects. The modified phase space as implied by equations (8) and (9) would again have effects on particle decays. That is, we would have: pLpxdd w , which could have drastic effects on decay times of unstable particles at these energies. The decay rate of the particles is given by: dWdNHd fi … (11)Where the density of the final state is given by: dpVpdWdN … (12) W is the total energy of the final state given by: cpmpW The density of the final state is:
36 2222 c dppVmpWdWdN eee … (13)The total rate is then given by:
273 722 mc pMgd … (14)With the modified phase space, the density of final state will be modified from that given by equation (12) to: