Localization of gauge fields in a tachyonic de Sitter thick braneworld
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In this work we show that universal gauge vector fields can be localized on the recently proposed 5D thick tachyonic braneworld which involves a de Sitter cosmological background induced on the 3-brane. Namely, by performing a suitable decomposition of the vector field, the resulting 4D effective action corresponds to a massive gauge field, while the profile along the extra dimension obeys a Schrödinger-like equation with a Pöschl–Teller potential. It turns out that the massless zero mode of the gauge field is bound to the expanding 3-brane and allows us to recover the standard 4D electromagnetic phenomena of our world. Moreover, this zero mode is separated from the continuum of Kaluza–Klein (KK) modes by a mass gap determined by the scale of the expansion parameter. We also were able to analytically solve the corresponding Schrödinger-like equation for arbitrary mass, showing that KK massive modes asymptotically behave like plane waves, as expected.
KeywordsGauge Field Warp Factor Tachyon Field Cosmological Background Teller Potential
1 Introduction and review of the thick tachyonic braneworld
The original proposal that extra dimensions can be non-compact [1, 2, 3, 4, 5] or even large [6, 7, 8] supplied new insights when attempting to solve the long standing cosmological constant problem [1, 2, 3, 5, 9, 10] and the intriguing gauge hierarchy problem [7, 8, 11, 12]. Subsequently, after the surprising success of the Randall–Sundrum models in addressing the mass hierarchy and 4D gravity localization problems [13, 14], during the last years, several braneworld configurations have been proposed in order to address different open problems in modern physics. Most of the braneworlds that generalize the original Randall–Sundrum setup make use of bulk scalar fields. In this framework, different kinds of scalar fields give rise to a wide range of scenarios [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In particular, some authors have choosen a bulk tachyonic scalar field in order to address problems like the gauge hierarchy, the stabilization of the corresponding brane separation, and the localization of gravity and other matter fields [35, 36, 37, 38, 39, 40]. However, many attempts to solve the highly non-linear field equations that arise in these scalar–tensor systems lead to imaginary tachyon field configurations [41, 42] or to delocalization of gravity and other kinds of matter, like scalar and gauge vector fields (see, for instance, the tachyonic braneworlds constructed in [36, 37, 41] and references therein).
An interesting issue within the braneworld paradigm is related to the question of whether bulk matter fields can be confined to the brane by a natural mechanism. Usually, scalar fields and gravitons, as well as fermions, can be localized on branes of different types. However, within most of the so far proposed thin braneworlds, gauge vector fields cannot be localized on the 3-brane that represents our Universe by a gravitational trapping mechanism [43, 44, 45, 46]. Quite recently some new braneworlds endowed with different mechanisms for localizing gauge vector fields, either on Weyl thick branes [47, 48], on purely geometrical expanding (de Sitter) branes  or in anisotropically curved branes supported by phantom  and real scalar fields , were constructed. In particular, vector fields can be localized on the RS model in some higher-dimensional cases .
In this paper we use a recently proposed thick tachyonic braneworld configuration with a de Sitter metric induced on the brane and a decaying warp factor that enables the localization of 4D gravity , and show that universal gauge vector fields can also be localized in it. Moreover, when analyzing the dynamics of the vector field fluctuations, we see that their spectrum contains a single bound state corresponding to the 4D massless photon of the model, separated from a continuum of massive KK excitations by a mass gap.
On the other hand, this kind of de Sitter 3-branes embedded in a higher dimensional manifold are also interesting from the point of view of cosmology. Since the aforementioned field configuration describes a 3-brane with 4D de Sitter symmetry (i.e. an expanding \(dS_4\) brane) embedded in a positive definite and asymptotically Minkowski spacetime \(M_5\), it qualitatively describes some aspects of the inflationary period of our own universe [53, 54]. Moreover, since the second derivative of the scale factor with respect to time is positive, it can describe some aspects of the accelerated expansion of the universe, related to its present epoch. It is known that the cosmic inflation theory is in accordance with the temperature fluctuation properties observed in the cosmic microwave background radiation. This agreement and other observational successes, together with the fact that inflation likely occurs at very high temperatures and its study involves relevant assumptions regarding the physical phenomena that take place at such high energies, lead to several serious attempts to find inflationary configurations within the framework of string theory and supergravity [55, 56, 57].
The second therm in (1) was originally proposed in [58, 59] when studying a tachyon field living on the world volume of a non-BPS brane and has found interesting applications in braneworld cosmology  and in string cosmology [65, 66, 67]. In  it was realized that this action can, in fact, be used for any relativistic scalar field. Moreover, solar system constraints were analyzed in  by using this tachyonic action as a scalar tensor theory.
Thus, the full solution corresponds to a non-trivial 5D configuration that describes a thick braneworld supported by a real tachyonic scalar field in which the induced metric on the 3-brane is described by a de Sitter 4D cosmological background.
It is evident that the warp factor decays and asymptotically vanishes along the extra dimension, whereas the tachyon scalar possesses a kink or antikink-like profile. In contrast with previously found solutions for this system, see for instance , the self-interaction tachyon potential \(V(T)\) can be explicitly expressed in terms of the tachyonic scalar field \(T\) and has a maximum/minimum at the position of the brane, but it is positive/negative definite as can be seen from (5). Moreover, since the tachyon field is bounded, the self-interaction potential \(V(T)\) is bounded as well and remains always real.
2 Localization of gauge vector fields
3 Conclusions and discussion
We have shown that gauge field localization in the de Sitter thick braneworld generated by gravity coupled to a tachyonic scalar field is feasible since the normalizable vector zero mode can be localized on the brane. The spectrum consists of a massless zero mode (the ground state) and a series of continuous massive KK gauge modes separated by a mass gap from the massless one, as also happens in the KK spectrum of the metric fluctuations . The mass gap is proportional to \(H\), and hence, proportional to the square root of the 4D cosmological constant. Thus, the height of the mass gap, which determines the energy scale at which the KK fluctuations can be excited, somehow is defined by the 4D cosmological constant and is big/small if the Hubble parameter is big/small.
It would be interesting to consider the localization of fermion fields in the tachyonic braneworld model under consideration. This study could lead to the possibility of computing the corrections to Coulomb’s law due to the KK massive modes of the gauge vector field within the model. It is also interesting to perform the analysis of one-loop corrections to brane-to-brane propagators as in  in order to study possible divergent pathologies related to the existence of the mass gap of the model under consideration. These two issues are currently under consideration, and we do hope to overcome the difficulties we found and report on them in the near future.
The authors acknowledge useful and illuminating discussions with Raúl Henríquez. AHA is grateful to A.S. Adamidou for her valuable support. AHA and ADR are grateful to MCTP, UNACH for hospitality, and thank SNI for support. ADR acknowledges a postdoctoral grant from CONACYT while the research of AHA was supported from grant “Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica” (PAPIIT) UNAM, IN103413-3, Teorías de Kaluza–Klein, inflación y perturbaciones gravitacionales.
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