{'fraction_non_alphanumeric': 0.03137534539620287, 'fraction_numerical': 0.008556912380782601, 'mean_word_length': 4.446601941747573, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Reconsidering extra time-like dimensions

In this study we reconsider the phenomenological problems related to tachyonic modes in the context of extra time-like dimensions. First we reconsider a lower bound on the size of extra time-like dimensions. Next we discuss the issues of spontaneous decay of stable fermions through tachyonic decays and disappearance of fermions due to tachyonic contributions to their self-energies. We find that the tachyonic modes due to extra time-like dimensions are less problematic than the tachyonic modes in the usual 4-dimensional setting because the most troublesome Feynman diagrams are forbidden once the conservation of momentum in the extra time-like dimensions is imposed.

Extra spatial coordinates are considered thoroughly in recent years. A glance at ArXiv shows that there are hundreds of papers on extra dimensions in the last five years and almost all of them being wholly or mainly on spatial extra dimensions. From the theoretical point of view the scarcity of studies involving extra time-like dimensions [1,2,3,4,5,6,7] is mainly due to the existence of tachyonic modes in such models, which are problematic because of the violation of causality and unitarity and lack of an adequate field theoretic description of tachyonic fields [2] while from the phenomenological point of view the most serious problems are the extremely small empirical lower bound in literature on the size(s) of extra time-like dimensions [8], the spontaneous decay of stable particles induced by negative energy tachyons [2,9], the imaginary self energy for charged fermions induced by tachyonic photon modes, which in turn, seems to cause disappearance of the fermion into nothing in a very short time [2]. In this study we will focus on the phenomenological difficulties and try to seek if one may moderate the phenomenological problems mentioned above with the hope that a thorough consistent formulation of the field theory of tachyons and their interactions with the usual particles may be formulated in future (if tachyons exist at all). The first phenomenological problem that will be considered here is the extremely small lower bound derived from the lower bound on the lifetime of proton [8]. In the light of this extremely small lower bound on the size of extra time-like dimension(s), in the order of a tenth of the Planck scale, either one should dare to employ such ( unnaturally) small dimension(s)or should use brane models where our physical world is a brane with an infinitesimal width in the extra time-like direction [2] or a scheme where tachyonic modes are not allowed to be produced [5,6]. A possible relaxation of the bound on the size of extra time dimension(s) would give more freedom to the model constructions with extra time-like dimension(s). So we reconsider the lower bound obtained from the lower bound on the proton lifetime and the calculation of a tree level Feynman diagram. We find that the calculation leads to no bound on the size of extra time-like dimensions. In fact we just repeat the calculations in [8] except we notice the fact that there is a cutoff momentum in the Fourier transform. In other words the difference between our result and the original study results from the naive application of the Fourier transform in [8] to get the non-relativistic potential corresponding to the scattering of protons inside a nucleus by tachyonic photon modes. In the original study the effect of tachyonic modes on fermion self-energies are neglected and no cutoff was taken, the integration is from minus infinity to plus infinity in momenta while one should take a cutoff corresponding to the maximum momentum available to the protons inside the nucleus. One obtains the same result as the one obtained in [8] when one lets the cutoff momentum go infinity and neglects the self-energy contributions. Next we consider the problems of the spontaneous decay of the particles through release of negative energy tachyons and the imaginary mass induced through self energy diagrams of fermions. We argue that these problems may be evaded by imposing conservation of momentum in the extra time direction provided that the standard model particles are identified as the zero modes of the Kaluza-Klein tower ( that is the standard identification). First consider the following tree level diagram for the electromagnetic scattering of two protons inside a nucleus [10,11].  The scattering cross section corresponding to this diagram may be obtained from the scattering amplitude of elastic fermion-fermion scattering. The differential cross section for elastic fermion-fermion scattering is related to scattering amplitude T by where M is the matrix element given by where and u's are 4-component Dirac spinors,γ µ are gamma matrices. One should also include the exchange scattering where p , but we are only interested in the order of magnitude results and the crossed term of (2) gives a similar contribution as (2) itself and does not alter the conclusion. So it is sufficient to consider (2). In the non-relativistic limit [10] the zero component of the proton 4-momenta p 01 , p 02 and the photon 4-momentum transfer k are approximated by Hence in the strict non-relativistic limit (i.e. where we have introduced the cut-off R which should be taken in the order of the momentum corresponding to the binding energy of the nucleus. This cut-off is explicitly written in Eq. (5) because k 2 ≃ −| k| 2 is not enough to indicate that T in (5) is the non-relativistic expression since the photon is off-shell in the propagator and one may take k 2 ≃ −| k| 2 for relativistic values of | k| as well provided that k 0 << | k|. In other words the strict non-relativistic limit implies k 2 = −| k| but k 2 ≃ −| k| 2 does not necessarily imply the strict non-relativistic limit.
Therefore the explicit expression of the conserve is not true that is | k| < | R| is necessary.
In non-relativistic quantum mechanics the scattering amplitude for the elastic scattering of a particle from a potential V, in the Born approximation may be written as [10,11,12] T After comparing (5) and (6) one notices that where We take the wave function of two protons inside a nucleus be where m π denotes the mass of pions. Then the decay width is obtained as The evaluation of Γ = Ψ|V (r)|Ψ is done in the appendix and found to be where m = 2im π , β = n L . One notices that Ψ|V (r)|Ψ is real if β > R (which is the most natural choice). Otherwise it means that the tachyonic photon masses are in the order of MeV . (In fact one obtains the result of [8] when one lets R → ∞.) In other words the tachyonic photon modes can not lead to decay of proton through processes given in Fig.1   So there would be an imaginary contribution to Eq.(12) even in the case R < β, that is, even in the case the size of the extra dimension(s) are much smaller than nuclear sizes. So we will impose conservation of momentum in extra time-like dimensions in the paragraph after the next paragraph to forbid fermion self-energy diagrams with tachyonic photons. In that way the processes similar to Fig.1 are forbidden as well as the processes as in Fig.2.
One may question if the calculation of that process (given in Eq. (12) ) is unnecessary or redundant once conservation of momentum is imposed in extra dimensions. In fact it is not.
The result of (12) gives more flexibility in model building. For example, one may consider a process similar to the one given in Fig.1 One might think that the scattering of high energy free protons ( e.g. in cosmic rays ) through processes similar to the one given in Fig.1 may change the bound given above. The cross section in that case can be directly found from (5) and is seen to be real. So the decay width of a two free nucleon system due to a process similar to fig.1 is zero. One may notice this fact without doing the calculation of the corresponding decay width explicitly. The decay width, i.e. the imaginary part of Ψ|V (r)|Ψ is due to the mixture of the arguments of the real exponent in Ψ and the complex exponential in V (r). If one takes Ψ be wave function of two free protons ( which is expressed by a complex exponential) then in the evaluation of Ψ|V (r)|Ψ the overall complex exponentials cancel and Ψ|V (r)|Ψ results in a real number so it has no imaginary component. In other words the decay width of two free protons due to tachyonic photon modes is always zero. However for confined particles one may expect a wave function of the form of (10), which results in a non-zero decay width.
Hence the quarks inside the nucleons may give such a non-zero decay width. On the other hand we do not know the wave functions of quarks inside nucleons so it is impossible to obtain an exact lower bound on the size of extra time-like dimensions by considering the quarks inside nucleons. However one may expect this wave function not be drastically different from (10). In that case one would expect the lower bound on the size of extra dimensions be in the order of ( cut-off momentum) −1 , that is, O( 1 1GeV ). In the same way one may put a still smaller lower limit if quarks are made of composites of some other particles (preons). If this generalization is reliable then one may relate the lower limit on the size of extra timelike dimension(s) and the binding energy. In this case one may speculate that, if an extra time-like dimension of the size much larger than the (Planck mass) −1 is discovered then it excludes possibility of stable bound states with energies much higher than the inverse of the size of the extra time-like dimension.
Next we consider the problem of the spontaneous decay of a particle (e.g. electron) into a tachyon and the original particle, and the problem of imaginary mass contribution to the stable fermions (e.g. electron or proton) through self energy diagrams involving a tachyon. The decay of a particle ( say an electron) into another electron and a negative energy tachyonic photon is kinematically allowed. It is difficult to identify these negative energy tachyons with anti-tachyons because negative energy tachyons may be made positive energy by a simple Lorentz boost [9]. So the result of such decays can be catastrophic because the where m, e, µ, Λ stand for the fermion mass, the electric charge of the fermion, the mass of the tachyonic photon, the Pauli-Villars regularization cut-off scale; respectively and we have modified the propagator of the tachyonic photon mode ( in the Pauli-Villars regularization scheme) by 1 By definition Λ > µ so Eq. there are still some points to be studied further. We hope that this study will facilitate more freedom in model building in future studies.

Acknowledgments
We would like to thank Dr. Gregory Gabadadze for his valuable comments and suggestions.

APPENDIX A
In this appendix we give the details of the evaluation of the integral given in (12).
The denominator of the integral may be written as We use the identity to write (A2) as The second term in (A5)is then (A6) becomes 1 After combining (A6), (A10) and using the explicit values of x 1 , x 2 , x 3 , x 4 ; and letting ǫ → 0 one obtains The evaluation of the integral (A1) by the use of (A11) gives