EXPLORING VARIATIONS IN THE GAUGE SECTOR OF A SIX-DIMENSIONAL FLAVOUR MODEL
J.-M. Frère'1*, M. Libanovh'c**, S. Mollet0***, S. Troitskyh****
" Service de Physique Théorique. Université Libre de Bruxelles 1050, Brussels, Belgium
bInstitute for Nuclear Research of the Russian Academy of Sciences 117312, Moscow, Russia
€Moscow Institute of Physics and Technology 141700, Dolgoprudny, Moscow Region, Russia
Received October 1, 2014
In the context of extra-dimensional models which describe three families of fermions, including their masses and mixings in terms of a single 6-dimensional family, we explore possible variations, including in the geometry of the extra dimensions, and argue that the apparent plethora of variants does not lead to drastic changes in the expected phenomenology.
Cwitribvtiwi for the JETP spécial issue in honor of V. A. Rubakov's 60th birthday
The wonderful world of largo and infinite extra dimensions (ED), where low-energy excitations of multidimensional fields ("zero modes") are bound to a (3^1)-dimensional manifold ("the brane") representing our world, was discovered for theoretical physicists in independent works of Rubakov and Shaposhnikov fl], Akama , and Vissor  more than four decades ago. Since then, enlarged symmetries of multidimensional worlds have been exploited in field-theory frameworks to address various fine-tuning and hierarchy problems of the Standard Model (SM) of particle physics (see, e.g., reviews [4, 5] and references therein). One of the approaches transfers geometric symmetries of the ED into flavour symmetries of our world, explaining in an elegant way the hierarchy of masses and mixings of SM quarks and charged loptons [6 8] and leading to rich testable phenomenology [9 12]. The same model explains as well a very different pattern of neutrino masses
* E-mail: frere'fflulb.ac.be
**** E-mail: sergey.troitsky'fflgmail.com
and mixing, the difference with quarks being caused by the Majorana form of the neutrino mass term  (see Rof.  for a recent update). The purpose of the present work is to explore some ways beyond the simplest model and to sketch how robust its predictions are.
In ED models that hope to embed the SM, some vector fields must be introduced which will play the role of usual gauge fields at low energy. Their (almost) massloss "zero" modes appear as the usual (3^1 ^dimensional (4D) gauge bosons. The way of implementing a mechanism responsible for that is not always an easy task for there are further requirements to build a realistic model. Indeed, while we want the gauge zero mode to interact properly with the formionic ones, we know that there will also exist a set of heavier (excited) modes which should not talk too much with this low energy sector, i.e., either there must exist a mass gap or these modes must only interact very weakly with the low-energy sector . On the other hand, these new modes could manifest themselves at higher energy (in collider experiments for instance) or in (very) rare processes (e.g., flavour-changing neutral currents), thus providing hints for this kind of models.
In this note, we would like to provide with a short update of the constraints from these experiments for
various models of this kind. Wo will focus on a particular class in (4—2) dimensions where a Nielsen Oloson vortex-like defect plays the role of our 4D world fC 8,15,16]. We know that, quite generally in this background, we can get several localized (chiral) formion zero modes from a single spinor in 6D , each of them associated with a different winding in ED1) (<?mvi<?i(«.-+i)vi<?i(«.-+2)vi...). They can acquire (small) masses through the vacuum expectation value (vov) of a Brout Englort Higgs (BEH) field H. In a certain range of parameters , the particular shape of this vov in ED (nonzero in the core, almost zero outside) leads to a hierarchical pattern of masses. This idea was exploited in different contexts to reproduce the three SM generations and their spectrum. Here however, we will only be interested in their interactions with gauge bosons (both zero and heavy modes).
In Sec. 2, we come back on some possible ways of introducing gauge bosons in the model and try to convince the reader that the expected phenomenology should not change drastically from one realization to the other. In particular, we will recall the existence of heavy localized modes whose IIlclSS scale is set by the geometry. Unlike the zero mode, the former possess nonzero windings and can therefore be responsible for flavour changing processes (even in the absence of mixing in the fermionic sector) [10, 11]. In Sec. 3, we comment on these processes and provide with some numerical results for the precise realization of . Finally, we conclude in Sec. 4.
2. SOME GENERIC EXAMPLES
Let us here quickly remind some general results. We will focus on models with 4D Poincare invariance and 4D flat space. The most general metrics of such a kind can be written as 
ds2 = G,\B(lxAdxB = a(y)rilll,dx'1 dx1'
With the following choice of gauge: c?o W"o - 0, I \= 0,
we have the obvious separation of variables in the equation of motion for vector modes,
11 The exact values of the windings are not important. What will really be relevant for us are the difference in windings between two modes.
with the modal wavefunctions Pn satisfying
a 1m2P = 0.
There always exists a zero mode (m2 = 0) with a constant transverse wavefunction (P(y) = const), but we cannot conclude, at this level, if it is normalizable or not.
Two ways to ensure the normalizability are (i) to deal with compact ED whose finite volume renders the integral with the constant delocalized wavefunction bounded, or (ii) to make use of warp factors [19 22] which will sufficiently "dilute" the wavefunction, yet yield to a finite integral [23, 24]. Note that in the latter case, we can also consider effective wavefunctions in flat space which include warp factors and are thus localized from this point of view . We will provide realizations of these two scenarios in the further simplified metrics, which is a particular case of (1):
ds2 = cr(u),)illi,dx11 dx" — du2 — j(tt)dv2.
A simple example of the first way (compact space) is the 2-sphere [8, 10, 11] of radius R which corresponds to a = 1, u = R9, v = Rip, and 7 = sin2 8. The modal equation becomes then the equation for spherical harmonics with R2m2 = ('(('+ 1). As expected, we have a (normalizable) zero mode (' = 0 with constant wave-function P = l/s/iitR. Heavier modes appear to be normalizable, too. The IIlclSS scale is dictated by the size of ED. In particular, there is a mass gap of the order of 1 /R. For each value of i, there are degenerate modes with windings —(' < m < t The wavefunctions oscillate 011 a scale of order of R. for the lightest modes.
If we opt instead for the warped case, the warp metrics can be parametrized  as u = r, v = u9, a = eA^r\ and 7 = The precise behavior of the
,4 and B functions are determined by the exact realization of the defect, but we can establish general features of their asymptotics by requiring (i) the metrics to be a regular solution of the 6D Einstein equations where a negative bulk cosmological constant balances a positive string tension (in the core)2) and (ii) the gravity to be localized*^. What we get is [15, 16] .4'(0) = 0 and B(r —¥ 0) ~ 2 In (r/u) around the origin and
2) Note that at 4D level, we ask for a zero cosmological constant to have a flat space.
i.e., ask for a normalizable zero mode for the graviton .
A = B = —2rc outside the core (c is a dimensional constant related to the bulk cosniological constant) which correspond to an AdSg geometry. We still have the arbitrariness of normalization and choose ,4(0) = 0. The dimonsionfull constant, which will play an important role later on, u is not a free parameter but is determined by an interplay between the gravity and the vortex scales. With these asymptotics it is easy to realize that the two ED are a warped plane in polar coordinates and it is then obvious to further develop the P wavofunctions on a Fourier basis:
Pn(r,e) = X>nt(»Vi0.
With this, the equation for p becomes
p" + + ^ p' + (^ni2e-A - p = 0.
Outside the core, the solutions are classified in terms of 112 = m2 — (2¡u2. For //. = 0, we have a constant solution, while for //. ^ 0, it reads
For m = i/u (corresponding to localized mode //. = 0 at infinity), we have (note that here, the metric factor is simply r)
Po(r) ~ J£ (i-) .
For (. = 0, we get the usual constant solution (which matches with the constant solution at infinity, since we know that p = const is an exact solution for the all range of r). For nonzero i, we cannot get an exact solution, but we see that (at least for the first modes) we have oscillating functions with a scale of oscillation of order u.
In conclusion, we have a pattern which looks very much like the spherical case: discrete (localized) modes with mass scale 1/u and this same scale giving also an idea of the oscillation scale for the associated wave-functions. On the other hand, there are (associated to each of these bounded modes) a continuum, starting just above, but the derealization should kill the overlaps with localized profiles. Of course, this should be computed properly to be more quantitative.
p(r) = P3"/'2 [C1J3/2 (£<°rr) + C2Y3/2 (V'-)] ,
where J and 1" are Bessel functions, and C, are arbitrary constants. The boundary conditions (absence of the flux at infinity) lead to a continuous spectrum for //. > 0 . If we use the expression of J and Y in terms of elementary functions, it is easy to show that p behaves as r/e"' at sufficiently large r, where j/ is some oscillating and bounded function. Now remember that, in the initial action, we have a factor ~ \/b
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