except for the $\Phi_{x}$ and $\Phi_{y}^{\ast}$ states due to derivatives, so
we need a list of matrices $\mathbf{M}^{2}_{S,G}(\Phi_{x},\Phi_{y}^{\ast})$
for all relevant pairs of indices $x,y$ compatible with gauge symmetry. We
permute a basis of each $\mathbf{M}^{2}_{S,G}(\Phi_{x},\Phi_{y}^{\ast})$ in
such a way that it is block diagonal. Then, we traverse the entire list of
blocks for all $x,y$ and collect only those which are formally different
(treating field labels $\Phi_{x}$ and $\Phi_{y}^{\ast}$ as dummy variables).
Hence, the evaluation of $\mathcal{V}$ on each formal block needs to be
computed only once, provided we keep track of their multiplicities for each
fixed $x$ and $y$.

To recapitulate, our calculation of one-loop masses for a particular parameter
point thus involves the computation of $\partial_{x}\mathcal{V}(v_{0})$ and
$\partial_{x}\partial_{y}^{\ast}\mathcal{V}(v_{0})$ for a few thousand
formally different blocks of various sizes. If the diagonalization of Eq. (39)
for that block is smooth at $\Phi=v_{0}$, we compute $\lambda_{i}$,
$\lambda_{i,x}$, and $\lambda_{i,xy}$ with the algorithm described in Appendix
A.2. They are then inserted into Eqs. (40)–(42) evaluated at $v_{0}$, with
proper logarithm treatment described in Appendix A.3.

Only a few small blocks were found not to be smoothly diagonalizable at
$v_{0}$, all of them located in the gauge mass matrix
$\mathbf{M}^{2}_{G}(\Phi_{x},\Phi_{y}^{\ast})$. Fortunately, it was possible
to compute the expression $\mathcal{V}$ for these blocks
analytically.232323There exists an analytically computable smooth Jordan
decomposition for them.

###  A.2 Details of the numerical procedure

####  A.2.1 Preliminaries and notation

As discussed in Appendix A.1, we are interested in finding a procedure where
the starting point is a $\Phi$-dependent matrix $\mathbf{\hat{A}}$, and the
desired result are the eigenvalues and their first- and second-order
derivatives evaluated in the tree-level vacuum $v_{0}$: $\lambda_{i}$,
$\lambda_{i,x}$, and $\lambda_{i,xy}$. Equivalently, expanding the diagonal
field-dependent matrix $\mathbf{\mathbf{\hat{\Lambda}}}$ in Eq. (39) into a
Taylor series around vacuum via

$\displaystyle\mathbf{\hat{\Lambda}}=\mathbf{\Lambda}+\mathbf{\Lambda}_{,x}\,(\Phi-
v_{0})_{x}+\tfrac{1}{2!}\mathbf{\Lambda}_{,xy}\,(\Phi-v_{0})_{x}(\Phi-
v_{0})_{y}+\ldots,$ (43)

we seek the diagonal-matrix coefficients $\mathbf{\Lambda}$,
$\mathbf{\Lambda}_{,x}$, and $\mathbf{\Lambda}_{,xy}$. Note that $\Phi_{x}$
goes over all real degrees of freedom when the indices $x,y$, etc., are summed
over.

Obtaining the eigenvalues $\lambda_{i}$ is not difficult, since the tree-level
vacuum can simply be inserted into $\mathbf{\hat{A}}$ and then the numeric
matrix $\mathbf{\hat{A}}(v_{0})$ can easily be diagonalized. Finding the
eigenvalue derivatives, however, is complicated in the most general case by
possible degeneracies. In particular, suppose we have an eigenspace of
$\mathbf{\Lambda}$ that corresponds to a particular eigenvalue. While any
basis of this eigenspace can be used for diagonalizing $\mathbf{\Lambda}$,
derivatives such as $\mathbf{\Lambda}_{,x}$ may impose a particular preferred
basis in this eigenspace — in other words, the transition matrix
$\mathbf{\hat{R}}$ is field dependent and higher-order coefficients in its
$\Phi$-expansion matter. The proper ($\Phi$-dependent) eigenbasis therefore
gets progressively resolved only when higher derivatives of degenerate
eigenvalues are taken into account. For a more detailed discussion, we refer
the interested reader to the literature [46, 47, 48]. We mimic the procedure
from [48], in particular to derive the algorithm for the numeric evaluation of
the expansion coefficients in Eq. (43) up to 2nd order.

Before describing the algorithm and providing a quick derivation, we set up
our compact but convenient notation and definitions. As part of the algorithm,
we shall obtain a sequence of progressive basis transformations
$\mathbf{P}_{n}$. We denote the cumulative transformation up to the $n$-th
level by $\mathbf{R}_{n}$:

$\displaystyle\mathbf{R}_{n}$
$\displaystyle:=\mathbf{P}_{1}\ldots\mathbf{P}_{n}.$ (44)

For any matrix $\mathbf{X}$ from the set

$\displaystyle\mathbf{X}$
$\displaystyle\in\\{\mathbf{A},\mathbf{A}_{,x},\mathbf{A}_{,y},\mathbf{A}_{,xy}\\},$
(45)

we define its form in a different basis by

$\displaystyle\mathbf{X}_{0}$ $\displaystyle:=\mathbf{\hat{X}}(v_{0}),$ (46)
$\displaystyle\mathbf{X}_{n}$
$\displaystyle:=\mathbf{R}_{n}^{-1}\mathbf{X}_{0}\mathbf{R}_{n}=\mathbf{P}_{n}^{-1}\mathbf{X}_{n-1}\mathbf{P}_{n}.$
(47)

We combine the basis and derivative labels in the subscript via, e.g.,
$\mathbf{A}_{n,x}$, and refer to such a matrix as being in the $n$-basis. The
matrix $\mathbf{P}_{n}$ transitions from the $n$-basis to the $(n-1)$-basis.

It turns out each matrix $\mathbf{P}_{n}$ is obtained in the algorithm by
diagonalizing some particular numeric matrix; cf. Eqs. (54) and (56)–(58). We
write this diagonalization generically as

$\displaystyle\mathcal{A}_{n-1}$
$\displaystyle=\mathbf{P}_{n}\mathcal{D}_{n}\mathbf{P}_{n}^{-1},$ (48)

where book-keeping subscripts under $\mathcal{A}$ and $\mathcal{D}$ correspond
also to the underlying basis of the matrix, but the transformations among them
analogous to Eq. (47) do not apply. The diagonal matrices $\mathcal{D}_{n}$
are the matrix coefficients of interest in the expansion of
$\mathbf{\hat{\Lambda}}$. Explicitly, for fixed $x$ and $y$ we have

$\displaystyle\mathcal{D}_{1}$ $\displaystyle=\mathbf{\Lambda},$
$\displaystyle\mathcal{D}_{2}$ $\displaystyle=\mathbf{\Lambda}_{,x},$
$\displaystyle\mathcal{D}_{3}$ $\displaystyle=\mathbf{\Lambda}_{,y},$
$\displaystyle\mathcal{D}_{4}$ $\displaystyle=\mathbf{\Lambda}_{,xy}.$ (49)

Crucially, at each step $n$ there is an associated structure that
$\mathcal{A}_{n-1}$ possesses.

  * •

There exists a partition of the $(n-1)$-basis already from the previous step.
It is referred to as the $(n-1)$-partition, and the associated block structure
is composed of $(n-1)$-blocks. Edge case: The initial $0$-basis has the
trivial partition corresponding to a single block.

  * •

The matrix $\mathcal{A}_{n-1}$ turns out to be $(n-1)$-block diagonal.
Consequently, the transition matrix $\mathbf{P}_{n}$ that diagonalizes it can
also be taken as $(n-1)$-block diagonal.

  * •

The transition matrix $\mathbf{P}_{n}$ defines the $n$-basis. Since
$\mathbf{P}_{n}$ is $(n-1)$-block diagonal, the $(n-1)$-partition also applies
to the $n$-basis. We order the $n$-basis so that within each $(n-1)$-block the
eigenvectors belonging to the same eigenspace of $\mathcal{D}_{n}$ are grouped
together. This subdivision of the $n$-basis defines the $n$-partition, which
is clearly a refinement of the $(n-1)$-partition of the same basis. The
transition matrix $\mathbf{P}_{n}$ is arbitrary only up to basis changes
within $n$-blocks, within which $\mathbf{P}_{m}$ for $m>n$ can make changes to
the basis in later steps.

  * •

Incidentally, knowledge of $\mathbf{P}_{n}$ suffices to obtain the full
transition matrix $\mathbf{R}_{n}=\mathbf{R}_{n-1}\mathbf{P}_{n}$ transforming
the $n$-basis to the initial $0$-basis, where for $n=1$ we have
$\mathbf{R}_{1}=\mathbf{P}_{1}$. Intuitively, $\mathbf{R}_{n}$ simultaneously
diagonalizes $\mathcal{A}_{k}$ for all $k=0,\ldots,(n-1)$ if the $\mathcal{A}$
matrices are rewritten in the $0$-basis. Note that $\mathbf{R}_{n}$ is in
general not block diagonal.

The above underlying structure therefore yields a progressive series of basis
partitions, where the $n$-partition acts on the $n$-basis. The $n$-partition
is obtained in the diagonalization procedure of $\mathcal{A}_{n-1}$ and is
subordinate to the $m$-partition for any $m<n$. Consequently, $\mathbf{P}_{n}$
is $m$-block diagonal for $m<n$. The inductively defined structure is self-
perpetuating from step to step as long as $\mathcal{A}_{n}$ is indeed
$n$-block diagonal, which is true by construction.

For any matrix $\mathbf{X}_{n}$ in the $n$-basis, we denote the block-diagonal
and block-off-diagonal parts of the $n$-partition in the matrix by $[n]$ and
$(n)$ in the superscript, respectively,

$\displaystyle(\mathbf{X}_{n}^{[n]})_{ij}$
$\displaystyle:=\begin{cases}(\mathbf{X}_{n})_{ij}&\mbox{if }i\in I,j\in
J,I=J\\\ 0&\mbox{if }i\in I,j\in J,I\neq J\end{cases},$ (50)
$\displaystyle\mathbf{X}_{n}^{(n)}$
$\displaystyle:=\mathbf{X}_{n}-\mathbf{X}_{n}^{[n]},$ (51)

where capital indices $I$ and $J$ are $n$-block labels. For $m>n$ we further
define the notation

$\displaystyle\mathbf{X}_{m}^{[n]}$
$\displaystyle:=(\mathbf{R}_{n}^{-1}\mathbf{R}_{m})^{-1}\;\mathbf{X}_{n}^{[n]}\;(\mathbf{R}_{n}^{-1}\mathbf{R}_{m}),$
(52) $\displaystyle\mathbf{X}_{m}^{(n)}$
$\displaystyle:=(\mathbf{R}_{n}^{-1}\mathbf{R}_{m})^{-1}\;\mathbf{X}_{n}^{(n)}\;(\mathbf{R}_{n}^{-1}\mathbf{R}_{m}),$
(53)

which represents taking the $n$-block (off-)diagonal part of $\mathbf{X}_{n}$
and further transforming it from the $n$-basis into the $m$-basis.
Equivalently, we can obtain $\mathbf{X}^{[n]}_{m}$ or $\mathbf{X}^{(n)}_{m}$
by first fully rotating $\mathbf{X}$ to the $m$-basis, and taking the
(off)-diagonal part of the $n$-partition only in the end. This works due to
the hierarchy in block structure; i.e., the $n$-partition can be applied to
any $m$-basis for $m>n$.

####  A.2.2 The algorithm

Starting with a field-dependent matrix
$\mathbf{\hat{A}}\equiv\mathbf{A}(\Phi)$, we want to obtain $\lambda_{i}$,
$\lambda_{i,x}$, $\lambda_{i,y}$, and $\lambda_{i,xy}$ for fixed $x$ and $y$.
Following the notation introduced in Appendix A.2.1, perform the following
steps:

  1. 1.

Given the matrix $\mathbf{\hat{A}}$, compute (in the tree-level vacuum) the
numeric matrices $\mathbf{A}_{0}$, $\mathbf{A}_{0,x}$, $\mathbf{A}_{0,y}$,
$\mathbf{A}_{0,xy}$.

  2. 2.

Diagonalize $\mathbf{A}_{0}$ to obtain $(\mathbf{\Lambda},\mathbf{P}_{1})$:

$\displaystyle\mathbf{A}_{0}$
$\displaystyle=\mathbf{P}_{1}\mathbf{\Lambda}\mathbf{P}_{1}^{-1}.$ (54)

This gives us $\lambda_{i}$ and the $1$-partition, as well as
$\mathbf{R}_{1}=\mathbf{P}_{1}$.

  3. 3.

Compute an auxiliary matrix $\mathbf{\Omega}$ from $\mathbf{\Lambda}$:

$\displaystyle(\mathbf{\Omega})_{ij}$
$\displaystyle=\begin{cases}(\lambda_{j}-\lambda_{i})^{-1}&\mbox{if
}\lambda_{i}\neq\lambda_{j}\\\ 0&\mbox{if
}\lambda_{i}=\lambda_{j}\end{cases}.$ (55)

Clearly, $\mathbf{\Omega}=\mathbf{\Omega}^{(1)}$.

  4. 4.

Compute $\mathbf{A}_{1,x}=\mathbf{R}_{1}^{-1}\mathbf{A}_{0,x}\mathbf{R}_{1}$
and diagonalize its diagonal $1$-blocks to obtain
$(\mathbf{\Lambda}_{,x},\mathbf{P}_{2})$:

$\displaystyle\mathbf{A}^{[1]}_{1,x}$
$\displaystyle=\mathbf{P}_{2}\mathbf{\Lambda}_{,x}\mathbf{P}_{2}^{-1}.$ (56)

This gives us $\lambda_{i,x}$ and the $2$-partition.

  5. 5.

Compute $\mathbf{A}_{2,y}=\mathbf{R}_{2}^{-1}\mathbf{A}_{0,y}\mathbf{R}_{2}$,
where $\mathbf{R}_{2}=\mathbf{P}_{1}\mathbf{P}_{2}$. Take its diagonal
$2$-blocks and diagonalize to get $(\mathbf{\Lambda}_{,y},\mathbf{P}_{3})$:

$\displaystyle\mathbf{A}^{[2]}_{2,y}$
$\displaystyle=\mathbf{P}_{3}\mathbf{\Lambda}_{,y}\mathbf{P}_{3}^{-1}.$ (57)

This also gives the $3$-partition.

  6. 6.

Obtain $\mathbf{\Lambda}_{,xy}$ by diagonalizing the matrix on the left side
of the expression

$\displaystyle\left(\mathbf{A}_{3,xy}-(\mathbf{\Omega}\cdot\mathbf{A}_{3,x})\;\mathbf{A}_{3,y}^{(1)}-(\mathbf{\Omega}\cdot\mathbf{A}_{3,y})\;\mathbf{A}_{3,x}^{(1)}\right)^{[3]}$
$\displaystyle=\mathbf{P}_{4}\mathbf{\Lambda}_{,xy}\mathbf{P}_{4}^{-1}.$ (58)

The dot $(\cdot)$ denotes entry-wise multiplication. This gives us the sought-
after $\lambda_{i,xy}$, as well as the incidental transition matrix
$\mathbf{P}_{4}$ and the corresponding $4$-partition.

####  A.2.3 Derivation of the algorithm

To derive the algorithm we have specified in Appendix A.2.2, we start by
considering the field-dependent diagonalization in Eq. (39), which we
rearrange into

$\displaystyle\mathbf{\hat{A}}\mathbf{\hat{R}}$
$\displaystyle=\mathbf{\hat{R}}\mathbf{\hat{\Lambda}}.$ (59)

The matrix $\mathbf{\hat{A}}$ is provided, while both $\mathbf{\hat{R}}$ and
$\mathbf{\hat{\Lambda}}$ are not known explicitly, but their full form is not
necessary to compute the desired
$\Lambda,\Lambda_{,x},\Lambda_{,y},\Lambda_{,xy}$ for fixed $x$ and $y$. We
remind the reader that all hatted quantities imply field dependence, while the
non-hatted quantities are evaluated in the tree-level vacuum and are numeric
for given input parameters. We assume all notation defined so far in Appendix
A.2.1, in particular, Eqs. (43)–(53). For later convenience, we further amend
the definitions with

$\displaystyle\mathbf{\hat{\Gamma}}_{n}$
$\displaystyle:=\mathbf{P}_{n}\mathbf{P}_{n+1}\ldots\mathbf{P}_{N-1}\mathbf{\hat{P}}_{N},$
(60) $\displaystyle\mathbf{\hat{S}}_{;X}$
$\displaystyle:=\mathbf{\hat{R}}^{-1}\mathbf{\hat{R}}_{,X},$ (61)
$\displaystyle\mathbf{\hat{S}}_{n;X}$
$\displaystyle:=\mathbf{\hat{\Gamma}}_{n+1}\mathbf{\hat{S}}_{;X}\mathbf{\hat{\Gamma}}_{n+1}^{-1},$
(62) $\displaystyle\mathbf{S}_{n;X}$
$\displaystyle:=\mathbf{\hat{S}}_{n;X}(v_{0})=\mathbf{\hat{\Gamma}}_{n+1}(v_{0})\,\mathbf{\hat{S}}_{;X}(v_{0})\,\mathbf{\hat{\Gamma}}_{n+1}^{-1}(v_{0}),$
(63)

where $X$ denotes any combination of derivative variables $X\in\\{x,y,xy\\}$.
It turns out we need $N=5$ progressive transformations in total, with all
field dependence then pushed to $\mathbf{\hat{P}}_{5}$, while the prior
$\mathbf{P}$ transformations are used in the process of determining
$\Lambda,\Lambda_{,x},\Lambda_{,y},\Lambda_{xy}$. The above definitions imply

$\displaystyle\mathbf{\hat{R}}=\mathbf{R}_{n}\mathbf{\hat{\Gamma}}_{n+1}$ (64)

for any $n=1,2,3,4$, along with the trivial edge case
$\mathbf{\hat{\Gamma}}_{1}=\mathbf{\hat{R}}$. Intuitively, $\mathbf{R}_{n}$ is
the rotation from the initial into the $n$-basis, while the remaining part of
the full $\mathbf{\hat{R}}$ diagonalization is represented by
$\mathbf{\hat{\Gamma}}_{n+1}$. The useful quantities $\mathbf{\hat{S}}_{n;X}$
represent the $X$-derivative of $\mathbf{\hat{R}}$ unrotated by
$\mathbf{\hat{R}}$ from the left, with the result taken in the $n$-basis,
while the unhatted $\mathbf{S}_{n;X}$ is the same quantity evaluated in the
tree-level vacuum. Lastly, we define
$\mathbf{P}_{5}=\mathbf{\hat{P}}_{5}(v_{0})$.

To perform algebraic manipulations in the subsequent derivation, the following
set of commutators will likely prove useful to the reader:

$\displaystyle 0$
$\displaystyle=[\mathbf{\Lambda},\mathbf{\hat{\Gamma}}_{n\geq
2}(v_{0})]=[\mathbf{\Lambda},\mathbf{P}_{n\geq 2}],$ (65) $\displaystyle 0$
$\displaystyle=[\mathbf{\Lambda}_{,x},\mathbf{\hat{\Gamma}}_{n\geq
3}(v_{0})]=[\mathbf{\Lambda}_{,x},\mathbf{P}_{n\geq 3}],$ (66) $\displaystyle
0$ $\displaystyle=[\mathbf{\Lambda}_{,y},\mathbf{\hat{\Gamma}}_{n\geq
4}(v_{0})]=[\mathbf{\Lambda}_{,y},\mathbf{P}_{n\geq 4}],$ (67) $\displaystyle
0$ $\displaystyle=[\mathbf{\Lambda}_{,xy},\mathbf{\hat{\Gamma}}_{n\geq
5}(v_{0})]=[\mathbf{\Lambda}_{,xy},\mathbf{P}_{n\geq 5}].$ (68)

The commutators’ identities with the $\mathbf{P}_{n}$ matrices follow directly
from the block structure of $\mathbf{P}_{n}$ and the hierarchies among
$n$-partitions described in Appendix A.2.1. The $\mathbf{P}_{m}$ is $n$-block
diagonal for $m>n$, while the matrices $\mathcal{D}_{n}$ identified in Eq.
(49) are diagonal with the same eigenvalues in a given $n$-block, leading to
vanishing $\mathbf{P}$-commutators in Eqs. (65)–(68) in particular.

As for commutators with $\mathbf{\Gamma}_{n}$, the argument needs to be a bit
more subtle, since they are used in deriving the $\mathcal{A}_{n}$ matrix at a
step where $\mathbf{P}_{m}$ for $m>n$ has not yet been defined. Consider the
lowest non-trivial $\mathbf{\Gamma}$, which is $\mathbf{\Gamma}_{2}$. Since
$\mathcal{A}_{0}=\mathbf{A}_{0}$, we have both
$\mathbf{R}=\mathbf{\hat{R}}(v_{0})$ and $\mathbf{R}_{1}=\mathbf{P}_{1}$
diagonalizing it according to Eq. (59) evaluated in the vacuum and the
definition of $\mathbf{P}_{1}$ in Eq. (48), respectively. Thus,

$\displaystyle\mathcal{A}_{0}=\mathbf{R}\mathcal{D}_{1}\mathbf{R}^{-1}=\mathbf{R}_{1}\mathcal{D}_{1}\mathbf{R}_{1}^{-1},$
(69)

with the two rotation matrices related via
$\mathbf{R}=\mathbf{R}_{1}\mathbf{\Gamma}_{2}$ due to Eq. (64). The matrix
$\mathbf{\Gamma}_{2}$ thus represents the arbitrariness in the transition
matrix, which must commute with the matrix being diagonalized (cf. [48]):
$[\mathbf{\Lambda},\mathbf{\Gamma}_{2}]=0$. The matrices $\mathbf{\Gamma}_{m}$
for $m>2$ are related to $\mathbf{\Gamma}_{2}$ recursively via
$\mathbf{\Gamma}_{n}=\mathbf{P}_{n}\mathbf{\Gamma}_{n+1}$, so their
commutators with $\mathbf{\Lambda}$ can be derived from the
$\mathbf{\Gamma}_{2}$\- and $\mathbf{P}$-commutators. Analogous arguments can
be used to derive $\mathbf{\Gamma}$-commutators with derivatives of
$\mathbf{\Lambda}$, since $\mathbf{\Gamma}_{n}$ represents the arbitrariness
in the simultaneous diagonalization of the matrices $\mathcal{A}_{m}$ for all
$m$ between $0$ and $n-2$.

We now perform a sequence of steps.

  1. 1.

Evaluating Eq. (59) in vacuum and taking the first transformation matrix as
$\mathbf{P}_{1}=\mathbf{\hat{R}}(v_{0})$ yields
$\mathbf{A}_{0}=\mathbf{P}_{1}\mathbf{\Lambda}\mathbf{P}_{1}^{-1}$. Numeric
diagonalization of $\mathbf{A}_{0}$ thus gives $\mathbf{P}_{1}$ and
$\mathbf{\Lambda}$. Formally, this also implies the expression
$\mathbf{\hat{\Gamma}}_{2}=\mathbf{\hat{R}}(v_{0})^{-1}\mathbf{\hat{R}}$.

  2. 2.

Take the derivative $\partial_{x}$ of Eq. (59):

$\displaystyle\mathbf{\hat{A}}_{,x}\mathbf{\hat{R}}+\mathbf{\hat{A}}\mathbf{\hat{R}}_{,x}$
$\displaystyle=\mathbf{\hat{R}}_{,x}\mathbf{\hat{\Lambda}}+\mathbf{\hat{R}}\mathbf{\hat{\Lambda}}_{,x}.$
(70)

Multiply from the left with $\mathbf{\hat{R}}^{-1}$, rearrange and insert
suitable definitions for $\mathbf{\hat{S}}$ to obtain

$\displaystyle\mathbf{\hat{R}}^{-1}\mathbf{\hat{A}}_{,x}\mathbf{\hat{R}}-\mathbf{\hat{\Lambda}}_{,x}$
$\displaystyle=\mathbf{\hat{S}}_{;x}\mathbf{\hat{\Lambda}}-\mathbf{\hat{\Lambda}}\mathbf{\hat{S}}_{;x}.$
(71)

We now formally multiply Eq. (71) with $\mathbf{\hat{\Gamma}}_{3}$ from the
left and $\mathbf{\hat{\Gamma}}_{3}^{-1}$ from the right, thus undoing the
field-dependent rotation $\mathbf{\hat{R}}$ up to the as-yet-unknown
$\mathbf{R}_{2}=\mathbf{R}_{1}\mathbf{P}_{2}$. We then evaluate that
expression in vacuum and multiply it with $\mathbf{P}_{2}$ from the left to
obtain

$\displaystyle\mathbf{A}_{1,x}\mathbf{P}_{2}-\mathbf{P}_{2}\mathbf{\Lambda}_{,x}=\mathbf{P}_{2}\mathbf{S}_{2;x}\mathbf{\Lambda}-\mathbf{P}_{2}\mathbf{\Lambda}\mathbf{S}_{2;x}.$
(72)

We consider this matrix equation in terms of $1$-blocks, using labels $K_{1}$
and $L_{1}$ for $1$-block rows and $1$-block columns on each side,
respectively. Since $\mathbf{P}_{2}$ is $1$-block diagonal, we can write
$(\mathbf{P}_{2})_{K_{1}L_{1}}=(\mathbf{P}_{2})_{K_{1}K_{1}}\delta_{K_{1}L_{1}}$.
Similarly,
$(\mathbf{\Lambda})_{K_{1}L_{1}}=\lambda_{K_{1}}\,\delta_{K_{1}L_{1}}$; i.e.,
each diagonal $1$-block in $\mathbf{\Lambda}$ is proportional to the identity
matrix, while the off-diagonal blocks are zero.

We first examine the $1$-block-diagonal part of Eq. (72). For $K_{1}=L_{1}$,
the right-hand side vanishes, since $\mathbf{\Lambda}$ acts as an identity
block multiplied with the same proportionality factor in both terms, i.e.,
$\lambda_{K_{1}}=\lambda_{L_{1}}$. Hence,

$\displaystyle(\mathbf{A}_{1,x}\mathbf{P}_{2})_{K_{1}K_{1}}$
$\displaystyle=(\mathbf{P}_{2}\mathbf{\Lambda}_{,x})_{K_{1}K_{1}}.$ (73)

Since $\mathbf{P}_{2}$ and $\mathbf{\Lambda}_{,x}$ are already block diagonal
with respect to the $1$-partition, the equation can be written in full-matrix
form as

$\displaystyle\mathbf{A}^{[1]}_{1,x}\;\mathbf{P}_{2}$
$\displaystyle=\mathbf{P}_{2}\mathbf{\Lambda}_{,x}.$ (74)

Diagonalizing $\mathbf{A}_{1,x}^{[1]}$ yields $\mathbf{P}_{2}$ (and thus
$\mathbf{R}_{2}$) and the first derivatives $\mathbf{\Lambda}_{,x}$.

We now return to the $1$-block off-diagonal parts in Eq. (72), i.e.,
$K_{1}\neq L_{1}$. Since $\mathbf{P}_{2}$, $\mathbf{\Lambda}$, and
$\mathbf{\Lambda}_{,x}$ are all $1$-block diagonal, the second term on the
left-hand side vanishes, and we get

$\displaystyle(\mathbf{A}_{1,x})_{K_{1}L_{1}}(\mathbf{P}_{2})_{L_{1}L_{1}}$
$\displaystyle=(\lambda_{L_{1}}-\lambda_{K_{1}})(\mathbf{P}_{2})_{K_{1}K_{1}}\,(\mathbf{S}_{2;x})_{K_{1}L_{1}}.$
(75)

Solving for $(\mathbf{S}_{2;x})_{K_{1}L_{1}}$ gives

$\displaystyle(\mathbf{S}_{2;x})_{K_{1}L_{1}}$
$\displaystyle=\frac{1}{(\lambda_{L_{1}}-\lambda_{K_{1}})}(\mathbf{P}_{2}^{-1})_{K_{1}K_{1}}(\mathbf{A}_{1,x})_{K_{1}L_{1}}(\mathbf{P}_{2})_{L_{1}L_{1}}=(\mathbf{\Omega}\cdot\mathbf{A}_{2,x})_{K_{1}L_{1}},$
(76)

where the dot $(\cdot)$ denotes entry-wise multiplication and the auxiliary
matrix $\mathbf{\Omega}$ is defined in Eq. (55). This can be written as a
full-matrix equation via

$\displaystyle\mathbf{S}_{2;x}^{(1)}$
$\displaystyle=\mathbf{\Omega}\cdot\mathbf{A}_{2,x}.$ (77)

Note that the knowledge of the $1$-block off-diagonal parts of
$\mathbf{S}_{2;x}$ is sufficient for all further steps in our derivation of
the quantities of interest.

  3. 3.

Perform the following sequence of operations: Take the derivative
$\partial_{y}$ of Eq. (59), multiply with $\mathbf{\hat{R}}^{-1}$ from the
left, then formally multiply with $\mathbf{\hat{\Gamma}}_{4}$ from the left
and $\mathbf{\hat{\Gamma}}_{4}^{-1}$ from the right, and finally evaluate in
vacuum. These steps give the analogue of Eq. (72):

$\displaystyle\mathbf{A}_{2,y}\mathbf{P}_{3}-\mathbf{P}_{3}\mathbf{\Lambda}_{,y}=\mathbf{P}_{3}\mathbf{S}_{3;y}\mathbf{\Lambda}-\mathbf{P}_{3}\mathbf{\Lambda}\mathbf{S}_{3;y}.$
(78)

Similar to the case of $\partial_{x}$, we consider this equation in terms of
$2$-blocks. For diagonal blocks, we get

$\displaystyle(\mathbf{A}_{2,y}\mathbf{P}_{3})_{K_{2}K_{2}}=(\mathbf{P}_{3}\mathbf{\Lambda}_{,y})_{K_{2}K_{2}},$
(79)

yielding the full-matrix equation

$\displaystyle\mathbf{A}_{2,y}^{[2]}\mathbf{P}_{3}$
$\displaystyle=\mathbf{P}_{3}\mathbf{\Lambda}_{,y}$ (80)

indicating that diagonalization of $\mathbf{A}_{2,y}^{[2]}$ produces the
transition matrix $\mathbf{P}_{3}$ and the eigenvalue derivatives
$\mathbf{\Lambda}_{,y}$. The block off-diagonal part of Eq. (78) analogous to
the $\partial_{x}$ case gives

$\displaystyle(\mathbf{A}_{2,y})_{K_{1}L_{1}}\,(\mathbf{P}_{3})_{L_{1}L_{1}}$
$\displaystyle=(\lambda_{L_{1}}-\lambda_{K_{1}})\;(\mathbf{P}_{3})_{K_{1}K_{1}}(\mathbf{S}_{3;y})_{K_{1}L_{1}}$
(81)

in terms of the coarser $1$-blocks with $K_{1}\neq L_{1}$, resulting in

$\displaystyle(\mathbf{S}_{3;y})_{K_{1}L_{1}}$
$\displaystyle=(\mathbf{\Omega}\cdot\mathbf{A}_{3,y})_{K_{1}L_{1}}.$ (82)

In full-matrix notation, this can be written as

$\displaystyle\mathbf{S}_{3;y}^{(1)}$
$\displaystyle=\mathbf{\Omega}\cdot\mathbf{A}_{3,y}.$ (83)

  4. 4.

We now turn to the 2nd derivative and take $\partial_{x}\partial_{y}$ of Eq.
(59):

$\displaystyle\mathbf{\hat{A}}_{,xy}\mathbf{\hat{R}}+\mathbf{\hat{A}}_{,x}\mathbf{\hat{R}}_{,y}+\mathbf{\hat{A}}_{,y}\mathbf{\hat{R}}_{,x}+\mathbf{\hat{A}}\mathbf{\hat{R}}_{,xy}=\mathbf{\hat{R}}_{,xy}\mathbf{\hat{\Lambda}}+\mathbf{\hat{R}}_{,x}\mathbf{\hat{\Lambda}}_{,y}+\mathbf{\hat{R}}_{,y}\mathbf{\hat{\Lambda}}_{,x}+\mathbf{\hat{R}}\mathbf{\hat{\Lambda}}_{,xy}.$
(84)

Multiplying it by $\mathbf{\hat{R}}^{-1}$ on the left and inserting the
$\mathbf{\hat{S}}$-definitions gives

$\displaystyle\mathbf{\hat{R}}^{-1}\mathbf{\hat{A}}_{,xy}\mathbf{\hat{R}}+\mathbf{\hat{R}}^{-1}\mathbf{\hat{A}}_{,x}\mathbf{\hat{R}}\mathbf{\hat{S}}_{;y}+\mathbf{\hat{R}}^{-1}\mathbf{\hat{A}}_{,y}\mathbf{\hat{R}}\mathbf{\hat{S}}_{;x}+\mathbf{\hat{\Lambda}}\mathbf{\hat{S}}_{;xy}=\mathbf{\hat{S}}_{;xy}\mathbf{\hat{\Lambda}}+\mathbf{\hat{S}}_{;x}\mathbf{\hat{\Lambda}}_{,y}+\mathbf{\hat{S}}_{;y}\mathbf{\hat{\Lambda}}_{,x}+\mathbf{\hat{\Lambda}}_{,xy}.$
(85)

Multiplying this equation from the left with $\mathbf{\hat{\Gamma}}_{5}$, from
the right with $\mathbf{\hat{\Gamma}}_{5}^{-1}$, evaluating it in vacuum,
making use of the commutators in Eq. (68) and expressing
$\mathbf{\Lambda}_{,xy}$ yields

$\displaystyle\begin{split}\mathbf{\Lambda}_{,xy}&=\mathbf{P}_{4}^{-1}\mathbf{A}_{3,xy}\mathbf{P}_{4}+\mathbf{P}_{4}^{-1}\mathbf{A}_{3,x}\mathbf{P}_{4}\mathbf{S}_{4;y}+\mathbf{P}_{4}^{-1}\mathbf{A}_{3,y}\mathbf{P}_{4}\mathbf{S}_{4;x}-\mathbf{S}_{4;x}\,\mathbf{\Lambda}_{,y}-\mathbf{S}_{4;y}\,\mathbf{\Lambda}_{,x}+[\mathbf{\Lambda},\mathbf{S}_{4;xy}].\end{split}$
(86)

Since $\mathbf{\Lambda}_{,xy}$ is diagonal (due to $\mathbf{\hat{\Lambda}}$
being diagonal by definition), knowledge of diagonal $3$-blocks in Eq. (86)
suffices:

$\displaystyle(\mathbf{\Lambda}_{,xy})_{K_{3}K_{3}}$
$\displaystyle=\big{(}\mathbf{P}_{4}^{-1}\mathbf{A}_{3,xy}\mathbf{P}_{4}+(\mathbf{P}_{4}^{-1}\mathbf{A}_{3,x}\mathbf{P}_{4}-\mathbf{\Lambda}_{,x})\mathbf{S}_{4;y}+(\mathbf{P}_{4}^{-1}\mathbf{A}_{3,y}\mathbf{P}_{4}-\mathbf{\Lambda}_{,y})\mathbf{S}_{4;x}\big{)}_{K_{3}K_{3}}$
$\displaystyle=\big{(}\mathbf{P}_{4}^{-1}\mathbf{A}_{3,xy}\mathbf{P}_{4}+\mathbf{P}_{4}^{-1}[\mathbf{S}_{3;x},\mathbf{\Lambda}]\mathbf{S}_{3;y}\mathbf{P}_{4}+\mathbf{P}_{4}^{-1}[\mathbf{S}_{3;x},\mathbf{\Lambda}]\mathbf{S}_{3;x}\mathbf{P}_{4}\big{)}_{K_{3}K_{3}}.$
(87)

To derive the first line, we used the fact that the diagonal matrices
$\mathbf{\Lambda}$, $\mathbf{\Lambda}_{,x}$, and $\mathbf{\Lambda}_{y}$
respect the $3$-partition; i.e., each of their diagonal blocks is proportional
to an identity matrix block, and thus they commute with any matrix on the
$3$-block-diagonal part. The second line was derived by taking Eqs. (72) and
(78) into account, properly transformed to the $3$-basis.

Eq. (87) gives the diagonal $3$-blocks of $\mathbf{\Lambda}_{,xy}$ in terms of
an expression that is sandwiched between $\mathbf{P}_{4}^{-1}$ and
$\mathbf{P}_{4}$. Since $\mathbf{P}_{4}$ is $3$-block diagonal, and
$(\mathbf{\Lambda})_{K_{3}L_{3}}=\lambda_{K_{3}}\,\delta_{K_{3}L_{3}}$, we can
write the result more explicitly as

$\displaystyle\begin{split}(\mathbf{\Lambda}_{,xy})_{K_{3}K_{3}}&=(\mathbf{P}_{4}^{-1})_{K_{3}K_{3}}\Big{(}(\mathbf{A}_{3,xy})_{K_{3}K_{3}}+(\lambda_{L_{3}}-\lambda_{K_{3}})\big{(}(\mathbf{S}_{3;x})_{K_{3}L_{3}}(\mathbf{S}_{3;y})_{L_{3}K_{3}}+\\\
&\quad+(\mathbf{S}_{3;y})_{K_{3}L_{3}}(\mathbf{S}_{3;x})_{L_{3}K_{3}}\big{)}\Big{)}(\mathbf{P}_{4})_{K_{3}K_{3}},\end{split}$
(88)

with an implied sum over $L_{3}$ on the right-hand side. For $K_{3}$ and
$L_{3}$ belonging to the same $1$-block, i.e., $K_{3},L_{3}\in K_{1}$, the
factor $\lambda_{L_{3}}-\lambda_{K_{3}}$ becomes zero, since the eigenvalues
in $1$-blocks are identical. Hence, only the $1$-block off-diagonal entries of
$\mathbf{S}_{3;x}$ and $\mathbf{S}_{3;y}$ need to be considered. These are
available from Eqs. (76) and (83) transformed to the $3$-basis if necessary
(which is subordinate to the $1$-partition of $\mathbf{\Omega}$). This finally
gives

$\displaystyle\begin{split}(\mathbf{\Lambda}_{,xy})_{K_{3}K_{3}}&=\
(\mathbf{P}_{4}^{-1})_{K_{3}K_{3}}\Big{(}(\mathbf{A}_{3,xy})_{K_{3}K_{3}}+(\lambda_{L_{3}}-\lambda_{K_{3}})\times\big{(}(\mathbf{\Omega}\cdot\mathbf{A}_{3,x})_{K_{3}L_{3}}(\mathbf{\Omega}\cdot\mathbf{A}_{3,y})_{L_{3}K_{3}}+\\\
&\quad+(\mathbf{\Omega}\cdot\mathbf{A}_{3,y})_{K_{3}L_{3}}(\mathbf{\Omega}\cdot\mathbf{A}_{3,x})_{L_{3}K_{3}}\big{)}\Big{)}(\mathbf{P}_{4})_{K_{3}K_{3}}.\end{split}$
(89)

The $\lambda_{L_{3}}-\lambda_{K_{3}}$ factor removes one entry-wise
$\mathbf{\Omega}$ multiplication while preserving only the $1$-block off-
diagonal parts. The entire equation for diagonal $3$-blocks of
$\mathbf{\Lambda}_{,xy}$ can then be extended to a full-matrix equation by
using the block-diagonal extraction operator $[3]$ on both sides. By taking
into account that $\mathbf{\Lambda}_{,xy}$ and $\mathbf{P}_{4}$ are $3$-block
diagonal already, we get

$\displaystyle\mathbf{\Lambda}_{,xy}$
$\displaystyle=\mathbf{P}_{4}^{-1}\Big{(}\mathbf{A}_{3,xy}-(\mathbf{\Omega}\cdot\mathbf{A}_{3,x})\,\mathbf{A}_{3,y}^{(1)}-(\mathbf{\Omega}\cdot\mathbf{A}_{3,y})\,\mathbf{A}_{3,x}^{(1)}\Big{)}^{[3]}\mathbf{P}_{4}.$
(90)

We have thus derived the final result of Eq. (58). Note that this expression
is manifestly symmetric with respect to $x$ and $y$.

###  A.3 Detailed handling of logs

The procedure of Appendix A.1 computes one-loop corrections only to the mass
parameters in the scalar potential, so the resulting quantity is the _one-loop
effective mass_ rather than the _physical mass_ (also referred to as the _pole
mass_). The latter requires also knowledge of momentum-dependent contributions
coming from self-energy diagrams, which we do not consider. It is possible,
however, to improve upon the effective-mass calculation by carefully
considering its logarithmic contributions. This improved version is referred
to as the _regularized effective mass_.

One-loop effective scalar masses in Eq. (38) inevitably contain contributions
of the form

$\displaystyle\log\left[\frac{m^{2}_{l}}{\mu_{R}^{2}}\right],$ (91)

where $m_{l}$ is the tree-level mass of a scalar or gauge field in the loop
part of a Feynman diagram. The following situations require further attention:

  1. 1.

The particles in the loop are much lighter than the rest of the spectrum,
e.g., when their masses are near the intermediate scale. In such a case, the
regime $m^{2}_{l}\ll\mu_{R}^{2}$ causes logarithmic contributions to be
unphysically large. This effect is cancelled in physical masses by self-energy
contributions, which we do not have access to.

  2. 2.

For WGBs, $m_{l}^{2}=0$, so the logs contain an IR divergence unless their
prefactor expressions vanish as well.242424Some of the diverging logarithms of
Eq. (91) are tamed by a $m^{2}_{l}$-proportional prefactor.

  3. 3.

At least one of the PGBs has a tree-level tachyonic instability, i.e.,
$m_{l}^{2}<0$, which causes the argument of the log to be negative.

We prepare the ground for solving these issues in Appendices A.3.1 and A.3.2
and then present a comprehensive list of how to practically treat all the
cases in Appendix A.3.3.

####  A.3.1 The regularized effective mass

We define a quantity called _the regularized one-loop effective scalar mass_
$M^{2}_{S,reg}$ : We take the expression for the one-loop effective scalar
mass $M^{2}_{S}$ but solve the problem of IR divergences by emulating the
shift from the effective to the physical mass.

In particular, we take inspiration from the Abelian Higgs model [49], which
suggests using the replacement

$\displaystyle\log\left[\frac{m^{2}_{l}}{\mu_{R}^{2}}\right]$
$\displaystyle\mapsto\int_{0}^{1}dx\log\left[\frac{|m^{2}_{l}-m^{2}_{S}x(1-x)|}{\mu_{R}^{2}}\right]=\log\left[\frac{m^{2}_{S}}{\mu_{R}^{2}}\right]+\log\left[c\right]+I\left(c\right),$
(92)

where $m_{S}$ is the tree-level mass of the particle on the outer legs (i.e.,
the particle whose one-loop mass we are computing) and
$c:=\frac{m^{2}_{l}}{m^{2}_{S}}$. The term $I\left(c\right)$ includes
integration and is rather time-consuming to be evaluated in all one-loop
corrections. The effect of the integral $I\left(c\right)$ is shown in Fig. 14.

Figure 14: Logarithmic contributions to the one-loop effective scalar masses
with taming of the IR divergences [$\log(c)+I(c)$] and without it [$\log(c)$].
The parameter $c=\frac{m^{2}_{l}}{m^{2}_{S}}$, where $m_{l}$ and $m_{S}$ are
tree-level masses of fields in the loop and on the outer legs, respectively.

We simplify the approximation in the replacement rule of Eq. (92) by following
the normal log (blue curve) at high $c$ until $I\left(c_{0}\right)=0$ for
$c_{0}\approx 0.0763$ (the intersection of the blue and orange curves), then
take a constant value for $c$ smaller than $c_{0}$. The regularized one-loop
effective scalar mass is thus defined as the one-loop effective mass with all
logarithmic terms of Eq. (91) substituted by

$\displaystyle
M^{2}_{S,reg}:\log\left[\frac{m^{2}_{l}}{\mu_{R}^{2}}\right]\mapsto\begin{cases}\log\left[\frac{m^{2}_{l}}{\mu_{R}^{2}}\right]&;\frac{m^{2}_{l}}{m^{2}_{S}}>c_{0},\\\
\log\left[c_{0}\frac{m^{2}_{S}}{\mu_{R}^{2}}\right]&;\frac{m^{2}_{l}}{m^{2}_{S}}\leq
c_{0}.\end{cases}$ (93)

We refer to this procedure as _taming the logs_.

####  A.3.2 Physical vs. regularized effective mass

The regularized one-loop effective mass is the best possible approximation to
the actual physical mass without knowledge of self-energy contributions. A
crucial observation for inferring the non-tachyonicity of the physical
spectrum is that in the perturbative regime, the non-tachyonicity of the
regularized effective mass implies that the physical mass is also non-
tachyonic. We schematically argue this point below.

Let us consider the continuous Fourier-transformed two-point one-particle
irreducible (OPI) Green’s function
$\Gamma^{(2)}_{\overline{\text{MS}}}(p^{2})$ evaluated in the
$\overline{\text{MS}}$ renormalization scheme, where $p^{2}$ is square of the
outer leg particle’s four-momentum. In what follows, we suppress the matrix
structure in mass expressions for simplicity. In the effective potential
approach, we obtain the effective mass of a field, which is simply the zero-
momentum value of the Green’s function:

$\displaystyle M^{2}_{S}=-\Gamma^{(2)}_{\overline{\text{MS}}}(p^{2}=0).$ (94)

Subsequently,

$\displaystyle M^{2}_{S,reg}=M^{2}_{S}-\text{IR divergences}$ (95)

schematically holds for the regularized effective mass. The physical mass
though is defined as a solution to the equation

$\displaystyle\Gamma^{(2)}_{\overline{\text{MS}}}(p^{2}=M^{2}_{phys})=0.$ (96)

In addition, the behaviour of the two-point Green’s function is constrained in
such a way that

$\displaystyle\left.\frac{dZ\Gamma^{(2)}_{\overline{\text{MS}}}}{dp^{2}}\right|_{p^{2}=M^{2}_{phys}}=1,$
(97)

where $Z$ is the field-strength renormalization. We require that there exists
only one solution to Eq. (96), and we stay in the perturbative regime, where
higher-loop corrections are subdominant with respect to the tree-level values.
The latter tree-level case corresponds to

$\displaystyle M^{2}_{S,reg}=M^{2}_{phys},$ $\displaystyle\quad\quad Z=1.$
(98)

Figure 15: Four different schematic depictions of two-point OPI Green’s
function $\Gamma^{(2)}_{\overline{\text{MS}}}$ behaviour demonstrating four
combinations of tachyonic or non-tachyonic physical $M^{2}_{phys}$ or
regularized effective $M^{2}_{S}$ masses.

Schematically, there are four different qualitative solutions that can occur;
see Fig. 15. The solid lines show what must qualitatively happen in
perturbative scenarios, while the black dashed lines illustrate what might
occur in non-perturbative situations significantly deviating from the tree-
level case. The red color labels the case with a perturbative non-tachyonic
$M^{2}_{S}$ mass, while the blue color is used for the tachyonic $M^{2}_{S}$
mass scenario. The non-perturbative cases are not of interest here since they
prevent us from using perturbative expansions of OPI Green’s functions. Hence,
one can conclude that in the perturbative regime

$\mathbf{M^{2}_{phys}}$ is non-tachyonic $\mathbf{\Leftrightarrow}$
$\mathbf{M^{2}_{S,reg}}$ is non-tachyonic.

For valid parameter points, which pass the perturbativity and non-tachyonicity
check, the situation for all mass eigenstates thus corresponds to the red-line
scenario.

####  A.3.3 Practical aspects of the one-loop scalar-mass calculations

Table 6:  Summary of potential issues related to vastly different scales (associated with GUT-scale and intermediate-scale sectors at play) appearing in the one-loop mass calculations, and their practical treatment. Light field type |  A: Potential issues related to the calculation of loop contributions to their masses |  B: Potential issues related to their presence in other fields’ loop mass corrections |  Practical treatment of A and/or B
---|---|---|---
WGBs (would-be Goldstone bosons) |  None. WGBs are massless at all loops if calculated exactly. |  WGBs produce IR-diverging logarithms, such as $\log\left[\frac{m_{WGB}^{2}}{\mu_{R}^{2}}\right]$ with $m_{WGB}=0$, that get cancelled in physical masses. |  A: The WGBs’ zero masses will be recovered no matter the order of the perturbation series. B: The shift from the effective to physical masses is modelled by substitution (93).
Fields associated with intermediate-symmetry breaking |  Loop corrections to the light fields should be calculated using intermediate-symmetry effective-field theory, not the full $\mathrm{SO}(10)$. |  If the light mass is way lighter than $m^{2}_{S}$ (denoting the tree-level mass the loop correction is calculated to) potentially large logs may emerge. |  A: The tree-level mass will be used for all practical purposes. B: Large logs will get replaced (as for WGBs) via substitution (93).
PGBs (pseudo-Goldstone bosons), such as those discussed in Sec. II.1.4 |  Light fields in the loops may produce large logs as they are not properly tamed by using substitution (93) with tree-level PGB mass inserted in. |  The same as above for the light fields associated with intermediate-symmetry breaking. |  A: An iterative approach is employed,_a_ with the PGB tree-level masses in (93) substituted by one-loop PGB masses calculated in the previous iteration; cf. Appendix B.1.3. B:  Via Eq. (93).
_a_ Some potentially large logs vanish in the two limits of interest due to
the $T_{i}$ and $O_{i}$ prefactors of Ref. [26], Appendix B.

All potential issues related to the presence of vastly different scales in the
one-loop mass calculation (due to lighter fields in the loop), as well as
methods of their resolution are identified in Table 6. This resolves cases 1
and 24 in Appendix A.3 proper.

The arguments of the logs are always taken in absolute values, so that the
issues with a negative log argument are avoided (case 3 of Appendix A.3
proper). Since we check the non-tachyonicity of the spectrum for every point
at the one-loop level, and our perturbativity requirements demand the loop
contributions to GUT-scale particles to be smaller than the tree-level values,
the absolute value is relevant only for PGB fields in the loop. Furthermore,
since the PGB contribution is tamed in the computation of masses for _heavy_
fields, we only need to worry about this problem when computing corrections to
PGBs coming from PGBs in the loop. In such a case, the tree-level PGB
approximation is considered bad anyway, and we perform a further iterative
procedure with an associated perturbativity check; cf. item 4 in Appendix
B.1.3.

##  Appendix B Details of the numerical analysis

In this appendix, we address the technical details of the viability criteria
formulated in Sec. III. In particular, we outline the implementation of
various parameters describing the quality of fits presented in Sec. IV.2 along
with the penalties associated with the potential violation of the tachyonicity
and perturbativity criteria in the step-by-step numerical procedure.

###  B.1 The viability criteria of Sec. III

####  B.1.1 Non-tachyonicity of the scalar spectrum

Non-tachyonicity of the mass spectrum is an essential consistency criterion.
To this end, we use the regularized one-loop effective mass defined in
Appendix A.3 for all the scalars, except for those fields which are
parametrically lighter due to their association with the intermediate-symmetry
scale $\sigma$. Since none of these states is a pseudo-Goldstone boson, we can
afford using the tree-level formulae for their masses instead; see Table 6.

The contribution to the penalty of a parameter-space point from the possible
scalar spectrum tachyonicity is defined as

$\displaystyle p_{1}$
$\displaystyle=A_{1}\arctan\left[\sqrt{\sum_{j}\frac{1-\operatorname{sign}\left(M^{2}_{j}\right)}{2}\frac{|M^{2}_{j}|^{2}}{s_{j}}}\;\right],$
(99)

where $A_{1}\in\mathbb{R}^{+}$ is a weight parameter, $j$ runs over all the
scalar fields, and $s_{j}$ stands for the expected scale of $M^{2}_{j}$, i.e.,
$s_{j}=\omega_{max}^{2}$
($\omega_{max}:=\max\left[|\omega_{BL}|,|\omega_{R}|\right]$) for the _heavy_
fields associated with the GUT scale, $s_{j}=\tfrac{\omega_{max}^{2}}{10}$ for
PGBs, and $s_{j}=\sigma^{2}$ for the fields which should be naturally around
the intermediate scale. The only exceptions are the SM multiplets
$(3,1,-\tfrac{1}{3})$ and $(1,2,-\tfrac{1}{2})$, for which the mass and sign
factors $|M^{2}_{j}|$ and $\operatorname{sign}\left(M^{2}_{j}\right)$ are
replaced by the leading principal minors of the relevant mass matrices and
their signatures, respectively252525For these fields, the typical scale factor
$s_{j}$ is defined as $\omega_{max}^{2k}$ where $k$ stands for the
dimensionality of the corresponding minor.; see Sec. III.1. Note that the
first factor in the sum above ensures that there is no penalty for non-
tachyonic masses.

####  B.1.2 Unification of gauge couplings

In this study, the SM gauge couplings are expected to be unified at the one-
loop level; cf. Sec. III.2. The evolution of the usual $\alpha$-factors

$\displaystyle\alpha_{i}$ $\displaystyle=\frac{g_{i}^{2}}{4\pi}$ (100)

with the EW-scale boundary conditions is then driven by the equations

$\displaystyle\frac{d}{dt}\alpha_{i}^{-1}$ $\displaystyle=-a_{i}\,.$ (101)

The dimensionless running parameter $t$ is defined as

$\displaystyle t$ $\displaystyle=\frac{1}{2\pi}\log\frac{\mu_{R}}{M_{Z}},$
(102)

with $\mu_{R}$ denoting the running renormalization scale, and

$\displaystyle a_{i}$
$\displaystyle=-\frac{11}{3}\sum_{G}T(G_{i})D(G_{i})+\frac{4}{3}\sum_{F}\kappa_{F}T(F_{i})D(F_{i})+\frac{1}{3}\sum_{S}\eta_{S}T(S_{i})D(S_{i}),\qquad
i=1,2,3$ (103)

correspond to the beta functions of the “reduced” one-loop SM gauge couplings.
The three sums above run over the gauge, fermion, and scalar fields,
respectively, $T(R_{i})$ are the Dynkin indices of representations $R_{i}$
with respect to the group factor
$G_{i}~{}\in~{}\\{\mathrm{SU}(3)_{c},\mathrm{SU}(2)_{L},\mathrm{U}(1)_{Y}\\}$,
and $D(R_{i})$ denote the relevant multiplicities of $R_{i}$. We take
$\kappa_{F}=\tfrac{1}{2}$ or $1$ for Weyl or Dirac fermions, respectively;
likewise $\eta_{S}=\tfrac{1}{2}$ or $1$ for real or complex scalar fields.

The solution of Eq. (101) is trivial:

$\displaystyle\alpha_{i}^{-1}(t)$
$\displaystyle=\alpha_{i}^{-1}-\left(a_{i}^{(SM)}t+\sum_{t_{k}<t}\Delta
a_{i}^{(k)}(t-t_{k})\right),$ (104)

where we used the abbreviation $\alpha_{i}^{-1}(0)=\alpha_{i}^{-1}$ and
$t_{k}~{}=~{}\tfrac{1}{2\pi}\log\tfrac{M_{k}}{M_{Z}}$. In the expression
above, the Standard Model contributions to Eq. (103) have been collected in
$a_{i}^{(SM)}$ with values
$(a_{3}^{(SM)},a_{2}^{(SM)},a_{1}^{(SM)})=(-7,-\tfrac{19}{6},\tfrac{41}{10}),$
while the $\Delta a_{i}^{(k)}$ terms listed in Tables 8 and 9 encompass the
effects of gradual addition of the beyond-SM fields at scales $t$ exceeding
their individual masses $t_{k}$.

The gauge-coupling-unification constraint thus reads
$\alpha_{i}^{-1}(t_{GUT})=\alpha^{-1}(t_{GUT}),~{}i=1,2,3,$ where $t_{GUT}$ is
any scale at or above the heaviest threshold non-trivially influencing the
gauge coupling evolution. Thus, relation (104) can be rewritten as

$\displaystyle\begin{split}\alpha_{i}^{-1}&=\alpha^{-1}(t_{GUT})+a_{i}^{(SM)}t_{GUT}+\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad\sum_{\;\;\;\;\;\;\;k\in\text{all
fields}\backslash SM}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Delta
a_{i}^{(k)}(t_{GUT}-t_{k})-\frac{1}{4\pi}\log\frac{\alpha^{-1}(t_{\mu_{R}})}{\alpha^{-1}(t_{GUT})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\,\,\,\sum_{\;\;\;\;\;\;\;j\in\text{gauge
bosons}\backslash SM}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Delta
a_{i}^{(j)},\\\ \end{split}$ (105)

where the last subleading term takes care of the fact that the gauge boson
masses are calculated with the $\mathrm{SO}(10)$ gauge coupling evaluated at
the renormalization scale $\mu_{R}$, which is optimally chosen for each point
separately (to be specified in Appendix B.2).

In addition to all details of the bosonic spectrum, a good unification pattern
is characterized by $3$ main parameters, namely, $t_{GUT}$,
$t_{\sigma}~{}=~{}\tfrac{1}{2\pi}\log\tfrac{|\sigma|}{M_{Z}}$, and
$\alpha^{-1}$. These three quantities correspond to three basic degrees of
freedom which can be manipulated by: (i) scaling all dimensionful quantities
$\\{\omega_{BL},\omega_{R},\sigma,\mu_{R}\\}$ by a common factor, (ii)
shifting the intermediate-scale VEV $\sigma$, and (iii) modifying the initial
condition for the $\mathrm{SO}(10)$ gauge coupling. Parametrizing the
associated shifts by

$\displaystyle\\{\Delta t_{GUT},\Delta t_{\sigma},\Delta\alpha^{-1}\\},$ (106)

Eqs. (105) become the following set of three independent conditions for these
parameters262626Here we assume that the $\mathrm{SO}(10)$ fields (except the
SM field content) can be divided into two categories according to mass scale:
(i) the _heavy_ fields and PGBs, and (ii) the intermediate-scale fields. Their
masses are governed by $\omega_{max}$ and $\sigma$, respectively.:

$\displaystyle\begin{split}\alpha_{i}^{-1}&=\phantom{+}\alpha^{-1}(t_{GUT})+\Delta\alpha^{-1}+a_{i}^{(SM)}(t_{GUT}+\Delta
t_{GUT})-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\;\;\;\;\;\;\;k\in\text{all
fields}\backslash SM}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Delta
a_{i}^{(k)}(t_{k}-t_{GUT})-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\;\;\;\;\;\;\;l\in\text{intermediate
fields}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad\Delta
a_{i}^{(l)}\Delta t_{\sigma}-\\\
&\quad-\frac{1}{4\pi}\log\frac{\alpha^{-1}(t_{\mu_{R}})}{\alpha^{-1}(t_{GUT})+\Delta\alpha^{-1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;\;\sum_{\;\;\;\;\;\;\;j\in\text{gauge
bosons}\backslash
SM}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\quad\Delta
a_{i}^{(j)}.\\\ \end{split}$ (107)

These are rather easy to solve for most of the initial choices of $t_{GUT}$,
$t_{\sigma}$, and $\alpha^{-1}$, which together with the resulting shifts
yield the desired gauge unification pattern.

####  B.1.3 Perturbativity constraints

As far as the perturbativity constraints of Sec. III.3 are concerned, we
implement four simple tests that quantify the level of our
satisfaction272727Recall that perturbativity constraints are the kind of
criteria which, practically by definition, require human (and hence, biased)
input; cf. Sec. III.3. with the overall perturbativity. The associated
considerations in the order of Sec. III.3 are as follows:

  1. 1.

Global-mass-perturbativity test
The penalty $p_{2}$ associated with the Global-mass-perturbativity test is
defined as

$\displaystyle p_{2}$
$\displaystyle=\frac{2}{\pi}A_{2}\cdot\arctan\left[H(\overline{\Delta}-1)\right],$
(108)

where $A_{2}\in\mathbb{R}^{+}$ is a weight factor, $H(x):=x\,\theta(x)$,
$\theta(x)$ is the Heaviside step function, $\overline{\Delta}$ is defined in
a repeat of Eq. (22) by

$\displaystyle\overline{\Delta}$
$\displaystyle:=\frac{\max_{i,j\in\text{\emph{heavy}
fields}}[|M^{2}_{ij,\text{one-
loop}}-M^{2}_{ij,\text{tree}}|]}{\overline{M^{2}}_{\rm heavy}},$ (109)

with $M^{2}_{ij,\text{one-loop}}$ and $M^{2}_{ij,\text{tree}}$ denoting one-
loop and tree-level scalar-mass-matrix elements, respectively. The symbol
$\overline{M^{2}}_{\rm heavy}$ denotes the average (over real degrees of
freedom) of the _heavy_ tree-level scalar masses. In physical terms, a penalty
is awarded if the maximal one-loop correction of the mass square is larger
than the average tree-level mass square.

  2. 2.

Perturbativity of the RG evolution
For a given individual point in the parameter space, all calculations are done
at a certain “optimal” renormalization scale $\mu_{R}$. For measurable
quantities (such as pole masses), the specific choice of $\mu_{R}$ should not
matter. However, due to our use of the effective potential approach and proxy
quantities such as regularized effective masses, the $\mu_{R}$-dependence is
technically not eliminated [50, 51].

The residual $\mu_{R}$-dependence needs to be kept under control to ensure
that the perturbativity constraints will not get out of hand once the
renormalization scale is changed. For that purpose, we perform one-loop RG
running of dimensionless parameters and check for the stability of the vacuum
position under loop corrections (cf. next point) at a renormalization scale
other than $\mu_{R}$ as well (specifics are given later in Appendix B.2).

For further convenience, we introduce some auxiliary quantities, repeating
here the definitions of Eqs. (23) and (24) for completeness:

$\displaystyle t_{\pm}$ $\displaystyle=\log_{10}\frac{\mu_{R\pm}}{\mu_{R}},$
(110)

where $\mu_{R}$ is the starting “optimal” renormalization scale specific to
every point in the parameter space, and $\mu_{R+}$ ($\mu_{R-}$) denotes the
renormalization scale for which the couplings hit a Landau pole when running
upward (downward). The geometric average

$\displaystyle\overline{t}=\sqrt{t_{-}t_{+}}$ (111)

encodes, roughly speaking, how many orders of magnitude the scalar couplings
can run up and down before encountering a Landau-pole-type singularity.
Finally we define an optional penalty $p_{5}$ in order to test levels of
robustness with respect to the RG running:

$\displaystyle
p_{5}=\frac{2}{\pi}A_{5}\cdot\arctan\left[H\left(\frac{1}{\bar{t}}-\frac{1}{\bar{t}_{thr}}\right)\right],$
(112)

where $A_{5}\in\mathbb{R}^{+}$ is a weight factor, $H(x)$ is defined below Eq.
(108), and $\bar{t}_{thr}$ is an acceptance threshold for $\bar{t}$.

  3. 3.

Stability of the vacuum position
We start by defining a vector of dimensionful scalar parameters
$\vec{w}=\\{\mu^{2},\nu^{2},\tau^{2}\\}$. It is connected to the VEVs via one-
loop stationarity conditions, which can be written in the form

$\displaystyle\vec{w}=\vec{w_{0}}+f(\vec{w}),$ (113)

where $\vec{w_{0}}$ denotes the tree-level part from Eqs. (12)–(14), and the
function $f(\vec{w})$ represents one-loop corrections. Eq. (113) is solved
iteratively via

$\displaystyle\vec{w}^{(k+1)}=\vec{w}_{0}+f(\vec{w}^{(k)}),~{}k\geq 1,$ (114)

taking $\vec{w}^{(0)}=\vec{w_{0}}$. The iterative procedure is stopped, and
$\vec{w}$ is deemed to not have converged if

$\displaystyle\max_{i\in\\{1,2,3\\}}\sqrt{\left|\frac{w^{(k)}_{i}}{w_{0i}}\right|}$
$\displaystyle>1+\zeta,$ (115)

where we set $\zeta=0.3$, and the result of the iterative procedure is
labelled as $\vec{w}_{iter}$. The maximal number of iterations is chosen to be
$30$.

The penalization $p_{3}$ that assesses vacuum stability is the following:

if $\displaystyle\sqrt{\frac{\lVert\vec{\lambda}\rVert^{2}}{12}}>10:$
$\displaystyle p_{3}$
$\displaystyle=\frac{1}{3}\left(A_{3}^{(1)}+A_{3}^{(2)}\right)+\frac{2}{3\pi}A_{3}^{(3)}\arctan\left[\sqrt{\frac{\lVert\vec{\lambda}\rVert^{2}}{12}}-10\right],$
else if $\displaystyle\vec{w}_{iter}\text{ does not converge}:$ $\displaystyle
p_{3}$
$\displaystyle=\frac{1}{3}A_{3}^{(1)}+\frac{2}{3\pi}A_{3}^{(2)}\arctan\Bigg{[}H\left(\frac{\lVert\vec{w}_{iter}-\vec{w}_{0}\rVert}{\lVert\vec{w}_{0}\rVert}-1\right)+$
$\displaystyle\qquad+(30-\\#_{\text{iterations}})\Bigg{]},$ else
$\displaystyle\vec{w}_{iter}\text{ converges:}$ $\displaystyle p_{3}$
$\displaystyle=\frac{2}{3\pi}A_{3}^{(1)}\arctan\left[H\left(\frac{\lVert\vec{w}_{iter}-\vec{w}_{0}\rVert}{\lVert\vec{w}_{0}\rVert}-1\right)+\right.$
$\displaystyle\left.\quad\quad+H\left(\frac{\lVert\vec{w}_{iter}-\vec{w}^{(1)}\rVert}{\lVert\vec{w}^{(1)}\rVert}-1\right)\right],$
(116)

where $A_{3}^{(i)}\in\mathbb{R}^{+}$ are weight factors, $H(x)$ is defined
below Eq. (108), and the vector of scalar couplings is

$\displaystyle\vec{\lambda}$
$\displaystyle:=\\{a_{2},a_{0},\lambda_{0},\lambda_{2},\lambda_{4},\lambda_{4}^{\prime},\alpha,\beta_{4},\beta_{4}^{\prime},\gamma_{2},\eta_{2}\\}.$
(117)

The penalization in Eq. (116) is constructed so that each parameter point
falls under exactly one of three conditions. The first condition and the
associated $A_{3}^{(3)}$ weight penalize points which fall well outside the
$\mathcal{O}(1)$ circle. The second condition and $A_{3}^{(2)}$ penalize
points for which the vector of couplings $\vec{\lambda}$ is sufficiently
small, but $\vec{w}_{iter}$ does not converge under the criterion in Eq.
(115); i.e., the iterative procedure is stopped prematurely. Analogously, the
third condition and weight $A_{3}^{(1)}$ penalize points if a converged
$\vec{w}_{iter}$ differs too much from the tree-level $\vec{w}_{0}$ and the
initial approximation of the one-loop result $\vec{w}^{(1)}$. Note that the
expressions are such that the penalization is smaller if a later condition
applies, effectively ranking the criteria in descending order of importance.

  4. 4.

Iterative pseudo-Goldstone masses
It turns out the computation of regularized one-loop effective masses for PGBs
involves certain subtleties. As laid out in Table 6, the presence of particles
with masses much below the GUT scale in the loop is handled by the replacement
in Eq. (93) in order to tame unphysical large-log contributions [52]. When
computing mass corrections to PGBs, Eq. (93) implies replacing log arguments
involving the mass of lighter (e.g., intermediate-scale) particles with the
PGB mass, leading to an equation of the form

$\displaystyle M^{2}_{PGB}$
$\displaystyle=C_{1}+C_{2}\,\log(M^{2}_{PGB}/\mu_{R}^{2}),$ (118)

where $C_{1}$ and $C_{2}$ are numeric coefficients independent of the
expression $M^{2}_{PGB}$. The tree-level masses of the PGBs are accidentally
small and not very close to the chosen renormalization scale. This implies
that the log contribution is additionally enhanced, and the $C_{1}$ term,
usually dominated by the tree-level mass, is reduced. The relative importance
of the $C_{2}$ term is thus increased, and the solution of Eq. (118) can be
far away from $C_{1}$. This is problematic conceptually, since the $C_{2}$
term itself is merely an approximation arising from taming the logs. Note that
the computation of _heavy_ masses does not suffer from the same problem,
despite an analogous equation, since their tree-level masses are large and the
$C_{1}$ term there dominates.

In practice, we solve Eq. (118) iteratively via

$\displaystyle M^{2}_{PGB,(i+1)}$
$\displaystyle=C_{1}+C_{2}\,\log(|M^{2}_{PGB,(i)}|/\mu_{R}^{2})$ (119)

starting with the value $M^{2}_{PGB,(0)}$ as the tree-level mass and requiring
that all iterations after $i\geq 1$ are non-tachyonic. Note the absolute value
in the argument of the log placed there to deal with the possible initial
tachyonic instability. We perform a maximum of $30$ iterations. We deem that
convergence has been achieved and stop the process if the relative size of the
shift between $M^{2}_{PGB,(i+1)}$ and $M^{2}_{PGB,(i)}$ of two consecutive
steps is smaller than $10^{-3}$, a size corresponding to corrections at two-
loop level. We denote the result of the iterative process as
$M^{2}_{PGB,iter}$, while $M^{2}_{PGB,(1)}$ is what we refer to in Appendix
A.3 as the _regularized one-loop effective mass_.

Because of the issues discussed above, a point is considered valid not only if
convergence in Eq. (119) is achieved, but also that the $C_{2}$-proportional
log contribution is sufficiently small, ensuring the log-taming approximation
works as intended and the computed regularized effective mass is close to the
physical mass. Hence, we demand that the relative difference between the
regularized one-loop effective PGB mass $M^{2}_{PGB,(1)}$ and the iterative
solution $M^{2}_{PGB,iter}$ is less than282828Smaller differences are at the
level of two-loop mass corrections. Moreover, this choice assures that the
replacement of $M^{2}_{PGB,(1)}$ with $M^{2}_{PGB,iter}$ in gauge-coupling
unification causes changes comparable to two-loop running effects. $10\%$. We
define the penalty $p_{4}$ associated with these considerations, applied now
to multiple PGBs, by

if $\displaystyle M^{2}_{PGB,iter}\ \text{does not converge}:$
$\displaystyle\qquad
p_{4}=\frac{1}{2}A_{4}^{(1)}+\frac{1}{\pi}A_{4}^{(2)}\arctan\left[H(\Delta
M^{2}-10^{-1})\right.\left.+H(\delta M^{2}_{iter}-10^{-3})\right],$ else
$\displaystyle M^{2}_{PGB,iter}\ \text{converges}:$ $\displaystyle\qquad
p_{4}=\frac{1}{\pi}A_{4}^{(1)}\arctan\left[H(\Delta M^{2}-10^{-1})\right],$
(120)

where $A_{4}^{(i)}\in\mathbb{R}^{+}$ are weight factors, $H(x)$ is defined
below Eq. (108), and

$\displaystyle\Delta M^{2}$ $\displaystyle:=\max_{x\in
PGBs}\left|\frac{\left(M^{2}_{PGB,(1)}\right)_{x}-\left(M^{2}_{PGB,iter}\right)_{x}}{\left(M^{2}_{PGB,(1)}\right)_{x}}\right|,$
(121) $\displaystyle\delta M^{2}_{iter}$ $\displaystyle:=\max_{x\in
PGBs}\left|\frac{\left(M^{2}_{PGB,(i_{max})}\right)_{x}-\left(M^{2}_{PGB,(i_{max}-1)}\right)_{x}}{\left(M^{2}_{PGB,(i_{max})}\right)_{x}}\right|,$
(122)

with $i_{max}=30$.

###  B.2 Implementation of the numerical analysis: Step-by-step procedure

Eventually, the criteria discussed above are used to find viable points, i.e.,
those which pass all the consistency checks and hence have zero penalization.
Conceptually, the numerical procedure for assessing a single parameter point
consists of two steps:

  1. (a)

First, a point in the parameter space is selected. The VEVs and unified gauge
coupling $g$ are adjusted so that the resulting one-loop scalar spectrum is
consistent with gauge unification requirements.

  2. (b)

Second, the penalization of the adjusted point is calculated. It is used for
determining its viability.

The procedure of choosing new candidate points in the parameter space for
assessment, i.e., a scan of the parameter space, follows the differential
evolution algorithm [32] as the minimization algorithm of choice. We use a
stochastic implementation with a mutation factor $F$ randomly sampled from the
interval $[0.5,2]$. The scanning process first discovers viable points by
minimizing the penalization function until it reaches zero, and then it
explores the zero penalty region. In this way, datasets of viable points from
Table 3 are produced.

Returning to part (a), the detailed steps are the following:

  1. 1.

The input parameters for a point consist of

$\displaystyle\vec{\lambda},\quad
g,\quad\omega_{BL},\quad\omega_{R},\quad\sigma,$ (123)

where $\vec{\lambda}$ is defined in Eq. (117) and $g$ is the unified gauge
coupling. These parameters are deemed to be the running values at the yet to
be determined “optimal” renormalization scale $\mu_{R}$. For later
convenience, we label the $\mathrm{SO}(10)$-breaking VEV as $\omega_{max}$
($\omega_{R}$ or $\omega_{BL}$ in the regime $\omega_{BL}\to 0$ or
$\omega_{R}\to 0$, respectively), and the subdominant induced VEV as
$\omega_{min}$ ($\omega_{BL}$ or $\omega_{R}$ in the $\omega_{BL}\to 0$ or
$\omega_{R}\to 0$ limit, respectively).

  2. 2.

We initially fix $g=0.5$ and $\omega_{max}=10^{16}\,\mathrm{GeV}$. The
$\vec{\lambda}$, $\sigma$, and $\omega_{min}$ inputs are provided by the
selection of the parameter point, for which we demand that they conform to Eq.
(21).

  3. 3.

With all the initial input parameters now set, the tree-level gauge and scalar
spectra are calculated.

  4. 4.

The “optimal” renormalization-scale square $\mu^{2}_{R}$ is determined as the
arithmetic mean (over real degrees of freedom) of the _heavy_ tree-level
scalar-mass squares.

  5. 5.

With $\mu_{R}$ in hand, the one-loop scalar-mass spectrum for _heavy_ and PGB
fields is computed using the method described in Appendix A, replacing their
tree-level values. The corrections to gauge fields and the intermediate-scale
scalars are not calculated. We refer to this collection of our best available
results for all gauge and scalar masses as the _initial spectrum_.

  6. 6.

The _initial spectrum_ is used for thresholds in the one-loop gauge evolution
analysis.292929For tachyonic masses, we take their absolute values. Such
points are later penalized due to tachyonicity anyway. Note that the tachyonic
property is generally preserved even after adjustment described in this step.
The complete SM-gauge-coupling unification with the EW-scale boundary
conditions [53]

$\displaystyle\alpha_{EM}^{-1}\,(M_{Z})$ $\displaystyle=127.952\pm 0.009,$
(124) $\displaystyle\sin^{2}\theta_{W}\,(M_{Z})$ $\displaystyle=0.23121\pm
0.00004,$ (125) $\displaystyle\alpha_{3}\,(M_{Z})$ $\displaystyle=0.1179\pm
0.0010$ (126)

is achieved as described in Appendix B.1.2. Technically, this is accomplished
by three simultaneous actions: (i) rescaling of all dimensionful quantities
$\\{\omega_{BL},\omega_{R},\sigma,\mu_{R}\\}$ by a common factor, (ii) further
adjustment303030The VEVs $\sigma$ and $\omega_{min}$ are adjusted in such a
way that the ratio $\chi$ of Eq. (15) stays constant. of $\sigma$ and
$\omega_{min}$, and (iii) optimizing the value of $g$. This procedure provides
new values for $g$, $\mu_{R}$, and the three VEVs — defining the _adjusted
parameter point_.

  7. 7.

For the _adjusted parameter point_ , the tree-level spectrum for gauge and
scalar fields is computed. Furthermore, the _heavy_ and PGB masses are
improved with the one-loop correction. This collection of the best available
mass values is referred to as the _updated spectrum_.

In part (b), the remaining consistency criteria of Appendices B.1.1 and B.1.3
(tachyonicity and perturbativity) are imposed on the _adjusted parameter
point_ and the relevant penalizations are calculated. The detailed steps for
part (b) are as follows:

  1. 1.

The stability of the vacuum position is investigated — the penalization
$p_{3}^{(1)}$ is computed. For a viable point, the one-loop corrections should
not shift the position of the tree-level vacuum by more than $100\%$.

  2. 2.

All the dimensionless parameters are RG run half-an-order of magnitude upward
using one-loop beta functions (cf. Appendix C). Note that this running
distance in scale turns out to be consistent with the spread of $\mu_{R}$
values for viable points (see Sec. IV), implying that all viable points could
be adjusted to a common scale for comparison if desired.

  3. 3.

RG-run couplings are used to recheck the stability of the vacuum position —
the penalization $p_{3}^{(2)}$ is acquired. All viable points satisfy
$t_{+}>0.5$, and their shift of the tree-level vacuum position due to the one-
loop corrections is still, even after running, not more than $100\%$.

  4. 4.

The tachyonicity of the _updated spectrum_ is inspected — the penalization
$p_{1}$ is obtained.

  5. 5.

Global mass perturbativity is examined — the penalization $p_{2}$ is computed.
For viability, the maximal one-loop scalar-mass correction is required to be
smaller than the average of _heavy_ tree-level scalar masses, i.e.,
$\overline{\Delta}\leq 1$.

  6. 6.

The iterative pseudo-Goldstone masses are investigated — the penalization
$p_{4}$ is determined. For a viable point, the relative difference between the
initial and final values in the iterative procedure for the PGB one-loop mass
is constrained to less than $10\%$, i.e., $\Delta M^{2}<0.1$.

  7. 7.

(Optional) Stricter RG-perturbativity criterion is imposed — the penalization
$p_{5}$ is added. All viable points satisfy $\bar{t}>\bar{t}_{thr}$, where
$\bar{t}_{thr}$ is an acceptance threshold of a given search (see Table 3 in
Sec. IV.2).

  8. 8.

The overall penalization of a point is the following:

$\displaystyle p$
$\displaystyle=p_{1}+p_{2}+p_{3}^{(1)}+p_{3}^{(2)}+p_{4}+p_{5}.$ (127)

The relative significance of a particular criterion in the minimization
algorithm, such as tachyonicity or perturbativity, can be tuned by changing
the value of the $A_{i}$ weight factors present in penalizations $p_{i}$. We
chose the configuration $A_{i}=1$, which leads to $p_{i}\in[0,1]$ for all
$i=1,2,3,4,5$. If the point has zero penalization, it satisfies all the
imposed criteria and belongs to the set of viable points obtained in the
parameter-space scan. Note that the shape of the viable part of the parameter
space is independent of the values $A_{i}$.

##  Appendix C Beta functions for scalar parameters

The one-loop beta functions for scalar-potential parameters can be extracted
from the field-dependent effective potential (30) following the procedure
outlined in [44].

###  C.1 General principles

Suppose that at tree level a scalar coupling $\lambda$ can be written as a
derivative of the scalar potential via

$\displaystyle\lambda$
$\displaystyle=\frac{\partial^{4}V_{0}(\Phi)}{\prod_{i=1}^{4}\partial\Phi_{i}}$
(128)

for a suitable choice of scalar fields $\Phi_{i}.$ The complete one-loop
effective potential (30) has to satisfy the Callan–Symanzik renormalization
group equation

$\displaystyle\left(\frac{\partial}{\partial\log\mu_{R}}+\beta_{\lambda}\frac{\partial}{\partial\lambda}+\sum_{j}\gamma_{\Phi_{j}}\Phi_{j}\frac{\partial}{\partial\Phi_{j}}\right)V(\Phi)$
$\displaystyle=0,$ (129)

where $\mu_{R}$ is the running renormalization scale, and the $j$ index runs
over all scalar fields. Consequently,

$\displaystyle\left(\frac{\partial}{\partial\log\mu_{R}}+\beta_{\lambda}\frac{\partial}{\partial\lambda}+\sum_{j}\gamma_{\Phi_{j}}\frac{\partial}{\partial\Phi_{j}}+\sum_{i=1}^{4}\gamma_{\Phi_{i}}\right)\frac{\partial^{4}V(\Phi,\mu_{R})}{\prod_{i=1}^{4}\partial\Phi_{i}}$
$\displaystyle=0.$ (130)

Equations (128), (130), and (30) then imply

$\displaystyle\beta_{\lambda}$
$\displaystyle=-\frac{\partial^{4}}{\prod_{i=1}^{4}\partial\Phi_{i}}\frac{\partial
V_{1}(\Phi,\mu_{R})}{\partial\log\mu_{R}}-\sum_{i=1}^{4}\gamma_{\Phi_{i}}\frac{\partial^{4}V_{0}(\Phi)}{\prod_{i=1}^{4}\partial\Phi_{i}}=\frac{1}{32\pi^{2}}\frac{\partial^{4}}{\prod_{i=1}^{4}\partial\Phi_{i}}(\text{Tr}\left[\mathbf{M}^{4}_{S}(\Phi)\right]+3\text{Tr}\left[\mathbf{M}^{4}_{G}(\Phi)\right])+\beta_{\lambda,FS},$
(131)

where $\mathbf{M}^{4}_{S}(\Phi)$ and $\mathbf{M}^{4}_{G}(\Phi)$ are defined in
Eqs. (32) and (33), $\beta_{\lambda}$ denotes the desired one-loop quartic
scalar beta function satisfying

$\displaystyle\beta_{\lambda}$ $\displaystyle=\frac{d\lambda}{d\log\mu_{R}},$
(132)

and

$\displaystyle\beta_{\lambda,FS}$
$\displaystyle=-\lambda\sum_{i=1}^{4}\gamma_{\Phi_{i}}$ (133)

is the field-strength-dependent part.

###  C.2 Field-strength-dependent part

The $\beta_{\lambda,FS}$ part of Eq. (131) is closely connected to the
$\Phi$-field anomalous dimension defined as

$\displaystyle\gamma_{\Phi}$
$\displaystyle=\frac{1}{2}\frac{1}{Z_{\Phi}}\frac{\partial
Z_{\Phi}}{\partial\log\mu_{R}},$ (134)

where

$\displaystyle Z_{\Phi}$
$\displaystyle=1+\left.\frac{\partial\Sigma^{\overline{MS}}_{\Phi}(p^{2})}{\partial
p^{2}}\right|_{p^{2}=m^{2}_{\Phi}}$ (135)

is the field-strength-renormalization factor of $\Phi$ in the
$\overline{\text{MS}}$ renormalization scheme, and
$\Sigma^{\overline{MS}}_{\Phi}(p^{2})$ is the corresponding self-energy. The
momentum-squared-proportional part of $\Sigma^{\overline{MS}}_{\Phi}(p^{2})$
originates from the diagram

which yields

$\displaystyle\Sigma^{\overline{MS}}_{\Phi}(p^{2})$
$\displaystyle=C_{2}(R)\cdot
p^{2}\cdot\frac{3g^{2}}{16\pi^{2}}\log\mu_{R}^{2}+\ldots,$ (136)

where $C_{2}(R)$ is the quadratic Casimir of the irreducible representation
$R$ that $\Phi$ belongs to. Hence,

$\displaystyle\gamma_{\Phi}$ $\displaystyle=C_{2}(R)\frac{3g^{2}}{16\pi^{2}}.$
(137)

Combining Eqs. (133) and (137), the $\beta_{\lambda,FS}$ term in Eq. (131)
becomes

$\displaystyle\beta_{\lambda,FS}$
$\displaystyle=-\lambda\frac{3g^{2}}{16\pi^{2}}\sum_{i=1}^{4}C_{2}(R_{\Phi_{i}}).$
(138)

###  C.3 Resulting expressions

With all this in hand, the one-loop beta functions for the dimensionless
scalar-potential couplings are readily obtained:

$\displaystyle\begin{split}16\pi^{2}\beta_{\beta_{4}^{\prime}}&=\phantom{+}16\alpha\beta_{4}^{\prime}+16a_{0}\beta_{4}^{\prime}+2a_{2}\beta_{4}-4a_{2}\beta_{4}^{\prime}-\beta_{4}^{2}-28\beta_{4}\beta_{4}^{\prime}+2\beta_{4}\lambda_{2}+6\beta_{4}\lambda_{4}+\\\
&\quad+80\beta_{4}\lambda_{4}^{\prime}-124\beta_{4}^{\prime
2}+4\beta_{4}^{\prime}\lambda_{0}-12\beta_{4}^{\prime}\lambda_{2}+20\beta_{4}^{\prime}\lambda_{4}-144\beta_{4}^{\prime}\lambda_{4}^{\prime}+16\left|\gamma_{2}\right|^{2}-\\\
&\quad-3g^{4}-123\beta_{4}^{\prime}g^{2},\\\ \end{split}$ (139)
$\displaystyle\begin{split}16\pi^{2}\beta_{\alpha}&=\phantom{+}8\alpha^{2}+508\alpha\lambda_{0}+1220\alpha\lambda_{2}+1340\alpha\lambda_{4}+2480\alpha\lambda_{4}^{\prime}+376\alpha
a_{0}+80a_{0}\beta_{4}+\\\ &\quad+160a_{0}\beta_{4}^{\prime}+76\alpha
a_{2}+16a_{2}\beta_{4}+32a_{2}\beta_{4}^{\prime}+4\beta_{4}^{2}+16\beta_{4}\beta_{4}^{\prime}+112\beta_{4}\lambda_{0}+\\\
&\quad+272\beta_{4}\lambda_{2}+288\beta_{4}\lambda_{4}+512\beta_{4}\lambda_{4}^{\prime}+144\beta_{4}^{\prime
2}+224\beta_{4}^{\prime}\lambda_{0}+544\beta_{4}^{\prime}\lambda_{2}+576\beta_{4}^{\prime}\lambda_{4}+\\\
&\quad+1024\beta_{4}^{\prime}\lambda_{4}^{\prime}+64|\gamma_{2}|^{2}+12g^{4}-123\alpha
g^{2},\\\ \end{split}$ (140)
$\displaystyle\begin{split}16\pi^{2}\beta_{\lambda_{0}}&=\phantom{+}90\alpha^{2}+40\alpha\beta_{4}+80\alpha\beta_{4}^{\prime}+10\beta_{4}^{2}+80\beta_{4}^{\prime
2}+520\lambda_{0}^{2}+2440\lambda_{0}\lambda_{2}+2680\lambda_{0}\lambda_{4}+\\\
&\quad+4960\lambda_{0}\lambda_{4}^{\prime}+3460\lambda_{2}^{2}+7880\lambda_{2}\lambda_{4}+12320\lambda_{2}\lambda_{4}^{\prime}+4660\lambda_{4}^{2}+13280\lambda_{4}\lambda_{4}^{\prime}+\\\
&\quad+16960\lambda_{4}^{\prime 2}+\frac{135g^{4}}{2}-150g^{2}\lambda_{0},\\\
\end{split}$ (141)
$\displaystyle\begin{split}16\pi^{2}\beta_{a_{0}}&=\phantom{+}126\alpha^{2}+56\alpha\beta_{4}+112\alpha\beta_{4}^{\prime}+424a_{0}^{2}+152a_{0}a_{2}+12a_{2}^{2}+\frac{33\beta_{4}^{2}}{2}+26\beta_{4}\beta_{4}^{\prime}+\\\
&\quad+106\beta_{4}^{\prime
2}-56|\gamma_{2}|^{2}+\frac{9g^{4}}{2}-96a_{0}g^{2},\\\ \end{split}$ (142)
$\displaystyle\begin{split}16\pi^{2}\beta_{a_{2}}&=\phantom{+}96a_{0}a_{2}+76a_{2}^{2}-5\beta_{4}^{2}+60\beta_{4}\beta_{4}^{\prime}-100\beta_{4}^{\prime
2}+560|\gamma_{2}|^{2}+3g^{4}-96a_{2}g^{2},\\\ \end{split}$ (143)
$\displaystyle\begin{split}16\pi^{2}\beta_{\lambda_{2}}&=-4\beta_{4}^{2}-32\beta_{4}^{\prime
2}+24\lambda_{0}\lambda_{2}-180\lambda_{2}^{2}-584\lambda_{2}\lambda_{4}-160\lambda_{2}\lambda_{4}^{\prime}-656\lambda_{4}^{2}-\\\
&\quad-800\lambda_{4}\lambda_{4}^{\prime}-2560\lambda_{4}^{\prime
2}-1264|\eta_{2}|^{2}-24g^{4}-150g^{2}\lambda_{2},\\\ \end{split}$ (144)
$\displaystyle\begin{split}16\pi^{2}\beta_{\lambda_{4}}&=\phantom{+}2\beta_{4}^{2}+16\beta_{4}^{\prime
2}+24\lambda_{0}\lambda_{4}+16\lambda_{2}^{2}+112\lambda_{2}\lambda_{4}+128\lambda_{2}\lambda_{4}^{\prime}+268\lambda_{4}^{2}+\\\
&\quad+640\lambda_{4}\lambda_{4}^{\prime}+1408\lambda_{4}^{\prime
2}+1328|\eta_{2}|^{2}+12g^{4}-150g^{2}\lambda_{4},\\\ \end{split}$ (145)
$\displaystyle\begin{split}16\pi^{2}\beta_{\lambda_{4}^{\prime}}&=\phantom{+}4\beta_{4}\beta_{4}^{\prime}-4\beta_{4}^{\prime
2}+24\lambda_{0}\lambda_{4}^{\prime}-4\lambda_{2}^{2}-8\lambda_{2}\lambda_{4}-16\lambda_{2}\lambda_{4}^{\prime}+4\lambda_{4}^{2}+112\lambda_{4}\lambda_{4}^{\prime}-\\\
&\quad-240\lambda_{4}^{\prime
2}+32|\eta_{2}|^{2}-3g^{4}-150g^{2}\lambda_{4}^{\prime},\\\ \end{split}$ (146)
$\displaystyle\begin{split}16\pi^{2}\beta_{\beta_{4}}&=\phantom{+}16\alpha\beta_{4}+16a_{0}\beta_{4}+16a_{2}\beta_{4}^{\prime}+48\beta_{4}^{2}+80\beta_{4}\beta_{4}^{\prime}+4\beta_{4}\lambda_{0}-8\beta_{4}\lambda_{2}+\\\
&\quad+32\beta_{4}\lambda_{4}+16\beta_{4}\lambda_{4}^{\prime}+16\beta_{4}^{\prime
2}+16\beta_{4}^{\prime}\lambda_{2}+48\beta_{4}^{\prime}\lambda_{4}+640\beta_{4}^{\prime}\lambda_{4}^{\prime}+64|\gamma_{2}|^{2}+\\\
&\quad+12g^{4}-123\beta_{4}g^{2},\\\ \end{split}$ (147)
$\displaystyle\begin{split}16\pi^{2}\beta_{\gamma_{2}}&=\phantom{+}8\alpha\gamma_{2}+14\beta_{4}\gamma_{2}+28\beta_{4}^{\prime}\gamma_{2}+440\eta_{2}\gamma_{2}^{*}-123\gamma_{2}g^{2},\\\
\end{split}$ (148)
$\displaystyle\begin{split}16\pi^{2}\beta_{\eta_{2}}&=\phantom{+}\frac{32\gamma_{2}^{2}}{3}+24\eta_{2}\lambda_{0}+160\eta_{2}\lambda_{2}+600\eta_{2}\lambda_{4}+640\eta_{2}\lambda_{4}^{\prime}-150\eta_{2}g^{2}.\\\
\end{split}$ (149)

It is perhaps worth noting that several parts of these results, in particular,
those corresponding to Green’s functions with four adjoint scalar outer legs
[such as the $a_{0}^{2}$-, $a_{0}a_{2}$-, and $a_{2}^{2}$-proportional terms
in Eqs. (142) and (143)], have been independently cross-checked by direct
diagrammatic methods.

##  Appendix D Masses in limits

In this appendix the analytical expressions for the tree-level spectrum
(Appendices D.2 and D.3) and the one-loop masses of pseudo-Goldstone bosons
(Appendix D.4) are collected for both symmetry-breaking patterns of interest
($|\omega_{R}|\gg|\sigma|\gg|\omega_{BL}|$ and
$|\omega_{BL}|\gg|\sigma|\gg|\omega_{R}|$). The presented formulas are merely
approximate, since they contain only the leading tree-level contributions,
while the radiative corrections are computed exclusively in the analytical
$a_{2}\to 0$, $\gamma_{2}\to 0$, $\sigma\to 0$ limit. It should be stressed
that in the actual scans all the masses have been computed numerically at the
one-loop level for an arbitrary choice of couplings and VEVs, i.e., not only
in the aforementioned limit. Nonetheless, the expressions introduced below can
be quite useful in providing some qualitative insight into the structure of
non-tachyonicity regions in Figs. 1 and 2.

###  D.1 Representation decompositions

The decompositions of all the representations that constitute the scalar and
gauge sectors of the model can be found in Table 7. They correspond to the
two-stage breaking scenarios in which either the value $\omega_{R}$ or
$\omega_{BL}$ spontaneously breaks $\mathrm{SO}(10)$ GUT symmetry to the
$\mathrm{SU}(4)_{C}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{R}\
$or
$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{SU}(2)_{R}\times\linebreak[0]\mathrm{U}(1)_{B-L}\
$intermediate symmetry, respectively, while $|\sigma|\ll M_{\text{GUT}}$
breaks it further to the Standard Model gauge group.

Table 7: Decomposition of the (a) $45$-dimensional adjoint and (b)
$126$-dimensional self-dual representations of $\mathrm{SO}(10)$ into
multiplets of
$\mathrm{SU}(4)_{C}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{R}\
$(the $\omega_{BL}\to 0$, $\sigma\to 0$ limit) in the left columns and
$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{SU}(2)_{R}\times\linebreak[0]\mathrm{U}(1)_{B-L}\
$(the $\omega_{R}\to 0$, $\sigma\to 0$ limit) in the right columns and their
subsequent breaking into Standard Model representations after engaging non-
zero $\sigma$; cf. Table 2. The color scheme indicates whether a
representation is _real_ (blue), _complex_ (black) or a _complex conjugate_
(red) of some other multiplet within the same $\mathrm{SO}(10)$
representation. Since the $\mathbf{126}$ is a complex representation, all the
multiplets that belong to it are also complex, while the $\mathbf{45}$ is real
and thus consists of real multiplets and complex-conjugate pairs. The multiple
copies of the same SM multiplets in the $\mathbf{126}$, e.g., the
$(1,2,+\tfrac{1}{2})$ and $(1,2,-\tfrac{1}{2})$ weak doublets, therefore
represent independent degrees of freedom and not just the complex-conjugate
counterparts of each other.

(a) $\mathbf{45}$ of $\mathrm{SO}(10)$

$4_{C}2_{L}1_{R}$ | $\begin{array}[]{l}3_{c}2_{L}1_{Y}\end{array}$
---|---
${\color[rgb]{0,0,1}(1,3,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(1,3,0)}\end{array}$
${\color[rgb]{0,0,1}(15,1,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(8,1,0)}\\\ {\color[rgb]{0,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})}\\\ {\color[rgb]{1,0,0}(3,1,+\tfrac{2}{3})}\\\ {\color[rgb]{0,0,1}(1,1,0)}\end{array}$
${\color[rgb]{0,0,0}(6,2,-\tfrac{1}{2})}$ | $\begin{array}[]{l}{\color[rgb]{0,0,0}(3,2,-\tfrac{5}{6})}\\\ {\color[rgb]{1,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{1}{6})}\end{array}$
${\color[rgb]{1,0,0}(6,2,+\tfrac{1}{2})}$ | $\begin{array}[]{l}{\color[rgb]{1,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,+\tfrac{5}{6})}\\\ {\color[rgb]{0,0,0}(3,2,+\tfrac{1}{6})}\end{array}$
${\color[rgb]{0,0,0}(1,1,+1)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,0}(1,1,+1)}\end{array}$
${\color[rgb]{1,0,0}(1,1,-1)}$ | $\begin{array}[]{l}{\color[rgb]{1,0,0}(1,1,-1)}\end{array}$
${\color[rgb]{0,0,1}(1,1,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(1,1,0)}\end{array}$

$3_{c}2_{L}2_{R}1_{BL}$ | $\begin{array}[]{l}3_{c}2_{L}1_{Y}\end{array}$
---|---
${\color[rgb]{0,0,1}(1,3,1,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(1,3,0)}\end{array}$
${\color[rgb]{0,0,1}(8,1,1,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(8,1,0)}\end{array}$
${\color[rgb]{0,0,0}(3,2,2,-\tfrac{2}{3})}$ | $\begin{array}[]{l}{\color[rgb]{0,0,0}(3,2,-\tfrac{5}{6})}\\\ {\color[rgb]{0,0,0}(3,2,+\tfrac{1}{6})}\end{array}$
${\color[rgb]{1,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,2,+\tfrac{2}{3})}$ | $\begin{array}[]{l}{\color[rgb]{1,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,+\tfrac{5}{6})}\\\ {\color[rgb]{1,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{1}{6})}\end{array}$
${\color[rgb]{0,0,1}(1,1,3,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,0}(1,1,+1)}\\\ {\color[rgb]{1,0,0}(1,1,-1)}\\\ {\color[rgb]{0,0,1}(1,1,0)}\end{array}$
${\color[rgb]{0,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,1,-\tfrac{4}{3})}$ | $\begin{array}[]{l}{\color[rgb]{0,0,0}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})}\end{array}$
${\color[rgb]{1,0,0}(3,1,1,+\tfrac{4}{3})}$ | $\begin{array}[]{l}{\color[rgb]{1,0,0}(3,1,+\tfrac{2}{3})}\end{array}$
${\color[rgb]{0,0,1}(1,1,1,0)}$ | $\begin{array}[]{l}{\color[rgb]{0,0,1}(1,1,0)}\end{array}$

(b) $\mathbf{126}$ of $\mathrm{SO}(10)$

$4_{C}2_{L}1_{R}$ | $\begin{array}[]{l}3_{c}2_{L}1_{Y}\end{array}$
---|---
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$10$\kern-1.00006pt}}}\hskip 1.00006pt,1,+1)$ | $\begin{array}[]{l}(1,1,+2)\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{4}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{2}{3})\end{array}$
$(10,3,0)$ | $\begin{array}[]{l}(1,3,-1)\\\ (3,3,-\tfrac{1}{3})\\\ (6,3,+\tfrac{1}{3})\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$10$\kern-1.00006pt}}}\hskip 1.00006pt,1,-1)$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{4}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})\\\ (1,1,0)\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$10$\kern-1.00006pt}}}\hskip 1.00006pt,1,0)$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{1}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})\\\ (1,1,+1)\end{array}$
$(15,2,+\tfrac{1}{2})$ | $\begin{array}[]{l}(1,2,+\tfrac{1}{2})\\\ (3,2,+\tfrac{7}{6})\\\ (8,2,+\tfrac{1}{2})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{1}{6})\end{array}$
$(15,2,-\tfrac{1}{2})$ | $\begin{array}[]{l}(1,2,-\tfrac{1}{2})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{7}{6})\\\ (8,2,-\tfrac{1}{2})\\\ (3,2,+\tfrac{1}{6})\end{array}$
$(6,1,0)$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})\\\ (3,1,-\tfrac{1}{3})\end{array}$

$3_{c}2_{L}2_{R}1_{BL}$ | $\begin{array}[]{l}3_{c}2_{L}1_{Y}\end{array}$
---|---
$(1,1,3,+2)$ | $\begin{array}[]{l}(1,1,+2)\\\ (1,1,+1)\\\ (1,1,0)\end{array}$
$(1,3,1,-2)$ | $\begin{array}[]{l}(1,3,-1)\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,3,+\tfrac{2}{3})$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{4}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})\end{array}$
$(3,3,1,-\tfrac{2}{3})$ | $\begin{array}[]{l}(3,3,-\tfrac{1}{3})\end{array}$
$(6,3,1,+\tfrac{2}{3})$ | $\begin{array}[]{l}(6,3,+\tfrac{1}{3})\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,3,-\tfrac{2}{3})$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{4}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{1}{3})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{2}{3})\end{array}$
$(1,2,2,0)$ | $\begin{array}[]{l}(1,2,+\tfrac{1}{2})\\\ (1,2,-\tfrac{1}{2})\end{array}$
$(3,2,2,+\tfrac{4}{3})$ | $\begin{array}[]{l}(3,2,+\tfrac{7}{6})\\\ (3,2,+\tfrac{1}{6})\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,2,-\tfrac{4}{3})$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{7}{6})\\\ (\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,2,-\tfrac{1}{6})\end{array}$
$(8,2,2,0)$ | $\begin{array}[]{l}(8,2,+\tfrac{1}{2})\\\ (8,2,-\tfrac{1}{2})\end{array}$
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,1,+\tfrac{2}{3})$ | $\begin{array}[]{l}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})\end{array}$
$(3,1,1,-\tfrac{2}{3})$ | $\begin{array}[]{l}(3,1,-\tfrac{1}{3})\end{array}$

###  D.2 Gauge boson masses

Here the tree-level masses of gauge bosons are gathered. Their leading
contributions for both scenarios of interest are also shown explicitly. In
addition to $33$ non-zero states listed in Table 8, there exist $12$ massless
would-be Goldstone modes corresponding to the $12$ generators of the Standard
Model gauge group.

Table 8: Tree-level spectrum of massive gauge bosons produced in $\mathrm{SO}(10)\to\;$$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{Y}\ $breaking is listed for the most general case and for both perturbative limits of interest. In the last column, the changes of the corresponding one-loop $\beta$ coefficients $\Delta a_{i}$ for gauge-coupling running are gathered; cf. Eq. (103). Note that massive gauge contributions implicitly include also those of their longitudinal components treated as adding the WGB scalar counterparts at the same scale.  Mass | General case |  $\omega_{BL}\to 0$ |  $\omega_{R}\to 0$ | $(\Delta a_{3},\Delta a_{2},\Delta a_{1})$
---|---|---|---|---
|  |  |  |
$M^{2}_{G}(3,2,-\tfrac{5}{6})$ |  $\frac{1}{2}g^{2}\left(\omega_{R}-\omega_{BL}\right)^{2}$ |  $\tfrac{1}{2}g^{2}\omega_{R}^{2}$ |  $\tfrac{1}{2}g^{2}\omega_{BL}^{2}$ | $(-7,-\tfrac{21}{2},-\tfrac{35}{2})$
$M^{2}_{G}(3,2,+\tfrac{1}{6})$ |  $\tfrac{1}{2}g^{2}(\omega_{R}+\omega_{BL})^{2}+2g^{2}|\sigma|^{2}$ |  $\tfrac{1}{2}g^{2}\omega_{R}^{2}$ |  $\tfrac{1}{2}g^{2}\omega_{BL}^{2}$ | $(-7,-\tfrac{21}{2},-\tfrac{7}{10})$
$M^{2}_{G}(3,1,+\tfrac{2}{3})$ |  $2g^{2}\omega_{BL}^{2}+2g^{2}|\sigma|^{2}$ |  $2g^{2}|\sigma|^{2}$ |  $2g^{2}\omega_{BL}^{2}$ | $(-\tfrac{7}{2},0,-\tfrac{28}{5})$
$M^{2}_{G}(1,1,+1)$ |  $2g^{2}\omega_{R}^{2}+2g^{2}|\sigma|^{2}$ |  $2g^{2}\omega_{R}^{2}$ |  $2g^{2}|\sigma|^{2}$ | $(0,0,-\tfrac{21}{5})$
$M^{2}_{G}(1,1,0)$ |  $10g^{2}|\sigma|^{2}$ |  $10g^{2}|\sigma|^{2}$ |  $10g^{2}|\sigma|^{2}$ | $(0,0,0)$

###  D.3 Scalar masses

The tree-level scalar masses are presented in the $\omega_{BL}\to 0$ and
$\omega_{R}\to 0$ limits, which should be properly understood as
$|\omega_{R}|\gg|\sigma|\geq\sqrt{|\omega_{R}||\omega_{BL}|}\gg|\omega_{BL}|$
and
$|\omega_{BL}|\gg|\sigma|\geq\sqrt{|\omega_{BL}||\omega_{R}|}\gg|\omega_{R}|$,
respectively, satisfying the perturbativity condition (21). For the
corresponding mass matrices in the general breaking scenario, the interested
reader is referred to [26].

The mass squares are, in general, linearly proportional to the $\tau$
parameter, whose value is determined by the vacuum stationarity condition (14)

$\displaystyle\tau$
$\displaystyle=2\beta_{4}^{\prime}(3\omega_{BL}+2\omega_{R})+a_{2}\chi(\omega_{BL}+\omega_{R})$
(150)

implying an implicit dependence of the spectrum on the $\chi$ parameter
defined in Eq. (15). Its contribution should not be neglected despite the
$|\chi|\leq 1$ constraint from perturbativity (and thus, $|\tau|\lesssim
M_{\text{GUT}}$) and a small $a_{2}$ coefficient in front of it (with values
$|a_{2}|\lesssim 0.05$ produced in scans; see Fig. 9). It turns out that the
value of the $\beta_{4}^{\prime}$ coupling is also rather small for valid
points (see Fig. 7), which makes both terms in $\tau$ comparable. Furthermore,
$\tau$ is always found to be negative, but its relative size (compared to the
GUT scale), and thus its impact on the masses, is typically larger in the
$\omega_{BL}\to 0$ case due to larger $|\beta_{4}^{\prime}|$.

Because of the relation in Eq. (15), the effect of the smallest VEV — either
$\omega_{BL}$ or $\omega_{R}$ — on tachyonicity of the scalar spectrum should
not be immediately dismissed. The distribution of $\chi$ presented in Fig. 16
shows a slight inclination for negative values in the $\omega_{BL}\to 0$ case,
while favouring positive values for $\omega_{R}\to 0$. In other words, the
first scenario prefers $\omega_{BL}$ and $\omega_{R}$ of different sign, while
in the other limit, both VEVs are more often of the same sign.

Figure 16: Distribution of values of $\chi$ parameter for all the points in
the main unbiased datasets $B_{+}$ ($\omega_{BL}\to 0$) and $R_{+}$
($\omega_{R}\to 0$); see Table 3.

The eigenvalues of tree-level mass matrices for scalars are presented in both
limits in Table 9. We show only their leading-order terms, which are
proportional either to the $\mathrm{SO}(10)$-breaking VEV ($|\omega_{R}|$ or
$|\omega_{BL}|$) or the intermediate-symmetry-breaking scale $|\sigma|$. When
the dominant contributions to masses are $M_{\text{GUT}}^{2}$-proportional,
the corresponding states belong to approximate
$\mathrm{SU}(4)_{C}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{R}\
$or
$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{SU}(2)_{R}\times\linebreak[0]\mathrm{U}(1)_{B-L}\
$intermediate-symmetry multiplets listed in Table 7. Their subleading
$|\sigma|^{2}$-proportional tree-level terms can be safely neglected due to
the large hierarchy between the scales ($|\sigma|$ is either of order
$10^{-4}M_{\text{GUT}}$ or even $10^{-10}M_{\text{GUT}}$; see Fig. 8). These
$|\sigma|^{2}$ terms cannot compete even with the leading one-loop
contributions of the order $\sim M_{\text{GUT}}^{2}/(16\pi^{2})$, which become
dominant when the tree-level $M_{\text{GUT}}^{2}$-proportional terms are
accidentally small (e.g., for PGBs in the $a_{2}\to 0$ regime). The only
exception are those states whose leading-order mass contributions as well as
their loop corrections are proportional to $|\sigma|^{2}$. One can thus
distinguish several types of states (cf. also Sec. IV.4):

  * •

The _would-be Goldstone_ scalars $(3,2,-\tfrac{5}{6})$, $(1,1,+1)_{1}$,
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip
1.00006pt,1,-\tfrac{2}{3})_{1}$, $(3,2,+\tfrac{1}{6})_{1}$, and $(1,1,0)_{1}$
are massless. They get absorbed into the longitudinal components of their
massive gauge boson companions listed in Table 8 through the Higgs mechanism.

  * •

The masses of _generic (heavy) states_ can be accurately estimated by their
dominant $M_{\text{GUT}}^{2}$-proportional tree-level contributions. These are
$\omega_{R}^{2}$\- or $\omega_{BL}^{2}$-proportional in the $\omega_{BL}\to 0$
and $\omega_{R}\to 0$ limits, respectively.

  * •

A separate category from the generic states are the _pseudo-Goldstone bosons_
, whose tree-level masses are accidentally small, since their leading
contributions are proportional to the suppressed value of $a_{2}$. Therefore,
their one-loop corrections can be of comparable size or even dominate. As in
the generic case, the subleading $|\sigma|^{2}$-proportional terms can still
be safely neglected, and they are in fact not present at all at tree level for
the triplet and octet. The states in this category consist of $(8,1,0)$,
$(1,3,0)$, and $(1,1,0)_{3}$, as well as $(\hskip 1.00006pt\hbox{\vbox{\hrule
height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip
1.00006pt,1,-\tfrac{2}{3})_{2}$ for $\omega_{BL}\to 0$ and $(1,1,+1)_{2}$ for
$\omega_{R}\to 0$.

  * •

The _intermediate-scale states_ (including the intermediate-symmetry-breaking
Higgs field) have masses that are $|\sigma|^{2}$-proportional to all orders in
a perturbative expansion, since they are protected by symmetry. Therefore,
their tree-level terms suffice. These states are the $(1,1,0)_{2}$ and either
$(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip
1.00006pt,1,-\tfrac{4}{3})$ for $\omega_{BL}\to 0$ or $(1,1,+2)$ for
$\omega_{R}\to 0$.

Table 9: The dominant contributions to tree-level scalar masses in both perturbative scenarios and the associated contributions $\Delta a_{i}$ to one-loop $\beta$ coefficients for gauge-coupling running.  Mass |  $\omega_{BL}\to 0$ |  $\omega_{R}\to 0$ | $(\Delta a_{3},\Delta a_{2},\Delta a_{1})$
---|---|---|---
|  |  |
$M^{2}_{S}(1,3,0)$ |  $+4a_{2}\omega_{R}^{2}$ |  $-2a_{2}\omega_{BL}^{2}$ | $(0,\tfrac{1}{3},0)$
$M^{2}_{S}(8,1,0)$ |  $-2a_{2}\omega_{R}^{2}$ |  $+4a_{2}\omega_{BL}^{2}$ | $(\tfrac{1}{2},0,0)$
$M^{2}_{S}(1,1,+2)$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $+4(2\lambda_{2}+2\lambda_{4}+8\lambda_{4}^{\prime}-a_{2}\chi^{2})|\sigma|^{2}$ | $(0,0,\tfrac{4}{5})$
$M^{2}_{S}(1,3,-1)$ |  $-2(2\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $-6(6\beta_{4}^{\prime}+a_{2}\chi)\omega_{BL}^{2}$ | $(0,\tfrac{2}{3},\tfrac{3}{5})$
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{4}{3})$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{1}{6},0,\tfrac{16}{15})$
$M^{2}_{S}(3,3,-\tfrac{1}{3})$ |  $-2(2\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $+2(\beta_{4}-8\beta_{4}^{\prime}-2a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{1}{2},2,\tfrac{1}{5})$
$M^{2}_{S}(6,3,+\tfrac{1}{3})$ |  $-2(2\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $-2(2\beta_{4}^{\prime}+a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{5}{2},4,\tfrac{2}{5})$
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{4}{3})$ |  $+4(2\lambda_{2}+2\lambda_{4}+8\lambda_{4}^{\prime}-a_{2}\chi^{2})|\sigma|^{2}$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{5}{6},0,\tfrac{32}{15})$
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{1}{3})$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{R}^{2}$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{5}{6},0,\tfrac{2}{15})$
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{2}{3})$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{R}^{2}$ |  $-4(4\beta_{4}^{\prime}+a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{5}{6},0,\tfrac{8}{15})$
$M^{2}_{S}(1,2,+\tfrac{1}{2})_{1,2}$ |  $+\tfrac{1}{2}\left(\beta_{4}-10\beta_{4}^{\prime}-4a_{2}\chi\right.\mp$ |  $+\tfrac{1}{2}(7\beta_{4}-18\beta_{4}^{\prime}-6a_{2}\chi\mp 4|\gamma_{2}|)\omega_{BL}^{2}$ | $(0,\tfrac{1}{6},\tfrac{1}{10})$
|  $\phantom{+\tfrac{1}{2}.}\mp\left.2\sqrt{4|\gamma_{2}|^{2}+(4\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{R}^{2}$ |  |
$M^{2}_{S}(3,2,+\tfrac{7}{6})_{1,2}$ |  $+\tfrac{1}{2}\left(\beta_{4}-10\beta_{4}^{\prime}-4a_{2}\chi\right.\mp$ |  $+\tfrac{1}{2}\left(\beta_{4}-26\beta_{4}^{\prime}-6a_{2}\chi\right.\mp$ | $(\tfrac{1}{3},\tfrac{1}{2},\tfrac{49}{30})$
|  $\phantom{+\tfrac{1}{2}.}\mp\left.2\sqrt{4|\gamma_{2}|^{2}+(4\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{R}^{2}$ |  $\phantom{+\tfrac{1}{2}.}\mp\left.4\sqrt{|\gamma_{2}|^{2}+(6\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{BL}^{2}$ |
$M^{2}_{S}(8,2,+\tfrac{1}{2})_{1,2}$ |  $+\tfrac{1}{2}\left(\beta_{4}-10\beta_{4}^{\prime}-4a_{2}\chi\right.\mp$ |  $+\tfrac{1}{2}(\beta_{4}-18\beta_{4}^{\prime}-6a_{2}\chi\mp 4|\gamma_{2}|)\omega_{BL}^{2}$ | $(2,\tfrac{4}{3},\tfrac{4}{5})$
|  $\phantom{+\tfrac{1}{2}.}\mp\left.2\sqrt{4|\gamma_{2}|^{2}+(4\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{R}^{2}$ |  |
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})_{1,3}$ |  $+(\beta_{4}-4\beta_{4}^{\prime}-2a_{2}\chi\mp 4|\gamma_{2}|)\omega_{R}^{2}$ |  $+\left(\beta_{4}-10\beta_{4}^{\prime}-3a_{2}\chi\right.\mp$ | $(\tfrac{1}{6},0,\tfrac{1}{15})$
|  |  $\phantom{+.}\mp\left.\sqrt{16|\gamma_{2}|^{2}+(6\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{BL}^{2}$ |
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,+\tfrac{1}{3})_{2}$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{R}^{2}$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{1}{6},0,\tfrac{1}{15})$
$M^{2}_{S}(1,1,+1)_{2}$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{R}^{2}$ |  $-2a_{2}\omega_{BL}^{2}$ | $(0,0,\tfrac{1}{5})$
$M^{2}_{S}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})_{2}$ |  $-2a_{2}\omega_{R}^{2}$ |  $+2(\beta_{4}-2\beta_{4}^{\prime}-a_{2}\chi)\omega_{BL}^{2}$ | $(\tfrac{1}{6},0,\tfrac{4}{15})$
$M^{2}_{S}(3,2,+\tfrac{1}{6})_{2,3}$ |  $+\tfrac{1}{2}\left(\beta_{4}-10\beta_{4}^{\prime}-4a_{2}\chi\right.\mp$ |  $+\tfrac{1}{2}\left(\beta_{4}-26\beta_{4}^{\prime}-6a_{2}\chi\right.\mp$ | $(\tfrac{1}{3},\tfrac{1}{2},\tfrac{1}{30})$
|  $\phantom{+\tfrac{1}{2}.}\mp\left.2\sqrt{4|\gamma_{2}|^{2}+(4\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{R}^{2}$ |  $\phantom{+\tfrac{1}{2}.}\mp\left.4\sqrt{|\gamma_{2}|^{2}+(6\beta_{4}^{\prime}+a_{2}\chi)^{2}}\right)\omega_{BL}^{2}$ |
$M^{2}_{S}(1,1,0)_{2}$ |  $+\left(8\lambda_{0}-\frac{8((\alpha+\beta_{4}^{\prime})^{2}+(\alpha+\beta_{4}^{\prime})a_{2}\chi)}{4a_{0}+a_{2}}+\right.$ |  $+\left(8\lambda_{0}-\frac{12((\alpha+\beta_{4}^{\prime})^{2}+(\alpha+\beta_{4}^{\prime})a_{2}\chi)}{6a_{0}+a_{2}}+\right.$ | $(0,0,0)$
|  $\phantom{+.}\left.+\frac{4(6a_{0}+a_{2})a_{2}\chi^{2}}{4a_{0}+a_{2}}\right)|\sigma|^{2}$ |  $\phantom{+.}\left.+\frac{(24a_{0}+a_{2})a_{2}\chi^{2}}{6a_{0}+a_{2}}\right)|\sigma|^{2}$ |
$M^{2}_{S}(1,1,0)_{3}$ |  $-2a_{2}\omega_{R}^{2}$ |  $-2a_{2}\omega_{BL}^{2}$ | $(0,0,0)$
$M^{2}_{S}(1,1,0)_{4}$ |  $+4(4a_{0}+a_{2})\omega_{R}^{2}$ |  $+4(6a_{0}+a_{2})\omega_{BL}^{2}$ | $(0,0,0)$

As an interesting aside, note that the actual mass eigenvalues are not
necessary to check for either non-tachyonicity of the spectrum or gauge-
coupling unification — in both cases, the quantities of interest can be
directly extracted from mass matrices by computing their characteristic
polynomial or from knowledge of their principal minors. Positive
(semi-)definiteness can be examined by applying Sylvester’s criterion on the
matrices; cf. Sec. III.1. This, for example, leads to conditions in Eqs. (25)
and (26) for the $a_{2}\to 0$ limit. Alternatively, alternating signs of
coefficients in the characteristic polynomial also imply positive definiteness
of the spectrum. For positive semi-definite matrices with $n$ zero-
eigenvalues, the $n$ lowest-order coefficients in the characteristic
polynomial vanish. Unification constraints at one loop, on the other hand,
require only knowledge of the product of non-zero eigenvalues
$\prod_{k=1}^{n_{R}}m^{2}_{R_{k}}$ for every SM representation $R$ with
multiplicity $n_{R}$. This number can be extracted as the absolute value of
the lowest degree coefficient in the characteristic polynomial that does not
vanish. The reason that this product suffices is that all $n_{R}$ eigenvalues
of multiplets in the representation $R$ contribute with the same $\beta$
coefficient $\Delta a_{i}^{(R)}$ to Eq. (105).

###  D.4 One-loop pseudo-Goldstone masses

An accurate assessment of the PGB masses in the small $a_{2}$ regime requires
also the computation of their one-loop corrections. In addition to the
recurring octet, triplet, and singlet pseudo-Goldstone fields, the two limits
$\omega_{BL}\to 0$ and $\omega_{R}\to 0$ provide another PGB in the form of
the $(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip
1.00006pt,1,-\tfrac{2}{3})_{2}$ or $(1,1,+1)_{2}$ state, respectively.

In the $\omega_{BL}\to 0$ scenario with
$\mathrm{SU}(4)_{C}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{R}\
$intermediate stage, the new PGB state $(\hskip 1.00006pt\hbox{\vbox{\hrule
height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip
1.00006pt,1,-\tfrac{2}{3})_{2}$ belongs to the $(15,1,0)$ multiplet of that
symmetry, together with the octet and singlet PGBs; see Table 7(a). This
confirms its PGB character, since its mass must be degenerate with the octet
and singlet up to subdominant corrections of the intermediate-symmetry-
breaking VEV $\sigma$. Similarly, in the $\omega_{R}\to 0$ scenario with the
$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{SU}(2)_{R}\times\linebreak[0]\mathrm{U}(1)_{B-L}\
$intermediate stage, the new PGB state $(1,1,+1)_{2}$ is embedded together
with the SM singlet into a $(1,1,3,0)$ representation.

An interesting observation is that the weak-triplet and SM-singlet PGBs
receive the same one-loop _gauge_ contributions to their masses in the
$\omega_{R}\to 0$ limit. Note that the two mentioned states belong to
different representations of the intermediate _left-right_ symmetry, namely,
to $(1,3,1,0)$ and $(1,1,3,0)$, respectively. However, these intermediate-
stage representations are connected by _D-parity_ ($L\leftrightarrow R$
exchange), which corresponds to a transformation $\omega_{BL}\to-\omega_{BL}$
at the VEV level. In the $\omega_{R}\to 0$ case, the dominant one-loop gauge
contributions to masses must be proportional to $g^{4}\omega_{BL}^{2}$ and are
hence the same for pairs of states that exchange under D-parity. The one-loop
contributions from scalars, on the other hand, are not parity invariant due to
the implicit presence of massive scalar-potential parameters $\mu$, $\nu$, and
$\tau$ obtained by solving stationarity conditions, where a change of sign in
$\omega_{BL}$ does have an effect.

The one-loop contributions to masses of PGB states in the $a_{2}\to 0$,
$\gamma_{2}\to 0$ limit313131Note that in this limit, the tree-level PGB
masses exactly vanish (see Table 9), so the one-loop corrections actually
represent their entire masses at the one-loop level. are presented in Table
10. Although this limit is apparently unphysical since the numerical scans
show that $|\gamma_{2}|\gtrsim 0.1$ for valid points (cf. Fig. 9), it is well
suited for obtaining analytical expressions. For both the $\omega_{BL}\to 0$
and $\omega_{R}\to 0$ scenario, we show only the dominant,
$\mathrm{SO}(10)$-breaking terms proportional to the largest VEV, while
neglecting those with $\sigma$. The qualitative features of PGBs discussed in
the beginning of this section are confirmed by the explicit formulae in Table
10, at least in the given limit. For more general formulae with an arbitrary
$\omega_{BL}$ and $\omega_{R}$ hierarchy, but still in the $a_{2}\to 0$,
$\gamma_{2}\to 0$, $\sigma\to 0$ limit, the interested reader is referred to
[26, 43].

The one-loop expressions in Table 10 are organized as follows: The
_polynomial_ contributions from gauge bosons and scalars are followed by gauge
and scalar _logarithmic_ terms. The arguments of the logarithms in the
$\omega_{BL}\to 0$ scenario correspond to the mass squares of the
$\mathrm{SU}(4)_{C}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{U}(1)_{R}\
$multiplets ordered as

$\displaystyle M^{2}_{G}(6,2,-\tfrac{1}{2}),$ $\displaystyle
M^{2}_{G}(1,1,+1),$ $\displaystyle M^{2}_{S}(\hskip
1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$10$\kern-1.00006pt}}}\hskip 1.00006pt,1,0),$
$\displaystyle M^{2}_{S}(15,2,+\tfrac{1}{2})_{1},$ $\displaystyle
M^{2}_{S}(15,2,+\tfrac{1}{2})_{2},$ $\displaystyle M^{2}_{S}(\hskip
1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$10$\kern-1.00006pt}}}\hskip 1.00006pt,1,+1),$
$\displaystyle M^{2}_{S}(10,3,0),$ $\displaystyle M^{2}_{S}(6,1,0),$

and for $\omega_{R}\to 0$ to mass squares of the
$\mathrm{SU}(3)_{c}\times\linebreak[0]\mathrm{SU}(2)_{L}\times\linebreak[0]\mathrm{SU}(2)_{R}\times\linebreak[0]\mathrm{U}(1)_{B-L}\
$multiplets arranged as

$\displaystyle M^{2}_{G}(3,2,2,-\tfrac{2}{3}),$ $\displaystyle
M^{2}_{G}(3,1,1,+\tfrac{4}{3}),$ $\displaystyle M^{2}_{S}(1,3,1,-2),$
$\displaystyle M^{2}_{S}(3,2,2,+\tfrac{4}{3})_{1},$ $\displaystyle
M^{2}_{S}(3,2,2,+\tfrac{4}{3})_{2},$ $\displaystyle M^{2}_{S}(\hskip
1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$6$\kern-1.00006pt}}}\hskip
1.00006pt,1,3,-\tfrac{2}{3}),$ $\displaystyle M^{2}_{S}(6,3,1,+\tfrac{2}{3}),$
$\displaystyle M^{2}_{S}(3,1,1,-\tfrac{2}{3})_{1},$ $\displaystyle
M^{2}_{S}(3,1,1,-\tfrac{2}{3})_{2},$ $\displaystyle
M^{2}_{S}(3,3,1,-\tfrac{2}{3}),$ $\displaystyle M^{2}_{S}(\hskip
1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern
1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip
1.00006pt,1,3,+\tfrac{2}{3}),$ $\displaystyle M^{2}_{S}(1,2,2,0),$
$\displaystyle M^{2}_{S}(8,2,2,0).$

The tree-level masses in the log arguments are taken in the $a_{2}\to 0$,
$\gamma_{2}\to 0$ limit, as confirmed by cross-referencing Tables 7, 8, and 9.
Note that for all terms where the logarithm diverges, e.g., when the argument
of the log contains the mass of a PGB in the $a_{2}\to 0$ limit or a would-be
Goldstone boson, the coefficient in front of the log vanishes, so these terms
do not contribute and are omitted.

Table 10: The dominant one-loop contributions to masses of PGB scalars in the $a_{2}\to 0$, $\gamma_{2}\to 0$ limit for both perturbative scenarios and containing polynomial as well as logarithmic terms with gauge bosons and scalar fields running in the loop.  Mass |  $\omega_{BL}\to 0$ | $\omega_{R}\to 0$
---|---|---
|  |
$M_{S}^{2}(1,3,0)$ |  $\frac{\omega_{R}^{2}}{16\pi^{2}}\Big{(}16g^{4}+40(\beta_{4}^{2}+36\beta_{4}^{\prime 2})$ | $\frac{\omega_{BL}^{2}}{16\pi^{2}}\Big{(}19g^{4}+(31\beta_{4}^{2}+60\beta_{4}\beta_{4}^{\prime}+2900\beta_{4}^{\prime 2})$
|  $+18g^{4}\log\left[\frac{g^{2}\omega_{R}^{2}}{2\mu_{R}^{2}}\right]-12g^{4}\log\left[\frac{2g^{2}\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $+21g^{4}\log\left[\frac{g^{2}\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]-24g^{4}\log\left[\frac{2g^{2}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $+40(-\beta_{4}^{2}+2\beta_{4}\beta_{4}^{\prime})\log\left[\frac{2(\beta_{4}-2\beta_{4}^{\prime})\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $-72(\beta_{4}\beta_{4}^{\prime}+6\beta_{4}^{\prime 2})\log\left[-\frac{36\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $+15(\beta_{4}^{2}+14\beta_{4}\beta_{4}^{\prime})\log\left[\frac{(\beta_{4}-18\beta_{4}^{\prime})\omega_{R}^{2}}{2\mu_{R}^{2}}\right]$ | $+(7\beta_{4}^{2}-136\beta_{4}\beta_{4}^{\prime}-1100\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-50\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  $+15(\beta_{4}^{2}-6\beta_{4}\beta_{4}^{\prime}+8\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-2\beta_{4}^{\prime})\omega_{R}^{2}}{2\mu_{R}^{2}}\right]$ | $+(7\beta_{4}^{2}-40\beta_{4}\beta_{4}^{\prime}+52\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  $-1280\beta_{4}^{\prime 2}\log\left[-\frac{16\beta_{4}^{\prime}\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $-768\beta_{4}^{\prime 2}\log\left[-\frac{16\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $+80(-\beta_{4}\beta_{4}^{\prime}+12\beta_{4}^{\prime 2})\log\left[-\frac{4\beta_{4}^{\prime}\omega_{R}^{2}}{\mu_{R}^{2}}\right]\Big{)}$ | $+48(-\beta_{4}\beta_{4}^{\prime}+14\beta_{4}^{\prime 2})\log\left[-\frac{4\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-(23\beta_{4}^{2}-376\beta_{4}\beta_{4}^{\prime}+128\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-16\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(\beta_{4}^{2}-8\beta_{4}\beta_{4}^{\prime}+16\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-4\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+12(3\beta_{4}^{2}-24\beta_{4}\beta_{4}^{\prime}+64\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-8\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+24(-\beta_{4}^{2}+\beta_{4}\beta_{4}^{\prime}+2\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(-\tfrac{53}{3}\beta_{4}^{2}+4\beta_{4}\beta_{4}^{\prime}+108\beta_{4}^{\prime 2})\log\left[\frac{(7\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  | $+(\tfrac{56}{3}\beta_{4}^{2}+128\beta_{4}\beta_{4}^{\prime}+864\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]\Big{)}$
$M_{S}^{2}(8,1,0)$ |  $\frac{\omega_{R}^{2}}{16\pi^{2}}\Big{(}13g^{4}+(19\beta_{4}^{2}+60\beta_{4}\beta_{4}^{\prime}+1340\beta_{4}^{\prime 2})$ | $\frac{\omega_{BL}^{2}}{16\pi^{2}}\Big{(}22g^{4}+(\tfrac{172}{3}\beta_{4}^{2}+3000\beta_{4}^{\prime 2})$
|  $+9g^{4}\log\left[\frac{g^{2}\omega_{R}^{2}}{2\mu_{R}^{2}}\right]-12g^{4}\log\left[\frac{2g^{2}\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $+12g^{4}\log\left[\frac{g^{2}\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]-6g^{4}\log\left[\frac{2g^{2}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $-24(\beta_{4}\beta_{4}^{\prime}-6\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-2\beta_{4}^{\prime})\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $-1296\beta_{4}^{\prime 2}\log\left[-\frac{36\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $+(\tfrac{23}{2}\beta_{4}^{2}+38\beta_{4}\beta_{4}^{\prime}+54\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-18\beta_{4}^{\prime})\omega_{R}^{2}}{2\mu_{R}^{2}}\right]$ | $+4(\beta_{4}^{2}+29\beta_{4}\beta_{4}^{\prime}-350\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-50\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  $+(\tfrac{23}{2}\beta_{4}^{2}-30\beta_{4}\beta_{4}^{\prime}+126\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-2\beta_{4}^{\prime})\omega_{R}^{2}}{2\mu_{R}^{2}}\right]$ | $+4(\beta_{4}^{2}-7\beta_{4}\beta_{4}^{\prime}+10\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  $-64(\beta_{4}\beta_{4}^{\prime}+11\beta_{4}^{\prime 2})\log\left[-\frac{16\beta_{4}^{\prime}\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $+48(-3\beta_{4}\beta_{4}^{\prime}+14\beta_{4}^{\prime 2})\log\left[-\frac{16\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $-48(\beta_{4}\beta_{4}^{\prime}-9\beta_{4}^{\prime 2})\log\left[-\frac{4\beta_{4}^{\prime}\omega_{R}^{2}}{\mu_{R}^{2}}\right]$ | $-12(3\beta_{4}\beta_{4}^{\prime}-38\beta_{4}^{\prime 2})\log\left[-\frac{4\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  $-2(9\beta_{4}^{2}-34\beta_{4}\beta_{4}^{\prime}-24\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-4\beta_{4}^{\prime})\omega_{R}^{2}}{\mu_{R}^{2}}\right]\Big{)}$ | $+(\beta_{4}^{2}+22\beta_{4}\beta_{4}^{\prime}-32\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-16\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(\beta_{4}^{2}-14\beta_{4}\beta_{4}^{\prime}+40\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-4\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-6(\beta_{4}^{2}-46\beta_{4}\beta_{4}^{\prime}+16\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-8\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-6(\beta_{4}^{2}+8\beta_{4}\beta_{4}^{\prime}-20\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-(\tfrac{224}{9}\beta_{4}^{2}-64\beta_{4}\beta_{4}^{\prime})\log\left[\frac{(7\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  | $+(\tfrac{152}{9}\beta_{4}^{2}-88\beta_{4}\beta_{4}^{\prime}+1296\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]\Big{)}$
Continued on next page
Table 10 — continued from previous page
---
Mass |  $\omega_{BL}\to 0$ | $\omega_{R}\to 0$
|  |
$M_{S}^{2}(1,1,0)_{3}$ |  same as $M_{S}^{2}(8,1,0)$ | $\frac{\omega_{BL}^{2}}{16\pi^{2}}\Big{(}19g^{4}+(31\beta_{4}^{2}+60\beta_{4}\beta_{4}^{\prime}+2900\beta_{4}^{\prime 2})$
|  $(9\beta_{4}^{2}-34\beta_{4}\beta_{4}^{\prime}-24\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-4\beta_{4}^{\prime})\omega_{R}^{2}}{\mu_{R}^{2}}\right]\Big{)}$ | $+21g^{4}\log\left[\frac{g^{2}\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]-24g^{4}\log\left[\frac{2g^{2}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-1296\beta_{4}^{\prime 2}\log\left[-\frac{36\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(7\beta_{4}^{2}+104\beta_{4}\beta_{4}^{\prime}-1100\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-50\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  | $+(7\beta_{4}^{2}+8\beta_{4}\beta_{4}^{\prime}+52\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  | $+192(-\beta_{4}\beta_{4}^{\prime}+8\beta_{4}^{\prime 2})\log\left[-\frac{16\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+96\beta_{4}^{\prime 2}\log\left[-\frac{4\beta_{4}^{\prime}\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(\beta_{4}^{2}-8\beta_{4}\beta_{4}^{\prime}-128\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-16\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-(23\beta_{4}^{2}-88\beta_{4}\beta_{4}^{\prime}-16\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-4\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $-24(\beta_{4}^{2}-10\beta_{4}\beta_{4}^{\prime}+16\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-8\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+12(3\beta_{4}^{2}-12\beta_{4}\beta_{4}^{\prime}+28\beta_{4}^{\prime 2})\log\left[\frac{2(\beta_{4}-2\beta_{4}^{\prime})\omega_{BL}^{2}}{\mu_{R}^{2}}\right]$
|  | $+(-\tfrac{53}{3}\beta_{4}^{2}+100\beta_{4}\beta_{4}^{\prime}+108\beta_{4}^{\prime 2})\log\left[\frac{(7\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]$
|  | $+(\tfrac{56}{3}\beta_{4}^{2}-256\beta_{4}\beta_{4}^{\prime}+864\beta_{4}^{\prime 2})\log\left[\frac{(\beta_{4}-18\beta_{4}^{\prime})\omega_{BL}^{2}}{2\mu_{R}^{2}}\right]\Big{)}$
$M_{S}^{2}(\hskip 1.00006pt\hbox{\vbox{\hrule height=0.5pt\kern 1.72218pt\hbox{\kern-1.00006pt$3$\kern-1.00006pt}}}\hskip 1.00006pt,1,-\tfrac{2}{3})_{2}$ |  same as $M_{S}^{2}(8,1,0)$ | not a PGB
$M_{S}^{2}(1,1,+1)_{2}$ | not a PGB |  same as $M_{S}^{2}(1,1,0)_{3}$

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