Lepton Electric Charge Swap at the 10 TeV Energy Scale

We investigate the nature of the dark matter by proposing a mechanism for the breaking of local rotational symmetry between ordinary third family leptons and proposed non-regular leptons at energy scales below 10 TeV. This symmetry breaking mechanism involves electric charge swap between ordinary families of leptons can and produces highly massive non-regular leptons of order O (1 TeV) mass unobservable at energy scales below 10 TeV (the scale of LEP I, II and neutrino oscillation experiments). Electric charge swap between ordinary families of leptons produces heavy neutral non-regular leptons with order O (1 TeV) masses, which may form cold dark matter. The existence of these proposed leptons can be tested once the Large Hadron Collider (LHC) becomes operative at 10 TeV energy-scales. This proposition may have far reaching applications in astrophysics and cosmology.

The main goal of the present paper is to investigate the possibility that, at the 10 TeV energy scale, the W ± boson decays to a neutral non regular lepton, of zero mass mode 1784 MeV, and a charge non regular lepton, of zero mass mode 35 MeV. In doing so, this paper formulates a new rotational symmetry between the ordinary lepton and proposed new leptons, with many potential applications in astrophysics and cosmology. The existence of these proposed leptons can be tested once the Large Hadron Collider (LHC) becomes operative at the 10 TeV energy-scales.

The set-up of the electric charge swap (ECS) symmetry
Το formulate the electric charge of swapped particles, we have to look for symmetry that characterizes swap processes in the framework of 2-extra dimensions with compactification scale 10 TeV.
Let us consider the 6-dimensional spacetime with signature ( , , , , , )       . Einstein's equations in this spacetime have the form 4
To split the 6-dimensional space-time into 4-dimensional and 2-dimensional parts we use the metric ansatz [38]  to the brane at point 0   . When  changes from 0 to π, therefore, the geodesic distance into the extra dimensions shifts from the north to the south pole of the 2-spheroid. For 1 b  in equation (2), the extra 2-surface is exactly a 2-sphere with radius  (0.1TeV -1 ).
The six dimensional massless Dirac equation in the 2-spheroid becomes a system of first order partial differential equations [38], [39], [40]. (2) the internal 2-surface is a sphere 2 S . For this case there exists only one zero mode with angular quantum number ( 0 l  ). This zero mode corresponds to a massless lepton of the third family, of either ordinary or swapped electric charge. This suggests that only the third lepton family propagates in the six dimensional bulk, while the second and first families of leptons are confined to the 3-brane. The normalizable solution of 6-dimensional Dirac equation (3) is given by where 0 A and 0 B are integration constants with the dimensions of mass, and 0 () x   is the one lepton zero mode . The expression for the determinant of our ansatz (2), which will be used often in what follows, is given by where (4) g  is the determinant of 4-dimensional space-time.
This can be understood from the point of view that this state has a non-zero effective wave function on the brane ( 0)   , and this has an overlap with the scalar field. (By effective wave function we mean the combination from (4) and the square root of the determinant from (5).
In this way the singular sin term cancels out).
This fundamental representation is given by [ , ] j k jkl l Τhe generators are denoted as are the isospin versions of Pauli matrices.
The latter act to the new leptons states represented by 0 10 , 01 To link the two distinct sectors, the ordinary and ECS leptons, we assume that neither ordinary L nor ECS swap L lepton numbers are conserved, while the overall lepton number must be conserved.
The next step is to define the group transformation responsible for the swap of electric charges between the tau and neutrino of tau particles. The ECS transformation must given by a transformation from  Here, since the two extra dimensions are endowed with the Fubini-Study [1] metric [46], [47], then not all Möbius transformations (e.g. dilations and translations) are isometries. Therefore the automorphism from the 2 (2) / (1) S SU U  to itself, which brings the electric charge swap between the tau and neutrino of tau particles, is given by the isometries that form a proper subgroup of the group of projective linear transformations 2 ( arg ) ()  [46], [47], which is the isometric group of the unit sphere in three-dimensional real space 3 R . The automophism of the Riemann sphere [2] C is given by: The universal cover of () . This group it is also differomorphic to the unit 3-sphere S 3 .
We regard ordinary and ECS leptons as different electric charge states of the same particleanalogous, that is, to the proton-neutron isotopic pair.
Finally, in terms of rotational symmetry between the original lepton and the proposed ECS leptons, the 2-extra dimensional sphere 2 S is given by [1] The round metric of the 2-extra dimensional sphere can be expressed in stereographic coordinates as 22 12 22 (1 ) 22 12 yy   . The metric g is Fubini-Study metric of the 2-sphere and [46], [47]. [2] Recall that stereographic projection is a conformal bijection from the round sphere 2 S to the Riemann sphereĈ .
An automorphism of Ĉ that corresponds under stereographic projection to a rotation of 2 S is called a rotation of Ĉ . The group of rotation of Ĉ is denoted ( ) Rot C . Thus under stereographic projection, Hence the normalizable combine solution of 6-dimensional Dirac equation (3) becomes A 0 and B 0 are integrating constants (for details see [38]), 0 () x   is the one zero mode which combining the third family massless ordinary lepton doublet 0 () lx  and the massless electric charge swap (ECS) lepton doublet 0 () lx   respectively.
The Higgs fields propagate in the bulk, the Vacuum Expectation Value (VEV) of the Higgs zero-mode, the lowest lying KK state, generates spontaneous symmetry breaking and gives mass to charge ECS lepton (for details see [38]) .
Extra dimensions allow one to reduce the fundamental scale of the theory down to where the corresponds warp factor. Βrane is localized in different points of extra dimension with different values of warp factor are given in (Table.1) Βrane is localized in the specific point of extra dimension, the mass of neutral ECS and the corresponds warp factor are given in (Table.2) The quantum numbers of the new non regular leptons, of zero-mode mass 1784 MeV and 35MeV respectively, are given in (Table 3).  L  for ordinary antileptons, respectively) and their electric charges (positive or neutral for ordinary leptons and negative or neutral for ordinary antileptons, respectively).

The W + decays to ECS leptons
In this paper, we investigate the particular category of (W, Z) decay processes that have additional ECS leptons as daughter particles Let us call this process electric charge swap. We are beginning with a familiar decay process: where W  is the positive charged boson, and ( ,    ) are the positive charged tau and neutral tau neutrino. In this particular decay, mass and energy, momentum and angular momentum, electric charge and the overall leptonic number overall L (equation (10)) are always conserved.
Neither ordinary leptonic number L , nor ECS leptonic number s L are conserved, in the presence of ECS symmetry. The ordinary leptonic number pair is given by By substituting (10), (11) and (12) for the ordinary pair of leptonic numbers (20) we obtain the ECS pair of leptonic numbers: In the presence of W + decays, the electric charge swap rotation in the 3-dimensional vector space 3 V is given by: Since electric charge is conserved, there is a particle with non-swapped electric charge This yields the electric charge swap rotation with respect to the z-axis The electric charge of ECS particles can be derived as a linear combination of the electric charge of the ordinary particles: The conservation of electric charge then becomes corresponding to a swap of electric charge between the tau and the tau neutrino during decay Equations (21), (26) yields to the variation of the ordinary decay process (equation (19)) to the following decay : In which the products are a hypothetical particle -a zero-charged version of the tau, 0  (swap-antilepton) -and a positive charged version of the tau neutrino,    (swap-lepton) .

The process of electric charge swap at the 10TeV energy scale
In addition to zero mass modes, there are massive KK modes, with mass at the order1/ 10TeV   . With mass of these values, these massive KK modes lie above the (W, Z) zero mass modes. In this model, the internal space ( ) must be small enough to permite. Therefore, assuming that the standard Model (SM) remains valid up to a cut-off of order the LHC centra -of-mass energy 10TeV. However, the SM with a cut-off of order the LHC energy would be 10% fine tuned, and so we should expect to see new physics at the LHC. The search for new physics involves measuring the deviation from the SM. Here, this deviation is small and a precise measurement may be needed. For this reason will assume a ECS physics program when LHC running at 14 TeV center -of-mass energy, and integrated luminosity of 10fb -1 per year.
The SM prediction: The   term arises from the (t,b) contribution to the W and Z self-energies and can be written to one loop:

The ECS invariant
This swap of electric charge does not violate any conservation law. In addition, the electric charge swap does not affect the transition rates of neither ordinary (19), nor electric charge swapping processes (28). The rate dΓ for the electric charge swapping processes (28) where Ε W is the energy released to the swap lepton pair and θ is the opening angle between the two leptons and the swap-antilepton velocity u s in our approximation.
Substituting (33) The rate dΓ for the ordinary processes (19) is given by where Ε W is the energy released to the lepton pair and θ is the opening angle between the two leptons and the antitau velocity u in our approximation. Substituting (36) to (35) By comparing equations (33), (34), (36) and (37) we conclude that there is an electric charge The electric charge swapping invariant with at the 10TeV energy scale is given by In the presence of electric charge swap symmetry, the rate of Z decays to ECS leptons is given by  Table. 3.

Table. 4 Electric charge, axial and vector couplings of ECS leptons ECS leptons
Using equation (39) and the values given in Table.4, we calculate the Z decay to ECS lepton partial widths: Taken together, equations (38) and (40) predict that ordinary third family leptons and ECS leptons couple equally with (W, Z) bosons at collision energy at the scale of M C ≈ 10TeV. This prediction can be tested when LHC becomes operative at such energy scales.
The invariant energy-averaged annihilation cross-section at 10 TeV scale of energy is given by (flavor or mass basis) independent of any flavor mixing since the annihilation mechanism is a neutral current process.

The ECS leptons in the LHC detectors
These propositions can only be tested at a higher energy linear collider with high integrated luminosity >> >50 fb −1 , such as the LHC when it becomes operative at 10TeV energy scale.
Since the proposed non regular ( 0  ) lepton is neutral, it will not be detectable by the LHC detectors. This lepton could still be detected indirectly: if leptons escape the accelerator, they will manifest as missing energy. In a toroidal LHC apparatus (ATLAS) inner detector experiment [61] the charged non regular lepton (    ) with zero mass mode of 35 MeV carries the electric charge. Its orbit must, therefore, be different from that of ordinary tau neutrinos (   ). The deviation of this orbit will have a significant value ( [48], [49]). This signature track is unexplainable by any of the observed particles, as it streaks across the LHC's detectors.

Current collider signatures of ECS leptons
The above equation yields to changes of Fermi constant: The ordinary leptons coupling to (W, Z) gauge bosons by Being with the relation: where (experimentally ρ≈1), we obtain a change of 2 g . 2 cos 8 22 Substituted equations (43) and (44) Calculations of the contribution of ECS leptons at collision energy scale (0.1TeV≤M s ≤10TeV) are given in Table.5.  By the values given in  (50) where 167MeV   is the Z invisible decay into one neutrino species [51]. We find that the contribution of the non regular neutral lepton Z  ≈3.1eV, at LEP.2 collision energy scale is required to be ( 0  )that is a dark matter particle.
Since ECS lepton number s L is conserved, ECS leptons can be created or annihilated in pairs through (Z, γ   Table. 6. By the values given in Table.6, we conclude that the contribution of ECS leptons (equations (52), (54)), at the current LHC collision energy scale (M s ≈7TeV) [52] and LEP.2 measurements of cross-section for electron-positron annihilation [50], [51] is too small to be detected. The E821 experiment at the Brookhaven National Laboratory (BNL) studied the precession of muon and anti-muon in a constant external magnetic field as they circulated in a confining storage ring [50], [51]. For two extra dimensions of radius (1/10TeV), the additional contribution of the proposed non regular charge lepton with zero mass mode of 35MeV to anomalous magnetic moments: which is too small to be detected by this experiment ( [51], [53], [54], [55], [56], [57]). For this reason, neither the current LHC [52] collision scale of energy nor the LEP2 measurements of the cross-section for electron-positron annihilation [50], [51] and of anomalous magnetic moments of the electron and muon [50], [51] can prove the existence of the proposed ECS lepton with a zero mass mode of 35 MeV. The energy scale we propose here is large compared to the electroweak energy scale. These propositions, and the predicted level of the standard model (SM) loop corrections, can only be tested at a higher energy linear collider with high integrated luminosity >>> 50 fb −1 , such as the LHC.

Discussion
The proposition of electric charge swap predicts the occurrence of a non regular neutral The interactions of the predicted non regular lepton ( 0  ) freeze out at a temperature such that cosmic radiation for the energy range 10 GeV to 100 GeV [62], [63] and no excess in / pp  from the theoretical calculations. And also very recently, (Fermi-LAT) [64] and (HESS) [65] data showed clear excess of () ee   spectra in the multi-hundred GeV range above the convential model [66], although they do not confirm the previous (ATIC) [67] peak. The phenomenology of ECS dark matter at PAMEL/FERMI is under investigation. [3] At the freeze out temperature of ( The amount of 7 Li predicted by the SBBN, for instance, is about 2-3 times larger than the observational value from stellar atmospheres of the low metalisity halo stars ( [68], [69], [70]). This statistically significant discrepancy is known as the 'Lithium problem'.
The proposed charge ECS lepton is a heavy electron. In analogy with the muon, the charge lepton could catalyze high temperature fusion of lithium and beryllium nuclei. The proposed charge lepton could thus provide an explanation for the missing lithium in the SBBN model [68].

Conclusions
The (W ± , Z) decay to neutral non regular lepton (zero mass mode 1784 MeV) and charge non regular lepton (zero mass mode 35 MeV) proposed here is strictly a phenomenon of the 10TeV energy scale. Its proposition is formulated by reference to a 2-extra dimensional sphere with a global isometric group, the electric charges-swapping group, (3) ECS SO .
Instead of introducing ad hoc new particles, our proposition introduces new particles from ordinary ones, using an alternative interpretation of the distribution of lepton electrical charge.
We suggest that (W, Z) decay beyond SM comes from the proposed new leptons. The existence of these non regular leptons is testable once the LHC becomes operative. The neutral non regular lepton (zero mass mode 1784 MeV) is a possible cold dark matter candidate.
Furthermore, we find that the contribution of the proposes non regular leptons on scale of energies below the compactification scale is suppressed. Therefore the new non regular leptons are not subject of bounds comes from both current colliders: Large Electron Positron (LEP) 2, which reached a central of mass energy of 209 GeV [50], [51] and current LHC, which reached a central of mass energy of 7 TeV [52] and from the E821 experiment at Brookhaven National Laboratory reported a measurement of the muon's magnetic moment [50], [51].