text:
\section{Introduction}

Grand unified theories (GUTs)~\cite{Georgi:1974sy}, and in particular their non-supersymmetric (non-SUSY) $\mathrm{SO}(10)$ realizations~\cite{Fritzsch:1974nn}, embed of the Standard Model (SM) group in a unifying gauge group. Thereby, they lead to frameworks for physics beyond the SM which may address some of the outstanding problems, both phenomenological and aesthetic. In particular, $\mathrm{SO}(10)$ models account for the generation of neutrino mass in a natural way through the type-I~\cite{Minkowski:1977sc,GellMann:1980vs,Mohapatra:1979ia,Yanagida:1979,Schechter:1980gr} and type-II~\cite{Magg:1980ut,Lazarides:1980nt,Mohapatra:1980yp} seesaw mechanisms. To verify the viability of these models, one must attempt to fit their parameters to the known observables of the SM in order to find whether or not the models in question allow the observed low-energy values.

Fits of the Yukawa sector of $\mathrm{SO}(10)$ models to the observables of the SM have previously been presented in the literature, with various levels of detail. As neutrino data became available to the level of precision that allowed fits to be made, the initial attempts to accommodate neutrino masses and leptonic mixing parameters were performed for SUSY models~\cite{Bertolini:2004eq,Babu:2005ia,Bertolini:2006pe,Bajc:2008dc,Altarelli:2010at,Fukuyama:2015kra} with the type-I or type-II seesaw mechanism, or a mixture thereof. There have also been numerous fits of the Yukawa sector of various non-SUSY models with type-I seesaw~\cite{Altarelli:2013aqa,Dueck:2013gca,Meloni:2014rga,Babu:2015bna,Meloni:2016rnt,Ohlsson:2018qpt,Boucenna:2018wjc}, with the conclusion that these fits are possible depending on the specifics of the Yukawa sector and the symmetry breaking pattern. Furthermore, fits with type-II seesaw or a combination of types I and II seesaw have been considered in non-SUSY scenarios~\cite{Joshipura:2011nn,Babu:2016bmy}, which concluded that models with type-II seesaw only do not yield acceptable fits, but that type-II seesaw in combination with type-I seesaw provides good fits.

The works mentioned above can be classified based on the level of sophistication of the procedure used to perform the fits. Most of the previous fits have been performed with a \emph{bottom-up} approach, by first evolving the experimental values of the SM observables up to the GUT scale $M_\mathrm{GUT}$ using the renormalization group equations (RGEs) and then fitting the $\mathrm{SO}(10)$ Yukawa sector to the evolved data at that scale. This procedure involves several approximations, since the renormalization group (RG) evolution in general depends on parameters which are not known a priori, such as the mass scales of right-handed neutrinos (where they are integrated out) and the matching conditions at intermediate breaking steps. The more complete procedure is to use a \emph{top-down} approach, which involves randomly sampling the parameters of the $\mathrm{SO}(10)$ Yukawa sector at $M_\mathrm{GUT}$ and evolving each parameter down to the electroweak scale $M_\mathrm{Z}$ using the RGEs, where they are compared to experimental values of the observables, as has been done in Refs.~\cite{Dueck:2013gca,Meloni:2016rnt,Ohlsson:2018qpt,Boucenna:2018wjc}. Related to this is the treatment of the RG evolution of the neutrino sector. As the parameters are evolved down from $M_\mathrm{GUT}$ to $M_\mathrm{Z}$, a complete analysis should integrate out the right-handed neutrinos at their respective mass scales, which has been carried out in Refs.~\cite{Dueck:2013gca,Boucenna:2018wjc}. Other works either performed the fits at $M_\mathrm{GUT}$ or assumed that all right-handed neutrinos were integrated out simultaneously during the RG evolution.

In this work, we consider fits to non-SUSY $\mathrm{SO}(10)$ models with neutrino mass being generated by either the type-I or type-II seesaw mechanism or a combination of both, similar to Ref.~\cite{Babu:2016bmy}. The procedure used is similar to that of Ref.~\cite{Dueck:2013gca}, which involves sampling the parameters of the models at $M_\mathrm{GUT}$, evolving them down to $M_\mathrm{Z}$ using the RGEs, and comparing the resulting values to data of the observables. The novelty of this work is the combination of the type-I and type-II seesaw mechanisms with a proper and complete treatment of the RG evolution, including integrating out right-handed neutrinos at their respective mass scales. Furthermore, we use updated data for all fermion observables.

This paper is structured as follows. First, in Sec.~\ref{sec:model}, we present the model that we investigate and the origin of the seesaw mechanism in $\mathrm{SO}(10)$. Next, in Sec.~\ref{sec:running}, we describe how the parameters of this model are related to those of the SM. Then, in Sec.~\ref{sec:procedure}, we discuss the parametrization and the numerical procedure used. Finally, in Sec.~\ref{sec:results}, we present the results before summarizing our findings and concluding in Sec.~\ref{sec:summary}.

\section{Model}\label{sec:model}

The model that we consider is a non-SUSY $\mathrm{SO}(10)$ model with each generation of fermions and right-handed neutrinos belonging to a $\rep{16}_F$ representation, whereas the Higgs scalars reside in the $\rep{10}_H$ and $\repb{126}_H$ representations. We also introduce a global $\mathrm{U}(1)_\mathrm{PQ}$ symmetry which has a double purpose. Firstly, it solves the strong CP problem and provides the QCD axions~\cite{Peccei:1977hh,Peccei:1977ur,Weinberg:1977ma,Wilczek:1977pj}. Secondly, and more importantly for the Yukawa sector, it allows us to complexify the real $\rep{10}_H$ representation without introducing additional couplings~\cite{Bajc:2005zf}, as described in Sec.~\ref{sec:SO10}.

We assume that the $\mathrm{SO}(10)$ symmetry is broken at $M_\mathrm{GUT}=2\times 10^{16}\,\mathrm{GeV}$ in one step to the SM, similarly to Ref.~\cite{Dueck:2013gca}. Since we focus on the fermion observables, the exact details of symmetry breaking and the exact value of $M_\mathrm{GUT}$ are irrelevant (and we check this explicitly in Sec.~\ref{sec:results}). Therefore, the gauge couplings in our model do not unify. As in Ref.~\cite{Dueck:2013gca}, we assume that this is taken care of by some new physics between $M_\mathrm{Z}$ and $M_\mathrm{GUT}$. For examples of such models, see Refs.~\cite{Frigerio:2009wf,Parida:2016hln,Boucenna:2018wjc}. At $M_\mathrm{Z}$, the electroweak symmetry is further broken according to the usual Higgs mechanism, so that the complete breaking chain is
\begin{equation}
\mathrm{SO}(10) \xrightarrow[]{M_\mathrm{GUT}}\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y
\xrightarrow[]{\ \ M_\mathrm{Z}\ \ } \mathrm{SU}(3)_C\times\mathrm{U}(1)_Q.
\end{equation}

\subsection{SO(10) Lagrangian}\label{sec:SO10}
The model described above uniquely defines the Lagrangian of the Yukawa sector to be
\begin{equation}
\mathcal{L}_Y = \rep{16}_F(Y_{10}\rep{10}_H + Y_{126}\repb{126}_H)\rep{16}_F,
\end{equation}
where $Y_{10}$ and $Y_{126}$ are $3\times3$ matrices in flavor space. The representation $\repb{126}_H$ is complex in $\mathrm{SO}(10)$, but the representation $\rep{10}_H$ is real. This means that the two $\mathrm{SU}(2)_L$ doublets in the $\rep{10}_H$ necessarily take the same vacuum expectation value (vev), which would imply certain mass relations that contradict data~\cite{Bajc:2005zf}. Thus, we complexify it, which allows the two components to take different vevs. This in effect introduces  a $\rep{10}_H^*$, which may couple to the fermions in the representation $\rep{16}_F$ with a new Yukawa matrix, thereby decreasing the predictability of the model. However, if we assign the global PQ  charges such that
\begin{equation}
\rep{16}_F\rightarrow\mathrm{e}^{\mathrm{i}\alpha}\rep{16}_F,\quad \rep{10}_H\rightarrow\mathrm{e}^{-2\mathrm{i}\alpha}\rep{10}_H, \quad \repb{126}_H\rightarrow\mathrm{e}^{-2\mathrm{i}\alpha}\repb{126}_H,
\end{equation}
where $\alpha$ is some real parameter, then this additional coupling of the $\rep{10}_H^*$ to the $\rep{16}_F$ is forbidden.

After symmetry breaking, the SM Higgs doublet is a combination of the four $\mathrm{SU}(2)_L$ doublets found in the complexified $\rep{10}_H$ and the $\repb{126}_H$. The Yukawa matrices of the SM are determined by combinations of the Yukawa matrices $Y_{10}$ and $Y_{126}$, weighted by the vevs of these four $\mathrm{SU}(2)_L$ doublets $v^u_{10}$, $v^d_{10}$, $v^u_{126}$, and $v^d_{126}$. Note that these vevs do not exist at the $\mathrm{SO}(10)$ breaking scale, since they arise from electroweak symmetry breaking, but their values enter as parameters in the matching conditions. Thus, the SM fermion observables may be extracted from the following combinations of $\mathrm{SO}(10)$ Yukawa matrices $Y_{10}$ and $Y_{126}$~\cite{Altarelli:2013aqa,Dueck:2013gca,Babu:2015bna,Dueck:2013gca,Joshipura:2011nn}
\begin{equation}\label{eq:params}
\begin{split}
Y_u &= \frac{1}{v_\mathrm{SM}}(v^u_{10} Y_{10} + v^u_{126}Y_{126}), \\
Y_d &= \frac{1}{v_\mathrm{SM}}(v^d_{10} Y_{10} + v^d_{126}Y_{126}), \\
Y_\nu &= \frac{1}{v_\mathrm{SM}}(v^u_{10} Y_{10} -3 v^u_{126}Y_{126}), \\
Y_\ell &= \frac{1}{v_\mathrm{SM}}(v^d_{10} Y_{10} -3 v^d_{126}Y_{126}),
\end{split}
\end{equation}
where $Y_u$, $Y_d$, $Y_\nu$, and $Y_\ell$ are the Yukawa matrices for the up-type quarks, the down-type quarks, the neutrinos, and the charged leptons, respectively, while $v_\mathrm{SM}\simeq174\,\mathrm{GeV}$ is the SM Higgs vev. The relative signs and factors of $3$ come from Clebsch--Gordan coefficients.

\subsection{Seesaw from SO(10)}
To generate neutrino mass via the type-I seesaw mechanism~\cite{Minkowski:1977sc,GellMann:1980vs,Mohapatra:1979ia,Yanagida:1979,Schechter:1980gr}, the right-handed neutrinos must obtain Majorana masses, which occurs naturally in $\mathrm{SO}(10)$ models~\cite{Lazarides:1980nt}. The right-handed neutrinos reside in the $\rep{16}_F$ and couple to an SM singlet contained in the $\repb{126}_H$. This singlet takes a vev $v^R_{126}$, which generates a Majorana mass matrix
\begin{equation}\label{eq:M_R}
M=v^R_{126}Y_{126}.
\end{equation}
Since $v^R_{126}$ is close to $M_\mathrm{GUT}$, the Majorana masses for the right-handed neutrinos are large. Through their Yukawa coupling to the light neutrinos, the type-I seesaw mechanism provides small neutrino masses.

The type-II seesaw mechanism~\cite{Magg:1980ut,Lazarides:1980nt,Mohapatra:1980yp} also naturally occurs in $\mathrm{SO}(10)$ models~\cite{Bajc:2002iw,Bajc:2004fj} and contributes to neutrino mass. In addition to the SM singlet, the $\repb{126}_H$ contains an $\mathrm{SU}(2)_L$ triplet $\Delta$, which interacts with the light neutrinos with Yukawa coupling
\begin{equation}\label{eq:YDelta}
Y_\Delta = Y_{126}.
\end{equation}
After electroweak symmetry breaking, the interactions between this scalar triplet and the Higgs boson induces a vev $v^L_{126}$ for the triplet. This vev is inversely proportional the square of the scalar triplet mass. Hence, a heavy scalar triplet generates small masses for the neutrinos.

During RG evolution from $M_\mathrm{GUT}$ to $M_\mathrm{Z}$, the right-handed neutrinos and the scalar triplet are integrated out at their respective mass scales. This results in the effective dimension-5 operator for neutrino mass.

\section{SM Observables from SO(10)}\label{sec:running}

In this section, we describe how the sampled parameter values at $M_\mathrm{GUT}$ are evolved down to the values of the SM observables at $M_\mathrm{Z}$, as well as the procedure how to integrate out right-handed neutrinos and the scalar triplet at their respective mass scales.

\subsection{Renormalization Group Equations}\label{sec:RGEs}

To perform the RG evolution of the sampled parameters from $M_\mathrm{GUT}$ to $M_\mathrm{Z}$, we need to solve the RGEs numerically. The parameters which exhibit RG evolution are the gauge couplings $g_1$, $g_2$, and $g_3$, the Yukawa coupling matrices $Y_u$, $Y_d$, $Y_\nu$, $Y_\ell$, and $Y_\Delta$, the right-handed neutrino Majorana mass matrix $M$, the Higgs quartic coupling $\lambda$, and the coupling matrix of the dimension-5 effective neutrino mass operator $\kappa$ (see Secs.~\ref{sec:typeI} and \ref{sec:typeII}). The general set of RGEs, assuming the presence of the right-handed neutrinos and the scalar triplet, i.e.~with both type-I and type-II seesaw, is~\cite{Jones:1981we,Machacek:1983tz,Machacek:1983fi,Machacek:1984zw,Dueck:2013gca,Antusch:2002rr,Antusch:2005gp,Schmidt:2007nq,Chao:2006ye}
\begin{align}
16\pi^2 \beta_{g_1} &= \frac{41}{10}g_1^3 + \frac{3}{5}g_1^3 = \frac{47}{10} g_1^3,\label{eq:g1}\\
16\pi^2 \beta_{g_2} &= -\frac{19}{6}g_2^3 + \frac{2}{3}g_2^3 = -\frac{5}{2} g_2^3,\label{eq:g2}\\
16\pi^2 \beta_{g_3} &= -7g_3^3, \\
16\pi^2 \beta_{\lambda} &= 6\lambda^2 - 3\lambda \left(3g_2^2 + \frac{3}{5}g_1^2 \right) + 3g_2^4 + \frac{3}{2}\left(\frac{3}{5}g_1^2 + g_2^2 \right)^2 \nonumber \\ & + 4\lambda \mathrm{Tr} \left[ Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u \right] \nonumber \\ &- 8\mathrm{Tr}\left[ Y_\ell^\dagger Y_\ell Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u Y_u^\dagger Y_u \right], \\
16\pi^2 \beta_{Y_u} &= Y_u \Bigg( \frac{3}{2} Y_u^\dagger Y_u - \frac{3}{2} Y_d^\dagger Y_d - \frac{17}{20} g_1^2 - \frac{9}{4} g_2^2 - 8 g_3^2  +  \mathrm{Tr}\left[ Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u\right] \Bigg),\\
16\pi^2 \beta_{Y_d} &= Y_d \Bigg( \frac{3}{2} Y_d^\dagger Y_d - \frac{3}{2} Y_u^\dagger Y_u - \frac{1}{4} g_1^2 - \frac{9}{4} g_2^2 - 8 g_3^2  +  \mathrm{Tr}\left[ Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u\right] \Bigg),\\
16\pi^2 \beta_{Y_\nu} &= Y_\nu \Bigg( \frac{3}{2} Y_\nu^\dagger Y_\nu - \frac{3}{2} Y_\ell^\dagger Y_\ell + \frac{3}{2}Y_\Delta^\dagger Y_\Delta - \frac{9}{20} g_1^2 - \frac{9}{4} g_2^2  +  \mathrm{Tr}\left[ Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u\right] \Bigg),\\
16\pi^2 \beta_{Y_\ell} &= Y_\ell \Bigg( \frac{3}{2} Y_\ell^\dagger Y_\ell - \frac{3}{2} Y_\nu^\dagger Y_\nu + \frac{3}{2}Y_\Delta^\dagger Y_\Delta - \frac{9}{4} g_1^2 - \frac{9}{4} g_2^2 +  \mathrm{Tr}\left[ Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_d^\dagger Y_d + 3 Y_u^\dagger Y_u\right] \Bigg),\\
16\pi^2 \beta_{Y_\Delta} &= \left[ \frac{1}{2} Y_\nu^\dagger Y_\nu + \frac{1}{2} Y_\ell^\dagger Y_\ell + \frac{3}{2} Y_\Delta^\dagger Y_\Delta\right]^T Y_\Delta + Y_\Delta \left[\frac{1}{2} Y_\nu^\dagger Y_\nu + \frac{1}{2} Y_\ell^\dagger Y_\ell + \frac{3}{2} Y_\Delta^\dagger Y_\Delta \right] \nonumber \\ &+ \left[-\frac{3}{2}\left(\frac{3}{5}g_1^2 + 3g_2^2 \right) + \mathrm{Tr}\left( Y_\Delta^\dagger Y_\Delta\right)\right] Y_\Delta, \\
16\pi^2 \beta_{M} &= (Y_\nu Y_\nu^\dagger) M + M(Y_\nu Y_\nu^\dagger)^T, \\
16\pi^2 \beta_{\kappa} &= \frac{1}{2}(Y_\nu^\dagger Y_\nu - 3Y_\ell^\dagger Y_\ell + 3Y_\Delta^\dagger Y_\Delta )^T \kappa + \frac{1}{2}\kappa (Y_\nu^\dagger Y_\nu - 3Y_\ell^\dagger Y_\ell + 3Y_\Delta^\dagger Y_\Delta)  \nonumber \\ &+  2 \mathrm{Tr}\left[Y_\ell^\dagger Y_\ell + Y_\nu^\dagger Y_\nu + 3Y_u^\dagger Y_u + 3Y_d^\dagger Y_d \right]\kappa -3g_2^2\kappa + \lambda\kappa. \label{eq:kappaRGE}
\end{align}

The RGEs in Eqs.~\eqref{eq:g1}--\eqref{eq:kappaRGE} are for the general case in which the set of fields that survive below $M_\mathrm{GUT}$ contains both the right-handed neutrinos and the scalar triplet. We are also interested in the situations in which there is no scalar triplet or no right-handed neutrinos below $M_\mathrm{GUT}$ (corresponding to pure type-I seesaw or pure type-II seesaw, respectively). In these cases, the equations can be easily modified by removing the irrelevant contributions as follows:
\begin{itemize}
\item {\bf Pure type-I seesaw:} For the case with type-I seesaw only, one removes the second terms from Eqs.~\eqref{eq:g1} and \eqref{eq:g2} as well as any contributions from $Y_\Delta$ in the RGEs. This results in a set of RGEs with no contribution from the scalar triplet $\Delta$.
\item {\bf Pure type-II seesaw:} For the case with type-II seesaw only, the gauge couplings remain unchanged (since the right-handed neutrinos are neutral under the SM gauge group) and one simply has to remove any contributions from $Y_\nu$ in the RGEs, as well as the RGE for $M$.
\end{itemize}

\subsection{Right-Handed Neutrino Mass Thresholds}\label{sec:typeI}

For the scenarios in which the right-handed neutrinos are a part of the set of fields below $M_\mathrm{GUT}$, the energy scale during the RG evolution of the parameters at some point coincides with the mass of the heaviest right-handed neutrino $N_3$, $\mu=M_3$. At that threshold, we integrate out $N_3$ following the procedure outlined in Refs.~\cite{Antusch:2002rr,Antusch:2005gp}. The procedure entails removing the last row of the matrix $Y_\nu$ (which corresponds to the coupling of the three light neutrinos to $N_3$) and also removing the last row and column of the right-handed neutrino mass matrix $M$. Note that this is basis-dependent, and it is therefore crucial to work in a basis in which the matrix $M$ is diagonal (with the corresponding basis transformation applied to $Y_\nu$). The information regarding the interactions of $N_3$ that is removed from $Y_\nu$ and $M$ is placed in the effective $3\times3$ neutrino mass matrix $\kappa$ as
\begin{equation}
\kappa \rightarrow \kappa + \frac{2}{M_3}\left( Y_\nu^{(3)}\right)^T \left(Y_\nu^{(3)} \right),
\end{equation}
where $Y_\nu^{(3)}$ is the removed row from $Y_\nu$. If this is the first contribution to the effective neutrino mass, $\kappa$ is initially a $3\times 3$ zero matrix. Otherwise, it may be non-zero if there are other contributions to the effective neutrino mass operator that have already been integrated at a higher scale (for example, if we have a scalar triplet $\Delta$ that is heavier than $N_3$).

Between $M_3$ and the next threshold, we again solve the same equations as above, but with the new matrices $Y_\nu$ and $M$, which are now $2\times 3$ and $2\times 2$ matrices, respectively. Additionally, we have an RGE for $\kappa$ as in Eq.~\eqref{eq:kappaRGE}.

At the threshold of the second right-handed neutrino mass, $\mu=M_2$, we integrate out $N_2$ following the same prescription and updating $\kappa$ as
\begin{equation}
\kappa \rightarrow \kappa + \frac{2}{M_2}\left( Y_\nu^{(2)}\right)^T \left( Y_\nu^{(2)} \right),
\end{equation}
where $Y_\nu^{(2)}$ is the removed row from $Y_\nu$. After this, $Y_\nu$ is a $1\times 3$ matrix and $M$ is a scalar. The same RGEs apply below $M_2$ as above.

At the last right-handed neutrino threshold, $\mu=M_1$, the lightest right-handed neutrino $N_1$ is integrated out and $\kappa$ is updated to
\begin{equation}
\kappa \rightarrow \kappa + \frac{2}{M_1}\left(Y_\nu^{(1)}\right)^T \left( Y_\nu^{(1)} \right),
\end{equation}
where $Y_\nu^{(1)}$ is the final remaining row from $Y_\nu$. After this threshold, the parameters $Y_\nu$ and $M$ are no longer present in the RG evolution.

\subsection{Triplet Mass Threshold}\label{sec:typeII}

Similarly to the situation described in Sec.~\ref{sec:typeI}, if the scalar triplet $\Delta$ is involved, it has an RGE above its mass threshold. As the RG evolution reaches its mass scale $M_\Delta$, it is integrated out and its interactions with the neutrinos will be encoded in the effective neutrino mass matrix $\kappa$. This contribution takes the form~\cite{Schmidt:2007nq}
\begin{equation}\label{eq:kappaII}
\kappa \rightarrow \kappa -4\frac{v^L_{126}}{v_\mathrm{SM}^2}Y_\Delta.
\end{equation}
Below this mass threshold, $Y_\Delta$ is no longer a parameter of the model and therefore is no longer present in the RG evolution. The RGEs for $g_1$ and $g_2$ are modified accordingly and any contribution from $Y_\Delta$ in the set of RGEs is removed, as noted in Sec.~\ref{sec:RGEs}.

\section{Fitting Procedure}\label{sec:procedure}
In this section, we describe the numerical procedure used, including the parametrization of the $\mathrm{SO}(10)$ parameters and the input data that we fit to. The input data are based on the experimental values of the SM observables, see Sec.~\ref{sec:ID}.

\subsection{Parametrization}\label{sec:params}

Following the conventions used in Ref. \cite{Dueck:2013gca}, we define the following parameters
\begin{equation}
H\equiv\frac{v^d_{10}}{v_\mathrm{SM}} Y_{10},\quad F\equiv\frac{v^d_{126}}{v_\mathrm{SM}}Y_{126},\quad r\equiv \frac{v^u_{10}}{v^d_{10}}, \quad s\equiv \frac{1}{r}\frac{v^u_{126}}{v^d_{126}} = \frac{v^d_{10}}{v^u_{10}}\frac{v^u_{126}}{v^d_{126}},\quad r_R\equiv v^R_{126}\frac{v_\mathrm{SM}}{v^d_{126}},\quad r_L \equiv  \frac{v_\mathrm{SM}}{v^d_{126}}.
\end{equation}
Using these parameters, we can rewrite Eqs.~\eqref{eq:params}--\eqref{eq:YDelta} as
\begin{equation}\label{eq:matching}
\begin{gathered}
 Y_u = r(H+sF),\quad
Y_d = H+F,\quad
Y_\nu =r(H-3sF),\quad
Y_\ell = H-3F,\\
M_R = r_RF,\quad
Y_\Delta = r_L F.
\end{gathered}
\end{equation}

The $\mathrm{SO}(10)$ symmetry implies that both $Y_{10}$ and $Y_{126}$ (and hence $H$ and $F$) are complex symmetric matrices. One can choose to work in a basis in which $H$ is diagonal and real. In this basis, $F$ will in general be any complex symmetric matrix. Since $r$, $r_L$, and $r_R$ are just multiplicative factors, their complex phases will have no relevance and they can be chosen to be real. Finally, $s$ remains a complex parameter.

For type-II seesaw, this parametrization suggests that it is enough to sample $r_L$, or equivalently $v^d_{126}$. However, to perform the matching when integrating out the scalar triplet according to Eq.~\eqref{eq:kappaII}, we also need the value of $v^L_{126}$. Therefore, we sample both $v^L_{126}$ and $v^d_{126}$. Furthermore, we need to sample the mass $M_\Delta$ of the triplet in order to determine at what mass scale the triplet should be integrated out. Note that, although $v^L_{126}$ and $M_\Delta$ are related, there is another parameter in this relation which does not enter elsewhere in our situation. Therefore, we sample $v^L_{126}$ and $M_\Delta$ separately and keep their relationship in mind when deciding the bounds to sample the parameters within.

The total number of parameters for both type-I and type-II seesaw is thus $3(H) + 12(F) + 1(r) + 2(s) + 1(r_R) + 1(v^L_{126}) + 1(v^d_{126}) + 1(M_\Delta) = 22$. If we have type-I seesaw only, we have three parameters less ($v^L_{126}$, $v^d_{126}$, and $M_\Delta$), resulting in 19 parameters, whereas if we have type-II seesaw only, we have one parameter less ($r_R$), resulting in 21 parameters. In principle, the Higgs quartic coupling $\lambda$ at $M_\mathrm{GUT}$ should also be included as a parameter in the fit. However, it was observed to consistently be very close to zero and we therefore set $\lambda(M_\mathrm{GUT})=0$ throughout the fits.

There is a constraint on the vevs from the mass of the W boson, which must be fulfilled. That is, they must add in quadrature to the SM Higgs vev. Without type-II seesaw, we have the constraint
\begin{equation}
(v_{10}^u)^2 + (v_{10}^d)^2 + |v_{126}^u|^2 + |v_{126}^d|^2 = v_\mathrm{SM}^2.
\label{eq:normVEV}
\end{equation}
Since the vevs are considered constant in energy, this relation applies at all scales, but is used in our procedure during the sampling at $M_\mathrm{GUT}$. Using the definitions of $r$ and $s$, we can rewrite Eq.~\eqref{eq:normVEV} as
\begin{equation}
\left(\frac{v_{10}^d}{v_\mathrm{SM}}\right)^2 \left(1+r^2\right) + \left(\frac{v_{126}^d}{v_\mathrm{SM}}\right)^2 \left(1+r^2s^2\right) = 1.
\end{equation}
Given any parameter values for $r$ and $s$, one can choose $v_{10}^d$ and $v_{126}^d$ such that Eq.~\eqref{eq:normVEV} is satisfied. The only lower bound on the vevs is from perturbativity of the Yukawa couplings. That is, $Y_{10} = v_\mathrm{SM} H / v_{10}^d < \mathcal{O}(1)$ and $Y_{126} = v_\mathrm{SM} F / v_{126}^d < \mathcal{O}(1)$. Although this is usually satisfied in the fits, it should be checked after the parameter values have been obtained.

Including the vev of the scalar triplet, the constraint becomes
\begin{equation}
(v^u_{10})^2 + (v^d_{10})^2 + |v^u_{126}|^2 + |v^d_{126}|^2 + 2(v^L_{126})^2= v_\mathrm{SM}^2,
\end{equation}
where the factor of $2$ comes from the fact that $v^L_{126}$ stems from an $\mathrm{SU}(2)_L$ triplet. Note that in this case, we also need to sample $v_{126}^d$, and thus, we need to make sure that this satisfies any constraints. Since $v_{10}^d$ is still a free parameter and $v^L_{126}$ will be small, we have an absolute limit $\left(v_{126}^d / v_\mathrm{SM}\right)^2 \left(1+r^2s^2\right) < 1$. For the parameter values found, $r = \mathcal{O}(100)$ and $s = \mathcal{O}(0.1)$, this limit implies $|v_{126}^d| \lesssim v_\mathrm{SM}/10$. As mentioned above, these constraints should be checked once the parameter values have been found from the fits. For the parameters related to type-II seesaw, we have experimental bounds $v^L_{126} \lesssim 1\,\mathrm{GeV}$ and $M_\Delta\gtrsim 1\,\mathrm{TeV}$ \cite{Perez:2008ha,Ferreira:2019qpf}.

\subsection{Input Data}\label{sec:ID}

In Tab.~\ref{tab:data}, the 19 input data for the SM observables used in the fits are listed. The masses of the quarks and charged leptons are taken from Ref.~\cite{Deppisch:2018flu}, and the Higgs quartic coupling $\lambda$ is calculated from parameters therein. The neutrino mass-squared differences and the leptonic mixing angles are taken from the global fits presented in Ref.~\cite{deSalas:2017kay}. The CKM parameters have been computed from those listed in the ICHEP 2016 update by the CKMFitter Group~\cite{Charles:2004jd}. For the observable that have a higher precision than $5~\%$, we have chosen to set them to $5~\%$ in order to aid the numerical fitting procedure, as done for example in Ref.~\cite{Dueck:2013gca}. This approach of choice for the errors has the undesired effect of exaggerating the errors on the charged-lepton masses, which are extremely well-known compared to the other observables. However, a very small error on a given observable would mean that any deviation from its central value would cause a large effect on the fit and thereby render the fit almost impossible. The ideal treatment of this issue would be to use the exact values of the charged-lepton masses, as done in some previous fits, see for example Refs.~\cite{Joshipura:2011nn,Altarelli:2013aqa}. However, it is not possible to use such a treatment while solving the RGEs from $M_{\rm GUT}$ to $M_{\rm Z}$ and taking into account the three mass thresholds.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{l l l l}
\hline
\hline

Observable													& Value & Error 	& Reference	\tabularnewline
\hline
$m_u\,\mathrm{(MeV)}$								&  $1.36$	&  $0.15$ 	&\cite{Deppisch:2018flu}\tabularnewline
$m_c\,\mathrm{(MeV)}$								&  $635$		& $32$ 		&\cite{Deppisch:2018flu}\tabularnewline
$m_t\,\mathrm{(GeV)}$									&  $172$		& $8.7$ 		&\cite{Deppisch:2018flu}\tabularnewline
$m_d\,\mathrm{(MeV)}$								&  $2.90$	& $0.15$		&\cite{Deppisch:2018flu}\tabularnewline
$m_s\,\mathrm{(MeV)}$								&  $54.1$	& $2.8$ 		&\cite{Deppisch:2018flu}\tabularnewline
$m_b\,\mathrm{(GeV)}$								&  $2.87$	& $0.15$		&\cite{Deppisch:2018flu}\tabularnewline

$m_e\,\mathrm{(MeV)}$								&  $0.487$	& $0.025$ 	&\cite{Deppisch:2018flu}\tabularnewline
$m_\mu\,\mathrm{(MeV)}$							&  $103$		& $5.2$ 		&\cite{Deppisch:2018flu}\tabularnewline
$m_\tau\,\mathrm{(GeV)}$							&  $1.75$	& $0.088$ 	&\cite{Deppisch:2018flu}\tabularnewline

$\Delta m_{21}^2\,(10^{-5}\mathrm{eV^2})$	&  $7.55$	&  $0.38$		&\cite{deSalas:2017kay}\tabularnewline
$\Delta m_{31}^2\,(10^{-3}\mathrm{eV^2})$ (NO)	&  $2.50$	&  $0.13$ 	&\cite{deSalas:2017kay}\tabularnewline
$\Delta m_{32}^2\,(10^{-3}\mathrm{eV^2})$ (IO)	&  $-2.42$	&  $0.13$ 	&\cite{deSalas:2017kay}\tabularnewline

$\sin \theta^q_{12}$										&  $0.225$& $0.012$ 	  &\cite{Charles:2004jd}\tabularnewline
$\sin \theta^q_{13}$										&  $0.00372$&$0.00019$  &\cite{Charles:2004jd}\tabularnewline
$\sin \theta^q_{23}$										&  $0.0418$&$0.0021$  &\cite{Charles:2004jd}\tabularnewline
$\delta_\mathrm{CKM}$												&  $1.14$&$0.058$  		  &\cite{Charles:2004jd}\tabularnewline

$\sin^2 \theta^\ell_{12}$								&  $0.320$	&  $0.020$	&\cite{deSalas:2017kay}\tabularnewline
$\sin^2 \theta^\ell_{13}$	(NO)						&  $0.0216$&  $0.0011$	&\cite{deSalas:2017kay}\tabularnewline
$\sin^2 \theta^\ell_{13}$	(IO)						&  $0.0222$&  $0.0012$	&\cite{deSalas:2017kay}\tabularnewline
$\sin^2 \theta^\ell_{23}$	(NO)						&  $0.547$	&  $0.030$ 	&\cite{deSalas:2017kay}\tabularnewline
$\sin^2 \theta^\ell_{23}$	(IO)						&  $0.551$	&  $0.030$ 	&\cite{deSalas:2017kay}\tabularnewline

$\lambda$													&  $0.516$	&  $0.026$ 	&\cite{Deppisch:2018flu}\tabularnewline

\hline
\hline
\end{tabular}
\caption{\label{tab:data} Data for the SM observables at $M_\mathrm{Z}$ and their corresponding errors used in the fits. The abbreviations NO and IO stand for normal neutrino mass ordering and inverted neutrino mass ordering, respectively.}
\end{center}
\end{table}

\subsection{Numerical Procedure}

The numerical procedure to fit the $\mathrm{SO}(10)$ parameters to the SM observables consists of two components. One component transforms the $\mathrm{SO}(10)$ parameters to SM parameters and performs the RG evolution down to $M_\mathrm{Z}$. The other component is a numerical optimization algorithm, which iterates this procedure by sampling different sets of the $\mathrm{SO}(10)$ parameters with the objective of fitting the derived parameter values to the data.

To relate the $\mathrm{SO}(10)$ parameters to the fermion observables of the SM, we employ the following procedure:
\begin{enumerate}
\item The required GUT scale parameters are randomly sampled given prior distributions as described below. The number of parameters that are sampled depends on the scenario which is investigated (19 for type-I seesaw, 21 for type-II seesaw, or 22 for type-I+II seesaw).
\item These parameter values are transformed into the parameters of the SM via the matching conditions in Eq.~\eqref{eq:matching}.
\item The SM parameters are evolved from $M_\mathrm{GUT}$ down to $M_\mathrm{Z}$ using the RGEs and the matching conditions at each mass threshold. To do this, the following steps are iterated until the parameters have been evolved down all the way to $M_\mathrm{Z}$:
\begin{enumerate}
\item The RGEs for the scenario of interest are used to evolve the parameters down to the first mass threshold, which is either the heaviest right-handed neutrino mass or the scalar triplet mass.
\item At this threshold, the corresponding particle is integrated out following the procedure outlined in Secs.~\ref{sec:typeI} and~\ref{sec:typeII}. This changes the number of parameters, for example by removing some of the parameters associated with the right-handed neutrinos and adding the effective neutrino mass matrix.
\item The RGEs of the new set of parameters are solved to the next threshold.
\end{enumerate}
\item The fermion masses and mixing parameters are calculated from the SM observables at $M_\mathrm{Z}$.
\item These parameter values are compared to the 19 data listed in Tab.~\ref{tab:data} to compute the $\chi^2$ goodness of fit function given by
\begin{equation}\label{eq:chi2}
\chi^2 = \sum_{i=1}^{19} \left(\frac{X_i - \overline{x_i}}{\sigma_i}\right)^2
\end{equation}
in which the current prediction $X_i$ for the $i$th observable is compared to the actual value $\overline{x_i}$ with error $\sigma_i$. Although the $\chi^2$ function usually carries a statistical interpretation, this is non-trivial in problems such as this in which the model is highly non-linear~\cite{Bjorkeroth:2017ybg,Deppisch:2018flu} and may not always be possible.
\end{enumerate}

The above steps are repeated until we converge to a set of parameter values that minimize the $\chi^2$ function. In order to perform the numerical minimization, we link the procedure to the differential evolution algorithm \texttt{Diver} from the \texttt{ScannerBit} package~\cite{Workgroup:2017htr}. We run this parallelized software package on a computing cluster. During this procedure, the parameters are sampled from predetermined distributions with ranges determined from parameter bounds and results of preliminary investigations. For the matrix elements of $H$ and $F$ as well as the vev ratio $r_R$, the vev $v^L_{126}$, and the scalar triplet mass $M_\Delta$, we sample from logarithmic distributions. The parameters $r$, $s$, and $v^d_{126}$ have better known orders of magnitude and are therefore sampled from uniform priors. The parameter ranges are given by
\begin{equation}
\begin{gathered}
H_{11}\in [10^{-7},10^{-5}]_\pm,\quad H_{22}\in [10^{-5},10^{-3}]_\pm,\quad H_{33}\in [10^{-3},10^{-1}]_\pm, \\
|F_{11}| \in [10^{-7},10^{-5}],\quad |F_{12}| \in [10^{-6},10^{-4}], \quad |F_{13}| \in [10^{-5},10^{-3}], \\
|F_{22}| \in [10^{-5},10^{-3}],\quad |F_{23}| \in [10^{-4},10^{-2}], \quad |F_{33}| \in [10^{-4},10^{-2}], \\
r \in [-100,100], \quad |s| \in [0.0,0.5], \quad r_R\in [10^{14},10^{17}], \\
M_\Delta \in [10^6\,\mathrm{GeV},M_\mathrm{GUT}], \quad v^L_{126}\in [10^{-9},10^{-1}]\,\mathrm{GeV}, \quad v^d_{126} \in [0.0,10]\,\mathrm{GeV},
\end{gathered}
\end{equation}
where the subscript ``$\pm$" signifies that the parameter is allowed to be positive or negative. The parameters for which the bounds are given as absolute values are complex and their phases are sampled uniformly over $[0,2\pi)$.

After this algorithm has converged, we link the procedure to the basin-hopping algorithm~\cite{bh1997} in the \texttt{SciPy} library~\cite{scipy2001} to further improve the fit, starting from the previously found parameter values. The reason that the algorithms are run in this order is that \texttt{Diver} is more efficient in exploring a large and high-dimensional parameter space, whereas the basin-hopping algorithm improves the fit by perturbing the point in parameter space around the starting point. Finally, we also use a Nelder--Mead simplex algorithm~\cite{Press:1992zz} to further minimize the $\chi^2$ function.

Using these three algorithms provides some confidence that a reasonable minimum has been found. Note, however, that it is impossible to guarantee that a global minimum has been obtained. To increase our confidence in the minimum found, we run the optimization several times to verify that our set of parameters provides the best fit.

\section{Results and Discussion}\label{sec:results}

The results of the fits show that the known observables of the SM with normal neutrino mass ordering are well accommodated by the model with the type-I seesaw mechanism ($\chi^2 \simeq 14.8$), and that including the type-II seesaw mechanism improves the fit by a small amount to $\chi^2 \simeq 14.7$. The fact that the combination of the two mechanisms provides a better fit than pure type-I seesaw is expected, since it introduces more freedom in the fit. Pure type-II seesaw does not provide as good a fit. With inverted neutrino mass ordering, all fits are much worse than the corresponding ones with normal ordering, in agreement with results of previous fits~\cite{Joshipura:2011nn,Dueck:2013gca,Boucenna:2018wjc} and global fits of neutrino parameters~\cite{deSalas:2017kay}. In Tab.~\ref{tab:chi2}, the resulting values of the $\chi^2$ function in Eq.~\eqref{eq:chi2} are displayed for the six different fits that have been performed.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{ccccccc}
\hline
\hline
\multicolumn{3}{c}{Normal ordering} & & \multicolumn{3}{c}{Inverted ordering}\tabularnewline

Type-I+II & Type-I & Type-II & \hspace{0.5cm} & Type-I+II & Type-I & Type-II\tabularnewline
\hline
14.7 & 14.8 & 119 & & \multicolumn{3}{c}{$\mathcal{O}(1000)$}\tabularnewline
\hline
\hline
\end{tabular}
\caption{\label{tab:chi2} Values of the $\chi^2$ function for the six different cases considered.}
\end{center}
\end{table}

We find that the best fit with the combination of the type-I and type-II seesaw mechanisms is given by a situation in which the dominant contribution to neutrino mass is given by the type-I seesaw mechanism. This is achieved by having the scalar triplet mass $M_\Delta$ close to $M_\mathrm{GUT}$ and a very small value of $v^L_{126}$. The parameter values for the two cases of type-I+II seesaw and pure type-I seesaw, respectively, are the following
\begin{equation}\label{eq:tI+II}
\begin{gathered}
H =
\begin{pmatrix}
1.00002\times 10^{-7} && 0 && 0 \\
0 && 5.56015\times10^{-5} && 0 \\
0 && 0 && 6.51100\times 10^{-3}
\end{pmatrix}\,,\\
F =
\begin{psmallmatrix}
5.55836\times10^{-6}-3.17854\times 10^{-6}\mathrm{i} && -1.13049\times10^{-5}-1.20803\times 10^{-5}\mathrm{i} && -3.54614\times10^{-5}-1.45941\times 10^{-4}\mathrm{i} \\
-1.13049\times10^{-5}-1.20803\times 10^{-5}\mathrm{i} && -1.63916\times10^{-4}+3.47085\times10^{-5}\mathrm{i} && -2.56495\times10^{-4}+2.55822\times10^{-4}\mathrm{i} \\
-3.54614\times10^{-5}-1.45941\times 10^{-4}\mathrm{i} && -2.56495\times10^{-4}+2.55822\times10^{-4}\mathrm{i} && -9.19624\times10^{-4}-5.02769\times10^{-4}\mathrm{i} \\
\end{psmallmatrix}\,,\\
r = -65.9350, \quad s = 0.391447+0\mathrm{i}, \quad r_R = 2.04454\times10^{15}\,\mathrm{GeV}, \\
M_\Delta = 1.99986\times10^{16}\,\mathrm{GeV}, \quad v^L_{126} = 1.01639\times10^{-6}\,\mathrm{GeV}, \quad v^d_{126} = 4.10530\,\mathrm{GeV}
\end{gathered}
\end{equation}
for ``Type-I+II'' and
\begin{equation}\label{eq:tI}
\begin{gathered}
H =
\begin{pmatrix}
1.00000\times 10^{-7} && 0 && 0 \\
0 && 5.55975\times10^{-5} && 0 \\
0 && 0 && 6.51243\times 10^{-3}
\end{pmatrix}\,,\\
F =
\begin{psmallmatrix}
5.55892\times10^{-6}-3.17760\times 10^{-6}\mathrm{i} && -1.13026\times10^{-5}-1.20776\times 10^{-5}\mathrm{i} && -3.54522\times10^{-5}-1.45970\times 10^{-4}\mathrm{i} \\
-1.13026\times10^{-5}-1.20776\times 10^{-5}\mathrm{i} && -1.63938\times10^{-4}+3.46916\times10^{-5}\mathrm{i} && -2.56583\times10^{-4}+2.55861\times10^{-4}\mathrm{i} \\
-3.54522\times10^{-5}-1.45970\times 10^{-4}\mathrm{i} && -2.56583\times10^{-4}+2.55861\times10^{-4}\mathrm{i} && -9.19847\times10^{-4}-5.02620\times10^{-4}\mathrm{i} \\
\end{psmallmatrix}\,,\\
r = -66.0472, \quad s = 0.391339+0\mathrm{i}, \quad r_R = 2.05408\times10^{15}\,\mathrm{GeV}
\end{gathered}
\end{equation}
for ``Type-I''. Note that although $s$ is a complex parameter, the fit favored a value with a negligibly small imaginary part. As can be observed, the parameter values are very close to each other in the two cases. In fact, the parameter values for type-I+II seesaw were found by starting from the parameter values for pure type-I seesaw and adding a very small contribution from the scalar triplet $\Delta$, i.e.~a large $M_\Delta$ and a small $v^L_{126}$, and using the basin-hopping and simplex algorithms to improve the parameter values. This biased procedure was found to produce a better fit than the one which started from unknown values of the parameters. Hence, the results of the two fits suggest that the best fit is provided by a set of parameter values for which the type-I seesaw mechanism is the dominant contribution to neutrino mass, but the type-II seesaw mechanism has a small contribution.

In Tab.~\ref{tab:predictions}, the resulting values of the observables for the ``Type-I+II'' parameter values in Eq.~\eqref{eq:tI+II} and the ``Type-I'' parameter values in Eq.~\eqref{eq:tI} are shown. The corresponding pulls are also shown, which are the quantities whose squares are summed to give the value of the $\chi^2$ function in Eq.~\eqref{eq:chi2}. In Fig.~\ref{fig:pulls}, these pulls are further displayed for ease of comparison. Since the parameter values and $\chi^2$ values are close in the two cases, their pulls are also similar. The largest contribution to the $\chi^2$ values comes from $\sin^2\theta^\ell_{23}$, for which a lower value than the measured one is predicted. In fact, the fit results suggest a value of $\theta^\ell_{23}$ in the lower octant, whereas the data favors a value in the higher octant~\cite{deSalas:2017kay}. This tension in the values of $\sin^2\theta^\ell_{23}$ has been observed in similar fits to ours~\cite{Dueck:2013gca,Boucenna:2018wjc}. If future neutrino experiments shift the value of $\theta^\ell_{23}$ to be in the lower octant, the goodness of the fits presented in this work would be greatly improved. In fact, the octant of $\theta^\ell_{23}$ the lower octant is allowed at $1\sigma$ in one of the global fits to neutrino data~\cite{Capozzi:2018ubv} and consistently allowed at $3\sigma$~\cite{deSalas:2017kay,Capozzi:2018ubv,Esteban:2018azc,nu-fit18}.

\begin{table}[!htbp]
\begin{center}
\begin{tabular}{l  l l  l l}
\hline
\hline
Observable/ & \multicolumn{2}{c}{Type-I+II} & \multicolumn{2}{c}{Type-I}  \tabularnewline
Parameter\\
	& Value & Pull & Value & Pull \tabularnewline

\hline

$m_u$				& $1.37\,\mathrm{MeV}$ 	& $0.0473$	& 	$1.37\,\mathrm{MeV}$&$0.0422$ \tabularnewline
$m_c$				& $646\,\mathrm{MeV}$	& $0.351$		& 	$645\,\mathrm{MeV}$&$0.314$ \tabularnewline
$m_t$				& $161\,\mathrm{GeV}$	& $-1.24$		&$161\,\mathrm{GeV}$	& $-1.26$	  \tabularnewline
$m_d$				& $2.94\,\mathrm{MeV}$	&$0.248$		& $2.93\,\mathrm{MeV}$& $0.199$\tabularnewline
$m_s$				& $55.6\,\mathrm{MeV}$	&$0.528$		& $55.4\,\mathrm{MeV}$&$0.473$ \tabularnewline
$m_b$				& $2.69\,\mathrm{GeV}$	& $-1.18$		& $2.69\,\mathrm{GeV}$&$-1.23$ \tabularnewline
$m_e$				& $0.489\,\mathrm{MeV}$& $0.0613$	& $0.489\,\mathrm{MeV}$&$0.0613$ \tabularnewline
$m_\mu$  			& $103\,\mathrm{MeV}$	& $-0.0421$	&	$103\,\mathrm{MeV}$&$-0.0482$ \tabularnewline
$m_\tau$  		& $1.79\,\mathrm{GeV}$	& $0.411$		&	$1.79\,\mathrm{GeV}$&$0.408$ \tabularnewline

$\Delta m_{21}^2$	& $7.74\times10^{-5}\,\mathrm{eV}^2$&$0.511$& $7.73\times10^{-5}\,\mathrm{eV}^2$ &$0.470$ \tabularnewline
$\Delta m_{31}^2$	& $2.42\times10^{-3}\,\mathrm{eV}^2$&$-0.608$& $2.41\times10^{-3}\,\mathrm{eV}^2$&$-0.665$ \tabularnewline

$\sin \theta^q_{12}$			& $0.236$	&$0.942$		& 	$0.236$&$0.950$ \tabularnewline
$\sin \theta^q_{13}$			& $0.00376$&  $0.210$	& $0.00376$&$0.215$ \tabularnewline
$\sin \theta^q_{23}$			& $0.0393$	&$-1.20$		&$0.0393$ & $-1.20$ \tabularnewline
$\delta_\mathrm{CKM}$	& $1.10$		& $-0.618$	& $1.11$ &$-0.598$ \tabularnewline

$\sin^2 \theta^\ell_{12}$	& $0.332$	& $0.589$	& $0.331$ & $0.565$ \tabularnewline
$\sin^2 \theta^\ell_{13}$	& $0.0203$	& $-1.18$		&$0.0203$ & $-1.17$\tabularnewline
$\sin^2 \theta^\ell_{23}$	& $0.474$	& $-2.43$		& $0.474$& $-2.44$ \tabularnewline

$\lambda$						& $0.522$	& $0.221$		& $0.522$ & $0.222$ \tabularnewline
\hline
$\chi^2$							&  	& $14.7$   	& 	& $14.8$ \tabularnewline
\hline
$m_1$ 								&  $3.70\times10^{-3}\,\mathrm{eV}$&   --& 	$3.70\times10^{-3}\,\mathrm{eV}$& --\tabularnewline
$m_2$ 								& $9.55\times10^{-3}\,\mathrm{eV}$&   --& 	$9.54\times10^{-3}\,\mathrm{eV}$& --\tabularnewline
$m_3$					 			&  $4.93\times10^{-2}\,\mathrm{eV}$&   --& 	$4.93\times10^{-2}\,\mathrm{eV}$& --\tabularnewline
$M_1$ 								&$1.87\times10^{10}\,\mathrm{GeV}$&   --& 	$1.88\times10^{10}\,\mathrm{GeV}$& --\tabularnewline
$M_2$						 		&$4.46\times10^{11}\,\mathrm{GeV}$&   --& $4.48\times10^{11}\,\mathrm{GeV}$& --\tabularnewline
$M_3$						 		&$2.34\times10^{12}\,\mathrm{GeV}$&   --& $2.36\times10^{12}\,\mathrm{GeV}$& --\tabularnewline
$m_{ee}$						 	&$1.56\times10^{-3}\,\mathrm{eV}$	&   --& $1.56\times10^{-3}\,\mathrm{eV}$& --\tabularnewline
$\delta_\mathrm{CP}$ 		&$0.441$											&   --& $0.447$ & --\tabularnewline
\hline
\hline
\end{tabular}
\caption{\label{tab:predictions} Predicted values of the observables that are included in the fits together with the associated pulls for the two cases of type-I+II seesaw and pure type-I seesaw, respectively, and normal neutrino mass ordering. Shown are also predicted values for some of the unknown parameters of the neutrino sector, which are not included in the fits, namely the small neutrino masses $m_1$, $m_2$, and $m_3$, the large right-handed neutrino masses $M_1$, $M_2$, and $M_3$, the effective neutrinoless double beta decay mass $m_{ee}$, and the leptonic CP-violating phase $\delta_\mathrm{CP}$.}
\end{center}
\end{table}

Also shown in Tab.~\ref{tab:predictions} are predictions for some parameters not included in the fits, namely the small neutrino masses $m_1$, $m_2$, and $m_3$, the large right-handed neutrino masses $M_1$, $M_2$, and $M_3$, the effective neutrinoless double beta decay mass $m_{ee}$, and the leptonic CP-violating phase $\delta_\mathrm{CP}$. From the small neutrino masses, we note that their sum is below the cosmological upper limit~\cite{Tanabashi:2018oca}, and further that $m_{ee}$ is within the allowed region~\cite{Deppisch:2014zta,Pas:2015eia}. As for $\delta_\mathrm{CP}$, the predicted value is far from the value favored by global fits~\cite{deSalas:2017kay}. Although this seems like a failure of the fits, one should keep in mind that it is possible that including it as an observable in the fits may still yield an acceptable value and further that its exact value has not been directly measured.

\begin{figure}[t]
\includegraphics[clip,width=0.8\textwidth]{tI_I+II_pulls.pdf}
\caption{\label{fig:pulls} Pulls corresponding to the SM observables for the two cases of type-I+II seesaw and pure type-I seesaw, respectively, and normal neutrino mass ordering. The sum of the pulls squared gives the $\chi^2$ value.}
\end{figure}

In Fig.~\ref{fig:running}, the RG evolution for some of the quantities are displayed using the parameter values in Eq.~\eqref{eq:tI+II}. First, the upper-left panel shows the RG evolution of the quark Yukawa couplings. These are the singular values of the Yukawa matrices $Y_u$ and $Y_d$. Second, the upper-right panel shows the charged-lepton Yukawa couplings, calculated in the same way from the Yukawa matrix $Y_\ell$. Third, the lower-left panel shows the RG evolution of the neutrino mass-squared differences, calculated from the differences of the squared singular values of the neutrino mass matrix. This is taken as $\kappa$ plus the type-I seesaw contributions from the neutrino Yukawa couplings that have not yet been integrated out. Finally, the lower-right panel shows the RG evolution of the leptonic mixing angles, which are calculated by computing the leptonic mixing matrix as the mixing matrix between the charged-lepton Yukawa matrix and the neutrino mass matrix.

As can be seen in the two lower panels, the right-handed neutrino mass thresholds considerably affect the RG evolution of the parameters related to neutrino masses. This enhanced RG evolution between $M_1$ and $M_3$ can be understood by noting that in order to derive these quantities, one must consider $\kappa_\mathrm{eff}=\kappa+2Y_\nu^TM^{-1}Y_\nu$, since some but not all right-handed neutrinos have been integrated out in this energy region. The RG evolution of the neutrino mass-squared differences are thus due to the RG evolution of both terms in $\kappa_\mathrm{eff}$. This causes an enhancement of the RG evolution inside the energy region compared to outside it, since its RG evolution is given by
\begin{equation}\label{eq:kappa_eff}
\frac{d\kappa_\mathrm{eff}}{d \ln\mu} = \frac{d\kappa}{d \ln\mu} + 2 \frac{dY_\nu^T}{d \ln\mu}M^{-1}Y_\nu + 2 Y_\nu^T M^{-1} \frac{dY_\nu}{d \ln\mu} - 2Y_\nu^T M^{-1} \frac{dM}{d\ln\mu}M^{-1} Y_\nu.
\end{equation}
Between $M_1$ and $M_3$, the second and third terms in the right-hand side of Eq.~\eqref{eq:kappa_eff} are about three to five larger than the first term. Below $M_1$, only the first term contributes. Its contribution is smaller than above $M_1$ due to the absence of the terms with $Y_\nu$ in Eq.~\eqref{eq:kappaRGE}, since $Y_\nu$ is not a parameter of the theory after the right-handed neutrinos have been integrated out. Above $M_3$, only the last three terms in the right-hand side of Eq.~\eqref{eq:kappa_eff} contribute, but they are suppressed by $M_3$, while below $M_3$ they are only suppressed by $M_2$. Furthermore, note that the parameters related to neutrino masses (i.e.~the neutrino mass-squared differences and the leptonic mixing angles) should only be considered effective parameters between $M_1$ and $M_3$, i.e.~before the actual light neutrino mass matrix is formed.

\begin{figure*}[t]
\begin{tabular}{c c}
\includegraphics[width=0.495\textwidth]{q_yukawas.pdf} &

\includegraphics[width=0.495\textwidth]{l_yukawas.pdf}\\

\includegraphics[width=0.495\textwidth]{dm.pdf}&

\includegraphics[width=0.495\textwidth]{l_angles.pdf}
\end{tabular}

\caption{\label{fig:running}Renormalization group evolution of some quantities for the fit with type-I+II seesaw and normal neutrino mass ordering as functions of the energy scale $\mu$ between $M_\mathrm{Z}$ and $M_\mathrm{GUT}$. \emph{Upper-left panel:} Quark Yukawa couplings (singular values of $Y_u$ and $Y_d$). \emph{Upper-right panel:} Charged-lepton Yukawa couplings (singular values of $Y_\ell$). \emph{Lower-left panel:} Neutrino mass-squared differences (Differences of squared singular values of the neutrino mass matrix, multiplied by $v_\mathrm{SM}^2/4$). \emph{Lower-right panel:} Sine squareds of the leptonic mixing angles calculated from the mixing between the charged-lepton Yukawa matrix and the neutrino mass matrix.}
\end{figure*}

Since the model presented is not a complete $\mathrm{SO}(10)$ model in the sense that we have not imposed gauge coupling unification, it is relevant to consider the effect on the $\chi^2$ function by changing the value of $M_\mathrm{GUT}$. We assume that gauge coupling unification is taken care of by some new physics between $M_\mathrm{Z}$ and $M_\mathrm{GUT}$. Such a model can be found in e.g.~Refs.~\cite{Frigerio:2009wf,Parida:2016hln,Boucenna:2018wjc}. Our aim in this work is to present general results of fits for the seesaw mechanisms of types I and II, for which we need to verify that the results are not sensitive to the exact unification scale. Indeed, we find that the $\chi^2$ value is fairly insensitive to changes in $M_\mathrm{GUT}$, as shown in Fig.~\ref{fig:MGUT}. The blue curve shows the value of the $\chi^2$ function for various values of $M_\mathrm{GUT}$ in the type-I+II seesaw case with the parameter values of Eq.~\eqref{eq:tI+II}. These $\chi^2$ values will, of course, not be the ones found from fits performed with the given $M_\mathrm{GUT}$. After running the basin-hopping algorithm starting from those parameter values, the points shown in red were found.

Thus, one can conclude that if a fit was performed with a given $M_\mathrm{GUT}$, the resulting $\chi^2$ values would be at most the ones given by the red dots. Since these are very close to the $\chi^2$ value corresponding to our choice of $M_\mathrm{GUT}$, one can conclude that the ability to find an acceptable fit applies also to realistic models.\footnote{This neglects the fact that changes in $M_\mathrm{GUT}$ will in general be accompanied by changes in the RG evolution of the Yukawa couplings due to new physics between $M_\mathrm{Z}$ and $M_\mathrm{GUT}$.} The results should not be interpreted to suggest that the value $M_\mathrm{GUT}=2\times 10^{16}\,\mathrm{GeV}$ provides the best fit. The reason why this has the lowest $\chi^2$ value is simply that the original fit was performed with that value and the other points were found by perturbing the corresponding parameter values.

\begin{figure}
\includegraphics[clip,width=0.8\textwidth]{chi2vsMGUT.pdf}
\caption{\label{fig:MGUT} Variation of the $\chi^2$ function with $M_\mathrm{GUT}$. The blue curve shows the result of varying $M_\mathrm{GUT}$, while keeping the same best-fit parameter values. The red dots show the result of performing local optimization around the best-fit parameter values for some selected values of $M_\mathrm{GUT}$.}
\end{figure}

\section{Summary and Conclusions}\label{sec:summary}

The parameters of minimal non-SUSY $\mathrm{SO}(10)$ models have been fitted to the known observables of the SM with neutrino mass generated via type-I or type-II seesaw, or a combination thereof. We have refrained from specifying a complete model by not imposing gauge coupling unification, which we have assumed can be taken care of by some new physics between $M_\mathrm{Z}$ and $M_\mathrm{GUT}$. As opposed to many similar works, we have performed the fits by sampling the parameters of the $\mathrm{SO}(10)$ model, and evolving them down to $M_\mathrm{Z}$ using the RGEs, where the corresponding parameter values have been compared to the known values of the observables. This method allows for a proper treatment of mass thresholds due to the masses of the right-handed neutrinos. Furthermore, we have used updated data at $M_\mathrm{Z}$.

The results have shown that it is possible to find acceptable fits only in the case of normal neutrino mass ordering, in agreement with hints from global fits to neutrino data. The combination of the seesaw mechanisms of types I and II provides the best fit, which is marginally better than that with the type-I seesaw mechanism only. This, together with the parameter values of the best fit, suggests that the type-I seesaw mechanism is the dominant contributor to neutrino mass.

From the values of best-fit parameters, it can be observed that the largest contribution to the $\chi^2$ function is coming from $\sin^2\theta^\ell_{23}$, which is favored to be below $0.5$ by our fits, thus predicting the leptonic mixing angle $\theta^\ell_{23}$ in the lower octant in contrast to the latest neutrino data. The best-fit parameters also provide predictions on some of the unknown neutrino parameters, such as the absolute neutrino masses, the effective neutrinoless double beta decay mass, and the leptonic CP-violating phase.

Finally, to verify the robustness of these fits to model-dependent changes of the unification scale, we have investigated the effect of varying $M_\mathrm{GUT}$ around our chosen value of $2\times 10^{16}\,\mathrm{GeV}$. After performing local minimization around our best-fit parameter values, we have found that the effect of such variations of $M_\mathrm{GUT}$ is small. Therefore, the results presented can be considered to be somewhat general for minimal non-SUSY $\mathrm{SO}(10)$ models, and thus, we expect the fits of specific models, which impose gauge coupling unification by new low-scale physics, to be similar.

In general, future experiments will continue to improve the precision of the observables of the SM, which will affect future fits of the kind presented in this paper. In particular, the value of $\theta^\ell_{23}$ has been shown to have a large effect on the fits. Determination of the effective neutrinoless double beta decay mass as well as the leptonic CP-violating phase will also provide invaluable information to determine the ability of different models to reproduce the observables of the SM.

\begin{acknowledgments}
The authors wish to thank Sofiane M.~Boucenna for collaboration in early stages of this project. T.O.~acknowledges support by the Swedish Research Council (Vetenskapsrådet) through contract No.~2017-03934 and the KTH Royal Institute of Technology for a sabbatical period at the University of Iceland. M.P.~thanks "Stiftelsen Olle Engkvist Byggmästare" and "Roland Gustafssons Stiftelse för teoretisk fysik" for financial support. Numerical computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC Center for High Performance Computing (PDC-HPC) at KTH Royal Institute of Technology in Stockholm, Sweden under project numbers PDC-2018-49 and SNIC 2018/3-559.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}


Home Football CD Montesinos A well deserved point for Monte.
A well deserved point for Monte.
A dedicated band of supporters made the long trip (without getting lost!!) to Monovar, on an extremely cold Sunday afternoon. The ground was spectacular! Two playing fields, made as if cut from the sides of a mountain, with a covered stand between the two pitches.
The Monte team was well short of it's normal members for all manner of reasons, and therefore having to rely on players from the youth teams. And to be honest they put up a spirited performance.
The first half, Monovar were clearly on top, and obviously intent on attacking from the off, but they didn't account for the magnificent defence of the Monte's back 5 who never wavered from their duties. Fernando and Dimitry commandeered the defence into an almost perfect straight line, and with back up from the the midfield, and almost non-existant forwards, all attacks were repelled.
Having got the ball Monte played the ball forward, but Omar, Roberto, and Macia were constantly caught offside, whether this was due to Monovar pulling forward, or Monte too eager I don't know, but the referee's arm must have been aching at the end of the game, especially with fouls, and abundant yellow cards to both sides. So for the first half, honours were even.
The second half was completely different with Monte taking the game to Monovar, and within 15 minutes, following a gorgeous cross ball from the right wing by Roberto, the ball fell to Dario, who belted the ball with a loping drive over the Monovar's keeper's head for a superb goal.
Monte continued in this vein, Roberto receiving a long ball from his defence, just put the ball over the Monovar net. Monte were clearly on top until on about 30 minutes and following a foul in the Monte goal area, Monovar earned a penalty which was easily put away, 1 – 1.
On about 40 minutes of the second half, the referee missed an obvious offside by Monovar, the player ran forward on a one on one with the Monte goalie, but Carlos saved what should have been a goal by a thundering shot to his face, clearing the ball to a Monte layer, panic over.
On almost the last kick of the game, Monovar were awarded a free-kick near the half-way line, only for a Monte player to stop the ball with his hand, to the great annoyance of Fernando and fans. Monovar were awarded another free-kick on the edge of the box, but Monte cleared the ball and the game ended 1 – 1.
Sub's used were Adrian for David, Fran for Omar, German for Diego, and Silverwood for Roberto. My man of the match would have been Roberto.
Potential trouble at the end of the game was thwarted by Fernando and officials when a Monovar "fan" tried to cause trouble with Dimitry and other Monte players.
TEAM:- Carlos, Dimitry, Fernando, Manuel, Daniel, Angel, Omar, Roberto, Dario, Macia, David. SUB's :- German, Fran, Silverwood, Adrian.
Monte's next game is at home against Bahia Santa Pola next weekend, details to be forwarded when known.
Next articleNetworks Theatre Orihuela – BE A PIRATE!


Are you interested in utilising Fedora on Raspberry Pi? Then join Peter Robinson and Linuxing in London for this month's meetup!
Learn tips and tricks for making better use of Gource to visualize your source code repositories and other data. Demos will feature examples from the Linux kernel.
Last year, the ODI found themselves wanting to use the code behind gov.uk for a new project. In this talk James and Sam from the ODI tech team will share their experience of picking up a codebase which was open source, but never really designed for reuse, and what they learned along the way.


text:
Better transport connectivity for southeast region needed for unleashing full potentials: Experts
Weak regional connectivity across land, air and sea has hampered the growth of southeast region of Vietnam. Economists believe for the region develop, it needs to change its mind set and strategy.
Tan Cang – Cai Mep Thi Vai Port in the southeastern province Ba Ria-Vung Tau. — Photo courtesy of the Ministry of Trade
Economist Tran Dinh Thien, member of the Economic Advisory Team of Prime Minister Nguyen Xuan Phuc, made the remarks during a conference on transport infrastructure connection in the region.
Comprising HCM City and six provinces of Ba Ria-Vung Tau, Dong Nai, Binh Duong, Binh Phuoc, Tay Ninh, and Binh Thuan, the area covers 31,300sq.km and has a population of 19.6 million.
Bordering the East Sea, the area is rich in resources and enjoys favourable climate conditions, making it the country's most economically developed region.
Thien said the Communist Party of Vietnam has paid significant attention to the development of the south-east region, especially HCM City, which has helped to maintain the region's lead in national GDP shares (50.8 per cent), State budget contributions, and other economic factors.
However, its 'driver of growth' status is under threat as the GDP growth in 2016-18 reached 6.72 per cent against the 9.08 per cent growth of the Northern Key Economic Zone, he remarked, attributing to weak connectivity between areas in the region, while ring roads and highways are insufficient or slow to implement.
If important projects like soon-to-be underway Long Thanh International airport in Dong Nai or the upgrading of Cai Mep-Thi Vai port suffer setbacks and delays, the impact would be felt on a national scale.
Thien stressed that transport is a "severe bottleneck" of the southeast region – the Southern Key Economic Zone only has 91km of highway (11 per cent of the country's total), Tan Son Nhat International Airport in HCM City is increasingly overloaded, and Cai Mep-Thi Vai sea terminal is currently operating over capacity while a synchronous system of roads and railways leading to and from the terminal is lacking.
He proposed the region receive a prioritised budget as well as "vanguard" development mechanisms to become a "modern locomotive" unleashing its full potential.
The vision of a synchronous development between air, road, rail and sea needing to be a well-oiled cooperation between the State and local administrations are not new ideas, but must be thoroughly implemented, Thien said.
He said he has floated the idea of direct monetary rewards for businesses that implement well infrastructure projects in the region based on the "benefits brought to the country" and its leadership.
In the northern region, the investment to infrastructure is mostly sourced from the State budget, but the benefits the southeast region can bring are still huge, mechanisms are needed to entice private businesses – especially encouraging Public-Private Partnership (PPP) model, according to Thien.
Nguyen Duc Kien, deputy chair of National Assembly's Economic Committee, said PM Phuc himself has noted the lack of connectivity in the region which lead to three trips to the area in a year, plus several other meetings and resolutions to boost the development here – especially repeatedly urging the completion of land clearance of Long Thanh International Airport project so that construction could start on time.
While experts at the conference agreed that localities have been proactive in seeking diverse investment sources for intra-provincial and inter-regional transport systems for development given the constrained State budget.
Binh Duong Province, for example, has called for capitals both inside and outside the province to invest in its infrastructure, becoming the first locality in the country to implement the build-operate-transfer (BOT) model in transport and achieve unexpected success.
Similar undertakings have also been done in Dong Nai, resulting in much "brighter picture" of transport in the province.
Kien, however, opined that the action plans of the localities in the region lack cohesion and coordination, and suffer from a clear "commander role."
Tran Hoang Ngan, Director of the Institute for Development Research of HCM City, also pointed out that the awareness of the benefits of transport infrastructure connectivity seems to be lacking, and are still limited to an individual province or city.
A shared database that the entire region can make use of has not been set up, and regional common infrastructure is not available due to the lack of appropriate mechanisms and consensus, Ngan said, adding that the interests between localities in the whole region have not been "harmonised".
He urged the Government to authorise local governments – especially HCM City, Binh Duong, Dong Nai, and Ba Ria-Vung Tau – to decide matters related to key projects in their jurisdiction.
Nguyen Danh Huy, Director of Department of Planning and Investment under the Ministry of Transport, stated the implementation of infrastructure projects in the region is "too slow."
There are 11 highways planned for the region, with a total length of 970km, Huy said.
According to the plan to 2020, 497km will be operational but currently only 122km have been put into use, while another 278km are still being built.
Airway-wise, by 2030, the southeast region will have three international airports – Tan Son Nhat, Long Thanh, and Con Dao – with total capacity of 103 million passengers a year, with Tan Son Nhat and Long Thanh shouldering about 50 million each.
The Ministry of Transport has submitted to the Government a proposed project on regional transport infrastructure connectivity for the southeast region worth VND54 trillion, which will require some deliberations as to the coordination between public and private sources for these estimated capital, Huy said. VNS
South-east, Mekong Delta regions to prioritise transport infrastructure
Minister of Planning and Investment Nguyen Chi Dung has urged localities in the south-east and Mekong Delta regions to develop transport infrastructure by diversifying the sources of and effectively using capital over the next five years.
Vietnam promotes and protects human rights in Covid-19 epidemic
A special flight to guarantee citizens' right to life
Building a specialized monitoring agency to increase effectiveness in ensuring the rights of ethnic minorities
Law on Support for and Development of Ethnic Minorities and Mountainous Areas should be issued
The government accepts economic damage to guarantee the right to health care for everyone
Human rights viewed from illegal sales of personal information
After 5 years, our country is really better, all people enjoy development results
Human rights considerations: Dialogue and cooperation are the only effective way
Resolution No. 28-NQ/TW: A progressive resolution promoting the right to social security
Despite difficulties, Vietnam constantly promotes the right to social security
Accepting economic losses, protecting people's lives at all costs
Vietnam - a typical case of integrating human rights protection
Achievements in ensuring human rights, the premise for Vietnam's positions at international forums
Ministry of Information and Communications strives to contribute to protecting children's rights in cyberspace
Stock market plummets, $10 billion lost

redpajama_set_name: RedPajamaCommonCrawl


I can absolutely not believe that my years as a student sitting in lecture all day are ending this week. I've been dreaming of this day for a long time. Seven years of spending long days in the classroom will soon be replaced by largely clinical rotations. Sure, I do have some that are more lecture or discussion based, such as animal welfare and nutrition. However, there will be many large blocks of time when I don't have to sit in a chair if I don't want to!
I'm trying hard to make this a good round of finals and end with a bang to try and boost my GPA as much as possible, since this is my last chance to get actual grades. Finals start this Friday and carry on for the next two weeks. We're lucky this time around that we have only eight finals! Quite a breeze after having finals numbering in the double digits for the past couple semesters.
I know that these are not the last exams I will ever take (the NAVLE and licensing boards are yet to come). But these last few finals are a drop in the bucket compared to the number of exams I've taken over the years. The countless pens that have died in my hands trying to rewrite notes, the hours spent, the sleep lost, all have led to this end of an era. I will finally be practicing veterinary medicine!
So, so proud of you! You have wanted to do this since you were 5 years old. You are living the dream, and about to realize all this hard work will have paid off!


Foil is a great way to have your designs stand out. To achieve awesome looking results using foil, we use a low mesh count screen and adhesives added to the ink during the screen printing process. The foil transfer is then laid over the top of the printed garment and heat and pressure is applied. Peel the foil and BANG, instant bling.
Pro : Creates a unique shine that you cannot get using inks alone.
Pro : Foil comes in several different styles and colors.
Con : Can dull out, flake and peal after repeated washing.
Con : Has the least longevity of any other print process.


text:
Designer Spotlight: The Sexy Yellow Night Watchers
DotYeti keeps the creative train running for hours on end, even when the world starts sleeping. The Sexy Yellow Team (team name credits to Art Director Ezra) are the mighty designers that come aboard and create fantastic work.
Meet the Night Shift Team: The Sexy Yellow
The Creative Yetis: Bryan and Bea
Bryan is the designer that epitomizes the idea of a chill guy that's lived life to the fullest. He's worked a variety of odd jobs, but always had his mind and creative passion that drives him forward. Besides turning to him for top-notch layout and design projects, he's the type of guy that will tell you what he knows and feels and leave you wondering if you were at a life coaching conference. His insights extend to his podcast and video projects that he makes as a hobby and an extension to his creativity.
Pro tip from Bryan: when you're stressed, give yourself "marinating time." Work a bit, rest, then get back to it and you'll find yourself more productive than ever.
A true master in her craft, Bea has been working in the creative industry for nearly 5 years. This design genius also likes to play video games, produce her own music and song covers in her free time.
Her can-do and friendly attitude has the ability to make anyone feel comfortable instantly. One thing she always says to graphic designers who are just starting out in the industry is to never compare their work to others — that they should always be proud of the designs they created. She is such an inspiration to everyone around her which makes us proud that she's part of our team.
The Motion Yeti: Alyssa
One of the easiest people to get along with has got to be Alyssa. Her calm attitude and wisdom will make you feel like you can talk to her for hours even though you've just met.
Aside from creating amazing graphic designs, she's also a fitness enthusiast. She regularly joins Spartan Races before the pandemic. To give you a gist of what she does there — she effortlessly carries heavy sacks, climbs ropes, and runs back and forth from one base to the next. Today she still challenges herself physically by doing push-up challenges to maintain her strong physique during her free time.
The Speedy Yeti: Mark
Mark is one of the pioneer designers that helped establish DotYeti. His nickname at work is "Terminator" which was given by our Top Yeti Roy because of his ability to work extremely fast without sacrificing the quality of the designs — and he is somewhat robotic.
Even though he may seem like a tough guy on the outside, he's really approachable and polite. He continues to help our team become better designers by challenging everyone with his amazing art skills and unique perspective on each design.
Creative Design Company Built on Fun and Joy
We nurture talent and creativity here at DotYeti and work with the best designers you'd ever find. If you want a shot at becoming part of our community, you can visit DotYeti's careers page for more information.
You can also check out the work of our design team here.
Excellent Names for Graphic Design Companies 2023
Are you looking for an appealing graphic design company name? Graphic design companies do have creative and witty names so they would be easy to remember. As a business in the creative field, the name of the graphic design company you choose should center on...
Top 10 Graphic Design Tips For Your Event
Whether launching a new product or hosting a grand birthday party, you need to captivate your target audience's attention. Your event graphic design can make or break your event. That's why you need to be strategic when it comes to graphic design for events. So how do...


for everything from meetings and networking events to private parties.
Hire the Mercat Grill for your next event.
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We're located only eight miles from Edinburgh City Centre – just off the A1, making us ideally located for your private event, away from the hustle and bustle of the big city and with the advantage of lots of free, onsite parking.
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For drinks, dining, get togethers with friends, family or work colleagues to private functions.
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Enjoy a superb selection of handpicked beverages, from wonderful wines, exclusive whiskies, craft gins and craft beers as well as excellent quality coffee, tea and soft drinks.


Installation of Exhibition - Sarcelles, 2017 from Camilo Leon-Quijano on Vimeo.
Installation, vernissage and exhibition of #lesrubywomen project in Collège Chantereine of Sarcelles.
180 linear meters of prints, 35 liters of glue and a two days in a nacelle at 12m high for installing the exhibition. In total: 22 photos (3,60x2,40m) of the rugbywomen.


text:
Calling all innovators and trailblazers to participate in a global entrepreneurship competition and become next generation disruptors
Category: July 2017 - Internet of Things (IoT) & Wearables
Global entrepreneurship	, next generation disruptors	,
Global Entrepreneurship Movement (GEM) is calling all entrepreneurs, startups, and innovators to participate in its first global entrepreneurship competition, ColossusINNO2017, where one winner will be awarded with a grand prize of USD 100,000.
Recently launched in June 2017, this competition was created by GEM, a non-government, not-for-profit organization, in order to recognize entrepreneurs with world class innovations that will help transform ASEAN's socio-economic community. ColussusINNO2017 competition kicks off in anticipation to help aspiring entrepreneurs develop innovative ideas into global businesses to create positive societal change in ASEAN
GEM's mission and purpose is to democratize entrepreneurship – to create zero barriers for aspiring entrepreneurs to spurn new enterprises, to be easily educated in the field of entrepreneurship and to connect the Malaysian and the ASEAN entrepreneurship ecosystem to other global ecosystems. The aim is to make innovation and entrepreneurship ubiquitous.
In line with GEM's mission, ColossusINNO2017 was incepted to inspire breakthrough thinking among entrepreneurs to identify and build ventures that will result in providing solutions to solve regional problems in developing countries in ASEAN – be it the problem of poverty, governance, education, health or environment.
According to GEM's patron Tan Sri Irwan Serigar, "It is a known fact that there is no lack of innovation in the ASEAN region, as the entrepreneurial ecosystem has evolved tremendously over the years. What we need now is greater confidence amongst our entrepreneurs to think big, take on bigger challenges and direct their imagination on innovating in ASIA for ASIA. Thus, the creation of ColossusINNO2017 – as the name suggests, we want the ideas to be 'born global."
Tan Sri Irwan added that ColossusINNO2017 is on the lookout for the next big solutions that help address the market gaps, which in turn will propel ASEAN nations closer towards first world status – such as innovation in the areas of governance, poverty eradication, education, health, environment and cleanliness. Furthermore, the timing for ColossusINNO2017 is just right as it is also the time where Malaysia celebrates the year of startup, and this global competition is one of the programs organized by GEM under Startup and SME Promotion Year (SSPY 2017).
President of GEM, Dash Dhakshinamoorthy also shared his enthusiasm for ColossusINNO2017 and was confident that it will attract top notch startups and ecosystem players from around the world.
"We are positive that a global competition of this scale and stature will attract the support from top notch ecosystem players – such as accelerators and early stage funders. And with its global support network, GEM aims to attract regional and global startups to launch their ventures in Malaysia through ColossusINNO2017, in hopes of garnering more interest from the international funding community and support network," commented Dash.
He also added that ColossusINNO2017 is working closely with a Silicon-Valley-based competition platform, You Noodle, to be its competition engine – to call for submissions, shortlist and select winners. The platform will also help engage with the judges to make the selection independently based on the parameters set together with global requirements. The panel of judges consists of world-class industry leaders and experts, including Director of Global Education at Georgetown University in Washington D.C., Sam Potolicchio, Advisor to the Vice President for the MENA Region at World Bank in Washington D.C., Olivier Lavinal, and Director and Leader of Digital Transformation at Cisco in India, Lux Rao.
This process of judging is to ensure that the winners meet the global standards in terms of knowledge, solution to problems and other requirements of a scalable startup.
The ColussusINNO2017 competition will occur in several stages. Submissions will be opened to challengers from across the world from June 21 to end of October, followed by the announcement of the top ten finalists in November. These finalists will be then invited to present their winning stories to ASEAN and global entrepreneurs, policymakers, industry leaders, investors and other startup community participants at the Global Entrepreneurship Community 2017 (GECommunity 2017). This will also give them an opportunity to connect on a global level via specially-curated programs, designed to build strategic linkages and drive real change.
ColossusINNO2017 is a collaborative effort by GEM and its partners – Ministry of Finance (MOF) and Malaysian Global Innovation & Creativity Centre (MaGIC), The group will announce the winner and hand over a cash fund prize worth USD 100,000 at the GECommunity 2017 on December 5 and 6 2017. Subsequently, the winner will be exposed to a variety of mentorship, resources and skills to aid the growth of the idea, amongst which collaboration with the local government and corporations will be made possible in aiding a stronger market entry.
By MediaBUZZ
July 2017 - Internet of Things (IoT) & Wearables
Older generations savvier about IoT security, Irdeto's new study revealed
What marketers want to achieve by implementing Artificial Intelligence (AI)
Boon or bane: Sharing contextual data via the Internet of Things
Why IoT and Industry 4.0 need professional marketing
The Internet of Things and Wearables for innovative, interactive Marketing
Wearable Tech: A growing payment opportunity
element14 community launches 'IoT on Wheels' Design Challenge with STMicroelectronics
IoT turns the security industry upside down
IoT, AI, and VR will determine the future of customer experiences
CtrlShift reorganizes its business around new SaaS offering – The Hub
AccuWeather introduces plain language AI weather bot for Facebook Messenger
Convergence 005 | SWITCH: A Consumer Revolution
Frost & Sullivan celebrated Asia Pacific's leading ICT firms at annual ICT Awards
Appointments & Announcements
Tweets by @MediaBUZZ_Asia


Candy Crush Level 496 Cheats, Tips, and Strategy. This is how you beat level 496 of Candy Crush Saga. Follow these Candy Crush cheats, tips and strategy and you will pass level 496 in no time! The goal of level 496 is to clear 33 jelly squares in 45 moves and get at least 65,00 points. Clear the meringues to form a path to the bottom of the board.
Candy Crush Level 593 Cheats and Tips August 16, 2014 By Candy Crush Level Cheats — Leave a Comment We currently have no written hints and tips for Candy Crush Level 593, but feel free to leave us your own experiences below, and if you're really struggling, you can request that we write a walkthrough for this level.
Candy Crush Soda Level 34 Cheats, Tips & Help November 1, 2014 By Candy Crush Level Cheats — Leave a Comment • In this level you need to get the bear above the candy string.
Nov 26, 2018· This is the first stage whereby luck plays an important part. 1. The rule is always start from the bottom 2. However when you can create specials (striped or wrapped) on top, just do them first.
Level 18 is the eighth level in Candy Factory and the 9th jelly level. To pass this level, you must clear 32 single and 32 double jelly squares in 25 moves or fewer. When you complete the level, Sugar Crush is activated and will score you additional points.
Swanky tips for Shuffle Cats. News card image. Meet the Star Cats! Bubble Witch 3 Saga. These furry friends are here to make levels easier! Check this video to find out how! 1 / 4. 109. open positions. ... Satisfy your sweet tooth with Official Candy Crush Merchandise! Go to the store . King.com.
Level 24 is the fourth level in Lemonade Lake and the 6th ingredients level. To pass this level, you must collect 1 hazelnut and 1 cherry in 20 moves or fewer. When you complete the level, Sugar Crush is activated and will score you additional points.
Candy Crush Jelly Level 33 Tips Requirement: Get more Pufflers than the Cupcake Carl [Pufflers boss mode] You have only 12 moves. Jelly Level 33 guide and cheats: This level has medium difficulty. For this level try to play near to special candy to break them to break more frosting blockers.
Candy Crush Saga is all about strategy, much like chess, so take a leaf out of the classic game. Tip #4: Restart a Level Without Losing Lives Every time you start a new level, the order of the jellies is completely randomized.
Welcome to Candy Crush level 34! Congratulations on making this far, and I trust you're having tons fun crushing your way to success. I hope you're ready to collect some ingredients because that's what this level is all about.
Other than Candy Crush level 130, Candy Crush level 140 is one very tricky level in Wafer Wharf. You are someone who played 139 levels in Candy Crush Saga so you should be able to know how and why the board is set in this way. But, you still cannot skip the tips and cheats provided here in order to win in this level.
Candy Crush Level 3832 Tips or Cheats: For ingredients level you need to play from bottom or where more candies are available to play to make special candy by mixing four or mare same color candies. Or try to crush the candies near to ingredients to move the ingredients to take them out from the table.
The Candy Crush level 33 is one of the most challenging levels in this game. An ordinary person with IQ less than 50 will find the puzzle in level 33 a hard thing to solve. An ordinary person with IQ less than 50 will find the puzzle in level 33 a hard thing to solve.
The goal of Candy Crush Saga level 33 is to remove 16 Jelly squares and reach 32,000 points in 17 moves or fewer.Read the tips, watch the video and get an idea of what you're supposed to do on Candy Crush level 33. Candy Crush Saga Level 33 Video. Demonstrated in the video below is how I completed the level.
Candy crush level 149 is an alphabet level because the level makes candies in plus, T or L shapes. The five same color candies combine and make plus forms or l shapes. This level principal object has removed all candies.


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Turkish Denialism is Security Threat for Armenia – FM Mnatsakanyan
Home Armenia EU-Armenia Parliamentary Cooperation Committee Ends in Strasbourg
EU-Armenia Parliamentary Cooperation Committee Ends in Strasbourg
An Appeal to End the Suffering in Syria
STRASBOURG — On January 21, the next sixteenth meeting of the EU-Armenia Parliamentary Cooperation Committee (PCC) ended in Strasbourg, where the RA NA Delegation composed of the NA deputies Samvel Farmanyan (Head of Delegation), Alexander Arzumanyan, Aghvan Vardanyan, Nikol Pashinyan, Vahe Enfiajyan, Edmon Marukyan and Levon Dokholyan, took part.
There were discussions on the perspectives of Armenia-EU relations, the new legal document regulating those relations, regional security, the Nagorno Karabakh conflict and Armenia-Turkey relations. At the European Parliament Committee on Foreign Affairs, moderated by the Committee Chairman Elmar Brok, an exchange of thoughts took place over agenda items, and the members of the Armenian Delegation answered the questions concerning the MPs of the European Parliament.
At the end of the two-day debates a joint Final Statement and Recommendations were adopted. In the adopted document, among other problems the concerning debates going on over the Nagorno Karabakh conflict in other inter-parliamentary organisations were also touched upon. In that connection, the European Parliament has stated that "it fully supports the negotiating process going on under the auspices of the OSCE Minsk Group Co-Chairs, which is the only internationally recognized format of the Nagorno Karabakh conflict settlement, and in this direction greatly encourages the Pan-European and other international organisations and institutions to be in consonance with the OSCE Minsk Group Co-Chairs' statements in their motions and resolutions on Nagorno Karabakh in order to contribute to the peaceful resolution in that way."
EU-Armenia Parliamentary Cooperation Committee , Sixteenth Meeting Final Statement
1. welcomes the high-level opening of EU-Armenia negotiations on 7 December 2015, in view of a new comprehensive framework agreement that would cover political, economic and sectoral cooperation; recalls that the PCC had already recommended, on 20 March 2015, to use the experience acquired in the scoping exercise to establish a legally binding contractual basis in order to replace the outdated 1999 Partnership and Cooperation Agreement;
2. recalling the work carried out in negotiating the AA/DCFTA trusts that the technical result of these talks, which was safeguarded for future reference at the time, will facilitate the swift completion of the EU / Armenia negotiations which are now re-launched;
3. expects to see firmly enshrined in the new agreement a strong commitment to rule of law, democracy and human rights, which should be cornerstones of the text; further underlines that the respect for these shared values can only enhance and further facilitate cooperation in all other issues expected to be covered by the agreement, from energy, transport, or the environment to investment and trade; reminds that Armenia remains covered by the preferential GSP+ trade scheme, with an enhanced role played by the EP in its reformed monitoring procedure;
4. believes that a new, ambitious EU/Armenia framework agreement is compatible with the progressive differentiation approach promoted within the Eastern Partnership, particularly taking into account the revised European Neighbourhood Policy; reminds that enhancing citizens mobility in a secure and well managed environment remains a core objective of the Eastern Partnership, and welcomes the overall good implementation by Armenia of the Visa facilitation and Readmission Agreements, stressing that continued and sustained progress is crucial in order to further proceed with a Visa Liberalisation Dialogue;
5. underlines the need for a strong parliamentary dimension to be included in any future agreement, encouraging legislators to develop procedures allowing them to be duly informed throughout the negotiation process, and fully involved in its subsequent implementation and monitoring;
6. recalls the outmost importance and value of the annual EU-Armenia Human Rights Dialogue, as a unique format allowing to constructively focus on issues such as fundamental freedoms, including the freedom of assembly and association, the development of civil society, the rule of law, mobility, judicial reform, the continued fight against corruption, the independence of the media and the fight against torture and inhuman and degrading treatment;
7. welcomes the considerable effort recently undertaken by the Armenian National Assembly to align national legislation with the UN Convention Against Torture and looks forward to further progress on the ancillary measures currently under examination to avoid torture and fight impunity; further welcomes the review of the Judicial Code and the strengthening of the independence of the judiciary provided by the Constitutional Amendments as supported by the Venice Commission, particularly the Council of Justice; stresses the importance the EU attaches to ICC ratification worldwide;
8. takes note of the 6 December Constitutional Referendum results, recalling the first and the second opinions delivered by the Venice Commission on the draft Amendments to the Constitution, and the local EU Statement issued on 10 December 2015 which urged the authorities to fully investigate in a transparent manner the election fraud allegations;
9. underlines that the full implementation of existing recommendations already issued by previous OSCE/ODIHR election observation missions remains a key to increase the confidence in the integrity of the electoral process, and expects that the necessary amendments, particularly pertaining to the electoral code, will be adopted well before the 2017 general elections;
10. recognizes that the EU's support over the years has been instrumental for the effective implementation and sustainability of Armenia's reform process and institutional capacity building; thus welcomes the EU adopting, on 16 December 2015, a new support programme of €30 million to enhance further efforts in the areas of quality employment, fiscal governance and civil society participation in the democratic decision making process; notes the decisive role civil society can play in enhancing transparency and accountability in all fields of public life, but also in monitoring GSP+ implementation, through dedicated EDIHR-funded projects; supports and encourages Armenian full participation in the COSME and Horizon 2020 programmes ;
11. deeply regrets the continued -and rising-human cost of the Nagorno-Karabakh conflict, and the increasingly serious violations of the cease-fire agreement, including by the use of heavy artillery/mortars, which repeatedly led to the loss of life even amongst the civilian population; stresses there can be no justification for the death and injury of innocent civilians on all sides, supporting the creation of a mechanism to investigate ceasefire violations, as repeatedly called for by the OSCE Minsk Group Co-Chairs; reminds that the Minsk Group Co-Chairs noted on 26 September that Armenia has agreed to discuss the details of this mechanism, urging Azerbaijan to do the same; expects from the EU to play a more active role in supporting further confidence building measures to spread the ideas of peace, reconciliation and trust;
12. believes that the recent escalation proves beyond possible doubt that the status quo has become unsustainable and that th ere can be no alternative to a peaceful solution in accordance with the UN Charter, Helsinki Final Act and relevant principles of international law;
13. firmly supports the negotiation process under the auspices of the OSCE Minsk Group Co-Chairs, as the only internationally agreed format for the settlement of the Nagorno-Karabakh conflict, in this regard, strongly encourages pan-European and other international institutions and bodies to align with the OSCE Minsk group co-chairs statements in their motions and resolutions on Nagorno-Karabakh in order to contribute to the peaceful resolution of the conflict; calls for a greater political commitment from all parties involved; calls on the European Union to include in its exchange and mobility programmes students and teachers hailing from conflict areas, such as Nagorno-Karabakh, as outlined in the Bologna Process communiqué of May 2015[1] adopted also by all EU member states and the European Commission;
14. welcomes the steps taken by the Government of Armenia in accepting around 20,000 Syrian refugees and calls on the European Commission and the EU member states to consider in a positive light any support request from the Armenian authorities in carrying out relevant programmes aimed at the further social integration of the Syrian refugees in Armenia.
15. recalls the commemorations of the Centennial of the Armenian Genocide which took place in Yerevan in April 2015, and commends the participating EU institutions and their Office-Holders ; encourages all Member States and EU Institutions to contribute further to the recognition of the Armenian Genocide, noting that the issue has, in recent years, become the focus of open and public debate in Turkey itself;
16. echoes the call of the EP Plenary, which, on 10 June 2015, urged the normalisation of Armenia – Turkey relations by the ratification, without preconditions, of the protocols on the establishment of diplomatic relations, and by the opening of the unilaterally closed border; supports initiatives aimed at promotion of regional cooperation, in the belief they can contribute to the ending of isolation of any country in the region; strongly hopes such developments will help to overcome the legacy of the past and face it courageously.
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