Cluster algebras in scattering amplitudes with special 2D kinematics

We study the cluster algebra of the kinematic configuration space $Conf_n(\mathbb{P}^3)$ of a n-particle scattering amplitude restricted to the special 2D kinematics. We found that the n-points two loop MHV remainder function found in special 2D kinematics depend on a selection of \XX-coordinates that are part of a special structure of the cluster algebra related to snake triangulations of polygons. This structure forms a necklace of hypercubes beads in the corresponding Stasheff polytope. Furthermore in $n = 12$, the cluster algebra and the selection of \XX-coordinates in special 2D kinematics replicates the cluster algebra and the selection of \XX-coordinates of $n=6$ two loop MHV amplitude in 4D kinematics.Comment: 22 page

Scalar Dark Matter in light of LEP and ILC Experiments

In this work we study a scalar field dark matter model with mass of the order of 100 MeV. We assume dark matter is produced in the process $e^-+e^+\to \phi +\phi^*+\gamma$, that, in fact, could be a background for the standard process $e^-+e^+\to \nu +\bar\nu+\gamma$ extensively studied at LEP. We constrain the chiral couplings, $C_L$ and $C_R$, of the dark matter with electrons through an intermediate fermion of mass $m_F=100$ GeV and obtain $C_L=0.1(0.25)$ and $C_R=0.25(0.1)$ for the best fit point of our $\chi^2$ analysis. We also analyze the potential of ILC to detect this scalar dark matter for two configurations: (i) center of mass energy $\sqrt{s}=500$ GeV and luminosity $\mathcal{L}=250$ fb$^{-1}$, and (ii) center of mass energy $\sqrt{s}=1$ TeV and luminosity $\mathcal{L}=500$ fb$^{-1}$. The differences of polarized beams are also explored to better study the chiral couplings.Comment: 15 pages, 6 figures and 1 table. New references added and improvements in the text. Conclusions unchange