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\begin{document}
\title{\bf Viable quark-lepton Yukawa ratios and nucleon decay predictions in $SU(5)$ GUTs with type-II seesaw}

\author[]{Stefan Antusch,}
\author[]{Kevin Hinze,}
\author[]{and Shaikh Saad}

\affiliation[]{Department of Physics, University of Basel, Klingelbergstrasse\ 82, CH-4056 Basel, Switzerland}

\emailAdd{stefan.antusch@unibas.ch, kevin.hinze@unibas.ch, shaikh.saad@unibas.ch}
\abstract{
We investigate the viability of predictive schemes for quark-lepton Yukawa ratios and nucleon decay in non-supersymmetric SU(5) Grand Unified Theories (GUTs) where neutrino masses are generated by a type II seesaw mechanism. The scalar sector of the considered scenario contains 5-, 24- and 45-dimensional representations plus a 15-dimensional representation for realising the type II seesaw. Predictions for the ratios of the quark and lepton Yukawa couplings emerge when the relevant entries of the Yukawa matrices are generated from single joint GUT operators (i.e.\ under the condition of \textit{single operator dominance}).  Focusing on the 2nd and 3rd family and hierarchical Yukawa matrices, we show that only two sets of predictions, $\frac{y_\tau}{y_b}=\frac{3}{2}$, $\frac{y_\mu}{y_s}=\frac{9}{2}$ and $\frac{y_\tau}{y_b}=2$, $\frac{y_\mu}{y_s}=6$ are viable. To further investigate both options, we extend the minimal scenarios to two ``toy models'', including the 1st family of charged fermions,
and calculate the models' predictions, e.g.\ for the nucleon decay rates and for the masses of the light relics that are potentially within reach of colliders.
}
\maketitle

\section{Introduction}
Grand Unified Theories (GUTs)~\cite{Pati:1974yy, Georgi:1974sy, Georgi:1974yf, Georgi:1974my, Fritzsch:1974nn} present a fascinating framework for physics beyond the Standard Model (SM). On top of the unification of gauge couplings, they (partially) unify the fermions into joint GUT representations, providing an interesting framework to address the flavor puzzle, i.e. the origin of the observed fermion masses, mixing angles and CP violating phases.

The simplest model based on the unifying group $SU(5)$, proposed by Georgi and Glashow in 1974\ \cite{Georgi:1974sy}, predicts between the charged lepton and down-type quark Yukawa matrices the GUT scale relation
\begin{align}\label{eq-Ye=YdT}
Y_e=Y_d^T,
\end{align}
which in particular results in a GUT scale unification of tau and bottom Yukawa couplings, as well as a unification of muon and strange Yukawa couplings. It is however well-known that these results disagree with the low energy experimental data of fermion masses, as long as only the known SM particles are taken into account in the renormalization group (RG) evolution\  \cite{Arason:1991ic}.
Typical approaches for dealing with this shortcoming can be subdivided into two classes:
\begin{enumerate}
\item In the first class of approaches any GUT prediction for the GUT scale quark-lepton Yukawa coupling ratios is removed by introducing new degrees of freedom, which basically make these GUT scale ratios freely adjustable parameters. This can be achieved, for example, by adding additional renormalizable (see e.g.\  \cite{Dorsner:2014wva, Tsuyuki:2014xja, FileviezPerez:2016sal, Saad:2019vjo, Dorsner:2019vgf, Dorsner:2021qwg}) or non-renormalizable (see e.g.\ \cite{Ellis:1979fg, Dorsner:2006hw, Bajc:2006ia, Bajc:2007zf}) GUT operators, such that the Yukawa couplings emerge as linear combinations from these operators. While, on the one hand, consistency with the low energy observable data can be restored this way, on the other hand the predictivity for the Yukawa ratios is lost.
\item The second class of approaches maintains predictivity for the GUT scale Yukawa ratios by considering models which feature alternative GUT relations. They have in common that only a single GUT operator dominates each relevant entry in the GUT Yukawa matrices, a concept we refer to as \textit{single operator dominance}\ (cf.\ \cite{Antusch:2009gu, Antusch:2013rxa,Antusch:2019avd}). Different types of GUT operators can result in different GUT predictions for the respective quark-lepton Yukawa ratio. Their consistency with the low energy experimental data imposes constraints on the model parameters, such as the value of the GUT scale or the couplings to additional fields introduced e.g.\ for explaining the observed light neutrino masses.
\end{enumerate}

We follow in this paper an approach of the second type. With each Yukawa entry being dominated by a different single non-renormalizable GUT operator, different GUT scale charged lepton and down-type quark ratios (cf.\ \cite{Antusch:2009gu, Antusch:2013rxa,Antusch:2019avd}) can be considered in the different Yukawa entries, maintaining a high predictivity while potentially having a better viability with the experimental data.\footnote{Examples of models employing the concept of \textit{single operator dominance} can be found, e.g. in\  \cite{Antusch:2012fb, Antusch:2013kna, Antusch:2017ano, Antusch:2018gnu}.} It has been recently suggested~\cite{Antusch:2021yqe}, that in non-SUSY $SU(5)$ GUTs the GUT scale ratios
\begin{align}\label{eq-ytau/yb=3/2}
\frac{y_\tau}{y_b}=\frac{3}{2}\qquad\text{and}\qquad \frac{y_\mu}{y_s}=\frac{9}{2},
\end{align}
where $y_i$ denotes the low-scale singular value of the Yukawa matrices, and $i$ its fermion flavor, are in good agreement with the experimental data, if the neutrino masses originate from a type I seesaw mechanism\ \cite{Minkowski:1977sc}. We will in this paper show that on top of the GUT scale relations in Eq.~\eqref{eq-ytau/yb=3/2}, the ratios
\begin{align}\label{eq-ytau/yb=2}
\frac{y_\tau}{y_b}=2\qquad\text{and}\qquad \frac{y_\mu}{y_s}=6
\end{align}
can also be viable if instead the neutrino masses are generated by a type II seesaw mechanism\ \cite{Magg:1980ut,Schechter:1980gr} in a non-SUSY $SU(5)$ GUT. Both of these possibilities of GUT scale charged lepton and down-type quark Yukawa ratios can be realized if the Georgi-Glashow model is extended such that the Higgs sector contains the representations \textbf{5}, \textbf{24} and \textbf{45}. Moreover, adding a 15-dimensional Higgs representation allows for a type II seesaw mechanism\ \cite{Dorsner:2005fq}.

In this paper we investigate the viability of the two combinations of GUT scale Yukawa coupling ratios listed in Eq.~\eqref{eq-ytau/yb=3/2} and Eq.~\eqref{eq-ytau/yb=2} in a GUT scenario in which the neutrino masses are generated by a type II seesaw mechanism and the Higgs sector consists of the representations \textbf{5}, \textbf{24}, \textbf{45} and \textbf{15}.\footnote{A model with the same Higgs sector was already considered in\ \cite{Dorsner:2007fy}. However, the focus of the present work lies on the viability of the predicted GUT scale Yukawa ratios.} Extending this GUT scenario to two ``toy models'' (following \cite{Antusch:2014poa}) we compute the nucleon decay rates for 13 different decay channels using the \texttt{ProtonDecay} package\ \cite{Antusch:2020ztu} and calculate the allowed ranges of the masses of the added intermediate-scale scalar fields, hinting various possibilities to test our models by future experiments\  \cite{Acciarri:2015uup, Abe:2018uyc, An:2015jdp}.

The paper is organised as follows: In Section~\ref{sec-model} we introduce our GUT scenario and investigate the viability of the GUT scale ratios $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, respectively the GUT scale relations $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$, w.r.t. the experimental low-scale fermion masses. We further extend our GUT scenario to two ``toy models'' both predicting one of these relations. We explain the numerical procedure in Section~\ref{sec-numerical analysis}, before discussing the numerical results in Section~\ref{sec-results} and concluding in Section~\ref{sec-conclusions}. We list the renormalization group equations (RGEs) of the gauge and Yukawa couplings in Appendix~\ref{app-rge gauge and yukawa} and show in Appendix~\ref{app-RGE kappa} how the added scalar fields change the running of the effective neutrino mass operator. Finally, in Appendix~\ref{sec-perturbativity of Ydelta above the GUT scale} we discuss the perturbativity of our ``toy models'' by approximating the running of the neutrino couplings above the GUT scale.

\section{Model}\label{sec-model}
\subsection{Particle content}\label{sec-particle content}
We investigate an extension of the Georgi-Glashow (GG) model\ \cite{Georgi:1974sy} in which neutrino masses are generated by a type II seesaw mechanism\ \cite{Magg:1980ut} and in which charged lepton and down-type quark Yukawa ratios are predicted at the GUT scale through the concept of \emph{single operator dominance} \cite{Antusch:2009gu, Antusch:2013rxa}. On top of the usual GG particle content, this SU(5) GUT scenario contains the Higgs field $\bm{15_H}$ to allow for a type II seesaw mechanism\ \cite{Dorsner:2005fq}. Furthermore, a Higgs field $\bm{45_H}$ is used in order to allow for gauge coupling unification and to produce some of the operators which predict fixed GUT scale charged lepton and down-type quark Yukawa ratios\ \cite{Antusch:2009gu, Antusch:2013rxa}. In summary, the Higgs sector consists of the representations $\bm{5_H}$, $\bm{24_H}$, $\bm{45_H}$, and $\bm{15_H}$, whereas the fermion sector is given by ${\bm{\overline{5}_F}}_i$, and ${\bm{10_F}}_i$, where $i=1,2,3$. We will show that within this scenario, depending on which particles are light and which are heavy, two different combinations of GUT scale Yukawa ratios are possible: (i) $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, respectively (ii) $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$.

As usual, the GUT representations ${\bm{\overline{5}_F}}_i$, and ${\bm{10_F}}_i$ contain the SM fermions
\begin{align}
&{\bm{\overline{5}_F}}_i= \ell_i (1,2,-\frac{1}{2}) \oplus d_i^c (\overline{3},1,\frac{1}{3}),  \label{GG1}\\
&{\bm{10_F}}_i= q_i (3,2,\frac{1}{6}) \oplus u_i^c (\overline{3},1,-2/3) \oplus e_i^c (1,1,1). \label{GG2}
\end{align}
Under the SM gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$ the Higgs fields have the following decomposition:
\begin{align}
{\bm{5_H}} =&\; T_1(\overline{3},1,-\frac{1}{3}) \oplus H_1(1,2,\frac{1}{2}),\\
{\bm{45_H}} =&\; \phi_1(8,2,\frac{1}{2}) \oplus \phi_2(6,1,-\frac{1}{3}) \oplus \phi_3(3,3,-\frac{1}{3}) \oplus \phi_4(\overline{3},2,-\frac{7}{6}) \nonumber\\
&\oplus \phi_5(\overline{3},1,\frac{4}{3}) \oplus T_2(3,1,-\frac{1}{3}) \oplus H_2(1,2,\frac{1}{2}),  \label{GG4}\\
{\bm{24_H}}=&\; \Sigma_8 (8,1,0) \oplus \Sigma_1 (1,3,0) \oplus \Sigma_0 (1,1,0) \oplus \Sigma_3(3,2,-\frac{5}{6}) \oplus \Sigma_{\overline{3}}(\overline{3},2,\frac{5}{6}), \label{GG5}\\
\bm{15_H}=&\;\Delta_1\left(1,3,1\right)\oplus\Delta_3(3,2,\frac{1}{6})\oplus\Delta_6(\overline{6},1,-\frac{2}{3}).
\end{align}
A linear combination of $H_1$ and $H_2$ gives the electroweak breaking Higgs doublet $h$. In the following we will assume that the second Higgs doublet $h^\perp$ which is given by a linear combination of $H_1$ and $H_2$ orthogonal to $h$ has its mass at the GUT scale $M_\text{GUT}$. Moreover, the color triplets $T_1$ and $T_2$ mix and we denote the corresponding mass eigenstates by $t_1$ and $t_2$. Due to nucleon decay restrictions the masses of these two fields have to be above $10^{11}$\ GeV \cite{Nath:2006ut}.

\iffalse
\begin{align}
\bm{\overline{5}_F}=\begin{pmatrix}
d^c_1\\d^c_2\\d^c_3\\e\\ -\nu
\end{pmatrix},\;\;\;\bm{10_F}=\frac{1}{\sqrt{2}} \begin{pmatrix}
0&u^c_3&-u^c_2&u_1&d_1\\
-u^c_2&0&u^c_1&u_2&d_2\\
u^c_2&-u^c_1&0&u_3&d_3\\
-u_1&-u_2&-u_3&0&e^c\\
-d_1&-d_2&-d_3&-e^c&0
\end{pmatrix}.
\end{align}
\fi
\iffalse
$15$ is symmetric under $\alpha\leftrightarrow \beta$:
\begin{align}
\bm{15^{\alpha\beta}_H}=\Delta_1\left(1,3,1\right)\oplus\Delta_3\left(3,2,\frac{1}{6}\right)\oplus\Delta_6\left(6,1,-\frac{2}{3}\right)
\end{align}

\begin{align}
\Delta^{\alpha\beta}= \begin{pmatrix}
\Delta_6^{\mu\nu}&\frac{1}{\sqrt{2}}\Delta_3^{\mu a}\\
\frac{1}{\sqrt{2}}\Delta_3^{\mu a}&\Delta_1^{ab}
\end{pmatrix}
\end{align}
$\alpha, \beta=1-5$, $\mu,\nu=1-3$, $a,b=4,5$.
\fi

\subsection{Yukawa sector}\label{sec-Yukawa sector}
\subsubsection{Possible Yukawa couplings}\label{sec-possible Yukawa couplings}
With the particle content specified above the following Yukawa couplings are possible
\begin{align}
    (Y_{\bm{\overline{5}}})_{ij}: \quad {\bm{10_F}}_i{\bm{\overline{5}_F}}_j\,\bm{X}\quad &\supset \quad (Y_d)_{ij},\,(Y_e)_{ij},\label{eq:5F10F}\\
    (Y_{\bm{10}})_{ij}: \quad {\bm{10_F}}_i{\bm{10_F}}_j\,\bm{Y}\quad &\supset \quad (Y_u)_{ij},\\
    \frac{1}{\sqrt{2}}(Y_{\Delta})_{ij}: \quad {\bm{\overline{5}_F}}_i{\bm{\overline{5}_F}}_j\,\bm{Z}\quad &\supset \quad  \frac{1}{\sqrt{2}}(Y_1)_{ij},\,(Y_3)_{ij},\, \frac{1}{\sqrt{2}}(Y_6)_{ij},\label{eq:5F5F}
\end{align}
where $\bm{X}$, $\bm{Y}$, and $\bm{Z}$ represent one or multiple Higgs fields and where $Y_u$, $Y_d$, and $Y_e$ are the typical SM up-type, down-type and charged lepton Yukawa couplings, whereas $Y_1$, $Y_3$, and $Y_6$ are new quasi-like Yukawa couplings defined by the interactions $\frac{1}{\sqrt{2}}(Y_1)_{ij} \Delta_1 \ell_i\ell_j$, $(Y_3)_{ij} \Delta_3 d^c_i\ell_j$, and $\frac{1}{\sqrt{2}}(Y_6)_{ij} \Delta_6 d^c_i d^c_j$. These quasi-Yukawa matrices are allowed to have entries of the order of $\mathcal{O}(1)$. Moreover, the SU(5) operator ${\bm{10_F}}_i{\bm{\overline{5}_F}}_j\,\bm{X}$ generates after the GUT symmetry breaking interactions between the SM fermions and the Higgs fields $\phi_1$, $\phi_2$, $\phi_3$, $\phi_4$, and $\phi_5$. For all of these interactions, the largest entry in the corresponding quasi-Yukawa matrix is of the order of $\mathcal{O}(10^{-2})$, since the quasi-Yukawa matrices are up to Clebsch-Gordan (GC) coefficients identical to $Y_d$. Furthermore, if $\bm{Y}$ contains $\bm{5_H}$, the operator ${\bm{10_F}}_i{\bm{10_F}}_j\,\bm{Y}$ only generates the up-type Yukawa coupling. If $\bm{Y}$ was further allowed to also contain $\bm{45_H}$, then couplings between the SM fermions and the Higgs fields $\phi_1$, $\phi_2$, $\phi_3$, $\phi_4$, and $\phi_5$ of the order of $\mathcal{O}(1)$ would be allowed. We will in the following assume that there exists a symmetry which forbids the latter case.

\subsubsection{Neutrino mass generation}\label{sec-neutrino mass generation}
After the GUT symmetry breaking the following interactions are relevant for the neutrino mass generation\ \cite{Magg:1980ut}:
\begin{align}
    \mathcal{L}\supset - \frac{1}{\sqrt{2}}Y_1 \ell \ell \Delta_1 - \mu hh\Delta_1 - M_{\Delta_1} \Delta_1^\dagger \Delta_1,
\end{align}
where $Y_1$ stems from the GUT operator defined in Eq.~\eqref{eq:5F5F}, $\mu$ is a trilinear scalar coupling   coming from the GUT operator $\bm{5_H}\bm{5_H}\bm{15_H}$, and $M_{\Delta_1}$ describes the mass of the scalar field $\Delta_1$. Integrating out $\Delta_1$ below its mass $M_{\Delta_1}$ gives the effective dimension five neutrino mass operator $\kappa$
\begin{align}
    \kappa=-2\frac{\mu Y_1 }{M_{\Delta_1}^2},
\end{align}
from which the neutrino mass matrix $m_\nu$ is obtained by
\begin{align}
    m_\nu=-\frac{v^2}{4}\kappa,
\end{align}
after the electroweak breaking Higgs doublet $h$ takes its vacuum expectation value $v=246\,$GeV\ \cite{ParticleDataGroup:2020ssz}.

\subsubsection{Viable GUT scale Yukawa ratios}\label{sec-viable GUT scale Yukawa ratios}
From now on we assume that (i) the Yukawa matrices $Y_{\bm{\overline{5}}}$ and $Y_{\bm{10}}$ are hierarchical and (ii) the concept of \emph{single operator dominance}  in the Yukawa sector. The latter assumption means that each Yukawa entry is dominated by a singlet GUT operator. With these two assumptions the charged lepton and down-type quark masses stem dominantly from the GUT operators $\mathcal{O}_2$, and $\mathcal{O}_3$ for the second, and third family, respectively, if the GUT scale Yukawa matrix $Y_{\bm{\overline{5}}}$ is given by
\begin{align}
Y_{\bm{\overline{5}}}=
\begin{pmatrix}
0 & 0 & 0 \\
0 & \mathcal{O}_2 & 0 \\
0 & 0 & \mathcal{O}_3
\end{pmatrix}
+\dots \;,
\end{align}
where the dots represent subdominant contributions. On the one hand, if the operators $\mathcal{O}_2$, and $\mathcal{O}_3$ are chosen as (specifying the contraction between different brackets by the SU(5) representation in the index position)
\begin{align}
&\mathcal{O}_2 = \left({\bm{10_F}}_2 \bm{45_H}^*\right)_{\bm{5}}\left({\bm{5_F}}_2 \bm{24_H}\right)_{\bm{\overline{5}}},    \\
&\mathcal{O}_3 = \left({\bm{10_F}}_3 \bm{5_H}^*\right)_{\bm{45}}\left({\bm{5_F}}_3 \bm{24_H}\right)_{\bm{\overline{45}}},
\end{align}
then the GUT scale Yukawa ratios $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$ are predicted. On the other hand, a second interesting choice of the operators $\mathcal{O}_2$ and $\mathcal{O}_3$ giving rise to the GUT scale Yukawa ratios $\frac{y_\tau}{y_b}=2$ and  $\frac{y_\mu}{y_s}=6$ is given by
\begin{align}
&\mathcal{O}_2 = \left({\bm{10_F}}_2 \bm{45_H}^*\right)_{\bm{5}}\left({\bm{5_F}}_2 \bm{24_H}\right)_{\bm{\overline{5}}},    \\
&\mathcal{O}_3 = \left({\bm{10_F}}_3 \bm{45_H}^*\right)_{\bm{5}}\left(\bm{24_H}\right)_{\bm{\overline{5}}\otimes\bm{5}}\left({\bm{5_F}}_3 \bm{1_H}\right)_{\bm{\overline{5}}}.
\end{align}
In the following we will show that the two possibilities for GUT scale charged lepton and down-type quark Yukawa ratios presented above are indeed viable if the particle content introduced in Section~\ref{sec-particle content} is assumed. For this analysis we investigate the RGE evolution of the SM parameters together with the quasi-Yukawa couplings $Y_1$, $Y_3$ and $Y_6$. At 1-loop the RGEs of the charged lepton and down-type quark Yukawa couplings read (cf. Appendix~\ref{app-rge gauge and yukawa})
\begin{align}
&16\pi^2 \mu \frac{dY_d}{d\mu}=\beta_d^{SM} + \left(\mathcal{H}(\mu,m_{\Delta_3})Y_3Y^\dagger_3\right)Y_d + \Big(\mathcal{H}(\mu,m_{\Delta_6})2Y_6Y^\ast_6 \Big)Y_d,
\\
&16\pi^2 \mu \frac{dY_e}{d\mu}=\beta^{SM}_e + Y_e \left( \frac{3}{2}\mathcal{H}(\mu,m_{\Delta_1})Y^\ast_1Y_1  \right) + Y_e \left( \frac{3}{2}\mathcal{H}(\mu,m_{\Delta_3})Y^\dagger_3Y_3  \right),
\end{align}
where $\beta_d^{SM}$ and $\beta_e^{SM}$ are the SM beta functions, and where $\mathcal{H}(\mu,m)$ is the Heaviside step function defined by
\begin{align}\label{eq-Heaviside step function}
    \mathcal{H}(\mu,m)=\left\{
\begin{array}{lll}
1, && \mu>m \\
0, && \, \mu\leq m \\
\end{array}.
\right.
\end{align}
Now, considering a fixed GUT scale and comparing the running in the SM with the running in which also the effects of $Y_1$, $Y_3$ and $Y_6$ are taken into account, we make the following observations for the GUT scale Yukawa charged lepton and down-type quark Yukawa ratios: If $m_{\Delta_6}$ is well below the GUT scale, while $m_{\Delta_1}$ is around the GUT scale, smaller GUT scale charged lepton and down-type quark Yukawa ratios become viable due to the term $\left(2Y_6Y^\ast_6 \right)Y_d$ ``speeding up'' the running of $Y_d$. Similarly, if $m_{\Delta_1}$ is well below the GUT scale, while $m_{\Delta_6}$ is around the GUT scale, larger GUT scale charged lepton and down-type quark Yukawa ratios become viable due to the term $Y_e \left( \frac{3}{2}Y^\ast_1Y_1  \right)$ ``speeding up'' the running of $Y_e$.

In order to investigate the viability of the GUT scale Yukawa ratios $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, respectively $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$, we considered the bottom-up RG evolution performing a Markov chain Monte Carlo (MCMC) analysis for which we varied the SM parameters around their experimental central value and weighted the points according to the $\chi^2$-function, while also varying the masses $m_{\Delta_1}$, $m_{\Delta_3}$ and $m_{\Delta_6}$. As argued in Appendix~\ref{sec-perturbativity of Ydelta above the GUT scale}, we stopped the integration if the largest singular value of $Y_1$, $Y_3$ and $Y_6$ got larger than 1.9. We then computed on a narrow grid the 1-$\sigma$ HPD intervals of the two ratios $\frac{y_\tau}{y_b}$ and $\frac{y_\mu}{y_s}$. Figure~\ref{fig-yukawa ratio running} shows that for GUT-scales above $10^{15.5}$\,GeV within the 1-$\sigma$ uncertainties one of the two possibilities of GUT-scale Yukawa ratios $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, or $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$ can be realized.

\begin{figure}
    \centering
    \includegraphics[width=10cm]{figures/YukawaRatioRunningTypeII.pdf}
    \caption{Estimated RG evolution of the 1-$\sigma$ ranges of the Yukawa ratios $y_\tau/y_b$ (blue) and $y_\mu/y_s$ (red) taking into account the effect of $Y_1$, $Y_3$ and $Y_6$. The horizontal grid lines indicate the values $\frac{3}{2}$, 2, $\frac{9}{2}$ and 6.}\label{fig-yukawa ratio running}
\end{figure}

\subsection{Toy models}\label{sec-toy models}
After having argued that there are two viable possibilities for GUT scale charged lepton and down-type quark Yukawa ratios for the second and third generation, namely (i) $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, and (ii) $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$, we now extend the two scenarios to ``toy models'' which also include the first family. Assuming the concept of \emph{single operator dominance}, Table~\ref{tab-dominant operators} shows which operator is dominant in which entry of the Yukawa matrices $ Y_{\bm{\overline{5}}}$, $ Y_{\bm{10}}$, and $ Y_{\Delta}$ which were defined in Section~\ref{sec-possible Yukawa couplings}.

\begin{table}[h!]
    \centering
    \renewcommand{\arraystretch}{1.2}
    \begin{tabu}{ccc}
    \tabucline[1.1pt]{-}
       operator  & model 1 & model 2 \\
       \hline
       $(Y_\Delta)_{ij}$ & ${\bm{\overline{5}_F}}_i{\bm{\overline{5}_F}}_j\bm{15_H}$ & ${\bm{\overline{5}_F}}_i{\bm{\overline{5}_F}}_j\bm{15_H}$ \\
       $(Y_{\bm{10}})_{ij}$ & ${\bm{10_F}}_i{\bm{10_F}}_j{\bm{5_H}}$ & ${\bm{10_F}}_i{\bm{10_F}}_j{\bm{5_H}}$ \\
       $(Y_{\bm{\overline{5}}})_{11}$ & $({\boldsymbol{10_F}}_1\boldsymbol{5_H}^*)_{\boldsymbol{5}}(\boldsymbol{24_H})_{\boldsymbol{24}}(\boldsymbol{1_H})(\boldsymbol{24_H})_{\boldsymbol{24}}({\boldsymbol{\overline{5}_F}}_1\boldsymbol{1_H})_{\bm{\overline{5}}}$ & $({\boldsymbol{10_F}}_{1} \boldsymbol{45_H}^*)_{\boldsymbol{\overline{45}}}({\boldsymbol{\overline{5}_F}}_1 \boldsymbol{24_H})_{\boldsymbol{45}}$ \\
       $(Y_{\bm{\overline{5}}})_{22}$  & $({\bm{10_F}}_2{\bm{45_H}^*})_{\bm{5}}({\bm{\overline{5}_F}}_2\bm{24_H})_{\bm{\overline{5}}}$ & $({\bm{10_F}}_2{\bm{24_H}})_{\bm{10}}({\bm{\overline{5}_F}}_2\bm{5_H})_{\bm{\overline{10}}}$  \\
       $(Y_{\bm{\overline{5}}})_{33}$  & $({\bm{10_F}}_3{\bm{5_H}^*})_{\bm{45}}({\bm{\overline{5}_F}}_3\bm{24_H})_{\bm{\overline{45}}}$ & $({\bm{10_F}}_3{\bm{45_H}^*})_{\bm{5}} (\bm{24_H})_{\bm{24}} ({\bm{\overline{5}_F}}_3\bm{1_H})_{\bm{\overline{5}}}$  \\
       \tabucline[1.1pt]{-}
    \end{tabu}
    \caption{Dominant operators in the respective Yukawa entries in the two toy models.}
    \label{tab-dominant operators}
\end{table}
From the renormalizable operator ${\bm{\overline{5}_F}}{\bm{\overline{5}_F}}\bm{15_H}$ we get the GUT scale relation
\begin{align}
    Y_1=Y_3=Y_6.
\end{align}
Moreover, since $Y_{\bm{10}}$ is dominated by the operator ${\bm{10_F}}{\bm{10_F}}{\bm{5_H}}$, the up-type Yukawa matrix is symmetric, i.e. $Y_u=Y_u^T$. We assume that the operator ${\bm{10_F}}{\bm{10_F}}{\bm{45_H}}$, which would yield anti-symmetric contributions, vanishes as a consequence of an appropriate family symmetry (see e.g.\ \cite{Antusch:2021yqe} for an example of such a symmetry).
Furthermore, at the GUT scale  the Yukawa matrices $Y_e$ and $Y_d$ are in the two ``toy models'' related via\ \cite{Antusch:2009gu, Antusch:2013rxa}
\begin{align}
&\text{model 1:}\qquad Y_e=\text{diag}\left(
\frac{4}{9},\frac{9}{2},\frac{3}{2}\right)\cdot Y_d^T, \label{eq-GUT scale Yukawa ratio 1}\\
&\text{model 2:}\qquad Y_e=\text{diag}\left(
\frac{1}{2},6,2\right)\cdot Y_d^T.\label{eq-GUT scale Yukawa ratio 2}
\end{align}

\section{Numerical analysis}\label{sec-numerical analysis}
\subsection{Implementation}\label{sec-implementation}
The models 1 and 2 defined in Section~\ref{sec-toy models} are implemented at the GUT scale. Since they only differ in their corresponding charged lepton sector, all other sectors can be implemented identically in both of the two ``toy models''. The symmetric Yukawa matrices $Y_u$ and $Y_\Delta$ can be diagonalized by a Takagi decomposition:
\begin{align}
    Y_u &= U_{u}^\dagger Y_u^\text{diag} U_{u}^*, \\
    Y_\Delta &= U_{\Delta} Y_\Delta^\text{diag} U_{\Delta}^T,
\end{align}
where $U_u$ and $U_\Delta$ are unitary matrices and $Y_u^{\text{diag}}$ as well as $Y_\Delta^{\text{diag}}$ are defined as
\begin{align}
    Y_u^\text{diag} &= \text{diag}(y_1^u,y_2^u,y_3^u), \\
    Y_\Delta^\text{diag} &= \text{diag}(y_1^\Delta,y_2^\Delta,y_3^\Delta),
\end{align}
with $y_i^u$ and $y_i^\Delta$ being real entries larger or equal to zero for $i=1,2,3.$ The unitary matrices $U_g$ ($g=u,\Delta$) can be parametrized by the physical parameters $\theta_{12}^g$, $\theta_{13}^g$, $\theta_{23}^g$, $\delta^g$, $\beta_1^g$, and $\beta_2^g$ as well as the unphysical parameters $\alpha_1^g$, $\alpha_2^g$, and $\alpha_3^g$. The latter will be set to zero for the analysis. The matrices $U_g$ then read
\begin{align}\label{eq-UDelta}
    U_g =
    \begin{pmatrix}
    e^{i\alpha_1^g}&0&0\\
    0&e^{i\alpha_2^g}&0\\
    0&0&e^{i\alpha_3^g}
    \end{pmatrix}
    \begin{pmatrix}
    1 & 0 & 0 \\
    0 & c_{23}^g & s_{23}^g \\
    0 & -s_{23}^g & c_{23}^g
    \end{pmatrix}
    \begin{pmatrix}
    c_{13}^g & 0 & s_{13}^ge^{-i \delta^g}  \\
    0 & 1 & 0 \\
    -s_{13}^ge^{i \delta^g} & 0 & c_{13}^g
    \end{pmatrix}
    \begin{pmatrix}
    c_{12}^g & s_{12}^g & 0 \\
    -s_{12}^g & c_{12}^g & 0 \\
    0 & 0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    e^{i\beta_1^g}&0&0\\
    0&e^{i\beta_2^g}&0\\
    0&0&1
    \end{pmatrix},
\end{align}
where $s_{ij}^g=\sin(\theta_{ij}^g)$, $c_{ij}^g=\cos(\theta_{ij}^g)$, $i,j=1,2,3$. Then the matrices $Y_1$, $Y_3$ and $Y_6$ are given by the relation
\begin{align}
    Y_1 = Y_3 = Y_6 = Y_\Delta.
\end{align}
The down-type Yukawa matrix $Y_d$ is implemented as a diagonal matrix with real positive entries
\begin{align}
    Y_d=\text{diag}(y_1^d,y_2^d,y_3^d).
\end{align}
The charged lepton Yukawa matrix is for the two models then given by Eq.~\eqref{eq-GUT scale Yukawa ratio 1} and Eq.~\eqref{eq-GUT scale Yukawa ratio 2}, respectively, and implemented accordingly.
\subsection{Parameters and observables}\label{sec-input parameters}
After having discussed the implementation of the Yukawa matrices, we now give the GUT scale input parameters. In total, there are 36 parameters, some of which can be ignored for some parts of the numerical analysis (cf. Section~\ref{sec-fit to experimental data}). These parameters decompose into the GUT scale $M_{\text{GUT}}$, the unified gauge coupling $g_{\text{GUT}}$, the intermediate-scale masses $m_{\Delta_1},$ $m_{\Delta_3},$ $m_{\Delta_6},$ $m_{\phi_1},$ $m_{\phi_2},$ $
m_{\phi_3},$ $m_{\phi_4},$ $m_{\phi_5},$ $m_{t_1},$ $m_{t_2},$ $m_{\Sigma_1},$ $m_{\Sigma_8}$, the scalar coupling constant $\mu$, the singular values $y_1^u,$ $y_2^u,$ $y_3^u,$ $y_1^d,$ $y_2^d,$ $y_3^d,$ $y_1^\Delta,$ $y_2^\Delta,$ $y_3^\Delta$ as well as the angles $\theta_{12}^u,$ $\theta_{13}^u,$ $\theta_{23}^u,$ $\theta_{12}^\Delta,$ $\theta_{13}^\Delta,$ $\theta_{23}^\Delta$ and phases $\delta^u,$ $\beta_1^u,$  $\beta_2^u,$ $\delta^\Delta,$ $\beta_1^\Delta,$ $\beta_2^\Delta$ of the Yukawa matrices. The following list summarizes all the input parameters as well as their allowed ranges:\footnote{The upper bound for the parameters $y_i^\Delta$, where $i=1,2,3$, is discussed in Appendix~\ref{sec-perturbativity of Ydelta above the GUT scale}.}\footnote{The cubic coupling $\mu$ cannot be arbitrarily large. Large values typically drives the potential to a deeper charge-breaking (or color-breaking, depending on the theory) minimum \cite{Frere:1983ag,Alvarez-Gaume:1983drc,Gunion:1987qv,Barroso:2005hc}. This is why we adopt a conservative bound of  $\mu<3m_{\Delta_1}$ for our numerical study.}
\begin{align}\label{eq-input parameters}
M_\text{GUT}&< M_{Pl},  \nonumber\\ m_{\Delta_1},m_{\Delta_3},m_{\Delta_6},m_{\phi_1},m_{\phi_2},
m_{\phi_3},m_{\phi_4},m_{\phi_5},m_{\Sigma_1},m_{\Sigma_8} &\in [500\,\text{GeV},M_\text{GUT}],  \nonumber\\
m_{t_1},m_{t_2}&\in [10^{11}\,\text{GeV},M_{\text{GUT}}], \nonumber\\
\mu&<3m_{\Delta_1}, \nonumber \\
g_{\text{GUT}},\,y_1^u,\,y_2^u,\,y_3^u,\,y_1^d,\,y_2^d,\,y_3^d&\in[0,1],   \\
y_1^\Delta,\,y_2^\Delta,\,y_3^\Delta &\in[0,1.9],   \nonumber\\
\theta_{12}^u,\,\theta_{13}^u,\,\theta_{23}^u,\,\theta_{12}^\Delta,\,\theta_{13}^\Delta,\,\theta_{23}^\Delta &\in[0,\pi/2],   \nonumber\\
\delta^u,\,\beta_1^u,\,\beta_2^u,\,\delta^\Delta,\,\beta_1^\Delta,\,\beta_2^\Delta &\in[-\pi,\pi). \nonumber
\end{align}
The two models are fitted to the following 22 low energy observables (as well as to the nucleon decay rates for 13 different decay channels, see Table~\ref{tab-nucleon decay channels experimental bounds}):
\begin{align}
&g_1,\,g_2,\,g_3\nonumber\\
&y_u,\,y_c,\,y_t,\,y_d,\,y_s,\,y_b,\,\theta_{12}^{\text{CKM}},\,\theta_{13}^{\text{CKM}},\,\theta_{23}^{\text{CKM}},\,\delta^{\text{CKM}},\,y_e,\,y_\mu,\,y_\tau,\,\hspace{0.6cm}\\
&\Delta m_{21}^2,\,\Delta m_{31}^2,\,\theta_{12}^{\text{PMNS}},\,\theta_{13}^{\text{PMNS}},\,\theta_{23}^{\text{PMNS}},\,\delta^{\text{PMNS}}.\nonumber
\end{align}
The GUT scale $M_\text{GUT}$, the unified gauge coupling $g_\text{GUT}$ as well as the intermediate-scale masses are used to fit the SM gauge couplings $g_1$, $g_2$ and $g_3$. Moreover, since the quasi-Yukawa couplings $Y_1$, $Y_3$ and $Y_6$ effect the RG evolution of the SM Yukawa couplings $Y_e$ and $Y_d$, the masses $m_{\Delta_1}$, $m_{\Delta_3}$ and $m_{\Delta_6}$ are further used to fit the singular values of the charged-lepton Yukawa matrix $y_e$, $y_\mu$ and $y_\tau$. The singular values $y_i^u$ and $y_i^d$ ($i=1,2,3$) are used to fit the up- and down-type Yukawa couplings, while the angles $\theta_{12}^u$, $\theta_{13}^u$, $\theta_{23}^u$, and the phase $\delta^u$ are used to fit the CKM parameters. The singular values $y_i^\Delta$ ($i=1,2,3$) together with the trilinear coupling $\mu$ as well as the angles $\theta_{12}^\Delta$, $\theta_{13}^\Delta$, $\theta_{23}^\Delta$, $\delta^\Delta$ are used to fit the neutrino sector. Furthermore, the phases $\beta_1^\Delta$ and $\beta_2^\Delta$ are related to the Majorana phases in the PMNS matrix. Finally, the phases $\beta_1^u$ and $\beta_2^u$ are the so-called GUT phases\ \cite{Ellis:1979hy, Ellis:2019fwf}. They have an impact on the nucleon decay rates, but do not change the predictions for the SM parameters.

\begin{table}[ht!]
\centering
\begin{tabu}{llll}\tabucline[1.1pt]{-} \noalign{\vskip 2mm}
& decay channel  &  $\tau/\mathcal{B}$ [year] & $\Gamma_{\text{partial}}$ [GeV]
\\\noalign{\vskip 1mm}\hline\noalign{\vskip 2mm}
Proton: & $p\rightarrow \pi^0\,e^+$  &  $>\,2.4\cdot 10^{34}$  &  $>\,8.7\cdot 10^{-67}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow \pi^0\,\mu^+$  &  $>\,1.6\cdot 10^{34}$  &  $>\,1.3\cdot 10^{-66}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow \eta^0\,e^+$  &  $>\,4.1\cdot 10^{33}$  &  $>\,5.1\cdot 10^{-66}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow \eta^0\,\mu^+$  &  $>\,1.2\cdot 10^{33}$  &  $>\,1.7\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow K^0\,e^+$  &  $>\,1.1\cdot 10^{33}$  &  $>\,1.9\cdot 10^{-65}$  \\ \noalign{\vskip 2mm}
&  $p\rightarrow K^0\,\mu^+$  &  $>\,1.6\cdot 10^{34}$  &  $>\,1.3\cdot 10^{-66}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow \pi^+\,\overline{\nu}$  &  $>\,2.8\cdot 10^{32}$  &  $>\,7.4\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $p\rightarrow K^+\,\overline{\nu}$  &  $>\,6.6\cdot 10^{33}$  &  $>\,3.2\cdot 10^{-66}$
\\\noalign{\vskip 1mm}\hline\noalign{\vskip 2mm}
Neutron:  &  $n\rightarrow \pi^-\,e^+$  &  $>\,2.1\cdot 10^{33}$  &  $>\,1.0\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $n\rightarrow \pi^-\,\mu^+$  &  $>\,9.9\cdot 10^{32}$  &  $>\,2.1\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $n\rightarrow \pi^0\,\overline{\nu}$  &  $>\,9.9\cdot 10^{32}$  &  $>\,2.1\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $n\rightarrow \eta^0\,\overline{\nu}$  &  $>\,5.6\cdot 10^{32}$  &  $>\,3.7\cdot 10^{-65}$  \\\noalign{\vskip 2mm}
&  $n\rightarrow K^0\,\overline{\nu}$  &  $>\,1.2\cdot 10^{32}$  &  $>\,1.7\cdot 10^{-64}$  \\\noalign{\vskip 1mm}
\tabucline[1.1pt]{-}
\end{tabu}
\caption{Current experimental bounds \cite{Dev:2022jbf} on the decay widths $\Gamma_{\text{partial}}$ (respectively lifetime bound $\tau/\mathcal{B} $, where $\mathcal{B}$ is the branching ratio for the decay channel) for various nucleon decay channels at 90 \% confidence level. The data stems from Figure 5-3 of \cite{Brock:2012ogj} and was updated with the findings from \cite{Mine:2016mxy, Takenaka:2020vqy, Bajc:2016qcc, Miura:2016krn, Abe:2014mwa, 1205.6538}.}\label{tab-nucleon decay channels experimental bounds}
\end{table}

\subsection{Fit to experimental data}\label{sec-fit to experimental data}
As described in Section~\ref{sec-implementation} we implement the input parameters given in Eq.~\eqref{eq-input parameters} at the GUT scale $M_\text{GUT}$ and compute the running to the $Z$ scale $M_Z$. The running of the SM gauge couplings is computed at 2-loop, while the running of the (quasi-) Yukawa matrices as well as the effective neutrino mass operators is accounted for at 1-loop. Then, at $M_Z$ the singular values of the SM Yukawa matrices as well as the CKM and PMNS parameters and the neutrino squared mass differences are determined. Furthermore, as described in \cite{Antusch:2021yqe}, the nucleon decay widths for the 13 different decay channels listed in Table~\ref{tab-nucleon decay channels experimental bounds} are computed, using the Mathematica package \texttt{ProtonDecay}\ \cite{Antusch:2020ztu} for the final step of this computation.

At the low scale $M_Z$ we define the total $\chi^2$-function by the sum of the pulls $\chi_i^2$ for each observable $i$
\begin{align}
\chi^2(\vec{x})=\sum_i \chi_i^2(\vec{x}),
\end{align}
where $\vec{x}$ represents a vector which consists of the input parameters listed in Eq.~\eqref{eq-input parameters}. While we use the exact pull $\chi_i^2$ given by NuFIT~5.1~\cite{Esteban:2020cvm} for the observables $\theta_{23}^\text{PMNS}$ and $\delta^\text{PMNS}$, we compute the pull $\chi_i^2$ for all other observables as
\begin{equation}
\chi_i^2(\vec{x})=\left(\frac{f_i(\vec{x})-y_i}{\sigma_{i}}\right)^2,
\end{equation}
where $f_i(\vec{x})$ is our prediction for the observable $i$, while $y_i$ represents the experimental central value and $\sigma_i$ its corresponding standard deviation.\footnote{Note that we take the experimental central value and standard deviations of the SM gauge and Yukawa couplings as well as CKM observables at $M_Z$ from\ \cite{Antusch:2013jca} while the corresponding data for the PMNS observables and neutrino mass squared differences are taken from\ \cite{Esteban:2020cvm}.} Using a differential evolution algorithm the $\chi^2$-function is minimized delivering a benchmark point. Afterwards, starting from that benchmark point, an MCMC analysis is performed using an adaptive Metropolis-Hastings algorithm\ \cite{Metropolis-Hastings-algorithm}. For both models 16 independent chains with $2.5\times 10^5$ data points are computed with a flat prior probability distribution. Using these data points the posterior densities of various observables are calculated.

To simplify the analysis we do not vary all the input parameters listed in Eq.\ \eqref{eq-input parameters} for the minimization procedure. Keeping the masses $m_{\phi_2},$ $m_{\phi_4},$ $m_{\phi_5},$ $m_{t_1},$ $m_{t_2},$ $m_{\Sigma_1}$ and $m_{\Sigma_8}$ at the GUT scale still leaves enough freedom to fit the gauge couplings. Moreover, the correct neutrino mass squared differences can still be obtained if $y_1^\Delta$ is set to zero, which then implies a strong normal hierarchy of the light neutrino masses. Furthermore, since the CKM parameters are almost constant during the RG evolution, the parameters $\theta_{12}^u$, $\theta_{13}^u$, $\theta_{23}^u$, $\delta^u$ are not varied (and the observables $\theta_{12}^\text{CKM}$, $\theta_{13}^\text{CKM}$, $\theta_{23}^\text{CKM}$, $\delta^\text{CKM}$ are not considered in the $\chi^2$-function). Since the GUT phases $\beta_1^u$ and $\beta_2^u$ do not change the predictions for the SM parameters and usually only have a small effect on the nucleon decay predictions in non-SUSY models (see e.g.~\cite{Antusch:2021yqe}) they can be set to zero. Finally, the Majorana phases are also set to zero since they only have a small effect on the running of the SM parameters. In summary, for the fitting procedure the following 20 input parameters are varied
\begin{align}
M_\text{GUT},g_{\text{GUT}},m_{\Delta_1},m_{\Delta_3},m_{\Delta_6},m_{\phi_1},m_{\phi_3},\mu,  \nonumber\\
y_1^u,\,y_2^u,\,y_3^u,\,y_1^d,\,y_2^d,\,y_3^d,y_2^\Delta,\,y_3^\Delta,\theta_{12}^\Delta,\,\theta_{13}^\Delta,\,\theta_{23}^\Delta,\delta^\Delta,
\end{align}
giving a benchmark point. From this benchmark point the MCMC analysis --- for which, on the other hand, all 36 input parameters are varied --- is started.

\section{Results}\label{sec-results}
In this section we present our numerical results for the minimization of the $\chi^2$-function as well as for the MCMC analysis. We are interested in the predictions for the low-scale charged lepton and down-type quark Yukawa ratios which give a direct hint on the viability of the GUT scale Yukawa ratios. Furthermore, we give the predictions of the highest posterior densities (HPD) for the masses of the added scalar fields as well as the nucleon decay widths. All of these predictions may be used to test our models by upcoming experiments.

\subsection{Benchmark points}\label{sec-benchmark points}
Minimizing the $\chi^2$-function under the constraints presented in Section~\ref{sec-fit to experimental data} gives for both models a benchmark point. The GUT scale input parameters for the benchmark points of both models are listed in Table~\ref{tab-input parameters benchmark points}, while Table~\ref{tab-chi squared benchmark points} shows the respective total $\chi^2$ as well as the dominant pulls $\chi_i^2$. With a total $\chi^2$ of 1.86 model 2 gives a better fit than model 1 ($\chi^2=4.35$). In both models the dominant pull comes from the Yukawa coupling of the strange quark with $\chi_{y_s}^2=1.55\,(2.75)$ in model 2 (1).

Regarding the GUT scale input parameters, the main difference between the two models are the choices of the masses of the intermediate-scale scalar fields. The reason for this is that --- as discussed in Section~\ref{sec-viable GUT scale Yukawa ratios} --- if $m_{\Delta_1}$ is well below the GUT scale, while $m_{\Delta_6}$ is close to the GUT scale the running of $Y_e$ is ``speeded up'' yielding larger charged lepton and down-type quark GUT scale ratios, while conversely, if $m_{\Delta_6}$ is well below the GUT scale, while $m_{\Delta_1}$ is close to the GUT scale the running of $Y_d$ is ``speeded up'' resulting in smaller charged lepton and down-type quark GUT scale ratios. This fixes the masses $m_{\Delta_1}$, $m_{\Delta_3}$ and $m_{\Delta_6}$ in the two models. The remaining masses have then to be chosen such that the gauge couplings can unify which is the reason why they differ in the two models.

Moreover, an interesting difference between the two models is the chosen GUT scale value for $\theta_{23}^\Delta$ which is, apart from renormalization group effects, equal to the observable $\theta_{23}^\text{PMNS}$. The current experimental 1-$\sigma$ range of this observable consists of two disjoint intervals. The benchmark point of model 1 predicts $\theta_{23}^\text{PMNS}$ to lie in the lower interval, while the benchmark point of model 2 suggests that $\theta_{23}^\text{PMNS}$ should lie in the upper interval. Therefore, a more precise measurement of $\theta_{23}^\text{PMNS}$ could distinguish between the two models (cf.\ Section\ \ref{sec-MCMC}).

\begin{table}[ht]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabu}{|c|ccc|ccc|}
\tabucline[1.1pt]{-}
&&Model 1&&&Model 2&\\
\hline
$g_\text{GUT}\;/\;10^{-1}$ & & 6.12 & & & 6.39 & \\
$\log_{10}(M_{\text{GUT}}\;/\;\text{GeV})$ & & 15.7 & & & 15.8 &\\
$\log_{10}(m_{\Delta_1}\;/\;\text{GeV})$ & & 15.1 & & & 9.52 &\\
$\log_{10}(m_{\Delta_3}\;/\;\text{GeV})$ & & 10.4 & & & 15.0 &\\
$\log_{10}(m_{\Delta_6}\;/\;\text{GeV})$ & & 5.79 & & & 13.1 &\\
$\log_{10}(m_{\phi_1}\;/\;\text{GeV})$ & & 9.97 & & & 2.70 &\\
$\log_{10}(m_{\phi_3}\;/\;\text{GeV})$ & & 6.72 & & & 9.56 &\\
$\log_{10}(\mu\;/\;\text{GeV})$ & & 15.6 & & & 4.34 &\\
$y_1^u\;/\;10^{-6}$ & & 2.69 & & & 2.24 &\\
$y_2^u\;/\;10^{-3}$ & & 1.31 & & & 1.10 &\\
$y_3^u\;/\;10^{-1}$ & & 4.05 & & & 3.35 &\\
$y_1^d\;/\;10^{-6}$ & & 6.03 & & & 5.11 & \\
$y_2^d\;/\;10^{-4}$ & & 1.39 & & & 0.96 & \\
$y_3^d\;/\;10^{-3}$ & & 7.48 & & & 4.82 & \\
$y_2^\Delta\;/\;10^{-1}$ & & 1.92 & & & 1.28 & \\
$y_3^\Delta\;/\;10^{-1}$ & & 19.0 & & & 9.46 & \\
$\theta^\Delta_{12}\;/\;10^{-1}$ & & 5.81 & & & 5.82 & \\
$\theta^\Delta_{13}\;/\;10^{-1}$ & & 1.15 & & & 1.33 & \\
$\theta^\Delta_{23}\;/\;10^{-1}$ & & 6.00 & & & 8.00 & \\
$\delta^\Delta$ & & 3.39 & & & 3.39 &\\
\tabucline[1.1pt]{-}
\end{tabu}
\caption{The GUT scale input parameters of the benchmark points for both models.}\label{tab-input parameters benchmark points}
\end{table}

\begin{table}[ht]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabu}{|c|ccccccc|}
\tabucline[1.1pt]{-}
& $\chi^2$ & $\chi^2_{y_s}$ & $\chi^2_{y_b}$ & $\chi^2_{y_\mu}$ & $\chi^2_{y_\tau}$ & $\chi^2_{\theta_{23}^\text{PMNS}}$ & $\chi^2_{\Gamma({p\rightarrow \pi^0e^+})}$ \\
\hline
model 1 & 4.35 & 2.75 & 0.23 & 0.09 & 0.23 & 0.72 & 0.21 \\
model 2 & 1.86 & 1.55 & 0.09 & 0.05 & 0.09 & 0.00 & 0.03 \\
\tabucline[1.1pt]{-}
\end{tabu}
\caption{The total $\chi^2$ as well as the dominant pulls $\chi_i^2$ for the benchmark points of both models.}\label{tab-chi squared benchmark points}
\end{table}

\subsection{MCMC analysis}\label{sec-MCMC}
We obtain the highest posterior densities of different quantities using an MCMC analysis varying the input parameters around the benchmark points. For this analysis all the input parameters listed in Eq.~\eqref{eq-input parameters} are varied.

\begin{enumerate}
\item \textbf{Charged lepton and down-type quark Yukawa ratios}\newline
Figure~\ref{fig-HPD Yukawa} shows the 1-$\sigma$ (dark) and 2-$\sigma$ (light) HPD intervals of the charged lepton and down-type quark Yukawa ratios at the low scale $M_Z$ for model 1 (red) and model 2 (green). The experimental 1-$\sigma$ range is indicated by the white region, while the dashed line represents the current experimental central value. As it could already be seen from the benchmark points (cf.\ Section~\ref{sec-benchmark points}), the ratio $\frac{y_\mu}{y_s}$ is predicted to be above the experimental central value in both models. However, there is an intersection between the 1-$\sigma$ HPD interval and the experimentally allowed 1-$\sigma$ region. Future experimental measurements of the three ratios can further test the two models.

\begin{figure}[th]
    \centering
    \includegraphics[width=4.5cm]{figures/yukawa1TypeII.pdf}
    \includegraphics[width=4.5cm]{figures/yukawa2TypeII.pdf}
    \includegraphics[width=4.5cm]{figures/yukawa3TypeII.pdf}
    \caption{The predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals of the charged lepton and down-type quark Yukawa ratios at $M_Z.$}
    \label{fig-HPD Yukawa}
\end{figure}

\item \textbf{PMNS mixing angle $\theta_{23}^\text{PMNS}$}\newline
As it was pointed out in Section\ \ref{sec-benchmark points} the benchmark points indicate distinct predictions for the observable $\theta_{23}^\text{PMNS}$. Figure\ \ref{fig-HPD theta23} shows that the predicted HPD intervals for model 1 lie within the lower experimentally allowed region, while the predicted HPD intervals of model 2 span over both allowed regions. Therefore, a measurement of $\theta_{23}^\text{PMNS}$ in the upper interval would favor model 2 over model 1.

\begin{figure}
    \centering
    \includegraphics[width=4.5cm]{figures/theta23PMNStpeII.pdf}
    \caption{The predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals at $M_Z$ of the PMNS mixing angle $\theta_{23}^\text{PMNS}$.}
    \label{fig-HPD theta23}
\end{figure}

\item \textbf{Intermediate-scale scalar masses}\newline
The prediction for the 1-$\sigma$ and 2-$\sigma$ HPD intervals of the masses of the added scalar fields are presented in Figure~\ref{fig-HPD masses}. The ranges for most of the masses are quite large. However, some of the fields could be observed by future experiments. Especially interesting is the predicted range of $m_{\phi_1}$ in model 2. The upper bound of the predicted 1-$\sigma$ and 2-$\sigma$ HPD interval is given by 13.5\ TeV and 207\ TeV, respectively. Moreover, a possible observation of one of the fields $\Delta_6$, $\phi_1$ or $\phi_3$ could distinguish between the two models.

\item \textbf{Nucleon decay widths}\newline
Figure~\ref{fig-HPD nucleon decay rates} visualizes the predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals for the nucleon decay widths of various decay channels (cf.\ Table\ \ref{tab-nucleon decay channels experimental bounds}). The horizontal blue lines indicate the current experimental bound at a 90\ \% confidence level. For the dominant decay channel,  $p\rightarrow \pi^0 e^+$, also the expected bounds of the planned DUNE\ \cite{Acciarri:2015uup} and Hyper-Kamiokande experiment\ \cite{Abe:2018uyc} are shown. Of particular interest will be the planned Hyper-Kamiokande experiment which will test the full 1-$\sigma$ region of this decay channel predicted by model 1.

Moreover, as Figure\ \ref{fig-HPD MGUT} shows, model 1 predicts a lower GUT scale than model 2.
Therefore, model 1 generally predicts larger decay widths than model 2. The 1-$\sigma$ HPD intervals do not intersect for any of the decay channels. This fact can be used to distinguish between the two models if nucleon decay is observed in at least one of the 13 different decay channels. However, as it was pointed out in\ \cite{Antusch:2021yqe} the Yukawa texture has a non-trivial effect on the nucleon decay predictions. This means  that if different Yukawa textures were considered (which still predict the GUT scale relations $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, respectively $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$) the two scenarios might no-longer be distinguished so easily.

\begin{figure}
    \centering
    \includegraphics[width=4.5cm]{figures/MGUTtypeII.pdf}
    \caption{Predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals of the GUT scale $M_\text{GUT}$ for both models.}
    \label{fig-HPD MGUT}
\end{figure}

\end{enumerate}

\begin{figure}[p]
    \centering
    \includegraphics[width=15cm]{figures/massesTypeII.pdf}
    \caption{The predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals of the intermediate-scale scalar masses.}
    \label{fig-HPD masses}
\vspace{1cm}
    \includegraphics[width=15cm]{figures/decayratesTypeII.pdf}
    \caption{The predicted 1-$\sigma$ and 2-$\sigma$ HPD intervals of the decay widths for various different nucleon decay channels. The current experimental bounds at 90\ \% confident level are indicated by blue lines (cf. Table\ \ref{tab-nucleon decay channels experimental bounds}). Moreover, the future sensitivity of DUNE\ \cite{Acciarri:2015uup} for the decay channel $p\rightarrow \pi^0e^+$ is represented by a cyan line, while the future sensitivity of Hyper-Kamiokande\ \cite{Abe:2018uyc} with one tank (two tanks) for this decay channel is indicated by a red (orange) line.}
    \label{fig-HPD nucleon decay rates}
\end{figure}

It is interesting to note that part of the parameter space, as depicted in Fig.~\ref{fig-HPD masses}, is already ruled out by existing collider searches.  In the following, we briefly summarize the current collider limits.

The color-octet scalar $\phi_1$ has both neutral $\phi_1^0$ and charged $\phi_1^+$ components. The charged partner provides a stronger bound since it cannot decay to gluons. These states can be efficiently produced via gluon-fusion $pp\to \phi_1^+\phi_1^-, \overline t b \phi_1^+$ at the LHC, which lead to $pp\to \overline t b t \overline b$. The analysis performed in Ref.~\cite{Miralles:2019uzg} shows that the current LHC data provides a lower bound of $m_{\phi_1^+}> 800$ GeV. The neutral components can also be pair-produced through gluon-fusion,  and for our scenario, this would lead to $pp\to \overline b b\overline b b$. LHC currently provides a lower bound~\cite{Hayreter:2017wra} on its mass of $m_{\phi_1^0}>530$ GeV, which is weaker than its charged partner.

Concerning the scalar leptoquark $\phi_3$, which couples predominantly to the second (third) generation quark-lepton pairs for model-1 (model-2), current LHC data limits its mass to be greater than $\gtrsim 1.7$ TeV~\cite{ATLAS:2020dsk} ($\gtrsim 1.4$ TeV~\cite{ATLAS:2019qpq,ATLAS:2021oiz}) from pair-produced leptoquarks decaying into $jj\mu\overline \mu$ (or $bb\tau\overline \tau$) final state.

For both models, the sextet diquark $\Delta_6$ that couples to right-handed down-type quarks $d_{Ri}d_{Rj}$,  has Yukawa coupling of the form $\sim U_\textrm{PMNS}\; \textrm{diag}\{0,\mathcal{O}(0.1),\mathcal{O}(1)\}\; U^T_\textrm{PMNS}$. Subsequently, the coupling to $d_Rd_R$ is somewhat small; however, due to $\mathcal{O}(0.1)$ couplings with $d_Rs_R$ and $s_Rs_R$, it can still be produced through the annihilation  of a pair of quarks via an $s$-channel resonance. By considering  equal diquark couplings of
electromagnetic strength $y_{12,13,23}\sim 0.3$, and using the CMS measurements of narrow dijet resonances at $\sqrt{13}$ TeV~\cite{CMS:2018mgb}, a lower
bound on its mass $\gtrsim 2.5$ TeV is obtained in Ref.~\cite{Pascual-Dias:2020hxo}. This bound is expected to get relaxed in our scenario due to additional non-zero diagonal couplings and the somewhat smaller Yukawa couplings involved in the production. On the other hand, a search for pair-produced resonances in four-jet final states finds diquark mass $\gtrsim 610$ GeV if the decay is into a $b$-quark and a light-quark~\cite{ATLAS:2017jnp}. To obtain the exact bound, however, a dedicated collider study would be required.

Future LHC upgrades have the potential to discover some of these colored particles that could be used to distinguish between these two models.

\section{Conclusions}\label{sec-conclusions}
In this work we considered a rather minimal $SU(5)$ GUT scenario which contains a 45-dimensional as well as a 15-dimensional Higgs representation on top of the Georgi-Glashow $SU(5)$ particle content and in which neutrino masses are generated by a type II seesaw mechanism. We showed that within the concept of \textit{single operator dominance} in the third and second family this scenario can lead to two different GUT scale predictions for charged lepton and down-type Yukawa ratios, namely (i) $\frac{y_\tau}{y_b}=\frac{3}{2}$ and $\frac{y_\mu}{y_s}=\frac{9}{2}$, respectively (ii) $\frac{y_\tau}{y_b}=2$ and $\frac{y_\mu}{y_s}=6$. We investigated the compatibility of both of these two combinations of GUT scale Yukawa ratios with the low-energy experimental data by computing the RG evolution and taking into account the contribution of new arising Yukawa interactions on the beta functions of the charged lepton and down-type quark Yukawa matrices (cf.\ Figure\ \ref{fig-yukawa ratio running}). The outcome of this analysis motivated the formulation of two ``toy models'' which also contain the first family. We investigated various predictions of these ``toy models'' in Section\ \ref{sec-results} and showed that there are several possibilities to distinguish between them in future experiments, namely the nucleon decay rates, the masses of the fields $\Delta_6$, $\phi_1$ or $\phi_3$, which could be discovered at colliders, as well as the prediction for the PMNS 2-3 mixing angle $\theta_{23}^\text{PMNS}$.

redpajama_set_name: RedPajamaArXiv


text:
Caerhun – St Mary's Church
plan of the church according to the Royal Commission on The Ancient and Historical Monuments
Church of St. Mary in Caerhun was probably built in the 13th century by the Cistercians from the Maenan abbey. It was erected within the earth ramparts of the Roman fort of Canovium. In the fifteenth century, the temple was rebuilt and enlarged. In 1591, the chapel was added to the building, founded by Edward Williams and his wife Grace, according to a prominent Latin inscription (Edward was the Sheriff of Caernarfon in 1570). In the 19th century and then in the second half of the 20th century, the church underwent renovations.
The church was built of local, unworked stone, but some of red sandstone blocks, possibly of Roman origin from the ruins of Fort Canovium were also used. The roof, on the other hand, was thatched, later replaced with slate. Originally, the building consisted only of a long nave without an externally separated chancel. It was illuminated by narrow, pointed windows, grouped in a triad in the eastern wall. The entrance was traditionally located in the western part of the southern wall, where a semicircular portal was created.
In the fifteenth century, the church was extended from the east by the presbytery part, keeping the same width of the building. The new eastern wall probably had older windows transferred from the dismantled eastern wall. Additionally, a small porch was built on the south side. In the 16th century, the western facade of the church was rebuilt, then equipped with a small bellcote set on a shallow projection supported by four corbels. At the end of the 16th century, a four-sided chapel was built on the southern side of the eastern part of the church, illuminated from the south and east with late Gothic windows with trefoils (one three-light and one two-light).
Inside, the nave and the presbytery were topped with a magnificent 14th or 15th-century wooden truss with arched collar beams. The secular part of the church was probably separated from the priestly section with the main altar by a wooden rood screen. On the side of the altar, a niche was created, most likely housing a statue of Mary until the Reformation.
Today, the medieval church has an early modern sacristy on the south – eastern side, built on the site of a medieval porch. Most of the windows were also transformed, except for the three eastern ones in the presbytery and the windows of the southern chapel. The southern portal is original, currently hidden inside the vestry. A stone baptismal font has survived from the oldest furnishings, but the most valuable element of the interior is the roof truss from the 14th or 15th century. In the early modern period, it was reinforced with short timber supports, probably when the thatch was replaced with a heavier slate roof.
north-west side of the church, photo: N.Kaye, Flickr
church from the south-west side, photo: N.Walley, Wikimedia Commons
church from the south, photo: N.Kaye, Flickr
interior of the nave, photo N.Kaye, Flickr
Salter M., The old parish churches of North Wales, Malvern 1993.
The Royal Commission on The Ancient and Historical Monuments and Constructions in Wales and Monmouthshire. An Inventory of the Ancient and Historical Monuments in Caernarvonshire, volume I: east, the Cantref of Arllechwedd and the Commote of Creuddyn, London 1956.

redpajama_set_name: RedPajamaCommonCrawl


How to Move Within the Town with Packers & Movers?
› Discussion › Discuss From Your Inner › How to Move Within the Town with Packers & Movers?
This topic contains 0 replies, has 1 voice, and was last updated by rohitsingh April 12, 2018 at 2:48 PM.
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text:
46 Hammersmith Broadway, London, Greater London, W6 0DZ 020 8748 1043
THANKS FOR RAISING A GLASS WITH US THIS ST PATRICK'S DAY!
We hope you had a craic'ing time celebrating St Patrick's Day Weekend with us at The Swan.
Pints and pints of perfectly poured Guinness were enjoyed by all, while we watched the drama and action unfold from the final round of the Six Nations. There was also exciting FA Cup quarter-final action, as Wolves hosted Manchester United and table toppers Leeds played against fellow promotion hopefuls Sheffield United in the Championship.
From great beer to delicious seasonal dishes plus all the action from the games that mattered, there really was something for everyone.
Sign up today and we'll keep you up-to-date on all our plans for St Patrick's Day 2020.
46 Hammersmith Broadway, London, Greater London, W6 0DZ


text:
Posts Tagged 'Elon in NYC'
Ethnography Study: Key Findings
Culture is defined as "the quality in a person or society that arises from a concern for what is regarded as excellent in arts, letters, manners, scholarly pursuits, etc" (dictionary.com). In this way, Madison Square Garden serves as a place for people of a similar culture to congregate in support of something. This is different than most areas in New York City.
Every neighborhood in NYC has its own culture, an aspect unique to the area, that is both defined by the people who live there and that defines those people. The Upper East Side is ritzy. Soho is trendy. Chelsea is gay-friendly. Greenwich Village is artsy. Madison Square Garden is distinct because its "culture" changes daily. Depending on the event taking place in the Graden and the people who choose to attend the event, the culture will change.
In studying Madison Square Garden, I was faced with many different types of information – both subjective and objective. Everyone who has an experience or a memory regarding Madison Square Garden adds to the ethnographic makeup of the arena. However, each of these is only a small aspect of the study. These are subjective elements because they all believe that their experience is the most important or valuable. It is my job, as the researcher, to understand how these subjective aspects relate to the ethnography as a whole.
These stories and memories do make up a huge part of the ethnography of Madison Square Garden. Because it is such a diverse space, it's important to know what types of experiences have been had in the arena. It is also important, however, to do objective research about the space. Why was it built above Penn Station? What types of entertainment is it designed to accommodate? How has technology changed MSG? These are objective questions that were answered either through research or by speaking with someone who works at Madison Square Garden.
The combination of information gained through both objective and subjective sources allowed me to get a full understanding of the ethnography of Madison Square Garden. Having one of these types of understanding without the other would leave the research only partially complete and lacking important elements.
Those who have a unique experience or memory create knowledge. The combination of all of these different elements together forms knowledge. Each memory is valuable and represents a different aspect of our understanding so, with each that is added, a greater knowledge is achieved. It is through the gathering of many different experiences with Madison Square Garden that I began to piece together my understanding of the arena. With each interview or bit of research that I did, my ethnographic knowledge was enhanced. Particularly with a location that is as diverse and variable as Madison Square Garden, each new bit of knowledge goes a long way in helping me understand the magnitude of Madison Square Garden.
As mentioned, MSG's culture varies immensely with each day. Because the Garden hosts everything from the Westminster Kennel Club Dog Show to a Lady Gaga concert to a New York Knicks basketball game, the culture is always changing. Because of this, it was my job as the ethnographer to understand each individual aspect of MSG. I could not simply learn about the concerts that are hosted or the basketball games that are played. In order to do the arena justice, I had to research all of the different elements that are present in MSG. Simple research would not cut it, either. It was my responsibility to experience MSG for myself. I had to understand the history of the arena and what it strives to achieve. I had to understand it as more than just an arena, but as a cultural and historical icon.
Tags:Elon in NYC, Elon University, ethnography, Madison Square Garden, New York City, research
Madison Square Garden: Informative Flyer
My "street" is Madison Square Garden. Though I cannot provide a guide through the actual arena, I can provide an informative flyer that could be available in the MSG concourse and online for viewers to see.
Click here to see the flyer of Madison Square Garden
Tags:Elon in NYC, Elon University, ethnography, flyer, Madison Square Garden, New York City
Nine weeks in a nutshell
In all honesty, I didn't think I would be able to handle this city. I thought it would take me the entire summer to learn the subway system. I thought I would never feel completely comfortable in the city. I thought I would be so ready to go back to North Carolina, back to the familiar and the easy-going lifestyle.
And yes, I am ready to go home. I'm ready to be able to drive my car and get some real sweet tea. But I am going to miss this place. I'm going to miss the hustle-and-bustle of the city and the constant excitement.
I'm still not ready to graduate. But my time in New York has made me both more confident and more excited to graduate. I know now that I can live on my own and take care of myself.
I've always been very close with my family and anticipated moving back to Raleigh after graduation. That's definitely still an option, but I know now that I am not limited. I took on New York City and came out (relatively) unscathed. If I can do that in NYC, I feel confident that I can live anywhere.
For me, this summer was so much more than just an internship and a class. It was a chance for me to push myself and test my boundaries. And while I'm still not keen on the idea of growing up and being a real person, I am excited to know that I can and will survive.
Tags:Elon in NYC, Elon University, New York City
Final thoughts on first impressions
In my first post, I expressed feelings of anxiety and excitement, hope and uncertainty. I was excited to learn about the city and about myself. I am pleased to say that I did. A lot of the anxiety that I mentioned has proved to be unfounded. The people are wonderful, the city is less scary than expected and my time here couldn't have been better.
MSG in the Guggenheim
I got frustrated walking through the Guggenheim. I was having a hard time finding a piece of art that reminded me of Madison Square Garden. There was nothing related to athletics or entertainment. I realized, however, that I was thinking far too literally.
Then I walked through the exhibit called Broken Forms: European Modernism from the Guggenheim Collection. This exhibit was made up a many different pieces of art. They all consisted of vibrant colors and exciting shapes. There wasn't a clear top or bottom;the pieces could be interpreted anyway the viewer saw fit.
Franz Marc, Broken Forms (Zerbrochene Formen), 1914 (detail). Oil on canvas, 111.8 x 84.4 cm. Solomon R. Guggenheim Museum, New York 50.1240
These pieces reminded me of Madison Square Garden because of their ability to be seen in different ways. MSG is a large space that can accommodate anything from sporting events to concerts to the circus to dinosaur exhibits. It is flexible and it's true purpose can be debated.
I felt that the pieces in the Broken Forms exhibit were similar. Their meaning could be debated and the meaning might change depending on how the viewer chose to look at the art.
Tags:Broken Forms, Elon in NYC, Eruopean Modernism, Guggenheim, Madison Square Garden
Ethnography Study: Interview Report
I spent a few hours Monday walking around Madison Square Garden and searching for interviews for my ethnography study. Our assignment is to tell a story, based on these interviews, that is representative of our "street".
Because it was the middle of the work day, MSG was slightly less hectic than usual. There wasn't the madness that typically surrounds Penn Station during rush hours. There weren't the mobs of tourists admiring the arena in awe. Most of the people in the area were working – leaving them with little motivation or time to talk to me. But I did find a few who were willing to indulge a student.
I first went to Brother Jimmy's, a sports bar, that sits adjacent to Madison Square Garden. I spoke to one hostess and one bartender who gave me insight as to the impact that MSG has on their business. They know, for instance, that when there is a game or concert, they can expect an influx of people before and after the event. The mentioned repeatedly how much their business would have benefitted if LeBron James had come to the Knicks.
Following my conversation with these two women, I headed into Madison Square Garden to speak with an employee. I approached one security guard and she directed me to one of her colleagues. He was very friendly and had been working in MSG for 26 years. Because of this, he was able to give me information about how the arena has changed over the years. Unfortunately, because of confidentiality agreements, he was not able to go into too much detail, but he provided me with some good information.
I am planning on speaking with a coworker who has attended events in Madison Square Garden and I look forward to getting her story. I am hoping that I will be able to use these different stories to tell the full experience that comes with Madison Square Garden.
Tags:Brother Jimmy's, Elon in NYC, Elon University, LeBron James, Madison Square Garden, The Streets of New York
What is this feeling? I think it's called obsession.
"You deserve each other, this hat and you. You're both so…smart."
I can't tell you how many times those lyrics from the Wicked song "Dancing through Life" have been on repeat in my head this summer. Mind you, I had never seen the show and I had no context for the lyrics. One of my roommates is a Wicked fanatic and would play the songs incessantly. After awhile, they just stuck.
There was a lot of build up to Wicked. I've been hearing about since high school, when one of my best friends became obsessed. Now, I can hardly walk into my New School dorm without hearing one of the many amazing songs.
Having heard so much about Wicked made me that much more excited to see the show. I wanted to know who was singing the songs. What were they talking about? And what in the world caused the feelings of unadulterated loathing?
I've never been a huge fan of "The Wizard of Oz" so I was a little nervous about seeing Wicked. I was afraid that it wouldn't live up to hype that surrounded it. But, I am happy to report that this uncertainty was completely unwarranted. I was taken aback by the scale of the show and how magnificent the performances were. Katie Rose Clark as Glinda and Jennifer DiNoia as Elphaba were amazing. The costumes were intricate and the scenery made the audience believe it was in Oz with the two witches.
Following the show, we were able to check out the stage. There was scenery stored on either side, above and below the stage. We say where some costume changes take place and the tracks where the scenery comes into and off of the stage. We saw where Elphaba gets her green make up touched up between acts and where the munchkin heads hang when they're not being worn. It was an insiders look at what takes place behind the scenes of a Broadway production.
After I got back from the show last night, I went straight to youtube.com and watched scenes from the show over and over…and over.
I think it's safe to say that I have a full-blown case of Wicked fever.
Tags:Broadway, Elon in NYC, Elphaba, Glinda, The Wizard of Oz, Wicked
"Anything less than 87% is Fraud-way"
I love Broadway. I don't (or should I say, didn't) know a thing about it, but I love it. It is a magical thing to watch a plain, wooden stage transformed into a story and I can't help but let myself fall into it.
Costumes, lighting, actors, props, music, etc. The show cannot exist without each of these elements. No matter how minuscule one may seem, they are all crucial aspects and add something that the show needs.
I didn't realize how true this was until our visits on Monday. From seeing the intricate detail that goes into every stage of the costuming process at William Ivey Long's studio to hearing about the daily preparation and repairs that our panelists spoke about to seeing the stories and stories of practice space, workshops and costume storage that exist at the Metropolitan Opera.
I would have never suggested that putting together a Broadway production was any small task, but I could have never imagined just how much goes into each and every performance. One thing that was consistent with all of those we spoke with was their love for it.
It's hard work and long hours. It requires going on the road for weeks at a time. There are hours of rehearsal and shows on holidays. It isn't always secure work or the extremely lucrative. But, when the curtain rises and the show begins, it makes all the work well worth it.
All of those who work on the shows understand the importance of each other. While the actors on stage get most of the glory as they are the faces of the show, they are supportive of the crew that works backstage. When IATSE, the union for professional stagehands, motion pictures technicians and allied crafts went on strikes, the actors were holding signs and protesting with them. The show won't succeed without all of these people, so they must work together and support each other.
We were able to go backstage after seeing Wicked and speak with Lindsay K. Northern who has a role in the ensemble as well as the understudy for Glinda. She said "anything less that 87 percent is Fraud-way." If those working on the show aren't able to give at least that much, they are detracting from the performance. But, luckily for us viewers, most of the time, these people are happy to give 110 percent and we get to see an amazing show.
Tags:Broadway, Elon in NYC, Lindsay K. Northern, Metropolitan Opera, Wicked, William Ivey Long
Unfamiliar Territory
As a print journalism major, I never saw a future for myself in either the advertising or public relations field. However, after our visits to Young & Rubicam and Ketchum, I see that there is certainly a place for journalism in these industries.
We heard from a young woman at Y&R, Kristen, who is a copywriter. I took a copywriting class last semester and it was interesting to see how traditional copywriting could apply to a position in an advertising agency.
It was made very evident at Ketchum how much correspondence goes into a successful public relations campaign. Journalists have an important place in PR because of their ability to use the power of language to convey a message.
Though I love journalism, I often see it as a more serious, more formulaic type of writing. Because of this, I didn't believe that I was learning the skills necessary to thrive in a more creative workplace. I am happy to report, however, that the visits to these two powerhouse companies have completely changed my mind.
Tags:advertising, Elon in NYC, Elon University, journalism, Ketchum, public relations, Young & Rubicam


On Thursday, November 17, 2016 Unified co-hosted its 3rd annual Corporate Intellectual Property Strategy Conference (CIPSC). Members were also invited to attend the morning session to receive updates on recent activities.
The afternoon session was open to the public and featured leading IP executives from as well as multiple panel discussions on IP management strategies and developing issues at the USPTO.
This year's event sold out. Details about the event and presentations linked below.


text:
package com.google.gwt.user.client.rpc.core.com.google.gwt.core.shared;

import com.google.gwt.core.shared.SerializableThrowable;
import com.google.gwt.user.client.rpc.CustomFieldSerializer;
import com.google.gwt.user.client.rpc.SerializationException;
import com.google.gwt.user.client.rpc.SerializationStreamReader;
import com.google.gwt.user.client.rpc.SerializationStreamWriter;

/**
 * Custom field serializer for SerializableThrowable.
 *
 * Modified by Prithviraj for Android at Version 0.5 to handle null return of
 * Throwables for stack trace
 */
public final class SerializableThrowable_CustomFieldSerializer extends
		CustomFieldSerializer<SerializableThrowable> {

	public static void deserialize(SerializationStreamReader streamReader,
			SerializableThrowable instance) throws SerializationException {
		String type = streamReader.readString();
		boolean typeIsExact = streamReader.readBoolean();
		instance.setDesignatedType(type, typeIsExact);
		instance.setStackTrace((StackTraceElement[]) streamReader.readObject());
		instance.initCause((Throwable) streamReader.readObject());
	}

	public static SerializableThrowable instantiate(
			SerializationStreamReader streamReader)
			throws SerializationException {
		return new SerializableThrowable(null, streamReader.readString());
	}

	/**
	 * Handles a NPE thrown by the Android OS when there is no Stack trace.
	 * Android for whatever reason resets the stack trace to null instead of
	 * leaving it as an empty array as defined by
	 * libcore.util.EmptyArray#STACK_TRACE_ELEMENT.
	 * https://android.googlesource.
	 * com/platform/art/+/master/runtime/native/java_lang_Throwable.cc #32
	 */
	public static void serialize(SerializationStreamWriter streamWriter,
			SerializableThrowable instance) throws SerializationException {
		streamWriter.writeString(instance.getMessage());
		streamWriter.writeString(instance.getDesignatedType());
		streamWriter.writeBoolean(instance.isExactDesignatedTypeKnown());
		try {
			streamWriter.writeObject(instance.getStackTrace());
		} catch (NullPointerException npe) {
			streamWriter.writeObject(new StackTraceElement[0]);
		}
		streamWriter.writeObject(instance.getCause());
	}

	@Override
	public void deserializeInstance(SerializationStreamReader streamReader,
			SerializableThrowable instance) throws SerializationException {
		deserialize(streamReader, instance);
	}

	@Override
	public boolean hasCustomInstantiateInstance() {
		return true;
	}

	@Override
	public SerializableThrowable instantiateInstance(
			SerializationStreamReader streamReader)
			throws SerializationException {
		return instantiate(streamReader);
	}

	@Override
	public void serializeInstance(SerializationStreamWriter streamWriter,
			SerializableThrowable instance) throws SerializationException {
		serialize(streamWriter, instance);
	}
}

redpajama_set_name: RedPajamaGithub


Why should we need to have computers in education? How would it help in learning? Introducing computers in early education lays the foundation of the major competitive careers in the industry. Computers play a vital role in one's personal and professional life. All of us can admit that there's no single day passes without opening our computers for work or for communication. These inventions have changed not only the way we work but also the way we take in the information.
Let's talk about the relation between computers and education. Several people say that computers should never be part of education because it might destroy their children's future, not to mention those teenagers spend too much time in it.
Computers play a significant role in education and it can't be denied in any industry or field. It also facilitated different industries, used in various educational institutions and even in the medical field. Computer technology influenced education by making a large amount of information readily available and young minds enjoy it.
The computer technology has a deep impact on the education industry. It forms a part of the school curriculum and it is important in every individual's lives today. Computer teaching also plays a role in the modern systems of education. Students find it easier to reference on the internet to search for information rather than doing research in reference books. Today, most people can satisfy their thirst for knowledge with the help of the internet.
Most of the teachers also consider the computer to be one of the cornerstones of modern education. They promoted the appearance of computers in teaching because they want the education to be not limited only in the classroom. One of the easiest ways to teach a lesson is by using a presentation through PowerPoint. The lesson is delivered through slides and students find it comfortable for them to take down notes.
Teachers also benefit in computer technology because it helps them to make their teaching more comfortable. They share presentations, reproducible worksheets, activities, and ideas. A computer helps them to easily input grades and student data into programs that will quickly compute the percentages and scores of their students accurately.
With access to the internet today, it is now possible to teach students from far remote places, provided that their places have internet signals. When barriers to communication will not be a problem, the possibilities for education will be endless. In this, teachers can reach and teach more learners. It is important for both teachers and students to learn and be familiar with using a computer to gather information and distribute ideas.
Unfortunately, access to a laptop or computer in general is not provided to everyone despite the best efforts of those leading the charge. If you are forced to purchase a machine yourself you may find yourself in need of financing either through a credit card or through the distributor. Qualifying for either of these can be a pain if you have a negative history, especially if you happen to be in the United States. If you do have this problem and need to repair a poor credit history we recommend a Phoenix credit repair company backed by a team with years and years of experience. We trust them to take care of our readers and haven't heard a bad word since.
As we progress every day, computer in education can be beneficial in several ways. It can increase the productivity of students in making their assignments. We can't deny the fact that a computer has made our lives easier. It facilitates speedy access to information and faster communication. To keep the increasing pace of our technology today, the computer is extremely important in education.
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elaborate tactics, experienced squad leaders and focus on team work. Enlist here!
Between now and end of this year CMP has planned 1 more campaign with more or less 10 battles. The campaign is located in the pacific. We will play battles like; Philippines, Sittang Bridge, Tarawa, Guam, Peleliu, and Iwo Jima.
The campaign forums are open for registration now, there you can choose to become axis or allied. I would love to welcome you all with the Japanese, since I will be leading the axis side as first time in HQ together with Matsku. The allied side is also taken care of by Heiea Safari and Druidx. Either way, we try to get some good gameplay and nice teamwork going on.
And we plan to have the pacific maps soon back as a new release of the Community Mappack too.
Viel Erfolg / Good luck!
Set topic as global announcement.
The 762-Crew wish you a good Tournament!
Last chance for some Pacific action.
We're playing Iwo Jima tomorrow.
Last battle, teams are basically tied.


I love Halloween- it is one of those holidays that you can really treat like a theme party. However, I'm not exactly a scary movie and skulls kind of decorator. I'd rather mix the more cutesy stuff (seriously, a Le Creuset pumpkin cocotte?) with spooky (but not creepy) décor.
1. Le Creuset Pumpkin Cocotte. A pumpkin shaped stoneware dish? How perfect for Halloween and autumn in general. I would definitely use this for dips!
2. Horribly Sweet Candy Bowl. I love that this bowl says boo at the bottom- as your candy grows low, you don't loose the Halloween charm.
3. Hemlock Tea Cup. I Need a full set of these, yesterday! I love that the tea cups are so 'normal' looking on the outside then add a bit of a Halloween pop saying 'Hemlock' on the inside!
4. Bat Wine Bottle Stopper. I may be a bit obsessed with wine bottle stoppers. I started my collection the first time I went to Italy, and now I can't pass up a fab stopper! This flying bat is perfect for a spooky touch at the dinner table.
5. Halloween Copper Cookie Cutters. Cookie decorating is one of my all time favorite holiday activities. This collection of cookie cutters is perfect parts spooky and cute!
6. Bat Halloween Décor Wall Sign. I love this for an entry way. Understated and still festive, this is a great piece to set the Halloween mood!
What would you add to this list? Let me know in the comments below!


Chicagoans can start signing up today for health insurance through the Affordable Care Act signed into law by President Obama.
This year's open enrollment period for 2018 coverage is shorter, running from Nov. 1 to Dec. 15, so now is the time to sign up for health insurance if you don't have insurance through your job or aren't covered under Medicaid or Medicare.
To see the individual plans available in Illinois, go to www.getcoveredillinois.gov. The website is updated, so you can look at all individual plans that are both on and off the exchange.
You can also call licensed agents at (866) 311-1119 Monday through Friday from 7 a.m. to 8 p.m. except Thanksgiving Day, and Saturday and Sunday from 8 a.m. to 6 p.m.
Cost should not be the only factor when selecting a plan. For example, look at a plan's provider network by visiting getcoveredillinois.gov, which lets consumers search by ZIP code to find free local help available in Spanish and English.
Consumers can also use the "connector" to schedule in-person assistance appointments with navigators and certified application counselors. Throughout open enrollment, there will also be in-person assistance from licensed navigators available at numerous places, including the James R. Thompson Center in Downtown Chicago.
You can also try your doctor's office. Many West Side local community health centers and hospitals also have people who can help you sign up for insurance.
For those covered by Medicaid, there are some changes for some Illinois patients. Some managed care organizations will no longer be operating next year, so you may be getting a letter soon regarding what your options will be to change to another plan. Make sure to confirm your doctor takes your Medicaid managed care insurance.
For those covered under Medicare, this is open enrollment time, too; sign up for 2018 coverage began Oct. 15 and runs through Dec. 7.
During open enrollment, eligible adults can adjust Medicare Advantage or Medicare prescription drug coverage; move from original Medicare to a Medicare Advantage plan – or vice versa; switch between Medicare Advantage plans; join a Medicare Prescription Drug Plan; or switch or drop existing drug plans.
The Illinois Department on Aging's (IDoA) Senior Health Insurance Program (SHIP) is reminding seniors of the free counseling assistance available to Medicare-eligible individuals and their caregivers.
SHIP counselors help seniors and people with disabilities make the best choices for their individual situation. Contact SHIP by calling (800) 252-8966 week days 8:30 a.m. to 5 p.m.
To compare 2018 drug and health plans on your own and to find out more about Medicare options, including preventive services now covered, visit the federal website, at www.medicare.gov or call (800) 633-4227. Medicare phone lines are open 24/7.
Enrollment ends Friday, Dec. 15 if you need coverage. You can enroll online and call for help, too.


As Mobile Developer, you will be responsible for the development of new & existing mobile apps across REA Group Asia. You will monitor, maintain and continuously resolve issues in live apps. You will also continuously research for process improvement and be responsible for the training and mentorship of junior team members.


text:
San Martino di Finita () is a town and comune in the province of Cosenza in the Calabria region of southern Italy.

It was founded in the late 15th-early 16th centuries by Arbëreshë refugees.

References

Arbëresh settlements
Cities and towns in Calabria

redpajama_set_name: RedPajamaWikipedia