text:
\section{Introduction}
Grand unified theories (GUTs) in general~\cite{Georgi:1974sy}, and in particular models based on the $\text{SO}(10)$ gauge symmetry~\cite{Fritzsch:1974nn}, are popular extensions of the Standard Model (SM) of particle physics. They can provide solutions to a number of open questions in the SM, such as the nature of charge quantization, anomaly cancellation, and the existence of three separate gauge groups~\cite{Langacker:1980js}. Of a more phenomenological nature, $\text{SO}(10)$ models naturally account for the generation of small neutrino masses through the type I~\cite{Minkowski:1977sc,GellMann:1980vs,Mohapatra:1979ia,Yanagida:1979,Schechter:1980gr} or type II~\cite{Magg:1980ut,Lazarides:1980nt,Mohapatra:1980yp} seesaw mechanisms.

A prerequisite of grand unification is that the evolution of the SM gauge couplings with energy scale, governed by the renormalization group equations (RGEs), must be such that they converge to each other. It is well-known that this does not occur in non-supersymmetric (non-SUSY) models, but that invoking SUSY can allow for successful gauge coupling unification~\cite{Amaldi:1991cn,Giunti:1991ta,Langacker:1991an}. On the other hand, $\text{SO}(10)$ models allow for a symmetry breaking chain in multiple steps, due to it having rank five, which is one larger than the SM gauge group~\cite{Georgi:1979ga,delAguila:1980qag,Rizzo:1981su,Rizzo:1981dm,Buccella:1980qb,Rajpoot:1980xy,Yasue:1980qj,Yasue:1980fy,Anastaze:1983zk,Gipson:1984aj,Babu:1984mz,Chang:1984qr,Bertolini:2009qj,Bertolini:2009es}. This can allow gauge coupling unification even without invoking SUSY.

Currently, the only experimentally relevant prediction of GUTs is the instability of protons. The additional leptoquark scalar and gauge bosons that reside at the scale of unification $M_\mathrm{GUT}$ in general mediate proton decay. This, together with the non-observation of proton decay, places a lower bound on the value of $M_\mathrm{GUT}$. In turn, this can disfavor some of the possible intermediate gauge groups since they predict a value of $M_\mathrm{GUT}$ that is too low~\cite{Deshpande:1992au}.

Threshold corrections~\cite{Weinberg:1980wa,Hall:1980kf} are one-loop corrections arising from fields lying at and around the scale of symmetry breaking that modify the matching conditions of the gauge couplings of the models above and below the energy scale of symmetry breaking. This can in turn modify the value of $M_\mathrm{GUT}$  and thereby save some of the models that were previously disfavored~\cite{Parida:1989an,Rani:1993pp,Mohapatra:1992dx,Bertolini:2009qj,Babu:2015bna,Parida:2016hln,Chakrabortty:2019fov}. Furthermore, since threshold corrections modify the matching conditions of the gauge couplings, they can allow for unification in models where the gauge couplings do not unify~\cite{Lavoura:1993su,Ellis:2015jwa,Schwichtenberg:2018cka}.

In this work, we consider the direct breaking of $\text{SO}(10)$ to the SM as well as all relevant models with one intermediate symmetry breaking scale. For the model with direct breaking to the SM, we study how the threshold corrections may allow for unification without the addition of intermediate symmetries. For the models with an intermediate symmetry breaking step in which unification is achieved, we investigate how the threshold corrections affect $M_\mathrm{GUT}$ and the intermediate scale $M_\mathrm{I}$. Thereby, we quantify how large threshold corrections are required in order to save the models that \textit{a priori} predict a proton lifetime that is too short. The renormalization group (RG) running is performed at two-loop level, with the one-loop level result given for comparison.

In Sec.~\ref{sec:models}, we discuss the models that are analyzed in this work and present the solutions to the RGEs. We comment on which models achieve gauge coupling unification and the prediction for the proton lifetime in each of the models. Then, in Sec.~\ref{sec:thresholds}, we describe the computation of threshold corrections. In Sec.~\ref{sec:results}, we present the results of the threshold corrections for the different models. Finally, in Sec.~\ref{sec:conclusion}, we summarize our findings and conclude.

\section{Models}\label{sec:models}
In this section, we discuss the eight models that we investigate in this work. Furthermore, we give some details on the particle content that is involved in each model and comment on the RG running.

The most minimal non-SUSY $\text{SO}(10)$-based model is one in which the gauge symmetry is broken directly to the SM. Following this logic, the next-to-minimal breaking chains are those with one intermediate gauge symmetry. The possible intermediate breaking chains may be seen, for example, in Refs.~\cite{Chang:1984qr,Ferrari:2018rey}. Here, we consider the direct breaking of $\text{SO}(10)$ to the SM as well as all models with one intermediate symmetry breaking scale with at least two group factors in the intermediate symmetry. The reason that we require at least two group factors is that if there is only one, \textit{e.g.} $\,\text{SU}(5)$, then this does not help enable gauge coupling unification.

In all models, the fermionic particle content consists of three generations of the spinorial $\rep{16}_F$. In order to accommodate realistic fermion mass and mixing parameters, the scalar sector contains a complexified $\rep{10}_H$ representation~\cite{Babu:1992ia,Bajc:2005zf} and a $\repb{126}_H$ representation. In order to retain some predictivity of the Yukawa sector of $\text{SO}(10)$, we impose a Peccei--Quinn (PQ) symmetry~\cite{Peccei:1977hh,Peccei:1977ur}, which forbids one of the two independent couplings between the fermions and the $\rep{10}_H$.

We assume that below the intermediate symmetry breaking scale $M_\mathrm{I}$, only the SM particle content survives and all other multiplets have masses around either $M_\mathrm{GUT}$ or $M_\mathrm{I}$. This is in accordance with the ``Survival Hypothesis", namely that scalars acquire the largest possible mass that is compatible with the symmetry breaking~\cite{Georgi:1979md,delAguila:1980qag,Mohapatra:1982aq}.

To one-loop order in perturbation theory, gauge couplings evolve from one scale $M$ to another scale $\mu$ according to
\begin{equation}
\alpha_i^{-1} (\mu) = \alpha_i^{-1} (M) - \frac{a_i}{2\pi}\ln\left(\frac{\mu}{M} \right),
\end{equation}
where the index $i$ denotes the group to which the gauge coupling corresponds. The coefficient $a_i$, known as the $\beta$ coefficient, is determined by the particle content that exists in the relevant energy regime. It is given by~\cite{Jones:1981we,Machacek:1983tz}
\begin{equation}
a_i = -\frac{11}{3}C_2(G_i) +\frac{4}{3}\kappa_F S_2(F_i) + \frac{1}{3} \kappa_S S_2(S_i),
\end{equation}
where $C_2(r)$ is the quadratic Casimir and $S_2(r)$ [sometimes also denoted $C(r)$] is the Dynkin index of the representation $r$, related to the quadratic Casimir by
\begin{equation}
S_2(r) = \frac{d(r)}{d(G)}C_2(r),
\end{equation}
where $G$ refers to the adjoint representation and $d(r)$ denotes the dimension of representation $r$. Furthermore, $F_i$ is the representation that the fermions belong to and $S_i$ is the representation of the scalars. The coefficient $\kappa_F$ is $1$ for Dirac fermions and $1/2$ for Weyl fermions and $\kappa_S$ is $1$ for complex scalars and $1/2$ for real scalars.

To two-loop order in perturbation theory, the gauge couplings obey the differential equation
\begin{equation}
\frac{\text{d}\,\alpha_i^{-1}(\mu)}{\text{d}\ln\mu} = -\frac{a_i}{2\pi} - \sum_j \frac{b_{ij}}{8\pi^2 \alpha_j^{-1}(\mu)},
\end{equation}
where the two-loop coefficients are given by~\cite{Jones:1981we,Machacek:1983tz}
\begin{equation}\label{eq:2loop}
b_{ij} = -\frac{34}{3}\left[ C_2(G_i)\right]^2 \delta_{ij} + \kappa_F\left[4C_2(F_j) + \frac{20}{3} C_2(G_i)\delta_{ij} \right]S_2(F_i) + \kappa_S \left[4C_2(S_j) + \frac{2}{3} C_2(G_i)\delta_{ij} \right]S_2(S_i).
\end{equation}
There is also a contribution from the Yukawa coupling to the two-loop $\beta$ function above. However, since the RG running of the Yukawa couplings is somewhat model-dependent, we neglect that term in Eq.~\eqref{eq:2loop}. The $\beta$ coefficients $a_i$ and $b_{ij}$ for the models considered are listed in Tab.~\ref{tab:beta} in App.~\ref{app:beta}.

Given the values of the gauge couplings at the electroweak scale $M_Z \simeq 91.1876\,\mathrm{GeV}$,
\begin{equation}
\left(\alpha^{-1}_3(M_Z),\, \alpha^{-1}_2(M_Z),\, \alpha^{-1}_1(M_Z)\right) = (8.50, \,29.6,\, 59.0),
\end{equation}
the system of RGEs can be solved. Depending on the $\beta$ functions, precise gauge coupling unification may be obtained. If it is possible, then that model is an allowed model for grand unification.

The relevant experimental prediction of grand unification related to the scale of unification is proton decay. From the scale of grand unification and the coupling strength $g_\text{GUT}$ at that scale, the proton lifetime in the most constraining channel can be computed as~\cite{Nath:2006ut,Babu:2010ej}
\begin{equation}
\Gamma(p\rightarrow e^+\pi^0 ) \simeq \frac{m_p}{64\pi f_\pi^2} \frac{g_\text{GUT}^4}{M_\mathrm{GUT}^4}A_L^2 \alpha_H^2 F_q,
\end{equation}
where $f_\pi\simeq 139\,\text{MeV}$ is the pion decay constant, $A_L\simeq 2.726$ is a renormalization factor, $\alpha_H\simeq 0.012\,\mathrm{GeV} ^3$ is the hadronic matrix element, and $F_q\simeq 7.6$ is a quark-mixing factor. With these numerical factors, the proton lifetime in this channel can be estimated by
\begin{equation}
\tau(p\rightarrow e^+\pi^0)\simeq (7.47\times 10^{35}\, \text{yr}) \left(\frac{M_\mathrm{GUT}}{10^{16}\,\mathrm{GeV}}\right)^4 \left(\frac{0.03}{\alpha_\text{GUT}}\right)^2.
\end{equation}
Since proton decay has not been experimentally observed, there is a lower bound on the lifetime of the proton. The most constraining one is from Super-Kamiokande~\cite{Mine:2016mxy,Miura:2016krn,Abe:2014mwa} with the bound $\tau(p\rightarrow e^+\pi^0)>1.67\times 10^{34}\,\text{yr}$ at $90~\%$ confidence level. Any model must be able to accommodate a proton lifetime longer than the experimental bound.

In what follows, we employ the conventions that gauge couplings, $\beta$ coefficients, and representations of fields appear in the order in which the gauge group is written. For example, in $\,\text{SU}(3)_\text{C}\times \,\text{SU}(2)_\text{L}\times\text{U}(1)_Y$, the first entry corresponds to $\,\text{SU}(3)_\text{C}$, the second to $\,\text{SU}(2)_\text{L}$, and the third to $\text{U}(1)_Y$. For representations of fields, Abelian charges are always listed as subscripts.

\subsection{No Intermediate Symmetry}
The direct breaking of the $\text{SO}(10)$ symmetry to the SM gauge group $\G{321} = \,\text{SU}(3)_\text{C}\times \,\text{SU}(2)_\text{L}\times\text{U}(1)_Y$ can be achieved with a $\rep{144}_H$ taking a vacuum expectation value (vev) in the appropriate direction~\cite{Babu:2005gx}. We then assume that all multiplets from within the $\rep{144}_H$ have masses at $M_\mathrm{GUT}$. Assigning a non-zero PQ charge to the $\rep{144}_H$ allows it to also break the PQ symmetry at $M_\mathrm{GUT}$. Further, we assume that from the $\rep{10}_H$ and the $\repb{126}_H$, only one combination of the $\,\text{SU}(2)_\text{L}$ doublets survives below $M_\mathrm{GUT}$. This is the SM Higgs doublet~\cite{Bajc:2005zf}, such that the SM particle content is recovered below $M_\mathrm{GUT}$. The other fields that are not part of the SM field content reside at $M_\mathrm{GUT}$. These are listed in Tab.~\ref{tab:SM} in App.~\ref{app:fields}.

From the particle content described above, the $\beta$ coefficients $a_i$ and $b_{ij}$ may be computed. They are listed in Tab.~\ref{tab:beta} in App.~\ref{app:beta}. The resulting evolution of the SM gauge couplings is shown in Fig.~\ref{fig:SM}. As is well known, the gauge couplings fail to unify.

\subsection{$\,\text{SU}\text{(4)}\times\,\text{SU}\text{(2)}\times\,\text{SU}\text{(2)}$}\label{sec:422}
A popular model of intermediate symmetry in the breaking of $\text{SO}(10)$ is the Pati--Salam (PS) model, based on the gauge symmetry $\G{422} = \,\text{SU}(4)_\text{C}\times \,\text{SU}(2)_\text{L}\times \,\text{SU}(2)_\text{R}$~\cite{Pati:1974yy}. It is a maximal subgroup of $\text{SO}(10)$ and contains the SM gauge group $\G{321}$ as a subgroup. The fermions are embedded in this model as $(\rep{4},\rep{2},\rep{1})\oplus (\repb{4},\rep{1},\rep{2})$.

Models based on a Pati--Salam intermediate symmetry have been investigated extensively in the literature~\cite{Babu:1992ia,Lavoura:1993vz,Buccella:2012kc,Meloni:2014rga,Meloni:2016rnt,Babu:2016bmy,Ohlsson:2018qpt}. In this work, we follow the model described in Ref.~\cite{Altarelli:2013aqa}, in which the $\text{SO}(10)$ symmetry is broken by a vev in the $\rep{210}_H$. To break the PS symmetry as well as the PQ symmetry down to $\G{321}$, we employ a vev in the $\repb{126}_H$ together with a complex $\rep{45}_H$ with a non-zero PQ charge.

The reason that two separate vevs are needed even though the $\repb{126}_H$ has a non-zero PQ charge is to break the linear combination of PQ, $B-L$, and $T_{3,\text{R}}$ which is otherwise left invariant~\cite{Mohapatra:1982tc,Holman:1982tb,Bajc:2005zf}. Although the minimal choice of an extra representation for the breaking of the PQ symmetry could be considered to be a singlet, we will not use this. The reason is that singlets have mass terms that are unprotected by any symmetry and this choice would therefore introduce unnecessary fine-tuning. Thus a $\rep{45}_H$ is introduced.

Between $M_\mathrm{GUT}$ and $M_\mathrm{I}$, the scalar fields are $(\rep{1},\rep{2},\rep{2})$ from the $\rep{10}_H$, $(\rep{15},\rep{2},\rep{2})\oplus(\rep{10},\rep{1},\rep{3})$ from the $\repb{126}_H$, and $(\rep{1},\rep{1},\rep{3})$ from the $\rep{45}_H$. From these, the $\beta$ coefficients, listed in App.~\ref{app:beta}, can be computed. The fields that lie at $M_\mathrm{GUT}$ and $M_\mathrm{I}$ are given in Tab.~\ref{tab:PS} in App.~\ref{app:fields}.

To compute the RG running in this model, one also requires the matching conditions between the SM and the PS models. The gauge couplings of the model based on $\G{422}$ at $M_\mathrm{I}$ are derived from the gauge couplings of the SM by
\begin{align}
\alpha_4^{-1}(M_\mathrm{I}) &= \alpha_3^{-1}(M_\mathrm{I}),  \label{eq:match3}\\
\alpha_\text{2L}^{-1}(M_\mathrm{I}) &= \alpha_2^{-1}(M_\mathrm{I}), \label{eq:match2}\\
\alpha_\text{2R}^{-1}(M_\mathrm{I}) &= \frac{5}{3} \alpha_Y^{-1}(M_\mathrm{I}) - \frac{2}{3}\alpha_3^{-1}(M_\mathrm{I}) .\label{eq:matchY}
\end{align}
The resulting RG evolution is shown in Fig.~\ref{fig:PS}. At two-loop level, the solutions for the scales gives an intermediate scale of $M_\mathrm{I} \approx 2.64\times 10^{9}\,\mathrm{GeV}$ and a unification scale of $M_\mathrm{GUT} \approx 3.72\times 10^{16}\,\mathrm{GeV}$, corresponding to a proton lifetime of $\tau_p\approx 1.2\times 10^{38}\,\text{yr}$.\footnote{In comparison, at one-loop level, the scales are $M_\mathrm{I} \approx 1.28\times 10^{11}\,\mathrm{GeV}$ and $M_\mathrm{GUT} \approx 1.96\times 10^{16}\,\mathrm{GeV}$, corresponding to a proton lifetime of $\tau_p\approx 1.3\times 10^{37}\,\text{yr}$.} Hence, it is clear that the PS model is allowed by the proton lifetime limit.

\subsection{$\,\text{SU}\text{(4)}\times\,\text{SU}\text{(2)}\times\,\text{SU}\text{(2)}\times \text{D}$}
This model is based on a PS gauge symmetry with an additional left-right $D$ parity which acts on the fields such that $(\rep{r_4},\rep{r_L},\rep{r_R})\rightarrow(\repb{r_4},\rep{r_R},\rep{r_L})$~\cite{Mohapatra:1974hk,Mohapatra:1974gc,Senjanovic:1975rk}. For a previous analysis of a similar model see \textit{e.g.} Ref.~\cite{Babu:2015bna}.

In order to preserve $D$ parity when breaking the $\text{SO}(10)$ symmetry, a $\rep{54}_H$ can be used. The breaking of the $\G{422D}$ symmetry can be achieved using the $\repb{126}_H$. A $\rep{45}_H$ is used to also break the PQ symmetry, as described in Sec.~\ref{sec:422}.

Between $M_\mathrm{GUT}$ and $M_\mathrm{I}$, we have $(\rep{1},\rep{2},\rep{2})$ from the $\rep{10}_H$ and $(\rep{15},\rep{2},\rep{2})\oplus (\rep{10},\rep{1},\rep{3})\oplus (\repb{10},\rep{3},\rep{1})$ from the $\repb{126}_H$. Note that the latter is needed only to preserve $D$ parity of the model. From the $\rep{45}_H$, we have $(\rep{1},\rep{1},\rep{3})\oplus (\rep{1},\rep{3},\rep{1})$, where, again, the latter representation only serves to conserve $D$ parity. Note that the difference between this model and the one in Ref.~\cite{Babu:2015bna} is that they do not include the $\rep{45}_H$ and that they place $(\rep{6},\rep{1},\rep{1})$ from the $\repb{126}_H$ at $M_\mathrm{I}$ due to considerations of the scalar potential, which are beyond the scope of this work. The fields that lie at $M_\mathrm{GUT}$ and $M_\mathrm{I}$ are given in Tab.~\ref{tab:PSD} in App.~\ref{app:fields}.

With the particle content described, we can compute the $\beta$ coefficients which are given in App.~\ref{app:beta}. The matching conditions are the same as for the PS model, namely Eqs.~\eqref{eq:match2}--\eqref{eq:matchY}, with the additional constraint that $\alpha_\text{2L}^{-1}=\alpha_\text{2R}^{-1}$ due to $D$-parity. With these, one can calculate the RG evolution of the gauge couplings and the required intermediate scale for their unification, as shown in Fig.~\ref{fig:PSD}. At two-loop level, the intermediate scale is $M_\mathrm{I}\approx 4.34\times 10^{13}\,\mathrm{GeV}$ and the unification scale is $M_\mathrm{GUT}\approx 7.45\times 10^{14}\,\mathrm{GeV}$, giving a proton lifetime of $\tau_p\approx 3\times 10^{31}\,\text{yr}$, below the experimental lower limit, as previously noted in \textit{e.g.}~\cite{Deshpande:1992au}.\footnote{At one-loop level, the scales are $M_\mathrm{I}\approx 5.00\times 10^{13}\,\mathrm{GeV}$ and $M_\mathrm{GUT}\approx 1.40\times 10^{15}\,\mathrm{GeV}$, resulting in a proton lifetime of $\tau_p\approx 3.7\times 10^{32}\,\text{yr}$, which is also too short.}

\subsection{$\,\text{SU}\text{(4)}\times\,\text{SU}\text{(2)}\times\text{U(1)}$}
The gauge group $\G{421}=\,\text{SU}(4)_\text{C}\times\,\text{SU}(2)_\text{L}\times\text{U}(1)_\text{R}$ is a subgroup of the PS gauge group, but it may be reached directly by breaking the $\text{SO}(10)$ symmetry. This is possible by assigning a vev to the appropriate direction of the $\rep{45}_H$. Models based on this gauge group have been previously analyzed in \textit{e.g.} Refs.~\cite{Parida:1991sj,Rani:1993pp}. The breaking of $\G{421}$ down to $\G{321}$ can then be done using a vev of $(\repb{10},\rep{1})_1$ from the $\repb{126}_H$. Since now the $\rep{45}_H$, which carries a PQ charge, is used to break the symmetry at $M_\mathrm{GUT}$, the PQ symmetry will also be broken at that scale. Contrary to the  previously discussed models, we do not need to include two separate vevs to break the remaining linear combination of charges, since now both $B-L$ and $\text{R}$ remain unbroken at $M_\mathrm{GUT}$.

To compute the $\beta$ coefficients between $M_\mathrm{I}$ and $M_\mathrm{GUT}$, we first need to list the various fields that are present between those two scales. For the fermions, we have $(\rep{4},\rep{2})_0\oplus (\repb{4},\rep{1})_{1/2}\oplus (\repb{4},\rep{1})_{-1/2}$. The scalars are $(\rep{1},\rep{2})_{-1/2}$ from the $\rep{10}_H$ and $(\rep{15},\rep{2})_{-1/2}\oplus (\rep{10},\rep{1})_{-1}$ from the $\repb{126}_H$. The resulting $\beta$ coefficients are found in App.~\ref{app:beta} and the resulting gauge coupling unification can be seen in Fig.~\ref{fig:421}. The matching conditions between $\G{321}$ and $\G{421}$ are identical to those given in Eqs.~\eqref{eq:match2}--\eqref{eq:matchY}, with the replacement $\alpha_\text{2R}^{-1}(M_\mathrm{I}) \rightarrow \alpha_\text{1R}^{-1}(M_\mathrm{I})$. The fields that lie at $M_\mathrm{GUT}$ and $M_\mathrm{I}$ are given in Tab.~\ref{tab:421} in App.~\ref{app:fields}.

At two-loop level, the resulting scales are $M_\mathrm{I}\approx 1.57\times 10^{11}\,\mathrm{GeV}$ and $M_\mathrm{GUT} \approx 2.69\times 10^{14}\,\mathrm{GeV}$, giving a proton lifetime of $\tau_p\approx 5.8\times 10^{29}\,\text{yr}$.\footnote{At one-loop level, the scales are $M_\mathrm{I}\approx 1.35\times 10^{11}\,\mathrm{GeV}$ and $M_\mathrm{GUT}\approx 4.60\times 10^{14}\,\mathrm{GeV}$, corresponding to a proton lifetime of $\tau_p\approx 5.1\times 10^{30}\,\text{yr}$.} Thus, as noted previously in the literature~\cite{Deshpande:1992au}, this model is ruled out by proton decay bounds.

\subsection{$\,\text{SU}\text{(3)}\times\,\text{SU}\text{(2)}\times\,\text{SU}\text{(2)}\times \text{U(1)}$}\label{sec:3221}
Another subgroup of the PS gauge group is $\G{3221}=\,\text{SU}(3)_\text{C}\times\,\text{SU}(2)_\text{L}\times\,\text{SU}(2)_\text{R}\times \text{U}(1)_{B-L}$, as studied in \textit{e.g.} Refs.~\cite{Chang:1983fu,Chang:1984uy,Chakrabortty:2017mgi}. This may be reached by direct breaking of the $\text{SO}(10)$ symmetry by \textit{e.g.} a vev in the $\rep{45}_H$. Similar to the $\G{421}$ model, the PQ symmetry is broken at $M_\mathrm{GUT}$ by the $\rep{45}_H$ so that only one vev is required to break the symmetry at $M_\mathrm{I}$. The breaking of $\G{3221}$ down to $\G{321}$ can be achieved with $(\rep{1},\rep{1},\rep{3})_2$ from the $\repb{126}_H$.

Computing the $\beta$-functions between $M_\mathrm{I}$ and $M_\mathrm{GUT}$, we first note that the fermions are embedded as $(\rep{3},\rep{2},\rep{1})_{1/3}\oplus (\rep{1},\rep{2},\rep{1})_{-1}\oplus (\repb{3},\rep{1},\rep{2})_{-1/3}\oplus (\rep{1},\rep{1},\rep{2})_1$. The scalars that are between those two scales are $(\rep{1},\rep{2},\rep{2})_0$ from the $\rep{10}_H$ as well as $(\rep{1},\rep{2},\rep{2})_0\oplus (\rep{1},\rep{1},\rep{3})_{-2}$ from the $\repb{126}_H$. Based on these fields, the $\beta$ coefficients can be found in App.~\ref{app:beta}. The fields that lie at $M_\mathrm{GUT}$ and $M_\mathrm{I}$ are given in Tab.~\ref{tab:3221} in App.~\ref{app:fields}.

In this model, the matching condition at $M_\mathrm{I}$ is more involved than in the above-discussed models due to the fact that the $B-L$ needs to be appropriately normalized. Before normalization, the hypercharge $Y$ may be expressed as
\begin{equation}
Y = \frac{B-L}{2} - T_{3R}.
\end{equation}
In order to normalize these charges, the hypercharge $Y$ is multiplied by the GUT normalization factor of $\sqrt{3/5}$ and the $B-L$ charge is multiplied by $\sqrt{3/8}$. From this, one can derive the matching conditions of the appropriately normalized gauge couplings, namely
\begin{equation}
\alpha_Y^{-1} = \frac{2}{5} \alpha_{B-L}^{-1} + \frac{3}{5}\alpha_{R}^{-1}.
\end{equation}
In order to invert this relation, we face the issue that we are matching three gauge couplings to four. Thus, we introduce the parameter $x$ such that $\alpha_{B-L}^{-1}(M_\mathrm{I}) = x\alpha_\text{R}^{-1}(M_\mathrm{I})$. This parameter is then solved for together with the scales $M_\mathrm{GUT}$ and $M_\mathrm{I}$ such that gauge coupling unification is achieved. The resulting matching conditions are
\begin{align}
\alpha_3^{-1}(M_\mathrm{I}) &= \alpha_3^{-1}(M_\mathrm{I}),\\
\alpha_{2\text{L}}^{-1}(M_\mathrm{I}) &= \alpha_2^{-1}(M_\mathrm{I}), \\
\alpha_\text{2R}^{-1}(M_\mathrm{I}) &= \left(\frac{2}{5} x + \frac{3}{5}\right)^{-1} \alpha_Y^{-1}(M_\mathrm{I}), \\
\alpha_{B-L}^{-1}(M_\mathrm{I}) &= x\left(\frac{2}{5} x + \frac{3}{5}\right)^{-1} \alpha_Y^{-1}(M_\mathrm{I}).
\end{align}
The RG running is shown in Fig.~\ref{fig:3221}.

Solving for the scales that result in unification, we obtain at two-loop level $M_\mathrm{I} \approx 1.57\times 10^{10}\,\text{GeV}$ and $M_\mathrm{GUT} \approx 5.18\times 10^{15}\,\text{GeV}$ with the parameter $x\approx1.38$, resulting in a proton lifetime of $\tau_p\approx 9.4\times 10^{34}\,\text{yr}$.\footnote{At one-loop level, the result is $M_\mathrm{I} \approx 6.59\times 10^{9}\,\text{GeV}$ and $M_\mathrm{GUT} \approx 1.39\times 10^{16}\,\text{GeV}$ with the parameter $x\approx1.43$, resulting in $\tau_p\approx 5.0\times 10^{36}\,\text{yr}$.} Therefore, this model is allowed by proton lifetime considerations.

\subsection{$\,\text{SU}\text{(3)}\times\,\text{SU}\text{(2)}\times\,\text{SU}\text{(2)}\times \text{U(1)}\times \text{D}$}
A similar model to the one in Sec.~\ref{sec:3221} but with a surviving $D$ parity may be constructed. In this case, the $\text{SO}(10)$ symmetry is broken down to the group $\G{3221D}=\,\text{SU}(3)_\text{C}\times\,\text{SU}(2)_\text{L}\times\,\text{SU}(2)_\text{R}\times \text{U}(1)_{B-L}\times D$ by a vev in the $\rep{210}_H$, which conserves the $D$ parity. Such models have been previously studied in \textit{e.g.} Ref.~\cite{Mohapatra:1974gc,Senjanovic:1975rk,Buccella:1987hq}. Then, $\G{3221D}$ is broken to $\G{321}$ using $(\rep{1},\rep{1},\rep{3})_2$ from the $\repb{126}_H$. The PQ symmetry is broken at $M_\mathrm{I}$ by $(\rep{1},\rep{1},\rep{3})_0$ from the $\rep{45}_H$.

The fermion embedding is the same as in Sec.~\ref{sec:3221}. For the scalars, between $M_\mathrm{GUT}$ and $M_\mathrm{I}$, there is $(\rep{1},\rep{2},\rep{2})_0$ from the $\rep{10}_H$. From the $\repb{126}_H$, we have $(\rep{1},\rep{1},\rep{2})_0$ for the SM Higgs and $(\rep{1},\rep{1},\rep{3})_{-2}$ for the symmetry breaking. To conserve $D$ parity, we also need to have $(\rep{1},\rep{3},\rep{1})_{2}$. From the $\rep{45}_H$, we have $(\rep{1},\rep{1},\rep{3})_0$ which is used to break the PQ symmetry as well as $(\rep{1},\rep{3},\rep{1})_0$ in order to conserve $D$ parity. The fields that lie at $M_\mathrm{GUT}$ and $M_\mathrm{I}$ are given in Tab.~\ref{tab:3221D} in App.~\ref{app:fields}.

From these fields, the $\beta$ coefficients may be calculated and are given in App.~\ref{app:beta}. Although this model has a similar gauge structure to the one in Sec.~\ref{sec:3221}, the matching conditions become somewhat simpler due to the requirement that $\alpha_{2\text{L}}^{-1}=\alpha_{2\text{R}}^{-1}$. This removes the extra freedom introduced by the parameter $x$ above and the matching conditions simply read
\begin{align}
\alpha_3^{-1}(M_\mathrm{I}) &= \alpha_3^{-1}(M_\mathrm{I}),\\
\alpha_{2\text{L}}^{-1}(M_\mathrm{I}) &= \alpha_2^{-1}(M_\mathrm{I}), \\
\alpha_\text{2R}^{-1}(M_\mathrm{I}) &= \alpha_2^{-1}(M_\mathrm{I}), \\
\alpha_{B-L}^{-1}(M_\mathrm{I}) &= \frac{5}{2}\alpha_Y^{-1}(M_\mathrm{I}) + \frac{3}{2} \alpha_2^{-1}(M_\mathrm{I}).
\end{align}
The resulting gauge coupling running is shown in Fig.~\ref{fig:3221D}.

The scales that result in gauge coupling unification with at two-loop level are $M_\mathrm{I}\approx 3.13\times 10^{11}\,\mathrm{GeV}$ and $M_\mathrm{GUT} \approx 6.31\times 10^{14}\,\mathrm{GeV}$, resulting in a proton lifetime of $\tau_p\approx 1.9\times 10^{31}\,\text{yr}$.\footnote{At one-loop level, the scales are $M_\mathrm{I}\approx 1.69\times 10^{11}\,\mathrm{GeV}$ and $M_\mathrm{GUT}\approx 1.29\times 10^{15}\,\mathrm{GeV}$, with a proton lifetime of $\tau_p\approx 3.3\times 10^{32}\,\text{yr}$.} This model is therefore disfavored due to its prediction of the proton lifetime.

\subsection{$\,\text{SU}\text{(3)}\times\,\text{SU}\text{(2)}\times\text{U}\text{(1)}\times \text{U(1)}$}
A subgroup of the $\G{3221}$ group is $\G{3211} = \,\text{SU}(3)_\text{C}\times\,\text{SU}(2)_\text{L}\times\text{U}(1)_\text{R}\times \text{U}(1)_{B-L}$, which may also be reached directly by breaking the $\text{SO}(10)$ symmetry. In order to do so, the singlet inside the $\rep{210}_H$ which also breaks $\G{421}$ and $\G{3221}$ must take a vev. For models based on this gauge group, see \textit{e.g.} Refs.~\cite{Chang:1984uy,Parida:1989an,Parida:1991sj}. The breaking of $\G{3211}$ down to $\G{321}$ can then be achieved using $(\rep{1},\rep{1})_{1,2}$ in the $\repb{126}_H$. In order to also break the remaining combination of the Abelian charges and the PQ symmetry, a vev must be taken by one of the singlets in the $\rep{45}_H$.

The fermions are embedded as $(\rep{3},\rep{2})_{0,1/3}\oplus (\rep{1},\rep{2})_{0,-1}\oplus (\repb{3},\rep{1})_{-1/2,-1/3}\oplus (\repb{3},\rep{1})_{1/2,-1/3}\oplus (\rep{1},\rep{1})_{1/2,1}\oplus (\rep{1},\rep{1})_{-1/2,1}$. The scalars that contribute to the RG running of the gauge couplings are $(\rep{1},\rep{2})_{-1/2, 0}$ from the $\rep{10}_H$ and $(\rep{1},\rep{2})_{-1/2,0}$ as well as $(\rep{1},\rep{1})_{-1,-2}$ from the $\repb{126}_H$. The field from the $\rep{45}_H$ does not contribute since it is a singlet.

From this, one can compute the $\beta$ coefficients, given in App.~\ref{app:beta}, as well as the RG running using the same matching conditions as in Sec.~\ref{sec:3221}, replacing $\alpha_\text{2R}^{-1}$ by $\alpha_\text{1R}^{-1}$. The result is shown in Fig.~\ref{fig:3211}, from which it is clear that unification is not achieved in this model. The reason is that the slopes of the two lines corresponding to the Abelian gauge couplings are too similar, meaning that they do not converge. Therefore, this model is disfavored on that ground and no prediction of the proton lifetime can be made.

\subsection{$\,\text{SU}\text{(5)}\times\text{U(1)}$}
The final model considered is the breaking of $\text{SO}(10)$ to a model of the $\,\text{SU}(5)$ type. The reason that $\,\text{SU}(5)$ is not considered on its own is that an intermediate symmetry that is a simple group does not help in achieving gauge coupling unification and instead changes the problem to requiring unification of the three gauge couplings at $M_\mathrm{I}$. We therefore consider the flipped $\,\text{SU}(5)$ model, \textit{i.e.} $\G{51}=\,\text{SU}(5)\times\text{U}(1)_X$~\cite{DeRujula:1980qc,Barr:1981qv,Derendinger:1983aj,Antoniadis:1987dx,Ellis:1988tx}, in which the mixing between the external $\text{U}(1)_X$ and the Abelian charge from inside $\,\text{SU}(5)$ to produce the hypercharge has the potential to help achieve gauge coupling unification.

The model which we construct, motivated by minimality, is one in which the $\text{SO}(10)$ symmetry is broken by a vev in the $\rep{45}_H$. In order to break the symmetry down to $\G{321}$, one can use the $\rep{24}_0$ from within the $\rep{45}_H$ together with the $\rep{1}_{10}$ from within the $\repb{126}_H$.

Between $M_\mathrm{GUT}$ and $M_\mathrm{I}$, the fermions are embedded as $\rep{10}_1\oplus \repb{5}_{-3}\oplus\rep{1}_5$. The scalars are $\rep{24}_0$ from the $\rep{45}_H$ for the breaking together with $\repb{50}_{-2}$ for the breaking and $\rep{45}_{-2}$ for the $\,\text{SU}(2)_\text{L}$ doublet from within the $\repb{126}_H$. From the $\rep{10}_H$, we have $\rep{5}_{-2}$ also for the SM Higgs. The resulting $\beta$ coefficients can be found in App.~\ref{app:beta}.

To compute the RG running, the Abelian charge must be normalized by a factor of $1/\sqrt{40}$. Normalizing also the hypercharge by its usual GUT factor, the matching conditions for the gauge couplings at $M_\mathrm{I}$ can be derived. One must also take into account that since the $\,\text{SU}(5)$ group contains the $\,\text{SU}(3)_\text{C}\times\,\text{SU}(2)_\text{L}$ part of $\G{321}$, these must match at $M_\mathrm{I}$. The unification of $\alpha_2^{-1}$ and $\alpha_3^{-1}$ therefore determines $M_\mathrm{I}$. Hence, the matching conditions read
\begin{align}
\alpha_5^{-1}(M_\mathrm{I}) &= \alpha_3^{-1}(M_\mathrm{I}) = \alpha_2^{-1}(M_\mathrm{I}), \\
\alpha_{1X}^{-1}(M_\mathrm{I}) &= \frac{25}{24}\alpha_Y^{-1}(M_\mathrm{I}) - \frac{1}{24} \alpha_2^{-1}(M_\mathrm{I}).
\end{align}

Using these matching conditions, the RG running may be computed and is displayed in Fig.~\ref{fig:51}. As is shown, gauge coupling unification is not achieved in this model due to the diverging lines of $\alpha_5^{-1}$ and $\alpha_{1X}^{-1}$. To rectify this, the model would need to be made more complicated in order to either significantly change the RG running between $M_\mathrm{I}$ and $M_\mathrm{GUT}$ or to change the RG running in the SM region so as to change $M_\mathrm{I}$.

\subsection{Gauge Coupling Running}
The RG running of the gauge couplings for all models discussed in Sec.~\ref{sec:models} are displayed in Fig.~\ref{fig:RGrunning}. In this figure, the inverse gauge couplings $\alpha_i^{-1}$ are plotted as functions of the energy scale $\mu$. Results to two-loop (one-loop) order are shown by the solid (dashed) lines and the corresponding scales that result in gauge coupling unification are displayed as vertical lines. For the two models in which gauge coupling unification is not achieved, namely the models based on $\G{3211}$ and $\G{51}$, representative intermediate scales are chosen. Particularly for the model based on $\G{51}$, this corresponds to the scale at which the gauge couplings corresponding to $\,\text{SU}(3)_\text{C}$ and $\,\text{SU}(2)_\text{L}$ unify. The gauge coupling that each color corresponds to is given by the label in the figure. The subscript of each $\alpha^{-1}$ denotes which of the gauge group factors it corresponds to.

\begin{figure}[t]
\centering
\subfloat[\label{fig:SM}$\G{321}$]{\includegraphics[width=.48\linewidth]{figures/SM1vs2loops.pdf}}\,
\subfloat[\label{fig:PS}$\G{422}$]{\includegraphics[width=.48\linewidth]{figures/PS1vs2loops.pdf}}\,
\subfloat[\label{fig:PSD}$\G{422D}$]{\includegraphics[width=.48\linewidth]{figures/PSD1vs2loops.pdf}}\,
\subfloat[\label{fig:421}$\G{421}$]{\includegraphics[width=.48\linewidth]{figures/4211vs2loops.pdf}}\,

\caption{\label{fig:RGrunning}RG running of the inverse gauge couplings in the eight different models considered. The results to two-loop (one-loop) order are shown as solid (dashed) lines. For the models without gauge coupling unification, representative values for the scales were chosen. GUT normalization of hypercharge is used throughout. (Continued on next page)}
\end{figure}

\begin{figure}\ContinuedFloat
\centering
\subfloat[\label{fig:3221}$\G{3221}$]{\includegraphics[width=.48\linewidth]{figures/32211vs2loops.pdf}}\,
\subfloat[\label{fig:3221D}$\G{3221D}$]{\includegraphics[width=.48\linewidth]{figures/3221D1vs2loops.pdf}}\,
\subfloat[\label{fig:3211}$\G{3211}$]{\includegraphics[width=.48\linewidth]{figures/32111vs2loops.pdf}}\,
\subfloat[\label{fig:51}$\G{51}$]{\includegraphics[width=.48\linewidth]{figures/511vs2loops.pdf}}

\caption{(Continued from previous page) RG running of the inverse gauge couplings in the eight different models considered. The results to two-loop (one-loop) order are shown as solid (dashed) lines. For the models without gauge coupling unification, representative values for the scales were chosen. GUT normalization of hypercharge is used throughout.}
\end{figure}

\section{Threshold Corrections}\label{sec:thresholds}
The results presented in Sec.~\ref{sec:models} assume that the matching of two models occurs at tree-level, meaning that the gauge couplings of the subgroup are equal to a linear combination of the gauge couplings of the group from which it originates. At higher-loop orders, the matching conditions are modified by threshold corrections.

For the symmetry breaking of a group $\mathcal{G}_m$ to another group $\mathcal{G}_n$ at a scale $M_{m\rightarrow n}$, the matching condition with threshold corrections reads
\begin{equation}\label{eq:matching}
\alpha_n^{-1}(M_{m\rightarrow n}) = \alpha_m^{-1}(M_{m\rightarrow n}) - \frac{\lambda^m_n}{12\pi},
\end{equation}
where the one-loop threshold corrections $\lambda^m_n$ are given by~\cite{Hall:1980kf,Weinberg:1980wa}
\begin{equation}\label{eq:thresholds}
\lambda^m_n = \sum_{i\, \in\, \text{vectors}}k_{V_i} S_2(V_i) + \sum_{i\, \in\, \text{scalars}} \kappa_{S_i} k_{S_i} S_2(S_i) \ln\left( \frac{M_{S_i}}{M_{m\rightarrow n}}\right).
\end{equation}
Here, the $\kappa_{S_i}$ are $1$ or $2$ for real or complex representations, while $k_{V_i}$ and $k_{S_i}$ are the multiplicities of the vector and scalar field, respectively, taking into account the dimension of the representation under the other gauge group factors. Note that we assume that all superheavy vector bosons have masses which coincide with the symmetry breaking scale so that there is no scale-dependent term for vectors. If there were superheavy fermions, they would also contribute to the threshold corrections. In what follows, we use the shorthand notation $\eta_i = \ln (M_{S_i}/M_{m\rightarrow n})$.

Note that Eq.~\eqref{eq:matching} holds when the matching of the gauge couplings at tree-level is such that they are equal at the scale of symmetry breaking. This is not the case, for example, when breaking the $\G{422}$ symmetry to $\G{321}$, in which case the hypercharge generator is a linear combination of one generator from $\,\text{SU}(4)_\text{C}$ and one from $\,\text{SU}(2)_\text{R}$. The matching of the gauge couplings in that and similar models involves forming a linear combination of the gauge couplings of the broken symmetry group. To each such term, one adds the threshold correction. For more explicit details, we refer the reader to Ref.~\cite{Bertolini:2009qj}.

Given the fields that lie around each energy scale as given in App.~\ref{app:fields}, the threshold corrections to each of the gauge couplings may be computed using Eq.~\eqref{eq:thresholds}. Since there is one threshold correction corresponding to each gauge group factor in the unbroken symmetry at the symmetry breaking scale, one needs to use the representations under the unbroken group of all the fields that lie at the symmetry breaking scale. For each of the models in Sec.~\ref{sec:models} that achieve gauge coupling unification and the model without an intermediate symmetry group, the threshold corrections to the matching conditions have been computed and are presented in App.~\ref{app:thresholds}. Note that, in this work, we do not analyze the effects of threshold corrections for the models based on $\G{3211}$ and $\G{51}$ since these do not achieve gauge coupling unification.

\section{Results}\label{sec:results}
Given the threshold corrections given in App.~\ref{app:thresholds}, we randomly sample the masses of the scalars around the symmetry breaking scale and thereby find how large the deviations from the symmetry breaking scale are required to be in order to save the models.

In the case of no intermediate symmetry, the impact of the threshold effects on the matching conditions is such that they can compensate for the difference between the gauge couplings and therefore allow gauge coupling unification~\cite{Ellis:2015jwa,Schwichtenberg:2018cka}. That is, since threshold corrections are meant to account for the failure of gauge coupling unification, we can compare the difference of the gauge couplings with the size of the threshold corrections. To this end, we define
\begin{equation}
\Delta\lambda_{ij}(\mu) = \alpha^{-1}_i(\mu) - \alpha^{-1}_j(\mu) = \frac{\lambda_j^{10} - \lambda_i^{10}}{12\pi}.
\end{equation}
From the three gauge couplings in the SM, the failure of gauge coupling unification can be demonstrated by the two differences $\Delta\lambda_{32}$ and $\Delta\lambda_{21}$. For each energy scale, we can plot the correlation of these two quantities, as is shown by the red lines in Fig.~\ref{fig:SMfailure}, in which the solid line shows the result with RG running at two-loop level and the dashed line at one-loop level. These lines demonstrate the size of the threshold corrections that would be required in order to obtain gauge coupling unification.

From the expressions for the threshold corrections given in App.~\ref{app:thresholds_SM}, one may compute the size of the difference of the threshold corrections, given values of the parameters $\eta_i$. We randomly sample these parameters independently according to uniform distributions in the regions $\eta_i\in[-1,+1]$, $\eta_i\in[-2,+2]$, and $\eta_i\in[-3,+3]$. These regions demonstrate the possible size of threshold corrections that can be obtained by allowing each scalar mass to vary within the determined ranges.

\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{figures/SM_thresholds.pdf}
\caption{\label{fig:SMfailure} Threshold effects to allow for unification of the SM gauge couplings. The difference between gauge couplings $\alpha_3^{-1}-\alpha_2^{-1}$ and $\alpha_2^{-1}-\alpha_1^{-1}$ is shown in the solid (dashed) red line at two-loop (one-loop) level. Numbers above the lines are the energy scales in units of $\text{GeV}$. The blue shaded regions show the size of the threshold corrections. Intersections between the blue shaded regions and the red lines correspond to successful gauge coupling unification.}
\end{figure}

Computationally, each scan was performed on a computing cluster utilizing 48 cores each sampling $8\times 10^{8}$ points. Following this, the convex hull was computed using the \verb|ConvexHull| routine from the \verb|SciPy|~\cite{scipy2001} package version \verb|1.3.2| in \verb|Python 3.7.0|, which produced the blue shaded regions shown in Fig.~\ref{fig:SMfailure}. This was performed by first sampling $10^4$ points for each region and using the \verb|ConvexHull| routine to find the smallest convex polygon containing the given points. From there, only the points on the boundary were saved. When sampling the points, only those that fell outside the initial boundary were saved and were used to create the new boundary. This produced the regions shown in Figs.~\ref{fig:SMfailure} and~\ref{fig:thresholds}. These figures were used to illustrate the results rather than scatter plots, since they better demonstrate the fact that the sampled threshold corrections span a region.

The intersection of the blue shaded regions and the red lines shows the size of the threshold corrections that lead to unification of the SM gauge couplings at a particular scale. For example, with RG running at two-loop level, threshold corrections with $\eta_i\in[-2,+2]$ can provide unification, but only at a range of scales around $M_\mathrm{GUT}\simeq (5\times 10^{12} - 7\times 10^{14})\,\mathrm{GeV}$, with a slightly narrower range at one-loop level. The largest $M_\mathrm{GUT}$ that is possible with $\eta_i\in[-3,+3]$ is about $5\times 10^{15}\,\mathrm{GeV}$, which leads to a proton lifetime of $\tau_p\approx 5\times 10^{34}\,\text{yr}$, which is allowed but close to the current bound.

For the models with an intermediate symmetry that achieve gauge coupling unification without threshold corrections, the effect of threshold corrections on the scales $M_\mathrm{I}$ and $M_\mathrm{GUT}$ were found. This was performed by solving the matching conditions to determine the two scales as functions of the $\eta_i$ parameters. Then, the $\eta_i$ were individually sampled in the same way as described above for Fig.~\ref{fig:SMfailure}. Regions of possible scales for $\eta_i\in[-1,+1]$, $\eta_i\in[-2,+2]$, and $\eta_i\in[-3,+3]$ are plotted in Fig.~\ref{fig:thresholds}, again using the \verb|ConvexHull| routine. The red shaded regions denote the range of possible scales with RG running at two-loop level and the black contours show the same at one-loop level. The scales obtained in the absence of threshold corrections are denoted by a ``$\times$" (``$\star$") for the two-loop (one-loop) result.

\begin{figure}[!htbp]
\centering
\subfloat[\label{fig:thr_422}$\G{422}$]{\includegraphics[width=.48\linewidth]{figures/PS.pdf}}
\subfloat[\label{fig:thr_422D}$\G{422D}$]{\includegraphics[width=.48\linewidth]{figures/PSD.pdf}}\\[-1ex]
\subfloat[\label{fig:thr_421}$\G{421}$]{\includegraphics[width=.48\linewidth]{figures/421.pdf}}\\[-1ex]
\subfloat[\label{fig:thr_3221}$\G{3221}$]{\includegraphics[width=.48\linewidth]{figures/3221.pdf}}
\subfloat[\label{fig:thr_3221D}$\G{3221D}$]{\includegraphics[width=.48\linewidth]{figures/3221D.pdf}}

\caption{\label{fig:thresholds}Variations in the scales due to threshold corrections for the models which achieve gauge coupling unification. The red shaded regions show the results with RG running at two-loop level, while the full, dashed, and dotted contours show the results at one-loop level. Concentric regions show the scales for the thresholds in the range $\eta_i\in [- 1,+1], [-2,+2]$, and $[-3,+3]$, respectively. The scales without threshold corrections are marked with a ``$\times$" (``$\star$") for the two-loop (one-loop) result. The grey shaded regions are forbidden by a too short proton lifetime and $M_\mathrm{I} > M_\mathrm{GUT}$.}
\end{figure}

To investigate if the models which were originally disfavored due to a too short proton lifetime can be saved by threshold corrections, the nearly horizontal grey shaded region denotes a too short proton lifetime. Thus, points in the shaded region are ruled out. This line was computed by first calculating the proton lifetime for the randomly sampled points and then using a Support Vector Machine (SVM) from the \verb|scikit-learn| package~\cite{scikit-learn} version \verb|0.19.2| in \verb|Python 3.7.0| to find the equation of the line that best separates the two classes. This line separates the two classes well and was observed to be nearly identical for the one-loop and two-loop RG running. The slanted shaded region is the forbidden region corresponding to $M_\mathrm{I} > M_\mathrm{GUT}$. Note that there also exist bounds on $M_\mathrm{I}$, such as right-handed neutrino masses in the type I seesaw mechanism or leptogenesis. However, these are model dependent and therefore neglected.

For the model based on $\G{422}$, it was already allowed without threshold corrections. As shown in Fig.~\ref{fig:thr_422}, they have quite a large effect on the scales. The model based on $\G{422D}$ has smaller threshold corrections, as displayed in Fig.~\ref{fig:thr_422D}, due to the absence of the $\rep{210}_H$. This model was disfavored without threshold corrections and is saved by threshold corrections with $\eta_i\in[-1,+1]$. Turning to the model based on $\G{421}$ shown in Fig.~\ref{fig:thr_421}, it was disfavored without threshold corrections and is still disfavored by a too short proton lifetime with $\eta_i\in[-3,+3]$. It requires $\eta_i\in[-4,+4]$ in order to predict a long enough proton lifetime. The model based on $\G{3221}$ shown in Fig.~\ref{fig:thr_3221} was also allowed without threshold corrections. Lastly, the model based on $\G{3221D}$, shown in Fig.~\ref{fig:thr_3221D}, predicted a too short proton lifetime in the absence of threshold corrections. Only a small amount of threshold corrections are required to save it, since part of the region with $\eta_i\in[-1,+1]$ is above the grey shaded region. We summarize these findings in Tab.~\ref{tab:summary}.

\begin{table}[t]
\caption{\label{tab:summary}Comparison of the viability of the models considered. The second column (``GCU") denotes whether or not gauge coupling unification is achieved without threshold corrections. The next four columns show if the predicted proton lifetime is above the lower bound for various sizes of threshold corrections. A checkmark (``\cmark ") denotes an allowed model whereas a cross (``\xmark ") denotes a disfavored model. The dashes in the last two rows signify that we did not investigate threshold corrections in those models. The results given are with RG running at two-loop level. Results at one-loop level may be found in the main text and the figures.}
\centering
\begin{tabular}{l c c c c c}
\hline
\hline
Model & GCU & $\eta_i\in[-3,+3]$ & $\eta_i\in[-2,+2]$ & $\eta_i\in[-1,+1]$ & no thresh.\\
\hline
$\G{321}$ & \xmark & \cmark & \xmark & \xmark & \xmark\\
$\G{422}$ & \cmark & \cmark & \cmark & \cmark & \cmark \\
$\G{422D}$ & \cmark & \cmark & \cmark & \cmark & \xmark \\
$\G{421}$ & \cmark & \xmark & \xmark & \xmark & \xmark\\
$\G{3221}$ & \cmark & \cmark & \cmark & \cmark & \cmark \\
$\G{3221D}$ & \cmark & \cmark & \cmark & \cmark & \xmark \\
$\G{3211}$ & \xmark & {\bf --} & {\bf --} & {\bf --} & {\bf --}\\
$\G{51}$ & \xmark & {\bf --} & {\bf --} & {\bf --} & {\bf --}\\
\hline
\hline
\end{tabular}
\end{table}

\section{Summary and Conclusions}\label{sec:conclusion}
Unification in models in which the $\text{SO}(10)$ gauge symmetry breaks to the SM directly or with one intermediate symmetry breaking have been investigated. Particularly, we have considered non-SUSY models, in which it is known that achieving gauge coupling unification is more difficult than in their SUSY counterparts.

We have solved the RG running in these models and studied whether or not gauge coupling unification is achieved and, if so, whether or not the prediction for the proton lifetime is above the experimental lower bound. Gauge coupling unification was achieved in models with $\G{422}$, $\G{422D}$, $\G{421}$, $\G{3221}$, or $\G{3221D}$ as intermediate symmetries. Of these, only $\G{422}$ and $\G{3221}$ predicted a proton lifetime that is above the experimental lower bound, with $\tau_p\approx 1.2\times 10^{38}\,\text{yr}$ and $\tau_p \approx 9.4\times 10^{34}\,\text{yr}$, respectively. These two models are the well-studied PS and left-right models.

Threshold corrections have been computed for the model with direct breaking as well as the models in which gauge coupling unification occurred. For the model with direct breaking, we have found that gauge coupling unification can be achieved with a long enough proton lifetime if $\eta_i\in[-3,+3]$. This holds with RG running at both one-loop and two-loop level.

The models with intermediate symmetries for which the predicted proton lifetimes are too short can be saved by invoking threshold corrections, given that they are large enough. For the models based on $\G{422D}$ and $\G{3221D}$, it is sufficient to have $\eta_i\in[-1,+1]$, both with one-loop and two-loop level RG running. The model based on $\G{421}$, on the other hand, requires larger threshold corrections. With RG running at one-loop level, the proton decay bound can be evaded with $\eta_i\in[-3,+3]$, while at two-loop level, $\eta_i\in [-4,+4]$ is required.

The results in this work assume the specific model details as described in Sec.~\ref{sec:models}. It should be noted that it is possible to modify some of these details while still achieving the same symmetry breaking chain. However, the results reported in this work can be seen as representative of these models. Furthermore, in the construction of the models, we have not taken into account any constraints from the scalar potential. It would be interesting to investigate this since there can be correlations between the masses of the scalars that could impact the results.

Additionally, we have not taken into account any bounds on physics related to $M_\mathrm{I}$. This may in some of the models be related to neutrino masses and/or leptogenesis. Furthermore, we have neglected the effect that perturbing the scalar fields around $M_\mathrm{I}$ has on the RG running, as investigated for example in Ref.~\cite{Babu:2015bna}. For larger perturbations, this effect may become substantial and have an effect on the conclusions. The above mentioned points, together with the investigation of the two models that were not considered here, namely those based on $\G{3211}$ and $\G{51}$, would make for an interesting future study.

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urer, but there are gold locks that such a big child can bring, not too big, just fine." Sumi never believed in life before. God, it's just that now, even through the corpse of the corpse, she can't say anything and don't believe it. Doesn't it all mean that the lock can still lock the child? Sumi now treats two children as if they were life-like. With this condition, I remembered that I wanted to buy two locks for my two children. "Is it for Xiao Gongzi and Miss Xiao? Is there any, there are still a lot of styles, you are waiting here, I let the guys come up with some to show you." Obviously, like Sumi The person who thought was not in a minority. After Hu Yu told a guy, but for a few minutes, the little guy took a box and opened it, all of them were all kinds of locks. Big small, golden silver, beautiful, simple, dazzling. Sumi and Zhang Yuyu picked and picked them for a long time and bought two very small gold locks. One is a small lock with a pattern in the cat's arms. It is very embarrassing and cute. The other is a lock with a bamboo pattern on it. I asked Cheng Zhi, he said that he liked it, and Suomi and Zhang Yuyu decided to come down. The two golden locks have not yet had the weight of Zhang Yuyu's sea otter, but they have more than 30 silver. However, Sumi still feels value for money, because the ingenuity of these two locks is so amazing that Sumi, who is used to modern goods, feels very amazing. After paying the money, both Zhang and Zhang Yuyu's nephews were wrapped in brocade boxes. The two were like children. They put the scorpion close to themselves and they could see it. They liked it very much, especially Zhang's mouth. I have to get to the side of my ear. After all, these things, when Sumi was selected, they said that they let Sumi take the lead, but Sumi also chose to look at their faces, and their own things, how can they not be happy. As for the locks of Cheng Zhi and Duo Duo, Sumi put them on the spot. However, Sumi still knows about the fact that the money is not leaking. Now I just
Posted on 2019年7月16日 2019年7月7日 by njwho
rugged, walked over and pulled up the hand of his wife Xue Li. "Wife, I am going abroad first, you are waiting for me, I will soon find a way to help you get a visa, then we can be together!" "Well, I am waiting for you, don't let me wait too long." "Reassured, rest assured!" Shen Jialei and Xue Li hugged goodbye. Shen Jialei and Shen Jiahai also embraced together. Shen Jialei also concealed the five brothers' cancer. He did not tell Xue Li, and he did not dare to mention it in the face of Xue Li. If Xue Li knew it, it would definitely make a big noise. Things must be stalked by Xue Li, so Shen Jialei did not say much, just simply bid farewell to Shen Jiahai. 85cm tall, wearing a black jacket, beige trousers under the feet, Adidas sneakers under the feet, with a black large suitcase, after passing the security check, got on the plane. The flight that Shen Jialei took took off at 10:00 in the morning. The plane drawn a beautiful arc in the blue sky, with good hope, flew into the distance until it disappeared into the sky and could not be seen. Shen Jialei left, Xue Li said nothing and Shen Jiahai said that she knew that Shen Jiahai had her opinion on her and had been guarding her. She refused to help her to apply for a visa for her abroad. Shen Jialei had previously sent a message in the middle. Xue Li also expected Shen Jiahai to see it. In the relationship between relatives, she helped her go abroad. Now Shen Jialei has gone abroad. No one has sent a message to her. Xue Li faces Shen Jiahai and feels uncomfortable. She has nothing to say, but she is polite and polite. She said that she still If something happens, you have to go first. Shen Jiahai just nodded politely, did not stop, watching Xue Li wearing a black long trench coat, stepping on high-heeled shoes at the foot, high-heeled shoes with a rhythmic tapping on the marble floor tiles in the airport hall to make a crisp sound. Shen Jiahai squinted at the background of this charming woman who was still charming, thoughtful. "Five brothers, what are you looki
st two steps, she opened Sumi and actually wanted to break into the house. in! Sumy and Zhang Yuyu did not think that Wang Guihua would suddenly be so arrogant, and suddenly they did not react and they were in the same place. Cheng Zhi and Du Duo stood opposite Wang Guihua, and the small body blocked the fat osmanthus like the sow. "You two will give me a roll!" Wang Guihua has now been overwhelmed by food, and what Sumi has done is really delicious. Now I want to directly follow Cheng Zhi and blossoming! "Looking for death!" Long Qing did not know when he stood at the door and saw the claws lifted by Wang Guihua. One of them jumped over and grabbed Wang Guihua's wrist. When he slammed, Wang Guihua flew out! "Hey! It hurts me! Li Laiqing, you don't hurry to help me!" Wang Guihua wolf cried and looked like the people in the yard had returned to God. Su Mi and Zhang Yuyu in Li Laiqing When there was no action, the tacit rushed up and pressed Wang Guihua to the ground and moved his hand! ☆, the fourth fifty-one chapter: Long Qing shot Long Qing, a light blue brocade came over against the light, watching Wang Guihua's eyes are clearly want to peel the cramps of the eyes! "Who gave you the courage to let you dare to do two children?" Long Qing looked at Wang Guihua, who was hit by Sumi and Zhang Yuyu and wanted to fight back. The cold light flashed in his eyes and he squatted on his knees. On the arm of Wang Guihua. Suomi clearly heard a slam, and after a moment of squatting, Zhang Yuyu stood up. "Have I told you that no matter who is coming to Zhangjia, I will not let her be better? Is Wang Guihua?" Long Qing looked at the ground, the clothes were messy and the soil was constantly mourning. Wang Guihua, the corner of his mouth sneered a sneer: "You really should be glad that you are a woman. If you are a man today, you are afraid that you can't walk out of this yard!" Long Qing has always been reluctant to be old and weak. Women and children started, but today Wang Guihua actually wants to work on Cheng Zhi and blos
" Like pigs. Can you eat so much if you are uncomfortable?" Xia Yiyi: "…" This is so active to send his artists out, it seems that this agent lady is not a good bird. He looked up at Zhang Qingyu, but he saw Zhang Qingyu looking at himself. The two eyes were on the same side, and Xia Yi was as scared as he was, and immediately turned his head and looked away. The guilty look was no longer obvious. Only then broke the relationship of ex, if the same room at night, Xia Yiyi really do not know how to face Zhang Qingyu. There was a sigh in the ear. It seems that there is no more. He heard Zhang Qingyu not have the slightest feeling of emotion. "Since the body is uncomfortable, it is good to raise. Today is even better." "Zhang Shao!" The agent anxiously explained, "Don't be angry, summer and summer are just…" Xia Yi Back sneaked into Zhang Qingyu, and found that the latter's eyes actually placed on himself. This time he did not turn his face, but looked back at the scalp. "I am not angry." Zhang Qingyu ridiculed his lips. "How do I…" Halfway through the sentence, the statement paused and disappeared. He turned around, tall and long legs, and a few steps into the crowd, he could never find it again. The agent hated iron and couldn't make steel. "Look, Zhang Shao obviously moved his true feelings. If you don't like him, you can find a decent reason. What kind of ghost is uncomfortable? You mean that today is too messy to avoid suspicion. It's better." Xia Yi's heart spurs and licks his lips. What did Zhang Qingyu just want to say? 'How can I be angry? ' 'How can I be angry? ' 'How can I be angry? 'No matter which kind, it makes Xia Yi back to heart. His hiding posture is really hurting. However, he suddenly told him that he had been abandoned and that he had to give him some time to buffer and think. Seeing the appearance of Xia Yihui, the broker has no way. In this place of entertainment, the most taboo is to talk about feelings. Because there is no talk, talking about it is more than anyone else. The crowd d
of bullets in the live broadcast room, such a barrage is also conspicuous: Ma Zhe is a bit sweet: Dad! Summer daddy! Before I was wrong, I actually suspected that your QAQ didn't have time to see the son of the barrage. Xia Yiyi knew that the copy was about to end soon. I don't know if there is any chance to see Zhang Dashen. After all, it is a comrade-in-arms comrade, the first companion in the true sense of the copy, the feelings are more or less a little bit. Xia Yi thought about it and smiled. "Thank you for these days, I have a chance to repay." Zhang Qingyu's eyes were crystal clear, and for a few seconds, he reached out and pinched his face in the summer, calmly saying: Nothing." Pinching his face again? ! I squeezed it last time, this time? Xia Yizheng was preparing to open Zhang Qingyu's hand, and he saw that the latter slightly licked his lips and smiled lightly. "We will meet again soon." "Ah?" Xia Yi back. What does it mean? Zhang Qingyu can't decide a copy. Why is it so sure? I couldn't wait to think about it. Xia Yi's head was dizzy, and the environment around him had changed a lot. It seems that he returned to the cultivation space. The touch on the face still seems to exist, but the person who pinched his face did not disappear. At the same time, all players who pass through the copy are sent away. The eliminated players also left only a cold body, unable to tell the cruelty of the copy game. When the horizontal line rises and floods, everything that happens here will only be buried, and no one knows the truth. As for "GO OUT OF HELL", who knows what is in the part covered by the picture frame? Chapter 53 Entertainment Werewolf Kill (1) [Player Xiayi successfully completed a copy of the campus glade. [You are in the college entrance examination, the ranking is no.2, the rating is excellent +, and the additional reward experience value is 10000. You have survived a total of 7 days in the copy. Reward 700 currency. [Congratulations to the player successfully upgraded to S11 (0/1100) in the s
ss?" Muze Yun asked Zhou Mou. Zhou abandoned his face and didn't look at him for a day. He looked down and ate his head. After he swallowed his mouth, he slowly said: "Yes, what do you think?" After a long time in Ozawa, people also became Poisoned tongue. I don't know if Muze Cloud is too indulgent, but it is still too much work during this time, making Zhou Ai Mo become like this. The specific reasons are not known, but the result is like this. "But I would like to ask, how much money can you support in your hands now?" Muzeyun was silent once, and smiled bitterly: "At most three months, no more. "I didn't expect to have such a long time." Zhou Xiaomo laughed. "If you can really have such a long time, you don't have to worry about it. I still have a way to support it, but…" Zhou Ai Mo looked up at the oysters. Zeyun, I looked down at the dishes in the bowl: "You still get Jincheng as soon as possible, so everyone saves strength. "I try my best." "Ozawa Yundao. Lin Xiangyu took Jin Rui to see a doctor and performed regular psychotherapy. Although the effect was minimal, even if it had only a little bit of effect, it was worthwhile for them to stick to it. It was recommended by a friend and the confidentiality was very good. Therefore, when Lin Xiangyu saw Jiang Yue, he was just amazed. When he thought about the professional conduct of the psychiatrist, he was relieved. It is probably a coincidence. After all, Jiang Yue looks like It is also a disease. "What is wrong with you? Lin Xiangyu asked cautiously. Jiang Yue was thinking about things in concentration, suddenly heard a person call her, looked up at the man in horror. Focus, see who it is. "Xiang language sister, how are you?" Lin Xiangyu nodded and touched the head of Jin Rui and smiled. "Is this not a child to see a doctor?" What about you? Jiang Yue's face just showed a smile: "Me too." Lin Xiangyu did not expect that although Jiang Yue answered it before, he almost guessed it. But she said that she was still unacceptable. In her eyes, Jiang Yue is a
ist, a tassel on her back, a small satin tube top on her upper body, a small pink coat on the outside, and wide cuffs, revealing a white and delicate wrist. Gu Chengyan did not turn his eyes, and itchy dryness in his throat. … unhealthy piglet hooves. Where come so many small skirts that are tempting. Qin Youyin received a phone call from the roommate when he changed his clothes. Cheng Jia's big voice is very high: "The sound, where? I will give you makeup!" She said: "I just changed clothes, I used to look for you. Gu Chengyan was going to meet, but I thought of the expression of the little girl coming over and bumping into him, and there was a bitter and refreshing back. Abandoned. Not to be seen. It is his own. Still have to go back. Gu Chengyan pretends that nothing happens, while aiming at the movement of Qin Youyin, while making himself look like a normal laborer, moving things around the hospital to get things. After a while, the person in charge ran over and panted and folded his hands together: "Inflammation brother, help, come The original five people who sent the piano, one of them had just used too much force on the ankle. Would you please take the hand? Gu Chengyan followed the past, watching the other four are very strong, wearing short-sleeved overalls, full of arm tattoos. He did not speak, endured the injury to help lift a corner, because the squad is big enough, attracted a lot of attention when entering the background, also Including Qin Yuyin. Gu Chengyan aimed at her, and suddenly she found that she was staring at the eight large flower arms next to her eyes, and she couldn't conceal her worship. The second time! It's really fucking strange, this is awkward. Good-looking! OK, Qin Youyin, you will look at it one by one. When you are my turn, you should never hide! Gu Chengyan waited for Qin Youyin to turn to him, but she controlled it with special precision. It was his first second, fast. Twisted at the beginning, Yu Guang did not give one. … Good. Lemon tree under the lemon essence,
, and Sumi ate the cake while teasing the forest. I only thought that the time was too fast. When the palace gate was reached, Sumi sat full and full of energy. Qing also opened the micro-closed scorpion, because the faint smile of listening to Sumi's speech also faded. "Well, the palace has arrived." Long Qing got off the car first, then Zilin supported Sumi. Originally there was a ladder and a slave to serve next to the carriage. Even though the clothes worn by Sumi were not very convenient, they would not fall, but the dragon. The engine did not care about the people who came and went to salute him. He directly reached out and took Sumi to the ground and placed it firmly on the ground. . . So many people look at it, what are you doing here! Sumi said a little embarrassed. If this is known to the emperor, will it be said that the mistakes are too frivolous and do not know the rules? "Nothing, let's go, come." "Dragon Qing didn't care at all, just took the hand of Sumi directly." "I told you that after you enter the palace, you will be next to me. You can sit on the side of my body." But your niece has no choice, you are not willful, your body can not stand. "Long Qing stuck to Sumi's ear and said finely. "Okay, you." Don't worry, I am not a fool. What can I do? Can't I do it, can I still know? Even I am an accessory. It is very good to be able to sit down. Can I still ask you to sit down with the accessory of the accessory? That is too difficult for people. "Sumi is still very understanding of these things, and will not make things that make Long Qing embarrassed. ☆, Chapter 199: I have never seen "I know that my Susu is the most I am sensible, let's go. Long Qing smiled and nodded, and led Sumi to walk in the inner courtyard of the palace. Sumi took the time to look back and gave Zilin a look that must be followed, and quickly turned his head, for fear that others would see it later. What is the meeting? "Don't be so careful, what to do in a while, how to do it, I will tell you." "Long Qing can feel the cold
hing. Today, even Li's father did not speak. It seems that Sumi's today's work. It really made him anxious. "I don't want to go back to our old Li family!" Long can bear this! When Zhou said this, his face was very ugly. Compared with Li Laiqing, Li Laiwang was ugly ten times as many as eight times. People in this room also know why Zhou's crying is such an expression. Because at Sumi just got started At some time of development, Zhou once said that if she let Sumi come back now, Sumi would definitely be happy to climb back, not only at home, but also outside, and when she said, she did not consider the actual situation. Sumi is said to be one, as long as she proposes to Sumi to give her a fingertip. In fact, when Zhou first started talking at home, it was just a fast-paced mouth. Later, when he ran outside, he wanted to satisfy his vanity. He wanted the big guy to admire her two women like Sumi. In addition, in fact, she had already moved to let Sumi come back. This idea is certainly not because she is lost in her heart, but she has realized how great the benefits Sumi can bring. So she deliberately went outside to quarrel and wanted others to bring this news to Sumi. The purpose was to let Sumy know that as long as Sumigen came to the door to ask for help, she could consider giving her to come back with mercy. Naturally, the reason why Sumi said that he was so unbearable to put his position on the bottom is that she thinks that Sumi is inseparable from the old Li family. This time I saw that Li's father and her three sons had suffered very little in Sumi, and she realized that Sumi didn't seem to want to come back. How does this make the proud Zhou feel? Let Sumi come back to give Sumi a face, so that the daughter-in-law who is not sensible should not climb back from the old Zhang's family with the two wild species along the way. "You all mean it. Let's just listen to it. Don't use those who are useless. There are no outsiders in this room. If you have a hurry, you can say it." Li said, sighing. This sen
showed the color of appreciation. This younger brother, really talented, strong and weak, is not inferior to him. He is still young, likes to show off, but also has to temper his heart. Seeing Yuan Tianqi nod, likes to say: "The younger brother is right? Li Yufeng sighed a bit, and some regretted: "I see this person, some eyes, do not call him, lest he go the wrong way, send his life!" Yuan Tianyi immediately said: "And slow! Hurricane, smashed! Life and death have a life, can you intervene? Yuan Tianyi looked at Li Yu deeply and said: "He is uncertain, and he is undecided. It is not God's will, let him make his own choices. You and I should not interfere." Let's go! Yuan Tianqi kicked and kicked, the handlebar style slammed a whip, and the old oxen slowly took a step. *** *** *** *** *** *** *** Li In the heart of the fish, I didn't know where to go. I thought about the past, I was sour from my nose, my eyes were wet, and my shoulder was suddenly shot. Li Yu turned to look at it, actually the ship's boss Liu Yuntao. Liu Yuntao said with joy to Li Yudao: Little brother, why not go home? Li Yu said: "Liu boss, have you left yet?" Liu Lao Avenue: "I went to find a friend and borrowed a bit of it. After a while, I went to the bridge, took a boat, sailed eastward, and went back to Luoyang." Where is your home, can you be with me? "Li Yu's heart is moving, since there is nowhere to go, why not go with Liu Laos, first follow him for a few days to eat and drink, to be familiar with the world, and then make plans not too late, maybe, can also find back to the future world." The way to go. But when I got to the lips, a "familiar" figure suddenly appeared in his mind, that is the mother of Li Yu. Only on the day when his prisoner was escorted into Beijing, I only saw her side. It was raining heavily, she ran in the rain, fell, climbed, fell again, climbed again, covered with mud, tears and rain, and the miserable cry still seemed to linger in his ear. Li Yu suddenly had a bit of panic in his heart. Li Yu hesitated a mom
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redpajama_set_name: RedPajamaCommonCrawl


text:
<?php
namespace app\modules\wechat\services;
/*
 * To change this license header, choose License Headers in Project Properties.
 * To change this template file, choose Tools | Templates
 * and open the template in the editor.
 */

/**
 * Description of MemberService
 *
 * @author qinyangsheng
 */
class MemberService {

//保存或者更新用户的送货地址
    public static function saveProAddress(UserProudctAddress $userprodaddress, $user_id) {

        $_POST['UserProudctAddress']['province'] = 'xx';
        $_POST['UserProudctAddress']['city'] = 'xx';
        $_POST['UserProudctAddress']['area'] = 'xx';
        $_POST['UserProudctAddress']['sysaddress'] = 'xx';
        if (isset($_POST['province'])) {
            $_POST['UserProudctAddress']['province'] = $_POST['province'];
            $provincename = Province::model()->find("provinceID=:proid", array(":proid" => $_POST['UserProudctAddress']['province']));
            if ($provincename) {
                $_POST['UserProudctAddress']['sysaddress'] = $provincename->province;
            }
        }
        if (isset($_POST['city'])) {
            $_POST['UserProudctAddress']['city'] = $_POST['city'];
            $cityname = City::model()->find("cityID=:cityid", array(":cityid" => $_POST['UserProudctAddress']['city']));
            if ($cityname) {
                $_POST['UserProudctAddress']['sysaddress'] .= $cityname->city;
            }
        }
        if (isset($_POST['area'])) {
            $_POST['UserProudctAddress']['area'] = $_POST['area'];
            $areaname = Area::model()->find("areaID=:areaid", array(":areaid" => $_POST['UserProudctAddress']['area']));
            if ($areaname) {
                $_POST['UserProudctAddress']['sysaddress'] .= $areaname->area;
            }
        }
        $_POST['UserProudctAddress']['user_id'] = $user_id;
        $userprodaddress->setAttributes($_POST['UserProudctAddress']);
        foreach ((array) $_POST['UserProudctAddress'] as $key => $value) {
            if (trim($value) == '') {
                $userprodaddress->addError($key, "字段不能为空");
                break;
            }
        }
        if (!$userprodaddress->getErrors()) {
            $userprodaddress->setAttribute('addtime', time());
            $userprodaddress->setAttribute('addip', Yii::app()->request->userHostAddress);
            if ($userprodaddress->validate()) {
                if ($userprodaddress->isNewRecord) {
                    $result = $userprodaddress->save();
                } else {
                    $result = $userprodaddress->update();
                }
                if (!$result) {
                    $userprodaddress->addError("realname", "更新失败");
                }
            } else {
                $userprodaddress->addError("realname", "更新失败");
            }
        }
        return $userprodaddress;
    }

    /**
     * 保存用户的银行卡或者更改
     * @param Bankcard $bankCard
     * @param type $user_id
     * @return \Bankcard
     */
    public static function saveBankCard(Bankcard $bankCard, $user_id) {
        $_POST['Bankcard']['bank'] = 'xx';
        if (isset($_POST['bank'])) {
            $_POST['Bankcard']['bank'] = $_POST['bank'];
            $bank = Linkage::getValueChina($_POST['bank'], "account_bank");
            if ($bank) {
                $_POST['Bankcard']['bank_name'] = $bank;
            }
        }
        $_POST['Bankcard']['bank_type'] = 'xx';
        if (isset($_POST['bank_type'])) {
            $_POST['Bankcard']['bank_type'] = $_POST['bank_type'];
        }
        $_POST['Bankcard']['area'] = '0';
        if (isset($_POST['province'])) {
            $_POST['Bankcard']['province'] = $_POST['province'];
        }
        if (isset($_POST['city'])) {
            $_POST['Bankcard']['city'] = $_POST['city'];
        }
        $_POST['Bankcard']['user_id'] = $user_id;
        $bankCard->setAttributes($_POST['Bankcard']);
        foreach ((array) $_POST['Bankcard'] as $key => $value) {
            if (trim($value) == '') {
                $bankCard->addError($key, "字段不能为空");
                break;
            }
        }
        if (!$bankCard->getErrors()) {
            $bankCard->setAttribute('addtime', time());
            $bankCard->setAttribute('addip', Yii::app()->request->userHostAddress);
            if ($bankCard->validate()) {
                if ($bankCard->isNewRecord) {
                    $result = $bankCard->save();
                } else {
                    $result = $bankCard->update();
                }
                if (!$result) {
                    $bankCard->addError("realname", "更新失败");
                }
            } else {
                $bankCard->addError("realname", "更新失败");
            }
        }
        return $bankCard;
    }

}

redpajama_set_name: RedPajamaGithub


The Gospel is not only a free ticket to Heaven, purchased by Jesus and delivered by God's grace through the Holy Spirit. It is that, but not just that - there is more. When the Holy Spirit indwells us and revives our spiritually dead hearts, we are given the power to overcome sin and live lives that are pleasing and glorifying to God. This is the mark of a Christian.


text:
USA Topics
Europe Update
californiatoday
Venezuela opposition leader Guaido vows to lead parliament despite having been blocked from building
Venezuelan opposition denounces parliamentary coup as Guaido rival names himself speaker
New Yorkers march against anti-semitism following a wave of violent attacks on Jews
New email evidence shows Trump directly ordered maintaining freeze on Ukraine aid
Are cell phone calls on airplane flights inevitable?
Libyas Humanitarian Coordinator underlines the need to restore essential services for people affected by conflict
New Report: Israel Aims to Change Jerusalem Identity by Shutting Down Institutions and Confiscating Land
Tunisian Authorities must meet protesters demands and respect their right to peaceful demonstration
Massive rains trigger flash floods in Djibouti, more than 150,000 people affected
Euro-Med Monitor Condemns US Announcement on Israeli Settlements
Home Europe Update 11 of Budapest's best festivals
11 of Budapest's best festivals
(CNN) — Straddling the picturesque Danube, Budapest provides the perfect backdrop for a festival and this city definitely knows how to put on a show.Barely a month goes by when the Hungarian capital isn't playing host to some sort of event celebrating food and drink, music, dance or the arts.
For those keen to go and join the party, we've rounded up some of the most entertaining festivities happening in Budapest throughout the year.
Rosalia Festival
Rosalia Festival is dedicated to rosé wines, sparkling wines and champagnes.
Courtesy Rosalia Festival
Each year, Budapest jumps the gun on summer over a weekend in May for the Rosalia Festival.
Created by the organizers of September's wine festival, it's Hungary's only event dedicated to celebrating rosé wine, as well as champagne and sparkling wines.
Taking place over three days, it features a Rosé Garden, tastings, jazz concerts, Hungarian food stalls and special events for children.
Dates: May 31 to June 2, 2019
Sziget Festival
One of the biggest music festivals in Europe — Sziget Festival takes place every August.
Attila Kisbenedek/AFP/Getty Images
For more than 25 years, the week-long Sziget Festival has been taking over the Danube river island of Óbudai-sziget every August, showcasing more than 1,000 performers and drawing tens of thousands of people from all over the world.
It's one of Europe's biggest music festivals, attracting performers including 2019 headliners Foo Fighters and Ed Sheeran.
Revelers soak up the lively ambience as dance artists put on theatrical performances on the site and everyone goes for a dip in the Danube along the sandy beach.
Dates: Aug 1 to 13, 2019
Budapest Summer Festival
Held throughout June, July and August, the Budapest Summer Festival brings some of the world's top classical musicians and ballet dancers to Margaret Island, located in the heart of Budapest.
There's a varied program of opera, ballet and classical music — with a bit of jazz and pop thrown in for good measure — most of which takes place in the enchanting setting of the Margaret Island Open-Air Stage.
Look out for the performances held in the open-air stage set up in the shadow of Margaret Island's historic water tower.
Dates: June to September
Budapest Summer Festivals, Open-Air Theatre, 1122 Budapest, Városmajor; +36 1 375-5922
Budapest Christmas markets
The Christmas market on St. Stephen's Square is one of Budapest's top draws in winter.
Courtesy Hungarian National Tourist Office
Prepare to be utterly charmed by Budapest's Advent Christmas fair, which is held annually in the square in front of St. Stephen's Basilica.
From late November to early January, the area is filled with market stalls selling trinkets, toys, crafts and plenty of irresistible Hungarian food and drink.
Those who visit on Sunday can watch the Advent candles being lit.
To top it all off, there's a small but perfectly formed ice rink in the center, adding a further dollop of festive magic.
Even more treats are on offer at Vorosmarty Square, where the city's main Christmas market is held.
There are more than 100 stalls selling gifts and food — all of which have been personally vetted by a jury — ensuring the quality is high.
Budapest Wine Festival
Every September, Buda Castle becomes one giant civilized party in the late summer sun when scores of wine producers show off their latest vintages in a relaxed, yet convivial atmosphere.
Buy a glass and take it round for tastings at the various stalls, picking up Hungarian snacks along the way.
Four festival stages take turns with music and entertainment throughout the four-day event and there's also a Harvest Parade around Buda Castle celebrating folk music and dancing.
Dates: September 5 to 8, 2019
Budapest Fish Festival
Traditional Hungarian cuisine meets international creations at the Budapest Fish Festival.
Courtesy Budapest Fish Festival
Hungarians spend the winter months keeping warm with a dish called halászlé — a red hot fisherman's soup brimming with paprika and river fish.
When early March comes round, many head to the three-day Budapest Fish Festival to feast on this spicy dish and plenty of other types of fish.
Heroes' Square is the setting for cooking contests, wine tastings, folklore music and fun for the kids — not to mention stall after stall of mouthwatering dishes.
Dates: March 2, to 4, 2019
Budapest Fish Festival, Heroes' Square, Budapest, Hősök tere, 1146
Danube Carnival
The Margaret Island Open-Air Stage and a host of other open-air venues around the Danube become filled with color during this week-long festival of folk dance in June.
Several hundred international dancers and musicians bring their own cultural sounds and dances to mingle with traditional Hungarian styles at the annual event.
The Carnival Parade that goes along the Danube Promenade to Vorosmarty Square is one of the festival's main highlights.
Dates: TBC
Festival of Folk Arts
Festival of Folk Arts brings top Hungarian craftsmen to Buda Castle.
Janos Peter photography
Craftspeople from all around Hungary descend on Buda Castle every August for a three-day celebration of crafts made in the country for hundreds of years.
Visitors can take part in workshops and watch the experts in action as they spin, weave, carve, paint, demonstrating skills that have been handed down over the generations.
The festival includes folk dances and performances and — this being Hungary — plenty of food stalls offering delectable traditionalRead More – Source
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text:
Melbourne media become arm of the police
Police say Melbourne's "society killer" has admitted his guilt. Hugo Kelly looks at how the media helped them get their man.
Police say Melbourne's "society killer" has admitted his guilt. Hugo Kelly looks at how the media helped them get their man.
His arrest on Saturday and his apparent confession to the murder of his mother and her invalid husband is a triumph of psychological policing and media manipulation by the Victoria Police.
Large sections of the media were complicit in the process "partners" in the investigation, aiding and abetting the police tactics of selectively leaking and suppressing information.
In the three weeks since it became clear this was a murder investigation, not a missing persons' case, Matthew and Maritza Wales have been placed under intense pressure by police.
Direct pressure, when the homicide squad dug up their back yard early in the investigation and conducted a search of the house, then mounted overt surveillance of the property.
And indirect, but probably just as effective, pressure as some police apparently selectively leaked information to media outlets during the course of their investigation, helping to shape media coverage. Just enough information was revealed to let Matthew and his wife Maritza know they were on to them. But not too much they hoped to compromise their case when it eventually came to court.
The leaking game was a fine balance, and the media lapped it all up, as journalists and editors asserted their own role in a murder investigation that had the whole town talking.
The Herald Sun took the lead, quickly dubbing the killings "The Society Murders", complete with their own dinkus, and devoting front page treatment to developments that pointed towards Matthew and Maritza as the suspects – like the police calling in Maritza for questioning on 6 May.
The electronic media followed suit. Last Tuesday, Channel Seven's Claire Brady revealed an apparently significant piece of evidence. Matthew had hired a trailer from a nearby service station the morning after the disappearance. She interviewed the servo employee who conducted the $30 EFTPOS transaction for Matthew and his male friend. Police had impounded the trailer and were conducting forensic tests.
So how did Claire jag this yarn? Is it because she's partnered with someone working at the Homicide Squad.
The story was followed up on the front pages next morning. The media drip feed must have felt like Chinese water torture to Matthew, Maritza and their uneasy consciences.
We're happy to attribute the trailer scoop to Claire Brady's investigative skills. But it's most likely the leak came from police.
Senior homicide cop Charlie Bezzina is a smart operator not unknown to give judicious off-the-record briefings to reporters, and with links to Channel Seven reporters going back nearly 20 years when Seven's legendary Lena Kaneva had the police beat sewn up. Charlie was one of the many aces in her flamboyantly-cut suit pockets.
And during this investigation, although police publicly and repeatedly refused to provide information on the record to the media, good journalists with established police sources were able to get information through unofficial channels.
These off-the-record briefings pointed in one direction only, and were reflected in media coverage. As Age editor Michael Gawenda testily told 3LO's Jon Faine on Friday: "Some of the other media have clearly decided who is the guilty party in this murder. We have not decided. It is not up to us to decide."
In response, Hun editor Peter Blunden told Faine this morning: "It's interesting when the Age says it would never convict anyone through their paper. I suppose Geoff Clarke's one interesting case to put that theory to rest."
Faine: "The relationship your reporters have with the police. Do you think there's anything improper? Do they go too far in working with the police virtually?"
Blunden: "Far from it. In fact, the police took out a suppression order on us last week from publishing quite a bit of information to do with this case…The police haven't exactly set there and run a circus giving out information, quite frankly. They've gone about their job and we've gone about ours."
Faine: "You singled out Matthew Wales for special attention as much as a week ago, didn't you?'
Blunden: "Oh yes. Yes we did. Only because there was information worth publishing.It was very fair for us to run material on it. And everybody was given their chance to have a say and we reported it accordingly."
Michael Gawenda would have also worried that the Hun was beating the Age to key developments in the story. On Friday, the Hun continued the trend, revealing the murdered matriarch's lawyer had taken Supreme Court action to freeze her assets.
Meanwhile, after the discovery of the couples' bodies, homicide squad head Inspector Brian Rix took an extraordinary step. With the Hun about to publish information about items found in the shallow grave containing the bodies, he sought, and obtained, a suppression order from the acting state coroner, Iain West, prohibiting the publication of details of evidence found with the bodies.
The media would normally be expected to raise a hue and cry about press freedom under such circumstances. Not so in this case. The Sunday Age's experienced scribbler Paul Heinrichs wrote that the suppression order was obtained by police to "protect the integrity of their investigation".
It was in line with the police strategy of selective leaking and censorship, aimed at placing their suspects under intense scrutiny, and flushing out a quick, clean confession.
Heinrichs wrote: "With only limited information available from police, some sections of the media began reporting evidence and interviewing witnesses independent of the formal investigation."
"In the past week the media have mounted a virtual round-the-clock vigil outside the home of Matthew Wales." Heinrichs reported police "have been surprised by the media interest".
During the media frenzy, journalists tested ethical limits. On Saturday, one Sunday Herald Sun reporter trawled through Prudence Reed's mail, reporting: "a card in the open letterbox indicated flowers had been left at a neighbour's two days ago".
With the Melbourne media going berserk on the story, interstate media took an interest, particularly since Margaret Wales-King's sister is Sydney art dealer Di Yeldham.
In Sydney, the Daily Telegraph took a bold line, with headlines like: "Mourning and suspicion" plastered over page three. AAP's Mike Hedge was by-lined in The Daily Telegraph writing as follows: "The family of a murdered Melbourne couple united in mourning yesterday as police continued to investigate one of the sons of victim Margaret Wales-King."
By then, Matthew and Maritza could really have done with a spin doctor. Maybe the PR-flak-of-last-resort, Mike Smith, could have helped them massage their testy relationship with journalists and get their spin out. As the Hun reported on Friday: "Matthew appeared to glare at the media" after his mother's memorial service.
As pressure mounted, police swooped, taking Matthew in for questioning on Saturday morning, then delivering him to a bail justice to be charged with murder just in time for the evening TV bulletins.
One can imagine what happened when Matthew Wales was taken to police headquarters. After placing him in a very public pressure-cooker for three weeks, experienced detectives probably only had to probe very gently before the hairdresser with a shaved head opened up to them.
Heinrichs wrote: "He is believed to have become emotional while questioned." One imagines him desperately trying to avoid implicating his wife: Maritza, the older woman, the foreigner with dark exotic good looks. The mother of their young child, arrested last night, following Matthew's confession, as an accessory after the fact, but bailed to go home for Mother's Day.
So is the media's complicity necessarily a bad thing? After all, according to police, "full admissions" to the crime have now been made.
Matthew and Maritza Wales' fate will be decided by a jury. A jury that's been subject to a media campaign cleverly directed in part by police that had a clear aim: pointing to the guilt of two particular suspects.
Maybe a tainted jury in a murder trial is the price of a free press. It's a global village now, baby, and we can't escape the media. But if sections of the media act in effect as arms of law enforcement agencies, at least we need to be aware of the competing agendas.
It's been an interesting month for murders in Melbourne. In a sense, this is the tale of two killings. The juxtaposition of the Wales-King killings with the execution of gangster Victor Peirce two weeks ago is intriguing.
Peirce was one of four men acquitted of the 1980s Walsh Street murders of Constables Eyre and Tynan. Every police officer in Victoria believes Peirce and his co-accused did it except the jury didn't buy it. Now Peirce joins in the grave two others police believe were linked to the Tynan/Eyre killings – Gary Abdallah and Jedd Houghton. Houghton and Abdallah were killed by operational police shortly after the Walsh St killings, back when Victoria Police shot first and asked questions later.
In the days following Peirce's Mayday murder, senior police told the media they weren't too upset by his death. The strong implication was they wouldn't waste too many resources chasing down the killer of their enemy.
Police Commissioner Christine Nixon eventually stepped in and promised the force would investigate the Peirce shooting thoroughly.
But if the same resources that were marshalled during the Wales-King circus had been placed into finding the killers of Victor Peirce, would police also now be hot on the trail of his assassins?
Feedback to Hugo Kelly at [email protected]
The media and gangland timeline
This day in Crikey: Tuesday, March 23, 2004
Ganglands: Three criminal trials and a closure


text:
From: Alveston (ST635880)
To: Whitfield (ST671913)
Via: Thornbury
Distance: 4.7 miles (7.6 km)
Meets: A38, B4461, A38
Former Number(s): A38
Old route now: A38
B4061 Alveston – Whitfield
B4061 (Spur) Eastwood Lodge – Buckover
The B4061 is a loop off the A38 to serve the town of Thornbury.
It starts at traffic lights at Alveston and heads north out of the village, quickly meeting the B4461. It then descends to Thornbury and goes through the town. It runs along the High Street before bending sharply right into The Plain, after which it turns left at a mini-roundabout.
After leaving town the road heads eastwards through Upper Morton and presently arrives at the A38 again at Whitfield, where the road ends. Shortly before this junction a spur heads southwards to provide access to the southbound A38, although there is no restriction of movement at the northern junction.
The B4061 in 1923
Originally the A38 went through Thornbury and the B4061 bypassed it to the east. However, the southern end of the B4061 was further north and the northern end further west (the street names Old Gloucester Road show the original route of the B4061). It was soon decided that it would be better to have the main road bypass the town and so the B4061 and A38 swapped numbers in this area in the late 1920s. Later still, the corner was cut off in Alveston (the old road there is now unclassified but was part of the B4461 for some time). The northern end of the ex-B4061 was bypassed later and has reverted to its original number (this is the south-facing spur).
B4000 – B4099
B4000 • B4001 • B4002 • B4003 • B4004 • B4005 • B4006 • B4007 • B4008 • B4009 • B4010 • B4011 • B4012 • B4013 • B4014 • B4015 • B4016 • B4017 • B4018 • B4019
B4060 • B4061 • B4062 • B4063 • B4064 • B4065(W) • B4065(E) • B4066 • B4067 • B4068 • B4069 • B4070 • B4071 • B4072 • B4073 • B4074 • B4075 • B4076 • B4077 • B4078 • B4079
B4080 • B4081 • B4082(W) • B4082(E) • B4083 • B4084 • B4085 • B4086 • B4087 • B4088 • B4089 • B4090 • B4091 • B4092 • B4093 • B4094 • B4095 • B4096 • B4097 • B4098 • B4099
Earlier iterations: B4004 • B4007 • B4008 • B4014 • B4018 • B4021(W) • B4021(E) • B4023 • B4024 • B4025(W) • B4025(E) • B4027 • B4028 • B4029
B4043 • B4044 • B4045 • B4047 • B4053 • B4063 • B4064 • B4065 • B4066 • B4068 • B4069(W) • B4069(E) • B4070 • B4076 • B4088 • B4094 • B4098
Retrieved from 'https://www.sabre-roads.org.uk/wiki/index.php?title=B4061&oldid=582867'
B4000-B4999
Roads which start and finish on the same road
Roads numbered B4061