arXiv subjects

R. Omar Rodriguez

Publications and source records attributed to R. Omar Rodriguez.

12 recordsLinked to original sources

arXiv2026

Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses

We derive exact analytic representations for the perimeter of a Lam\'e superellipse of degree $s>0$. The result is expressed in terms of two branches defined by series whose terms are Gauss hypergeometric functions: a negative branch for $0 1$. For the positive branch, the convergence condition follows from the Leibniz test; the negative branch, although divergent in the ordinary sense, is shown to be Abel-summable. Consistently with the symmetry under interchange of the semi-axes, the formula is invariant under axis permutation. As $s$ varies, the family interpolates between the Lam\'e cross and the rectangle, while the case $s=1$ corresponds to the rhombus, which acts as the transition curve with the shortest perimeter within the family.

math.CA↗

arXiv2026

On the Arc Length of a Supercircle and a Hypergeometric Formulation of $\pi$

We obtain an infinite-series representation for the arc length of a supercircle in terms of the scale parameter $a$ and the shape parameter $n$. The resulting expression is constructed by means of generalized binomial coefficients and Gauss hypergeometric functions, distinguishing two regimes associated with the value of $n$. We also analyze the absolute convergence of the resulting series. We verify the consistency of the formulation from limiting cases and particular configurations of the family of supercircles: when $n\to0^+$ and $n\to\infty$, the length converges to the value $8a$, corresponding to the limiting rectilinear geometries, whereas for $n=1$ we recover the perimeter of the rhombus with diagonals of length $2a$. In addition, as a validation against supercircles with exact arc length, the formulation reproduces with high numerical precision the arc length of the parabolic star, the astroid, and the circle. Finally, by specializing the circular case $n=2$ and normalizing the length by the diameter $2a$, we obtain a series representation, in terms of hypergeometric functions, for the constant $\pi$.

math.CA↗

arXiv2019

Fermions localization on de Sitter branes

Spin 1/2 fields localization on an asymmetric dS${}_4$ scenario, where the brane interpolates between two spacetimes dS${}_5$ and AdS${}_5$, is determined. The bulk spinor is coupled to scalar field of the brane by a nonminimal Yukawa term compatible with the scenario's geometry. We show that, independently of wall's thickness, only one massless chiral mode is localized on the wall. The massive chiral modes follow a Schrödinger equation, whose potential has a mass gap determined by Yukawa constant, which is a generic property of this system. The fermions spectrum is defined bellow the gap, by bound states of both chiralities with the same mass, and above the gap, by a continuous spectrum with local and global resonant modes of both chiralities and different mass.

hep-th↗

arXiv2019

Massless fermions localization on domain walls

Massless fermions on scalar domain walls are considered. Two walls are established, corresponding to 5-dimensional static spacetime asymptotically Anti de-Sitter, differentiated by the symmetry around the wall, and in each case massless chiral fermions are coupled to the wall by a Yukawa term. We identify a normalizable state associated to the migration of fermions toward the edge of the wall. This effect is generated by the competition between the Yukawa interaction and the gravitational repulsion on the matter fields, and it is independent of the $Z_2$ symmetry of the wall.

hep-th↗

arXiv2019

A $SU(5)\times Z_2$ kink solution and its local stability

A non-abelian kink inducing asymptotically the breaking pattern $SU(5)\times Z_2\rightarrow SU(4)\times U(1)/Z_4$ is obtained. We consider a fourth order Higgs potential in a $1+1$ theory where the scalar field is in the adjoint representation of $SU(5)$. The perturbative stability of the kink is also evaluated. A Schrödinger-like equation for the excitations along each $SU(5)$ generator is determined, and in none of the cases negative eigenvalues compromising the stability of solution are found. In particular, several bounded scalar states are found, being one of them the translational zero mode of the flat space $SU(5)\times Z_2$ kink.

hep-th↗

arXiv2017

$SO(10)$ thick branes and perturbative stability

Three self-gravitating $SO(10)$ domain walls in five dimensions are obtained and their properties are analyzed. These non-abelian domain walls interpolate between AdS${}_5$ spacetimes with different embedding of $SU(5)$ in $SO(10)$ and they can be distinguished, among other features, by the unbroken group on each wall, being either $SO(10)$, $SO(6)\otimes SU(2)\otimes U(1)/Z_2$ or $SU(4)\otimes SO(2)\otimes U(1)/Z_4$. We show that, unlike Minkowskian versions, the curved scenarios are perturbatively stable due to the gravitational capture of scalar fluctuations associated to the residual orthogonal subgroup in the core of the walls. These stabilizer modes are additional to the four-dimensional Nambu-Goldstone states found in two of the three gravitational sceanarios.

hep-th↗

arXiv2013

Including gauge symmetry in the localization mechanism of massive vector fields

On the four-dimensional sector of an AdS$_5$ warped geometry the standard electromagnetic interaction can be simulated by massive vector fields via the Ghoroku - Nakamura localization mechanism. We incorporate gauge symmetry to this theory by finding the required interaction terms between the vector bosons and the gravitational field of the scenario. The four-dimensional effective theory defined by a Maxwell term and a tower of Stueckelberg fields is obtained after expanding the vector fields on a massive eigenstates basis where the zero mode is uncoupled from the rest of the spectrum. The corrections generated by the massive gauge fields set to the electrostatic potential are also calculated.

hep-th↗

arXiv2011

Newtonian gravity and resonance on de-Sitter branes

A dS brane on the boundary between two five-dimensional spacetimes is determined. We consider asymmetric scenarios with AdS${}_5$ vacua at each side of the dS brane; and as a result, a resonant mode inside of the spectrum of the gravitational fluctuations is found. We analyze the deviations to the Newton potential generated by the gravitational excitations, finding that, for scenarios with large values of the cosmological constants, the contribution of the resonant mode is exponentially suppressed. However, when one of the vacua is null, the resonant mode belongs to the light states set of the gravitational fluctuations and five-dimensional gravity is recovered on the dS brane.

hep-th↗

arXiv2009

Gauge field localization on brane worlds

We consider the effects of spacetime curvature and brane thickness on the localization of gauge fields on a brane via kinetic terms induced by localized fermions. We find that in a warped geometry with and infinitely thin brane, both the infrared and the ultraviolet behavior of the electromagnetic propagator are affected, providing a more stringent bound on the brane's tension than that coming from the requirement of four-dimensional gravity on the brane. On the other hand, for a thick wall in a flat spacetime, where the fermions are localized by means of a Yukawa coupling, we find that 4-dimensional electromagnetism is recovered in a region bounded from above by the same critical distance appearing in the thin case, but also from below by a new scale related to the brane's thickness and the electromagnetic couplings.

hep-th↗

arXiv2007

De Sitter and double irregular domain walls

A new method to obtain thick domain wall solutions to the coupled Einstein scalar field system is presented. The procedure allows the construction of irregular walls from well known ones, such that the spacetime associated to them are physically different. As consequence of the approach, we obtain two irregular geometries corresponding to thick domain walls with $dS$ expansion and topological double kink embedded in $AdS$ spacetime. In particular, the double brane can be derived from a fake superpotential.

gr-qc↗

arXiv2006

Close to the edge: hierarchy in a double braneworld

We show that the hierarchy between the Planck and the weak scales can follow from the tendency of gravitons and fermions to localize at different edges of a thick double wall embedded in an $AdS_5$ spacetime without reflection symmetry. This double wall is a stable BPS thick-wall solution with two sub-walls located at its edges; fermions are coupled to the scalar field through Yukawa interactions, but the the lack of reflection symmetry forces them to be localized in one of the sub-walls. We show that the graviton zero-mode wavefunction is suppressed in the fermion edge by an exponential function of the distance between the sub-walls, and that the massive modes decouple so that Newtonian gravity is recuperated.

hep-th↗

arXiv2005

De-Sitter and double asymmetric brane worlds

Asymmetric brane worlds with dS expansion and static double kink topology are obtained from a recently proposed method and their properties are analyzed. These domain walls interpolate between two spacetimes with different cosmological constants. In the dynamic case, the vacua correspond to dS and AdS geometry, unlike the static case where they correspond to AdS background. We show that is possible to confine gravity on such branes. In particular, the double brane world host two different walls, so that the gravity is localized on one of them.

hep-th↗