Distances on the tropical line determined by two points

Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This line is a balanced unrooted semi--labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p'$ and $q'$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also prove that $L(p,q)$ is caterpillar. We prove that every vertex in $L(p,q)$ belongs to the tropical linear segment joining $p$ and $q$. A vertex, denoted $pq$, closest (w.r.t tropical distance) to $p$ exists in $L(p,q)$. Same for $q$. The distances between pairs of adjacent vertices in $L(p,q)$ and the distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the matrix $|F|$. In addition, if $p$ and $q$ are generic, then the tree $L(p,q)$ is trivalent. The entries of $F$ are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of $A$.Comment: New corrected version. 31 pages and 9 figures. The main result is theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7 was obtained with A. Jim\'enez; see Arxiv 1205.416

Electroweak Vacuum Stability and the Seesaw Mechanism Revisited

We study the electroweak vacuum stability in Type I seesaw models for 3 generations of neutrinos in scenarios where the right-handed neutrinos have explicit bare mass terms in the Lagrangian and where these are dynamically generated through the mechanism of spontaneous symmetry breaking. To best highlight the difference of the two cases we concentrate on the absolute stability of the scalar potential. We observe that for the first scenario, the scale at which the scalar potential becomes unstable is lower from that within the Standard Model. In addition the Yukawa couplings $\mathbf{Y}_\nu$ are constrained such that \Tr{[\mathbf{Y}^{\dagger}_\nu \mathbf{Y}_{\nu}}] \lesssim10^{-3}. In the second scenario the electroweak stability can be improved in a large region of parameter space. However, we found that the scalar used to break the lepton number symmetry cannot be too light and have a large coupling to right-handed neutrinos in order for the seesaw mechanism to be a valid mechanism for neutrino mass generation. In this case we have \Tr [\mathbf{Y}^\dagger_{\nu} \mathbf{Y}_\nu]\lesssim 0.01

Long-lived Colored Scalars at the LHC

We study the collider signatures of a long-lived massive colored scalar transforming trivially under the weak interaction and decaying within the inner sections of a detector such as ATLAS or CMS. In our study, we assume that the colored scalar couples at tree-level to a top quark and a stable fermion, possibly arising from a dark sector or from supersymmetric extensions of the Standard Model. After implementing the latest experimental searches for long-lived colored scalars, we observe a region of parameter space consistent with a colored electroweak-singlet scalar with mass between $\sim200-350$ GeV and a lifetime between $0.1-1$ $\text{mm}/c$ together, with a nearly degenerate dark fermion that may be probed at the $\sqrt{s}=13$ TeV LHC. We show that a search strategy using a combination of cuts on missing transverse energy and impact parameters can exclude regions of parameter space not accessed by prompt searches. We show that a region of parameter space within our simplified model may naturally arise from the light-stop window regime of supersymmetric extensions of the Standard Model, where a light mostly right-handed stop has a mass slightly larger than the lightest neutralino and decays through a four-body process

A Singlet Extension of the MSSM with a Dark Matter Portal

The minimal extension of the MSSM (NMSSM) has been widely studied in the search for a natural solution to the $\mu$ problem. In this work, we consider a variation of the NMSSM where an extra singlet is added and a Peccei-Quinn symmetry is imposed. We study its neutralino sector and compute the annihilation cross section of the lightest neutralino. We use existent cosmological and collider data to constrain the parameter space and consider the lightest neutralino, which is very light, as a dark matter candidate.Comment: 26 pages, 8 figures . v4: minor corrections; version accepted for publicatio

Tropical conics for the layman

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.Comment: 19 pages, 4 figures. Major rewriting of formerly entitled paper "Metric invariants of tropical conics and factorization of degree--two homogeneous tropical polynomials in three variables". To appear in Idempotent and tropical mathematics and problems of mathematical physics (vol. II), G. Litvinov, V. Maslov, S. Sergeev (eds.), Proceedings Workshop, Moscow, 200

Top Quark as a Dark Portal and Neutrino Mass Generation

We present a new model for radiatively generating Majorana active neutrino masses while incorporating a viable dark matter candidate. This is possible by extending the Standard Model with a single Majorana neutrino endowed with a dark parity, a colour electroweak singlet scalar, as well as a colour electroweak triplet scalar. Within this framework, the $up$-type quarks play a special role, serving as a portal for dark matter, and a messenger for neutrino mass generation. We consider three benchmark scenarios where the abundance of dark matter can match the latest experimental results, while generating neutrino masses in the milli-electronvolt range. We show how constraints from lepton flavour violation, in particular the branching fraction of $\mu\to e\gamma$, can place lower bounds on the coupling between our dark matter candidate and top quarks. Furthermore, we show that this coupling can also be constrained using collider data from the Tevatron and the LHC.Comment: 8 captions, 10 figure