Universal extra dimension

In particle physics, the ADD model, also known as the model with large extra dimensions, is an alternative scenario to explain the weakness of gravity relative to the other forces. This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale. The model was proposed by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali in 1998cite journal|author= N. Arkani-Hamed, S. Dimopoulos, G. Dvali|title=The Hierarchy problem and new dimensions at a millimeter |journal=Phys. Lett. B|volume= 436|pages= 263–272|year= 1998] cite journal|author= N. Arkani-Hamed, S. Dimopoulos, G. Dvali|title= Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity|journal=Phys. Rev. D|volume= 59|pages=086004|year= 1999|doi= 10.1103/PhysRevD.59.086004] .

Proponents' views

Traditionally in theoretical physics the Planck scale is the highest energy scale and all dimensionful parameters are measured in terms of the Planck scale. There is a great hierarchy between the weak scale and the Planck scale and explaining the ratio of $G\_F/G\_N=\; 10^\{-32\}$ is the focus of much of beyond the Standard Model physics. In models of large extra dimensions the fundamental scale is much lower than the Planck. This occurs because the power law of gravity changes. For example, when there are two extra dimensions of size $d$, the power law of gravity is $1/r^4$ for objects with $scriptstyle\; r\; ll\; d$ and $1/r^2$ for objects with $scriptstyle\; r\; gg\; d$. If we want the Planck scale to be equal to the next accelerator energy (1 TeV), we should take $d$ approximately 1mm. For larger numbers of dimensions, fixing the Planck scale at 1TeV, the size of the extra-dimensions become smaller and as small as 1 femtometer for six extra dimensions.

By reducing the fundamental scale to the weak scale, the fundamental theory of quantum gravity, such as string theory, might be accessible at colliders such as the Tevatron or the LHC.cite journal|author= I Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali|title=New dimensions at a millimeter to a Fermi and superstrings at a TeV|journal=Phys. Lett. B|volume= 429|pages= 257–263|year= 1998] There has been recent progress in generating large volumes in the context of string theory.cite journal|author= O. DeWolfe, A. Giryavets, S. Kachru, W. Taylor|title=Type IIA moduli stabilization|journal=JHEP|volume= 0507|pages= 066|year= 2005] Having the fundamental scale accessible allows the production of blackholes at the LHC,cite journal|author= S. Dimopoulos, G. Landsberg|title=Black holes at the LHC|journal=Phys. Rev. Lett.|volume= 87|pages= 161602|year= 2001|doi=10.1103/PhysRevLett.87.161602] cite journal|author=S. Giddings, S. Thomas|title=High-energy colliders as black hole factories: The End of short distance physics|journal=Phys. Rev. D|volume= 65|pages= 056010|year= 2002|doi=10.1103/PhysRevD.65.056010] though there are constraints on the viability of this possibility at the energies at the LHC cite journal|author=G. Giudice, R. Rattazzi, J. Wells|title=Transplanckian collisions at the LHC and beyond|journal=Nucl. Phys. B|volume= 630|pages= 293–325|year= 2002|doi=10.1016/S0550-3213(02)00142-6] . There are other signatures of large extra dimensions at high energy colliders.cite journal|author= D. Bourilkov|title=Analysis of Bhabha scattering at LEP2 and limits on low scale gravity models|journal=Journal of High Energy Physics|volume= 9908|pages= 006|year= 1999] cite journal|author= K. Cheung, G. Landsberg|title=Drell-Yan and diphoton production at hadron colliders and low scale gravity models|journal=Phys. Rev. D|volume= 62|pages= 076003|year= 2000|doi=10.1103/PhysRevD.62.076003] cite journal|author=T. Rizzo|title=Using scalars to probe theories of low scale quantum gravity|journal=Phys. Rev. D|volume= 60|pages= 075001|year= 1999|doi=10.1103/PhysRevD.60.075001] cite journal|author=G. Shiu, R. Shrock, S. Tye|title=Collider signatures from the brane world|journal=Phys. Lett. B|volume= 458|pages= 274–282|year= 1999|doi=10.1016/S0370-2693(99)00609-7] cite journal|author=C. Balazs, H-J. He, W. Repko, C. Yaun, D. Dicus|title=Collider tests of compact space dimensions using weak gauge bosons|journal=Phys. Rev. Lett.|volume= 83|pages= 2112–2115|year= 1999|doi=10.1103/PhysRevLett.83.2112]

Many of the mechanisms that were used to explain the problems in the Standard Model used very high energies. In the years after the publication of ADD, much of the work of the beyond the Standard Model physics community went to explore how these problems could be solved with a low scale of quantum gravity. Almost immediately there was an alternate explanation to the see-saw mechanism for the neutrino mass.cite journal|author= N. Arkani-Hamed, S. Dimopoulos, G. Dvali, J. March-Russell|title=Neutrino masses from large extra dimensions|journal=Phys. Rev. D|volume= 65|pages= 024032|year= 2002] cite journal|author= G. Dvali, A. Yu. Smirnov|title=Probing large extra dimensions with neutrinos|journal=Nucl. Phys. B|volume= 563|pages= 63–81|year= 1999|doi=10.1016/S0550-3213(99)00574-X] Using extra dimensions as a new source of small numbers allowed for new mechanisms for understanding the masses and mixings of the neutrinos.cite journal|author= Y. Grossman, M. Neubert|title=Neutrino masses and mixings in nonfactorizable geometry|journal=Phys. Lett. B|volume= 474|pages= 361–371|year= 2000|doi=10.1016/S0370-2693(00)00054-X] cite journal|author=N. Arkani-Hamed, L. Hall, H. Murayama, D. R. Smith, N. Weiner|title=Neutrino masses at v**(3/2)|journal=hep-ph/0007001|pages=4 |year= 2000]

Another major issue with having a low scale of quantum gravity was the existence of possibly TeV-suppressed proton decay, flavor violating, and CP violating operators. These would be disastrous phenomenologically. It was quickly realized that there were novel mechanisms for getting small numbers necessary for explaining these very rare processes.cite journal|author= N. Arkani-Hamed, M. Schmaltz|title=Hierarchies without symmetries from extra dimensions|journal=Phys. Rev. D|volume= 61|pages= 033005|year= 2000|doi=10.1103/PhysRevD.61.033005] cite journal|author= N. Arkani-Hamed, Y. Grossman, M. Schmaltz|title=Split fermioons in extra dimensions and exponentially small cross-sections at future colliders|journal=Phys. Rev. D|volume= 61|pages= 115004|year= 2000|doi=10.1103/PhysRevD.61.115004] cite journal|author= D. E. Kaplan, T. Tait|title= New tools for fermion masses from extra dimensions|journal=JHEP|volume= 0111|pages= 051|year= 2001] cite journal|author= G. Branco, A. de Gouvea, M. Rebelo|title=Split fermioons in extra dimensions and CP violation|journal=Phys. Lett. B|volume= 506|pages= 115–122|year= 2001|doi=10.1016/S0370-2693(01)00389-6] cite journal|author=N. Arkani-Hamed, L. Hall, D. R. Smith, N. Weiner|title=Flavor at the TeV scale with extra dimensions|journal=Phys. Rev. D|volume=61|pages=116003 |year= 2000]

Opponents' views

In the traditional view, the enormous gap in energy between the mass scales of ordinary particles and the Planck mass is reflected in the fact that virtual processes involving black holes or gravity are strongly suppressed. The suppression of these terms is the principle of renormalizability--- in order to see an interaction at low energy, it must have the property that its coupling only changes logarithmically as a function of the Planck scale. Nonrenormalizable interactions are weak only to the extent that the Planck scale is large.

Virtual gravitational processes don't conserve anything except gauge charges, because black holes decay into anything with the same charge. So it is difficult to suppress interactions at the gravitational scale. The only way to do it is by postulating new gauge symmetries.

In electromagnetism, the electron magnetic moment is described by perturbative processes derived in the QED Lagrangian::

And it is calculated and measured to one part in a trillion. But it is also possible to include a Pauli term in the Lagrangian:

:

and the magnetic moment would change by A. The reason the magnetic moment is correctly calculated without this term is because the coefficient A has the dimension of inverse mass. The mass scale is at most the Planck mass. So A would only be seen at the 20th decimal place with the usual Planck scale.

Since the electron magnetic moment is measured so accurately, and since the scale where it is measured is at the electron mass, a term of this kind would be visible even if the Planck scale were only about $10^9$ electron masses, which is 1000TeV. This is much higher than the proposed Planck scale in the ADD model.

QED is not the full theory, and the standard model does not have many possible Pauli terms. A good rule of thumb is that a Pauli term is like a mass term--- in order to generate it the Higgs must enter. But in the ADD model, the Higgs vacuum expectation value is comparable to the Planck scale, that's the reason for the model. So the Higgs field can contribute to any power without any suppression. One coupling which generates a Pauli term is the same as the electron mass term, except with an extra $Y^\{mu\; u\}sigma\_\{mu\; u\}$ where Y is the U(1) gauge field. This is dimension 6, and it contains one power of the Higgs expectation value, and is suppressed by two powers of the Planck mass. This should start contributing to the electron magnetic moment at the sixth decimal place. A similar term should contribute to the muon magnetic moment at the third or fourth decimal place.

The neutrinos are only massless because the following dimension five operator does not appear::

Involving L, the left-handed weak SU(2) doublet and H, the Higgs expectation value which is also a doublet. But neutrinos have a mass scale of approximately $10^\{-2\}\; eV$, which is 14 orders of magnitude smaller than the scale of the Higgs expectation value of 1TeV. This means that the term is suppressed by a mass M such that:

:$\{H^2over\; M\}\; =\; .01\; eV,$

substituting H=1TeV, gives $M=10^\{26\}eV=10^\{17\}GeV$. So this is where the neutrino masses suggest new physics--- at close to the traditional GUT scale, a few orders of magnitude less than the traditional Planck scale. The same term in a large extra dimension model would give a mass to the neutrino in the MeV-GeV range, comparable to the mass of the other particles.

In the view of opponents, authors working on ADD miscalculate the neutrino masses because they assume for no reason that the mass is due to interactions with a hypothetical right-handed partner. The only reason to introduce a right-handed partner is to produce neutrino masses in a renormalizable GUT. If the Planck scale is small so that renormalizability is no longer an issue, there are many neutrino mass terms which don't require extra particles.

For example, at dimension 6, there is a Higgs-free term which couples the lepton doublets to the quark doublets.

:

which is a coupling to the strong interaction quark condensate. Even with a relatively low energy pion scale, this type of interaction could conceivably give a mass to the neutrino of size $scriptstyle\; \{f\_pi\}^3/TeV^2$, which is only a factor of $10^7$ less than the pion condensate itself at 200MeV. This would be some 10 eV of mass, about than a thousand times bigger than what is measured.

This term also allows for lepton number violating pion decays, and for proton decay. In fact in all operators with dimension greater than four, there are CP, baryon, and lepton-number violations. The only way to suppress them is to deal with them term by term, which nobody has done.

Despite its serious flaws, this model has gained currency in some circles. One reason might be because it predicts black holes at LHC, which is exciting to many people. Another reason is that most of the truly fatal difficulties come from terms which involve the Higgs, and the Higgs mechanism is still a completely unknown quantity. In the more recent Randall-Sundrum models, many of the same problems arise, although there the size of the extra dimensions might appear infinite because of the redshift, when it is finite and small.

ee also

* Kaluza-Klein theory
* Randall-Sundrum model
* DGP model

References